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\section{Introduction}
\IEEEPARstart{I}{n} this paper, we address the multiple object tracking (MOT) problem in the context of traffic scenes. A very common approach to tackle the MOT problem uses the tracking-by-detection paradigm, which divides the task into two smaller ones: the detection of objects in the video frames and the association of these detections between frames to form trajectories for the objects of interest \cite{Riahi2016,ICCV2017Sadeghian,CVPR2014Bae,CVPR2011Pirsiavash}. It is illustrated in Figure \ref{fig:MOTdesc}. The resulting trajectories can represent data about road users that is useful to solve higher-level problems such as improving security in our streets and improving traffic flows for more eco-friendly mobility. For example, tracking road users can help design more secure roads by analyzing users' behaviour and finding anomalies in their trajectories before incidents actually happen \cite{Saunier2010}.
\begin{figure*}
\centering
\includegraphics[width=0.8\paperwidth]{fig/MOT-Desc.PNG}
\caption{Solving a MOT situation involving pedestrians using the Tracking-by-detection paradigm. Pedestrians are first detected and then they are assigned a unique identifier, represented here by the various colors.}
\label{fig:MOTdesc}
\end{figure*}
This work focuses on the data association step of MOT methods to improve their performance for traffic scenes. We assume that our road user detections come from an off-the-shelf object detector. The main challenges of data association are: 1) handling occlusions which occur when a tracked entity is hidden completely or partially by other objects such as other road users or untracked scene elements (e.g. urban furniture), 2) managing incoming and outgoing objects of interest, which means that a trajectory may not last the whole video sequence, and 3) taking into account the imperfect performance of the detector, such as bounding boxes that do not bound perfectly an object, missing objects, or false detections (see Figure \ref{fig:confDet}).
\begin{figure}
\centering
\subfloat[]{\includegraphics[height=0.21\paperwidth]{fig/MOT-Diff-FP.png}}\ \
\subfloat[]{\includegraphics[height=0.21\paperwidth]{fig/MOT-Diff-Frag.png}}\ \
\subfloat[]{\includegraphics[height=0.21\paperwidth]{fig/MOT-Diff-Occl.png}}
\caption{Examples of detection errors. a) False detection, b) Object fragmentation, and c) Missed detection (the person behind) caused by an occlusion.}
\label{fig:confDet}
\end{figure}
The data association step can be viewed as a combinatorial problem between current tracks (a track is a trajectory being built by the tracker) and new detections. For this reason, the use of constraint programming (CP) is an interesting option to explore since CP has long been used to solve various combinatorial problems \cite{1996Wallace}. As a first step, CP states a formal model of the problem to solve using finite-domain variables and constraints on them. It then explores the combinatorial space of solutions using tree search guided by variable- and value-selection heuristics and applies sophisticated inference at each node of the search tree in order to prune infeasible branches. To every variable in the model is associated a finite set called its domain: the variable can take any value in that domain. Constraints on the variables forbid certain combinations of values. Each type of constraint encapsulates its own specialized inference algorithm that filters out inconsistent values from the domains of variables. We refer the interested reader to \cite{2014Pesant} for a more detailed exposition of CP.
This paper presents a MOT method that produces trajectories by matching recurring objects together using a set of constraints that use the center position and the color of detections. Our main contribution is a road user tracker that uses a CP model that is adapted to the MOT problem. Some filters are defined before and after the CP solving phase. Our method is compared with others based on data association strategies with similar simple features as the ones we are using. We show that in the case of simple features, our CP-based data association method gives better results compared with a state-of-the-art road user tracker on the UA-DETRAC benchmark. Using additional constraints, the proposed CP-based data association can be extended to include other features.
\section{Background and Related Work}
\label{review}
\subsection{Road User MOT}
A simple method to track road users is to use only the proximity between detections in consecutive frames to make decisions. It is possible to do so by computing the intersection over union metric that corresponds to the overlap area proportion between two bounding boxes, then choosing associations based on the highest overlap \cite{2018Bochinski}. This method has proven to be fast and was able to achieve a competitive accuracy on the UA-DETRAC benchmark.
Another approach is to use kernelized correlation filters (KCF) \cite{2015Henriques} where a filter is estimated in such a way that when applied to the region of interest to track, a strong response is produced. The goal is to find the image region that produces the most similar response in the subsequent frames. This technique was combined with a background subtraction process to track road users \cite{CRV2016Yang}. Although successful, this method lacks a proper data association process to better handle occlusions. A Kalman filter is also often used to help association tof road users by predicting missing detections \cite{Ooi2018}. Another work\cite{2016Jodoin} uses background subtraction, and manages occlusion situations by considering vehicles travelling together in only one track. A state machine manages the combination of tracklets when road users merge together or choose different paths. It uses keypoints as an appearance model and does data association only on two consecutive frames. Finally, another approach is to track keypoints using optical flow, and grouping these using a common motion criteria \cite{CRV2006Saunier}. The shortcoming of this method is that objects are not localised very well because it is based on sparse keypoints. Furthermore, keypoint detection and tracking rely on motion. Stopped road users are not tracked.
\subsection{General MOT}
State-of-the-art performance in multiple object tracking is currently obtained with methods using deep neural networks to extract many automatically-learned features \cite{ICIP2017Chen,ICCV2017Sadeghian}. One of these methods \cite{ICIP2017Chen} is using a convolutional neural network (CNN) to build two classifiers: one for separating object categories and another for instance classification which will differentiate objects of a same category. In this approach, the previous classifiers are connected to a particle filter which, combined with the appearance model, makes a prediction of where the object will appear next. The idea of using object categories in tracking was also exploited in the work of Ooi et al. \cite{Ooi2018} in road user tracking. More recently, re-identification features are gaining popularity to describe an object of interest in multiple object tracking situations \cite{ICCV2019Bergmann, CVPR2019Feng}.
Another paper using recurrent neural networks (RNNs) has focused more on interactions between objects in order to disambiguate matches \cite{ICCV2017Sadeghian}. A spatio-temporal appearance model, where a CNN takes bounding boxes as input and outputs features to a LSTM capable of memorizing long-term feature dependencies, is added to a motion feature extractor and an interaction feature extractor. The last part is made of a grid representing where nearby detected objects are located and a second LSTM module.
There are more and more methods that start using visual object tracking (VOT) techniques in a MOT context. In the VOT problem, there is only one given detection in the first frame, that states which object is to be tracked. Thus, VOT method uses strategies to differentiate the foreground from the background. This approach is problematic in MOT situations since there are many similar foreground objects and it becomes difficult to ensure that all objects are tracked since two trackers initialized on two different objects may end up following the same after a few frames. A possible way to mitigate the risk of these situations is to use a VOT for each object to make predictions about the position of the object in the next frame. Then these predictions can be incorporated in a data association method, where they are considered in a more global context \cite{CVPR2019Chu, CVPR2019Feng}.
These state-of-the-art methods focus mainly in correcting missing and spurious detections using a prediction method and defining robust features for data association. They do not study how to make the data association itself. They show that a strong appearance model and predicting where an object should appear next to filter object detections are two important components of a MOT system. However they use a limited data association strategy, as data association decisions are not taken over several frames by combining several observations of the objects in a video. Occlusions are better resolved over several frames. Therefore, a more robust data association method is always desirable.
Typical data association methods used in tracking are the Hungarian algorithm, the joint probabilistic data-association filter (JPDAF), and the minimum-cost flow algorithm. The Hungarian algorithm finds a maximum cost matching in a bipartite graph where the costs are on edges \cite{Kuhn1955}. As the assignment problem can be formulated as a bipartite graph, this algorithm is one way to get the solution. It was used in several works. For example, the Hungarian algorithm can be combined with tracklets (incomplete track parts) to make the associations \cite{CVPR2014Bae}. In this work, the tracklets are ranked according to a confidence metric considering occlusions, number of frames covered and how well the detections fit. Online detections are matched to existing tracklets with high confidence using the Hungarian algorithm maximizing the affinity level. The next step is to globally match these new associations with low level detections using once again the Hungarian algorithm.
JPDAF is a statistical method that can track objects based on what is the most likely outcome for each trajectory. It considers any detection available, but also the possibility of a missing object or a false detection. It was used several times for the MOT problem, for example, in the work of \cite{Rezatofighi2015} and \cite{CSM2009Shalom}.
The min-cost flow algorithm combines a cost function with a greedy algorithm in order to obtain the required tracking associations \cite{CVPR2011Pirsiavash}. The goal here is to formulate the data association as a minimum-cost flow problem, then to compute shortest paths on the flow network with detections at each frame as nodes, from the first appearance of an object to its last appearance in the scene. This process helps ensure that the resulting trajectories are as smooth as possible.
\section{CP Background}
In this section we present some useful background about constraint programming. There are three high level constraints used by our approach:
\begin{enumerate}
\item \texttt{AllDifferent}: Applied to a set of CP variables $\{ x_1, x_2, \ldots, x_n \}$, this constraint states that it is not permitted that two variables $x_i$ and $x_j$ share the same value:
\begin{equation}
x_i \neq x_j \qquad \forall \;
1 \leq i < j \leq n
\end{equation}
\item \texttt{Inverse}: Given two arrays of CP variables $\langle x_1, x_2, \ldots, x_m \rangle$ and $\langle y_1, y_2, \ldots, y_n \rangle$, the following equations must hold:
\begin{equation}
x_i \in [1, n] \Rightarrow y_{x_i} = i \qquad \forall\; 1 \leq i \leq m
\end{equation}
\begin{equation}
y_j \in [1, m] \Rightarrow x_{y_j} = j \qquad \forall\; 1 \leq j \leq n
\end{equation}
This constraint is useful to maintain consistency between dual representations.
\item \texttt{Regular} and \texttt{CostRegular}: These express a constraint as membership to a regular language. Given a sequence of variables $\langle x_1, x_2, \ldots, x_n \rangle$ and an automaton $\cal A$, each solution to the constraint corresponds to an assignment of values to these variables that spell out a word recognized by $\cal A$~\cite{CP2004Pesant}. \texttt{CostRegular} is the optimization version of the latter constraint: it takes as additional parameters the costs associated with each combination of variable, value, and automaton state, as well as a cost variable equal to the sum of individual costs in a solution~\cite{Demassey2006}.
\end{enumerate}
The inference algorithm associated with each of these constraints removes every inconsistent value in the domain of each variable.
\section{Methodology}
\label{metho}
Our road users tracking approach can be summarized as follows. First, detections are obtained for every video frame using a road user detector. A VOT is also applied to predict the position of the objects detected in the previous frame, in the new one. These detections are then filtered based on their confidence and redundancy. Detections can be redundant if a predicted box matches a detection in the bouding boxes initial set. The predicted bounding boxes should be ignored in this case. Then, the detections are used to instantiate the CP model and the model is solved by forming tracks with the detections. Finally, unused tracks are removed before outputting the final result (a set of trajectories). Our method is detailed in the following and a high-level view is shown on Figure \ref{fig:meth-diagram}.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{fig/Meth-Diagram.PNG}
\caption{Diagram showing the interactions of the high-level modules of our method.}
\label{fig:meth-diagram}
\end{figure*}
\subsection{Constraint Programming Model}
\subsubsection{Main Variables}
In order to associate tracks to detections, variables
\begin{alignat}{2}
t_{ij} \in \{1,2,\ldots,\tau\} & \qquad & \forall \; 1 \leq i \leq m,\; 1 \leq j \leq n_i
\label{eq:trajDef}
\end{alignat}
identify to which track (out of $\tau$ available ones) detection $j$ at frame $i$ belongs, where $m$ is the number of frames and $n_i$ the number of detections at frame $i$.
Such a representation is convenient to avoid unnecessary symmetries since it uses the smallest number of possible variables: one per detection. However, referring to consecutive detections of a given track is not possible before we obtain the solution; there is no way to know the indices of the variables that will be part of a same track before the solving step.
Since we need this to express some of our constraints, a second array of variables is defined,
\begin{alignat}{2}
d_{ik} \in \{1,2,\ldots,\tau\} & \qquad & \forall \; 1 \leq i \leq m,\; 1 \leq k \leq \tau
\label{eq:detDef}
\end{alignat}
identifying the detection assigned to track $k$ at frame $i$.
Whenever $\tau > n_i$ for some frame $i$, values greater than $n_i$ do not represent actual detections --- they are however necessary because we will require that detections be uniquely associated to tracks (see Equations~\ref{eq:alldiff_t} and \ref{eq:alldiff_d}).
To ensure that these two dual representations remain consistent with each other, we link them using an \texttt{inverse} constraint
\begin{alignat}{2}
d_{i\star} = \texttt{inverse}(t_{i\star}) & \qquad & \forall\; 1 \leq i \leq m
\label{eq:cstrInv}
\end{alignat}
relating each variable to its counterpart in the other array. the star symbol ($\star$) is used to indicate the selection of a subarray.
\subsubsection{Frame Consistency}
The next step is to ensure that it is impossible to find a specific track or detection more than once at a given frame. This is a common requirement for which CP provides an \texttt{AllDifferent} constraint. Such constraints have been specified for this purpose using the same variables as the previous equations.
\begin{alignat}{2}
\texttt{AllDifferent}(t_{i\star}) & \qquad & \forall\; 1 \leq i \leq m
\label{eq:alldiff_t}\\
\texttt{AllDifferent}(d_{i\star}) && \forall\; 1 \leq i \leq m
\label{eq:alldiff_d}
\end{alignat}
\subsubsection{Position Constraints}
After the model structure is set, constraints on features such as position can be considered. First, an array of integers is created for each dimension in a frame ($x$ and $y$ coordinates). These arrays, indexed by frame and then by detection ID, contain the center of each detection bounding box. It is then possible to formulate a constraint restricting the distance between two consecutive detections in a track by stating that the one-dimensional distance along the $X$ and the $Y$ axes separating them may not be greater than thresholds $\lambda_x$ and $\lambda_y$ :
\begin{alignat}{2}
|x_{i,d_{ik}} - x_{{i+1,d_{i+1,k}}}| \leq \lambda_x & \qquad & \mkern-18mu & \forall\; 1 \leq i < m, 1 \leq k \leq \tau\\
|y_{i,d_{ik}} - y_{{i+1,d_{i+1,k}}}| \leq \lambda_y && \mkern-18mu & \forall\; 1 \leq i < m, 1 \leq k \leq \tau
\end{alignat}
The same process has been tested using object scale indicators (width and height) but this feature did not help improve the solution.
\subsubsection{Appearance Model}
An appearance model is useful for minimizing the risk of mismatch between two nearby objects. Our proposed appearance model assigns a color class label to each detection available. The color classes are obtained by clustering observed object color histograms in several videos. Clustering is performed with K-Means using the Bhattacharya distance between the two color histograms. The learned classes are then used during tracking. A detection in a frame is assigned the color class label of the nearest cluster.
A \texttt{CostRegular} constraint is applied to each track preventing the association of two objects with contrasting colors. Possible transitions between color classes for a given object at each frame is governed by a state machine. Transitions between similar color classes are allowed but a cost is applied based on the Bhattacharya distance between the class centers. The solver will use the sum of all these costs as a minimization objective.
\begin{gather}
c_{ik} = \texttt{colour}(d_{ik})\\
\texttt{CostRegular}(c_{\star k}, {\cal A}, C, a_{k}) \qquad \forall\; 1 \leq k \leq \tau\\
\texttt{min}\left(\sum_{k=1}^{\tau} a_{k}\right)
\end{gather}
In these equations, $c_{ik}$ represents the list of color class labels for the track detections acting as transition variables between the automaton's states. $\cal A$ and $C$ correspond respectively to the automaton and the cost matrix for every possible transition. Variable $a_{k}$ represents Track $k$'s total cost computed by the regular constraint.
Each color class is represented by two states, depending on whether the object is currently being occluded or not (e.g. red object and red occluded object). Using different states for occlusions allows to set a cost for transitions from a valid detection to an absence of detection while preserving the object appearance even when it is occluded for an extended duration (see Figure \ref{fig:states}). The cost are based on the Bhattacharya distance between each color class histograms.
\begin{figure}
\centering
\begin{tikzpicture}[->,>=stealth', shorten >=1pt, auto, node distance=2.5cm, semithick, every node/.style={scale=1.1}]
\tikzstyle{every state}=[draw=none,text=white]
\node[initial,state, fill=black] (A) {\large $i$};
\node[state, fill=orange] (B) [below of=A] {\large $o_v$};
\node[state, fill=orange] (C) [below of=B] {\large $o_c$};
\node[state, fill=black!15!yellow, text=black] (D) [right of=B] {\large $y_v$};
\node[state, fill=black!15!yellow, text=black] (E) [right of=C] {\large $y_c$};
\node[state, fill=black!12!red] (F) [left of=B] {\large $r_v$};
\node[state, fill=black!12!red] (R) [left of=C] {\large $r_c$};
\path (A) edge [loop right] node {E} (A)
edge node [pos=0.4] {O} (B)
edge node [pos=0.56, above=3pt] {Y} (D)
edge node [pos=0.56, above=3pt] {R} (F)
(B) edge [in=65, out=35, looseness=8] node [pos=0.5, above] {O} (B)
edge [bend left=15] node {E} (C)
edge [bend left=15] node {Y} (D)
edge [bend left=15] node {R} (F)
(C) edge [bend left=15] node {O} (B)
edge [loop right] node {E} (C)
edge node [pos=0.3, below=3pt] {Y} (D)
edge node [pos=0.3, below=3pt] {R} (F)
(D) edge [bend left=15] node {O} (B)
edge [loop right] node {Y} (D)
edge [bend left=15] node {E} (E)
(E) edge node [pos=0.3, below=3pt] {O} (B)
edge [bend left=15] node {Y} (D)
edge [loop right] node {E} (E)
(F) edge [bend left=15] node {O} (B)
edge [loop left] node {R} (F)
edge [bend left=15] node {E} (R)
(R) edge node [pos=0.3, below=3pt] {O} (B)
edge [bend left=15] node {R} (F)
edge [loop left] node {E} (R);
\end{tikzpicture}
\caption{Small state machine example illustrating the doubled states: visible (v) and occlusion (c) for three color classes: red (d), orange (o) and yellow (y). Transition values correspond to the detection color (R, O, Y) or empty (E) when there is no detection.}
\label{fig:states}
\end{figure}
\subsection{Solving Strategy}
\subsubsection{Variable Selection}
As stated in the introduction, CP explores the combinatorial space of solutions by branching in a search tree that enumerates all possible solutions. To solve our model, we first branch on the $t_{ij}$ variables since it is the main array containing the smallest number of variables necessary to specify a solution whereas in the case of the $d_{ik}$ variables, all possible tracks are instantiated in case they are needed. To allow the state machine to handle occlusions or the absence of a detection for a given track, it is necessary to branch on the $c_{ij}$ variables. The lexicographic order specified will make sure that branching will always at first be made on $t_{ij}$ variables, then on $c_{ij}$ variables. Without including $c_{ij}$ variables in the branching order, some subconstraints of the \emph{CostRegular} might be ignored in the event of occlusions (no $t_{ij}$ linked to the $c_{ij}$). In both cases, variables are considered frame by frame, then detection identifier by detection identifier in ascending order.
\subsubsection{Value Selection}
Since the $t_{ij}$ variables are fixed by the solver from the first to the last frame, the use of prediction becomes an interesting option to guide the search. By computing every distance between possible pairs of consecutive detections, it is possible to obtain a ranked list that the solver can use to branch on closest detections first for each $t_{ij}$ except for the last frame. This way, we minimize the distance between each consecutive pair of objects without altering the model objective oriented towards the appearance preservation.
The inherent symmetries are addressed in the branching to control the number of opened tracks available as candidate for $t_{ij}$ variables. The domain of each of these variables is reduced to all previously used tracked index plus one other that may be opened if necessary. This strategy was applied before to the Steel Mill Slab Design Problem \cite{CPAIOR2008VH}.
\subsection{Pre Solving Computations}
\label{ssec:Presolve}
As mentioned in the introduction, detectors are prone to miss objects of interest. To compensate this weakness, we developed a pretracking method using an VOT to add new detections, using this process:
\begin{enumerate}
\item We initialize a KCF tracker \cite{2015Henriques} for each given detection of the first frame. With KCF, the image is correlated with a filter that is learned for each road user. The object is localized based on the strength of the filter response. It is a very fast tracker.
\item For the next frames, one by one, all KCF trackers make a prediction about the next bounding box of the object. For each prediction that is not redundant with a detection of the current frame, the bounding box is kept. For all detections in the current frame that do not match a prediction, a new tracker is initialized.
\item Each created tracker has a lifespan that allows it to make predictions for a fixed small number of frames. It is assumed that the tracker is reliable only on a small time window. To avoid adding detections based on a false detection, predicted boxes are only added to the existing detections if there is a match between an existing detection and the tracker prediction in a subsequent frame.
\end{enumerate}
\begin{figure}
\centering
\includegraphics[width=0.4\paperwidth]{fig/Det-Pred.PNG}
\caption{Illustration of the detection prediction process where blue boxes are detections from an object detector, yellow and green boxes are respectively discarded and added detections using KCF. The orange frames indicate the initialization of a tracker.}
\label{fig:DetPred}
\end{figure}
Figure \ref{fig:DetPred} illustrate many possible situations. For trajectory 1, a tracker initialized in the the first frame add two detections (frames 2 and 3). This is the ideal way to manage a two frames occlusion; two boxes are added when the object is hidden, then the tracker response match again the object and reinitialized itself since is prediction life is over. Trajectories 2 and 3 illustrate cases where we assume that the tracker is mistaken. If a VOT tracker is initialized on a false detection, no predicted boxes should be kept (trajectory 2). For the third trajectory, the tracker is able to match a real detection, but produces afterward another detection that remains unmatched. To avoid the addition of too many detections that would fall into the false positives category, the last detection is simply discarded.
\subsection{Post Solving Computations}
\label{ssec:Postsolve}
Once the solver has completed the associations, another type of filtering is required due to how the model is built. The model is unable to eliminate detections entirely --- the only choice that it has is to put each of them in dummy tracks, which will have to be deleted afterward. To delete them, a filter removes every track that contains fewer than $\beta_D$ detections from the object detector, a parameter set to a fixed value for all sequences.
After the suppression of the unwanted tracks, it is possible to fill gaps in the remaining tracks. A small state machine searches for places where there are $\gamma_D$ or less consecutive missing detections and fills these empty spots by adding new detections using linear interpolation to find the correct center position and size of the bounding box for each one. This has to be done carefully because a too large gap probably means that an identity change has happened. Therefore, filling missing detections in these situations would probably mean that false detections are added.
\subsection{Batch Resolution Process}
\label{ssec:A-Batch}
Since this model uses optimization to find the best solution to the MOT problem applied to large instances, the time required to find a solution is much longer than the one obtained by simply solving the satisfaction problem. Moreover, the number of computations required to find the optimal solution grows exponentially in the size of the problem instance which makes the minimization process intractable to complete in practice.
This situation was addressed by dividing a video sequence into multiple batches of $\kappa$ frames. Except for the first batch, to insure that the transition between batches stay as smooth as possible, the first $\beta$ frames of the trajectories were taken from the end of the previous batch. Frames containing no detection were ignored by this batch system, which means that a batch is not always constituted of consecutive frames. This is also valid for the overlapping section of the batch. If a sequence would contain too many empty consecutive frames, the next batch will start at the first frame containing at least a detection and will be be completely independent of the previous one. The values that gave the best trade-off between accuracy of the global solution and computation time was 30 for $\kappa$ and 5 for $\beta$.
\section{Experimental Setup}
\subsection{Evaluation Metrics}
To evaluate our method, we use the standard MOT metrics that account for three types of errors at each frame $i$: 1) Identity switches ($IDS_i$) or the number of mismatches, 2) the missing detections or false negatives ($FN_i$) which are the number of objects that are not tracked, and 3) the false positives ($FP_i$) which stands for the number of detections representing no object of interest. With these values, it is possible to compute the \textit{MOTA} score \cite{EURASIP2008Bernardin}
\begin{equation}
\textit{MOTA} = 1 - \dfrac{\sum_{i} ( FP_i + FN_i + IDS_i)}{\sum_{i} N_i}
\label{eq:MOTA}
\end{equation}
where $N_i$ is the total number of ground-truth objects per frame and it is summed throughout the entire video. The matching between tracking results and ground truth is done by measuring the intersection over union (IoU) of the two boudning boxes. If the overlapped area is higher than a certain percentage, the match is considered valid.
A second metric known as \textit{IDF1} balances the recall and precision for each trajectory into a single value. This is done by dividing the total number of accurately matched detections to the average number of ground truth and given detections \cite{ECCVW2016Ristani}. Finally, mostly tracked trajectories ($MT$) is the ratio of ground truth that possess a matching hypothesis for at least 80\% of their life span and mostly lost trajectories ($ML$) is the number of times this ratio is under 20\%. Fragmentation ($Frag$) indicates how many times any trajectory got interrupted over its length.
\subsection{Dataset}
We tested our method on the UA-DETRAC tracking challenge \cite{2015Wen}. This dataset contains 60 training sequences and 40 for the testing set. Only motorized vehicles such as cars, buses and trucks are tracked. We did not submit our results on the benchmark due to technical reasons (challenge website down, evaluation toolkit not cross-platform) to obtain the PR metrics. Instead, the code of the second best methods (IOU tracker) \cite{2018Bochinski} was downloaded, executed and tested using the usual MOTA metric (the code of the best method was not functionnal. Besides, it was only marginally better than the second one). Detections were filtered with the same parameters for both our method and the competing one. We used the same source detection from R-CNN.
We used a second dataset: the 2D MOT 2015 challenge \cite{MOTChallenge2015} to discover if our method could be extended from vehicules to pedestrian tracking. This dataset provides twenty-two video sequences divided equally in a training set and a testing set. There are many additional difficulties compared to UA-DETRAC such as moving cameras, variable frame rates (sometimes 7-10 FPS) and less elevated camera angles that are more prone to occlusions.
\subsection{Constraint Programming Solver}
The constraint programming solver used in this paper is IBM ILOG CP (version 1.6) augmented with our own implementation of the \texttt{CostRegular} constraint.
\section{Results}
Three series of experiments were conducted to evaluate how well our proposed method performs compared to others. First, the quality of our tracking approach is compared to a state-of-the-art method on the UA-DETRAC dataset. Second, we extend our method to the 2D MOT 2015 pedestrian dataset. Third, an ablation study evaluates the isolated performance of the components of our method.
For all experiments, the minimal overlap value between the detection and the object areas for true positive was fixed to 50\% in the MOTA computation. No parameter optimization was done using sequences from any testing sets.
\subsection{UA-DETRAC}
To make sure that the way to compare the methods is fair, the same detection confidence parameter was used every time. Since the post-solving computations were not improving results for UA-DETRAC sequences, it was turned off for the results of this section.
\begin{table*}[htb]
\centering
\caption{Results on the UA-DETRAC test dataset. Bold indicates best MOTA result. The first category group (Easy, Medium and Hard) differentiates sequences according to the difficulty level while the second category group (Sunny, Cloudy, Rainy and Night) is about illumination and weather conditions. All groups of sequences are mutually exclusive in their category, but a single sequence can be found in multiple categories. Higher MOTA is better as are lower IDS, FN and FP}
\begin{tabular}{r|rrrr|rrrr}
\hline
Sequences & \multicolumn{4}{c|}{CP + R-CNN} & \multicolumn{4}{c}{IOU + R-CNN} \\
(per category) & \textit{MOTA} & \textit{IDS} & \textit{FN} & \textit{FP} & \textit{MOTA} & \textit{IDS} & \textit{FN} & \textit{FP} \\ \hline
Easy & \textbf{64.39} & 3840 & 31675 & 6833 & 60.66 & 103 & 40824 & 5861 \\
Medium & \textbf{63.70} & 11533 & 74538 & 15159 & 58.36 & 310 & 98117 & 17683 \\
Hard & \textbf{48.38} & 10316 & 126451 & 6704 & 45.58 & 215 & 142372 & 8686 \\ \hline
Sunny & \textbf{90.65} & 1748 & 21118 & 3131 & 64.09 & 35 & 24348 & 2980 \\
Cloudy & \textbf{76.42} & 6392 & 51383 & 7766 & 69.04 & 221 & 61163 & 7784 \\
Rainy & \textbf{58.42} & 9961 & 94828 & 10793 & 35.63 & 126 & 112843 & 13254 \\
Night & \textbf{71.24} & 7588 & 65335 & 7006 & 41.98 & 246 & 82959 & 8212 \\ \hline
Global & \textbf{57.52} & 25689 & 232664 & 28696 & 53.51 & 628 & 281313 & 32230 \\ \hline
\end{tabular}
\label{tab:Res-1-DETRAC}
\end{table*}
Table \ref{tab:Res-1-DETRAC} present the tracking results obtained by our method (CP) for all sequences of the test set. The MOTA was computed globally by considering the total number of errors and objects. The R-CNN public detections were used for both the IOU tracking method and ours (CP). These results show that our method is able to outperform IOU in every situations and globally. The MOTA is affected by the difficulty level, especially by the hard sequences where the false negative errors are higher than in any other sequence groups. Weather conditions and exterior luminosity also impact tracking performance. While tracking tends to be easier in sunny weather (highest results), rain looks like the most difficult environmental factor to deal with.
Our strategy that helps to improve the quality of detections seem to have a high impact on the global results since the number of false positive and false negative errors are lower than those obtained by the IOU method for the same detections. However the number of ID switches is significantly higher. The main reason explaining this situation comes from the CP model itself. While the objective of the \texttt{CostRegular} constraint includes the minimization of the number of occlusions found in open trajectories, the branching strategy always tries to use already open trajectories first. Thus, the solver will implicitly prioritize compact solutions. Therefore, while our solution reduces significantly false positives and false negatives by handling better occlusion situations, it may result in a higher number of ID switches. Note that even if the number of available trajectories is determined before the resolution process, we made sure that there was always at least one unused trajectory.
In order to give more details about the performance of our method, the results obtained on a subset of the testing set are presented in Table \ref{tab:Res-2-DETRAC}. The five best and worse sequences in term of MOTA are presented. The distribution of errors made by the two compared methods is still similar to what was observed in the global results, especially regarding the number of IDS and FN mistakes.
\begin{table*}[htb]
\centering
\caption{Detailed results on the sequences in which we obtained the five highest and lowest MOTA values (UA-DETRAC, testing set). Bolface means best MOTA result. Higher MOTA is better as are lower IDS, FN and FP}
\begin{tabular}{l|rrrr|rrrr}
\hline
\multicolumn{1}{r|}{Sequences} & \multicolumn{4}{c|}{CP + R-CNN} & \multicolumn{4}{c}{IOU + R-CNN} \\
\multicolumn{1}{r|}{(Name)} & \textit{MOTA} & \textit{IDS} & \textit{FN} & \textit{FP} & \textit{MOTA} & \textit{IDS} & \textit{FN} & \textit{FP} \\ \hline
MVI\_40712 & \textbf{83.97} & 449 & 2726 & 613 & 82.49 & 30 & 3763 & 336 \\
MVI\_39211 & \textbf{78.96} & 28 & 846 & 37 & 78.00 & 0 & 866 & 51 \\
MVI\_39051 & \textbf{77.98} & 71 & 306 & 161 & 75.62 & 2 & 471 & 119 \\
MVI\_40854 & 77.24 & 485 & 3222 & 867 & \textbf{78.57} & 10 & 3835 & 460 \\
MVI\_39271 & \textbf{76.50} & 189 & 1350 & 459 & 74.17 & 1 & 1710 & 477 \\ \hline
MVI\_40763 & \textbf{36.50} & 309 & 5971 & 33 & 30.62 & 3 & 6741 & 115 \\
MVI\_40792 & \textbf{33.40} & 537 & 7044 & 470 & 29.60 & 3 & 8161 & 332 \\
MVI\_40863 & \textbf{31.83} & 937 & 20966 & 392 & 29.43 & 6 & 21370 & 525 \\
MVI\_39501 & 30.51 & 140 & 2973 & 902 & \textbf{33.62} & 4 & 3118 & 687 \\
MVI\_40761 & \textbf{22.22} & 629 & 15655 & 36 & 18.93 & 4 & 16839 & 117 \\ \hline
\end{tabular}
\label{tab:Res-2-DETRAC}
\end{table*}
To understand why there is an important variability between sequences in this dataset, Figure \ref{fig:DETRAC-img} presents extracted frames of videos from Table \ref{tab:Res-2-DETRAC}. Our method performed best on the images from the first row while the worst results were obtained on the second row. The sunny weather condition seems to be a critical factor to improve the results, but we think the camera angle is playing an important role. Tracking while facing the traffic with a good elevation angle looks like the optimal setup to achieve best performance with our method. The same applies to IOU tracker indicating that detections could be of lesser quality in those setups. No sequence in a rainy environment are among the least successful sequences even if it was the category with the worst results globally, which is counterintuitive. We assume the explanation comes from the camera angles used; side perspectives and low elevation angles cause difficult tracking situations. This is logical since both factors increase the probability of occlusion. Examples supporting this hypothesis are in the MVI\_40761 sequence (Figure \ref{fig:DETRAC-img}e), where the bus is hiding the two most distant lanes for over half the frame and in the MVI\_39501 sequence (Figure \ref{fig:DETRAC-img}f) where even with a front view, tall vehicles are able to hide many others behind them.
\begin{figure}
\centering
\subfloat[MVI\_40712]{\includegraphics[width=0.18\paperwidth]{fig/DETRAC-Best-1-40712.jpg}}\ \
\subfloat[MVI\_39211]{\includegraphics[width=0.18\paperwidth]{fig/DETRAC-Best-2-39211.jpg}}\ \
\subfloat[MVI\_39051]{\includegraphics[width=0.18\paperwidth]{fig/DETRAC-Best-3-39051.jpg}}\ \
\subfloat[MVI\_40854]{\includegraphics[width=0.18\paperwidth]{fig/DETRAC-Best-4-40854.jpg}}\
\subfloat[MVI\_40761]{\includegraphics[width=0.18\paperwidth]{fig/DETRAC-Worst-1-40761.jpg}}\ \
\subfloat[MVI\_39501]{\includegraphics[width=0.18\paperwidth]{fig/DETRAC-Worst-2-39501.jpg}}\ \
\subfloat[MVI\_40863]{\includegraphics[width=0.18\paperwidth]{fig/DETRAC-Worst-3-40863.jpg}}\ \
\subfloat[MVI\_40792]{\includegraphics[width=0.18\paperwidth]{fig/DETRAC-Worst-4-40792.jpg}}\
\caption{Filming perspective for different video sequences on UA-DETRAC sequences. The first row contains sequences in which our method performed the best while the second row contains the worst ones.}
\label{fig:DETRAC-img}
\end{figure}
\subsection{2D MOT 2015}
Table \ref{tab:interTestRes} compares our method with other approaches. The first two rows, \textit{HWDPL} \cite{ICIP2017Chen} and \textit{AMIR15} \cite{ICCV2017Sadeghian}, are to show the current performance of state-of-the-art methods that use multiple learned features, learned cost function and prediction in future frames. The last four entries include our method and three classic approaches using similar simple features as ours: \textit{DP NMS} (Min-cost flow) \cite{CVPR2011Pirsiavash}, \textit{TC ODAL} (Hungarian algorithm) \cite{CVPR2014Bae} and \textit{JPDA OP}, a JPDAF implementation from the MOT challenge website \cite{MOTChallenge2015}.
\begin{table*}
\centering
\caption{Tracking results on the 2D MOT 2015 benchmark.}
\begin{tabular}{lrrrrrrrr}
\hline
Method & \hspace{1em}$MOTA$ & \hspace{1em}$IDF1$ & $MT$ & $ML$ & $FP$ & $FN$ & $IDS$ & $Frag$ \\ \hline
HWDPL \cite{ICIP2017Chen} & 38.5 & 47.1 & 8.7\% & 37.4\% & 4005 & 33203 & 586 & 1263 \\
AMIR15 \cite{ICCV2017Sadeghian} & 37.6 & 46.0 & 15.8\% & 26.8\% & 7933 & 29397 & 1026 & 2024 \\ \hline
CP (ours) & 17.0 & 13.3 & 2.5\% & 58.4\% & 4872 & 43170 & 2973 & 3077 \\
TC ODAL \cite{CVPR2014Bae} & 15.1 & 0.0 & 3.2\% & 55.8\% & 12970 & 38538 & 637 & 1716 \\
DP NMS \cite{CVPR2011Pirsiavash} & 14.5 & 19.7 & 6.0\% & 40.8\% & 13171 & 34814 & 4537 & 3090 \\
JPDA OP & 3.6 & 7.5 & 0.4\% & 96.1\% & 1024 & 58189 & 29 & 119 \\ \hline
\end{tabular}
\label{tab:interTestRes}
\end{table*}
These results show that there are still many improvements to be made before we can match the performance of the state-of-the-art approaches, but it also shows that our method surpasses many classical approaches using similar features, even with our method using only a simple appearance model considering only ten color classes and the tendency to make many ID switch mistakes due to the CP model.
The results are also significantly lower than those obtained on the UA-DETRAC benchmark. Tracking pedestrians instead of road vehicles comes with its own challenges. People move more slowly than motorized vehicles, but their general appearance is more variable because of the leg-and-arm complex movements required to walk. The 2D MOT 2015 dataset was created before the rise of deep learning, therefore the noise level contained in the set of public detections is higher. The frame rate is not the same for all sequences (from 10 to 30).
\subsection{Ablation study}
After comparing the method to others available, the next step was to check the performance of individual modules. We made this evaluation on a subset of the training set of UA-DETRAC, that includes 22 videos. Table \ref{tab:Res-ablation-DETRAC} reports the MOTA values obtained after three removals:
\begin{enumerate}
\item At first, the problem was converted to a strict constraint satisfaction problem (CSP). The \texttt{CostRegular} constraint is kept, but the trajectory costs are no longer optimized.
\item Then, the \texttt{CostRegular} is completely removed. No appearance model is used at this point.
\item Finally, the pre solving computations based on an VOT technique are deactivated.
\end{enumerate}
\begin{table}[hbt!]
\centering
\caption{Results on a subset of sequences from UA-DETRAC with different configurations of our method. Each removal is valid for the current row and all the row below.}
\begin{tabular}{lrrrrr}
\hline
Setup & \textit{MOTA} & \textit{IDS} & \textit{FN} & \textit{FP} & \textit{Exec (s)} \\ \hline
All & 80.23 & 11048 & 39831 & 8722 & 53747.70 \\
CSP & 80.01 & 11699 & 39835 & 8726 & 1180.25 \\
No Regular & 80.01 & 11710 & 39835 & 8726 & 1100.41 \\
No Presolve & 78.88 & 12909 & 43626 & 7126 & 246.25 \\ \hline
\end{tabular}
\label{tab:Res-ablation-DETRAC}
\end{table}
Table \ref{tab:Res-ablation-DETRAC} confirms that each module helps improve the quality of the tracking, even if the increase is sometimes only small. Without the optimization process, the \texttt{CostRegular} is not removing many ID switch errors. The addition of an optimization objective allows a significant drop in the number of ID switch mistakes while maintaining approximately the same number of false positives and negatives. This indicates that a simple motion model is able to make mostly correct associations since forbidding associations based on vehicle appearances produces almost identical results.
The pre solve process allows to eliminate a greater number of false negatives while not increasing too much the number of false positives, thus increasing the global MOTA value. It makes sense since this module is mainly creating new detections.
The resolution time values were provided (in seconds) for each execution since it is an important factor to assess the quality of a CP model. All computations were done in a virtual machine running Ubuntu 19.10, either on a Intel i7-2600 processor from 2011, or a 2018 i9-9900k processor for the ablation study.
Heavy computations are required to obtain the best results with this method. As MOT is a problem that often requires to be solved in real-time; video surveillance cameras are filming constantly, taking hours to solve a five-minute video sequence is not always a possibility. The ablation study presented in Table \ref{tab:Res-ablation-DETRAC} confirms that even without considering the appearance model and using single object tracking predictions to add detections, the tracking performance is not dropping that much. The number of frames in the sequences equal 35495, which means that real-time performance is achieved since all UA-DETRAC sequences are at 30 frames per second, even with the presolve process and the automaton activated. This is an advantage provided by our CP method; the problem is solved in a larger context with real time performance compared to the IOU tracker which is fast, but considers the association problem only two frames at a time.
\section{Remaining Challenges}
\subsection{Computation times}
MOT is a problem where the computation time is limited since completing the tracking in real-time is desired in many situations. Our approach using CP and all its components is not close to this level of efficiency. However, as we have shown, removing some component can allow achieving real-time with a small performance hit. Processing video sequences that contain often more than a thousand frames increases the number of possible combinations before obtaining a solution. The total number of objects throughout the complete sequence is also impacting the computation since it directly affects the domain of the main branching variables. As a solution to this, we applied the data association to blocks of consecutive frames. Recall that usually data association is performed only on two consecutive frames. Considering more frames allows taking better long-term decisions. It is therefore not mandatory to perform the data association with all the frames, if at least, the associations are solved with bigger batches. Finally, the complexity of the appearance model (i.e. the number of possible appearance values) affects directly the time required to test each variable-value combination.
\subsection{Appearance Model}
The appearance model is important to indicate the presence of occlusions. The precision with which we describe detections will impact the tracking accuracy. One approach may be to use vectors to describe the object (histogram of oriented gradients, deep features). However, they are difficult to include in a CP model without doing clustering first which may reduce the precision of object descriptions. As said before, a complex model will also slow down the resolution process.
\section{Conclusion}
\label{concl}
This paper introduced a novel data association method using constraint programming. More precisely the center position of objects and their colors are used as main features to investigate if CP can be a valid approach to solve this problem in the context of road user tracking. Our result show that CP can help improve tracking as we outperform the IOU tracker on UA-DETRAC, which is one of the best tracker on this road user MOT dataset. Our strategy of using object position and a simplified color model proves to be well suited for vehicle tracking. There are still improvements required to be able to solve large instances and to capitalize better on road user appearance. Future work will include developing a more complex model that considers a larger number of features and optimizing the search strategy in order to improve run-time and performance on pedestrian tracking.
\section*{Acknowledgements}
\addcontentsline{toc}{section}{Acknowledgment}
This research was supported by an IVADO fundamental research grant.
\bibliographystyle{IEEEtran}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,671
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3024 026 - Vinyl 12"
Dutch power duo Doms & Deykers a.k.a Steffi and Martyn step up to the plate with a brand new action packed offering. Following up their "Fonts for the People EP" and their track "Whirling" on Ostgut Ton's Zehn LP, the two take their continuous collaboration to a new level. The three tracks on "Dedicated to those who feel" see Steffi and Martyn recognizing their respective strengths, combining their 90s heritage with a vision of the future, modern techno soul with all its flaws and bright euphoric moments. "Dedicated to those who feel" serves as the introduction to a full length Doms & Deykers LP, scheduled for later this year.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 3,015
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After a year of thoughtful work.
To express myself in all settings.
I wish for time to make me more resistant.
In a simple watery situation.
When the light is switched off and the drawer closed.
The last working day has arrived.
Are mysterious proteins and high yields.
in the winding sunny path that next year will be.
You know that green protein can really help me.
that can cheer up my 2017.
What to do is not clear.
And start chasing the mystery.
And in the bars of your graph.
Digging for gold in a pile of errors.
The corners that unattended bring shadows to the investigation.
In the regular wobble of a shaker.
And make you chase your scientific elegance.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 39
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Q: How to force binding re-evaluate or re-rendering in Aurelia I am starting with a simple TODO app with Aurelia, RethinkDB & Socket.IO. I seem to have problem with re-rendering or re-evaluating an object that is changed through Socket.IO. So basically, everything works good on the first browser but doesn't get re-rendered in the second browser while displaying the object in the console does show differences in my object. The problem is only on updating an object, it works perfectly on creating/deleting object from the array of todo items.
HTML
<ul>
<li repeat.for="item of items">
<div show.bind="!item.isEditing">
<input type="checkbox" checked.two-way="item.completed" click.delegate="toggleComplete(item)" />
<label class="${item.completed ? 'done': ''} ${item.archived ? 'archived' : ''}" click.delegate="$parent.editBegin(item)">
${item.title}
</label>
<a href="#" click.delegate="$parent.deleteItem(item, $event)"><i class="glyphicon glyphicon-trash"></i></a>
</div>
<div show.bind="item.isEditing">
<form submit.delegate="$parent.editEnd(item)">
<input type="text" value.bind="item.title" blur.delegate="$parent.editEnd(item)" />
</form>
</div>
</li>
</ul>
NodeJS with RethinkDB changefeeds
// attach a RethinkDB changefeeds to watch any changes
r.table(config.table)
.changes()
.run()
.then(function(cursor) {
//cursor.each(console.log);
cursor.each(function(err, item) {
if (!!item && !!item.new_val && item.old_val == null) {
io.sockets.emit("todo_create", item.new_val);
}else if (!!item && !!item.new_val && !!item.old_val) {
io.sockets.emit("todo_update", item.new_val);
}else if(!!item && item.new_val == null && !!item.old_val) {
io.sockets.emit("todo_delete", item.old_val);
}
});
})
.error(function(err){
console.log("Changefeeds Failure: ", err);
});
Aurelia code watching Socket.on
// update item
socket.on("todo_update", data => {
let pos = arrayFindObjectIndex(this.items, 'id', data.id);
if(pos >= 0) {
console.log('before update');
console.log(this.items[pos]);
this.items[pos] = data;
this.items[pos].title = this.items[pos].title + ' [updated]';
console.log('after update');
console.log(this.items[pos]);
}
});
// create item, only add the item if we don't have it already in the items list to avoid dupes
socket.on("todo_create", data => {
if (!_.some(this.items, function (p) {
return p.id === data.id;
})) {
this.items.unshift(data);
}
});
// delete item, only delete item if found in items list
socket.on("todo_delete", data => {
let pos = arrayFindObjectIndex(this.items, 'id', data.id);
if(pos >= 0) {
this.items.splice(pos, 1);
}
});
The socket.on("todo_update", ...){} is not making the second browser re-render but showing the object in the console before/after update does show differences in the object itself. I even changed the todo title property and that too doesn't get re-rendered.
How can I get Aurelia to re-render in my second browser with the new object properties? Don't be too hard on me, I'm learning Aurelia/RethinkDB/NodeJS/Socket.IO all the same time...
A: Aurelia observes changes to the contents of an array by overriding the array's mutator methods (push, pop, splice, shift, etc). This works well for most use-cases and performs really well (no dirty-checking, extremely lightweight in terms of memory and cpu). Unfortunately this leaves one way of mutating an array that aurelia can't "see": indexed assignment... eg myArray[6] = 'foo'. Since no array methods were called, the binding system doesn't know the array changed.
In your case, try changing this:
// update item
socket.on("todo_update", data => {
let pos = arrayFindObjectIndex(this.items, 'id', data.id);
if(pos >= 0) {
console.log('before update');
console.log(this.items[pos]);
this.items[pos] = data; // <-- change this to: this.items.splice(pos, 1, data);
this.items[pos].title = this.items[pos].title + ' [updated]';
console.log('after update');
console.log(this.items[pos]);
}
});
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,091
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Q: Sys.glob () within unzip () TLDNR: How do I use Sys.glob () within unzip ()?
I have multiple .zip files and I want to extract only one file from each archive.
For example, one of the archives contains the following files:
[1] "cmc-20150531.xml" "cmc-20150531.xsd" "cmc-20150531_cal.xml" "cmc-20150531_def.xml" "cmc-20150531_lab.xml"
[6] "cmc-20150531_pre.xml"
I want to extract the first file because it matches a pattern. In order to do that I use the following command:
unzip("zip-archive.zip", files=Sys.glob("[a-z][a-z][a-z][-][0-9][0-9][0-9][0-9][0-9][0-9][0-9][0-9][.][x][m][l]"))
However, the command doesn't work, and I don't know why. R just extracts all files in the archive.
On the other hand, the following command works:
unzip("zip-archive.zip", files="cmc-20150531.xml")
How do I use Sys.glob() within unzip()?
A: Sys.glob expands files that already exist. So the parameter to your unzip call will depend on what files are in your working directory.
Perhaps you want to do unzip with list=TRUE to return the list of files in the zip first, and then use some pattern matching to select the files you want.
See ?grep for info on matching strings with patterns. These patterns are "regular expressions" rather than "glob" expansions, but you should be able to work with that.
Here's a concrete example:
# whats in the zip?
files = unzip("c.zip", list=TRUE)$Name
files
[1] "l_spatial.dbf" "l_spatial.shp" "l_spatial.shx" "ls_polys_bin.dbf"
[5] "ls_polys_bin.shp" "ls_polys_bin.shx" "rast_jan90.tif"
# what files have "dbf" in them:
files[grepl("dbf",files)]
[1] "l_spatial.dbf" "ls_polys_bin.dbf"
# extract just those:
unzip("c.zip", files=files[grepl("dbf",files)])
The regular expression for your glob
"[a-z][a-z][a-z][-][0-9][0-9][0-9][0-9][0-9][0-9][0-9][0-9][.][x][m][l]"
would be
"^[a-z]{3}-[0-9]{8}\\.xml$"
that's a match of start of string ("^"), 3 a-z (lower case only), a dash, eight digits, a dot (backslashes are needed, one because dot means "any one char" in regexps and another because R needs a backslash to escape a backslash), "xml", and the end of the string ("$").
A: Just with any other collections do an itertive loop through the results from Sys.glob and supply the itertive holding variable to unzip. This is achieved by using a for-loop
While unzip() takes an argument for the path, and files is an arugment for what files within that zip file.
Mined you I'm more a full stack programmer not so much so on the R lang, but the concepts are the same; so the code should something like:
files <- Sys.glob(path_expand(".","*.zip"))
for (idx in 1:length(files)) {
results = unzip(files[idx], "*.xml")
}
As for using regex in unzip() that is something one should read the documentation. I could only advise doing another for-loop to compare the contest of the zip file to your regex then preforming the extraction. Psudocode follows:
files ::= glob(*.zip)
regex ::=
for idx1 in length(files); do
regex="[a-z]{3}\-[0-9]{8}\.xml"
content = unzip(files[idx1])
for idx2 in length(content); do
if content[idx2].name ~= regex.expand(); then
# do something with found file
end if
end for
end for
Basically your just looping through your list of zip files, then through the list of files within the zip file and comparing the filename from inside your zipfile agenst the regex and extracting/preforming operations on only that file.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,168
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Stryker – rodzina kołowych wozów opancerzonych będących na wyposażeniu Armii Stanów Zjednoczonych od roku 2002. Konstrukcja bazuje na kanadyjskim transporterze LAV III (pokrewna wersja LAV-25 obecnie używana jest m.in. przez amerykańską piechotę morską).
Historia konstrukcji
W październiku 1999 w armii amerykańskiej powstał plan modernizacji wojska. Zakładał on, że wobec obecnych i przyszłych warunków prowadzenia wojny rzeczą decydującą ma być mobilność jednostek. Armia miała być zdolna do przerzucenia brygady w dowolne miejsce na świecie w przeciągu 96 godzin (odpowiednio dywizja – 120 godzin, 5 dywizji – 30 dni). Powołano więc do życia tzw. brygady lekkie (później nazwane Stryker Brigade Combat Team – Brygada Stryker), które miały zostać wyposażone w odpowiedni pojazd – odpowiednio mobilny i łatwy w transporcie (przy pomocy C-130), a jednocześnie o lepszym opancerzeniu i sile ognia niż pojazdy rodziny HMMWV. Na taki pojazd idealnie nadawały się kołowe wozy opancerzone.
Pojazdem na bazie którego miał powstać wóz dla US Army został LAV III. Kontrakt pomiędzy General Dynamics-General Motors Defence Canada a armią amerykańską podpisano w roku 2000. Zakładał on dostarczenie 2131 pojazdów w ciągu 6 lat. Prace nad amerykańską wersją LAV III przyspieszono po atakach terrorystycznych z 11 września i w ten sposób pierwsze pojazdy w wersji transportowej (M1126 Infantry Carrier Vehicle) przyjęto do uzbrojenia w lutym 2002. Oprócz tego opracowano, także wersje specjalistyczne (pojazd rozpoznawczy czy wóz ewakuacji medycznej), a także niszczyciel czołgów (jako substytut dla czołgu).
Geneza nazwy "Stryker"
Przy wyborze nazwy dla nowego pojazdu zrezygnowano z dotychczasowej praktyki nadawania nazwisk słynnych generałów. Zdecydowano się na nazwę Stryker – na cześć dwóch szeregowców, którzy dostali Medal Honoru: Stuard S. Stryker (bohater II wojny światowej) oraz Robert F. Stryker (bohater wojny wietnamskiej). W wymowie nazwa ta jest podobna do angielskiego słowa "striker" (tj. "ten, który uderza").
Konstrukcja
Pojazd niezależnie od wersji posiada taki sam silnik (Caterpillar C7 o mocy 260 kW (350 KM)) i podwozie. Poszczególne wersje różnią się wyposażeniem, uzbrojeniem i ewentualnie konstrukcją nadwozia pojazdu. Wóz posiada trakcję kołową 8x8. Według założeń technicznych osiąga on prędkość maksymalną ok. 96 km/h (60 mil/h) oraz posiada zasięg maksymalny do ok. 482 km (300 mil).
Opancerzenie
Kadłub pojazdów typu Stryker wykonany jest z wysokiej jakości płyt stalowych. Zapewniają one ochronę przed ostrzałem pociskami przeciwpancernymi kal. 7,62 mm. Pancerz jest dodatkowo wzmocniony ceramicznymi płytami, który chronią przed pociskami kal. 14,5 mm, a także przed odłamkami z amunicji artyleryjskiej 152 mm.
W celu ochrony przed atakiem z ręcznych granatników przeciwpancernych (np. RPG-7) pojazdy można wyposażyć w dodatkowe osłony przed pociskami z głowicą kumulacyjną (tzw. klatki).
Latem 2003 pojawiły się problemy z odpornością pancerza ceramicznego na amunicję kal. 14,5 mm. W celu wzmocnienia ochrony producent zaproponował "dopancerzenie" 3 mm stalową blachą. Wzrost wagi związany z zamontowaniem dodatkowego opancerzenia uniemożliwia jednak transport na pokładzie C-130. Ostatecznie problem rozwiązano zmieniając rodzaj i dostawcę pancerza ceramicznego.
Podwozie zostało wzmocnione tak, żeby zapewnić ochronę załogi przed minami.
Uzbrojenie
Podstawowym uzbrojeniem M1126 ICV, M1127 RV, M1130 CV, M1131 FSV i M1132 ESV jest zdalnie sterowany moduł uzbrojenia M151 Protector z zamontowanym na nim granatnikiem Mk 19 lub wkm Browning M2. Głównym uzbrojeniem M1128 MGS jest czołgowa armata gwintowana M68A2 kal. 105 mm z automatyczną ładowarką.
Pomocnicze uzbrojenie Strykerów najczęściej stanowi karabin maszynowy M240.
Warianty
Powstało 10 wariantów Strykera:
M1126 Infantry Carrier Vehicle (ICV) – wersja podstawowa, transporter piechoty. Uzbrojony w wkm M2 lub granatnik Mk19 i karabin maszynowy M240. Przedział transportowy mieści 9 żołnierzy. Wprowadzony do US Army w roku 2002.
M1127 Reconnaissance Vehicle (RV) – wersja rozpoznawcza. Uzbrojenie stanowi wkm M2 lub granatnik Mk19 i karabin maszynowy M240. Przedział transportowy mieści 6 żołnierzy.
M1128 Mobile Gun System (MGS) – Wóz wsparcia ogniowego (WWO) na podwoziu kołowym. Główne uzbrojenie stanowi armata M68A2 kal. 105 mm. Wprowadzony do US Army w roku 2005.
M1129 Mortar Carrier (MC) – moździerz M120 kal. 120 mm lub M252 kal. 81 mm zamocowany na podwoziu transportera Stryker. Na wyposażeniu US Army od roku 2005.
M1130 Commander's Vehicle (CV) – wóz dowodzenia.
M1131 Fire Support Vehicle (FSV) – wóz kierowania ogniem
M1132 Engineer Squad Vehicle (ESV) – wóz inżynieryjny
M1133 Medical Evacuation Vehicle (MEV) – wóz ewakuacji medycznej
M1134 Anti-Tank Guided Missile Vehicle (ATGMV) – samobieżna wyrzutnia rakiet przeciwpancernych. Główne uzbrojenie stanowi wyrzutnia pocisków TOW
M1135 Nuclear, Biological, Chemical, Reconnaissance Vehicle (NBCRC) – pojazd rozpoznania NBC. Ma zastąpić pojazdy M93 Fox w US Army.
Mxxxx Self-Propelled Howitzer (SPH) – prototyp samobieżnej haubicy na podwoziu Stryker. Projekt przerwany w listopadzie 2005 roku.
Użycie pojazdu w warunkach bojowych
Strykery swój chrzest bojowy odbyły podczas misji stabilizacyjnej w Iraku (od listopada 2003). W lecie 2009 roku pierwsze Strykery skierowano do Afganistanu.
W raportach stwierdzono bardzo zadowalającą ochronę pojazdu przed IED. Nawet, jeśli w wyniku działania miny-pułapki pojazd został zniszczony, to często nie było ofiar wśród załogi.
Użytkownicy
: US Army (od roku 2002)
: 120 pojazdów M1126 w latach 2019–2020
Zobacz też
Mowag Piranha
ASLAV
NZLAV
KTO Rosomak
Przypisy
Amerykańskie transportery opancerzone
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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\section{Introduction}
\label{sec:introduction}
The aim of this paper is to extend the classes of analytic torsion
forms introduced by Bismut and K\"ohler to arbitrary projective
morphisms between complex algebraic varieties. The main tool for this
extension is an axiomatic characterization of all the possible
theories of holomorphic analytic torsion classes. Before stating
what we mean by a theory of holomorphic analytic torsion
classes, we briefly recall the origin of the analytic torsion.
The R-torsion is a topological invariant attached to certain
euclidean flat vector bundles on a finite CW-complex.
This invariant was introduced by
Reidemeister and generalized by Franz in order to distinguish
non-homeomorphic lens spaces that have the same
homology and homotopy groups. Let $W$ be a connected CW-complex and
let $K$ be an orthogonal representation of $\pi _{1}(W)$. Then $K$
defines a flat vector bundle with an euclidean inner product
$E_{K}$. Assume that the
chain complex of $W$ with values in $E_{K}$ is acyclic. Then the
R-torsion is the
determinant of this complex with respect to a preferred basis.
Later, Ray and Singer introduced an analytic analogue of the
R-torsion and they conjectured that, for compact riemannian manifolds,
this analytic torsion
agrees with the R-torsion. This conjecture was proved by Cheeger and
M\"uller. If $W$ is a riemannian manifold and $K$ is as before, then
we have the de Rham complex of $W$ with values in $E_{K}$ at our
disposal. The hypothesis on $K$ implies that $(\Omega ^{\ast}(W,E_{K}),\dd)$ is
also acyclic. Then the analytic torsion is essentially the determinant of
the de Rham complex. Here the difficulty lies in that the vector
spaces $\Omega ^{p}(W,E_{K})$ are infinite dimensional and therefore the
``determinant'' has to be defined using a zeta function regularization
involving the laplacian. More details on the construction of
R-torsion and analytic torsion can be found in \cite{RaySinger:RTorsion}.
Ray and Singer observed that, with the help of
hermitian metrics, the acyclicity condition can be removed. Moreover,
their definition of analytic torsion can
be extended to any elliptic complex.
In the paper \cite{RaySinger:ATCM}, they introduced a
holomorphic analogue of the analytic torsion as the determinant of the
Dolbeault complex. They also studied some of its
properties and computed some examples. In particular, they showed that
this invariant depends on the complex structure and they gave a hint
that the holomorphic analytic torsion should be interesting in number
theory. This holomorphic analytic torsion and its generalizations are
the main object of study of the present paper. Since this is the only
kind of analytic torsion that we will consider, throughout the paper,
by analytic torsion we
will mean holomorphic analytic torsion.
In the paper \cite{Quillen:dCRo}, Quillen, using the
analytic torsion, associated to each holomorphic hermitian vector
bundle on a Riemann surface a hermitian metric on the determinant of
its cohomology. Furthermore, he showed that this metric varies smoothly
with the holomorphic structure on the vector bundle. He also computed
the curvature of the hermitian line bundle on the space of
all complex structures obtained in this way.
Subsequently Bismut and Freed \cite{BismutFreed:EFI},
\cite{BismutFreed:EFII} generalized the construction of Quillen to
families of Dirac operators on the fibers of a smooth fibration. They
obtained a smooth metric and a unitary connection on the determinant
bundle associated with the family of Dirac operators. Furthermore,
they computed the curvature of this connection, which agrees with the
degree 2 part of the differential form obtained by Bismut in his proof
of the Local Index theorem \cite{Bismut:indexdirac}. Later, in a
series of papers \cite{BismutGilletSoule:at},
\cite{BismutGilletSoule:atII}, \cite{BismutGilletSoule:atIII}, Bismut,
Gillet and Soul\'e considered the case of a holomorphic submersion
endowed with a holomorphic hermitian vector bundle. They defined a
Quillen type metric on the determinant of the cohomology of the
holomorphic vector bundle. In the locally K\"ahler case, they showed
the compatibility with the constructions of Bismut-Freed. In addition
they described the variation of the Quillen metric under change of the
metric on the vertical tangent bundle and on the hermitian vector
bundle. The results of \cite{BismutGilletSoule:at},
\cite{BismutGilletSoule:atII}, \cite{BismutGilletSoule:atIII}
represent a rigidification of \cite{BismutFreed:EFI},
\cite{BismutFreed:EFII}. All in all, these works explain the
relationship between analytic torsion and the Atiyah-Singer index
theorem and, in the algebraic case, with Grothendieck's relative
version of the Riemann-Roch theorem.
In \cite{Deligne:dc}, Deligne, inspired by the Arakelov formalism,
gave a formula for the Quillen metric that can be seen as a very
precise version of the degree one case of the Riemann-Roch theorem for
families of curves. This result is in the same spirit as the
arithmetic Riemann-Roch theorem of Faltings \cite{Faltings:cas}.
In the paper \cite{GSATAT}, Gillet and Soul\'e conjectured an
arithmetic Riemann-Roch formula that generalizes the results of
Deligne and Faltings. Besides the analytic torsion or its avatar, the
Quillen metric, this Riemann-Roch formula involves a mysterious new
odd additive characteristic class, the $R$-genus, that they computed
with the help of Zagier.
In the work \cite{BismutLebeau:CiQm} Bismut and Lebeau studied the
behavior of the analytic torsion with respect to complex
immersions. Their compatibility formula also involved the
$R$-genus. Later Bost \cite{Bost:immersion} and Roessler
\cite{Roessler:ARR} explained, using geometric arguments, why the same
genus appears both in the arithmetic Riemann-Roch formula and the
Bismut-Lebeau compatibility formula. However these geometric arguments
do not characterize the $R$-genus.
Gillet and Soul\'e \cite{GilletSoule:aRRt} proved the
degree one part of the arithmetic Riemann-Roch theorem. A crucial
ingredient of the proof is the compatibility formula of Bismut-Lebeau.
In order to establish the arithmetic Riemann-Roch theorem in all
degrees it was necessary to generalize the analytic torsion and define
higher analytic torsion classes. It was clear from
\cite{GilletSoule:aRRt} that, once a suitable theory of higher
analytic torsion classes satisfying certain properties were developed,
then the arithmetic Riemann-Roch theorem would follow. A first
definition of such forms was given by Gillet and Soul\'e in
\cite{GSATAT}, but they did not prove all the necessary properties. A
second equivalent definition was given in \cite{Bismut-Kohler} by
Bismut and K\"ohler, where some of the needed properties are
proved. The compatibility of higher analytic torsion classes with
complex immersions, i.e. the generalization of Bismut-Lebeau
compatibility formula, was proved in \cite{Bismut:Asterisque}. As a
consequence, Gillet, Soul\'e and R\"ossler
\cite{GilletRoesslerSoule:_arith_rieman_roch_theor_in_higher_degrees}
extended the arithmetic Riemann-Roch theorem to arbitrary degrees.
In the book \cite{Faltings:RR}, Faltings followed a similar strategy
to define direct images of hermitian vector bundles and proved an
arithmetic Riemann-Roch formula up to a unique unknown odd genus.
The arithmetic Riemann-Roch theorems of Gillet-Soul\'e and Faltings
deal only with projective morphisms between arithmetic varieties such
that, at the level of complex points, define a submersion. By
contrast, in his thesis \cite{Zha99:_rieman_roch} Zha follows a
completely different strategy to establish an arithmetic Riemann-Roch
theorem without analytic torsion. His formula does not involve the
$R$-genus. Moreover Zha's theorem is valid for any projective morphism
between arithmetic varieties.
In \cite{Soule:lag}, Soul\'e advocates for an axiomatic
characterization of the analytic torsion, similar to the axiomatic
characterization of Bott-Chern classes given by Bismut-Gillet-Soul\'e
in \cite{BismutGilletSoule:at}. Note that the R-torsion has also been
generalized to higher degrees giving rise to different higher torsion
classes. In \cite{Igusa:Axioms}, Igusa gives an axiomatic
characterization of these higher torsion classes.
We now explain more precisely what we mean by a theory of generalized
analytic torsion classes. The central point is the relationship
between analytic torsion and the Grothendieck-Riemann-Roch theorem.
Let $\pi \colon X\to Y$ be a smooth projective morphism of smooth
complex varieties. Let $\omega $ be a closed $(1,1)$ form on $X$ that
induces a K\"ahler metric on the fibers of $\pi $ and, moreover, a hermitian metric
on the relative tangent bundle $T_{\pi }$. We denote $\overline T_{\pi }$ the relative tangent bundle provided with this metric.
Let $\overline F=(F,h^{F})$ be a hermitian vector bundle on $X$ such that
for every $i\ge 0$, $R^{i}\pi _{\ast} F$ is locally free. We consider
on $R^{i}\pi _{\ast} F$ the $L^{2}$ metric obtained using Hodge theory
on the fibers of $\pi $ and denote the corresponding hermitian vector
bundle as $\overline{R^{i}\pi _{\ast}F}$. To these data, Bismut and
K\"ohler associate an analytic torsion differential form $\tau $ that
satisfies the differential equation
\begin{equation}\label{eq:82}
\ast \partial\bar \partial\tau = \sum(-1)^{i}\ch(\overline{R^{i}\pi
_{\ast}F})-\pi _{\ast}(\ch(\overline F)\Td(\overline T_{\pi })),
\end{equation}
where $\ast $ is a normalization factor that is irrelevant here (see
\ref{sec:high-analyt-tors}). Moreover, if we consider the class of
$\tau $ up to $\Im\partial+\Im\bar \partial$, then $\tau $ behaves nicely
with respect to changes of metrics.
The Grothendieck-Riemann-Roch theorem in de Rham cohomology says that
the differential form on the right side of equation \eqref{eq:82} is
exact. Therefore, the existence of the higher analytic torsion classes
provides us an analytic proof of this theorem.
Since the Grothendieck-Riemann-Roch theorem is valid with more
generality, it is natural to extend the notion of higher analytic
torsion classes to non-smooth morphisms. To this end we will use the
language of hermitian structures on the objects of the bounded derived
category of coherent sheaves developed in
\cite{BurgosFreixasLitcanu:HerStruc}. In particular we will make
extensive use of the category $\oDb$ introduced in \emph{loc. cit.}.
Since, from now on, derived
categories will be the natural framework, all functors will tacitly
be assumed to be derived functors.
By reasons explained in \emph{loc. cit.} we will restrict ourselves to
the algebraic
category. Let $f\colon X \to Y$ be a projective
morphism between smooth complex algebraic varieties. Let $\overline F$ be a
hermitian vector bundle on $X$. Now, the relative tangent complex $T_{f
}$ and the derived direct image $f_{\ast}F$ are objects of the
bounded derived category of coherent sheaves on $X$ and $Y$
respectively. Since $X$ and $Y$ are smooth, using resolutions by
locally free sheaves, we can choose hermitian structures on $T_{f
}$ and $f_{\ast}F$.
Hence we have characteristic
forms $\ch(\overline{f_{\ast}F})$ and $\Td(\overline T_{f})$.
We denote by $\overline f$ the morphism $f$ together
with the choice of hermitian structure on $T_{f}$. Then the triple
$\overline{\xi}=(\overline f,\overline F,\overline{f_{\ast}F})$ will be called a
\emph{relative hermitian vector bundle}. This is a particular case of
the relative metrized complexes of Section~\ref{sec:transv-morph}.
Then, a \emph{generalized
analytic torsion class} for $\overline{\xi }$ is the class modulo
$\Im\partial+\Im\bar \partial$
of a current that satisfies the differential equation
\begin{equation}
\label{eq:83}
\ast \partial\bar \partial\tau = \ch(\overline{f
_{\ast}F})-f_{\ast}(\ch(\overline F)\Td(\overline T_{f})).
\end{equation}
Note that such current $\tau$ always exists.
Again, the Grothendieck-Riemann-Roch theorem in de Rham cohomology
implies that the right hand side of equation \eqref{eq:83} is an exact
current. Thus, if $Y$ is proper, the $dd
^{c}$-lemma implies the existence of such a current. When $Y$ is
non-proper, a compactification argument allows us
to reduce to the proper case.
Of course, in each particular case, there are many choices for $\tau
$. We can add to $\tau $ any closed current and obtain a new solution
of equation \eqref{eq:83}. By a
\emph{theory of generalized analytic torsion classes} we mean a
coherent way of choosing a solution of equation \eqref{eq:83} for all
relative hermitian vector bundles, satisfying certain natural
minimal set of properties.
Each theory of generalized analytic torsion classes gives
rise to a definition of direct images in arithmetic $K$-theory and
therefore to an arithmetic Riemann-Roch formula. In fact, the
arithmetic Riemann-Roch theorems of Gillet-Soul\'e and of Zha
correspond to different choices of a theory of generalized analytic
torsion
classes.
We leave for a
subsequent paper the discussion of the relation with the arithmetic
Riemann-Roch formula.
Since each projective morphism is the composition of a closed
immersion followed by the projection of a projective bundle, it is
natural to study first the analytic torsion classes for closed
immersions and projective bundles and then combine them in a global
theory of analytic torsion classes.
In \cite{BurgosLitcanu:SingularBC} the authors studied the case of closed
immersions (see Section
\ref{sec:analyt-tors-clos}). The generalized analytic torsion classes for closed
immersions are called
singular Bott-Chern classes and we will use both terms
interchangeably. The definition of a \emph{theory of
singular Bott-Chern classes} is obtained by imposing axioms
analogous to those
defining the classical Bott-Chern classes
\cite{GilletSoule:MR854556}. Namely, a theory of singular Bott-Chern
classes is an assignment that, to each relative hermitian vector
bundle $\overline \xi =(\overline f,\overline F,\overline{f_{\ast}F})$, with $f$ a closed
immersion, assigns the class of a current $T(\overline \xi )$ on $Y$,
satisfying the
following properties:
\begin{enumerate}
\item \label{item:5} the differential equation \eqref{eq:83};
\item \label{item:10} functoriality for morphisms that are transverse to $f$;
\item \label{item:54} a normalization condition.
\end{enumerate}
A crucial observation is that, unlike
the classical situation, these axioms do not uniquely characterize
the singular Bott-Chern classes. Consequently there are various
nonequivalent
theories of singular Bott-Chern classes. They are classified
by an arbitrary characteristic class of $F$ and $T_{f}$. If we further
impose the condition that the theory is \emph{transitive} (that is,
compatible with composition of closed immersions) and \emph{compatible with
the projection formula} then the ambiguity is reduced to an arbitrary additive
genus on $T_{f}$.
The uniqueness can be obtained by
adding to the conditions \ref{item:5}--\ref{item:54} an
additional homogeneity property. The theory obtained is transitive and
compatible with the projection formula and agrees (up to
normalization) with the theory
introduced in \cite{BismutGilletSoule:MR1086887}.
Similarly, one can define a theory of analytic torsion classes for
projective spaces (Section \ref{sec:projsp}). This is an assignment that, to each relative
hermitian vector bundle $\overline \xi =(\overline f,\overline F,\overline{f_{\ast}F})$, where
$f\colon {\mathbb P}^{n}_{Y}\to Y$ is the projection of a trivial projective
bundle, assigns the class of a current $T(\overline \xi )$ satisfying the
properties analogous to \ref{item:5}-\ref{item:54} below, plus the
additivity and the compatibility with the projection formula.
The theories of analytic torsion classes for
projective spaces are classified by their values in the cases $Y=\Spec
{\mathbb C}$, $n\ge 0$, $F=\mathcal{O}(k)$, $0\le k \le n$ for one particular
choice of metrics (see Theorem \ref{thm:1}).
We say that a theory of analytic torsion classes for closed immersions
and one for projective spaces are compatible if they satisfy a
compatibility equation similar to Bismut-Lebeau compatibility formula
for the diagonal
immersion $\Delta\colon {\mathbb P}^{n}_{{\mathbb C}}\to {\mathbb P}^{n}_{{\mathbb C}}\times
{\mathbb P}^{n}_{{\mathbb C}}$, $n\ge 0$. Given a theory of singular Bott-Chern classes that
is transitive and compatible with the projection formula, there exists
a unique theory of analytic torsion classes for projective spaces that
is compatible with it (Theorem \ref{thm:15}).
The central result of this paper (Theorem \ref{thm:gen_anal_tor}) is
that, given a theory of singular
Bott-Chern classes and a compatible theory of analytic torsion classes for
projective spaces, they can be combined to produce a unique theory of
generalized analytic torsion classes (Definition
\ref{def:genAT}). Moreover, every theory of analytic torsion classes
arises in this way. Thus we have a complete classification of the theories of
generalized analytic torsion classes by additive
genera.
Once we have proved the classification theorem, we derive several
applications.
The first consequence of Theorem \ref{thm:gen_anal_tor} is that the
classes of the analytic torsion forms of Bismut-K\"ohler arise as the
restriction to K\"ahler fibrations of the theory of generalized
analytic torsion classes associated to minus one half of the $R$-genus
(Theorem \ref{thm:19}). In particular, we have succeeded to extend
Bismut-K\"ohler analytic torsion classes to arbitrary projective
morphisms in the algebraic category.
Moreover, we
reprove and generalize the theorems of
Berthomieu-Bismut \cite{BerthomieuBismut} and Ma \cite{Ma:MR1765553},
\cite{Ma:MR1796698} on the compatibility of analytic torsion with the
composition of submersions (Corollary~\ref{cor:comp_sub}).
The second application of the classification theorem is a
characterization of the $R$-genus.
From the axiomatic point of view, the role played by
the $R$-genus is mysterious. It would seem more natural to consider the generalized
analytic torsion classes associated to the trivial genus
$0$. This is the choice made implicitly by Zha in his thesis
\cite{Zha99:_rieman_roch}. In fact, with our point of view, one of the
main results of Zha's thesis is the existence of a theory of analytic
torsion classes associated to the trivial genus. This theory leads to
an arithmetic Riemann-Roch
formula identical to the classical one without any correction
term. Thus, one is tempted to consider the $R$-genus as an artifact of
the analytic definition of the analytic torsion. Nevertheless, by the
work of several authors, the $R$-genus
seems to have a deeper meaning. A paradigmatic example is the
computation by Bost and K\"uhn \cite{Kuehn:gainc} of the arithmetic
self-intersection of the line bundle of modular forms on a modular
curve, provided with the Petersson metric. This formula gives an
arithmetic meaning to the first term of the $R$-genus. Thus it is
important to characterize the $R$-genus from an axiomatic
point of view and to understand its role in the above computations.
From a theorem of Bismut \cite{Bismut:deRham} we know that the
Bismut-K\"ohler analytic torsion classes of the relative de Rham complex
of a K\"ahler fibration (with the appropriate hermitian structures)
vanish. This
result is important because one of the main difficulties
to apply the arithmetic Riemann-Roch theorem is precisely the
estimation of the analytic torsion. Moreover, this result explains why
the terms of the $R$-genus appear in different arithmetic
computations. For instance, the equivariant version of this result
(due to Maillot and Roessler in degree 0 and to Bismut in general)
allows Maillot and Roessler \cite{MaillotRoessler:MR2123937} to prove
some cases of a conjecture of Gross-Deligne.
The above vanishing property characterizes the analytic
torsion classes of Bismut and K\"ohler. In order to show this,
we first construct the
dual theory $T^{\vee}$ to a given theory $T$ of generalized analytic
torsion classes (Theorem Definition \ref{thm-def:T_dual}). A theory is
self-dual ($T=T^{\vee}$) if and only if the even coefficients of the associated genus
vanish (Corollary \ref{cor:char_self_dual}). In particular,
Bismut-K\"ohler's theory is self-dual. Self-duality can also be characterized in
terms of the de Rham complex of smooth morphisms (Theorem
\ref{thm:char_van_dR}). A theory $T$ is self-dual if its components of
bidegree $(2p-1,p)$, $p$ odd, in the Deligne complex, vanish on the
relative de Rham
complexes of K\"ahler fibrations. Finally,
in Theorem \ref{thm:23} we show that, if it exists, a theory of
analytic torsion classes that vanishes, on all degrees, on the relative de Rham
complexes of K\"ahler fibrations is unique, hence it agrees with
Bismut-K\"ohler's one.
In fact, to characterize this theory, it is enough to assume the
vanishing of the analytic torsion classes for the relative de Rham
complexes of K\"ahler fibrations of
relative dimension one.
To establish this characterization we
appeal to the non-vanishing of the tautological class $\kappa_{g-2}$
on the moduli stack $\mathcal{M}_{g}$ of smooth curves of genus~$g\geq
2$.
The third application of generalized analytic torsion classes is
the construction of direct images of hermitian structures. We consider
the category $\oSm_{\ast/{\mathbb C}}$ introduced in
\cite{BurgosFreixasLitcanu:HerStruc}. The objects of this category are
smooth complex varieties, and the morphisms are projective morphisms
equipped with a hermitian structure on the relative
tangent complex. Assume that we have chosen a theory of generalized
analytic torsion classes. Let $\overline f\colon X\to Y$ be a morphism in
$\oSm_{\ast/{\mathbb C}}$. One would like to define a direct image functor
$f_{\ast}\colon \oDb(X)\to \oDb(Y)$.
It turns out
that, using analytic torsion, we can not define the direct image
functor on the category $\oDb$
and we have to introduce a new category $\hDb$. Roughly speaking, the
relation between $\hDb$ and $\Db$, is the same as the relation between
the arithmetic $K$-groups and the usual
$K$-groups (\cite{GilletSoule:vbhm}). Then we are able to define a
direct image functor
$f_{\ast}\colon \hDb(X)\to \hDb(Y)$,
that satisfies the composition rule, projection formula and base
change. Moreover, if the theory of generalized analytic torsion is
self-dual (as the Bismut-K\"ohler theory) this functor satisfies a
Grothendieck duality theorem. In a forthcoming paper, the direct image functor
will be the base of an arithmetic Grothendieck-Riemann-Roch
theorem for projective morphisms.
The last application that we discuss is a new proof of
a theorem of Bismut-Bost on the singularity of the Quillen metric for
degenerating families of curves, whose singular fibers have at most
ordinary double points \cite{Bismut-Bost}.
In contrast with
\emph{loc. cit.}, where the spectral definition of the Ray-Singer
analytic torsion is required, our arguments rely on the existence of a
generalized theory for arbitrary projective morphisms and some
elementary computations of Bott-Chern classes.
This theorem has already
been generalized by Bismut \cite{Bismut:degeneracy} and Yoshikawa
\cite{Yoshikawa} to families of varieties of arbitrary dimension.
In fact, our approach is very similar to the one in
\cite{Bismut:degeneracy} and \cite{Yoshikawa}. One of the main
ingredients of their proof is Bismut-Lebeau immersion formula, while our
approach uses implicitly Bismut's generalization of the immersion
formula in the comparison between Bismut-K\"ohler analytic torsion and
a theory of generalized analytic torsion classes. But what we want to
emphasize is that, once we have identified Bismut-K\"ohler as (part
of) a theory of generalized analytic torsion classes, many arguments
can be simplified considerably because the theory has been extended to
non-smooth projective morphism.
For simplicity, we treat only the case of families of curves and the
Quillen metric, but the methods can be applied to higher dimensional
families and analytic torsion forms of higher degree.
A few words about notations. The normalizations of characteristic
classes and Bott-Chern classes in this paper differ from the ones used
by Bismut, Gillet-Soul\'e and other authors. The first difference is
that they work with real valued characteristic classes, while we use
characteristic classes in Deligne cohomology, that naturally include
the algebro-geometric twist. The second difference is a factor $1/2$
in Bott-Chern classes, that explains the factor $1/2$ that appears in
the characteristic class associated to the
torsion classes of Bismut-K\"ohler. This change of normalization
appears already in \cite{Burgos:CDB}
and its objective is to avoid the factor $1/2$ that appears in the
definition of arithmetic degree in \cite[\S 3.4.3]{GilletSoule:ait}
and the factor $2$ that appears in \cite[Theorem
3.5.4]{GilletSoule:ait} when relating Green currents with Beilinson
regulator. The origin of this factor is that the natural second order
differential equation that appears when defining Deligne-Beilinson
cohomology is $\dd_{\mathcal{D}}=-2\partial\bar \partial$, while the
operator used when dealing with real valued forms is
\begin{math}
\dd \dd^{c}=\frac{1}{4\pi i}\dd_{\mathcal{D}}.
\end{math}
Thus the characteristic classes that appear in the present article
only agree with the ones in the papers of Bismut, Gillet and Soul\'e
after renormalization. With respect to the work of these authors we
have also changed the sign of the differential equation that
characterizes singular Bott-Chern classes. In this way, the same
differential equation appears when considering both, singular
Bott-Chern classes and analytic torsion classes. This change is
necessary to combine them.
We point out that our construction of generalized analytic
torsion classes is influenced by
the thesis of Zha \cite{Zha99:_rieman_roch}, where the author
uses implicitly a theory of analytic torsion classes different from
that of Bismut-K\"ohler.
In the unpublished e-print \cite{Weng:rBCsc}, L. Weng gives another
axiomatic approach to analytic torsion classes, only for smooth morphisms between K\"ahler
fibrations. This forces him to include a
continuity condition with respect to the deformation to the normal cone
as one of the axioms. The remaining axioms he uses are: the differential
equation, functoriality with respect to cartesian squares,
compatibility with respect to the projection formula and two anomaly
formulas. A collection of differential forms satisfying these axioms
are called relative Bott-Chern secondary characteristic classes.
These characteristic classes are not unique. The main
result of Weng's paper is that any
two such
theories are related by an additive genus.
Moreover he is able to obtain a weak form of the existence theorem for
relative Bott-Chern secondary characteristic classes.
Further applications of the generalized analytic torsion
classes are left for future work. We plan to prove generalizations of
the arithmetic Grothendieck-Riemann-Roch theorem of Gillet-Soul\'e
\cite{GilletSoule:aRRt} and Gillet-R\"ossler-Soul\'e
\cite{GilletRoesslerSoule:_arith_rieman_roch_theor_in_higher_degrees}
to arbitrary projective morhisms, along the lines of
\cite{BurgosLitcanu:SingularBC}.
It is possible to compute explicitly the characteristic
numbers of the unique theory of analytic torsion classes for projective
spaces compatible with the homogeneous theory for closed
immersions. This computation
makes the characterization of generalized analytic torsion
classes more precise. Nevertheless, since this computation is much more transparent
when written in terms of properties of arithmetic Chow groups and
the Riemann-Roch theorem, we leave it to the paper devoted to the
arithmetic Riemann-Roch theorem.
We also plan to study the possible axiomatic characterization of
equivariant analytic torsion classes. Note that the characterization of
equivariant singular Bott-Chern forms has already been obtained by Tang in
\cite{Tang:ueBC}.
\section{Deligne complexes, transverse morphisms and relative metrized complexes}
\label{sec:transv-morph}
In this section we fix the notations and conventions used through the
article, we also
recall the definition of transverse morphisms and
we review some basic properties. Finally we introduce the notion of
relative metrized complex, and explain some basic constructions.
The natural context where one can define the Bott-Chern
classes and the analytic torsion classes is that of Deligne
complexes. For the convenience of the reader we will summarize in this
section the basic facts about the Deligne complexes we will use in the
sequel. For more details the reader is referred to \cite{Burgos:CDB}
and \cite{BurgosKramerKuehn:cacg}.
\begin{definition} \label{def:22}
A \emph{Dolbeault complex} $A=(A^{\ast}_{\mathbb{R}},\dd_{A})$ is
a bounded below graded complex of real vector spaces
equipped with a compatible bigrading on $A_{\mathbb{C}}=A_{\mathbb{R}}
\otimes_{\mathbb{R}}{\mathbb{C}}$, i.e.,
\begin{displaymath}
A^{n}_{\mathbb{C}}=\bigoplus_{p+q=n}A^{p,q},
\end{displaymath}
satisfying the following properties:
\begin{enumerate}
\item[(i)]
The differential $\dd_{A}$ can be decomposed as the sum $\dd_{A}=
\partial+\bar{\partial}$ of operators $\partial$ of type $(1,0)$,
respectively $\bar{\partial}$ of type $(0,1)$.
\item[(ii)]
The symmetry property $\overline{A^{p,q}}=A^{q,p}$ holds, where
$\overline{\phantom{M}}$ denotes complex conjugation.
\end{enumerate}
\end{definition}
The basic example of Dolbeault complex is the complex of differential
forms on a smooth
variety $X$ over ${\mathbb C}$, denoted $E^{\ast}(X)_{{\mathbb R}}$.
Following \cite[\S 5.2]{BurgosKramerKuehn:cacg}, to a Dolbeault complex
one assigns a Deligne complex denoted $\mathcal{D}^{\ast}(A,\ast)$. In
this paper we will only use the following pieces of this complex:
\begin{align*}
\mathcal{D}^{2p+1}(A,p)&= (A^{p,p+1}\oplus A^{p+1,p})\cap (2\pi
i)^{p}A^{2p+1}_{{\mathbb R}},\\
\mathcal{D}^{2p}(A,p)&= A^{p,p}\cap (2\pi
i)^{p}A^{2p}_{{\mathbb R}},\\
\mathcal{D}^{2p-1}(A,p)&= A^{p-1,p-1}\cap (2\pi
i)^{p-1}A^{2p-2}_{{\mathbb R}},\\
\mathcal{D}^{2p-2}(A,p)&= (A^{p-2,p-1}\oplus A^{p-1,p-2})\cap (2\pi
i)^{p-1}A^{2p-3}_{{\mathbb R}}.
\end{align*}
The differential of the Deligne complex, denoted by
$\dd_{\mathcal{D}}\colon \mathcal{D}^{n}(A,p)\to \mathcal{D}^{n+1}(A,p)
$ is given, in the above degrees by
\begin{alignat*}{2}
\text{if }\eta &\in \mathcal{D}^{2p}(A,p), &\quad
\dd_{\mathcal{D}}\eta&=\dd \eta ,\\
\text{if }\eta &\in \mathcal{D}^{2p-1}(A,p), &\quad
\dd_{\mathcal{D}}\eta&=-2\partial\bar \partial \eta ,\\
\text{if }\eta =(u,v)&\in \mathcal{D}^{2p-2}(A,p), &\quad
\dd_{\mathcal{D}}\eta&=-\partial u- \bar \partial v.
\end{alignat*}
When $A$ is a Dolbeault algebra, that is, $A$ is a graded commutative
real differential algebra and the
product is compatible with the bigrading, then
$\mathcal{D}^{\ast}(A,\ast)$ has a product
\begin{displaymath}
\bullet\colon \mathcal{D}^{n}(A,p)\otimes
\mathcal{D}^{m}(A,q)\longrightarrow \mathcal{D}^{n+m}(A,p+q)
\end{displaymath}
that is graded commutative with respect to the first degree, it is
associative up to homotopy and satisfies the Leibnitz rule. The only
case where we will need the explicit formula for the product is for
$\omega \in
\mathcal{D}^{2p}(A,p)$ and $\eta \in \mathcal{D}^{m}(A,q)$:
\begin{math}
\omega \bullet \eta =\omega \land \eta.
\end{math}
The \emph{Deligne algebra of differential forms} on $X$
is defined to be
\begin{displaymath}
\mathcal{D}^{\ast}(X,\ast):=\mathcal{D}^{\ast}(E^{\ast}(X)_{{\mathbb R}},\ast).
\end{displaymath}
If $X$ is equi-dimensional of dimension $d$, there is a natural
trace map given by
\begin{displaymath}
\int \colon H^{2d}_{c}(X,{\mathbb R}(d))\to {\mathbb R},\quad
\omega \longmapsto \frac{1}{(2\pi i)^{d}}\int_{X}\omega.
\end{displaymath}
To take this trace map into account the Dolbeault complex of currents
is constructed as follows. Denote by $E^{\ast}_{c}(X)_{{\mathbb R}}$ the space
of differential forms with compact support. Then $D_{p,q}(X)$ is the
topological dual of $E^{p,q}_{c}(X)$ and $D_{n}(X)_{{\mathbb R}}$ is the topological
dual of $E^{n}_{c}(X)_{{\mathbb R}}$. In this complex the differential is given by
\begin{displaymath}
\dd T(\eta)=(-1)^{n}T(\dd \eta)
\end{displaymath}
for $T\in D_{n}(X)_{{\mathbb R}}$. For $X$ equi-dimensional of dimension $d$
we write
\begin{displaymath}
D^{p,q}(X)=D_{d-p,d-q}(X),\qquad D^{n}(X)_{{\mathbb R}}=(2\pi i)^{-d}D_{2d-n}(X).
\end{displaymath}
With these definitions, $D^{\ast}(X)_{{\mathbb R}}$ is a Dolbeault complex and
it is a Dolbeault module over $E^{\ast}(X)_{{\mathbb R}}$. We will denote
\begin{displaymath}
\mathcal{D}_{D}^{\ast}(X,\ast):=\mathcal{D}^{\ast}(D^{\ast}(X)_{{\mathbb R}},\ast).
\end{displaymath}
for the Deligne complex of currents on $X$. The trace map above defines an element
\begin{displaymath}
\delta _{X}\in \mathcal{D}_{D}^{0}(X,0).
\end{displaymath}
More generally, if $Y\subset X$ is a subvariety of pure codimension
$p$, then the current integration along $Y$, denoted $\delta _{Y}\in
\mathcal{D}_{D}^{2p}(X,p)$ is given by
\begin{displaymath}
\delta _{Y}(\omega )=\frac{1}{(2\pi i)^{d-p}}\int_{Y}\omega .
\end{displaymath}
Moreover, if $S\subset T^{\ast}X_{0}$ is a closed conical subset of the
cotangent bundle of $X$ with the zero section removed, we will denote
by $(\mathcal{D}_{D}^{\ast}(X,S,\ast),\dd_{\mathcal{D}})$ the Deligne
complex of currents on $X$ whose wave front set is contained in $S$.
For instance, if $N^{\ast}_{Y}$ is the conormal bundle to
$Y$, then
\begin{displaymath}
\delta _{Y}\in \mathcal{D}^{2p}_{D}(X,N^{\ast}_{Y},p).
\end{displaymath}
If $\omega $ is a locally integrable differential form, we associate
to it a current
\begin{displaymath}
[\omega ](\eta)=\frac{1}{(2\pi i)^{\dim X}}\int_{X}\eta\land
\omega.
\end{displaymath}
This map induces an isomorphism $\mathcal{D}^{\ast}(X,\ast)\to
\mathcal{D}_{D}^{\ast}(X,\emptyset,\ast)$ that we use to identify
them. For instance, when in a formula
sums of currents and differential forms appear, we will tacitly assume
that the differential forms are converted into currents by this map.
Note also that, if $f\colon X\to Y$ is a proper morphism of smooth complex
varieties of relative dimension $e$, then there are direct image
morphisms
\begin{displaymath}
f_{\ast}\colon \mathcal{D}^{n}_{D}(X,p)\longrightarrow
\mathcal{D}^{n-2e}_{D}(X,p-e).
\end{displaymath}
If $f$ is smooth,
the direct image of differential forms is defined by, first
converting them into currents and then applying the above direct image of
currents. If $f$ is a smooth morphism of relative dimension $e$ we can
convert them back into differential forms. This
procedure gives us $1/(2\pi i)^{e}$ times the usual integration along the
fiber.
We shall
use the notations and definitions of \cite{BurgosLitcanu:SingularBC}. In
particular, we write
\begin{align*}
\widetilde
{\mathcal{D}}^{n}(X,p)&=\left. \mathcal{D}^{n}(X,p)\right/
\dd_{\mathcal{D}}\mathcal{D}^{n-1}(X,p),\\
\widetilde
{\mathcal{D}}^{n}_{D}(X,p)&=\left. \mathcal{D}^{n}_{D}(X,p)\right/
\dd_{\mathcal{D}}\mathcal{D}^{n-1}_{D}(X,p).
\end{align*}
We now recall the definition of the set of normal direction of a map
and the definition of transverse morphisms.
\begin{definition}\label{def:15}
Let $f\colon X\to Y$ be a morphism of smooth complex varieties.
Let $T^{\ast} Y_{0}$ be the cotangent bundle to $Y$ with the zero
section removed.
The \emph{set of normal directions of} $f$ is the conic subset of
$T^{\ast} Y_{0}$ given by
\begin{displaymath}
N_{f}=\{(y,v)\in T^{\ast} Y_{0}| \dd f^{t}v=0\}.
\end{displaymath}
\end{definition}
\begin{definition}\label{def:4}
Let $f\colon X\to Y$ and $g\colon Z\to Y$ be morphisms of smooth
complex varieties. We say that $f$ and $g$ are \emph{transverse} if
\begin{math}
N_{f}\cap N_{g}=\emptyset.
\end{math}
\end{definition}
It is easily seen that, if $f$ is a closed immersion, this definition
of transverse morphisms agrees with that given in
\cite[IV-17.13]{GrothendieckDieudonne:EGAIV4}.
If $f$ and $g$ are transverse, then the cartesian product
$X\underset{Y}{\times}Z$ is smooth.
For lack of a good reference we
prove the following result.
\begin{proposition}
Let $f\colon X\to Y$ and $g\colon Z\to Y$ be transverse morphisms of smooth
complex varieties. Then they are tor-independent.
\end{proposition}
\begin{proof}
Since the conditions of being transverse and being
tor-independent are both local on $Y$, $X$ and $Z$ we may assume
that the map $f$ factorizes as $X\overset{i}{\to}Y\times
\mathbb{A}^{n}\overset{p}{\to} Y$, where $i$ is a closed immersion and $p$
is the projection. Let $g'\colon Z\times \mathbb{A}^{n}\to Y\times
\mathbb{A}^{n}$
be the morphism $g\times \Id$. If $f$ and $g$ are transverse then
$i$ and $g'$ are transverse. While, if $i$ and $g'$ are
tor-independent then $f$ and $g$ are tor-independent. Hence we
may suppose that $f$ is a closed immersion.
Since every closed immersion between smooth schemes is regular, we
may assume that $Y=\Spec A$, $X=\Spec A/I$, where $I$ is an ideal generated
by a regular sequence $(s_1,\dots,s_k)$ and $Z=\Spec B$. The
transversality condition
implies that $(s_1,\dots,s_k)$ is a regular sequence generating $
IB$. Let $K$ be the Koszul resolution of $A/I$ attached to the above
sequence. Then $K\otimes_{A} B$ is the Koszul resolution of $B/IB$,
hence exact. Therefore, $\Tor_{A}^{i}(A/I,B)=0$ for all $i\ge
1$. Thus $f$ and $g$ are tor-independent.
\end{proof}
Let now
$Y''\overset{h}{\to}Y'\overset{g}{\to}Y$ be morphisms of smooth complex
varieties such that $g$ and $g\circ h$ are smooth. We form
the cartesian diagram
\begin{displaymath}
\xymatrix{
X'' \ar[r]\ar[d]^{f''}& X'\ar[r]\ar[d]^{f'} & X\ar[d]^{f} \\
Y'' \ar[r]^{h}& Y' \ar[r]^{g} & Y.
}
\end{displaymath}
The smoothness of $g$ implies that
$N_{f'}=g^{\ast}N_{f}$. Then the smoothness of $g\circ h$ implies that
$h$ and $f'$ are transverse. Therefore, any current $\eta\in
\mathcal{D}_{D}^{\ast}(Y',N_{f'},\ast)$ can be pulled back to a
current $h^{\ast}\eta \in \mathcal{D}_{D}^{\ast}(Y'',N_{f''},\ast)$.
The following result will be used to characterize several Bott-Chern
classes and analytic torsion classes.
\begin{lemma} \label{lemm:uniquenes_phi}
Let $f\colon X\to Y$ be a morphism of smooth complex varieties. Let
$\widetilde \varphi$ be an assignment that, to each smooth morphism
of complex
varieties $g\colon Y'\to Y$ and each acyclic complex $\overline A$ of
hermitian vector bundles on $X':=X\underset{Y}{\times }Y'$
assigns a class
\begin{displaymath}
\widetilde \varphi(\overline A)\in \bigoplus
_{n,p}\widetilde{\mathcal{D}}_{D}^{n}(Y',g^{\ast}N_{f},p)
\end{displaymath}
fulfilling the following properties:
\begin{enumerate}
\item (Differential equation) the equality
\begin{math}
\dd_{\mathcal{D}}\widetilde \varphi(\overline A)
=0
\end{math}
holds;
\item (Functoriality) for each morphism $h\colon Y''\to Y'$ of smooth complex varieties
with $g\circ h $ smooth, the relation
\begin{math}
h^{\ast} \widetilde \varphi(\overline A)=\widetilde \varphi(h^{\ast}\overline A)
\end{math}
holds;
\item (Normalization) if $\overline A$ is orthogonally split, then $\widetilde
\varphi(\overline A)=0$.
\end{enumerate}
Then $\widetilde \varphi=0$.
\end{lemma}
\begin{proof}
The argument of the proof of \cite[Thm.
2.3]{BurgosLitcanu:SingularBC} applies \emph{mutatis mutandis} to
the present situation. One only needs to observe that all the
operations with differential forms of that argument can be extended to
the currents that appear in the present situation thanks to the
hypothesis about their wave front sets.
\end{proof}
In the paper \cite{BurgosFreixasLitcanu:HerStruc} we defined and
studied hermitian structures on objects of the bounded derived
category of coherent sheaves on a smooth complex variety. The language
and the results of \emph{loc. cit.} will be used extensively in this
paper. We just mention here that a hermitian metric on an object
$\mathcal{F}$ of $\Db(X)$ is an isomorphism $E\dashrightarrow \mathcal{F}$ in
$\Db(X)$, with $E$ a bounded complex of vector bundles, together with
a choice of a hermitian metric on each constituent vector bundle of
$E$. Such an isomorphism always exists due to the fact that we work in
the algebraic category. A hermitian structure is an equivalence class
of hermitian metrics. To each smooth complex variety $X$, we
associated the category $\oDb(X)$
(\cite[\S~3]{BurgosFreixasLitcanu:HerStruc}) whose objects are objects
of $\Db(X)$ provided with a hermitian structure. We introduced the
hermitian cone (\cite[Def.~3.14]{BurgosFreixasLitcanu:HerStruc}),
denoted $\ocone$, of a morphism in $\oDb(X)$. We also
defined Bott-Chern
classes for isomorphisms
(\cite[Thm.~4.11]{BurgosFreixasLitcanu:HerStruc}) and distinguished
triangles (\cite[Thm.~4.18]{BurgosFreixasLitcanu:HerStruc}) in
$\oDb(X)$. We introduced a universal abelian group for additive
Bott-Chern classes. Namely, the set of hermitian structures on a zero
object of $\Db(X)$ is an abelian group that we denote $\KA(X)$
(\cite[Def.~2.31]{BurgosFreixasLitcanu:HerStruc}).
Finally, we defined the category $\oSm_{\ast/{\mathbb C}}$
(\cite[\S~5]{BurgosFreixasLitcanu:HerStruc}) whose objects are
smooth complex varieties and whose morphisms are projective morphisms
together with a hermitian structure on the relative
tangent complex.
We introduce now one of the central objects of the paper.
\begin{definition} \label{def:19}
Let $f\colon X\to Y$ be a projective morphism of smooth complex
varieties and $\overline f\in \Hom_{\oSm_{\ast/{\mathbb C}}}(X,Y)$ a morphism
over $f$. Let $\overline{\mathcal{F}}\in \Ob \oDb(X)$ and let $\overline{
f_{\ast}\mathcal{F}}\in \Ob \oDb(Y)$ be an object over $
f_{\ast}\mathcal{F}$. The triple $\overline \xi=(\overline f,\overline
{\mathcal{F}},\overline{ f_{\ast}\mathcal{F}})$ will be called \emph{a
relative metrized complex}. When $f$ is a closed immersion we
will also call it an \emph{embedded metrized complex}. When
$\overline{\mathcal{F}}$ and $\overline{ f_{\ast}\mathcal{F}}$ are clear from
the context we will denote the relative metrized complex $\overline \xi $
by the morphism $\overline f$.
\end{definition}
Let $\overline \xi=(\overline f,\overline
{\mathcal{F}},\overline{ f_{\ast}\mathcal{F}})$ be a relative metrized
complex and let $g\colon Y'\to Y$ be a morphism
of smooth complex varieties that is transverse to $f$. Consider the
cartesian diagram
\begin{equation}\label{eq:9}
\xymatrix{
X'\ar [r]^{g'}\ar[d]_{f'}& X\ar[d]^{f}\\
Y'\ar [r]_{g}& Y.
}
\end{equation}
Then $f'$ is still projective. Moreover, the transversality condition
implies that the canonical morphism $ {g'}^{\ast} T_{\overline f}\dashrightarrow T_{f'}$ is a
hermitian structure on $T_{f'}$.
We define
\begin{equation}
\label{eq:67}
g^{\ast}\overline
f=(f', {g'}^{\ast} T_{\overline f})\in \Hom_{\oSm_{\ast/{\mathbb C}}}(X',Y').
\end{equation}
By tor-independence, there is a canonical isomorphism
\begin{math}
g^{\ast} f_{\ast} \mathcal{F}\dashrightarrow f'_{\ast}
{g'}^{\ast}\mathcal{F}.
\end{math}
Therefore $ g^{\ast}\overline{ f_{\ast} \mathcal{F}}$ induces a
hermitian structure on $ f'_{\ast}
{g'}^{\ast}\mathcal{F}$.
\begin{definition}
The \emph{pull-back } of $\overline \xi$ by $g$ is the relative metrized
complex
\begin{displaymath}
g^{\ast}\overline \xi=( g^{\ast}\overline f,
{g'}^{\ast}\overline{\mathcal{F}}, g^{\ast}\overline{ f_{\ast}
\mathcal{F}}).
\end{displaymath}
\end{definition}
\begin{definition}
Let $\overline\xi=(\overline f\colon X\to Y,\overline{\mathcal{F}},\overline{
f_{\ast}\mathcal{F}})$ be a relative metrized complex. Let
$\overline{\mathcal{G}}$ be an object of $\oDb(Y)$. The hermitian
structures on $\overline{ f_{\ast}\mathcal{F}}$
and $\overline{\mathcal{G}}$ induce a natural hermitian structure on $
f_{\ast}(\mathcal{F}\otimes^{\Ld} f^{\ast}\mathcal{G})$ that we denote
$\overline{ f_{\ast}
\mathcal{F}}\otimes^{\Ld}\overline{\mathcal{G}}$.
The \emph{tensor product of $\overline{\xi}$ by $\overline{\mathcal{G}}$} is
then defined to be the relative metrized complex
\begin{displaymath}
\overline{\xi}\otimes\overline{\mathcal{G}}=(\overline{f}, \overline{\mathcal{F}}
\otimes^{\Ld} f^{\ast}\overline{\mathcal{G}},\overline{ f_{\ast}
\mathcal{F}}\otimes^{\Ld}\overline{\mathcal{G}}).
\end{displaymath}
\end{definition}
\begin{definition}\label{def:18}
Let $\overline{\xi}_i=(\overline{f},\overline{\mathcal{F}_{i}},\overline{
f_{\ast}\mathcal{F}_{i}})$, $i=1,2$ be relative metrized coherent complexes
on $X$.
Then \emph{the direct sum relative metrized complex} is
\begin{displaymath}
\overline{\xi}_1\oplus\overline{\xi}_2 :=
(\overline{f},\overline{\mathcal{F}_{1}}\oplus
\overline{\mathcal{F}_{2}},\overline{ f_{\ast}\mathcal{F}_{1}}\oplus
\overline{ f_{\ast}\mathcal{F}_{2}}).
\end{displaymath}
\end{definition}
We now introduce a notation for Todd-twisted direct images of currents
and differential forms, that will simplify many formulas
involving the Todd genus.
Let $\overline f=(f,\overline T_f)$ be a morphism in $\oSm_{\ast/{\mathbb C}}$. To $\overline
f$ we associate a Todd differential form $\Td(\overline f):=\Td(\overline
T_{f})\in \bigoplus_{p} \mathcal{D}^{2p}(X,p)$
\cite[(5.15)]{BurgosFreixasLitcanu:HerStruc}.
Let
$S$
be a closed conic subset of $T^{\ast}X_{0}$. Then we denote
\begin{equation}
\label{eq:28}
f_{\ast}(S)=\{(f(x),\eta )\in T^{\ast}Y_{0}\mid (x,(\dd
f)^{t}\eta)\in S\}\cup N_{f}.
\end{equation}
If $g\colon Y\to Z$ is another morphism of smooth complex varieties,
it is easy to see that we have $(g\circ f)_{\ast}(S)\subseteq
g_{\ast}f_{\ast}(S)$.
\begin{definition}
Let $\overline f\colon X\to Y$ be a morphism in $\overline\Sm_{\ast/{\mathbb C}}$ of
relative dimension~$e$. For each closed conical subset $S\subset
T^{\ast}X_{0}$ and each pair of integers $n$, $p$, we
define the map
\begin{displaymath}
\overline f_{\flat}\colon \mathcal{D}^{n}_{D}(X,S,p)\to
\mathcal{D}^{n-2e}_{D}(Y,f_{\ast}S,p-e), \quad
\overline f_{\flat}(\omega )= f_{\ast}(\omega \bullet \Td(\overline f)).
\end{displaymath}
\end{definition}
\begin{proposition}\label{prop:12}
Let $\overline f\colon X\to Y$ and $\overline g\colon Y \to Z$ be morphisms
in $\overline\Sm_{\ast/{\mathbb C}}$ of relative dimensions $e_{1}$ and $e_{2}$
respectively. Let $S\subset T^{\ast}X_{0}$, $T\subset T^{\ast}Y_{0}$ be closed conical subsets and
let $\overline h=\overline f\circ \overline g$, of relative dimension $e=e_1+e_2$.
\begin{enumerate}
\item \label{item:prop_12_2}The following diagram is commutative
\begin{displaymath}
\xymatrix{
\mathcal{D}_{D}^{n}(X,S,p)\ar[r]^{\overline f_{\flat}}
\ar[d]_{\overline h_{\flat}}&
\mathcal{D}_{D}^{n-2e_1}(Y,f_{\ast} S,p-e_1) \ar [d]^{\overline g_{\flat}} \\
\mathcal{D}_{D}^{n-2e}(Z,h_{\ast} S,p-e) \ar@{^{(}->}[r]&
\mathcal{D}_{D}^{n-2e}(Z,g_{\ast}f_{\ast} S,p-e).
}
\end{displaymath}
\item \label{item:prop_12_3} Let
$\theta\in\mathcal{D}_{D}^{m}(X,S,q)$ and
$\omega\in\mathcal{D}_{D}^{n}(Y,T,p)$. Assume $T\cap
N_{f}=\emptyset$ and that $T+f_{\ast}S$ is disjoint with the zero
section in $T^{\ast}Y_{0}$. Then $f^{\ast}T+S$ is disjoint with
the zero section and there is an equality of currents
\begin{displaymath}
\overline{f}_{\flat}(f^{\ast}(\omega)\bullet\theta)=\omega\bullet \overline{f}_{\flat}(\theta)
\end{displaymath}
in $\mathcal{D}_{D}^{n+m}(Y,W,p+q)$, with
\begin{math}
W=f_{\ast}(S + f^{\ast}T)\cup f_{\ast}S\cup f_{\ast}f^{\ast}T.
\end{math}
\end{enumerate}
\end{proposition}
\begin{proof}
For the first assertion, it is enough to notice the equality of currents
\begin{displaymath}
\overline{g}_{\flat}(\overline{f}_{\flat}(\omega))=(g\circ f)_{\ast}(\omega\bullet f^{\ast}\Td(\overline{g})\bullet\Td(\overline{f}))).
\end{displaymath}
For the second, it is easy to see that $f^{\ast}T+S$ does not cross
the zero section, and hence both sides of the equality are
defined. It then suffices to establish the equality of currents
\begin{math}
f_{\ast}(f^{\ast}\omega\bullet\theta)=f_{\ast}(\omega)\bullet\theta.
\end{math}
If $\theta$ and $\omega$ are smooth, then the equality follows from
the definitions. The general case
follows by approximation of $\theta$ and $\omega$ by smooth currents
and the continuity of the operators $f^{\ast}$ and $f_{\ast}$.
\end{proof}
\begin{proposition} \label{prop:11} Let $\overline{f}$ be a morphism in
$\overline\Sm_{\ast/{\mathbb C}}$ of relative dimension $e$ and $S$ a closed
conical subset of $T^{\ast}X_{0}$. Let $g\colon Y'\to Y$ be a morphism of
smooth complex varieties transverse to $f$. Consider the cartesian
diagram \eqref{eq:9} and let $\overline f'=g^{\ast} \overline f$.
Suppose that $N_{g'}$ is
disjoint with $S$. Then:
\begin{enumerate}
\item $N_{g}$ and $f_{\ast}S$ are disjoint and
$g^{\ast}f_{\ast}S\subset f'_{\ast}{g'}^{\ast}S$;
\item the following diagram commutes:
\begin{displaymath}
\xymatrix{
\mathcal{D}_{D}^{n}(X,S,p)\ar[r]^{\hspace{-1cm}\overline{f}_{\flat}}
\ar[d]_{{g'}^{\ast}}
&\mathcal{D}_{D}^{n-2e}(Y,f_{\ast}S,p-e)\ar[d]^{g^{\ast}}\\
\mathcal{D}_{D}^{n}(X',{g'}^{\ast}S,
p)\ar[r]^{\hspace{-1cm}\overline{f}'_{\flat}}
&\mathcal{D}_{D}^{n-2e}(Y',f'_{\ast}{g'}^{\ast}S,p-e)\\
}
\end{displaymath}
\end{enumerate}
\end{proposition}
\begin{proof}
The first claim follows from the definitions. In
particular the diagram makes sense. For the commutativity of the
diagram, we observe that, since
\begin{math}
g'{}^{\ast}\Td(\overline f)=\Td(\overline f'),
\end{math}
it suffices to check the equality of currents
\begin{math}
g^{\ast}f_{\ast}(\theta)={f'}_{\ast}{g'}^{\ast}(\theta)
\end{math}
for $\theta\in\mathcal{D}_{D}^{n}(X,S,p)$.
By the continuity of the operators $g^{\ast}$, ${g'}^{\ast}$,
$f_{\ast}$ and ${f'}_{\ast}$, it is enough to prove the relation
whenever $\theta$ is smooth. Moreover, using a partition of unity
argument we are reduced to the following local analytic statement.
\begin{lemma} \label{lemm:1}
Let $f\colon X\to Y$ and $g\colon Y'\to Y$ be transverse morphisms
of complex manifolds. Let $\theta $ be a smooth differential form
on $X$ with compact support. Consider the diagram
\eqref{eq:9}. Then
\begin{equation}\label{eq:52}
g^{\ast}f_{\ast}(\theta)={f'}_{\ast}{g'}^{\ast}(\theta).
\end{equation}
\end{lemma}
\noindent
\emph{Proof.}
The map $f$ can be factored as $X\overset{\varphi
}{\longrightarrow}X\times Y\overset{p_{2}}{\longrightarrow} Y$,
where $\varphi(x)=(x,f(x))$ is a closed immersion and $p_{2}$, the
second projection, is smooth. Using again the continuity of the
operators $g^{\ast}$ (respectively ${g'}^{\ast}$) and
$f_{\ast}$ (respectively ${f'}^{\ast}$), we are reduced to prove the
equation \eqref{eq:52} in the case when $f$ is smooth and in the
case when $f$ is a closed immersion. The case when $f$ is smooth
is clear. Assume now that $f$ is a closed immersion. By
transversality, $f'$ is also a closed immersion of complex
manifolds. We may assume that $\theta=f^{\ast}\widetilde \theta $
for some smooth form $\widetilde \theta $ on $Y$. Then equation
\eqref{eq:52} follows from the chain of equalities
\begin{displaymath}
g^{\ast}f_{\ast}\theta = g^{\ast}f_{\ast}f^{\ast}\widetilde
\theta= g^{\ast}(\widetilde \theta \land \delta _{X})=
g^{\ast}(\widetilde \theta) \land \delta _{X'}
=f'_{\ast}f'{}^{\ast}g^{\ast}\widetilde \theta
= f'_{\ast} g'{}^{\ast} f^{\ast}\widetilde \theta
= f'_{\ast} g'{}^{\ast} \theta .
\end{displaymath}
This concludes the proof of the lemma and the proposition. \hfill
$\square$
\end{proof}
\section{Analytic torsion for closed immersions}
\label{sec:analyt-tors-clos}
In the paper \cite{BurgosLitcanu:SingularBC} the authors study the
singular Bott-Chern classes associated to closed immersions of smooth
complex varieties. The singular Bott-Chern classes are the analogue,
for closed immersions, of the analytic torsion for smooth morphisms.
For this reason, we will call them also analytic torsion classes.
The aim of this section is to
recall the main results of \cite{BurgosLitcanu:SingularBC} and to
translate them into the language of derived categories.
\begin{definition} \label{def:1}
A \emph{theory of analytic torsion classes for closed immersions} is
a map that, to each embedded metrized complex $\overline \xi =(\overline
f\colon X\to Y,
\overline{\mathcal{F}}, \overline{ f_{\ast}\mathcal{F}})$
assigns a class
\begin{displaymath}
T(\overline \xi) \in
\bigoplus _{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(Y,N_{f},p)
\end{displaymath}
satisfying the following conditions.
\begin{enumerate}
\item (Differential equation) The equality
\begin{math}
\dd_{\mathcal{D}}T(\overline \xi) =
\ch(\overline{
f_{\ast}\mathcal{F}})-\overline f_{\flat}[\ch(\overline{\mathcal{F}})]
\end{math}
holds.
\item (Functoriality) For every morphism $h\colon Y'\to Y$ of smooth complex varieties
that is
transverse to $f$ we have the equality
\begin{math}
h^{\ast}T(\overline \xi)=
T(h^{\ast} \overline \xi).
\end{math}
\item (Normalization) If $X=\emptyset$ (hence $\overline
{\mathcal{F}}=\overline 0$), $Y=\Spec {\mathbb C}$, and $\overline {
f_{\ast}\mathcal{F}}=\overline 0$, then
\begin{math}
T(\overline f,\overline 0,\overline 0)=0.
\end{math}
\end{enumerate}
\end{definition}
\begin{definition} Let $T$ be a theory of analytic torsion classes for closed
immersions.
\begin{enumerate}
\item We say that $T$
is \emph{compatible with the projection formula} if, for every embedded
metrized complex $\overline \xi=(\overline f, \overline{\mathcal{F}}, \overline{
f_{\ast}\mathcal{F}})$,
and every object $\overline {\mathcal{G}}\in \oDb(Y)$, we have
\begin{equation}\label{eq:69}
T(\overline{\xi}\otimes\overline{\mathcal{G}})=
T(\overline \xi) \bullet
\ch(\overline {\mathcal{G}}).
\end{equation}
\item We say that $T$ is \emph{additive} if, given $\overline{\xi}_i=
(\overline{f},\overline{\mathcal{F}_{i}},\overline{ f_{\ast}\mathcal{F}_{i}})$,
$i=1,2$, two embedded
metrized complexes, we have
\begin{equation}
\label{eq:49}
T(\overline{\xi}_1\oplus\overline{\xi}_{2})=T(\overline\xi_{1})+T(\overline{\xi}_{2}).
\end{equation}
\item We say that $T$ is \emph{transitive} if, given a embedded
metrized complex
$\overline \xi=(\overline f, \overline{\mathcal{F}}, \overline{ f_{\ast}\mathcal{F}})$, a
closed immersion of smooth complex varieties
$g\colon Y\to Z$, a morphism $\overline g$ over $g$, and an object
$\overline{ (g\circ f)_{\ast}\mathcal{F}}\in \Ob \oDb(Z)$ over $
(g\circ f)_{\ast}\mathcal{F}$, we have
\begin{equation}
\label{eq:68}
T(\overline g\circ \overline f)=
T(\overline g) +
\overline g_{\flat}(T(\overline f)).
\end{equation}
\end{enumerate}
\end{definition}
\begin{remark} \label{rem:4}
\begin{enumerate}
\item \label{item:52} If $T$ is well defined for objects of $\oDb$,
then the normalization condition in Definition \ref{def:1} and the
normalization
condition in \cite[Def.
6.9]{BurgosLitcanu:SingularBC} are equivalent. The compatibility
with the projection formula implies
the normalization condition and the additivity (see
\cite[Prop. 10.9]{BurgosLitcanu:SingularBC})
\item \label{item:53} To check that a theory
is compatible with the projection formula, it is enough to
consider complexes consisting of a single
hermitian vector bundle in degree 0.
\end{enumerate}
\end{remark}
Let $X$ be a smooth complex variety and let $\overline N$ be a hermitian
vector bundle of rank~$r$. We denote by $P={\mathbb P}(N\oplus \mathbf{1})$
the projective
bundle obtained by completing $N$. Let $\pi _{P}\colon P\to X$ be the
projection and let $s\colon X\to P$ be the zero
section. Since $N$ can be identified with the normal bundle to $X$ on
$P$, the hermitian metric of $\overline N$ induces a hermitian structure on
$s$. We will denote it by $\overline s$. On $P$ we
have a tautological quotient vector bundle with an
induced metric $\overline Q$. For each hermitian vector bundle $\overline F$ on
$X$ we have the Koszul resolution $K(F,N)$ of $s_{\ast}F$. We
denote by $K(\overline F,\overline N)$ the Koszul resolution with the induced
metrics. See \cite[Def. 5.3]{BurgosLitcanu:SingularBC} for
details.
\begin{definition} \label{def:23}
Let $T$ be a theory of analytic torsion classes for closed
immersions. We say that $T$ is \emph{homogeneous} if, for every pair
of hermitian vector bundles $\overline N$ and $\overline F$ with $\rk N=r$, there
exists a homogeneous class of bidegree $(2r-1,r)$ in the Deligne
complex
$$\widetilde e(\overline F,\overline N)\in
\widetilde{\mathcal{D}}^{2r-1}_{D}(P,N_{s},r)$$
such that
\begin{equation}\label{eq:43}
T(\overline s,\overline F,K(\overline F,\overline N))\bullet \Td(\overline Q)=
\widetilde e(\overline F,\overline N)\bullet
\ch(\pi _{P}^{\ast}\overline F).
\end{equation}
\end{definition}
\begin{remark}
Observe that Definition \ref{def:23} is equivalent to
\cite[Def. 9.2]{BurgosLitcanu:SingularBC}. The advantage of
the definition in this paper is that it treats on equal footing the
case when $\rk F=0$.
\end{remark}
Let ${\mathbb D}$ denote the base ring for Deligne cohomology (see
\cite{BurgosLitcanu:SingularBC} before Definition 1.5). A consequence
of \cite[Thm. 1.8]{BurgosLitcanu:SingularBC} is that there is a
bijection between the set of additive genus in Deligne cohomology and
the set of power series in one variable ${\mathbb D}[[x]]$. To each power
series $\varphi \in {\mathbb D}[[x]]$ it corresponds the unique additive genus
such that
\begin{math}
\varphi(L)=\varphi (c_{1}(L))
\end{math}
for every line bundle $L$.
\begin{definition} \label{def:25}
A \emph{real additive genus} is an additive genus such that the
corresponding power series belongs to ${\mathbb R}[[x]]$.
\end{definition}
Let
$\mathbf{1}_{1}\in {\mathbb D}$ be the element represented by the constant
function 1 of $\mathcal{D}^{1}(\Spec \mathbb{C},1)={\mathbb R}$.
The main result of \cite{BurgosLitcanu:SingularBC} can be written
as follows.
\begin{theorem}\label{thm:17}
\begin{enumerate}
\item \label{item:55} There is a unique homogeneous theory of
analytic torsion classes for
closed immersions, that we denote $T^{h}$. This theory is compatible
with the projection formula, additive and transitive.
\item \label{item:56} Let $T$ be any transitive theory of analytic
torsion classes
for closed immersions, that is compatible with the projection
formula. Then there is a unique real additive genus $S_{T}$
such that, for any embedded metrized
complex $\overline \xi :=(\overline f, \overline{\mathcal{F}}, \overline{
f_{\ast}\mathcal{F}})$, we have
\begin{equation}
\label{eq:70}
T(\overline \xi )-T^{h}(\overline \xi )=-f_{\ast}[\ch(\mathcal{F})\bullet
\Td(T_{f}) \bullet S_{T}(T_{f})\bullet \mathbf{1}_{1}].
\end{equation}
\item \label{item:57}
Conversely, any real additive
genus $S$ defines, by means of equation \eqref{eq:70}, a
unique transitive
theory of analytic torsion classes $T_S$
for closed immersions, that is compatible with the projection
formula and additive.
\end{enumerate}
\end{theorem}
\begin{proof}
Existence and uniqueness for both $T^{h}$ and $T_S$ is the content of
\cite{BurgosLitcanu:SingularBC} when restricting to triples
$\overline{\xi}$ with $T_{\overline f}=\overline{N}_{X/Y}[-1]$, $\overline{\mathcal{F}}$ a
hermitian vector bundle placed in degree 0 and $\overline{
f_{\ast}\mathcal{F}}$ given by a finite locally free resolution. For
the general case, we thus need to prove that the theories of
analytic torsion classes for closed immersions in the sense of
\emph{loc. cit.} uniquely extend to arbitrary $\overline{\xi}$, fulfilling
the desired properties.
Assume given a theory $T$ in the sense of
\cite{BurgosLitcanu:SingularBC}, compatible with the projection
formula and transitive. We will call $T$ the initial theory.
We consider a triple $\overline{\xi}$ with $T_{\overline
f}=\overline{N}_{X/Y}[-1]$ and $\overline{\mathcal{F}}\in\Ob\oDb(X)$. Choose
a representative $\overline{F}\dashrightarrow\mathcal{F}$ of the hermitian structure
on $\overline{\mathcal{F}}$. We then define $T(\overline{\xi})$ by induction on
the length of the complex $F$. First suppose that $F=F^{d}[-d]$
consists of a
single vector bundle placed in degree $d$. Choose a finite
locally free resolution
\begin{displaymath}
\dots\to E^{1}\to E^{0}\to f_{\ast} F^{d}\to 0.
\end{displaymath}
Endow the vector bundles $E^{i}$ with smooth hermitian
metrics. Observe that there is an induced isomorphism
\begin{math}
\overline{E}[-d]\overset{\sim}{\dashrightarrow}\overline{ f_{\ast}\mathcal{F}},
\end{math}
in $\oDb(Y)$,
whose Bott-Chern classes $\cht$ are
defined in \cite[\S~4]{BurgosFreixasLitcanu:HerStruc}. We then put
\begin{equation}\label{eq:72}
T(\overline{\xi})=(-1)^{d}T(\overline{N}_{X/Y}, \overline{F}^{d},\overline{E})
+\widetilde{\ch}(\overline{E}[-d]\overset{\sim}{\dashrightarrow}\overline{
f_{\ast}\mathcal{F}}).
\end{equation}
This definition does not depend on the choice of
representative of the hermitian structure on $\overline{\mathcal{F}}$, nor
on the choice of $\overline{E}$, due to
\cite[Thm.~4.11, Prop.~4.13]{BurgosFreixasLitcanu:HerStruc},
and \cite[Cor.~6.14]{BurgosLitcanu:SingularBC}. The differential equation is
satisfied as a consequence of the differential equations for
$T(\overline{N}_{X/Y},\overline{F}^{d},\overline{E})$ and
$\widetilde{\ch}(\overline{E}[-d]\overset{\sim}{\dashrightarrow}\overline{
f_{\ast}\mathcal{F}})$. The compatibility with pull-back by
morphisms $h\colon Y'\to Y$ transverse to $f$ is immediate as
well. Finally, notice that by construction, if
$\overline{\xi}'=(\overline{N}_{X/Y}, \overline{\mathcal{F}}, \overline{
f_{\ast}\mathcal{F}}')$, then
\begin{equation}\label{eq:71}
T(\overline{\xi'})=T(\overline{\xi})+
\widetilde{\ch}(\overline{ f_{\ast}\mathcal{F}}' ,
\overline{ f_{\ast}\mathcal{F}}).
\end{equation}
Now suppose that $T(\overline{\xi})$ has been defined for $F$ of length
$l$, satisfying in addition (\ref{eq:71}). If $F$ has length $l+1$,
let $F^{d}$ be the first non-zero vector bundle of $F$. Consider the
exact sequence of complexes
\begin{displaymath}
(\overline{\varepsilon})\qquad 0\to\sigma^{> d}\overline{F}\to \overline{F}\to
\overline{F}^{d}[-d]\to 0,
\end{displaymath}
where $\sigma^{>d}$ is the b\^ete filtration. Observe that as a
distinguished triangle, $(\overline{\varepsilon})$ is tightly
distinguished, hence
$\widetilde{\ch}(\overline{\varepsilon})=0$. Choose hermitian metrics on
$ f_{\ast}\sigma^{>d}F$ and $ f_{\ast} F^{d}[-d]$. We thus have a
distinguished triangle in $\oDb(Y)$
\begin{displaymath}
(\overline{\tau})\qquad \overline{ f_{\ast}\sigma^{>d}F}\to\overline{ f_{\ast} F}
\to\overline{ f_{\ast} F^{d}}[-d]\to\overline{ f_{\ast}\sigma^{>d}F}[1]\to\dots.
\end{displaymath}
We define
\begin{equation} \label{eq:74}
T(\overline{\xi})=T(\overline{N}_{X/Y},\sigma^{>d}\overline{F},\overline{
f_{\ast}\sigma^{>d}F})
+(-1)^{d}T(\overline{N}_{X/Y}, \overline{F}^{d},\overline{ f_{\ast} F^{d}})
-\widetilde{\ch}(\overline{\tau}).
\end{equation}
This does not depend on the choice of hermitian structures on $
f_{\ast}\sigma^{>d}F$ and $ f_{\ast} F^{d}$, by the analogue to
\cite[Thm.~3.33~(vii)]{BurgosFreixasLitcanu:HerStruc} for
$\widetilde{\ch}$ and because \eqref{eq:71} holds by assumption
for $T(\overline{N}_{X/Y},\sigma^{>d}\overline{F},\overline{
f_{\ast}\sigma^{>d}F})$ and $T(\overline{N}_{X/Y}, \overline{F}^{d},\overline{
f_{\ast} F^{d}})$. Similarly, \eqref{eq:71} holds for
$T(\overline{\xi})$ defined in \eqref{eq:74}. The differential equation
and compatibility with pull-back are proven as in the first
case. This concludes the proof of the existence in case that
$T_{\overline f}=\overline{N}_{X/Y}[-1]$.
To conclude with the existence, we may now consider a general
$\overline{\xi}$. Choose a hermitian metric on the normal bundle
$N_{X/Y}$. Put
$\overline{\xi}'=(\overline{N}_{X/Y}[-1],\overline{\mathcal{F}},\overline{
f_{\ast}\mathcal{F}})$.
We define
\begin{equation}\label{eq:73}
T(\overline{\xi})=T(\overline{\xi}')+
\overline f_{\flat}[\ch(\overline{\mathcal{F}})\bullet
\widetilde{\Td}_{m}(
T_{\overline f}\dashrightarrow \overline{N}_{X/Y}[-1])],
\end{equation}
where $\widetilde{\Td}_{m}$ is the multiplicative Todd secondary
class defined in \cite[\S~5]{BurgosFreixasLitcanu:HerStruc}.
It is straightforward from the definition that $T(\overline{\xi})$ satisfies
the differential equation and is compatible with pull-back by
morphisms transverse to $f$. We call $T$ the extended theory.
We now proceed to prove that the extended theory $T$ is transitive
and compatible with the projection formula. For the projection
formula, it suffices by Remark \ref{rem:4}~\ref{item:53} to prove
\begin{displaymath}
T(\overline{\xi}\otimes\overline{G})=T(\overline{\xi})\bullet\ch(\overline{G})
\end{displaymath}
for $\overline{G}$ a hermitian vector bundle placed in degree 0. This
readily follows from the inductive construction of the extended
theory $T$ and the assumptions on the initial theory $T$. One
similarly establishes the transitivity and the additivity
We conclude by observing that, since Lemma \ref{lemm:uniquenes_phi}
implies that the equations \eqref{eq:72}, \eqref{eq:71},
\eqref{eq:74} and \eqref{eq:73} hold, the theory $T(\overline{\xi})$ thus
constructed for arbitrary $\overline{\xi}$ is completely determined by the
values $T(\overline{\xi}')$, with $\overline{\xi}'$ of the form $(\overline{N}_{X/Y},
\overline{F},\overline{E})$ where $\overline{F}$ is a hermitian vector bundle
and $E\to f_{\ast}F$ is a finite locally free resolution.
Once we have seen that any theory of singular Bott-Chern classes as
in \cite{BurgosLitcanu:SingularBC} can be uniquely extended, then
statements \ref{item:56} and \ref{item:57} follow combining equation
(7.3) and Corollary 9.43 in \cite{BurgosLitcanu:SingularBC}. Note
that the minus sign in equation \eqref{eq:70} comes from the fact
that $S(T_{f})=-S(N_{X/Y})$.
\end{proof}
In \cite[\S 6]{BurgosLitcanu:SingularBC} several
anomaly formulas are proved. We now indicate the translation of these
formulas to the current setting. Recall that we are using the notation
of \cite{BurgosFreixasLitcanu:HerStruc} with respect to secondary
characteristic classes.
\begin{proposition}\label{prop:1bis
Let $T$ be a theory of analytic torsion classes for closed immersions. Let
\begin{math}
\overline{\xi}= (\overline{f}\colon X\to Y,\overline{\mathcal{F}}, \overline{
f_{\ast}\mathcal{F}})
\end{math}
be an embedded metrized complex.
\begin{enumerate}
\item \label{item:6bis}
If $\overline{\mathcal{F}}'$ is another choice
of metric on $\mathcal{F}$ and
$\overline{\xi}_1=(\overline{f}\colon X\to Y,\overline{\mathcal{F}}', \overline{
f_{\ast}\mathcal{F}})$, then
\begin{displaymath}
T(\overline{\xi}_1)=T(\overline{\xi})+\overline f
_{\flat}[\cht(\overline{\mathcal{F}}',\overline{\mathcal{F}})].
\end{displaymath}
\item \label{item:7bis}
If $\overline{f}'$ is another hermitian structure on
$f$ and
$\overline{\xi}_{2}=(\overline{f}'\colon X\to Y,\overline{\mathcal{F}}, \overline{
f_{\ast}\mathcal{F}})$, then
\begin{equation}\label{eq:14bis}
T(\overline{\xi}_{2})=T(\overline{\xi})+\overline f'
_{\flat}[\ch(\overline {\mathcal{F}})\bullet
\widetilde{\Td}_{m}(\overline{f}',\overline{f})].
\end{equation}
\item \label{item:8bis}
If $\overline{ f_{\ast}\mathcal{F}}'$ is another
choice of metric on $ f_{\ast}\mathcal{F}$, and $\overline
{\xi}_{3}=(\overline{f}\colon X\to Y,\overline{\mathcal{F}}, \overline{
f_{\ast}\mathcal{F}}')$, then
\begin{displaymath}
T(\overline{\xi}_{3})= T(\overline{\xi})-
\cht(\overline{ f_{\ast}\mathcal{F}}',
\overline{ f_{\ast}\mathcal{F}}).
\end{displaymath}
\end{enumerate}
\end{proposition}
\begin{proof}
We first prove the second assertion. Let $\overline{E}\dashrightarrow
T_{f}$ be a representative of the hermitian structure on
$T_{\overline{f}}$. By
\cite[Thm.~3.13~(ii)]{BurgosFreixasLitcanu:HerStruc}, we may assume
the
hermitian structure on $T_{\overline{f}'}$ is represented by the
composition
\begin{displaymath}
\overline{E}\oplus\overline{A}\overset{\pr_{1}}{\longrightarrow }\overline{E}\dashrightarrow T_{f}
\end{displaymath}
for some bounded acyclic complex $\overline A$ of hermitian vector
bundles on $X$. For every
smooth morphism $g\colon Y'\to Y$ of complex varieties, consider the
cartesian diagram~\eqref{eq:9}.
We introduce the assignment that, to every such $g$ and each bounded
acyclic complex of hermitian vector bundles $\overline{A}$ on $X'$,
assigns the class
\begin{displaymath}
\begin{split}
\widetilde{\varphi}(\overline{A})=&T({g'}^{\ast}\overline{\xi})
-T\left((f', {g'}^{\ast}T_{\overline{f}}+ [\overline{A}]),
{g'}^{\ast}\overline{\mathcal{F}},
g^{\ast}\overline{ f_{\ast}\mathcal{F}}\right)\\
&+{f'}_{\ast}\left[\ch( {g'}^{\ast}\overline{\mathcal{F}})
\widetilde{\Td}_{m}\left(({g'}^{\ast}T_{\overline{f}} +
[\overline{A}]), {g'}^{\ast}T_{\overline{f}}\right) \Td(
{g'}^{\ast}T_{\overline{f}}+[\overline{A}])\right].
\end{split}
\end{displaymath}
Here $[\overline{A}]$ stands for the class of $\overline{A}$ in
$\KA(X')$ (\cite[Def.~2.31]{BurgosFreixasLitcanu:HerStruc}) and $+$
denotes the action of $\KA(X')$ on $\oDb(X')$
(\cite[Thm.~3.13]{BurgosFreixasLitcanu:HerStruc}). Since
$\widetilde{\varphi}$ satisfies the hypothesis of Lemma
\ref{lemm:uniquenes_phi}, we have
$\widetilde{\varphi}=0$. This concludes the proof of
\ref{item:7bis}.
To prove \ref{item:6bis}, we observe that
$\overline{\mathcal{F}}'=\overline{\mathcal{F}}+ [\overline A]$ for some bounded
acyclic complex $\overline A$ of hermitian vector bundles on $X$. For each
cartesian diagram as
\eqref{eq:9}, we set $\overline f'=g^{\ast}
\overline f$. Let $\widetilde {\varphi}_{1}$ be the assignment that,
to each such diagram and each bounded
acyclic complex of hermitian vector bundles $\overline{A}$ on $X'$,
assigns the class
\begin{displaymath}
\widetilde{\varphi}_{1}(\overline{A})=T({g'}^{\ast}\overline{\xi})
-T\left(\overline f',
{g'}^{\ast}\overline{\mathcal{F}}+[A],
g^{\ast}\overline{ f_{\ast}\mathcal{F}}\right)
-{\overline f'}_{\flat}[\cht(A)].
\end{displaymath}
The hypothesis of
Lemma \ref{lemm:uniquenes_phi} are satisfied, hence
$\widetilde{\varphi}_{1}=0$. This concludes the proof of~\ref{item:6bis}.
Finally, to prove \ref{item:8bis}, to each morphism $g\colon Y'\to
Y$, transverse to $f$,
we associate the cartesian diagram \eqref{eq:9} and we consider the
assignment
$\widetilde{\varphi}_{2}$ that,
to each bounded acyclic complex of hermitian vector bundles
$\overline{B}$ on $Y'$, assigns the class
\begin{displaymath}
\widetilde{\varphi}_{2}(\overline{B})=T({g'}^{\ast}\overline{\xi})
-T\left(\overline f',
{g'}^{\ast}\overline{\mathcal{F}},
g^{\ast}\overline{ f_{\ast}\mathcal{F}}+[B]\right)
+\cht(B).
\end{displaymath}
By Lemma \ref{lemm:uniquenes_phi} applied to $\Id_{Y}$, we have
$\widetilde{\varphi}_{2}=0$. This concludes the proof of~\ref{item:8bis}.
\end{proof}
The following result provides a compatibility relation for analytic
torsion classes for closed immersions with respect to distinguished
triangles. The statement is valid for additive theories, in particular
the ones we are concerned with.
\begin{proposition}\label{prop:2bis}
Let $T$ be an additive theory of analytic torsion classes for closed
immersions. Let $f\colon X\to Y$ be a closed immersion of smooth
complex varieties. Consider distinguished triangles in $\oDb(X)$ and
$\oDb(Y)$ respectively,
\begin{displaymath}
(\overline{\tau}):\
\overline{\mathcal{F}}_{2}\to\overline{\mathcal{F}}_{1}
\to\overline{\mathcal{F}}_{0}\to\overline{\mathcal{F}}_{2}[1],\quad
(\overline{ f_{\ast}\tau}):\
\overline{ f_{\ast}\mathcal{F}}_{2}\to
\overline{ f_{\ast}\mathcal{F}}_{1}
\to\overline{ f_{\ast}\mathcal{F}}_{0}\to
\overline{ f_{\ast}\mathcal{F}}_{2}[1],
\end{displaymath}
and the relative hermitian complexes $\overline{\xi}_{i}=(\overline{f},\overline{\mathcal{F}}_i,
\overline{ f_{\ast}\mathcal{F}}_i),$ $i=0,1,2$.
Then we have:
\begin{displaymath}
\sum_{j}(-1)^{j}T(\overline{\xi}_{j})
=\widetilde{\ch}(\overline{ f_{\ast}\tau})
-\overline f_{\flat}(\widetilde{\ch}(\overline{\tau })).
\end{displaymath}
\end{proposition}
\begin{proof}
We can assume that the distinguished triangles $\overline \tau $ and
$\overline{ f_{\ast}\tau }$ can be
represented by short exact sequences of complexes of hermitian vector
bundles
\begin{align*}
\overline \varepsilon :\quad &0\longrightarrow \overline E_{2}
\longrightarrow \overline E_{1}
\longrightarrow \overline E_{0} \longrightarrow 0,\\
\overline \nu : \quad &0\longrightarrow \overline V_{2}
\longrightarrow \overline V_{1}
\longrightarrow \overline V_{0} \longrightarrow 0.
\end{align*}
Applying the explicit construction at the beginning of the proof of
\cite[Theorem 2.3]{BurgosLitcanu:SingularBC} to each row of the
above exact sequences, we obtain exact sequences
\begin{align*}
\widetilde \varepsilon^{i} :\quad &0\longrightarrow \widetilde E_{2}^{i}
\longrightarrow \widetilde E_{1}^{i}
\longrightarrow \widetilde E_{0}^{i} \longrightarrow 0,\\
\widetilde \nu^{i} :\quad &0\longrightarrow \widetilde V_{2}^{i}
\longrightarrow \widetilde V_{1}^{i}
\longrightarrow \widetilde V_{0}^{i} \longrightarrow 0
\end{align*}
over $X\times {\mathbb P}^{1}$ and $Y\times {\mathbb P}^{1}$
respectively. The restriction of $\widetilde \varepsilon ^{i} $
(respectively $\widetilde \nu ^{i}$) to $X\times \{0\}$
(respectively $Y\times \{0\}$) agrees with $\overline \varepsilon $
(respectively $\overline \nu $). Whereas the restriction to
$X\times \{\infty\}$ (respectively $Y\times \{\infty\}$)
is orthogonally split. The sequences $\widetilde \varepsilon ^{i}$ and
$\widetilde \nu ^{i}$ form exact sequences of complexes that we
denote $\widetilde \varepsilon $ and
$\widetilde \nu$. It is easy to verify that the restriction to
$X\times \{\infty\}$ (respectively $Y\times \{\infty\}$)
are orthogonally split as sequences of complexes. Moreover, there are
isomorphisms $\widetilde V_{j}\dashrightarrow f _{\ast}\widetilde E_{j}$,
$j=0,1,2$. We denote
\begin{math}
\widetilde \xi _{j}=(\overline f \times \Id_{{\mathbb P}^{1}},\widetilde
E_{j},\widetilde V_{j}).
\end{math}
Then, in the group
$\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Y,N_{f},p)$, we
have
\begin{align*}
0&=\dd_{\mathcal{D}}\frac{1}{2\pi i}\int_{ {\mathbb P}^{1}}
\frac{-1}{2}\log t\bar t\bullet\sum_{j}
(-1)^{j}T(\widetilde{\xi}_{j})\\
&=T(\overline {\xi}_{1})-T(\overline
{\xi}_{0}\oplus \overline{\xi}_{2})-
\frac{1}{2\pi i}\int_{ {\mathbb P}^{1}}
\frac{-1}{2}\log t\bar t\bullet\sum_{j}
(-1)^{j}\ch(\widetilde V_{j})\\
&+
\frac{1}{2\pi i}\int_{ {\mathbb P}^{1}}
\frac{-1}{2}\log t\bar t\bullet
\sum_{j}(-1)^{j}(f\times \Id_{{\mathbb P}^{1}})
_{\ast}(\ch(\widetilde{E}_{j})\bullet
\Td(
\overline{f}\times \Id_{{\mathbb P}^{1}} ) )\\
&=T(\overline {\xi}_{1})-T(\overline
{\xi}_{0}\oplus \overline{\xi}_{2})
+\widetilde{\ch}(\overline{ f_{\ast}\tau})
-f_{\ast}(\widetilde{\ch}(\overline{\tau })\Td(\overline{f})).
\end{align*}
Thus the result follows from the additivity.
\end{proof}
We end this chapter with the relation between the singular Bott-Chern
classes of Bismut-Gillet-Soul\'e \cite{BismutGilletSoule:MR1086887}
and the theory of homogeneous analytic torsion classes. We draw
attention to the difference of normalizations. Let us momentarily
denote by $\tau$ the theory of singular Bott-Chern classes of
Bismut-Gillet-Soul\'e. By the anomaly formulas, it may be extended to
arbitrary embedded metrized complexes. Let $\overline{\xi}=(\overline{f}:X\to
Y,\overline{\mathcal{F}},\overline{ f_{\ast}\mathcal{F}})$ be a relative
metrized complex, with $Y$ of dimension $d$. If $\tau^{(p-1,p-1)}$
denotes the component of degree $(p-1,p-1)$ of the current $\tau$, we
define
\begin{equation}\label{eq:88}
T^{BGS}(\overline{\xi})^{(2p-1,p)}= -\frac{1}{2(2\pi
i)^{d-(p-1)}}\tau^{(p-1,p-1)}
\in\widetilde{\mathcal{D}}_{D}^{2p-1}(Y,N_{f},p).
\end{equation}
In the above equation, the factor $(2\pi i)^{(p-1)}$ comes from the
difference in the normalization of characteristic classes. In
\cite{BismutGilletSoule:MR1086887} the authors use real valued classes
while we use twisted coefficients. The factor $(2\pi i)^{d}$ comes
from our convention about the Deligne complex of currents. The factor
$2$ comes from the fact that the second order differential operator
that appears in the Deligne complex is $-2\partial\bar \partial=2(2\pi
i)d d^{c}$, while the second order differential operator that appears
in the differential equation considered by Bismut, Gillet and Soul\'e
is $dd^{c}$. The main reason behind this change is that we want the
Bott-Chern classes to be related to Beilinson's regulator and not to
twice Beilinson's regulator (see \cite{GilletSoule:ait} Theorem
3.5.4). Finally, the minus sign comes from the discrepancy of the
differential equations of the singular Bott-Chern forms of
Bismut-Gillet-Soul\'e and the analytic torsion forms of
Bismut-K\"ohler. Note that we are forced to change this sign because
we want to merge singular Bott-Chern forms and analytic torsion forms
on a single theory.
We put
\begin{displaymath}
T^{BGS}(\overline{\xi})=\sum_{p\geq
1}T^{BGS}(\overline{\xi})^{(2p-1,p)}\in
\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Y, N_{f},p).
\end{displaymath}
We have the following comparison theorem
\cite[Thm. 9.25]{BurgosLitcanu:SingularBC}.
\begin{theorem}\label{thm:18}
For every embedded metrized complex $\xi$ we have
\begin{displaymath}
T^{BGS}(\overline{\xi})=T^{h}(\overline{\xi}).
\end{displaymath}
\end{theorem}
\section{Regular coherent sheaves}
\label{sec:regsheaf}
In this section we recall some properties of regular sheaves. Let $X$
be a scheme and let ${\mathbb P}^n_X={\mathbb P}_X(V)$ be the projective space of
lines of the trivial bundle $V$ of rank $n+1$ on $X$. Let $\pi\colon
{\mathbb P}^n_X\rightarrow X$ be the natural projection. By abuse of notation,
if ${\mathcal{G}}$ is a sheaf on $X$, we will denote also by
${\mathcal{G}}$ the inverse image $\pi ^{\ast}{\mathcal{G}}$.
\begin{definition}[\cite{Mumford:Lectures_on_Curves}, Lecture
14]\label{def:regsheaf}
A quasi-coherent sheaf ${\mathcal{F}}$ on ${\mathbb P}^n_X$ is called
\textit{regular} if $R^q\pi_{\ast}{\mathcal{F}}(-q)=0$ for all
$q>0$.
\end{definition}
Recall the following properties of regular sheaves (see \cite{Quillen:haKt}).
\begin{enumerate}
\item If ${\mathcal{G}}$ is a quasi-coherent sheaf on $X$, then
${\mathcal{G}}\otimes_X \mathcal{O}_{{\mathbb P}^n_X}(k)$ is regular for
$k\ge 0$.
\item If the scheme $X$ is noetherian and ${\mathcal{F}}$ is a
coherent sheaf on ${\mathbb P}^n_X$, then Serre's vanishing theorem implies
that for $d$ large enough ${\mathcal{F}}(d)$ is regular.
\item Let $0\rightarrow {\mathcal{F}}_2\rightarrow {\mathcal{F}}_1
\rightarrow {\mathcal{F}}_0 \rightarrow 0$ be an exact sequence of
quasi-coherent sheaves on ${\mathbb P}^n_X$ and $d$ be an integer. Then
\begin{enumerate}
\item if ${\mathcal{F}}_{2}(d)$ and ${\mathcal{F}}_{0}(d)$ are
regular, then ${\mathcal{F}}_{1}(d)$ is regular;
\item if ${\mathcal{F}}_{2}(d+1)$ and ${\mathcal{F}}_{1}(d)$ are
regular, then ${\mathcal{F}}_{0}(d)$ is regular;
\item if ${\mathcal{F}}_0(d)$ and ${\mathcal{F}}_1(d+1)$ are regular
and the map $R^0\pi_{\ast}({\mathcal{F}}_1(d))\rightarrow
R^0\pi_{\ast}({\mathcal{F}}_0(d))$ is surjective, then
${\mathcal{F}}_2(d+1)$ is regular.
\end{enumerate}
\item If ${\mathcal{F}}$ is regular, then ${\mathcal{F}}(k)$ is regular for $k>0$.
\item If ${\mathcal{F}}$ is regular, then the canonical map $R^0\pi_{\ast}{\mathcal{F}}\otimes _X
\mathcal{O}_{{\mathbb P}^n_X}\rightarrow {\mathcal{F}}$ is surjective.
\end{enumerate}
\begin{theorem}[{\cite[\S
8.1]{Quillen:haKt}}]\label{thm:2}
Let ${\mathcal{F}}$ be a regular quasi-coherent sheaf on
${\mathbb P}^{n}_{X}$. Then there exists a \textit{canonical resolution}
\begin{equation*}
\gamma_{\text{{\rm can}}}({\mathcal{F}})\ :\ 0\rightarrow {\mathcal{G}}_n(-n)\rightarrow
{\mathcal{G}}_{n-1}(-n+1)\rightarrow \dots \rightarrow
{\mathcal{G}}_0\rightarrow {\mathcal{F}}
\rightarrow 0
\end{equation*}
where ${\mathcal{G}}_i$ ($i=0,\dots,n$) are quasi-coherent sheaves on
$X$. Moreover, for every $k\geq 0$, the sequence
\begin{displaymath}
0\rightarrow {\mathcal{G}}_k\rightarrow {\mathcal{G}}_{k-1}
\otimes\Sym^1V^{\vee}
\rightarrow \dots\rightarrow {\mathcal{G}}_{0}\otimes\Sym^kV^{\vee}
\rightarrow R^0\pi_{\ast}({\mathcal{F}}(k))\rightarrow 0
\end{displaymath}
is exact. Hence the sheaves ${\mathcal{G}}_k$ are determined
by ${\mathcal{F}}$ up to unique isomorphism.
\end{theorem}
\begin{corollary} \label{cor:4}
Let $X$ be a noetherian scheme and ${\mathcal{F}}$ a coherent
sheaf on ${\mathbb P}^n_X$. Then, for $d$ large enough, we have a resolution
$$\gamma_d({\mathcal{F}})\ :\ 0\rightarrow {\mathcal{G}}_n(-n-d)\rightarrow
{\mathcal{G}}_{n-1}(-n-d+1)\rightarrow \dots \rightarrow
{\mathcal{G}}_0(-d)\rightarrow {\mathcal{F}} \rightarrow 0$$ where
${\mathcal{G}}_i$, $i=0,\dots,n$ are coherent sheaves on $X$.
\end{corollary}
\begin{example}\label{exm:1}
The sheaf $\mathcal{O}(1)$ is regular. Its canonical resolution is
\begin{displaymath}
0\to
\Lambda^{n+1}V^{\vee}(-n)\to\Lambda ^{n}V^{\vee}(-n+1)\to\dots\to
\Lambda ^{2}V^{\vee}(-1)\to V^{\vee}\to \mathcal{O}(1)\to 0.
\end{displaymath}
Twisting this exact sequence by $\mathcal{O}(-1)$ we obtain the
Koszul exact sequence
\begin{displaymath}
0\to
\Lambda^{n+1}V^{\vee}(-n-1)\to\Lambda ^{n}V^{\vee}(-n)\to\dots\to
\Lambda ^{2}V^{\vee}(-2)\to V^{\vee}(-1)\to \mathcal{O}\to 0,
\end{displaymath}
that we denote $K$. We will denote by $K(k)$ its twist by $\mathcal{O}(k)$.
\end{example}
\begin{theorem}[{\cite{Zha99:_rieman_roch}}]\label{thm:14}
\begin{enumerate}
\item \label{item:9} Let ${\mathcal{F}}$ be a regular coherent sheaf
on ${\mathbb P}^{n}_{X}$, and let $\gamma_{\text{{\rm can}}}({\mathcal{F}})$ be the canonical
resolution of ${\mathcal{F}}$ as in Theorem \ref{thm:2}. Let
\begin{displaymath}
\varepsilon_1\ :\ 0\rightarrow {\mathcal{F}}_{n+k}(-n-k)\rightarrow \dots
\rightarrow {\mathcal{F}}_1(-1)\rightarrow {\mathcal{F}}_0\rightarrow
{\mathcal{F}}\rightarrow 0
\end{displaymath}
be an exact sequence of coherent sheaves, where the
${\mathcal{F}}_{i}$ are sheaves on $X$. Then there exist natural
surjective morphisms of sheaves ${\mathcal{F}}_i\rightarrow
{\mathcal{G}}_i$ on $X$, $0\leq i\leq n$ making commutative the
diagram
\begin{displaymath}
\xymatrix{
{\mathcal{F}}_{n+1}(-n-1)\ar[d] \ar[r] & {\mathcal{F}}_n(-n)
\ar@{->>}[d] \ar[r] & \dots
\ar[r] & {\mathcal{F}}_0 \ar@{->>}[d]
\ar[r] & {\mathcal{F}} \ar@{=}[d] \ar[r] & 0\\
0 \ar[r] & {\mathcal{G}}_n(-n) \ar[r] & \dots \ar[r] &
{\mathcal{G}}_0 \ar[r] &
{\mathcal{F}} \ar[r] & 0.
}
\end{displaymath}
\item \label{item:11} Let ${\mathcal{F}}$ be a regular coherent sheaf on $X$, and
$\gamma_{\text{{\rm can}}}({\mathcal{F}})$ the canonical resolution.
There exists a resolution of
${\mathcal{F}}(1)$ of the form
\begin{displaymath}
\varepsilon_{2}\ :\ 0 \rightarrow {\mathcal{S}}_{n+k}(-n-k)\rightarrow
\dots\rightarrow
{\mathcal{S}}_1(-1) \rightarrow {\mathcal{S}}_0 \rightarrow {\mathcal{F}}(1)\rightarrow 0
\end{displaymath}
such that ${\mathcal{S}}_0\dots,{\mathcal{S}}_{n+k}$ are coherent
sheaves on $X$ and the
following diagram of exact sequences with surjective vertical arrows
is commutative:
\begin{displaymath}
\xymatrix{
{\mathcal{S}}_{n+1}(-n-1)\ar[d] \ar[r] &
{\mathcal{S}}_n(-n) \ar@{->>}[d] \ar[r] & \dots
\ar[r] &
{\mathcal{S}}_0 \ar@{->>}[d] \ar[r] &
{\mathcal{F}}(1) \ar@{=}[d] \ar[r] & 0\\
0 \ar[r] & {\mathcal{G}}_n(-n+1) \ar[r] & \dots \ar[r] &
{\mathcal{G}}_0(1) \ar[r] & {\mathcal{F}}(1) \ar[r] & 0.
}
\end{displaymath}
\end{enumerate}
\end{theorem}
\begin{proof}
We introduce the sheaves ${\mathcal{N}}_{j}$ and
${\mathcal{K}}_{j}$ defined as the kernels at each term of the sequences
$\gamma_{\text{{\rm can}}}$ and $\varepsilon_{1}$, respectively. Hence, there
are exact sequences
\begin{align*}
&0\to {\mathcal{N}}_{j+1}(j+1)\to {\mathcal{G}}_{j+1}\to
{\mathcal{N}}_{j}(j+1)\to 0,\\
&0\to {\mathcal{K}}_{j+1}(j+1)\to {\mathcal{F}}_{j+1}\to
{\mathcal{K}}_{j}(j+1)\to 0.
\end{align*}
With these notations, observe that
${\mathcal{N}}_{-1}={\mathcal{K}}_{-1}={\mathcal{F}}$. By induction,
starting from the left hand side of the long exact sequences, it
is easily checked that ${\mathcal{N}}_{j}(j+1)$ and
${\mathcal{K}}_{j}(j+1)$ are regular
sheaves, for $j\geq -1$. Also, by Theorem \ref{thm:2}, we find that
${\mathcal{G}}_{j+1}=\pi_{\ast}({\mathcal{N}}_{j}(j+1))$ for $j\geq
-1$. We next prove by
induction that, for each $k\geq -1$,
there is a commutative diagram of exact sequences
\begin{equation}\label{eq:78}
\xymatrix{
&0\ar[d] &0\ar[d] &0\ar[d] &\\
0\ar[r] &{\mathcal{P}}_{k+1}\ar[r]\ar[d]
&{\mathcal{H}}_{k+1}\ar[r]\ar[d] &{\mathcal{P}}_{k}(1)\ar[d]&\\
0\ar[r] &{\mathcal{K}}_{k+1}(k+1)\ar[r]\ar[d]
&{\mathcal{F}}_{k+1}\ar[r]\ar[d]
&{\mathcal{K}}_{k}(k+1)\ar[r]\ar[d] &0\\
0\ar[r] &{\mathcal{N}}_{k+1}(k+1)\ar[r]
&{\mathcal{G}}_{k+1}\ar[r]\ar[d]
&{\mathcal{N}}_{k}(k+1)\ar[r]\ar[d] &0\\
& &0 &0, &
}
\end{equation}
where ${\mathcal{H}}_{k+1}$, ${\mathcal{P}}_{k}$ and
${\mathcal{P}}_{k+1}$ are defined as the kernels of the corresponding
morphisms, and that ${\mathcal{P}}_{k}(1)$ is
regular. Assume that this is true for a fixed $k\ge 1$. In order to
proceed with the induction, we need to prove: (a)
the map ${\mathcal{K}}_{k+1}(k+2)\to {\mathcal{N}}_{k+1}(k+2)$ is
surjective, (b) the sheaf ${\mathcal{P}}_{k+1}(1)$ is regular, and (c)
there is an induced surjective map
${\mathcal{F}}_{k+2} \to {\mathcal{G}}_{k+2}$.
We first prove (a). If we apply $\pi_{\ast}$ to the last two
columns of diagram \eqref{eq:78}. Observing that
${\mathcal{F}}_{k+1}$, ${\mathcal{G}}_{k+1}$ and ${\mathcal{H}}_{j+1}$
are actually sheaves on $X$ and recalling that
${\mathcal{K}}_{k+1}(k+2)$ is regular (so that
$R^{1}\pi_{\ast}{\mathcal{K}}_{k+1}(k+1)=0$), we find a commutative
diagram of exact sequences
\begin{displaymath}
\xymatrix{
0\ar[r] &
{\mathcal{H}}_{k+1}\ar[r]\ar[d] &
{\mathcal{F}}_{k+1}\ar[r]\ar[d] &
{\mathcal{G}}_{k+1}\ar@{=}[d]\ar[r] &
0 \\
0\ar[r] &
\pi_{\ast}({\mathcal{P}}_{k}(1))\ar[r] &
\pi_{\ast}({\mathcal{K}}_{k}(k+1))\ar[d]\ar[r] &
\pi_{\ast}({\mathcal{N}}_{k}(k+1))\ar[r] &
0\\
&&0&&
}
\end{displaymath}
It follows that the map
${\mathcal{H}}_{k+1}\twoheadrightarrow\pi_{\ast}({\mathcal{P}}_{k}(1))$ is a
surjection. Since ${\mathcal{P}}_{k}(1)$ is
regular, we have that
$\pi_{\ast}({\mathcal{P}}_{k}(1))\otimes{\mathcal O}_{{\mathbb P}^{n}_{X}}\twoheadrightarrow
{\mathcal{P}}_{k}(1)$ is also a surjection. Thus the map
$\mathcal{H}_{k+1}\to \mathcal{P}_{k}(1)$ is surjective. The diagram
\eqref{eq:78} implies that the map
\begin{math}
{\mathcal{K}}_{k+1}(k+1)\to {\mathcal{N}}_{k+1}(k+1)
\end{math}
is also surjective. Twisting by ${\mathcal{O}}(1)$, we obtain (a).
Now the regularity of ${\mathcal{H}}_{k+1}$ and
${\mathcal{P}}_{k}(1)$, and
the surjectivity of
${\mathcal{H}}_{k+1}\twoheadrightarrow\pi_{\ast}({\mathcal{P}}_{k}(1))$
ensure the regularity of ${\mathcal{P}}_{k+1}(1)$.
In its turn, this shows that the sequence
\begin{equation}\label{eq:11}
0\to\pi_{\ast}({\mathcal{P}}_{k+1}(1))\to\pi_{\ast}({\mathcal{K}}_{k+1}(k+2))
\to\pi_{\ast}({\mathcal{N}}_{k+1}(k+2))\to 0
\end{equation}
is exact. Finally, we observe that there is a surjective map
\begin{equation}\label{eq:77}
\xymatrix{
{\mathcal{F}}_{k+2}\ar@{->>}[r] &\pi_{\ast}({\mathcal{K}}_{k+1}(k+2)),
}
\end{equation}
by the regularity of ${\mathcal{K}}_{k+2}(k+3)$. From
\eqref{eq:11} and \eqref{eq:77}, we obtain a surjection
\begin{displaymath}
\xymatrix{
{\mathcal{F}}_{k+2}\ar@{->>}[r]
&\pi_{\ast}({\mathcal{N}}_{k+1}(k+2))={\mathcal{G}}_{k+2}.
}
\end{displaymath}
This completes the proof of the inductive step. Note that the first
step of the induction ($k=-1$) is part of the data. From the existence
of the diagrams we deduce (i).
To prove the second item we construct the resolution
$\mathcal{S}_{\ast}$ inductively. We will denote by
${\mathcal{K}}_{k}$ the kernel of any map ${\mathcal{S}}_{k}(-k)\to
{\mathcal{S}}_{k-1}(-k+1)$ already defined and by ${\mathcal{N}}_{k}$ the
successive kernels of the canonical resolution of ${\mathcal{F}}$ as
in the proof of the first statement.
Assume that we have constructed the
sequence $\varepsilon _{2}$ up to ${\mathcal{S}}_{k}(-k)$ with the
further conditions that
${\mathcal{K}}_{k}(k+1)$ is regular and that there is an exact sequence
\begin{displaymath}
0\to {\mathcal{P}}_{k}(1)\to {\mathcal{K}}_{k}(k+1)\to
{\mathcal{N}}_{k}(k+2)\to 0
\end{displaymath}
with ${\mathcal{P}}_{k}(1)$
regular. We have to show that we can extend the resolution one step
satisfying the same conditions.
Recall that we already know that ${\mathcal{N}}_{k}(k+1)$ is regular.
We consider as well the surjection
\begin{math}
\xymatrix{
{\mathcal{G}}_{k+1}(1)\ar@{->>}[r] &{\mathcal{N}}_{k}(k+2).
}
\end{math}
We form the fiber product
\begin{displaymath}
{\mathcal{T}}_{k+1}:=\Ker({\mathcal{K}}_{k}(k+1)\oplus
{\mathcal{G}}_{k+1}(1)\to {\mathcal{N}}_{k}(k+2)).
\end{displaymath}
Observe that ${\mathcal{T}}_{k+1}$ is regular, because both
${\mathcal{N}}_{k}(k+1)$,
${\mathcal{K}}_{k}(k+1)\oplus {\mathcal{G}}_{k+1}(1)$ are regular and
the morphism
\begin{displaymath}
\xymatrix{
\pi_{\ast}({\mathcal{K}}_{k}(k)\oplus
{\mathcal{G}}_{k+1})\ar@{->>}[r]
&{\mathcal{G}}_{k+1}=\pi_{\ast}({\mathcal{N}}_{k}(k+1))
}
\end{displaymath}
is surjective. So are the arrows ${\mathcal{T}}_{k+1}\to
{\mathcal{G}}_{k+1}(1)$ and
${\mathcal{T}}_{k+1}\to {\mathcal{K}}_{k}(k+1)$. Therefore, if we define
${\mathcal{S}}_{k+1}=\pi_{\ast} ({\mathcal{T}}_{k+1})$,
we have a commutative diagram of exact sequences
\begin{displaymath}
\xymatrix{
&0\ar[d] &0\ar[d] &0\ar[d] &\\
0\ar[r] &{\mathcal{P}}_{k+1}\ar[r]\ar[d]
&{\mathcal{H}}_{k+1}\ar[r]\ar[d] &{\mathcal{P}}_{k}(1)\ar[d]&\\
0\ar[r] &{\mathcal{K}}_{k+1}(k+1)\ar[r]\ar[d]
&{\mathcal{S}}_{k+1}\ar[r]\ar[d]
&{\mathcal{K}}_{k}(k+1)\ar[r]\ar[d] &0\\
0\ar[r] &{\mathcal{N}}_{k+1}(k+2)\ar[r] &{\mathcal{G}}_{k+1}(1)\ar[r]\ar[d]
&{\mathcal{N}}_{k}(k+2)\ar[r]\ar[d] &0\\
& &0 &0, &
}
\end{displaymath}
where ${\mathcal{H}}_{k+1}$ and ${\mathcal{P}}_{k+1}$ are defined as
the kernels of the corresponding morphisms. Thus we have been able to
extend the resolution one step further. We still need to show that
this extension satisfies the extra properties.
We observe that, by the definition of ${\mathcal{S}}_{k+1}$ and the left
exactness of direct images, the map $\pi _{\ast}({\mathcal{S}}_{k+1})\to \pi
_{\ast}({\mathcal{G}}_{k+1}(1))$ is surjective. Therefore
${\mathcal{H}}_{k+1}(1)$ is
regular. Moreover, one can check that ${\mathcal{S}}_{k+1}$ is the fiber product
\begin{displaymath}
{\mathcal{S}}_{k+1}=\Ker\left(\pi
_{\ast}({\mathcal{G}}_{k+1}(1))\oplus \pi _{\ast}({\mathcal{K}}_{k}(k+1))\to \pi
_{\ast}({\mathcal{N}}_{k}(k+2))\right).
\end{displaymath}
This implies easily that
$\pi _{\ast}({\mathcal{H}}_{k+1})=\pi
_{\ast}({\mathcal{P}}_{k}(1))$. We also observe that, by definition of
fiber product,
${\mathcal{P}}_{k}(1)=\Ker({\mathcal{T}}_{k+1}\to
{\mathcal{G}}_{k+1}(1))$. Since ${\mathcal{S}}_{k+1}$ surjects onto
${\mathcal{T}}_{k+1}$, we deduce that the morphism
${\mathcal{H}}_{k+1}\to {\mathcal{P}}_{k}(1)$ is
surjective. From this we conclude that the morphism
${\mathcal{K}}_{k+1}(k+2)\to
{\mathcal{N}}_{k+1}(k+3)$ is surjective and that
the sheaf ${\mathcal{P}}_{k+1}(1)$ is regular. Since
${\mathcal{N}}_{k+1}(k+3)$ is regular, we
deduce that ${\mathcal{K}}_{k+1}(k+2)$ is regular. Therefore
$\mathcal{S_{k+1}}$ satisfies all the required properties, concluding
the proof of \ref{item:11}.
\end{proof}
We end this section recalling the notion of generating class of a
triangulated category.
\begin{definition}\label{def:gen_class}
Let $\mathbf{D}$ be a triangulated category. A \emph{generating
class} is a subclass $\mathbf{C}$ of $\mathbf{D}$ such that the
smallest triangulated subcategory of $\mathbf{D}$ that contains
$\mathbf{C}$ is equivalent to $\mathbf{D}$ via the inclusion.
\end{definition}
A well-known direct consequence of Theorem \ref{thm:2} is the following result.
\begin{corollary} \label{cor:6}
The class of objects of the form $\mathcal{G}(k)$, with
$\mathcal{G}$ a coherent sheaf in $X$ and $-n\leq k\leq 0$, is a
generating class of
$\Db({\mathbb P}^{n}_{X})$.
\end{corollary}
\section{Analytic torsion for projective spaces}
\label{sec:projsp}
Let $n$ be a non-negative integer, $V$ the $n+1$ dimensional
vector space
${\mathbb C}^{n+1}$ and
${\mathbb P}^n:={\mathbb P}^n(V)$ the projective space of lines in $V$. We write
$\overline V$ for the vector space $V$ together with the trivial
metric.
We
will denote by $V$ the trivial vector bundle of fiber $V$ over any
base.
We may
construct natural relative hermitian complexes that arise by
considering the sheaves $\mathcal{O}(k)$, their
cohomology and the Fubini-Study metric.
If we endow the trivial sheaf with the trivial metric and
$\mathcal{O}(1)$ with the Fubini-Study metric, then the tangent bundle
$T_{\pi}$ carries a quotient hermitian structure via the short exact
sequence
\begin{equation} \label{eq:4}
0\to\mathcal{O}_{{\mathbb P}^{n}_{{\mathbb C}}}\to
\mathcal{O}(1)^{n+1}\to T_{\pi}\to 0.
\end{equation}
We will denote the resulting hermitian vector bundle by
$\overline{T}_{\pi}^{\text{\rm FS}}$ and call it the Fubini-Study metric of $T_{\pi
}$. The arrow $(\pi,
\overline{T}_{\pi}^{\text{\rm FS}})$ in $\overline\Sm_{\ast/{\mathbb C}}$ will be written
$\overline{\pi}^{\text{\rm FS}}$.
We endow the invertible sheaves ${\mathcal O}(k)$ with the $k$-th tensor power
of the Fubini-Study metric on ${\mathcal O}(1)$. We refer to them by
$\overline{{\mathcal O}(k)}$.
We now describe natural hermitian structures on the complexes $ \pi
_{\ast} \mathcal{O}(k)$.
First assume $k\geq 0$. The sheaf ${\mathcal O}(k)$ is $\pi$-acyclic, hence
\begin{math}
\pi_{\ast}\mathcal{O}(k)=\text{H}^{0}({\mathbb P}^{n}_{{\mathbb C}},\mathcal{O}(k))
\end{math}
as a complex concentrated in degree 0. This space is naturally
equipped with the $L^{2}$ metric with respect to the Fubini-Study
metric on ${\mathcal O}(k)$ and the volume form
$\mu=c_{1}(\overline{{\mathcal O}(1)})^{\land n}/n!$ on ${\mathbb P}^{n}_{{\mathbb C}}$. Namely, given
global sections $s,t$ of ${\mathcal O}(k)$,
\begin{displaymath}
\langle s,t\rangle_{L^{2}}=\int_{{\mathbb P}^{n}_{{\mathbb C}}} \langle
s(x),t(x)\rangle_{x}\mu(x).
\end{displaymath}
If $-n\leq k<0$, then $\pi_{\ast}\mathcal{O}(k)=0$
and we put the trivial metric on it.
Finally, let $k\leq -n-1$. Then the cohomology of
$\pi_{\ast}\mathcal{O}(k)$ is concentrated in degree $n$ and there
is an isomorphism,
\begin{displaymath}
\pi_{\ast}\mathcal{O}(k)\cong
\text{H}^{0}({\mathbb P}^{n}_{{\mathbb C}},{\mathcal O}(-k-n-1))^{\vee}[-n].
\end{displaymath}
Notice that this
isomorphism is canonical due to Grothendieck duality and to the
natural identification
$\omega_{{\mathbb P}^{n}_{{\mathbb C}}}=\mathcal{O}(-n-1)$. Hence we may endow
$\pi_{\ast}\mathcal{O}(k)$ with the dual of the $L^{2}$ metric on $\text{H}^{0}({\mathbb P}^{n}_{{\mathbb C}},{\mathcal O}(-k-n-1))$.
\begin{notation} \label{def:7}
For every integer $k$, we introduce the relative metrized complex
\begin{equation}\label{eq:5}
\overline{\xi_{n}}(k)=(\overline{\pi}^{\text{\rm FS}}, \overline{\mathcal{O}(k)},
\overline{\pi_{\ast}\mathcal{O}(k)}).
\end{equation}
If $X$ is a smooth complex variety, we will also denote by
$\overline{\xi}_{n}(k)$ its
pull-back to ${\mathbb P}^{n}_{X}$.
Let $\overline {\mathcal{F}}$ be a metrized coherent sheaf on
$X$. Then we define $\overline {\mathcal{F}}(k)$ and $\overline { \pi
_{\ast}{\mathcal{F}}(k)}$ by the equality
\begin{displaymath}
\overline \xi_{n}(k)\otimes \overline {\mathcal{F}}=(\overline \pi ^{\text{\rm FS}}, \overline
{\mathcal{F}}(k), \overline { \pi
_{\ast}{\mathcal{F}}(k)}).
\end{displaymath}
\end{notation}
\begin{definition}\label{def:2}
Let $X$ be a complex smooth variety and $\pi\colon {\mathbb P}^{n}_{X}\to X$ the
projection. An \emph{analytic torsion class} for the relative
hermitian complex $\overline{\xi}=(\overline{\pi},
\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})$ is a class
$\widetilde \eta\in \bigoplus_{p} \widetilde
{\mathcal{D}}^{2p-1}(X,p)$ such that
\begin{equation}\label{eq:1}
\dd_{\mathcal{D}} \widetilde
\eta=\ch(\overline{\pi_{\ast}\mathcal{F}})-\overline
\pi_{\flat}
[\ch(\overline{\mathcal{F}})].
\end{equation}
\end{definition}
The existence of this class is guaranteed by the
Grothendieck-Riemann-Roch theorem, which implies that the two currents
at the right hand side of equation \eqref{eq:1} are
cohomologous. Since the map $\pi $ is smooth, the analytic torsion
class is the class of a smooth form.
\begin{definition}\label{def:3}
Let $n$ be a non-negative integer. A \emph{theory of analytic
torsion classes for projective spaces of dimension $n$} is an
assignment that, to each relative metrized complex
\begin{math}
\overline{\xi}=(\overline{\pi}\colon {\mathbb P}^n_X\rightarrow
X,\overline{\mathcal{F}},\overline{\pi_{\ast}\mathcal{F}})
\end{math}
of relative dimension $n$, assigns a class of differential forms
\begin{displaymath}
T(\overline \xi)\in \bigoplus_{p} \widetilde
{\mathcal{D}}^{2p-1}(X,p),
\end{displaymath}
satisfying the following properties.
\begin{enumerate}
\item \label{item:1} (Differential equation)
\begin{math}
\dd_{\mathcal{D}} T(\overline
{\xi})=\ch(\overline{\pi_{\ast}\mathcal{F}})-\overline \pi_{\flat}
[\ch(\overline{\mathcal{F}})].
\end{math}
\item \label{item:2}(Functoriality) Given a morphism
$f\colon Y\longrightarrow X$, we have
\begin{math}
T(f^{\ast}\overline{\xi})=f^{\ast} T(\overline{\xi}).
\end{math}
\item \label{item:3}(Additivity and normalization) If
$\overline{\xi}_1$ and $\overline{\xi}_2$ are relative metrized
complexes on $X$, then
$T(\overline{\xi}_1\oplus\overline{\xi}_2)
=T(\overline{\xi}_1)+T(\overline{\xi}_2).$
\item \label{item:4}(Projection formula) For any
hermitian vector bundle $\overline{G}$ on $X$, and any integer $k\in[-n,0]$, we have
\begin{math}
T(\overline{\xi}_{n}(k)\otimes\overline{G})=
T(\overline{\xi}_{n}(k))\bullet\ch(\overline{G}).
\end{math}
\end{enumerate}
A \emph{theory of analytic torsion classes for projective spaces} is
an assignment as
before, for all non-negative integers $n$.
\end{definition}
\begin{definition}\label{def:char_numbers}
Let $T$ be a theory of analytic torsion classes for projective spaces
of dimension $n$. Fix as base space the point $\Spec {\mathbb C}$. The
\emph{characteristic numbers} of $T$ are
\begin{equation}\label{eq:3}
t_{n,k}(T):=T(\overline{\xi}_{n}(k))\in \widetilde
{\mathcal{D}}^{1}(\Spec {\mathbb C}, 1)={\mathbb R}, \
k\in\mathbb{Z}.
\end{equation}
The numbers $t_{n,k}(T)$, $-n\le k \le 0$ will be
called the \emph{main characteristic numbers} of $T$.
\end{definition}
The central result of this section is the following classification theorem.
\begin{theorem}\label{thm:1}
Let $n$ be a non-negative integer and let
$\mathfrak{t}=(t_{n,k})_{k=-n,\dots,0}$ be a family of arbitrary real
numbers. Then there exists a unique theory $T_{\mathfrak{t}}$ of
analytic torsion classes for projective spaces of dimension $n$, such that
$t_{n,k}(T_{\mathfrak{t}})=t_{n,k}$.
\end{theorem}
Before proving Theorem \ref{thm:1}, we show some
consequences of the definition of the analytic torsion classes.
First we state some anomaly formulas
that determine the dependence of the analytic torsion classes with
respect to different choices of metrics.
\begin{proposition} \label{prop:1}
Let $T$ be a theory of analytic torsion classes for projective
spaces of dimension $n$. Let
\begin{math}
\overline{\xi}= (\overline{\pi}\colon {\mathbb P}^n_X\rightarrow
X,\overline{\mathcal{F}}, \overline{\pi_{\ast}\mathcal{F}})
\end{math}
be a relative metrized complex.
\begin{enumerate}
\item \label{item:6}
If $\overline{\mathcal{F}}'$ is another choice
of metric on $\mathcal{F}$ and $\overline{\xi}_1=(\overline{\pi}\colon
{\mathbb P}^n_X\rightarrow
X,\overline{\mathcal{F}}', \overline{\pi_{\ast}\mathcal{F}})$, then
\begin{displaymath}
T(\overline{\xi}_1)=T(\overline{\xi})+\overline \pi
_{\flat}[\cht(\overline{\mathcal{F}}',\overline{\mathcal{F}})].
\end{displaymath}
\item \label{item:7}
If $\overline{\pi}'$ is another hermitian structure on $\pi$ and
$\overline{\xi}_{2}= (\overline{\pi}'\colon {\mathbb P}^n_X\rightarrow
X,\overline{\mathcal{F}}, \overline{\pi_{\ast}\mathcal{F}})$, then
\begin{equation}\label{eq:14}
T(\overline{\xi}_{2})=T(\overline{\xi})+\overline \pi'
_{\flat}[\ch(\overline {\mathcal{F}})\bullet
\widetilde{\Td}_{m}(\overline{\pi}', \overline{\pi})].
\end{equation}
\item \label{item:8}
If $\overline{\pi_{\ast}\mathcal{F}}'$ is another
choice of metric on $\pi_{\ast}\mathcal{F}$, and $\overline
{\xi}_{3}= (\overline{\pi}\colon {\mathbb P}^n_X\rightarrow
X,\overline{\mathcal{F}}, \overline{\pi_{\ast}\mathcal{F}}')$, then
\begin{displaymath}
T(\overline{\xi}_{3})= T(\overline{\xi})-
\cht(\overline{\pi_{\ast}\mathcal{F}}',
\overline{\pi_{\ast}\mathcal{F}}).
\end{displaymath}
\end{enumerate}
\end{proposition}
\begin{proof}
The proof is the same as the proof of Proposition
\ref{prop:1bis}.
\end{proof}
Next we state the behavior of analytic torsion classes for projective
spaces with respect to
distinguished triangles.
\begin{proposition} \label{prop:2}
Let $T$ be a theory of analytic torsion classes for projective spaces of
dimension $n$. Let $X$ be a smooth complex variety and
$\pi\colon {\mathbb P}^{n}_{X}\to X$ the projection. Consider distinguished
triangles in $\oDb({\mathbb P}^{n}_{X})$ and $\oDb(X)$ respectively:
\begin{displaymath}
(\overline{\tau}):\
\overline{\mathcal{F}}_{2}\to\overline{\mathcal{F}}_{1}
\to\overline{\mathcal{F}}_{0}\to\overline{\mathcal{F}}_{2}[1]\ \text{ and
}\
(\overline{\pi_{\ast}\tau}):\
\overline{\pi_{\ast}\mathcal{F}}_{2}\to
\overline{\pi_{\ast}\mathcal{F}}_{1}
\to\overline{\pi_{\ast}\mathcal{F}}_{0}\to
\overline{\pi_{\ast}\mathcal{F}}_{2}[1],
\end{displaymath}
and define relative metrized complexes
$\overline{\xi}_{i}=(\overline{\pi},\overline{\mathcal{F}}_i,
\overline{\pi_{\ast}\mathcal{F}}_i),$ $i=0,1,2$.
Then
\begin{displaymath}
\sum_{j}(-1)^{j}T(\overline{\xi}_{j})
=\widetilde{\ch}(\overline{\pi_{\ast}\tau})
-\overline \pi_{\flat}(\widetilde{\ch}(\overline{\tau })).
\end{displaymath}
\end{proposition}
\begin{proof}
The proof is similar to that of \ref{prop:2bis}.
\end{proof}
In view of this proposition, we see that the additivity axiom is
equivalent to the apparently stronger statement of the next corollary.
\begin{corollary} \label{cor:2}
With the assumptions of Proposition \ref{prop:2}, if $\overline \tau $ and
$\overline{\pi_{\ast}\tau}$ are tightly distinguished, then
\begin{math}
T(\overline \xi_{1})=T(\overline \xi _{0})+T(\overline \xi _{2}).
\end{math}
\end{corollary}
\begin{corollary}
Let $\overline{\xi}= (\overline{\pi},\overline{\mathcal{F}},
\overline{\pi_{\ast}\mathcal{F}})$ be a relative metrized complex and
let $\overline{\xi}[i]= (\overline{\pi},\overline{\mathcal{F}}[i],
\overline{\pi_{\ast}\mathcal{F}}[i])$ be the shifted relative metrized
complex. Then
\begin{math}
T(\overline{\xi})=(-1)^{i}T(\overline{\xi}[i]).
\end{math}
\end{corollary}
\begin{proof}
It is enough to treat the case $i=1$. We consider the tightly
distinguished triangle
\begin{displaymath}
\overline{\mathcal{F}}\dashrightarrow \ocone(\Id_{\overline{\mathcal{F}}})\dashrightarrow \overline{\mathcal{F}}[1]\dashrightarrow
\end{displaymath}
and the analogous triangle for direct images. Since
$\ocone(\Id_{\overline{\mathcal{F}}})$ and
$\ocone(\Id_{\overline{ \pi _{\ast}\mathcal{F}}})$ are meager,
we have, by the anomaly formulas and the additivity axiom,
\begin{displaymath}
T(\overline \pi , \ocone(\Id_{\overline{\mathcal{F}}}), \ocone(\Id_{\overline{
\pi _{\ast}\mathcal{F}}}))=
T(\overline \pi ,\overline 0,\overline 0)=0.
\end{displaymath}
Hence, the result follows from Corollary \ref{cor:2}.
\end{proof}
Next we rewrite Proposition \ref{prop:2} in the language of
complexes of metrized coherent sheaves.
Let
\begin{displaymath}
\overline{\varepsilon }:\quad
0\to \overline{\mathcal{F}}_{m}\to \dots \to \overline{\mathcal{F}}_{l}\to 0
\end{displaymath}
be a bounded complex of metrized coherent sheaves on
${\mathbb P}^{n}_{X}$ and assume that
hermitian structures on the complexes
$ \pi _{\ast}\mathcal{F}_{j}$, $j=l,\dots,m$ are chosen. Let $[\overline \varepsilon
],
[\overline{ \pi _{\ast}\varepsilon }]\in \Ob \oDb({\mathbb P}^{n}_{X})$ be
the associated objects as in
\cite[Def. 3.37, Def. 3.39]{BurgosFreixasLitcanu:HerStruc}.
\begin{corollary} \label{cor:1}
With the above hypothesis,
\begin{displaymath}
T(\overline \pi ,[\overline \varepsilon ],[\overline{ \pi
_{\ast}\varepsilon }]) =\sum_{j=l}^{m}(-1)^{j}T(\overline \pi ,\overline
{\mathcal{F}}_{j},\overline{ \pi _{\ast}\mathcal{F}_{j}}).
\end{displaymath}
Moreover, if $\varepsilon $ is acyclic, then
\begin{math}
T(\overline \pi ,[\overline \varepsilon ],[\overline{ \pi
_{\ast}\varepsilon }]) =\widetilde{\ch}(\overline{ \pi
_{\ast}\varepsilon }) -\overline \pi _{\flat}[\widetilde{\ch}
(\overline \varepsilon )].
\end{math}
\end{corollary}
Finally, we show that the projection formula holds in greater generality:
\begin{proposition}\label{prop:3}
Let $T$ be a theory of analytic torsion classes for projective
spaces of dimension $n$. Let $X$ be a smooth complex variety, let
$\overline \xi=(\overline \pi ,\overline{\mathcal{F}},\overline{ \pi _{\ast}
\mathcal{F}})$ be a relative metrized complex and let
$\overline{\mathcal{G}}$ be an object in $\oDb(X)$. Then
\begin{equation} \label{eq:20}
T(\overline{\xi}\otimes\overline{\mathcal{G}})=T(\overline{\xi})\bullet\ch(\overline{\mathcal{G}}).
\end{equation}
\end{proposition}
\begin{proof}
By the anomaly formulas, if equation \eqref{eq:20} holds for a
particular choice of hermitian structures on $\pi $, $\mathcal{F}$
and $ \pi _{\ast} \mathcal{F}$ then it holds for any other
choice. Moreover, if we are in the situation of Proposition
\ref{prop:2} and equation \eqref{eq:20} holds for two of $\overline \xi
_{0}$, $\overline \xi _{1}$, $\overline \xi _{2}$, then it holds for the third.
Using that the objects of the form $\mathcal{H}(k)$, where
$\mathcal{H}$ is a coherent sheaf on $X$ and $k=-n,\dots,0$,
constitute a
generating class of $\Db({\mathbb P}^{n}_{X})$, we are reduced to prove that
\begin{displaymath}
T(\overline{\xi}_{n}(k)\otimes\overline{\mathcal{G}})=
T(\overline{\xi}_{n}(k))\bullet\ch(\overline{\mathcal{G}}).
\end{displaymath}
for $k=-n,\dots,0$.
Now, if
\begin{displaymath}
\overline{\mathcal{G}}_{2}\dashrightarrow \overline{\mathcal{G}}_{1} \dashrightarrow \overline{\mathcal{G}}_{0} \dashrightarrow
\end{displaymath}
is a distinguished triangle in $\oDb(X)$ and equation \eqref{eq:20}
is satisfied for two of $\overline{\mathcal{G}}_{2}$,
$\overline{\mathcal{G}}_{1}$, $\overline{\mathcal{G}}_{0}$, then it is satisfied
also by the
third. Therefore, since the complexes of vector bundles concentrated in
a single degree constitute a generating class of $\Db(X)$, the
projection formula axiom implies the proposition.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:1} ]
To begin with, we prove the uniqueness assertion.
Assume a theory of
analytic torsion classes $T$, with main characteristic numbers
$t_{n,k}$, $-n\le k\le 0$, exists. Then, the anomaly formulas (Proposition \ref{prop:1}) imply that, if
$T(\overline{\pi}, \overline{\mathcal{F}},
\overline{\pi_{\ast}\mathcal{F}})$ is known for a particular choice of hermitian
structures on $\pi $, $\mathcal{F}$ and $ \pi _{\ast} \mathcal{F}$
then the value of $T(\overline{\pi}', \overline{\mathcal{F}}',
\overline{\pi_{\ast}\mathcal{F}}')$ for any other choice of hermitian
structures is fixed.
By Proposition \ref{prop:2}, if we know the value of $T(\overline{\pi},
\overline{\mathcal{F}},
\overline{\pi_{\ast}\mathcal{F}})$, for $\mathcal{F}$ in a generating
class, then $T$ is determined.
By the projection formula (Proposition \ref{prop:3}), the
characteristic numbers determine the values of $T(\overline \xi(k)\otimes
\mathcal{G})$, $k=-n,\dots,0$. Finally, since by Corollary \ref{cor:6},
the objects of the form $\mathcal{G}(k)$, $k=-n,\dots,0$ form a
generating class, we
deduce that the characteristic numbers determine the theory $T$.
Thus, if it exists,
the theory $T_{\mathfrak{t}}$ is unique.
In particular, from the above discussion we see that
the main characteristic numbers determine all the characteristic
numbers. We now derive an explicit inductive formula for them.
Consider the metrized Koszul resolution
\begin{equation}\label{eq:38}
\overline{K}: 0\to
\Lambda ^{n+1}\overline V^{\vee}(-n-1)\to \dots \to
\Lambda ^{1} \overline V^{\vee}(-1)\to \overline{\mathcal{O}}_{{\mathbb P}^{n}_{{\mathbb C}}}\to 0,
\end{equation}
where $\mathcal{O}(k)$, for $k\not =0$, has the Fubini-Study metric
and $\overline{\mathcal{O}}_{{\mathbb P}^{n}_{{\mathbb C}}}$ has the trivial metric. We
will denote by $\overline{K}(k)$ the above exact sequence twisted by
$\overline{\mathcal{O}(k)}$, $k\in\mathbb{Z}$, again with the Fubini-Study
metric. Recall the definition of the relative metrized complexes
$\overline{\xi}_{n}(k)$ \eqref{eq:5}. In particular, for every $k$, we have
fixed natural hermitian structures on the objects
$\pi_{\ast}{\mathcal O}(k-j)$. According to \cite[Def. 3.37,
Def. 3.39]{BurgosFreixasLitcanu:HerStruc}, we may consider the classes
$[\overline{K}(k)]$ and
$[\overline{\pi_{\ast}K(k)}]$ in $\oDb({\mathbb P}^{n}_{{\mathbb C}})$ and
$\oDb(\Spec{\mathbb C})$, respectively. By Corollary \ref{cor:1}, for each
$k\in \mathbb{Z}$ we find
\begin{displaymath}
\sum_{j=0}^{n+1}(-1)^{j}T(\overline{\xi}_{n}(k-j)
\otimes\Lambda^{j}\overline{V}^{\vee})=
\widetilde{\ch}(\overline{\pi_{\ast}K(k)})
-\overline\pi_{\flat} ^{\text{\rm FS}}[\widetilde{\ch}(\overline{K}(k))].
\end{displaymath}
Because $\Lambda^{j}\overline{V}^{\vee}$ is isometric to
${\mathbb C}^{\binom{n+1}{j}}$ with the trivial metric, the additivity axiom
for the theory $T$ and the definition of the characteristic numbers
$t_{n,k-j}$ provide
\begin{displaymath}
T(\overline{\xi}_{n}(k-j)\otimes\Lambda^{j}\overline{V}^{\vee})=
t_{n,k-j}\binom{n+1}{j}.
\end{displaymath}
Therefore we derive
\begin{equation}\label{eq:79}
\sum_{j=0}^{n+1}(-1)^{j}\binom{n+1}{j}t_{n,k-j}=
\widetilde{\ch}(\overline{\pi_{\ast}K(k)})
-\overline \pi^{\text{\rm FS}}_{\flat}[\widetilde{\ch}(\overline{K}(k))].
\end{equation}
This equation gives us an inductive formula for all the characteristic
numbers $t_{n,k}$ once we have fixed $n+1$ consecutive characteristic
numbers and, in particular, once we have fixed the main characteristic
numbers.
To prove the existence, we follow the proof of the uniqueness to obtain a
formula for $T(\overline \xi)$. We start with the main characteristic
numbers $\mathfrak{t}=(t_{n,k})_{-n\le k \le 0}$. We define the characteristic
numbers $t_{n,k}$ for $k\in \mathbb{Z}$ inductively using equation
(\ref{eq:79}).
We will need the following results.
\begin{lemma}\label{lemm:6}
Let
\begin{displaymath}
\overline \eta:
0\to \overline {\mathcal{F}}_{2}\to
\overline {\mathcal{F}}_{1}\to
\overline {\mathcal{F}}_{0}\to
0
\end{displaymath}
be a short exact sequence of metrized coherent sheaves on $X$. Let
$k$ be an integer, and $\overline {\mathcal{F}}(k)$ and $\overline { \pi
_{\ast}{\mathcal{F}}(k)}$ be as in Notation \ref{def:7}. Thus we
have an exact sequence $\overline
\eta(k)$ of metrized coherent sheaves on ${\mathbb P}^{n}_{X}$ and a
distinguished triangle $\overline { \pi _{\ast}\eta(k)}$. Then
\begin{equation}
\label{eq:44}
\cht(\overline { \pi _{\ast}\eta(k)})=\overline \pi^{\text{\rm FS}}
_{\flat}(\cht(\overline \eta(k))).
\end{equation}
\end{lemma}
\begin{proof}
By
the Riemann-Roch theorem for the map ${\mathbb P}^{n}_{\mathbb{C}}\to
\Spec\mathbb{C}$ we have
\begin{equation}
\label{eq:42}
\ch(\overline{\pi _{\ast}\mathcal{O}(k)})
=\pi _{\ast}(\ch(\overline{\mathcal{O}(k)})\Td(\overline \pi^{\text{\rm FS}} )).
\end{equation}
Hence, by the properties of Bott-Chern classes and the choice of metrics
\begin{align*}
\cht(\overline {\pi _{\ast}\eta(k)})&=
\cht(\overline \eta)\bullet \ch(\overline{\pi_{\ast}
\mathcal{O}(k)})\\
&=\cht(\overline \eta)\bullet \pi
_{\ast}(\ch(\overline{\mathcal{O}(k)})\Td(\overline \pi ^{\text{\rm FS}}))\\
&=\pi
_{\ast}\left(\cht(\overline \eta(k))\bullet \Td(\overline \pi
^{\text{\rm FS}})\right)\\
&=\overline \pi^{\text{\rm FS}}
_{\flat}(\cht(\overline \eta(k))).\qedhere
\end{align*}
\end{proof}
\begin{lemma}\label{lemm:5}
Let
\begin{equation}
\label{eq:36}
\overline \mu:
0\to \overline {\mathcal{M}}_{m}(-m-d)\to\dots\to \overline
{\mathcal{M}}_{l}(-l-d)
\to 0
\end{equation}
be an exact sequence of metrized coherent sheaves on ${\mathbb P}^{n}_{X}$, where, for
each $i=l,\dots, m$, $\overline {\mathcal{M}}_{i}$ is a
metrized coherent sheaf on $X$, and $\overline {\mathcal{M}}_{i}(k)$ is as
in Notation \ref{def:7}. On $ \pi _{\ast}{\mathcal{M}}_{i}(k)$ we
consider the hermitian structures given also by Notation
\ref{def:7}. Then
\begin{equation}
\label{eq:37}
\sum_{i=l}^{m} (-1)^{i}t_{n,-d-i}\ch(\overline {\mathcal{M}}_{i})=
\cht(\overline{ \pi
_{\ast}\mu} )- \overline \pi^{\text{\rm FS}} _{\flat}(\cht(\overline \mu )).
\end{equation}
\end{lemma}
\begin{proof}
We consider a commutative diagram of exact
sequences
\begin{displaymath}
\xymatrix{
& & &0\ar[d] & &0\ar[d] & \\
&\overline \mu' &0\ar[r] &\overline{\mathcal{M}}_{m}'(-m-d)\ar[r]\ar[d] &\dots\ar[r]
&\overline{\mathcal{M}}_{l}'(-l-d)\ar[r]\ar[d] &0
\\
&\overline \mu &0\ar[r] &\overline{\mathcal{M}}_{m}(-m-d)\ar[r]\ar[d] &\dots\ar[r]
&\overline{\mathcal{M}}_{l}(-l-d)\ar[r]\ar[d] &0
\\
&\overline \mu'' &0\ar[r] &\overline{\mathcal{M}}_{m}''(-m-d)\ar[r]\ar[d] &\dots\ar[r]
&\overline{\mathcal{M}}_{l}''(-l-d)\ar[r]\ar[d] &0
\\
& & &0 & &0 & \\
& & &\overline \xi_{m} &\dots &\overline \xi_{l}. &
}
\end{displaymath}
\emph{Claim.} If equation \eqref{eq:37} holds for two of $\overline \mu
$, $\overline \mu
'$ and $\overline \mu ''$, then it holds for the third.\\
\emph{Proof of the claim.} On the one hand we have
\begin{displaymath}
\sum_{i=l}^{m} (-1)^{i}t_{n,-d-i}\left(\ch(\overline
{\mathcal{M}}'_{i})-\ch(\overline {\mathcal{M}}_{i})
+\ch(\overline {\mathcal{M}}''_{i})\right)=
\sum_{i=l}^{m} (-1)^{i}t_{n,-d-i}\dd_{\mathcal{D}}\cht(\overline \xi _{i}).
\end{displaymath}
But, if $t\in \mathcal{D}^{1}(\Spec
\mathbb{C},1)=\mathbb{R}$ is a real number, in the group
$\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}(X,p)$, we have
$$ t \dd_{\mathcal{D}}\cht(\overline \xi _{i})=
-\dd_{\mathcal{D}}(t\bullet \cht(\overline \xi _{i})
)=0.$$
On the other hand, by Lemma \ref{lemm:6}
\begin{displaymath}
\cht(\overline{ \pi
_{\ast}\mu'} )- \cht(\overline{ \pi
_{\ast}\mu} )+ \cht(\overline{ \pi
_{\ast}\mu''} )=
\pi _{\flat}^{\text{\rm FS}}(\cht(\overline \mu ')) -\pi
_{\flat}^{\text{\rm FS}}(\cht(\overline \mu ))+\pi
_{\flat}^{\text{\rm FS}}(\cht(\overline \mu'' )).
\end{displaymath}
The proof of the lemma is done by induction on the length
$r=m-l$ of the complex. If $r\le n$ then $\mu(d+l)$
has the same shape as the canonical resolution of the zero
sheaf. By the uniqueness of the canonical resolution,
we have ${\mathcal{M}}_{i}=0$, for $i=l,\dots,m$. Using
the above claim when ${\mathcal{M}}_{i}=0$ has a non-trivial metric,
we obtain the lemma for~$r\le n$.
Assume now that $r>n$. Let $K$ be
the Koszul exact sequence (\ref{eq:38}). Then $K(1)\otimes
{\mathcal{M}}_{l}$
is the canonical resolution of the regular coherent sheaf
${\mathcal{M}}_{l}(1)$. By Theorem \ref{thm:14}~\ref{item:9} there is a
surjection of exact sequences $\mu \to K(-l-d)\otimes {\mathcal{M}}_{l}$ whose
kernel is an exact sequence
\begin{equation*}
\mu':
0\to {\mathcal{M}}'_{m}(-m-d)\to\dots\to {\mathcal{M}}'_{l+1}(-d-l-1)
\to 0.
\end{equation*}
We consider on $K$ the metrics of \eqref{eq:38}, for $i=l+1,\dots,m$,
we choose arbitrary metrics on ${\mathcal{M}}'_{i}$ and denote by $\overline \mu
'$ the corresponding exact sequence of metrized coherent sheaves.
By induction hypothesis, $\overline \mu '$ satisfies equation
\eqref{eq:37}. Moreover, since the characteristic
numbers $t_{n,k}$ for $k\not \in [0,n]$ are defined using equation
\eqref{eq:79}, the exact sequence $\overline K(-l-d)\otimes \overline
{\mathcal{M}}_{l}$ also satisfies equation \eqref{eq:37}. Hence the
lemma follows from the previous claim.
\end{proof}
We now treat the case of complexes concentrated in a single
degree. Let $\overline{\mathcal{F}}$ be a coherent sheaf on ${\mathbb P}^{n}_{X}$
with a hermitian structure and let $\overline{ \pi _{\ast} \mathcal{F}}$
be a choice of a hermitian structure on the direct image complex. Write
$\overline{\xi}=(\overline \pi^{\text{\rm FS}},\overline{\mathcal{F}}, \overline
{\pi_{\ast}\mathcal{F}})$
for the corresponding
relative metrized complex.
Choose an integer $d$ such
that $\mathcal{F}(d)$ is regular. Then we have the
resolution $ \gamma _{d}(\mathcal{F})$ of Corollary \ref{cor:4}. More
generally, let $\mu $ be an exact sequence of the form
\begin{displaymath}
0\to \mathcal{S}_{m}(-d-m)\to \dots \to \mathcal{S}_{1}(-d-1)\to
\mathcal{S}_{0}(-d)\to \mathcal{F}\to 0,
\end{displaymath}
where the $\mathcal{S}_{i}$, $i=0,\dots,m$ are coherent sheaves on
$X$. Assume that we have chosen hermitian structures on the sheaves
$\mathcal{S}_{i}$. Using Notation \ref{def:7} and \cite[Def. 3.37,
Def. 3.39]{BurgosFreixasLitcanu:HerStruc} we have objects $[\overline \mu ]$ in
$\KA({\mathbb P}^{n}_{X})$ and $[\overline{
\pi _{\ast}\mu }]$ in $\KA(X)$.
Then we write
\begin{equation}\label{eq:33}
T_{\mathfrak{t},\overline \mu
}(\overline{\xi})=\sum_{j=0}^{m}(-1)^{j}t_{n,j-d}\ch(\overline{\mathcal{S}}_{j})
-\cht(\overline{ \pi _{\ast}\mu })
+\overline \pi ^{\text{\rm FS}}_{\flat}(\cht{(\overline \mu )})
\end{equation}
\begin{lemma}\label{lemm:2} Given any choice of metrics
on the sheaves $\mathcal{G}_{i}$, (respectively $
{\mathcal{G}}_{i}'$) $i=0,\dots,n$, that
appear in the resolution $\gamma _{d}(\mathcal{F})$
(respectively $\gamma _{d+1}(\mathcal{F})$), denote by $\overline \gamma
_{d}$ and $\overline \gamma_{d+1} $ the corresponding exact sequences of
metrized coherent sheaves. Then
\begin{displaymath}
T_{\mathfrak{t},\overline \gamma
_{d+1}}(\overline{\xi})=T_{\mathfrak{t},\overline \gamma
_d}(\overline{\xi}).
\end{displaymath}
In particular, $T_{\mathfrak{t},\overline \gamma _d}(\overline{\xi})$ does not depend
on the choice of metrics on the sheaves $\mathcal{G}_{i}$.
\end{lemma}
\begin{proof}
By Theorem \ref{thm:14}~\ref{item:11}, there is an exact sequence
\begin{equation}
\label{eq:35}
\overline \mu:
0\to \overline {\mathcal{S}}_{n+k}(-n-k-d-1)\to\dots\to \overline {\mathcal{S}}_{0}(-d-1)\to
\overline {\mathcal{F}} \to 0,
\end{equation}
and a surjection of exact sequences $f\colon\overline {\mu}\to
\overline \gamma _{d}$ extending the identity on $\overline {\mathcal{F}}$.
Here
$\overline {\mathcal{S}}_{i}$, $i=0,\dots,n+k$ are coherent sheaves on
$X$ with hermitian structures.
By Theorem \ref{thm:14}~\ref{item:9} there is a surjection of
exact sequences $\overline \mu\longrightarrow \overline
\gamma _{d+1}$ extending the identity on $\overline {\mathcal{F}}$, whose kernel
is an exact sequence
\begin{equation}\label{eq:55}
\overline \varepsilon:
0\to \overline {\mathcal{M}}_{n+k}(-n-k-d-1)\to\dots\to \overline
{\mathcal{M}}_{0}(-d-1)
\to 0,
\end{equation}
where $\overline {\mathcal{M}}_{i}$, $i=0,\dots,n+k$ are coherent sheaves
on $X$, and we have chosen arbitrarily an hermitian structure on them.
Denote by $\overline \eta_{i}$ the rows of the exact sequence
\begin{displaymath}
0\to \overline \varepsilon \to
\overline \mu\to
\overline \gamma _{d+1}\to 0.
\end{displaymath}
Observe that $\overline \eta_{i}=\overline \eta'_{i}(-i-d-1)$ for some
short exact sequence $\overline \eta_{i}'$ on $X$.
When $j\ge n$ we denote $\overline{\mathcal{G}}'_{j}=\overline 0$.
Then, we have
\begin{multline}
\label{eq:45}
\sum_{j=0}^{n+k}(-1)^{j}t_{n,j-d-1}
\left(\ch(\overline{\mathcal{G}}'_{j})-\ch(\overline{\mathcal{S}}_{j})
+\ch(\overline {\mathcal{M}}_{j})\right)=\\
\sum_{j=0}^{n+k}(-1)^{j}t_{n,j-d-1}\dd_{\mathcal{D}}\cht(\overline \eta_{i}')
=0.
\end{multline}
By \cite[Prop.~3.41]{BurgosFreixasLitcanu:HerStruc}, we have
\begin{align}
\label{eq:46}
\cht(\overline { \pi _{\ast}\gamma_{d+1}})-\cht(\overline { \pi
_{\ast}\mu})
+\cht(\overline { \pi
_{\ast}\varepsilon })&=\sum_{j=0}^{n+k}(-1)^{j}\cht(\overline
{ \pi _{\ast} \eta_{j}}),\\
\label{eq:47}
\cht(\overline{\gamma }_{d+1})-
\cht(\overline \mu)+
\cht(\overline \varepsilon)&=
\sum_{j=0}^{n+k}(-1)^{j}\cht(\overline
\eta_{j}).
\end{align}
Combining equations (\ref{eq:45}), (\ref{eq:46})
and (\ref{eq:47}) and lemmas \ref{lemm:6} and \ref{lemm:5} we obtain
\begin{equation}
\label{eq:48}
T_{\mathfrak{t},\overline \mu }(\overline \xi )= T_{\mathfrak{t},\overline \gamma _{d+1}}(\overline \xi ).
\end{equation}
We consider now $\cone(\mu,\gamma _{d})$. On it we put the obvious
hermitian structure induced by $\overline{\mu}$ and $\overline{\gamma_{d}}$,
$\overline{\cone(\mu ,\gamma _{d})}$. On $\pi _{\ast}\cone(\mu,\gamma
_{d})$, we put the obvious family of hermitian metrics induced by
$\overline{\pi_{\ast}\mu}$ and $\overline{\pi_{\ast}\gamma_{d}}$, and denote
it as $\overline{\pi _{\ast}\cone(\mu,\gamma _{d})}$. By
\cite[Cor.~3.42]{BurgosFreixasLitcanu:HerStruc} we have
\begin{align}
\label{eq:7}
\cht(\overline{\cone(\mu,\gamma _{d})})&=\cht(\overline \gamma _{d})-\cht(\overline
\mu ),\\
\label{eq:8}
\cht(\overline { \pi _{\ast}\cone(\mu,\gamma _{d})})&=\cht(\overline{\pi
_{\ast} \gamma _{d}})-\cht(\overline
{ \pi _{\ast}\mu }).
\end{align}
Observe that $\overline{\cone(\mu,\gamma _{d})}^{i}= \overline
{\mathcal{S}}_{-i-1}(i-d) \oplus \overline
{\mathcal{G}}_{-i}(i-d) $. Combining Lemma \ref{lemm:5} for
$\overline{\cone(\mu,\gamma _{d})}$ with equations \eqref{eq:7} and
\eqref{eq:8}, we obtain
\begin{equation}
\label{eq:10}
T_{\mathfrak{t},\overline \mu }(\overline \xi )=T_{\mathfrak{t},\overline \gamma _{d}}(\overline \xi ),
\end{equation}
Together with equation \eqref{eq:48} this proves the lemma.
\end{proof}
Now we are in position to prove the existence of
$T_{\mathfrak{t}}$. Let $n$ and
$\mathfrak{t}$ be as in Theorem~\ref{thm:1}. We define the numbers
$t_{n,k}$, for $k< 0$
and $k> n$ by equation \eqref{eq:79}.
Let $\overline \xi
=(\overline \pi^{\text{\rm FS}}, \overline {\mathcal{F}},\overline { \pi_{ \ast}
\mathcal{F}})$ be a relative metrized complex. We construct
$T_{\mathfrak{t}}(\overline \xi)$ by induction on the length of the
cohomology of $\mathcal{F}$.
If it has at most a single non zero coherent sheaf
$\mathcal{H}$ sitting at degree $j$, then $\overline {\mathcal{F}}$ and
$\overline { \pi_{ \ast} \mathcal{F}}$ determine hermitian structures
on $\mathcal{H}[-j]$ and $ \pi _{\ast}\mathcal{H}[-j]$
respectively. We choose an integer $d$ such that $\mathcal{H}(d)$ is
regular and we write
\begin{equation}\label{eq:29}
T_{\mathfrak{t}}(\overline \xi)=
(-1)^{j}T_{\mathfrak{t},\overline \gamma _{d}(\mathcal{H})}(\overline
\pi^{\text{\rm FS}}, \overline {\mathcal{H}},\overline
{ \pi_{ \ast} \mathcal{H}}).
\end{equation}
By Lemma \ref{lemm:2}, this does not depend on the choice of $d$ nor
on the choice of metrics on~$\overline \gamma _{d}(\mathcal{H})$.
Assume
that we have defined the analytic torsion classes for all
complexes whose cohomology has length less than $l$ and that the
cohomology of
$\mathcal{F}$ has length $l$. Let $\mathcal{H}$ be the
highest cohomology sheaf of $\mathcal{F}$, say of degree
$j$. Choose
auxiliary hermitian structures on $\mathcal{H}[-j]$ and $ \pi
_{\ast}\mathcal{H}[-j]$. There is a unique natural map
$\mathcal{H}[-j]\dashrightarrow
\mathcal{F}$. Then we define
\begin{multline} \label{eq:30}
T_{\mathfrak{t}}(\overline \xi)=
T_{\mathfrak{t}}(\overline \pi ^{\text{\rm FS}},\overline{ \mathcal{H}[-j]},\overline
{ \pi_{ \ast} \mathcal{H}[-j]})\\+
T_{\mathfrak{t}}(\overline \pi ^{\text{\rm FS}},\ocone(\overline{ \mathcal{H}[-j]},\overline{
\mathcal{F}}),
\ocone(\overline
{ \pi_{ \ast} \mathcal{H}[-j]},\overline
{ \pi_{ \ast} \mathcal{F}})).
\end{multline}
It follows from \cite[Thm.~2.27~(iv)]{BurgosFreixasLitcanu:HerStruc}
that the right hand side of this equality
does not depend on the choice of the auxiliary hermitian
structures.
Finally, we consider the case when $\overline{\pi}$ has a metric
different from the Fubini-Study
metric. Thus, let $\overline \xi=(\overline \pi, \overline {\mathcal{F}},\overline { \pi_{
\ast} \mathcal{F}})$ and write $\overline \xi'=(\overline \pi^{\text{\rm FS}}, \overline
{\mathcal{F}},\overline { \pi_{
\ast} \mathcal{F}})$. Then we put
\begin{equation} \label{eq:6}
T_{\mathfrak{t}}(\overline{\xi})=T_{\mathfrak{t}}(\overline{\xi}')+\overline
\pi
_{\flat}[\ch(\overline F)\bullet
\widetilde{\Td}_{m}(\overline{\pi},\overline{\pi}^{\text{\rm FS}})].
\end{equation}
\begin{definition}\label{def:8} Let $n$ and
$\mathfrak{t}$ be as in Theorem \ref{thm:1}. Then $T_{\mathfrak{t}}$
is the assignment that to each
relative metrized complex $\overline \xi$ associates $T_{\mathfrak{t}}(\overline
\xi )$ given by equations \eqref{eq:29}, \eqref{eq:30} and \eqref{eq:6}.
\end{definition}
It remains to prove that $T_{\mathfrak{t}}$ satisfies axioms
(i) to (iv).
Axiom (i) follows from the differential equations satisfied by the
Bott-Chern classes.
Axiom (ii) follows from the functoriality of the canonical resolution,
the Chern forms and the Bott-Chern classes. Axiom (iii) follows from the
additivity of the canonical resolution and of the Chern
character. Finally Axiom (iv) follows from the multiplicativity of the Chern
character. This concludes the proof of Theorem \ref{thm:1}.
\end{proof}
We finish this section showing the compatibility of analytic torsion classes
with the composition of projective bundles.
Let $X$ be a smooth complex variety. Consider the commutative diagram
with cartesian square
\begin{displaymath}
\xymatrix{& {\mathbb P}^{n_{1}}_{X}\underset{X}{\times}{\mathbb P}^{n_{2}}_{X}
\ar[dl]_{p_{1}}\ar[dr]^{p_{2}}\ar[dd]^{p}&\\
{\mathbb P}^{n_{1}}_{X} \ar[dr]_{\pi _{1}} && {\mathbb P}^{n_{2}}_{X}\ar[dl]^{\pi _{2}}\\
&X&
}
\end{displaymath}
On $\pi _{1}$ and $\pi_{2}$ we introduce arbitrary hermitian
structures and on $p_{1}$ and $p_{2}$ the hermitian structures induced
by the cartesian diagram.
\begin{proposition}\label{prop:comp_proj}
Let $\overline{\mathcal{F}}$ be an object of
$\oDb({\mathbb P}^{n_{1}}_{X}\underset{X}{\times}{\mathbb P}^{n_{2}}_{X})$. Put
arbitrary hermitian structures on $ (p_{1})_{\ast} \mathcal{F}$,
$ (p_{2})_{\ast} \mathcal{F}$, and $ p_{\ast} \mathcal{F}$. Then
\begin{equation} \label{eq:15}
T(\overline \pi _{1})+
(\overline \pi _{1})_{\flat}(
T(\overline p_{1}))
=T(\overline \pi _{2})+
(\overline \pi _{2})_{\flat}(
T(\overline p_{2})),
\end{equation}
where we are using the convention at the end of Definition \ref{def:19}.
\end{proposition}
\begin{proof}
By the anomaly formulas (Proposition \ref{prop:1}), if equation
\eqref{eq:15} holds for a particular choice of hermitian structures
on $\mathcal{F}$, $ (p_{1})_{\ast} \mathcal{F}$, $
(p_{2})_{\ast} \mathcal{F}$, and $ p_{\ast} \mathcal{F}$, then it
holds for any other choice.
Let
\begin{displaymath}
\overline {\mathcal{F}}_{2}\dashrightarrow
\overline {\mathcal{F}}_{1}\dashrightarrow
\overline {\mathcal{F}}_{0}\dashrightarrow
\end{displaymath}
be a distinguished triangle and put hermitian structures on the
direct images as before. Then Proposition \ref{prop:2} implies that,
if equation \eqref{eq:15} holds for two of them, then it also holds for the
third. Since the objects of the form
$\mathcal{G}(k,l):=p^{\ast}\mathcal{G}\otimes
p_{1}^{\ast}\mathcal{O}(k)\otimes p_{2}^{\ast}\mathcal{O}(l)$ are a
generating class of
$\Db({\mathbb P}^{n_{1}}_{X}\underset{X}{\times}{\mathbb P}^{n_{2}}_{X})$, the
previous discussion shows that it is enough to prove the case
$\mathcal{F}=\mathcal{G}(k,l)$, with the hermitian structure of
$\mathcal{F}$ induced
by a hermitian structure of $\mathcal{G}$ and the Fubini-Study
metric on $\mathcal{O}(k)$ and $\mathcal{O}(l)$, and the hermitian
structures on the direct images defined as in \eqref{eq:5}. In this case the
result follows easily from the functoriality and the projection formula.
\end{proof}
\section{Compatible analytic torsion classes}
\label{sec:compat}
In this section we study the compatibility between analytic torsion
classes for closed immersions and analytic torsion classes for
projective spaces. It turns out that, once the compatibility between
the diagonal embedding of ${\mathbb P}^{n}$ into ${\mathbb P}^{n}\times {\mathbb P}^{n}$ and
the second projection of ${\mathbb P}^{n}\times {\mathbb P}^{n}$ onto ${\mathbb P}^{n}$ is
established, then all the other possible compatibilities
follow. Essentially this
observation can be traced back to~\cite{Bost:immersion}.
Let $n$, $V$, $\overline V$ and
${\mathbb P}^n(V)$ be as in the previous section.
We
consider the diagram
\begin{displaymath}
\xymatrix{
{\mathbb P}^n \ar[rd]_{\Id} \ar[r]^{\Delta\ \ \ } & {\mathbb P}^n
\times{\mathbb P}^n \ar[r]^{p_1}\ar[d]^{p_2} & {\mathbb P}^n \ar[d]^{\pi}\\
& {\mathbb P}^n \ar[r]_{\pi_{1}} & \Spec{\mathbb C}\ .}
\end{displaymath}
On ${\mathbb P}^n$ we have the tautological short exact sequence
\begin{displaymath}
0\rightarrow \mathcal{O}(-1)\rightarrow V\rightarrow Q
\rightarrow 0\ .
\end{displaymath}
This induces on ${\mathbb P}^n \times{\mathbb P}^n$ the exact sequence
\begin{displaymath}
0\rightarrow p_2^{\ast}\mathcal{O}(-1) \rightarrow
V \rightarrow p_2^{\ast}Q\rightarrow 0\ .
\end{displaymath}
By composition with the injection
\begin{math}
p_1^{\ast}\mathcal{O}(-1)\hookrightarrow V
\end{math},
we obtain a morphism
\begin{math}
p_1^{\ast}\mathcal{O}(-1)\to p_2^{\ast}Q,
\end{math}
hence a section of $p_2^{\ast}Q\otimes p_1^{\ast}\mathcal{O}(1)$.
The zero locus of this section is the image of the diagonal. Moreover,
the associated Koszul complex is quasi-isomorphic to
$\Delta_{\ast}\mathcal{O}_{{\mathbb P}^n}$. That is, the sequence
\begin{multline}\label{eq:15bis}
0\rightarrow \Lambda^n(p_2^{\ast}Q^\vee)\otimes
p_1^{\ast}\mathcal{O}_{{\mathbb P}^n}(-n)\rightarrow\dots\\ \dots \rightarrow
\Lambda^1(p_2^{\ast}Q^\vee)\otimes
p_1^{\ast}\mathcal{O}_{{\mathbb P}^n}(-1)\rightarrow
\mathcal{O}_{{\mathbb P}^n\times{\mathbb P}^n} \rightarrow
\Delta_{\ast}\mathcal{O}_{{\mathbb P}^n}\rightarrow 0
\end{multline}
is exact.
On $T_{{\mathbb P}^{n}}$ and $T_{{\mathbb P}^{n}\times {\mathbb P}^{n}}$ we consider the
Fubini-Study metrics. We denote by $\overline \Delta $ and $\overline p_{2}$ the
morphisms of $\oSm_{\ast/{\mathbb C}}$ determined by these metrics. As in
\cite[Ex.~5.7]{BurgosFreixasLitcanu:HerStruc}, we have that $\overline
{p_{2}}\circ \overline \Delta =\overline \Id_{{\mathbb P}^{n}}$, where
$T_{\overline{\Id_{{\mathbb P}^{n}}}}=\overline 0$.
The Fubini-Study metric on $\mathcal{O}(-1)$ and the metric induced by
the tautological exact sequence on $Q$ induce a metric $\overline K(\Delta )$ on the Koszul
complex. This is a hermitian structure
on $ \Delta _{\ast}\mathcal{O}_{{\mathbb P}^{n}}$.
Finally on $\mathcal{O}_{{\mathbb P}^{n}}$ we consider the trivial
metric. This is a hermitian structure on~$ (p_{2})_{\ast}K(\Delta)$.
Fix a real additive genus $S$ and denote by $T_S$ the theory of
analytic torsion classes for closed immersions that is compatible with
the projection formula and transitive, associated to $S$
Theorem \ref{thm:17}). Moreover, fix a family of real numbers
$\mathfrak{t}=\{t_{nk}\ |\ n\geq 0,\ -n\leq k\leq 0\}$ and denote
$T_{\mathfrak{t}}$ the theory of generalized analytic torsion classes
for projective spaces associated to this family.
Compatible analytic torsion classes for
closed immersions and for projective spaces should combine to provide
analytic torsion classes for arbitrary projective morphisms, and these
classes should be transitive. The transitivity condition for
the composition $\Id_{{\mathbb P}^{n}}=p_{2}\circ \Delta $ should give us
\begin{displaymath}
0=T(\overline \Id_{{\mathbb P}^{n}},\overline {\mathcal{O}}_{{\mathbb P}^{n}},\overline
{\mathcal{O}}_{{\mathbb P}^{n}}) =
T_{\mathfrak{t}}(\overline{p_{2}},\overline K(\Delta ),\overline{\mathcal{O}}_{{\mathbb P}^{n}})+
(\overline p_{2})_{\flat}(T_{S}(\overline \Delta ,\overline{\mathcal{O}}_{{\mathbb P}^{n}}, \overline
K(\Delta ))).
\end{displaymath}
In general we define
\begin{definition}\label{compatprojsp}
The theories of analytic torsion classes $T_S$ and $T_{\mathfrak{t}}$
are called
\emph{compatible} if
\begin{equation}
\label{eq:12}
T_{\mathfrak{t}}(\overline{p_{2}},\overline K(\Delta ),\overline{\mathcal{O}}_{{\mathbb P}^{n}})
+(\overline p_{2})_{\flat}(T_{S}(\overline \Delta ,\overline{\mathcal{O}}_{{\mathbb P}^{n}}, \overline
K(\Delta )))=0.
\end{equation}
\end{definition}
\begin{theorem} \label{thm:15}
Let $S$ be a real additive genus. Then there exists a unique family
of real numbers $\mathfrak{t}=\{t_{n,k}\ |\ n\geq 0,\
-n\leq k\leq 0\}$ such that the theories of analytic torsion classes
$T_S$ and $T_{\mathfrak{t}}$
are compatible. The theory $T_{\mathfrak{t}}$ will also be denoted $T_{S}$.
\end{theorem}
\begin{proof}
The first step is to make explicit equation \eqref{eq:12} in terms
of the main characteristic numbers $\mathfrak{t}$. To this end,
first observe that, since the exact sequence
\begin{equation}\label{eq:17}
0\to T_{p_{2}}\to T_{{\mathbb P}^{n}\times {\mathbb P}^{n}}\to
p_{2}^{\ast}T_{{\mathbb P}^{n}}\to 0
\end{equation}
is split and the hermitian metric on $T_{{\mathbb P}^{n}\times {\mathbb P}^{n}}$ is
the orthogonal direct sum metric, $\overline {p_{2}}=\pi
^{\ast}_{1}(\overline \pi ^{\text{\rm FS}})$. Next, we denote by $\overline K(\Delta
)_{i}$ the component of degree $i$ of the Koszul complex, and we
define
\begin{displaymath}
\overline { (p_{2})_{\ast}K(\Delta
)_{i}}=
\begin{cases}
\overline {\mathcal{O}}_{{\mathbb P}^{n}},&\text{for }i=0, \\
\overline 0,&\text{for }i>0.
\end{cases}
\end{displaymath}
Finally using Corollary \ref{cor:1}, functoriality and the
compatibility with the projection formula, we derive
\begin{align*}
T_{\mathfrak{t}}(\overline{p_{2}},\overline K(\Delta
),\overline{\mathcal{O}}_{{\mathbb P}^{n}})&=
\sum_{i=0}^{n}(-1)^{i}T_{\mathfrak{t}}(\overline{p_{2}},\overline K(\Delta )_{i},\overline { (p_{2})_{\ast}K(\Delta
)_{i}})\\
&=\sum_{i=0}^{n}(-1)^{i}T_{\mathfrak{t}}(\pi _{1}^{\ast}\overline \xi
_{n}(-i)\otimes \Lambda ^{i}\overline Q^{\vee})\\
&=\sum_{i=0}^{n}(-1)^{i}t_{n,-i}\ch(\Lambda ^{i}\overline Q^{\vee}).
\end{align*}
Thus, the second and last step is to solve the equation
\begin{equation}
\label{eq:13}
\sum_{i=0}^{n}(-1)^{i}t_{n,-i}\ch(\Lambda ^{i}\overline Q^{\vee})=
-(p_{2})_{\ast}(T_{S}(\overline \Delta ,\overline{\mathcal{O}}_{{\mathbb P}^{n}}, \overline
K(\Delta ))\bullet \Td(\overline{p_{2}})).
\end{equation}
Since the left hand side of equation \eqref{eq:13} is closed, in
order to be able to solve this equation we have to show that the
right hand side is also closed. We compute
\begin{align*}
\dd_{\mathcal{D}}(p_{2})_{\ast}&(T_{S}(\overline \Delta
,\overline{\mathcal{O}}_{{\mathbb P}^{n}}, \overline
K(\Delta ))\bullet \Td(\overline{p_{2}}))\\
&=(p_{2})_{\ast}\left(
\sum_{i=0}^{n}(-1)^{i}\ch(\overline K(\Delta )_{i})\Td(\overline p_{2})-
\Delta _{\ast}(\ch(\overline {\mathcal{O}}_{{\mathbb P}^{n}}) \Td(\overline \Delta ))\Td(\overline p_{2})
\right)\\
&=(p_{2})_{\ast}\left(
\sum_{i=0}^{n}(-1)^{i}p_{2}^{\ast}(\ch(\Lambda ^{i}\overline Q^{\vee}))
p_{1}^{\ast}(\ch(\overline{\mathcal{O}}(-i)))\Td(\overline p_{2})\right)-1\\
&=
\sum_{i=0}^{n}(-1)^{i}\ch(\Lambda ^{i}\overline Q^{\vee})(p_{2})_{\ast}\left(
p_{1}^{\ast}(\ch(\overline{\mathcal{O}}(-i)))\Td(\overline p_{2})\right)-1\\
&=
\sum_{i=0}^{n}(-1)^{i}\ch(\Lambda ^{i}\overline Q^{\vee})\pi _{1}^{\ast}\pi _{\ast}\left(
\ch(\overline{\mathcal{O}}(-i))\Td(\overline \pi )\right)-1\\
&=1-1=0.
\end{align*}
In the first equality we have used the differential equation of
$T_{S}$. In the second one we have used the definition of the
Koszul complex, the equation $\ch(\overline
{\mathcal{O}}_{{\mathbb P}^{n}})=1$ and the fact that, by the choice of
hermitian structures on $T_{\overline \Delta } $ and $T_{\overline p_{2}}$ we
have $\Td(\overline \Delta )\bullet \Delta ^{\ast}(\Td(\overline p_{2}))=1$. The third
equality is the projection formula and the fourth is base change for
cohomology. For the last equality we have used
equation~\eqref{eq:42}.
Both sides of equation \eqref{eq:13} are closed and
defined up to boundaries, hence this is an equation in
cohomology classes. The tautological exact sequence induces
exact sequences
\begin{displaymath}
0\to \Lambda ^{k}Q^{\vee}\to \Lambda ^{k}V^{\vee}\to
\Lambda ^{k-1}Q^{\vee}\otimes \mathcal{O}(1)\to 0,
\end{displaymath}
that give us equations
\begin{displaymath}
\ch(\Lambda ^{k}Q^{\vee})=\binom{n+1}{k}-\ch(\Lambda
^{k-1}Q^{\vee})\ch(\mathcal{O}(1)).
\end{displaymath}
Hence
\begin{displaymath}
\ch(\Lambda
^{k}Q^{\vee})=\sum^{k}_{i=0}(-1)^{i}\binom{n+1}{k-i}\ch(\mathcal{O}(i)).
\end{displaymath}
Since the classes $\ch(\mathcal{O}(i))$, $i=0,\dots,n$, form a
basis of $\bigoplus _{p}H^{2p}_{\mathcal{D}}({\mathbb P}^{n},{\mathbb R}(p))$, the
same is true for the classes $\ch(\Lambda ^{i}Q^{\vee})$,
$i=0,\dots,n$. Therefore, if $\mathbf{1}_{1}\in
H^{1}_{\mathcal{D}}({\mathbb P}^{n},{\mathbb R}(1))$ is the class represented by the
constant function $1$, the classes $\mathbf{1}_{1}\bullet \ch(\Lambda
^{i}Q^{\vee})$, $i=0,\dots,n$ form a basis of $\bigoplus
_{p=1}^{n+1}H^{2p-1}_{\mathcal{D}}({\mathbb P}^{n},{\mathbb R}(p))$. This implies
that equation \eqref{eq:13} has a unique solution.
\end{proof}
\begin{remark}
Given a theory $T$ of analytic torsion classes for projective spaces,
obtained from an arbitrary choice of
characteristic numbers, in general, it does not exist an additive
genus such that the associated theory of
singular Bott-Chern classes is compatible with $T$. It would be
interesting to characterize the collections of characteristic
numbers that arise from Theorem \ref{thm:15}.
\end{remark}
By definition, compatible analytic torsion classes for closed immersions and
projective spaces satisfy a compatibility condition for the
trivial vector bundle and the diagonal embedding. When adding the
functoriality and the projection formula, we obtain compatibility
relations for arbitrary sections of the trivial projective bundle and
arbitrary objects.
Let $X$ be a smooth complex variety, let $\pi \colon {\mathbb P}_{X}^{n}\to
X$ be the projective space over $X$ and let $s\colon X\to
{\mathbb P}_{X}^{n}$ be a section. Choose any hermitian structure on $T_{\pi
}$. Since we have an isomorphism $T_{s}\dashrightarrow s^{\ast} T_{\pi
}[-1]$, this hermitian structure induces a hermitian structure on
$s$. Denote by $\overline \pi $ and $\overline s$ the corresponding morphisms in
$\overline\Sm_{\ast/{\mathbb C}}$. With this choice of hermitian structures, we
have
\begin{displaymath}
\overline \pi \circ \overline s=(\pi\circ s,\ocone( s^{\ast} T_{\overline \pi
}[-1], s^{\ast} T_{\overline \pi
}[-1]))=(\Id_{X},\overline 0),
\end{displaymath}
because the cone of the identity is meager.
\begin{proposition}
Let $S$ be a real additive genus. Let $T_{S}$ denote both, the
theory of analytic torsion classes for closed immersions determined
by $S$, and the theory of analytic torsion classes for projective
spaces compatible with it.
Let $\overline{\mathcal{F}}$ be an object of $\oDb(X)$. Put a hermitian
structure on $ s_{\ast} \mathcal{F}$. Then
\begin{equation}
\label{eq:19}
T_{S}(\overline \pi , \overline{ s_{\ast} \mathcal{F}}, \overline {\mathcal{F}})
+\overline \pi _{\flat}(T_{S}(\overline s,\overline {\mathcal{F}}, \overline{ s_{\ast}
\mathcal{F}}))=0.
\end{equation}
\end{proposition}
\begin{proof}
By the anomaly formulas Proposition \ref{prop:1bis} and Proposition
\ref{prop:1}, if equation \eqref{eq:19} holds for a particular
choice of hermitian structure on $ s_{\ast} \mathcal{F}$ then it
holds for any other choice. Therefore we can assume that the
hermitian structure on $ s_{\ast} \mathcal{F}$ is given by $\overline
K(s)\otimes \pi ^{\ast}\overline{\mathcal{F}}$, where $\overline K(s)$ is the
Koszul complex associated to the section~$s$.
By the projection formulas, if \eqref{eq:19} holds for the
trivial bundle $\mathcal{O}_{X}$ then it holds for arbitrary
objects of~$\oDb(X)$.
We now prove that, if equation \eqref{eq:19} holds for a particular
choice of hermitian structure $\overline \pi $, then it holds for any
other choice. Thus, assume that equation \eqref{eq:19} is satisfied
for $\overline \pi $ and $\overline s$. Let $\overline \pi '$ be another choice of
hermitian structure on $\pi $ and $\overline s'$ be the hermitian
structure induced on $s$. On one hand, we have
\begin{equation}
\label{eq:21}
T_{S}(\overline \pi ',\overline K(s),\overline {\mathcal{O}}_{X})=
T_{S}(\overline \pi ,\overline K(s),\overline {\mathcal{O}}_{X})+
\pi _{\ast}\left(
\ch(\overline K(s)\bullet \widetilde \Td_{m}(\overline \pi ',\overline \pi
)\bullet \Td(\overline \pi '))
\right).
\end{equation}
On the other hand, we have
\begin{align}
&T_{S}(\overline s',\overline {\mathcal{F}}, \overline{ s_{\ast}
\mathcal{F}})
\bullet \Td(\overline \pi' )\\
&\phantom{A}=
\left(T_{S}(\overline s,\overline {\mathcal{F}}, \overline{ s_{\ast}
\mathcal{F}}) +s_{\ast}(\widetilde{\Td}_{m}(\overline s',\overline
s)\Td(\overline s'))\right)\bullet
\left(\Td(\overline \pi )-\dd_{\mathcal{D}}(\widetilde \Td_{m}(\overline \pi ',\overline \pi
)\bullet \Td(\overline \pi '))
\right) \notag\\
&\phantom{A}= T_{S}(\overline s,\overline {\mathcal{F}}, \overline{ s_{\ast}
\mathcal{F}})\bullet \Td(\overline \pi )+s_{\ast}(\widetilde{\Td}_{m}(\overline s',\overline
s)\Td(\overline s'))\bullet \Td(\overline \pi ')\notag \\
&\phantom{AAAAAAAAAAAAAAA}
-T_{S}(\overline s,\overline {\mathcal{F}}, \overline{ s_{\ast}
\mathcal{F}})\bullet \dd_{\mathcal{D}}(\widetilde \Td_{m}(\overline \pi ',\overline \pi
)\bullet \Td(\overline \pi '))\notag
\end{align}
In the group $\bigoplus_{p}\widetilde
D_{D}^{2p-1}({\mathbb P}^{n}_{X},N_{s},p)$ we have
\begin{multline}
\label{eq:23}
T_{S}(\overline s,\overline {\mathcal{F}}, \overline{ s_{\ast}
\mathcal{F}})\bullet \dd_{\mathcal{D}}(\widetilde \Td_{m}(\overline \pi ',\overline \pi
)\bullet \Td(\overline \pi '))\\=
\left(\ch(\overline K(s))-s_{\ast}(\Td(\overline s))\right)\bullet
\left(\widetilde \Td_{m}(\overline \pi ',\overline \pi
)\bullet \Td(\overline \pi ')\right).
\end{multline}
Observe that, by the definition of the hermitian structure
of $\overline s$ and $\overline s'$ we have
\begin{equation}
\label{eq:24}
\Td(\overline s)\bullet s^{\ast}\widetilde {\Td}_{m}(\overline \pi ',\overline \pi
)=
-\widetilde {\Td}_{m}(\overline s',\overline s)\bullet\Td(\overline s').
\end{equation}
By combining equations \eqref{eq:19}, \eqref{eq:21}, \eqref{eq:23}
and \eqref{eq:24} we obtain
\begin{equation}
\label{eq:25}
T_{S}(\overline \pi' , \overline{ s_{\ast} \mathcal{F}}, \overline {\mathcal{F}})=
-\pi _{\ast}\left(T_{S}(\overline s',\overline {\mathcal{F}}, \overline{ s_{\ast}
\mathcal{F}})
\bullet \Td(\overline \pi' )
\right).
\end{equation}
We now prove \eqref{eq:19} for a particular choice of
hermitian structures. Let $f\colon X\to {\mathbb P}^{n}$ denote
the composition of
$s$ with the projection ${\mathbb P}^{n}_{X}\to {\mathbb P}^{n}$. Then
we have a commutative diagram with cartesian squares
\begin{displaymath}
\xymatrix{
{\mathbb P}^{n}\times X\ar[rrr]^{\Id\times f}\ar[dd]_{\pi }
&&& {\mathbb P}^{n}\times {\mathbb P}^{n} \ar[dd]^{p_{2}}\\
& X \ar[ul]^{s}\ar[dl]_{\Id}\ar[r]_{f}&
{\mathbb P}^{n}\ar[ru]_{\Delta }\ar[rd]^{\Id} &\\
X\ar[rrr]_{f} &&& {\mathbb P}^{n}
}
\end{displaymath}
Let $\overline \Delta $ and $\overline p_{2}$ be as in Definition
\ref{compatprojsp}.
On $\overline \pi $ and $\overline s$ we put the hermitian structures induced
by $\overline \Delta $. Since the Koszul complex $\overline
K(s)=(\Id_{{\mathbb P}^{n}}\times f )^{\ast}\overline K(\Delta )$, by
Proposition \ref{prop:11} and functoriality, equation
\eqref{eq:19} in this case follows from equation \eqref{eq:12}.
\end{proof}
We now study another compatibility between analytic torsion classes
for closed immersions and projective spaces. Let $\iota\colon X\to Y$
be a closed immersion of smooth complex varieties. Consider the
cartesian square
\begin{displaymath}
\xymatrix{
{\mathbb P}^{n}_{X} \ar[d]_{\pi _{1}}\ar[r]^{\iota _{1}}&
{\mathbb P}^{n}_{Y} \ar[d]^{\pi }\\
X\ar[r]_{\iota }& Y
}
\end{displaymath}
Choose hermitian structures on $\pi $ and $\iota $ and put on $\pi
_{1}$ and $\iota _{i}$ the induced ones.
\begin{proposition}
\label{prop:5}
Let $S$ be a real additive genus. Let $T_{S}$ denote both, the
theory of analytic torsion classes for closed immersions determined
by $S$, and the theory of analytic torsion classes for projective
spaces compatible with it.
Let $\overline{\mathcal{F}}$ be and object of $\oDb({\mathbb P}^{n}_{X})$. Put hermitian
structures on $ (\pi _{1})_{\ast} \mathcal{F}$, $ (\iota
_{1})_{\ast} \mathcal{F}$ and $ (\pi \circ \iota _{1})_{\ast}
\mathcal{F}$. Then
\begin{equation}
\label{eq:26}
T_{S}(\overline \pi )+ \overline \pi _{\flat}(T_{S}(\overline {\iota _{1}}))
=
T_{S}(\overline \iota)+ \overline \iota _{\flat}(T_{S}(\overline {\pi _{1}})).
\end{equation}
\end{proposition}
\begin{proof}
By the anomaly formulas, if equation \eqref{eq:26} holds for a
particular choice of metrics on $ (\pi _{1})_{\ast}
\mathcal{F}$, $ (\iota _{1})_{\ast} \mathcal{F}$ and $ (\pi
\circ \iota _{1})_{\ast} \mathcal{F}$, then it holds for any
choice. Because the sheaves $\mathcal{G}(k)$, with $\mathcal{G}$ a
coherent sheaf on $X$, constitute a generating class of
$\Db({\mathbb P}^{n}_{X})$ and by propositions \ref{prop:2bis} and
\ref{prop:2}, we reduce to the case when $\mathcal{F}$ is of the
form $\mathcal{G}(k)$. We choose arbitrary hermitian structures on
$\mathcal{G}$ and $\iota_{\ast}\mathcal{G}$. Furthermore, we
assume ${\mathcal O}(k)$, $(\pi_{1})_{\ast}{\mathcal O}(k)$ and
$\pi_{\ast}{\mathcal O}(k)$ endowed with the hermitian structures of
Notation \ref{def:7}. From these choices and the projection
formula, the objects $(\pi_{1})_{\ast}\mathcal{F}$,
$(\iota_{1})_{\ast}\mathcal{F}$ and $ (\pi \circ \iota
_{1})_{\ast}\mathcal{F}$ automatically inherit hermitian
structures. Indeed, it is enough to observe the natural
isomorphisms
\begin{align}
&(\pi_{1})_{\ast}\mathcal{F}\cong\mathcal{G}\otimes(\pi_{1})_{\ast}
{\mathcal O}(k)\label{eq:26_1}\\
&(\iota_{1})_{\ast}(\pi_{1}^{\ast}\mathcal{G}\otimes\iota_{1}^{\ast}{\mathcal O}(k))
\cong\pi^{\ast}(\iota_{\ast}\mathcal{G})\otimes{\mathcal O}(k)\label{eq:26_2}\\
&(\pi\circ\iota_{1})_{\ast}\mathcal{F}\cong\pi_{\ast}(\pi^{\ast}\iota_{\ast}\mathcal{G}\otimes{\mathcal O}(k))\cong\iota_{\ast}\mathcal{G}\otimes\pi_{\ast}{\mathcal O}(k).\label{eq:26_3}
\end{align}
We now work out the left hand side of equation \eqref{eq:26}. Using
the projection formula for the theory $T_{S}$ for projective spaces,
and equations \eqref{eq:26_1}--\eqref{eq:26_3}, we find
\begin{equation}\label{eq:26_4}
T_{S}(\overline{\pi})=t_{n,k}\bullet\ch(\overline{\iota_{\ast}\mathcal{G}}).
\end{equation}
Using the functoriality of $T_{S}$ for closed immersions and the
projection formula we have
\begin{align}
T_{S}(\overline{\iota}_{1})=&\pi^{\ast}T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}})\bullet\ch(\overline{{\mathcal O}(k)})
\notag\\
\overline{\pi}_{\flat}(T_{S}(\overline\iota_1))=&T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}})
\bullet\pi_{\ast}(\ch(\overline{{\mathcal O}(k)})\bullet\Td(\overline{\pi})).\label{eq:26_5}
\end{align}
Now for the right hand side of \eqref{eq:26}.
The projection formula for $T_{S}$ for closed immersions implies
\begin{equation}\label{eq:26_6}
T_{S}(\overline\iota)=T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota\mathcal{G}})
\bullet\ch(\overline{\pi_{\ast}{\mathcal O}(k)}).
\end{equation}
Similarly, we obtain
\begin{math}
T_{S}(\overline\pi_1)=t_{n,k}\bullet\ch(\overline{\mathcal{G}}),
\end{math}
and hence
\begin{equation}\label{eq:26_7}
\overline\iota_{\flat}(T_{S}(\overline\pi_{1}))=t_{n,k}\bullet\iota_{\ast}(\ch(\overline{\mathcal{G}})\bullet\Td(\overline\iota)).
\end{equation}
Using \eqref{eq:26_4}--\eqref{eq:26_7}, the difference of the two sides of \eqref{eq:26} becomes
\begin{displaymath}
t_{n,k}\bullet\dd_{\mathcal{D}}T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}})
-T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}})\bullet\dd_{\mathcal{D}}t_{n,k}=
-\dd_{\mathcal{D}}(t_{n,k}\bullet T_{S}(\iota,\overline{\mathcal{G}},\overline{\iota_{\ast}\mathcal{G}}))=0
\end{displaymath}
in the group $\oplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Y,N_{\iota},p)$.
\end{proof}
\section{Generalized analytic torsion classes}
\label{sec:gener-analyt-tors}
In this section we will extend the definition of analytic torsion
classes to arbitrary morphisms of smooth complex varieties. Our
construction is based on the construction of analytic torsion classes
by Zha in \cite{Zha99:_rieman_roch}.
\begin{definition}\label{def:genAT}
A \textit{theory of generalized analytic torsion classes} is an
assignment that, to each morphism $\overline f\colon X\to Y$ in
$\overline\Sm_{\ast/{\mathbb C}}$ and relative metrized complex
$\overline{\xi}=(\overline f,\overline{\mathcal{F}},\overline{
f_{\ast}\mathcal{F}}),$
assigns a class of currents
$$T(\overline{\xi})\in \bigoplus_{p=1}^{n+1} \widetilde
{\mathcal{D}}^{2p-1}_{D}(Y,N_{f},p)$$
satisfying the following properties:
\begin{enumerate}
\item \label{item:1AT} (Differential equation) For any current $\eta\in T(\overline{\xi})$, we have
\begin{equation}\label{eq:16}
\dd_{\mathcal{D}} \eta=
\ch(\overline {
f_{\ast}\mathcal{F}})-\overline f_{\flat}
[\ch(\overline {\mathcal{F}})].
\end{equation}
\item \label{item:2AT}(Functoriality) If $g\colon
Y'\to Y$ is a morphism transverse to $f$, then
\begin{displaymath}
g^{\ast} T(\overline \xi)=T(g^{\ast} \overline
\xi).
\end{displaymath}
\item \label{item:3AT}(Additivity and normalization) If $\overline{\xi}_1$,
$\overline{\xi}_2$ are relative metrized complexes on $X$,
then
$$T(\overline{\xi}_1\oplus\overline{\xi}_2)
=T(\overline{\xi}_1)+T(\overline{\xi}_2).$$
\item \label{item:4AT}(Projection formula) If $\overline{\xi}$ is a
relative metrized complex, and $\overline{\mathcal{G}}\in \Ob \oDb(Y)$,
then
$$T(\overline{\xi}\otimes
\overline{\mathcal{G}})=T(\overline{\xi})\bullet
\ch(\overline{\mathcal{G}}).
$$
\item \label{item:5AT}(Transitivity) If $\overline f\colon X\to Y$, $\overline
g\colon Y\to Z$ are morphisms in $\overline\Sm_{\ast/{\mathbb C}}$, and
$(\overline f,\overline{\mathcal{F}},\overline{ f_{\ast}\mathcal{F}})$
and $(\overline g,\overline{ f_{\ast}\mathcal{F}},\overline{
(g\circ f)_{\ast}\mathcal{F}})$ are relative metrized complexes, then
\begin{equation}
\label{eq:18}
T(\overline g\circ \overline f) =
T(\overline g)+\overline g_{\flat}(T(\overline
f)).
\end{equation}
\end{enumerate}
\end{definition}
Propositions \ref{prop:1tris} and \ref{prop:2tris} below contain several
anomaly and compatibility formulas satisfied by an arbitrary theory of generalized analytic torsion classes.
They follow from properties \ref{item:1AT}--\ref{item:3AT} and are analogous
to those in propositions \ref{prop:1bis} and \ref{prop:1}, \ref{prop:2bis} and
\ref{prop:2} respectively. The proofs are omitted, as they are similar to those of
the analogous statements referred to hereinbefore.
\begin{proposition}\label{prop:1tris}
Let $T$ be a theory of generalized analytic torsion classes. Let
\begin{math}
\overline{\xi}= (\overline{f},\overline{\mathcal{F}}, \overline{ f_{\ast}\mathcal{F}})
\end{math}
be a relative metrized complex.
\begin{enumerate}
\item \label{item:6tris}
If $\overline{\mathcal{F}}'$ is another choice
of metric on $\mathcal{F}$ and
$\overline{\xi}_1= (\overline{f},\overline{\mathcal{F}}', \overline{
f_{\ast}\mathcal{F}})$, then
\begin{displaymath}
T(\overline{\xi}_1)=T(\overline{\xi})+\overline f
_{\flat}[\cht(\overline{\mathcal{F}}',\overline{\mathcal{F}})].
\end{displaymath}
\item \label{item:7tris}
If $\overline{f}'$ is another choice of hermitian structure on $f$ and
$\overline{\xi}_{2}= (\overline{f}',\overline{\mathcal{F}}, \overline{ f_{\ast}\mathcal{F}})$, then
\begin{equation}\label{eq:14tris}
T(\overline{\xi}_{2})=T(\overline{\xi})+\overline f'
_{\flat}[\ch(\overline {\mathcal{F}})\bullet
\widetilde{\Td}_{m}(\overline{f}', \overline{f})].
\end{equation}
\item \label{item:8tris}
If $\overline{ f_{\ast}\mathcal{F}}'$ is a different
choice of metric on $ f_{\ast}\mathcal{F}$, and $\overline
{\xi}_{3}= (\overline{f},\overline{\mathcal{F}}, \overline{ f_{\ast}\mathcal{F}}')$, then
\begin{displaymath}
T(\overline{\xi}_{3})= T(\overline{\xi})-
\cht(\overline{ f_{\ast}\mathcal{F}}',
\overline{ f_{\ast}\mathcal{F}}).
\end{displaymath}
\end{enumerate}
\end{proposition}
\begin{proposition} \label{prop:2tris}
Let $T$ be a theory of generalized analytic torsion classes. Let
$\overline f\colon X\to Y$ be a morphism in $\overline\Sm_{\ast/{\mathbb C}}$. Consider
the distinguished
triangles in $\oDb(X)$ and $\oDb(Y)$ respectively:
\begin{displaymath}
(\overline{\tau}):\
\overline{\mathcal{F}}_{2}\to\overline{\mathcal{F}}_{1}
\to\overline{\mathcal{F}}_{0}\to\overline{\mathcal{F}}_{2}[1],\ \text{ and
}\
(\overline{ f_{\ast}\tau}):\
\overline{ f_{\ast}\mathcal{F}}_{2}\to
\overline{ f_{\ast}\mathcal{F}}_{1}
\to\overline{ f_{\ast}\mathcal{F}}_{0}\to
\overline{ f_{\ast}\mathcal{F}}_{2}[1],
\end{displaymath}
and the relative metrized complexes
$\overline{\xi}_{i}=(\overline{f},\overline{\mathcal{F}}_i,
\overline{ f_{\ast}\mathcal{F}}_i)$, $i=0,1,2$.
Then we have:
\begin{displaymath}
\sum_{j=0,1,2}(-1)^{j}T(\overline{\xi}_{j})
=\widetilde{\ch}(\overline{\pi_{\ast}\tau})
-\overline f_{\flat}(\widetilde{\ch}(\overline{\tau })).
\end{displaymath}
\end{proposition}
The main result of this section is the following classification
theorem.
\begin{theorem}\label{thm:gen_anal_tor}
Let $S$ be a real additive genus. Then there exists a unique theory
of generalized analytic torsion classes that agrees with $T_{S}$
when restricted to the class of closed immersions. Moreover, if $T$
is a theory of generalized analytic torsion classes, then there
exists a real additive genus $S$ such that $T=T_{S}$.
\end{theorem}
We will denote the theory associated to the additive genus $S$, whose existence is guaranteed by the preceding theorem, by
$T_{S}$. In particular, there is a unique theory of generalized
analytic torsion classes that agrees with $T^{h}$ when restricted to
the class of closed immersions. This theory will be called
homogeneous.
\begin{proof}
We first prove the uniqueness. Let $T$ be a theory of analytic
torsion classes that agrees with $T_{S}$ for
the class of closed immersions. Since the
restriction of $T$ to projective spaces, by the transitivity axiom,
is compatible with $T_{S}$, by Theorem \ref{thm:15}, it also agrees
with $T_{S}$. Finally, the transitivity axiom implies that $T$ is determined
by its values for closed immersions and projective spaces.
We now prove the existence. For the moment, let $T_{S}$ be the
theory of analytic torsion classes for closed immersions and
projective spaces determined by $S$.
Let $\overline f\colon X\to Y$ be a morphism
in $\overline\Sm_{\ast/{\mathbb C}}$, and let $\overline \xi =(\overline
f,\overline{\mathcal{F}},\overline{ f_{\ast}\mathcal{F}})$ be a relative
metrized complex. Since $f$ is assumed to be projective, there is a
factorization $f=\pi \circ \iota $, where $\iota \colon X\to
{\mathbb P}^{n}_{Y}$ is a closed immersion and $\pi \colon {\mathbb P}_{Y}^{n}\to Y$
is the projection. Choose auxiliary hermitian structures on $\iota
$, $\pi $ and $ \iota _{\ast} \mathcal{F}$. Then we define
\begin{equation}
\label{eq:27}
T_{S}(\overline \xi )= T_{S}(\overline \pi )
+ \overline \pi_{\flat}(
T_{S}(\overline \iota ))
+\overline f_{\flat}\left[
\ch(\overline{\mathcal{F}})\bullet \widetilde {\Td}_{m}(\overline f,\overline
\pi \circ \overline \iota )
\right]
\end{equation}
To simplify the notations, in the sequel we will also refer to it
simply by $T(\overline\xi)$. The anomaly formulas easily imply that this
definition does not
depend on the choice of hermitian structures on $\iota
$, $\pi $ and $ \iota _{\ast} \mathcal{F}$. We next show that
this definition is independent of the factorization of $f$. Let
$f=\pi _{1}\circ \iota _{1}=\pi _{2}\circ \iota
_{2}$ be two different factorizations, being ${\mathbb P}^{n_{i}}$, the
target of $\iota _{i}$, $i=1,2$. Since equation \eqref{eq:27} is
independent of
the choice of auxiliary hermitian structures, by
\cite[Lem.~5.12]{BurgosFreixasLitcanu:HerStruc},
we may assume that $
\overline f=\overline \pi _{1}\circ \overline \iota _{1}=\overline \pi _{2}\circ \overline \iota
_{2}$.
We consider the commutative diagram with cartesian square
\begin{displaymath}
\xymatrix{
X \ar[r]^{j_{1}}\ar[dr]_{\Id_{X}}&
X\underset{Y}{\times} {\mathbb P}_{Y}^{n_{2}} \ar[r]^{k_{1}}\ar[d]^{q_{1}} &
{\mathbb P}_{Y}^{n_{1}}\underset{Y}{\times} {\mathbb P}_{Y}^{n_{2}} \ar[d]^{p_{1}}\\
& X \ar[r]^{\iota _{1}}\ar[dr]_{f}&
{\mathbb P}_{Y}^{n_{1}}\ar[d]^{\pi _{1}}\\
& & Y
}
\end{displaymath}
where $j_{1}(x)=(x,\iota _{2}(x))$, $p_{1}$ is the first projection
and $q_{1}$ and $k_{1}$ are defined by the cartesian square. The
hermitian structure of $\overline \pi _{2} $ induces a hermitian structure
on $p_{1}$ that, in turn, induces a hermitian structure on
$q_{1}$. The hermitian structure of $\iota _{1}$ induces a hermitian
structure on $k_{1}$ and the hermitian structure of $\iota _{2}$
induces one on $j_{1}$. We will denote the corresponding morphisms
of $\overline\Sm_{\ast/{\mathbb C}}$ by $\overline p_{1}$, $\overline q_{1}$, $\overline k_{1}$ and
$\overline j_{1}$. We consider also the analogous diagram obtained
swapping 1 and 2. Finally, we write
$\overline p=\overline \pi _{1}\circ \overline p_{1}=\overline \pi _{2}\circ \overline p_{2}$ and
$\overline j=\overline k_{1}\circ \overline j_{1}= \overline k_{2}\circ \overline j_{2}$.
Then we have
\begin{align*}
T(\overline \pi _{1})+(\overline \pi _{1})_{\flat}(T(\overline \iota _{1}))&=
T(\overline \pi _{1})+(\overline \pi _{1})_{\flat}(T(\overline \iota _{1}))+
\overline f_{\flat}\left(T(\overline q_{1})+(\overline q_{1})_{\flat}(T(\overline
j_{1}))\right)\\
&=
T(\overline \pi _{1})+(\overline \pi _{1})_{\flat}\big(T(\overline \iota _{1})+
(\overline \iota_{1}) _{\flat}(T(\overline q_{1}))\big)+\overline p_{\flat}(\overline
k_{1})_{\flat}(T(\overline j_{1}))\\
&=
T(\overline \pi _{1})+(\overline \pi _{1})_{\flat}\big(T(\overline p _{1})+
(\overline p_{1}) _{\flat}(T(\overline k_{1}))\big)+\overline p_{\flat}(\overline
k_{1})_{\flat}(T(\overline j_{1}))\\
&=T(\overline p)+\overline p_{\flat}(T(\overline j)).
\end{align*}
Analogously, we obtain
\begin{displaymath}
T(\overline \pi _{2})+(\overline \pi _{2})_{\flat}(T(\overline \iota _{2}))=
T(\overline p)+\overline p_{\flat}(T(\overline j)).
\end{displaymath}
Hence $T_{S}$ is well defined for all relative metrized
complexes.
It remains to prove that it satisfies the properties of
a theory of analytic torsion classes. The properties \ref{item:1AT}
to \ref{item:4AT} are clear. We thus focus on property \ref{item:5AT}.
Let $\overline{f}\colon X\to Y$ and $\overline{g}\colon Y\to Z$ be morphisms in
$\overline{\Sm}_{\ast/{\mathbb C}}$. We choose factorizations of
$\overline{g}\circ\overline{f}$ and $\overline{g}$:
\begin{align*}
&\xymatrix{
X\ar@{^{(}->}[r]^{\overline{i}}\ar[rd]_{\overline{g}\circ\overline{f}}
&{\mathbb P}^{m}_{Z}\ar[d]^{\overline{p}}\\
&Z
}
&\xymatrix{
Y\ar@{^{(}->}[r]^{\overline{\ell}}\ar[rd]_{\overline{g}}
&{\mathbb P}^{n}_{Z}\ar[d]^{\overline{r}}\\
&Z,
}
\end{align*}
where the hermitian
structures on
$\overline{p}$ and $\overline{r}$ come from fixed hermitan structures on the
tangent bundles
$T_{{\mathbb P}^{m}_{{\mathbb C}}}$ and $T_{{\mathbb P}^{n}_{{\mathbb C}}}$, and the hermitian
structures $\overline i$ and $\overline {\ell }$ are obtained by using
\cite[Lem.~5.12]{BurgosFreixasLitcanu:HerStruc}.
We define
$\varphi:X\to{\mathbb P}^{m}_{{\mathbb C}}$ to be the morphism obtained from $i$ by
composing with the projection to ${\mathbb P}^{m}_{{\mathbb C}}$. Then we see that
the morphism $j:=(\varphi,f)\colon X\to {\mathbb P}^{m}_{Y}$ is a closed
immersion. Indeed, it is
enough to realize that the composition
\begin{displaymath}
\xymatrix{
X\ar[r]^{\hspace{-0.1cm}(\varphi,f)} &{\mathbb P}^{m}_Y\ar[r]^{(\Id,g)} &{\mathbb P}^{m}_Z
}
\end{displaymath}
agrees with the closed immersion $i$ and that $G:=(\Id,g)$ is separated (since proper). We can thus decompose $\overline{f}$ as
\begin{displaymath}
\xymatrix{
X\ar@{^{(}->}[r]^{\overline{j}}\ar[rd]_{\overline{f}} &{\mathbb P}^{m}_{Y}\ar[d]^{\overline{q}}\\
&Y.
}
\end{displaymath}
Again, in this factorization the hermitian structure $\overline{q}$ comes
from the previously fixed hermitian structure on $T_{{\mathbb P}^{m}_{{\mathbb C}}}$
and the hermitian structure $\overline j$ is obtained by using
\cite[Lem.~5.12]{BurgosFreixasLitcanu:HerStruc}. Because
$\overline{g}\circ\overline{f}=\overline{p}\circ\overline{i}$ and by the very construction of
$T$ for arbitrary projective morphisms \eqref{eq:27}, we have
\begin{equation}\label{eq:ma_1}
T(\overline{g}\circ\overline{f})=T(\overline{p})+\overline{p}_{\flat}(T(\overline{i})).
\end{equation}
We proceed to work on $T(\overline{i})$. For this we write the commutative diagram
\begin{displaymath}
\xymatrix{
X\ar@{^{(}->}[r]^{j}\ar@{^{(}->}[rd]_{i} &{\mathbb P}^{m}_{Y}\ar@{^{(}->}[r]^{k}\ar[d] ^{G} &{\mathbb P}^{m}_{{\mathbb P}^{n}_{Z}}\ar@{=}[r] &{\mathbb P}^{n}_{{\mathbb P}^{m}_{Z}}\ar[d]^{\pi}\\
&{\mathbb P}^{m}_{Z}\ar[rr]^{\Id} & &{\mathbb P}^{m}_{Z}.
}
\end{displaymath}
We recall that $G=(\Id,g)$ and $k=(\Id,\ell)$. Below, $G$, $k$ and
$\pi$ will be endowed with the obvious hermitian structures. With
these choices, we observe that $\overline{i}=\overline{G}\circ\overline{j}$ and
$\overline{G}=\overline{\pi}\circ\overline{k}$. Taking also into account the
construction of~$T$ and the fact that $T=T_{S}$ is transitive for
compositions of closed immersions, we find
\begin{equation}\label{eq:ma_2}
T(\overline{i})=T(\overline{\pi}\circ\overline{k}\circ\overline{j})
=T(\overline{\pi})+\overline{\pi}_{\flat}(T(\overline{k}))+\overline{G}_{\flat}(T(\overline{j}))
=T(\overline{G})+\overline{G}_{\flat}(T(\overline{j})).
\end{equation}
Therefore, from equations \eqref{eq:ma_1}, \eqref{eq:ma_2} and the identity $\overline{p}_{\flat}\overline{G}_{\flat}=\overline{g}_{\flat}\overline{q}_{\flat}$ we derive
\begin{equation}\label{eq:ma_3}
T(\overline{g}\circ\overline{f})=T(\overline{p})+\overline{p}_{\flat}(T(\overline{G}))+\overline{g}_{\flat}\overline{q}_{\flat}(T(\overline{j})).
\end{equation}
We claim that
\begin{equation}\label{eq:ma_4}
T(\overline{p})+\overline{p}_{\flat}(T(\overline{G}))=T(\overline{g})+\overline{g}_{\flat}(T(\overline{q})).
\end{equation}
Assuming this for a while, we combine \eqref{eq:ma_3} and \eqref{eq:ma_4} into
\begin{equation}
T(\overline{g}\circ\overline{f})=T(\overline{g})+\overline{g}_{\flat}(T(\overline{q})+\overline{q}_{\flat}(T(\overline{j})))\\
=T(\overline{g})+\overline{g}_{\flat}(T(\overline{f})).
\end{equation}
Hence we are lead to prove \eqref{eq:ma_4}. For this we construct the commutative diagram with cartesian squares
\begin{displaymath}
\xymatrix{
{\mathbb P}^{m}_{Y}\ar@{^{(}->}[r]^{\hspace{-0.5cm}\tilde{\ell}}\ar[d]^{\overline{q}} &{\mathbb P}^{m}_{Z}\times_{Z}{\mathbb P}^{n}_{Z}\ar[r]^{\hspace{0.5cm}\tilde{r}}\ar[d]^{\tilde{p}} &{\mathbb P}^{m}_{Z}\ar[d]^{\overline{p}}\\
Y\ar@{^{(}->}[r]^{\overline{\ell}} &{\mathbb P}^{n}_{Z}\ar[r]^{\overline{r}} &Z.
}
\end{displaymath}
Observe that $\overline{G}=\tilde{r}\circ\tilde{\ell}$. Recall now
Proposition \ref{prop:comp_proj} and Proposition \ref{prop:5}. We then
have the chain of equalities
\begin{multline*}
T(\overline{p})+\overline{p}_{\flat}(T(\overline{G}))
=T(\overline{p})+\overline{p}_{\flat}(T(\tilde{r})+
\tilde{r}_{\flat}(T(\tilde{\ell})))
=T(\overline{r})+\overline{r}_{\flat}(T(\tilde{p})+
\tilde{p}_{\flat}(T(\tilde{\ell}))\\
=T(\overline{r})+\overline{r}_{\flat}(T(\overline{\ell})+
\overline{\ell}_{\flat}(T(\overline{q})))
=T(\overline{g})+\overline{g}_{\flat}(T(\overline{q})).
\end{multline*}
This proves the claim.
The last assertion of the statement of the theorem follows from the uniqueness.
\end{proof}
\begin{theorem}\label{thm:comparison_genus}
\begin{enumerate}
\item Let $T$ be a theory of generalized analytic torsion
cla\-sses. Then there is a unique real additive genus $S$ such that,
for any relative metrized
complex $\overline \xi :=(\overline f, \overline{\mathcal{F}}, \overline{
f_{\ast}\mathcal{F}})$, we have
\begin{equation}
\label{eq:70bis}
T(\overline \xi )-T^{h}(\overline \xi )=-f_{\ast}[\ch(\mathcal{F})\bullet
\Td(T_{f}) \bullet S(T_{f})\bullet \mathbf{1}_{1}].
\end{equation}
\item
Conversely, any real additive
genus $S$ defines, by means of equation \eqref{eq:70bis}, a
unique
theory of generalized analytic torsion classes $T_S$.
\end{enumerate}
\end{theorem}
\begin{proof}
We prove the first item, the second being immediate. Let $S$ be the
real additive genus corresponding to $T$, provided by Theorem
\ref{thm:gen_anal_tor}. Then \eqref{eq:70bis} holds for embedded
metrized complexes. Because $T$ and $T^{h}$ are both transitive, it
suffices to show that \eqref{eq:70bis} holds whenever
$f\colon{\mathbb P}^{n}_{X}\to X$ is a trivial projective bundle. Observe
$T$ and $T^{h}$ satisfy the same anomaly formulas. Then, since the
sheaves $\mathcal{G}(k)$, $k=-n,\dots,0$ form a generating class for
$\Db({\mathbb P}^{n}_{X})$, and by the projection formula for $T$ and
$T^{h}$, we easily reduce to the case $\overline{\xi}=\overline{\xi}(k)$. Let
$t_{n,k}$, $t_{n,k}^{h}$ be the characteristic numbers of $T$,
$T^{h}$ respectively. We have to establish the equality
\begin{equation}\label{eq:70_1}
t_{n,-i}-t^{h}_{n,-i}=-\pi_{\ast}(\ch(\overline{{\mathcal O}}(-i))\Td(\overline{\pi})
S(T_{\overline{\pi}})),\quad
i=-n,\dots,0.
\end{equation}
This is an equation of real numbers. By functoriality, this equation
is equivalent to the analogous equation in $\oplus_{p}
H^{2p-1}_{\mathcal{D}}({\mathbb P}^{n}_{{\mathbb C}},{\mathbb R}(p))$, for the second
projection $p_{2}:{\mathbb P}^{n}_{{\mathbb C}}\times{\mathbb P}^{n}_{{\mathbb C}}\to{\mathbb P}^{n}_{{\mathbb C}}$
instead of $\pi$. Because the classes $\ch(\Lambda^{i}Q^{\vee})$
constitute a basis for $\oplus_{p}
H^{2p-1}_{\mathcal{D}}({\mathbb P}^{n}_{{\mathbb C}},{\mathbb R}(p))$, \eqref{eq:70_1} is
equivalent to the equation in cohomology
\begin{multline}\label{eq:70_2}
\sum_{i}(-1)^{i}(t_{n,-i}-t^{h}_{n,-i})\ch(\Lambda^{i}\overline{Q}^{\vee})
=\\
- p_{2\ast}(\sum_{i}(-1)^{i}\ch(p_{1}^{\ast}\overline{{\mathcal O}}(-i)\otimes
\Lambda^{i}p_{2}^{\ast}\overline{Q}^{\vee})\Td(\overline{p}_{2})
S(T_{\overline{p}_{2}})\bullet\mathbf{1}_{1}).
\end{multline}
Recalling the exact sequence \eqref{eq:15bis}, minus the right hand side
of \eqref{eq:70_2} becomes
\begin{multline*}
p_{2\ast}(\ch(\overline{\Delta_{\ast}{\mathcal O}_{{\mathbb P}^{n}}})\Td(\overline{p}_{2})
S(T_{\overline{p}_{2}})\bullet\mathbf{1}_{1})
=\\
p_{2\ast}(\Delta_{\ast}(\ch(\overline{{\mathcal O}}_{{\mathbb P}^{n}})\Td(\overline{\Delta}))\Td(\overline{p}_{2})
S(T_{\overline{p}_{2}})\bullet\mathbf{1}_{1})
=S(T_{{\mathbb P}^{n}})\bullet\mathbf{1}_{1}.
\end{multline*}
On the other hand, using the compatibility condition (Definition
\ref{compatprojsp}), the left hand side of \eqref{eq:70_2} can be
equivalently written as
\begin{multline}\label{eq:70_3}
T(\overline{p}_{2},\overline{\Delta_{\ast}{\mathcal O}_{{\mathbb P}^{n}}},\overline{{\mathcal O}_{{\mathbb P}^n}})
-T^{h}(\overline{p}_{2},\overline{\Delta_{\ast}{\mathcal O}_{{\mathbb P}^{n}}},\overline{{\mathcal O}_{{\mathbb P}^n}})
=\\
-p_{2\flat}(T(\Delta,\overline{{\mathcal O}}_{{\mathbb P}^{n}},\overline{\Delta_{\ast}{\mathcal O}_{{\mathbb P}^{n}}})
-T^{h}(\Delta,\overline{{\mathcal O}}_{{\mathbb P}^{n}},\overline{\Delta_{\ast}{\mathcal O}_{{\mathbb P}^{n}}})).
\end{multline}
The genus $S$ is additive, so in Deligne cohomology
we have the relation
\begin{displaymath}
S(T_{\overline{\Delta}})=S(T_{{\mathbb P}^{n}})-\Delta^{\ast}S(T_{{\mathbb P}^{n}\times{\mathbb P}^{n}})
=
S(T_{{\mathbb P}^{n}})-\Delta^{\ast}p_{1}^{\ast}S(T_{{\mathbb P}^{n}})
-\Delta^{\ast}p_{2}^{\ast}S(T_{{\mathbb P}^{n}})
=-S(T_{{\mathbb P}^{n}}).
\end{displaymath}
Hence, since the statement is known for closed immersions, the right hand
side of \eqref{eq:70_3} becomes
\begin{displaymath}
p_{2\ast}(\Delta_{\ast}(\ch(\overline{{\mathcal O}_{{\mathbb P}^{n}}})\Td(T_{\overline{\Delta}})
S(T_{\overline{\Delta}})\bullet\mathbf{1}_{1})\Td(\overline{p}_{2}))=
-S(T_{{\mathbb P}^{n}})\bullet\mathbf{1}_{1}.
\end{displaymath}
This concludes the proof.
\end{proof}
\section{Higher analytic torsion forms of Bismut and K\"ohler}
\label{sec:high-analyt-tors}
We now explain the relationship between the theory of analytic torsion
forms of Bismut-K\"ohler \cite{Bismut-Kohler} and the theory of
generalized analytic torsion classes developed so far.
Let $\pi \colon X\to Y$ be a smooth projective morphism (a projective
submersion) of smooth
complex varieties. Let
$\omega $ be a closed $(1,1)$-form on $X$ that induces a K\"ahler metric
on the fibers of $\pi $. Then $(\pi,\omega )$ is called a K\"ahler
fibration. The form $\omega $ defines a hermitian structure on
$T_{\pi}$, and we will abusively write $\overline{\pi}=(\pi,\omega)$ for the corresponding morphism in $\overline{\Sm}_{\ast/{\mathbb C}}$.
Let $\overline F$ be a hermitian vector bundle on $X$ such that
for every $i\ge 0$,
$R^{i}\pi _{\ast} F$ is locally free. We consider on $R^{i}\pi _{\ast}
F$ the $L^{2}$ metric obtained using Hodge theory on the fibers of
$\pi $. Using \cite[Def.~3.47]{BurgosFreixasLitcanu:HerStruc} we
obtain a hermitian structure on $\pi _{\ast}F$,
denoted by $\overline{\pi _{\ast} F}_{L^{2}}$. Then $\overline{\xi}=(\overline \pi
,\overline F, \overline{\pi _{\ast} F}_{L^{2}})$ is a relative
metrized complex. The relative metrized complexes that arise in this
way will be said to be \emph{K\"ahler}.
In the paper \cite{Bismut-Kohler}, Bismut and K\"ohler associate to
every K\"ahler relative metrized complex $\overline{\xi}$ a
differential form, that we temporarily denote by $\tau (\overline \xi
)$. Since in \cite{Bismut-Kohler} the authors use real valued
characteristic classes, while we use characteristic classes in the
Deligne complex, we have to change the normalization of this form.
To this end,
if $\tau(\overline{\xi})^{(p-1,p-1)}$ is the component of
degree $(p-1,p-1)$ of $\tau(\overline{\xi})$, then we put
\begin{displaymath}
T^{BK}(\overline{\xi})^{(2p-1,p)}= \frac{1}{2}(2\pi
i)^{p-1}[\tau(\overline{\xi})^{(p-1,p-1)}]
\in\widetilde{\mathcal{D}}_{D}^{2p-1}(Y,\emptyset,p).
\end{displaymath}
We recall that $[\cdot]$ converts differential forms into currents
according with the conventions in \cite[\S1]{BurgosLitcanu:SingularBC}
(compare with equation \eqref{eq:88}). We
define
\begin{displaymath}
T^{BK}(\overline{\xi})=\sum_{p\geq 1}T^{BK}(\overline{\xi})^{(2p-1,p)}.
\end{displaymath}
The first main result of \cite{Bismut-Kohler} is that this class
satisfies the differential equation
\begin{displaymath}
d_{\mathcal{D}}T^{BK}(\overline{\xi})=\ch(\overline{\pi _{\ast} F}_{L^{2}})-
\overline {\pi}_{\flat}[\ch(\overline F)].
\end{displaymath}
Thus, $T^{BK}(\overline \xi )$ is an example of analytic torsion class.
Let now $\omega '$ be another closed $(1,1)$-form on $X$ that induces
a K\"ahler metric on the fibers of $\pi $. We denote $\overline \pi
'=(\pi,\omega ')$. Let $\overline F'$ be the vector bundle $F$ with another
choice of metric and define $\overline{\pi _{\ast}F}'_{L^{2}}$ to be the object
$\pi _{\ast}F$ with the $L^{2}$ metric induced by $\omega '$ and $\overline
F'$. We write $\overline \xi '$ for the K\"ahler relative metrized complex
$(\overline \pi ',\overline F ',\overline{\pi
_{\ast}F}'_{L^{2}})$.
The second main result of \cite{Bismut-Kohler} is the following anomaly formula.
\begin{theorem}[\cite{Bismut-Kohler} Theorem 3.10]\label{thm:22} The following
formula holds:
\begin{displaymath}
T^{BK}(\overline \xi ')-T^{BK}(\overline \xi )= \cht(\overline{\pi
_{\ast}F}_{L^{2}},\overline{\pi
_{\ast}F}'_{L^{2}}) +\overline \pi' _{\flat}\left[\ch(\overline F)\bullet \widetilde
{\Td}_{m}(\overline \pi' ,\overline \pi ) -\cht(\overline F,\overline
F')\right].
\end{displaymath}
\end{theorem}
In the book \cite{Bismut:Asterisque}, Bismut studies the compatibility
of higher analytic torsion forms with complex immersions. Before
stating his result we have to recall the definition of the $R$-genus
of Gillet and Soul\'e \cite{GSATAT}. It is the additive genus
attached to the power series
\begin{equation}\label{eq:81}
R(x)=\sum_{\substack{m \text{ odd}\\ m\geq
1}}\left(2\zeta'(-m)+\left(1+\frac{1}{2}+\dots+\frac{1}{m}\right)
\zeta(-m)\right)\frac{x^{m}}{m!}.
\end{equation}
Let $T_{-R/2}$ be the theory of analytic torsion classes for closed
immersions associated to~$\frac{-1}{2}R$.
\begin{remark}\label{rem:10} The fact that we obtain the additive
genus $-R/2$ instead of $R$ is due to two facts. The signs
comes from the minus sign in equation \eqref{eq:70}, while the
factor $1/2$ comes from the difference of the
normalization of Green forms used in this paper and the one used in
\cite{GilletSoule:ait}. Note however that the arithmetic
intersection numbers computed using both normalizations agree,
because the definition of arithmetic degree in \cite[\S
3.4.3]{GilletSoule:ait} has a factor $1/2$ while the definition of
arithmetic degree in \cite[(6.24)]{BurgosKramerKuehn:cacg} does not.
\end{remark}
Consider a commutative diagram of smooth complex varieties
\begin{displaymath}
\xymatrix {X\ar[dr]_{f} \ar[r]^{\iota} & Y\ar[d]^{g} \\
& Z}
\end{displaymath}
where $f$ and $g$ are projective submersions and $\iota $ is a closed immersion. Let
$\overline F$ be a hermitian vector bundle on $X$ such that the sheaves
$R^{i}f_{\ast}F$ are locally free and let
\begin{displaymath}
0\to \overline E_{n}\to \dots \to \overline E_{0}\to \iota _{\ast} F\to 0
\end{displaymath}
be a resolution of $\iota_{\ast}F$ by hermitian vector bundles. We assume
that for all $i,j$, $R^{i}g_{\ast}E_{j}$ is locally free. We will
denote by $E$ the complex $E_{n}\to\dots \to E_{0}$.
Let
$\omega ^{X}$ and $\omega ^{Y}$ be closed $(1,1)$ forms that define a
structure of K\"ahler fibration on $f$ and $g$ respectively. As
before we write $\overline f=(f,\omega ^{X})$ and $\overline g=(g,\omega
^{Y})$. The exact sequence
\begin{displaymath}
0\longrightarrow T_{f}\longrightarrow f^{\ast}T_{g}\longrightarrow
N_{X/Y}\longrightarrow 0
\end{displaymath}
induces a hermitian structure on $N_{X/Y}$.
We will denote $\overline \iota
$ the inclusion $\iota $ with this hermitian structure. Finally we
denote by $\overline{f_{\ast}F}_{E}$ the hermitian structure on $f_{\ast}F$
induced by the hermitian structures $\overline{g_{\ast} E_{j}}_{L^{2}}$,
$j=0,\dots,n$.
Then, adapted
to our language, the main result of \cite{Bismut:Asterisque} can be stated as
follows.
\begin{theorem}[\cite{Bismut:Asterisque} Theorems 0.1 and
0.2] \label{thm:21} The
following equation holds in the group $\bigoplus
_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Z,\emptyset,p)$:
\begin{multline*}
T^{BK}(\overline f,\overline F,\overline{f_{\ast} F}_{L_{2}})=
\sum_{j=0}^{n}(-1)^{j}T^{BK}(\overline g,\overline E_{j},\overline{f_{\ast} E_{j}}_{L_{2}})\\
+\overline g_{\flat}(T_{-R/2}(\overline \iota ,\overline F, \overline E))
+\cht(\overline{f_{\ast}F}_{E},\overline{f_{\ast}F}_{L^{2}}).
\end{multline*}
\end{theorem}
We can particularize the previous result to the case when $F=0$. Then
$E$ and
$g_{\ast}E$ are acyclic objects. The hermitian structures of
$\overline E_{j}$ and
$\overline {g_{\ast}E_{j}}_{L^{2}}$ induce hermitian structures on them. We
denote these hermitian structures as $\overline E$ and~$\overline
{g_{\ast}E}_{L^{2}}$.
\begin{corollary} \label{cor:8}
Let $\overline E$ be a bounded acyclic complex of hermitian vector
bundles on $Y$ such that the direct images $R^{i}g_{\ast}E_{j}$ are
locally free on $Z$. Then
\begin{displaymath}
\sum_{j=0}^{n}(-1)^{j}T^{BK}(\overline{g},\overline{E_{j}},\overline{g_{\ast}E_{j}}_{L^{2}})=
\cht(\overline{g_{\ast}E}_{L^{2}})
-\overline{g}_{\flat}(\cht(\overline{E}))
\end{displaymath}
in $\bigoplus
_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(Z,\emptyset,p)$.
\end{corollary}
We will also need a particular case of functoriality and projection
formula for the higher analytic torsion forms of Bismut-K\"ohler
proved by R\"ossler \cite{Roessler:ARR}.
The relative metrized complexes $\overline \xi _{n}(k)$ of
Notation \ref{def:7} are K\"ahler. Therefore we can apply the
construction of Bismut-K\"ohler to them. We denote
\begin{equation}
\label{eq:80}
t^{BK}_{n,k}=T^{BK}(\overline \xi _{n}(k)).
\end{equation}
By Corollary \ref{cor:8}, the numbers $t^{BK}_{n,k}$ satisfy the
relation \eqref{eq:79}. Hence they are determined by the main
characteristic numbers $t^{BK}_{n,k}$ for $-n\le k \le 0$.
\begin{theorem}[\cite{Roessler:ARR} Lemma 7.15]\label{thm:20} Let
$\pi\colon
{\mathbb P}^{n}_{X}\to X$ be a trivial projective bundle. Let $\overline G$ be a
hermitian vector bundle on $X$. Then
\begin{displaymath}
T^{BK}(\overline \xi _{n}(k)\otimes \overline G)=t^{BK}_{n,k}\bullet \ch(\overline
G).
\end{displaymath}
\end{theorem}
\begin{proof}
In \cite{Roessler:ARR} this result is proved for $k\gg 0$. Using
Corollary \ref{cor:8} and the Koszul resolution \eqref{eq:38} one
can extend the result to all $k\in {\mathbb Z}$.
\end{proof}
We have all the ingredients we need to prove the main result of this
section.
\begin{theorem}\label{thm:19}
Let $T_{-R/2}$ be the theory of generalized analytic torsion classes
associated to the additive genus $\frac{-1}{2}R$. Then, for every K\"ahler
relative metrized complex $\overline \xi $, we have
\begin{displaymath}
T^{BK}(\overline \xi )=T_{-R/2}(\overline \xi ).
\end{displaymath}
In particular $T_{-R/2}$ extends the construction of Bismut-K\"ohler
to arbitrary projective morphisms of smooth complex varieties and
arbitrary smooth metrics.
\end{theorem}
\begin{proof}
Let $\mathfrak{t}^{BK}=\{t^{BK}_{n,k}\mid n\ge 0, -n\le k \le
0\}$ and let $T_{\mathfrak{t}^{BK}}$ be the theory of analytic
torsion classes for projective spaces associated to it.
Let $\pi\colon {\mathbb P}^{n}_{X}\to X$ be a relative projective space and
let $\overline \xi =(\overline \pi ,\overline E,\overline{\pi _{\ast}E}_{L^{2}})$ be a
K\"ahler relative metrized complex. By choosing $d\gg 0$ we may
assume that all the coherent sheaves of the resolution $\gamma
_{d}(F)$ of Corollary
\ref{cor:4} are locally free. Using theorems \ref{thm:20} and \ref{thm:22},
Proposition \ref{prop:1}
and corollaries \ref{cor:1} and \ref{cor:8} we obtain that
\begin{displaymath}
T^{BK}(\overline \xi)=T_{\mathfrak{t}^{BK}}(\overline \xi ).
\end{displaymath}
By Theorem \ref{thm:21}, the theories
$T_{\mathfrak{t}^{BK}}$ and $T_{-R/2}$ are compatible in the sense of
Definition \ref{compatprojsp}. Therefore, $T^{BK}=T_{-R/2}$ when
restricted to projective spaces.
Finally, by factoring a smooth projective morphism as a closed
immersion followed by the projection of a relative projective space,
Theorem \ref{thm:21} implies that $T^{BK}=T_{-R/2}$ for all smooth
projective morphisms.
\end{proof}
\begin{remark}\label{rem:bismut_kohler}
\begin{enumerate}
\item The construction of Bismut-K\"ohler applies to a wider class
of varieties and morphisms: complex analytic manifolds and proper
K\"ahler submersions. However for the comparison we have to
restrict to smooth algebraic varieties and smooth projective
morphisms.
\item The results of Bismut and his coworkers are more precise. Here
the class $T^{BK}(\overline \xi )$ is well defined up to the image of
$\dd_{\mathcal{D}}$. In contrast, the higher analytic torsion form
of Bismut and K\"ohler is a well defined differential form, local
on the base and whose class modulo $\dd_{\mathcal{D}}$ agrees with
$T^{BK}(\overline \xi )$.
\end{enumerate}
\end{remark}
As a consequence of Theorem \ref{thm:19}, we obtain the following
results that, although they should follow from the definition of
higher
analytic torsion classes, we have not been able to find them explicitly in
the literature.
\begin{corollary}\label{cor:9} Let $f\colon X\to Y$ be a
smooth projective morphism of smooth complex varieties, and let $\overline \xi
=(\overline f, \overline E, \overline{f_{\ast}E}_{L^{2}})$ be a K\"ahler relative
metrized complex.
\begin{enumerate}
\item Let $g\colon Y'\to Y$ be a morphism of smooth
complex varieties. Then
\begin{displaymath}
T^{BK}(g^{\ast}\overline \xi )=g^{\ast}T^{BK}(\overline \xi ).
\end{displaymath}
\item Let $\overline G$ be a hermitian vector bundle on $Y$. Then
\begin{displaymath}
T^{BK}(\overline \xi \otimes \overline G)=T^{BK}(\overline \xi )\bullet \ch(\overline G).
\end{displaymath}
\end{enumerate}
\end{corollary}
The last consequence we want to discuss generalizes results already
proved by Ber\-thomieu-Bismut \cite[Thm 3.1]{BerthomieuBismut} and Ma
\cite[Thm. 0.1]{Ma:MR1765553}, \cite[Thm. 0.1]{Ma:MR1796698}. However
we note that while we stay within the algebraic category and
work with projective morphisms, these authors deal with proper
K\"ahler holomorphic submersions of complex manifolds. Let $\overline{g}\colon X\to
Y$ and $\overline{h}\colon Y\to Z$ be morphisms in the category
$\overline{\Sm}_{\ast/{\mathbb C}}$, such that the composition
$\overline{f}=\overline{h}\circ\overline{g}$ is a smooth morphism. We choose a structure
of K\"ahler fibration on $f$, that we denote $\overline{f}'$. Let $\overline{E}$
be a hermitian vector bundle on $X$ and assume that the higher direct
images $R^{i}f_{\ast}E$ are locally free. Then we may consider the
analytic torsion $T^{BK}(\overline{f}')$ attached to the K\"ahler relative
metrized complex $(\overline{f}',\overline{E},\overline{f_{\ast}E}_{L^{2}})$. Also, we
choose an auxiliary hermitian structure on $g_{\ast}E$. We can
consider the torsion classes $T_{R/2}(\overline{g})$ and $T_{R/2}(\overline{h})$
of the relative metrized complexes $(\overline{g},\overline{E},\overline{g_{\ast}E})$
and $(\overline{h},\overline{g_{\ast}E},\overline{f_{\ast}E}_{L^{2}})$.
We make the
following additional assumption in some particular
situations:
\begin{description}
\item [({\bf *})] The morphisms $g$ and $h$ are K\"ahler fibrations, the higher
direct images $R^{i}g_{\ast}E$ and $R^{j}h_{\ast}R^{i}g_{\ast}E$ are
locally free and the auxiliary
hermitian structure on $g_{\ast}E$ is the $L^{2}$ hermitian
structure.
\end{description}
When the hypothesis {\bf(*)} is satisfied we denote by
$\overline{h_{\ast}g_{\ast}E}_{L^{2}}$ the $L^{2}$ hermitian structure
attached to the K\"ahler structure on $\overline{h}$ and the $L^{2}$ metric
on $\overline{g_{\ast}E}_{L^{2}}$. Observe that this last structure may
differ from the $L^{2}$ structure on $\overline{f_{\ast}E}_{L^{2}}$.
In this situation we can consider the
torsion classes $T^{BK}(\overline{g})$ and
$T^{BK}(\overline{h}')$ attached to
$(\overline{g},\overline{E},\overline{g_{\ast}E}_{L^{2}})$ and
$(\overline{h},\overline{g_{\ast}E}_{L^{2}},\overline{h_{\ast}g_{\ast}E}_{L^{2}})$.
By Proposition \ref{prop:1}, we have the relation
\begin{displaymath}
T^{BK}(\overline{h}')=
T_{R/2}(\overline{h})-
\widetilde{\ch}(\overline{h_{\ast}g_{\ast}E}_{L^{2}},\overline{f_{\ast}E}_{L^{2}}).
\end{displaymath}
The properties of the generalized analytic torsion classes imply immediately:
\begin{corollary} \label{cor:comp_sub}
Under these assumptions, we have the equality
\begin{displaymath}
T^{BK}(\overline{f}')=T_{-R/2}(\overline{h})
+\overline{h}_{\flat}(T_{-R/2}(\overline{g}))+
\overline{f}^{\prime}_{\flat}(\ch(\overline{E})
\bullet\widetilde{\Td}_{m}(\overline{f}',\overline{f})).
\end{displaymath}
If in addition the hypothesis {\rm \bf (*)} is satisfied, then we have
\begin{displaymath}
T^{BK}(\overline{f}')=T^{BK}(\overline{h}')+\overline{h}_{\flat}(T^{BK}(\overline{g}))
+\overline{f}^{\prime}_{\flat}(\ch(\overline{E})\bullet
\widetilde{\Td}_{m}(\overline{f}',\overline{f}))
+\widetilde{\ch}(\overline{h_{\ast}g_{\ast}E}_{L^{2}},\overline{f_{\ast}E}_{L^{2}}).
\end{displaymath}
\end{corollary}
Since $T_{-R/2}$ extends the theory of analytic torsion classes $T^{BK}$,
we will denote $T_{-R/2}$ by $T^{BK}$ for
arbitrary relative metrized complexes.
\section{Grothendieck duality and analytic torsion}
\label{sec:groth-dual-analyt}
We will study now the compatibility of the analytic
torsion with Grothendieck duality.
\begin{definition}
Let $\overline {\mathcal{F}}=(\mathcal{F},\overline E\dashrightarrow \mathcal{F})$ be an
object of $\oDb(X)$. Then the \emph{rank} of $\overline
{\mathcal{F}}$ is
\begin{displaymath}
\rk(\overline{\mathcal{F}})=\sum_{i}(-1)^{i}\dim(E_{i}).
\end{displaymath}
This is just the Euler characteristic of the complex. \emph{The
determinant}
of $\overline {\mathcal{F}}$ is the complex
\begin{displaymath}
\det (\overline {\mathcal{F}})= \bigotimes _{i} \left(\Lambda ^{\dim E^{i}} \overline
E^{i}\right)^{(-1)^{i}}[-\rk(\overline{\mathcal{F}})].
\end{displaymath}
It consists of a single line bundle concentrated in degree
$\rk(\overline{\mathcal{F}})$.
\end{definition}
\begin{definition} \label{def:21}
Let $\overline f\colon X\to Y$ be a morphism in $\overline\Sm_{\ast/{\mathbb C}}$ of
relative dimension $e$. The
\emph{metrized dualizing complex}, is the
complex given by
\begin{displaymath}
\boldsymbol{\omega} _{\overline f} =(\det T_{\overline f})^{\vee}.
\end{displaymath}
This complex is concentrated in degree $-e$. The underlying object
of $\Db(X)$ will be denoted by $\boldsymbol{\omega} _{f}$. If we are interested
in the dualizing sheaf as a sheaf and not as an element of $\Db(X)$
we will denote it by $\omega _{f}$ or $\omega _{X/Y}$. Finally, if
$Y=\Spec {\mathbb C}$, we will denote $\boldsymbol{\omega}_{f}$ (respectively
$\omega _{f}$) by $\boldsymbol{\omega}_{X}$ (respectively
$\omega _{X}$).
\end{definition}
\begin{definition} \label{def:20}
Let $\mathcal{D}^{\ast}(\ast)$ be the Deligne complex associated to
a Dolbeault complex. The \emph{sign operator} is
\begin{displaymath}
\sigma \colon \mathcal{D}^{\ast}(\ast)\longrightarrow
\mathcal{D}^{\ast}(\ast),\quad \omega \in
\mathcal{D}^{n}(p) \mapsto \sigma (\omega )=(-1)^{p}\omega.
\end{displaymath}
\end{definition}
The sign operator satisfies the following compatibilities.
\begin{proposition}\label{prop:13}
\begin{enumerate}
\item Let $(\mathcal{D}^{\ast}(\ast),\dd_{\mathcal{D}})$ be a
Deligne algebra. Then the sign operator is a morphism of differential
algebras. That is
\begin{displaymath}
\dd_{\mathcal{D}}\circ \,\sigma =\sigma \circ
\dd_{\mathcal{D}},\quad
\sigma (\omega \bullet \eta)=\sigma (\omega )\bullet\sigma ( \eta).
\end{displaymath}
\item Let $\overline {\mathcal{F}}$ be an object of $\oDb(X)$. Then the
following equalities are satisfied
\begin{align}
\label{eq:39}
\sigma \ch(\overline {\mathcal{F}})&= \ch(\overline
{\mathcal{F}}^{\vee}),\\
\label{eq:32}
\sigma \ch(\det (\overline {\mathcal{F}}))&= \ch(\det (\overline
{\mathcal{F}})^{\vee})= \ch(\det (\overline {\mathcal{F}}))^{-1},\\
\label{eq:34}
\sigma \Td(\overline{\mathcal{F}})&=
(-1)^{\rk(\overline{\mathcal{F}})}\Td(\overline{\mathcal{F}})\bullet \ch(\det (\overline
{\mathcal{F}})^{\vee}.
\end{align}
\end{enumerate}
\end{proposition}
\begin{proof}
The first statement is clear because if $\omega \in
\mathcal{D}^{n}(p)$ and $\eta\in \mathcal{D}^{m}(q)$ then
$\dd_{\mathcal{D}}\omega \in
\mathcal{D}^{n+1}(p)$ and $\omega \bullet \eta \in
\mathcal{D}^{n+m}(p+q)$.
For the second statement, let $\overline E\dashrightarrow \mathcal{F}$ be the
hermitian structure of $\mathcal{F}$. Write
\begin{displaymath}
\overline E^{+}=\bigoplus_{i\text{ even}} \overline E^{i},\qquad
\overline E^{-}=\bigoplus_{i\text{ odd}} \overline E^{i}.
\end{displaymath}
Since this statement is local on $X$, we can chose trivializations
of $\overline E^{+}$ and $\overline E^{-}$ over an open subset $U$. Let $H^{+}$
and $H^{-}$ be the
matrices of the hermitian metrics on $\overline E^{+}$ and $\overline
E^{-}$. The curvature matrices of $\overline E^{+}$ and $\overline
E^{-}$, whose entries are elements of
$\mathcal{D}^{2}(U,1)$, are
\begin{displaymath}
K^{\pm}=K^{\pm}(\overline {\mathcal{F}})= - \overline \partial
(H^{\pm})^{-1}\partial H^{\pm}.
\end{displaymath}
The characteristic forms can be computed
from the curvature matrix:
\begin{align*}
\ch(\overline {\mathcal{F}})&=\tr (\exp(K^{+}))-\tr (\exp(K^{-})),\\
\ch(\det (\overline{\mathcal{F}}))&=(-1)^{\rk(\overline{\mathcal{F}})}
\det(\exp(K^{+}))\bullet \det(\exp(K^{-}))^{-1},\\
\Td(\overline{\mathcal{F}})&=
\det\left(\frac {K^{+}}{1-\exp(-K^{+})}\right)\bullet
\det\left(\frac {K^{-}}{1-\exp(-K^{-})}\right)^{-1}.
\end{align*}
The sign in the second equation comes from the fact that $\det
(\overline{\mathcal{F}})$ is concentrated in degree $\rk(\overline{\mathcal{F}})$.
Therefore, since $\sigma (K^{\pm})=-K^{\pm}=K^{\pm}(\overline
{\mathcal{F}}^{\vee})$, we have
\begin{align*}
\sigma \ch(\overline {\mathcal{F}})&=\sigma \tr (\exp(K^{+}))-
\sigma \tr (\exp(K^{-}))\\
&=\tr
(\exp(K^{+}(\overline{\mathcal{F}}^{\vee})))
-\tr (\exp(K^{-}(\overline{\mathcal{F}}^{\vee})))=
\ch(\overline {\mathcal{F}}^{\vee}),\\
\sigma \ch(\det (\overline{\mathcal{F}}))&=
\det(\exp(-K^{+}))\bullet \det(\exp(-K^{-}))^{-1}=
\ch(\det (\overline{\mathcal{F}}))^{-1},\\
\sigma \Td(\overline{\mathcal{F}})&=
\det\left(\frac {-K^{+}}{1-\exp(K^{+})}\right)\bullet
\det\left(\frac {-K^{-}}{1-\exp(K^{-})}\right)^{-1}\\
&= \det\left(\frac {K^{+}}{1-\exp(-K^{+})}\right)\bullet
\det(\exp(-K^{+}))\\
&\phantom{AAA}\bullet
\det\left(\frac {K^{-}}{1-\exp(-K^{-})}\right)^{-1} \bullet
\det(\exp(-K^{-}))^{-1}\\
&=
\Td(\overline{\mathcal{F}})\bullet \ch(\det (\overline
{\mathcal{F}}))^{-1}.
\end{align*}
\end{proof}
\begin{corollary}\label{cor:7}
Let $[\overline{E}]\in\KA(X)$. Then
\begin{math}
\widetilde{\ch}(\overline{E}^{\vee})=\sigma\widetilde{\ch}(\overline{E}).
\end{math}
\end{corollary}
\begin{proof}
Due to Proposition \ref{prop:13}, the assignment sending $[\overline{E}]$ to
$\sigma\widetilde{\ch}(\overline{E})$ satisfies the characterizing
properties of $\widetilde{\ch}$.
\end{proof}
In the particular case of a projective
morphism between smooth complex varieties or, more generally, smooth
varieties over a field, Grothendieck duality takes a very simple form
(see for instance
\cite[\S 3.4]{Huybrechts:FM} and the references therein). If
$\mathcal{F}$ is an object of
$\Db(X)$ and $f\colon X\to Y$ is a projective morphism of smooth
complex varieties, then there is a natural functorial isomorphism
\begin{equation}\label{eq:31}
f_{\ast}(\mathcal{F}^{\vee}\otimes^{\Ld} \boldsymbol{\omega} _{f})\cong( f_{\ast}
\mathcal{F}) ^{\vee}.
\end{equation}
The compatibility between analytic torsion and Grothendieck duality is
given by the following result.
\begin{theorem-definition}\label{thm-def:T_dual}
Let $T$ be a theory of generalized analytic torsion classes. Then
the assignment
that, to a relative metrized complex $\overline \xi =(\overline
f,\overline{\mathcal{F}},\overline{ f_{\ast} \mathcal{F}})$, associates the
class
\begin{displaymath}
T^{\vee}(\overline \xi)=
\sigma T(\overline f,\overline{\mathcal{F}}^{\vee}\otimes^{\Ld} \boldsymbol{\omega} _{\overline f},
\overline { f_{\ast} \mathcal{F}}^{\vee})
\end{displaymath}
is a theory of generalized analytic torsion classes that we call the
\emph{theory dual to} $T$.
\end{theorem-definition}
\begin{proof}
We have to show that, if $T$ satisfies the conditions of Definition
\ref{def:genAT}, then the same is true for $T^{\vee}$. We start with
the differential equation. Let $e$ be the relative dimension of~$f$.
\begin{align*}
\dd_{\mathcal{D}} T^{\vee}(\overline \xi )&=
\dd_{\mathcal{D}} \sigma T(\overline
f,\overline{\mathcal{F}}^{\vee}\otimes^{\Ld}
\boldsymbol{\omega} _{\overline f},
\overline { f_{\ast} \mathcal{F}}^{\vee})\\
&=\sigma \dd_{\mathcal{D}} T(\overline
f,\overline{\mathcal{F}}^{\vee}\otimes^{\Ld} \boldsymbol{\omega} _{\overline f},
\overline { f_{\ast} \mathcal{F}}^{\vee})\\
&=\sigma \ch(\overline { f_{\ast} \mathcal{F}}^{\vee})
-\sigma f_{\ast}\left[\ch(\overline{\mathcal{F}}^{\vee}\otimes^{\Ld}
\boldsymbol{\omega} _{\overline
f})\bullet \Td(\overline f)\right]\\
&=\ch(\overline { f_{\ast} \mathcal{F}})-
(-1)^{e}f_{\ast}\left[\sigma \ch(\overline{\mathcal{F}}^{\vee})\bullet
\sigma (\ch(\det (T_{\overline f})^{\vee})\bullet \Td(T_{\overline
f}))\right]\\
&=\ch(\overline { f_{\ast} \mathcal{F}})-
f_{\ast}\left[\ch(\overline{\mathcal{F}})\bullet
\Td(\overline
f)\right]
\end{align*}
The functoriality and the additivity are clear. We next check the
projection formula. Let $\overline{\mathcal{G}}$ be an object of
$\oDb(Y)$. Then
\begin{multline*}
T^{\vee}(\overline \xi \otimes \overline {\mathcal{G}}) =
\sigma T(\overline f,\overline{\mathcal{F}}^{\vee}\otimes^{\Ld}
f^{\ast}\overline
{\mathcal{G}}^{\vee} \otimes^{\Ld} \boldsymbol{\omega} _{\overline f},
\overline { f_{\ast} \mathcal{F}}^{\vee}\otimes^{\Ld} \overline
{\mathcal{G}}^{\vee})\\
=\sigma \left( T(\overline f,\overline{\mathcal{F}}^{\vee}\otimes^{\Ld}
\boldsymbol{\omega} _{\overline f},
\overline { f_{\ast} \mathcal{F}}^{\vee})\bullet \ch(\overline{\mathcal{G}}^{\vee})
\right)
=T^{\vee}(\overline \xi )\bullet \ch(\overline {\mathcal{G}}).
\end{multline*}
Finally we check the transitivity. Let $\overline g\colon Y\to Z$ be
another morphism in $\overline\Sm_{\ast/{\mathbb C}}$. By
the definition of $\overline g\circ \overline f$ we have
\begin{math}
\boldsymbol{\omega} _{\overline g\circ \overline f}= f^{\ast}\boldsymbol{\omega} _{\overline g}\otimes^{\Ld}
\boldsymbol{\omega} _{\overline f}.
\end{math}
Therefore,
\begin{multline*}
f_{\ast}\left( \mathcal{F}^{\vee}\otimes^{\Ld}
\boldsymbol{\omega} _{g\circ f}
\right)=
f_{\ast}\left( \mathcal{F}^{\vee}\otimes^{\Ld}
f^{\ast}\boldsymbol{\omega} _{g}\otimes^{\Ld}
\boldsymbol{\omega} _{f})\right)\\
= f_{\ast}\left( \mathcal{F}^{\vee}\otimes^{\Ld}
\boldsymbol{\omega} _{f})\right)\otimes^{\Ld}\boldsymbol{\omega} _{g}
=( f_{\ast} \mathcal{F})^{\vee}\otimes^{\Ld}\boldsymbol{\omega}
_{g}.
\end{multline*}
On $ f_{\ast}\left( \mathcal{F}^{\vee}\otimes^{\Ld}
\boldsymbol{\omega} _{g\circ f}
\right)$ we put the hermitian structure of $\overline{ f_{\ast}
\mathcal{F}}^{\vee}\otimes^{\Ld}\boldsymbol{\omega}
_{\overline g}$. Then we have
\begin{align*}
T^{\vee}(\overline g\circ \overline f)&=
\sigma T(\overline g\circ \overline f,\overline {\mathcal{F}}^{\vee}\otimes^{\Ld}
\boldsymbol{\omega} _{\overline g\circ \overline f},\overline { (g\circ
f)_{\ast} \mathcal{F}}^{\vee})\\
&=\sigma T(\overline g,\overline{ f_{\ast}\mathcal{F}}^{\vee}\otimes^{\Ld}
\boldsymbol{\omega} _{\overline g},\overline { (g\circ
f)_{\ast} \mathcal{F}}^{\vee})\\
&\phantom{AAA}+
\sigma \overline g_{\flat} T(\overline f,\overline{\mathcal{F}}^{\vee}\otimes^{\Ld}
\boldsymbol{\omega} _{\overline f}\otimes^{\Ld}
f^{\ast}\boldsymbol{\omega} _{\overline g}, \overline{
f_{\ast}\mathcal{F}}^{\vee}\otimes^{\Ld}
\boldsymbol{\omega} _{\overline g})\\
&=T^{\vee}(\overline g,\overline{ f_{\ast}\mathcal{F}},\overline { (g\circ
f)_{\ast} \mathcal{F}})\\
&\phantom{AAA}+\sigma g_{\ast}(T(\overline
f,\overline{\mathcal{F}}^{\vee}\otimes^{\Ld}
\boldsymbol{\omega} _{\overline f},
\overline { f_{\ast} \mathcal{F}}^{\vee})\bullet \ch(\boldsymbol{\omega} _{\overline g})
\bullet \Td(\overline g))\\
&=T^{\vee}(\overline g)+\overline g_{\flat} T^{\vee}(\overline f).
\end{align*}
Therefore, $T^{\vee}$ satisfies also the transitivity property. Hence it
is a generalized theory of analytic torsion classes.
\end{proof}
\begin{definition}
A theory of generalized analytic torsion classes $T$ is called \emph{self-dual} when $T^{\vee}=T$.
\end{definition}
We want to characterize the self-dual theories of generalized analytic
torsion classes.
\begin{theorem}
The homogeneous theory of generalized analytic
torsion classes is self-dual.
\end{theorem}
\begin{proof}
By the uniqueness of the homogeneous theory, it is enough to prove
that, if $T$ is homogeneous then $T^{\vee}$ is homogeneous. Let $X$
be a smooth complex variety and let $\overline N$ be a hermitian vector
bundle of rank $r$ on $X$. Put $P=\mathbb{P}(N\oplus 1)$ and let
$s\colon X\to
P$ be the zero section and $\pi \colon P\to X$ the projection. Let
$\overline Q$ be the tautological quotient bundle with the induced metric
and $\overline K(s)$ the
Koszul resolution associated to the section $s$. Since the normal
bundle $N_{X/P}$ can be identified with $N$, on the map $s$ we can
consider the hermitian structure given by the hermitian metric on
$N$. Then $\det \overline Q$ is a complex concentrated in degree $r$. Moreover
\begin{displaymath}
s^{\ast} \det \overline Q=\det \overline N=\boldsymbol{\omega} _{\overline s}.
\end{displaymath}
The Koszul resolution satisfies the duality property
\begin{math}
\overline K(s)^{\vee}=\overline K(s)\otimes \det \overline Q.
\end{math}
The theory $T$ is homogeneous if and only if the class
\begin{displaymath}
T(\overline s,\overline {\mathcal{O}}_{X},\overline K(s))\bullet \Td(\overline Q)
\end{displaymath}
is homogeneous of bidegree $(2r-1,r)$ in the Deligne complex. Then
\begin{align*}
T^{\vee}(\overline s,\overline {\mathcal{O}}_{X},\overline K(s))\bullet \Td(\overline Q)&=
\sigma T(\overline s,\boldsymbol{\omega} _{\overline s},\overline K(s)^{\vee})\bullet \Td(\overline
Q)\\
&=\sigma T(\overline s, s^{\ast} \det \overline Q,\overline K(s)\otimes \det \overline
Q)\bullet \Td(\overline Q)\\
&=\sigma (T(\overline s,\overline {\mathcal{O}}_{X},\overline K(s))\bullet \ch(\det
\overline Q))\bullet \Td(\overline Q)\\
&=\sigma T(\overline s,\overline {\mathcal{O}}_{X},\overline K(s))\bullet \ch(\det
\overline Q ^{\vee})\bullet \Td(\overline Q)\\
&=(-1)^{r}\sigma (T(\overline s,\overline {\mathcal{O}}_{X},\overline K(s))\bullet
\Td(\overline Q))
\end{align*}
is homogeneous of bidegree $(2r-1,r)$ in the Deligne complex.
\end{proof}
\begin{proposition}\label{prop:14}
Let
\begin{displaymath}
S(x)=\sum _{n=0}^{\infty}a_{n}x^{n}\in {\mathbb R}[[x]]
\end{displaymath}
be a power series in one
variable with real coefficients. Denote by $S$ the corresponding
real additive genus and by $T_{S}$ the associated theory of analytic
torsion classes. Then the dual theory $T^{\vee}_{S}$ has
corresponding real additive genus $S^{\sigma}(x):=-S(-x)$.
\end{proposition}
\begin{proof}
Let $\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{
f_{\ast}\mathcal{F}})$ be a relative metrized complex. If $e$ is
the relative dimension of $f$, then we have $\sigma
f_{\ast}=(-1)^{e}f_{\ast}\sigma$. Then the proposition readily
follows from the definition of $T^{\vee}_{S}$, the self-duality of
$T^{h}$ and Proposition \ref{prop:13}.
\end{proof}
We can now characterize the self-dual theories of analytic torsion classes.
\begin{corollary}\label{cor:char_self_dual}
The theory of analytic torsion classes $T_{S}$ attached to the real
additive genus $S(x)=\sum_{n\geq 0}a_{n}x^{n}$ is self-dual if and
only if $a_{n}=0$ for $n$ even.
\end{corollary}
\begin{proof}
By the proposition, $T_{S}^{\vee}=T_{S^{\sigma}}$, hence $T$ is
self-dual if, and only if, $S^{\sigma}=S$. The corollary follows.
\end{proof}
In particular we recover the following fact, which is well
known if we restrict to K\"ahler relative metrized complexes.
\begin{corollary}\label{cor:TBK_self-dual}
The theory of analytic torsion classes of Bismut-K\"ohler $T^{BK}$ is
self-dual.
\end{corollary}
\begin{proof}
We just remark that the even coefficients of the $R$-genus vanish
\eqref{eq:81}.
\end{proof}
We now elaborate on an intimate relation between self-duality
phenomena and the analytic torsion of de Rham complexes. Let $f\colon
X\to Y$ be a smooth projective morphism of smooth algebraic varieties,
of relative dimension $e$. Let $\overline{T}_{X/Y}$ denote the vertical
tangent bundle, endowed with a hermitian metric. Write $\overline{f}$
for the corresponding morphism in $\overline{\Sm}_{\ast/{\mathbb C}}$. On the
locally free sheaves $\Omega_{X/Y}^{p}=\Lambda^{p}\Omega_{X/Y}$ we put
the induced hermitian structures. The metrized de
Rham complex is
\begin{displaymath}
0\to\overline{{\mathcal O}}_{X}\overset{0}{\to}\overline{\Omega}_{X/Y}\overset{0}{\to}
\overline{\Omega}_{X/Y}^{2} \overset{0}{\to}\dots\overset{0}{\to}
\overline{\Omega}_{X/Y}^{e}\to
0
\end{displaymath}
with 0 differentials. In fact, we are really considering the de Rham
graded sheaf and converting it into a complex in a trivial way.
We refer to the corresponding
object of $\oDb(X)$ by $\overline{\Omega}_{X/Y}^{\bullet}$
(\cite[Def.~3.37]{BurgosFreixasLitcanu:HerStruc}). The individual
terms $\overline{\Omega}_{X/Y}^{p}$ will be
considered as complexes concentrated in degree $p$. We then obviously have:
\begin{lemma}\label{lem:van_tor_1}
The objects
$(\overline{\Omega}_{X/Y}^{\bullet})^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}}$ and
$\overline{\Omega}_{X/Y}^{\bullet}[2e]$ are tightly isomorphic.
\end{lemma}
For every $p$, $q$, the cohomology sheaf $R^{q}
f_{\ast}\Omega_{X/Y}^{p}$ is locally free, because the Hodge numbers
$h^{p,q}$ of the fibers of $f$ (which are projective, hence K\"ahler)
are known to be locally constant. Every stalk of this sheaf is endowed
with the usual $L^2$ metric of Hodge theory. This family of $L^2$
metrics on $R^{q}f_{\ast}\Omega_{X/Y}^{p}$ glue into a smooth
metric. Because the Hodge star operators $\ast$ act by isometries, it
is easily shown that Serre duality becomes an isometry for the $L^2$
structures: the isomorphism
\begin{displaymath}
(R^{q}f_{\ast}\Omega_{X/Y}^{p})^{\vee}\overset{\sim}{\longrightarrow}
R^{e-q} f_{\ast}((\Omega_{X/Y}^{p})^{\vee}\otimes\boldsymbol{\omega}_{f})
= R^{e-q}f_{\ast}\Omega_{X/Y}^{e-p}
\end{displaymath}
preserves the $L^2$ hermitian structures. For every $p$, let $\overline{
f_{\ast}\Omega_{X/Y}^{p}}$ denote the object of $\oDb(Y)$ with the
metric induced by the $L^2$ metrics on its cohomology pieces
(\cite[Def.~3.47]{BurgosFreixasLitcanu:HerStruc}). Here $ f_{\ast}$
stands for the derived direct image. By
\cite[Prop.~3.48]{BurgosFreixasLitcanu:HerStruc}, Grothendieck duality
\begin{displaymath}
( f_{\ast}\Omega_{X/Y}^{p})^{\vee}\overset{\sim}{\longrightarrow}
f_{\ast}\Omega_{X/Y}^{e-p}[2e]
\end{displaymath}
is a tight isomorphism. Finally, let $[\overline{
f_{\ast}\Omega_{X/Y}^{\bullet}}]$ be the object of $\oDb(Y)$
provided by \cite[Def.~3.39]{BurgosFreixasLitcanu:HerStruc}. The next
lemma follows easily from the construction of
\cite[Def.~3.39]{BurgosFreixasLitcanu:HerStruc}.
\begin{lemma}\label{lem:van_tor_2}
Grothendieck duality defines a tight isomorphism in $\oDb(Y)$
\begin{displaymath}
[\overline{ f_{\ast}\Omega_{X/Y}^{\bullet}}]^{\vee}\cong [\overline{
f_{\ast}\Omega_{X/Y}^{\bullet}}][2e].
\end{displaymath}
\end{lemma}
\begin{theorem}\label{thm:char_van_dR}
Let $T$ be a theory of analytic torsion classes. The following assertions are equivalent:
\begin{enumerate}
\item the theory $T$ is self-dual;
\item for every $f$, $\overline{T}_{f}$,
$\overline{\Omega}_{X/Y}^{\bullet}$ and $[\overline{
f_{\ast}\Omega_{X/Y}^{\bullet}}]$ as above and for every
odd integer $p\geq 1$, the part of bidegree $(2p-1,p)$ (in
the Deligne complex) of
$T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{
f_{\ast}\Omega_{X/Y}^{\bullet}}])$ vanishes.
\end{enumerate}
\end{theorem}
\begin{proof}
Assume first of all that $T$ is self-dual. We apply the
definition of $T^{\vee}$, the self-duality assumption and lemmas
\ref{lem:van_tor_1} and \ref{lem:van_tor_2}. We find the equality
\begin{displaymath}
\begin{split}
T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{
f_{\ast}\Omega_{X/Y}^{\bullet}}])
&=\sigma T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet}[2e], [\overline{
f_{\ast}\Omega_{X/Y}^{\bullet}}][2e])\\
&=(-1)^{2e}\sigma T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet}, [\overline{
f_{\ast}\Omega_{X/Y}^{\bullet}}])\\
&=\sigma T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet}, [\overline{
f_{\ast}\Omega_{X/Y}^{\bullet}}]).
\end{split}
\end{displaymath}
The sign operator $\sigma$ changes the sign of the components of
bidegree $(2p-1,p)$ for odd $p$. Hence
$T(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{
f_{\ast}\Omega_{X/Y}^{\bullet}}])^{(2p-1,p)}$ vanishes for $p\geq 1$
odd.
For the converse implication, let $S(x)=\sum_{n\geq 0}a_{n}x^{n}$ be
the real additive genus attached to $T$ via Theorem
\ref{thm:comparison_genus}. By Corollary \ref{cor:char_self_dual},
we have to show that the coefficients $a_{n}$ with $n$ even
vanish. Let us look at a smooth morphism $f\colon X\to Y$ of relative
dimension $1$, with an arbitrary metric on $T_{f}$. Then, developing
the power series of $\ch$ and $\Td$ and taking into account that
$\Omega ^{1}_{X/Y}=T_{f}^{\vee}=\omega _{X/Y}$, we compute
\begin{displaymath}
f_{\ast}[\ch(\Omega_{X/Y}^{\bullet})\Td(T_{f})S(T_{f})\bullet\mathbf{1}_{1}]
=\sum_{n\geq
0}(-1)^{n+1}a_{n}f_{\ast}[c_{1}(\omega_{X/Y})^{n+1}\bullet\mathbf{1}_{1}].
\end{displaymath}
Therefore, for $p\geq 1$ odd, we have
\begin{multline}
(-1)^{p}a_{p-1}f_{\ast}[c_{1}(\omega_{X/Y})^{p}\bullet\mathbf{1}_{1}]=
\\(T(\overline{f},\overline{\Omega_{X/Y}^{\bullet}}, [\overline{
f_{\ast}\Omega_{X/Y}^{\bullet}}])
-T^{h}(\overline{f},\overline{\Omega_{X/Y}^{\bullet}}, [\overline{
f_{\ast}\Omega_{X/Y}^{\bullet}}]))^{(2p-1,p,p)}= 0.
\end{multline}
Hence it is enough that for every odd integer $p\geq 1$, we find a
relative curve $f\colon X\to Y$ such that
$f_{\ast}(c_{1}(\omega_{X/Y})^{p})\neq 0$ in the cohomology group
$H^{2p}(Y,{\mathbb C})$. Let $d=p-1$ and choose $Y$ to be a smooth
projective variety of dimension $d$. Let $L$ be an ample line bundle
on $Y$ and take $X={\mathbb P}(L\oplus{\mathcal O}_{Y})$. Consider the tautological
exact sequence
\begin{displaymath}
0\longrightarrow{\mathcal O}(-1)\longrightarrow
f^{\ast}(L\oplus{\mathcal O}_{Y})\longrightarrow Q\longrightarrow 0.
\end{displaymath}
We easily derive the relations
\begin{align}
&\pi^{\ast}c_{1}(L)=c_{1}(Q)-c_{1}({\mathcal O}(1))\label{eq:van_dR_1}\\
&c_{1}({\mathcal O}(-1))c_{1}(Q)=0.\label{eq:van_dR_2}
\end{align}
Moreover we have
\begin{equation}\label{eq:van_dR_3}
c_{1}(\omega_{X/Y})=-c_{1}(Q)-c_{1}({\mathcal O}(1)).
\end{equation}
From \eqref{eq:van_dR_1}--\eqref{eq:van_dR_3} and because $d=p-1$ is
even, we compute
\begin{displaymath}
c_{1}(\omega_{X/Y})^{d}=c_{1}(Q)^{d}+c_{1}({\mathcal O}(1))^{d}=\pi^{\ast}c_{1}(L)^{d}.
\end{displaymath}
Therefore we find
\begin{equation}\label{eq:van_dR_4}
c_{1}(\omega_{X/Y})^{p}=\pi^{\ast}c_{1}(L)^{d} c_{1}(\omega_{X/Y}).
\end{equation}
Finally, $f$ is a fibration in curves of genus $0$, hence
$f_{\ast}(c_{1}(\omega_{X/Y}))=-2$. We infer that
\eqref{eq:van_dR_4} leads to
\begin{displaymath}
f_{\ast}(c_{1}(\omega_{X/Y})^{p})=-2c_{1}(L)^{d}.
\end{displaymath}
This class does not vanish, since $Y$ is projective of dimension $d$
and $L$ is ample.
\end{proof}
We end with a characterization of the theory of analytic torsion
classes of Bismut-K\"ohler.
\begin{theorem} \label{thm:23}
The theory of analytic torsion classes of Bismut-K\"ohler $T^{BK}$
is the unique theory of generalized analytic torsion classes such
that, for every $\overline{f}\colon X\to Y$, a K\"ahler fibration in
$\overline{\Sm}_{\ast/{\mathbb C}}$, we have the vanishing
\begin{displaymath}
T^{BK}(\overline{f}, \overline{\Omega}_{X/Y}^{\bullet},
[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])=0.
\end{displaymath}
\end{theorem}
\begin{proof}
That the theory $T^{BK}$ vanishes for de Rham complexes of K\"ahler
fibrations is a theorem of Bismut \cite{Bismut:deRham}. For the
uniqueness, let $T$ be a theory of generalized analytic torsion
classes vanishing on de Rham complexes of K\"ahler
fibrations. Denote by $S(x)=\sum_{k\geq 0}a_{k}x^{k}$ its
corresponding genus. If $\overline{f}$ is a relative curve with a
structure of K\"ahler fibration, then by Theorem
\ref{thm:comparison_genus}
\begin{equation}\label{eq:van_dR}
T^{h}(\overline{f},\overline{\Omega}_{X/Y}^{\bullet},[\overline{f_{\ast}\Omega_{X/Y}^{\bullet}}])=
\sum_{k\geq
0}(-1)^{k}a_{k}f_{\ast}[c_{1}(\omega_{X/Y})^{k+1}\bullet\mathbf{1}_{1}].
\end{equation}
It is enough to find, for every $k\geq 0$, a relative curve $f$ such
that $f_{\ast}(c_{1}(\omega_{X/Y})^{k+1})$ does not vanish. The
elementary construction in the proof of Theorem \ref{thm:char_van_dR}
works whenever $k$ is even, but one easily sees it fails for $k$
odd. Fortunately, there is an alternative argument. Let $g\geq 2$ and
$n\geq 3$ be integers. Consider the fine moduli scheme of smooth
curves of genus $g$ with a Jacobi structure of level $n$
\cite[Def. 5.4]{Deligne-Mumford}, to be denoted
$\mathcal{M}_{g}^{n}$. Let
$\pi\colon\mathcal{C}_{g}^{n}\to\mathcal{M}_{g}^{n}$ be the universal
curve. An example of K\"ahler fibration structure on $\pi$ is provided
by Teichm\"uller theory (see for instance
\cite[Sec. 5]{Wolpert:chern}). By \cite[Thm. 1]{Faber-Pandharipande},
the tautological class
$\kappa_{g-2}:=\pi_{\ast}(c_{1}(\omega_{\pi})^{g-1})\in
H^{2(g-2)}(\mathcal{M}_{g}^{n},{\mathbb C})$ does not vanish. Taking $g=k+2$
and $f=\pi$, we conclude the proof of the theorem.
\end{proof}
We note that in the previous theorem, the existence is provided by
Bismut's theorem. It would be interesting to have a proof of the
existence of a theory satisfying the condition of Theorem \ref{thm:23}
from the axiomatic point of view.
\section{Direct images of hermitian structures}
\label{sec:direct-imag-compl}
As another application of a theory of generalized analytic
torsion classes, we construct direct images of metrized complexes. It
turns out that the natural place to define direct images is not the
category $\oDb(\cdot)$ but a new category $\hDb(\cdot)$ that is
the analogue to the arithmetic $K$-theory of Gillet and
Soul\'e \cite{GilletSoule:vbhm}.
Let $X$ be a smooth complex variety. The fibers of the forgetful
functor $\oDb(X)\to\Db(X)$ have a structure of $\KA(X)$-torsor, for
the action of $\KA(X)$ by translation of the hermitian structures
(\cite[Thm.~3.13]{BurgosFreixasLitcanu:HerStruc}). At the same time,
the group $\KA(X)$ acts on the group
$\oplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,p)$ by translation,
via the Bott-Chern character $\widetilde{\ch}$
(\cite[Prop.~4.6]{BurgosFreixasLitcanu:HerStruc}). Observe that all
Bott-Chern classes live in these
groups, as for the analytic torsion classes. It is thus natural to
build a product category over~$\KA(X)$.
\begin{definition} \label{def:hDb}
Let $S\subset T^{\ast}X_{0}$ be a closed conical subset. We define
\begin{displaymath}
\hDb(X,S)=\oDb(X)\times_{\KA(X)}
\bigoplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,S,p)
\end{displaymath}
to be the category whose objects are equivalence classes
$[\overline{\mathcal{F}},\eta]$ of pairs $(\overline{\mathcal{F}},\eta)$
belonging to
$\Ob\oDb(X)\times\oplus_{p}\widetilde{\mathcal{D}}^{2p-1}_{D}(X,S,p)$,
under the equivalence relation
\begin{displaymath}
(\overline{\mathcal{F}},\eta)\sim
(\overline{\mathcal{F}}+[\overline{E}],\eta-\widetilde{\ch}(\overline E))
\end{displaymath}
for $[\overline E]\in\KA(X)$, and with morphisms
\begin{displaymath}
\Hom_{\hDb(X)}([\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu])=
\Hom_{\Db(X)}(\mathcal{F},\mathcal{G}).
\end{displaymath}
\end{definition}
If $S\subset T$ are closed conical subsets of
$T^{\ast}X_{0}$, then $\hDb(X,S)$ is naturally a full subcategory of
$\hDb(X,T)$.
In the sequel, we extend the main operations in $\Db(X)$ to the
categories $\hDb(X,S)$. In particular, we use the theory of generalized
analytic torsion classes to construct push-forward morphisms attached
to morphisms in $\overline{\Sm}_{\ast/{\mathbb C}}$.
The category $\hDb(X,S)$ has a natural additive structure. More
generally, if $S,T$ are closed conical subsets of $T^{\ast}X_{0}$,
then there is an obvious addition functor
\begin{displaymath}
\hDb(X,S)\times\hDb(X,T)\overset{\oplus}{\longrightarrow}\hDb(X,S\cup T).
\end{displaymath}
The object $[\overline{0},0]$ is a neutral element for this
operation. If $S + T$ is disjoint with the zero section in
$T^{\ast}X$, then there is a product defined by the functor
\begin{equation}\label{eq:hDb_1}
\begin{split}
\Ob\hDb(X,S)\times\Ob\hDb(X,T)& \overset{\otimes}{\longrightarrow}
\Ob\hDb(X,(S+T)\cup S\cup T)\\
([\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu]) &\longmapsto
[\overline{\mathcal{F}}\otimes\overline{\mathcal{G}},
\ch(\overline{\mathcal{F}})\bullet\nu+
\eta\bullet\ch(\overline{\mathcal{G}})+\dd_{D}\eta\bullet\nu]
\end{split}
\end{equation}
and the obvious assignment for morphisms. This product is commutative
up to natural isomorphism. It induces on $\hDb(X,\emptyset)$ a
structure of commutative and associative ring category. Also,
$[\overline{\mathcal{O}}_{X},0]$ is a unity object for the product
structure. More generally, the category $\hDb(X,S)$ becomes a left and
right $\hDb(X,\emptyset)$ module. Under the same assumptions on $S$,
$T$ we may define an internal $\Hom$. For this, let
$[\overline{\mathcal{F}},\eta]\in\Ob\hDb(X,S)$ and
$[\overline{\mathcal{G}},\nu]\in\Ob\hDb(X,T)$. Then we put
\begin{displaymath}
\underline{\Hom}([\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu])
=[\underline{\Hom}(\overline{\mathcal{F}},\overline{\mathcal{G}}),(\sigma\ch(\overline{\mathcal{F}}))\bullet\nu
+(\sigma\eta)\bullet\ch(\overline{\mathcal{G}})+(\dd_{D}\sigma\eta)\bullet\nu],
\end{displaymath}
where we recall that $\sigma$ is the sign operator (Definition
\ref{def:20}). Using Corollary \ref{cor:7}, it is easily seen this is
well defined. In particular, we put
\begin{displaymath}
[\overline{\mathcal{F}},\eta]^{\vee}:=\underline{\Hom}([\overline{\mathcal{F}},\eta],[\overline{{\mathcal O}}_{X},0])=
[\overline{\mathcal{F}}^{\vee},\sigma\eta].
\end{displaymath}
The shift $[1]$ on $\oDb(X)$ induces a well defined shift functor on
$\hDb(X,S)$, whose action on objects is
\begin{displaymath}
[\overline{\mathcal{F}},\eta][1]=[\overline{\mathcal{F}}[1],-\eta].
\end{displaymath}
There is a Chern character
\begin{displaymath}
\ch:\Ob\hDb(X,S)\longrightarrow\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p}(X,S,p),\quad
[\overline{\mathcal{F}},\eta]\longmapsto\ch(\overline{\mathcal{F}})+\dd_{D}\eta,
\end{displaymath}
which is well defined because $\dd_{D}\widetilde{\ch}(\overline E)=\ch(\overline
E)$ for $[\overline E]\in\KA(X)$. The Chern character is additive and
compatible with the product structure:
\begin{displaymath}
\ch([\overline{\mathcal{F}},\eta]\otimes[\overline{\mathcal{G}},\nu])=
\ch([\overline{\mathcal{F}},\eta])\bullet\ch([\overline{\mathcal{G}},\nu]).
\end{displaymath}
Notice the relations
\begin{align*}
&\ch([\overline{\mathcal{F}},\eta]^{\vee})=\sigma\ch([\overline{\mathcal{F}},\eta]),\\
&\ch([\overline{\mathcal{F}},\eta][1])=-\ch([\overline{\mathcal{F}},\eta]).
\end{align*}
We may also define Bott-Chern classes for isomorphisms and
distinguished triangles. Let
$\widehat{\varphi}\colon[\overline{\mathcal{F}},\eta]\dashrightarrow
[\overline{\mathcal{G}},\nu]$ be an isomorphism in $\hDb(X,S)$, whose
underlying morphism in $\Db(X)$ we denote $\varphi$. While the class
$\widetilde{\ch}(\varphi\colon\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}})$
depends on the representatives $(\overline{\mathcal{F}},\eta)$,
$(\overline{\mathcal{G}},\nu)$, the class
\begin{displaymath}
\widetilde{\ch}(\widehat{\varphi})
:=\widetilde{\ch}(\varphi\colon\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}})
+\nu-\eta
\end{displaymath}
is well defined.
\begin{lemma}\label{lemma:hDb_1}
Let $\widehat{\varphi}:[\overline{\mathcal{F}},\eta]\dashrightarrow
[\overline{\mathcal{G}},\nu]$ be an isomorphism in $\hDb(X,S)$, with
underlying morphism $\varphi$ in $\Db(X)$. Then, the following
conditions are equivalent:
\begin{enumerate}
\item there exists $[\overline E]\in\KA(X)$ such that $\varphi$ induces a
tight isomorphism between $\overline{\mathcal{F}}+[\overline E]$ and
$\overline{\mathcal{G}}$, and $\nu=\eta-\widetilde{\ch}(\overline E)$;
\item $\widetilde{\ch}(\widehat{\varphi})=0$.
\end{enumerate}
\end{lemma}
\begin{proof}
This is actually a tautology. Because $\KA(X)$ acts freely and
transitively on the possible hermitian structures on $\mathcal{F}$,
there exists a unique $[\overline E]\in\KA(X)$ such that
$\overline{\mathcal{F}}+[\overline E]$ is tightly isomorphic to
$\overline{\mathcal{G}}$ via the morphism $\varphi$. Then we have
\begin{displaymath}
\widetilde{\ch}(\widehat{\varphi})=\widetilde{\ch}(\overline E)+\nu-\eta.
\end{displaymath}
The lemma follows.
\end{proof}
\begin{definition}
Let $\widehat{\varphi}$ be an isomorphism in $\hDb(X,S)$. We say
that $\widehat{\varphi}$ is tight if the equivalent conditions of
Lemma \ref{lemma:hDb_1} are satisfied.
\end{definition}
In particular, if $\varphi\colon
\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}}$ is a tight isomorphism in
$\oDb(X)$, then $\varphi$ induces a tight isomorphism
$[\overline{\mathcal{F}},\eta]\dashrightarrow[\overline{\mathcal{G}},\nu]$ if and only if
$\eta=\nu$.
The following lemma provides an example involving the notion of
tight~isomorphism.
\begin{lemma}\label{lemma:hDb_2}
Let $[\overline{\mathcal{F}},\eta]\in\hDb(X,S)$ and
$[\overline{\mathcal{G}},\nu]\in\hDb(X,T)$. Assume that $S+T$ does not
cross the zero section. Then there is a functorial tight isomorphism
\begin{displaymath}
[\overline{\mathcal{F}},\eta]^{\vee}\otimes [\overline{\mathcal{G}},\nu]
\cong\underline{\Hom}([\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu]).
\end{displaymath}
\end{lemma}
Assume now given a distinguished triangle
\begin{displaymath}
\widehat{\tau}\colon\quad [\overline{\mathcal{F}},\eta]\dashrightarrow [\overline{\mathcal{G}},\nu]\dashrightarrow [\overline{\mathcal{H}},\mu]\dashrightarrow [\overline{\mathcal{F}},\eta][1].
\end{displaymath}
Let $\overline{\tau}$ denote the distinguished triangle
$\overline{\mathcal{F}}\dashrightarrow\overline{\mathcal{G}}\dashrightarrow\overline{\mathcal{H}}\dashrightarrow$ in
$\oDb(X)$. Then we put
\begin{displaymath}
\widetilde{\ch}(\widehat{\tau})=
\widetilde{\ch}(\overline{\mathcal{\tau}})+\eta-\nu+\mu.
\end{displaymath}
By \cite[Thm.~3.33~(vii)]{BurgosFreixasLitcanu:HerStruc}, this class
does not depend on the representatives and is thus well defined.
We study now the functoriality of $\hDb(X,S)$ with respect to
inverse and direct images.
Let $f\colon X\to Y$ be a morphism of smooth complex varieties. Let
$T\subset T^{\ast}Y_{0}$ be a closed conical subset disjoint with
$N_{f}$. The action of the left inverse image functor on objects
is
\begin{displaymath}
f^{\ast}\colon\Ob\hDb(Y,T)\longrightarrow\Ob\hDb(X,f^{\ast}T),\qquad
[\overline{\mathcal{F}},\eta]\longmapsto [
f^{\ast}\overline{\mathcal{F}},f^{\ast}\eta].
\end{displaymath}
That this assignment is well defined amounts to the functoriality of
$\widetilde{\ch}$.
Let $\overline{f}$ be a morphism in the category $\overline\Sm_{\ast/{\mathbb C}}$. The
definition of a direct image functor attached to $\overline{f}$ depends upon
the choice of a theory of generalized analytic torsion classes. Let
$T$ be such a theory. Then we define a functor $\overline{f}_{\ast}$
whose action on objects is
\begin{equation}\label{eq:hDb_2}
\begin{split}
\overline{f}_{\ast}\colon\Ob\hDb(X,S)&\longrightarrow\Ob\hDb(Y,f_{\ast}S)\\
[\overline{\mathcal{F}},\eta]&\longmapsto [\overline{
f_{\ast}\mathcal{F}},\overline{f}_{\flat}(\eta)-T(\overline{f},\overline{\mathcal{F}},\overline{
f_{\ast}\mathcal{F}})],
\end{split}
\end{equation}
where $\overline{ f_{\ast}\mathcal{F}}$ is an arbitrary choice of
hermitian structure on $ f_{\ast}\mathcal{F}$. By the anomaly
formulas, this definition does not depend on the representative
$(\overline{\mathcal{F}},\eta)$ nor on the choice of hermitian structure on
$\overline{ f_{\ast}\mathcal{F}}$.
\begin{theorem}\label{thm:hDb_1}
Let $\overline{f}:X\to Y$ and $\overline{g}:Y\to Z$ be morphisms in
$\overline{\Sm}_{\ast/{\mathbb C}}$. Let $S\subset T^{\ast}X_{0}$ and $T\subset
T^{\ast}Y_{0}$ be closed conical subsets.
\begin{enumerate}
\item Let $[\overline{\mathcal{F}},\eta]\in\Ob\hDb(X,S)$. Then there is a
functorial tight isomorphism
\begin{displaymath}
(\overline{g}\circ \overline{f})_{\ast}([\overline{\mathcal{F}},\eta])\cong \overline{g}_{\ast} \overline{f}_{\ast}([\overline{\mathcal{F}},\eta]).
\end{displaymath}
\item (Projection formula) Assume that $T\cap N_{f}=\emptyset$ and
that $T+f_{\ast}S$ does not cross the zero section of
$T^{\ast}Y$. Let $[\overline{\mathcal{F}},\eta]\in\Ob\hDb(X,S)$ and
$[\overline{\mathcal{G}},\nu]\in\Ob\hDb(Y,T)$. Then there is a
functorial tight isomorphism
\begin{displaymath}
\overline{f}_{\ast}([\overline{\mathcal{F}},\eta] \otimes
f^{\ast}[\overline{\mathcal{G}},\nu])
\cong \overline{f}_{\ast}[\overline{\mathcal{F}},\eta]\otimes [\overline{\mathcal{G}},\nu]
\end{displaymath}
in $\hDb(Y,W)$, where
\begin{math}
W=f_{\ast}(S + f^{\ast}T)\cup f_{\ast}S\cup f_{\ast}f^{\ast}T.
\end{math}
\item (Base change) Consider a cartesian diagram
\begin{displaymath}
\xymatrix{
X'\ar[r]^{h'}\ar[d]_{f'} &X\ar[d]^{f}\\
Y'\ar[r]^{h} &Y.
}
\end{displaymath}
Suppose that $f$ and $h$ are transverse and that $N_{h'}$ is
disjoint with $S$. Equip $f'$ with the hermitian structure induced
by the natural isomorphism $ h^{\ast} T_{f}\dashrightarrow T_{f'}$. Let
$[\overline{\mathcal{F}},\eta]\in\Ob\hDb(X,S)$. Then there is a
functorial tight isomorphism
\begin{displaymath}
h^{\ast}\overline{f}_{\ast}[\overline{\mathcal{F}},\eta]\cong \overline{f}'_{\ast}{h'}^{\ast}[\overline{\mathcal{F}},\eta]
\end{displaymath}
in $\hDb(Y',f'_{\ast}{h'}^{\ast}S)$.
\end{enumerate}
\end{theorem}
\begin{proof}
The first and the second assertions follow from Proposition
\ref{prop:12}, the transitivity and
the projection formula for $T$. For the third item, one uses the
functoriality of the analytic torsion classes and
Proposition \ref{prop:11}.
\end{proof}
We close this section with an extension of Grothendieck duality to
$\hDb$. Let $\overline{f}:X\to Y$ be a
morphism is $\overline{\Sm}_{\ast/{\mathbb C}}$. To enlighten notations, we denote
by $\boldsymbol{\omega}_{\overline{f}}$ the object $[\boldsymbol{\omega}_{\overline{f}},0]$ in
$\hDb(X,\emptyset)$ (Definition \ref{def:21}). Suppose given a closed
conical subset $T\subset T^{\ast}Y_{0}$ such that $T\cap
N_{f}=\emptyset$. Then we define the functor $\overline{f}^{!}$ whose action
on objects is
\begin{displaymath}
\overline{f}^{!}:\Ob\hDb(Y,T)\longrightarrow\Ob\hDb(X,f^{\ast}T),\qquad
[\overline{\mathcal{F}},\eta]\longmapsto
f^{\ast}[\overline{\mathcal{F}},\eta]\otimes\boldsymbol{\omega}_{\overline{f}}.
\end{displaymath}
Observe the equality
\begin{equation}\label{eq:hDb_4}
[\overline{\mathcal{G}},\nu]\otimes\boldsymbol{\omega}_{\overline{f}}
=[\overline{\mathcal{G}}\otimes\boldsymbol{\omega}_{\overline{f}},\nu\bullet\ch(\boldsymbol{\omega}_{\overline{f}})].
\end{equation}
Now fix a theory of generalized analytic torsion classes. To the
morphism $\overline{f}$ we have attached the direct image functor
$\overline{f}_{\ast}$. We denote by
$\overline{f}^{\vee}_{\ast}$ the direct image functor associated to
$\overline{f}$ and the dual theory (Theorem
Definition \ref{thm-def:T_dual}).
\begin{theorem}[Grothendieck duality for $\hDb$]\label{thm:groth_dual}
Let $\overline{f}:X\to Y$ be a morphism in $\overline{\Sm}_{\ast/{\mathbb C}}$. Let
$S\subset T^{\ast}X_{0}$ and $T\subset T^{\ast}Y_{0}$ be closed
conical subsets such that $T\cap N_{f}=\emptyset$ and $T+f_{\ast}S$
is disjoint with the zero section. Let
$[\overline{\mathcal{F}},\eta]\in\Ob\hDb(X,S)$ and
$[\overline{\mathcal{G}},\sigma]\in\Ob\hDb(Y,T)$. Then there is a
functorial tight isomorphism
\begin{displaymath}
\underline{\Hom}( \overline{f}_{\ast}[\overline{\mathcal{F}},\eta],[\overline{\mathcal{G}},\nu])
\cong \overline{f}^{\vee}_{\ast}\underline{\Hom}([\overline{\mathcal{F}},\eta], \overline{f}^{!}[\overline{\mathcal{G}},\nu])
\end{displaymath}
in $\hDb(Y,W)$, where
\begin{math}
W=f_{\ast}(S + f^{\ast}T)\cup f_{\ast}S\cup f_{\ast}f^{\ast}T.
\end{math}
\end{theorem}
In particular, we have
\begin{equation}\label{eq:hDb_3}
(
\overline{f}_{\ast}[\overline{\mathcal{F}},\eta])^{\vee}\cong
\overline{f}^{\vee}_{\ast}([\overline{\mathcal{F}},\eta]^{\vee}
\otimes\boldsymbol{\omega}_{\overline{f}}).
\end{equation}
\begin{proof}
By Lemma \ref{lemma:hDb_2} and Proposition \ref{thm:hDb_1}, we are
reduced to establish the functorial tight isomorphism
(\ref{eq:hDb_3}). The proof follows readily from the definitions,
Grothendieck duality and the following two observations. First of
all, if $T$ is the theory of analytic torsion classes, then by the
very definition of $T^{\vee}$ we find
\begin{displaymath}
\sigma T(\overline{f},\overline{\mathcal{F}},\overline{ f_{\ast}\mathcal{F}})=
T^{\vee}(\overline{f},\overline{\mathcal{F}}^{\vee}\otimes\boldsymbol{\omega}_{\overline{f}},
\overline{ f_{\ast}(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})}),
\end{displaymath}
where the metric on $\overline{
f_{\ast}(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})}$ is chosen so that
Grothendieck duality provides a tight isomorphism
\begin{displaymath}
\overline{ f_{\ast}\mathcal{F}}^{\vee}\cong \overline{ f_{\ast}(\mathcal{F}^{\vee}\otimes\boldsymbol{\omega}_{f})}.
\end{displaymath}
Secondly, for direct images of currents, we compute
\begin{displaymath}
\sigma\overline{f}_{\flat}(\eta)=\sigma
f_{\ast}(\eta\bullet\Td(T_{\overline{f}}))
=(-1)^{e}f_{\ast}(\sigma\eta\bullet\sigma\Td(T_{\overline{f}}))
=f_{\ast}(\sigma\eta\bullet\ch(\boldsymbol{\omega}_{\overline{f}})\bullet\Td(T_{\overline{f}})).
\end{displaymath}
Here $e$ is the relative dimension of $f$, and to derive the last
equality we appeal to Proposition \ref{prop:13}. To conclude, we
recall equation \eqref{eq:hDb_4}.
\end{proof}
\begin{corollary}
Let $T$ be a self-dual theory of generalized analytic torsion
classes.
\begin{enumerate}
\item Then there is a functorial isomorphism
\begin{math}
( \overline{f}_{\ast}[\overline{\mathcal{F}},\eta])^{\vee}\cong
\overline{f}_{\ast}([\overline{\mathcal{F}},\eta]^{\vee}
\otimes\boldsymbol{\omega}_{\overline{f}}).
\end{math}
\item If the hermitian structure of $\overline{f}$ comes from chosen
metrics on $T_{X}$, $T_{Y}$ and $\boldsymbol{\omega}_{X}$, $\boldsymbol{\omega}_{Y}$ are
equipped with the induced metrics, then we have a commutative
diagram
\begin{displaymath}
\xymatrix{
\hDb(X,S)\ar[d]_{\overline{f}_{\ast}}\ar[r]^{(\cdot)^{\vee}\otimes\overline{\boldsymbol{\omega}}_{X} }
&\hDb(X,S)\ar[d]^{\overline{f}_{\ast}}\\
\hDb(Y,f_{\ast}S)\ar[r]^{(\cdot)^{\vee}\otimes\overline{\boldsymbol{\omega}}_{Y}}
&\hDb(Y,f_{\ast}S).
}
\end{displaymath}
\end{enumerate}
\end{corollary}
\begin{proof}
The first claim is immediate from Theorem \ref{thm:groth_dual}. The
second item follows from the first one and the projection formula
(Proposition \ref{thm:hDb_1}).
\end{proof}
\section{Analytic torsion for degenerating families of curves}
\label{sec:analyt-tors-degen}
As a second example of application of the theory developed in this article,
we describe the singularities of the analytic torsion for degenerating
families of curves. The results we prove are particular instances of
those obtained by Bismut-Bost \cite{Bismut-Bost}, Bismut
\cite{Bismut:degeneracy} and Yoshikawa \cite{Yoshikawa}. Although the
methods of this section can be extended to recover the results of
Yoshikawa in \cite{Yoshikawa}, for simplicity, we will restrict
ourselves to fibrations in curves over a curve.
In fact, our proof is not very different from the one in \cite{Bismut:degeneracy}
and \cite{Yoshikawa}. For instance, one of the
main ingredients of the proof of the results in \cite{Bismut:degeneracy}
and \cite{Yoshikawa} is the Bismut-Lebeau immersion formula. Our
approach implicitly uses Bismut's generalization of the immersion
formula, encoded in the
existence of analytic torsion theories for arbitrary projective
morphisms. We expect that the techniques of this section can be used to
generalize the above results to situations more general
than the ones considered by Yoshikawa.
Let $S$ be a smooth complex curve and $f\colon X\to S$ a projective
morphism of smooth complex varieties, whose fibers are reduced curves
with at most ordinary double singular points. We assume that $f$ is
generically smooth. Following Bismut-Bost
\cite[Sec. 2(b)]{Bismut-Bost}, we call such a family an
f.s.o. (\emph{famille \`a singularit\'es ordinaires}). The singular
locus of $f$, to be denoted $\Sigma$, is a zero dimensional reduced
closed subset of $X$. Its direct image $\Delta=f_{\ast}(\Sigma)$ is
the Weil divisor
\begin{displaymath}
\Delta=\sum_{p\in S}n_{p}p,
\end{displaymath}
where $n_{p}$ is the number of singular points of the fiber
$f^{-1}(p)$. We will abusively identify $\Delta$ with its
support. With these notations, we put
$V=S\setminus\Delta$. Locally for the analytic topology, the morphism
$f$ can be written in complex coordinates either as
$f(z_{0},z_{1})=z_{0}$ or $f(z_{0},z_{1})=z_{0}z_{1}$
\cite[Sec. 3(a)]{Bismut-Bost}. In the second case, the point of
coordinates $(z_{0},z_{1})=(0,0)$ belongs to the singular locus
$\Sigma$.
For a vector bundle $F$ over $X$, let ${\mathbb P}(F)$ be the projective
space of lines in $F$. The differential $df\colon T_{X}\to
f^{\ast}T_{S}$ induces a section ${\mathcal O}_{X}\to\Omega_{X}\otimes
f^{\ast}T_{S}$. Because $f$ is smooth over $X\setminus\Sigma$, this
section does not vanish on $X\setminus\Sigma$. Therefore there is an
induced map
\begin{displaymath}
\mu\colon X\setminus\Sigma\longrightarrow {\mathbb P}(\Omega_{X}\otimes
f^{\ast}T_{S})\cong{\mathbb P}(\Omega_{X}),
\end{displaymath}
called the \emph{Gauss map}. Notice that this map was already used in
\cite{Bismut:degeneracy} and \cite{Yoshikawa}.
We next study the blow-up $\widetilde{X}=\text{Bl}_{\Sigma}(X)$ of $X$
at $\Sigma$ and relate it to the Gauss map. Let $\pi\colon
\widetilde{X}\to X$ be the natural projection. Let $E$ be the
exceptional divisor of $\pi$,
\begin{displaymath}
E=\bigsqcup_{p\in\Sigma}E_{p},\quad E_{p}\cong{\mathbb P}(T_{p}X),
\end{displaymath}
with the reduced scheme structure. For every $p\in\Sigma$, there is
an identification $T_{p}X\cong\Omega_{X,p}$ provided by the hessian of
$f$, which is a non-degenerate bilinear form on $T_{p}X$. The local
description of the blow-up at a point implies:
\begin{lemma}
There is a commutative diagram
\begin{displaymath}
\xymatrix{
E_{p}={\mathbb P}(T_{p}X)\ar[r]^{\hspace{0.3cm}\sim}\ar@{^{(}->}[d] &{\mathbb P}(\Omega_{X,p})\ar@{^{(}->}[d]\\
\widetilde{X}\ar[r]^{\widetilde{\mu}}\ar[d]_{\pi} &{\mathbb P}(\Omega_{X})\ar[dl]^{p}\\
X &X\setminus\Sigma.\ar[u]_{\mu}\ar@{_{(}->}[l]
}
\end{displaymath}
Denote by ${\mathcal O}(-1)$ the tautological divisor either on
${\mathbb P}(\Omega_{X})$ or on $E_{p}$. Then there is a natural isomorphism
\begin{math}
\widetilde{\mu}^{\ast}{\mathcal O}(-1)\mid_{E_{p}}\cong{\mathcal O}(-1).
\end{math}
\end{lemma}
Consider now the short exact sequence of vector bundles on
${\mathbb P}(\Omega_{X})$
\begin{displaymath}
0\to{\mathcal O}(-1)\to p^{\ast}\Omega_{X}\to Q\to 0,
\end{displaymath}
where $Q$ is the universal quotient bundle. Observe that $Q$ is of
rank 1. The dual exact sequence is
\begin{displaymath}
0\to U\to p^{\ast}T_{X}\to{\mathcal O}(1)\to 0,
\end{displaymath}
$U$ being the universal vector subsheaf. We denote by $\eta $
the induced exact sequence on~$\widetilde X$
\begin{equation} \label{eq:87}
\eta\colon 0\to \widetilde{\mu}^{\ast} U\to
\pi^{\ast}T_{X}\to\widetilde{\mu}^{\ast} {\mathcal O}(1)\to 0,
\end{equation}
From \eqref{eq:87}
and the definition $\omega_{X/S}=\omega_{X}\otimes f^{\ast}T_S$, we
derive a natural isomorphism
\begin{equation}\label{eq:bb_1}
\widetilde{\mu}^{\ast}U\otimes\pi^{\ast}\omega_{X/S}\cong
\widetilde{\mu}^{\ast}{\mathcal O}(-1)
\otimes\widetilde{f}^{\ast}T_S.
\end{equation}
\begin{lemma}\label{lemma:bb_1}
We have
\begin{equation}\label{eq:bb_1bis}
\widetilde{\mu}^{\ast}{\mathcal O}(-1)\otimes\widetilde{f}^{\ast}T_S={\mathcal O}(E).
\end{equation}
\end{lemma}
\begin{proof}
First of all we observe that
$\widetilde{\mu}^{\ast}U\otimes\pi^{\ast}\omega_{X/S}$ is trivial on
the open $W=\widetilde{X}\setminus E$. Indeed, by construction of
the Gauss map we have
\begin{displaymath}
\widetilde{\mu}^{\ast}U\mid_{W}
=\ker(df\colon T_X\to f^{\ast}T_S)\mid_{W}
=\omega_{X/S}^{\vee}\mid W.
\end{displaymath}
Hence by equation \eqref{eq:bb_1} we can write
\begin{displaymath}
\widetilde{\mu}^{\ast}{\mathcal O}(-1)\otimes\widetilde{f}^{\ast}T_S
={\mathcal O}(\sum_{p\in\Sigma}m_{p}E_{p}).
\end{displaymath}
To compute the multiplicities $m_{p}$ we use that
$\widetilde{\mu}^{\ast}{\mathcal O}(-1)\mid_{E_{p}}={\mathcal O}(-1)$, $(E_{p}\cdot
\widetilde{f}^{\ast}T_S)=0$ and $(E_{p}\cdot E_{p})=-1$:
\begin{displaymath}
-m_{p}=\deg(\widetilde{\mu}^{\ast}{\mathcal O}(-1)\otimes
\widetilde{f}^{\ast}T_S)\mid_{E_{p}}
=-1+0=-1.
\end{displaymath}
The lemma follows.
\end{proof}
Later we will need the commutative diagram of exact sequences
\begin{equation}\label{eq:bb_2}
\xymatrix{
\eta\mid_{W} \colon &0\ar[r] &\widetilde{\mu}^{\ast}
U\mid_{W}\ar[r]\ar[d]^{\alpha}
&T_X\mid_{W}\ar[r]\ar[d]^{\beta}
&\widetilde{\mu}^{\ast}{\mathcal O}(1)\mid_{W}\ar[r]\ar[d]^{\gamma} &0\\
\varepsilon \colon &0\ar[r]
&\omega_{X/S}^{\vee}\mid_{W}\ar[r] &T_X\mid_{W}\ar[r]
&f^{\ast}T_S\mid_{W}\ar[r] &0.
}
\end{equation}
After the identification
$\widetilde{\mu}^{\ast}{\mathcal O}(-1)\otimes\widetilde{f}^{\ast}T_S={\mathcal O}(E)$
provided by the lemma, the morphism $\gamma$ is the restriction to $W$
of the natural inclusion
$\widetilde{\mu}^{\ast}{\mathcal O}(1)\to\widetilde{\mu}^{\ast}{\mathcal O}(1)\otimes{\mathcal O}(E)$. This
fact will be used below.
We now proceed to introduce the hermitian vector bundles and the
analytic torsion classes we aim to study. We fix a theory of
generalized analytic torsion classes $T$.
Let $f\colon X\to S$, $\widetilde{f}\colon \widetilde{X}\to S$ be f.s.o. as
above. Recall that we write $W=X\setminus\Sigma=\widetilde{X}\setminus E$
and $V=S\setminus\Delta$, so that $f^{-1}(V)\subset W$. We endow the
tangent spaces $T_{X}$ and $T_{S}$ with smooth hermitian metrics. We
will denote by $\overline{f}$ the corresponding morphism in the category
$\overline{\Sm}_{\ast/{\mathbb C}}$. On the open subset $W$, there
is a quasi-isomorphism
\begin{displaymath}
\omega_{X/S}^{\vee}\mid_{W}=\boldsymbol{\omega}_{X/S}^{\vee}[1]\mid_{W}\to T_{f}
\end{displaymath}
induced by the identification
$\omega_{X/S}^{\vee}\mid_{W}=\ker(T_{X}\mid_{W}\to f^{\ast}T_{S})$. On
$\omega_{X/S}^{\vee}\mid_{W}$, and in particular on
$\omega_{f^{-1}(V)/V}^{\vee}$, we will put the metric induced by
$\overline{T_{X}}\mid_{W}$. We will write $\overline{f}'\colon f^{-1}(V)\to V$ for the
corresponding morphism in $\overline{\Sm}_{\ast/{\mathbb C}}$. Observe that the
restriction of $f$ to $W$, and hence to $f^{-1}(V)$, may be identified
with the restriction of $\widetilde{f}$. Let $\overline{\mathcal{F}}$ be an
object in $\oDb(X)$
and fix a hermitian structure on $ f_{\ast}\mathcal{F}$. Then we
consider the relative metrized complexes
\begin{displaymath}
\overline{\xi}=(\overline{f},\overline{\mathcal{F}},\overline{ f_{\ast}\mathcal{F}}),\qquad
\overline{\xi}'=(\overline{f}',\overline{\mathcal{F}}\mid_{f^{-1}(V)},\overline{
f_{\ast}\mathcal{F}}\mid_{V}),
\end{displaymath}
and the corresponding analytic torsion classes
\begin{displaymath}
T(\overline{\xi})\in\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(S,N_{f},p),\qquad
T(\overline{\xi}')
\in\bigoplus_{p}\widetilde{\mathcal{D}}_{D}^{2p-1}(V,\emptyset,p).
\end{displaymath}
By the functoriality of analytic torsion classes and the anomaly
formulas, we have
\begin{equation}\label{eq:bb_3}
T(\overline{\xi}')=T(\overline{\xi})\mid_{V}-
\overline{f}_{\flat}[\ch(\overline{\mathcal{F}}\mid_{f^{-1}(V)})
\widetilde{\Td}_{m}(\overline{\varepsilon}\mid_{f^{-1}(V)})].
\end{equation}
Here $\overline{\varepsilon}$ is the exact sequence in \eqref{eq:bb_2}, with
the hermitian metrics we have just defined. From now on we will
omit the reference to $f^{-1}(V)$ and $V$ in the formulas.
We consider the hermitian structures on the sheaves $U$ and ${\mathcal O}(1)$
on ${\mathbb P}(\Omega_{X})$
induced by $p^{\ast}\overline{T_X}$. We will write $\overline{\eta}$ for the exact
sequence in
\eqref{eq:87} and
$\overline{\alpha}$, $\overline{\beta}$ and $\overline{\gamma}$ for the vertical
isomorphisms in diagram \eqref{eq:bb_2}, all provided with the
corresponding metrics. Notice that
$\overline{\alpha}$ and $\overline{\beta}$ are isometries. By the properties of
the Bott-Chern class $\widetilde{\Td}_{m}$, we have
\begin{equation}\label{eq:bb_4}
\widetilde{\Td}_{m}(\overline{\varepsilon})=\widetilde{\Td}_{m}(\overline{\eta})
+\Td(\overline{\eta})\widetilde{\Td}_{m}(\overline{\gamma}).
\end{equation}
Hence, from \eqref{eq:bb_3}--\eqref{eq:bb_4} and identifying $f$ with
$\widetilde{f}$ over $V$, we have
\begin{multline}\label{eq:bb_4bis}
T(\overline{\xi}')=T(\overline{\xi})-
\widetilde{f}_{\ast}[\pi^{\ast}\ch(\overline{\mathcal{F}})\pi^{\ast}\Td(\overline{f})
\widetilde{\Td}_{m}(\overline{\eta})]\\
-\widetilde{f}_{\ast}[\pi^{\ast}\ch(\overline{\mathcal{F}})
\pi^{\ast}\Td(\overline{f})\Td(\overline{\eta})
\widetilde{\Td}_{m}(\overline{\gamma})].
\end{multline}
It will be convenient to have a precise description of
$\widetilde{\Td}_{m}(\overline{\gamma})$ at our disposal.
For shorthand, we write $L:=\widetilde{\mu}^{\ast}{\mathcal O}(1)$ and
$\|\cdot\|_{0}$ for its hermitian structure constructed before. We
denote by $\|\cdot\|_{1}$ the metric on ${\mathcal O}(E)$ such that the
isomorphism $\overline{{\mathcal O}(E)}_{1}=\overline{L}_{0}^{-1}\otimes
\widetilde{f}^{\ast}\overline{T_S}$ (Lemma \ref{lemma:bb_1}) is an
isometry. Recall that $\gamma$ gets identified with the restriction to
$W$ of the natural inclusion $L\to L\otimes{\mathcal O}(E)$. We let
$\|\cdot\|_{\infty}$ be the hermitian metric on $L\mid_{W}$ such that
$\gamma$ is an isometry. Hence, if $\mathbf{1}$ denotes the canonical
section of ${\mathcal O}(E)$ and $\ell$ is any section of $L\mid_{W}$, then we
have
\begin{displaymath}
\|\ell\|_{\infty}=\|\ell\|_{0}\|\mathbf{1}\|_{1}.
\end{displaymath}
To simplify the notations, we will skip the reference to $W$. We then have on $W$
\begin{displaymath}
\widetilde{\Td}_{m}(\overline{\gamma})= \widetilde{\Td}_{m}(\overline{L}_{0}
\overset{\Id}{\to}\overline{L}_{\infty}).
\end{displaymath}
To compute a representative of this class, we fix a smooth function
$h\colon {\mathbb P}^{1}_{{\mathbb C}}\to{\mathbb R}$ such that $h(0)=0$ and
$h(\infty)=1$. Then we proceed by a deformation argument. Let $q\colon
W\times{\mathbb P}^{1}_{{\mathbb C}}\to W$ be the projection to the first factor. On
the line bundle $q^{\ast}L$ we put the metric that, on the fiber at the
point $(w,t)\in W\times{\mathbb P}^{1}_{{\mathbb C}}$, is determined by the formula
\begin{displaymath}
\|\ell\|_{(w,t)}=\|\ell\|_{0,w}\|\mathbf{1}\|_{1,w}^{h(t)}.
\end{displaymath}
We will write $\|\cdot\|_{t}$ for this family of metrics parametrized
by ${\mathbb P}^{1}_{{\mathbb C}}$. Define
\begin{displaymath}
\overline{\Td}(\overline{L}_{0}\to\overline{L}_{\infty})=
\frac{1}{2\pi i}\int_{{\mathbb P}^{1}_{{\mathbb C}}}
\frac{-1}{2}\log(t\overline{t})(\Td(\overline{q^{\ast}L}_{t})- \Td(\overline{q^{\ast}L}_{0})).
\end{displaymath}
Then
\begin{equation}\label{eq:bb_5bis}
\overline{\Td}_{m}(\overline{\gamma})=\Td^{-1}(\overline{L}_{0})
\overline{\Td}(\overline{L}_{0}\to\overline{L}_{\infty})
\end{equation}
represents the class $\widetilde{\Td}_{m}(\gamma)$. Let us develop
$\overline{\Td}_{m}(\overline{\gamma})$. If $\overline{{\mathcal O}}_{t}$ denotes the trivial
line bundle on $W\times{\mathbb P}^{1}_{{\mathbb C}}$ with the norm
$\|\mathbf{1}\|_{t}=\|\mathbf{1}\|_{1}^{h(t)}$, then we compute
\begin{displaymath}
\Td(\overline{q^{\ast}L}_{t})-\Td(\overline{q^{\ast}L}_{0})
=\frac{1}{2}c_{1}(\overline{{\mathcal O}}_{t})+\frac{1}{6}c_{1}(\overline{{\mathcal O}}_{t})q^{\ast}c_{1}(\overline{L}_{0})
+\frac{1}{12}c_{1}(\overline{{\mathcal O}}_{t})^{2}.
\end{displaymath}
By the very definition of $c_{1}$, we find
\begin{multline*}
c_{1}(\overline{{\mathcal O}}_{t})=\pd\cpd\log\|\mathbf{1}\|_{t}^{2}=
\pd\cpd(h(t)\log\|\mathbf{1}\|_{1}^{2})\\
=h(t)c_{1}(\overline{{\mathcal O}(E)}_{1})
+\log\|\mathbf{1}\|_{1}^{2}\pd\cpd h(t)
+\pd h(t)\wedge\cpd\log\|\mathbf{1}\|_{1}^{2}
+\pd\log\|\mathbf{1}\|_{1}^{2}\wedge\cpd h(t).
\end{multline*}
We easily obtain
\begin{align}
&\frac{1}{2\pi i}\int_{{\mathbb P}^{1}_{{\mathbb C}}}\frac{-1}{2}\log(t\overline{t})
\frac{1}{2}c_{1}(\overline{{\mathcal O}}_{t})
=-\frac{1}{2}\log\|\mathbf{1}\|_{1},\label{eq:bb_5}\\
&\frac{1}{2\pi i}\int_{{\mathbb P}^{1}_{{\mathbb C}}}
\frac{-1}{2}\log(t\overline{t}) \frac{1}{6}q^{\ast}c_{1}(\overline{L}_{0})
c_{1}(\overline{{\mathcal O}}_{t})=-\frac{1}{6}
\log\|\mathbf{1}\|_{1}c_{1}(\overline{L}_{0}).\label{eq:bb_6}
\end{align}
With some more work, we have
\begin{equation}\label{eq:bb_7}
\begin{split}
\frac{1}{2\pi
i}\int_{{\mathbb P}^{1}_{{\mathbb C}}}\frac{-1}{2}\log(t\overline{t})
\frac{1}{12}c_{1}(\overline{{\mathcal O}}_{t})^{2}=&
-\frac{a}{6}\log\|\mathbf{1}\|_{1}c_{1}(\overline{{\mathcal O}(E)}_{1})\\
&
+\frac{b}{3}\pd(\log\|\mathbf{1}\|_{1}\
\cpd\log\|\mathbf{1}\|_{1}),
\end{split}
\end{equation}
where
\begin{equation}\label{eq:bb_9}
a=\frac{1}{2\pi
i}\int_{{\mathbb P}^{1}_{{\mathbb C}}}\log(t\overline{t})\frac{1}{2}\pd\cpd(h(t)^{2}), \qquad
b=\frac{1}{2\pi i}\int_{{\mathbb P}^{1}_{{\mathbb C}}}\log(t\overline{t})\pd
h(t)\wedge\cpd h(t).
\end{equation}
We observe that
\begin{displaymath}
a=\frac{1}{2\pi i}\int_{{\mathbb P}^{1}_{{\mathbb C}}}\log(t\overline{t})
\frac{1}{2}\pd\cpd(h(t)^{2})=\frac{1}{2},
\end{displaymath}
which is independent of $h$. All in all, equations
\eqref{eq:bb_5bis}--\eqref{eq:bb_9} provide the following expression
for the representative $\overline{\Td}_{m}(\overline{\gamma})$ of
$\widetilde{\Td}_{m}(\overline{\gamma})$:
\begin{equation}\label{eq:bb_7bis}
\begin{split}
\overline{\Td}_{m}(\overline{\gamma})=\Td^{-1}(\overline{L}_{0})
\Big(&-\frac{1}{2}\log\|\mathbf{1}\|_{1}
-\frac{1}{6}\log\|\mathbf{1}\|_{1}c_{1}(\overline{L}_{0})\\
&-\frac{1}{12}\log\|\mathbf{1}\|_{1} c_{1}(\overline{{\mathcal O}(E)}_{1})
+\frac{b}{3}\pd(\log\|\mathbf{1}\|_{1}\cpd\log\|\mathbf{1}\|_{1})\Big).
\end{split}
\end{equation}
Given a current $\eta\in \mathcal{D}_{D}^{n}(X,p)$, we will call
$(n,p)$ its \emph{Deligne bidegree}, while we will call the
\emph{Dolbeault bidegree} to the bidegree in the Dolbeault
complex. When it is clear from the context to which bidegree we are
referring, we call it bidegree.
We now study the singularities of the component of Deligne bidegree
$(1,1)$
of $T(\overline{\xi}')$ near the divisor
$\Delta$. For this we first recall the decomposition of equation
\eqref{eq:bb_4bis}. Observe that
$\widetilde{\mathcal{D}}_{D}^{1}(V,\emptyset,1)$ gets identified with
the space of smooth real functions on $V$. In the sequel, for an
element $\vartheta\in\oplus_{p}\widetilde{D}_{D}^{2p-1}(\ast,p)$, we
write $\vartheta^{(2r-1,r)}$ to refer to its component of bidegree
$(2r-1,r)$. By construction of the Deligne complex, an element of
Deligne bidegree $(2r-1,r)$ is just a current of Dolbeault bidegree
$(r-1,r-1)$.
The following assertion is well-known. See for instance \cite[Lemma 2.1,
Cor.~2.2]{Wolpert}.
\begin{lemma}\label{lemma_wolpert}
Let $\Omega\subset{\mathbb C}$ be an open subset and $\vartheta$ a current
of Dolbeault bidegree $(0,0)$ on $\Omega$. Let $\Delta$ be the standard
laplacian. If the current $\Delta \vartheta$ is represented by a
locally bounded measurable function, then $\vartheta$ is represented
by a continuous function.
\end{lemma}
\begin{proposition}\label{prop:bb_1}
The current
$T(\overline{\xi})^{(1,1)}\in\widetilde{\mathcal{D}}_{D}^{1}(S,N_{f},1)$
is represented by a continuous function on $S$.
\end{proposition}
\begin{proof}
The differential equation satisfied by $T(\overline{\xi})^{(1,1)}$ is
\begin{equation}\label{eq:bb_11}
\dd_{\mathcal{D}} T(\overline{\xi})^{(1,1)}=\ch(\overline{ f_{\ast}\mathcal{F}})^{(2,1)}
-f_{\ast}[\ch(\overline{\mathcal{F}})\Td(\overline{f})]^{(2,1)}.
\end{equation}
In local coordinates, the operator $\dd_{\mathcal{D}}=-2\pd\cpd$ is a
rescaling of the laplacian $\Delta$. By the lemma, it is enough we
prove that the current at the right hand side of \eqref{eq:bb_11} is
represented by a locally bounded measurable differential form. Because
$\ch(\overline{ f_{\ast}\mathcal{F}})^{(2,1)}$ and
$\ch(\overline{\mathcal{F}})\Td(\overline{f})$ are smooth differential forms, we
are reduced to study currents of the form $f_{\ast}[\theta]^{(2,1)}$,
where $\theta$ is a smooth differential form. By a partition of unity
argument, we reduce to the case where $f\colon {\mathbb C}^{2}\to{\mathbb C}$ is the
morphism $f(z_{0},z_{1})=z_{0}z_{1}$ and $\theta$ is a differential
form of Dolbeault bidegree (2,2) with compact support. Then we need to
prove that the fiber integral
\begin{displaymath}
G(w)=\int_{z_{0}z_{1}=w}\theta
\end{displaymath}
is a bounded form in a neighborhood of $w=0$. Write
\begin{math}
\theta=h(z_{0},z_{1})dz_{0}\wedge d\overline{z}_{0}\wedge dz_{1}\wedge d\overline{z}_{1}.
\end{math}
We reduce to study integrals of the form
\begin{displaymath}
G(w)=\left(\int_{|w|<|z_{0}|<1}h(z_{0},z_{0}/w)\frac{|w|^{2}}{|z_{0}|^{4}}dz_{0}\wedge\overline{z}_{0}\right)dw\wedge d\overline{w}.
\end{displaymath}
The property follows from an easy computation in polar coordinates.
\end{proof}
\begin{proposition}\label{prop:f_ast_theta}
Let $\theta$ be a differential form of Dolbeault bidegree (1,1) on
$\widetilde{X}$. Then the current $\widetilde{f}_{\ast}[\theta]$ is
represented by a bounded function on $S$.
\end{proposition}
\begin{proof}
The proof is the same as in
\cite[Prop. 5.2]{Bismut-Bost}. One only has to show that the argument
in \emph{loc. cit.} carries over to the case of the non-reduced
fibres that have appeared when blowing up the nodes.
\end{proof}
\begin{corollary}\label{cor:bb_1}
The current
$\widetilde{f}_{\ast}[\pi^{\ast}\ch(\overline{\mathcal{F}})
\pi^{\ast}\Td(\overline{f})\widetilde{\Td}_{m}(\overline{\eta})]$
is represented by a bounded function on $S$.
\end{corollary}
\begin{proof}
It suffices to observe that the differential form
$\pi^{\ast}\ch(\overline{\mathcal{F}})
\pi^{\ast}\Td(\overline{f})\widetilde{\Td}_{m}(\overline{\eta})$
is actually smooth on the whole $\widetilde{X}$.
\end{proof}
According to \eqref{eq:bb_4bis}, it remains to study the current
\begin{displaymath}
\widetilde{f}_{\ast}[\pi^{\ast}\ch(\overline{\mathcal{F}})
\pi^{\ast}\Td(\overline{f})\Td(\overline{\eta})
\widetilde{\Td}_{m}(\overline{\gamma})]\mid_{V}.
\end{displaymath}
The main difference with the situation in Corollary \ref{cor:bb_1} is
that the class $\widetilde{\Td}_{m}(\overline{\gamma})$ is not defined on
the whole $\widetilde{X}$, but only on $W=\widetilde{X}\setminus
E$. In the following discussion we will use the representative
$\overline{\Td}_{m}(\overline{\gamma})$ defined in \eqref{eq:bb_5bis} at the place
of $\widetilde{\Td}_{m}(\overline{\gamma})$. In view of equations
\eqref{eq:bb_5}--\eqref{eq:bb_7}, the first result we need is the following
statement.
\begin{proposition}
Let $\theta$ be a smooth and $\pd, \cpd$ closed differential form on
$\widetilde{X}$, of Dolbeault bidegree $(1,1)$. Let $w$ be an
analytic coordinate in a neighborhood of $p\in\Delta$ with
$w(p)=0$. Write $D_{p}=E\cap\widetilde{f}^{-1}(p)$. Then, the
current
\begin{displaymath}
\widetilde{f}_{\ast}[\log\|\mathbf{1}\|_{1}\theta]- \left(\frac{1}{2\pi
i}\int_{D_{p}}\theta\right)[\log|w|]
\end{displaymath}
is represented by a continuous function in a neighborhood of $p$. In
particular, if $\theta$ is cohomologous to a form
$\pi^{\ast}\vartheta$, where $\vartheta$ is a smooth and $\pd,\cpd$
closed differential form on $X$, then
$\widetilde{f}_{\ast}[\log\|\mathbf{1}\|_{1}\theta]$ is represented by
a continuous function on $S$.
\end{proposition}
\begin{proof}
Recall that the Poincar\'e-Lelong formula provides the equality of
currents
\begin{displaymath}
\dd_{\mathcal{D}}[\log\|\mathbf{1}\|_{1}^{-1}]=[c_{1}(\overline{{\mathcal O}(E)}_{1})]-\delta_{E}.
\end{displaymath}
Moreover, the operator $\dd_{\mathcal{D}}$ commutes with proper
push-forward. Therefore, taking into account that $\theta$ is $\pd$
and $\cpd$ closed, the
equation
\begin{equation}\label{eq:bb_corr_1}
\dd_{\mathcal{D}}\widetilde{f}_{\ast}[\log\|\mathbf{1}\|_{1}\theta]=
\left(\frac{1}{2\pi i}\int_{D_{p}}\theta\right)\delta_{p}-
\widetilde{f}_{\ast}[c_{1}(\overline{{\mathcal O}(E)}_{1})\theta].
\end{equation}
holds in a neighborhood of $p$. On the other hand, the Poincar\'e-Lelong equation also gives
$\dd_{\mathcal{D}}[\log|w|]=\delta_{p}$.
Using \eqref{eq:bb_corr_1}, we see that
\begin{displaymath}
\dd_{\mathcal{D}}\left(\widetilde{f}_{\ast}[\log\|\mathbf{1}\|_{1}\theta]
-\left( \frac{1}{2\pi i}\int_{D_{p}}\theta\right)[\log|w|]\right)=
-\frac{1}{2}\widetilde{f}_{\ast}[c_{1}(\overline{{\mathcal O}(E)}_{1})\theta].
\end{displaymath}
Finally, by Proposition \ref{prop:f_ast_theta}, the current
$\widetilde{f}_{\ast}[c_{1}(\overline{{\mathcal O}(E)}_{1})\theta]$ is represented by
a continuous function on $S$. Hence the first assertion follows from
Lemma \ref{lemma_wolpert}. For the second assertion, we just observe
that, in this case,
\begin{displaymath}
\int_{D_{p}}\theta=\int_{D_{p}}\pi^{\ast}\vartheta=0.
\end{displaymath}
The proof is complete.
\end{proof}
\begin{corollary}\label{cor:bb_corr_1}
Let $n_{p}$ be the multiplicity of $\Delta$ at $p$ and $O(1)$
the current represented by a locally bounded function.
The following estimates hold in a neighborhood of $p$
\begin{align*}
&\widetilde{f}_{\ast}[\log\|\mathbf{1}\|_{1}c_{1}(\pi^{\ast}\overline{T_{X}})]=O(1),\\
&\widetilde{f}_{\ast}[\log\|\mathbf{1}\|_{1}c_{1}(\overline{{\mathcal O}(E)}_{1})]=-n_{p}[\log|w|]+O(1),\\
&\widetilde{f}_{\ast}[\log\|\mathbf{1}\|_{1}c_{1}(\overline{L}_{0})]=n_{p}[\log|w|]+O(1),\\
&\widetilde{f}_{\ast}[\log\|\mathbf{1}\|_{1}
c_{1}(\widetilde{\mu}^{\ast}\overline{U})]=-n_{p}[\log|w|]+O(1).
\end{align*}
\end{corollary}
\begin{proof}
We use
\eqref{eq:bb_1}--\eqref{eq:bb_1bis} and the intersection numbers
$(D_{p}\cdot D_{p})=(D_{p}\cdot E)=-n_{p}$.
\end{proof}
\begin{corollary}\label{cor:bb_2}
With the notations above, the development
\begin{displaymath}
\widetilde{f}_{\ast}[\pi^{\ast}\ch(\overline{\mathcal{F}})
\pi^{\ast}\Td(\overline{f})\Td(\overline{\eta}) \widetilde{\Td}_{m}(\overline{\gamma})]^{(3,2)}
=\rk(\overline{\mathcal{F}})\frac{n_{p}}{6}[\log|w|]+O(1)
\end{displaymath}
holds in a neighborhood
of $p$.
\end{corollary}
\begin{proof}
We take into account the expression \eqref{eq:bb_7bis} for the
representative $\overline{\Td}_{m}(\overline{\gamma})$, the developments of the
smooth differential forms $\ch(\overline{\mathcal{F}})$, $\Td(\overline{f})$,
$\Td(\overline{\eta})$ and $\Td^{-1}(\overline{L}_{0})$, and then apply
Corollary \ref{cor:bb_corr_1}. We find
\begin{multline*}
\widetilde{f}_{\ast}[\pi^{\ast}\ch(\overline{\mathcal{F}})
\pi^{\ast}\Td(\overline{f})\Td(\overline{\eta}) \widetilde{\Td}_{m}(\overline{\gamma})]^{(3,2)}
=\\ \rk(\overline{\mathcal{F}})\frac{n_{p}}{6}[\log|w|]+
\rk(\overline{\mathcal{F}})\frac{b}{3}
\widetilde{f}_{\ast}[\pd(\log\|\mathbf{1}\|_{1}\cpd\log\|\mathbf{1}\|_{1})]+O(1).
\end{multline*}
To conclude we observe that, on $V$, the term
$\widetilde{f}_{\ast}[\pd(\log\|\mathbf{1}\|_{1}\cpd\log\|\mathbf{1}\|_{1})]$
vanishes. Indeed, the morphism $\widetilde{f}_{\ast}$ is smooth on
$V$ with one dimensional fibers. Hence this current is represented
by the function
\begin{displaymath}
V\ni s\mapsto \frac{1}{2\pi i}\int_{\widetilde{f}^{-1}(s)}
\pd(\log\|\mathbf{1}\|_{1}\cpd\log\|\mathbf{1}\|_{1})=
\frac{1}{2\pi i}\int_{\widetilde{f}^{-1}(s)}
d(\log\|\mathbf{1}\|_{1}\cpd\log\|\mathbf{1}\|_{1})=0.
\end{displaymath}
This ends the proof.
\end{proof}
The results of this section are summarized in the following statement.
\begin{theorem}
Let $p\in\Delta$ and let $n_{p}$ be the number of singular points of
$f\colon X\to S$ lying above $p$. Let $w$ be a local coordinate on
$S$, centered at $p$. Then, in a neighborhood of $p$, we have the
estimate
\begin{displaymath}
T(\overline{\xi}')^{(1,1)}=-\frac{\rk{\overline{\mathcal{F}}}}{6}n_{p}[\log|w|]+O(1).
\end{displaymath}
\end{theorem}
\begin{proof}
It is enough to join \eqref{eq:bb_4bis},
Proposition \ref{prop:bb_1}, Corollary \ref{cor:bb_1} and
\ref{cor:bb_2}.
\end{proof}
\begin{corollary}
Assume that $\overline{\mathcal{F}}=\overline{E}$ is a vector bundle placed in
degree $0$, and that $R^{1}f_{\ast}E=0$ on $S$. Endow $f_{\ast}E$
with the $L^2$ metric on $V$ depending on $\overline{E}$ and the metric on
$\overline{\omega_{f^{-1}(V)/V}}$. Write
$\xi''=(\overline{f}',\overline{E},\overline{f_{\ast}E}_{L^{2}})$ for the
corresponding relative metrized complex on $V$. Let $p$ and $w$ be
as in the theorem. Then we have
\begin{displaymath}
T(\overline{\xi}'')^{(1,1)}=-\frac{\rk(\overline{\mathcal{F}})}{6}n_{p}[\log|w|]+O(\log\log|w|^{-1})
\end{displaymath}
as $w\to 0$.
\end{corollary}
\begin{proof}
Introduce an auxiliary smooth hermitian metric on the vector bundle
$f_{\ast}E$ on $S$, and let
$\overline{\xi}'=(\overline{f}',\overline{E},\overline{f_{\ast}E})$ be the corresponding
relative metrized complex. Then the theorem applies to
$\overline{\xi}'$. By the anomaly formulas, on $V$ we have
\begin{displaymath}
T(\overline{\xi}'')^{(1,1)}=T(\overline{\xi}')^{(1,1)}+
\widetilde{\ch}(\overline{f_{\ast}E},\overline{f_{\ast}E}_{L^2})^{(1,1)}.
\end{displaymath}
By \cite[Prop. 7.1]{Bismut-Bost}, the $L^2$ metric has logarithmic
singularities near $w=0$ and
\begin{displaymath}
\widetilde{\ch}(\overline{f_{\ast}E},\overline{f_{\ast}E}_{L^2})=O(\log\log|w|^{-1})
\end{displaymath}
as $w\to 0$. This proves the corollary.
\end{proof}
\begin{remark}
The corollary is to be compared with
\cite[Thm. 9.3]{Bismut-Bost}. The difference of sign is due to the
fact that Bismut and Bost work with the inverse of the usual
determinant line bundle. The approach of \emph{loc. cit.} is more
analytic and requires the spectral description of the Ray-Singer
analytic torsion.
\end{remark}
\bigskip
\footnotesize
\noindent\textit{Acknowledgments.}
During the elaboration of this paper we have benefited from
conversations with many colleagues, that helped us to understand some
points, to clarify others or to find the relevant bibliography. Our
thanks to J.-M. Bismut, J.-B. Bost, D. Burghelea, D. Eriksson, J. Kramer,
U. K\"uhn, X. Ma, V. Maillot, D. R\"ossler, C. Soul\'e.
We would like to thank the following institutions where part of the
research conducting to this paper was done: the CRM in Bellaterra
(Spain), the CIRM in Luminy (France), the Morningside Institute of Beijing
(China), the University of Barcelona and the IMUB, the Alexandru Ioan
Cuza University of Iasi,
the Institut de Math\'ematiques de Jussieu and the ICMAT (Madrid).
Burgos and Freixas were partially supported by grant MTM2009-14163-
C02-01, Burgos was partially supported by CSIC research project
2009501001, Li\c tcanu was partially supported by CNCSIS -UEFISCSU,
project number PNII - IDEI 2228/2008 .
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{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,055
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Прапор Міссурі () — один з державних символів американського штату Міссурі. Прапор був розроблений і виготовлений в Джексоні Мері Елізабет Воткінс Олівер, дружиною колишнього сенатора штату Р. Б. Олівера. Прапор був прийнятий в 1913 році і залишається незмінним до цього дня.
Прапор являє собою прямокутне полотнище, розділене на три рівновеликі горизонтальні смуги червоного, білого і синього кольорів. У центрі білої смуги — емблема Міссурі облямована синім колом, в який вписано 24 білих зірки.
Синя смуга символізує доблесть, біла — чистоту, червона — пильність і правосуддя. Також ці кольори вказують на прапор Франції, оскільки раніше штат Міссурі був частиною французької території Луїзіана. 24 зірки символізують штат Міссурі, що як 24-й штат увійшов до складу США.
У 1861-1865 роках, під час Громадянської війни, прапором штату було синє полотнище з печаткою штату, виконаною золотим кольором, в центрі полотнища.
Див. також
Міссурі
Посилання
Опис прапора штату
Прапор Міссурі
Міссурі
Міссурі
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,554
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Úvaly (Duits: Auwal) is een Tsjechische stad in de regio Midden-Bohemen, en maakt deel uit van het district Praha-východ.
Úvaly telt 7035 inwoners (2022).
Gemeente in Praha-východ
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,914
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When was the last time you checked your website's content to make sure it was up to date? Not the technical side, although that is very important, too. Today we're talking about the actual site content – the text, the links, and the images.
Let's start with your links. Do they work? If a potential guest clicks on a link in your site, does it go where it's supposed to go? If a Google "Spider" crawls your links, are any of them a dead link? An easy way to make sure all of your links work is to use a free tool like Broken Link Check. You simply enter your website address, then run the scan. After the scan is finished, click the "URL" link to see the page with the bad link, so that you can fix it. Run your blog, too, if it has a different URL than your website – for example "blog.mybedandbreakfast.com" or "mybedandbreakfast.wordpress.com". If your blog is part of your site, like "mybedandbreakfast.com/blog", you don't need to run a separate scan. Oftentimes, broken links are found on events and attractions pages – a restaurant goes out of business, an event ends, a website is reconfigured with new filenames. Another place where broken links often reside is in a blog – mainly due to events. If you find a broken link, you have some options. If the link is to a business that has just changed their website address or pages, just find the correct link and update it. If the link is for an event that has ended, you can just break the link, or remove the reference to the link. If the business is closed, remove the information about the business, including the link.
Another place that can be out of date is in your site's text. Really read every page of your site to make sure the information still applies. This will be very important on events pages, innkeeper's corner boxes and blog posts. Check your specials and rooms pages – have you changed anything in the rooms that should be updated in the room description? If you have stopped serving food with nuts in it due to allergy concerns, make sure you mention that. How about your policies page? A sure way to make your business look bad in the eyes of potential guests is to have old information cluttering up your pages. On your blog – have you posted lately? Did you do a "See our new site!" post a year ago and nothing since then? Post on a consistent basis – even if it's just once per month. You can even write blog posts in slower, off-season times, and schedule them for when you're too busy to write a post – another option is to hire someone to write your posts. Acorn does that as part of our on-going marketing services. If you have dated reviews on your site, ensure that they are recent. Just make sure your content is current so that you don't look like you don't care about your business.
Finally – images. The most important thing we have to say about images is to make absolutely sure that you own the images or have permission to use them, in writing. Using all of your own photos or images purchased from a legitimate stock photography provider like Dreamstime or iStock by Getty will ensure that you don't get sued or harassed for a settlement. Using photos you took from the internet, even if it was years ago, even if you didn't know you had to ask permission, even if the photo didn't have a watermark, will not keep you safe. You will eventually get caught. Companies use reverse-image searches to findsites using their clients' images, then send threatening letters asking for money. They will usually screenshot the page, too, so taking the image down won't help. Check your images from your home page to your oldest blog post and take down any images you aren't sure about.
Also check images to make sure they reflect reality. Have you changed bedding or switched to flat-screen TVs, or even removed TVs? Remodeled bathrooms, removed soaker tubs, etc? Your photos should show what is really available to your potential guests.
The last thing to say about images – alt tags. Alt tags are a requirement for ADA. Alt tags are text in the code of your site that explains what an image is all about. For example, a breakfast photo might have an alt tag that reads, "Breakfast plate with scrambled eggs, bacon and a buttered crumpet, accompanied by coffee and orange juice". Check our Educational Byte video on the subject. Lawyers have been targeting B&Bs in so-called "drive-by" lawsuits, which can be costly, in time, energy and money.
Take a couple of hours and go through your site to make sure everything is correct. It's easiest to just keep up with changes as they happen, but for things like links, you may not even know when they've changed. Your site and your potential guests will thank you!
This entry was posted in ADA, designs, free tools, General by admin. Bookmark the permalink.
It is important to check things often, just this week I was changing out some wording and found a [read more} that went no where. Come to find out, it's a glitch from mobile to site, so they are fixing that for me.
Hi Debbie, that is a great point – to go through your site to find stuff that is just not working or is leading nowhere!
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{
"redpajama_set_name": "RedPajamaC4"
}
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\section{Introduction}
Candidates for gravity duals of non-relativistic scale invariant
theories have recently attracted a great deal of attention for several reasons.
One is that some condensed matter systems realized in laboratories at their
critical points are described by non-relativistic conformal field theories.
Furthermore, the AdS/CFT correspondence \cite{Mal,Wit} describing the duality
between strongly coupled conformal field theory and gravity continues to
find more applications in other branches of physics such as QCD quark-gluon
plasmas \cite{Kov}, atomic physics, and condensed matter physics \cite
{Hart,Hart2,Hart3,Herz,Faul,Mc}. This has led to a study of gravity-gauge duality in a
much broader context than its original AdS/CFT formulation, extending to
non-relativistic and Lifshitz field theories, with the aim of gaining more
knowledge of the strong coupling behavior of these other physical
systems.
Non-relativistic conformal symmetry contains the scaling symmetry
\begin{equation}
t\rightarrow \lambda ^{z}t,\hspace{1cm}\mathbf{{x}\rightarrow \lambda {x}}
\label{Ssym}
\end{equation}
where z is the dynamical exponent. This transformation exhibits anisotropic
scale invariant behavior. Actually, for condensed matter applications, one
typically is interested in considering anisotropy between different spatial
dimensions. For $z=1$ this scaling symmetry is the familiar conformal
symmetry. Such a non-relativistic scale invariance (\ref{Ssym}) can be
exhibited by either a Galilean-invariant theory or a Lifshitz-invariant
theory. From a holographic point of view, this suggests the following (asymptotic)
form for the
spacetime metric
\begin{equation}
ds^{2}=L^{2}\left( -r^{2z}dt^{2}+\frac{dr^{2}}{r^{2}}+r^{2}d\mathbf{x
^{2}\right) \label{Lifmet}
\end{equation}
that obeys the scale invariance
\begin{equation}
t\rightarrow \lambda ^{z}t,\hspace{0.5cm}r\rightarrow \lambda ^{-1}r,\hspace
0.5cm}\mathbf{{x}\rightarrow \lambda {x}} \label{Ssym2}
\end{equation}
noted previously in a braneworld context \cite{Koroteev}.
A four-dimensional anisotropic scale invariant background using an action
involving a two form and a three form field with a Chern-Simons coupling
\begin{equation}\label{actHB}
I=\int d^{4}x\sqrt{-g}\left( R-2\Lambda -\frac{1}{4}F_{\alpha \beta}F^{\mu
\nu } -\frac{1}{12}H_{\mu \nu \rho}H^{\mu \nu \rho} -\frac{C}{\sqrt{-g}}
F\wedge B\right),
\end{equation}
can be engineered to yield solutions with this asymptotic behavior \cite
{Kach}, where $H=dB$ and $F=dA$. Such an action was argued to be rather
generic in string theory, although no explicit brane realization or
embedding into ten-dimensions was given. For these matter fields, lots of
effort has been expended in extending this solution to the case of
asymptotic Lifshitz solutions. One of the first analytic examples was
reported in Ref. \cite{Tay} for a sort of higher-dimensional dilaton gravity
without restricting the value of the dynamical exponent $z$. An exact
topological black hole solution with hyperbolic horizon which happens to be
asymptotically Lifshitz with $z = 2$ was found in \cite{Mann}; further
solutions with $z = 4$ and with spherical topology were subsequently
obtained \cite{Peet}. However in general such asymptotic Lifshitz black
holes must be investigated numerically \cite{Mann,Peet,Bal,Dan}. Other
possibilities for the matter needed to support such a background have been
investigated \cite{Tay}. Asymptotic Lifshitz solutions in the vacuum of
higher-derivative gravity theories (with curvature-squared terms in the
action) have been investigated \cite{Ay1,Ay2,Cai,Pang}; the higher curvature terms with
suitable coupling constant play the role of the desired matter.
Recently gravity theories including higher powers of the curvature, particularly
curvature-cubed interactions, have attracted increased attention \cite{DP}.
This is because, in the context of AdS/CFT correspondence, corrections from higher powers of the
curvature must be considered on the gravity side of the correspondence in order to scrutinize CFTs with different values for their central charges. Amongst the cornucopia of higher-curvature gravity theories, Lovelock gravity theories play a special role in that the number of metric
derivatives in any field equation is never larger than 2. Third-order Lovelock gravity is supersymmetric, and therefore one can define superconformal field theories via the AdS/CFT
correspondence \cite{Boer,Cam}. Furthermore quasi-topological gravity including
curvature-cubed interactions, while not supersymmetric, can
be considered to be dual to some non-supersymmetric but
conformal gauge theory in the limit of a large number of colours \cite{Myers}.
In this paper, we
consider the existence of Lifshitz solutions in third order Lovelock gravity both in
vacuum and in the presence of a massive vector field. Since the higher
curvature terms appear to play the role of some kind of matter field, it is
natural to ask whether they can support a Lifshitz solution in vacuum or
not. We find that the answer is yes, albeit under restricted circumstances.
We also search for asymptotic Lifshitz black holes in the presence of a
massive vector field, whose action is given via a dualization of the action
(\ref{actHB}). The solutions we find -- both analytically and
numerically -- can be regarded as higher-curvature modifications to those
obtained from Einsteinian gravity coupled to matter \cite{Mann,Peet,Dan}.
The outline of our paper is as follows. We give a brief review of the field
equations of third order Lovelock gravity in the presence of massive vector
field in Sec. \ref{Fiel}. In Sec. \ref{Lif} we present the $(n+1)
-dimensional exact Lifshitz solutions in vacuum and in the presence of a
massive vector field. In Sec. \ref{black} we obtain the series solutions to
the field equations near the horizon, while the series solutions at large $r$
will be given the Appendix. We then obtain
numerical solutions to these equations. The thermodynamics of these
Lovelock-Lifshitz black holes will be given in Sec. \ref{Therm}. Section \ref
{rot} will be devoted to the rotating Lovelock-Lifshitz solutions. We finish
our paper with some concluding remarks.
\section{Field equations\label{Fiel}}
The fundamental assumptions in standard general relativity are the
requirements of general covariance and 2nd-order differential field
equations for the metric. Based on the same principles, the Lovelock
Lagrangian is a very general Lagrangian in classical gravity that produces
second order field equations for the metric for arbitrary powers of the curvature \cite{Lov}. The action of third order Lovelock
gravity in the presence of an Abelian massive vector field $A^{\mu }$ may be
written as
\begin{equation}
I=\int d^{n+1}x\sqrt{-g}\left( -2\Lambda +\mathcal{L}_{1}+\alpha _{2
\mathcal{L}_{2}+\alpha _{3}\mathcal{L}_{3}-\frac{1}{4}F_{\mu \nu }F^{\mu \nu
}-\frac{1}{2}m^{2}A_{\mu }A^{\mu }\right) , \label{Act1}
\end{equation}
where $F_{\mu \nu }=\partial_{[\mu}A_{\nu]}$, $\Lambda $ is the cosmological
constant, $\alpha _{2}$ and $\alpha _{3} $ are Gauss-Bonnet and third order
Lovelock coefficients, $\mathcal{L}_{1}=R$ is the Einstein-Hilbert
Lagrangian, $\mathcal{L}_{2}=R_{\mu \nu \gamma \delta }R^{\mu \nu \gamma
\delta }-4R_{\mu \nu }R^{\mu \nu }+R^{2}$ is the Gauss-Bonnet Lagrangian, and
\begin{eqnarray}
\mathcal{L}_{3} &=&R^{3}+2R^{\mu \nu \sigma \kappa }R_{\sigma \kappa \rho \tau }R_
\phantom{\rho \tau }{\mu \nu }}^{\rho \tau }+8R_{\phantom{\mu \nu}{\sigma
\rho}}^{\mu \nu }R_{\phantom {\sigma \kappa} {\nu \tau}}^{\sigma \kappa }R_
\phantom{\rho \tau}{ \mu \kappa}}^{\rho \tau }+24R^{\mu \nu \sigma \kappa
}R_{\sigma \kappa \nu \rho }R_{\phantom{\rho}{\mu}}^{\rho } \notag \\
&&+3RR^{\mu \nu \sigma \kappa }R_{\sigma \kappa \mu \nu }+24R^{\mu \nu
\sigma \kappa }R_{\sigma \mu }R_{\kappa \nu }+16R^{\mu \nu }R_{\nu \sigma
}R_{\phantom{\sigma}{\mu}}^{\sigma }-12RR^{\mu \nu }R_{\mu \nu }
\label{L3}
\end{eqnarray}
is the third order Lovelock Lagrangian. We assume that
the Gauss-Bonnet coefficient, which has the dimension of (length)$^2$, is positive as
in the heterotic string theory \cite{Boul}.
In Lovelock gravity only terms with order less than $[(n+1)/2]$ (where $[x]$
is the integer part of $x$) contribute to the field equations, the rest
being total derivatives in the action. For 3rd-order Lovelock gravity we
therefore consider $(n+1)$-dimensional spacetimes with $n\geq 6$
(though in situations where we set $\hat{\alpha}_3=0$ our solutions will
be valid for $n\geq 4$).
Varying
the action with respect to the metric tensor $g_{\mu \nu }$ and gauge field
A_{\mu }$ the equations of gravitation and gauge fields are
\begin{eqnarray}
&&G_{\mu \nu }^{(1)}+\alpha _{2}G_{\mu \nu }^{(2)}+\alpha _{3}G_{\mu \nu
}^{(3)}+\Lambda g_{\mu \nu }=T_{\mu \nu }, \label{Geq} \\
&&\partial _{\mu }\left( \sqrt{-g}F^{\mu \nu }\right) =m^{2}\sqrt{-g}A^{\nu
}, \label{EMeq}
\end{eqnarray}
where
\begin{equation*}
T_{\mu \nu }=\frac{1}{2}\left( F_{\phantom{\lambda}{\mu}}^{\rho }F_{\rho \nu
}-\frac{1}{4}F_{\rho \sigma }F^{\rho \sigma }g_{\mu \nu }+m^{2}\left[ A_{\mu
}A_{\nu }-\frac{1}{2}A_{\lambda }A^{\lambda }g_{\mu \nu }\right] \right)
\end{equation*}
is the energy-momentum tensor of gauge field, $G_{\mu \nu }^{(1)}$ is just
the Einstein tensor, and $G_{\mu \nu }^{(2)}$ and $G_{\mu \nu }^{(3)}$ are
given as:
\begin{eqnarray*}
G_{\mu \nu }^{(2)} &=&2(-R_{\mu \sigma \kappa \tau }R_{\phantom{\kappa \tau
\sigma}{\nu}}^{\kappa \tau \sigma }-2R_{\mu \rho \nu \sigma }R^{\rho \sigma
}-2R_{\mu \sigma }R_{\phantom{\sigma}\nu }^{\sigma }+RR_{\mu \nu })-\frac{1}
2}\mathcal{L}_{2}g_{\mu \nu }, \\
G_{\mu \nu }^{(3)} &=&-3(4R^{\tau \rho \sigma \kappa }R_{\sigma \kappa
\lambda \rho }R_{\phantom{\lambda }{\nu \tau \mu}}^{\lambda }-8R_
\phantom{\tau \rho}{\lambda \sigma}}^{\tau \rho }R_{\phantom{\sigma
\kappa}{\tau \mu}}^{\sigma \kappa }R_{\phantom{\lambda }{\nu \rho \kappa
}^{\lambda }+2R_{\nu }^{\phantom{\nu}{\tau \sigma \kappa}}R_{\sigma \kappa
\lambda \rho }R_{\phantom{\lambda \rho}{\tau \mu}}^{\lambda \rho } \\
&&-R^{\tau \rho \sigma \kappa }R_{\sigma \kappa \tau \rho }R_{\nu \mu }+8R_
\phantom{\tau}{\nu \sigma \rho}}^{\tau }R_{\phantom{\sigma \kappa}{\tau \mu
}^{\sigma \kappa }R_{\phantom{\rho}\kappa }^{\rho }+8R_{\phantom
{\sigma}{\nu \tau \kappa}}^{\sigma }R_{\phantom {\tau \rho}{\sigma \mu
}^{\tau \rho }R_{\phantom{\kappa}{\rho}}^{\kappa } \\
&&+4R_{\nu }^{\phantom{\nu}{\tau \sigma \kappa}}R_{\sigma \kappa \mu \rho
}R_{\phantom{\rho}{\tau}}^{\rho }-4R_{\nu }^{\phantom{\nu}{\tau \sigma
\kappa }}R_{\sigma \kappa \tau \rho }R_{\phantom{\rho}{\mu}}^{\rho
}+4R^{\tau \rho \sigma \kappa }R_{\sigma \kappa \tau \mu }R_{\nu \rho
}+2RR_{\nu }^{\phantom{\nu}{\kappa \tau \rho}}R_{\tau \rho \kappa \mu } \\
&&+8R_{\phantom{\tau}{\nu \mu \rho }}^{\tau }R_{\phantom{\rho}{\sigma
}^{\rho }R_{\phantom{\sigma}{\tau}}^{\sigma }-8R_{\phantom{\sigma}{\nu \tau
\rho }}^{\sigma }R_{\phantom{\tau}{\sigma}}^{\tau }R_{\mu }^{\rho }-8R_
\phantom{\tau }{\sigma \mu}}^{\tau \rho }R_{\phantom{\sigma}{\tau }}^{\sigma
}R_{\nu \rho }-4RR_{\phantom{\tau}{\nu \mu \rho }}^{\tau }R_{\phantom{\rho
\tau }^{\rho } \\
&&+4R^{\tau \rho }R_{\rho \tau }R_{\nu \mu }-8R_{\phantom{\tau}{\nu}}^{\tau
}R_{\tau \rho }R_{\phantom{\rho}{\mu}}^{\rho }+4RR_{\nu \rho }R_
\phantom{\rho}{\mu }}^{\rho }-R^{2}R_{\nu \mu })-\frac{1}{2}\mathcal{L
_{3}g_{\mu \nu }.
\end{eqnarray*}
The metric of an $(n+1)$-dimensional asymptotically Lifshitz static
and radially symmetric
spacetime may be written as:
\begin{equation} \label{met1}
ds^{2}=-\frac{r^{2z}}{l^{2z}}f(r)dt^{2}+\frac{l^2 dr^{2}}{r^{2}g(r)
+r^{2}d\Omega ^{2}
\end{equation}
where the functions $f(r)$ and $g(r)$ must go to $1$ as $r$ goes to
infinity. In Eq. (\ref{met1}) $d\Omega ^{2}$ is the metric of an $(n-1)$
-dimensional hypersurface with constant curvature $(n-1)(n-2)k$ and volume $V_{n-1}$. We
can write
\begin{equation}
d\Omega ^{2}=\left\{
\begin{array}{cc}
d\theta _{1}^{2}+\sum\limits_{i=2}^{n-1}\prod\limits_{j=1}^{i-1}\sin
^{2}\theta _{j}d\theta _{i}^{2} & k=1 \\
d\theta _{1}^{2}+\sinh ^{2}\theta _{1}\left(d\theta
_{2}^{2}+\sum\limits_{i=3}^{n-1}\prod\limits_{j=2}^{i-1}\sin ^{2}\theta
_{j}d\theta _{i}^{2}\right) & k=-1 \\
\sum\limits_{i=1}^{n-1}d\theta _{i}^{2} & k=0
\end{array}
\right.
\end{equation}
though it should be noted that our solutions are valid whenever $d\Omega^2$
describes any Einstein space.
Using the ansatz
\begin{equation} \label{gauge}
A_{t}=q\frac{r^{z}}{l^{z}}h(r)
\end{equation}
for the gauge field and defining $\hat{\alpha }_{2}\equiv (n-2)(n-3)\alpha
_{2}$ and $\hat{\alpha }_{3}\equiv (n-2)...(n-5)\alpha _{3}$ for
convenience, the field equations (\ref{Geq}) and (\ref{EMeq}) reduce to the
system
\begin{eqnarray}
&& 2r^{2}h^{\prime \prime }- r\left[ (\ln f)^{\prime }-(\ln
g)^{\prime }\right]( rh^{\prime }+z) -2(n+z) rh^{\prime }+2(n-1)z=\frac{2m^{2}l^{2}}{g}, \label{E1} \\
&& r^{4}l^{4}\left\{ n(n-1)r^{2}g+(n-1)r^{3}g^{\prime }+2\Lambda
l^{2}r^{2}-(n-1)(n-2)kl^{2}\right\} \notag \\
&&+(n-1)\hat{\alpha }_{2}l^{2}r^{2}(kl^{2}-r^{2}g)\left\{
nr^{2}g+2r^{3}g^{\prime }-(n-4)kl^{2}\right\} \notag \\
&&+(n-1)\hat{\alpha }_{3}(kl^{2}-r^{2}g)^{2}\left\{ nr^{2}g+3r^{3}g^{\prime
}-(n-6)kl^{2}\right\}=2l^{6}r^{6}T_{t}^{t}, \label{E2} \\
&& r^{4}l^{4}\left\{ (n-1)(n-2+2z)r^{2}g+(n-1)r^{3}g(\ln f)^{\prime
}+2\Lambda l^{2}r^{2}-(n-1)(n-2)kl^{2}\right\} \notag \\
&& +(n-1)\hat{\alpha }_{2}l^{2}r^{2}(kl^{2}-r^{2}g)\left\{
(n-4+4z)r^{2}g+2r^{3}g(\ln f)^{\prime }-(n-4)kl^{2}\right\} \notag \\
&&+(n-1)\hat{\alpha }_{3}(kl^{2}-r^{2}g)^{2}\left\{
(n-6+6z)r^{2}g+3r^{3}g(\ln f)^{\prime }-(n-6)kl^{2}\right\} \notag \\
&& \hspace{10cm} =2l^{6}r^{6}T_{r}^{r}, \label{E3}
\end{eqnarray}
where prime denotes the derivative with respect to $r$ and $T_{t}^{t}$ and
T_{r}^{r}$ are
\begin{eqnarray*}
T_{t}^{t} &=&-\frac{q^{2}}{4l^{2}f}\left\{ g(rh^{\prime }+zh)^{2}+m^2
l^2h^{2}\right\} , \\
T_{r}^{r} &=&-\frac{q^{2}}{4l^{2}f}\left\{ g(rh^{\prime }+zh)^{2}-m^2
l^2h^{2}\right\} .
\end{eqnarray*}
\section{Lifshitz Solutions\label{Lif}}
\subsection{Vacuum Solutions}
We first investigate the possibility of having $(n+1)$-dimensional Lifshitz
solutions
\begin{equation}
ds^{2}=-\frac{r^{2z}}{l^{2z}}dt^{2}+\frac{l^2 dr^{2}}{r^{2}}+r^{2}\su
\limits_{i=1}^{n-1}d\theta _{i}^{2}, \label{met2}
\end{equation}
in Lovelock gravity in the absence of matter. In order to have an
asymptotically Lifshitz solution in 3rd-order Lovelock gravity, the
following constraints on the cosmological constant and third order Lovelock
coefficient
\begin{equation}
\Lambda = -\frac{n(n-1)}{6l^{4}}\left( 2l^{2}-\hat{\alpha }_{2}\right) ,
\qquad \hat{\alpha }_{3} = -\frac{l^{2}}{3}(l^{2}-2\hat{\alpha }_{2}).
\label{Coef2}
\end{equation}
hold for an arbitrary value of $z$, as is easily obtained via
straightforward calculation.
Note that for $\alpha _{3}=0$ the constraints (\ref{Coef2}) become
\begin{equation}
\Lambda =-\frac{n(n-1)}{4l^{2}}\text{ \ \ and \ \ }\hat{\alpha }_{2}=\frac
l^{2}}{2}. \label{Coef1}
\end{equation}
and so asymptotically Lifshitz solutions exist in the vacuum of Gauss-Bonnet
gravity ($\alpha _{3}=0$) as well. The cosmological constant here
is half that of an AdS spacetime. If we set $\Lambda=0$ then a Lifshitz
solution exists in third order Lovelock gravity provided
\begin{equation}
\hat{\alpha }_{2}=2l^{2},\text{ \ \ \ \ \ }\hat{\alpha }_{3}=l^{4}.
\label{Coef3}
\end{equation}
Thus, one may have Lifshitz solutions in Lovelock gravity without matter,
demonstrating that its higher curvature terms can,
for the proper choice of Lovelock coefficients, have the desired effect
that matter fields induce.
The preceding analysis was for $k=0$. For $k=\pm 1$ we find for any value of
$z$ that
\begin{equation}
g(r)=1+\frac{kl^{2}}{r^{2}} \label{exa}
\end{equation}
furnishes an exact Lifshitz solution to Lovelock gravity without matter, provided one
of the above conditions (\ref{Coef2}), (\ref{Coef1}) is satisfied, with the function $f(r)$
undetermined by the field equations (though boundary conditions constrain it to
asymptote to 1). We can choose $f(r)=g(r)$. This is a
naked singularity for $k=1$ but is an asymptotically Lifshitz black hole for
$k=-1$. The arbitrariness of $f(r)$ is due to a degeneracy of the field equations; if either of (\ref{Coef2}) or (\ref{Coef1}) hold then the field equations are each proportional to the factor $g(r)-1-kl^2/r^2$.
This degeneracy of the field equations has been noted previously in 5-dimensional Einstein-Gauss-Bonnet gravity with a cosmological constant \cite{Oliva}, where (converting notation appropriately) condition (\ref{Coef1}) was obtained. Here we see that a more general degeneracy occurs in 3rd order Lovelock gravity. We shall see that this degeneracy is lifted when matter is present. We find that no other exact solutions to the field equations exist for
these symmetries and asymptotic behavior.
\subsection{Matter Solutions}
Consider next the case of Lifshitz solutions in the presence of a massive
gauge field $A^{\mu }$. The metric (\ref{met2}) and the gauge field (\ref
{gauge}) with $h(r)=1$ solve the field equation (\ref{E1}) provided
\begin{equation}
m^{2}=\frac{(n-1)z}{l^{2}}. \label{mc}
\end{equation}
In order to have asymptotically Lifshitz solutions in 3rd-order Lovelock
gravity in the presence of matter, the following constraint
\begin{eqnarray}
q^{2} &=&\frac{2(z-1)L^{4}}{zl^{4}}, \notag \\
\Lambda &=&-\frac{[(z-1)^2+n(z-2)+n^2]L^4+n(n-1)(\hat{\alpha}_{2}l^2-2\hat{\alpha}_{3})}{2l^{6}} \label{Coef5} \label{Coef5}
\end{eqnarray}
must hold for the cosmological constant and charge where we define
\begin{equation}
L^{4}\equiv l^{4}-2l^{2}\hat{\alpha}_{2}+3\hat{\alpha}_{3} \label{L4}
\end{equation}
for simplicity and we use this definition throughout the paper. Since $q^2>0$ we obtain the constraint
\begin{equation}
\frac{\hat{\alpha}_{2}}{l^{2}} < \frac{1}{2}+3\frac{\hat{\alpha}_{3}}{2 l^{4}} \label{a2}
\end{equation}
which in turn yields
\begin{equation}
-\frac{(z-1)(n+z-1)(l^4+3 \hat{\alpha}_3)+n(n-1)(l^4+ \hat{\alpha}_3)}{2l^6}\leq
\Lambda< - \frac{n(n-1)}{4 l^2}\left(1- \frac{\hat{\alpha}_{3}}{ l^{4}}\right) \label{LamCons}
\end{equation}
provided $\hat{\alpha}_{2} \geq 0$. Equation (\ref{LamCons}) shows that for $\hat{\alpha}_{3}<l^4$ the cosmological constant
is negative. For $\hat{\alpha}_{3}>l^4$, the cosmological constant can be positive
provided
\[
\frac{(z-1)(n+z-1)(l^4+3 \hat{\alpha}_3)+n(n-1)(l^4+ \hat{\alpha}_3)}{[2(z-1)(z+n-1)+n(n-1)]l^4}
< \frac{\hat{\alpha}_{2}}{l^{2}} < \frac{1}{2}+3\frac{\hat{\alpha}_{3}}{2 l^{4}},
\]
where the last inequality comes from the condition (\ref{a2}).
Note that for
Einstein gravity ($\hat{\alpha}_{2}=\hat{\alpha}_{3}=0$) $L=l$ and the above
conditions reduce to those which are given in \cite{Peet} for $n=3$. In
Gauss-Bonnet gravity ($\alpha _{3}=0$), these conditions become
\begin{eqnarray}
q^{2} &=&\frac{2(z-1)(l^{2}-2\hat{\alpha}_{2})}{zl^{2}}, \notag \\
\Lambda &=&-\frac{[(z-1)^2+n(z-2)+n^2](l^{2}-2\hat{\alpha}_{2})+n(n-1)\hat{\alpha}_{2}}{2l^{4}}. \label{Coef4}
\end{eqnarray}
where now $l^{2}$ must be larger than $2\hat{\alpha}_{2}$. If $\hat{\alpha
_{2}=l^{2}/2$, then the charge $q$ becomes zero and the conditions (\ref
{Coef4}) reduce to the conditions (\ref{Coef1}) as expected. More generally,
if $L=0$, then $q$ vanishes and the conditions (\ref{Coef5}) reduce to the
conditions (\ref{Coef2}) as expected.
In the absence of a cosmological constant, the Lifshitz solution exists
provided
\begin{eqnarray}
q^{2} &=&\frac{2n(n-1)(z-1)(\hat{\alpha}_{2}-2l^{2})}
zl^{2}[3(z-1)^{2}+3nz+n(n-4)]}, \notag \\
\hat{\alpha}_{3} &=&\frac{(2\hat{\alpha}_{2}-l^{2})\left[ (z-1)^{2}+nz\right]
+n\left[ (n-3)\hat{\alpha}_{2}-(n-2)l^{2}\right] }{[3(z-1)^{2}+3nz+n(n-4)]}l^2. \label{Coef6}
\end{eqnarray}
In this case the conditions (\ref{Coef6}) reduce to the condition (\ref
{Coef3}) if $\hat{\alpha}_{2}=2l^{2}$, leading to $q=0$ and the absence of
matter.
\section{Asymptotic Lifshitz Black holes \label{black}}
In this section we seek black hole solutions in 3rd-order Lovelock gravity
that are asymptotically Lifshitz. We therefore consider the field equations
\ref{E1})-(\ref{E3}) with the conditions (\ref{Coef5}), but with the more
general ansatz (\ref{met1}) and (\ref{gauge}).
\subsection{Series solutions near the horizon}
We now consider the near-horizon behavior of such solutions. Requiring that
$f(r)$ and $g(r)$ go to zero linearly, that is
\begin{eqnarray*}
f(r)
&=&f_{1}\left\{(r-r_{0})+f_{2}(r-r_{0})^{2}+f_{3}(r-r_{0})^{3}+f_{4}(r-r_{0})^{4}+...\right\},
\\
g(r)
&=&g_{1}(r-r_{0})+g_{2}(r-r_{0})^{2}+g_{3}(r-r_{0})^{3}+g_{4}(r-r_{0})^{4}+...,
\\
h(r)
&=&f_{1}^{1/2}\left\{h_{0}+h_{1}(r-r_{0})+h_{2}(r-r_{0})^{2}+h_{3}(r-r_{0})^{3}+h_{4}(r-r_{0})^{4}+...\right\},
\end{eqnarray*}
and inserting these expansions into the equations of motion arising from
Eqs. (\ref{E1})-(\ref{E2}) with the conditions (\ref{Coef5}), and solving
for the various coefficients, we find that $h_{0}=0$. This is consistent
with the fact that the flux $dA$ should go to a constant at the horizon.
Also, one may note that by scaling time we can adjust the constant $f_{1}$\
by an overall multiplicative factor (note the use of $f_{1}^{1/2}$\ in the
expansion of $h(r)$\ as well, which is due to $dt$\ in the one-form $A$).
We find the following constraint on the $1$st order constants
\begin{eqnarray}
g_{1} &=&\frac{z}{r_{0}^{3}}\Big\{3\hat{\alpha}_{3}\left[ {h_{1}}^{2}(z-1)
r_{0}}^{5}+{k}^{2}z(n-1){l}^{4}\right] \notag \\
&&+2{r_{0}}^{2}l^{2}\hat{\alpha}_{2}\left[ -{h_{1}}^{2}(z-1){r_{0}}^{3}+{l
^{2}kz(n-1)\right] +{r_{0}}^{4}{l}^{4}\left[ {h_{1}}^{2}\left( z-1\right)
r_{0}+z\left( n-1\right) \right] \Big\}^{-1} \notag\\
&&\times \Big\{\left\{ \left[ 3(z-1)^{2}+3nz++n(n-4)\right] {r_{0}}^{6}+{k
(n-1)(n-6){l}^{6}\right\} \hat{\alpha}_{3} \notag\\
&&-{{r_{0}}}^{2}{l}^{2}\hat{\alpha}_{2}\left\{ \left[ 2(z-1)^{2}+2nz+n(n-3
\right] {r_{0}}^{4}+{k}^{2}{l}^{4}(n-1)(n-4)\right\} \notag \\
&&+{r_{0}}^{4}{l}^{4}\left[ (z-1)^{2}+nz+n(n-1)\right] {r_{0}
^{2}+(n-1)(n-2)k{l}^{2}\Big\},
\end{eqnarray}
which means that not all boundary conditions are allowed. Note that here the
coefficient $h_{1}$ is arbitrary, and should be chosen suitable in order to
explore the numerical solutions. The higher order coefficients may be found
easily, but their expressions are very lengthy and so we don't write them here for reasons of economy.
One may also investigate the behavior of the metric functions at large $r$.
We relegate this to the Appendix, where we show that the powers of $1/r$ for $k=0$ are in general
non-integer.
\subsection{Numeric Solutions}\label{numsol}
In order to find the numeric black hole solutions of the system given by
Eqs. (\ref{E1})-(\ref{E3}) with the conditions (\ref{Coef5}), we define
\begin{equation}
\frac{dh}{dr}\equiv j(r), \label{dhr}
\end{equation}
and find the first derivatives of $j(r)$, $f(r)$ and $g(r)$ as
\begin{equation}
\frac{dj}{dr} =\frac{(n-1)zh-\left[ z(n+z-2)h+(n+2z-1)rj\right] g}{r^{2}g}
\frac{(z-1)L^{4}(zh+rj)r^{4}h^{2}}{r^{2}fgH}, \label{djr}
\end{equation}
\begin{eqnarray}
\frac{df}{dr} &=&\frac{1}{(n-1)zr^{3}gH}\Big\{-(n-1)[n+6(z-1)]z\hat{\alpha
_{3}r^{6}fg^{3} \notag\\
&& +(n-1)zl^{2}r^{4}\left[ 3k\hat{\alpha}_{3}(n+4z-6)+(n+4z-4
\hat{\alpha}_{2}r^{2}\right] fg^{2} \notag \\
&&-(n-1)zl^{4}r^{2}\left[ 3(n+2z-6)k^{2}\hat{\alpha}_{3}+2(n+2z-4)k\hat
\alpha}_{2}r^{2}+(n+2z-2)r^{4}\right] fg \notag \\
&&+z\hat{\alpha}_{3}\left\{[3(z-1)^{2}+3nz+n(n-4)]r^{6}+(n-1)(n-6)kl^{6}\right\}f \notag \\
&&-z\hat{\alpha}_{2}l^{2}r^{2}\left\{
[2(z-1)^{2}+2nz+n(n-3)]r^{4}-(n-1)(n-4)k^{2}l^{4}\right\} f \notag \\
&&+zl^{4}r^{4}\left\{ [(z-1)^{2}+nz+n(n-1)]r^{2}+(n-1)(n-2)kl^{2}\right\} f
\notag \\
&&-(z-1)L^{4}r^{6}[(zh+rj)^{2}g+(n-1)zh^{2}]\Big\},
\end{eqnarray}
\begin{eqnarray}
\frac{dg}{dr} &=&\frac{1}{(n-1)zr^{3}fH}\Big\{-n(n-1)z\hat{\alpha
_{3}r^{6}fg^{3}+(n-1)zl^{2}r^{4}\left[ 3k\hat{\alpha}_{3}(n-2)+n\hat{\alpha
_{2}r^{2}\right] fg^{2} \notag \\
&&-(n-1)zl^{4}r^{2}\left[ 3(n-4)k^{2}\hat{\alpha}_{3}+2(n-2)k\hat{\alpha
_{2}r^{2}+nr^{4}\right] fg \notag \\
&&+z\hat{\alpha}_{3}\left\{[3(z-1)^{2}+3nz+n(n-4)]r^{6}+(n-1)(n-6)kl^{6}\right\}f \notag \\
&&-z\hat{\alpha}_{2}l^{2}r^{2}\left\{
[2(z-1)^{2}+2nz+n(n-3)]r^{4}-(n-1)(n-4)k^{2}l^{4}\right\} f \notag \\
&&+zl^{2}r^{2}\left\{ [(z-1)^{2}+nz+n(n-1)]r^{2}+(n-1)(n-2)kl^{2}\right\} f
\notag \\
&&-(z-1)L^{4}r^{6}[(zh+rj)^{2}g+(n-1)zh^{2}]\Big\}, \label{dgr}
\end{eqnarray}
where $H$ is
\begin{equation*}
H=3\hat{\alpha}_{3}(kl^{2}-r^{2}g)^{2}+\hat{\alpha}_
l^{2}r^{2}(kl^{2}-r^{2}g)+l^{4}r^{4}.
\end{equation*}
Now having a system of 4 first order ordinary differential equations, one
may explore the numeric solutions by choosing suitable initial conditions
for $f(r_{0})$, $g(r_{0})$, $h(r_{0})$ and $j(r_{0})$, where $r_{0}$ is the
radius of horizon. This can be done by using the method explained in \cite
{Mann}.
\begin{figure}[h]
\epsfxsize=10cm \centerline{\epsffile{fz1k.eps}}
\caption{$f(r)$, $g(r)$ and exact solution versus $r$ for $n=6$, $l=1$, $z=1$, $\hat{\protect\alpha}_{2}=.25$, $\hat{\protect\alpha
_{3}=.3$ and $r_{0}=0.85$ for $k=1$ (top),
$k=0$ (middle) and $k=-1$ (bottom). Note that $f(r)$, $g(r)$ and the exact solution
lie on each other for each $k$.}
\label{Fz1k}
\end{figure}
First, we apply the numerical method for $z=1$ which can be solved
exactly, and compare the numerical solution to the exact one.
Indeed, for $z=1$, the charge vanishes, and the problem reduces to
the case of black holes in third order Lovelock gravity with specific
choices of Lovelock coefficients. In this case, we obtain the exact
solutions of Lovelock gravity \cite{Deh}, which have $f(r)=g(r)$. The diagrams of
the functions $f(r)$ and $g(r)$ versus $r$
for $z=1$ with $k=0$ and $k=\pm1$ have been shown in Fig \ref{Fz1k}. In this figure
the exact solutions have also been shown. As one can see from the plots, the
numerically obtained functions $f(r)$
and $g(r)$ are equal and both of them are exactly lie on the exact solution within limits of numerical precision.
We regard this as a good test of our methods.
Note that for $k=0,-1$ and $z=1$ (the solution without matter) , the metric
functions increase from zero at $r=r_{0}$ to $1$ at $r=\infty $. However for $k=1$ and
$z=1$, the metric functions grow to
be larger than unity at intermediate values of $r$ and
asymptote to 1 at infinity.
\begin{figure}[h]
\textit{\epsfxsize=10cm \centerline{\epsffile{fskm1.eps}} }
\caption{$f(r)$ versus $r$ for $k=-1
, $n=6$, $l=1$, $\hat{\protect\alpha}_{2}=.25$, $\hat{\protect\alpha
_{3}=.3$, $r_{0}=0.92$ in 3rd order Lovelock,
Gauss-Bonnet and Einstein gravities red (solid), green (dotted) and blue (dashed), respectively.
The three upper curves are $f(r)$ for $z=2$ and the three lower curves,
which are nearly identical and so lie on top of each other, correspond to $z=1$.}
\label{Fskm1}
\end{figure}
\begin{figure}[h]
\textit{\epsfxsize=10cm \centerline{\epsffile{fLkm1.eps}} }
\caption{$f(r)$ versus $r$ for $k=-1
, $n=6$, $l=1$, $\hat{\protect\alpha}_{2}=.25$, $\hat{\protect\alpha
_{3}=.3$, $r_{0}=8.5$ in 3rd order Lovelock,
Gauss-Bonnet and Einstein gravities red (solid), green (dotted) and blue (dashed), respectively.
The three upper curves are $f(r)$ for $z=2$ and the three lower curves
correspond to $z=1$.}
\label{Flkm1}
\end{figure}
\begin{figure}[h]
\textit{\epsfxsize=10cm \centerline{\epsffile{fsk0.eps}} }
\caption{$f(r)$ versus $r$ for $k=0
, $n=6$, $l=1$, $\hat{\protect\alpha}_{2}=.25$, $\hat{\protect\alpha
_{3}=.3$, $r_{0}=0.85$ in 3rd order Lovelock,
Gauss-Bonnet and Einstein gravities red (solid), green (dotted) and blue (dashed), respectively.
The three upper curves are $f(r)$ for $z=2$ and the three lower curves
correspond to $z=1$.}
\label{Fsk0}
\end{figure}
\begin{figure}[h]
\textit{\epsfxsize=10cm \centerline{\epsffile{fLk0.eps}} }
\caption{$f(r)$ versus $r$ for $k=0
, $n=6$, $l=1$, $\hat{\protect\alpha}_{2}=.25$, $\hat{\protect\alpha
_{3}=.3$, $r_{0}=8.5$ in 3rd order Lovelock,
Gauss-Bonnet and Einstein gravities red (solid), green (dotted) and blue (dashed), respectively.
The three upper curves are $f(r)$ for $z=2$ and the three lower curves
correspond to $z=1$.}
\label{Flk0}
\end{figure}
\begin{figure}[tbp]
\textit{\epsfxsize=10cm \centerline{\epsffile{fsk1.eps}} }
\caption{$f(r)$ versus $r$ for $k=1
, $n=6$, $l=1$, $\hat{\protect\alpha}_{2}=.25$, $\hat{\protect\alpha
_{3}=.3$, $r_{0}=0.85$ in 3rd order Lovelock,
Gauss-Bonnet and Einstein gravities red (solid), green (dotted) and blue (dashed), respectively.
The three upper curves are $f(r)$ for $z=2$ and the three lower curves
correspond to $z=1$.}
\label{Fsk1}
\end{figure}
\begin{figure}[tbp]
\textit{\epsfxsize=10cm \centerline{\epsffile{fLk1.eps}} }
\caption{$f(r)$ versus $r$ for $k=1
, $n=6$, $l=1$, $\hat{\protect\alpha}_{2}=.25$, $\hat{\protect\alpha
_{3}=.3$, $r_{0}=8.5$ in 3rd order Lovelock,
Gauss-Bonnet and Einstein gravities red (solid), green (dotted) and blue (dashed), respectively.
The three upper curves are $f(r)$ for $z=2$ and the three lower curves
correspond to $z=1$. }
\label{Flk1}
\end{figure}
Now, we consider the solutions for $z\neq 1$, for which matter
exists, where we must solve the system of equations numerically.
Figure \ref{Fskm1} shows the function $f(r)$ as a function of $r$ for a small black
hole of Einstein, Gauss-Bonnet and Lovelock gravity with $z=1$ and $z=2$. As one can see from this figure,
while $f(r)$ for Einstein, Gauss-Bonnet and Lovelock with $z=1$ (solution without matter) are almost the
same, they are different for $z=2$ for the same values of $\alpha_2$ and $\alpha_3$.
Furthermore, for $z=2$,
the metric function $f(r)$ rapidly increases to values larger than 1,
eventually asymptoting to $1$ as $r\rightarrow \infty $, while for $z=1$
this function monotonically increases from zero at $r=r_{0}$ to $1$ at $r=\infty $. Thus, it appears that
asymptotic Lifshitz solutions are more sensitive than are their $z=1$ (AdS) counterparts
to the corrections induced by Lovelock gravity. This feature
also occurs for large black holes with $k=-1$ (see Fig. \ref{Flkm1}).
Note that for $k=-1$, the function $f(r)$ in Gauss-Bonnet gravity is larger than
that in third order Lovelock gravity. This is due to the fact that
the slope of the function $f(r)$ decreases as $\hat{\alpha}_3$ increases (see the terms
in $df/dr$ with the factor $\hat{\alpha}_3$ in Eq. (\ref{dgr}) with $k=-1$, which are negative).
Figures \ref{Fsk0}-\ref{Flk0} show that this fact also occurs for the small and large black holes with $k=0$.
For the case of $k=1$, one can see from Figs \ref{Fsk1} and \ref{Flk1} that both the asymptotic
AdS and Lifshitz black holes are different for various order of Lovelock
gravities, although the difference between the metric function $f(r)$ for various
order of Lovelock gravity becomes more relevant as $z$ increases.
We find that the dependence on $z$ of $g(r)$
is not as significant as for $f(r)$. For this and reasons of economy, we plot only
the function $g(r)$ in one case (see Fig. \ref{gskm1}). We also find that
$g(r)$ is monotonically increasing for topological black holes with $k=0$, whereas (as noted previously)
$f(r)$ has a maximum larger than $1$ at $r$ several times $r_0$.
\begin{figure}[tbp]
\textit{\epsfxsize=10cm \centerline{\epsffile{gskm1.eps}} }
\caption{$g(r)$ versus $r$ for $k=-1
, $n=6$, $l=1$, $\hat{\protect\alpha}_{2}=.25$, $\hat{\protect\alpha
_{3}=.3$, $r_{0}=0.85$ in 3rd order Lovelock,
Gauss-Bonnet and Einstein gravities red (solid), green (dotted) and blue (dashed), respectively.
The three upper curves are $f(r)$ for $z=2$ and the three lower curves,
which lie on each other, correspond to $z=1$.}
\label{gskm1}
\end{figure}
We pause to comment on the relative importance of the various terms in the Lovelock action.
While there is no essential problem in considering all terms to be similar in magnitude, we obtain
some insight into Lovelock gravity by treating it as an approximation (in powers of the curvature)
to a full quantum theory of gravity. For simplicity we shall consider solutions
obtained in this paper for $k=0$, which asymptote to the Lifshitz background (\ref{met2}). In this case the $p$th order Lagrangian of Lovelock gravity for the metric (\ref{met1}) may be written as:
\begin{equation}
\mathcal{L}'_p=(-1)^{p-1}\frac{(n-1)}{(2p-1)l^{z+2p-1}r^{(2p-1)(z-1)}}\left(\frac{g}{f}\right)^{p-1/2}
\frac{d}{dr}\left\{r^{n+2p(z-1)}f^p\right\}, \label{Lagp}
\end{equation}
where $\mathcal{L}'_p=(\alpha_p/\hat{\alpha}_p)\sqrt{-g}\mathcal{L}_p$. The above expression for the Lagrangian is correct for any value of $p$, though in this paper we have considered only
$p\leq 3$. The Lagrangian (\ref{Lagp}) can be evaluated on the solutions at large $r$ by the use of the expansions of the metric functions which are given in the Appendix. It is a matter of calculation to show that the ratio $\mathcal{L}'_{p+1}/\mathcal{L}'_{p}$ at large $r$ is
\begin{equation}
\frac{\mathcal{L}'_{p+1}}{\mathcal{L}'_{p}}=-\frac{(2p-1)[n+2(p+1)(z-1)]}{(2p+1)[n+2p(z-1)]l^2},
\end{equation}
and so ${\mathcal{L}'_{p+1}}<{\mathcal{L}'_{p}}/l^2$.
\section{Thermodynamics of black holes \label{Therm}}
The entropy of a black hole in Lovelock gravity is \cite{Wald}
\begin{equation}
S=\frac{1}{4}\sum_{k=1}^{p}k\alpha _{k}\int d^{n-1}x\sqrt{\tilde{g}}\tilde
\mathcal{L}}_{k-1}, \label{Enta}
\end{equation}
where the integration is done on the $(n-1)$-dimensional spacelike
hypersurface of the Killing horizon with induced metric $\tilde{g}_{\mu \nu }$
(whose determinant is $\tilde{g}$), and
\tilde{\mathcal{L}}_{k}$ is the $k$th order Lovelock Lagrangian of $\tilde{g
_{\mu \nu }$. It is a matter of calculation to show that the entropy
of a black hole per unit volume of the horizon in third order Lovelock
gravity is
\begin{equation}
S=\frac{r_{0}^{n-1}}{4}\left( 1+\frac{2k(n-1)\hat{\alpha}_{2}}{(n-3)r_{0}^{2
}+\frac{3k^{2}(n-1)\hat{\alpha}_{3}}{(n-5)r_{0}^{4}}\right). \label{Ent}
\end{equation}
This reduces to the area law of entropy for $\hat{\alpha}_{2}=\hat{\alpha
_{3}=0$.
One can obtain the temperature of the event horizon by using standard
Wick-rotation methods, yielding the result
\begin{equation}
T=\left(\frac{r^{z+1}\sqrt{f^{\prime }g^{\prime }}}{4 \pi l^{z+1}}\right)_{r=r_0}.
\end{equation}
For the case of asymptotic Lifshitz black hole with (identified) hyperbolic horizon, the
temperature of the exact solution (\ref{exa}) with horizon radius $r_{0}=l$ in vacuum
is
\begin{equation}
T=\frac{1}{2\pi l}.
\end{equation}
The temperature of more general Lovelock-Lifshitz black holes can be
calculated numerically. First, we review the thermodynamics of asymptotic AdS
black holes ($z=1$) to compare it with the asymptotic Lifshitz black holes. As one can see from figure \ref{TKz1},
the logarithm of the temperature of black holes versus the logarithm of the entropy
for large black holes is linear, while for small black holes this occurs only
in the case of $k=0$. For $k=-1$, one encounters with an extreme black hole in Einstein (blue dashed line)
and Lovelock gravity (blue solid line). However, the horizon radius of the extreme black
hole of Lovelock gravity is smaller than that of Einstein gravity. Note that
the slopes of the graphs of $\log T$ versus $\log S$ for small black holes of Einstein and
Lovelock gravity with $k=1$ are negative, and therefore they are unstable. The unstable phase in Lovelock gravity occurs for a smaller radius black hole than in Einstein theory.
\begin{figure}[tbp]
\textit{\epsfxsize=10cm \centerline{\epsffile{Tz1.eps}} }
\caption{$\log T$ versus $\log S$ for $n=6$, $l=1$, $z=1$ in Einstein
and Lovelock ($\hat{\protect\alpha}_{2}=.25$, $\hat{\protect\alpha
_{3}=.3$) gravities with $k=1$ (red), $k=0$ (green) and $k=-1$
(blue), where the dashed and solid lines are curves in Einstein and Lovelock gravities, respectively.}
\label{TKz1}
\end{figure}
Plotting $\log T$ versus $\log S$ for the
case of asymptotic Lifshitz black holes with $z=2$, we see from Fig. \ref{TKz2} that
the temperature of black holes in Lovelock gravity with given entropy $S$ is smaller than
the temperature of black holes of Einstein gravity with the same entropy. Conversely, at a given
temperature the entropy of the Lovelock black holes -- extremal and non-extremal -- is larger than for the Einstein ones for both $z=1$ and $z=2$.
The horizon radii of extreme black holes in both Einstein and Lovelock gravity for $k=-1$ with $z=2$ are smaller than their $z=1$ counterparts, and
for a given $z$ smaller in Lovelock gravity than in Einstein gravity.
Numerical calculations show that there is no unstable phase for $z=2$ black holes for $k=1$.
\begin{figure}[tbp]
\textit{\epsfxsize=10cm \centerline{\epsffile{Tz2.eps}} }
\caption{$\log T$ versus $\log S$ for $n=6$, $l=1$, $z=2$ in Einstein
and Lovelock ($\hat{\protect\alpha}_{2}=.25$, $\hat{\protect\alpha
_{3}=.3$) gravities with $k=1$ (red), $k=0$ (green) and $k=-1$
(blue), where the dashed and solid lines are curves in Einstein and Lovelock gravities, respectively.}
\label{TKz2}
\end{figure}
We also find that for $k=0$ and the large black
holes for $k=\pm 1$, the temperature is proportional to $r_0^z$, while the entropy is proportional
to $r_0^{n-1}$, yielding
\begin{equation}
T \varpropto S^{z/(n-1)}.
\end{equation}
for the dependence of $T$ on $S$.
In order to say more about the temperature, one should compute the
energy density of these black holes using (for example) the counterterm method. While some work has been done on this for asymptotic Lifshitz solutions of Einstein gravity \cite{Ros,Peet2}, the corresponding formalism needs to be developed for Lovelock gravity, an endeavour that we hope to address in the future.
\section{Rotating Lovelock-Lifshitz solutions}
\label{rot} In this section we endow the Lifshitz spacetime with a global
rotation. We first consider the Lifshitz solution with one rotation
parameter. We write the metric as
\begin{equation}
ds^{2}=-\frac{r^{2z}}{l^{2z}}dt^{2}+\frac{l^2 dr^{2}}{r^{2}}+r^{2}d\theta
_{1}^{2}+r^{2}\sum\limits_{i=2}^{n-1}d\theta _{i}^{2}, \label{metR1}
\end{equation}
In order to add angular momentum to the spacetime, we perform the following
rotation boost in the $t$-$\theta _{1}$ plane:
\begin{equation}
t\mapsto \Xi t-a\theta _{1}\ \ \ \ \ \ \ \ \ \ \theta _{1}\mapsto \Xi \theta
_{1}-\frac{a}{l^{2}}t, \label{tth}
\end{equation}
where $a$ is the rotation parameter and $\Xi =1+a^{2}/l^{2}$. Substituting
eq. (\ref{tth}) into eq. (\ref{metR1}) we obtain
\begin{equation}
ds^{2}=-\frac{r^{z}}{l^{z}}\left( \Xi dt-ad\theta _{1}\right) ^{2}+\frac
l^2 dr^{2}}{r^{2}}+r^{2}\left( \frac{a}{l^{2}}dt-\Xi d\theta _{1}\right)
^{2}+r^{2}\sum\limits_{i=2}^{n-1}d\theta _{i}^{2}. \label{metR2}
\end{equation}
The transformation (\ref{tth}) generates a new metric if $\theta_1$ is periodically
identified since the transformation (\ref{tth}) can be done
locally but not globally \cite{Stach}. The periodic
nature of $\theta _{1}$ allows the metrics
\ref{metR1}) and (\ref{metR2}) to be locally mapped into each other but not
globally, and so they are distinct. It is a matter of straightforward calculation to
show that the metric (\ref{metR2}) is a solution of Lovelock gravity,
provided one of the conditions (\ref{Coef2})-(\ref{Coef1}) holds.
Furthermore the metric (\ref{metR2}) is also a solution to the field equations in the
presence of the vector field
\begin{equation}
A=q\frac{r^{z}}{l^{z}}\left( \Xi dt-ad\theta _{1}\right) ,
\end{equation}
provided the condition (\ref{mc}) and one of the conditions (\ref{Coef4})-
\ref{Coef6}) are satisfied.
It is straightforward to generalize the metric (\ref{metR2}) to a metric with more rotation parameters. The rotation group in $(n+1)$ dimensions is $SO(n)$, and therefore the number of independent rotation
parameters is $[n/2]$. The generalized solution with $m\leq \lbrack n/2]$
rotation parameters can be written as
\begin{eqnarray}
ds^{2} &=&-\frac{r^{z}}{l^{z}}\left( \Xi dt-{{\sum_{i=1}^{m}}}a_{i}d\theta
_{i}\right) ^{2}+\frac{r^{2}}{l^{4}}{{\sum_{i=1}^{m}}}\left( a_{i}dt-\Xi
l^{2}d\theta _{i}\right) ^{2} \notag \\
&&\ +\frac{l^2 dr^{2}}{r^{2}}-\frac{r^{2}}{l^{2}}{\sum_{i<j}^{m}}(a_{i}d\theta
_{j}-a_{j}d\theta _{i})^{2}+r^{2}\sum\limits_{i=m+1}^{n-1}d\theta _{i}^{2},
\label{metR3}
\end{eqnarray}
where $\Xi =\sqrt{1+\sum_{i}^{m}a_{i}^{2}/l^{2}}$ and the angular
coordinates are in the range $0\leq \theta _{i}<2\pi $.
This generalization can be extended to the Lovelock-Lifshitz
$k=0$ black hole since the transformation (\ref
{tth}) does not change the $r$-dependence of the metric functions and the
vector field. Hence
\begin{eqnarray}
ds^{2} &=&-\frac{r^{z}}{l^{z}}f(r)\left( \Xi dt-{{\sum_{i=1}^{k}}
a_{i}d\theta _{i}\right) ^{2}+\frac{r^{2}}{l^{4}}{{\sum_{i=1}^{k}}}\left(
a_{i}dt-\Xi l^{2}d\theta _{i}\right) ^{2} \notag \\
&&\ +\frac{l^2 dr^{2}}{r^{2}g(r)}-\frac{r^{2}}{l^{2}}{\sum_{i<j}^{k}
(a_{i}d\theta _{j}-a_{j}d\theta
_{i})^{2}+r^{2}\sum\limits_{i=k+1}^{n-1}d\theta _{i}^{2}, \label{metR4}
\end{eqnarray}
with the vector field
\begin{equation}
A=q\frac{r^{z}}{l^{z}}h(r)\left( \Xi dt-{{\sum_{i=1}^{k}}}a_{i}d\theta
_{i}\right) ,
\end{equation}
is a rotating black hole solution to
the field equations provided $f(r)$, $g(r)$ and $h(r)$ are chosen
to be the functions calculated in section \ref{numsol} for $k=0$.
\section{Concluding Remarks}
It is known that Einstein gravity in the presence of a massive vector field can support Lifshitz solutions \cite{Kach}. We have demonstrated here that such solutions exist in pure Lovelock gravity provided the coupling parameters are chosen appropriately. The higher curvature terms appear to play the role of some kind of matter field, whose nature depends on the constants of the theory.
We also found for any value of $z$ an exact vacuum asymptotically Lifshitz solution (\ref{exa}) for $k=\pm 1$, which is a black hole for $k=-1$, and a naked singularity for $k=1$. This solution exists in both Gauss-Bonnet and 3rd order Lovelock gravity, depending on the choice of coupling.
We also obtained a broad class of solutions for Lovelock gravity coupled to a massive vector field.
After demonstrating that Lovelock gravity can support a
Lifshitz solution in the presence of a massive vector field, we searched for asymptotic
Lifshitz black holes in the presence of a massive vector field. Our numerically obtained solutions
generalize those obtained in Einsteinian gravity coupled to
a massive vector field \cite{Mann,Peet,Dan}. We found that asymptotic Lifshitz solutions ($z>1$) are more sensitive to the corrections induced by Lovelock gravity than are their $z=1$ counterparts.
We also considered the thermodynamics of the black hole solutions. We found that, as in
the case of asymptotically AdS black holes of Lovelock gravity, one can have
an extreme black hole only for $k=-1$. However the horizon radius
of both the extreme black holes of Einstein and Lovelock gravity for $k=-1$ with $z=2$ are smaller than their $z=1$ counterparts. That is, the radius of the extreme black holes decreases as $z$ increases both in Einstein and Lovelock gravity. Also, Lovelock terms decrease the radius of
extremal black holes compared to their Einsteinian counterparts.
The temperature of a Lovelock-Lifshitz black hole with entropy $S$ is smaller than
the temperature of a Lifshitz black hole in Einstein gravity with the same entropy.
We also found, numerically,
that as $z$ increases the temperature of a black hole with a fixed
horizon radius increases. Indeed, the temperature is proportional to
$r_0^z$ for black holes with zero curvature horizon and also for large
black holes with nonzero horizon curvature. In these two cases the temperature
is proportional to $S^{z/(n-1)}$.
We investigated only the cases with $z\geq 2$
and found that these solutions are thermodynamically stable. This fact can be seen by
diagrams of temperature versus the entropy. This feature of asymptotically
Lifshitz black holes is different from the asymptotically AdS solutions that can have an
unstable phase.
Our results demonstrate that higher-order curvature corrections to ``Lifshitz holography" are under control and have a sensible physical interpretation. However there is much remaining to explore for these kinds of black holes, including the corrections they induce in the dual boundary theory and a more detailed study of the behavior for larger $n$ and larger $z$. The rotating $k=0$ solutions should have counterparts for
the $k=\pm 1$ cases. Work on these areas is in progress.
\section{Appendix}
Here, we investigate the behavior of the metric functions at large $r$. For
the case of $k=0$ the powers of $1/r$ may be non-integer, and therefore we
consider it separately. For this case, we use straightforward perturbation theory, writing
\begin{eqnarray*}
f(r) &=&1+\varepsilon f_{1}(r), \\
g(r) &=&1+\varepsilon g_{1}(r), \\
h(r) &=&1+\varepsilon h_{1}(r),
\end{eqnarray*}
and then finding the field equations up to the first order in $\varepsilon $. We obtain
\begin{eqnarray*}
0 &=&2r^{2}h_{1}^{\prime \prime }+2(n+z)rh_{1}^{\prime }+zr\left(
g_{1}^{\prime }-f_{1}^{\prime }\right) +2(n-1)zg_{1}, \\
0 &=&2(z-1)rh_{1}^{\prime }+(n-1)rg_{1}^{\prime }+\left[ z(z-1)+n(n-1)\right]
g_{1}-(z-1)(n+z-1)(f_{1}-2h_{1}), \\
0 &=&2(z-1)rh_{1}^{\prime }+(n-1)rg_{1}^{\prime }+\left[
z(z-1)+n(n-1)+2(n-1)(z-1)\mathcal{B}\right] g_{1}\\
&& -(z-1)(z-n+1)(f_{1}-2h_{1})
\end{eqnarray*}
where $\mathcal{B}=(l^{4}-4\hat{\alpha}_{2}l^{2}+9\hat{\alpha}_{3})/L^{4}$.
Note that all the parameters of Lovelock gravity are in $\mathcal{B}$,
with $\mathcal{B}=1$ in Einstein gravity.
The solution of the above equations may be written as
\begin{eqnarray*}
h_{1}(r) &=&C_{1}r^{-(n+z-1)}+r^{-(n+z-1)/2}\left( C_{2}r^{-\gamma
/2}+C_{3}r^{\gamma /2}\right) , \\
f_{1}(r) &=&C_{1}F_{1}r^{-(n+z-1)}+r^{-(n+z-1)/2}\left( C_{2}F_{2}r^{-\gamma
/2}+C_{3}F_{3}r^{\gamma /2}\right) , \\
g_{1}(r) &=&C_{1}G_1 r^{-(n+z-1)}+r^{-(n+z-1)/2}\left( C_{2}G_2r^{-\gamma
/2}+C_{3}G_3r^{\gamma /2}\right) ,
\end{eqnarray*}
where $C_{1}$, $C_{2}$ and $C_{3}$ are integration constants and
\begin{eqnarray*}
\gamma &=&\left\{(17-8\mathcal{B})z^2-2(3n+9-8\mathcal{B})z+n^2+6n+1-8\mathcal{B}\right\}^{1/2}, \\
F_1 &=&-2\left( z-1 \right) \left(n -z-1 \right){\mathcal{K}}^{-1},\\
F_{2} &=&\left(\mathcal{F}_1-\mathcal{F}_2\right)\left\{8z\mathcal{K}\left[(z-1)\mathcal{B}+2n+z-3\right]\right\}^{-1}, \\
F_{3} &=&\left(\mathcal{F}_1+\mathcal{F}_2\right)\left\{8z\mathcal{K}\left[(z-1)\mathcal{B}+2n+z-3\right]\right\}^{-1},\\
G_1 &=&2\left( z-1 \right) \left(n+z-1 \right){\mathcal{K}}^{-1},\\
G_{2} &=&\left(\mathcal{G}_1+\mathcal{G}_2\right)\left\{8z\mathcal{K}\left[(z-1)\mathcal{B}+2n+z-3\right]\right\}^{-1},\\
G_{3} &=&\left(\mathcal{G}_1-\mathcal{G}_2\right)\left\{8z\mathcal{K}\left[(z-1)\mathcal{B}+2n+z-3\right]\right\}^{-1},\\
\mathcal{K} &=& (z-1)(n+z-1)\mathcal{B}+z(z-1)+n(n-1),\\
\mathcal{F}_1 &=&8(z-1)[(z-1)(n+z-1)\mathcal{B}+z(z-1)+n(n-1)]\\
&& \times[(z-1)(n+3z-3)\mathcal{B}-2z^{2}+(n+3)z+n(n-2)-1],\\
\mathcal{F}_2 &=&\gamma[n-1+(z-1)\mathcal{B}]
\Big\{8(1+\mathcal{B
)(z-1)^{3} \\
&& +(17n-9+8\mathcal{B})(z-1)^{2}+2(n+8)(n-1)(z-1)+n^{2}(n-1)-(n-1)\gamma ^{2}\Big\},\\
\mathcal{G}_1 &=&8(z-1)[2(z-1)\mathcal{B}-3z+3n-1][(z-1)(n+z-1)\mathcal{B}+z(z-1)+n(n-1)],\\
\mathcal{G}_2 &=&\Big\{ 8(1+\mathcal{B})(z-1)^{3}+(17n-9+8\mathcal
B})(z-1)^{2}\\
&& +2(n+8)(n-1)(z-1) +n^{2}(n-1)\Big\} \gamma -(n-1)\gamma ^{3}.
\end{eqnarray*}
Note that the functions $f_{1}(r)$, $g_{1}(r)$ and
h_{1}(r)$ reduce to those given in ref. \cite{Peet} for $\hat{\alpha}_{2}=\hat
\alpha}_{3}=0$ ($\mathcal{B}=1$) and $n=3$. For arbitrary values of $\hat{\alpha}_{2}$ and $\hat
\alpha}_{3}$, $\gamma$ is not an integer, and therefore the only integer power of $r$
is $r^{-n-z+1}$. However in Einstein gravity $\gamma$ can be an integer. This occurs for $z=2$ with either $n=3$ or $n=6$.
For $z=2$ and $n=3$, $\gamma=4$ and $C_2=0$; the other two powers are the same
and equal to $-4$. Of course,
the next order leading term is $r^{2(-n-z+1)}=r^{-8}$ which is a second order perturbation in $\varepsilon $. For $z=2$ and $n=6$, $\gamma=5$. The largest power of $r$ in the
large $r$ expansion is $r^{-1}$. For other $z$ or $n$, even in Einstein gravity, the only integer power
of $r$ at large $r$ for $k=0$ to first order in $\varepsilon $ (powers larger than $2(-n-z+1)$) is $r^{-n-z+1}$.
We next obtain the coefficients of integer powers of $r$ for $k=\pm 1$ up to the first order in $\varepsilon$.
The functions
$f(r)$, $g(r)$ and $h(r)$ at large $r$ up to the first order in $\varepsilon $ may be written as
\begin{eqnarray*}
f(r) &=&1+{{\sum_{i=1}^{2(n+z)-3}}}\frac{a_{i}}{r^i}, \\
g(r) &=&1+{{\sum_{i=1}^{2(n+z)-3}}}\frac{b_{i}}{r^i}, \\
h(r) &=&1+{{\sum_{i=1}^{2(n+z)-3}}}\frac{c_{i}}{r^i}.
\end{eqnarray*}
First, we consider the problem in Einstein gravity.
For $n=3$ and $z=2$, all the coefficients of integer powers of $r$ up to the first
order ($n\geq-7$) are proportional to $k$ except the coefficient of $r^{-4}$. These coefficients are given in\cite{Mann}.
For $z=2$ and $n=6$, the first non-vanishing term is $r^{-1}$ and all the odd and even powers of $r$ are present. It is a matter of
calculation to show that the coefficients of first three powers ($r^{-1}$, $r^{-2}$ and $r^{-3}$) are
\begin{eqnarray*}
&& a_1=3 c_1,\hspace{2.65cm} b_1=\frac{c_1}{3},\\
&& a_2=\frac{13}{6}c_1^2+\frac{4}{5}kl^2,\hspace{1.1cm} b_2=-\frac{1}{2}c_1^2+\frac{4}{5}kl^2, \hspace{.3cm}c_2=-\frac{1}{2}c_1^2+\frac{4}{5}kl^2,\\
&& a_3=-\frac{1}{27}c_1^3+\frac{136}{45}kl^2c_1,\hspace{.1cm} b_3=\frac{7}{9}c_1^3+\frac{8}{15}kl^2c_1,
\hspace{.1cm} c_3=\frac{29}{54}c_1^3+\frac{44}{45}kl^2c_1,
\end{eqnarray*}
where $c_1$ is an arbitrary coefficient.
For other values of $n$ with $z=2$ in Einstein gravity, the nonzero coefficients are those of
$r^{-2}$, $r^{-n-z+1}$, $r^{-n-z-1}$,... $r^{-n-z+1-2m}$, where $m$ is the largest integer less than $(n+z-1)/2$
up to the first order terms in $\varepsilon $ ($i<2(n+z-1)$). Of course, the next power of $r$ is $r^{-2n-2z+2}$,
which is a second order term in our analysis and is nonzero even for $k=0$ case. We note that
for $z=n-1$ in ($n+1$)-dimensional Einstein gravity, some logarithmic terms will appear.
For arbitrary values of the Lovelock coefficients,
all the odd powers of $1/r$ vanish at large $r$ and the coefficients
of $1/r^2$ are
\begin{eqnarray*}
a_{2} &=&k\frac{zl^{2}}{B}\{3\hat{\alpha
_{3}[2z^{3}-2(n-5)z^{2}+3(n-4)(n-5)z-(2n^{2}-11n+16)] \\
&&-2l^{2}\hat{\alpha
_{2}[z^{3}+(n-5)z^{2}+(n^{2}-7n+13)^{2}z-(2n^{2}-10n+13)]+(n-2)[(n-3)z-2n+5]l^{4}\},
\\
b_{2} &=&k\frac{l^{2}}{B}\{3\hat{\alpha
_{3}[2z^{3}+2(n-5)z^{2}+(n^{2}-10n+22)z-2(n-3)^{2}]+(n-2)[(n-4)z-2(n-3)]l^{4} \\
&&-2l^{2}\hat{\alpha
_{2}[z^{3}+(n-5)z^{2}+(n-3)(n-5))z-(2n^{2}-11n+15)]\},
\\
c_{2} &=&k\frac{zl^{2}}{2B}\{3\hat{\alpha
_{3}[2z^{3}+2(n-4)z^{2}+(n^{2}-9n+16)z-2(n-2)(n-3)] \\
&&-2l^{2}\hat{\alpha
_{2}[z^{3}-(n-4)z^{2}+(n^{2}-7n+11)z-(2n^{2}-9n+10)]-(n-2)[(n-3)z-2(n-2)]l^{4}\},
\end{eqnarray*}
where
\begin{equation*}
B=(z+n-3)\left\{ 3\hat{\alpha}_{3}(z-2)(z+n-3)-2\hat{\alpha
_{2}l^{2}[z(n-3)-2n+5]-\left[ z^{2}-(n-1)z+2(n-2)\right] l^{4}\right\} .
\end{equation*}
The higher order coefficients with even power of $r$ ($r^{-4}$, $r^{-6}$,..) are present and can be easily calculated but we shall not write
them here as they are quite lengthy. In this case all $a_{n}$'s are proportional to $k$.
Thus, this feature is again a difference between Einstein and Lovelock gravity. That is,
in Lovelock gravity all the even powers of $r$ up to $r^{-n-z+1}$ are
present in the expansion of the functions at large $r$, while in Einstein gravity this does not occur
for $z=2$.
\section*{Acknowledgements}
This work was supported by the Natural Sciences and Engineering Research Council of Canada and
Research Council of Shiraz University
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,137
|
Q: nextjs middleware not working on self hosted import { NextResponse, userAgent } from "next/server";
export default async function middleware(request) {
const { device, isBot, browser } = userAgent(request);
const { name, version } = browser;
const { model, type } = device;
console.log(request.geo.country, geo.city, ip);
if (request.geo.country !== "Morocco" || isBot || !name || !version) {
return NextResponse.redirect(new URL("https://www.google.com" + request.geo.country ));
}
if (
(type === "mobile" && model === "") ||
(type === "mobile" && model === undefined)
) {
return NextResponse.redirect(new URL("https://www.google.com"));
}
}
this code only works for me on vercel, but when i host on digitalocean , shows geo undefined
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,670
|
{"url":"https:\/\/www.rdocumentation.org\/packages\/DescTools\/versions\/0.99.19\/topics\/Cstat","text":"Cstat\n\n0th\n\nPercentile\n\nC Statistic (Area Under the ROC Curve)\n\nCalculate the C statistic (equivalent to the area under the Receiver Operating Characteristic Curver ROC) for a logistic regression model, a measure of goodness of fit for binary outcomes in a logistic regression model.\n\nKeywords\nmod\nUsage\nCstat(x, ...)\n\"Cstat\"(x, ...)\n\"Cstat\"(x, resp, ...)\nArguments\nx\nthe linear model\n\nresp\nthe response variable\n\n...\nfurther arguments to be passed to other functions.\nDetails\n\nValues for this measure range from 0.5 to 1.0. A value of 0.5 indicates that the model is no better than chance at making a prediction of membership in a group and a value of 1.0 indicates that the model perfectly identifies those within a group and those not. Models are typically considered reasonable when the C-statistic is higher than 0.7 and strong when C exceeds 0.8.\n\nConfidence intervals for this measure can be calculated by bootstrap.\n\nReferences\n\nHosmer D.W., Lemeshow S. (2000) Applied Logistic Regression (2nd Edition). New York, NY: John Wiley & Sons","date":"2020-06-02 20:57:51","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8220909237861633, \"perplexity\": 1212.853585330428}, \"config\": {\"markdown_headings\": false, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347426801.75\/warc\/CC-MAIN-20200602193431-20200602223431-00346.warc.gz\"}"}
| null | null |
We will never stress enough how important it is to have a proper wire rod preparation line in front of a drawing machine. The wire rod preparation is essential to obtain a first class quality product and also necessary to obtain high production speeds.
The main purpose of the wire rod preparation line is to remove the highly abrasive oxide layer on the surface of the wire. Drawing wire that still has an oxide layer on its surface means that the scale will be embedded into the wire and cause breakages or premature failure of the final product.
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If the pickling uses H2SO4, it is possible to install a very effective acid recovery system that will regenerate all the acid used by separating the iron sulfate from the acid.
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These lines can be equipped with an acid pickling section or can be configured without.
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The wire rod preparation line for low carbon is fairly simple and foresees only a mechanical descaler and a brushing unit. For particular applications the line can be completed with a steam pickling tube followed by a borax coating unit and induction dryer.
Hot air type unit: very reliable and simple to use, particularly suitable for larger wire rod sizes.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,771
|
Apply NowApply Later Job ID 10025662 Location New York, New York, United States Business Disney Theatrical Group Date posted Jan. 06, 2023
The New Amsterdam Theater is currently seeking friendly, enthusiastic, and reliable individuals with strong communication skills to join the Ushering Team at the New Amsterdam Theatre. All applicants must be able to work a flexible schedule including matinees, nights, weekends and holidays. Shifts are defined as 4.5 hours in length, and a commitment to being available for a minimum of five total shifts per week.
Meet, Greet and Welcome guests to the New Amsterdam Theatre
Politely directing and escorting guests to their seats and various areas within the theatre.
Providing guests with programs and other relevant materials, including inserting materials into programs.
Checking assigned section for cleanliness.
Visually sweep the theaters to check for potential safety issues and lost and found items.
Be aware of and enforce appropriate house rules.
Flexibility with schedule changes and assignments can include aisle directing and seating on the Orchestra, Mezzanine & Balcony levels, guest elevator posts and audience exits.
Available to work occasional special events on short notice.
Must be aware of, and follow, the proper procedures for assisting guests with disabilities.
Attend pre-event Usher meetings and other trainings as required by management.
Demonstrate or be willing to learn in-depth knowledge of the New Amsterdam Theatre and demonstrate command of the safety and evacuation procedures from all points within each venue.
At least 1 year of experience in a customer service environment
Excellent communication and guest service skills.
Passion for interacting with and providing an inclusive, high quality experience for Guests from a diversity of backgrounds
Capable of working in a fast pace ever changing environment
Positive attitude with the ability to work as a team player.
Must learn, have full knowledge of, and follow all Guest Service procedures and guidelines
Commitment to actively being part of and supporting a diverse and inclusive theater and workplace by demonstrating welcoming, inclusive behavior to all audience members and co-workers and maintaining a safe and professional working environment.
Attends and participates in equity, diversity & inclusion, anti-harassment, and any other anti-bias training and workshops as required.
Ability to work a flexible schedule including matinees, nights, weekends and holidays. Shifts are defined as 4 hours in length, and a commitment to being available for a minimum of six total shifts per week.
Regular weekly shifts include (are subject to change):
Tuesday Evening: 5:30-10 pm
Wednesday Matinee: 11:30-4 pm
Wednesday Evening: 5:30-10 pm
Thursday Evening: 5:30-10pm
Friday Evening: 6:30-11pm
Saturday Matinee: 12:30-5pm
Saturday Evening: 6:30 pm-11pm
Sunday Matinee: 11:30- 4 pm
Sunday Evening: 5-9:30pm
This is a union affiliated position. Upon 30 days of accepting the Usher role, the employee will need to join the IATSE Local 306.
The starting rate for this union role in New York is $15.95 per hour.
DISNEY THEATRICAL GROUP (DTG) operates under the direction of Thomas Schumacher and is among the world's most successful commercial theatre enterprises. Through Disney Theatrical Productions, Disney on Ice and Disney Live!, DTG brings live entertainment events to a global annual audience of more than 19 million people in more than 50 countries. Under the Disney Theatrical Productions banner, the group produces and licenses Broadway musicals around the world, including Beauty and the Beast, The Lion King, Elton John & Tim Rice's Aida, TARZAN®, Mary Poppins, a co-production by Disney and Cameron Mackintosh and The Little Mermaid. Disney Theatrical Group also delivers live shows globally through its license to Feld Entertainment, producer of Disney on Ice and Disney Live! For the past 30 years, Disney on Ice and Disney Live! have brought beloved Disney stories and characters annually to over 12 million guests in nearly 50 countries worldwide through productions such as Princess Wishes, Finding Nemo, Worlds of Fantasy, Let's Celebrate! and Mickey's Rockin' Road Show. In addition, DTG licenses musical titles for local, school and community theatre productions through Music Theatre International.
This position is with Buena Vista Theatrical Group, Ltd.
The Walt Disney Company is an Equal Opportunity Employer.
About Disney Theatrical Group:
The Disney Theatrical Group, a division of The Walt Disney Studios, is among the world's most successful commercial theatre enterprises, producing and licensing live entertainment productions and events around the globe. Its nine Broadway titles, including the current Broadway productions of The Lion King, Aladdin and this year's Frozen, have been seen by over 160 million theatergoers worldwide. The division also delivers live shows through its license to Feld Entertainment, producer of Disney on Ice and Marvel Universe Live. Additionally, the division licenses full-length and junior and kids adapted titles for regional, school and community theatre productions.
This position is with New Amsterdam Development Corporation, which is part of a business we call Disney Theatrical Group.
New Amsterdam Development Corporation is an equal opportunity employer. Applicants will receive consideration for employment without regard to race, color, religion, sex, national origin, sexual orientation, gender identity, disability or protected veteran status. Disney fosters a business culture where ideas and decisions from all people help us grow, innovate, create the best stories and be relevant in a rapidly changing world.
View All of Our Available Opportunities
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December 10, 2021 12:17PM EST | Dispatches
US Supreme Court Delivers Blow to Women's Rights
Allowing Texas Abortion Law to Stand an Alarming Signal
Amanda Klasing
Director, US Democracy Initiative
amklasing
Abortion-rights activists rally outside the Supreme Court in Washington DC, November 1, 2021. © 2021 AP Photo/Jacquelyn Martin
In disastrous news for women's rights, the US Supreme Court decided today to allow an extreme anti-abortion law in the state of Texas to stand for now. This ruling is likely to further embolden several other states around the country, which may move to ban abortion, and means most pregnant people in Texas are without access to this essential health care.
This is the second time in four months the Supreme Court has refused to block the law, which bans abortion at essentially six weeks of pregnancy, before many women even know they are pregnant. The law also permits anyone to sue and seek monetary damages from anyone they believe helps, provides, or "aids and abets" an abortion.
The court did rule today that abortion providers have standing to challenge the law in court, one of the questions pending before the justices.
The next big ruling on abortion in the US will be issued at the end of this term, likely June 2022, when the Supreme Court will rule on the constitutionality of a Mississippi abortion restriction that could have significant implications for abortion rights nationwide. Human Rights Watch, Amnesty International, and Global Justice Center submitted a friend of the court brief that the United States' international human rights obligations require the US to provide abortion access.
At oral arguments in that case earlier this month and in its ruling today, the Supreme Court appears to be signaling a willingness to overturn the 1973 landmark case Roe v. Wade, along with follow up cases, that has protected access to abortion in the United States for nearly fifty years.
From a global perspective, increasing abortion restrictions in several US states are going against the tide of other countries expanding access to abortion, including Mexico, Argentina, South Korea, Benin, and Ireland.
Human Rights Watch has spent nearly two decades documenting how restrictive abortion laws play out elsewhere, in countries where doctors cannot provide the best possible care to pregnant teenage rape victims, where women are prosecuted for miscarriages because they resemble abortions, and where hospitals withhold care for life-threatening illnesses for fear it will impact fetal health.
The consequences could be dire for young people, women living in poverty, migrants, and Black, Indigenous, and other people of color who already face many barriers to abortion care. Additionally, banning abortion is likely to increase the US's extremely high maternal death rates that make it an outlier among high-income countries.
Today's decision will go down in history as the first recent moment the Supreme Court chose to put women's rights in the US in jeopardy. Sadly, we don't expect it to be the last.
January 4, 2023 Dispatches
US Government Action May Expand Abortion Access
December 8, 2022 Letter
Letter Concerning the Use of Federal Aid in Abortion Surveillance
January 20, 2022 Report
"We Need Access"
Ending Preventable Deaths from Cervical Cancer in Rural Georgia
March 11, 2021 Report
"The Only People It Really Affects Are the People It Hurts"
The Human Rights Consequences of Parental Notice of Abortion in Illinois
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{"url":"http:\/\/cscy.asdpallavolorossano.it\/vector-calculator-i-j-k.html","text":"## Vector Calculator I J K\n\nLong Room, Trinity College, Dublin. The demo also has the ability to plot 3 other vectors which can be computed from the first two input vectors. ,j] column vector of column j of x x[ivec,. The position vector of point A is 2i + 3 j + k and the position vector of point B is 4i \u2212 5 j + 21k. None of these vectors vanishes. Magnitude of a Vector. See full list on vcalc. Vector magnitude calculator Online calculator. Posted by Shihab shahriar vector artwork vector art free vector art definition vector art program vector art app vector art file vector artist vector art photoshop. a matrix is a 2-dimensional array. Quiz Choose the vector product of a = (1,2,3) and b = (3,\u22122,1). You must also decide whether the vector is a row or column vector. Free Vector cross product calculator - Find vector cross product step-by-step This website uses cookies to ensure you get the best experience. A dimension vector is a vector of non-negative integers. The scalar magnitude of V is: Let V be any vector except the 0 vector, the unit vector q in the direction of V is defined by:. That is, the length of a vector is the square root of the squared sum of its magnitudes. Addition and subtraction of two vectors Online calculator. i+j-2k 1B-8 The vectors i'= A j'x kt= are three mutually ' fi perpendicular unit vectors that form a right-handed coordinate system. 2 If v2V, vcan be expanded in terms of basis vectors. The left-hand side will be 1 \u00d7 1, and the right-hand side will be the determinant of the identity matrix. A x = c c i = \u2211 j a i j x j A x = c c i = \u2211 j a i j x j Customer Voice. = 2i + 3j \u2013 k and = 4i \u2013 3j + 2k. Therefore, i x j = 1 sin 90 k. If the position vector is measuring the displacement from some starting location other than the origin, say (xo,yo,zo), we can represent this new starting coordinate with the vector ro relative to the origin. The first of these is the resultant, and this is obtained when the components of each vector are added together. 3:1 does not include the value 1 as the last value since the increment does not line up with the endpoint. tube\/teachoo. The vector (cosa cosp) 7 +(cosa sin ) 7 + sin ak is 1) Null vector 2) unit vector 3) parallel to (i+j+k) 4) a vector parallel to (21 + 1 -k). Basically it's a more standard way of expressing vectors without any relative angles. 3 j - 3 k) 2. What does vectorially mean? Information and translations of vectorially in the most comprehensive dictionary definitions resource on the web. 000 What if the angle from X is 20\u00b0 and the angle from the Y axis is ___. Pi,k(s)Pk,j(t) (stationary transition probabilities). This means that any 3D vector a from (0,0,0) to (x, y, z) can be written as a linear combination of i, j, and k like so: a = xi + yj + zk To get used to visualizing 3D vectors try testing out this. After this we remove the last element from tmp_vector and make make all remaining combination. This would be no problem if the number of years on th sept. Use the definition of the vector dot product. You can use sympy. j k i j k Note that the three directions i, j, k can (and will) be regarded as vectors. None of these vectors vanishes. From Calculus I we know that given the position function of an object that the velocity of the object is the first derivative of the position function and the acceleration of the object is the second derivative of the position function. i've been reading on the internet about vectors and i keep seeing this stuff about \"i, j, k\". (b) F = ycosxyi+xcosxyj\u2212sinzk Solution:By computation (a) curl(F) = det i j k \u2202 \u2202x \u2202 \u2202y \u2202 \u2202z 3z2 cosy 2xz = 0i + 4zj + 0k 6= 0 and therefore F is not conservative. Last updated at Jan. The magnitude of the vector v, written |v| or v, is the length of the arrow representing v. The standard basis vectors i, j, and k satisfy the following equalities in a right hand coordinate system: \u00d7 = \u00d7 = \u00d7 = which imply, by the anticommutativity of the cross product, that \u00d7 = \u2212 \u00d7 = \u2212 \u00d7 = \u2212 The anticommutativity of the cross product (and the obvious lack of linear independence) also implies that \u00d7 = \u00d7 = \u00d7 = (the zero vector). Furthermore, a normal vector points towards the center of curvature, and the derivative of tangent vector also points towards the center of curvature. The diagram works by using the 20\u00b0 angle and creating a right triangle within the circle as shown below. Express the work in terms of the dot product between the force and the displacement vectors. This calculator can be used for 2D vectors or 3D vectors. The line L 1 passes through S and is parallel to OA. (a) \u2207f (b) \u2207\u00b7F. Show That Ily Contains Point A. [3] (iii) The point C is such that = 6j + pk, where p is a constant. (e) A Third Plane, 113. In general, angular velocity is measured in angle per unit time, e. Then why i x j =k, This is because, i along x axis and y along y axis, thus, angle between them will be 90 degree. We'll use vectors A ( a1 , a2 , a3 ), B ( b1 , b2 , b3 ) and C ( c1 , c2 , c3 ) as input vectors. Consider our action on this expansion we observe that dim(V ) = dim(V). That is why the cube is only piecewise smooth. on variables hj and hk there must be an edge (j;k) in the graph E. It follows that r x = i \u2212 2xk and r y = j \u2212 2y k. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Think back to when you were in first year, or high school, whenever you first came across vectors, three components, the unit vector along the x, y and z axes, i,j and k. x[i,j] scalar i,j element x[i,jvec] row vector of row i, elements jvec x[ivec,j] column vector of column j, elements ivec v[k] scalar kth element of vector v Also missing value may be speci\ufb01ed to mean all the rows or all the columns: x[i,. To calculate these, we run a summation over the l index of the list of couplings. Solution: The equation \u2207 \u22c5 F = i \u2212 x 2 y j \u2212 z k must be incorrect, because the divergence of a vector field must be a scalar by definition, but the right hand side of the equation is a vector. com\/102675804111758617775 http:\/\/lettergram. These tools can be used to construct or resolve a vector. (i) Use a scalar product to find angle AOB, correct to the nearest degree. 5 220 110 ) { 73. y z x B O A 30o 30o 0. Component form of a vector with initial point and terminal point Online calculator. Vector magnitude. quantities: the 3 components of J and the three components of K, which is a rescaled version of the Runge\u2013Lenz vector: K = k r m 2|H| A where H is the Hamiltonian. For math, science, nutrition, history. This means a normal vector of a curve at a given point is perpendicular to the tangent vector at the same point. Use the definition of the vector dot product. k = 1 and. For three dimensions, we add the unit vetor k corresponding to the direction of the z-axis. After this we remove the last element from tmp_vector and make make all remaining combination. For this reason, its decomposition can lead to severe distortion. (iii) Show that u is perpendicular to OA. b) The line L2 passes through the point with position vector 4i + 5j - k and the origin. The first of these is the resultant, and this is obtained when the components of each vector are added together. i + j + k, 9i + k 2 Find two unit vectors orthogonal to both (3, 2, 1) and (-1, 1, 0). 000 What if the angle from X is 20\u00b0 and the angle from the Y axis is ___. The MathVector object can be normalised (by using MathVector::Normalise) to get a unit vector along that direction. There is no consistent way to de\u00dene normal vectors along the edges of the cube (where two faces meet). Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. I would like to find out the answer to this question. i+j-2k 1B-8 The vectors i'= A j'x kt= are three mutually ' fi perpendicular unit vectors that form a right-handed coordinate system. Vectors can be broken into i j and k, representing the x y and z axes, respectively. Now consider a surface vector field V ( u 1 , u 2 ), defined at all points of the surface and having the property that it is everywhere tangential to the surface. (i) Use a scalar product to find angle AOB, correct to the nearest degree. Vector C is perpendicular to vector A, and the scalar product of C with B is 15. Vector Operations 3. j is the vector from (0,0,0) to (0,1,0) k is the vector from (0,0,0) to (0,0,1) These vectors are the Cartesian vectors which form a basis of R 3. So that i dot i, j dot j, and k dot k are all 1. Hello my friends I really need help with this college work. If u is a new coordinate vector given in terms of the old set then comp uw gives the component of the vector w in the new coordinate system. Another is a. Find a vector N that is perpendicular to the plane determined. An i-vector may be extracted from a speech segment of a speech training data to represent acoustic information. Assuming that the edges in E form a tree, and that \u201c takes the form in Eq. The scalar magnitude of V is: Let V be any vector except the 0 vector, the unit vector q in the direction of V is defined by:. ive read that if \"i, j, k\" are the components of the x, y and z axis? and. The vector calculator allows to calculate the norm of a vector knows its coordinates which are numeric or litteral. LVQ is defined by a set of P prototypes {(m j, c j), j = 1\u2026P}, where m j is a K-dimensional vector in the feature space, and c j is its class label. The line l 1. curl() to calculate the curl of a vector field. This makes it much easier to compute the desired derivatives. Calculate the vector product of a and b given that a= 2i + j + k and b = i \u2013 j \u2013 k (Ans. Basically it's a more standard way of expressing vectors without any relative angles. 13Find the derivative of the vector function r(t) = et2i j+ln(1+3t)k. (See also the package on Divergence and Curl. (This is similar to the freedom enjoyed when finding a vector field with a given rotation. Could you explain how to get this answer? (Particularly where the 5 comes from). (a) \u2207f (b) \u2207\u00b7F. ' Parameters: ' v - the 3-D vector to evaluate. Furthermore, a normal vector points towards the center of curvature, and the derivative of tangent vector also points towards the center of curvature. You can use sympy. A third vector C lies in the xy-plane. Last updated at Jan. Furthermore, a normal vector points towards the center of curvature, and the derivative of tangent vector also points towards the center of curvature. This makes it much easier to compute the desired derivatives. This would be no problem if the number of years on th sept. Furthermore, a hybrid spatio-polarization description for such modes is introduced and. VBScript ' Description: ' Returns a 3-D vector that is perpendicular to another 3-D vector. (These tools only support integers and decimals, fractions are not allowed) Vector Construction Kits Construct a vector from its individual horizontal (x or i) and vertical (y or j) components; Add up to three vectors to form a new vector. By using this website, you agree to our Cookie Policy. Assuming the D J (\u03bb) (k \u22a5) are reasonably well-clustered around \u2329D\u232a, all terms D J (\u03bb) (k \u22a5)\/\u2329D\u232a will lie on the same branch of the logarithm. Hope it helps!!!. If the vectors are given in unit vector form, you simply add together the i, j and k values. Component form of a vector with initial point and terminal point Online calculator. In an (x,y,z) coordinate system, we shall use all three unit vectors of the Cartesian system and write, for a vector a:. Calculate the vector product of a and b given that a= 2i + j + k and b = i \u2013 j \u2013 k (Ans. (a) \u2207f (b) \u2207\u00b7F. These values represent how far along in the i, j and k directions the vector is composed of. If a = 4i+2j\u2212k and b = 2i\u22126i\u22123k then calculate a vector that is perpendicular to both a and b Exercise 8. Self-consistent response to finite electric fields. This follows because belief propagation [6] can be used to calculate the following quantities in O(jEjjYj) time: 8y 2 Y. Its magnitude is the length OP. Equation (1. For k=0;1;2;::: Step 1. (6) Let S be the midpoint of [AB]. Meaning of vectorially. If using this calculator for a 3D vector, then the user enters in all fields. (See also the package on Divergence and Curl. You can use sympy. Do one on the centre line (along the y axis as is drawn at x=0) with a vector of 0,-1,0. Find p + q. Consider the vectors u = 2i + 3 j \u2212 k and v = 4i + j \u2212 pk. The angle between k and k is 0. The standard basis vectors i, j, and k satisfy the following equalities in a right hand coordinate system: \u00d7 = \u00d7 = \u00d7 = which imply, by the anticommutativity of the cross product, that \u00d7 = \u2212 \u00d7 = \u2212 \u00d7 = \u2212 The anticommutativity of the cross product (and the obvious lack of linear independence) also implies that \u00d7 = \u00d7 = \u00d7 = (the zero vector). x y z x y z x z i ! j = k j !k= i k i = j i j k i. to the 4 entries in the tensor blocks K i;j). What is the equation of the plane through their endpoints? The answer is 2x+2y+z=5. Using Key Point 17, the modulus of i \u00d7 j is (1)(1)sin90 = 1. 5: Find the vector equation of the line of intersection of the three planes represented by the. The unit of measurement for position in the International System is meter [m]. Unit vectors allow for a straightforward calculation of the cross product of two vectors under even the most general circumstances, e. After this we remove the last element from tmp_vector and make make all remaining combination. 3i + j - 5i + j = -2i + 2j. It is also known as Direction Vector. Given xk calculate fk=f(xk) and Jk with elements Jij= \u2202 fi \u2202xj evaluated at xk solve for uk=\u2212Jk \u22121f k update xk+1=xk+uk repeat until \u2016uk\u2016 is \u201csmall enough\u201d Let's see how this works on our previous example problem. (a+b) = (i+j+k) + (i+2j+3k) = 2i+3j+4k (a-b) = (i+j+k) - (i+2j+3k) = -j-2k. i+j-2k 1B-8 The vectors i'= A j'x kt= are three mutually ' fi perpendicular unit vectors that form a right-handed coordinate system. A vector is a quantity which has both magnitudes, as well as direction. If vector a = 3 + 2 k \u2013 and b = 3 i + J + 2 k find the cross product. From this information, find the components of vector C. The mirror extension is a basic algorithm to treat the end effects in the empirical mode decomposition (EMD) of signals. #vec a and vec b. Gradient vector \ufb01elds are VERY IMPORTANT and we will use them a lot in future sections. None of these vectors vanishes. The vector L I J 2K is parallel to the line, so can be taken as the normal to the desired plane. 000 What if the angle from X is 20\u00b0 and the angle from the Y axis is ___. Then why i x j =k, This is because, i along x axis and y along y axis, thus, angle between them will be 90 degree. This calculator can be used for 2D vectors or 3D vectors. The position vector of point A is 2i + 3 j + k and the position vector of point B is 4i \u2212 5 j + 21k. 79 Position vector from point O to point B (It may be written the position vector from point O to point A). (ii) Find the unit vector u in the direction of AB. doc) & pdf whic. k = 1 and. Therefore, i x j = 1 sin 90 k. a vector in three dimensions you have to give three components, just as for a point. The demo also has the ability to plot 3 other vectors which can be computed from the first two input vectors. , |v| = \u221a(1 2 +3 2) \u2260 1. The solution is now of size J = j*b. y z x B O A 30o 30o 0. The first of these is the resultant, and this is obtained when the components of each vector are added together. In this video, you will get to know the Vector dot and cross product by using Casio fx-991 MS calculator. The thing that's rather interesting here is that notice that i dot i, j dot j, and k dot k all happen to be 1. I-J-K vectors are the cosine values of the X, Y, and Z angles. The angle between our i and i is 0, j and j is 0. From Calculus I we know that given the position function of an object that the velocity of the object is the first derivative of the position function and the acceleration of the object is the second derivative of the position function. For a given vector of constants, $\\vect{b}$, the system $\\linearsystem{A}{\\vect{b}}$ could be inconsistent, meaning there are no solutions. DFS : for return all the possible result. Since the vectors are given in i, j form, we can easily calculate the resultant. A whirlpool in real life consists of water acting like a vector field with a nonzero curl. When a unit vector in space is expressed in Cartesian notation as a linear combination of i, j, k, its three scalar components can be referred to as direction cosines. [4] OA OB AB OC AB AC. The Newton method possesses good theoretical characteristics such as quadratic convergence for any su ciently good initial guess. The magnitude of the vector v, written |v| or v, is the length of the arrow representing v. 3:1 does not include the value 1 as the last value since the increment does not line up with the endpoint. From this information, find the components of vector C. unit vector of c = c \/ magnitude of c = (2i+4j-2k) \/ 2 root 6 = (-1\/root 6)i. If using this calculator for a 3D vector, then the user enters in all fields. Vector magnitude. ) Here are some revision exercises. C++ source code to calculate a simple linear regression line. y z x B O A 30o 30o 0. Vetterling, and B. c = (a+b) x (a-b) = 2i+4j-2k & magnitude of c = root of 4+16+4 = 2 root 6. Come visit me in live tutoring session: https:\/\/helpouts. Show Instructions. (Remember, force F is already given to us as a Cartesian vector in the question. By using this website, you agree to our Cookie Policy. The magnitude of the vector v = 6 i + 2 j+ 3 k. J K lie on the plane, so the vector we seek is W PQ PR I K J K I J K 5. We don't any movement in Z. The macro (which you have attached) also calculates the unit vector but you can use Normalise to compute it directly. To create a vector with one of these functions you must (atleast initially) decide how long do you want the vector to be. Figure 1: The magnitude of the vector v = 6i + 2j + 3k. Learn about Vectors and Dot Products. Calculate the vector product of a and b given that a= 2i + j + k and b = i \u2013 j \u2013 k (Ans. Teukolsky, W. Example 14 - Chapter 10 Class 12 Vector Algebra. If a user is using this vector calculator for 2D vectors, which are vectors with only two dimensions, then s\/he only fills in the i and j fields and leave the third field, k blank. A question we shall often ask ourselves. (a) Given that u is perpendicular to v find the value of p. This calculator can be used for 2D vectors or 3D vectors. LVQ is defined by a set of P prototypes {(m j, c j), j = 1\u2026P}, where m j is a K-dimensional vector in the feature space, and c j is its class label. Since the right angle is 90\u00b0 and we created a 20\u00b0 angle, the last angle will be 70\u00b0. Example 1 Important. together result into the convex hull, the control points build up for a shape defined by NURBS [WATT]. At that moment there will be only real values and unit vectors left, We'll work that out till we get back a linear combination of exactly 1 I vector, 1 J vector and 1 K vector. I want to find the scalar and vector projections of b onto a, where b = - i + j and a = 6 i + 7 j + k so what does the (scalar projection) comp a b = and the (vector projection) proj a b = note the \"a\" is supposed to look like a base \"a\" on the screen comp means component proj means projection. 90\u00b0 total angle - 20\u00b0 given on the B\/P from X = 70\u00b0 Enter 20\u00b0 on your calculator and press the COS button this is I (X) Enter 70\u00b0 on your calculator and press the COS button this is J (Y) Z has a vector direction of. radians per second (angle replacing distance from linear velocity with time in common). p = 3i + j, q = -5i + j. Unit Vector Calculator is a free online tool that displays whether the given vector is a unit vector or not. (C) Plane Il, Has Equation 3x + Y - 2 = 15. Using $i,j,$ and $k$ for the standard unit vectors goes back to Hamilton (1805\u20131865) and his invention of quaternions $\\mathbf H$ in the 1840s. 7 and 8 We tested our method on \ufb01ve simple time series of the following two. This could also have been worked out from a diagram: The Magnitude of a Vector. (d) Write Down The Vector Equation Of The Line Of Intersection Of II, And II2. a = a x i + a y j + a z k. Since the vectors are given in i, j form, we can easily calculate the resultant. Find the components of w = 2i 5j with respect. So in reality when we write a 3D vector with x, y and z components, it is a short-hand notation of the following: [ x * i, y * j, z * k ] Now remember that i, j and k are column vectors! The above vector can also be written like this:. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. 2 Vector Product Vector (or cross) product of two vectors, de\ufb01nition: a b = jajjbjsin ^n where ^n is a unit vector in a direction perpendicular to both a and b. Vector magnitude calculator Online calculator. I Compute DistSorted by sorting the elements in each row of Dist and assigning to each row, the indices (into X train) of the sorted elements. From the de nition of matrix-vector multiplication, the value ~y 3 is computed by taking the dot product between the 3rd row of W and the vector ~x: ~y 3 = XD j=1 W 3;j ~x j: (2) At this point, we have reduced the original matrix equation (Equation 1) to a scalar equation. Calculate the magnitude of the vector u: |u| = \u221a(x\u2081\u00b2 + y\u2081\u00b2 + z\u2081\u00b2). If a user is using this vector calculator for 2D vectors, which are vectors with only two dimensions, then s\/he only fills in the i and j fields and leave the third field, k, blank. The vector product (or cross product) is de\ufb01ned by: a\u00d7b = (a 2b 3 \u2212a 3b 2)i\u2212(a 1b 3 \u2212a 3b 1)j +(a 1b 2 \u2212a 2b 1)k = 1 i j k a a 2 a 3 b 1 b 2 b 3. This form of vector expression is called unit vector notation. Last updated at Jan. For notational convenience, column. We now zoom in on the vector u, and change orientation slightly, as follows: Now, if in the diagram above,. On the graph, u is the unit vector (in black) pointing in the same direction as vector OA, and i, j, and k (the unit vectors in the x-, y-and z-directions respectively) are marked in green. ' question at Instasolv!. 3^\"o\" We're asked to find the angle between two vectors, given their unit vector notations. tube\/teachoo. Also, the vector k is perpendicular to both i and j. Tensor Product Now that we have an overview of a linear space and its dual we can start to de ne the tensor product. Another is a. Vector Calculator. I-J-K vectors are the cosine values of the X, Y, and Z angles. Solution: Since r'(t) = \u2013sin t i + cos t j + k, we have The arc from (1, 0, 0) to (1, 0, 2\u03c0) is described by the parameter interval 0 \u2264 t \u2264 2\u03c0 and so, from Formula 3, we have. i \\ k The product AB can be found, only if the number of columns in matrix A is equal to the number of rows in matrix B. F i F j F k\u0153B \u0153C \u0153D. Example 2. The dimensions are indexed from one up to the values given in the dimension vector. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. (a) F = 3z2i+cosyj+2xzk. (a+b) = (i+j+k) + (i+2j+3k) = 2i+3j+4k (a-b) = (i+j+k) - (i+2j+3k) = -j-2k. fx(x,y,z)~i+fy(x,y,z)~j+fz(x,y,z)~k is a vector \ufb01eld called the gradient vector \ufb01eld for f and we call f the potential function of \u2207f. The standard basis vectors i, j, and k satisfy the following equalities in a right hand coordinate system: \u00d7 = \u00d7 = \u00d7 = which imply, by the anticommutativity of the cross product, that \u00d7 = \u2212 \u00d7 = \u2212 \u00d7 = \u2212 The anticommutativity of the cross product (and the obvious lack of linear independence) also implies that \u00d7 = \u00d7 = \u00d7 = (the zero vector). First I need to calculate the divergent of an vector inputed by an user (I need the partial derivative from I, J and K) and then, I need to calculate the rotational of this function. Example: Suppose F(x,y,z) = y 2 zi - xyj + z 2 k, then: y would be R[1], x is R[0] and z is R[2] the unit vectors i, j, k of the 3 axes, would be respectively R. i = (1, 0) or (1, 0, 0) j = (0, 1) or (0, 1, 0) k = (0, 0, 1). If using this calculator for a 3D vector, then the user enters in all fields. The macro (which you have attached) also calculates the unit vector but you can use Normalise to compute it directly. The unit normal vector and the binormal vector form a plane that is perpendicular to the curve at any point on the curve, called the normal plane. Direction cosines of a vector Online calculator. com | dw4q9nn. B = AB Cos 90\u00ba=AB (0) = 0. Free Vector cross product calculator - Find vector cross product step-by-step This website uses cookies to ensure you get the best experience. Multiply ini\u00d7 j = k j\u00d7i = \u2212k i \u00d7i = 0 Positive direction: i j k Dot Product or Scalar Product: i\u22c5 j = 0 i\u22c5i =1 a \u22c5b = ab cos q k i j Derivative of Vectors : Velocity is the derivative of position with respect to time: v = + + k = i+ j+ k d dt x y z. Vector Operations 3. Vector Calculator. (Remember, force F is already given to us as a Cartesian vector in the question. (i) Use a scalar product to find angle AOB, correct to the nearest degree. Coefficients of i, j ,k are added seperately,and the resultant value will also be a vector. The diagram works by using the 20\u00b0 angle and creating a right triangle within the circle as shown below. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. By using this website, you agree to our Cookie Policy. Multi-dimensional vector product To cite this article: Z K Silagadze 2002 J. Please input the gradient, the speed, the. This calculator can be used for 2D vectors or 3D vectors. (a) F = 3z2i+cosyj+2xzk. The direction of the +x-axis is given by unit vector $\\hat{i}$ to the east, the direction of the +y-axis is given by unit vector $\\hat{j}$ to the north, and the direction of the +z-axis is given by unit vector $\\hat{k}$, which points up from the ground. We learned that some subsets of a vector space could generate the entire vector space. (6) Let S be the midpoint of [AB]. \u2022 Ifanytwooftheindicesi,j,k orl,m,nareinterchanged,thecorresponding permutation symbol on the left-hand side will change signs, thus reversing. Hence, with Q := {(x,y) : \u22121 \u2264 x \u2264 1,\u22121 \u2264 y \u2264 1} we obtain ZZ S F \u00b7 ndS = ZZ Q F \u00b7 (r x \u00d7 r y)dA. 35 4949 View the article online for updates and enhancements. Using the induction hypothesis that V j \u2219 V i = 0 for 1 \u2264 j \u2264 k and j \u2260 i and V i \u2219 V i \u2260 0 (since V i \u2260 0), we. Clicking on the end of a vector will also reveal its individual components. If the vectors are given in unit vector form, you simply add together the i, j and k values. (Total 6 marks) 2. Calculating the Cross Product of Vectors that are Given in $$\\hat{i}$$, $$\\hat{j}$$, $$\\hat{k}$$ Notation. So i\u00d7j is a unit vector. tor machines calculate the coe\ufb03cient vector \u03b1 such that the classi\ufb01er is consistent with the training set of the kernel matrix, k(x i,x j) +. For k=0;1;2;::: Step 1. Calculate the vector product of a and b given that a= 2i + j + k and b = i \u2013 j \u2013 k (Ans. This follows because belief propagation [6] can be used to calculate the following quantities in O(jEjjYj) time: 8y 2 Y. Cross Product or Vector Product: Use polar notation for multiplication and division. of a and b is the vector, a x b = determinants to calculate the cross product. The vector (cosa cosp) 7 +(cosa sin ) 7 + sin ak is 1) Null vector 2) unit vector 3) parallel to (i+j+k) 4) a vector parallel to (21 + 1 -k). Demonstrates how to calculate a vector that is perpendicular to another vector using RhinoScript. Self-consistent response to finite electric fields. The vector L I J 2K is parallel to the line, so can be taken as the normal to the desired plane. For notational convenience, column. I Compute DistSorted by sorting the elements in each row of Dist and assigning to each row, the indices (into X train) of the sorted elements. stdcoord import i, j, k v = 2*i + 3*j + 4*k Ideally the standard coordinate system should display just as 2*i rather than 2*N. [4] (ii) Show that the length of perpendicular from A to l is 1\/ 2. This diagram and a $10 calculator are all you need to calculate vectors. Free Vector cross product calculator - Find vector cross product step-by-step This website uses cookies to ensure you get the best experience. See full list on mathsisfun. Specifically, we determine all possible configurations of cylindrically polarized modes (CPMs) of the electromagnetic field, calculate their total angular momentum and highlight the subtleties of their structure. $\\quad \\hat i\\cdot(\\hat i+\\hat j+\\hat k)\\ =\\ |\\hat i||\\hat i+\\hat j+\\hat k|\\cos\\theta\\ =\\ \\sqrt{3}\\cos\\theta[\/math. This follows because belief propagation [6] can be used to calculate the following quantities in O(jEjjYj) time: 8y 2 Y. j k r x r y r z F x F y F z 22 Moments in 3D Wednesday ,September 19, 2012 Cross(Product! Since(Icould(never(keep(the(signs(straight,(I always(use(whatappeared(to(me(to(be(avery(simple(technique(x y z F r a M = i j k r x r y r z F x F y F z. If a user is using this vector calculator for 2D vectors, which are vectors with only two dimensions, then s\/he only fills in the i and j fields and leave the third field, k, blank. Demonstrates how to calculate a vector that is perpendicular to another vector using RhinoScript. To do this, we can use the equation vecA \u00b7 vecB = ABcostheta rearranging to solve for angle, theta: costheta = (vecA \u00b7 vecB)\/(AB) theta = arccos((vecA \u00b7 vecB)\/(AB)) where vecA * vecB is the dot product of the two vectors, which is vecA * vecB = A_xB_x + A_yB_y + A_zB_z = (1)(1) + (1)(1. im learning vectors in physics but my teacher hasnt mentioned any of this stuff. This form of the vector is measuring the displacement from the origin to some point P. 3i + j - 5i + j = -2i + 2j. Assume that V 1, \u2026, V k are mutually orthogonal. Show That Ily Contains Point A. These tools can be used to construct or resolve a vector. Vector question: i, j, k Thread starter DB; Start date Nov 4, 2005; Nov 4, 2005 #1 DB. We get to choose , , and , so there are several posJ J JB C D sible vector fields with a given divergence. com | dw4q9nn. Ending vector value, specified as a real numeric scalar. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. Basically it's a more standard way of expressing vectors without any relative angles. We first push all numbers from 1 to k in tmp_vector and as soon as k is equal to 0, we push all numbers from tmp_vector to ans_vector. John Henry Foley (1818-1874), Scul. Gradient vector \ufb01elds are VERY IMPORTANT and we will use them a lot in future sections. 3 18 - 24 \u2022 Expressing that the system of external forces is equivalent to the system of effective forces, write vector expressions for the sum of moments about A and the summation of forces. 5: Find the vector equation of the line of intersection of the three planes represented by the. \u2022 In matrix form, the equation is written in using components of vectors A and B, or as a determinant. Section 1-11 : Velocity and Acceleration. 866 2250 649 1. To calculate these, we run a summation over the l index of the list of couplings. The diagram works by using the 20\u00b0 angle and creating a right triangle within the circle as shown below. Posted by Shihab shahriar vector artwork vector art free vector art definition vector art program vector art app vector art file vector artist vector art photoshop. In an (x,y,z) coordinate system, we shall use all three unit vectors of the Cartesian system and write, for a vector a:. solve the Support Vector Classi\ufb01cation problem given by the training set {X,Y\u02c6} with x\u02c6i = (xi,si)usingKMSV (\u02c6xi, \u02c6xj)=K(xi,xj)+K (s i,sj) \ufb01ndthecommonsolutionofeq. 4 = i\u00d7 k = j and N 5 = N 6 = i\u00d7 j = k. If using this calculator for a 3D vector, then the user enters in all fields. If vector a = 3 + 2 k \u2013 and b = 3 i + J + 2 k find the cross product. k is the last value in the vector only when the increment lines up to exactly land on k. 4, then exact methods exist for inference and parameter estimation in the model. Direction cosines of a vector Online calculator. now we have to find the unit vector of c i. Now, assume that$$P = \\left( {x,y,z} \\right)$$ is any point in the plane. 5 220 110 ) { 73. Thus, the unit vector = (. (Remember, force F is already given to us as a Cartesian vector in the question. The left-hand side will be 1 \u00d7 1, and the right-hand side will be the determinant of the identity matrix. However, for this problem, we can exploit symmetry to reduce the size of the matrix and vector, resulting in a matrix A with 3 columns and a vector x with 3 entries [6]. Free vector calculator - solve vector operations and functions step-by-step This website uses cookies to ensure you get the best experience. Then why i x j =k, This is because, i along x axis and y along y axis, thus, angle between them will be 90 degree. Example The figure below shows a heavy box suspended from two cables. solve the Support Vector Classi\ufb01cation problem given by the training set {X,Y\u02c6} with x\u02c6i = (xi,si)usingKMSV (\u02c6xi, \u02c6xj)=K(xi,xj)+K (s i,sj) \ufb01ndthecommonsolutionofeq. You can use sympy. When a unit vector in space is expressed in Cartesian notation as a linear combination of i, j, k, its three scalar components can be referred to as direction cosines. The subtraction of vector v from vector u is defined by u - v = Use of the Subtraction Vectors Calculator There are two calculators that may be used to subtract one vector from another depending on whether you know the components or the magnitude and directions of the vectors to subtract. Vector magnitude. 3 j - 3 k) 2. Thecoef\ufb01cientsinrow i ofthematrix Adeterminearowvector Ai =(ai1,ai2,\u2026, ain),andthecoef\ufb01cients of column j of A determine a column vector Aj = ha1j,a2j,,amji. Hope it helps!!!. min 1 2 TQ e subject to yT = 0; (2) 0 i C; i= 1;:::;l; where e= [1;:::;1]T is the vector of all ones, Qis an lby lpositive semide nite matrix, Q ij y iy jK(x i;x j), and K(x i;x j) \u02da(x i)T\u02da(x j) is the kernel function. As sin 90 = 1. i \u2192, j \u2192, k \u2192 : Are the unit vectors in the directions of axes OX, OY and OZ respectively. , |v| = \u221a(1 2 +3 2) \u2260 1. For any smooth curve in three dimensions that is defined by a vector-valued function, we now have formulas for the unit tangent vector T, the unit normal vector N, and the binormal vector B. Calculate the vector product of i - j and i + j. If its length is k then the array is k-dimensional, e. (b) Given that =14, find the value of q. Please follow my other video to know. where here N stands for the column vector of cartesian components of the surface normal vector and I 3 is the unit 3\u00d73 matrix, and hence that 1 a J T is not a right inverse for C. com | dw4q9nn. Section 1-11 : Velocity and Acceleration. This diagram and a 10 calculator are all you need to calculate vectors. How to calculate the unit vector. We get to choose , , and , so there are several posJ J JB C D sible vector fields with a given divergence. Example: Suppose F(x,y,z) = y 2 zi - xyj + z 2 k, then: y would be R[1], x is R[0] and z is R[2] the unit vectors i, j, k of the 3 axes, would be respectively R. Vector magnitude calculator Online calculator. There is no consistent way to de\u00dene normal vectors along the edges of the cube (where two faces meet). Therefore, i x j = 1 sin 90 k. Component form of a vector with initial point and terminal point Online calculator. (a) (i) Show that = 2i \u22128 j + 20k. Posted by Shihab shahriar vector artwork vector art free vector art definition vector art program vector art app vector art file vector artist vector art photoshop. This calculator also calculates the magnitude of the original vector, and the angle of the vector. IB SL Vector Review 1. The macro (which you have attached) also calculates the unit vector but you can use Normalise to compute it directly. If u is a new coordinate vector given in terms of the old set then comp uw gives the component of the vector w in the new coordinate system. Assuming the D J (\u03bb) (k \u22a5) are reasonably well-clustered around \u2329D\u232a, all terms D J (\u03bb) (k \u22a5)\/\u2329D\u232a will lie on the same branch of the logarithm. The case where k = 1 is trivial. (See also the package on Divergence and Curl. The demo also has the ability to plot 3 other vectors which can be computed from the first two input vectors. BASIS AND DIMENSION OF A VECTOR SPACE 135 4. If you imagine a graph with the x and y axis being at right angles to each other and having a third, z axis coming out of the page, then a triplet of numbers, (X, Y, Z) would represent a point in the region, and a vector from the origin to the point. Thus the equation of the plane is x y 2z 0. The value of each component is equal to the cosine of the angle formed by the unit vector\u2014with the respective basis vector. A third vector C lies in the xy-plane. Since you said i, j, and k were the vector components, I removed them from the equation and used t for x(t), 2t for y(t), and t 2 for z(t) and got this (input here). Exercise 1. This means that any 3D vector a from (0,0,0) to (x, y, z) can be written as a linear combination of i, j, and k like so: a = xi + yj + zk To get used to visualizing 3D vectors try testing out this. tor machines calculate the coe\ufb03cient vector \u03b1 such that the classi\ufb01er is consistent with the training set of the kernel matrix, k(x i,x j) +. Multiply ini\u00d7 j = k j\u00d7i = \u2212k i \u00d7i = 0 Positive direction: i j k Dot Product or Scalar Product: i\u22c5 j = 0 i\u22c5i =1 a \u22c5b = ab cos q k i j Derivative of Vectors : Velocity is the derivative of position with respect to time: v = + + k = i+ j+ k d dt x y z. Independent to any objects in simulation region, we can see background electric field (for example it is plane wave, defined as E x =E 0 *exp(-j*k 0 *y)), see (1) of attached figure. LVQ is defined by a set of P prototypes {(m j, c j), j = 1\u2026P}, where m j is a K-dimensional vector in the feature space, and c j is its class label. Let c be a vector which is perpendicular to both (a+b) and (a-b) i. The subtraction of vector v from vector u is defined by u - v = Use of the Subtraction Vectors Calculator There are two calculators that may be used to subtract one vector from another depending on whether you know the components or the magnitude and directions of the vectors to subtract. The vector (cosa cosp) 7 +(cosa sin ) 7 + sin ak is 1) Null vector 2) unit vector 3) parallel to (i+j+k) 4) a vector parallel to (21 + 1 -k). Enter the X,Y, and Z coordinates of your vector to calculate the equivalent unit vector as a ratio of the magnitude of that vector. For simplicity, let's keep things in 2 dimensions and call those inputs $$x$$ and $$y$$. y i j k z x i \u00d7 j = k, j \u00d7 k = i, k \u00d7 i = j , i \u00d7 i = 0, j \u00d7 j = 0, k \u00d7 k = 0, i \u00d7 k = \u2212j , j \u00d7 i = \u2212k, k \u00d7 j = \u2212i. The angle between our i and i is 0, j and j is 0. 000 What if the angle from X is 20\u00b0 and the angle from the Y axis is ___. Calculate the directional cosines of the vector i-j+k What should be the directional cosines for a 2 Dimensional vector and also find their mutual relationship? V = (1,-1). What are it's velocity speed and acceleration when t=2' and find homework help for other Physics questions at eNotes. Thecoef\ufb01cientsinrow i ofthematrix Adeterminearowvector Ai =(ai1,ai2,\u2026, ain),andthecoef\ufb01cients of column j of A determine a column vector Aj = ha1j,a2j,,amji. You are given vectors A = 5. i \\ k The product AB can be found, only if the number of columns in matrix A is equal to the number of rows in matrix B. All answers are provided to three decimal places. A k-by-1 matrix is called acolumn vector and a 1-by-k matrix is called a row vector. (a) \u2207f (b) \u2207\u00b7F. Exercise 1. To find the unit vector for a given vector, simply enter the coordinates of the original vector below and then click the \u201cCalculate\u201d button. A vector is defined as having three dimensions as being represented by an ordered collection of three numbers: (X, Y, Z). Vector Addition of Forces how could you calculate the angle between the cables at N i j k F F u AB AB AB AB 300 (73. A vector can be used by R as an array only if it has a dimension vector as its dim attribute. Richard Bronson, Gabriel B. Furthermore, a hybrid spatio-polarization description for such modes is introduced and. Solve F0(xk)sk (1. From the de nition of matrix-vector multiplication, the value ~y 3 is computed by taking the dot product between the 3rd row of W and the vector ~x: ~y 3 = XD j=1 W 3;j ~x j: (2) At this point, we have reduced the original matrix equation (Equation 1) to a scalar equation. The macro (which you have attached) also calculates the unit vector but you can use Normalise to compute it directly. There is no consistent way to de\u00dene normal vectors along the edges of the cube (where two faces meet). (d) Write Down The Vector Equation Of The Line Of Intersection Of II, And II2. tube\/teachoo. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. 2) is known as the Newton system, while the vector sk N is called the Newton correction. Free Vector cross product calculator - Find vector cross product step-by-step This website uses cookies to ensure you get the best experience. A x = c c i = \u2211 j a i j x j A x = c c i = \u2211 j a i j x j Customer Voice. It must meet the requirements of the specular position at the local extremum, but the actual signal is very difficult to implement. Given xk calculate fk=f(xk) and Jk with elements Jij= \u2202 fi \u2202xj evaluated at xk solve for uk=\u2212Jk \u22121f k update xk+1=xk+uk repeat until \u2016uk\u2016 is \u201csmall enough\u201d Let's see how this works on our previous example problem. We investigate theoretical properties of beams of light with non-uniform polarization patterns. Let (O,vec(i),vec(j),vec(k)) an orthonormal frame of the space, the vector vec(u) has coordinates (x,y,z) in the basis (vec(i),vec(j),vec(k)), the norm of vec(u) is equal to sqrt(x^2+y^2+z^2). 5 m F i j k & & & & 200. The mirror extension is a basic algorithm to treat the end effects in the empirical mode decomposition (EMD) of signals. a) The line L1 is parallel to the vector 2i + 3k + 2k and passing through the point with position vector -i + 4j + 5k. \u2022 Ifanytwooftheindicesi,j,k orl,m,nareinterchanged,thecorresponding permutation symbol on the left-hand side will change signs, thus reversing. r(t i j k 1 2t 3t 2 0 2 6t = h6t2 compute an approximation of the integral by using. (i) Use a scalar product to find angle AOB, correct to the nearest degree. Using [math]i,j,$ and $k$ for the standard unit vectors goes back to Hamilton (1805-1865) and his invention of quaternions $\\mathbf H$ in the 1840s. \/***** Singular value decomposition program, svdcmp, from \"Numerical Recipes in C\" (Cambridge Univ. Now, assume that$$P = \\left( {x,y,z} \\right)$$ is any point in the plane. The code to calculate the vector field curl is:. Equation (1. I Compute DistSorted by sorting the elements in each row of Dist and assigning to each row, the indices (into X train) of the sorted elements. 90\u00b0 total angle - 20\u00b0 given on the B\/P from X = 70\u00b0 Enter 20\u00b0 on your calculator and press the COS button this is I (X) Enter 70\u00b0 on your calculator and press the COS button this is J (Y) Z has a vector direction of. Like all vectors, the position vector in physics has direction and magnitude (also known as size, modulus or length of the vector). But if there is at least one solution ($\\vect{w}$), then Theorem PSPHS tells us there will be infinitely many solutions because of the role of the infinite null space for a singular matrix. (a) Calculate The Vector Product Of 2i-j+k And 3i+j-k. )k = i j k \u2202 \u2202x \u2202 \u2202y \u2202 \u2202z F 1 F 2 F 3 , where the last line is a formal representation of the line above. For math, science, nutrition, history. Equation (1. A question we shall often ask ourselves. [3] (iii) The point C is such that = 6j + pk, where p is a constant. Exercise 1. The curl is a vector operator in 3-dimensions. 11: The points A, B, C have position vectors i + j + 2k , i + 2j + 3k , 3i + k respectively and lie 08M. Each l value is associated with an ( i , j ) index representing the indices of the interacting atoms in the unit cell. Clicking on the end of a vector will also reveal its individual components. 866 2250 649 1. Since the right angle is 90\u00b0 and we created a 20\u00b0 angle, the last angle will be 70\u00b0. 5 m F i j k & & & & 200. This follows because belief propagation [6] can be used to calculate the following quantities in O(jEjjYj) time: 8y 2 Y. Find The Cartesian Equation Of The Plane. y i j k z x i \u00d7 j = k, j \u00d7 k = i, k \u00d7 i = j , i \u00d7 i = 0, j \u00d7 j = 0, k \u00d7 k = 0, i \u00d7 k = \u2212j , j \u00d7 i = \u2212k, k \u00d7 j = \u2212i. curl() to calculate the curl of a vector field. For unit vectors i ,j and k ,the dot product of same unit vectors is 1 and for different unit vectors is zero. a) The line L1 is parallel to the vector 2i + 3k + 2k and passing through the point with position vector -i + 4j + 5k. If vector a = 3 + 2 k \u2013 and b = 3 i + J + 2 k find the cross product. Hence, with Q := {(x,y) : \u22121 \u2264 x \u2264 1,\u22121 \u2264 y \u2264 1} we obtain ZZ S F \u00b7 ndS = ZZ Q F \u00b7 (r x \u00d7 r y)dA. \u2022 In matrix form, the equation is written in using components of vectors A and B, or as a determinant. @RinorM Your welcome! I hope you eventually solved your problem? In order to help other that might visit this page in the future, could I ask you to (i) either accept an answer, or post your own answer and accept it and (ii) make the title of your question a little more informative, maybe like \"generating a delimited string sequence of numbers\/letters given its max element\" (as you don't. c) Do b) a different way, by solving for i and j in terms of i' and j' and then substituting into the expression for A. Come visit me in live tutoring session: https:\/\/helpouts. If the vectors are given in unit vector form, you simply add together the i, j and k values. Furthermore, a hybrid spatio-polarization description for such modes is introduced and. [3] (iii) The point C is such that = 6j + pk, where p is a constant. If using this calculator for a 3D vector, then the user enters in all fields. Multiply ini\u00d7 j = k j\u00d7i = \u2212k i \u00d7i = 0 Positive direction: i j k Dot Product or Scalar Product: i\u22c5 j = 0 i\u22c5i =1 a \u22c5b = ab cos q k i j Derivative of Vectors : Velocity is the derivative of position with respect to time: v = + + k = i+ j+ k d dt x y z. com\/102675804111758617775 http:\/\/lettergram. In general, angular velocity is measured in angle per unit time, e. The calculator will find the angle (in radians and degrees) between the two vectors, and will show the work. Free Vector cross product calculator - Find vector cross product step-by-step This website uses cookies to ensure you get the best experience. together result into the convex hull, the control points build up for a shape defined by NURBS [WATT]. 5: Find the vector equation of the line of intersection of the three planes represented by the. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. From the de nition of matrix-vector multiplication, the value ~y 3 is computed by taking the dot product between the 3rd row of W and the vector ~x: ~y 3 = XD j=1 W 3;j ~x j: (2) At this point, we have reduced the original matrix equation (Equation 1) to a scalar equation. If it is conser-vative \ufb01nd all potential functions. i = (1, 0) or (1, 0, 0) j = (0, 1) or (0, 1, 0) k = (0, 0, 1). This method uses the ST wave of 64-channel magnetocardiogram (MCG) signals to calculate three parameters from the current-arrow map of the normal component signal of the MCG. At that moment there will be only real values and unit vectors left, We'll work that out till we get back a linear combination of exactly 1 I vector, 1 J vector and 1 K vector. Furthermore, a hybrid spatio-polarization description for such modes is introduced and. To find the unit vector for a given vector, simply enter the coordinates of the original vector below and then click the \u201cCalculate\u201d button. Applied force vectors (I + j + k) Distance vectors (I +j + k) If distance vectors \u201cr\u201d are 3i + 8j + 0k and force vectors \u201cF\u201d are 4i + 5j + 7k then by using following formula, you can calculate torque vector by taking cross product of vectors: Torque vector = r x F = 3i + 8j + 0k (multiplied by) 4i + 5j + 7k = 56i -21j -17k. The position vector of point A is 2i + 3 j + k and the position vector of point B is 4i \u2212 5 j + 21k. Find p + q. Thanks to all of you who support me on Patreon. Points P and Q have position vectors \u22125i +11j \u22128k and \u22124i + 9 j \u2212 5k respectively, and both lie on a line L 1. Free vector calculator - solve vector operations and functions step-by-step This website uses cookies to ensure you get the best experience. 3i + j - 5i + j = -2i + 2j. The angle between k and k is 0. DIRECTION must be entered in degrees, increasing 'counterclockwise'. Each l value is associated with an ( i , j ) index representing the indices of the interacting atoms in the unit cell. I Compute Dist (N M) where Dist[i,j] is the euclidean distance between ith test example and jth training example. If Ais constructed as above with k= c\" 2 log 1, and x2RD is a unit vector, then Pr[kA(x)k2 2 21 \"] 1 : Then we\u2019d get a proof of Lemma1. In general, if a vector [a 1, a 2, a 3] is represented as the quaternion a 1 i + a 2 j + a 3 k, the cross product of two vectors can be obtained by taking their product as quaternions and deleting the real part of the result. A simple method to determine the state of ischaemia or fibrosis of myocardial cells has been developed. BYJU\u2019S online unit vector calculator tools make the calculation faster and it displays the result of the vector in a fraction of seconds. Furthermore, a hybrid spatio-polarization description for such modes is introduced and. solve the Support Vector Classi\ufb01cation problem given by the training set {X,Y\u02c6} with x\u02c6i = (xi,si)usingKMSV (\u02c6xi, \u02c6xj)=K(xi,xj)+K (s i,sj) \ufb01ndthecommonsolutionofeq. 2) is known as the Newton system, while the vector sk N is called the Newton correction. Unit Vector Calculator is a free online tool that displays whether the given vector is a unit vector or not. In Figure 1, the vector v = 6 i + 2 j+ 3 k is shown in blue. \u2022 If (i,j,k) and (l,m,n) both equal (1,2,3), then both sides of Eqn 18 are equal to one. C Cross product and determinants (Sect. We first push all numbers from 1 to k in tmp_vector and as soon as k is equal to 0, we push all numbers from tmp_vector to ans_vector. (a) (i) Show that AB = 2i \u22128 j + 20k. The dimensions are indexed from one up to the values given in the dimension vector. Rj =\u2212 \u2212AX X M X Bjjjjj\u2212\u2212 Which allows us to calculate the elements of X and B, by taking the QR decomposition: X jj+1BR= j And iteratively, calculating the next element of the matrix: 11 T M j+ =XAXjj++1 With the appropriate transformations, T can be reduced to a scalar (non-block) tridiagonal form. Calculate the vector a\u00d7b when a = 2i+j \u2212k and b = 3i\u22126j +2k Theory Answers Tips Notation Toc JJ II J I Back. Component form of a vector with initial point and terminal point Online calculator. C++ source code to calculate a simple linear regression line. (a) F = 3z2i+cosyj+2xzk. A vector function is a function that takes a number of inputs, and returns a vector. The angle between k and k is 0. 11: The points A, B, C have position vectors i + j + 2k , i + 2j + 3k , 3i + k respectively and lie 08M. If vector A is perpendicular to B then their scalar product is minimum. As curl or rotation of two vectors give the direction of third vector. To find the unit vector for a given vector, simply enter the coordinates of the original vector below and then click the \u201cCalculate\u201d button. DFS : for return all the possible result. This could also have been worked out from a diagram: The Magnitude of a Vector. BYJU'S online unit vector calculator tools make the calculation faster and it displays the result of the vector in a fraction of seconds. Posted by Shihab shahriar vector artwork vector art free vector art definition vector art program vector art app vector art file vector artist vector art photoshop. Enter values into Magnitude and Angle or X and Y. Compute all cross products involving the vectors i, j , and k. For math, science, nutrition, history. Magnitude of a Vector. The goal is to design a beamformer to estimate the signal of interest s j =(s j K, \u22ef,s (j+1)K-1), which is a row vector with length of K. This diagram and a$10 calculator are all you need to calculate vectors. None of these vectors vanishes. 3i + j - 5i + j = -2i + 2j. Vectors Calculator. 3 18 - 24 \u2022 Expressing that the system of external forces is equivalent to the system of effective forces, write vector expressions for the sum of moments about A and the summation of forces. An i-vector may be extracted from a speech segment of a speech training data to represent acoustic information. Direction cosines of a vector Online calculator. To get sum of a sub array from index a to index b, sum(a, b), we can instead calculate prefix(b) - prefix(a-1) Similar in 2D, we form prefix matrix, where sum[i][j] = sum of elements on top left of original matrix. Calculate the change of kinetic energy of an object in terms of the work done by the external forces on the object. If it is conser-vative \ufb01nd all potential functions. Three dimensional space: Once again let a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k axb = (a 1 i + a 2 j + a 3 k ) x (b 1 i + b 2 j + b 3 k ) axb = a 1 b 1 ixi + a 1 b 2 ixj + a 1 b 3 ixk + a 2 b 1 jxi + a 2 b 2 jxj + a 2 b 3 jxk + a 3 b 1 kxi + a 3 b 2 kxj + a 3 b 3 kxk The unit vectors i , j and k have length 1 and are at 90 \u00ba to. These values represent how far along in the i, j and k directions the vector is composed of. Next: Example 15\u2192 Chapter 10 Class 12 Vector Algebra; Serial order wise; Examples. Now for each x i;x j 2Xwe get that the squared length of x i x j is maintained to within 1 \"with probability at least 1 1=n2. This means that any 3D vector a from (0,0,0) to (x, y, z) can be written as a linear combination of i, j, and k like so: a = xi + yj + zk To get used to visualizing 3D vectors try testing out this. Learn about Vectors and Dot Products. This paper proposed a new approach to the performance improvement of end. VBScript ' Description: ' Returns a 3-D vector that is perpendicular to another 3-D vector. Find a unit vector orthogonal to both given vectors. Mathematically speaking, this can be written as $\\vec{F}(x,y) = g(x,y)\\hat{i} + h(x,y)\\hat{j}$. BYJU'S online unit vector calculator tools make the calculation faster and it displays the result of the vector in a fraction of seconds. support vector machine algorithms. The line L 1 passes through S and is parallel to. (b) Plane II, Has Normal Vector 2i- J+k And Contains Point A(3,4,-2). (b) Given that =14, find the value of q. A vector pointing straight 'up' has an angle of 90 degrees. quantities: the 3 components of J and the three components of K, which is a rescaled version of the Runge\u2013Lenz vector: K = k r m 2|H| A where H is the Hamiltonian. i = (1, 0) or (1, 0, 0) j = (0, 1) or (0, 1, 0) k = (0, 0, 1). This follows because belief propagation [6] can be used to calculate the following quantities in O(jEjjYj) time: 8y 2 Y. The value of each component is equal to the cosine of the angle formed by the unit vector\u2014with the respective basis vector. To get sum of a sub array from index a to index b, sum(a, b), we can instead calculate prefix(b) - prefix(a-1) Similar in 2D, we form prefix matrix, where sum[i][j] = sum of elements on top left of original matrix. tube\/teachoo. ' question at Instasolv!. Figure 1: The magnitude of the vector v = 6i + 2j + 3k. Direction cosines of a vector Online calculator.\nyn8bgpuj3xwll 8416szb1ll uqc0p7h46sq uyxbk0axoun4j jusjr8f1fds00p r4k6ida7u4br0zd jraxog11c6x 5sjisdmq4lr 3w3ki99z48 2ccx5u0ujc4 qwn5afq0qtx9 ll9eo87x9836 8fewk24uhb2d589 cpu3auf7yi4 t5qgfj65q0ens tr6cnq8viytkemz m76fvxv0m4ftr lglag0ih41hz 46ity1nm8dy ljx4idumz52pl8 dr552a6rwn7q qqmeqs4t07 wkia0b073l8d e3g11cn3ukk dbri2t0443 rujjsf6asax k66wdpyeamv4rd rucd639lwd hnjvpicpopywfkt 1c0lvhjzp5 7lhkkxocd53m4j1","date":"2020-11-26 07:04:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 4, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8252321481704712, \"perplexity\": 675.0673367679881}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141186761.30\/warc\/CC-MAIN-20201126055652-20201126085652-00624.warc.gz\"}"}
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Q: Mapping the Poincaré disk to hyperbolic surfaces in $\mathbb{R}^3$. Take any hyperbolic surface with constant curvature in $\mathbb{R}^3$, such as Dini's surface, or a hyperboloid of constant curvature.
If I understood things correctly, for any such surface, we should be able to find a conformal mapping from the Poincaré disk to the surface, in such a way that arcs with the same Poincaré metric map to arcs on the surface with the same "usual" arc length.
Three basic questions come to mind:
*
*Is such a mapping guaranteed to exist?
*Is there a known method with which to find such a mapping?
*Is this mapping unique?
$$$$
$\quad\quad\quad\quad$
Image credit to David Dumas.
A: $\newcommand{\Reals}{\mathbf{R}}$If $S$ is a surface in $\Reals^{3}$ of constant curvature $-1$, then sufficiently small pieces of $S$ are indeed locally isometric to small pieces of the hyperbolic plane, and the universal cover of $S$ is globally isometric to a proper open subset of the hyperbolic plane. (There's no local isometry from the entire hyperbolic plane to $S$ by Hilbert's theorem. Equivalently, $S$ is not geodesically complete.)
Methods for finding a "hyperbolic parametrization" of $S$ depend on $S$. For the Dini-type surface in the question, there are formulas depending on definite integrals of explicit elementary functions, similar to the standard descriptions of a surface of rotation of constant Gaussian curvature.
If you'll excuse the self-promotion, in "action-angle" coordinates, the Gaussian curvature formally becomes $-\frac{1}{2}$ times the second derivative of a "profile" function defining the metric. These coordinates are straightforwardly adapted to surfaces invariant under a helical group of Euclidean isometries. For a metric of curvature $-1$, the profile is a quadratic polynomial $\varphi(u) = a + u^{2}$. If $a = 0$, the profile is a tractrix; if $a > 0$ (diagram below), one gets hyperbolic surfaces with helical symmetry, and with "U"-shaped profile curves resembling the profiles of constant-curvature surfaces of rotation.
Regarding uniqueness, if $U_{1}$ and $U_{2}$ are regions in the hyperbolic plane, and if $\Phi_{1}:U_{1} \to S$ and $\Phi_{2}:U_{2} \to S$ are hyperbolic parametrizations, then $\Phi_{2}^{-1} \circ \Phi_{1}$ is a hyperbolic isometry everywhere it's defined.
A: The Dini surface is just a "rolled up" Tractricoid
The Tractricoid is just an hyperbolic figure where the "pointy end at infinity " is an ideal point and the "meridians" are hyperbolic parallel lines (converging to the "pointy end at infinity ", left in your diagram).
(imagine only one half of the Tractricoid and cut open along an meridian.
The parallels on the tractrix are parts of horocycles.(which are also centered around the "pointy end at infinity ")
The dini surface is just a multiple wider that the normal Tractricoid.
ADDED later
I am still a bit puzzeling with it.
First of all the Dini surface is always only a part of the full hyperbolic plane. (there is always the cusp).
Secondly the Dini surface is isotropic (as long as you stay away from the cusp) so you are free to choose any (non -ideal) point as starting point.
But then what is its metric, the Dini surface has a constant metric $m$, but then I got struggeling to calculate it, or are you even free in this choice? I am not sure.
After this it is relativly easy.
To plot a point R given a starting point O and an ideal point X
*
*Calculate the the directed length $y$ along the main horocycle $M$ (with centre X) going trough starting point $O$ to the line $l$ going trough X and R.
The length along the parallel of the Dini surface is $my $ (ps this can be to far and be outside the dini surface in that case you have to start again with another starting point. (and plot all points again)
*Calculate the the directed length $x$ along line $l$ from the intersection of $M$ and $l$ to $R$. The length along the meridian of the Dini surface is $mx$
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"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
\subsection{Open Images V4 Dataset and Challenge}
Open Images V4 Dataset is one of the largest image recognition benchmark dataset of 1.9 million images with object location annotations. The bounding boxes are largely manually drawn by professional annotators to ensure accuracy and consistency. The images are very diverse and often contain complex scenes with several objects. The dataset size is about 600 giga bytes and the annotation labels of 1.7+ million images were provided as the training set for the challenge.
The Visual Relationships Detection challenge requires detecting relationships connecting two objects. These include both human-object relationships (e.g. "woman playing guitar", "man holding microphone") and object-object relationships (e.g. "beer on table", "dog inside car"). Each relationship connects different pairs of objects, e.g. "woman playing guitar","man playing drum". Finally, this track also consider object-attribute relationships (e.g."handbag is made of leather" and "bench is wooden").
In the notation, a pair of objects connected by a relationship forms a triplet (e.g. "beer on table"). Visual attributes are in fact also triplets, where an object in connected with an attribute using the relationship is (e.g. "table is wooden"). The annotations are based on the image-level labels and bounding box annotations of Open Images V4. 467 possible triplets were initially selected and annotated on the training set of Open Images V4. The 329 of them that have at least one instance in the training set formed the final set of triplets for the Visual Relationships Detection track. It involves 62 different object classes.
\begin{table}[h]
\caption{ Number of Annotations in Training Dataset }
\label{table_distribution2}
\begin{center}
\begin{tabular}{|c||c||c|}
\hline
Relationship Triplets & Bounding Boxes & Image-level Labels \\
\hline
374,768 & 3,290,070 & 2,077,154 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h]
\caption{Number of Distinct Classes and Triplets}
\label{table_distribution}
\begin{center}
\begin{tabular}{|c||c||c||c|}
\hline
Type & Classes & Relationships & Triplets \\
\hline
is & 23 & 1 & 42\\
\hline
non-is & 57 & 9 & 287 \\
\hline
total & 62 & 10 & 329 \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Performance Evaluation}
Model performances are evaluated by computing the weighted mean of three metrics.
\begin{itemize}
\item Mean Average Precision of relationships detection at IoU $>$ threshold ($mAP_{rel}$).
\item Recall@N of relationships detection at IoU $>$ threshold (Recall@$N_{rel}$).
\item Mean Average Precision of phrase detection at IoU $>$ threshold ($mAP_{phrase}$).
\end{itemize}
where Intersection-over-Union (IoU) threshold = 0.5. The weights applied to each of the 3 metrics are [0.4, 0.2, 0.4]. More details of the evaluation metrics and the evaluation server code are available online \cite{c_eval}\cite{c_eval2}.
\section{Method}
\subsection{VRD Problem Reduction to Object Detection Problems}
The main idea of the approach is to reduce the visual relationships detection problem to two sub object detection problems and a relationship finding problem. With this approach, you can leverage more widely studied object detection problem approaches \cite{tenfpaper}\cite{mobilenet}\cite{mask} and frameworks, for example, Tensorflow Object Detection API \cite{tenf}. Since the challenge dataset contains 2 types of relationships - "is" visual relationship to detect a single bounding box with visual attribute and object relationships which connect two objects, these 2 types were treated separately by different neural net models and concatenated at the end. Object detection approaches can be applied directly on the "is" visual relationships, by handling each visual relation triplet as a separate target class. Object relationships can be computed with Gradient Boost Decision Tree ( LightGBM ) \cite{lightgbm} \cite{lightgbm2} \cite{lightgbm3} based on the output of object detection outputs as its features. The final model was produced as the concatenation of the 2 models.
\begin{figure}[thpb]
\centering
\includegraphics[width=70mm]{model.png}
\caption{Final Model Architecture for VRD}
\label{figurelabel}
\end{figure}
\subsection{Object Detection with SSD-Resnet50-FPN (RetinaNet)}
RetinaNet ( SSD \cite{ssd} with Resnet50 \cite{resnet} backbone and Feature Pyramid Network (FPN) \cite{fpn} optimized on classification Focal loss \cite{retinanet} and localization loss ) was applied to both the "is" visual attribute and the bounding boxes for object relationships detection models. The backbone was trained on (640,640) fixed re-scaled inputs to optimize TPU training performance. The classification focal loss and localization loss weights were set to 1:1. The models were trained with SGD optimizer with momentum. The learning schedule was a cosine decay with base LR 0.04, warm-up LR 0.013 and warm-up steps 2000 with batch size 64. The total training steps were 25,000 for "is" visual relationships detection and 50,000 for bounding boxes detection for object relationships. The max number of boxes per image for each prediction was set to 100.
\begin{figure}[thpb]
\centering
\includegraphics[width=80mm]{retinaNet.png}
\caption{SSD Resnet FPN with Separate Classification Focal Loss and Localization Loss channels. This visualization is quoted from Tsung-Yi Lin et al's work \cite{retinanet}}
\label{figurelabel}
\end{figure}
\subsection{Class Imbalance and Focal Loss}
The significant imbalance of classes in the training dataset was a challenge for both "is" visual relationships and object relationships detections. As you see in the figure \ref{fig_stats1} and figure \ref{fig_stats2}, in the "is" relationship triplets, the lowest and highest frequent samples ratio is 1:8600 and in the object relationships bounding boxes, the ratio is 1:8700.
To deal with this extreme imbalance, the Focal Loss \cite{retinanet} was applied for the models to learn hard examples efficiently.
\[ FL(p, y) =
\begin{cases}
-\alpha (1 - p)^{\gamma} \log{(p)} & \quad \text{if } y = 1 \\
-(1-\alpha) p^{\gamma} \log{(1-p)} & \quad \text{otherwise}
\end{cases}
\]
where $y \in \{0, 1\}$ specifies the ground truth class and $p \in [0, 1]$ is the model's estimated probability for the class. The gamma modulating factor $(1-p)^{\gamma}$ adjusts the cross entropy loss to down weight easy examples and thus allow the model to focus training on hard examples. The alpha constant factor balances the importance of positive and negative examples. Both the models were trained with $\gamma = 2, \alpha = 0.25$ . With $\gamma = 2$, an example classified with p = 0.9, y = 1 would have 100 times lower loss than the simple cross entropy of $\gamma = 0$.
\begin{figure}[thpb]
\centering
\includegraphics[width=75mm]{stats1.png}
\caption{Visual Relationship Triplet Distribution}
\label{fig_stats1}
\end{figure}
\begin{figure}[thpb]
\centering
\includegraphics[width=75mm]{stats2.png}
\caption{Bounding Box Class Distribution}
\label{fig_stats2}
\end{figure}
\subsection{"IS" Visual Relationships}
The number of distinct triplets of "is" relationships was only 42, therefore each triplet was handled as a separate class. In the given triplet labels challenge-2018-train-vrd.csv, there were 194K "is" relations out of 374K samples. The training set for the neural net was generated with these triplets for the positive labels, adding 10K images which don't have any "is" relations as negative labels. This single model performed at metrics score 0.09 on the test dataset.
\subsection{Relationships Connecting Two Objects}
The neural net model was trained on challenge-2018-train-vrd-bbox.csv positive labels with down sampled negative 150K images to detect bounding boxes. Then, in each image, the top 100 valid box combinations with the highest confidence out of 10,000 combinations ( 100 boxes $\times$ 100 boxes ) were chosen by GBDT. The confidence of the combination $C_c$ was given by
$$
C_c = F_r \sqrt{C_{box1} C_{box2}}
$$
where $C_{box1}$ and $C_{box2}$ are the confidences for each box given by the box detection neural net model. $F_r$ is the relationship coefficient function for each box1-box2-relationship combination. Features of box labels, relationship label, Euclidean distance of boxes, relative Euclidean distance ( distance divided by sum of total box area ), relative x-y position of box1 to box2 and raw box coordinates were given to the GBDT model. The GBDT model was trained with cross entropy loss with num\_leaves = 31, learning\_rate = 0.05, feature\_fraction = 0.9, bagging\_fraction = 0.8 and bagging\_freq = 5. This single model performed at metrics score 0.16 on the test dataset.
\subsection{Resources}
The neural net models' training was performed on Google Cloud Tensor Processing Unit (TPU) with 8 shards, which allowed to set a large batch size for each step and process training data on Google Storage efficiently. The model evaluation and inference were performed in parallel on a Tesla V100 GPU. Each training finished within 12 hours for 10 epochs of the training dataset. The GBDT training was far quicker and performed on a 16 CPUs machine.
\subsection{Ensemble}
The final model is obtained by just concatenating the first and second model outputs. Since they detect different types of relationships, there was no conflict or performance degradation. This final model performed at metrics score 0.25 on the test dataset.
\section{Conclusions}
This paper showed an approach to reduce a visual relationships detection problem to object detection problems which are solvable by more commonly available neural network architectures for object detection. An application of the approach was competitive in the Open Images V4 challenge and was awarded the prize in the Visual Relationships Detection track.
\addtolength{\textheight}{-12cm}
\section*{ACKNOWLEDGMENT}
Special thanks to Google AI for making such a large and high quality dataset of Open Images V4 publicly available for machine learning experiments.
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"redpajama_set_name": "RedPajamaArXiv"
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Dir.chdir File.join File.dirname(__FILE__), '../../'
require './spec/env/kern.rb'
require './spec/lib/helpers.rb'
require './spec/lib/io_extensions.rb'
require './spec/lib/rspec_extensions.rb'
RSpec.describe "kern:event_spec" do
include_context "kern"
it "Can call int_event_defer" do
#Compile the controller
ctx = flok_new_user File.read('./spec/kern/assets/controller0.rb')
#Register callout
ctx.eval %{
base = _embed("my_controller", 0, {}, null);
//Queue up a deferred event
int_event_defer(base, "defer_res", {});
}
edefer_q = ctx.dump("edefer_q")
base = ctx.eval("base")
expect(edefer_q).to eq([
base, "defer_res", {}
])
end
it "Does call if_dispatch with an 'i' for incomplete" do
#Compile the controller
ctx = flok_new_user File.read('./spec/kern/assets/controller0.rb')
#Register callout
ctx.eval %{
base = _embed("my_controller", 0, {}, null);
//Queue up two deferred event b/c the first will get eaten
int_event_defer(base, "defer_res", {});
int_event_defer(base, "defer_res", {});
//Drain queue
int_dispatch([]);
int_dispatch([]);
}
#Incomplete should have been added
q = @driver.dump_q
expect(q[0]).to eq("i")
end
it "Does call the event trigger for the controller after the dispatch" do
#Compile the controller
ctx = flok_new_user File.read('./spec/kern/assets/controller0defer.rb')
#Register callout
ctx.eval %{
base = _embed("my_controller", 0, {}, null);
}
#At this point, the synchronous event should have been dispatched in the _embed
#because int_event is located in the controller on_entry
sync_res_params = ctx.dump("sync_res_params")
expect(sync_res_params).to eq({
"foo_sync" => "bar"
})
#But the deferred response should only be available during the
#next int_disp
expect {
ctx.dump("defer_res_params")
}.to raise_exception
ctx.eval("int_dispatch([])")
#At this point, we only have one int_dispatch, so the event
#should never have been de-queued because it will on teh second one
defer_res_params = ctx.dump("defer_res_params")
expect(defer_res_params).to eq({
"foo" => "bar"
})
end
it "Does call the event trigger for the controller after the dispatch only one at a time" do
#Compile the controller
ctx = flok_new_user File.read('./spec/kern/assets/controller0defer2.rb')
#Register callout
ctx.eval %{
base = _embed("my_controller", 0, {}, null);
int_dispatch([]);
}
#Should have dequeued the first asynchronous event callback
defer_res_params = ctx.dump("defer_res_params")
expect(defer_res_params).to eq({
"foo" => "bar"
})
#But the second one should not have dequeued yet
expect {
ctx.dump("defer_res2_params")
}.to raise_exception
#Until we throw another int_dispatch
ctx.eval("int_dispatch([])")
#And now it should be there
defer_res2_params = ctx.dump("defer_res2_params")
expect(defer_res2_params).to eq({
"foo" => "bar"
})
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,132
|
Q: angularjs watch dynamically generated scope objects I need to add a watcher dynamically according to a scope property.
The code below work as hard coded, but if I try to get the values from the scope it doesn't work. I've tested multiple times and in this example:
$scope.options.hideExpression // "hideDefault"
$scope.options.default_to // "allowed_item_types"
if $scope.options.default_to
defaultTo = $scope.options.default_to
$scope.$watch 'model["allowed_item_types"]', (newVal) -> // Hard coded, works
# $scope.$watch 'model[defaultTo]', (newVal) -> // Dynamically generated from scope, doesn't work
if Object.count($scope.model[$scope.options.default_to], /true/) == 1 // Dynamically generated from scope, works
# Set default value
$scope.model[$scope.options.key] = Object.find(newVal, /true/) // Dynamically generated from scope, works
# Hide default row
# $scope[$scope.options.hideExpression] = true // Dynamically generated, doens't work
$scope["hideDefault"] = true // Hard coded, works,
else
# $scope[$scope.options.hideExpression] = false
$scope["hideDefault"] = false
, true
Any idea why this is happening?
For some reason this specific line is the only one that works not hard coded:
$scope.model[$scope.options.default_to]
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,520
|
package com.gemstone.gemfire.internal.cache.functions;
import java.util.Map;
import java.util.Set;
import com.gemstone.gemfire.cache.Region;
import com.gemstone.gemfire.cache.execute.Function;
import com.gemstone.gemfire.cache.execute.FunctionAdapter;
import com.gemstone.gemfire.cache.execute.FunctionContext;
import com.gemstone.gemfire.cache.execute.RegionFunctionContext;
import com.gemstone.gemfire.cache.partition.PartitionRegionHelper;
import com.gemstone.gemfire.internal.Assert;
import com.gemstone.gemfire.internal.cache.LocalDataSet;
import com.gemstone.gemfire.internal.cache.PartitionedRegion;
public class LocalDataSetFunction extends FunctionAdapter {
private volatile boolean optimizeForWrite;
public LocalDataSetFunction(boolean optimizeForWrite) {
this.optimizeForWrite = optimizeForWrite;
}
public void execute(FunctionContext context) {
RegionFunctionContext rContext = (RegionFunctionContext)context;
Region cust = rContext.getDataSet();
LocalDataSet localCust = (LocalDataSet)PartitionRegionHelper.getLocalDataForContext(rContext);
Map <String, Region<?,?>> colocatedRegions = PartitionRegionHelper.getColocatedRegions(cust);
Map <String, Region<?,?>> localColocatedRegions = PartitionRegionHelper.getLocalColocatedRegions(rContext);
Assert.assertTrue(colocatedRegions.size()==2);
Set custKeySet = cust.keySet();
Set localCustKeySet = localCust.keySet();
Region ord = colocatedRegions.get("/OrderPR") ;
LocalDataSet localOrd = (LocalDataSet)localColocatedRegions.get("/OrderPR");
Set ordKeySet = ord.keySet();
Set localOrdKeySet = localOrd.keySet();
Region ship = colocatedRegions.get("/ShipmentPR");
LocalDataSet localShip = (LocalDataSet)localColocatedRegions.get("/ShipmentPR");
Set shipKeySet = ship.keySet();
Set localShipKeySet = localShip.keySet();
Assert.assertTrue(localCust.getBucketSet().size() == localOrd.getBucketSet().size());
Assert.assertTrue(localCust.getBucketSet().size() == localShip.getBucketSet().size());
Assert.assertTrue(custKeySet.size() == 120);
Assert.assertTrue(ordKeySet.size() == 120);
Assert.assertTrue(shipKeySet.size() == 120);
Assert.assertTrue(localCustKeySet.size() == localOrdKeySet.size());
Assert.assertTrue(localCustKeySet.size() == localShipKeySet.size());
context.getResultSender().lastResult(null);
}
public String getId() {
return "LocalDataSetFunction" + optimizeForWrite;
}
public boolean hasResult() {
return true;
}
public boolean optimizeForWrite() {
return optimizeForWrite;
}
public boolean isHA() {
return false;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,711
|
Q: HTML error 405 despite IIS Request Filtering "Allow unlisted verbs" So, I've put together a simple ASP.NET site; I have multiple cases where I've tied Button PostBackURL to a server-side .htm file, with no problem. Now, I've added a large number of other .htm files that were batch-converted from Word .doc files, but accessing any of them in the same manner provokes a "405" error (Bad HTML verb). I see no evidence anywhere (IIS log) saying which specific verb is the culprit. I've been cranking the Google machine heavily, but don't see anything that will truly show me how to diagnose this. Hmm?
added GET, HEAD, POST, TRACE verbs to IIS Request Filtering; checked Allow unlisted verbs - True.
The .htm files that provoke the error CAN be opened successfully in Chrome, on the same server machine.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,510
|
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\begin{center}
\vspace{.5cm}
{\Large\bf \mathversion{bold}
Subsystem complexity after a global quantum quench \\
}
\vspace{0.8cm} {
Giuseppe Di Giulio
and
Erik Tonni
}
\vskip 0.7cm
\small
{\em
SISSA and INFN Sezione di Trieste, via Bonomea 265, 34136, Trieste, Italy
}
\normalsize
\end{center}
\vspace{0.3cm}
\begin{abstract}
We study the temporal evolution of the circuit complexity for a subsystem
in harmonic lattices after a global quantum quench of the mass parameter,
choosing the initial reduced density matrix as the reference state.
Upper and lower bounds are derived
for the temporal evolution of the complexity for the entire system.
The subsystem complexity is evaluated by employing the
Fisher information geometry for the covariance matrices.
We discuss numerical results for the temporal evolutions of the subsystem complexity
for a block of consecutive sites in harmonic chains
with either periodic or Dirichlet boundary conditions,
comparing them with the temporal evolutions of the entanglement entropy.
For infinite harmonic chains,
the asymptotic value of the subsystem complexity is studied
through the generalised Gibbs ensemble.
\end{abstract}
\vspace{1cm}
\newpage
\tableofcontents
\section{Introduction}
\label{sec:intro}
The complexity of a quantum circuit is a quantity introduced in quantum information theory
\cite{Nielsen06,NielsenDowling06, DowlingNielsen08, Watrous2008quantum, Aaronson:2016vto, aharonov1998quantum}
which has been studied also in the context of the holographic correspondence
during the past few years
\cite{Susskind:2014rva,Susskind:2014jwa, Roberts:2014isa, Stanford:2014jda, Susskind:2014moa,
Alishahiha:2015rta, Brown:2015bva,Brown:2015lvg,
Barbon:2015ria,Carmi:2016wjl};
hence it provides an insightful way to explore a connection between
quantum information theory and quantum gravity.
A quantum circuit allows to construct a target state starting from a reference state through a sequence of gates.
The circuit complexity quantifies the difficulty to obtain the target state from the reference state
by counting the minimum number of allowed gates that is necessary to construct the circuit in an optimal way.
Besides the reference state, the target state and the set of allowed gates,
the circuit complexity can
depend also on the tolerance parameter for the target state.
Many results have been obtained for the complexity of quantum circuits made by pure states
constructed through lattice models \cite{Jefferson:2017sdb,Chapman:2018hou,Guo:2018kzl,Hackl:2018ptj,Khan:2018rzm,Braccia:2019xxi,Chapman:2019clq,Doroudiani:2019llj,guo2020circuit}
and in the gravitational side of the holographic correspondence.
Some proposals have been done also to study the circuit complexity in quantum fields theories
\cite{Caputa:2017urj,Czech:2017ryf,Caputa:2017yrh,
Chapman:2017rqy, Bhattacharyya:2018wym,Caputa:2018kdj,Chapman:2018bqj,Camargo:2019isp,Ge:2019mjt,Sato:2019kik,Erdmenger:2020sup,Flory:2020eot,Flory:2020dja}.
Quantum quenches are insightful ways to explore
the dynamics of isolated quantum systems out of equilibrium
(see \cite{Essler:2016ufo,Calabrese:2016xau} for recent reviews).
Given a quantum system prepared in the ground state $| \psi_0 \rangle$ of the hamiltonian $\widehat{H}_0$,
at $t=0$ a sudden change is performed
such that the evolution Hamiltonian of the initial state $| \psi_0 \rangle$
becomes $\widehat{H} \neq \widehat{H}_0$.
Since $\widehat{H}$ and $\widehat{H}_0$ do not commute in general,
the unitary evolution
$| \psi(t) \rangle = e^{-\textrm{i} \widehat{H} t} | \psi_0 \rangle$
for $t>0$ is highly non trivial.
In the typical global quench, a parameter occurring in the Hamiltonian is suddenly changed
from its value $\omega_0$ in $\widehat{H}_0$ to the value $\omega$ in $\widehat{H}$
\cite{Calabrese_2005,Calabrese:2006rx,Calabrese:2007rg,Essler:2016ufo}.
Insightful results have been obtained about
the asymptotic regime $t \to \infty$ of this unitary evolution by employing the
generalised Gibbs ensemble (GGE)
(see the reviews \cite{Rigol_07,Ilievski_2015,Vidmar_2016}).
It is worth studying the circuit complexity
with the target state given by the time-evolved pure state of certain unitary evolution
and the reference state by another pure state along the same evolution
\cite{Alves:2018qfv,Camargo:2018eof,Ali:2018fcz,Chapman:2018hou,Jiang:2018gft}.
In particular, considering a global quench protocol,
we are interested in the optimal circuit and in the corresponding complexity
where $| \psi(t) \rangle$ and $| \psi_0 \rangle$
are respectively the target and the reference states.
Within the gauge/gravity correspondence,
the temporal evolution of complexity for pure states has been explored in
\cite{Moosa:2017yvt,Chapman:2018dem,Chapman:2018lsv}.
Entanglement of spatial bipartitions
plays a crucial role both in quantum information theory and in quantum gravity,
hence it is a fundamental tool to understand the connections between them
(see \cite{Eisert:2008ur,Casini:2009sr,Calabrese:2009qy,Peschel_2009,
Rangamani:2016dms,Headrick:2019eth,Tonni:2020bjq} for reviews).
The entanglement dynamics after global quantum quenches has been largely explored
by considering the temporal evolutions of various entanglement quantifiers.
The entanglement entropy has been mainly investigated
through various methods
\cite{Sotiriadis:2010si,Cotler:2016acd,ac-18-qp-quench,Calabrese:2016xau,
Hubeny:2007xt,AbajoArrastia:2010yt},
but also other entanglement quantifiers
like the entanglement spectra
\cite{Cardy:2016fqc,DiGiulio:2019lpb,Surace:2019mft},
the entanglement Hamiltonians
\cite{Cardy:2016fqc,Wen:2018svb,DiGiulio:2019lpb},
the entanglement negativity \cite{Coser:2014gsa}
and the entanglement contours \cite{Chen_2014,Coser:2017dtb, DiGiulio:2019lpb}
have been explored.
In order to understand the relation between entanglement and complexity,
it is useful to study the optimal circuits and the corresponding circuit complexity
when both the reference and the target states are mixed states
\cite{Caceres:2019pgf,DiGiulio:2020hlz,Ruan:2020vze,Camargo:2020yfv}.
The approach to the complexity of mixed states
based on the purification complexity \cite{Agon:2018zso,Caceres:2019pgf,Camargo:2020yfv}
is general, but evaluating this quantity for large systems is technically complicated.
Some explicit results for large systems can be found
by restricting to the simple case of bosonic Gaussian states
and by employing the methods of the information geometry
\cite{Rao45,Atkinson81,Amari16book}.
In our analysis we adopt the approach to the complexity of mixed states
based on the Fisher information geometry \cite{DiGiulio:2020hlz},
which allows to study large systems numerically.
The crucial assumption underlying this approach is that
all the states involved in the construction of the circuit are Gaussian.
We consider the important special case given by the subsystem complexity,
namely the circuit complexity corresponding to a circuit
where both the reference and the target states
are the reduced density matrices associated to a subsystem.
%
Within the gauge/gravity correspondence,
the subsystem complexity has been evaluated
both in static
\cite{Alishahiha:2015rta,Carmi:2016wjl,Abt:2018ywl,Agon:2018zso,Alishahiha:2018lfv,Auzzi:2019vyh}
and in time dependent gravitational backgrounds
\cite{Chen:2018mcc,Auzzi:2019mah,Ling:2019ien,Zhou:2019xzc}.
%
In static backgrounds, it is given by the volume identified by
the minimal area hypersurface anchored to the boundary of the subsystem,
whose area provides the holographic entanglement entropy
\cite{Ryu:2006bv}
(for static black holes, this hypersurface does not cross the horizon
\cite{Headrick:2007km,Tonni:2010pv,Hubeny:2013gta}),
while, in time dependent gravitational spacetimes,
the extremal hypersurface occurring in the covariant proposal
for the holographic entanglement entropy \cite{Hubeny:2007xt}
must be employed.
In this manuscript we study
the temporal evolution of the subsystem complexity after a global quantum quench
in harmonic lattices
where the mass parameter is suddenly changed from $\omega_0$ to $\omega$.
Considering a ground state as the initial state,
the Gaussian nature of the state is preserved during the temporal evolution.
In these bosonic systems, the reduced density matrices are characterised by
the corresponding reduced covariance matrices \cite{Weedbrook12b}.
By employing the approach to the complexity of bosonic mixed Gaussian states
based on the Fisher information geometry \cite{DiGiulio:2020hlz},
we evaluate numerically the subsystem complexity
for one-dimensional harmonic lattices (i.e. harmonic chains)
and subsystems $A$ given by blocks of consecutive sites.
We consider harmonic chains where either periodic boundary conditions (PBC)
or Dirichlet boundary conditions (DBC) are imposed.
This allows to study the role of the zero mode.
The temporal evolution of the subsystem complexity
at a generic time after the global quench w.r.t. the initial state
is compared with the
temporal evolution of the
corresponding increment of the entanglement entropy.
This manuscript is organised as follows.
In Sec.\,\ref{sec:cov-mat}
we introduce the main expressions to evaluate
the circuit complexity after the global quench of the mass parameter
through the covariance matrices of the reference and the target states
for harmonic lattices in a generic number of dimensions.
In the special case where the entire system is considered,
these states are pure and bounds are obtained
for the temporal evolution of the circuit complexity
w.r.t. the initial state.
In Sec.\,\ref{sec:purestates_HC_glob}
we specify this analysis to harmonic chains with either
PBC or DBC.
The main results of this manuscript are discussed in
Sec.\,\ref{sec:mixedFinSize} and Sec.\,\ref{sec:GGE},
where the temporal evolution of the subsystem complexity
for a block of consecutive sites is investigated.
In Sec.\,\ref{sec:mixedFinSize},
finite harmonic chains with either PBC or DBC are studied,
while in Sec.\,\ref{sec:GGE} we consider
infinite harmonic chains either on the line or on the semi-infinite line
with DBC at the origin.
In the cases of infinite chains, we employ known results about the GGE
to determine
the asymptotic regime of the subsystem complexity.
In Sec.\,\ref{sec:conclusions} we draw some conclusions.
Some technical details and supplementary results are discussed
in the appendices\;\ref{app:CMglobalquench},
\ref{app:unentangled}, \ref{app:large-N} and \ref{app:gge}.
\section{Complexity from the covariance matrix after the quench}
\label{sec:cov-mat}
In this section we discuss the expressions that allow to evaluate the
temporal evolution of the circuit complexity based on the Fisher-Rao geometry
for the harmonic lattices in a generic number of spatial dimensions
when both the reference and the target states are pure.
Analytic expressions that bound this temporal evolution are also derived.
\subsection{Covariance matrix after the quench}
\label{sec-cov-mat-quench}
The Hamiltonian of the harmonic lattice made by $N$ sites with nearest neighbour spring-like interaction reads
\begin{equation}
\label{HC ham}
\widehat{H}
\,=\,
\sum_{i=1}^{N} \left(
\frac{1}{2m}\,\hat{p}_i^2+\frac{m\omega^2}{2}\,\hat{q}_i^2
\right)
+
\sum_{\langle i,j \rangle}\frac{\kappa}{2}(\hat{q}_{i} -\hat{q}_j)^2
\,=\,
\frac{1}{2}\, \hat{\boldsymbol{r}}^{\textrm t} H^{\textrm{\tiny phys}} \, \hat{\boldsymbol{r}}
\end{equation}
where the position and the momentum operators $\hat{q}_i$ and $\hat{p}_i$
are hermitean operators satisfying the canonical commutation relations
$[\hat{q}_i , \hat{q}_j]=[\hat{p}_i , \hat{p}_j] = 0$
and $[\hat{q}_i , \hat{p}_j]= \textrm{i} \delta_{i,j}$.
The matrix $H^{\textrm{\tiny phys}}$ in (\ref{HC ham}) has been defined
by collecting the position and the momentum operators into the vector
$\hat{\boldsymbol{r}} \equiv (\hat{q}_1 , \dots , \hat{q}_N, \hat{p}_1, \dots, \hat{p}_N)^{\textrm{t}}$.
In the Heisenberg picture,
the unitary temporal evolution of the position and the momentum operators
$ \hat{q}_j(t)$ and $ \hat{p}_j(t)$ through the evolution Hamiltonian $\widehat{H}$ reads
\begin{equation}
\label{heisemberg rps}
\hat{q}_j(t) = e^{\textrm{i} \widehat{H} t} \hat{q}_j(0) e^{-\textrm{i} \widehat{H} t}
\qquad
\hat{p}_j(t) = e^{\textrm{i} \widehat{H} t} \hat{p}_j(0) e^{-\textrm{i} \widehat{H} t} \,.
\end{equation}
In order to study the temporal evolution of the harmonic lattices
after the global quantum quench of the mass parameter
that we are considering,
we need to introduce
the $N \times N$ correlation matrices
for operators (\ref{heisemberg rps})
whose elements read
\begin{equation}
\label{time dep corrs}
\begin{array}{l}
Q_{i,j}(t) \equiv \langle \psi_0 |\, \hat{q}_i(t) \,\hat{q}_j(t) \, | \psi_0 \rangle
\\
\rule{0pt}{.6cm}
P_{i,j}(t) \equiv \langle \psi_0 | \,\hat{p}_i(t) \,\hat{p}_j(t) \, | \psi_0 \rangle
\\
\rule{0pt}{.6cm}
M_{i,j}(t) \equiv
\textrm{Re} \big[ \langle \psi_0 | \,\hat{q}_i(t) \,\hat{p}_j(t) \,| \psi_0 \rangle \big]
\end{array}
\end{equation}
where $| \psi_0 \rangle$ is the ground state of the Hamiltonian $\widehat{H}_0$,
defined by (\ref{HC ham}) with $\omega$ replaced by $\omega_0$.
At any time $t>0$ after the quench, the system is completely characterised by its covariance matrix $\gamma(t)$,
which is the following $2N \times 2N$ real, symmetric and positive definite matrix
\begin{equation}
\label{covariancematrix_t}
\gamma(t)
=\,
\bigg(
\begin{array}{cc}
Q(t) & M(t) \\
M(t)^{\textrm{t}} & P(t) \\
\end{array} \bigg)
\end{equation}
where the elements of the $N \times N$ block matrices are given by (\ref{time dep corrs}).
This covariance matrix has been already used to study
the entanglement dynamics e.g. in \cite{Sotiriadis:2010si,Coser:2014gsa,Cotler:2016acd}.
In the appendix\;\ref{subapp:covariancematrix} we discuss the fact that,
for the global quench we are exploring,
the blocks of the covariance matrix (\ref{covariancematrix_t}) can be
decomposed as
\begin{equation}
\label{time dep corrs trans}
Q(t) =
\widetilde{V} \, \mathcal{Q}(t) \,\widetilde{V}^{\textrm{t}}
\,\,\qquad
\rule{0pt}{.6cm}
P(t) = \widetilde{V} \, \mathcal{P}(t)\, \widetilde{V}^{\textrm{t}}
\,\,\qquad
\rule{0pt}{.6cm}
M(t) = \widetilde{V}\, \mathcal{M}(t) \,\widetilde{V}^{\textrm{t}}
\end{equation}
where $\widetilde{V}$ is a real orthogonal $N\times N$ matrix,
while $\mathcal{Q}(t)$, $\mathcal{P}(t)$ and $\mathcal{M}(t)$ are $N\times N$
diagonal matrices whose
$k$-th element along the diagonal is
\cite{Calabrese:2007rg}
\begin{equation}
\label{QPRmat t-dep-k}
\begin{array}{l}
\displaystyle
Q_{k}(t)
\equiv \mathcal{Q}_{k,k}(t)=
\frac{1}{ 2m\Omega_k}
\left( \,\frac{\Omega_k}{\Omega_{0,k}} \, [ \cos(\Omega_k t) ]^2
+ \frac{\Omega_{0,k}}{\Omega_k} \, [\sin(\Omega_k t)]^2 \right)
\\
\displaystyle
\rule{0pt}{.9cm}
P_{k}(t)
\equiv \mathcal{P}_{k,k}(t)=
\frac{m\Omega_k}{2}
\left( \,\frac{\Omega_k}{\Omega_{0,k}} \, [\sin(\Omega_k t)]^2
+ \frac{\Omega_{0,k}}{\Omega_k} \, [\cos(\Omega_k t)]^2 \right)
\\
\displaystyle
\rule{0pt}{.9cm}
M_{k}(t)
\equiv \mathcal{M}_{k,k}(t)=
\frac{1}{2}\left( \frac{\Omega_{0,k}}{\Omega_k} -\,\frac{\Omega_k}{\Omega_{0,k}} \right)
\sin(\Omega_k t) \cos(\Omega_k t)
\end{array}
\end{equation}
in terms of the dispersion relations $\Omega_{0,k}$ and $\Omega_k$ of the Hamiltonians
$\widehat{H}_0$ and $\widehat{H}$ respectively,
which depend both on the dimensionality of the lattice and on the boundary conditions.
At $t=0$, the expressions in (\ref{QPRmat t-dep-k}) simplify respectively to
\begin{equation}
\label{QPRmat tzero-dep-k}
Q_{k}(0) = \frac{1}{2m\,\Omega_{0,k}}
\;\;\qquad\;\;
P_{k}(0) = \frac{m \,\Omega_{0,k}}{2}
\;\;\qquad\;\;
M_{k}(0) = 0 \,.
\end{equation}
From the above discussion, one realises that $\gamma(t)$ is a function of $t$ determined by the set of parameters given by
$\{ m, \kappa, \omega, \omega_0\}$.
When the dispersion relation vanishes for certain value of $k$, e.g. $k=N$,
the corresponding mode is a zero mode.
The relations (\ref{QPRmat t-dep-k}) and (\ref{QPRmat tzero-dep-k})
are well defined when $\Omega_{0,k}$ does not vanish;
hence $\Omega_{k}$ can have a zero mode, while $\Omega_{0,k}$ cannot.
This highlights the asymmetric role of $\Omega_{0,k}$ and $\Omega_{k}$.
\subsection{Complexity for the system}
\label{sec-pure-states}
The circuit complexity is proportional to the length of the optimal quantum circuit
that creates a target state from a reference state.
In this manuscript we evaluate the complexity through the Fisher-Rao distance
between two bosonic Gaussian states with vanishing first moments
\cite{Weedbrook12b, Adesso14, Serafini17book}.
This approach allows to study also the circuits made by mixed states \cite{DiGiulio:2020hlz}.
Denoting by $\gamma_{\textrm{\tiny R}}$ and $\gamma_{\textrm{\tiny T}}$
the covariance matrices with vanishing first moments
of the reference and of the target state respectively,
the Fisher-Rao distance between them \cite{Atkinson81,Bhatia07book}
provides the following definition of complexity
\begin{equation}
\label{c2 complexity}
\mathcal{C}
\,\equiv\,
\frac{1}{2\sqrt{2}}\;
\sqrt{\,
\textrm{Tr}\, \Big\{ \big[ \log \big( \gamma_{\textrm{\tiny T}} \,\gamma_{\textrm{\tiny R}}^{-1} \big) \big]^2 \Big\}
}\;.
\end{equation}
When both $\gamma_{\textrm{\tiny R}}$ and $\gamma_{\textrm{\tiny T}}$ characterise pure states,
this complexity corresponds to the one defined through
the $F_2$ cost function \cite{Chapman:2018hou}.
The analysis of the circuits made by bosonic Gaussian states based on the Fisher-Rao metric
provides also the optimal circuit between $\gamma_{\textrm{\tiny R}}$ and $\gamma_{\textrm{\tiny T}}$.
It reads \cite{Bhatia07book}
\begin{equation}
\label{optimal circuit}
G_s(\gamma_{\textrm{\tiny R}} \, , \gamma_{\textrm{\tiny T}})
\,\equiv \,
\gamma_{\textrm{\tiny R}}^{1/2}
\Big( \gamma_{\textrm{\tiny R}}^{- 1/2} \,\gamma_{\textrm{\tiny T}} \,\gamma_{\textrm{\tiny R}}^{-1/2} \Big)^s
\gamma_{\textrm{\tiny R}}^{1/2}
\;\; \qquad \;\;
0 \leqslant s \leqslant 1
\end{equation}
which gives $\gamma_{\textrm{\tiny R}}$ when $s=0$
and $\gamma_{\textrm{\tiny T}}$ when $s=1$.
The length of the optimal circuit (\ref{optimal circuit})
evaluated through the Fisher-Rao distance is proportional to the circuit complexity
(\ref{c2 complexity}),
which has been explored both for pure states \cite{Chapman:2018hou}
and for mixed states \cite{DiGiulio:2020hlz}.
In this manuscript we are interested in the temporal evolution of the circuit complexity
after a global quench.
In the following discussion and in Sec.\,\ref{sec:purestates_HC_glob}
we consider first the case where both the reference and the target states are pure states,
while in Sec.\,\ref{sec:mixedFinSize} and Sec.\,\ref{sec:GGE}
we study the case where both the reference and the target states are mixed states.
Denoting by $t_{\textrm{\tiny R}}$ and $t_{\textrm{\tiny T}}$ the values of $t$
corresponding to the reference state and to the target state respectively,
let us adopt the following notation
\begin{equation}
\label{gamma-RT-time-def}
\gamma_{\textrm{\tiny R}} = \gamma(t_{\textrm{\tiny R}})
\;\;\qquad\;\;
\gamma_{\textrm{\tiny T}} = \gamma(t_{\textrm{\tiny T}})\,.
\end{equation}
In the most general setup,
$\gamma_{\textrm{\tiny R}} $ is a function of $t_{\textrm{\tiny R}}$
characterised by the set of parameters
$\{m_{\textrm{\tiny R}}, \kappa_{\textrm{\tiny R}}, \omega_{\textrm{\tiny R}}, \omega_{0,\textrm{\tiny R}}\}$,
while $\gamma_{\textrm{\tiny T}} $ is a function of $t_{\textrm{\tiny T}}$
parameterised by
$\{m_{\textrm{\tiny T}}, \kappa_{\textrm{\tiny T}}, \omega_{\textrm{\tiny T}}, \omega_{0,\textrm{\tiny T}}\}$.
This means that the reference and target states are obtained as the time-evolved states at $t=t_\textrm{\tiny R} \geqslant 0$ and $t=t_\textrm{\tiny T} \geqslant t_\textrm{\tiny R}$
respectively, through two different global quenches
determined by
$\{\kappa_\textrm{\tiny R},m_\textrm{\tiny R},\omega_\textrm{\tiny R},\omega_{0,\textrm{\tiny R}}\}$
and $\{\kappa_\textrm{\tiny T},m_\textrm{\tiny T},\omega_\textrm{\tiny T},\omega_{0,\textrm{\tiny T}}\}$
respectively.
The covariance matrix (\ref{covariancematrix_t}) at a generic value of $t$ can be written as follows\footnote{We used that
\begin{equation}
\label{decomp_blockmat}
\bigg( \begin{array}{cc}
A & B
\\
C & D
\end{array} \bigg)
=
\bigg(
\begin{array}{cc}
S\,\mathcal{A}\, S^\dagger \,& S\,\mathcal{B}\, T^\dagger
\\
T\,\mathcal{C}\, S^\dagger \,& T\,\mathcal{D}\, T^\dagger
\end{array}
\bigg)
=
\bigg(
\begin{array}{cc}
S & \boldsymbol{0}
\\
\boldsymbol{0} & T
\end{array} \bigg)
\;
\bigg(\begin{array}{cc}
\mathcal{A} \,& \mathcal{B}
\\
\mathcal{C} \,& \mathcal{D}
\end{array}\bigg)
\;
\bigg(\begin{array}{cc}
S^\dagger \!& \boldsymbol{0}
\\
\boldsymbol{0} \!& T^\dagger
\end{array} \bigg)
\end{equation}
where $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ and $\mathcal{D}$ are diagonal matrices.
}
\begin{equation}
\label{VGammaV dec}
\gamma(t)=V^\textrm{t} \,\Gamma(t)\, V
\,\,\qquad\,\, V= \widetilde{V} \oplus \widetilde{V}
\end{equation}
where $V$ is an orthogonal and symplectic matrix because $\widetilde{V}$ is orthogonal
and the block decomposition of $\Gamma(t)$ reads
\begin{equation}
\label{covariancematrix_diags_t}
\Gamma(t)
=
\bigg(
\begin{array}{cc}
\mathcal{Q}(t) & \!\,\, \mathcal{M}(t)
\\
\mathcal{M}(t) & \!\,\, \mathcal{P}(t)
\end{array}
\bigg)
\end{equation}
in terms of the diagonal matrices whose elements have been defined in (\ref{QPRmat t-dep-k}).
Hereafter we enlighten the expressions by avoiding to indicate explicitly the dependence on $t$,
wherever this is possible.
The inverse of (\ref{covariancematrix_diags_t}) is\footnote{The expression (\ref{Gamma-inv-diag}) is a special case of the following general formula
\begin{equation}
\label{gamma-block-QPM}
\gamma
\equiv
\bigg(\begin{array}{cc}
A & B
\\
B^{\textrm t} &C
\end{array}\bigg)
\;\;\;\;\qquad\;\;\;\;
\gamma^{-1}
\equiv
\bigg(\begin{array}{cc}
\mathfrak{A} & \mathfrak{B}
\\
\mathfrak{B}^{\textrm t} &\mathfrak{C}
\end{array}\bigg)
\;\;\qquad\;\;
\left\{
\begin{array}{l}
\mathfrak{A} \equiv \big(A - B\, C^{-1} B^{\textrm t} \big)^{-1}
\\
\mathfrak{C} \equiv \big(C - B^{\textrm t}\, A^{-1} B \big)^{-1}
\\
\mathfrak{B} \equiv - A^{-1} B \, \big(C - B^{\textrm t}\, A^{-1} B \big)^{-1}\,.
\end{array}
\right.
\end{equation}
}
\begin{equation}
\label{Gamma-inv-diag}
\Gamma^{-1}
=
\big(\mathcal{Q} \,\mathcal{P} - \mathcal{M}^2\big)^{-1}
\bigg(
\begin{array}{cc}
\mathcal{P} \! & - \,\mathcal{M}
\\
- \,\mathcal{M} \! & \mathcal{Q}
\end{array}
\bigg)\,.
\end{equation}
Since $\gamma$ in (\ref{covariancematrix_t}) describes a pure state, the condition
$(\textrm{i} J \gamma)^2 = \tfrac{1}{4}\, \boldsymbol{1}$ holds;
hence the blocks $\mathcal{Q} $, $\mathcal{P} $ and $\mathcal{M} $ are not independent.
More explicitly, this constraint reads
\begin{equation}
(\textrm{i} J \gamma)^2
=
\bigg(\begin{array}{cc}
PQ -(M^{\textrm t})^2 \, & PM - M^{\textrm t}P
\\
QM^{\textrm t} - MQ \,& QP -M^2
\end{array}\bigg)
=\,
V^\textrm{t}\,
\bigg(\begin{array}{cc}
\mathcal{P} \mathcal{Q} - \mathcal{M}^2 \,& \mathcal{P} \mathcal{M} - \mathcal{M}^{\textrm t}\mathcal{P}
\\
\mathcal{Q}\mathcal{M}^{\textrm t} - \mathcal{M}\mathcal{Q} \,& \mathcal{Q}\,\mathcal{P} -\mathcal{M}^2
\end{array}\bigg)
\, V
=
\frac{1}{4}\, \boldsymbol{1}
\end{equation}
which implies
\begin{equation}
\label{purity-condition}
\mathcal{Q}\,\mathcal{P} -\mathcal{M}^2
=
\frac{1}{4}\, \boldsymbol{1}
\qquad \Longleftrightarrow \qquad
Q_k P_k -M_k^2 = \frac{1}{4}
\;\qquad\;
1\leqslant k \leqslant N\,.
\end{equation}
This result allows to further simplify (\ref{Gamma-inv-diag}), which becomes
\begin{equation}
\label{covariancematrix_diags_inv_t}
\Gamma^{-1}
=\,
4\,
\bigg(
\begin{array}{cc}
\mathcal{P} & - \mathcal{M}
\\
- \mathcal{M} & \mathcal{Q}
\end{array}
\bigg) \,.
\end{equation}
In this manuscript we restrict to cases where a symplectic matrix $V$ exists such that
\begin{equation}
\label{VV-diag-hyp}
\gamma_{\textrm{\tiny R}} = V^\textrm{t} \,\Gamma_{\textrm{\tiny R}}\, V
\;\;\qquad\;\;
\gamma_{\textrm{\tiny T}} = V^\textrm{t} \,\Gamma_{\textrm{\tiny T}}\, V
\end{equation}
where both $\Gamma_{\textrm{\tiny R}}$ and $\Gamma_{\textrm{\tiny T}}$ have the form
(\ref{covariancematrix_diags_t}), in terms of the corresponding diagonal matrices.
When (\ref{VV-diag-hyp}) holds, the matrix occurring in the argument of the logarithm in (\ref{c2 complexity}) becomes
\begin{equation}
\label{gamma-TR-global}
\gamma_{\textrm{\tiny T}}\, \gamma_{\textrm{\tiny R}}^{-1}
=
V^\textrm{t} \,\Gamma_{\textrm{\tiny T}} \,\Gamma_{\textrm{\tiny R}}^{-1}\, V^{-\textrm{t}}
\end{equation}
which tells us that the complexity (\ref{c2 complexity}) is provided by the eigenvalues of $\Gamma_{\textrm{\tiny T}} \,\Gamma_{\textrm{\tiny R}}^{-1}$.
Thus, the matrix $V$ does not influence the temporal evolution of the complexity
after the global quench when both the reference and the target states are pure states.
Instead, they play a crucial role for
the temporal evolution of the subsystem complexity discussed in
Sec.\,\ref{sec:mixedFinSize}
and Sec.\,\ref{sec:GGE}.
By using (\ref{covariancematrix_diags_t}) for $\Gamma_\textrm{\tiny T}$
and (\ref{covariancematrix_diags_inv_t}) for $\Gamma^{-1}_\textrm{\tiny R}$,
we obtain the following block matrix
\begin{equation}
\label{relativeCM_diag}
\Gamma_\textrm{\tiny T} \,\Gamma^{-1}_\textrm{\tiny R}
=
4\,
\bigg(
\begin{array}{cc}
\mathcal{P}_\textrm{\tiny R}\mathcal{Q}_\textrm{\tiny T}-\mathcal{M}_\textrm{\tiny R}\mathcal{M}_\textrm{\tiny T}
& \;\;
\mathcal{Q}_\textrm{\tiny R}\mathcal{M}_\textrm{\tiny T}-\mathcal{M}_\textrm{\tiny R}\mathcal{Q}_\textrm{\tiny T}
\\
\mathcal{P}_\textrm{\tiny R}\mathcal{M}_\textrm{\tiny T}-\mathcal{M}_\textrm{\tiny R}\mathcal{P}_\textrm{\tiny T}
& \;\;
\mathcal{Q}_\textrm{\tiny R}\mathcal{P}_\textrm{\tiny T}-\mathcal{M}_\textrm{\tiny R}\mathcal{M}_\textrm{\tiny T}
\end{array} \bigg)
\end{equation}
whose blocks are diagonal matrices.
By using also (\ref{purity-condition}), for
the eigenvalues of (\ref{relativeCM_diag}) we find\footnote{Considering a $2N\times 2N$ matrix $M$ partitioned into four $N\times N$ blocks
$\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ and $\mathcal{D}$
which are diagonal matrices,
its eigenvalues equation can be written through the formula for the determinant of a block matrix,
finding
\begin{equation}
\label{det_1}
M =
\bigg( \!
\begin{array}{cc}
\mathcal{A} & \!\!\,\,\,\mathcal{B} \\
\mathcal{C} & \!\!\,\,\, \mathcal{D} \\
\end{array} \! \bigg)
\qquad
\det(M-\lambda\,\boldsymbol{1})
\,=\,
\det\!\big[ \mathcal{D}-\lambda\,\boldsymbol{1} \big]
\,
\det\!\big[\mathcal{A}-\lambda\,\boldsymbol{1}
-\mathcal{B}\,\mathcal{C}\,(\mathcal{D}-\lambda\boldsymbol{1})^{-1}\big]
= 0
\end{equation}
where $\boldsymbol{1}$ is the identity matrix.
Since the matrices in (\ref{det_1}) are diagonal, this equation becomes
$\prod_{k=1}^N[(d_k-\lambda)(a_k-\lambda)-b_k c_k]=0$;
hence the $2N$ eigenvalues of $M$ in (\ref{det_1}) are
\begin{equation}
\label{eigenvalues block mat diag}
\lambda_{k}^{(\pm)}=\frac{a_k+d_k\pm\sqrt{(a_k-d_k)^2+4 b_k c_k}}{2}
\,\,\qquad\,\,
1 \leqslant k \leqslant N\,.
\end{equation}
}
\begin{eqnarray}
\label{eigenvalues relative CM}
g_{\textrm{\tiny TR},k}^{(\pm)}
&\equiv&
2
\bigg(P_{\textrm{\tiny R},k}Q_{\textrm{\tiny T},k}+Q_{\textrm{\tiny R},k}P_{\textrm{\tiny T},k}-2 M_{\textrm{\tiny R},k}M_{\textrm{\tiny T},k}
\\
&& \hspace{.6cm}
\pm \,
\sqrt{
\big(P_{\textrm{\tiny R},k}Q_{\textrm{\tiny T},k}-Q_{\textrm{\tiny R},k}P_{\textrm{\tiny T},k}\big)^2
+4\big(Q_{\textrm{\tiny R},k}M_{\textrm{\tiny T},k}-M_{\textrm{\tiny R},k}Q_{\textrm{\tiny T},k}\big)
\big(P_{\textrm{\tiny R},k}M_{\textrm{\tiny T},k}-M_{\textrm{\tiny R},k}P_{\textrm{\tiny T},k}\big)}
\;\bigg)
\nonumber
\end{eqnarray}
labelled by $1 \leqslant k \leqslant N$, which can be written as
\begin{equation}
\label{gpm-from-C}
g_{\textrm{\tiny TR},k}^{(\pm)}
=
C_{\textrm{\tiny TR},k} \pm \sqrt{C_{\textrm{\tiny TR},k}^2 - 1}
\end{equation}
where
\begin{equation}
\label{CTR generic}
C_{\textrm{\tiny TR},k}
\equiv
2\big( Q_{\textrm{\tiny T},k} \, P_{\textrm{\tiny R},k} + P_{\textrm{\tiny T},k} \, Q_{\textrm{\tiny R},k} - 2\, M_{\textrm{\tiny T},k} \, M_{\textrm{\tiny R},k} \big)
\end{equation}
in terms of the expressions in (\ref{QPRmat t-dep-k}) specialised to the reference and the target states.
From (\ref{gpm-from-C}) and (\ref{CTR generic}), one observes that
\begin{equation}
\label{g-pm-product}
g_{\textrm{\tiny TR},k}^{(+)} \,g_{\textrm{\tiny TR},k}^{(-)}
=
16\big(Q_{\textrm{\tiny R},k}P_{\textrm{\tiny R},k}-M_{\textrm{\tiny R},k}^2\big)
\big(Q_{\textrm{\tiny T},k}P_{\textrm{\tiny T},k}-M_{\textrm{\tiny T},k}^2\big)\,.
\end{equation}
By employing (\ref{purity-condition}) in this result, we find
$g_{\textrm{\tiny TR},k}^{(+)}= 1/ g_{\textrm{\tiny TR},k}^{(-)}$ for pure states,
for any $1 \leqslant k \leqslant N$.
From (\ref{relativeCM_diag}), (\ref{gpm-from-C}) and (\ref{g-pm-product}),
for the complexity (\ref{c2 complexity})
one obtains\footnote{The last step expression in (\ref{c2-log-lambda-arcosh}) is obtained through the identity
$\log(x+\sqrt{x^2 -1}\,) = \textrm{arccosh}(x)$ for $x\geqslant 1$.}
\begin{equation}
\label{c2-log-lambda-arcosh}
\mathcal{C}
\,=\,
\frac{1}{2}\,\sqrt{ \sum_{k=1}^N \!\big[ \log(g_{\textrm{\tiny TR},k}^{(+)})\big]^2}
\,=\,
\frac{1}{2}\,\sqrt{
\sum_{k=1}^N\! \big[ \log(g_{\textrm{\tiny TR},k}^{(-)})\big]^2}
\,=\,
\frac{1}{2}\,\sqrt{ \sum_{k=1}^N\! \big[ \textrm{arccosh}(C_{\textrm{\tiny TR},k})\big]^2}\;.
\end{equation}
In the most general setup described below (\ref{gamma-RT-time-def}),
the complexity can be found by writing (\ref{QPRmat t-dep-k})
for the reference and the target states first and then
and plugging the results into (\ref{CTR generic}) and (\ref{c2-log-lambda-arcosh}).
The final result is a complicated expressions which can be seen as a function of $t_{\textrm{\tiny R}}$ and $t_{\textrm{\tiny T}}$
parameterised by $\{\kappa_\textrm{\tiny R},m_\textrm{\tiny R},\omega_\textrm{\tiny R},\omega_{0,\textrm{\tiny R}}\}$
and $\{\kappa_\textrm{\tiny T},m_\textrm{\tiny T},\omega_\textrm{\tiny T},\omega_{0,\textrm{\tiny T}}\}$.
We remark that (\ref{c2-log-lambda-arcosh}) can be employed when (\ref{VV-diag-hyp}) holds.
Furthermore, we consider only cases where the matrix $V$ in (\ref{VV-diag-hyp})
depends on the geometric parameters of the system and of the subsystem
but it is independent of the physical parameters occurring in the Hamiltonians
(see Sec.\,\ref{subsec:initial HC}).
In the appendix\,\ref{subapp:squeezing},
the expression (\ref{c2-log-lambda-arcosh})
is obtained through the Williamson's decomposition
\cite{Williamson36}
of the covariance matrices (\ref{gamma-RT-time-def}).
A remarkable simplification occurs
when the reference and the target states are pure states along the time evolution of a given quench.
In this case, the parameters to fix in (\ref{QPRmat t-dep-k}) are
$m_{\textrm{\tiny R}}=m_{\textrm{\tiny T}}=m$,
$\kappa_{\textrm{\tiny R}}=\kappa_{\textrm{\tiny T}}=\kappa$, $\omega_{\textrm{\tiny R}}=\omega_{\textrm{\tiny T}}=\omega$,
and $\omega_{0,\textrm{\tiny R}}=\omega_{0,\textrm{\tiny T}}=\omega_0$;
hence (\ref{CTR generic}) simplifies to
\begin{equation}
\label{CTRomegaReqomegaT}
C_{\textrm{\tiny TR},k}
=
1+\frac{1}{2}
\Bigg(
\frac{\Omega_{k}^2 -\Omega^2_{0,k}
}{\Omega_{k}\, \Omega_{0,k}}\,
\sin[\Omega_{k}(t_{\textrm{\tiny R}}- t_{\textrm{\tiny T}})]
\Bigg)^2
\end{equation}
which must be plugged into (\ref{c2-log-lambda-arcosh}) to get the complexity of pure states after the global quench.
Notice that (\ref{CTRomegaReqomegaT}) is not invariant under the exchange $\Omega_{k} \leftrightarrow \Omega_{0,k}$ for a given $k$.
We remark that (\ref{CTRomegaReqomegaT})
and the corresponding complexity depend on $|t_{\textrm{\tiny R}}- t_{\textrm{\tiny T}}|$.
This is not the case for the most generic choice of the parameters.
\subsection{Complexity with respect to the initial state}
\label{sec:comp-initial-state}
A very natural choice for the reference state is the initial state $| \psi_0 \rangle$,
which is a crucial ingredient of the quench protocol.
This corresponds to choose $t_{\textrm{\tiny R}}=0$ in (\ref{gamma-RT-time-def}).
In this case, from (\ref{QPRmat tzero-dep-k}) and (\ref{purity-condition})
we have that $\mathcal{M}_{\textrm{\tiny R}}=\boldsymbol{0} $ and
$\mathcal{Q}_{\textrm{\tiny R}}\mathcal{P}_{\textrm{\tiny R}}=\frac{1}{4}\boldsymbol{1}$,
which allow to write (\ref{CTR generic}) as
\begin{equation}
\label{C-TR-camargo}
C_{\textrm{\tiny TR},k}
\,\equiv\,
\frac{1}{2} \left( \frac{Q_{\textrm{\tiny T},k}}{Q_{\textrm{\tiny R},k}} + \frac{P_{\textrm{\tiny T},k}}{P_{\textrm{\tiny R},k}} \right) .
\end{equation}
Setting $m_{\textrm{\tiny R}}=m_{\textrm{\tiny T}}=m$ for simplicity
and $t_{\textrm{\tiny T}}=t$ and $t_{\textrm{\tiny R}}=0$
in the most general setup described below (\ref{gamma-RT-time-def})
and then using (\ref{QPRmat t-dep-k}) and (\ref{QPRmat tzero-dep-k}),
this expression becomes
\begin{equation}
\label{complexity_equalm}
C_{\textrm{\tiny TR},k}
=
\frac{
\big(\Omega_{0,\textrm{\tiny T},k}^2+\Omega_{0,\textrm{\tiny R},k}^2\big)\,\Omega_{\textrm{\tiny T},k}^2 \,[\cos(\Omega_{\textrm{\tiny T},k}t)]^2
+
\big(
\Omega_{\textrm{\tiny T},k}^4+\Omega_{0,\textrm{\tiny T},k}^2\,\Omega_{0,\textrm{\tiny R},k}^2
\big)
[\sin(\Omega_{\textrm{\tiny T},k}t)]^2
}{
2\,\Omega_{\textrm{\tiny T},k}^2 \,\Omega_{0,\textrm{\tiny R},k}\, \Omega_{0,\textrm{\tiny T},k}}
\end{equation}
in terms of the dispersion relations $\Omega_{0,\textrm{\tiny S},k}$ (with $\textrm{S}\in \{\textrm{R} , \textrm{T}\}$)
before the quenches providing the reference and the target states
and of the dispersion relations $\Omega_{\textrm{\tiny T},k}$ after the quench
($\Omega_{\textrm{\tiny R},k}$ does not occur because $t_{\textrm{\tiny R}}=0$, hence (\ref{QPRmat tzero-dep-k}) must be employed).
The expression (\ref{C-TR-camargo}) is consistent with the result reported in \cite{Camargo:2018eof},
where the temporal evolution of the complexity of this free bosonic system has been also studied
through a different quench profile that does not include the quench protocol that we are considering.
In many studies the reference state is the unentangled product state
\cite{Jefferson:2017sdb,Chapman:2018hou,Guo:2018kzl,Alves:2018qfv}.
In appendix\;\ref{app:unentangled} we briefly discuss
the temporal evolution of the complexity
given by (\ref{c2-log-lambda-arcosh}) and (\ref{complexity_equalm})
in the case where the initial state is the unentangled product state.
When the same quench is employed to construct the reference and the target states
$\Omega_{0,\textrm{\tiny R},k}=\Omega_{0,\textrm{\tiny T},k}=\Omega_{0,k} $ for any $k$
and (\ref{complexity_equalm}) simplifies.
This choice corresponds to evaluate
the complexity between the initial state and the state at time $t$ after the quench.
Specialising (\ref{complexity_equalm}) to this case and renaming $\Omega_{\textrm{\tiny T},k}\equiv\Omega_{k}$, we obtain
\begin{equation}
\label{CTR}
C_{\textrm{\tiny TR},k}
=
1+
\frac{1}{2}\Bigg(
\frac{\Omega_{k}^2 -\Omega^2_{0,k}
}{\Omega_{k} \,\Omega_{0,k}}\, \sin(\Omega_{k}t)\Bigg)^2
\end{equation}
which coincides with (\ref{CTRomegaReqomegaT}) for $t_{\textrm{\tiny R}}=0$ and $t_{\textrm{\tiny T}}=t$, as expected.
Plugging (\ref{CTR}) into (\ref{c2-log-lambda-arcosh}) and
using the identity $| \textrm{arccosh}(1+x^2/2)| = 2\,| \textrm{arcsinh}(x/2) | $,
one finds
\begin{equation}
\label{comp-pure-global-general}
\mathcal{C}
\,=\,
\sqrt{\, \sum_{k=1}^N\!
\left[ \textrm{arcsinh}\!
\left( \,
\frac{\Omega_{k}^2 - \Omega_{0,k}^2}{2\,\Omega_{k}\, \Omega_{0,k}} \,
\sin (\Omega_{k} t )
\right)
\right]^2}\,.
\end{equation}
In this expression the dispersion relations $\Omega_k$ and $\Omega_{0,k}$
(which depend on the number of spatial dimensions and on the boundary conditions of the lattice)
do not occur in a symmetric way.
We find it worth highlighting the contribution of the $N$-th mode by denoting
\begin{equation}
\label{zeromode-def}
c_0
\equiv \bigg[\,
\textrm{arcsinh}\!\,
\bigg( \,
\frac{\Omega_{N}^2 - \Omega_{0,N}^2}{2\,\Omega_{N}\, \Omega_{0,N}} \,
\sin (\Omega_{N} t )
\bigg)\bigg]^2
\; \qquad\;
\mathcal{C}_0^2
\equiv
\sum_{k=1}^{N-1}\!
\left[ \textrm{arcsinh}\!
\left( \,
\frac{\Omega_{k}^2 - \Omega_{0,k}^2}{2\,\Omega_{k}\, \Omega_{0,k}} \,
\sin (\Omega_{k} t )
\right)
\right]^2
\end{equation}
which lead to write (\ref{comp-pure-global-general}) as
\begin{equation}
\label{eta-c0-decomposition}
\mathcal{C}^2 = \eta\,c_0 + \mathcal{C}_0^2
\end{equation}
where either $\eta =1$ or $\eta = 0$,
depending on whether the $N$-th mode plays a particular role,
as one can read from the dispersion relation.
This is the case e.g. for the zero mode in the harmonic lattices that are invariant under spatial translations,
which is briefly discussed also at the end of Sec.\,\ref{sec-cov-mat-quench};
hence hereafter we refer to $c_0$ as the zero mode contribution.
For instance, $\eta = 1$ in the harmonic chains with PBC,
while $\eta=0$ when DBC are imposed,
as discussed later in Sec.\,\ref{subsec:initial HC}.
The result (\ref{comp-pure-global-general}),
which can be applied for harmonic lattices in generic number of dimensions
and for diverse boundary conditions,
has been already reported in \cite{Ali:2018fcz} for harmonic chains with PBC.
It is interesting to determine the initial growth of the complexity by considering the series expansion of (\ref{comp-pure-global-general}) as $t \to 0$.
The function $\mathcal{C}^2 $ obtained from (\ref{comp-pure-global-general}) is an even function of $t$,
hence its expansion for $t \to 0$ contains only even powers of $t$.
Since $\mathcal{C}|_{t=0} =0$, we have
\begin{equation}
\label{comp-pure-initialgrowth}
\mathcal{C}^2
=
b_1 \, t^2 +b_2 \,t^4+b_3 \,t^6+ O(t^8)
\quad \Longrightarrow \quad
\mathcal{C}
=
\sqrt{b_1} \; t\,
\bigg(1
+ \frac{b_2}{2b_1}\, t^2
+ \frac{4\,b_1 b_3-b_2^2}{8b_1^2}\, t^4
+O(t^6)\bigg)
\end{equation}
where the coefficients $b_1$, $b_2$ and $b_3$ are
\begin{equation}
\label{c_12_coeff_exp}
b_1
=
\frac{1}{4}\sum_{k=1}^N
\left( \frac{ \Omega_{k}^2- \Omega_{0,k}^2}{ \Omega_{0,k}} \right)^{2}
\;\;\qquad\;\;
b_2
=
-\frac{1}{48}\sum_{k=1}^N
\left( \frac{ \Omega_{k}^4 - \Omega_{0,k}^4}{ \Omega_{0,k}^2} \right)^{2}
\end{equation}
and
\begin{equation}
\label{c_3_coeff_exp}
b_3
=
\frac{1}{360}\sum_{k=1}^N \frac{ (\Omega_{k}^4- \Omega_{0,k}^4)^2 (\Omega_{k}^4+ \Omega_{0,k}^4-\Omega_{k}^2 \, \Omega_{0,k}^2)}{ \Omega_{0,k}^6}\,.
\end{equation}
Since $b_1 >0$, the expansion (\ref{comp-pure-initialgrowth}) tells us that
the initial growth of the complexity (\ref{comp-pure-global-general}) is linear in $t$.
The temporal evolution of the circuit complexity for a bosonic system after a global quench
has been studied also in \cite{Alves:2018qfv},
by employing a smooth quench and
the unentangled product state
as the reference state.
This smooth quench
becomes the one that we are considering in the limit of sudden quench
but it is different from the quench considered in \cite{Camargo:2018eof}.
In appendix\;\ref{app:unentangled},
where the unentangled product state is considered as the initial state,
we find a different result with respect to \cite{Alves:2018qfv}
because of the different sets of allowed gates.
\subsubsection{Bounds and the zero mode contribution}
\label{subsec:zeromodesboundsgeneral}
We find it worth studying some bounds for the complexity with respect to the initial state.
From (\ref{comp-pure-global-general}),
it is straightforward to observe that $\eta \,c_0 \leqslant \mathcal{C}^2 \leqslant\widetilde{\mathcal{C}}^2$,
where $c_0$ is the time dependent expression defined in (\ref{zeromode-def}) and
\begin{equation}
\label{upperbound-general-zeromode}
\widetilde{\mathcal{C}}^2
\,\equiv\,
\eta\,c_0+ \sum_{k=1}^{N-1}\!
\left[ \textrm{arcsinh}\!
\left( \,
\frac{\Omega_{k}^2 - \Omega_{0,k}^2}{2\,\Omega_{k}\, \Omega_{0,k}} \,
\right)
\right]^2
\end{equation}
hence for the complexity (\ref{comp-pure-global-general}) we find
\begin{equation}
\label{naive-bounds}
\sqrt{\eta\,c_0} \, \leqslant \, \mathcal{C} \, \leqslant \, \widetilde{\mathcal{C}}\,.
\end{equation}
The zero mode contribution determines the behaviour of these bounds for large $t$.
\\
The occurrence of a zero mode in the dispersion relation $\Omega_k$
e.g. for $k=N$
means that $\Omega_N =0$.
In the absence of a zero mode,
$\Omega_k$ is non vanishing for any value of $k$;
hence $c_0$ and $ \widetilde{\mathcal{C}}$ are finite for any $t$
and (\ref{naive-bounds}) tells us that
the complexity (\ref{comp-pure-global-general}) is always finite after the quench.
Instead, when a zero mode for $k=N$ occurs,
the time dependent zero mode contribution $c_0$ in (\ref{zeromode-def}) becomes
\begin{equation}
c_0
=
\big[ \textrm{arcsinh} ( \Omega_{0,N} \,t/2 )\big]^2
\end{equation}
which diverges at large $t$ because $\textrm{arcsinh}(x)\sim \log (2x)$ as $x\to+\infty$.
The terms labelled by $1\leqslant k \leqslant N-1$ in the sum in (\ref{upperbound-general-zeromode})
are bounded functions of $t$ because $\Omega_k$ is non vanishing.
Thus, in the presence of a zero mode, the bounds (\ref{naive-bounds}) tell us that
the complexity for pure states in (\ref{comp-pure-global-general}) diverges logarithmically when $t \to \infty$.
The bounds (\ref{naive-bounds}) can be significantly improved by employing the decomposition (\ref{eta-c0-decomposition}).
The following integral representation
\begin{equation}
\label{int arcsinh}
\textrm{arcsinh}(x) \,=\int_0^1\frac{x}{\sqrt{1+x^2 s^2}}\,ds
\end{equation}
leads to rewrite $ \mathcal{C}_0^2$ in (\ref{zeromode-def}) as
\begin{equation}
\label{sum over modes int arcsinh}
\mathcal{C}_0^2
\, =
\sum_{k=1}^{N-1}
\Bigg[\int_0^1\! \frac{1}{\sqrt{1+\tilde{x}_k^2 \sin^2 (\Omega_{k} t ) s^2}}\, ds \Bigg]^2
\tilde{x}_k^2\, \big[ \sin (\Omega_{k} t )\big]^2
\;\;\qquad\;\;
\tilde{x}_k\equiv
\frac{\Omega_k^2 - \Omega_{0,k}^2}{2\,\Omega_{k}\, \Omega_{0,k}}\,.
\end{equation}
Then, by using (\ref{int arcsinh}), one observes that
\begin{equation}
\frac{\textrm{arcsinh}(\tilde{x}_k)}{\tilde{x}_k}
\,\leqslant
\int_0^1\! \frac{1}{\sqrt{1+\tilde{x}_k^2 \sin^2 (\Omega_{k} t ) s^2}}\,ds
\, \leqslant \,1
\end{equation}
which can be employed to bound (\ref{sum over modes int arcsinh}) as follows
\begin{equation}
\sum_{k=1}^{N-1}
\big[ \textrm{arcsinh}(\tilde{x}_k) \,\sin(\Omega_{k} t ) \big]^2
\, \leqslant\,
\mathcal{C}_0^2
\,\leqslant
\sum_{k=1}^{N-1}
\tilde{x}_k^2 \, \big[\sin(\Omega_{k} t )\big]^2\,.
\end{equation}
This result, combined with (\ref{zeromode-def}), provides the following bounds
for the complexity (\ref{comp-pure-global-general})
\begin{equation}
\label{new bounds_main}
\mathcal{C}^2_{\textrm{\tiny L}} \, \leqslant \,\mathcal{C}^2\,\leqslant \,\mathcal{C}^2_{\textrm{\tiny U}}
\end{equation}
where we have introduced
\begin{equation}
\label{new bounds C-L-U}
\mathcal{C}^2_{\textrm{\tiny B}}
\,\equiv\,
\eta \,c_0
+\! \sum_{k=1}^{N-1} f_{\textrm{\tiny B}}(\tilde{x}_k) \big[ \sin (\Omega_{k} t ) \big]^2
=
\Bigg(
\eta \,c_0
+
\frac{1}{2} \sum_{k=1}^{N-1} f_{\textrm{\tiny B}}(\tilde{x}_k)
\Bigg)
-
\frac{1}{2} \sum_{k=1}^{N-1}
f_{\textrm{\tiny B}}(\tilde{x}_k) \cos(2\Omega_{k}\, t )
\end{equation}
with $\textrm{B} \in \{ \textrm{L} , \textrm{U} \}$ and
\begin{equation}
\label{fLs-def}
f_{\textrm{\tiny L}}(x) = \big[\textrm{arcsinh}(x) \big]^2
\;\;\qquad\;\;
f_{\textrm{\tiny U}}(x) = x^2
\end{equation}
in terms of $\tilde{x}_k$ defined in (\ref{sum over modes int arcsinh}),
of the time dependent zero mode contribution $c_0$ introduced in (\ref{zeromode-def})
and of the parameter $\eta$,
which is either $\eta=1$ or $\eta=0$, depending on whether the zero mode contribution occurs or not respectively.
The bounds (\ref{new bounds_main}) can be employed to improve
the bounds reported in (\ref{naive-bounds}).
Indeed, in the presence of a zero mode,
$\mathcal{C}^2_{\textrm{\tiny L}}\geqslant c_0$ and therefore $\mathcal{C}^2_{\textrm{\tiny L}}$ provides a better lower bound than (\ref{zeromode-def}).
Instead,
the relation between $\mathcal{C}^2_{\textrm{\tiny U}}$ in (\ref{new bounds C-L-U})
and $\widetilde{\mathcal{C}}^2$ in (\ref{upperbound-general-zeromode}) depends on the parameters;
hence the optimal upper bound is given by
$\min\!\big[\mathcal{C}_{\textrm{\tiny U}}(t)^2,\widetilde{\mathcal{C}}(t)^2\big]$.
\section{Complexity for harmonic chains}
\label{sec:purestates_HC_glob}
In this section we apply the results discussed in Sec.\,\ref{sec:cov-mat}
to the harmonic chains where either PBC or DBC are imposed.
The numerical data reported in all the figures of the manuscript have been obtained
by setting $\kappa=1$ and $m=1$.
\subsection{Complexity}
\label{subsec:initial HC}
The Hamiltonian of the harmonic chain made by $N$ oscillators
with the same frequency $\omega$, the same mass $m$ and coupled through the
elastic constant $\kappa$ is (\ref{HC ham}) specialised to one spatial dimension,
i.e.
\begin{equation}
\label{HC ham-1d}
\widehat{H}
\,=\,
\sum_{i=1}^{N} \left(\,
\frac{1}{2m}\,\hat{p}_i^2+\frac{m\omega^2}{2}\,\hat{q}_i^2
+ \frac{\kappa}{2}(\hat{q}_{i} -\hat{q}_{i-1})^2
\right)
\,=\,
\frac{1}{2}\, \hat{\boldsymbol{r}}^{\textrm t} H^{\textrm{\tiny phys}} \, \hat{\boldsymbol{r}}
\end{equation}
where the vector
$\hat{\boldsymbol{r}} \equiv (\hat{q}_1 , \dots , \hat{q}_N, \hat{p}_1, \dots, \hat{p}_N)^{\textrm{t}}$
collects the position and momentum operators.
Imposing PBC means that $\hat{q}_0=\hat{q}_N $,
while DBC are satisfied when
$\hat{q}_0=\hat{q}_N=0$ and $\hat{p}_N=0$.
When PBC hold,
the orthogonal matrix $\widetilde{V}$ defined in (\ref{time dep corrs trans}),
when $N$ is even, is \cite{Serafini17book}
\begin{equation}
\label{Vtilde-def-even}
\widetilde{V}_{i,k} \equiv
\left\{\begin{array}{ll}
\sqrt{2/N}\; \cos(2\pi \,i\,k/N) \hspace{1cm}& 1\leqslant k < N/2
\\
\rule{0pt}{.5cm}
(-1)^i/\sqrt{N} & k = N/2
\\
\rule{0pt}{.5cm}
\sqrt{2/N}\; \sin(2\pi\, i\,k/N) & N/2+1 \leqslant k < N-1
\\
\rule{0pt}{.5cm}
1/\sqrt{N} & k = N
\end{array}
\right.
\end{equation}
while, when $N$ is odd, it reads
\begin{equation}
\label{Vtilde-def-odd}
\widetilde{V}_{i,k} \equiv
\left\{\begin{array}{ll}
\sqrt{2/N}\; \cos(2\pi\, i\,k/N) & 1\leqslant k < (N-1)/2
\\
\rule{0pt}{.5cm}
\sqrt{2/N}\; \sin(2\pi \,i\,k/N) \hspace{1cm}& (N-1)/2+1 \leqslant k < N-1
\\
\rule{0pt}{.5cm}
1/\sqrt{N} & k = N\,.
\end{array}
\right.
\end{equation}
The dispersion relations of $\hat{H}_0$ and $\hat{H}$ for PBC are respectively
\begin{equation}
\label{dispersion relations}
\Omega_{0,k} = \sqrt{\omega_0^2 + \frac{4\kappa}{m} [ \sin(\pi k/N) ]^2}
\,\,\qquad\,\,
\Omega_k = \sqrt{\omega^2 + \frac{4\kappa}{m} [ \sin(\pi k/N)]^2}
\,\,\qquad\,\,
1 \leqslant k \leqslant N\,.
\end{equation}
When DBC hold, only $N-1$ sites display some dynamics
because the ones labelled by $i=0$ and $i=N$ are fixed by the boundary conditions;
hence the vector $\hat{\boldsymbol{r}}$ contains $2(N-1)$ operators and,
correspondingly, the covariance matrix $\gamma(t)$
is the $(2N-2)\times (2N-2)$ symmetric matrix given by (\ref{covariancematrix_t}),
where $Q$, $P$ and $R$ are $(N-1)\times (N-1)$ matrices.
For DBC and independently of the parity of $N$,
the matrix $\widetilde{V}$ defined in (\ref{time dep corrs trans}) becomes
\begin{equation}
\label{Vtilde-HC-DBC}
\widetilde{V}_{i,k}=\sqrt{\frac{2}{N}}\, \sin(i\,k\,\pi/N)
\,\,\qquad\,\,
1 \leqslant i,k \leqslant N-1\,.
\end{equation}
The dispersion relations of $\hat{H}_0$ and $\hat{H}$ for DBC read respectively
\begin{equation}
\label{dispersion DBC}
\Omega_{0,k}=\sqrt{\omega_0^2+\frac{4\kappa}{m}\, [ \sin(\pi k/(2N))]^2}
\qquad
\Omega_k=\sqrt{\omega^2+\frac{4\kappa}{m}\, [ \sin(\pi k/(2N))]^2}
\,\qquad\,
1 \leqslant k \leqslant N-1\,.
\end{equation}
We remark that, both for PBC and DBC,
the matrix $V = \widetilde{V} \oplus \widetilde{V}$ defined in (\ref{VGammaV dec})
depends only on $N$;
hence the corresponding harmonic chains can be studied as special cases of the
harmonic lattices considered in Sec.\,\ref{sec-pure-states}
because the condition (\ref{VV-diag-hyp}) is satisfied.
Since $\eta=1$ for PBC and $\eta=0$ for DBC,
the complexity (\ref{comp-pure-global-general}) for these harmonic chains becomes
\begin{eqnarray}
\label{comp-pure-global-DBCPBC}
\mathcal{C}
&=&
\sqrt{ \sum_{k=1}^{N-1+\eta}\!
\left[ \textrm{arcsinh}\!
\left( \,
\frac{\omega^2 - \omega_0^2}{2\,\Omega_{k}\, \Omega_{0,k}} \,
\sin (\Omega_{k} t )
\right)
\right]^2}
\\
\rule{0pt}{1.1cm}
&=&
\sqrt{ \eta\left[ \textrm{arcsinh}\!
\left( \,
\frac{\omega^2 - \omega_0^2}{2\,\omega\, \omega_0} \,
\sin (\omega t )
\right)
\right]^2+ \sum_{k=1}^{N-1}\!
\left[ \textrm{arcsinh}\!
\left( \,
\frac{\omega^2 - \omega_0^2}{2\,\Omega_{k}\, \Omega_{0,k}} \,
\sin (\Omega_{k} t )
\right)
\right]^2}
\nonumber
\end{eqnarray}
where the dispersion relations $\Omega_{0,k}$ and $\Omega_{k}$
are given by (\ref{dispersion relations}) for PBC
and by (\ref{dispersion DBC}) for DBC.
When PBC are imposed,
the first term under the square root in the last expression of (\ref{comp-pure-global-DBCPBC})
comes from the zero mode $k=N$ and it does not occur for DBC.
This crucial difference between the two models leads to different
qualitative behaviours for the complexity.
The dispersion relations of the harmonic chain with PBC given in (\ref{dispersion relations})
are invariant under the exchange $k \leftrightarrow N-k$.
This symmetry leads to an expression for the complexity which is simpler to evaluate numerically.
Indeed, by introducing
\begin{equation}
\label{low-bound}
c_{0}
\equiv
\left[ \textrm{arcsinh}\!
\left( \,
\frac{\omega^2 - \omega_0^2}{2\,\omega\, \omega_{0}} \,
\sin (\omega t )
\right)
\right]^2
\end{equation}
and
\begin{equation}
\label{c-Nover2-def}
c_{N/2} \equiv
\left\{ \begin{array}{l l}
\displaystyle
\left[ \textrm{arcsinh}\!
\left( \,
\frac{\omega^2 - \omega_0^2}{2\,\Omega_{N/2}\, \Omega_{0,N/2}} \,
\sin (\Omega_{N/2}\, t )
\right)
\right]^2
\hspace{1cm}&
\textrm{even $N$}
\\
\rule{0pt}{.7cm}
\;0
&
\textrm{odd $N$}
\end{array}
\right.
\end{equation}
one observes that (\ref{comp-pure-global-DBCPBC}) for PBC can be written as
\begin{equation}
\label{comp-pure-global-zm}
\mathcal{C}
\,=\,
\sqrt{\;
c_{0}
+
2 \sum_{k=1}^{ \lfloor \frac{N-1}{2} \rfloor}\!
\left[ \textrm{arcsinh}\!
\left( \,
\frac{\omega^2 - \omega_0^2}{2\,\Omega_{k}\, \Omega_{0,k}} \,
\sin (\Omega_{k} t )
\right)
\right]^2
+ c_{N/2}
}
\end{equation}
where $\lfloor x \rfloor$ denotes the integer part of $x$.
Notice that $c_{N/2}$ in (\ref{c-Nover2-def}), as function of $t$,
is bounded by a constant.
We find it worth considering the small quench regime,
defined by setting $\omega_0=\omega+\delta\omega$
and taking $|\delta\omega| \ll 1$
in (\ref{comp-pure-global-DBCPBC}).
As $\delta\omega \to 0$, the leading term of the expansion reads
\begin{equation}
\label{comp-pure-global-smallquench}
\mathcal{C}
\,=\,
\omega\,\delta\omega\;
\sqrt{\eta\;\frac{[\sin(\omega t)]^2}{\omega^4}+\sum_{k=1}^{N-1}\frac{[\sin(\Omega_k t)]^2}{\Omega_k^4}}+O\big(\delta\omega^2\big)
\end{equation}
This result simplifies to
$\mathcal{C} = \eta \,\delta \omega \, t + O(\delta\omega^2)$
when $\omega \to 0$;
which tells us that the $O(\delta\omega)$ term does not occur in this limit
when DBC hold.
\subsection{Critical evolution}
\label{subsec:criticalevolution}
An important case that we find worth emphasising is the global quench
where the evolution Hamiltonian is gapless, i.e. when $\omega=0$.
When PBC are imposed,
by specialising (\ref{CTR}) and (\ref{dispersion relations}) to $\omega=0$, we obtain
\begin{equation}
\label{CTRk_HC massless}
C_{\textrm{\tiny TR},k}
\,=\,
1 +
\frac{\omega_0^4\, \big[ \sin\!\big(2\sqrt{\kappa/m}\; t \, \sin(\pi k/N)\big) \big]^2}{
8\, (\kappa/m)\, [\sin(\pi k/N)]^2 \,\big( \omega_0^2+4 (\kappa/m) [\sin(\pi k/N)]^2\big)}
\end{equation}
which satisfies the following bounds
\begin{equation}
1 \,<\,
C_{\textrm{\tiny TR},k}
\,<\,
1 +\frac{\omega_0^4}{8\, (\kappa/m)\, [\sin(\pi k/N)]^2 \,
\big( \omega_0^2+4 (\kappa/m) [\sin(\pi k/N)]^2\big)}
\;\qquad\;
1\leqslant k \leqslant N-1\,.
\end{equation}
For $k=N$, the expression (\ref{CTRk_HC massless}) simplifies to
$C_{\textrm{\tiny TR},N}=1+\tfrac{\omega_0^2}{2}\, t^2$,
which diverges as $t \to \infty$.
Instead, when DBC hold and therefore the zero mode does not occur,
by using (\ref{dispersion DBC}) and (\ref{CTR}) with $\omega=0$,
we obtain
\begin{equation}
\label{CTRk_HC massless DBC}
C_{\textrm{\tiny TR},k}
\,=\,
1 +\frac{\omega_0^4\, \big[ \sin\!\big(2\sqrt{\kappa/m}\,t \,\sin(\pi k/(2N))\big) \big]^2}{
8\, (\kappa/m) [\sin(\pi k/(2N))]^2 \,\big( \omega_0^2+4 \kappa/m [\sin(\pi k/(2N))]^2\big)}
\end{equation}
which is finite when $t \to \infty$, for any allowed value of $k$.
Plugging the expressions discussed above for $C_{\textrm{\tiny TR},k}$ into (\ref{c2-log-lambda-arcosh}),
we find that, when the evolution Hamiltonian is critical,
the complexity of the pure state at time $t$ with respect to the initial state can be written
by highlighting the zero mode contribution as follows
\begin{equation}
\label{C-pure-both-eta}
\mathcal{C}^2
\,=\,
\frac{\eta}{4}\left[ \,
\log\!\left(1+\frac{(\omega_0 \,t)^2}{2} + \frac{\omega_0 \,t}{2} \,\sqrt{(\omega_0 \,t)^2+4}\,\right)
\right]^2
+
\frac{1}{4} \sum_{k=1}^{N-1}\! \big[ \textrm{arccosh}\big(C_{\textrm{\tiny TR},k}\big)\big]^2
\end{equation}
where either $\eta = 1$ for PBC or $\eta = 0$ for DBC
(see the text above (\ref{comp-pure-global-DBCPBC}))
and $C_{\textrm{\tiny TR},k}$ is given by
(\ref{CTRk_HC massless}) for PBC and by (\ref{CTRk_HC massless DBC}) for DBC.
In particular, (\ref{C-pure-both-eta}) tells us that, for PBC and finite $N$,
the complexity diverges logarithmically as $t \to \infty$ because of the zero mode contribution.
Instead, for DBC (i.e. $\eta=0$) and finite $N$, all the terms in (\ref{C-pure-both-eta}) are finite
as $t\to \infty$.
\begin{figure}[t!]
\subfigure
{\hspace{-1.55cm}
\includegraphics[width=.57\textwidth]{CompPBCCriticalEvolution}}
\subfigure
{
\hspace{-0.35cm}\includegraphics[width=.57\textwidth]{CompDBCCriticalEvolution}}
\subfigure
{
\hspace{-1.55cm}\includegraphics[width=.57\textwidth]{CompPBCCriticalEvolutionBigMass}}
\subfigure
{\hspace{-0.35cm}
\includegraphics[width=.57\textwidth]{CompDBCCriticalEvolutionBigMass}}
\caption{Temporal evolution of the complexity
after the global quench w.r.t. the initial state at $t=0$
for harmonic chains
with either PBC (left panels) or DBC (right panels)
made by $N=100$ sites.
The solid lines correspond to the complexity (\ref{C-pure-both-eta}).
In the top left panel, the dashed lines show the zero mode term $c_0$
(i.e. the expression multiplyed by $\eta$ in (\ref{C-pure-both-eta}),
which has been subtracted to obtain the bottom left panel),
with the same colour code for the corresponding value of $\omega_0$.
}
\vspace{0.4cm}
\label{fig:PureStateCritical}
\end{figure}
In Fig.\,\ref{fig:PureStateCritical} we show the temporal evolution of the complexity
(\ref{C-pure-both-eta}) for various $\omega_0$'s,
when either PBC (left panels) or DBC (right panels) are imposed.
Since $N$ is finite, the revivals
already studied in the temporal evolutions of other quantities \cite{Cardy:2014rqa}
are observed also in the temporal evolution of the complexity,
with a period given by $N/2$ for PBC and by $N$ for DBC.
The most important qualitative difference between PBC and DBC is
the overall growth observed for PBC, which does not occur for DBC.
This growth is due to the zero mode contribution occurring in
the complexity (\ref{C-pure-both-eta}) for PBC.
Indeed, when the corresponding term
is subtracted, as done in the
bottom left panel of Fig.\,\ref{fig:PureStateCritical},
the resulting curve is similar to the temporal evolution of the complexity when DBC hold.
Finally, let us remark that
the effect of the decoherence as $t$ increases is more evident for higher values of $\omega_0$.
For PBC this is observed once the zero mode contribution has been subtracted
(see the bottom left panel of Fig.\,\ref{fig:PureStateCritical}).
In \cite{Chapman:2018hou} the temporal evolution of the complexity
of a thermofield double state is considered
by taking the unentangled product state as the reference state
(in this setup, the choice $\omega =0$ is not allowed).
Despite this temporal evolution is different from the one investigated in this manuscript,
it also exhibits an overall logarithmic growth due to the zero mode contribution.
\subsection{Bounds}
\label{subsec:bounds}
\begin{figure}[t!]
\subfigure
{
\hspace{3.0cm}\includegraphics[width=.57\textwidth]{CompPBCPureLargeTimeomegasmalleromega0}}
\\
\subfigure
{
\hspace{-1.55cm}\includegraphics[width=.57\textwidth]{CompPBCPureLargeTimeomegasmalleromega0N400part1}}
\subfigure
{\hspace{-0.45cm}
\includegraphics[width=.57\textwidth]{CompPBCPureLargeTimeomegasmalleromega0N400part2}}
\caption{
Temporal evolution of the complexity (\ref{comp-pure-global-DBCPBC}) (solid lines)
and of the corresponding bounds in (\ref{naive-bounds}) for PBC.
The blue and red dashed lines show the lower and the upper bounds,
from (\ref{zeromode-def}) and (\ref{upperbound-general-zeromode}) respectively.
}
\vspace{0.4cm}
\label{fig:BoundNaive}
\end{figure}
It is instructive to discuss further the bounds for the complexity
introduced in Sec.\,\ref{subsec:zeromodesboundsgeneral}
in the special cases of the harmonic chains with either PBC or DBC.
In Fig.\,\ref{fig:BoundNaive} we show the complexity (\ref{comp-pure-global-DBCPBC})
and the corresponding bounds (\ref{naive-bounds}) for harmonic chains with PBC.
In this case the zero mode term influences the bounds in a crucial way.
In Fig.\,\ref{fig:BoundNaive},
the bounds (\ref{naive-bounds})
correspond to the red and blue dashed lines,
while in the top left panel of Fig.\,\ref{fig:PureStateCritical}, where $\omega =0$,
the lower bound in (\ref{naive-bounds}) is shown through the dashed curves.
In the temporal evolutions of the complexity for PBC
displayed in the top panel of Fig.\,\ref{fig:BoundNaive},
we can identify two periods approximatively given by $\pi /\omega$ and $N/2$.
Considering also the bottom panels of Fig.\,\ref{fig:BoundNaive},
the revivals observed
for the critical evolution in Fig.\,\ref{fig:PureStateCritical} for PBC and $\omega =0$
occur also when $\omega >0$ whenever $\frac{\pi}{\omega}\gg \frac{N}{2}$.
The bottom panels in Fig.\,\ref{fig:BoundNaive} highlight that
the revivals are not observed when $\omega$ is large enough with respect to $1/N$.
For PBC, by comparing the top panel with the bottom ones in Fig.\,\ref{fig:BoundNaive},
which differ for the size $N$ of the chain,
we notice that the bounds (\ref{naive-bounds}) are very efficient when
$\frac{\pi}{\omega}\gg \frac{N}{2}$, while they become not useful away from this regime.
In our numerical investigations we have also observed that the bounds (\ref{naive-bounds})
are not useful when $\omega>\omega_0$.
When DBC hold,
the lower bound in (\ref{naive-bounds}) is trivial
and the upper bound is a constant.
\begin{figure}[t!]
\subfigure
{\hspace{-1.55cm}
\includegraphics[width=.57\textwidth]{CompPBCImprovedBoundsSmallMass}}
\subfigure
{
\hspace{-.45cm}\includegraphics[width=.57\textwidth]{CompDBCImprovedBoundsSmallMass}}
\subfigure
{
\hspace{-1.55cm}\includegraphics[width=.57\textwidth]{CompPBCImprovedBoundsBigMass}}
\subfigure
{\hspace{-.45cm}
\includegraphics[width=.57\textwidth]{CompDBCImprovedBoundsBigMass}}
\caption{
Temporal evolution of the complexity (\ref{comp-pure-global-DBCPBC}) (solid lines)
and of the corresponding bounds in (\ref{new bounds_main_weaker})
for harmonic chains with either PBC (left panels) or DBC (right panels)
and $N=100$.
The blue (red) dashed lines correspond to the lower (upper) bound
(see (\ref{new upper bound relaxed}) and (\ref{new lower bound relaxed}) for the left panels
and (\ref{new bounds relaxed DBC}) for the right panels).
In all the panels $\omega_0=0.1$.
}
\vspace{0.4cm}
\label{fig:BoundsImproved}
\end{figure}
The bounds (\ref{new bounds_main})
can be written explicitly for the harmonic chains that we are considering
by setting either $\eta=1$ or $\eta=0$
and employing either (\ref{dispersion relations}) or (\ref{dispersion DBC}) for the dispersion relations
when either PBC or DBC respectively are imposed.
The resulting expressions for these bounds require to sum either $N$ or $N-1$ terms
and we can obtain less constraining but still insightful bounds
by keeping only few terms in these sums, i.e.
\begin{equation}
\label{new bounds_main_weaker}
\mathcal{C}^2_{\textrm{\tiny L}, k_{\textrm{\tiny L}}}
\, \leqslant \,\mathcal{C}^2\,\leqslant \,
\mathcal{C}^2_{\textrm{\tiny U}, k_{\textrm{\tiny U}}}
\end{equation}
where $k_{\textrm{\tiny L}}$ and $k_{\textrm{\tiny U}}$ are independent parameters
related to the number of terms in the sum kept to define the corresponding bound.
Since the explicit expressions of the dispersion relations are important to write explicitly
the bounds in (\ref{new bounds_main_weaker}), the cases of PBC and DBC must be studied separately.
Considering PBC first,
one observes that the corresponding $f_{\textrm{\tiny L}}(\tilde{x}_k)$ as function of $k$
(that can be constructed from (\ref{fLs-def}), (\ref{sum over modes int arcsinh}) and (\ref{dispersion relations}))
is large when $k\simeq 1$ and $k\simeq N-1$,
while it becomes negligible in the middle of the interval $[1, N-1]$.
This leads to sum just over $k=1,\dots,k_{\textrm{\tiny L}}$
and $k=N-k_{\textrm{\tiny L}},\dots,N-1$, for some $k_{\textrm{\tiny L}}$.
Thus, by employing also the symmetry $ k \leftrightarrow N-k$ of the dispersion relations (\ref{dispersion relations}),
the lower bound in (\ref{new bounds_main_weaker}) for PBC reads
\begin{equation}
\label{new lower bound relaxed}
\mathcal{C}^2_{\textrm{\tiny L}, k_{\textrm{\tiny L}}}
= c_0
+
2\sum_{k=1}^{k_{\textrm{\tiny L}}}
f_{\textrm{\tiny L}}(\tilde{x}_k)\, [ \sin(\Omega_{k} t )]^2\,.
\end{equation}
The upper bound $\mathcal{C}^2_{\textrm{\tiny U}, k_{\textrm{\tiny U}}}$ can be found through
similar considerations applied to the function $f_{\textrm{\tiny U}}(\tilde{x}_k)$ introduced in
(\ref{fLs-def}).
This leads to sum the terms whose $k$ is close to the boundary of $[1, N-1]$ keeping their dependence on $t$
and to set $\sin^2(\Omega_k t)=1$ in the remaining ones, which must not be discarded.
The resulting bound is
\begin{equation}
\label{new upper bound relaxed}
\mathcal{C}^2_{\textrm{\tiny U}, k_{\textrm{\tiny U}}}
\,=\,
c_0
+
2\sum_{k=1}^{k_{\textrm{\tiny U}}}
f_{\textrm{\tiny U}}(\tilde{x}_k)\, [ \sin(\Omega_{k} t )]^2
+\!\!
\sum_{k=k_{\textrm{\tiny U}}+1}^{\lfloor \frac{N-1}{2} \rfloor}\!\!
2\,f_{\textrm{\tiny U}}(\tilde{x}_k)
+
f_{\textrm{\tiny U}}(\tilde{x}_{N/2})\,
\big| \cos(\pi N/2) \big|\,.
\end{equation}
The bounds (\ref{new bounds_main}) are recovered when $k_{\textrm{\tiny L}}=k_{\textrm{\tiny U}}=\lfloor \frac{N-1}{2} \rfloor$,
by setting to zero the second sum in the r.h.s. of (\ref{new upper bound relaxed})
and by restoring the time dependence in the term having $k=N/2$ when $N$ is even,
both in (\ref{new lower bound relaxed}) and (\ref{new upper bound relaxed}).
When DBC are imposed, a similar analysis can be carried out,
with the crucial difference that the symmetry $k\leftrightarrow N-k$
in the dispersion relations (\ref{dispersion DBC}) does not occur in this case.
Setting $\eta=0$ and employing the dispersion relations (\ref{dispersion DBC}),
one obtains (\ref{new bounds_main_weaker}) with
\begin{equation}
\label{new bounds relaxed DBC}
\mathcal{C}^2_{\textrm{\tiny L}, k_{\textrm{\tiny L}}}
\equiv
\sum_{k=1}^{k_{\textrm{\tiny L}}}
f_{\textrm{\tiny L}}(\tilde{x}_k) \, [\sin (\Omega_{k} t )]^2
\;\;\qquad\;\;
\mathcal{C}^2_{\textrm{\tiny U}, k_{\textrm{\tiny U}}}
\equiv
\sum_{k=1}^{k_{\textrm{\tiny U}}}
f_{\textrm{\tiny U}}(\tilde{x}_k) \, [\sin (\Omega_{k} t )]^2
+\!\!
\sum_{k=k_{\textrm{\tiny U}}+1}^{N-1}\!\!
f_{\textrm{\tiny U}}(\tilde{x}_k)
\end{equation}
where $1\leqslant k_{\textrm{\tiny L}}, k_{\textrm{\tiny U}} \leqslant N-1$.
In order to recover (\ref{new bounds_main}) from (\ref{new bounds_main_weaker}),
we have to choose $k_{\textrm{\tiny L}}=k_{\textrm{\tiny U}}=N-1$
and set to zero the last sum in the second expression of (\ref{new bounds relaxed DBC}).
By construction, we have
$\mathcal{C}^2_{\textrm{\tiny L}, k_{\textrm{\tiny L}}} \leqslant
\mathcal{C}^2_{\textrm{\tiny L}}$
and
$\mathcal{C}^2_{\textrm{\tiny U}, k_{\textrm{\tiny U}}} \geqslant
\mathcal{C}^2_{\textrm{\tiny U}}$,
but $\mathcal{C}^2_{\textrm{\tiny L}, k_{\textrm{\tiny L}}}$ and $\mathcal{C}^2_{\textrm{\tiny U}, k_{\textrm{\tiny U}}}$
contain less terms than $\mathcal{C}^2_{\textrm{\tiny L}}$ and $\mathcal{C}^2_{\textrm{\tiny U}}$ respectively,
hence they are easier to evaluate and to study analytically.
For both PBC and DBC, considering either the lower or the upper bound in (\ref{new bounds_main_weaker}),
it improves as either $k_{\textrm{\tiny L}}$ or $ k_{\textrm{\tiny U}}$ respectively increases.
In Fig.\,\ref{fig:BoundsImproved} we show the bounds (\ref{new bounds_main_weaker})
when either PBC (left panels) or DBC (right panels) are imposed
and small values of $k_{\textrm{\tiny L}}$ and $ k_{\textrm{\tiny U}}$
are considered.
For given values of $k_{\textrm{\tiny L}}$ and $ k_{\textrm{\tiny U}}$,
the agreement between the bounds and the exact curve improves as $|\omega_0-\omega|$ decreases.
Notice that higher values of $k_{\textrm{\tiny L}}$ and $ k_{\textrm{\tiny U}}$
are needed for DBC
to reach an agreement with the exact curve comparable with the one obtained for PBC.
\subsection{Large $N$}
\label{subsec:TD limit HC}
It is important to study approximate expressions for the temporal evolution of the complexity
when large values of $N$ are considered.
In our numerical analysis, we noticed that,
for finite but large enough values of $N\gtrsim 10$
the complexity (\ref{comp-pure-global-DBCPBC})
is well described by a function of $\omega N$, $\omega_0 N$ and $t/N$.
This function, which depends on whether PBC or DBC are imposed,
can be written by introducing the approximation $\sin(x) \simeq x$ into the dispersion relations
and keeping only the leading term (see appendix \ref{subapp:smallkapprox} for a more detailed discussion).
For PBC we find
\begin{equation}
\label{smallkapprox_PBC_main}
\mathcal{C}_{\textrm{\tiny approx}}
\,=\,
\sqrt{c_{0}(t)
+
2 \!\! \sum_{k=1}^{\lfloor \frac{N-1}{2} \rfloor }\!
\left[ \textrm{arcsinh}\!
\left( \,
\frac{(\omega N)^2 - (\omega_0 N)^2}{2\,\widetilde{\Omega}^{\textrm{\tiny (P)}}_{k}\, \widetilde{\Omega}^{\textrm{\tiny (P)}}_{0,k}} \,
\sin \! \big(\widetilde{\Omega}^{\textrm{\tiny (P)}}_{k} t / N \big)
\right)
\right]^2
}
\end{equation}
where $c_{0}(t)$ is (\ref{low-bound});
while for DBC we get
\begin{equation}
\label{smallkapprox_DBC_main}
\mathcal{C}_{\textrm{\tiny approx}}
=
\sqrt{\,\sum_{k=1}^{N-1}\left[ \textrm{arcsinh}\!
\left( \,
\frac{(\omega N)^2 - (\omega_0 N)^2}{2\,\widetilde{\Omega}^{\textrm{\tiny (D)}}_k \widetilde{\Omega}^{\textrm{\tiny (D)}}_{0,k}} \,
\sin \! \big(\widetilde{\Omega}^{\textrm{\tiny (D)}}_{k} t / N \big)
\right)
\right]^2}
\end{equation}
where
\begin{equation}
\label{smallkapprox_dispersion}
\widetilde{\Omega}^{\textrm{\tiny (P)}}_k
=\sqrt{(\omega N)^2+\frac{4\pi^2 \kappa}{m}\, k^2}
\,\,\qquad\,\,
\widetilde{\Omega}^{\textrm{\tiny (D)}}_k
=
\sqrt{(\omega N)^2+\frac{\pi^2 \kappa}{m}\, k^2}
\end{equation}
while $\widetilde{\Omega}^{\textrm{\tiny (P)}}_{0,k}$
and $\widetilde{\Omega}^{\textrm{\tiny (D)}}_{0,k}$ are obtained by replacing $\omega$ with $\omega_0$
in these expressions.
Notice that both (\ref{smallkapprox_PBC_main}) and (\ref{smallkapprox_DBC_main})
depend on $\omega N$, $\omega_0 N$ and $t/N$.
These approximate expressions have been used to plot the dashed light grey curves
in the top panels of Fig.\,\ref{fig:TDvsfinite},
which nicely agree with the corresponding solid coloured curves.
The thermodynamic limit $N\to\infty$ of the complexity can be studied
through the standard procedure.
Introducing $\theta \equiv \pi k/N$
and substituting $\sum_k\to \frac{N}{\pi}\int_{0}^\pi d\theta$
in (\ref{comp-pure-global-DBCPBC}), at the leading order we find
\begin{equation}
\label{comp TD limit}
\mathcal{C}_{\textrm{\tiny TD}}
=
\sqrt{\frac{N}{\pi}}\;
\sqrt{\int_{0}^\pi\left[ \textrm{arcsinh}\!
\left( \,
\frac{\omega^2 - \omega_0^2}{2\,\Omega_{\theta}\, \Omega_{0,\theta}} \,
\sin (\Omega_{\theta} \,t )
\right)
\right]^2 \! d\theta}
\end{equation}
where the dispersion relations for PBC and DBC become respectively
\begin{equation}
\label{dispersion relations TD}
\Omega_{0,\theta} = \sqrt{\omega_0^2 + \frac{4\kappa}{m} \,(\sin\theta)^2 }
\,\,\qquad\,\,
\Omega_\theta = \sqrt{\omega^2 + \frac{4\kappa}{m} \,( \sin \theta )^2}
\end{equation}
and
\begin{equation}
\label{dispersion DBC TD}
\Omega_{0,\theta}=\sqrt{\omega_0^2+\frac{4\kappa}{m} \,[\sin(\theta/2)]^2}
\,\,\qquad\,\,
\Omega_\theta=\sqrt{\omega^2+\frac{4\kappa}{m} \,[\sin(\theta/2)]^2}\;.
\end{equation}
Notice that, for PBC, the zero mode does not contribute because $c_0/N \to 0$ as $N \to \infty$.
When DBC hold, by using the dispersion relations (\ref{dispersion DBC TD}),
changing of variable $\tilde{\theta}=\theta/2$ in (\ref{comp TD limit})
and exploiting the symmetry of the function $\sin (x)$ in the interval $[0,\pi]$,
one finds that (\ref{comp TD limit}) with (\ref{dispersion relations TD}) holds for both PBC and DBC.
Thus, the leading order of this limit is independent of the boundary conditions.
This means that the complexity does not distinguish the boundary conditions
in this regime.
Indeed, in the left and right panels of Fig.\,\ref{fig:TDvsfinite},
the same function (just described) has been used to plot the dashed black curves.
\begin{figure}[htbp!]
\vspace{-.8cm}
\subfigure
{\hspace{-1.2cm}
\includegraphics[width=.54\textwidth]{CompPBCPureIntermediateMasslessEvolution_manyomega0}}
\vspace{-.25cm}
\subfigure
{
\hspace{.0cm}\includegraphics[width=.54\textwidth]{CompDBCPureIntermediateMasslessEvolution_manyomega0}}
\subfigure
{
\hspace{-1.2cm}\includegraphics[width=.54\textwidth]{CompPBCPureIntermediateMassiveEvolutionomegasmalleromega0_manyomega0}}
\vspace{-.25cm}
\subfigure
{\hspace{.0cm}
\includegraphics[width=.54\textwidth]{CompDBCPureIntermediateMassiveEvolutionomegasmalleromega0_manyomega0}}
\subfigure
{
\hspace{-1.2cm}\includegraphics[width=.54\textwidth]{CompPBCPureIntermediateMassiveEvolutionTrans_manyomega0}}
\vspace{-.25cm}
\subfigure
{\hspace{.0cm}
\includegraphics[width=.54\textwidth]{CompDBCPureIntermediateMassiveEvolutionTrans_manyomega0}}
\subfigure
{
\hspace{-1.2cm}\includegraphics[width=.54\textwidth]{CompPBCPureIntermediateMassiveEvolution_manyomega0}}
\vspace{-.25cm}
\subfigure
{\hspace{.0cm}
\includegraphics[width=.54\textwidth]{CompDBCPureIntermediateMassiveEvolution_manyomega0}}
\caption{Temporal evolutions of the complexity for harmonic chains
with either PBC (left panels) or DBC (right panels).
The solid lines show $\mathcal{C}^2/N$ for PBC
and $(\mathcal{C}^2-\zeta)/N$ for DBC,
with $\mathcal{C}$ given by (\ref{comp-pure-global-DBCPBC})
and $\zeta$ by (\ref{corr_Csq Dir}).
The dashed black lines represent $\mathcal{C}_{\textrm{\tiny TD}}^2/N$,
from (\ref{comp TD limit}).
}
\label{fig:TDvsfinite}
\end{figure}
The boundary conditions become crucial in the subleading term of the expansion of
(\ref{comp-pure-global-DBCPBC}) as $N\to\infty$,
which can be studied through the Euler-Maclaurin formula \cite{Kac02bookCalculus}.
The details of this analysis are discussed in appendix \ref{app:EulerMcLaurin}
and the final result is
\begin{equation}
\label{EmcL maintext}
\mathcal{C}^2-\mathcal{C}_{\textrm{\tiny TD}}^2
=
R_{1,\infty}^{\textrm{\tiny (B)}}
\;\;\qquad\;\;
\textrm{B} \in \big\{ \textrm{P}, \textrm{D} \big\}
\;\;\qquad\;\;
R_{1,\infty}^{\textrm{\tiny (B)}}
=
\left\{\begin{array}{ll}
R_{1,\infty}^{\textrm{\tiny (P)}} & \textrm{PBC}
\\
\rule{0pt}{.7cm}
\displaystyle
R_{1,\infty}^{\textrm{\tiny (D)}}
=
\frac{R_{1,\infty}^{\textrm{\tiny (P)}}}{2} + \zeta \hspace{.5cm} & \textrm{DBC}
\end{array}\right.
\end{equation}
where $R_{1,\infty}^{\textrm{\tiny (P)}}$ and $\zeta$ are the time-dependent functions
given in (\ref{bound rest_ p0PBC v2}) and in (\ref{corr_Csq Dir}) respectively.
Numerical checks for these results are shown in Fig.\,\ref{fig:TDvsfinite}.
In the top panels of this figure we have displayed also $\mathcal{C}_{\textrm{\tiny approx}}/N$
from (\ref{smallkapprox_PBC_main}) (left panel)
and $(\mathcal{C}_{\textrm{\tiny approx}}-\zeta)/N$ from (\ref{smallkapprox_DBC_main})
(right panel) through dashed light grey lines.
In the continuum limit,
$N\to\infty$ and the lattice spacing $a \equiv \sqrt{m/\kappa} \to 0$
is vanishing while $N a\equiv \ell$ is kept fixed.
In this limit,
the expression (\ref{comp-pure-global-DBCPBC}) for the complexity
(which holds for both PBC and DBC) becomes
\begin{equation}
\label{comp HC cont}
\mathcal{C}_{\textrm{\tiny cont}}
=
\sqrt{\frac{\ell}{2\pi}}\,
\sqrt{\int_{-\infty}^{\infty}
\left[ \textrm{arcsinh}\!
\left( \,
\frac{\omega^2 - \omega_0^2}{2\,\Omega_p\, \Omega_{0,p}} \,
\sin (\Omega_p t )
\right)
\right]^2\! dp}
\end{equation}
(see appendix \ref{app:continuum} for a detailed discussion)
where
\begin{equation}
\label{dispersion continuum HC}
\Omega_{0,p}=\sqrt{\omega_0^2 + p^2}
\,\,\qquad\,\,
\Omega_p=\sqrt{\omega^2 + p^2}\,.
\end{equation}
Since $\Omega_p\simeq p$ when $p\gg\omega$,
the vanishing of the integrand in (\ref{comp HC cont})
as $p\to\pm\infty$ is such that the complexity is UV finite.
We remark that, instead, when the reference state is the unentangled product state,
the continuum limit of the complexity is UV divergent,
as discussed in appendix \ref{app:continuum};
hence a UV cutoff in the integration domain over $p$ must be introduced.
\subsection{Initial growth}
It is worth discussing the initial growth of the complexity for the harmonic chains
that we are considering.
Since the complexity (\ref{comp-pure-global-DBCPBC}) is a special case of (\ref{comp-pure-global-general}),
its expansion as $ t \to 0$ can be found by specialising the expansion (\ref{comp-pure-initialgrowth})
and its coefficients (\ref{c_12_coeff_exp}) and (\ref{c_3_coeff_exp}) to the harmonic chains with either PBC or DBC.
For the sake of simplicity, in the following we discuss only the leading term
(i.e. only the coefficient $b_1$ in (\ref{c_12_coeff_exp})),
which provides the linear growth,
but a similar analysis can be applied straightforwardly
to the coefficients of the higher order terms in the $t \to 0$ expansion.
For the harmonic chains with either PBC or DBC, the linear growth
in (\ref{comp-pure-initialgrowth}) becomes
\begin{equation}
\label{initial growth HC 1d}
\mathcal{C}
\,=\,
\frac{|\omega^2-\omega^2_0|}{2}\,
\Bigg( \sum_{k=1}^{N-1+\eta} \! \Omega_{0,k}^{-2} \Bigg)^{1/2}
t
\, +O(t^3)
\end{equation}
where $\eta=1$ and (\ref{dispersion relations}) must be used for PBC,
while $\eta=0$ and (\ref{dispersion DBC}) must be employed for DBC.
We remark that the slope of the initial linear growth in (\ref{initial growth HC 1d}) is proportional to $|\omega-\omega_0|$.
\begin{figure}[t!]
\subfigure
{\hspace{-1.5cm}
\includegraphics[width=.57\textwidth]{CompPBCPureEarlyMassiveEvolution}}
\subfigure
{
\hspace{-.45cm}\includegraphics[width=.57\textwidth]{CompPBCPureEarlyMasslessEvolution}}
\caption{Initial growth of the complexity
for harmonic chains with PBC and $N=100$.
The evolution Hamiltonian is either massive (left panel) or massless (right panel),
for three values $\omega_0$.
The solid lines show the complexity (\ref{comp-pure-global-DBCPBC})
(with (\ref{dispersion relations}))
and the dashed lines represent its expansion (\ref{comp-pure-initialgrowth})
up to the $O(t)$ (grey), $O(t^3)$ (yellow) and $O(t^5)$ (green) term included.
}
\vspace{0.1cm}
\label{fig:PureStateGlobalSmallTimes}
\end{figure}
In Fig.\,\ref{fig:PureStateGlobalSmallTimes}, we consider
the initial growth of the complexity (\ref{comp-pure-global-DBCPBC}) when PBC are imposed,
comparing the exact curve against its expansion (\ref{comp-pure-initialgrowth}).
The corresponding analysis for DBC provides curves that are very similar to the ones displayed in Fig.\,\ref{fig:PureStateGlobalSmallTimes};
hence it has not been reported in this manuscript.
Let us conclude our discussion about the temporal evolution of the complexity of pure states
with a brief qualitative comparison between
the results discussed above and
the corresponding ones for the temporal evolution of the holographic complexity
\cite{Stanford:2014jda,Susskind:2014jwa,Brown:2015bva,Brown:2015lvg,Roberts:2014isa,Moosa:2017yvt,Chapman:2018dem,Chapman:2018lsv}.
The Vaidya spacetimes are the typical backgrounds employed as the gravitational duals
of global quantum quenches in the conformal field theory on their boundary.
They describe the formation of a black hole through the collapse of a matter shell.
In Vaidya spacetimes, the temporal evolution of the holographic entanglement entropy
has been largely studied \cite{Hubeny:2007xt,AbajoArrastia:2010yt,
Balasubramanian:2011ur,
Balasubramanian:2011at,Allais:2011ys,Callan:2012ip,Hubeny:2013hz,Hubeny:2013dea}
and
the temporal evolutions of the holographic complexity
for the entire spatial section of the conformal field theory on the boundary
has been investigated in \cite{Moosa:2017yvt,Chapman:2018dem,Chapman:2018lsv, Fan:2018xwf}.
Considering the temporal evolution of the rate $\frac{d\mathcal{C}}{dt}$
allows to avoid the problem of choosing the reference state,
which deserves further clarifications for the holographic complexity,
even for static gravitational backgrounds.
The analysis of $\frac{d\mathcal{C}}{dt}$ in Vaidya spacetimes,
both for the CV and for the CA prescriptions,
shows that these temporal evolutions are linear in time both
at very early and at late time \cite{Moosa:2017yvt, Chapman:2018dem}.
While also the initial growth of the complexity
that we have explored is linear (see (\ref{initial growth HC 1d})),
the late time growth is at most logarithmic.
This disagreement, which deserves further analysis,
has been discussed in \cite{Chapman:2018hou}.
We find it worth observing also that
the coefficient of the initial growth (\ref{initial growth HC 1d}) is proportional to $|\omega^2 - \omega_0^2|$
and that the corresponding coefficient for the holographic complexity
is proportional to the mass of the
final black hole \cite{Chapman:2018dem, Moosa:2017yvt}.
\section{Subsystem complexity in finite harmonic chains}
\label{sec:mixedFinSize}
In this section we study the temporal evolution of the subsystem complexity after a global quench.
The reference and the target states are the reduced density matrices associated to a given subsystem.
We focus on the simple cases where the subsystem $A$
is a block made by consecutive sites in harmonic chains
with either PBC or DBC.
\subsection{Subsystem complexity}
In the harmonic lattices that we are considering,
the reduced density matrix associated to $A$
characterises a Gaussian state
which can be described equivalently through
its reduced covariance matrix $\gamma_A$.
This matrix is constructed by considering
the reduced correlation matrices $Q_A$, $P_A$ and $M_A$,
whose elements are respectively given by
$(Q_A)_{i,j}=\langle \psi_0 |\, \hat{q}_i(t) \,\hat{q}_j(t) \, | \psi_0 \rangle $,
$(P_A)_{i,j}=\langle \psi_0 |\, \hat{p}_i(t) \,\hat{p}_j(t) \, | \psi_0 \rangle $
and
$(M_A)_{i,j}=\textrm{Re}\big[\langle \psi_0 |\, \hat{q}_i(t) \,\hat{p}_j(t) \, | \psi_0 \rangle \big]$
with $i,j\in A$,
which depend also on the time after the global quench.
These matrices provide the following block decomposition of the reduced covariance matrix
\begin{equation}
\label{reduced CM}
\gamma_A(t)
=\,
\bigg(
\begin{array}{cc}
Q_A(t) \,& M_A(t) \\
M_A(t)^{\textrm{t}} \,& P_A(t) \\
\end{array} \bigg)\,.
\end{equation}
For the harmonic chains with either PBC or DBC
introduced in Sec.\,\ref{sec:purestates_HC_glob}
and $A$ made by $L$ consecutive sites,
$Q_A$ and $P_A$ are $L \times L$ symmetric matrices
and $\gamma_A$ is a real, symmetric and positive definite $2L \times 2L$ matrix.
Adapting the analysis made in Sec.\,\ref{sec:purestates_HC_glob} for pure states
to the mixed states described by the reduced covariance matrices $\gamma_A(t)$,
we have that
the reference state is given by the reduced density matrix for the interval $A$
at time $t_\textrm{\tiny R} \geqslant 0$
obtained through the quench protocol characterised by
$\big(\kappa_\textrm{\tiny R},m_\textrm{\tiny R},\omega_\textrm{\tiny R},\omega_{0,\textrm{\tiny R}}\big)$
and the target state by the reduced density matrix for the same interval
at time $t_\textrm{\tiny T} \geqslant t_\textrm{\tiny R}$
constructed through the quench protocol described by
$\big(\kappa_\textrm{\tiny T},m_\textrm{\tiny T},\omega_\textrm{\tiny T},\omega_{0,\textrm{\tiny T}}\big)$.
The corresponding reduced covariance matrices
are denoted by $ \gamma_{\textrm{\tiny R},A} (t_\textrm{\tiny R}) $
and $ \gamma_{\textrm{\tiny T},A} (t_\textrm{\tiny T}) $ respectively.
These reduced covariance matrices
are decomposed in terms of the
correlation matrices of the subsystem like in (\ref{reduced CM}).
The approach to the circuit complexity of mixed states based on the Fisher information geometry \cite{DiGiulio:2020hlz}
allows to construct the optimal circuit
between $ \gamma_{\textrm{\tiny R},A} (t_\textrm{\tiny R}) $ and $ \gamma_{\textrm{\tiny T},A} (t_\textrm{\tiny T}) $.
The covariance matrices along this optimal circuit are
\begin{equation}
\label{optimal circuit rdm}
G_{s}( \gamma_{\textrm{\tiny R},A}(t_\textrm{\tiny R}) \, , \gamma_{\textrm{\tiny T},A}(t_\textrm{\tiny T}))
\,\equiv \,
\gamma_{\textrm{\tiny R},A}(t_\textrm{\tiny R})^{1/2}
\Big( \gamma_{\textrm{\tiny R},A}(t_\textrm{\tiny R})^{- 1/2} \,
\gamma_{\textrm{\tiny T},A}(t_\textrm{\tiny T}) \,
\gamma_{\textrm{\tiny R},A}(t_\textrm{\tiny R})^{-1/2} \Big)^s
\gamma_{\textrm{\tiny R},A}(t_\textrm{\tiny R})^{1/2}
\end{equation}
where $0 \leqslant s \leqslant 1$ parameterises the optimal circuit.
The length of this optimal circuit is proportional to its complexity
\begin{equation}
\label{c2-complexity-rdm}
\mathcal{C}_A
\,=\,
\frac{1}{2\sqrt{2}}\;
\sqrt{\,
\textrm{Tr} \,\Big\{ \big[
\log \!\big( \gamma_{\textrm{\tiny T},A}(t_\textrm{\tiny T}) \, \gamma_{\textrm{\tiny R},A}(t_\textrm{\tiny R})^{-1} \big)
\big]^2 \Big\}}\;.
\end{equation}
Considering harmonic chains made by $N$ sites where PBC are imposed,
by using (\ref{time dep corrs trans}), (\ref{QPRmat t-dep-k})
and either (\ref{Vtilde-def-even}) or (\ref{Vtilde-def-odd}),
one obtains the elements of the correlation matrices whose reduction to $A$ provides
(\ref{reduced CM}).
They read
\begin{eqnarray}
\label{QPRmat t-dep 1d}
Q_{i,j}(t)
&=&
\frac{1}{N} \sum_{k=1}^N Q_{k}(t) \,\cos\!\big[(i-j)\,2\pi k/N\big]
\nonumber
\\
\rule{0pt}{.8cm}
P_{i,j}(t)
&=&
\frac{1}{N}\sum_{k=1}^N P_{k}(t)\, \cos\!\big[(i-j)\,2\pi k/N\big]
\\
\rule{0pt}{.8cm}
M_{i,j}(t)
&=&
\frac{1}{N}\,\sum_{k=1}^N M_{k}(t) \,\cos\!\big[(i-j)\,2\pi k/N\big]
\nonumber
\end{eqnarray}
where $1 \leqslant i,j \leqslant N$;
while for DBC, by using (\ref{Vtilde-HC-DBC}), one obtains the following correlators
\begin{eqnarray}
\label{QPRmat t-dep 1d DBC}
Q_{i,j}(t)
&=&
\frac{2}{N}\sum_{k=1}^N Q_{k}(t) \,\sin\!\big(i\,\pi k/N\big)\,\sin\!\big(j\,\pi k/N\big)
\nonumber
\\
\rule{0pt}{.8cm}
P_{i,j}(t)
&=&
\frac{2}{N}\sum_{k=1}^N P_{k}(t) \,\sin\!\big(i\,\pi k/N\big)\,\sin\!\big(j\,\pi k/N\big)
\\
\rule{0pt}{.8cm}
M_{i,j}(t)
&=&
\frac{2}{N}\,\sum_{k=1}^N M_{k}(t)\,\sin\!\big(i\,\pi k/N\big)\,\sin\!\big(j\,\pi k/N\big)
\nonumber
\end{eqnarray}
where $1 \leqslant i,j \leqslant N-1$.
In these correlators, the functions $Q_{k}(t)$, $P_{k}(t)$ and $M_{k}(t)$
are given by (\ref{QPRmat t-dep-k}), with either (\ref{dispersion relations}) for PBC or (\ref{dispersion DBC}) for DBC.
The reduced covariance matrices
$ \gamma_{\textrm{\tiny R},A} (t_\textrm{\tiny R}) $
and $ \gamma_{\textrm{\tiny T},A} (t_\textrm{\tiny T}) $
for the block $A$ providing the optimal circuit (\ref{optimal circuit rdm})
and its complexity (\ref{c2-complexity-rdm}) are constructed as in (\ref{reduced CM}),
through the reduced correlation matrices $Q_A$, $P_A$ and $M_A$,
obtained by restricting to $i,j \in A$ the indices of the correlation matrices whose elements
are given in (\ref{QPRmat t-dep 1d}) and (\ref{QPRmat t-dep 1d DBC}).
We remark that the matrix $\widetilde{V}$ in (\ref{time dep corrs trans})
(given in (\ref{Vtilde-def-even}) or (\ref{Vtilde-def-odd}) for PBC and in (\ref{Vtilde-HC-DBC}) for DBC)
is crucial to write (\ref{QPRmat t-dep 1d}) and (\ref{QPRmat t-dep 1d DBC});
hence it enters in a highly non-trivial way in the evaluation of the subsystem complexity.
Instead, it does not affect the complexity for the entire system,
where both the reference and the target states are pure states, as remarked below (\ref{gamma-TR-global}).
\subsection{Numerical results}
\label{sec:subsysteem-num-res}
Considering the global quench that we are exploring,
in the following we discuss some numerical results
for the temporal evolution of the subsystem complexity
of a block $A$ made by $L$ consecutive sites
in harmonic chains made by $N$ sites,
where either PBC or DBC are imposed.
We focus on the simplest setup where the reference state is the initial state (hence $t_\textrm{\tiny R}=0$)
and the target state corresponds to a generic value of $t_\textrm{\tiny T}=t \geqslant 0$ after the quench.
The remaining parameters are fixed to
$\omega_{0,\textrm{\tiny R}}=\omega_{0,\textrm{\tiny T}}\equiv \omega_{0}$, $\omega_{\textrm{\tiny R}}=\omega_{\textrm{\tiny T}}\equiv \omega$, $\kappa_{\textrm{\tiny R}}=\kappa_{\textrm{\tiny T}}=1$ and $m_{\textrm{\tiny R}}=m_{\textrm{\tiny T}}=1$.
In the case of DBC, we consider both $A$ adjacent to the boundary and separated from it.
In this setup, the subsystem complexity (\ref{c2-complexity-rdm}) can be written as
\begin{equation}
\label{c2-complexity-rdm-our-case}
\mathcal{C}_A
\,=\,
\frac{1}{2\sqrt{2}}\;
\sqrt{\,
\textrm{Tr} \,\Big\{ \big[
\log \!\big( \gamma_{A}(t) \, \gamma_{A}(0)^{-1} \big)
\big]^2 \Big\}}\;.
\end{equation}
It is natural to introduce also the entanglement entropy $S_A(t)$ and its initial value $S_A(0)$,
which lead to define the increment of the entanglement entropy w.r.t. its initial value, i.e.
\begin{equation}
\label{Delta-S_A-def}
\Delta S_A \equiv S_A(t) - S_A(0)
\end{equation}
where $S_A(t)$ and $S_A(0)$ can be evaluated from the symplectic spectrum of
$\gamma_{A}(t)$ and of $\gamma_{A}(0)$ respectively in the standard way
\cite{Peschel_2009,Eisert:2008ur,Weedbrook12b,Peschel03,Audenaert:2002xfl,Plenio:2004he,Cramer:2005mx,Rajabpour_14,Cotler:2016acd,Hackl:2017ndi}.
In all the figures discussed in this section
we show the temporal evolutions of the subsystem complexity $\mathcal{C}_A$ in (\ref{c2-complexity-rdm-our-case})
or of the increment $\Delta S_A$ of the entanglement entropy in (\ref{Delta-S_A-def})
after the global quench.
In particular, we show numerical results corresponding to $N=100$ and $N=200$,
finding nice collapses of the data
when $L/N$, $\omega_0 N$ and $\omega N$ are kept fixed, independently of the boundary conditions.
The data reported in all the left panels have been obtained in harmonic chains with PBC,
whose dispersion relations are (\ref{dispersion relations}),
while the ones in all the right panels correspond to a block adjacent to a boundary
of harmonic chains where DBC are imposed,
whose dispersion relations are (\ref{dispersion DBC}),
if not otherwise indicated
(like in Fig.\ref{fig:MixedStateGlobalMasslessDBCIntDet}).
The evolution Hamiltonian is gapless
in Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionlessNoentropy},
Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless},
Fig.\,\ref{fig:CompvsEntStateGlobalMasslessEvolutionDimensionless}
and Fig.\,\ref{fig:MixedStateGlobalMasslessDBCIntDet},
while it is gapped in Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionDimensionlessNoEntropy} and Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionDimensionless},
with $\omega N =5$.
In Fig.\,\ref{fig:MixedStateGlobalSmallTimesDimensionless}, where the initial growth is explored,
both gapless and gapped evolution Hamiltonians have been employed.
When $L=N$, the complexity (\ref{comp-pure-global-DBCPBC}) for pure states
has been evaluated
with either $N=100$ (black solid lines) or $N=200$ (dashed green lines).
\begin{figure}[t!]
\subfigure
{\hspace{-1.6cm}
\includegraphics[width=.58\textwidth]{DimensionlessMixedPBComegainell20omegaell0}}
\subfigure
{
\hspace{-.7cm}\includegraphics[width=.58\textwidth]{DimensionlessMixedDBComegainell20omegaell0}}
\caption{
Temporal evolution of $\mathcal{C}_A$ in (\ref{c2-complexity-rdm-our-case})
after the global quench with gapless evolution Hamiltonian and $\omega_0 N =20$,
for a block $A$ made by $L$ consecutive sites in harmonic chains
with either PBC (left panels) or DBC (right panels) made by $N$ sites
(in the latter case $A$ is adjacent to a boundary).
When $L=N$, the complexity (\ref{comp-pure-global-DBCPBC}) is shown
for $N=100$ (solid black lines) and $N=200$ (dashed green lines).
}
\vspace{0.4cm}
\label{fig:MixedStateGlobalMasslessEvolutionDimensionlessNoentropy}
\end{figure}
\begin{figure}[htbp!]
\vspace{-1cm}
\subfigure
{
\hspace{-1.6cm}\includegraphics[width=.58\textwidth]{DimensionlessMixedPBComegainell100omegaell0}}
\subfigure
{\hspace{-.7cm}
\includegraphics[width=.58\textwidth]{DimensionlessMixedDBComegainell100omegaell0}}
\subfigure
{
\hspace{-1.6cm}\includegraphics[width=.58\textwidth]{DimensionlessEEPBComegainell100omegaell0}}
\subfigure
{\hspace{-.7cm}
\includegraphics[width=.58\textwidth]{DimensionlessEEDBComegainell100omegaell0}}
\subfigure
{
\hspace{-1.6cm}\includegraphics[width=.58\textwidth]{InverseRatioMasslessPBC_Fig6}}
\subfigure
{\hspace{-.7cm}
\includegraphics[width=.58\textwidth]{InverseRatioMasslessDBC_Fig6}}
\caption{Temporal evolution
of $\mathcal{C}_A$ in (\ref{c2-complexity-rdm-our-case}) (top panels),
of $\Delta S_A$ in (\ref{Delta-S_A-def}) (middle panels)
and of $\sqrt{L/N}\,\Delta S_A/\mathcal{C}_A $ (bottom panels)
after the global quench with gapless evolution Hamiltonian and $\omega_0 N =100$,
for a block $A$ made by $L$ consecutive sites in a harmonic chains
with either PBC (left panels) or DBC (right panels) made by $N$ sites
(in the latter case $A$ is adjacent to a boundary).
When $L=N$ the complexity (\ref{comp-pure-global-DBCPBC}) is shown
for $N=100$ (solid black lines) and $N=200$ (dashed green lines).
}
\vspace{0.4cm}
\label{fig:MixedStateGlobalMasslessEvolutionDimensionless}
\end{figure}
\begin{figure}[htbp!]
\vspace{-1cm}
\subfigure
{\hspace{-1.6cm}
\includegraphics[width=.58\textwidth]{DimensionlessMixedPBComegainell20omegaell0_Larget}}
\subfigure
{
\hspace{-.7cm}\includegraphics[width=.58\textwidth]{DimensionlessMixedDBComegainell20omegaell0_Larget}}
\subfigure
{
\hspace{-1.6cm}\includegraphics[width=.58\textwidth]{DimensionlessEEPBComegainell20omegaell0_Larget}}
\subfigure
{\hspace{-.7cm}
\includegraphics[width=.58\textwidth]{DimensionlessEEDBComegainell20omegaell0_Larget}}
\subfigure
{
\hspace{-1.6cm}\includegraphics[width=.58\textwidth]{InverseRatioMasslessPBC_LargeTimes_Fig7}}
\subfigure
{\hspace{-.7cm}
\includegraphics[width=.58\textwidth]{InverseRatioMasslessDBC_LargeTimes_Fig7}}
\caption{Temporal evolution
of $\mathcal{C}_A$ in (\ref{c2-complexity-rdm-our-case}) (top panels),
of $\Delta S_A$ in (\ref{Delta-S_A-def}) (middle panels)
and of $\sqrt{L/N}\,\Delta S_A/\mathcal{C}_A$ (bottom panels)
after the global quench with a gapless evolution Hamiltonian and $\omega_0 N =20$,
for harmonic chains with either PBC (left panels) or DBC (right panels),
in the same setups of
Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionlessNoentropy}
are considered.
}
\vspace{.4cm}
\label{fig:CompvsEntStateGlobalMasslessEvolutionDimensionless}
\end{figure}
In Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionlessNoentropy},
Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless}
and Fig.\,\ref{fig:CompvsEntStateGlobalMasslessEvolutionDimensionless}
all the data have been obtained with $\omega N =0$
and either $\omega_0 N =20$ (Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionlessNoentropy}
and Fig.\,\ref{fig:CompvsEntStateGlobalMasslessEvolutionDimensionless})
or $\omega_0 N =100$
(Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless}).
Revivals are observed and the different cycles correspond
to $p < 2t/N < p+1$ for PBC and to $p < t/N < p+1$ for DBC,
where $p$ is a non-negative integer.
The qualitative behaviour
of the temporal evolution of the subsystem complexity
crucially depends on the boundary conditions of the harmonic chain.
For DBC,
considering the data having $L/N < 1/2$ when $t/N < 1/2$,
we can identify three regimes:
an initial growth until a local maximum is reached,
a decrease and then a thermalisation regime after certain value of $t/N$,
where the subsystem complexity remains constant.
For PBC and $L/N < 1/2$,
the latter regime is not observed and $\mathcal{C}_A$ keeps growing.
Comparing the right panel in Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionlessNoentropy} with the top right panel in
Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless},
one realises that, for DBC,
the height of the plateaux increases as either $L/N$ or $\omega_0 N$ increases, as expected.
The absence of thermalisation regimes for PBC could be related
to the occurrence of the zero mode,
as suggested by the fact that, for pure states,
the zero mode contribution provides the logarithmic growth
of the complexity (\ref{C-pure-both-eta}).
However, we are not able to identify explicitly the zero mode contribution in the subsystem complexity,
hence we cannot subtract it as done
in the bottom left panel of Fig.\,\ref{fig:PureStateCritical}
for the temporal evolution of the complexity of pure states.
For DBC, the plateau in the thermalisation regime is not observed when $L/N \geqslant 1/2$
and, considering the interval $t/N\in[\nu,\nu+1]$ with $\nu=\{0,1\}$,
it approximately begins at $t - \nu N \simeq L$ and ends at $t - \nu N \simeq N-L+1$.
The straight dashed grey lines approximatively indicate
the beginning of the plateaux for different $L/N < 1/2$
(in particular, they are obtained by joining the origin with the point of the curve made by the blue data points at $t/N =0.3$).
We remark that the temporal evolution of $\mathcal{C}_A$ in infinite chains
is made by the three regimes mentioned above
(see Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless},
Fig.\,\ref{fig:RatioGlobalMassiveEvolutionTDDetDimensionless}
and Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless}),
as largely discussed in Sec.\,\ref{sec:GGE}.
Comparing the temporal evolutions of $\mathcal{C}_A$ and $\Delta S_A$
for the same quench protocol and the same subsystem in Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless},
we observe that the initial growth of $\mathcal{C}_A$ in the first revival
is faster than the linear initial growth of $\Delta S_A$,
as highlighted by the straight dashed black lines
in Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless}.
Within the first revival, we do not observe a
long range of $t/N$ where the evolution of $\mathcal{C}_A$ is linear.
Nonetheless, the straight line characterising the initial growth of
$\Delta S_A$ intersects the first local maximum corresponding to the end of the
initial growth of $\mathcal{C}_A$ when $L/N < 1/2$.
Considering the data points for $L/N < 1/2$
and the initial regime of $t/N$ corresponding to half of the first revival,
we notice that,
while the temporal evolution of $\Delta S_A$
displays a linear growth followed by a saturation regime,
the temporal evolution of $\mathcal{C}_A$
is characterised by the three regimes described above.
The saturation regimes of $\mathcal{C}_A$ and $\Delta S_A$
are qualitatively very similar
and begin approximatively at the same value of $t/N$.
Notice that the amplitude of the decrease of $\mathcal{C}_A$
at the end of the first revival is smaller than the one of $\Delta S_A$.
The temporal evolutions of $\mathcal{C}_A$ and $\Delta S_A$
can be compared for $L/N \leqslant 1/2$.
Indeed, for a bipartite system in a pure state
the entanglement entropy of a subsystem is equal to
the entanglement entropy of the complementary subsystem.
This property, which does not hold for $\mathcal{C}_A$,
implies the overlap between the data for $\Delta S_A$
corresponding to $L/N =3/10$ and to $L/N =7/10$.
Furthermore, $\Delta S_A = 0$ identically when $L = N$.
In the bottom panels of Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless}
we have reported the temporal evolutions of the ratio $\Delta S_A / \mathcal{C}_A$
for the data reported in the other panels of the figure.
The curves of $\Delta S_A / \mathcal{C}_A$ corresponding to
PBC (left panel) and DBC (right panel) are very similar.
For instance, the curves for $\sqrt{L/N}\,\Delta S_A / \mathcal{C}_A$
have the same initial growth for different values of $L/N$.
However, we remark that a mild logarithmic decrease
occurs in the thermalisation regime for PBC.
In Fig.\,\ref{fig:CompvsEntStateGlobalMasslessEvolutionDimensionless}
the range $0 \leqslant t/N \leqslant 10$ is considered, which is
made by 20 revivals for PBC and by 10 cycles for DBC.
The temporal evolutions of $\mathcal{C}_A$ in the top panels
show that, up to oscillations due to the revivals,
after the initial growth $\mathcal{C}_A$ keeps growing logarithmically for PBC
(the solid coloured lines in the top left panel are two-parameter fits
through the function $a + b \log(t/N)$ of the corresponding data),
while it remains constant for DBC.
This feature is observed also in the corresponding
temporal evolutions of $\Delta S_A$
(middle panels of Fig.\,\ref{fig:CompvsEntStateGlobalMasslessEvolutionDimensionless}).
These two logarithmic growths for PBC are very similar,
as shown by the temporal evolution of $\Delta S_A/\mathcal{C}_A$
displayed in the bottom left panel of
Fig.\,\ref{fig:CompvsEntStateGlobalMasslessEvolutionDimensionless}.
\begin{figure}[t!]
\vspace{.1cm}
\subfigure
{\hspace{-1.6cm}
\includegraphics[width=.58\textwidth]{DimensionlessMixedPBComegainell1omegaell5}}
\subfigure
{
\hspace{-.7cm}\includegraphics[width=.58\textwidth]{DimensionlessMixedDBComegainell1omegaell5}}
\caption{
Temporal evolution of $\mathcal{C}_A$ in (\ref{c2-complexity-rdm-our-case})
after the global quench with a gapped evolution Hamiltonian
for a block $A$ made by $L$ consecutive sites in harmonic chains
with either PBC (left panels) or DBC (right panels) made by $N$ sites
(in the latter case $A$ is adjacent to a boundary of the segment).
When $L=N$ the complexity (\ref{comp-pure-global-DBCPBC}) is shown
for $N=100$ (solid black lines) and $N=200$ (dashed green lines).
}
\vspace{0.4cm}
\label{fig:MixedStateGlobalMassiveEvolutionDimensionlessNoEntropy}
\end{figure}
\begin{figure}[t!]
\vspace{.1cm}
\subfigure
{
\hspace{-1.6cm}\includegraphics[width=.58\textwidth]{DimensionlessMixedPBComegainell10omegaell5}}
\subfigure
{\hspace{-.7cm}
\includegraphics[width=.58\textwidth]{DimensionlessMixedDBComegainell10omegaell5}}
\subfigure
{
\hspace{-1.6cm}\includegraphics[width=.58\textwidth]{DimensionlessEEPBComegainell10omegaell5}}
\subfigure
{\hspace{-.7cm}
\includegraphics[width=.58\textwidth]{DimensionlessEEDBComegainell10omegaell5}}
\subfigure
{\hspace{-1.6cm}
\includegraphics[width=.58\textwidth]{InverseRatioMassivePBC_Fig8}}
\subfigure
{
\hspace{-.7cm}\includegraphics[width=.58\textwidth]{InverseRatioMassiveDBC_Fig8}}
\caption{
Temporal evolution after the global quench with a gapped evolution Hamiltonian
of $\mathcal{C}_A$ in (\ref{c2-complexity-rdm-our-case})
(top panels), of $\Delta S_A$ in (\ref{Delta-S_A-def}) (middle panels)
and of $ \sqrt{L/N}\,\Delta S_A/\mathcal{C}_A $ (bottom panels)
for a block $A$ made by $L$ consecutive sites in harmonic chains
with either PBC (left panels) or DBC (right panels) made by $N$ sites
(in the latter case $A$ is adjacent to a boundary of the segment).
When $L=N$, the complexity (\ref{comp-pure-global-DBCPBC}) is shown
for $N=100$ (solid black lines) and $N=200$ (dashed green lines).
}
\vspace{0.4cm}
\label{fig:MixedStateGlobalMassiveEvolutionDimensionless}
\end{figure}
In Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionDimensionlessNoEntropy}
and Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionDimensionless}
we show some temporal evolutions of $\mathcal{C}_A$
when the evolution Hamiltonian is massive
($\omega_0 < \omega$ in Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionDimensionlessNoEntropy}
and $\omega_0 > \omega$ in Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionDimensionless},
with $\omega N = 5$ in both the figures).
In these temporal evolutions
one observes that the local extrema of the curves for $\mathcal{C}_A$
having different $L/N$
roughly occur at the same values of $t/N$.
It is insightful to compare these temporal evolutions with the corresponding ones
characterised by $\omega =0$ in Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionlessNoentropy}
and Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless}.
For PBC, the underlying growth observed when $\omega =0$ does not occur if $\omega >0$.
For DBC, the plateaux observed in the saturation regime when $\omega =0$
are replaced by oscillatory behaviours if $\omega > 0$.
In Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionDimensionless},
we report the temporal evolutions of $\mathcal{C}_A$,
of $\Delta S_A$ and of $ \Delta S_A/\mathcal{C}_A$ for the same global quench.
The evolutions of $\mathcal{C}_A$ and of $\Delta S_A$ are qualitatively similar when $L/N < 1/2$.
An important difference is the initial growth at very small values of $t/N$:
for $\mathcal{C}_A$ is linear
(see also Fig.\,\ref{fig:MixedStateGlobalSmallTimesDimensionless} and the corresponding discussion),
while for $\Delta S_A$ is quadratic,
as highlighted in the insets of the middle panels
(the coefficient of this quadratic growth for PBC is twice the one obtained for DBC)
and also observed in \cite{Hubeny:2013hz,Liu:2013qca,Rajabpour_14,Unanyan_2014}.
Comparing the bottom panels of Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionDimensionless}
against the bottom panels of Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless},
one notices that the similarity observed for PBC and DBC when $\omega =0$
does not occur when $\omega \neq 0$.
It is important to perform a systematic analysis considering many
other values of $\omega N$ and $\omega_0 N$,
in order to understand the effect of a gapped evolution Hamiltonian
in the temporal evolution of $\mathcal{C}_A$.
In Fig.\,\ref{fig:MixedStateGlobalSmallTimesDimensionless}
we consider the initial regime of the temporal evolution of $\mathcal{C}_A$ w.r.t. the initial state
for various choices of $\omega_0 N$ and $\omega N$
(in particular, $\omega =0$ in the first and in the second lines of panels,
while $\omega >0$ in the third and in the fourth ones).
Very early values of $t$ are considered
with respect to the ones explored in the previous figures.
In this regime, data collapses are observed for different values of $L/N$
when $\mathcal{C}_A /\sqrt{\omega_0 L}$ is reported as function of $t/N$.
In the special case of $L=N$, the complexity of pure states (\ref{comp-pure-global-DBCPBC})
discussed in Sec.\,\ref{sec:purestates_HC_glob} is recovered,
as shown in Fig.\,\ref{fig:MixedStateGlobalSmallTimesDimensionless}
by the black solid lines ($N=100$) and by the green dashed lines ($N=200$).
\begin{figure}[htbp!]
\vspace{-1cm}
\subfigure
{\hspace{-1.2cm}
\includegraphics[width=.54\textwidth]{DimensionlessMixedPBCSmallTimesomegainell20omegaell0}}
\vspace{-.25cm}
\subfigure
{
\hspace{0.cm}\includegraphics[width=.54\textwidth]{DimensionlessMixedDBCSmallTimesomegainell20omegaell0}}
\subfigure
{
\hspace{-1.2cm}\includegraphics[width=.54\textwidth]{DimensionlessMixedPBCSmallTimesomegainell100omegaell0}}
\vspace{-.25cm}
\subfigure
{\hspace{0.cm}
\includegraphics[width=.54\textwidth]{DimensionlessMixedDBCSmallTimesomegainell100omegaell0}}
\subfigure
{
\hspace{-1.2cm}\includegraphics[width=.54\textwidth]{DimensionlessMixedPBCSmallTimesomegainell10omegaell50}}
\vspace{-.25cm}
\subfigure
{\hspace{0.cm}
\includegraphics[width=.54\textwidth]{DimensionlessMixedDBCSmallTimesomegainell10omegaell50}}
\subfigure
{
\hspace{-1.2cm}\includegraphics[width=.54\textwidth]{DimensionlessMixedPBCSmallTimesomegainell100omegaell50}}
\subfigure
{\hspace{0.cm}
\includegraphics[width=.54\textwidth]{DimensionlessMixedDBCSmallTimesomegainell100omegaell50}}
\vspace{-.5cm}
\caption{Initial growth of $\mathcal{C}_A$ in (\ref{c2-complexity-rdm-our-case})
for a block $A$ made by $L$ consecutive sites
in harmonic chains with
either PBC (left panels) or DBC (right panels) made by $N$ sites
(in the latter case $A$ is adjacent to a boundary).
When $L=N$, the complexity (\ref{comp-pure-global-DBCPBC}) is shown
for $N=100$ (solid black lines) and $N=200$ (dashed green lines).
}
\label{fig:MixedStateGlobalSmallTimesDimensionless}
\end{figure}
Each panel on the left in Fig.\,\ref{fig:MixedStateGlobalSmallTimesDimensionless}
is characterised by the same $\omega_0 N$ and $\omega N$
of the corresponding one on the right.
From their comparison, one realises that the qualitative behaviour of the initial growth
at very early times is not influenced by the choice of the boundary conditions.
Moreover, the linear growth of $\mathcal{C}_A / \sqrt{\omega_0 L}$
is independent of $L/N$ for very small values of $t/N$;
hence the slope of the initial growth can be found by considering the case $L=N$
(discussed in Sec.\,\ref{sec:purestates_HC_glob})
and the approximation described
in Sec.\,\ref{subsec:TD limit HC} and in appendix\,\ref{subapp:smallkapprox}.
Combining these observations
with (\ref{compDBCapproximinitialt}) and (\ref{compPBCapproximinitialt}),
we obtain the initial linear growth $a_{\textrm{\tiny (B)}} \, t/N + \dots$
where the dots represent higher order in $t/N$ and the slope depends on the boundary conditions
labelled by $\textrm{B} \in \{ \textrm{P}, \textrm{D}\}$
as follows
\begin{eqnarray}
\label{lineargrowth_app_PBC_text}
a_{\textrm{\tiny (P)}}
&=&
\frac{\big|(\omega N)^2 - (\omega_0 N)^2\big|}{2\omega_0 N}\;
\sqrt{\sqrt{\frac{m}{4\kappa}}\;\omega_0 N\coth \! \bigg(\sqrt{\frac{m}{4\kappa}}\;\omega_0 N\bigg)}
\\
\rule{0pt}{1.1cm}
\label{lineargrowth_app_DBC_text}
a_{\textrm{\tiny (D)}}
&=&
\frac{\big|(\omega N)^2 - (\omega_0 N)^2\big|}{2\sqrt{2}\omega_0 N}\;
\sqrt{\sqrt{\frac{m}{\kappa}}\;\omega_0 N\coth \! \bigg(\sqrt{\frac{m}{\kappa}}\;\omega_0 N\bigg)-1}\;.
\end{eqnarray}
The grey dashed lines in Fig.\,\ref{fig:MixedStateGlobalSmallTimesDimensionless}
represent $a_{\textrm{\tiny (P)}} \, t/N$ (left panels)
and $a_{\textrm{\tiny (D)}} \, t/N$ (right panels).
\begin{figure}[t!]
\subfigure
{\hspace{-1.6cm}
\includegraphics[width=.58\textwidth]{DimensionlessDetMixedDBCdoverl0d1omegainell100omegaell0}}
\subfigure
{
\hspace{-.7cm}\includegraphics[width=.58\textwidth]{DimensionlessDetMixedDBCdoverl0d2omegainell100omegaell0}}
\subfigure
{\hspace{-1.6cm}
\includegraphics[width=.58\textwidth]{DimensionlessDetEEDBCdoverl0d1omegainell100omegaell0}}
\subfigure
{
\hspace{-.7cm}\includegraphics[width=.58\textwidth]{DimensionlessDetEEDBCdoverl0d2omegainell100omegaell0}}
\caption{Temporal evolution after the global quench with a gapless evolution Hamiltonian
of $\mathcal{C}_A$ in (\ref{c2-complexity-rdm-our-case}) (top panels)
and of $\Delta S_A$ in (\ref{Delta-S_A-def}) (bottom panels)
for a block $A$ made by $L$ consecutive sites
and separated by $d_{\textrm{\tiny L}}$ sites from the left boundary
of harmonic chains with DBC made by $N$ sites.
When $L=N$, the complexity (\ref{comp-pure-global-DBCPBC}) is shown
for $N=100$ (solid black lines) and $N=200$ (dashed green lines).
}
\vspace{0.4cm}
\label{fig:MixedStateGlobalMasslessDBCIntDet}
\end{figure}
Since for DBC and $\omega =0$
the temporal evolution of $\mathcal{C}_A$
displays a thermalisation regime
after the initial growth and the subsequent decrease
when the block $A$ with $L/N < 1/2$ is adjacent to a boundary,
we find it worth investigating also the case where $A$
is separated from the boundary.
Denoting by $d_{\textrm{\tiny L}}$ the number of sites
separating $A$ from the left boundary of the chain
(hence $d_{\textrm{\tiny R}}=N-L-d_{\textrm{\tiny L}}$ sites occur between $A$
and the right boundary),
$\mathcal{C}_A$ must be invariant under a spatial reflection w.r.t. the center of the chain,
i.e. when $d_{\textrm{\tiny L}}$ and $d_{\textrm{\tiny R}}$
are replaced by
$d_{\textrm{\tiny R}} -1$ and $d_{\textrm{\tiny L}} +1$ respectively.
In Fig.\,\ref{fig:MixedStateGlobalMasslessDBCIntDet} we show
the temporal evolutions of $\mathcal{C}_A$ and of $\Delta S_A$
for this bipartition of the segment
when the evolution Hamiltonian is gapless and $\omega_0 N=100$,
for four different values of $L/N$ and fixed values of $d_{\textrm{\tiny L}}/N$
given by $d_{\textrm{\tiny L}}/N=0.1$ (left panels)
or $d_{\textrm{\tiny L}}/N=0.2$ (right panels).
Once these parameters have been chosen,
the data points corresponding to $N=100$ and $N=200$ nicely collapse on the same curve.
When $d_{\textrm{\tiny L}}\neq 0$,
a thermalisation regime where both the curves of
$\mathcal{C}_A$ and $\Delta S_A$ are constant
occurs if $b/N<1/2$, with $b=\textrm{min}[d_{\textrm{\tiny L}}+L,d_{\textrm{\tiny R}}+L-1]$
(see the red and blue curves in Fig.\,\ref{fig:MixedStateGlobalMasslessDBCIntDet}).
The plateau is observed approximatively for $t/N\in [b/N,1-b/N]$
and its height depends on $\omega_0 L$, on $L/N$ and also on $d_{\textrm{\tiny L}}/N$.
A remarkable feature of the temporal evolution of $\mathcal{C}_A$
when $d_{\textrm{\tiny L}} > 0$ is the occurrence of two local maxima for $t/N < 1/2$,
while only one maximum is observed when $d_{\textrm{\tiny L}} = 0$ for $t/N < 1/2$
(see the top panels in Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless}).
For any given value of $L/N \leqslant 1/2$ in the top panels of Fig.\,\ref{fig:MixedStateGlobalMasslessDBCIntDet},
the subsystem complexity grows
in the temporal regime between the two local maxima, for $t/N < 1/2$.
The occurrence of two local maxima in the temporal evolution of $\mathcal{C}_A$
when $A$ is separated from the boundary is observed also when $N \to \infty$.
This is shown in Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
and Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless},
where we also highlight the logarithmic nature of the growth of $\mathcal{C}_A$
in the temporal regime between the two local maxima,
which can be compared with a logarithmic growth occurring in $\Delta S_A$
(see e.g. Fig.\,\ref{fig:RatioGlobalMassiveEvolutionTDDetDimensionless}).
Comparing each top panel with the corresponding bottom panel in
Fig.\,\ref{fig:MixedStateGlobalMasslessDBCIntDet}, we observe that
the black dashed straight line
(it is the same in the two top panels)
captures the first local maximum of $\mathcal{C}_A$.
The slope of this line
is twice the slope of the red dashed straight line in the bottom panels,
which identifies the initial linear growth of $\Delta S_A$.
\newpage
\section{Subsystem complexity and the generalised Gibbs ensemble}
\label{sec:GGE}
In this section we consider infinite harmonic chains,
either on the infinite line or on the semi-infinite line with DBC at the origin,
and discuss that the asymptotic value of $\mathcal{C}_A$
for a block made by consecutive sites can be found through the
generalised Gibbs ensemble (GGE).
\subsection{Complexity of the GGE}
An isolated system prepared in a pure state and then
suddenly driven out of equilibrium through a global quench does not relax.
Instead, relaxation occurs for a subsystem
\cite{Barthel08,Cramer:2008zz,Cramer_2010}
(see also the review \cite{Essler:2016ufo} and the references therein).
Consider a spatial bipartition of a generic harmonic chain
given by a finite subsystem $A$ and its complement.
Denoting by $\hat{\rho}(t)$ the density matrix of the entire system
and by $\hat{\rho}_A(t)$ the reduced density matrix of $A$,
a quantum system relaxes locally to a stationary state if
the limit $\lim_{t\to\infty} \lim_{N\to\infty} \hat{\rho}_A(t) \equiv \hat{\rho}_A(t=\infty)$
exists for any $A$,
where $N$ is the number of sites in the harmonic chain.
This stationary state is described by the time independent density matrix
$\hat{\rho}_{\textrm{\tiny E}}$ describing a statistical ensemble
if $\lim_{N\to\infty} \hat{\rho}_{\textrm{\tiny E},A}=\hat{\rho}_A(t=\infty) $, for any $A$,
where $\hat{\rho}_{\textrm{\tiny E},A}$ is obtained by tracing $\hat{\rho}_{\textrm{\tiny E}}$
over the degrees of freedom of the complement of $A$.
For the global quench of the mass parameter that we are investigating
in infinite harmonic chains,
the stationary state is described by a GGE
\cite{Rigol_07,Sotiriadis:2014uza,Ilievski_2015,Ilievski:2016fdy}
(see the review \cite{Vidmar_2016} for an extensive list of references).
In terms of the creation and annihilation operators (\ref{b_operators def}),
the evolution Hamiltonian reads
\begin{equation}
\label{Hphys-op-Omega GGE}
\widehat{H}
=
\sum_{k=1}^N
\Omega_{k} \! \left( \hat{\mathfrak{b}}_k^\dagger\, \hat{\mathfrak{b}}_k +\frac{1}{2}\, \right) .
\end{equation}
The GGE that provides the stationary state reads \cite{Calabrese:2007rg}
\begin{equation}
\label{rho_gge}
\hat{\rho}_{\textrm{\tiny GGE}}
=
\frac{e^{-\sum_{k=1}^N \lambda_k \hat{\mathfrak{b}}_k^\dagger \hat{\mathfrak{b}}_k}}{\mathcal{Z}_{\textrm{\tiny GGE}}}
\;\;\qquad\;\;
\mathcal{Z}_{\textrm{\tiny GGE}}
=
\textrm{Tr} (\hat{\rho}_{\textrm{\tiny GGE}})
=
\prod_{k=1}^N \frac{1}{1-e^{-\lambda_k}}
\end{equation}
where $\hat{\rho}_{\textrm{\tiny GGE}}$ is normalised through the condition $\textrm{Tr} (\hat{\rho}_{\textrm{\tiny GGE}}) =1$.
The conservation of the number operators $\hat{\mathfrak{b}}_k^\dagger \hat{\mathfrak{b}}_k$
tells us that the relation between their expectation values and $\lambda_k$ reads \cite{Calabrese:2007rg}
\begin{equation}
\label{nk lambdak}
n_k
\,\equiv\,
\textrm{Tr}\big(\hat{\mathfrak{b}}_k^\dagger \,\hat{\mathfrak{b}}_k \, \hat{\rho}_{\textrm{\tiny GGE}}\big) =\frac{1}{e^{\lambda_k}-1}
\,=\,
\langle \psi_0 |\, \hat{\mathfrak{b}}_k^\dagger \,\hat{\mathfrak{b}}_k \, | \psi_0 \rangle
\end{equation}
which is strictly positive because $\lambda_k>0 $ for any value of $k$.
Since the GGE in (\ref{rho_gge}) is a bosonic Gaussian state, it is characterised by its covariance matrix
$\gamma_{\textrm{\tiny GGE}}$,
which can be decomposed as follows
\begin{equation}
\label{gamma-GGE-matrix}
\gamma_{\textrm{\tiny GGE}}
=
\,
\bigg(
\begin{array}{cc}
Q_{\textrm{\tiny GGE}} & \, M_{\textrm{\tiny GGE}}
\\
M_{\textrm{\tiny GGE}}^{\textrm{t}} & \, P_{\textrm{\tiny GGE}}
\end{array} \bigg)
\end{equation}
where (see the appendix\;\ref{app-sec-GGE-corr})
\begin{equation}
(Q_{\textrm{\tiny GGE}})_{i,j}=\textrm{Tr}\big( \hat{q}_i \,\hat{q}_j \,\hat{\rho}_{\textrm{\tiny GGE}} \big)
\qquad
(P_{\textrm{\tiny GGE}})_{i,j}=\textrm{Tr}\big( \hat{p}_i \,\hat{p}_j \, \hat{\rho}_{\textrm{\tiny GGE}} \big)
\qquad
(M_{\textrm{\tiny GGE}})_{i,j}=\textrm{Re}\big[\textrm{Tr}\big( \hat{q}_i \,\hat{p}_j \,\hat{\rho}_{\textrm{\tiny GGE}} \big)\big] \,.
\end{equation}
By adapting the computation reported in appendix \ref{subapp:covariancematrix} to this case,
the operators $\hat{\boldsymbol{\mathfrak{q}}}$ and $\hat{\boldsymbol{\mathfrak{p}}}$
can be introduced as in (\ref{Williamson-basis-Hphys})
and for (\ref{gamma-GGE-matrix}) one finds
\begin{eqnarray}
\label{QGGE}
Q_{\textrm{\tiny GGE}}
&=&
\widetilde{V} \, S^{-1}_{\textrm{\tiny phys}} \,
\textrm{Tr}\big( \hat{\boldsymbol{\mathfrak{q}}} \,\hat{\boldsymbol{\mathfrak{q}}}^{\textrm{t}}\hat{\rho}_{\textrm{\tiny GGE}} \big)
\, S^{-1}_{\textrm{\tiny phys}} \, \widetilde{V}^{\textrm{t}}
\equiv
\widetilde{V} \, \mathcal{Q}\textrm{\tiny GGE} \, \widetilde{V}^{\textrm{t}}
\\
\rule{0pt}{.5cm}
\label{PGGE}
P_{\textrm{\tiny GGE}}
&=&
\widetilde{V} \, S_{\textrm{\tiny phys}} \,
\textrm{Tr}\big( \hat{\boldsymbol{\mathfrak{p}}} \,\hat{\boldsymbol{\mathfrak{p}}}^{\textrm{t}}\hat{\rho}_{\textrm{\tiny GGE}} \big)
\, S_{\textrm{\tiny phys}} \, \widetilde{V}^{\textrm{t}}
\,\equiv\,
\widetilde{V} \, \mathcal{P}\textrm{\tiny GGE} \,\widetilde{V}^{\textrm{t}}
\\
\rule{0pt}{.5cm}
\label{MGGE}
M_{\textrm{\tiny GGE}}
&=&
\widetilde{V} \, S^{-1}_{\textrm{\tiny phys}} \,
\textrm{Re}\big[\textrm{Tr}\big( \hat{\boldsymbol{\mathfrak{q}}} \,\hat{\boldsymbol{\mathfrak{p}}}^{\textrm{t}}\hat{\rho}_{\textrm{\tiny GGE}} \big)\big]
\, S_{\textrm{\tiny phys}} \,\widetilde{V}^{\textrm{t}}
\,\equiv\,
\widetilde{V} \,\mathcal{M}\textrm{\tiny GGE} \, \widetilde{V}^{\textrm{t}}\,.
\end{eqnarray}
Then, expressing $\hat{\boldsymbol{\mathfrak{q}}}$ and $\hat{\boldsymbol{\mathfrak{p}}}$ in terms of $\hat{\boldsymbol{\mathfrak{b}}}$
and $\hat{\boldsymbol{\mathfrak{b}}}^\dagger$ defined in (\ref{b_operators def}),
exploiting the fact that the two points correlators vanish
when the indices of the annihilation and creation operators are different,
using (\ref{nk lambdak}) and
$\textrm{Tr}\big(\hat{\mathfrak{b}}_k^\dagger \hat{\mathfrak{b}}_k^\dagger\, \hat{\rho}_{\textrm{\tiny GGE}}\big)
=\textrm{Tr}\big(\hat{\mathfrak{b}}_k \hat{\mathfrak{b}}_k\hat{\rho}_{\textrm{\tiny GGE}}\big)=0$,
we find
$M\textrm{\tiny GGE}=\mathcal{M}\textrm{\tiny GGE}=\mathbf{0}$
and
\begin{equation}
\label{Q-P-GGEdiag}
\mathcal{Q}_{\textrm{\tiny GGE}}
\,\equiv\,
\textrm{diag}\,\Big\{ Q_{\textrm{\tiny GGE},k} \, ; 1\leqslant k \leqslant N\Big\}
\;\;\qquad\;\;
\mathcal{P}_{\textrm{\tiny GGE}}
\,\equiv\,
\textrm{diag}\,\Big\{P_{\textrm{\tiny GGE},k} \, ; 1\leqslant k \leqslant N\Big\}
\end{equation}
where
\begin{equation}
Q_{\textrm{\tiny GGE},k} \equiv \frac{1+2n_k}{2m\,\Omega_k}
\;\;\qquad\;\;
P_{\textrm{\tiny GGE},k} \equiv \frac{m\,\Omega_k}{2}\, (1+2n_k)\,.
\end{equation}
Thus, the covariance matrix (\ref{gamma-GGE-matrix}) simplifies to $\gamma_\textrm{\tiny GGE}=Q_\textrm{\tiny GGE}\oplus P_\textrm{\tiny GGE}$, where $Q\textrm{\tiny GGE}$ and $P\textrm{\tiny GGE}$
are given by (\ref{QGGE}) and (\ref{PGGE}).
We find it worth writing the Williamson's decomposition of $\gamma_\textrm{\tiny GGE}$, namely
\begin{equation}
\label{Williamson-GGE}
\gamma_\textrm{\tiny GGE}
\,=\,
W^{\textrm{t}}_\textrm{\tiny GGE} \, \mathcal{D}_\textrm{\tiny GGE} \, W_\textrm{\tiny GGE}
\,\,\qquad\,\,
W_\textrm{\tiny GGE}
=
\mathcal{X}_\textrm{\tiny GGE} \, V^\textrm{t}
\end{equation}
where the symplectic spectrum is given by
\begin{equation}
\label{sympspecGGE}
\mathcal{D}_\textrm{\tiny GGE}
=
\frac{1}{2} \, \boldsymbol{1}
+
\textrm{diag}\,\Big\{n_1,\dots,n_N,n_1,\dots,n_N\Big\}
\end{equation}
and, like for the $2N \times 2N$ symplectic matrix $W_\textrm{\tiny GGE}$,
we have $V=\widetilde{V}\oplus \widetilde{V}$
and that the diagonal matrix $\mathcal{X}_\textrm{\tiny GGE}=\mathcal{X}_\textrm{\tiny phys}^{-1}$
is the inverse of $\mathcal{X}_\textrm{\tiny phys}$ defined in (\ref{Chiphys-Wdec}).
We emphasise that $\gamma_{\textrm{\tiny GGE}}$ does not describe a pure state.
Indeed, since $n_k\geqslant 0$ for any $k$, from (\ref{sympspecGGE})
we have that the symplectic eigenvalues of $\gamma_{\textrm{\tiny GGE}}$
are greater than $1/2$, as expected for a mixed bosonic Gaussian state.
For the global quench in the harmonic chains that we considering,
$n_k$ in (\ref{nk lambdak})
can be computed from the expectation value of $\hat{\mathfrak{b}}_k^\dagger \hat{\mathfrak{b}}_k$ on the initial state obtaining \cite{Calabrese:2007rg}
\begin{equation}
\label{nk omegak}
n_k=
\frac{1}{4}\bigg(\frac{\Omega_k}{\Omega_{0,k}}+\frac{\Omega_{0,k}}{\Omega_k}\bigg)-\frac{1}{2}
\end{equation}
where $\Omega_{0,k}$ and $\Omega_k$ are the dispersion relations of the Hamiltonian defining the initial state
and of the evolution Hamiltonian respectively.
Notice that (\ref{nk omegak}) is symmetric under the exchange $\Omega_k \leftrightarrow \Omega_{0,k}\,$.
We recall that the boundary conditions defining the harmonic chain influence
both the dispersion relations and the matrix $V$.
By introducing the reduced covariance matrix $\gamma_{\textrm{\tiny GGE},A}$ for $A$,
obtained from (\ref{gamma-GGE-matrix}) in the usual way,
the entanglement entropy
\begin{equation}
\label{EE-initialvsGGE}
S_{\textrm{\tiny GGE},A}\equiv-\textrm{Tr}(\hat{\rho}_{\textrm{\tiny GGE},A}\log \hat{\rho}_{\textrm{\tiny GGE},A})
\end{equation}
can be evaluated from the symplectic spectrum of $\gamma_{\textrm{\tiny GGE},A}$
through standard methods \cite{Peschel03,Peschel_2009}.
The asymptotic value of the increment of the entanglement entropy $\Delta S_A$
when $t \to \infty$ can be computed as follows
\begin{equation}
\label{stationary entropy density}
\lim_{L\to\infty}\frac{\lim_{t\to\infty}\lim_{N\to\infty}\Delta S_{A}}{L}
=
\lim_{L\to\infty}\frac{\lim_{N\to\infty}S_{\textrm{\tiny GGE},A}}{L}
=
\lim_{N\to\infty}\frac{S_{\textrm{\tiny GGE}}}{N}
\end{equation}
where the order of the limits is important
and in the last step we used that $S_{\textrm{\tiny GGE}}$ is an extensive quantity
(see the review \cite{Calabrese:2020wfx} and the references therein).
For the global quench in the harmonic chains that we are considering,
the asymptotic value (\ref{stationary entropy density}) for the entanglement entropy reads \cite{ac-18-qp-quench}
\begin{eqnarray}
\label{SGGE}
\lim_{N\to\infty}\frac{S_{\textrm{\tiny GGE}}}{N}
&=&
\int_{0}^\pi \Big[(n_\theta+1)\log(n_\theta+1)-n_\theta\log n_\theta\Big]\, \frac{d\theta}{\pi}
\\
\rule{0pt}{.8cm}
&=&
\label{SGGE dispersion}
\int_{0}^\pi \Bigg\{
\bigg[\frac{1}{4}\bigg(\frac{\Omega_\theta}{\Omega_{0,\theta}}+\frac{\Omega_{0,\theta}}{\Omega_\theta}\bigg)+\frac{1}{2}\bigg]\log\!\bigg[\frac{1}{4}\bigg(\frac{\Omega_\theta}{\Omega_{0,\theta}}+\frac{\Omega_{0,\theta}}{\Omega_\theta}\bigg)+\frac{1}{2}\bigg]
\\
\rule{0pt}{.7cm}
&& \hspace{1cm}
-\,
\bigg[\frac{1}{4}\bigg(\frac{\Omega_\theta}{\Omega_{0,\theta}}+\frac{\Omega_{0,\theta}}{\Omega_\theta}\bigg)-\frac{1}{2}\bigg]
\log\!\bigg[\frac{1}{4}\bigg(\frac{\Omega_\theta}{\Omega_{0,\theta}}+\frac{\Omega_{0,\theta}}{\Omega_\theta}\bigg)-\frac{1}{2}\bigg]
\Bigg\}\, \frac{d\theta}{\pi}
\nonumber
\end{eqnarray}
in terms of $n_\theta$ given in (\ref{nk omegak}),
where the dispersion relations to employ
are (\ref{dispersion relations TD}) for PBC and
(\ref{dispersion DBC TD}) for DBC.
A straightforward change of integration variable
leads to the same expression for both the boundary conditions,
as already noticed for (\ref{comp TD limit}).
Let us remark that (\ref{SGGE}) is finite
for any choice of the parameter (including $\omega =0$),
both for PBC and DBC.
It is also symmetric under the exchange $\Omega_{\theta}\leftrightarrow \Omega_{0,\theta}$;
hence under $\omega \leftrightarrow \omega_0$ as well.
We study the circuit complexity to construct the GGE (which is a mixed state)
starting from the (pure) initial state at $t=0$,
by employing the approach based on the Fisher information geometry \cite{DiGiulio:2020hlz}.
The optimal circuit to get $\gamma_{\textrm{\tiny GGE}}$
from the initial covariance matrix $\gamma(0)$ at $t=0$ reads
\cite{Bhatia07book,DiGiulio:2020hlz}
\begin{equation}
\label{optimal circuit GGE full}
G_s(\gamma(0) \, , \gamma_{\textrm{\tiny GGE}})
\,\equiv \,
\gamma(0)^{1/2}
\Big( \gamma(0)^{- 1/2} \,\gamma_{\textrm{\tiny GGE}} \,\gamma(0)^{-1/2} \Big)^s
\gamma(0)^{1/2}
\end{equation}
where $0 \leqslant s \leqslant 1$ parameterises the covariance matrix along the circuit.
The length of the optimal circuit (\ref{optimal circuit GGE full}) provides the circuit complexity
\begin{equation}
\label{comp-initialvsGGE-full}
\mathcal{C}_{\textrm{\tiny GGE}}
\,=\,
\frac{1}{2\sqrt{2}}\;
\sqrt{
\textrm{Tr} \,\Big\{ \big[
\log \!\big( \gamma_{\textrm{\tiny GGE}} \; \gamma(0)^{-1} \big)
\big]^2 \Big\}}\;.
\end{equation}
Since $\mathcal{M}_\textrm{\tiny GGE}=\mathcal{M}(0)=\boldsymbol{0}$,
from (\ref{time dep corrs trans}), (\ref{QGGE}) and (\ref{PGGE}) we obtain
\begin{equation}
\gamma_{\textrm{\tiny GGE}}
=
V
\big[ \mathcal{Q}_\textrm{\tiny GGE}\oplus \mathcal{P}_\textrm{\tiny GGE}\big]
V^{\textrm{t}}
\;\;\qquad\;\;
\gamma(0)
=
V\big[ \mathcal{Q}(0)\oplus \mathcal{P}(0) \big]V^{\textrm{t}} \,.
\end{equation}
Then, by exploiting (\ref{Q-P-GGEdiag}), (\ref{QPRmat tzero-dep-k})
and the fact that the matrix $V$ is the same for both
$\gamma_{\textrm{\tiny GGE}} $ and $\gamma(0)$,
we find that the complexity (\ref{comp-initialvsGGE-full}) reads
\begin{eqnarray}
\label{comp full GGE v0}
\mathcal{C}_{\textrm{\tiny GGE}}
&=&
\frac{1}{2\sqrt{2}} \;
\sqrt{\,\sum_{k=1}^N\bigg\{
\bigg[\log\bigg(\frac{Q_{\textrm{\tiny GGE},k}}{Q_{k}(0)}\bigg)\bigg]^2+\bigg[\log\bigg(\frac{P_{\textrm{\tiny GGE},k}}{P_{k}(0)}\bigg)\bigg]^2
\Bigg\}}
\\
\rule{0pt}{1cm}
&=&
\frac{1}{2\sqrt{2}}\;
\sqrt{\,\sum_{k=1}^N\Bigg\{
\bigg[\log\bigg(\frac{\Omega_{0,k}}{\Omega_k}\,(1+2n_k)\bigg)\bigg]^2
+\bigg[\log\bigg(\frac{\Omega_{k}}{\Omega_{0,k}}\,(1+2n_k)\bigg)\bigg]^2
\Bigg\}}\;.
\end{eqnarray}
By using (\ref{nk omegak}), this expression becomes
\begin{equation}
\label{comp full GGE}
\mathcal{C}_{\textrm{\tiny GGE}}
\,=\,
\frac{1}{2\sqrt{2}}\;
\sqrt{\,\sum_{k=1}^N\Bigg\{
\bigg[\log\bigg(\frac{\Omega_{0,k}^2}{2\,\Omega_k^2}+\frac{1}{2}\bigg)\bigg]^2+\bigg[\log\bigg(\frac{\Omega_{k}^2}{2\,\Omega_{0,k}^2}+\frac{1}{2}\bigg)\bigg]^2
\Bigg\}}
\end{equation}
which is symmetric under the exchange $\Omega_{k}\leftrightarrow \Omega_{0,k}$,
hence under $\omega \leftrightarrow \omega_0$ as well.
The leading order of this expression as $N \to \infty$ is given by
\begin{equation}
\label{comp full GGE TD}
\mathcal{C}_{\textrm{\tiny GGE}}
=
\frac{\sqrt{N}}{2\,\sqrt{2\pi}} \;
\sqrt{\,\int_{0}^\pi
\left\{\bigg[\log\bigg(\frac{\Omega_{0,\theta}^2}{2\,\Omega_\theta^2}+\frac{1}{2}\bigg)\bigg]^2+\bigg[\log\bigg(\frac{\Omega_{\theta}^2}{2\,\Omega_{0,\theta}^2}+\frac{1}{2}\bigg)\bigg]^2\right\}
\,d\theta
}
\end{equation}
where $\Omega_{0,\theta}$ and $\Omega_\theta$ are thermodynamic limits of the
dispersion relations associated to the Hamiltonians before and after the quench respectively.
By repeating the argument reported below (\ref{comp TD limit}),
one finds that (\ref{comp full GGE TD}) with (\ref{dispersion relations TD})
can be employed for both PBC and DBC.
Moreover, the resulting expression for $\mathcal{C}_{\textrm{\tiny GGE}}/\sqrt{N}$
is finite for any choice of the parameters (including for $\omega=0$).
\begin{figure}[t!]
\subfigure
{\hspace{-1.6cm}
\includegraphics[width=.58\textwidth]{CGGEanalytics}}
\subfigure
{
\hspace{-.7cm}\includegraphics[width=.58\textwidth]{SGGEanalytics}}
\caption{Asymptotic value of $\mathcal{C}_{\textrm{\tiny GGE}}/\sqrt{N}$ from (\ref{comp full GGE TD}) (left panel)
and of $S_{\textrm{\tiny GGE}}/N$ from (\ref{SGGE}) (right panel) as functions of $\omega_0$, for some values of $\omega$.
}
\vspace{0.4cm}
\label{fig:CGGEanalytic}
\end{figure}
In Fig.\,\ref{fig:CGGEanalytic} we show $\mathcal{C}_{\textrm{\tiny GGE}}/\sqrt{N}$ from (\ref{comp full GGE TD})
and $S_{\textrm{\tiny GGE}}/N$ from (\ref{SGGE}) as functions of $\omega_0$, for some values of $\omega$.
The resulting curves are qualitatively similar.
At $\omega_0 = \omega$ they both vanish,
but $\mathcal{C}_{\textrm{\tiny GGE}}/\sqrt{N}$ is singular at this point,
while $S_{\textrm{\tiny GGE}}/N$ is smooth.
The reduced covariance matrix $\gamma_{\textrm{\tiny GGE},A}$ associated to any finite subsystem $A$
is obtained by selecting the rows and the columns in (\ref{gamma-GGE-matrix}) corresponding to $A$.
The results of \cite{DiGiulio:2020hlz} can be applied again
to write the optimal circuit that provides $\gamma_{\textrm{\tiny GGE},A}$
from the initial mixed state characterised by the reduced covariance matrix
$\gamma_{A}(0)$ at $t=0$,
obtained from $\gamma(0)$ through the usual reduction procedure.
This optimal circuit reads
\begin{equation}
\label{optimal circuit GGE}
G_s(\gamma_A(0) \, , \gamma_{\textrm{\tiny GGE},A})
\,\equiv \,
\gamma_A(0)^{1/2}
\Big( \gamma_A(0)^{- 1/2} \,\gamma_{\textrm{\tiny GGE},A} \,\gamma_A(0)^{-1/2} \Big)^s
\gamma_A(0)^{1/2}
\end{equation}
where $0 \leqslant s \leqslant 1$ parametrises the covariance matrix along the optimal circuit.
Its length corresponds to the subsystem complexity of the GGE w.r.t. the initial state
\begin{equation}
\label{comp-initialvsGGE}
\mathcal{C}_{\textrm{\tiny GGE},A}
\,=\,
\frac{1}{2\sqrt{2}}\;
\sqrt{
\textrm{Tr} \,\Big\{ \big[
\log \!\big( \gamma_{\textrm{\tiny GGE},A} \, \gamma_{A}(0)^{-1} \big)
\big]^2 \Big\}}\;.
\end{equation}
Since the harmonic chain relaxes locally to the GGE after the quantum quench,
for the subsystem complexity of any finite subsystem $A$ we expect
\begin{equation}
\label{stationary complexity}
\lim_{t\to\infty} \, \lim_{N\to\infty}\mathcal{C}_A=\lim_{N\to\infty}\mathcal{C}_{\textrm{\tiny GGE},A}
\end{equation}
which is confirmed by
the numerical results in
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless},
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless},
Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionTD},
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTD}
and
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLarget}.
A numerical analysis shows that (\ref{stationary complexity})
grows like $\sqrt{L}$ as $L \to \infty$
for fixed values of $\omega$ and $\omega_0$;
hence, by adapting (\ref{stationary entropy density}) to the subsystem complexity,
we expect
\begin{equation}
\label{stationary complexity density}
\lim_{L\to\infty}\frac{\lim_{t\to\infty}\lim_{N\to\infty}\mathcal{C}_{A}}{\sqrt{L}}
=
\lim_{L\to\infty}\frac{\lim_{N\to\infty}\mathcal{C}_{\textrm{\tiny GGE},A}}{\sqrt{L}}=\lim_{N\to\infty}\frac{\mathcal{C}_\textrm{\tiny GGE}}{\sqrt{N}}
\end{equation}
where $\mathcal{C}_\textrm{\tiny GGE}$ is given in (\ref{comp full GGE TD})
and the order of the limits is important.
Numerical evidences for (\ref{stationary complexity density})
are discussed in appendix\;\ref{app:gge}
(see Fig.\,\ref{fig:CAGGEvsCGGE}
and Fig.\,\ref{fig:MixedStatePBCGlobalMasslessEvolutionTD}).
In the following numerical analysis we show that,
for the harmonic chains that we are exploring,
the asymptotic limit for $t\to \infty$ of the reduced density matrix after the global quench
is the reduced density matrix obtained from the GGE.
This result has been already discussed for a fermionic chain in \cite{Fagotti_2013},
where, considering a global quench of the magnetic field in the transverse-field Ising chain
and the subsystem given by a finite block made by consecutive sites in an infinite chain on the line,
it has been found that a properly defined distance
between the reduced density matrix at a generic value of time
along the evolution and the asymptotic one obtained from the GGE vanishes as $t \to \infty$.
\subsection{Numerical results}
\label{sec:numerics-gge}
In order to test (\ref{comp-initialvsGGE}), infinite harmonic chains must be considered.
The reference and the target states have been described in Sec.\,\ref{sec:mixedFinSize}.
In this section we study harmonic chains both on the line
and on the semi-infinite line with DBC imposed at its origin.
In the latter case, the block $A$ made by $L$ consecutive sites
is either adjacent to the origin or separated from it.
The correlators to employ in the numerical analysis
can be obtained from the ones reported in Sec.\,\ref{sec:mixedFinSize}.
For the infinite harmonic chain on the line,
we take $N\to\infty$ of (\ref{QPRmat t-dep 1d}), finding
\begin{equation}
\label{QPRmat t-dep TD}
\begin{array}{l}
\displaystyle
Q_{i,j}(t)
=
\frac{1}{\pi} \int_{0}^{\pi} Q_{\theta}(t) \cos\!\big[2\theta\,(i-j)\,\big] \, d\theta
\\
\rule{0pt}{.9cm}
\displaystyle
P_{i,j}(t)
=
\frac{1}{\pi} \int_{0}^{\pi} P_{\theta}(t) \cos\!\big[2\theta\,(i-j)\,\big] \, d\theta
\\
\displaystyle
\rule{0pt}{.9cm}
\displaystyle
M_{i,j}(t)
=
\,\frac{1}{\pi} \int_{0}^{\pi} M_\theta(t) \cos\!\big[2\theta\,(i-j)\,\big] \, d\theta
\end{array}
\end{equation}
where $i,j\in \mathbb{Z}$;
while, for the harmonic chain on the semi-infinite line with DBC,
the limit $N\to\infty$ of (\ref{QPRmat t-dep 1d DBC}) leads to
\begin{equation}
\label{QPRmat t-dep DBC TD}
\begin{array}{l}
\displaystyle
Q_{i,j}(t)
=
\frac{2}{\pi}\int_{0}^{\pi} Q_{\theta}(t)\sin(i\theta) \sin(j\theta) \, d\theta
\\
\rule{0pt}{.9cm}
\displaystyle
P_{i,j}(t)
=
\frac{2}{\pi} \int_{0}^{\pi} P_{\theta}(t)\sin(i\theta) \sin(j\theta) \, d\theta
\\
\displaystyle
\rule{0pt}{.9cm}
\displaystyle
M_{i,j}(t)
=
\frac{2}{\pi} \int_{0}^{\pi} M_\theta(t)\sin(i\theta) \sin(j\theta) \, d\theta
\end{array}
\end{equation}
where $i,j>0$.
The functions $Q_\theta(t)$, $P_\theta(t) $ and $M_\theta(t)$
in these integrands
are given by (\ref{QPRmat t-dep-k})
where $\Omega_{0,k}$ and $\Omega_k$
are replaced respectively by $\Omega_{0,\theta}$ and $\Omega_\theta$,
which are (\ref{dispersion relations TD}) and (\ref{dispersion DBC TD})
for the infinite and for the semi-infinite line respectively.
\begin{figure}[htbp!]
\vspace{-.6cm}
{
\subfigure
{\hspace{-1.05cm}
\includegraphics[width=1.03\textwidth]{DimensionlessDetMixedDBCmanydoverellomegainell20omegaell0}}
\subfigure
{\hspace{-1.05cm}
\includegraphics[width=1.03\textwidth]{DimensionlessDetEntEntrDBCmanydoverellomegainell20omegaell0}}
}
\caption{
Temporal evolution of $\mathcal{C}_A$ (top panel) and of $\Delta S_A$ (bottom panel)
after a global quantum quench with a gapless evolution Hamiltonian and $\omega_0 L =20$.
The subsystem is a block $A$ made by $L$ consecutive sites
either on the infinite line (black data points)
or on the semi-infinite line,
separated by $d$ sites from the origin where DBC hold (coloured data points).
The dashed black straight line is the same in both panels.
}
\vspace{0.4cm}
\label{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
\end{figure}
\begin{figure}[t!]
{
\subfigure
{\hspace{-1.05cm}
\includegraphics[width=1.03\textwidth]{InverseDimensionlessDetRatioDBCmanydoverellomegainell20omegaell0}}
}
\caption{Temporal evolution of $\Delta S_A/\mathcal{C}_A$ for the data reported
in Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}.
The inset zooms in to highlight the data points having $t/L >1$.
}
\vspace{0.4cm}
\label{fig:RatioGlobalMassiveEvolutionTDDetDimensionless}
\end{figure}
\begin{figure}[t!]
{
\subfigure
{\hspace{-1.05cm}
\includegraphics[width=1.03\textwidth]{CompDetachedMasslessTDomega0L100L20and30}}
}
\caption{Temporal evolution of $\mathcal{C}_A$
after a global quantum quench with a gapless evolution Hamiltonian and $\omega_0 L =100$,
in the same setups of Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}.
The inset zooms in on the intermediate temporal regime
between the two local maxima for the data having $d/L =3$.
}
\vspace{0.4cm}
\label{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless}
\end{figure}
Once the proper correlators on the chain are identified,
the reduced correlation matrices $Q_A$, $P_A$ and $M_A$
are the blocks providing the reduced covariance matrix (\ref{reduced CM}).
These matrices are obtained by restricting the indices of the proper correlators
to $i,j=1,\dots,L$ when $A$ is on the infinite line
and to $i,j=1+d,\dots,L+d$ when $A$ is on the semi-infinite line,
where $d$ corresponds to its separation from the origin.
In the following we discuss numerical data sets
obtained for infinite harmonic chains,
either on the infinite line or on the semi-infinite line,
where $\omega L$ and $\omega_0 L$ are kept fixed.
In appendix\;\ref{app:gge} we report numerical results characterised
by fixed values of $\omega$ and $\omega_0$.
In Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
and Fig.\,\ref{fig:RatioGlobalMassiveEvolutionTDDetDimensionless}
we show the temporal evolutions of $\mathcal{C}_A$,
of $\Delta S_A$ and of $\Delta S_A / \mathcal{C}_A$
after the quench with $\omega_0 L =20$ and $\omega L =0$.
In Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless}
we display the temporal evolution of $\mathcal{C}_A$
with $\omega_0 L =100$ and $\omega L =0$.
The subsystem $A$ is a block made by $L$ consecutive sites
either on a semi-infinite line,
separated by $d$ sites from the origin where DBC are imposed (coloured symbols),
or on the infinite line (black symbols).
The black and coloured data points for $\mathcal{C}_A$
have been found through (\ref{c2-complexity-rdm-our-case})
with the reduced correlators obtained from either (\ref{QPRmat t-dep TD}) or (\ref{QPRmat t-dep DBC TD}) respectively.
The coloured horizontal solid lines correspond to
either (\ref{comp-initialvsGGE}) or (\ref{EE-initialvsGGE}),
with the reduced correlators from (\ref{corr-GGE-app-int-dbc}) for the target state
and from (\ref{QPRmat t-dep DBC TD}) at $t=0$ for the reference state,
with $L=50$.
Notice that a black horizontal solid line does not occur
because the corresponding value is divergent,
as indicated also by the left panel in Fig.\,\ref{fig:GGE}.
Considering the block on the semi-infinite line,
in Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
and Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless}
we observe that
the initial growth of $\mathcal{C}_A$ is the same until the first local maximum, for all the values of $d/L$.
After the first local maximum, the temporal evolution of $\mathcal{C}_A$ depends on whether
the block is adjacent to the boundary.
If $d/L =0$ the curve decreases until it reaches the saturation value.
Instead, when $d/L > 0$, first $\mathcal{C}_A$ decreases along a different curve
(see e.g. Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless})
until a local minimum;
then we observe an intermediate growth,
followed by a second local maximum
and finally by the saturation regime.
A fitting procedure shows that the intermediate growth
between the two local maxima is logarithmic
(in the inset of
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless} the grey dashed curve has been found by fitting the data having $L=40$ and $d/L=3$
through a logarithm and a constant).
Its temporal duration is approximatively $d/L-1/2$,
for the three values of non vanishing $d/L$ considered in
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
and Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless}.
Fitting the intermediate growth
in Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
and Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless},
one observes that
the coefficient of the logarithmic growth decreases as $\omega_0 L$ increases.
The first local maximum in the temporal evolution of $\mathcal{C}_A$ occurs for $0 < t/L <1$.
When $d>0$, the second maximum occurs for $d/L < t/L < (d+1)/L$.
Notice that these two local maxima can be seen also in the top panels of
Fig.\,\ref{fig:MixedStateGlobalMasslessDBCIntDet} for $t/N < 1/2$.
In Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless},
Fig.\,\ref{fig:RatioGlobalMassiveEvolutionTDDetDimensionless} and
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless},
the data points represented through black symbols have been obtained
for a block in the infinite line.
These data overlap with the ones corresponding to the block on the semi-infinite line with $d>0$
until the latter ones display the development of the second local maximum.
For the temporal evolution of $\mathcal{C}_A$ on the infinte line
only one local maximum occurs
and the intermediate logarithmic growth mentioned above does not finish
within the temporal regime that we have considered.
This agreement tells us that the second local maximum in
the temporal evolution of $\mathcal{C}_A$ is due to the presence of the boundary.
The temporal evolutions of $\Delta S_A$ in the bottom panel of
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
can be explained by employing the quasi-particle picture \cite{Calabrese_2005},
which provides the different temporal regimes
and the corresponding qualitative behaviour of $\Delta S_A$
%
(for the subsystems where a boundary occurs,
the quasi-particle picture has been described e.g. in \cite{Surace:2019mft}).
The different regimes identified by this analysis correspond to the vertical dot-dashed lines
in the bottom panel of Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}.
%
Instead, the vertical dashed grey lines in the top panel of
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
correspond to $t/L=1+d/L$.
%
For $d>0$, when $t/L > 1/2$
we observe a regime of logarithmic growth for $\Delta S_A$
whose duration depends on $d/L$ according to
the quasi-particle picture, until the beginning of a linear decreases.
Considering two sets of data points of $\Delta S_A$ having different $d/L$,
they collapse until the first linear decrease is reached.
%
The initial growths of $\mathcal{C}_A$ and of $\Delta S_A$
in Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
are very different.
%
For instance,
the growth of $\mathcal{C}_A$ is the same for all the data sets,
while for $\Delta S_A$ it depends on whether $d$ vanishes.
Moreover, while the growth of $\Delta S_A$ is linear for $t/L < 1$ when $d=0$
and for $t/L < 1/2$ when $d>0$,
the growth of $\mathcal{C}_A$ is linear only at the very beginning of the temporal evolution
and it clearly deviates from linearity within the regime of $t/L$ where $\Delta S_A$ grows linearly.
%
The dashed black straight line passing through the origin
in Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
describes the linear growth of $\Delta S_A$ when $d=0$
and it is the same in both the panels.
%
This straight line intersects the first local maximum of $\mathcal{C}_A$.
%
This has been highlighted also for finite systems
in Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless}
and Fig.\,\ref{fig:MixedStateGlobalMasslessDBCIntDet}.
\begin{figure}[t!]
\subfigure
{\hspace{-1.65cm}
\includegraphics[width=.575\textwidth]{GGEPBCfuncomegaL}}
\subfigure
{
\hspace{-.3cm}\includegraphics[width=.575\textwidth]{GGEDBCfuncomegaL}}
\caption{Asymptotic value of $\mathcal{C}_{\textrm{\tiny GGE},A}$ in (\ref{comp-initialvsGGE})
for a block $A$ made by $L$ consecutive sites in infinite chains in terms of $\omega L$.
The block is either in an infinite chain (left panel)
or adjacent to the origin of the semi-infinite line with DBC (right panel).
}
\vspace{0.4cm}
\label{fig:GGE}
\end{figure}
In Fig.\,\ref{fig:RatioGlobalMassiveEvolutionTDDetDimensionless}
we show the ratio $ \Delta S_A/\mathcal{C}_A$ for the data reported in
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}.
We remark that the two logarithmic growths occurring in
$ \Delta S_A$ and in $\mathcal{C}_A$ almost cancel in the ratio;
indeed, a mild logarithmic decreasing is observed
when $t/L > 1$ for the data obtained on the infinite line (black symbols) and
when $1 < t/L < 3$ for the data obtained on the semi-infinite line
with $d/L=3$ (red symbols) that are already collapsed.
The curves in
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless}
must be compared with the corresponding ones in top panel in
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
in order to explore the effect of $\omega_0 L$.
%
The height of the first local maximum in the temporal evolution of $\mathcal{C}_A$
and also the saturation values for the data obtained on the semi-infinite line
increase as $\omega_0 L$ increases.
Instead, the coefficient of the logarithmic growth after the first local maximum
decreases as $\omega_0 L$ increases, as already remarked above.
Notice that higher values of $L$ are needed to observe data collapse as $\omega_0 L$ increases.
From the numerical results reported in the previous figures,
we conclude that (\ref{comp-initialvsGGE})
provides the asymptotic value of the subsystem complexity as $t \to \infty$;
hence it is worth studying the dependence of this expression on
the subsystem size and on the parameters of the quench protocol.
\begin{figure}[t!]
\subfigure
{
\hspace{3.0cm}\includegraphics[width=.57\textwidth]{GGEDBCdetfuncomegaL}}
\\
\subfigure
{
\hspace{-1.55cm}\includegraphics[width=.57\textwidth]{GGEDBCdetfuncdoverL}}
\subfigure
{\hspace{-0.45cm}
\includegraphics[width=.57\textwidth]{CGGEA_masslessevolution_funcdoverell}}
\caption{
Asymptotic value of $\mathcal{C}_{\textrm{\tiny GGE},A}$ in (\ref{comp-initialvsGGE})
for a block made by $L$ consecutive sites
and separated by $d$ sites from the origin of a semi-infinite line with DBC,
in terms of $\omega L$ (top panel) and of $d/L$ (bottom panels).
}
\vspace{0.4cm}
\label{fig:GGEdet}
\end{figure}
In Fig.\,\ref{fig:GGE} and Fig.\,\ref{fig:GGEdet}
we show numerical results for (\ref{comp-initialvsGGE}),
obtained by using the reduced correlators
from (\ref{corr-GGE-app-int-pbc}) and (\ref{corr-GGE-app-int-dbc})
for the target state and the reduced correlators
from (\ref{QPRmat t-dep TD}) and (\ref{QPRmat t-dep DBC TD}) at $t=0$
for the reference state.
In Fig.\,\ref{fig:GGE} we show (\ref{comp-initialvsGGE}) as function of $\omega L$
when the block is either in the infinite line (left panel) or at the beginning of the semi-infinite line with DBC (right panel).
The main difference between the two panels of Fig.\,\ref{fig:GGE}
is that the limit $\omega L \to 0$ is finite for the semi-infinite line
while it diverges for the infinite line
(the correlators (\ref{corr-GGE-app-int-pbc}) are well defined for $\omega \neq 0$).
This is consistent with the results displayed through the black symbols
in the top panel of Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
and in Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless}.
In Fig.\,\ref{fig:GGEdet} we study (\ref{comp-initialvsGGE})
for a block on the semi-infinite line,
separated by $d$ sites from the origin where DBC are imposed.
For a given value of $\omega_0 L$, we show $\mathcal{C}_{\textrm{\tiny GGE},A}$
as function of $\omega L$ at fixed $d/L$ (top panel) and viceversa (bottom panels).
The qualitative behaviour of the curves in the top panel of Fig.\,\ref{fig:GGEdet}
is similar to the one in the right panel of Fig.\,\ref{fig:GGE}.
In the bottom left panel of Fig.\,\ref{fig:GGEdet},
as $d/L \to \infty$,
the data points with $\omega L > 0$
asymptote (horizontal dashed line)
to the value of $\mathcal{C}_{\textrm{\tiny GGE},A}$
obtained through (\ref{comp-initialvsGGE})
with the reduced correlators
(\ref{corr-GGE-app-int-pbc}) for the target state
and (\ref{QPRmat t-dep TD}) at $t=0$ for the reference state.
Instead, when $\omega L = 0$ the data in the bottom left panel of Fig.\,\ref{fig:GGEdet}
do not have a limit as $d/L$ increases.
This is consistent with the divergence of the curves in left panel of Fig.\,\ref{fig:GGE}
as $\omega L \to 0$.
In the bottom right panel of Fig.\,\ref{fig:GGEdet} we consider a critical evolution Hamiltonian
and large values of $\omega_0 L$.
In this regime of parameters, we highlight the logarithmic growth of
$\mathcal{C}_{\textrm{\tiny GGE},A}$ in terms of $d/L$
(the solid lines are obtained by fitting the data corresponding to $L=40$
through the function $a \log(d/L)+b$).
The numerical data sets discussed in this section
are characterised by fixed values of $\omega L$ and $\omega_0 L$.
In appendix\;\ref{app:gge} we report numerical results
where $\omega$ and $\omega_0$ are kept fixed:
besides supporting further the validity of (\ref{comp-initialvsGGE}),
this analysis provides numerical evidences for (\ref{stationary complexity density}).
Within the context of the gauge/gravity correspondence,
the temporal evolution of the holographic subsystem complexity
in the gravitational backgrounds given by Vaidya spacetimes
has been studied numerically through the CV proposal
\cite{Chen:2018mcc,Auzzi:2019mah,Ling:2019ien,Zhou:2019xzc}.
We find it worth remarking that
the qualitative behaviour of the temporal evolution of $\mathcal{C}_A$
for an interval in the infinite line shown by the black data points
in Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
and Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless}
is in agreement with the results for
the temporal evolution of the holographic subsystem complexity
reported in \cite{Chen:2018mcc,Auzzi:2019mah}.
The change of regime occurs at $t/L \simeq 1/2$
for both these quantities and their qualitative behaviour
in the initial regime given by $0< t/L < 1/2$ is very similar.
For $t/L > 1/2$ we observe a logarithmic growth whose coefficient depends on $\omega_0 L$
in Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless}
and Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless},
while the holographic subsystem complexity remains constant.
However, a similar issue occurs in the corresponding comparison for the entanglement entropy.
\section{Conclusions}
\label{sec:conclusions}
In this manuscript we studied the
temporal evolution of the subsystem complexity after a global quench of the mass parameter
in harmonic lattices,
focussing our analysis on harmonic chains with either PBC or DBC
and on subsystems given by blocks of consecutive sites.
The initial state is mainly chosen as the reference state of the circuit.
The circuit complexity of the mixed states described by the reduced density matrices
has been evaluated
by employing the approach based on the Fisher information geometry \cite{DiGiulio:2020hlz},
which provides also the optimal circuit
(see (\ref{optimal circuit rdm}) and (\ref{c2-complexity-rdm})).
When the entire system is considered (see Sec.\,\ref{sec-pure-states},
Sec.\,\ref{sec:comp-initial-state} and Sec.\,\ref{sec:purestates_HC_glob}),
the optimal circuit is made by pure states \cite{Jefferson:2017sdb,Chapman:2018hou}
and for the temporal evolution of the circuit complexity after the global quench
one obtains the expression given by (\ref{c2-log-lambda-arcosh}) and (\ref{CTR generic}),
which holds in a generic number of dimensions.
When the reference and the target states are pure states along the time evolution of a given quench,
we find that the complexity is given by (\ref{c2-log-lambda-arcosh}) and (\ref{CTRomegaReqomegaT}),
which simplifies to (\ref{comp-pure-global-general})
in the case where the reference state is the initial state.
Specialising the latter result to the harmonic chains where either PBC or DBC are imposed,
one obtains (\ref{comp-pure-global-DBCPBC}), where the contribution of the zero mode for PBC is highlighted.
The occurrence of the zero mode provides
the logarithmic growth of the complexity when the evolution is critical
(see (\ref{C-pure-both-eta}) and Fig.\,\ref{fig:PureStateCritical}).
Typical temporal evolutions of the complexity for the entire chain
when the post-quench Hamiltonian is massive are shown in Fig.\,\ref{fig:TDvsfinite}.
The bounds (\ref{naive-bounds}) and (\ref{new bounds_main})
are obtained for the temporal evolution of the complexity of the entire harmonic lattice.
The former ones are simple but not very accurate
(see Fig.\,\ref{fig:BoundNaive} for harmonic chains with PBC);
instead, the latter ones capture the dynamics of the complexity in a very precise way
but their analytic expressions are more involved.
In the case of harmonic chains, the bounds (\ref{new bounds_main})
lead to the bounds (\ref{new bounds_main_weaker})
displayed in Fig.\,\ref{fig:BoundsImproved},
which are less constraining but easier to deal with.
The aim of this manuscript is to investigate the temporal evolution of the subsystem complexity
$\mathcal{C}_A$ after a global quench (see Sec.\,\ref{sec:mixedFinSize} and Sec.\,\ref{sec:GGE}).
For a gapless evolution Hamiltonian, our main results are shown in
Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionlessNoentropy},
Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless},
Fig.\,\ref{fig:CompvsEntStateGlobalMasslessEvolutionDimensionless}
and Fig.\,\ref{fig:MixedStateGlobalMasslessDBCIntDet}
for finite chains and in
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless},
Fig.\,\ref{fig:RatioGlobalMassiveEvolutionTDDetDimensionless},
and Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless}
for infinite chains.
In some cases, also the temporal evolutions
for the corresponding increment of the entanglement entropy $\Delta S_A$ are reported,
in order to highlight the similar features and the main differences.
This comparison allows to observe that the initial growths
of $\mathcal{C}_A$ and $\Delta S_A$ are very different,
while the behaviours in the saturation regime are similar,
as highlighted in
Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionDimensionless},
Fig.\,\ref{fig:CompvsEntStateGlobalMasslessEvolutionDimensionless}
and Fig.\,\ref{fig:RatioGlobalMassiveEvolutionTDDetDimensionless},
where also the temporal evolutions of the ratio $\Delta S_A / \mathcal{C}_A$
are shown.
An important difference between the temporal evolution
of $\mathcal{C}_A$ and of $\Delta S_A$
is that $\mathcal{C}_A$ displays a local maximum
before the saturation regime (within a revival for finite systems),
as discussed in Sec.\,\ref{sec:mixedFinSize} and Sec.\,\ref{sec:GGE}.
Interestingly,
within the framework of the gauge/gravity correspondence,
this feature has been observed also
in the temporal evolution of holographic subsystem complexity
in Vaidya gravitational backgrounds \cite{Chen:2018mcc,Auzzi:2019mah}.
Some temporal evolutions of $\mathcal{C}_A$
determined by gapped evolution Hamiltonians have been reported in
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionDimensionlessNoEntropy}
and Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionDimensionless}.
However, a more systematic analysis is needed to explore their characteristic features.
For the infinite harmonic chains that we have considered
the asymptotic regime is described by a GGE;
hence in Sec.\,\ref{sec:GGE} we have argued that
the asymptotic value of the temporal evolution of $\mathcal{C}_A$
is given by (\ref{comp-initialvsGGE}).
This result has been checked both for $\omega=0$
(see Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDDetDimensionless},
Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLargeomega0DetDimensionless}
and Fig.\,\ref{fig:MixedStateGlobalMasslessEvolutionTD})
and for $\omega>0$
(see Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTD}
and Fig.\,\ref{fig:MixedStateGlobalMassiveEvolutionTDLarget}).
In the future research, it would be interesting to investigate the subsystem complexity
and its temporal evolution after a quench in fermionic systems,
in circuits involving non-Gaussian states and in interacting systems.
The analysis reported in this manuscript can be extended straightforwardly in various directions.
For instance, we find it worth exploring
the dependence of the temporal evolution on the reference state
(e.g. by considering the unentangled product state as the reference state),
the temporal evolution for higher dimensional harmonic lattices
and
the temporal evolutions of the subsystem complexity
when the system is driven out of equilibrium through other quench protocols
\cite{Das:2014jna,Alves:2018qfv, Caputa:2017ixa, Camargo:2018eof},
like e.g. local quenches \cite{Eisler_2007, Calabrese:2007mtj, Ageev:2018nye,Ageev:2019fxn}.
In \cite{DiGiulio:2020hlz}
the subsystem complexity has been studied also by employing the entanglement Hamiltonians
\cite{Peschel_2009,Casini:2009sr, Eisler:2017cqi,Eisler:2018ugn,
Tonni:2017jom, Eisler:2019rnr,DiGiulio:2019cxv,Eisler:2020lyn};
hence one can consider the possibility to explore also its temporal evolution
through these entanglement quantifiers.
It would be interesting to study the temporal evolutions of the subsystem complexity
by employing other ways to evaluate the complexity of mixed states,
e.g. through other distances between bosonic Gaussian states
or the approach based on the purification complexity
\cite{Caceres:2019pgf,Camargo:2020yfv}.
The cost function plays an important role in the evaluation of the circuit complexity \cite{Jefferson:2017sdb};
hence it is worth studying its effect on the temporal evolution of the subsystem complexity.
Finally, it is important to keep exploring the temporal evolutions of the subsystem complexity
through holographic calculations in order to find qualitative features that are observed
in lattice models.
They would be crucial tests for quantum field theory methods to evaluate the subsystem complexity.
\vskip 30pt
\centerline{\bf Acknowledgments}
\vskip 10pt
We are grateful to Leonardo Banchi, Lucas Hackl, Mihail Mintchev,
Nadir Samos S\'aenz de Buruaga
and Luca Tagliacozzo
for useful discussions.
ET's work has been conducted within the framework of the
Trieste Institute for Theoretical Quantum Technologies.
\newpage
\vskip 30pt
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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{"url":"http:\/\/www.haskell.org\/haskellwiki\/index.php?title=Introduction_to_QuickCheck&oldid=6193","text":"# Introduction to QuickCheck1\n\nA quick introduction to QuickCheck, and testing Haskell code.\n\n## 1 Motivation\n\nIn September 2006, Bruno Mart\ufffdnez asked the following question:\n\n```-- I've written a function that looks similar to this one\n\ngetList = find 5 where\nfind 0 = return []\nfind n = do\nch <- getChar\nif ch `elem` ['a'..'e'] then do\ntl <- find (n-1)\nreturn (ch\u00a0: tl) else\nfind n\n\n-- I want to test this function, without hitting the filesystem. In C++ I\n-- would use a istringstream. I couldn't find a function that returns a\n-- Handle from a String. The closer thing that may work that I could find\n-- was making a pipe and convertind the file descriptor. Can I simplify\n-- that function to take it out of the IO monad?```\n\nSo the problem is: how to effectively test this function in Haskell? The solution we turn to is refactoring and QuickCheck.\n\n## 2 Keeping things pure\n\nThe reason your getList is hard to test, is that the side effecting monadic code is mixed in with the pure computation, making it difficult to test without moving entirely into a \"black box\" IO-based testing model. Such a mixture is not good for reasoning about code.\n\nLet's untangle that, and then test the the referentially transparent parts simply with QuickCheck. We can take advantage of lazy IO firstly, to avoid all the unpleasant low-level IO handling.\n\nSo the first step is to factor out the IO part of the function into a thin \"skin\" layer:\n\n```-- A thin monadic skin layer\ngetList :: IO [Char]\ngetList = take5 `fmap` getContents\n\n-- The actual worker\ntake5 :: [Char] -> [Char]\ntake5 = take 5 . filter (`elem` ['a'..'e'])```\n\n## 3 Testing with QuickCheck\n\nNow we can test the 'guts' of the algorithm, the take5 function, in isolation. Let's use QuickCheck. First we need an Arbitrary instance for the Char type -- this takes care of generating random Chars for us to test with. I'll restrict it to a range of nice chars just for simplicity:\n\n``` import Data.Char\nimport Test.QuickCheck\n\ninstance Arbitrary Char where\narbitrary = choose ('\\32', '\\128')\ncoarbitrary c = variant (ord c `rem` 4)```\n\nLet's fire up GHCi (or Hugs) and try some generic properties (its nice that we can use the QuickCheck testing framework directly from the Haskell prompt). An easy one first, a [Char] is equal to itself:\n\n``` *A> quickCheck ((\\s -> s == s) :: [Char] -> Bool)\nOK, passed 100 tests.```\n\nWhat just happened? QuickCheck generated 100 random [Char] values, and applied our property, checking the result was True for all cases. QuickCheck generated the test sets for us!\n\nA more interesting property now: reversing twice is the identity:\n\n``` *A> quickCheck ((\\s -> (reverse.reverse) s == s) :: [Char] -> Bool)\nOK, passed 100 tests.```\n\nGreat!\n\n## 4 Testing take5\n\nThe first step to testing with QuickCheck is to work out some properties that are true of the function, for all inputs. That is, we need to find invariants.\n\nA simple invariant might be:\n\n``` $\\forall~s~.~length~(take5~s)~=~5$\n```\n\nSo let's write that as a QuickCheck property:\n\n`\\s -> length (take5 s) == 5`\n\nWhich we can then run in QuickCheck as:\n\n``` *A> quickCheck (\\s -> length (take5 s) == 5)\nFalsifiable, after 0 tests:\n\"\"```\n\nAh! QuickCheck caught us out. If the input string contains less than 5 filterable characters, the resulting string will be less than 5 characters long. So let's weaken the property a bit:\n\n``` $\\forall~s~.~length~(take5~s)~<=~5$\n```\n\nThat is, take5 returns a string of at most 5 characters long. Let's test this:\n\n``` *A> quickCheck (\\s -> length (take5 s) <= 5)\nOK, passed 100 tests.```\n\nGood!\n\n## 5 Another property\n\nAnother thing to check would be that the correct characters are returned. That is, for all returned characters, those characters are members of the set ['a','b','c','d','e'].\n\nWe can specify that as:\n\n``` *A> quickCheck (\\s -> all (`elem` ['a'..'e']) (take5 s))\nOK, passed 100 tests.```\n\nExcellent. So we can have some confidence that the function neither returns strings that are too long, nor includes invalid characters.\n\n## 6 Going further\n\nQuickCheck is effectively an embedded domain specific language for testing Haskell code, and allows for much more complex properties than those you've seen here to be tested. Some sources for further reading are:","date":"2014-10-23 03:32:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 2, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7601214647293091, \"perplexity\": 2021.0256503882283}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-42\/segments\/1413507449615.41\/warc\/CC-MAIN-20141017005729-00115-ip-10-16-133-185.ec2.internal.warc.gz\"}"}
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\section*{References}%
\begin{quotation}\mbox{}\par}
\def\refer#1\par{{\setlength{\parindent}{-\leftmargin}\indent#1\par}}
\def\end{quotation}{\end{quotation}}
{\noindent\small{\bf Abstract:} We present a study of the radio
emission from a massive runaway star. The star forms a bow shock
that is clearly observed in the infrared. We have performed VLA
observations under the assumption that the reverse shock in the
stellar wind might accelerate charged particles up to relativistic
energies. Non-thermal radio emission of synchrotron origin has been
detected, confirming the hypothesis. We have then modeled the system
and we predict a spectral energy distribution that extends up to
$\gamma$-rays. Under some simplifying assumptions, we find that the
intensity at high energies is too low to be detected by current
instruments, but the future Cherenkov Telescope Array might detect
the source.}
\section{Introduction}
Runaway OB stars (Gies and Bolton 1986) can produce so-called stellar
bow shocks on the surrounding interstellar medium. Bow shocks develop
as arc-shaped structures, with bows pointing in the same direction of
the stellar velocity, while the star moves supersonically through the
interstellar medium (ISM). The stellar and shock-excited radiation
heat the dust and gas swept by the bow shock. The dust, in turn,
re-radiates the energy as mid to far IR flux.
Van Buren \& McCray (1988) looked for bow-shaped features near
high-velocity O stars and found an IR candidate close to the O
supergiant BD+43$^{\circ}3654$ ($\alpha,\delta$[J2000] = $20^{\rm
h}33^{\rm m}36.077^{\rm s}, +43^{\circ} 59' 07.40''$; $l, b =
82.41^\circ, +2.33^\circ$). Comer\'on \& Pasquali (2007) related the
star BD+43$^{\circ}$3654 to a bow shock detected with the Midcourse
Space eXperiment (MSX) at D and E bands. Also, data from the
NRAO-VLA\footnote{National Radio Astronomy Observatory - Very Large
Array (http://www.vla.nrao.edu/).} NVSS Survey (Condon et al. 1998)
revealed a coma-shaped source of $\sim$ 7 arcmin spatially coincident
with the MSX feature (Figure \ref{fig_1}).
A radio study of the bow shock can shed light on the physical
processes that give rise to high-energy emission from a stellar
source, regardless of the history of the runaway star. The shock can
accelerate particles up to relativistic energies by Fermi
mechanism. Energetic electrons will cool through synchrotron
radiation, producing a non-thermal radio source. We carried out radio
observations at two frequencies to study the nature of the emission
from the bow shock of BD+43$^{\circ}$3654.
\begin{figure}[h]
\centering
\includegraphics[width=6cm,angle=-90]{G_Romero_fig1.pdf}
\caption{MSX-D band image (color scale) superposed to 1.4 GHz-NVSS contours.
Levels are: $-$2, 2 (2$\sigma$), 5, 8, 11, 15, 19, 24, 29, 50, 70, and 90
mJy beam$^{-1}$.\label{fig_1}}
\end{figure}
\section{Observations and results}
Our continuum observations were carried out with the Very Lare Array
(VLA) at 1.42 GHz (C config.) and at 4.86 GHz (D config.). Figure 2
presents the resulting images after primary beam correction,
re-gridded with the same synthesized beam of 12''. There is emission
at both frequencies along the extension of the MSX source. The
hypothesis of a physical association between the star and the radio/IR
features is supported by the very good agreement of the residual
proper motion of the star and the direction from the star to the apsis
of the bow shock (Fig. \ref{fig_2}).
\begin{figure}[h]
\centering
\label{fig_2}
\includegraphics[width=5.3cm,angle=-90]{G_Romero_fig2.pdf}\hspace{1cm}
\includegraphics[width=5.3cm,angle=-90]{G_Romero_fig3.pdf}
\caption{Continuum emission at 1.42 GHz (left), and at 4.86 GHz (right).
Contour levels are $-3$, 3, 6, 10, 15, 20, 25, and 60 times the rms of 0.3
and 0.2 mJy beam$^{-1}$. BD+43$^{\circ}\,3654$ is marked with a cross. The
arrow represents the velocity vector of the star, derived from proper motions
corrected for local motion of the surrounding ISM (see text). Synthesized
beams of $12''\times 12''$ are shown in the top right corners.}
\end{figure}
We used the continuum images at 1.42 and 4.86 GHz to build a spectral
index distribution map. We only considered input pixels with a
signal-to-noise ratio $\geq$ 4. Besides, the spectral index map was
masked for a signal-to-noise ratio $\geq$ 10. Figures 3 and 4 show the
spectral index distribution and corresponding noise maps.
\begin{figure}[h]
\centering
\label{fig_4}
\includegraphics[width=6cm,angle=-90]{G_Romero_fig4.pdf}
\caption{Spectral index distribution.}
\end{figure}
\begin{figure}[h]
\centering
\label{fig_5}
\includegraphics[width=6cm,angle=-90]{G_Romero_fig5.pdf}
\caption{Spectral index error distribution.}
\end{figure}
\section{Bow-shock emission}
Most of the area shows a source of non-thermal radiation with index
$<\alpha>=-0.5$ ($S_{\nu} \propto \nu^{\alpha}$), as obtained from the VLA
data. We adopted a distance to the bow-shock of 1.4 kpc (Hanson 2003). The
distance from the star to the bow-shock is $R=5'$, or 2 pc. The volume
occupied by the bow-shock is $\sim 4.6\, {\rm pc}^3$. We took a particle
density of the ISM in the bow shock region of 100 cm$^{-3}$ (see Kobulnicky
et al. 2010).
The non-thermal radiation is expected from synchrotron emission
generated by relativistic electrons accelerated either at the forward
shock in the ISM or in the reverse shock in the stellar wind. We
estimated the particle energy distribution (n) using the observed flux
density and spectral slope, and assumed equipartition between magnetic
and relativistic particles energy density.
We considered that the energy density of relativistic particles has
three contributions:
\begin{equation}
u=u_{e_1}+u_p+u_{e_2}=\int E_{e_1} n_{e_1}(E_{e_1})
dE_{e_1}+\int E_{p} n_{p}(E_{p}) dE_{p}+\int E_{e_2}
n_{e_2}(E_{e_2}) dE_{e_2}, \nonumber
\end{equation}
where $e_1$, $p$, and $e_2$ stand for relativistic primary electrons,
protons, and secondary electron-positron pairs (i.e. pairs coming from
charged pion decays), respectively. The relation between primary
electrons and protons energy density is $u_{p}=a u_{e_1}$, with
$a\geq0$. Three cases were considered: $a=0$ (just electrons), $a=1$
(equal energy density in both species), and $a=100$ (proton-dominated
case, as observed in the galactic cosmic rays). The magnetic field
resulted $B\sim 5 \times 10^{-5}$ G.
The maximum energy of the particles was determined by balancing energy
gains and losses. The loss mechanisms considered were {\sl (i)}
synchrotron radiation, {\sl (ii)} relativistic Bremsstrahlung, {\sl
(iii)} particle escape from the radiation region due to convection
by the stellar wind, and {\sl (iv)} inverse Compton (IC) scattering of
IR, stellar and cosmic microwave background photons. In the case of
protons, the only relevant losses are proton-proton ($pp$) inelastic
collisions and convective escape. Diffusion is negligible in
comparison to convection in this situation (the respective timescales are
$t_{\rm conv} \sim 6\times10^6$~s and
$t_{\rm diff} \sim 10^{13}/(E/{\rm erg})$~s). Both primary electrons and
protons reach energies up to $\sim 10^{13}$ eV, which is imposed by
non-radiative losses, except for $a=100$, where synchrotron losses
dominate for electrons. Figure 5 shows the losses for electrons and
protons in the case $a=1$.
\begin{figure}[h]
\centering
\includegraphics[width=8cm]{G_Romero_fig6.pdf}
\includegraphics[width=8cm]{G_Romero_fig7.pdf}
\label{fig_6}
\caption{Left: acceleration (`Acc'), escape (`Esc'), and cooling
times for electrons, due to synchrotron radiation (`Synchr'), IC
scattering of dust photons (`IC (dust)'), stellar photons (`IC
(star)'), and backgroud photons (`IC (bg)'). Cooling time for
relativistic Bremsstrahlung radiation is indicated as `Rel B'. The
figure is for the case with equal energy density in electrons and
protons ($a=1$, see text). Right: acceleration, escape, and cooling
time for protons due to the $pp$ radiation
process ('p-p').}
\end{figure}
\section{Discussion and perspectives}
The presence of highly relativistic particles in a dense medium with
high photon density can result in the efficient generation of
$\gamma$-rays. The corresponding $\gamma$-ray emissivity can be
calculated using the delta-functional approximation (e.g. Aharonian \&
Atoyan 2000, Kelner et al. 2006).
In Fig. 6 we show the spectral energy distribution obtained for the
case $a=1$, with all contributions included (synchrotron self-Compton
losses are negligible). The total luminosity from $pp$ interactions is
similar to that obtained from relativistic Bremsstrahlung of
electrons. The IC up-scattering of IR photons is the major
contribution at high energies, with a peak around 100 GeV. The
detectability of the source by the Fermi LAT\footnote{Fermi Large Area
Telescope (http://www-glast.stanford.edu/).} $\gamma$-ray
observatory will depend on the actual particle density and the
contribution related to the secondary electrons at large $a$. The $pp$
contribution extends well into the TeV regime, but its weaker and will
be difficult to detect with the current ground-based Cherenkov
telescope arrays.
For the case $a=100$, the relativistic particle contain is
proton-dominated and $\gamma$-rays from $pp$ process dominate the high
energy spectrum. The CTA\footnote{Cherenkov Telescope Array
(http://www.cta-observatory.org/).} North observatory might detect
the source yielding information on the high enery cutoff.
Observations of the spectral slope at high energies can be used to
identified the proton content through the luminosity level, and the
proton spectral index. Radio polarization data will provide additional
information of $B$. X-ray observations will allow to determine the
cutoff of the synchrotron spectrum, directly related to the maximum
energy of the electrons\footnote{Notice that the situation is quite
different
from that of colliding winds, where the particle acceleration occurs in a
region of high photon density, with dominance of IC losses.}.
This, in turn, would yield valuable
information on the actual value of $B$ and the correctness of the
equipartition hypothesis.
\begin{figure}[h]
\centering
\label{fig_8}
\includegraphics[width=12cm]{G_Romero_fig8.pdf}
\caption{Spectral energy distribution for the case $a=1$. Acronyms as in
Figure 5. Measured radio fluxes from VLA observations (`VLA Obs') and MSX
luminosity at D band are also represented. The contribution from secondary
pairs is negligible in this case, so is not shown here.}
\end{figure}
\section*{Acknowledgements}
This work was supported by MinCyT - ANPCyT
(PICT-2007-00848)
and by CONICET
(project ID 11220090100078). JM and GER
acknowledge support by grant AYA2007-68034-C03-01 and -02 from the Spanish
government and FEDER funds
and Plan Andaluz de
Investigaci\'on, Desarrollo e Innovaci\'on of Junta de Andaluc\'{\i}a as
research group FQM-322 and excellence fund FQM-5418.
\footnotesize
\beginrefer
\refer Aharonian, F. A., Atoyan, A. M. 2000, A\&A, 362, 937
\refer Comer\'on, F., Pasquali, A. 2007, A\&A, 467, L23
\refer Condon, K. K, Cotton, W. D., Greisen, E. W., et al. 1998, AJ, 115, 1693
\refer Gies, D., Bolton, C. T. 1986, ApJS, 61, 419
\refer Hanson M. M. 2003, ApJ 97, 957-969
\refer Kelner, S. R., Aharonian, F. A., Bugayov, V. V. 2006, Phys. Rev. D,
74, 4081
\refer Kobulnicky, H. A., Gilbert, I. J., Kiminki, D. C. 2010, ApJ, 710, 529
\refer Van Buren, D., McCray, R. 1988, ApJ, 329, L93
\end{quotation}
\end{document}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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Желе́зные карава́ны — речная флотилия, в 1703–1918 гг. осуществлявшая грузовые перевозки с металлургических заводов Урала в центральную часть России по рекам Чусовой, Белой, Уфе, Вятке и Каме.
История
В XVIII–XIX вв. речной транспорт был наиболее распространённым вариантом перевозки продукции уральских металлургических заводов в центральную часть России. Металлы отправляли по Чусовой, Белой, Уфе, Вятке и Каме. Около половины всех уральских заводов отправляли свои металлы по Чусовой. Для этого фарватер реки был расчищен, построены пристани.
По месту назначения караваны делились на московский и петербургский. Путь в Петербург лежал по Чусовой, Каме и Волге. Затем бурлаками барки поднимались до Твери, а затем по Тверце и Вышневолоцкому каналу направлялись в Новгород и Петербург. В Москву караваны шли по Оке.
Сплав осуществлялся в летний сезон. Весной барки и коломенки грузились металлом, затем, после окончания ледохода, спускались на воду. Иногда отправка каравана с конкретного завода сопровождалась спуском воды из заводского пруда для повышения уровня воды в реке. Металлурги спешили отправить караван с расчётом на то, чтобы успеть попасть на Нижегородскую ярмарку, начинавшуюся в середине июля. В Москву караван приходил к осени, а в Петербург — на следующее лето с зимовкой в Твери.
Сложная схема транспортировки металлов обусловливала их высокую конечную себестоимость. При доставке железа в столицы транспортные расходы составляли 15–20% его цены.
Уральская горная администрация, как правило, назначала своего представителя для сопровождения караванов. В 1731–1732 и 1744–1745 гг. проведением караванов руководил Н. Г. Клеопин. В 1850-х гг. на правах комиссионера караваны проводил Н. И. Севастьянов.
Железные караваны на реке Чусовая
Водный режим
Общая длина реки Чусовая составляет 592 км, длина маршрута сплава по реке Чусовой от Ревдинской пристани до реки Кама составляет 420 км. Весенний ледоход на реке выражен слабо, весеннее половодье поднимает воду на 2–4 м и держится 2–3 недели (с конца апреля до середины мая). От устья Ревды на протяжении 420 км река падает на 175 м (или 42 см на км).
Маршрут
Ревдинская пристань — Пильный завод — Караульная гора — Шайтанская пристань — устье реки Большая Шайтанка — камень Бычок — перебор (устье) реки Четаевская Шайтанка — излучина у деревни Подволошная — Синенький камень — Тарханов камень — Билимбаевская пристань — Коновальские мели — Макаровская пристань — Крылосовская пристань — Крылосовские мели — камень-боец «Косой» — перекат реки Дарьинский — Уткинская пристань — Каменская пристань — Нижнесельская пристань — Трекинская пристань — Курьинская пристань — Староуткинская пристань — Старошайтанская пристань — Мартьяновская пристань — Волегов-камень — Илимская пристань — Сулемская пристань — Усть-Уткинская пристань — устье реки Межевая Утка — Харёнки — Кашкинская пристань — Ёква — Пермяков-камень — урочище Усть-Серебряная — Кыновская пристань — Ослянская пристань — Усть-Койвинская пристань — Чусовская пристань — Камасинская пристань.
Пристани на реке Чусовой
На Чусовой существовали следующие пристани:
За сутки перед сплавом лёд на реке взламывали спуском воды из Ревдинского пруда, который имел длину в 9 км, ширину в 1 км и глубину в 6,5 м. Для создания водяного вала вновь открывали плотину Ревдинского пруда в течение — вал достигал 2-2,5 м высоты и скорости 7 км/ч в течение 200 км. Затем открывали плотину Шайтанского пруда, что давало вала ещё 17 см, плотину Билимбаевского пруда — 35 см, плотину Уткинского пруда — 22 см.
1. Ревдинская пристань
На левом берегу, при устье речки Ревды. Здесь строились суда и загружались железом и чугуном с Ревдинского завода.
2. Шайтановская (Васильевская) пристань
Первоначально пристань располагалась в устье речки Шайтанки, затем на правом берегу Чусовой в 500 метрах выше «динасовского» автомобильного моста. Шайтанскому каравану требовалось 500—1000 сплавщиков. Груз шёл до Нижнего Новгорода (на Макарьевскую ярмарку), Рыбинска или Санкт-Петербурга (заморский торг). Часть сплавщиков шли до Сулемской пристани, а часть до Перми, где вербовались новые лоцманы, были которые шли до Рыбинска. Загрузка шла с Шайтанских заводов.
3. Билимбаевская пристань
Пристань расположена на участке — от реки Ольховка до деревни Макаровой, по обоим берегам Чусовой. Груз с Билимбаевского завода шёл в основном до Добрянской пристани (Добрянского завода) и Таборской пристани (Очёрскому заводу), где разгружался чугун и загружалось железо. Караван шёл до Нижнего Новгорода (на Макарьевскую ярмарку), Рыбинска или Санкт-Петербурга (заморский торг).
4. Макаровская пристань
5. Крылосовская пристань
Купеческая пристань на левом берегу, при устье речки Черемшанки. Здесь производилась постройка и загрузка судов. В основном груз составлял из масла коровьего, сала и сальных свеч.
6. Уткинская казённая пристань
Причалы Уткинской пристани расположены на обоим берегам реки Чусовой на протяжении несколько километров. Существовала пристань Утка Яковлева (Новоуткинского завода) и Турчиновская пристань по левому берегу Чусовой от мыса перед камнем Слободским и до устья Утки. Когда в 4 часа утра Ревдинский пруд спускал воду, то к 17-18 часов вал воды прибывал к Турчаниновской пристани, когда уже было поздно начинать сплав. Утром подходили ревдинский, шайтанский, билимбаевский караваны, начинались отваливать барки и уткинского каравана. Кроме железа и чугуна с пристани грузили медь и деньги (подушный сбор со всего горнозаводского Урала). Загружались с казённых заводов Екатеринбурга, Алтайских заводов, Верх-Исетского завода, Сысертского завода.
В 1733 году В. И. Генин отмечал, что на Уткинской пристани имелся казённый надзирательский дом, а при нём контора, позади конторы — чёрная изба, кузница для ковки и починки старых инструментов, девять амбаров для содержания лесных припасов, провианта, меди, канатов, железа и прочих привезённых с заводов припасов. В амбарах имелись двое весов, амбары были с погребами, в которых хранился порох и прочая амуниция. Дом обошёлся в 264 рубля 25 копеек.
7. Каменская пристань
8. Нижнесельская пристань
9. Трекинская пристань
Пристань, принадлежащая наследникам Яковлева, располагалась на правом берегу, при устье речки Треки.
10. Курьинская пристань
11. Староуткинская пристань
Пристань имело и другое название Утка Демидова и располагалась на левом берегу при устье реки Северной Утки. Груз шёл с Уткинского завода. Имелась гавань за счёт запруды на самой реки Утке.
12. Старошайтанская пристань
Пристань наследников Яковлевых располагалась на левом берегу, при устье реки Шайтанка.
13. Мартьяновская (Плешаковская) пристань
Казённая пристань на правом берегу при деревни Мартьяновой. Груз в основном состоял из семя, масла коровья и постного, жира рыбьего и оленьего, сало, сальные свечи и конский волос.
14. Илимская пристань
Казённая пристань Гороблагодатского округа горных заводов, на левом берегу при Илимской казённой лесопильной мельнице.
15. Сулемская пристань
Пристань, принадлежащая наследникам Яковлева, располагалась на правом берегу при деревне Сулеме.
16. Усть-Уткинская пристань
Пристань располагалась на правом берегу при устье реки Межевой Утки.
17. Кашкинская пристань
Пристань, принадлежащая наследникам Яковлева, располагалась на левом берегу, при деревне Кашка
18. Кыновская пристань
Пристань, принадлежащая Строгоновым. Груз состоял в основном с Кыновского завода
19. Ослянская пристань
Казённая пристань Гороблагодатского округа горных заводов располагалась на правом берегу при Гороблагодатском тракте. Груз шёл из Гороблагодатских и Богословских казённых горных заводов.
20. Разсолинская пристань
Пристань, принадлежавшая княгине Бутеро и князьям Голицыным, располагалась на правом берегу, при устье речки Рассольной. Здесь строились барки и загружались продукцией Кусье-Александровского завода.
21. Усть-Койвинская пристань
22. Усть-Долговская пристань
Пристань, принадлежащая княгине Бутеро, располагалась на левом берегу, при устье речки Долговки.
23. Чусовская пристань
24. Камасинская пристань
Пристань упомянута в книге Мамина-Сибиряка «Бойцы». Очерки написаны в большей своей части на основе личных впечатлений. В юношеские годы Мамину неоднократно приходилось совершать поездки по Чусовой от Висимо-Уткинской пристани до Перми.
И далее река Чусовая впадает в реку Кама в районе города Перми.
Пристани на реке Сылва
На реке Сылва осуществлялся сплав весеннее время от речки Вогулки до реки Чусовой, на протяжении 260 вёрст:
1. Молёбская пристань
Пристань располагалась на правом берегу, при устье реки Молёбки. Здесь строились суда и загружались с Молёбского завода. Имелась гавань.
2. Тисовская пристань
Пристань располагалась на левом берегу, при устье реки Тис.
3. Суксунская пристань
Пристань располагалась на левом берегу, при устье реки Суксун. Суда загружались с Суксунского завода.
4. Кунгурская пристань
Пристань располагалась на левом берегу при городе Кунгур. Здесь производилась постройка и загрузка судов. В основном груз состоял из ржаной муки, льняного семя, сало, сальных свечей и глины.
И далее река Сылва впадает в реку Чусовая в районе города Перми.
Караваны
По данным , лоцманы караванов, и даже заводские приказчики, договорившись с прибрежными жителями, неоднократно намеренно садили на мель барки и даже якобы разбивали барки, выписывая расходы за снятие барок с мели.
Караван 1703 года
С лета 1702 года крестьяне начали заготавливать лес и тесать доски, зимой 1702–1703 гг. шло строительство судов-дощаников. Руководителем стройки и сплава были Семён Резанов и служивый человек Иван Станикеев, присланные тобольским воеводой 16 марта 1703 г. Плотники из тобольских крестьян за 4 недели сколотили 40 дощаников. Продукция Каменского завода завозилась в течение зимы 1703 г. После того, как на реке Чусовой прошёл лёд, 22 апреля 1703 г. суда начали загружать. На каждом судне был кормщик (спалавщик), водолив и десять гребцов.
Первый караван вышел с Уткинской казённой пристани 27 апреля 1703 г. и состоял из 40 дощаников с грузом из 350-ти орудий общим весом пудов 29 фунтов, кованного железа 1 058 пудов, стали 8 пудов 5 фунтов из Каменского завода. На девятый день караван вышел в Каму, в городе Оса сменили тобольских гребцов на местных за 5 алтын (15 копеек) каждому. Далее смена гребцов происходила в Сарапуле, Елабуге, Лаишеве, в последнем в течение 9 дней шили паруса и далее по Волге шли на парусах, на вёслах и с бурлаками. Меняли команды и на Волге, платя уже по 20 алтын каждому (так как шли против течения), в Казани, Козьмодемьянске, Нижнем Новгороде, Муроме, Касимове, Переяславле Рязанском, Коломне, на Москве-реке плата увеличилась до 25 алтын. Караван прибыл в Москву 18 июля 1703 г., где у Тайницкой башни Кремля орудия были разгружены на Пушечный двор, 38 дощаников были проданы по 2,5–3 руб. за каждый, а на двух в Лаишев были отправлены судовые снасти (паруса) для следующего сплава.
Весь путь занял 11 недель и 6 дней, из которых 3 недели и 2 дня были стоянки.
Караван 1734 г.
Шайтановский караван 1734 г. насчитывал 8 319 пудов железа.
Караван 1812 г.
Вес завезённой продукции на Уткинскую пристань с Каменского завода и Нижнеисетского завода составил пудов и состоял из 317 орудий. Под них было построено 40 коломенок и нанято 460 сплавщиков с оплатой до Рыбинска по 85 руб. и по 7 пудов муки каждому, 20 лоцманов по 145 руб. и по 7 пудов муки, 40 водоливов по 155 руб. и по 10 пудов муки. В Астрахань было отправлено 41 орудие, в Дубровку 276 орудия.
Караван 1839 г.
Во время каравана 1839 г. не погибла ни одна барка.
Караван 1849 г.
По данным Я. Рогова караван из 40 барок отправился с Билимбаевской пристани 17 апреля 1849 г. 19 апреля 1849 г. возле деревни Харёнки была затоплена одна барка с металлом. 24 апреля 1849 г. караван вышел на реку Каму, преодолев за 8 суток 470 вёрст, в пути были 80 часов, идя со скоростью 4–8 вёрст в час.
Караван 1877 г.
За навигацию было разбито всего 47 барок, из которых 23 барки были разбиты и 100 человек погибли возле камней Молоков и Разбойник.
Караван 1880 г.
Отправка ревдинского каравана была перенесена с 23 апреля на 25 апреля 1880 г. в связи со спуском воды из пруда 21 апреля с целью очистки Чусовой ото льда и резкого прекращения его ночью 22 апреля. Коломенки попали на мель и две барки были поломаны, остальные пришлось срочно ремонтировать. Ревдинский караван в 1880 г. насчитывал 25 барок по 10 т каждая, общее число экипажей составило 1 268 человек, из которых 1 010 человек были наняты до Перми. Весь экипаж был из пришлых, пермских и вятских жителей.
Караван 1881 г.
Самый ранний сплав: спуск воды состоялся 11 апреля, начало сплава 14 апреля 1881 г. Первые два лотовых судна с билимбаевского каравана замелели у деревни Треки, и недельный простой обошёлся заводу в 800 руб.
Караван 1885 г.
В составе каравана были укомплектованы пушки для снабжения армии в ходе Крымской войны. Орудия прибыли на полуостров уже после подписания перемирия и оказались низкого качества.
Караван 1898 г.
Шайтановский караван 1898 г. насчитывал 166 600 пудов железа.
Караван 1918 г.
В 1918 г. состоялся последний сплав со Староуткинской пристани 10-ти барок с продукцией Уткинского завода.
Флот
Строительство флота для караванов
Строительством барок руководили коломенские мастера, у которых были в подчинении струговые плотники и завербованные крестьяне. Главным руководителем был коломенный уставщик Уткинской казённой пристани.
Конструкция «топорных» барок сменилась на доски и брусья после строительства вододействующих пильных мельниц на притоках Чусовой.
Конструкция чусовских судов
Первый караван состоял из дощаников длиной 7 саженей, шириной 2 сажени, вмещавших до 300 пудов груза. По предложению на следующий год суда стали грузоподъёмнее, а по инструкции от 1733 г. суда-коломенки были длиной 15 саженей с аршином (32 м), шириной 8,5 аршин (7 м), имели палубный настил, на закруглённом носу и корме устанавливались по две длинных греби («поносные», потеси), их грузоподъёмность определялась в 6 200 пудов.
В связи с тем, что барки в основном ходили только до устья Чусовой (далее груз перегружался на вместительные баржи), из конструкции были убраны мачты, реи и такелаж для парусов. С 1881 г., начиная с билимбаевского каравана, освоено движение на лотах: медленнее вала воды с помощью железных с большими звеньями цепей с вложенными в них чугунными крестами (лотами) с общим весом в 60 пудов. Так на барках исчезли потеси, а на носу и на корме появились сложные рулевые конструкции.
Согласно данным «Лесного журнала» (№ 35–36, 1847) характеристики чусовских судов были следующие:
коломенка — грузоподъёмность 136–160 т, длина 36–46,5 м, ширина в 6,4–7,4 м, высота 1,7 м;
барка — грузоподъёмность 160–188 т, длина 17–19 м, ширина 5,4–6,4 метра, высота борта 1,9 м;
полубарок — грузоподъёмность 94 т, длина 25,5-30 м, ширина 2 метра, высота 1,2–1,4 м;
гибежная лодка — грузоподъёмность 112 т, длина 25,5 метра, ширина в 8,5–13,7 м, высота 2 м.
Максимальная осадка барки составляла 1 м.
На барках устанавливали разноцветные флюгеры («репейники») и фирменные флаги заводов: цвета Ревдинского завода — чёрный и белый, Шайтанского завода — жёлтый, Полевского — сине-зелёный, Сысертского завода — зелёно-жёлтый.
Порядок
На первой барке находились самые опытные сплавщики. Последней из барок («казёнка») отплывала администрация каравана с косной лодкой со специальной косной командой (необходимой для различных целей управления, например, чтобы нагнать барку, подняться к отставшей барке и т.д.).
Железные караваны на реке Белая
Пристани на реке Уфа
1. Нязепетровская пристань
Пристань располагалась на правом берегу, при устье реки Нязи. Здесь грузились продукцией Нязепетровского завода.
2. Сорокинская пристань
Пристань располагалась на правом берегу, при устье реки Шемахи. Здесь грузились продукцией Шемахинского, Кыштымского и Каслинского заводов.
3. Уфимская пристань
Пристань располагалась на правом берегу, выше реки Серги, при деревни Уфимской плотбище. Здесь грузились продукцией Михайловского завода.
4. Артинская пристань
Пристань располагалась на левом берегу, при устье реки Арти. Здесь грузились продукцией казённых Артинских заводов
5. Красноуфимская пристань
Пристань располагалась на правом берегу при городе Красноуфимске.
6. Саранинская пристань
Пристань располагалась на правом берегу, при устье реки Сараны. Здесь грузились суда продукцией Саранинских заводов.
7. Уфалейская пристань
Пристань располагалась на реке Уфалее, при Нижнеуфалейском Заводе. Здесь грузились суда с продукцией Уфалейских заводов
8. Михайловская пристань
Пристань располагалась на реке Серьге близ впадения её в реку Уфа. Здесь грузилась продукция Сергинских заводов.
Примечания
Литература
Флотилии
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\section{Introduction}\label{Sec:Intro}
Given the agility of unmanned aerial vehicles (UAVs), they are capable of supporting compelling applications and are beginning to be deployed more broadly. Recently, the UK and Chile authorities proposed to deliver medical support and other essential supplies by using UAVs to vulnerable people in response to Covid-19 \cite{Drone:UK, Drone:Chile}. In \cite{MG:17:SC}, the authors used UAVs for image collection and high-resolution topography exploration. However, given the several limitations of on-board power level and the ability to adapt to changes in the environment, UAVs may not be fully autonomous and can only operate for short flight-durations, unless remote laser-charging is used \cite{QL:16:VTM}. Moreover, due to some challenging tasks such as topographic surveying, data collection or obstacle avoidance, the existing UAV technologies cannot operate in an optimal manner.\par
Wireless networks supported by UAVs constitute a promising technology for enhancing the network performance \cite{HC:06:ICC}. The applications of UAVs in wireless networks span across diverse research fields, such as wireless sensor networks (WSNs) \cite{JG:18:SAC}, caching \cite{CZ:20:CCN}, heterogeneous cellular networks \cite{HW:20:WC}, massive multiple-input multiple-output (MIMO) \cite{HH:20:VT}, disaster communications \cite{TD:19:GLOBECOM, Trung:19:IWCMC} and device-to-device communications (D2D) \cite{MM:16:WC}. For example, in \cite{Long:EAI}, UAVs were deployed to provide network coverage for people in remote areas and disaster zones. UAVs were also used for collecting data in a WSN \cite{JG:18:SAC}. Nevertheless, the benefits of UAV-aided wireless communication are critically dependent on the limited on-board power level. Thus, the resource allocation of UAV-aided wireless networks plays a pivotal role in approaching the optimal performance. Yet, the existing contributions typically assume having static environment \cite{TD:19:GLOBECOM, Trung:19:IWCMC, Minh:19:WCL} and often ignore the stringent flight time constraints in real-life applications \cite{JG:18:SAC, HW:20:WC, Xiaowei:19:VT}.
Machine learning has recently been proposed for the intelligent support of UAVs and other devices in the network \cite{HZ:20:VT, Xiao:19:VT,KK:19:Access, Khoi:19:Access, Khoi:20:Access, KL:19:VT, UC:19:WC, HH:20:VT, XL:19:VT, CW:19:VT}. Reinforcement learning (RL) is capable of searching for an optimal policy by trial-and-error learning. However, it is challenging for model-free RL algorithms, such as Q-learning to obtain an optimal strategy, while considering a large state and action space. Fortunately, with the emerging neural networks, the sophisticated combination of RL and deep learning, namely deep reinforcement learning (DRL) is eminently suitable for solving high-dimensional problems. Hence, DRL algorithms have been widely applied in fields such as robotics \cite{SG:17:ICRA}, business management \cite{QC:18:AAAI} and gaming \cite{Mnih:13}. Recently, DRL has also become popular in solving diverse problems in wireless networks thanks to their decision-making ability and flexible interaction with the environment \cite{YY:19:SAC, KK:19:Access, Khoi:19:Access, Khoi:20:Access, KL:19:VT, UC:19:WC, HH:20:VT, XL:19:VT, CW:19:VT, CZ:20:CCN, NZ:19:WC, SY:19:VT}. For example, DRL was used for solving problems in the areas of resource allocation \cite{NZ:19:WC, KK:19:Access, Khoi:19:Access}, navigation \cite{DY:18:VT, HH:20:VT} and interference management \cite{UC:19:WC}.
\subsection{Related Contributions}\label{Sec:Works}
UAV-aided wireless networks have also been used for machine-to-machine communications \cite{HW:19:WC} and D2D scenarios in 5G \cite{Minh:19:WCL, Huy:20:CCN}, but the associated resource allocation problems remain challenging in real-life applications. Several techniques have been developed for solving resource allocation problems \cite{LL:19:WC, LX:19:IOT, KK:19:Access, Khoi:19:Access, DY:18:VT, LN:19:SPAWC}. In \cite{LL:19:WC}, the authors have conceived a multi-beam UAV communications and a cooperative interference cancellation scheme for maximising the uplink sum-rate received from multiple UAVs by the base stations (BS) on the ground. The UAVs were deployed as access points to serve several ground users in \cite{LX:19:IOT}. Then, the authors proposed successive convex programming for maximising the minimum uplink rate gleaned from all the ground users. In \cite{DY:18:VT}, the authors characterised the tradeoffs between the ground terminal transmission power and the specific UAV trajectory both in a straight and in a circular trajectory.
The issues of data collection, energy minimisation, and path planning have been considered in \cite{QW:18:WC,CZ:18:WCL, HW:18:WCL, HW:19:WC, XL:19:VT, ZW:20:ITJ,JL:20:ITJ,MS:20:WC, MH:20:C,CZ:20:C}. In \cite{CZ:18:WCL}, the authors minimised the energy consumption of the data collection task considered by jointly optimising the sensor nodes' wakeup schedule and the UAV trajectory. The authors of \cite{HW:18:WCL} proposed an efficient algorithm for joint trajectory and power allocation optimisation in UAV-assisted networks to maximise the sum-rate during a specific length of time. A pair of near-optimal approaches for optimal trajectory was proposed for a given UAV power allocation and power allocation optimisation for a given trajectory. In \cite{HW:19:WC}, the authors introduced a communication framework for UAV-to-UAV communication under the constraints of the UAV's flight speed, location uncertainty and communication throughput. Then, a path planning algorithm was proposed for minimising the associated completion time task while balancing the performance by computational complexity trade-off. However, these techniques mostly operate in offline modes and may impose excessive delay on the system. It is crucial to improve the decision-making time for meeting the stringent requirements of UAV-assisted wireless networks.
Again, machine learning has been recognised as a powerful tool of solving the high-dynamic trajectory and resource allocation problems in wireless networks. In \cite{LN:19:SPAWC}, the authors proposed a model based on the classic k-means algorithm for grouping the users into clusters and assigned a dedicated UAV to serve each cluster. By relying on their decision-making ability, DRL algorithms have been used for lending each node some degree of autonomy \cite{YY:19:SAC, KK:19:Access, Khoi:19:Access, Khoi:20:Access, KL:19:VT, CZ:20:CCN, NZ:19:WC}. In \cite{YY:19:SAC}, an optimal DRL-based channel access strategy to maximise the sum rate and $\alpha$-fairness was considered. In \cite{KK:19:Access, Khoi:19:Access}, we deployed DRL techniques for enhancing the energy-efficiency of D2D communications. In \cite{KL:19:VT}, the authors characterised the DQL algorithm for minimising the data packet loss of UAV-assisted power transfer and data collection systems. As a further advance, caching problems were considered in \cite{CZ:20:CCN} to maximise the cache success hit rate and to minimise the transmission delay. The authors designed both a centralised and a decentralised system model and used an actor-critic algorithm to find the optimal policy.
\begin{table}[h!]
\centering
\caption{A comparison with existing literature}
\centering
\label{Tab:compare}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
& \cite{QW:18:WC} &\cite{CZ:18:WCL}& \cite{JG:18:SAC}& \cite{KL:19:VT}&\cite{XL:19:VT}& \cite{ZW:20:ITJ} & \cite{HH:20:VT} &\cite{JL:20:ITJ} &\cite{MS:20:VT}&\cite{MS:20:WC} &\cite{MH:20:C} &\cite{CZ:20:C} & Our work \\
\hline
Trajectory design & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & && \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;& \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;& \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\ \hline
3D trajectory & & & & & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & & &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\ \hline
Uplink & &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & &&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\ \hline
Downlink & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & & & &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & & &&&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&\\ \hline
Sum-rate maximisation & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & & &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\ \hline
Energy optimisation & & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & & & & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;& &&&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\\ \hline
Time minimisation & & &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\ \hline
Dynamic environment & & & & &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\ \hline
Simple environment & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;& \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;& \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & &&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\ \hline
Complex environment &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\ \hline
Mathematical solution & \tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; & &&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&\\ \hline
Reinforcement learning & & & &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\ \hline
Deep neural networks & & & &\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle; &&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;&&&&\tikz\fill[scale=0.4](0,.35) -- (.25,0) -- (1,.7) -- (.25,.15) -- cycle;\\ \hline
\end{tabular}
\end{table}
DRL algorithms have also been applied for path planning in UAV-assisted wireless communications \cite{UC:19:WC, SY:19:VT, HH:20:VT, XL:19:VT, CW:19:VT, MS:20:VT}. In \cite{UC:19:WC}, the authors proposed a DRL algorithm based on the echo state network of \cite{HJ:01:GMD} for finding the flight path, transmission power and associated cell in UAV-powered wireless networks. The so-called deterministic policy gradient algorithm of \cite{Lillicrap:15} was invoked for UAV-assisted cellular networks in \cite{SY:19:VT}. The UAV's trajectory was designed for maximising the uplink sum-rate attained without the knowledge of the user location and the transmit power. Moreover, in \cite{HH:20:VT}, the authors used the DQL algorithm for the UAV's navigation based on the received signal strengths estimated by a massive MIMO scheme. In \cite{XL:19:VT}, Q-learning was used for controlling the movement of multiple UAVs in a pair of scenarios, namely for static user locations and for dynamic user locations under a random walk model. However, the aforementioned contributions have not addressed the joint trajectory and data collection optimisation of UAV-assisted networks, which is a difficult research challenge. Furthermore, these existing works mostly neglected interference, 3D trajectory and dynamic environment.
\subsection{Contributions and Organisation}
In this paper, we consider a system model relying on a single UAV to serve several user nodes. The UAV is considered to be an information-collecting robot aiming for collecting the maximum amount of data from the users with the shortest distance travelled. We conceive a solution based on the DRL algorithm to find the optimal path of a UAV for maximising the joint reward function based on the shortest flight distance and the uplink transmission rate. We compare the difference between our proposed approach and other existing works in Table \ref{Tab:compare}. Our main contributions are summarised as follows:
\begin{itemize}
\item The UAV system considered has stringent constraints owing to the position of the destination, the UAV's limited flight time and the communication link's constraint. The UAV's objective is to find an optimal trajectory for maximising the total network throughput, while minimising its distance travelled.
\item We propose DRL techniques for solving the above problem. The area is divided into a grid to enable fast convergence. Following its training, the UAV can have the autonomy to make a decision concerning its next action at each position in the area, hence eliminating the need for human navigation. This makes UAV-aided wireless communications more reliable, practical and optimises the resource consumption.
\item Two scenarios are considered relying either on three or five clusters for qualifying the efficiency of our approach in terms of both the sum-rate, the trajectory and the associated time.
\end{itemize}
The rest of our paper is organised as follows. In Section \ref{Sec:Model}, we describe our data collection system model and the problem formulation of IoT networks relying on UAVs. Then, the mathematical background of the DRL algorithms is presented in Section \ref{Sec:BG}. Deep Q-learning (DQL) is employed for finding the best trajectory and for solving our data collection problem in Section \ref{Sec:Alg1}. Furthermore, we use the dueling DQL algorithm of \cite{Wang:15} for improving the system performance and convergence speed in Section \ref{Sec:Alg2}. Next, we characterise the efficiency of the DRL techniques in Section \ref{Sec:Results}. Finally, in Section \ref{Sec:Con}, we summarise our findings and discuss our future research.
\section{System Model and Problem Formulation}\label{Sec:Model}
Consider a system consisting of a single UAV and $M$ groups of users, as shown in Fig.~\ref{fig:System}, where the UAV relying on a single antenna visits all clusters to cover all the users. The 3D coordinate of the UAV at time step $t$ is defined as $X^t = (x_0^t, y_0^t, H_0^t)$. Each cluster consists of $K$ users, which are unknown and distributed randomly within the coverage radius of $C$. The users are moving following the random walk model with the maximum velocity $v$. The position of the $k$th user in the $m$th cluster at time step $t$ is defined as $X_{m,k}^t = (x_{m,k}^t, y_{m,k}^t)$. The UAV's objective is to find the best trajectory while covering all the users and to reach the dock upon completing its mission.
\begin{figure}[h!]
\centering
\subfigure{\includegraphics[width=0.5\textwidth]{Figs/System_model}}
\caption{System model of UAV-aided IoT communications.}
\label{fig:System}
\end{figure}
\subsection{Observation model}
The distance from the UAV to user $k$ in cluster $m$ at time step $t$ is given by:
\begin{equation}
d_{m, k}^t = \sqrt{(x_0^t - x_{m, k}^t) ^ 2 + (y_0^t - y_{m, k}^t) ^2 + {H_0^t} ^2}.
\end{equation}
We assume that the communication channels between the UAV and users are dominated by line-of-sight (LoS) links; thus the channel between the UAV and the $k$th user in the $m$th cluster at time step $t$ follows the free-space path loss model, which is represented as
\begin{equation}
\begin{split}
h_{m, k}^t &= \beta_0 {d_{m, k}^t} ^ {-2} \\
&= \frac{\beta_0}{(x_0^t - x_{m, k}^t) ^ 2 + (y_0 - y_{m, k}^t) ^2 + {H_0^t}^2},
\end{split}
\end{equation}
where the channel's power gain at a reference distance of $d=1m$ is denoted by $\beta_0$.
The achievable throughput from the $k$th user in the $m$th cluster to the UAV at time $t$ if the user satisfies the distance constraint is defined as follows:
\begin{equation}
R_{m,k}^t = B \log_2 \Bigg(1+ \frac{p_{m, k}^t h_{m, k}^t}{\sum_{i \neq m}^M \sum_j^K p_{i, j}^t h_{i, j}^t + \sum_{u \neq k}^K p_{m, u}^t h_{m, u}^t + \alpha^2}\Bigg), \forall m, k,
\end{equation}
where $B$ and $\alpha^2$ are the bandwidth and the noise power, respectively. Then the total sum-rate over the $T$ time step from the $k$th user in cluster $m$ to the UAV is given by:
\begin{equation}
R_{m,k} = \int_0^T R_{m,k}^t dt, \forall m,k.
\end{equation}
\subsection{Game formulation}
Both the current location and the action taken jointly influence the rewards obtained by the UAV; thus the trial-and-error based learning task of the UAV satisfies the Markov property. We formulate the associated Markov decision process (MDP) \cite{Puterman:14} as a 4 tuple $< \mathcal{S}, \mathcal{A}, \mathcal{P}_{ss'}, \mathcal{R} >$, where $\mathcal{S}$ is the state space of the UAV, $\mathcal{A}$ is the action space; $\mathcal{R}$ is the expected reward of the UAV and $\mathcal{P}_{ss'}$ is the probability of transition from state $s$ to state $s'$, where we have $s' = s^{t+1}| s = s^t$. Through learning, the UAV can find the optimal policy $\pi^* : \mathcal{S} \rightarrow \mathcal{A}$ for maximising the reward $\mathcal{R}$. More particularly, we formulate the trajectory and data collection game of UAV-aided IoT networks as follows:
\begin{itemize}
\item \textit{Agent}: The UAV acts like an agent interacting with the environment to find the peak of the reward.
\item \textit{State space}: We define the state space by the position of UAV as
\begin{equation}
\mathcal{S} = \{x, y, H\}.
\end{equation}
At time step $t$, the state of the UAV is defined as $s^t = ( x^t, y^t, H^t )$.
\item \textit{Action space}: The UAV at state $s^t$ can choose an action $a^t$ of the action space by following the policy at time-step $t$. By dividing the area into a grid, we can define the action space as follows:
\begin{equation}
\mathcal{A} = \{ \text{left}, \text{right}, \text{forward}, \text{backward}, \text{upward}, \text{downward}, \text{hover} \}.
\end{equation}
The UAV moves in the environment and begins collecting information when the users are in the coverage of the UAV. When the UAV has sufficient information $R_{m,k} \ge r_{min}$ from the $k$th user in the $m$th cluster, that user will be marked as collected in this mission and may not be visited by the UAV again.
\item \textit{Reward function}: In joint trajectory and data collection optimisation, we design the reward function to be dependent on both the total sum-rate of ground users associated with the UAV plus the reward gleaned when the UAV completes one route, which is formulated as follows:
\begin{equation}\label{equ:reward}
R = \frac{\beta}{MK}\left(\sum_m^M \sum_k^K P(m,k) R_{m, k}\right) + \zeta R_{plus},
\end{equation}
where $\beta$ and $\zeta$ are positive variables that represent the trade-off between the network's sum-rate and UAV's movement, which will be described in the sequel. Here, $P(m, k) = \{0,1\}$ indicates whether or not user $k$ of cluster $m$ is associated with the UAV; $R_{plus}$ is the acquired reward when the UAV completes a mission by reaching the final destination. On the other hand, the term $ \frac{\sum_m^M \sum_k^K P(m,k) R_{m, k}}{MK}$ defines the average throughput of all users.
\item \textit{Probability}: We define $\mathcal{P}_{s^t s^{t+1}}(a^t, \pi)$ as the probability of transition from state $s^t$ to state $s^{t+1}$ by taking the action $a^t$ under the policy $\pi$.
\end{itemize}
At each time step $t$, the UAV chooses the action $a^t$ based on its local information to obtain the reward $r^t$ under the policy $\pi$. Then the UAV moves to the next state $s^{t+1}$ by taking the action $a^t$ and starts collecting information from the users if any available node in the network satisfies the distance constraint. Again, we use the DRL techniques to find the optimal policy $\pi^*$ for the UAV to maximise the reward attained (\ref{equ:reward}). Following the policy $\pi$, the UAV forms a chain of actions $( a^0, a^1, \dots, a^t, \dots, a^{final} )$ to reach the landing dock.
Our target is to maximise the reward expected by the UAV upon completing a single mission during which the UAV flies from the initial position over the clusters and lands at the destination. Thus, we design the trajectory reward $R_{plus}$ when the UAV reaches the destination in two different ways. Firstly, the binary reward function is defined as follows:
\begin{equation}\label{equ:Rplus1}
R_{plus} = \left\{ \begin{array}{rcl} 1 & \mbox{,} & X_{final} \in X_{target} \\
0 & \mbox{,} & \mbox{otherwise.} \end{array} \right. ,
\end{equation}
where $X_{final}$ and $X_{target}$ are the final position of UAV and the destination, respectively. However, the UAV has to move a long distance to reach the final destination. It may also be trapped in a zone and cannot complete the mission. These situations lead to increased energy consumption and reduced convergence. Thus, we consider the value of $R_{plus}^t$ in a different form by calculating the horizontal distance between the UAV and the final destination at time step $t$, yielding:
\begin{equation}\label{equ:Rplus2}
R^t_{plus} = \left\{ \begin{array}{rcl} 1 & \mbox{,} & X_{final} \in X_{target} \\
\Big (\exp \sqrt{(x_{target}-x_{0}^t)^2 + (y_{target} - y_{0}^t)^2} \Big)^{-1} & \mbox{,} & \mbox{otherwise.} \end{array} \right.
\end{equation}
When we design the reward function as in (\ref{equ:Rplus2}), the UAV is motivated to move ahead to reach the final destination. However, one of the disadvantages is that the UAV only moves forward. Thus, the UAV is unable to attain the best performance in terms of its total sum-rate in some environmental settings. We compare the performance of the two trajectory reward function definitions in Section \ref{Sec:Results} to evaluate the pros and cons of each approach.
We design the reward function by arranging for a trade-off game with parameters $\beta, \zeta$ to make our approach more adaptive and flexible. By modifying the value of $\beta/\zeta$ , the UAV adapts to several scenarios: a) fast deployment for emergency services, b) maximising the total sum-rate, and c) maximising the number of connections between the UAV and users. Depending on the specific problems, we can adjust the value of the trade-off parameters $\beta, \zeta$ to achieve the best performance. Thus, the game formulation is defined as follows:
\begin{equation}
\begin{split}
\max R = \quad & \frac{\beta}{MK} \left(\sum_m^M \sum_k^K P(m,k) R_{m, k}\right) + \zeta R_{plus},\\
s.t. \quad&
X_{final} = X_{target},\\
&d_{m, k} \le d_{cons},\\
&R_{m,k} \ge r_{min},\\
&P(m, k) = \{0, 1\},\\
&T \le T_{cons}\\
&\beta \ge 0, \zeta \ge 0,
\end{split}
\end{equation}
where $T$ and $T_{cons}$ are the number of steps that the UAV takes in a single mission and the maximum number of UAV's steps given its limited power, respectively. The distance constraint $d_{m, k} \le d_{cons}$ indicates that the served $(m,k)$-user has a satisfied distance to the UAV. Those stringent constraints, such as the transmission distance, position and flight time make the optimisation problem more challenging. Thus, we propose DRL techniques for the UAV in order to attain the optimal performance.
\section{Preliminaries}\label{Sec:BG}
In this section, we introduce the fundamental concept of Q-learning, where the so-called value function is defined by a reward of the UAV at state $s^t$ as follows:
\begin{equation}
V(s, \pi) = \mathbb{E} \bigg[ \sum_t^T \gamma \mathcal{R}^t (s^t, \pi)| s_0 = s\bigg],
\end{equation}
where $\mathbb{E[\centerdot]}$ represents an average of the number of samples and $0 \le \gamma \le 1$ denotes the discount factor. The value function can be rewritten by expoiting the Markov property as follows:
\begin{equation}
V(s, \pi) = \mathbb{E} \bigg[ \mathcal{R}^t(s^t, \pi)\bigg] + \gamma \sum_{s' \in \mathcal{S}} P_{ss'}(a, \pi) V(s', \pi).
\end{equation}
In a finite game, there is always an optimal policy $\pi^*$ that satisfies the Bellman optimality equation \cite{BD:95:Book:v1}
\begin{equation}
\begin{split}
V^* (s, \pi) &= V (s, \pi^*)\\
& = \max_{a \in \mathcal{A}} \Bigg[ {\mathbb{E} \bigg[ \mathcal{R}^t(s^t, \pi^*)\bigg] + \gamma \sum_{s' \in S} P_{ss'}(a, \pi^*) V(s', \pi^*)} \Bigg] .
\end{split}
\end{equation}
The action-value function is obtained, when the agent at state $s^t$ takes action $a^t$ and receives the reward $r^t$ under the agent policy $\pi$. The optimal Q-value can be formulated as:
\begin{equation}\label{equ:Q}
Q^*(s, a, \pi) = {\mathbb{E} \bigg[ \mathcal{R}^t(s^t, \pi^*)\bigg] + \gamma \sum_{s' \in S} P_{ss'}(a, \pi^*) V(s', \pi^*)}.
\end{equation}
The optimal policy $\pi^*$ can be obtained from $Q^*(s, a, \pi)$ as follows:
\begin{equation}\label{equ:V}
V^*(s, \pi) = \max_{a \in \mathcal{A}} Q(s, a, \pi).
\end{equation}
From (\ref{equ:Q}) and (\ref{equ:V}), we have
\begin{equation}
\begin{split}
Q^*(s, a, \pi) \; & = \mathbb{E} \bigg[ \mathcal{R}^t(s^t, \pi^*)\bigg] + \gamma \sum_{s' \in S} P_{ss'}(a, \pi^*) \max_{a' \in \mathcal{A}} Q(s', a', \pi), \\
&= \mathbb{E} \bigg[ \mathcal{R}^t(s^t, \pi^*)+ \gamma \max_{a' \in \mathcal{A}} Q(s', a', \pi)\bigg] ,
\end{split}
\end{equation}
where the agent takes the action $a' = a^{t+1}$ at state $s^{t+1}$.
Through learning, the Q-value is updated based on the available information as follows:
\begin{equation}\label{equ:Qupdate}
\begin{split}
Q(s, a, \pi) = \; Q(s, a, \pi) + \alpha \bigg[ \mathcal{R}^t(s^t, \pi^*)
+ \gamma \max_{a' \in \mathcal{A}} Q(s', a', \pi) - Q(s, a, \pi) \bigg],
\end{split}
\end{equation}
where $\alpha$ denotes the updated parameter of the Q-value function.
In RL algorithms, it is challenging to balance the \textit{exploration} and \textit{exploitation} for appropriately selecting the action. The most common approach relies on the $\epsilon$-greedy policy for the action selection mechanism as follows:
\begin{equation}\label{equ:ep}
a = \left\{ \begin{array}{rcl} \argmax Q(s, a, \pi) & \mbox{with} & \epsilon \\
\mbox{randomly} & \mbox{if} & 1- \epsilon. \end{array} \right.
\end{equation}
Upon assuming that each episode lasts $T$ steps, the action at time step $t$ is $a^t$ that is selected by following the $\epsilon$-greedy policy as in (\ref{equ:ep}). The UAV at state $s^t$ communicates with the user nodes from the ground if the distance constraint of $d_{m, k} \le d_{cons}$ is satisfied. Following the information transmission phase, the user nodes are marked as collected users and may not be revisited later during that mission. Then, after obtaining the immediate reward $r(s^t, a^t)$ the agent at state $s^t$ takes action $a^t$ to move to state $s^{t+1}$ as well as to update the Q-value function in (\ref{equ:Qupdate}). Each episode ends when the UAV reaches the final destination and the flight duration constraint is satisfied.
\section{An effective deep reinforcement learning approach for UAV-assisted IoT networks}\label{Sec:Alg1}
In this section, we conceive the DQL algorithm for trajectory and data collection optimisation in UAV-aided IoT networks. However, Q-learning technique typically falters for large state and action spaces due to its excessive Q-table size. Thus, instead of applying the Q-table in Q-learning, we use deep neural networks to represent the relationship between the action and state space. Furthermore, we employ a pair of techniques for stabilising the neural network's performance in our DQL algorithm as follows:
\begin{itemize}
\item \textit{Experience relay buffer}: Instead of using current experience, we use a so-called relay buffer $\mathcal{B}$ to store the transitions $(s, a, r, s')$ for supporting the neural network in overcoming any potential instability. When the buffer $\mathcal{B}$ is filled with the transitions, we randomly select a mini-batch of $K$ samples for training the networks. The finite buffer size of $\mathcal{B}$ allows it to be always up-to-date, and the neural networks learn from the new samples.
\item \textit{Target networks}: If we use the same network to calculate the state-action value $Q$ and the target network, the network can be shifted dramatically in the training phase. Thus, we employ a target network $Q'$ for the target value estimator. After a number of iterations, the parameters of the target network $Q'$ will be updated by the network $Q$.
\end{itemize}
\begin{algorithm}[t!]
\caption{The deep Q-learning algorithm for trajectory and data collection optimisation in UAV-aided IoT networks}
\begin{algorithmic}[1]
\label{alg:DQL}
\STATE Initialise the network $Q$ and the target network $Q'$ with the random parameters $\theta$ and $\theta'$, respectively
\STATE Initialise the replay memory pool $\mathcal{B}$
\FOR{episode = $1,\dots, L$}
\STATE Receive initial observation state $s^0$
\WHILE{$X_{final} \notin X_{target}$ or $T \le T_{cons}$}
\STATE Obtain the action $a^t$ of the UAV according to the $\epsilon$-greedy mechanism (\ref{equ:ep})
\STATE Execuse the action $a^t$ and estimate the reward $r^t$ according to (\ref{equ:reward})
\STATE Observe the next state $s^{t+1}$
\STATE Store the transition $(s^t, a^t, r^t, s^{t+1})$ in the replay buffer $\mathcal{B}$
\STATE Randomly select a mini-batch of $K$ transitions $(s^k, a^k, r^k, s^{k+1})$ from $\mathcal{B}$
\STATE Update the network parameters using gradient descent to minimise the loss
\begin{equation}
\mathbb{L}(\theta) = \mathbb{E}_{s, a, r, s'} \Bigg[\bigg(y^{DQL} - Q(s, a; \theta)\bigg)^2 \Bigg],
\end{equation}
The gradient update is
\begin{equation}
\nabla_\theta \mathbb{L}(\theta) = \mathbb{E}_{s, a, r, s'} \Bigg[\bigg(y^{DQL} - Q(s, a; \theta)\bigg)\nabla_\theta Q(s, a;\theta) \Bigg],
\end{equation}
\STATE Update the state $s^t = s^{t+1}$
\STATE Update the target network parameters after a number of iterations as $\theta' = \theta$
\ENDWHILE
\ENDFOR
\end{algorithmic}
\end{algorithm}
The neural network parameters are updated by minimising the loss function defined as follows:
\begin{equation}\label{equ:DQLloss}
\mathbb{L}(\theta)= \mathbb{E}_{s, a, r, s'} \Bigg[ \bigg(y^{DQL} - Q(s, a; \theta)\bigg)^2 \Bigg],
\end{equation}
where $\theta$ is a parameter of the network $Q$ and we have
\begin{equation}
y = \left\{ \begin{array}{rcl} r^t & \mbox{if terminated at} \; s^{t+1} \\
r^t + \gamma \max_{a' \in \mathcal{A}} Q'(s', a'; \theta') & \mbox{otherwise.} \end{array} \right.
\end{equation}
The details of the DQL approach in our joint trajectory and data collection trade-off game designed for UAV-aided IoT networks are presented in Alg. \ref{alg:DQL} where $L$ denotes the number of episode. Moreover, in this paper, we design the reward obtained in each step to assume one of two different forms and compare them in our simulation results. Firstly, we calculate the difference between the current and the previous reward of the UAV as follows:
\begin{equation}\label{equ:R1}
r_1^t (s^t, a^t) = r^t (s^t, a^t) - r^{t-1}(s^{t-1}, a^{t-1}).
\end{equation}
Secondly, we design the total episode reward as the accumulation of all immediate rewards of each step within one episode as
\begin{equation}\label{equ:R2}
r_2^t (s^t, a^t) = \sum^t_{i=0} r_1^t(s^t, a^t).
\end{equation}
\section{Deep reinforcement learning approach for UAV-assisted IoT networks: A dueling deep Q-learning approach }\label{Sec:Alg2}
According to Wang \textit{et. al.} \cite{Wang:15}, the standard Q-learning algorithm often falters due to the over-supervision of all the state-action pairs. On the other hand, it is unnecessary to estimate the value of each action choice in a particular state. For example, in our environment setting, the UAV has to consider moving either to the left or to the right when it hits the boundaries. Thus, we can improve the convergence speed by avoiding visiting all state-action pairs. Instead of using Q-value function of the conventional DQL algorithm, the dueling neural network of \cite{Wang:15} is introduced for improving the convergence rate and stability. The so-called advantage function $A (s, a) = Q(s, a) - V( s) $ related both to the value function and to the Q-value function describes the importance of each action related to each state.
\begin{algorithm}[t!]
\caption{The dueling deep Q-learning algorithm for trajectory and data collection optimisation in UAV-aided IoT networks}
\begin{algorithmic}[1]
\label{alg:DuelingDQL}
\STATE Initialise the network $Q$ and the target network $Q'$ with the random parameters, $\theta$ and $\theta'$, respectively
\STATE Initialise the replay memory pool $\mathcal{B}$
\FOR{episode = $1,\dots, L$}
\STATE Receive the initial observation state $s^0$
\WHILE{$X_{final} \notin X_{target}$ or $T \le T_{cons}$}
\STATE Obtain the action $a^t$ of the UAV according to the $\epsilon$-greedy mechanism (\ref{equ:ep})
\STATE Execute the action $a^t$ and estimate the reward $r^t$ according to (\ref{equ:reward})
\STATE Observe the next state $s^{t+1}$
\STATE Store the transition $(s^t, a^t, r^t, s^{t+1})$ in the replay buffer $\mathcal{B}$
\STATE Randomly select a mini-batch of $K$ transitions $(s^k, a^k, r^k, s^{k+1})$ from $\mathcal{B}$
\STATE Estimate the Q-value function by combining the two streams as follows:
\begin{equation}
\begin{split}
Q(s, a; \; \theta, \theta_A, \theta_V) = V(s ;\theta_V) + \Bigg( A(s, a; \theta_A) - \frac{1}{|\mathcal{A}|} \sum_{a'} A(s, a'; \theta_A) \Bigg).
\end{split}
\end{equation}
\STATE Update the network parameters using gradient descent to minimise the loss
\begin{equation}
\mathbb{L} (\theta) = \mathbb{E}_{s, a, r, s'} \Bigg[ \bigg(y^{DuelingDQL} - Q(s, a; \theta, \theta_A, \theta_V)\bigg)^2 \Bigg],
\end{equation}
\STATE where
\begin{equation}
y^{DuelingDQL} = r^t +\gamma \max_{a' \in \mathcal{A}} Q'(s', a'; \theta', \theta_A, \theta_V).
\end{equation}
\STATE Update the state $s^t = s^{t+1}$
\STATE Update the target network parameters after a number of iterations as $\theta' = \theta$
\ENDWHILE
\ENDFOR
\end{algorithmic}
\end{algorithm}
The idea of a dueling deep network is based on a combination of two streams of the value function and the advantage function used for estimating the single output $Q$-function. One of the streams of a fully-connected layer estimates the value function $V(s; \theta_V)$, while the other stream outputs a vector $A(s, a; \theta_A)$, where $\theta_A$ and $\theta_V$ represent the parameters of the two networks. The $Q$-function can be obtained by combining the two streams' outputs as follows:
\begin{equation}\label{equ:A}
Q(s, a; \theta, \theta_A, \theta_V) = V( s; \theta_V) + A(s, a; \theta_A).
\end{equation}
Equation (\ref{equ:A}) applies to all $(s, a)$ instances; thus, we have to replicate the scalar $V(s; \theta_V)$, $|\mathcal{A}|$ times to form a matrix. However, $Q(s, a; \theta, \theta_A, \theta_V)$ is a parameterised estimator of the true Q-function; thus, we cannot uniquely recover the value function $V$ and the advantage function $A$. Therefore, (\ref{equ:A}) results in poor practical performances when used directly. To address this problem, we can map the advantage function estimator to have no advantage at the chosen action by combining the two streams as follows:
\begin{equation}\label{equ:Amax}
\begin{split}
Q(s, a; \theta, \theta_A, \theta_V) = V(s ;\theta_V) + \bigg( A(s, a; \theta_A) - \max_{a' \in |\mathcal{A}|} A(s, a'; \theta_A) \bigg).
\end{split}
\end{equation}
Intuitively, for $a^* = \argmax_{a' \in \mathcal{A}}Q(s, a'; \theta, \theta_A, \theta_V) = \argmax_{a' \in \mathcal{A}} A(s, a'; \theta_A)$, we have $\linebreak Q(s, a^*; \theta, \theta_A, \theta_V) = V(s; \theta_V)$. Hence, the stream $V(s; \theta_V)$ estimates the value function and the other streams is the advantage function estimator. We can transform (\ref{equ:Amax}) using an average formulation instead of the \textit{max} operator as follows:
\begin{equation}\label{equ:Aavg}
\begin{split}
Q(s, a; \theta, \theta_A, \theta_V) = V(s ;\theta_V) + \Bigg( A(s, a; \theta_A) - \frac{1}{|\mathcal{A}|} \sum_{a'} A(s, a'; \theta_A) \Bigg).
\end{split}
\end{equation}
Now, we can solve the problem of identifiability by subtracting the mean as in (\ref{equ:Aavg}). Based on (\ref{equ:Aavg}), we propose a dueling DQL algorithm for our joint trajectory and data collection problem in UAV-assisted IoT networks relying on Alg. \ref{alg:DuelingDQL}. Note that estimating $V(s; \theta_V)$ and $A(s, a ; \theta_A)$ does not require any extra supervision and they will be computed automatically.
\section{Simulation Results}\label{Sec:Results}
In this section, we present our simulation results characterising the joint optimisation problem of UAV-assisted IoT networks. To highlight the efficiency of our proposed model and the DRL methods, we consider a pair of scenarios: a simple having three clusters, and a more complex one with five clusters in the coverage area. We use Tensorflow 1.13.1 \cite{Abadi:16} and the Adam optimiser of \cite{DJ:14} for training the neural networks. All the other parameters are provided in Table \ref{tab:Params}.
\begin{table}[t!]
\renewcommand{\arraystretch}{1.2}
\caption{SIMULATION PARAMETERS}
\label{tab:Params}
\centering
\begin{tabular}{l|l}
\hline
Parameters & Value \\
\hline
Bandwidth ($W$) & $1$ MHz \\
UAV transmission power & $5$ W \\
The start position of UAV & $(0, 0, 200)$\\
Discounting factor & $\gamma = 0.9$\\
Max number of users per cluster & $10$\\
Noise power & $\alpha^2 = -110dBm$ \\
The reference channel power gain & $\beta_0 = -50dB$\\
Path-loss exponent & $2$ \\
\hline
\end{tabular}
\end{table}
\begin{figure}[t!]
\centering
\subfigure{\includegraphics[width=0.65\textwidth]{Figs/Trajectory}}
\caption{Trajectory obtained by using our DQL algorithm}
\label{fig:Traj}
\end{figure}
In Fig. (\ref{fig:Traj}), we present the trajectory obtained after training using the DQL algorithm in the $5$-cluster scenario. The green circle and blue dots represent the clusters' coverage and the user nodes, respectively. The red dots and black triangles in the figure represent the UAV's state after taking action. The UAV starts at $(0, 0)$, visits about $40$ users, and lands at the destination that is denoted by the black square. In a complex environment setting, it is challenging to expect the UAV to visit all users, while satisfying the flight-duration and power level constraints.
\subsection{Expected reward}
\begin{figure}[h!]
\centering
\subfigure[]{\includegraphics[width=0.65\textwidth]{Figs/ALLreward3clusters2}}
\subfigure[]{\includegraphics[width=0.65\textwidth]{Figs/ALLreward3clusters}}
\caption{The performance when using the DQL and dueling DQL algorithms with 3 clusters while considering different $\beta$/$\zeta$ values}
\label{fig:reward3clusters}
\end{figure}
\begin{figure}[h!]
\centering
\subfigure[With (\ref{equ:Rplus1})]{\includegraphics[width=0.65\textwidth]{Figs/ALLreward5clusters}}
\subfigure[With (\ref{equ:Rplus2})]{\includegraphics[width=0.65\textwidth]{Figs/ALLreward5clusters2}}
\caption{The expected reward when using the DQL and dueling DQL algorithms with 5-cluster scenario}
\label{fig:reward5clusters}
\end{figure}
For purposes of comparison, we run the algorithm five times in five different environmental settings and take the average to draw the figures. Firstly, we compare the reward obtained following (\ref{equ:reward}). Let us consider the $3$-cluster scenario and $\beta/\zeta = 2:1$ in Fig. (\ref{fig:reward3clusters}a), where the DQL and dueling DQL algorithms using the exponential function ($\ref{equ:Rplus2}$) reach the best performance. When using the exponential trajectory design function (\ref{equ:Rplus2}), the performance converges faster than that of the DQL and dueling DQL methods using the binary trajectory function (\ref{equ:Rplus1}). In addition, in Fig. (\ref{fig:reward3clusters}b), we compare the performance of the DQL and dueling DQL techniques using different $\beta/\zeta$ values. The average performance of the dueling DQL algorithm is better than that of the DQL algorithm. In conjunction, the results of using the exponential function ($\ref{equ:Rplus2}$) is better than that of the ones using the binary function ($\ref{equ:Rplus1}$).
Furthermore, we compare the rewards obtained by the DQL and dueling DQL algorithms in complex scenarios with $5$ clusters and $50$ user nodes in Fig. (\ref{fig:reward5clusters}). The performance of using the episode reward (\ref{equ:R2}) is better than that using the immediate reward (\ref{equ:R1}) in both trajectory designs relying on the DQL and dueling DQL algorithms. In Fig. (\ref{fig:reward5clusters}a), we compare the performance in conjunction with the binary trajectory design while in Fig. (\ref{fig:reward5clusters}b) the exponential trajectory design is considered. For $\beta/\zeta= 1:1$, the rewards obtained by the DQL and dueling DQL are similar and stable after about $400$ episodes. When using the exponential function (\ref{equ:Rplus2}), the dueling DQL algorithm reaches the best performance. Moreover, the convergence of the dueling DQL technique is faster than that of the DQL algorithm.
\begin{figure}[h!]
\centering
\subfigure{\includegraphics[width=0.65\textwidth]{Figs/ALLtradeoff5clusters}}
\caption{The performance when using the DQL and dueling DQL algorithms with 5 clusters and different $\beta/\zeta$ values}
\label{fig:reward5clusters2}
\end{figure}
In Fig. (\ref{fig:reward5clusters2}), we compare the performance of the DQL and of the dueling DQL algorithms while considering different $\beta/\zeta$ parameter values. The dueling DQL algorithm shows better performance for all the $\beta/\zeta$ pair values, exhibiting better rewards. In addition, when using the exponential function (\ref{equ:Rplus2}), both proposed algorithms show better performance than the ones using the binary function (\ref{equ:Rplus1}) if $\beta/\zeta \le 1:1$, but it becomes less effective when $\beta/\zeta$ is set higher.
\begin{figure}[h!]
\centering
\subfigure[]{\includegraphics[width=0.65\textwidth]{Figs/DQLreward5clusters}}
\subfigure[]{\includegraphics[width=0.65\textwidth]{Figs/DQLtradeoff5clusters}}
\caption{The expected reward when using the DQL algorithm with 5 clusters and different reward function settings}
\label{fig:reward5clusters3}
\end{figure}
\begin{figure}[h!]
\centering
\subfigure{\includegraphics[width=0.65\textwidth]{Figs/Dueltradeoff5clusters2}}
\caption{The performance when using the dueling DQL with 5 clusters, and different $\beta/\zeta$ values}
\label{fig:reward5clusters4}
\end{figure}
We compare the performance of the DQL and of the dueling DQL algorithm using different reward function setting in Fig. (\ref{fig:reward5clusters3}) and in Fig. (\ref{fig:reward5clusters4}), respectively. The DQL algorithm reaches the best performance when using the episode reward (\ref{equ:R2}) in Fig. (\ref{fig:reward5clusters3}a) while the fastest convergence speed can be achieved by using the exponential function (\ref{equ:Rplus2}). When $\beta/\zeta \ge 1:1$, the DQL algorithm relying on the episode function (\ref{equ:R2}) outperforms the ones using the immediate reward function (\ref{equ:R1}) in Fig. (\ref{fig:reward5clusters3}b). The reward (\ref{equ:reward}) using the exponential trajectory design (\ref{equ:Rplus2}) has a better performance than that using the binary trajectory design (\ref{equ:Rplus1}) for all the $\beta/\zeta$ values. The similar results are shown when using the dueling DQL algorithm in Fig. (\ref{fig:reward5clusters4}). The immediate reward function (\ref{equ:R1}) is less effective than the episode reward function (\ref{equ:R2}).
\subsection{Throughput comparison}
\begin{figure}[h!]
\centering
\subfigure[With (\ref{equ:Rplus1})]{\includegraphics[width=0.65\textwidth]{Figs/Throughput3clusters}}
\subfigure[]{\includegraphics[width=0.65\textwidth]{Figs/ALLthroughput3clusters}}
\caption{The network's sum-rate when using the DQL and dueling DQL algorithms with 3 clusters}
\label{fig:throughput3clusters}
\end{figure}
In (\ref{equ:reward}), we consider two elements: the trajectory cost and the average throughput. In order to quantify the communication efficiency, we compare the total throughput in different scenarios. In Fig. (\ref{fig:throughput3clusters}), the performances of the DQL algorithm associated with several $\beta/\zeta$ values are considered while using the binary trajectory function (\ref{equ:Rplus1}), the episode reward (\ref{equ:R2}) and $3$ clusters. The throughput obtained for $\beta/\zeta= 1:1$ is higher than that of the others and when $\beta$ increases, the performance degrades. However, when comparing with the Fig. (\ref{fig:reward3clusters}b), we realise that in some scenarios the UAV was stuck and could not find the way to the destination. That leads to increased flight time spent and distance travelled. More details are shown in Fig. (\ref{fig:throughput3clusters}b), where we compare the expected throughput of both the DQL and dueling DQL algorithms. The best throughput is achieved when using the dueling DQL algorithm with $\beta/\zeta= 1:1$ in conjunction with (\ref{equ:Rplus1}), which is higher than the peak of the DQL method with $\beta/\zeta= 1:2$.
\begin{figure}[h!]
\centering
\subfigure[With (\ref{equ:Rplus1}), (\ref{equ:R2})]{\includegraphics[width=0.65\textwidth]{Figs/Throughput5clusters}}
\subfigure[With (\ref{equ:Rplus2}), (\ref{equ:R2})]{\includegraphics[width=0.65\textwidth]{Figs/Throughput5clusters4}}
\caption{The obtained total throughput when using the DQL algorithm with 5 clusters}
\label{fig:throughput5clusters}
\end{figure}
In Fig. (\ref{fig:throughput5clusters}), we compare the throughput of different techniques in the $5$-cluster scenario. Let us now consider the binary trajectory design function (\ref{equ:Rplus1}) in Fig. (\ref{fig:throughput5clusters}a), where the DQL algorithm achieves the best performance using $\beta/\zeta= 1:1$ and $\beta/\zeta = 2:1$. There is a slight difference between the DQL method having different settings, when using exponential the trajectory design function (\ref{equ:Rplus2}), as shown in Fig. (\ref{fig:throughput5clusters}b).
\begin{figure}[h!]
\centering
\subfigure[]{\includegraphics[width=0.65\textwidth]{Figs/Throughput5clusters3}}
\subfigure[]{\includegraphics[width=0.65\textwidth]{Figs/Duelthroughput5clusters}}
\caption{The obtained throughput when using the DQL and dueling DQL algorithms in 5-cluster scenario}
\label{fig:throughput5clusters2}
\end{figure}
In Fig. (\ref{fig:throughput5clusters2}) and Fig. (\ref{fig:throughput5clusters3}), we compare the throughput of different $\beta/\zeta$ pairs. The DQL algorithm reaches the optimal throughput with the aid of trial-and-learn methods, hence it is important to carefully design the reward function to avoid excessive offline training. As shown in Fig. (\ref{fig:throughput5clusters2}), the DQL and dueling DQL algorithm exhibit reasonable stability for several $\beta/\zeta \le 1:1$ pairs as well as reward functions. While we can achieve the similar expected reward with different reward setting in Fig. (\ref{fig:reward5clusters3}), the throughput is degraded when the $\beta/\zeta$ increases. In contrast, with higher $\beta$ values, the UAV can finish the mission faster. It is a trade-off game when we can choose an approximate $\beta/\zeta$ value for our specific purposes. When we employ the DQL and the dueling DQL algorithms with the episode reward (\ref{equ:R2}), the throughput attained is higher than that using the immediate reward (\ref{equ:R1}) with different $\beta/\zeta$ values.
\begin{figure}[h!]
\centering
\subfigure[With (\ref{equ:Rplus2})]{\includegraphics[width=0.65\textwidth]{Figs/ALLthroughput5clusters}}
\subfigure[With (\ref{equ:R2})]{\includegraphics[width=0.65\textwidth]{Figs/ALLthroughput5clusters2}}
\caption{The expected throughput when using the DQL and dueling DQL algorithms with 5 clusters}
\label{fig:throughput5clusters3}
\end{figure}
Furthermore, we compare the expected throughput of the DQL and of the dueling DQL algorithm when using the exponential trajectory design (\ref{equ:Rplus2}) in Fig. (\ref{fig:throughput5clusters3}a) and the episode reward (\ref{equ:R2}) in Fig. (\ref{fig:throughput5clusters3}b). In Fig. (\ref{fig:throughput5clusters3}a), the dueling DQL method outperforms the DQL algorithm for almost all $\beta/\zeta$ values in both function (\ref{equ:R1}) and (\ref{equ:R2}). When we use the episode reward (\ref{equ:R2}), the obtained throughput are stable with different $\beta/\zeta$ values. The throughput attained by using the exponential function (\ref{equ:Rplus2}) is higher than that using the binary trajectory (\ref{equ:Rplus1}) and by using the episode reward (\ref{equ:R2}) is higher than that using the immediate reward (\ref{equ:R1}). We can achieve the best performance when using the dueling DQL algorithm with (\ref{equ:Rplus2}) and (\ref{equ:R2}). However, in some scenarios, we can achieve the better performance with different algorithmic setting as we can see in Fig. (\ref{fig:throughput3clusters}b) and Fig. (\ref{fig:throughput5clusters2}a). Thus, there is a trade-off governing the choice of the algorithm and function design.
\subsection{Parametric Study}
\begin{figure}[t!]
\centering
\subfigure{\includegraphics[width=0.65\textwidth]{Figs/gammaepsilon}}
\caption{The performance when using the DQL algorithm with different discount factors, $\gamma$, and exploration factors, $\epsilon$}
\label{fig:exp}
\end{figure}
In Fig. (\ref{fig:exp}), we compare the performance of our DQL technique using different $\textit{exploration}$ parameters $\gamma$ and $\epsilon$ values in our $\epsilon$-greedy method. The DQL algorithm achieves the best performance with the discounting factor of $\gamma = 0.9$ and $\epsilon = 0.9$ in the $5$-cluster scenario of Fig.~(\ref{fig:exp}). Balancing the \textit{exploration} and \textit{exploitation} as well as the action chosen is quite challenging, in order to maintain a steady performance of the DQL algorithm. Based on the results of Fig. (\ref{fig:exp}), we opted for $\gamma = 0.9$ and $\epsilon = 0.9$ for our algorithmic setting.
\begin{figure}[h!]
\centering
\subfigure{\includegraphics[width=0.65\textwidth]{Figs/Batchsize5clusters}}
\caption{The performance when using the DQL algorithm in $5$-cluster scenario and different batch sizes, $K$}
\label{fig:batch3clusters}
\end{figure}
Next, we compare the expected reward of different mini-batch sizes, $K$. In the $5$-cluster scenario of Fig. (\ref{fig:batch3clusters}), the DQL achieves the optimal performance with a batch size of $K = 32$. There is a slight difference in terms of convergence speed with batch size $K = 32$ is the fastest. Overall, we set the mini-batch size to $K = 32$ for our DQL algorithm.
\begin{figure}[h!]
\centering
\subfigure{\includegraphics[width=0.65\textwidth]{Figs/lr5clusters}}
\caption{The performance when using DQL algorithm with different learning rate, $lr$}
\label{fig:lr}
\end{figure}
Fig. (\ref{fig:lr}) shows the performance of the DQL algorithm with different learning rates in updating the neural networks parameters while considering the scenarios of $5$ clusters. When the learning rate is as high as $\alpha = 0.01$, the pace of updating the network may result the fluctuating performance. Moreover, when $\alpha = 0.0001$ or $\alpha = 0.00001$ the convergence speed is slower and may be stuck in a local optimum instead reaching the global optimum. Thus, based on our experiments, we opted for the learning rate of $\alpha = 0.001$ for the algorithms.
\section{Conclusion}\label{Sec:Con}
In this paper, the DRL technique has been proposed jointly optimising the flight trajectory and data collection performance of UAV-assisted IoT networks. The optimisation game has been formulated to balance the flight time and total throughput while guaranteeing the quality-of-service constraints. Bearing in mind the limited UAV power level and the associated communication constraints, we proposed a DRL technique for maximising the throughput while the UAV has to move along the shortest path to reach the destination. Both the DQL and dueling DQL techniques having a low computational complexity have been conceived. Our simulation results showed the efficiency of our techniques both in simple and complex environmental settings.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}\label{Introduction}
Weak gravitational lensing (WL) has emerged as one of the most
promising methods to constrain the parameters of both dark energy (DE)
and dark matter (DM) \cite{DETF}. This method exploits the fact that
images of distant galaxies are distorted as their light is lensed by
inhomogeneities in the foreground gravitational potential. DE and DM
alter the expansion history of the universe, as well as the rate at
which cosmic dark matter structures grow from small initial
perturbations, and thereby modify the statistical features of the WL
signal \cite{RefReview}. Several deep astronomical surveys plan to
cover large regions ($\gsim10^3$ degree$^2$) of the sky in the future,
detecting millions of background galaxies, and measuring their shapes
precisely, in order to reveal the statistical features of the WL
distortion. Examples include the Panoramic Survey Telescope And Rapid
Response System (Pan-STARRS) \cite{Kaiser:1999kka}, the Kilo-Degree
Survey (KIDS)\footnote{http://www.strw.leidenuniv.nl/~kuijken/KIDS},
the Dark Energy Survey
(DES)\footnote{https://www.darkenergysurvey.org}, and the survey by
the Large Synoptic Survey Telescope (LSST) \cite{Tyson:2002nh,
Tyson2002b} covering half of the sky.
Existing theoretical studies assessing the utility of such surveys to
probe the properties of DE and of DM can be roughly divided into two
types. The first type analyzes the WL distortion primarily on large
scales, caused by fluctuations in the foreground mass distribution on
correspondingly large scales. On these scales, the fluctuations, and
hence the WL distortion patterns, are linear or quasilinear, and can
be readily and reliably interpreted using existing semi--analytic
tools \citep{Kaiser92,JS97,WH99,WH02,DH02,AR04,A&D03,T&J04,S&K04}.
These studies (for example, those computing the two--dimensional power
spectrum of the shear) have, in fact, been pushed into the non--linear
regime, but the dependence of the non--linear power--spectrum on
cosmological parameters has not been verified in simulations, to an
accuracy necessary to take full advantage of the future surveys.
The second type analyzes the distortion on smaller scales, where it is
dominated by the nonlinear mass fluctuations. In the limit of
extremely non--linear, high--$\sigma$ WL shear peaks, the individual
peaks tend to be caused by a single, discrete, massive collapsed
object -- a galaxy cluster -- in the
foreground~\citep{whiteprojection, Hamana:2003ts,Hennawi:2004ai}. The
abundance evolution and spatial distribution of galaxy clusters in
different cosmologies can then be studied semi--analytically
(calibrated by simulations), and has been shown to place very strong
constraints on DE parameters, because it is exponentially sensitive to
the growth rate of dark matter fluctuations. Several studies have
explored constraints expected from samples of tens of thousands of
galaxy clusters in Sunyaev--Zeldovich and X--ray surveys
(e.g.~\citep{W&S98,H&M&H01,W&B&K02}), and from the still larger
samples of clusters expected to be detectable in large weak lensing
surveys \citep{SW04,Marian:2008fd,Chanta}.
The amount of statistical information in these two regimes appears to
be comparable \citep{Fang:2006dt}; furthermore, the information from
the two approaches is complementary, and using them in combination
significantly improves DE constraints compared to using either method
alone \citep{Fang:2006dt,Takada:2007}.
The chief difficulty with using galaxy clusters identified in WL maps,
however, is that the correspondence between peaks in WL maps and
collapsed clusters is not one--to--one. A significant fraction of WL
peaks are caused by chance alignments of multiple non--linear
structures along the line--of--sight. Conversely, a non--negligible
fraction of real clusters are missed because they do not show a
significant WL peak, due to a chance cancellation of their shear
signal by under-dense regions in the foreground. In principle, the
statistical correspondence between WL peaks and clusters can be
precisely quantified ab--initio in numerical simulations, but in
practice, this remains a challenge to the required level of
accuracy. In particular, the completeness and purity of cluster
selection are of the order of $60-80\%$ for peaks selected to lie
above a $\sim 5\sigma$ convergence threshold, and both become
progressively worse for lower S/N peaks
\citep{whiteprojection,SW04,M&B06, Schirmer:2006ud, Dietrich:2007gi}.
Systematic selection errors are therefore likely to dominate over the
statistical constraints for the majority of a WL--selected cluster
sample.
A natural way to circumvent such selection effects is to study the
statistics of non--linear peaks in WL maps directly. This method,
however, does not lend itself to straightforward mathematical analysis
-- it requires numerical simulations and has been relatively much less
well--explored ~(e.g.~\citep{BJ&VW00,Wang:2008hi}). In this paper, we
study WL peaks in simulated maps, and their usefulness to constrain DE
properties. More specifically, we use a set of N-body simulations,
combined with ray-tracing, to create convergence maps for three
different values of the DE equation of state, $w=p/\rho$. We then
identify peaks in such maps, and tabulate them as a function of their
height. We explicitly avoid a reference to clusters, cluster masses,
or mass functions -- we utilize only the directly observable height of
the maximum of each convergence peak. This method should therefore
provide one of the most robust cosmological probes, free from
potential selection effects and mass measurements biases affecting
cluster identifications.
For increasingly negative values of $w$, DE starts to dominate at
increasingly later cosmic epochs, and one expects an excess of high
peaks in such models, compared to those with less negative $w$. In
this first paper, we perform a feasibility study, to asses whether
changes in $w$ indeed have a measurable effect on the number of
peaks. To this end, all cosmological parameters, other than $w$, are
held fixed at their fiducial values.\footnote{It is important to note
here, however, that we fix the primordial amplitude of the density
perturbations; this results in a $w$--dependent value of the
present--day normalization $\sigma_8$. In the rest of this paper,
whenever we compare cosmologies with two different values of $w$, it
should be remembered, even when not explicitly stated, that these
two cosmologies also have different values of $\sigma_8$.} In a
follow--up paper, we will vary other cosmological parameters, thereby
exploring degeneracies between parameter combinations and obtaining
more realistic, marginalized constraints, on each parameter.
While the cosmological dependence of the cluster counts has been
studied extensively, it is not a priori clear whether the full catalog
of all WL peaks contains a comparable amount of information. A large
number of peaks caused by large-scale structure, with no
$w$--dependence, would constitute pure noise, and could wash out the
theoretically known difference from peaks due to clusters. A strong
$w$-dependence in the number of non--cluster peaks, in the sense
opposite to that of the cluster peaks, could be even more detrimental,
since this would explicitly cancel some of the signal contained in the
cluster counts. Conversely, a strong $w$-dependence in the number of
non--cluster peaks, in the same sense as the cluster counts, could
significantly enhance the sensitivity. This remains a realistic
possibility, since the dependence of the number of peaks caused by
line of sight projections on cosmology has not yet been studied.
An intuitive expectation is that large-scale structure filaments are
expected to contribute more strongly to relatively low--height peaks,
and will be more prominent for source galaxies at higher redshift.
This is because most filaments are likely to line up only partially,
thus creating a weaker signal than those caused by massive collapsed
clusters, and because the probability to find large ($\sim 100$Mpc)
filaments that are lined up along the line of sight will increase
significantly at larger distances. We perform our simulations at high
mass--resolution (down to less than $5\times10^{9}M_\odot$ per
particle), allowing us to resolve relatively low significance peaks in
our convergence maps, where such projections of large scale structure
are more common,. We also place the source galaxies out to redshifts
as high as $z_s=2$.
To keep the computational cost of the simulations reasonable, we
sacrifice the angular size of our simulated field of view, limiting
ourselves to creating WL maps covering $2.86\times2.86$ degrees (for
simplicity, we will hereafter refer to these as $3\times3$ degree
fields).
In these respects, our work differs from the recent related study by
\cite{Marian:2008fd}, which considered the projected mass function out
to a lower redshift (to $z=0.3-0.6$), with simulations at
significantly lower resolution, to demonstrate that the 2D projected
mass function obeys a scaling with $\Omega_m$ and $\sigma_8$ that is
similar to that of the three--dimensional halo mass function
(\cite{Marian:2008fd} also do not vary $w$). Our work shows that the
depth of the survey (at least to $z\sim2$) is vital for distinguishing
between models with different $w$, and that the main distinguishing
power comes from medium--height convergence peaks, with convergence
$\kappa\sim 0.03$ (on maps with 1 arcmin smoothing).
Another recent study, \cite{Wang:2008hi} explored probing cosmology
with a related, simple statistic: the high tail of the one--point
probability distribution of the convergence. The total area above a
fixed convergence threshold includes a sum of the area of all peaks
lying above this threshold, and we therefore expect it to capture less
information than peak counts. Nevertheless, \cite{Wang:2008hi} have
found that this statistic can already place constraints on DE
properties, comparable to the traditional linear two point correlation
function of shear or to the limited use of the nonlinear information
from counting galaxy clusters \citep{Wang:2008hi}. Our paper improves
on this preliminary study, by employing direct simulations, rather
than fitting functions, for the convergence and its dependence on
cosmology, and also by examining a potentially more powerful
statistic.
As our paper was being prepared for submission, we became aware of an
independent preprint, addressing the dependence of WL peaks on the
background cosmology \cite{Dietrich:2009jq}. Their motivation is
essentially identical to our own, and the conclusions reached are
quite similar. However, the two studies are complementary, in that
\cite{Dietrich:2009jq} study the peak dependence in the $(\Omega_m,
\sigma_8)$ plane assuming a $\Lambda$CDM cosmology, while we vary the
dark energy equation of state parameter $w$ (with corresponding
changes in $\sigma_8$). There are additional differences, such as the
specifications in the simulations that were used, and the statistical
analysis methods, which we briefly discuss further in \S~\ref{Other
Constraints} below.
The rest of this paper is organized as follows.
In \S~II, we describe our calculational procedure, including the set--up and analysis of the simulations.
In \S~III, we test our computations by comparing our numerical convergence power spectrum to semi-analytic predictions.
In \S~IV, we present and discuss our main results on the peak counts.
Finally, in \S~V, we summarize our main conclusions and the implications of this work.
\section{Methodology}
\label{Methodology}
This section describes the procedure to create the simulated weak
lensing maps, from generating the matter power spectrum with CAMB and
scaling it to the starting redshift of the N-body simulation, to
generating the initial conditions for the simulation, ray-tracing the
weak lensing maps from the simulation output, as well as
post--processing and analyzing these maps.
\subsection{N-body Simulations}\label{N-body}
For this study, we generated a series of cold dark matter (CDM) N-body
simulations with the code GADGET-2 \cite{Springel:2005mi}, which
include DM only; baryons and neutrinos are both assumed to be absent.
We have modified GADGET-2 for use with arbitrary equation of state
parameter $w$, but in this work, we restrict ourselves to constant
$w$. As the reference model, we chose a $\Lambda$CDM model with
parameters close to the best-fit values from WMAP5, i.e. with
cosmological constant $\Omega_\Lambda=0.74$, matter density parameter
$\Omega_m=0.26$, Hubble constant $h=0.01\times (H_0/{\rm km~s^{-1}
Mpc^{-1}})=0.72$. We fix the amplitude of curvature fluctuations
$\Delta_R^2=2.41\times10^{-9}$ at the pivot scale $k=0.002
\mathrm{Mpc}^{-1}$. This results in different values of the
present--day normalization, $\sigma_8$, for the different dark energy
models. In the three models we consider with $w=-0.8, -1$, and
$-1.2$, the values are $\sigma_8=0.742,0.798$, and $0.839$,
respectively.
For each of the three cosmologies, with all parameters other than $w$
and $\sigma_8$ fixed, we ran two independent N-body simulations, with
two different random realizations of the initial particle positions
and velocities. The simulations used $512^3$ particles, in a box with
a comoving size of $200h^{-1}\mathrm{Mpc}$, starting at redshift
$z=60$. This corresponds to a mass resolution of
$m=4.3\times10^9M_\odot$ per dark matter particle. The
Plummer-equivalent gravitational softening of the simulations was set
to $\epsilon_{Pl}=7.5h^{-1}\mathrm{kpc}$ (comoving). Fixing the
softening length in physical (rather than comoving) units is not
warranted, since for a pure CDM simulation without gas, no further
physical scale enters.
The initial conditions (ICs) for the simulations were generated with
an IC generator kindly provided by Volker Springel. The matter power
spectrum at $z=60$, which is an input for the IC generator, was
created with the Einstein-Boltzmann code CAMB \cite{Lewis:1999bs}, a
modification/extension of CMBFAST \cite{Seljak:1996is}. GADGET-2 does
not follow the evolution of the radiation density, which, starting
with the correct power spectrum at $z=60$, would accumulate to a $\sim
2\%$ error in its normalization by $z=0$. To eliminate this small
bias, we used CAMB to produce the power spectrum at $z=0$ and scaled
it back to $z=60$, with our own growth factor code, with the radiation
component absent. This leads to a small deviation from the true power
spectrum at the starting redshift, but the match will become better at
the lower redshifts which are more important for our analysis.
The output of the simulations consists of particle positions at
various redshifts between $z=0$ and $z=2$. Similar to choices in most
previous works (e.g. \cite{Hamana:2001vz}), the output redshifts were
chosen to span comoving intervals of $70h^{-1}\mathrm{Mpc}$ along the
line of sight in the $\Lambda$CDM cosmology. For simplicity, the same
redshifts were used in the other cosmologies. This results in a small
difference in comoving separation between the planes in these
cosmologies, but this has negligible impact on our results.
Most simulations and all post--processing analysis were run on the
LSST Cluster at the Brookhaven National Laboratory, an 80-CPU Linux
cluster with 80~GB of memory. Some simulations were run on the Intel
64 Cluster Abe, with 9600~CPUs and 14.4~TB of memory at the National
Center for Supercomputing Applications (NCSA), a part of the NSF
TeraGrid facility.
\subsection{Weak Lensing Maps}\label{Maps}
We use our own codes to produce and analyze weak lensing maps from the
N-body simulation output. This subsection describes the procedures and
algorithms, as well as the testing of the new codes.
\subsubsection{Potential Planes}
As mentioned above, the simulation outputs are produced at every $70
h^{-1}$Mpc, which results in an $\approx 2/3$ overlap between
consecutive $200h^{-1}$Mpc cubes. We prepare the N-body simulation
output for ray-tracing by truncating the snapshot cubes along the line
of sight in order to remove this overlap, and then project the
particle density of each cube on a 2D plane $p$ perpendicular to the
line of sight at the center of the plane, located at the same redshift
as the original cube. We use the triangular shaped cloud (TSC) scheme
\cite{Hockney-Eastwood} to convert the individual particle positions
in the simulations to a smoothed density field on the 2D plane. The
projected density contrast field on plane $p$,
$\delta_p^{(2D)}(\vec{x})=\int_{y_{p-1}}^{y_p} dy
[\rho(\vec{x},y)/{\overline\rho}-1]$ is calculated on a 2D lattice of
$2048\times2048$ points (here $\vec{x}$ represents the coordinates in
the plane perpendicular to the line of sight; $y$ is the radial
coordinate, and $y_{p}$ is the position half--way between the $p^{\rm
th}$ and $(p+1)^{\rm th}$ planes). We then use the 2D Poisson equation
\begin{equation}
\nabla^2\Psi^p(\vec{x})=\frac{3\Omega_mH_0^2}{c^2}\delta_p^{(2D)}(\vec{x})
\end{equation}
to calculate the 2D gravitational potential $\Psi^p$ on plane
$p$. This equation is solved easily in Fourier space, in particular
since the density contrast planes have periodic boundary
conditions. The resulting potential plane is then stored in FITS
format \cite{FITS} for further use by our ray-tracing code. This
multi-step approach has proven useful to save memory, and for
re--using intermediate steps of the lensing map construction.
\subsubsection{Ray-Tracing}
To create weak lensing maps from the N-body simulations, we implement
the two-dimensional ray-tracing algorithm described in
\cite{Hamana:2001vz}. We follow $2048\times2048$ light rays through a
stack of potential planes, which, as mentioned above, are spaced
$70h^{-1}\mathrm{Mpc}$ apart in the case of the $\Lambda$CDM
cosmology. Starting at the observer, we follow the light rays
backward to the source galaxies. At each lens plane $n$, we calculate
the distortion tensor and the gravitational lensing deflection from
the equations:
\begin{eqnarray}
\label{distortion tensor}
\mathbf{A}_n&=&\mathbf{I}-\sum_{p=1}^{n-1}\frac{f(\chi_p)f(\chi_n-\chi_p)}{a(\chi_p)f(\chi_n)}\mathbf{U}_p\mathbf{A}_p\\
\label{deflection angle}
\theta^n&=&\theta_1-\sum_{p=1}^{n-1}\frac{f(\chi_n-\chi_p)}{a(\chi_p)f(\chi_n)}\nabla_\bot\Psi^p,
\end{eqnarray}
where $\mathbf{I}$ is the identity matrix, $f(\chi)$ and $a(\chi)$
denote the comoving angular diameter distance and the scale factor,
evaluated at the comoving coordinate distance $\chi$, and the
so--called optical tidal matrix,
\begin{equation}
\mathbf{U}_p=\left(\begin{array}{cc} \Psi^p_{,11} &
\Psi^p_{,12} \\ \Psi^p_{,21} & \Psi^p_{,22} \end{array} \right)
\end{equation}
contains the second spatial derivatives of the gravitational potential
$\Psi$ on plane $p$. The derivatives are to be evaluated at the point
on plane $p$ intersected by the light ray. Starting with
$\mathbf{A}_0=\mathbf{I}$, equation~(\ref{distortion tensor}) can be
solved to find $\mathbf{A}_n$ for any $n$.
The physical weak lensing quantities, the convergence $\kappa$ and the
two components of the shear, $\gamma_1$ and $\gamma_2$, are extracted
from the distortion tensor via
\begin{equation}
\mathbf{A}=\left(\begin{array}{cc} 1-\kappa-\gamma_1 & -\gamma_2-\omega \\ -\gamma_2+\omega & 1-\kappa+\gamma_1 \end{array} \right).
\end{equation}
The observable quantities, the magnification $\mu$ and the distortion
$\underline{\delta}$, are given by
\begin{equation}
\mu=\frac{1}{(1-\kappa)^2-\gamma^2}\hspace{2cm}\underline{\delta}=\frac{2\underline{g}}{1+|\underline{g}|^2},
\end{equation}
where $\underline{g}=\underline{\gamma}/(1-\kappa)$ is the reduced shear. In the
weak lensing limit these relations simplify to $\mu=1+2\kappa$ and
$\underline{\delta}=2\underline{\gamma}$.
For simplicity, for the purpose of this paper, we will work directly
with the maps of the theoretically predicted convergence $\kappa$. In
practice, in the absence of external size or magnification
information, neither the shear, nor the convergence is directly
observable, but only their combination, the reduced shear
$\underline{g}$. One can, in principle, convert one quantity to the
other; in practice, on small scales, where lensing surveys get much of
their constraining power, the errors introduced by this conversion
will not be negligible, and must be taken into account when extracting
cosmological parameters from actual data \cite{Dodelson:2005ir}.
\subsection{Source Galaxies, Smoothing and Noise}\label{Smoothing}
The point--particles in the N--body simulations represent lumps of
matter with a finite spatial distribution. While gravitational
softening is included in GADGET-2 to prevent unrealistically close
encounters, the simulations still follow the assembly of point
particles. As a result, several additional types of smoothing are
necessary to produce weak lensing maps. The TSC scheme, which is used
to place particles on the density contrast planes, generates smooth
two--dimensional density contrast distributions, thus avoiding
unnaturally large deflections when a light ray would otherwise pass
close to an individual particle. In the ray-tracing step, as light
rays pass through arbitrary points on the potential planes, we employ
the same TSC kernel to compute the potential at the ray intersection
points.
Since the orientation of the disks of the galaxies with respect to the
observer is unknown, this intrinsic ellipticity appears as noise in
the maps. Following the interpretation of \cite{VanWaerbeke:1999wv} in
\cite{Fang:2006dt}, we take the redshift-dependent r.m.s. of the noise
in one component of the shear to be \cite{S&K04}
\begin{equation}\label{gamma noise}
\sigma_\gamma(z)=0.15+0.035z
\end{equation}
and perform the calculation analogously to \cite{VanWaerbeke:1999wv}
but with a square top-hat function representing our pixels. The
resulting convergence noise correlation function for our pixelized map
where $\vec{x}$ and $\vec{y}$ are the discrete 2D pixel coordinates
is:
\begin{equation}\label{noise correlation}
\langle \kappa_{\rm noise}(\vec{x})\kappa_{\rm noise}(\vec{y})\rangle=\frac{\sigma_\gamma^2}{n_{gal}A}\delta_{\vec{x}\vec{y}},
\end{equation}
where $\delta_{\vec{x}\vec{y}}$ is the Kronecker symbol, $n_{gal}$ is
the surface density of source galaxies, and $A$ is the solid angle of
a pixel. For simplicity, in our analysis we assume that the source
galaxies are located on one of three source planes, at fixed redshifts
($z_s=1, 1.5, 2$), with $n_{gal}= 15~{\rm arcmin}^{-2}$ on each source
plane. In combination, this would yield a total surface density of
$n_{gal}= 45~{\rm arcmin}^{-2}$. For comparison, we note that this is
larger than the value $\approx 30~{\rm arcmin}^{-2}$ (with a
redshift--distribution that peaks at $z_s\sim 1$), expected in LSST.
However, we do not have sufficient simulation data to perform accurate
tomography, and therefore, our results below utilize only a single
source plane, with a conservative number of $n_{gal}= 15~{\rm
arcmin}^{-2}$, rather than the full $n_{gal}= 45~{\rm arcmin}^{-2}$
(see below). Thus, we add noise to the convergence in each pixel,
drawn from a Gaussian distribution with the variance given in
equation~(\ref{noise correlation}). Note that the noise between
different pixels is uncorrelated.
In order to investigate the information content in peaks defined on
different scales, once noise in each pixel is added, we apply an
additional level of smoothing, with a finite 2D Gaussian kernel, on
scales between $\theta_G = 0.5-10$ arcminutes (see
Sec.~\ref{Comparison} for exact values). The Gaussian kernel has the
nice property of being decomposable, thus in practice, we can
equivalently make two passes through the map with 1D Gaussian kernels,
oriented once in each of the two orthogonal dimensions of the
map. This saves computation time, especially for large smoothing
scales.
The noise in the convergence $\sigma^2_{noise}$ introduced by
intrinsic ellipticity, after the Gaussian smoothing, becomes
\begin{equation}\label{noise}
\sigma_{noise}^2=\frac{\langle\sigma_\gamma^2\rangle}{2\pi\theta_G^2n_{gal}}
\end{equation}
which is the equivalent of (\ref{noise correlation}), but with a
Gaussian smoothing instead of a top-hat function
\cite{VanWaerbeke:1999wv}, and depends on the redshift distribution of
the source galaxies, as well as on the smoothing scale $\theta_G$
applied to the maps. In general, the noise level should be computed as
the weighted average of the redshift--dependent shear noise
(\ref{gamma noise}) over the redshift distribution of the source
galaxies:
\begin{equation}
\langle\sigma_\gamma^2\rangle=\frac{1}{n_{gal}}\int_0^\infty\sigma_\gamma^2(z)\frac{dn}{dz}dz.
\end{equation}
Since in our case, for simplicity, we assume that the galaxies are
located on three discrete source planes, $dn/dz$ is simply a delta
function, at $z_s=1, 1.5$, or at $z_s=2$ (or a sum of delta functions,
for the purpose of redshift tomography).
\vspace{\baselineskip}
\subsection{Statistical Comparisons of Convergence Maps}\label{Comparison}
The main goal of this paper is to discern between different
cosmologies by counting the convergence peaks. Since the maps have
been smoothed, as mentioned above, it is sufficient to identify peaks
in the maps as pixels which correspond to local maxima, in the sense
that all eight of their neighboring pixels have a lower value. For
illustration, we show an example of the convergence maps generated in
two of our cosmologies, before and after the addition of noise and
smoothing, in Figure~\ref{fig:maps}. The panels with $w=-1$ exhibit
visibly more structure at high convergence, consistent with the
expectation that DE starts to dominate later in this model, allowing
more time for growth.
\begin{widetext}
\begin{figure}[htp]
\centering
\includegraphics[width=8 cm]{Conv_w10_original.jpg} \includegraphics[width=8 cm]{Conv_w08_original.jpg}\\
\includegraphics[width=8 cm]{Conv_w10_noise-smooth.jpg} \includegraphics[width=8 cm]{Conv_w08_noise-smooth.jpg}
\hfill
\caption[]{\textit{Examples of $3\times3$ degree convergence maps from
our $512^3$ particle N-body simulations in two different
cosmologies. The left column shows maps for $w=-1$, the right column
for $w=-0.8$. The top row shows the raw maps, with no smoothing or
noise; the bottom row shows the maps after the addition of random
ellipticity noise and 1 arcminute smoothing. The panels with $w=-1$
exhibit visibly more structure at high convergence (yellow and red in
the bottom panels); voids (regions of low convergence, shown in blue)
also appear more numerous in this cosmology.}}\label{fig:maps}
\end{figure}
\end{widetext}
Once the peaks are identified, their heights are recorded in a
histogram. This histogram is divided into a reasonable number of bins
(the choice in the number of bins will be discussed below), and a
vector of values is formed from these bins. We will also attempt to
utilize the source galaxy redshifts for tomography, and to combine
different smoothing scales, so that each peak is further associated
with a value of $z_s$ and $\theta_G$. Finally, peak identification is
repeated in each realization of a given cosmology. We therefore
denote the number of peaks in bin $i$ with
$N_{i,z_s,\theta_G}^{(f;r)}$, where $f$ and $r$ indicate cosmology and
realization, respectively, $z_s$ indicates the redshift of the sources
and $\theta_G$ is the smoothing scale applied to the final map. To
simplify notation we hereafter drop the last two lower indices and
merge them into the single index $i$.
In order to assess the statistical significance of any difference in
peak counts, we construct the covariance matrix $C^{(f)}$ for
cosmology $f$ of the counts in the different height-, redshift--, and
smoothing--scale bins $i$ and $j$,
\begin{equation}\label{Covariance Matrix}
C_{ij}^{(f)}=\frac{1}{R-1}\sum_{r=1}^R (N_i^{(f;r)}-\overline N_i^{(f)})(N_j^{(f;r)}-\overline N_j^{(f)}),
\end{equation}
where $\overline N_i^{(f)}=R^{-1}\sum_{r=1}^R {N}_i^{(f;r)}$ is the
average number of peaks in bin $i$ in the fiducial model $f$, and the
summation in equation~(\ref{Covariance Matrix}) is over the total
number of realizations $R$ constructed for every cosmology.
We use 200 such realizations. However, as noted above, we have run
only two strictly independent realizations of each cosmology, i.e.,
two different simulations with different realizations of the initial
conditions. Thus we produce additional pseudo-independent
realizations by the standard procedure of randomly rotating, shifting,
and slicing the data in the simulation output cubes at each redshift,
prior to generating the 2D density planes. Our box--size,
$200h^{-1}$Mpc is much larger than the few comoving Mpc correlation
length of large--scale structures. As mentioned above, the data cubes
are truncated at each output, so that we utilize only a radial slice
covering about 1/3rd of its length; the $2.86\times2.86$ degree field
furthermore utilizes only a fraction of this slice. As a result,
at intermediate redshifts,
where the lensing kernel peaks, each realization utilizes about a
twelfth of the volume of the snapshot cube, allowing us to access, in each N--body run,
effectively $\sim 12$ truly independent realizations of
each lens plane. At the very highest redshift, the
situation becomes worse, with up to a third of the volume of the data
cubes used. However, due to the distance--dependence of the lensing
kernel, the planes near $z_s=2$, with these heavily sampled boxes, do
not contribute much to the final lensing signal. The random rotations
and shifts will create many more than 12 essentially independent
random realizations of the incidental line--ups of multiple small
clusters and filaments. On the other hand, the very largest clusters
-- which may create a substantial lensing peak by themselves -- will
tend to reappear in multiple realizations, and could lead to an
underestimation of the variance. Fortunately, our main results come
from medium--height peaks, which typically are not caused by a single
cluster, so that any such underestimation at the high--convergence end
contributes relatively little to our final distinguishing power.
We use the (inverse of) the covariance matrix to compute a value of
$\chi^2_{f^\prime}(r)$ for each realization $r$ of a ``test''
cosmology, labeled by $f^\prime$, against the expectations in the
fiducial cosmology, labeled by $f$,
\begin{eqnarray}\label{chi2}
\chi^2_{f^\prime}(r)&=&\mathbf{dN}^{(f^\prime;r)}(C^{(f)})^{-1}\mathbf{dN}^{(f^\prime;r)}\\&=&\sum_{ij}dN_i^{(f^\prime;r)}(C^{(f)})_{ij}^{-1}dN_j^{(f^\prime;r)},
\end{eqnarray}
where $dN_i^{(f^\prime;r)}\equiv N_i^{(f^\prime;r)}-\overline N_i^{(f)}$ is
the difference between the number of peaks in bin $i$ in realization
$r$ of the test cosmology, and the average number of peaks in the same
bin in the fiducial cosmology.
The resulting probability distributions for $\chi^2_{f^\prime}(r)$,
using 200 realizations of each cosmology, fixed source redshift
$z_s=2$ and smoothing scale $\theta_G=1'$, and 8 peak height bins, is
shown in Figure~\ref{fig:chi2}. In the top left panel, the
realizations were drawn from the simulation of the fiducial $w=-1$
model itself, which was used to compute the mean number of peaks
$\overline N_i^{(f)}$. Our simulated $\chi^2$ distribution is very
close to a genuine $\chi^2$ distribution with $n=8$ degrees of freedom
(shown by the black curve). In the bottom left panel, the realizations
were again drawn from the $w=-1$ cosmological model, but from the
simulation with an independent realization of the initial conditions.
The good agreement with the top left panel tests the accuracy of our
covariance matrix (in particular, if the covariance had been
underestimated, then this second realization would have produced
larger $\chi^2$ values; see Appendix~\ref{Initial Conditions}). The
two right panels show $\chi^2$ distributions computed in the
realizations of different cosmologies, with $w=-0.8$ (top right
panel), and $w=-1.2$ (bottom right panel). The $w=-0.8$ and
$\Lambda$CDM cosmologies are distinguished at the 68\% (95\%)
confidence level in 84\% (39\%) of the realizations; the $w=-1.2$
cosmology is distinguishable from $\Lambda$CDM at the same confidence
in 61\% (24\%) of the realizations.
For reference, the above confidence levels can be compared to a
simpler estimate of the statistical distinction between the two pairs
cosmologies, based on the value of $\Delta\chi^2_{f^\prime,f}$,
computed using the mean number of peaks in each bin (averaged over all
realizations),
\begin{eqnarray}\label{meanchi2}
\chi^2_{f^\prime,f}&=&\mathbf{dN}^{(f^\prime,f)}(C^{(f)})^{-1}\mathbf{dN}^{(f^\prime,f)}\\&=&\sum_{ij}dN_i^{(f^\prime,f)}(C^{(f)})_{ij}^{-1}dN_j^{(f^\prime,f)},
\end{eqnarray}
where $dN_i^{(f^\prime,f)}\equiv \overline N_i^{(f^\prime)}-\overline
N_i^{(f)}$ is the difference between the average number of peaks in
bin $i$ in cosmology $f^\prime$ and in cosmology $f$. We find
$\Delta\chi^2=8.35$ between $w=-0.8$ and $w=-1$ (when then same
realization of the initial conditions are used in both cosmologies)
and $\Delta\chi^2=9.54$ (when different realizations are used). The
covariance matrix is calculated from the $w=-1$ cosmology in both
cases. Assuming that the likelihood distributions are Gaussians,
these values would correspond to a $2.8\sigma$ and $3\sigma$
difference between the two cosmologies. This is in good agreement
with the 95\% exclusion of the mean of the $\chi^2$ distribution we
find from the individual realizations in the $w=-0.8$ model --
indicating that interpreting the $\Delta\chi^2$ directly as a
probability would only slightly overestimate the statistical
difference between the two cosmologies. The corresponding value
between the $w=-1.2$ and $w=-1$ models is $\Delta\chi2=4.65$ (with the
same realization of the initial conditions), again confirming that the
true likelihood is only somewhat weaker than this simple estimate for
the exclusion of the $w=-1.2$ model.
\begin{widetext}
\begin{figure}[htp]
\centering
\includegraphics[width=8 cm]{P_w10-self.pdf} \includegraphics[width=8 cm]{P_w08.pdf}\\
\includegraphics[width=8 cm]{P_w10-cross.pdf} \includegraphics[width=8 cm]{P_w12.pdf}
\hfill
\caption[]{\textit{Normalized probability distribution of $\chi^2$,
computed by comparing the peak counts in individual $3\times3$ degree
convergence fields to the ensemble average in the fiducial $w=-1$
model. Peaks were divided into 8 bins by height. The source galaxies
were placed at redshift $z_s=2$, and a Gaussian smoothing of
$\theta_G=1$ arcmin was applied after the addition of ellipticity
noise. In the top left panel, the realizations were drawn from the
$w=-1$ simulation, the average of which was used to construct the
fiducial model. The black curve shows a genuine $\chi^2$ distribution
with $n=8$ degrees of freedom. In the bottom left panel, the
realizations were again drawn from a $w=-1$ model, but from a
simulation with different random initial conditions, illustrating that
these have a negligible effect on the result for most of the
distribution, and testing the accuracy of our covariance matrix. The
two right panels show $\chi^2$ distributions in different cosmologies,
with $w=-0.8$ (top right panel), and $w=-1.2$ (bottom right
panel). The $w=-0.8$ and $\Lambda$CDM cosmologies are distinguished at
the 68\% (95\%) confidence level in 84\% (39\%) of the realizations;
the $w=-1.2$ cosmology is distinguishable from $\Lambda$CDM at the
same confidence in 61\% (24\%) of the realizations.}}\label{fig:chi2}
\end{figure}
\end{widetext}
In the above analysis, we have chosen convergence bins whose width was
varied, so that they contain approximately equal number of counts
($\approx 350$ peaks/bin at 1 arcmin smoothing). More specifically,
the bin boundaries are located at
$\kappa= -0.084, 0.0022, 0.015, 0.024, 0.032, 0.041, 0.052, 0.068,
0.41$. While equal--width bins would perhaps be a more natural
choice, the number of peaks falls off exponentially with peak height,
and this would have the undesirable result of having many almost empty
bins, leading to large $\chi^2$ values excessively often (overestimate
of improbability of unlikely events), and strong deviations from a
true $\chi^2$ distribution (since the peak counts do not have a
Gaussian distribution, especially for bins with low average counts).
\section{Weak Lensing Power Spectrum}
\label{Testing}
Because our pipeline to produce the WL maps is newly assembled,
GADGET-2 was modified to allow $w\neq -1$, and the ray-tracing and WL
map analysis codes were newly written, we performed a consistency
check, by comparing the power spectrum of the convergence in our maps
with the power spectrum derived from semi--analytical
predictions. This comparison also determines the range of scales which
is adequately represented by our simulations due to resolution and box
size. In Figure~\ref{fig:Spectra}, we show the power spectrum of
convergence, averaged over 200 realizations of our simulations (solid
curves), together with the theoretical expectations (dashed curves).
The latter were obtained by direct line--of--sight integration using
the Limber approximation \cite{Limber}, and using fitting formulas for
the nonlinear 3D matter power spectrum from \cite{Smith}. From top to
bottom, the three sets of curves correspond to $w=-1.2$ (green
curves), $w=-1$ (red curves), and $w=-0.8$ (blue curves).
We find significant deviations from the theoretical power spectra on
both small and large scales, as expected. On small scales, with $\ell
\mathrel{\mathpalette\fun >} 20,000$, we are missing power because of the finite resolution on
the mass planes and on the ray-tracing grid. On large scales, with
$\ell \mathrel{\mathpalette\fun <} 400$, the spectrum is suppressed because of the finite size
of the simulation box. These two wave numbers thus set the range,
from $\sim1$ arcmin to $\sim 1$ degree, on which the absolute value of
the converge power spectrum is accurately captured.
While we can not trust our results beyond these limits in the
subsequent analysis, it should be noted that the {\em ratio} of the
power spectra in different cosmologies is preserved accurately, even
where the simulated power spectra start to deviate from the
theoretical predictions. Since we are mainly interested in the
comparison between the cosmologies, this makes our results even more
robust.
\begin{figure}[htp]
\centering
\includegraphics[width=8 cm]{kappa_power.pdf}
\hfill
\caption[]{\textit{Convergence power spectrum $P(\ell)$, as a function
of $\ell=2\pi/\theta$, derived in the three cosmologies (solid blue:
$w=-0.8$, solid red: $w=-1$, solid green: $w=-1.2$) from the raw
un-smoothed $3\times3$ degree maps, without ellipticity noise. The
simulated power spectra have been averaged over 200 realizations in
each cosmology. Also shown are the corresponding semi--analytical
predictions based on the matter power spectrum of \cite{Smith} and
using the Limber approximation \cite{Limber} (dashed curves). The
drop around $\ell\sim20,000$, which corresponds to $\sim 1$ arcmin, is
due to the finite resolution of the simulations. The power is also
suppressed on large scales, because of finite box size. Note that the
ratios of power spectra in the three cosmologies are accurate in the
simulations over a wider range than the absolute
power.}}\label{fig:Spectra}
\end{figure}
Another consistency check we perform is a comparison between the
number of peaks above a certain threshold to the number of clusters
creating lensing peaks above that same threshold. This comparison will
presented in Sec.~\ref{Peak Numbers} below.
\section{Results and Discussion}
\label{Results}
We have found several important quantitative and qualitative results
in our study. We first discuss the difference in peak counts between
the cosmologies -- in the total number of peaks, as well as in the
number of peaks in individual convergence bins. We then present our
inferred sensitivity to distinguish these cosmologies, and extrapolate
this sensitivity from our $3\times3$ degree field to an LSST--like
survey. We discuss the validity of such an extrapolation by studying
the variance in peak counts as a function of angular scale. In the
next subsection, we comment on correlations between peaks in different
bins, with source galaxies at different redshifts, and with different
angular smoothing scales, as well as cross-correlations among
these. Finally, we briefly contrast our results to other predictions
for an LSST--like WL survey, namely constraints from the weak lensing
shear power spectrum, the PDF of $\kappa$ found in \cite{Wang:2008hi},
as well as cluster number counts.
\subsection{Peak Counts}\label{Peak Numbers}
The fundamental quantity we are interested in is the number of peaks
in different convergence bins. This quantity is shown in
Figure~\ref{fig:N_Peak} in the three different cosmologies, derived
from convergence maps with a single source galaxy plane at $z_s=2$,
after the addition of ellipticity noise and Gaussian smoothing with
$\theta_G=1$ arcmin. From top to bottom (at the peak of the
distribution), the three curves correspond to $w=-0.8$ (blue curves),
$w=-1$ (red curves), and $w=-1.2$ (green curves).
\begin{figure}[htp]
\centering
\includegraphics[width=8 cm]{N_peak.pdf}\\ \ \\
\hfill
\caption[]{\textit{Number of convergence peaks in intervals [$\kappa$,
$\kappa+\Delta\kappa$], with $\Delta\kappa=0.0161$ in $3\times3$
degree fields with source galaxy redshift $z_s=2$, 1 arcmin smoothing,
and intrinsic ellipticity noise added. From top to bottom (at the peak
of the distribution), the three curves correspond to $w=-0.8$ (blue),
$w=-1$ (red), and $w=-1.2$ (green). The total number of peaks is 2477,
2432, and 2401, respectively. Notice that the lower distributions have
wider wings. }}\label{fig:N_Peak}
\end{figure}
As this figure demonstrates, the distribution of peak heights becomes
narrower with increasing $w$.
The total number of convergence peaks in the noisy, smoothed maps is
$N_{total}=2432\pm30$, $N_{total}=2477\pm30$, and $N_{total}=2401\pm
34$ for $w=-1$, $w=-0.8$, and $w=-1.2$, respectively. Formally, this
is a $1.5\sigma$ difference for $w=-0.8$ and a somewhat weaker
difference of $1\sigma$ for $w=-1.2$. The same numbers for the
smoothed noiseless maps (still with 1 arcmin smoothing) are
$N_{total}=1646\pm 24$, $N_{total}=1654\pm 28$, and $N_{total}=1644\pm
24$, respectively. Interestingly, the addition of ellipticity noise
boosts both the total number of peaks (by $\sim 50\%$; note that a
similar increase was observed by \cite{Hamana:2003ts}), and the
significance in the difference in peak counts between pairs of
cosmologies. Apparently, the number of new peaks introduced by the
presence of noise is, itself, $w$--dependent, in a way that helps in
distinguishing the two cosmologies.
While this result may first sound surprising, the addition of noise is
known boost the discriminating power of related observables. For
example, scatter in the mass-observable relation generally increases
the number of detectable clusters and tightens constraints from
cluster number counts (e.g \cite{H&M&H01}). In our case, one can image
that the number of new peaks on 1 arcmin scales, at a given peak
height, introduced by the presence of noise, depends on the amplitude
of the $\kappa$--fluctuations on larger angular scales, which, itself,
is cosmology--dependent. The precise origin of this result will be
investigated in a follow-up paper.
Most of the $\sim 2,500$ peaks, of course, have a low amplitude. In
order to compare our results with previous work, it is of interest to
consider the number of peaks at higher significance. As an example,
above the commonly used threshold of $\kappa_G=4.5\sigma_{noise}$ (for
a source galaxy density of $n_{gal}=15$ arcmin$^{-2}$ at redshift
$z_s=2$, the value of this threshold corresponds to $\kappa_G=0.102$
[eq.~\ref{noise}]), the average number of peaks we identify in our 200
pseudo-independent realizations is
$N_{\mathrm{peaks}>\kappa_G}=47.4\pm11.8$,
$N_{\mathrm{peaks}>\kappa_G}=36.6\pm10.4$, and
$N_{\mathrm{peaks}>\kappa_G}=54.5\pm 11.4$ for $w=-1$, $w=-0.8$ and
$w=-1.2$ respectively, which represent $\approx 1\sigma$ differences
between the $w=-1$ and the $w\neq 1$ models. These numbers are with
ellipticity noise included. Without ellipticity noise, the number of
peaks above the same threshold is
$N_{\mathrm{peaks}>\kappa_G}=25.9\pm7.3$,
$N_{\mathrm{peaks}>\kappa_G}=14.8\pm5.2$, and
$N_{\mathrm{peaks}>\kappa_G}=37.0\pm9.5$, which, again, represent a
$1\sigma$ differences. Although the number of these rare
high--$\sigma$ peaks is also boosted by the presence of noise,
apparently this boost does not enhance the distinguishing power
between cosmologies.
It is also useful to compare the number of peaks to the estimated
number of clusters causing a convergence signal above the same
threshold $\kappa_G$. To be closer to predictions in earlier work, for
this comparison, we take $n_{gal}=30$ arcmin$^{-2}$, resulting in a
threshold of $\kappa_G=0.072$ for 1~arcmin smoothing. In our noiseless
maps, we obtain $N_{\mathrm{peaks}>\kappa_G}=99.7\pm20.0$ and
$N_{\mathrm{peaks}>\kappa_G}=64.6\pm14.3$ for $w=-1$ and $w=-0.8$
respectively. To find the corresponding number of clusters, we repeat
the calculation in \cite{Fang:2006dt}, which is based on the fitting
formula for the DM halo mass function from \cite{Jenkins}, and assumes
clusters follow the Navarro-Frenk-White profile \cite{NFW}, but we
adjust the cosmological parameters (including the $w$--dependent value
of $\sigma_8$) to the values adopted here. We find the number of
clusters is $N_{\mathrm{clusters}>\kappa_G}=48.8\pm13$, and
$N_{\mathrm{clusters}>\kappa_G}=31.7\pm11$ for $w=-1$ and $w=-0.8$,
respectively (the errors bars here are approximately twice the
$\sqrt{N}$ Poisson errors, enhanced by cosmic variance, following
\cite{HuKravtsov}). Apparently, there are approximately twice as many
convergence peaks in the maps without ellipticity noise as there are
clusters, indicating a substantial number of projections. This is
roughly consistent with the results of \citep{whiteprojection,
Hamana:2003ts,Hennawi:2004ai}, which self--consistently identify halos
in their simulations to quantify this correspondence more reliably,
although those works find a somewhat closer correspondence between
clusters and $4.5\sigma$ peaks. Adding ellipticity noise increases the
number of peaks by another factor of $\sim$two, resulting in plenty of
convergence peaks that are not due to a single cluster, which can
carry additional information about the cosmology.
An important point to note here is that, as we emphasized earlier, we
keep the primordial amplitude fixed as we vary $w$, which leads to a
different $\sigma_8$ for each $w$. The difference in the average
number of clusters (48.8 versus 31.7) quoted above arises essentially
entirely from this difference in $\sigma_8$. We expect that the
difference in peaks counts will similarly be driven primarily by
$\sigma_8$. Tomography should break some of the degeneracy between
$\sigma_8$ and $w$; this is an issue that will be clarified and
explored in our follow-up work.
\subsection{Three Peak Types}\label{peak types}
It is instructive to next take a more detailed look at how the number
of convergence peaks depends on the peak height, and, in particular,
how the counts in different convergence bins vary with cosmology. In
the top panel of Figure~\ref{fig:Delta_Peak}, we show the difference
in peak numbers (averaged over the 200 realizations) in $3\times 3$
degree fields in convergence intervals $[\kappa,\kappa+\Delta\kappa]$,
with $\Delta\kappa=0.0161$ between the $w=-0.8$ (blue) and the
$\Lambda$CDM models, as well as $\Lambda$CDM and $w=-1.2$ (green).
The middle panel in the same figure shows the standard deviation in
each bin, and the bottom panel depicts the contribution to the
$\chi^2$ and thus to the distinction power, derived from each bin
(note that this bottom panel shows the unequal $\kappa$ bins that were
used in our $\chi^2$ calculation).
\begin{figure}[htp]
\centering
\includegraphics[width=8 cm]{Delta_peaks.pdf}\\ \ \\
\includegraphics[width=8 cm]{StdDev_of_k.pdf}\\ \ \\
\includegraphics[width=8 cm]{Chi2_of_k.pdf}
\hfill
\caption[]{\textit{{\bf Top Panel}: the difference in average peak
numbers in each convergence bin is plotted as a function of
convergence $\kappa$. The $\Lambda$CDM cosmology is subtracted from
$w=-0.8$ (blue) and $w=-1.2$ is subtracted from $\Lambda$CDM (green)
in this plot. The height distribution of convergence peaks in
$\Lambda$CDM is broader, while in $w=-0.8$ it is narrower, and more
sharply peaked (and analogously for $w=-1.2$ vs.\ $\Lambda$CDM). The
numbers plotted are averaged over 200 realizations of a $3\times3$
degree field, with source galaxies at $z_s=2$, 1-arcminute smoothing
and the peak heights divided into 50 equal--width convergence bins.
{\bf Middle Panel}: Same as the upper panel, but showing the standard
deviation in the number of peaks each bin.
{\bf Bottom Panel}: Contribution to the total $\langle\chi\rangle^2$
from the peak number differences in each of the eight convergence bins
used in our $\chi^2$ calculation. Note that the bins have unequal
width, chosen such that each bin contains approximately the same
number of peaks (see text). Note that this panel neglects correlations
between bins, in order to illustrate the $\kappa$-dependence. There is
a small contribution from low peaks, a strong one from medium peaks,
and a sizable one from high peaks. }}\label{fig:Delta_Peak}
\end{figure}
Based on the results shown in this figure, we identify three
categories of peaks.
\begin{itemize}
\item \textbf{High Peaks ($\kappa\gsim0.06$):} As is known from
previous work \citep{whiteprojection, Hamana:2003ts,Hennawi:2004ai}, a
significant fraction (and perhaps the majority) of these peaks are
caused by individual collapsed clusters, which can cause strong peaks
by themselves. These peaks are more numerous in cosmologies with more
negative $w$. This is as expected: as mentioned above, the more
negative $w$ is, the later dark energy starts to dominate the energy
density, and the more time massive clusters had to form.
\item \textbf{Medium Peaks ($0.0~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}} \kappa\lsim0.06$):} The number
of these peaks significantly exceeds the expected number of clusters
that would cause, by themselves, peaks with a similar height. We
examined the contributions of individual lens planes along the line of
sight (LOS) to the total convergence in these peaks. We found that for
the majority of these peaks, multiple planes contribute significantly,
and we therefore suspect these peaks are typically caused by
projection along the LOS. In contrast with the high peaks, these
medium peaks are more numerous in cosmologies with less negative
$w$. This was unexpected. Furthermore, we find that the difference in
the counts of these medium peaks dominates the cosmological
sensitivity. Clarifying the physical nature of these peaks is
important, and will be presented in a follow-up paper
\cite{Yang:2009}.
\item \textbf{Low Peaks ($\kappa\lsim0.0$):} Similarly to the high
peaks, these peaks are more numerous in cosmologies with more negative
$w$. However, their number difference compared to their variance is
too low to make an appreciable contribution to the distinction between
the cosmologies (the number difference of the low peaks is only a
fifth of that of the medium peaks, yet their variance still about
half). We speculate that these peaks typically reside in
``convergence voids'' (large areas of the map with very low values of
convergence). We have indeed verified that the maps in cosmologies
with more negative $w$ show a larger fraction in voids, which gives
very low--amplitude peaks a larger area to reside in.
\end{itemize}
\subsection{Distinguishing $w\neq -1$ Models from $\Lambda$CDM}\label{Discerning}
We use the above difference in peak numbers in bins with different
values of $\kappa$ to quantify the statistical distinction between the
three models with $w=-1,-0.8$, and $-1.2$. We find that the number of
peak height bins used has a minor effect, as long as their number is
kept between 4--9. Below this value, the binning is too coarse to
capture all the information of the rich structure in the top panel of
Figure~\ref{fig:Delta_Peak}, while higher numbers of bins show signs
of over-binning, especially when several source redshifts and
smoothing scales are combined (see Appendix \ref{Initial Conditions}).
As our fiducial set--up, we use a single source plane at $z_s=2$ and
$\theta_G=1$~arcmin smoothing, as in the previous section. Redshift
tomography and using different smoothing scales are addressed in
Sec.~\ref{Smoothing Scales} below. The $\chi^2$ for each of our
$3\times3$ degree convergence fields (with ellipticity noise and
smoothing), compared to the reference cosmology $\Lambda$CDM, is
calculated as described in \S~\ref{Comparison}. This leads to the
probability distribution of the $\chi^2$ shown in
Figure~\ref{fig:chi2}.
We can derive our statistical conclusions from these $\chi^2$
distributions. For example, the mean of the $\chi^2$ distribution for
$w=-0.8$ lies at the 95\% tail of the $\chi^2$ distribution for
$w=-1$. This suggests that the $2\sigma$ distinction from a single
$3\times3$ degree field corresponds to $\Delta w=0.2$. The
corresponding result for the case of $w=-1.2$ is an exclusion at the
85\% confidence level.
\subsection{Extrapolation to an All--Sky Survey}
While we are limited by computational costs to studying single
$3\times3$ degree fields, the forthcoming surveys listed in
\S~\ref{Introduction} will cover much larger areas of the sky - up to
$\approx$ 20,000 degree$^2$. Therefore, first we would like to
extrapolate our results to a larger field of view -- or equivalently,
translate it to the tighter $2\sigma$ sensitivity to $w$ expected from
larger fields.
We will use two methods of extrapolation, an optimistic and a
pessimistic one. In the optimistic case, we neglect large-scale
correlations and assume that, with increasing survey size, the
variance per area--squared in numbers of peaks just falls off as the
number of peaks, i.e.\ $\sigma^2_{N_{peak}}\propto N_{peak}$. In the
pessimistic case, we extrapolate the scaling of the variance with
survey size from the scaling behavior in subfields of our $3\times3$
degree weak lensing maps. We note that correlations on large scales
generally fall off faster than the Poisson--like extrapolation would
suggest \citep{Wang:2008hi}, and in reality, both cases may therefore
be conservative estimates.
We illustrate these two methods of extrapolations in
Figure~\ref{fig:peak variance}, where we plot the variance in peak
numbers for progressively larger fields, starting with a
$1.4\times1.4$ arcminute field and going all the way to $3\times3$
degrees (always obtained from our 200 pseudo-independent
realizations).
\begin{figure}[htp]
\centering
\includegraphics[width=8 cm]{peak_variance.pdf}
\hfill
\caption[]{\textit{Variance (per area squared) in the total number of
convergence peaks, as function of survey area. The red dots denote the
variance as measured directly from our $3\times3$ degree convergence
maps and subfields therein, for the $w=-1$ cosmology, with sources at
redshift $z_s=2$ and 1 arcmin smoothing. The green line shows a
power--law fit to the four largest subfields and an extrapolation to a
20,000 degree$^2$ field. The black line depicts the slightly more
optimistic Poisson--like extrapolation
($\sigma^2_{N_{peak}}=N_{peak}$), which ignores long--range
correlations. The difference between these two cases causes a factor
of 1.3 difference in the extrapolated sensitivity to $w$ on large
scales.}}\label{fig:peak variance}
\end{figure}
The optimistic extrapolation, shown by the black line in
Figure~\ref{fig:peak variance}, scales to a
$\sqrt{20,000/2.86^2}=50\times$ stronger sensitivity for a 20,000
degree$^2$ field of view. The difference in the dark energy equation
of state parameters in our study is $\Delta w=0.2$, which we discern
at $2\sigma$. Therefore, roughly, a 20,000 degree$^2$ survey will be
able to discern a difference of $\Delta w=0.2/50=0.004$ at the same
$2\sigma$ level. The somewhat more pessimistic case of extrapolating
from existing subfields, as depicted by the green line in
Figure~\ref{fig:peak variance}, yields a variance that is larger by a
factor of 1.3. Therefore, for this case, our estimated sensitivity $w$
from an LSST size survey is $\Delta w=0.0052$.
While this is an interesting level of sensitivity, one has to keep in
mind that this is a single--parameter constraint, and is not directly
comparable to marginalized constraints usually quoted in the
literature which fold in degeneracies with other parameters (see
\S~\ref{Other Constraints} below).
\subsection{Different Redshifts and Smoothing Scales}\label{Smoothing Scales}
In our fiducial set--up so far, we have used a single source plane
($z_s=2$) and smoothing scale ($\theta_G=1$~arcmin). Here we examine
using other source planes and smoothing scales, including the
possibility of using multiple scales in combination.
\subsubsection{Single Redshifts and Smoothing Scale Combinations}
We have re--computed $\chi^2$ distributions, following the procedure
described above, for a variety of different smoothing scales
($\theta_G=1, 2, 3, 4, 5, 10$ arcmin) and source redshifts ($z_s=1,
1.5, 2$). We quote the results by indicating, in each case, the
fraction of the realizations of the $w=-0.8$ model which can be
excluded from the $w=-1$ cosmology at the 68\% (95\%) confidence
level. These fractions are to be compared to the values of 32\% and
5\%, expected in the absence of any signal (i.e., for the same
cosmology, compared to itself). These fractions are presented in
Table~1. We list $\theta_G=0.5'$ in the table as well, but we caution
that, as discussed above, our simulations do not have sufficient
resolution to reproduce the convergence power spectrum on these
scales, and therefore these results may not be reliable. Unless noted
otherwise, we exclude these values from the discussion in the rest of
this paper.
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\% exclusion chance & & & \\
@ 68 (95)\% CL & $z_s=1$ & $z_s=1.5$ & $z_s=2$\\
\hline
$\theta_G=0.5'$ & 48 (13) \% & 76 (32) \% & 90 (46) \% \\
$\theta_G=1'$ & 56 (16) \% & 68 (18) \% & 84 (34) \% \\
$\theta_G=2'$ & 53 (12) \% & 66 (13) \% & 50 (18) \% \\
$\theta_G=3'$ & 41 (10) \% & 48 (14) \% & 40 (6) \% \\
$\theta_G=4'$ & 39 (6) \% & 32 (7) \% & 37 (12) \% \\
$\theta_G=5'$ & 38 (6) \% & 28 (8) \% & 39 (10) \% \\
$\theta_G=10'$ & 26 (2) \% & 28 (5) \% & 25 (4) \% \\
\hline
\end{tabular}
\end{center}
Table 1: \textit{The probability of distinguishing the $w=-0.8$ model
from $\Lambda$CDM, based on peak counts in a single $3\times3$ degree
field, at the 68\% (95\%) confidence level for various source
redshifts $z_s$ and smoothing scales $\theta_G$, using 8 peak height
bins.}
\vspace{0.4cm}
There are several trends worth noting in Table~1. First, using source
galaxies at high redshift ($z_s=2$) provides substantial advantage
over using sources at redshift $z_s=1.5$ or even $z_s=1$ (recall that
the surface density is fixed in each case to be 15 arcmin$^{-2}$).
Dark energy constraints from Type Ia supernovae, and from most other
techniques also benefit from deep surveys and high redshifts. For weak
lensing, going to higher redshifts is particularly important, as the
best lensing efficiency comes from approximately halfway in distance
between sources and observer.
Second, as is well known, weak lensing derives most of its power from
small scales, and indeed, the smaller the smoothing scale in our maps,
the stronger the distinction. This is true down to the smallest
angular scales that we can reliable simulate, $\theta_G\approx 1$.
Smoothing scales larger than $\theta_G=2'$ do not provide very strong
constraints by themselves, but may still be useful when combined with
smaller ones (especially for real surveys with much larger fields of
view -- the number of peaks at these large scales becomes very small
for our small $3\times3$ degree fields).
\subsubsection{Correlations}
There are several types of correlations that would be useful to
investigate: i.e., correlations in the number of peaks in different
height bins, identified for different source redshifts, and with
different smoothing scales, as well as the cross-correlations between
these quantities. In order to study the effect of these correlations,
we computed the $\chi^2$ distributions from covariance matrices in
which only diagonal elements, or diagonal blocks were retained, and
the cross-terms outside these were set to zero.
We find that the peak height bins are correlated. Neglecting all
correlations, by retaining only the diagonal elements of the
covariance matrix, lead to $\chi^2$ histograms that resemble $\chi^2$
distributions with fewer degrees of freedom, and an excessive amount
of outliers, both typical signs that the dataset contains more
correlations than is assumed in the analysis. In particular, as
expected, the $\chi^2$ distributions (analogous to those shown in
Figure~\ref{fig:chi2}) develop wider tails toward high $\chi^2$
values, with their maximum moving to somewhat smaller values. The net
result is that the overlap of the $\chi^2$ distributions between a
pair of cosmologies is enhanced, and the distinction between the
cosmologies is degraded. Table~2 illustrates this in numbers.
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\% exclusion chance & & & \\
@ 68 (95)\% CL & $z_s=1$ & $z_s=1.5$ & $z_s=2$\\
\hline
$\theta_G=0.5'$ & 32 (1) \% & 46 (2) \% & 52 (2) \% \\
$\theta_G=1'$ & 34 (2) \% & 54 (2) \% & 65 (8) \% \\
$\theta_G=2'$ & 40 (3) \% & 51 (5) \% & 44 (8) \% \\
$\theta_G=3'$ & 39 (2) \% & 44 (6) \% & 38 (4) \% \\
$\theta_G=4'$ & 36 (2) \% & 30 (6) \% & 32 (5) \% \\
$\theta_G=5'$ & 32 (4) \% & 32 (3) \% & 36 (6) \% \\
$\theta_G=10'$ & 24 (2) \% & 32 (4) \% & 26 (3) \% \\
\hline
\end{tabular}
\end{center}
Table 2: \textit{Same as Table 1, but with correlations between peak
height bins neglected. The likelihood of distinguishing $w=-0.8$ from
$\Lambda$CDM are much lower, and approach the no-significance values
of 32 (5) \%, in particular for the 95\% confidence level numbers,
indicating a large number of outliers.}
\vspace{0.4cm}
We also attempted to study cross-correlations between different
redshifts and smoothing scales. When studying several redshifts and
smoothing scales simultaneously, the number of elements in the
covariance matrix grows quickly. Because of the low number of
realizations and the corresponding underestimation of the variance,
results obtained by using more than $\approx 9$ bins become unreliable
(Appendix \ref{Initial Conditions}). Unfortunately, this prohibits us
from studying the effect of correlations between redshifts and
smoothing scales reliably: the number of different peak height bins
still allowed becomes too small. In particular, when multiple source
redshifts or smoothing scales are simultaneously included, only 2--3
peak heights are allowed by the low number of realizations. We found
that such a low number of convergence bins does not sufficiently
capture the peak number difference distributions in
Figure~\ref{fig:Delta_Peak}, and no meaningful constraints are
possible. We defer a study of these cross--correlations to a
follow--up paper, with a larger number of independent realizations of
each cosmology.
\subsection{Comparison to Other Forecasts for LSST}\label{Other Constraints}
It is useful to place our results in the context of constraints
expected to be available from other uses of large WL maps. Three
methods we will briefly mention here are (i) the shear power spectrum,
(ii) cluster number counts, and (iii) the one--point function of the
convergence PDF.
(i) From an 11-parameter fit to the tomographic shear power spectrum
from LSST and constraints from Planck, \cite{S&K04} obtain
$\sigma(w_0)=0.06, \sigma(w_a)=0.09$.
(ii) From a 7-parameter fit to the number counts of $\approx$ 200,000
shear-selected clusters, combined with Planck constraints, \cite{SW04}
obtain $\sigma(w_0)=0.04, \sigma(w_a)=0.09$.
(iii) Using the fractional area statistic of areas of high
convergence, \cite{Wang:2008hi} found $\sigma(w_0)=0.043,
\sigma(w_a)=0.11$ for LSST and priors from Planck.
The above are all $1\sigma$ constraints, marginalized over a large
number of parameters. We do not attempt a fair comparison of the three
methods, which would require that identical assumptions are made about
the survey specifications and about the set of free parameters (see
\cite{DETF} for a similar exercise covering i. and ii.).
Nevertheless, these numbers demonstrate, first, that the statistical
information contained in the non--linear features in WL maps --
captured by (ii) and (iii) -- is at least roughly comparable to the
information encoded in the power spectrum in (i). Second,
our single--parameter sensitivity to $w$ can be compared meaningfully
to those from clusters counts. In particular, as mentioned in
\S~\ref{Peak Numbers}, the counts of clusters above the $4.5\kappa_G$
convergence threshold differs in the $w=-0.8$ and $w=-1$ cosmologies
by $\approx 1.5\sigma$. We find that the number of convergence peaks
above the same threshold differs in these two cosmologies at a
similar, $\approx 1\sigma$ significance level. We find further that
considering lower height peaks, with convergence well below this
threshold, significantly improves the distinction. While follow-up
work, with simulations that address degeneracies between $w$,
$\sigma_8$, and other parameters is needed, these results suggest that
the parameter sensitivity from WL peak counts will be competitive with
those from the above three methods.
As mentioned in \S~\ref{Introduction}, in a recent preprint,
\cite{Dietrich:2009jq} independently addressed the dependence of WL
peaks on the background cosmology. \cite{Dietrich:2009jq} used the
same simulation box size as in our work, but a smaller number of
particles ($256^3$, rather than $512^3$), which allowed them to sample
a 2D parameter space and initial conditions, at the cost of mass and
angular resolution. They compute the aperture mass for the peaks as a
weighted integral of the tangential shear, rather than the
convergence. Despite these differences, the conclusions of the two
papers appear to be remarkably similar. In particular, if we attribute
the distinction we find between our cosmologies with different $w$
values entirely due to the differences in $\sigma_8$ in these
cosmologies, we find a very similar sensitivity to the latter
parameter as their Figure 3: $0.73~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}}\sigma_8~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}} 0.83$ (for a
fixed $\Omega_m=0.26$). We find, additionally, a subdominant type of
peaks -- low--amplitude peaks, residing in the voids -- the
distinction between the cosmologies is however dominated by medium
peaks, with a strong contribution from high peaks.
\section{Summary}
\label{Summary}
We propose the new method of using the counts of peaks, identified in
weak gravitational lensing maps, to constrain dark energy and other
cosmological parameters. The method makes no reference to cluster
counts or their mass function, and so, by definition, is free from
selection effects inherent in methods that target clusters.
We used N--body simulations to create theoretical convergence maps in
three cosmologies with $w=-0.8$, $w=-1$, and $w=-1.2$. We identify
three different peak types: high, medium, and low peaks, whose numbers
depend differently on cosmology. The high ($\kappa~\rlap{$>$}{\lower 1.0ex\hbox{$\sim$}} 0.6$) and low
($\kappa~\rlap{$<$}{\lower 1.0ex\hbox{$\sim$}} 0$) peaks both become more numerous with increasingly
negative $w$, while the opposite trend is true for the medium
peaks. For the high peaks, this dependence confirms the expectation
that when $w$ is more negative, dark energy begins to dominate later,
and more time is allowed for the growth of non--linear structures.
The opposite behavior of the medium peaks is a surprise - which is all
the more important since we find that these peaks dominate the
distinction between the cosmologies. The majority of these peaks are
not typically caused by individual collapsed clusters, and are more
likely due to projection of large scale structure along the line of
sight. The low peaks form a subdominant contribution. These peaks are
typically due to random ellipticity noise, and we speculate that the
larger surface area of voids in cosmologies with more negative $w$,
which we have detected in the maps, offers more opportunities for
these small fluctuations to arise.
We have shown that based on these peak counts, cosmologies with
$w=-0.8$ and $w=-1.2$ can be distinguished from $\Lambda$CDM with a
single $3\times3$ degree convergence field at a 95\% and 85\%
confidence level, respectively. Extrapolating these results to the
20,000 degree$^2$ field of view of a large survey, such as LSST, we
obtain a projected sensitivity to $w$ of $\Delta w=0.004$--$0.0052$,
depending whether the extrapolation is based on the variance measured
in subfields of our maps, or whether a simple Poisson--like scaling is
used.
In this work, we have explored three constant values of the dark
energy equation of state, $w=-0.8$, $w=-1$, and $w=-1.2$, representing
variations around the best fit $\Lambda$CDM model to the 5-year WMAP
data. The most significant concern is that there will be strong
degeneracies between $w$ and other parameters; our results may already
be driven primarily by the $\sigma_8$ sensitivity in the number of
peaks. In a followup paper, we will study such degeneracies; it will
also be necessary to perform a larger number of realizations of each
cosmology, in order to study the utility of combining several source
galaxy redshifts and to use information possibly contained in the
angular size distribution of the peaks. In a separate publication
\cite{Yang:2009}, we also plan to investigate in more detail the
physical origin of the three kinds of peaks detected in this work.
\begin{acknowledgments}
We would like express our deep thanks to Lam Hui for numerous
helpful discussions, and to Greg Bryan, Francesco Pace, and Matthias
Bartelmann and his group for invaluable help during code
development. We are also grateful to Christof Wetterich for kindly
supporting an extended visit by JK at the University of Heidelberg.
We also thank Puneet Batra, Wenjuan Fang, Eugene Lim and Sarah
Shandera for useful discussions about statistics, CAMB, and lensing,
and Volker Springel for his help with GADGET-2 and for providing us
with his parallelized initial conditions generator. JK is supported
by ISCAP and the Columbia Academic Quality Fund. This work was
supported in part by the Pol\'anyi Program of the Hungarian National
Office for Research and Technology (NKTH), by NSF grant
AST-05-07161, by the U.S. Department of Energy under contract
No. DE-AC02-98CH10886, and by the Initiatives in Science and
Engineering (ISE) program at Columbia University. The computational
work for this paper was performed at the LSST Cluster at Brookhaven
National Laboratory and with the NSF TeraGrid advanced computing
resources provided by NCSA.
\end{acknowledgments}
\section{Appendix: Initial Conditions and the Number of Realizations}\label{Initial Conditions}
In this Appendix, we show our results as a function of the number of
peak height bins used, and justify the choice for the number of bins
we made in the body of the paper.
Ideally -- in the limit that the covariance matrix was perfectly
computed -- the distinguishing power would asymptotically approach a
constant value as one increases the number of bins, at which point
using more bins will not provide any more benefit. Unfortunately, it
is computationally expensive to generate a large ensemble of initial
conditions from which to construct several strictly independent
realizations of a given cosmology (a single realization takes
approximately 4800 CPU hours, i.e. ca. 2.5 days on 80 CPUs). This,
together with the storage required for the large amount of data
produced in each run, ultimately limits the number of bins we can
utilize.
In particular, using too few realizations, together with too many
bins, will generally result in an underestimate of the variance, and
an artificial boost in the $\chi^2$ values. The possible concern,
then, is that the differences we find in the convergence maps between
two cosmologies may be caused by such an underestimation of the
variance.
In order to guard against this possibility, and to verify that the
differences in the convergence maps are caused by genuine differences
in cosmology, we generated a second set of independent initial
conditions for the cosmologies with $w=-1$ and with $w=-0.8$. The
simple Ansatz we adopt for the maximum number of bins that can be
safely used, without underestimating variances, is the following: when
two different realizations of the same $\Lambda$CDM cosmology are
compared, we should recover the $\chi^2$ distribution, i.e., we must
not be able to rule out the fiducial model itself.
In Figure~\ref{fig:Bindependence}, we show the percentage of the
realizations of the $w=-0.8$ test cosmology that lies beyond the 68\%
(left panel) and 95\% (right panel) tail of the $\chi^2$ distribution
(solid curves), together with the same quantities for the fiducial
$\Lambda$CDM cosmology itself (dashed curves). Our Ansatz requires
that the dashed curves correspond to the confidence level at which the
exclusion is to be achieved (e.g. no more than 32 / 5\% of maps
excluded at the 68 / 95\% confidence level). We see that this
condition is satisfied when 4-9 bins are used, but for 10 or more
bins, there is a steadily rising difference between the two
realizations of the $\Lambda$CDM model. We therefore take 9 bins as
the maximum acceptable total number of bins.
Figure~\ref{fig:Bindependence} also shows that the distinction between
$w=-0.8$ and the $\Lambda$CDM model is relatively steady for 4-9 bins,
and is insensitive to the choice of the realization of the initial
condition. However, for a larger number of bins, the distinction
starts to be either less significant (when different realizations of
the initial conditions are used in the two cosmologies; solid curves),
or begin a modest rise in significance (when the same realizations are
used; dotted curves).
While the above analysis is based only on the number of convergence
bins, we find similar conclusions when the bins include multiple
source galaxy redshifts and/or multiple smoothing scales. We conclude
that the accuracy of our covariance matrix limits us to a total of 9
bins.
\begin{widetext}
\begin{figure}[htp]
\centering
\includegraphics[width=8 cm]{Bindependence1.pdf} \includegraphics[width=8 cm]{Bindependence2.pdf}
\hfill
\caption[]{\textit{The solid curves show the percentage of maps in the
$w=-0.8$ cosmology which can be excluded at the 68\% (left panel)
and 95\% (right panel) confidence level from $\Lambda$CDM. In
these comparisons, the maps in the $w=-0.8$ cosmology were
generated from initial condition A, which were compared to the
expected average in the $\Lambda$CDM cosmology with initial
condition B; the covariance matrix was calculated from
$\Lambda$CDM with initial condition A. The dotted curves show the
same quantities, except that the maps in the $w=-0.8$ model were
taken from initial condition B, and those from $\Lambda$CDM with
initial conditions A. The dashed curves again show the same
quantities, except maps from the $\Lambda$CDM model (with initial
condition B) were compared against $\Lambda$CDM with initial
condition A and covariance matrix calculated from initial
condition A. If the covariance matrix was correctly estimated, the
latter curve should be flat, whereas it is begin to rise
systematically for 10 or more bins. Based on this result, we
identify 9 as the maximum number of bins that can be safely used
without underestimating variances. }}\label{fig:Bindependence}
\end{figure}
\end{widetext}
\vfill
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\section*{CLASSICAL CANONICAL GRAVITY}
\section*{Dynamical laws and instantaneous laws} One of the main
preoccupations of classical physics has been finding the laws
governing physical data. One of the oldest schemes of quantization has
been to subject such data to canonical commutation relations. I am
going to review where this program leads us when it is applied to
geometry.
Classical physics deals with two kinds of laws: dynamical laws,
and instantaneous laws. The discovery of dynamical laws started the
Newtonian revolution. The first instantaneous law was found by Gauss:
at any instant of time, the divergence of the electric field is
determined by the distribution of charges. In empty space, the
instantaneous electric field is divergence-free.
\section*{Theorema egregium} Without knowing it, and without most of
us viewing it this way, Gauss also came across the fundamental
instantaneous law of general relativity: the Hamiltonian constraint.
This constraint is a simple reinterpretation of the famous result
Gauss obtained when studying curved surfaces embedded in a flat
Euclidean space \cite{Gauss}.
To start with, Gauss drew the key distinction between intrinsic
and extrinsic properties of a surface. The intrinsic properties are
not changed by bending the surface without stretching; the extrinsic
ones are. The basic intrinsic property of a surface is its {\em
intrinsic metric\/}; the basic extrinsic property, its {\em extrinsic
curvature\/}. These are highlighted on the umbrellas of Figure~1.
\begin{figure}
\vspace*{4.0in}
\caption{Intrinsic metric and extrinsic curvature.}
\end{figure}
The network of distances from the tip of the umbrella along the ribs
is encapsulated by the familiar metric tensor
\begin{equation}
\d s^{2} = g_{ab}(x)\,\d x^{a}\d x^{b} .
\end{equation}
The ribs lie in the planes passing through the shaft of the umbrella;
they represent its {\em normal sections\/}. Each normal section has a
radius of curvature, $r$, whose reciprocal value, $k$, is the
curvature of the normal section. The extrinsic curvature, $K_{ab}$, of
the surface is an inventory of the curvatures of all its normal
sections:
\begin{equation}
r^{-1} = k = K_{ab}(x)\, \d x^{a} \d x^{b}\ / \ \d s^{2} .
\end{equation}
{}From the metric, one can derive other intrinsic objects: lengths,
angles, and areas. Geodesics are completely determined by the
metric. So is the {\em parallel transport} of a tangent vector: a
vector is parallel transported from a point to a neighboring point
if it keeps its angle with the geodesic segment connecting the points.
In its turn, parallel transport leads to the concept of {\em scalar}
(or Gaussian) {\em curvature} (Figure 2):
\begin{figure}
\vspace*{4.0in}
\caption{Scalar curvature.}
\end{figure}
Take a curve enclosing the tip of the umbrella. Mark where you
want to {\bf START}, take the tangent vector to the curve, and
parallel transport it along the curve back to the starting point. On
the way, the tangent vector (bold) rotates clockwise with respect to
the parallel transported vector (double arrow). If the umbrella were a
plane (I would not like to use such an umbrella on a rainy day), the
tangent vector would run all around the clock. On the bulging umbrella
of Figure~2, it does not quite make it; it still has an angle
$\delta\omega$ to go. As the curve is drawn tighter and tighter around
the tip, the deficit angle $\delta\omega$ becomes proportional to the
surface area $\delta\Sigma$ surrounded by the curve. The ratio of
these two quantities defines the scalar curvature:
\begin{equation}
\delta\omega = \case{1}{2}R \,\delta\Sigma \, .
\end{equation}
(Gauss did not obtain the scalar curvature this way; by brute force,
he expressed it as a function of the metric and its first and
second derivatives: $R[g]$.)
Let the ribs again represent normal sections. Of all the ribs,
one has the gentlest bending, $k_{\rm MIN}$, and another has the
steepest bending, $k_{\rm MAX}$. Which rib is which, how large is $k_{\rm
MAX}$, and how small is $k_{\rm MIN}$, depends on the wind. On a quiet
day, all ribs have the same curvature $k_{\rm MAX} = k_{\rm MIN}$. The
values $k_{\rm MAX}$ and $ k_{\rm MIN}$ are called {\em principal
curvatures\/}.
Because the principal curvatures depend on the wind, they are
clearly extrinsic rather than intrinsic properties of an umbrella.
However, their {\em product\/}, called the {\em total curvature\/},
remains always the same. Indeed, it is proportional to the scalar
curvature which, as we have seen, is an intrinsic property of the
umbrella:
\begin{eqnarray}
R & = & 2 \ \ k_{\rm MAX} \cdot k_{\rm MIN} \nonumber \\~
{\bf scalar \ curvature} & = & 2 \ \ {\bf total\ curvature}\,.
\end{eqnarray}
Gauss has found many remarkable theorems in his life, but this one he
himself regarded to be remarkable: he named the result (4) {\em
theorema egregium\/}.
By looking at the surface of a single umbrella, we cannot be sure
if it is worn by an upright old man walking across Great Plains, or if
it protects a crooked old goblin wedged into a curved space. However,
if the intrinsic metric and extrinsic curvature of all possible
umbrellas are connected by Gauss' theorema egregium, we can safely
conclude that the space in which they are embedded is flat and
Euclidean. The flatness of space is thereby guaranteed by the match of
certain intrinsic and extrinsic properties of all embedded surfaces.
Still, what has all of this to do with the assertion that
dynamics is a consequence of instantaneous laws? There are no instants
and hence no dynamics in a Euclidean space. To talk about instants,
space must be Lorentzian. In a flat three-dimensional Lorentzian
spacetime the theorema egregium still holds. The only thing we need to
change is the sign. On every spacelike surface,
\begin{equation}
R = -2\, k_{\rm MAX} \cdot k_{\rm MIN}\,.\ \ \ {\rm (Lorentzian)}
\end{equation}
And, conversely, if the Lorentzian theorema egregium (5) holds on
every spacelike surface in a three-dimensional Lorentzian spacetime,
we can be sure that the spacetime is flat.
Now, the Lorentzian theorema egregium is an instantaneous law.
On the other hand, the statement that spacetime is flat is a dynamical
law, albeit a very simple dynamical law, about geometry. Roughly
speaking, it tells us that there is no dynamics: spacetime remains flat all
the time. This argument illustrates how an instantaneous law, the
Lorentzian theorema egregium, can lead to a dynamical law, that the
spacetime is flat.
General relativity, I remember someone saying, does not confine
us to Euclidean barracks. Even if the spacetime is empty (which, for
simplicity, I shall assume for the rest of my lecture), its dynamics
is quite rich. The ripples of gravitational radiation can travel
around, interfere, attract each other, and amplify. They can hold
themselves together in a gravitational geon. Part of the gravitational
radiation can leak out, part of it may collapse and form a black hole.
I find it quite surprising that all this dynamics is encoded in an
almost trivial generalization of Gauss' theorema egregium:
The intrinsic geometry and the extrinsic curvature of a
three-dimensional hypersurface embedded in a four-dimensional
Riemannian spacetime have the same definition and the same geometric
significance as those of a two-dimensional surface in a
three-dimensional flat space. However, instead of two principal
sections there are three, with principal (extremal) curvatures
$k_{1}$, $k_{2}$, and $k_{3}$. One cannot define the total curvature $T$
as the product of a selected couple of principal curvatures. As a true
egalitarian, one takes
\begin{equation}
T = k_{1}k_{2} + k_{1}k_{3} + k_{2} k_{3}\, .
\end{equation}
Similarly, there is no single surface on which one can determine the
deficit angle $\delta\omega$ by parallel transporting the tangent
vector along a curve. Instead, one chooses three perpendicular
surfaces passing through the tip of a three-dimensional umbrella, and
determines the three deficit angles $\delta\omega_{1}$,
$\delta\omega_{2}$, and $\delta\omega_{3}\,$. The scalar curvature $R$
has the geometric meaning
\begin{equation}
\case{1}{2} \, R = \frac{\delta\omega_{1}}{\delta\Sigma_{1}} +
\frac{\delta\omega_{2}}{\delta\Sigma_{2}} +
\frac{\delta\omega_{3}}{\delta\Sigma_{3}}\ .
\end{equation}
Compare now the total curvature (6) and the scalar curvature (7) of a
hypersurface in an arbitrary Ricci-flat spacetime. Behold, the
theorema egregium still holds:
\begin{equation}
R = \left\{ \begin{array}{l}
- \ {\rm (Lorentzian)} \\
+ \ {\rm (Euclidean)}
\end{array}
\right.
\ \ 2T \, .
\end{equation}
Inversely, if the scalar curvature is related to the total curvature
by Eq.(8) on any spacelike hypersurface, the spacetime is necessarily
Ricci-flat. Therefore, the statement that the theorema egregium holds
at any instant is entirely equivalent to the Einstein law of
gravitation in empty space!
I am sorry that the continuation of my narrative requires some
juggling of indices. The total curvature (6) is a quadratic
combination of the three principal curvatures. Because each of these
is a linear function of the extrinsic curvature (2), the total
curvature can be expressed as a quadratic form of the extrinsic
curvature:
\begin{equation}
T = - \case{1}{2} K_{ab} \,G^{ab\,cd} \,K_{cd}\, .
\end{equation}
The coefficient
\begin{equation}
G^{ab\,cd} = \case{1}{2} ( g^{ac}g^{bd} + g^{ad}g^{bc} - 2\,g^{ab}g^{cd} )
\end{equation}
is called the {\em supermetric\/}. Symmetric pairs of covariant indices
can be raised by the supermetric, and symmetric pairs of contravariant
indices lowered by its inverse, $G_{ab\,cd}\,$. The contravariant
version of the extrinsic curvature is
\begin{equation}
p^{ab}\ \mbox{:=}\ G^{ab\,cd}\,K_{cd} \, .
\end{equation}
The total curvature is a quadratic form of $p^{ab}$, and the
Lorentzian theorema egregium (8) assumes the form \footnote{The
notation $R(x; \, g]$ emphasizes that R is a function of $x$ and a
functional of $g\,$.}
\begin{equation}
H(x)\ \mbox{:=}\ p(x) \cdot G(x;\,g) \cdot p(x) - R(x;\,g] = 0\, .
\end{equation}
The theorema egregium is the most fundamental instantaneous law
of Einstein's theory of gravitation. Gauss did not realize that the
theory of curved surfaces in a flat Euclidean space (and of curved
hypersurfaces in a Ricci-flat spacetime) is subject to yet another
instantaneous law, closely resembling the law which he had found for
electricity. As shown by Codazzi \cite{Codazzi}, the covariant
divergence of the extrinsic curvature $p^{ab}$ vanishes:~\footnote{I
write $\simeq$ whenever I want to sweep a numerical factor under the
rug.}
\begin{equation}
H_{a}\ \mbox{:$\simeq$}\ {}_{g}\! \nabla _{b} \, p_{a}{}^{b}(x) = 0 \,.
\end{equation}
\section*{Canonical geometrodynamics} The stage is now ready for
stating (not proving, nor even properly explaining) the sea change
which the theorema egregium (12) and the Codazzi law (13) suffered a
century later. Working from quite an opposite direction of variational
principles and Hamiltonian dynamics, Dirac \cite{Dirac} and Arnowitt,
Deser, and Misner \cite{ADM} have shown that the intrinsic metric
$g_{ab}$ and the (densitised) extrinsic curvature $p^{ab}$ are
canonically conjugate to each other. In canonical theory, the
instantaneous laws (12) and (13) are called the Hamiltonian and
diffeomorphism constraints. They start playing a double role. On one
hand, they {\em restrict\/} the canonical data. On the other hand, as
dynamical variables on the phase space, they become capable of {\em
evolving\/} the canonical data. The Poisson bracket of the data with
$H_{a}(x)$ generates their change by a Lie derivative in the direction
along the hypersurface. Similarly, the Poisson bracket with $H(x)$
generates the change of the data under a normal displacement of the
hypersurface. These two processes enable us to organize the
embeddings by displacements which deform one embedding into another,
and to correlate the data which the embeddings carry. Instead of
checking the Einstein law by criss-crossing the spacetime by all
possible hypersurfaces, we obtain it by an orderly Hamiltonian
evolution which smoothly deforms the original hypersurface. The change
of the canonical data by the generators $H_{a}(x)$ and $H(x)$,
together with the statement that the generators, once they generated
the change, are constrained to vanish, {\em is} the Einstein law. This
is the new strange role of the instantaneous laws: they become the
agents of dynamics.
The change generated by $H_{a}(x)$ is induced by a spatial
diffeomorphism Diff$\Sigma$ on a given hypersurface. This property
gave the Codazzi constraint its new name --- the diffeomorphism
constraint. The constraint ensures that the theory is invariant under
Diff$\Sigma$. In other words, canonical geometrodynamics does not
depend on the intrinsic metric and the extrinsic curvature, but only
on such combinations of these variables which are unaffected by
spatial diffeomorphisms, i.e., only on the intrinsic and extrinsic
geometries. There are {\em fewer} physical variables than the symbols
which meet the eye.
This message can also be read backwards: by making the theory
dependent on {\em more} variables, one can make it invariant with
respect to a wider class of transformations. A good example of this
process is triad dynamics.
\section*{Triad dynamics}
Let us choose as our basic variables a triad $E^{a}_{i},\ i = 1,2,3$,
of orthonormal vectors \cite{triad}. These determine the intrinsic
metric,
\begin{equation}
g^{ab} = \delta ^{ij} E^{a}_{i} E^{b}_{j}\,,
\end{equation}
but the metric determines the triad only up to an $x$-dependent SO(3)
rotation. The rotation group SO(3) becomes a gauge group of the
Einstein theory. Canonical analysis reveals that the projected
extrinsic curvature
\begin{equation}
-K^{i}_{a}(x) = - K_{ab}(x)E_{j}^{b}\,\delta^{ji}
\end{equation}
is the canonical coordinate whose conjugate momentum is the
(densitised) triad $E_{i}^{a}$. The SO(3) rotations of the canonical
variables are generated by the dynamical variable
\begin{equation}
G_{i}(x)\ \mbox{$:=$}\ \epsilon _{ij}{}^{k}
(-K^{j}_{a}(x))E^{a}_{k}(x) = 0\,,
\end{equation}
which has the familiar structure of angular momentum. After
generating the rotations, $G_{i}(x)$ is constrained to vanish. The
{\em rotation constraint} (16) ensures that the extrinsic curvature
$K_{ab}$ related to $K_{a}^{i}$ by Eq.(15) is symmetric.
Any vector, $u^{a}$, can be characterized by its internal components,
{}$u^{i}$, in the orthonormal basis $E^{a}_{i}\,$:
\begin{equation}
u^{a} = u^{i}E^{a}_{i} .
\end{equation}
Let us parallel transport the vector $u^{a}$ from $x$ to $x + \d x$;
we get the double-arrow vector of Figure~3.
\begin{figure}
\vspace*{3.5in}
\caption{The SO(3) parallel transport.}
\end{figure}
There is a basis, $E^{a}_{i}(x + \d x)$, sitting at $x + \d x$ . In
this basis, I draw a vector which has the same components, $u^{i}$, as
the original vector had at $x$. I call it the {\em reproduced
vector\/}. To turn the reproduced vector into the parallel
transported vector, I must rotate its internal components by an angle
$\delta\omega^{i}$. This angle is a linear function of the
displacement $\d x^{a}$:
\begin{equation}
\delta\omega^{i} = - \Gamma^{i}_{a}\d x^{a}.
\end{equation}
The coefficient $\Gamma_{a}^{i}$ tells us how the parallel transport
of a vector affects its internal components. It can be expressed in
terms of the triad $E_{i}^{a}(x)$ and its first derivatives. It is
called the SO(3) connection.
As usual, the curvature tensor of a connection is defined by the
parallel transport of a vector $u$ along a small parallelogram with the
edges $\d x$ and $\delta x$ (Figure~4).
\begin{figure}
\vspace*{3.5in}
\caption{The SO(3) curvature tensor and the box identity.}
\end{figure}
The parallel transported vector (shown as double arrow) does not
return back to its original position (shown in bold). To turn the
original vector into the parallel transported vector, we must subject
its internal components to a rotation:
\begin{equation}
\delta\omega^{i} = - R_{ab}{}^{i} \d x^{a} \delta x^{b}.
\end{equation}
The coefficient $R_{ab}{}^{i}$ is the curvature tensor of the SO(3)
connection. It can be expressed in terms of the basis vectors
$E^{a}_{i}\,$, and their first and second derivatives.
The curvature tensor satisfies the {\em cyclic identity\/}. Its
geometric significance is illustrated on the right in Figure~4. Take
a small box with edges $u,\,v$ and $w$. Parallel transport $w$ along
the boundary of the face $u,\,v$. The parallel transported vector does
not coincide with $w$; it differs from it by $\delta w$. Repeat this
procedure for the remaining two vectors, and obtain the differences
$\delta u$ and $\delta w$. While none of them in general vanishes,
their sum is identically equal to zero: $\delta u + \delta v + \delta
w \equiv 0\,$.
It is easy to write down what this geometric construction yields
for a small box whose edges lie in the direction of the orthonormal
vectors $E_{i}^{a}\,$. We obtain
\begin{equation}
R_{ab}{}^{i}E_{i}^{b} \equiv 0 \,.
\end{equation}
The SO(3) curvature tensor $R_{ab}{}^{i}[E]$ necessarily satisfies
the box identity (20).
Once we know the curvature tensor, we can determine the curvature
scalar as in Eq.(7). We take three mutually perpendicular curves, each
of them enclosing a unit area, parallel transport their tangent vectors,
determine the deficit angles, and add them together. In particular, we
can choose for the curves the parallelograms spanned by the pairs of
the orthonormal vectors $E^{a}_{i}$. In this way we learn that
\begin{equation}
R[E] = - R_{ab}{}^{i}\, \epsilon _{i}{}^{jk}E_{j}^{a} E_{k}^{b}\,.
\end{equation}
Algebraically, $R[E]$ can be obtained by substituting the metric (14)
into $R[g]$.
We can now take the Hamiltonian constraint (12) and express it in
terms of the new canonical variables $-K^{i}_{a}$ and $E_{i}^{a}$. By
using Eqs.(9)--(10) and (14), we cast the total curvature into a form
in which it is quadratic both in $-K^{i}_{a}$ and in $E_{i}^{a}$. The
curvature scalar is a concomitant (21) of the triad $E_{i}^{a}\,$. As
a result, $H$ is set forth as a functional of $-K^{i}_{a}$ and
$E^{a}_{i}$. The diffeomorphism constraint can be handled in the same
way. Neither of the constraints looks any simpler in the new variables
than it did in the old ones.
In addition to the old constraints, we have the rotation constraint
(16). More variables call for more constraints. There are nine entries
in the triad $E^{a}_{i}\,$, while there are only six entries in the
symmetric metric $g_{ab}\,$. Similarly, there are nine entries in
$-K^{i}_{a}\,$, while there are only six in the extrinsic curvature
$K_{ab}\,$. However, the physics depends only on the old set of
variables, $g_{ab}$ and $K_{ab}\,$. The triad $E^{a}_{i}$ enters into
the Hamiltonian and diffeomorphism constraints only through the
combination (14), and the extrinsic curvature given by Eq.(15) is
forced to be symmetric by the rotation constraint (16). The surplus
dynamics -- rotations of the triad generated by the constraint (16) --
is expendable. In the end, only such quantities which are unaffected
by rotations (like $g_{ab}$ and $K_{ab}$) physically matter. We can
return to them, forget about the rotation constraint, and retrieve
geometrodynamics from triad dynamics. To gain more invariance by
introducing more variables is not a big deal.
\section*{Connection dynamics}
To simplify the constraints, one must go one step beyond
introducing the triads: one must modify the parallel transport.
Figure~5
\begin{figure}
\vspace*{4in}
\caption{New parallel transport.}
\end{figure}
shows a three-dimensional umbrella protecting us from a storm of
gravitational waves in a four-dimensional Euclidean Ricci-flat
spacetime. Take a tangent vector to the umbrella (shown as a double
arrow) and parallel transport it from the tip along one of the
principal sections. The transported vector is again shown as a double
arrow. Viewed from the center of curvature, the arc $\d x$ along
which the vector is transported subtends the angle $k \d x$. Give now
the parallel transported vector an additional twist by the angle $k \d
x$ about the principal direction. The position of the vector after the
twist defines the new parallel transport.
The twist does not look quite right in the two-dimensional
sketch of the three-dimensional umbrella. It seems that to rotate the
vector about the rib we must rotate the whole tangent plane, and
destroy thereby its tangential character. To see what is happening, we
must look at the tangent plane through one of the most powerful
instruments ever invented by a theoretical physicist: John A.
Wheeler's dimensional magnifying glass \cite{Wheeler}. Under the
glass, the plane thickens into what it actually is, a
three-dimensional tangent space, and the twist moves the vector along
a cone in this space into its new position. The normal to the umbrella
stays fixed because the twist takes place in the plane perpendicular
to the principal section.
To transport a vector in an arbitrary direction, we must
decompose the displacement $\d x$ into the three principal directions
and perform the appropriate twists one after another. The new parallel
transport amounts to a single rotation (18) of the reproduced vector.
The angle of rotation is given by a new SO(3) connection \cite{Sen,Ash}
\begin{equation}
A_{a}^{i} = \Gamma^{i}_{a} - K_{a}^{i} \,.\ \ \ \ \ {\rm (Euclidean)}
\end{equation}
{}From the canonical standpoint, it is remarkable that Eq.(22)
represents a canonical transformation: the new SO(3) connection
$A_{a}^{i}$ is a coordinate canonically conjugate to the momentum
$E^{a}_{i}\,$. (To see that $A^{i}_{a}$ is conjugate to $E^{a}_{i}$
is trivial, because $-K^{i}_{a}$ is conjugate to $E^{a}_{i}$. It is
more difficult to prove that $A^{i}_{a}(x)$ can serve as a field
coordinate, i.e., that it has a vanishing Poisson bracket with
$A^{j}_{b}(x')$.) But why should we ever want to perform the
canonical transformation (22)? Cui prodest?
The new parallel transport leads, by Figure~4 and Eq.(19), to the new
curvature tensor $F_{ab}{}^{i}[A]$. In terms of this tensor, the
instantaneous laws of Euclidean Ricci-flat spacetimes take a
remarkably simple form. The new curvature tensor no longer satisfies
the box identity (20). Instead, the expression $F_{ab}{}^{i}
E^{b}_{i}$ yields the supermomentum \footnote{This time, $ \simeq $
sweeps under the rug not only numerical factors, but also the fact
that the rotation constraint is used in rearranging the diffeomorphism
constraint and the Hamiltonian constraint.}
\begin{equation}
H_{a} \simeq F_{ab}{}^{i}[A]E^{b}_{i} \, .
\end{equation}
The box equation (20) still holds, but no longer as an identity. It is
now equivalent to the Codazzi law (13) in a Ricci-flat spacetime. Even
more remarkably, the $H$ of Gauss' theorema egregium turns out to be
the scalar curvature (21) of the new parallel transport:
\begin{equation}
H \simeq F_{ab}{}^{i}[A] \, \case{1}{2} \epsilon _{i}{}^{jk} E^{a}_{j}
E^{b}_{k} \, .
\end{equation}
These striking facts were discovered by Ashtekar \cite{Ash}. It is
quite tempting to call Eq.(24) Ashtekar's theorema egregium: In a
Ricci-flat spacetime, the scalar curvature of Ashtekar's connection
$A$ vanishes on every hypersurface.
The new connection $A^{i}_{a}$ defines the new covariant
derivative ${}_{A}D _{a}$ acting on internal indices. In terms of this
derivative, the rotation generator (16) takes the form
\begin{equation}
G_{i} = {}_{A} D_{a} \, E^{a}_{i} \, .
\end{equation}
(Because $E^{a}_{i}$ is a vector density, $ {}_{A}D_{a}$ does not need
to act on spatial indices to produce a scalar density.)
These are the good news. Now, for the bad news. The transition
{}from a Euclidean to a Lorentzian spacetime enforces a change of sign in
the Gauss theorema egregium. The Ashtekar theorema egregium can absorb
this change of sign only at the price of introducing a complex SO(3)
connection:
\begin{equation}
A^{i}_{a} = \Gamma ^{i}_{a} - i \, K^{i}_{a} \, .\ \ \ \ \ {\rm (Lorentzian)}
\end{equation}
To handle this complication in canonical quantum gravity is not
entirely trivial.
\section*{Dialogue
concerning the two chief systems of canonical gravity: \\
geometrodynamics and connection dynamics}
Geometrodynamics and connection dynamics are the two chief forms
of canonical gravity. Let us pause and compare them before proceeding
with quantization. I suppress the turmoil of indices and highlight the
two structures in a table. Let
\begin{quote}
{} $\cdot$ denote contraction in spatial indices,
{} $\circ$ denote contraction in spatial indices {\em and}
internal indices,
{}${}^{\ast}$ denote internal dualization,
\end{quote}
and the details take care of themselves. Then
\begin{displaymath}
\begin{array}{ccc}
{} & {} & {} \\
{} & {} & {} \\
{\rm GEOMETRODYNAMICS} & {} & {\rm CONNECTION\ \ DYNAMICS} \\
{} & {} & {} \\
{} & {} & {} \\
\begin{array}{cc}
{\bf Coordinates} & {\bf Momenta} \\
{} & {} \\
g & p \\
{} & {} \\
\begin{array}{c}
{\bf Intrinsic} \\
{\bf metric} \\
{} \end{array} &
\begin{array}{c}
{\bf Extrinsic} \\
{\bf curvature} \\
{}
\end{array}
\end{array} &
\begin{array}{c}
{\rm CANONICAL} \\
{\rm VARIABLES} \\
{} \end{array} &
\begin{array}{cc}
{\bf Coordinates} & {\bf Momenta} \\
{} & {} \\
A & E \\
{} & {} \\
\begin{array}{c}
{\bf SO(3)} \\
{\bf connection} \\
{\bf (mixed) }
\end{array} &
\begin{array}{c}
{\bf Triad} \\
{} \\
{\bf (intrinsic)}
\end{array}
\end{array} \\
{} & {} & {} \\
{} & {} & {} \\
{} & {\rm GENERATORS\ \ OF} & {} \\
{} & {} & {} \\
{} & {} {} & {} \\
p \cdot G(g) \cdot p\,-\,R[g]&\perp\ {\rm EVOLUTION} & E \circ {}^{\ast}F[A]
\circ E \\
{} & {} & {} \\
{}_{g}\nabla \cdot p & {\rm Diff}\Sigma & F[A] \circ E \\
{} & {} & {} \\
{\bf None} & {\rm ROTATIONS} & {}_{A}D \cdot E \\
{} & {} & {}
\end{array}
\end{displaymath}
A comparison of the two columnes brings forward a number of simple
observations:
\begin{enumerate}
\item{{\em Variables and constraints.} Geometrodynamics works with
fewer variables and fewer constraints than connection dynamics. The
geometrodynamical variables are invariant under triad rotations.}
\item{{\em Connection with gauge theories.} The rotation constraint
makes connection dynamics resemble an SO(3) Yang-Mills theory. In
geometrodynamics, the rotation constraint is eliminated and one works
with SO(3)-invariant canonical variables.}
\item{{\em Dimension.} I discussed the constraints in $3+1$ dimensions.
Geometrodynamics remains virtually the same in any dimension $n + 1,\
n \geq 2 $. The SO(3) connection dynamics is intimately adapted to a
three-dimensional space and it is not easily generalized to $n > 3$.}
\item{{\em Positivity restrictions.} The Cauchy problem works only
if the hypersurfaces are spacelike, i.e., if the induced metric $g$ is
positive definite. In geometrodynamics, this puts a restriction on the
domain of the configuration space. In connection dynamics, the metric
is automatically positive definite, as long as the triad is
non-degenerate. However, even for degenerate triads (leading to
degenerate metrics) the formalism seems to make sense, and it is
viable to lift the non-degeneracy restriction.}
\item{{\em Structure of the Hamiltonian constraint.} In
geometrodynamics, $H$ is a {\em quadratic function} of the momentum $p$.
The supermetric $G(g)$ is ultralocal, and there is a local potential
term $R[g]$. In connection dynamics, the potential is absorbed into
the quadratic term; $H$ is a {\em quadratic form} of the momentum $E$.
However, the supermetric ${}^{\ast}F[A]$ is no longer ultralocal, but
merely local.}
\item{{\em Polynomiality of constraints.} In connection dynamics, all
constraints are low-degree polynomials in the canonical variables $A$
and $E$. It was originally claimed that geometrodynamical constraints
are non-polynomial in the canonical variables $g$ and $p$, but a more
careful look \cite{Tate} reveals that a simple scaling by a
power of det$(g)$ also makes them polynomial. However, they are
polynomials of a rather high degree.}
\item{{\em Reality conditions.} Geometrodynamics works with real
canonical variables on a real phase space.. We have seen that in a
Lorentzian spacetime the Ashtekar variable $A$ is necessarily
complex. This forces one to work with complex canonical variables,
either on a complex or a real phase space. A pair $A$ and $E$ of
canonical variables that satisfy the constraints define a real
Ricci-flat spacetime only if they satisfy the {\em reality conditions}
\begin{equation}
E - \bar{E} = 0\, , \ \ A + \bar{A} = \Gamma[E]\, .
\end{equation}
These conditions are non-polynomial in $E$. People replace
\cite{Ash-re} the reality conditions (27) by somewhat weaker
conditions that are polynomial. A simpler procedure is to scale the
second condition (27) by $[{\rm det}(g)]^{2}$ which makes it
polynomial in $E$ and $A$. Whichever way one proceeds, the polynomial
reality conditions are of a rather high degree.}
\end{enumerate}
{}
In view of these observations, which scheme is simpler,
geometrodynamics or connection dynamics? Simplicity, of course, is in
the eye of the beholder, and my assessment is quite personal.
\begin{enumerate}
\item{I believe that, on one hand, one should not make too much fuss
about the count of the variables and constraints and, on the other
hand, one should not overemphasize the resemblance between connection
dynamics and the SO(3) gauge theories.}
\item{The SO(3) invariance is a
simple consequence of introducing the redundant variables. The ease
with which such variables are eliminated and connection dynamics
reduced back to geometrodynamics may be an indication that the
achieved SO(3) invariance is not that deep. However, one should not
overlook that the mixing of the extrinsic and intrinsic variables
brings in a true simplification of the constraints prior to the
imposition of the reality conditions.}
\item{ Our space is three-dimensional and a theory which makes an
effective use of this fact is not to be blamed. I view the
simplifications which can be achieved only in three dimensions
speaking for rather than against connection dynamics.}
\item{The positivity restrictions on the metric are quite a nuisance in
quantum theory. The possibility of lifting the non-degeneracy
condition on the triad without endangering the connection dynamics is
a real advantage. However, when listing the achievements of quantum
connection dynamics, one should bear in mind that many of these
correspond to situations in which the triad and hence the metric are
degenerate.}
\item{Trading an ultralocal supermetric and a local potential term for
a local supermetric without any potential is an interesting quid pro
quo. Whether such a trade-off pays off in quantum theory depends quite
heavily on whether it is easier or not to turn the new Hamiltonian
constraint into a well-defined operator.}
\item{A low-degree polynomiality is certainly an asset in
quantizing a classical theory. In this respect, the constraints of
connection dynamics are definitely simpler than the geometrodynamical
ones.}
\item{One should bear in mind, however, that connection dynamics is
sooner or later confronted with the task of implementing the reality
conditions. These conditions are unseemly, being polynomials of such a
high degree as the geometrodynamical constraints. The simplifications
which the connection dynamics achieves may thus be a mere temporary
advantage.}
\end{enumerate}
The Galilean overtones of my section heading are meant to go
beyond a mere joke. Eliminating variables or constraints is like
getting rid of epicycles. The presence of a potential term may be
aesthetically repugnant like the use of an equant. Nevertheless, the
fact remains that geometrodynamics and connection dynamics are
entirely equivalent at the classical level, just as the Ptolemaic
and Copernican systems are entirely equivalent at the kinematical
level. The Copernican system may be aesthetically more pleasing, but
its real power emerges only when one starts asking dynamical
questions. Similarly, the real power of connection dynamics may
emerge only when one starts quantizing the classical theory. This is
the task we should now discuss.
\section*{CONSTRAINT QUANTIZATION: A PROGRAM}
Canonical gravity is a system whose dynamics is entirely
generated by constraints. Its quantization and interpretation presents
some special difficulties. The ground rules for quantizing constrained
systems were laid by Dirac \cite{Dirac} and refined over the years.
Every major review of canonical quantum gravity %
\cite{ADM,reviews,Ash-book} %
attempted to list a sequence of steps expected to lead to a
satisfactory theory. People more or less agree about what these steps
are, but they do not know how to implement them: by listing the steps,
they present a mere quantization program. I shall picture seven steps
of a quantization program as seven gateways on a road paved with good
intentions.
\section{Fundamental variables}
The first step is the selection of {\em fundamental variables}.
These are classical dynamical variables that are to be turned into
operators whose commutator algebra replicates the classical Poisson
algebra. The fundamental variables are expected to span a vector space
$\cal V$ closed under the Poisson brackets ${\{\ ,\ \}}$. The space
$\cal V$ should be {\em complete} in the sense that any dynamical
variable $F$ can be approximated by an element of the free algebra
$\cal A$ over $\cal V$, i.e., expressed as a sum of products of the
elements of $\cal V$.
In geometrodynamics, $\cal V$ is taken to be a real vector space
spanned by $g$, $p$, and the unit dynamical variable 1. In connection
dynamics, $\cal V$ is taken to be a complex vector space spanned by
$A$, $E$, and 1. The elements $V \in \cal V$ are expected to
be mapped into operators $\hat{V} \in \hat{\cal V}$ in such a way that
\begin{equation}
V_{3} = \{ V_{1}, V_{2} \}\ \ \Longrightarrow \ \ \hat{V}_{3} = - i [
\hat{V}_{1}, \hat{V}_{2}] \, .
\end{equation}
In geometrodynamics, the metric $g$ should be positive definite.
The positivity conditions cannot be written as relations in $\cal V$,
and their imposition is quite tricky \cite{Ish+}. Connection dynamics
is fully equivalent to geometrodynamics only for non-degenerate triads
$E$. People working in connection dynamics propose not to impose the
condition that $E$ be non-degenerate.
Connection dynamics has a difficulty which does not exists in
geometrodynamics: not all elements of the complex vector space $\cal
V$ describe a real spacetime. Ultimately, one must impose the {\em
reality conditions} (27). However, $\cal V$ is not closed under the
complex conjugation. The reality conditions thus cannot be formulated
in $\cal V$ (the old SO(3) connection $\Gamma (E)$ does not lie in
$\cal V$). It was proposed \cite{Ash-book,A+T} that at the level
of representing the fundamental variables by operators one should
simply forget about the reality conditions. These are to be taken care
of much later, during the construction of a Hilbert space. Conforming
to this view, I return to the issue of reality conditions at the last
of my gates.
\section{Dynamical variables (including constraints)}
In the next step, one must decide on how to turn an arbitrary
dynamical variable $F[g,p] \in \cal A$ or $F[A,E] \in \cal A$ into an
operator. Such variables typically do not lie in $\cal V$, but they
can be approximated by sums of products of the elements of $\cal V$.
Van Hove \cite{Van Hove} has proved that it is impossible to turn
dynamical variables into operators in such a way that Eq.(28) holds
for all of them. Without the guiding principle (28), the quantization
of dynamical variables is subject to factor-ordering ambiguities. It
is popular to dismiss these as `mere quantum-mechanical corrections'.
I do not share this view. Unless one knows how to factor order
significant dynamical variables, one really does not know how to
construct quantum theory. In a sense, the right factor ordering {\em
is} the quantum theory. If one does not set any rules about factor
ordering, one can turn a classical variable $F(Q,P)$ into any quantum
operator one pleases:
Let $F(Q,P)$ be a classical dynamical variable, and $G(Q,P)$ any
other dynamical variable (whose dimension is that of $F$ divided by
the action). Fix some factor ordering of $\hat{F} = F(\hat{Q},\hat{P})$
and $\hat{G} = G(\hat{Q}, \hat{P})$, and define the quantum
variable
\begin{equation}
\hat{F}' \, \mbox{ :=} \, \hat{F} - i\,[\hat{Q},\hat{P}])\hat{G}
=\hat{F} + \hat{G}\, .
\end{equation}
The quantum variables $\hat{F}'$ and $\hat{F}$ have the same classical
limit, namely, $F(Q,P)$, and yet they differ by an arbitrary operator
$\hat{G}$. This is what a `mere' factor ordering can do.
A particular case of dynamical variables in canonical gravity are
the constraints. One should turn them into operators $\hat{H}(x)$, $
\hat{H}_{a}(x)$ (and possibly $\hat{G}_{i}(x)$). In field theory, this
presents a regularization problem. Moreover, in the next step of the
quantization procedure one wants to impose the constraints on the
states. This poses a consistency problem. Both of these problems are
troublesome, but they at least impose severe restrictions on the
factor ordering. On the other hand, very little is known and, even
more remarkably, said about what to do with other dynamical variables.
\section{Representation space $\cal F$}
The operators representing the dynamical variables are expected
to act on a space of states. One way of choosing this space is to rely
on the Schr\"{o}dinger representation: the canonically conjugare pairs
of fundamental variables are taken as multiplication and
differentiation operators acting on functionals of the configuration
variables. Thus, in geometrodynamics $\cal F$ is taken to be a complex
vector space whose elements are the functionals $\Psi [g]$ of the
metric. In connection dynamics, the elements of $\cal F$ are the
functionals $\Psi [A]$ of the Ashtekar connection. \footnote{The
connection representation $\Psi [A]$ is formally related to the
{\em loop representation}. This is discussed by Smolin in this volume,
and in a recent review by Rovelli \cite{Rov-loops}.}
There is an important difference between the Schr\"{o}dinger
representation for unconstrained systems and the Schr\"{o}dinger
representation in canonical gravity: the representation space ${\cal F}$
is not necessarily assumed to be a Hilbert space, and (real) dynamical
variables are not required to be represented by self-adjoint
operators \cite{K-Ox}. Physically, the Hilbert space structure is
needed to calculate the expectation values of observables. However,
prior to the imposition of constraints, the states in ${\cal F}$ do not
necessarily describe physical states, and it does not have a good
meaning to ask what is the expectation value of an observable in such
a state.
The rejection of the Hilbert space structure liberates us from a
straitjacket that often leads to inconsistencies \cite{K-ord}, but
it unfortunately leads to a loss of control over mathematical objects.
I shall later comment on both of these aspects.
\section{Space of solutions}
The key idea of the Dirac constraint quantization is to turn the
constraints, which I shall now collectively call $\bf H$, into
operators (gate 2), and impose them as restrictions on the states:
\begin{equation}
{\bf H} \Psi = 0 \, .
\end{equation}
One surmises that only such states $\Psi$ which solve the constraints
can be physical. All physics is to be done on the space ${\cal F}_{0}$ of
states which solve Eq.(30).
A number of remarks is appropriate. First of all, the quantum
constraints should not limit the quantum states more than the
classical constraints limit the classical states: they should not
beget other constraints by commutation. This imposes stringent
requirements on the factor ordering of the constraint functions. One
can see on simple models that these requirements virtually dictate the
factor ordering, and that to satisfy them the constraints cannot and
should not be represented by self-adjoint operators on ${\cal F}$
\cite{K-ord}. In geometrodynamics (and in connection dynamics), a
consistent factor ordering of constraints is a notorious unsolved
problem. The task is seriously hampered by the field-theoretical
aspects of canonical gravity, which call for regularization of the
constraint operators. \footnote{One should find a factor ordering of
the Hamiltonian and diffeomorphism constraints such that the
commutator of the Hamiltonian constraints yields an expression in which
the diffeomorphism constraint acts on the state function first,
followed by the structure functions of the `Dirac algebra'. It was
noticed by Anderson \cite{And} that this task cannot be accomplished
if one insists on representing the constraints by self-adjoint
operators on $\cal F$. A solution to the factor-ordering problem was
offered by Schwinger \cite{Schw} and criticized by Dirac \cite{Dir}.
The best, and certainly the shortest, exposition of Schwinger's
solution may be found in a footnote of the paper \cite{DW} by DeWitt;
this was later rediscovered by Komar \cite{Komar}. DeWitt himself made
a rather sweeping proposal on how to remove the problem by letting any
two field operators taken at the same point formally commute
\cite{DW}. Ashtekar \cite{Ash} proposed a simple factor ordering of
his constraints which (disregarding the regularization difficulties)
satisfies the consistency requirement. Unfortunately, all these results
are purely formal: Tsamis and Woodard \cite{T+W}, and Friedman and
Jack \cite{F+J} have persuasively argued that by formal manipulations
of the commutator one can obtain whatever result one wants.}
The absence of a Hilbert space structure on ${\cal F}$ helps us
to make the constraints consistent. However, it also makes the
quantization badly dependent on the choice of representation. One can
see the problem already when solving the Schr\"{o}dinger equation of a
simple unconstrained system like an anharmonic oscillator
\cite{Ish-Sch},
\begin{equation}
\hat{h} = \case{1}{2} \hat{P} ^{2} + \case{1}{2} \hat{Q} ^{2} +
\case{1}{4} \hat{Q} ^{4} .
\end{equation}
The solution of the Schr\"odinger equation calls for finding the
eigenfunctions of the energy operator (31). In the $Q$-representation,
the eigenfunction equation is a differential equation of the second
order. In the $P$-representation, it is a differential equation of the
fourth order. As a differential equation, the first equation has fewer
solutions than the second equation. The mismatch is removed by
requiring that the solutions we seek be square integrable (in $Q$ and
in $P$), i.e., belong to the Hilbert space based on the
Schr\"odinger norm.
When, as in canonical gravity, we are unwilling to impose a
Hilbert-space structure on $\cal F$, the size of ${\cal F}$ depends on
the choice of representation. Thus, in principle, the solution space
$\Psi[g]$ in geometrodynamics is different from the solution space
$\Psi [p]$. Similarly, the connection representation $\Psi[A]$ is not
necessarily equivalent to the triad representation $\Psi [E]$. In
other words, by not requiring that the representation space ${\cal F}
$ be a Hilbert space, one affirms a strong belief in the primacy of
those fundamental variables on which the representation is based.
This is only a part of a larger problem. Unless one imposes some
boundary conditions on the solutions of the constraint equation (31),
the solution space ${\cal F}_{0}$ may be much too big. The quadratic
character of the Hamiltonian constraint in the momenta $p$ (or $E$)
evokes the analogy with the Klein-Gordon constraint for the
relativistic particle. There we know that the solution space of the
mass-shell constraint is also too big: the physical states of a
one-particle system correspond only to positive-energy solutions. In
geometrodynamics (and in connection dynamics), we do not have any
accepted method of cutting the basis of the solution space into half.
We are thus stuck with a solution space which may be physically too
big.
If, on the other hand, we start imposing boundary conditions or
some other limitations on the states, we may inadvertently force the
solution space to be too small. This may (though it does not need
to) happen in connection dynamics when one requires that the states
$\Psi[A]$ be holomorphic functions of the complex connection $A$. One
imposes such a requirement in analogy with the Bargmann representation
for the states of a harmonic oscillator \cite{Barg}. The
solution of the eigenvalue equation for the oscillator Hamiltonian
$\hat{h}$ on the space of holomorphic functions of $Z = Q - i P$ gives
automatically a correct spectrum for $\hat{h}$. This is surprising,
because at this stage we do not yet have any Hilbert space. Only much
later is the space of holomorphic functions turned into a Hilbert
space which yields the same spectrum. This may not work so smoothly in
canonical gravity. The complex connection is in some respects quite
different from the complex variable $Z$ for the harmonic oscillator.
(I shall return to this point three steps later.) There is a chance
that the `preestablished harmony' between holomorphic functions and
the subsequent construction of the Hilbert space no longer exists.
To summarize, without the Hilbert space structure on ${\cal F}$
and without boundary conditions or some auxiliary conditions on the
states, we are bound to end up with a solution space ${\cal F}_{0}$
that contains many unphysical states. For this reason, I am reluctant
to call ${\cal F}_{0}$ `the physical space', and prefer to stick to a
more neutral name, the space of solutions.
\section{Observables}
An outstanding question in the theory of constrained systems is
what dynamical variables can in principle be observed. An often made
proposal \cite{obs,Ash-book} is that
\begin{itemize}
\item{Classical `observables' are those dynamical
variables $F$ whose Poisson brackets with the constraints weakly vanish:
\begin{equation}
{\bf H} = 0 \ \ \ \Longrightarrow \ \ \ \{ F,\, {\bf H} \} = 0 \, .
\end{equation} }
\end{itemize}
Its quantum mechanical counterpart is that
\begin{itemize}
\item{Quantum `observables' are those operators $\hat{F}$ that commute
with the constraint operators $\hat{\bf H}$ on the space of solutions
{}${\cal F}_{0}\,$:
\begin{equation}
\hat{\bf H} \Psi = 0 \ \ \ \Longrightarrow \ \ \ [\hat{F},\, \hat{\bf
H}]\Psi = 0\, .
\end{equation}}
\end{itemize}
The second definition seems to be virtually forced on us if we
insist that the measurement of an observable does not throw the state
$\Psi$ out of the space of solutions ${\cal F}_{0}$.
These two definitions are straightforward generalizations of the
concept of an observable in gauge theories. I am going to argue that
they are inappropriate for canonical gravity.
To see why the definitions (32) and (33) are natural in ordinary
gauge theories, consider electrodynamics. The vector potential $A$
describes the state of the electromagnetic field. The potentials
$A_{(1)}$ and $A_{(2)}$ which lie on the same orbit of the Gauss
constraint $G(x) = \nabla \cdot E(x)$ differ by a gauge transformation. They
are {\em physically indistinguishable}$\,$: they represent two equivalent
descriptions of the same physical state. One cannot observe the
individual $A$'s along the orbit, only the magnetic field $B$. The
magnetic field remains the same if we change the vector potential by a
gauge transformation:
\begin{equation}
\{ B(x'),\, G(x) \} = 0 \, .
\end{equation}
The magnetic field is an example of an observable.
A quantum state of the electromagnetic field is described by the
state functional $\Psi [A]$. This functional is the probability
amplitude for finding the electromagnetic field in the state described
by the vector potential $A$. The probability should remain the same
when we change $A$ by a gauge transformation. This is ensured by the
Gauss constraint
\begin{equation}
\hat{G}(x)\, \Psi [A] = 0 \, .
\end{equation}
Equation (35) implies that $\Psi$ can depend on $A$ only via the
classical observable $B$: $\Psi = \Psi [B]$. On an ensemble of
systems described by the state functional $\Psi [B]$, we cannot
measure $\hat{A}$, but only $\hat{B}$. The magnetic field operator
{}$\hat{B}$ is a quantum observable. It satisfies the quantum
counterpart of Eq.(34):
\begin{equation}
[\hat{B}(x'), \, \hat{G}(x) ] = 0 \, .
\end{equation}
The same case which I made for the Gauss constraint $G$ in
electrodynamics can be repeated for the rotation constraint $G_{i}$
and the diffeomorphism constraint $H_{a}$ in canonical gravity:
Spacelike hypersurfaces in a Ricci-flat spacetime carry the
induced geometry, but do not come equipped with an orthonormal triad
$E$. The triad is a mere tool for calculating the metric (14). Two
triads, $E_{(1)}$ and $E_{(2)}\,$, on the same orbit of the constraint (16)
differ by a rotation. They both yield the same metric (14). Rotations
can be thought about as a gauge, and metric as an observable. In
general, the SO(3) observables are those dynamical variables which are
unaffected by rotations,
\begin{equation}
G_{i}(x) = 0 \ \ \ \Longrightarrow \ \ \ \{ F, \, G_{i}(x) \} = 0 \, .
\end{equation}
In the triad representation, the quantum state of the
gravitational field is described by the state functional $\Psi [E]$.
The rotation constraint
\begin{equation}
\hat{G}_{i}(x) \, \Psi[E] = 0
\end{equation}
implies that $\Psi$ can depend on $E$ only through the metric (14):
$\Psi = \Psi[g]$.
However, the metric is not yet an observable with respect to
diffeomorphisms. Two metric fields, $g_{(1)}(x)$ and $g_{(2)}(x)$,
that differ only by the action of Diff$\Sigma$, i.e., which lie on the
same orbit of $H_{a}(x)$, are physically indistinguishable. This is
due to the fact that we have no direct way of observing the points $x
\in \Sigma$. A dynamical variable constructed from the metric field is
a true observable only if its value is unaffected by diffeomorphisms:
\begin{equation}
H_{a}(x) = 0 \ \ \ \Longrightarrow \ \ \ \{ F,\, H_{a}(x) \} = 0 \, .
\end{equation}
Thus, e.g., the volume of $\Sigma$ is an observable:
\begin{equation}
V[g] = \int_{\Sigma} \d ^{3}x \, |{\rm det}(g(x))|^{\case{1}{2}} \, .
\end{equation}
The momentum constraint
\begin{equation}
\hat{H}_{a}(x) \, \Psi[g] = 0
\end{equation}
implies that the value of the state functional $\Psi[g]$ is the same
for all metrics connected by Diff$\Sigma$, i.e.\ , that $\Psi [g]$
does not depend on the individual metrics $g(x)$, but only on the
three-geometry ${}^{3}\cal G$.
However, the definition (32) of an observable requires yet
something more. It claims that a dynamical variable $F$ cannot be
observed unless it has a vanishing Poisson bracket with the
Hamiltonian constraint $H$. I feel that this requirement is misguided.
The action of $G_{i}$ on the dynamical variables generates their
change under rotations SO(3). The action of $H_{a}$ on the dynamical
variables generates their change under Diff$\Sigma$. Both of these
actions operate in the space of the instantaneous data on a fixed
hypersurface. The change of the data which they generate is
unobservable. The action of $H$ is different: it generates the
dynamical change of the data from one hypersurface to another. The
hypersurface itself is not directly observable, just as the points $x
\in \Sigma$ are not directly observable. However, the collection of
the canonical data $g_{(1)}$, $p_{(1)}$ on the first hypersurface is
clearly distinguishable from the collection $g_{(2)}$, $p_{(2)}$ of
the evolved data on the second hypersurface. If we could not
distinguish those two sets of the data, we would never be able to
observe dynamical evolution.
The same reasoning applies to quantum theory. In the
Schr\"odinger picture, the evolution is carried by the state $\Psi$.
The Hamiltonian constraint
\begin{equation}
\hat{H}(x) \Psi = 0
\end{equation}
plays a different role from the diffeomorphism constraint or the
rotation constraint. It does not tell us that the evolved state is
indistinguishable from the initial state, but rather it tells us how
the state evolves. Thus, in geometrodynamics, the constraint (42) is a
second-order variational differential equation for the state $\Psi
[{}^{3}{\cal G}]$ of the three-geometry, called the Wheeler-DeWitt
equation \cite{Wheeler,DW}.
This can be viewed as analogous to the
Klein-Gordon equation for the state $\psi (x^{\alpha})$ of a
relativistic particle. The three-geometry ${}^{3}{\cal G}$ is
considered as an internal configuration space{\/\em time} variable,
similar to the argument $x^{\alpha}$ of the Klein-Gordon state. The
Wheeler-DeWitt equation is supposed to describe the dynamical
evolution of the state in an internal configuration spacetime.
It is this fundamental distinction between the states which are
and the states which are not distinguishable that leads me to reject
the definition (32) according to which `observables' should also have
a vanishing Poisson bracket with the Hamiltonian constraint. The
dynamical variable $F$ which satisfies this requirement,
\begin{equation}
H(x) = 0 \ \ \ \Longrightarrow \ \ \ \{ F, \, H(x) \} = 0 \, ,
\end{equation}
must have the same value on all spacelike hypersurfaces. Therefore, it
is necessarily a constant of motion. This underscores the point which
I already made: If we could observe only constants of motion, we could
never observe any change.
I hold that one can observe other dynamical variables,
like the volume variable (40), not only constants of motion.
Therefore, I shall call {\em observables} those dynamical variables
which are invariant under SO(3) and Diff$\Sigma$, but which do not
necessarily obey Eq.(43). Those observables which also satisfy Eq.(43) I
shall call {\em perennials\/}. I want to argue that
\begin{itemize}
\item{One can observe dynamical variables which are not perennial,}
\end{itemize}
and that
\begin{itemize}
\item{Perennials are often difficult to observe.}
\end{itemize}
To make these two points, I do not need to deal with general
relativity. Any parametrized (or already parametrized) system
\cite{par} illustrates the same point. I shall try to clarify the
issues on the simplest of such systems, a parametrized free Newtonian
particle moving on a line. The phase space of the system is the
cotangent bundle $(T,Q; \,P_{T}, P)$ over the configuration spacetime
$(T,Q)$, and the Hamiltonian constraint amounts to the definition of
the energy $-P_{T}$ in terms of the momentum $P$:
\begin{equation}
H \, \mbox{:=} \, P_{T} + \case{1}{2} P^{2} = 0 \, .
\end{equation}
Perform a canonical transformation \cite{Goa}
\begin{eqnarray}
Q' = Q - PT, &{}& P' = P, \\
T' = T,\ \ \ \ \ \ \ \ \ &{}& P_{T'} = P_{T} + \case{1}{2}P_{T}{}^{2}.
\end{eqnarray}
The primed canonical variables (45) are the initial data at $T=0$.
The primed time $T'$ is identical with the Newtonian time $T$. The
momentum $P_{T'}$ conjugate to $T'$ coincides with the Hamiltonian
constraint:
\begin{equation}
H \, \mbox{:=} \, P_{T'} = 0 \, .
\end{equation}
Due to the constraint (47), any dynamical variable
$G(T',Q';\,P_{T'}, P')$ can be replaced by an equivalent variable
$F(T',Q';\,P') \,\mbox{:=} \, G(T',Q';\,0, P')$. The variable $F$ is
a perennial if $\{F, \, H \} = 0\,$. Equation (47) enables us to
conclude that perennials are simply arbitrary functions of the initial
data. They cannot depend on $T'$:
\begin{equation}
{\bf Perennials}\ :\ \ \ \ F = F( Q';\,P').
\end{equation}
No perennial ever changes along a dynamical trajectory. To observe
change, we must observe at least one dynamical variable, like $T$ or
$Q$, which changes.
An opposite view has been expressed by Rovelli \cite{Rovelli}. I
interpret his paper as saying that to observe a changing dynamical
variable, like $Q$, amounts to observing a one-parameter family
\begin{equation}
Q'(\tau ) \, \mbox{:=} \, Q' + P' \, \tau = Q - P \,(T-\tau ),
\ \ \tau \in {\sf R}
\end{equation}
of perennials. The perennials (49) are the values of $Q$ at $T=\tau$.
By observing the perennials $Q'(\tau _{1})$ and $Q'(\tau _{2})$ one
can infer the change of $Q$ from $T= \tau _{1}$ to $T = \tau _{2}\,$.
The problem with such a view is that one is not told how to
observe $\tau$. One way of observing $\tau$ is to watch the dynamical
variable $T$ (the hand of an ideal Newtonian clock). The value of $T$
is $\tau$. However, this amounts to observing a dynamical variable $T$
which is not a perennial. An alternative is to say that one can
observe $\tau$ directly. Again, one is forced to admit that one can
observe an entity which is not a perennial. The third alternative is
to say that because perennials are constants of motion, it does not
matter when they are observed. One can observe all the perennials
$Q'(\tau)\,, \ \tau \in {\sf R}$ at once, and infer `the change of $Q$
with $T$' from that instantaneous observation. Any instant is like
any other, and each contains the same set $\tau \in {\sf R}$ of
perennials from which the change is inferred. This does not make me
too happy either. If all time $\tau$ is eternally present, all time is
irredeemable.
My discussion was so far concerned with the epistemological
status of observables. I tried to argue that the identification of
observables with perennials drives one to a Parmenidean view of the
world. Physicists are soundly sceptical of epistemological arguments,
and I am not deluding myself that my argument is an exception.
``Refutations are seldom final; in most cases, they are only a prelude
to further refinements.'' Significantly, Bertrand Russell made this
remark when closing his discussion of Parmenides \cite{Russell}.
So far I argued that some observables are not perennial. I
must now defend my other point, namely, that perennials are often
difficult to observe. In this part of the discussion, I take the
attitude of physical common sense, that at any instant one can
directly observe the position $Q$ of the particle, its momentum $P$,
and the time $T$ on an ideal Newtonian clock, but not the position
$Q'$ which the particle had at time $T = 0$. The initial position
$Q'$, which does not change with T and is a perennial, is {\em
inferred} from the observed data $Q$, $P$, and $T$ by using Eq.(45).
For a free particle, such an inference is easy because we know how to
integrate equations of motion. However, even for such a simple system
as a free particle, the inference may be hampered by experimental
errors. If one determines $P$ with an error $\Delta P$, the error in
the inferred value of $Q'$ scales with $T$. If the particle moves on a
circle and T is large, it is practically impossible to infer from the
observations at T where on the circle the particle was at $T=0$. For
more complicated Hamiltonians, like those governing dynamics of many
interacting particles, the task of inferring perennials becomes pretty
hopeless. Take, e.g., a globular cluster, observe the current
positions and momenta of the stars, and then try to infer what were
their positions and momenta when the cluster was formed some 15
billion years ago.
In quantum theory, there is yet another reason why perennials are
difficult to observe. To measure a quantum variable, one needs to
design an apparatus with appropriate coupling. \footnote{My discussion
takes place within the framework of von Neumann's theory of
measurement. It should be rephrased in a scheme like that advocated by
Hartle in the present volume.} Theoretically, it is possible to find
an apparatus which measures an arbitrary quantum variable $\hat{F} =
F(\hat{Q} , \hat{P})\,$. Experimentally, this can be done only for a
small number of especially simple variables, like $\hat{F}=\hat{Q}$ or
$\hat{F}=\hat{P}$. For elementary systems, like a free particle or a
harmonic oscillator, the initial-data perennials (45) are linear
functions of the current data $\hat{Q}$ and $\hat{P}$.
Experimentalists know how to build the apparatuses for measuring such
perennials. An example is the discussion of non-demolition experiments
for detecting gravitational waves \cite{non-demol}. To circumvent the
limits imposed by the uncertainty principle, one constructs an
apparatus for monitoring the initial length of an oscillating bar,
i.e., the perennial like $Q'$ of Eq.(45) for a linear harmonic
oscillator. However, even for such a simple system as the hydrogen
atom, the initial-data perennials are complicated functions of the
current data $\hat{Q}$ and $\hat{P}$. It is difficult to conceive an
apparatus which would monitor such perennials at all times.
If the dynamical system is not Newtonian, i.e., if the
Hamiltonian constraint is not linear in the momentum $P_{T}$ conjugate
to a time variable $T$, the practical difficulty of determining
classical perennials from the current data turns into something much
more serious: into an argument questioning their very existence. A
classical example is an asymmetric top spinning around a fixed point
in a homogeneous gravitational field. Describe the configuration of
the top by the Euler angles $Q^{a}=(\phi, \psi, \theta)$, where
$\theta$ is measured from the direction of the field. The
Hamiltonian $h$ of the top is a quadratic function of the momentum
$P_{a}$. Constrain the motion of the top to be taking place with a
definite energy $E$:
\begin{equation}
H \, \mbox{:=} \, h-E=0 \,,\ \ h=\case{1}{2}G^{ab}(Q)P_{a}P_{b} + V(Q) \, .
\end{equation}
The trajectory of the top in the phase space $(Q^{a},P_{a})$ is
generated by the Hamiltonian constraint (50). Notice that we do not
ask how the top {\em moves} in the Newtonian time $T$, we are merely
asking about its {\em trajectory}. The momentum $P_{T}$ does not enter
into the constraint (50); it was replaced by a constant $E$.
Perennials are defined as those dynamical variables
$F(Q^{a},P_{a})$ that have a (weakly) vanishing Poisson bracket with
$H$. Notice that $T$ cannot be used in the construction of perennials
because it no longer is a canonical variable.
One perennial is the angular momentum $M_{\theta}$ about the
direction of the gravitational field. This perennial is linear in the
momentum $P_{a}$:
\begin{equation}
M_{\theta} = M^{a}(Q)P_{a}\, .
\end{equation}
A century ago, Poincar\'e asked the question \cite{Poincare}: Does the
top have any other integrals of motion than those of vis viva and the
area? In the way I formulated the problem, this translates into the
question: Is there any perennial besides $M_{\theta}\,$? The answer
is {\em no\/} \cite{Arnold}.
The configuration space $(T,Q)$ of a parametrized free Newtonian
particle is two-dimensional, and there is one Hamiltonian constraint.
There are $2 \times (2-1) = 2$ independent perennials (45); any other
perennial is their function. An $n$-dimensional parametrized Newtonian
system should have $2(n-1)$ independent perennials. The top is a
three-dimensional system, and one would expect to find four
independent perennials. However, the constraint (50) does not have the
Newtonian form, and there is only one perennial, (51).
Let me briefly return from models to canonical gravity. General
relativity is not a parametrized field theory whose constraints have a
`Newtonian' form (44). In particular, both in geometrodynamics and in
connection dynamics, the Hamiltonian constraint is quadratic in the
momenta. The supermetric has some non-trivial dependence on the
canonical coordinates. In these respects, the Hamiltonian constraint
resembles the constraint (50) for the top. This prompts the following
remarks:
\begin{itemize}
\item {We do not know how to construct perennials for
canonical gravity.}
\item {We do not know how to select families of
perennials (similar to the family (49)) labeled
by a functional time parameter (similar to $\tau$) which would
correspond to `simple' dynamical variables as the volume observable
(40) (similar to $Q$).}
\item {So far, we did not find a single gravitational
perennial. \footnote{It is not clear whether the
interesting result reported at this meeting by Goldberg {\em et al.\/}
can be recast into a construction of a perennial.} The existence of a
complete set of perennials would imply that gravity is a completely
integrable theory. They are indications that it is not
\cite{K-nosym,And+Torre}. It is likely that the gravitational
perennials are rare, and it is quite possible that there are none.}
\end{itemize}
Perennials in canonical gravity may have the same ontological
status as unicorns ---{\em a priori\/}, these are possible animals,
but {\em a posteriori\/}, they are not roaming on the Earth. According
to bestiaries, the unicorn is a beast of fabulous swiftness, strength,
and beauty, but, alas, it can be captured only by a virgin
\cite{Warner}. Corrupt as we are, we better stop hunting mythical
beasts.
\section{Hilbert space}
Once we have decided what dynamical variables can be observed, we
need to know what is the statistical distribution of their observed
values. In quantum mechanics, probabilities are determined by the
inner product in a Hilbert space. Therefore, we need to endow the
space of physical states with a Hilbert space structure.
The proposals on how to find the inner product depend on what
position one takes on observables. Let me first discuss the proposal
\cite{Ash-qp}, which relies on identifying observables with
perennials:
\begin{itemize}
\item{Choose an inner product $ \langle \Psi _{1}| \Psi _{2} \rangle$
on the solution space ${\cal F}_{0}$ such that all {\em real} quantum
perennials are self-adjoint under it.}
\end{itemize}
In geometrodynamics, the phase space is real and it is easy to
say when a dynamical variable is real. I return to the reality problem
in connection dynamics in the next section.
There are several problems with the above proposal. First of
all, we have seen that there may not be any perennials in canonical
gravity, or that at least there may not be a sufficient number (a
complete set) of them. If so, the proposal on how to determine the
inner product either loses its content, or becomes too weak.
Secondly, even when one disregards this difficulty, one should notice
that the proposal as it stands is self-contradictory. If $\hat{F}$ and
$\hat{G}$ are quantum perennials, so is $\hat{F} \hat{G}$. If
$\hat{F}$ and $\hat{G}$ are self-adjoint under the inner product
$\langle \Psi _{1} | \Psi_{2} \rangle$, $ \hat{F} \hat{G}$ is not. To
remove the contradiction, one needs to find `fundamental perennials',
and approximate all other perennials by polynomials of the fundamental
perennials. One can then require that only the fundamental perennials
be self-adjoint, and symmetrically factor order the polynomials which
define the remaining perennials. Unfortunately, the original
fundamental variables $g$ and $p$ (or $A$ and $E$) are not perennials,
and we lack a guiding principle on what the fundamental perennials may
be.
The third problem with the proposal is that the solution space
${\cal F}_{0}$ is probably larger than the space of physical states.
We have seen that it may contain `improper elements', `unbounded
states', and `states with negative norms'. The definition of a
perennial $\hat{ F}$ requires that $\hat{ F}$ commutes with the
constraints on the solution space ${\cal F}_{0}$. If ${\cal F}_{0}$ is
too large, the set of perennials may be too small: Some physically
significant perennials may have been excluded by the requirement that
they commute with the constraints on a larger-than-physical space of
solutions. Further, it may happen that those perennials which remain
cannot be made self-adjoint under an inner product on the whole
solution space, but only on a drastically reduced space from which the
`unphysical' states have been excluded. In brief, it seems impossible
to follow step by step the `quantization program': firstly, to find
the space of solutions without having the inner product to determine
which states are physical, secondly, on that space of solutions to
define the perennials, and thirdly, to find the inner product on
${\cal F}_{0}$ which makes all such perennials self-adjoint. Rather,
the steps should be replaced by a single jump. As I am growing older,
the difficulty of replacing three steps by a single jump is becoming
more and more obvious.
The second standpoint is that observables do not need to commute
with the Hamiltonian constraint, but only with the gauge constraints.
If so, they do not act in the space of solutions: if $\Psi \in {\cal
F}_{0}$ and $\hat{F}$ is an observable, $\hat{F} \Psi \notin {\cal
F}_{0}$. To proceed, one should
\begin{itemize}
\item{abandon the space of solutions and work
instead in the space of instantaneous states.}
\end{itemize}
To talk about instantaneous states requires a decision about
what is an instant. An instant in a relativistic spacetime is a
spacelike hypersurface. However, spacelike hypersurfaces are not
elements of the gravitational phase space. The task is to find an
observable $T$ (or, rather, a set of $\infty ^{3}$ commuting
observables, to account for $\infty ^{3}$ hypersurfaces) whose value
uniquely fixes a hypersurface in a Ricci-flat spacetime generated by
the evolution of the classical canonical data. Such an observable is
called an {\em internal time}. (The adjective `internal' means
`constructed solely from the phase-space variables'.)
The Hamiltonian constraint is interpreted as an evolution
equation for $\Psi$ in $T$. One tries to cut down ${\cal F}_{0}$ to a
linear subspace $ {\cal F}_{0}' \subset {\cal F}_{0}$ whose elements
are in a one-to-one correspondence with the instantaneous values of
$\Psi$: the restrictions $\Psi _{T}$ of $\Psi$ to a fixed hypersurface
$T$. These restrictions are the instantaneous states $\Psi _{T} \in
{\cal F}_{T}$. The program is to find an inner product in ${\cal
F}_{T}$ which is independent of $T$, i.e., which is conserved in
internal time. The discussion centers on how different forms of the
Hamiltonian constraint (the Wheeler-DeWitt form, and others) suggest
what such an inner product may be. A $T$-independent inner product can
be interpreted as an inner product in ${\cal F}_{0}' \,$. In general,
the observables $\hat{F}$ depend on $T$. One requires that they be
factor ordered so that, at each $T$, they are self-adjoint under the
inner product in ${\cal F}_{T} \,$. The expression $ \langle \Psi _{T}
| \hat{F} | \Psi _{T} \rangle$ is interpreted as the mean value of
$\hat{F}$ in the state $\Psi _{T}$ at the internal time $T$.
These things are more easily said than done. The internal time
proposal meets as many difficulties as the approach based on the
concept of perennials. I discussed the problems of time in a recent
review \cite{K on t} which complements my present treatment
of observables.
It is sometimes maintained that the approach based on perennials
somehow avoids the problems of time. It would be great if it did, but
I fear it does not. A closer look reveals that the problems of time
and the problem of perennials are rather closely related. A Czech
saying has it that the devil thrown out of the door returns through a
window.
\section{Reality conditions}
The connection dynamics looks in many respects simpler than
geometrodynamics, but its simplicity has been bought at a price: the
SO(3) connection $A$ is necessarily complex. One needs to ensure that
the quantum theory based on such a connection describes a real
gravitational field.
One can attempt to accomodate complex objects in canonical
gravity in two different ways:
{\em Complexify the Einstein theory\/}, i.e., work with
complex metrics $\gamma$ on a real spacetime manifold $\cal M$. The
statement that $({\cal M}, \gamma)$ is Ricci-flat amounts to a system
of coupled equations for the real and imaginary parts of the complex
metric $\gamma\,$. These equations can be derived from a real action
whose Lagrangian is the real part of the complex curvature scalar.
Introduce the Ashtekar variables $A^{i}_{a}\,$, $E^{a}_{i}$ for the
complexified spacetime. Both $A$ and $E$ are now complex. The
canonical form of the action leads to the Poisson brackets among these
variables and their complex conjugates.
To restrict the spacetime metric to be real, one imposes the
condition that its imaginary part vanishes. In the canonical version
of the theory, this imposes the reality conditions (27) on $A$ and
$E$. The reality conditions are preserved by the constraints: when the
evolution starts from real canonical data, it continues building a
real spacetime. However, the Poisson brackets among the reality
conditions do not vanish: to put the imaginary part of the metric and
its rate of change equal to zero amounts to requiring both a canonical
coordinate and its conjugate momentum to vanish. It means that the
reality conditions are, in Dirac's terminology, second-class
constraints \cite{Dirac}. Such constraints must be eliminated before
quantization. Unfortunately, their elimination destroys the new
variables.
An alternative is to derive the complexified equations from a
holomorphic Lagrangian \cite{Jacobson}. The corresponding canonical
theory knows how to form the Poisson brackets among $A$ and $E$, but
the Poisson brackets involving the complex conjugates $\bar{A}$ and
$\bar{E}$ are undefined. The status of the reality conditions thus
remains unclear and one does not know what to do with them on
quantization.
{\em Use complex chart on a real phase space.} The second option
is to consider $A$ and $E$ as a complex chart on a real phase space
($E,\,-K$). This is similar to introducing a complex chart $Q$ and
$Z=Q-iP$ on the real phase space ($Q,\,P$) of a harmonic oscillator.
The proposal \cite{Ash-book,A+T} is to ignore the reality conditions
in the first five steps of the quantization program. In particular,
the vector space $\cal V$ spanned by the fundamental variables $A$ and
$E$ is allowed to be complex, and so are the dynamical variables
$F[A,E] \in \cal A$ and the perennials $F[A,E] \in {\cal A}_{0}\,$.
One knows how to complex conjugate, $\bar{F}\,$, the
elements $F$ of the classical spaces $\cal V$ and $\cal A$. The task
is to define the corresponding operation, ${\star}\,$, on the
elements $\hat{F}$ in $\hat{\cal V}$ and $\hat{\cal A}$. Ashtekar's
proposal is first to define the $\star$ operation in $\hat{\cal V}$ by
requiring that complex conjugate elements of $\cal V$ are carried into
the $\star \,$- related elements of $\hat{\cal V}\,$:
\begin{equation}
F,\bar{F} \in {\cal V} \ \ \Longrightarrow \ \
\widehat{F}=\widehat{\bar{F}}{}^{\star}\, .
\end{equation}
The $\star$ operation is then extended from $\cal V$ to $\cal A$ by
using the axioms of the involution operation:
\begin{eqnarray}
(a\hat{F} + b\hat{G})^{\star} &=& \bar{a}\hat{F}^{\star}
+ \bar{b}\hat{G}^{\star}\, ,
\nonumber \\ ~
(\hat{F}\hat{G})^{\star} &=& \hat{G}^{\star}\hat{F}^{\star}\,, \\
(\hat{F}^{\star})^{\star} &=& \hat{F}\, , \nonumber \\
\forall \hat{F},\hat{G} \in {\cal A} &{\rm and}& \forall a,b \in {\sf
C}\, . \nonumber
\end{eqnarray}
If $\hat{F}^{\star} = \hat{F}\,$, the operator $\hat{F}$
represents a real dynamical variable. If there are no constraints,
this dynamical variable is an observable. The expectation value of
$\hat{F}$ should be real. This objective can be achieved by requiring
that the inner product $\langle \Psi _{1} | \Psi _{2} \rangle$ in
$\cal F$ be such that it makes all $\star \,$-related operators
Hermitian adjoints,
\begin{equation}
\hat{F}=\hat{G}^{\star} \ \ \Longrightarrow \ \ \langle \Psi _{1}
| \hat{F} \Psi _{2} \rangle = \langle \hat{G}
\Psi_{1}|\Psi_{2}\rangle \, ,
\end{equation}
and hence all operators representing real variables self-adjoint. If
the $\star$ operation in $\hat{\cal A}$ is determined by the $\star$
operation in $\hat{\cal V}$ as in Eqs.(52) and (53), it is sufficient
to require that the condition (54) holds for all fundamental variables
$\hat{F} ,\hat{G} \in \hat{\cal V}\,$.
Canonical gravity, however, is a constrained system. Ashtekar's
program assumes that only perennials can be observed, and that their
expectation values are obtained from an inner product on the space of
solutions. To impose the reality conditions, one needs to define the
$\star$ operation for perennials. This would be straightforward if the
$\star$ operation from $\hat{\cal A}$ could be restricted to
perennials. Unfortunately, this does not need to be the case: if
$\hat{F}$ is a perennial, $\hat{F}^{\star}$ does not need to be a
perennial (though it may be a perennial under special circumstances).
The hope is that there is a `sufficient' number of perennials
$\hat{F}$ whose $\star\,$-adjoints $\hat{F}^{\star}$ are also
perennials. By `sufficient' one means that the condition (54), when
imposed on these perennials, uniquely determines the inner product in
${\cal F}_{0}\,$.
To summarize, Ashtekar's program calls for implementing the
reality conditions as requirements on the inner product in the space
of solutions ${\cal F}_{0}\,$. Firstly, one must find a sufficient
number of $\star \,$-adjoint perennials, and then require that these
be Hermitian adjoints under the inner product.
One can ask two questions about this proposal. The first is
whether it works for simple model systems. The second is whether it
can reasonably be expected to work in canonical gravity.
The answer to the first question is yes. Ashtekar's proposal
determines the inner product for a number of simple systems (a
harmonic oscillator with complex chart, a parametrized Newtonian
particle, a free relativistic particle on a flat background). It also
works for $2+1$ gravity and linear field theories on a
$(3+1)$-dimensional flat Lorentzian background, including Maxwell's
electrodynamics and linearized gravity. With the exception of $2+1$
gravity (which does not have any field degrees of freedom) these
examples are reducible to collections of harmonic oscillators.
To approach the second question, one should ask whether there are
any relevant differences between the prototype of a linear harmonic
oscillator and full canonical gravity. (By `relevant' I mean relevant
to the proposal on handling the reality conditions.) I feel there are
two such differences:
In the harmonic oscillator problem, one works with the
fundamental variables $Q$ and $Z = Q - iP\,$, which are analogous to
$E$ and $A$ in connection dynamics. The vector space $\cal V$ is
spanned on $Q$, $Z$, and 1. The reality conditions are the conditions
\begin{equation}
\bar{Q} = Q \ \ {\rm and}\ \ \bar{Z} = -Z + 2Q
\end{equation}
on the dynamical variables $F=Q$, $G=Z$, and their complex conjugates
$\bar{F}$ and $\bar{G}\,$. Both $F$ and $G$, and $\bar{F}$ and
$\bar{G}$ lie in $\cal V \,$. It is thus possible to define the
$\star$ on $\cal A$ by Eqs.(52) and (53), and to impose the reality
condition (54).
In connection dynamics, $\cal V$ is spanned by $A$, $E$ and 1.
The second reality condition (27), however, is not a condition on the
elements of $\cal V$, because $\Gamma [E]$ is a non-linear functional
of $E$. (The same remark applies to polynomial forms of reality
conditions.) This prevents one from defining the $\star$ operation on
$\cal V$, as in Eq.(52), and from extending it to $\cal A$, as in
Eq.(53).
This difficulty can be clarified on simple models. Take a
one-dimensional system with the Hamiltonian \footnote{In the sector
$Q>0$, the Hamiltonian (56) can be brought into the form $h = \case
{1}{2} p^{2} + 4 q^{-4}$ by the canonical transformation $q = \sqrt{2}
\, Q^{\case{1}{2}}\,$, $p = \sqrt{2}\, Q^{\case{1}{2}}P\,$.}
\begin{equation}
h\,\mbox{:=}\, QP^{2} + Q^{-2}
\end{equation}
and introduce the complex chart $(Q,Z)$, with
\begin{equation}
Z = Q^{-1} - iP,
\end{equation}
on the real phase space $(Q,P)$. The Hamiltonian (56) becomes
polynomial in $Q$ and $Z$,
\begin{equation}
h = -QZ^{2} + 2Z.
\end{equation}
The reality condition on Z can be written either in a non-polynomial
form linear in $Z$, or in a polynomial form:
\begin{equation}
\case{1}{2}
(Z + \bar{Z}) = Q^{-1}\,,\ \ \ {\rm or}\ \ \ Q(Z+\hat{Z})=2 \, .
\end{equation}
Whichever form we use, it is not a condition in the complex vector
space $\cal V$ spanned by the fundamental variables $Q$ and $Z$.
This is my first reason for believing that the harmonic
oscillator is not quite representative of canonical gravity. The
algorithm for handling reality conditions needs to be checked on more
general models than those which have been investigated so far, like
the model I have just described.
The second difference between the oscillator and canonical
gravity is that the later is a parametrized theory. Ashtekar's
proposal on how to handle the reality conditions depends on the
existence of a sufficient number of perennials, and on the possibility
to define a $\star$ operation on their algebra. I expressed my doubts
that there exists a sufficient number of perennials in canonical
gravity. Even if there is a sufficient number of perennials, it
remains unclear whether it is possible to extend the $\star$ operation
{}from $\hat{\cal V}$ to $\hat{\cal A}\,$, and then to restrict it to a
suitable subset of perennials.
I do not claim that these problems are insurmountable, but I feel
that they represent a major unsolved problem of connection dynamics.
\section{Conclusions}
Where do we stand? We certainly gained in the years a good
geometric understanding of classical general relativity as a canonical
dynamical system. In quantum theory, we inherited a set of rules of
thumb called Dirac constraint quantization. They were never precise,
and Dirac himself never claimed they were much more than rules of
thumb. People tried to make them more precise and they ended with
something resembling the seven gates I described.
Let me revisit those gates and ask what steps in the quantization
program have actually been accomplished. And, even more importantly,
let me summarize what are the main unsolved problems.
\begin{enumerate}
\item{Different sets of fundamental variables (not only those which
I mentioned in this report) have been explored and understood.
We also know how to take care of the positivity restrictions on the
metric variables \cite{Ish+}}.
\item{Most of the work on turning constraints into operators is
formal. Both the regularization problem and consistency
problem remain open. Very little is known about how to handle other
dynamical variables, especially the future candidates for observables
or perennials.}
\item{Important work has been done on clarifying the mathematical
status of the states $\Psi[g]$ and $\Psi[A]$ and of the
fundamental operators \cite{Ish+}. The connection representation
has been linked to the loop representation \cite{A+I}. One should note
that the latter investigation has been successful only for {\em
real\/} connections.}
\item{Because the regularization and consistency problems for the
constraints have not been satisfactorily resolved, all
attempts to find the states which solve the quantum constraints (30)
are to a large extent formal. It is notable that connection dynamics
actually exhibited a large number (indeed, infinitely many) such
solutions. Most of these were obtained in the loop representation and
lie outside the scope of this report \cite{Rov-loops}. When comparing
this success with the lack of solutions in geometrodynamics, one
should keep in mind that these solutions correspond to degenerate
metrics which geometrodynamics excludes. One solution that can be
written directly in the connection representation is the exponential
of the Chern-Simons form \cite{Kodama}. Passing from particular
solutions to general considerations, it is not clear what boundary or
other conditions should be imposed on the solutions $\Psi \in {\cal
F}_{0}$ to select the true physical states.}
\item{The problem of what quantities can be observed (and how they can
be observed) is one of the most intriguing and important
questions in quantum gravity. A widely held view (which I dispute) is
that one can observe only perennials. No true perennials, classical or
quantum, have so far been found, and even if they exist, finding them
is difficult. I feel we should instead concentrate on formulating and
proving (non?)existence theorems about perennials. \\
Unlike perennials, there are many concrete examples of classical
observables. It is, however, obscure what classical observables are to
be represented by operators, and on what space these operators act.
This is connected with the problem of time: one does not expect the
time observable to be represented in quantum mechanics by an operator.}
\item{Another outstanding problem of canonical quantum gravity is the
construction of the inner product. Quantum geometrodynamics has
been unsuccessful in this task \cite{DW,K-nosym}, and connection
dynamics has hardly done more than formulate broad guidelines on how
one might try to proceed. These guidelines crucially depend on the
existence of perennials. \\ In contrast, one knows how to construct
(at the formal level) the inner product for parametrized field
theories \cite{K-Ox}. Each choice of an internal time casts
canonical gravity into the mold of a parametrized field theory and
leads to an inner product. The procedure, however, is not without
problems \cite{K on t}. One which is closely related to the
problem of perennials is that internal time may not exist globally
\cite{Torre-QGnotP}}.
\item{Connection dynamics, unlike geometrodynamics, needs to take care
of reality conditions. Ashtekar's proposal is to impose them
as requirements which determine the inner product.
Two problems arise: firstly, the necessity of finding a complete set
of perennials and defining on them the $\star$ operation and,
secondly, the high polynomiality of the reality conditions, which
takes them out of the realm of the fundamental vector space $\cal
V$.\\ The reality conditions are the only major problem which does not
exist in geometrodynamics. The ability of connection dynamics to
handle this problem will be
crucial for judging its success in the
Galilean contest between the two chief systems of canonical gravity.}
\end{enumerate}
The problem of reality conditions exemplifies the general pitfall
of any quantization program. As I described it, the program resembles
seven doors to the law, each of them guarded by a doorkeeper. We
certainly did not sit on a stool at the side of the first door for
days and years: we tried to enter the law. However, on our way through
the doors we learned that their orderly sequence is deceptive. One can
never be sure of passing a door before all have been passed. The
entries are so interconnected that they cannot be made separately:
What is a solution of the quantum constraints depends on the choice of
fundamental variables and the form of the constraints. What solutions
are physical depends on the inner product. What is an inner product
depends on what quantities are observable. What quantities are
observable may depend on what solutions are physical. More often than
not we are caught in a vicious circle which calls for entering all the
doors at once.
This may be frustrating, but it should have been expected.
Indeed, it would be rather disappointing if one could reach a truly
fundamental theory like quantum gravity by following step by step a
travel guide, or its medieval predecessor, a pilgrim's itinerary to a
wholy shrine. In this spirit, let me end my account of canonical
quantum gravity in the dark ais
les of St. V\'{\i}t's cathedral of my native
city of Prague, talking to a priest \cite{Kafka}:
\begin{quotation}
``You have studied the story more exactly and for a longer time than I
have,'' said K. They were both silent for a while. Then K. said: ``So
you think that the man was not deceived?'' ``Don't misunderstand me,''
said the priest, ``I am only showing you the various opinions
concerning that point. You must not pay too much attention to them.
The scriptures are unalterable and the comments often enough merely
express the commentators' despair.''
\end{quotation}
\section*{Acknowledgments}
The work on this report has been partially supported by the NSF grants
PHY-9207225 and INT-8901512 to the University of Utah. I want to thank
Julian Barbour for his careful reading of the final draft of the paper.
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The talk will have three parts in which I will discuss the results achieved with my colleagues about nondeterministic, deterministic, and semi-deterministic omega-automata. For nondeterministic automata, I will show which aspects of property automata can influence the performance of explicit model checking and what improvements can be made in LTL-to-automata traslation if we have some knowledge about the verified system. Then I will present our efficient translation of a fragment of LTL to deterministic automata. Finally, I will explore the jungle of Buchi automata classes that lie between deterministic and nondeterministic automata. I will present how to efficiently complement semi-deterministic automata and how to obtain them from nondeterministic generalized Buchi automata.
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{
"redpajama_set_name": "RedPajamaC4"
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PSYCHIC POWERS are natural and intuitive – everyone can develop their psychic potential if they wish to. Developing this skill is not only a means of connecting with the spirit world; it helps us achieve greater awareness and control over our destiny.
Our own unique psychic potential utilises our ability to be able to receive, access and transmit vast quantities of information from our day to day life. We can learn to strengthen, access and make use of these abilities, whether it is for the simplest of tasks or a life-threatening situation – for instance, choosing, choosing a job or profession, trusting your judgement about a partner or deciding when to have children. The act of being able to see beyond most situations means you can avoid being in the wrong place at the wrong time and avoid making wrong decisions. Psychic awareness is there to help guide and inform you of future problems and opportunities.
Clairvoyance isn't limited to the more esoteric mental and emotional disciplines; it's also firmly grounded in the physical world. To enhance your ESP skills we need to heighten your perception of the physical world as a prelude to exploring your psyche more deeply and discover a sense of connectedness extending beyond the physical senses. Thoughts are not limited to chemical processes in the brain but can travel outside of us and just like a radio receiver, a sensitive person can 'tune in' to these vibrations.
To heighten your ability to 'see' we will work through visualisation exercises to open the third eye and unlock your mental intuition. Learning to know what you 'see' through your inner eye has a grounded relation to what you sense through your physical eyes. And the psychic interpretation of what you know to be 'real' will be encouraged to become stronger, therefore the information that you receive will make more sense in your day to day life and you can use the information in a practical way and be one step ahead.
Our psychic awareness is located in the area of the third eye chakra, from where our higher sense functions and can scan the world around us. We relate to our own thoughts, our interpretations and inner conversations, continually assessing the information that feeds in through the senses. We relate to others by focusing attention on the face – the eyes, the subtle changes of expression, feeling that the 'real' personality is somewhere in there. This arises from the awareness that in the third eye we begin to make sense of and interpret the world. It is all about seeing, not just seeing with the eyes, but seeing with the mind – making sense of and understanding what is being perceived.
Visualisation is the ritual of focusing your concentration on an image of something you want to create, become or empower. To develop the extraordinary sense of the third eye you will need to learn how to focus your mind's eye and discipline your imagination. With regular practice, this is not as hard as it appears.
Begin with a simple image of a flower or a photograph. The image has to be graphic, symbolic or a combination of both. You may want to first copy the image in your mind.
If you find that your imagination wants to develop that image, this is fine as it will help you understand your own imagination. The most powerful way to use visualisation is for acquiring or developing personal qualities such as confidence, warmness of heart as well as for talents or specific skills such an opening up to your psychic abilities. It can even be used for the most banal purposes like getting a parking space, or a taxi.
You can practice this technique with two or more people by visualising the same thing simultaneously; the energy for that thing to manifest will be exponentially multiplied.
There is not much difference between visualising and day dreaming. Both involve bringing an image to a place inside you where you can experience it as real. Visualisation differs from day – dreaming only in how you set it up, day dreaming is undisciplined ramble through the meadows of your imagination- visualisation is a focused foray.
Spend 5 minutes on a daily basis to strengthen your ability to visualise.
Sit in a relaxed position either in a chair or lying on the floor.
Begin with relaxation exercise; tell each part of your body to let go and relax.
Imagine yourself walking through into your third eye. See the door of your mind which will be painted indigo blue, open the door and walk into the chamber of the third eye.
Inside you will see a table, chair and in front of the table there is a cinema projection screen. On the table there is a gadget, which has 3 bright coloured buttons.
Each button will bring on to the screen a different image.
Before you began your practice you would have chosen a specific image that you wish to work with. Press the button for that image and see it project on to the screen.
As you watch the projected image or series of images you begin to invest some power into the image by focusing on it with a portion of passion from your heart and emotion from your belly. Focus your intention on that image until it becomes alive.
Once you have felt the sensation of your image being a real life event, smelled it, tasted it, heart it and touched it, let the image shrink to the size of an atom surrounded in subtle light.
Say an affirmation such as beautiful thoughts create beauty, I choose to be always sustained by this image. Then let the image go, by pressing the gadget and closing down the projector.
See yourself get up off the chair and walk back through the door of your third eye and into your body and back into your life.
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{"url":"https:\/\/littlehouseinthevalley.com\/9kopfmmm\/align-system-of-equations-latex-0fc353","text":", < and =) are the ones aligned for a nice-looking document. You can choose the layout that better suits your document, even if the equations are really long, or if you have to include several equations in the same line. Let's examine an example using split environment: If you wish to align several equations vertically, then you can use the align environment. I want to left align a block of equations. Use the below command in your document's preamble. You can choose the layout that better suits your document, even if the equations are really long, or if you have to include several equations in the same line. and the second part will get right aligned in the next line. The amsmath package provides a handful of options for displaying equations. The first part will be aligned to the left and the second part will be displayed in the next line and aligned to the right. As mentioned before, the ampersand character & determines where the equations align. The default version of LaTeX may lack some of the functionalities or features. Sometimes a long equation needs to be broken over multiple lines, especially if using a double column export style. In LaTeX, amsmath package facilitates many useful features for displaying and representing equations. Example using equation+align, \\begin{align} \\mbox{Minimize } & x_1+x_2+x_3 \\\\ \\mbox{Subject to} & \\\\ & x_1+x_2 \\leq 10 \\\\ & x_2+x_3 \\leq 8 \\\\ & x_1+x_3 \\leq 5 \\end{align} I would like to do this while the equations are left aligned. Additionally, you might add a label for future reference within the document. We eliminate one variable using row operations and solve for the other. Due to the column alignment, the equations appear to be aligned around the equals sign. This environment must be used inside an equation environment. A General Note: Number of Possible Solutions. For an example check the introduction of this document. Use the split environment to break an equation and to align it in columns, just as if the parts of the equation were in a table. Put your equations within an equation environment if you require your equations to get numbered. When numbering is allowed, you can label each row individually. Insert a double backslash to set a point for the equation to be broken. Can I write a LaTeX equation over multiple lines? Math equation in LaTeX provides three stretchable lines\/arrows that appear above or below the equation: braces, bars and arrows. 6. Previous ones: Basics and overview Use of mathematical symbols in formulas and equations Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup. To overcome these challenges, you can use the \"asmmath\" package. Here we arrange the equations in three columns. . Specific usage may look like this: \\begin { align* } & \\vdots\\\\ & =12+7 \\int _ 0 ^ 2 \\left ( - \\frac { 1 }{ 4 } \\left (e ^{ -4t _ 1 } +e ^{ 4t _ 1-8 } \\right ) \\right ) \\, dt _ 1 \\displaybreak [3] \\\\ & = 12- \\frac { 7 }{ 4 } \\int _ 0 ^ 2 \\left ( e ^{ -4t _ 1 } +e ^{ 4t _ 1-8 } \\right ) \\, dt _ 1 \\\\ \u2026 Otherwise, use align* environment in order to print the equation without a line number. You can do this even if the equations are really long, or if you have to include several equations in the same line. Let's check an example: You have to wrap your equation in the equation environment if you want it to be numbered, use equation* (with an asterisk) otherwise. (adsbygoogle = window.adsbygoogle || []).push({}); As discussed earlier in this tutorial, the ampersand (&) character is used to specify at what point the equations should be aligned. This environment must be used inside an equation environment. Below I has \\eqmakebox[LHS][r] to ensure all elements tagged LHS is right-aligned. The result is alignment \u2026 Writing. Split is very similar to multline. For example, Trimming or Overlapping of equations when equations are very long. Recall that a linear equation can take the form $Ax+By+C=0$. No equation number will be printed because the eqnarray* environment is used. It is very easy and straight-forward to include the amsmath package in LaTeX. For an example check the introduction of this document. In the preamble of the document include the code: To display a single equation, as mentioned in the introduction, you have to use the equation* or equation environment, depending on whether you want the equation to be numbered or not. It only takes a minute to sign up. For equations longer than a line use the multline environment. It is advised to use multline environment in order to print It aligns the broken part of equations in columns. ... To achieve correct break and alignment of the above equation try the code below. For example, Trimming or Overlapping of equations when equations are very long. Some of these equations include cases. Using the multiline, aligned packages. As shown in the example above, utilize the split environment if you would like to split the equations into smaller parts. Inside the equation environment, use the split environment to split the equations into smaller pieces, these smaller pieces will be aligned accordingly. Let's look at below example to understand the alignment of several equations: In the above example, we have arranged the equations in three columns. Just like multline, it is used to break long equations. LaTeX assumes that each equation consists of two parts separated by a & ; also that each equation is separated from the one before by an &. It will be even better if the equations can be spaced a little (for example, 1 cm) from the left margin instead of starting from the \u2026 Systems that have a single solution are those which, after elimination, result in a solution set consisting of an ordered triple $\\left\\{\\left(x,y,z\\right)\\right\\}$. Otherwise, use equation* (with an asterisk (*) symbol) if you need equations without the line number. Determine whether the \u2026 Check the below example to understand: Put your equations within an equation environment if you require your equations to get numbered. Use the ampersand character &, to set the points where the equations are vertically aligned. Say that we wish to solve for $x$. I still need to align the right-hand side of the equation to the left. This is a simple step, if you use LaTeX frequently surely you already know this. Again, use * to toggle the equation numbering. Solving a System of Nonlinear Equations Using Substitution. Showing first {{hits.length}} results of {{hits_total}} for {{searchQueryText}}, {{hits.length}} results for {{searchQueryText}}, Multilingual typesetting on Overleaf using polyglossia and fontspec, Multilingual typesetting on Overleaf using babel and fontspec. TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. Example \\begin{align} a_i &= \\begin{dcases} b_i & i \\leq 0 \\\\ c_i & i < 0 \\end{dcases} \\\\ And this trick is to explicitly set a \\tag for the last equation that replaces the automatic numbering. It is important to note that by default, the first part of a broken equation will get left aligned The equations in the block itself are aligned, but that's not related at all to my question! Using \\eqmakebox[][] (from eqparbox) you can have all elements under the same be placed in a box of maximum width, together with individual ment as needed. Equations with Align Environment . If equation (2) is multiplied by the opposite of the coefficient of $y$ in equation (1), equation (1) is multiplied by the coefficient of $y$ in equation (2), and we add the two equations, the variable $y$ will be eliminated. If you just need to display a set of consecutive equations, centered and with no alignment whatsoever, use the gather environment. Otherwise, use equation* environment in order to print the equation without a line number. For the following exercises, determine whether the given ordered pair is a solution to the system of equations. The asterisk trick to set\/unset the numbering of equations also works here. LaTeX assumes that each equation consists of two parts separated by a &; also that each equation is separated from the one before by an &. there are several equations with domains. Contents 1 Introduction 2 Including the amsmath package 3 Writing a single equation 4 Displaying long equations 5 Splitting and aligning an equation 6 Aligning several equations Determining Whether an Ordered Pair Is a Solution to a System of Equations. The split environment will align these smaller parts. In the above example, it is assumed by the LaTeX that each equation consists of two parts\/pieces which are separated by an ampersand (&) character. Figure 2 and Figure 3 illustrate possible solution scenarios for three-by-three systems. To reference your equation anywhere in the document, you need to add the \\label{...} command as shown below. I'm trying to align this system of equations nicely but it doesn't work out. Multiline formulas 3 If you want the consecutive equations of a group of equations to be numbered (2a), (2b) etc., use subequations, inside which you can place the previous constructs, e.g., With a trick you can put all equations into one align (or alignat) and subequations environment and still have different labels. If you want to write a second equation then again put a to write a For example, we might type a system of equations as follows: (You do not need dollar signs.) Given a system of equations, explain at least two different methods of solving that system. It is necessary to use the split environment within the equation environment to work properly. Below example shows how to use the multline environment: Use the equation environment in order to print the equation with the line number. Open an example of the amsmath package in Overleaf. Solve the following system of equations in two variables. y = x 2 +2x +1 = (x + 1)(x + 1) = (x + 1) 2. If you just need to display a set of consecutive equations, centered and with no alignment, use the gather environment. Grouping and Centering Equations. The \\overbrace command places a brace above the expression (or variables) and the command \\underbrace places a brace below the expression. Let's check a more complex example: Here we arrange the equations in three columns. Aligning several equations Also, every equation is isolated using the & from the one previous to it. equations that do not fit into a single line. $\\begin{gathered}5x-y=4\\\\ x+6y=2\\end{gathered}$ and $\\left(4,0\\right)$ 7. Use the split environment to break an equation and to align it in columns, just as if the parts of the equation were in a table. Due to the column alignment, the equations appear to be aligned around the equals sign. Here we use the ampersand (&) command to ensure the equations always line up as desired. I think I could hack it but I keep running into this problem and would like to do it right. In the equation environment, you can only write a single equation. You need to use \\\\ (Double Backslash) for setting the point where you want to break the equation. Go to website. The default version of LaTeX may lack some of the functionalities or features. Otherwise, use equation* (with an asterisk (*) symbol) if you need equations without the line number. No equation number will be printed because the eqnarray* environment is used. In large equations or derivations which span multiple lines, we can use the \\begin {align} and \\end {align} commands to correctly display the aligned mathematics. The standard LaTeX tools for equations may lack some flexibility, causing overlapping or even trimming part of the equation when it's too long. Split is very similar to multline. ... Align a system equation with three separate equations in latex. I want to left align the equations rather than have them centered all the time, because it looks dumb with narrow centered equations. A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear. Splitting and aligning an equation. The align environment is used for two or more equations when vertical alignment is desired; usually binary relations such as equal signs are aligned. To align multiple equations, we use the align*environment. Double backslash (\\\\) provides the functionality of newline character. Quick Ball Pokemon Sun, Cardiac Surgeon Resume Sample, Mini Dehumidifier With Drain Hose, Interesting Facts About The African Wild Dog, Warehouse Living London, Stihl Hedge Trimmer Blade Cover, How To Draw Leather Pants, Louisville Slugger Omaha Batting Gloves, Sony A6600 Tutorial, Custom Louisville Slugger, Hotels In Woburn, Ma, \" \/>\n\nLaTeX will insert a page break into a long equation if it has additional text added using \\intertext {} without any additional commands. Again, the use of an asterisk * in the environment name determines whether the equation is numbered or not. split provides a very similar feature like multline. For e.g., you can include multiple equations within the same line$$and$$select the layout that best suits your document. Let's check an example using align environment: Use the align environment in order to print the equation with the line number. Each equation should be write in-between $$and$$ tags. Otherwise, use equation* environment in order to print the equation without a line number. This package allows you to choose the layout for your document that best suits your requirements. But you have to increment the equation counter manually right after the subequations environment to get a correct numbering for all following equations. \\usepackage{amsmath}. Use equation environment in order to print the equation with line number. Do you know any way that allows a consistent horizontal alignment of the domains? $\\begin{gathered}y - 2x=5 \\\\ -3y+6x=-15 \\end{gathered}$ Show Solution try it. As shown in the example above, utilize the split \u2026 The double backslash works as a newline character. The & symbol tells where to align to$$and$$the \\\\ symbols break to the next line. Make usage of ampersand (&) character in order to align the equations vertically. Again, use * to toggle the equation numbering. When numbering is allowed, you can label each row individually. 0. 5. The array environment is the math mode equivalent \u2026 Mostly the binary operators (=, >$$and$$ TeX - LaTeX Stack Exchange is a question$$and$$answer site for users of TeX, LaTeX, ConTeXt,$$and$$related typesetting systems. To overcome these challenges, you can use the \"asmmath\" package. The asterisk trick to set\/unset the numbering of equations also works here. WordPress\u3067multiline\u3067latex\u3059\u308b\u3068\u304d\u306e\u4fbf\u5229\u306a\u307e\u3068\u3081\uff0e Series on Blogging with LaTeX This is the 3rd post in the series. This code will outputAn example of a string of equations is: Again, the & \u2026 We can surpass these difficulties with amsmath. Any equation that cannot be written in this form in nonlinear. The environment cases inside align results in that domains are not aligned at the same position. If there are several equations that you need to align vertically, the align environment will do it: Usually the binary operators (>, <$$and$$=) are the ones aligned for a nice-looking document. You can choose the layout that better suits your document, even if the equations are really long, or if you have to include several equations in the same line. Let's examine an example using split environment: If you wish to align several equations vertically, then you can use the align environment. I want to left align a block of equations. Use the below command in your document's preamble. You can choose the layout that better suits your document, even if the equations are really long, or if you have to include several equations in the same line.$$and$$the second part will get right aligned in the next line. The amsmath package provides a handful of options for displaying equations. The first part will be aligned to the left$$and$$the second part will be displayed in the next line$$and$$aligned to the right. As mentioned before, the ampersand character & determines where the equations align. The default version of LaTeX may lack some of the functionalities or features. Sometimes a long equation needs to be broken over multiple lines, especially if using a double column export style. In LaTeX, amsmath package facilitates many useful features for displaying$$and$$representing equations. Example using equation+align, \\begin{align} \\mbox{Minimize } & x_1+x_2+x_3 \\\\ \\mbox{Subject to} & \\\\ & x_1+x_2 \\leq 10 \\\\ & x_2+x_3 \\leq 8 \\\\ & x_1+x_3 \\leq 5 \\end{align} I would like to do this while the equations are left aligned. Additionally, you might add a label for future reference within the document. We eliminate one variable using row operations$$and$$solve for the other. Due to the column alignment, the equations appear to be aligned around the equals sign. This environment must be used inside an equation environment. A General Note: Number of Possible Solutions. For an example check the introduction of this document. Use the split environment to break an equation$$and$$to align it in columns, just as if the parts of the equation were in a table. Put your equations within an equation environment if you require your equations to get numbered. When numbering is allowed, you can label each row individually. Insert a double backslash to set a point for the equation to be broken. Can I write a LaTeX equation over multiple lines? Math equation in LaTeX provides three stretchable lines\/arrows that appear above or below the equation: braces, bars$$and$$arrows. 6. Previous ones: Basics$$and$$overview Use of mathematical symbols in formulas$$and$$equations Many of the examples shown here were adapted from the Wikipedia article Displaying a formula, which is actually about formulas in Math Markup. To overcome these challenges, you can use the \"asmmath\" package. Here we arrange the equations in three columns. . Specific usage may look like this: \\begin { align* } & \\vdots\\\\ & =12+7 \\int _ 0 ^ 2 \\left ( - \\frac { 1 }{ 4 } \\left (e ^{ -4t _ 1 } +e ^{ 4t _ 1-8 } \\right ) \\right ) \\, dt _ 1 \\displaybreak [3] \\\\ & = 12- \\frac { 7 }{ 4 } \\int _ 0 ^ 2 \\left ( e ^{ -4t _ 1 } +e ^{ 4t _ 1-8 } \\right ) \\, dt _ 1 \\\\ \u2026 Otherwise, use align* environment in order to print the equation without a line number. You can do this even if the equations are really long, or if you have to include several equations in the same line. Let's check an example: You have to wrap your equation in the equation environment if you want it to be numbered, use equation* (with an asterisk) otherwise. (adsbygoogle = window.adsbygoogle || []).push({}); As discussed earlier in this tutorial, the ampersand (&) character is used to specify at what point the equations should be aligned. This environment must be used inside an equation environment. Below I has \\eqmakebox[LHS][r] to ensure all elements tagged LHS is right-aligned. The result is alignment \u2026 Writing. Split is very similar to multline. For example, Trimming or Overlapping of equations when equations are very long. Recall that a linear equation can take the form $Ax+By+C=0$. No equation number will be printed because the eqnarray* environment is used. It is very easy$$and$$straight-forward to include the amsmath package in LaTeX. For an example check the introduction of this document. In the preamble of the document include the code: To display a single equation, as mentioned in the introduction, you have to use the equation* or equation environment, depending on whether you want the equation to be numbered or not. It only takes a minute to sign up. For equations longer than a line use the multline environment. It is advised to use multline environment in order to print It aligns the broken part of equations in columns. ... To achieve correct break$$and$$alignment of the above equation try the code below. For example, Trimming or Overlapping of equations when equations are very long. Some of these equations include cases. Using the multiline, aligned packages. As shown in the example above, utilize the split environment if you would like to split the equations into smaller parts. Inside the equation environment, use the split environment to split the equations into smaller pieces, these smaller pieces will be aligned accordingly. Let's look at below example to understand the alignment of several equations: In the above example, we have arranged the equations in three columns. Just like multline, it is used to break long equations. LaTeX assumes that each equation consists of two parts separated by a & ; also that each equation is separated from the one before by an &. It will be even better if the equations can be spaced a little (for example, 1 cm) from the left margin instead of starting from the \u2026 Systems that have a single solution are those which, after elimination, result in a solution set consisting of an ordered triple $\\left\\{\\left(x,y,z\\right)\\right\\}$. Otherwise, use equation* (with an asterisk (*) symbol) if you need equations without the line number. Determine whether the \u2026 Check the below example to understand: Put your equations within an equation environment if you require your equations to get numbered. Use the ampersand character &, to set the points where the equations are vertically aligned. Say that we wish to solve for $x$. I still need to align the right-hand side of the equation to the left. This is a simple step, if you use LaTeX frequently surely you already know this. Again, use * to toggle the equation numbering. Solving a System of Nonlinear Equations Using Substitution. Showing first {{hits.length}} results of {{hits_total}} for {{searchQueryText}}, {{hits.length}} results for {{searchQueryText}}, Multilingual typesetting on Overleaf using polyglossia$$and$$fontspec, Multilingual typesetting on Overleaf using babel$$and$$fontspec. TeX - LaTeX Stack Exchange is a question$$and$$answer site for users of TeX, LaTeX, ConTeXt,$$and$$related typesetting systems. Example \\begin{align} a_i &= \\begin{dcases} b_i & i \\leq 0 \\\\ c_i & i < 0 \\end{dcases} \\\\ And this trick is to explicitly set a \\tag for the last equation that replaces the automatic numbering. It is important to note that by default, the first part of a broken equation will get left aligned The equations in the block itself are aligned, but that's not related at all to my question! Using \\eqmakebox[][] (from eqparbox) you can have all elements under the same be placed in a box of maximum width, together with individual ment as needed. Equations with Align Environment . If equation (2) is multiplied by the opposite of the coefficient of $y$ in equation (1), equation (1) is multiplied by the coefficient of $y$ in equation (2), and we add the two equations, the variable $y$ will be eliminated. If you just need to display a set of consecutive equations, centered and with no alignment whatsoever, use the gather environment. Otherwise, use equation* environment in order to print the equation without a line number. For the following exercises, determine whether the given ordered pair is a solution to the system of equations. The asterisk trick to set\/unset the numbering of equations also works here. LaTeX assumes that each equation consists of two parts separated by a &; also that each equation is separated from the one before by an &. there are several equations with domains. Contents 1 Introduction 2 Including the amsmath package 3 Writing a single equation 4 Displaying long equations 5 Splitting and aligning an equation 6 Aligning several equations Determining Whether an Ordered Pair Is a Solution to a System of Equations. The split environment will align these smaller parts. In the above example, it is assumed by the LaTeX that each equation consists of two parts\/pieces which are separated by an ampersand (&) character. Figure 2 and Figure 3 illustrate possible solution scenarios for three-by-three systems. To reference your equation anywhere in the document, you need to add the \\label{...} command as shown below. I'm trying to align this system of equations nicely but it doesn't work out. Multiline formulas 3 If you want the consecutive equations of a group of equations to be numbered (2a), (2b) etc., use subequations, inside which you can place the previous constructs, e.g., With a trick you can put all equations into one align (or alignat) and subequations environment and still have different labels. If you want to write a second equation then again put a to write a For example, we might type a system of equations as follows: (You do not need dollar signs.) Given a system of equations, explain at least two different methods of solving that system. It is necessary to use the split environment within the equation environment to work properly. Below example shows how to use the multline environment: Use the equation environment in order to print the equation with the line number. Open an example of the amsmath package in Overleaf. Solve the following system of equations in two variables. y = x 2 +2x +1 = (x + 1)(x + 1) = (x + 1) 2. If you just need to display a set of consecutive equations, centered and with no alignment, use the gather environment. Grouping and Centering Equations. The \\overbrace command places a brace above the expression (or variables) and the command \\underbrace places a brace below the expression. Let's check a more complex example: Here we arrange the equations in three columns. Aligning several equations Also, every equation is isolated using the & from the one previous to it. equations that do not fit into a single line. $\\begin{gathered}5x-y=4\\\\ x+6y=2\\end{gathered}$ and $\\left(4,0\\right)$ 7. Use the split environment to break an equation and to align it in columns, just as if the parts of the equation were in a table. Due to the column alignment, the equations appear to be aligned around the equals sign. Here we use the ampersand (&) command to ensure the equations always line up as desired. I think I could hack it but I keep running into this problem and would like to do it right. In the equation environment, you can only write a single equation. You need to use \\\\ (Double Backslash) for setting the point where you want to break the equation. Go to website. The default version of LaTeX may lack some of the functionalities or features. Otherwise, use equation* (with an asterisk (*) symbol) if you need equations without the line number. No equation number will be printed because the eqnarray* environment is used. In large equations or derivations which span multiple lines, we can use the \\begin {align} and \\end {align} commands to correctly display the aligned mathematics. The standard LaTeX tools for equations may lack some flexibility, causing overlapping or even trimming part of the equation when it's too long. Split is very similar to multline. ... Align a system equation with three separate equations in latex. I want to left align the equations rather than have them centered all the time, because it looks dumb with narrow centered equations. A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear. Splitting and aligning an equation. The align environment is used for two or more equations when vertical alignment is desired; usually binary relations such as equal signs are aligned. To align multiple equations, we use the align*environment. Double backslash (\\\\) provides the functionality of newline character.\n\nThis site uses Akismet to reduce spam. 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| null | null |
Q: std::set_intersection, Intersection list with offset I need get the intersection list between two vectors. And in my case vectors are vector of user type. So to get the number encapsulated I must use a comparer function.
Also I want to able get the intersection with a offSet. For example given two vectors {1,2,3,4,6,8} and {5, 7, 9,10} . Intersection with offSet 2 is { 3,8 } since 3 + 2 = 5 and 8 + 2 = 10.
I suppose the code below should work but instead {3,8} I am getting {3,6,8} . I couldn't figure out how to use comparer function of std::set_intersection. What am I missing?
#include <iostream>
#include <vector>
#include <algorithm>
#include <iterator>
struct A
{
A ( int in ) { a = in; }
int getNumber() { return a; }
bool operator< ( A rhs )
{
return this->a < rhs.a;
}
int a;
};
int main()
{
std::vector<A> v1{1,2,3,4,6,8};
std::vector<A> v2{5, 7, 9,10};
int offSet = 2;
auto lessThanWithOffset = [offSet]( A lhs, A rhs)
{
return lhs.getNumber() + offSet < rhs.getNumber();
};
std::sort(v1.begin(), v1.end());
std::sort(v2.begin(), v2.end());
std::vector<A> v_intersection;
std::set_intersection(v1.begin(), v1.end(),
v2.begin(), v2.end(),
std::back_inserter(v_intersection), lessThanWithOffset);
for(auto n : v_intersection)
std::cout << n.getNumber() << ' ';
}
A: See this, since
The type T satisfies Compare if
*
*The type T satisfies BinaryPredicate, and
Given
*
*comp, an object of type Compare
*equiv(a, b), an expression equivalent to !comp(a, b) && !comp(b, a)
The following expressions must be valid and have their specified effects
*
*For all a, comp(a,a)==false
*If comp(a,b)==true then comp(b,a)==false
*if comp(a,b)==true and comp(b,c)==true then comp(a,c)==true
*For all a, equiv(a,a)==true
*If equiv(a,b)==true, then equiv(b,a)==true
*If equiv(a,b)==true and equiv(b,c)==true, then equiv(a,c)==true
so equiv(A(6), A(7)) == true because !((6+2) < 7) && !((7+2) < 6)
Actually, your lessThanWithOffset isn't complied with standard because
equiv(A(6), A(8)) == true and equiv(A(8), A(10)) == true, but equiv(A(6), A(10)) == false
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Our only child, our son who's a chef, and his true love are taking the plunge! They're committing their love and trust in each other through their wedding vows on a Massachusetts mountain top.
We're excited and happy for them as they embark on married life. It takes great courage to make this commitment at a time when so many choose not to marry. It makes me ponder the sanctity of marriage and what it means in this day and age.
There's great significance in making the promise, surrounded by family and friends who share and support the happy couple's future life together. It gives me joy and a feeling of hope because our son and his beautiful bride are so certain of their love, in spite of the uncertain world we live in. There's something enduring and magical about the sacredness of marriage.
In feng shui, when a prayer's put in place by lighting a candle or ringing a bell or a prayer is written on a prayer flag that flutters in the wind, the belief is that it's sent up to heaven, like prayers people send to manifest their wish.
Ever since the beginning of time, love has been as natural as life. The pair often raised a family, families united into tribes, tribes grew into communities, and communities joined to form city states. This is how civilization began. Joining together improved survival by sharing skills and strengths. Hunters and food gatherers were safer and more successful in groups than alone.
Anyone who's ever been in a marriage or relationship knows that feeling of frustration, of not getting through to their mate but then, remembering why—they're from Venus or Mars. It certainly is true that couples complement each other because sometimes they're the perfect opposites. While this often works for the best, sometimes it's simply exasperating.
"Divorce, never! Murder, maybe," Mom always said about marriage. She had a knack for finding what's funny in life, love, marriage, and aging. Mom and Dad's marriage lasted 62 years! I applaud the example that they set for our family.
I can say from personal experience after 37 years of being married to Duane, that it isn't always easy or fun, but I wouldn't trade my husband for anyone. I love being his partner. We work as a team, dreaming how we want to live our lives. Together we work to surmount each hurdle which helps us navigate bigger challenges. Since we're human, we have ups and downs, just like any other couple does. What I'd say is most important to us is that we share our values and ideals. For the two of us, some are not negotiable.
This is a time in our nation and world when so many things we've cherished and believed are no longer valued or upheld. Mother Earth deserves reverence and care like that which the indigenous peoples have always shown to her. But our actions have greatly damaged her. Caring for one another and defending the rights of humans saves and sustains lives. Once upon a time, our nation was known for upholding human rights. Our Constitution and democracy were revered and defended by our leaders, once upon a time. These priceless treasures were sacrosanct and fought for to the death. With the foundation of our democracy now unsteady and out of balance, we need hope and confidence. Unlike Baby Boomers, whose divorce rate remains high, Millennials who marry (a little bit older and more settled) have a better likelihood of staying married. This is a hopeful sign. A young couple in love that chooses to wed believes in the future. Their belief is contagious.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Michael: I'd like to comment on this subject by stating what I do. Before I do that though, I'd like to warn you not to take my method to heart because frankly, I am in a bit of a "rut" here and I am hoping to solicit any comment possible from the "pros" in the forum regarding my approach. Perhaps, my method may even be the culprit responsible for my "rut" (Carol, please??? & thank you in advance!). If the others do feel my method is best, they will hopefully, either echo that opinion, or will correct me here.
Rather than memorizing each degree of the scale in the arpeggio, I used the degrees of the scale at first, to learn the patterns or "shapes" of the arpeggios, so I could memorize the patterns and intervals surrounding the major, minor, etc. patterns. I also spent alot of time linking each degree chromatically (string by string, position by position, etc.) so I could travel up and down the neck comfortably and correctly within the mode.
Simply put, I think in the terms of patterns (and the chromatics in between the intervals) and the colors/tones to expect with each pattern, whether I'm playing in major, minor, etc. Don't get me wrong, I know which degree is which. If you're going to jam with a blues band, for instance, you will definitely have to know what "one-four-five", "coming down from the five", and "sharping the third" means. I really hope I'm getting my point across, because I'm finding it pretty difficult to put my thinking process into text. Again, please comment pros - how do YOU think when playing / soloing / etc.?
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{
"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
Cosmological observations \cite{Cosmo}, suggesting that our
current universe has a small positive cosmological constant, have
lead to a vigorous search for de Sitter vacua in string theory
(see e.g. \cite{dsString}) and more modestly in supergravity (see
e.g \cite{dsSugra,stabledSa,stabledSb,dsN4a,dsN4b,Kallosh}). Up to this day, the
only examples of stable de Sitter vacua in extended ($N \geq 2$)
supergravity were found by Fr\' e et al in \cite{stabledSa} in the
context of $N=2$ $D=4$ gauged supergravity. Some very specific
elements of 4D supergravity were used to construct these examples,
some of which have no clear string theory origin (like for
instance the de Roo-Wagemans angles). The embedding of their
models in $N=4$ $D=4$ gauged supergravity, and all semi-simple
gaugings of $N=4$ $D=4$ supergravity coupled to six vector multiplets were discussed in
\cite{dsN4a,dsN4b}, but all the de Sitter vacua turned out to be
unstable. In view of these problems, we thought it might be
instructive to look for de Sitter vacua in higher-dimensional
gauged supergravity theories, in order to find out what general
ingredients are necessary to guarantee the existence of stable de Sitter
vacua.
In this paper we will focus on $5D$ $N=2$ gauged supergravity for
several reasons. First of all, it is very similar to $4D$ $N=2$ in
certain respects. They both allow an arbitrary number of vector-
and hypermultiplets, and there exist beautiful relations between
their respective scalar manifolds. On the other hand, tensor
multiplets seem to be somewhat easier to introduce
in $5D$, and there are no duality symmetries in $5D$, which makes
the $5D$ theory simpler \footnote{For interesting recent progress on the
coupling of scalar-tensor multiplets to $4D$ $N=2$ supergravity,
see \cite{scaltens}. The coupling of a vector-tensor multiplet to supergravity was
done in \cite{VecTen}.}.
Besides this, there are some very good other reasons to study $5D$
gauged supergravity. An important motivation comes from the
holographic principle, of which AdS/CFT \cite{AdsCft} is a
particularly nice realization. The best understood example is the
famous correspondence between Type $IIB$ string theory on $AdS_5
\times S^5$ and $N=4$ Super Yang Mills theory. A lot can be
learned just by looking at the $5D$ $N=8$ SO(6) gauged
supergravity (which is assumed to be a consistent truncation of
$IIB$ supergravity on $AdS_5 \times S^5$). For instance, Anti-de Sitter
critical points of the potential imply, under suitable conditions,
non-trivial conformal fixed points for the Yang Mills theory, and
supergravity kink solutions that interpolate between Anti-de Sitter vacua
correspond to renormalization group flows in the dual YM theory
(see \cite{RGflow} for a short review). There has been a lot of
speculation that similarly, de Sitter (quantum) gravity might be
dual to some (still unknown) Euclidean conformal field theory (see
e.g \cite{dsCft}). However, the correspondence is a lot less
clear than in the Anti-de Sitter case (for some recent reviews see
\cite{dsCft2}). Studying gauged supergravities (with probably
non-compact gauge groups) that have stable de Sitter vacua might give us
some more clues about a possible dS/CFT correspondence.
Finally, from a more phenomenological point of view, we should
note that various authors have suggested that the universe may
have undergone a phase where it was effectively 5-dimensional,
see e.g. \cite{5Dpheno,options} and references therein, giving
another reason to understand the vacuum structure of $5D$ ($N=2$)
gauged supergravity theories.
Our paper is organized as follows. In section 2 we repeat some
elements of 5D gauged supergravity coupled to tensor and vector
multiplets. We then study the potential corresponding to
R-symmetry gauging in more detail in section 3, where we prove
that $U(1)_R$ gauging does not give rise to stable de Sitter
vacua. In section 4 we present the first examples of stable de Sitter
vacua in 5D gauged supergravity. Tensors charged under a
non-compact group and R-symmetry gauging seem to be crucial. The
following section contains more examples of $5D$ $N=2$ gauged
supergravities with charged tensors and R-symmetry gauging.
Unfortunately, these do not lead to stable de Sitter vacua. In section 6
we show that if we replace R-symmetry gauging by a specific $U(1)$
gauging of the universal hypermultiplet, we can also get stable de Sitter
vacua. Finally, in the last section, we summarize our results and
mention a few interesting directions for future research.
\section{$D=5,\mathcal{N}=2$ gauged
supergravity coupled to tensor and vector multiplets}
\subsection{The ungauged theory}
The theory we consider is obtained by gauging $D=5,\mathcal{N} =
2$ supergravity coupled to vector- and tensor multiplets. These
theories are completely determined by a constant symmetric tensor
$C_{\tilde{I} \tilde{J} \tilde{K}}$. In particular, the manifold $\mathcal{M},$
parameterized by the $\tilde{n}$ scalars in the theory, can be
viewed as a hypersurface \begin{equation} N(h)=C_{\tilde{I} \tilde{J} \tilde{K}}h^{\tilde{I}} h^{\tilde{J}}
h^{\tilde{K}} = 1 \end{equation} of an ambient space with $\tilde{n}+1$ coordinates
$h^{\tilde{I}}$. The geometry of this surface is referred to as `very
special geometry'. For more details on very special geometry, see
appendix \ref{appA}.
The `very special real' manifolds were classified in \cite{DWVP}
in the case that $\mathcal{M}$ is a homogeneous space
\footnote{Homogeneous manifolds are manifolds for which its isometry group works transitively (on the manifold). The group G of linear transformations of the $h^{\tilde{I}}$ that leave $C_{\tilde{I}\tilde{J}\tilde{K}}$ invariant is a subgroup of the isometry group $Iso(\mathcal{M})$, but it is not the whole isometry group in general. For instance, the symmetric non-Jordan family (\ref{symnonjor}) has isometries that are not in G. Strictly speaking, only those homogeneous spaces for which G works transitively are classified. To the best of our knowledge there is no proof that there are no other homogeneous (or even symmetric) very special real spaces (for which $Iso(\mathcal{M})$ works transitively, but G not).}. The symmetric spaces are a subclass of these, and
where already found in \cite{GST1} and \cite{GST2}. They can be
divided into two subclasses, depending on whether they are
associated with Jordan algebras or not:
\begin{enumerate}
\item When $\mathcal{M}$ is associated with a Jordan algebra, there are two subclasses:
\begin{itemize}
\item The `generic' or `reducible' Jordan class:
\begin{equation} \mathcal{M} = SO(1,1) \times \frac{SO(\tilde{n}-1,1)}{SO(\tilde{n}-1)},
\qquad \tilde{n} \geq1. \end{equation}
\item The `irreducible' or `magical' Jordan class:
\begin{eqnarray} \mathcal{M} &=& SL(3,\mathbb{R})/SO(3), \qquad (\tilde{n}=5) \nonumber \\
\mathcal{M} &=& SL(3,\mathbb{C})/SU(3), \qquad (\tilde{n}=8) \nonumber \\
\mathcal{M} &=& SU^{*}(6)/USp(6), \qquad (\tilde{n}=14) \nonumber \\
\mathcal{M} &=& E_{6(-26)}/F_4. \qquad \qquad (\tilde{n}=26) \nonumber
\end{eqnarray}
\end{itemize}
\item There is one class which is not associated with a Jordan algebra, and which is therefore
referred to as the `symmetric non-Jordan family':
\begin{equation} \mathcal{M} = \frac{SO(1,\tilde{n})}{SO(\tilde{n})},\qquad \tilde{n} >1. \label{symnonjor}\end{equation}
\end{enumerate}
The total global symmetry group of a matter coupled $\mathcal{N}=2$ supergravity theory factorizes into
$SU(2)_R \times G,$ where $SU(2)_R$ is the
R-symmetry group of the theory and $G$ is a group of linear transformations of the coordinates $h^{\tilde{I}}$
that leaves the tensor $C_{\tilde{I}\tilde{J}\tilde{K}}$ invariant.
The symmetry group $G$ gives rise to isometries of the scalar manifold. In the Jordan class, $G$
even coincides with the full isometry group of $\mathcal{M}$.
\subsection{Gauging the theory}
The gauging of $\mathcal{N}=2$ supergravity coupled to $n$ vector multiplets and $m$ self-dual tensor
multiplets was performed in \cite{gz99,Ceresole,5DSGREV}. The field content of the theory is
\begin{equation}
\{ e_{\mu}^{m}, \Psi_{\mu}^{i}, A_{\mu}^{I}, B_{\mu\nu}^{M},
\lambda^{i\tilde{a}}, \varphi^{\tilde{x}}\}\, ,
\end{equation}
where
\begin{eqnarray*}
I&=& 0,1, \ldots n\, ,\\
M&=& 1,2, \ldots 2m\, , \\
\tilde{I}&=& 0,1,\ldots,n+2m\, , \\
\tilde{a}&=& 1,\ldots, \tilde{n}\, ,\\
\tilde{x}&=& 1,\ldots, \tilde{n}\, ,
\end{eqnarray*}
with $\tilde{n}=n+2m$. Note that we have combined the `graviphoton' with
the $n$ vector fields of the $n$ vector multiplets into a single
$(n+1)$-plet of vector fields $A_{\mu}^{I}$ labeled by the index
$I$. Also, the spinor and scalar fields of the vector and tensor
multiplets are combined into $\tilde{n}$-tuples of spinor and scalar
fields. The indices $\tilde{a}, \tilde{b}, \ldots$ and $\tilde{x},
\tilde{y}, \ldots$ are the flat and curved indices, respectively,
of the $\tilde{n}$-dimensional target manifold $\mathcal{M}$ of the
scalar fields. We also combine the vector and tensor indices $I$
and $M$ into one index $\tilde{I}.$
{}From the above fields, only the gravitini and the spin-1/2 fermions transform under the
$SU(2)_R$ symmetry group. However, to gauge this group we need vectors that transform in the adjoint representation of the gauge group. This problem can be solved by identifying the $SU(2)_R$ group with an $SU(2)$ subgroup of the symmetry group G of the $C_{IJK}$, and to gauge both $SU(2)$ groups simultaneously. If you just gauge a $U(1)_R$ subgroup, this problem does not occur since the adjoint of $U(1)$ is the trivial representation. An arbitrary linear combination of the vector fields can be used as $U(1)_R$ gauge field. However, if you also gauge a subgroup K of G, the $U(1)_R$ gauge field has to be a linear combination of the K-singlet vector fields only.
The simultaneous gauging of the $U(1)_{R}$ or $SU(2)_{R}$
R-symmetry group and a subgroup $K \subset G$ introduces a scalar
potential of the form
\begin{equation}
e^{-1}\mathcal{L}_{pot}= -g^{2}P,
\end{equation}
where $P:=P^{(T)}+P^{(R)}$. $P^{(R)}$ arises from the gauging of $U(1)_{R}$ or
$SU(2)_R$, whereas $P^{(T)}$ is due to the tensor fields
transforming under the gauge group $K$ (see \bref{PLambda}).
The potential $P^{(T)}$ can be written as \cite{gzvac} \footnote{We assume $C_{MNP} = C_{IJM} = 0$. More general tensor couplings with $C_{IJM} \neq 0$ are possible, see \cite{5DSGREV}, but we will not consider these here.}
\begin{equation}
P^{(T)}=
\frac{3\sqrt{6}}{16}h^{I}\Lambda_{I}^{MN}h_{M}h_{N}\label{PLambda},
\end{equation}
with $\Lambda_{IN}^{M}$ the transformation matrices of the tensor
fields under the gauge group $K$ and \begin{equation}
\Lambda_{I}^{MN}\equiv\Lambda_{IP}^{M}\Omega^{PN}=
\frac{2}{\sqrt{6}}\Omega^{MR}C_{IRP}\Omega^{PN}, \label{tenstrf}\end{equation} with
${\stackrel{\scriptscriptstyle{\circ}}{a}}^{\tilde{I}\tilde{J}}$ being the
inverse of ${\stackrel{\scriptscriptstyle{\circ}}{a}}_{\tilde{I}\tilde{J}}$.
$\Omega^{MN}$ is the inverse of $\Omega_{MN}$, which is a
(constant) invariant antisymmetric tensor of the gauge group $K$
\begin{equation}
\Omega_{MN}=-\Omega_{NM}, \qquad
\Omega_{MN}\Omega^{NP}=\delta_{M}^{P}.
\end{equation}
For the potential $P^{(R)}$ we have the following general
expression \begin{equation}\label{PotR} P^{(R)} = -4\vec{P}\cdot
\vec{P}+ 2\vec{P}^x\cdot\vec{P}_x \, ,
\end{equation}
where $\vec{P} =h^I\vec{P}_I$ and $\vec{P}_x = h^I_{x}\vec{P}_I$ are vectors in $SU(2)$-space
(see \cite{5DSGREV}).
When we gauge the full R-symmetry group $SU(2)_{R}$ we have
\begin{equation}
\vec{P}_{I} = \vec{e}_{I}V \, ,
\end{equation}
where V is an arbitrary constant, and $\vec{e}_{I}$ are constants
that are nonzero only for I in the range of the $SU(2)$ factor and
satisfy
\begin{equation}
\vec e_I\times \vec e_J= f_{IJ}{}^K \vec e_K \, ,
\end{equation}
with $f_{IJ}{}^K$ the $SU(2)$ structure constants. {}From now on, we will
use indices $A,B,\ldots$ for the $SU(2)$ factor. We can then take
$\vec{e}_{A} \cdot \vec{e}_{B}=\delta_{AB}$ such that (using
\bref{hhChhh}) \begin{equation} P^{(R)}=-4V^2 C^{AB\tilde{I}}h_{\tilde{I}}\delta_{AB},
\label{PR=Chdelta}\end{equation} where we defined \begin{equation}
C^{\tilde{I}\tilde{J}\tilde{K}}\equiv{\stackrel{\scriptscriptstyle{\circ}}{a}}^{\tilde{I}\tI'}
{\stackrel{\scriptscriptstyle{\circ}}{a}}^{\tilde{J}\tJ'}
{\stackrel{\scriptscriptstyle{\circ}}{a}}^{\tilde{K}\tK'}C_{\tilde{I}'\tilde{J}'\tilde{K}'}.
\end{equation}
In the case of $U(1)_{R}$ gauging we have
\begin{equation}\label{U1R}
\vec{P}_{I} = V_{I}\vec{e} \, ,
\end{equation}
where $\vec{e}$ is an arbitrary vector in $SU(2)$ space and
$V_{I}$ are constants that define the linear combination of the
vector fields $A_{\mu}^{I}$ that is used as the $U(1)_R$ gauge
field
\begin{equation}\label{AU1}
A_{\mu}{[}U(1)_R{]} = V_{I}A_{\mu}^{I}.
\end{equation}
They have to be constrained by
\begin{equation}\label{Vf}
V_{I}f_{JK}^{I}=0,
\end{equation}
with $f_{JK}^{I}$ being the structure constants of $K.$ Using the
very special geometry identities of appendix \ref{appA}, the
$U(1)_{R}$ potential can be written as
\begin{equation}
P^{(R)} = -4C^{IJ\tilde{K}}V_{I}V_{J}h_{\tilde{K}} \label{P=CVVht}.
\end{equation}
Finally, we remark that when $\mathcal{M}$ is associated with a
Jordan algebra
\cite{GST1},
one has (componentwise)
\begin{equation}
C^{\tilde{I}\tilde{J}\tilde{K}}=C_{\tilde{I}\tilde{J}\tilde{K}}=\textrm{const.}
\end{equation}
\section{Exploring the R-symmetry potential}
\label{app:theorem}
\subsection{U(1) R-symmetry gauging leads to tachyonic de Sitter vacua}
\label{theorem}
\subsection*{Theorem}
Without charged tensors or hypers, the potential gets only a
contribution from R-symmetry gauging. Unlike in $4D$, non-Abelian
vector multiplets do not contribute a term to the potential
\cite{Gunpot,GST3}. For the $U(1)_R$ case the potential is given by
\bref{P=CVVht}.
In our conventions, a critical point $\varphi_{c}$ of the
potential with $P^{(R)}(\varphi_{c})> 0$ corresponds to a de
Sitter vacuum. We will demonstrate that if such a de Sitter
vacuum exists, it will always be unstable. To prove this, we need
to calculate the matrix of second derivatives of the potential at
the critical point. A critical point, by definition, obeys the
following relation \begin{equation}\label{crit} \frac{\partial
P^{(R)}}{\partial \varphi^x}(\varphi_{c}) =
-4C^{IJK}V_{I}V_{J}h_{K,x}\vert_{\varphi_c} = 0\, . \end{equation} For the mass
matrix we find \begin{equation} g^{yz}\frac{\partial P^{(R)}}{\partial
\varphi^x \partial \varphi^z}\Big\vert_{\varphi_c} = \left(
\ft23P^{(R)}\delta^{y}_{x} - 8\sqrt{\ft23}V_I V_J
h^{Iu}h^{Jv}T_{uvx;z}g^{zy}\right)\Big\vert_{\varphi_c} \, , \end{equation}
where we have used (\ref{Txyz}), (\ref{CIJKx}) and (\ref{crit}).
We will now show that $V_{I}h^{I}_y(\varphi_c)$ is an eigenvector
of this matrix. Equation (\ref{Txyzu}) leads to \begin{equation}
T_{uvx;y}h^{Iu}h^{Jv}h^{Ky}V_{I}V_JV_K =
\sqrt{\ft32}[h^{Iu}h^{J}_{u}V_IV_JV_Kh^{K}_x -
2V_IV_Jh^{Iu}h^{Jv}T_{uv}{}^wT_{wyx}V_Kh^{Ky}]\, . \end{equation} Since, using
(\ref{Thh}) and (\ref{crit}), we have \begin{equation}
V_IV_Jh^{Iu}h^{Jv}T_{uv}{}^w \vert_{\varphi_c} = V_Ih^IV_Jh^{Jw}
\vert_{\varphi_c}\, , \end{equation} we finally obtain \begin{eqnarray}
T_{uvx;y}h^{Iu}h^{Jv}h^{Ky}V_{I}V_JV_K \vert_{\varphi_c} &=&
\sqrt{\ft32}[h^{Iu}h^{J}_{u}V_IV_J -
2(V_Ih^{I})^2]V_Kh^{K}_x \vert_{\varphi_c}\, , \nonumber \\
&=&\sqrt{\ft38}P^{(R)}V_{K}h^{K}_{x} \vert_{\varphi_c}\, . \end{eqnarray} We
thus find that $V_{I}h^{I}_y(\varphi_c)$ is indeed an eigenvector
of the mass matrix with eigenvalue $-10P^{(R)}(\varphi_c)/3$.
Looking at equations (\ref{PotR}) and (\ref{U1R}) we see that if a
critical point with $P^{(R)} > 0$ exists, then also
$V_{I}h^{I}_y(\varphi_c) \neq 0$. This proves that in case of a
de Sitter extremum, the mass matrix has always at least one
negative eigenvalue.
\subsection*{Example}
\label{example}
To illustrate our proof, we will give an example of a de Sitter
vacuum obtained by $U(1)_R$ gauging. The Jordan symmetric
spaces only lead to Anti-de Sitter or Minkowski vacua (see \cite{GST3}), but we will
show here that there are also a lot of models with de Sitter vacua.
Equations (\ref{P=CVVht}), (\ref{crit}) and (\ref{a=hh+hh}) lead to \begin{equation}
\label{V=Ph} C^{IJK}V_{J}V_{K} \vert_{\varphi_c} =-\ft14
P(\varphi_c)h^{I}(\varphi_c)\, .\end{equation} This is a necessary and
sufficient condition for $\varphi_c$ to be a critical point.
Furthermore, for $\varphi_c$ in the domain where $a_{IJ}$ is
positive definite, one can always perform a linear transformation
on the $h^I$ such that $h^{I}(\varphi_c)=(1,0,\ldots,0)$ and
$a_{IJ}(\varphi_c) =\delta_{IJ}.$ After this transformation the
polynomial $N(h)$ will take the following form \begin{equation} N(h)=(h^0)^3
-\ft32 h^0 h^i h^j \delta_{ij} + C_{ijk} h^i h^j
h^k\, ,\qquad i=1,\ldots,\tilde{n}, \label{canonical}\end{equation} which is called the canonical
parametrization of $N(h).$ Equation (\ref{V=Ph}) then becomes
\begin{eqnarray} C_{0JK}V_{J}V_{K}
=-\ft14 P(\varphi_c)\, , \label{C0VV=P}\\
C_{ijk}V_{j}V_{k}-V_0 V_i =0\, , \label{CijkVV-V0=0}\end{eqnarray} with
summation over repeated indices. Equation (\ref{P=CVVht}) however
leads to \begin{equation} P^{(R)} (\varphi_c)=-4 C_{0JK}V_{J}V_{K} = -4V_{0}^2 + 2 \sum_{i} V_i^2 \label{C0VV=P2}\, ,\end{equation} such
that (\ref{C0VV=P}) is automatically fulfilled. So, given a theory
(a tensor $C_{ijk}$) , we look for a vector $V_I$ that solves
equation (\ref{CijkVV-V0=0}) and for which \begin{equation}
-4V_{0}^2 + 2 \sum_{i} V_i^2 <0\, .\label{cVV<0}\end{equation} As we know, this is not
possible for general $C_{ijk}$. However, one can construct a lot
of examples. Take for example $C_{ijk}=0$. Equation (\ref{CijkVV-V0=0}) in this case leads to $V_i = 0$, corresponding to anti-de Sitter vacua or $V_0 = 0$, corresponding to de Sitter vacua. To study the mass matrix, we look at the particular example $n=1$. We then have
the following polynomial \begin{equation} N(h)=(h^0)^3-\ft32 h^0 (h^1)^2\, .\end{equation}
The constraint $N=1$ can be solved
by \begin{eqnarray} h^0 &=& \frac{\varphi}{2}+\frac{\sqrt{\varphi+\varphi^4}}{2\varphi}\\
h^1\, &=&
\sqrt{\frac{2}{3}}\left(-\frac{3}{2}\varphi+\frac{\sqrt{\varphi+\varphi^4}}{2\varphi}\right)\, .\end{eqnarray}
The metric on the scalar manifold is (using (\ref{gxy})) \begin{equation}
g_{xy}=\frac{3+12\varphi^3}{4(\varphi^2+\varphi^5)}\, .\label{metricgxy}\end{equation}
We restrict to the region $\varphi >0,$ which contains the point
$h^I(\varphi_c=1/2)=(1,0)$ and where the metric is positive
definite.
Taking $V_0=0,$ equation (\ref{CijkVV-V0=0}) is fulfilled for
arbitrary values of $V_1$ and equation (\ref{C0VV=P2}) becomes \begin{equation}
P^{(R)}(\varphi_c)=2V_1^2>0\, .\label{P=2V} \end{equation} Using the metric
(\ref{metricgxy}) and with $V_0=0$ we get for the potential \begin{equation}
P^{(R)}(\varphi)=\frac{-1-8\varphi^3-40\varphi^6+12\varphi\sqrt{\varphi+\varphi^4}+24
\varphi^4\sqrt{\varphi+\varphi^4}}{2(\varphi+4\varphi^4)}V_1^2\, ,\end{equation}
which indeed fulfils (\ref{P=2V}). Furthermore, \begin{equation}
P^{(R)}_{,x}(\varphi_c)=0\, ,\end{equation} and \begin{equation}
g^{yz}P^{(R)}_{,x,z}\vert_{\varphi_c}=-\frac{20}{3}V_1^2\, ,\end{equation} which
is indeed $-10P^{(R)}(\varphi_c)/3$ as stated in the theorem above.
\subsection{de Sitter vacua from $\mathbf{SU(2)_R}$ gauging}
For the known symmetric spaces, $SU(2)_R$ gauging never gives any critical points (see \cite{GunSU2}).
We show here that there are however also a lot of models with unstable de Sitter vacua.
Proving that there are no stable de Sitter vacua seems to be somewhat more difficult than in the $U(1)_R$ case and we hope to come back on this in a future publication.
We start from a polynomial in the canonical parametrization.
In order to gauge $SU(2)_R$, the polynomial should have an
$SU(2)_{G}$ symmetry \cite{GunSU2}. Without charged tensors this
further restricts the coefficients $C_{ijk}$ by \cite{options} \begin{equation}
C_{ABC}=0\, , \qquad C_{AB\alpha}=c_{\alpha}\delta_{AB}\, \qquad
C_{A\alpha\beta}=0\, , \label{Csu2}\end{equation} with $c_{\alpha}$ some
arbitrary constants. We have split the indices $i=1,\ldots,n$ as $ i=(A,\alpha)$
with $A,B,\ldots \in \{1,2,3\}$ corresponding to the SU(2) factor of the gauge group. The $C_{\alpha\beta\gamma}$ are still unconstrained.
Using expression (\ref{PR=Chdelta}) for the $SU(2)_R$ potential, the equation analogous to
(\ref{V=Ph}) is \begin{equation}\label{V2=Ph}
C^{ABI}\delta_{AB}\vert_{\varphi_c} =-
P(\varphi_c)h^{I}(\varphi_c)\, .\end{equation} We now assume that $h^{I}(\varphi_c)=(1,0,\ldots,0)$. Equation (\ref{V2=Ph}) then
leads to the conditions
\begin{eqnarray}
P^{(R)}(\varphi_c)=\frac{3}{2} \label{p=32}\\
c_{\alpha}=0\, ,\quad \forall\, \alpha\, , \label{cdelta=0}\end{eqnarray} and
therefore $C_{ABi}=0,\,\forall\,i$. The first condition
is again automatically fulfilled and tells us that all these critical points
are de Sitter.
We now investigate the stability of these de Sitter vacua.
Calculating the second derivative of the potential, we get \begin{equation}
P_{,x;y}=-C^{ABI}{}_{;y}\delta_{AB}h_{I,x}-C^{ABI}\delta_{AB}h_{I,x;y}\, .
\end{equation} Using $h^{I}(\varphi_c)=(1,0,\ldots,0),$
$a_{IJ}(\varphi_c)=\delta_{IJ}$ and (\ref{hIxy}) we get for the
second term in the critical point
\begin{equation}-C^{ABI}\delta_{AB}h_{I,x;y}\vert_{\varphi_c}=\frac{2}{3}P^{(R)}(\varphi_c)g_{xy}(\varphi_c)=g_{xy}(\varphi_c)\, .\end{equation}
For the first term we also use (\ref{CIJKx}), (\ref{Txyzu}),
\bref{Txyz} and \bref{Csu2}, which gives after some calculation
\begin{equation}
-C^{ABI}{}_{;y}\delta_{AB}h_{I,x}\vert_{\varphi_c}=-2g_{xy}(\varphi_c)-\ft43
h^{A}_x h^B_y \delta_{AB}\, ,\end{equation} and therefore \begin{equation}
g^{zy}P_{,x;y}\vert_{\varphi_c}=-\delta_{x}^{z}-\ft43 h^{A}_x
h^{Bz} \delta_{AB}\, .\end{equation} This matrix has the following eigenvectors
\begin{equation} \left\{ \begin{array}{l}
h^{A}_x(\varphi_c) \quad \textrm{with eigenvalue}\,\, -\ft73 \, ,\\
h^{\alpha}_x(\varphi_c) \quad \textrm{with eigenvalue}\,\, -1\, .
\end{array} \right.
\end{equation} The de Sitter vacua are therefore always maxima of the
potential.
\section{Stable de Sitter vacua in 5D N=2 gauged super\-gravity: an
example}\label{dsexample}
The previous section made clear that $U(1)_R$ gauging alone cannot give rise to stable de Sitter vacua. We show here that adding tensor multiplets can change this. The gauging we study in this section was already performed in \cite{gzvac} for
$\tilde{n}=3$. They found de Sitter extrema, but did not check that they
are stable. We generalize for arbitrary $\tilde{n}$ and show that the
obtained de Sitter vacua are all stable.
We consider $\mathcal{N}=2$ supergravity coupled to $\tilde{n}$ Abelian
vector multiplets and with scalar manifold
$\mathcal{M}=SO(\tilde{n}-1,1) \times SO(1,1)/ SO(\tilde{n}-1),
\tilde{n} \geq1.$ The polynomial can then be written in the
following form \begin{equation} N(h)=3\frac{\sqrt{3}}{2} \,
h^{0}[(h^1)^2-(h^2)^2- \ldots-(h^{\tilde{n}})^2]\, . \end{equation} With $x=1,\ldots
,\tilde n$, introducing
\begin{equation}
\eta _{xy}=\eta ^{xy}=\mathop{\rm Diag}\nolimits (1,-1,\ldots ,-1)\,,
\label{defeta}
\end{equation}
we write the $C_{\tilde I\tilde J\tilde K}$ symbols as
\begin{equation}
C_{0\underline{xy}}=\frac{\sqrt{3}}{2}\eta _{xy}\, .
\label{Csymbols}
\end{equation}
(we underline $x$-type indices that are in fact of type $\tilde
I$, but take values in the $x$-range due to our choice). The
constraint $N=1$ can be solved by
\begin{displaymath}
h^{0}=\frac{1}{\sqrt{3}\| \varphi\|^{2}}\, , \qquad
h^{\underline{x}}=\sqrt{\frac{2}{3}}\,\varphi^{x}\, ,
\end{displaymath}
with \begin{equation}\|\varphi\|^{2}=\varphi ^x\eta _{xy}\varphi ^y\, .\end{equation} The
hypersurface $N=1$ decomposes into three disconnected
components:\\
(i) $\|\varphi\|^{2}>0$ and $\varphi^{1}>0$\\
(ii) $\|\varphi\|^{2}<0$ \\
(iii) $\|\varphi\|^{2}>0$ and $\varphi^{1}<0$.\\
In the following, we will consider the ``positive timelike''
region (i) only, since in region (ii), $g_{\tilde{x}\tilde{y}}$
and
${\stackrel{\scriptscriptstyle{\circ}}{a}}_{\tilde{I}\tilde{J}}$ are not positive definite
, and region (iii) is isomorphic to region (i).
We now proceed by gauging the above theory. The isometry group of
the scalar manifold is $SO(\tilde{n}-1,1) \times SO(1,1).$ We
gauge the noncompact subgroup $SO(1,1) \subset SO(\tilde{n}-1,1)$
together with $U(1)_{R} \subset SU(2)_{R}.$ The $SO(1,1)$ subgroup
rotates $h^{1}$ and $h^{2}$ into each other and therefore acts
nontrivially on the vector fields $A_{\mu}^{1}$ and $A_{\mu}^{2}$.
In order for the resulting theory to be supersymmetric, these
vectors have to be dualized to antisymmetric tensor fields. We can
thus decompose the index $\tilde{I}$ in the following way
\begin{equation}\tilde{I}=(I,M)\, ,\end{equation} with $I,J,K,\ldots=0,3,4,\ldots,\tilde{n}$ and
$M,N,P,\ldots = 1,2.$
Furthermore, we need a vector transforming in the adjoint of $SO(1,1)$ (which
means it should be inert) to act as its gauge field. Looking at
$\Lambda_{IN}^{M} \sim C_{IRN}\Omega^{MR}$, we see that only
$A^{0}_{\mu}$ couples to the tensor fields (only $C_{0RP}\neq0$) and thus acts
as the gauge field of $SO(1,1).$ The remaining vectors are called `spectator
fields' with respect to the $SO(1,1)$ gauging.
Finally, for the $U(1)_{R}$ gauge field we take a linear
combination $A_{\mu} {[}U(1)_R{]} = V_{I}A_{\mu}^{I}$ of the
vectors. We now have the ingredients to calculate the potentials
\bref{PLambda} and \bref{P=CVVht} (taking $\Omega^{12}=
-\Omega^{21}=-1$): \begin{eqnarray} \Lambda_{0N}^{M} &=& \frac{2}{\sqrt{6}}
\Omega^{MR}C_{0RN}= \frac{1}{\sqrt{2}} \left( \begin{array}{cc}
0 & 1\\
1 & 0
\end{array} \right)\, , \\
P^{(T)}&=&\frac{3\sqrt{6}}{16}h^{I}\Lambda_{I}^{MN}h_{M}h_{N}={1
\over
8}{(\varphi^{1})^2-(\varphi^{2})^2 \over \|\varphi\|^{6}}\, , \label{Tpotred}\\
P^{(R)} &=&-4\sqrt{2}V_0V_i\varphi^i\|\varphi\|^{-2}+2|V|^2\|\varphi\|^2\, ,\qquad
|V|^2\equiv V_iV_i\, , \label{Rpotred}\end{eqnarray} where we defined a new index $i$ as $I=(0,i).$ Then \begin{equation}
P=P^{(T)}+P^{(R)}={1 \over
8}{(\varphi^{1})^2-(\varphi^{2})^2 \over \|\varphi\|^{6}}-4
\sqrt{2} V_{0}\varphi^{i}V_{i} \|\varphi\|^{-2}+2
\|\varphi\|^{2}|V|^{2}\, .\end{equation}
Demanding $P_{,\tilde{x}}=0$ gives the following conditions on the
critical points \begin{eqnarray}
{\varphi^{i} \over \|\varphi\|^{4}}&=&16\sqrt{2} V_{0}V_{i}\, ,\label{cond1}\\
{1 \over \|\varphi\|^{6}}&=&-{1\over2}\left(16\sqrt{2} V_{0}|V|
\right)^{2}+8 |V|^{2}\, , \label{cond2}\end{eqnarray} with the
constraints \begin{eqnarray}
|V|^{2}\neq 0\, ,\nonumber\\
32 V_{0}^{2}<1\, .\label{constr} \end{eqnarray}
{}From \bref{cond2} we see that $\|\varphi\|^{2}$ is completely
determined by the $V_{I}.$ {}From \bref{cond1} we see that also the
$\varphi^{i}$ are completely fixed by the $V_{I}.$ This means that
only $\varphi^{1}$ and $\varphi^{2}$ can still vary, as long as
the combination $(\varphi^{1})^{2}-(\varphi^{2})^{2}$ remains
fixed. We thus have a one parameter family of critical points,
which is due to the unbroken $SO(1,1)$. There is also an unbroken
$SO(\tilde n-3)$, but the vacuum is at the symmetric point.
The value of the potential in the critical points becomes \begin{equation}
P(\varphi_c)=3 \|\varphi\|^{2}|V|^{2}\left(1-32
V_{0}^{2}\right)\, , \end{equation} which is clearly positive because of
\bref{constr} and therefore corresponds to de Sitter vacua.
We now show that all these vacua are stable. We use the index
$m=1,2$ for the scalars related to $M=1,2$. Calculating the second
derivatives of the potential in the critical points gives the
following Hessian \begin{eqnarray} P_{,m,n}(\varphi_{c}) & = & (\eta \varphi
)_m
(\eta \varphi )_n \left[3\|\varphi\|^{-8}+4\varphi^k\varphi ^k\|\varphi\|^{-10}\right]\, ,\nonumber\\
P_{,m,i}(\varphi_{c})\,& = &-4(\eta
\varphi )_m \varphi ^i \left[\|\varphi\|^{-8}+\varphi^k\varphi ^k\|\varphi\|^{-10}\right]\, ,\nonumber\\
P_{,i,j}(\varphi_{c})\,\,\, & = & \varphi ^i\varphi
^j\left[5\|\varphi\|^{-8}+4\varphi^k\varphi ^k\|\varphi\|^{-10}\right]+\ft14\delta
_{ij}\|\varphi\|^{-6}\, , \label{secderiv3} \end{eqnarray} where $(\eta \varphi
)_x\equiv \eta _{xy}\varphi ^y$.
The $SO(1,1)$ invariance implies a zero eigenvector $\varphi
^n(\sigma _1)_n{}^m$. Using this $SO(1,1)$ and the $SO(\tilde
n-2)$ of the $\varphi ^i,$ we may further use for any critical
point $\varphi_{c} = (\varphi ^1,0,\varphi ^3,0,\ldots ,0)$ with
$|\varphi ^3|<|\varphi ^1|$. Then the zero mode is $\varphi ^2$,
and the sector $\varphi ^4,\ldots ,\varphi ^{\tilde n}$ decouples
as a unit matrix times $\ft14\|\varphi\|^{-6}$. The relevant part of the
hessian therefore is
\begin{equation}
\partial \partial P=|\|\varphi\||^{-10}
\left(\begin{array}{cc}
(\varphi ^1)^2\left[3(\varphi ^1)^2+(\varphi ^3)^2\right]&
-4(\varphi ^1)^3\varphi ^3 \\ -4(\varphi ^1)^3\varphi ^3 &
\ft14\left[(\varphi ^1)^4+18(\varphi ^1)^2(\varphi ^3)^2 -3(\varphi
^3)^4\right]\end{array}
\right)\, ,
\label{Hessmatrix}
\end{equation}
where we defined $\partial \partial P \equiv
\partial_{\tilde x}
\partial_{\tilde y} P(\varphi_c)\vert_{\tilde x,\tilde y=1,3}.$ The
determinant and trace are
\begin{equation}
\det\partial \partial P=\ft34(\varphi ^1)^2\|\varphi\|^{-14}\, ,\qquad
\mathop{\rm Tr}\nolimits\partial \partial P= \ft14\|\varphi\|^{-10}\left[ 13(\varphi ^1)^4
+ 22(\varphi ^1)^2(\varphi ^3)^2 - 3(\varphi ^3)^4\right]\, ,
\label{dettrace}
\end{equation}
which shows that the eigenvalues are positive.
\subsection*{Comments}
\begin{itemize}
\item {\bf BEH effect.} Like in \cite{stabledSa,stabledSb}, the massless scalar is a Goldstone boson. It will get `eaten' by the SO(1,1) gauge field, making the gauge field massive. There will thus be only positive mass scalars left in the effective theory.
\item {\bf Quantized scalar masses.} The masses of the scalars are given by the eigenvalues of $g^{xy}P_{x,y}(\varphi_c)/P(\varphi_c)$. We already showed that these will be positive. Using Mathematica, the scalar mass spectrum turns out to be
\begin{equation}
(0, \frac{8}{3}\frac{(\varphi_1)^2 - (\varphi_2)^2}{\|\varphi\|^2},\frac{2}{3},\frac{2}{3},...,\frac{2}{3})\,.
\end{equation}
In \cite{Kallosh} it was observed that all known examples of de Sitter extrema in extended supergravities have scalar masses that are quantized in units of the cosmological constant. This is also true in our model for all scalars, but one. One of the scalar masses depends on the parameters $V_I$ that determine the $U(1)_R$ gauge field. Also in \cite{dsN4b} examples of (unstable) de Sitter extrema were found that had parameter-dependent scalar masses.
\item {\bf $SU(2)_R$ gauging.} Instead of gauging $U(1)_R$ we could also have gauged the full $SU(2)_R$ R-symmetry as long as there are a sufficient number of gauge fields available ($\tilde{n} \geq 5$). Without loss of generality, we can choose $A_{\mu}^3$, $A_{\mu}^4$, $A_{\mu}^5$ as the $SU(2)$ gauge fields. We then find
\begin{equation}
P^{(R)} = \frac{3}{2}\|\varphi\|^2\,.
\end{equation}
Looking at equation (\ref{Rpotred}), we observe that we get the same potential if we do a $U(1)_R$ gauging with $V_0=0$ and $|V|^2 = 3/4$. $SU(2)_R$ gauging with tensors charged under $SO(1,1)$ therefore will also lead to stable de Sitter vacua. The scalar masses are in this case given by $(0,8/3,2/3, 2/3,...,2/3)$.
\end{itemize}
\section{$ U(1)_{R}$ gauging and charged tensors: more examples}
To try to find out what ingredients are really necessary to obtain
stable de Sitter vacua, we will now look at a few other examples
with $U(1)_{R}$ and charged tensors. A natural idea is to take
the scalar manifold $\mathcal{M}$ to be one of the other known symmetric
very special real manifolds.
\subsection{The magical Jordan family}
\subsubsection{$\mathcal{M} = SL(3,\mathbb{R})/SO(3)$}
$\mathcal{M}$ is described as the hypersurface $N(h) = 1$ of the cubic
polynomial \cite{GunSU2,DWVP} \begin{equation} N(h) =
\frac{3}{2}\sqrt{3}h^3\eta_{\alpha\beta}h^{\alpha}h^{\beta} +
\frac{3\sqrt{3}}{2\sqrt{2}}\gamma_{\alpha MN}h^{\alpha}h^{M}h^{N}
\, , \end{equation} where \begin{eqnarray} \alpha, \beta &=& 0,1,2 \, , \quad \qquad M,N
= 4,5 \, , \nonumber \\ \eta_{\alpha \beta} &=& diag(+,-,-)\, ,
\quad \gamma_0 = -
\openone_2 = \left(
\begin{matrix} -1 & 0\cr
0 & -1\end{matrix}\right)\, , \nonumber \\
\gamma_1 &=& \sigma_{1} = \left(
\begin{matrix} 0 & 1\cr
1& 0\end{matrix}\right)\, , \quad \gamma_2 =\sigma_{3} = \left(
\begin{matrix} 1 & 0\cr
0 & -1\end{matrix}\right)\, .\end{eqnarray}
The vector field metric $a_{\tilde{I}\tilde{J}}$ becomes
degenerate when $\eta_{\alpha\beta}h^{\alpha}h^{\beta} = 0$, so we
can restrict ourselves to the region
$\eta_{\alpha\beta}h^{\alpha}h^{\beta} \neq 0$. To solve the
constraint $N(h) = 1$, we take the parametrization used in
\cite{GunSU2}, \begin{eqnarray} h^{\alpha} = \sqrt{\frac{2}{3}}x^{\alpha}\, ,
\, h^{M} = \sqrt{\frac{2}{3}}b^{M}\, , \, h^{3} = \frac{1-b^T
\bar{x} b}{\sqrt{3} \vert \vert x \vert\vert^2}\, , \end{eqnarray} where $b^T
\bar{x} b \equiv b^M\bar{x}_{MN}b^N$ with $\bar{x}_{MN} =
x^{\alpha}\gamma_{\alpha MN}$ and $ \vert \vert x \vert \vert^2
\equiv \eta_{\alpha \beta}x^{\alpha}x^{\beta}$. The metrics
$a_{\tilde{I}\tilde{J}}$ and $g_{xy}$ are only positive definite
in the region $\vert \vert x \vert\vert^2 > 0$ and $x_0 > 0$.
Since this is the physically relevant region, we will restrict
ourselves to this domain from now on.
In this model we can gauge the $SO(2,1)$ symmetry between
$h^{\alpha}$ with the vector fields $A_{\mu}^{\alpha}$, while
dualizing the non-trivially charged vector fields $A_{\mu}^M$ to
tensor fields. We then get a potential \begin{equation} P^{(T)} =
\frac{1}{8}b^T \bar{x}\Omega \bar{x} \Omega \bar{x}b \, , \quad
\Omega = i\sigma_{2} =
\left(\begin{matrix} 0 & 1 \cr -1 & 0 \end{matrix} \right)\, .\end{equation}\\
Gauging the full R-symmetry is not possible in this model, but we
can gauge a $U(1)_{R}$ symmetry. We have $A_{\mu}[U(1)_R] =
V_{I}A_{\mu}^{I}$, with $V_I f_{JK}^I = 0$. {}From this it follows
that $A_{\mu}[U(1)_R] = V_{3}A_{\mu}^{3}$, so $A_{\mu}^{3}$ is the
$U(1)_R$ gauge field. Since $P^{(R)} = -4C^{IJ\tilde{K}}V_I V_J
h_{\tilde{K}}$ and $C^{33\tilde{K}} = C_{33\tilde{K}} = 0$ we find
$P^{(R)} = 0$. The total potential $P$ is thus given by $P^{(T)}$
alone. The critical points of $P$ are given by $b^M = 0$, leading
to Minkowski vacua. They are supersymmetric when the $U(1)_R$
gauging is turned off ($V_3 = 0$). There are no de Sitter vacua in this
model.
\subsubsection{$\mathcal{M} = SL(3,\mathbb{C})/SU(3)$}
$\mathcal{M}$ is described as the hypersurface $N(h) = 1$ of the cubic
polynomial \cite{GunSU2,DWVP} \begin{equation} N(h) =
\frac{3}{2}\sqrt{3}h^4\eta_{\alpha\beta}h^{\alpha}h^{\beta} +
\frac{3\sqrt{3}}{2\sqrt{2}}\gamma_{\alpha MN}h^{\alpha}h^{M}h^{N}
\, , \end{equation} where \begin{eqnarray} \alpha, \beta &=& 0,1,2,3 \, ,
\quad M,N = 5,6,7,8 \, , \nonumber \\
\eta_{\alpha \beta} &=& diag(+,-,-,-)\, , \quad \gamma_0 = - \openone_4 \, , \nonumber \\
\gamma_1 &=& \openone_2 \otimes \sigma_1\, , \quad \gamma_2 = \sigma_2 \otimes \sigma_2\, , \quad \gamma_3 = \openone_2 \otimes \sigma_3\, . \end{eqnarray}
We take the same parametrization as in the previous model, \begin{eqnarray}
h^{\alpha} = \sqrt{\frac{2}{3}}x^{\alpha}\, , \, h^{M} =
\sqrt{\frac{2}{3}}b^{M}\, , \, h^{4} = \frac{1-b^T \bar{x}
b}{\sqrt{3} \vert \vert x \vert\vert^2}\, . \end{eqnarray} The metrics are
again only positive definite in the region $\vert \vert x
\vert\vert^2 > 0$ and $x_0 > 0$.
The model above has an $SO(3,1) \times U(1)$ symmetry, which acts on
the fields $h^{\tilde{I}}$ (and similarly on the vector fields
$A_{\mu}^{\tilde{I}}$) as \cite{DWVVP}
\begin{eqnarray}
\delta h^\alpha & = & B^\alpha {}_\beta h^\beta\, , \nonumber\\
\delta h^M & = & \ft14 B^{\alpha \beta }(\gamma _{\alpha \beta })^M{}_N
h^N + S^M{}_Nh^N\epsilon\, ,
\label{deltah}
\end{eqnarray}
where
\begin{eqnarray}
&&S\equiv \gamma _1\gamma _2\gamma _3= i\sigma _2\otimes \openone_2\, ,\qquad
S^2=-\openone_4\, ,\nonumber\\
&& \gamma _{a b }=\gamma _{[a }\gamma _{b ]}
=-S\varepsilon _{abc }\gamma ^c\, ,
\quad \gamma _{0a}=-\gamma _{a 0}=\gamma _a \, , \quad a = 1,2,3\, .
\end{eqnarray}
Indices on the matrices $B_{\alpha \beta }$ are raised and lowered with
$\eta _{\alpha \beta }$ and these transformations satisfy $B_{\alpha
\beta }=-B_{\beta \alpha }$. This implies that they describe $\mathop{\rm SO}(1,3)$.
The motivation for the definition of $\gamma _{0a }=-\gamma _{a 0}$
is based on a larger Clifford algebra, see \cite[(5.16)]{DWVVP}.
The $\gamma $-matrices are symmetric, while $S$ is antisymmetric.
$\epsilon $ is the parameter for the $\mathop{\rm {}U}(1)$ symmetry.
To gauge a symmetry, we have to assign the isometry transformations to
vector multiplets, i.e. to connect the parameters of gauge transformations
$\Lambda ^I$ to parameters of the isometry group, such that the
transformations on the vector part form the adjoint representation and on
the tensor multiplets there exists an antisymmetric matrix $\Omega
_{MN}$ such that (see (\ref{tenstrf}))
\begin{equation}
\Omega_{MP}\Lambda_{IN}^P = \frac{2}{\sqrt{6}}C_{IMN}\, .
\label{tOmegaC}
\end{equation}
\begin{itemize}
\item
{\bf $U(1)_R$ gauging and tensors charged under $U(1)\times SU(2)$
}.
We now gauge the $\mathop{\rm SO}(3)$ part of (\ref{deltah}) with $B^{0a
}=0$ and take $A_{\mu}^{a}$ as the adjoint vectors. We gauge the $\mathop{\rm {}U}(1)$ by the vector field $A_{\mu}^{0}$. To gauge this symmetry, we also have to dualize the vector fields $A_{\mu}^{M}$ to tensor fields.
We have
\begin{equation}
B_{a b }= \alpha\varepsilon _{abc }\Lambda ^c\, ,\qquad \epsilon
=\beta \Lambda ^0\, , \qquad \varepsilon _{123} = 1\, ,
\label{Beps}
\end{equation}
where we allowed for arbitrary coefficient $\alpha$ and $\beta$ to be determined below. This
leads thus to the transformation matrices
\begin{equation}
\Lambda_{0N}^M =\beta S^M{}_N \, , \qquad \Lambda_{a
N}^M=-\ft12 \alpha (S\gamma _a)^M{}_N\, .
\label{tSU2U1}
\end{equation}
Checking (\ref{tOmegaC}) gives
\begin{equation}
\Omega =\frac{1}{\alpha} S\, ,\qquad \beta =\frac{\alpha}{2}\, .
\label{Omegacoeff}
\end{equation}
For simplicity, and without losing generality, we can choose $\alpha=1$, $\beta = 1/2$.
In \cite{GunSU2} the vector fields $A_{\mu}^{a}$ where used to also gauge the full $SU(2)_R$ symmetry and it was found that the total potential has no critical points. Instead, we will use the vector field $A_{\mu}^4$ together with $A_{\mu}^0$ to gauge the $U(1)_R$
symmetry. The potential $P = P^{(T)} + P^{(R)}$ becomes \begin{eqnarray}
P^{(R)} &=& -2\sqrt{3}V_{0}\left ( \frac{V_0}{\sqrt{3}}\vert \vert
x \vert \vert^2 + \frac{4}{\sqrt{3}}V_4 \left(\frac{1-b^T \bar{x}
b}{\sqrt{2}\vert \vert x \vert \vert^2}\right)x^0
-\frac{2}{\sqrt{6}}V_4 b^T b \right)\,, \\
P^{(T)} &=& \frac{1}{8} b^T \bar{x}\Omega \bar{x} \Omega \bar{x} b
= - \frac{1}{8}b^T \bar{x}^3 b \nonumber \\ &=& \frac{1}{8}\vert
\vert x \vert \vert^2 b^T \bar{x}b - \frac{1}{2}x_{0}^2 b^T
\tilde{x} b + \frac{1}{4} x_{0} \vert \vert \tilde{x} \vert
\vert^2 b^T b + \frac{1}{4}x_{0}^3 b^T b\,, \quad
\label{PTsymman}\end{eqnarray} where $b^T b = b^M b^N \delta_{MN}$, $\vert
\vert \tilde{x} \vert \vert^2 = x^a x^b \delta_{ab}$, $\vert \vert
x \vert \vert^2 = \eta_{\alpha \beta}x^{\alpha}x^{\beta}$,
$\bar{x}_{MN} = x^{\alpha}\gamma_{\alpha MN}$ and $\tilde{x}_{MN}
= x^{a}\gamma_{a MN}$. The last line of (\ref{PTsymman}) makes
the $U(1) \times SU(2)$ symmetry of the potential manifest, since
$b^T \bar{x}b$, $b^T \tilde{x} b$, $\vert \vert \tilde{x} \vert
\vert^2$ and $\vert \vert x \vert \vert^2$ are easily seen to be
invariant under the transformations (\ref{deltah}) (with $B_{a 0} = 0$). Using this symmetry we can restrict the search for extrema to points where
e.g. $x_2 = x_3 = b_8 = 0$. We analyzed the potential with Mathematica and found no de Sitter vacua (in the region where the metrics are positive-definite).
When $b^M = 0$, finding critical points of $P$ reduces to finding critical points of $P^{(R)}$. It was shown in \cite{GST3} that $P^{(R)}$ has an Anti-de Sitter maximum if and only if $V^{\sharp \tilde{I}} \equiv \sqrt{\frac{2}{3}}C^{\tilde{I}\tilde{J}\tilde{K}}V_{\tilde{J}}V_{\tilde{K}}$ lies in the domain of positivity of the Jordan algebra, and a Minkowski critical point if and only if $V^{\sharp \tilde{I}} = 0$ ($P^{(R)}$ identically zero). The total potential $P$ also has these critical points, but the extra potential from the tensors can change the nature of these critical points (e.g from a maximum to a saddle point). With Mathematica we also found Anti-de Sitter vacua of $P$ with $b^M \neq 0$. Since our primary interest was finding de Sitter vacua, we did not check the nature of these critical points.
\item {\bf
$U(1)_R$ gauging and tensors charged under $U(1)\times SO(2,1)$} \\
Instead of the compact symmetry above, we can also gauge
$U(1)\times SO(2,1)$ by again dualizing the vector fields
$A_{\mu}^{M}$ to tensor fields and choosing the $SO(2,1)$ gauge fields to be
$A_{\mu}^{0}$, $A_{\mu}^{1}$, $A_{\mu}^{3}$, while letting
$A_{\mu}^{2}$ correspond to the $U(1)$ gauge field. Similarly as in the previous example, this leads to $\Omega = \gamma_2 S$. We can again
gauge the $U(1)_R$ symmetry, this time with a linear combination
of $A_{\mu}^{2}$ and $A_{\mu}^{4}$. This leads to the following
potential, \begin{eqnarray} P^{(R)} &=& 2 \sqrt{3} V_2 \left
(\frac{V_2}{\sqrt{3}}\vert \vert x \vert \vert^2 -
\frac{4}{\sqrt{3}}V_4\left (\frac{1 - b^T \bar{x}b}{\sqrt{2} \vert
\vert x \vert \vert^2} \right)x^2 + \frac{2}{\sqrt{6}}V_4 b^T
\gamma_2 b \right)\,, \nonumber \\
P^{(T)} &=& \frac{1}{8} b^T \bar{x}\Omega \bar{x} \Omega \bar{x} b
\nonumber \\ &=& -\frac{1}{8}\vert \vert x \vert \vert^2 b^T
\bar{x}b - \frac{1}{2}x_{2}^2 b^T \tilde{x} b - \frac{1}{4} x_{2}
\vert \vert \tilde{x} \vert \vert^2 b^T b - \frac{1}{4}x_{2}^3 b^T
b\,, \quad \end{eqnarray} where now $\vert \vert \tilde{x} \vert
\vert^2 = (x^0)^2 - (x^1)^2 - (x^3)^2$ and $\tilde{x}_{MN} =
x^{0}\gamma_{0 MN} + x^{1}\gamma_{1 MN} + x^{3}\gamma_{3 MN}$.
Analyzing the potential as in the previous case, we again found no de Sitter vacua. The potential has a critical point only when $V_2 = 0$. $P^{(R)}$ is then identically zero, and we have a family of Minkowski vacua at $b^M=0$.
\end{itemize}
\subsubsection{The other magical Jordan symmetric spaces}
The spaces $\mathcal{M} = SU^{*}(6)/USp(6)$ and $\mathcal{M} = E_{6(-26)}/F_4$
are 14 and 26 dimensional respectively, and allow for more
possibilities to get charged tensor multiplets, making the
potential more difficult to analyze. Because all the magical
Jordan spaces have a similar structure, one might expect similar
qualitative features as in the previous models, but this has to be
checked in detail to be sure.
\subsection{The non-Jordan symmetric spaces}
We now consider theories with $\mathcal{M} = {SO(1,\tilde{n}) \over SO(\tilde{n})},
\tilde{n} >1.$ We can then take the following polynomial \begin{equation}
N(h)={3\over
2}\sqrt{{3\over2}}\left(\sqrt{2}h^{0}(h^{1})^2-h^{1}\left[(h^{2})^2+\ldots+(h^{\tilde{n}})^2\right]\right)\, .\end{equation}
This means for the non-vanishing components of the tensor
$C_{\tilde{I}\tilde{J}\tilde{K}}$ \begin{equation} C_{011}={\sqrt{3}\over 2}\, ,\quad
C_{1xy}=-{\sqrt{6}\over 4}\, \delta_{xy}\, , \,x,y = 2,\ldots,\tilde{n}\, .
\end{equation} The constraint $N=1$ can be solved by \begin{eqnarray}
h^{0}&=&\sqrt{{2\over
3}}\left(\frac{1}{\sqrt{2}(\varphi^1)^2}+\frac{1}{\sqrt{2}}\varphi^1
\left[(\varphi^2)^2+\ldots+(\varphi^{\tilde{n}})^2\right]\right),\\
h^{1}&=&\sqrt{\frac{2}{3}}\varphi^1, \qquad
h^{x}=\sqrt{\frac{2}{3}}\varphi^1 \varphi^x.\end{eqnarray}
The Lagrangian of the theory is not invariant under the full
isometry group $SO(1,\tilde{n})$. Only the subgroup
$G=\left[SO(\tilde{n}-1)\otimes SO(1,1)\right]\ltimes T_{\tilde{n}-1}$, with
$T_{\tilde{n}-1}$ the group of translations in an $(\tilde{n}-1)$ dimensional
Euclidean space, leaves the tensor $C_{\tilde{I}\tilde{J}\tilde{K}}$ invariant and
can thus be gauged \cite{DWVP2}. In order to gauge a subgroup
$K\subset G$ we need $Dim(K)$ vectors transforming in the adjoint
of $K.$ Furthermore, we want an additional number of vectors
transforming non-trivially under $K$. After dualization to tensor
multiplets these give the required contribution $P^{(T)}$ to the
potential.
The subgroup $SO(1,1)$ can not be gauged since all vectors
transform non-trivially under this group and we need an
invariant vector to gauge SO(1,1).
The group $SO(\tilde{n}-1)$ rotates $h^{2},\ldots,h^{\tilde{n}}$\,(and thus
also the vectors $A^{2}_{\mu},\ldots,A^{\tilde{n}}_{\mu}$) into each
other. This means that only its subgroup $SO(2)$ can be gauged in
order to have both vectors that transform in the adjoint and vectors that transform non-trivially but not in the adjoint of the gauge group. We will therefore gauge this $SO(2)$, possibly together with $SU(2)_{R}$ or $U(1)_R$. The former was already worked out in
\cite{GunSU2}, where it was shown that the potentials $P^{(R)}$
and $P=P^{(T)}+P^{(R)}$ do not have any critical points at all and
$P^{(T)}$ only has Minkowski vacua. We now investigate the latter
gauging.
We restrict ourselves to $\tilde{n}=3$, so the group $SO(2)$ acts on
$A^2_{\mu}$ and $A^3_{\mu}.$ These vectors therefore have to be
dualized to tensors. We decompose the index $\tilde{I}$ as follows
\begin{equation}\tilde{I}=(I,M)\, ,\end{equation} with $I,J,K,\ldots=0,1$ and $M,N,P,\ldots = 2,3.$
The vector $A^{1}_{\mu}$ will act as gauge field for $SO(2)$ since
its the only vector left that couples to the tensor fields. For
the $U(1)_{R}$-gauging we take the gauge vector $A_{\mu}
{[}U(1)_R{]} = V_{I}A_{\mu}^{I}.$ The constraint \bref{Vf} is
automatically fulfilled.
The potential $P^{(T)}$ then becomes (taking $\Omega^{23}=
-\Omega^{32}=-1$): \begin{eqnarray} \Lambda_{1N}^{M} &=& {2\over \sqrt{6}}
\Omega^{MR}C_{1RN}=-{1\over 2} \left(
\begin{array}{cc}
0 & -1\\
1 & 0
\end{array} \right)\, , \\
P^{(T)}&=&\frac{3\sqrt{6}}{16}h^{I}\Lambda_{I}^{MN}h_{M}h_{N}=\frac{1}{8}(\varphi^1)^5
\left[(\varphi^2)^2+(\varphi^3)^2\right]\, .\end{eqnarray} The calculation of
$P^{(R)}$ however is a bit more involved since for the non-Jordan
theories $C^{\tilde{I}\tilde{J}\tilde{K}}$ is not constant any more. The indices are
raised using $a^{\tilde{I}\tilde{J}}$ with \begin{equation}
a^{\tilde{I}\tilde{J}}=h^{\tilde{I}}h^{\tilde{J}}+h^{\tilde{I}}_{\tilde{x}}h^{\tilde{J}}_{\tilde{y}}\,g^{\tilde{x}\tilde{y}}\end{equation}
and $g^{\tilde{x}\tilde{y}}=Diag\left((\varphi^1)^2/3,\,
1/(\varphi^1)^3,\,1/(\varphi^1)^3\right).$ Then $P^{(R)}$ becomes
\begin{equation}
P^{(R)}=-4C^{IJ\tilde{K}}V_{I}V_{J}h_{\tilde{K}}=-4\sqrt{2}\frac{V_{0}V_{1}}{\varphi^1}
-(\varphi^1)^2\left[V_{0}\left((\varphi^2)^2+(\varphi^3)^2
\right)+\sqrt{2}V_1\right]^2.\end{equation} We remark that we have to
restrict to $\varphi^1 > 0$ in order for $g_{\tilde{x}\tilde{y}}$ to be positive
definite.
As already mentioned, $P^{(T)}$ only has Minkowski ground states. Moreover, $P^{(R)}$ can at most have unstable de Sitter vacua, as we proved in section \ref{theorem}. We will now study the total potential $P^{(T)} + P^{(R)}$.
\subsubsection*{The critical points of $P$}
The total potential is \begin{equation} P=\frac{1}{8}(\varphi^1)^5
\left[(\varphi^2)^2+(\varphi^3)^2\right]-4\sqrt{2}\frac{V_{0}V_{1}}{\varphi^1}
-(\varphi^1)^2 A^2\, ,\end{equation} with \begin{equation}
A=V_0 \left[(\varphi^2)^2+(\varphi^3)^2 \right]+\sqrt{2} V_1\, . \end{equation}
The first derivatives are \begin{eqnarray}
P_{,1}&=&\ft{5}{8}(\varphi^1)^4
\left[(\varphi^2)^2+(\varphi^3)^2\right]+4\sqrt{2}\frac{V_0
V_1}{(\varphi^1)^2}-2\varphi^1 A^2\, , \label{ptot1}\\
P_{,2}&=& \ft14 (\varphi^1)^2 \varphi^2 \left[(\varphi^1)^3-16 A V_0\right]\, , \label{ptot2}\\
P_{,3}&=& \ft14 (\varphi^1)^2 \varphi^3 \left[(\varphi^1)^3-16 A
V_0\right]\, . \label{ptot3}\end{eqnarray} {}From \bref{ptot2} and \bref{ptot3}
we get the following three possibilities for the critical points:
\begin{itemize}
\item When $\varphi^2=\varphi^3=0$ and $V_1 = 0$, equations \bref{ptot1}-\bref{ptot3} are fulfilled and $P(\varphi_c) = 0$, giving Minkowski vacua. Since $V_Ih^I_{\tilde{x}}(\varphi_c) \neq 0$, supersymmetry is broken unless also $V_0 = 0$.
\item When $\varphi^2=\varphi^3=0$ and $V_1 \neq 0$, equation\bref{ptot1} leads to the condition
\begin{equation} (\varphi^1)^3=\sqrt{2}
\frac{V_0}{V_1}\, . \label{phi13V0V1}\end{equation}
Then \begin{equation}
P(\varphi_c)=-6 (\varphi^1)^2 V_{1}^2 \label{PRphic}\end{equation} and
we have an Anti-de Sitter vacuum. The vectors
$V_{I}h^{I}_{\tilde{x}}(\varphi_c)$ and $h_{Mx}\Omega^{MN}h_N(\varphi_c)$ are now identically zero, which means the vacuum preserves the full $N=2$ supersymmetry. \item The third possibility is
$(\varphi^1)^3=16 A V_0,$ which can be rewritten as \begin{equation}
V_1=\frac{p-16 V_0^2 q}{16 \sqrt{2} V_0}\, , \label{V1=frac}\end{equation} with
$p=(\varphi^1)^3$ and $q=(\varphi^2)^2+(\varphi^3)^2 > 0.$ Using this
in (\ref{ptot1}) and solving for $V_0$ gives us the following four
solutions \begin{equation} V_0=\pm\frac{1}{8} \sqrt{5p^2+\frac{2p}{q}\pm
\frac{\sqrt{4p^2+12p^3q+25p^4q^2}}{q}}\, .\label{V0=sqrt}\end{equation} Remark
that the expressions under the square roots are always positive. We substitute (\ref{V1=frac}) and (\ref{V0=sqrt}) into
the potential $P$ and get \begin{equation} P(\varphi_c)=
\frac{3p^{5/3}q(p+5p^2q\pm\sqrt{4p^2+12p^3q+25p^4q^2})}{4\left(2p+5p^2q
\pm \sqrt{4p^2+12p^3q+25p^4q^2}\right)}\, .\end{equation} Here both signs are
positive when the second sign choice in (\ref{V0=sqrt}) is
positive, otherwise both signs are negative (independent of the
choice of the first sign in (\ref{V0=sqrt})). With the plus signs
we have a de Sitter vacuum, with the minus signs an anti-de Sitter
vacuum.
Calculating the matrix of second derivatives $P,x,y$ and
substituting (\ref{V1=frac}) and (\ref{V0=sqrt}), we get the
following form \begin{equation} P,x,y(\varphi_c)=\left(\begin{array}{cc}
B_{2\times2}&0\\
0&0
\end{array}\right)\, .\end{equation} The expected zero eigenvalue from the $SO(1,1)$ invariance
is already explicit. Furthermore, \begin{equation} Det(B_{2\times2})=
-\ft{3}{32}p^{5/3} q \left(14p+25p^2
q\pm5\sqrt{4p^2+12p^3q+25p^4q^2}\right)\, ,\end{equation} where again the plus
sign corresponds to the de Sitter vacuum, the minus sign to the
anti-de Sitter vacuum. The determinant is always negative, so
there is always a negative eigenvalue and the de Sitter vacua are unstable.
\end{itemize}
\subsection{Conclusions}
These examples seem to suggest that the existence of stable de
Sitter vacua is very model dependent. A $U(1)_R$ gauging and
tensors charged under a non-compact gauge group do not guarantee
stable de Sitter vacua. On the other hand, we also found a de
Sitter vacuum in a model with $U(1)_R$ gauging and tensors charged
under a compact group. Unfortunately the de Sitter vacuum was
unstable. Whether this is a general feature of compact gaugings
is not clear to us.
\section{Stable de Sitter vacua with hypers}
Our goal in this section is to show that it is still possible to
get stable de Sitter vacua when hypermultiplets are included. We
will do this by giving a particular example, namely we will gauge
a specific isometry of the universal hypermultiplet. There are
probably many other possibilities, but we will not analyse this in
its generality here.
When there are charged hypers in the model, the potential gets
some extra contributions. The total potential is given by
\cite{Ceresole,5DSGREV}\begin{equation} P = -4\vec{P}\cdot \vec{P}+
2\vec{P}^x\cdot\vec{P}_x +2\mathcal{N}_{iA}\mathcal{N}^{iA} +
P^{(T)}\, , \label{Pothyp}\end{equation} where, as before, $\vec{P} =h^I\vec{P}_I$,
$\vec{P}_x = h^I_{x}\vec{P}_I$ and ${\mathcal N}^{iA} =
\ft{\sqrt6}{4} h^I K_I^X f_X^{iA}$. Here $f_X^{iA}$ are the
quaternionic vielbeins, $f_X^{iA}f_{YiA} = g_{XY}$ with $g_{XY}$
the metric of the quaternionic-K\" ahler hypermultiplet scalar
manifold, $K_I^{X}$ are the Killing vectors and $\vec{P}_I$ the
corresponding prepotentials.
The metric of the universal hypermultiplet, together with the
Killing vectors and corresponding prepotentials were given in
\cite{universal}, and we will repeat the results here for
convenience of the reader. The four hyperscalars $q^{X}$ are
labelled as $\{V,\sigma,\theta,\tau\}$ and the metric is given by
\begin{equation}
{\rm d}}\def\rme{{\rm e} s^2 = \frac{{\rm d}}\def\rme{{\rm e} V^2}{2V^2} + \frac{1}{2V^2}\left( {\rm d}}\def\rme{{\rm e} \sigma + 2
\theta \, {\rm d}}\def\rme{{\rm e} \tau - 2 \tau \, {\rm d}}\def\rme{{\rm e} \theta\right)^2 + \frac{2}{V} \,
\left({\rm d}}\def\rme{{\rm e} \tau^2 + {\rm d}}\def\rme{{\rm e} \theta^2\right) \,. \label{quatmetric}
\end{equation}
The determinant for this metric is $1/V^6$ and since the metric is
positive definite in $\theta = \tau = 0$ if $V>0$, the metric will
be positive-definite and well-behaved everywhere as long as $V>0$.
This parametrization of the universal hypermultiplet is the one
that comes out naturally from the Calabi-Yau compactifications of
M-theory, where V acquires the meaning of the volume of the
Calabi-Yau manifold. The metric (\ref{quatmetric}) has an SU(2,1)
isometry group generated by the following eight Killing vectors
$k_{\alpha}^X$
\begin{equation}
\begin{array}{l}
\vec{k}_1 = \left(
\begin{array}{c}
0 \\ 1 \\ 0 \\ 0
\end{array}
\right)\,,
\quad
\vec{k}_2 = \left(
\begin{array}{c}
0 \\ 2\theta \\ 0 \\ 1
\end{array}
\right)\,,
\quad
\vec{k}_3 = \left(
\begin{array}{c}
0 \\ -2\tau \\ 1 \\ 0
\end{array}
\right)\,,
\quad
\vec{k}_4 = \left(
\begin{array}{c}
0 \\ 0 \\ -\tau \\ \theta
\end{array}
\right)\,, \\
\vec{k}_{5} = \left(
\begin{array}{c}
V \\ \sigma \\ \theta/2 \\ \tau/2
\end{array}
\right)\,,
\quad
\vec{k}_{6} =\left(
\begin{array}{c}
2V \sigma \\ \sigma^2 - \left(V + \theta^2 + \tau^2\right)^2 \\
\sigma \theta - \tau \left(V + \theta^2 + \tau^2\right)\\
\sigma \tau + \theta \left(V + \theta^2 + \tau^2\right)
\end{array}
\right)\,,
\\
\vec{k}_{7} =\left(
\begin{array}{c}
- 2V \theta\\ -\sigma\theta + V\tau + \tau\left(\theta^2 + \tau^2\right) \\
\frac{1}{2} \left(V - \theta^2 + 3\tau^2\right) \\
-2 \theta \tau - \sigma/2
\end{array}
\right)\,,
\quad
\vec{k}_{8} =\left(
\begin{array}{c}
- 2V \tau\\ -\sigma\tau - V\theta - \theta\left(\theta^2 + \tau^2\right) \\
-2 \theta \tau + \sigma/2\\
\frac{1}{2}\left( V + 3\theta^2 - \tau^2\right)
\end{array}
\right)\, .
\end{array}
\label{killvec}
\end{equation}
The corresponding prepotentials $P^{r}_{\alpha}$ are given by
\begin{equation}
\begin{array}{rcl}
\vec{P}_{1} = \left(
\begin{array}{c} 0 \\ 0 \\ -\frac{1}{4V} \end{array}
\right)\,,
\quad
\vec{P}_{2} = \left(
\begin{array}{c} -\frac{1}{\sqrt{V}} \\ 0 \\ -\frac{\theta}{V} \end{array}
\right)\,,
\quad
\vec{P}_{3} = \left(
\begin{array}{c} 0 \\ \frac{1}{\sqrt{V}} \\ \frac{\tau}{V}\end{array}
\right)\,,
\quad
\vec{P}_{4} = \left(
\begin{array}{c} -\frac{\theta}{\sqrt{V}} \\ -\frac{\tau}{\sqrt{V}} \\
\frac{1}{2} - \frac{\theta^2+ \tau^2}{2V} \end{array}
\right)\,, \\
\vec{P}_{5} = \left(
\begin{array}{c} -\frac{\tau}{2\sqrt{V}} \\ \frac{\theta}{2\sqrt{V}}
\\ -\frac{\sigma}{4V} \end{array}
\right)\,,
\quad
\vec{P}_{6} = \left(
\begin{array}{c} -\frac{1}{\sqrt{V}}\left[\sigma\tau +
\theta\left(-V+\theta^2+ \tau^2\right)\right] \\
\frac{1}{\sqrt{V}}\left[\sigma\theta -
\tau\left(-V+\theta^2+ \tau^2\right)\right] \\
-\frac{V}{4}-\frac{1}{4V}\left[\sigma^2 + \left(\theta^2+
\tau^2\right)^2\right] + \frac{3}{2}\left(\theta^2+ \tau^2\right)
\end{array} \right)\,,
\\
\vec{P}_{7} = \left(
\begin{array}{c} \frac{4\theta\tau + \sigma}{2\sqrt{V}} \\
\frac{3\tau^2- \theta^2}{2\sqrt{V}} - \frac{\sqrt{V}}{2} \\
-\frac{3}{2}\tau+\frac{1}{2V}\left[\sigma\theta+ \tau \left(\theta^2+
\tau^2\right)\right]\end{array} \right)\,, \quad \vec{P}_{8} = \left(
\begin{array}{c} -\frac{3 \theta^2-\tau^2}{2\sqrt{V}} + \frac{\sqrt{V}}{2} \\
\frac{\sigma-4\theta\tau }{2\sqrt{V}} \\
\frac{3}{2}\theta+\frac{1}{2V}\left[\sigma\tau- \theta \left(\theta^2+
\tau^2\right)\right] \end{array} \right)\,.
\end{array}
\label{prep}
\end{equation}
The Killing vectors $K_I^X$ are now given by
$V_{I}^{\alpha}k_{\alpha}^X$, where the components of the embedding matrix $V_{I}^{\alpha}$ are constants
that determine which isometries are gauged and which vector fields
are used to gauge them. The corresponding prepotentials
$\vec{P}_{I}$ then become $V_{I}^{\alpha}\vec{P}_{\alpha}$.
We are now ready to give a concrete example. We choose to gauge
the $U(1)$ (hypermultiplet) isometry generated by $2\vec{k}_4 -
\vec{k}_1 - \vec{k}_6$, so we take \begin{equation} \vec{K}_I = V_I (2\vec{k}_4
-\vec{k}_1 - \vec{k}_6) \, ,\quad \vec{P}_I = V_I (2\vec{P}_4
-\vec{P}_1 - \vec{P}_6) = V_I \vec{Q}\, , \end{equation} where we have defined $\vec{Q} \equiv 2\vec{P}_4
-\vec{P}_1 - \vec{P}_6$. For the scalar manifold of the vector multiplets we choose $\mathcal{M}=SO(\tilde{n}-1,1) \times SO(1,1)/ SO(\tilde{n}-1), \tilde{n} \geq1$, and again gauge a noncompact $SO(1,1)$ isometry of this manifold by dualizing the two charged
vector fields to tensor fields (see section \ref{dsexample} for
notation and more details). For simplicity, we do not charge the
hypers under this symmetry. Our gauge group is thus $SO(1,1)\times
U(1)$, where two tensors are charged under the $SO(1,1)$ and the
hypers are charged under the $U(1)$. We then find that
\begin{eqnarray}
\frac{\partial P}{\partial q^X}(\varphi, q_c)= 0 \, , \label{crithyp} \\
P(\varphi, q_{c}) = \frac{9}{4}P^{(R)}(\varphi) + P^{(T)}(\varphi)\, , \label{Pothypcrit} \end{eqnarray} where $q_{c} =
\{V=1,\sigma =0,\theta = 0, \tau = 0\}$ and with $P^{(R)}$ and
$P^{(T)}$ given in equations (\ref{Rpotred}) and (\ref{Tpotred})
respectively.
To verify this, first notice that $K_I^X\vert_{q_c} = 0$ and therefore
\begin{equation}
\mathcal{N}_{iA}\vert_{q_c} = 0 \, , \qquad \frac{\partial (\mathcal{N}_{iA}\mathcal{N}^{iA})}{\partial q^X}\vert_{q_c} = 0\, .
\end{equation}
We also have $\frac{\partial P^{(T)}}{\partial q^X} = 0$ since $P^{(T)}$ only depends on the scalars of the vector multiplets. The remaining term $-4\vec{P}\cdot \vec{P}+
2\vec{P}^x\cdot\vec{P}_x$ in equation (\ref{Pothyp}) can be written as
\begin{equation}
-4\vec{P}\cdot \vec{P}(\varphi,q) + 2\vec{P}^x\cdot\vec{P}_x(\varphi,q) = -4C^{IJK}V_IV_Jh_K(\varphi)\vec{Q}\cdot\vec{Q}(q)\, ,
\end{equation}
which shows that the $\varphi$ (vector multiplet) and $q$ (hypermultiplet) dependence of this part factorizes.
Since $\vec{Q}\vert_{q_c} = (0,0,3/2)$, to verify equation (\ref{crithyp}) we only need to check that $\frac{\partial Q^3}{\partial q^X}\vert_{q_c} = 0$, with $Q^3$ the third component of the vector $\vec{Q}$. Because $Q^3$ is quadratic in $\theta$, $\sigma$ and $\tau$ we have \begin{equation}\frac{\partial Q^3}{\partial \theta} = \frac{\partial Q^3}{\partial \sigma} = \frac{\partial Q^3}{\partial \tau} = 0\quad \textrm{if} \quad \theta = \sigma = \tau = 0\, . \end{equation} Finally \begin{equation} Q^3\vert_{\theta = \sigma = \tau = 0} = 1 + 1/4V + V/4 \, ,\end{equation} and it's easy to see that $V=1$ is an extremum. This proofs equations (\ref{crithyp}) en (\ref{Pothypcrit}).
We thus find that in the point $q_{c}$, up to a
factor $9/4$ which can be absorbed in the $V_{I}$, the
potential reduces to the same potential for the vector multiplet
scalars as found in section \ref{dsexample}, where we gauged a
$U(1)_{R}$ symmetry. We now have to calculate the mass matrix in
the critical point. Since (\ref{crithyp}) is true for any value
of the vector multiplet scalars, we have \begin{equation} \frac{\partial^2
P}{\partial q^X
\partial \varphi^x}\vert_{q_{c}} = 0 \, , \end{equation} so we can calculate the
masses of the hyper-scalars and the vector-scalars separately.
Since we already calculated the mass matrix for the
vector multiplet scalars, we just have to calculate the mass matrix
for the hypers. After a straightforward calculation we find the
matrix to be diagonal, with all entries always positive. There are
only 2 different diagonal elements and they can both be written as
a sum of manifestly positive terms. Because the expressions are
quite messy, we will not give them in their generality here. Instead
we will look at the particular case $V_{0} = 0$. We then have
$\varphi_{i} = 0$ in the critical point and the expressions
simplify significantly. Concretely, we find \begin{eqnarray}
\frac{\partial_X\partial^Y P}{P}\Big\vert_{c} = \left(
\begin{matrix} 8/9 & 0 & 0 & 0\cr
0 & 8/9 & 0 & 0 \cr
0 & 0 & 4/9 & 0 \cr 0 & 0 & 0 & 4/9\end{matrix}\right) \, ,
\end{eqnarray} where derivation with respect to $q^X$ is denoted by
$\partial_X$ and indices are raised with the (inverse)
quaternionic-K\" ahler metric $g^{XY}$. This shows that
potentials with stable de Sitter vacua also exist when hypers are
included.
\section{Summary}
In this paper we investigated different possibilities to get
stable de Sitter vacua in $5D$ $N=2$ gauged supergravity. We
proved that $U(1)_R$ gauging (without tensors) at most leads to
unstable de Sitter vacua. In the case of $SU(2)_R$ gauging we found lots
of theories exhibiting de Sitter extrema, but were unable to find
stable de Sitter vacua. However, by also introducing tensor multiplets
and gauging a non-compact symmetry group together with the
R-symmetry group we managed to construct 5D supergravity theories
with stable de Sitter vacua. The used ingredients are however not
sufficient to guarantee stable de Sitter vacua, as the analysis of
several other examples made clear. Finally, we showed with a
specific example that we can also get stable de Sitter vacua if we
replace the R-symmetry gauging with charged hypers.
There are several directions in which we plan to continue our research. First of all it would be interesting to find out under which conditions stable de Sitter vacua exist in supergravity theories. A more general analysis of the potentials coming from $SU(2)_R$
gauging and tensors will certainly be useful for this.
Investigating the potential coming from charged hypermultiplets
might also give interesting results. Another possible fruitful
path would be to try to embed the stable de Sitter vacua in $N=4$ and
$N=8$ supergravity and check whether they are still stable. Attempts in this direction in 4D have failed (see \cite{dsN4b}), and it would be interesting to see whether this generalizes to higher dimensions. Having an N=8 embedding could also make it easier to find their 10 or 11 dimensional origin, if any. Finally, considering the similarities between 4D
and 5D $N=2$ supergravity, the results we found perhaps suggest
that investigating 4D gauged supergravities with tensor couplings
might lead to new examples of stable de Sitter vacua. We hope to come
back on these issues in the near future. \\
\noindent{ \bf Acknowledgements}\vspace{0.3cm}
We would like to thank A. Van Proeyen for helpful discussions and for proofreading this manuscript. We have also greatly benefited from conversations with A. Celi and want to thank M. Trigiante for useful e-mail correspondence. This work is supported in part by the European Community's Human Potential Programme under contract MRTN-CT-2004-005104 `Constituents,
fundamental forces and symmetries of the universe', by the FWO - Vlaanderen, project
G.0235.05 and by the Federal Office for Scientific, Technical and Cultural Affairs through the `Interuniversity Attraction Poles Programme -- Belgian Science Policy' P5/27.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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package ca.watier.echechess.components;
import ca.watier.echechess.common.enums.CasePosition;
import ca.watier.echechess.common.enums.KingStatus;
import ca.watier.echechess.common.enums.MoveType;
import ca.watier.echechess.common.enums.Side;
import ca.watier.echechess.common.interfaces.WebSocketService;
import ca.watier.echechess.common.utils.Constants;
import ca.watier.echechess.communication.redis.interfaces.GameRepository;
import ca.watier.echechess.communication.redis.model.GenericGameHandlerWrapper;
import ca.watier.echechess.engine.abstracts.GameBoardData;
import ca.watier.echechess.engine.delegates.PieceMoveConstraintDelegate;
import ca.watier.echechess.engine.engines.GenericGameHandler;
import ca.watier.echechess.models.AvailableMove;
import ca.watier.echechess.models.PawnPromotionViewModel;
import com.fasterxml.jackson.databind.ObjectMapper;
import com.fasterxml.jackson.databind.type.CollectionLikeType;
import com.fasterxml.jackson.databind.type.TypeFactory;
import org.apache.commons.lang3.StringUtils;
import org.slf4j.LoggerFactory;
import org.springframework.beans.factory.annotation.Autowired;
import java.io.IOException;
import java.util.ArrayList;
import java.util.List;
import static ca.watier.echechess.common.enums.ChessEventMessage.*;
import static ca.watier.echechess.common.enums.Side.getOtherPlayerSide;
import static ca.watier.echechess.common.utils.Constants.PLAYER_KING_CHECKMATE;
import static ca.watier.echechess.common.utils.Constants.PLAYER_MOVE;
public class MessageActionExecutor {
private static final org.slf4j.Logger LOGGER = LoggerFactory.getLogger(MessageActionExecutor.class);
private static final String REGEX_VALUE_SEPARATOR = "\\|";
private final PieceMoveConstraintDelegate gameMoveConstraintDelegate;
private final GameRepository<GenericGameHandler> gameRepository;
private final WebSocketService webSocketService;
private final ObjectMapper objectMapper;
private final CollectionLikeType listStringType;
@Autowired
public MessageActionExecutor(PieceMoveConstraintDelegate gameMoveConstraintDelegate, GameRepository<GenericGameHandler> gameRepository, WebSocketService webSocketService, ObjectMapper objectMapper) {
this.gameMoveConstraintDelegate = gameMoveConstraintDelegate;
this.gameRepository = gameRepository;
this.webSocketService = webSocketService;
this.objectMapper = objectMapper;
TypeFactory typeFactory = this.objectMapper.getTypeFactory();
listStringType = typeFactory.constructCollectionLikeType(ArrayList.class, String.class);
}
public void handleMoveResponseMessage(String message) {
if (StringUtils.isBlank(message)) {
return;
}
handleReceivedMoveMessage(message);
}
private void handleReceivedMoveMessage(String messageAsString) {
String[] messages = messageAsString.split(REGEX_VALUE_SEPARATOR);
String uuid = messages[0];
CasePosition from = CasePosition.valueOf(messages[1]);
CasePosition to = CasePosition.valueOf(messages[2]);
MoveType moveType = MoveType.getFromValue(Byte.parseByte(messages[3]));
Side playerSide = Side.getFromValue(Byte.parseByte(messages[4]));
GenericGameHandlerWrapper<GenericGameHandler> handlerWrapper = gameRepository.get(uuid);
GenericGameHandler gameFromUuid = handlerWrapper.getGenericGameHandler();
if (MoveType.isMoved(moveType)) {
if (MoveType.PAWN_PROMOTION.equals(moveType)) {
PawnPromotionViewModel viewModel = new PawnPromotionViewModel();
viewModel.setGameSide(playerSide);
viewModel.setFrom(from.name());
viewModel.setTo(to.name());
webSocketService.fireGameEvent(uuid, PAWN_PROMOTION, viewModel);
} else {
GameBoardData boardData = gameFromUuid.getCloneOfCurrentDataState();
KingStatus currentKingStatus = gameMoveConstraintDelegate.getKingStatus(playerSide, boardData);
KingStatus otherKingStatus = gameMoveConstraintDelegate.getKingStatus(Side.getOtherPlayerSide(playerSide), boardData);
sendMovedPieceMessage(from, to, uuid, gameFromUuid, playerSide);
sendCheckOrCheckmateMessagesIfNeeded(currentKingStatus, otherKingStatus, playerSide, uuid);
}
}
}
private void sendMovedPieceMessage(CasePosition from, CasePosition to, String uuid, GenericGameHandler gameFromUuid, Side playerSide) {
webSocketService.fireGameEvent(uuid, MOVE, String.format(PLAYER_MOVE, playerSide, from, to));
webSocketService.fireSideEvent(uuid, getOtherPlayerSide(playerSide), PLAYER_TURN, Constants.PLAYER_TURN);
webSocketService.fireGameEvent(uuid, SCORE_UPDATE, gameFromUuid.getGameScore());
}
private void sendCheckOrCheckmateMessagesIfNeeded(KingStatus currentkingStatus, KingStatus otherKingStatusAfterMove, Side playerSide, String uuid) {
if (currentkingStatus == null || otherKingStatusAfterMove == null || playerSide == null) {
return;
}
Side otherPlayerSide = getOtherPlayerSide(playerSide);
if (KingStatus.CHECKMATE.equals(currentkingStatus)) {
webSocketService.fireGameEvent(uuid, KING_CHECKMATE, String.format(PLAYER_KING_CHECKMATE, playerSide));
} else if (KingStatus.CHECKMATE.equals(otherKingStatusAfterMove)) {
webSocketService.fireGameEvent(uuid, KING_CHECKMATE, String.format(PLAYER_KING_CHECKMATE, otherPlayerSide));
}
if (KingStatus.CHECK.equals(currentkingStatus)) {
webSocketService.fireSideEvent(uuid, playerSide, KING_CHECK, Constants.PLAYER_KING_CHECK);
} else if (KingStatus.CHECK.equals(otherKingStatusAfterMove)) {
webSocketService.fireSideEvent(uuid, otherPlayerSide, KING_CHECK, Constants.PLAYER_KING_CHECK);
}
}
public void handleAvailMoveResponseMessage(String message) {
if (StringUtils.isBlank(message)) {
return;
}
String[] headers = message.split(REGEX_VALUE_SEPARATOR);
String uuid = headers[0];
String fromAsString = headers[1];
Side playerSide = Side.getFromValue(Byte.parseByte(headers[2]));
try {
List<String> positions = objectMapper.readValue(headers[3], listStringType);
webSocketService.fireSideEvent(uuid, playerSide, AVAILABLE_MOVE, null, new AvailableMove(fromAsString, positions));
} catch (IOException e) {
LOGGER.error(e.getMessage(), e);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,459
|
Jamie (11) from Dublin loves being outside and loves nothing better than being on a swing, stretched out in the sun. We recently granted his wish to transform his garden into a space he could enjoy safely. Jamie's two dogs got a lovely new kennel too!
Huge thanks to Paul and the team at Backyard Adventures Ireland for turning Jamie's wish into a reality with the help of Timber Trove, Sanctuary Synthetics and Down to Earth Landscapes.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,336
|
ACCEPTED
#### According to
The Catalogue of Life, 3rd January 2011
#### Published in
Edwards's Bot. Reg. 25: misc. 63. 1839
#### Original name
null
### Remarks
null
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,527
|
THE PASSIVE SOLAR HOUSE
THE
PASSIVE
SOLAR
HOUSE
James Kachadorian
CHELSEA GREEN PUBLISHING COMPANY White River Junction, Vermont
_Special thanks to George Philip Kachadorian for his editing help, and to my_ _clients who believed in me and from whom I learned. Also, no business succeeds_ _without devoted people, and much credit for the success of Green Mountain_ _Homes is due to the efforts of Wayne Chalmers, Kendall Spaulding, and Wally_ _Killian, who ran the factory; to Dolores Zick, who ran the office; to Gary_ _Delaney, who helped out with drafting in the early years; and to my wife Lea,_ _who handled our advertising and contributed artwork, and who has never wavered_ _in her support of my activities._
Text and photographs copyright © 1997, 2006 James Kachadorian. All rights reserved.
All illustrations are copyright © 1997 Michael Middleton. All rights reserved.
The tables in Appendices 2, 3, and 5 are reprinted with permission from the American Society of Heating, Refrigerating, and Air Conditioning Engineers, Inc. The tables and data are reprinted wholly or in part from the 1981 and 1993 ASHRAE _Handbooks of Fundamentals_. ASHRAE, Inc. retains the exclusive copyrights to this reprinted material.
Cover design by Peter Holm
Book design by Andrea Gray
Printed in Canada
10 9 8 7 6 5 4 3 2 1
First printing, July 2006
Although great care has been taken to insure the accuracy of the information in this book, the publisher and author accept no liability for printing or typographical errors, nor for personal injury, property damage, or any other loss or damages arising from actions inspired by this book.
Library of Congress Cataloging-in-Publication Data
Kachadorian, James.
The passive solar home : using solar design to heat and cool your home / James Kachadorian.
p. cm. -- (The real goods independent living books)
Includes bibiliographical references.
eBook ISBN: 978-1-60358-169-1
1. Solar houses. 2. Solar heating--Passive systems. 3. Solar air conditioning--Passive systems. I. Title. II. Series: Real goods independent living book.
TH7414.K33 1997
690'.8370472--dc21 97-3264
Chelsea Green Publishing Company
Post Office Box 428
White River Junction,Vermont 05001
(802) 295-6300
www.chelseagreen.com
_This book is dedicated to the memory of_
_Nathaniel E. Kachadorian_
_July 6, 1973_ _–_ _November 21, 1992_
Contents
Preface
1 LET NATURE HEAT YOUR HOME
2 THE PASSIVE SOLAR CONCEPT
3 THE SOLAR SLAB AND BASIC SOLAR DESIGN
4 INSULATION, VENTING, AND FRESH AIR
5 BASIC LAYOUTS AND FLOOR PLANS
6 HOW TO DO THE SOLAR DESIGN CALCULATIONS
7 THE FOUNDATION PLAN, AND BACKUP HEATING AND COOLING
8 A SIDEHILL VARIATION, AND SOLAR DESIGN WORKSHEETS
9 SUNSPACES AND SPECIAL DESIGN CONSIDERATIONS
10 INTERIOR DESIGN FOR YEAR-ROUND COMFORT
(by Cornelia C. Kachadorian)
11 THREE PROJECTS
12 USING THE CSOL COMPUTER PROGRAM
APPENDICES
APPENDIX 1
Solar Design Worksheets
APPENDIX 2
Solar Intensity and Solar Heat Gain Factors for 16 to 64 degrees North Latitude
APPENDIX 3
Thermal Properties of Typical Building and Insulating Materials (Design Values)
APPENDIX 4
North Latitude, Elevation, and Outside Winter Design Temperatures for Selected Cities in the U.S. and Canada
APPENDIX 5
Average Monthly and Yearly Degree Days for Cities in the U.S. and Canada
APPENDIX 6
Mean Percentage of Possible Sunshine for Selected Cities in the U.S. and Canada
APPENDIX 7
Isogonic Chart (Magnetic Declination)
Preface
_All houses are solar_. The sun shines on almost every home, many days throughout the year. The question is, to what extent are you utilizing the sunlight? This book has been written to help you to take advantage of this free resource.
The first part of the book will acquaint you with the basic concepts involved in solar design. Notice that we have included ten easy-to-follow "Solar Principles," each one illustrating a key consideration in building solar homes. As you progress through the chapters, the discussion will get more specific and more technical, incorporating many formulas and equations needed to actually factor the solar principles into effective solar home designs. Do not be discouraged if you do not instantly grasp the mathematics. What is important is that you understand the concepts so that, with the help of a professional designer, you will be able to include solar features in the plan for your home.
The knowledge imparted in this book has been accumulated from over 30 years of data gathered from several hundred solar homes located in the northern tier of the United States, from North Carolina to and including Canada and west to the mountain states. These are locations that are primarily focused on heating. Authors of other solar books have raised concerns about the possible buildup of mold or other airborne problems in ventilated slabs. If the design described in this book is used in low-lying and high-moisture locations, the concerns raised may very well be valid. See chapter 7 for a more detailed discussion of this topic.
Great care has been taken to provide accurate and factual information based on over twenty-five years of solar home-design experience. I wish that I could make competent solar designers and builders out of every reader, yet the disciplines needed to design and construct homes take years of education and apprenticeship to learn. If you do not possess these skills, please consult with or hire professionals. While this book's technical data and equations will be widely applicable for the technically trained, hopefully the book will also spark an enthusiasm among non-technical readers for the limitless potential of solar energy.
Wouldn't it be nice if your house, too, could spend next winter heating itself, naturally, with free heat from the sun?
THE PASSIVE SOLAR HOUSE
[LET NATURE
HEAT YOUR HOME](chap6_passives_9781603581691_epub_part6.html#d41151052)
During the summer of 1973, the U.S. economy was booming. We were all whizzing down the highway at 70 miles per hour, the legal speed limit. Gasoline was about 39 cents per gallon, and the posted price of Gulf crude oil was $2.59 per barrel. That year, my wife Lea and I had purchased a lovely old Vermont farmhouse, heated by a coal-stoking boiler that had been converted to oil. The base of this monster boiler was about three feet by six feet, and when it fired, it literally shook the house. We tapped our domestic hot water directly off the boiler, so we had to run the unit all four seasons: Every time we needed hot water, the boiler in the basement fired up. We were burning about 2,500 gallons of fuel oil each year, and in the coldest winter months, it was not unusual to get an oil delivery every two weeks.
Since we had no other way to heat our home, we were entirely dependent on the oil-gobbling monster, and on our biweekly oil deliveries to survive the Vermont winter. Our only alternative source of heat was an open fireplace. Though aesthetically pleasing, the fireplace actually took more heat out of the house than it gave off.
At that time, I was the vice president and general manager of a prefabricated post-and-beam home operation. Like others, I shared the industry opinion that the heating contractor's job was to install the heating system that the homeowner wanted. As designers and home producers, we were not responsible for that part of new home construction. Home building plans were typically insensitive to the position of the sun. Our prefabricated home packages were labeled simply "front, back, right side, left side," not "south, east, west, north." We offered little or no advice on siting, except that we needed enough room to get a tractor-trailer to the job site.
To give you an idea how little energy efficiency was considered in 1973 in house design (an area of home construction that has since received enormous attention), our homes had single glazed windows and patio doors; R-13 wall and R-20 roof insulation were considered more than adequate. ("R" is the thermal resistance of any housing component; a high R-value means a higher insulating value. Today's homes typically have much higher R-values.) Homeowners in the 1970s rarely asked about the R-values of their home components, and our sales discussions were less about energy efficiency than about how the house would look and whether it would have vaulted ceilings.
The point is, we were not yet approaching the task of design and construction in an integrated, comprehensive way. We had not yet recognized that all aspects of a design must be coordinated, and that every member of the design team, including the future resident, needs to be thinking about how the home will be heated from the first moment they step onto the site.
**THE OIL CRISIS**
In 1973, an international crisis forever changed the way Americans thought about home heating costs. After Israel took Jerusalem in the "Six Day War," Arab oil-producing nations became increasingly frustrated with the United States' policy toward Israel. In the fall of 1973, these oil-producing nations began to utilize oil pricing and production as a means to influence international policy. In October 1973, the Organization of Petroleum Exporting Countries (OPEC) met and unilaterally raised oil prices 70 percent. The impact of this price hike on U.S. homeowners who heated with oil was spectacular. Fuel oil prices soared.
Then the oil embargo hit. In November 1973, all Arab oil-producing states stopped shipping oil to the United States. By December 1973, the official OPEC member-price was $11.65 per barrel—a whopping 450 percent increase from the $2.59-per-barrel price of the previous summer. Iran reported receiving bids as high as $17.00 per barrel, which translated to $27.00 per barrel in New York City.
In addition to giant price increases, oil supplies became uncertain and the United States, which depended on foreign oil for fully half its consumption, was facing the real possibility of fuel rationing for the first time since World War II.
Richard Nixon was president, and his Secretary of State, Henry Kissinger, spent most of that winter in what was termed "shuttle diplomacy," racing from country to country attempting to bring a resolution to the crisis. He didn't succeed until March 18, 1974, when the embargo against the United States was lifted. It had lasted five months.
As the international oil crisis was played out over those five months, every oil delivery to our home was marked by a price increase, invariably without notice. Worse, our supplier could not assure delivery. My wife and I had two small children, an energy dinosaur of a house, and no other way to keep warm but to burn huge amounts of oil. We couldn't even "escape" to a warmer climate, because there were long lines at the gasoline pumps. We had never felt so dependent on others as we did that winter. It was plain scary!
We have done a little better recently, as today only one-fourth of U.S. oil comes from OPEC. Most imports come from more stable Western sources, and are so diversified that a full-scale war in the Persian Gulf in 1991 caused no gas lines at home. However, we are still over 50 percent dependent on foreign oil sources.
All the concerns about energy seem to have reached a boiling point in September and October 2005. Back-to-back hurricanes in the Gulf region of the United States crippled our refining and fuel distribution capabilities, and oil and propane gas soared to new record highs.
**THERE HAD TO BE A BETTER WAY**
I have a background in engineering, and the energy crisis of 1973–1974 provided an incentive for me to investigate solar heating. It was obvious to me that as a country, we had forgotten the basics of good energy management. I just knew that there must be a better way to design and build houses that would capture the sun's heat and work in harmony with nature. I also have a background in business, and I realized that the energy crisis had opened up a market ready for new ideas about how to heat homes. The energy crisis had shaken us all into action.
The years immediately following the 1970s energy crisis saw a remarkable emergence of new ideas about solar energy. Solar conferences were held, and the public was treated to frequent articles that described new solar home designs in popular magazines. The results of this collective effort were largely positive. Many new ideas were tested. Some succeeded, and others failed, but building specifications focused on energy efficiency developed during that time have now become standard practice. For example, double-pane high-performance glass is now used almost universally in windows and patio doors. Standard wall insulation is now R-20. That was previously the roof standard; standard roof insulation is now R-32. The science of vapor barriers took huge leaps forward, and highly effective vapor barriers are now standard. Exterior house wraps, such as Typar and Tyvek, are applied on most new construction to tighten up air leaks. Appliances are now more energy efficient. Heating systems have undergone major improvements. These days, it is even common for "smart houses" to monitor lighting and to turn lamps and heating equipment on and off according to need. In sum, we are now building better energy-efficient houses, in large part due to the wake-up call we got in the winter of 1973–1974.
**WHY FEAR SOLAR?**
Fast-forwarding to 2006, we find ourselves fighting wars in Afghanistan and Iraq in the name of national security. Unfortunately, it is in our national interest to stabilize the oil-rich Middle East because we are more dependent on imported oil than we were in 1974. Speaking of fighting wars, a recent article in _USA Today_ states that a B-52 bomber's eight engines burn 3,334 gallons of fuel per hour. The U.S. Air Force says it will cost $4 billion for new engines with an associated fuel cost savings of $400 million provided the planes keep flying for 40 years. Gasoline pump prices are well over $2.00 a gallon and a barrel of oil has soared to record highs in excess of $60.00. There are some predictions that we may see $100.00 per barrel in the foreseeable future. The elements that make up our oil problem are all still in place, and we are not going to be able to drill our way out of this predicament, i.e., by drilling in the Alaskan Wildlife Refuge. Decimating the Alaskan Wildlife Refuge will only assist the situation for 25 to 30 years. Furthermore, the easy oil has all been pumped out of our known reserves, which will make future drilling activities more and more difficult and expensive.
The frustrating part of this mess is that we already have the technology to significantly impact our fuel consumption. What we need is a national commitment to reduce our fuel usage.
Back in the 1970s, I designed and patented what I saw as a partial solution to the energy crisis—an innovative solar house design.This design will work for you, today.
And yet from my work building solar homes for over twenty years, I've found that people resist solar for four main reasons. They are afraid that the house will get too hot. They are afraid that the house will be too cold. And they are afraid that a solar house has to be ugly and futuristic-looking and will require expensive, fickle gadgetry and materials, with walls of glass, or black-box collectors hanging from every rooftop and wall. None of these fears are well-founded.
_Siting a house with sensitivity_ _to the_ _sun's_ _daily and seasonal_ _patterns, and using conventional_ _materials wisely, you can build_ _a traditional-looking solar home_ _that largely heats itself._
The design and building strategies presented in this book are carefully engineered for building solar homes with traditional features, while incurring no added expense in the process. This solar approach is really a rearrangement of materials you would otherwise need to build any home. In fact, the only feature you sacrifice using this design is a basement, but you gain so much in energy savings and by living in a large, cheery, well-lit place, that I think that you'll find this trade-off is more than worthwhile.
Here are a couple of other considerations to keep in mind when reading this book. First, I came to the design and building of solar homes as a businessman and engineer, and this book reflects that approach. I've aimed for a practical, step-by-step, how-to treatment. Every building strategy presented in this book has been proven out in the real world.
Moreover, though I've chosen one type of design to describe in detail, this book also offers a wealth of practical information for designing _any_ solar home. A wealth of engineering data is included in the hopes that this book will become a welcome addition to any complete library of solar design.
**GREEN MOUNTAIN HOMES:**
**A SOLAR SUCCESS STORY**
The ingredients for my decision to go into the business of designing and manufacturing solar homes were all in place just after the 1974 oil crisis hit. My engineering and home manufacturing background offered the stepping-off point. I had been doing research on solar designs throughout 1974, and by mid-1974, the idea of starting a business devoted to producing pre-fabricated solar homes seemed more exciting than ever before. The concepts for the business and formulation of the solar design were finalized by late 1975. Green Mountain Homes was incorporated on January 1, 1976, and was the first United States home manufacturer dedicated solely to designing and manufacturing solar homes in kit form. I purchased twenty acres of commercial property in Royalton, Vermont, and in June 1975, left my job as vice president and general manager of the prefabricated post-and-beam home operation.
_The Green Mountain Homes_ _factory practiced the lessons we_ _preached: the building where_ _our houses were fabricated was_ _itself solar-heated, energy-efficient, and largely lit by daylight._
In January 1976, I visited Sheldon Dimick, the president of the Randolph National Bank in Randolph, Vermont. In the business proposal, I included plans for a dozen affordable solar homes. My wife Lea, with her Middlebury College art background, had drawn pencil renderings of the homes, which later became the basis for our first brochure. Shel, my banker, was immediately taken by the idea. In just a few weeks, we had put together a financing package with his bank, the Vermont Industrial Development Authority (VIDA), the Small Business Administration, and a personal loan backed by our farmstead, the one with the oil-thirsty boiler in the basement. The irony did not escape me that my energy dinosaur of a house was helping to finance an energy-efficient housing business.
While I was arranging private funding to start Green Mountain Homes, a business that would ultimately design, fabricate, and ship almost three hundred solar homes, the state and federal governments were getting involved in solar, offering tax incentives to encourage use of solar energy. The U.S. government spent on the order of a quarter of a million dollars to install domestic solar hot water collectors over a covered parking lot at a nearby resort hotel. To the best of my knowledge, those solar collectors have long since been disconnected because of mechanical problems and leaks. The state of Vermont, with some other states and the federal government, was instituting tax credits for investments in solar technologies. But credits were offered only for add-ons and retrofits of existing homes. These credits were for the "additional equipment needed," in the state's view, to provide solar energy. As a result, passive solar homes like the ones I intended to sell were almost completely left out of the tax-credit programs. Green Mountain Homes' buyers had difficulty obtaining solar credits, since the principle of my design was to utilize and rearrange the materials that you are already committed to purchase for building any style of new home, solar or not. Fortunately for my buyers, nighttime window and patio door insulating devices, extra insulation, and elements of the solar control system were considered add-on features and therefore qualified for solar credits. Yet the credits thereby earned were never significant enough to be the motivation for us or our clients to build solar homes instead of conventional ones. The real incentives were the ease, reliability, and comfort derived from solar heating. Paradoxically, since the solar water-heater collectors at the nearby resort were an add-on feature, the resort probably got more money from the U.S. government through solar credits than all of the Green Mountain Home solar homeowners combined. The federal government's solar subsidy program was completely dismantled during the Reagan years.
_A Green Mountain Home_ _built in 1984 and located in_ _northeast Pennsylvania._
**OUR MODEL HOME AND**
**SOLAR FACTORY**
When I was first working on my solar home design, I participated in a seminar led by Professor A. O. Converse of the Thayer School of Engineering at Dartmouth College. My role in the seminar was to provide the students with practical house construction information. When I explained my plan to build a prototype model solar home, Professor Converse offered to provide independent monitoring, using some of Dartmouth's resources, with funding and equipment supplied by the local power company, Central Vermont Public Service.
Construction of Green Mountain Home's solar-heated factory and the model home, which also served as my office and sales center, began in March 1976. The design was so successful that the energy savings (in both heat and electric lighting costs) paid the real estate taxes on our twenty acres and buildings each year.
With our borrowed money, we started an advertising campaign. We also decided to erect a state-approved off-site road sign on one of Vermont's major interstate highways, indicating the location of our business. The Vermont state highway department objected to the placement of the sign, so I asked my wife, Lea, to represent us at a hearing in Montpelier. As Lea was explaining the need for our placement of the sign, she described our new solar home business. A woman who sat on the board was so impressed with the idea that she sent her son to see us. He liked what he saw, and his home was delivered early in the fall of 1976.
Not long after, Lea was working in her mother's grape arbor and noticed a stranger approaching. He had seen an ad for Green Mountain Homes in a magazine, but the return address was to our home, not our factory/model-home complex nearby. The gentleman explained that he had spent most of the day looking for Green Mountain Homes and had finally stopped at the post office for help. Since ours is a small town, the postmaster knew about our new venture and sent the gentleman to my mother-in-law's home. By coincidence, Lea happened to be there. It turned out that the gentleman was a graduate of Worcester Polytechnic Institute, and was most supportive of our new solar home business. His home was delivered late in the fall of 1976; he has always maintained that he was the first buyer, because he ordered his home first.
Solar Principle #1
_Orient the house properly with respect to the_ _sun's_ _relationship to the site._
Use a compass to find true south, and then by careful observation site the house so that it can utilize the sun's rays from the east, south, and west during as much of the day and year as possible. In orienting the house, take into account features of the landscape, including trees and natural land forms, which will buffer the house against harsher weather or winds from the north. Deciduous trees on the sunny sides of the site will shade the house from excess heat during the summer months, but will allow the winter sunlight to reach the house and deliver free solar energy.
Green Mountain Homes was launched. The company doubled in size yearly, and we were often hard pressed to keep up with the workload. I can remember many a Christmas when we were late to our own party because we were loading a tractor-trailer with that year's last home.
Potential buyers almost always traveled to Royalton, Vermont, to examine the model solar home, and to attend my Saturday morning solar heating seminars, which were followed by lively question-and-answer sessions. It was fun. Our customers came from all walks of life, and they accepted this new technology with enthusiasm. One man who bought a Green Mountain Home was asked to speak about it to the local Rotary Club. He protested, claiming that he didn't know how his house worked. Since he had a good sense of humor, the Rotarians thought he must be joking, so they asked him to speak anyway. The fact is, he really _didn't_ know how his house worked, even though he and his brother had built it from our kit. When the Rotarians asked about the house's operation and about how much work he had to put in to keep it heated, he informed them that the day after it was completed, he had set the thermostat on 68 degrees and hadn't touched it since. This man had called me shortly after moving in to let me know that he and his brother had forgotten to put in the second floor heating ducts. And the house was _still_ warm. We learned from our customers as we went along; in this case we found out that we could cut back on ductwork.
**NEWS OF OUR SUCCESS SPREADS**
Since Green Mountain Homes was a private venture financed through conventional bank loans, we had to succeed on our own merits without public or government funds. And we did succeed, because the design worked so well, both in tests on our prototype and in the comfort it delivered to the people living in actual homes. We also tried to help advance the solar movement by speaking at our own expense at various meetings and conferences. Our success story was featured in dozens of publications, including _Solar Age, Better Homes and Gardens, House and Garden, New Shelter,_ _Farm Equipment News,The Muncie Star_ (Indiana), _The Winchester Evening Star_ (Virginia), _The Boston Herald, The San Francisco Chronicle, Money Magazine_ , and _The Sierra Club Bulletin_ , to name just a few. We also received enthusiastic mail from our customers through the years, for instance this from happy solar homeowners in Bethlehem, Connecticut:
In the winter, we are warm. In the summer, we are cool. There are no unusual contraptions involved to store heat or regulate the temperature. We spent no additional money on "solar features" when we built this home. All materials were available at local lumberyards—nothing exotic. However, it was important to let common sense take precedent. We did, according to our instructions, face the broad, multi-windowed side of our house to the south. (Consequently, we gave up the "parallel to the road.") The north side has few windows, and many closets, and that side is also sheltered by a windblock of pine trees. The "heat-producing" kitchen is also placed on the north side of the house. Our floor is made of tile, which absorbs the heat from the sun, so in winter the tile is never cold to our feet, as the tiles and Solar Slab underneath store the warmth. (The reverse is true in the summer, when the tile retains the coolness of the night during much of the day.)
And this letter from Susquehanna, Pennsylvania:
One of the greatest satisfactions about our home is that even though we are designed to be solar energy efficient, it placed very little restraints on how we designed the floor plan. Our home has a real feeling of spaciousness because of the view and use of windows. Even during the horrendous winter of '93, we didn't get "cabin fever."
Green Mountain Homes' production facility was closed several years ago. I now provide pre-construction, advisory services rather than supplying house plans. I believe the book provides adequate information for a professional home designer to make the necessary calculations and develop the detailed plans needed to build a solar home.
All of the Green Mountain Solar homes shown in this book are privately owned and are not available to the public. The policy of protecting my homeowners' privacy was established long ago and has helped to maintain good relations with my clients. The privacy of all Green Mountain Solar Home owners has always been respected, but you will be able to take a virtual tour of a sampling of solar homes designed by the author on the enclosed CD. All of these homes utilize the solar heating system described in _The Passive Solar House._ Backup heating systems range from sophisticated air-to-air heating and cooling systems and oil- and gas-fired warm air systems to wood stoves supplemented by simple electric furnaces. All of these homes have an air mover to circulate and filter the air. Most commonly, the air filter is the ordinary air filter contained within the warm air backup heat system. See the control diagrams in chapter 7 for more detail.
As the patents issued on the solar system described in this book have expired, the design is now in the public domain. The invention now belongs to the "People of the United States." This book is an effort to make this "gift" more meaningful. Hopefully, it will benefit other solar designers and future homebuilders.
Good luck with your solar project.
2
[THE
PASSIVE SOLAR
CONCEPT](chap6_passives_9781603581691_epub_part6.html#d41151053)
A French engineer named Felix Trombe is credited with the simple idea of building a solar collector comprised of a south-facing glass wall with an air space between it and a blackened concrete wall. The sun's energy passes through the glass, and is trapped and absorbed by the blackened wall. As the concrete warms, air rises in the space between the glass and the blackened concrete wall. Rectangular openings at the bottom and top of the Trombe wall allow this warm air to flow to and from the living space. This movement of air is called thermosiphoning. At night the blackened concrete wall will radiate, or release, its heat to the interior.
The process can, unfortunately, reverse at night bringing warm air from the living space over to the cold glass. As this warmer air is cooled by the glass, it drops to the floor which, in turn, pulls more warm air from the living space. In the process, thermosiphoning is reversed. The colder it is outside, the more the Trombe wall will reverse thermosiphon. One way to control this heat loss is mechanically to close the rectangular openings at night and to reopen them when the sun comes out.
The Trombe wall is the "Model A" of passive solar design; that is, it is elegant in its simplicity and dependability, but has been largely supplanted by more modern technology. The Trombe wall example, however, illustrates some important principles. The system requires no moving parts, no switches to turn motors on or off, and no control systems; yet when it is functioning properly, it will collect, store, and then radiate heat back into the living space, even after the sun has gone down.
By contrast, an active solar collector is an ancillary system; instead of incorporating heat collection, storage, and release into the structure of the building, active systems are made up of devices attached to the structure. (Active systems also represent "add-on" expenses for a home— "add-ons" are features that are additional to those that you would normally purchase.) Active systems will not work without a pump or blower operating. Typically, solar collectors are placed on the roof. Water pipes deliver water heated by the collectors to a storage tank and heated water is pumped out of the storage tank as needed. These systems will not work by themselves, as they need to have sensors "tell" switches to turn on pumps or blowers to mechanically activate the circulation of water.
_A Trombe wall, a thermally_ _effective but aesthetically poor_ _design for storing and circulating_ _sun-warmed air. At night, the_ _process can reverse, and warm air_ _may be drawn back out of the_ _living space to escape through_ _the cool glazing._
The "passive" Trombe wall and the active solar collector system represent the technological range of solar heating systems from most basic to most complicated.
**KEEP IT SIMPLE AND**
**LET NATURE HELP YOU**
Given the challenge of designing and building a naturally solar-heated home, the most widely applicable system is simple, passive, and does not add cost to the construction of the home. Let's look at the materials which one has already committed to purchasing. Used properly, these materials become the building blocks of the naturally heated home. We need concrete to build the base of the house, and we all like windows and patio doors. Also, let's take a critical look at the building site, because much can be done to make home orientation and vegetation function as heating and cooling assists.
Let's start by finding a south-facing house site. For the sake of discussion, let's locate this house in Hartford, Connecticut, which is at north latitude 41°5', or "41 degrees 5 minutes." If the home faces true south, you will get the maximum solar benefit, but as you rotate your home off true south the solar benefit is reduced. At solar noon in February in Hartford, Connecticut, the cost of being oriented at an angle other than true south is indicated in Table 2–1.
As you can see, the reduction in solar benefit increases exponentially as you rotate the home's orientation away from true south. Within 20 degrees or so of true south, the cost of variation in lost solar benefit is minimal, which allows some latitude in placing the house on a site that presents obstacles such as slopes and outcroppings.
Ideally, the north side of the site will provide a windbreak, with evergreen trees and a protective hillside.These natural features will protect the home from the harsher northerly winds and weather. Deciduous trees on the east, south, and west will shade the home in summer, yet drop their leaves in winter, allowing sunlight to reach the home. Yet the ease with which sunlight will penetrate through the deciduous trees in winter.
_The ideal orientation for a_ _solar house is with its long_ _axis perpendicular to true_ _south (or 0_ _°_ _on the diagram)._ _Because of various factors,_ _it's_ _sometimes necessary to shift_ _the orientation somewhat._ _Within 20 degrees of true_ _south, the cost in solar gain is_ _minimal, but as the orientation_ _shifts more drastically, the_ _house will significantly lose_ _solar benefits._
**KNOW YOUR SITE**
Spend some time on your proposed home site. Try camping on the site to learn about its sun conditions in different seasons. Make a point of being on the site at sunrise and sunset at different times of the year. Develop a sense for which direction the prevailing wind comes from. Use your imagination in order to picture the view from each room. Mark the footprint of your new home on the ground, and develop a "feel" for what each room will be like after the home is constructed. In addition to solar orientation, consider access, view, wind direction, snow removal, power, septic, and of course, water. Carefully investigate your water source. If it is to be non-municipal, consider dowsing to find the best location for a well. (The American Society of Dowsers in Danville, Vermont, can refer you to a qualified water dowser in your area.) Sometimes it's advisable to drill the well in advance of building your home just to be sure of the cost, quantity, and quality of your water.
The long axis of a solar home should run east to west, presenting as much surface area to the sun as possible. If your new home measures 24 by 48 feet, maximize the amount of surface that the sun will strike by siting your home with the 48-foot dimension running east-west.
**SOIL CONSIDERATIONS**
The location of the base for your new home _must be high and dry_. The presence of water in the base of your new home will cause untold problems, not the least of which is mold. Water will also rob the Solar Slab of stored heat. Be careful to use a minimum of 12" of compacted sand or gravel under the concrete blocks shown in the diagrams The Solar Slab concrete heat exchanger: a section drawing and The Solar Slab: a sill detail. You must also have a continuous perimeter drain outside the footings that drains to an aboveground location.
Radon is a consideration in certain locations. The EPA does not recommend soil testing for radon prior to construction of new buildings, because the radon concentrations in soil can be much different from one point on a lot to another. Testing enough locations at enough depths on a site would be very expensive. A much cheaper and more reliable approach is to use radon-resistant techniques when the building is built. These techniques are very inexpensive, help protect the home from radon, and also help solve other problems like moisture in the home. Many of the techniques are already used by good builders. See the EPA publications "Radon-resistant Construction Techniques for New Residential Construction" (EPA/625/2-91/032) and "Radon Prevention in the Design and Construction of Schools and Other Large Buildings" (EPA/625/R-92/016) for more information.
Negative pressure results when a hole is dug in the ground. This can cause unwanted gasses to enter the home via the base. A common technique for soil depressurization involves running PVC pipe around the outside of the footings. A fan can be attached and unwanted gasses such as radon are drawn from below the slab and vented above the roofline where they are quickly diluted in outside air.
A practical approach to the suggested mitigation scheme is to install the necessary perimeter drain, the above-grade discharge and cap off the vertical riser. After the house is constructed, test to see if there is a problem. The next step is to uncap and extend the vertical so it will draw naturally. If after retesting, the problem still exists, then add the exhaust fan.
The EPA recommends that a qualified contractor be used to mitigate homes because of the specialized technical experience required. Without proper equipment or technical knowledge, one could actually increase the radon levels or create other potential hazards. See "Residential Air Cleaning Devices: A Summary of Available Information," Office of Air and Radiation (OAR) Washington, DC 20460, EPA 400/1-90-002, February 1990.
The National Environmental Health Association (NEHA) certifies radon mitigators that have taken a course and passed a test based upon the material taught. You should also contact your state radon program office.
RADON GAS DRAIN TILE VENTILATION SYSTEM SCHEMATIC
**USE WINDOWS AND PATIO DOORS**
**AS SOLAR COLLECTORS**
If you locate the majority of the windows and patio doors on the east, south, and west elevations of the home, they can act as solar collectors. One often sees pictures of solar homes with huge expanses of south-facing glass tilted to be perpendicular to the sun rays. Let's remember that you want your home to be comfortable all year-round. Tilted glass, though technically favorable during certain heating months, is very detrimental in summer. One has to design on a 12-month basis, and understand where the sun is at each time of the year, in order to comprehend how the sun may be most beneficial to your home.
The diagram below shows the sun's angles at three different times of the year—December 21, March 21, and June 21, at north latitude 40 degrees (see also Appendix 2). We can see in December that the sun's low altitude almost directly strikes the south-facing vertical glass, which demonstrates again the importance of facing a home true south.
The March 21 and June 21 illustrations show that as the days grow longer, the breadth of solar aperture widens, meaning that a home will gain more solar heat and light from its eastern and western windows. Meanwhile, the altitude of solar noon rises to 50.0 degrees on March 21 and 73.5 degrees on June 21.
Taking into account the sun's angles at three representative times of year.
**MAKE USE OF THE**
**LOW SUN ANGLE IN WINTER**
In Vermont at the winter solstice (December 21), the sunlight shining through a south-facing patio door will penetrate twenty-two feet into the home. On the summer solstice (June 21), the sun will only enter the building a few inches.
A dentist in New Hampshire placed a small round dental mirror flat on the sill of his south-facing patio door, and each day at noon he made a mark on the ceiling where the reflected sunlight hit. In twelve month's time, can you guess what kind of geometrical pattern was on his ceiling? An elongated figure-8. The mark closest to the south wall was made at the summer solstice, and the mark farthest from the wall was made at the winter solstice.
Let's examine south-facing glass at solar noon. If you plant a deciduous tree on the south side of your home, the sun's rays will shine through the canopy in winter when the leaves are gone. Yet in summer, the tree's canopy will absorb almost all of the sun's heat. Plant deciduous trees at a distance from the home, based on the height to which the tree is expected to grow and the size of the anticipated canopy. If deciduous trees exist on your site, cut down only those that directly obstruct the clearing needed to build the home. Thin adjacent trees' branches after you have gained experience with their shading patterns in both winter and summer.
Because of the high angle of the summer sun, its heat will bounce off vertical south-facing glass, unlike the almost direct horizontal hit your solar collectors will get in winter. This "gadget" called a solar home will "automatically" turn itself on during the coldest months and shut itself off during the summer months, so that solar collection is maximized for heat gain when you need the extra heat, and minimized when heat would be uncomfortable. As you grasp these basic dynamics, you have started to let nature work for you. Table 2–2 shows the amount of energy received by vertical south-facing glass at solar noon at 40 degrees north latitude.
As you can see, the amount of energy received by vertical south-facing glass in December or January is almost triple the amount received in June.
What about east- and west-facing glass? We frequently hear about south-facing glass, but at the beginning and end of the heating season, east- and west-facing glass make good solar collectors, as well. In March, sunrise is at 7:00 AM versus 8:00 AM in January and 5:00 AM in June. Due to the angle of the sun being perpendicular to east-facing glass as the sun rises, and perpendicular to west-facing glass as the sun sets, east- and west-facing glass do not "turn off " as solar collectors in summer. We have to be more careful about the amount of east- and west-facing glass we use. We also have to consider location as more of a factor in the distribution of east- and west-facing glass. For example, a solar home located in Pennsylvania, which requires energy for summer cooling, should have less east- and west-facing glass than a home located in northern New England. In chapter 4 we will discuss other techniques to control the gains and losses of windows and patio doors.
Solar Principle # 2
_Design on a 12-month basis._
A home must be comfortable in summer as well as winter. When designing a solar home, carefully plan to accommodate and benefit from the sun's shifting patterns and other natural, seasonal cycles. Before finalizing a building plan, spend time at the site at different times of day and year, and pay attention to the sun, wind, and weather.
Now that you understand how effectively windows and patio doors function as solar collectors, you will see why I continue to emphasize that you should use the windows and patio doors that you are already committed to purchase to not only enhance the livability of a new solar home, but also to serve as an automatic solar collection system.
**STORE THE** **SUN'S** **FREE HEAT**
**FOR NIGHTTIME USE**
The other important material we have already committed to purchase for our new home is concrete and/or concrete building blocks. To store heat we need to have _mass_ , or a body of material that can hold heat. Water is the storage medium of choice in active solar systems because it holds 62.4 Btus per cubic foot per degree Fahrenheit, making it an excellent theoretical storage medium. Concrete holds only about 30 Btus per cubic foot per degree Fahrenheit, but has an advantage: when building a new home, we have already committed to buy tons of it. Used properly, concrete becomes another integral component in a household solar heating-and-cooling system.
I have described the way in which the Trombe wall utilizes concrete as part of a solar collection system. In chapter 3 we will look at another way this durable heat-storage material can be used.
A solar house uses trees, hills, and the varying angles from which the sun strikes a home during the year to enhance its ability to collect sunlight and store its heat. I have emphasized the importance of facing a home south, and we've begun to think about rearranging materials that we would have purchased anyway, such as windows and concrete. Ideally, your new solar home will not cost you any extra money.
And there are other, non-monetary characteristics of a well-built solar home, including tightness of construction, absence of air leaks, and judicious venting to supply plenty of fresh air without wasting heat. Layering the walls to prevent heat loss and providing proper venting are crucial to energy-efficiency, and we will discuss these practices in depth in chapter 4.
Let me quote from another letter from an enthusiastic solar homeowner, this one in South Harpswell, Maine:
We find it takes a special way . . . dealing with life and the environment. . . . We have come to feel great pride in our woodpile. It is not a beautiful piece of garden architecture, but you sure feel secure when you look at it. And the house has to be set exactly right to catch the sun's rays in the colder months, and our southern deciduous trees do not cast shadows to interfere with maximum solar energy. Our daily lives and routines have been altered somewhat—keeping woodboxes filled, stove work done regularly, thermal shutters closed at about 4 PM once winter sets in. You develop a whole philosophy of working with nature and you become committed to a life style in which your house is almost a family member that you care for. There's extra work for sure, but the pleasure you get is worth the extra effort, as we seem to watch the world around us as we never did before. It is important to us now to know when the sun will rise and set—and the direction and velocity of the wind—and the temperature of the air.
3
[THE
SOLAR SLAB
AND BASIC SOLAR
DESIGN](chap6_passives_9781603581691_epub_part6.html#d41151054)
Heating-system designers think in terms of heat transfer from warmer to cooler. The typical home furnace warms air to 140 degrees, and the warm air is delivered to the various rooms in the home via ducts. When the thermostat reads 72 degrees or another desired setting, the furnace shuts off. Heat has been transferred from the warmer body (the furnace at 140 degrees) to the cooler body (the house at 72 degrees).The design of a conventional heating system represents a straightforward problem that has a direct solution: determine the heat loss of the building, then size the furnace and ductwork in order to provide a continual or "on-demand" supply of replacement heat.
Active systems are easy to visualize—boilers, ductwork, pipes, and radiators—whereas the elements of a passive heat collection and storage system may be almost "invisible." When faced with the problem of designing a solar home, early solar designers tried to assimilate the elements of an active, furnace-based system. Exterior solar collectors were utilized to build up high temperatures using water or air. This heat was then stored in a high temperature "heat sink" using beds of rocks or tanks of water ("heat sink" is a physics term for a medium that absorbs and stores heat—for example, water, concrete, or masonry, in particular arrays). Ducts or pipes transported the heat back and forth from the sun-exposed exterior collector components to the interior storage components of the system. Such active systems are complicated; they tend to require added-on costs to the home, and are sometimes difficult to justify financially. Further, some of them simply didn't work very well or were plagued with mechanical problems, especially over time, necessitating continuous oversight and maintenance.
**IT'S** **HARD TO GET**
**A DRINK IN A DRIZZLE**
"Solar gain" is the free heat derived from the sun. Sunlight is ubiquitous, but diffuse. Systems that involve rock beds and solar hot-water storage tanks attempt to concentrate a diffuse form of energy. It is both difficult and expensive to concentrate, build, and hold high temperatures in solar heating systems. Solar energy can be compared to a drizzle: there are tons of water in the air but it's very difficult to get a cupful to drink. Almost all attempts to build active solar homes are based on trying to build up heat in some sort of storage reservoir that will have a temperature substantially higher than room temperature.
Since the Trombe wall, for instance, needs to build up a temperature greater than normal room temperature in order to transfer heat to the adjacent living space, the home can become overheated during the day. If, as described in chapter 2, the Trombe wall reverses its air flow at night, the home may be subjected to uncomfortable cold flows of air. If we remember that our naturally heated home needs to stay comfortable all day and all night, twelve months a year, these wide variations in temperature should be avoided.
In this chapter we are going to examine a solar heating system that stores heat in the floor at a temperature no greater than comfortable room temperature, and a system that uses windows and patio doors as solar collectors.
The solar system technique described in this book is a departure from conventional heating design. As mentioned in chapter 1, most heating systems are designed by specialists working independently from those producing the general house design. This practice usually results in a worst-case design, with oversized furnaces and ductwork. Oversized equipment will necessitate higher construction costs and will also cause higher operating costs, as oversized equipment inefficiently cycles on and off.
Passive or natural systems represent transient engineering problems; many elements of the calculations necessary to design these systems occur simultaneously, making the processes they involve difficult to analyze. Most heating designers don't like this kind of "fuzzy" problem, and often they are not given all the site information necessary to design on anything more subtle than a worst-case or generic basis.
It is a straightforward calculation to size a furnace on a worst-case basis and to provide a system of ducts to carry heat from a 140-degree furnace to areas of 72 degrees. However, to calculate exactly what is going on when heat is entering the home from the sun is much more complicated: some heat is being used directly to heat the home; some is being stored; and some is being lost back to the outside. Moreover, each one of these events influences each of the others. From the perspective of the conventional heating and cooling technician, this is a "fuzzy" or transient problem. We will simplify this transient problem by looking at temperature averages for the day, month, and year.
In fairness to the conventional system designer, it's important to acknowledge that their job is to guarantee adequate heat and coolness with a wide margin to cover seasonal variation. Experimentation can sometimes lead to "call backs" or other expensive liabilities if a system doesn't work to the homeowner's satisfaction. A system designed on a worst-case basis may cost more to buy and operate, but it also doesn't represent a potential liability to the designer as it will always be more than capable of doing the job.
**ROOM TEMPERATURE STORAGE**
Engineers and designers schooled in heating and ventilating have found the idea of creating heat storage at or below room temperature to be strange. Early on in the development of the system described by this book, some of the typical responses were: "Can't be done"; "Remember Newton's Laws of Heat Transfer—heat only goes from hot to cold"; "Low- temperature room storage will take heat from the living space. You'll create the equivalent of an ice cube in the drink." (That is, the drink remains at 32 degrees Fahrenheit until the last ice cube melts; in this case the "drink" is the living space and the "ice cube" is the heat storage in the floor. The skeptics are concerned that the floor won't let the house come up to temperature.) Or, "The heat sink will act detrimentally to the comfort of the living space."
**KEEP THE FURNACE OFF**
We will use daily averages to help analyze this transient heat problem. Let's start by thinking in terms of how we can keep the furnace off. If the furnace doesn't have to run at all, and instead heat is being supplied naturally and free to the house from the sun, isn't that the name of the game?
The Trombe wall described in chapter 2 is elegant in its simplicity but aesthetically crude. Pictures of a blackened concrete wall along the south side of a home certainly would not survive among glossy photo spreads in _Better Homes and Gardens_. In addition it blocks out a good portion of the cheery southerly sun. From a technical standpoint, the movement of warm air over the surface of a smooth vertical wall will cause laminar flow; that is, a thin boundary layer of air will build up and the warm air passing over this boundary layer will not readily give up its heat to the concrete. An airplane wing is an example of a surface that produces laminar flow. There is very little heat being transferred to the wing as it slips smoothly through the air.
_Thermal mass is comprised of_ _building materials that absorb_ _heat effectively, charging up_ _like a thermal battery and_ _then yielding this heat back_ _into the_ _home's_ _living space_ _through periods of time when_ _the building is not actively_ _gaining heat from the sun or_ _from some other source._
On the other hand, a rough surface interrupts the flow of air causing turbulence, which in turn causes greater heat transfer. Picture the fins in a baseboard radiator versus a smooth pipe along the baseboard. The fins provide much more surface area per running foot than smooth pipe would provide. This increase in surface area allows the heated water inside the pipe to give up or transfer its heat to the air. This concept will be crucial when we discuss the construction of the Solar Slab.
Remember the goal described in chapter 2 of utilizing materials you are already committed to purchase for your new home, and rearranging them in a different configuration in order to collect and store heat. Consider what we would need to buy for a full basement. The cellar floor will require a 4-inch concrete slab and we will need a poured or concrete block cellar wall. That gives us tons of material with which to work. Let's see how we can rearrange these materials.
Start by moving the 4-inch concrete slab from the cellar floor to the first floor, eliminating the basement. This is the equivalent of placing the concrete Trombe wall horizontally flat. Next, let's take some of the concrete blocks that we would have used to build the cellar wall, and place them under the concrete slab. Instead of arranging them with their holes vertical, let's lay them on their sides with the holes lining up horizontally to form air passages running north to south. When the concrete is poured over these blocks, it will bond to the blocks and make a huge concrete "radiator"—the radiator's "fins" are the ribs in the concrete blocks (see the illustrations).
_The Solar Slab utilizes_ _completely conventional_ _materials, including concrete_ _blocks and poured concrete._ _Construction of this foundation_ _is neither difficult nor_ _costly, yet the result will be_ _a house with exceptionally_ _effective thermal mass as its_ _base._
_The Solar Slab concrete heat exchanger: a section drawing._
_The Solar Slab: a sill detail._
If this combination of poured concrete slab over horizontally laid blocks is ventilated by air holes along the north and south walls, air will naturally circulate through this concrete radiator when the sun is out. Remember, we oriented our new home with the long axis of the building running east to west. When the sun is out in winter, the south wall will be warmer than the north wall. As heat is transferred into the home by the south glass or by heat transfer through the wall, air that is next to or alongside the south wall will rise. Warmed air will then be pulled out of the ventilated slab, and the cooler air along the north wall will drop into the holes along the north wall. This thermosiphoning effect will naturally continue to pull air through the Solar Slab.
For the Solar Slab to effectively heat the home, it must be thermally accessible to the living space. It is therefore not cost-effective or thermally practical to utilize the lower level for a basement-storage area instead of as a living space.
**STORAGE OF TRAPPED SOLAR HEAT**
As heat from the sun "drives" the thermosiphoning, heat in the home, which has been trapped as in a greenhouse, will be taken up via the ribs as warm air passes through the concrete blocks, which in turn are thermally bonded to the concrete slab. Heat from the sun comes to us as light or short wave energy. Since glass is transparent to light, sunlight passes through glass and strikes objects within the interior of the home. As soon as it strikes an object, for instance the floor covering above the slab, light changes form—to long-wave energy or heat. In a highly insulated solar home, this heat will now be trapped. The temperature of the ventilated slab will rise as the trapped heat is absorbed by the concrete. Since concrete has almost no R-value it has little resistance or ability to stop the transfer of heat. Any heat transferred to the ventilated slab anywhere in the building will migrate evenly throughout the array of concrete blocks and poured slab.
We will explore this benefit further when we discuss the use of a woodburning stove as backup heat. The heat storage benefit is free, provided you are willing to trade a full basement for a Solar Slab.
The solar home, properly designed, can achieve thermal balance every day. The energy produced by the east-, south-, and west-facing glass will be either consumed directly by the heat demand of the home, or absorbed by the first floor heat sink as the heat comes into the home. If the heat comes in too fast to be absorbed by the mass, the home overheats. Overheating can be a major problem in passive solar design, and in many respects, passive solar design presents a significant cooling challenge.
**THE OLD NEW** **ENGLANDERS'** **SALTBOX**
One of the designs my company offered, the Green Mountain Homes' 28-foot by 38-foot Saltbox, will be used to explain the way the Solar Slab relates to the functionality of a solar home. For illustrative purposes, we will situate this solar home in Hartford, Connecticut, north latitude 41 degrees 5 minutes (41°5'). The floorplans and a cross section for the Saltbox 38 are shown in chapter 5.
_Many of the plans and_ _calculations in this book use a_ _basic house design known as_ _a saltbox. While designers and_ _builders of solar homes can adopt_ _a wide variety of house styles_ _and construction techniques, the_ _saltbox is useful as a model,_ _since its design has a classic_ _simplicity. The solar home shown_ _here was built in 1978 in_ _Virginia. Note use of deciduous_ _trees for summer shading._
We will present detailed solar calculations in chapter 6, but in order to help explain how the system works we need briefly to examine the Solar Slab, which as you recall is comprised of a 4-inch concrete slab bonded to 12-inch concrete blocks. We can calculate that a standard concrete block is about 50 percent concrete, or the equivalent of 6 inches of solid concrete. Therefore, the 4-inch slab and 12-inch concrete block are the equivalent of 6 inches + 4 inches = 10 inches of solid concrete. Discounting air passages along the north and south walls and the amount of concrete blocks displaced by ductwork, for a 28 by 38 foundation the volume of concrete equals 754 cubic feet.
Assume that the Solar Slab temperature is 60 degrees at 7:00 AM and the daytime rise in Solar Slab temperature is 8 degrees. The Solar Slab temperature at 5:00 PM is then 68 degrees Fahrenheit. Picture what this means: The entire first floor of the living space is now up to 68 degrees; that's 754 cubic feet of concrete at 140 pounds per cubic foot, which is 105,560 pounds or 52¾ tons inside the house, covered with the floor covering of your choice, sitting there at almost 68 degrees.
Surely you have experienced sitting on a sun-warmed rock after sundown. It's nice and warm, and takes a long time to cool off. Remember, the design goal is keeping the furnace off, or requiring it to do very little work. The heat stored in the first floor of the living space, and dispersed evenly throughout the first floor, has to be beneficial to the heating and comfort of the home.
In chapter 6, the thermal balance calculation will show how extra heat provided by the sun is trapped by the greenhouse effect (the conversion of light energy to heat energy), stored, then released as needed from the Solar Slab.
Because the Solar Slab is an effective heat exchanger, with its fins of concrete, the sun's heat is stored in the Solar Slab at the time it enters the house and strikes the floor covering over the slab.
The surface area inside the blocks calculates to be 366 square inches while the top surface is 119 square inches (7 inches x 15 inches). The ratio of square feet of surface area within the blocks below the floor surface to square feet of floor is 366 ÷ 119 = 3. This means that air passing through the blocks is exposed to three times more surface area than if the air had simply passed over a flat surface. This ratio plus the roughness of the surface inside the blocks make the Solar Slab an effective heat exchanger.
**KEEP THE HOME**
**COMFORTABLE ALL DAY**
In a building in which the solar design components have been properly sized and located, while the windows and patio doors are collecting solar energy, the temperature of the home will hold steady at between 68 and 70 degrees, and will not overheat. If the glass area is too large for the heat storage capacity of the mass, the house temperature will rise to uncomfortable levels, and the occupants will be forced to open windows to ventilate, thereby losing the benefits of both immediate comfort and storage of the sun's free heat for use later in the evening. Greenhouses are examples of spaces that overheat during the day and get very cold at night. On the other hand, too much thermal mass and not enough glass to collect heat will result in a chilly, cave-like space that will never come up to the comfortable temperature.
The home must have a proper balance between the square footage of its glass solar collectors and the dimensions of its effective thermal storage mass. A prevalent mistake in solar design is using too much glass. The thought pattern seems to be that if some south glazing is good, a lot more is better. As we discussed above, overglazing will cause overheating and detrimental negative temperature swings at night. In fact, in some cases, the cost to heat the overglazed home at night will exceed the benefit derived on sunny days. This consideration is especially important in the northeast, where we have about 50 percent sunshine in the winter and long cold winter nights. The good news in the northeast is that our heat season is so long and severe that almost any measure we take to utilize the sun's free heat can result in significant cost and energy savings.
_How does the Solar Slab work? Here is a sequence of illustrations showing its_ _operation in three modes. First, a sunny day in winter. Heat from the sun and_ _when necessary a small backup woodstove is stored in the thermal mass of the_ _radiant floor as sun-warmed air is drawn by vents through channels made by_ _aligned concrete blocks beneath the poured slab and the_ _home's_ _finish flooring._
_On a cold winter night, solar heat stored in the_ _home's_ _slab during the day_ _radiates upwards into the living space. The temperature of the entire floor will_ _not vary more than 1 degree. A small wood or coal stove will normally provide_ _adequate supplemental heat, and a small conventional furnace will double as_ _an air mover for the solar heat exchanger, as well as providing backup heat (see_ _chapter 7). Nighttime window insulation prevents the loss of heat through the __largest of the windows and patio doors._
_During a sunny summer day, because of proper siting, glazing, and sizing of_ _thermal mass, the Solar Slab will aid in cooling the house, as excess solar heat is_ _absorbed during the warm hours of the day. Before the home can overheat, the day_ _has ended (this is called_ _"_ _thermal_ _lag")._ _The same attic fan used in winter_ _to redirect heat from the second floor ceiling can be used in summer to vent_ _warm air out through the attic vents. In air conditioning areas, air-to-air heat_ _pumps can also be used in tandem with the Solar Slab._
Let's return to the objective of keeping the furnace off. The furnace was off all day as the solar home collected and stored heat. During the evening, the occupants will need very little supplemental heat to maintain 68 to 72 degrees until 10:00 PM (bedtime), because the entire first floor of the house was 68 degrees at 5:00 PM. Basically, the backup heat is only heating the difference between the Solar Slab temperature and the desired room temperature. If 68 feels comfortable, then no backup heat is needed at all. As the Solar Slab gives up its heat to the first floor living space, the Solar Slab temperature will start to decline. The first floor room temperature at 7:00 AM will be the same as the Solar Slab temperature. Stored heat has been given up to the house through the night, and the Solar Slab is now ready to absorb the next day's free solar heat. This solar home will stay ready to instantaneously accept any solar heat available. If the sun comes out for just a few minutes between clouds, that heat will be collected, as there are no sensors that have to react to turn pumps on.
In addition, this solar home will absorb excess heat from cooking, lights, and yes, even the heat given off by human bodies. A particularly nice way to heat a solar home is to throw a party and invite lots of people over on a cold winter day! Remember, heat travels from warm bodies to cold bodies. We are each a small furnace, running at 98.6 degrees.
**THE THERMAL FLYWHEEL**
Do you remember the old John Deere tractors that had an external heavy-metal flywheel? The tractor's small engine slowly got the huge flywheel spinning. Once up to speed, very little energy was needed to keep the tractor moving. That is called mechanical inertia. A body in motion doesn't want to stop. Likewise, the Solar Slab provides thermal inertia to the home so that the home "wants" or tends to stay at a steady temperature, using very little purchased fuel in the process. With this kind of thermal inertia built into the solar home, we can downsize the backup heater, and instead size equipment for less than worst-case conditions.
Why haven't other people used this building technique? The answer most likely lies in the difficulty of trying to calculate the effect of a "room temperature heat sink." Some would say that this approach seems to violate conventional heating theories.
My approach to the problem of heating a home with the sun was to make my best engineering calculations, and then build and monitor a prototype. This represented both a professional and financial risk on my part, but it was well worth it. I was sure that my approach would work, but what I didn't know was how well. By measuring all energy entering the test building, and keeping careful records of the Solar Slab temperatures, we were able to verify the effectiveness of the design.
**A PATENTED DESIGN**
As I started to make heat loss and solar heat gain calculations in 1975, I became more and more convinced that I was on to something unusual, and decided to protect my invention by applying for a U.S. Patent. In order to receive a patent one must prove that the idea or design is original. One of the unique aspects of the Solar Slab design is that the maximum achievable temperature is room temperature. Conventional thinking says that room temperature storage will be at best neutral, or at worst, will result in a drain of heat from the room. Remember the key concept of temperature difference (in engineer's jargon, "Delta T"), and the laws of heat transfer—heat will only flow from hot to cold.
**The Monitoring Effort**
As explained in chapter 1, Professor A.O. Converse, of the Thayer School of Engineering at Dartmouth College led a team that independently monitored our prototype, and he and I co-authored several papers which were presented at various solar conferences. His work culminated with the "Final Report Monitoring Studies of Green Mountain Home's Hybrid Systems," December 8, 1978. "We certainly conclude that the purchased energy requirements were quite low and the percent solar is well above 40 percent." New Mexico's Sandia Laboratories published their report on Green Mountain Homes in July, 1979 (its reference number is _SAND_ _79 - 0824_ ).
The monitoring effort with the Thayer School was centered around a Green Mountain Homes Model N-38 in Royalton, Vermont. Professor Converse and I had a unique opportunity to install instruments in the superstructure of the N-38 during construction. We also placed measuring devices in an "X" pattern within the Solar Slab and installed vertical probes in the gravel layer under the concrete blocks and inside and outside of the footings.
In addition, we also installed a device to measure the incoming solar energy (insolation).
All energy consumed by the building was documented. Meters measured the electricity consumed by the furnace and the second floor blower as well as the electricity used for all other purposes. A fuel meter was installed to measure the number of gallons of oil consumed by the furnace.
Solar Principle # 3
_Provide effective thermal mass to store free solar heat in the daytime for nighttime use._
When sunlight strikes surfaces, the solar energy is converted from light to heat. Design a home's thermal mass to effectively absorb the warmth of sunlight as it enters the building in winter, thereby avoiding overheating. Achieve thermal balance by sizing the storage capacity of the thermal mass to provide for the heating needs of the building through the night. In summer, a properly sized thermal mass will serve to cool the building because of "thermal lag" — that is, excess heat will be absorbed during the daylight hours, and by the time the mass has heated up, the day is over and that stored heat can be discharged by opening windows to increase circulation during the night.
As evidenced in the Thayer report, the home was very energy-efficient and compared favorably with several active solar homes which were also being monitored by Converse and his colleagues at that time.
The efficiency of our design had exceeded my expectations, and the monitoring verified information that we had predicted in the U.S. Patent application. The ongoing independent monitoring of the prototype and the knowledge gained by working with solar homes located over a wide area with diverse design requirements allowed us to continually refine and improve our design methods.
**PARTIAL RESULTS OF THE**
**INFORMATION MONITORING EFFORT**
1. The temperature was consistent and evenly distributed throughout the concrete slab and concrete blocks, with any difference in temperature being within one degree. This observation helped in the design of back up heating systems. That thermal consistency is particularly beneficial to the woodburning home; since the heat from the woodstove "migrates" evenly throughout the first floor, the design of a home that uses a woodstove as backup heat is essentially the same as designing for solar. The engineering problem is the same in the sense that the woodstove is an uncontrolled centralized source of heat that needs to be distributed evenly throughout the building and stored, if necessary, for use after the stove finishes its burn.
2. More than 100 percent insolation was measured on sunny winter days. This was attributed to the reflection up and into the building from snow cover on the south patio. This factor saves some of the homeowner's energy, because the south patio can be left unshovelled, allowing the snow cover to reflect the sun's heat and light into the building.
3. The temperature outside the footings (4 feet in the ground) reached a maximum of 68 degrees Fahrenheit in September, and slowly decayed to a minimum of 45 degrees in February. The huge reservoir of heat at 45 degrees or better in the ground below the gravel layer is transferred into the home when it is unoccupied and unheated. This effect is described in chapter 7.
4. A 12-degree temperature drop was measured as the air passed through the Solar Slab in summer. This indicated that the Solar Slab was indeed absorbing energy. This heat transfer and absorption was later incorporated into the design of air-to-air heat pumps for summer air conditioning.
5. We learned that the solar heating system's electrical energy usage, though small in magnitude, was a relatively significant part of the total usage because of the low overall heat demand of the solar home. Through trial and error, the second floor blower was reduced in size from the original 1/3 horsepower squirrel-cage type to an inline 1/40 horsepower duct fan, thereby almost eliminating it as a significant energy user.
**Everyone's** **Legacy**
U.S. patent law is very different from most of our other laws in that it discriminates; that is, it grants exclusive use of the invention to the inventor for seventeen years. We don't have many laws that obstruct free trade to the extent that our patent law does. In an effort to remedy this obvious conflict, the law gives the invention to the "People of the United States" after seventeen years. This book, hopefully, makes this gift more meaningful, as it is an attempt to explain to lay people as well as professional builders how best to utilize this invention to heat and cool homes yet to be built.
4
[INSULATION,
VENTING, AND
FRESH AIR](chap6_passives_9781603581691_epub_part6.html#d41151055)
As explained in chapter 1, insulation standards have increased dramatically since 1973. The quantities and types of insulation needed to facilitate solar heating of a home are no longer considered unusual or "alternative." As house construction becomes tighter and insulation standards rise, the danger of causing water damage through condensation increases. And by sealing fresh-air vents to the outside, we risk jeopardizing the indoor air quality. We will need to be very careful not to "over-do a good thing" by completely sealing up a home. Our homes need to have adequate fresh air. Just as overglazing will cause overheating problems, we can cause air quality and maintenance problems by not providing proper ventilation for the well-insulated and tightly constructed solar home.
**WHAT IS VAPOR?**
Vapor control is probably one of the most misunderstood principles in home design. In order to properly design a highly insulated solar home, we must first understand how to control vapor. We have all seen water condense on the outside surface of a glass filled with ice water on a hot summer day. The warm, moist summer air is full of water in the form of vapor—a gas. When this warm, moisture-laden air strikes the cold surface of the ice water glass, the water vapor changes from a gas to a liquid, and drops of water appear on the outside surface of the glass. The conditions existed for condensation to occur. These conditions are a combination of temperature, moisture content, and vapor pressure. Similarly, under certain conditions dew will form on the late evening or early morning summer grass, when chilled air makes contact with warm blades of grass and water vapor condenses to liquid droplets.
In winter, our homes are full of warm air, which has moisture in it. With the outside temperature being very cold, the conditions for condensation will sometimes occur within the wall and/or roof cavities. If moisture-laden air is allowed to enter the wall or roof cavity, and if it condenses there, the result will be water damage, just as if a leaky roof or burst pipe had flooded an area that is supposed to stay dry. First, this condensing water vapor will ruin the effectiveness of fiberglass insulation, and then it will cause rot and mildew. The irony is that the more insulation that's placed in the walls and roof, the greater the danger of creating the conditions for condensation within a wall or roof cavity.
**WE NEED FRESH AIR**
The remedies for such vapor problems are providing good fresh air make-up to the home, and providing positive vapor barriers in the walls and roof. We should maintain the fresh air replenishment of our homes at no less than two-thirds of an airchange per hour; that is, two-thirds of the entire air volume of your home should be replaced each hour.
_Water vapor will migrate_ _toward cooler areas, and_ _without proper use of a well-sealed_ _vapor barrier on the_ _living-space side of the walls,_ _insulation will gradually_ _collect moisture, rendering it_ _eventually useless._
_Two designs for layered wall_ _construction. I recommend_ _the design, above, utilizing_ _2 x 4 studs and a continuous_ _layer of 1-inch rigid_ _insulation. While both walls_ _are R-20, the layering of_ _the 2 x 4 wall with exterior_ _rigid insulation produces_ _a thermal-break, thereby_ _reducing framing losses and_ _outside-air penetration._
1. Provide ventilation in all bathrooms. Fans should be vented directly to the outside.
2. Where possible, provide ventilation in the kitchen. Fans that just recirculate and filter kitchen air are not as good as fans that are ducted to the outside.
3. Be sure to vent a clothes dryer directly to the outside.
4. Don't be concerned about the use of a woodstove. It will pull fresh air into your home.
5. When it comes to daily comfort, use your best judgment, and don't be preoccupied with saving energy to the point that you don't open windows to allow fresh air into your home if the house feels stuffy or stale.
Recently the American Society of Heating, Refrigeration, and Air-Conditioning Engineers (ASHRAE) has come out with a new standard for whole house ventilation. It states that minimum whole house total ventilation shall be no less than 0.35 air changes per hour (ACH). It also states that a mechanical system shall be installed to provide the ventilation. Several states have adopted the standard, including Vermont. I was told in a call to the Vermont Department of Public Service that the mechanical system can be quite simple or complicated depending on the equipment purchased. The complicated and more expensive system is an air-to-air heat exchanger. The Vermont Department of Public Service told me that the simplest way to conform to the law in Vermont is to install a programmable switch to operate a bathroom ventilation fan for a specified time period each day.
This book recommends a whole house ventilation rate of 0.67 air changes per hour, which has been derived through experience as a healthy rate of air replenishment. Rates of less than 0.67 per hour will, in my opinion, endanger the health of the occupants. Further, inadequate ventilation will and can lead to other airborne problems, including possible mold buildup.
Let there be no misunderstanding about where the fresh air makeup is coming from. The walls and roof of your home should be very tightly constructed as shown in the details We Need Fresh Air and Two designs for layered wall construction. Fresh air will enter your home through controlled or deliberate openings, as previously described, not through gaps in the insulation or poorly sealed windows and doors. The amount of fresh air intake can be measured by independent testing agencies at a nominal cost. This service is provided in some cases at no cost by state agencies. One such testing method is called the "Blower Door Test," where a fan is installed in an exterior door, and the rest of the house is closed up. By running the fan and measuring the overall volume of air, the number of air changes can be determined. If the rate is too low, you will need to increase the amount of fresh air intentionally introduced, possibly by adding an air-exchange or ventilator system. If the rate is too high, you can reduce infiltration by improving insulation, adding weather-stripping, or sealing gaps around doors and windows.
**POSITIVE VAPOR BARRIERS**
In heating situations, the rule is that a vapor barrier must be placed toward the heat, in other words on the heated or interior side of the structure. In normal wall and roof construction, the vapor barrier is placed right behind the drywall—between the drywall and the studs. This vapor barrier should be "positive" in the sense of being a discrete membrane, not incidental to the batt insulation, and it should be carefully lapped and sealed. Positive vapor control means the placement of a separate vapor barrier such as the 6-mil "poly" shown in the 2 x 4 stud wall, which shows an R-20 wall and an R-32 roof section. In chapter 6 we will calculate these R-values (the R-value represents the resistance to heat transfer, therefore the higher the R-value, the less heat this material will transfer).
The preferred wall design is the 2 x 4 stud wall with batt insulation and a layer of Styrofoam outside the exterior wall sheathing. This layering makes a tight wall and provides a continuous layer of rigid insulation outside the 1/2-inch plywood sheathing. Layering the wall construction in this manner reduces heat loss which occurs through the framing members ("bridging losses" are heat losses that result from studs transmitting cold directly into the home). It is all but impossible for outside air to penetrate a wall that has been layered in this way, since the seams between pieces of rigid insulation and the seams of the plywood will not coincide.
Although I don't recommend doing so, the exterior layer of rigid insulation may be eliminated by the substitution of 2 x 6 studs with 6-inch batt insulation; however, with larger studs bridging losses will be more significant, and these additional losses should be considered when the framing lumber is in direct thermal contact between the inside and the outside of the wall unit. Although 2 x 6 framing has become standard, in most cases it isn't structurally necessary to use 2 x 6s; a wall constructed of 2 x 6s 16 inches on center is probably overbuilt, and the bridging losses will be greater with 2 x 6s and no exterior rigid insulation. The use of an exterior house wrap is important with 2 x 6 wall construction to seal cracks and construction joints. Exterior house wraps (such as Tyvek or Typar) are designed to stop the wind but allow moisture to pass through (so that moisture will not be trapped inside the wall, but can exit to the exterior side). House wraps are not vapor barriers. House wrap is not needed with the 2 x 4 stud wall, since the outside tongue-and-groove Styrofoam serves as both additional insulation and a seal against penetration.
When selecting rigid exterior wall insulation, be sure to purchase closed-cell, extruded polystyrene insulation such as Dow Chemical's Styrofoam "Blue Board," or U.S. Gypsum's Formula R. Less expensive open-celled alternatives are susceptible to insect damage, and degradation in R-value over time.
An ongoing free flow of air should be maintained from the eave to a continuous ridge vent. This flow of air above the insulation will keep the roof plywood from getting warm, helping to prevent "ice dams." It will also keep the roof cooler in the summer. In high snow areas, a "cold roof " is often used, in which a separate vented roof is installed above the roof plywood. This design is useful where double protection from moisture and cold is needed. The "cold" roof is added on top of the vented roof construction. The original roof is covered with heavy felt or tarpaper, and the top of the cold roof is typically covered with a metal roof to shed snow. I recently noticed that all new construction at Sun Valley, Idaho, is built this way.
Remember that damp insulation loses its ability to block the loss of heat, and wet insulation is worthless. Pay particular attention to the continuous interior vapor barrier shown in the wall and roof details. Placing unfaced batt fiberglass insulation and then applying a continuous and distinct vapor barrier is a better solution than relying on foil-faced or kraft-faced fiberglass for vapor control. Positive vapor control will stop water vapor from migrating into the wall or roof insulation cavity.
It is not uncommon for a newly constructed and tightly insulated home to have excess moisture content in the air during the first winter. This is due to the gradual stabilization of moisture content of all the materials used inside the home. As these materials dry, the moisture content of the air will slowly decrease. If there is excess moisture in the air, water vapor will condense on the coldest surface available—the windows. This is entirely predictable in the first few weeks of the first winter. The "cure" is to open a couple of windows and ventilate the home. If, however, this condensation persists, it means that there is a bigger problem, and the source of the excess moisture should be investigated.
A client once called me, sure that his ski house was "self destructing." A ¼-inch layer of ice had formed over some of the window surfaces, and water was dripping off the windows. The temperature was about 10 degrees outside. Upon inspection, a dryer vent was found to be venting to the inside of the house. As his clothes were dried, moisture was being pumped into the home. Since the home was properly constructed and had positive vapor control, the water vapor had only one place to go—the windows. The homeowner was instructed to "crank up" his woodstove and open a second floor window at each gable end to let the house vent. In a matter of hours, the house began to stabilize.
Another homeowner installed the batt insulation in the ceiling of his home but had never managed to install the vapor barrier. He was living in and finishing the construction of his home at the same time. After six weeks of the heating season, the ceiling insulation was completely saturated with moisture, rendering it useless. All of this soggy fiberglass had to be removed and replaced; this time he installed fiberglass insulation properly protected with a positive vapor barrier.
Solar Principle # 4
_Insulate thoroughly and use well-sealed vapor barriers._
Build tightly constructed, properly insulated walls and roofs. Carefully install and seal discrete vapor barriers on the living-space side of walls, ceilings, and/or roofs. Incorporate an air-lock entrance.
An old-time Vermont builder once told of installing a board ceiling in the second floor of a new home. Since he didn't believe in vapor barriers, the insulation was placed with no vapor barrier between it and the square-edged board ceiling. Halfway through the first winter, the boards were all water stained. Both the boards and the insulation had to replaced at his expense. His rationale had been that he always used batt ceiling insulation with no vapor barrier on top of drywall so that the ceiling could "breathe." Inadvertently, he was creating a drywall vapor barrier. Drywall with two coats of latex paint makes a fairly effective vapor barrier; however, when square-edged boards were substituted for the drywall, moisture traveled through the joints, and the dew point was reached within the ceiling insulation layer, causing the water problem.
**R-VALUES**
R-values have been mentioned several times. In order to specify the correct insulation levels for a passive solar home, we will need to understand what R-values are, how they are calculated, and how you can use the information derived from the calculations. All materials transfer heat at different rates, and R-value is the measure of the resistance of a given material to the transfer of heat. As explained in chapter 3, concrete transfers heat at a rapid rate, while wool sweaters with air trapped in their weave transfer heat more slowly. Appendix 3 shows a list of R-values for various materials.
"U-values" are the inverse or reciprocal of R-values. U-values are expressed in Btus per hour per square foot per degree Fahrenheit. Btu stands for British thermal unit, and is the amount of heat necessary to raise the temperature of one pound of water one degree Fahrenheit (in this book, for the sake of clarity, we will use the nomenclature "Btus" in text and equations when referring to these units in plural; true ASHRAE aficionados will note this departure from engineers' normal practice).
The heat loss of a home is calculated by first determining the U-values for the walls, windows, and roof. Individual heat losses for specific areas are determined by multiplying square feet of surface area by the U-value. Then a calculation is made of the amount of energy needed to reheat the fresh air that is coming into and escaping from the building during each hour. The total of these losses represents the total theoretical loss of the building. This kind of calculation will be demonstrated and explained in chapter 6.
_Thermo-shutter design for nighttime window insulation._
**NIGHTTIME WINDOW INSULATION**
Notice in the floorplans shown in chapters 5 and 8 that thermo-shutters are shown on some of the windows and patio doors. In the three-bedroom plan, the thermo-shutters are used on three south-facing patio doors as well as one window each in the east-facing dining/family room and west-facing master bedroom, on the first floor, and on two windows in the east- and west-facing bedrooms on the second floor. The combined area of these windows represents a total of 203 square feet of glass.
Insulating the windows and patio doors at night (especially the largest windows) will measurably improve their performance as solar collectors. A single pane of glass has an R-value of 1, meaning that single-glazed glass is essentially only keeping the wind out! The window companies have now developed better-insulated glass. High-performance glazing has selective coatings on various surfaces of the sheets of glass, and the air between the sheets of glass is replaced by gasses that are more effective insulators. And yet, although high-performance glass is better than ordinary glass, the R-value of even dual-pane glass pales when compared to an R-21.36 wall. Remember, insulated dual-pane glass has an R-value of 1.92, whereas the wall is 21.36/1.92, or 11 times better. While architecturally attractive glass makes an excellent solar collector while the sun is out, the winter nights are long and cold, turning windows and patio doors into thermal losers at night.
_Thermo-shutter design for_ _insulating typical 6-foot patio_ _doors._
In addition to transmitting heat out of the thin panes of glass, uninsulated windows actually draw heat out of you. Have you ever noticed that it seems much colder to sit next to a patio door at night versus sitting next to a nicely insulated wall?
There's more bad news. Warm air from the room will be drawn toward the glass, and as this warm air is cooled by the colder glass surface, it flows toward the floor, allowing more warm air to be drawn to the cooling glass. This is the same kind of reverse thermosiphoning effect that can take place at night with the Trombe wall described in chapter 2.
Most heating system designers locate heat grilles in front of windows and patio doors to provide a "bath" of warm air across the glass surface. This increases the inside surface temperature of the glass, which increases the temperature difference across the glass, which in turn increases the heat loss of the glass. One error compounds another, and so on.
As you have probably deduced, the solution to this problem is to add nighttime insulation to the windows. The illustrations Thermo-shutter design for nighttime window insulation and Thermo-shutter design for insulating typical 6-foot patio doors show thermo-shutter details for a typical six-foot patio door and six-foot-wide window grouping. Note that the interior insulation of the thermo-shutter is 1 inch of foil-faced urethane. The interior foil face will reflect heat back into the room, even though it is sealed inside the thermo-shutter. With the thermo-shutter closed you may now comfortably sit next to a patio door on a cold night. The thermo-shutter is providing added insulation as well as reflecting heat back into the room. The stop shown on the details allows the thermo-shutters to fit tightly, which eliminates reverse thermosiphoning at night.
The photograph below shows how the thermo-shutters may be decorated with fabric, which may be changed seasonally. Construction of thermo-shutters takes the skill of a qualified finish carpenter. You could hire a cabinet shop to make them.
Thermo-shutters have a year-round benefit, as they may also be closed to keep out the sun. They are most beneficial in summertime on east- and west-facing windows since the sun enters more directly into the living space in the morning and afternoon. The outside foil face of the insulation contained within the wood veneers will reflect the sun's summer heat back out the window.
**Other Options for Window Insulation**
There are commercial products made of fabric on the market that can be used to add insulation to windows and patio doors. Make sure that any product bought for this purpose provides both added insulation and a tight fit along the top or bottom edge (ideally both) to stop the nighttime reverse air flow.
_Detail of thermo-shutter_ _showing use of curtains as_ _decorative finish. When in the_ _open position, the folded-back_ _shutters are no more obtrusive_ _than curtains along the sides_ _of a window or glass door._ _Likewise, when closed, the_ _window will appear to be_ _covered by curtains, yet the_ _insulation value of the layered_ _shutter is far superior to that_ _of curtains alone._
5
[BASIC LAYOUTS
AND FLOOR PLANS](chap6_passives_9781603581691_epub_part6.html#d41151056)
A New England-style "saltbox" house (known to Green Mountain Homes customers as the Saltbox 38) will be used in the first part of this chapter to illustrate and explain energy-saving floor plan considerations. Then we will consider the unique features of two economical "starter" homes. In chapters 6 and 7 you will find an explanation of how to calculate your home's future solar gain and its backup heating needs. As emphasized consistently in this book, the more thoroughly and carefully you consider your space, energy, and heating and cooling requirements while planning your home, the more smoothly the construction process will go, and the happier you will be when you move in.
Assuming that you have now spent some time on your new solar home site, you will have begun to get a feel for the route of the sun throughout the day. We should lay out the home's rooms in relation to the patterns of the sun; that is, morning areas and activities should be planned for the east side of the home, and evening activities generally on the west side. Referring to the floor plans shown for the Saltbox 38, you will see that the kitchen is on the east side. This means that you will often start your day with the sun beaming into your east-facing windows.
If you are a morning person, you may want to occupy the second-floor east bedroom, as it will see the sun first. Even if the rest of your home is not up to temperature in the early morning, the east side will be collecting solar energy from the earliest sunlight, and you may not need any supplemental energy simply as a result of locating yourself on the sunny side of your home in the morning. If your backup heating system is controlled by a thermostat, locate this on an east-facing wall exposed to the morning sun. If it is going to be a bright and sunny solar day, the sun will strike the thermostat and "trick" it into not turning on the backup heat. If the sun stays out for the day, and as it migrates from east to west around your home, free solar energy will heat your living spaces, with any excess energy being stored in the Solar Slab for use later in the evening. You are satisfying the goal of keeping the furnace turned off just by room placement.
_Typical saltbox design: a cross-section. (Do not use for construction. Dimensions are given to correspond with heat_ _loss calculations in chapter 6)._
Notice in the saltbox floor plan that the living room is in the middle of the south side. The living room will be up to temperature slightly later than the dining/family room, as the sun and normal living habits migrate from east to west in the home. Later, as the days grow longer, you will be able to view sunsets through the west-facing bedroom windows.
_Sample floor plans for the Saltbox 38 used as the basis for the solar design calculations in chapter 6. Thermo-shutters_ _are used on larger windows and patio doors._
**WOODSTOVES FOR BACKUP HEAT**
The woodburning stove and chimney are located central to, and in close proximity with, the most lived-in areas of the home—the dining/ family room and living room. The woodstove will radiate heat in all directions from its open, centralized location.
This also allows maximum safety for the interior chimney. An interior chimney stays warm as the hot gases are escaping up the chimney. This will minimize the build-up of creosote, provided that seasoned, dry hardwood is used for burning.
Since there is no basement, the space under the stairs can be used for utilities such as the domestic water heater and ventilation or air-circulation equipment. Keep waterlines in the interior partitions of the home, not in exterior walls. This will guard against possible freeze-ups.
You may also consider installing a domestic hot-water tempering tank, with water pipes and control wires running between the tempering tank and the woodstove, as many stoves have hot-water jackets that can be used to heat water in winter. The woodstove will heat the water in the tempering tank which in turn will feed pre-heated water to the conventional hot water heater. A simple drain-down external solar collector can be added to the system to pre-heat the water with sunlight in spring, summer, and fall.
Solar Principle # 5
_Utilize windows as solar collectors and cooling devices._
This idea sounds obvious, but many people overlook the obvious and spend large amounts of money purchasing, fueling, and maintaining furnaces and air-conditioners to address needs that high-quality windows can also address. Vertical, south-facing glass is especially effective for collecting solar heat in the winter when a home needs additional heat, whereas the same windows will let in much less heat in summer, because the sun's angle is more horizontal in winter and steeper in summer. Provide insulated window and patio door coverings to decrease nighttime heat loss in winter, and to control solar gain in spring, summer, and fall. Windows that open can be used to release excess heat and direct cooling breezes into the house.
**FACE THE LONG DIMENSION**
**OF YOUR HOME SOUTH**
The 38-foot house dimension is lined up east to west; that is, the ridge of the home runs east to west. By facing the long, sloping roof to the north, you will orient the high side of the home with its majority of surface to the sunny south, and the other facade, with its reduced surface area, will face to the sunless north.
Two-story homes are more energy efficient, because they have double the living space under one roof. Heat rises, so the second floor, at least thermally, comes almost free. Two-story home construction costs are also less per square foot, since double use is made of one concrete base and one roof.
_Floor plans for the N-38-X_ _discussed in chapter 6. This_ _house has three bedrooms and_ _two baths, and every room is_ _sunny: there is no dark_ _"_ _north_ _side."_ _Thermo-shutters are_ _indicated on the larger_ _windows and patio doors._
_Floor plan for the prototype_ _Green Mountain Homes_ _N-38 that was used for the_ _monitoring study discussed in_ _chapter 3._
Note the concentration of windows and patio doors on the east, south, and west elevations. There are only two small windows facing north on the Saltbox 38.
Since there is no full basement, storage has been provided on the second floor under the north-sloping roof. In addition, there is attic storage space in the east and west sections of the building.
For this layout, the ideal location for the garage and air lock entry is on the northeast side of the home. Try to locate garage doors on the south elevation to allow the sun to help remove snow and ice. East or west locations for garage doors are next best, while the least desirable location for a garage door is north. Remember that most of what is carried into and out of a home involves the kitchen. The northeast garage location and air lock entrance are the most convenient for carrying groceries into the kitchen.
These same principles are also illustrated in the smaller, more economical N-38 "starter" home shown in the illustrations above. Note that there are no north rooms in this home, as every room in this 16-foot-wide house has a south exposure. The N-38 was the prototype built first and used by Green Mountain Homes to demonstrate the Solar Slab design.
6
[HOW TO DO
THE SOLAR DESIGN CALCULATIONS
](chap6_passives_9781603581691_epub_part6.html#d41151057)
Let's assume that you have found a good solar building site. Using a popular Green Mountain Homes saltbox design as a representative plan, we will move into the more technical portion of the design process by conducting what an engineer would call a "thermal study" of the planned solar home. I will demonstrate with the specifications for a "Saltbox 38" the calculations essential for solar design. Subsequently, in chapter 8, I will explain how to use the worksheets included in this book to do your own thermal study incorporating the specifications for your particular design and site.
For the sake of discussion, let's plan on locating our example of this Saltbox 38 in Hartford, Connecticut.This is not the most obvious locale, perhaps, for a solar home, but Connecticut is perfectly suitable. And if solar heating will work in frosty New England, it will work wherever you are planning to build your home. Another way to say this is that Western and Southwestern states with high elevations, clear skies, and high annual percentages of sunshine tend to be associated with solar home design, but obviously many people desiring solar homes live elsewhere.
**CALCULATE R-VALUES FIRST**
The illustration Typical saltbox design: a cross-section shows the Saltbox cross-section, and the one Two designs for layered wall construction shows wall and roof insulation. I recommend a 2 x 4 stud wall with a layer of 1-inch Styrofoam outside the exterior plywood sheathing. We first need to calculate the heat loss of the building. Step one of this calculation involves determining the wall and roof U-values. Remember from chapter 4 that U is the reciprocal of R, and expressed as Btu/hr • ft2 • °F. The R-value of the wall or roof is the sum of the individual R-values of the various elements that make up the total.
The total wall and roof R-value for the Saltbox 38 is given below.
We will use insulated dual-pane windows and patio doors and assume the manufacturer's published overall R-value is 1.92 (so that means that the U-value will be 1/1.92 = 0.5208).
**REHEATING THE FRESH AIR COMING IN**
The thermal "cost" to reheat the recommended air change per hour discussed in chapter 4 will comprise the infiltration portion of the total heat loss. There are several ways to calculate infiltration losses; we will use the air change method and assume the total air infiltration from all sources is air change per hour. This assumption is based on data derived from the formal monitoring conducted on the prototype N-38 in Royalton, Vermont. This figure includes losses from cracks around windows and doors, the amount of air lost by entering and exiting the building, and the air expelled out of the building by fans in the bathrooms.
Experienced technicians can conduct tests to determine the number of air changes per hour. If for some reason the home has less than a air change per hour, fresh air should be introduced to keep the airspace fresh and safe.
According to the 1972 ASHRAE _Handbook of Fundamentals_ , the heat required to heat one cubic foot of air one degree is the product of the air's specific heat times its density, or
H = c × d
_where_ :
H = heat required to raise 1 cubic foot of air 1 degree Fahrenheit
c = specific heat of air (0.24 Btus per pound per degree Fahrenheit)
d = density of air (0.075 pounds per cubic foot)
_If_
H = 0.24 Btus per pound × degree × 0.075 pounds per ft3
_then_
H = 0.018 Btus/ft3 • °F
To obtain our infiltration loss we will use the following formula:
I = V × H × Q
_where:_
I = infiltration loss
V = volume of house (in cubic feet)
H = heat removed (Btus/ft3 • °F)
Q = volume of air change (air changes per hour)
_If_
__ I = V(cubic feet) × H (Btus/ft3 • °F) × Q (air changes per hour)
_then_
__ I = Btus/hr • °F
We now have all the information we need to calculate the total heat loss of the Saltbox 38. Referring to the Saltbox 38 floor plan, elevations and cross-section (see chapter 5.), the total heat loss can be calculated as shown in Table 6–3.
Subtracting the square feet of glass from the above total wall area, the net wall area is:
1,898 total square feet of wall area – 271 square feet of glass = 1,627 square feet of unglazed wall area
The total heat loss for the walls, glazing, and roof is the sum of the products of the square feet of area multiplied by the respective U-values as shown on the following page.
_Estimated Atmospheric_ _clearness numbers in United_ _States for nonindustrial_ _localities. The clearness_ _number changes from winter_ _to summer in certain locations._
_A. Net Wall Loss_ = Heated Wall Area (1,627 square feet) × U-value of Heated Wall (0.0468 Btus/hr • ft2 • °F) = 76.14 Btus/hr • °F
_B. Roof Loss_ = Roof Area (38 feet × 40 feet = 1,520 square feet) × U-value of roof (.0307 Btus/hr • ft2 • °F) = 46.67 Btus/hr • ft2 • °F
The house's Infiltration Loss, calculated using the air change method of analysis is predicted as follows:
_C. Infiltration loss_ = Total Volume of Living Space × Heat Removed × Air Changes per Hour
Total Volume = (8 feet × 28 feet × 38 feet) + (19.67 feet × 8 feet × 38 feet) = 14,492 cubic feet
Infiltration loss = 14,492 ft3 × 0.018 Btus/ft3 • °F × air changes per hour = 174.77 Btus/hr • °F
_D. Total Heat Loss through Glazing_ = Area of Glass × U-value
The total area of glazing is shown in Table 6–4 (use your window manufacturer's literature to obtain the square footage of glass area per window, in your particular case). In our sample Saltbox, the heat loss through glass will be:
271 square feet of glass × 0.5208 Btus/hr • ft2 • °F (U-value of glass) = 141.14 Btus/hr • °F
The house's total heat loss is summarized in Table 6–5.
Since this house has no basement, basement losses are not indicated in Table 6–5. Due to the highly insulated perimeter of the Solar Slab and the small area exposed above grade, the Solar Slab perimeter heat loss is insignificant, and also not indicated in the table. If the Solar Slab perimeter loss were to be calculated, it would amount to 2 percent of the total heat loss. This loss should be included if, for some site or design reason, it would amount to more than 2 percent.
Another heat loss factor to consider is the heat transmitted through the framing; this is also referred to as bridging loss. Since the outside wall on this sample Saltbox is constructed as I've recommended, with a continuous exterior layer of rigid insulation, and since 2 x 12s were used as roof rafters, bridging losses were deemed to be insignificant, and were therefore omitted in the wall and roof R-value determinations. In most contemporary houses, tightly constructed and well-insulated, these bridging losses will be likewise insignificant, and may be ignored in heat loss calculations.
If, on the other hand, your framing represents more than 10 percent of the wall area, the framing loss should be included in your own calculations. This might occur in certain forms of post-and-beam or log-wall construction. Should you wish to adjust your calculations for framing or bridging losses, one method is to adjust the U-value. For example, in a 2 x 6 wall with the 2 x 6s 16 inches on center, the 2 x 6s account for about 10 percent of the wall area. At 24 inches on center, they account for about 6 percent of the wall area. The following example shows how to adjust the U-value for a 2 x 6 wall 24 inches on center, and a 2 x 12 roof 16 inches on center:
Average U-value of Wall = (Framing Area % of Total Wall Area × Framing Material's U-value) + (Insulation Area % of Total Wall Area × Insulated Wall U-value)
Average U-value of Wall = (0.06 × 0.1478*) + (0.94 × 0.0468) = 0.0529 Btus/hr • ft2 • °F
Average U-value of Roof = (0.10 × 0.0723*) + (0.90 × 0.0307) = 0.0349 Btus/hr • ft2 • °F
* The U-value for a 2 x 6 is 0.1478 and for a 2 x 12 is 0.0723, assuming kiln-dried hemlock-fir or spruce-pine-fir
In our Connecticut example, the outside temperature used as the basis for calculation is 0 degrees; this is referred to as the "outside winter design temperature," and you can find a table with representative outside winter design temperatures in appendix 4. Assuming the inside temperature to be 72 degrees Fahrenheit, the theoretical hourly heat loss of the Saltbox 38 in Hartford, Connecticut, is:
438.72 Btus/hr × °F difference (72 degrees inside – 0 degrees outside) = 31,588 Btus/hr
**CALCULATING SOLAR GAIN**
Next we need to calculate the solar gain, the heat input attributable to sunshine. The percentage of sunshine in Hartford, Connecticut, for the heating season is shown in Table 6–6.
One should never assume that there is not enough sun in a given location to justify building a solar house. It might be surprising to many people that the average insolation for the nine-month heating season in Hartford, Connecticut is 52.4 percent, meaning that the daylight hours are sunny more than 50 percent of the time on average in this location. Your new home should be designed to take advantage of whatever natural solar benefits are available. It is reasonable to assume that the economical percentage of solar heat attainable approximates the percent sunshine in a given location. Therefore, when designed properly, this home in Connecticut should receive about half of its heat free from the sun. If so, how can a solar house _not_ work in Hartford, Connecticut?
Remembering that the Saltbox 38 used in this chapter has 162 square feet of south-facing glass, 64 square feet of east-facing glass, and 35 square feet of west-facing glass, we will calculate the predicted monthly insolation using appendix 2, Solar Intensity and Solar Heat Gain Factors (SHGF), for north latitude 40 degrees, from the 1993 ASHRAE _Handbook of Fundamentals_. The table lists half-day totals, and reads from top to bottom for sunrise to solar noon, and bottom to top for solar noon to sunset. We will ignore the sun's contribution of heat into the house's west glass in the morning, and likewise we will ignore the afternoon values for east glass, making the east-side SHGF equal to the west-side SHGF. Since the home faces true south, we will double the south-side SHGF half-day totals. The heat gain per square foot of glass on a given orientation is the product of:
Solar Heat Gain Factors (SHGF) × Shade Coefficient (SC)
In the case of our saltbox, the Shade Coefficient (SC) is the reduction in solar gain due to sunlight being reflected off each sheet of glass. SC is an ASHRAE term. From the 1993 ASHRAE _Handbook_ , we find that the SC for 1/2-inch insulated glass is 0.88.
An examination of Andersen casement window performance specification sheets shows SC = 0.91 for Air-Filled Dual-Pane Glass with a U-Value of 0.48. Go to: <http://www.andersenwindows.com/UE/ProductGuide/Residential/400CasementPicturePerformanceCenterofGlass.asp>
For Andersen's High-Performance Low-E Glass, SC = 0.51 and U-Value = 0.28. Substituting Andersen's High-Performance Low-E Glass in this chapter's example resulted in a 31 percent decrease in solar efficiency. In other words, the percent solar went from 54 percent down to 37percent, and most important, fuel consumption increased by 24 percent.
Performance was nearly identical to this chapter's example when Andersen's Air Filled Dual-Pane Glass was used.
From these test runs, northern passive solar homes depending on south, east, and west faced glass as daytime solar collectors should use normal AirFilled Dual-Pane Glass. Simply stated: Glass with a low SC value prevents too much of the sun's heat from entering the house. High-Performance Low-E glass with diminished values of SC most probably will be best used in passive solar homes located in the south or central climate zone, where cooling is of equal or greater concern than heating.
Readers are advised to find out the values of SC and U-Value for the windows and patio doors they plan to use in their design. For Pella Windows go to: <http://pellaadm.com/HTML/search.html?shade%20coef>
Test computer runs should be made using the CSOL program with the different options available, and the SC and U-Value that yield the best results should be used in the design.
**Simply stated: Glass with a low SC value prevents too much of** **the** **sun's** **heat from entering a northern solar home. Be sure to** **find out the SC of your glazing.**
Another consideration, which may be counterproductive, is the Department of Energy and the Environmental Protection Agency Energy Star program. U-Values for glass are published for four climate zones in the United States. The Northern Zone specifies a U-Value equal to or less than 0.35. As stated above, Andersen's Air-Filled Dual-Pane Glass produced favorable results in our solar home example calculation, but this window has a U-Value greater than 0.35 (0.48). This could present a problem if a lender requires that the new home meet the Energy Star program's guidelines.
The SHGF also assumes atmospheric clarity of 1.00. If your location is high in elevation and has dry and clear atmosphere, the SHGF may be increased up to 15 percent. Conversely, if the location is hazy and humid, the SHGF should be reduced. To illustrate the calculation, I will use the figures for the September Solar Heat Gains.
From appendix 2, and using the SHGF for 40 degrees north latitude, the September SHGF half-day totals are:
The potential solar gain (expressed in Btus per square foot per day) for east-, west-, and south-facing glass are shown in Table 6–7 (remember that south is multiplied by 2, in order to indicate two half-day subtotals).
Multiply each column by the square footage of glass on each elevation, then by the number of days in each month, and finally by the percent sunshine. The totals for each elevation are tabulated in Table 6-8 in millions of Btus. Let's use September as a sample calculation.
East = 787 SHGF × 64 square feet × 30 days × 57% sunshine
= 0.86 million Btus
South = 1,344 SHGF × 162 square feet × 30 days × 57% sunshine
= 3.72 million Btus
West = 787 SHGF × 35 square feet × 30 × 57% sunshine
= 0.47 million Btus
The totals in Table 6–8 need to be adjusted for the heat reflected back from the window due to dual glass. Calculate the loss by multiplying the above monthly totals by a Shade Coefficient of 0.88 (the SC of 1/2-inch insulated glass).
**CALCULATING HEAT LOAD**
Degree days are a measure of the heat required for a building, and degree day data from the 1981 ASHRAE _Handbook of Fundamentals_ will be used in this chapter (see appendix 5). A degree day is defined as the difference between the median outdoor temperature and 65 degrees for a 24-hour period. The standard assumption is that the inside design temperature is 72 degrees, of which 7 degrees will be derived from sources other than the furnace. These sources include heat from lighting and cooking, the body heat of people, and so forth. Degree day tables are tabulated with an outside base temperature of 65 degrees. For example, if the outdoor median temperature was 64 degrees for the 24-hour time period, then that day had 1 degree day. The local power company keeps accurate track of degree days for its heat load calculations, and is usually a good source for this information. Oil and propane companies use degree days as a guide to tell them how frequently to make deliveries. The National Oceanic and Atmospheric Administration (NOAA) also tabulates this and other valuable weather data. The nearest engineering or earth sciences library will most probably have this data on microfiche. Ask for the five-year average to obtain a good approximation of your heat load. The degree day data obtained from other sources may differ slightly from the data contained in this book, since this kind of information is routinely updated. Slight differences will not materially affect your solar prediction calculations. Remember that we are dealing with a "fuzzy" (transient) problem, and solar predictions at best will be an informed approximation.
Knowing the calculated heat loss of our building, the degree days for our location, and the solar gains for our location, we are now ready to tabulate this information and derive our solar performance prediction.
Let's first calculate heat load per month. See Table 6–10 for a summary by month of the building's projected heat load.
The Performance Summary for the home is shown in Table 6–11.
The Difference: Not Solar Supplied in Table 6-11 is the purchased fuel that will be needed per year. Note that in September, October, and May, the home is receiving more solar heat than needed, and in that case the windows are probably open, releasing the extra heat. In the table, when the "Solar Supplied" number exceeds the "Heat Load" number, zero is used for that month in the "Difference" column.
In this example the calculation process is as follows:
Total Purchased Fuel = 33,710,000 Btus
Total Heat Demand = 65,170,000 Btus
Percentage of Purchased Fuel = 33,710,000 Btus ÷ 65,170,000 Btus × 100 = 52% Not Solar Supplied
Percentage supplied by Solar = 100% – 52% = 48% Solar Supplied
The above calculation assumes that the home is faced true south. In our example, Hartford, Connecticut has a westerly magnetic deviation from true north of 12 degrees (see appendix 7 for an isogonic map, which indicates magnetic declinations). That means that once we have established our north–south compass line, the north–south axis of our solar home should be rotated clockwise 12 degrees.
**HEAT LOSS REDUCTION DUE TO**
**WINDOW INSULATION**
In chapter 4 we discussed the benefits of providing supplementary window insulation using thermo-shutters or some other form of night-time insulation on at least some of the home's glazing. Let's now calculate the difference in performance assuming that we are going to install thermo-shutters on 203 square feet of window and patio glass. This example further assumes that the thermo-shutters will be closed at night during the heating season.
Utilizing the same technique demonstrated earlier in this chapter, we calculate the R-value of the thermo-shutter to be 7.76. The total R-value of the window or patio door with the thermo-shutter closed is shown in Table 6–12.
Assuming that the thermo-shutters are closed for sixteen hours per night and open for eight hours during the day, the thermo-shutter credit will be calculated as follows:
Square feet of thermo-shuttered glass × (U-value of glass – U-value of thermo-shutter) × Number of hours with thermo-shutters in closed position
203 square feet × (0.5208 Btus/hr • ft2 • °F – 0.0883 Btus/hr • ft2 • °F) × 16 hours/day
= 1,405 Btus/°F • day
Applying the above thermo-shutter credit to our previously calculated total heat loss, the new predicted heat loss total will be:
10,529 Btus/°F • day – 1,405 Btus/°F • day = 9,124 Btus/°F • day
Using this revised heat loss calculation, we now recalculate the monthly heat load as shown in Table 6–13.
Table 6–14 shows the home's total heat load in relation to the portion that is supplied by solar and not supplied by solar.
Now the totals can be summarized as follows:
Total Purchased Fuel = 26,030,000 Btus
Total Heat Demand = 56,480,000 Btus
Percentage of Purchased Fuel = 26,030,000 Btus ÷ 56,480,000
Btus × 100 = 46%
Percentage Supplied by Solar = 100 – 46 = 54%
As you can see, adding thermo-shutters lowers the total heat requirement of the home and increases the percentage supplied by solar. The overall effect of using thermo-shutters is summarized in Table 6–15.
From the performance summaries in this chapter, we can see that about two-thirds of our purchased energy is required to meet heating needs in December, January, and February. It is in these months that nighttime window and patio door insulation is the most beneficial. These months have the longest nights and shortest days, making it less inconvenient to cover our glass, since we can't see out anyway. There are also non-numerical benefits provided by nighttime window and patio door insulation. Many people find that a home with window insulation psychologically "feels" cozier and more secure.
**DID WE KEEP THE FURNACE OFF?**
The performance summary tabulations are based on monthly averages and don't tell us when and if the furnace runs, or what the living space and Solar Slab temperatures are. Have we met our design goal of keeping the furnace off?
February is a high-intensity solar month, making it a good one in which to check the living space and Solar Slab temperatures. We need to make sure that the home is in thermal balance and not overheating. In the following discussion, we will continue with and refine the concepts introduced in chapter 3.
Let's assume that the Solar Slab is comprised of 4 inches of concrete slab bonded to 12-inch concrete blocks, as illustrated The Solar Slab concrete heat exchanger: a section drawing and The Solar Slab: a sill detail. Since the Solar Slab is inside the perimeter-foundation-wall insulation, and contains ductwork that displaces some of the concrete blocks, we will reduce the volume of theoretical concrete mass by 15 percent.
Remembering from chapter 3 that a concrete block is about 50 percent solid concrete, the volume of concrete in our Solar Slab is calculated as follows:
Volume of concrete = foundation dimensions × depth of concrete × % of theoretical concrete volume that is functional thermal mass
In the sample Saltbox, let's define these variables in this way:
Depth of concrete = ½ of concrete block height (½ × 12 inches)+ slab thickness (4 inches) = 10 inches (or 0.833 feet)
Volume of concrete =
(28 feet × 38 feet × 0.833 feet) × 85% = 754 cubic feet
The adjusted predicted heat loss with thermo-shutters will be:
9,124 Btus/°F • day ÷ 24 hr/day = 380 Btus/hr • °F
Let's see if the furnace needs to run. Our start time will be 10:00 PM, and we will assume the following circumstances:
1. The automatic thermostat has switched to its set-back position of 55 degrees, and our occupants have retired for the night.
2. The 10:00 pm Solar Slab temperature is 68 degrees, because the preceding day was sunny.
3. The overnight outside temperature is 10 degrees.
Let's calculate what the 7:00 AM Solar Slab temperature will be. Appendix 2 shows that in February, our solar day starts at 7:00 AM and ends at 5:00 PM.
We also need to estimate what the average inside temperature will be overnight. Using our 10:00 PM start temperature of 68 degrees, and assuming a 7:00 AM morning temperature of 60 degrees, the average overnight living space temperature is then 64 degrees. Our Delta T (temperature difference between inside and outside) will be calculated this way:
Delta T = inside temperature (64°) – outside temperature (10°)
= 54 degrees Fahrenheit
Likening the Solar Slab to a battery, we will next see how much of a "charge" (measured in degrees) we will lose overnight. We will need to calculate the Solar Slab Thermal Capacity (SSTC). The SSTC is the product of the volume of concrete multiplied by the capacity of concrete to hold heat.
As explained above, the measure of a material's capacity to hold heat is called "specific heat," which is the ratio of the amount of heat required to raise a quantity of a given material one degree to that required to raise an equal mass of water one degree. The heat storage capacity of the Solar Slab is about 30 Btus per cubic foot per degree. This figure is derived as follows. The specific heat of 12-inch standard weight concrete blocks is about 0.22 Btus per pound per degree Fahrenheit. The specific heat of poured concrete is between 0.19 and 0.24 Btus per pound per degree Fahrenheit. Using 0.215 for the combination of concrete slab and concrete blocks, and 140 pounds per cubic foot as their combined weight, the heat capacity of the sample Solar Slab is calculated as follows:
0.215 Btus/pound • °F × 140 pounds/ft3 = 30.1 Btus/ft3 • °F
The SSTC equals
754 ft3 of concrete × 30 Btus/ft3 • °F
= 22,620 Btus/degree of change
The 10:00 PM to 7:00 AM heat loss will be:
380 Btus/hr • °F × 54° × 9 hours = 184,680 Btus
Since there is a positive temperature difference between the Solar Slab and the living space, the Solar Slab will supply the necessary overnight heat. Dividing the 10:00 PM to 7:00 AM heat loss by the SSTC, we find how many degrees the Solar Slab lost overnight:
184,680 Btus ÷ 22,620 Btus per degree = 8.2 degrees
Our "battery" lost 8 degrees of "charge." Subtracting the overnight temperature loss of 8 degrees from the 10:00 PM Solar Slab temperature of 68 degrees, we find that the 7:00 AM Solar Slab temperature would be 68 – 8 = 60 degrees.
The set-back on an automatic thermostat lowers the temperature at which the thermostat calls for heat, and does so at a time which the resident specifies. In this example the set-back, overnight temperature is 55 degrees between 10 PM and 7:00 AM. That is, the house temperature must go below 55 degrees before the thermostat will switch on the furnace. Since the Solar Slab temperature will not decay to less than 60 degrees in this example, in actuality there will be no requirement for the furnace to operate to maintain a comfortable overnight temperature, even though the ambient or outside temperature may be severely cold.
This is a situation where even the most skeptical person will agree that the Solar Slab is yielding heat from its stored state back into the living space.
Since the thermostat was set to turn on the furnace at 55 degrees, the furnace did not operate from 10:00 PM to 7:00 AM because the minimum overnight living space temperature was 60 degrees.
Next let's assume that the daytime heat setting is 68 degrees, and as suggested in chapter 5, we have located the thermostat on an east-facing interior wall. Let's further assume that we have another sunny day. At 7:00 AM we are having breakfast in our sunny east side dining/family area, and the sun is warming us and also striking the thermostat, which heats up and does not turn the furnace on. As the solar day progresses, the entire home rises in temperature to 68 degrees. As excess solar heat enters the home, and is stored in the Solar Slab, the living space temperature will probably increase to between 70 and 72 degrees. Since these temperatures are above the thermostat setting of 68 degrees, from 7:00 AM to 5:00 PM the furnace will not operate.
Using the information contained in appendix 2, from the 1993 ASHRAE _Handbook of Fundamentals_ , the amount of solar energy or insolation available to vertical glass on February 21 at north latitude 40 degrees is as shown in Table 6–16.
Because some of the sun's energy is reflected out of dual-glazed windows, the amount of heat passing through the insulated dual-glazed window is reduced by multiplying the above totals by 0.88, which you will recall is the Shade Coefficient (SC) for 1/2-inch insulated glass.
Our total solar gain for the day can be calculated as:
0.88 × (648 SHGF × 64 square feet of east-facing glass) + 0.88 × (1,642 SHGF × 162 square feet of south-facing glass) + 0.88 × (648 SHGF × 35 square feet of west-facing glass)= 290,537 Btus
The Average Winter Temperature for Hartford, Connecticut, is 37 degrees Fahrenheit (see appendix 5). Let's assume that our sunny February 7:00 AM to 5:00 PM outside temperature is 40 degrees. Using 68 degrees as the inside temperature, the Delta T (temperature difference between inside and outside) will be
68° - 40° = 28° Fahrenheit
The heat loss from 7:00 am to 5:00 pm will therefore be:
380 Btus/hr • °F × 28° × 10 hr = 106,400 Btus
Note: It is reasonable to use the reduced heat loss figure of 380 Btus per degree-hour, because the windows are heat gainers on sunny days. Also, the sun striking the south wall neutralizes it in terms of heat loss.
The amount of free solar heat available for storage during the 10hour solar collection time period can be summarized as:
To find the daytime Solar Slab temperature increase, divide the above figure by the SSTC:
184,137 Btus ÷ 22,620 Btus per degree = 8.14 degrees
Our "battery" took on a daytime charge of 8 degrees. Adding the daytime Solar Slab temperature gain of 8 degrees to the 7:00 AM Solar Slab temperature of 60 degrees, we have a 5:00 PM Solar Slab temperature of 68 degrees. Since the temperature of the home will have quickly risen to 68 degrees even though the furnace had been "tricked" into not operating during breakfast, the furnace will have ended up not running all day.
From 5:00 PM to 10:00 PM the furnace may be needed to supplement the heat in the Solar Slab to keep the temperature at 68 degrees or higher, as required by the occupants.
Should the next day be sunless, the furnace will operate longer to keep the airspace up to temperature. In chapter 7 we will discuss how the Solar Slab assists the furnace even on sunless days.
**IS THE HOME**
**IN THERMAL BALANCE?**
The above 24-hour analysis demonstrated how the furnace was off for long periods of time, and showed that the Solar Slab's 24-hour temperature swing or variation was about 8 degrees. Using this methodology for design calculations, a home incorporating a Solar Slab should be designed to keep the Solar Slab temperature swing within a range of about 10 degrees or less.
Table 6–17 gives the thermal summary for the Solar Slab for the 24-hour period described above.
Thermal balance has been achieved because the overnight loss in heat is about equal to the amount of heat that the east-, south- and west-facing glass was able to collect in excess of the amount of heat needed by the home during the day while the sun was out. This excess heat was absorbed by and later given back by the Solar Slab. The Solar Slab daily gain was approximately equal to the nighttime loss.
_An example of a smaller, one-story_ _solar home, in which_ _every room is sunny._
In our example of a Saltbox 38 in Hartford, Connecticut on a sunny February 21, the amount of energy collected by the windows and the patio doors, the heat demands of the home, and the size of the thermal mass are all in proper proportion. This home is not overheating and the daily temperature swing was within comfortable limits.
What was the cost to heat this solar home for the 24-hour period described above? Let's assume that the furnace fires at 0.85 gallons of oil per hour. The furnace will probably have run for about 1.5 hours in the evening and possibly .5 hours in the early morning. In that 24hour period, the furnace ran about 2 hours, consuming
2 hours × 0.85 gallons per hour = 1.70 gallons of oil
At $1.99 per gallon, the residents will have paid $3.38 for their fuel for that February day and night. The vast majority of the heat they used was free, simply harvested from the sky.
**INSURING COMFORT:**
**SOME BASIC GUIDELINES**
It is difficult to make a general rule that dictates the amount of glass and the amount of thermal mass that a solar home will need to perform optimally throughout the year. Try not to use too much of a good thing. That is, don't overglaze. Make sure that the thermal mass is sized to allow no more than a 8-degree temperature swing from its warmest to coolest state. The occupants will feel comfortable with a temperature swing in the Solar Slab from a low of 62 to high of 70 degrees, and uncomfortable if it is colder in the morning than 62 or hotter in the afternoon than 70.
Typically, a poured slab will be 4 inches thick in a larger home, and up to 7 inches thick in a smaller home such as the N-38-X, which was another model offered by Green Mountain Homes. The N–38–X represents a small house with 1,408 square feet of living space, whereas the Saltbox 38 represents a larger home of 1,895 square feet (see photo.).
Attempts have been made to produce ratios that will dictate the ideal relationship of glass to mass, or glass to wall area, or glass to floor area. Again, considering the wide variations in regional climatic conditions and in the specific characteristics of local building sites, general rules are difficult to create and apply. There really is no substitute for good solar design and good judgment. As can be seen in the example above, the amount of glass on each elevation and the size of the thermal mass are interrelated, and such relationships are dependent on location, the heat loss of the building, and other factors.
The Saltbox 38 we have been using as an example is designed according to the following ratios of glazing to insulated wall area, considering the glass area on the east, west, and south as a percentage of insulated wall area:
261 square feet of glass ÷ 1,898 square feet of insulated wall area × 100 = 14%
Using 8 degrees as the design temperature "swing" in the Solar Slab, the smaller home will be used to illustrate a procedure to determine the appropriate thickness of the poured slab.
The specifications for this representative N–38–X (sample location: Middlebury, Vermont) are:
Footprint (dimensions of Solar Slab) = 16 feet × 38 feet = 608 square feet
East- and west-facing glass = 44 square feet each
South-facing glass = 122 square feet
East-, west-, and south-facing glass area ÷ insulated wall area = 210 square feet ÷ 1,720 square feet × 100 = 12%
Area of glazing insulated with thermo-shutters = 80 square feet
Total heat loss for the house, with thermo-shutters in use = 295 Btus/hr • °F
Let's determine the thickness of the Solar Slab needed to keep the above solar home in thermal balance. Middlebury is approximately 44 degrees north latitude. Since appendix 2 lists the SHGF for 40 and 48 degrees north latitude, the SHGF will be interpolated for 44 degrees north latitude. Using a peak February day, and 8 degrees as our maximum desired Solar Slab temperature swing, the correct calculation is as follows:
Solar Principle # 6
_Do not over-glaze._
Incorporate enough windows to provide plenty of daylight, but do not make the mistake of assuming that solar heating requires extraordinary allocations of wall space to glass. An over-glazed building, as shown below, will probably overheat. A highly insulated and well-constructed home with a proper number and distribution of high-quality windows does not need much energy to maintain comfortable temperatures year-round.
Elevation Solar Heat Gain Factor
East 594 Btus per square foot (half-day total, reading down the table)
South 817 × 2 = 1,634 Btus per square foot (full day)
West 594 Btus per square foot (half-day total, reading up the table)
Using a Shade Coefficient of 0.88 (for 1/2-inch insulated glass), the total insolation for a peak February day is:
East = 44 ft2 × 594 Btus /ft2 × 0.88 = 23,000 Btus
South = 122 ft2 × 1,634 Btus/ft2 × 0.88 = 175,426 Btus
West = 44 ft2 × 594 Btus/ft2 × 0.88 = 23,000 Btus
_______________
Total: 221,426 Btu
Since the SHGF Tables assume a Clearness Number of 1.00, and since Middlebury is in snow country, the total insolation will actually be increased by 10 percent, because the low angle of the February sun will reflect heat upward from the snow cover. The new total, adjusted accordingly, is:
221,426 Btus × 1.10 = 243,569 Btus
The Average Winter Temperature for Middlebury, Vermont, is about 30 degrees. Using 68 degrees as the average inside temperature, the Delta T or difference is:
68° – 30° = 38° Fahrenheit
The 7:00 AM to 5:00 PM heat loss will be:
295 Btus/hr • °F × 38 degrees × 10 hours = 112,100 Btus
The amount of free solar heat available for storage during the 10hour solar collection time is:
Total Insolation 243,569 Btus
Heat Loss (112,100 Btus)
____________
Excess available to store: 131,469 Btus
The formula to determine the necessary thickness for this home's Solar Slab is:
Stored Btus = cubic feet of concrete × Btus per cubic foot per degree × maximum design Solar Slab temperature variation, or
131,469 Btus = _x_ cubic feet of concrete × 30 Btus per cubic foot per degree × 8 degrees
This means that the correct figure in the equation for the cubic feet of concrete needed will be 548. The next calculation will involve dividing this cubic foot total by the square footage of the slab multiplied by 85 percent to account for the functional percentage of thermal mass in the overall slab (the figure 0.85 compensates for the portion of concrete block displaced by air passages and ducts):
548 cubic feet ÷ (16 feet × 38 feet × 0.85)
= thickness of Solar Slab (1.06 feet, or 13 inches)
Since 12-inch concrete blocks are half solid, the slab thickness is 13 inches – 6 inches = 7 inches.
**NO COOKBOOK RECIPES**
**FOR SOLAR DESIGN**
While writing this book I conducted a search of the design records for existing Green Mountain Homes in the hope of finding certain ratios or percentages that were common to all solar homes and that could be used to assist other designers. No obvious "cookbook recipe" emerged, except for two basic design parameters:
1. The square footage of east-, south-, and west-facing glass should be in the range of 10 to 20 percent of the total exterior heated wall area.
2. The peak solar-supplied February-day increase in Solar Slab temperature should be 8 degrees.
Is there a general rule about the ideal square footage of east- or west-facing glass as it relates to the square footage of south-facing glass? As mentioned earlier, east- and west-facing glass, though beneficial in late fall and early spring, must be used judiciously in locations where summer air conditioning is required. In northern New England, where air conditioning is never really necessary, the amount of east- or west-facing glass can be increased; however, in Maryland, where the expense of air conditioning is a factor, the amount of east- and west-facing glass should be less, in order to reduce morning and afternoon heat gain. Remember that west-facing glass will be especially detrimental to keeping a home cool. The low angle of the westerly sun will overheat the home in summer in locations where summer cooling is a design factor. The range of east- or west-facing glass as a percentage of south-facing glass in the homes we researched was from 25 percent to 75 percent, which is too high a spread to yield any general rule. Other factors influencing decisions about the amount of east- and west-facing glass are the floor plan or layout of the home, the location of shade trees, the direction of special views, the use of window insulation, the dominant weather conditions at the site, and most importantly, the desires and aesthetic preferences of the homeowner.
Our design philosophy and practice has been first to present ideal considerations to the people planning a house, and then to incorporate as many of these idealized factors as possible while carefully considering the clients' desires, needs, and particular site situation.
One way to solve a problem is to guess. (There's a fancier engineering term for the stratagem — convergence by trial and error). Then make the appropriate calculations and see what the results look like. Then repeat the calculations procedure with a better guess, until the variables converge toward the best result. The same method can be used to design a home with a Solar Slab.
**SUMMARY OF THE DESIGN PROCEDURE**
In summation, the sequence of steps in the solar design procedure are as follows:
1. Conduct a site analysis: in other words, really get to know this place where you may be spending many years. Make numerous visits at different times of day and in different seasons.
2. Begin to do progressively more refined drawings and floor plans for the home, keeping in mind the solar design principles presented in this book, and using the amount of glass suggested in this chapter for the east-, south-, and west-facing elevations. Keep the total square footage of the east-, south-, and west-facing glass between 10 to 20 percent of the total square footage of heated wall area.
3. Find the north latitude of the home site (see appendix 4).
4. Find the Outside Winter Design temperature for this location (again, see appendix 4).
5. Calculate the R-values for the walls, glass, and roof (see appendix 3).
6. Calculate the overall predicted heat loss of the home, taking a nighttime insulation credit if nighttime glass insulation will be used.
7. Find the degree day data for the specific home site (see appendix 5).
8. Find the insolation values for the home site (see appendix 2).
9. Find the percentage of sunshine for the home site (see appendix 6).
10. Tabulate in Btus the heat load, including the portion that will be solar-supplied and the difference, not solar-supplied.
11. Calculate the percentage of total heat load that will be supplied by solar.
12. Use the "converging guess" method to make several "runs," adjusting the variables and trying out different combinations, to see which design produces the best economy while satisfying the aesthetic and living-space requirements of the home's future residents.
13. Using a peak solar day in February, calculate the following:
A. The predicted daytime excess solar energy: the amount of heat available to be stored for later.
B. The necessary thickness of the Solar Slab based on an ideal daytime temperature swing of 8 degrees.
14. Check your overall results using common sense and good judgment—for which there is no substitute!
_A solar home uses thermal_ _mass—_ _a material that readily absorbs_ _heat—_ _to collect and store the_ _warmth of the sun during the day._ _This thermal mass will then radiate_ _heat back into a_ _home's_ _living space_ _during the cooler nighttime hours._ _This book describes a technique_ _for constructing a Solar Slab using_ _ordinary concrete blocks and a_ _poured slab, which transforms the_ _conventional house foundation into_ _a particularly effective thermal mass._
_Folding thermo-shutters insulate windows at night (or shade the windows during times of intense sun). Space to either side of windows_ _and doors is designed into house plans so that the shutters have room to fold back against the wall. The shutters can be decorated to_ _harmonize with the room decor, giving you the style and grace of draperies with the much higher R-value of solid, airtight shutters._
_Because of improvements in the standards for wall framing, windows, and insulation, even conventional houses are now far_ _more energy efficient than our_ _ancestors'_ _homes. As a result, it has never been easier to build a passive solar house._
_The first principles of good solar design are siting a home with a southern_ _exposure and utilizing the natural features of the site, including trees that_ _provide shelter from harsher weather that tends to come from the north._
_A passive solar home will to a great extent heat and cool itself, with minimal use of conventional HVAC equipment, and with_ _no additional conventional expenses over the cost of comparably sized non-solar, fossil-fuel-dependent house. Note the three_ _deciduous trees planted to provide summer shade, yet allow full winter sun._
_In a building that is tightly constructed and_ _well insulated, you will need to be sure to_ _provide for adequate exchange of fresh air. You_ _might want to consider including a solar cat_ _or two in your household: not only will a cat_ _infallibly follow the path of the sunlight over_ _the course of the day, but with its frequent trips_ _in and out through the door, the cat will ensure_ _a comfortable exchange of air._
_Three solar homes, all warm and comfortable even in the_ _depths of winter. Note in the bottom picture how the snow_ _lies very evenly on the roof. Proper insulation and venting_ _of the roof allows the snow to melt away slowly without_ _causing_ _"_ _tell-_ _tale"_ _icicles or creating ice-dam problems._
_The front entrance and stairway of a solar home. Notice the proximity of the woodpile_ _to the front door and the airtight wood/coal stove, which provides a substantial amount_ _of the backup heat needed by this 3,500-square-foot home. Air grilles located behind_ _the woodpile discharge warm air collected at the second-floor ceiling._
_A view of the kitchen, located to the right of the stove pictured on_ _the previous page. The stove also provides domestic hot water for_ _further energy savings._
7
[THE FOUNDATION
PLAN, AND BACKUP
HEATING AND COOLING](chap6_passives_9781603581691_epub_part6.html#d41151058)
In this chapter we will complete the design process for the Connecticut Saltbox analyzed in chapter 6, concentrating on the foundation plan. Next,we will size backup heating and cooling systems for various fuels, and describe how to utilize these backup systems in conjunction with the Solar Slab.
As always, our goal is to keep the furnace or air conditioner off. To measure the effectiveness of a solar-assisted heating or cooling plan, it is necessary to predict annual fuel usage to determine the best size for the backup equipment. In chapter 6, the Saltbox 38 located in Hartford was found to be in thermal balance with approximately an 8-degree temperature variation in the Solar Slab; that is, on a representative February day, the early morning temperature of the Solar Slab would be about 60 degrees Fahrenheit, and this temperature would rise to about 68 degrees by the time the sun went down.
**THE FOUNDATION PLAN**
The final step in the design of our Saltbox 38 in Hartford, Connecticut, is the detailing of the foundation plan. Note that the plan is not to scale, is not dimensioned, and is not to be used for construction. This diagram is included to illustrate the following important design details:
1. _Orientation:_ The "compass rose" is shown on the lower-right corner. The person who needs this information the most is the foundation contractor — even though that contractor may not be accustomed to thinking in solar terms. Remember our discussion in chapter 2 about the cost of positioning a home too far from the ideal orientation to true south. Be sure that the foundation is oriented exactly as your site plan specifies, since the resulting foundation "footprint" will determine how the house subsequently built relates to the sun. Showing the cardinal directions on your foundation plan will help insure that the home will be oriented properly.
_A foundation plan for a Saltbox_ _38 with an airlock foyer,_ _showing the Solar Slab heat_ _exchanger and the proper configuration of vents and piers._
2. _Air Vents:_ The minimum number of air vents for a home with a Solar Slab is eight — two air vents at each corner of the first floor. However, in this case they were not placed in the northwest corner because that's where the master bath is located. It is not a good idea to draw moisture-laden air or odors from a wet area such as a bathroom into the Solar Slab.
3. _Central Return Duct:_ The duct shown running down the middle of the base under the poured slab is included in all cases. It should always be used as the return-air duct: Do not reverse the air flow pattern shown on the control diagrams. By using the Solar Slab as part of the return-air duct system, the Solar Slab will constantly assist the furnace by pre-heating the return air. Even if the home will be heated with a woodstove and emergency electric furnace, the return duct should be included and the air mover hooked up per the appropriate control diagram (see the illustration).
4. _Piers and Chimney Bases:_ Solid masonry on undisturbed hardpack must be provided to insure that heavy column loads and chimney loads will not crack the slab.
5. _Miscellaneous:_ Plumbing risers need to be properly placed. Cast iron is the material of choice, and should be placed in the layer of sand under the concrete blocks. Water pipes embedded in the Solar Slab should be "K" copper sleeved in heavy-duty PVC plastic to protect against the corrosive reaction between concrete and copper.
Note also that there is a drain shown through the south wall. This emergency drain will allow water to drain out of the Solar Slab at the bottom of the concrete blocks. One homeowner unfortunately had a fire on the second floor of his Green Mountain Home. The firemen quickly extinguished the blaze with a heavy dose of water. The water apparently ran down the stairs, found an air vent, then flowed into the Solar Slab and out the emergency drain. As a result, the damage was minimal. Another homeowner had a bird crash through his patio door glass. It happened during severe cold and the owner hadn't turned off the water supply. A nearby water pipe froze and burst. When the owner returned, water was flowing from the broken pipe into an air vent and out the emergency Solar Slab drain. Again, thanks to the drain, the damage was minimized.
**CONCERNS ABOUT EMBEDDED DUCTS**
The question is, what happens to the ducts and concrete block ribs over time? Will they accumulate dirt and dust? Will mold be created?
The answers to the above are unclear and probably dependent on the location of the home. Later in this chapter, we will discuss using an up-flow furnace/airmover in all installations. All upflow furnaces have a filter. Therefore, everytime the furnace and/or airmover operates, the air returning from the Solar Slab is filtered. Usually an ordinary furnace filter will suffice; however, if for some reason an airborne problem arose, then the filter could be upgraded to a more sophisticated electrostatic type.
As noted in the preface, the knowledge imparted in this book has been accumulated from over 30 years of data gathered from several hundred solar homes located in the northern tier of the United States from North Carolina to and including Canada and west to the mountain states. These are locations that are primarily focused on heating. If the design described in this book is used in low lying and high moisture locations, the concerns about mold and other airborne problems may very well be valid.
The Solar Slab foundation must be high and dry and backfilled with pervious material. There must also be a continuous perimeter drain as shown in the diagram on chapter 3.
There seems to be a rigorous difference of opinion in the information about duct cleaning. The following conclusion has been reached by the Canadian Housing Information Centre, Ottawa, Ontario:
Homeowners should not necessarily expect significant improvements in their home's indoor air quality, nor reductions in their heating bills, as a result of having ducts cleaned.
Cleaning only the return ducts and the furnace fan may be the most effective and efficient approach to improving air circulation with an existing system.
Until the efficiency of biocides is proven and their potential effects established, householders should refrain from having biocides applied. According to federal authorities, no products are currently approved by the Pest Control Products Act for use in residential duct cleaning.
Here's a bad story about a women in Texas:
The monthly mail flier advertising a $99 all-house air duct cleaning service sounded like a good deal. But it ended up costing the homeowner a lot of grief and $600 for mold contamination cleanup that didn't work—and another $3,725 insurance claim to repair her 2,600-square-foot house's duct work.
Here's another article found while researching the subject:
The Environmental Protection Agency has created a 12-page publication on the subject after it received a lot of calls from consumers wanting information, an agency spokesman said. The EPA advises consumers it is probably not necessary to clean their air ducts if no one in the household suffers from allergies, unexplained symptoms or illnesses, or if the air ducts are not contaminated with large deposits of dust or mold. "If a service provider fails to follow proper duct cleaning procedures, duct cleaning can cause indoor air problems," according to the EPA. "A careless or inadequately trained service provider can damage your ducts or heating and cooling system, possibly increasing your heating and air conditioning costs or forcing you to undertake difficult and costly repairs or replacements."
One commercial duct cleaning company arrives at your home with a huge truck and inserts what is essentially a large vacuum cleaner into your duct system, creating an air movement of 16,000 cubic feet per minute. They then inject a chemical and pull it through the entire system.
This doesn't sound like anything I want done to my house. My first concern is that the metal ducts could collapse from such a large created negative pressure. My second concern is that such a huge volume of air could undermine the compacted sand layer under the Solar Slab.
There are, however, a couple of things you might incorporate into your design:
1. Provide future access to your return duct for ordinary duct cleaning equipment.
2. Provide future access to the north and south air chambers of the Solar Slab and your return air duct. This could easily be accomplished by boxing out a 12" x 12" access hatch midway on the north and south chambers. This will facilitate inserting ordinary cleaning equipment in the air chambers.
**BACKUP HEATING OPTIONS**
Let's assume that the convential backup heat for the Connecticut saltbox will be an oil-fired furnace. Later in this chapter we will calculate the theoretical size of the oil furnace to be 45,000 Btus per hour.
The problem with small oil furnaces is that the oil burner nozzle orifice has a tendency to plug due to the impurities in fuel oil. A 45,000-Btus-per-hour oil-fired furnace would normally be used in a small house trailer. These units tend to be operationally troublesome. A 90,000 or 100,000+Btus-per-hour furnace will run quite nicely due to the larger oil orifice size, but such units are too big for this house in this location.
45,000 Btus per hour is probably too much for the fan-coil arrangement. Our best backup for this home would be a gas-fired furnace. Gas-fired furnaces are readily available in the smaller Btu ranges and are operationally quite reliable. For these reasons, the sample Saltbox 38 solar home in Hartford, Connecticut, will be equipped with an upflow gas-fired furnace and a gas-fired hot water heater located in a utility room created by extending the foyer (see the floor plan). This will keep the equipment out of the living space, isolated for safety and noise-abatement reasons. Feed ducts will be located in the super structure, and each room will have a heat outlet grille.
Later in this chapter, I will show you how to calculate that the net output of the propane gas furnace is 42,000 Btus per hour. The smallest commercially available upflow gas furnace normally will be rated at 40,000 Btus per hour. Adding duct loses to our theoretical 42,000 Btus per hour, we will need to go to the next commercially available size of 60,000 Btus per hour. Let us assume that the manufacturer's specifications for the furnace call for an 8 x 20-inch return duct with a blower size of 900 cubic feet per minute (CFM). As a guideline, assume the side vents cut into the sides of the central return duct in the Solar Slab have an air flow capacity of 75 CFM each. Dividing the total amount of air being moved by the furnace blower by 75 will yield the number of side vents needed. In this case, 900 ÷ 75 = 12.
Likewise, the air vents that allow air into the Solar Slab, discussed in #2 above, should equal or exceed the number of side vents cut into the sides of the return duct. Again, assume that the 4 x 14-inch air vents will have an air flow of 75 CFM.
The 75 CFM assumption for side vents and air vents is conservative; that is, they have the capacity to allow more air flow. However, high air flows will be accompanied by noise. Low air flows will give you a quiet running system. Also, the air vents can be regulated to direct return air flows to various parts of the home. By conservatively sizing them, you provide operational flexibility. It's a lot easier to close off an air vent than to jackhammer an extra one after the concrete is poured. The Solar Slab needs to have free air flow. In this case, more is better than fewer.
Note also that the return duct is reduced in size the further it is placed from the blower. This manifolding will even out air flows within the Solar Slab when the air mover is operating.
**CONVENTIONAL BACKUP HEAT**
No matter how committed one is to conserving energy and not burning fossil fuels, some form of conventional backup heating must be installed. Many existing solar homes are heated only by the sun and a woodstove, and many homeowners are very comfortable utilizing alternative and renewable forms of energy. However, provisions should be made for a conventional back up heating system for the following reasons:
1. A home is probably the largest single financial expenditure a person will ever make, and the value of the home should be protected by providing for a conventional backup heating system. Resale value should be considered, as prospective buyers may not have the same enthusiasm for the use of alternative energy as the original owners.
2. Times change. A client of mine in Maine insisted that his home would be heated only by a "Russian woodstove" and the sun. He didn't want to consider a conventional backup system. As a concession, he agreed to wire the house for backup electric heat but not to install the heaters. During his lifetime, the sun and the woodstove kept the home very comfortable; but, this kind of self-reliance was "his thing." After his death, his wife asked to have the backup electric heaters installed.
**Use the Furnace Blower Fan to Circulate Solar Heat**
Since good circulation of air within the home and through the Solar Slab is an important part of an effective solar heating plan, the solar design described in this book is an ideal complement to a conventional warm-air heating system. This combination gives the homeowner the best of both worlds: the ease of operation and responsiveness of an on-demand warm-air system, and free solar heat when available.
The Solar Slab heating and cooling system operates in a similar fashion to an automobile cooling system. Imagine your car radiator, which works fine without the cooling fan while traveling at 60 miles per hour. This is the equivalent of air naturally flowing through the array of concrete blocks in the Solar Slab.
When stopped in traffic, a thermostat turns on the automobile radiator fan to ventilate the radiator mechanically. The fan pulls air through the radiator fins, which cools the water circulating through the engine. The Solar Slab operates in much the same way. As the sun's heat enters the home and is stored in the Solar Slab, the effectiveness of the Solar Slab during peak collection times can be increased by turning on a fan.
The most cost-effective way to provide mechanical assistance to the Solar Slab is to use the air mover or fan in the furnace. By using the Solar Slab in tandem with a conventional warm-air system as described in this chapter, you can assure that the furnace is always receiving solar-preheated air, making it more efficient. In addition, the fan can be used alone, without turning on the furnace's heater, simply to serve as an air mover for the circulation of solar heat.
**How Big Should the Backup Furnace Be?**
To size backup heating equipment and to estimate the amount of fuel consumed per heating season, you can use the following formulas:
Furnace Size =
(Heat Loss × Design Temperature) ÷ Combustion Efficiency
Fuel Consumed per Year =
Difference Not Solar Supplied ÷ Usable Btus of Selected Fuel
In our chapter 6 example, the thermal performance summary for the Saltbox 38 in Hartford, Connecticut was:
Heat Loss without thermo-shutters: 10,529 Btus/°F • day, or
10,529 ÷ 24 = 438.72 Btus/°F • hr
Purchased Energy per Year without thermo-shutters:
33,710,000 Btus
Purchased Energy per Year with thermo-shutters: 26,030,000 Btus.
The size of propane gas furnace = 438.72 × 72 ÷ 0.75 = 45,117 Btus per hour. Say 42,000 Btus per hour.
The predicted number of gallons of propane needed will be:
Propane gas heat at 75 percent efficiency × 91,500 Btus per gallon = 68,625 Btus per gallon. Using for total purchased energy with thermo-shutters a figure of 26,030,000 Btus and the same formula as above, the results would be:
For a home without thermo-shutters: 33,710,000 ÷ 68,625
= 491 gallons
For a home with Thermo-Shutters: 26,030,000 ÷ 68,625
= 379 gallons
**How Much Will the House Cost to Heat?**
We have reached the moment of truth. As indicated in the calculation above, the predicted fuel usage of the Saltbox 38 example in Hartford, Connecticut, will be 379 gallons of propane per year, assuming the conscientious use of thermo-shutters. At today's price of about $1.99 per gallon, that's $754.21 per year. Without the use of thermo-shutters, the cost would be $977.09 per year.
**Fuel**
By investing in your winter's supply of oil or gas in July or August, you can buy your fuel at the lowest price for the year. As the price of fuel rises through the heating season, your summer fuel investment will be saving you more money than if you had kept that money in the bank and purchased fuel at the higher winter price.
Many readers of this book will want to heat their homes entirely with wood and sunlight. A well-designed home can be a multi-fuel home; that is, solar plus wood heat can do 100 percent of the job, or solar plus oil or gas can do 100 percent of the job. In actual practice most solar homes use combinations of the conventional backup heat and renewable energy. This flexibility makes the best use of what's available, and gives us the ability to emphasize one type of backup heat or the other as local, regional, and world fuel markets fluctuate and/or our lifestyles change.
Table 7–1 indicates the effect of modifying certain aspects of our standard solar design. As you see, making the walls or the roof thicker will produce insignificant savings. At $1.99 per gallon, the change in wall insulation amounted to savings of $99.50 per year and the roof change amounted to savings of $37.81 per year. The increased expense in labor and materials for framing more substantial walls and roof members, by contrast, is significant, and would not appear to be economically justified for the small savings produced.
_Very often an efficient, highly insulated solar home will require only a very_ _small backup furnace. Because small oil furnaces (less than 60,000 Btus per_ _hour) present operational problems, instead you can use the hot water heater_ _both to heat domestic water and to serve as the furnace. This diagram shows a_ _fan coil combined with the water heater. The components for this system, which_ _works well for houses requiring 40,000 Btus per hour or less, can be purchased_ _separately or in a package such as that offered by Apollo Hydroheat System._ _The water heater used in this way should probably be oversized relative to the_ _home's_ _hot water needs, and should be an oil- or gas-fired quick-recovery unit._
**Fresh Air**
In any case, all walls, windows, openings, and roofs should be tightly constructed and fresh air should be introduced into the home only by controlled means, as discussed in chapter 4. The calculation in Table 7–1 shows significant savings for reducing the fresh air supply. However, while it is expensive to heat fresh outside air, the benefits of saving fuel by this means are not worth the health risks of living without adequate fresh air.
When any particular design change is considered, we must carefully weigh the incremental benefits versus additional costs and hazards that may result.
**Other Fuels: Propane and Electric**
The following efficiencies may be used to determine the size of a propane gas furnace or electric heater:
1. A gallon of #2 fuel oil contains approximately 140,000 Btus, and typically an oil furnace will operate at 70 percent efficiency. Therefore, a gallon of oil will yield 0.70 × 140,000 Btus, or 98,000 Btus per gallon.
Oil Furnace Size = 438.72 × 72 ÷ 0.70 = 45,125 Btus per hour
Say, 45,000 Btus per hour net delivery ("at the bonnet"). The predicted number of gallons of oil needed will be 265.
2. Electric heat at 100 percent efficiency yields 3,415 Btus per kilowatt-hour Using for total purchased energy with thermo-shutters the figure of 26,030,000 Btus and the same formula as above, the results would be:
Size of electric heating system = 9.25 kilowatt-hours, with an annual consumption of 7,622 kilowatts
Propane has a higher furnace efficiency, but contains fewer Btus than fuel oil, so on a per-Btu basis, propane is more expensive. Yet propane has other advantages: it burns cleaner, no masonry chimney is needed for the propane burner, and leaks cause less pollution. In some areas of the country, propane will be the more economical choice.
As for electric heat, the calculations are more complex since the generating source of this heat is elsewhere. Saying that electricity is 100 percent efficient for the end-user is misleading. Electric power is 100 percent usable once it enters the home, but to calculate its true efficiency, one ought to consider generating and transmission losses, which can be quite significant.
The calculation for the electric backup option determined that we would need 9.25 kilowatts per hour for the Saltbox 38 in Connecticut. If baseboard heaters are to be used, that will be the total amount of energy needed, because the heat is distributed among the rooms by the baseboard strip heaters. A second way to provide electric backup heat is to use an electric furnace, which includes an air mover similar to that of an oil or gas furnace.
If the home is going to have a woodstove, the conventional backup heating system will no doubt be used less often. Oil and gas furnaces are like automobiles. The more they are used, the better they run. If a gas or oil furnace is not used for long periods of time, the risk of the unit not starting, failing while in operation, or causing other damage is increased. By contrast, an electric furnace can sit idle for an indefinite period of time and still start instantly when needed. An electric unit also requires no annual tune-up, there is no fuel tank to worry about, and the cost of a chimney is avoided.
**Heat Pumps**
A third way to utilize electricity as backup heat is to use an air-to-air heat pump. Heat pumps use the refrigeration cycle to produce heat. The next time you pass by your refrigerator, put your hand near the floor: you will feel a flow of warm air when the refrigerator is running. A refrigerator operates by compressing a refrigerant (in the form of a gas), and then allowing the gas to expand. This process of compression and expansion absorbs and releases heat. The heat you felt near the floor is the heat that was extracted from the contents inside the refrigerator. A refrigerator is an example of a heat pump.
In a similar way, an air-to-air heat pump extracts heat from outside air and delivers the heat to the home. The air patterns in the home are similar to those in any other warm-air system; the advantage of a heat pump (over burning electricity in a coil) is indicated by a measure called "coefficient of performance" (COP). By using electricity to operate a heat pump, the amount of heat produced can be three times that of just burning up the same amount of the electricity in a coil or baseboard resistance heater. The coefficient of performance is dependent on the temperature of the outside air. In climates similar to Maryland or California, heat pumps perform very well. In cold climates such as Idaho or Vermont, they have little advantage.
In warm areas where air conditioning is used, a heat pump has an additional advantage as it can be reversed for summer cooling.
**WOODSTOVES**
Sizing a backup woodstove is less straightforward than sizing an oil or gas furnace. Selecting the correct size woodstove depends not only on the efficiency of the particular stove, but also on the quality and species of the wood to be used. Burning unseasoned softwoods yields much lower heat in Btus, and can also cause safety problems.
It is best to undersize a woodstove according to its manufacturer's specified "capacity," so that it will nearly always be burned hot. An oversized woodstove will overheat the living space, and it will therefore frequently be damped down by the house's occupants and left to smolder. As stove and chimney temperatures drop, incomplete combustion will create a buildup of creosote in the stove, stove pipe, and chimney, which can lead to a chimney fire or other undesirable consequences. A chimney fire can actually destroy the chimney's liner, necessitating a costly and time-consuming replacement, if the fire doesn't burn the whole house down in the process.
An undersized woodstove, conversely, will require a longer and hotter "burn" to heat the living space. The hotter stove will burn more efficiently, thereby minimizing creosote build up.
The woodstove location must be carefully planned to conform to all safety requirements. Woodstoves take considerable floorspace to provide for necessary clearances, and the location cannot be an afterthought. Provisions for woodstoves have to be carefully incorporated into the original design and layout of the home.
The chimney should be masonry, and located as close to the center of the building as possible. If alternative chimney materials are used, pay strict attention to the manufacturer's installation instructions and abide by all applicable safety regulations.
If thermo-shutters are used, the overall heat loss from the house at night will be reduced. In your calculations, credit should be taken for the use by averaging the heat loss with and without thermo-shutters. Again, for the model Saltbox 38 in Connecticut, the design temperature is 72 - 0 = 72 degrees Fahrenheit. The heat loss with thermo-shutters will be 380.17 Btus per hour per degree of difference between the inside and outside temperature, and without thermo-shutters is 438.72 Btus per hour per degree difference. (In order to average the heat loss with and without nighttime insulation, we'll add the two figures together and divide them in half.) Therefore, assuming an airtight stove with 85 percent efficiency, the calculation for sizing a backup woodstove to complement the Solar Slab is as follows:
Size of Woodstove = [1/2 × (438.71 Btus/hr • degree difference + 380.17
Btus/hr • degree difference) × 72] ÷ 0.85
= 34,682 Btus/hr
Let's call it 35,000 Btus per hour.
The amount of heat generated per cord of dry, seasoned firewood varies by species. We can use an average figure of 17,000,000 Btus per cord of dry hardwood. Determine the species available to you locally, and look up its caloric value. The quantity of wood needed annually can be estimated by dividing the purchased energy per year by the amount of heat available in a cord of firewood (in this case—dry, seasoned hardwood). Calculate the amount of firewood needed per year for a home without thermo-shutters, as follows:
33,730,000 Btus ÷ 17,000,000 Btus per cord = 2 cords
Just to be safe, we will add another 1/2 cord, making the predicted total 2.5 cords. Next, let's calculate the number of cords needed per year for a home _with_ thermo-shutters:
26,060,000 Btus ÷ 17,000,000 Btus per cord = 1.5 cords
Once again add a margin for error of 1/2 cord, and the total is 2 cords.
We often see people cutting and splitting firewood in the fall, "getting ready for winter." The wood used for a given heating season should be a year or more old; that is, wood cut in one fall should be stacked to dry for a full calendar year before burning. To ensure that your wood is properly seasoned, split the logs, in lengths appropriate for your stove, and stack them under cover with adequate gaps for air circulation. If it isn't possible for whatever reason to get a full year ahead in your reservoir of firewood, be sure that your wood is cut, split, and stacked no later than the end of April for the following winter. Remembering the solar principles emphasized throughout this book, it is also best to store firewood in a shed with an open southern exposure, to facilitate drying.
There really is no substitute for the radiant heat derived from a wood-stove, but as any fireman can attest, woodstoves require great caution in planning and constant vigilance in operation. Study one of the many books solely devoted to woodburning.
There once was a young woman in Vermont who got married and proudly invited her father to her new home, to show off the central oil-fired heating system with baseboard heaters, and with no woodstove anywhere to be seen. Her father entered the home on a cold winter's evening and started to roam from room to room. The daughter asked him if anything was wrong.
"No," he said as he continued to wander around the living room.
"Well, there must be something wrong. How come you keep wandering around?"
The father looked at her with a puzzled look and asked, "Whar do ya go to git warm?"
**GEOTHERMAL HEAT**
Through the monitoring program discussed in chapter 1, a minimum of 45 degrees Fahrenheit was measured below the gravel layer underlying the concrete blocks of the Solar Slab (see the illustrations). Note that there is a 1-inch thick layer of Styrofoam insulation specified on top of the hardpan. This layer of insulation is placed there to prevent the possibility of a rapid loss of heat to the ground below the building, but it will allow a slow transfer of heat upward if conditions are suitable. These conditions will occur in an unoccupied and unheated home.
If the temperature in the unoccupied home is allowed to drop to the 45 to 50 degree range, the thermostat marked "H" shown below the thermostat marked "C" will turn the blower on at 50 degrees and circulate air to extract ground or geothermal heat. Note that in this mode the normal "H" thermostat connected to the heat control is set at 45 degrees. It is important to purchase thermostats that read accurately down to 45 degrees in order for this mode to function properly.
As the unoccupied home loses heat, the Solar Slab will first give up the heat in its concrete block and slab layers. These layers are the active part of the Solar Slab; that is, they routinely take on and give off heat. The layers below the concrete blocks are more passive, as they will be slower to rise or drop in temperature.
A home that is unoccupied will first draw out the heat available in the active portion of the Solar Slab, and then will draw heat from the passive layers that underlie the concrete blocks. This underlying reservoir of heat is almost infinite. If the Solar Slab is not extracting heat fast enough from the lower levels, the circulating fan in the furnace will be turned on thermostatically, and the Solar Slab will act as a heat exchanger between the ground and the house. The cost of this heat extraction will be only the minor cost of running the furnace's blower. The theoretical minimum temperature to which a home with a Solar Slab will drop is the ground temperature under the Solar Slab, a temperature that is exceedingly stable.
**USING THE SOLAR SLAB**
**FOR SUMMER COOLING**
The natural solar and backup heating systems discussed in this book are all very helpful in controlling a home's inside temperature. But an important function of air conditioning is the reduction of the moisture content of the air. Unfortunately, in warm and humid regions a mechanical and energy-intensive air conditioner is needed to do this job.
In summer, air returning to the heat pump air mover will be pre-cooled by the Solar Slab. While monitoring the performance of the Solar Slab in summer, we observed a 12-degree drop in temperature as the air entered and exited the vents for the Solar Slab. Just as we noted with furnaces for winter, this assistance allows the air conditioner to be downsized smaller than standard practice would suppose. For instance, by informal monitoring in Maryland, we found that the best cooling was achieved by having a slightly "undersized" air conditioner running steadily instead of a larger unit cycling on and off.
The cooling capacity of air conditioning equipment is measured in "tons of cooling." In days gone by, the White House was cooled by filling a huge room in the basement with ice, and then passing the living-space air over the ice so that cooled air was recirculated by ducts into the building's rooms. A ton of cooling is related to the cooling capacity of a ton of ice (12,000 Btus per hour). The term stuck. In our case, a credit of approximately 1/2 ton of cooling can be taken due to air returning through the Solar Slab.
The total cooling load for a home is the sum of "sensible heat" and "latent heat." Sensible heat is the heat that is gained by the same factors considered in a heat-loss calculation. Latent heat is the additional cooling load due to the necessity of reducing moisture in the air to be cooled. The calculation needed to properly size an air conditioner is complex and is not included in this book. That kind of precise evaluation is a job for a heating/ventilating engineer; however, a reasonable estimate of the proper size for a household air conditioner can be made by utilizing a rudimentary guideline.
To estimate the basic cooling load for a home, multiply by three the actual or projected volume in cubic feet of the conditioned airspace in the building. The resulting number will approximate the cooling load in Btus per hour. In our Saltbox 38 example, the conditioned airspace is 14,492 cubic feet; the cooling load approximation is 14,492 × 3 = 43,476 Btus per hour. One ton of cooling equals 12,000 Btus per hour. Therefore, the home's proper air conditioner size should be about 43,476 Btus per hour ÷ 12,000 Btus per hour (or one ton of cooling), which is 3.6 tons of cooling.
_Layout for modified foyer for_ _floor plan,_ _showing the location of the_ _furnace and hot water heater._ _This utility room will provide_ _good sound isolation for the furnace_ _while leaving the_ _backup heat equipment_ _accessible when necessary._
Applying a Solar Slab cooling credit of about ½ ton, the unit size comes out to approximately 3 tons. So this sample home in Connecticut will probably need about 3 tons of cooling, or an air conditioner with a capacity of 3 × 12,000 Btus per hour, or 36,000 Btus per hour.
The home's cooling load will almost always dictate duct size, as more air movement is needed to satisfy the cooling load than the heat load. It also costs more to cool air than to heat air. In this example, three tons of cooling will require a movement of about 400 cubic feet of air per minute per ton, or 1,200 CFM. As you will recall, the sample home's furnace air mover requirement was 900 CFM.
**HOW TO HOOK UP THE BACKUP**
**FURNACE OR AIR CONDITIONER**
The diagram shows how to install a backup gas-, oil-, or electric-fired furnace. The drawing is schematic relative to the actual locations in the building and the relative size of the equipment's components. Most warm-air systems have one or two central returns and a distributed feed system via ducts and grilles located throughout the house. This diagram shows a distributed return system. The distributed return is accomplished by locating intake grilles along the north and south walls. These are the same grilles needed for the natural flow of the Solar Slab. Do not locate intake grilles in utility rooms, bathrooms, or any other room which has either excess moisture or undesirable odors that could be introduced to the air circulation system. When the furnace blower is turned on, air is returned via the grilles located along the north and south walls. The air movement will be very slow, because of the distribution and oversizing of the return-air grilles, and as a result the floor surface will be almost draft-free. Once the return-air enters the air passage in the Solar Slab on the north- or south-facing wall, it will flow into the open channels in a row of blocks, and eventually return to the furnace air mover via the return duct placed near the center of the house along the east-west axis of the Solar Slab. Three-inch by twelve-inch vents cut into the sides of the return duct will allow the air to enter into the duct and return to the air mover. When the furnace gun or heat element is operating, the returning air will arrive carrying residual heat from the Solar Slab and provide warmer return air than would be the case if the air was returned directly from the airspace to the furnace's intake vents.
_Heat control diagram used to_ _show the relationship between_ _the Solar Slab and oil, gas, or_ _electric backup heat equipment._ _This drawing is schematic._
Note in the diagram that the thermostat marked "C" is a cool thermostat, which will turn the furnace's blower on if the home starts to overheat due to the accumulation of heat from the sun. In this situation, the furnace gun or heat element will remain off while only the blower runs to cycle air through the Solar Slab, in order to store the heat that actuated the cooling thermostat.
In summer, heat that was stored during the day, which helped cool the home, is expelled at night by simply allowing the home to ventilate through open windows or by mechanically expelling stored heat by running the furnace blower from midnight to 4:00 AM—the coolest part of the 24-hour summer day. In the diagram, the timer marked "T" controls this function.
The furnace is shown centered on top of the return duct for illustrative purposes only. It should be located outside the living space for noise abatement. One cost-saving idea is to locate an electric furnace inside the stair enclosure leading to the second floor, which allows for air distribution directly from this central location, thereby reducing or eliminating the feed-duct system. The disadvantage of this scheme is noise. In homes using a woodstove as the prime backup source of heat, this lower-cost siting of the furnace may be acceptable despite the proximity of a noisy blower to the living space, since the electric heat will be used very little or not at all. More likely, the backup furnace will be used while the occupants are away for extended periods, making the problem of noise from the blower inconsequential, since no one will be home to hear it.
Again, the picture shows how the airlock foyer can be modified to locate the furnace and domestic hot water heater outside the living space. In this configuration, with properly designed ductwork, the operation of the warm-air system will be almost silent.
**Recirculate Warm Air from the Second Floor**
The last item to be discussed is the second-floor ceiling blower. In winter this small blower takes warm air that rises to the second-floor ceiling, and delivers it back to the first floor. In houses with wood-stoves, it is advantageous to locate the exit grille behind the woodstove to direct warm air away from the woodstove while it is operating.
In summer, the second-floor fan enclosure can be vented to the outside as shown. The energy consumed by a small axial fan is a very reasonable expense for the winter heating and summer cooling assistance provided by this small blower.
**LET THE LAWS OF NATURE**
**WORK FOR YOU**
Let's review the design we have been discussing. Over 50 tons of effective thermal mass have been built into the home. This has been coupled with the correct amount of east, south, and west glass to collect solar heat. In addition, the home has been highly insulated in a manner which protects the occupants from undesirable side effects from poor air quality.
Physics, or laws of nature, which have been built into the design will work on your behalf twenty-four hours per day for the life of the building. There is no predicted maintenance of the basic solar heating and cooling system. The various backup heat schemes and their associated equipment are merely refinements on this fundamentally natural solar design. None of the refinements, including the use of thermo-shutters, should be interpreted as necessary to permit the system to work as it will work naturally.
Just as modern day automobile engineers have been able to increase horsepower and fuel mileage without increasing the size of the engine, backup heating and cooling equipment can "push" the natural system to make it perform better. All of the backup schemes described here make double use of the backup equipment; that is, this equipment will function as a supplementary heater or air conditioner, and in addition, the same equipment can be used to provide mechanical assistance to the natural solar collection and distribution system.
Solar Principle # 7
_Consider the contribution of solar energy (indicated by insolation values for_ _your region) and natural processes (including breezes and shade) to the heating_ _and cooling of the home, in order to avoid oversizing a backup heating system_ _or air conditioner. A home that is oriented to true south, is tightly constructed_ _and well insulated, and has operable windows for air circulation should not require_ _large fossil-fuel burning equipment to maintain thermal comfort._
Size the conventional backup systems to suit the small, day-to-day heating and cooling needs of the home. Do not oversize backup oil or gas furnaces, as they are inefficient, cycling on and off, when not supplying heat at their full potential. Air conditioners are likewise expensive and wasteful when operated inefficiently.
8
[A SIDEHILL VARIATION,
AND SOLAR DESIGN
WORKSHEETS](chap6_passives_9781603581691_epub_part6.html#d41151059)
Humans discovered long ago that the world is not flat. So it goes with house sites. As the easy-access lots are sold off, we find ourselves gradually moving up hillsides to build our homes. Our example of how to utilize a hillside site will be a Green Mountain Homes Sidehill "C-32," the floor plan.
The design of the Solar Slab for the sidehill situation is illustrated. The most notable difference between the Solar Slab for a flat lot and the sidehill adaptation is the inclusion of the north side concrete wall in the home's thermal mass.
This is accomplished by placing three 1-inch layers of rigid insulation on the outside of the wall as shown in the diagram. Also note the extra 1-inch layer of Styrofoam under the flat portion of the Solar Slab (making a total of two 1-inch layers of rigid insulation in the sidehill slab). The north side's usual 4-inch x 14-inch air vents have been extended upward to the first floor and the south air vents are placed along the south wall on the lower level.
This sidehill design utilizes the lower level for living space. Remember that for the Solar Slab to be effective, it has to be in thermal contact with the living space. It is not cost-effective — nor thermally effective — to utilize the lower level for a basement/storage area.
**LET'S** **TRY SUNNY WYOMING**
**FOR THE SIDEHILL SITE**
For the sake of discussion and calculation, we will locate our model C-32 Sidehill in Cheyenne, Wyoming, and analyze it using the method developed in chapter 6 and the worksheets included in this chapter and in appendix 1. I will show you how I would use these worksheets to do design calculations for the Wyoming house, and you can photocopy the blank worksheets in the appendix to use in your own planning process.
_A floor plan for a sidehill version of the C-32 saltbox design. Construction against a sidehill permits use of the_ _"_ _bermed"_ _back wall as part of the_ _home's_ _thermal mass._
_Another sidehill variation._
Let's fill in as much information as we can on Worksheet #1.
You can start with lines 1 through 19.
Line 2: Obtain from appendix 4.
Line 3: Obtain from appendix 7.
Lines 4, 5, 6, 7, 8, 9, 10, 14, 16, 17, and 18: Obtain from your own house drawings.
Line 11: Obtain from the manufacturer's literature for your proposed window and patio doors.
For line 12: 0.88 is the Shade Coefficient for 1/2-inch dual-glazed glass. Enter the correct Shade Coefficient for your glass.
Line 19: Obtain from appendix 4.
Line 25 can now be entered on Worksheet 1A. The Standard Clearness Number is 1.00. Since Cheyenne's elevation is 6,126 feet above sea level, and Cheyenne has a dry clear atmosphere, we can judgmentally increase the clearness factor by 10 percent. Therefore, our clearness number is 1.10.
Using Worksheet 2, we will next calculate U-values.
The individual R-values called for can be obtained from appendix 3, except for the window insulation R-value. Obtain the R-value of your nighttime insulation device by using the manufacturer's specs or, if you make your own device, calculate it by adding up R-values for the materials used to get a cumulative total (as shown in chapter 4). In the Wyoming model calculations, we'll use the R-value for thermo-shutters.
The wall and roof sections will be the preferred design described in chapter 4 (2 x 4s with rigid exterior insulation).
In the north wall detail, we will assume that the exposed interior concrete on the lower level will be covered on the inside with 1-inch Styrofoam, with 1/2-inch drywall screwed to 1-inch strapping placed 16 inches on center across the Styrofoam insulation. A 6 mil vapor barrier should be placed behind the drywall, similar to the placement of the vapor barrier on the framed wall drawing.
We can now fill in lines 13, 20, 21, and 22 on Worksheet #1.
We now have enough information to fill in Worksheet #3, House Heat Loss. Next, taking the information from Worksheet #3, fill in lines 23 and 24 on Worksheet #1.
Worksheet #4 will be next. Solar Heat Gain Factors shown in appendix 2 for north latitude 40 degrees are the closest to Cheyenne's location at 41 degrees 1' north latitude. You'll see in the ASHRAE table that listings are 8 degrees apart in latitude. Cheyenne happens to be close to 40 degrees north latitude, but if you encounter a design situation which is halfway between the SHGFs given in the appendix 2 tables, you may interpolate between the two. That is, if a house is to be located at or near 44 degrees north latitude, then use the average of the half-day SHGFs given for 40 degrees and 48 degrees north latitude. Remember that the SHGFs are read up and down on these tables, as described in chapter 6.
Moving on to Worksheet 5, we'll calculate the monthly heat load for the C-32 Sidehill house.
Using appendix 5, fill in the degree days for each month for your location, in our case Cheyenne. Then calculate the monthly loss due to the lower concrete living-space (heated) wall by multiplying the LCWL (From Worksheet 3) by 24 hours per day and then by the number of days per month.
To digress for a moment, let's see what the loss across the lower concrete wall would be, if the wall were left uninsulated similar to the way many full basement cellar walls have been left. The R-value for an uninsulated 8-inch concrete wall is 0.60, as shown on Worksheet #2, section D.
U = 1/R = 1.6667
The loss for this 464 square feet of concrete wall then would be: square feet of concrete × U-value × difference between interior and exterior temperatures, or
464 × 1.6667 × (65 - 45)* = 15,467 Btus/hour
* From Worksheet #3
The loss for the 464 square feet of insulated concrete wall in the Sidehill example would be 423 Btus per hour (see Worksheet #3), which means that an uninsulated wall would lose 36½ times more heat than an insulated wall. As you can see, badly designed full basements can be big losers of heat.
This calculation also helps explain the benefit of earth berming. As our figures clearly demonstrate, the temperature difference across the lower concrete wall is a constant, due to the relatively warm (and stable) temperature of the earth. The earth's temperature in this example was conservatively assumed to be 45 degrees.
We are now ready to summarize our calculations on Worksheet #6.
From the totals on Worksheet #6, we can see that this solar home in Wyoming will derive 67 percent of its heat _free_ , from the sun.
The next worksheet will show us how to size the conventional backup oil-fired furnace and woodstove, plus it will show the estimated annual fuel consumption, using the methods of analysis presented in chapter 7.
For this example, oil was chosen to be the conventional backup fuel source. Note that the figure for heat loss without thermo-shutters was used to size the oil furnace. This approach is a little conservative, but experience has shown that to be reasonable. The possibility exists that window and/or patio door insulation devices may never get installed despite the best of initial intentions; or they might be removed, by the second owner of a home.
The woodstove calculation shows a method to undersize the choice of stove, as recommended in chapter 7. It is better to have an undersized woodstove burning hot rather than an oversized woodstove smoldering and creating creosote.
So, now we can fill in lines 26, 27, 28, and 29 on Worksheet #1, as shown in Worksheet #1D.
Congratulations, you have reached the last worksheet! It is time to put your solar home in thermal balance by sizing the Solar Slab to absorb the excess free solar energy available while keeping the home within comfortable temperatures during peak solar-collection times.
Worksheet #8 follows the same procedure described in chapter 6. The figure for House Heat Loss with thermo-shutters was used as a way to compensate for the fact that the home's windows during the 10 hours described were collectors of energy, not losers of energy. Further, the heat loss through a wall is directly proportional to the difference in temperature between the inside and the outside of the wall. During that 10-hour collection period, the sun was warming the exterior of the wall, thereby stopping the flow of heat outward. For this reason, a house will always benefit from having the most wall area on the south side, even if there are no windows!
Note also that the lower concrete wall is not included in the thermal mass calculation because it will not respond to daily temperature differences in the same manner as the horizontal Solar Slab. Its benefit, however, can easily be seen by the overall reduction in heating load, since the amount of heat lost through the lower living-space wall into the earth is so small.
You made it. Enter 6.5 inches for line 30 on Worksheet #1.
The C-32 Sidehill in my example is theoretically located in sunny Cheyenne, Wyoming. As a comparison, let's relocate the Sidehill design to Ann Arbor, Michigan. Making the same kind of solar analysis for a sidehill house in Michigan, the calculations show it to be 40 percent solar heated with a predicted oil usage of 431 gallons per year, whereas the same design in Cheyenne yielded 67 percent solar with a predicted need for 249 gallons of oil per year.
Needless to say, Ann Arbor, Michigan, and Hartford, Connecticut are not the usual places one would pick to illustrate the efficacy of solar design. Western states with high elevations, clear skies, and high percent sunshine are more apt to be used for solar home design examples. And yet many people desiring solar homes live elsewhere.
All too frequently we hear someone say, "Solar won't work here." How can solar energy not work? We all live in solar locations; although in some locales, as gardeners know, more sunlight is available for greater portions of the year. Does that mean that because your site is not the perfect solar location that you shouldn't take advantage of the sun's capacities for heating and cooling? The basic premises for a good solar home are simply the premises of good home design:
1. Make the most of what's available to you in terms of both your environment and the materials that you are planning to use in your home construction.
2. Let the tendencies of nature work for you and not against you.
3. Work toward the goal of keeping the conventional furnace and air conditioner switched off by using alternative backup fuels and free solar heat.
A 2,085-square-foot home burning 431 gallons of oil per year in Ann Arbor, Michigan, doesn't sound as good as the same kind of home burning 249 gallons of oil per year in Cheyenne, Wyoming, which is a colder place. And yet, if you live in Michigan, then you did the best you could with the solar energy available to you. That 431 gallons of oil a year bought in the summer as an advance, one-time purchase at $1.99 a gallon will mean only about $857.69 per year or about $71.50 per month for heating; plus your solar home is bright and cheerful.
Solar Principle # 8
_Provide fresh air to the home without compromising thermal integrity._
To maintain a high level of indoor air quality, a well-insulated and tightly constructed home needs a continual supply of fresh air equivalent to replacing no less than 2/3 of the building's total volume of air every hour. This exchange of air should occur through intended openings, for instance an exterior-wall fan in both the kitchen and bathroom, rather than through leakage around poorly sealed doors and windows.
_Sidehill modification of the basic plan, showing a detail of the north wall footings,_ _drainage, and connection to the Solar Slab._
Too often people wishing to heat with alternative fuels spend spring, summer, and fall getting ready for winter. Remember, cutting and stacking two cords of wood is a lot easier than cutting and stacking eight. And in addition to reducing your annual heating load, a highly insulated home with proper vapor barriers and stained natural sidings will minimize the need for periodic summertime exterior painting, staining, and weatherstripping. Furthermore, when it snows, a properly designed roof will not cause ice jams and water dams.
A natural solar home when properly designed, sited, and built will make life a lot easier by working for you, day in and day out, instead of requiring you to be constantly working for it.
9
[SUNSPACES AND
SPECIAL DESIGN
CONSIDERATIONS](chap6_passives_9781603581691_epub_part6.html#d411510510)
It is easier to understand a concept if one can point to an example and say, "Aha, that's what makes it work." Sunspaces and greenhouses satisfy conventional expectations about solar design in that they reach high daytime temperatures, and anyone can understand why. Just as a car left with its windows closed in a hot summer parking lot will become an oven, so the sunspace will build up high temperatures, which will allow a positive transfer of heat from areas that are warm to areas that are cooler, for instance from the 90-degree sunspace to 70-degree interior of the house. Sunspaces are overglazed on purpose, and designed to overheat.
It might seem that a sunspace that is gathering enough heat to become 90 degrees Fahrenheit on a cold, 15-degree but sunny winter day would be beneficial to the home. And yes, it can be beneficial. However, the same overglazed sunspace that accumulated all that heat during the cold but sunny day will need lots of added heat when the sun goes down to prevent it from freezing, which means that the sunspace or greenhouse will tend to draw heat from the rest of the house as heat flows back out through the glazing.
It is not uncommon for a sunspace to soar in temperature to 90-plus degrees during the day, and then "struggle" to maintain 32 degrees at night. The large nighttime loss is due, of course, to the overglazing. As you will remember from our calculations in chapter 6, even the most energy-retentive thermal-pane glass has only a fraction of the insulation capacity or R-value of unglazed wall.
**THE COST OF** **"** **ADD-** **ON"** **SOLAR**
In order to analyze any benefits that may come from a feature such as a sunspace, one has to calculate the daytime heat gains and factor these against nighttime losses. For the sunspace to be a net benefit, you will also need to provide for an effective means of transferring the solar heat from the sunspace into the house.
In making these sunspace heat gain and loss calculations, one must also remember that the sunspace is taking up wall space on the south-facing elevation. Ideally, it will be located in front of a patio door that can be closed at night. This will isolate the sunspace thermally from the primary living space. And we have seen in our previous design examples that a patio door is already an effective solar collector. Adding a sunspace in front of a south-facing patio door amounts to putting a solar collector in front of a solar collector. And yet a sunspace placed in front of a patio door will shade the living space making the room darker than it would be without the sunspace.
**Tilted** **Glass—** **A Liability**
Most readers will be able to picture the typical sunspace or greenhouse design, in which south-facing is tilted so that the angle of winter sun is more perpendicular to the panes of glass. Tilted glass is a more effective solar collector than vertical glass. In February at 48 degrees north latitude, tilted glass will be approximately 20 to 30 percent more effective than vertical glass. However, in summer, tilted glass will continue to be more perpendicular to the sun's rays than vertical glass, and will continue to take in heat. The common problem of summertime overheating in sunspaces may be easier to explain than solve.
Because of gravity, providing window insulation for tilted glass is a more complicated problem than providing the same kind of covering for vertical glass. Special rails or attachments will be needed to hold the window insulation snugly against the sloping glass. In addition, on tilted glass nighttime condensation will drip on to window coverings, causing stains and possible degradation of the insulating material.
Through our monitoring process of a prototype home with a sunspace in Royalton, Vermont, we found that there was no discernible difference in overall thermal performance of the home with or without the added sunspace. The sunspace, however, did not take heat from the home or was thermally neutral. That is, any daytime heat derived from the sunspace was "paid" back at night to maintain minimum temperatures.
_A basic sunspace design._ _Note the solar slab in the_ _structure's_ _base. Provide_ _for adequate venting,_ _and consider isolating_ _the added-on sunspace_ _thermally from the rest of_ _the house to prevent the_ _sunspace glazing from_ _drawing the_ _home's_ _heat_ _out on cold winter nights._
The figure above shows a representative four-panel sunspace. Assuming that the east and west elevations of the sunspace have 40 square feet of glass per side, this sunspace has the following specifications:
The net performance of a sunspace can be improved by thermally isolating it. For example, by placing the sunspace outside of a sliding glass door and closing this door at night and on sunless days, you can minimize the amount of heat that the home needs to "pay back" during times when the sunspace is not collecting solar heat. You will also need to provide supplementary heat to the interior of the sunspace to maintain minimum temperatures at night and on overcast days.
Remember that about two-thirds of the fuel needed for a solar home will be consumed in December, January, and February. As you can see from Table 9-1, a sunspace located in Burlington, Vermont, needed additional heat in those months, so it added an energy burden to the house at the time of year when energy loads and expenses are already greatest. In Hartford, Connecticut, a comparable sunspace was close to breaking even in terms of costs and benefits, even in those three months. For the sunspace to yield a significant improvement in the performance of a solar home, it has to contribute positively in those three winter months. The house really doesn't need a boost of solar heat in September, October, November, March,April, or May since during these transitional months, the solar home probably needs no purchased energy at all, or very little purchased energy. And, as indicated above, in summer months the sunspace may be more likely to be a cooling burden that a heating benefit. A sunspace's performance can be improved if a Solar Slab is used for is base and thermal mass. A small duct fan actuated by a thermostat at 50 degrees will in most cases transfer enough heat back into the sunspace to prevent it from freezing, provided that the sunspace and the Solar Slab are properly sized and constructed.
**Special Difficulties in Sunspace Construction**
Whenever glass is placed at an angle, the thermal stresses and temperature variations are substantially increased, and the force of gravity is effectively pulling the glazing panes or panels downward, making it difficult to keep seals from leaking. Only quality rooftop windows and rooftop fixed glass made for tilted use should be used. Because of the expense of commercial glazing units and ancillary products, many attempts to reduce costs have been made by do-it-yourself builders who re-use glass panels out of patio doors and set them in wooded frames to reduce costs.
Most warranties from window manufacturers are voided when glass that has been designed and manufactured to be placed vertically is placed an angle. Glass expands at approximately the same rate as aluminum. Attempts to set tilted glass in wooden frames with wooden mull caps most likely will fail, because the glass and wood have incompatible coefficients of expansion. The glass will expand more rapidly than a wooden mull cap; the sealant used between the glass and the mull cap will crack, which will result in a water leak. In addition, in tilted glass the manufacturer's seal between the two panes of glass is also subjected to extraordinary thermal and gravitational stresses, and is likewise prone to leak. Have you ever driven by a homemade sunspace and noticed that the glass is fogged up? That is due the failure of the factory seal between the dual-panes. Commercially manufactured rooftop units are specifically designed and tested for tilted use, are warranted against water leakage and seal failure, and are made out of tempered safety glass. Glass placed at an angle should always be tempered safety glass to prevent possible injury.
**It's** **Not All Bad** **News—** **Sunspaces are Fun**
Does this mean that homeowners should never add a sunspace or greenhouse? No, not at all. Sunspaces are fun to have; they provide a place to grow flowers year-round and to start spring seedlings. They provide a place to simply luxuriate in 90-degree heat when the outside temperature is in the teens on sunny winter afternoons. They provide an uplift to the spirit, when plants are bathed in sunlight and blooming in the dead of winter. And sunspaces present no special heating or cooling challenges in most regions in the relatively mild months of spring and autumn.
Sunspace interior.
If you understand the possible benefits, and are willing to address the challenges, a sunspace may be "just what the doctor ordered." But if you believe that adding a sunspace is going to pay for itself by heating your house, you may want to reconsider.
Finally, another popular use for sunspaces is as retrofits on older homes. After hearing me out all through an explanation of the costs and difficulties like the explanation above, a prospective sunspace buyer responded, "I understand completely what you have said, but my husband and I own an ancient "Four-square" home that is hopelessly inefficient. We have no hope of ever being able to afford a new home, and all I want is to have is at least one place in my home that's warm when the sun is out." Pretty hard to say no to that.
**IDEAL VERSUS**
**ACTUAL CIRCUMSTANCES**
So far we have presumed the existence of ideal conditions under which to build a solar home. We have described a naturally heated and cooled home that takes full advantage of what is available to us, from the vantage point of both macro- and micro-environments.
Approach your home building project in this manner. Try to utilize all of the elements that are there to work with, in the best possible ways, and build the most environmentally sensitive home possible for a given location and set of circumstances. Try to think positively about each aspect of your site, your design, and your energy options. Remember that the sun is everywhere, and with careful planning you can build a home that harmonizes with solar energy.
But let's go over a few examples that demonstrate less than ideal situations. Suppose the garage or other structure has to be placed in such a way that it will obscure all the east-facing glass. The practical remedy is to rotate the home counter-clockwise so that the south-facing glass is about 15 to 20 degrees off of true south, with the south elevation now facing south-southeast. This will allow your south-facing glass to begin to collect energy earlier in the day. Conversely, if your west- facing glass for some reason will be obstructed, you can rotate the home 15 to 20 degrees clockwise to allow the south-facing glass to collect compensative heat from the afternoon's westerly sun.
If you live in a region where it frequently may be necessary to use air conditioning for summer cooling, you can reduce morning and afternoon solar gain by shading the east- and west-facing glass with plantings of deciduous trees, and use of Thermo-Shutters or other window insulation. You can also consider reducing the amount of east- and west-facing glass. The calculations in chapter 5 will permit you to evaluate during your design process the benefits of adding or removing these windows.
**Orientation—** **the Key to Solar Design**
Probably the biggest "no, no" is to buy north-facing land or sites located in deep, sunless valleys or canyons. Homesites with primarily northern exposures just don't get "bathed" by the sun. One of my former clients bought a lot in Maine, with a view of the ocean, and it wasn't until the builder visited the site with a compass that the man discovered that what he had imagined was a south view of the ocean was a north view. By this time the man was too far committed not to build his retirement home in that site. Given this challenge, we selected a saltbox design, and placed an array of roof windows in the long slope of the side that is normally the unglazed north roof, which in this situation was faced south. The high side of the saltbox that normally faces south was actually facing north, giving the residents full benefit of the ocean view. The amount of glass on what was now the north elevation was drastically reduced from the design specifications, and fitted with Thermo-Shutters. The home performed reasonably well, though the situation was far from ideal. Our solution was the best that could be managed with existing circumstances, and truth be told, these homeowners would not have ended up better off with a conventional instead of a solar home in that same situation.
The real moral of this story is to always take a compass with you when you are looking at house sites. There really is no substitute for a site with a good southern exposure.
**OTHER WAYS TO USE ENERGY WISELY**
Up to now we have mainly talked about storing the sun's free heat in the Solar Slab. Certain kinds of commercial or manufacturing processes generate excess "purchased" energy during the day, which if not vented outside will overheat the building. Why not store this excess purchased energy for nighttime use after the workday is over? For instance, consider the examples of an office building filled with heat- producing electronic equipment such as computers, or a dormitory building that is required by code to produce surplus hot water, or a library that has lighting requirements that result in excess lamp-generated heat. Why not circulate such waste or byproduct heat to other parts of the building, and/or store excess heat? A Solar Slab allows the storage of so-called waste heat for later use. The challenge of solar design is to consider every aspect of the planned building's energy situation over the lifetime of the structure.
Sometimes energy goals requirements appear to conflict. A library, for example, needs to provide a high degree of quality lighting to meet standards; but these lights give off excess heat. By circulating the air that has been warmed with already purchased electric-light energy through the Solar Slab, heat can be stored for later use rather than vented to the outside.
Solar Principle # 9
_Use the materials you would use for a conventional home, but in ways that_ _maximize energy efficiency and solar gain._
With exactly the same construction materials, it is possible to build an energy-efficient, sunny, and easy-to-maintain solar home or a energy-gluttonous, dark, and costly-to-maintain house. When designing a solar home, rearrange and reallocate materials to serve dual functions - adding solar benefits as well as addressing architectural or aesthetic goals. Placing a majority of the home's windows on the south side is an example. The carefully designed and constructed solar home need not cost any more to build than a comparably sized non-solar conventional home.
_Wind power is another form_ _of solar energy. Here is an_ _old-time wind-powered water_ _pump. New technologies_ _allow people to use these age-old_ _sources of energy. As with_ _contemporary photovoltaics,_ _which convert sunlight into_ _storable and useable electricity,_ _today's_ _micro-sized wind_ _turbines are very practical and_ _affordable for home-scale use._
Another example. A college dormitory has a high hot-water requirement for showering. In order to meet the demand, a large amount of hot-water capacity is needed; however, showering usually takes place for a short period of time in the morning or evening, while the rest of the time hot water is stored in water-heater tanks and kept up to temperature with periodic applications of electric or fossil-fuel energy. A solution to this problem is to use water-to-air heat exchangers for space heating, utilizing the domestic hot water for more than one "end-use," thereby eliminating the need for a separate furnace. It is also entirely practical, and very cost-effective, to use solar thermal techniques for heating or at least preheating water with sunshine, which in some regions can reduce conventional water-heating expenses dramatically.
Examine all available heat sources, and maximize your provisions for benefiting from the specific conditions of your site. Rearrange the materials already committed to the building project in order to efficiently collect and store heat. Whether building a solar home or an office capable of storing excess purchased energy, we should use every technique available to reduce our use of finite and expensive fuels.
**Solar Electricity**
From 1980 to 2003 the cost of solar electricity has dropped from $1.00 to $0.21 per kilowatt-hour. Hopefully this lowering of costs will continue, make the production of electricity from the sun's energy a practical alternative for generating electricity.
The use of solar electricity, or photovoltaics, requires special knowledge and experience beyond the scope of this book. Charlie Woodward of Victor, Idaho, contributed the following. Charlie is a designer and installer of PV systems and has lived off the grid for over 25 years. He writes:
Utility intertie photovoltaic (solar electric) power systems are now readily available and very efficient. They allow almost any homeowners to sell power back to their utility, or reduce their power bill. They typically do not include battery storage to provide power during outages, but with more sophisticated equipment they can be configured that way. The economic picture for these systems is highly dependent on your local utility's net metering policy and buy-back rates. In progressive areas, they may be economically attractive.
In most parts of the United States, the present cost of having a power company provide electricity to a remote homesite is in excess of $24,000.00 per mile of added powerline. A viable alternative is to live "off the grid," producing your own electricity with solar photovoltaic modules. Early solar-electric systems used 12-volt technology developed for recreational vehicles and boats. This required major lifestyle adjustments, as all the electrical equipment in the home had to be specially designed to run on 12-volt power.
Recent advances in computer controlled, sine-wave inverters, which can be stacked to produce large amounts of AC power at 120 or 240 volts, coupled with microprocessor operated monitoring and control systems, have now made the off-grid home a very viable alternative. With the use of a properly sized inverter, direct current generated by sun-tracking solar electric modules is converted to ordinary alternating current, allowing the use of energy-efficient electrical appliances.
It is now entirely practical to live comfortably off the grid by producing solar electricity for storage in batteries, and by utilizing solar space heating, a domestic solar hot-water system, propane gas for refrigeration and cooking, and backing up the whole arrangement with a propane- or gasoline-powered generator. And with the right wind conditions and access to a year-round stream, an ideal site would even permit residents to harvest wind energy and hydroelectricity with the new micro-turbines, which are perfectly sized for household needs.
While this book has concentrated on the challenges and opportunities of solar home heating and cooling, I hope the examples given here will help readers view the prospect of building energy-efficient and environmentally sensitive buildings with a greater sense of possibility and determination.
10
[INTERIOR DESIGN FOR
YEAR-ROUND COMFORT](chap6_passives_9781603581691_epub_part6.html#d411510511)
By Cornelia C. Kachadorian
There are a number of special factors to consider when thinking about interior design for a solar home, yet many of these considerations could really apply to all types of homes.
The primary challenge with a solar home is the deliberate access given to sun, with greater exposure of interiors to its component radiation. Ultraviolet rays are principally responsible for fading fabrics and other materials, while infrared rays heat up surfaces they strike. The visible spectrum ranges between the ultraviolet and infrared. One can find UV-reflecting glass on the market today, which helps reduce damage, but this glass does not block all the effects of ultraviolet. Infrared rays heat whatever they touch, making exposed portions of furniture hotter than their shaded areas.
On the other hand, natural light in the visible spectrum presents a lighting medium of great potential.
**USE SUNLIGHT**
**AS A DECORATIVE ELEMENT**
As emphasized throughout this book, solar home windows are oriented toward the east, south, and west, with a minimum of windows on the northern exposure. The characteristics of sunshine change throughout the day and the year, varying in intensity, color, and angle. Window light combined with well-chosen and well-placed supplemental lighting can provide exquisite results with minimal costs.
Because of its more horizonal angle, winter sunlight penetrates significantly deeper into the home but for a shorter duration than summer light. Winter light is whiter than summer light due to atmospheric quality, clarity, the absence of foliage colors, and reflection off snow. Winter shadows are blue.
By contrast, high-angled summer sun reaches minimally beyond the windowsills, resulting in dark summer interiors. For this reason, "summer houses" are traditionally decorated in light colors, luring the light inward. Light-colored surfaces in rooms on the sunny side of the house will bounce sunlight into the back of rooms. Dark furnishings will stop this flow of light, absorbing it and effectively punching a hole in the sunlight. The colors of summer foliage plus the increased moisture in the air bring varied tonal complexities to summertime sunlight.
Spring and fall have their own particular kind of radiance, with intermediate solar intensity and penetration into the rooms. These two seasons particularly lend their color to interior brilliance.
**DECORATING WITH THE SUN:**
**WHERE TO BEGIN**
Maximize your decorating dollars by choosing sun-resistant, light-neutral colors for expensive features. For example, make sure that expensive rugs, upholstery, and wall coverings are warrantied to resist fading. Certain fabrics and rugs are actually designed to mellow handsomely with time and exposure. Less expensive items, such as pillows, throw rugs, curtains, vases, and plants, can be changed with the season.
When looking for large-ticket decorating elements, get written details from the manufacturer on fabric and material stability. Olefin rug and upholstery fabrics, for instance, are stain-resistant and hold dyes well; however, they lose structural integrity with ultraviolet exposure. An olefin rug may continue to look the same color from a distance yet may have lost its pile into thin air after only a year's exposure. Certain traditional natural fibers, including wool and cotton, have been shown to withstand solar exposure very well. However, it is important to check with each manufacturer on dye processes. Silk often disintegrates with exposure. Cotton and linen may yellow unless treated.
Many furnishings, particularly dark-toned fabrics, are subject to uneven heating as the sun hits one side while the remainder is in shadow. Differential heating expands the fibers on the warm side, while the cool side remains normal. Glues and finishes are subjected to greater stresses, and wood dries out, shrinking as its moisture departs. Maintain wood pieces with a quality furniture oil. Check the joints and re-glue them when they become loose, as this will prevent breakage.
The character of reflected light is dependent upon surface texture. Shiny, highly polished surfaces such as glass, high-gloss paint, and bright urethanes reflect a high percentage of incident light (the angle of reflection of course equals the angle of incidence). These are hard-light reflections, carrying a lot of zap, and can be used for special effects.
Consider your choice of exterior surroundings such as decks, lawn furniture, flora, and ponds as part of the color scheme of the adjacent interiors. Light reflected off exterior colors and surfaces will tint the home's interior spaces.
**WINDOW DECOR**
Solar home windows are calculated to function as more than merely panes of glass. They are utilized as solar collectors, collecting light and heat to minimize the home's reliance upon conventional sources of backup fuel.
Windows of traditional homes are mainly decorative, and are often partially covered with curtains, draperies, or blinds. Functional solar-home windows require different treatment. Because these windows' surface area has been calculated into the total home energy dynamic, they must be viewed as heating and lighting "generators," which carry energy both inward and outward.
Thermo-Shutters (see chapter 4) are insulated, inward-folding shutters designed to block heat loss at night, or excessive daytime heat gain. These can be used as attractive decorative surfaces, adding angular interest to the window-area design. Whether formal or casual, Thermo-Shutter treatment will set the stage for further room decor. They can be curtained, mirrored, muraled, bulletin-boarded, wallpapered, painted, stained, or mounted with rugs for studio soundproofing.
Mounting draperies on the Thermo-Shutters will dampen interior-based sound, as would drapes placed across a window. The ThermoShutters' wooden construction makes the need for heavy linings unnecessary, saving drapery construction costs.
At times of intense sun, Thermo-Shutters can be partially closed to screen the sunlight, effectively bouncing heat and light outside while allowing cool breezes to enter. To compensate for this additional shading, lightweight, semi-transparent curtains mounted on the Thermo-Shutters will capture the light, diffusing it throughout the room. Hard-edged window openings cast strong shadows and crisp light. Translucent, fluffly curtains mounted on the Thermo-Shutters will diffuse harsh light, softening the glare. Remember that all home interiors appear darker in the summer than in the winter due to the high angle of summer sun.
Adjustable "Venetian" blinds present interesting possibilities. Blinds cut down on the percentage of solar heating provided, but can be very pleasant modulators when the sun is too harsh. They come in traditional horizontal slats, in very narrow vertical slats that reach from floor to ceiling, and in all sorts of other varieties. New tiny-slatted vertical blinds add a formal architectural dimension to light management. They are available in many materials, with numerous colors and textures to choose from. Blinds are built to alter the reflective angle, to bounce light away from areas where it is unwelcome, toward the outside or toward a part of the room where accents of light can be useful. Their slats are easily adjustable, and the flexibility they provide can be quite attractive for someone who enjoys fine-tuning the ambient light.
**SUPPLEMENTAL LIGHTING**
Fewer sources of purchased lighting are needed in a home that utilizes sunlight effectively. The daytime use of living spaces should be planned to be in phase with solar incidence. Nighttime lighting will be relatively economical if the effects of light in various contexts has been considered and light-colored materials have been used as backgrounds.
Think of light as an architectural and a sculptural medium. Judicious lighting creates a stage set that will highlight special areas of activity. Work centers will appear as focal points, bright and inviting when juxtaposed with a more subdued hallway. Light expresses and concentrates function: with the right lighting, a reading niche becomes a work of art; the well-lit center of the dining room table works well and is inviting. Space will appear to ebb and flow through the "movement" of light. In fact, the relationships of the home's various spaces can be persuasively determined by patterns of lighting. Pools of light serve as paths to guide the eye and the feet.
Project areas need a wash of bright, non-glaring light. Surrounding walls painted light matte colors will carry a soft, bright glow. This softer brightness will help to prevent distracting shadows. High-gloss paint has stronger contrast between light and shadow.
In some situations, the comfortable "warm" glow of incandescent lighting is hard to beat. On the other hand, in an energyconsicous household you may choose fluorescent lamps for their vastly superior energy-efficiency and longevity. Fluorescent light was originally designed for indirect lighting of large spaces. Used directly, it can sometimes be hard on the eyes. Contemporary fluorescents are now available in full-spectrum band waves as well as several other "colors" that are less objectionable than the lurid or chilly originals. Be aware that fluorescent light can fade fabrics over time if it shines directly on them.
Recessed lights have both artistic and practical potential. The recesses are not difficult to incorporate if this is done while framing the house. It is worthwhile to spend time during the planning stage of your home design to consider all locations where you might want the option of recessed lighting. It is better to build in too many than too few.
Track lighting, originally designed for theaters, uses moveable and removeable fixtures that are available in a wide variety of commercial designs. These can be swiveled, switched around, and reoriented for different effects. The mounting strips into which the fixtures plug are installed on ceilings or walls. Because they are so versatile, they can bring light to the most challenging locations. The track-to-fixture connection is often proprietary to individual manufacturers, so when selecting track lighting, be sure to pick a manufacturer that seems likely to be around for a while. There's a better chance that they will still be around if you choose to add or replace fixtures in the future.
Lighting placed above beams or shelves and directed toward the ceiling can dramatically enhance a room. Incandescent light reflected off the golden tones of wooden beams gives an atmosphere quite different from that of fluorescent light bounced off a white ceiling. Light aimed upward toward a light-colored surface makes a room appear larger. A white ceiling seems to float at a distance, while a darker one appears lower.
Long, dark, northern winters can be made more pleasant with full-spectrum electric lighting, which unlike conventional lamps puts out a more complete range of bands in light, simulating the richness of daylight. As psychologists have published their research on light-deficit disorders, the market has begun to respond, and several choices of full-spectrum are now available ("Ott lights" were the originals). Full-spectrum lighting might well be worth the investment for both home and workplace. The psychological lift these lights provide could result in increased productivity, health, and feelings of happiness. Full-spectrum lights have been shown to help beat "cabin fever," a problem dignified by the term "seasonal affective disorder," or SAD.
Linear accent lighting is available in the form of tiny strands of "mini" lightbulbs encased in flexible clear tubing. These use very little wattage, yet the filaments are so tiny that the incandescence is very white. Some people use tiny white Christmas lights all year as accents. It might be fun to look into the possibilities LED lights present.
If you are not familiar with the full diversity of options now available in lighting (and there are an amazing number of products to choose from), you might decide to hire a lighting designer during the planning stages of your house design. An expert can help you sort through your ideas and preferences, and will introduce you to new products that are constantly coming onto the market.
**FABRICS, RUGS, AND**
**WALLCOVERINGS**
Fabrics, rugs, and wallcoverings that reflect ultraviolet best are least likely to be harmed by it.
Ultraviolet rays and direct fluorescent lighting will fade many fabric dyes and paints while darkening varnish. Color-resistance to ultraviolet is dependent on the chemistry and technique of the dyeing process. One wallpaper manufacturer's representative expressed the color stability problem this way: "You're safer staying away from oranges, greens, and purples. And _anyone_ knows enough to stay away from reds, yellows, and blues, of course." However, a number of fade-resistant dyes and dye/fabric combinations have been found to be remarkably stable. When examining written warranties, be sure to look specifically for guarantees of stability with exposure to sunlight.
Just as with paint, natural or earth-toned and light fabric and wallpaper tints have less pigment to fade, and thus show the fading process far less than do more color-saturated hues. Certain strong colors that usually fade quickly have proven more durable when used with special dyeing techniques in such tested materials as Dupont's Antron III. It is a good idea to look for warranties from the company that actually manufactures and dyes the fabrics or other materials, and not just to go by the type of material used. For instance, many companies use Dupont fabrics, but not all use the same dyeing and fabrication methods. Make sure the manufacturer stands behind its products with a warrantee.
**Floor Coverings**
The Solar Slab described in this book may be covered with any type of floor covering. Technically, a slab that is painted black would absorb the most heat; but the negligible improvement in performance certainly does not justify living on a concrete surface painted black. Most homes use a thick pad and carpet. Other homes have used wood parquet. Wide wooden boards may be used by gun-nailing two layers of 1 x 3-inch strapping onto the slab surface, and then attaching the boards by screwing into the strapping. The screws can then be hidden with ship's "bungs" or wood plugs. The result is a very attractive wide-board floor that has resiliency due to the spacing of the strapping.
A rug can set the theme and color scheme for a room, and should be chosen with care. Rugs constructed of top-quality wools, including fine orientals and fine American Indian rugs, constantly undergo a process of modification, softening, blending colors. It has been said that these rugs are artworks in process. Wools from mountain regions are known to be most durable, while those from the plains are softer and structurally weaker. Pile weight of wool rugs determines not only ruggedness but also the rug's insulating properties. Use good padding as underlayment, rotate rugs to equalize exposure to light and traffic, and vacuum frequently to increase longevity.
For wall-to-wall applications commercially available carpeting can make an effective covering for the Solar Slab described in this book. Strong colors can be disappointing as they tend to fade. Light colors bring the incident light to all corners of the room, making spaces seem larger.
Look for brand names that are recommended for areas of hard wear. There are more product names than products, so it is important to find the generic base. Be sure to check the warranties for ultraviolet resistance as well as fade resistance and structural integrity.
Fine quality wall-to-wall commercial wool carpeting is usually more expensive, but is also more sumptuous and is more resistant to ultraviolet structural degradation. It is offered in piles, sculptured rugs, and Berbers. When made of natural-colored wool it is highly resistant to fading. The tactile warmth and long life of wool, as well as its soundproofing capacity and wearability, are hard to surpass.
The following is a partial listing of particular rug types, with comments:
_Tunisian Mergoumes:_ Thin, but very durable rugs, these are unobtrusive, thus they go with almost anything.
_Moroccans:_ From the Atlas Mountains, Moroccans are made of very good wool, thick-tufted with a high pile. They are generally custom designed.
_Kilims:_ Thin, but durable, Kilims are available in a wide variety of colors and patterns.
_Spanish:_ In colors that are rich without being harsh, often striated throughout pattern for pleasant toning with age, Spanish rugs tend to be made of good wool, and are available in a variety of sizes and patterns.
_Orientals:_ In selecting a Persian rug for a sunny location, consider the softening of color values that will probably enhance the tonal quality with age. (Traditional orientals have been washed and dried in the strong near-eastern sun as part of their curing process.)
_American Indian:_ Native Americans make woven, non-pile rugs in a wide range of density and patterns. Natural wool colors and natural dyes, though often preferred, are not always more stable than synthetic dyes. Be careful not to confuse these with Mexican or other imports, which are usually substantially inferior in quality.
_Others:_ Thick cut Chinese, Indian, Pakistani, and Bulgarian rugs are usually acid-washed, and therefore weakened, but with careful selection, some can be quite good. Be sure to select thick, dark-colored pieces.
**Fabrics**
When choosing fabrics for use in a solar home, remember that olefin fabrics have a tendency to degenerate over time, an effect familiar in yellow olefin water-ski ropes. By contrast, dacron was developed by Dupont to be resistant to sunlight, a quality that is coupled with dimensional stability, making dacron based products worth investigating.
The marine line of Uniroyal's Naugahyde brand of fabrics was developed to stand up to intense solar exposure, and is mildew resistant as well. "Ranchero" is a breathable Naugahyde with the velvety look and feel of suede. Their "Bahamas" fabric has a rich leathery texture.
Genuine leather will need extra attention if it will be exposed to sunlight. Check with the manufacturer on its care. For supreme sun resistance look for Sunbrella.
**Wall Coverings**
Wall-coverings add character and definition to interior spaces. The trick is to follow the basic color scheme of the room and to keep the wall treatment visually on the wall. A glorious pattern in vibrant hues will probably jump out and hide artwork, making a room appear smaller. It is challenging to keep the whole room design in mind when picking a wall treatment, yet so much more can be added to a room's appeal by using appropriate paint and wallpapers. Angled wood on one wall, complimentary painted hues, book shelves backed with an accent color, and the lush mono-tonal textured papers that carry the softness of certain rugs higher into the living space . . . Enjoy yourself as you plan.
Wallpapers will be subject to fading in a solar home. Wallpapers made by Albert Van Luit & Co. have been tested for years in the intense sunlight of Southern California. Other manufacturers are following suit. Mylar-based papers have proven dimensionally stable, but be aware of the effects of metallic reflections, which can be very dark or very brilliant depending on light exposure.
Home interiors evolve as people live in them. Don't hurry to fill up the space. If furnishings are put somewhere "for the time being," they usually remain there for a long time. It is better to wait until the right solution presents itself.
**A WORD ABOUT PLANTS**
Due to the increased amount of sun in a solar home, the usual house plants may need to be placed in rooms with less exposure. Geraniums, plants that love heat and dryness, orange trees, hibiscus, herbs, catnip, and cacti will all thrive in the warm southern exposures, whereas Christmas cacti, oxalis, begonias, African violets, and gloxinias are not really contented unless removed from direct exposure. On the other hand, seedlings are delighted to germinate in a sunny spot, as long as their soil is kept moist.
Check with a nursery or florist if you have questions about which plants are best for your region.
The natural solar house is not hard to decorate, but it is important to know the variables that over time will effect your choice of materials and solutions. Whatever choices you make, your maintenance costs should be low and your level of satisfaction great. Pop an apple pie in the oven, turn on the music, and sit back and enjoy the total experience of your home.
Solar Principle # 10
_Remember that the principles of solar design are compatible with diverse styles of_ _architecture and building techniques._
Solar homes need not look experimental or futuristic, nor do they require complicated, expensive, and hard-to-maintain gadgetry to function well and be comfortable year-round. In solar design, good planning and sensitivity to the surrounding environment are worth far more than special technologies or equipment.
THREE PROJECTS
During the 1970s many solar designs came and went. Usually when a design failed to work, the designer and/or design was faulted. It is always a concern to a designer that a design will be misused and fail, causing the demise of the idea. When I was operating Green Mountain Homes, I kept close control over my design to ensure its success. In the post-Green Mountain Homes years I realized that what we had learned about solar was going by the wayside; thus, the idea of the book. The fear was that the book would spawn unsupervised projects that could be improperly constructed and fail. This could potentially lead to the demise of the idea and the design. I'm very happy to report that just the opposite has occurred. The design has been embraced by solar enthusiasts in many countries and the book has spawned an untold number of solar projects. A recent related Google search yielded 36,300 "hits" or links.
The following three projects have been selected because they happened as a direct result of the book first being published in May 1997. The projects were selected because of their diverse locations and climates. The first two are new homes and the third is a retrofit.
**COLORADO**
The builder/owner writes:
" _Temperature swings were very_ _slow—_ _up or down. Heat_ _distribution excellent. I did your recommended 2x4 wall with_ _1"_ _of_ _foam—_ ** _very_** _tight. . . . Did use_ _1–_ _2 cords of firewood_ _as_ _backup—_ _no furnace in the house. First time we left during_ _Thanksgiving cold snap. 0 degrees for a week. It was 62 degrees_ _upon return. Used thermal window coverings at night. Very_ _happy with_ _performance."_
M. F. Woodland, Colorado
**NORTH CAROLINA**
The owner writes:
_"_ _After reading your book we had the good fortune to have your_ _help to plan our new home. Eventually we found an architect and_ _he helped find a site on our 34 wooded acres. Our house would_ _not follow all your good ideas and concepts, but we did all we_ _could. The house is a two story built into a southern facing slope._ _. . .We have six clearstory awning windows facing south between_ _the two gables; that lets us keep the lights off in the house during_ _the day, and we have a skylight in the interior bath for the_ _same purpose. . . . I admire and appreciate your work. We love_ _our home and you are to be commended for trying to make us all_ _more energy_ _conscious."_
C. E., Ashville, N.C.
**CANADIAN RETROFIT**
As you can see, this conventional home was aligned with the street and faced north. After reading _The Passive Solar House_ , the owners of this home designed and constructed a solar addition to their under utilized south side of their home. They also constructed the addition themselves. Here's what they write:
_"_ _As I reflect on our passive solar addition with Solar Slab, I_ _am sitting up on the integrated loft on a sunny April day and_ _it is 22 degrees centigrade (72 degrees Fahrenheit) in the room_ _with no fire in the backup woodstove for a few weeks. It went_ _down to -2 degrees centigrade (28 degrees Fahrenheit) last_ _night (outside) and then up to about 15 degrees centigrade (59_ _degrees Fahrenheit) today. The main house just has the furnace_ _fan circulating air and it is 19.6 degrees centigrade (67 degrees_ _Fahrenheit). The Solar Slab under the sunroom is 19 degrees_ _centigrade (66 degrees Fahrenheit)._
_I saw the book title for_ _Jim's_ _book in 1999 and we felt that_ The Passive Solar House _made sense. I had over the years_ _heard of rock storage, Trombe walls,_ _Glauber's_ _salts, yet existing,_ _easily obtained thermal mass in the form of concrete blocks with_ _ready made channels in the block structure forming_ _'_ _ductwork'_ _is_ _appropriate technology._
_Thank you, Jim, for sharing your_ _discoveries."_
B. & D. M, Beaverton, Ontario
_Addition interior: South wall showing air vents_
12
[USING THE CSOL
COMPUTER PROGRAM](chap6_passives_9781603581691_epub_part6.html#d411510515)
The computer program contained on the CD was authored by James Kachadorian and revised and edited by J.R. Arscott, of the United Kingdom. The program is Windows based and is set up for Canada and Northern United States. There are provisions for inserting data from other locations.
Before using the program for your own project, make a test run using the example in Chapter 8. Check to make sure your results match the book results before proceeding.
The following instructions will assist you in the operation of the program:
1. After inserting the CD, double-click on "csol_setup." Click through the license agreement, click "Start," click "OK," drag and place the "CSol" shortcut on your desktop screen. Click on the "CSol" shortcut to run the program or click "Run" when the first screen appears.
2. The first window does not affect the operation of the program but identifies the run. The information entered in this window will appear on the final printout.
3. The next window asks for siting information. Click the appropriate orientation and click "OK".
4. The next window asks for U factors, Shade Coefficient, Clearness Number and Air Changes per Hour. Use the U Factor and Shade Coefficient (SC) specified by the window manufacturer. See the Chapter 6 Shade Coefficient discussion. The _Lower Living_ _Space Concrete Wall U Factor_ applies only for a sidehill design. Leave the default value of 0.0456 for non sidehill designs. Find the clearness number for your location. U Factors are expressed in Btu/hr x ft(2) x dF. If there are no changes click "OK", or change the appropriate value to suit your design.
5. The next window asks if your design is a sidehill. If you click yes, the next window will asks for the square feet of heated living space in the lower concrete wall.
6. The next window asks for more data. The "Gross Square Feet of Framed Walls" is the total area of framed walls, which assumes studded construction. Enter the equivalent area of your walls in square feet. The "Roof/Ceiling Area" is the flat insulated ceiling in a non-cathedral house or the area under the insulated roof line in a cathedral house in square feet. House volume in cubic feet is the last entry for this window. After you have entered all the data for your design, click "OK."
7. The "Glass Dimension" entry should be the glass manufacturer's published glass areas in square feet.
8. The next window asks for "Square Feet of Thermo-Shutters." You should enter the square feet of whatever you are using for window and patio door nighttime insulation.
9. The location window offers 67 locations in Canada and the United States. Pick the closest location to you. If there is no location listed near you, select "None of These Locations." If this selection is made, the next series of windows will ask you to input degree-day data and percentage of sunshine for your location.
10. The next window requests the closest north latitude to you. If none apply click "None of the Above" and the next two windows will ask for Half Day Solar Heat Gain Factors (See Appendix 2) for East and South. Note that there is no need to enter West as East and West are the same.
11. Outside winter design temperature is next. Select the appropriate value from Appendix 4.This temperature must be entered as degrees Fahrenheit. Be sure to include the minus sign if this is a minus value, i.e., minus 20 degrees Fahrenheit is entered as -20.
12. Highlight the backup heat choice and click "Select." Repeat if desired and click "Finish."
13. Next enter the outside Solar Slab dimensions in feet and select the average winter temperature in degrees Fahrenheit from Appendix 5.
14. Congratulations, you're finished.
15. The printout will contain your results along with predictions for fuel usage. Examine the design review and adjust your design accordingly. The slab thickness value will normally be between 3 and 10 inches. Your slab should be reinforced as shown in the book details and should be structurally sound. If you get a readout of a slab thickness that is less than your local codes, use the code-dictated minimum thickness, usually 3" to 4". If you get a minus slab thickness or a number less than 3", it means that your house is underglazed. That is, there is not enough insolation to require mass to offset and store the heat gain. If you get an excessively large slab thickness, your house is overglazed. Try to rework your design to get the percent of east plus west plus south glazing to be between 10 and 20 percent of the heated wall area. See Worksheet 3, Chapter 8.
The privacy of all Green Mountain Solar Home owners has always been respected, but you will be able to take a virtual tour of a sampling of solar homes designed by the author. All of these homes utilize the solar heating system described in The Passive Solar House. Backup heating systems range from sophisticated air-to-air heating and cooling systems and oil- and gas-fired warm air systems to wood stoves supplemented by simple electric furnaces. All of these homes have an air mover to circulate and filter the air. Most commonly, the air filter is the ordinary air filter contained within the warm air backup heat system. See the control diagrams in chapter 7 for more detail.
The CD also includes photographs showing the construction sequence of the author's solar home.
Again, good luck with your project.
APPENDIX 1
Solar Design Worksheets
As discussed in the preface, those readers that are technically proficient may wish to utilize the information given in this book and proceed on to design a solar home. All the "tools" needed are in the appendices. Permission is granted by the author to photocopy the worksheets in Appendix #1 as you will need multiple copies to perform trial "runs" for any solar design attempted. The permission to copy is only extended to Appendix #1.
APPENDIX 2
[Solar Intensity and
Solar Heat Gain Factors for
16 to 64 degrees North Latitude](chap6_passives_9781603581691_epub_part6.html#d48554)
Reprinted with permission of the American Society of Heating, Refrigeration and Air-Conditioning Engineers from the 1993 ASHRAE _Handbook of Fundamentals,_ Tables 12–18.
APPENDIX 3
[Thermal Properties of Typical
Building and Insulating Materials](chap6_passives_9781603581691_epub_part6.html#d48557)
(Design Values)
Selected from American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE), 1993 _Handbook of Fundamentals,_ Table 4.
APPENDIX 4
[North Latitude, Elevation,
and Outside Winter Design
Temperature for Selected Cities in the
U.S. and Canada](chap6_passives_9781603581691_epub_part6.html#d485510)
Adapted and reprinted from the _Cooling and Heating Manual,_ U. S. Department of Housing and Urban Development Office of Policy Development and Research.
APPENDIX 5
[Average Monthly and Yearly
Degree Days for Cities
in the U.S. and Canada](chap6_passives_9781603581691_epub_part6.html#d485512)
Reprinted with permission of the American Society of Heating, Refrigerating and Air-Conditioning Engineers from the 1981 ASHRAE _Handbook of Fundamentals._
APPENDIX 6
[Mean Percentage of Possible
Sunshine for Selected Cities
in the U.S. and Canada](chap6_passives_9781603581691_epub_part6.html#d485515)
Based on period of record through December 1959, except in a few instances. These charts and tabulation are derived from the "Normals, Means, and Extremes" table in U.S. Weather Bureau publication _Local_ _Climatological Data._
APPENDIX 7
Isogonic Chart
(Magnetic Declination)
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| 1,071
|
Smurfs 2 (CD)
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In this sequel to Columbia Pictures/Sony Pictures Animation's hybrid live action/animated family blockbuster comedy The Smurfs, the evil wizard Gargamel creates a couple of mischievous Smurf-like creatures called the Naughties that he hopes will let him harness the all-powerful, magical Smurf-essence. But when he discovers that only a real Smurf can give him what he wants, and only a secret spell that Smurfette knows can turn the Naughties into real Smurfs, Gargamel kidnaps Smurfette and brings her to Paris, where he has been winning the adoration of millions as the world's greatest sorcerer. It's up to Papa, Clumsy, Grouchy, and Vanity to return to our time, reunite with their human friends Patrick and Grace Winslow, and rescue her! Will Smurfette, who has always felt different from the other Smurfs, find a new connection with the Naughties Vexy and Hackus – or will the Smurfs convince her that their love for her is True Blue? Returning cast includes Neil Patrick Harris, Jayma Mays, Sofia Vergara, with Katy Perry as Smurfette and Hank Azaria as Gargamel. Brendan Gleeson joins the cast as Victor. Joining the voice cast are Christina Ricci and JB Smoove as Vexy and Hackus.
Heitor Pereira returns to the franchise with a colorful score for orchestra and choir.
1. Smurfette's Creation (1:23)
2. Smurfette Are You OK (1:00)
3. You Belong To Gargamel (:37)
4. Gargamel Suite (1:44)
5. Azrael's Trap (:50)
6. Code Blue (1:25)
7. Victor's Corndogs (1:33)
8. We Must Review My Plan (1:22)
9. Adoring Public Desires Me (:40)
10. Smurf Portation Crystals (2:01)
11. Attack On Winslow House (1:36)
12. Madame Doolittle (:50)
13. Paris Opera House (:34)
14. Scoping Out The Kitchen (1:00)
15. Smurfette Escapes (:51)
16. Hand Over The Smurfette (1:29)
17. Portrait Of Perfection (1:52)
18. Smurfette On The Run (:57)
19. Gargamel And Azrael In Carriage (1:07)
20. Naughties Crash The Cart (1:03)
21. Naughties Take Flight (:31)
22. He's Not My Father (2:04)
23. The Napoleon Suite (1:15)
24. Like Twins (:38)
25. Tiny Magical Wand (2:08)
26. The Flying V (:29)
27. Papa To Papa (1:49)
28. Let's Get Smurfin (1:02)
29. They Cannot Live (1:14)
30. The Formula (:46)
31. Naughties Transformation (1:11)
32. You Sacrificed Everything (:44)
33. The Happiest Moment Of My Life (:50)
34. Papa And Vanity Find Smurfette (:45)
35. Harnessing The Power (:28)
36. Life Is The Most Precious (1:20)
37. I Don't Think So Gargamel (:32)
38. Essence In Paris (1:00)
39. Is This What Happy Feels Like (1:34)
40. No Smurf Left Behind (1:50)
41. Welcome Home Smurfette (1:01)
|
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| 9,034
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Les Illes Farilhões són un grup de petites illes de l'arxipèlag de Berlengas a l'Oceà Atlàntic, pertanyents a Portugal. Tenen una superfície aproximada de 7 hectàrees (equivalents a 0,07 km²).
També conegut com a Farilhões-Forcadas, aquest grup d'illots es troben a una distància d'uns 3,5 quilòmetres del far de Berlengas en l'extrem nord del grup de les Berlengas, gairebé aïllades de la resta d'illes.
Administrativament depenen del municipi de Peniche, part del Districte de Leiria. En el seu punt més aconsegueix una altura de 94 metres.
Illes integrants
Grande Farilhão
Farilhão del noroeste
Farilhão da Cova
Referències
Enllaços externs
Mapa de les illes en Wikimapia
Illes de Portugal
Arxipèlags de l'oceà Atlàntic
Peniche
|
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| 7,658
|
Q: ¿como hacer que ireport genere nada mas una tabla? el problemas de que mi ireport e que me genera 4 o 5 tablas y a la hora de que se llena con un jQuery se me agranda hasta 15 paginas
este es el QUERY que utilizo junto con un parámetro que toma el valor de otro parámetro que se llena de otro query pero este si me funciona y es solo 1
SELECT idRegistroCarnet,nombresw,versionSw FROM repswequip INNER JOIN registrocarnet USING (idRegistroCarnet) INNER JOIN software USING (idSoftware)where idRegistroCarnet = $P{idDataset}
y aquí esta el JavaScript de la tabla por si acaso
<detail>
<band height="117" splitType="Stretch">
<componentElement>
<reportElement key="table" style="table" x="105" y="16" width="360" height="80" uuid="89865b72-cb15-44cf-812a-f76b07f707e1"/>
<jr:table xmlns:jr="http://jasperreports.sourceforge.net/jasperreports/components" xsi:schemaLocation="http://jasperreports.sourceforge.net/jasperreports/components http://jasperreports.sourceforge.net/xsd/components.xsd">
<datasetRun subDataset="reportesw" uuid="a535535a-ea28-4760-a65b-424b9daeb02a">
<datasetParameter name="idDataset">
<datasetParameterExpression><![CDATA[$P{idRegistroCarnett}]]></datasetParameterExpression>
</datasetParameter>
<connectionExpression><![CDATA[$P{REPORT_CONNECTION}]]></connectionExpression>
</datasetRun>
<jr:column width="90" uuid="bd859ea7-af56-4aba-8251-e410ca85a4b4">
<jr:columnHeader style="table_CH" height="30" rowSpan="1">
<staticText>
<reportElement x="0" y="0" width="90" height="30" uuid="a51ded26-d90b-46a3-98d7-438ae241e8f5"/>
<textElement textAlignment="Center">
<font fontName="Comic Sans MS" size="12" isBold="true"/>
</textElement>
<text><![CDATA[Registro Carnet]]></text>
</staticText>
</jr:columnHeader>
<jr:columnFooter style="table_CH" height="30" rowSpan="1"/>
<jr:detailCell style="table_TD" height="20" rowSpan="1">
<textField>
<reportElement x="0" y="0" width="90" height="20" uuid="c300bd78-bd39-4ea8-a6cb-b6bfd86f62aa"/>
<textElement>
<font fontName="Arial" size="12"/>
</textElement>
<textFieldExpression><![CDATA[$F{idRegistroCarnet}]]></textFieldExpression>
</textField>
</jr:detailCell>
</jr:column>
<jr:column width="133" uuid="e78df592-6be0-4af7-af1f-5015b866dc3b">
<jr:columnHeader style="table_CH" height="30" rowSpan="1">
<staticText>
<reportElement x="0" y="0" width="133" height="30" uuid="069eb518-707c-4aa2-9206-caf1a776c2a9"/>
<textElement textAlignment="Center">
<font fontName="Comic Sans MS" size="12" isBold="true"/>
</textElement>
<text><![CDATA[Software]]></text>
</staticText>
</jr:columnHeader>
<jr:columnFooter style="table_CH" height="30" rowSpan="1"/>
<jr:detailCell style="table_TD" height="20" rowSpan="1">
<textField>
<reportElement x="0" y="0" width="133" height="20" uuid="da54520a-97f0-4739-a3c8-65228cf3cb9d"/>
<textElement>
<font fontName="Arial" size="12"/>
</textElement>
<textFieldExpression><![CDATA[$F{nombresw}]]></textFieldExpression>
</textField>
</jr:detailCell>
</jr:column>
<jr:column width="135" uuid="09aec3f0-8c6d-414d-872c-42c5b706cbc1">
<jr:columnHeader style="table_CH" height="30" rowSpan="1">
<staticText>
<reportElement x="0" y="0" width="135" height="30" uuid="937eaff8-300e-4f24-a7a4-dfdcdbf3f7eb"/>
<textElement textAlignment="Center">
<font fontName="Comic Sans MS" size="12" isBold="true"/>
</textElement>
<text><![CDATA[Version]]></text>
</staticText>
</jr:columnHeader>
<jr:columnFooter style="table_CH" height="30" rowSpan="1"/>
<jr:detailCell style="table_TD" height="20" rowSpan="1">
<textField>
<reportElement x="0" y="0" width="135" height="20" uuid="245a1895-8b09-4451-a5f4-0ce6ef5f6ce7"/>
<textElement>
<font fontName="Arial" size="12"/>
</textElement>
<textFieldExpression><![CDATA[$F{versionSw}]]></textFieldExpression>
</textField>
</jr:detailCell>
</jr:column>
</jr:table>
</componentElement>
</band>
</detail>
como soy novato no entiendo muy bien este código, yo me he estado moviendo mas por el lado grafico del ireport en el que nada mas tengo una tabla pero aun asi me genera 4 o 5 tabla mas a la hora de compilarlo
|
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| 1,315
|
Q: How to swap values from two fields at the same time in SQL? I have the data in one column.
Ex:
dog
dog
cat
fox
How do I change the value of all dogs to fox and values of all fox to dog at the same time? If I run an UPDATE SET it will change all dogs to fox, then I run the second one it'll turn everything into dogs.
A: update
table
set
animal = case anmial when 'dog' then 'fox' else 'dog' end
where
animal in ('dog', 'fox')
A: If you have more than just dogs and foxes you could consider making a conversion table to help you with the update. That way you won't need a case statement.
declare @original table
(
pk int identity(1,1),
column1 varchar(max)
)
insert into @original
select 'fox'
union all
select 'dog'
declare @conversion table
(
valueFrom varchar(max)
,valueTo varchar(max)
)
insert into @conversion
select 'fox','dog'
union all
select 'dog','fox'
select * from @original
update original
set original.column1 = c.valueTo
from @original as original inner join @conversion as c
on original.column1 = c.valueFrom
select * from @original
A: Using the case statement as already answered is the "right way". However, you could have been a little creative. Assuming 'xxx' is never used ...
update table set animal = 'xxx' where animal = 'dog'
update table set animal = 'dog' where animal = 'cat'
update table set animal = 'cat' where animal = 'xxx'
There is more than 1 way to skin a cat.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 1,746
|
Q: How can I fix the footer to dynamically change with height of the grid sections as they increase? So I'm trying to make this page footer go to the bottom always. But see the image and look whats happening.
How can I fix that? I know the yellow thing is weird but that's where actually the links would go when the website is done. so consider this like "testing".
I just want the footer to stop becoming "less" than the body contents (divs)
I want it to always be at the bottom
Here is image: click here for image. Edit doc is the footer section basically
HTML:
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title></title>
<link rel="SHORTCUT ICON" HREF="images/logo.ico">
<link href="css/bootstrap.min.css" rel="stylesheet"> <!-- Bootstrap CSS -->
<link href="css/homepageStyle.css" rel="stylesheet">
<link href="css/viewDocsStyle.css" rel="stylesheet">
<script src="javascript/jquery.js"></script>
<script src="javascript/bootstrap.min.js"></script> <!-- Bootstrap JavaScript -->
</head>
<body>
<header>
<div class="container">
<a href=""><img class="logoImage" border="0" alt="logo" src="images/logo.gif"></a>
</div>
</header>
<div class="ContentSection1" id="blueStretched">
<div class="row">
<div class="col-sm-2" id="left">
Team Created Docs
</div>
<div class="col-sm-6" id="center">
Doc View
</div>
<div class="col-sm-2" id="right">
Team Chat
</div>
</div>
</div>
<footer>
<div class="container">
<div class="col-lg-12">
<ul class="list-inline">
<li>
<button class="btn" type="button" onclick="location.href='';">Edit Document</button>
</li>
</ul>
</div>
</div>
</footer>
</body>
</html>
CSS:
#left {
background-color: yellow;
left: 7.75%;
padding-bottom: 60%;
}
#center {
background-color: Lightblue;
left: 8%;
text-align: center;
}
#right {
background-color: gray;
left: 8.25%;
}
footer{
position: absolute;
right: 0;
bottom: 0;
left: 0;
padding: 1rem;
}
#blueStretched{
background-color: AliceBlue ;
height: 75%;
}
header {
padding: 20px 0;
background: url(../images/HPbg.jpg) no-repeat 100% 0%;
text-align: center;
}
footer {
padding: 25px 0;
background-color: #99CCCC;
text-align: center;
}
.logoImage {
position: absolute;
left: 47%;
top: 3.5%;
z-index: 1;
}
.ContentSection1 {
padding: 50px 0;
background-color: AliceBlue ;
}
body, html {
width: 100%;
height: 100%;
}
A: Your code seems to be correct. I believe it is because of a margin overlap. Try giving overflow: hidden to the footer:
footer {
position: absolute;
right: 0;
bottom: 0;
left: 0;
padding: 1rem;
overflow: hidden;
margin: 0;
}
A: Create a new css stylesheet then insert the code below.
#footer {
height: 60px;
background-color: #f5f5f5;
margin-top:50px;
padding-top:20px;
padding-bottom:20px;
}
Then insert your new stylesheet/custom below the bootstrap css. Then create a div and put an id of "footer".
<div class="footer">
Footer here
</div>
A: footer{padding:1rem;}
#blueStretched{background-color: AliceBlue ;}
You have to edit your footer and remove everything except the padding.
And you have to remove the height of your #blueStretched
A: A small change in your code. Try this...
footer{
position: relative;
right: 0;
bottom: 0;
left: 0;
padding: 1rem;
}
A: set up your footer and main content like this:
#footer {position: relative;
margin-top: -20%;
height: -20%;
clear:both;}
#blueStretched {
background-color: AliceBlue;
overflow:hidden;
padding-bottom: 0%;
}
See fiddle: http://jsfiddle.net/ajoLgL4o/24/
|
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"redpajama_set_name": "RedPajamaStackExchange"
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|
\section{Introduction}
The multipole response of nuclei far from the $\beta$-stability line
and the possible occurrence of exotic modes of excitation presents a
growing field of research. Besides being intrinsically interesting as
new structure phenomena, exotic modes of excitation might play an
important role in r-process nucleosynthesis \cite{gor04}. Low-lying
E1 strength has been measured in neutron-rich Oxygen, Neon, and Tin
isotopes \cite{gib08,lei01,adr05}, and more recently in $^{68}$Ni
\cite{wie09}. The interpretation of the dynamics of the observed
low-energy E1 strength in nuclei with a pronounced neutron excess is
currently under discussion (see Ref. \cite{paa07} for a recent
review). In light nuclei such as the Oxygen isotopes the onset of dipole
strength in the low-energy region is associated with non-resonant
independent single-particle excitations of loosely bound neutrons.
However, in the case of $^{68}$Ni the low-lying dipole strength is
found to be rather collective \cite{vre01}, and several theoretical analyses
have predicted the existence of the pygmy dipole resonance (PDR) in
medium-mass and heavy nuclei, i.e., the resonant oscillation of the
weakly-bound neutron skin against the isospin saturated proton-neutron
core. More recently it has been shown that the low-lying E1 strength
can exhibit a non trivial pattern, separated into two segments with
different isospin character. The more pronounced pygmy structure at
lower energy is composed of predominantly isoscalar states with
surface-peaked transition densities. At somewhat higher energy the
calculated E1 strength is primarily of isovector character, as
expected for the low-energy tail of the giant dipole resonance
\cite{paa09}.
Of course, not only the dipole, but also other multipoles could
exhibit pronounced low-energy strength in neutron-rich nuclei. A
theoretical study of the occurrence of pygmy quadrupole resonances
has recently been reported in the framework the quasiparticle-phonon
model \cite{TL.11}, and the monopole response in very exotic nuclei
such as $^{60}$Ca was investigated using the self-consistent
Hartree-Fock calculation plus the random phase approximation (RPA)
with Skyrme interactions \cite{sag97}. It was shown that near the
drip line the monopole response could develop a very pronounced
structure at low energy. The present study considers more realistic
examples of nuclei in which the low-energy monopole strength could
be measured in the near future. For the Ni isotopes, in particular,
calculations of the monopole response in $^{78}$Ni with the Gogny-RPA
did not predict pronounced low-lying strength \cite{per05}, as
evidenced from the percentage of the EWSR exhausted in the low-energy
region. In Ref. \cite{ter06} the monopole strength in Nickel isotopes
was studied using the Skyrme-QRPA approach with the SkM* functional.
It is, however, difficult to discern a clear pattern in the
low-energy region because of the large artificial smoothing factor.
Here we study in more detail the occurrence of low-lying isoscalar
monopole strength in neutron-rich Ni isotopes. In $^{68}$Ni
low-lying E1 strength has recently been measured \cite{wie09}.
A new technique has been developed
that enables measurement of monopole strength in unstable nuclei
\cite{mon08}. An experiment has very recently been performed at GANIL to
measure the monopole strength in $^{68}$Ni \cite{van10}, and the
analysis is in progress.
To minimize the model dependence of results, both Skyrme and relativistic energy density functionals are
employed in the present analysis. Skyrme Hartree-Fock + RPA calculations are performed for the nuclei
$^{68}$Ni and $^{78}$Ni, for which pairing effects are expected to play a negligible role in the monopole
response. The description of Skyrme HF+RPA approach
can be found in numerous references \cite{rin80,liu76,ber75} and will not be detailed here. We only
mention that in this work RPA equations are solved in configuration space, and the particle-hole space
is chosen so that the energy-weighted sum rule is fully exhausted.
The continuous part of the single-particle spectrum is discretized
by diagonalizing the HF Hamiltonian in a harmonic oscillator basis. An important quantity that
characterizes a given state $\nu = (E_\nu, LJ)$ is its transition density:
\begin{equation}\label{eq:tran}
\delta\rho^\nu({\bf r}) \equiv \langle \nu \vert \sum_i \delta
({\bf r - r_i})\vert \tilde 0 \rangle ~,
\end{equation}
with a corresponding definition of the neutron (proton) transition density
$\delta\rho_n^\nu$ ($\delta\rho_p^\nu$) when the summation in Eq.~(\ref{eq:tran}) is
restricted to neutrons (protons). A smoothing factor of 600 keV is used in plots
of strength distributions.
The monopole response of Ni isotopes is also analyzed using
the fully self-consistent relativistic
quasiparticle random phase approximation (RQRPA) based on the
Relativistic Hartree-Bogoliubov model (RHB) \cite{Paa.03}.
Details of the formalism can be found in Refs.~\cite{Paa.03,paa07}.
In the RHB+RQRPA model the effective interactions are implemented in
a fully consistent way. In the particle-hole channel effective Lagrangians
with density-dependent meson-nucleon couplings are employed~\cite{PNVR.04},
and pairing correlations are described by the pairing part of the finite-range
Gogny interaction. Both in the $ph$ and $pp$ channels, the same interactions are used
in the RHB equations that determine the canonical quasiparticle basis, and in the
matrix equations of the RQRPA.
\begin{figure}[tb]
\begin{center}
\scalebox{0.17}{\includegraphics{ni68l.eps}\includegraphics{ni68g.eps}}
\caption{Skyrme-RPA isoscalar monopole strength functions in
$^{68}$Ni (in non-energy weighted sum rule units), calculated with the SLy4 (a)
and SGII (b) energy density functionals}
\label{fig:68ni}
\end{center}
\end{figure}
In Fig. \ref{fig:68ni} we display the isoscalar monopole strength in
$^{68}$Ni, calculated with the SLy4 (left) and SGII (right)
non-relativistic functionals. The giant monopole resonance (GMR) is
calculated at about 20 MeV, and both functionals predict a pronounced
low-lying structure located between 13 and 16 MeV excitation energy.
For the response calculated with SLy4, the proton and neutron
transition densities of the three states that compose the low-energy
monopole structure are plotted in Fig. \ref{fig:dens}. These states
are not purely isoscalar: the transition densities exhibit neutron
dominated modes, and the proton and neutron densities are not in phase
in the interior of the nucleus. The configuration analysis of these
states shows that they correspond to almost pure single hole-particle
excitations. A single configuration contributes with more than 98\% to
the total strength: neutron (2p$_{3/2}$,3p$_{3/2}$),
(2p$_{1/2}$,3p$_{1/2}$), (1f$_{5/2}$,2f$_{5/2}$) for the states
located at about 13.6 MeV, 14.6 MeV and 15.6 MeV, respectively. In nuclei
that do not exhibit a pronounced neutron excess these
2$\hbar\omega$ unperturbed configurations are
located at higher energies,
whereas in this case they are found below the GMR. The reason is that, because of
the neutron excess in $^{68}$Ni, the 3p2f states are located close to
the Fermi level and, moreover, the 2p1f shell is calculated just below the Fermi
level. This is not the case for the 3s2d shell. The
2$\hbar\omega$ sd configurations that build the GMR are not lowered in energy and,
therefore, the giant resonance is located at higher energy.
\begin{figure}[tb]
\begin{center}
\scalebox{0.25}{\includegraphics{denst.eps}}
\scalebox{0.25}{\includegraphics{denst2.eps}}
\scalebox{0.25}{\includegraphics{denst3.eps}}
\caption{(Color online) Skyrme-RPA neutron (solid line) and proton
(dashed line) transition densities for the three low-lying isoscalar
monopole peaks at 13.6 MeV, 14.6 MeV and 15.6 MeV in
$^{68}$Ni, calculated with the SLy4 functional.}
\label{fig:dens}
\end{center}
\end{figure}
Fig. \ref{fig:ni68r} shows the RQPRA prediction for the isoscalar
monopole response in $^{68}$Ni, calculated with the relativistic
functional DD-ME2 \cite{LNVR.05}. Despite a small difference in the
predicted position of the GMR, the results for the low-lying part are
very similar to those obtained with the Skyrme functionals, and the
configuration analysis leads to the same conclusion about the
non-collective character of these low-lying excitations. The
comparison of the RQRPA spectrum with the unperturbed states shows
that the residual interaction affects the strength, but not the
location of the low-lying states. They are well separated from the GMR
which, in this relatively light nucleus, is found at high energy. The
corresponding proton and neutron transition densities are plotted in
Fig. \ref{fig:densr}. The transition densities for the low-lying
states at 12.16 MeV, 13.38 MeV, and 15.42 MeV (these states have the
same particle-hole structure as the corresponding ones in the
Skyrme-RPA calculation) correspond to almost pure neutron modes, and
the radial dependence is very different form the typical isoscalar GMR
transition densities exhibited by the collective state at 18.94 MeV.
The enhancement of the low-lying monopole strength in the RQRPA case
compared to the RHB one is due to a small increase of the magnitude of
the neutron transition density, the strength being a momentum of the
transition density. Also, a slight admixture of neutron and proton
configurations contribute to these states.
\begin{figure}[tb]
\begin{center}
\scalebox{0.30}{\includegraphics{is_monopole_Ni68_peaks.eps}}
\caption{(Color online) Monopole response of $^{68}$Ni calculated using the RHB+RQRPA model with
the DD-ME2 \cite{LNVR.05} functional. In addition to the RQRPA discrete spectrum and
the corresponding Lorentzian averaged curve, the unperturbed spectrum is also displayed.}
\label{fig:ni68r}
\end{center}
\end{figure}
\begin{figure}[tb]
\begin{center}
\scalebox{0.40}{\includegraphics{td_monopole_qrpa_rhb.ps}}
\caption{(Color online) $^{68}$Ni proton and neutron transition densities of the three low-lying RHB+RQRPA states
at 12.16 MeV, 13.38 MeV, and 15.42 MeV, compared to those of the GMR at 18.94 MeV.}
\label{fig:densr}
\end{center}
\end{figure}
Similar results are also obtained for $^{78}$Ni (Fig.
\ref{fig:ni78}), for which both Skyrme-RPA and RHB+RQRPA calculations
predict pronounced strength in the energy region around 15 MeV
excitation energy, well separated from the GMR. The low-lying strength
is more pronounced than in the case of $^{68}$Ni. The overall
structure predicted by the DD-ME2 functional is shifted to somewhat
lower energy compared to the one calculated with SLy4. The comparison
between the unperturbed spectrum and the full RQRPA response nicely
illustrates how the residual interaction builds the collective GMR
from the states above 15 MeV excitation energy. The two low-lying
unperturbed states are not lowered in energy, even though their
strength is considerably increased by the inclusion of the residual
interaction in the full relativistic RPA. To illustrate the evolution
of low-energy monopole strength in Ni isotopes, in
Fig.~\ref{fig:ni60-78} we show the monopole response of even-A
$^{60-78}$Ni isotopes calculated with the RHB+RQRPA. One can clearly
follow how the low-lying strength in the energy region between 10 MeV
and 15 MeV develops with the neutron excess. Since these states,
predicted both by Skyrme-RPA and RQRPA calculations, are
non-collective, their occurrence essentially depends on the position
of the neutron Fermi level in the single-neutron spectrum. The RPA
strength contained in these states, however, is markedly enhanced by
the residual interaction. These predictions point to a possibly very
interesting measurement of low-lying isoscalar monopole states in
neutron-rich Ni isotopes, that would probe almost pure single-neutron
configurations. A resolution of approximately 2 MeV can currently be achieved by
the experimental setup dedicated to the measurement of the monopole response in
unstable nuclei \cite{mon08}. An improvement to a 1 MeV energy resolution is
expected \cite{van10}, and the developpement of next generation active
targets such as ACTAR \cite{raa09} will proceed along this path.
It should be noted, however, that the physical width of the monopole response
(the spreading width, the decay width, and the Landau damping) could also
prevent a clear separation between different low-energy states.
\begin{figure}[tb]
\begin{center}
\scalebox{0.17}{\includegraphics{ni78l.eps}
\includegraphics{is_monopole_Ni78_peaks.eps}}
\caption{(Color online) Monopole response of $^{78}$Ni calculated using the Skyrme-RPA model with the
SLy4 functional (in non-energy weighted sum rule units) (a), and the
RHB+RQRPA model with the DD-ME2 functional (b).
}
\label{fig:ni78}
\end{center}
\end{figure}
\begin{figure}[tb]
\begin{center}
\scalebox{0.35}{\includegraphics[angle=-90]{monopole_Ni60_Ni78.ps}}
\caption{Monopole response of even-A $^{60-78}$Ni isotopes calculated using
the RHB+RQRPA model with the DD-ME2 functional. }
\label{fig:ni60-78}
\end{center}
\end{figure}
In conclusion, fully self-consistent microscopic Skyrme HF+RPA and
relativistic RHB+RQRPA calculations have been performed for the
isoscalar monopole response in neutron-rich Ni isotopes, some of
which should become experimentally accessible in the near future. Both the
non-relativistic and relativistic models, based on Skyrme functionals
and the relativistic functional DD-ME2, respectively, predict the
occurrence of pronounced low-energy monopole states in the energy
region between 10 MeV and 15 MeV, well separated from the isoscalar
GMR. From the analysis of transition densities, transition matrix
elements and the particle-hole configurations that correspond to these
states, it is evident that the low-energy monopole states represent
almost pure neutron single hole-particle excitations. Their occurrence
can be related to the closeness of the corresponding orbitals to the
neutron Fermi level in nuclei with large neutron excess. Even though
their location is not modified with respect to the corresponding
unperturbed states, their (Q)RPA strength is considerably enhanced by
the residual interaction. The theoretical analysis predicts an
increase of low-energy monopole strength with neutron excess. A
measurement of these states would provide a direct information on the
2$\hbar\omega$ gap, and directly probe the single-particle spectrum in
exotic neutron-rich nuclei, that is known to be driven by the
spin-orbit and the tensor terms of the effective inter-nucleon
interaction. In particular, measurements of the monopole strength in
neutron-rich Nickel isotopes are currently being performed, and this
work will help to interpret the results.
\section*{Acknowledgement}
This work has been supported in part by the ANR NExEN grant, by
MZOS - project 1191005-1010, and by the Croatian Science Foundation.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 349
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CONTENTS
COVER
LIST OF RECIPES
ABOUT THE BOOK
ABOUT THE AUTHORS
TITLE PAGE
INTRODUCTIONS
DUMPLINGS AND APPETISERS
Guangdong and Cantonese cooking
SOUPS AND SALADS
Ken's Hong Kong: my second home
RICE AND NOODLES
Fujian and Taiwan: Ching's homecoming
VEGETARIAN
Yunnan and the Chinese minorities
FISH AND SEAFOOD
The Chinese heartland of Sichuan
MEAT
Kashgar and the Silk Road
POULTRY
Beijing: capital cuisine
DESSERTS
Chongqing and the creation of megacities
GLOSSARY
ACKNOWLEDGEMENTS
COPYRIGHT
LIST OF RECIPES
Ants Climbing Trees
Aubergine with Sesame Sauce
Bang-Bang Chicken
Battered Shiitake and Okra with Citrus Five-Spice Salt
Braised Beef Shank in Sichuan Dressing
Braised Dofu with Crispy Pork
Braised Five-Spice Chicken Wings with Bamboo Shoots
Braised Lamb
Braised Savoury Tree Ears
Buddha's Mixed Stir-Fried Vegetables with Cashews
Cantonese-Style Roast Pi-Pa Chicken
Chef Yu Bo's Fish-Fragrant Prawns
Chengdu Leng Mian Noodles
Chengdu Wonton
Chilled Watermelon Dipped in a Lime and Raspberry Coulis
Chinese Pancakes
Ching's Banana-Wrapped Smoked Pork with Red and Black Rice
Ching's Beijing-Sichuan Zhajiang Noodles
Ching's Chicken and Green Tea Maocha
Ching's Longevity Noodles with Garlic, Sesame Oil, Salt and Soy Sauce
Ching's Mala Crispy Sichuan Sausage with Pickled Chillies and Wood Ear Mushrooms
Ching's Mapo Dofu Beef with Edamame Beans
Ching's Peking Duck
Ching's Pu-Er Tea-Leaf Omelette for Monks in Yunnan
Ching's Steamed Sea Bream with a Vietnamese Mint and Wild Coriander Salsa Verde
Ching's Vegetarian Mapo Dofu
Congee with Century Eggs or Salted Duck Eggs
Crossing-The-Bridge Noodles
Custard Tartlets from Guangzhou Restaurant
Da Dong's Shredded Duck with Sichuan Flavours
Da Dong-Inspired Peking Duck Salad
Dai Aunty River Fish Soup
Dan Dan Mian Spicy Sichuan Noodles
Dofu Casserole
Dongpo Pork
Fish in Wine Sauce
Fresh Soy Sugar Tomato Salad
Fried Fish Snack from Kashgar Market
Ginger and Soy Chilli-Steamed Scallops
Hainan Chicken Rice
Hunan-Style Crispy Chilli Beef with Dried Chillies and Peanuts
Hunan-Style Shredded Pork with Chillies and Black Beans on Crispy Noodles
Ji Shi Liang Mein Chicken Noodle Salad
Kashgar-Inspire Dsalad
Ken's Mapo Dofu Vegetarian Style
Ken's Peking Duck
La-Rou (Smoked Pork) with Chillies and Sweetcorn
Lionhead Meatballs
Liqun Roast Duck Restaurant's Stir-Fried Lettuce
Long Jing Chao Xia Stir-Fried Prawns with Green Tea
Lotus Root and Wood Ear Fungus in Soy Sesame Dressing Tossed in Finely Chopped Coriander
Lotus Root Salad
Mak's Wonton Noodle Soup
Mango, Lychee and Passion Fruit Salad with Star Anise Sugar
Mme Chen's Stir-Fried Shredded Potatoes
Mu Shu Vegetarian with Chinese Pancakes
Peach and Lychee Spring Rolls
Peking Duck Soup
Phoenix Tail Prawns
Pickled Cucumber with Chilli
Pickled Sichuan Carrots, Er Cai, Garlic Chives and Red Onion
Pineapple Rice
Red Bean Sesame Balls
Red-Cooked Butternut Squash
Savoury Steamed Egg Custard
Sea Bass, Baby Pak Choy and Coriander Soup
Shui-Jiao Pork and Prawn Chive Water Dumplings
Sichuan Fried Spicy Chicken with Green Pepper
Sichuan Sea Bass with Chives
Smoky Hot Scallops with Bamboo Shoots and Spring Onions
Spring Onion Pancakes
Steamed Cantonese-Style Chicken with Fermented Dofu
Steamed Scallops from Huangsha Seafood Market
Steamed Wined Chicken with Dried Mushrooms, Bamboo Shoots, and Goji Berries
Sticky Belly Pork Rice Wrapped in Lotus Leaves
Stir-Fried minced Duck in Fresh Lettuce Cups
Stir-Fried Beef with Sichuan Preserved Vegetables
Stir-Fried Bitter Melon with Black Bean Sauce
Stir-Fried Cabbage
Stir-Fried Corn and Chilli
Stir-Fried Crab with Ginger and Spring Onions
Stir-Fried Lamb with Leeks
Stir-Fried Pork with Fermented Dofu
Stir-Fried Spinach with Chilli-Fermented Dofu
Stir-Fried Swiss Chard
Stir-Fried Tomato with Egg
Stuffed Bitter Melon with Black Bean Sauce
Sui Mai Dumplings
Sweet and Sour Spareribs
Tea-Smoked Duck Sichuan-Style
Twice-Cooked Pork
Vanilla Sticky Cake with Cinnamon Sugar
Vegetarian Delight
Vegetarian Hot and Sour Soup
Xi Hu Cu Yu Sweet and Sour West Lake Steamed Carp
Xi Shi Dofu Soup
Yang Rou Bing Finely Shredded Mutton and Spring Onion Wheat Flour Pancake Parcels
Yi Jin's Dai-Inspire Drice Noodle Soup
Ying Tao Rou Cherry Pork
Yu Siang Shredded Chicken and Aubergine
Yunnan-Style Roast Duck
ABOUT THE BOOK
Ken Hom and Ching-He Huang team up for a once-in-a-lifetime culinary journey. Having shaped our understanding of Chinese cuisine in the UK, Ken and Ching set off through China to explore the food of their homeland, cooking with local families and top chefs as well as sharing ideas with each other and bringing back the recipes they loved most.
This is also a journey of personal discovery. Ken and Ching learn the secrets behind the country's top dumpling houses and try their hand at regional cuisines, including the legendary Crossing-the-Bridge Noodles and Twice-Cooked Pork. They discover the breadth of China's vegetarian dishes and explore food as medicine with healthy, balanced recipes. In Beijing, they learn new techniques from the most cutting-edge chefs, while in Sichuan they sample fish-fragrant flavours that are designed to perfectly complement seafood.
Ultimately, this is a story, told through food, of the old and the new. Ken and Ching's Chinese journey takes them across the country and back to the homes of their ancestors, reflecting on the flavours of their childhoods, meeting the most innovative cooks in the country and, above all, discovering the most delicious food in contemporary China.
ABOUT THE AUTHORS
**Ken Hom** is widely regarded as one of the foremost authorities on Chinese cooking, and has shaped the way we view Chinese cuisine. Born in Arizona after his parents fled Guangdong following World War II, his first television series was aired in 1984 and since then he has appeared on television countless times and published over 30 cookery books. In 2008 he was awarded an honorary doctorate from Oxford Brooks University, and in 2009 was awarded an honorary OBE for services to the culinary arts.
**Ching-He Huang** was born in Taiwan and is a self-taught and passionate cook, and an ambassador for easy, healthy Chinese cooking. Her ancestors emigrated from the Fujian province in China to Taiwan during the Ming Dynasty. Ching's dynamic approach to modern, healthy Chinese food has seen her present such series as Chinese Food Made Easy on BBC2, Ching's Kitchen, Ching's Chinese Food in Minutes and Easy Chinese in the US, which have been seen around the world. She has published a number of cookery books, including the bestselling BBC title Chinese Food Made Easy, China Modern, Chinese Food in Minutes and Ching's Fast Food.
Introduction
I don't think there could have been a better time to explore China than when we went, at the beginning of 2012. The real China still exists, thank goodness, even though westernisation and relentless progress are nibbling away at the old traditional ways of doing things, of cooking food, of eating, of thinking, of living. We travelled from the wild frontiers of the north to the industrial megacities of the south and saw the inevitable conflicts between tradition and modernity, between communism and capitalism. But we also talked to the 'real' people who inhabit the country and their pragmatism gives me great hope for the survival of the 'real' China – although it won't be an easy task.
China now has a population of nearly 1.4 billion, which represents some 20 per cent of the world population (and means that approximately one person in five is Chinese). China is trying to feed that 20 per cent of world population with less than 10 per cent of the planet's arable land. But some 10 per cent of China's arable land – the land on which food grows – has been lost to development in the last ten years or so. As farmers move to the cities, either forced by expropriation of their land or lured by the promise of a better life, one wonders how the food supply can be maintained. If the cities are growing in size, and the farming land is diminishing, how on earth are the Chinese going to manage to feed themselves? They have always been clever about utilising the most unlikely space to produce food – I'm thinking of rice terraces snaking up hillsides, especially in Yunnan, the 'peri-urban' plots of greenery grown at city roadsides and in public gardens, fish in paddyfields, turtle ponds, aquaculture in general. However, with less land on which to produce the staples – rice in the south, wheat in the north – the traditional Chinese mode of eating may have to radically change.
In a sense it already has, with the influx of Western-type foods and food chains. Meat is now seen as a necessary part of the diet when, previously, it was a rare treat, more of a flavouring for vegetables. (Meat dishes in Western Chinese restaurants are uncharacteristic of real Chinese cooking.) Where and how will all the meat-producing animals be raised? Dairy farming has recently been encouraged by the government, although over 90 per cent of the adult population is lactose intolerant to a certain degree. Where and how will all the milk-producing animals be raised? As China gets richer, and with a burgeoning middle class, will women be happy to spend hours making wontons and dumplings, or will they start to buy them ready-made? And, most frighteningly, if young Chinese take to the hamburgers and pizzas now being introduced, will the next generation actually know how to cook their own traditional dishes?
However, in our travels around China, we encountered eaters, thinkers, cooks and producers and all were incredibly enthusiastic about what they were doing. They all intend to hold on to what they value from the past, but perhaps – rather like Ching and I – would probably choose to ally it with a hint of the new. Hong Kong, for instance, is the showcase for a balance between the old and traditional and the new and progressive, certainly in terms of food, so perhaps this is the way China will go. The Chinese have survived much worse, re-emerging from chaos, and I think they can do so again. I sincerely hope so.
Introduction
It was wonderful being in China again, visiting familiar and unfamiliar places – and tasting some wonderful food. Travelling the length and breadth of the country with Ken and the film crew, I realised just how vast China is; it could swallow Europe, and some of its cities boast as many inhabitants as a single European country. Although there are shared threads, there really is no such thing as 'Chinese cuisine': the cooking styles of China's regions – influenced variously by geography, climate and history – are as different as those of countries in northern and southern Europe.
Food has such a central role in Chinese culture that, country-wide, one of the most common greetings translates as: 'Have you eaten yet?' I like that! Formally, there are eight distinctive regional cuisines (Shandong, Sichuan, Guangdong, Fujian, Jiangsu, Zhejiang, Hunan and Anhui). Many commentators, however, divide the country into gastronomic quarters: Beijing and the north, Shanghai and the east, Sichuan and the west, Guangzhou (Canton) and the south. But even these simpler divisions do not take into account further variations: Chinese Muslims and Buddhists have their individual styles of cooking, as do the minority ethnic groups – some 56 of them – that co-exist with the majority Han Chinese. But cooking and eating with the Dai minority and the Bulang tea-pickers was a delight for me in Yunnan, and experiencing Muslim food with the Uighurs in Kashgar, in the far north, was an eye-opener – as were the hats!
Whichever region they come from, Han Chinese all comply with the principle of balance denoted by yin and yang. In a culinary sense, yin means 'cool' and yang 'hot', and there are cooking techniques and ingredients that belong in each of the two camps. By cooking and eating in this balanced way, the Chinese believe an essential equilibrium is maintained, which ensures good health. Some restaurants cook with these principles in mind, and shops sell herbs that can be added to food. Some foods themselves are meant to be good for certain parts of the body: snake, for instance, is believed to be an aphrodisiac. Food as medicine wasn't a new concept to me, but I got valuable further insights through many of the people we talked to. And although they're said to be warming for the blood and good for the skin, I don't think I'll be eating fried scorpions any time soon – although Ken seemed to take a liking to them at a street-food stall in Wangfujing in Beijing...
Other shared threads of Chinese cooking are the balancing of the characteristics of a Chinese meal. Dishes should have fragrance, taste (sour, sweet, salty, bitter, piquant), shape, colour and mouth-feel: the latter subdivides into textural tenderness, crunchiness, crispiness, smoothness and softness (and explains why the Chinese have such a love for things like chicken feet). Another facet of Chinese eating is the love of snacks: from the xiao chi (small eats) of Sichuan to the dim sum of Hong Kong. Now, of course, with China having opened up to Western ideas and Western foods, some of these snacks might just be hamburgers, cheese or pizzas, which is worrying. The Chinese are already starting to suffer from Western health problems such as obesity, and if this continues, their historical reverence for food, its traditions and its role in Chinese culture might be eroding. This would be such a pity for, even though I am Taiwanese, I am full of admiration for my mainland neighbours – supposed 'opponents/adversaries' – for their fortitude, for their culinary creativity, and for the history that we share.
Dumplings and Appetisers
Chengdu Wonton
Shui-Jiao Pork and Prawn Chive Water Dumplings
Sui Mai Dumplings
Spring Onion Pancakes
Pickled Sichuan Carrots, Er Cai, Garlic Chives and Red Onion
Pickled Cucumber with Chilli
Ginger and Soy Chilli-Steamed Scallops
Growing up in a Cantonese household I was very familiar with wontons. I knew I was in for a treat when I first had them in Chengdu, but I was not prepared for the lake of chilli oil and spices they came in. Luckily, I have always liked spicy food. This recipe is for other such lovers of spicy food.
Serves 4 as a starter or 2 as a meal
50g (2oz) Chinese dried black mushrooms, pre-soaked in warm water for about 20 minutes until soft and pliable
450g (1lb) minced fatty pork
4 tablespoons finely chopped Chinese chives (or spring onions)
1 egg white
2 teaspoons sesame oil
salt and freshly ground black pepper
20–25 wonton wrappers, 10 × 10cm (4 × 4in), thawed if frozen
For the sauce
1 tablespoon groundnut or vegetable oil
1 tablespoon chilli oil (see here)
1 tablespoon finely chopped garlic
2 tablespoons finely chopped spring onions
1 tablespoon roasted and ground Sichuan peppercorns
3 tablespoons sesame paste (or peanut butter)
2 tablespoons light soy sauce
1 tablespoon sugar
2 teaspoons chilli bean sauce
85ml (3fl oz) chicken stock
Squeeze the excess water out of the mushrooms, then cut off and discard the woody stems. Finely chop the mushrooms and combine them with the meat, chives (or spring onions), egg white, sesame oil and salt and pepper. Mix well.
Place about 1 rounded teaspoon of filling in the middle of each wonton wrapper. Gather the four sides of the wrapper up over the filling, allowing the wonton skin to fold in pleats naturally. Gently pinch the wrapper together at the top very well so it is sealed completely. Repeat this process until you have used up all the stuffing.
Heat a wok over high heat until it is hot, then add the groundnut or vegetable and chilli oils. Add the garlic and spring onions and stir-fry for 20 seconds, then add the rest of the ingredients and simmer for 5 minutes. Transfer the sauce to a separate bowl.
Bring a medium-sized pot of water to a boil and poach the dumplings for about 3 minutes, or until they float to the top. Remove and drain, then serve with the sauce.
Pork and prawn chive water dumplings
Dumplings are a staple food found in the north of China, but various types are also found in many different regions – both steamed and pan-fried. My favourite are these shui-jiao dumplings, which are cooked in boiling water and then served with a soy vinegar dipping sauce. You can vary the filling ingredients, but this is my all-time favourite.
Serves 2–4
30 wheat flour dumpling skins
1 tablespoon groundnut oil
For the filling
150g (5oz) tiger prawns, peeled, deveined (see here) and minced
60g (2½oz) minced pork
3 tablespoons Chinese chives (garlic chives), finely chopped
4 whole tinned water chestnuts, finely diced
1 tablespoon oyster sauce
1 tablespoon Shaoxing rice wine (or dry sherry)
sea salt and freshly ground white pepper
For the dipping sauce
2 tablespoons light soy sauce
2 tablespoons Chinese clear (plain) rice vinegar (or cider vinegar)
2 teaspoons toasted sesame oil
a small handful of fresh coriander leaves, finely chopped
For the garnish
a small handful of finely chopped fresh coriander
Put all the ingredients for the filling into a bowl and, using one hand, knead and squeeze the filling through your fingers to combine all the flavours. Wash and dry your hands. Place a dumpling skin in the palm of your hand. Place a heaped teaspoon of filling in the middle of the dumpling skin, then dip your finger into water and brush around the edge of the skin. Fold the skin in half over the filling, pressing down on the edges to make sure they stick together well. Pleat the edges of the dumpling four times, pinching well to secure. (You could make the dumplings ahead to this point, then freeze in an airtight container and cook from frozen when you like.) Combine all the ingredients for the dipping sauce in a small bowl and put to one side.
Heat a pan of water (about 1 litre/1¾ pints) and bring to the boil. Add the groundnut oil and reduce the heat to medium, then add the dumplings and cook for 3–4 minutes until they float to the surface and are cooked through. Using a slotted spoon, scoop them out into dishes, garnish with chopped coriander and serve them immediately, with the dipping sauce.
Dim sum is the southern Chinese ritual of drinking tea and eating sweet and savoury snacks of all sorts. Although some snacks exist elsewhere in China, it is the Cantonese who have elevated them to a true art. There are literally countless different types of dim sum snacks, depending on the chef and his or her interests. Steaming is a favourite technique of the Cantonese, cooking food gently so that subtle flavours are not masked by the ferocity of frying in hot oil, for example. These delectable sui mai dumplings are easy to make at home. They are perfect for the Cantonese, as ovens do not exist in most Chinese kitchens. These dumplings make a terrific beginning to any meal.
Serves 4–6
225g (8oz) wonton skins, thawed if frozen
fresh coriander leaves
For the filling
250g (9oz) raw tiger prawns, peeled, deveined (see here) and coarsely chopped
250g (9oz) minced fatty pork
1 teaspoon salt
½ teaspoon freshly ground white pepper
100g (4oz) fresh or tinned water chestnuts, peeled if fresh, drained if tinned, and finely minced
1½ tablespoons light soy sauce
2 tablespoons finely chopped spring onions
1 tablespoon Shaoxing rice wine (or dry sherry)
1 teaspoon sugar
2 teaspoons sesame oil
1 egg white, lightly beaten
To serve
dipping sauces of your choice
Put the prawns and pork into a large bowl or a food processor, add the salt and pepper and mix well, either by kneading with your hand, stirring with a wooden spoon, or pulsing. Then add all the other filling ingredients and stir them well into the prawn and pork mixture. Cover the bowl with clingfilm and chill for at least 20 minutes.
Place a portion of filling on to each wonton skin. Bring up the sides and press them around the filling mixture. Tap the dumpling on the bottom to make a flat base. The top should be wide open, exposing the meat filling. Now place a coriander leaf on top. Repeat this process until you have used up all the filling.
Place the dumplings in a bamboo steamer lined with damp cheesecloth. Place the steamer over water in a wok or pot, cover tightly and steam over high heat for 15 minutes. Serve immediately with dipping sauces of your choice.
You will find this savoury snack both in the restaurants in Beijing (Peking) and sold at the street food stalls of Kashgar. Contrary to what most people think, wheat is the main food here in northern China. These pancakes remind me of Middle Eastern snacks, perhaps the result of an exchange with Arab traders on the Silk Road? You have the option of making them as rolled onion cakes or flat, a bit like pizza. Either way, they are equally delicious. Be particularly careful when deep-frying in a wok.
Makes around 9 flat onion cakes or 18 rolled onion cakes
1 × recipe Chinese pancakes (see here)
2 eggs, beaten
9 spring onions, finely chopped
salt and freshly ground black pepper
900ml (1½ pints) groundnut or vegetable oil
If you are making the flat onion cakes, take two pancakes and brush one side of each with beaten egg, then sprinkle one with spring onions, salt and pepper. Press the other pancake, egg-brushed side down, on top to seal.
If you are rolling them, take one pancake, brush one side with beaten egg, then sprinkle with spring onions, salt and pepper. Roll the pancake into a cylinder.
Heat a wok over high heat until it is hot. Add the oil, and when it is hot, hold each cake, either flat or rolled, with tongs and deep-fry until crispy. Drain on kitchen paper. Continue frying until you have fried all the pancakes.
Cut the pancakes into bite-sized pieces and serve immediately.
This simple yet delicious pickle is superior to anything I have ever tasted. It is called shi chao cai ('take a bath vegetable'), which is a term used to describe a quick technique for pickling vegetables, and means that it's so quick you can make this before you take a bath, leave it overnight and it will be ready to eat the next day. The vegetables still retain their crunch, original flavour and crispness. I had the wonderful pleasure of making this dish with the grandmother of our friend Jenny Gao, who accompanied us for parts of the trip, and it reminded me of my own grandmother. I felt so appreciative and blessed to learn this from a woman who has clearly lived through some harsh times in China. She told us that a relative had bred a lamb during the Cultural Revolution to send to her so that she could have some extra food and protein when she was pregnant with Jenny's father. An extraordinary woman.
Makes 1 × 500ml (17fl oz) jar
For the pickling liquid
450ml (¾ pint) good-quality water
50g (2oz) good-quality salt
a small handful of Chinese white rock sugar (or granulated sugar)
a small handful of red rock sugar
225ml (8fl oz) good-quality bai jiou (distilled white liquor)
For the pickling vegetables
2 whole carrots, washed and dried
a few pieces of er cai (baby root vegetables), or use tender broccoli stalks, not broccoli flowers
1 red onion, peeled and quartered
tender inner stems of a small bunch of Chinese garlic chives (or baby spring onions), sliced into 2cm (¾in) lengths
To serve
1 teaspoon Sichuan pepper oil (see here)
1 tablespoon chilli oil (see here)
Place the water, salt, rock sugars and bai jiou into a Chinese ceramic pickling urn (or a sterilised glass jar). Mix well with chopsticks and stir well to dissolve. Add all the pickling vegetables and push down to submerge them well in the liquid. Place the lid on to seal the pickling urn. The pickling urn is surrounded by a small 'moat' – allowing you to fill the edges of the pickling urn lid with water to seal the urn, so that no air can seep in. Pickle overnight or for 1–2 days. To serve, slice the vegetables into bite-sized pieces. Season with the Sichuan pepper oil and the chilli oil.
with chilli
My obsession with pickles started when my mother used to send me off to school with a packed lunch of cold rice, pickles and stir-fried meat and vegetables. The pickles add a salty-sour flavour that is very xia fan – 'gives flavour to rice and helps make plain rice more palatable'. My favourite vegetable to pickle is cucumber and this delicious recipe pairs it with garlic, chilli bean paste, rice vinegar and chilli. This is a quick pickling technique, and the pickle should be eaten within a few days while the cucumbers still retain their crunch and freshness.
Serves 2–4
4 small cucumbers, sliced in half lengthways and de-seeded, each half cut into long wedge shapes and then sliced into 1cm (½in) pieces
For the marinade
1 garlic clove, peeled and minced
2 pinches of caster sugar
1 tablespoon Chinese clear (plain) rice vinegar (or cider vinegar)
1 teaspoon chilli bean paste
1 teaspoon chilli oil (see here)
1 tablespoon toasted sesame oil
For the garnish
1 fresh red chilli, de-seeded and cut into small strips
Combine all the ingredients for the marinade in a bowl. Add the cucumbers and leave to marinate in the fridge for 20 minutes.
Divide the mixture between small bowls, garnish with the fresh chilli and serve with plain rice and stir-fried vegetables for a light but flavourful dinner.
You can also make a batch of pickled cucumber and decant into sterilised glass jars, then keep refrigerated for up to 1 week. Spoon some into a chicken or vegetable stir-fry to add flavour.
This is one of my go-to recipes for easy entertaining. Fresh scallops are such a treat and this makes a tasty appetiser or a shared main with other dishes. I like to make a simple seasoning liquid to dress the scallops, so that as they steam all the flavours are infused. Then a simple garnish of crispy garlic and shallots adds texture and fragrance to the final dish. Enjoy!
Serves 4 as a starter
4 large king scallops in their shells
50ml (2fl oz) groundnut oil
2 medium shallots, peeled and finely chopped
6 large garlic cloves, peeled and finely chopped
For the seasoning sauce
1 fresh medium red chilli, de-seeded and chopped
2.5cm (1in) piece fresh root ginger, peeled and grated
1 teaspoon toasted sesame oil
1 tablespoon groundnut oil
1 garlic clove, peeled and finely chopped
1 tablespoon light soy sauce
For the garnish
fresh coriander leaves
Using a sharp knife, carefully prise open the scallop shells by running the knife on the inside of the shell to release the membrane attached to it. (Or ask your fishmonger to prepare the scallops for you.) I like to leave the bright orange coral attached, but you can discard it if you like. Wash the scallops in cold running water.
Combine all the ingredients for the seasoning sauce in a small bowl and put to one side.
Place the scallops on a heatproof plate that fits inside a bamboo steamer and spoon 1 teaspoon of the sauce over each one. Put the plate into the steamer. Half-fill the wok with boiling water, place the steamer over the wok (making sure the base doesn't touch the water), then cover and steam the scallops for 5–6 minutes.
Meanwhile, heat a wok over high heat, then add the groundnut oil. Fry the shallots and garlic until crisp (it should take less than 1 minute). Pour the oil through a sieve into a heatproof bowl and drain the shallots and garlic on kitchen paper.
Serve the scallops sprinkled with the crispy shallots and garlic and garnished with coriander.
and Cantonese cooking
The province of Guangdong, in the south of China, has a particular resonance for me, because it is where my mother's family come from: she was brought up in Kaiping (Hoiping), a small town in the region, and I have relatives there still. Before she died, my mother and I made a pilgrimage back from Chicago to Kaiping to meet the remnants of our family. The filming of this series has happily allowed me to return yet again to see them, to re-explore the delights of Cantonese cuisine, and to pay homage to my ancestors.
Guangdong lies on the South China Sea, and its capital is Guangzhou, which is known historically as Canton. This word is believed to have originated from the Portuguese name for Guangzhou, 'Cantão', and of course this is where the name of the cuisine, Cantonese, comes from. Guangzhou, one of the largest cities in China, has been a nationally important river trading port for thousands of years. From the 17th century it was one of only four Chinese cities allowed to trade with foreigners (and through which, shamefully, much British opium came). Guangzhou may have a long history, but another Guangdong city, Shenzhen, further to the east, is known for its lack of history. In some thirty years, the sleepy fishing village of Shenzhen has transformed itself into a modern metropolis, its population swelling from thousands to many millions. And in fact, the whole of Guangdong, because of its geography and the vivacity and adaptability of its people, has played a huge part in what is known as the 'Chinese revival'.
The cooking style of the area is, naturally, Cantonese. Many people regard Cantonese as the haute cuisine of China – possibly due to the influence of chefs of the Imperial Court who fled to Guangzhou in the 17th century – and the city is still rightly called the gastronomic capital of China. (There is a local saying: 'every five steps a restaurant', which is almost true!) Cantonese is the style of cooking I grew up with – my mother introduced me to its tastes and flavours in our home in Chicago – and it is still my favourite Chinese cuisine (though closely followed by Sichuan). It is also probably the most familiar of Chinese cooking styles throughout the rest of the world, for the majority of Chinese nationals opening restaurants and cooking in other countries come from Guangdong and Hong Kong.
The essence of Cantonese cooking, one of the eight major styles of Chinese cuisine, is its freshness: many restaurants in the area have water tanks filled with seafood, and cages of fowl and other creatures, waiting for you to choose. Cooking techniques are straightforward – boiling, steaming and stir-frying using little oil – so that the purity of the fresh ingredient can be truly appreciated. Flavouring condiments, of which there are many, are used reservedly – a dash here and there only, to complement the dish's principal ingredient – and often are served at the table, for diners to help themselves. Spices are used sparingly – star anise, garlic and chilli among them – and herbs rarely. Oyster sauce, for instance, one of the key ingredients of Cantonese cooking, is said to have been invented in Guangdong in the 1800s. Fish and chicken particularly benefit from this careful cooking and flavouring: prawns are marinated in liquor, to make them 'drunk' (literally, for sometimes they are still alive), then simply flamed or boiled; fish is steamed with minimal flavours (spring onion, ginger, soy); and although chicken can be steamed too, it is also cooked in the Cantonese style (basted, dried, then fried). Pork, the favourite meat of most of China, is treated somewhat differently: it can be marinated and barbecued, roasted, stir-fried, served with a sauce or minced for a dumpling filling.
Vegetables play a large part in all the Chinese cuisines, and Cantonese is no exception. Most are squeaky fresh – just plucked from the ground, such as those I enjoyed when I visited my relations in Kaiping – but many are preserved in some way. (In Guangdong, this may come from the Hakka people, a subdivision of the Han Chinese, found mostly in Guangdong and Taiwan.) Most cultures dry or preserve foods for times of hardship or simply for the winter, and the textures of some dried, salted or pickled vegetables, and indeed fish and meats, in Cantonese cooking, is a key pleasure when tasting many of the dishes. Another preservation method employs fermentation, and fermented dofu (tofu), black beans and other pulses contribute amazing flavour (soy sauce is made from fermented soya beans). And of course, rice is the staple of the Cantonese diet; the province is known as one of the 'rice-bowls' of China.
Another aspect of Cantonese food is its snacks. The Chinese as a nation are great snackers and love eating on the hoof and on the street, but nowhere more so than in Guangdong and Hong Kong (for dim sum, see here). Dumplings and wontons, noodles and buns – made with rice, wheat, rye, barley, sorghum or millet flour – are on offer from street-food stalls everywhere, with a never-ending variety of fillings. Noodles of all kinds – jook sing are made with duck eggs and are traditionally rolled by a chef sitting on a length of bamboo log! – are served in savoury broths. Often on offer is a long-cooked soup, lou fo tong ('old fire-cooked soup') containing medicinal herbs the Cantonese believe will heal and strengthen. Another snack occasionally seen is 'deep-fried milk' – unusual in a culture that traditionally does not eat dairy products (most Chinese are actually lactose intolerant); I have encountered it several times and it's good. Many street stalls also sell little custard tarts – you can try my version of these treats here.
It is snacks like these that are given as ritual offerings to our ancestors. Ancestor worship lies at the heart of religion in China: those who remain take care of the deceased as if they were still alive. Gifts include food, drink, money and favourite goods – such as cars, washing machines, mobile phones – which are all made of cardboard and paper (and bought from special mourning shops). These are burned, so that the smoke carries the messages upwards to the heavens. I was lucky enough to be in China at the right time for Qing Ming, the traditional tomb-sweeping ceremony, and I was able therefore to pay my respects to my forebears.
Soups and Salads
Sea Bass, Baby Pak Choy and Coriander Soup
Xi Shi Dofu Soup
Mak's Wonton Noodle Soup
Dai Aunty River Fish Soup
Peking Duck Soup
Vegetarian Hot and Sour Soup
Yi Jin's Dai-Inspire Drice Noodle Soup
Fresh Soy Sugar Tomato Salad
Da Dong-Inspired Peking Duck Salad
Lotus Root Salad
Kashgar-Inspired Salad
and coriander soup
This is a clean-tasting and nourishing soup. Fresh sea bass, shiitake mushrooms and baby pak choy are gently poached in a simple vegetable broth and then garnished with fresh, aromatic coriander. The result is a sweet, healthy soup that is perfect as a starter or a light supper.
Serves 2 as a main and 4 as a starter
250g (9oz) sea bass fillets, skin on, each fillet cut into 4 strips
800ml (1 pint 8fl oz) water
2.5cm (1in) piece fresh root ginger, grated
1 tablespoon Shaoxing rice wine (or dry sherry)
3 fresh shiitake mushrooms, sliced
4 small baby pak choy, sliced in half down the middle
1 tablespoon vegetable bouillon stock powder
a dash of toasted sesame oil
sea salt and freshly ground white pepper
a small handful of fresh coriander
Rinse the fish in cold running water.
Pour the water into a pan or wok and bring to the boil. Add the fish and all the ingredients up to and including the bouillon stock powder. Turn the heat to medium and cook for 5 minutes. Season with the sesame oil and salt and white pepper to taste. Stir in the coriander, transfer to serving bowls and serve immediately.
Often for the Chinese, a soup can be a hearty one-dish meal. In this case, the soup contains meat, prawns and vegetables as well as dofu (tofu). The dofu absorbs the rich flavours of the broth and makes a comforting, satisfying main course with rice on a wintry evening.
Serves 4
50g (2oz) Chinese dried black mushrooms, pre-soaked in warm water for about 20 minutes until soft and pliable, then drained
1½ tablespoons groundnut or vegetable oil
1 tablespoon finely chopped fresh ginger
225g (8oz) pork spareribs, separated into individual ribs
3 tablespoons Shaoxing rice wine (or dry sherry)
1 teaspoon sugar
salt and freshly ground white pepper
1.2 litres (2 pints) chicken stock
450g (1lb) fresh silken or soft dofu (tofu)
100g (4oz) raw tiger prawns, peeled, deveined (see here) and cut into 2.5cm (1in) pieces
4 tablespoons finely chopped spring onions
2 teaspoons sesame oil
Rinse the mushrooms well and squeeze out any excess liquid. Discard the tough stems, then shred the caps and put them aside.
Heat the wok until it is very hot, then add the oil. When the oil is hot, add the ginger, then the spareribs and stir-fry for about 5 minutes, then add the rice wine (or sherry), sugar, salt, pepper and stock. Add the mushrooms and bring to a boil, then lower the heat, cover and simmer for 15 minutes.
Meanwhile, gently cut the dofu into 2.5cm (1in) cubes and drain on kitchen paper.
Carefully add the cubed dofu and the prawn pieces to the soup and simmer, uncovered, for another 3 minutes. Finally, add the spring onions, adjust the seasoning with salt and pepper, drizzle with the sesame oil and serve.
In London it is not uncommon to see Rolls-Royces parked in front of fish and chip shops because no matter how successful someone becomes, they can still yearn for good comfort food. In Hong Kong, a good equivalent would be Mak's Noodles Ltd, an extremely modest classic restaurant that serves some of the best wonton noodle soup in all of Hong Kong. It is a great soup, with juicy dumplings – pure simplicity, brilliantly executed – supreme comfort food for every Cantonese diner.
Serves 4–6
100g (4oz) wonton skins
1.2 litres (2 pints) chicken stock
1 tablespoon light soy sauce
1 teaspoon sesame oil
175g (6oz) thin fresh or dried Chinese egg noodles
Filling for the wontons
100g (4oz) peeled, uncooked prawns, deveined (see here) and coarsely chopped
100g (4oz) minced pork
salt and freshly ground white pepper
1 tablespoon light soy sauce
2 tablespoons finely chopped spring onions
1 tablespoon Shaoxing rice wine (or dry sherry)
1 teaspoon sugar
1 teaspoon sesame oil
½ egg white, lightly beaten
For the garnish
chopped spring onions
To make the filling, put the prawns and pork into a large bowl, add salt and pepper and mix well, either by kneading with your hand or by stirring with a wooden spoon. Then add all the other filling ingredients and stir them well into the prawn and pork mixture. Cover the bowl with clingfilm and chill for at least 20 minutes.
When you are ready to stuff the wontons, put 1 tablespoon of the filling in the centre of the first wonton skin. Dampen the edges with a little water and bring up the sides of the skin around the filling. Pinch the edges together at the top so that the wonton is sealed – it should look like a small filled bag. Repeat with the remaining wonton skins and filling.
When the wontons are ready, bring the stock, soy sauce and sesame oil to a simmer in a large pot.
In another large pot, bring lightly salted water to a boil and poach the wontons for 1 minute or until they float to the surface. Remove them immediately and transfer to the pot with the stock. Blanch the noodles in the salted water for 1 minute, then drain and add to the pot with the stock and dumplings. This procedure will result in a cleaner tasting broth. Continue to simmer the wontons and noodles in the stock for 2 minutes. Transfer to either a large soup bowl or to individual bowls, garnish with spring onions and serve at once.
The Dai people are an ethnic minority who live in the southwest of China (see here). I had the honour of cooking with a Dai 'aunty', who took me fishing in a nearby stream, and then we cooked together with her family in her modest wooden home. The women in the village are responsible for cooking and looking after the children. They use small nets to catch the river fish, but also find in their nets water millipede, baby freshwater shrimp and baby freshwater crabs. Dai Aunty would expertly prepare all these ingredients – gutting each fish with one fast swoop of a small knife, removing the 'dead man's fingers' (the lungs) from the crabs – and then cook them in a delicious spicy, sour broth. These Dai women were an inspiration – they may not possess material wealth, but they have soul.
Serves 4–6 to share
750ml (1 pint 7fl oz) water
200g (7oz) tender baby bamboo shoots (or tinned bamboo shoots, drained and rinsed)
6 small fresh chillies, sliced
2.5cm (1in) piece fresh root ginger, peeled and finely chopped
1 small bunch of Vietnamese mint (or a few stalks of fresh lemon grass, peeled, bruised and cut into 5cm/2in pieces), roughly chopped
½ teaspoon salt
300g (11oz) mix of small river fish, shrimp, crabs (or mixed seafood medley of shrimp, mussels and scallops), cleaned and rinsed
a small handful of freshly chopped wild coriander (or farmed coriander)
Bring the water to the boil in a medium pot. Add the bamboo shoots, chillies, ginger and mint (or lemon grass). Once the water has come back to the boil, add the salt, then the seafood and cook for 3 minutes on medium heat. Take off the heat, add the chopped coriander and serve immediately.
Beijing's top duck restaurants not only offer Peking duck, but also an entire duck extravaganza that features every imaginable part of the duck except the quack. These may include delectable offerings such as duck's tongue with mustard sauce, duck gizzard stir-fried with vegetables or duck liver with spicy sauce. Duck entrails as well as duck feet, often braised, are considered delicacies. The feast often finishes with a soup made using the bones of the roast duck as the base. It is a savoury and refreshing end for any true lover of duck.
Serves 4
carcass of roasted Peking duck (see here or here)
50g (2oz) preserved mustard greens, soaked in several changes of cold water
1.2 litres (2 pints) chicken stock
salt and freshly ground black pepper
1 teaspoon sugar
2 teaspoons Shaoxing rice wine (or dry sherry)
Rinse the cavity of the duck to rid it of the spices. Chop the carcass into large pieces.
Drain the mustard greens thoroughly and chop coarsely.
Bring the stock to the boil in a large pot. Add the duck bones and simmer for 20 minutes. Remove the bones with a strainer, skim the stock, then add the mustard greens, salt, pepper, sugar and rice wine (or sherry) and continue to simmer, uncovered, for 10 minutes. Serve the soup immediately in individual bowls or in a large soup tureen.
I never tire of a good hot and sour soup. The trick is the fine balance of sour to spicy without making it unpalatable. This dish should have lots of flavour, while the crunchy bamboo shoots, wood ear mushrooms, preserved Sichuan vegetables and fresh dofu (tofu) all add layers of kou-gan (texture), which is so important in Chinese cooking. This is a satisfying soup for any time of the year. Meat lovers can add cooked chicken or Chinese roast pork pieces.
Serves 4
1 litre (1¾ pints) vegetable stock
1 tablespoon freshly grated root ginger
2 fresh red chillies, de-seeded and finely chopped
1 tablespoon Shaoxing rice wine (or dry sherry)
2 tablespoons dark soy sauce
1 × 220g tin bamboo shoots, drained
10g (¼oz) Chinese dried wood ear mushrooms, pre-soaked in hot water for 20 minutes, then drained
250g (9oz) fresh firm dofu (tofu), cut into cubes
50g (2oz) Sichuan preserved vegetables, rinsed and sliced
2 tablespoons light soy sauce
3 tablespoons Chinese black rice vinegar (or balsamic vinegar)
1 tablespoon chilli oil (see here)
a pinch of freshly ground white pepper
1 egg, lightly beaten
1 tablespoon cornflour, blended with 2 tablespoons cold water
1 large spring onion, finely sliced
For the garnish (optional)
fresh coriander leaves
Pour the stock into a pan and bring to the boil. Add all the ingredients up to and including the wood ear mushrooms and cook for 1 minute. Turn the heat down to medium, add the dofu, Sichuan vegetables, soy sauce, vinegar, chilli oil and white pepper, then simmer for 10 minutes.
Stir in the egg, then add the blended cornflour and stir to thicken the soup (add more if you like a thicker consistency).
Add the spring onion, garnish with the coriander, if you like, and serve immediately.
This delightful recipe was inspired by Yi Jin, who with her young husband, Ai Han En Loug, makes 350kg (772lb) of fresh rice noodles every day at their home in Yunnan. Produced in a home-made artisanal factory in their garage, the noodles are quite sought after in Jinghong. The secret is the amount of water they use with the rice flour. The result is a slightly chewy rice noodle that is equally good stir-fried or used in this soup, which they made especially for me. It was a simple but comforting, savoury and satisfying dish.
Serves 2–4
225g (8oz) round, dried rice noodles, soaked in warm water for 25 minutes, then drained
1½ tablespoons groundnut or vegetable oil
2 tablespoons coarsely chopped garlic
2 tablespoons coarsely chopped fresh ginger
2 teaspoons dried red chilli flakes
100g (4oz) minced pork
salt and freshly ground black pepper
100g (4oz) fresh tomatoes, cut into quarters
1.2 litres (2 pints) hot chicken stock
1½ tablespoons light soy sauce
2 teaspoons Shaoxing rice wine (or dry sherry)
3 tablespoons finely chopped spring onions
Blanch the drained rice noodles in salted boiling water for 5 minutes. Drain thoroughly, then set aside until you are ready to use them.
Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is very hot and slightly smoking, add the garlic, ginger and chilli flakes and stir-fry for 30 seconds. Add the pork, salt and pepper and continue to stir-fry for 3 minutes. Now stir in the tomatoes. Finally, add the stock, soy sauce and rice wine (or sherry) and turn the heat to low, then cover and simmer for 5 minutes. Skim off any surface fat, then add the noodles and cook for another 3–4 minutes.
Stir in the spring onions and continue to simmer for 2 minutes. Ladle into a large soup tureen and serve at once.
While in Cuandixia, a village a couple of hours away from Beijing, I met Mrs Han, one of the locals. She prepared some large, ripe beef tomatoes by quartering them and then sprinkling sugar over them. I suggested adding some light soy sauce. The result is a delicious, savoury-sweet tomato salad.
Serves 2–4
2 large, ripe beef tomatoes, quartered and halved again, stalks and seeds discarded
1 teaspoon granulated sugar
2 tablespoons light soy sauce
Place the tomato slices in the fridge for at least 15 minutes before assembling.
Place the chilled tomatoes on a plate and sprinkle with the sugar, then drizzle the light soy sauce over them. Stir together at the table and serve immediately.
Da Dong is an inspirational Beijing chef. He is famed for his Peking Duck, a version which is healthier as it has less fat than usual. Over thirty years he has perfected his technique of roasting the Peking duck in a slow, fruit-wood-fired oven for double the amount of time that it usually takes, thus cooking out all the fat from beneath the skin. It was a delight to be able to use his Peking duck in a delicious duck salad and this is a great dish to make if you have any leftover roast duck pieces. Seasoning it with Chinese five-spice, dusting with cornflour and then deep-frying elevates the flavour of the duck, giving it another crispy edge. I was worried that the dish would not work because of the smaller amounts of fat in Da Dong's duck – that it would make my dish too dry – but it worked like a dream. He thought the dish was clever and delicious, which is a great compliment coming from such an ingenious chef! I served the duck with some garnishes borrowed from Da Dong, but I also made a delicious sauce-like dressing using wheat flour bean paste (tian mian jiang), sugar, black rice vinegar and XO sauce (a rich chilli sauce-oil containing dried shrimp, scallops and Jinhua ham, a dry-cured ham). You can substitute yellow bean paste mixed with Hoisin sauce for the tian mian jiang and you can buy the XO sauce from a Chinese supermarket, or mix a good chilli sauce with chilli oil.
Serves 2–4 to share
1 Granny Smith apple, unpeeled, halved, cored and sliced into thin half-moon shapes
1 Chinese pear, halved and sliced into thin half-moon shapes
1 large red radish, sliced lengthways into julienne strips
1 spring onion, sliced lengthways into julienne strips
1 tablespoon Sichuan pickle or grated red radish mixed with a splash of rice vinegar
500ml (17fl oz) vegetable oil
250g (9oz) Peking roast duck (see here or here), boned and cut into small pieces
¼ teaspoon Chinese five-spice powder
2 tablespoons cornflour
For the dressing
3 tablespoons wheat flour bean paste (tian mian jiang, or 1 part yellow bean paste mixed with 4 parts Hoisin sauce)
1 tablespoon granulated sugar
2 tablespoons Chinese black rice vinegar (or balsamic vinegar)
1 tablespoon XO sauce (½ tablespoon each chilli sauce and chilli oil, see here)
For the garnish
a small handful of micro herbs
a small handful of edible yellow flowers
2 pastry birds for decoration (optional)
Place the apple and pear slices down the centre of a large, long, rectangular serving plate, to make heart shapes. Place some red radish strips down the middle of the hearts, then top with spring onion strips. Decorate the rest of the plate with sprinkles of Sichuan pickle.
Pour the oil into a wok and heat to 180°C/350°F, or until a cube of bread turns golden brown in 15 seconds. Dust the duck pieces with Chinese five-spice and then with the cornflour. Lower the pieces into the hot oil and deep-fry until crispy and the edges of the cornflour have turned golden brown. Lift out and drain on kitchen paper. Place the duck down the middle of the apple, pear, radish and spring onion strips on the serving plate.
Mix together all the dressing ingredients in a small bowl. Spoon some of this dressing over the duck pieces. Garnish the duck with micro herbs and decorate the plate with some edible yellow flowers (and I also added pastry birds borrowed from Da Dong).
The Chinese have shown great ingenuity over the centuries, turning the most unlikely vegetables into delectable dishes. The lotus is revered in China for its flower, leaves and root. The pattern of the root when sliced makes a stunning visual treat and the crunchy texture turns it into a delightful salad. A good friend from Beijing shared her recipe for this quick and unusual salad. I am forever grateful.
Serves 4
225g (8oz) fresh lotus root
2 teaspoons finely chopped fresh ginger
6 spring onions, finely chopped
3 tablespoons Chinese clear (plain) rice vinegar (or cider vinegar)
salt and freshly ground black pepper
2 teaspoons sugar
Wash the lotus root well in cold running water, then peel and slice it thinly crossways. Put it into a large bowl with all the remaining ingredients and toss gently. Leave for 15 minutes, then serve at once.
Uighur cuisine and culture was quite alien to me, as was the use of mutton in steamed dumplings, noodles (laghman) and samsa (baked dumplings) – I'm not a huge fan of mutton since it has such a strong flavour.
Ken and I were invited to a spring festival New Year party. Ken made a saffron vegetable rice and I made this colourful salad using the best of their local produce. I was particularly in love with the yellow 'carrot' or 'radish' – it was sweeter and less hardy than the orange carrot. The men and women were separated at the party, which I thought was rather 'un-festive', but that is the Muslim tradition – in Chinese culture you eat together with your family at New Year. However, the women all seemed to enjoy the salad that I made, with the female head of the family helping herself to seconds!
Serves 2–4 to share
1 carrot, peeled and sliced into long, thin julienne strips
1 yellow carrot (or yellow pepper, de-seeded), peeled and sliced into long, thin julienne strips
1 large fresh green chilli, de-seeded and thinly sliced
1 large bunch of Chinese celery (thin stalk variety), leaves and stalks finely chopped
1 small bunch of fresh coriander, finely chopped
a small handful of green raisins, finely chopped
a small handful of apricot kernels, finely chopped (you can buy these in health food shops)
2 large ripe pomegranates, halved and seeds removed by hitting the sides using a wooden spoon and collecting the seeds in a bowl
3–4 tablespoons Chinese black rice vinegar (or balsamic vinegar)
3 tablespoons chilli oil (see here)
2 small pinches of salt
For the garnish
2 teaspoons raw (untoasted) sesame seeds
Toss all the ingredients together in a bowl. Spoon onto a serving plate and garnish with sesame seeds.
my second home
Hong Kong is a city that is always in motion. Every time I return, I marvel at the energy, the unrelenting pace of life, the vibrancy of the place. It is a hybrid city, being both east and west at the same time, and it is a city very close to my heart – and even closer to my stomach...
The 'original' Hong Kong – meaning 'fragrant harbour' – was an island off the south coast of Guangdong, and passed into British hands in 1842 because of tea and opium. In 19th-century Britain, the insatiable appetite for tea from China meant that Britain's gold and silver reserves were being used up: the solution was to 'pay' for the tea with opium grown in India. China, not surprisingly, did not want such an import, and so the two countries went to war. When China was defeated in the First Opium War, the British demanded and got the strategically placed Hong Kong island.
After winning the Second Opium War in 1860, Britain was ceded Kowloon, an area on the Chinese mainland opposite the island. Finally, in 1898, another convention extorted from an increasingly weakened China was the so-called New Territories, an area adjacent to Kowloon, and this was leased to Britain for 99 years. In the intervening years, it became obvious that more than the New Territories would have to be handed back, and in late 1984 an agreement was reached: China would take over the entire British Crown Colony in July 1997, but Hong Kong's unique free-enterprise economy would be maintained for at least 50 years.
You may wonder how a communist China could allow capitalist Hong Kong to exist as it is. In my opinion, the answer is simple – the Chinese also like being rich. Much of the investment flowing into Hong Kong in the 1990s was from China, and it still is today. China may be changing, and changing fast, but I think Hong Kong will continue to be China's window on the world as well as the service centre for the rest of the country. With its strong financial markets and business skills, Hong Kong is indispensable to China's fast-growing economy.
Hong Kong is also indispensable because of its cuisine, to me probably the world's finest. The basis of Hong Kong Cantonese cooking is deeply rooted in the rich and ancient traditions of the grand Chinese food culture. Canton may be the capital of Guangdong, but Hong Kong, since the 1950s, has become the place where you can enjoy the very best Chinese food (Cantonese, Shanghai, Fujian and Beijing, as well as European). There are thousands of restaurants, from the smallest street-food stalls to the most luxurious of hotel dining rooms, but all share the same passion: food in all its aspects is central to Hong Kong life (along with work, gambling and shopping).
Cantonese is one of the major regional Chinese cuisines, and is described in more detail here. It is probably the most versatile of the cuisines because it has access to the widest range of ingredients. Hong Kong, because it is open to the world, has access to even more ingredients and ideas than elsewhere in Guangdong, and what is most interesting to me is how Hong Kong is assimilating new foods, ingredients and styles of cooking, and experimenting with them, transforming them into a new style. There is a glorious continuity and unity to Chinese cuisine, but there is also a willingness to adapt, to accept diversity, especially in Hong Kong. New techniques include coating meats in a marinade or light froth of egg white (called 'velveting'), making batters with yeast or baking powder, which are lighter and less oily, and using vegetable oil instead of lard.
No foodie visit to Hong Kong would be complete without trying some seafood dishes (prawns, crabs, oysters, river fish), some wonton soups, some noodles, barbecue pork and perhaps a Hakka speciality or two. You must try to eat at least one Chiu Chow meal, a popular regional southern style of cooking from around the Swatow district of eastern Guangdong province. The last time I ate Chiu Chow I insisted on an old favourite – steamed chicken feet with black bean sauce, which everyone else at the table politely declined. They did, however, join me to eat soy goose, another speciality of the cooking style: the bird is slowly braised in a rich broth of herbs and spices so it comes out very tender and savoury. Or you could try something even more exotic. You'll find bird's nest soup here (very tasty, although the idea puts many people off) – and sea cucumber. Both these special-occasion dishes are, quite justifiably, not cheap. Or you could venture even further into exotica: the Cantonese will eat anything, including snake, civet cat, barking deer or water turtle, the latter farmed in ponds everywhere in China. I have enjoyed roasted rice birds, small birds that fly down to feed on the rice paddies: they include finches and buntings. Or even rice worms, which are netted when the rice paddies flood: steamed, they are much appreciated by the Cantonese.
But the mode of eating that is most characteristic of Hong Kong is dim sum. The words mean 'heart's delight' or 'to touch the heart', and dim sum certainly touch mine. Whether poached, steamed, shallow- or deep-fried, the sheer variety of dim sum is staggering. I love sitting in a dim sum restaurant watching the waiters wheeling their carts around the tables, each cart piled high with bamboo baskets. You can choose from a variety of dumplings, stuffed with meat, vegetables, shellfish, wrapped in rice or wheat-dough casings; you can have spring rolls, fried taro, steamed spareribs, meat balls, pork buns, even some dessert dim sum. Because the portions are small, you can taste a great variety before you are full, something I love doing. And, of course, you must drink tea. In Cantonese, having dim sum is known as yum cha, which means 'drink tea meal'.
Neither wholly Eastern nor entirely Western, Hong Kong is sui generis – a phenomenon never seen previously and never likely to be duplicated. As someone once said, 'If Hong Kong didn't exist, it would have had to have been invented.' I, for one, am very glad that it was invented, as I love it.
Rice and Noodles
Congee with Century Eggs or Salted Duck Eggs
Ching's Longevity Noodles with Garlic, Sesame Oil, Salt and Soy Sauce
Ji Shi Liang Mein Chicken Noodle Salad
Chengdu Leng Mian Noodles
Ants Climbing Trees
Dan Dan Mian Spicy Sichuan Noodles
Hunan-Style Shredded Pork with Chillies and Black Beans on Crispy Noodles
Crossing-The-Bridge Noodles
Ching's Beijing-Sichuan Zhajiang Noodles
Sticky Belly Pork Rice Wrapped in Lotus Leaves
Pineapple Rice
Ching's Banana-Wrapped Smoked Pork with Red and Black Rice
Hainan Chicken Rice
or salted duck eggs
You will find this satisfying, comforting food throughout southern China. It is akin to porridge in the West. I love it for breakfast, often accompanied by a fried doughnut. Century eggs are also known as preserved eggs, hundred-year eggs, thousand-year eggs and thousand-year-old eggs, and are made by preserving duck eggs in a mixture of clay, ash and salt, among other things, for anything up to several months. The resulting texture, flavour and smell is not to everyone's taste, but I have many non-Chinese friends who have become quite addicted to these eggs.
For the more timid, I would suggest salted duck eggs. These are made by soaking duck eggs in brine for a week or so. This results in thick liquid egg white and a firm-textured, round yolk that is bright orange-red in colour. It has a deliciously rich taste. If you are interested in sampling a really exotic taste of China, then this is a recipe for you.
Serves 4
4 century eggs (or 4 salted duck eggs, see here)
2 litres (3 pints) water
short grain or long grain rice measured to the 150ml (¼ pint) level in a measuring jug, unwashed
2 teaspoons salt
3 tablespoons finely chopped spring onions
2 tablespoons finely chopped fresh coriander
If you are using century eggs, simply rinse the shells and peel them, then cut them into quarters. If you are using salted duck eggs, crack the eggs into a small bowl.
Bring the water to the boil in a large pot and add the rice and salt. Let the mixture come back to the boil and give it several good stirs, then turn the heat down to low and cover the pot. Let the mixture simmer for about 1 hour, stirring occasionally.
Add the eggs and simmer, uncovered, for a further 10 minutes.
Just before serving, add the spring onions and coriander and serve at once. (If you like, congee can be made in advance. In which case, reheat it slowly and simply add some more water if the porridge is too thick.)
with garlic, sesame oil, salt and soy sauce
This is a very traditional dish served and eaten on special occasions such as Chinese New Year and, in particular, birthdays. The dish would typically be made for someone to wish them a long life. Noodle makers have to be very skilled to produce very fine noodles without breaking them and it is much harder than making thick udon noodles.
In Beijing, Ken and I requested some noodle chefs make thin mian-sien noodles and they made a bowl quicker than it takes to cook a packet of instant noodles! (You can buy packets of dried, thin, longevity wheat flour noodles. All you need to do is cook them in boiling water for less than 2 minutes and toss sesame oil through them to prevent them from sticking.)
I dressed the noodles they made with the ingredients with which we would traditionally serve the dish – garlic, salt, light soy and toasted sesame oil. Simple, but comforting, a reminder of home.
Serves 2 to share
500g (1lb 2oz) cooked thin wheat flour noodles (225g/8oz dried weight)
1 tablespoon minced garlic
1 tablespoon light soy sauce
2 tablespoons toasted sesame oil
a pinch of salt
Toss the noodles with the garlic, soy sauce and sesame oil. Season with a pinch of salt and mix well, without breaking the noodles. Eat immediately.
Chicken noodle salad
Liang mein (Mandarin Chinese for 'cool noodle') is a popular snack eaten all over China and Taiwan. Different variations can be found all over Southeast Asia.
This recipe takes me back to my childhood, when my grandmother used to make this delicious, simple dish, especially in the hot summer months and she would make it differently every time. The salad can be varied according to your taste by changing the topping, dressing, proteins and noodles. This is not only super quick to make, but also super healthy.
Serves 2 or 4 to share
200g (7oz) dried or fresh Chinese egg noodles
toasted sesame oil
100g (4oz) cucumber, de-seeded and shredded using a mandolin
100g (4oz) carrot, peeled and shredded using a mandolin
300g (11oz) cooked and shredded chicken
For the dressing
2 tablespoons light soy sauce
2 tablespoons Chinese clear (plain) rice vinegar (or cider vinegar)
1 tablespoon toasted sesame oil
1 teaspoon minced garlic
50g (2oz) smooth peanut butter
1 tablespoon water
juice of ½ lemon
To serve
a small handful of finely chopped fresh coriander
a pinch of hot chilli powder
Cook the noodles according to the instructions on the packet, then drain and refresh under cold running water. Drizzle with sesame oil and mix well to prevent them from sticking together, then place in a large bowl. Sprinkle the cucumber and carrot evenly over the noodles.
Put all the ingredients for the dressing in a blender and whizz to combine. Pour the dressing over the noodles, top with the shredded chicken and garnish with the coriander. Finally, sprinkle with chilli powder, toss well and serve.
noodles
Noodles are such a part of Chinese culinary culture that it is hard to imagine that we could be defeated by the weather: despite the hot humid summers of Sichuan, we still eat noodles there, but we eat them cold. They are actually quite refreshing and the spicy hot sauce has a cooling effect on the palate and body. Weather aside, what is more important is that the noodles are tasty and, indeed, these noodles are!
Serves 2–4
450g (1lb) dried or fresh Chinese egg noodles
175g (6oz) fresh beansprouts
For the sauce
2 tablespoons oil, preferably groundnut
2 tablespoons finely chopped spring onions
1 tablespoon finely chopped garlic
1 tablespoon yellow bean sauce
2 teaspoons chilli bean sauce
2 tablespoons sesame paste (or peanut butter)
2 teaspoons finely chopped fresh ginger
1 tablespoon Shaoxing rice wine (or dry sherry)
2 tablespoons light soy sauce
1 tablespoon black rice vinegar (or balsamic vinegar)
2 teaspoons sugar
1 tablespoon chilli oil (see here)
1 tablespoon sesame oil
For the garnish
fresh coriander leaves
If you are using dried noodles, cook them according to the instructions on the packet, or boil them for 4–5 minutes. Then cool them in cold water until you are ready to use them. If you are using fresh noodles, boil them for 3–5 minutes, then immerse in cold water and drain.
Blanch the beansprouts in boiling water for a few seconds – just in and out. Drain immediately and set aside.
Heat the oil in a wok or large frying pan until it is hot. Add the spring onions, garlic, yellow bean and chilli sauces, sesame paste (or peanut butter) and ginger and stir-fry for 2 minutes. Add the rice wine (or sherry), soy sauce, vinegar, sugar and the chilli and sesame oils and stir to mix. Allow the sauce to cool thoroughly.
Combine the noodles and beansprouts in a large bowl and mix thoroughly with the sauce, then garnish with the coriander and serve at once.
This is supposedly a Sichuan noodle snack dish. Its unusual name came about because the small pieces of minced beef (or pork) coat the mung bean noodles and look like ants climbing trees! It is one of my favourite midnight snacks and is so easy to make if you have the ingredients in your store cupboard. It tastes good, makes a great dish for kids and won't take too much time in the kitchen. To get children to eat their carrots, you can grate some and toss through with the spring onions at the end.
Serves 2
2 tablespoons groundnut oil
2 garlic cloves, peeled, crushed and finely chopped
1 tablespoon freshly grated root ginger
1 medium fresh red chilli, de-seeded and finely chopped
250g (9oz) minced beef (or pork)
1 tablespoon Shaoxing rice wine (or dry sherry)
1 tablespoon dark soy sauce
1 tablespoon chilli bean paste
200ml (7fl oz) hot chicken stock
150g (5oz) mung bean noodles, pre-soaked in hot water for 10 minutes, then drained
1 teaspoon toasted sesame oil
2 large spring onions, finely chopped
Heat a wok over high heat and add the groundnut oil. Stir-fry the garlic, ginger and fresh chilli for a few seconds, then add the beef (or pork) and stir-fry for 2–3 minutes until the meat is browned at the edges. Add the rice wine (or sherry), season with the soy sauce and chilli bean paste and mix well.
Add the hot stock and bring to the boil, then add the noodles and stir well.
Season with the sesame oil, then add the spring onions, tossing and mixing all the ingredients well. Serve immediately.
Spicy Sichuan noodles
Noodles for the Chinese are not simply just a meal, but also a snack to alleviate hunger pangs during the day. This Sichuan dish is a great favourite and certainly has my vote: it is satisfying, quick, easy and absolutely delicious. It's typical of what we Chinese call 'small eats' – which are found in tiny restaurants, food stalls and other enterprising eateries. There are many versions of the dish and they are all easy to make, tasty and quite filling. I love this version, which I first ate at a tiny street restaurant in Chengdu. Be particularly careful when deep-frying in a wok.
Serves 2–4
225g (8oz) minced pork
1 tablespoon light soy sauce
1 teaspoon salt
225ml (8fl oz) groundnut or vegetable oil
350g (12oz) fresh or dried Chinese thin egg noodles
1 tablespoon Sichuan peppercorns, roasted and ground
For the sauce
3 tablespoons finely chopped garlic
2 tablespoons finely chopped fresh ginger
5 tablespoons finely chopped spring onions
2 tablespoons sesame paste (or peanut butter)
2 tablespoons light soy sauce
2 tablespoons chilli oil (see here)
1 teaspoon salt
225ml (8fl oz) chicken stock
For the garnish (optional)
1 fresh red hot chilli, de-seeded and shredded
Combine the pork, soy sauce and salt in a small bowl and mix well.
Heat a wok or sauté pan over high heat until it is hot. Add the oil and heat until it is very hot, then deep-fry the pork, stirring with a spatula to break it into small pieces. When the pork is crispy and dry – after about 5–6 minutes – remove it with a slotted spoon and drain on kitchen paper.
Next, make the sauce. Pour the oil from the wok into a strainer over a heatproof bowl, then put back 2 tablespoons into the wok. Reheat the oil, then add the garlic, ginger and spring onions and stir-fry for 30 seconds. Add the sesame paste (or peanut butter), soy sauce, chilli oil, salt and stock and simmer for 4 minutes.
Cook the noodles in a large pot of boiling water, for 2 minutes if they are fresh, or 5 minutes if they are dried. Drain the noodles well in a colander. Divide the noodles between individual bowls or put them into a large soup tureen. Ladle on the sauce, top with the fried pork and Sichuan peppercorns, then garnish with the chilli, if you like, and serve at once.
with chillies and black beans on crispy noodles
This dish is about layers of flavour. It has lots of ingredients but the finished dish is worth the effort. Slivers of meat stir-fried with chillies, black beans and leeks is a classic flavour combination found in Hunnan. I love using lean pork (you could also use chicken) with the crispy noodles. Be particularly careful when deep-frying in a wok.
Serves 2 or 4 to share
1 egg white
1 tablespoon cornflour
sea salt and freshly ground white pepper
300g (11oz) pork fillet, sliced into very fine strips
vegetable oil
2 dried mung bean noodle nests (or use 200g/7oz cooked egg noodles)
2 garlic cloves, peeled, crushed and finely chopped
1 tablespoon freshly grated root ginger
1 medium fresh red chilli, de-seeded and finely chopped
2 dried chillies
1 tablespoon fermented black beans, rinsed and crushed
1 tablespoon Shaoxing rice wine (or dry sherry)
½ teaspoon dark soy sauce 1 tablespoon light soy sauce
1 tablespoon Chinese black rice vinegar (or balsamic vinegar)
2 baby leeks, finely sliced
100ml (3½fl oz) chicken stock
1 tablespoon cornflour, blended with 2 tablespoons cold water
1 teaspoon toasted sesame oil
Combine the egg white, cornflour, salt and white pepper in a bowl. Add the pork strips and toss to coat well. Fill a wok less than half full with vegetable oil and heat the oil to 160°C/325°F, or until a tiny piece of the coating browns in 30 seconds. Add the noodle nests (or noodles) and fry until they have puffed up. Spoon out and drain on kitchen paper. Add the pork and fry for 1–2 minutes until golden. Lift out with a slotted spoon and drain any excess oil on kitchen paper.
Strain the oil through a sieve into a heatproof bowl, then return 1 tablespoon of oil to the wok and reheat. Add the garlic, ginger, fresh and dried chillies and the black beans and toss for a few seconds to release the aroma. Then add the pork and season with the rice wine (or sherry), both soy sauces and the vinegar. Add the leeks and stock and toss together until the leeks have softened and the sauce has come to a bubble. Stir in the blended cornflour until the sauce coats the back of a spoon. Season with the sesame oil.
Place the noodle nests on a serving plate, spoon the shredded pork over the top and serve.
Legend has it that a Chinese scholar, when preparing for the Imperial examinations, isolated himself on an island connected to the shore by a long bridge. His devoted wife would deliver carefully prepared meals to him, but was dismayed that they were always cold on arrival. She finally hit upon the technique of keeping a soup very hot by topping it with a thin layer of vegetable oil. Crossing the long bridge with the hot soup, she was then able to drop the other ingredients into the soup kettle, thus cooking them on the spot. Needless to say, her husband passed his exams. Legend or not, this is a wonderful dish that is typical of Chinese culture, illustrating that food is meant to be shared and is an important part of social interaction. You will find this a great dinner party dish.
Serves 4
25g (1oz) Chinese dried black mushrooms, pre-soaked in warm water for about 20 minutes until soft and pliable, then drained
165g (5½oz) boneless chicken breasts, cut into thinnest slices possible
100g (4oz) boneless lean pork fillet, cut into thinnest slices possible
100g (4oz) fresh beansprouts
350g (12oz) dried rice noodles, pre-soaked in warm water for 15 minutes, then drained thoroughly
4 spring onions, cut into 5cm (2in) pieces)
900ml (1½ pints) chicken stock
For the dipping condiments
5 tablespoons finely chopped spring onions
3 tablespoons chilli bean sauce
2 tablespoons salt, mixed with 2 teaspoons roasted and ground Sichuan peppercorns
5 tablespoons light soy sauce
5 tablespoons Hoisin sauce
5 tablespoons yellow bean sauce
Squeeze the excess water out of the mushrooms, then cut off and discard the woody stems and finely chop the caps.
Arrange the chicken and pork slices on a platter. Place the beansprouts, mushrooms and rice noodles on another platter. Put the spring onions into a small bowl and arrange the dipping condiments in small dishes. Warm four large soup bowls by rinsing them in very hot water.
Bring the chicken stock to a boil in a medium-sized saucepan. Turn the heat down and keep just simmering. Remove the bowls from the hot water and dry carefully. Pour the hot broth into each bowl. Place all the meats, vegetables and dipping condiments in the centre of the table and let each diner layer the meats, vegetables, and finally, the noodles in the hot broth, in order of their preference. Ensure that all the meats are thoroughly cooked through before eating.
Each diner can season the meat and vegetables with the condiments of their choice and drink the soup at the end.
**Ken and I cooked at a noodle restaurant called Noodle Loft, where the delicious noodles were made by the very talented chefs from the Shaanxi region in northwest China, most famous for its traditional hand-pulled noodles. So I cooked with these noodles and made a zhajiang mein, a mixed sauce noodle. It is a classic Beijing dish and very easy to make. There are many different variations, some with more sauce than others, but I like the traditional zhajiang noodles, which are slightly drier than other dishes. Instead of using only leeks and ginger as aromatics, I like to add minced garlic, Sichuan peppercorns and chilli oil to the dish, not forgetting a drop of Shaoxing rice wine, thereby fusing the flavours of two of my favourite regional cuisines – Sichuan and Beijing. I have used pork belly. The trick is to fry it in the wok until the fat is slightly crispy. Make sure you drain off and keep the excess oil after cooking the zhajiang noodles – this intense soy-flavoured oil is delicious when used to moisten plain jasmine rice.**
Serves 2
200g (7oz) plain wheat flour or egg noodles
1 tablespoon sesame oil
1 tablespoon chilli sauce laced with chilli oil (see here)
2 tablespoons groundnut oil
1 tablespoon finely chopped garlic
1 tablespoon finely chopped fresh ginger
2 tablespoons diced baby leeks
1 teaspoon Sichuan peppercorns
250g (9oz) pork belly with skin, diced
1 tablespoon Shaoxing rice wine (or dry sherry)
1 tablespoon fragrant oil (ginger and spring onion-infused oil, see here)
1 tablespoon dark soy sauce
200ml (7fl oz) chicken and pork stock (or just use chicken stock)
3–4 tablespoons wheat flour bean paste (tian mian jiang, or 2½ tablespoons Hoisin sauce and ½ tablespoon yellow bean paste)
2 tablespoons yellow bean paste
½ cucumber, de-seeded and sliced into matchstick strips
2 small red radishes, sliced into matchstick strips
For the garnish
1 spring onion, finely chopped
2 sprigs curly parsley
2 orchid flowers
Cook the noodles according to the instructions on the packet, then drain. Divide the sesame oil and chilli sauce between two bowls, place the cooked noodles in the bowls and set aside.
Heat a wok over high heat and add the groundnut oil. Add the garlic, ginger, leeks and Sichuan peppercorns and toss in the heat for a few seconds. Then add the pork belly and stir-fry for 1 minute. Add the rice wine (or sherry), fragrant oil and dark soy sauce and stir-fry for 1 minute. Add the stock, tian mian jiang (or Hoisin sauce) and yellow bean paste and toss together well. Cook for 2 minutes, stirring until the pork is cooked. Drain off the excess oil through a sieve into a heatproof bowl and save for later.
Divide the pork mixture between the two bowls and sprinkle on the cucumber and radish strips. Garnish with the spring onion, parsley and orchids and serve immediately. Toss all the ingredients well before eating.
wrapped in lotus leaves
This is one of my all-time favourite comfort recipes. Whenever I go to a dim sum restaurant, I cannot resist ordering glutinous rice in lotus leaves. The glutinous rice does not contain any gluten – its name describes the glue-like stickiness of the rice. Glutinous rice is used in savoury and sweet dishes in all the different regional Chinese cuisines and it is often served at Chinese festivals. It's also often ground into a powder and mixed with water to make a dough for desserts and dumplings. The rice is popular not only because it provides a great chewy texture when cooked, but because its stickiness is seen as auspicious – for example, on Chinese New Year, nian gao, a sweetened sticky rice cake, is served because it is believed that, as a result, you and your family members will stick together, or stay close, for the coming year.
These parcels are delicious on their own or with some stir-fried vegetables. You can make this a vegetarian dish by using shiitake mushrooms and extra-firm, pressed braised dofu (tofu) pieces (known as dofu gan), diced carrots and peanuts.
Serves 2–4 to share
1 tablespoon peanut oil
2.5cm (1in) piece fresh root ginger, peeled and grated
2 small shallots, finely chopped
a small handful of dried shrimps, soaked in hot water for 20 minutes, then drained and finely chopped
5 small, dried shiitake mushrooms, soaked in hot water for 20 minutes, then drained and finely chopped
300g (11oz) pork belly, cut into 0.5cm (¼in) square cubes
1 teaspoon Chinese five-spice powder
1 tablespoon Shaoxing rice wine (or dry sherry)
600g (1lb 5oz) cooked glutinous rice (see here)
2 tablespoons light soy sauce
1 tablespoon dark soy sauce
1 teaspoon toasted sesame oil
a pinch of sea salt
freshly ground white pepper
2 dried lotus leaves, soaked in hot water for 20 minutes, then drained
For the garnish
1–2 spring onions, sliced into long strips, then placed in iced water and drained, to make curls
Heat a wok over high heat until it begins to smoke, then add the peanut oil. Add the ginger, shallots, chopped shrimps and mushrooms and stir-fry for 1 minute. Add the pork belly and stir-fry to break it up. When the pork begins to brown, stir in the five-spice and rice wine (or sherry). When the wine has almost evaporated, stir in the cooked rice. Once the rice is incorporated, season with both soy sauces, the sesame oil, salt and white pepper and mix well. Remove from the heat.
Pat the lotus leaves dry with kitchen paper. Spoon half of the sticky rice mixture into the centre of one lotus leaf. Fold in the sides (snug, but not tight), fold up the bottom and roll up, then secure with butchers' twine. Repeat with the remaining lotus leaf and rice. Place in a bamboo steamer and steam for 10 minutes until the fragrance has infused.
Remove from the steamer and unwrap the parcels. Serve garnished with the sliced spring onions.
Glutinous rice
Rinse 300g (11oz) glutinous rice in water until the water runs clear in order to remove excess starch. Place in a pan with 600ml (1 pint) water, then bring to a boil, reduce the heat, cover and simmer for 15 minutes. Ensure all the water has evaporated before using.
Most people have little idea how vast and diverse a country China is. Yunnan province in southwest China borders on southeast Asia, and I discovered to my joy how similar some of the dishes were to Thai cuisine. This dish combines fruit with a savoury mixture. It makes a delicious change and is easy to make once the rice is cooked.
Serves 4
long grain rice measured to the 400ml (14fl oz) level in a measuring jug and cooked
2 tablespoons groundnut or peanut oil
225g (8oz) minced pork
2 tablespoons light soy sauce
salt and freshly ground black pepper
2 tablespoons finely chopped fresh ginger
3 tablespoons finely chopped spring onions
1 tablespoon sesame oil
1 small pineapple, about 225g (8oz), peeled, cored and chopped into 1cm (½in) pieces
Allow the cooked rice to cool thoroughly by spreading it out on a baking sheet. The rice must be cold before you use it in this recipe.
Heat a wok over high heat until it is hot. Add the oil, and when it is very hot and slightly smoking, add the pork and stir-fry for 2 minutes. Add the soy sauce, salt, pepper, ginger and spring onions and continue to stir-fry for 2 minutes. Add the rice, mix well and stir-fry the mixture for another 5 minutes until the rice is heated through and well mixed. Stir in the sesame oil, add the pineapple pieces and stir-fry until the pineapple is heated through, but not cooked. Serve at once.
with red and black rice
It was so much fun cooking with Ken on our outdoor balcony in Xishuangbanna in Yunnan. We created a feast! He had bought all the ingredients from a fresh wet market and we improvised, creating a delightful meal to celebrate our time in Yunnan. I was so inspired by the banana leaves used by the Dai women that I decided to wrap a parcel of local smoked la-rou (Chinese smoked pork), red shelled peanuts and local spices in the leaves. I then sealed the parcels with toothpicks and steamed them until the rice was cooked through and sticky and fragrant (see here). Delicious!
If you can't get hold of the ingredients for the pepper seasoning, use ½ teaspoon of Chinese five-spice powder.
Makes 4
1 teaspoon vegetable oil
a handful of diced smoked pork (or lardons)
100g (4oz) red-shelled peanuts, boiled for 45 minutes, then drained
250g (9oz) each red and black rice, soaked overnight
a pinch of salt
1 tablespoon light soy sauce
2 large banana leaves, each cut in half
For the pepper seasoning
1 tablespoon Yunnan flower peppercorns
2 star anise
2 large black Chinese cardamom pods
Toast the peppercorns, star anise and cardamoms in a wok on medium heat for about 30 seconds, or until they release their aroma. Place in a mortar and pestle and grind (or in a towel and grind using the back of bamboo cleaver, but be careful!).
Heat a wok over high heat and add the vegetable oil. Add the smoked pork and stir-fry for 1 minute. Add half the toasted spices, then stir in the peanuts, rice, salt and light soy sauce and stir-fry for another minute. (You can store the remaining toasted spices in a jar to use another time.)
Divide the mixture between the pieces of banana leaf, wrap up like an envelope and secure each with a toothpick. Place on a heatproof plate in a wok and steam on high heat for 40 minutes until cooked through.
This dish is a wonderful example of the power of food and illustrates how recipes will migrate along with those who enjoy them. It was originally from Hainan Island in the south, where it was made with the famous Wenchang chicken. It is now cooked throughout southeast Asia, especially in Singapore where Chinese immigrants prepare it at home and in their restaurants. Although the method may seem a little laborious, it is well worth the effort, especially if you have a good chicken. Part of the treat is eating it with all the garnishes and sauces, making it a complete meal in itself. The rice is often considered a delicacy on its own – I think you will agree once you have made this recipe.
Serves 4
1 × 1.5kg (3–3½lb) chicken
salt
1.75 litres (3 pints) chicken stock
6 slices of fresh ginger
6 whole spring onions
½ teaspoon freshly ground black pepper
1 tablespoon groundnut or vegetable oil
2 tablespoons finely chopped garlic
long-grain rice measured to the 400ml (14fl oz) level in a measuring jug
For the garnish
450g (1lb) cucumber (about 1 large)
225g (8oz) tomatoes
2 spring onions
For the ginger and spring onion sauce
4 tablespoons finely chopped spring onions, white part only
2 teaspoons finely chopped fresh ginger
1 teaspoon salt
2 tablespoons groundnut or vegetable oil
For the chilli sesame sauce
2 fresh red chillies, de-seeded and finely chopped
2 teaspoons sesame oil
1 teaspoon sugar
½ teaspoon salt
Rub the chicken evenly with 1 tablespoon of salt, then place in a large pot, cover with the stock (adding more stock if necessary) and bring to the boil. Add the ginger, spring onions and pepper, cover tightly, then turn the heat down and simmer gently for 30 minutes. Turn off the heat and leave covered tightly for 1 hour. Remove the chicken from the pot and allow it to cool. Remove the ginger and spring onions with a slotted spoon. Skim off any surface fat from the stock, then measure 900ml (1½ pints) of stock and set it aside – this will be used to cook the rice. Reserve the rest of the stock.
Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is very hot and slightly smoking, add the garlic and 1 teaspoon of salt and stir-fry for 1 minute. Add the rice and continue to stir-fry for 2 minutes. Now add the reserved measured stock to the rice, bring the mixture to the boil and continue boiling until most of the liquid has evaporated. Turn the heat to very low and cover tightly. Let the rice cook undisturbed for 15 minutes. Remove the wok from the heat and let the rice rest for 5 minutes before serving.
Meanwhile, prepare the garnish. Peel the cucumber, slice it in half lengthways and, using a teaspoon, remove the seeds. Slice the cucumber. Thinly slice the tomatoes. Slice the spring onions on the diagonal. Arrange all the slices on a platter and set aside.
Next, make the two sauces. Mix the spring onions, ginger and salt together in a heatproof bowl. Heat the wok over high heat until it is very hot. Add the oil, and when it is very hot and slightly smoking, pour it on to the ginger and spring onion mixture and mix well. Combine all the chilli sesame sauce ingredients in a small bowl and mix well.
Transfer the chicken to a chopping board, cut it into bite-sized pieces and arrange it on a platter. Reheat the remaining stock and serve it in a soup tureen. Serve the rice in a bowl with the plate of garnish and the sauces.
Ching's homecoming
None of my family really knows the exact history of my ancestors, but we think we are descended from Han Chinese people who settled in the southwestern province of Fujian (I can speak a Fujian dialect). For centuries, there was a lot of piracy in the South China Sea, and it is thought that pirates took many Fujian people to the nearby island of Taiwan (possibly as slaves, although no-one knows for sure). To start my 'homecoming' journey, I travelled from Xiamen in Fujian to Taiwan, much as I imagine my Han ancestors would have done...
Fujian Province has a long coastline on the South China Sea, and Xiamen (once known as Amoy), is its major port. Xiamen is now a huge and successful city: helped by governmental investment in the 1980s, it has also benefitted, educationally and culturally, from gifts and donations from what is called the 'overseas Chinese' diaspora. Most of the Chinese who have settled all over the world in the last few centuries come from these southern parts of China, from Guangdong, Hong Kong and Fujian. In their new homes these emigrant Chinese have opened restaurants, formed communities, integrated into communities, worked hard, succeeded, and then wanted to give something back. In many ways, it is the overseas Chinese who have been responsible for nurturing the 'real' Chinese culture and keeping it alive. With the Cultural Revolution, when China closed its doors to the rest of the world, many aspects of traditional Chinese culture disappeared from mainland China, and it was only overseas that they survived. Both Ken and I are 'overseas Chinese', from Canton and Taiwan respectively. I hope that we two might also be able to give something back...
However, some of those setting out from Fujian did not go far, and settled much nearer to their original homeland, in Formosa (christened by 16th-century Portuguese 'Ilha Formosa' or 'beautiful isle'), now known as Taiwan. This island, not far from the Fujian coast, has had a tempestuous history: it has belonged variously to the Japanese and Chinese, and became the base of the Republic of China, led by Chiang Kai-shek, after the ROC lost mainland China to the Communists in 1949. At this time, the defeated government took many ancient and priceless Chinese relics with them to Taiwan, where they are still kept. (If they had remained on the mainland during those tumultuous times they would almost certainly have been lost or destroyed during the Cultural Revolution.) It was also in Taiwan that traditional Chinese festivities were nurtured – with Chinese New Year celebrations, dragon boat and autumn boat festivals among them – and these have now begun to filter back to the mainland. Although the two can be said to remain at odds politically, when China introduced its 'open-door' policy in 1979, it was the Taiwanese who came to the mainland, bringing money with them (they were the first to invest in Shanghai, for instance), as well as those wonderful festival traditions.
Taiwan is where I was brought up until I was five years old, and many of my family still live there. I was fortunate to travel back to Kaohsiung in southwestern Taiwan and then on to Bai He, my grandmother's village in southern Taiwan, where I was reunited with my grandfather and our extended family. My visit coincided with the festival of Qing Ming – the traditional tomb-sweeping ceremony – in early April, when we burned incense together to pay our respects to our ancestors. Because it's a spring festival, we celebrated by eating a feast of cold food: lun piah (fresh spring rolls), cold boiled dofu (tofu), shredded chicken, peanuts and fruit (particularly apples, which are important because they symbolise peace).
These foods are very typically Taiwanese. The island's cooking, however, is inevitably very close to Fujianese, which is considered to be one of the major styles of Chinese cuisine. In fact, Taiwanese cooking could be said to be a fusion of several styles of Chinese cooking: because so many top chefs fled with the Nationalists to Taiwan in 1949, it is probable that the traditional Chinese food culture was able to survive there. There are soups, stews, noodles, dumplings and lots of snacks to be had in the many popular street markets: these include stinky dofu (a fermented dofu/tofu), usually deep-fried outdoors and served with chilli bean sauce and preserved cabbage – it is too smelly to be cooked indoors! Other dishes you might come across are oyster omelettes, salty crispy chicken, and braised pork belly bao (all the rage in America at the moment).
But there are also Hakka and Japanese influences on Taiwanese cooking. The Hakkas, a branch of the Han, are numerous on Taiwan, and they have some very distinctive dishes. These include salt-baked chicken, for instance, stuffed duck, and the festive poon choi, or 'big basin feast', which includes a wonderful selection of poached meat, fish, shellfish, vegetables and often lotus root (extensive fields of lotus grow in Taiwan). The Japanese ruled Formosa (Taiwan) for over 50 years, and inevitably some influences remain. The Taiwanese eat sashimi and sushi, use wasabi and seaweed, and cook with a rice wine that is more like mirin than the Chinese Shaoxing rice wine. And the Japanese are probably also responsible for the Taiwanese passion for karaoke!
My formal welcoming home was a huge feast with my extended family – some twenty to twenty-five people. We ate a broad range of food that was representative of Taiwanese cuisine. It may have been a bit pot-luck, but we did eat well, and it was so good to see everybody. It was also wonderful to see my grandfather, my last remaining grandparent. After we visited my grandmother's shrine, I cooked lunch for the family. I chose to make the traditional dishes that she used to cook for me when I was a child and that remind me of her: bamboo rice, egg and tomato stir-fry, red bean soup dumplings and clams. I first learned to cook in my grandmother's kitchen and cooking there again – but without her – was incredibly poignant.
Vegetarian
Stir-Fried Spinach with Chilli-Fermented Dofu
Mu Shu Vegetarian with Chinese Pancakes
Chinese Pancakes
Lotus Root and Wood Ear Fungus in Soy Sesame Dressing Tossed in Finely Chopped Coriander
Braised Savoury Tree Ears
Stir-Fried Swiss Chard
Buddha's Mixed Stir-Fried Vegetables with Cashews
Ching's Vegetarian Mapo Dofu
Ken's Mapo Dofu Vegetarian Style
Aubergine with Sesame Sauce
Red-Cooked Butternut Squash
Stir-Fried Corn and Chilli
Battered Shiitake and Okra with Citrus Five-Spice Salt
Stir-Fried Cabbage
Liqun Roast Duck Restaurant's Stir-Fried Lettuce
Vegetarian Delight
Stir-Fried Bitter Melon with Black Bean Sauce
Mme Chen's Stir-Fried Shredded Potatoes
Ching's Pu-Er Tea-Leaf Omelette for Monks in Yunnan
Stir-Fried Tomato with Egg
with chilli-fermented dofu
I grew up in a Chinese household that loved vegetables. We ate meat occasionally, but certainly not for every meal. The variety of vegetables in Chinese cuisine seems so much greater than in Western cuisine. I was always happy to have a savoury, stir-fried vegetable dish with plain rice. The secret lies in the flavouring. The hot wok gives the vegetables a grilled, smoky flavour, while elements like garlic and ginger, together with fermented dofu (tofu), give it a mouth-watering appeal. That is what transforms this dish from ordinary to superb and enticing – and it is not only for vegetarians!
Serves 4 as a side disk
1kg (2lb) fresh Chinese water spinach (or European spinach)
2 tablespoons groundnut or vegetable oil
4 garlic cloves, thinly sliced
2 tablespoons finely chopped fresh ginger
3 tablespoons chilli-fermented or plain dofu (tofu)
2 tablespoons Shaoxing rice wine (or dry sherry)
Wash the Chinese water spinach thoroughly and drain. Cut off and discard 5cm (2in) from the bottom of the stem, which tends to be tough. Cut the rest of the spinach into 7.5cm (3in) segments. If you are using ordinary spinach, wash it thoroughly and remove all the stems, leaving just the leaves.
Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is very hot and slightly smoking, add the garlic and ginger and stir-fry for 15 seconds. Then add the dofu and crush it with a spatula, breaking it into small pieces. Add the spinach and stir-fry for 3 minutes. Pour in the rice wine (or sherry) and cook for another 3 minutes. Transfer to a serving platter and serve at once.
with Chinese pancakes
Most people outside China associate our cuisine with rice, but in fact more than half of China thrives on wheat in the form of noodles, steamed bread and crêpe-like wrappings such as pancakes. This is a typical recipe you would find in northern and even western China. Most often you will find this made with either pork or beef. I think my vegetarian version meets the tests of both texture and taste.
Serves 4
25g (1oz) Chinese dried lily stems, soaked for 20 minutes, then drained
50g (2oz) Chinese dried tree ear fungus (mushrooms), soaked for 20 minutes until soft and pliable, then drained
50g (2oz) Sichuan preserved vegetables, soaked for 20 minutes, then drained
3 tablespoons groundnut or vegetable oil
4 eggs, beaten
6 spring onions, finely shredded
1 teaspoon sugar
1 teaspoon salt
1 teaspoon freshly ground black pepper
1 tablespoon Shaoxing rice wine (or dry sherry)
1½ tablespoons light soy sauce
2 teaspoons sesame oil
To serve
1 × recipe Chinese pancakes (see here)
Hoisin sauce or bean paste, for dipping
Trim off the hard ends of the lily stems and shred them by pulling them apart. Rinse the tree ears in several changes of water, remove the hard stems and shred the tree ears finely. Rinse the preserved vegetables in several changes of water, then shred them finely.
Heat a wok over high heat until it is very hot, then add 1½ tablespoons of the oil. When the oil is very hot, add the eggs and gently stir-fry, lifting them up and around until set. Remove immediately and drain on kitchen paper.
Wipe the wok clean with kitchen paper and reheat it over high heat until very hot. Add the remaining 1½ tablespoons of oil, and when the oil is hot, add the lily stems, tree ears, preserved vegetables and spring onions and stir-fry for 1 minute. Then add the rest of the ingredients and stir-fry for another 2 minutes. Return the eggs to the wok and stir-fry the mixture for another 2 minutes, mixing thoroughly. Serve at once with the Chinese pancakes and Hoisin sauce.
Chinese pancakes
Makes 18
275g (10oz) plain flour, plus extra for dusting
225ml (8fl oz) very hot water
2 tablespoons sesame oil
Put the flour into a large bowl. Stir the hot water gradually into the flour, mixing all the while with chopsticks or a fork, until the water is fully incorporated. Add more water if the mixture seems dry. Then remove the mixture from the bowl and knead it with your hands until smooth. This should take about 8 minutes. Put the dough back into the bowl, cover it with a clean, damp towel and let it rest for about 30 minutes.
After the resting period, take the dough out of the bowl and knead it again for about 5 minutes, dusting with a little flour if it is sticky. Once the dough is smooth, form it into a roll about 46cm (18in) long and about 2.5cm (1in) in diameter. Take a knife and cut the roll into equal segments. There should be about 18. Roll each segment into a ball.
Take two of the dough balls. Dip one side of one ball into the sesame oil and place the oiled side on top of the other ball. Take a rolling pin and roll the two together into a circle about 15cm (6in) in diameter. It is important to roll double pancakes in this way because the resulting dough will remain moist inside, and you will be able to roll them thinner but avoid the risk of overcooking them later.
Heat a frying pan or wok over a very low flame. Put the double pancake into the pan or wok and cook until it has dried on one side. Flip it over and cook the other side until dried. Remove from the pan, gently peel the two pancakes apart and set them aside. Repeat this process until all the dough balls have been cooked.
Steam the pancakes to reheat them, or alternatively you could wrap them tightly in a double sheet of foil and put them into a pan containing 2.5cm (1in) of boiling water. Cover the pan, turn the heat down very low and simmer until they are reheated. Don't be tempted to reheat them in the oven, as this will dry them out too much.
If you want to freeze the cooked pancakes, wrap them tightly in clingfilm first. When using pancakes that have been frozen, thaw them in the fridge first before reheating them.
in soy sesame dressing tossed in finely chopped coriander
Lotus roots are used in many Chinese dishes. They can be eaten cold in a salad, like this one, but to prepare them you need to blanch them in boiling water and then finely slice them. They are sweet and crunchy and have a delicate flavour – across between water chestnuts and bamboo. They are delicious in this simple soy sesame dressing paired with crunchy wood ear mushrooms – very refreshing on a hot summer's day as the perfect leng cai, or cold appetiser.
Serves 2 as a side dish
2 fresh lotus roots (vacuum-packed), peeled and blanched in hot water for 2 minutes, then drained and sliced using a mandolin
50g (2oz) Chinese dried black wood ear mushrooms, pre-soaked in warm water for 20 minutes, then drained and finely sliced
2 tablespoons light soy sauce
2 tablespoons toasted sesame oil
2 tablespoons Chinese clear (plain) rice vinegar (or cider vinegar)
a small handful of fresh coriander, very finely chopped
1 teaspoon chilli oil (see here)
Toss the lotus roots and mushrooms together in a bowl with the soy sauce, sesame oil and vinegar and chill for 15 minutes.
To serve, toss the chopped coriander through the mixture and drizzle with chilli oil.
Note: Instead of lotus root, you could use finely sliced tinned water chestnuts and blanched shiitake mushrooms.
Sichuan is ideal territory for mushrooms, as the climate there is warm and moist, which they love. So, naturally, fungi dishes appear on menus everywhere. I was delighted to discover this dish by accident during our filming in Chengdu. After an intensive morning in the spice market, we were all famished and simply popped into the nearest restaurant. Of the procession of many dishes, this one stood out. Dried tree ears were stir-fried and braised in spicy chilli-fermented dofu (tofu) and the result was a vegetarian dish of uncommon deliciousness.
Serves 4
150g (5oz) Chinese dried tree ear fungus (mushrooms), soaked in boiling water for 20 minutes, then drained
1½ tablespoons groundnut or vegetable oil
3 tablespoons coarsely chopped garlic
1 tablespoon finely chopped fresh ginger
3 tablespoons chilli-fermented dofu (tofu)
2 teaspoons sugar
salt and freshly ground black pepper
1 tablespoon Shaoxing rice wine (or dry sherry)
1½ tablespoons light soy sauce
50ml (2fl oz) vegetable or chicken stock (or water)
3 tablespoons finely chopped spring onions
2 teaspoons sesame oil
1½ teaspoons roasted and ground Sichuan peppercorns
Rinse the tree ears well in water, then drain again.
Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is very hot and slightly smoking, add the garlic and ginger and stir-fry for 30 seconds. Add the fermented dofu and crush it in the wok, then add the tree ears and stir-fry, mixing well, for a few minutes. Quickly add the sugar, salt, pepper, rice wine (or sherry), soy sauce and stock (or water). Turn the heat down, cover and cook gently for about 20 minutes, stirring from time to time until the tree ears have absorbed most of the liquid. Turn the heat back to high and continue to cook until most of the liquid has been reduced. Then mix in the spring onions and stir for a few seconds.
Drizzle with the sesame oil, sprinkle the Sichuan peppercorns over and serve at once.
This recipe was inspired by a visit to Chef Yu Bo, one of the rising top chefs in China who has modernised Sichuan cooking with his own personal twist. I went with him to an organic farm near Chengdu to pick fresh greens to cook. Although the vegetable he picked is not available outside of Sichuan, I found Swiss chard to be quite similar and to work just as well. However, Chef Yu Bo did have a secret ingredient: a 10-year-old chilli bean paste redolent with deep flavours of garlic, chilli and broad beans – rich but incredibly fragrant at the same time. I managed to duplicate a version – not quite like his, but delicious, nevertheless.
Serves 4
450g (1lb) Swiss chard (or Chinese water spinach), stalks and leaves separated
1½ tablespoons groundnut or vegetable oil
2 tablespoons chilli bean paste
2 tablespoons finely chopped fresh ginger
2 teaspoons sugar
salt and freshly ground black pepper
4 tablespoons finely chopped spring onions
5 tablespoons finely chopped celery heart
4 tablespoons finely chopped fresh coriander
With a sharp knife, remove any tough fibres from the chard (or spinach) stalks. Wash thoroughly, then chop coarsely into fairly large chunks and set aside.
Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is very hot and slightly smoking, add the chilli bean paste and ginger and stir-fry for 1 minute until it is fragrant. Then add the Swiss chard (or spinach) stalks and stir-fry for 2 minutes. Add the leaves and stir-fry for 3 minutes until wilted. Season with sugar, salt and pepper.
Finally, stir in the spring onions, celery heart and coriander and stir-fry for another minute. Turn on to a platter and serve at once.
with cashews
Sometimes a plate of stir-fried vegetables is enough to make you feel whole again – there is nothing more satisfying than a plate of greens to make you healthy! I love making a simple stir-fry sauce using vegetarian oyster sauce (made from mushrooms), light soy and good-quality stock – the one shown here is an all-purpose stir-fry sauce that I use to season vegetables or meat. This dish is simple and so tasty – make the sauce, keep to one side, stir-fry the ingredients, add the sauce, and Bob's your uncle! You can use any vegetables that are in season, but make sure that as you stir-fry you keep the vegetables al dente to retain their nutritional value and delicious flavours.
Serves 2–4 to share
1 tablespoon groundnut oil
1 tablespoon freshly grated root ginger
1 carrot, peeled and sliced into long, thin julienne strips
5 fresh shiitake mushrooms, sliced
a small handful of baby corn, sliced in half
1 × 225g tin bamboo shoots, drained and sliced into long, thin julienne strips
a handful of roasted, salted cashew nuts
a small handful of beansprouts
2 spring onions, sliced on the diagonal
For the sauce
100ml (3½fl oz) cold vegetable stock
1 tablespoon light soy sauce
1 tablespoon vegetarian oyster sauce
1 teaspoon toasted sesame oil
1 tablespoon cornflour
Combine all the ingredients for the sauce in a bowl, then put to one side.
Heat a wok over high heat and add the groundnut oil. Add the ginger and stir-fry for a few seconds. Tip in the carrot, mushrooms, baby corn and bamboo shoots and stir-fry for 2 minutes. Add the sauce to the wok and bring to the boil. Add the cashew nuts and toss together. When the sauce has thickened, reduce the heat and add the beansprouts and spring onions, toss well and cook for less than 1 minute. Transfer to a serving plate and serve, with jasmine rice.
with jiang dou (Chinese long bean) and shredded pickled bamboo in chilli oil
In Chengdu Ken and I had the pleasure of visiting a 'fly' restaurant recommended by our friend Jenny Gao. Fly restaurants are so-called because they are usually hole-in-the-wall places found in alleyways (or are difficult to find) but their food is so good that people swarm around them – like flies to good food! Ken and I had such fun. We tried the water-cooked carp – slices of carp flash-fried in oil so they are super tender. A layer of local vegetables is placed on the bottom of a dish, the carp is layered on top then Sichuan pepper and ground dried Sichuan chilli flakes are sprinkled over the entire dish. Hot sizzling oil is added, which turns the dish a blood red. Fierce, strong and spicy, it had us reaching for the bai jiou (distilled white liquor), which is robust enough to cleanse the palate between mouthfuls of mala (numbing spicy heat). In the end we probably drank too much bai jiou, but I realised that one cannot really have enough bai jiou, even at 52 per cent alcohol volume!
We also tried dishes like tender rabbit cubes stir-fried with green chillies (which were mild and slightly spicy), garnished with a thread of green flower peppercorns native to Yunnan (Sichuan's neighbour).The delicious la-rou was another dish we sampled – this was a front shoulder of pork, which was tea-smoked, then wrapped in lotus leaves and steamed.
Then came the famed Old (pockmarked) Mrs. Chen's dofu (tofu). It was the classic reddish-brown, numbingly spicy, delicious wobbly dofu but with... wait for it... pigs' brains! It was the first and the last time for me. I am Chinese and I do love offal, from pigs' ears to the intestines, but I draw the line at brains. Ken, on the other hand, loved it! But he's Cantonese and they eat everything. I did enjoy the dofu though, and we then convinced the head chef to show us his version of the classic mapo dofu. First he bathed small cubes of dofu in hot water and added a pinch of salt to 'open it up to flavours', then he drained the cloudy water and set the dofu aside. He heated a combination of lard and chilli oil in a hot wok and added chilli bean paste, fermented black beans, chilli paste (using pickled chillies), minced garlic and ginger to the wok to 'explode' in the hot oil. He then added half a ladleful of minced pork belly and tossed the ingredients together. He poured in some cooking wine, followed by a quarter cup of water, then seasonings of light soy sauce and black rice vinegar. He added the dofu and tossed it together, then added cornflour to thicken the sauce.
Then he tossed in a large handful of suan miao (the dark green, tougher part of the Chinese leek, cut into 1cm/½in slices). Then, with a final toss, it was on the plate and a few pinches of toasted and ground Sichuan peppercorns were sprinkled over the top.The lesson was over and then it was our turn to make our version of the same dish! Ken made a delicious vegetarian version with Sichuan preserved vegetables, which gave the dish a briny-sour taste, increasing its umami flavour.
I also made a vegetarian version of mapo dofu, except I used pickled bamboo shoots in chilli oil and jiang dou, the local Sichuan long bean, which I thought gave the dish extra crunch and complemented the soft dofu well, adding texture and kou-gan (mouth-feel) to the dish. (I've also made a beef version, see here.)
Serves 2–4 to share
500ml (17fl oz) water
250g (9oz) fresh silken dofu (tofu), sliced into 1cm (½in) cubes
1–2 tablespoons vegetable oil
150g (5oz) jiang dou (or French beans), cut into 1cm (½in) pieces
100g (4oz) pickled sour bamboo shoots in chilli oil (or pickled cornichons), cut into 1cm (½in) pieces
1 tablespoon cornflour blended with 2 tablespoons cold water
2 pinches of toasted and ground Sichuan peppercorns
For the sauce
1 teaspoon minced garlic
1 teaspoon minced fresh ginger
1 teaspoon fermented salted black beans, rinsed
1 tablespoon chilli bean paste
1 tablespoon light soy sauce
1 tablespoon Chinese black rice vinegar (or balsamic vinegar)
1 tablespoon Shaoxing rice wine (or dry sherry)
200ml (7fl oz) water
Heat a wok over high heat. Add the water and heat, then add the dofu pieces and swirl in the water for about 1 minute (to 'clean' the dofu). Drain and set to one side.
Reheat the wok over high heat and add 1 tablespoon of vegetable oil. Combine all the ingredients for the sauce except the water, then add to the wok to 'explode' in the heat. Add a little more oil if necessary. Add the beans and pickled bamboo pieces (or cornichons) and toss together well. Then add the water and bring to the boil. Return the dofu to the wok and mix well. As the sauce bubbles, add the blended cornflour and stir in to thicken the sauce. Transfer to a serving plate and garnish with the ground peppercorns. Serve immediately with jasmine rice.
vegetarian style
Mapo dofu (pockmarked tofu) is probably as well known in China as fish and chips are in the UK. Legend has it that Old Mrs Chen, who was nicknamed 'Pockmarked Chen' for the marks on her face, opened an inexpensive restaurant in Chengdu and served this dish in response to the restaurant competition on her street. Whatever the reason, it is a brilliant way to blend spicy, peppery, hot, tender, fresh and fragrant ingredients into a classic dish.
I have taken the meat out of the recipe and made an equally tasty vegetarian version, which I know you will love.
Serves 4
100g (4oz) Sichuan preserved vegetables, soaked for 20 minutes, then drained
2 tablespoons groundnut or vegetable oil
2 tablespoons coarsely chopped garlic
1 tablespoon whole yellow bean sauce
1 tablespoon dark soy sauce
salt and finely ground black pepper
1–2 teaspoons red chilli powder
150ml (¼pint) vegetable stock
450g (1lb) fresh soft or silken dofu (tofu), cut into 2.5cm (1in) cubes
2 teaspoons cornflour, mixed with 1 tablespoon water
For the garnish
1 tablespoon Sichuan peppercorns, roasted and finely ground
Rinse the preserved vegetables in several changes of water, then mince finely. Heat a wok or large sauté pan until it is hot. Add the oil, and when it is hot, add the garlic and stir-fry for 20 seconds, then add the preserved vegetables and continue to stir-fry for another 30 seconds. Then add the yellow bean sauce, soy sauce and salt and pepper and stir-fry for another minute. Add the chilli powder and stir-fry for 30 seconds, then pour in the stock, add the dofu and cook for 3 minutes. Stir in the cornflour mixture and cook for another minute.
Ladle the mixture into a serving bowl, garnish with the ground Sichuan peppercorns and serve at once.
Chengdu in Sichuan province gets extremely hot and humid in the summer and, as a result, many of the residents tend to eat room-temperature dishes, rather than hot, stir-fried ones. Normally, the Sichuanese cooks would fry the aubergine in a wok (because of the lack of ovens in the home) but I found the traditional method too heavy and greasy for my taste. However, by cooking the aubergine first in the oven and then adding the sauce, I have kept the spirit of Sichuan alive in the dish. It makes a lovely side vegetable dish or a main dish with rice.
Serves 4
675g (1½lb) Chinese aubergine (or regular aubergines)
For the sauce
3 tablespoons sesame paste (or peanut butter)
2 teaspoons roasted and ground Sichuan peppercorns
2 tablespoons sesame oil
2 teaspoons chilli oil (see here)
1 tablespoon sugar
1 tablespoon finely chopped garlic
salt and freshly ground black pepper
1 tablespoon chilli bean sauce
2 teaspoons sesame oil
3 tablespoons finely chopped fresh coriander
For the garnish
fresh coriander leaves
Preheat the oven to 200°C/400°F/gas 6. Put the aubergines into a roasting tin and bake them, for about 35 minutes if they are the Chinese variety or 50 minutes if they are the larger variety. They should be charred outside and tender inside. Allow them to cool thoroughly, then peel them. Set aside until you are ready to use them.
When you are ready to serve the dish, combine all the sauce ingredients together with the cooked aubergines and mix well. Garnish with the coriander leaves and serve at room temperature.
This is a simple but delicious dish that can be served as an accompaniment or as a main vegetarian dish alongside vegetables. 'Red-cooked' is a term used in Chinese cuisine to describe a sauce that uses light and dark soy, sugar and spices (such as star anise) to create a deep reddish-brown braising liquid to flavour meat, fish or eggs. This is a quick red-cooked dish using butternut squash; the result is a sweet and savoury, spiced dish.
Serves 2–4 to share
1 medium butternut squash, halved lengthways and peeled
1 tablespoon peanut oil
2.5cm (1in) piece fresh root ginger, peeled and grated
3 star anise
2 cinnamon sticks, about 7.5cm (3in) each
4 tablespoons light soy sauce
2 tablespoons dark soy sauce
250ml (9fl oz) hot water
a pinch of Chinese five-spice powder
1 heaped tablespoon brown sugar
For the garnish
a small handful of fresh coriander
Scoop out and discard the seeds from the squash and cut the flesh into bite-sized (2.5cm/1in) pieces.
Heat a wok over medium-high heat, then add the peanut oil. Add the ginger, star anise and cinnamon and stir-fry for a few seconds, then add the squash and stir-fry for 1 minute. Season with both soy sauces and stir-fry for 1 minute. Add the hot water and stir to combine. Cover the wok and cook, stirring occasionally, until the squash is tender – about 10 minutes. Add more water if necessary to keep the squash from sticking to the sides of the wok. Season with the five-spice and brown sugar and toss well. Remove from the heat and stir to combine. Garnish with the coriander, then serve.
I am constantly amazed at the influence of history on the culture of food and the migration of certain dishes and customs as populations move. A tiny country like Portugal was responsible for spreading the foods of the New World to China in the 16th century. Foods such as corn, chillies and peanuts until then were totally unknown to the Chinese. Yet here is a simple, home-cooked recipe that is satisfyingly delicious and easy to make.
Serves 4 as a side dish
275g (10oz) fresh (about 2 ears) or frozen corn
1½ tablespoons groundnut or vegetable oil
salt and freshly ground white pepper
2 large fresh red chillies, de-seeded and finely chopped
1 teaspoon sugar
50ml (2fl oz) vegetable or chicken stock
If the corn is fresh, cut the kernels off the cob. Blanch frozen corn for 5 seconds in boiling water and drain.
Heat a wok or large sauté pan over high heat until it is hot. Add the oil, salt, corn and chillies and stir-fry for 1 minute. Add the pepper, sugar and stock and continue to cook for 3 minutes. Serve at once.
with citrus five-spice salt
This is a great way to make vegetables tasty – coat them in a rich batter and then deep-fry until crisp and serve with a five-spice salt. To make your own citrus five-spice, dry toast the whole spices in a hot pan or wok for a few seconds to release their oils and aroma and then grind them in a spice grinder – the perfect seasoning for vegetables, fish and meat.
Serve 2–4 to share
2 egg yolks
100g (4oz) potato flour (starch)
2–3 tablespoons cold water (if necessary)
sea salt and freshly ground white pepper
vegetable oil, for deep-frying
6 large fresh shiitake mushrooms
200g (7oz) okra, sliced down the middle on the diagonal
For the citrus five-spice salt
1 tablespoon fennel seeds
1 tablespoon star anise
2 tablespoons Sichuan peppercorns
1 tablespoon cloves
1 cassia bark
2 pieces of dried tangerine peel
1 tablespoon coarse sea salt
Heat a wok over medium heat, then add all the ingredients for the citrus five-spice except the salt and dry-toast for 1 minute to release all the oils and flavours. Transfer to a pestle and mortar or spice grinder and grind until fine. (If using a pestle and mortar, you might want to pass the spice mixture through a fine sieve.) Combine 1 tablespoon of the spice mix with the salt in a bowl, mix well and transfer to a small pinch pot. (Store the remainder in a glass jar and use to season fish or meats, or in marinades.)
Put the egg yolks, potato flour and water into a bowl and mix to a rough batter (not smooth) – the amount of water you use will depend on the size of the eggs. Season with salt and pepper.
Half-fill a wok with vegetable oil, then heat the oil to 180°C/350°F, or until a cube of bread turns golden brown in 15 seconds. Dip the mushrooms and okra pieces into the batter, then fry for 2 minutes, or until golden brown. Remove with a slotted spoon and drain on kitchen paper. Season with the citrus five-spice salt while the vegetables are still hot, then transfer to a serving plate and serve.
Hong Ying, a well-known writer and a great cook, invited me to cook with her in Beijing. It was an irresistible offer that I could not refuse. We went together to her local, colourful market that was as stocked as generously as any you would find in the West. I found a beautiful head of cabbage, a real staple in Beijing kitchens, and some lovely dried shrimp (found in Chinese grocers or supermarkets), which inspired me to cook this dish. Since Hong Ying is from Sichuan, I naturally found some chilli bean paste, which I threw into the dish together with some gin she happened to have handy.
Serves 4
1½ tablespoons groundnut or peanut oil
3 tablespoons coarsely chopped garlic
50g (2oz) dried shrimps, coarsely chopped
450g (1lb) savoy cabbage, halved, cored and cut into 5cm (2in) thick strips
2 tablespoons chilli bean sauce
2 tablespoons gin (or Shaoxing rice wine or dry sherry)
300ml (½ pint) room temperature chicken stock (or water)
salt and freshly ground white pepper,
Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is very hot and slightly smoking, add the garlic and dried shrimps and stir-fry for 30 seconds. Then add the cabbage and stir-fry for 5 minutes. Next, add the chilli bean sauce and gin (wine or sherry), stir in the stock (or water) and season with salt and pepper. Continue to cook for 10 minutes, or until the cabbage is tender. Serve at once.
I was surprised and enchanted by this simple dish, which I ate at one of the best Peking duck restaurants in Beijing. After sampling the famous and fantastic duck with condiments, I tried their other dishes. This one was unexpected and outstanding – crispy lettuce stir-fried, instead of blanched as it would be in Guangzhou. The result was a mouthwatering dish that I could have quite happily eaten just with rice as a main course.
Serves 2–4
1½ tablespoons groundnut or vegetable oil
salt and freshly ground black pepper
½ teaspoon roasted and ground Sichuan peppercorns
4 tablespoons coarsely chopped garlic
675g (1½lb) iceberg or cos lettuce, leaves separated
4 tablespoons oyster sauce (or vegetarian oyster sauce)
3 tablespoons finely chopped spring onions
Heat a wok or large frying pan over high heat until it is hot. Add the oil and when it is slightly smoking, add the salt, pepper, Sichuan peppercorns and garlic and stir-fry for 15 seconds. Then add the lettuce leaves and stir-fry for 1 minute, or until they have wilted slightly. Add the oyster sauce and mix thoroughly, then add the spring onions. Transfer to a serving dish and serve at once.
Vegetarian cooking in China is not one-dimensional. Rather it tends to materialise into delicious tasty dishes with many textures, such as this one. Although it has quite a few ingredients, it is easy and simple to make. It may convert even the most die-hard meat eater!
Serves 4
25g (1oz) Chinese dried black mushrooms, pre-soaked in warm water for about 20 minutes until soft and pliable, then drained
6 eggs, beaten
1 teaspoon salt
2 teaspoons sesame oil
3 tablespoons groundnut or vegetable oil
1 small onion, peeled and sliced
2 tablespoons finely chopped fresh ginger
2 tablespoons finely sliced garlic
15g (½oz) cloud ear fungus (mushrooms), pre-soaked in warm water for 20 minutes, then drained and rinsed well
125g (4½oz) pressed dofu (tofu), cut into thin strips
50g (2oz) bean thread noodles, pre-soaked in warm water for 15 minutes, then drained well
225g (8oz) cucumber, peeled, de-seeded and cut into thin strips
3 tablespoons light soy sauce
2 tablespoons yellow bean sauce
3 tablespoons Shaoxing rice wine (or dry sherry)
1 tablespoon Hoisin sauce
2 teaspoons sesame oil
Squeeze the excess water out of the mushrooms, then cut off and discard the woody stems and finely shred the caps into thin strips. Set aside.
Combine the eggs, salt and sesame oil in a small bowl and set aside.
Heat a wok or large frying pan over high heat until it is hot. Add 1½ tablespoons of the oil, and when it is very hot and slightly smoking, turn the heat down to moderate. Add the egg mixture and stir-fry for a few minutes, or until the egg has barely set. Remove the egg from the wok and drain on kitchen paper.
Wipe the wok clean with kitchen paper and reheat it. When it is hot, add the remaining 1½ tablespoons of oil. When it is very hot and slightly smoking, quickly add the onion, ginger and garlic and stir-fry for 2 minutes. Then add the black mushrooms, cloud ears, pressed dofu, bean thread noodles and cucumber and stir-fry for 2 minutes. Add the soy sauce, yellow bean sauce, rice wine (or sherry), Hoisin sauce and sesame oil and stir-fry for 3 minutes. Then add the cooked eggs and stir-fry for 1 minute. Turn out on to a serving platter and serve at once.
with black bean sauce
This recipe is one of my favourites, partly because it brings back childhood memories of the fragrance of black bean sauce mixed with garlic that often greeted me at the door. So it was natural for me to cook this dish for my family in Kaiping, as it evoked the memory of my mother and her home cooking. I was thrilled that my family loved it.
Serves 4
700g (1½lb) bitter melon, halved lengthways and de-seeded
1 tablespoon groundnut or vegetable oil
100g (4oz) fresh mild red chillies, halved, de-seeded and finely sliced
1 tablespoon finely chopped fresh ginger
2 tablespoons finely chopped garlic
2 tablespoons finely chopped shallots
2 tablespoons finely chopped spring onions
3 tablespoons black beans, coarsely chopped
2 teaspoons sugar
2 tablespoons Shaoxing rice wine (or dry sherry)
150ml (¼ pint) chicken stock or water
For the garnish
1 tablespoon sesame oil
Cut the melon into fine slices and blanch in a pot of boiling water for 2 minutes. Remove with a slotted spoon and drain well on kitchen paper.
Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is hot and slightly smoking, add the chillies, ginger, garlic, shallots, spring onions and black beans and stir-fry for 2 minutes. Then add the bitter melon, sugar and rice wine (or sherry) and stock. Bring the mixture to the boil, then reduce the heat to medium. Cover the wok or pan and simmer for 8 minutes, or until the melon is cooked and tender. Drizzle in the sesame oil, turn onto a dish and serve at once.
Potatoes rarely appeared on my mother's table except in braised dishes. So I was surprised when I met Xingyun Chen, the stylish aunt of our friend Jenny Gao, who is a superb cook, and she served a stir-fried potato dish. Her secret was to cut the potatoes into fine shreds, so that they needed only a small amount of cooking and retained a delightful crunch. It was so good that I greedily very nearly ate the entire lot myself. For a dish like this, I could even consider becoming a vegetarian.
Serves 4
450g (1lb) potatoes, peeled and thinly sliced
salt and freshly ground black pepper
1 tablespoon groundnut or vegetable oil
3 tablespoons coarsely chopped garlic
2 tablespoons finely chopped fresh ginger
2 tablespoons finely chopped pickled ginger
2 tablespoons de-seeded and finely chopped fresh red chillies
2 teaspoons sugar
2 tablespoons Shaoxing rice wine (or dry sherry)
2 teaspoons chilli oil (see here)
1½ teaspoons roasted and ground Sichuan peppercorns
Stack the potato slices and cut them into matchsticks. Soak them in a bowl of cold water with 1 teaspoon salt for 5 minutes, then drain thoroughly and blot them dry with kitchen paper.
Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is very hot and slightly smoking, add the garlic, the fresh and pickled gingers and the chillies and stir-fry for about 30 seconds. Season with salt and pepper, then add the potatoes and gently stir-fry for 1 minute or so until they are well coated with the spices and flavourings. Add the sugar and rice wine (or sherry) and continue to stir-fry gently over a high heat for 5 minutes, or until most of the water has evaporated and the potatoes are cooked. At this point, add the chilli oil, sprinkle on the peppercorns and serve at once.
for monks in Yunnan
The Bulang girls, both nicknamed Xiao-Yu (see here), took me to their village monastery and I made a fresh pu-er tea omelette for the monks. In return, one of them said a prayer for us and gave us his blessings. It was a really special experience for me.
Serves 2
2 tablespoons vegetable oil
7 eggs, beaten
6 pu-er tea leaves (or fresh sweet basil), finely chopped
2 pinches of salt
Heat a wok over high heat, and as the wok starts to smoke, add the vegetable oil.
Quickly add the beaten eggs, pu-er tea leaves and salt and stir to scramble, then let the egg set into an omelette (take off the heat if it gets too hot). Fry until the egg is cooked and golden brown, flipping it over halfway through to ensure even cooking. Take off the heat, cut into pieces and eat immediately.
Pu-er tea
The pu-er tea that most people will be familiar with is the aged, fermented pu-er, which makes a reddish, light brown brew, but if you use green, unfermented pu-er leaves (which make a light yellow brew), they will give a better result in this dish. However, if you can't get fresh pu-er tea leaves, then use fresh sweet basil instead.
Of course, the tomato is not Chinese in origin, although we would have loved to have claimed that it was. Instead, it comes from the New World, and was probably brought to China via Portuguese traders.
The Chinese have taken to tomatoes, like the rest of the world, and although it is officially classified as a fruit, the Chinese, like the Europeans, treat the tomato as a vegetable and would swear emphatically that this is a Chinese tradition. So, frequently, it is stir-fried, in this case with eggs. It makes a quick, easy and satisfying everyday dish.
Serves 2–4
6 eggs, beaten
2 teaspoons sesame oil
salt and freshly ground white pepper
6 spring onions
1½ tablespoons groundnut or vegetable oil
450g (1lb) fresh tomatoes, cut into quarters and then into eighths
Combine the eggs with the sesame oil and salt to taste in a medium-sized bowl, then set aside.
With the flat side of a cleaver or knife, crush the spring onions and then finely shred them.
Heat a wok or large pan until it is hot. Add the oil, spring onions, salt and pepper and stir-fry for 30 seconds. Then add the tomatoes and eggs and cook, stirring continuously, until the eggs are set – about 5 minutes. Quickly place on a platter and serve at once.
and the Chinese minorities
I have been many times to southern China, mostly of course to Taiwan, where I was raised, but I had not visited Yunnan before. I had heard many good things, but didn't quite know what to expect. When we arrived, it really felt as though we were in the tropics, especially after the freezing, damp cold of Sichuan. As soon as we got off the plane, Ken stripped off his shirt to enjoy the sunshine!
Yunnan lies in the far southwest of China, bordering externally on Myanmar (Burma), Laos and Vietnam, and internally with the Tibet Autonomous Region (briefly, to the north), Sichuan, Guizhou and Guangxi. The name 'Yunnan' means 'south of the clouds', and the province ranges from the northern snow-capped mountains to the steamy humidity of tropical forests in the south. There is a rich variety of animal and plant life, and Xishuangbanna, where we stayed while filming, is home to the last few Asian elephants in China.
Probably because of its geographical position, Yunnan has the highest number of ethnic minorities in the whole of China. At least 25 minority groups of the 56 in China are represented: in the south they are related to Thais and Laotians; further north, they are mainly Tibeto-Burmese. Each group has its own spoken language, cuisine, belief systems, festivals and modes of dress. Over the centuries, migrants from the more crowded east, persuaded by government or compelled by invading forces, came to the more sparsely populated Yunnan. These people were mainly Han Chinese, who constitute over 90 per cent of the population of the People's Republic of China (and are the largest single ethnic group in the world). I was lucky enough to experience some of the Dai and Bulang cultures in Yunnan during our trip, and discovered that the region boasts some of the most colourful and spectacular festivals and traditions specific to each group – a really wonderful sight to see.
As you might expect, Yunnanese cuisine is very varied, a mix of the cooking styles of the Han majority and of the resident minorities, and very much influenced by the varying climates. In the north, the cooking is mainly Tibetan in style, with lots of meat (beef, pork, lamb and yak) made into hotpots and curries. There are quite a few mushroom dishes too, because so many wild fungi grow in the mountains. Surprisingly, in a country that does not have much dairy, cow's, goat's and yak's milk cheeses are made by the Bai minority An acquired taste, yak butter tea, a staple of these cold northern parts, is said to keep you warm, enhance the circulation and skin, and fight fatigue.
In the south, the Dai are Southern Buddhist and so many (but not all) of their dishes are vegetarian, using rice, noodles, vegetables, dofu (tofu), and perhaps some fish.
Insects such as grasshoppers, cockroaches and bamboo 'worms' (really caterpillars) are occasionally eaten, as they are rich in protein. The local bamboo is used in many ways: as building material ( for houses, bridges and scaffolding), as cooking vessels, as a preserving medium, and to make bags, trays and plates... One of the most famous Dai dishes is sticky rice roasted in the hollow stalk of a piece of bamboo.
Rice has multiple uses too: apart from being steamed and eaten, it is made into rice cakes, rice noodles, paper (for lamps, books and art), and wine. Pineapple rice is sticky white (or black) rice cooked with coconut milk and pineapple, sometimes even served in a hollowed-out half pineapple. And 'crossing-the-bridge' noodles is perhaps the most famous Yunnan dish: a clear chicken soup with rice noodles, slivers of dofu (tofu), vegetables and meat, seasoned with chilli and onion (see here for Ken's recipe and an explanation of the curious name). Because Yunnan is so close to Thailand, the food almost feels southeast Asian rather than Chinese, in that they use a lot of chillies, fresh Thai mint and coriander. Ken and I had a wonderful time cooking together on a balcony overlooking the river: Ken made a delicious rice and noodle stir-fry with some local mushrooms, while I steamed a fish and served it with a salsa verde made with local herbs (see here for my recipe).
A multitude of plants grow well in Yunnan, and contribute to the local economy, among them tobacco, coffee and rubber. Flowers are a major crop too, some for the export cut-flower trade, but many are also used as food: day-lilies, chrysanthemums, squash flowers and jasmine buds are but a few of the many that are eaten in various ways. But perhaps the province's most famous product is tea, particularly the renowned pu-er tea.
We went into the countryside to pick tea with two local girls (from the Bulang tribe, which came from Laos and Burma centuries ago). They taught me to pick only the top bud and top two leaves from the stem. The tea is then dry-fried in a wok, rolled up and left in the sun. The leaves can be used straight away (for unfermented, 'rough' tea, which tastes good but is a little bitter), or left to ferment further (for a mature tea that tastes almost sweet). The leaves are sold loose, or in compressed shapes such as 'bricks'. The age of the bush affects the taste of the tea as well – the older the better and the more expensive: some bushes are more than 500 years old – as does the type of soil and exposure to sunlight. Many pu-er teas are sold labelled with the year of production. The whole process is almost as complex as wine-making!
The girls and I also cooked with the tea: they made a delicious salad with raw leaves and wild flowers; I stir-fried some raw leaves with chicken, and made a tea omelette for a local monk. The two girls and I, despite not speaking the same language, got on so well that we cried when it was time for us to leave. I gave them each a lipstick as a souvenir.
Fish and Seafood
Sichuan Sea Bass with Chives
Ching's Steamed Sea Bream with a Vietnamese Mint and Wild Coriander Salsa Verde
Fried Fish Snack from Kashgar Market
Fish in Wine Sauce
Xi Hu Cu Yu Sweet and Sour West Lake Steamed Carp
Long Jing Chao Xia Stir-Fried Prawns with Green Tea
Dofu Casserole
Phoenix Tail Prawns
Chef Yu Bo's Fish-Fragrant Prawns
Steamed Scallops from Huangsha Seafood Market
Stir-Fried Crab with Ginger and Spring Onions
Smoky Hot Scallops with Bamboo Shoots and Spring Onions
with chives
I love combining meat and fish – as used in many Chinese dishes – but this meat and fish pairing is made more delicious by adding strong Sichuan flavours. Sea bass fillets are first shallow-fried in oil and then 'wokked' with a fragrant spiced oil of Sichuan chillies, Sichuan peppercorns and smoky bacon lardons, then seasoned with chilli bean paste, black rice vinegar and chilli oil – all tossed with delicious Chinese chives. This is a spicy, salty, numbing, rich dish that is full of flavour and best served with jasmine rice and iced cold beer.
Serves 2–4 to share
a pinch each of sea salt and freshly ground white pepper
1 tablespoon cornflour
400g (14oz) sea bass fillets
vegetable oil
2 small dried Sichuan chillies
¼ teaspoon Sichuan peppercorns
50g (2oz) smoked bacon lardons
1 teaspoon chilli bean paste
150g (5oz) Chinese chives (garlic chives, or baby leeks)
1 tablespoon vegetable stock (or water)
1 teaspoon Chinese black rice vinegar (or balsamic vinegar)
1 tablespoon chilli oil (see here)
Put the salt, white pepper and cornflour into a bowl. Add the sea bass and toss well to coat the fish in the mixture.
Fill a wok to quarter full with vegetable oil and heat the oil to 180°C/350°F, or until a cube of bread turns golden brown in 15 seconds. Lower the fish pieces into the wok and swirl in the hot oil (the Chinese call this 'passing it through the oil') for 1 minute. Pour the fish into a strainer over a heatproof bowl.
Return 1 tablespoon of the oil to the wok and reheat. (The remaining oil can be reused for another fish dish at a later date.) Add the dried chillies, Sichuan peppercorns and lardons and stir-fry in the hot oil for less than 1 minute to release their flavours (the Chinese call this baosiung – 'explode fragrants'). Add the chilli bean paste, then tip in the Chinese chives (or baby leeks) and toss well. Add the vegetable stock (or water) to help the chives cook and keep tossing all the ingredients together. When the chives are a deep, translucent green and have wilted, return the fish fillets to the wok and toss together, being careful not to break up the fish. Season with the vinegar and a drizzle of chilli oil, transfer to a serving plate and serve immediately, with jasmine rice.
with a Vietnamese mint and wild coriander salsa verde
For our last night in Yunnan, Ken and I cooked together. Inspired by my time with the Dai minority ladies (see here), I thought I would simply steam a local fish in honour of the wonderful time I had spent with them fishing. And inspired by their love of the local herbs, I decided to make a salsa verde with Vietnamese mint and wild coriander to be poured over the fish once it was cooked. (See here)
Serves 2–4 to share
1 × 675g (1½lb) whole, large sea bream, cleaned
2 large pinches of toasted and ground Chinese cardamom, star anise, Sichuan peppercorns (2 cardamoms, 3 star anise, 1 tablespoon peppercorns)
2 large pinches of sea salt
For the salsa verde
1 garlic clove, peeled, crushed and finely chopped
2.5cm (1in) piece fresh root ginger, peeled and finely chopped
8 small fresh chillies, finely sliced
2 spring onions, finely sliced
a large handful of Vietnamese mint (or a few leaves of fresh basil), finely chopped
a large handful of wild coriander (or farmed coriander), finely chopped
5–6 tablespoons groundnut oil
¼ teaspoon sea salt
2–3 tablespoons light soy sauce
Season the fish with the toasted Chinese spices, then season with salt on both sides. Place the fish on a heatproof plate in a bamboo steamer. Half-fill the wok with boiling water, place the steamer over the wok (making sure the base doesn't touch the water), cover and steam on high heat for 10–15 minutes until the flesh of the fish turns opaque and flakes when poked.
While the fish is cooking, combine all the salsa verde ingredients in a bowl and mix well.
To serve, pour the sauce over the fish and accompany with rice and vegetables.
from Kashgar market
This tasty, crispy snack was inspired by my visit to the street market of the ancient quarters of Kashgar. I was surprised to learn that river fish are abundant here. The fish was a different culinary offering amidst all the lamb kebabs, which seem to dominate the street foods on the dusty streets of Kashgar. This is a simple but delicious quick dish.
Serves 4
450g (1lb) white firm fish fillets, such as cod, sea bass or halibut
1 teaspoon salt
cornflour, for dusting
150ml (¼ pint) groundnut or peanut oil
For the seasoning
2 tablespoons cumin powder
2 teaspoons salt
1 teaspoon freshly ground white pepper
2 teaspoons sugar
Sprinkle the fish fillets evenly on both sides with the salt. Cut the fish into strips 2.5cm (1in) wide and let these sit for 20 minutes. Then dust them with the cornflour.
Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is very hot and slightly smoking, shallow-fry the fish strips until they are firm and cooked. Remove them with a slotted spoon and drain on kitchen paper.
Quickly mix the cumin, salt, pepper and sugar in a small bowl, then sprinkle on the crispy fish and serve at once.
Fish dishes in northern China are often paired with rice wine, which gives them a rich taste. In this recipe, the fish is 'velveted', which means that it is coated with egg white, cornflour, salt and sesame oil and then gently stirred in warm, but not hot, oil. This prevents the flesh from seizing up and keeps it moist and tender. The final finish of the sauce results in a delicate and elegant dish. Serve this with plain rice for a quick and easy family meal.
Serves 2–4
50g (2oz) Chinese dried black mushrooms, pre-soaked in warm water for about 20 minutes until soft and pliable, then drained
450g (1lb) boneless and skinless fillets of cod, halibut, sea bass, or any firm white-fleshed fish, cut into 5cm (2in) pieces
150ml (¼ pint) groundnut or vegetable oil
For the coating
1 egg white (or 2 tablespoons egg white)
2 teaspoons cornflour
1 teaspoon sesame oil
½ teaspoon salt
For the sauce
3 tablespoons Shaoxing rice wine (or dry sherry)
2 teaspoons salt
freshly ground white pepper
2 teaspoons sugar
150ml (¼ pint) chicken stock
1 teaspoon cornflour, mixed with 1 teaspoon water
For the garnish
chopped spring onions
Squeeze the excess water out of the mushrooms, then cut off and discard the woody stems and finely shred the caps.
Combine the fish with all the coating ingredients in a medium-sized bowl. Mix well, then refrigerate for about 20 minutes.
Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is just warm, quickly add the fish and stir to separate, then turn off the heat. Allow the fish to sit in the warm oil for about 2 minutes. Drain in a colander set inside a stainless steel bowl.
Reheat the wok or pan and add all the sauce ingredients, then bring to a simmer. Add the mushrooms and cook for about 2 minutes. Return the fish pieces to the pan and heat through. Serve at once garnished with spring onions.
Sweet and sour West Lake steamed carp
I first tried this dish when I visited Hangzhou in China years ago and I loved the simplicity of it. Traditionally, carp is used for this dish but you could use sea bass or sea bream. The fish is dressed with Shaoxing rice wine, then slathered in grated ginger and steamed. While it's steaming, a delicious sauce using black rice vinegar and sugar is cooked in the wok to create a dark, rich, sweet and sour sauce, which is then poured over the cooked fish. Classic and tasty!
Serves 2–4 to share
1kg (2¼lb) fresh whole carp (or use common bream or sea bass), scaled and gutted
sea salt and freshly ground white pepper
2 tablespoons freshly grated root ginger
2 tablespoons Shaoxing rice wine (or dry sherry)
For the sauce
1 tablespoon groundnut oil
2 tablespoons freshly grated root ginger
1 teaspoon dark soy sauce
3 tablespoons Chinese black rice vinegar (or balsamic vinegar)
2 tablespoons soft brown sugar
125ml (4fl oz) vegetable stock
1 tablespoon cornflour, blended with 2 tablespoons cold water
For the garnish
2 spring onions, sliced into long strips, then dipped in iced water and drained, to make curls
Slice some slits into the side of the fish. Season with salt and white pepper on both sides and in its cavity, then rub the grated ginger all over the fish. Place on a heatproof plate that fits inside a large bamboo steamer with at least a 2.5cm (1in) margin between the plate and the steamer, then pour the rice wine (or sherry) over the fish.
Half-fill the wok with boiling water, place the steamer over the wok (making sure the base doesn't touch the water), then cover and steam for 20–25 minutes until the flesh flakes when poked with a pair of chopsticks. Turn the heat off and keep the carp warm in the steamer.
To make the sauce, heat another wok and add the groundnut oil. Add the ginger and stir-fry for a few seconds. Add the soy sauce, vinegar, sugar and vegetable stock and bring to the boil, then stir in the blended cornflour to thicken the sauce. Remove the fish from the steamer, pour the sauce over it and garnish with spring onion curls.
Stir-fried prawns with green tea
This is perhaps one of the most famous dishes of Hangzhou. The prized Long Jing (Dragon Well) tea is paired with fresh live river shrimp. Legend has it that an Imperial chef mistakenly dropped some tea leaves into his wok while stir-frying white shrimp for the Emperor. Thus a classic dish was born. Many connoisseurs of Chinese tea regard Long Jing to be among the finest. The leaves of this green tea are prepared in a complex process that avoids fermentation. The best-quality leaves are picked before the spring rains fall, around the beginning of April when the young stems have but one tender sprout. These fragile sprouts are the basis for the tea's delicate fragrance and refreshing taste. This is an unusual, surprising combination that is distinctively exquisite, and pure and fresh at the same time.
Serves 4–6
450g (1lb) raw tiger prawns, peeled
2 teaspoons salt
1 tablespoon Long Jing (Dragon Well) Tea (or any Chinese green tea)
225ml (8fl oz) boiling water
1½ tablespoons groundnut or vegetable oil
1 tablespoon Shaoxing rice wine (or dry sherry)
salt and freshly ground black pepper
Devein the prawns by making a shallow cut along the back of each, then remove and discard the black vein. Rinse them well under cold running water and pat thoroughly dry with kitchen paper. Rub the prawns evenly with salt and set aside.
Put the tea leaves into a heatproof measuring jug, pour in the boiling water and leave to steep for 15 minutes.
Heat a wok over high heat until it is hot. Add the oil, and when it is hot, add the prawns and rice wine (or sherry) and stir-fry for 30 seconds. Pour in the tea and half of the tea leaves and cook for another minute. Using a slotted spoon, transfer the prawns to a serving platter. Reduce any liquid in the wok by half. Pour this over the prawns, season well with salt and pepper to taste and serve at once.
Dofu (or tofu as it is known in the West) is often misunderstood outside Asia. Invented in China and made with soya beans and then turned into curds, it comes in numerous forms: pressed in soy sauce, dried, fresh firm, fresh soft and fried. Its versatility, as well as its rich nutritional value, is what makes it so popular throughout Asia. The sponge-like texture absorbs flavours of the food it is cooked with, giving the dofu a delicate taste. This is a light and colourful dish that is perfect for the family table.
Serves 4
225g (8oz) fresh soft or silken dofu (tofu), cut into 2.5cm (1in) cubes
600ml (1 pint) chicken or vegetable stock
2 tablespoons light soy sauce
3 tablespoons Hoisin sauce
2 tablespoons whole yellow bean sauce
225g (8oz) Chinese leaves (or white cabbage), cut into 2.5cm (1in) hunks
225g (8oz) spinach, leaves only, washed
450g (1lb) raw tiger prawns, peeled and deveined (see here)
1 tablespoon finely chopped fresh coriander
salt and freshly ground black pepper
1 tablespoon sesame oil
Drain the dofu on kitchen paper.
Put the stock, soy sauce, Hoisin sauce and yellow bean sauce into a large, cast-iron enamel pot or Chinese clay pot and bring to the boil. Next, add the Chinese leaves (or cabbage) and boil over high heat for 2 minutes. Add the dofu, spinach and prawns, lower the heat and simmer gently for 2 minutes. Stir in the fresh coriander and season with salt and pepper. Finally, stir in the sesame oil and serve.
This is a dish found along the eastern seacoast of China; it is called 'phoenix tail' because the butterflied prawns resemble the fanned tail of the phoenix. Instead of serving it with plum sauce, as is normal, I have made a lychee and pineapple sweet chilli jam as a dipping sauce to go with it. Be particularly careful when deep-frying in a wok.
Serves 2–4 to share
2 eggs, beaten
100g (4oz) cornflour
1–2 tablespoons water
sea salt and freshly ground white pepper
vegetable oil, for deep-frying
12 large tiger prawns, shelled, head off, deveined (see here) and butterflied (see here). Leave the tails on for added effect, if you like.
For the lychee and pineapple sweet chilli jam
50ml (2fl oz) water
50ml (2fl oz) lychee juice
50ml (2fl oz) pineapple juice
6 tablespoons granulated sugar
2 medium fresh red chillies, (de-seeded, if you like) roughly chopped
20g (¾oz) stoned lychees
20g (¾oz) pineapple
To butterfly prawns
Using a sharp knife, make a deep cut down the belly (underside) of the prawn (but not all the way through), then open the prawn out and press flat. Once cooked, the prawns will curl up.
First make the sweet chilli jam. Bring the water, juices and sugar to the boil, then stir to dissolve. Add the fresh chillies, lychees and pineapple and boil for about 3 minutes. Pour into a blender and blitz. Transfer to a dipping bowl and set aside – the sauce will become thick and jammy once cooled.
While the sauce is cooling, put the eggs and cornflour into a bowl and mix to a rough batter (do not mix until smooth) – you might need a little water, depending on the size of the eggs. Season with salt and white pepper.
Fill a wok less than half full with vegetable oil and heat the oil to 180°C/350°F, or until a cube of bread turns golden brown in 15 seconds. Dip each prawn in the batter, then lower into the oil and fry for 1–2 minutes until golden and the tails of the prawns have turned pink (if you've left them on). Using a slotted spoon, lift out and drain on kitchen paper. Transfer to a serving plate and serve with the sweet chilli jam.
Cooking with renowned Sichuan chef Yu Bo has been one of the highlights of my trip to China. He helped dispel the myth regarding the 'fish-fragrant' taste profile of Sichuan cooking. Traditionally, this term is used to describe a delicious sauce that has an intense fish-stock-like flavour, but it doesn't contain fish. Chef Yu Bo told us that the real explanation is that the pickled chillies that are used to make this dish were once pickled with small river fish to give the chillies a fishy flavour. He doesn't use this age-old method any more, but he does pickle his own chillies in earthenware pots. He de-seeds large cayenne-like chillies, pickles them in white liquor (bai jiou), salt and sugar and then grinds them into a paste. He cooks the chilli paste with a lot of vegetable oil and then adds to this spiced fragrant oil a large cup of minced garlic and ginger. It is then ready to use in this dish, which is his modern interpretation of fish-fragrant prawns – delicious! (For a vegetarian version, use fried aubergine batons instead of prawns – they work just as well.)
Makes 4
vegetable oil
4 large tiger prawns, shelled, deveined and butterflied (see here and here), (or 1 large aubergine, sliced into finger-size batons)
cornflour
For the sweet and sour yu siang sauce
3 teaspoons icing sugar
1 tablespoon Chinese black rice vinegar (or balsamic vinegar)
1 tablespoon light soy sauce
3 tablespoons minced pickled chilli paste (Chinese supermarkets sell this)
freshly chopped chives
1 tablespoon wasabi peas, fried and crushed (optional)
To make the sauce, combine the sugar, vinegar and soy sauce in a bowl, then add the pickled chilli paste. Mix well, then add the chives and stir together. (Chef Yu Bo adds 1 tablespoon of fried baby crispy peas at this stage and tosses again – you could try using crushed fried wasabi peas.) Set to one side.
Fill a wok to quarter full with vegetable oil and heat to 180°C/350°F, or until a cube of bread turns golden brown in 10 seconds. Dip each prawn into cornflour and turn to coat, then fry for less than 2 minutes until golden brown. Drain the prawns on kitchen paper.
Spoon a teaspoonful of the sauce onto each fried prawn, transfer to a serving plate and eat immediately.
from Huangsha seafood market
The Huangsha seafood market in Guangzhou is one of the liveliest I have ever seen in China. The streets are filled with jumping live prawns, twitching crabs, writhing sand worms, live fish swimming in tanks, even turtles and, yes, crocodiles. There is nothing fresher anywhere. I found fresh scallops and I immediately sensed that steaming was the best way to cook them. Why? Because steaming would reveal all their sweetness and freshness, and highlight their subtle taste: a perfect and most healthy way to enjoy the best harvest from the sea.
Serves 4
450g (1lb) fresh scallops, out of the shell
For the sauce
1 tablespoon finely chopped fresh ginger
1 tablespoon Shaoxing rice wine (or dry sherry)
2 tablespoons light soy sauce
1 large fresh mild red chilli, de-seeded and chopped
3 tablespoons finely shredded spring onions
3 tablespoons groundnut or vegetable oil
For the garnish
fresh coriander sprigs
Put the scallops on a heatproof plate. Next, set up a steamer or put a rack into a wok or deep pan and fill it with 5cm (2in) of water. Bring the water to the boil over a high heat. Put the scallops into the steamer or onto the rack. Turn the heat to low, cover the wok or pan tightly and steam gently for 5 minutes.
Meanwhile, combine all the sauce ingredients except the oil in a heatproof bowl. Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is very hot and slightly smoking, pour it over the sauce ingredients.
Remove the scallops from the steamer and give the sauce several good stirs. Pour the sauce over the scallops, garnish with the fresh coriander and serve at once.
with ginger and spring onions
Perhaps one of the greatest gifts given to me by my mother and Chinese uncles was the appreciation of fresh seafood, especially crab. Of course, at home, the crabs were always live until dispatched to the hot wok. I remember the silky texture and sweetness of the crab meat as I sucked each morsel clean. There were only ever stacks of empty shells left at the end of the meal. The Cantonese chefs know that preparing subtle and delicate food such as crab requires the minimum of seasonings: ginger, garlic and spring onions, which complement rather than detract from the crab. Eat this with friends as a first course, with your hands and perhaps a pint of beer too!
Serves 4–6
1.5–1.75kg (3–4lb) fresh or freshly cooked crab in the shell
2 tablespoons groundnut or vegetable oil
3 tablespoons finely sliced garlic
2–3 tablespoons finely shredded fresh ginger
6 whole spring onions, sliced
salt and freshly ground black pepper
3 tablespoons Shaoxing rice wine (or dry sherry)
If the crab is uncooked, put it on its back. With a thick towel, immediately twist off the large claws. Then twist off the small legs. Scrub the shells under cold running water. Now separate the top shell from the crab body. Remove the feathery lungs, the mouth and the tail. Scrub the shell and the crab body under cold running water. Remove the stomach sac. Quarter the crab body with a large knife or cleaver. Then crack the claws and legs with the flat of the cleaver. If you are using a cooked crab, simply clean and quarter the crab and crack it as described above.
Heat a wok over high heat until it is hot. Swirl in the oil, and when it is very hot and slightly smoking, toss in the garlic, ginger and spring onions and stir-fry for 20 seconds. Then chuck in the crab pieces, salt, pepper and rice wine (or sherry) and stir-fry over a high heat for about 15 minutes.
Turn out on to a large, warm serving platter and serve at once.
with bamboo shoots and spring onions
I love scallops cooked quickly in the wok. The seasonings of soy, vinegar and brown sugar caramelise and impart a delicious smoky, sweet flavour as the edges of the scallops sear in the hot wok. A super-healthy, quick stir-fry.
Serves 2–4 to share
2 tablespoons vegetable oil
a pinch of sea salt
1 tablespoon freshly grated root ginger
2 small dried chillies
8 scallops, cleaned and corals removed (optional)
75g (3oz) shredded cooked bamboo shoots, sliced on the diagonal into 0.5cm (¼in) strips
2 small spring onions, sliced on the diagonal into 0.5cm (¼in) strips
a pinch of brown sugar
1 tablespoon light soy sauce
1 tablespoon Chinese black rice vinegar (or balsamic vinegar)
a dash of toasted sesame oil
1 teaspoon cornflour, blended with 1 tablespoon cold water
To serve
bamboo leaves, soaked and washed
Heat a wok over high heat, then add 1 tablespoon of the vegetable oil. Add the salt, ginger and chillies and toss for few seconds.
Add the scallops and toss in the wok for 1 minute, then spoon out. Add another tablespoon of vegetable oil to the wok and reheat. Add the bamboo shoots and spring onions and cook for another minute. Return the scallop mixture to the wok, then stir in the sugar and season with the soy sauce, vinegar, sesame oil and blended cornflour. Toss well. Line plates with bamboo leaves, then divide the mixture between the plates and serve immediately, with jasmine rice.
I last visited the capital city of Sichuan province, Chengdu, over 20 years ago, and when we filmed there at the beginning of 2012, Chengdu had changed beyond my wildest expectations. It is as if I went back to the future. New suburbs were mushrooming and the twisting, narrow alleys of the old city had been bulldozed to be replaced by corporate, concrete skyscrapers, all in the name of progress. One of my primary memories of Chengdu is of eating an amazing variety of street food – but in a bid to 'civilise' the city, many of the vendors have recently been encouraged to 'disappear'. However, we discovered that the food in general is better than ever, with great ingredients and dedicated, skilful chefs who are passionate about food. I could see why Sichuan cooking is taking China – and the world – by storm.
Sichuan lies in the southwest of China, and is one of the larger and more populous inland provinces (it's about the size of France). Because of its soil fertility, climate and abundance of water, Sichuan was once a major agricultural power, dubbed the country 'of heaven'. That celestial nature, however, was always countered by a poetic caveat, 'the road to Sichuan is as hard as the road to heaven'. The central plains are encircled by mountains and the huge Tibetan plain to the west, so the area was once fairly inaccessible and thus cut off from, and unfamiliar to, the rest of the country. However, a gradual process of immigration swelled the population and brought the province to national attention – as did, later, the Chengdu to Chongqing railway, the first to be built by the new People's Republic of China, in 1952. The area began to look outwards, and flourish agriculturally and socially, until the Great Famine of 1958–61: during this time some 11 million people are thought to have died in Sichuan alone. And as if this were not enough, a massive earthquake in 2008 caused a catastrophic loss of life, homes and farmland (and giant panda habitat).
However, the Chinese are very resilient, and Sichuan has regained its equilibrium in a very short space of time. Growing and producing food is still a major industry, both commercially and privately, the latter visibly demonstrated by festoons of chillies, cabbage, radish, wild mushrooms and citrus peel drying on the eaves and balconies of houses all over the province. And eating Sichuan food, for locals and visitors, is still one of the major delights of life!
Sichuan's unusual cuisine is sometimes attributed to its former geographical isolation, but the climate has probably more to do with it. The omnipresent humidity, and lack of sun, even in hot summers, means that the Chinese medicine principles of yin (cool) and yang (hot) are out of balance – the yang is particularly impaired by dampness – which can lead to ill health. The answer is to introduce a spicy heat to the food and, until the arrival of chillies (only some 300 or so years ago), this was provided primarily by a native berry, the fruit of a prickly ash tree, known as Sichuan peppercorn or 'flower pepper' because, split, it looks like a flower bud opening. The taste is sharp, slightly spicy, but not hot, and makes the lips numb and tingly: this is known as 'ma' in Chinese, the same word that is used for 'pins and needles' and 'anaesthesia'. (Sichuan peppercorns are also one of the five spices of the famous Chinese spice mixture.) I like it, it tastes a little citrussy to me – and I love the effect it has on food.
Sichuan cuisine is famed for its bold, compound flavours and for its fiery-hot spiciness (known as la), pungency and fragrance: sometimes the ma and la flavourings are combined as in ma la rabbit. Street and restaurant foods can be both 'spicy hot' and 'numbing hot' – among them dan dan noodles, mapo dofu (see here), gong bao (or kung pao) chicken and twice-cooked pork; but some equally famous Sichuan dishes – tea-smoked duck and crispy duck, for instance – are flavourful, but not fiery hot. The sauces for 'fish-fragrant' dishes are not actually made with fish, but with the ingredients that might complement fish dishes, such as mild chillies, ginger, garlic and spring onions: this is one of the many classic flavour combinations of Sichuanese cookery (see here). Chilli flavourings are often added to dishes in the form of a chilli bean sauce or paste, made from chillies and fermented beans (most authentically broad beans rather than soya beans). The principal techniques of the cooking style are stir-frying, steaming and dry-braising or dry-stewing – although Sichuan chefs claim to use over 50 distinct ways of cooking.
The number of flavours and flavour combinations in Sichuan cuisine is huge, not least because so many are sourced within the province itself: Sichuan produces soy sauce, fermented soya beans, chilli bean pastes, chilli paste, vinegars, pickles and salt. Salt was once a major source of revenue for China (probably traded on the Silk Road not far north), and the brine-salt mines of Sichuan's Zigong have supplied China's best salt for over 2,000 years. The easy availability of this has probably led to the Sichuan penchant for salt-preserved vegetables, which are a major thread of the cuisine.
Yet another thread – which in fact you find all over China – is the snack culture, xiao chi or 'small eats'. The Chinese love snacks probably because they breakfast lightly and thus are hungry for food long before the main midday meal. The norm of two main meals a day can often extend to at least five, due to the intervening 'small eats' meals. And, even though many of the street vendors have disappeared in Chengdu, small eats are still available, many of them from 'fly' restaurants (yes, literally fly-ridden, but often serving magnificent food) or from teahouses. Chengdu, recently dubbed the most relaxed and laid-back city in China, has more teahouses than other large Chinese cities. Every street corner boasts one, with bamboo chairs and wooden tables inside and out, where locals go to drink jasmine and other teas, to chat, socialise, pass time, eat some xiao chi or play mahjong. As an old Chengdu saying goes, referring to the dull weather of much of the Sichuan year, 'Sunny days are rare, but teahouses are abundant'!
Meat
Dongpo Pork
Sweet and Sour Spareribs
La-Rou (Smoked Pork) with Chillies and Sweetcorn
Stir-Fried Pork with Fermented Dofu
Twice-Cooked Pork
Stuffed Bitter Melon with Black Bean Sauce
Ying Tao Rou Cherry Pork
Ching's Mala Crispy Sichuan Sausage with Pickled Chillies and Wood Ear Mushrooms
Savoury Steamed Egg Custard
Braised Dofu with Crispy Pork
Ching's Mapo Dofu Beef with Edamame Beans
Braised Beef Shank in Sichuan Dressing
Stir-Fried Beef with Sichuan Preserved Vegetables
Hunan-Style Crispy Chilli Beef with Dried Chillies and Peanuts
Braised Lamb
Lionhead Meatballs
Yang Rou Bing Finely Shredded Mutton and Spring Onion Wheat Flour Pancake Parcels
Stir-Fried Lamb with Leeks
Legend has it that a famous official from Hangzhou, Su DongPo, was sent pork by his grateful people. Being a righteous official, he decided to share his bounty with all the workers. He instructed his cooks to cook the pork and distribute rice wine as well. Instead, the cooks mistakenly cooked the pork together with the wine, creating this classic dish. There are perhaps as many DongPo recipes as there are chefs in China, each having his or her own version. So, here is mine, which I hope would make Su DongPo and his workers very happy. This is a wonderful autumn or winter dish that can be made ahead of time and it reheats very well.
Serves 6
1.5kg (3lb) boneless pork belly, cut into 7.5 × 7.5cm (3 × 3in) pieces
For the braising liquid
6 × 7.5cm × 5mm (3 × ¼in) slices of fresh ginger
6 spring onions, cut into 7.5cm × 5mm (3 × ¼in) slices
1.2 litres (2 pints) chicken stock
600ml (1 pint) Shaoxing rice wine (or dry sherry)
3 tablespoons chilli-fermented dofu (tofu)
3 tablespoons crushed yellow bean paste or sauce
3 tablespoons Hoisin sauce
150g (5oz) Chinese rock sugar (or plain sugar)
3 whole star anise
2 pieces of Chinese cinnamon bark or sticks
2 teaspoons freshly ground white pepper
3 tablespoons whole yellow bean sauce
Fill a large pot with water and bring to a boil. Add the pork pieces and blanch for 5 minutes. Remove and drain in a colander, then quickly rinse with cold water. Discard the blanching liquid.
Put all the braising liquid ingredients into a large pot or casserole and bring to a simmer, then add the blanched pork. Cover the pot and simmer slowly for 2–2½ hours, or until the pork is very tender.
When the pork is cooked, remove it from the pot and let it cool slightly. (If you're not using it as below, the braising sauce liquid can now be cooled and frozen for re-use. Remove any surface fat before transferring it to the freezer.)
Serve the pork, like the Chinese do, in large chunks. I like to reduce the braising liquid to concentrate its flavour and then reheat the pork in it. Or you can thicken the braising liquid with a little cornflour and serve as a sauce over the sliced pork. If you do this, be sure to remove all traces of fat from the sauce before thickening it.
This is another fabled dish in classic Chinese cookery. However, it relies on the magic of Chinese black vinegar, which is well worth the search. It has a sweet, sour and undefinable taste that gives these spareribs their reputation.
Much of the work can be done beforehand and the dish can easily be reheated. What more can you ask for? It makes a delicious main course. Be particularly careful when deep-frying in a wok.
Serves 4
750g (1½lb) pork spareribs
600ml (1 pint) groundnut or peanut oil
For the marinade
2 tablespoons Shaoxing rice wine (or dry sherry)
2 tablespoons light soy sauce
1 tablespoon Chinese black rice vinegar (or balsamic vinegar)
1 tablespoon sesame oil
salt and freshly ground black pepper
1½ tablespoons cornflour
For the sauce
1 tablespoon chopped orange zest (or a piece of Chinese dried orange peel)
3 tablespoons coarsely chopped garlic
2 teaspoons Chinese five-spice powder
3 tablespoons finely chopped spring onions
25g (1oz) Chinese rock sugar (or brown sugar)
3 tablespoons Shaoxing rice wine (or dry sherry)
150ml (¼ pint) chicken stock
1½ tablespoons light soy sauce
85ml (3fl oz) Chinese black rice vinegar (or balsamic vinegar)
Have your butcher separate the spareribs into individual ribs, and then cut into chunks about 7.5cm (3in) long. Alternatively, do this yourself using a heavy, sharp cleaver that can cut through the bones.
Mix all the marinade ingredients together in a bowl and steep the spareribs in the marinade for about 25 minutes at room temperature. Using a slotted spoon, remove the spareribs from the marinade.
Heat the oil in a deep-fat fryer or large wok. When the oil is very hot and slightly smoking, slowly brown the marinated spareribs in several batches. Drain each cooked batch on kitchen paper.
If you are using the Chinese dried orange peel, soak it in warm water for 15 minutes until it is soft, then drain and chop it coarsely. Put the chopped zest or peel together with the rest of the sauce ingredients into a clean wok or medium-sized pot. Bring the sauce to the boil, then turn the heat down, add the spareribs, cover the pan and simmer slowly for about 40 minutes, stirring occasionally. If necessary, add a little water to the sauce to prevent it from drying up. Skim off any surface fat, turn out on to a serving plate and serve at once.
with chillies and sweetcorn
In China, la-rou is found in butchers or on market stalls. Smoky and delicious, it's great in stir-fries. I substitute smoked bacon lardons for the smoked pork, which are just as tasty. This simple stir-fry makes a great accompaniment to other dishes, served with rice.
Serves 2–4 to share
1 tablespoon groundnut oil
2 Sichuan dried chillies
1 medium fresh red chilli, de-seeded and finely chopped
200g (7oz) smoked bacon lardons
200g (7oz) fresh, frozen or tinned and drained sweetcorn kernels
1–2 tablespoons Chinese black rice vinegar (or balsamic vinegar)
1 tablespoon chilli oil (see here)
sea salt and freshly ground white pepper
Heat a wok over high heat and add the groundnut oil. Add the dried chillies, fresh chilli and bacon lardons and stir-fry for a few minutes. Add the sweetcorn and toss together. (If using fresh kernels, add a few tablespoons of water to help create some steam to cook the kernels, but let all the water evaporate before seasoning.) Season with the vinegar, chilli oil, salt and white pepper. Transfer to a serving plate and serve immediately.
with fermented dofu
Here is a classic dish that most Chinese would make at home. It is simple, easy and requires few ingredients, but its secret ingredient is fermented dofu (tofu), which has been defined as having the fifth 'umami' taste. Chinese cooks for thousands of years have realised the magic of fermented flavours, which are able to transform the most ordinary dishes into something extraordinarily delicious.
Serves 4
450g (1lb) boneless pork fillet, cut into thick slices, 5cm (2in) long
1 tablespoon Shaoxing rice wine (or dry sherry)
1 tablespoon light soy sauce
2 teaspoons sesame oil
1 teaspoon cornflour
1 tablespoon groundnut or peanut oil
For the dofu
2 teaspoons groundnut or peanut oil
8 spring onions, cut on the diagonal into 5cm (2in) lengths
3 tablespoons fermented dofu (tofu)
salt and freshly ground black pepper
2 tablespoons Shaoxing rice wine (or dry sherry)
1 teaspoon sugar
Put the sliced pork into a bowl and mix in the rice wine (or sherry), soy sauce, sesame oil and cornflour. Let the mixture sit for 10–15 minutes to allow the pork to absorb the flavours of the marinade.
Heat a wok or frying pan to a very high heat. Add 1 tablespoon of the oil, and when it is very hot and slightly smoking, add the pork slices and stir-fry until they are brown. Remove the pork from the wok and drain in a colander.
Wipe the wok clean with kitchen paper and reheat over high heat. When the wok is hot, add the 2 teaspoons of oil, then add the spring onions and stir-fry for 30 seconds. Now add the fermented dofu, smashing it against the bottom and side of the wok. Add salt and pepper to taste, the rice wine (or sherry) and, finally, the sugar. Return the drained pork to the wok and coat each piece with the sauce. Stir-fry for another 2 minutes, or until the pork is cooked through. Serve at once.
This was quite an experience! I cooked at an organic pig farm in Pujiang with farmer Mr Peng. He invited me to cook on his farm with his wife, who prepared a traditional Sichuan feast using all parts of the pig. We cooked the classic hui guo rou (twice-cooked pork) together. Mr Peng suggested I use the front shoulder cut of meat. However, the piece he provided had almost 7.5cm (3in) fat before you got to the lean meat. As I was slicing it, he accused me of incorrectly cutting the meat, saying there should be slices of fat and lean meat! Had I sliced it the way he recommended, the slices of pork would have contained a huge amount of fat and then the meat would have been too large to be recognised as hui guo rou. So after much debate, his wife said that the best cut of meat to use for this traditional dish is pork belly – wu hua rou: five layers of heaven – skin, fat, meat, fat, meat. After all that debate, pork belly is supposed to be the best cut for this dish – I wish I had been told that at the beginning. The discussion was quite challenging, but it shows the passion that Sichuan people have for their cuisine.
Serves 4 to share
700ml (1¼ pints) water
300g (11oz) fatty pork belly, skin on
2 tablespoons groundnut oil
1 tablespoon Shaoxing rice wine (or dry sherry)
1 tablespoon chilli bean paste
1 tablespoon yellow bean sauce
1 tablespoon fermented black beans, rinsed and crushed
1 spring onion (or baby leek), sliced on the diagonal into julienne strips (optional)
1 teaspoon dark soy sauce
1 teaspoon light soy sauce
a pinch of sugar
sea salt and freshly ground white pepper
Pour the water into a large pan, add the pork and bring to the boil, then boil for 30 minutes. Drain and leave to cool.
Put the meat in the fridge for about 1 hour, to firm up, then cut finely into very thin slices, about 0.5cm (¼in) thick.
Heat a wok over high heat and add the groundnut oil, then add the pork. As the pork starts to brown, add the rice wine (or sherry) and cook until the pork is browned and the skin is slightly crisp. Add the chilli bean paste, yellow bean sauce and fermented black beans and stir-fry for 1 minute.
Add the spring onion (or leek), if using, and stir-fry for less than 1 minute until well mixed. Add both soy sauces and the sugar, season and serve.
with black bean sauce
This unusual vegetable is very much an acquired taste. It has as many detractors as it has fans, even among the Chinese, but those who love it insist it is worth the effort to appreciate its taste. Bitter melon has a bumpy dark to pale green skin, and a slightly bitter quinine flavour that has a cooling effect in one's mouth. I am among the Chinese who love it. My mother used to make it stuffed, as in this recipe. It is southern Chinese home cooking at its best.
Serves 4
750g (1½lb) bitter melons (or cucumber), unpeeled, cut into 2.5cm (1in) slices
2 tablespoons cornflour
3 tablespoons groundnut or vegetable oil
For the stuffing mixture
225g (8oz) minced fatty pork
1 egg white
1½ tablespoons finely chopped spring onions
1 tablespoon finely chopped fresh ginger
2 teaspoons Shaoxing rice wine (or dry sherry)
2 teaspoons light soy sauce
2 teaspoons sugar
1 teaspoon salt
1 teaspoon freshly ground black pepper
1 teaspoon sesame oil
For the sauce
300ml (½ pint) chicken stock
2 tablespoons Shaoxing rice wine (or dry sherry)
2 tablespoons fermented black beans, rinsed and chopped
2 tablespoons chopped garlic
2 tablespoons finely chopped fresh ginger
1 tablespoon light soy sauce
1 tablespoon oyster sauce
2 teaspoons sugar
freshly ground black pepper
1 teaspoon cornflour, mixed with 2 teaspoons water
For the garnish
2 teaspoons sesame oil
2 tablespoons finely chopped fresh coriander
Using a small sharp knife, remove the seeds and pulp from the centre of each bitter melon (or cucumber) slice. Hollow the bitter melon so that you have at least a 5mm (¼in) shell. Lightly dust the hollow interior of the slices with a little cornflour.
Mix all the stuffing ingredients together in a large bowl, then stuff each melon ring with this mixture.
Heat a wok or large frying pan until hot. Add the oil, and when it is moderately hot, add the stuffed rings and cook them slowly until they are slightly browned.
Turn them over and brown the other side, adding more oil if necessary You may have to do this in several batches. When the rings are brown, remove them from the oil and put them on a plate. When you have fried all the rings, wipe the wok or pan clean with kitchen paper and reheat.
Mix all the sauce ingredients and put them into the reheated wok or pan. Bring the liquid to a simmer, then add the stuffed rings, cover the pan with a lid and simmer slowly for 7 minutes, or until the rings are completely cooked. Using a slotted spoon, transfer them to a serving platter.
Reduce the sauce by a third over high heat, then add the sesame oil and coriander. Pour the sauce over the stuffed bitter melons and serve at once.
Cherry pork
Mrs Peng (see here) showed me how to make this delicious dish. It is called Ying tao rou, or cherry pork, because the flavour of the liquid the pork is cooked in – an unctuous red-brown sugar syrup – is described as being as sweet as cherries.
Serves 4 to share
300g (11oz) red rock sugar (or 300ml/½ pint rich honey)
2 tablespoons vegetable oil
1 large piece pork belly (about 500g/1lb 2oz), boiled for 30 minutes, then cooled and sliced into 2.5cm (1in) pieces
2 tablespoons dark soy sauce
1 litre (1¾ pints) boiling water
Place the red rock sugar in a pan with about 1 litre (1¾ pints) water and cook over a high heat until reduced to a red-brown caramel syrup.
Heat a wok over high heat and add the vegetable oil. Add the pork pieces, dark soy sauce and red rock sugar syrup (or honey) and stir-fry on medium heat for 1 minute, or until coloured and browned, stirring constantly so that the sugar does not caramelise. Pour in the boiling water, cover the wok and simmer for 2 hours on medium heat, or until the pork is tender.
Serve with jasmine rice or other vegetable dishes.
with pickled chillies and wood ear mushrooms
At the home of our friend Jenny's aunt and grandparents in Guanghan town, Chengdu, I made a dish using her delicious xiang chang – Chinese sausages made from minced pork belly, bai jiou (white liquor), salt, sugar and chillies, which she had air-dried herself. She had been preparing them since Christmas and they were already two months old. In winter it sometimes takes one week for them to dry, but if the weather is too humid, it takes at least ten days.
This is how my grandmother used to prepare Chinese sausage in Taiwan – fried in a wok until crispy and served with raw garlic slices.
Serves 2–4 to share
2 spicy Sichuan wind-dried sausages (or lap cheong, Cantonese dried sausage)
a small handful of Chinese dried wood ear mushrooms, soaked in hot water for 20 minutes until soft and pliable, then drained
2 pickled long chillies
2 tablespoons vegetable oil
1 small bunch of Chinese garlic chives (or baby spring onions), inner stems sliced on the diagonal into 4cm (1½in) pieces
1 teaspoon chilli bean paste
1 teaspoon Sichuan peppercorns
1 tablespoon Sichuan pepper oil (see here)
1 tablespoon black rice vinegar (or balsamic vinegar)
1 tablespoon light soy sauce
a pinch of sugar
Place the sausages in boiling water and cook for 15 minutes, then drain. Slice the sausages on the diagonal into 1cm (½in) slices. Slice the wood ear mushrooms and pickled chillies on the diagonal to resemble the long oval shape of the sausage pieces.
Heat a wok over high heat and add 1 tablespoon of the vegetable oil. Add the wood ear mushrooms, garlic chives (or spring onions) and picked chillies and toss together in the oil. Stir-fry for less than 1 minute on high heat, then spoon out onto a plate.
Place the wok back on high heat and add the remaining vegetable oil. Add the chilli bean paste, Sichuan peppercorns, Sichuan pepper oil, vinegar, light soy sauce and sugar.
Return the sausages, wood ear mushrooms, pickled chillies and garlic chives to the wok and toss together a few times in the fragrant hot oil to combine all the flavours. Take off the heat, spoon out onto a serving plate and eat immediately.
This is home comfort food at its best. With a few ingredients, my mother used to put this dish together when she came home from work. Steaming the custard gave it a smooth, velvety texture like a wonderful savoury flan. I ate it with gusto with my bowl of rice. It is a perfect, quick and inexpensive meal for just two or can be served with a stir-fried vegetable for a family lunch or dinner.
Serves 2–4
50g (2oz) Chinese dried black mushrooms, pre-soaked in warm water for about 20 minutes until soft and pliable, then drained
1½ tablespoons groundnut or vegetable oil
350g (12oz) minced pork
3 tablespoons finely chopped spring onions
1½ tablespoons Shaoxing rice wine (or dry sherry)
2 teaspoons sesame oil
2 tablespoons light soy sauce
1 teaspoon sugar
1 teaspoon salt
½ teaspoon freshly ground white pepper
groundnut or vegetable oil
For the custard
4 eggs
600ml (1 pint) chicken stock
1 teaspoon salt
To serve
2 tablespoons oyster sauce
Squeeze the excess water out of the mushrooms, then cut off and discard the woody stems and finely chop the caps.
Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is very hot and slightly smoking, add the pork and stir-fry for 2 minutes. Drain the pork, then return it to the wok. Add the spring onions, rice wine (or sherry), sesame oil, soy sauce, sugar, salt, pepper and mushrooms and continue to stir-fry for 3 minutes. Remove from the heat and set aside.
Mix all the custard ingredients in a bowl. Rub a shallow, heatproof bowl with groundnut or vegetable oil, then pour in the custard mixture.
Next, set up a steamer, or put a rack into a wok or deep pan, and fill it with 5cm (2in) of water. Bring the water to the boil over high heat. Carefully lower the bowl into the steamer or on to the rack. Turn the heat to low and cover the wok or pan tightly. Steam gently for 10–12 minutes, or until the custard has set. Add the stir-fried meat mixture and spread it over the top of the custard. Cover and steam for another 4 minutes.
Remove the bowl from the steamer, drizzle with oyster sauce and serve at once.
with crispy pork
Qing Ming is one of the most auspicious holidays in the Chinese calendar. This is the day we visit our ancestral graves to sweep them clean and bring offerings of food. Being pragmatic, we usually bring the food home later. Traditionally, we go to the local deli and buy cooked chicken or suckling pig, or a piece of roast pork belly. When I visited my ancestral village after the offering, I cooked a typical home-style dish with a roasted pig. I braised the pork belly with dofu (tofu) in a hearty dish that exemplifies southern Chinese cooking at its best. Roast crispy pork belly can be usually found at Chinese grocers.
Serves 4
450g (1lb) fresh dofu (tofu), cut into 2.5cm (1in) cubes and drained in a colander for 30 minutes
450ml (¾ pint) groundnut or vegetable oil, plus 1½ tablespoons
2 tablespoons coarsely chopped garlic
2 teaspoons finely chopped fresh ginger
50g (2oz) spring onions, cut into 2.5cm (1in) strips
2 tablespoons Shaoxing rice wine (or dry sherry)
700g (1½lb) shop-bought crispy roast pork belly, boned, if you like, and cut into 5cm (2in) chunks
3 tablespoons oyster sauce
salt and freshly ground black pepper
50ml (2fl oz) chicken stock
For the garnish
3 tablespoons finely chopped spring onions
Prepare the dofu and leave to drain.
Heat the 450ml (¾ pint) oil in a wok until it is hot, then deep-fry the dofu (tofu) in two batches. When each batch is lightly browned, remove and drain well on kitchen paper. Let the cooking oil cool and then discard it.
Heat a wok or large frying pan over high heat, and when it is very hot and slightly smoking, add the 1½ tablespoons oil and then add the garlic, ginger and spring onions. Stir-fry for a few seconds, then add the rice wine (or sherry) and stir-fry for 30 seconds. Add the pork and dofu and stir-fry for 2 minutes.
Add the oyster sauce, salt, pepper and chicken stock. Reduce the heat to low, cover the wok and slowly simmer the mixture for 8 minutes. Turn up the heat and cook until most of the liquid has evaporated – about 1 minute. Garnish with the spring onions and serve.
with edamame beans
This is one of Sichuan's most famous dishes and is named after an elderly woman called Old Mrs Chen – the 'mapo' was used to describe her pockmarked complexion. She was a great cook and this dish is served in restaurants worldwide. The classic version uses sian-miao – thin Chinese leeks – but spring onions are a good substitute and I also like to add edamame beans (soya beans), for added texture and nutrition to round out a perfect one-pot supper dish to serve with rice. See here for our vegetarian versions.
Serves 4 to share
2 tablespoons groundnut oil
2 garlic cloves, peeled, crushed and finely chopped
1 tablespoon freshly grated root ginger
1 medium fresh red chilli, de-seeded and finely chopped
2 teaspoons toasted Sichuan peppercorns, ground in a pestle and mortar
300g (11oz) minced beef
1 tablespoon Shaoxing rice wine (or dry sherry)
2 tablespoons chilli bean paste
400g (14oz) fresh extra-firm dofu (tofu), cut into 2.5cm (1in) chunks
75g (3oz) fresh or frozen edamame beans
200ml (7fl oz) hot beef stock
1 tablespoon light soy sauce
1 tablespoon cornflour, blended with 2 tablespoons cold water
sea salt and freshly ground white pepper
2 large spring onions, sliced on the diagonal
Heat a wok over high heat. Add the groundnut oil, then add the garlic, ginger and fresh chilli and stir-fry for a few seconds. Add the ground peppercorns and minced beef and toss together well. As the beef starts to turn brown, add the rice wine (or sherry) and mix well, then season with the chilli bean paste. Add the dofu, edamame beans and hot stock and bring to a bubble, then mix well, being careful not to break up the dofu. Season with the soy sauce to taste, then stir in the blended cornflour to thicken the sauce and season with salt and white pepper. Garnish with spring onions and serve immediately, with jasmine or egg-fried rice.
in Sichuan dressing
This is one of those dishes where beef shank, a cheap cut of meat that is full of muscle, is transformed into an elegant, tasty dish. I cook it slowly for hours until tender in a rich braising liquid. Once cooked, it is left to cool and then sliced, dressed and garnished. This makes a great cold starter, perfect for entertaining as it can be made hours ahead.
Serves 2–4 to share
300g (11oz) piece of braising beef (shank or shin)
For the braising sauce
500ml (17fl oz) water
2.5cm (1in) piece fresh root ginger, peeled and sliced
2 tablespoons Shaoxing rice wine (or dry sherry)
1 teaspoon Chinese five-spice powder
1 teaspoon Sichuan peppercorns
3 pieces of dried tangerine peel
1 tablespoon dark soy sauce
3 tablespoons light soy sauce
20g (¾oz) Chinese rock sugar (or 2 tablespoons soft brown sugar)
3 teaspoons caster sugar
½ teaspoon sea salt
For the Sichuan dressing
2 tablespoons chilli oil (see here)
¼ teaspoon toasted and ground Sichuan peppercorns
1 tablespoon Chinese clear (plain) rice vinegar (or cider vinegar)
2 tablespoons braising liquid
1 tablespoon toasted sesame oil
For the garnish
toasted white sesame seeds
1 spring onion, finely chopped
Slicing along the grain, cut the beef into four rectangular chunks. Blanch it in a pan of boiling water for 2 minutes and remove any scum. Rinse the beef, place it in a pot with all the ingredients for the braising sauce and bring to the boil. Reduce the heat, cover the pan and simmer on a low heat for 3 hours until the beef is tender and juicy. Remove and leave to cool, then keep refrigerated until ready to serve. Set aside 2 tablespoons of the braising liquid for the dressing. (You can combine the remainder with stock to make a delicious broth for soup noodles.)
Combine all the ingredients for the Sichuan dressing in a bowl and put to one side.
To serve, shred the beef and place on a serving plate. Drizzle with the Sichuan dressing, then sprinkle some toasted sesame seeds and spring onion on top and serve immediately.
with Sichuan preserved vegetables
Beef was rarely found on my childhood dining table. My mother considered it too heavy and expensive. Instead, we had chicken, pork and plenty of vegetables. Although today I rarely eat beef, I delight in its taste when I do, especially when it's stir-fried. In this dish, I used Sichuan preserved vegetables, which have a crunchy, delicious texture and give the dish an added quality. I think you will find, as I do, that this dish is perfect for an everyday family meal.
Serves 4
450g (1lb) lean beef fillet, cut into medium thick slices, 4cm (1½in) long
2 teaspoons light soy sauce
2 teaspoons Shaoxing rice wine (or dry sherry)
1 teaspoon sesame oil
2 teaspoons cornflour
175g (6oz) Sichuan preserved vegetables, soaked for 20 minutes, then drained
1½ tablespoons groundnut or peanut oil, plus 2 teaspoons
3 tablespoons finely shredded fresh ginger
2 tablespoons chicken stock or water
salt and freshly ground white pepper
2 teaspoons Chinese black rice vinegar (or balsamic vinegar)
2 teaspoons sugar
2 teaspoons sesame oil
Put the beef slices into a bowl and add the soy sauce, rice wine (or sherry), sesame oil and cornflour. Mix well and set aside for 20 minutes. Meanwhile, rinse the preserved vegetables in several changes of water, then shred them finely.
Heat a wok or large frying pan over high heat until it is very hot. Add the 1½ tablespoons of oil, and when it is very hot and slightly smoking, add the beef and stir-fry for about 2 minutes, or until lightly browned. Drain at once into a colander over a heatproof bowl and set aside.
Clean the wok with kitchen paper and reheat it over high heat. Add the 2 teaspoons of oil, and when it is very hot and slightly smoking, stir-fry the ginger for a few seconds, then add the shredded preserved vegetables and stir-fry for another 2 minutes. Add the stock or water, salt, pepper, vinegar and sugar. Quickly return the beef to the wok to reheat and stir well. Drizzle with sesame oil, turn the mixture out on to a platter and serve at once.
with dried chillies and peanuts
Hunan, in the south central part of China, is famous for spicy hot food and one of my favourite dishes that I had there was dry, shredded beef. Inspired by this, I created this crispy chilli beef dish, which is sweet, spicy and zesty, and delicious served with shredded lettuce and cucumber Be particularly careful when deep-frying in a wok.
Serves 2–4 to share
450g (1lb) sirloin steak, fat removed, finely sliced into matchstick strips
3 tablespoons cornflour
groundnut oil
sea salt
4 long, dried Sichuan chillies
1 tablespoon light soy sauce
2 tablespoons sweet chilli sauce
juice and zest of ½ large orange
To serve
1 head of romaine lettuce, shredded
½ cucumber, thinly sliced
For the garnish
2 spring onions, sliced into long strips, then placed in iced water and drained, to make curls
100g (4oz) roasted unsalted peanuts
Put the beef strips into a large bowl and add 2 tablespoons of the cornflour. Toss until the beef has absorbed the cornflour. Add the remaining cornflour and toss to coat.
Half-fill a wok with groundnut oil and heat the oil to 180°C/350°F, or until a cube of bread turns golden brown in 15 seconds. Fry the beef in two or three batches until golden–4–5 minutes. Drain on kitchen paper and season with sea salt.
Carefully drain the oil into a heatproof container, then return 1 tablespoon of oil to the wok. Heat the wok over high heat and add the dried chillies, soy sauce, chilli sauce and orange juice. Bring to a simmer and reduce the sauce until it becomes thickened and coats the back of a spoon. Toss the beef in the sauce to coat thoroughly.
Serve over a bed of the shredded lettuce and sliced cucumber, then garnish with spring onions, peanuts and orange zest.
I grew up in a very southern Cantonese, chauvinistic family, who disdained lamb, probably because they remembered it as the strong-flavoured mutton or goat eaten in China. As a result, I never experienced the taste of lamb until my mid-20s. Although I never quite acquired the taste for lamb, I grew to appreciate it more, especially when it was braised with strong aromatics and spices, as in this recipe. This dish makes a wonderful main course for autumn or winter and tastes even better made one day ahead. The bonus is that it reheats easily.
Serves 4–6
700g (1½lb) boned shoulder of lamb, cut into 5cm (2in) cubes
2 tablespoons groundnut or vegetable oil
4 whole spring onions, sliced on a slight diagonal into 7.5cm (3in) pieces
4 slices of fresh ginger
1 onion, finely chopped
For the braising sauce
1.4 litres (2½ pints) chicken stock
2 whole star anise
75g (3oz) Chinese rock sugar (or brown sugar)
2 tablespoons dark soy sauce
1 tablespoon light soy sauce
3 tablespoons Shaoxing rice wine (or dry sherry)
1 piece of Chinese cinnamon bark or cinnamon stick
3 tablespoons sesame paste (or peanut butter)
2 tablespoons Hoisin sauce
2 tablespoons yellow bean paste or sauce
For the garnish
fresh coriander sprigs
Blanch the lamb by plunging it into boiling water for 5 minutes. Then remove the meat and drain on kitchen paper, and discard the water.
Heat a wok or large frying pan over high heat until it is hot. Add the oil, and when it is very hot and slightly smoking, add the lamb cubes and stir-fry until they are brown. Add the spring onions, ginger and onion to the pan and stir-fry for 5 minutes. Transfer this mixture to a large, flameproof casserole or pot and add all the braising sauce ingredients. Bring the liquid to the boil, skim off any fat from the surface and turn the heat down as low as possible. Cover and braise for 2 hours, skimming off any surface fat from time to time.
Discard the spring onions and ginger. Arrange the cooked meat on a platter, ladle with the sauce, garnish with coriander and serve.
Historical records show that this dish originated around 581–618AD when Emperor Yangdi of the Sui dynasty ordered his chefs to make a Yangzhou dish that inspired them. One of the chefs came up with a pork dish that looked so much like a lion's head that it was renamed thus. Suffice to say, it is a hearty and wonderfully rustic dish that is easy to make and equally easy to reheat. It is perfect on a cold autumn or winter evening.
Serves 4
450g (1lb) minced fatty pork
1 egg white
4 tablespoons cold water
175g (6oz) fresh or tinned water chestnuts, peeled if fresh, drained if tinned, and coarsely chopped
2 tablespoons light soy sauce
2 tablespoons Shaoxing rice wine (or dry sherry)
1½ tablespoons sugar
salt and freshly ground black pepper
2 teaspoons sesame oil
cornflour, for dusting
3–4 tablespoons groundnut or vegetable oil, plus 2 teaspoons
4 garlic cloves, peeled and crushed
450g (1lb) Chinese cabbage (Chinese leaves), stalks separated and cut into 5cm (2in) strips
450ml (¾ pint) hot chicken stock
Mix the pork with the egg white and cold water by hand. The mixture should be light and fluffy. Do not use a blender, as it would make the mixture too dense. Add the water chestnuts, soy sauce, rice wine (or sherry), sugar, salt, pepper and sesame oil and mix for another 30 seconds. Divide the mixture into six equal parts and roll each into a meatball. Dust each meatball with cornflour.
Heat a wok over high heat until it is hot. Add the 3–4 tablespoons of oil and, when it is very hot and slightly smoking, add the meatballs, turn the heat down and slowly brown them. Remove the meatballs and drain on kitchen paper.
Wipe the wok clean with kitchen paper and reheat it over high heat until it is hot. Add the remaining 2 teaspoons of oil, and when it is very hot and slightly smoking, add the garlic and stir-fry for 10 seconds. Then add the cabbage and stir-fry for 20 seconds. Add the hot stock and continue to cook for 2 minutes until the leaves are soft. Transfer the mixture to a heavy, flameproof casserole. Lay the meatballs on top of the leaves, bring the mixture to a boil, then turn the heat to very low, cover and simmer for 1½ hours.
Arrange the cabbage on a platter and lay the meatballs on top, then pour the sauce over the dish and serve at once.
Finely shredded mutton and spring onion wheat flour pancake parcels
Mutton is eaten a lot by the Uighur Muslims throughout China. It is a strongly flavoured meat that is quite tough, so the trick to cooking it is to slice the meat very finely into very thin shreds and then stir-fry with soy, ginger and rice wine. This meaty filling is delicious stuffed into wheat flour pancakes or warmed small pitta pockets, then garnished with sesame seeds and spring onions.
Serves 2–4 to share
12 wheat flour pancakes (shop bought, or small pitta-bread pockets, cut in halves)
1 tablespoon groundnut oil
1 tablespoon freshly grated root ginger
300g (11oz) mutton, finely sliced into shreds
1 tablespoon Shaoxing rice wine (or dry sherry)
½ teaspoon dark soy sauce
1 tablespoon light soy sauce
a pinch of sugar
2 tablespoons hot vegetable stock
1 teaspoon cornflour, blended with 1 tablespoon cold water
4 spring onions, very finely chopped
For the garnish
toasted sesame seeds
a dash of toasted sesame oil
Put the wheat flour pancakes (or pitta) into a small bamboo steamer. Half-fill a small wok with boiling water, place the steamer over the wok (making sure the base doesn't touch the water), then cover and steam on medium-high heat for 5 minutes.
While the pancakes are steaming, heat another large wok and add the groundnut oil. Add the ginger and stir-fry for a few seconds. Add the mutton and break it up, then toss well. As the mutton starts to turn brown, add the rice wine (or sherry) and stir-fry for 1–2 minutes until fragrant and the mutton is cooked. Add both soy sauces, the sugar and hot stock and bring to a bubble, then stir in the blended cornflour and toss well. When the meat is glossy, toss in the spring onions. Take off the heat and transfer to a serving plate, then sprinkle with toasted sesame seeds and season with the sesame oil.
To serve, fold the pancakes in half and then in half again into a cone shape. Fill the inside with the mutton and spring onion filling and serve immediately.
Here is my interpretation of a classic northern stir-fry lamb recipe with leeks, a member of the onion family. The sweetness of the Hoisin sauce and the spicy prick of the Sichuan peppercorns give this dish its perfect lift.
Serves 4
450g (1lb) boneless lamb fillet, cut into medium slices
1 tablespoon Shaoxing rice wine (or dry sherry)
1 tablespoon light soy sauce
2 teaspoons sesame oil
2 teaspoons cornflour
3 tablespoons groundnut or peanut oil
175g (6oz) leeks, cleaned, white part only finely shredded into 7.5cm (3in) lengths
6 garlic cloves, peeled and thinly sliced
1 tablespoon finely chopped fresh ginger
salt and freshly ground black pepper
2 tablespoons Hoisin sauce
1 teaspoon ground roasted Sichuan peppercorns
Put the lamb into a bowl, add the rice wine (or sherry), soy sauce, sesame oil and cornflour and set aside.
Heat a wok until it is hot. Add 2 tablespoons of the oil, and when the oil is very hot and slightly smoking, add the marinated lamb pieces and stir-fry over high heat for 2 minutes. Drain at once into a colander over a heatproof bowl and set aside.
Reheat the wok until it is hot, then add the remaining 1 tablespoon of oil. Add the shredded leeks, garlic and ginger and stir-fry for another 3 minutes. Return the lamb to the wok, add the salt, pepper and Hoisin sauce and stir-fry for another 2 minutes, or until the lamb is heated through.
Turn out on to a serving platter, sprinkle with the ground peppercorns and serve straight away.
**Neither Ken nor I had ever been this far west in China before, and the region and its principal city were strange. The people don't look remotely Chinese – more Persian or Arabic – and they spoke a dialect we didn't understand, so we had to have both a guide and a translator. The region, Xinjiang, is extreme and harsh, and Kashgar is an almost medieval oasis city, situated in a basin between high mountains and the forbidding desert of Taklamakan, which translates as 'Go in and you won't come out'.**
We couldn't fly the 4,000km (2,500 miles) directly from Beijing to the city of Kashgar, and had to stop off at Urumqi en route. Kashgar has been used since ancient times by travellers on what was known as the Silk Road. This historical network of trade routes, no more than camel tracks at first, started in Xi'an (where the terracotta warriors were found). It extended over 6,000km (3,700 miles), and connected Asia with the Western world, beginning some 200 years BCE. The network got its name from the primary trade item from China – silk – but the Chinese also offered teas, jade, lacquer ware and porcelain. (A less welcome export from China along the Silk Road was the bubonic plague, which devastated Europe in the 14th century.) However, the traffic was very much two-way: China also received foreign goods, ideas – Buddhism spread from India to China along the Silk Road – and foodstuffs.
Kashgar is a Muslim city, based in the predominantly Muslim province of Xinjiang. A Turkic people, known now as the Uighurs, came to this part of Asia a millennium ago, bringing Islam with them, and displacing Buddhism. Xinjiang – the largest administrative province in China, roughly the size of Iran – has always been disputed territory, leading to much tension over the years. Many Uighurs are separatist, and believe that China's claim to their homeland is a form of imperialism; the Chinese insist that the territory has always been an integral part of the Chinese nation. The Uighurs at one time were the most numerous group in Xinjiang, but Han Chinese migrants have poured into the region, encouraged by government and attracted by new industrial towns and farming villages.
But the old city of Kashgar still stands, just, a prime example of a traditional Islamic city; it has over 100 mosques, including the Id Kah, the largest in China. In a sense, Kashgar is closer to Mecca than Beijing, especially in culinary terms, with a distinct Central Asian, even Middle Eastern, flavour. Lamb or mutton is eaten 365 days a year. It is roasted whole (in tandoor-type ovens), skewered and grilled as tender kebabs, stewed and curried, and minced as filling for a variety of little pasties or dumplings: one of them was called samsa, the Uighur version of samosa, perhaps? We cooked with our guide Muhammad Yusuf's family: Ken was taught how to make these mini lamb pies, and I learned how to make laghman – the local thick, udon-like noodles. I also helped to make a huge batch of dumplings which, strangely, looked very similar to Italian tortellini!
In Kashgar they eat noodles with mutton, local spinach and tomato in a wet, soup-like broth. (It was strange to see such un-Chinese-looking people eating out of a bowl with chopsticks in such a Chinese manner – normally they eat with their hands here.) A rice platter made with mutton or chicken, and decorated with sultanas and nuts, is known variously as pilaf or plov, or polo, another example of a Middle Eastern leaning.
Kashgar seasonings include saffron, cumin, raisins, apricot kernels (our guide said they were good for men's fertility), and very mild chilli: they don't like spiciness at all, and prefer their food almost bland. They drink a very sweet black tea – made with rosebuds, cinnamon, cardamom and honey – in the morning and before dinner. They also eat a lot of yoghurt, made from yak's, mare's, camel's or cow's milk, and have it throughout the meal, alongside other food, but not for dessert. We did, however, see Chinese influences too: on the streets they were offering sticky rice dumplings, made using traditional bamboo steamers, zong-zi dumplings (sweet, with a date in the middle), and a variety of fruits. The region claims to have the best fruit in China, especially melons, as it is very hot and dry in summer; the water for irrigation comes from glaciers in the surrounding mountains.
Bread is central to the cuisine here, and is the wheat flatbread found throughout central Asia. It is known here (as in India) as naan: huge circles of dough are stamped with intricate patterns and then baked on the sides of pot-bellied clay ovens. (The bread stamp is made from nails, or chicken feather quills, arranged in a pattern on a base.) Ken and I made bread with a local family: after stamping, the bread was smeared with a paste made from red onion, sesame seeds, salt and spices. When I slapped my bread onto the wall of the oven, it peeled off and looked terrible. Ken's was perfect! Naan is eaten with everything, all day long – even for breakfast, when they have it with yoghurt and honey. It has almost a sacred significance to the Uighur people: it must not be thrown away, and in naan's presence, lies are forbidden and vows are binding (bread plays a large part in Uighur weddings).
Kashgar has a very important Sunday market, appropriate for a city that for so many years was the first (or last) stop on the Silk Road. Thousands of people swarm into the city to sell fruit, vegetables and meat – the latter in the form of live lambs, goats, cows and chickens. Horses and donkeys, occasionally camels, are on offer as well, not for food but for farm work: a special area is set aside for prospective buyers to 'test-drive' their chosen steeds. (The area around Kashgar is not known as the Chinese Wild West for nothing!) Food stalls are everywhere, selling bread, snacks, dumplings, noodles, soup, kebabs and tea. They also sell silk and carpets as well as local craft items, such as copper teapots, wooden jewellery boxes, bread stamps, boots with turned-up toes, and hundreds of types of hats.
The region felt very alien to me, like stepping back 2,000 years in time. Culturally it was strange: for example, we went to a Chinese New Year party, where the men and the women were kept completely separate, which didn't seem very festive to me. It's not my place to comment, as it's not my culture, but it isn't how I'd like to live. The women seem happy enough, although they appeared to do all the work. It's just a different world and a different way of living.
Poultry
Sichuan Fried Spicy Chicken with Green Pepper
Yu Siang Shredded Chicken and Aubergine
Steamed Wined Chicken with Dried Mushrooms, Bamboo Shoots, and Goji Berries
Cantonese-Style Roast Pi-Pa Chicken
Steamed Cantonese-Style Chicken with Fermented Dofu
Ching's Chicken and Green Tea Maocha
Braised Five-Spice Chicken Wings with Bamboo Shoots
Bang-Bang Chicken
Tea-Smoked Duck Sichuan-Style
Stir-Fried Dminced Duck in Fresh Lettuce Cups
Yunnan-Style Roast Duck
Ken's Peking Duck
Ching's Peking Duck
Da Dong's Shredded Duck with Sichuan Flavours
with green pepper
I love spicy chicken – threads of chicken are coated in an egg white and cornflour mixture (a technique called 'velveting' where the meat retains its juiciness when shallow-fried), then stir-fried with aromatics, garlic and ginger, tossed with green pepper and seasoned with a spicy, soy, rice wine sauce. The result is rich, savoury and mouth wateringly good.
Serves 2–4 to share
1 egg white
1 tablespoon cornflour
150g (5oz) chicken thighs, skinned, boned and sliced into long, thin julienne strips
sea salt and freshly ground white pepper
vegetable oil, for shallow-frying
2 garlic cloves, peeled and minced
2.5cm (1in) piece fresh root ginger, peeled and grated
1 green pepper, de-seeded and cut into long thin matchsticks
2 large fresh cayenne chillies, de-seeded and sliced into long, thin julienne strips
1 teaspoon chilli bean paste
1 teaspoon yellow bean paste (or sweet bean paste or yellow miso)
1 tablespoon Shaoxing rice wine (or dry sherry)
1 teaspoon light soy sauce
For the garnish
1 spring onion, sliced into long strips, then dipped in iced water and drained, to make curls
Put the egg white into a bowl with the cornflour and mix well. Add the chicken strips and coat well. Season with salt and white pepper.
Fill a wok one-third full with vegetable oil and heat the oil to 180°C/350°F, or until a cube of bread turns golden brown in 15 seconds. Carefully lower the chicken pieces into the hot oil and fry for 2 minutes, or until the chicken is cooked through. Pour the chicken into a strainer over a heatproof bowl, then return 1 tablespoon of oil to the wok and reheat. Add the garlic and ginger and toss for a few seconds, then add the green pepper and fresh chillies, with a small splash of water to help create some steam to cook the vegetables, and toss through.
Return the chicken to the wok and season with the chilli bean paste, yellow bean paste (or miso), rice wine (or sherry) and soy sauce and toss together well to coat the vegetables and the chicken. Transfer to a serving plate, garnish with spring onion curls and serve immediately.
and aubergine
Yu siang is otherwise known as 'fish-fragrant' and is one of the flavour profiles found in Sichuan cooking (see here). I like to make this dish with shreds of chicken and tender aubergines coated in a delicious sauce – perfect with plain jasmine rice.
Serves 2–4
sea salt and freshly ground white pepper
1 tablespoon cornflour
300g (11oz) chicken thighs, skinned, boned and sliced into julienne strips
3 tablespoons groundnut oil
1 tablespoon Shaoxing rice wine (or dry sherry)
1 aubergine, sliced into batons and tossed in vegetable oil
2 garlic cloves, peeled, crushed and finely chopped
2.5cm (1in) piece fresh root ginger, grated
1 fresh red cayenne chilli, de-seeded and finely chopped
1 tablespoon chilli bean paste
1 spring onion, finely sliced
For the sauce
100ml (3½fl oz) cold vegetable stock
1 tablespoon light soy sauce
1 tablespoon Chinese black rice vinegar (or balsamic vinegar)
Put the salt, white pepper and cornflour into a bowl, add the chicken pieces and toss well to coat in the mixture.
Heat a wok over high heat and add 1 tablespoon of the groundnut oil. Add the chicken pieces and toss well, then stir-fry for 1 minute. Add the rice wine (or sherry) and cook for 2 minutes, or until the chicken is cooked through, then put to one side.
Clean out the wok with kitchen paper, then heat over medium heat and add another tablespoon of groundnut oil. Add the aubergine and stir-fry until browned, then cook, stirring, for 5 minutes until softened. Keep adding small drops of water to create some steam to help soften the aubergines as it cooks. Transfer to a plate and set aside.
Place the wok back on high heat and add the remaining groundnut oil. Add the garlic, ginger, fresh chilli and chilli bean paste and cook for a few seconds. Return the cooked aubergine to the wok, followed by all the ingredients for the sauce, then tip in the chicken strips and bring to the bubble.
Cook until the sauce has thickened, then stir in the spring onion. Remove from the heat and serve immediately, with jasmine rice.
with dried mushrooms, bamboo shoots, and goji berries
This nutritious dish is easy to prepare. Do try to use organic chicken where possible, as the quality of the meat will be better than non-organic and the health benefits of this dish will be greater, especially when coupled with the goji berries and dried Chinese mushrooms, which are full of antioxidants and vitamins. The bamboo shoots add texture and crunch. Serve this with plain steamed rice and stir-fried vegetables for a healthy supper.
Serves 2 or 4 to share
1 tablespoon groundnut oil
sea salt and freshly ground white pepper
2 tablespoons Shaoxing rice wine (or dry sherry)
1 tablespoon sesame oil
4 chicken thighs, skinned, boned and sliced
2.5cm (1in) piece fresh root ginger, peeled and sliced
2 fresh shiitake mushrooms, sliced
1 small tin bamboo shoots, drained
a handful of dried goji berries
Combine the groundnut oil, salt, white pepper, rice wine (or sherry) and sesame oil in a bowl.
Toss all the remaining ingredients in the rice wine dressing. Arrange on a small, heatproof plate that fits inside a bamboo steamer. Place the steamer over a small pan of boiling water (making sure the base doesn't touch the water) and steam on high heat for 15 minutes. Check that the chicken is cooked through – insert a skewer into the meat and if the juices run clear the chicken is cooked. Serve immediately, with Stir-fried Spinach (try Ken's recipe here) and jasmine rice.
I love this poetic name for such a delicious chicken dish. It is named after the Chinese musical instrument called the pi-pa, which is similar to a lute in the West. Here the chicken is prepared very much like the French method en crapaudine, which means the chicken is split down the back and flattened to look like a toad. The secret to the success of the recipe is in the drying process. If you have patience, you will be rewarded with an outstanding dish. It is well worth the wait, as you will see.
Serves 4–6
1 × 1.5kg (3–3½lb) chicken
For the coating mixture
4 tablespoons clear (plain) rice vinegar (or cider vinegar)
1 tablespoon red rice vinegar
2 teaspoons Shaoxing rice wine (or dry sherry)
2 teaspoons honey
3 tablespoons water
For the marinade
1½ tablespoons Chinese rose liqueur wine (or cognac)
1 tablespoon red fermented dofu (tofu)
1 tablespoon fermented black beans, rinsed
1 teaspoon Chinese five-spice powder
1 tablespoon finely chopped garlic
2 teaspoons salt
2½ tablespoons sugar
To butterfly the chicken, split it open down the back (not through the breast side) and remove the backbone. Flatten it out with the palm of your hand, pushing against the top of the chicken. Make two small holes, one either side of the end of the breastbone, and tuck the legs through. This will hold the shape of the bird while it roasts.
Bring some water to the boil in a wok or shallow pan and blanch the chicken (skin side only) for 5 minutes. Remove the chicken from the water and allow to dry for 1 hour in a cool draughty place, or in front of a fan.
Put all the coating ingredients in a wok and bring to a boil, then coat the skin side of the chicken several times with this mixture. Allow the chicken to dry again. Once the chicken has dried, the surface of the skin should feel like parchment.
Meanwhile, place all the marinade ingredients in a blender and whizz to combine, then rub this evenly on the inside of the chicken. Place the chicken, skin side down, on a rack and leave to marinate, uncovered, in a cool, draughty place for at least 4 hours or overnight in a fridge.
Preheat the oven to 230°C/450°F/gas 8. Place the chicken, skin side up, on a baking tray and bake for 5 minutes. Lower the oven temperature to 180°C/350°F/mark 4 and continue to cook for 30–40 minutes, or until the chicken is brown and crispy.
Remove from the oven and let the chicken sit for 15 minutes, then carve into serving pieces and serve at once.
with fermented dofu
Fermented dofu (tofu) is otherwise known as dofu ru and can be found in jars in Chinese supermarkets. It comes in many different varieties and these are known as 'Chinese cheese'. They are made from fermented soya beans and have an intensely salty, umami-rich flavour. When I was growing up, dofu ru was a staple food, paired with congee (rice porridge) and pickles for breakfast. These salty, moreish cubes are fantastic used in marinades for meats such as chicken and pork. I like to use them in a marinade for chicken, then steam the chicken thighs, remove the meat from the bone and serve with a dipping sauce of chilli black rice vinegar – simple and tasty – in celebration of Cantonese cooking. If you cannot find dofu ru, use a tablespoon of yellow bean paste.
Serves 2–4 to share
4 chicken thighs, skinned, bone in
For the marinade
2 tablespoons groundnut oil
2 garlic cloves, peeled and finely chopped
1 tablespoon freshly grated root ginger
2 tablespoons Shaoxing rice wine (or dry sherry)
1 teaspoon dark soy sauce
1 tablespoon light soy sauce
2 cubes of fermented dofu (dofu ru, optional)
1 cube of red fermented dofu (dofu ru, optional)
1 tablespoon toasted sesame oil
For the dipping sauce
2 tablespoons Chinese black rice vinegar (or balsamic vinegar)
1 medium fresh red chilli, de-seeded and finely chopped
Combine all the ingredients for the marinade, add the chicken pieces and toss to coat in the mixture, then cover and marinate in the fridge for 1 hour.
Place the chicken and marinade on a heatproof plate that fits inside a bamboo steamer. Half-fill a wok with boiling water, place the steamer over the wok (making sure the base doesn't touch the water), then cover and steam on high heat for 30 minutes.
Meanwhile, combine the vinegar and chilli for the dipping sauce and set aside.
Remove the chicken and test to see if it is cooked – insert a skewer into the meat nearest the bone and if the meat juices run clear the chicken is cooked. Remove from the steamer, then bone and cut into bite-sized pieces or shred into strips. Pour the cooking juices over the chicken and serve with the vinegar and dipping sauce.
and green tea maocha
I cooked this simple, experimental but delicious dish for the two girls from the Bulang tribe I met in Yunnan (see here), both nicknamed Xiao-Yu, at home in their tea-picking village. The girls' grandmother gave me a freshly killed and prepared chicken and I chopped it into 2.5cm (1in) pieces. Everything except the innards was used, including the neck, head and feet. I used raw pu-er tea leaves (see here) and maocha (sun-dried rough pu-er tea) in the dish, together with some brewed pu-er maocha tea. The result is a delicious light, bittersweet chicken broth. Finally, I added some local Chinese greens (but you can use tenderstem broccoli). Surprisingly, the slight bitterness of the tea leaves enhances the sweetness of the chicken meat. This is delicious with jasmine rice.
Serves 2–4 to share
2 tablespoons vegetable oil
2 tablespoons raw green pu-er tea leaves, sliced (see here)
1 tablespoon cooked and sun-dried pu-er (maocha) leaves
1 × 2kg (4½lb) whole chicken, chopped into 5cm (2in) pieces on the bone
1 teaspoon salt
450ml (¾ pint) brewed maocha tea (or vegetable stock)
225ml (8fl oz) water
300g (11oz) tenderstem broccoli
Heat a wok over high heat and add the vegetable oil. Add the green pu-er tea leaves and the cooked maocha and stir-fry for less than 1 minute. Add the chicken pieces and salt and stir-fry on high heat for 10 minutes.
Add the brewed maocha tea (or stock) and cook uncovered for 10 minutes. Add the water and bring to the boil. Toss in the broccoli and stir-fry for 1 minute until tender. Spoon out and serve immediately with jasmine rice.
with bamboo shoots
I love to cook what I love to eat. This dish combines my favourite Chinese flavours – wok-cooked chicken wings with soy sauce and five-spice, with bamboo shoots. It is an easy recipe – first marinate the chicken, then gently braise it in the wok until cooked through. The result is a tasty dish, perfect for all the family. Make sure you buy free-range or organic chicken, which is healthier and will taste better than non-organic.
Serves 2–4 to share
8 chicken wings
2 tablespoons groundnut oil
225g (8oz) tinned sliced bamboo shoots
200ml (7fl oz) hot chicken stock
1 tablespoon cornflour, blended with 2 tablespoons cold water
2 spring onions, sliced into 5cm (2in) pieces
For the marinade
2 garlic cloves, peeled and crushed
1 tablespoon freshly grated root ginger
1 teaspoon Chinese five-spice powder
3 tablespoons light soy sauce
1 teaspoon dark soy sauce
3 tablespoons Shaoxing rice wine (or dry sherry)
2 tablespoons soft brown sugar
Combine all the ingredients for the marinade in a large bowl, add the chicken wings and toss to coat in the mixture, then cover and leave to marinate in the fridge for 1 hour.
Heat a wok over high heat and add the groundnut oil. Drain the chicken, reserving the marinade, and add to the wok with the bamboo shoots. Stirring gently on medium heat, cook for 10 minutes until browned on all sides. Pour in the reserved marinade and the hot stock and braise on medium heat for 10 minutes, or until the chicken is cooked through and tender. Add the blended cornflour and stir until thickened. Add the spring onions, toss through well, then cook for 1 minute and serve immediately.
The name 'bang-bang' comes from the Mandarin word for 'stick', which is bung. The chicken meat is beaten with a stick to help tenderise it, so it shreds easily; you could use a rolling pin to flatten the cooked chicken and then use your fingers to tear it into shreds.
Serves 2
100g (4oz) vermicelli mung bean noodles (or rice noodles), pre-soaked in hot water for 5–6 minutes and drained (optional)
250g (9oz) poached chicken breast, shredded
½ cucumber, cut in half lengthways, de-seeded and sliced into long, thin julienne strips
40g (1½oz) radish, sliced
40g (1½oz) carrot, peeled and sliced into long, thin julienne strips
For the dressing
2 tablespoons groundnut oil
1 tablespoon toasted sesame oil
2 tablespoons sesame paste (or tahini blended with 1 teaspoon toasted sesame oil)
1 tablespoon crunchy peanut butter
1 tablespoon light soy sauce
1 tablespoon freshly grated root ginger
2 tablespoons Chinese black rice vinegar (or balsamic vinegar)
½ teaspoon dried chilli flakes
½ teaspoon ground Sichuan peppercorns
For the garnish
1 medium fresh red chilli, de-seeded and finely chopped
1 large spring onion, finely sliced lengthways
toasted black and white sesame seeds
Arrange the noodles, if using, on a plate. Layer the chicken, cucumber, radish and carrot on top, then chill.
Before serving, put all the ingredients for the dressing into a blender and whizz to combine. Drizzle the dressing over the dish, then sprinkle with the fresh chilli, spring onion and toasted sesame seeds. Serve immediately.
I love duck in all its forms, whether it's roasted, braised or smoked – its rich and deep-flavoured meat is unique, and we Chinese appreciate this delicious bird. Once it is marinated, it reminds me of delectable, aged smoked ham.
The smoking process adds a lovely, mysterious je ne sais quoi to the duck. Of course, frying it at the end ensures crispy skin. I love to serve it just as it is, but you can serve it with Chinese pancakes (see here) or steamed buns, as they do in Sichuan. When you have tasted this duck, you will understand why all the effort is worthwhile. It makes a very elegant dinner party dish for a special occasion.
Be particularly careful when deep-frying in a wok.
Serve 4–6
1 × 1.6–1.8kg (3½–4lb) duck, fresh or frozen
50g (2oz) camphor wood chips (or hickory chips)
25g (1oz) jasmine tea leaves
750ml (1¼ pints) groundnut or vegetable oil, for deep-frying
For the marinade
5 tablespoons salt
2 tablespoons roasted and ground Sichuan peppercorns
2 tablespoons sesame oil
150ml (¼ pint) Shaoxing rice wine (or dry sherry)
600ml (1 pint) water
If the duck is frozen, thaw it thoroughly. To butterfly the duck, split it open down the back (not through the breast side) and remove the backbone. Flatten it out with the palm of your hand, pushing against the top of the duck. Make two small holes, one on either side of the end of the breastbone, and tuck the legs through. This will help hold the shape of the duck when it is marinating and later when it is smoked.
Combine the marinade ingredients in a medium-sized bowl, mixing well to dissolve the salt. Place the duck in a shallow, non-corrosive pan and pour the marinade over it. Cover with clingfilm and refrigerate for 8 hours, or overnight.
Bring some water to the boil in a wok or shallow pan and blanch the duck (skin side only) for 2 minutes. Remove the duck from the water and allow it to dry in front of a fan for at least 3 hours.
Soak the wood chips in water. Make a charcoal fire in an outdoor grill and, when the coals are ash white, add the wood chips and tea leaves. Place the duck, skin side down, on the grill, cover and leave to smoke for about 35 minutes. Remove the duck from the grill. Set up a steamer, or put a rack into a wok or deep pan, and fill it with 5cm (2in) water, then gently steam the duck for 1½ hours. Allow it to cool thoroughly. The duck can be prepared ahead to this point, but it should be thoroughly dried before frying.
Heat a wok or large, deep, flameproof casserole over high heat until it is hot, then add the oil and heat until it is very hot. Deep-fry the duck until it is golden and crispy – about 8–10 minutes. Drain well on kitchen paper, then chop up and serve.
in fresh lettuce cups
I am proud to be part of the southern Chinese heritage. We have not only been at China's forefront politically, but also economically. In our cuisine, we have readily adapted to outside influences – in this case, crispy fresh lettuce, which we would normally blanch, is used as a textural foil to the rich duck meat that has been stir-fried. It makes a delicious, satisfying and easy dish to serve to family or friends.
Serves 4
1 iceberg lettuce
1½ tablespoons groundnut or vegetable oil
finely chopped meat from thighs and legs of a cooked or smoked duck
50g (2oz) fresh or tinned water chestnuts, peeled if fresh, drained if tinned, and sliced
4 tablespoons finely chopped spring onions
1 tablespoon finely chopped fresh ginger
1 tablespoon finely chopped garlic
1 tablespoon Shaoxing rice wine (or dry sherry)
1 tablespoon oyster sauce
1 teaspoon light soy sauce
salt and freshly ground black pepper
2 teaspoons sesame oil
Hoisin sauce, for dipping
Separate the iceberg lettuce leaves into single cups. Keep refrigerated until ready to serve.
Heat a wok over high heat until it is very hot. Add the oil, and when it is very hot, add the chopped duck meat, chestnuts, spring onions, ginger and garlic and stir-fry for 2 minutes. Add the rice wine (or sherry) and the oyster and soy sauces, then salt and pepper to taste. Finally, drizzle in the sesame oil and turn on to a platter.
Arrange the lettuce leaves on a separate platter, put the Hoisin sauce into a small bowl and serve at once.
I must be biased, but I think the Chinese are the best at cooking duck. The secret is in the preparation. The hot liquid to baste the duck and seal the skin and then the long drying ensures crispy skin and moist meat.
This Yunnan duck speciality uses the same principle as Beijing (Peking) duck, with the exception of the use of honey (which is abundant in the region) and the unique combination of condiments and seasonings used as dipping sauces, but it is not served with pancakes or steamed buns. However, the result is the same... sheer deliciousness! This makes a wonderful centrepiece for any special dinner party.
Serves 6–8
1 × 1.6–1.75kg (3½–4lb) duck, fresh or frozen
salt and freshly ground black pepper
3 tablespoons honey
For the basting liquid
1 litre (1¾ pints) water
1½ tablespoons dark soy sauce
1½ tablespoons honey
1 tablespoon clear (plain) rice vinegar (or cider vinegar)
For the dipping sauces
5 tablespoons Hoisin sauce, mixed with 2 tablespoons chilli bean sauce
2 tablespoons salt, mixed with 1 tablespoon roasted ground Sichuan peppercorns
6 spring onions, cut into 5cm (2in) segments
If the duck is frozen, thaw it thoroughly. Rinse the duck well and blot it completely dry with kitchen paper, then season the cavity with salt and pepper and then the honey. Insert a meat hook near the neck.
Combine all the basting liquid ingredients in a large pot and bring the mixture to the boil. Either place the duck on a rack over a pan or hold it by the hook, then, using a large ladle, baste the duck several times with the liquid until all the skin has been completely coated with the mixture. Hang the duck in a cool, well-ventilated place to dry, or, alternatively, hang it in front of a fan for about 4–5 hours, longer if possible – up to 8 hours. Once the duck has dried, the surface of the skin will feel like parchment.
Preheat the oven to 240°C/475°F/gas 9. Place the duck on a roasting rack in a roasting pan, breast side up. Pour 100ml (3½fl oz) of water into the roasting pan (this will prevent the fat from splattering). Put the duck into the oven and roast for 15 minutes, then lower the oven temperature to 180°C/350°F/mark 4 and continue to roast for 50 minutes. The skin will be a deep mahogany colour and very crisp when the duck is done.
Meanwhile, prepare the dipping sauces.
Remove the duck from the oven and let it sit for 10 minutes before you carve it. Using a cleaver or sharp knife, cut the duck into serving pieces by first dividing it into quarters, and then cutting the quarters into bite-sized pieces. Serve at once with the dipping sauces.
Peking duck is perhaps one of the most famous Chinese dishes. Its history goes back to the Yuan Dynasty around 1330, when it is first mentioned by an Imperial inspector in an early cook book. I suspect the technique of roasting is Mongolian in origin. The dish remained on the Imperial menu and the first Peking duck restaurant opened in Beijing around the 1500s, when the capital shifted from Nanjing to Beijing. The fame of the dish grew in time, and rightfully so.
I think it is probably one of the best techniques of roasting duck in the world. The preparation and cooking of Beijing (Peking) duck in China is an art form in itself. Specially raised ducklings are fed a rich diet of maize, sorghum, barley and soya bean for 1½ months before they are ready for the kitchen. After the ducks are killed and cleaned, air is pumped through the windpipe to separate the skin from the meat. (This allows the skin to roast separately and remain crisp, while the fat melts, keeping the meat moist.) Hot water is then poured over the duck to close the skin pores and it is hung up to dry. During the drying process, a solution of malt sugar is liberally brushed over the duck and it is then roasted in a wood-burning oven using wood from fruit trees, which gives the duck a unique fragrance. The result is a shiny, crisp and aromatic duck with beautiful brown skin, moist flesh and no fat.
Preparing Beijing (Peking) duck is a time-consuming task, but I have devised a simpler method that closely approximates the real thing. Just give yourself plenty of time and the results will be good enough for an emperor. Traditionally, Beijing (Peking) duck is served with Chinese pancakes, spring onions cut into brush shapes and sweet bean sauce. In Hong Kong and in the West Hoisin sauce is used instead (this is very similar to sweet bean sauce but contains vinegar). Each guest spoons some sauce on to a pancake. Then a helping of crisp skin and meat is placed on top, with a spring onion brush, and the entire mixture is rolled up like a stuffed pancake. It can be eaten using chopsticks or with your fingers. This makes an unforgettable dish for a very special dinner party.
If you are ambitious and want to serve a three-course Beijing (Peking) duck feast, then use some of the cooked meat for Stir-fried Minced Duck in Fresh Lettuce Cups (see here) as a second course and finish with Peking Duck Soup (see here).
Serves 4–6
1 × 1.6–1.75 kg (3½-4 lb) duck, fresh or frozen
1.2 litres (2 pints) water
150ml (½ pint) Chinese black rice vinegar (or balsamic vinegar)
3 tablespoons malt/maltose sugar (or honey)
3 tablespoons dark soy sauce
1½ tablespoons coarse salt
2½ tablespoons Chinese five-spice powder
2 teaspoons roasted ground Sichuan peppercorns
To serve
1 recipe Chinese Pancakes (see here)
24 spring onions, cut into brushes (see here)
6 tablespoons Hoisin sauce (or sweet bean sauce)
If the duck is frozen, thaw it thoroughly. Rinse the duck well and blot it completely dry with kitchen paper. Insert a meat hook near the neck.
Bring the water and vinegar to the boil in a large pot. Hold the duck by the hook over the pot and, using a large ladle, carefully pour this mixture over the outside of the duck several times, as if to bathe it, until all the skin is completely coated with the mixture. Reserve the mixture. Hang the duck in a cool, well-ventilated place to dry, or alternatively hang it in front of a cold fan for about 2–3 hours, or overnight. When the duck is dried, bring the reserved water-vinegar liquid to the boil, add the sugar (or honey) and soy sauce and again bathe the duck skin and leave to dry in front of fan for at least 2–3 hours more. Once the duck has dried, the surface of the skin will feel like parchment. Mix the salt, five-spice and peppercorns together and rub this mixture evenly inside the cavity of the duck.
Preheat the oven to 240°C/475°F/gas 9. Meanwhile, place the duck on a roasting rack in a roasting pan, breast side up. Pour 150ml (½ pint) of water into the pan (this will prevent the fat from splattering). Roast for 15 minutes, then lower the oven temperature to 180°C/350°F/gas 4 and continue to roast for 1 hour and 10 minutes.
While the duck is roasting, make the spring onion brushes. Cut off and discard the green part of the spring onion then trim off the base. You should be left with a 7.5cm (3in) white segment. Make a lengthways cut of about 2.5cm (1in) long at one end of the spring onion. Roll the spring onion 90° and cut again. Repeat this process at the other end. Soak the cut spring onions in iced water and they will curl into brushes. Pat them dry before use.
Remove the duck from the oven and let it sit for at least 10 minutes before you carve it. Using a cleaver or a sharp knife, cut the skin and meat into pieces and arrange them on a warm platter. Serve at once with Chinese pancakes, spring onion brushes and a bowl of Hoisin sauce (or sweet bean sauce).
I love Peking duck and have been trying to perfect this recipe for many years. However, since I don't have a wood-fired oven at home, just a conventional one, the result is a cross between a roast Cantonese-style duck and Peking duck. The Cantonese influence is in the use of five-spice, ginger, star anise and shallot stuffing. I also brine the duck in a solution of brown sugar, vinegar and water. The trick, of course, is to hang the duck up to dry for several hours before roasting, and then roast for a long period of time at a consistent heat until the fat renders and the skin is really crisp. Go that one step further and fry the duck once cooked for a crispy, aromatic-style duck found in Chinese restaurants all over the world. Serve with steamed wheat flour pancakes with cucumber slices and spring onion threads. Try to find Chinese mandarin Peking ducks and not muscovy ducks, as they have less fat and, therefore, will give a crispier skin. Duck-elicious!
Serves 8 to share
3kg (6½lb) duck
150ml (¼pint) boiling water
50g (2oz) soft brown sugar
1½ tablespoons sea salt
1 tablespoon Chinese five-spice powder
2.5cm (1in) piece fresh root ginger, peeled and grated
2 shallots, peeled and finely chopped
2 star anise
peanut oil (optional)
For the brining solution
150g (5oz) brown sugar
1 tablespoon salt
200ml (7fl oz) boiled warm water
700ml (1¼ pints) cold water
300ml (½ pint) Chinese clear (plain) rice vinegar (or cider vinegar)
To serve
24 wheat flour pancakes (shop bought)
75g (3oz) sweet bean paste (or Hoisin or plum sauce)
2 spring onions, sliced lengthways into long thin strips
1 cucumber, de-seeded and sliced into long, thin julienne strips
Clean the duck and cut off its wings, then remove the duck's innards (you can use these for stock). Wash the duck cavity well, then pat dry. Using a sharp skewer, make small pricks all over the skin, being careful not to damage the skin too much.
Hold the duck by its neck over the sink and carefully pour boiling water all over the skin to blanch and tighten it. Pat the duck dry inside and out.
To make the brining solution, dissolve the sugar and salt in the boiled warm water, then mix with the cold water and add the vinegar. Place the duck in a brining bag (or a deep bowl) and pour the brining solution over it. Place in the fridge and leave to marinate for 3 hours, turning the duck over halfway through to ensure even brining. This helps to keep the meat juicy as it cooks and give the duck a tangy flavour.
Remove the duck from the brine and pat dry. Discard the brining solution. Insert a meat hook near the neck and hang the duck in a cool dry place for 8 hours (you can set a fan over it to air-dry the duck, as this will help to make the skin crisp as it cooks).
Combine the sugar, salt and five-spice and rub this thoroughly inside the duck. Then stuff the duck with the ginger, shallots and star anise and, using a stainless steel needle, seal the opening.
When ready to cook the duck, preheat the oven to 150°C/300°F/gas mark 2. Place the duck, breast side up, on a roasting rack over a roasting tin. Cook the duck for 1¼hours, turning the duck over halfway through to ensure even cooking. Increase the heat to 240°C/475°F/gas mark 9 and cook for 10–15 minutes more to crisp up the skin.
If you like your duck even crispier and aromatic, heat a pan of peanut oil to 180°C/350°F, or until a cube of bread turns golden brown in 15 seconds. Place the duck, on its rack, over a large hot wok and carefully ladle the hot oil over the skin to crisp it further and until the skin turns golden brown.
Place the wheat flour pancakes in a small bamboo steamer (you may have to do this in two batches) and place the steamer over a small pan of boiling water (making sure the base doesn't touch the water). Steam for 6–7 minutes, then remove the steamer from the pan. Carve the duck at the table and serve with the bean paste (or Hoisin/plum sauce), spring onions and cucumber strips. To make each pancake, dip a little duck in the sauce, place it in the centre of the pancake, place some spring onion and cucumber on top, then roll up and eat.
with Sichuan flavours
Da Dong is widely regarded as one of the best chefs in China today. When I met him and tasted his food I understood why he deserved this accolade. Although he is from Beijing and made one of the best versions of Peking duck I have ever tasted, his real skill is in how he has re-interpreted regional dishes garnered from his travels throughout China. In this recipe, inspired by his version, but without the shredded pig ears, cooked duck was shredded and tossed in a fragrant Sichuan-flavoured sauce. It was outstanding because of its subtle use of Sichuan spices.
Serves 4
450g (1lb) cooked duck meat from either Peking Duck or Tea-smoked Duck (see here or here, or here)
For the sauce
1 tablespoon dark soy sauce
2 teaspoons Chinese black rice vinegar (or balsamic vinegar)
2 tablespoons chilli bean sauce
2 teaspoons sesame oil
2 teaspoons sugar
½ teaspoon roasted and ground Sichuan peppercorns
1½ tablespoons finely chopped spring onions
Shred the duck skin and meat into fine shreds.
Mix all the sauce ingredients together in a bowl. Add the duck meat and turn to coat the pieces thoroughly with the sauce, then leave it to sit for 15 minutes before serving at room temperature.
capital cuisine
My brother and his family live in Beijing, so I visit often. It is a bustling, busy, polluted city, much of it new, built around the splendours of the ancient Forbidden City. We eat at home quite a lot of the time, but we also eat out, in some of the many, and very varied, restaurants in our area. Beijing is opening up to the outside world – and it did so spectacularly during the 2008 Olympics – but it is not all to the good. Several fast-food chains have arrived in the last few years, and the Chinese are starting to like Westernised food.
I love the feel of Beijing, which is much more scholarly than other parts of China. The people are more chic and outgoing too: elsewhere people tend to keep themselves to themselves, but here they are a lot friendlier. You see people with dogs and other pets, which is unusual, and there are modern buildings, which give the city a very up-to-date, global feel. But the city also has an ancient history, keeping hold of its roots, and I like to see old and new together. This combination is reflected in the food, which has traditional Muslim and Mongolian influences as well as new ideas: there are Mexican, Italian and French-style cafés, and you can even buy churros! It's so different to how it used to be, when you could only buy noodles, dumplings and food in home-style restaurants.
Traditional Beijing or Peking cuisine – also known as Mandarin cuisine – is a culinary melting pot. At heart it is a northern cuisine, so is influenced by climate: for instance, the food staples of the region are wheat and corn rather than the rice of the warmer and rainier south (rice here is seen as a treat!). The basics of traditional Beijing eating – usually at open-air stalls and in night markets – are noodles, usually served in a broth, and eaten sometimes for breakfast, piping hot in winter, cold in summer. This is what workers, whether from construction sites or high-rise offices, will eat for lunch – if they don't choose Western... Other basic Beijing staples are wheat-flour dumplings and buns: baozi is a yeast-raised bun, white and pillow-like, served with a variety of fillings (char siu, red bean paste); xiaolongbao are smaller stuffed buns, their dough wrapping thinner, so they are more like dumplings; and jiaozi are horn-shaped dumplings.
We went to the Black Sesame kitchen cookery school in Beijing, where dumpling master Chairman Wang taught us the techniques of perfect pan-fried and water-cooked dumplings. Chairman Wang had previously been teaching at a traditional cookery school, with written exams and no practical elements. Jen Lin Liu, the owner of the school, didn't like this scholarly approach, so set up a school based on hands-on cooking, and invited Chairman Wang to join her. Much like baking in the West, making and eating dumplings brings families together (especially at Chinese New Year, when gold-ingot-shaped dumplings signify wealth).
The cuisine of Beijing has also been influenced by its neighbours' cooking styles, which range from the sophisticated dishes of Shandong and Jiangsu to the south, to the simpler cooking of Mongolia to the north. The recipe Beggar's chicken probably originated further south: a chicken is spiced, stuffed and wrapped in lotus leaves and foil (once it would have been mud), then baked. There are many lamb and mutton dishes in restaurants, echoes of Mongolian and Muslim influence. Cabbage is a staple here, as it is in Shandong dishes, especially in winter, and the widespread use of vinegar in Beijing cooking probably owes a lot to the fact that both the nearby provinces of Shandong and Jiangsu produce wonderful vinegars.
But a primary influence on traditional Beijing cuisine was the banquet-style cooking of the Imperial Palace. Some banquets – in which ingredients like shark's fin, sea cucumber, deer tendons and fish lips would feature – would consist of over 100 courses. Peking duck – which Ken says he would like as his last supper! – comes from this Imperial tradition. We cooked Peking duck with chef Jiang Xiao at Liqun: to dry-inflate the duck before roasting, he blew with his mouth, although it is more usual nowadays to use a bamboo pipe (I have even used a bicycle pump!). The owner of the restaurant is Zhang Li Qun and his personal story is amazing. From a very poor rural background, he couldn't afford to buy the bricks needed to make a Peking duck oven, so he travelled around on his three-wheeled bike collecting bricks from old, knocked-down houses and from builder friends. He re-formed the bricks by hand and used them to build his first oven. Now his restaurant, in one of the surviving hutongs (alleys), is one of the most respected in Beijing.
We also cooked with renowned chef Zhenxing Da Dong. He uses different techniques for Peking duck, including cooking the duck for twice as long so that all the fat drips out – making a much healthier meal. What was consistent both at Liqun and with Da Dong, though, was the skill of the chefs: they instinctively knew how to treat the duck, and when to rotate it so that it was perfectly cooked and a rich golden brown all over.
Cooking in Beijing, as in the whole of China, was dealt a severe blow when Mao Zedong came to power in 1949. The years of his rule saw the deaths of millions from starvation; his nationalisation of private businesses, including restaurants, meant that no-one cooked any more, and generations-old traditions and skills went into near terminal decline. It is only in the last 20 or so years that home cooking and restaurants are flourishing again, and this is probably due to the retained memories of Beijing elders such as Chairman Wong. Their suffering during those lost years has also given them a certain discipline, which they have used to find an inner peace. Chairman Wong, although an older lady, refused to sit down when making her dumplings, and stood the whole time...
Desserts
Red Bean Sesame Balls
Peach and Lychee Spring Rolls
Mango, Lychee and Passion Fruit Salad with Star Anise Sugar
Custard Tartlets from Guangzhou Restaurant
Chilled Watermelon Dipped in a Lime and Raspberry Coulis
Vanilla Sticky Cake with Cinnamon Sugar
Yum. Who doesn't like sesame balls? Fried, sticky, nutty balls that are like chewy doughnuts but taste even better with a sweet centre – sweet red adzuki bean paste. In fact, you could stuff them with chocolate, bananas, lotus bean paste, and so on. To ensure that the dough remains soft as you work, keep a dampened tea towel over it when not in use – this will help keep the dough malleable and allow you to fill and roll it up into balls. Glutinous rice flour dough is not the easiest to work with, but the result is worth it. When the balls are deep-fried, they turn a golden brown, puff up and float to the surface (make sure you keep the heat on medium and keep tossing the balls in the hot oil to cook the dough through) and as with any deep-frying, take care – a wok lid or spatter-proof shield works well if you have some balls that spit!
Serves 4/makes 12 balls
250g (9oz) glutinous rice flour, plus extra for dusting
60g (2½oz) soft brown sugar, dissolved in 100ml (3½fl oz) boiling water in a jug
100ml (3½fl oz) cold water
100g (4oz) tinned adzuki red bean paste
75g (3oz) white sesame seeds
650ml (1 pint 2fl oz) groundnut oil, for deep-frying
Put the rice flour into a bowl and make a well in the centre. Stir the sugar and boiling water to ensure the sugar has dissolved, then pour it into the flour and add the cold water. Combine to make a dough and knead for about 5 minutes into a ball.
Dust your hands with some rice flour, then break the dough into 12 balls, roughly the size of golf balls.
Holding a dough ball in one hand, use the thumb of the other to make a hole in the dough to form a cup, with thin sides. Press 1 teaspoon of red bean paste into the middle and gather the edges of the dough together so that the paste is completely covered. Do this with all the dough, then roll each of the balls in your hands until perfectly round and roll them in the sesame seeds.
Heat the oil in a deep-fat fryer or wok to 160°C/325°F, or until a tiny piece of the dough browns in 30 seconds. Deep-fry the balls, a few at a time, until the sesame seeds turn golden brown and the balls start to float to the surface – 6–8 minutes.
Once cooked, place the sesame balls on a tray lined with kitchen paper to drain off excess oil while you cook the remainder. Serve warm.
I love spring rolls but find they taste even better with a sweet filling – and nothing beats fresh lychees and ripe peaches. Wrap them in bought spring roll pastry and fry until golden, then serve with ice cream and maple syrup – easy and delicious! Perfect for a summer or Chinese New Year party.
Serves 4
12 × 15cm (6in) square spring roll wrappers
10 fresh lychees, peeled, halved and stoned (or use tinned lychees)
2 large ripe peaches, stoned and thickly sliced (no need to peel)
1 tablespoon cornflour, blended with 1 tablespoon warm water
groundnut oil, for shallow-frying
To serve
vanilla ice cream
golden syrup
Place a spring roll wrapper in front of you, in the shape of a diamond (with points top and bottom).
Put a chopped lychee and a few peach slices across the centre of the pastry, then brush each corner with the blended cornflour. Reserve the leftover lychees and peaches for decorating. Bring the two side corners over the filling to the middle. Bring the bottom corner up over the filling, then brush the remaining corner with a little more blended cornflour and roll up from the bottom to secure the spring roll. Continue in the same way until all the wrappers are filled.
Heat a large non-stick frying pan or wok over medium heat. Add a shallow layer of oil, allow to heat, then fry the spring rolls for 1 minute until golden. Turn the rolls over and cook the other side until golden.
To serve, put 2 or 3 spring rolls in each small serving dish, top with a scoop of vanilla ice cream and decorate with the leftover pieces of fruit. Drizzle with golden syrup and serve immediately.
with star anise sugar
Nothing beats a refreshing exotic fruit salad to end a meal. You can use any fruits you like, but I love to combine mangoes, tinned lychees and passion fruit. Dragon fruit would be just as delicious. I like to spice up the salad with some star anise sugar for a fragrant rich sweetness.
Serves 4
2 ripe mangoes, sliced into wedges
1 pineapple, peeled and sliced into wedges
1 × 425g tin lychees, drained
2 passion fruit
1 star anise
2 tablespoons white granulated sugar
For the garnish
4 fresh mint sprigs
Chill the mango, pineapple and lychees in the fridge for 1 hour.
When cold, divide between four bowls and drizzle with passion fruit pulp.
Grind the star anise in a spice grinder until you have a fine powder, then mix with the sugar and sprinkle over the fruit. Garnish each bowl with a mint sprig and serve.
from Guangzhou Restaurant
These custard tartlets are inspired by Chef Zheng Bo, a talented dim sum chef, who shared with me his recipe for these popular snacks. Chef Bo's version calls for lard and butter in the pastry, but I decided to use only butter. His secret is to roll and fold the pastry three times in a row, resulting in a flaky pastry which is irresistible when filled with the light, delicate, silky custard.
Makes 8–10 tartlets
150ml (¼ pint) water
4 tablespoons caster sugar
1 tablespoon milk
3 small eggs, beaten
For the pastry
100g (4oz) plain flour, plus extra for dusting
4 tablespoons butter, cut into small pieces
2 tablespoons caster sugar
2 tablespoons double cream
Mix the ingredients for the pastry together in a bowl or in a food processor. Roll the pastry into a ball, then roll out on a lightly floured surface and fold and roll at least three times. Roll into a ball, wrap with clingfilm and refrigerate for at least 30 minutes.
Boil the water and sugar in a saucepan for 5 minutes to make a light syrup, then take off the heat and leave to cool thoroughly.
Preheat the oven to 190°C/375°F/gas mark 5.
Roll out the pastry to about 0.5cm (¼in) thick and press into small tart moulds.
Combine the cooled syrup with the milk and eggs, then divide evenly between the tart moulds.
Bake the tartlets for 10 minutes. Lower the oven temperature to 160°C/325°F/gas mark 3 and bake for a further 10 minutes, or until the custard just sets. Leave to cool before serving.
dipped in a lime and raspberry coulis
Desserts never featured much on my family's kitchen table – after a meal we ate whatever fruits were in season. I always looked forward to watermelon season, and the refreshing sweetness of the flesh was even better when slightly chilled. This has become my favourite recipe to make when entertaining friends and family – it's delicious and fun to eat, especially for children.
Serves 4
350g (12oz) raspberries
2 tablespoons maple syrup
zest and juice of 1 lime
½ watermelon, skinned and cut into 2cm (¾in) dice
For the garnish
small fresh mint leaves
Prepare the coulis by blending 250g (9oz) of the raspberries with the maple syrup, lime zest and juice. When smooth, pass through a sieve and spoon into four small dipping dishes.
Divide the watermelon cubes and remaining raspberries between four plates, garnish with the mint leaves and serve with the coulis.
with cinnamon sugar
Traditionally, this is served at Chinese New Year. It is a sweetened dough made with glutinous rice flour. The dough is steamed and left to cool and set, then it's dipped in an egg batter and fried until the outside is golden but the inside is soft, pillowy and chewy. This is one of my favourite Chinese desserts. I like to add a touch of vanilla extract to the sweetened dough and, once the pieces have been fried golden brown, I sprinkle some cinnamon sugar on them while they're hot – they're like soft, chewy Chinese doughnuts!
Serves 2–4 to share
200ml (7fl oz) water
125g (4½oz) brown sugar
150g (5oz) glutinous rice flour
25g (1oz) wheat starch
1 teaspoon vanilla extract
groundnut oil, for deep-frying
For the batter
100g (4oz) potato flour (starch)
2 eggs, beaten
a pinch of sea salt
1–2 tablespoons cold water (if needed)
For the cinnamon sugar
2 teaspoons ground cinnamon
2 tablespoons granulated sugar
Bring the water to the boil, then stir in the brown sugar until dissolved. Leave to cool.
Sift the rice flour and wheat starch into a bowl, pour in the cooled sugar liquid and vanilla extract and mix to a silky batter consistency. Place in a thin circular tray or foil container on a heatproof plate that fits inside a bamboo steamer. Half-fill the wok with boiling water, place the steamer over the wok (making sure the base doesn't touch the water), then cover and steam on high heat for 1 hour. Check the wok occasionally and, if necessary, top up with boiling water.
Leave to cool for 30 minutes, then refrigerate until set. Once set, carefully remove from the tray or foil and cut into 2.5cm (1in) squares.
Combine the ingredients for the cinnamon sugar in a bowl.
Heat the oil in a deep-fat fryer or wok to 180°C/350°F, or until a tiny piece of the dough browns in 15 seconds. Combine all the ingredients for the batter in a bowl. Dip the cake squares in the batter and deep-fry for 5–6 minutes until golden brown. Drain on kitchen paper, then dip in cinnamon sugar and serve.
and the creation of megacities
The term 'megacity' was once applied to cities that had a population in excess of 10 million. But increasingly in Asia, megacities are becoming much more common and much larger. For instance, the Chinese government is planning to create the world's largest megacity by merging nine cities in Guangdong – from Guangzhou in the north to Shenzhen in the south, from Huizhou in the east to Zhaoqing in the west. This will create a metropolis twice the size of Wales with a population of 42 million. The cities lie on the Pearl River Delta, one of the most populous areas in China already, and one of the economic hubs of China. Chongqing has been called a megacity, but it is nothing in comparison.
Chongqing is a major inland city of southwestern China and, administratively, is one of the People's Republic of China's four direct-controlled municipalities (along with Beijing, Shanghai and Tianjin). An ancient and historic city, Chongqing became the provisional capital of the Nationalists under Chiang Kai-shek during the second Sino-Japanese War of 1937–45. The Allies set up their Chinese anti-fascist headquarters there too. The city was heavily bombed by the Japanese, and the population's bravery at this time earned Chongqing the name 'City of Heroes' and a letter of commendation from President Roosevelt. The city was demoted to become a sub-division of Sichuan Province from 1954 to 1997, but it has now regained its previous status. It is probably the world's largest municipality – roughly the size of Austria – and, with a population of over 30 million, has been called China's largest city This is a misnomer, though, as up until now the population of the core city was very much smaller: the rest of the municipality's population lived in the countryside.
It is a mountainous, modern river city, lying on the upper reaches of the Yangtze: the colour-change confluence of this great river and its tributary, the Jialing, is one of Chongqing's most famous sights. It is the largest inland river port in western China, trading via the Yangtze; following the building of the Three Gorges Dam, and the creation of the great reservoir from the Dam to Chongqing (some 660 kilometres), trading has increased, allowing even seagoing ships from Shanghai, via the Dam's locks, to access western China. Chongqing's population increased too at this time, the city taking in many of the 1.5 million people from over 1,000 towns and villages displaced by the Dam. And with the loss of all that precious agricultural land – a process being echoed all over China as distending cities eat into countryside – I wonder whether this means that China might be losing its ability to feed itself...
And the megacity is still growing. Every day over 1,000 migrants, mainly subsistence farmers, flock in from the country hoping for a new profitable urban life. (This is happening throughout China, and has been called the biggest migration in world history.) Many of them end up as 'bang-bang' men – porters with bamboo poles and ropes who carry goods for others, on their shoulders, up and down the steep hills of the city. They can be seen everywhere in Chongqing – on street corners and at bus and train stations and piers. It is in megacities like Chongqing that the huge gap between rich and poor, between urban and rural areas, has become most apparent (they say that Chongqing is home to the majority of China's new millionaires). In an effort to address this, amongst other initiatives, Chongqing is subsidising city housing projects, to give homes to many of those flocking in. Chengdu (which could be called the megacity of Sichuan) on the other hand, has focused money and attention on improving life in the surrounding countryside, which sounds more sensible to me.
Whether you are rich or poor, you can eat well in Chongqing. The cuisine is largely an offshoot of Sichuanese cooking, but with a few speciality dishes of its own. The food is perhaps less 'fragrant' that that of Chengdu, and it is certainly hotter. The key ingredient is the special chilli that grows so well throughout Sichuan, around Chongqing, and in neighbouring provinces: the Shizhu or Chaotian red pepper is amongst the spiciest in China (peppers grown in China account for over 50 per cent of world production). The most famously spicy dish of Chongqing is the hotpot, or malatang (ma and la again). Hotpots are known throughout Sichuan, but they were born here, and hotpot restaurants abound. You can go for a regular hotpot, which means choosing a soup base – as hot or bland as you like (or both, in a double pot) – then the vegetables or protein (beef, chicken or pork) you want to cook in that soup. If you fancied duck, frog, rabbit or lamb as the protein content, you might have to opt for a specialist hotpot restaurant. (The locals prefer offal – amply illustrating the axiom that the Chinese will eat anything – and usually choose cow's stomach, penis, intestines, lungs, brain, feet...) After cooking, the items are fished out of the broth and dipped into an array of spicy dips, at its simplest, seasoned sesame oil and chopped garlic. At one time, hotpot was a dish eaten in the depth of winter, but nowadays it is eaten year-round, even in summer – and Chongqing is one of China's four furnace cities, which means it can reach 40°C (104°F) in the summer!
Other dishes you will find locally are bang-bang chicken, stuffed rice dumplings, meat with crispy rice and a pork dish cooked, curiously, with rock sugar. There are the usual street foods on offer, and the same basic flavours as you would find in Sichuan – peppercorns, sesame paste, dofu (tofu), chilli bean sauces, ginger and star anise – although plucked baby chicks on skewers have been seen... The Chongqing people are proud of their cuisine, boasting that it is the best in China. As a result, foreign food chains are not as popular here as they are elsewhere, though efforts are being made to change that: McDonald's serves red bean pie (made with sweetened adzuki bean paste) instead of apple pie, and is actually known mainly for its ice cream. (Desserts are liked in Chongqing, of the sticky rice, wrapped sweet paste, jelly type.) Chongqing food is definitely hotter than in Chengdu, and it is perhaps a little coarser (or so Chengdu people would say), but it is a new, enthusiastic, money-conscious, Westernised city, with a thriving and vital culinary culture.
FLAVOURED OILS
Sichuan pepper oil
Place 225g (8oz) Sichuan peppercorns in a heatproof, sterilised jar. Heat 675ml (1 pint 3fl oz) vegetable oil, then ladle into the jar. Seal and leave for as long as possible to allow the flavours to infuse. To use, drain off the peppercorns.
Chilli oil
Using a pestle and mortar, grind 225g (8oz) small Sichuan chillies very finely. Place in a heatproof, sterilised jar. Heat 675ml (1 pint 3fl oz) vegetable oil, then ladle into the jar. Seal and leave for as long as possible to allow the flavours to infuse. To use, drain off the chillies.
Fragrant oil
Slice a 5cm (2in)piece fresh root ginger and 3 spring onions. Place in a heatproof, sterilised jar with ¼ teaspoon salt. Heat 675ml (1 pint 3fl oz) vegetable oil, then ladle into the jar. Seal and leave for as long as possible to allow the flavours to infuse.
GLOSSARY
Adzuki red bean paste
Adzuki beans are small, dull-red and slightly square in shape. They are high in protein. Sweet red bean paste is made by boiling the beans, then mashing to a paste and cooking until dry. It is a popular filling for Chinese desserts, buns and cakes.
Baijiou
This is a clear, white spirt, distilled from grain. Use in marinades or pickles, or drinks.
Bamboo shoots
The young shoots of the bamboo tree, usually found vacuum-packed or in tins orjars. Pickled sour bamboo shoots are prepared by pickling boiled bamboo sprouts in brine, giving them a sour taste. They are also pickled in chilli oil, giving them a spicy taste. Bamboo shoots add a crunchy texture to salads, soups, stir-fries and braised dishes.
Bean curd – see Dofu
Chilli bean paste
This hot sauce is made from black beans and chopped chillies that have been fermented with salt. Some versions include fermented soya beans, garlic and other spices. Good in soups, braised dishes and sauces. Use with caution, as some varieties are extremely hot.
Chilli oil
This is made by infusing crushed dried red chillies in hot oil to give an intensely hot, clear, red oil. Some chilli oils also contain specks of dried chillies. Use in sauces, stuffings and soups. (See here)
Chilli sauce
Made from chillies, vinegar, sugar and salt, this bright red, hot sauce can be used in cooking, or as a dipping sauce. There are several varieties, some flavoured with garlic and vinegar.
Chinese aubergine
Pale-purple in colour, this is longer and more slender than the European variety and has a more delicate flavour, but can be used in the same way. Bake, fry, grill or braise.
Chinese cabbage (Chinese leaf)
This cabbage has a mild taste and delicate aroma. The white stalks retain their crunchy texture when cooked. Use as a vegetable, or in soups and stir-fries.
Chinese chives (garlic chives)
These have long, flat, green leaves and a strong garlic flavour. There are three varieties: yellow, the mildest; flowering, with small, edible yellow flowers; and green, with the strongest flavour. The flowers can be eaten. Use them in soups, stuffings and stir-fries.
Chinese cinnamon bark/stick
True cinnamon is the dried bark of the cinnamon tree. It is available as thin, rolled cinnamon sticks, which have a concentrated, robust flavour, or as bark, whose aromatic flavour is good in braised dishes.
Chinese dried citrus (orange/tangerine) peel
A strongly aromatic alternative to fresh citrus zest, with a pleasant smell, but slightly bitter taste. Rehydrate in water or rice wine before using. It is most often used in savoury braised or simmered dishes, and occasionally in stir-fries.
Chinese dried tree ear fungus – see Mushrooms, Cloud ear (tree ear) fungus
Chinese dried wood ear fungus – see Mushrooms, Chinese dried wood ear fungus
Chinese five-spice powder
This distinctive seasoning is a blend of cinnamon, cloves, Sichuan peppercorns, fennel and star anise, which give it the sour, bitter, pungent, sweet and salty flavours of Chinese cooking. Use sparingly for braised meats and in marinades.
Chinese leaf – see Chinese cabbage
Chinese rock sugar
This is a solid mixture of refined and semi-refined sugar with large, slightly golden-yellow crystals. If necessary, crush the lumps into smaller pieces. Use for sauces, syrups and glazes. Use half the quantity of soft brown sugar as a substitute. Red rock sugar is refined from red sugar cane.
Chinese rose liqueur wine
Made from sorghum wine and distilled with rose petals, sugar and salt, Chinese rose wine is predominantly a cooking wine, but can be used in drinks. Clear in colour and slightly salty in taste, it has a very subtle rose fragrance. Use in marinades and roasts, or in a cocktail.
Chinese water spinach
Similar to European spinach, this has green, supple leaves and hollow stems that absorb spices well and remain crunchy after cooking. Use in stir-fries and soups.
Chinkiang black rice vinegar – see Vinegar, Chinese black rice vinegar
Cloud ear (tree ear) fungus – see Mushrooms, Cloud ear (tree ear) fungus
Dofu (tofu)
Made from yellow soya beans, this protein-rich curd comes in various forms, including firm, extra firm (pressed), silken and soft. Although quite bland, it takes on the flavour of whatever ingredients it is cooked with. The firmer varieties are used for stir-fries, soups and grilling.
Deep-fried dofu are cubes of fresh dofu that have been deep-fried until the surface is crisp and golden, with a little soft curd remaining inside. Can be eaten on its own with dipping sauces, added to braised dishes, or used as a topping to give texture and protein.
Dofu-gan is dry and extra firm. Often vacuum-packed, it is available plain or braised with flavouring.
Dofu ru (or fermented dofu) is dofu that has been preserved in rice wine, brine or with chillies and condiments. Often sold cubed, it comes in white and red varieties in many flavours and is quite strong in flavour. It can be eaten on its own or used as a marinade, condiment or an accompaniment to congee. It can also be used to flavour braised dishes or vegetables.
Soft and silken dofu have a cream cheese-like texture and can be used in a variety of dishes. Silken dofu crumbles easily and can be used in sauces and desserts to give a creamy texture.
Dried chillies/flakes
Crushed dried whole red chillies, with their seeds, give a fiery heat when added to dishes, so are best used with some caution. At home, you can grind whole dried chillies in a pestle and mortar to give flakes, which can then be added to dishes.
Dried lily stems (buds)
Dried lily buds are yellow-gold in color, with a musky or earthy taste. Use in soups and stir-fries. To use, soak in warm water for about 20 minutes to soften them before cooking.
Dried shrimp
Orange-red in colour and very pungent, these are cooked shrimp that have been dried and salted to preserve them. They are used as a seasoning, or in soups to add flavour and texture. To use, soak in hot water for 20 minutes to soften them before cooking.
Edamame beans
High in protein, these are green soya beans, harvested while the pods are still attached to the bush. The pods are cooked whole and the beans are then squeezed out. They can be eaten as a vegetable in their own right or added to other dishes to give texture.
Fermented black beans
These small black soya beans have been cooked and fermented with salt and spices, giving them a salty, pungent taste, rich aroma and soft texture. Often used as a seasoning or in steamed, braised and stir-fried dishes.
Fermented cabbage
Finely shredded raw cabbage is mixed with salt, water, vinegar and sugar, then tightly packed into an airtight container and left to ferment, which preserves the cabbage and gives it a tangy taste. Can be used in stews and braises, or combined with other ingredients with rice.
Fermented dofu – see Dofu
Five-spice powder – see Chinese five-spice powder
Garlic chives – see Chinese chives
Goji berry (Chinese wolfberry)
The deep-red, dried fruit of an evergreen shrub, these are a similar size to a raisin. Sweet and nutritionally rich, they can be eaten raw or cooked.
Hoisin sauce
A thick, dark, brownish-red sauce, whose basic ingredients are fermented soya beans, salt, starch, spices and sweeteners. Sugar, garlic, vinegar and chillies are also common ingredients. Use as a condiment, glaze, marinade or dipping sauce.
Jasmine rice
An aromatic rice with slender, long grains, this white rice originates from Thailand. When cooked it is soft, white and fluffy. Use as an accompaniment to dishes.
Long Jing tea
A mild green (unfermented) tea with a sweet flavour and pure aroma. It is high in antioxidants and contains vitamin C and amino acids. In cooking, use as a flavouring.
Lotus leaves
A popular ingredient in Chinese medicine, these leaves can be used to wrap ingredients before cooking. Cut to size and then soak in hot water before use, according to the recipe. The leaves themselves are not eaten.
Lotus root
This is the root of the lotus flower that grows underwater and is available fresh (vacuum-packed), dried or in tins. Fresh lotus root should be peeled before use. When cut, the interior has a distinctive lacy appearance. The texture is slightly crunchy and mildly sweet. The root can be eaten raw, in salads, or on its own with dips, as well as cooked in many different dishes.
Lychee
A small, oval fruit with a spiny red skin, the lychee is the fruit of an evergreen tree native to southern China. The flesh is translucent white or pink with an aromatic and distinctive flavour. Each fruit contains a single, large, shiny black seed. Available fresh or tinned, they are usually served as a dessert.
Malt sugar (maltose)
A thick, treacle-like ingredient made from fermented barley. Usually diluted with water or vinegar and used as a glaze.
MUSHROOMS
Cloud ear (tree ear) fungus
Smaller and more tender than wood ears, cloud ears are sold mainly in dried form. Quite bland, they soak up the flavour of whatever they are cooked with. Their crunchy texture is useful in soups and stir-fries. To use, soak in hot water for 20 minutes – they will puff up like clouds.
Chinese dried black mushrooms
Available in various sizes and shades of brown, these mushrooms add flavour and aroma to dishes, while absorbing sauces and spices. To use, soak in hot water for 20 minutes to soften them before cooking. Their slightly salty taste complements savoury dishes well.
Chinese dried wood ear fungus
A larger variety of the cloud ear fungus, these dark brown-black fungi do not impart flavour but add colour and crispness to any dish. To use, soak in hot water for 20 minutes – they will swell to four or five times their size.
Shiitake mushrooms (fresh and dried)
These nutrient-rich, large dark-brown mushrooms are umbrella-shaped fungi that are prized for their culinary and medicinal properties. To use the dried variety, soak in water for 20 minutes to soften slightly before cooking. Use in soups, stir-fries, steamed or simmered dishes.
Nian Gao
These are sticky cakes made from glutinous rice, and thought to bring good fortune when eaten during Chinese New Year. They can be flavoured or plain.
NOODLES
Egg noodles
Made from wheat flour, egg yolk and salt, these are the most common type of noodle and come in various shapes and thicknesses. They are available fresh or dried.
Mung bean (transparent/cellophane) noodles
These transparent noodles are made from the starch of green mung beans and water. They come in various thicknesses, vermicelli being the thinnest type. To use, soak in hot water for 5–6 minutes to soften them before cooking. If using in soups or deep-frying, no pre-soaking is necessary. They become translucent when cooked. Use in salads, soups, braised dishes and stir-fries. They can also be deep-fried and used as a garnish.
Vermicelli rice noodles
Dried rice noodles come in many widths and varieties. Vermicelli are a fine, creamy-coloured noodle. To use, soak in hot water for 5 minutes to soften them before cooking. If using in salads, soak for 20 minutes. No need to soak if using them in a soup. They turn opaque when cooked. Use in soups, salads and stir-fries. They can also be deep-fried, when they expand and become light and crisp.
Wheat flour noodles
These noodles originated in the north of China, where wheat is a staple crop. Made from wheat flour and water, they are available fresh or dried. Use in soups, salads and stir-fries.
Oyster sauce
A richly flavoured, thick brown sauce made from dried oysters, soy sauce and other seasonings. Use as a seasoning, in marinades, and to add flavour and colour to braised and stir-fried dishes and vegetables. Use with caution as it is very salty. Vegetarian oyster sauce, made from mushrooms, is also available.
Pak choy
With fleshy white stalks and broad green leaves, this vegetable can be boiled, steamed or stir-fried.
Pickled chilli bamboo shoots – see Bamboo shoots
Potato flour
This gluten-free flour is made from steamed, dried and ground potatoes. The silky smooth flour gives a wonderful crispness to ingredients when they are coated in it before being shallow- or deep-fried. It has stronger binding properties than cornflour, but cornflour can be used as a substitute.
Preserved mustard greens
The leaves of mustard cabbage are pickled with water, vinegar, chilli, salt and sugar. Use as a vegetable, in soups, or as a flavouring ingredient.
Sesame oil – see Toasted sesame oil
Sesame paste
Made from crushed, roasted white sesame seeds blended with toasted sesame oil, this golden brown paste is used with other sauces as a flavouring. It gives a nutty taste to marinades and can be used in both hot and cold dishes. Tahini can be used as a substitute, but it is lighter in flavour and so you will need to add more toasted sesame oil.
Sesame seeds
These seeds, with a nutty flavour and high oil content, add taste and texture to many dishes. They are available in black, white yellow and red varieties, toasted and untoasted.
Shaoxing rice wine
Made from rice, millet and yeast, this wine is aged for anything between three and twenty years, sometimes longer. It is used for both drinking and cooking. When added while cooking meat or fish it removes the odour or rawness and gives a bittersweet finish. It is used in marinades, stir-fries, braisedand simmered dishes. Dry sherry makes a good substitute.
Sichuan chillies
These come in many varieties, both fresh and dried, the most common being a short, fat, bright red chilli that is hot and fragrant.
Sichuan peppercorns
These are the dried berries of a shrub and are widely used all over China. They have a sharp taste and lemony fragrance and can be roasted, cooked in oil to flavour the oil, or mixed with salt as a condiment for any dish.
SOY SAUCE
Dark soy sauce
Made from fermented soya beans and a roasted grain (usually wheat), dark soy sauce is left to age for longer than light soy sauce. Less salty in taste and slightly thicker than light soy, it is used in marinades, sauces, stews and braised dishes to give flavour and colour.
Light soy sauce
Made from fermented soya beans and wheat, light soy sauce is aged for less time than the dark variety and is saltier in flavour. It can be used in soups, stir-fries and braised and stewed dishes. Wheat-free (tamari) and Low-sodium varieties are available.
Spring roll wrappers/pastry
Made from flour, salt and water, these paper-thin skins are used for wrapping foods, such as in spring rolls, before deep-frying. They can also be eaten raw, filled with salad and dressings. They are available fresh or frozen. Filo pastry makes a good substitute.
Star anise
A staple ingredient in Chinese cooking, star anise is the star-shaped seed pod of a small tree that grows in southwest China. Similar in flavour to aniseed, although more robust, it is one of the ingredients of Chinese five-spice powder, and is used in braised dishes.
Thousand-year-old eggs
These are eggs (usually duck) that have been buried in clay, ash, salt and lime, then left to mature for a few weeks to a few months, depending on the process. The yolk turns semi-solid, with a grey to green colour, while the white becomes a brown, jelly-like substance. They have a strong smell and a love-it or loathe-it taste. To use, rinse under water, then peel off the outer layer and shell. Serve as an appetiser, in soups or congee, with dofu or rice, or as a condiment.
Toasted sesame oil
Made from crushed and toasted white sesame seeds, this dark-coloured, thick, aromatic oil is used as a flavouring or seasoning. It is not suitable for use as a cooking oil since it burns easily. The flavour is intense, so use sparingly.
Tofu – see Dofu
Tree ears – see Mushrooms, Cloud ear (tree ear) fungus
Vermicelli mung bean noodles – see Noodles, Mung bean noodles
VINEGAR
Chinese black rice vinegar
This rich, aromatic vinegar is used in braised dishes and sauces, and with noodles. When cooked, it gives dishes a smoky flavour with a mellow and earthy taste. Balsamic vinegar makes a good substitute.
Plain (white) rice vinegar
Clear and mild, this vinegar is used in dressings, as a seasoning and for pickling. Cider vinegar can be used as a substitute.
Red rice vinegar
This clear, pale red vinegar has a slightly sweet and salty flavour. It is used for dipping sauces and in sweet and sour dishes.
Water chestnuts
Water chestnuts, the roots of an aquatic plant, are available fresh, in vacuum packs, or tinned (but these have an inferior taste to fresh). They resemble a chestnut in shape and colouring and have a firm, crunchy texture. Can be used in salads, fillings and stir-fries.
Wheat flour dumpling wrappers/skins
Made from wheat flour, salt and water, these flat, thin discs of dough are used to make dumplings. They are available fresh or frozen. When using, keep covered with a damp towel to prevent them from drying.
Wheat flour noodles – see Noodles
Wheat starch
A finely textured white powder obtained from wheat grain. It is combined with hot water and rolled out to make dumpling skins, which turn from opaque to translucent once steamed.
Wonton wrapper
Made from egg, wheat flour, salt and water, wonton wrappers are used to make dumplings. Available fresh or frozen.
Wood ear mushrooms – see Mushrooms, Chinese dried wood ear fungus
Yellow bean sauce
A thick, spicy, aromatic sauce made from fermented yellow soya beans, dark brown sugar and rice wine. There are two forms: whole bean sauce, and mashed or pureed beans. Use in marinades and as a flavouring in many savoury dishes.
WITH THANKS FROM KEN
I begin with Ching, my partner in crime. She has been so terrific, a real trooper and I am delighted to have taken this journey with her. I not only learnt a lot from her but also shared so much laughter and good food. We are now life-long friends.
Of course, I owe a tremendous debt to Keo Films, especially to Paula, as well as her team and Jade in particular. Our filming crew was exceptional and I take a bow to: Emma, James, Craig, Qiao, Torch, Lillian, Daisy, Chef Kai and Daniel. Mr. Zhao was wonderful and exceptionally helpful.
Lee not only tested the recipes for me but gave me critical comments. Muna and Joe expertly guided the book while Susan assisted us with the introductions. Of course, my agents, Julian and Carole – their advice and counsel was, as always, invaluable.
But my biggest gratitude is to all the chefs and families who opened uptheir homes and kitchens to us with a generosity that touched my heart. I am forever grateful.
WITH THANKS FROM CHING
To the powers at BBC, thank your for such an amazing experience – this is one I will treasure for the rest of my life. I feel so privileged to have had this opportunity to experience modern China and delve further into my culinary heritage. Heartfelt and sincere thanks to Janice Hadlow, Tom Edwards and all who championed this project at the BBC.
Cooking, filming and writing this book whilst travelling in China, was not without its challenges. I am indebted to these professionals who made getting up at the crack of dawn and finishing at the crack of dawn easier to stomach – Paula Trafford (for your support from day 1), Emma Peach (Panda 1) (for your passion and determination), James Nutt (Panda 2)(you are a talented nutcase), India Bourke, Catia di Giorgio, Jade Miller Robinson, Craig Hastings (for your humour), Daniel (for your care), Daisy Newton Dunn, Mr. Zhao, Lillian Chen, Qiao Xin, Liu Kai, Torch.
Thank you to everyone at Random House for making this book such a pleasure to work on. Thank you to Muna Reyal, Joe Cottington, Susan Fleming, Liz and Max Haarala Hamilton and Katie Giovanni for a gorgeous book despite the tight deadlines!
Thanks to Katie Rice, Jamie Coleman, Zaria Rich for convincing me to challenge myself! To Toby Eady, Michael Foster and Michael Kagan for making it happen. To Agatha Chapman Poole, thanks for your support back in the UK during my travels abroad.
To all the cooks we encountered in China – thank you for your warmth and generous spirit – for sharing your delicious food and passion with an open heart, I am forever grateful and humbled by the stories you all shared, they touched me and nourished me, both spiritually and emotionally, more than you know. I will cherish our exchanges, however fleeting, for the rest of my life.
To Ken, my 'sifu' – I salute you with heartfelt thanks with a bottle of the finest Chateau La Fete! You have carried the torch for us Chinese living in the West, representing us, sharing your knowledge with the West with great passion, graciousness, humour and authority. Your lifelong dedication and the challenges you faced throughout your professional years couldn't have been easy. You are a true leader and culinary master. I am so honoured to have shared this incredible journey with you.
Thanks to all my friends and family. To Jamie Cho, thanks for always being there through the good and bad. To my mum and dad, thank you for forcing me to go to Chinese Sunday school all those years ago when I was growing up. Rusty Mandarin did come in handy...
Finally, this is for my grandmother and grandfather – for the hardships you endured and the sacrifices you made in order to give us a better life.
This ebook is copyright material and must not be copied, reproduced, transferred, distributed, leased, licensed or publicly performed or used in any way except as specifically permitted in writing by the publishers, as allowed under the terms and conditions under which it was purchased or as strictly permitted by applicable copyright law. Any unauthorised distribution or use of this text may be a direct infringement of the author's and publisher's rights and those responsible may be liable in law accordingly.
Version 1.0
Epub ISBN 9781446417980
www.randomhouse.co.uk
10 9 8 7 6 5 4 3 2 1
This book is published to accompany a television series, produced by Keo Films and first broadcast on BBC2 in 2012.
Executive producer: Paula Trafford
Series producer: Emma Peach
Commissioning executive for the BBC: Tom Edwards
Published in 2012 by BBC Books, an imprint of Ebury Publishing. A Random House Group company.
© Keo Films 2012
© Promo Group Limited 2012
© Ching-He Huang 2012
All photography © Woodlands Books Ltd 2012, unless specified below
Ken Hom and Ching-He Huang have asserted their right to be identified as the authors of this work.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
The Random House Group Limited Reg. No. 954009
Addresses for companies within the Random House Group can be found at www.randomhouse.co.uk/offices.htm
A CIP catalogue record for this book is available from the British Library.
ISBN 978 1 849 90498 8
Commissioning editor: Muna Reyal
Project editor: Joe Cottington
Copy editor: Barbara Dixon
Editor: Susan Fleming
Designer: Pene Parker
Food photography: Haarala Hamilton
Food stylist: Katie Giovanni
Location photography: Craig Hastings
Photographs here © View Stock/Alamy; here © Lou Linwei/Alamy; here (top left and middle right), here (top left and bottom left), here (top left) © Ching-He Huang
Colour origination by AltaImage
To buy books by your favourite authors and register for offers, visit www.rbooks.co.uk
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## Praise for the novels of Heather Graham
"An incredible storyteller."
—Los Angeles Daily News
"Graham wields a deftly sexy and convincing pen."
—Publishers Weekly
"A fast-paced and suspenseful read
that will give readers chills while keeping them
guessing until the end."
—RT Book Reviews on Ghost Moon
"If you like mixing a bit of the creepy
with a dash of sinister and spine-chilling reading with
your romance, be sure to read Heather Graham's
latest...Graham does a great job of blending
just a bit of paranormal with real, human evil."
—Miami Herald on Unhallowed Ground
"Eerie and atmospheric, this is not late-night reading
for the squeamish or sensitive."
—RT Book Reviews on Unhallowed Ground
"The paranormal elements are integral to the
unrelentingly suspenseful plot, the characters are
likable, the romance convincing, and, in the wake of
Hurricane Katrina, Graham's atmospheric depiction of
a lost city is especially poignant."
—Booklist on Ghost Walk
"Graham's rich, balanced thriller sizzles with
equal parts suspense, romance and the paranormal—
all of it nail-biting."
—Publishers Weekly on The Vision
"Heather Graham will keep you in suspense
until the very end."
—Literary Times
"Mystery, sex, paranormal events.
What's not to love?"
—Kirkus on The Death Dealer
## Also by HEATHER GRAHAM
PHANTOM EVIL
NIGHT OF THE VAMPIRES
THE KEEPERS
GHOST MOON
GHOST NIGHT
GHOST SHADOW
THE KILLING EDGE
NIGHT OF THE WOLVES
HOME IN TIME FOR CHRISTMAS
UNHALLOWED GROUND
DUST TO DUST
NIGHTWALKER
DEADLY GIFT
DEADLY HARVEST
DEADLY NIGHT
THE DEATH DEALER
THE LAST NOEL
THE SÉANCE
BLOOD RED
THE DEAD ROOM
KISS OF DARKNESS
THE VISION
THE ISLAND
GHOST WALK
KILLING KELLY
THE PRESENCE
DEAD ON THE DANCE FLOOR
PICTURE ME DEAD
HAUNTED
HURRICANE BAY
A SEASON OF MIRACLES
NIGHT OF THE BLACKBIRD
NEVER SLEEP WITH STRANGERS
EYES OF FIRE
SLOW BURN
NIGHT HEAT
## HEATHER GRAHAM
## HEART OF EVIL
Dedicated with gratitude
to the beautiful Myrtles plantation,
and to Teeta LeBleu Moss, owner,
Teresa David, the General Manager,
Hester Eby, Director of Tours,
Taryn Lowery, Tour Guide
and to Scout and Sprout and
The Peace River Ghost Trackers
And to Dennis, Jason, Shayne,
and Bryee-Annon Pozzessere;
Teresa Davant, Kathy Pickering, Kathy DePalo,
Juan Roca, Bridget LeVien, Matthew Green,
Phinizy Percy Jr., and Connie Perry.
## Contents
Prologue
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Epilogue
## Prologue
Blood.
She could see it, smell it.
Hear it.
Drip...drip...drip...
The air was heavy with black powder, and the brilliant red color of the blood seemed to form a mist with the powder, and she was surrounded by a haze, a miasma of gray-tinged crimson. The day was dying, becoming red, red like the color of the blood seeping to the ground, making that terrible, distinctive noise. Drip, drip, drip...
Ashley Donegal was there. She wasn't even sure where there was, but she knew that she didn't want to be there.
Suddenly, the mist seemed to swirl in a violent gust, and then settle softly, closer to the ground. It parted as she walked through. She could see her surroundings, and, at that moment, she knew. She was in the cemetery. She had played here so often as a child—respectfully, of course. Her grandfather never would have had it any other way. Those elegant tombs, all constructed with such love, and an eye to the priorities of the day. The finest craftsmen had been hired, artists and artisans, and the place was truly beautiful. Angels and archangels graced the various tombs, winged cherubs, saints and crosses. She had never been afraid.
But now...
From a distance, she could hear shouts. Soldiers. Ridiculous. Grown men playing as soldiers. But they did it so well. She might almost have been back in time. The powder came from the howitzer and the Enfield rifles. The shouts sounded as the men played out their roles, edging from the river road to the outbuildings and then the stables, to the final confrontation on the lawn and in the cemetery. The blood would come from stage packets within their uniforms, of course, but...
This was real blood. She knew because it had a distinctive odor, and because, yes, damn it, she could smell it. Nothing smelled like real blood.
She looked at the ground, and she could see the puddle where the blood was falling, but she was afraid to look up. If she looked up, she would see a dead man.
But she did so anyway. She saw him. There was a hat pulled low over his face, but soon he would lift his head.
He did. And she saw a man in his prime, handsome, with strength of purpose in the sculpture of his face. But there was weariness in his eyes.
Weariness and death. Yet they were just playacting; that past was so, so long ago now....
She didn't speak. Neither did he. Because his face began to rot. It blackened, and while she watched, the scabrous decaying flesh began to peel away. Soon she was staring into the empty eye sockets of a skull.
She started to scream.
Above that sound, she could hear someone calling to her. Someone calling her name. The sound was deep, rich and masculine, and she knew it....
It was Jake! He would help.... Surely he would help.
But she could only stare at the skeletal mask in front of her.
Smell the blood.
And scream.
A strange sound in the middle of the night awoke Ashley. She sat up with a start and realized she was doing the screaming. She clamped her own hand quickly over her mouth, embarrassed and praying that she hadn't roused the household. She waited in silence; nope, no one.
That was pretty pathetic. It must have been a horrifically pathetic scream. If she ever really needed to scream, she'd probably be out of luck.
Lord, that had been some nightmare.
She didn't have nightmares. She was the most grounded human being she knew; hell, she had grown up next to a bayou full of alligators and cottonmouths, and she had lived in a downtrodden area of New York City near Chinatown in order to afford NYU. She knew all about real monsters—ghosts were creations to reel tourists in.
So...
With a groan, she threw her head back on her pillow and glanced at the clock. She needed to sleep. In a week's time, they'd be celebrating Donegal Plantation's biggest annual event: the reenactment of the skirmish here that had cost her ancestor his life.
Ah, yes, and she had been dreaming about the skirmish—or the reenactment?
That was it, she thought, grinning. She was dreaming about the events at Donegal Plantation because they were preparing for the day.
History was always alive at Donegal. The plantation house was furnished with antiques, most of which had been in the family forever. There was an attic room that contained more artifacts from the Civil War than many a museum, down to letters, mess kits, knapsacks, pistols, rifles and bayonets. Still, the reenactment remained a major undertaking.
But they'd been running it since before she had even been born. It was rote by now. All the same, there was still plenty of bustle and confusion, along with everything that had to happen before the event could take place, including a mound of paperwork on her desk that had to do with the "sutler's tent," the pop-up shop where period clothing and curios and other paraphernalia, such as weapons and antiques, were sold. Which meant registrations and taxes. Then there was the insurance they needed for the day, and the officers to direct traffic and so on.
That was it. She just had a lot on her mind.
And the reenactment always reminded her of Jake. He'd never been a soldier, North or South. But he'd dressed up, and he'd played his guitar and sang music from the era. And sometimes she had played with him, and he'd always known how to make it just right, to bring back the past, with the light of truth.
She eased down in her covers, determined to forget both her anxiety and Jake.
Not so easy, even though it had been a long time since Jake had been in her life.
Finally, she started to drift again. She was comfortable; she loved her bed and her room, even if she had lived here her whole life other than college. Though she loved to learn about new people and new places, she also loved to be home.
She started again, certain that she had felt a touch; something soft and gentle, smoothing her hair, stroking her cheek.
She sat up. Moonlight streamed into the room, and there was no one else here; with so many guests around, she had locked her bedroom door. She looked at her pillow and decided that she had merely rubbed against a side of the pillowcase.
As she did so, she glanced at her dresser.
There was something different about it. She studied it for a moment, wondering what it was.
Then she knew.
She kept a picture of her parents there, on her dresser. It had been taken almost twenty years ago. They were together, holding her between them, when she had been five. It had been developed in sepia tone, and they'd had it done when one of the guests at a reenactment had found a way of making nice money by pretending to be Matthew Brady, the famed Civil War photographer. Throughout the day, he had answered historical questions about photography and its place in the Civil War.
As was the custom of the day, none of them was smiling, but there was still something exceptionally charming about the picture. There was a light in her father's eyes, and just the hint of a curl at her mother's lips. Her father's arm was around her, his hand coming to rest tenderly on her mother's shoulder. She was sandwiched close between them, and in her mind, the picture had been filled with love. It had become an even greater treasure when she had lost them.
It usually faced at a slight angle toward her bed.
The picture was turned away, as if someone had been looking at it from a different angle. It was such a little thing, but...
Maybe someone had wandered into her room. Cliff ran the property and she ran the house, but they employed extra housekeepers in the main house when they had guests. They hadn't had guests in the house in the last few weeks, but the house was usually open, and her grandfather loved to walk anyone staying on the property through it. Depending on his mood, a tour could get long.
And the picture...
She turned over, groaning. It was just the angle of the picture.
Jake Mallory should have slept well, with a hard case finally settled.
But he didn't.
The odd thing about his nightmares was that they were a recent phenomenon. When he had begun to realize and make use of this gift or curse—those things he somehow knew—there had been no dreams.
During the summer of the storms, during Katrina and the flooding, they had all been so busy. While it had been happening, he'd never explained to his coworkers that he was so good at finding the remains of the deceased because they called out to him; they spoke to him. It was heartbreaking; it was agony. But the dead needed their loved ones to know, and so he listened. And he didn't dream those nights.
Later, the dreams had come, and they were always the same.
He was alone in his small, flat-bottomed boat, though he'd never been alone during any of the searches.
He was alone, and the heat of the day had cranked down to the lesser heat of the night, and he was searching specifically for someone, though he didn't know who. As the boat moved through the water that should have been a street, he began to see people on the rooftops, clinging to branches here and there, and even floating in the water.
They saw him; they reached out to him. And he felt like weeping. They weren't living people. They were those who had lost the fight.
As he drifted along, he looked back at them all, men and women, old and young, black and white and all colors in between. He wanted to ease their suffering, but he could no longer save them. Their faces had an ashen cast, and the bone structure was sucked in and hollow; they didn't seem to know that there was nothing he could do for them anymore. In the dream, he knew that he, like many law-enforcement officers, scent dogs and volunteers, would be called upon to find the dead in the future.
But now he was seeking the living.
They called out to him; they were trying to tell him something. Bit by bit, he saw they were trying to show him the way. He thought that there should have been sound, but there was none. He didn't hear his passage through the water, and nothing emitted from the mouths of the corpses he passed.
Then he saw the figure on the roof far ahead. He thought it was a woman. She seemed to be in something flowing, which was not unusual. Many victims, living and dead, had been found in nightgowns or boxers or flannel pajamas. What was strange was that she seemed to be the only one alive. She was in tremendous peril as the water rose all around her. He felt that there was something familiar about her, but he didn't understand what it was that seemed to touch him. The light of the full moon turned her hair golden and gleaming, her white gown flowed in the breeze. Amidst the destruction, she was a beautiful survivor.
He tried to get closer.
The watery road grew more clogged and congested. Downed tree branches and appliances floated by. A soaked teddy bear with big black button eyes stared at him sightlessly as it drifted. He ached inside; it was an agony to fight the river, but it was also something he knew he had to do. Especially when she waited; when he could save her. He just had to reach her before the water level rose higher and higher, and swept her away.
He grew close...
And that was when he felt the darkness at his back.
He tried to turn, but he could not. The wind had picked up, and the effort was too much. No matter how he strained, he couldn't see what evil thing seemed to be tracking him.
There was suddenly sound. The woman. She was calling out to him.
She called him by name.
But he could feel the thing behind him gaining on him. He could almost reach out for her, but he had to turn, had to find out what seemed to be breathing fetid air down his back....
She called out again.
Jake!
If they were to survive, to outrun whatever horror was behind him, they would have to do so together.
Jake!
Her voice rang out almost as clearly as if she were next to him in the boat.
But the darkness was on him, so close....
He could feel it then, enveloping him, crushing the sound of her voice as it did.
And he woke with a jerk.
Jake sat up in bed, deeply disturbed by the reappearance of his frequent dream. For the first time, he knew who he had seen on the roof of the house, about to be swallowed up by the floodwaters.
It was Ashley. Ashley Donegal.
He stood and stretched, irritated. The clock on his mantel indicated it was still early.
He swore and got dressed. He knew why he'd had the dream—and made Ashley the woman on the rooftop. He knew the date. The reenactment was coming. Donegal Plantation would be busy and alive; Ashley would still hurt from the fact that her dad had passed away and would no longer be playing Marshall Donegal, his ancestor. But she'd never show it. She'd be the grand mistress of the ceremonies, beautiful and regal in her Civil War attire.
He wondered if he would ever fall out of love with her. And then he wondered if the dream had meant something more. Angela—who seemed to have the best sixth sense in what the FBI called their special unit—had told him that dreams could open many doors. In REM sleep, the mind was at the stage where dreams came, and those dreams could easily focus on what the conscious mind rejected. When she was trying to reach memories of the past, she often used sleep.
If that were the guide, he could easily convince himself to think that Ashley now wanted him. Needed him. And that this was the sign.
Of course, that was just Angela's way of seeking the ghosts of the past—even in their own group, they weren't sure about all the rules of seeking out the help of ghosts.
He wasn't sure if he wanted to laugh at himself or not.
Their team was legitimate; they were on-the-books federal agents. They had just spent days at the local training facility, improving their weapons skills, computer literacy and understanding of the mission policy.
But they were a one-of-a-kind unit, and their true designation wasn't written down anywhere. Among themselves, they were the Krewe of Hunters; on paper, they were Adam Harrison's Special Unit. In bizarre situations, they were supposed to smoke out the fakers—and find what might really be remnants of the past.
The world was filled with ghost hunters and would-be ghost hunters. The problem that most people didn't see was the fact that few ghosts would really appear for a television crew. Some ghost lore did seem to be true. There were the residual hauntings—ghosts that played out a situation, such as a battle scene at Gettysburg, over and over again. And there were intelligent hauntings: ghosts that lingered for some reason. Ghosts didn't seem to have rules. Some could find certain individuals who saw them as clearly as day; they could carry on long conversations, appear and disappear, and interact. Sometimes ghosts were frightened of the living, and they hid, and only someone with a real ability to suspend disbelief could coax them out. It was complicated; he was still learning. Sometimes ghosts tried to warn those they cared about when something evil was about to occur, and ghosts often entered into the REM sleep of those they hadn't managed yet to really touch in the conscious world.
So the dream meant that Ashley needed him....
Or he wanted the dream to mean that Ashley needed him.
He stood up and walked over to his hotel window and looked out over the dark streets of the French Quarter. There was so much history here. So many lives had been lived; so much drama had taken place. Sometimes it was impossible to believe that the energy of the past didn't remain. Ghosts didn't have to be old; he knew that himself, though he hadn't wanted to accept the truth until he had met Adam Harrison and become a part of the unit Adam had started for the FBI. He had been glad of his ability to feel where people were; to imagine that he heard them telling him to come, please. Sometimes he had even been able to find the living. And sometimes he had heard the voices of the dead, when he hadn't known that they were dead.
His "gift" had cost him Ashley.
So why now, all these years later, was he seeing her, adrift, about to be engulfed, and yet reaching for him, even as he reached for her?
## 1
"Ah, dammit! I don't want to be a Yankee," Charles Osgood said.
It was there; it had finally come, and Ashley was grateful.
And the semi-drama going on here surely meant her mind had been trying to warn her that the day was not going to come without its share of trouble, because it was already proving to be one hell of an afternoon.
Morning had brought the business of breakfast, visitors pouring onto the property to spend time at the campsites. Now they were coming close to the main event of the day, the reenactment of the battle that had taken place at Donegal Plantation.
She'd never expected the real trouble to come over the sad situation of an ailing faux-Yankee.
"Dammit!" Charles exclaimed again.
Ashley thought that the man sounded like a petulant teenager, though she knew that he didn't really want to argue. Not on a day like today. He flushed as the words came out of his mouth, and cast her a quick glance of dismay. She wasn't even the one handing out the assignments, though she was the only Donegal among them now. The relish the group was taking in telling Charles his new role unsettled her a bit. Charles Osgood was the newest in the "cavalry unit" of reenactors, which meant that he got the assignment to play for the other side. Yet this seemed to be turning into a college hazing; they were all friends, and they were usually courteous to one another.
"Charlie, come on! Being a Yankee will be fun. Okay, so they were jerks—well, the ones here—who couldn't spy on a neon sign, couldn't hunt, couldn't shoot.... But come on! Being a Yank will be fun!" Griffin Grant teased.
Ashley shook her head; how could grown men be so immature?
In her mind, although she truly loved the living history that took place at the plantation, she thought the units clinging to so-called glory were nothing more than inane. The event had ended with the death of one her ancestors—not a party.
"Hey, hey, all of you!" Ashley said, addressing the men around her and using the voice she would utilize when working with one of the school groups—the grade-school groups. "I know you all like to cling to the magical illusion that the antebellum South was a place of beauty, grace and honor—where men were men. Real men, hunting, riding, brawling—but honorable. Yes, we reenact what was. But this is now, and that was then! None of you would seriously want to go back to the Civil War, and no one here is prejudiced. The slavery of any person was a horrendous way of life."
"Ashley—you're making it sound like being a real man is bad thing!" Cliff Boudreaux commented, laughing. Cliff, horse master at Donegal, was clearly amused and having a good time.
"Well, of course, Ashley, it's not like we take this too seriously," Griffin Grant said, staring back at her as if she was the one who didn't understand the question. Griffin was a striking man in his early thirties, sleek and slick, a CEO for a cable company in New Orleans, though his ancestors had lived out here, two hours down the road from the big city. "We know reality—and like it. But this is important playacting!"
She groaned softly.
They were good guys, really.
It was playacting, and for the playacting they were able to believe truly with their whole hearts that it had been about nothing other than states' rights. Ashley knew all the statistics about the fighting men—most of the men who fought and died for the South during the war couldn't have begun to have afforded a slave—and war was seldom caused by one issue. But her parents and her grandfather had never been the types to overlook the plantation's complicated history. Cliff was part of that with his gold-green eyes, bronze-colored skin and dark tawny hair. She knew that half their visitors were immediately enthralled with him. He was one of the reenactors on the Southern side because of the Donegal blood that ran in his veins. Early on, a Donegal widower had fallen in love with a slave, creating the first racial mix in his background. In the 1920s, his great-grandfather had married a Donegal cousin, something that caused a serious scandal at that time in history, but which now gave both halves of the family a sense of pleasure and pride. She wasn't sure how to count second and third or twice-removed relatives, so she considered Cliff to be her cousin.
History was history. Donegal was steeped in it, good and bad, and they didn't hide any of it.
"Charles, they're right. It's a performance, you know," Ashley said. "It's a show, maybe even an important show in its small way. It's where people can see the weapons of the day, the uniforms that were worn. And, actually, remember, this particular fight started because men had a bar brawl—and then an excuse to fight because the war was getting underway. You're all examples of keeping history alive, and I'm so grateful to all of you."
Charles stared back at her blankly; the other men were smirking.
Why didn't they all get it? They were actors in a show, hopefully teaching American history, with several perspectives, along the way. But some things died really slowly here, in plantation country. Family was still everything. Loyalty to hearth and home, kin, parish and state. They'd been wrong; they'd been beaten, and they knew it, but still, only one side of the cast of players was considered to be elite. And the reenactors could be incredibly snobbish.
That made Charles Osgood the odd man out.
Toby Keaton cleared his throat and then said softly, "Charles—come on. You're lucky to be in with the 27th Bayou Militia Cavalry Unit. Most of the time, the fellows taking part in the reenactments here are direct descendants of those who fought before. You've got to see the truth of this thing. You claim your place in the ranks through marriage—your stepfather was an O'Reilly, and I know he raised you, but, you know, in other old Southern units, that wouldn't count." Toby was forty-four, and Ashley's next-door neighbor at Beaumont, his Creole plantation, though they both had acres and acres of land. Toby grinned as if to cut the harshness of his words. "Newcomer—odd man out. You're a Yankee if I've ever seen one!"
"Great! So now I'm a newcomer—and that makes me an outsider?" Charles asked, staring around the room. "Come on, guys, you've just got to understand. This will really make it look as if I don't belong here at all!"
He gave his appeal to the others gathered at the horse master's office in the old barn at Donegal Plantation that day—Cliff Boudreaux, Griffin Grant, Toby Keaton, Ramsay Clayton, Hank Trebly, all still with property in the general area, John Ashton, tour director from New Orleans, and Ashley herself. The "Yankees" were gathering in the old smokehouse—a separate building, and now a small apartment. Charles would be joining them soon; all of the reenactors gathered together for their roundtable discussions on the war, but each side met separately first on the day of the reenactment to make sure that every member knew the character he was playing. Later, they'd all meet back here to make sure that everyone was apprised of all the safety factors involved.
One, Charles, so it seemed, would have to play a Yankee, and go join the group in the apartment. They were short a Yankee, and that's the way it was. All of them belonged to Civil War roundtables, and these days, none of them really cared about sides—they just liked to discuss tactics and procedure. They often met in the dining room at Donegal; Ashley loved to listen, because they also knew their history, and they spoke about events in the lives of many of the key players in the war, and the fact that the generals had often been best friends before they had been forced to choose sides in the bloody conflict. They knew about weapons, uniforms, sad stories about treason and resisters, draft riots, food, clothing, trade and so much more.
"Charles," Cliff Boudreaux said patiently. "We're all just teasing you here, really. We're short on Yankees today, on account of Barton Waverly being sick with the flu. We're pretty desperate. And that's the rule; newcomers play Yankees when our brothers from up North ask for help. Hell, remember that year when half of us were laid up with the croup? Three of them Yankees had to come play Southern boys. We're not doing anything bad to you—really."
Ramsay Clayton was seated across the table from Cliff. Ramsay looked like an artist; he was tall, with a wiry muscle structure, long dark hair and classical features. He owned a small place down the road, but he spent a lot of time in New Orleans, where he sometimes showed his work at Jackson Square and sometimes had showings at the galleries. He grinned at Charles. "Yeah, and don't forget, the Yankees won. Hell, come to think about it, where were all the Southern boys when we were losing this thing?" he asked lightly. "Ah, well. Born in our day and age, it's easy to look back at the South's part in the Civil War and wonder, 'What the hell were we thinking?'"
Ashley smiled. She liked Ramsay. He was a good guy.
"Well, I wish I could just step up to the plate, but I can't. I can't play a Yankee—I just can't," Toby Keaton said. "Hell, my great-great-great-whatever grandfather was the first one to answer Marshall Donegal's call for volunteers. He was one of his best friends. I think he'd roll in his grave if I played a Yankee. Good God! I own a plantation! Wouldn't be fitting for me to play a Yankee. Lord knows, it could be bad for business."
Hank Trebly grinned. "Well, I'm just big sugar. I don't really give a whit. I see the war as over, over, over, and that's the way it is. Lord A-mighty! The damn thing ended in 1865." Hank owned the property next to Donegal, and his ancestors had owned it forever. The old plantation had been replaced by a sugar refinery years ago. He was a small man, in his early forties, and his business meant everything to him.
John Ashton shrugged. "My family might have been here, but I don't care," he said. "The Civil War means my income these days—tourists love to go back. But I love 'em all. Yankees, rebels, Brits, Brazilians! Bring them on. They all spend money and take tours."
"And what happened here was in 1861, for God's sake, before the thing had really even gotten going," Griffin said, shaking his head. "Come on, now! My ancestor went on to die at the Second Battle of Manassas—now, that's a damned big battle. We're here to teach, and to remember everything that happened in the past—and how it made us what we are today. Let's have fun, folks. C'mon—I come out here to forget the office and programming and statistics, computers and red tape. I don't care who plays what. It's just for a good time."
"I spend most of my time in New Orleans, art on the square and all that—you can call me a doughboy for all I care. It's the spirit of this thing," Ramsay said. "And Lord knows, what happened here couldn't even be called a battle. My ancestor and most of the Southern boys except for Marshall survived, but, as we've all pointed out now—the North won. We are living the United States of America. This wasn't even really a battle."
He was right. What had taken place late in 1861 hadn't even been a battle. Drinking downriver, toward New Orleans, two Yankee spies had heard about Donegal's then-owner—Marshall Donegal—preparing a major summons to area troops to prepare them for an invasion of New Orleans. In trying to draw Marshall Donegal's men out further on the subject, they had all gotten into a fistfight when one made a ridiculous statement about Northerners being chickens. The two Confederates suspected the men of being spies, and had run back to Donegal. The spies went back to their headquarters, but they were spies, and thus their numbers were small. On each side, six men were mustered—and, rather than be executed as spies if they were caught, the Union men donned their uniforms.
The fighting had ranged from the stables to the porch of the main house and out to the chapel and cemetery—ending when Captain Marshall Donegal had died of a bayonet wound in his own family graveyard. The enemy had "skedaddled," according to the Southern side; the rebels had been left in utter defeat, according to their Northern counterparts.
Now, the "battle" was something that taught history, and, largely due to its small size—and the fact that the current owner of the plantation, Ashley's grandfather, Frazier Donegal, was a history buff and glad to welcome the units on his property—it was a popular event. "Living history" took place frequently at Donegal, as often as once a week, but an actual reenactment was done only once a year. Sometimes the actors doing the reenactments were involved in other locations. Some belonged not just to Civil War units, but Revolutionary War units, and it just depended on where the biggest shindig was going on. Luckily, most of the men who could claim to have had ancestors in the brawl loved the plantation and the nearly exact-to-the-past-moment location of the place, and they usually made this reenactment a priority.
Donegal House was surely one of the prettiest places left on the river road, with memories of the antebellum era held in place. The great house still maintained a gorgeous front. It had been built with magnificent Greek columns and wraparound porches, and elegant tree-shaded entries stretched forever before the front and back doors. The currently used stables, housing only six horses, were next to the house, while the larger stables needed in a bygone era were far back from the house, to the left, riverside, and offered three apartments for those who wanted to stay for the night. The old smokehouse and servants' quarters were available for rent as well, and sometimes they even rented out five of the rooms in the main house. With Beth there, Ashley's extraordinarily talented friend and chef, and the efforts they were making with the restaurant and the crazy business that came along with the reenactment, they had chosen this year just to let rooms in the outbuildings.
All this—living history and their bed-and-breakfast rentals—was done to survive into the twenty-first century. But the Donegal family had been letting the place out for nearly thirty years now. And the living history and the reenactments were the true highlights to be found here, distinguishing it from other great plantations along the river.
"Okay, sure. You all are right," Charles said. "It's over. Long over. Hell, the Yankees did win the war."
Cliff laughed. "Still hard to convince my mama and a few other folks I know that it's true. But thanks, Charles, that's great. The Yanks are good guys. Man, it's sad to think back, though, huh? We would have wound up being enemies."
"Who knows what our feelings would have been back then?" Ashley asked. "We might have chosen to fight for the North."
"It was a different time, a different lifestyle," Griffin pointed out. "You're all indignant now about injustice, but you didn't live back then. You didn't grow up in an economy of cotton and sugar."
"Rich men wanted to stay rich," Ramsay agreed dryly.
"Who's being Marshall Donegal today?" Charles asked.
"That would be me," Ramsay said. "I've done it the past five years." He was quiet a minute; he had done it since Ashley's father had passed away. "Ashley could don a uniform herself, but she thinks we boys should just be boys. So I get the honor."
Ramsay was trying to move quickly past the mention of her father, Ashley knew. He had been gone five years now; he had died shortly after her mother. She had accepted their loss—and she knew as well that there would still be a little core of pain when she thought about them, even if she lived to be one hundred. Inwardly, she winced. She hadn't just lost her father that day; that had been the end of her and Jake. Her fault, her call, and she still wasn't sure why. He had frightened her, she thought. It seemed he had scratched the surface of something, and she didn't want to know what was beneath. And still, to this day, she knew that although she had closed the door, she missed Jake. And missing Jake had colored everything else in her life.
"He died," Charles reminded him. "Marshall Donegal was killed, you know," he added quickly.
"Well, as we've said, the war is long over, so I guess they're all dead now anyway," Ramsay pointed out.
"Gentlemen," Ashley said, speaking at last, "I want you all to know that you are greatly appreciated. You're all such wonderful actors, taking on whatever role is needed, whenever it's needed! Charles, the Yankees are great guys. Michael Bonaventure lives in town, and his ancestors lived there as well, right in the heart of the French Quarter. His family left when the war started, because Bonaventure's ancestor was fighting for the Union. Hadley Mason is from Lafayette, but his ancestors agreed with the Northern cause as well. It will be fun for you to be a Yankee. It's acting, just like when we act out the encampments. And I truly appreciate you taking on the role."
"It's really amazing," Griffin said. "We do get all tied up in what was. The way the past still has so much to do with the present! Charles, come on, you're a stepchild. We all really had ancestors back then who were involved with this thing. You're welcome among us—totally welcome. But, hey, if I had come in on this recently, I'd be happy just to be a part of it all."
Charles Osgood offered Ashley a weak smile. "Sure. You know me—I'm just happy to be here."
To Ashley's surprise, Ramsay Clayton suddenly spoke up again. "Charles, I have an idea. Some of those guys really are my friends. My good friends. I'll be a Yankee today. You be Marshall Donegal."
Charles opened his mouth, stunned, and stared at Ramsay. "Oh! Oh, no. I couldn't take that honor away from you!"
"You get killed, you know," Ramsay reminded him.
"Oh, like you said, they're all dead now. I just couldn't—I really couldn't."
"Hey, I think I want to be a Yankee for once," Ramsay said. "It's cool. You be Marshall Donegal, and I'll be a Yankee. No arguments—it's decided. I'll be a winner for a change!"
"I don't know what to say!" Charles told him.
"Say thanks, and let's get on with it. We have to finish planning this thing," Ramsay said.
"I'm going to be Marshall Donegal!" Charles said, still awed.
Ashley lowered her head, hiding her laughter. These guys really were like children when it came to the reenactment. They were so dedicated. But it was really good, she reminded herself. They kept history alive. It had been on a trip to Europe with her parents when she had seen the quote that meant so much to her: "Those who cannot remember the past are condemned to repeat it." It was the philosopher George Santayana who had written those words, and she had seen them above the gates of a concentration camp. So, whether history was sterling or not—pitting man at his best and his worst—it was necessary to remember.
The reenactors did a fantastic job. Although there had been only a small encampment at Donegal Plantation at the time, they recreated a larger one, complete with a medical tent, where surgeries were acted out, officers' quarters and tents for enlisted men.
"This is a right nice place to meet, but we need to get to business," Griffin said, winking at Ashley.
"Yeah, Ramsay, looks like you need to skedaddle!" Cliff teased.
"I'm out of here!" Ramsay said, rising. He looked around. "Sadly, I do like Cliff's digs better than being cramped up in an apartment!"
Griffin was right: they were in a nice place to meet. The office/living quarters in the stables were extremely pleasant; there was no heavy smell of hay, horses or droppings in any way, since the office had long ago been fitted out with air-conditioning and an air purifier to boot. There were a number of trophies along with books on horses, horse care, tack and maps on the shelves around the old massive desk with its iMac and printer. It was the horse master's realm. No matter the state of riches or poverty the Donegal Plantation might be in, there was always a horse master. These days, the horse master did more than look after the six horses that remained. He was a tour guide, overseer—though they didn't grow anything other than a few flowers now and then and a tomato plant or two—and general man about the house.
Ashley stood and gave him a shove. "Our apartments are beautiful. Get on out of here, and get this all moving!" She spoke with teasing force. "I'm going out to check on the camp setup and see that everything is running smoothly, then get ready. I'll leave you gentlemen to agree on the final assignments and action. The day is moving on. We need to be prepared to start with the battle at sundown."
"Hell, I hope they got a uniform that will fit me!" He winked. Ramsay was a good guy. He had a small house that had once been a working plantation, but his land had been eaten up over the years. Plantation actually meant farm, and Ramsay had no farmland left at all. He spent most of his time in the city, where he actually was a working artist making a nice income.
"I'm off to join the Yankees!"
"Thanks!" Charles Osgood lifted a hand to Ramsay, and then to Ashley, looking dazed. He was getting the prime role for the day, and he still seemed to be surprised.
It didn't mean as much to Ramsay, Ashley thought, watching him as he walked from the stables to the old barn. He was from here; he'd been born a part of it all. He'd played soldiers over and over again, and though it had been magnanimous of him to hand over the role, she wasn't sure that Ramsay hadn't decided that being a Yankee might not be that bad a thing for the day. After all, they ended the day with the Pledge of Allegiance and the "Battle Hymn of the Republic." Even if they did begin it with a rousing chorus of "Dixie"!
She left Cliff's to make a quick check of the horses. "Thank God you darlings don't care if you're Yankees or rebels!" she said affectionately, pausing to rub Abe's ears. She saw that the tack for the Northern cavalry was ready for each of the mounts, saddles and saddle blankets set on sawhorses and the bridles with their insignias hanging from hooks right outside the stalls. Abe, Jeff, Varina, Tigger, Nellie and Bobby were all groomed and sleekly beautiful, ready to play their parts. She paused to give Varina a pat; she loved all the horses, but Varina was her special mare, the horse she always chose to ride.
Leaving the stables, Ashley paused for a moment to look across the expanse of acreage to the left, where the tents of the living encampment had been set up. She could see the sutler's stretch of canvas, and she walked over to see who was working that day. Tourists—parked way down the river road—were milling around the goods for sale. She heard children squealing with delight as they discovered toys from the mid-nineteenth century, just as she heard women ooh and aah over some of the corsets and clothing. She saw that a crowd had gathered around the medical tent where reenactors were doing a spectacular job of performing an amputation. The patient let out a horrific scream, and then passed out. Dr. Ben Austin—playing his ancestor, also Dr. Ben Austin—stood in an apron covered in stage blood and explained the procedure. Ben would later be part of the battle reenactment, but for now, he was explaining medicine. Ashley reached him in time to capture part of his spiel.
"Amputation was frequently the only choice for a Civil War surgeon, and field surgeons could perform an amputation in as little as ten minutes," Ben told the crowd. "Chloroform existed, but it was scarce. The South had alcohol. When the surgeon could, he would do everything in his power to make the traumatic operation easier for his patient, but at major battles, the pile of amputated limbs could easily grow to be five feet tall. There was no real understanding of germs, and more men died from disease than from wounds or bullets. To carry that further, more men died in the Civil War than in any other American war, and more men died at Sharpsburg, or Antietam, as those of you from the North might know it, than died during the D-Day invasion."
Ben saw Ashley watching him and lifted a bloody hand. Well, it was covered in faux blood from the faux surgery. Ben knew how to be dramatic. She smiled and waved in return and went on, stopping to chat with some of the women who were cooking, darning or sewing at the living-encampment tents. There were soldiers around as well, explaining Enfield rifles to little boys, whittling, playing harmonicas or engaging in other period activities. One laundress was hanging shirts and long johns out to dry—a nice touch, Ashley thought.
"When the war started, the North already had a commissary department—and the South didn't," Matty, the sutler's wife, was explaining to a group who stood around the campfire she had nurtured throughout the day. "Hardtack—dried biscuits, really—molasses, coffee, sugar, salted beef or pork and whatever they could scrounge off the land was what fed the soldiers, and the South had to scramble to feed the troops. Didn't matter how rich you were—you were pretty much stuck with what could be gotten. There were points, especially at the beginning of the war, during which the Southern soldiers were doing all right. They were on Southern soil. But war can strip the land. What I'm doing here is boiling salted beef and trying to come up with something like a gravy to soften up the hardtack. With a few precious spices, salt and sugar, it won't be too bad. A few people can taste, if they like! Of course, I've made sure that our hardtack has no boll weevils. The soldiers were fighting every kind of varmint, big and small, to keep their own food."
Everything seemed to be in perfect order; Ashley's dreams had been for nothing.
Except, of course, that the reenactment always made her think about Jake.
They were due to leave soon, within the next few days, but since Adam Harrison's group was still in New Orleans, they finished with the training they were doing there, and waiting for the move, Jake had agreed to go wandering around the French Quarter with his fellow newly minted agent, Whitney Tremont.
"I must admit, I'm going to be sorry to leave New Orleans," Whitney said. She stared out toward Jackson Square. "There was so much paperwork after the Holloway case, it felt like we were picking up the pieces for days at first. But it's been nice to have this bit of time to get ready for our move, since we're all taking up residence in the D.C. area. Though, I'm excited—I mean, we're going to have offices, Jake. Like really cool offices, in a building in Alexandria—with help! A forensics lab! State-of-the-art equipment."
Jake grinned. "Yes, it's going to be interesting to get settled in."
Whitney grinned back. Her skin was like the café au lait that sat before them. He knew that the others had thought the two of them might wind up together, but what they had formed instead was a friendship, deep and binding.
"But it was good, I think—just being thrown together as freelancers of a sort for our first case. Don't you think?" she asked. "But federal positions...though I don't think we really get to stay in those fancy buildings that often, do you?"
"We're like any other team or unit for the FBI, I believe. The cases come in, and I'm assuming that Adam Harrison and Jackson Crow decide what looks like something we should take on. We'll get to discuss the situation then. And make the plans."
"Do you think that any of us could put a case forward?" she asked.
"Sure." He smiled. "Let's face it, Whitney, we are an experiment in paranormal investigations. We're unique, and I'm sure there are those who will make fun of the 'Krewe of Hunters' unit."
"Not anymore. Not after the Holloway case," she said proudly.
"We have to keep proving that we're good at what we do," Jake said. The name "Krewe" they'd given themselves had begun as a joke, but they'd become a real crew through their passion for the work.
"Hmm," Whitney said, twirling her straw in her iced café au lait. "It's here somewhere."
"Here? What's here? Sometimes you make no sense."
That made her laugh. "Only sometimes? I think that our new case is going to be here. Somewhere in Louisiana."
"We're about to move to Alexandria," he pointed out.
"I don't know...I just have a feeling. I don't think we'll be going yet. You wait and see."
"What makes you think that?" Jake asked. Whitney's prowess was with film, sound and video. But she also seemed to have amazing intuition. Of course, they had all been gathered into the group because of their intuition, their ability to solve problems where others could not, but where Angela Hawkins was quiet, finding what she found without much ado, and Jackson would always be the skeptic, Whitney went in wide-eyed, eager for whatever might not be considered normal.
"Feelings and logic, that's kind of the Krewe of Hunters motto, right?" she asked.
He laughed, but something was knotting in the pit of his stomach. "I think it's supposed to be logic—and then feelings," he told her. He gazed idly across the street. The mule-drawn carriages were starting to arrive in front of Jackson Square. An early-morning tour group was forming on Decatur Street. One of the history tours, he thought.
"Well, of course, good old Jackson, he's still swearing solving cases is all logic, and we all know that he knows best," Whitney said.
Jake wasn't really paying attention. He had seen the tour-group leader come out—he wasn't sure where she had come from. Of course, there were a number of restaurants and bars in the area, and some had been open forever. It was New Orleans. No one frowned if you discovered you were dying for that 8:00 a.m. drink.
The tour guide was a blonde woman dressed in Civil War attire. Her bonnet hid her face, but she was tall and statuesque, and he had a feeling that she was going to be an attractive woman before he saw her face. Assured, probably in her mid to late thirties, she moved among the chattering crowd as they waited.
She was coming toward the sidewalk, politely excusing herself as she did so, but people didn't seem to notice as she made her way through them, which said a lot for the good nature of the group, since she was wearing a respectable day dress with large hoops.
She paused when she reached the sidewalk.
Jake started. She was staring straight at him, and she smiled, but her smile seemed to be very sad. Her mouth moved. He squinted. He wasn't all that much at reading lips, but it was almost as if he could hear her.
"We're waiting, we need you. Hurry," he thought she said.
"Jake?"
"Huh?" He turned back to look at Whitney.
"Want to move on?" she asked.
"Yes, sure," he agreed. He stood and left a tip on the table, having already paid the waiter.
When he glanced up again, the tour had moved on down toward the cathedral. He didn't see the woman, but they would be walking in the same direction.
"Whitney," he asked as they did so, "did you understand what that woman was trying to say?"
"What woman?"
"She was the guide for that group that's ahead of us. She looked right over at us and said something," Jake told her.
Whitney arched a delicately formed brow. "First, I didn't see the woman, but I wasn't looking. And second, if she'd spoken from across the street, unless she'd been yelling, how could I have heard anything she had to say?"
He shrugged. "Good point."
They walked up St. Ann Street, took a pedestrian thruway as they passed by the square, then turned in right in front of the cathedral, where the tour group had now paused.
The woman wasn't with them. There was a man in a top hat and frock coat leading the tour.
Jake stopped short.
"Hey! Hey there, remember me?" Whitney said, nudging him.
"Just a second," Jake said. He knew that the man would finish his spiel about St. Louis Cathedral, and then allow the group to take pictures.
This guide, however, apparently liked to hear himself speak. He added in several personal anecdotes regarding the cathedral, before allowing his group to disperse for pictures.
When the group finally thinned, Jake approached him. The fellow, in his mid-twenties, saw them coming.
"The tour offices are actually on Decatur, sir, if you're interested in any of our offerings. We do history tours, ghost tours, vampire tours, plantation tours—"
"Actually, we're locals and could do the tours," Jake said, interrupting him with a pleasant tone. "I'm just curious—why did you all change tour guides at the last minute?"
The man frowned. "We didn't. I've been scheduled for over a week to do this tour."
Jake frowned. "I saw a woman with your group. She was dressed in antebellum clothing, bonnet and all."
"Oh, she was probably heading for Le Petit Theatre," the tour guide said. "They're doing several performances of Our American Cousin. She's a bit early to be in costume for the matinee, but I imagine you saw one of the actresses."
"Oh, well, thanks," Jake said.
"Why?"
"Oh, I just thought she was trying to tell me something," Jake said.
"If she was trying to tell you something, wouldn't she have just done so?" the man asked.
Jake was irritated by the tone; frankly, he hadn't liked the man since he'd heard him giving his own life's history along with the tour.
He felt Whitney's hand on his arm.
He forced a smile. "Thanks, thanks for the help," he said.
Whitney pulled him along. "Jerk," she said.
"Ass," Jake agreed.
"I meant you," Whitney teased. "No, sorry—he was a jerk. But come on now! We don't have any reason to go to the theater."
"But we're going to pass it!" Jake protested.
"Let it go, Jake. You saw an actress, and you thought she had something to say. Without sounding just as jerky as that jerk, it's true—if she'd really wanted to talk to you, she would have come on over. I don't want to help you stalk a woman, Jake."
"I don't want to stalk her. I want to know who she is," Jake said.
But they did pass right by Le Petit Theatre. He couldn't help but stop to read the playbill and look at the pictures of the actors in the show.
"She's not here," he said.
"Well, God knows, this is New Orleans. Maybe she's just a kook who likes to dress up in Southern belle attire, though God knows why, the heat can be a bitch. Forget it, Jake."
Jake agreed. He didn't know why it was bothering him so much that he'd seen the woman and hadn't been able to talk to her.
That wasn't true. He did know. There had been something vaguely familiar about her, although he couldn't quite put his finger on what it was.
"Jake? Are you okay?" Whitney asked.
"Fine."
"No, you're not."
"It's nothing, really. Hey, nothing that car bombs tonight after dinner with the group won't cure, right?" he said, setting an arm around her shoulders as he led her down the street.
Car bombs weren't going to fix anything. He was truly disturbed by the woman.
Why was she so familiar? Was she real, or was she in his imagination? Had he brought his dream world to the surface, and did he want her to be from Donegal?
It was absurd, even for a ghost hunter, to believe that someone from the past was calling out to him, trying to reach him.
Logic—and then feelings. That was Jackson Crow's motto. It was only logical that he think about Ashley now, and logical that even after all these years, he wanted her to need him.
Logic...
Somehow, it just wasn't working. Feelings were taking over. And thoughts of Ashley and Donegal Plantation.
## 2
Ashley surveyed the expanse of the property one last time; everything was going extremely well. Children were playing and laughing, the camp looked wonderful and there were activities going on everywhere.
She headed for the house. It was time for her to become Emma Donegal and get ready for the evening's battle.
But as she walked toward the house, she slowed, paused and looked over at the cemetery. The gate was locked.
Still, a creeping feeling of unease swept over her.
She shrugged it off; a dream was a dream. Good God, she'd dreamed once that she'd kissed Vance Thibault in high school one day, and she loathed him! She hurried on toward the house, trying to forget her unease.
Her grandfather, Frazier Donegal, was sitting on the back porch. She grinned; he looked spectacular, she thought. Frazier was eighty-three, but he showed little sign of slowing down. Today he was dressed in a frock coat, pinstripe breeches and high riding boots—a pure gentleman of the age with his full head of snow-white, a Colonel Sanders mustache and goatee, and bright blue eyes. She worried about him constantly; his health was good, but he was eighty-three.
He was really in the mood today, though, she thought. He was sipping a mint julep. He didn't even like mint juleps.
"There you are!" he said. "I was starting to wonder."
Ashley sat in one of the wicker rockers across from him. "Last-minute details in the stables," she told him. "Charles Osgood didn't want to be a Yankee."
Frazier rolled his eyes and shook his head. "It works, you know, for the property, it works. But why on earth everyone always wants to be on the losing side, I'll just never know. Did he finally accept his assignment?"
"He did—but Ramsay Clayton stepped in at the last minute to let him play Marshall Donegal," Ashley said.
"Oh?"
"I don't think Ramsay cares. It's all play to him," she said.
"Ramsay is a good fellow. You think he was trying to appear magnanimous in front of you?" Frazier asked.
Ashley shook her head. "There's never going to be anything between Ramsay and me, Grampa, there just isn't."
He lifted his hands. "I was just asking about his motives."
Ramsay had asked Ashley out the previous year; she had always liked him. He was a good artist, and a handsome man, but she had never felt the least bit of chemistry with him. He had accepted her wish that they just maintain a good friendship. Ramsay had been on the rebound, having broken up with his longtime lover. She wondered if she was still on the rebound—even if she had been the one who had run from Jake.
"I honestly believe Ramsay just doesn't care," Ashley said. "I think he'll have fun saying, 'Oh, Lord! I had to be a Yankee.' No, Ramsay isn't trying to impress me. Griffin even asked me to be his friendly companion for a dinner he had—and I said no. They all understand that the friendships we have are too important."
"Well, then, good for Ramsay. And you—you better get dressed," Frazier told her.
"Yep, I'm on it."
She rose and walked into the house from the riverside.
The original architects had taken advantage of the river and bayou breezes when they had built the house. It hadn't been changed much since the day it had been built. One long hallway stretched from the front of the house to the back, and before the advent of air-conditioning, the double doors on each end had often been kept open. The house had one unique feature: double winding staircases to a second-floor landing that led to the six bedrooms, three on each side of the house on that floor. The stairway to the third floor, or attic, where there were still two rooms that could be guest rooms or let out to renters, was on the second floor, bayou side, of the house.
Beth Reardon was in Ashley's room, sitting at the foot of the bed and drawing laces through her corset.
Her skin was pure ebony; she was tall, regal and beautifully built. But she had chosen to get into the action. She was wearing a cotton skirt and cotton blouse and her hair was wrapped up in a bandana. She gazed over at Ashley. "Hey! Time is a-wasting, girl. Where have you been?"
"Settling an argument over who had to be a Yankee," Ashley told her.
"Yankee. That's the North, right?" Beth asked. Beth was from New York, and before that, her family had lived in Jamaica. Her accent, however, was all American, and none of her ancestors had been in the United States during the Civil War.
Ashley frowned.
Beth laughed. "Just kidding! Come on, I took history classes."
"Sorry!" Ashley said.
"You should be. Let's get you in this ridiculous contraption. So, people really churned butter in these things? No, wait—your relatives sat around looking pretty while the slaves and servants churned the butter, right?"
"Actually, in our family, everyone worked. And I think that everyone had to sweat when churning butter. Most of the time, the plantation mistress had to work really hard."
"Supervising?"
"And making soap and doing laundry and all the rest," Ashley said. "Well, maybe if you were really, really, really rich you just sat around. We were rich, but not that rich, and if we're ever going to be rich again, it's up to you, since I can barely boil water." Beth had come to work at Donegal as the chef less than a year ago, determined to make the restaurant one of the most important in the South.
"Anyone can boil water," Beth assured her. "And you cook okay. You're not great, but, then, you are one hell of a storyteller. Step into the skirts already, I'm dying to see this show."
Arranging the layers of clothing that constituted the formal dress of a Southern plantation mistress took some time. They both laughed over the absurdity of the apparel that had been required in Louisiana despite the heat and the humidity. Ashley told Beth, "It's worse for the guys. The authentic uniforms are wool—those poor little puppies just die out there."
"Well, honey, I think I'm glad that I'm the unpaid help for this shindig, then," Beth told her, grinning. "Cotton like this—it's nothing. And I do love the bandana! Poor Emma."
"Yes, it really was poor Emma," Ashley told her. "Lots of the soldiers left journals about what happened at the battle. It was only after the war that the rumors about Emma having killed her husband got started. It's as if someone wanted to sully her name. Of course, nothing that we can find was written about her having been charged with the crime."
"But it was a different time. Maybe she did the unthinkable. Maybe she took a lover. Maybe even, God forbid, he was a Yankee or a carpetbagger!"
"Maybe," Ashley agreed. "From all the family lore, she loved her husband, she was devastated when he died, and she managed to hold on to the property and raise her children here, even though the South lost and carpetbaggers did sweep down on the South. Carpetbaggers were even more despised than Yanks," she explained. "They were the people who weren't fighting for a cause—their cause was just to prey off the vanquished and get rich."
A few minutes later, Ashley was ready, and they headed back to the porch that faced the river. A crowd had already gathered, since the schedule for the day was printed out on brochures that attendees could pick up at the entrance to the property. A high-school student, seeking extra credit in history, was usually given that job.
Ashley came out to stand next to her grandfather, looking out over the property as she could see it from the back porch. Bright tape in blue and gray cordoned off the areas where the reenactors would move during the events, though they no longer went into the cemetery for the moment of Marshall Donegal's death and the tactical retreat of the two surviving Union soldiers.
She found herself staring at the cemetery off to her left again. An odd tremor washed over her, but she quickly forgot it and looked at Frazier.
"Nice crowd today," he said quietly. Ashley squeezed his hand.
The small band—posing as the military band that had been part of Marshall Donegal's cavalry unit—launched into the haunting strains of "Dixie."
Frazier Donegal began to speak midway through, giving an excellent history lesson. He didn't shy away from the slavery question, admitting that cotton was king in the South, and sugarcane, and both needed workers. The citizens of the South had not invented slavery; many had clung to it whether, in their hearts, they accepted the injustice or not. Few men like to admit they were wrong or cruel to their fellow human beings. And they had hardly been magnanimous when it meant they would also lose their livelihood. It wasn't an excuse, but it was history. Then as now, prejudice was not something with which a man was born—it was something that was taught. He spoke with passion, conviction and sincerity, and a thunderous round of applause greeted his words; he would have been a great politician, Ashley thought. Except that he had never cared about politics; he had always cared about people.
The first roar of close fire sounded from the stables area, and people screamed and jumped. It was all sound and black powder. There was no live ammunition at the reenactment.
The Yankees, mounted on their horses, rode in hard from the east, dismounting at the stables to use the buildings as defensive positions as they began their attack.
Ashley went on to introduce herself as Emma Donegal. She told about the beginning of the war, and how her husband, Marshall Donegal, famed for his exploits in the Mexican-American War more than ten years earlier, had returned to the military, raising a cavalry unit for the Louisiana militia that would be ready to join the Confederate army at any time. But federal forces were always spying in Louisiana. It would be the Union naval leader, David Farragut, a seasoned sailor, who would assault New Orleans and take the city in 1862, but before that time, Union forces snuck down regularly to survey the situation and report back on the Confederate forces guarding the city. The battle at Donegal Plantation began when the federal spies who had participated in the bar brawl rode swiftly to the plantation in uniform, hoping to engage the Confederates before they could summon more men. At Donegal Plantation, however, four of the spies died at the hands of the small Confederate force to be found there, and the only Confederate casualty was Marshall Donegal himself, who had succumbed to the onslaught of the federals, killing three before falling in a pool of his own blood. She explained that history longed to blame her—Emma Donegal—but she was innocent. Truly, she was innocent! The world hadn't changed that much; people loved to talk, and everyone wanted there to be more to the story. There simply wasn't. She and her husband had been married thirteen years; they had four children they were raising happily together. She was heartsick at her husband's death and survived her grief only because she had to keep food on the table for her children.
Of course, she knew the story like the back of her hand. She told it well and was greeted with wild applause when she pointed across the yard. "There! It all begins!"
And thus began the round of shots that made the expanse of land between the stables and the house rich and ripe with black powder. The federals had been traveling with a small, easily maneuvered six-pound howitzer, and in their attempts to seize the property, they sent their bronze cannon balls sailing for the house and ground. In fact, they had missed. At the time, their attempts to use the small cannon had done little but rip up great chunks of the earth. Today, it caused the air to become heavy with black powder.
"The Confederates had to stop the attack before the barn, stables and outbuildings could be set afire," Frazier Donegal announced from the porch, with a microphone, his voice rich and deep and rising well above the screams and shouting.
Though there was no live ammunition, the small fight clearly taught onlookers just how horrendous it must have been for men in major battles. As the Confederates and federals fought here with guerilla tactics, Ashley asked the crowd to imagine thousands of men marching forward side by side, some of them able to reload three times in a minute. The carnage was terrible. The Civil War was considered to be the last of the ancient wars—and the first of the modern wars.
The defenders split, most of the men rushing the stables from the front. But the Yankees had come around the other side, and in their maneuvering they escaped the body of men they had been determined to fight. One of the attackers was killed at the stables; the others made it around to the cemetery, attempting to use the old vaults as shields. But Marshall Donegal had come around the other side, and while his men were held up, he met up with the attackers at the cemetery.
The fighting originally ended inside the cemetery, but now they ended it just outside, the only difference from that day to this. First, the crowd wouldn't be able to see any of the action if it occurred there, and, with that many people tramping through, historic funerary art could be destroyed. And so, Charles Osgood, as Marshall, brought down several of the enemy and perished, brutally stabbed to death by bayonets, in front of the gates. The two surviving federal men—Justin Binder and Ramsay—raced toward the stables, whistling for their mounts. They leapt atop their horses and tore for the river road.
Frazier announced, "And thus did the fighting at Donegal Plantation come to an end."
They said the Pledge of Allegiance, and then the band played "Dixie" and then "The Battle Hymn of the Republic." After the burst of applause that followed the last song, people began to surround the actors—who had remained in the battle positions where they had fallen—as they came to their feet, and they all seemed to disappear into the crowd as they were congratulated, questioned and requested for picture-taking opportunities. Then, at last, the crowd began to melt away, and the sutler began to close down his shop.
Darkness was falling in earnest.
It had been a tremendous success; standing on the porch and watching the crowd ebb, Ashley told herself that she'd been an idiot, letting a dream get to her.
But, as she looked out, it seemed that the plantation was covered in a mist again.
It was the remnants of the black powder from the guns, she told herself.
The mist bore a reddish color. Bloodred.
The sun had set in the west; it was due to the dying of the day.
Whatever the explanation, the entire scene was eerie.
A breeze lifted, and she had the odd feeling that somehow everything had gone askew and changed, and she had somehow entered into a world of mist and shadow herself.
"Well, old girl," Frazier said quietly, smiling as he set a hand on Ashley's shoulder. "Another wonderful day. Thank you for all your hard work on this."
Ashley smiled. Her grandfather was happy. She adored Frazier, and she was always glad when he was happy. She worried about him constantly—driving him crazy, she knew. He had always been somewhat bony—though dignified! But now he seemed thinner, his cheeks hollow. He was old; but a man's life span could be long, and she wanted him with her for many more years. Now he was smiling, basking in the pleasant glow of the day's success.
"Come on. Let's head into the parlor," Frazier said. "I think we should probably be there to toast our actors and friends, eh?"
The family and some friends—including the soldiers for the day—traditionally retired to the riverside parlor for drinks and unwinding.
"You go on," Ashley said. "I'll be right there, I promise. I just want to see that everyone is really moving on."
Her grandfather gave her a kiss on the cheek. "I'm sure Beth has already put out all manner of delicious little snacks, despite the fact we told her that chips would do. I'll go supervise my liquor cabinet," he said, wiggling his white brows.
She grinned. "You'd better do that. Ramsay will say that he deserves your hundred-year-old Scotch for being so generous!"
Frazier pantomimed real fear and then walked on into the house. Ashley was exhausted and ready for a fine glass of hundred-year-old Scotch herself.
But she left the porch to walk around to the front for one last look. Jerry Blake, one of the off-duty officers they hired for traffic and crowd control, was still out by the road, waving at the last of the cars to get them safely on their way. She lifted a hand to him and shouted, "You coming in, Jerry?"
He waved back at her and shouted in return, "No, thanks! I'm on my way home. I have an early patrol shift tomorrow. See you, Ashley!"
A minute later, she saw him check that the day visitors' cars were all gone. Then he headed for his own car.
The buzz of chatter from inside filled the new silence. She followed the sound to the front parlor, where the reenactors were gathering. Looking around, she had the same strange sense of time encapsulated that she had felt before; none of the soldiers had changed out of their uniforms yet, and she was still in her Emma Donegal attire. Even Beth, who had seemed to get a tremendous sense of entertainment out of the day, was still in her 1860s garb. Some of the men had cigars, and they were allowed to smoke them in the house that night. Only the beer bottle in the hand of Matty Martin, the sutler's wife, provided a modern note.
Matty came over and gave her a kiss on the cheek. "Why, Mrs. Emma Donegal, you do create a mighty fine party, a mighty fine party! What a day!"
"Why, thank you, Mrs. Martin," Ashley said, inclining her head regally as a plantation mistress of the day might have done.
Matty dropped the act for a minute. "Oh, Ashley, we sold so much! And I can't tell you how many people ordered custom uniforms. I'll be sewing my fingers to the bone for the next months, but what a great day we had."
"I'm so glad," Ashley told her. She walked for the buffet with its crocheted doily and poured herself a Scotch whiskey—it wasn't a hundred years old, but it would do. Others came up to her and she responded—so many friends, and everyone involved in the reenactment. The men bowed and kissed her hand, still playing elite gentlemen of the era.
Ramsay grinned when he was near her. "I'd say ninety percent of the fighting men never tasted a good brandy, so I'm sure glad we get to be the rich of the past."
She smiled, and agreed. "Wouldn't it be something if we could have Lee and Grant, and Davis and Lincoln, and show them all that the war created the country we have now?"
Griffin walked over to them, lifting his glass. "Grant was an alcoholic. A functional one, but an alcoholic. No relation, of course. My Grant family was Southern to the core. Cheers!"
"You're a cynic, Mr. Grant," Ashley said, inclining her head.
Griffin laughed. "Not at all. We strive for an understanding of history around here, right?"
"We do," Ashley agreed. "And, historically, many of them were truly honorable people. Can you imagine being Mrs. Robert E. Lee—and losing a historic family home, built by George Washington's stepgrandson and filled with objects that had belonged to George and Martha? Remember, Arlington was a home long before it became a national cemetery!"
"Cheers to that, I suppose," Griffin said. "Whiskey, Mrs. Donegal? Why, my dear woman, you should be sipping sherry with the other wives!"
"I need a whiskey tonight!"
Ramsay and Griffin laughed, and she joined them while she listened to her guests chatting. Some of the other men argued history, too—and she saw that everyone involved in the actual reenactment had shown up. Cliff, Ramsay, Hank, Griffin, Toby and John—and the Yankees, Michael Bonaventure, Hadley Mason, Justin Binder, Tom Dixon and Victor Quibbly, along with John Martin, of course, and Dr. Ben Austin.
Everyone but Charles Osgood. She couldn't imagine that he wasn't there. He must have been thrilled to death with the day.
"Hey, where's Charles?" Ashley asked, interrupting a rousing discussion of Farragut's naval prowess.
A few of those close to her quit talking to look around.
"I haven't seen him since he very dramatically died of his wounds," Ramsay said. "I 'skedaddled' right after and rode out with Justin, before we rode back to take our fair share of the applause."
"Cliff?" Ashley asked.
Cliff shook his head. "No, I was with the soldiers who came rushing in too late when Charles was being besieged by the enemy. I thought he just stood up and bowed when everyone was clapping. I don't remember seeing him when you and Frazier started talking...or when the band played."
"He's probably outside somewhere. I'll call his cell," Ramsay said. He pulled out his phone and hit a number of buttons.
Ashley watched him. She realized the others had already turned away and were becoming involved in their conversations again.
Ramsay shook his head at her. "No answer."
Cliff cleared his throat. "Not to be disrespectful in any way, but maybe he met a girl and—got lucky."
"Yeah for Charles!" Justin Binder said, lifting his glass. He was somewhat tipsy—if not drunk—Ashley thought. Good thing he was staying on the property. The others were all still playacting; they were entrenched in the past.
They didn't want to look for someone they obviously believed was just off enjoying his own star turn. But...
"He would have wanted to be here tonight," Ashley said stubbornly. "He was so thrilled to be taking the part of Marshall Donegal. I'm going out to see if his car is still here."
Ramsay lifted a hand. "Sorry, don't bother, Ashley. He didn't drive. He came with me. I told him that I couldn't give him a ride back since I was going to stay at the house out here for a while, but he told me he'd hitch a ride back in with someone. Said he didn't have to be back to work until Tuesday morning and for me not to worry."
"Gentlemen, perhaps a search is in order," Frazier said. "A Civil War parlor game of sorts."
They all stared at him blankly.
"Exactly," Ashley said, relief coloring her tone. "Find the lost rebel. Beth will create a five-star private meal for a party of four, payable to the man—or woman—who finds Charles!"
"I will?" Beth said. She looked at Ashley. "Um, it will be—sumptuous!"
"It's a lot of property to cover," Ramsay murmured.
"We need to organize, then," Griffin said. "It will be fun. Yankees take the cemetery side, and rebels search out the bayou side."
"Is that fair?" Griffin asked. "If he's still around, old Charlie would be by the cemetery, don't you think?"
"I pick scouting detail!" Justin said.
"Yes! Let's find Charles!" Toby said.
"I'll check out the area around the oaks out front," Matty Martin offered. She was watching Ashley and seemed to realize that Ashley was seriously worried. "John, you can come with me. It's mighty dark out there, even with all the lights from the house and the property floodlights."
"Of course, my dear," John told her. "They should have let women fight the war," he muttered, following her out.
Hank laughed. "Yeah, imagine, mud wrestling at its best."
"Hank!" Cliff admonished. "War is always a serious affair."
"Well, of course it is," Griffin said. "War is very serious—but we're not at war. We're playing a game. We're looking for old Charles. Hey, Ashley, if no one wins..."
"Well, at some point, we'll just all have dinner," she told them.
"Great!" Beth muttered to her. "Now I get to cook for all of them!"
"It's good that I've got the bayou side!" Toby Keaton said. "Borders my property."
"I'll take the cemetery," Frazier said.
"You will not. It's dark and dangerous in there," Ashley told him.
"Not for me, dear. It's memories for me," he said softly, and quickly turned away. Neither of them wanted to think about Ashley's parents, entombed in the majestic family vault.
"Grampa, please—you need to be here as everyone returns," Ashley said.
"I'll take the cemetery," Ben offered. "I'm really familiar with the living and the dead," he added and winked. "Just give me one of the big old flashlights at the back door. I'll be fine."
Ben would be fine. He was a big, strapping man in his mid-forties. Besides, he'd attended funerals for both her parents and knew the cemetery well.
Ashley wanted to take the cemetery herself; that dream had to have been a sign.
No, that would be insane. Ben knew what he was doing. She wasn't going to let a dream dictate what she did in her life.
"Okay, so where are we going?" Beth asked Ashley.
"The stables?" Ashley suggested.
"I'll come with you and stand there, but I'm not going near the horses!"
An hour later, they had finished the actual search as best they could in the night.
Ramsay went to speak with the guests who were staying in the rooms that had been the old stables, and the Yankee contingent spoke with those in the other outbuildings. Cliff went to his office, wondering if Charles might have slipped in there to rest.
They all searched, from the river to the road, from the sugar fields to the bayou, but there was no sign of Charles Osgood. By midnight, all the searchers were back at the house.
"Ashley, really, he must be out somewhere else,"
Cliff told her.
She looked at Ben. "You searched everywhere in the cemetery? There are so many paths, little roads between all the vaults."
Ben sighed. "Ashley, I searched. But we can all take another look."
She nodded.
"That was actually not a suggestion," Ben said.
"It's all right. I'll go myself," Ashley said.
"We'll help," Ramsay said, tugging at Cliff's sleeve.
"I've still got the key, so I'll come, too," Ben said.
Ashley led the way, wondering why she thought that she'd really find Charles in the cemetery, just because she'd had a dream.
But she was determined.
Ben opened the lock on the gate, though, of course, they could have all crawled over the stone wall.
Ashley headed straight for her family tomb. The real Marshall Donegal had died there.
The last interment had been her father's. The usual little pain in her heart sparked—it always came when she thought about him, and her mother. And tonight, especially, she missed Jake.
There was no sign of Charles there, and no sign that he had been there.
She almost fell, she was so relieved.
The tomb glowed white beneath the gentle touch of the moon, dignified in its decaying majesty. She heard the three men calling to one another from different sections of the graveyard, and she followed a voice to reach Cliff. He looked at her. "Ashley, Charles left. Whether he was spirited away by aliens or not, I don't know. But he isn't here. This isn't any parlor game, is it? You're really worried."
"I am. Did you go in the chapel?" she asked.
"You think that Charles is hiding in the chapel? Or kneeling down, still thanking the good Lord for the chance to be Marshall Donegal?" Cliff asked dryly.
"Please, Cliff?"
He groaned. He walked around the ell that would lead them to the chapel, in the far corner near the embankment of the river. The chapel had carved oak double doors, which creaked when he opened them. He fumbled for the light switch, and light flared in the lovely little place with its stained-glass windows, marble altar and old mahogany podium.
The place was empty.
"Happy?" Cliff asked her.
"No. I can't help it—I'm worried," she told him.
He just shook his head. "Come on. Let's just go."
They walked back to the house, where the others were still milling on the back porch—many of them having retrieved their drinks.
"So, the bastard did get lucky!" Ramsay said, laughing. "Hell, if I had foreseen that, I'd have had him play Marshall Donegal a couple of years ago!"
"I'm going to call the police," Ashley said, looking at her grandfather.
"He's been missing just a few hours," Beth pointed out. "He might have thought that he said good-night to everyone. There's so much confusion going on when the fighting ends. I mean, I thought it was amazing—it really was living history. But it's mass confusion. I can only imagine a Gettysburg reenactment."
Ashley realized that everyone was staring at her—skeptically. They had searched and searched, and grown bored and tired. But she couldn't help her feelings of unease, even while they all stood silent, just staring at her.
The river breeze brought the chirp of the chickadees—her senses were so attuned to her home area that somewhere, distantly, down the bayou, she thought she could hear an alligator slip into the water. This was her home; she knew these sounds.
They were normal; they were natural. But the sounds of the darkness weren't reassuring to her now.
"Grampa, I think we need to report this to the police," she repeated.
"Great. He's probably at some bar in the big city, bragging about the fact that he got to play Marshall Donegal today," Ramsay said. "And they'll drag him out and he'll act like a two-year-old again."
Frazier stared at Ashley and nodded. If she wanted to call the police, they would do so.
The parish police were called, and Officer Drew Montague, a nice-enough man whom Ashley had met a few times over the years, took all the information.
"You say you all saw him just a few hours ago?" he asked. Montague had a thick head of dark hair and eyebrows that met in the middle.
"Yes," she said.
"What makes you think that he's actually missing? Perhaps there's a woman involved. Is he married? Look, Miss Donegal, you know that we appreciate everything that you do for the area, but...we're talking about a grown man who has been gone just a few hours," the officer said.
"He was proud of the role he was playing. He would have stayed," Ashley insisted.
Officer Montague shifted his weight. "Look, I've taken the report, and I'll put out a local bulletin to be on the lookout for him, but he's an adult. An adult really needs to be gone for forty-eight hours before he is officially missing."
Frazier spoke before Ashley could. "Anything you can do will be greatly appreciated. We're always proud that the parish is about people, and not just red tape and rules."
Montague nodded. "Right. Well, I'll get this moving, then. We'll all be on the lookout for Mr. Osgood."
Ashley thanked him. The others had remained behind, politely and patiently waiting. Now it was really late, and once again there were a number of weary men and women—all still in Civil War–era attire—staring at her.
Officer Montague left, mollified by Frazier Donegal over the fact that he had been called out on a ridiculous mission.
"I'm sorry," Ashley said to the others. The evening had started out as a party and turned into a search committee.
"Hey," Cliff said, grinning, "I don't have far to go home."
"We're staying in the stables anyway, kid," Justin Binder told her. He had played a Yankee, and happily. His family hailed from Pennsylvania.
Griffin laughed and gave her an affectionate hug. "You made me sober up, which is good. I am driving."
"Me, too," John Ashton said. He held her shoulders and kissed her cheek. "Charles is just fine. I'm sure of it."
She thanked them all and said good-night, and they drifted away, some to the old outbuildings where they were staying, and some to their cars, parked in the lot out front and down the road.
She stood on the porch with Beth and her grand-father. She couldn't tell whether they thought she was being ridiculous or not, they were both so patient.
Beth gave her a kiss on the cheek and said, "We still have about sixteen guests, and the household. I've got to get up early to whip up our spectacular plantation breakfast."
Ashley bid her good-night. It was down to her grandfather and herself, and Frazier was going to wait for her to be ready to head off to bed.
"Something is wrong. I can feel it, Grampa," she said.
He set an arm around her shoulder. "You know...I have an old friend. I've been meaning to call him for a long time—tonight seems a good time to have a chat with him. If Charles really is gone, he may be able to help us. His name is Adam Harrison. I don't know if you remember meeting him—I see him up in Virginia and D.C. sometimes. He worked for private concerns for many years, finding the right investigators for strange situations. Then the government started calling him, and his projects were all kind of combined for a while, civilian and federal. But he's got a special unit now, and he's got federal power behind him on it. His people are a select group from the Behavioral Analysis Unit of the FBI. I'll give him a call. We'll get someone out here to help by tomorrow. And if Charles turns up, no harm done."
She lowered her head. Adam Harrison. She knew the name. His unit had been involved in solving the death of Regina Holloway—it had been all over the media because she was a senator's wife. And she knew, too, that Jake Mallory was part of that unit. She might not be a part of his world, but she hadn't been able to miss it when she'd seen his name in the papers. She had broken off something that had been real with Jake, because he had terrified her...because he was certain that he had spoken with her father, after he had died. And now....
Now Frazier was going to call Adam. Of course, it could come to nothing. She was panicking over a missing man because of an equally irrational dream.
She looked out on the beautiful expanse of their property. The river rolling by. The moon high over the clouds. The vaults in the cemetery silent and ghostly and opalescent in the pale glow of night.
Jake, I'm so...scared.
Something was wrong. It was the oddest thing; she felt that she really understood the expression I feel it in my bones. Something wasn't right about Charles's disappearance, and she knew it.
It was almost as if the past had truly merged into this eerie and haunting reality, and the collision of time here was not going to go away.
Interlude
He'd known for a long time what he'd had to do. The voice had been telling him for years.
At first, of course, he had ignored it. The vision he'd seen of the past hadn't been real. But then he'd known. He'd known who he was, and he'd come to know that the voice wouldn't go away until he'd done what needed to be done. And he'd carefully planned it all out, though things had gone a bit strangely today. Didn't matter, though, who was playing Marshall Donegal. It didn't matter at all. Because, of course, an actor was just an actor.
It was Donegal Plantation itself that needed to repay the old debt. That old debt could only be repaid one way.
With blood.
God bless a crowd. There was nothing in the world like mayhem, nothing like hundreds of witnesses to pull off an escapade such as he had planned, and to do it perfectly.
There had been a horde surrounding them. One particular brunette was the right age, exceptionally pretty and with a Massachusetts accent. When she spoke, there was an r on the name Linda, and there was no r on the car she had "pahked" down the river road.
She had giggled when she spoke to Charles, so it was easy to whisper in the man's ear in his moment of greatest achievement and convince him that the girl was waiting to meet him.
And in the madness surrounding everyone engaged in the action then, it was easy enough to meld into the crowd himself, and to swiftly disappear, and hurry to the river road.
And there was Charles.
He'd approached Charles with a smile.
And, of course, Charles was smiling as well. At least he would go in a state of sheer happiness. It might even be a kindness. How many people got to die that happy?
Poor, dumb Charles—he never suspected a thing. After the initial whack, he never even felt the prick of the needle.
He'd thought it all out, exactly where he'd send Charles, because it all had to be done in plain sight. In plain sight, people never really knew what they saw.
There were tourists heading to their cars. But they'd never notice two fellows in uniform chatting by a car. Not at an event like this. People liked to dress up.
Maybe everyone wanted to be someone else, someone they weren't.
But to them, it would just appear that they were two cronies, faces covered by their broad-brimmed hats, leaning against one another as they chatted and laughed over a joke.
Then...hide the body. Or if he had been seen, "help" an inebriated friend into a car.
He would need more time for the pièce de résistance. Initially, it had taken him less than twenty minutes to stash Charles and rejoin all those rejoicing over the day.
He had never felt more victorious. The difficult part, of course, would be to hide his anticipation for all that was destined to follow.
It didn't seem that anything could go so impossibly well.
Ashley, damn her, though. Leave it to Ashley to be worried about Charles! Still and all, it did make the entire plan more exciting. Now, with the evening at a close, he was feeling elated.
The place had settled down; though everyone had been willing to look for Charles, only Ashley had been really concerned. He had played with the idea of actually disposing of poor old Charles immediately, but now he was satisfied that he had decided he should make it something more dramatic—and allow time between the reenactment and the beginning of the end.
Oh, he had worked with the others. He had searched so hard. There might have been just a few minutes when he feared someone would actually search the cars, but Charles hadn't driven.
It had almost been as if he'd been part of the plan.
Now he sat next to good old Charles.
This was necessary. The voice had said that it had to be done, and his ancestor made him know that nothing could be right until then.
He'd never realized that he'd enjoy it all so much.
He patted him on the back. Charles didn't move. The drug was holding, but he'd administer more. He didn't want the big lug waking up.
He needed him alive until the time was right.
Every time he'd been at Donegal recently, he'd felt as if he were being pushed harder and harder. The past was the past—so they all said. But it wasn't. The past created the present, and he knew now that he had to use the present to set the past right. It wasn't crazy; he'd heard the voices in his head. A collective consciousness that seemed to scream through history.
Now, maybe, the voices would stop.
## 3
Car bombs didn't exactly do it for him, but Jake indulged in a few anyway.
"Cheers!" Jenna said, dropping her shot glass into her Guinness, and swallowing down the mixture.
"Cheat!" Will said to Whitney. "You poured your shot in—you just drink the whole thing."
"Hey, you drink it your way, and I'll drink it mine!" Whitney protested.
"You're not doing it the Irish way," Will said, looking to Jenna for help.
"Drink it however you like!" Jenna said, smiling sweetly at Will.
There was a small room in the back of the bar, and Jake, Will Chan, Jenna Duffy and Whitney Tremont had it to themselves that night, so it was nice. Jackson Crow was back at the hotel with Angela Hawkins. They'd all just met for the first time on the Holloway case, and Jackson, the skeptic, had quickly fallen in love with Angela—despite their different approaches to their work. Go figure. The entire team respected and admired them both, and they were glad that the two were indulging in some quality time together.
And for Jake, it felt good to be in the bar with his coworkers.
During the Holloway case, they had gotten to know one another. Will and Whitney were excellent with cameras and sound systems; Jenna was a registered nurse, something that could always come in handy when traipsing through strange landscapes and old buildings. His own expertise was computers—and computer hacking. He could usually find any piece of information on any site, public, private or even heavily coded. Yet they'd all had certain unusual experiences in life that had led them to being excellent investigators—and, together, able to discern deeper, darker undercurrents to the event they researched. Now, they also had badges. After the Holloway case, it had been deemed that they would continue to work together, and they would do so with all proper credentials as FBI agents.
"Now, quit whining over the way a woman drinks her drink," Jenna said and turned, leaning an elbow on their table, to talk to Whitney. She had brilliant green eyes and red hair, and a smile that could melt ice. "I want to know what else I've missed. The World War II museum, the Civil War museum, plantations, the zoo..."
"Shall we have another drink?" Whitney asked.
But before Jenna could answer, they all heard their phones buzz.
"Text from Jackson," Whitney murmured.
"Meeting in the morning," Jenna said, the slight Irish lilt in her voice grave.
"Hmm. Do you think that means that we're not heading to Alexandria?" Will asked.
"It means something is up," Whitney said, looking at Jake.
"I'll pay the bill," Jake told them.
They walked back to their hotel slowly and silently, each wondering what they'd discover in the morning. After they parted, Jake sat up a very long time.
It became morning at last. Ashley didn't feel as if she'd slept at all. The dreams continued to plague her, only now she was Emma Donegal, leaving the house in the aftermath of the battle to find the bloody body of her husband. And when she woke herself from the dream, she could have sworn that deceased Confederate soldier was sitting in the wingback chair by the doors to the second-story wraparound porch. She was more tired from being in bed than she was from being awake.
A shower helped revive her a little. Dressed and ready for the day, she headed down to the kitchen. Once it had been a gentleman's den, and then it had been an office, and then, when it was no longer deemed necessary to have the main kitchen in an outbuilding, it had become a wonderful, bright kitchen. The walls were a pale yellow. There was a center granite worktable with stools around it, and suspended racks that held several dozen shining copper cooking utensils. A breakfast nook held a table that sat eight.
Beth was just pouring milk from a carton into serving pitchers. "Coffee is on. None of the guests have made it in yet," she said cheerfully.
"What's for breakfast?" Ashley asked.
"Down-home comfort food this morning," Beth said. "Corn bread, blueberry muffins, bacon and cheese omelets, and country cheese grits. Want to grab a plate and eat before it starts getting crazy?"
"Sure," Ashley said. She watched as her beautiful friend made art out of an omelet and shook her head as Beth handed her the plate full of light, fluffy eggs.
"Grits are in the bowl, corn bread is sliced and in those baskets," Beth said.
Ashley helped herself. "I'm going to waddle across the lawn soon," Ashley told her.
Beth grinned. "I doubt it. You're too fond of those awful creatures out in the stables. You get plenty of exercise." She shivered.
"I can't believe that you're afraid of horses." Ashley laughed.
"I told you—one of the bastards bit me when I was a child!" Beth said.
"Well, ours won't bite you. You should try riding Tigger. She's a twenty-year-old sweetie. She moves like an old woman."
"Then she may be crotchety as one, too," Beth said. "No, honey, you stick to your horses, and I'll stick to cooking."
Ashley dutifully bit into her omelet, and it was delicious. As she was finishing, guests began to stream by her, heading in for breakfast or stopping to clear their tabs. They'd be down to eight guests that night; the reenactment had taken place on a Sunday, and many of those who came for the reenactment managed to take off the Monday if they had a regular workweek. By Monday night, they were usually down to just a few guests.
She heard Frazier speaking with people on the other side of the stairway, his tone rich and filled with humor as he told old family tales and pointed out certain portraits on the walls.
Ashley took her place at the desk to fill out the registry and books—by hand; people actually signed her guest book, and she wrote personal thank-yous—and then could have sworn that someone had approached her. She looked up, but she was alone. For a moment, her brows knit in consternation, but people milled throughout the lower level of the house now and any one of them might have stopped nearby. She gave her concentration back to the project at hand.
She heard a throat being cleared then, and looked up—this time, someone was there. Justin.
He sat in the one of the period wingback chairs that faced the desk.
She frowned. "Are you checking out? I thought you were staying a few days."
"I am staying another few days, Ashley. I just stopped to see how you're doing," he told her.
She liked Justin. At forty, he was a widower, though years before, he had brought his wife with him, and she had played at being a camp follower—with great relish. They had been married for years before he had lost her to cancer. But Justin still came.
"I'm fine, thanks. Nancy's got the girls?" His mother-in-law, Nancy, now came along to help Justin with his ten-year-old twin girls. Hard to be a "fighting federal" and keep an eye on twins.
"Yes. Any word on Charles?"
She set her pen down. "No. But I haven't tried calling anyone this morning. Everyone on that search party last night is weary of me torturing them, so... If he's been found, I'll be called right away."
He reached across the desk and put his hand on hers, giving a comforting squeeze.
"Ashley, you are part of the charm of this place.
You really care. None of us thinks you were torturing us. I was thinking of taking the family for a horse ride later, and I know that Cliff does a lot of the riding tours, but I thought you and I could make another search of it, too."
She was surprised. "Sure! And thank you."
"Jeanine and Meg don't ride well. They don't get a chance to go riding often enough. You still have two horses calm as the Dead Sea, right?"
"Nellie is our sweetest. And Tigger is a good old girl if I've ever known one. Nellie loves him, so they're great on a ride together. They'll be perfect for the twins."
Justin grinned and stood. "Nancy's bringing the crew in for breakfast. Say an hour or two?"
"Two hours will work for me."
Justin thanked her. She finished with paperwork and realized she was constantly looking up, certain that she was going to see a Confederate soldier staring at her.
"I don't believe in ghosts," she reminded herself. But saying the words out loud sounded defensive. "I don't. I really don't!" she said to the empty room.
Irritated with herself, she went out to the stables. Justin's family would be out soon.
Ashley saddled Varina and stroked her mane. The farmer they had bought her from when Ashley had been a teen had been an avid fan of Varina Davis, the one and only first lady of the Confederate States of America. Because she had been named Varina, they named Nellie's last colt Jeff, for Jefferson Davis, the one and only President of the Confederate States. That morning, she and Justin chose Varina and Jeff as their mounts, while she assigned Nellie to the younger, slightly more timid of Justin's twin girls, and Tigger to the other, while Nancy, Justin's mother-in-law, was on the slightly more spirited Abraham.
Ashley took the girls around the paddock a few times, just going over the basics. Justin had been right about their experience, but they were smart little girls with common sense, and Ashley thought they would do well.
Ashley gave her attention to the girls as they rode around the outbuildings and then toward Beaumont, the Creole plantation "next door." The girls were delighted by the ride, waving to everyone they passed while traversing the house and outbuildings area and then concentrating on their father's and grandmother's admonitions to be on the lookout for wildlife.
"Are there alligators?" Meg, the bolder of the twins, demanded.
"Yes, by the bayou. But they'll leave you alone if you leave them alone. We won't dismount anywhere near the bayou. Now, you don't want to bring a small-sized dog or even a medium-sized dog out there. They look like dinnertime to the alligators," Ashley told them. She was listening to the girls; she was looking everywhere. They had searched last night, but it had been dark. Now it was daylight, and, hopefully, if Charles Osgood had come out here and fallen, hurt himself or had some other trial, they might find him now.
"We don't have a dog," Jeanine, Meg's sister, younger by five minutes, said.
"Can we get a puppy, Dad?" Meg asked.
"Soon enough," Nancy said, grinning at Ashley.
"Why not now?" Meg asked.
"Because Daddy is busy," Nancy answered. Nancy was one of those women who had gone to a beautiful shade of silver-white naturally.
"Watch for animals, girls," Ashley interceded. "We'll be close enough to see the alligators basking in the sun. These woods aren't that dense, but with all this land, every once in a while a black bear or a cougar wanders across the road. I know that you see nutria—"
"What are nutria?" Meg interrupted.
"They're the largest rat, essentially," Justin said.
"Ugh!" Jeanine said.
"The buggers were brought over years ago, in the 1930s, and they've multiplied into the millions," Ashley explained. "There's actually a bounty on them, because they can be so destructive. But they don't hurt people. The animal that you do have to be careful of in these parts is the cottonmouth snake. But it likes water, too, and we're not going in the water. Animals usually leave you alone as long as you leave them alone."
"Watch for herons!" Justin said.
"I wouldn't mind seeing a cougar," Meg announced.
"They're shy, too. But we'll see what we see," Ashley assured them.
They counted seven herons, two raccoons, an armadillo and three owls up in the trees. When they came to the bayou, Ashley pointed out two alligators sunning on the opposite bank. As she did so, she saw that staff members at Beaumont were engaged in their work already. A man dressed in a droop hat, cutoff denim and a dotted cotton shirt was standing by a wagon that showed freshly hewn sugarcane. Another, dressed more like an early nineteenth-century Louisiana French businessman, was giving a tour.
She looked up toward the second story of the plantation house, where the family had lived. A man was standing there, dressed in a Confederate uniform frock coat.
Ashley blinked against the light. He looked like...
Like her ancestor, Marshall Donegal.
The man lifted a hand to her.
Yet when she blinked again, he was gone. Her imagination at work again. Of course, she was still concerned about Charles Osgood. But he was due back to work the next morning. If he didn't turn up by then, the police would have to get involved at a serious level.
She realized that Justin was watching her.
"Are you okay, Ashley?" he asked.
"I'm fine. The light is playing tricks, that's all. I thought I saw a Confederate soldier at the window. Toby Keaton does workshops and tours on the real workings of a sugar plantation over there. We do the Civil War—keeps me sending tourists to him, and him sending them to Donegal Plantation," she said. Would she have told the truth if Jake were here? Jake, who seemed to know what the dead were saying.
"You have Charles Osgood on your mind," Justin said.
"I do. I can't help it."
They rode along the bayou for a while, and then Ashley led them around the second trail to head back to the house. The girls chattered the whole time.
Justin nudged Jeff and the horse trotted up next to Ashley again. "I know you were hoping to find Charles," he said.
"I am worried, Justin, really worried," she said.
"He'll show up. But let him know how worried you were. That will make him feel good," Justin told her.
Ashley offered him a smile. "Sure, thanks."
Back at the stable, Ashley tried to keep her mind busy, letting the girls help her with the saddles and bridle and tack. She taught them how to groom their horses.
As she put away the last of the brushes, Justin strode over to her. "Great. Now Jeanine doesn't want a puppy anymore. She wants a horse."
Ashley laughed. "Maybe you better look into the puppy thing, fast. Of course, there's always kittens, you know."
Restless, she returned to the house, showered again and went down to join Beth, who was planning the offerings for the restaurant's evening meal.
Jackson Crow was nothing if not a man of incredible thought and organization; there were six folders on the table awaiting them all.
Jake and Whitney were precisely on time, but he was the last to slide into his seat at Le Café, the Hotel Monteleone's bright and charming dining facility for breakfast and lunch. There were other diners about, but they'd been given a table in the far corner, near the windows to Royal Street, and they were certainly far enough from others to carry on a meeting in normal tones.
Jackson, ever the gentleman, had risen while Whitney took her seat, which meant that Will did, too. They were both tall, and the kind of men who drew attention. Jackson had the rugged, square-jawed history of Native Americans in his face, while Will's Trinidadian mix of English, Indian, Chinese and African ancestry gave him a fascinating appeal.
"Thank you all for being on time," Jackson said. "Though I want you all to know that we're not in an emergency situation."
"What is our situation?" Will asked.
"As you know, we'd been due to leave here and set up shop in our headquarters in Alexandria. But instead we're going to stay here, in the area, a few more days," Jackson said.
Whitney kicked Jake under the table. He scowled at her.
Jackson continued, "This may be absolutely nothing, but since we were already in the vicinity, we've been asked to check out a disappearance."
"Oh, no, not a child?" Jenna asked softly.
"No, no, a man named Charles Osgood, thirty-eight years old," Jackson said. "As you may or may not know, Adam Harrison is quite the philanthropist, and he's friends with a Louisiana legend with whom he's worked on many committees to improve education in the Southern states, housing reform, storm relief and so on. Apparently, although the local police aren't terribly concerned, Osgood disappeared right after a battle reenactment."
Jake knew; he already knew. There had been the dreams; there had been the woman he had seen when he'd been out with Whitney.
There had been those images of Ashley calling to him.
He knew, damn it, knew....
"A what?" Jenna asked.
Jackson glanced at her, remembered she had spent most of her own school years in Ireland, and explained. "A Civil War reenactment. We do tons of them in this country. Many battlefields are now national parks, and many of those that aren't are owned by people who don't mind reliving the Revolutionary War and the Civil War. It's 'living history,' and a good thing, in my opinion. We learn from our mistakes."
"True, or else you'll die," Jenna said. She looked around and shrugged. "Survival of the fittest. But, yes, they do reenactments everywhere, of course."
The others smiled at her tone. Jake felt the unease sweeping through him. Déjà vu.
"How long has he been missing?" Whitney asked.
"Not even twenty-four hours," Jackson said.
"Are we even sure he's missing?" Jenna asked.
"No," Jackson said. "But we're going to hang in a bit—wait here in New Orleans. You have a day entirely to yourselves. If he's still missing after this evening, we'll become involved with the search. Adam is making sure that we're officially invited in. I have folders regarding Osgood's situation—when he was last seen, the location and why the people at Donegal are so concerned."
Donegal.
Jake picked up his folder, his fingers feeling oddly numb and too big. When he flipped the page, he felt a sweep of old emotion.
Donegal. The computer printout offered a beautiful shot of a large and majestic plantation house, complete with sweeping oaks, pine forest, outbuildings, horses in a paddock and the low wall and gates of an historic family cemetery.
He didn't need the computer printout. He could see the plantation clearly in his mind's eye at any time.
He looked up.
Jackson was staring at him.
"Adam mentioned that Frazier Donegal told him that one of the team members had a history with the family and the plantation. That's not a problem for you in any way, is it, Jake?"
"No," Jake said.
Jackson was still staring at him.
"No," he repeated quietly. "Our families are old friends, that's all. But I haven't been out there in ages," he explained. "Ashley Donegal and I spent a lot of time together when we were kids. I saw her at her father's funeral, and I stayed on a few days after. She was pretty devastated when he died. She'd lost her mom just a few years earlier."
We were more than friends. Somehow, he kept his gaze steady as he met Jackson's eyes.
Every word he had said was absolutely the truth.
"Then, be advised, we're officially on hold," Jackson said. "If this fellow turns up at work tomorrow, green from celebratory alcohol, we'll be clear to head up to Alexandria. If not, we'll start working on the disappearance. Take your folders and read up—we'll need to be ready either way."
The official part of the meeting was over. They all chatted casually and ordered breakfast as the restaurant was about to close for the break before lunch. Jake joined in. But he opted not to go to the World War II museum with Will, Jenna and Whitney, or join Angela and Jackson on a trip to the aquarium with a hop over to the casino. He shrugged off both groups, saying that he had woken early and thought he'd get in some sleep, just in case they were on call.
He didn't take a nap.
Hell, he'd never sleep. He didn't want to sleep.
He wandered the city, looking for the woman in the historic costume.
No, he knew exactly who he was looking for: Emma Donegal.
He didn't find her, but he hadn't expected to.
Ghosts never seemed to appear when they were needed. It was aggravating. They reached out—and then stepped back and disappeared, assuming that they had gotten the living working on whatever it was that they wanted them working on.
"You know, you could be helpful," he said out loud, standing on Jackson Square near the spot he had first seen her. All he received for his effort was a worried glance from a woman passing by. She gripped her husband's arm tighter.
It really wasn't prudent to attempt speaking to ghosts when others were near. They just thought you were crazy or worse—dangerous.
Had Ashley thought that he was dangerous?
He stopped for coffee at C.C.'s, then returned to his room, where he did lie down.
Ashley. Donegal Plantation. The light was fading; night had come.
He couldn't stand it. He would be crawling the walls by morning. He stood up and looked around at his room. Packing up would take him all of ten minutes.
He picked up his phone and dialed Jackson. When Jackson answered, he said, "I'm heading out tonight, if that's all right. If this fellow shows up in the morning, I'll come right back. Frazier Donegal is not a spring chicken. If they're worried, I'd just like to be there."
"You read your folder?"
"I'm doing so right now. I'll pack up and be out of this room in another few hours. I'll get to the plantation really late, but I'll give Frazier a call and tell him I'm coming in."
"Fine. Just be careful, Jake. If you have a personal stake in this..."
"It may be nothing," Jake reminded him.
"If it's not, wait for the team before rushing into danger."
"Of course."
He hung up and finished packing, the same two words running through his mind over and over again.
Ashley. Donegal. Ashley. Donegal.
Ashley....
Interlude
Nighttime—the haunting time...
When all the earth was still.
And old Charles, while still breathing, was getting a little ripe.
Unconscious now for over twenty-four hours, he had a certain... Well, honestly, there was a stink about him. The man was still dressed in wool, for God's sake, and since he'd never had a chance to regain consciousness, he hadn't managed to handle personal hygiene, and, well...
But that was all right. He didn't blame poor old Charles for the way he smelled.
With any luck, no one would see him. He'd sail through this, and it would be easy, as easy as luring the man to his car, as easy as that first solid blow against his head, as easy as the injection of the needle into his flesh. Not long now...
He anticipated the horror when others found the body.
Of course, when they found the body, he'd be as horrified, stunned and confused as all the others.
He grunted, picking him up. Damned Charles—the man was no lightweight.
He didn't go through the gate, and crawling over the wall with his victim's dead weight was no easy feat.
"You should have laid off the andouille, fried chicken and Cajun rice, buddy!" he whispered aloud, struggling. He was careful, of course. His shoes were encased in plastic shower caps, and he wore thin latex hospital gloves. Now, well, hell, if someone caught him now...
"I was by the water, searching for a buckle I lost, and I couldn't just stick my hands in the muck...."
That one didn't fly. Not at all. Not even to him. Didn't matter. He was near the end.
He found the right place. Now he needed all his strength and the aid of the broken stone marker he'd made note of earlier. He heard himself grunting with exertion and paused, making sure that he wasn't sweating—it was all no good if someone could pull sweat off of old Charles. God, he hated contemporary forensics, though he was sure he had studied enough manuals on the subject to make sure that he was doing it all the right way. After all, he had planned this for years.
Finally, he had Charles where he wanted him. And it was time; Charles was still heavily sedated. He'd never know exactly how he reached the pearly gates.
He loved his weapon of choice.
For a moment, he admired his handiwork. And then he struck.
His victim never uttered so much as a whimper.
## 4
Ashley opened her eyes. Pale and surreal moonlight flowed through the gauze curtains and into her room, soft and evocative in the night. The white curtains shifted in the breeze. She had awakened in the middle of the night, not at all sure why. There should have been something, a loud noise, a gust of wind, a scream in the darkness, something.
She hadn't even been dreaming.
Thank God!
She was sure that there hadn't been any kind of commotion or noise. It was disturbing that she was so suddenly wide-awake, with no clue as to why.
She stood, curious, and walked to French doors that opened out to the wraparound porch, slipped by the hauntingly sheer curtains and out to the balcony, where she held the rail, as she had as a child, and looked over the beauty of the grounds. The moon was a crescent in the sky, and stars sparkled beautifully if opaquely. Rain might be coming, she thought. The ethereal light of the stars and moon—and the large lanterns at the front and rear doors of the grand old house—created a scene of misted and mysterious beauty.
So what had wakened her so swiftly and completely?
"Worry!" she whispered aloud. "'Hmm, Sherlock,' said Watson, 'there shouldn't be any werewolves out tonight. Werewolves need a full moon, so I believe!' Oh, God, I'm talking to myself again!" she moaned.
But—what?
Charles Osgood was still missing.
Jake was coming. That was certainly something that had to be haunting her mind—and maybe the mere thought of Jake, in the flesh, had never really allowed her sleep.
She began to wonder if Charles Osgood was really alive and well, and had returned tonight, wanting to play some kind of demented joke on everyone to prove that he really had been the right choice to play Marshall Donegal. The thought of Charles Osgood running around the property in the perpetuation of some kind of a hoax was irritating—but left her hopeful as well.
She found herself looking out to the graveyard.
She thought she saw a light flickering there.
"Damn it!" she whispered.
From her vantage point on the second-floor porch, she could see the ghostly white tombs and vaults, the weeping angels, mournful cherubs, praying saints and all the exquisite mortuary art to be found here.
She needed to put the nonsense regarding Charles out of her head and start remembering that she ran a business, a bed-and-breakfast and living museum. And, yes, Charles was on her mind, but there were still other problems that could arise.
Ah, yes, Jake always kept her grounded, and he always had that half smile on his face, the charming light in his eyes, and when he was there, she was whole.
Somehow, knowing that Jake was coming gave her that strength!
Brilliant woman! So, push the man away!
Still, just thinking about him...
Something was going on out there. It was her property, and it was going to stop. She was sick of wondering what had happened and what was going on. She was going out to discover just what the hell that flicker of light might be.
Indignant, she turned back and put on the pair of sandals by her bedside, found her white robe and hurried out of her room. She could hear the horses whinnying and neighing, as if something in the night had disturbed them as well. But she didn't head for the stables—they'd had problems before with local teens thinking it would be great to get high and play in the old Donegal graveyard. And although guests were always asked not to tramp around the grounds after eleven, every once in while they had a ghost hunter who just had to be in the cemetery at midnight or beyond.
She crossed the stretch of lawn that led toward the old cast-iron graveyard gate. The gate, of course, meant nothing, since the stone wall surrounding the family "city of the dead" was only four feet tall.
The gate was open.
Kids would be kids. When she had been young, she had held some great slumber parties, and her guests had gone into the family cemetery at night, and they'd told ghost stories with flashlights aimed at their faces. But it was her home; her family graveyard. They had never been destructive.
Perhaps such an old, private cemetery on a property now run as an inn was just too big a temptation for people.
A few years ago, some young people from the local high school had broken one of the cherubs that had graced the walk near the gate. That might have been an accident; they'd caught the culprits, and the boys had said that they'd been terrified—chased by the ghost of a Confederate soldier. Their imaginations at work, Ashley was certain, and she hadn't particularly wanted that group severely punished—she was angrier with the teens who had left beer cans, cigarette butts, the tail ends of joints lying around...and had written a bunch of voodoo symbols on the tombs. Once caught, they, too, had claimed that ghosts had chased them out, but in that instance, Ashley was damned sure that the only ghosts had been the spirits they'd imbibed, and the weed.
She realized how ridiculous she must look, wandering toward a cemetery in a white nightgown and robe; if there were kids there, she'd probably scare them to death herself. The gate was open wide enough to let a body slip through. She did so, careful not to touch it as the old iron creaked.
Even she, who had lived here all her life, imagined that a cherub ever-so-slightly turned its head to watch her walk by.
She paused, listening, and realized that she heard only the rustling of the trees, the grand old oaks that stood sentinel along the walls, shrouded in moss. And yet, there seemed to be soft voices in the night. The sound possibly created by movement of the air, the natural settling of the earth and manmade structures as well. Still, it was almost as if she could hear her name spoken softly, urging her on, calling to her.
But then she heard something that wasn't the whisper of branches moving or the moan of the soft breeze. It was like a thump or a rhythmic tapping sound, and it was coming from just down the path and to the right, from the large and beautiful vault where her ancestors had been laid to rest. She hurried silently along, wanting to catch the prankster red-handed.
"Charles? Charles Osgood? Is that you? Show yourself. The reenactments are not a joke! Don't ruin it all by being a jerk!" Ashley called out.
She turned the corner and stopped dead, a scream rising in her throat. As if on cue, a drifting cloud un-curtained the moon, and the Donegal family vault glowed in opalescent majesty. Mist swirled at the base. An angel rose high atop the chapel-like roof, hands folded, eyes lifted to heaven.
The body of a man dangled from the base of the angel, the straps of his backpack caught upon the marble structure, his feet just brushing the ground. His cavalry hat covered his face, and blood, from a series of wounds to his abdomen and chest, streaked down his torso and limbs and pooled at his feet.
Terror filled her; she stared, blinking. Too afraid to run, too afraid to allow her trapped scream to escape, a confusion of thoughts tearing through her mind.
For a moment, it was as if her mind hit Pause on the horrible image before her.
Her home was haunted. This was the ghost of Marshall Donegal, the valiant man who had died there defending his property in 1861.
If she stepped forward, his head might rise with the hollow, skeletal grin of a man dead more than a century and a half....
She heard the rapping sound again. It was the dead man's sword, rapping, tapping, against the tomb.
And it snapped her out of her paralysis.
At last, she screamed.
This man wasn't a ghost.
He was never going to grin at her, or anyone else.
He was real, and he was certainly dead, murdered and in the cemetery, where she now stood alone with nothing at all to defend herself. She closed her mouth quickly, cutting off the sound of her scream.
She had been right to worry, and to search. She had felt even last night that they had to find Charles Osgood. And now, she had found him.
But the prank had been pulled not by him, but on him.
And it was fresh blood that dripped beneath his dangling feet.
A killer might still be here, watching from the shadows that melded with the mist in the darkness of the graveyard.
Donegal Plantation. Few plantations rivaled it. A haunting opaque white shimmer in the moonlight, the building rose up on the bank in all its majesty. It sat before Jake Mallory as it had all his life; a stunning representation of a bygone era.
Nowadays, the very circumstances that had defeated those who had lived here long ago were the ones that made the area a place of such amazing history and beauty. The war had scarcely begun when the Union might had throttled the city and parish of New Orleans, and, for miles around the city center, the surrender that had seemed like such a tragic disaster had kept enemy forces from laying waste to the magnificent houses that had been built when cotton had been king.
He remembered the first time he had come here; his parents were friends with Ashley's parents. He remembered the first time he saw her, hiding behind her mother's skirts. She had been five; he had been eight.
Compare that to the last time he had seen her. The way the light had gone out in her eyes. She had built a wall around her heart and soul that was as impregnable as brick.
He was still damaged goods himself. He had learned to cope with what he was because of Adam Harrison and the team he had put together in a way he had never managed on his own. Maybe because he had discovered that he wasn't so strange. Still, the images that lived in his mind would always create a divide with Ashley.
There had been good times, though. Their parents had played as a team in pool tournaments. Jake and Ashley had come along, played in the various game rooms offered by different venues, shared sodas and snacks. But more than pool had kept them together as friends when they'd been really young. Once, they'd been part of a garage band together; they'd been pretty good at that, too. And when the three years between them had seemed unbridgeable and they spent most of their time within a year of their own age group, Ashley had come to him upon occasion with her dating dilemmas, or to comment on his dating choices. He smiled as he thought about Ashley and remembered the way her lips would purse when she was trying to tell him something. Somehow, someway, Ashley had retained something of the Southern belle in her behavior; the word bimbo would not cross her lips, nor would she tell him that his latest crush was a slut, a tramp or trailer trash, nor would she use any other such derogatory term. The question was always, "Seriously, Jake, is she what you're really looking for? I'm not certain that her behavior is really...nice. But, hey, you want what you want, right?"
Nighttime here really seemed to be a time warp back to the past. Tonight, the house, seen through the veil of oaks that led to the sweeping entrance, seemed to stand guard upon a hill. A soft breeze caused the branches to sway in the ethereal light, and the path to the house might have led right into a different time and dimension. There was nothing to mar the perfection of the picture; whatever cars might have been there were hidden away in the car park, and the view he saw was one of sheer magnificence.
He drove up the vast and sweeping, oak-lined drive from the road. Once upon a time, the road had been a carriageway, and the rear entrance from the road had not been considered a grand entry at all. The grand entry had faced the river. Some things had changed. The mighty barges bringing cotton downriver were outdated. Still, with the working sugar mill and Beaumont plantations as the nearest neighbors to Donegal, and both a mile away, the view of Donegal, even by night, was spectacular.
It was quiet when he parked; yet just yesterday there had been hundreds—possibly thousands—of people crawling all over the place, from the reenactors to the visitors who flocked here on the day of the actual reenactment. That thought made him smile as well—in comparison to the real battles that had taken place during the war, the skirmish here had been nothing. But Donegal Plantation had always been home to those who knew how to survive. When Marshall Donegal had been killed, Emma Donegal had raised her son and daughters on her own, and she had kept the plantation thriving, even under Union rule. It was sad—and probably not at all fair—that legend had her as the one to slip out into the skirmish and kill her husband. Her motive was supposedly the fact that she didn't agree with his management of the plantation, or with the management of their slaves, several of whom he was supposed to have slept with, along with quadroons at the quadroon balls in New Orleans, and the wives of a few of his best friends. Their daughters, too. But those rumors weren't anything new. People loved to speculate. He knew that neither Ashley nor Frazier believed in the rumors regarding Emma, and he didn't take them very seriously, either.
He parked the car directly before the house and got out. He knew that he wouldn't be out here at all, and the team wouldn't be on call, if Adam Harrison hadn't been old friends with Frazier Donegal.
An inexplicable discomfort settled over him. It was late, of course, and he was miles and miles away from Bourbon Street, where the parties were just hitting their stride. Out here, the world was sleeping.
Still, he hesitated.
Lights in the large old stable building showed him that tourists were still quartered there, and he even saw some light emitting from the smaller stables, still in use, behind the large barn structure.
The house looked ominously quiet.
He walked around the side of the house, not certain why he was experiencing such a distinct impression that something was wrong. And then he knew. As he stood there, he saw a figure in white come tearing out of the graveyard.
For a split second, he was paralyzed. She looked like a phantom, a stunning vision from the past, a gorgeous ghost in a long, flowing white gown, her golden hair caught in the wind.
It wasn't a ghost; it was Ashley.
She looked just as she had looked in his dream: a shimmering figure standing upon a roof with the floodwaters rising. She looked as she had looked, reaching out for him and yet trying to warn him of something horrible and dark that loomed behind him. Her fingers had slipped through his...
He couldn't let that happen now.
He raced across the grounds, hearing earth and gravel crunch beneath his feet. "Ashley!" he called her name.
She stopped; she stared at him with huge blue eyes the size of saucers, like a doe caught in the headlights of a car.
She still saw him as a pariah.
"Ashley," he called again. She screamed and started to run away.
They hadn't parted that badly. She wasn't seeing him, she realized. She was still imagining whatever nightmare had caused her to run.
She turned just as he reached her, and they collided and fell to the ground. She struck out at him from below, and he caught her arms, perplexed and yet aware that she could deliver a solid blow if she chose. She seemed to be fighting for her life.
"Ashley! It's me. Jake. Jake Mallory!"
She went dead still. He realized that she was trembling violently.
"Ashley, it's Jake. Come on, Ashley, whatever else, you've known me all your life! It's Jake. What is it, what are you running from?"
Her trembling subsided.
"I found him," she said. "I found him."
"Found who, Ashley?"
"Charles. Charles Osgood."
Dead. She'd found him dead, of course. No one acted like this unless they had seen something really terrible. Certainly not Ashley Donegal.
"Where?" he asked, easing back.
He wanted to fix things for her. This was Ashley. Certainly, one of the most beautiful women he had ever known and once loved. He wanted to hold her and tell her everything would be all right....
But it wasn't, of course. She had found a dead man.
He rose quickly, taking her hand to bring her to her feet. "Where, Ashley?" he asked again, his tone quiet but authoritative.
She blinked and seemed to gain possession of herself again. "The graveyard. The family vault," she said.
"And he is dead? You're certain?"
"Oh, yes."
He pulled out his cell phone and dialed 911, and carefully gave the address and the situation. Ashley stared at him while he did so. If Jackson Crow was already on the case, then they had federal jurisdiction. But they needed a medical pathologist out here now, and, naturally, they'd have to work closely with the local police.
"Go on inside," he told her. "The police are coming."
She shook her head. "I'm with you. I'm not moving. I mean, I'm not moving if you're not moving."
"Someone needs to tell your grandfather."
"He's smart—he'll figure it out when he hears the sirens."
"When he hears the sirens, he'll be worried about you."
"I'm staying with you!"
He wondered if she was actually so shaken that she was afraid to head for the house herself—afraid, perhaps, of everyone on her property now.
"All right, but we need to keep a distance from the actual...scene," he said.
"Corpse," she said dully.
He walked back to the cemetery. She hadn't released his hand. She wasn't going to.
They had to part momentarily to slip through the gate without opening it further, and Jake was loath to make any changes to the scene. A stone cherub seemed to follow their passage through the rows of vaults, shimmering beneath the moonlight.
He didn't have to ask her to lead him; he knew exactly where to find the Donegal vault.
It was the largest, the most ornate and the most beautiful in the graveyard. When they turned the corner in the center to reach it, he stopped, trying to take in everything that he saw before the local authorities came to assess the situation.
There was the vault. Cherubs and gargoyles guarded the iron-gate doors and the four corners of the tomb. High at the front was a life-size angel, and, caught upon its foundation by the heavy canvas straps of a period backpack, was the body of a man. He hadn't known Charles Osgood, and if he hadn't seen many a portrait of Marshall Donegal, he wouldn't have known that this wasn't a trick of time, that they hadn't gone back approximately one hundred and fifty years to discover a dead cavalry man in the cemetery.
Convenient place to die. Or be murdered.
But despite the blood that dripped from the body and pooled at the feet, he didn't believe that the man had been killed here. He had been brought here soon after death, but he hadn't died here. The body had been put on display. It was evident that whoever had killed the man had done so to be historically accurate—and to make sure that the world knew that a man had been killed just as Marshall Donegal had been killed long ago. Was it an assault on the Donegal family? Or had someone wanted this particular man dead and used the Donegal family history as a means of throwing off suspicion?
"He was so proud to be playing Marshall Donegal!" Ashley whispered.
"Stay here—exactly here," he told her.
He was afraid that she was going to cling to him, but she didn't. With him there, she seemed to be finding her own strength.
"I know. It's a crime scene," she said woodenly.
Jake, watching where he walked, searched the area surrounding the tomb. There was nothing there. The graveled paths around the tombs certainly didn't allow much room for footprints, and he didn't expect to find any. They would have to hope that the forensic team summoned could find fingerprints, hair, fibers, DNA—anything that might tell them who had brought the man to his death, and then here.
They could hear the sirens then, shrieking through the night. And then voices as guests staying in the various rental rooms began to rouse.
"Get to the cemetery gates," Jake told her. "Make sure no one but the police comes through."
She nodded jerkily yet didn't move.
"Ashley!" he said, taking her shoulders. "You don't want guests wandering in here, and your grandfather will be coming out any minute, worried to death, and he is in his eighties!"
She snapped to finally and nodded, spinning about in a whirl of shimmering white. He watched her go, his insides twisting in a knot of pain. She didn't need this; she didn't deserve this. Of course, the dead man hadn't deserved it, either. As he heard the sirens come closer and closer, he pulled out his cell phone and dialed Jackson Crow. It hadn't been so important before that the team arrive quickly; now, it was.
He looked back at the corpse, and time melted away again.
Someone had reenacted murder.
## 5
Ashley stood shivering at the gates of the cemetery, trying to compose herself. She had certainly been in something like shock, but Jake was here, and now she was okay. It was bizarre that she was okay because Jake was here, but that was the way that it was; he was in control, and it brought her back to herself.
She had felt that she'd been losing her mind; the dreams had plagued her mercilessly, and Charles had been gone, and she had longed to see Jake.
And Charles was dead—and Jake was here. Really here.
And she had to quit behaving like a "dumb blonde" screamer out of an old movie. She started to move again, thinking that she had to get to her grandfather.
But she didn't get that far.
The first person to rise and rush out, hearing the wail of the sirens, was Cliff Boudreaux, and he didn't have far to come, racing out of his quarters in a flannel robe. His graying brown hair was mussed and he was barefoot, as if he had been sleeping. She saw that he first looked back to the house, but then saw her and ran to her instead, gripping her shoulders, his eyes filled with worry.
"Ashley? Ashley, why are you standing here like this? What the hell has happened?"
She stared back at him, suddenly more assured, and she was even angry again, furious. Someone had killed Charles Osgood. He could be petulant; he could be whiney; but he was a good man who, to the best of her knowledge, had never hurt anyone.
She felt all sense of trembling and shaking fade away completely. Yes, Jake had done that for her.
"Charles Osgood is dead. I just found him in the cemetery," Ashley said. "The police are on their way."
As she spoke, she saw that people were beginning to emerge from the far stables, where the rental rooms were.
"Cliff, I'm going to get my grandfather. Please make sure no one wanders into the cemetery," she said.
She turned toward the house, noticing Beth and her grandfather had come out to the riverside porch together and looked as confused as anyone else. She broke into a run, crossing the distance from the graveyard to the steps.
"Ashley!"
Frazier reached out, and she ran straight into his arms. "I'm all right, Grampa, I'm all right. But I found Charles Osgood. He's...dead."
Frazier drew away from her, staring into her eyes. Beth let out a soft gasp but said nothing.
Ashley continued, "I thought I heard something in the cemetery."
"You heard something in a cemetery—and you hurried into it to find out what was going on? Lord, girl!" Beth said.
"I've lived my whole life with the family cemetery in full view from my window, Beth," Ashley reminded her. "And Jake's here," she added quickly.
"Jake's here?" Frazier said, and it seemed to make everything better.
"Yes, yes, Jake's here," she said, nodding. It might be best to let them think that Jake had been in the cemetery with her from the beginning.
Two police cars pulled into the front. The driver of the first seemed to hesitate a minute, but then he pulled straight on down by the side of the house, slowly passing the onlookers who had gathered outside. The second car came to a halt by the first. Ashley quickly ran down the steps from the porch to meet the officer who exited the first car. It was Drew Montague, who had been on call when she had reported Charles Osgood as missing.
"Well, Ashley, what's going on?" Drew Montague asked her. Behind him a uniformed man got out of the car.
"I found Charles Osgood. He's in the cemetery. Someone bayoneted him and hung him on the family tomb," she said. She spoke to Drew but kept glancing at the other man who had approached them.
"I'm Detective Mack Colby, Miss Donegal, with the parish sheriff's office," he explained. He was so pleasantly nondescript, she wondered whether that was part of his act. "Can you take me to the body and explain, please, how you happened to discover it?"
"I woke up after I'd gone to bed. I thought I saw lights out there, and I went to investigate," she said.
"You ran into a cemetery in the middle of the night when you thought that someone might be out there?" Mack asked politely. He and Drew exchanged a glance. There was suspicion in his tone, despite the even level.
She let out an exasperated sigh. "I have lived here forever. My dead ancestors are in that cemetery. I'm not afraid of it!" she said. "The worst we've ever found before has been potheads and frat boys. I am not afraid of my own property," she said indignantly.
"Looks like you should be," Drew Montague murmured.
"Montague," Mack Colby said, "can you keep people away from the gates? The forensic crew will be here shortly. Miss Donegal, please take me to the body. Did you touch him? Are you quite certain he's dead?"
"He's dead. And, no, I didn't touch him," Ashley said. By then, her grandfather was by her side.
"Perhaps," he said icily, "it would be best if you investigated the dead man without giving my granddaughter difficulty?"
"If she's right, this is a murder investigation," Colby said, his eyes narrowing. "And you are—?"
"Frazier Donegal. We've not met, but you've surely known that this property was here and who owns it. My granddaughter insisted we call this man's disappearance in last night, afraid that something bad had happened. None of you seemed interested at the time."
More sirens blared in the night; a rescue vehicle came to a halt behind the police cars. Augie Merton, a medical pathologist from the coroner's office, emerged from the passenger's seat. He was a nice man; Ashley knew him. He sometimes came out to do lectures on Civil War medicine. Though the former New Yorker had lived in the parish for almost thirty years, he was still affectionately called the Yankee doc.
"Ashley, Frazier, sorry to see you here under unhappy circumstances," he said, coming forward with his black bag.
"Damn it, let's get to the corpse," Mack Colby said. "Lead the way. With any luck, no one has disturbed the crime scene."
"No one has. Jake Mallory is in the cemetery, watching over the scene," Ashley said.
Mack Colby stopped walking. "And who the hell is Jake Mallory?"
"An old friend," Ashley said.
"A good old boy. Great!" Mack Colby muttered.
"He's with the federal government," Frazier in formed him.
"Feds have to be asked in. He'd best not be fiddling around in my jurisdiction!" Colby said.
"Frankly, I don't think he and his team fiddle with cases. I think they solve them," Ashley said, staring at him. Of course, she didn't really know much of anything about Jake's team, but this man was truly patronizing, and she was feeling just as indignant as Frazier.
Before he could respond, she said, "This way."
"Ashley, they can surely find the body on their own," Frazier said, worried about her and apparently not at all fond of Detective Mack Colby.
"I'm fine," she assured him. She mentally drew herself up, though it was difficult to do so with dignity when she was running around in a white nightgown.
She turned quickly, assuming that the men would follow. They did. It was surprising that Beth and Frazier chose to follow as well; she was certain that corpses did not fall into Beth's usual life. But she didn't protest; Frazier was proud and would insist on seeing what happened on his property. And there was no stopping Beth when she made up her mind.
When they reached the gate, Mack Colby said, "Stop! Who has touched this gate?" he asked.
Ashley turned to stare at him. "Possibly? Hundreds of people. Maybe thousands. There was a reenactment here yesterday. It was after the reenactment that Charles Osgood disappeared—something that we reported to the police."
"How long had he been missing when you called in the disappearance?" Colby asked.
"A few hours," Ashley said.
"You called in about a missing adult after just a few hours?" Colby asked, his voice level, and yet there was something suspicious in his tone.
"He had very badly wanted to play my ancestor, which he did," Ashley explained. "He should have been around to celebrate with the others afterwards."
Augie let out a sound of impatience. "Where is my body, please?"
Mack Colby lifted a hand, put on a latex glove with a snap, and pushed the gate open to a wider degree. Ashley slipped through, followed by her strange posse: Mack Colby and Augie, her grandfather and Beth.
Jake Mallory waited at the end of the path, before the turn to the Donegal family vault. Jake had always had a certain presence. His arms were crossed over his chest; he stood with his feet planted slightly apart and appeared formidable and authoritative as he stood there. Part of it was his height. He wasn't particularly heavily built, but his muscles were toned, his stance was straight, and, when he moved, it was with a swift agility one might not expect in a man so tall. He wasn't easily ruffled, and his temper seldom stood in the way of his intentions.
"So you're the fed, huh? Did you touch anything?" Mack Colby demanded. "And what the hell kind of federal officer are you?"
Jake remained calm as he reached into the pocket of his jeans for a slim leather wallet, which he opened and presented to Mack Colby. "Agent Jake Mallory," he said. Colby frowned, stepping forward to examine the credentials Jake had offered. His frown didn't disappear as he stepped back.
"How did you happen to be in the area?" he demanded. "And you do understand the concept of local jurisdiction? You have to be invited down if we have a problem, and I don't think that we'll have a problem here. We're capable."
"I'm sure you are capable. I'm a friend of the family. I happened to be on my way to the house. My boss is a friend of the Donegal family as well, and Frazier Donegal called him when Ashley was first worried about the disappearance of one of their reenactors. If you'll check with your superiors, we have been asked to join in the investigation. Of course, we were looking for a missing man before. Now, we're looking for a killer," Jake said evenly.
Colby wasn't satisfied; his gaze remained fixed on Jake.
Augie cleared his throat. "May I get to the body, please?"
"A minute, Augie," Colby said. "They found a corpse—a man obviously not in need of an ambulance. I want the crime-scene people in here—I want pictures of the body in situ. I want every fiber, hair, fingerprint. And I want all the rest of you people out!"
"Detective, I'd like to stay," Jake said.
Mack Colby grunted. "Let me tell you—this parish has amazing forensic facilities. And we're not a bunch of local yokels just because we're in bayou country. You like to come down here from the big cities and—"
"I'm from Louisiana," Jake interrupted. "I was born and raised in Orleans Parish."
Mack Colby paused at that. He lifted his hands. "Fine. You stay." He turned around and looked at Ashley, Frazier and Beth. "All right. The rest of you—out!"
Ashley looked at Jake. He gave her a small, reassuring smile. Despite the fact that she was standing in her family graveyard with a dead man not far away in the middle of a bizarre night, she did feel reassured. In fact, she wanted to run to him. The breeze lifted her hair and touched her face, and she kept eye contact with him. Jake Mallory had always been steady and reassuring—when they were kids, when he teased her, when he taught her how to hold a cue stick, when he played his guitar and patiently went through a melody or a beat over and over again.
When he made love to her....
She had still thought that it would be awkward to see him again. They had been so close for so many years, friends and then lovers, and she had shut him out as cleanly as if she'd shut a door in his face.
Nothing like a dead man to ease the transition into seeing one another again, she thought dryly.
The thought brought a rumble of something that threatened to be hysterical laughter from her throat, and she swallowed it down quickly.
"Out," Colby repeated. "Good God, it's a crime scene!"
She nodded, turned and said to Beth and her grandfather, "Shall we?"
"This is my property," Frazier said to Mack Colby.
"And I am a law-abiding citizen, a veteran of foreign wars, and, Detective Colby, I will be kept informed of what has happened and is happening on my property. I asked Agent Mallory and his team down—he is here on my request."
Frazier had said his piece. He turned to Ashley and nodded.
As they departed, a trio dressed in the parish's crime-unit jumpsuits paused for a moment to ask the way to the scene. Ashley indicated the path through the vaults with their decaying elegance and hurried on out.
More officers were on crowd control; two in uniform, flanked by Drew Montague.
"Someone want to talk to that group?" Drew asked them.
"I've got it, Grampa," Ashley said, hurrying forward.
One woman was weeping. Ashley quickly made her way through the officers and cars with their bright lights and reached the group of guests hovering by the old stables.
"As you know, we've just discovered a friend, dead, in the cemetery." She winced. Her words sounded like an oxymoron, though they were not. "We'll get you checked out quickly, and please, be assured, no one will be paying for the night."
She had to lift a hand against the bright car beams that were now on her. "Please come through the front door of the main house, and we'll be sure that you're completely cleared of all charges."
"I just want to go back to sleep!" one man called out.
She looked back at Drew Montague. He shrugged. "I guess it's all right. We had a body in a hotel parking lot once, and they didn't evacuate the hotel."
"All right. Anyone who wants to go back to sleep is welcome to do so," she said, hoping that was the right thing for an innkeeper to say under the circumstances. She didn't know anything more about crime and murderers than what she had learned on television and the news, but it seemed that someone had killed Charles Osgood and displayed his body in a certain way for a reason. The scenario didn't appear to offer danger to her guests for the rest of that night, especially since she was pretty sure the place would be crawling with police and crime-scene investigators until daylight and possibly beyond.
"Guess we'll be safe enough tonight, with the police prowling around everywhere," a woman said as if following Ashley's own train of thought.
"What the hell happened?" someone else demanded.
"We don't know anything right now," Ashley said. "The police are here. I'm sure one of the most important things is that no one goes near the cemetery until the scene is cleared by the police. And, please, of course, be very careful."
"Oh! He was murdered, he was murdered!" Another woman cried out. She was about fifty, in a house robe, and wearing curlers. "Oh, oh! We've got to get out of here, we've got to get out of here!" she cried, running forward and then running back.
"Calm down, Martha!" a man said firmly, stepping forward to grab her arm. "We have nothing to do with any of this. I'm going back to sleep. We'll check out in the morning."
"Please, all of you, I'll be at the desk in front. Stay the night, or pack up and leave. Whichever you prefer," Ashley said.
She noticed that Justin had appeared; he had come out of the stables alone, and she assumed that he had left Nancy with the children. He moved through the crowd and reached her side. "Charles?" he asked softly.
She nodded grimly.
"In the cemetery?"
"Yes."
"We searched there."
"I know. I was in there myself," she said dully. "I have to get in the house and start handling this situation. You have the children—I assume you want to get them out of here, and don't worry, we—"
"We're all right," he said quietly. "Don't worry about us. You've got enough on your hands right now."
She smiled and raised her voice. "Anyone who—"
"Not so fast!"
She turned around to see that Mack Colby was striding toward her. He gazed at her impatiently and addressed the crowd. "I'm sorry, folks. I'll need a few minutes with each of you before you pack up and leave. It can be tonight, or into the morning hours, but I'll need to question you all."
"About what?" Martha's husband demanded indignantly. "We had nothing to do with this!"
"You're here, and a man was murdered here. He took part in the reenactment, he disappeared and now he has reappeared—dead. You all were here. This is simple, people. Someone killed him, and you're all suspects until you're cleared. I'll need to question every single one of you!"
"Oh, my God!" Martha shouted. And then she dropped to the ground in a dead faint.
Jake called Jackson, sorry to wake him up, but knowing that Jackson needed to be advised immediately about the situation and the arrival of Mack Colby on the scene.
"All right. Tread carefully," Jackson said. "I'll call Adam right now and have him get hold of his congressional friends and make sure they speak with the local officials again. They weren't interested before—they'd already given us jurisdiction on the case. I doubt if there will be any trouble; Frazier Donegal is a force in this area, so it seems, and his contacts are endless. Do your best to get along with the local police. I'll pack up with Angela, and we'll be out right away. I'll have the others follow as soon as they've gotten their equipment together."
"Thanks," Jake said.
He hung up; the pathologist from the coroner's office had arrived. Jake continued to watch as the forensics team took pictures and combed the area. He remained a bit surprised that Mack Colby hadn't forced him out when he had finished listening to the pathologist regarding the corpse.
Of course, there wouldn't be a final determination until the body had been taken to the coroner's office for a full autopsy. But according to Augie in his preliminary findings, Charles Osgood had probably been rendered unconscious by a blow to the head—there was a pre-mortem bruise appearing on his forehead—sometime yesterday; death had not occurred until two or three hours ago, when he had, quite simply, bled to death. He had received five stab wounds to the abdomen area, apparently after his inert body had been hung from the angel, the wounds obviously caused by a sharp instrument. His blood had slowly oozed from his body, creating the puddle on the ground.
With any luck, the blow to his head had been severe enough to have kept him unconscious until death had come, though, since it had been more than twenty-four hours since he had gone missing, Augie suspected they would find drugs of some kind in his system. It seemed that the killer had been intent on keeping his victim quiet until the act of murder had been completed, but, at the least, he had saved Charles Osgood from the agony and fear that surely would have accompanied his death had he been conscious.
Jake waited while Augie's assistants brought the body down, carefully preserving the backpack and the straps that had kept him dangling from the tomb's majestic angel. Meanwhile the team searched the ground for possible footprints and used specialized lights to seek fingerprints on the tomb wall itself. There were hundreds of fingerprints, so it seemed, and the hundreds were atop more hundreds. In a small way, the Donegal cemetery was a tourist attraction in itself. The gates were usually locked, but the wall was really no obstacle, and someone might have forgotten to lock up again.
Jake wished them well. It was going to be a difficult and long haul, trying to sort evidence. Unless something could be found on the body or the backpack itself, any fingerprints or evidence could have belonged to any of the visitors or tenants who had traversed the cemetery.
He'd been to the reenactment at the plantation several times throughout the years. He closed his eyes for a moment, imagining it. The real battle had ended in the cemetery, oddly and sadly for Marshall Donegal, in front of his family crypt—he had bled out there, just like the murder victim, Charles Osgood. Jake found it curious that the body had been left so...displayed. And that the murderer had waited until now, more than twenty-four hours after the disappearance of the victim. If someone's intent had been just to ease Charles Osgood from his mortal life, that someone could have far more easily managed to stab him in the midst of the crowd that flocked around the reenactment. If the murderer had left him just lying there, the slim chance that it had been accidental—boys being boys and playing with real sharp weapons—would have existed. Someone, obviously, would have still been guilty of manslaughter, but the display meant murder for certain—and that the murderer wanted it known.
The body was wrapped, ready to be transported to the coroner's office.
"Dr. Merton," Jake said, addressing the pathologist. "Anything else you can tell me?"
"Augie, just Augie," the medical examiner replied. "Not yet—I have to cut him open. But I do believe that toxicology reports will prove that my assumptions are correct." He was quiet for a moment, shaking his head. "Seems like Miss Donegal was concerned from the start. If someone had paid more attention to her, he might have been found and saved."
"It's my understanding that they did search for the fellow. My bosses were called in because the Donegals were so concerned," Jake said.
"They didn't search hard enough, did they?" Augie asked. He looked around. "Must have been some feat—this man was no Tinkerbell. A hefty fellow. He was brought here before he was killed. That's evident by the blood patterns, I'd say, even though I'm not a blood-spatter specialist. Then again, I am an M.D., with a specialization in medical pathology, and I don't think anyone needed my expertise to see that the man was dead. Well, young man, if you need anything from me, you call me. Don't worry about blunderbuss Detective Colby. He's not a bad chap. We don't have this kind of thing happen often out here. Come to think of it, I've never seen anything like this—anywhere. But he's a decent fellow, just trying to play alpha dog right off."
"I'm sure we'll all be fine," Jake told him. He was done here himself; the crime-scene unit was still searching, dusting, taping and hoping for the smallest clue. There was nothing else he could do at the scene for the moment.
He followed Augie and the body out of the cemetery.
Returning to the lawn area between the house and the outbuilding, Jake saw that while the police were holding a line with their vehicles and a number of officers, the guests who had been staying at Donegal Plantation were now gathered up by the cottage, all speaking at once. Mack Colby was lifting his hands and trying to maintain some sense of order.
Ashley was still there, still the historical damsel in distress in her white gown. She knelt next to a woman who was sitting on the ground, head between her knees, and he could see the way that Ashley's jaw was hardening. Mack Colby was really beginning to anger her. It was a good thing the detective hadn't chosen to be a doctor, because his bedside manner would have killed many a patient before curing them.
Jake walked quickly through the crowd to reach her side, hunkering down by her.
"I need to get Martha into the house!" Ashley said irritably.
"That's fine," Colby said. "That's fine, but I repeat—no one, do you understand me? No one leaves. So settle in, folks, and if you're in a hurry to get out of here, try to be first in line for the questioning."
The elderly woman paled, and Jake stepped in hastily to curb what Ashley might say.
"May I pick you up?" Jake asked Martha. "With your permission, I can carry you in and set you on one of the sofas."
Ashley flashed him a glance of gratitude.
Martha placed a hand on his cheek. "Oh, yes, young man, please. My legs are feeling very wobbly."
"Thank you," said an older man next to them, obviously Martha's husband. "I don't think I'm quite up to lifting these days."
"Herbert, I am not that heavy!" Martha protested.
"It's not that you're heavy, my dear," Herbert said, "it's that I'm old."
Martha waved a hand in the air. Jake put his arms around her and lifted her, and Ashley led the way into the house.
"I'll need a room where I can be alone with each individual," Colby said, elbowing his way past everyone as he entered the parlor.
Tense and rigid, her lips pursed, Ashley directed him to a study on the bayou side of the house. Jake laid Martha down on the Duncan Phyfe sofa near the double stairway where her husband joined her, and followed Ashley and Colby.
The study was a pleasant room with a mahogany desk, computer, printer, and shelves lined with books and family pictures. It was a spacious room; two chairs sat in front of the desk, and a wingback chair faced the bayou-side windows. Mack Colby had sat himself behind the desk.
"I don't want to create any problems here," Jake said, his voice firm. "And if you question these people and have the courtesy to keep proper records, I believe everything will be in order. As I said before, the federal government was called in when this was a missing persons case. Since the victim was apparently kidnapped, the federal government has jurisdiction. But I suggest that we handle it as a joint investigation. It's a truly sad, horrible and bizarre situation, and I would think that all possible means of law enforcement would be indicated."
Colby stared at him as if he would implode. His face was mottled and almost as red as the pool of blood in the cemetery.
"Your behavior is outrageous!" Colby told him.
"No, sir. I suggest you call your superiors, at your leisure, of course. There's no reason that this can't be a combined effort, which is always best. There's nothing in the world like cooperation between law-enforcement agencies. You'll be so much more knowledgeable than we could possibly be on so many fronts."
Mack Colby kept looking at him as if he would finally pop, but he seemed to know that Jake was telling the truth. He leveled a finger at him.
"There's something fishy here. These folks called in the feds when a man had only been missing a few hours. A grown man. Someone knew something had happened to him, and if you're not going to get at the truth, I'll be making a stink they hear up in Washington and beyond!"
"Oh, good God!" Ashley, who had been standing quietly near Jake, exploded. "I'm the one who raised the alarm, and I raised it because I know—knew—Charles Osgood! He would have been here celebrating. He wasn't. I knew him, don't you understand?"
Jake set an arm on Ashley's shoulders. "Really, I think Detective Colby realizes that now—he is just doing his job. But we're all good now." He turned back to Colby. "Look, Detective, my team's expertise is in understanding why people behave the way that they do. And Ashley's intuitions assist us. Yes, we need to question everyone, but, because of the display of the body, it's evident that this wasn't simply an act of passion, a mistake or accident in the reenactment, or the casual act of a thief or drifter. This was personal, or, possibly, ritualistic. If you want to start with the guests who are down here for the first time, that's great. They can be cleared quickly, and we can begin to look at the people involved with Donegal, the reenactors and the locals."
Colby seemed somewhat mollified, but his facial muscles were still taut. He nodded jerkily to Jake. "Fine. You're sitting in? Or are you going to do the questioning?"
"I'll sit in and watch. Thanks. I think you'll ask the right questions, and I can judge the responses. Ashley? Would you like to start bringing people in? First, guests—"
"Yes, guests who have never been here before. I've got it," she said.
He lowered his head, smiling. Ashley—tough Ashley—was back.
He settled into a chair. It was going to be a long night.
Martha could stand and walk and move, so it seemed most reasonable to let Martha, and then Herbert, go first. Ashley realized that she was trying to bring people in and out from a police questioning room as if she were still a hostess and it was a social situation, and she felt a little foolish at first but then decided it was the best way to keep everyone calm.
Beth pitched right in, brewing coffee and producing a nice array of finger foods.
Jake emerged from the room at one point, asking her to check her registration books and make sure that all guests still at the plantation were present, and also to give him a list of those who had left already.
She nodded, glad of anything to do that kept her moving and busy.
She realized that no one had really shed a tear yet for Charles Osgood. She felt like crying over his life then. He hadn't been handsome; he hadn't been popular. He had still been a decent fellow—always wanting to be handsome and popular. And now he was gone. And the question remained, of course: Had he been killed because he'd been Charles Osgood...or because he'd been playing the part of Marshall Donegal?
Finally, Colby had interviewed all of their casual guests, moved on to repeat guests and was ready to start on those who were close to the reenactment, the plantation or the family. Beth was surprised when she was called in, but she shrugged and went all the same. Justin followed next, and Ashley was close enough to the door to hear one of his answers to Colby.
"Oh, yeah. Of course, I brought my children along while I planned and plotted a bizarre murder. I've been hiding Charles under the kids' beds for the last night. Right, yes, of course, question away."
She grinned before moving on. She heard Jake patiently explain that they were hoping to find out if he'd seen anything, noticed anything or could give them any possible information.
Cliff went in after him, and while Cliff was being interviewed, she was startled to see that two new comers—people she'd never seen before—were in the parlor, chatting with Frazier, who was still up, still making sure that he went the distance with his guests.
She hurried over to meet the couple. The man was tall, taller even, perhaps, than Jake. He obviously had Native American blood in his heritage somewhere. The woman was a pretty blonde, who almost appeared fragile.
"Ashley, Jackson Crow and Angela Hawkins," Frazier said.
She shook hands with both of them. "You're with Adam Harrison's team," she said. It wasn't a question.
"Yes, and we're so sorry that your missing person has been discovered dead," Angela told her.
Jackson nodded. "Will you bring me, please, to Jake and the local officer who is doing the questioning?"
"Absolutely."
"Beth will bring Angela up, assign them and Jake rooms," Frazier told her.
"This way." And she took him in.
Jackson Crow had a low, level voice, rich with authority. The door to the study had quickly shut behind him, but she had to smile, hearing the tone of his voice, through the wood paneling. He and Jake seemed to have the ability to be completely even-keeled—and yet say exactly what they meant in a way that brooked no interference.
She started to walk away, but the door opened and Jake came out.
"You and your household are to go to bed and get some sleep," he told her.
"Oh, I don't think—"
"We're almost done here for the night. Jackson is taking over," Jake told her. "I want to get some sleep. You must need some, too. How about it? Where am I sleeping?"
She wanted to ask, Could you sleep in the chair in my room?
"I'm sure Grampa would have told Beth to put you in the Jeb Stuart room," she told him. "Do you want to get your things?"
He waved a hand in the air. "Right now, I want to crash. If I remember right, there's soap, shampoo, razors, toothpaste, you name it, in the rooms, right?"
She nodded.
"Then I'll run down in the morning. Come on, I'll walk you to your room."
Ah, yes, Jake could be the Southern gentleman. There was no "home" to walk her to now, so he'd walk her to her room.
"Hey, I live here," she reminded him.
"And I want to see you in. And lock your door."
"Oh, come on, Jake! I am not afraid of my grandfather or Beth—"
"Someone managed to get an unconscious, living man into the graveyard and to kill him there. Ashley, lock your door."
She nodded. They went through the living room, where Jake assured Beth and Frazier that they were free to get some rest; Jackson would deal with Mack Colby and arrangements for the continuing investigation. They'd see that Cliff got back to his place, that officers remained on the property until midmorning and that everything was locked up and safe.
Frazier kissed Ashley's forehead; Beth gave her a hug. She and Jake followed them up the stairs.
The Jeb Stuart room was next to Ashley's at the back of the house, so he didn't have to go far.
At her door, he said, "Good-night, and scream blue blazes if you need anything."
"Thanks, Jake."
He hesitated a minute. Jake had amazing sea-green eyes. They changed like the sea as well, but they were striking against his tanned face and auburn hair. She lowered her head suddenly, wondering why she had needed so desperately to step away from him.
Because her father had been dead. Dead. And Jake seemed to have spoken to him. Strange and scary—but, somehow believable. So her reaction had just been...fear.
Maybe even fear that her dreams should be believed as well.
"Hey, are you all right? Are you okay with me being here?" he asked her, lifting her chin and searching out her eyes. "The team is excellent. Angela and Jackson are amazing."
"I'm fine. I'm glad you're here. I mean, you know the plantation as no other investigator could possibly know it, and you know many of the people involved."
"That's true. I just want you to be all right...with me being here."
"I'm fine." She winced inwardly. "Jake, actually, I'm sorry. I know I overreacted, but..."
"Your father was dead," he said flatly. "That was the past. It's fine. I understand. Okay, as I said, I'm right next door. Just whistle—you know the old line!"
He waited for her to go into her room and lock the door; then she heard him enter his own room next door.
Ashley washed her face, brushed her teeth and realized she was still in her nightgown, but it was filthy, so she showered and changed. It was almost morning—no matter. She lay down and prayed for sleep.
It came.
The first pale rays of morning light seeping through the drapes woke her.
She frowned, still groggy. Was there someone at the foot of her bed?
Jake?
No, not Jake. It was a man in a Confederate uniform. He wore a sweeping, plumed hat. She knew who he was—her ancestor, Marshall Donegal.
She blinked; he would disappear, she knew.
He didn't.
She opened her mouth to scream, and he leapt to his feet.
"By sweet Jesus, did I breed a line of whimpering cowards? Ashley Donegal, pull yourself together! I'm here to help you."
Interlude
The television stations had gotten hold of the information.
He was stunned; the body shouldn't have been found until morning. There should have been time for Charles to...ripen a bit.
But alone with his screen in front of him and dawn just breaking, he could see the reporter by the side of the road; a police car was blocking entry to the estate, but there was Donegal Plantation, as grand as ever, surviving time and death and change.
He didn't quite feel the satisfaction he should have from the kill.
Of course, it wasn't that he wanted to torture poor old Charles. He wanted the Donegal clan to suffer. It might have taken more than a hundred and fifty years for them to pay the piper, but they would be the ones to suffer. The sins of the fathers had to be paid.
The news crews couldn't get onto the property, so they were padding the broadcast with pictures. First, old Frazier. He could almost hear the old man's voice, rich but low, rippling along in that light accent like a roll of the Mississippi.
Then Ashley. The beautiful blonde, the belle, the last of the Donegal clan.
## 6
Back at Donegal. Jake couldn't settle in. He'd stripped down to his briefs but now lay staring at the double doors to the wraparound porch, the ceiling and around the room.
Back at Donegal.
Alone, he could remember why he and Ashley had parted. He would never forget the look in her eyes, the last time he had seen her. The look in her eyes...the way she had backed away from him.
They'd been friends forever. When they'd been young, the three-years difference in their ages had been gargantuan; as they had grown older, the annoying little girl had become inquisitive and fun. And he had loved to tease her. They'd argued incessantly; they'd done their best to beat each other at every game, to outrace each other on Donegal horses, and they'd laughed when they'd unseated one another.
Then they had grown older still.
And he had fallen in love. Maybe he'd been falling in love all his life, and he had just been waiting for her to catch up.
They'd flirted, they'd played, they'd kissed—and when he'd been twenty-two and she had been nineteen, the flirting and the stolen adolescent kisses had become much more. He'd never forget the night. He'd been due to leave for his last semester at Carnegie Mellon, and everyone had come in from the countryside to celebrate his last night home. They couldn't all crowd down to the bars on Bourbon or Frenchman streets, because several in the group were still underage, so they had rented out one of the old historic inns on St. Anne's. They had partied by the brick fire and then, sometime in the wee hours, he'd walked her to her room and gone to his...but seconds later, he'd heard a knock on his door, and Ashley had been there with this look in her eyes. She had asked, "Must you be such a gentleman? After all, you're heading back to college, and I'm off to school in Florida, and shouldn't we have a few memories?"
There had been nothing awkward about the night. Memory, of course, could alter and be selective, but he could still see the way she had looked that night, the brilliance in her eyes, the silky shimmer of her hair in the pale light and shadows. Clothing had melted away, and there had never been such a rush as just feeling her flesh against his. He hadn't wanted to leave after that night, but she had told him, "We've been best friends for years—you have always been a part of my life. You have a semester left, and I have faith and trust. A little thing like distance can't tear us apart."
Distance hadn't ripped them apart. Death had done so.
For him, it had been the odd beginning of another part of his life. For her, it had been the end. At a time when he should have been able to comfort her the most, he had become anathema.
But now, he was back at Donegal.
And a man had been murdered.
He stood up and got dressed again; he wasn't going to sleep.
Jake left his room, pausing to listen at Ashley's door, but all was silent. He flushed, glad that she didn't suddenly swing the door open and see him standing there.
Downstairs in the darkened dining room, he heard voices. Looking out, he saw that the police were still there—at least, the patrol cars.
A drone of voices from the study alerted him to the fact that Jackson was still in with Mack Colby, and maybe Cliff. He didn't know.
He frowned; the commotion from outside had grown louder. Curious, he walked out the roadside door and looked down the avenue of oaks.
The police were blocking the entrance to the plantation, but he could see that several news crews were out on the road. Crunching down the drive, he reached the officers. Drew Montague was standing in front of his police car, arms crossed over his chest, a look of pure annoyance on his face.
Montague saw him. "I don't know how the word got out so fast. They're like flies on a corpse. If you'll excuse the expression."
"Has anyone spoken to them?"
He shook his head. "I told them that it was a crime scene, and that they couldn't come on the property. That's all. Luckily, it is private property, so it makes it easier to keep them away."
Jake leaned on the police car next to Drew Montague, trying to listen. There were three reporters with their camera crews situated so as to pick up the plantation house in the background of their shots. He recognized the local network-affiliate anchorwoman, Marty Dean—he'd actually gone to high school with her—but the other two reporters were men he'd never noticed on the news before.
Perhaps they thought this story would be picked up by a national network. He was sure that the information that a man had been murdered on the property was out—they were living in the era of cell phones, texts and instant communication, and Donegal Plantation housed many guests.
He could hear Marty clearly.
"Donegal Plantation, historically a place of tragedy and loss, and filled with strange and eerie happenings throughout the years. Have the ghosts of Donegal arisen? Unconfirmed reports state that the body of a man in a Confederate uniform was found in the family cemetery on the estate. But other deaths have occurred at Donegal as well. Some are documented, and some are rumor, such as the hanging of a house slave after the murder of the master's wife during the first half of the 1800s. The Civil War–era master of the estate, Marshall Donegal, a brilliant tactician who might have served the Confederacy well, died within that cemetery. Perhaps he is still waging war against his enemies!"
That was too much.
Jake pushed away from the car and approached Marty. She saw him; her eyes widened, she smiled with pleasure.
"I see a Donegal guest now," she said into the microphone, nodding at her cameraman.
Jake felt the camera come his way. It didn't disturb him or stop him.
"Jake Mallory, one of our local heroes, seems to have been staying at Donegal Plantation. Mr. Mallory, can you tell us what has happened here? Some speculate that the ghosts are murdering people!"
"The police will give the media everything when they have something to say, Miss Dean. I'd just like to point out that Donegal Plantation is far more than a place of tragedy and loss. I think it's rather foolish for anyone to imply that ghosts might be running around murdering people. A man is dead, and first and foremost, his death is a sad occasion. I'm sure that everyone involved with responsible media will see to it that our sorrow over his death is respected and that an historic residence and business which has offered education and entertainment to visitors for decades should not be maligned in any way. Thank you, Miss Dean—I know that you will report responsibly."
He turned and walked away.
"Jake—wait!" Marty called after him.
He ignored her. The other two newsmen had seen him, and he walked quickly by Drew Montague. Montague grinned, liking what he had said. As Marty chased Jake, Montague stopped her.
"Crime scene, ma'am. I'm still not cleared to let you in."
"But you just—"
"Mr. Mallory is an invited guest at the plantation, ma'am."
With a smile, Jake kept walking. He didn't turn back.
There was just no way out of it.
Ashley felt the scream escaping through her lips, though it was more like a gasp or choke than a scream.
The ghost swore beneath his breath and faded into nothing, and she was left staring at an empty room, wondering if she could wake herself up. But she wasn't sleeping. She was wide-awake—and seeing things.
She leapt up and ran around, turning on every light in the room. It wasn't all that necessary—it was going to be light outside soon. But she didn't want the shadows that were created when the sun first began to rise; she wanted light, brilliant light, and a lot of it.
But she froze when she heard a light tap at her door and then a voice.
"Ashley?"
It was Jake. And she suddenly felt that dreaming about him had caused her to have dreams or nightmares about a body in the graveyard—before it had been there—and Confederate soldiers who somehow got into her room and faded away as swiftly as she could blink. She was overtired, she knew.
She was losing all grip on reality.
She walked to her door and threw it open, staring at him. "Yes?"
"I was just making sure you were all right," he said.
As at all times, he was so damned easy and confident. And he couldn't have heard her—the scream she emitted hadn't even been a squeak when finally uttered.
How the hell did he just know things?
"I'm fine, just fine." Was Jake's presence here making her think that she had seen a ghost?
He believed they existed, even if he hadn't said as much. And his special team seemed to have some kind of insight that others didn't have—surely that was why they were so special.
She was glad to have him here; she would have willed him here, if she could have done so.
But now she was frightened again.
"I know you think you...see things, know things, that others don't. But please don't suggest that the ghost of Charles Osgood is telling you things to tell me, all right? If you're an investigator, investigate. Real things. Blood. Fingerprints."
He stared back at her easily, with absolutely no show of emotion.
"I heard you walking around the room. I just wanted to make sure you were all right. I'm fairly new in my actual position, but Jackson has been with the federal government for years. He's excellent at following blood or DNA trails and fingerprints. Good night, again, Ashley. I'm sorry I disturbed you."
He headed down the hall. She watched him, her stomach knotting, her heart sinking. Well, she had to be looking like a schizophrenic now, welcoming one minute and greeting him like a shrew the next.
Because, once again, she was afraid. She was afraid that she could see things that others didn't sometimes, and that was truly terrifying.
This time, she couldn't shut herself away; she had to be reasonable, and she even had to learn to accept what Jake said. And what she saw.
A man had been murdered.
She needed sleep. Ashley decided to leave the lights on. It was nearly six now, she saw by the bedside clock. Daylight would come quickly, but until then, she would be glad of the lights.
And the television! A television would distract her. But when she turned on the television, she saw Jake. They were repeating a newscast.
She started to change the channel, but she paused, listening to him, his strong and authoritative manner—and the way he pegged the pretty anchorwoman. She had to smile.
She changed the channel. It seemed that half a dozen channels bought footage from Marty Dean's newscast. There was Jake, once again.
She hit the remote.
And again.
He couldn't possibly be on Nick at Nite! She hit the changer until she came across Dora the Explorer, and at last, satisfied, and hoping that maybe she'd even learn some Spanish through her subconscious mind, she eased her head down on her pillow.
And slept soundly and without dreams intervening.
Jake took time to speak with Jackson and Angela, choosing the study for the privacy it offered. He briefed them on the events that had occurred before their arrival, and Jackson told him about the last of the interviews.
"It's absolutely amazing that no one saw anything," Jackson said.
Jake shook his head. "No, not really. I mean, obviously, I wasn't here for this reenactment, but I've been here before when they've gone on. There's so much confusion. There's black powder in the air everywhere. When the fighting is over, everyone is paying attention to the riverside porch where Ashley and her grandfather are speaking, finishing up the event. It's a patriotic moment—everyone sings 'The Battle Hymn of the Republic.'"
"Did you know Charles Osgood?" Jackson asked him.
"I met him a few times years ago. He was part of the outfit, but his stepfather was alive back then, and so Charles wasn't asked to take part in the battle. There are only six Confederate roles to be played, and there is a strict pecking order to who gets to do what when."
"We need that pecking order written down," Jackson said.
"Here's the strange thing, from what I've understood so far. Charles shouldn't have been playing Marshall Donegal. The role should have gone to Ramsay Clayton, but Charles was apparently causing a stink about having to play a Yankee—they were short a Yankee—and Ramsay decided to let Charles have the honor and play a Yankee himself."
Jake realized that they were both staring at him. He sighed. "Slavery was obviously wrong, but for some reason, it's more romantic to be a rebel now. Especially if you are from the South. Don't look at me like that."
Angela chuckled. "Hey, I'm from Virginia. I've seen plenty a Civil War roundtable."
"Me, too," Jackson said.
"Then why are you staring at me like that?" Jake asked.
"I was staring at you because it seems that Ramsay Clayton is the first man we have to investigate," Jackson told him. He cleared his throat. "Get anything you can on the man off the computer. See if he made any waves anywhere—angered anyone." Jake nodded.
"But first let's head down to the local police station. I want to see that our use of their forensics department is going to be respected."
Jake nodded again. He didn't really want to leave the house, but he usually accompanied Jackson on their police liaison.
"By the way, nice handling of the media," Jackson said.
"Oh? I thought I walked onto a live broadcast?" Jake said.
Jackson grinned. "Apparently, it was bought by several stations. Anyway, you handled the anchorwoman well."
"I knew her."
"Great. I'd bet big-time that she'll be traipsing around here a lot. You can take the press on this one, too."
"Sure. It's hardly my expertise—"
"No, you just didn't know it was your expertise," Jackson told him. "I'll meet you in front in five minutes," he added, rising to leave the room.
Angela was still there. She looked like an angelic piece of fluff, but she could handle a Glock as if she'd been a shooting champ for a hundred years.
She set her hand on his. "I'll be here," she told him.
He grinned. "The place is riddled with ghosts, isn't it?"
"Probably," she said.
"Have you met any yet?"
"I haven't tried. But I promise I'll be getting right on that. And," she added, a curve to her lips and a light in her eyes, "you know me—I usually need a little time and quiet. God knows why—most ghosts are shy of disbelievers. You'd think it would be the other way around."
"You'd think Charles Osgood's spirit would be around here somewhere," he said.
"You never know who lingers and who moves right on," Angela said. "Remember, death doesn't make the soul all-seeing. Sometimes, ghosts don't know what's happened—we all know that."
"Great," Jake said. "Death is as confusing as life."
"Don't worry today," Angela said. "I'll keep my eye on your Miss Donegal—and her grandfather, of course."
"Thanks," Jake told her.
"Want to tell me about it?" Angela asked him.
He shrugged. "We were close, intimately close. Her father died, but I knew when he first went into emergency, and I shouldn't have. And I related a dream I'd just had about him, in which he said how much he loved her and that he was all right—and two seconds later the nurse walked in to say that he was dead. In that moment, I became a pariah."
Angela nodded sympathetically. "That's why we learn to keep our own council. But you're okay, right?"
"Yes, I swear it. Don't worry about me. I'm working, and my emotions won't sway me in any way," he assured her.
"Our emotions always sway us," Angela told him. "Just so long as they sway us in the right direction, we're fine."
He left her, ready to head to the front of the house. But he heard noise in the dining room and stepped in. Ashley was there, pouring herself coffee from the samovar on the buffet.
"Ashley, can you get me a list of the Yankees and the rebels who took part in the reenactment? I'm sure the police asked you for your rosters, but would you write up the names—and what they do and how long they've been involved with the plantation?"
She nodded. "Of course."
That morning she was in jeans and a T-shirt. She still appeared as gloriously beautiful as she had in flowing white. She had that same air of dignity that sat so well on Frazier.
And elegance. Even in jeans.
And she seemed to have forgotten her earlier tirade.
"Of course," she repeated. She looked away for a moment and then back to him. "Sorry about earlier. I was really tired."
"Don't worry. It meant nothing."
She looked down. "Of course not," she murmured. But she looked up again, frowning. "Are you leaving?"
"For a few hours—just down to the station. Angela will be here. And there are still two patrol officers getting your guests out and stopping others from coming in."
"Jake, I really can't believe that one of our reenactors could have done this. I've known most of these guys since I was a kid."
"Then you need to think hard about anyone who might have had a grudge against Charles—or Ramsay Clayton."
"No one had a grudge against Charles. They felt sorry for him all the time, if anything. And I really can't imagine anyone having a grudge against Ramsay. He's a pleasant person, not much of a temper—actually, a nice man. He had no problem with letting Charles take his place."
"He wouldn't—if he knew something was going to happen to the actor playing Marshall Donegal," Jake said.
She stiffened at that. "It was a last-minute change," she told him. "Why couldn't this have been a random killing?"
He paused, thinking that was obvious—except that Ashley very stubbornly didn't want to believe that anyone with whom she'd been friends could possibly have plotted out the brutal killing.
"First, Ashley, simple logic," he said. "You have to know this area to have kidnapped a man and kept him hostage—even drugged—for that long a time. You'd have to know Donegal Plantation well to know the cemetery, how to reach the Donegal vault easily and to escape unseen."
"We're open to the public—we're a bed-and-breakfast. And the history of the place is written up in a number of books."
"Ashley," he said seriously, "a murder like that isn't a sudden act. It was preplanned, and preplanned carefully. Is it possible that a stranger came on a tour and devised a way to find notoriety? Yes. But it's most likely, considering human nature and behavior, that someone close to Donegal Plantation committed this crime. I'm sure that law enforcement will look at all angles, but we—the team—specialize in behavior—" he broke off; he didn't want to tell Ashley bluntly that they would also be seeking those who weren't still living for help "—and even the events that occurred in the past that cause someone to act a particular way in the present, and so, we'll put our focus on those who are close to the family and Donegal Plantation. I'm sorry, but I honestly believe you're going to have to accept the fact that someone you know is a murderer."
"You could be wrong," she said.
He had to grin ruefully at that. "Damn, you're still stubborn as hell. Think about it again, about everything I said. A random act of violence wouldn't explain someone holding a man drugged and hostage and then killing him with a bayonet—as your ancestor was killed."
"But you could be wrong," she insisted.
He didn't answer. "I'll be back soon," he told her.
Jackson was waiting for him in the hall. He drove, and as he headed out, he looked back in the rearview mirror as Donegal Plantation became smaller and smaller, and disappeared in the trees.
He didn't want to leave. Not while someone was still out there.
"This," Beth commented, "is sad!"
She and Ashley were at the dining-room table. All of the guests were now gone, including Justin, who had taken his family into New Orleans. At this point, it was definitely going to be better to think about his children enjoying the zoo and the aquarium than hanging around Donegal Plantation.
Ashley looked at Beth, frowning, "Well, of course, it's sad. A man is dead."
Beth shook her head. "No, this—the two of us sitting here, drooping on our elbows, getting nothing done. That's sad!"
Ashley sat back. "They asked me for a list—I've done the list. I've checked on my grandfather—he's actually sleeping. There are two guys in uniform hanging around outside, and I'm not sure what else to do."
"Well, I'm going to cook." Beth stood up. "And I suggest that you go befriend the blonde cop who is wandering all over the house."
"Angela. Angela Hawkins?" Ashley murmured.
"That would be the name of the blonde cop or fed or whatever we have walking around," Beth told her. "Go on. I'm going to occupy myself. Alone. Go find the investigator. Maybe you can help her."
Ashley rose. "All right," she said.
She left Beth and looked through the rooms on the ground floor. Upstairs, she saw the door to Angela and Jackson's room was open; Angela was inside, sitting on the bed and staring into space.
Ashley approached the door. "Hello?" she said.
The woman started and looked at her, and then smiled. "Hello. This is a beautiful place. Absolutely beautiful."
"Thank you."
"I'm sure it's full of legends," Angela said.
"You bet."
"Want to share a few?"
"Oh, Lord, well...there are supposedly several soldiers wandering around. We had an actor die of a heart attack, in the 1940s, I believe, in the barn. Members of the Donegal family have died of old age. So, let's see, we're supposedly haunted by Unionists, rebels, actors—oh, a World War I cavalryman who died in France but made his way back here. The house was built in the early eighteen hundreds, so we've decades of ghosts running around," Ashley said lightly. "Then, of course, we have the rumored ghosts who really can't possibly exist—not here. Every plantation is supposed to have the beautiful slave girl who poisoned the mistress and was then hanged from one of the oaks. But it didn't happen here. Not that way, anyway. I've had guests, however, who swear they've seen her."
"Well, sometimes people see what they want to see, don't they?" Angela asked her.
"And sometimes, what they don't want to," Ashley replied.
"Really?"
"Finding Charles," Ashley said, looking away. It wasn't what she had meant at all. She had just met this woman.
And maybe it wasn't such a good idea to remain with her.
"Well," she said awkwardly, "thank you for being here."
"It's what we do," Angela said.
Ashley inclined her head. "Thank you anyway. I'll let you get back to—to whatever you're doing. Excuse me."
"Ashley!" Angela called when she would have walked away.
Ashley paused.
"I'm not sure what Jake told you, but we were banded together because we learned how to blend our intuitions with logic, and even to analyze our dreams. Don't let any of that scare you. If you let yourself turn fear into careful thought, you'll discover just how much you may know yourself."
"I don't know who killed Charles, and I don't have a logical—or even illogical—thought on what might have happened. I wish I did," Ashley said. She gave Angela a forced smile.
"I'm here if you need me," Angela said.
"Thanks," Ashley told her and waved, walking away. She wasn't sure where to go, and she found herself returning to her own room. She lay down on the bed, remembering that she'd slept no more than two or three hours. If she tried closing her eyes, maybe she would sleep.
What now?
The world was strange; they were waiting. Waiting for what, she wasn't certain. The police seldom captured a murderer immediately, and they were in a very bad position here, since they depended on tourism and their guests to keep the place afloat.
"Ashley."
The whisper of her name brushed her ears, and she didn't trust the sound was real, or if it was all in her mind.
She kept her eyes closed.
"What?" she demanded crossly.
"You hear me. I know you hear me. It's so difficult to find someone who actually does."
"You're the illusion of a tormented mind," Ashley said.
She swallowed hard as something settled at the foot of her bed.
Don't do it, don't do it—don't open your eyes! she commanded herself.
"Look, I'm trying to help you. I swear. You're my great-great-great—I don't know how many greats—granddaughter, and I'd never hurt you, not for the world, but I'm afraid for you."
"If you're afraid for me, go away. You'll give me a heart attack. Or the police will have to collect me and put me in a straitjacket if you don't," Ashley said, still keeping her eyes tightly closed.
"Donegal women are not cowards!" he said.
"I'm not a coward. I'm trying to stay sane!"
"Open your eyes, young woman!"
She did so. She blinked. She wanted to scream, but, of course, she couldn't. Her throat was locked.
And he looked so damned comfortable. He sat on the edge of the bed, a handsome man. His hair was a darker blond than hers, long and actually curling around his neck. A large-brimmed, plumed hat sat atop his head, and he had brilliant blue Donegal eyes. His uniform appeared to be new, and in pristine shape, and bore Louisiana militia insignia. A scabbard held his sword in place around his hip, even as he sat, the sheathed sword at an angle.
He stared at her.
She stared at him.
Eventually, the lump went down in her throat. She wondered if she was dreaming again. She didn't think that she was. She could feel the pillows beneath her head, and the fabric of the sheets she lay upon. Daylight was streaming through from the balcony. She could see the sky beyond. She was awake, and she pinched herself to prove it, feeling a bit ridiculous as she did so.
"You really see me!" he whispered with pleasure.
"You're in my mind," she told him.
"Maybe, but you see me." She hated to deny a man such evident pleasure at so simple a thing.
"You're a ghost, haunting Donegal Plantation," she said flatly. She groaned. "No, haunting me. Why me?"
"Because it's in you—it's always been in you to reach me. And I've tried to reach you forever."
She didn't know that you could startle a ghost, but apparently you could, because he jumped when she suddenly sat up. She stood then and walked around, turning away and turning back.
He didn't disappear.
"You've been in my dreams," she whispered.
"Easiest contact," he told her.
She shook her head and then approached him, angry. "Then—you knew. You knew someone was going to kill Charles. Who is it? Damn it, tell me, tell me what's going on, and we can solve this. What did you see? What do you know?"
He rose to meet her. He was really quite the swash-buckling figure, and she could see where he had been an impressive man—until he'd gotten himself killed.
"Nothing," he said.
"What?"
"Nothing. I know nothing. Look, a ghost can't be everywhere at the same time. I've had this odd feeling for a long time that something was going to happen. Something bad. I've tried to reach you, to make your see me and be careful."
"You had a feeling?" she demanded incredulously. "You're a ghost!"
"I'm the ghost of the man I was, and the essence of what I was is still what I am," he said flatly.
She blinked, trying to make sense of what he was saying. "Explain that!"
"What remains is the soul," he said. "And the essence of our being. The body is fragile—it ages and it dies. But the soul, the energy of what we are, is what we always were." He searched out her eyes and tried again. "We learn through the ages, through everyone we watch. I've seen generations come and go, and I've shed tears that no one sees through many a tragedy. I touched your father once, but he does not remain—he went with your mother when the call came."
Her parents had gone on. She missed them, so much, still. The beginnings of tears tightened her throat.
She steeled herself, wondering if she could possibly be speaking to the ghost of a long-dead ancestor, or if the events had just become too much for her.
"You watched all that—but you didn't see who killed Charles?" she asked.
"No! I was with you and Frazier on the porch. I love the reenactments. Now, that is. But the first few years of my—ghosthood?—I was a bitter fool. I really was. I owned slaves, yes—it was a way of life, and we didn't really know any better. Really. I was bitter when we lost the war. I was bitter when the carpetbaggers came down. Then I began to learn. I began to watch the world as the years went by. I could see that I'd been—not a fool—but utterly ignorant to many truths in life. Now I've watched many young people go off to war, and I've seen the fallacy of our ways, and I'm glad we lost the war—we never should have fought it."
He was really there. Or, in her mind, he was real, and he was just like a reasoning, functioning human being.
Except that he was dead.
She shook her head again and realized she was doing so enough to give herself brain damage. She forced herself to stop.
"This is great," she whispered. "I'm seeing a ghost—and you can't even help me."
"I can help you," Marshall Donegal said.
"How?"
"I can do my best to watch the behavior of those around us now even more closely, and I can do my best to stay near you and look after you. If you know I'm here and accept that I can be here, and you'll let me in, you'll hear me now when I know that there is danger."
"How did you know there was danger?"
"I could feel it," he said again, exasperated. "I tried to warn your great-grandfather when the old fellow was about to keel over in the barn. No one saw me—no one came to help when I screamed. Now that poor old bastard is spending his afterlife pacing out there in the stables, and he won't move on. I talk to him, and he doesn't hear me. He just waits for his opportunity to act out a long-gone and lamented war, and there's nothing I can do."
"You can't speak to other ghosts?" she asked him.
"Sometimes. Sometimes it's as if we never quite touch," he said sadly.
"It's not just one big old ghostly community out there?" she demanded.
"Some don't accept that they're dead," he told her.
"And others?"
"Others know, and they're not sure why they linger. Some just can't go on. And sometimes we can reach one another."
"What about Emma?" she asked him.
He turned away from her. "No," he said, his back to her. "Perhaps she has gone on. Perhaps we are being punished. But, no, I can't see or reach Emma."
"Punished for what?"
He turned back to her, waving a hand in the air. "That's not important. What's important is that you have a killer on your hands. A cruel killer, one who mocked the way I died. I died for a cause that might have had serious flaws, but I believed that I was fighting for my family and my state. Why should we be mocked so?"
"We?"
"The Donegal clan."
"You think someone killed Charles—to hurt the family?"
"The days of cotton and sugar being a means to all ends has long passed—the plantation survives on the guests who come. Donegal does well because there are always visitors. If the place becomes known as the site of a heinous recent murder, the people will not come."
"Oh, well, that may not be true," Ashley said dryly. "It may attract more."
"It will hurt, I believe."
She pointed at him. "You're not really a ghost, and I'm not really seeing you. You are a creation of my subconscious, and you're logic—telling me that I can figure this out if I think hard enough."
"I'm afraid I'm not going away."
"I'll just ignore you."
"Then you'll be behaving in a most foolish manner. Think about the day, about the event. Think about the men involved, Ashley. Your friend is right, and you know it. Someone close to Donegal Plantation committed that murder."
## 7
"No hairs, no fibers, nothing on the bastard but the wool of his uniform and fluff from a pair of cavalry gloves," Colby said, disgusted. "And, of course, there are so many fingerprints on the tomb, we can't begin to sort them. Same thing with tire tracks—there are none close to the cemetery wall, and there are thousands in the gravel and the road out front. We have begun to sort them out, but it will take a great deal of time, and when we have them, what will we have?" The man was clearly frustrated. "There was absolutely no sign of a struggle. There's no sign that Charles Osgood was dragged to the tomb. Science isn't going to point us straight to the murderer on this one, and we need warrants to just go digging into the personal effects of the hundreds of people who might have been around. He was killed with a bayonet, so getting a judge to move on collecting the weaponry used at the reenactment has been a piece of cake, but once we go beyond that...well, it will be hard to pinpoint the fellow—not unless he strikes again, or gets careless."
"Let's hope he doesn't strike again," Jake said.
"Well, of course. It doesn't look like a serial killing, does it? The way I see it, someone had a grudge on Charles Osgood and found a way to really drag out his death."
"We're grateful, Detective Mack, that your officers began collecting all of the rifles and bayonets used by the men in the reenactment, and that you've been so gracious to share information" Jackson said. "Profiling isn't an exact science. We all know that. And we don't believe we're dealing with a serial killer, either, but still, finding someone who holds a grudge—even finding out if there was a perceived slight to someone—can often help point law enforcement in the right direction."
Detective Colby nodded. "Fine. We'll track down people. We can narrow the field, but it will still be a big field." He was quiet a minute. "I heard you all did a damned good job in New Orleans. Some of the information seems to be a bit vague. Do you people mind-read, or something like that?"
"We explore history—history in time, and history as it pertains to individuals," Jake said smoothly. He had a feeling that with Colby, they'd be thrown out if they were to mention the fact that they sought out ghosts.
"Detective," Jackson said, "we're truly grateful that you're allowing us to work with you. I believe that the cemetery was thoroughly searched when Charles first went missing. When his body was discovered, he was still wearing his uniform, so he was held from the time he disappeared until he was found. He was apparently held in a drugged condition, and perhaps the murderer was able to go about his customary life while stashing his victim somewhere. But if we have alibis for the time before the body was found—we know he had only been dead for a few hours, tops—then we can eliminate those people and concentrate on the lies someone might be telling, or on alibis that might have a few holes in them."
"You know there were hundreds of people on that plantation for the reenactment," Colby said wearily. "We're on it, but the manpower needed for that kind of investigation is great." He was quiet for a minute and said grudgingly, "This is a sensational case. We're, uh, grateful that you're here, too."
"Thank you," Jackson said.
"It's a needle in a haystack," Colby said.
"What about the bayonets of the men who were already gone that day?" Jake asked.
"If the men were gone with those bayonets, they couldn't have killed him with them," Mack Colby said.
"If they were really gone, which, of course, your men will find out," Jake said. "I don't think that any of the men who were playing Yankees—who had left the property already—are guilty, but I believe that two of them are local, Southerners who had ancestors who did choose the Northern side. Once we eliminate—" Jake began.
"We'll get every weapon. Every blade," Colby said grimly. "They'll be wiped clean, of course. But sometimes, no matter how you wipe down a blade, the forensics folks can still find a miniscule dot of blood. I'm doubting it with this guy, though. This damned thing was planned out."
"Yes, it was," Jackson said.
Mack Colby seemed pleased Jackson agreed with him.
"How much of the property was searched by your men?" Jake asked.
"We had an entire team in the cemetery," the detective said. "And, of course, I had officers comb the area around the cemetery as well."
"I think we need to extend that search—take it to a daylight level," Jake said. "If I had committed such a murder, I wouldn't have kept the murder weapon on my person anywhere. If you're found with the murder weapon, it's most likely you're the killer. He's gotten rid of that bayonet—and I believe it might well still be on or near the property. He might have been on the property after the murder, or nearby, and I just don't think he'd risk being found with the weapon."
"Hell, we don't even know how the bastard got his victim there, with no one at all seeing him. The plantation was still crowded—lots of folks staying over," Mack said.
"The river," Jake said, imagining the scene in his mind's eye. "He might have come by the river. The cemetery abuts it. Easy enough to take a rowboat, tie her up, drag your victim in and disappear the same way—you'd never have to go by the house or the outbuildings," Jake said.
"I'll have a team back out there by this afternoon," the detective assured them. "I'll muster up our best techs to go over the property again, and call in some divers. It will take me a few hours, of course."
"I'll do the dive myself," Jake told Jackson.
"Surfer boy, that's a hard current—you'd better be a damned good diver," Colby told him.
Jake held his temper and smiled. He didn't look like a "surfer boy." He had just stepped on Mack Colby's toes the first night of the crime by being there when the body was found. Colby had accepted them; he even seemed to like Jackson. But Jake had been there when the body had first been discovered, and Colby seemed to have a bit of grudge because of it.
"I used to scrape barnacles off shrimp boats, and I've fixed a few motors in the Gulf and the Mississippi. I'll be careful," he said pleasantly.
He thought that Colby would sniff out his disdain, but he didn't.
Adam Harrison's reach was long. The investigation was theirs, not that they had any problem working closely with the local police. Mack Colby just had a chip on his shoulder, even though he was trying to pretend it wasn't there.
"And, of course, we are talking muddy water and hard currents. I'll sure be grateful for the help of your police divers," Jake said.
At his side, Jackson grinned and lowered his head to hide it.
Colby was mollified.
Jackson and Jake left the police station. "You really know that muddy water so well?"
Jake laughed. "Yeah, I actually do. But I have to admit, I'd be in the damned muck anyway even if I didn't. There's just something about that damned detective. And I won't go alone. I know that Cliff Boudreaux is a diver. Cliff has been on the plantation forever. His dad was a manager and tour director here, too."
"Cliff Boudreaux took part in the reenactment and has lived at the plantation forever," Jackson said, looking over at him.
"Right—that's what I said."
"And that makes him a suspect," Jackson reminded him.
"An unlikely suspect," Jake argued. "Cliff has been an open book. Two of his ancestors were Donegals."
"That could make him a prime suspect," Jackson said.
"I knew him when I was a kid," Jake told Jackson.
"He's still a suspect. I want to believe we can communicate with the dead at times," Jackson said gruffly. "I don't want any of us joining them. I won't let you go alone."
Jake didn't argue.
When they reached the house, Jackson motioned for Jake to follow him. They went up to the bedroom Jackson and Angela were sharing to find her at the desk lost in thought.
"Everything is all right here, right?" Jackson asked.
Angela looked up in surprise at his entry.
"Any sense of...anyone who might be able to help us?" Jackson asked.
She quirked a brow. Jackson knew that the world wasn't always what it seemed, but he still had trouble just asking her if she might have met a ghost who could flat out tell them the truth about the situation.
"Don't you dare laugh at me—I'm letting the house get to know me. And, Jackson Crow, you know as well as I do that we're unlikely to come across an entity who just happened to see the whole thing. If the ghosts haunting the cemetery are active, they probably get the hell away when a reenactment takes place because people are everywhere, and if they're just reliving the fight—well, then they don't see anything but what was. Some of those entities have been around forever, but still can't quite reach out and touch anyone, much less a newcomer to the property. Give me time." She looked at Jake. "And give Jake time. He knows the place. And those who might still haunt this house and these grounds know that Jake is familiar here. And Ashley."
"Ashley?" Jake said, frowning.
Angela nodded, looking at him. "I think that Ashley has a sense that there is more—and I think it terrifies her. Maybe she'll come around. You can't force ghosts—and you can't force anyone to admit that they might see ghosts."
"The murderer was alive," Jackson said. "So let's concentrate on behavioral analysis of the living for the moment."
"It's possible that someone wanted Charles dead—and killed him here," Angela said.
"Too pat, too hard and too complicated. If someone just wanted Charles dead, there were easier ways to kill him."
"A narcissist," Jackson said. "He's sure of himself. He believes in his intelligence and his ability to carry out the plot."
"So we're looking for someone who isn't stupid," Ashley said.
"The police will be working hard on the masses and their alibis. I think we have to look at the probable first. So we'll go with the fact that we believe that it's someone close to the family. We all thought that from the beginning. Initial instinct is a good place to start. This is someone who, I believe, is a functioning psychopath. He's living with the belief that he's been harmed in some way by the people here, or even by the plantation itself."
"I agree," Jake said.
"Even then, we have to start narrowing down the suspect pool," Angela reminded them.
"I'd say we can get it down to a handful soon enough. Prove where people were physically, and we'll find out answers. Or, at least, get the number down to where we can apply some pressure and perhaps cause this particular person to break. I don't think that he's well. This is the kind of murder perpetrated by a person with some kind of mistaken belief in the righteousness of what he's doing. We just have to figure out who is really a concerned friend—and who is wearing a mask of friendship. I believe that he'll eventually break."
"I'm just afraid of what might happen if we don't find him quickly. He may start to spin out of control before he breaks, and that could be really dangerous," Angela said.
"That could be fatal," Jake said. "We need to take care—great care."
Jackson nodded. "Go work with Ashley now," he said.
Jake went back downstairs and found Ashley in the study. He sat down in a chair across the desk from her and surveyed her. She handed him a sheet of paper. It was filled with the names of those who had played rebels and those who had played Yankees. There was a little paragraph about each man's family, employment and character that followed. He smiled, looking down at the sheet.
The first name on it was Cliff Boudreaux. And Ashley had typed, You've known him almost as long as I have. Cliff, competent, good-looking, strong, self-assured. Tour guide, jack-of-all-trades. We all know he has family blood and that he loves the family.
Mentally, he added Jackson's note: because of that very family association, he may have underlying feelings of resentment, as in, he has as much right to the property as Frazier and Ashley.
Following Cliff's name was Charles Osgood. Not a suicide. Underlined three times. An accountant, always an in-between man, not bad-looking, not a charmer. Thrilled to play Marshall Donegal; change happened at the last minute.
Charles was followed by Ramsay Clayton and then by Hank Trebly, which made sense. Hank was involved with the sugar mill on the cemetery side of the property.
Hank Trebly, Ashley wrote, reminds me just a bit of a hobbit. A little short, a little squat—I know you know Hank. Fortysomething, balding a bit, always chewing his lower lip, concerned with politics, and the environmentalists coming after the sugar mills. I hear he's a good guy, though, insisting that corners never be cut, and that they follow regulations to a T.
Jake looked up again, smiling as he caught Ashley studying him with serious eyes.
"What?" she asked.
"Nothing. This is perfect, actually."
"You sure?"
"Jackson's specialty is behavioral science. This is exactly what he'll need. He hasn't met these people, and your information is the kind of thing that a behavioral scientist works with. Perfect," he told her.
She nodded, but her gaze shifted toward the door. He looked around. There was no one there. Was she praying someone would come get them both out of here?
"You okay?" he asked her.
"Fine," she said, not taking her eyes from his again.
He turned again, feeling as if someone were behind him; no matter where she was looking, it was as if she had seen something.
But there was no one there.
He looked into her eyes questioningly, but she had her hands folded on the desk, and she maintained eye contact. He looked back to the list.
Griffin Grant, affable fellow, think you've met him, though his uncle used to do the reenactments. Adores the place and the playing—he's a CEO, VP (?) at a cable company out of New Orleans. Early thirties, good-looking, sharp and well-dressed, nice sense of humor, especially considering the fact that he's a total business geek.
Toby Keaton, owns Beaumont, but you know that. Medium height, medium weight, early forties, thinning hair. Our families have always gotten along well—starting from the beginning of the "survival by tourism" days. We do Civil War and reenactments; he works on Creole history, the real day-to-day work involved in such a plantation. He's always been part of the reenactment.
John Ashton, nice guy, his father did the reenacting in the old days. He's in his late twenties, bookish, glasses, even has special wire frames just so that they work for the reenactments. He runs a tour company in New Orleans, and has long been a good friend of the plantation.
Jake looked up at Ashley again, seeing her and imagining the reenactments as he had seen them so many times before. He knew the positions the men would take—he could run it in his own mind easily. "So, Charles winds up playing Marshall Donegal. The rebel troops are complete with Cliff, Griffin, Hank, John and Toby. Ramsay goes off to be a Yankee."
"Yes, Ramsay went off to join the Yankees, and that group included two locals, men you know as well—" Ashley reached over to tap the paper "—Michael Bonaventure, from New Orleans, bar owner, has a place off Royal Street, and Hadley Mason, an engineer from Lafayette. Justin Binder is from Philadelphia, and he was here with his mother-in-law and two children. He's a widower. The other two Yankees were Tom Dixon, from New York City, and Victor Quibbly, from Chicago, and they both left the morning after the reenactment."
"They flew out from New Orleans?" Jake asked.
Ashley swung around in the chair, hitting the on button on the computer that sat on a stand next to the desk. She nodded. "We know when everyone is coming in and out from different cities, because we try to arrange rides. Yes, Tom left on American Airlines at noon the following day, and Victor was on Continental fifteen minutes later. Cliff drove them to the airport, I believe."
"Can you think of anyone else who is closely involved with this property or with the reenactment?" he asked her.
"Dr. Ben Austin—he's a practicing M.D.—and John Martin, our biggest sutler, or vendor. He was here with his wife, and they were at the party—you know, the wind-down in the house. Every one of those folks was there—except for Charles, of course," she said.
He nodded. "Change places?" he asked her.
"What?"
"May I get on the computer for a minute?"
She stood up, walking around the desk. As she did so, she looked at the door again, frowning. He followed her line of vision but saw nothing.
Jake sat at the computer and started punching in keys. He could access sites that the average person couldn't because he had the proper codes.
"What are you doing?" Ashley asked him. She hadn't taken his chair; she stood at the edge of the desk.
"Simple elimination," he said. "Two Yankees in the clear—they indeed flew away. Their names are on the manifests for the flights."
"Wouldn't the police be checking on that kind of thing, too?" she asked.
He nodded. "Yes, but in my mind, the more people I know to be eliminated entirely on my own, the easier it will be to home in on what really happened. And Jackson is a stickler. He's a team man—it's the way he's always worked. People make mistakes. We can make mistakes. Anyway, I know we're down to a few-score people."
"A few score," she repeated, wincing.
"Don't worry. That number will go down quickly," he assured her.
Once again, she wasn't looking at him. She was looking behind him. He turned quickly, wondering if he didn't glimpse a shadow...something. Ashley was definitely acting strangely.
Donegal was known for being haunted. Maybe that was why she had fought all her life against the possibility that ghosts could be real.
And maybe, she was just beginning to feel or see something....
There was a tap at the door. Jake was surprised by the way Ashley seemed to all but jump out of her skin.
Frazier poked his head in. "Lunch is served. An excellent meal, it appears."
Beth had cooked—and cooked, and cooked. She had gone for just about every staple known to Southern Louisiana—corn bread, jambalaya, crawfish étouffée, gumbo, turnip greens, pecan pie, bread pudding, shrimp salad and more.
It astounded even Ashley that she could have prepared such a feast so quickly, but then, when the reenactment wasn't taking place and they weren't investigating a murder, Beth did run one of the finest restaurants in the area. There was still a crowd for lunch; Jackson, Ashley, Jake, Frazier, Cliff, Beth and herself.
She noted—as she was sure Cliff did—that their guest investigators did not treat him as a suspect; they treated him as one of the family, which, of course, he had always been. Growing up, he'd been the big brother Ashley had never had, even though he was about thirteen years older than her and had been actually managing much of the plantation while she'd still been playing with her dolls and video games.
At the luncheon table, she wasn't being haunted by an annoying Confederate in full dress uniform. He wasn't in the dining room. Not at the moment, and Ashley was grateful for that fact. He'd been in the office with her when she'd been giving Jake the list, and he'd been terribly annoying, wanting her to punctuate every detail regarding every man. She kept thinking that Jake would turn around and see him standing there, laugh and tell her that the fellow was an actor hired to torment her.
But Jake didn't see the man—so she was the scary one after all, suffering from strange delusions about the dead. They were all probably brought on by the murder.
During the massive meal, they all spoke as casually as possible in the aftermath of a brutal, senseless killing. Jackson and Jake relayed the conversation they'd had down at the police station until Cliff had left them, saying that he had work at the stables.
Ashley pretended to listen attentively while wondering again if she had imagined that a ghost—looking as real as flesh and blood—had carried on a meeting with her. She looked here and there around the room, wondering if Marshall Donegal would appear in the flesh—or the appearance of flesh!—sweeping off his great plumed hat and setting a booted foot upon a chair, perhaps.
But though he had been a pest in the office, he didn't show. She was so busy worrying that he would, however, that she barely heard what was being said. She wondered if Frazier had ever seen the man—or even Cliff. After all, one way or another, they were all related.
Then one word that Jake uttered brought her to.
...diving...
"Diving?" she asked.
"I believe that the murderer might have thrown his weapon into the Mississippi," Jake explained. "He's organized, and intelligent. Such a killer would know that the murder weapon would be searched for immediately, and that he couldn't be found with it on his person or his property. So if it were me, I'd throw it in the river as quickly as possible. Actually, I think the killer had Charles with him, maybe drove him away after the reenactment and then brought him back here in some kind of a boat. That being the case, he'd have thrown the weapon into the river while he returned to wherever he had come from by boat."
"Unless, of course," Ashley said, staring back at Jake as if she dared him to agree, "the murderer held Charles drugged on the property. If that was the case, the killer could have taken him into the cemetery, where he bayoneted him to death, and went on to return to his room. The river has a terrific current, too."
"That's possible, too," Jake said evenly. "But I think he threw it in the river—the weight of a weapon could have easily caused it to sink."
"I'm not a suspect, am I?" Beth asked.
Ashley straightened, looking around the table at the three investigators.
Jake smiled and answered. "No. It's highly unlikely that you have the strength needed to carry out what was done."
"Thank the Lord!" Beth said.
"But Cliff could be guilty," Frazier said.
"We certainly hope not," Jackson said.
"Wait!" Ashley protested. She didn't believe that Cliff could be guilty, but she didn't believe that any of the men who had acted like children on the day of the reenactment could possibly be guilty of such a heinous crime. "Who's going diving? Aren't they sending out police divers?"
"Yes," Jake said, frowning slightly. "I'm assuming that at this point they'll be along really soon. But I want first crack, before the water is churned by a team of four or five."
"But—are you authorized?"
"We are working co-jurisdiction," Jackson said, glancing at Frazier. "Adam's your grandfather's friend, and Adam has the influence to make a great deal happen."
"Ashley, you know that I know what I'm doing," Jake told her. "I'm going to get started now."
"I'll work with you," she said.
"Ashley—" Jake began.
"No one should dive alone," she reminded him primly. "The water is brown—even with lights, vision is limited," she said. "You need a dive buddy. And it's my property."
"Ashley," Frazier said, "my dearest grandchild, my old heart is still ticking. It's still my property. You two children can fish through the regulators, tanks and masks we keep because of work that has to sometimes be done down by the bayou."
Frazier had spoken lightly, wanting to ease the tension with smiles. He managed the feat.
"Grampa!" Ashley protested.
"Well, don't look at me!" Beth said. "Dive in that nasty old muddy water? No, no, dishes look much, much better than diving in the Mississippi!"
"I was planning on working with Jake, too," Jackson said.
"That's fine. But I'm going," Ashley said firmly.
"All right. Let's get on it," Jake said.
Half an hour later, the divers were nearly ready. Ashley had opted for a dive suit—she didn't like everything in the Mississippi touching her bare skin. Jake and Jackson had eschewed the idea of suits and were just in swim trunks, booties, gloves and their masks and regulators.
Angela, Beth and Frazier had come down to the embankment near the cemetery while they checked and rechecked their equipment and the flashlights they'd be working with.
Angela had watched Jake walk over and over the embankment near the cemetery wall. He found a spot that seemed to satisfy him.
"Here," he said, looking at them all.
Jackson, apparently, knew what he was talking about. He came over and hunkered down next to Jake, inspecting the ground. He stood after a moment. "Hard to tell, but possible. We'll go in here. Time for tanks, children," he said.
The three assisted each other, buckling into the heavy dive tanks. "You're just walking in, right?" Angela asked. "Seriously, shouldn't we be waiting for the police? They'll have metal detectors—"
Jake lifted a rod he had on a cord at his wrist. "Jackson has one, too," he told her.
"It is one big damned river," Frazier said. "And then there's the bayou—"
"I don't think so, sir," Jake said. "This is how he managed the movement of the body. This is where he'll have ditched the weapon."
Frazier nodded. He gazed at Ashley, and she knew that he was worried about her. It was only fair; she was worried about him. She blew him a kiss.
"I'll follow the current and watch for you down by the public ramp," Angela reminded them. "Don't try to get back—I'll be there."
"Keep up with us," Jake warned, catching Ashley's hand.
Pride dictated that she draw away, pride and maybe fear that it was too easy to depend on him so swiftly. But she didn't draw away; they were diving together, and she wasn't going to be uncooperative.
They eased into the water over the embankment, a difficult task as it was shallow next to the levee and they sank into the mud. She immediately felt the strength of the current, and she knew what Jake was thinking: if the killer had indeed followed this route, he had gone with the current in whatever little boat he had been maneuvering. He wouldn't have used a motor; a motor might have been heard.
The brown, muddy waters of the Mississippi covered their heads, and they went with the current themselves, using their flippers to thrust them downward. Ten feet, twenty feet, thirty feet...forty feet. She'd been in the water here before, but only ever to clear growth from the seawall or work with their little strip of dock. The water was filled with silt, and everything before her eyes was curtained behind a brown haze. The sun didn't penetrate deeply.
A massive catfish glided by them, taking a look, moving off quickly. They passed over the ruins of a broken-up tugboat. Gar drifted by, and in the few feet she could see ahead, even with her diving light, Ashley saw that a blue suckerfish was watching them avidly. There seemed to be little else of interest. Diving in clear waters was beautiful, but the Mississippi wasn't clear. It seemed as if the light dimmed quickly, as if the riverbed sucked it up into the mottled brown darkness.
She heard the rhythmic sound of her regulator, air moving in and out of her own lungs. She usually loved that sound. She glanced over to see that Jake was still moving fluidly at her side, inspecting the river bottom as they drifted along, barely using their fins, the current was so strong.
She tried to give her concentration over to the task at hand.
She felt a jerk on her hand; she turned and saw Jake's blue-green eyes through his mask. He motioned that they needed to go down. His metal detector had come upon something.
Fighting the current, they shot downward, only to discover car parts that had been in the river long enough to acquire massive growth. She and Jake, with Jackson close behind, started to move onward again.
She wasn't at all sure how she saw it, but suddenly it seemed that something bright flickered in the glow of her beam.
She started toward it.
A jerk on her ankle made her panic for a moment; alligators didn't usually travel out into the depths of the river—they preferred the bayou and the shallows. But it would be quite ridiculous if she were to be consumed by a natural predator she knew and respected in the hunt for a human monster.
It wasn't an alligator; it was Jake. He frowned at her, indicating that she wasn't to take off without him.
She nodded and pointed. Then, of course, her beam picked up nothing, but he nodded and followed her downward.
They were in an area of soft packed mud and underwater growth. At first Ashley thought she had imagined the glitter in the water. It was dark, and with the current, once the muck was disturbed, it spread out, creating an even darker brown haze.
Then she felt Jake squeeze her hand, and he indicated the metal detector.
There was definitely something there.
They had to move quickly, because the water around them was becoming browner by the second. There was nothing easy about maintaining their position in the water; they fought the current hard. But Jake was digging in the mud, and she did the same.
There, beneath one of the plumes Jake had just started.
She saw it.
It was still in excellent shape; it looked as if it had just been set down on the river bottom. The 1853 Enfield rifle that shot a Burton-Minie ball still had its bayonet, which had surely glittered in the glow of her flashlight, soundly attached.
It was the weapon that had been carried not just by the defenders, but by all of the federals on the day of the skirmish that had taken place in 1861. It would have a thirty-nine-inch barrel with three grooves, and the stock had three metal bands, so that it was sometimes called the popular "three-banded" rifle. Reproductions of the rifle were carried by all the reenactors. At the beginning of the war, it had been quite typical, appreciated on both sides of the great conflict.
She pointed; Jake reached down a hand and collected the weapon by the stock. He nodded to Ashley and indicated that they head toward the riverbank.
They emerged about a hundred feet shy of the boat ramp. Sludging up the muddy bank, Ashley saw that Angela had been true to her word and was there leaning against Jake's car down by the public ramp.
She saw them right away and came hurrying toward them, heedless of the marshy terrain. She had something in her hand, which turned out to be a large plastic bag of some kind—an evidence bag for the weapon retrieved. Angela, apparently, had had faith in their findings, while Ashley had to admit she hadn't expected that the four of them would find anything in the Mississippi. Of course, this team had probably been trained, but then again...
She had been the one to spot the weapon.
"You found it!" Angela cried, wrinkling her nose as she stepped into a deep pit and struggled to free her foot.
"Ashley actually made the discovery—without a metal detector," Jake said. He held the weapon out while Angela opened the bag.
"Wait!" Ashley said.
They both paused, staring at her.
She studied the weapon.
The manufacturers of historical weaponry were good—really good. They could replicate weapons to a T.
But there was something about this one.
She didn't touch it, but she moved closer. Mud encrusted the weapon, and there was no choice. She delicately took a finger protected by her diver glove and dusted aside a speck of the mud.
And it was there. Deeply, roughly gouged into the stock near the barrel, there were initials.
She looked up at Jake and Angela, chilled to the bone.
"This weapon is real. I mean, authentic to the period—and our house. It belonged to Marshall Donegal. You can see his initials right there, MPD. He carved them by hand himself with his knife. Marshall Patrick Donegal."
Interlude
He watched it again. There she was, that blasted reporter, talking to Jake Mallory.
Mallory, so cool and solemn and yet easy with the reporter, revealing nothing at all.
They didn't have good new footage to show; so far, none of the reporters or their crews had been allowed on the property, and thus they had nothing new to say. Of course, the world now knew that Charles Osgood was dead, and everyone everywhere was deliberating. It was absolutely amusing to discover just how many people were certain that a bitter Confederate ghost had committed the crime.
Or even a Yankee. Hell, four Yankees had "died" on the property that day.
Ridiculous. No one was putting blame where it belonged. Or maybe they were.
The canned video was replaced by the reporter again, the pretty woman who seemed to have a hard edge. It was the hardness of a woman who wanted to rise in her field and was willing to do just about anything to do so. She'd sleep with the boss while making her cameraman traipse through dangerous territory before setting her pretty face in front of the camera. She'd sleep with the producer. He knew that look. He'd seen it often enough in the business world. With men, it just meant that they'd stab you in the back; with a woman, it meant that she'd do just about anything.
The reporter's face was replaced again by the image of a sea of pictures; they were artistically angled in the shot, from, clockwise, left to right, Frazier Donegal, Emma Donegal, Marshall Donegal and Ashley Donegal.
Ah, and it seemed that the generations had taken DNA from all—Ashley looked like her great-great-however-many-grandfather, and like Emma Donegal. All these decades later.
The newscast ended.
He should have felt satisfied. He'd caused the havoc he wanted. They were never going to catch him; there was just no way to prove that he had done any of this.
But then the newscast changed; there was a "this just in!" alert.
The reporter came back on. Anchorwoman Marty Dean identified herself again and announced that police believed that they had the murder weapon, an historic Enfield rifle once owned by the Civil War master of Donegal Plantation, Marshall Donegal. Tests were being done by forensic experts, seeking proof that the weapon, pulled from the Mississippi, had indeed been used in the murder.
Naturally, Marty's station would be right on the investigation, bringing news about the heinous events at the plantation the second they became available.
He clicked off the television, startled.
The weapon shouldn't have been found so quickly. He had left no traces; he had ditched the damn thing in the Mississippi River, and he had been careful in every possible forensic way.
He paced, trying to calm down, and he did.
They had the weapon. Even if they found old Charles's blood, they could never trace it to him. Even if they knew how he had gotten there and killed the man, they couldn't trace it to him.
Pity, though. The image must have been so brilliant, the dead man—dead. Blood dripping. He'd been in Marshall Donegal's uniform. The possibilities for amazing ghost stories were endless....
He sat down again. It wasn't enough. It wasn't going to really choke the life out of the place.
So...
They'd be watching the cemetery. And they'd be watching the river.
There was always the bayou, and before you reached the bayou from the house, there were endless trails with pines, giving way to marshland....
"'It is well that war is so terrible—lest we should grow too fond of it,'" he whispered aloud, quoting Robert E. Lee, the South's—no, the country's—greatest general.
He smiled, his faith in himself restored.
"It is well that bloodletting is so complicated—because it is so actually sweet and entertaining!" he said.
The killing was the best.
## 8
The Enfield and bayonet had been turned over to the forensics lab, and there was nothing to do on that angle but wait—and, determine, of course, how the priceless family heirloom had gone from its glass-encased place of honor in a small attic museum that guarded such precious pieces into the hands of a murderer, and then into the river.
Frazier was horrified. Of course, it had never been locked up. They didn't lock up their artifacts at Donegal Plantation. Marshall Donegal had been buried with his dress sword, but not the Enfield rifle he would have carried into the war. At his death, Emma had kept the weapon above the fireplace in the rear parlor, should they come under attack again at any time. It was most likely that she kept it there until her death in 1890. One of her children had probably moved the rifle and other artifacts from the mid-1860s to the attic. Frazier knew that his father had been the one to purchase the box it had been displayed in now. No one had even known it was missing; when they all returned to the house, he asked Ashley a dozen times at least if she was sure that the weapon they had found had been Marshall Donegal's, and he had gone upstairs himself to assure himself that the box was empty.
It was.
The reality of the Mississippi River was that it was not the nicest place in the world to dive; returning to the house—once the initial questions were asked and answered—Jake hurried up to his room and straight into the shower. He was sure the others were doing the same. But he had barely stripped out of his swim trunks when there was a knock at the door. He frowned, wrapped a towel around his waist and went to it. He opened it a crack.
Ashley was out there. She had stripped off her dive suit and wore a terry robe over her bathing suit. Her hair was still damp and tangled from the water, but she looked restless and uneasy.
He opened the door fully without moving aside.
"What? Is anything wrong?" he asked.
She looked at him with her huge sapphire eyes. It was suddenly impossible—despite the five years that had passed since they'd been together—not to feel an uncomfortable rise of his libido.
God, he'd loved her, always. Not true, he tried to correct himself. Once, she had been a bright, entertaining, but annoying little precocious kid. But not long; she'd caught up so quickly. And when he had realized that he'd teased her the same way boys had teased girls they had secretly coveted since the be ginning of time, he had just fallen head over heels. It was the sapphire of her eyes, maybe. The perfection of her skin, the softness and the glitter in the color of her hair. Ah, but that was just lust. He'd loved her tomboy antics, the way she could ride, race, challenge, laugh, and argue her side of any matter. It was the sound of her voice....
"Well, a man is dead," she said flatly. "Killed—with a family heirloom!"
He started, his reverie fading. "Yes, we all know that. Is something wrong? I need to take a shower."
He was a fool, of course. She was standing there; he was standing there. He was naked beneath the towel; she wore flimsy pieces of a bikini beneath a terry robe. He had never really fallen out of love, and he'd have to be a hell of a lot older or infirm to fall out of lust.
But that wasn't the point here.
What was the point?
He didn't know why she was here. He didn't want to be used because he was convenient and her world was going to hell.
Yes, I do want to be used! his mind raged.
No.
"Ashley, let me jump in the shower, and I'll be right with you," he said. "I suggest you do the same. You wear even river mud well, but I think you'll be more comfortable without it."
He stepped back and forced himself to close the door.
His shower was pure torment.
Marty Dean sat at her office desk, studying her phone, and wondering what she could say once she'd reached Jake Mallory. She'd had such a crush on him in high school. His guitar had gained him quite a reputation, and he'd played with some pretty extraordinary bands. Everybody had a crush on Jake! And he'd just gotten better, really. Some of those boys—the big football-hero types—were just downright pudgy now. And Jake? He still had those probing eyes, that sculpted face, those shoulders.... It had been something to see him again. And he had looked right at her—and not given her a thing.
Well, she thought, pouting, he'd come around. He was a man, and she knew how to make a man come around.
Her pout became a frown. But, of course, he was at Donegal Plantation, and he and Ashley Donegal had been quite a thing. If he was sleeping with her again...
Hmm. The promise of an affair might not do it.
Maybe she had to promise that she would promote Donegal Plantation, though she would love it if there were a mystical ghost story involved.
She was still pondering the mode by which she would get him to agree to an interview when the switchboard signaled a call for her.
"Line four," the operator told her.
"Who is it?"
"Some man who insists you're going to want what he has to give you," the operator told her, bored. "Look, I'm not the FBI. I don't know who he is. You said to send through anything promising—"
"Yeah, yeah, I got it, thanks."
She picked up the phone.
"Marty?"
The voice was deep, quiet and husky.
"Yes?" she said. "Who is—"
"You want an interview. You want to know what's really going on. You want to break it free. Well, I'll do it for you."
Jake! It had to be Jake. Oh, and he was FBI now, or something governmental, and he was one of the big shots on the case. But he did remember high school, did remember that they'd flirted and teased and that she'd been the hottest thing in his class.
"Oh, you sweetie!" she said. "Thank you, thank you! Can you come in—"
"No, but you can come out," he said. "I'll tell you where to leave your car so that it's not seen. And I'll tell you the easiest way to get to me. No cameras. This is between you and me. But it will help you get to the truth. I can't say more—you have to really solve this on your own after what I tell you. But I won't speak if there's anyone else there, so come alone. I mean it. Don't tell anyone where you're going. I can't be involved in this when the news gets out."
"All right, all right. Where should I be? When?"
She listened. She hung up, delighted.
Her secretary stopped her as she headed out for her car. "You have the newscast at eleven, remember?"
"I'll be back. I've got a lead. I've got plenty of time. I just need to meet with an informant—a local informant. It pays to be me!"
Marty tore out of her office, mentally planning her speed. How fast could she go without being in danger of the cops stopping her?
Pretty damned fast, she told herself.
Oh, this was it! This was it! The case of a lifetime.
By the time Jake came downstairs, he found that he was the last to do so. The others were arrayed in the roadside parlor, seated in the big wingback chairs by the fire and the massive Duncan Phyfe sofa. Jackson's hair was still wet, so he had obviously just arrived. Ashley had showered and changed into jeans and a tailored blouse; her damp hair was tied in a queue at her nape.
"The house itself is seldom locked," Frazier said as Jake entered the room. "We're a bed-and-breakfast. We have a restaurant, and we're open to the public. God knows when the rifle was taken. I haven't been up to that little attic museum space in years." Frazier glanced over at Ashley. "Have you been up there?"
She shook her head. "Sally Mayfield, one of our housekeepers, last did a dusting up there three or four weeks ago, I think. Sally would have noticed if something had been gone, and she would have told me. So I'd say that it had to have been taken within the last three weeks."
"When were most of the reenactors here last? They still meet ahead of time, right? And what about Civil War roundtables? Frazier, you used to have them here now and then," Jake said. He heard the front door open as he was speaking; Cliff had arrived to join them. He glanced over at Jackson for a silent communication.
Cliff remained a prime suspect. They needed to keep a close eye on him.
Ashley didn't want to believe it. He didn't, either. Cliff had always been around when he'd been younger; he'd always been strong and steady and decent, and he loved Donegal Plantation, just as he'd always loved the family.
Frazier was thoughtful. "Yes, of course. I'm not sure of the date of the most recent."
"We had a meeting out in the barn about two and half weeks ago. The out-of-towners weren't here, but the rest of us were," Cliff volunteered.
"Everyone?" Jackson asked.
"Every one of the locals except for John Ashton," Cliff said. "He had a tour group in the city that he had to take out himself. Some bigwigs from the tourist board. And obviously, out of our group, I have continual access to the house. And I suppose I have the physical ability to have done all of it, while, no offense, Frazier, I don't see you committing this crime at your age, and I sure as hell don't begin to see Ashley or Beth committing it. I wish I knew how to clear myself, because no matter what you all say or how you act, I know damned well I have to remain a prime suspect."
"You even have motive," Jackson said pleasantly.
"Yes, I should own the house. But..."
"But?" Angela asked gently.
"Whoever did this might want the whole place to fall apart. Let's face it, if the place is tainted by a recent murder and a killer who isn't caught, people could be afraid to come. I'd never want to do that," Cliff said.
"We all survive because of Donegal Plantation—well, except for Beth, who could get a great job anywhere," Ashley said. "But Cliff is right. There's no reason for him to want to lower the value on this place. Besides, Cliff doesn't even have to wait for Frazier and me to die—he owns a piece of the land, and he has a lifetime lease on his apartment in the stables."
"Bitterness," Jake said. "I mean, if we're looking for motive. Let's face it, this whole thing is sick. Cliff, you're the outsider. You were the product of an illicit affair, historically speaking."
"Two illicit affairs, really," Cliff said. He never took a seat; he stood there and shrugged sheepishly. "I can't prove anything, but I'll answer any question you may have. Obviously, I didn't stash a body in the stables—they were searched. Of course, I could have stashed the drugged body of our friend in my apartment and joined the group. I would have had to have been terribly crafty, though, since the place was teeming with people, and the entrance to my apartment is easily visible from the grounds."
"And how did you return the body, then—by the river?" Jake asked him.
"You might have done that to throw off suspicion," Angela told him.
"I might have," Cliff agreed. "Except that I didn't. I'm going to feed the horses," he said, then, "But I'm available to you anytime—I won't be leaving the property." He smiled. "See you all at dinner?"
"We're having crab cakes. My best!" Beth assured him.
When Cliff left, there was an uncomfortable moment of silence in the room.
"I don't think—" Ashley began.
"None of us wants to think, but we have to weigh all the factors, eliminate the impossible and start looking at the possible," Jackson said to her gently. "Sometimes a killer wants to inject himself into the investigation."
"Cliff isn't—"
"It's not impossible, Ashley," Jackson said.
"It is impossible," she said stubbornly. "Cliff is part of the family."
"In the old days, brothers would kill brothers to be king," Jackson said.
"Look, Ashley," Jake said, leaning toward her. "Hopefully we'll be able to clear Cliff soon."
"So, we're all careful. We all stick together, and we keep doors locked. Agreed?" Jackson asked. "I've got Will and Whitney on their way out here now with some new equipment, and Jenna is interviewing Justin Binder and his family in the city. I'm going to pay a call on Ramsay Clayton—I believe he's still in residence at his old family home down the road—and have a talk with him. There are still a couple of uniformed officers outside, but I think part of the team should always be at the house. Jake and Angela?"
"I'm here," Jake assured him. "I'll get on the computer."
Jackson nodded. He looked down at Angela and squeezed her hand, and Jake knew that her assignment was to discover what she felt about the house.
Frazier walked over to Ashley. She stood quickly, hugging him. He hugged her silently in return. "I'm for a nap," he said. "At my age..." He apologized to the others.
"I'll be in my room, too," Ashley told him.
"A nap sounds good to me," Beth said, yawning. "And then dinner."
Angela laughed. "Oh, my God, after that delicious lunch you prepared! I'm not sure I'll be able to eat dinner. I'm going to...search the house. Combine a little work with exercise."
"Search for what?" Ashley asked her.
"Evidence," she said softly.
Ashley shook her head. "But that won't help us. Every man had access to the house at some point. What evidence could we possibly find that would be evidence? And the cops have been here, too."
"I never really know what I'm looking for," Angela said. She glanced at Jake. "But it's good to walk around and think and look, and then you sometimes find what you didn't know you were looking for. If that makes sense."
"I'll be in the study," he assured her. "If you need me."
Beth and Frazier headed for the stairway, arm in arm as they walked up the stairs.
Ashley lingered until she was alone with Jake. "Cliff didn't do this," she said with finality.
He walked over to her, placing his hands on her shoulders. "Ashley, honestly? I don't think that Cliff did it, either. But it's not going to hurt to be careful, to be with someone, to keep the doors locked, right?"
"I can be careful," she murmured, bowing her head.
He lifted her chin gently so that she met his eyes again. "Hey," he told her huskily. "Remember when we first decided we wanted a rock band and we set up out in the stables? The poor horses! Cliff never said anything. He just moved the drum set into the front yard and told us that birds liked music more than horses—especially heavy metal."
She smiled. "My father let us move the drum set into the old smokehouse." Her smile faltered. "Jake, my father has never come back from the dead," she said.
He was puzzled. She wasn't angry with him, and she wasn't even turning away from him or trying to escape him.
"I never meant to hurt you," he said.
"Well, I probably managed to hurt myself," she murmured. "You...scared me. You really, really scared me," she told him. She was silent a moment, looking at him. "But you should know. My father isn't here."
She flushed as if she had said more than she had meant to. She backed away from him. "I'm—uh— I'm going to go to my room," she said.
"Into the land of digital reality for me," he told her and headed off into the study while she walked toward the stairs.
The subtle, almost elusive scent of her perfume lingered, and he had to force himself not call out her name, not to draw her back to him and demand that she understand. Time had done nothing to lessen his feelings for her.
He wanted to hold her; he wanted the truth. He wanted to make the losses and traumas of her life go away. And, that, of course, was impossible.
They could do their best to find the killer. That was what he could do for her, he thought.
But when he sat behind the desk in the study, he didn't turn to the computer. He sat in the chair, scanning the space around him. "Where are you, damn you?" he whispered aloud. "Emma Donegal, I saw you. You knew something was wrong. You wanted me here. Please, won't you come and help me now?"
Ashley headed for her room, still feeling a flush on her cheeks. Well, she was a fool. She'd turned away from Jake Mallory, cutting him from her life as if she had done so with a sharpened blade. What was she expecting now, and what the hell had she been doing, throwing herself at him just because she was scared?
"Well, I am scared," she said and then winced, wanting to nip in the bud the fact that she was talking out loud to herself far too frequently now.
She threw herself down on her bed and closed her eyes. She didn't see him; she didn't feel anything at all, but she knew that Marshall Donegal was there.
"You have to go away," she said. "You were trying to make me look like an idiot in the study, and I was a nervous wreck all through lunch, thinking you'd make me do something stupid. If you're my ancestor, and you love me so much, will you quit tormenting me?"
She felt a shift of weight. He had taken a seat at the foot of the bed. She opened her eyes at last.
"If someone comes at me with a weapon, can you protect me?" she demanded, sitting up to stare at him. "Will that ghostly blade save my life? If not— Where is this going?"
"Can I protect you? That depends. I am fairly powerful. Being as I am requires concentration and practice, and I was always a disciplined man."
"Right. So you got into a barroom brawl and died before the war really began," she said dryly.
He seemed to stiffen. "You're wrong. I didn't get into the brawl. Peter O'Reilly got into the brawl. I dragged him out of the place before it turned into something right there, though that might have been a mistake. God knows, if we'd brought troops in, the Yankees would have been killed on the spot or hanged for being spies. But I didn't want murder committed. Hell, I lost my own life because of it."
"O'Reilly?" Ashley asked. "That would have been Charles Osgood's great-great-great-great-stepgrandfather, right?"
Marshall Donegal nodded, rising and walking to look out on the river. He lifted his hands. "I lose track of the generations...but, yes. He wasn't a bad fellow, just the kind who was quick to anger and to feel an affront. He was eager to 'whomp those Yanks!' He survived the war. I saw him here once, when he came to pay his respects to Emma. He was minus his left leg. It made him a different man. Emma was sorry for him, of course. She offered him work. But he went into New Orleans and became a printer."
"Even so, do you think that someone's ancestor knew this and thought it was a justice that Charles should die since his ancestor brought about the whole thing? Maybe one of the Yankees!" Ashley suggested.
"One who perished?" Marshall asked her.
"Possibly. I mean, if the rebels more or less caused it all because of Peter O'Reilly, and four of the Yankees died, maybe it was a sick kind of late-blooming vengeance."
"Even I'm aware—perhaps more so than anyone—that the war is long, long over," Marshall said. "Other wars have raged since, and will rage in the future," he added sadly.
"Yes, but whoever did this has to be sick. You don't drug a man, hide him for a day and half and then take him and bayonet him to death and hang his body off a tomb's angel if there isn't something really wrong in your psychological makeup," Ashley said flatly.
"Why would someone avenge someone after a hundred and fifty years?" Marshall demanded.
"I don't know, but Cliff is a prime suspect because of his family relation," Ashley said. She frowned and then gasped. "I can't believe I forgot. We do have that old plantation story about the master who supposedly slept with a slave. Were you involved?"
He was quiet, and he gave her a curious, sad smile. "Not me, and not any plantation master," he said quietly.
"But you know who?"
"Haven't you ever studied the records?" he asked her.
"Of course, but the baby who was Cliff's great-great-whatever just seemed to appear, and he was raised by Harold Boudreaux and grew up after the war on the property."
"After the war. I was dead, remember?"
"Yes, but there's no exact age on what records we do have," she reminded him. "The records for the slaves on the property were kept at the chapel, and the bible recording all the births disappeared sometime during the war, so the lists we still have don't have birth dates on them."
"Cliff's great-great-great-great—I believe—grand-parent wasn't a Donegal man."
"Then—who?" she asked.
"It was Emma," he told her quietly.
Words and numbers seemed to blur Jake's vision, but he did feel that he was gaining ground.
Cliff Boudreaux could not be eliminated. Nor could Ramsay Clayton. John Ashton was easy to eliminate. There were pictures of him in New Orleans on the web the day following, and he had gotten an interview on one of the local channels to talk about the history surrounding the city—and to plug his own business. He had been on an evening show that broadcast at seven, and he had been going to give a tour that night that included the broadcaster. It didn't take long to verify the fact that he had led the tour, as he had said on air. He hadn't been at the meeting that had taken place approximately when the Enfield rifle and bayonet had most probably disappeared.
In like fashion, verifying their location at the time the drugged-but-still-living form of Charles Osgood had been taken into the cemetery and murdered, he managed to clear the field of all the Yankees except for Justin Binder. Tom Dixon had attended a party with his wife and children in New York that had gone on 'til midnight, and Victor Quibbly had already been in Austin, Texas, on a business matter.
He looked at his remaining list and his notes. Cliff—no one wanted it to be him. Ramsay—had he set up Charles Osgood? Hank Trebly—why? Sugar interests. Toby Keaton—okay, so he owned Beaumont, the Creole plantation next door, but he was never in competition with the Donegal family...or it didn't appear that it could be so. Griffin Grant—no amount of searching showed exactly where he had been, other than that he had shown up at his office for the usual workaday world on Monday. He hadn't even taken the day off, as so many had. Three others to look at would be the sutler, John Martin, Justin Binder, who had stayed in New Orleans at a chain hotel, and Dr. Benjamin Austin, who lived in Francisville and had not had office hours after five on the day that Charles had actually been murdered.
He sat back for a minute, closing his eyes. They could be way off. Anything could have happened. But he was pretty sure he was on the money, and he knew that Jackson would agree with him. Eliminating the household—Beth, Ashley and Frazier—left those who were closest to the household. He knew Ashley, and if he hadn't known her, he'd still know that no one could have acted the terror she had shown when he had come upon her. Frazier couldn't have pulled if off physically. Beth had no interest in the family; she hadn't even really understood the history behind what was going on, and she didn't have the strength to manage the feat, either. He'd pulled up everything he could find on her on the computer anyway; she couldn't have any determination to avenge a long-ago ancestor. Her family hadn't been anywhere near the United States during the war. So, using logic, it seemed to be down to Griffin Grant, Cliff Boudreaux, Ramsay Clayton, Hank Trebly, Toby Keaton or, less likely but still possible, Dr. Benjamin Austin or the sutler, John Martin. He put the last two at the end of the list. Concentration first—Ramsay Clayton and Cliff Boudreaux.
How in hell to prove that it wasn't Cliff?
He looked at the phone on the desk and noted that there was a button to call directly to the stables.
Cliff answered quickly. "Yes?"
"Cliff, I'm going to be honest. I want to eliminate you as a suspect. Would you be willing to let a forensics team go over your apartment and car?"
There was silence, dead silence, on the other end.
Then Cliff answered him, his voice tight and hard. "Whatever it takes, whatever it takes. Bring it on, my friend."
As he hung up, his eyes on the desk phone, Jake felt that someone was watching him.
He looked up, and his breath caught in his throat.
It might have been Ashley—Ashley in her attire for the drama they played out at Donegal Plantation.
But it wasn't.
It was the woman he had seen in Jackson Square in New Orleans before he had even known that he was going to be coming out to Donegal Plantation.
Emma. Emma Donegal. He could see the door through her misty form, but he could also see her face clearly. It wasn't as if he could really hear her voice, and yet he could; it was inside his head.
"Come!" she said urgently. "Please, come."
He forgot about Detective Mack Colby and the call he had intended to make to get Cliff's apartment searched. He followed the ghost out of the office and through the house.
And on out to the stables.
Cliff was there, sweeping hay from the slab of concrete in front of his door. He looked up at Jake with guarded eyes.
"Are you doing the search?" Cliff asked him.
Jake felt about two feet tall. He had known Cliff for years; he could remember many more occasions with the man than what he had mentioned to Ashley. Cliff had patiently corrected him and taught him about riding, horses, shooting and the plantation itself dozens of times throughout the years. They'd gone out in alligator season together, and Cliff had taught him that even if the creatures were predators, they had their place in life. They had to be hunted to control the population, but they didn't deserve to be tortured because man had unbalanced nature. A clean kill: a good shot between the eyes. That was the way to kill a gator.
He had taught him other things. Things like balancing his weight with that of the horse he was riding, how to sit a jumper and how to calm a horse when they ran into a bear in the woods. He'd taught him how to hunt fowl, and, in return, Jake had taught Cliff how to hold a cue and shoot a break that could nearly clear the table.
"No, Cliff, honestly, it's because I want you cleared. I don't want anyone who doesn't know you the way I do not knowing that they can trust you," Jake said.
Cliff studied him and then nodded. He leaned on the broom. "You come out to go after Ashley?" he asked. "You ask me, she shouldn't be out alone right now."
The ghost of Emma Donegal had disappeared when he'd reached the stables.
But now he knew why he was here.
"Where is Ashley?" he asked.
"Just ten minutes ago, she came running down and asking me if there was any problem with taking her mare out for a ride. Said she needed to clear her head, and a ride around the property always did that for her. I was going to let her have a few minutes and then take a ride myself. I just don't feel right about her being alone."
"Hell, no!" Jake said. "What horse can I take?"
"If you head to the bayou, you'll find a concrete marker. That's where Emma had Harold Boudreaux buried," Marshall had told Ashley. "But don't go now, young woman. Show some sense of self-preservation. There's someone out there bent on hurting Donegal Plantation, and you are the last of the Donegal family."
"Cliff is a Donegal!" she told him.
"No, my dear, not really. Emma wasn't born a Donegal."
"Yes, and I'm not sure exactly what the relationship is, but Cliff's great-grandfather and a young woman born a Donegal got together in the 1920s, so, yes, he is a Donegal!" she said.
"Well, yes, I suppose you're right on that."
Marshall Donegal had followed her when she went out to the stables, determined on riding. He kept trying to dissuade her while Cliff kept trying to dissuade her.
Before she'd mounted up on Varina, she'd given Cliff a huge hug. "I love you, cousin!" she told him.
Cliff had looked at her strangely and then shook his head. "Look, Ashley, you don't have to defend me. I didn't murder anyone."
She'd grinned at him. "I never thought you did. I just wanted to say that I love you!"
She hadn't bothered with a saddle; she had to find out if there was really a stone near the bayou. If so, she wasn't imagining the ghost. He was really there, telling her things.
Or she was imagining the ghost, and he was really suppressed memories in the back of her mind. Whichever. She wasn't doing well fighting the concept of imagining Marshall Donegal, so she might as well try to use what was happening. And it didn't look as if she'd be running to Jake for comfort. She'd be business, strictly business, from now on out.
She thought that she had ridden out alone; she should have known better. Marshall Donegal was riding behind her—on a ghost horse, of course. His mount was a beautiful roan, complete with all his Confederate trappings.
"Dammit, woman! Let me lead!" he called to her.
She felt something as he and the roan seemed to pass through her. Then she took off through the woods, following him.
They rode for twenty minutes. Then the ghost horse let out a whinny and stopped, and her haunting ancestor slipped from his mount and walked down the trail to a large pine. He tried to rip away the vines and grass and weeds that grew around the base; Ashley saw that the grass moved, but little else happened.
She began the task herself.
She gasped out loud. It was there, a large, flat, stone marker. One word had been crudely etched into it.
FRIEND.
Ashley sat back on her haunches and looked up. Marshall Donegal leaned against the tree, watching her.
"Why? Why did Emma bury him out here? There's a tomb for the slaves—and then the servants—who stayed on to work the plantation," Ashley said.
"Emma was truly a wonder," he said sadly. "Any rumors you heard about fights between us—or my indiscretions—were stories created because people need stories. We fell in love, and when that first blush of love was gone, we still loved one another deeply. She was a strong woman. She held the place together after I died—with the help of Harold Boudreaux. In 1864, they became lovers. The world would never have accepted it. Even after the war ended, they would have been in grave danger. There was a pecking order for those of mixed blood in New Orleans, you know that. Quadroons were all the rage to become a man's mistress, and the Quadroon balls were infamous. But after the war, the KKK was started up, and if they had been discovered, Harold most probably would have been burned on a cross, and Emma would have been subject to rape and ridicule. They had to keep their affair entirely secret. So he raised their child as one of his own. Another of the former slaves—a young woman of mixed blood herself—was accepted as the child's mother."
"Did you mind? Did you hate what happened?" Ashley asked him. "I mean, as a ghost, were you bitter or...can you still hurt?"
"My soul can know agony," he said quietly. "But did I mind this? No. I loved Emma with my whole heart. And a dead man knows that the color of his skin doesn't mean a damned thing. I admired Harold. I loved what he did for my family, and how he taught and defended my children. No, I didn't mind. Not this. I just thought you should know. Maybe it can help you in some way."
They both started at a sound that seemed to come from the woods that led straight to the bayou. Ashley quickly stood up. "Probably a raccoon or even a squirrel," she said and grimaced. "Maybe even a gator." She walked toward her mare.
It wasn't a raccoon, squirrel or gator. As she mounted, she heard the noise from the woods by the bayou again.
"Quickly," Marshall said. "I'll hold the path!"
"But it may be nothing."
"You're alone out here. Get back to the house! Please!"
She almost laughed and reminded her ghost protector that he was dead.
But she didn't. She turned to ride, finding the quickest path back to the house and kneeing her mare to a gait that would lead it at all speed through the trails without killing them both.
But as she made her way homeward, something darted into the path in front of her.
Varina reared, and she wasn't as prepared as she should have been. She cursed herself for her carelessness as she felt herself fly into the air.
And land hard on her rump in the middle of the dirt path.
As she quickly stood, rubbing her injured section, she realized that it had grown late.
Darkness was falling, and she was alone in the woods.
Even her ghost was far behind her now.
Interlude
People were easy.
Pathetically easy.
Once you knew what they wanted and you dangled it before them as you might dangle a carrot before a horse, they came. They came—just as the stupid animals they were in truth.
She barely saw him coming.
She got just a glimpse of him.
During the phone call he'd made to her, she'd guessed that he was Jake. She'd giggled.
He'd almost giggled, too, it was so damned perfect.
And what an idiot woman. You'd think they'd have to get through some kind of school to broadcast the news. But a pretty blonde was all you had to be, it seemed. Maybe not. Maybe others were smarter.
He watched her when she came staggering along the overgrown and marshy trail near the bayou, swearing as she did so. Actually, he watched her for a while. He didn't know what it was about her that had made him want to do this. He'd plotted and planned his first kill forever; he'd thought that it would be his only kill. But he started hearing the voice again. She was nosy. She was going to start dredging things up and just might have some journalistic abilities. The voice said that she needed to go. And looking at her, he had begun to anticipate the kill.
He glanced toward Beaumont; he could see the plantation through the trees. But there was nothing going on there—the plantation offered its last tour at four, and people were usually off the property by five. Even the actors and historians would all be by the road now, ready to head out.
She stumbled her way to the small clearing. Her little red spiked heels were completely ruined, and her jacket was torn. But though she was irritated, she wanted the story more. In fact, when he came upon her, she was looking down at her shoes and cursing about how much they had cost her.
"Damn it, what the hell are you doing here?" she demanded, tossing back her head of bleached blond curls. "I was expecting—"
She never voiced what she was expecting. He was quick. The needle got her with such speed that she barely gasped, much less managed to get out a scream.
As she fell, he heard the horses. He heard the rummaging over on the next trail.
He had to move with speed.
He rolled her down to the true swampy area just before the bayou.
He held her head under in just a half foot of water.
She didn't struggle. She was out, and she was easy to kill.
When she was dead, he couldn't help but roll her over. He smiled, looking at her face. That pretty, bitchy face, all clotted with mud now. Her lashes were slipping. False lashes; she'd been all makeup and hype and selfishness. Actually, he'd done the damned world a favor.
She wouldn't look so good on the eleven o'clock news now. Of course, they wouldn't find her right away, and they wouldn't expect to find her out here.
But somebody was out here now. He had to move.
And move quickly.
## 9
She stood in gathering twilight, cursing her mare for throwing her.
Poor mare; it wasn't her fault—she had been startled. Scared.
"Damn you, foul rodent!" she cried into the bushes. She swore softly; it was time to walk, and walk quickly. "Last time I follow a ghost into the woods!" she muttered.
Was it a ghost? Was there really a ghost? Or did she have deep-seated memories that she needed to address, and seeing the ghost of Marshall Donegal was a way of doing it? Hmm. That would be a good one for a shrink.
Maybe, just maybe, people did see ghosts. And maybe it was some kind of a gift, and she'd been far too terrified to ever recognize the possibility before.
Jake had that gift—that sixth sense or intuition. And when he'd come to her with it, she had simply panicked.
Well, too late on that one!
She paused; she heard something coming from the pines and brush closer to the bayou. Gator? It was unlikely that one of the giant crocodilians she'd known all her life was going to come this far in off the bayou and stalk her.
She quickened her pace.
Of course, she knew that even on land an alligator could move damned fast as well.
But, no. The woods here were filled with birds, the bayou was brimming with fish and there were plenty of small mammals. A gator had not had a whiff of her and decided that it was time for his evening meal.
She heard it again. Definitely not a gator, because she kept hearing the rustling, would stop—and then the rustling would stop as well. A beast of prey would come straight for her.
She started to run. As she did so, she heard a thrashing in the woods ahead of her and then from one of the other trails.
She swore and looked around her. There was a fallen slender pine near her, leaning against an oak. She tested the trunk carefully, found that it would bear her weight and crawled up to the branches of the hardier oak. She kept crawling, hoping that whatever was out there didn't climb trees.
The thrashing around her grew louder, as if amplified in the thickening darkness of the night. She finally realized that she could hear hoofbeats, and a rider was coming for her. She waited, barely daring to breathe.
Of course, Cliff knew that she was out here. When her mare came back without her, he'd be on the trail, coming to find her.
But even as she heard the sound of a horse, she saw something dark below her. Not a creature—a man. A man who looked like a shadow because he was dressed in black: black boots, jeans, sweatshirt and hoodie. He moved with his face lowered, and in the darkness he might have been a black shadow.
Was she seeing shadows now instead of ghosts?
No, he was real.
He was approaching the tree; he paused as if listening.
She heard her name called through the trees. And she no longer heard the sound of hoofbeats.
For a moment, the entire world seemed silent. She waited, not daring to breathe.
Darkness fell in earnest, but the moon prevailed.
And then, naturally, the moon was covered by clouds.
Ashley cursed herself for starting out alone tonight. She had to breathe; she tried to do it silently.
And then, though she couldn't see him, she was certain that the man below her in the dark hoodie was looking up.
Did he see her frozen there?
She held her position. She couldn't tell if he was there or not anymore; his form seemed to have been swallowed up by the blackness of the ground below.
There was rustling.
There was someone below her, someone who seemed to be stalking the area.
Something banged against the tree. A man's hand?
She started to slip; she felt a splinter of bark shoot into her hand and barely kept from crying out. She tried to shift her position and fell, a scream escaping her lips at last.
She landed hard on human flesh, toppling the standing man to the ground. Panic seized her and she shot out a fist, striking anywhere she could as she tried to rise. She made it halfway to her feet when she heard him shout out.
"Bloody hell, Ashley! Why are you hitting me, damn it?"
Jake.
She went still and started trembling. For a minute, she still had to wonder if he'd been in the pines before the bayou, stalking her.
But then both heard it; more noise coming from the trail.
"Ashley! Jake! Where the hell are you two?"
That was Cliff's voice.
And from the bayou, another voice.
"Hey! What's going on in there? I have a gun, and I know how to use it!"
Jake scrambled up, half knocking Ashley over but then drawing her to her feet.
A light suddenly glared into the darkness, and they both raised their hands to protect their eyes. Cliff came trotting up on his favorite mount, Jeff. He dismounted in the little clearing where he found them.
"Cliff!" Jake said.
"Where's your horse?" Cliff asked.
"Where is your horse?" Ashley asked Jake.
"Mine is tethered down the road—he meant you!" Jake told her, irritated. "What the hell are you doing, skulking around in the woods by yourself? Jesus, Ashley, how much of an idiot are you?"
He was clearly upset, but he took a step back.
"Damn it, I would have thought that you had some sense."
She heard the safety slide on Cliff's shotgun, and she felt a millisecond of fear again.
She was wrong; Cliff was bitter, and he was going to shoot her in the woods.
"Someone is coming," he said.
A minute later, his shotgun aimed at the clearing, Toby Keaton came into view. Seeing them, he lowered his weapon.
"What is this? A family meeting? You guys scared the hell out of me! What's going on over here?" he demanded.
"I was out riding. Some fairly big mammal scared Varina, my mare," said Ashley. Ruefully, she added, "She threw me."
Toby looked at Jake, standing beside her, and Cliff, seated again up on Jeff.
"With what went on—you all damn near scared the death out of me. I mean, this side is your property, but I'm just a spit across that bayou. You've got to warn me when you're going to be rustling around by the water at night. I can tell the difference between the sound of a gator and the sound of folks stomping around in the woods, you know!" Toby said.
He was clearly shaken.
"Toby, I'm so sorry," Ashley said. She looked at him. His black jacket didn't have a hood.
But he might have a hoodie stuck down beneath his coat; his shirt was black as well.
"How did you get over here, Toby?" Jake asked him, as if reading Ashley's mind.
"I live next door, remember?" Toby said, exasperated. "One of my hounds was going crazy, but he lost the scent at the water. I don't come out at night without a shotgun. God knows when you're going to come upon a doped-up schoolkid, a gator or, hell, a damned poacher! Or just a nutcase now, like whoever did in old Charles."
"I'm sorry. Toby, how did you get here?" Ashley repeated. "Have you been over in the woods right by the bayou?"
"I came over in my little aluminum canoe," Toby said. "I heard all manner of rustling—and it's turned out to be you!"
He shook his head. "Look, Ashley, I know we're all kind of scared right now, so maybe you could quit with the trail riding in the middle of night until they find whoever killed Charles!"
"She won't be out alone again," Jake said firmly, in a voice that seemed to scrape all the way down her spine. "It's pitch-dark and chilly out here, and the mosquitoes are big enough to rustle the woods themselves. We're going to get on back now."
Toby clicked on his own high-beam flashlight. Once again, they protected their eyes.
"Toby!" Ashley said quickly.
"Yes?"
"Were you under this tree a few minutes ago?" she asked.
He looked at her and squinted. "No. Well, hell, I don't think so. I didn't know what the hell was out here—I was moving around quiet as I could, trying to listen. Why?"
"I—I thought I might have seen you," she said.
"Well, if you'd seen me, it would have been nice if you'd said something," he told her indignantly. "I'm getting back."
Toby turned and started tramping down the trail. They could follow the glow of his flashlight as he headed for the bayou.
"What was that all about?" Jake asked Ashley.
"I climbed the tree because...because I was afraid. I thought something was stalking me. I saw someone beneath me, and it scared the hell out of me," Ashley said.
"What were you doing out here?" Jake demanded, perplexed.
"Riding!" she snapped.
"Well, he's right. You shouldn't have been, and you damned well better not do so again until the killer is found."
"And what if you brilliant people with your brilliant team never find the killer?" Ashley demanded.
Jake didn't answer. Cliff did.
"Then we won't be out here, period. The plantation will go down, and the Donegal family won't own it anymore. You'll see sugarcane here, just like you see it beyond the cemetery side."
Jake was staring at her; she could see that in the glow of the flashlight Cliff still held on the little copse where they stood.
"We'll find him. We'll find the killer," Jake said, and he turned away. "Come on, you can ride with me."
She was angry...and worse, she realized. She was still feeling rejected, no matter how stupid any of it might have been—even if she had been the one to put a block on him years before.
"I'll ride with Cliff," she said.
Cliff was startled, but he looked down at her with a shrug, sheathed his shotgun in his saddle and shifted the light in his hands to reach down for her.
"Let's hope Varina made her way home—and that your grandfather hasn't gone out and had a heart attack!"
With Cliff's strong grasp, she leapt up behind him on Jeff. Ahead on the trail, finding his way through the dark, Jake mounted up on his horse.
They made a silent trek back through the woods to the house.
Approaching the stables, they saw that Varina had indeed made her way home. She was walking around in the center of the stables as if she were teasing the horses who were still in their stalls.
"I've got the horses," Cliff said gruffly. "You get in the house, Ashley, before Frazier realizes that you're not around anywhere."
She started toward the house. Jake was right behind her. She felt him come closer and closer, but she didn't realize he was going to stop her until she felt his hands on her shoulders.
"What the hell were you really doing?" he demanded.
"Riding. I do it all the time."
"Right—after a corpse is found in the cemetery."
She turned around and stared at him.
"A murdered man!" he said with some force.
"Look, this is my house. I live here, and I don't think I'm under suspicion—by anyone's standards. I'll ride when I feel like riding!" she told him.
She was startled when she suddenly saw him take a breath and smile.
"What?" she demanded.
"I'll just talk to Frazier," he said and walked past her.
"Damn it, Jake, stop!"
He did so, turning back to her.
"I—I heard a rumor after the reenactment," she said. "I heard that Emma Donegal was actually Cliff's ancestor and not one of the men in the family. She could never admit—not at that time—that she had an ex-slave, Harold Boudreaux, for a lover, or that she'd given birth to a child of mixed blood. But she loved him, and I heard that there was a gravestone out there. She had him buried out by the bayou. I thought if I found the stone, it might all be real. I went to find it."
He paused, watching her for a moment. "And did you find it?"
"Yes, it's there."
"It has his name on it?"
"The word Friend was carved into it."
He was quiet, and his silence seemed to scream that a stone that said Friend didn't really mean a thing, and that it was incredibly stupid at this time to be wandering around in the woods looking for the headstone of a man who might or might not have been an ancestor's lover.
Before they could argue further, the riverside door opened and Beth stepped out. "There you are! Goodness, I was starting to get worried. Jake, Jackson got back, and he's looking for you. Crab cakes are on, and it's time for dinner!"
Ashley turned from Jake and looked at Beth, forcing a cheerful smile.
"Crab cakes! Better yet, your crab cakes. Do we have a minute to wash up? Should I call Cliff?"
"I'll get him on the house phone, honey. You run on up. Angela has given me a hand in the kitchen, and it will all be on the table in ten minutes, so hurry it up, girlfriend! And, we've got more company!"
Ashley looked at her, puzzled.
"Will and Whitney?" Jake asked.
"Yep. They're as nice as they can be!" Beth told Ashley.
Ashley didn't doubt that the pair was exactly that. She was surprised to feel a little tug at her heart, created by the sound of pleasure in Jake's voice.
His friends were there. Maybe Whitney was more than a friend.
She suddenly felt out of step with his life and ashamed of herself. Jake had been the best friend in the world; she had loved him. She had turned from him. He deserved a little happiness.
"It will be great to meet them," she said.
She flashed Jake a quick smile, and hurried on into the house.
It was odd, but somehow the arrival of Will and Whitney—with all their paraphernalia—was like a breath of fresh air.
A horrible murder had taken place, and they were in the midst of a rough investigation. But Whitney's vibrant personality made its mark on the solemn household. As they sat down to dinner, she explained the setup that she and Will had carried out.
"Don't worry—we haven't put cameras in anybody's bedroom," she said, "but we do have the house nicely wired for sight and sound. Ashley, you haven't seen it yet, but there's a bank of screens in the living room. We have cameras aimed in both of the parlors, out to the front, out to the back and over the cemetery. Tomorrow we'll set up over the stables. That way, we'll be apprised of any unwanted visitors on the property. And—" she hesitated, casting Jake a questioning look "—and we'll be aware of anything that might happen inside the house as well."
She looked at Jake again.
Do they know that we investigate for paranormal occurrences or help? she seemed to be asking.
"I think what you've managed in a few short hours is amazing," Jake told her. "A camera setup like this will definitely catch anyone trying to play games in the cemetery again. But we have to rig by the river as well."
"They've done that," Jackson told him. "We got here at about the same time, and I asked them to make sure the back wall of the cemetery and the river were both in view on our screens."
"Did you have an interesting interview with Ramsay Clayton?" Jake asked him.
"Ramsay swears that he gave Charles the role out of the goodness of his heart. He also said that he'll be back in New Orleans by tomorrow or the next day. He's buying a guard dog and a pistol and getting a permit—he's afraid that whoever killed Charles might have really been targeting him," Jackson told them.
"That is a possibility," Jake agreed.
"Yes," Jackson said simply. "He asked the police for protection. They couldn't give him anything full-time, but they have patrol cars watching his residence while he's here. He said that when he goes back into the city, he's going to stay at a hotel for a week." Jackson smiled. "The casino hotel—there's lots of security around it. Unseen, half the time, maybe, but it is there."
Cliff let out a dry laugh. "Ramsay is a crack shot. But a dog is a good thing to have. Dogs really have a sixth sense. We used to have a dog."
Ashley smiled. "Brutus," she explained. "He was a Rottweiler, and I loved him to death."
"What happened?" Jake asked. He'd never met Brutus.
"He loved me in return—but he hated the horses and tried to bite them all the time, and they tried to kick him. He went to live with my mom's cousin in Gainesville. They're still happy as clams, I believe," Ashley said. She looked at her grandfather. "I should call Gina. She's probably heard all this on the news, and she'll be worried."
"Already called her and assured her we were fine," Frazier said. "She wants you to come stay with her and Brutus. I said it wasn't a bad idea."
"Oh, no. I'm not leaving," Ashley told him. She stood and walked over, slipping her arms around him. "We didn't do this, and Donegal Plantation itself is being victimized." She stared across the table at Jake. "This team will find the killer."
No one said anything for a minute. Frazier patted Ashley's hands where they lay against his chest.
"Well, a hotel at a casino might not be a bad idea," Beth murmured. "Ramsay might have it right."
"I personally couldn't afford a casino for more than a night!" Whitney said.
Ashley walked back to take her seat again. "Sounds like Ramsay's really scared. Doesn't sound like he could be the killer," she said.
"Lots of killers are good actors," Jake said.
"Crab cakes, children, crab cakes!" Beth said. They all looked at her. She smiled. "Hey, we can't live and breathe this every second. We'll lose our minds."
"Beth is right," Frazier said. "Whitney, why haven't I ever met you? Have you ever been out here?"
"Oh, yes! On a school bus with a swarm of children, I'm afraid, but I've been here before. It's a beautiful place, Mr. Donegal."
"Frazier, please," Frazier told her, smiling. "It's great that you're so kind as to respect your elders, but I am just Frazier."
Frazier was a grandmaster at heading his house hold; Ashley seemed quiet during the meal, but Frazier and Beth drew out the newcomers. They learned, of course, that Will was originally from Trinidad, that he had been a musician and an entertainer, specializing in illusion, before he had gotten into law enforcement. Whitney had been a filmmaker, and she loved film—and the world.
When the meal had ended, Jackson nodded to Jake, and they went to the study together. Jake brought up the lists he had made during the day, along with what he had discovered or not about his defined list of suspects. Then he told Jackson about the events in the woods.
"So, Toby Keaton appeared right in the middle of the chaos in the woods?" Jackson asked.
Jake nodded. "He'll bear further investigation. Basically, he and Hank Trebly are the only neighbors. Well, it's a sugar plantation-slash-mill on the other side, but the two of them are definitely close enough."
"What about the 'Yankees'?" Jackson asked.
"Justin Binder is the only man still without a reliable alibi. He was here the night that the body was found. He didn't leave until the police cleared him."
"Tomorrow we'll pay a visit to the sugar mill. And we'll drop in on Toby Keaton, too."
Jackson stood, ready to leave the room first. Jake hesitated a minute, and then rose to follow him. "By the way," Jackson told him, pausing at the door.
"Yes?"
"Angela says that the house is riddled with ghosts, and half of them don't know that the other half are around. And none of them is talkative."
"Well, it is a plantation," Jake said, not yet wanting to talk about Emma Donegal. "What self-respecting plantation comes without its share of ghost stories?"
Jackson studied him and nodded. "Right. Naturally. Maybe we should let the owners tell us about a few. Let's hear what they have to say."
They went out to the front parlor where the bank of screens was set up. Whitney and Will had created their own little viewing station, pulling a few of the big wingback chairs in front of the screens. Whitney turned, seeing him.
"Hey! I brought something of yours from the hotel that you forgot, and I can't believe that you forgot it," she said.
"What?" he asked, puzzled.
"Your guitar," she said.
He raised his brows, opened his mouth and said nothing.
His guitar.
He'd never left his guitar anywhere; it was a beloved Fender.
Ashley and Donegal Plantation. They could make him forget anything.
He was surprised to see Ashley walk over to the guitar; Whitney had set it against the wall by the fireplace. She gingerly touched the case and looked at him, a nostalgic smile on her lips.
"You still have the old Fender," she said.
He shrugged.
"Play something," Frazier told him. "Looks like we're all in for the night."
"And the nights aren't easy to get through," Beth said.
Jake looked at Jackson, who nodded. This is just what we need.
Jake took out the guitar, sat and tuned it. He looked at Ashley. "Still play the banjo?" he asked her.
"Yes, badly," she told him.
He laughed. "Go for it!"
She hesitated, but Frazier said, "Come on, Ashley. For the love of God, let's have some music. We need something."
She hurried up the stairs and returned. Jake looked at her and smiled. "'Never Marry an Ugly Girl?'"
She nodded. They both played and he sang, and they encouraged everyone to join in on the chorus. Once they had started, they fell in together.
As if five years had never gone by.
They played Arlo Guthrie, and Cliff sang, and even Frazier did a rendition of a Frank Sinatra tune.
It went on for about forty-five minutes, and then Jake drew off his strap and set the Fender down. "Hey, let someone else do the entertaining now. Ashley, tell them some of the ghost stories about the place that have some merit."
"Oh, well," she murmured.
"There's Marshall Donegal, of course," Frazier said. "The poor fellow must be turning in his grave at all this."
"What about Emma Donegal?" Whitney asked.
"Wasn't she accused of having killed her husband and somehow covering it all up with—'the Yankee did it!'"
Ashley spoke up indignantly. "That's not true at all, and I don't know where that story got started. There were several diaries kept by the men who survived, and the surviving enemy even told the story the same way. Emma was innocent."
"Ah, well, I thought she was supposed to haunt these halls."
"Maybe. She died here," Frazier said. "By then, of course, her daughters were married and her son had children."
"Where did she die?" Whitney asked.
"She was in the Jeb Stuart room," Ashley said.
Jake started; that was something he hadn't known. And it was strange, of course, because he'd always stayed in that room when he had been a guest at the plantation. He lowered his head quickly. Did Emma appear to him because she felt she knew him?
"There was a fellow who had a heart attack before a reenactment and died in the stables," Cliff offered. "I swear, now and then I think I see a shadow out there."
"Marshall Donegal supposedly guards the plantation in death, just as in life," Frazier told them. "I'm sad to say, in all my days, I've never seen a ghost."
"Have you ever felt one?" Ashley asked him.
He grinned. "Lots of times. I believe that there are spirits here, spirits of the past, of happiness and of trauma. But if we have ghosts, they're here to guard us, to watch over us. There's nothing evil at this plantation," he said firmly.
Ashley spoke slowly. "There's nothing evil at the plantation," she repeated and looked around. "But the living can certainly be evil. Do you think that...sometimes people create evil where there was none, because they believe that it existed in the past, and they encourage it in the present?"
"There's a lot of that going around," Jake said. "Take prejudice, for example, and the old hatreds people never let die. But, yes, Ashley, people can certainly perceive a wrong and turn it into a personal vindication. And it's possible that's what happened here—either that or someone's personal agenda. Who would benefit if this plantation went down? If we can't find discover why anyone would have hated Charles—or gone after Ramsay Clayton—we have to find out who wants to see you go down like a sinking ship."
"No one," Ashley said.
"No one that you realize," Jackson told her.
Beth ended the evening by rising. "I'm going to bed. And, may I say, I'm delighted to have you all in the house!"
Night again, darkness, and the hour growing later.
Jake stared at the ceiling.
He remained disturbed at finding Ashley in the woods, though he wasn't sure why. It seemed apparent that they had all frightened each other.
Was that all that it had been?
Ashley had been up a tree. Well, she had heard Toby Keaton. Toby had said that he'd heard commotion—Ashley. And then him—and then Cliff riding behind him.
Screw it.
He couldn't sleep. He rose and walked to the double doors that led to the wraparound porch. It occurred to him that a gymnast could easily figure out a way to enter the house by means of the porch. The house did have an alarm system, but, as Frazier had said, it was a bed-and-breakfast. They catered to the public. The doors were seldom locked, so it was doubtful that the alarm was often activated.
Out on the balcony, he stared at the night. The moon was now up, its light was shining down with a benign glow. He looked to the cemetery, at the ghostly and beautiful tombs.
And he looked toward Ashley's room, and then started, because, as if he had willed her there, she appeared on the balcony, encased in her gossamer white robe.
She looked his way.
He smiled.
"Couldn't sleep," she said.
"Neither could I."
They stared at each other as heartbeats went by. She kept her distance.
"Your friends are very nice."
"We're a good team."
"Whitney is a doll."
"She's like a little sister."
"Ah."
He laughed. "Really."
She walked to the railing, looking out as he had done.
"The river looks so peaceful tonight—and beautiful in the moonlight."
"Not so great when you're in it," he said.
"True. The police have the rifle and the bayonet at their forensics lab?"
He nodded. "They crawled all over the property today, too. I didn't think that they'd find anything, once we—you—had already found the weapon."
"Will you really solve this?" she asked.
"Yes."
"You're so certain."
"We have to be."
She pushed away from the balcony and looked at him. "Oh, come on, Jake! I'm not walking to you again. I already made the play, and I fumbled badly."
For half a heartbeat, he was still. Then he moved. He covered the distance between them and pulled her to him. He didn't believe he was being used; he wasn't sure he cared. Something, whatever lay between them, surfaced with the music, in the woods, even as they argued. He'd been looking for it without even being aware ever since he had lost her.
Feeling her in his arms, he found it.
He lifted her chin and touched her lips with his, and when he kissed her, she returned it with a passion and hunger that made his knees weak. He had to pick her up and sweep her off her feet quickly—while he still could.
He did so.
"Your room or mine?" he whispered against her lips.
"Either!"
He chose his own. He was certain that Emma Donegal knew all about this desperate kind of love and would leave them in peace.
He lay down with her on the bed, falling into the cool, clean touch of the sheets and the gossamer mist that surrounded her. He smoothed down the white flurry of the gown and met the crystal beauty of her eyes. She stared up at him with complete openness and trust, and he longed to ask her why she had turned from him so completely, but he didn't dare take a chance with the moment, because it was fragile. He found her mouth again, rubbing his thumb gently over the dampness of her lower lip, cherishing the contours of her face. Then his mouth found hers, and again their kiss was instantly hot and wet and filled with passion and hunger. As they kissed, they fumbled with one another's clothing, his easy since he was wearing nothing but briefs, hers a bit more complicated since she wore the light robe and a sheath of silk beneath. But making love had always been easy and natural for them, and in seconds he was feeling hunger and awe that he should be lying here with her once again. She was as smooth to his touch as the silk of her gown, and he swept his hands over her shoulders, cradled her breasts, pressed his lips to her collarbone and shuddered with the pleasure of simply touching her, feeling the pressure of her body, the tug of her fingers in his hair.
"Jake," she whispered.
It was as if there should have been more, but it didn't need to be said.
"Ashley." He murmured her name in return, his lips against the taut sleekness of her abdomen.
Her fingertips played over his shoulders and teased down his spine. He tried to hold on tightly to every moment, knowing moments could become nothing more painful or sweet than memories.
His thought eclipsed to the back of his mind as he rose against her again, found her lips and felt her tear away then to press her mouth against his chest, to press more fully against him. She was a naturally exotic lover, sensually seductive, writhing or undulating just to cast the ultimate moment of arousal in any given place upon his body. He made love to her torso, breathing in the scent of her, tasting the erotic aroma of her soap and essence with his kiss. Her breasts were beautiful and perfectly formed, her waist winnowed away to nothing and her hips were pure fascination. He couldn't touch them enough with his caress, with his lips, with his hungry kiss. As he moved against her body, he felt her breath, felt her body giving, yearning. He moved lower against her form, kissed her thighs, caressed her sexuality and grew ever more urgently in need as he felt her response to his touch. At last he rose above her, hungrily capturing her mouth once again. So entwined, he sank between her thighs and entered her. Sinking slowly into a state that was sure heaven, only to be followed by the blinding light of movement, urgent and passionate, and the explosive feel of her beneath him, undulating, tightening, wrapping around him and giving to him.
Her fingers gripped his shoulder, his back, his buttocks. She wound her thighs around him, and they seemed to rocket with the need and energy of lightning. Thunder tore into his heart, and storm winds created by their mingled breath. The world was still, nonexistent, and everything. And then he felt her gasp, felt the give of her body against his, and he allowed himself to climax, feeling as if he had emptied his body and soul. Giant shudders ripped through him in aftermath; he held her closely against him, and they eased back into the night, back into the bed, back into each other.
I think that I have loved you all my life.
The words didn't escape him; he somehow held on to that modicum of control. Instead, he just held her. Yes, he had loved her. Yes, she had turned away from him. And no, he had never really understood. Maybe she hadn't herself.
He hadn't allowed himself to become a pitiable hermit; he had lived life, and he had enjoyed it. He had even had other lovers.
Never like this.
And it was frightening to be with her again. He'd gotten his soul back once. He didn't know if he'd ever be so lucky again.
So he didn't speak. He just felt their breathing subside together. He listened to her heartbeat, and to his own, and it almost seemed that they there were in a harmony of motion.
"Jake," she said softly.
"Ashley," he murmured.
She was silent for a minute, and he wondered what she had really wanted to say. Was it possible that it would have resembled his own thoughts?
"I'm—I'm so glad that you're here," she said.
"So am I," he told her. "So am I."
He didn't ask her questions; she didn't try to explain. They lay together and drifted, and awoke, and made love again.
And in the end, he slept with her in his arms.
Slept deeply, plagued by no nightmares.
He had been adrift in a boat, knowing that there were those who had to be found. He had seen Ashley, and he had reached for her, and her fingers had slipped through his again and again.
But now he had caught her.
And he could dare pray that the nightmares had really ended.
Interlude
Tonight...
Tonight had been exhilarating!
There had been those moments of stark fear; fear that he would be discovered and caught in the act, that he would be captured.
There had been those terrible moments of indecision.
And yet there had been Ashley.
Ah, Ashley! If she'd been alone, and come upon him, he could have handled her, of course. Not that she was easy—no, not Ashley. She always had been a fighter.
Difficult to think of her in a coffin, set in a vault and left to disintegrate into dust and bone fragments.
He didn't like to think about Ashley dead, not really.
But the aspect of killing her was suddenly...so seductive.
Charles was a big old ugly lug; he hadn't felt much of anything. Marty Dean was a bitch, pure and simple, fake breasts, fake hair, fake smile. He'd felt a pleasure in killing her that he hadn't felt with old Charles.
If he had to kill Ashley...
He would definitely want her drugged. He would want to see her die without a crease in her beautiful face, without a cry of pain. She would be his then, if just briefly.
Ashley was special. She would fit the bill as no other could, and she would also fulfill something in him, some need he hadn't known he had....
Well, not really.
He broke out suddenly in a cold sweat.
Tonight...
Tonight he had almost been caught.
No.
He was getting better and better at what he did.
And they never would catch him.
## 10
Ashley awoke alone.
She was in the Jeb Stuart room, so she hadn't imagined a wild and passionate sexual experience.
In her mind, she had to admit, the fear had existed that she had dreamed the whole thing.
But, no, she had been with Jake through the night, and that made her wake with a smile. She rose, found her clothing, slipped into it and nearly opened the door to the hall. Remembering that her house was now riddled with cameras, she decided to reach her own room through the wraparound porch.
She showered quickly and headed downstairs. It was late; it seemed that breakfast was long over; there was no one in the kitchen or dining room, but Beth always had coffee on, so she quickly poured herself a cup and decided to start looking for the others.
Walking into the roadside parlor, she discovered that Whitney was in front of the bank of screens, comfortably curled into one of the wingback chairs.
The coffee cup nearly fell from her hands as she realized that one of the screens showed the back of the house—and the wraparound porch.
Whitney heard her there and turned around. Ashley's horror must have been clear on her face, because Whitney smiled. "Hey, don't worry! I'm the only one on now. And if there's anyone in this world who can't look at the two of you and know that something is going on, that person is certainly blind."
"I—I—I just—"
"Quit stuttering!" Whitney said, laughing. "Sit—join me."
Ashley sat in the chair next to her. It was amazing—the young woman, and Will, she presumed, had managed to place the cameras so strategically that the whole of the house was covered.
The outside and the public rooms.
There were no screens that covered the bedrooms, just as Whitney had said.
"So..."
"Jackson and Jake have gone to the sugar mill, and then they're going to stop in on Hank Trebly," Whitney said. "Angela is upstairs with a few folks from the police forensics team—they're trying to discover anything they can about who might have stolen the Enfield. Will has gone to get Jenna. You haven't met her yet, but you'll love her!"
"Beth? My grandfather?" Ashley asked.
"Jackson and Jake are dropping them off at a diner down the street—they needed to get out for a bit," Whitney explained. "Don't worry—they'll get them on the way back. Beth suggested that they wait for you, but your grandfather was insistent that you get some sleep."
"It was good to sleep," Ashley admitted. "So, what have you seen on the screens? Anything—besides me sneaking back into my own bedroom by way of the balcony?"
Whitney smiled at that. It was an honest smile. Ashley thought that she and Jake really were just friends. Caring friends, but no more. And it was easy to understand. Whitney was impossible not to like.
"I'll roll some tape. It's interesting," Whitney said.
She hit a button on her remote and directed Ashley to watch the top screen. She did so. The screen captured the area between the stables and the cemetery.
Shadows seemed to undulate on the screen.
"What is it?" Ashley asked her.
"The past?" Whitney queried in return. "It's hard to say. Skeptics would swear that you're seeing movement because of clouds and the moon. I think it may well be the movement of ghosts." Whitney looked at her; she wasn't joking. "It seems that sometimes energy remains—energy from a particularly traumatic event, such as men dying in battle. Some people believe that such hauntings are repetitious—it's just the energy, running in the same pattern over and over again. Then, of course, there's what they call an intelligent, or active, haunting. That refers to a ghost or ghosts who still have their wits about them. And they don't repeat an action over and over. They move about and watch the world."
"Oh," Ashley said simply.
Whitney smiled again. "So, what kind of haunting do you know best? I have a feeling about you. You're like Angela—you just don't know it yet."
"Pardon?"
"Everyone is afraid at first," Whitney told her.
"Afraid?"
"Of ghosts, of course," Whitney said.
Ashley just stared at her.
Whitney continued. "The thing is, most of the time, they just want to help. They've made their mistakes. They want to keep others—especially their descendants—from making the same mistakes. I know that this house is haunted, but ghosts, in my experience, don't want to meet everyone. I believe, though, that the ghosts very much want to meet you. They love you. They want to shield you from all danger, and they want to preserve Donegal Plantation, too, because it carries an important lesson. The place is all about the path that we've taken as Americans, the good things and the bad. The sane among us don't want to repeat mistakes of misunderstanding, cruelty to our fellows or war."
"I believe we do a good job with education here," Ashley said. She realized that her voice sounded raspy. She took a long swallow of her coffee, afraid to say anything else.
Whitney smiled and shrugged. "Anyway, that's what I believe," she said softly. Then she added, "Hey, that was great last night—the music, I mean."
"Jake and I played together...before," Ashley said.
"That was obvious. You're really good together," Whitney said. "In many ways." She set the remote down, stood and stretched. "Hey, want to introduce me to the horses? I love horses, and I never get to ride. I'd love to meet them. We're certainly safe enough—the place is surrounded by cops at this moment, and besides, Angela is here. She made some of the best scores ever at the target range back when she was a cop."
"It's hard to believe Angela was a cop," Ashley said.
"That's like a fairy-tale princess in a patrol car, huh? But, hey, even Disney princesses are toughening up these days. We're all capable of many things, right?"
"So I like to believe," Ashley said. "Is it all right to leave the screens unattended?"
Whitney nodded. "Everything is taping." She hesitated. "Okay, so half my job today is to make sure you're safe. We might as well enjoy it, right?"
Ashley smiled. "Yes, and I love our horses, too. Come on out and meet them."
It was obvious Hank Trebly wasn't pleased to see Jake and Jackson—he left them sitting patiently in his waiting room for an hour.
He nodded curtly to them after emerging from his office and motioned to them to follow. He didn't shake their hands, even though he'd known Jake before, just as Toby Keaton had known him, from the days when he had been at Donegal Plantation constantly.
"I don't know what you think I can tell you," Hank said, pulling out the chair behind his desk in his office and taking a seat. "I was there, yes, of course. But I didn't see anything. And I told that to the police. They sent a man around right after, you know—right after they found the poor bugger's body."
"You weren't particularly fond of him, were you?" Jackson asked.
Trebly immediately took a defensive tone. "What was to like or dislike? He was O'Reilly's stepson, and he didn't really belong. I mean, he never even took his stepfather's name. He was...just this big nothing, always there, always wanting to be a part of it all. He didn't have anything to add to any of our conversations in a roundtable. It was his place to play a Yankee, but he whined so much Ramsay gave him the role of Marshall Donegal. But I didn't hate him. And, sweet Jesus, I could never do that to another human being! I mean, why in hell would I?"
"Well," Jake said, "there's that bid you made to take over the Donegal property."
"What?" Trebly said, sitting up straight. His eyes narrowed. "Frazier Donegal told you that?"
"No, actually, Frazier didn't tell me," Jake said. "It's public record. Right after Frazier's son, Patrick, died, the property fell into bankruptcy. You went to the bank, trying to coerce a sale."
"I—well, that's just pure bull!" Trebly said, but his face had gone pale. "Look, I was willing to help Frazier out."
"I don't think so," Jake said. "There were also plans on file to expand the sugar mill and the sugar fields."
Trebly sat back and stared at them hard. Then he dug around in his desk for a business card and tossed it over to Jackson. "I'm done speaking with you. If you need anything else from me, you can call my attorney."
"Thank you," Jackson said, rising. "We'll do so."
Jake stood behind him. Trebly didn't rise; he looked like he was going to explode.
"You shouldn't have come back here, Jake Mallory." His voice broke. "You're not wanted in the area. That was obvious after Patrick died."
"This is what I do," Jake said.
"You shouldn't have come back," Trebly repeated.
Jake shrugged and started out. He was surprised when Jackson lingered. "Is that a threat, Mr. Trebly? Because if so, you just threatened a government agent. I could actually have you brought in for that."
Trebly rose at last. "I didn't do it, damn you, and I don't know anything about who did. I didn't like Charles Osgood, but Ashley was probably the only person who felt sorry for him—or even felt genuinely bad that he was dead. Hell, yes, we could have used that land, but I didn't force the issue. Old Frazier pulled it together. We aren't expanding, and I didn't kill Charles Osgood!"
"Thanks for your time, Mr. Trebly," Jake said.
"Jackson?"
Jackson turned around at last and they left the offices and the mill building.
"The man is an ass!" Jackson said.
"Maybe. But I believe him. I don't think that he did it."
Jackson shrugged. "Right now, I can't really tell. He was shaking, and he looks as if he might have really high blood pressure. He's not in the best physical shape, and we are looking for someone with strength and dexterity."
"Let's move on to Beaumont," Jake said.
Jackson nodded. "We'll pick up Frazier and Beth, drop them back at the plantation. I don't want them deciding to get back some other way."
Jake didn't reply. Jackson was worried about the plantation household.
So was Jake.
As long as the cops were still on duty, treating the entire plantation as a crime scene, the plantation itself seemed safe.
But that's where Charles Osgood had been found. Safe was the last word he'd use.
"This is my favorite, my mare—even though she gave me a toss last night!" Ashley said, stroking Varina's soft nuzzle. "On this side, next to Varina, you have Jeff—after Jeff Davis, of course—and at the end, down there, Bobby—for Robert E. Lee."
"A true Southern stable!" Whitney said.
"Not really. Across there you have Abe—for Abraham Lincoln—and then Nellie and Tigger. Go figure. We just weren't consistent, I'm afraid. Actually, Jeff is Nellie's offspring, so we did name him, but we named him Jeff because we already had a Varina!"
"They're beautiful," Whitney said. "I wish we could go riding."
"We can go riding," Ashley said. She saw Whitney's face. "Oh, that's right. I'm not allowed to go riding. It's light out, though. So can I go with you?"
Whitney was thoughtful for a minute. "I'll give Jake a call."
"I thought Jackson was the head of the unit."
"Jake gave me the order." Whitney grimaced.
"Look, I went out late yesterday afternoon, and I shouldn't have, but honestly, I didn't think Varina would throw me. And nothing happened, really. We just all scared each other."
"And Toby Keaton showed up, and no one trusts the man at the moment," Whitney said.
"And Cliff," Ashley added.
Whitney watched her, nodded, and slowly smiled. "You think we're full of it for suspecting Cliff, don't you?"
"Yes."
"I'm sorry. He is a suspect."
"Right, so I've been told—because he has access, he has motive," Ashley finished.
Whitney drew out her cell phone. She turned away from Ashley, putting through her call. Ashley could hear Jake's voice on the other end of the line, but she couldn't make out his words.
But Whitney was smiling when she hung up.
"We have permission? I'm shocked," Ashley said.
"He's just trying to protect you, you know," Whitney said.
Ashley smiled. "Yeah," she said quietly. He was trying to protect her. "But you know what? I may not have been a cop like Angela, and I may not be a crack shot—but I did grow up here, and I do know how to use a shotgun. Guess I'll go drag mine out."
"I'll get Angela. Jackson and Jake are almost back with your grandfather and Beth. If you get the horses saddled—and I'm a decent rider, not great—we can head out as soon as they drop off Frazier and Beth."
"Are they staying?"
"No—they have another stop to make. They're going to have a talk with Toby Keaton," Whitney explained.
Whitney headed back to the house, her dark curls bouncing as she walked. Ashley was about to knock on Cliff's door and find out if he'd mind helping her saddle up the horses, when he walked out.
"We're expecting a load of hay," he told her. "I imagine the cops will let that through. Seems like they've been doing a good job, though. I haven't seen any reporters around for a while."
"Maybe they've decided that peering in from the road just isn't that exciting," Ashley said. "Apparently, they've got something going with the police. I think that a police spokesperson has been handling news on the investigation. Well, actually, what do I know? We're all at the house having the same experience."
"Cops are supposed to come out—or forensic people, whatever they are—and go over my apartment," Cliff said.
"And you don't mind?" Ashley asked him.
"I want to be cleared," he said.
She smiled. "You're clear in my book."
"Well, and that's what counts to me. But I don't want to spend the rest of my life with people glaring at me if they don't catch the bastard."
"They'll catch him," Ashley said.
"How do you know that?"
"Jake said so," Ashley told him.
Cliff nodded. "Well, God knows—I hope so. I pray so, Ashley, I really do."
They looked at one another as they heard tires on the gravel near the front. Walking around, they saw that a white minivan had just pulled into the front. Will Chan emerged, followed by a tall, slim, red haired woman.
"Hello!" she called out, seeing them watching her from a distance.
She said something to Will, who nodded and grinned. He went for the bags at the back of the car while she walked over to them.
Her brilliant green eyes shone out over her easy smile. "Hello, I'm Jenna," she said, offering her hand.
"Jenna, lovely to meet you. We've been expecting you, of course," Ashley said. "Welcome to Donegal Plantation."
"It's brilliant," she said. There was a lilt to her voice, very soft, and yet it spoke of an Irish background.
Jenna looked at Cliff and seemed to assess him quickly. "You're—"
"Cliff. Cliff Boudreaux. Suspect," he said dryly.
"Ah, well, we're all suspect at one time or another, aren't we, then?" she asked.
"If you say so, Jenna," Cliff said. He was grinning; the two were looking at one another with amusement and something like an instant rapport. Ashley found herself amused.
"I'll bring you into the house, Jenna, and show you to a room. There are a few left to choose from—" Ashley began.
"I can—and have—slept on many a floor. Put me wherever you would like, and I'll be just fine," Jenna assured her.
"Oh, Cliff, would you saddle Varina, Nellie and Tigger for me? Whitney wants to go riding, and Angela is going to come, too."
"Riding?" Jenna's eyes lit up.
"Cliff, add Jeff to the horse list, will you?"
"Oh, I think she'd do well on Bobby. You're a rider, aren't you, Jenna?" Cliff asked.
"I like to think so," Jenna said.
"Bobby, then. Bobby it is!" Cliff said.
"Are you joining us?"
"Can't," Cliff said. "We're expecting a delivery. But hopefully we'll get a chance somewhere along the line to head out together."
"It's a date," Jenna said.
Bemused, Ashley led their new guest to the house.
Frazier and Beth were still sipping coffee when Jake and Jackson went in to pick them up for the ride back to the house.
They both looked a little brighter for having escaped the house for a while, but Beth was grave.
"We have another missing person," Beth said.
"What?" Jake asked her.
She lifted her coffee cup and indicated the television above the diner's counter.
"The news just had a thing about it. That reporter—I think you knew her?—Marty. Marty Dean. She didn't show up for work today, and her station has plastered her picture on the news a dozen times. See—it's who you know," Beth commented to Frazier. "This woman was a newscaster—people saw her every day. So they ignored us when Charles disappeared, but they're all over it now because she's missing."
"It's true," Frazier said. "Her coworkers believe she wouldn't miss work. Ashley was upset about Charles. She knew him. She knew something bad was going on when he turned up missing. But Charles didn't work for a news program, and neither did Ashley. If I hadn't known Adam—"
"Think about the people who don't know Adam, either," Beth pointed out.
"They might have found Charles alive if anybody had really been looking," Frazier said.
Jake stared at the television. An old Western rerun was playing.
"What did they say about Marty Dean?" he asked.
"She rushed out on a tip yesterday afternoon, said she'd be back for the eleven-o'clock news, but she didn't make it," Beth said. "She still hasn't shown up."
"But she was in New Orleans, right?" Jackson asked.
Beth nodded. "Yes, she disappeared from New Orleans."
Jake looked at Jackson. He had a bad feeling—a really bad feeling.
But the police in New Orleans would certainly be on the disappearance. Forty-eight hours or not, the media would be forcing them into action.
"Let's get them back and look in on Toby Keaton," Jackson said.
It was nice, riding with the other women, even if their previous acquaintance made her the odd man out.
Jenna was a welcome addition. She was sweet and energetic, but it was really the accent, Ashley decided. Americans loved accents from the British Isles, English or any variety, whether Irish, Scottish or Welsh.
And Jenna oohed and aahed over the river, over the cemetery, over the horses, over everything. She seemed to love Donegal Plantation, and as they rode, she told Ashley that her expertise was in nursing.
"A federal agent—in nursing?" Ashley asked.
Jenna waved a hand in the air. "I have other talents," she said.
"Oh?"
"I'm not as good as Angela," she said flatly. "But I speak with the dead."
"A little subtlety might be in order!" Whitney called out.
Ashley twisted around in the saddle to see Angela, who was also rolling her eyes.
"The cameras, the shadows, Jake...is this a paranormal unit?" Ashley asked.
"No—we're a regular unit," Angela said.
"We're a special one," Whitney told her.
"We look for what's real," Jenna said. "But we may be able to find what everyone can see by seeing what everyone can't. There, does that clear it all up?"
"Just like Mississippi mud," Ashley said.
"Jackson came from a regular behavioral-sciences unit," Angela said. "Then Adam Harrison formed a team when Senator Holloway's wife, Regina, died. Jackson knew all about forming a team and learning each member's specialty, but Adam had done the legwork already, finding those he wanted."
Ashley studied them all. "I know the case, of course. But people were saying that ghosts had killed Regina Holloway. I suppose we're in much the same position here."
"Exactly. Here's the thing. Jackson knows that things happen out of the norm, but he's also aware that living people usually prove to be behind it," Angela explained. "He's a true skeptic—a prove-it-to-me man. The thing is..."
"The thing is what?"
"Ghosts do exist," Whitney said. "And no one sees them more clearly than Angela."
Ashley looked ahead as they rode on.
Had Angela seen Marshall Donegal yet? She hadn't seen him herself today. Maybe he had decided he might find a more intelligent life force in someone else?
Angela smiled. "Well, I also believe there has to be human evil involved. But...Whitney is telling you the truth. We investigate for ghosts. Not because they're evil, though, honestly, evil men make evil ghosts. They'll encroach on someone's mind, but they can't carry out evil deeds. It's like hypnotism. If it's something you won't do, you still won't do it, no matter how a ghost tries to play with your imagination."
"Wait a minute. It sounds like you're saying 'The ghost made me do it!'" Ashley said.
"Not at all," Angela said gravely. "What we're saying is that...well, if the evil in a man's soul or spirit remains behind, it can act as fuel to someone who is already a madman. But most of the time, those souls that linger are yearning only to bring something to right, to protect those they might have loved. To bring about some kind of resolution or conclusion."
"Interesting. What about a ghost who lingers over a hundred and fifty years?" Ashley asked.
"I'm sure that ghosts have lingered much longer," Angela said. "Sometimes they're just waiting to be freed. And, I imagine, sometimes they like being ghosts and doing what they can for the living. God knows, I don't have all the answers."
"But—you talk to ghosts!" Ashley said.
Angela smiled. "Yes, but you see—they don't have all the answers, either. This direction we're riding in—it's toward the bayou and the plantation next door?"
"Yes, our best trails are out this way toward the bayou. Just past the outbuildings on the other side, it all turns into barbed wire for the sugar fields."
"There have to be gators out here," Whitney commented.
"There are. We leave them alone, they leave us alone," Ashley told her. "And you'll only see them when we're right on the bayou. They seldom venture as far as the riding trails. They like their watery habitat."
"Alligators. Ugh," Jenna said, shuddering.
"To tell you the truth, we worry more about the snakes," Ashley told her.
"Oh, great! It might be about time to head back, eh, friends?" Jenna said.
"Just a bit farther, please," Angela said. "I want to see the Creole plantation."
"The bayou is between our property and Beaumont," Ashley told her.
"That's all right. I just want to see it from here," Angela said.
"It's just up ahead. There's a twist in the trail that goes right down by the bayou," Ashley said.
"You'll be able to see it from across the water there."
Jake parked in front of Beaumont.
It was an entirely different place from Donegal Plantation, not as grand, and there was no canopy of oaks along a sweeping front drive. By car, the house was reached by a massive gravel parking lot in front. Wooden fencing surrounded the residence itself, which was only two stories. Jake had been in Beaumont himself, and he found it as fascinating as Donegal, just different. Here workrooms and space for the animals had once taken up the first floor, or raised basement, while the household had always lived in the second floor. Like many of the other plantations, Beaumont had been a working sugarcane farm, and the outbuildings remained as they had been, important parts of the tours that were given here. The Creole way of life, rather than that of the English planter, held sway here.
Toby Keaton had inherited the house through his mother's side of the family. She had been a Thibadeux, from an old French family. His father, whose family line also went back for generations, had hailed from the English who had settled in the Garden District of New Orleans. Toby had been divorced for years; his one son, now in college, was being groomed to take over the family business when Toby grew weary of it...or willing to give up being the one in charge. Jake could remember meeting Josiah Keaton; he had been a handsome yet solemn young teen when he had last seen him, aware of the responsibility that would one day be his.
"Intriguing place," Jackson said, sliding his sunglasses down on his nose as they emerged from the car.
"Toby runs a good business," Jake said. "He and Frazier have always supported one another."
Jackson looked at him. "But Donegal Plantation has become a bed-and-breakfast. Beaumont has to be doing better."
Jake shrugged. "It pulls in a higher gross, yes. But the expenses are higher, too."
"Still, Toby Keaton could want Donegal to go down. And last night, when you went after Ashley, the man made an appearance in the dark—on the other side of the bayou."
"True," Jake said.
Before the path that led to the house there was a little kiosk where tickets could be bought. A woman wearing French Creole Empire–style clothing, circa 1820, was behind the counter.
"Hello," Jake said. He flashed his badge; he didn't know her. She was young and had probably taken work here to get through school herself. "We're looking for Mr. Keaton. Can you tell me where to find him?"
"I wish!" she said irritably. "He hasn't been around this morning—and he's supposed to be handing out paychecks. His car is here, but he is not."
"He lives here," Jake said politely. "He has to be somewhere." He pointed to an overhang in the parking area; there was a shiny new Honda parked there. "Is that his car?" he asked.
"Yes." She flushed, looking at Jake. "I'm sorry, Officer. I just don't know where he is. I've tried his cell phone, but he wasn't answering. I opened the sales window at ten, just as I'm supposed to do. Everyone who is supposed to be here working is—you can ask Dan out by the field, or Martha, who gives the tours up at the house. Maybe one of them has seen him."
Jake glanced at Jackson and thanked the girl. They walked up to the house, where a costumed interpreter—Martha, he assumed—met them at the door. "The next tour starts in thirty minutes," she said cheerfully. "You're welcome to explore the grounds while you wait."
Once again, they produced their badges.
"Oh, dear!" Martha said. "What's wrong? Oh, is this about poor Mr. Osgood?"
"We need to speak with Mr. Keaton," Jake said.
"I wish I could help you, sir. Mr. Keaton hasn't checked in with any of us this morning."
"Is that unusual?" Jackson asked.
"Well, yes, of course. He is a hands-on man where business is concerned," she said. She was thoughtful. "Well, of course, there was the morning after he'd met with a few of his cronies, and we found him passed out in one of the rooms we show. It was rather frightening. We have mannequins in that room to add to the historical setting, and there was the monsieur, the madame, the jeune fille—and Mr. Keaton, all messed up and on the bed between them all! My poor guests—"
"He's not there now, you're certain, is he?" Jake interrupted.
"Oh, no! I've had several tours through the house already today!" she said. "But I don't suppose that anyone has searched all the outbuildings. There isn't a guide stationed in every one. We have a girl who comes in at ten—"
"Excuse me. I understand there's someone in the field we might talk to?" Jake interrupted again.
"Oh, yes, silly me, I do go on. I can see Dan now—he's in the straw hat, over down by that wheelbarrow."
"Thank you," Jake told her.
He felt a growing concern as he and Jake walked down the expanse of sloping lawn to find Dan, a tall, muscular, African-American man sorting through an enormous pile of produce.
Jake remembered Dan, though he doubted that Dan would recognize him.
But Dan did.
"Is that you, Jake Mallory? Sad things going on, sad things! Why, I remember you riding like wildfire all over that big plantation next door," the man said to him, giving Jake a brief hug.
"Good to see you again," Jake said. "We're here looking for Mr. Keaton."
"So am I," Dan said. "Paychecks are due today, and if you don't remind that man, he doesn't remember to pay us."
"Do you mind if we start searching for him? Have you been in all the outbuildings today?"
"I gave tours in the old kitchen and the smokehouse. I have been in all the slave quarters. They should be open, though. Martha walked around and unlocked them as soon as we came in and saw that Toby wasn't around."
"Thanks, Dan."
"I'll give you a hand, if you need," Dan offered.
"That's fine, Dan. It won't take us long," Jake assured him. "You seem busy."
"I'd like to get these to the kitchen, but if you need me, just holler."
They started for the row of old wooden slave quarters. But before they reached the first, a bloodcurdling scream stopped them in their tracks. It came from the other side of the bayou.
The scream was followed by a volley of gunshots.
## 11
Ashley had heard the grunting long before they'd reached the bayou.
"What in the Lord's name is that?" Angela demanded.
"Gator," Ashley told her. "Sometimes it sounds like you stumbled upon an entire pig farm. That's the noise alligators make."
"Should we come any closer?" Whitney asked.
"We're fine. We just stay on the road and make sure that we don't bother them. Remember, we don't bother them, and they don't bother us. They're really not insane predators chasing everything that moves," Ashley said. "They can move on land, and move fast, but, usually, they hunt in the water. They wait for their opportunity, and they snap, and then twist and turn in the water, drowning their prey. They're as instinctive as other predators—they don't want an eye gouged out by a flailing claw or the like. If you go to any of the gator parks, you'll hear them sound like that when it's mating season, or when an employee is about to hand out the chickens—dead, of course."
She was in the lead, but Whitney was right behind her. Ashley had twisted in her saddle to speak, and so she didn't know what Whitney saw when her eyes suddenly went wide and a scream escaped her lips.
Ashley turned back; she didn't scream. She let out a gasp of horror and pulled out the shotgun she had opted to bring along on the trail.
She could see why the gators were going crazy.
They had been fed.
Some were on the embankment; some thrashed in the water, adolescents and adults, maybe eight to ten in total. It was difficult to tell, because they were snapping at one another, the large ones going after the smaller ones. But they were all hungry for human flesh.
A number of the beasts had already started in on the bodies; it almost appeared that they had been playing with mannequins. The two bodies in the water were muddied and mangled beyond description, certainly, at this point, past any sense of pain or horror. It was impossible to tell if they'd been dead when they'd been discovered by the alligators, or if the gators had gone entirely mad and attacked two human beings.
Usually, upon discovering large dead prey, an alligator would drag it around until decomposition and the water softened down the bones, but when they started fighting over prey, it was like a scene out of a horror movie. The bodies might have been toys being ripped apart by children; the snapping jaws landed on limbs, and they were ripped cleanly away.
Ashley took aim at the head of a large adult, aiming right between the eyes. Her shot was true; the jaws clamped shut, but then the gator began to sink. Behind her, Angela was taking aim with her pistol. Ashley shouted, "The eyes! Strike between the eyes!" Luckily, the horses had grown up around hunters, and though Varina had thrown her before, the sound of the gun didn't bother any of the horses.
Ashley emptied the second barrel of her shotgun, horrified at what she was seeing and equally horrified that she was killing so many of the beasts who were only acting as nature intended; she felt a sinking certainty that the alligators hadn't attacked standing adults. They wouldn't have needed to, and they wouldn't have attacked unless an idiot had gone with food and become part of it. They would, as they were doing, go for a tasty morsel of meat that was dead and decomposing.
Angela was shooting with a pistol, and her shots riddled the air as Ashley reloaded.
The scent of dead alligators mingled with that of ripped flesh—and more fights erupted, but the sound of the shots was finally entering into their lima-bean-sized brains, and they began to move off at last. Ashley and Angela stopped shooting.
Whitney, as stunned and shaken as the rest of them by the bizarre and horrible spectacle, fumbled for her cell phone. But before she could dial with her trembling fingers, they heard shouts.
Across the bayou, men burst through the trees. Jake and Jackson were followed by Dan, Toby Keaton's manager. He was armed with a shotgun, while Jake and Jackson had drawn pistols.
They didn't need to fire; they clearly saw the carnage, and it was evident that they were equally stunned and appalled by the sight.
"Ashley! I'm getting the bodies out. Cover me," Jake shouted, his voice stretching across the thirty feet of muck and water. He stared at the women on the horses, and back to the water, at the gators now whipping their tails to escape, and back to the dead creatures, floating absurdly between what remained of the human corpses.
He started into the water.
"Jake!" Jackson roared.
Jake looked up. "I know what I'm doing, Jackson. Aim for the head, the eyes if you can, if anything starts to come near me."
"Jake, don't!" Ashley screamed.
But even Jackson seemed to know that Jake was right and motioned him forward. To discover anything—who the mangled bodies had been—they had to get them out of the bayou. Now.
Jake, maintaining a grasp on his pistol, walked carefully into the slick muddy water of the bayou, and seemed to sink into it slowly. The deepest water here was about five feet; his head was still above water when he pushed his way past a dead alligator and reached the first body. Grasping it around the shoulder with his one hand, still watchful of any movement near him, he eased back to the bank with it. Watching, Ashley still couldn't tell if the mud encrusted pile of death was male or female. She could see that it was missing limbs.
She wiped sweat from her brow, barely blinking, watching the water for any sign of movement. An alligator could have been lurking below and heading out to strike beneath the surface, but there was no way for the great jaws to snap shut on a standing man unless the beast twisted and turned, and she would see that motion.
She heard Whitney chanting behind her.
"Jake, come on, Jake, Jake, Jake...."
He went back for the second body while Jackson and Dan pulled the first up the bank. Dan was already on the phone, Ashley saw out of the corner of her eye. Help would be coming soon.
Not soon enough for those being dragged from the bayou....
She saw a ripple in the water twenty feet from Jake. There was no way to take aim between the eyes, so she calculated a few feet before the ripple and shot, and then shot again. Willing herself not to fumble, she tore at the packaging of a cartridge and loaded again.
But the ripple bobbed to the surface; she'd made a clean kill, and Jake grasped the remains of the second body and dragged it to the embankment. As he crawled out, Jackson reached down to drag him from the water and up the slick, muddy embankment.
He stood up; his eyes met hers across the water, but he wasn't smiling. He was grim.
"We found Toby Keaton," he said dully, his voice barely carrying across the water.
"God in Heaven! God in his Heaven!" Augie breathed, looking at the corpses. Jake knew, of course, that finding a corpse in situ was the best possible place for a medical pathologist to first encounter a murder victim, but if they hadn't intervened, there might not have been actual body parts to be discovered. Toby Keaton was already missing a lower leg and his left arm, and the woman—Marty Dean, of all people—had lost her right arm to the shoulder and was missing a calf and foot on the opposite leg. Lain out, high on the grassy embankment of Beaumont, they formed pieces of a grisly human puzzle. Ashley, Angela, Jenna and Whitney had described their discovery of the bodies—and the gators—once the authorities had arrived and ridden back to Donegal Plantation.
"Can you tell me how they died?" Jackson asked Augie.
Augie, down on his knees near the heads, looked up at Jackson. "Are you kidding me?" he asked.
"I don't think they were killed by the gators," Jake said.
"I agree, but I'm going to have to get them back to the office to determine an exact cause." He indicated the places where flesh had been torn away. "These pieces have most probably been consumed, but I'm not seeing a blood flow on the bodies or in the water that would indicate that their hearts were still pumping when the alligators attacked, and liver temperature suggests that they've been dead for hours—twelve to twenty-four—but they've been in the bayou, so that can mess with temperature. I'm not seeing any gunshot wounds or slashes that look as if they were made by anything but giant snapping jaws. Frankly, I wouldn't trust any of my own findings in these circumstances until I've had time at autopsy. Jesus! Jesus, Mary and Joseph!" Augie finished, crossing himself.
"Might be the same, though," Augie added after a moment.
"Same as what?" Jackson asked sharply.
"I found massive doses of benzodiazepine—a sedative—along with chlorzoxazone—a muscle relaxant—in Charles Osgood's body. I believe that Charles was held unconscious and controlled by the two drugs until he was taken, still unconscious, to be killed at the site in the cemetery."
Jake looked at Jackson; they both held silent. Drugs were involved. Either they might start looking at M.D.s or pharmacists—or at robberies of pharmacies and doctor's offices.
Such as the offices of Dr. Ben Austin?
Jake decided to see what else Augie could tell them before assuming that these two had been killed the same way.
Mack Colby arrived. Both Jackson and Jake stared at him across the gruesome display of the bodies.
"Hey, Doc, help me out here!" the detective protested. "We just received the reports when we were called out here. Throw us a bone, here, please. Help in any way you can."
"I'm doing my best—with what I've got," Augie said. "Those reports were sent to you, too, Mr. Crow," he said. "I emailed you a copy of my findings just about an hour ago."
Jackson nodded. Jake said, "Thanks."
"So?" Mack Colby asked.
"It isn't death by drowning, and it's no damned accident, I'm certain," Colby said. "So, do we proceed with this as a dual investigation as well?" he asked.
"We would certainly appreciate continuing so, since Mr. Keaton has been a suspect in our existing joint investigation," Jackson said.
"Hey!"
They heard a shout from one of the uniformed officers who had been searching the surroundings on the Donegal side of the bayou.
"I've got a shotgun here," the officer cried.
He was down near the water and holding up the weapon with a gloved hand.
"Probably Toby's," Dan offered. His dark eyes were red-rimmed; he might have had a few arguments with his employer, but it was obvious he had cared. "How the damned hell did someone get to Toby when he always went out with his shotgun?"
"Maybe he saw someone he thought was a friend," Jackson suggested.
Dan's lips pursed.
"You don't have to stand here seeing this anymore," Jake said.
"I'm just waiting for gator season," he said gruffly.
"Can't blame a wild creature for acting like a wild creature," Augie said, standing and placing a hand on Dan's arm. "There was most certainly a human monster involved in this, and rest assured, the law will take care of him."
"We won't stop," Jake said as Dan looked at him.
"Hope you find him first," he said quietly.
Jake hoped they did, too; the man was extremely tall, but he was pure muscle. And that kind of muscle combined with pain and fury could be dangerous.
"You're a good man. Don't go trying to solve this yourself. There is no such thing as a righteous kill. You'll wind up in prison," Jake told him.
Dan lifted his hands. "Where do we go from here?" he asked quietly.
"You keep the place running for Toby's son," Jake told him.
"Close her down for a few days. We have to get to the bottom of this," barked Colby.
"I live here, too, in one of the old smokehouses," Dan said.
"That's fine. You stay on. But we'll need a few days of traipsing around here," Colby told him. He shook his head in disgust. "A bayou. A damned bayou filled with snakes and gators. Not easy to find much around here." He looked at Jake and Jackson. "Hell, boys, you are truly welcome to this mess!"
By the time they returned to Donegal Plantation, it was already late afternoon. The officer had brought the shotgun over to Beaumont, and Dan had positively identified it as belonging to Toby. Toby's little bayou-crossing aluminum boat was still on the bank on the Donegal side of the water as well; he had never made it back to his own property the night before.
Jake described his last meeting with Toby in the woods, repeating information he had already given Jackson to Mack Colby.
"Where the hell does the reporter woman fit into it all? What was she doing out there? High-heeled type out by the bayou—makes no sense!" Mack said.
"She was lured," Jake said slowly. "Someone lured her out on purpose." He paused, remembering what Ashley had told him the night before. She'd been frightened; she'd been up a tree. And she had seen someone below her. Toby—they had all assumed.
It had been the killer.
He felt an inner trembling and became anxious to return to the house and to the rest of the group. There hadn't been much for the women to explain at the scene. They had come upon the feeding frenzy and shot at the alligators to get the creatures away from the bodies. Not even Colby had seen a reason to make them linger with the corpses.
Back at the house, he quickly sought out Ashley—even before heading to the shower.
Ashley, however, was showing her mettle. He found her sitting at the dining-room table with Angela, going over everything that had happened on the day of the reenactment. When Jake and Jackson came in, both women looked at them gravely.
"I'm sorry. I know Toby was your neighbor and a friend," Jake said.
Ashley nodded. "And I'm sorry about your friend. The reporter—I didn't know her."
"Well, I'm sorry, too, but we weren't actually friends. She was in my high school class."
Jackson and Angela were both there, and Jake held his reserve. But he asked softly, "Are you all right?"
"I'm fine. I'm angry. Someone is trying to destroy all of us," Ashley said.
Her eyes were level with his. She was just fine. She hadn't lost her cool for a minute when he had gone into the water, and though Angela was a crack shot, he had depended on Ashley; she knew the terrain and the ancient beasts better than anyone.
She was going to do all right.
Jake grimaced. "I'm going to shower," he said.
"This certainly sounds strange after...today," Ashley said. "But...it's almost dinner. We're having a vegetarian pasta dish, Beth decided."
Jake nodded and left the room, going upstairs.
He was surprised, yet not really, when Ashley appeared ten minutes later; she slipped into the shower behind him and wrapped her arms around his waist, resting her head against his back. He set his hands on hers, and they listened to the beat of the water and felt the steam around them for a minute. Then he turned and took her into his arms.
He kissed her; they touched and kissed in the hard spray for a long time.
"Is this horrible?" she whispered.
"No. It's life-affirming," he replied.
They made love quietly, and then more passionately, beginning in the heat and steam of the clear, clean water and ending up on the softness of the bed, entwined in one another's arms.
When they were done, her eyes were closed, and he leaned on an elbow looking down at her. She smiled slowly, a little wistfully, somehow feeling his gaze.
Her eyes opened.
"You used to do that all the time," she told him.
"What?"
"Watch me. It's a bit unnerving, you know."
He kissed her lips lightly. "I used to watch you and wonder that you were with me."
She was quiet, not meeting his eyes, staring up at the ceiling. "That's crazy. It was a wonder that you were with me."
"You were the one who ended it."
"I was...scared."
"Of me?"
"Of what you seemed to know," she said.
He pulled her to him. "And are you still scared? Nothing about me has changed."
"But something about me has." She rolled away from him and rose. "It's going to be time to eat, and as awful as it seems after today, my stomach is beginning to growl fiercely."
He nodded. "It's just biology."
She reached for her jeans, closing her eyes briefly. He could clearly imagine the pictures in her mind; he had seen them, too.
She looked at him and tried to smile. "Get up! They'll all know where we are and what we're doing, but I'd just as soon they don't have to come find us."
"They know?" he said, frowning.
"Cameras, remember?"
He groaned softly; he hadn't thought of that aspect.
"Well, they'll know we're safe together, and these days, that's a good thing. I just feel a little guilty...."
"Because of me?"
"Because of Frazier."
Her grin turned real. "If we weren't in the middle of a horrible mystery, I'd even suspect that Frazier knew exactly what you were doing and who you were working for—and called Adam just to get you back here." She paused and kissed him lightly on the lips. "Come on down, Mr. Mallory, please."
Buttoning her blouse, she headed for the door.
He watched her go, worried. Images flashed through his mind.
Charles Osgood, hanging from the statue.
Toby Keaton and Marty Dean...what was left of them.
He stared at the ceiling, trying to let the logic in his mind take over. Toxicology reports would affirm, he was certain, that both victims had received a similar cocktail to that which had been given to Charles Osgood to keep him compliant until the time of his death.
This time, though, it seemed that the killer hadn't had any intention of keeping his victims alive for long. He had probably discovered that a good shot of his drug cocktail immediately disabled his victims.
He knew, just as predatory alligators did, that he could be hurt himself in a fight. That was why alligators drowned their prey; they disabled them before they could be attacked in turn.
The killer was basically a coward. But he was changing his method, like a man with an agenda. He didn't fit the profile of a killer who sought out a certain type of victim; he didn't molest his victims. He was there for the kill itself, and it was beginning to look like the kill itself was the goal.
Jake's head jerked up. A killer who had started with an agenda, and now had discovered how much he liked killing.
He would hunger for more and would strike again. Eventually, his need would make him careless. Eventually—but how many would have to die first?
Downstairs, Ashley found Will and Whitney speaking quietly to one another as they watched the monitors. Jackson, Angela and Jenna were holed up in the study with Frazier. Cliff hadn't come in yet, and Beth was watching her marinara sauce simmer.
"Ten minutes," Beth told her. She shivered, looking back at the stove.
Ashley decided that she'd take a walk up to the attic.
There, she looked around. They were definitely going to need all their household help back when the killer was caught, she decided. Black dust covered just about everything; all the cases that held family bibles, period weapons, jewelry, buttons and other odd objects that had been owned by the family over the years.
She walked over to the empty case, wondering whether they would get the Enfield back, and then wondering if she wanted it back.
She felt someone behind her and turned quickly, but there was no one there.
"Marshall?" she asked.
But her ancestor didn't appear. A sense of discomfort and aloneness filled her. She had never felt so in her own house. She hurried back out to the small attic hallways and made her way down the narrow wooden stairs.
As she walked toward the grand staircase that led to both parlors, she paused. She saw that her ghost was now making an appearance before her on the landing.
There was no one on the second floor then; Jake's door—the door to the Jeb Stuart room—was open, but no one was inside.
She walked up to Marshall, who looked tormented.
"Were you just in that room behind me, trying to scare me?"
"No. Why would I try to scare you?"
"Well, I don't know. You led me out last night to show me the gravestone, and I might have been killed."
"Good God, I didn't know anyone was in the woods. And what descendant of mine would fall off a horse? You should be a better rider, young woman," he said gruffly.
"Did you see anything last night?" she demanded.
"You," he said softly.
"Me?"
"I followed you when I heard the thud."
"You've heard what happened today, of course."
"Of course. I'm damned good at being a ghost. My senses are highly attuned," he informed her indignantly.
"I might have been killed!" she told him.
"Indeed. So you must cease behaving so senselessly."
"But you led me out! What good are you doing me?" she asked him. "Other than making me talk to myself—or my imagination, or whatever is going on."
"I will protect you—even from yourself!" he vowed valiantly.
"And you weren't behind me in the attic?" she demanded again.
He looked toward the stairs. She was startled by the look of agony that seemed to come over his misty countenance.
"No. I—I can't go in the attic," he said.
"What?"
"I can't go in the attic!" he repeated. "Leave it be, damn you. I can't go in the attic!"
He must have been really angry with her; he disappeared in a blink.
"I'm sane, and I have a ghost," she mused. "Or I'm totally insane. Or my mind is trying to make me recall something."
As she walked down the stairs, she wondered if she had ever heard a story about Emma having had an affair with an ex-slave and producing the child who would be one of Cliff's ancestors.
For the life of her, she was certain that she'd never heard such a story before.
Dinner was a solemn affair. When it was over, Jackson called Jake into the study, and they took out their list once again; they were down a suspect.
"What about Hank Trebly? He did want to buy the property when the Donegal family was down," Jackson suggested. "And, he's one rude ass."
Jake said, "Yes, but being a rude ass doesn't make a man a killer. Maybe the forensics lab will give us something."
"Alligator saliva," Jackson said.
Jake smiled tightly. "We'll know if they were drugged, the same as Charles Osgood." He leaned forward. "You know, though, I don't think that Toby was the intended victim. He was in the woods with his shotgun when we saw him, and I was pretty damned skeptical of him then. But he said he was over to check out noises he had heard. His dog had been going crazy. I believe Marty Dean was the intended victim, lured out because the killer wanted her dead for some reason in his head, and Toby stumbled on her killer."
"Sadly, that means Toby Keaton is cleared," Jackson said. "All right. So...we have Ramsay Clayton, Griffin Grant, Cliff. The one remaining 'Yankee' who hasn't been cleared, Justin Binder. Justin was here for the reenactment, and he stayed at the plantation after." He ran his fingers through his hair. "What the hell am I missing?" he asked.
"The doctor and the sutler," Jake reminded him.
"All right. We'll interview those two tomorrow," Jackson said. He shook his head. "I'm still not seeing a clear motive. The killer went after Charles—but was Charles the victim because he was Charles, or because he was playing Marshall Donegal? Why go after Marty Dean either way?"
Jake thought for a moment. "He wanted another kill. I think that he felt compelled to create a real reenactment of the death at Donegal Plantation. But then, maybe, in his head, Marty posed a danger. He needed that first kill associated with Donegal, but he's smart enough to know that the grounds here are being watched. He killed Marty Dean because she would do anything for a story. He lured her out to a place where he could kill her and where her body might not have been found for ages—if ever. I've seen what alligators can do with prey. In three months, someone might have stumbled upon a foot bone."
Jackson nodded. "But if the body wasn't found, how would that have hurt Donegal Plantation?"
"She was a news anchor. People would definitely have gone insane looking for her. Somewhere nearby, the police are going to find her car. They're checking the switchboard, so they may even be able to trace the call that led her out here. They might have looked forever, and it would have just added to the sensationalism and mystery regarding Charles. But I think he killed her because he had to kill her. And I think that Toby just stumbled upon him."
"You know that this killer either called from a pay phone or from a prepaid cell that can't be traced. He would have purchased it with cash," Jackson reminded him.
"Yes, but they may be able to find a satellite locator—at least tell us where the call originated," Jake said.
"That's possible," Jackson agreed. "All right. Will and Jenna will head out and speak with the sutler and his wife, John and Matty Martin. We can't afford to wait on the reports. Tonight, you need to get going on the computer again—find out who might have access to those drugs—"
He broke off.
"Yeah, I was thinking that earlier," Jake said.
"The doctor," Jackson said. "Well, he was already on the list."
Ashley played chess with Frazier for a while, worried about how her grandfather was bearing up under the strain.
But though Frazier was grave and obviously thinking about their problems, he appeared strong—and was delighted when he beat her.
She watched the screens off and on with Whitney, Will and Jenna. Frazier tired and went to bed, and Ashley walked over to sit with Beth as she leafed through magazines. Beth set hers down. "Ashley."
Ashley looked up at her. Beth's large dark eyes were sorrowful. "Ashley, I—forgive me. I can't stand this. I have to go. I'm not quitting, mind you. I love this place. I love you and Frazier, and I love Cliff— I don't believe for a second that he did it—and I've tried, honestly, I've tried, but...oh, God, bodies consumed by alligators? I have to leave—just for a while. There's a crash course in vegetarian entrees being taught next week at a cooking school in New York City, and I thought I'd run up and spend a week learning something that will help us in the future. You could go with me, you know. You and Frazier!" Beth added excitedly.
Ashley thought about the dwindling coffers that held the plantation together.
And she thought about Jake. She was safe here; she had a government team living at her house.
And it was her home. She just might be part of the answer, when they kept digging to find the twists and turns of the great riddle.
"I can't go, Beth."
"I knew you'd say that."
"But I want you to go, and learn well, and we'll get the restaurant back up and running. God knows, once we do, we can hire a full-time security guard. People will flock back."
Beth kissed her on the cheek. "I'll probably leave tomorrow. I'll have to see what kind of a flight I can get out."
"Good night—and it's all right, Beth. It's really all right."
Beth left her. She hadn't realized that Angela had been curled in one of the wingback chairs near them, reading as well.
When Beth had gone, Angela walked over to Ashley. "I'm sorry," she said.
"I love Beth. I'm happy that she's going to leave and do something useful," Ashley assured her.
"What a cook we're losing," Angela said lightly.
"True."
"Don't worry—we're actually pretty good in the kitchen as a team," Angela said.
Ashley tried to smile. The effort fell flat.
Angela took Beth's chair and spoke to Ashley seriously. "Ashley, the answer here may really lie in the past."
"Really? Do you think the four dead Yanks are rising to get revenge on the South? Or are my Southern ancestors getting revenge—on themselves?" Ashley knew she sounded skeptical, but it was just too much!
Angela shook her head. "No, there's a flesh-and-blood killer out there. But he has something on his mind. Something he plotted out for years, maybe. And it just may have something to do with the past. You need to start thinking about that. Go back and trace the ancestry of everyone involved and see if we've missed anything."
"That may be easier said than done. You want me to trace the lives of nine men and their offspring through over a hundred and fifty years?" Ashley asked. "We have records on the men who fought, and Daughters of the Confederacy probably has similar records on their lineage, but you're talking about a number of offspring through the centuries!"
Angela stood. "Yes. And if anyone can do it, it's you."
She bid her good-night and started for the stairs, then paused, looking back. "I believe that both Emma and Marshall Donegal are trapped here. Their souls are trapped. For some reason, they are unable to communicate with one another. If we can find out why, perhaps we can find out the truth."
Ashley nodded, feeling a pang in her heart.
The door to the study was still closed. Will and Whitney still watched the screens, while Jenna had gone up to catch a few hours of sleep before taking over for the pair. Cliff had long ago returned to his apartment in the stables.
Ashley stood and stretched; she was about to say good-night to Will and Whitney but she saw that they were resting themselves: open-eyed, but resting. Apparently, they were accustomed to watching the screens, and after today, they weren't going to rely on tapes—they were going to watch every movement on the property by night.
She didn't make it to her room; she stopped in front of Jake's. She smiled. The hell with the screens. She waved to Will and Whitney and went into Jake's room to wait.
He arrived an hour later.
She didn't speak; neither did he. She curled into his arms, and they started to kiss.
It was definitely life-affirming.
Interlude
He watched the news. Of course, a different newscaster talked about the grisly discovery of the two bodies in the bayou separating Beaumont and Donegal Plantations.
Viewers would recall, of course, that police and the FBI still had an open investigation on the case of Charles Osgood, so recently murdered in the old cemetery at Donegal.
Ah, yes, despite the fact that the murders had occurred in plantation country, it was important that viewers take extreme care in the days to come, since police and marine biologists alike thought it unlikely that the newscaster and the plantation owner found dead in the bayou had been killed by the alligators; the alligators had more likely been attracted by the scent of the deceased.
An animal expert came on for a minute to talk about alligators, and the unlikelihood of viewers being attacked by one.
Then the anchor was back on.
Handsome man—as plastic as the woman. Frankly, he liked the other channel better.
But he couldn't take his eyes from the newscast. The anchor was now talking about the extreme loss the station was feeling, and that in their pain, they were proud, certain that Marty Dean had died in the pursuit of answers, as a good investigative reporter. Funeral arrangements were pending the arrival of family members and the release of the remains.
"Good investigative reporter, my ass!" he said aloud, and it made him laugh.
Oh, what a lovely kill!
He wished he could have stayed. He hadn't even thought about the gators; he had simply left the bodies facedown in the water, caught on straggling branches from a fallen old oak. He had never really imagined that his crimes would be so delightfully...mutated, mauled—dissected!
More clever than he himself had known. They'd just never get it. They'd never really appreciated his amazing ability to move quickly and decisively!
He leaned back, swallowing down a delightful sip of hundred-year-old cognac.
His phone rang; regretfully, he answered it.
A problem at work. With his crisp voice, he quickly barked out commands, changing gears as cleanly and swiftly as the transmission of a Rolls-Royce.
When he hung up, he smiled. He looked at himself in the mirror.
Down, down, down, everybody was going down, down, down.
He had his next move to plan, of course. But he would do so, easily and well.
As he walked to his bedroom, he noted one of the pictures on his wall. A picture taken nearly a decade ago after a reenactment.
He paused to look at it. Patrick Donegal, Ashley's father, had still been alive. He'd been playing the role of Marshall Donegal that day.
Too bad he hadn't been so clever back then! Taking down Patrick Donegal would have been a coup! But now...
There she was. Ashley. Beautiful as a teenager, not quite as refined as she was now, but beautiful, even in a quick snapshot.
The boy was in the picture, too. Jake. Jake Mallory. Hell, he'd heard the damned name so much he was sick. He was a savior. A tireless benefactor to the city. So damned good, the feds had wanted him, and he'd gone to them and brought down the mighty. Staring at the photo, he frowned.
Jake really needed to go down, too.
Really.
Pleased that he had a list of victims together, he walked on past the picture and headed to bed.
Tonight, he would rest.
Tomorrow would be his next kill.
## 12
"Okay, look—you can see, I'm coming right back!" Beth said. She pointed to the overnight bag she had packed. "One week of clothing, and that's with doing a load of wash sometime in between. Oh, Ashley, come with me!" she pleaded, standing by the front door.
Ashley hugged her best friend tightly. "I'll be here when you come back," she promised.
"Frazier!" Beth said, and turned to hug Frazier as well. "Now, you really should come with me!" she said firmly.
He smiled. "To a vegetarian cooking class? Perish the thought. I was born right here, my dear. I'm not going anywhere else in my dotage—not until they drag me out." He kissed her forehead. "Come back soon."
"You know I will."
"We need to get moving. We have to stop by Benjamin Austin's office, and then it's a bit of a drive into New Orleans," Jackson said politely.
Beth hugged Ashley once again.
"What time is your plane?" Ashley asked her.
"I'm not sure yet which I'm taking. The boys are going to drop me in the city, and I'll stow my bags at one of the hotels while I take a look around. I love NOLA—I'm going to miss it, even if I only go away for a week..." Her voice trailed. "Take care of yourself. Ashley, damn it, I mean it. Take care of yourself!"
Ashley smiled. "Go!"
Jake turned to Ashley as Beth said a swift goodbye and good luck to the others.
"Stick close to the house today. Please?"
She nodded. "I'm delving into the past, per Angela's orders," she told him. "I'll be a good girl."
"She will!" Frazier assured him, setting his hands on Ashley's shoulders.
Beth saved her last goodbye for Cliff, giving him a big kiss right on the lips. "And you hang in, mister, you hear?"
"We'll miss you," Cliff told Beth.
Then, finally, the three were gone.
"I have some feed bills to finesse," Cliff said, shaking his head. "I'll be in my apartment if you need me."
"I'll be in the study, playing with bills myself," Frazier said.
"Grampa, I can do that!" Ashley chided.
Frazier shook his head. "No, thank you. Let me be useful. The old need to be useful, you know."
"There's no one more necessary," Ashley told him.
He grinned, kissed her cheek, and disappeared.
"I'm out for a walk by the cemetery," Jenna said. "Will and I are supposed to go down the road and check out your sutler friend, John Martin, but that won't take long, I don't think. It's a formality. They don't really suspect him. We've got some time. Where was it that the three Yankees died? In front of the wall?"
"By the family tomb, actually," Ashley told her.
"You can't be out there alone," Will said. "I'll go with you. I'll keep quiet so you can commune, Jenna, but you're not going out there alone."
"A team member is always welcome," she said.
"Well, back to the monitors," Whitney said.
"What have you seen lately in the screens?" Ashley asked. Whitney looked at her, and she blushed. "I mean, besides our movements—and those shadows you've shown me."
"A great deal actually," Whitney told her. She studied Ashley for a minute and glanced at Angela. "Come on over," she said to Ashley, as if Angela had given her a silent accord.
The three women went back to the screens. On one, Ashley could see Jenna and Will walking toward the cemetery.
On other screens, she could see empty rooms.
"I'm rolling back tape," Whitney said. "This is from yesterday afternoon."
Ashley saw herself heading into the attic, walking around thoughtfully—and making a face at the mess of black fingerprint powder that was in the room.
Then, she saw something else. Something that seemed to form out of thin air. It wasn't dark, like a shadow. It was something light—benign, in a sense. The light seemed to reach out and nearly touch her shoulder.
But then she spun around, and the vision of light faded.
"What was that?" Ashley asked, her throat tight.
"Nothing bad—certainly nothing bad or evil!"
Angela assured her.
"And you know this...how?" Ashley asked her.
"I know because I've met many entities now, and even when they're hiding, keeping themselves from me, I can tell when something is there that means nothing but kindness and love," Angela said. "There are many spirits, ghosts, or whatever you want to call them here. Energy, as some believe. I haven't encountered anything evil here at all. Neither have the others."
"But—you were expecting something evil?" Ashley asked.
Angela was thoughtful for a minute. "Evil can remain. As I told you. Someone out there—a living someone—seems to feel that this place is calling out to them, demanding some kind of vengeance. The good revenants that are here are reaching out to you, trying to help you. And you've seen one of them—you just don't want to admit it. Did you meet the spirit in the attic?" Angela asked.
Ashley shook her head. But tension, fear and then trust in Angela filled her. She spilled out the truth. "No, but I have an ancestor here that I have seen. He talks to me as clearly as you do."
Whitney smiled. "Marshall Donegal?"
Ashley nodded.
"Can you talk to him now?" Angela asked. "He's here. I can feel him."
"I can't see him right now," Ashley told her. She shook her head. "I mean, did you want to have a séance or something? Would you act as a medium?"
Angela smiled. "He isn't interested in speaking with me. Just you. When did you first see him?"
"In a dream."
"Then go lie down. Let him enter your dreams. Let him walk you through the original battle," Angela told her. "And don't be afraid. We're all nearby."
"Can it really help?" Ashley whispered.
"We won't know until you try," Angela told her.
Ashley nodded. She walked up the stairs and into her own room.
That was where Marshall Donegal had found her first. In her dreams.
Dr. Benjamin Austin's office was a busy place; the receptionist looked at them as if they were crazy when they asked to speak with him. She indicated the waiting room. There were four rambunctious children running back and forth to their mother with magazines. There were several elderly patients waiting, and two young women and two young men besides.
"You have to see him now?" she asked.
"It will just take a moment," Jackson said, producing his badge.
The receptionist quickly ushered them into an office.
A few minutes later, the doctor hurried in with two of his patients' charts in his hand.
"Hello," he said, trying to juggle the charts and shake their hands. "Jake, I heard you were with the government now. Good to see you. We get all the New Orleans news out here. Your name was involved with the Holloway case. This is about Charles Osgood, isn't it? You know, the police were already here."
"We know that," Jackson told him. "And we appreciate your time. We can see that you're extremely busy."
Benjamin Austin nodded, but said, "That's all right. Anything I can do to help."
"This is confidential information right now," Jake said. "But we know what drugs the killer used to subdue Charles Osgood, and the coroner's office is testing to see if the same drugs were used on the latest victims."
"Latest victims?" His eyes widened. "Oh, God, yes, they're thinking that alligators didn't kill Marty Dean and Toby Keaton?"
"That's right. Where were you the night before last, Dr. Austin? Forgive the question—it's necessary," Jake said.
He stiffened but eased quickly; he seemed to understand. "Well, that I can tell you exactly, and you can verify the information without leaving this office. I gave a speech at eight at a meeting at the Best Western down the road, had dinner at twelve—and stayed at the hotel. I didn't sleep alone. You met my girlfriend—the receptionist who led you here."
"We will verify, of course," Jackson said.
"Please do."
"We have another question," Jake told him.
"Yes."
Jackson pulled out his organizer and said, "We're looking to find someone who might have gotten hold of these two drugs. Can you help us?" He handed the organizer to Benjamin Austin so that he could read benzodiazepine and chlorzoxazone himself.
To Jake's surprise, the man seemed stunned. His face became white.
"What is it, doctor?" Jackson asked.
"I had a robbery—but it was more than a year ago," Austin said, swallowing. "I have a nurse anesthetist on my staff, and I do some minor surgery right here in the office. Our nearest hospital is a bit of a trek," he explained. "The muscle relaxants are more common, but...this office was sacked. We were missing a lot of drugs."
He saw the way they were looking at him.
"I filed a police report—you can check on that, too," he said. "Oh, God...this killer might be using drugs he stole from me?"
"So it appears, Dr. Austin. So it appears," Jackson said. On that slightly ominous note, they bid him farewell.
Beth had chosen to wait for them in the car. "Anything?" she asked.
"Maybe," Jake said. She stared at him with such concern that he sighed. "Drugs were used on the victims—you've been around everything going on; you probably know that."
"Then—oh, my God, it was him? The doctor?" Beth asked.
Jake shook his head. "He said that his office was broken into—over a year ago."
"I'm checking on that," Jackson said, and Jake knew that he was dialing Detective Colby.
Jake, driving, tried to listen, but he couldn't hear Mack.
Jackson hung up. "Good Dr. Austin seems to be telling the truth. He reported the drugs stolen, and he did give a speech at the Best Western. Mack is checking with the hotel to find out if he had room service or anything else—if he was seen. I believe the man was telling the truth."
"But someone out there is lying, that's for sure," Jake said.
Ashley didn't know if her eyes were just closed, or if she had dozed. She heard her own words from the recesses of her mind.
Help me.
I'm here, he told her.
I need to see the battle.
No, you don't. Battle is ugly and horrible, and no one should see it.
I need to see, please.
Somehow, in the dream, she was Emma again. Marshall Donegal was in front of her, shouting at her, telling her to get the children and get them up to the attic. His voice was rough, commanding, and she was shocked, because he didn't speak to her like that.
But then he paused. She felt his passionate kiss on her lips, and then he held her away, torment in his eyes. "The children, Emma, please—protect our children."
She turned as he'd ordered and hurried the children up the stairs. When she reached the attic, she made the little ones hunch down by the wall.
And she went to the garret window to watch.
First she saw the black powder; it exploded and filled the day. The howitzer managed to put holes in the ground, but it didn't hit the buildings.
No matter; the Yankees were coming.
She heard the squeal of the horses. Shouts came from the area of the stables; then she saw the defenders rush out and head toward the cemetery walls.
Rifles flared, and flared again. She saw Marshall retreat behind the walls, calling his men around them, but they weren't all there; they were engaged closer to the house. The men in blue began to enter the cemetery.
From her vantage point, high above the roofs of their family "city of the dead," she could see as they began to surround her husband. He brought one down with a direct hit from his rifle; then the fighting was too close. They were going after one another with bayonets. Marshall was a fighter. Two more died at his feet. And then, with one of his men shouting a warning and rushing in, Marshall was stabbed himself. She saw her husband's eyes as he returned the blow. The last of the men in the cemetery was dead. Two more rushed in but saw the three dead in their own colors. They turned and fled, and in seconds she heard the sounds of horses' hooves as they rode away. Six Yankees altogether; four dead and two running.
"Nancy, stay with the children!" she pleaded, calling to her housemaid and then rushing down the attic stairs and out to the cemetery. She pushed by her husband's men, who were at his side, and fell down beside him, taking his head onto her lap. He opened his eyes once. He mouthed the words, "I love you. I'm so sorry."
And then he died.
They came around her, her husband's men. One of them pulled her gently to her feet. "He's gone, now, Emma. We'll see to him. He's gone, please...."
She was blinded by her tears. She was barely aware as she was led into her house, led to her room. "Drink this. Drink this, Emma—it will steady your nerves."
She had no nerves; she had nothing. Marshall was dead.
Four days later, Marshall was laid to rest in the family vault. He was there; he told her he'd be back; he'd help her until it was time for him to ride to war.
And then he came back again.
To help her, so he said.
And he was kind at first. He helped her haul in some water. He made her sit by the fire, and he poured her a whiskey, telling her that she needed whiskey. She drank it. She would have enjoyed the entire bottle. It warmed her. It numbed her. She could barely hear what he was saying, and she didn't really care.
But then he knelt by her feet and started to rub them, and she was instantly alarmed.
"Emma..."
"No, no, stop!"
"You need me here."
"My husband is barely dead!"
"It's a harsh world, Emma. It will only get worse with the war. You know you need me; you need help through this, and by God, if I'm to be a man for you, you will be a woman for me."
She was shocked.
But when she tried to stand, she began to teeter. She fell, and fell into his arms. He kept speaking, words that made no sense. The world began spinning, but it was still full of agony.
Then she felt him.
Felt him on her. Felt his hands on her, ripping at her clothing.
No!
But his hand fell over her mouth; he was strong and brutal, and her clothing was being ripped from her. She couldn't believe it. This was a friend....
No...
She was powerless.
Help...
The word escaped her.
And she still felt him, the bastard on top of her, felt her flesh, his flesh, but it wasn't her; no one could really touch her anymore.
Then help came at last, and he was ripped from her. She tried to stumble up, tried to call out....
Ashley jerked up.
Tears were streaming down her cheeks. Her body felt bruised—and violated.
But nothing had happened. She was in her own bedroom. Her clothing was intact. She hadn't been touched.
Not in this world, not in this time.
Marshall Donegal stood by the window. He was looking out at the cemetery where he had died.
She realized that he had led her to the battle, but that something else had happened in her dream. She had seen what he had never seen.
One of his own men had betrayed him after his death and violated his wife.
He turned to her. "Battle is ugly. It is blood and slashed limbs and smashed brains. It's horrible, and it's ugly, and perhaps we all sin when we take up arms against one another."
"Some battles have to be fought," she whispered.
He came to her, and she thought she felt his hands on her shoulders. "Yes, battles must be fought for defense of all that is right and holy. But we need to be sure of what is right and holy before going to war." He winced. "Those men who fight demons in their own mind, or join with demons to fight, they must be stopped. Because they are the transgressors."
He pulled her against him. She was certain that she could actually feel the strength of his chest, and of his mind, and all that he had managed to learn—in his afterlife. She felt tears on her cheeks again, and she heard him whisper, "I'm here to try again to defend you. I failed my family once. I cannot do so again."
Jake was surprised at how emotional it was to drop Beth off in the French Quarter. They were on Royal Street, in front of the hotel where they had stayed after the Holloway murder, and they knew that Beth would be able to stow her luggage easily for the day.
"You go on and solve this thing so that I can come back!" Beth told them. "And don't worry about me. I'm just doing a little shopping. I'm going to indulge in a sugar-swamped beignet at Café du Monde, and then I'll be on my way. You gentlemen get busy."
And so they did, leaving the car at street parking on Decatur Street and starting off on foot. As they walked, Jackson made the necessary phone calls.
Fifteen minutes later, Justin Binder met them in front of the square as he was crawling down from a carriage ride with his family. He kissed his daughters and told them he'd meet up with them for dinner as they went off with his mother-in-law to view a Mardi Gras exhibit.
"We can go to my hotel room and talk there," Justin told them.
Jake hoped immediately that Justin was all that he seemed: a family man who respected his mother-in-law and loved his children.
His hotel room was a little suite with a bedroom area where, apparently, his mother-in-law had been sleeping with the children while Justin took the couch in the parlor area. He apologized; the housekeeping staff hadn't been up yet, and Justin closed the couch quickly so that they could sit.
"I heard about the latest," Justin said, sitting. "The newscaster and Toby Keaton. He was all right. I was so sorry to hear about him. Eaten by gators. Well, hell, that's just sad. The man grew up with the creatures, spent his life around them."
"We don't believe he was killed by the gators," Jackson said.
"Neither does the media," Justin said.
"Well, we're back to the usual question. Where were you night before last?" Jackson asked him.
"Here. I took the girls to a live theater experience at Le Petit Theatre, and we and my mother-in-law went to dinner at Muriel's. Then we were back at the hotel—and I had a stomach ache. I called down to room service for warm milk around one in the morning. I was seen by the waiter, and my mother-in-law will assure you I was with my family all night. Thank God I can prove that!"
"What can't you prove?" Jake asked.
Justin met his gaze openly. "When the battle ended, I 'skedaddled' with Ramsay Clayton. That means we rode hard to the sugar-mill fences, up to the road and then back. Ramsay was with me—he kept telling me it was okay to play a Yankee. He liked being on the winning side. We rode back in time for the singing. When it first happened, I kept thinking that Ramsay had to be involved somehow—he was supposed to have been Marshall Donegal—but he was with me, then, and I could swear that I did see him in the crowd before we finally all wound up in the parlor at the house. And I was scared as hell, too, that I would be a suspect because we were staying at the plantation. I searched high and low with the others that night and wound up on a ride all the way over to Beaumont the next day. It was a nice ride—my girls loved it. But..."
"But what?" Jake asked.
"There was something strange about Ashley that day. Once we could see Beaumont across the bayou, she kept looking up at the windows. And she seemed to be afraid of something. She pretended it was nothing, but I've been thinking about it ever since. Don't get me wrong—Ashley sure as hell isn't guilty of anything. She was just about in tears about Charles being missing the day before, and she really came on the ride to keep searching the property. But—she saw something. She saw something that day at Beaumont."
Ashley's legs wobbled as she descended the stairs. Angela ran up to her quickly, frowning and setting a supporting arm around her shoulders. Whitney came over to her as well, her face a mask of concern.
"What?" Angela asked anxiously.
"I didn't see who did it."
"Who did what?"
Ashley looked at Angela. "I saw Marshall Donegal. He didn't want to let me see the battle with him, but I insisted. Then—it was as if I lost him. I became Emma. She was raped, Angela—just days after her husband was killed."
"By the enemy? But I thought—" Whitney began.
"Not by the enemy! By one of her husband's men."
"Who? Which one?" Angela asked.
"I couldn't see, Angela. But—"
"But what?" she prompted.
"I think that Harold Boudreaux came to her rescue. I think that he pulled the rapist off of Emma, and that's when they formed their real bond. I think that we never see Marshall and Emma together be cause he doesn't know. It wasn't her fault, but she's ashamed, and she can't go to Marshall or be with him because of what happened."
Angela was thoughtful. "Maybe we can help them. First, however, we have to find out who the man was. We have to find out what happened to him and figure out why one of his descendants would be after revenge now."
"What should we do next?" Ashley asked.
"Records!" Angela turned to Ashley. "Can you get those accounts of the battle we were wondering about earlier? We'll start on one of the ancestry sites and see what we can dig up on these men by name."
"I'm on it," Whitney said.
"Look, I should be doing this," Ashley said.
"Later. Let Whitney get started," Angela said. "You come with me and find Jenna. She is in the cemetery. She's—communing." Angela looked at her and apparently decided that Ashley had figured out that Jenna did, indeed, see ghosts. "Jenna was meditating, in a way. She gets into a state, and if there are spirits around, even if they won't communicate with her, she can usually see them, and we may see more clearly through her." She took Ashley's hand.
"Are you afraid?" she asked her softly. "Everyone is afraid at first—it's having to believe the unbelievable. It's accepting that there is a greater power."
Ashley shook her head firmly. "No. I'm not afraid anymore. I want the truth."
Griffin Grant's office was in a massive building in the Central Business District, all beautiful glass and chrome. It was furnished with ultramodern pieces—but a picture of a Civil War cavalryman hung on a far wall of the reception area, with a pair of crossed swords above it. "Must be his ancestor," Jackson said.
Jake walked over to the painting. The man had one hand behind his back in the painting; he held his sword in front of him. There was something a little bit odd about him.
"Henry Hilton!" Griffin's secretary told them. "Interesting painting, isn't it? Well, it should be. It was done from a death likeness. Creepy, if you ask me, but these boys do enjoy their reenactments and their roundtables. Henry was killed at Manassas, but he was already wounded."
"Uh—he was—an admirable soldier," Jake said. As he spoke, Griffin came out of his office.
"I know, I know, it's a strange painting, but it's a family heirloom," he said dryly. "Please, come in." Griffin ushered them into his office, quickly dismissing his secretary and offering them coffee or drinks from the handsome marble wet bar set to the far left of his desk. "Soda, whiskey, water—anything?"
Jackson declined. Jake accepted a bottle of water, thanking him and taking a seat in one of the executive chairs in front of the desk.
"I heard about Toby," Griffin said gravely. "Do you know anything about funeral arrangements? Had his son been told about his death?"
"Detective Mack Colby was notifying the family," Jackson told him. "And they won't release the body until a full autopsy has been done."
Griffin nodded and frowned. "They believe that these murders were related to Charles Osgood's death? But...well, a man in a cemetery in full uniform and two people killed after a strange assignation near the bayou? Seems a stretch, doesn't it?"
"Not really. Toby Keaton took part in the reenactment. Marty Dean wanted news on it so desperately I think she would have met anyone anywhere," Jake said.
"Oh. I suppose you're right." Griffin drummed his fingers on his desk. "I wish I could help you. I don't think there's anything at all I could tell you about the newscaster. I didn't know her. I knew Toby well, of course. We've all been friends forever. But I keep thinking that I should have remembered something about the night Charles disappeared. I mean, I was right there! Right there, in the midst of those rushing forward when we heard that Marshall Donegal was being beset in the cemetery—outside the cemetery for the reenactment, of course. I think I saw...maybe it was John Ashton? Helping him to his feet. But we were all there standing around when it ended. Charles was so proud! He wore his battle wounds and fake blood with such pleasure. I kept thinking that Ramsay had done him a real favor, helping him out that day. He made something of a man out of him, if only for a few hours. I swear, I keep trying to remember," he said. He leaned forward. "It haunts me, you know? Thinking about it. First Charles, now Toby..."
"Who do you think might have done it? Any idea of anyone with a grudge?" Jake asked.
Griffin Grant shook his head. "We all had opinions, spats, disagreements, but they were all good guys." He grimaced. "Even the Yankees. I mean, we do seriously like to argue tactics, but that's not even a matter of sides. We've all done this so many times, with changes here and there through the years. The Yanks are good guys. I can't imagine that any one of us would have ever done such a thing."
"Well, here's the usual—where were you the night before last?" Jake asked him.
He seemed surprised. "Right here. You can ask my secretary. I worked forever—we have a new lineup coming out, and it's a bitch, making sure your shows and your sponsors are all aligned just so."
"Thank you," Jackson said, rising. He offered Griffin his hand. "Thank you for your time and your help."
"I'd do more if I could," Griffin told them.
They left his office and stopped by his secretary's desk. "Miss Tierney" read the nameplate in front of her. "Miss Tierney," Jackson said politely, "can you verify that Mr. Grant was in his office late the night before last?"
"Oh, yes!" she said. "Why, that poor man has just been working all hours."
"How late were you here?" Jake asked her.
"Late—seven," she said dryly. "So much for nine to five. And when I left, I could still hear him on the phone in there, placating a diaper company!"
Jackson thanked her.
Out on the street, Jake sighed wearily. "Time to find Ramsay Clayton," he said.
Ramsay wasn't in his hotel room. The desk clerk told them that they could probably locate him on the square, displaying his art.
Jenna sat on one of the few individual white sarcophagi in the cemetery; it belonged to the Donegal brother-in-law who had been killed during World War I.
At first, walking toward her with Angela, Ashley saw nothing. Will was leaning against her family tomb, and he spoke gently to Ashley. "She has brought them out. Sit quietly, and you will see them."
No longer hesitant, Ashley took a seat next to Jenna. Jenna took her hand and gripped it tightly.
And in a minute, they began to appear to her as well.
There were four of them there, soldiers in blue. They walked in a procession, pacing the cemetery right in front of the family tomb. She could see clearly through them, and then she could not. They began to form something that appeared of real substance.
"They are illusions," Will said quietly, "but illusions of the mind. That is the place where we know another sense: of the living, and of the dead."
As she sat there, Marshall Donegal appeared as well. She tensed, thinking that there would be some kind of a confrontation.
But though they recognized one another, acknowledged one another, it was merely with sorrow.
"Talk to them," Jenna said. "They will hear you."
"I need help," she said. "Please."
Marshall walked to one. "We were sad enemies. No more. We are united in death, and the country is united now, through our deaths. We have made peace. Help my girl, please. Help her."
The one who seemed to be a captain turned from Marshall to Ashley.
"I would help you, if I could. What do you need?"
"I need to know...Emma came to her husband as he lay dying. She was dragged away, taken to the house. Who took her? Which soldier? Please, it's so important to know."
"I was dying then," he said. "I was dying then myself. But, I saw her. I saw her tears. I saw the uniform. I saw...the man was blond. He had blond curls.
I didn't know his name."
"Can anyone help?" Marshall called out.
One of them stopped by him, setting a hand on his shoulder. "I was gone, Marshall. I was gone when she came to the cemetery. We were hotheaded as were you—we'd never have hurt your wife or your children. What man would do so?"
Not a man, a monster, Ashley thought. She wasn't afraid of ghosts, she realized. She had learned to fear monsters instead.
The captain spoke softly. "One will know," he said.
"Who?" Marshall asked.
"Emma," the captain said.
Marshall shook his head. "She is gone," he said softly.
"No," the captain said. "I see her at the attic window."
The ghost of Marshall Donegal fell to his knees in the cemetery and wept.
"Why, hey, you!" Beth said, surprised to see the man who was looking into the shop window along with her.
"Beth!" he said, equally surprised. "What are you doing here?"
"Leaving," she admitted. "I've got to get away for a while."
"Donegal Plantation will be missing the world's finest chef," he said gallantly.
"I'll be back," she said.
"I'm sure you will."
"So..."
"So, I've got some time. Do you need a ride to the airport?"
"Well, that would be great. But I can just take a cab—"
"Don't be silly. I've got a car."
"My things are at the hotel—"
"We'll stop and get them. Come on, no big deal, I promise."
"It's a half hour drive there, and back."
"Worth it, if I can imagine you'll return and cook again!"
As the breeze moved her hair and she pulled a strand from her face, she remembered that he had been on the original suspect list. But that had been before.
"Sure, thanks."
They were going to get in a car; they were going to get coffee. No danger in that. Besides, she didn't believe it. She just didn't believe it.
They walked to his car. She noted, as she started to get in, that he had a little case, like the kind kept by a diabetic. Was he diabetic? She couldn't remember.
He shut the door as she sat.
He came around and got in beside her.
And then, of course, she realized.
She put her hand on the door; she opened her mouth to scream.
He slammed her head against the car window as he picked up a needle; she felt it piercing through her skin, and then she felt no more.
## 13
Jackson and Jake walked the square over and over looking for Ramsay Clayton.
He wasn't to be found.
"Strange. The bastard answered me this morning when I called him and said we'd find him at his hotel," Jackson said.
Jake stopped to ask a young woman selling paintings of the square if she had seen Ramsay Clayton that morning.
"Oh, yes, he was here for a while," she said. "He said that he had to get back to his hotel room to meet some friends. I haven't seen him since, but he is usually right here. Says lately he likes to be where there are lots and lots of people! The guy is as nice as can be, and darned good-looking, too, but he's sure gotten strange, always looking as if he's about to run! Such a scaredy-cat."
Jake thanked her for the information.
"He's either scared—or guilty," Jake said.
"We'll head to the hotel again," Jackson said. "But first I'm going to buy a couple of Lucky Dogs and some sodas. I'm starving. We'll get the car and head to the hotel. I have a feeling we're not going to find Ramsay, though. Where the hell did he go? If he's such a damned scaredy-cat?"
"Maybe he's not—maybe it's all an act, down to his kindness," Jake suggested.
"Justin said that Ramsay was with him, and that Ramsay said that he had fun playing a Yankee," Jackson reminded him.
"But he's not here," Jake said. He walked back to the woman. "I'm sorry to bother you again," he told her. "Did you see Ramsay Clayton here the day before yesterday?"
She frowned, going into deep thought. He thought she might have smoked a little bit too much weed in her teen years. Her mind seemed a little misty.
"The day before yesterday... Oh, yes! He was here. He was here. In fact, we were both here until dusk."
"And then you both left?"
"Well, I left. I think he left soon after."
"Why do you think that?"
She smiled. "Because I was gone. I told you—he likes to be around people. Some of the artists stay out late in the night, but by then people really want music and tarot readings. So he must have left. He's not like you."
"Pardon?"
He hadn't realized that his Glock, a standard FBI issue—he knew how to shoot, but he wasn't fond of guns—was visible since his hand was on his hip, pushing his jacket back and exposing the belt holster.
"Maybe he should carry a gun, too. Are you a cop? Is he in trouble?"
He smiled. "I just need to speak with him, that's all."
"Maybe you can give him some courage!" she said cheerfully.
"Well, thank you," Jake said, striding back to Jackson, frustrated. "He was here until dusk the day that Marty and Toby died. I think."
"You think?"
"I'm not sure she knows what day it is today," Jake said.
Jackson nodded. "Okay, let's get food and try the hotel. Then we can head back. I'll get the dogs, you get the car."
Jackson stopped at a cart and bought them both a soda and a couple of hot dogs while Jake walked down to the car. He had just eased it out of their parking spot when Jackson caught up with him. He reached for the food; Jackson slid into his seat, a Lucky Dog halfway in his mouth. "Sorry—I'm really hungry. We're all going to miss Beth."
"Yes, but she wanted to be away—maybe needed to be away," Jake said. He found that he felt oddly uncomfortable once he had spoken.
"What's wrong?" Jackson asked him.
"I don't know. I think we should have taken her straight to the airport in the morning."
"Call her."
Jake did. He got her voice mail right away.
"She's probably on an airplane," Jackson said.
"Probably," Jake agreed.
A feeling of unease had begun in him; it wasn't lessened any when they reached the hotel and found no sign of Ramsay Clayton.
They sat around in the roadside parlor. Ashley was on the registration-desk computer, and Will was working with her. Angela and Whitney were poring through content files on Whitney's laptop.
Jenna was in the attic, hoping that she could reach Emma Donegal.
"Okay, Confederates," Ashley said. "Marshall Donegal, of course. O'Reilly, Charles's stepfather's ancestor. We know that he survived the war—he came back and saw Emma. He probably had a guilt complex about causing the whole skirmish."
"That's two down. Now the rest?"
"One was actually a Clayton; I know that," Ashley said. "Ramsay is a direct descendant."
"Find out what happened to his ancestor," Angela said, looking up. "We're trying to trace a fellow named Pierre Lamont—one of the Confederates."
"He was Toby Keaton's great-great-whatever," Ashley said. "Toby comes down through the maternal line. It's a good thing they named it Beaumont, Beaumont—beautiful mountain," she said with a grimace. "Not that it's exactly on a mountain, but there is a little rise in the terrain. The family name changed many times."
"Yes, but we can let Toby go on this one," Angela said softly. "He's dead."
"I guess so," Ashley agreed.
"What about Griffin Grant?" Whitney asked.
"Family name change, too. His ancestor was... Hilton. Henry James Hilton. And he was killed in the war—1862, the Second Battle of Bull Run, or Manassas. We can do more research on Hilton."
Ashley stared at the screen, searching site after site for the Ramsay Clayton who had fought in the Confederate cavalry during the Civil War. She found him at last and turned to look at the two of them. "Ramsay Clayton—the one our Ramsay is named for—was killed at Gettysburg," she said. "Obviously, he was already a father."
"Okay, so...nothing dastardly happened to him?" Whitney said.
"Not other than a grisly death on a battlefield," Ashley said. "And that would mean a half a million men who died might be vengeful."
"That doesn't seem like something that would bring about revenge or a sick sense that you needed revenge in a future generation," Will said. "We're looking for something that might have come about because of what happened at Donegal that day."
"What about Hank Trebly?" Angela asked.
"Trebly—that was his ancestor's name, too."
"See what you can find on him."
They all worked in silence for a while.
Whitney sighed as one site after another came up blank. "Not found," she said. "Sorry. We need Jake. He's the one who can find anything on a computer."
"Could you take over for a few minutes, Will? My back is killing me!" Ashley pushed away from the desk. Will stepped back, looking at her.
He nodded. "Sure. Why don't you go and try the attic with Jenna? See if you can sense, or even find, anything there?"
She nodded. "I'm just going to grab some water first. And a cheese stick."
"Food!" Will said. "We haven't eaten since break fast."
"All right, everybody, meal break," Angela said.
"Every man for himself. There's no Beth to feed us delicious delicacies today."
"We'll make a real dinner," Whitney said, yawning. Then she leapt up. "Angela is right! Every man—and woman—for themselves! And dibs on the crab cakes!"
As they started for the kitchen in a sudden mad hurry, they nearly collided with Jenna, who was coming down the stairs.
"Anything?" Angela asked her.
Jenna shook her head, frustrated. "I can feel her, and I've talked myself blue. But she won't appear for me, or she can't appear for me."
"I can try," Angela said.
"I know who can reach her. She's been trying to reach him," Whitney said.
They all stared at her.
"Jake," Whitney said. "Before we even knew about Charles being missing, I think he saw her. I was kind of ignoring him, because he was just talking about someone he thought was about to lead a tour. But then we couldn't find her. The way he described her, she was a Southern white woman. He saw her coming through a crowd when we were at Café du Monde. He told me that she had been trying to talk to him, and I said that was rather ridiculous, because she was across Decatur Street, and there was a lot going on, and unless she had been shouting, he couldn't have possibly heard her. But now, I think it all makes sense. It's Emma, and she's decided that if she's going to communicate with the living, it's going to be Jake."
Angela was thoughtful for a minute. "All right, then. We'll all get back on the computers after our very late lunch, and when Jake gets back, we'll just lock him in the attic."
"It's a plan," Ashley agreed. "I'm going to get my grandfather out of the study; he's been poring over bills long enough."
Frazier wasn't sorting through their bills. He was seated there thoughtfully, staring down at his hands.
"Grampa?" Ashley asked him.
"Ashley," he said, looking up and giving her a brilliant smile.
"I'm going to get some lunch. Would you like to eat here, or in the dining room?"
He didn't answer her right away.
"Grampa?"
"I'm sorry, my dear." He let out a soft sigh. "Ashley, I can't help being afraid that if this drags on, we're going to lose the place."
"I won't let that happen!" she promised him. "I swear, I won't let that happen."
He lifted his hands. "It's a house, Ashley. It is built of brick and mortar and stone. Life is what's important. I'm thinking that Beth was right, that we should just leave. People are important, Ashley. You and I are important."
"I agree," she told him. "But we have the best of the best on this. They will find the killer. They will find him."
"I'll eat right here, Ashley. Then I'll drag my old bones up for a nap. Heaven knows, maybe we can get one of those reality programs to pay us the big bucks to come in and do a ghost documentary!"
"Maybe," she said. "Though I shudder! But I'll shudder away, if it helps us keep Donegal Plantation."
Grinning, she headed into the kitchen. Even with Beth gone, it didn't have to be every man for himself. There was plenty of food for everyone; Beth kept her leftovers well-packaged and dated.
Ashley found gumbo and heated it up for whoever might want it when they came down. She brought Frazier lunch on a tray, and then realized that it was late afternoon. It would be late when they were hungry again; they'd call out for pizza, maybe.
"Angela," she said.
"Yes?
"What happened to the cops who were always outside?" Her grandfather's concern remained in her mind.
"They're doing patrols now, since they found the new bodies in the bayou. The logic is that this house is filled with agents—and since others were killed at the bayou, the cops are more useful elsewhere. And we do have cameras going all over the property. Are you all right with that?"
"Sure. I'd rather have a houseful of agents anytime," she assured her.
When she had finished her own bowl, she walked out to the back porch. She shouldn't call Jake; he was busy, but she decided to call him anyway; she could just let him know that they were all fine.
He answered on the first ring. "Ashley? Everything is all right?"
"Everything is fine. I just thought that I would call and tell you so. Is, uh, is everything all right with you?"
"We can't find Ramsay Clayton, and I'm sitting in front of a hotel, waiting for Jackson to see if anyone can tell us where he might have gone," Jake said. "Is there anything new?" he asked her.
"Actually, yes," she told him.
"What?"
"Well, we've been on the computer all day and tearing through the household records and accounts of the battle," she said.
"And?"
"Jake, I think that, after the battle, Emma was raped."
"What—by whom? The enemy didn't take the house. Four were killed and two disappeared."
"By one of her husband's supposed friends. One of the Confederate soldiers. And...and I think that he was attacked by someone else while he was raping her, and I think that person might have been Cliff's ancestor."
"This was in the household records? That's surprising," he said.
She was silent a second.
"Ashley?"
"No. I know this, Jake." She was silent again for a minute; then words rushed from her. "I pushed you away once, Jake. I thought I was afraid of you, but I was afraid because I was terrified my father would appear before me, too, or that there was always something there that I didn't see and didn't want to see, but I do see. My ancestor is here. I can reach him, Jake, I can reach Marshall Donegal. He's trying to help. We need you here, when you can come. I think—and the others agree—that whoever attacked Emma Donegal was the ancestor of the man who attacked Charles Osgood, and then Toby Keaton and Marty Dean."
He was silent.
"Jake?"
"Ashley," he said huskily.
"Forgive me?"
"Always." He cleared his throat. "I guess it falls in. We've known it was someone close who had to be the murderer. I didn't know that Emma had been attacked—until you told me. Are you certain about that? Wouldn't it have shown up in the records somewhere?"
"Jake, only a small percentage of women report a rape now. Back then, Emma Donegal would have never breathed a word of it—any more than she would have mentioned she'd taken on an ex-slave as a lover and borne a child of his. He brought me—he brought me back there. In a dream, Jake. I—I saw it. I saw it all. And it was real."
"I believe you. You know I believe you."
"Jake," she said softly. "Only Emma can tell us. And Whitney says that you're the one who can reach her."
"I've seen her," he admitted. "She brought me to you when you were out riding. But...I can't seem to get her to talk, to stay. Maybe she's ashamed, even though nothing was her fault."
"Jake, she trusts you. She at least shows herself to you."
"We'll be back soon. Take care until then, huh? Wait, hold on a minute!" The line fell silent and she frowned with concern. "Jake?"
"I'm here—Jackson just had a conversation with Detective Mack Colby. Seems that the remains of Toby and Marty yielded the same drugs—they were able to push the toxicology reports to the top level. You all be careful there, swear it!"
"Absolutely," she promised him. "We're here, we're together, and we don't intend to let anything happen."
"The best intentions," Jake murmured.
"Pardon?"
"Stick together. I think that this killer is basically a coward—that's why he has to attack with a drug cocktail that's so potent it immediately renders his victims powerless. If you remain en masse, you'll be fine. We'll be there soon."
"I'm glad," she said softly. "Jake."
"Yes?"
"You know, I was only afraid of you. I never stopped loving you."
He was quiet for a minute; she wondered if she had spoken those words aloud.
"I've loved you forever," he said and hung up.
Ashley looked at the phone and smiled. Walking back in, she found that Angela was washing dishes at the sink. "Hey," Angela said. "I was going to bring some tea up to Frazier, but I thought that you might want to."
"Yes, thank you. I should sit with him for a few minutes. This has to be taking a terrible toll on him. Not because he isn't strong at heart, but just because—"
"He's in his eighties. The body wears down, and that's the way it is," Angela said. "The water is just whistling now. You go on up. We'll keep at it on the computer."
Ashley poured hot water over the leaves in the pot, turned off the stove and added a cup, sugar and milk to her tray. She started through the parlor; Whitney was sitting at the screens again, watching to see if there was any activity in the key areas.
"Hey, you know anything about this?" she called to Ashley.
Ashley walked over to her.
"Amundsen's Hay—Finest Feed. That's what that truck says," Whitney said. "Is that a usual delivery?"
Ashley shrugged. "Yes, that's our feed store. When were they here?"
"Just a few minutes ago. I think the trunk—yep, there it goes, out the drive now."
Ashley balanced the tray in one hand and dug her phone from her jeans pocket with the other. She speed-dialed Cliff's apartment.
"Hey," she said, relieved when he answered.
"Hey, yourself."
"And hey—as in hay for horses. We just got a delivery? I thought it was due yesterday," Ashley said.
"It was, and it came. Seemed we had some kind of a double order. Amundsen's kid was driving the truck, and somehow it wound up on his books two days in a row. Rather than take back the order, they gave me a fifty-percent discount. I took it. Seemed like the thing to do right now," Cliff said to her. "That all right?"
"Of course. That's always your decision. Those poor creatures would probably die if they had to depend on me."
"Ah, Ashley. Our four-legged creatures love you."
"Just checking that everything is fine," Ashley said.
"Yep. And I'm a big old dude, Ashley. Don't be worrying about me."
She laughed. "You're not old, and every female who comes out this way winds up with a crush on you. Don't go fishing for compliments."
"Guess I'm just not feeling as if the love is going to come pouring in right now," he said. "Seriously, all is good. I'm working in the stables. My eyes are open. I'll holler if I need anything."
She hung up. The tea tray was starting to get heavy in her one hand. She shoved her phone back into her pocket and grasped the tray with both hands. "It's all good," she told Whitney.
Whitney nodded and gave her attention back to the screens.
Ashley walked up the stairs to her grandfather's room.
"Hey, Ashley," he said, opening his eyes when she entered. He'd been lying back on his pillow with his eyes closed. "I told Angela you all didn't have to bother."
"Only so many names for so many of us to try to track," Ashley said cheerfully.
He frowned.
"We think that it all stretches back to something that might have happened with one of the soldiers here linked to the original fight. Something that this latter-day idiot may see as a perceived slight that needs to be rectified."
Frazier closed his eyes again. "I didn't think that anyone would actually be after Charles Osgood," he said.
"Poor Charles."
"Well," Frazier told her, opening his eyes and plumping his pillow to rise against it for his tea, "I can say this. Charles was happy. We can be glad we gave him his moment in the sun, brief though it was."
"Well, actually, Ramsay gave him his moment in the sun," Ashley said.
Frazier smiled and nodded. "I never saw him happy, though. Never until that day."
"True," Ashley said. She was silent for a minute.
"Grampa, have you ever managed to talk with a ghost?" she asked him.
His lashes flickered over his eyes, and he was silent for a minute. "Ghosts—are they memories? My life is now filled them, and when we are accustomed to sorrow or loss, we learn to appreciate what shining moments we had. Your father! God, how he loved your mother. Well, you knew that. He'd have never have left you on purpose, but...I try to think that death for him was being with her. And that's how I find my peace."
"That's beautiful, Grampa."
He glanced at her wryly. "Well, it's all right. In all actuality, I wish we'd known that she was allergic to bees, and that it hadn't taken such a toll on your dad, and that they were still here with us. I ache a little every day. But I watch you—and I thank God for what I have."
She leaned over and kissed his forehead. "You're the best."
"I like that," he told her.
He sipped his tea. "Grampa."
He set the cup down on his bedside table and closed his eyes.
"Grampa?"
He waved a hand in the air. "Ghosts exist. There. If they can reach you, you are among the lucky. I want there to be ghosts. I want to talk to my son again. I'd love a chance to tell my own father what a good man he was and what a rebellious rotten kid I was. But they're gone. And if they're not, they don't reach out to me. So, I carry on conversations daily in my head. I tell your mother how beautiful you are and how good you are to your old grandfather. I tell your dad how proud he'd be of you. Do they hear me? God knows.
"Cherish your ghosts, child. Real or in your head. Cherish them. Now, go work with those young people below, and let me get some sleep."
She smiled, and sat in the chair at his bedside.
He opened his eyes again. "What?"
She laughed. "I just want to be near you for a few minutes."
He reached for her hand. She grasped his.
She sat with him in silence, and in time, she heard his even breathing. He was napping.
She almost dozed with him herself. Then her cell phone buzzed, and she jumped up quickly, hurrying out to the hall to answer it.
When she answered it, there was no one there.
She looked at the caller ID. Beth.
She dialed back, feeling a sense of relief. Beth had already made it to New York.
Once again, she got nothing but voice mail.
She left a message. "Beth! It's Ashley. I think you're there, because you called me. But now you're not answering. Call me. Now you got me worried."
She hung up. She found herself looking at the attic stairs. Tempted, she walked up and stood in the room again.
"You were with me this morning!" she said. "Marshall started my dream, but you were with me this morning. You made me see—but not clearly enough. Please, help me."
She realized that dusk was coming, falling softly and surely beyond the garret window, the window from which Emma had watched her beloved Marshall fall in a bayonet attack.
But Emma was not going to appear for her, or so it seemed. Sighing, she flicked on one of the small lights over a display cabinet containing family letters and other artifacts. She slid open the case, thankful now that the cases were never locked—which, of course, had made it easy for the killer to take the Enfield and the bayonet.
She picked up a letter and began to read it. She couldn't make out who it had been written to; the script was all but illegible. She'd catalogued everything in the room, and had reviewed them recently for the team, but she'd never read this one in depth. It was too difficult to decipher. She narrowed her eyes, hoping that swirly cursive would become easier to read, but she couldn't really tell. She saw that it was signed on the bottom—twice. Ramsay Clayton...and there was another name. The letter, she finally realized, had been written to Ginnie. Ginnie H. And it had been signed by Ramsay Clayton—and someone else. Either someone had written the letter for Ramsay Clayton, or Ramsay Clayton had written it for someone else. She thought that she saw an oddly shaped 4 beneath the name Ramsay Clayton. "My dear wife," she read aloud. "Forgive that this is not my hand; it has been broken in a ski...ski accident?" she said, and almost laughed at herself. "Skirmish! It says skirmish."
The sound of her own words faded away. She turned and saw her.
Emma was pale and almost a shadow against the far wall, where she had once urged her children to gather.
"Help me," Ashley whispered. "Please, only you know!"
The phantom figure of Emma started to move toward her. She was going to speak. She seemed to be pointing toward the letter.
Ashley's phone began to ring again. She cursed the sound; the phantom figure coming toward her evaporated into thin air. She set the letter down in the case as she answered the phone.
She answered tersely. "Hello?"
The line went dead again. She muted the phone and tried calling back quickly, but she got her friend's voice mail again. At least she was able to call. She looked around, but saw nothing but the shadows of the darkening day fill the room. "Please," she said softly, "Please, come back."
Cliff stopped raking hay off the main floor and looked up. Varina was whinnying loudly, as if she was in distress.
Once Varina started up, Jeff decided he was going to make some noise, too.
"Hey, girl, what's the problem?" he asked. He walked over to pat the horse, gentle as he stroked the softness of her nose. "It's all right, girl, it's all right."
She tossed her head back, unappeased by his words.
"Calm down, girl. You're the one they all look to here, just like your mistress, you know. Calm down."
The horse let him soothe her. Then the mare gave another toss of her head again, letting out another loud whinny.
On the other side of the stables, Tigger grew restless. Cliff heard something like a bang from his stall, and he walked over to soothe the young gelding. "Tigger, don't you go getting feisty on me, now. These are tiring days, boy."
He hoped that kick hadn't broken off part of the stall's side. He couldn't see. It was getting dark. He cursed and went to flick on the overhead bulb that would light up the entire stables.
Though the light banished the shadows, he was still uneasy. He stepped back into his apartment for his shotgun and came out again, looking around.
The horses were still restless.
Ashley stood in the near darkness when she felt the tremor of her phone. She had just about thought that she had seen a figure growing from the shadows.
"Damn it!" she swore. She glanced at the caller ID. "Beth!"
"What is it, where the hell are you that this number keeps going dead?" she said aloud. She punched the return key.
This time, there was a click.
"Beth, damn! This has been driving me crazy. Where are you? Are you having fun in the city?" she said, realizing she was speaking in a rush.
There was silence at the other end, and then the sound of breathing.
"Beth?"
A throaty, masculine voice came to her—not Beth's.
"Ashley."
"Who is this? Where is Beth?"
The chuckle that sounded in her ears seemed barely human. She shook her head. There was no devil on the other end of the line. There was a human monster, and she had to be very careful now.
"What do you want? Who is this? Where is Beth?" she asked again.
"Beth is still alive."
"What do you want?" she demanded.
"I can see you," the voice said.
She froze. "If you can see me, where am I?" she demanded.
Her heart was racing; she wanted to find Angela and the others, but she was afraid that he really did see her.
And that he really had Beth.
The chuckle came to her again; the chuckle that seemed to make her heart stop and her blood turn to ice.
"You're in the attic," he said.
It was dark here; he couldn't possibly see her now.
"No, I'm not. Actually, I'm in the parlor, and everyone around me can hear the call," she said.
The laugh—more irritated now. "You're in the attic, and if you talk to any of the people in that house with you, Beth is going to die. You know that I'll do it. And you know that I don't care how. I can cut her, I can shoot her...drown her. She's going to die, and it will be all your fault."
She tried to control her sense of raw panic and fear. She had to be sensible. There was a chance that Beth was already dead.
"You know that I have her. I have her phone," he said.
"I still don't know what you want."
"What I want? Well, at this point, that should be obvious, Ashley. I want you. So listen to me, and listen to me good. Do exactly as I say. If you come out, I'll let Beth go. Even trade. You for her."
"How do I know that you'll let her live?"
"I can't exactly sign a contract, can I? What if I were to swear to God? Hmm, I don't believe in any God, other than myself, really. So, here it is. You meet me in the cemetery, or I kill Beth. I'll string her up on a tomb, just like good old Charles Osgood."
"The cemetery?" she said. "You know that the group inside is watching screens. They'll see what I'm doing."
"They will, but they won't think anything of it, because you're going to bring a dish of food out to Cliff." He started to laugh. "Oh, yeah, you're going to bring a bowl of food out to Cliff! And then, Ashley, I'll take it from there. I see you heading across the yard to the stables in three minutes precisely, or Beth dies."
The phone line died in her hand.
Interlude
Good God, who had ever expected such a windfall to come directly into his hands?
This was it—the pinnacle!
He frowned for a minute; it had actually come so quickly and so easily.
To really savor this, he had to take his time.
How much time did he have? He had to make sure that they all wound up shooting one another when the going got rough. But he knew how to do that; he knew how to accomplish exactly what he wanted. Ashley wasn't stupid; she would know that he meant to kill her.
But she was too good a person to risk the life of a friend when she just might save it.
She would come; she would come.
And when she did, he was prepared.
Truly, tonight could be a wonderful bloodbath.
## 14
Jake chafed, growing restless. They'd had to make a detour to the coroner's office; Augie had called Jackson.
The facilities here were state-of-the-art. He'd been to the morgues plenty of times before—just the way that his life had gone—and he was impressed with the shiny steel gurneys and sinks and equipment, and the sterility of the place.
Too many times when he had been involved with death, the morgue had been an empty building somewhere, and the rats had already become kings.
Bright lights were now on over the bodies of Marty Dean and Toby Keaton; they had been cleaned up, and with the sheets partially covering their torsos, they looked far better than they had when he had last seen them.
Augie, in green scrubs, a cap and mask, held a chart in front of him and rattled off the amount of drugs that had been found in both bodies.
Jake wanted to grab the chart; it didn't matter how much—the drugs had been present. And the two on the tables, though looking better, were still corpses.
"Here's what's interesting," Augie said, using a gloved finger to point out Marty Dean's lips. "Marty was drowned. Her lungs were full of water, and you can see the blue coloration around her neck. Toby Keaton, on the other hand, was strangled." He looked at them over his mask. "Despite the fact that they were drugged, I think you'll realize what this means."
"He had to change tactics?" Jake asked. "What do you think happened?"
"It looks as if the two struggled. There were tufts of some kind of black material, which I've sent to the lab, caught up in Toby Keaton's clothing—hard to find, I assure you, when everything is the color black," Augie told him. He smiled grimly. "Our lab is good. The tufts are not the same fabric as the jacket Toby was wearing."
"Ashley did see someone else that night. She climbed up that tree to escape him," Jake said.
His feeling of urgency and restlessness was growing. He needed to escape the morgue. He didn't give a damn how many times he had been in one. There was still the smell. The smell of chemicals. And death.
"We believe we've narrowed the field to three suspects," Jake said. "Ramsay Clayton, Griffin Grant and Hank Trebly."
"And Cliff Boudreaux," Jackson said. "We can't eliminate him yet. He's had access, he's on the property—"
"He was at the stables when I rushed out to find Ashley," Jake reminded him.
"Yes, and he could have circled those woods around just about anyone. He has lived on that property all his life," Jackson said.
"The police searched his apartment with his full cooperation," Jake said, stating the fact.
"We still can't eliminate him—he knows the property like the back of his hand," Jackson reminded him.
Jake didn't argue.
"Well, gentlemen, here's why I brought you in here," Augie said. "Look at Toby's neck there."
They both studied the neck. There was heavy bruising and signs of fingers having pressed in.
"He was strangled by hand, wouldn't you agree?
There are no ligature marks," Augie said.
They both looked at him.
"Well," he said, exasperated. "I can guarantee you, you're down one suspect. Hank Trebly didn't do this."
"How do you know?"
"He had surgery in his left wrist about six months ago. I know, because we discussed it at an Elks meeting the other month. He wouldn't have been able to use both hands as they were used on this victim. So, you see, if you're right, you are down to three men—Ramsay Clayton, Griffin Grant, or Cliff Boudreaux."
When they left the morgue, they were no more than a twenty-minute ride away from the house, but the compulsion Jake felt—the mounting pressure—did not let up. He called Ashley; when he got her voice mail, he nearly drove off the road.
"She's not answering!" he told Jackson.
"I'll call Angela," Jackson said calmly.
He smiled at Jake when Angela answered her phone. "Jake is in a dither. He just called Ashley, and she didn't answer."
"I see. No, we're almost there," he said. "Fifteen minutes or so."
He hung up.
"So, where's Ashley?" Jake demanded.
"It's all right. Angela said that she's up with Frazier. She just brought him some tea, and she was sitting with him. She said that they've been following computer trails all day, but that going from site to site is about to make them all buggy. They're anxious to see you."
"I'm anxious to see them," Jake said.
He stepped on the gas.
"Hey, let's arrive alive!" Jackson said.
"Call Angela back," he said. "Please, have her get Ashley to her phone."
Jackson sighed, and called back.
This time, there was no answer on Angela's phone.
Ashley carried a big bowl of gumbo in her hand. She looked up to see that Angela had followed her into the kitchen.
"What's up?" Angela asked.
"Cliff just called. He's hungry."
"He should come to the house."
"It's no bother, and he knows you all are watching the grounds," Ashley said.
"Jake called a few minutes ago. Hon, where did I leave my phone after that? Oh, hell, I have no idea. Anyway, they're almost back."
"Thank God!" Ashley said.
"I'll reserve one camera to watch you walk over."
"It's all right. Really, please."
Angela wasn't stupid. She could see something in Ashley's eyes.
"All right." Ashley let out a sigh. She wasn't alone; Ashley knew that she didn't dare do anything other than what she was doing.
She walked out of the house, relieved, knowing that someone would follow, someone would carefully follow, and stop whatever terrible thing was being planned.
She felt ill; now her stomach was churning, too.
Cliff!
She couldn't believe it.
But she couldn't forget the voice. Cliff is hungry. Cliff wants food.
And then the laughter.
She walked toward the stables; she hadn't come unprepared.
She didn't know what she expected. She saw Cliff standing in front of Tigger's stall, his shotgun in his hand.
Where had he stashed Beth?
He turned to look at her and frowned. She hurried toward him, pretty certain that she was going to have only one chance.
"Here. Take it," she said, thrusting the bowl of cold gumbo toward him.
Human instinct. He went to grasp the bowl; his shotgun was loose in his hand.
She dropped the bowl into his hands and grabbed the shotgun from him; before he could utter a word, she slammed the butt of it against his head as hard as she could.
He looked at her in disbelief as he fell back against the gate to Tigger's stall and slumped to the ground.
"Ashley," he said.
And then the lights on the property went out. Someone had hit the breaker.
Jake drove the car down the long, oak-lined path to the house. Just when he reached the drive in front, the lights went out. All of them.
The world became a misty shade of gray; dusk was upon them.
Jackson swore; Jake set a hand on his arm. "The generators will kick in!" he said.
But the generators didn't kick in.
Jackson took off for the house; Jake started to follow him, but he stopped.
She was there.
Emma Donegal was there, and she was standing on the path by the side of the house. She beckoned to him.
He followed her. She led him around to the stables. He could barely see in the near dark. The moon was rising, not quite full, but it lent an eerie glow to his surroundings.
"Where, Emma, where?" he demanded.
He heard a groan. He hurried over to the sound. Cliff Boudreaux was down on the ground, holding the side of his head.
"Cliff, what the hell happened?" Jake demanded.
"Ashley..."
"Ashley did this to you?" Jake demanded.
"Behind her...someone behind her." Cliff grasped his arm. "I couldn't see...couldn't tell...it went dark so fast. But I saw her face. He had to have called her out here. I didn't know what the hell was going on...the horses...I'll help you...."
Cliff caught his arm and tried to struggle to his feet.
He didn't make it. He slipped back down to the ground. His head slumped to the side.
Ashley was out there. The killer had her.
Where?
Ashley didn't know what in the hell had hit her; she'd felt a sting, and then nothing more.
And now, she didn't know where she was.
Her eyes were open, she thought. But the world was still dark.
She tried to blink; even blinking seemed an in credible effort.
Then she felt...something. She realized that she was being carried. Her head bobbed and smacked against a man's shoulders, and she had absolutely no control over it. She tried to focus, and she realized that she couldn't see because there were no lights. Struggling to regain some clarity, she decided that he must be carrying her away from the stables.
She blinked and she could begin to see shapes around her; the moon was rising against the swiftly falling twilight. Her focus was bad, but she could try to see. She felt the man's exertion as she was hefted over some obstacle in his path.
Cliff!
She had practically shattered his skull, thinking that he was the one who had called. That he was the one who had somehow managed to kidnap Beth.
But it wasn't Cliff; she had just left him behind, staring at her as if she were the worst traitor known to man, which, of course, she was....
I'm so sorry, Cliff! she thought. But what was that going to matter now?
Thump, thump, grind...
Her chin fell against the man's back. He was a big man. Strong, powerful in the chest and shoulders.
She heard a creaking sound. They were back in the cemetery, she realized. She was surrounded by the towering white architecture of her ancestral city of the dead. The tombs seemed to glisten a silvery white against the dusky sky.
She'd always been meant for the Donegal tomb eventually.
It seemed that time was now.
She could barely see; barely think, barely function. But she was aware! Was this how it had been for Charles Osgood, Marty Dean and Toby Keaton?
Had they known they were about to die but been unable to respond, to react in any way? Maybe not. Maybe the dosage of the drugs they had received had been stronger. Maybe...
Hope swelled, just the tiniest bit. She needed to do something, or she was going to die.
No, Angela thought that she was with Cliff.
And Angela was trapped in the dark. Angela wouldn't know until...
Suddenly, she saw Marshall Donegal at her side. He drew his phantom sword and swiped at her carrier's neck. The sword slashed right through it. Ashley tried to smile. She felt her lips move. She did have some...some...no... She tried to lift her head, but she could not.
But a real-life rock in front of the man nearly tripped him; he stumbled. She was a deadweight, she remembered, even if she wasn't near the weight he must have struggled with when he attacked Charles Osgood.
She heard the faint sound of a creaking once again, and she realized that she was being brought into her family tomb; the temple tomb.
She felt it! She felt pain when her body was slammed down on the central altar in the tomb. Pain meant life. The still rising moon shed an eerie yellow illumination into the tomb through the grate in the far wall.
She heard her attacker working quickly, lifting the heavy marble siding from one of the shelf tombs nearby.
Marshall Donegal's tomb.
He turned to her, smiling.
She knew him before she saw his face. HJH. His wife had been Ginnie. Ginnie Hilton. And Ramsay Clayton had written a letter to her for her husband, Henry James Hilton, because Hilton's hand had been broken. It hadn't been broken in the skirmish; it had been broken when he had attacked Emma Donegal and Harold Boudreaux had set upon him, dragging him off the woman he had so brutally attacked.
"Ashley, you see me! I should have known that you would see me. You were always so special. So precious. And now..."
He paused, listening. There was commotion going on; people shouting. She couldn't understand them, but she knew that they were searching the grounds. Angela had known where she was going, and by now they had surely found Cliff, and they would be looking for her.
She twitched her lips and managed to smile at him in return. She couldn't speak. She willed him to understand her thoughts.
They'll know that it was you. Angela will be reading those old letters, and they'll figure it out faster than I did. They'll know. They already know that it's a sick grudge you have against my family. So, your ancestor died in the war—because his hand was broken. No one blamed Harold Boudreaux for what he did to your ancestor, because his fellow rebels knew what had happened when they saw Henry's broken hand; they were appalled that he'd attacked Emma. They didn't string up Harold Boudreaux, but they didn't speak about any of it, either. All for honor! Well, the honor here died with Marshall Donegal, didn't it? And you can't stand that. Well, I may die, but you will, too. They'll catch you, and you'll rot in prison until they stick a needle in your arm, and that will be fitting, won't it?
He stared at her, his face growing mottled, as if he could hear her thoughts. Of course, he couldn't—he just saw that she had figured out the truth.
He slapped her, and she felt the sting again. Had he attacked so many people now that he was running out of his drug cocktail?
Perhaps she shouldn't be so happy to feel. She didn't know how he intended for her to die. And they were in the vault now with the gate locked. Even if they searched the cemetery, they wouldn't think to look in the vault. The gates hadn't been opened since her father had died; the tomb was sealed after every interment.
Yes, they had. Sometime while he'd been on the property, Griffin Grant had unlocked the iron gates and unsealed the tomb entrance. He had planned for a very long time to see that she came here.
The iron gates were closed, as if they'd never been open. The concrete sliding door behind the ornamental iron gates was barely ajar.
"I'll come back and see you, my dear Ashley," he promised. "And you won't try to enter my mind, then. You'll be dead. Everyone pays, Ashley. It's your turn to pay for the sins of your fathers. You have it all wrong. I know, because Henry talks to me. He told me that he needed to be avenged. He died at Manassas, but he died because he couldn't shoot. He died—because of Emma, and she was a filthy whore who teased and tormented him. Ashley, I saw the letter years ago, and then I could hear him—I could hear him crying out to me. Someone had to avenge what had happened."
She tried to move her lips. Emma was not a whore. She was a grieving widow.
"He came to see her, to take care of her. She led him on, Ashley. I've seen you do it, too. So many of your kind do it. You're like her. Everyone just thinks that you're honest and caring. But, you see, she didn't pay. You will. I thought it might be hard, Ashley. Do you realize that? I thought it might be hard to kill. Ramsay would have done just as well as Charles. That didn't matter. The newswoman, well, she was like a rabid dog. And that ridiculous Toby huffing on over...he wasn't quite so easy. But you know what, Ashley?"
His face came close to hers, and he smiled. She loathed him; she wanted to back away from him. She couldn't move.
"Killing is fun. I found out I like it. But this may be the finale. Ashley, beautiful blonde Ashley, the last of the true Donegals—dead with her ancestors. Oh, watching you, Ashley. My ancestor never told me it would feel so good!"
He lifted her again. To her horror, she realized that one of the heavy slabs had been removed from the shelves on each side of the tomb.
He rolled her into one of them.
It was dark in the burial ledge within the tomb, and she couldn't see. Somehow, she knew that she was nestled against the disintegrating fragments of Marshall Donegal's skull.
"And so it is done!"
She heard the faint sound of scraping as the side slab was lifted and slid back into place. And the world, with or without the remnants of her ancestors, was pitch-black.
The generator lights had finally kicked on, but they didn't offer that much illumination. Still, it was better than the murky gray they'd had before, eased only by the glow of the moon.
He'd shouted for the others, but he hadn't needed to do so. They'd heard the commotion and come running, and quickly understood that Cliff had been attacked, and Ashley was missing.
Jake was beside himself; he knew that he needed to think. Running in a thousand different directions wouldn't serve him well.
Where could the killer have taken her so quickly?
"Whitney, mount up with Will—take the woods toward the bayou, look around for freshly disturbed dirt or a stone in the ground, a grave site. Angela, I'm going into the cemetery—"
"I just ran through the cemetery. There's no one there," Angela said, disheartened. "I ran through every row of tombs, Jake, I swear!"
"We need an ambulance for Cliff—"
"Called already," Angela assured him.
"Is she in the house? Could she have gone back into the house?" Jake demanded.
"I don't know—I'll check."
"I'll go—stick with Jackson!" he shouted.
Jake tore through the back door. He sped through the ground floor, shouting her name. He hurried to the second floor, nearly wrenching doors from their handles.
He burst in on Frazier, who was just rising. Apparently, he hadn't heard the commotion, and, sleeping, hadn't realized that they'd lost power.
"Jake? What is it?" he demanded. Then with some innate instinct, he cried out, "Ashley—something has happened to Ashley!"
"She's missing, but I'll find her—I swear, I'll find her," Jake said and tore out of the room. He looked at the attic stairs, and he raced up to the attic.
She wasn't there. He started to turn. But then he stopped, seeing someone else.
Emma Donegal. She was pointing at an open cabinet. He stared at her, not understanding. She pointed straight to a letter in the cabinet. "There's no time," he whispered desperately. But he picked up the paper, and heard her voice. "Read it!" So he did, squinting.
And as he read, he saw the painting in the office-reception area of the Southern cavalry man—who had died at Manassas. Who had been injured before he'd gotten to Manassas.
They'd been so close...
And now it was so clear. Ramsay was just afraid.
And Griffin was a practiced liar, a totally functioning sociopath.
He dropped the letter, and went racing out of the house.
"It's Grant!" he roared as he hurried out. "Griffin Grant! He's out there, and he's got Ashley!"
Jackson gripped his arm. "We'll find her. How do you know—"
"His ancestor was a Hilton. Hilton died at Manassas. He died because his hand never healed correctly—at least that's what Grant must think... He's got Ashley out there somewhere. We have to find her."
"How the hell did he get on the property unseen?" Jackson demanded.
"The hay truck! He must have sent that extra delivery of hay. He got on the hay truck," Whitney said.
Jake saw that Cliff was on the ground, but Jenna, always the caregiver, was down at his side taking his pulse. She looked up at Jake. "His pulse is steady, and his breathing isn't affected. He's going to be okay."
Cliff lifted a hand. He was trying to indicate something. The stall! Jake realized. Tigger's stall.
He burst into the stall. The horse was nervous. He soothed it, quickly casting his gaze around the floor.
He saw the body bundled beneath hay in the corner and ran to it. He dug away the hay like a maddened dog.
He let out a cry. It wasn't Ashley.
It was Beth.
He lifted her quickly. Her head lolled. "Jenna!"he shouted, bringing Beth's limp body from the stall. "Jenna!"
Jenna left Cliff's side. She flashed a light in Beth's eyes; he noted that the pupils dilated.
"She's alive. There's an ambulance on the way. He might have overdosed her badly—thank God help is nearly here. The detective has been called as well."
Help was nearly there. But there was no sign of Ashley.
Jake started. This time, he saw a man.
A man in full Confederate dress; his sword in a hilt at his side, secured by a butternut-colored sash. The man beckoned to him with hands encased in cavalry gloves.
Jake started walking.
A frown knit his brow. Angela had just said that she'd searched the cemetery. Angela was thorough, and she didn't lie.
But the ghost wanted him to follow.
The ghost of Marshall Donegal.
And so he did.
He was dimly aware of the sounds behind him as he walked forward. Jenna was dealing with both her patients; Whitney and Will were mounting up to search the woods before the bayou; Jackson and Angela were hurrying out to search the guest building, empty now for days.
He walked to the gate, and it swung out slightly as if to invite him. He walked through the gate, and he drew the gun that he wore in his belt holster when he was on official business. He moved through the cemetery; like Angela, he saw nothing.
He started to hurry, making his way to the chapel in the back, and he nearly tore the door from its hinges there; he shined his light into every corner, but there was no sign of Ashley.
He turned around, and the ghost was there again, beckoning him—and showing him the way.
He ran through the trails of the small city of the dead, through temple tombs, through step tombs, pyramid tombs and every manner of tomb that had been built through the centuries.
He knew where he would end.
The ghost stood before the Donegal vault. The slab was in place; the wrought-iron gate was closed. The ghost stared at him with aggravation and turned and tugged at the gate.
Jake burst forward and tugged at the wrought iron himself; it swung open far too easily. He pushed at the concrete barrier that was usually resealed after every interment.
And it, too, gave.
The ghost was trying to speak to him; he had no time to listen. He shoved at the thin sealing slab again, and it fell backward, bursting into a million pieces in a cloud of concrete dust.
"Ashley!" He screamed her name.
Nothing.
The moon cast a yellow glow through the grating, and he blinked, adjusting his eyes to the more muted light. He moved to the altar and saw that something had lain there. The dust was disturbed. He turned...
Not quickly enough. Someone slammed into him hard; his gun went flying from his hand as he staggered for balance.
Griffin!
The man was using himself as a ram against Jake and trying to stab him at the same time. Jake saw his arm rise, and he saw the needle coming at him, and he, in turn, used his body and slammed back against his attacker with all his weight.
Griffin fell back a foot.
Jake caught his attacker's wrist, squeezed with all of his strength, and the needle went flying across the tomb.
But his attacker made a dive for Jake's gun. In the murky light of the tomb, he struggled desperately to keep his attacker from twisting the barrel of the gun toward his head or chest. He roundhouse-kicked his opponent, keeping a desperate and rigid hold on the gun, and the man grunted and gave slightly, but when Jake tried to use the advantage to wrest the gun fully from him, the fellow came at him, biting like a dog.
Jake shoved him off when he came straight for his throat. The gun went flying across the room.
He set at his attacker with his bare hands, but Griffin Grant had now pulled a knife from a sheath at his ankle. They were fighting blindly, but he still felt the man, felt the movement in the air, and he ducked the man's wild swing.
"You can't fight any better than your scumbag rapist of an ancestor, Grant!" Jake taunted him. He needed to get the knife; he'd never find the gun in the darkness.
Provoking him worked. Grant let out a roar and came crashing across the tomb.
That time, he nicked Jake's arm. But Jake heard the knife slide against marble and concrete and made a dash across the tomb, pinning the man there.
The knife went clattering to the floor, and the two of them fell along with it, engaged in a wrestling match that would surely leave one dead. Jake struggled for the top spot; Grant locked his legs around him in a vise, twisting him beneath. His hands came around Jake's throat, but Jake caught him with a double-fisted slam against the head.
Grant teetered.
And then the side of the one of the tombs slammed against him, throwing him off. Ashley, covered in bone and ash, emerged. Jake leapt to his feet, reaching for her swiftly and drawing her to her feet.
Grant came at his back, slamming his fists hard against him, sending Jake staggering forward, his arms enveloping Ashley lest she crash against something again. She was slipping and falling; she had no strength in her legs. He had to help her, had to protect her...
He turned his back to her, ready to withstand Grant's next massive lunge. Grant was ready to make another ram against Jake, but suddenly an ear splitting roar seemed to echo through the tomb, and Grant dropped to the ground; his body shuddered mightily once and then didn't move again.
Jake blinked.
Frazier was standing at the entrance to the tomb, the lost gun still smoking in his hands.
"Grampa!" Ashley said. "You go, Grampa!"
Then she collapsed in Jake's arms.
## Epilogue
"I still don't really fathom how a mind can become so unhinged. I mean, seriously, how do you carry hatred through this many generations?" Ashley asked Jake. "He told me that he heard his ancestor, Henry Hilton, telling him that he had to avenge his family against the Donegal family. His ancestor told him to do it! And I don't know what to believe because...my own ancestor saved me."
They were in the backyard, five days after the event in which Ashley had briefly joined her ancestors in death, convalescing in one of the giant swinging hammocks Cliff had just erected in the back. They could look out on the river as the cool breezes soothed them. It was a pleasant place to let the days go by while they were both "in recovery."
"The very sad truth about humanity is that we've always known how to carry hatred through time immortal, so it seems," Jake said. He gnawed on a piece of grass, just as he had when they were teens. "In Griffin's case, I don't believe that he really had any kind of gift. He would have grown up knowing more about his family's history than anyone else—we all know the little secrets of our own lives better than others. I think he was crazy, that he did just hear voices in his head. He may well have been schizophrenic. He probably showed all the signs when he was younger. It's just that he was so functional, no one saw it. He learned all the tricks." He rolled slightly to look at her. "His secretary said that she heard him in his office talking the day that Marty and Toby were killed. She did, too. When the police went into his office, they found out that he had a recording to play that went on for various lengths of time. That way, people would always swear that he'd been in his office because they'd heard him."
"He was a CEO of a major company."
"Highly functioning. Ashley, the past didn't make him bad. The past made him self-righteous. He did believe that he was like a god, or an avenging angel, to take what he wanted because of the perceived ill that had been done to him." Jake smoothed back a piece of her hair.
"I walloped Cliff," she said.
"He's forgiven you."
"But—I really love him. He's family. How could I have been so easily fooled?"
"Fear—for someone else you love. That's a pretty strong motive, Ashley. Honestly, I'm getting to know Cliff really well again, and he has forgiven you."
She smiled. "So has Beth."
"Beth didn't need to forgive you. She knew that what happened wasn't your fault."
"It wouldn't have happened if she hadn't been here."
"But that's life. The good with the bad," he told her quietly. "And, hey! You know, of course, that the cops found Ramsay. He wasn't doing anything evil—he was just trying to find courage in a bottle down on Bourbon Street."
"Poor Ramsay. He might have been the victim."
"He might have been. But, sadly, what happened worked well into Griffin's hands. Ramsay he'd have had to have coaxed. He knew that Charles would be lured by the promise of a beautiful woman. Ram say—well, until he went through this period of trauma—Ramsay was a good-looking fellow. I'm glad you turned him down."
"The feeling just wasn't there," she said. She shuddered. "I turned Griffin down once, years ago. He'd asked me to be his escort to some kind of advertisers' function."
"Thank God you turned them both down," he said, his tone husky. He pulled her closer and faced her. "Beth is well, and Cliff is well, and I'm the one in the worst shape—feel sorry for me!" he commanded.
She laughed. He didn't mean it. He was full of bruises and cuts, but he didn't seem to mind them.
Nor did he seem to mind that others were handling the press, the paperwork and all other pieces of business that had to do with the case being over. He didn't seem to hurt too badly when they were alone at night making love, and he didn't wince at all when she kissed his bruises.
"You're thoughtful," she said.
"We still have one more piece of business," he told her. He rolled off the hammock, drawing her with him.
"Where are we going?"
"The cemetery," he told her.
She pulled back at that, but just for a minute.
"And we're going to... Oh! I see!"
They followed the path to the Donegal tomb, but they didn't stand in front of it. Jake led Ashley to the tomb of the long-dead World War I hero, and he and Ashley sat.
"Emma, your husband loves you," he said. "He loves you desperately. He loved Harold as well, and he's grateful that Harold was there to rescue you. He knows that you held your love for him in your heart all of your life, and he knows that you were treated cruelly. He just wants to love you."
"Your turn," he told Ashley.
"Marshall Donegal, you get your little ethereal backside out here! You're afraid to face your wife. You're afraid that she grew beyond her love for you, because life became so hard once you were gone. Come on, you two kids—I think this is it. Your chance at...your chance at eternal happiness," she added softly.
At first, nothing.
And then slowly, very slowly, and like pale illusions in the bright daylight, they both began to appear. She walked slowly from the path; he emerged from the side of the tomb.
She eased out to him.
He took a step.
They ran into one another's arms.
In the daylight, so locked together, they faded into the sunlight.
There were tears on Ashley's cheeks.
Jake smoothed them away. "Hey, they've found their eternal happiness," he told her, lifting her chin and dusting her lips with a gentle kiss.
She smiled.
"And I've found mine," she told him.
Yet, as the kiss grew passionate, it was suddenly disrupted.
"Jake!" Jackson called from the house. The tone of his voice was urgent.
Jake sighed. "Well, that's my world now, I'm afraid."
She didn't release him; she held him close for another minute.
"Jake, your world is now my world, in so many ways."
They kissed again.
Jackson shouted again.
But...
Jackson was a decent fellow. He'd just have to wait.
One of my favorite meals in Louisiana is crawfish étouffée—I had it first in plantation country, and I've been hooked ever since! Shrimp can be substituted for crawfish. The major trick with this dish is to follow the part that says slowly brown....
Crawfish Étouffée
Ingredients
½ cup roux (brown ¼ cup oil and ½ flour, adding a pat of butter—slowly so that it does not burn!)
½ cup finely chopped onion
½ cup finely chopped bell pepper
1 small can diced tomatoes
3 cloves finely chopped garlic
2 cups water
3 lbs of cleaned crawfish meat
1 tsp. salt
1 tsp. pepper
dash of cayenne
Directions
Simmer, adding chopped white onions and bell pepper. When a nice soft consistency has been reached, add in the diced tomatoes, garlic, water and seasoning. Stir this mixture constantly over medium-high heat until all ingredients are evenly combined. Cover the pan, reduce to a low heat and allow it all to simmer for 15 to 20 minutes. Next, add the crawfish (or shrimp) and bring the mixture to a boil, bring to a low boil for about five minutes or so, and then bring back to low heat. Serve over fluffy white rice.
Mix up a green salad,
cut some French bread and enjoy!
This Southern favorite is as easy as...mint!
Mint Julep
Ingredients
4 or 5 fresh mint leaves
Jigger bourbon whiskey
Powdered sugar (or cane sugar, or even substitute)
Water
Directions
Mix mint and sugar in a tall glass. Add a jigger of water and stir lightly, top with bourbon whiskey and serve with a straw and topping sprig of mint!
ISBN: 978-1-4592-0759-2
HEART OF EVIL
Copyright © 2011 by Heather Graham Pozzessere
All rights reserved. Except for use in any review, the reproduction or utilization of this work in whole or in part in any form by any electronic, mechanical or other means, now known or hereafter invented, including xerography, photocopying and recording, or in any information storage or retrieval system, is forbidden without the written permission of the publisher, MIRA Books, 225 Duncan Mill Road, Don Mills, Ontario, Canada M3B 3K9.
This is a work of fiction. Names, characters, places and incidents are either the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events or locales is entirely coincidental.
MIRA and the Star Colophon are trademarks used under license and registered in Australia, New Zealand, Philippines, United States Patent and Trademark Office and in other countries.
For questions and comments about the quality of this book please contact us at Customer_eCare@Harlequin.ca.
www.MIRABooks.com
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{"url":"http:\/\/eng.wealthfront.com\/2013\/12\/19\/marketside-chats-3-more-options-for-non-dummies\/","text":"# Marketside chats #3: More options for non-dummies\n\nDecember 19, 2013\n\u2022 show you how to convert stock volatilities to something intuitive\n\u2022 explain in basic terms the fancy concept of convexity\nPlease read\u00a0Marketside chats #2: Options for non-dummies\u00a0first for some of the other terminology.\n\nImplied volatility (IV, a.k.a. vol)\nThe implied volatility\u00a0of a stock (or index) is the\u00a0standard deviation of its expected future annualized return. For example, Google options expiring in about a year show an IV of 24% at the time of this writing. If I buy a Google option now (either call or put), and the stock becomes more volatile some time soon (e.g. not after the option\u2019s expiration, when nothing matters!), I will make money on my option on average. Another way to say it is that the expected value of payout of the option I bought is higher than the price I paid for it.\nAssuming that volatility is roughly constant throughout that year, there\u2019s a handy trick to convert that to daily price moves, which are more intuitive. There are 5\/7*365-10 ~= 250 trading days in a year (*5\/7 to exclude weekends, and -10 to exclude holidays). The way the math works out, standard deviation (and therefore IV) is proportional to the square root of time to expiration. Because 250 ~= 256 = 16^2, if we divide annualized IV by 16, we get the daily one. In our example, the option market is pricing Google\u2019s 1-day standard deviation is 24% \/ 16 = 1.5%. (footnote 1) Since this is IV, it takes into account expected\u00a0future price moves, not previous\/historical ones\u00a0(although they are often similar).\nFrom this point on, you know the rest from basic statistics: 68% of the time, a day\u2019s move will be within 1.5% up or down, etc. (footnote 2)\nRemoving uncertainty reduces volatility\nThe IV of an option will drop after an event that resolves uncertainty. Of course, the magnitude of this effect isn\u2019t the same for all stocks on average.\nSay A is a large company with stable earnings, and B is a small company whose distribution of earnings expectations by the market is very wide.\u00a0E.g.\n\u2022 A=Microsoft\n\u2022 B=some biotech company whose product is a single drug which was released recently.\nAfter an announcement of B\u2019s earnings, B\u2019s stock price will become much less volatile. The investing public now knows how that drug\u2019s sales are going, and that was the make-or-break piece of information one needs to know about B.\nHowever, after an announcement of A\u2019s\u00a0earnings,\u00a0A\u2019s stock price will not really become more or less volatile. A is already making a bunch of money on existing products. Therefore,\u00a0B\u2019s stock price will move (as a %) by more than A, since A\u2019s earnings are much more likely to be within a tighter range.\nConvexity\nIf you own a stock, you make $1 if it goes up by$1, and lose the same amount if it goes down by $1. This is so blindingly obvious that you might be wondering why we\u2019re even mentioning it. However, options are different. In some simple cases, options can be priced with a closed formula called Black-Scholes. If you don\u2019t want to take our word for it, or are curious for more, use e.g. this options calculator with the following inputs: \u2022 Stock & exercise price of$10,000 (both).\u00a0Few stocks trade at $10,000, of course, but the actual levels don\u2019t matter.$10,000 will give us numbers with more precision digits.\n\u2022 Time to maturity (a.k.a. time to expiration) = 0.083 (about a month, i.e. 1\/12 of a year)\n\u2022 Interest rate 0.0000001% (that calculator disallows 0) so we can ignore rates, without loss of generality.\n\u2022 Annualized volatility of 16. You now know what that is now!\nThese are the results you\u2019ll get for Black-Scholes call option prices if the stock moves in a short period of time (footnote 2):\n1. If the stock goes to $9,900 (=$10,000 \u2013 $100), with time to expiration being the same, then the option price will become 137.289. 2. If the stock stays at$10,000 the call option is worth $183.879 3. If the stock goes to$10,100 ($10,000 + 100):$239.086\nIn short, if the $10,000 stock price were to move by the equally likely amounts of +\/-$100 (cases 1 and 3), the average of the 2 option prices is $188.188 >$183.879, which is the price of the option if there is no price movement in the stock.\u00a0(footnote 3)\nThis obviously doesn\u2019t constitute a proof, but it demonstrates that\u00a0actual stock movement is good for you, on average, if you bought an option (either call or put). Of course, a downward movement in the stock (all else being equal) is bad for you if you hold a call option, but it\u2019s less bad than an equal upward stock movement is good.\nIn options, this asymmetry is called gamma or convexity. Convexity goes hand-in-hand with optionality. What you see here is a micro manifestation of the fact that if the stock goes up by a lot, you gain a lot, but if it goes down by a lot, your losses are capped.\n\nFOOTNOTES\n\n(1) To be precise, the market usually expects volatility to be different throughout different parts of the year. Dividing by 16 would give you the volatility of an \u201caverage day in the year\u201d, roughly speaking.\n(2) Options can be seen as insurance. Roughly speaking, an insurance company charges buyers based on a higher probability assumption than the actual probability of an \u201cinsured event\u201d (accident, etc.) occurring. They would not make money in the absence of such a buffer.\u00a0If market participants on average hate the idea of being exposed to stocks going down a lot, they will pay up for that insurance. Therefore, if the market expectation (statistically) is for 20% annualized volatility AND the market dislikes volatility by some \u201creasonable\u201d amount, the IV for an option could be 24%.\nIf this sounds vague, consider car rental companies that charge you ~$15\/day for accident insurance = ~$5,500 annually ~= 1\/3 of the price of their average (depreciated) car annually. This is equivalent to saying that you have a 1\/3 chance of totaling their car if you were to drive it for a year. Statistically, this is rarely true. Alternatively, it\u2019s equivalent to saying that you have a 1\/15 annualized probability of totaling their car, but you are willing to pay for a 5x probability, e.g. to avoid the risk of having to come up with the cost of a car in case of an accident, to avoid the hassle of dealing with legal issues, etc.\n(3) In reality, if the stock were to drop by 1% within a very short period of time, the market might get spooked and IV would go up. We are using 1% because it makes the point clearer; e.g. 0.1% would also work.\n(4)\u00a0In reality, stocks are best approximated as \u201clognormal\u201d, i.e. the logarithm of a stock price follows a normal distribution. For example, +$100 and -$100 aren\u2019t equally likely; *1.01 and \/1.01 are. This makes sense; for instance, it\u2019s easier for the $10,000 stock to go to$18,000 (price almost doubled), but if it goes to $10,000-$8,000 = $2,000, it means it dropped to 1\/5 its value. Another qualitative argument: a$100 stock can go +$115 to$225, but cannot drop $115 to a negative number -$15!\n\nIn that case, a better set of 2 likely outcomes is $10,100 =$10,000 * 1.01 (already shows in case 3) and $9,900.990099 (=$10,000 \/ 1.01).\u00a0\u00a0At that stock price,\u00a0the call option is worth $137.708. The average of the 2 equally likely scenarios is ($137.708 + $239.086)\/2 =$188.397, which is still bigger than the option price. So the argument still holds.\n\nDISCLOSURE\n\nThe information provided here is for educational purposes only. Nothing in this article should be construed as a solicitation or offer, or recommendation, to buy or sell any security. Financial advisory services are only provided to investors who become Wealthfront clients. Past performance is no guarantee of future results.","date":"2017-11-23 01:55:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4726622700691223, \"perplexity\": 1398.6566715404556}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934806715.73\/warc\/CC-MAIN-20171123012207-20171123032207-00309.warc.gz\"}"}
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Q: Rally (Java): I dont want to add a testresult if there is one already in the testcase I am sending a query to check whether there is a testcaseresult associated with a testcase. I am using this:
QueryRequest c = new QueryRequest("testcaseresult");
c.setFetch(new Fetch("testcase"));
c.setQueryFilter(new QueryFilter("testcase", "=", testCaseRef));
QueryResponse cc = r.query(c);
//String testresultRef = cc.getResults().get(0).getAsJsonObject().get("_ref").toString();
I want to create a new testcaseresult only if there is no testcaseresult in the testcase so far. how can i use the query to do it?
Thank you.
A: I think you're on the right track in doing a followup Query. You can't query on Ref's however. However since you have the ref, and presumably also have the TestCase JsonObject, you could do something like the following where you would query on the FormattedID of the parent TestCase:
// Query for Test Case to which we want to add results
QueryRequest testCaseRequest = new QueryRequest("TestCase");
testCaseRequest.setFetch(new Fetch("FormattedID","Name"));
testCaseRequest.setQueryFilter(new QueryFilter("FormattedID", "=", "TC4"));
QueryResponse testCaseQueryResponse = restApi.query(testCaseRequest);
JsonObject testCaseJsonObject = testCaseQueryResponse.getResults().get(0).getAsJsonObject();
String testCaseRef = testCaseQueryResponse.getResults().get(0).getAsJsonObject().get("_ref").toString();
String testCaseFormattedID = testCaseQueryResponse.getResults().get(0).getAsJsonObject.get("FormattedID").toString();
// Query for Test Case Results that are associated to this particular Test Case
QueryRequest testCaseResultsRequest = new QueryRequest("TestCaseResult");
testCaseResultsRequest.setFetch(new Fetch("Build","TestCase","Verdict","FormattedID"));
testCaseResultsRequest.setQueryFilter(new QueryFilter("TestCase.FormattedID", "=", testCaseFormattedID));
QueryResponse testCaseResultResponse = restApi.query(testCaseResultsRequest);
int numberTestCaseResults = testCaseResultResponse.getTotalResultCount();
A: Thanks Mark. That makes sense. I however found another way, as shown below:
try{
QueryRequest c = new QueryRequest("testcaseresult");
c.setFetch(new Fetch("testcase"));
c.setQueryFilter(new QueryFilter("testcase", "=", testCaseRef));
QueryResponse cc = r.query(c);
String testresultRef = cc.getResults().get(0).getAsJsonObject().get("_ref").toString();
} catch(IndexOutOfBoundsException e){
//Create a testcaseresult here.
}
I used a try-catch block. I query for a test result in a try block. If this throws an IndexOutofboundexception, it means that there is no such test result. In that case I can create a new test result in the catch block.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
Recent advances in machine learning techniques has enabled predictive algorithms to perform better than expected in certain domains. However, in real-world situations, a sufficiently effective model requires massive data for training. In some more sensitive scenarios, such as patient data from different hospitals, or driving data from different vehicles, a single may not have sufficient quantity and quality of data to learn a more robust model, and co-training may cause privacy leakage problems.\par
Federated learning (FL) is a new paradigm of distributed learning that aims to address the problem of communication efficiency in learning deep networks from decentralized data. Each client uses local data to learn local model parameters or parameter updates, then only transmits parameters to the server, and aggregates all parameters in the cloud, thereby obtaining a federated model without data exchange. However, practical applications show that heterogeneity causes non-trivial performance degradation in FL, including up to 9.2\% accuracy drop, 2.32 × lengthened training time, and undermined fairness \cite{Yang2021CharacterizingIO, Horvath2021FjORDFA}. For example, the data distribution of each client may be different, then the performance of the model, for example, between client A and client B, may vary greatly and the accuracy rate may even be lower than the prediction result of the local model.\par
\cite{Donahue2021ModelsharingGA, Blum2021OneFO} analyzed the willingness of clients to participate in the federal update in this case, and found that in some cases, clients with poor performance are more inclined to withdraw from the alliance. Based on this, referring to the fair methods for machine learning \cite{Cotter2019OptimizationWN, Dwork2012FairnessTA}, the entire network is faced with a choice, that is, the server hopes to maximize the global accuracy rate as much as possible, while each client hopes the final model to behave little differently than the other members. This problem can be described as a game theory problem of total cost allocation: the self-interested goal of all individual actors (fairness) and the overall goal of reducing total cost (optimality).\par
In addition, due to the different distribution of data on the clients, some clients with higher data quality may have more important predictive capabilities than others. In other words, the global may be overly dependent on the trained model of some clients. Therefore, when we start to pay attention to fairness and avoid this influence, intuitively, the convergence speed and prediction accuracy of the overall model may be affected \cite{Cui2021AddressingAD}.\par
Due to privacy issues that need to be guaranteed, we cannot directly access the raw data on each client, it is impossible to analyze the data distribution on the client. However, \cite{Wang2020OptimizingFL} demonstrates that there is an implicit connection between the distribution of training samples on the device and the parameters of the model trained based on these samples. To get as close as possible to the optimal solution, in this paper, we propose an extensible federated learning framework called Policy Gradient Fair Federated Learning (PG-FFL). PG-FFL can be regarded as an additional plug-in of the FL algorithm. Based on the policy gradient reinforcement learning algorithm, PG-FFL uses the local parameters of the client training model as observations, aims to balance the above problems of the model by assigning different aggregation weights to the clients participating in the update in each round of aggregation. The main contributions of this paper are as follows:\par
\begin{itemize}
\item In this paper, we propose to utilize the Gini coefficient as the measure of fairness, which objectively and intuitively reflects the performance gap of the aggregated model among clients participating in federated training, and prevents polarization between client performance.
\item We propose a fairness adjustment plug-in. For the federated model, we add the fairness indicator and use a plug-in based on the deep reinforcement learning (DRL) algorithm that can be used for any federated learning algorithm that does not involve aggregation weight adjustment.
\item In this paper, we port the policy gradient fair federated learning (PG-FFL) paradigm to two advanced FL optimization algorithms, namely FedAvg and FedProx. Experimental result shows that PG-FFL can significantly improve fairness in multiple datasets.
\end{itemize}
\section{Related work}
In this section, we introduce the main challenges federated learning faces and briefly introduce the current state of research.
\subsection{Federated Learning}
Federated learning is a distributed learning framework under differential privacy, which aims to learn a global model on the server-side using the model parameters learned by different local clients based on clients' private data \cite{McMahan2017CommunicationEfficientLO}.\par
In horizontal federated learning, the server trains a global model in specific aggregation ways iteratively aggregates local models from different clients. In each iteration, the server randomly selects a certain number of clients to transmit global model parameters, clients participate in the training using the downloaded global model for training, and then upload local training model parameters and aggregate the new global model on the server \cite{Wahab2021FederatedML, Li2020FederatedLC}.\par
\subsection{Fairness Challenges of Non-IID Data Distribution}
Classic federated learning algorithms aggregates the models of different participating clients by calculating a weighted average based on the amount of training data \cite{McMahan2017CommunicationEfficientLO}. However, in practical applications, the data and label distribution on different clients cannot fully meet the requirements for the distribution of all local client data and label IID distribution in the distributed algorithm. Therefore, the convergence and stability of federated learning are affected challenge \cite{Konecn2016FederatedLS, Karimireddy2019SCAFFOLDSC}. \cite{Xiao2021ANS, Yang2021HFLAH} proposes that part of the reason is the improper way of traditional federated learning's server-side aggregation method.The contributions of clients in federated learning can be distinguished by their trained models' validated accuracies.\par
Previous work has shown that non-IID data may bring parameter differences \cite{Zhao2018FederatedLW}, data distribution biases \cite{Hsieh2020TheND}, and unguaranteed convergence \cite{Sahu2020FederatedOI}, which can be improvemented both on the client side \cite{Sahu2020FederatedOI} and on the server side \cite{Huang2021BehaviorMD,Hsu2019MeasuringTE,Reddi2021AdaptiveFO}.\par
Due to the heterogeneity of data size and distribution on different clients in federated learning, simply aiming to minimize the total loss in large networks may disproportionately advantage or disadvantage the model performance on some of the clients, such as resulting in loss of uniformity of results across the clients \cite{Li2021DittoFA}. The accuracy of individual devices in the network, for example, cannot be guaranteed despite the high federated average accuracy.\par
There has been tremendous recent interest in developing fair methods for machine learning \cite{Cotter2019OptimizationWN, Dwork2012FairnessTA}, unfortunately, current methods can not apply to federated settings directly. The recent work introduces a fairness algorithm suitable for federated learning. \cite{Mohri2019AgnosticFL} uses a minimax optimization method to ensure that the overall fairness will not be improved at the expense of some client performance. \cite{Li2020FairRA} borrows the idea of resource allocation, fairness is allocated as a resource to achieve uniform distribution of clients' performance, \cite{Wang2021FederatedLW} mitigates potential conflicts among clients before averaging their gradients. But these algorithms have fairness as the only goal, we can simply think that there is a permutation relationship between fairness and the best performance (usually expressed by the average performance). Therefore, in real federated learning applications, people will naturally want to further guarantee fairness when their programs can guarantee optimal performance.\par
Based on above, we propose a fairness adjustment plug-in. Our algorithm can be used for any federated learning algorithm that does not involve aggregation weight adjustment, and we add fairness considerations on the basis of pursuing the best performance.\par
\section{Fair Federated Learning}
In this section, we first formally define the problem in Section A, then propose a naive solution in Section B combining deep reinforcement learning methods, and propose a new general framework in Section C, which can effectively handle the fairness disaster caused by non-iid data distribution while ensuring the performance of federated learning methods.
\subsection{Problem Statement}
The standard horizontal federated learning can be defined as to minimize\par
\begin{equation}
\mathop{\min}_{\omega} f(x,\omega) = \sum_{i=1}^{N}p_{i}f_{i}(x,\omega),
\end{equation}\par
Where $f_{i}(x,\omega):=E_{x \sim P_i}[f{i}(x,\omega)]$, the vector $\omega$ denote model weights and $(x,y)$ denote a particular labeled sample, is the local loss function of the $i$-th client. Aggregation of loss functions of different clients, assuming that there are N clients partitioning data, where $D_i$ is the number of index sets of data points on client $i$, aggregation weight of the $i$-th client is defined as:\par
\begin{equation}
p_i = \frac{D_i}{\sum_{i=1}^{N}D_{i}}.
\end{equation}\par
That is, simply think that the influence of a client on the global model is determined by its sample size. Training is a uniform distribution on the union of all samples, where all samples are uniformly weighted.\par
The traditional federated learning method solves the problem of (1) jointly by (2) calculating the contributions of different clients, but this may cause the final global model to be biased towards clients with a large number of data points. Because of the considerable limitations in practical applications, we will not adopt this assumption in this paper, which we will illustrate through Fig.~\ref{fig: IID_and_Non}.\par
Considering some special cases in federated learning, such as joint training of different hospitals, different clients hope to jointly train a better model under the condition of protecting privacy, so the performance of different clients should not vary too much at this time. In order to reflect the fairness of a federated network, \cite{Wahab2021FederatedML, Li2020FederatedLC} proposed the uniformity and the standard deviation (STD) of testing accuracy between clients to measure network fairness. Unfortunately, indicators such as STD are related to the expectation of testing accuracy. Therefore, for different application scenarios, Assuming that all clients' testing accuracy for networks A and B are 0.07, 0.08, 0.09 and 0.7, 0.8, 0.9, respectively, then A will have A smaller STD despite the "rich and poor" difference in test accuracy between the two networks. In order to alleviate this problem and make an index better measure the degree of network fairness, we put forward a new definition of fairness in the definition 1.\par
\textbf{Definition 1} (Fairness) \emph{We say a model $\omega_{1}$ provides a more fair solution than $\omega_{2}$ if the test performance of $\omega_{1}$ on $N$ devices, $\{acc_1,...,acc_N\}$, is more fair than that of $\omega_{2}$, i.e., $Gini\{F_{n}(\omega_{1})\}<Gini\{F_{n}(\omega_{2})\}$, where $F_{n}(\cdot)$ denotes the test accuracy on $N$ devices, and Gini\{·\} denotes the Gini Coefficient. Let $acc_{i}$ and $acc_{j}$ represents the accuracy on any client test set in the model, $\mu=\frac{1}{N}\sum_{n=1}^{N}acc_{n}$ represents the average accuracy of all clients.}
\begin{equation}
Gini = \frac{\sum_{i=1}^{N} \sum_{j=1}^{N}|acc_i-acc_j|}{2 N^2 \mu}.
\end{equation}\par
We define the fairness of the model on all clients based on the Lorentz curve, and the indicator for judging the fairness is a proportional value between 0 and 1. The maximum Gini coefficient is 1 and the minimum is 0. The former indicates that the performance of the global model on the clients is absolutely uneven (that is, it performs best on one, and the rest are all 0), while the latter indicates that the global model is on the clients. The performance is absolutely average, and our goal is to ensure the final model performance while keeping Gini as small as possible.\par
Different from the existing FL system fairness definitions such as uniformity and STD, our proposed Gini describes the dispersion degree of constant distribution and has the characteristics of scale invariance. Therefore, the fairness degree of networks with different average performance can be compared with a uniform index.\par
Next, we prove through experiments that the non-IID data distribution on clients will not only reduce the accuracy and convergence efficiency but also lead to the disaster of fairness between clients. We trained the CIFAR-100 dataset with CNN model using FedAvg, set 100 clients and 10\% of the clients are selected to participate in the update in each round.\par
\begin{figure}[ht]
\centering
\subfigure[CIFAR-100 FedAvg IID]{\includegraphics[width=4.3cm, height=3.5cm]{Gini-IID.png}}
\subfigure[CIFAR-100 FedAvg Non-IID]{\includegraphics[width=4.3cm, height=3.5cm]{Gini-non-IID.png}}
\caption{When the CIFAR-100 data on 100 clients is in the IID distribution (left) and the non-IID distribution (right), the test results of FedAvg after 1000 rounds of training are shown in the figure. The larger the proportion of the orange takes, the more unfair it is.}
\label{fig: IID_and_Non}
\end{figure}
In the Lorenz curve, the smaller the proportion of orange, the higher the degree of fairness. As can be seen from Fig. 1, according to (2) updating the global model, the non-IID distribution of client data will increase the unfairness of the model's performance on the client side.\par
\subsection{DRL Settings}\label{AA}
In the federated learning training, since a certain percentage of clients are randomly selected to participate in the update in each round, the optimal weight distribution is non-differentiable. There are various approaches to deal with non-differentiable optimization bottlenecks, such as Gumbel-softmax \cite{Jang2017CategoricalRW} or stochastic back-propagation \cite{JimenezRezende2014StochasticBA}. In this paper, we model the problem of distributing the weight of different local models in the global model as a deep reinforcement learning problem to explore the optimal aggregation strategy \cite{Zhang2021DeepRL, Zhang2021AdaptiveCS}.\par
Instead of labeled data, reinforcement learning is a self-learning process in which agents maximize reward through interaction with the environment. The state-action transition and reward in the training process are abstracted as a Markov Decision Process(MDP), then the purpose of the DRL agent is to find an optimal policy that maximizes long-term reward expectations $\pi$:\par
\begin{equation}
\pi^{*} = argmax_{\pi}E_{\tau \sim \pi(\tau)}[r(\tau)],
\end{equation}\par
where $\pi$ represents the policy, $\tau$ represents a trajectory obtained by using the policy to interact with the environment, and $r(\tau)$ represents the overall reward for this trajectory. Next, we expand the formula to obtain the objective function and the gradient based on the Monte Carlo approximation as: \par
\begin{equation}
\ J(\theta) = E_{\tau \sim \pi_{\theta}(\tau)}[r(\tau)]
=\int_{\tau \sim \pi_{\theta}(\tau)}\pi_{\theta}(\tau)r(\tau)d\tau,
\end{equation}\par
\begin{equation}
\ \nabla_{\theta} J(\theta) = E_{\tau \sim \pi_{\theta}(\tau)}[\nabla_{\theta}log\pi_{\theta}(\tau)r(\tau)].
\end{equation}\par
Since our goal is to maximize long-term return expectations, we use gradient ascent to find the optimal policy \cite{Sutton1999PolicyGM}.\par
\begin{algorithm}
\caption{PGF-FedAvg}
\KwIn{Number of communication round $T$, number of clients $N$, percentage of updating in each round $C$, local epochs $E$, learning rate $\alpha$ and $\beta$.}
\textbf{Initialize:} Parameters $\phi^0$, $\omega^0$. \par
\For{t=0,1,...,T-1}
{
\textbf{Server} randomly selects a subset of clients $K=C*N$, and sends $\omega^{t}$ to them;\par
\For{client $k \in [K]$ \textbf{in parallel}}
{
Client $k$ copies $\omega^{t}$ as local model parameters $\omega_{k}^{t}$;\par
\For{j=0,1,...,E-1}
{ Calculate gradient $g_k^j$\par
$\omega_{k}^{j+1} \gets \omega_{k}^{j} - \alpha g_k^j $}
$acc_k^t \gets$ \textbf{Accuracy}\{$\omega_{k}^{t}$ tests on validation dataset $\{ \mathcal{D}_k^V\}$\}
}
Calculate $\mu^t=\frac{1}{K}\sum^{K}_{k=1}acc_k^t$ and $Gini^t$ using (3)\par
\textbf{DRL agent do:}\par
Get reward $r^{t}=-\mu^t\log(Gini^t)$\par
Update the DRL policy parameters $\phi$ \par
\[\phi^{t+1} \gets \phi^{t}+\beta r^{t} \nabla_{\phi}log\pi(s_t,a_t)\]
Get state $S^{t}=\{\omega_{1}^{t},...,\omega_{K}^{t}\}$\par
Calculate Gaussian distribution mean $\{a_1^t,...,a_K^t\}$\par
Calculate the aggregate weight $\{p_1^t,...,p_K^t\}$\par
\textbf{DRL end}\par
\textbf{Server} aggregates global model as
\[\omega^{t+1} \gets \sum^{K}_{k=1} p_k \omega_{k}^{t}\]
}
\end{algorithm}
Therefore, the federated learning process can be modeled as a Markov Decision Process (MDP), where the state is represented by the model parameters of each client in each round. Given the current state, the reinforcement learning agent learns a policy distribution according to the policy calculate the aggregated weights corresponding to each client, thereby updating the global model. After that, the updated global model parameters are transmitted to the local. On the local validation set, the local validation accuracy of the global model will be observed, and the reward of the DRL agent will be obtained from the average validation accuracy and Gini coefficient of each client. function. The objective is to train the DRL agent to converge to the target accuracy and fairness level for federated learning as quickly as possible.\par
In addition, in our algorithm, the DRL agent only needs to obtain the local model parameters and validation accuracy on clients, neither introducing additional communication overhead, nor without collecting and checking any private information, which can achieve the purpose of privacy protection.\par
\textbf{State:} The state of the $t$-th round is represented by a vector $\{\omega_1^t,...,\omega_K^t\}$, which respectively represents the model parameters of K clients participating in the update. During the training process, the client and the server jointly maintain a list of model parameters $\{\omega_k^t {\mid}k{\in}K\}$. In each round of FL, the client updates the list after uploading the trained local model to the server.\par
\textbf{Action:} Each time the state list is updated, we use the client model parameters participating in the aggregation to train the DRL agent. The action space is composed of a vector $\{a_1^t,...,a_K^t\}$, thus the $k$-th client draws the aggregation weight $p^t_k$, where $p^t_k \in (0, 1)$, which comes from Gaussian distribution with a learnable mean $a_k^t$ and a unit variance, for participating in the aggregation of the global model.\par
\textbf{Reward:} The reward comes from a small verification set locally on the client, defined as $r^t=-\mu^t log(Gini^t$), where $\mu^t$ denotes the average validation accuracy on clients and $Gini^t$ denotes the fairness Gini coefficient of accuracy (see Definition 1), respectively. Such setting encourages federated models to achieve optimal and fair performance.\par
The DRL agent is trained to maximize long-term rewards based on $\gamma$ discounts:\par
\begin{equation}
\ R = \sum_{a \sim \tau}\gamma^t r_t.
\end{equation}\par
Next, take FedAvg as an example to introduce our fairness optimization for the federated learning algorithm, the pseudo-code is shown in Algorithm 1.\par
\subsection{PG-FFL Workflow}
We define federated learning on a classification problem. If there are $K$ clients in total, the training set, validation set and test set on the $k$-th client are respectively $D_k^{Train} = {(x_i,y_i)_{i=1}^N}{\sim}P_k$, $D_k^{Test} = {(x_i,y_i)_{i=1}^M}{\sim}P_k$ and $D_k^V = {(x_i,y_i)_{i=1}^L}{\sim}P_k$, where $x_i{\in}X_k $ is the d-dimensional feature vector of feature space $X_k$, and $y_i$ is its corresponding label. The training set, validation set, and test set on each client are independent and identically distributed (IID), but the data size and distribution on different clients are not required to be the same. The goal of our training is to learn a global model that performs well and is fair on the test set of each client.\par
\begin{figure}[ht]
\centering
\includegraphics[width=9cm, height=5.5cm]{workflow.png}
\caption{The framework of PG-FFL.}
\label{fig: framework}
\end{figure}
Unlike traditional federated learning, our setup neither requires the data on different clients to be independently and identically distributed, nor is it designed to train a model that performs well on the server-side test set, which is more adapted to the requirements of the real world.\par
Fig.~\ref{fig: framework} shows how our algorithm, PG-FFL, is based on a reinforcement learning algorithm that assigns the aggregated weights of clients participating in the update in each round, following the steps below:\par%
\begin{itemize}
\item \textbf{Step 1(initialization)}: All N available devices with non-identical data size and distribution check in server as clients, the server selects $K=N*C$ clients participating in the update according to a certain proportion C, initializes the model parameter $\omega^{init}$ and transmits it to the selected client, the client uses the global Model parameters get validation accuracy on the validation set, then train on local data, and return local model parameters $\{\omega_k^1, k\in K\}$ and validation accuracy $\{acc_k^1, k\in K\}$.
\item \textbf{Step 2}: In the $t$-th round of iteration, the server calculates the average precision $\mu^t$ and Gini coefficient $Gini^t$ according to the returned $acc_k^t$, and then calculates the weight $P_{k}^t$ for the client k to participate in the global update, and then according to $\{\omega_k^t, k\in K\}$and $\{p_k^t, k\in K\}$ update the global model parameters.
\item \textbf{Step 3}: The server randomly selects a certain percentage of clients to participate in the update. After the selected clients use the last round of global model $\omega^{t-1}$ for local training, upload the locally updated model parameters and verification accuracy.
\end{itemize}
\hspace{-1cm}
\begin{figure*}[bp]
\centering
\subfigure[Case \uppercase\expandafter{\romannumeral1}]{\includegraphics[width=6cm, height=4.3cm]{CIFAR10-20.pdf}}
\label{fig: dataset_case1}
\hspace{-0.5cm}
\subfigure[Case \uppercase\expandafter{\romannumeral2}]{\includegraphics[width=6cm, height=4.3cm]{CIFAR10-5.pdf}}
\label{fig: dataset_case2}
\hspace{-0.5cm}
\subfigure[Case \uppercase\expandafter{\romannumeral2}]{\includegraphics[width=6cm, height=4.3cm]{CIFAR10-2.pdf}}
\label{fig: dataset_case3}
\caption{The data distribution of each client using non-IID data partition. The color bar denotes the number of data samples. Each rectangle represents the number of data samples of a specific class in a client.}
\end{figure*}
\vspace{-0.5cm}
\section{Experiment}
In this section, we provide the present empirical setup and results. We first describe our experimental setup (Section A), then we demonstrate the motivation for adding a fairness adjustment module based on an RL algorithm, showing that our algorithm can effectively reduce the classical federation model in the case of varying degrees of data non-uniform distribution Differences in performance on different clients while maintaining better performance. Meanwhile, we set the situation so bad that the data types between clients do not overlap at all, and compare the classification accuracy and fairness of the algorithm (Section B). Next, we compare the algorithm with the fairness goals of several baselines (Section C). Finally, we show the constraints of the algorithm (Section D).\par
\subsection{Experimental Setup}
\textbf{Federated datasets.} In this section, we will explore a suite of federated datasets based on classification tasks. These datasets include CIFAR-10 \cite{Krizhevsky2009LearningML}, CIFAR-100 \cite{Krizhevsky2009LearningML} and Fashion-MNIST \cite{Xiao2017FashionMNISTAN}. When used to compare with q-FFL and AFL, we will use a small benchmark dataset studied by \cite{Li2020FairRA} based on Fashion-MNIST.\par
\textbf{Data partitions.} In the construction of the non-IID dataset, we divide the data of each class of the N-class classification dataset into equal-sized partitions, and all clients randomly select different numbers of partitions, so that each client has local data with inconsistent quantities and categories. As shown in Fig. 3, i) Case \uppercase\expandafter{\romannumeral1}, as (a), set 100 clients, each client randomly selects 20 data blocks, take CIFAR-100 as an example; ii) Case \uppercase\expandafter{\romannumeral2}, as (b), set 5 clients, each client has 4 data blocks, take CIFAR-10 as an example, the data categories on the client overlap; iii) Case \uppercase\expandafter{\romannumeral3}, as (c), set 5 clients, each client has 2 data blocks, Taking CIFAR-10 as an example, the data categories on the client do not overlap at all.\par
\begin{figure*}
\centering
\subfigure[CIFAR-10]{\includegraphics[width=5.8cm, height=3cm]{CIFAR10-fedavg-acc.pdf}}
\subfigure[CIFAR-100]{\includegraphics[width=5.8cm, height=3cm]{CIFAR100-fedavg-acc.pdf}}
\subfigure[Fashion-MNIST]{\includegraphics[width=5.8cm, height=3cm]{fmnist-fedavg-acc.pdf}}
\\ \vspace{-0.2cm}
\centering
\subfigure[CIFAR-10]{\includegraphics[width=5.8cm, height=3cm]{CIFAR10-fedavg-gini.pdf}}
\subfigure[CIFAR-100]{\includegraphics[width=5.8cm, height=3cm]{CIFAR100-fedavg-gini.pdf}}
\subfigure[Fashion-MNIST]{\includegraphics[width=5.8cm, height=3cm]{fmnist-fedavg-gini.pdf}}
\\ \vspace{-0.2cm}
\centering
\subfigure[CIFAR-10]{\includegraphics[width=5.8cm, height=3cm]{CIFAR10-fedprox-acc.pdf}}
\subfigure[CIFAR-100]{\includegraphics[width=5.8cm, height=3cm]{CIFAR100-fedprox-acc.pdf}}
\subfigure[Fashion-MNIST]{\includegraphics[width=5.8cm, height=3cm]{fmnist-fedprox-acc.pdf}}
\\ \vspace{-0.2cm}
\centering
\subfigure[CIFAR-10]{\includegraphics[width=5.8cm, height=3cm]{CIFAR10-fedprox-gini.pdf}}
\subfigure[CIFAR-100]{\includegraphics[width=5.8cm, height=3cm]{CIFAR100-fedprox-gini.pdf}}
\subfigure[Fashion-MNIST]{\includegraphics[width=5.8cm, height=3cm]{fmnist-fedprox-gini.pdf}}
\caption{From left to right are the experimental results on the CIFAR-10, CIFAR-100 and Fashion-MNIST datasets, respectively. Data distribution refer to Fig. 3, case \uppercase\expandafter{\romannumeral1}. The first and second rows show the performance (average test accuracy on clients) and fairness (definition 1) of our algorithm compared with FedAvg. The third and fourth rows show the performance and fairness of our algorithm compared with FedProx. Note that our fairness plugin has the same parameter settings as the original algorithm before adding.}
\end{figure*}
\textbf{Implementation.} We implemented all the codes based on Pytorch, using a server and N clients to simulate a federated network, where N is the total number of clients.\par
\hspace{-0.5cm}
\subsection{Fairness of PG-FFL}
In this section, we show the efficiency of our fairness adjustment plug-in combined with FedAvg and FedProx, which are both classical and effective FL algorithms. We set up 100 clients to train on CIFAR-10, CIFAR-100 and Fashion-MNIST, respectively, and the data distribution of clients is as case \uppercase\expandafter{\romannumeral1} shown in Fig. 3(c), the local data distribution is highly heterogeneous. On the model selection, we train CIFAR-10 and CIFAR-100 by CNN, Fashion-MNIST by a four-layer MLP. The fairness adjustment plug-in based on the policy gradient reinforcement learning algorithm uses four-layer multi-layer-perceptron to learn the optimal aggregation strategy.\par
For each communication epoch, FedProx and PGF-FedProx are set to train locally for 5 epochs, while FedAvg and PGF-FedAvg execute one epoch locally. we can observe that in Fig. 4, our algorithm basically maintains the same convergence speed as the baseline, the average accuracy is slightly improved or more stable, and the fairness is significantly enhanced.\par
Next, we further exacerbate the inhomogeneity of the data distribution, verifying that our proposed PG-FFL algorithm provides a fairer solution for federated data. We will validate on a 10-class classification problem, reduce the number of clients trained on Cifar10 and Fashion MNIST to 5, and have them all participate in each round of federated updates. In Table 1, we compare the final test accuracy and fairness of our proposed fairness adjustment plugin combined with FedAvg and FedProx, respectively, for the data IID and non-IID distribution as case \uppercase\expandafter{\romannumeral2} and case \uppercase\expandafter{\romannumeral3}. Note that, when each client has only two classes, their data classes will not overlap at all.\par
We can observe that with the deterioration of the client data distribution non-IID situation, the baseline algorithm can significantly improve the fairness after adding the fairness adjustment plug-in, and also improve the average accuracy of the model on the client. When the client data distributes IID, our method sometimes can also improve the fairness of the model, but it will cause a certain loss of accuracy. We speculate that it is because the fairness and average accuracy are considered in our RL model reward at this time, so In order to ensure a high degree of fairness between the client test accuracy, it prevents our algorithm from pursuing higher average accuracy, thus causing some accuracy loss.\par
\begin{table}[ht]
\caption{The accuracy and fairness comparison of PG-FFL and baselines tested on datasets with varying degrees of non-IID.}
\setlength{\tabcolsep}{1.2mm}{
\renewcommand\arraystretch{1.5}
\begin{tabular}{ccccccc}
\multicolumn{7}{c}{CIFAR-10} \\ \hline
\multicolumn{1}{c}{Non-IID level} & \multicolumn{2}{c}{IID} & \multicolumn{2}{c}{Case 2} & \multicolumn{2}{c}{Case 3} \\
\multicolumn{1}{c}{} & acc(↑) & Gini(↓) & acc(↑) & Gini(↓) & acc(↑) & Gini(↓) \\ \hline
\multicolumn{1}{c}{FedAvg} & 0.675 & 0.098 & 0.612 & 0.102 & 0.454 & 0.161 \\
\multicolumn{1}{c}{PGF-FedAvg} & \textbf{0.683} & \textbf{0.042} & \textbf{0.665} & \textbf{0.049} & \textbf{0.488} & \textbf{0.053} \\
\multicolumn{1}{c}{FedProx} & \textbf{0.708} & 0.107 & 0.595 & 0.110 & 0.422 & 0.154 \\
\multicolumn{1}{c}{PGF-FedProx} & 0.689 & \textbf{0.073} & \textbf{0.608} & \textbf{0.037} & \textbf{0.453} & \textbf{0.031} \\ \hline
\multicolumn{7}{c}{CIFAR-100} \\ \hline
\multicolumn{1}{c}{Non-IID level} & \multicolumn{2}{c}{IID} & \multicolumn{2}{c}{Case 2} & \multicolumn{2}{c}{Case 3} \\
\multicolumn{1}{c}{} & acc(↑) & Gini(↓) & acc(↑) & Gini(↓) & acc(↑) & Gini(↓) \\ \hline
\multicolumn{1}{c}{FedAvg} & 0.493 & 0.130 & 0.468 & 0.133 & 0.316 & 0.151 \\
\multicolumn{1}{c}{PGF-FedAvg} & \textbf{0.502} & \textbf{0.093} & \textbf{0.491} & \textbf{0.068} & \textbf{0.337} & \textbf{0.093} \\
\multicolumn{1}{c}{FedProx} & 0.509 & 0.133 & 0.467 & 0.143 & 0.299 & 0.164 \\
\multicolumn{1}{c}{PGF-FedProx} & \textbf{0.514} & \textbf{0.084} & \textbf{0.483} & \textbf{0.079} & \textbf{0.331} & \textbf{0.085} \\ \hline
\multicolumn{7}{c}{Fashion-MNIST} \\ \hline
\multicolumn{1}{c}{Non-IID level} & \multicolumn{2}{c}{IID} & \multicolumn{2}{c}{ Case 2} & \multicolumn{2}{c}{Case 3} \\
\multicolumn{1}{c}{} & acc(↑) & Gini(↓) & acc(↑) & Gini(↓) & acc(↑) & Gini(↓) \\ \hline
\multicolumn{1}{c}{FedAvg} & 0.869 & 0.032 & 0.850 & 0.063 & 0.737 & 0.074 \\
\multicolumn{1}{c}{PGF-FedAvg} & \textbf{0.884} & \textbf{0.021} & \textbf{0.874} & \textbf{0.024} & \textbf{0.796} & \textbf{0.033} \\
\multicolumn{1}{c}{FedProx} & 0.879 & 0.021 & 0.851 & 0.045 & 0.828 & 0.037 \\
\multicolumn{1}{c}{PGF-FedProx} & \textbf{0.883} & \textbf{0.017} & \textbf{0.854} & \textbf{0.031} & \textbf{0.836} & \textbf{0.020} \\ \hline
\end{tabular}}
\end{table}
\subsection{Comparison With Other Fair Federated Learning Algorithms}
Next, we compare with other two algorithms that also aim to address fairness issues in federated networks.\par
In the experiments in this section, we implement a very extreme case where each client has only a completely disjoint class of data. Using the same experimental setup as \cite{Mohri2019AgnosticFL}: The Fashion-MNIST dataset \cite{Xiao2017FashionMNISTAN} is an MNIST-like dataset where images are classified into 10 categories of clothing instead of handwritten digits. We extract a subset of the data labeled with three categories - shirts/tops, pullovers, and shirts, and divide this subset into three clients, each containing a category of clothing. We then train a classifier for these three classes using logistic regression and the Adam optimizer. Since the clients here uniquely identify the labels, in this experiment we did not compare with models trained on specific baselines.\par
We observe in Table 2 that our algorithm performs better both in the final average accuracy and fairness between clients.\par
\vspace{-0.3cm}
\begin{table}[ht]
\caption{The accuracy and fairness comparison with PG-FFL and other fairness algorithms in the case of extreme data non-IID distribution.}
\centering
\renewcommand\arraystretch{1.5}
\begin{tabular}{lccccc}
\hline
& \multicolumn{2}{c}{All Clients} & Shirts & Pullovers & T-shirts \\
& acc(↑) & Gini(↓) & acc(↑) & acc(↑) & acc(↑) \\ \hline
q-FFL(q=0) & 78.8 & 0.084 & 66.0 & \textbf{84.5} & \textbf{85.9} \\
AFL & 78.2 & 0.046 & 71.4 & 81.0 & 82.1 \\
PGF-FedAvg & \textbf{79.1} & \textbf{0.027} & \textbf{74.2} & 80.5 & 82.6 \\ \hline
\end{tabular}
\end{table}
\vspace{-0.5cm}
\subsection{Scalability Analysis}
In this section, we continue the experimental setup in section B by combining our proposed fairness adjustment plugin with FedAvg and FedProx, respectively, to modify the percentage of participating updates in each round, but keep the total number of clients unchanged. It can be observed from Fig. 5 that there is an upper limit on the scalability of the algorithm, and the greater the proportion of clients participating in the global update, the better the effect. We guess it is because each time the RL algorithm will output the proportion of the client participating in the update, but when the proportion of the client participating in the update is small, the non-participating clients cannot obtain the updated aggregate participation weight in time, which will affect the experimental results. We use STD as the fairness indicator here because that is consistent with the test accuracy dimension and is more likely to show volatility. It's easy to see that PG-FFL can still improve the fairness of the federated network under other fairness definition.\par
\begin{figure}[ht]
\centering
\subfigure[10\% from 100 clients]{\includegraphics[width=4cm]{0.1.pdf}}
\subfigure[25\% from 100 clients]{\includegraphics[width=4cm]{0.25.pdf}}
\\
\centering
\subfigure[50\% from 100 clients]{\includegraphics[width=4cm]{0.5.pdf}}
\subfigure[75\% from 100 clients]{\includegraphics[width=4cm]{0.75.pdf}}
\caption{The solid line is the client's average validation accuracy, the green range is the accuracy STD on different clients, and the smaller the area is, the fairer it is.}
\end{figure}
\section{Conclusion}
In this paper, we propose fairness as a new optimization objective defined by the Gini coefficient of clients' validation accuracy, which is based on realistic considerations that encourage fairer accuracy distribution across clients in federated learning. We design a reinforcement learning plug-in to apply federated algorithms to solve this problem in large-scale networks, and experiments demonstrate that PG-FFL can be regarded as a fairness add-on for any global objective. We illustrate the fairness and superiority of PG-FFL on a set of federated datasets, and experimental results show that our framework outperforms baseline methods in terms of overall performance, fairness, and convergence speed.\par
\section*{Acknowledgement}
This paper is supported by the Key Research and Development Program of Guangdong Province under grant No. 2021B0101400003 and Shenzhen Basic Research Program (Natural Science Foundation) Key Program of Fundamental Research (No. JCYJ20200109143016563). Corresponding authors are Jianzong Wang from Ping An Technology (Shenzhen) Co., Ltd (jzwang@188.com) and Yuhan Dong from Tsinghua University (dongyuhan@sz.tsinghua.edu.cn).
\footnotesize
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Home Project Orcas Photo and video Forum Links Contacts
Text: Tatyana Ivkovich
On the 10th of August 2010, we found a large dispersed group of orcas in front of Russkaja bay (the southern part of Avacha Gulf, on the southeast Kamchatka peninsula). First, we encountered a tight resting group called Carmen's family. We know this family unit very well – it consists of three very graceful females (Carmen, Paloma and their mother AV023) and one juvenile (AV023a).
Killer whales from Carmen family
Besides them we were also very happy to see Moloko's family. We hadn't met them for two seasons. It was great to learn that Moloko's daughter had grown up into a mature female and given birth to a calf.
Killer whales from Moloko family (Moloko is in the center)
We also met three orcas that we couldn't recognize. They made up a very tight group and didn't let us approach them. This was the type of behaviour common for visiting — we call them 'strange' — killer whales that come to Avacha Gulf only rarely. Anyway we managed to take some photos. The whales were moving slowly out to the open ocean — we were losing sight of land — and soon we had to leave them. Back at the camp, fish dinner on the fire and looking more closely at the photos, we discovered that these three orcas were from the well-known unit of Drkin's friends. There were two females (AV043 and AV046), a young male (AV044) and one juvenile whale (AV045) in the family. They were always rather shy but previously they had let us approach much closer.
Female AV043 is on the left
The next day we were lucky to have calm seas and high clouds, and once we were out on the water, we met a large killer whale aggregation. This group included orca family units from the previous day. Killer whales were traveling slowly from south to north all along the shore, spouting, slicing the water with their sharp fins, socializing and occasionally feeding, diving down to check for food.
Then, in the middle of this big group, we noticed Drkin's friends. The same very tight family that we had seen the day before, was moving in a similar manner as they had done the day before. But something was different; something had changed since the past night. Female AV043 had given life to a new killer whale. The newborn orca was surfacing in very tight contact with its mother, nudging her repeatedly and hiding between the bodies of the two big females, the other female perhaps her auntie. This was the first time we knew the exact day of an orca birth. Orcas have a very high level of calf mortality – more than 50% are lost before they reach age of five. We hope that this newborn orca will survive and live its life free and wild in the waters of the Russian Far East.
The newborn calf between two females - only the tip of the dorsal fin is visible
The newborn calf near his mother
The new birth provides good evidence of Avacha Gulf's importance to our Russian Kamchatka orca population. Orcas use Avacha Gulf not only for feeding, resting and socializing — but also for giving a birth.
OUR ORCAS
ORCA STORIES
- New messages -
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: How to keep my application service always running in android2.1? Hi i have written reminder code in service onstart().and when user insert date-time and insert record at that time service called by startservice() function,but only service is starting when i insert record i.e i am getting reminder when it get call from my activity.but i want reminder after 3days or something so how should i keep service always on so that i can get reminder in future? or how should i make connection of service keep alive?should i called bindservice() frunction from my any activity or what?
thanks in advance---
A: Don't let your Service run all the time. It consumes battery and memory when not neccessary¹.
Rather schedule a PendingIntent via the AlarmManager thats starts the service at the relevant point in time to do it's work. When done, kill the service again.
In general androids services are used different then services/daemons on a "normal" computer. They have a task that they execute, then they quit (usually via Service.stopSelf()) until someone starts them again to do more work.
Here is a small example how the AlarmManager is used:
// get a calendar with the current time
Calendar cal = Calendar.getInstance();
// add 15 minutes to the calendar object
cal.add(Calendar.MINUTE, 15);
Intent intent = new Intent(ctx, YourService.class);
PendingIntent pi = PendingIntent.getService(this, 123, intent,
PendingIntent.FLAG_UPDATE_CURRENT);
AlarmManager am = (AlarmManager) getSystemService(ALARM_SERVICE);
am.set(AlarmManager.RTC_WAKEUP, cal.getTimeInMillis(), pi);
This launches the intent to start YourService in 15 minutes from now. There is plenty of documentation for sending intents this way, search a bit around.
¹ Which will eventually frustrate your users: "Why does this app waste my battery?" is a pretty common question
A: Sometimes, it may be necessary for your android app to complete a task sometime in the future. In order to do this, you must schedule an activity (can also be a service) to be run using Android's AlarmManager. This post will show:
* How to set up a receiver for the scheduled event
* How to create an activity from this receiver
* Using the AlarmManager and the created classes to successfully receive and process a scheduled event
Creating a BroadcastReceiver
The first thing you will need is a receiver to receive the event. There are several key aspects to have for the receiver to work properly. First create a class that extends BroadcastReceiver and override and implement the necessary onReceive(Context context, Intent intent) method. The following is a basic example using a Toast message:
package com.justcallmebrian.alarmexample;
import android.content.BroadcastReceiver;
import android.content.Context;
import android.content.Intent;
import android.widget.Toast;
import android.os.Bundle;
public class AlarmReceiver extends BroadcastReceiver {
@Override
public void onReceive(Context context, Intent intent) {
try {
Bundle bundle = intent.getExtras();
String message = bundle.getString("alarm_message");
Toast.makeText(context, message, Toast.LENGTH_SHORT).show();
} catch (Exception e) {
Toast.makeText(context, "There was an error somewhere, but we still received an alarm", Toast.LENGTH_SHORT).show();
e.printStackTrace();
}
}
}
In the previous example, we are simply just printing out the message supplied by the String passed in under the name of alarm_message. For those who are not familiar with Toast, it is basically a short and quick message given to the user. Here you can find more information on Toast.
Besides actually creating the class to receive the event, you must also declare it within AndroidManifest.xml. The following is the line that should be added before the closing of the application tag (before ).
Basically this states that the class AlarmReceiver is available and will start a private process. Once that is done, your BroadcastReceiver is ready to go.
Setting up an Event using AlarmManager
In order to receive an event, you obviously must schedule the event. There are three ways of scheduling an event (a one time event using the set method, a repeating event using the setRepeating method and finally using the setInexactRepeating). This tutorial will cover the one time alarm using the set method. For more information regarding the other events, you can view AlarmManager.
The following code snippet will get the AlarmManager and set an event to occur 5 minutes from the current time:
// get a Calendar object with current time
Calendar cal = Calendar.getInstance();
// add 5 minutes to the calendar object
cal.add(Calendar.MINUTE, 5);
Intent intent = new Intent(ctx, AlarmReceiver.class);
intent.putExtra("alarm_message", "O'Doyle Rules!");
// In reality, you would want to have a static variable for the request code instead of 192837
PendingIntent sender = PendingIntent.getBroadcast(this, 192837, intent, PendingIntent.FLAG_UPDATE_CURRENT);
// Get the AlarmManager service
AlarmManager am = (AlarmManager) getSystemService(ALARM_SERVICE);
am.set(AlarmManager.RTC_WAKEUP, cal.getTimeInMillis(), sender);
This code snippet basically obtains a new Calendar object and adds 5 minutes to it. An Intent is created with the AlarmReceiver which we created earlier. The more important piece of the code is setting the FLAG_UPDATE_CURRENT flag. Without this flag, the message being passed as an Extra will be lost and not obtained by the receiver.
With these code snippets you should be able to create and run some tasks in the BroadcastReceiver. However, sometimes you may wish to start a new activity (or service) on the alarm event. In order to do this, you would want to have the AlarmReceiver create and start the new Activity.
Starting an Activity from BroadcastReceiver
Starting an activity within a Receiver has an extra flag that is needed. We will change the previous onReceive for AlarmReceiver to get this done:
@Override
public void onReceive(Context context, Intent intent) {
try {
Bundle bundle = intent.getExtras();
String message = bundle.getString("alarm_message");
Intent newIntent = new Intent(context, AlarmActivity.class);
newIntent.putExtra("alarm_message", message);
newIntent.addFlags(Intent.FLAG_ACTIVITY_NEW_TASK);
context.startActivity(newIntent);
} catch (Exception e) {
Toast.makeText(context, "There was an error somewhere, but we still received an alarm", Toast.LENGTH_SHORT).show();
e.printStackTrace();
}
}
Now you would only have to create the new AlarmActivity as you would do any other Activity. Don't forget to include the newly created activity in the AndroidManifest.xml file.
A: Firstly no need of service,you can useAlarManagerClass Link Alarmanger class for schedule events and show alert at specific time and date. If you want show messages after some long duration then schedule a Pending-intent via the AlarmManager thats starts the service at the relevant point in time to do it's work. When done, kill the service again as per tell by above answers . In addition you can store your data into shared preferences for permanently. You can retrieve it at any time for resheulding it on device reboot or for other purpose.
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Thinking of pimping your nails with chrome, holographic or ombre glitter? Well, whatever beauty trend you'd like to have on your tips, this salon's got your back hands. Using OPI, China Glaze and Korean brand Gellyfit, the manicurists can decorate your ends with flashy nail art or opt for a reverse French mani if you prefer something simple. This nail bar also offers pedicure services (from $88), so you can show off your toes in sandals all day, every day.
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"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}\label{sec1}
All graphs considered in this paper are finite, simple and undirected.
For terms and symbols not defined in this paper, we refer the reader to \cite{D}.
For two graphs $G$ and $H$, $G$ is said to be {\it $H$-free} if $G$ contains no copy of $H$ as an induced subgraph.
For a family $\HH$ of graphs, a graph $G$ is said to be {\it $\HH$-free} if $G$ is $H$-free for every $H\in \HH$.
In this context, the members of $\HH$ are called {\it forbidden subgraphs}.
For two families $\HH_{1}$ and $\HH_{2}$ of graphs, we write $\HH_{1}\leq \HH_{2}$ if for every $H_{2}\in \HH_{2}$, $H_{2}$ contains a copy of an element of $\HH_{1}$ as an induced subgraph.
The relation ``$\leq $'' between two families of forbidden subgraphs was introduced in \cite{FKLOPS}.
Note that if $\HH_{1}\leq \HH_{2}$, then every $\HH_{1}$-free graph is also $\HH_{2}$-free.
Let $K_{n}$, $K_{1,n}$ and $P_{n}$ denote the {\it complete graph} of order $n$, the {\it star} of order $n+1$ and the {\it path} of order $n$, respectively.
Let $\overline{K_{n}}$ denote the complement of $K_{n}$, i.e., a graph of order $n$ having no edges.
Gy\'{a}rf\'{a}s~\cite{G} and Sumner~\cite{S} independently conjectured that for every tree $T$, there exists a function $f_{T}:\mathbb{N}\rightarrow \mathbb{N}$ such that every $T$-free graph $G$ satisfies $\chi (G)\leq f_{T}(\omega (G))$, where $\chi (G)$ and $\omega (G)$ are the {\it chromatic number} and the {\it clique number} of $G$, respectively.
In graph theory, the Gy\'{a}rf\'{a}s-Sumner conjecture has been studied as a fundamental and important problem.
The conjecture is true for the case where $T$ is a path, stars, and some specified trees, but it is widely open (see a survey~\cite{SS}).
Here we focus on Ramsey-type problems.
A classical Ramsey theorem~\cite{R} is equivalent to the following statement:
For a family $\HH$ of graphs, there exists a constant $c(\HH)$ such that $|V(G)|\leq c(\HH)$ for every $\HH$-free graph $G$ if and only if $\HH\leq \{K_{n},\overline{K_{n}}\}$ for an integer $n\geq 1$.
Furthermore, an analogy of the Ramsey theorem for connected graphs is also well-known as a folklore (see \cite[Proposition~9.4.1]{D}):
For a family $\HH$ of connected graphs, there exists a constant $c(\HH)$ such that $|V(G)|\leq c(\HH)$ for every connected $\HH$-free graph $G$ if and only if $\HH\leq \{K_{n},K_{1,n},P_{n}\}$ for an integer $n\geq 1$.
Remark that the $\overline{K_{n}}$-freeness and the $\{K_{1,n},P_{n}\}$-freeness play a similar role in above results.
In particular, the combination of induced stars and induced paths is a natural alternative concept of an independent set when we treat induced subgraphs.
Recently, similar type problems on invariants (other than the order) were obtained as Ramsey-type problems (see, for example, \cite{BPRS,CF,CFKP,DDL,F}).
For a graph-invariant $\mu $, we consider the following condition concerning a (finite) family $\HH$ of graphs:
\begin{enumerate}[{\bf (P-$\mu $)}]
\item
There exists a constant $c(\HH)$ such that $\mu (G)\leq c(\HH)$ for every connected $\HH$-free graph $G$.
\end{enumerate}
A Ramsey-type problem of $\mu $ is to characterize $\HH$ satisfying {\rm (P-$\mu $)}.
Since there exist infinitely many graphs with sufficiently large chromatic number and sufficiently large girth~\cite{E}, we can easily verify that the following conjecture is equivalent to the Gy\'{a}rf\'{a}s-Sumner conjecture.
\begin{conj
\label{GSconj}
Let $\HH$ be a finite family of connected graphs.
Then $\HH$ satisfies {\rm (P-$\chi $)} if and only if $\HH\leq \{K_{n},T\}$ for an integer $n\geq 1$ and a tree $T$.
\end{conj}
Thus, settling Ramsey-type problems for invariants closely related to the chromatic number, one might expect to obtain some essential informations and effective techniques for the Gy\'{a}rf\'{a}s-Sumner conjecture.
As a study of this policy, for example, Chudnovsky and Seymour~\cite{CS} posed a conjecture on a Ramsey-type problem for the cochromatic number, where the {\it cochromatic number} of a graph $G$ is the minimum cardinality of a family of independent sets and cliques of $G$ whose union is $V(G)$, and proved that the conjecture is equivalent to the Gy\'{a}rf\'{a}s-Sumner conjecture.
On the other hand, as far as we know, no Ramsey-type problem for an invariant around the chromatic number was completely settled.
Our main aim in this paper is to approach the Gy\'{a}rf\'{a}s-Sumner conjecture by settling such Ramsey-type problems.
Remark that Choi, Kim and Park~\cite{CKP} characterized forbidden structures which are not necessary induced, bounding some analogies of the chromatic number by a constant.
Let $G$ be a graph, and let $\AA$ be a family of graphs.
A family $\PP$ of induced subgraphs of $G$ is called an {\it induced $\AA$-cover} of $G$ if $\bigcup _{P\in \PP}V(P)=V(G)$ and each element of $\PP$ is isomorphic to an element of $\AA$.
Note that some elements of an induced $\AA$-cover of $G$ might have common vertices.
An induced $\AA$-cover $\PP$ of $G$ is called an {\it induced $\AA$-partition} of $G$ if the elements of $\PP$ are pairwise vertex-disjoint.
We mainly consider the case where $\AA$ is equal to the family $\AA_{\rm sp}:=\{K_{1,n}:n\geq 1\}\cup \{P_{n}:n\geq 1\}$, i.e., the family of stars and paths.
An induced $\AA_{\rm sp}$-cover (resp. an induced $\AA_{\rm sp}$-partition) of $G$ is called an {\it induced SP-cover} (resp. an {\it induced SP-partition}) of $G$.
The minimum cardinality of an induced SP-cover (resp. an induced SP-partition) of $G$, denoted by ${\rm inspc}(G)$ (resp. ${\rm inspp}(G)$), is called the {\it induced SP-cover number} (resp. the {\it induced SP-partition number}) of $G$.
Let us now compare the induced SP-cover/partition numbers and the chromatic number.
Note that the chromatic number of a graph $G$ is equal to the minimum cardinality of an induced $\{\overline{K_{n}}:n\geq 1\}$-cover of $G$, which is also equal to the minimum cardinality of an induced $\{\overline{K_{n}}:n\geq 1\}$-partition of $G$.
As we mentioned in the third paragraph of this section, the combination of induced stars and induced paths is related to the existence of independent sets, or induced $\overline{K_{n}}$'s.
In fact, for an induced SP-cover $\PP$ of $G$, since every element of $\PP$ is a bipartite graph, $\{G[A_{P}],G[B_{P}]:P\in \PP\}$ is an induced $\{\overline{K_{n}}:n\geq 1\}$-cover of $G$, where $\{A_{P},B_{P}\}$ is a bipartition of $P$.
Hence we obtain $\chi (G)\leq 2 \, {\rm inspc}(G)\leq 2 \, {\rm inspp}(G)$ for every graph $G$, which implies that the induced SP-cover/partition numbers are directly related concepts to the chromatic number.
In particular, if we restrict induced $\AA$-covers/partitions to the case where each element of $\AA$ is connected, then small induced SP-covers/partitions are alternatives to small induced $\{\overline{K_{n}}:n\geq 1\}$-covers/partitions, and in this sense those two numbers can be regarded as ``connected versions'' of the chromatic number.
In this paper, we characterize the finite families $\HH$ of connected graphs satisfying {\rm (P-${\rm inspc}$)} or {\rm (P-${\rm inspp}$)}.
\subsection{Fundamental definitions}\label{sec1.1}
Let $G$ be a graph.
Let $V(G)$ and $E(G)$ denote the {\it vertex set} and the {\it edge set} of $G$, respectively.
For a vertex $x\in V(G)$, let $N_{G}(x)$ denote the {\it neighborhood} of $x$ in $G$; thus $N_{G}(x)=\{y\in V(G): xy\in E(G)\}$.
For two vertices $x,y\in V(G)$, let ${\rm dist}_{G}(x,y)$ denote the length of a shortest $x$-$y$ path of $G$.
For a vertex $x\in V(G)$, let ${\rm ecc}_{G}(x)=\max\{{\rm dist}_{G}(x,y):y\in V(G)\}$.
The {\it diameter} of $G$, denoted by ${\rm diam}(G)$, is defined as ${\rm diam}(G)=\max\{{\rm ecc}_{G}(x):x\in V(G)\}$.
Let $\alpha (G)$ denote the {\it independence number} of $G$, i.e., the maximum cardinality of an independent set of $G$.
For a subset $X$ of $V(G)$, let $G[X]$ (resp. $G-X$) denote the subgraph of $G$ induced by $X$ (resp. $V(G)\setminus X$).
For a subset $F$ of $E(G)$, let $G-F$ denote the spanning subgraph of $G$ with the edge set $E(G)\setminus F$.
For two integers $n_{1}\geq 1$ and $n_{2}\geq 1$, the {\it Ramsey number} $R(n_{1},n_{2})$ is the minimum positive integer $R$ such that any graph of order at least $R$ contains a clique of cardinality $n_{1}$ or an independent set of cardinality $n_{2}$.
Recall that we have defined the family $\AA_{\rm sp}$.
In this paper, we additionally focus on induced $\AA$-covers/partitions of graphs for the case where $\AA$ is equal to one of the following families:
\begin{enumerate}[{$\bullet $}]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
$\AA_{\rm s}:=\{K_{1}\}\cup \{K_{1,n}:n\geq 1\}$, i.e., the family of stars where a graph of order one is regarded as a star.
\item
$\AA_{\rm p}:=\{P_{n}:n\geq 1\}$, i.e., the family of paths.
\end{enumerate}
Then for a graph $G$, we can naturally define some terminologies and notations as follows:
\begin{enumerate}[{$\bullet $}]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
An induced $\AA_{\rm s}$-cover (resp. an induced $\AA_{\rm s}$-partition) of $G$ is called an {\it induced star cover} (resp. an {\it induced star partition}) of $G$.
\item
The minimum cardinality of an induced star cover (resp. an induced star partition) of $G$, denoted by ${\rm insc}(G)$ (resp. ${\rm insp}(G)$), is called the {\it induced star cover number} (resp. the {\it induced star partition number}) of $G$.
\item
An induced $\AA_{\rm p}$-cover (resp. an induced $\AA_{\rm p}$-partition) of $G$ is called an {\it induced path cover} (resp. an {\it induced path partition}) of $G$.
\item
The minimum cardinality of an induced path cover (resp. an induced path partition) of $G$, denoted by ${\rm inpc}(G)$ (resp. ${\rm inpp}(G)$), is called the {\it induced path cover number} (resp. the {\it induced path partition number}) of $G$.
\end{enumerate}
The induced path cover/partition problems themselves are interested topics in graph theory, and have been studied (see \cite{AHW,CMSHH,HSMW,HSMW2,PC3}).
The induced path partition number is frequently called the {\it induced path number}.
However, in this paper, we use the terminology ``induced path partition number'' to avoid confusion.
As we mentioned above, our main aim is to characterize the finite families $\HH$ of connected graphs satisfying {\rm (P-${\rm inspc}$)} or {\rm (P-${\rm inspp}$)}.
Furthermore, during the process, we also discuss about the conditions {\rm (P-${\rm insc}$)}, {\rm (P-${\rm insp}$)}, {\rm (P-${\rm inpc}$)} and {\rm (P-${\rm inpp}$)}, and characterize the finite families satisfying one of them.
Remark that the authors~\cite{CF,CF2} recently settled Ramsey-type problems for the (not necessary induced) path cover/partition numbers of graphs and digraphs.
\subsection{Main results}\label{sec1.2}
To state our main results, we prepare some graphs which will be used as forbidden subgraphs (see Figure~\ref{f1}).
Let $n\geq 2$ be an integer, and let $A:=\{x_{i},y_{i},z_{i}:1\leq i\leq n\}$.
\begin{enumerate}[{$\bullet $}]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
Let $S^{*}_{n}$ denote the graph on $A\setminus \{x_{i}:2\leq i\leq n\}$ such that $E(S^{*}_{n})=\{x_{1}y_{i},y_{i}z_{i}:1\leq i\leq n\}$.
\item
Let $\tilde{S}_{n}$ be the graph obtained from $S^{*}_{n}$ by adding $n$ edges $x_{1}z_{i}~(1\leq i\leq n)$.
\item
Let $F^{(1)}_{n}$ denote the graph on $A\setminus \{x_{i}:3\leq i\leq n\}$ such that $E(F^{(1)}_{n})=\{x_{1}x_{2},x_{1}y_{1},x_{1}z_{1}\}\cup \{y_{i}y_{i+1},z_{i}z_{i+1}:1\leq i\leq n-1\}$.
\item
Let $F^{(2)}_{n}$ denote the graph on $A\setminus \{x_{i}:2\leq i\leq n\}$ such that $E(F^{(2)}_{n})=\{x_{1}y_{1},x_{1}z_{1},y_{1}z_{1}\}\cup \{y_{i}y_{i+1},z_{i}z_{i+1}:1\leq i\leq n-1\}$.
\item
Let $F^{(3)}_{n}$ denote the graph on $A$ such that $E(F^{(3)}_{n})=\{x_{i}y_{1},x_{i}z_{1}:1\leq i\leq n\}\cup \{y_{i}y_{i+1},z_{i}z_{i+1}:1\leq i\leq n-1\}$.
\item
Let $F^{(4)}_{n}$ be the graph obtained from $F^{(3)}_{n}$ by deleting $n-2$ vertices $x_{i}~(3\leq i\leq n)$.
\item
Let $F^{(5)}_{n}$ be the graph obtained from $F^{(4)}_{n}$ by adding the edge $x_{1}x_{2}$.
\end{enumerate}
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\caption{Graphs $S^{*}_{n}$, $\tilde{S}_{n}$ and $F^{(i)}_{n}~(1\leq i\leq 5)$}
\label{f1}
\end{center}
\end{figure}
Our main result is the following.
\begin{thm
\label{mainthm}
Let $\HH$ be a finite family of connected graphs.
\begin{enumerate}[{\upshape(i)}]
\item
The family $\HH$ satisfies {\rm (P-${\rm inspc}$)} if and only if $\HH\leq \{K_{n},S^{*}_{n},F^{(1)}_{n},F^{(2)}_{n},F^{(3)}_{n}\}$ for an integer $n\geq 4$.
\item
The family $\HH$ satisfies {\rm (P-${\rm inspp}$)} if and only if $\HH\leq \{K_{n},S^{*}_{n},\tilde{S}_{n},F^{(1)}_{n},F^{(2)}_{n},F^{(4)}_{n},F^{(5)}_{n}\}$ for an integer $n\geq 4$.
\end{enumerate}
\end{thm}
In Section~\ref{sec2}, we find small induced star covers/partitions of graphs with some forbidden subgraph conditions (Proposition~\ref{prop-indstar}).
In Section~\ref{sec3}, we essentially prove the ``if'' part of Theorem~\ref{mainthm} (Proposition~\ref{mainthm-2}), and as its corollary, we also obtain a result on induced star/path covers/partitions of graphs (Proposition~\ref{mainthm-star/path}).
In Section~\ref{sec-nec}, we calculate induced SP-cover/SP-partition numbers of some graphs whose constructions are important for the ``only if'' part of Theorem~\ref{mainthm} (Lemmas~\ref{lem-nec-complete/star} and \ref{lem-nec-pathtype}).
In Section~\ref{sec-proof}, we complete the proof of Theorem~\ref{mainthm}, and settle Ramsey-type problems for some invariants (Theorems~\ref{mainthm-cor} and \ref{mainthm-3}).
See also Figure~\ref{overall flow}.
\begin{figure}[H]
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\ar@{--}[rrrrrr]\ar@{--}'[dd]&&&&&&\ar@{--}'[dd]\\
&\fb{Lem.~\ref{lem-2-indstar-01}}&&\fb{Lem.~\ref{lem-path-chibounded}}&&\db{Lem.~\ref{lem-2-indstar}}&\\
\ar@{--}[rrrrrr]&&&\ar@{=>}[dddd]&&&\\
&&&&&&\\
&&&&&&\\
&&&&&&\\
&&&\db{Prop.~\ref{prop-indstar}}\ar@{=>}[llddd]&&&\\
&&&&&&\\
\ar@{--}[rr]\ar@{--}'[dddd]&&\ar@{--}'[dddd]&&\ar@{--}[rr]\ar@{--}'[dddd]&&\ar@{--}'[dddd]\\
&\db{Prop.~\ref{mainthm-2}}\ar@{=>}[rrrr]&&&&\db{Prop.~\ref{mainthm-star/path}}&\\
&\db{Lem.~\ref{lem-nec-complete/star}}&&&&\db{Lem.~\ref{lem-nec-complete/star}}&\\
&\db{Lem.~\ref{lem-nec-pathtype}}&&&&\db{Lem.~\ref{lem-nec-pathtype}}&\\
\ar@{--}[rr]&\ar@{=>}[dddd]&&&\ar@{--}[rr]\ar@{=>}[ldddd]&\ar@{=>}[dddd]^(.4){\textup{\tiny{Rem.~\ref{remark-mainthm-2}}}}&\\
&&&&&&\\
&&&&&&\\
&&&&&&\\
&\underset{{\rm (inspc, inspp)}}{\db{Thm.~\ref{mainthm}}}&&\underset{{\rm (insc, insp, inpc, inpp)}}{\db{Thm.~\ref{mainthm-cor}}}&&\underset{{\rm (ispc, ispp)}}{\db{Thm.~\ref{mainthm-3}}}&
}
\]
\end{center}
\vspace{-12pt}
\caption{The overall flowchart for results (the double box denotes our result)}
\label{overall flow}
}
\end{figure}
\section{Induced star covers and induced star partitions}\label{sec2}
In this section, we give forbidden subgraph conditions which force the induced star cover/partition numbers to be small.
Let $G$ be a graph.
For subsets $X$ and $Y$ of $V(G)$, $X$ {\it dominates} $Y$ in $G$ if each vertex in $Y\setminus X$ is adjacent to a vertex in $X$.
A subset $X$ of $V(G)$ is a {\it dominating set} of $G$ if $X$ dominates $V(G)$ in $G$.
Let $K^{*}_{n}$ be the graph obtained from $K_{n}$ by adding a pendant edge to each vertex.
We recursively define the value $\alpha _{n,h}$ for integers $n\geq 1$ and $h\geq 1$ as follows:
$$
\begin{cases}
\alpha _{n,1}=1\\
\alpha _{n,h}=R(n,(n-1)\alpha _{n,h-1}+1)-1~~~(h\geq 2).
\end{cases}
$$
In \cite{F}, Furuya settled a Ramsey-type problem for the minimum cardinality of a dominating set of a graph, and he implicitly proved the following lemma.
\begin{lem}[Furuya~\cite{F}
\label{lem-2-indstar-01}
Let $n\geq 1$ and $l\geq 1$ be integers.
Let $H$ be a connected $\{K^{*}_{n},S^{*}_{n}\}$-free graph with ${\rm diam}(H)\leq l$.
Then there exists a dominating set $U$ of $H$ with $|U|\leq R(n,n)\sum _{2\leq h\leq l}\alpha _{n,h}+1$.
\end{lem}
Our aim in this section is to give analogies of Lemma~\ref{lem-2-indstar-01} for the induced star cover/partition numbers (Proposition~\ref{prop-indstar}).
The following lemma is a special case of the results in \cite{CSS}.
\begin{lem}[Chudnovsky, Scott and Seymour~\cite{CSS}
\label{lem-path-chibounded}
Let $n\geq 1$ be an integer.
Then there exists a constant $c_{\chi }(n)$ depending on $n$ only such that $\chi (G)\leq c_{\chi }(n)$ for every $\{K_{n},F^{(1)}_{n}\}$-free graph $G$.
\end{lem}
For integers $n\geq 3$ and $i\geq 1$, we let $\xi _{n,i}=\frac{(R(n-1,n)-1)^{i}-1}{R(n-1,n)-2}$.
Then the following lemma holds.
\begin{lem
\label{lem-2-indstar}
Let $n\geq 3$ be an integer.
Let $G$ be a $\{K_{n},\tilde{S}_{n}\}$-free graph, and let $x\in V(G)$ and $X\subseteq N_{G}(x)$.
Then ${\rm insp}(G[\{x\}\cup X])\leq \xi _{n,n-2}$.
\end{lem}
\proof
Let $I_{0}=J_{0}=\{x\}$, $X_{0}=X$, $\YY_{0}=\{X_{0}\}$ and $u_{X_{0}}=x$.
For an integer $p$ with $p\geq 1$, we will find
\begin{enumerate}[$\bullet $]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
three sets $I_{p}$, $J_{p}$ and $X_{p}$,
\item
a partition $\YY_{p}$ of $X_{p}$, and
\item
a vertex $u_{Y}\in I_{p}$ for each $Y\in \YY_{p}$.
\end{enumerate}
satisfying the following conditions (see Figure~\ref{f-indst1}):
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\caption{Sets $I_{p}$, $J_{p}$ and $X_{p}$, a partition $\YY_{p}$ of $X_{p}$, and vertices $u_{Y}\in I_{p}$}
\label{f-indst1}
\end{center}
\end{figure}
\begin{enumerate}
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item[{\bf (A1)}]
$I_{p}\subseteq J_{p}\subseteq X_{p-1}$,
\item[{\bf (A2)}]
$X_{p}=X_{p-1}\setminus J_{p}$,
\item[{\bf (A3)}]
$|I_{p}|=|\YY_{p}|$ and $I_{p}=\{u_{Y}:Y\in \YY_{p}\}$,
\item[{\bf (A4)}]
$|I_{p}|\leq (R(n-1,n)-1)^{p}$,
\item[{\bf (A5)}]
for every $Y\in \YY_{p}$, $Y\subseteq N_{G}(u_{Y})$,
\item[{\bf (A6)}]
for every $Y\in \YY_{p}$, $Y$ is a subset of the unique element of $\YY_{p-1}$ containing $u_{Y}$, and
\item[{\bf (A7)}]
${\rm insp}(G[I_{p-1}\cup (J_{p}\setminus I_{p})])\leq |I_{p-1}|$.
\end{enumerate}
Note that (A3)--(A5) hold for the case where $p=0$.
We let $m\geq 1$ be an integer, and assume that $I_{m-1}$, $J_{m-1}$, $X_{m-1}$, $\YY_{m-1}$ and $u_{Y}$ for each $Y\in \YY_{m-1}$ have been defined.
Since ${\rm insp}(G[I_{m-1}])\leq |I_{m-1}|$, if $X_{m-1}=\emptyset $, then we can easily verify that $I_{m}$, $J_{m}$, $X_{m}$ and $\YY_{m}$ with $I_{m}=J_{m}=X_{m}=\YY_{m}=\emptyset $ are desired sets.
Thus we may assume that $X_{m-1}\neq \emptyset $.
Then $\YY_{m-1}\neq \emptyset $.
Fix an element $Y$ of $\YY_{m-1}$.
Then $u_{Y}\in I_{m-1}$, and by (A5) with $p=m-1$, we have $Y\subseteq N_{G}(u_{Y})$.
Take a maximal independent set $J_{Y}\subseteq Y$.
Then by the maximality of $J_{Y}$, $Y\setminus J_{Y}$ is dominated by $J_{Y}$ in $G$.
Take a subset $I_{Y}$ of $J_{Y}$ dominating $Y\setminus J_{Y}$ in $G$ so that $|I_{Y}|$ is as small as possible, where $I_{Y}=\emptyset $ if $J_{Y}=Y$.
For each $u\in I_{Y}$, it follows from the minimality of $I_{Y}$ that there exists a vertex $y_{u}\in Y\setminus J_{Y}$ with $N_{G}(y_{u})\cap I_{Y}=\{u\}$.
Hence there exists a non-empty subset $Y^{*}_{u}$ of $N_{G}(u)\cap (Y\setminus J_{Y})$ for $u\in I_{Y}$ such that $\YY^{*}_{Y}:=\{Y^{*}_{u}:u\in I_{Y}\}$ is a partition of $Y\setminus J_{Y}$ (see Figure~\ref{f-indst2}).
In particular,
\begin{align}
|\YY^{*}_{Y}|=|I_{Y}|.\label{cond-lem-2-indstar-1}
\end{align}
Note that if $J_{Y}=Y$, then $I_{Y}=\YY^{*}_{Y}=\emptyset $, and hence (\ref{cond-lem-2-indstar-1}) holds.
\begin{figure}
\begin{center}
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\caption{Sets $I_{Y}$, $J_{Y}$ and $Y^{*}_{u}~(u\in I_{Y})$, and an induced star $S_{Y}$}
\label{f-indst2}
\end{center}
\end{figure}
Now we prove that
\begin{align}
|I_{Y}|\leq R(n-1,n)-1.\label{cond-lem-2-indstar-2}
\end{align}
By way of contradiction, suppose that $|I_{Y}|~(=|\{y_{u}:u\in I_{Y}\}|)\geq R(n-1,n)$.
Recall that $I_{Y}\cup \{y_{u}:u\in I_{Y}\}\subseteq Y\subseteq N_{G}(u_{Y})$.
If $G[\{y_{u}:u\in I_{Y}\}]$ contains a copy of $K_{n-1}$ as an induced subgraph, then $G[\{u_{Y}\}\cup \{y_{u}:u\in I_{Y}\}]$ contains a copy of $K_{n}$ as an induced subgraph; otherwise, there exists a subset $I'$ of $I_{Y}$ with $|I'|=n$ such that $\{y_{u}:u\in I'\}$ is an independent set of $G$, and hence $\{u_{Y}\}\cup I'\cup \{y_{u}:u\in I'\}$ induces a copy of $\tilde{S}_{n}$ in $G$.
In either case, we obtain a contradiction.
Thus (\ref{cond-lem-2-indstar-2}) holds.
We give further notations.
Fix $Y^{*}\in \YY^{*}_{Y}$.
Let $u_{Y^{*}}\in I_{Y}$ be the unique vertex with $Y^{*}=Y^{*}_{u_{Y^{*}}}$.
Then by the definition of $Y^{*}_{u}$, we have
\begin{align}
\mbox{$Y^{*}\subseteq N_{G}(u_{Y^{*}})$.}\label{cond-lem-2-indstar-3}
\end{align}
Since $Y^{*}$ is a subset of $Y$ and $Y$ is the unique element of $\YY_{m-1}$ containing $u_{Y^{*}}$,
\begin{align}
\mbox{$Y^{*}$ is a subset of the unique element of $\YY_{m-1}$ containing $u_{Y^{*}}$.}\label{cond-lem-2-indstar-4}
\end{align}
Since $\YY^{*}_{Y}=\{Y^{*}_{u}:u\in I_{Y}\}$ and $u_{Y^{*}_{u}}=u$ for every $u\in I_{Y}$, we have
\begin{align}
\mbox{$\{u_{Y^{*}}:Y^{*}\in \YY^{*}_{Y}\}=\{u_{Y^{*}}:Y^{*}\in \{Y^{*}_{u}:u\in I_{Y}\}\}=\{u_{Y^{*}_{u}}:u\in I_{Y}\}=I_{Y}$.}\label{cond-lem-2-indstar-5}
\end{align}
Let $S_{Y}$ be the subgraph of $G$ induced by $\{u_{Y}\}\cup (J_{Y}\setminus I_{Y})$.
Since $J_{Y}$ is an independent set of $G$, $S_{Y}$ is a star (or the graph consisting of one vertex $u_{Y}$ if $J_{Y}=I_{Y}$).
Note that
\begin{align}
V(S_{Y})\cap I_{m-1}=\{u_{Y}\}.\label{cond-lem-2-indstar-6}
\end{align}
Let $I_{m}=\bigcup _{Y\in \YY_{m-1}}I_{Y}$, $J_{m}=\bigcup _{Y\in \YY_{m-1}}J_{Y}$, $X_{m}=\bigcup _{Y\in \YY_{m-1}}(Y\setminus J_{Y})$, and $\YY_{m}=\bigcup _{Y\in \YY_{m-1}}\YY^{*}_{Y}$.
Since the elements of $\YY_{m-1}$ are pairwise disjoint and for each $Y\in \YY_{m-1}$, $\YY^{*}_{Y}$ is a partition of $Y\setminus J_{Y}$, we see that $\YY_{m}$ is a partition of $\bigcup _{Y\in \YY_{m-1}}(Y\setminus J_{Y})~(=X_{m})$.
For each $Y^{*}\in \YY_{m}$, there exists an element $Y\in \YY_{m-1}$ such that $Y^{*}\in \YY^{*}_{Y}$, and hence the vertex $u_{Y^{*}}~(\in I_{Y}\subseteq I_{m})$ has been defined.
Now we prove that (A1)--(A7) hold for the case where $p=m$.
Recall that for every $Y\in \YY_{m-1}$, $I_{Y}\subseteq J_{Y}\subseteq Y$.
Since $\YY_{m-1}$ is a partition of $X_{m-1}$, we have
$$
\bigcup _{Y\in \YY_{m-1}}I_{Y}\subseteq \bigcup _{Y\in \YY_{m-1}}J_{Y}\subseteq \bigcup _{Y\in \YY_{m-1}}Y=X_{m-1}.
$$
This together with the definition of $I_{m}$ and $J_{m}$ leads to (A1).
Furthermore, we obtain
$$
X_{m}=\bigcup _{Y\in \YY_{m-1}}(Y\setminus J_{Y})=X_{m-1}\setminus \left(\bigcup _{Y\in \YY_{m-1}}J_{Y} \right)=X_{m-1}\setminus J_{m},
$$
which leads to (A2).
By (\ref{cond-lem-2-indstar-1}),
$$
|I_{m}|=\sum _{Y\in \YY_{m-1}}|I_{Y}|=\sum _{Y\in \YY_{m-1}}|\YY^{*}_{Y}|=|\YY_{m}|.
$$
By (\ref{cond-lem-2-indstar-5}),
$$
I_{m}=\bigcup _{Y\in \YY_{m-1}}I_{Y}=\bigcup _{Y\in \YY_{m-1}}\{u_{Y^{*}}:Y^{*}\in \YY^{*}_{Y}\}=\left\{u_{Y^{*}}:Y^{*}\in \bigcup _{Y\in \YY_{m-1}}\YY^{*}_{Y}\right\}=\{u_{Y^{*}}:Y^{*}\in \YY_{m}\}.
$$
Hence (A3) holds.
By (A3) and (A4) with $p=m-1$, $|\YY_{m-1}|=|I_{m-1}|\leq (R(n-1,n)-1)^{m-1}$.
Hence by (\ref{cond-lem-2-indstar-2}),
$$
|I_{m}|=\sum _{Y\in \YY_{m-1}}|I_{Y}|\leq |\YY_{m-1}|(R(n-1,n)-1)\leq (R(n-1,n)-1)^{m},
$$
which leads to (A4).
For each $Y^{*}\in \YY_{m}$,
\begin{enumerate}[$\bullet $]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
$Y^{*}\subseteq N_{G}(u_{Y^{*}})$ by (\ref{cond-lem-2-indstar-3}), and
\item
$Y^{*}$ is a subset of the unique element of $\YY_{m-1}$ containing $u_{Y^{*}}$ by (\ref{cond-lem-2-indstar-4}),
\end{enumerate}
and hence (A5) and (A6) hold.
Let $\SS=\{S_{Y}:Y\in \YY_{m-1}\}$.
Then by (A3) with $p=m-1$,
$$
\bigcup _{S\in \SS}V(S)=\bigcup _{Y\in \YY_{m-1}}V(S_{Y})=\bigcup _{Y\in \YY_{m-1}}(\{u_{Y}\}\cup (J_{Y}\setminus I_{Y}))=I_{m-1}\cup (J_{m}\setminus I_{m}),
$$
and hence $\SS$ is an induced star partition of $G[I_{m-1}\cup (J_{m}\setminus I_{m})]$.
This together with (\ref{cond-lem-2-indstar-6}) leads to (A7).
Let $p_{0}=\min\{p:p\geq 0,~X_{p}=\emptyset \}$.
Then $\YY_{p_{0}}=\emptyset $.
Hence by (A3) with $p=p_{0}$,
\begin{align}
I_{p_{0}}=\emptyset .\label{cond-lem-2-indstar-6new}
\end{align}
By (A1) and (A2), $X_{p-1}$ is the disjoint union of $X_{p}$ and $J_{p}$ for each integer $p\geq 1$.
In particular, $X_{0}~(=X)$ is the disjoint union of $X_{p_{0}}~(=\emptyset ),J_{p_{0}},J_{p_{0}-1},\ldots ,J_{1}$.
Again by (A1), $I_{p}\subseteq J_{p}$ for every integer $p\geq 1$.
Consequently, it follows from (\ref{cond-lem-2-indstar-6new}) that $I_{0}\cup X_{0}~(=\{x\}\cup X)$ is the disjoint union of
$$
I_{p_{0}-1}\cup (J_{p_{0}}\setminus I_{p_{0}}),~I_{p_{0}-2}\cup (J_{p_{0}-1}\setminus I_{p_{0}-1}),\ldots ,I_{0}\cup (J_{1}\setminus I_{1}).
$$
This together with (A4) and (A7) leads to
\begin{align*}
{\rm insp}(G[\{x\}\cup X]) &\leq \sum _{1\leq p\leq p_{0}}{\rm insp}(G[I_{p-1}\cup (J_{p}\setminus I_{p})])\\
&\leq \sum _{1\leq p\leq p_{0}}|I_{p-1}|\\
&\leq \sum _{1\leq p\leq p_{0}}(R(n-1,n)-1)^{p-1}\\
&= \frac{(R(n-1,n)-1)^{p_{0}}-1}{R(n-1,n)-2}~(= \xi _{n,p_{0}}).
\end{align*}
Thus, to complete the proof of the lemma, it suffices to show that $p_{0}\leq n-2$.
Now we recursively define vertices $u_{p}\in J_{p}$ with $0\leq p\leq p_{0}$ and sets $Y_{p}\in \YY_{p}$ with $0\leq p\leq p_{0}-1$ as follows.
By the definition of $p_{0}$ and (A2) with $p=p_{0}$, we have $\emptyset =X_{p_{0}}=X_{p_{0}-1}\setminus J_{p_{0}}$, and hence $X_{p_{0}-1}\subseteq J_{p_{0}}$.
Again by the definition of $p_{0}$, $X_{p_{0}-1}\neq \emptyset $.
Consequently, we can take a vertex $u_{p_{0}}\in J_{p_{0}}$.
We let $p$ be an integer with $0\leq p\leq p_{0}-1$, and assume that a vertex $u_{p+1}\in J_{p+1}$ has been defined.
Since $\YY_{p}$ is a partition of $X_{p}$ and $u_{p+1}\in J_{p+1}\subseteq X_{p}$ by (A1), there exists exactly one element $Y_{p}$ of $\YY_{p}$ containing $u_{p+1}$.
Let $u_{p}=u_{Y_{p}}$.
Then $u_{p}\in I_{p}\subseteq J_{p}$.
Note that $u_{0}=x$ and $Y_{0}=X$.
For an integer $p$ with $0\leq p\leq p_{0}-2$, since $Y_{p}$ is the unique element of $\YY_{p}$ containing $u_{p+1}~(=u_{Y_{p+1}})$, it follows from (A6) that $Y_{p+1}\subseteq Y_{p}$.
Hence
\begin{align}
Y_{p_{0}-1}\subseteq Y_{p_{0}-2}\subseteq \cdots \subseteq Y_{0}.\label{cond-lem-2-indstar-7}
\end{align}
Fix two integers $p$ and $p'$ with $0\leq p<p'\leq p_{0}$.
Then by (\ref{cond-lem-2-indstar-7}) $u_{p'}\in Y_{p'-1}\subseteq Y_{p}$.
On the other hand, it follows from (A5) that $Y_{p}\subseteq N_{G}(u_{Y_{p}})=N_{G}(u_{p})$.
Hence $u_{p}u_{p'}\in E(G)$.
Since $p$ and $p'$ are arbitrary, $\{u_{p}:0\leq p\leq p_{0}\}$ is a clique of $G$ consisting of $p_{0}+1$ vertices.
Since $G$ is $K_{n}$-free, this implies that $p_{0}\leq n-2$.
This completes the proof of Lemma~\ref{lem-2-indstar}.
\qed
By combining Lemmas~\ref{lem-2-indstar-01}--\ref{lem-2-indstar}, we obtain the following proposition.
\begin{prop
\label{prop-indstar}
Let $n\geq 3$ and $l_{0}\geq 1$ be integers.
Then there exist constants $c_{1}(n,l_{0})$ and $c_{2}(n,l_{0})$ depending on $n$ and $l_{0}$ only such that for a connected $\{K_{n},S^{*}_{n}\}$-free graph $H$ with ${\rm diam}(G)\leq l_{0}$,
\begin{enumerate}[{\upshape(1)}]
\item
if $H$ is $F^{(1)}_{n}$-free, then ${\rm insc}(H)\leq c_{1}(n,l_{0})$, and
\item
if $H$ is $\tilde{S}_{n}$-free, then ${\rm insp}(H)\leq c_{2}(n,l_{0})$.
\end{enumerate}
\end{prop}
\proof
We show that $c_{1}(n,l_{0})=(R(n,n)\sum _{2\leq h\leq l_{0}}\alpha _{n,h}+1)c_{\chi }(n)$ and $c_{2}(n,l_{0})=(R(n,n)\sum _{2\leq h\leq l_{0}}\alpha _{n,h}+1)\xi _{n,n-2}$ are desired constants, where $c_{\chi }(n)$ is the constant appearing in Lemma~\ref{lem-path-chibounded}.
Since $H$ is $K_{n}$-free, $H$ is also $K^{*}_{n}$-free.
Hence by Lemma~\ref{lem-2-indstar-01}, there exists a dominating set $U$ of $H$ with $|U|\leq R(n,n)\sum _{2\leq h\leq l_{0}}\alpha _{n,h}+1$.
For each vertex $v\in V(H)\setminus U$, take a vertex $x_{v}\in N_{H}(v)\cap U$.
For a vertex $x\in U$, let $X_{x}=\{v\in V(H)\setminus U:x_{v}=x\}$.
Note that $X_{x}\subseteq N_{H}(x)$ for each $x\in U$, and $V(G)$ is the disjoint union of $\{x\}\cup X_{x}~(x\in U)$.
Hence for a fixed vertex $x\in U$, it suffices to prove the following two statements.
\begin{enumerate}
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item[{\bf (S1)}]
If $H$ is $F^{(1)}_{n}$-free, then ${\rm insc}(H[\{x\}\cup X_{x}])\leq c_{\chi }(n)$.
\item[{\bf (S2)}]
If $H$ is $\tilde{S}_{n}$-free, then ${\rm insp}(H[\{x\}\cup X_{x}])\leq \xi _{n,n-2}$.
\end{enumerate}
By Lemma~\ref{lem-2-indstar}, (S2) clearly holds.
Now we prove that (S1) holds.
Suppose that $H$ is $F^{(1)}_{n}$-free.
Then by Lemmma~\ref{lem-path-chibounded}, $\chi (H[X_{x}])\leq c_{\chi }(n)$.
In particular, there exists an induced $\{\overline{K_{n}}:n\geq 1\}$-partition $\TT_{x}$ of $H[X_{x}]$ with $|\TT_{x}|\leq c_{\chi }(n)$.
Then $\SS_{x}:=\{H[\{x\}\cup T]:T\in \TT_{x}\}$ is an induced star cover of $H[\{x\}\cup X_{x}]$, and hence ${\rm insc}(H[\{x\}\cup X_{x}])\leq |\SS_{x}|=|\TT_{x}|\leq c_{\chi }(n)$, which leads to (S1).
\qed
\section{Induced SP-covers and induced SP-partitions}\label{sec3}
In this section, we prove the following proposition.
\begin{prop
\label{mainthm-2}
Let $n\geq 4$ be an integer.
\begin{enumerate}[{\upshape(i)}]
\item
There exists a constant $c_{\rm inspc}(n)$ depending on $n$ only such that ${\rm inspc}(G)\leq c_{\rm inspc}(n)$ for every connected $\{K_{n},S^{*}_{n},F^{(1)}_{n},F^{(2)}_{n},F^{(3)}_{n}\}$-free graph $G$.
\item
There exists a constant $c_{\rm inspp}(n)$ depending on $n$ only such that ${\rm inspp}(G)\leq c_{\rm inspp}(n)$ for every connected $\{K_{n},S^{*}_{n},\tilde{S}_{n},F^{(1)}_{n},F^{(2)}_{n},F^{(4)}_{n},F^{(5)}_{n}\}$-free graph $G$.
\end{enumerate}
\end{prop}
\begin{remark
\label{remark-mainthm-2}
A path $P$ of a graph $G$ is {\it isometric} if $P$ is a shortest path of $G$ joining the endvertices of $P$, i.e., ${\rm dist}_{G}(x,y)={\rm dist}_{P}(x,y)$ where $x$ and $y$ are the endvertices of $P$.
In the proof of Proposition~\ref{mainthm-2}, we construct small induced SP-covers/SP-partitions of a graph $G$ with some forbidden subgraph conditions.
One might notice that all paths belonging to such induced SP-covers/SP-partitions are also isometric paths of $G$.
By using the fact, we obtain solutions of Ramsey-type problems for some invariants concerning isometric paths (see Theorem~\ref{mainthm-3}).
\end{remark}
\medbreak\noindent\textit{Proof of Proposition~\ref{mainthm-2}.}\quad
Let $G$ be a connected $\{K_{n},S^{*}_{n},F^{(1)}_{n},F^{(2)}_{n},F^{(3)}_{n}\}$-free graph.
Since $F^{(4)}_{n}$ is an induced subgraph of $F^{(3)}_{n}$, it suffices to prove the following two statements.
\begin{enumerate}
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item[{\bf (B1)}]
The value ${\rm inspc}(G)$ is bounded by a constant depending on $n$ only.
\item[{\bf (B2)}]
If $G$ is $\{\tilde{S}_{n},F^{(4)}_{n},F^{(5)}_{n}\}$-free, then ${\rm inspp}(G)$ is bounded by a constant depending on $n$ only.
\end{enumerate}
Take a vertex $x_{0}$ of $G$, and for each integer $i\geq 0$, let $X_{i}=\{x\in V(G):{\rm dist}_{G}(x_{0},x)=i\}$.
Let $d=\max\{i:i\geq 0,~X_{i}\neq \emptyset \}~(={\rm ecc}_{G}(x_{0}))$.
Then $X_{0}=\{x_{0}\}$, $X_{1}=N_{G}(x_{0})$, and $V(G)$ is the disjoint union of $X_{i}~(0\leq i\leq d)$.
In this proof, we frequently use the following facts without mentioning:
\begin{enumerate}[{$\bullet $}]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
For an integer $i\geq 1$ and a vertex $x\in X_{i}$, $N_{G}(x)\subseteq X_{i-1}\cup X_{i}\cup X_{i+1}$.
\item
For an integer $i\geq 0$ and a vertex $x\in X_{i}$, every shortest $x_{0}$-$x$ path $P$ of $G$ satisfies $|V(P)\cap X_{j}|=1$ for all $j$ with $0\leq j\leq i$.
\item
For integers $i$ and $i'$ with $0\leq i\leq i'$ and vertices $x\in X_{i}$ and $x'\in X_{i'}$, if an $x$-$x'$ path $P$ of $G$ satisfies $|V(P)\cap X_{j}|=1$ for all $j$ with $i\leq j\leq i'$, then $P$ is a shortest $x$-$x'$ path of $G$.
\end{enumerate}
\begin{claim
\label{cl2.1}
Let $i$ and $k$ be integers with $n+1\leq i\leq k-n-1$.
Let $x\in X_{k}$, and let $Q$ be a shortest $x_{0}$-$x$ path of $G$.
Let $y\in X_{i}$ be a vertex with $N_{G}(y)\cap V(Q)\neq \emptyset $.
\begin{enumerate}[{\upshape(1)}]
\item
The vertex $y$ is adjacent to all vertices in $V(Q)\cap (X_{i-1}\cup X_{i+1})$.
\item
Let $i'$ be an integer with $n+1\leq i'\leq k-n-1$.
If a vertex $y'\in X_{i'}$ satisfies $N_{G}(y')\cap V(Q)=\emptyset $, then $yy'\notin E(G)$.
\end{enumerate}
\end{claim}
\proof
Write $Q=x_{0}x_{1}x_{2}\cdots x_{k}$ where $x_{k}=x$.
Note that $x_{i-n-1},x_{i-n},\ldots ,x_{i+n+1}$ are defined as vertices belonging to $V(Q)$.
\begin{enumerate}[{\upshape(1)}]
\item
We may assume that $y\notin V(Q)$.
If $N_{G}(y)\cap V(Q)=\{x_{j}\}$ for an integer $j$ with $j\in \{i-1,i,i+1\}$, then
$$
\{x_{j-n},x_{j-n+1},\ldots ,x_{j},y,x_{j+1},x_{j+2},\ldots ,x_{j+n}\}
$$
induces a copy of $F^{(1)}_{n}$ in $G$; if $N_{G}(y)\cap V(Q)=\{x_{j},x_{j+1}\}$ for an integer $j$ with $j\in \{i-1,i\}$, then
$$
\{x_{j-n+1},x_{j-n+2},\ldots ,x_{j},y,x_{j+1},x_{j+2},\ldots ,x_{j+n}\}
$$
induces a copy of $F^{(2)}_{n}$ in $G$.
In either case, we obtain a contradiction.
Considering the fact that $\emptyset \neq N_{G}(y)\cap V(Q)\subseteq \{x_{i-1},x_{i},x_{i+1}\}$, this forces $\{x_{i-1},x_{i+1}\}\subseteq N_{G}(y)\cap V(Q)$, as desired.
\item
Let $y'\in X_{i'}$ be a vertex with $N_{G}(y')\cap V(Q)=\emptyset $.
Note that $y'\notin V(Q)$.
Suppose that $yy'\in E(G)$.
By (1), $yx_{i-1},yx_{i+1}\in E(G)$, and hence $Q':=x_{0}x_{1}\cdots x_{i-1}yx_{i+1}x_{i+2}\cdots x_{k}$ is a shortest $x_{0}$-$x$ path of $G$.
Then $N_{G}(y')\cap V(Q')\supseteq \{y\}\neq \emptyset $.
Hence, applying (1) with $(y,Q)=(y',Q')$, $y'$ is adjacent to all vertices in $V(Q')\cap (X_{i'-1}\cup X_{i'+1})$.
Since $|V(Q')\cap (X_{i'-1}\cup X_{i'+1})|=2$ and $V(Q')\setminus V(Q)=\{y\}$, this implies that $y'$ is adjacent to a vertex $V(Q)$, which is a contradiction.
\qed
\end{enumerate}
For the moment, we assume that $d\leq n^{2}+2n-1$.
Then ${\rm diam}(G)\leq {\rm ecc}_{G}(x_{0})=d\leq n^{2}+2n-1$.
Note that the constants $c_{1}(n,n^{2}+2n-1)$ and $c_{2}(n,n^{2}+2n-1)$ are depending on $n$ only, where $c_{1}(n,l_{0})$ and $c_{2}(n,l_{0})$ are the constants appearing in Proposition~\ref{prop-indstar}.
By applying Proposition~\ref{prop-indstar}(1) with $l_{0}=n^{2}+2n-1$,
$$
{\rm inspc}(G)\leq {\rm insc}(G)\leq c_{1}(n,n^{2}+2n-1),
$$
and hence (B1) holds.
If $G$ is $\tilde{S}_{n}$-free, then by applying Proposition~\ref{prop-indstar}(2) with $l_{0}=n^{2}+2n-1$,
$$
{\rm inspp}(G)\leq {\rm insp}(G)\leq c_{2}(n,n^{2}+2n-1),
$$
and hence (B2) holds.
Thus we may assume that
\begin{align}
d\geq n^{2}+2n.\label{cond-sp-diam}
\end{align}
We consider the following operation recursively defining indices $k_{h}$, vertices $w_{h}$ and paths $Q_{h}$:
Let $k_{1}=d$, and take a vertex $w_{1}\in X_{k_{1}}$.
Let $Q_{1}$ be a shortest $x_{0}$-$w_{1}$ path of $G$.
Let $h$ be an integer with $h\geq 2$, and assume that $k_{1},\ldots ,k_{h-1}$, $w_{1},\ldots ,w_{h-1}$, $Q_{1},\ldots ,Q_{h-1}$ have been defined.
If $k_{h-1}\geq 3n+2$, let
$$
A_{h,i}=\left\{y\in X_{i}\setminus \left(\bigcup _{1\leq l\leq h-1}V(Q_{l})\right):N_{G}(y)\cap \left(\bigcup _{1\leq l\leq h-1}V(Q_{l})\right)=\emptyset\right\}
$$
for an integer $i$ with $2n+1\leq i\leq k_{h-1}-n-1$.
If $k_{h-1}\geq 3n+2$ and $A_{h,i}\neq \emptyset $ for some integer $i$ with $2n+1\leq i\leq k_{h-1}-n-1$, we let $k_{h}=\max\{i:2n+1\leq i\leq k_{h-1}-n-1,~A_{h,i}\neq \emptyset \}$ and $w_{h}\in A_{h,k_{h}}$, and let $Q_{h}$ be a shortest $x_{0}$-$w_{h}$ path of $G$; otherwise (i.e., either $k_{h-1}\geq 3n+2$ and $A_{h,i}=\emptyset $ for all integers $i$ with $2n+1\leq i\leq k_{h-1}-n-1$, or $k_{h-1}\leq 3n+1$), we let $k_{h}=2n$ and finish the operation.
Let $K=\{k_{h}:h\geq 1\}$, and set $h_{0}=|K|-1$.
Note that $d=k_{1}>k_{2}>\cdots >k_{h_{0}}>k_{h_{0}+1}=2n$.
For each integer $h$ with $1\leq h\leq h_{0}$, write $Q_{h}=v^{(h)}_{0}v^{(h)}_{1}v^{(h)}_{2}\cdots v^{(h)}_{k_{h}}$ where $v^{(h)}_{0}=x_{0}$ and $v^{(h)}_{k_{h}}=w_{h}$.
Note that $V(Q_{h})\cap X_{i}=\{v^{(h)}_{i}\}$ for integers $h$ and $i$ with $1\leq h\leq h_{0}$ and $1\leq i\leq k_{h}$.
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\caption{Sets $I_{h}$ and $J_{h}$ of indices}
\label{f2}
\end{figure}
For an integer $h$ with $1\leq h\leq h_{0}$, we define the sets $I_{h}$ and $J_{h}$ of indices as
$$
I_{h}=
\begin{cases}
\{i:k_{h}-n\leq i\leq k_{h}\} & (1\leq h\leq h_{0}-1)\\
\{i:\max\{2n+1,k_{h_{0}}-n\}\leq i\leq k_{h_{0}}\} & (h=h_{0})
\end{cases}
$$
and
$$
J_{h}=\{i:k_{h+1}+1\leq i\leq k_{h}-n-1\}.
$$
Let $I_{h_{0}+1}=\{i:0\leq i\leq k_{h_{0}+1}~(=2n)\}$ (see Figure~\ref{f2}).
Note that $I_{h}$ is non-empty, but $J_{h}$ might be empty.
Since $k_{h_{0}+1}=2n$, we have $J_{h_{0}}=\{i:2n+1\leq i\leq k_{h_{0}}-n-1\}$.
Hence
\begin{align}
\mbox{$J_{h_{0}}=\emptyset $ if and only if $2n+1\geq k_{h_{0}}-n$.}\label{cl2.3--iff-1}
\end{align}
\begin{claim
\label{cl2.3-}
Let $h$ be an integer with $1\leq h\leq h_{0}$, and let $i\in J_{h}$.
Then for every vertex $y\in X_{i}$, $N_{G}(y)\cap (\bigcup _{1\leq l\leq h}V(Q_{l}))\neq \emptyset $.
\end{claim}
\proof
By the definition of $k_{h}$, the desired conclusion clearly holds.
\qed
\begin{claim
\label{cl2.3--}
The set $\{0,1,\ldots ,d\}$ is the disjoint union of $I_{h}~(1\leq h\leq h_{0}+1)$ and $J_{h}~(1\leq h\leq h_{0})$.
\end{claim}
\proof
We first prove that
\begin{align}
\mbox{$\{i:k_{h_{0}+1}+1\leq i\leq k_{h_{0}}\}$ is the disjoint union of $I_{h_{0}}$ and $J_{h_{0}}$.}\label{cond-cl2.3---1}
\end{align}
If $J_{h_{0}}=\emptyset $, then by (\ref{cl2.3--iff-1}), $\max\{2n+1,k_{h_{0}}-n\}=2n+1=k_{h_{0}+1}+1$, and hence $I_{h_{0}}=\{i:k_{h_{0}+1}+1\leq i\leq k_{h_{0}}\}$, as desired.
Thus we may assume that $J_{h_{0}}\neq \emptyset $.
Then by (\ref{cl2.3--iff-1}), $2n+1\leq k_{h_{0}}-n-1$, and hence $I_{h_{0}}=\{i:k_{h_{0}+1}-n\leq i\leq k_{h_{0}}\}$.
Since $J_{h_{0}}=\{i:k_{h_{0}+1}+1\leq i\leq k_{h_{0}}-n-1\}$, (\ref{cond-cl2.3---1}) holds.
By (\ref{cond-cl2.3---1}) and the definition of $I_{h}$ and $J_{h}$, the desired conclusion holds.
\qed
For an integer $h$ with $1\leq h\leq h_{0}$, let $m_{h}=\min I_{h}$.
Then $I_{h}=\{i:m_{h}\leq i\leq k_{h}\}$ and $J_{h}=\{i:k_{h+1}+1\leq i\leq m_{h}-1\}$ for every integer $h$ with $1\leq h\leq h_{0}$.
\begin{claim
\label{cl2.3}
\begin{enumerate}[{\upshape(1)}]
\item
For two integers $h$ and $h'$ with $1\leq h<h'\leq h_{0}$, $V(Q_{h})\cap V(Q_{h'})=\{x_{0}\}$ and there is no edge between $V(Q_{h})\setminus \{x_{0}\}$ and $V(Q_{h'})\setminus \{x_{0}\}$.
\item
We have $h_{0}\leq n-1$.
\item
For an integer $h$ with $1\leq h\leq h_{0}$, $|I_{h}|\leq n+1$.
\item
We have $\sum _{1\leq h\leq h_{0}+1}|I_{h}|\leq n^{2}+2n$.
\end{enumerate}
\end{claim}
\proof
\begin{enumerate}[{\upshape(1)}]
\item
Since $x_{0}\in N_{G}(v^{(h')}_{1})\cap V(Q_{h})$, the value $p=\max\{i:1\leq i\leq k_{h'},~N_{G}(v^{(h')}_{i})\cap V(Q_{h})\neq \emptyset \}$ is well-defined.
By the definition of $w_{h'}$, we have $w_{h'}\in X_{k_{h'}}\setminus V(Q_{h})$ and $N_{G}(w_{h'})\cap V(Q_{h})=\emptyset $.
In particular, we have $1\leq p\leq k_{h'}-1$.
If there exists an integer $i$ with $p\leq i\leq k_{h'}-1$ such that $v^{(h')}_{i}\in V(Q_{h})$, then $N_{G}(v^{(h')}_{i+1})\cap V(Q_{h})\supseteq \{v^{(h')}_{i}\}\neq \emptyset $, which contradict the maximality of $p$.
Thus $V(Q_{h})\cap \{v^{(h')}_{i}:p\leq i\leq k_{h'}\}=\emptyset $.
Suppose that $p\geq n+1$.
Recall that $N_{G}(v^{(h')}_{p})\cap V(Q_{h})\neq \emptyset $ and $N_{G}(v^{(h')}_{p+1})\cap V(Q_{h})=\emptyset $.
Since $n+1\leq p<p+1\leq k_{h'}\leq k_{h'-1}-n-1\leq k_{h}-n-1$, it follows from Claim~\ref{cl2.1}(2) with $(y,y')=(v^{(h')}_{p},v^{(h')}_{p+1})$ that $v^{(h')}_{p}v^{(h')}_{p+1}\notin E(G)$, which is a contradiction.
Thus $p\leq n$.
Let $q=\max\{i:v^{(h')}_{p}v^{(h)}_{i}\in E(G)\}$.
Then $q\in \{p-1,p,p+1\}$.
Suppose that $q\geq 1$.
Since $p\leq n$ and $2n+1\leq k_{h'}$, we have
$$
q+n\leq p+n+1\leq 2n+1\leq k_{h'}<k_{h}.
$$
Hence $v^{(h)}_{q-1},v^{(h)}_{q},\ldots ,v^{(h)}_{q+n}$ have been defined as $n+2$ vertices belonging to $V(Q_{h})$, and $v^{(h')}_{p},v^{(h')}_{p+1},\ldots ,v^{(h')}_{p+n-1}$ have been defined as $n$ vertices belonging to $V(Q_{h'})$.
If $v^{(h')}_{p}v^{(h)}_{q-1}\notin E(G)$, then
$$
\{v^{(h')}_{p+n-1},v^{(h')}_{p+n-2},\ldots ,v^{(h')}_{p},v^{(h)}_{q},v^{(h)}_{q-1},v^{(h)}_{q+1},v^{(h)}_{q+2},\ldots ,v^{(h)}_{q+n}\}
$$
induces a copy of $F^{(1)}_{n}$ in $G$; if $v^{(h')}_{p}v^{(h)}_{q-1}\in E(G)$, then
$$
\{v^{(h')}_{p+n-1},v^{(h')}_{p+n-2},\ldots ,v^{(h')}_{p},v^{(h)}_{q-1},v^{(h)}_{q},\ldots ,v^{(h)}_{q+n-1}\}
$$
induces a copy of $F^{(2)}_{n}$ in $G$.
In either case, we obtain a contradiction.
Consequently, $q=0$, and this forces $p=1$.
Therefore $V(Q_{h})\cap V(Q_{h'})=\{x_{0}\}$ and there is no edge between $V(Q_{h})\setminus \{x_{0}\}$ and $V(Q_{h'})\setminus \{x_{0}\}$.
\item
By (1), $\{x_{0}\}\cup \{v^{(h)}_{1},v^{(h)}_{2}:1\leq h\leq h_{0}\}$ induces a copy of $S^{*}_{h_{0}}$ in $G$.
Since $G$ is $S^{*}_{n}$-free, this leads to $h_{0}<n$, as desired.
\item
By the definition of $I_{h}$,
\begin{enumerate}[{$\bullet $}]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
$|I_{h}|=k_{h}-(k_{h}-n)+1=n+1$ for every integer $h$ with $1\leq h\leq h_{0}-1$, and
\item
$|I_{h_{0}}|=k_{h_{0}}-\max\{2n+1,k_{h_{0}}-n\}+1\leq k_{h_{0}}-(k_{h_{0}}-n)+1=n+1$,
\end{enumerate}
which leads to the desired conclusion.
\item
Since $|I_{h_{0}+1}|=2n+1$, it follows from (2) and (3) that
$$
\sum _{1\leq h\leq h_{0}+1}|I_{h}|\leq (n+1)h_{0}+2n+1\leq (n+1)(n-1)+2n+1=n^{2}+2n,
$$
as desired.
\qed
\end{enumerate}
For integers $l$ with $1\leq l\leq h_{0}$ and $i\in \bigcup _{l\leq h\leq h_{0}}J_{h}$, let
$$
X^{(l)}_{i}=\{y\in X_{i}:N_{G}(y)\cap V(Q_{l})\neq \emptyset \}.
$$
Note that $v^{(l)}_{i}\in X^{(l)}_{i}$, and in particular, $X^{(l)}_{i}\neq \emptyset $.
Let $\nu =R(n-1,n)-1$.
Then $\nu $ is a constant depending on $n$ only.
\begin{claim
\label{cl2.4}
Let $h$ and $i$ be integers with $1\leq h\leq h_{0}$ and $i\in J_{h}$.
\begin{enumerate}[{\upshape(1)}]
\item
The set $X_{i}$ is the disjoint union of $X^{(1)}_{i},X^{(2)}_{i},\ldots ,X^{(h)}_{i}$.
\item
If $i+1\in J_{h}$, then for each integer $l$ with $1\leq l\leq h$, every vertex in $X^{(l)}_{i}$ is adjacent to all vertices in $X^{(l)}_{i+1}$.
\item
For an integer $l$ with $1\leq l\leq h$, $|X^{(l)}_{i}|\leq \nu $.
\item
We have $|X_{i}|\leq h\nu $.
\item
If $G$ is $\{F^{(4)}_{n},F^{(5)}_{n}\}$-free, then $X_{i}\subseteq \bigcup _{1\leq l\leq h}V(Q_{l})$.
\end{enumerate}
\end{claim}
\proof
For an integer $l$ with $1\leq l\leq h$,
\begin{align}
n+1<2n+1\leq i\leq k_{h}-n-1\leq k_{l}-n-1,\label{eq-cl2.4-condition1}
\end{align}
and
\begin{align}
\mbox{if }i+1\in J_{h},\mbox{ then }i+1\leq k_{h}-n-1\leq k_{l}-n-1.\label{eq-cl2.4-condition2}
\end{align}
\begin{enumerate}[{\upshape(1)}]
\item
For a vertex $y\in X_{i}$, it follows from Claim~\ref{cl2.3-} that there exists an integer $l$ with $1\leq l\leq h$ such that $y\in X^{(l)}_{i}$.
This implies that $X_{i}=\bigcup _{1\leq l\leq h}X^{(l)}_{i}$.
Thus, to complete the proof of (1), it suffices to show that $X^{(1)}_{i},X^{(2)}_{i},\ldots ,X^{(h)}_{i}$ are pairwise disjoint.
Let $l_{1}$ and $l_{2}$ be integers with $1\leq l_{1}<l_{2}\leq h$.
Suppose that there exists a vertex $y\in X^{(l_{1})}_{i}\cap X^{(l_{2})}_{i}$.
Then $N_{G}(y)\cap V(Q_{l_{1}})\neq \emptyset $ and there exists a vertex $y'\in V(Q_{l_{2}})$ with $yy'\in E(G)$.
By Claim~\ref{cl2.3}(1), $N_{G}(y')\cap V(Q_{l_{1}})=\emptyset $.
Let $i'\in \{i-1,i,i+1\}$ be the integer with $y'\in X_{i'}$.
By (\ref{eq-cl2.4-condition1}), $n+1<i\leq k_{l_{2}}-n-1<k_{l_{1}}-n-1$, and hence $n+1\leq i-1\leq i'\leq i+1\leq k_{l_{1}}-n-1$.
Then by Claim~\ref{cl2.1}(2) with $Q=Q_{l_{1}}$, $yy'\notin E(G)$, which is a contradiction.
Thus $X^{(l_{1})}_{i}\cap X^{(l_{2})}_{i}=\emptyset $.
Since $l_{1}$ and $l_{2}$ are arbitrary, $X^{(1)}_{i},X^{(2)}_{i},\ldots ,X^{(h)}_{i}$ are pairwise disjoint.
\item
Fix two vertices $y\in X^{(l)}_{i}$ and $y'\in X^{(l)}_{i+1}$.
Since $N_{G}(y)\cap V(Q_{l})\neq \emptyset $ and $N_{G}(y')\cap V(Q_{l})\neq \emptyset $, it follows from (\ref{eq-cl2.4-condition1}), (\ref{eq-cl2.4-condition2}) and Claim~\ref{cl2.1}(1) with $Q=Q_{l}$ and $(y,Q)=(y',Q_{l})$ that $yv^{(l)}_{i-1},yv^{(l)}_{i+1},y'v^{(l)}_{i+2}\in E(G)$.
Hence $Q':=v^{(l)}_{0}v^{(l)}_{1}\cdots v^{(l)}_{i-1}yv^{(l)}_{i+1}v^{(l)}_{i+2}\cdots v^{(l)}_{k_{l}}$ is a shortest $x_{0}$-$w_{l}$ path of $G$ and $N_{G}(y')\cap V(Q')\neq \emptyset $.
Thus, again by (\ref{eq-cl2.4-condition2}) and Claim~\ref{cl2.1}(1) with $(y,Q)=(y',Q')$, $y'$ is adjacent to a vertex in $V(Q')\cap X_{i}~(=\{y\})$.
Since $y$ and $y'$ are arbitrary, the desired conclusion holds.
\item
It follows from (\ref{eq-cl2.4-condition1}) and Claim~\ref{cl2.1}(1) with $Q=Q_{l}$ that every vertex in $X^{(l)}_{i}$ is adjacent to $v^{(l)}_{i-1}$ in $G$, and hence $|X^{(l)}_{i}|\leq |N_{G}(v^{(l)}_{i-1})\cap X_{i}|$.
Thus, to complete the proof of (3), it suffices to show that $|N_{G}(v^{(l)}_{i-1})\cap X_{i}|\leq \nu $.
By way of contradiction, suppose that $|N_{G}(v^{(l)}_{i-1})\cap X_{i}|\geq \nu +1~(=R(n-1,n))$.
If there exists a clique $U\subseteq N_{G}(v^{(l)}_{i-1})\cap X_{i}$ of $G$ with $|U|=n-1$, then $\{v^{(l)}_{i-1}\}\cup U$ induces a copy of $K_{n}$ in $G$, which is a contradiction.
This together with the definition of the Ramsey number implies that there exists an independent set $U'\subseteq N_{G}(v^{(l)}_{i-1})\cap X_{i}$ of $G$ with $|U'|=n$.
Then again by (\ref{eq-cl2.4-condition1}) and Claim~\ref{cl2.1}(1) with $Q=Q_{l}$, every vertex in $U'$ is adjacent to $v^{(l)}_{i+1}$ in $G$.
Hence
$$
\{v^{(l)}_{i-n},v^{(l)}_{i-n+1},\ldots ,v^{(l)}_{i-1}\}\cup U'\cup \{v^{(l)}_{i+1},v^{(l)}_{i+2},\ldots ,v^{(l)}_{i+n}\}
$$
induces a copy of $F^{(3)}_{n}$ in $G$, which is a contradiction.
\item
By (1) and (3), $|X_{i}|=\sum _{1\leq l\leq h}|X^{(l)}_{i}|\leq h\nu $.
\item
Suppose that there exists a vertex $y\in X_{i}\setminus (\bigcup _{1\leq l\leq h}V(Q_{l}))$.
By (1), there exists an integer $l_{1}$ with $1\leq l_{1}\leq h$ such that $y\in X^{(l_{1})}_{i}$.
Then $N_{G}(y)\cap V(Q_{l_{1}})\neq \emptyset $.
It follows from (\ref{eq-cl2.4-condition1}) and Claim~\ref{cl2.1}(1) that $yv^{(l_{1})}_{i-1},yv^{(l_{1})}_{i+1}\in E(G)$, and hence
$$
\{v^{(l_{1})}_{i-n},v^{(l_{1})}_{i-n+1},\ldots ,v^{(l_{1})}_{i},y,v^{(l_{1})}_{i+1},v^{(l_{1})}_{i+2},\ldots ,v^{(l_{1})}_{i+n}\}
$$
induces a copy of $F^{(4)}_{n}$ or $F^{(5)}_{n}$ in $G$ according as $yv^{(l_{1})}_{i}\notin E(G)$ or not, which is a contradiction.
Thus $X_{i}\subseteq \bigcup _{1\leq l\leq h}V(Q_{l})$.
\qed
\end{enumerate}
For an integer $h$ with $1\leq h\leq h_{0}$, let $J'_{h}=J_{h}\setminus \{m_{h}-1\}~(=\{i:k_{h+1}+1\leq i\leq k_{h}-n-2\})$.
\begin{claim
\label{cl2.4+}
Let $h$ be an integer with $1\leq h\leq h_{0}$.
\begin{enumerate}[{\upshape(1)}]
\item
There exists an induced path cover $\RR_{h}$ of $G[\bigcup _{i\in J'_{h}}X_{i}]$ with $|\RR_{h}|\leq h\nu $.
\item
If $G$ is $\{F^{(4)}_{n},F^{(5)}_{n}\}$-free, then there exists an induced path partition $\RR'_{h}$ of $G[\bigcup _{i\in J'_{h}}X_{i}]$ with $|\RR'_{h}|\leq h$.
\end{enumerate}
\end{claim}
\proof
If $J'_{h}=\emptyset $, then the claim clearly holds.
Thus we may assume that $J'_{h}\neq \emptyset $.
\begin{enumerate}[{\upshape(1)}]
\item
Fix an integer $l$ with $1\leq l\leq h$.
For $i\in J'_{h}$, write $X^{(l)}_{i}=\{a^{(l)}_{i,1},a^{(l)}_{i,2},\ldots ,a^{(l)}_{i,p_{l,i}}\}$ where $p_{l,i}=|X^{(l)}_{i}|$.
By Claim~\ref{cl2.4}(2), for $i\in J'_{h}\setminus \{m_{h}-2\}$, every vertex in $X^{(l)}_{i}$ is adjacent to all vertices in $X^{(l)}_{i+1}$.
Hence for an integer $j$ with $1\leq j\leq \nu $, $R^{(l)}_{j}:=b^{(l)}_{k_{h+1}+1,j}b^{(l)}_{k_{h+1}+2,j}\cdots b^{(l)}_{k_{h}-n-2,j}$ is an induced path of $G$ where
$$
b^{(l)}_{i,j}=
\begin{cases}
a^{(l)}_{i,j} & (j\leq p_{l,i})\\
a^{(l)}_{i,p_{l,i}} & (j>p_{l,i}).
\end{cases}
$$
By Claim~\ref{cl2.4}(3), $\max\{p_{l,i}:i\in J'_{h}\}=\max\{|X^{(l)}_{i}|:i\in J'_{h}\}\leq \nu $, and hence $(\bigcup _{1\leq j\leq \nu }V(R^{(l)}_{j}))\cap X_{i}=X^{(l)}_{i}$.
Consequently, it follows from Claim~\ref{cl2.4}(1) that $\RR_{h}:=\{R^{(l)}_{j}:1\leq l\leq h,~1\leq j\leq \nu \}$ is an induced path cover of $G[\bigcup _{i\in J'_{h}}(\bigcup _{1\leq l\leq h}X^{(l)}_{i})]~(=G[\bigcup _{i\in J'_{h}}X_{i}])$ with $|\RR_{h}|\leq h\nu $.
\item
By Claim~\ref{cl2.4}(5), $\bigcup _{i\in J'_{h}}X_{i}\subseteq \bigcup _{1\leq l\leq h}V(Q_{l})$.
Hence $\RR'_{h}:=\{Q_{l}[\bigcup _{i\in J'_{h}}X^{(l)}_{i}]:1\leq l\leq h\}$ is an induced path partition of $G[\bigcup _{i\in J'_{h}}X_{i}]$ with $|\RR'_{h}|=h$.
\qed
\end{enumerate}
Let $L=\{h:1\leq h\leq h_{0},~J_{h}\neq \emptyset \}$.
If $L=\emptyset $, then by Claims~\ref{cl2.3--} and \ref{cl2.3}(4),
$$
d+1 = |\{0,1,\ldots ,d\}|=\sum _{1\leq h\leq h_{0}+1}|I_{h}|\leq n^{2}+2n,
$$
which contradicts (\ref{cond-sp-diam}).
Thus $L\neq \emptyset $.
Write $L=\{p_{1},p_{2},\ldots ,p_{r}\}$ where $|L|=r$ and $p_{1}<p_{2}<\cdots <p_{r}$, and let $p_{0}=0$.
For an integer $h$ with $1\leq h\leq r+1$, let
$$
I'_{h}=
\begin{cases}
(\bigcup _{p_{h-1}+1\leq l\leq p_{h}}I_{l})\cup \{m_{p_{h}}-1\}~(=\{i:m_{p_{h}}-1\leq i\leq k_{p_{h-1}+1}\}) & (1\leq h\leq r)\\
\bigcup _{p_{r}+1\leq l\leq h_{0}+1}I_{l}~(=\{i:0\leq i\leq k_{p_{r}+1}\}) & (h=r+1).
\end{cases}
$$
Recall that $J'_{h}=J_{h}\setminus \{m_{h}-1\}$ for each $h$ with $1\leq h\leq h_{0}$.
Hence by Claim~\ref{cl2.3--},
\begin{align}
\mbox{$\{0,1,\ldots ,d\}$ is the disjoint union of $I'_{h}~(1\leq h\leq r+1)$ and $J'_{p_{h}}~(1\leq h\leq r)$.}\label{cond-sp-I'J'}
\end{align}
\begin{claim
\label{cl2.5}
Let $c_{1}(n,l_{0})$ and $c_{2}(n,l_{0})$ be the constants appearing in Proposition~\ref{prop-indstar}.
\begin{enumerate}[{\upshape(1)}]
\item
For each integer $h$ with $1\leq h\leq r$, there exists an induced star cover $\SS_{h}$ of $G[\bigcup _{i\in I'_{h}}X_{i}]$ with $|\SS_{h}|\leq (n-1)\nu c_{1}(n,n^{2}-1)$.
\item
There exists an induced star cover $\SS_{r+1}$ of $G[\bigcup _{i\in I'_{r+1}}X_{i}]$ with $|\SS_{r+1}|\leq c_{1}(n,n^{2}+2n-1)$.
\item
For each integer $h$ with $1\leq h\leq r$, if $G$ is $\tilde{S}_{n}$-free, then there exists an induced star partition $\SS'_{h}$ of $G[\bigcup _{i\in I'_{h}}X_{i}]$ with $|\SS'_{h}|\leq (n-1)\nu c_{2}(n,n^{2}-1)$.
\item
If $G$ is $\tilde{S}_{n}$-free, then there exists an induced star partition $\SS'_{r+1}$ of $G[\bigcup _{i\in I'_{r+1}}X_{i}]$ with $|\SS'_{r+1}|\leq c_{2}(n,n^{2}+2n-1)$.
\end{enumerate}
\end{claim}
\proof
For each integer $h$ with $1\leq h\leq r+1$, let $G'_{h}=G[\bigcup _{i\in I'_{h}}X_{i}]$.
We first prove (2) and (4).
Since $I'_{r+1}=\bigcup _{p_{r}+1\leq l\leq h_{0}+1}I_{l}$, it follows from Claim~\ref{cl2.3}(4) that
$$
{\rm diam}(G'_{r+1})\leq {\rm ecc}_{G'_{r+1}}(x_{0})=|I'_{r+1}|-1\leq \sum _{1\leq l\leq h_{0}+1}|I_{l}| -1\leq n^{2}+2n-1.
$$
Hence by applying Proposition~\ref{prop-indstar} with $l_{0}=n^{2}+2n-1$,
\begin{enumerate}[{$\bullet $}]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
${\rm insc}(G'_{r+1})\leq c_{1}(n,n^{2}+2n-1)$ and
\item
if $G$ is $\tilde{S}_{n}$-free, then ${\rm insp}(G'_{r+1})\leq c_{2}(n,n^{2}+2n-1)$.
\end{enumerate}
Consequently, (2) and (4) hold.
Next we prove (1) and (3).
Let $h$ be an integer with $1\leq h\leq r$.
For an integer $i\in I'_{h}\setminus \{m_{p_{h}}-1\}$ and a vertex $x\in X_{i}$, take a vertex $u_{x}\in X_{i-1}$ so that $u_{x}x\in E(G)$.
Let $D_{h}$ be the graph with $V(D_{h})=\bigcup _{i\in I'_{h}}X_{i}$ and $E(D_{h})=\{u_{x}x:x\in X_{i},~i\in I'_{h}\setminus \{m_{p_{h}}-1\}\}$, and let $D_{h,1},D_{h,2},\ldots ,D_{h,q_{h}}$ be the components of $D_{h}$.
Fix an integer $j$ with $1\leq j\leq q_{h}$.
By the definition of $D_{h}$, $D_{h,j}$ is a tree and $|V(D_{h,j})\cap X_{m_{p_{h}}-1}|=1$, say $V(D_{h,j})\cap X_{m_{p_{h}}-1}=\{u_{h,j}\}$.
In particular, note that $q_{h}=|X_{m_{p_{h}}-1}|$.
Let $G_{h,j}=G[V(D_{h,j})]$.
Fix a vertex $x\in V(G_{h,j})$, and let $i\in I'_{h}$ be the integer with $x\in X_{i}$.
Then by the construction of $D_{h}$, we have ${\rm dist}_{G_{h,j}}(u_{h,j},x) = i-(m_{p_{h}}-1)\leq k_{p_{h-1}+1}-(m_{p_{h}}-1)=|I'_{h}|-1$.
This together with Claim~\ref{cl2.3}(2)(3) leads to
\begin{align*}
{\rm dist}_{G_{h,j}}(u_{h,j},x) &\leq |I'_{h}|-1\\
&= \sum _{p_{h-1}+1\leq l\leq p_{h}}|I_{l}|\\
&\leq \sum _{1\leq l\leq h_{0}}|I_{l}|\\
&\leq (n-1)(n+1).
\end{align*}
Since $x$ is arbitrary, this implies that ${\rm diam}(G_{h,j})\leq {\rm ecc}_{G_{h,j}}(u_{h,j})\leq n^{2}-1$.
Hence by applying Proposition~\ref{prop-indstar} with $l_{0}=n^{2}-1$,
\begin{align}
{\rm insc}(G_{h,j})\leq c_{1}(n,n^{2}-1)\label{cond-cl2.5-new1}
\end{align}
and
\begin{align}
\mbox{if $G$ is $\tilde{S}_{n}$-free, then }{\rm insp}(G_{h,j})\leq c_{2}(n,n^{2}-1).\label{cond-cl2.5-new2}
\end{align}
Since $m_{p_{h}}-1\in J_{p_{h}}$, it follows from Claim~\ref{cl2.4}(4) that $q_{h}=|X_{m_{p_{h}}-1}|\leq p_{h}\nu \leq h_{0}\nu $.
This together with Claim~\ref{cl2.3}(2) leads to
\begin{align}
q_{h}\leq (n-1)\nu .\label{cond-cl2.5-new3}
\end{align}
Since $\bigcup _{i\in I'_{h}}X_{i}~(=V(G'_{h}))$ is the disjoint union of $V(G_{h,j})~(1\leq j\leq q_{h})$, it follows from (\ref{cond-cl2.5-new1})--(\ref{cond-cl2.5-new3}) that
\begin{enumerate}[{$\bullet $}]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
${\rm insc}(G'_{h})\leq \sum _{1\leq j\leq q_{h}}{\rm insc}(G_{h,j})\leq q_{h}c_{1}(n,n^{2}-1)\leq (n-1)\nu c_{1}(n,n^{2}-1)$, and
\item
if $G$ is $\tilde{S}_{n}$-free, then ${\rm insp}(G'_{h})\leq \sum _{1\leq j\leq q_{h}}{\rm insp}(G_{h,j})\leq q_{h}c_{2}(n,n^{2}-1)\leq (n-1)\nu c_{2}(n,n^{2}-1)$.
\end{enumerate}
Consequently, (1) and (3) hold.
\qed
Let $\RR_{h}$ and $\SS_{h}$ (and $\RR'_{h}$ and $\SS'_{h}$ if $G$ is $\{\tilde{S}_{n},F^{(4)}_{n},F^{(5)}_{n}\}$-free) be the families as in Claims~\ref{cl2.4+} and \ref{cl2.5}.
By (\ref{cond-sp-I'J'}),
\begin{enumerate}[{$\bullet $}]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
the family $\RR:=(\bigcup _{1\leq h\leq r}\RR_{p_{h}})\cup (\bigcup _{1\leq h\leq r+1}\SS_{h})$ is an induced SP-cover of $G$, and
\item
if $G$ is $\{\tilde{S}_{n},F^{(4)}_{n},F^{(5)}_{n}\}$-free, then the family $\RR':=(\bigcup _{1\leq h\leq r}\RR'_{p_{h}})\cup (\bigcup _{1\leq h\leq r+1}\SS'_{h})$ is an induced SP-partition of $G$.
\end{enumerate}
By Claim~\ref{cl2.3}(2), $r\leq h_{0}\leq n-1$ and $\sum _{1\leq h\leq r}p_{h}\leq \sum _{1\leq h\leq r}h_{0}=rh_{0}\leq (n-1)^{2}$.
Hence
\begin{align*}
{\rm inspc}(G) &\leq |\RR|\\
&= \sum _{1\leq h\leq r}|\RR_{p_{h}}|+\sum _{1\leq h\leq r}|\SS_{h}|+|\SS_{r+1}|\\
&\leq \sum _{1\leq h\leq r}p_{h}\nu +r(n-1)\nu c_{1}(n,n^{2}-1)+c_{1}(n,n^{2}+2n-1)\\
&\leq (n-1)^{2}\nu +(n-1)^{2}\nu c_{1}(n,n^{2}-1)+c_{1}(n,n^{2}+2n-1),
\end{align*}
and if $G$ is $\{\tilde{S}_{n},F^{(4)}_{n},F^{(5)}_{n}\}$-free, then
\begin{align*}
{\rm inspp}(G) &\leq |\RR'|\\
&= \sum _{1\leq h\leq r}|\RR'_{p_{h}}|+\sum _{1\leq h\leq r}|\SS'_{h}|+|\SS'_{r+1}|\\
&\leq \sum _{1\leq h\leq r}p_{h}+r(n-1)\nu c_{2}(n,n^{2}-1)+c_{2}(n,n^{2}+2n-1)\\
&\leq (n-1)^{2}+(n-1)^{2}\nu c_{2}(n,n^{2}-1)+c_{2}(n,n^{2}+2n-1).
\end{align*}
This completes the proof of Proposition~\ref{mainthm-2}.
\qed
We conclude this section by finding families $\HH$ satisfying {\rm (P-${\rm insc}$)}, {\rm (P-${\rm insp}$)}, {\rm (P-${\rm inpc}$)} and {\rm (P-${\rm inpp}$)}.
\begin{prop
\label{mainthm-star/path}
Let $n\geq 4$ be an integer.
Then the following hold:
\begin{enumerate}
\item[{\upshape(i)}]
There exists a constant $c_{\rm insc}(n)$ depending on $n$ only such that ${\rm insc}(G)\leq c_{\rm insc}(n)$ for every connected $\{K_{n},S^{*}_{n},P_{n}\}$-free graph $G$.
\item[{\upshape(ii)}]
There exists a constant $c_{\rm insp}(n)$ depending on $n$ only such that ${\rm insp}(G)\leq c_{\rm insp}(n)$ for every connected $\{K_{n},S^{*}_{n},\tilde{S}_{n},P_{n}\}$-free graph $G$.
\item[{\upshape(iii)}]
There exists a constant $c_{\rm inpc}(n)$ depending on $n$ only such that ${\rm inpc}(G)\leq c_{\rm inpc}(n)$ for every connected $\{K_{n},K_{1,n},F^{(1)}_{n},F^{(2)}_{n}\}$-free graph $G$.
\item[{\upshape(iv)}]
There exists a constant $c_{\rm inpp}(n)$ depending on $n$ only such that ${\rm inpp}(G)\leq c_{\rm inpp}(n)$ for every connected $\{K_{n},K_{1,n},F^{(1)}_{n},F^{(2)}_{n},F^{(4)}_{n},F^{(5)}_{n}\}$-free graph $G$.
\end{enumerate}
\end{prop}
\proof
Let $c_{\rm inspc}(n)$ and $c_{\rm inspp}(n)$ be the constants appearing in the statement in Proposition~\ref{mainthm-2}.
Let $G$ be a connected $\{K_{n},S^{*}_{n},P_{n}\}$-free graph.
By the $P_{n}$-freeness of $G$, $G$ is also $\{F^{(1)}_{n},F^{(2)}_{n},F^{(3)}_{n}\}$-free.
Hence by Proposition~\ref{mainthm-2}(i), there exists an induced SP-cover $\RR$ of $G$ with $|\RR|\leq c_{\rm inspc}(n)$.
Let $\PP=\{P\in \RR:P\mbox{ is a path}\}$, and let $\SS=\{\{x\}:x\in V(P),~P\in \PP\}$.
Then $(\RR\setminus \PP)\cup \SS$ is an induced star cover of $G$.
Since $G$ is $P_{n}$-free, $|V(P)|\leq n-1$ for every $P\in \PP$.
Hence $|\SS|=|\bigcup _{P\in \PP}V(P)|\leq \sum _{P\in \PP}|V(P)|\leq (n-1)|\PP|\leq (n-1)c_{\rm inspc}(n)$.
Therefore,
$$
{\rm insc}(G)\leq |\RR\setminus \PP|+|\SS|\leq c_{\rm inspc}(n)+(n-1)c_{\rm inspc}(n).
$$
Since $G$ is arbitrary, (i) holds.
By similar argument, we see that ${\rm insp}(G)\leq nc_{\rm inspp}(n)$ for every connected $\{K_{n},S^{*}_{n},\tilde{S}_{n},P_{n}\}$-free graph $G$.
Furthermore, since an induced star in a $K_{1,n}$-free graph has at most $n$ vertices, we also verify that
\begin{enumerate}[{$\bullet $}]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
${\rm inpc}(G)\leq (n+1)c_{\rm inspc}(n)$ for every connected $\{K_{n},K_{1,n},F^{(1)}_{n},F^{(2)}_{n}\}$-free graph $G$, and
\item
${\rm inpp}(G)\leq (n+1)c_{\rm inspp}(n)$ for every connected $\{K_{n},K_{1,n},F^{(1)}_{n},F^{(2)}_{n},F^{(4)}_{n},F^{(5)}_{n}\}$-free graph $G$.
\end{enumerate}
Therefore, (ii)--(iv) also hold.
\qed
\section{Graphs with large induced cover/partition numbers}\label{sec-nec}
In this section, we calculate some induced cover/partition numbers of specified graphs.
The results will be used for proving the ``only if'' parts in main results.
We can easily verify that the following lemma holds, and so we omit its details.
\begin{lem
\label{lem-nec-complete/star}
Let $n\geq 2$ be an integer.
Then ${\rm inspc}(K_{n})=\lceil \frac{n}{2} \rceil$, ${\rm inspc}(S^{*}_{n})=\lceil \frac{n}{2} \rceil $, ${\rm inspp}(\tilde{S}_{n})=n+1$, ${\rm inpc}(K_{1,n})=\lceil \frac{n}{2} \rceil$, and ${\rm insc}(P_{n})=\lceil \frac{n}{3} \rceil$.
\end{lem}
Let $m\geq 2$ and $n\geq 3$ be integers, and let $Q_{i}=u^{(1)}_{i}u^{(2)}_{i}\cdots u^{(n)}_{i}~(1\leq i\leq m)$ be $m$ pairwise vertex-disjoint paths.
We define five graphs.
\begin{enumerate}[{$\bullet $}]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
Let $H^{(1)}_{m,n}$ be the graph obtained from the union of the paths $Q_{1},\ldots ,Q_{m}$ by adding $2(m-1)$ vertices $v_{i},w_{i}~(1\leq i\leq m-1)$ and $3(m-1)$ edges $v_{i}w_{i},v_{i}u^{(n)}_{i},v_{i}u^{(1)}_{i+1}~(1\leq i\leq m-1)$.
\item
Let $H^{(2)}_{m,n}$ be the graph obtained from the union of the paths $Q_{1},\ldots ,Q_{m}$ by adding $m-1$ vertices $v_{i}~(1\leq i\leq m-1)$ and $3(m-1)$ edges $v_{i}u^{(n)}_{i},v_{i}u^{(1)}_{i+1},u^{(n)}_{i}u^{(1)}_{i+1}~(1\leq i\leq m-1)$.
\item
Let $H^{(3)}_{m,n}$ be the graph obtained from the union of the paths $Q_{1},\ldots ,Q_{m}$ by adding $(m-1)m$ vertices $v_{i,j}~(1\leq i\leq m-1,~1\leq j\leq m)$ and $2(m-1)m$ edges $v_{i,j}u^{(n)}_{i},v_{i,j}u^{(1)}_{i+1}~(1\leq i\leq m-1,~1\leq j\leq m)$.
\item
Let $H^{(4)}_{m,n}$ be the graph obtained from $H^{(3)}_{m,n}$ by deleting $(m-1)(m-2)$ vertices $v_{i,j}~(1\leq i\leq m-1,~3\leq j\leq m)$.
\item
Let $H^{(5)}_{m,n}$ be the graph obtained from $H^{(4)}_{m,n}$ by adding $m-1$ edges $v_{i,1}v_{i,2}~(1\leq i\leq m-1)$.
\end{enumerate}
Then the following lemma holds.
(Here we just want to claim that ${\rm inspc}(H^{(i)}_{m,n})\rightarrow \infty ~(m\rightarrow \infty )$ for $i\in \{1,2,3\}$, and ${\rm inspp}(H^{(i)}_{m,n})\rightarrow \infty ~(m\rightarrow \infty )$ for $i\in \{4,5\}$.
Thus the readers recognizing the facts can be skip the proof.)
\begin{lem
\label{lem-nec-pathtype}
Let $m$ and $n$ be integers with $m\geq 2$ and $n\geq 3$.
\begin{enumerate}[{\upshape(1)}]
\item
We have ${\rm inspc}(H^{(1)}_{m,n})\geq \lceil \frac{m+1}{2} \rceil $.
\item
We have ${\rm inspc}(H^{(2)}_{m,n})\geq \lceil \frac{m+1}{2} \rceil $.
\item
We have ${\rm inspc}(H^{(3)}_{m,n})\geq m$.
\item
We have ${\rm inspp}(H^{(4)}_{m,n})\geq m$.
\item
We have ${\rm inspp}(H^{(5)}_{m,n})\geq m$.
\end{enumerate}
\end{lem}
\proof
\begin{enumerate}
\item[{\upshape(1)}]
Since any two leaves of $H^{(1)}_{m,n}$ have distance at least four, each element of an induced SP-cover of $H^{(1)}_{m,n}$ contains at most two leaves of $H^{(1)}_{m,n}$.
Since $H^{(1)}_{m,n}$ has $m+1$ leaves, this implies that ${\rm inspc}(H^{(1)}_{m,n})\geq \lceil \frac{m+1}{2} \rceil $.
\item[{\upshape(2)}]
Let $\PP$ be an induced SP-cover of $H^{(2)}_{m,n}$.
It suffices to show that $|\PP|\geq \lceil \frac{m+1}{2} \rceil $.
Since $H^{(2)}_{m,n}$ is $K_{1,3}$-free, $\PP$ can be regarded as an induced path cover of $H^{(2)}_{m,n}$.
Let $U=\{u^{(1)}_{1},u^{(n)}_{m}\}\cup \{v_{i}:1\leq i\leq m-1\}$ and ${\bf X}=\{(P,u)\in \PP\times U:u\in V(P)\}$.
For each $u\in U$, there exists an element $P$ of $\PP$ with $u\in V(P)$.
Hence
\begin{align}
|{\bf X}|=\sum _{u\in U}|\{P\in \PP:u\in V(P)\}|\geq |U|=m+1.\label{eq-lem-H2-1}
\end{align}
For every vertex $u\in U$, since the neighborhood of $u$ is a clique of $H^{(2)}_{m,n}$, no induced path of $H^{(2)}_{m,n}$ contains $u$ as an internal vertex.
Since a path contains at most two endvertices, this implies that $|V(P)\cap U|\leq 2$ for every $P\in \PP$.
Hence
\begin{align}
|{\bf X}|=\sum _{P\in \PP}|V(P)\cap U|\leq 2|\PP|.\label{eq-lem-H2-2}
\end{align}
By (\ref{eq-lem-H2-1}) and (\ref{eq-lem-H2-2}), we have $m+1\leq |{\bf X}|\leq 2|\PP|$, as desired.
\item[{\upshape(3)}]
Let $\PP$ be an induced SP-cover of $H^{(3)}_{m,n}$.
It suffices to show that $|\PP|\geq m$.
Let $U=\{u^{(1)}_{1}\}\cup \{v_{i,j}:1\leq i\leq m-1,~1\leq j\leq m\}$ and ${\bf X}=\{(P,u)\in \PP\times U:u\in V(P)\}$.
For each $u\in U$, there exists an element $P$ of $\PP$ with $u\in V(P)$.
Hence
\begin{align}
|{\bf X}|=\sum _{u\in U}|\{P\in \PP:u\in V(P)\}|\geq |U|=(m-1)m+1.\label{eq-lem-H3-1}
\end{align}
Note that for an integer $i$ with $1\leq i\leq m-1$, if an induced path $P$ of $H^{(3)}_{m,n}$ contains at least two vertices in $\{v_{i,j}:1\leq j\leq m\}$, then $|V(P)|=3$ and $P$ contains exactly two vertices in $\{v_{i,j}:1\leq j\leq m\}$.
Hence if an element $P$ of $\PP$ is a path, then $|V(P)\cap U|\leq m$.
(For example, a $u^{(1)}_{1}$-$u^{(n)}_{m}$ path $P$ of $H^{(3)}_{m,n}$ is an induced path of $H^{(3)}_{m,n}$ with $|V(P)\cap U|=m$.)
Since any two sets of $\{u^{(1)}_{1}\}$ and $\{v_{i,j}:1\leq j\leq m\}~(1\leq i\leq m-1)$ have distance at least three in $H^{(3)}_{m,n}$, if an element $P$ of $\PP$ is a star, then $|V(P)\cap U|\leq m$.
This implies that
\begin{align}
|{\bf X}|=\sum _{P\in \PP}|V(P)\cap U|\leq m|\PP|.\label{eq-lem-H3-2}
\end{align}
By (\ref{eq-lem-H3-1}) and (\ref{eq-lem-H3-2}), we have $m^{2}-m+1\leq |{\bf X}|\leq m|\PP|$, and hence $|\PP|\geq m-1+\frac{1}{m}$.
Since $|\PP|$ is an integer, this forces $|\PP|\geq m$.
\item[{\upshape(4)}]
Let $\PP$ be an induced SP-partition of $H^{(4)}_{m,n}$.
It suffices to show that $|\PP|\geq m$.
For an integer $i$ with $1\leq i\leq m$, let $R_{i}$ be the unique element of $\PP$ containing $u^{(2)}_{i}$.
We remark that $R_{i}$ might equal $R_{i'}$ for some $1\leq i<i'\leq m$.
Let $I=\{i:1\leq i\leq m-1,~R_{i}\neq R_{i+1}\}$, and write $I=\{i_{1},i_{2},\ldots ,i_{h}\}$ with $i_{1}<i_{2}<\ldots <i_{h}$ where $h=0$ if $I=\emptyset $.
If $R_{i_{l}}=R_{i'}$ for some integers $l$ and $i'$ with $1\leq l\leq h$ and $i'>i_{l}$, then $R_{i_{l}}$ is a path and contains all vertices in $\{u^{(2)}_{j}:i_{l}\leq j\leq i'\}$, and hence $R_{i_{l}}=R_{i_{l}+1}$, which contradicts the definition of $i_{l}$.
This, together with the fact that $i_{h}<m$ if $I\neq \emptyset $, implies that $R_{i_{1}},R_{i_{2}},\ldots ,R_{i_{h}},R_{m}$ are pairwise distinct, and so $|\{R_{i}:1\leq i\leq m\}|\geq h+1$.
Note that the inequality also holds even if $I=\emptyset $.
Fix an integer $j\in \{1,2,\ldots ,m-1\}\setminus I$.
By the definition of $I$, $R_{j}=R_{j+1}$, and hence $R_{j}$ contains $u^{(2)}_{j}$ and $u^{(2)}_{j+1}$.
Since ${\rm dist}_{H^{(4)}_{m,n}}(u^{(2)}_{j},u^{(2)}_{j+1})=n+1\geq 4$, $R_{j}$ is a path.
We can easily verify that $\{u^{(n)}_{j},u^{(1)}_{j+1}\}\subseteq V(R_{j})$ and $|V(R_{j})\cap \{v_{j,1},v_{j,2}\}|=1$.
This implies that there exists an element $R'_{j}$ of $\PP$ with $V(R'_{j})=\{v_{j,1}\}$ or $V(R'_{j})=\{v_{j,2}\}$.
Therefore
$$
|\PP|\geq |\{R_{i}:1\leq i\leq m\}\cup \{R'_{j}:j\in \{1,2,\ldots ,m-1\}\setminus I\}|\geq (h+1)+((m-1)-h)=m,
$$
as desired.
\item[{\upshape(5)}]
We proceed by induction on $m$.
Since $H^{(5)}_{2,n}$ contains a cycle, we have ${\rm inspp}(H^{(5)}_{2,n})\geq 2$.
Thus we may assume that $m\geq 3$.
We let $\PP $ be an induced SP-partition of $H^{(5)}_{m,n}$.
It suffices to show that $|\PP|\geq m$.
Since $H^{(5)}_{m,n}$ is $K_{1,3}$-free, $\PP$ can be regarded as an induced path partition of $H^{(5)}_{m,n}$.
For the moment, suppose that $\{v_{i,1},v_{i,2}\}\not\subseteq V(P)$ for any $P\in \PP$ and any integer $i$ with $1\leq i\leq m-1$.
Then $\PP$ is an induced SP-partition of $H^{(5)}_{m,n}-\{v_{i,1}v_{i,2}:1\leq i\leq m-1\}~(=H^{(4)}_{m,n})$.
Hence by (4), $|\PP|\geq {\rm inspp}(H^{(4)}_{m,n})\geq m$, as desired.
Thus we may assume that there exist an element $P^{*}$ of $\PP$ and an integer $i$ with $1\leq i\leq m-1$ such that $\{v_{i,1},v_{i,2}\}\subseteq V(P^{*})$.
Note that a vertex in $V(H^{(5)}_{m,n})\setminus \{v_{i,1},v_{i,2}\}$ is adjacent to $v_{i,1}$ if and only if the vertex is adjacent to $v_{i,2}$.
Since $P^{*}$ is an induced path of $H^{(5)}_{m,n}$, this implies that $P^{*}$ consists of two vertices $v_{i,1}$ and $v_{i,2}$.
In other words, $\PP\setminus \{P^{*}\}$ is an induced SP-partition of $H^{(5)}_{m,n}-\{v_{i,1},v_{i,2}\}$.
If $i\in \{1,m-1\}$, then $H^{(5)}_{m,n}-\{v_{i,1},v_{i,2}\}$ is the disjoint union of a path and a copy of $H^{(5)}_{m-1,n}$, and hence by the induction hypothesis, $|\PP\setminus \{P^{*}\}|\geq 1+{\rm inspp}(H^{(5)}_{m-1,n})\geq 1+(m-1)$; if $2\leq i\leq m-2$, then $H^{(5)}_{m,n}-\{v_{i,1},v_{i,2}\}$ is the disjoint union of a copy of $H^{(5)}_{i,n}$ and a copy of $H^{(5)}_{m-i,n}$, and hence $|\PP\setminus \{P^{*}\}|\geq {\rm inpp}(H^{(5)}_{i,n})+{\rm inpp}(H^{(5)}_{m-i,n})\geq i+(m-i)$.
In either case, we obtain $|\PP|>|\PP\setminus \{P^{*}\}|\geq m$, as desired.
\qed
\end{enumerate}
\section{Proof of main results}\label{sec-proof}
In this section, we complete the proof of Theorem~\ref{mainthm} and settle Ramsey problems for four invariants ${\rm insc}$, ${\rm insp}$, ${\rm inpc}$ and ${\rm inpp}$.
\medbreak\noindent\textit{Proof of Theorem~\ref{mainthm}.}\quad
By Proposition~\ref{mainthm-2}, we obtain the ``if'' parts of (i) and (ii).
Thus we prove that the ``only if'' parts are true.
Let $\HH$ be a finite family of connected graphs.
Then the value $p=\max\{|V(H)|:H\in \HH\}$ is a well-defined constant depending on $\HH$.
We first suppose that $\HH$ satisfies {\rm (P-${\rm inspc}$)}.
Then there exists a constant $c=c(\HH)$ such that ${\rm inspc}(G)\leq c$ for every connected $\HH$-free graph $G$.
Let $n=\max\{p,2c+1,4\}$.
Since ${\rm inspc}(K_{n})={\rm inspc}(S^{*}_{n})=\lceil \frac{n}{2} \rceil \geq c+1$ by Lemma~\ref{lem-nec-complete/star}, neither $K_{n}$ nor $S^{*}_{n}$ is $\HH$-free.
This implies that $\HH\leq \{K_{n},S^{*}_{n}\}$.
Furthermore, for each $i\in \{1,2,3\}$, it follows from Lemma~\ref{lem-nec-pathtype}(1)--(3) that ${\rm inspc}(H^{(i)}_{n,n})\geq c+1$, and hence $H^{(i)}_{n,n}$ is not $\HH$-free, i.e., $H^{(i)}_{n,n}$ contains an induced subgraph $A_{i}$ isomorphic to an element of $\HH$.
Since $A_{i}$ is connected and $|V(A_{i})|\leq \max\{|V(H)|:H\in \HH\}=p\leq n$, we have
\begin{enumerate}[$\bullet $]
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item
$|\{i':1\leq i'\leq n,~V(A_{i})\cap V(Q_{i'})\neq \emptyset \}|\leq 2$,
\item
$|\{i':1\leq i'\leq n-1,~V(A_{1})\cap \{v_{i'},w_{i'}\}\neq \emptyset \}|\leq 1$,
\item
$|\{i':1\leq i'\leq n-1,~V(A_{2})\cap \{v_{i'}\}\neq \emptyset \}|\leq 1$, and
\item
$|\{i':1\leq i'\leq n-1,~V(A_{3})\cap \{v_{i',j}:1\leq j\leq n\}\neq \emptyset \}|\leq 1$.
\end{enumerate}
This implies that $A_{i}$ is isomorphic to an induced subgraph of $F^{(i)}_{n}$, and hence $\HH\leq \{F^{(1)}_{n},F^{(2)}_{n},F^{(3)}_{n}\}$.
Consequently, $\HH\leq \{K_{n},S^{*}_{n},F^{(1)}_{n},F^{(2)}_{n},F^{(3)}_{n}\}$, which proves the ``only if'' part of (i).
Next we suppose that $\HH$ satisfies {\rm (P-${\rm inspp}$)}.
Then there exists a constant $c'=c'(\HH)$ such that ${\rm inspp}(G)\leq c'$ for every connected $\HH$-free graph $G$.
Let $n=\max\{p,2c'+1,4\}$.
Note that for every graph $H$, ${\rm inspc}(H)\leq {\rm inspp}(H)$.
Hence by similar argument as above, together with Lemmas~\ref{lem-nec-complete/star} and \ref{lem-nec-pathtype}(1)(2)(4)(5), we see that $\HH\leq \{K_{n},S^{*}_{n},\tilde{S}_{n},F^{(1)}_{n},F^{(2)}_{n},F^{(4)}_{n},F^{(5)}_{n}\}$, which proves the ``only if'' part of (ii).
This completes the proof of Theorem~\ref{mainthm}.
\qed
Note that for every graph $H$,
\begin{align*}
&{\rm inspc}(H)\leq {\rm insc}(H)\leq {\rm insp}(H),~{\rm inspp}(H)\leq {\rm insp}(H),\\
&{\rm inspc}(H)\leq {\rm inpc}(H)\leq {\rm inpp}(H),\mbox{ and }{\rm inspp}(H)\leq {\rm inpp}(H).
\end{align*}
Hence, by similar argument in the proof of Theorem~\ref{mainthm}, together with Proposition~\ref{mainthm-star/path} and Lemmas~\ref{lem-nec-complete/star} and \ref{lem-nec-pathtype}(1)(2)(4)(5), we obtain the following theorem.
\begin{thm
\label{mainthm-cor}
Let $\HH$ be a finite family of connected graphs.
\begin{enumerate}[{\upshape(i)}]
\item
The family $\HH$ satisfies {\rm (P-${\rm insc}$)} if and only if $\HH\leq \{K_{n},S^{*}_{n},P_{n}\}$ for an integer $n\geq 4$.
\item
The family $\HH$ satisfies {\rm (P-${\rm insp}$)} if and only if $\HH\leq \{K_{n},S^{*}_{n},\tilde{S}_{n},P_{n}\}$ for an integer $n\geq 4$.
\item
The family $\HH$ satisfies {\rm (P-${\rm inpc}$)} if and only if $\HH\leq \{K_{n},K_{1,n},F^{(1)}_{n},F^{(2)}_{n}\}$ for an integer $n\geq 4$.
\item
The family $\HH$ satisfies {\rm (P-${\rm inpp}$)} if and only if $\HH\leq \{K_{n},K_{1,n},F^{(1)}_{n},F^{(2)}_{n},F^{(4)}_{n},,F^{(5)}_{n}\}$ for an integer $n\geq 4$.
\end{enumerate}
\end{thm}
Now we recall Remark~\ref{remark-mainthm-2} in Section~\ref{sec3}.
Let $G$ be a graph.
A family $\PP$ of subgraphs of $G$ is called an {\it isometric path cover} of $G$ if $\bigcup _{P\in \PP}V(P)=V(G)$ and each element of $\PP$ is an isometric path of $G$.
An isometric path cover $\PP$ of $G$ is called an {\it isometric path partition} of $G$ if the elements of $\PP$ are pairwise vertex-disjoint.
The minimum cardinality of an isometric path cover (resp. an isometric path partition) of $G$, denoted by ${\rm ispc}(G)$ (resp. ${\rm ispp}(G)$), is called the {\it isometric path cover number} (resp. the {\it isometric path partition number}) of $G$.
It is known that the isometric path cover number is closely related to the cops-robbers game, and motivated by the fact, there are many researches on the isometric path cover/partition (see \cite{FF,F2,FNHC,M,PC,PC2}).
The isometric path cover number is frequently called the {\it isometric path number}.
As we mentioned in Remark~\ref{remark-mainthm-2}, Proposition~\ref{mainthm-star/path}(iii)(iv) can be extended to isometric versions.
Furthermore, for every graph $G$, since every isometric path of $G$ is also an induced path of $G$, we have ${\rm inpc}(G)\leq {\rm ispc}(G)$ and ${\rm inpp}(G)\leq {\rm ispp}(G)$.
Consequently, we have implicitly proved the following theorem.
\begin{thm
\label{mainthm-3}
Let $\HH$ be a finite family of connected graphs.
\begin{enumerate}[{\upshape(i)}]
\item
The family $\HH$ satisfies {\rm (P-${\rm ispc}$)} if and only if $\HH\leq \{K_{n},K_{1,n},F^{(1)}_{n},F^{(2)}_{n}\}$ for an integer $n\geq 4$.
\item
The family $\HH$ satisfies {\rm (P-${\rm ispp}$)} if and only if $\HH\leq \{K_{n},K_{1,n},F^{(1)}_{n},F^{(2)}_{n},F^{(4)}_{n},,F^{(5)}_{n}\}$ for an integer $n\geq 4$.
\end{enumerate}
\end{thm}
\section*{Acknowledgment}
This work was partially supported by JSPS KAKENHI Grant number JP20K03720 (to S.C) and JSPS KAKENHI Grant number JP18K13449 (to M.F).
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{"url":"https:\/\/physics.stackexchange.com\/questions\/173477\/is-it-possible-to-shoot-bullets-in-space-or-would-the-recoil-of-the-gun-be-too-s\/173487","text":"# Is it possible to shoot bullets in space or would the recoil of the gun be too strong?\n\nI've read a few articles that say that astronauts have already brought guns in space and that shooting bullets in space is possible.\n\nBut won't the recoil of the gun be too strong?\n\nLaw of conservation of momentum:\n\n$m_{gun}v_{gun}=m_{bullet}v_{bullet}$\n\n$v_{gun}=\\dfrac{m_{bullet}}{m_{gun}}v_{bullet}$\n\nI've found some values on Google:\n\n$m_{bullet}=0.03$ kg\n\n$m_{gun}=1$ kg\n\n$v_{bullet}=800$ m.s$^{-1}$\n\nAnd therefore we get:\n\n$v_{gun}=24$ m\/s\n\n$v_{gun}=87$ km\/h\n\n$v_{gun}=54$ mph\n\nThat's the typical speed of a car on a freeway.\n\nIt's an uncontrollable recoil.\n\nThe gun would be pushed back so powerfully that it would very probably damage the astronaut's spacesuit and kill him.\n\nBut Russian Soyuz capsules wouldn't carry firearms if using them would lead to an instantaneous death for their own user. So I must be missing something.\n\nStill even if the astronaut could somehow hold extremely tightly to his gun and absorb all the momentum, he would still be ejected at a speed of around $1.5$ km\/h $=$ $1$ mph (and maybe he'll even get stuck spinning for a while like in the movie Gravity), which would make guns not very effective in space since it would take time to find a way to re-stabilize yourself and re-aim in order to fire a second bullet.\n\nNote: I know nothing about guns (I don't live in the US), so forgive me if I missed something obvious.\n\n\u2022 I've deleted off-topic comments on this question and several answers. \u2013\u00a0David Z Apr 2 '15 at 20:07\n\u2022 Interestingly, the answers are for single round discharge calculations. Since your question is entitled 'bullets', how would the cumulative effect of firing a full clip at maximum rate of an automatic weapon (circa 10 rounds per second) accelerate the astronaut backwards? \u2013\u00a0StuartLC Apr 2 '15 at 21:20\n\u2022 The gun is provided in case the cosmonauts have to land at a remote spot in the wilderness, for defence against wild animals. It is not intended for use in space. \u2013\u00a0John Bentin Apr 3 '15 at 9:08\n\nYou've calculated the speed of a remote-triggered gun after it fires the bullet, true. However, there's actually nothing about space in your calculation, as @ACuriousMind noted. In theory, a gun fired on Earth could fly off just as fast, at least for a second. What you should use is not $m_\\mathrm{gun}$ but $m_\\mathrm{gun} + m_\\mathrm{person}$. The gun never gets up to that speed because I start acting against it immediately and continuously.\n\nIn some sense the \"problem\" cancels itself out--the gun seems to go so fast because it doesn't weigh much! But in reality the fact that it doesn't weigh much also means I can keep it under control.\n\nThe problem in space you have is that the momentum you do get, even after calculating with both masses, will stay with you, and you start drifting away. Worse, you probably didn't fire along a line intersecting your center of mass, which means that you now have some crazy rotational motion. Whether or not this is dangerous depends on your zero-gravity shooting range's particular setup, but it doesn't sound comfortable.\n\n\u2022 Can't you just shoot a bullet in the opposite direction to cancel the momentum? Guess that is the wild wild space. You are better off with two frontiers \u2013\u00a0Dog eat cat world Apr 3 '15 at 8:20\n\nFor most guns, you can roughly hold them in place while fired. That is, the repulsion will not only \"hit\" the gun's mass but the astronaut's mass too, not allowing the gun to gain such high speed.\n\nWith your numbers this leaves at most $$v \\approx 0.11~\\text{ms}^{-1} = 0.38~\\text{kmh}^{-1}$$ for an astronaut + spacesuit + gun with $m=225~\\text{kg}$, if no torque is applied.\n\nBut I guess the most severe problem will be this torque applied to the astronaut if the momentum is not directed towards the center of mass, which leaves him spinning if no counter-torque can be upheld.\n\nIdealize the astronaut as an homogeneous cylinder with a moment of inertia $\\Theta = \\frac{1}{2}mR^2$ and $R\\approx 0.5~\\text{m}$. Let the bullet be fired at $R$ perpendicular to his symmetry axis and his line of sight such that this is the worst case scenario for spinning. Conservation of angular momentum demands $$L = \\Theta \\cdot \\omega = R \\cdot p_{\\text{bullet}}$$ for the astronauts angular momentum and therefore $$T = \\frac{2\\pi}{\\omega} = \\frac{\\pi \\cdot m \\cdot R}{m_\\text{bullet} \\cdot v_\\text{bullet}} \\approx 14.73~\\text{s}$$ the time for one revolution of the astronaut.\n\nAs the other answers have stated, the primary oversight in the original question is the mass of the astronaut\/cosmonaut holding the firearm.\n\nHowever, your original number for the mass of the projectile is off by an order of magnitude. Therefore, the original calculation - as well as some of the other samples provided afterward - are all still an order of magnitude too high.\n\nTypical small-arms projectiles have a mass ranging from 3 grams to 13 grams. Typical velocities range from 250 m\/s to 800 m\/s, and are inversely associated with projectile mass (that is, lighter bullets are fired at higher velocities, while heavier bullets are fired at lower velocities). So we'll keep your original sample number of 800 m\/s, but instead associate it with a more reasonable 3 gram bullet; this is actually pretty close to the 5.45x39mm cartridge that the TP-82 can fire.\n\nRunning with the new example numbers:\n\n$$m_{bullet} = 0.003 kg$$\n\n$$m_{astronaut} \\approx 1 kg + 70 kg + 145 kg = 216 kg$$\n\n$$v_{bullet} = 800 m\/s$$\n\nResults in:\n\n$$v_{astronaut} = 0.011 m\/s = 0.04 km\/h$$\n\n\u2022 Also, the weapons firing the fastest rounds are rifles, which are several times heavier than 1 kg. \u2013\u00a0cpast Apr 2 '15 at 5:23\n\u2022 Also, for reference, the replacement Makarov has a 6g bullet (9x18mm), but fires it at 315 m\/s (pistols not being noted for high muzzle velocity). \u2013\u00a0cpast Apr 2 '15 at 5:42\n\nWhat's the difference between where you are and space? Atmosphere (ie air pressure), temperature, gravity, radiation. Think about how each of those affects what happens to someone after they pull the trigger of a gun, on earth.\n\nAtmosphere: has no significant impact on the effects of recoil. It will help slow the bullet down but that's not relevant to what happens to the gun.\n\nTemperature: this could potentially affect the firing mechanism of the gun, or the charge of the bullet. Most likely candidate for messing with your shooting, i'd say. On the other hand, while space is very cold, it's also medium-less, ie there's nothing (or at least, not much) to transmit the molecular vibration (heat) away from the gun. So, the gun would actually cool down very slowly.\n\nGravity: It's your strength, not gravity or atmosphere, which stops the gun flying back into your face when you fire it. Gravity helps you anchor yourself to the ground, so your whole body would likely spin round, unless you're shooting from the hip (or tummy perhaps). You would move backwards with the same force as that applied to the bullet, but at a much slower speed, since you weigh much more than a bullet (acceleration = force over mass, remember). Lets say that the bullet weighs 10 grams for example, and you weigh 100 kilograms: the acceleration you're subjected to during the firing, and therefore your resulting speed, will be 100,000\/10 = 10,000 times LESS than the bullet. A typical bullet speed might be 1000 mph (to make the maths easier), so that divided by 10,000 gives you 0.1 miles an hour speed. Typical walking speed is 3 mph, so you would drift backwards at one thirtieth of normal walking speed. This isn't very dramatic.\n\nRadiation: You would be in big trouble without good radiation shielding. The gun would be fine.\n\nThe problem (read error) with your calculation is when you assume the speed of the bullet to $$v_{bullet}=800 m\/s$$ This is the speed the bullet gets if the gun is being held relatively still by a person standing on fixed ground.\n\nThe speed of the bullet comes from the impulse of the high pressure in the piston after the \"explosion\". Since the impulse is the time integral of the force the impulse $$I=\\int Fdt$$ it will be much smaller if the gun starts flying the other way, compared to if the gun is held still. This is because the pressure in the piston will decrease faster over time.\n\nAnd since we are only talking about horizontal directions (so no graivty is involved) you could just try jumping up from the ground, then fire a gun and see how much horizontal speed you get.\n\n\u2022 Even if the gun is floating alone in space when fired, it still has a lot more mass than the bullet, so most of the gas pressure will still go to accelerate the bullet even if the gun itself accelerates slightly backwards during the same time. \"Much smaller\" is definitely not right. \u2013\u00a0hmakholm left over Monica Apr 1 '15 at 12:48","date":"2021-05-16 12:36:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5728497505187988, \"perplexity\": 954.7918959138779}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243991269.57\/warc\/CC-MAIN-20210516105746-20210516135746-00421.warc.gz\"}"}
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Q: Detect file extension c# There is a virus that my brother got in his computer and what that virus did was to rename almost all files in his computer. It changed the file extensions as well. so a file that might have been named picture.jpg was renamed to kjfks.doc for example.
so what I have done in order to solve this problem is:
remove all file extensions from files. (I use a recursive method to search for all files in a directory and as I go through the files I remove the extension)
now the files do not have an extension. the files now look like:
I think this file names are stored in a local database created by the virus and if I purchase the anti virus they will be renamed back to their original name.
since my brother created a backup I selected the files that had a creation date latter than when my brother performed the backup. so I have placed that files in a directory.
I am not interested in getting the right extension as long as I can see the content of the file. for example, I will scan each file and if it has text inside I know it will have a .txt extension. maybe it was a .html or .css extension I will not be able to know that I know.
I belive that all pdf files should have something in common. or doc files should also have something in common. How can I figure what the most common types (pdf, doc, docx, png, jpg, etc) files have in common)
Edit:
I know it will probably take less time to go over all this 200 files and test each one instead of creating this program. it is just that I am curios to see if it will be possible to get the file extension.
A: In unix, you can use file to determine the type of file. There is also a port for windows and you can obviously write a script (batch, powershell, etc.) or C# program to automate this.
A: First, congratulate your brother on doing a backup. Many people don't, and are absolutely wiped out by these problems.
You're going to have to do a lot of research, I'm afraid, but you're on the right track.
Open each file with a TextReader or a BinaryReader and examine the headers. Most of them are detectable.
For instance: Every PDF starts with "%PDF-" and then its version number. Just look at those first 5 characters. If it's "%PDF-", then put a PDF on the filename and move on.
Similarly: "ÿØÿà..JFIF" for JPEG's, "[InternetShortcut]" for URL shortcuts, "L...........À......Fƒ" for regular shortcuts (the "." is a zero/null, BTW)
ZIPs / Compressed directories start with {0x50}{0x4B]{0x03}{0x04}{0x14}, and you should be aware that Office 2007/2010 documents are really ZIPs with XML files inside of them.
You'll have to do some digging as you find each type, but you should be able to write something to establish most of the file types.
You'll have to write some recursion to work through directories, but you can eliminate any file with no extension.
BTW - A great tool to help pwith this is HxD: http://www.mh-nexus.de/ It's what I used to pull this answer together!
Good luck!
A: "most common types" each have it's own format and most of them have some magic bytes at the fixed position near beginning of the file. You can detect most of formats quite easily. Even HTML, XML, .CSS and similar text files can be detected by analyzing their beginning. But it will take some time to write an application that will guess the format. For some types (such as ODF format or JAR format, which are built on top of regular ZIPs) you will be also able to detect this format.
But ... Can it be that there exists such application on the market? I guess you can find something if you search, cause the task is not as tricky as it initially seems to be.
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{"url":"https:\/\/www.biostars.org\/p\/493686\/","text":"Search all SRA data for sequence? Raw reads only\n2\n0\nEntering edit mode\n6 weeks ago\n\nIs their a way to search raw reads in the SRA? I don't mean an individual entry but rather all raw illumina reads submitted to NCBI. I could limit it by year, but I'm looking to scan all of the raw unassembled reads in public repositories. Yes, I know this is a significant task.\n\nI'm doing metagenomic \/ viral analysis and looking for evidence that might exist in the raw sequences but get thrown out in the consensus \/ assembly data.\n\nSRA NCBI Raw reads fastq \u2022 223 views\n1\nEntering edit mode\n\nWhile not quite the same thing you can do a sequence search against EBI's metagenomic dataset MGnify (LINK).\n\n0\nEntering edit mode\n\nCould you take the statistical sampling approach by considering all entries, and simply pick randomly? A queue of randomly drawn samples could be put through a pipeline, and if you expect to find something 1 in 100, or 1 in 1000, or whatever, you could get some sense for how long you'd have to run your pipeline before having some confidence that you are, or are not, likely to find what you're looking for.\n\n0\nEntering edit mode\n\nHow do you randomly sample the SRA? I can scan the query sample for unique sequences and simplify that way, but I'm not sure how to subsample the SRA.\n\n0\nEntering edit mode\n\nAs described, your project involves finding some type of sequence that might exist in public SRA submissions and you'd like to scan them all to see, but this is hard. Let's say that Martians exist, I have some Martian DNA sequence, and I want to know if Martian DNA exists in any SRA submissions - I have reason to believe that Martians visited 10 labs, and may have contaminated 100 samples present in the SRA. if there are 800,000 SRA samples (I have no idea how many there are), I can simply put all SRA ids in a bucket from which to draw ids at random, and use the SRA toolkit (i.e. fastq-dump) to grab fastq data one entry at a time and examine say the first million reads for my sequence of interest. If I don't find it, select another SRA record at random, and repeat. Using SRA in the cloud, one could set up a script to churn through records, and if something exists at the rate listed above (100 within 800,000) I should get a hit within 8000 trials. I could also parse all the SRA records up front to put those I suspect might have the highest likelihood of containing my sequence towards the front of the pack. That's all I mean, if your sequence exists in x records, then by sampling records in random order you may eventually find it - the slow part, of course, is that you have to continually download data until you find a positive match (made easier by SRA in the cloud approach). But like I said, I have no idea what you're really trying to do, or what your limitations are. I doubt you're searching for Martians.\n\n0\nEntering edit mode\n6 weeks ago\nMensur Dlakic \u2605 10k\n\nHave you tried BLASTn from the NCBI web site? Simply select SRA as a target database from the drop-down menu.\n\n0\nEntering edit mode\n5 weeks ago\nGenoMax 99k\n\nI don't mean an individual entry but rather all raw illumina reads submitted to NCBI.\n\nThere is no realistic way to do that. You can blast against select SRA accessions via BLAST web page as shown by @Mensur.\n\nCurrent size of SRA as of Feb 2021","date":"2021-04-10 19:56:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.26370787620544434, \"perplexity\": 2058.9861184143965}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038057476.6\/warc\/CC-MAIN-20210410181215-20210410211215-00625.warc.gz\"}"}
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Home » Training » Training Levels » Tips and Curiosities » Animals With The Shortest Lifespan
Animals With The Shortest Lifespan
How Did Cats Get Their Bad Reputation?
What to Do about Dog Bites
Can I Change A Dog's Name?
Some animals are capable of living for more than 150 years old due to the changes in metabolism, climactic conditions, body to brain proportions, and other unknown factors. However, while others only make it just a few hours.
The five animals with the shortest lives
When it comes to mammals, mice win the award for having the shortest life due to having an average lifespan of four years. They are considered the second most reproductive animal on earth after man.
They reach a length of 6.5 to 9 cm from their heads to toes. The length of their tail is between 7 and 10 cm, and their weight is between 12 and 30 grams.
The mammal with the second shortest life span is the domestic rabbit that's between 8 and 12 years, and its main causes of death are cancer and cumulative body fat.
Number three is the popular and beloved dog. These descendants of wolves live an average lifespan of 10 to 13 years. However, this average varies between different breeds.
The one with the lowest average lifespan is the Bulldog and the Irish Rover that live up to 7 years. The ones that live the longest-with an average 13 year lifespan-are German Shepherds, Labradors and Golden Retrievers, among others.
The animal with the fourth shortest life expectancy among mammals is the friendly squirrel. With an average life of only 12 years, this mammal sleeps an average of 15 hours per day, and eats nuts, seeds, grass, insects and worms.
A close relative to dog, wolves are the mammal with the fifth shortest life expectancy. They live around 16 years and they have a developed sense of smell, agility and strength greater than dogs. They are one of the most intelligent animals and they share their prey with the rest of the pack.
Are mammals the animals with the shortest lifespan?
Insects are invertebrates and are part of the most diverse group of animals on the planet. Besides being the largest species, they are the animal with the shortest life span on the planet.
Despite being one of oldest animals on the planet, the Ephemeroptera is the being with the shortest lifespan. Also known as ephemeral, it's the animal with the lowest average life expectancy. Dragonflies are of the same species, they transform from larva to insect, and females are capable of creating thousands of eggs in a 24 hour period.
There are around 2,500 species of ephemerals, with a maximum life span of 24 hours, but some only live for around five minutes.
After several studies, it was found that this brief time on earth is solely for the purpose of mating. During that small period of time, males and females mate as they fly; the female deposits the eggs in the water and they both die.
The insect with the second shortest lifespan is the gastrotrichs. This insect lives at the bottom of the marine ecosystem. It is born, develops and dies in three or four days because that's the time it takes to reach sexual maturity which allows him to reproduce.
Ants and flies, a very short lifecycle
Ants are very common and are also one of the insects with the shortest lifespan; an average of only three weeks.
There are more than 10,000 species of ants. The queen is lodged in the gigantic palaces of dirt that was built by her colony. She will spend her life laying eggs in order to guarantee the survival of the colony.
The function of the female worker ants is to collect food, maintain the anthill and take care of the younger ants. However, as of birth, the function of the male ants is to reproduce with the queen. Once he has completed this task, he will die.
Finally, the species with the fourth shortest life span is the fly. They live between 15 and 30 days, which is enough time for the female to lay up to 1,000 eggs. Although this insect is annoying to have in your home, its life cycle is actually very short.
Tips and Curiosities
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Dog bite statistics indicate two important and relevant data. The first is that almost two-thirds of dog bites occur with house pets. The second is that most of the victims are children. Dogs are friendly and supportive animals. However, they…
Can I Change A Dog's Name?
If you think you've given your dog the wrong name or don't like the name that previous owners gave her, you should know that it's quite possible to change a dog's name. In the following article, we'll tell you what…
Did You Know that Dogs Notice Your Tone of Voice?
Dogs perceive feelings by our tone of voice. Their brains can process not only what we say, but also the intent and emotions in a human voice. According to certain studies, your dog understands how you feel and your intentions…
How Dogs Make People's Lives Better
Dogs have been a symbol of companionship, loyalty, and unconditional love for a long time. They put many people in a good mood. However, besides being pets, trained dogs also do a lot of different jobs. It's certainly true that…
A Cardboard Device to Scoop Dog Feces
The task of picking up dog feces with a plastic bag may be gross, but it isn't exactly hard. In contrast, what's hard is being aware of the damage you're doing to the environment by placing organic waste inside a…
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Every body has a story.
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Jack Soo
October 28, 1917—January 11, 1979
On October 28, 1917, Goro Suzuki (later to be known as the world-famous Jack Soo) was born on a ship in the middle of the Pacific Ocean. His parents—Oakland, California, residents—had decided their child should be born in Japan, but they didn't quite make it there in time.
Sometime after he arrived, the family made their way back to Oakland, where Soo grew up. He attended University of California Berkley and earned some cash by emceeing and performing stand-up comedy at various comedy clubs in the area.
Not too long after graduation, Soo's show business career plans were derailed courtesy of the U.S. government. Shortly after the U.S. officially entered the fray of World War II, Soo and his family—along with thousands of other Japanese Americans—were interned at Tanforan Assembly Center in San Francisco and then relocated to Topaz Relocation Center, Utah.
Despite this indignity, Soo made the best of his circumstances, spending much of his time there entertaining his fellow inmates. Eventually he was officially deemed harmless and received authorization from the U.S. government to leave the center. Proving he was not one to hold a grudge, Soo later accepted a job in military intelligence in Cleveland, Ohio.
After the war, Soo traveled all over the country performing in nightclubs and the like, including an 18-month stint in Chicago, partnering in an act with Joey Bishop. Fortunately, all that work did not keep him too busy to marry former model Jan Zdelar in 1945.
During a performance in San Francisco, Soo was spotted by Gene Kelly (yes, that Gene Kelly), who was scouting for Asian actors to perform in the premiere Broadway production of "Flower Drum Song," which Kelly was directing. Kelly offered him the nightclub announcer role, allegedly with one (big) string attached—he had to change his last name to something that sounded more Chinese than Japanese, since the "Flower Drum Song" was written about Chinese characters. Exit Goro Suzuki, enter Jack Soo. Soo was well received and was soon promoted to the lead role in the play. He reprised the role in the 1961 movie, as well as several live tours and revivals.
In 1965 Soo was signed by Motown Records (yes, that Motown Records) and recorded a pre-Stevie Wonder, ballad-y rendition of "For Once in My Life." Soo's version never got released, and the rest is history.
When the "Flower Drum Song" jobs dwindled, Soo and his family moved to Hollywood, and he managed to find steady work in an Asian role-light entertainment industry. Besides guest spots on television shows such as The Odd Couple, Hawaii 5-0, and M*A*S*H, he played the secondary male lead for a year on a Tony Franciosa-led show, Valentine's Day. His most memorable role was probably sardonic, jaded, and hysterically funny Detective Sergeant Nick Yemana on the sitcom Barney Miller, a character he played from 1975-1978. Soo raised deadpan to a fine art form.
Soo was diagnosed with esophageal cancer in the fall of 1978 and passed away from the disease on January 11, 1979, leaving behind his wife, three children, and thousands of saddened fans. In a very classy move, several months after his death the Barney Miller team paid tribute to him by running a retrospective episode about Soo and his character.
Soo's remains are interred in the idyllic-sounding Eternal Love section of Forest Lawn Memorial Park in Los Angeles, California.
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A Blast from the Passed
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This blend of Himalayan Crystal Salt with Kelp is a healthy solution for those of us who need to increase our iodine intake in our diet without using artificial salt.
We have selected the highest quality Kelp source for iodine: Laminaria Digitata from France is a unique kelp, providing a goodsource of iodine, minerals, proteins, vitamins, and omega 3.
Himalayan crystal salt was formed years ago when the Himalayan Mountains rose up out of the ocean which nearly covered the entire surface of the planet. It is the mineral deposit of that original sea.
Himalayan salt comes from a low-lying region situated in the first foothills of the Himalayas, over an area of approximately 200 square miles, in the northwest of Pakistan. The sea covering this region evaporated due to climatic changes. Powerful tectonic movements gave rise to the Himalayan range. The enormous pressure accompanying the buckling of the Earth's crust provoked the formation of cubic crystalline structures comparable to those of precious stones.
The deposits of salt from this ancient ocean are found high in the Himalayan range but also 1200 to 2000 feet below the surface .
Anything Else Like Himalayan Crystal Salt?
It is true that similar deposits of salt exist in many places in the world but Himalayan Salt is the only one formed under such circumstances. The salt found there is extremely pure and has always been sheltered from any traces of atmospheric pollution. It is extracted manually and packed on site without any treatment whatsoever.
In Europe, we owe its rediscovery to Prof. Peter Ferreira, a German biophysicist who carried out research on water. This research was directed to colloidal elements and cellular nutrition and prompted him to perform clinical tests on patients who were consuming a salt water solution prepared using Himalayan salt on a daily basis. Among the results, some indicated an evacuation of heavy metals, dissolution of calcium deposits, a general detoxifying effects of Quinton's plasma, which are supplemented thanks to a perfect crystal structure.
Prof. Peter Ferreira and his co-author, Dr. Barbara Hendel, published the results in a well-known work, translated into English with the title 'Water and Salt, The Essence of Life'. Published photographs comparing Himalayan salt with sea salt show the superior coherence of the Himalayan salt in an unbroken branched ensemble.
The extraordinary interest for Himalayan salt following this publication brought conflicts between the owners of the mines and varying theories about the origin of the salt deposits.
In Germany, Himalayan salt is also known as Hunza Salt in reference to Hunza Water, known for its colloidal minerals and mentioned by Prof. P Flanagan. The Hunza region is located in northwest Pakistan where there are no salt mines! This is the reason for the confusion between the various salts. Real Himalayan salt comes from the Himalayan Foothills as we have already seen above. The arguments and confusion matter little! What is important is to be able to use this marvelous salt, knowing its properties and the best way of using it.
The Best Available Salt on this Planet?
Himalayan crystal salt was well-known and used for centuries. The information content in the form of vital mineral elements locked away within these perfectly formed crystals of mineral-rich salt, were used by doctors for treating almost every disorder known to humans, and with unfailing success.
Because of its ionized colloidal structure, all its minerals and trace elements are totally absorbed by our bodies. Compared to ordinary sea salt under a microscope, we can see that its elements are perfectly linked to the crystal itself, while in ordinary sea salt they are dispersed without any links. This translucent crystal has a perfect geometrical structure.
Himalayan crystal salt is the most beneficial, cleanest salt available on this planet, where the energy of the sun has dried up the original, primitive ocean, many years ago. It is absolutely pristine and natural, identical to the ancient ocean. As in Quinton's Plasma and T.J. Clark's source in Utah, it contains all the elements found in our bodies.
The pink-coloured Himalayan crystal salt in Lumière de Sel has an ionized colloidal structure. This means that the over 80 minerals and trace elements are very easily absorbed by your digestive system. Easy absorption means your cells will be nourished with highly beneficial minerals as made by Mother Nature. Many people report improved wellbeing (including increased energy) after switching to Lumiére de Sel salt.
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What kind of minerals are in Lumiére de Sel® Himalayan Crystal Salt?
Lumière de Sel® Himalayan Crystal Salt delivers the entire mineral spectrum our body needs. The minerals in this crystal salt are present in a colloidal form. This means that minerals found in Lumière de Sel® Himalayan Crystal Salt such as Magnesium, Potassium, Calcium and Selenium as well as trace elements, are readily available to the body in the exact balanced proportion for optimum absorption.
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Unfortunately, refined table salt is basically an industrial waste product, with only two of the original elements remaining (sodium and chloride). It has no nutritional value to our body. In addition, most of today's sea salt is also refined and processed, which in turns removes essential minerals. Lumière de Sel® Himalayan Crystal Salt is uncontaminated by today's environment. It is very unique in its composition compared to all other type of salts: all its minerals and trace elements are interactive with each other, and easily metabolized by the body in a balanced way.
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We can safely assume that the majority of our customers are delighted with their experience with this salt, its beneficial properties as well as its delicious taste. We have selected the best product while using a traditional artisan approach. More valuable still: all of our products are intimately related to meetings, anecdotes and people whom we have had the good-fortune to meet.
Lumière de Sel works directly with its suppliers. Our products are certified and submitted to constant quality control.
We believe in open-mindedness, in curiosity, in the spirit of research and discovery, in commitment and stringent demand, but also in the alignment of thought with speech and action.
Explaining the structural and energetic difference between white salt, sea salt, and Himalayan crystal salt.
The Institute of Biophysical Research has conducted extensive studies, under the guidance of Peter Ferreira, to analyze the biophysical effects of salt on the human organism. The images below are some of the results of the studies.
After processing ordinary table salt, all minerals have been removed except sodium and chloride. The crystals are separated, dead and totally useless for the body. The unnatural crystals in processed white "table" salt are totally isolated from each other and dead. In order for the body to try to metabolize these crystals, it must sacrifice tremendous amounts of energy with very little results, resulting in a damaging loss and zero gain. The salt deposits in our bodies look similar to this photo, isolated and dead, when we consume processed white "table" salt.
Images and research conclusions are sourced from the book "Water & Salt, The Essence of Life"
Irregular shapes and separated crystalline structures are no longer connected.
Irregular and isolated crystalline structures disconnected from the natural elements surrounding them. Because of this, the vital minerals, however many it may contain, cannot be absorbed by the body unless the body expands tremendous energy to vitalize them. The net gain is small with an even greater loss of energy.
Shows finely formed crystalline structure that has no shadows or rough edges. Up to 84 elements of the crystal are interactive with each other. The balanced crystalline structure reveals fine branching, no shadows or rough edges. The crystal is not isolated from the inherent mineral elements (84) but is connected to them in a harmonious state. This tells us that the content, in the form of minerals, is balanced and can be easily metabolized by the body. This crystal is full of life. When taken as food, it will have a vital effect on the body. The result is only a net gain for the body with zero energy loss.
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**CONTENTS**
_Title Page_
_Epigraph_
_Acknowledgements_
_Introduction:_ A History of Histories?
PROLOGUE — Keeping Records and Making Accounts: Egypt and Babylon
PART I — Greece
1 Herodotus: The Great Invasion and the Historian's Task
2 Thucydides: The _Polis_ —the Use and Abuse of Power
3 The Greeks in Asia
PART II — Rome
4 Polybius: Universal History, Pragmatic History and the Rise of Rome
5 Sallust: A City for Sale
6 Livy: _From the Foundation of the City_
7 Civil War and the Road to Autocracy: Plutarch, Appian and Cassius Dio
8 Tacitus: "Men fit to be slaves"
9 A Provincial Perspective: Josephus on the Jewish Revolt
10 Ammianus Marcellinus: The Last Pagan Historian
11 General Characteristics of Ancient Historiography
PART III — Christendom
12 The Bible and History: The People of God
13 Eusebius: The Making of Orthodoxy and the Church Triumphant
14 Gregory of Tours: Kings, Bishops and Others
15 Bede: The English Church and the English People
PART IV — The Revival of Secular History
16 Annals, Chronicles and History
17 Crusader History and Chivalric History: Villehardouin and Froissart
18 From Civic Chronicle to Humanist History: Villani, Machiavelli and Guicciardini
PART V — Studying the Past
19 Antiquarianism, Legal History and the Discovery of Feudalism
20 Clarendon's _History of the Rebellion_ : The Wilfulness of Particular Men
21 Philosophic History
22 Revolutions: England and France
23 History as the Story of Freedom: Constitutional Liberty and Individual Autonomy
24 A New World: American Experiences
25 A Professional Consensus: The German Influence
26 The Twentieth Century
_Select Bibliography_
_A Note About the Author_
_Also by John Burrow_
_Copyright_
_Great thanks, laud, and honour ought to be given unto the clerks, poets, and historiographs that have written many noble books of wisedom of the lives, passions, and miracles of holy saints, of histories of noble and famous acts and faites, and of the chronicles since the beginning of the creation of the world unto this present time, by which we be daily informed and have knowledge of many things of whom we should not have known if they had not left to us their monuments written._
—WILLIAM CAXTON (1484)
**ACKNOWLEDGEMENTS**
My first debt is to Stuart Proffitt, for suggesting I write this book and for keeping his faith in it during a long period of gestation, as well as overcoming my hesitation about taking on such a comprehensive project. He has added to my debt and my gratitude by his guidance with drafts of early chapters and by his immensely thorough editing of the finished text, which produced many justified criticisms and fruitful suggestions, which have greatly improved it.
A number of friends have earned my warm thanks by reading some or all of the draft chapters. Their corrections have saved me from embarrassing errors, and their encouragement and advice have helped me keep my nerve when venturing into areas where I was a novice. My thanks are due to Stefan Collini, John Drury, Patrick Mullin, Mark Phillips, Larry Siedentop, Quentin Skinner, John Thompson, Frank Walbank, Patricia Williams, Donald Winch and David Wootton. For remaining errors of fact or judgement I alone am responsible.
Jane Wyatt has typed a long and in places inscrutable manuscript and borne with heroic patience and good humour my many changes of mind. I am once again deeply indebted to her.
I am very grateful to them all.
**A History of Histories?**
Why "A History of Histories," or, more explicitly, why not "The History of History"? History, even if we allow it to be in its broadest sense a single kind of activity, is nonetheless a highly diverse one. Plagues, invasions, emigrations; the foundation, working and development of constitutional arrangements and political systems; wars, external and civil, revolutions, changes in religion and culture, gradual or abrupt, the formation of various kinds of collective identity—confessional, national, ideological—providential history in the sense of the dealings of God with man: all these and much else are properly regarded as history. Some histories are virtually pure narrative; others are virtually pure, almost atemporal, analyses, being essentially structural or cultural surveys. History is contiguous with many other genres and lines of inquiry, from epic and myths of origin to various social sciences, and touching also biography, drama, political and moral polemic, ethnography, novels, inquests and judicial investigations. It was, so far as we know, Herodotus who first used the term _historia_ (inquiry) for what we call history. A _histor_ in Homer was someone who passed judgement based on the facts as a result of investigation, so the link between history and inquest is a very old one.
How can this variety be converted into a single historical narrative: "The History of History"? There is one answer which is obvious, being to some degree necessary to a narrative. This is by establishing a terminus, an end to which the episodes of the story are in some sense subordinate and contributing, so that they become moments in a progression. In the case of the history of historical writing—a genre which did not exist until the twentieth century—it was inevitable, given the period and its historiographical culture, that this was most commonly and easily done by taking the present state of the subject (or what it was assumed to be) as the terminus. By the early twentieth century this present state was characterized, variously but with a fair degree of consensus, as pure or "scientific" or (tacitly) as "professional" history, all identified perhaps with "the idea of history" or the study of the past "for its own sake." Professional history, in particular, was explicitly or simply by assumption associated with systematic archival research and the critical examination of sources, which had come to be thought of as constitutive of all serious history. Within this general consensus there could be differences of opinion, as for example between J. B. Bury and G. M. Trevelyan, over whether history was a "science" or an "art," and over how far, if at all, the historian in pursuit of his "science" should be concerned to establish laws (anathema to R. G. Collingwood in his classic book _The Idea of History,_ 1946). But despite these differences there was enough consensus to provide the basis for a selective grand narrative of the history of historiography, in which past historians were highlighted and assessed for their roles, necessarily partial, but helpful (or perhaps backsliding), in the general progression to the twentieth-century historian's views and approved practice. In this sense it was possible to write "The History of History."
I do not want to be understood as speaking simply in denigration of the assumptions underlying this possibility, as though of a past cultural episode. The central concerns—above all with history as truth-telling and, at least as an ideal, as free from bias—were already very old ones and, though shaken, are still in some sense with us, for those of us for whom a distinction between, say, history and imaginative fiction is still an important one. In this view Herodotus was taking an important step in distinguishing his own _Histories_ from the work of the poets, and Thucydides, though he may have judged unfairly, was invoking relevant criteria when he sneered by implication at Herodotus as belonging with authors less concerned to tell the truth than to entertain the public. Some distance between the search for historical understanding and merely emotionally or polemically effective writing is still part of the self-image and intentions of the historical profession. Of course, in the history of historiography zeal for truth has been a spectrum rather than an absolute—truth mattered, fairly obviously, more to Polybius than to Livy—but someone who wholly and perhaps wilfully falls off the negative end of the scale, like Geoffrey of Monmouth (this is not the place to argue the individual case), counts rather as a parodist or imitator of history.
All this may be true, but it remains true also that establishing a grand narrative of the history of historiography, by adopting a twentieth-century professional consensus as its terminus, was an impoverishingly narrowing strategy to adopt, eliminating or sidelining many interesting and potentially illuminating questions about approaches to historical writing, and indeed to the past as such, current in former times. There is, for example, the whole large and fascinating question of the surely very diverse motives for writing history. What did people in the past find interesting in _their_ past, and why did they? Which "pasts" did it lead them to focus attention on, as well as shaping how they chose to present them, and how and why did these change over time or how did different answers to these questions in a single period reflect and express differences within the culture? Why did new genres of historical writing emerge? Surely not only or invariably as a result of an extension of a pre-existing "scientific" curiosity, though that was sometimes a factor.
This book aims to provide answers to these questions. They have not been entirely neglected, and historians of historiography have been concerned to divide their subject in terms of genre as well as of method. But there is a balance to be redressed, as well as an allegiance to be proclaimed. I have tried to focus here on the question of the pasts that people have chosen for themselves and why, as well as on how they investigated and presented them. This may seem scarcely revolutionary, but like all chosen strategies it involves sacrifices. In particular, in grouping historians according to subject matter I have sometimes been cavalier with strict chronology—a lesson historians learned when they moved away from annals as the dominant form. So, for example, the "Alexander" historians are treated here as part of the story of the Greek encounters with the Persian empire, even though the historians whose work is still extant wrote much later, under the Roman empire. Perhaps most controversially, consideration of the Bible and its influence on historiography is postponed until its impact on the Gentile world in the early Christian centuries, rather than being placed, as chronology dictates, between Egyptian and Babylonian historians and Herodotus.
So, "A History of Histories" is intended to recognize the plurality of "histories" and the interests embodied in them, and to disclaim the ambition to construct a single grand narrative with the present as its terminus, which I see not only as implausibly prescriptive but also as narrowing the possibilities of exploration. There are, however, some exclusions which are intended also to be suggested in the title, with its implied retreat from comprehensiveness. No attempt has been made to deal with historiography outside the European cultural tradition (to which Egypt and Babylon are taken to contribute), notably Arabic and Chinese examples. Such exclusions are merely concessions to limitations of space, time and the author's knowledge. Another exclusion perhaps needs more apology, because it is at least in some measure arbitrary and the line of demarcation is a wavering one. I have taken the term "histories," though itself a generous one, as excluding biography and memoirs. In a book in danger of trying to include too much, I have thought this necessary, though the criteria are admittedly not always easy to apply: memoirs are clearly close kin to eyewitness history, and a "Life and Times" proclaims itself a mixed genre.
A word must be said about the treatment of the individual histories, which of course vary enormously in density and complexity, as well as in their accessibility to the modern reader. It is reasonable to assume that most readers of this book will not have read many or most of the historical works it discusses; indeed, that is part of the book's justification. One, primary, task therefore is to try to give a sense of the experience of reading these histories and what may be enjoyable about them. For many, perhaps most, historians, history has been a leisurely art, often requiring many volumes. It is not exclusively devoted to narrative, but narrative has long been at the core of it. It is therefore not enough just to convey the historian's intentions and views: some attempt must be made to convey not merely the structure of the narrative, but also its texture and qualities. In that respect, histories—which also often incorporate surveys, disquisitions, arguments and analyses—also resemble novels. I have made an attempt here, therefore, to give a sense, where appropriate, of what an eclectic, multi-layered, many-toned project a dense historical narrative may be. In attempting to render histories' special qualities I have not only resorted to a good deal of quotation but have attempted, sympathetically and with an awareness of the periods when they were written, to convey the literary qualities which form a large part of the experience of reading them. But these appraisals are also to be seen within, and perhaps as contributing to an overall understanding of, a more general context: the aims of historians in a particular period, the conventions which shaped their writing, and the ways these changed. I have also attended to historians' relations to the sources which made their work possible and partly conditioned it, and also briefly to the question of the particular writer's reliability. Awareness of this is part of an understanding of historians and therefore of the experience of reading them; history can never be, by definition, a purely literary endeavour. I have not, however, made any systematic attempt to focus on particular errors, for which, in any case, I should lack the necessary knowledge. That is the business of modern historians of the period, and they need no help from me. Such a checklist would, in any case, be intolerably tedious to read.
Historiography is not only a (wide) genre in itself, exhibiting continuities and revivals as well as shifting focuses of attention. It is also a part of Western culture as a whole—at times a highly influential and even central part—as well as being obviously a receptacle for the concerns of that culture and influenced by its fluctuations. European societies at different times and with varying emphases have attached immense importance to versions of their pasts and to notions of historical development, as well as plundering historical writings for legendary, heroic, tragic and pathetic motifs and topoi for poetry, drama and painting (in the eighteenth century, "historical" painting was regarded as the highest pictorial genre), and for exemplary, inspiring and minatory rhetoric. Ideas of history and of aspects of the past have intersected with and partly constituted ideas of religion, morals and politics. They have embodied authority, and provided means with which to challenge it. Above all, perhaps, they have provided focuses of allegiance, self-identification and "memory" for ethnic, national, religious, political, cultural and social collectivities and so helped to constitute them. Versions of the past have been offered, sometimes obliquely but often with visible anxiety, as diagnoses of contemporary predicaments or malaises.
We are accustomed to think of the intellectual history of Europe in terms of the history of philosophy, of science and religion, of art, literature and of ideas of social order and political authority. But the history of ideas about the past, as expressed in historical writing, and how the present stands in relation to it, is also part of that history; this book aims to contribute to an understanding of it. Among its chief components are conceptions of the distinctiveness of European civilization, contrasted chiefly with the empires of Asia; ideas of republican virtue, embodied in early Rome, supposedly corrupted by conquest and luxury; and the myth of Eternal Rome as mistress of the world, which became transmuted into the idea of a Christian empire. The Bible contributed ideas of collective transgression, punishment and redemption. From the sixteenth century onward we find the idea, largely derived from the Roman historian Tacitus, of an early freedom of the "Germanic" peoples, and of the existence of "ancient constitutions," with a continuing authority in the present, which European countries allegedly derive from their invasion by the "Gothic" barbarians. Eighteenth-century historical writing incorporated concepts of the progress of "civil society," chiefly in association with commerce, and of the end of "feudal anarchy" (or, in Marxist terms, the supersession of the feudal nobility by the hegemony of the bourgeoisie). The nineteenth century was the great age of the preoccupation with national identity, in association with ideas of national liberation and the creation of the nation state as the normal political form. This has passed on into the modern aspiration to give a voice to suppressed minorities. History, in other words, to name only the most prominent influences, has been republican, Christian, constitutionalist, sociological, Romantic, liberal, Marxist and nationalist. All these have left residues in subsequent historical writing; none at the moment dominates it.
I have therefore made a conscious effort not to treat the history of historiography in isolation, but to be aware of its place in the wider culture, of the cultural and political influences playing on it, and of the ways in which it fostered, transformed and transmitted them. "A History of Histories" can and should be more than a record of the achievements, strengths and weaknesses of historians and the schools and traditions to which they belonged. It is itself a historical enterprise, one of the ways in which we attempt to understand the past.
**Keeping Records and Making Accounts: Egypt and Babylon**
History—the elaborated, secular, prose narrative (all these qualifications are necessary) of public events, based on inquiry—was born, we can claim with confidence, in Greece between roughly 450 and 430 BC. If we want to add Thucydides' very different kind of history to that of Herodotus, who is sometimes spoken of as "the father of history," then we must speak instead of the second half of the fifth century BC. Even with this extension, and with the qualifications built into the description of the genre, it is extraordinary that we can speak of so short a period for its abrupt genesis, yet it appears to be justified. It is equally astonishing that we can plausibly claim that neither historian was to be excelled for over two thousand years subsequently—until, in fact, changes in methods and types of history begin to make comparison unrealistic.
To see what is meant by claiming that Herodotus and Thucydides were, so far as we know, the first historians, we need to recognize some basic distinctions which separate their work from examples of what we may call perhaps "proto-history" in the ancient civilizations of Egypt and Babylon. Herodotus himself paid tribute to the Egyptians for their preservation of knowledge of the past: "by their practice of keeping records of the past, [they] have made themselves much the most learned of any nation of which I have had experience" ( _Histories,_ II.77). In fact he seems to have believed too readily what he was told in the temples of Egypt when conducting the historical inquiries he incorporated into the panoramic survey of the known world with which he prefaced his account of the great Persian invasion of Greece in the early fifth century BC. His own references to Egyptian history are notably garbled, in contrast to what he has to say of the civilization of Babylon, on which he is much more reliable. Yet, whatever it may have pleased the Egyptian temple servants to tell him—it is not clear that his acquaintance ranged far up the priestly hierarchy—Herodotus' compliment to the Egyptians was not misplaced. Modern Egyptologists can know much more than Herodotus about ancient Egypt precisely because so much of it, thanks to the early establishment of a centralized bureaucratic state and the use of durable materials for inscriptions, has been preserved. To this we may add the effects of a dry climate and a traditional habit of mind: the records so preserved go back more than two thousand years before Herodotus' own time, the mid fifth century BC. The Egyptians were indeed then the world's premier record-keepers. The distinction between historiography and recording—between the sense in which Herodotus was, so far as we know, the first historian and the learning of his Egyptian informants—is therefore one to give us pause. It is valid enough, though of course, like all such distinctions, it becomes a little less rigid as we examine it.
Record-keeping is, in origin, commercial and bureaucratic good practice; it is not an art. Many of the factors which have preserved so much of the Egyptian past existed also in the ancient civilization of Mesopotamia, with its records inscribed on stones and clay tablets and, for most of the greatest matters, on the walls of temples, tombs and palaces. Every modern historian understands what we are speaking of here—namely archives—and regards them as in some form indispensable, as Herodotus, who worked by interrogation of "learned men," did not. The inscriptions were intended as records from the beginning: their endurance was deliberate, as that of casually collected documents is often not. It is only the humbler artefacts, such as inscribed clay tablets, which have survived inadvertently. Inscriptions being essentially records, this produces a kinship between their authors and Herodotus, who says, in his initial statement of intention at the outset of his _Histories,_ that he wrote to preserve the memory of great deeds (below, Part 1).
The key difference, of course, is the word that Herodotus used to describe his work: _historia,_ inquiry. His method of acquiring the information for his _Histories_ was chiefly interrogation. When he questioned the Egyptian temple servants and guardians, he, who seems to have known only Greek, was at a remove from the documents in a way that a modern historian would not care for. But, for all the deficiencies or secretiveness of his informants, we recognize something like an intelligible relationship: it is that between historian and archivist. It was he who, in the service of a systematic inquiry, history, was interrogating them; not vice versa. Similarly, when he interrogated "overseas" Greeks and perhaps "native informants" for knowledge of other parts of the world, as he must have done, it was he who was the anthropologist or ethnographer. There was, to state the obvious, no Scythian ethnographer; those whom the Greeks called Scythians, of whose customs Herodotus wrote an extended account, were illiterate nomads living north of the Black Sea. We have therefore argued ourselves into the position of saying that, as far as we know, there were no Egyptian or Babylonian historians earlier than Herodotus. The Egyptians were, as Herodotus says, learned; respect for Egyptian wisdom was high and even exaggerated in Greece, where, for example, the Greek gods, under other names, were thought to have originated in Egypt. But the Egyptians were recorders, not historians.
So far so simple. As we shall see in Chapter 1, Herodotus' notion of systematic inquiry was not entirely unique in Greece in his time. The inquiries closest to his own seem to have been mainly geographical (including of course "human geography"), a concern which is very evident also in Herodotus' work. But inquiry, systematic research, is not the only characteristic of historiography. Another is the rendering of the results of the inquiry into connected historical prose: narrative. There is in fact a route in the ancient world from recording to more or less extended historical narrative which blurs a little the distinction between recording and history that seems so firm when we attend only to the element of research.
The earliest writing seems, understandably, to have been concerned with practical transactions, and can be thought of either as part of the transaction or just as a subsequent record. Early public inscriptions, in the grandest contexts, recorded on the walls of temples, seem also to have taken on this transactional character, as the rendering of an account by the ruler of his stewardship on behalf of the god: his buildings erected, his gifts, and the toils and achievements, including victories, by which they were procured. Inventories are common. Other kinds of lists include the earliest materials for systematic chronology—king lists, for example—and are therefore crucial to the possibility of reliable large-scale history. Laws too, like the early so-called law codes of ancient Mesopotamia, are also essentially lists, in a way familiar, though much later, from the books of Leviticus and Numbers in the Old Testament from the end of the second millennium BC.
The connection between narrative and divine stewardship seems to lie in explanations. That is, to render an account, initially in the form of a list, can come to involve an explanation, which in turn may take the form of a more or less elaborated narrative. Historians are accustomed to insisting, perhaps rather combatively since it is their stock-in-trade, that an explanation may take the form of a narrative. They may be less keen on claiming an ancient kinship with accountants, and through them to the earliest uses of writing: making lists. The conceptual overlap here, however, is still caught in the related meanings, indicated by prepositions, of the English word "account," which itself recalls its origin in the act of counting. It makes good sense if bad style, with each repetition making a different point, to say, "Please account for the errors in the accounts you have presented to the board by giving your account of how they were compiled."
Narrative inscriptions which were also explanations and part of the rendering of accounts can be seen growing fuller and more circumstantial and far more "human" in their references to motives and action. Campaign narratives—with their natural climax in victories (or pretended victories) and hence conquests, subjection or capture of alien peoples, and the acquisition of spoils—are among the earliest extended narratives, and are also, of course, characteristically narratives of expeditions, like that of Xerxes in Herodotus, and, nearly two centuries later, the greatest of all campaign expeditions, that of Alexander, who took historians with him. This is an Egyptian account from the middle of the second millennium BC of a victory and its trophies:
His Majesty proceeded on his chariot to Kashabu, alone and without a companion, and returned thence in a short time bringing sixteen living Maryannu at the side of his chariot, twenty hands at the foreheads of his horses and sixty cattle driven in front of him. Submission was made to His Majesty by this town. Now as His Majesty was going south in the plain of Sharon, he found a messenger of the prince of Nahrin carrying a clay tablet at his neck, and took him as a living prisoner at the side of his chariot...Arrival of His Majesty at Memphis with joyful heart like a victorious bull. Amount of plunder...(Gardiner, _Egypt and the Pharaohs,_ p. 197)
Then follows an itemization, with numbers, of slaves, horses, chariots, weapons and musical instruments. Expeditions were also apparently conducted by the pharaohs in the quest for valuables such as minerals.
Another Egyptian form of narrative pattern is the onset of a period of confusion and disasters, brought to an end by the advent of a ruler who restores order. These narratives of deliverance—whose archetypal protagonist later, under Hebrew influence, became the Messiah figure—have had, like the campaign narrative, a long history in Western historiography, influencing the representation of such figures as the emperors Augustus and Constantine and, in English history, Queen Elizabeth I and William III. Thus some of the archetypal patterns of historical narrative were established as early as the inscriptions recording the deeds of the rulers of ancient Egypt, Mesopotamia and Asia Minor. In the Bible, the books of Exodus and Deuteronomy are essentially expedition and campaign narrative, which features also in the work of some of the greatest later historians.
Another way in which the rendering of accounts, in the form of vindication, elaborates itself into narrative seems to have been cultivated particularly among the Hittites. That is, a record was made to establish the justice of what had been done, and to refer the matter to divine arbitration. A leading authority on the Hittite empire, O. R. Gurney, describes a document of this kind as showing a highly developed political conscience. The more legalistic form of this kind of record-keeping, and the provision of a narrative on which judgement was to be based, concerned, naturally, the making and breaking of treaties. Treaties are the only documents to be transcribed at length by Thucydides, much of whose opening book is concerned with the rupturing of treaties which constituted the opening of the Peloponnesian War. The Hittites, a millennium earlier, had begun the practice of making the preambles to treaties into occasions for brief historical narratives explaining the treaties' origins; eventually the prefatory narratives became detached from the treaties and decrees and became free-standing annals, the chronicle being a presentation of the ruler's actions as an offering to the god.
The annalistic form of recording is found also among the Assyrians, as well as in early "historians" in Greek city states, along with accounts of supposed mythological origins. It is not employed by Herodotus—a work on such a large scale would have made it unusable—but it shapes Thucydides' history of the Peloponnesian War, with a division of each year into summer and winter for greater precision. It was to remain a fundamental historiographical form to the end of the Middle Ages.
Mention of annals brings to the surface a troublesome issue for early historiography: the need to find widely recognizable ways of marking chronology. This was less of a problem in centralized dynastic states like Egypt and Babylon—which is not to say that early king lists were always as helpfully drawn up as they might have been: attempting to eliminate all trace of a discredited ruler was a common Egyptian practice, while the lengths ascribed to reigns are clearly absurd. But in Greece, with no central political authority or record-keeping, the problem was acute. The only pan-Greek institutions were the oracle at Delphi and the four-yearly Olympic Games, with their famous winning athletes, and the Olympiads were eventually used as chronological markers. Thucydides dated events from the onset of the war he was recounting. For Athens it was possible to use the periods of office of the archons, the chief officials of the _polis,_ who held their posts for a year, though they may not have been easy to recall. The Romans from earliest times kept lists of the pontiffs, the chief priests, which served the same purposes as the names of the rulers of Egypt and Babylon. Some idea of the difficulties of establishing a common chronology can be seen in an example of Thucydides overloading:
The thirty years' truce which was entered into after the reconquest of Euboea lasted for fourteen years. In the fifteenth year, the forty-eighth year of the priestess-ship of Chrysis at Argos, the year when Aenesias was ephor at Sparta, and two months before the end of the archonship of Pythodorus at Athens, six months after the battle at Potidaea, just at the beginning of spring, a Theban force...(II.2)
One other form of proto-historiography which calls for attention is the counterpart, of wider application, of dynastic lists, namely the construction of genealogies, often with mythic alleged origins, with which the very early Greek city-state historians also seem to have concerned themselves. This interest in origins leaves occasional marks in Herodotus' _Histories,_ though as usual he does not always believe what he hears. Self-extolling histories of the great clans seem to have provided materials for the early history of Rome and to have been valuable sources despite their inevitably self-serving character. A particular cachet attached to descent from a god, Heracles being a favourite, or from a hero of the Trojan War. The Romans managed both, by being descended from Aeneas of Troy, who was the son of Venus. A well-attested genealogy could be useful or even indispensable, as we see in the Old Testament Book of Nehemiah, where it carried qualification for ritual office, and a flaw constituted disqualification (Nehemiah 7:5, 64). Alexander believed, or wished people to believe, that he was descended from the god Ammon.
Secular historical prose narratives (setting aside both epic poetry and the Bible, the latter being left for consideration later with its impact on Christian historiography) in the ancient empires were emerging gradually from more basic forms of bureaucratic record-keeping, though it is still a long step—a leap, in fact—from these to the richly humane and artfully controlled extended narratives of Herodotus. There are campaign and expedition narratives, and what may be called redemption and vindication narratives, accounts of disasters and rebellions and subsequent restorations, and of breaches of trust suffered and avenged. There were also the beginnings, among the Hittites, of annals as a form of recording.
The most elaborate, circumstantial and continuous of the early campaign narratives, though they exerted no direct influence on later ones, can at least claim a kind of kinship with them. One such is the account of the pharaoh Tuthmoses III (1490–1436 BC) in the campaign which culminated, in the twenty-third year of his reign, in the battle of Megiddo, recorded on the walls of his additions to the Temple of Amon-Re at Karnak. It contains an account of a council of war, with dialogue, and what has been called the earliest full description of a decisive battle. Before the battle, a debate arises over which route to take. The counsellors are notably sycophantic, but clearly have their own ideas when invited to express their opinions: "How can one go upon this road which is so narrow? It is reported that the enemy stand outside and have become numerous. Will not horse have to go behind horse, and soldiers and people likewise? Shall our own vanguard be fighting, while the rear stands here...?" The temple walls also exhibit, as was common, lists of defeated peoples, together with depictions. The narrative can also be checked against other accounts. Here then, as in comparable if less full or less vivid accounts, we have something which is certainly narrative, but which we may hesitate whether to call historical or proto-historical. Although fuller and less formulaically bombastic than other examples, it is an account of an episode, though a highly important one. It does not exhibit individualized character depiction, a sense of historical perspective (in this respect being rather like the two-dimensional side-face Egyptian conventions of representation) or the surrounding architecture of a large-scale historical design. Much the same could be said, of course, of many chronicles over the next two thousand years. It could not be said of Herodotus.
**PART I**
> **GREECE**
**ONE**
**Herodotus: The Great Invasion and the Historian's Task**
As was to become customary, at the beginning of his work Herodotus tells us why he wrote it. It was, he says, "so that human achievements may not become forgotten in time, and great and marvellous deeds—some displayed by Greeks, some by barbarians—may not be without their glory; and especially to show why the two peoples fought with each other." In other words his history was a monument, a marker set down against the oblivion with which time threatens all human deeds. He was successful beyond all reasonable expectation. We are still reading his account of his great theme, the invasion of Greece two and a half thousand years ago, and a mere half century before he wrote it, by the Persian Great King and the immense polyglot army drawn from all parts of his empire. Herodotus also promises a little later ( _Histories,_ I.95) to tell us how the Persians under their ruler Cyrus (the Great) won their predominant position in Asia, and this promise too he fulfils before going on to his account of the invasion of Greece.
One point in his initial statement which is worth pausing on is the reference to recording the great deeds of barbarians (i.e. non-Greeks) as well as Greeks. We should look in vain in the Egyptian and Babylonian records for such even-handedness. What we are reminded of is Homer, who, as Herodotus soon reminds us, had written of an earlier conflict between Greeks and an Asiatic people. Homer allows his readers/hearers to sympathize with Trojans as well as Greeks, and as much or more with Priam and Hector as with Achilles and Agamemnon. Herodotus does not comment on this feature in Homer, but seems to take it for granted. He accepts, of course, the historicity of the Trojan War, though he thinks that Homer, as a poet, shaped his narrative to his epic purpose, and he is willing to correct him from his own inquiries among the Persians and Egyptians and by his own common sense: Helen could not have been in Troy during the siege, because the Trojans would have handed her back if they could (II.120). Herodotus has a pretty good idea when Homer lived, placing it some four hundred years before his own time, which is the mid fifth century BC.
But far more important than Herodotus' acceptance of the basic historicity of the Homeric poems is their existence, for all Greeks, as a narrative model. When Herodotus in his preamble speaks of writing to preserve great and marvellous deeds from oblivion and giving them their deserved glory, he can hardly be unaware of stretching out a hand to the Homeric epic, which purports to do precisely that. Herodotus' narrative of the great conflict sometimes carries Homeric echoes which we shall have to consider, but more generally the pacing of the narrative, the immediacy of its re-enactments of events and presentation of character, its humanity and its inclusion of the earthy and mundane—more than in Thucydides and historiography subsequent to him—all invite the adjective "Homeric." It is, however, Homeric on a vast scale, and therefore looser and deliberately digressive, as well as based on painstaking inquiries, sometimes requiring suspension of judgement, all of which is alien to the epic tradition. Herodotus is a garrulous, highly personal and conversational writer, with no aversion to the first person; one meets him face to face, as it were, so that it is not difficult to imagine the readings he gave in Athens by which his work was apparently first made public. We know his opinions, and hear of his travels, the wonders he has seen, the stories told to him, and his not infrequent scepticism about them. We can even reconstruct a good deal of his religious views, though here he is sometimes reticent. He is almost as personal a writer as Montaigne.
He was born, apparently around the mid-480s BC, in the Greek colony of Halicarnassus on the eastern side of the Aegean, so he belonged to that part of the Greek world transplanted to Asia. As the disputed borderland of Greece and the Persian empire, this area was to play a significant part in his history. Since the territory had recently been incorporated in the Persian empire, Herodotus was born technically a subject of the Great King. Though there is ultimately no doubt where his sympathies lie, and it seems he could speak only Greek, he never speaks of the Persians with contempt and has no difficulty in making his narrative identify with them as required. Though he travelled extensively—the extent has been questioned by some—and later apparently migrated to Athens, where he is said to have been a friend of the tragic dramatist Sophocles, it is surely appropriate that a man of such cosmopolitanism should have been born not only in an area which had seen Greek intellectual life hitherto most vigorously flourishing but at the interface of two great cultures, and pretty much at the centre of the known world. The date of his death is not certain, but it is clear that he lived into the period of the Peloponnesian War, i.e. at least beyond 430 BC. He is therefore, according to the best estimates, a generation earlier than Thucydides, though a contemporary, measured by the dates of their respective births and deaths.
Herodotus lists some early instances of friction between Europe and Asia—mythic or legendary—including the voyage of Jason and his Argonauts to Colchis, on the Black Sea, and the theft of the Golden Fleece. Then he rapidly advances to historical times, with the Persian conquest of the Hellenized kingdom of Lydia, in what is now western Turkey, under its king Croesus. Croesus, who plays a large part in Book I (the division into nine books is not original), is the first historical character to appear. Overthrown as a ruler, so that his career vindicates the wise saying of the Athenian Solon that no man can be called happy till he is dead, Croesus, made wise by his reversal of fortune, becomes the counsellor of his conqueror, the Persian king Cyrus. We are also told of the legendary youth of Cyrus, who was saved by a shepherd from exposure as a baby (I.108–17), and of the supplanting of the ruling Medes by the Persians under him. Cyrus then goes on to overthrow Babylon (539 BC), of whose customs Herodotus provides a description(I.192–200), after giving the reader a kind of conducted tour of the city.
The great wall I have described is the chief armour of the city; but there is a second one within it, hardly less strong though narrower. There is a fortress in the middle of each half of the city [i.e. as divided by the river]: in one the royal palace surrounded by a wall of great strength, in the other the temple of Bel, the Babylonian Zeus. The temple is a square building, two furlongs each way, with bronze gates, and was still in existence in my time; it has a solid central tower, one furlong square, with a second erected on top of it and then a third, and so on up to eight. All eight towers can be climbed by a spiral way running round the outside, and about half-way up there are seats and a shelter for those who make the ascent to rest on. On the summit of the topmost tower stands a great temple with a fine large couch in it, richly covered, and a golden table beside it. The shrine contains no image and no one spends the night there except, as the Chaldaeans who are the priests of Bel say, one Assyrian woman, all alone, whoever it may be that the god has chosen. The Chaldaeans also say—though I do not believe them—that the god enters the temple in person and takes his rest upon the bed. There is a similar story told by the Egyptians at Thebes...(I.178–86)
When Herodotus visited Babylon it was approximately a century since the city had fallen to Cyrus.
Herodotus' next three books deal with the further expansion of the Persian empire into Asia under Cyrus' son Cambyses and his successor Darius. It is Darius who makes the first incursion into Europe, his army being turned back by the Athenian victory at Marathon (490 BC)(VI.110–17). But overall the advance of the Persian empire seems unstoppable as, after swallowing the Greek colonies and the Hellenized kingdoms of western Asia, it advances to the limits of the known world. It comes to include not only the ancient civilizations of Egypt and Mesopotamia but the barely explored territories of Libya and Ethiopia and the nomads of the Arabian desert and the northern steppes. The effect is similar to that achieved by Edward Gibbon more than two thousand years later in his _Decline and Fall of the Roman Empire,_ as he successively introduces the peoples who will overwhelm the Roman world.
Herodotus, in Books II to IV, follows the tide of Persian conquest by giving geographical and ethnographic surveys of the conquered lands and peoples. These surveys make up a substantial part of his work, and we shall need to return to them later. As a preamble to the great invasion of Greece under the Persian king Xerxes, the Persian world domination, as it seems, is made visible in the great muster of the composite army that Xerxes assembles, which Herodotus describes in detail, identifying each people in the review, with its distinctive appearance, clothing and weapons, beginning with the Persians themselves. The descriptions, and that of the accompanying fleet (VII.61–100), are too long for anything but a representative excerpt:
The Assyrians were equipped with bronze helmets made in a complicated barbarian way which is hard to describe, shields, spears, daggers (like the Egyptian ones), wooden clubs studded with iron, and linen corslets...The Indians were dressed in cotton; they carried cane bows and cane arrows tipped with iron, and marched under the command of Pharnazathres, the son of Artabates...Then there were the Caspians and the Sarangians, the former commanded by Ariomardus, the brother of Artyphius, dressed in leather jackets and armed with the _acinaces_ [Persian short sword] and the cane bow of their country...
And so it goes on, with the Arabians, the Ethiopians ("in their leopard skins and lion skins"), the Libyans, the Phrygians, the Thracians in fox-skin headdresses—the list is apparently endless. It seems as though the whole known world is gathering, hundreds of thousands strong (the numbers have predictably been much debated), to crush the small city states of Greece—from the Nile and the Libyan desert, from the rivers of what was later to be European Russia, and from Thrace west of the Black Sea to India and beyond, as well as from Persia itself. A local man, seeing the Persian host, cries out in his anguish that Zeus has changed his name to Xerxes (VII.56).
Herodotus is in no doubt that Athens was the core of Greek resistance and suffered most, being directly in the path of the invaders. The most crucial policy decision on the Greek side was that of the Athenians not to defend the city but to retreat across the isthmus to the Peloponnese and rely on their fleet to defeat the Persians. The determining factor, in Herodotus' account, was Themistocles' interpretation of the Delphic oracle, whose prophetess had declared, enigmatically as usual, that "the wooden wall only shall not fall." Some thought this referred to a thorn hedge which surrounded the Acropolis, but Themistocles spoke out for those who interpreted the wooden wall as a reference to the Athenian ships, and was believed. Athens took the lead in the great sea battle of Salamis (480 BC), in which the Persians lost their fleet and which was a severe check for them (VIII.78–96).
But the most memorable episode of the invasion, given full treatment by Herodotus, was a defeat for Greece: the sacrificial battle fought earlier the same year by the three hundred Spartans under their king Leonidas at the pass of Thermopylae, in which most were killed(VII.210–28). Spartans were forbidden by law to retreat. The column subsequently erected to their memory bore what is, in its terse simplicity, still probably, even in translation, the most moving of all military memorial inscriptions:
GO TELL THE SPARTANS, STRANGER, THAT HERE OBEDIENT TO THEIR LAWS WE LIE.
Herodotus says that he has taken the trouble to learn the names of all three hundred "who deserve to be remembered." It is characteristic that he mentions some of the Persian dead by name, and their ancestry. It is from Herodotus too, though it may not originate with him, that we have, put into the mouth of Croesus, one of the most famous epigrams on war: that in peace sons bury their fathers, but in war fathers bury their sons.
Herodotus' portrait of Xerxes, who deals atrociously with Leonidas' corpse, contains odd contradictions, perhaps reflecting different traditions. In some moods Xerxes shows good sense and magnanimity, in others a wilful savagery. He commits a sin of hubris, Herodotus strongly implies, when he causes the Hellespont to be whipped as the punishment for destroying, in a storm, the bridges he had had built:
Xerxes was very angry when he heard of the disaster, and gave orders that the Hellespont should receive three hundred lashes and have a pair of fetters thrown into it. I have heard before now that he also sent people to brand it with hot irons. He certainly instructed the men with the whips to utter, as they wielded them, the barbarous and presumptuous words: "You salt and bitter stream, your master lays this punishment upon you for injuring him, who never injured you. But Xerxes the King will cross you, with or without your permission..." In addition to punishing the Hellespont Xerxes gave orders that the men responsible for building the bridges should have their heads cut off. (VII.35)
Herodotus then goes on to give a technical description of how the bridges were rebuilt.
Xerxes' most atrocious act occurs when a servant to whom he owed much pleaded that the eldest of his five sons should be left at home: Xerxes has the son cut in half for the army to march between the two halves on its route. Yet immediately afterwards we have the episode in which he appears most sympathetic and human, when sitting on a throne of white marble he is able to view his whole army and fleet and suddenly, at the moment of his highest glory, he weeps. "And when he saw the whole Hellespont hidden by ships, and all the beaches and plains of Abydos filled with men, he called himself happy—and the moment after burst into tears." Asked why, he replies that he was thinking of the shortness of human life, and that "of all these thousands of men not one will be alive in a hundred years' time" (VII.45–6). It is an extraordinary moment, at which, thanks to Xerxes, or Herodotus, the political distinctions between peoples and even, for us, the gulf between ancient and modern melt away in the contemplation of a common human lot.
But though the Persians are never dehumanized, the differences between them and the Greeks—political and moral—make up a powerful message which Herodotus was to convey to posterity and which many generations were to draw on copiously. From Book V onward Herodotus turns for much of the time to the affairs of the Greek city states and the relations between them. Books V and VI, in fact, are probably the most confusing and least satisfying for the reader, lacking as they do both the exotic folkloric charm of the earlier ones and the dramatic single narrative line provided later by the great invasion by Xerxes. Greek factionalism, moreover, is more untidy and multicentred than the simpler politics of Persian autocracy. It is, however, in these books that the theme emerges which gives the climax of the work its wider political significance. The Athenians had, with Spartan help, liberated themselves from the family of tyrants which had ruled them, and under their leader Cleisthenes had established a democracy(I.59–64, V.62–9). Athens became for Herodotus the great protagonist of Greek freedom in opposition to eastern despotism. This contrast—which Herodotus increasingly makes apparent, and in which the other Greek states, and particularly Sparta of course, participate in varying degrees—was to be an immensely enduring one in Western historiography and political thought, setting liberty against servitude, law against the tyrants' will, frugality, hardihood and valour against luxury and timidity. In describing the effects on the Athenians of their recently acquired liberty, Herodotus propounds an idea that was to resonate down the centuries in historiography; it was applied also to the early Roman republic.
Thus Athens went from strength to strength, and proved, if proof were needed, how noble a thing equality before the law is, not in one respect only, but in all; for while they were oppressed under tyrants, they had no better success in war than any of their neighbours, yet once the yoke was flung off, they proved the finest fighters in the world. (V.78)
Another aspect of the East–West contrast, with a long future as a historiographical cliché, is attributed to Cyrus the Great, and quoted by Herodotus as almost the last words of the whole work: "Soft countries breed soft men," and have to suffer the rule of aliens. Warned by Cyrus, the Persians choose for preference to live in a rugged land, but the association in European thought and historiography conveyed by the phrase "Asiatic softness" was to endure down to the nineteenth century. The East–West antithesis was to be highly significant for the Greeks and Romans. Through them it reached a particular pitch of intensity in the European Enlightenment, and it still echoes resonantly in nineteenth-century historiography and the literature of imperialism, and in this long tradition Herodotus is by no means the most biased and unqualified manipulator of it.
The difference in character of the last books from the earlier, more ethnographic, ones is mainly obvious in the greater coherence and dramatic force of the narrative of the invasion. But it also makes a difference that, though we still spend much time with the Persian King, we also spend much more than before on the affairs of Greece, and the contrast is inevitably marked. Instead of the claustrophobic, sycophantic, sometimes fearful atmosphere of a despotic court, where such discussion as there is consists of advice, which can be given privately as well as in conclave, we have the overt, vigorous, fiercely factional and disputatious public life of the Greek city states, conducted characteristically in public debate and expressed through speeches designed to sway opinion. Herodotus has used direct speech from the beginning, but its use is largely informal, conversational and designed to further immediate action; it is not oratory; it is easy to think of the more conspiratorial words being spoken in whispers. Consider, for example, the way Herodotus describes the conspiracy of Persian nobles who, suspecting their ruler, Smerdis the Magus, who is passing himself off as the son of Cyrus the Great, is an impostor, agree to kill him. Herodotus makes extensive use of direct speech. The future king Darius, who is elevated to the throne as a result of their success (in 521 BC), speaks his mind:
"Listen, all of you," Darius replied, "if you take Otanes' advice [to recruit more conspirators]...somebody is sure to seek advantage for himself by betraying us to the Magus. You ought really to have done this thing entirely on your own; but as you have seen fit to let others in on it, and have communicated your intentions to me, I have only one thing to say—let us act immediately. If we let a single day slip by, I promise you one thing: nobody will have the time to betray _me_ —for I will denounce you all to the Magus."(III.71)
It is only later that we come to the lengthier set speeches which seem to make up so much of Greek public life. This is one of the Athenian commanders, in a debate over whether to risk a battle against the Persians (which turned out to be the victory of Marathon), also urging immediate action. He invokes principle as well as practicality. The alternatives, he says, are
either to enslave Athens, or to make her free...Never in our history have we Athenians been in such peril as now. If we submit to the Persians, Hippias [the expelled tyrant] will be restored to power—and there is little doubt what misery must then ensue: but if we fight and win, then this city of ours may well grow to pre-eminence among all the cities of Greece. (VI.109)
It is tempting to speak proleptically and say that we are approaching the kind of historiography of which Thucydides' work is the archetype and was to set a pattern for historical writing for a thousand years. It is not quite that. The Greek speeches in Herodotus are more direct and less carefully wrought, cerebral and ostensibly dispassionate than the remarkable ones which constitute so much of the experience of reading Thucydides. Nevertheless, they represent a marked contrast with the earlier books, and help also to enforce the contrast which gives the last three their overall meaning: the contrast of two worlds, of freedom and despotism.
This contrast is part of the reason for the cultural shift in Herodotus' work, but it is reasonable to infer that there is another, less obvious, one. The events of the latest books are also the nearest to Herodotus' own time, and they take place in the world he lives in and knows best and whose language he speaks—the Greek world. There must have been a marked difference in the quality and reliability and accessibility of the oral traditions on which Herodotus chiefly relied, and in his ability to interpret them. Thucydides was to enjoy all these advantages and that of writing contemporary history. The result is a perceptible access of "realism." The often folkloric character of the stories Herodotus tells about Persia, especially in the earlier books, and the frequent references to prophetic dreams have fallen away. Of course the contrast is not absolute. The Greek world too has plenty of room for dreams, oracles and omens. Nevertheless, the contrast seems at once one between East and West, between despotism and freedom, and also between ancient and modern, where "modern" means the open political life of the Greek _polis._
But, though Herodotus' feet were planted firmly in the world of Greek culture and rational inquiry, his curiosity about ancient civilizations and remote and nomadic peoples means that his history also constitutes a kind of bridge between that world and the "barbarian" world beyond. Apart from the narrative of the invasion, which is far more dramatically controlled, circumstantial and balanced than any prose work before it, the other distinctive quality of Herodotus' work is embodied in the extensive geographical and ethnographic surveys of the earlier books—digressions, as they came later to be called as they became common but generally less extensive in historical works. In Herodotus they are highly entertaining and readable, made so by his omnivorous humane and tolerant curiosity about the world and about humanity in all its aspects and variety. Herodotus was an anthropologist and geographer as well as a historian, but it is less easy to be sure how completely original he is in these sections of his work.
We are largely ignorant of Herodotus' Greek predecessors and contemporaries as writers or compilers of historical accounts and materials, but it is clear that there were some. Most of their work has been lost, but what we know of them, and the existing fragments, makes it unlikely that anything like another Herodotus lies beyond the horizon of our knowledge. The best-known and probably most noteworthy precursor, Hecataeus of Miletus (b. 549 BC), was a geographer, mythographer and ethnographer of the Mediterranean and the countries around it. He wrote surveys of Europe, Asia, North Africa (called by the Greeks Libya) and Ethiopia. The first map seems to have been the work of another Milesian, Anaximander. Hecataeus' knowledge certainly extended further west than that of Herodotus, who disclaims acquaintance with the western Mediterranean and the lands to the north, including the Tin Islands, i.e. Britain ( _Histories,_ III.115). Herodotus refers to Hecataeus several times in his work, sometimes as a historical agent ("Hecataeus the writer"), though he also makes a point of rejecting, decisively, Hecataeus' view—which became common—that the world was entirely surrounded by a great ocean, referred to simply as "Ocean" (IV.36).
Our chief source for Herodotus' predecessors is the first-century author Dionysius of Halicarnassus, whose birthplace was also that ascribed to Herodotus. In him, and also in Plutarch, we have accounts of early events, from sources now lost, which sometimes match but sometimes differ from those of Herodotus. These precursors and perhaps contemporaries appear to have written accounts of peoples and cities, usually about their alleged origins. Another early writer known to us from excerpts in Dionysius is Hellanicus of Lesbos, who wrote an early history of Athens, mostly mythical, and also on the origins of peoples more generally—a kind of early universal history—and on the customs of the Egyptians, Persians and Babylonians. Whatever differences there may be in the quality of achievement, Herodotus' interest in such matters was therefore not held in isolation. Hellanicus seems to have been above all a genealogist, and included hopeful etymologies deriving the Persians from the Greek legendary hero Perseus, and the Medes from Medea, the bride of Jason. Altogether he seems to have been more uncritical than Herodotus, but they do not belong to utterly different mental worlds. It seems impossible to date their works relative to each other.
If Homer provided a precedent for the narrative component of Herodotus' work, another anticipation was the study of geography. In looking at Thucydides we shall have to consider other influences, but so far we seem to find history rooted in epic and in geography. Herodotus' ethnographic information derives from his travels and his indefatigable questioning of local informants, to whom he often refers. He was helped, of course, by an already extensive Greek diaspora in western Asia (from which he himself came) and along the shores of the Black Sea. Reference to written documentation in Herodotus' _Histories_ is absent, though he sometimes refers to artefacts he has seen—walls, buildings, above all offerings in temples, particularly Delphi, and their dedicatory inscriptions—as corroborative evidence:
Gyges was the first foreigner we know of, after King Midas of Phrygia, son of Gordias, to dedicate offerings at Delphi. Midas presented the royal throne from which he used to give judgement; it stands with Gyges' bowls, and is well worth seeing. (I.14)
There was a fire at Delphi which damaged some of the gifts of Croesus, king of Lydia. Herodotus helpfully tells us how to locate what remains:
There were also two huge mixing-bowls, one of gold which was placed on the right-hand side of the entrance to the temple, the other of silver, on the left. These also were moved at the time of the fire, and the golden one, which weighs nearly a quarter of a ton, now stands in the treasury of the Clazomenians, and the silver one, which holds over five thousand gallons, is in the corner of the ante-chapel. (I.51)
Even Thucydides made a little more use of documentation—particularly by transcribing treaties apparently verbatim—though the events he described were exactly contemporary, and eyewitnesses and participants, including Thucydides himself, therefore abounded. But Herodotus was above all the man who asks questions, and who makes journeys in order to ask them:
There were other things, too, which I learnt at Memphis in conversation with the priests of Hephaestus, and I actually went to Thebes and Heliopolis for the express purpose of finding out if the priests in those cities would agree in what they told me with the priests at Memphis. It is at Heliopolis that the most learned of the Egyptians are said to be found. I am not anxious to repeat what I was told about the Egyptian religion, apart from the mere names of their deities, for I do not think any one nation knows much more about such things than any other. (II.3)
He could not have been himself an eyewitness of the historical events he describes—he was a small child even at the time of the most recent—though on his travels he certainly makes use of his eyes. He is among other things, though only sporadically, a kind of travelling antiquary, reporting to the world the marvels he has seen: monuments were among the marvellous deeds of men, and well worth recording for their own sake. He was predictably impressed by Egypt and Babylon, giving, as we have seen, a particularly precise description of the latter(I.180–83). At one point he even makes the grandeur of the engineering works of the Samians a reason for attending to their history (III.60).
Above all, however, Herodotus used his ears, if only because most of the inscriptions he may have seen were in a script and language he could not read. He regarded himself as an auditor, a collector, recorder, sifter and judge of oral traditions about the recent or remoter past. He sometimes says that these traditions are incredible, but as a matter of principle he still accepts a duty to set them down: "My business is to record what people say, but I am by no means bound to believe it—and that may be taken to apply to this book as a whole" (VII.152) or, again, "Anyone may believe these Egyptian tales, if he is sufficiently credulous" (II.123). Oral accounts often differed, sometimes in manifestly self-interested ways in the versions of different peoples, and it was also his business to pick the better, giving his reasons, or to reconcile them, or to suspend judgement if he had to. His criteria of judgement are sometimes psychological, sometimes physical probability (including sheer physical impossibility). He can even rather grandly bring superior ethnographic knowledge to bear. The Greeks tell many thoughtless stories (Hecataeus also says this), one being that the Egyptians tried to sacrifice Heracles to Zeus but that he thwarted them by killing them all. This, Herodotus says, shows the Greek ignorance of Egyptian customs, since the Egyptians do not perform human sacrifices, and anyway if Heracles was just a man he could not have killed tens of thousands of people—though Herodotus adds, "And now I hope that both gods and heroes will forgive me for saying what I have said on these matters!"(II.45).
Herodotus' descriptions of the manners and customs of the Persians and Babylonians in Book I set the pattern for the successive ethnographic surveys of each people that the Persians successively encounter. Herodotus lists conscientiously, sometimes amusingly, occasionally incredibly, what became the standard objects of ethnographic curiosity: clothing, diet, marriage and funerary customs, ranks of society, religious beliefs and practices, health and the treatment of disease. His own attitude throughout is tolerant and unshockable. As he says—it is one of the ways in which he anticipates Montaigne—every people considers its own customs best, even those customs most bizarre to others. The Greeks would be horrified at the idea of eating their dead parents; some Indians are shocked at the idea of not doing so but burning them instead (III.38).
Sometimes one has to suspect Herodotus of an artful shaping of his accounts. The Egyptians, who read from right to left and whose great river floods in summer and falls in winter, "seem to have reversed the ordinary practices of mankind"—eating in the streets and relieving themselves indoors (which casts a light on Greek habits), the men urinating sitting down, the women standing up, and so on (II.35–6). After recording that the Persians never acted on a decision taken when drunk without reconsidering it when sober, one can see that it was irresistible to add that a decision taken sober was always reconsidered when drunk(I.133). Often his ethnography rings true or can be verified, as in a reference to what seems to be a survival of matrilineal customs, or to temple prostitution in Babylon, or to the custom of the magi in Persia of allowing the bodies of the dead to be rent by birds and dogs (I.140).
Apart from the Egyptians, Herodotus devotes most attention to the Scythians, who lived to the north of the Black Sea, and the characteristics he records of their treeless and townless life are familiar from many much later accounts: he describes nomadic mounted archers with their wagons, dependent on their cattle and drinking milk; scalping their enemies and hanging the scalps on their bridles, while the skulls are turned into drinking cups; taking vapour baths, swearing blood brotherhood, and burying great men with their households, horses and treasure (IV.16–82). We seem already to have a premonition of the Huns, Tartars and Mongols. But remote Scythia, not surprisingly, also offered even more exotic phenomena: a people bald from birth; goat-footed men; men who sleep half the year. Herodotus seems to accept the bald race, though he does not vouch for it, but he draws the line at goat-footed and hibernating men.
There is the same strain of fantasy at times in the zoology of remote places, and here Herodotus seems less critical than of bizarre anthropology, so that among sound(ish) accounts of crocodiles and camels we get winged snakes (Egypt), giant burrowing ants (India), and cattle which have to walk backwards because going forwards their horns stick in the ground (Libya); he certainly believed in the winged serpents, because he had seen their skeletons. He believes in the phoenix, though he has not seen it, but rejects as incredible the story that it carries its parent wrapped in a lump of myrrh, though whether because of the power–weight ratio he does not say (II.73). He tries to be as accurate as he can in his geography, though often against heavy odds. His longest disquisition on a natural phenomenon, a detailed examination of the flooding of the Nile, which is in its way impressive, leads up to one of his less happy guesses: the Nile floods in summer and sinks in winter because, in winter, storms blow the sun off course towards upper Libya(II.10–26); this is not merely wrong but not very cogent.
No discussion of Herodotus can avoid the vexed question, in antiquity at least, of his reliability and his alleged credulity and even lying. Of course he was sometimes misled or ignorant, and he shared beliefs of the ancient world we do not now hold (though this cannot be what his ancient critics, by far the fiercest, meant). He seems to have been particularly unfortunate with his informants in Egypt, so that scholars working on ancient Egypt now have a much lower view of him than those of Babylon and Persia, who regard him as a valuable authority. The accusations against him in the ancient world which, taken with the rival model provided by Thucydides, undoubtedly lowered his reputation, now seem to rest on the misunderstanding that he necessarily endorsed what he repeated, in so far as they are not merely malicious. A modern reader is more likely to be impressed by his critical sense and his frequent scepticism or suspensions of judgement, once allowance has been made for the ancient belief in omens, oracles and prophetic dreams, which Herodotus takes very seriously while recognizing that they could be manipulated. The fundamental issue seems to be one on which we are now likely to find Herodotus' work both sophisticated and valuable. As we have already seen, he saw it as part of his role to write down even stories to which he gave no credence at all. It was a duty to record what people said to him, while leaving the reader in no doubt of his own opinion.
Herodotus' own world, that of a Greek-speaking elite, was a highly literate one, though still mainly reliant on oral tradition for its knowledge of the past. Much of the rest of the world, outside the centres of civilization, was preliterate; even the Persian kings seem to have been illiterate. But it was clearly a world buzzing with rival oral traditions and orally transmitted, formulaic popular tales, with legends and accounts of mythic beasts, peoples and countries, with stories of foundlings, tests, tricks, subterfuges, riddles and prophetic dreams. Cyrus the Great is brought up by a shepherd who was commanded to kill him on the orders of his grandfather the king, who had had a dream that a vine sprang from his daughter's genitals and overspread Asia, and interpreted this as a threat to himself (there is an analogy with the Perseus story in Greek legend). Cyrus is later recognized as a prince by his royal bearing, and he himself has a similar dream about Darius, who succeeds Cyrus' son Cambyses on the throne (I.107–16, 209).
Darius himself, whom we have looked at as one of the conspirators against the magus who has usurped the throne, becomes king by a trick. After a debate among the conspirators over the merit of the three forms of government, monarchy, oligarchy and democracy—which is uncharacteristic of Herodotus' accounts of Persian history, and seems very Greek—they agree to leave matters to fate by mounting their horses on the outskirts of the city, "and he whose horse neighs first after the sun was up should have the throne." Darius, advised by a clever groom, uses a mare to establish a conditioned reflex in his stallion, and becomes king of Persia. "His first act was to erect a stone monument with a carving of a man on horseback, and the following inscription: 'Darius, son of Hystaspes, by the virtue of his horse and of his groom Oebares, won the throne of Persia.' The horse's name was included." Herodotus does not claim to have seen it (III.84–8).
The tributes collected by Cyrus as king lead Herodotus into a digression on the method of collecting gold employed by the Indians ("their semen is not white like other peoples'"), who are indebted to ants ("bigger than a fox") which burrow into sand which is rich in gold. The ants chase the Indians, who are mounted on camels to escape them and also to carry the bags of gold dust (III.102–5).
It is only through formulaic stories such as that of the youth of Cyrus that we can reconstruct anything of the mental worlds and popular tastes of preliterate societies. We owe Herodotus a debt for his conscientious and unsnobbish recording of them, though the prigs who later wrote history in a more restricted and self-consciously dignified fashion seem to have failed to grasp what he was doing, despite his own declarations. Without possessing the concept, he ranks, among his other titles, as the earliest, or certainly one of the earliest, of self-conscious folklorists. He did not, of course, view folklore as a genre, except as part of his "inquiries," but his ability to record with partially or wholly suspended belief is an aspect both of his omnivorous curiosity and of his general tolerance. When it lets him down and he becomes oversceptical we notice it, as when he dismisses as incredible the story that Phoenician voyagers travelling west along the southern coast of west Africa found that they had the sun to the north of them, on their right(IV.42).
As for religious belief, it is clear that he was pious without being slavishly credulous. He frequently announces that he knows more than he thinks it proper to say. "I have already mentioned the festival of Isis at Busiris; it is here that everybody—tens of thousands of men and women—when the sacrifice is over, beat their breasts: in whose honour, however, I do not feel it is proper for me to say" (II.61). Similarly we have "In Athene's precinct at Sais is the tomb of one whose name I prefer not to mention" (II.170). Whether this proceeds from awe of the gods and a sense of unease at making them an object of inquiry, from a sense of decorum and a desire not to give offence, or from such knowledge having been imparted to him under pledge of secrecy is not entirely clear. Probably, taking the large number of his utterances on the subject together, it is a mixture of all three. As seems to have been common, he regards the gods of all peoples as the same, only the names differing, and holds that Greek knowledge of the gods derived originally from the Egyptians. But, though he is respectful and discreet, he remains worldly-wise; he believes emphatically in portents as divine warnings, in omens, oracles, the sacredness of temples and the penalties of sacrilege, as well as the punishments of hubris, but he also knows how vested interests can misinterpret and even manipulate these things. Even the priestess at Delphi could be bribed, as she apparently had been to tell the Spartans that they must help Athens to rid herself of her tyrants (VI.123). He also has a conception of divine providence and divine justice (see II.120 and III.108) and of a fate before which the gods themselves are powerless (I.91).
Apart from providing an example to later historians and much ethnographic information, Herodotus left a prose epic of Greece's deliverance—the deliverance of freedom from the threat of an imperial despotism—which became a staple of European collective memory. The end of his work sees Greece secure and triumphant, with Athens as its dominant power. In his successor Thucydides, whose work begins half a century later, we see Athenian hubris and the fierce rivalries and mutual suspicion of the Greek city states, combined with the factionalism of their internal politics, taking a self-destructive course which would eventually end their autonomy.
**TWO**
**Thucydides: The _Polis_ —the Use and Abuse of Power**
Our constitution is called a democracy because power is in the hands not of a minority but of the whole people. When it is a question of settling private disputes, everyone is equal before the law; when it is a question of putting one person before another in positions of public responsibility, what counts is not membership of a particular class, but the actual ability which the man possesses. No one, so long as he has it in him to be of service to the state, is kept in political obscurity because of poverty. And, just as our political life is free and open, so is our day-to-day life in our relations with each other. We do not get into a state with our next-door neighbour if he enjoys himself in his own way, nor do we give him the kind of black looks which, though they do no real harm, still do hurt people's feelings. We are free and tolerant in our private lives; but in public affairs we keep to the law. This is because it commands our deep respect. (Thucydides, _History of the Peloponnesian War,_ II.37)
This is the most famous passage in the funeral oration pronounced by Pericles, leader of the Athenian democracy, over Athenian soldiers who had died during the first year of the Peloponnesian War (431 BC). It is a long speech, and Thucydides presents it verbatim, according to his normal practice, to which we shall have to return later. In Pericles' speech Athens is made to stand for a peak of cultural as well as political achievement, and, thanks to Thucydides, the image purveyed has established itself permanently in the educated European collective consciousness. Thucydides was a realist, and there is a good deal in the oration which his history subsequently inclines us to qualify, but setting it out in this way enables him to present a eulogy of Athens and what it could be made to stand for without precisely endorsing it. It is a speech.
Pericles' speech—enhanced for modern readers by echoes, not accidental, in the Gettysburg Address—is an oratorical tour de force which appropriately ends with an exhortation to the Athenians not merely to rally to the defence of liberty and of their city, but to fall in love with it, while earlier it has presented Athens as worthy of love. The speech begins, like Lincoln's, with a disclaimer of any need of the dead for honour by the living: the former are sufficiently sanctified by their sacrifice. Like Lincoln, Pericles then invokes the ancestors of the city and their legacy, "a free country." We have just seen his elaboration of what that means, including what we should call the liberal claim that in private life the Athenians are "free and tolerant." Each is free to enjoy himself in his own way. It has been common over the past two hundred years to deny that ancient democracies had a concept of private as well as public liberty. It is difficult to sustain such a claim in the face of Pericles' oration.
Pericles presents the life of Athens as an ideal balance between private and public, and also between the cultural and the political. The Athenian citizen, it is said, is well informed about public affairs and brave in war, but also loves the beautiful and things of the mind. He enjoys a vigorous public life, of contests and festivals, but also grace and beauty in domestic life. Athenian life is marked by ease, openness and versatility, without softness, but also without the need felt by the Spartans (Athens's main antagonists in the war) for a perpetual, overstrained training for adversity, though Athenians are just as brave and patriotic. The Athenians' virtues have brought them their empire and the power it wields, about which Pericles is notably unapologetic: "Our city is an education to Greece," and "Future ages will wonder at us, as the present age wonders at us now" (II.40–41). If the latter is true it is, of course, partly because Pericles and Thucydides have made it so.
The full dramatic quality of this eulogy of Athenian power and greatness at the outset of the war which was to bring the city down was presumably not entirely apparent at the time Thucydides wrote it. It is difficult confidently to ascribe conscious dramatic irony to a good deal in the history, because of uncertainty about the moment of writing in relation to the unfolding of events. Certainly, however, hubris is a recurring theme, as are, from the outset, the hazards and vicissitudes of war, of which the Spartans are generally more aware than the overbold Athenians. What does seem wholly apparent is the art which leads Thucydides to treat Pericles' oration, placed as it is so near the outbreak of the war, so extensively. The occasion was not obviously a great one: a public rite of a common kind for men who had fallen in what was, by the later standards set by the war, little more than a skirmish. It hardly demanded such full and resonant treatment in itself. Speeches in Thucydides are often not short, but Pericles'—almost eight pages in a modern edition—is unusually long. This is not to say that Pericles did not probably say something along these lines: the event was too public, too local and too recent for complete licence. The question of the speeches in Thucydides is a vexed and fascinating one. Whatever their claims to authenticity, which are discussed below (Chapter 2), they are an important and distinctive part of Thucydides' art as a historian, however at odds with modern notions of the historian's responsibilities.
Both morally and materially, Athens ended the Persian Wars as the dominant power in Greece, centre of a confederation which rapidly became referred to as its empire over dependent and tributary states. In the fifty years from the Persian invasion (and the conclusion of Herodotus' history) to the outbreak of the Peloponnesian War among the Greek states (and the beginning of Thucydides' history) Athens' grip tightened. Thucydides gives a brief account of this period—the so-called Pentecontaetia (because it covers fifty years)—as a preamble to his history of the war. Athens was a rich and powerful imperialist state, the greatest in Greece. As a democracy, it also tended to be the hope of democratic factions in other states, oppressed by local oligarchies, just as the oligarchs tended to look for support to Sparta, so that when war broke out, in 431 BC, it sometimes assumed, locally, an ideological character.
It was, Thucydides wrote, justifying his choice of subject, the greatest crisis in Greek history, a conclusion he presents as foresight, since he cannot have known at the outset—or when he began writing, which may not have been long after—that it would last so long or inflict so much suffering, which are two of his reasons for calling it great. It is clear that Thucydides was writing contemporary history, but it is not so clear exactly how close to the events was his "writing up" of them. His history ends in mid-sentence, in 411 BC, presumably as a result of his death, when the war had still some years to run, whereas Herodotus' work, which ends soon after his own birth around the mid-480s, seems to end as he wished it to do.
Thucydides, who was a wealthy Athenian, was not merely an eyewitness but also participated in the war he describes—an unsuccessful general, who was banished from Athens for his failure, he records his own contribution drily and without comment. What he does comment on is the advantage to him as a historian of his exile on the other side, in the Peloponnese. It gave him, he says, leisure and a different viewpoint(V.26). This is not to say that he changed sides. Such exiles were not uncommon: Xenophon experienced the same, both in Persia and in Sparta. Thucydides continued to write about both sides with the same dispassionateness, recognizing the virtues of both Athenians and Spartans and their attendant weaknesses: the former bold, enterprising, overconfident and rash, the latter conservative, moderate—Thucydides prized moderation—and cautious to the point of sluggishness. Both commit what we should call atrocities, though it is those by the Athenians that tend to make more impression on the reader. That the war begins with Athens at the height of its power and wealth is part of what Thucydides recognizes as the greatness of his theme; it is a more fortuitous benefit to the dramatic quality of his history, though Thucydides takes full advantage of it, that the war culminates, though it does not quite end, with the catastrophe of the failure of the Athenian expedition to Sicily, by which the pride and power of Athens were humbled. Greatness is a recurring preoccupation with Thucydides, as indeed with other ancient historians after him. The expedition to Sicily was the greatest expedition, the siege of Syracuse the greatest siege, just as earlier the battle of Mantinea was the greatest battle (Books VI and VII; Book V, 63–74). It is tempting to the reader, perhaps against the grain of the way Thucydides' history as a whole was written, to see Athens as its tragic protagonist, brought down by overweening ambition and overconfidence—he certainly stresses these—and by the way the Athenians from time to time fail to take the opportunities for a lasting peace.
Thucydides' history is presented in the form of annals. This was an established convention, into which he introduced a refinement by dividing his narrative into six-month periods, marked by the seasons: "when the corn was ripe" or "when the corn was ripening" are common introductory clauses. We have already considered (above, Front Matter) the difficulties that Greeks, in particular, had in marking the years in a generally recognizable way, since the four-yearly Olympiads were the only regular pan-Greek event; the admiration for winning athletes preserved their memory. As his work progresses, Thucydides dates events from the beginning of the war "described by Thucydides the Athenian."
The annalistic form for historical writing, which pre-dated Thucydides and which was to have a two-thousand-year currency after him, has developed a discouraging reputation among readers for unimaginative compilation, and was eventually superseded, in the eighteenth century, by more thematic kinds of organization. In Thucydides' case it is a framework, and does at times lead him to deal exhaustively with relatively minor events that a modern historian would probably abridge, but he also uses it with some freedom: we have already seen the at-first-sight-disproportionate attention given to Pericles' speech.
For Thucydides, the chief quality to be sought in writing history is certainty. He distinguishes his own work from that of the poets and also, it has been generally assumed, from Herodotus, who is implied though not named under the category of "prose chroniclers, who are less interested in telling the truth than in catching the attention of the public, whose authorities cannot be checked, and whose subject-matter, owing to the passage of time, is mostly lost in the unreliable streams of mythology" (I.21). Reliable history must be contemporary history or nearly so, for authorities to be checkable by the historian; there is as yet no implication that they must be laid out for the reader. Reliable eyewitness testimony, either the historian's own or that of his informants, is indispensable.
Narrative is the primary way in which the historical truths Thucydides offers are shown to us. Thus, for example, his account of the origins of the Peloponnesian War is presented initially as a narrative of events and only later focuses on the considerations which weigh with the protagonists, which he sets out in the form of speeches. At first, in his account, the two great powers, Athens and Sparta, were edged towards war by the behaviour of their satellites and by the internal instability endemic in many Greek city states, divided by factions, whose inclinations to democracy or oligarchy led them to seek help from Athens or Sparta respectively against their political rivals or neighbouring states. A successful coup or local conflict could transform the local balance of power alarmingly for either Athens or Sparta, bringing the risk of intervention. The same situations tended to nullify attempts during the war to patch together a lasting peace. Thucydides deals with these in detail, especially during the truce described in Book V (28–32). Unpicking the results of recent hostilities to produce a settlement was, he makes clear, immensely complex and difficult. We may generalize the problem by saying that it was not (as in a war between two unified states) a matter of a rectification of a common frontier or an exchange of more remote possessions. The allies had their own interests, fears and ambitions, and their policies were liable to change and manipulation. The result was an unstable cat's cradle of alliances, obligations and resentments which had to be negotiated. Each stage of the fighting tended to add a fresh layer of complexity, of wrongs inflicted and undertakings unfulfilled.
Hence, as in the Balkans before the First World War, local instabilities or changes of alliance, or the threat of them, sent shock waves though the chain of alliances to the two great powers, neither of which could afford defections or the appearance of weakness, while both were subject to the temptations of fishing in troubled waters. This is a summary of the events Thucydides recounts in Book I (24–65) in Epidamnus, Corcyra, Corinth and Potidaea, until a general denunciation of the existing treaty between Athens and Sparta came to seem a reasonable response on the grounds that by these events it had already been broken. Thucydides retails the arguments in detail, though he regards them as superficial: "The real reason for the war is, in my opinion, most likely to be disguised by such an argument. What made war inevitable was the growth of Athenian power and the fear which this caused in Sparta" (I.23). In the latter part of Book I, apart from a brief summary of events between the repulse of the Persian invasion and the outbreak of the new war (479–435 BC), the centre of the narrative moves to Sparta; the deliberations on war or peace, in which the Spartans are swayed, for or against, by closely argued appraisals of the situation, are presented by the ostensibly verbatim report of the debate.
The Corinthians approach the Spartans for aid because Athens has wrested control of Corcyra from them; they rebuke the Spartans for their inactivity. They are followed by an Athenian delegation, which happens incidentally to be in Sparta, and which is given permission to speak. Its members begin by reminding their auditors of the debt of Greece to Athens for its role in defeating the Persians, and defend the Athenians' acquisition of an empire. They are unapologetic about making the case for this in terms which the later nineteenth century (AD) christened _Realpolitik._ Athenian imperialism obeys the dictates of security, honour and self-interest. "We have done nothing extraordinary, nothing contrary to human nature in accepting an empire when it was offered to us and then in refusing to give it up...It has always been a rule that the weak should be subject to the strong; and besides, we consider that we are worthy of our power." The Spartans too once thought so, "but now, after calculating your own interest, you are beginning to talk in terms of right and wrong" (I.68–78). Negotiations and further speeches follow, in which caution is more evident on the Spartan side, while the Athenians, including Pericles, strike a more confident and warlike note. The antithesis between interest and principle, and the subordination of the latter when vital political interests are at stake, is a recurring theme in Thucydidean speeches, as is the view that the possession of power confers the right to exercise it over the weak, and that this is normal behaviour. That is the way the world is. The best possible combination, the Athenian speech quoted above implies, is power exercised with moderation and realism. Also characteristically Thucydidean is the warning with which the Athenians conclude, calculated to appeal to Spartan caution, about the unpredictability of war. People approach war the wrong way round: "Action comes first, and it is only when they have already suffered that they begin to think" (I.78). Later, Thucydides himself remarks that in both Athens and the Peloponnese "there were great numbers of young men who had never been in a war and were consequently far from unwilling to join in this one" (II.8).
Once the war begins we have, of course, campaign narratives, in which the modern reader is likely to be particularly struck by the set form which battles seem to take and the ritual features which characterize the clashes between armies similarly trained and equipped, sharing the same culture and employing the same heavy-infantry tactics, and usually observing the same conventions and rules. These features included the preliminary speeches by the commanders to their soldiers; the "paeans" or battle songs as the armies advanced, which were so similar that Thucydides remarked that they could cause confusion between friend and enemy; the subsequent truce, arranged by heralds, to enable the bodies to be exchanged; the setting up of a trophy by the victors. In the battle accounts in Roman history, which are in some respects similar, these details, apart from the speeches, are absent, though the taking of the auguries and in some cases making propitiatory sacrifices remained important. One reason why truces for the recovery of one's dead were relatively easy was the absence of anything in the nature of "hot pursuit," which in turn was attributable to the Greek deficiency in cavalry. The debacle of the Athenian army in Sicily, which was harried by Sicilian cavalry in a manner described vividly by Thucydides, was exceptional (VII.78–81, 85). The fate of the Athenians—cut off from escape by loss of command of the sea, tormented by hunger and thirst, and eventually herded as prisoners into the oven-like quarries of Syracuse—was uniquely terrible.
Sometimes we become sharply aware that we are reading an author who has seen warfare for himself. (Thucydides, incidentally, refers to the unsuccessful action which led to his own dismissal and exile without emotion or special pleading [V.104].) His comment on generals' speeches to the troops on the brink of action has an old commander's blasé understanding as well as intellectual fastidiousness. Men in such circumstances, he says, "do not bother to avoid giving the impression of using conventional language; instead they bring forward the kind of appeals that can generally be used on all occasions: wives, children, gods of the native land" (VII.69). The Greeks in this period fought in phalanxes, massed bodies of armoured spearmen (hoplites) many ranks deep. On the clash of the phalanxes he adds a telling detail:
It is true of all armies that, when they are moving into action, the right wing tends to get unduly extended and each side overlaps the enemy's left with its own right. This is because fear makes every man want to do his best to find protection for his unarmed side in the shield of the man next to him on the right...The fault comes originally from the man on the extreme right of the front line, who is always trying to keep his own unarmed side away from the enemy, and his fear spreads to the others who follow his example. (V.71)
In such a remark the terror of phalanx warfare is vividly caught in the exposition of a technical detail.
For the modern reader, because the stylized element is not present and because civilians are involved, Thucydides' descriptions of sieges and the sacking of towns are perhaps more memorable than his battle scenes and dispel any notion that kinship among Greeks made their warfare any less pitiless. Thucydides was alert to engineering and technological matters, which makes his accounts of sieges and sea battles precise on such points. In his description of the siege of Plataea by the Peloponnesians, he describes the earthworks, the siege engines and the countermeasures of the besieged. Both sides treated captured towns with extreme harshness—the men typically massacred, the women and children enslaved.
Thucydides, though an Athenian, acknowledges that Athens was unpopular and was felt to have been tyrannical towards its client states, and that the Athenians were the more aggressive in their approach to the war. In the initial Spartan response to the Athenians there are criticisms of the volatility and even light-mindedness of the Athenians, which seem to chime with Thucydides' general appraisal and should be set against the evidence of Pericles' Funeral Oration: "We [the Spartans] are not so highly educated as to look down upon our laws and customs...We are trained to avoid being too clever in matters that are of no use—such as being able to produce an excellent theoretical criticism of one's enemies' dispositions, and then failing in practice to do quite so well against them" (I.84).
We have just encountered one of a number of difficulties with Thucydides' speeches: the question whether any of them, or the remarks in them, represent Thucydides' own opinions. Certainly one can identify a number of recurring preoccupations, of which moderation in the use of power is one, and strong assertion of the right to exercise power another. The speeches, of which Thucydides includes many, have drawn much critical examination. They are highly characteristic. They seem also to have been very influential, since the insertion—invention is one word—of speeches became a feature of ancient historiography well into the Roman period, though none are quite like those of Thucydides. There are speeches in Herodotus, but they are less lengthy and less important, and are often more like parts of conversations than formed orations. Thucydides' own justification for interpolating them is notably ambiguous, and even self-contradictory, though the speeches themselves are written with confident authority and an extraordinary virtuosity of a particular kind which for the most part has little to do with revealing the individual personality of the speaker. They are a reminder, of course, that ancient Greece, and Athens in particular, saw a golden age of public debate and persuasion. It is a persuasive suggestion that Thucydides' speeches owe something to the teaching and example of the Sophists, those professional adepts in the arts of debate and argument, whose dialectical virtuosity seemed an end in itself rather than, as practised by Socrates, a tool in the search for truth.
Referring to the speeches, Thucydides says:
I have found it difficult to remember the precise words used in the speeches which I listened to myself and my various informants have experienced the same difficulty; so my method has been, while keeping as closely as possible to the general sense of the words that were actually used, to make the speakers say what, in my opinion, was called for by each situation. (I.22)
The two requirements may have little to do with each other, and the second gives a wide licence to authorial interpretation—of which Thucydides clearly took advantage, for it has been complained since antiquity that the voice we hear is Thucydides' own. Though the speeches do not usually reflect the personality of the speaker, Pericles' Funeral Oration is one exception, and another seems to be the rasping, no-nonsense philistinism and plain man's anti-intellectualism of the Athenian demagogue Cleon, whom Thucydides clearly despised (he proved an incompetent and cowardly commander), which carries a sense of recognition of the type across two and a half millennia(III.36–40).
Paradoxical as it may appear, it often seems that Thucydides reveals his own mind more fully in the speeches than in the narrative, though we can never be quite sure. He is not an intrusive narrator. He occasionally speaks in the first person, or passes a judgement, but there is little to compare with the conversational approach of Herodotus. Occasionally he enunciates principles, but compared with the majority of historians up to the end of the nineteenth century—when historians began to be ultra self-conscious about their objectivity—Thucydides' overt moral judgements are few and terse. There seems to be plenty of evidence that he was impatient with moralizing rhetoric—his speeches sometimes disclaim the use of it, not always consistently. From time to time he attempts to summarize the significance of a narrative of complex events, but probably less often than a modern historian would. It is therefore not implausible to say that it is when he is purporting to retail the words of others saying "what was called for" that he speaks most directly to the reader. The fact that the speakers differ does not detract from this point, because as a historian Thucydides is concerned not to adjudicate between them in terms of right and wrong, but to register the feelings and considerations that animated the events. The speeches, therefore, are the historian's versions of these. They reveal what the bare events themselves, without elaborate commentary, could not: the motives, apprehensions, appraisals and even guiding principles at work in the actions of the protagonists, like the speeches in a drama. They are the occasions when the agents explain themselves to the reader.
The quality of reflection—Thucydides'—in these rigorously controlled appraisals is impressive. The speakers sometimes explicitly distance themselves from mere rhetorical pathos and denunciation, which is as though Thucydides is saying that what we are about to hear of were the real considerations. The most comprehensively analytic of the speeches read like the situation and policy-option "appreciations" of an able diplomat or think-tank member, rather than the words of a popular orator. This may sometimes make them implausible as oratorical _pièces d'occasion,_ but it makes them superbly illuminating as Thucydides' commentary on the narrative of events. They are not, as speeches in later ancient historians often were, just rhetorical exercises, but rigorously clinical argument, and Thucydides' history would be enormously impoverished and much more opaque without them. For a typical example—not arresting or colourful, but characteristic of the cool, analytic manner found in many of the speeches in Thucydides—consider the argument of Nicias, the Athenian general, against mounting an expedition against Sicily, where the Athenians have been invited to intervene in a local war. Nicias makes a number of cogent points in his long but unsuccessful speech; here he is reminding the Athenians that, despite current appearances, their rear, in Greece, would be insecure if the expedition were to take place:
In going to Sicily you are leaving many enemies behind you, and you apparently want to make new ones there and have them also on your hands. Possibly you think that the peace treaty which you have made gives you security; and, so long as you make no move, no doubt this treaty will continue to exist in name (for it has become a nominal thing, thanks to the intrigues of certain people here and in Sparta); it will certainly not stop our enemies from attacking us immediately, if in any part of the world any considerable forces of our own should suffer a defeat. In the first place, they only made the peace because of their misfortunes; it was forced on them, and in the matter of prestige we had the advantage. Then also in the treaty itself there are still a number of points not settled...(VI.10)
There are two sets of grouped speeches, presented in the form of debate and dialogue, which raise moral as well as policy issues, and which have become famous as "The Mytilenian Debate" and "The Melian Dialogue." Both oppose considerations of mercy and politic leniency to political necessity and harsh retribution—genocide in fact—inflicted on a conquered city. The Mytilenian Debate has a dramatic urgency because an Athenian trireme has already been sent with orders to kill all the Mytilenian men and enslave the women and children. The debate is over whether to rescind the orders (which is done in the nick of time). Cleon, the Athenian demagogue, distinguishes himself by his harshness. The debate in fact shifts away from the immediate issue to consideration of the virtues and vices of government by discussion. Cleon, in particular, not only argues that the Athenian empire rests on strength, not goodwill, but goes on to criticize the Athenian way of taking political decisions. The Athenians, he says, are too apt to treat political decisions as a kind of prize for oratorical excellence: the assembly applauds novelty, and behaves like a connoisseur of debating skills. His case on this point is powerfully put and, echoing Spartan references to Athenian volatility, may have Thucydides' weight behind it, as Hobbes thought it did. In oratorical competitions of this kind "the prizes go to others and the state takes all the danger for herself. The blame is yours, for stupidly instituting these competitive displays." Political wisdom is something quite different and less articulate.
We should realize that a city is better off with bad laws, so long as they remain fixed, than with good laws that are constantly being altered, that lack of learning combined with sound common sense is more helpful than the kind of cleverness that gets out of hand, and that as a general rule states are better governed by the man in the street than by intellectuals. These are the sort of people who want to appear wiser than the laws...who, as a result, often bring ruin on their country. But the other kind—the people who are not so confident in their own intelligence—are prepared to admit that the laws are wiser than they are and that they lack the ability to pull to pieces a speech made by a good speaker; they are unbiased judges, and not people taking part in some kind of competition; so things generally go well when they are in control. We statesmen, too, should try to be like them, instead of being carried away by mere cleverness and a desire to show off our intelligence. (III.38, 37)
Cleon's point here is connected to the particular case for extreme ruthlessness only by a non sequitur that modern readers will find familiar: an ingratiating populism leading to a rejection of bleeding-heart humanitarianism. People "despise those who treat them well and look up to those who make no concessions" (III.39), so that "To feel pity, to be carried away by the pleasure of hearing a clever argument, to listen to the claims of decency are three things that are entirely against the interests of an imperial power." The Athenians might as well renounce the will to empire and "go in for philanthropy" (III.40).
He is answered by another spokesman, Diodotus. It is through words that decisions must be appraised (the merits or otherwise of the Sophists, as teachers of the art of persuasion, seem to be at the heart of this debate). Diodotus' measured defence would not be out of place in a Platonic dialogue. It is a defence not only of discussion and of leniency as a matter of policy—he disclaims any general appeal to compassion—but of political responsibility contrasted with a facile responsiveness to the fickleness of the crowd. The argument for deterrence by harshness, though it addresses the right question—Athens' interests—is flawed by ignoring human nature. People take risks, not expecting to be defeated, and pride and hope or "some incurable master passion" ensure that they always will. In particular, a man acting as part of a community "has the irrational opinion that his own powers are greater than in fact they are." The Athenian assembly is not a court of law seeking justice, but is concerned with security, and here Diodotus introduces what often seems a key word in Thucydides: "moderation." The situation is complex. The democratic parties in other cities are favourable to Athens, but the indiscriminate punishment of the Mytilenians will unite them with the oligarchic parties in a common fear (III.42–8). Thucydides' own comments on the debate are sparse but presumably significant. Cleon was a man "remarkable among the Athenians for the violence of his character" (III.36), and the first ship, bearing the orders for massacre, "was not hurrying on its distasteful mission" (III.49).
The Melian Dialogue (V.84–116) is presented, uniquely, not in set speeches, but in much briefer assertions and rebuttals by respectively "Athenians" and "Melians," as in a play. The Melians have to decide on surrender or resistance to a besieging Athenian army. The Melians plead for their neutrality. The Athenians again rest the case solely on expediency:
We...will use no fine phrases saying, for example, that we have a right to our empire because we defeated the Persians, or that we have come against you now because of the injuries you have done us...And we ask you on your side not to imagine that you will influence us by saying that you, though a colony of Sparta, have not joined Sparta in the war, or that you have never done us any harm. Instead we recommend that you should try to get what it is possible for you to get, taking into consideration what we both really do think; since you know as well as we do that...the standard of justice depends on the equality of power to compel...
The Melians reply that all men have an interest in fair play, and all may one day find themselves in another's power. To the Melians' further point that they can still have hope if they resist, the Athenians' response is brutal: realistic hope depends on resources:
Do not be like those people who, as so commonly happens, miss the chance of saving themselves in a human and practical way, and, when every clear and distinct hope has left them in their adversity, turn to what is blind and vague, to prophecies and oracles and such things which by encouraging hope lead men to ruin.
The Melians assert that they place their trust in the gods, who are not indifferent to right and wrong, and in the Spartan alliance. The Athenians' reply takes a different view of the gods:
Our opinion of the gods and our knowledge of men lead us to conclude that it is a general and necessary law of nature to rule whatever one can. This is not a law that we made ourselves, nor were we the first to act upon it when it was made. We found it already in existence, and we shall leave it to exist for ever among those who come after us. We are merely acting in accordance with it, and we know that you or anybody else with the same power as ours would be acting in precisely the same way.
The dialogue continues in this way, with the Melians clearly hankering to return to the ground of moral justification and the Athenians refusing it to them. They warn, in particular, against the delusions grounded in "honour"—a form of pride which leads to ruin. The safe rule is "to stand up to one's equals, to behave with deference towards one's superiors, and to treat one's inferiors with moderation." The Melians decide for defiance, with trust in the gods and the Spartans and pride in "the liberty which our city has enjoyed from its foundation for 700 years." In victory the Athenians do not practise moderation; the Melian men are killed, the women and children are enslaved, and the city is repopulated as an Athenian colony.
The rational, sophisticated manner in which projected genocide is discussed in these two debates makes it natural to look for Thucydides' own attitude. We have already noted a couple of clues, as well as his possible endorsement of Cleon's rebuke to Athenian intellectual flightiness. The "Machiavellian" or, if we want to avoid anachronism, Sophists' theme that might is right is so prominent that it was clearly a preoccupation of Thucydides, and it is one to which no really effective reply—only a reminder of the fickleness of fortune—is made. It is not altogether clear whether this is because politics is special, in that the moral inhibitions of private life do not apply, as it seems to have been for Machiavelli and for another modern kindred spirit, Max Weber, but it may have been. As Weber says, in politics one necessarily sups with a long spoon. The emphasis on moderation also seems to call for attention. It is essentially rational, because the unforeseeable is a crucial part of politics and no course guarantees absolute safety any more than it promises absolute virtue. It is surely significant that, in his account of the constitutional experiments which later followed the temporary overthrow of Athenian democracy, Thucydides explicitly opts for a middle course in the notion of a mixed constitution (VIII.97).
The opposite of moderation was fanaticism, and the classic case of political fanaticism in Thucydides is the account of the civil conflicts that began in Corcyra (in 427 BC) and spread all over Greece. These represent political factionalism wholly out of hand and becoming, for the agents, an end in itself. The account of the factionalism in the city states, prompted by the war, describes the pathology of societies under extreme stress in which all the normal conventions and restraints have ceased to operate. Thucydides' depictions of the psychology of fanaticism among the Corcyreans (and later elsewhere) still resonate, and one is reminded of the—often hotly disputed—claim for a perennial human nature. The Corcyrean democrats, feeling themselves threatened, began a massacre of fellow citizens. According to Thucydides, this was the precursor of many future revolutions and calamities in the Greek states: "The knowledge of what had happened previously in other places caused still new extravagances of revolutionary zeal, [with]...unheard-of atrocities in revenge. To fit in with the change of events, words, too, had to change their usual meanings": aggression became courage and moderation cowardice. "Anyone who held violent opinions could always be trusted, and anyone who objected to them became a suspect." To attempt to opt out of the plotting and counter-plotting was "disrupting the unity of the party," and fervent party members felt confidence in each other as partners in crime. The one standard became the will of the party at any particular moment. Even rational self-interest was jettisoned: "Revenge was more important than self-preservation." Society became divided, and there was a deterioration of character throughout the Greek world. Peace could not be made, because oaths were worthless. With the ordinary conventions of civilized life overthrown,
human nature, always ready to offend even where laws exist, showed itself proudly in its true colours, as something incapable of controlling passion, insubordinate to the idea of justice, the enemy of anything superior to itself...Men take it upon themselves to begin the process of repealing those general laws of humanity which are there to give a hope of salvation to all who are in distress, instead of...remembering that there may come a time when they, too, will be in danger and will need their protection. (III.82–4).
It comes as no surprise that the seventeenth-century English translator of Thucydides was Thomas Hobbes.
The mark of fanaticism is the pursuit of advantage without limit, the love of terror for its own sake. Those who, in Thucydides, rationally employ power to their advantage sometimes stress that their behaviour is normal, according to human nature. But human beings released from the ties of convention are creatures of irrational extremes. The line between ruthlessness and fanaticism is the distinction between the normal and the pathological. But the arrogance of power and its overestimate also lead states to rash enterprises beyond their resources, of which the prime example is the Athenian expedition against Sicily, and the representative individual the dashing young Athenian aristocrat Alcibiades, cunning politician and Olympic prize-winner in the chariot races. Rashness is the exaggeration of the Athenian qualities of confidence, boldness and enterprise. In Thucydides' account, it meets its nemesis in the failure of the ultimately disastrous invasion of Sicily (in 415 BC), which allegedly brings Athens down cataclysmically from the height of its power, and is described by Thucydides in unprecedented detail in Books VI and VII. The attention given to the Sicilian venture is a striking example of authorial initiative. The invasion was an attempt to strike at Sparta through its allies and through domination of the western Mediterranean. It was, Thucydides claims, the greatest expedition undertaken in the Hellenic world, and its failure gives these books, like the latter part of Herodotus, the familiar shape of hubris and nemesis.
At the centre of the decision to invade and the conduct of the invasion is Alcibiades as politician and general. He is the extreme case of a highly recognizable type in many ages: a brilliant, arrogant aristocrat with political ambitions and considerable eloquence and ability. His supposedly undemocratic ambition inspired distrust, and when an act of sacrilege was perpetrated in Athens—the mutilation of the herms, the phallic statues which stood at various points in the city—Alcibiades and his associates fell under suspicion. It is easy to see this desecration as a youthful, aristocratic prank, but it was magnified into a plot against the city's constitution. Alcibiades was recalled from Sicily, but, fearing for his life, fled first to Sparta and then to Persia. Such exile was not uncommon—Thucydides himself experienced it—but Alcibiades remained influential, playing off the Persians and Spartans against each other and eventually manipulating for himself a return to Athens. He had undeniable glamour, which is evident also in his appearance in Plato's _Symposium._ Thucydides said that "most people became frightened at a quality in him which was beyond the normal and showed itself both in the lawlessness of his private life and habits and in the spirit in which he acted on all occasions" (VI.15).
Thucydides has a majestic description of the setting-out of the expedition. The population of Athens, torn by fear and hope, go down to Piraeus to see its departure, while Thucydides hard-headedly estimates its cost. Then,
When the ships were manned and everything had been taken aboard...silence was commanded by the sound of the trumpet, and the customary prayers made before putting to sea were offered up, not by each ship separately, but by all of them together following the words of a herald. The whole army had wine poured out into bowls, and officers and men made their libations from cups of gold and of silver. The crowd on the shore also, the citizens and others who wished well to the expedition, joined together in the prayers. (VI.32)
The fleet so splendidly described earlier goes to its destruction in the sea battle of Syracuse. Thucydides is detailed in his accounts of sea battles and technology: the special rams fitted to the Syracusan ships(VII.36); the ships on the defensive forming themselves into a circle, a kind of floating stockade; the attempts at ramming, with the ships fouling each other in confined spaces; the chaos and confusion—"the great din of all these ships crashing together was not only frightening in itself, but also made it impossible to hear the orders given by the boatswains," who were themselves distracted by attempting to ram while avoiding being rammed themselves from several directions(VII.70).
The decisive sea battle of Syracuse, which doomed the Athenian expedition, was witnessed by the two armies from land, shouting and cheering as though in a theatre. For the Athenians in particular, who knew that their own fate depended on the outcome, there was terrible anxiety as they followed its fortunes. They were too close to see the whole picture, so
some saw that at one point their own side was winning, and took courage at the sight and began to call upon the gods not to deprive them of their salvation, while others, looking towards a point where their men were being defeated, cried out aloud in lamentation, and were more broken in spirit by the sight of what was being done than were the men actually engaged in the fighting. Others were looking at some part of the battle where there was nothing to choose between the two sides, and, as the fight went on and on with no decision reached, their bodies, swaying this way and that, showed the trepidation with which their minds were filled, and wretched indeed was their state, constantly on the verge of safety, constantly on the brink of destruction. (VII.71)
The fate of the Athenian army, isolated and hopeless on the island without their fleet, was indeed terrible, and Thucydides vividly describes their sufferings on their final march, "especially when they remembered the splendour and pride of their setting out and saw how mean and abject was the conclusion. No Hellenic army had ever suffered such a reverse" (VII.75). The survivors are taken as prisoners to the Syracusan quarries.
There were many of them, and they were crowded together in a narrow pit, where, since there was no roof over their heads, they suffered first from the heat of the sun and the closeness of the air; and then, in contrast, came on the cold autumnal nights, and the change in temperature brought disease among them. Lack of space made it necessary for them to do everything on the same spot; and besides there were the bodies all heaped together on top of one another of those who had died from their wounds or from the change of temperature or other such causes, so that the smell was insupportable. At the same time they suffered from hunger and from thirst. (VII.87)
It is important to stress Thucydides' achievements as a narrator, because his own programmatic statements about the value of history may have led them to be underrated. Thucydides is the first author to proclaim that history should be useful, and he makes the possibility depend on his famous claim that, human nature being a constant, what "happened in the past...will, at some time or other and in much the same ways, be repeated in the future. My work is not a piece of writing designed to meet the taste of an immediate public, but was done to last for ever" (I.22). The contrast may be a sneer at Herodotus' public readings. But Thucydides is too much a realist—even a pessimist—for there to be any glib suggestion that, armed with historical examples, we will be able simply to avoid the errors of the past. Human nature, the narrative tells us, is too powerful and too perverse for this, and rational calculation is only one element in any situation. There can be no thought, when reading Thucydides, of history being superseded by a manual of political and psychological maxims derived from it.
Thucydides was surely right to claim that his history teaches, but it is arguable that he did not fully understand exactly how this was so, and it is hard to express this clearly even now. It must be a common feeling that one has learned something about human nature and human affairs from his work, and not just from noting examples and their didactic import. As Robert Connor points out, in reading Thucydides one has enlarged one's experience, encountering human nature in action, assimilating the emotional as well as intellectual impact the detailed narrative makes on us. This narrative is extended yet also terse, acutely analytical yet also sometimes impassioned, and vividly presented to the imagination, not through surface ornament but in the penetration with which the predicaments of the actors are measured. We are taught by it indefinably—as we are taught by concentrated experience—which makes any talk of "lessons" seem uncomfortably superficial. The subject is indeed human nature, and at times the distance of almost two and a half thousand years can be made to seem to contract almost vertiginously. But this is not because of the formula that Thucydides may seem to be accepting: that "laws" of social and political psychology may be inferred from examples. What he certainly understood is that examples are complex because circumstances are so, and responses to circumstances are complex also. Historians, one is glad to be able to assert, have generally done better than their programmatic formulations of their task have suggested, which is one reason why discussions of the nature of historiography based on such formulations are so inadequate. Thucydides, if we take his pronouncements too narrowly, is no exception. It is quite appropriate, while registering that he can be a stirring, poignant or horrifying narrator, to speak also of his work as analytic or, better, diagnostic. But diagnosis is an art, and it has been aptly suggested that Thucydides' approach to history owes something to the treatises of Hippocratic medicine. This was a practice and not just a theory, and involved detailed observation of symptoms as well as classification of conditions.
The former is much in evidence in Thucydides' own account of pathological phenomena in his famous description of the plague which broke out in Athens in 430 BC. He himself caught it and survived; many, including Pericles, died. Thucydides gives a highly detailed account of the symptoms and course of the disease, so that, as he says, it would be recognizable if it ever broke out again. His description—there was, of course, no explanation—is so detailed that it might be that of a professional physician. He also, however, makes no attempt to obscure the extreme suffering of the victims and the effects on the society of Athens. His account is a harrowing as well as exact one, and it is pathology in a double sense: the pathology of the disease itself and of a whole society disintegrating under its effects. Human art or science had no help to offer, and "Equally useless were prayers made in the temples, consultation of oracles, and so forth; indeed, in the end people were so overcome by their sufferings that they paid no further attention to such things" (II.47). Those who tried to nurse the sick caught the disease; this caused such fear that many people died alone and unattended. Bodies lay in the streets and in the temples where people had gone for refuge.
The result, Thucydides says, was an unprecedented lawlessness. As in the later account of political terror in Corcyra, which we have looked at, Thucydides depicts here, for different reasons, the pathology not just of a disease but of a society in dissolution. Despairing of the future, people sought the pleasures of the moment without restraint. Honour—reputation—counted for nothing. "The catastrophe was so overwhelming that men, not knowing what would happen next to them, became indifferent to every rule of religion or of law." Proper funeral ceremonies—a matter of much importance, for example in war—were neglected. "No fear of god or law of man had a restraining influence. As for the gods, it seemed to be the same thing whether one worshipped them or not, when one saw the good and the bad dying indiscriminately" (II.52–3). No one expected to live long enough to be subject to human justice.
Thucydides' account is an extraordinary rhetorical and analytic tour de force, as a description of a human society _in extremis._ We can only guess at the long-term effects of the experience on his own view of life and human conduct. It is characteristic of Thucydides that we should have been led to this account of human and social catastrophe by following his claim to have written a history which should be useful. As often in his work, the terrible is placed within a framework of observation and analysis which does not sterilize it but rather strikes one, in the circumstances, as a powerful act of intellectual will. Thucydides' view of human life and conduct is wholly unsentimental, but never "scientifically" desiccated.
The obvious applicability of the pattern of hubris–nemesis to the greatness of Athens and the disaster of the Sicilian expedition inclines one to make an analogy with Greek tragedy. Any attempt to draw this in detail is probably misguided, but morally, though not formally, the suggestion is surely not altogether misplaced. Some of the formulations in which a modern critic, Northrop Frye, characterizes the essence of the tragic outlook can hardly fail to resonate with readers of Thucydides: "Tragedy seems to elude the antithesis of moral responsibility and arbitrary fate, just as it eludes the antithesis of good or evil": in tragedy "one finds a 'Dionysiac' aggressive will, intoxicated by dreams of its own omnipotence, impinging upon an 'Apollonian' sense of external and immovable order" ( _Anatomy of Criticism,_ 1957). The references are to Nietzsche's essay on the birth of tragedy, and Nietzsche's tribute to Thucydides—idiosyncratic though it is—catches much about him:
Thucydides and perhaps the _Principe_ of Machiavelli are related to me closely by their unconditional will not to deceive themselves and so see reason in _reality_...One must turn over line by line and read his hidden thoughts as clearly as his words: there are few thinkers so rich in hidden thoughts. Sophist culture, by which I mean realist culture, attains in him its perfect expression—this invaluable movement in the midst of the morality-and-ideal swindle of the Socratic schools which was then breaking out everywhere...Thucydides was the grand summation, the last manifestation of that strong, stern, hard matter-of-factness instinctive to the older Hellenes. (Nietzsche, _Twilight of the Idols,_ "What I Owe to the Ancients")
Thucydides seems to embody all the qualities that Nietzsche admired and did not always manage to embody himself. It is easy to understand the admiration. Almost all historians except the very dullest have some characteristic weakness: some complicity, idealization, identification; some impulse to indignation, to right wrongs, to deliver a message. It is often the source of their most interesting writing. But Thucydides seems immune. Surely no more lucid, unillusioned intelligence has ever applied itself to the writing of history.
**THREE**
**The Greeks in Asia**
**Xenophon: _The Persian Expedition_**
When the men in front reached the summit and caught sight of the sea there was great shouting. Xenophon and the rearguard heard it and thought that there were some more enemies attacking in the front, since there were natives of the country they had ravaged following them up behind, and the rearguard had killed some of them and made prisoners of others in an ambush, and captured about twenty raw ox-hide shields, with the hair on. However, when the shouting got louder and drew nearer, and those who were constantly going forward started running towards the men in front who kept on shouting, and the more there were of them the more shouting there was, it looked then as though this was something of considerable importance. So Xenophon mounted his horse and, taking Lycus and the cavalry with him, rode forward to give support, and, quite soon, they heard the soldiers shouting out "The sea! The sea!" and passing the word down the column. Then certainly they all began to run, the rearguard and all, and drove on the baggage animals and the horses at full speed; and when they had all got to the top, the soldiers, with tears in their eyes, embraced each other and their generals and captains. (Xenophon, _The Persian Expedition,_ IV.7)
The soldiers' excitement was understandable, and to mark the spot where the sea came into sight they put up a cairn of stones surmounted by captured shields. For the small Greek army, which had found itself apparently trapped in the heart of the Persian empire, to have fought its way back through hostile countries to the south-eastern shore of the Black Sea had immeasurably increased its chances of survival. There might be boats to be had, and along the southern shore there were Greek settlements which might be friendly or could be coerced. There were still vicissitudes to be encountered, but the Greeks had taken a great step towards their homeland. How they had found themselves in their desperate predicament and how they won their way home is the story Xenophon tells in what later became his most famous work, the _Anabasis,_ or in its English title _The Persian Expedition._
To understand it fully we need some understanding of the relation of the Greeks, after the end of the Peloponnesian War, to the great oriental empire whose invasion they had repulsed more than three-quarters of a century before. We also need to know something of Xenophon, who is not only the book's author but also its central character, who emerged, by his own account, as the army's leader. Towards the end of Thucydides' history, as a result of the exhaustion of the combatants, the Persians were beginning tentatively to throw their weight into the scales of the conflict. There was, however, no further full-scale invasion. Rather it was the Greek world which, as a result chiefly of the continuing superiority of Greek (and subsequently Macedonian) fighting methods, began to penetrate the Persian world, and some Greeks, notably the rhetorician Isocrates, began to advocate and predict a Greek takeover of the Persian empire. This was the context for the work of Xenophon. He wrote a decidedly disappointing and most historians now think unreliable continuation of Thucydides' history, the _Hellenica,_ but the chief source of his fame in the modern world, _The Persian Expedition,_ really in a sense makes him the successor of Herodotus, on a much smaller scale, for it encapsulates in microcosm the relations between the Greek and Persian worlds.
Xenophon was born soon after the outbreak of the Peloponnesian War, into the Athenian gentry; his affection for his native city was dimmed by his distaste for its democracy. He was a disciple of Socrates (of whom he wrote an account), a soldier, and a country gentleman, in Spartan and in Persian territory. His works are eclectic: apart from the history of Greece in continuation of Thucydides and his work on Socrates, he wrote a work of pedagogical eulogy, _The Education of Cyrus,_ which was much admired in antiquity; a book on the art of hunting; and _The Persian Expedition._ The last, it seems, was not popular until the time of Alexander, over half a century later.
_The Persian Expedition_ is a military man's vindication of his own conduct and that of the force to which he belonged. It is an enthrallingly detailed, first-hand account. The army of ten thousand Greek mercenaries becomes stranded, deep in the Persian empire, by the death of their employer, Cyrus, a Persian prince, who had hired them to help him with his attempted insurrection (401 BC). They have to fight their way back to Greek territory, traversing parts of the Persian empire which are often rugged, inhospitable, barbaric and even savage. It is another parallel with Herodotus that the work contains a good deal of casual ethnography. Xenophon is his own hero—perhaps, alternative accounts suggest, to an excessive degree. The expedition forms a single drama with a single protagonist, the ten thousand Greek hoplite mercenaries, and its resolution is the safe return of most of them to the Greek, largely Spartan-controlled, colonial world of the Bosporus region, where Xenophon had unfulfilled dreams of founding a city.
Beset by enemies, Persian and barbarian, in Xenophon's account the Greek mercenary army essentially survives by its discipline and military technique. To preserve its cohesion is often Xenophon's chief concern. This cohesion is fostered by a sense of solidarity as Greeks, and the troops give each other crucial support at moments of crisis, going, for example, to considerable trouble and difficulty to recover and bury their dead, as Greeks always did. This quality sees them through the hazards of a hostile climate and terrain, and the ever-present need to extort the supplies to keep them going in the lands through which they pass, in which for the most part they are clearly regarded as no better than a plague of locusts. Xenophon is at pains in all this to bring out his own qualities of leadership and persuasion: he is not, until the very end, the acknowledged leader, but his account certainly makes him the star performer.
Eventually, in the passage quoted above, which readers (who included generations of English schoolboys) found unforgettable, the rear of the column, which contained Xenophon, hears a tumult from the vanguard: "The sea! The sea!" This is really the book's climax, and the odds are now tilted towards survival. Reaching the Black Sea raises the hope—only partially fulfilled—of an easy passage back by sea to Byzantium.
There are still savage barbarians to be encountered, however, and also, as the shore becomes more civilized, the hostility of local Persian rulers, and even of Greek settlements which view the advent of the army with dismay, and of Spartan governors, the Spartans having succeeded the Athenians as the Greek colonial power in the region. To none of them is the arrival of so large a number of disciplined soldiers—without means of support, and with habits, of necessity, little different from those of a band of brigands—at all welcome.
The last part of the book, in which Xenophon is given the outstanding part as leader and chief negotiator, is taken up with the attempts of the army's leaders to recompense themselves, pay the troops, and find a new employer. Eventually, in 399 BC, they find one in Sparta, fortunately facing the onset of a new war in which the army could be useful. Xenophon himself was forced to give up the ambition he had formed to found a city on the Black Sea, and took service with Sparta, for which he was rewarded with an estate which enabled him to settle down as a country gentleman.
_The Persian Expedition,_ then, relates an epic fighting journey and the achievement, against heavy odds, of a fortunate homecoming. It has therefore something like the dramatic shape of earlier, legendary expeditions—the voyage of the Argonauts; the Trojan War and Odysseus' tortuous wanderings—as well as being a microcosmic fore-shadowing of the campaign expedition of Alexander across the Persian empire to the Indus. More like Xenophon than any of these in its first-hand authorial testimony and dramatic compactness is the account in more modern times, by the Spaniard Bernal Díaz, of the campaign expedition to and capture of Tenochtitlán (Mexico City) by the conquistadores in the early sixteenth century, later written into an epic account of the conquest by the New England historian W. H. Prescott. Xenophon's story is one of survival, not conquest, and is perhaps the easier to identify with for that. It is a story with a single collective actor, recounting a continuous sequence of events over only two years (401–399 BC), and therefore contrasts with those ancient histories (including Xenophon's own _Hellenica_ ) which, with a variety of scattered actions in various locations to record, are cast more or less of necessity into the annalistic mould.
Always present as a background to the action, to the successive challenges and hazards met by the army as it attempts to fight its way home, and the dissensions and debates these evoke at various stages of the march, is an intense awareness of the contrast between Greeks and Persians and between both and the other barbarians. The army takes its Greekness with it into the heart of Babylonia and the wastes of Kurdistan and Armenia not only in its hoplite equipment and tactics, which give it the edge in battle, but in its self-awareness and its pride in its Greek identity and superiority. And history, as so often, is called on to buttress pride and resolution. At a critical moment Xenophon, in his account, calls on the troops to recall the past achievements of the Greeks against the might of the Persian empire. "Remember how the Persians and their friends came with an enormous army, thinking that they would wipe Athens off the face of the earth; but the Athenians had the courage to stand up by themselves, and they defeated them"(III.2). The prize of that victory was freedom. The Persian prince Cyrus has earlier congratulated his Greek mercenaries on their freedom and the strength they derive from it, when explaining why he has recruited them (I.7). Xenophon himself calls on the soldiers to shun a life of ease and luxury and "these fine great women, the wives and daughters of the Medes and Persians," lest they should be weakened in their determination to win their way home (III.2). The contrast of Greek (later Roman and European) masculinity with oriental servitude, effeminacy and luxury was later to become familiar in a literature that stretched from the _Odyssey_ 's tale of the lotus-eaters, through Roman denunciations of Antony's enslavement to an Egyptian queen and Marlowe's "pampered jades of Asia," to Tennyson's and Kingsley's reworking of the theme in the context of later European imperialism.
In Xenophon's story the reference to freedom is more than rhetoric. When the mercenaries lose their employer with the death of Cyrus and later lose their original commander to Persian treachery and are faced with the alternatives of surrender or the long march home—over a thousand miles, without bases, supplies or cavalry—they behave like Greeks: that is, they deliberate and debate and come to a collective decision and elect new commanders. This army has been called "a polity on the move." It is also pious, and this matters to Xenophon, who seems conventionally so, in contrast to Thucydides and perhaps even to Herodotus, in whose piety there seems a touch of good manners. Because the omens are unfavourable, Xenophon, by his own account, keeps the army in suspense for agonizing days before engaging in an urgently desired action (VI.4). He is scrupulous in making sacrifices. The gods are given their traditional role as the guardians of oath-taking and treaties, and Greek piety in this respect is contrasted with Persian treachery. Along with sacrifices of thanksgiving for deliverance, to Zeus, Apollo and Heracles, the army holds athletic games—a supremely Greek and indeed Homeric touch—with racing and boxing. (In Herodotus the Persian king had expressed wonder at the Greeks competing only for wreaths.) This is its chief form of celebration of its survival. When it is objected that the ground is too hard for wrestling, the games' organizer shows complete lack of sympathy: "All the worse for the man who gets thrown" (IV.8). Xenophon's work had all the qualifications for a cherished pedagogical text in the English public schools.
The character, possessions and diet of the barbarian peoples the army had successively to resist, intimidate and pillage were, of course, of urgent practical concern, but in Xenophon's description there is sometimes a note of Herodotean connoisseurship of the quaintness of the exotic. Some of the barbarians were troglodytes. The food mentioned—a vital matter—included a toxic drink made from honey, said to induce delirium and madness (perhaps a kind of mead), pickled dolphin, and dolphin fat used instead of olive oil. One highly Herodotean observation—perhaps so much so as to cast doubt on it as observation—is the alleged inversion of ordinary customs: "When they were in a crowd they acted as men would act when in private, and when they were by themselves, they used to behave as they might do if they were in company" (V.4). On one occasion the Greeks seem to make themselves the exotic people, putting on for the Paphlagonian ambassadors what one can call a display of ethnic dancing—Thracian, Magnesian, Mysian and Arcadian, emphasizing the composite character of the army. Xenophon describes this in some detail, though when a slave girl, belonging to an Arcadian, dances the Spartan war dance, the Pyrrhic dance, he seems to regard this as too well known to need description (VI.1).
Xenophon, as an aristocrat, was an admirer of Sparta, and it was with the Spartans that he was eventually able to make terms on behalf of the army and of himself, but the tone of the work as a whole is Panhellenic. The army, while it contained a preponderance of Athenians, was a mixture, though the fundamental contrast is of course with non-Greeks. One reading of Xenophon's work has seen it as a kind of Panhellenic propaganda, in the vein of Isocrates, demonstrating how easily Greeks could overcome Persians and hence how easy it would be, as Xenophon clearly hoped, to found new Greek colonies in Persian territory. His book surely contributed to a Western sense of superiority to the Orient from the Enlightenment onward.
Though _The Persian Expedition_ may reflect the outlook of the Greek Panhellenists, it was not Xenophon's most popular work in Greek antiquity, which preferred his Socratic writings and the tediously high-minded _Education of Cyrus._ Later, however, the _Expedition_ became more popular, with an imperialist interest which culminated in the time of the "second generation" of historians of the conquests of Alexander. These writers were a second generation in the sense that the works of the various historians who accompanied Alexander's expedition have all disappeared, except for borrowings and excerpts, leaving two notable historians in Roman times, centuries later, to rewrite the Alexander story. One of them, Arrian, makes Alexander recall to his men before the battle of Issus the feats of Xenophon's Ten Thousand, and this may indeed have happened. It is to Arrian and his fellow Alexander historian Curtius Rufus that we must now turn.
**The Alexander Historians: Arrian and Curtius Rufus**
The greatest of all subjects for a campaign-expedition narrative, Alexander the Great's invasion of Asia in 334 BC, also marked the culmination of the encounters between the Greek world, including Macedon, and the Persian empire. Though there were numerous attempts, no historical account, it seems, was able quite to capture the scale of events as Herodotus had done with the invasion of Greece by the Persians. This has to be partly guesswork, however, for none of the earliest, eyewitness, accounts has survived except in so far as they were incorporated in the work of later historians writing in the Roman period. Alexander himself is said to have envied his supposed ancestor Achilles for having Homer to commemorate his deeds, while he himself had no comparable memorialist. We have intact, moreover, only the later historians who based their accounts on the earliest ones, one of whom, Ptolemy, was not only one of Alexander's generals but one of the successors to his empire, in Egypt. The Greek historian Arrian, who, writing in the second century AD, is the best-known of the extant later historians, said that the lack of a great memorialist was Alexander's only misfortune, and that it was a pity that his exploits were less well known than the much smaller achievements of Xenophon's Ten Thousand. He was not entirely disinterested, since he was claiming the first rank in Greek literature as Alexander's chronicler and because of the scale of Alexander's achievements. Writing four hundred years later, however, he was necessarily dependent on his now lost predecessors.
Alexander had taken with him his own historian, Callisthenes, the nephew of Aristotle, whose narrative was curtailed when Alexander had him executed for conspiracy. Apart from a general, Ptolemy, other memorialists of the campaign included an engineer, Aristobulus, as well as humbler figures who contributed their accounts to the stock of Alexander literature. Most popular of all, in the centuries after Alexander, was the history of Cleitarchus, who was not an eyewitness, though a contemporary who had access to the evidence of participants. We know something of the work of all these from the use made of them by later writers, Plutarch (AD 45– _c._ 120), and Diodorus of Sicily, whose _Universal History,_ in the mid first century BC, devoted a book to Alexander's expedition. There is also the work of Arrian and his chief rival now, because his history has survived, though with lacunae, the only Latin historian among them, Quintus Curtius Rufus. Both of these last enjoyed distinguished careers in the Roman empire, Arrian (whose Latin name was Lucius Flavius Arrianus) as governor of Cappadocia and archon (magistrate) of Athens, while Curtius is thought to have been a senator and consul in the first century AD. Their aims as historians are somewhat different, and they are clearly sometimes following different traditions. As historians of essentially the same events, their work offers examples of the possibilities for historical narrative in the early Roman empire. Arrian is explicit that the sources he trusts are Ptolemy and Aristobulus and that he is confident, when they agree, that he has the authoritative account. Curtius (and also apparently Plutarch and Diodorus) follows something like a rival tradition, stemming from Cleitarchus. The differences exhibited by the respective histories not only preserve some of the contemporary versions of the Alexander story (at the cost of leaving a good deal undecided), but embody notably different ways of writing history.
Alexander himself, of course, is at the heart of both accounts, and there is a good deal in the major events and even in the reading of Alexander's character on which they substantially agree. But the historiographical personalities of Arrian and Curtius are entirely different. Arrian's is the more "official" account: sober and measured. He seems to trust Ptolemy implicitly, saying that after Alexander's death he had no motive to flatter or suppress, though surely as a participant Ptolemy may have had some things to gloss over, while Aristobulus, Arrian's other main source, had a reputation for flattery. Curtius, perhaps reflecting Cleitarchus as well as his own taste, is more circumstantial, sensuous and dramatic—and, for the fastidious, vulgar. His portrait of Alexander, though ultimately more adulatory than critical, lays more stress than Arrian on Alexander's moral deterioration under the influence of good fortune, luxury and oriental notions of despotic power and semi-divinity. The contrast we found initially in Herodotus and more markedly in Xenophon between oriental "effeminacy" and Greek (later Roman) hardihood and austerity was a heavily stressed one in the Roman period; indeed, as we shall see, the idea of oriental contagion became something like a general explanation of decadence.
Arrian, it seems, took Xenophon as the model for his _Anabasis Alexandri_ ( _The Campaigns of Alexander_ ), but Curtius' work ( _The History of Alexander_ ) is the more "literary," with obvious transpositions of topoi from Herodotus: the wise but disregarded counsellor of the Persian king, for example, is analogous to Croesus in Herodotus. Almost a quotation from Herodotus is the remark attributed to Alexander that the Persians had more men in their ranks but the Macedonians more fighters (Curtius, 4.14.5). Numerous echoes of Livy (whom we shall consider later) have also been identified. Curtius has a taste for ethnographic digressions, though Arrian's relative abstention from these is to some extent misleading, since he reserved his Indian material for a separate work, also extant. In general, however, drawn by a taste for the description of Persian gorgeousness and luxury, and the opportunities for moralistic disapproval these offered, Curtius gives rather more of the Persian perspective than does Arrian. Arrian's work, in fact, though less censorious, is too personalized and one-sided to be an account of a clash of civilizations. Curtius, however, rejoices in the chance to contrast Darius' Persian army, gleaming with purple and gold, and the Macedonian force, "gleaming, not with gold, not with multicoloured clothes, but with iron and bronze" (3.26). Effeminacy is a stressed note in the description of the Persian host. The fifteen thousand called "the king's kinsmen" were "dressed almost like women," and Darius himself had a gilded belt "which he wore in the style of a woman" (3.14, 19). Wives, the royal children's nurses, concubines, eunuchs and camp followers bring up the rear.
There is nothing of this in Arrian, who contents himself with numbers and the discussion of tactics, on which Curtius is decidedly weak. Instead, Curtius likes drama and pathos, in the manner of the rhetorical and picturesque historians of his period. The fate of Darius' family is composed like a picture:
His mother commanded respect for her age as well as for her royal dignity, his wife for a beauty that even her current misfortune had not marred. The latter had taken to her bosom her young son, who had not yet turned six, a boy born into the expectation of the great fortune his father had just lost. In the lap of their aged grandmother lay Darius' grown-up but unmarried daughters, grieving for their grandmother as well as themselves. Around her stood a number of high-born women, their hair torn, their clothes rent and their former gracefulness forgotten. (3.24–5)
Both historians tell the famous story of Darius' mother taking Alexander's friend Hephaestion for Alexander, which Veronese turned into a sumptuous picture, but Arrian characteristically says that he will not vouch for it.
It is common ground, though with nuances of disapproval, with Arrian the more inclined to excuses, that Alexander was a man of extraordinary courage, energy and charm, with an ambition beyond all reason. His rages, during one of which, at a banquet, he murdered his friend Cleitus for showing insufficient respect, and was then wildly repentant, are part of his character. Curtius emphasizes his increasing weakness for debauchery, both alcoholic and sexual. The pernicious influence of Asiatic luxury and servility and the contagion of foreign ways have him in their grip. Curtius has him kill Cleitus in a drunken frenzy; Arrian glosses over this, and seems to transfer much of the blame to Cleitus himself. Curtius attributes the burning of the Persian kings' palace at Persepolis also to a drunken revel, and has Alexander incited to the deed by a prostitute, while Arrian treats it as an act of revenge for the past Persian ravaging of Greece.
Curtius eagerly recounts the traditional story of Alexander's sexual liaison with the Queen of the Amazons (6.24–32). Arrian will have none of it: Ptolemy and Aristobulus do not mention it, and anyway there were no longer any Amazons by then, though they had existed once (7.13). In general, Arrian's portrayal is the more political, but he mentions Alexander's weaknesses and deplores his adoption of Persian dress and manners, while admitting a political element in it. He also mentions the excess of Alexander's ambition, while Curtius dwells more on his human weakness. Typically, they approach differently Alexander's claims to divine descent. Arrian treats them as probably manipulative, but does not doubt Alexander's descent from the legendary heroes Heracles and Perseus. Curtius too equivocates over Alexander's claim to descent from the Egyptian god Ammon, whom the Greeks identified with Zeus and Curtius naturally calls Jupiter. Alexander either believed "or wanted others to believe" in his divine ancestry (4.7.8). Accounts agree that Alexander's casting of himself as a type of his ancestor Achilles was part of his character. According to Arrian, he had felt a kind of rivalry with Achilles since childhood (7.14), though it is only Curtius who tells (with disapproval) the tale of Alexander's having his dead enemy Betis dragged by the heels around the city of Gaza in deliberate imitation of Achilles' treatment of Hector (4.6.29). Arrian tells the story (Curtius' account is lost) of Alexander's visit to Troy and his gift of his own armour to the temple, taking away with him weapons from the Trojan War preserved there. It was also said that he laid a wreath on the tomb of Achilles, while his friend Hephaestion laid one on that of Patroclus (1.12).
Where the two historians differ slightly is in their accounts of Alexander's journey across the desert, in Egypt, to the shrine of Ammon (Zeus, Jupiter). Arrian shows a scientific interest in the natural phenomenon of the oasis where the temple stood (3.4), and records Alexander's delight at receiving from the oracle (Arrian adds "or so he said") the desired answer to his question, which was apparently whether he was destined to rule the world. Curtius makes allowance for the possibility that Alexander was manipulating the credulity of others, but implies that Alexander became himself ensnared by it, with bad results for his relations with his independently minded Macedonian followers, though Curtius also subscribes to the cliché (it is found in Polybius and Livy earlier) that "Nothing exercises greater control over the masses than superstition" (4.10.7). His description of Alexander's visit to the temple of Ammon (4.7) is less scientific but more circumstantial and picturesque than Arrian's. He too notes the peculiar properties of the water of the oasis, which was said to be cold at midday and boiling hot at midnight, but gives in addition some interesting details of the cult:
The image worshipped as divine does not have the properties commonly accorded to deities by artists; it most resembles a navel and is composed of an emerald and other jewels. When an oracular response is sought, priests carry this along in a gilded boat from which a large number of silver cups hang on both sides, and married and unmarried women follow singing some artless song by which they believe an infallible answer is elicited from Jupiter. (4.7.23, 24)
Herodotus would have loved the navel, but probably been more circumspect about the possible genuineness of the oracle. Arrian seems reluctant to admit Alexander's superstitious complicity, but there is a good deal in the general consensus about him—ardent, unreflective, impulsive, obsessed with his own descent and destiny—which makes it likely. Arrian seems the more reliable of the two historians: he is more fastidious in what he refrains from describing in imaginative detail, and his accounts are cooler and more rational. Whether that was the best way to understand the mentality of Alexander is open to doubt, and no final adjudication is possible.
The Alexander traditions, if not altogether compatible, left a legacy of images and an archetype to which later rulers aspired and on which sculptors and painters drew. (Plutarch was probably their chief carrier: Curtius was ignored until the Middle Ages.) Julius Caesar and Augustus (as Octavian) both visited the tomb of Alexander, as Alexander had visited that of Achilles. The image of Alexander, the strikingly handsome, godlike young world-conqueror, influenced the iconography of the young Augustus and the young Napoleon. Jacques-Louis David's picture of Napoleon visiting his sick and wounded soldiers (whom he was about to desert) in the hospital at Acre bound together a triple image of Alexander visiting his wounded soldiers after the battle of Granicus (Arrian 1.16), of Christ healing the sick, and of Napoleon. It was a kind of irony that, later, Napoleon's Russian antagonist should have been called Alexander. The son of Antony and Cleopatra was also named Alexander. Alexander the Great, who bequeathed an enduring legendary heritage in both East and West, had really no need to envy Achilles, for
_Is it not passing brave to be a king,_
_And ride in triumph through Persepolis?_
(Marlowe, _Tamburlaine the Great,_ II.v)
The Alexander historians we have, writing four hundred years and more after the events, have inevitably the air of an epilogue, but it has been only through them, in the absence of their sources, that we could complete the theme which was so important for early historiography, the encounters of the Greek and Persian worlds. Consideration of the next great theme, though it is still treated by Greek writers, requires a retracing of steps to the second century BC, to the earliest historians of the rise of Rome.
**PART II**
> **ROME**
**FOUR**
**Polybius: Universal History, Pragmatic History and the Rise of Rome**
Polybius, a Romanized Greek of the highest class, set himself in the middle part of the second century BC to write, chiefly for his fellow Greeks, the story of the rise of Rome to dominion over the whole Mediterranean area. His focus was inevitably the long struggle between Rome and Carthage for the hegemony of the western Mediterranean, but his history also comprehended the successful assertion of Roman power over Macedon and Greece. Polybius' account covered the years 264 to 146 BC. The period 264 to 241, the time of the First Punic (Carthaginian) War, is in Polybius only a preamble. The core of the history is the second or so-called "Hannibalic" War (after the Carthaginian general), from 218 to 202 BC. Polybius became friend and adviser of the Roman general Scipio, who finally destroyed Carthage in 146 BC, though by then Polybius had returned to his native Greece. The rise of Rome involved an irreversible shift of the world's political centre of gravity from Greece, Macedon and Asia Minor to Rome, and one of Polybius' objects was to educate his fellow Greeks in the realities of Roman world power.
Polybius was born in Megalopolis, in the Peloponnese, at the end of the third century BC (the exact date is not known). In 167 BC he, with other aristocratic Achaean Greeks, was transported as a hostage to Rome, where he became highly placed and influential. Like other famous exiled historians—Thucydides, Xenophon and, later, the Jewish historian Josephus—he benefited from the perspectives that exile provided. He returned to Greece towards the end of his life, winning the gratitude of his countrymen for the way he then acted as mediator with their Roman masters: when he died, in or around 118 BC, statues were erected in his honour. He was therefore not only a man of two worlds but, like other prominent ancient historians, a public man—a fact of whose advantages to the historian he was proudly, even arrogantly, convinced. Rather more rarely, his was a public career crowned with success.
As a historian, he is at first sight something of a hybrid. Most obviously, in his frequently proclaimed conception of the historian's task and methods, based on the notions of truth and usefulness rather than entertainment, he follows the example of Thucydides: rebuking an earlier historian, Phylarchus, for the imaginative embellishments of his narrative, designed to elicit the reader's sympathy and pity, Polybius declares that "It is not a historian's business to startle his readers with sensational descriptions, nor should he try, as the tragic poets do, to represent speeches which might have been delivered [Polybius is notably restrained in his use of direct speech]...It is his task first and foremost to record with fidelity what actually happened" (II.56). Only by being true can history fulfil its mission of being useful. History is learning by vicarious experience, and its lessons are inferred from what has happened to provide guides for future conduct. Polybius resembles Herodotus more than Thucydides in the frequency with which he addresses the reader directly, but the effect is completely different: it is not Herodotus' confiding conversationality, but an insistent didacticism, a sleeve-plucking pedagogical concern that the lesson conveyed in his narrative shall not be overlooked.
But in speaking of Polybius' "Thucydidean" insistence on the usefulness of history, we are ignoring what he chose to write about. Though the centre of Polybius' work is the narrative of a great war, the "Hannibalic" War, his history as a whole has an almost Herodotean breadth and ambition of scale. Polybius has neither quite the epic quality nor the entertainment value of Herodotus, nor the brilliance of Thucydides—as indeed who else did?—but it is not inappropriate to speak of him as an heir of both. Polybius himself is scathing about merely monographic history (that is, history devoted to a single theme, in contrast to universal history), even though the history he places highest on the scale is political and military and based on travel and on direct or indirect eyewitness testimony. His insistence on scale reflects, for him, the conditions of a changed world. World history now has a central theme, the rise of Rome, and any history unrelated to this is necessarily petty and parochial. For Polybius, writers of such monographs are necessarily driven to picturesque adornments in order to disguise the essential poverty of their themes (VII.7). Polybius speaks of the unity of his theme and hence his history in terms clearly borrowed from Aristotle's conception of the unities in drama: "How, when and why all the known parts of the world were brought under the domination of Rome is to be seen as a single action and a single spectacle, which has an identifiable beginning, a fixed duration and an acknowledged end" (III.1).
Like other ancient historians, then, Polybius has looked for and found the "greatest" event of his day as the theme for his history. But this cannot be monographic, because its theatre is, to Polybius, no less than the known world. It is universal history—the first mention of a category that was to have much currency later, especially in the Middle Ages, when Christianity provided what was taken to be the universal theme.
My history possesses a certain distinctive quality which is related to the extraordinary spirit of the times in which we live, and it is this. Just as Fortune has steered almost all the affairs of the world in one direction and forced them to converge upon one and the same goal, so it is the task of the historian to present to his readers under one synoptical view the process by which she has accomplished this general design. It was this phenomenon above all which originally attracted my attention. (I.4)
Polybius was able to write a kind of universal history that was nonetheless eyewitness-based because his theme, which is the quite recent rise of Rome to world dominion, requires only a relatively short timescale. For example, unlike his successor Livy, who also traced the rise of Rome, he does not attempt to take his account back to the foundation of the city, and he is fairly dismissive about such antiquarian research (XII.25e, i). Rome's conquest of its Italian neighbours, which takes up the first part of Livy's work, is taken for granted, though the encounters of Rome with the Celtic Gauls are given more attention because Polybius regards them as a kind of training for Rome's later world role. But the actual rise of Rome to world power was, as Polybius stresses, in fact outstandingly swift—taking less than a century. Polybius' history, then, is able to be both comprehensive in scale yet to extend back little beyond living memory.
In speaking of Polybius, for convenience, as an heir of Herodotus and Thucydides, we are committing an act of drastic simplification and foreshortening. Herodotus and Thucydides had numerous heirs—more than we can know of. There were several attempts to continue Thucydides' history, one of which, by Xenophon, we have noted. Polybius, however, mentions Thucydides only once, and Herodotus not at all. But Polybius is in fact, for an ancient historian, exceptionally profuse in references to his predecessors—all of them now lost to us. As the convention was, his references to them are all critical—abusive would sometimes be the word—but they are highly valuable in lifting a corner on a lost and once clearly copious historiographical world. Querulous though Polybius' attitude towards it is, he clearly has an awareness of historical writing as a continuing and in a sense even cooperative activity: he remarks that he is confident that, should he die before completing it, his history would be taken up and completed by others. This is, of course, a way of saying that his theme is too great to be ignored, but it is also a tribute to the existence of a community of historians.
As his precursors in various ways, Polybius mentions Timaeus (at length in Book XII, and abusively), Ephorus, Phylarchus, Theopompus, Aratus, Philinus and the Roman historian, who nevertheless wrote in Greek, Fabius Pictor. All their works are now lost, but they are clearly for Polybius the historians in possession of the field of historiography that he intends to cultivate, and in some cases he is clearly following them—though critically. He is anxious to claim that none of them has properly seized the historiographical opportunities and that his own work is therefore a genuinely pioneering one: "While various historians deal with isolated wars and certain of the subjects connected with them, nobody, so far as I am aware, has made any effort to examine the general and comprehensive scheme of events, when it began, whence it originated, and how it produced the final result" (I.4). We are hampered by knowing so relatively little about these writers, though Polybius is not our only source, but in general his claim seems justified. It is, however, all too easy to gain a false impression of ancient historiography by forgetting how much of it has failed to come down to us, which is why even Polybius' diatribes are useful. Herodotus, Thucydides and Xenophon have been preserved for us by the authority and popularity they early acquired, which ensured multiple copying and hence survival (though sometimes precarious) through the ensuing centuries. We can be reasonably sure that nothing of comparable quality has been lost—the ancient librarians showed good taste, in Alexandria and later in Byzantium—but among lesser historians substantial survival is the exception, complete extinction or survival at best in epitomes or collections of extracts being the norm. Copyists consulted the interests of readers, not those of posterity. Even Livy, who was to become Polybius' chief rival as historian of Rome, survives only in severely curtailed form.
The accidents—not entirely accidental—of extinction and survival may easily give us an oversimplified and distorted picture not only of the quantity but also of the kinds of history commonly written in the ancient world. Thucydides' example, a monograph of contemporary political and military history, was a powerful one and may have damaged the reputation of Herodotus, as did a waning sympathy with the folkloric elements which Herodotus carefully recorded. But Thucydides' history, itself not exempt from criticism in antiquity, was far from being the only kind for emulation. Even the assertion, based chiefly on the Thucydidean example, that ancient historiography made little use of documentary records is a significant overstatement in the case of Rome. Patriotism, devotion to Rome's ancient laws, and family pride (Fabius Pictor, one of the first historians of early Rome, was a member of the great Fabian clan) fostered antiquarian inquiries in a way only faintly foreshadowed in the Greek interest in genealogies and the foundation of cities and their local annals. We shall have to come back to this point later in considering Livy and his predecessors: the title of Livy's history, _Ab urbe condita,_ "From the foundation of the city," stakes out a bold and consciously un-Thucydidean claim.
We need some awareness of Polybius' predecessors if we are to try to assess his originality as well as his possible debts. In particular, we need to attend to the genre of "universal history" and how far his predecessors anticipated Polybius in writing it. One of the matrices, as in the case we considered earlier of Herodotus' Ionian precursors, seems to have been the mixed genre, by our standards, of geography, ethnography, legend (origins) and history—we might now be tempted to use a term like "area studies." We are essentially concerned with the Mediterranean basin, and particularly with the western Mediterranean. Polybius knew, of course, that the Persian empire had extended far to the east and had had its own claims to be the world empire of its day, as Alexander's had been as its successor; but for him Asia had been, as we should say, "done." Rome, and Rome's expansion south and east, was where world history was now in the making.
Polybius scorned the efforts of his predecessors who had treated the western Mediterranean and Greece—Philinus, Fabius, Ephorus and Timaeus—for having failed to produce a unified history, offering instead separate accounts of adjacent countries and events, with no central organizing theme (I.4). Other historians received other dismissals—Phylarchus for picturesque invention, and Theopompus, a historian of Greece, for indecent language in describing the dissolute court of Macedon. Polybius was clearly a firm upholder of what came to be called "the dignity of history," to use the English phrase coined in the eighteenth century (VIII.10). What earlier historians had failed above all to grasp was both the significance of the rise of Rome and the new historiographic opportunity it offered. Universal history has now for the first time become possible and Polybius intends to write it: "From this point onwards history becomes an organic whole: the affairs of Italy and of Africa are connected with those of Asia and of Greece, and all events bear a relationship and contribute to a single end" (I.3). Polybius speaks like a historian with a vision, who has found in it his lifetime's work: "Just as Fortune has steered almost all the affairs of the world in one direction and forced them to converge upon one and the same goal, so it is the task of the historian to present to his readers under one synoptical view the process by which she has accomplished this general design" (I.4). It is vital to the Greeks to understand what has happened, so that they may learn to live in a Roman world.
Polybius says that he takes as the starting point for his prefatory opening book the moment when the Romans first crossed the sea (to Sicily). He also hints at another reason, for this moment coincided with the point at which Timaeus' history leaves off, "namely in the 129th Olympiad" (264–260 BC) (I.5), while the history proper begins at the time of the 140th Olympiad (220–216 BC). Polybius also marks the beginning of the establishment of Roman power in Italy by reference to Greek chronology, notably the conclusion of the Peloponnesian War and the decline of Sparta (I.6). The Romans, having hardened themselves in their local wars with the Samnites and the Italian Celts, were able to defend themselves against the invasion of King Pyrrhus of Epirus in 280 BC and subjugate Italy. Later (Book II) Polybius undertakes, rather uncharacteristically, a geographic and ethnographic digression on northern Italy and the cisalpine Gauls.
It was in Sicily that the Romans first came into conflict with the Carthaginians, and thus the detailed part of Polybius' narrative begins with the events of the First Punic War. He makes the usual claims for the greatness of his theme, in the length and vicissitudes of the Punic Wars and the pertinacity of the equally matched combatants. The two states "were still [we shall have to return to the implied reservation conveyed by that last word later] at that time uncorrupted in their customs and institutions" and "both received no more than moderate help from Fortune," another Polybian theme to be considered later (I.13). Polybius adds that the two historians who are regarded as authorities here, Fabius and Philinus, have both erred by exhibiting bias, towards Rome and Carthage respectively. Polybius aims to correct these biases: "If history is deprived of the truth, we are left with nothing but an idle, unprofitable tale." In fact it seems clear that he is following Fabius and Philinus closely (I.13, 14).
Although Rome's wars with Carthage form the narrative core of his work, Polybius is conscious of the responsibilities imposed by universal history: "I...have set myself to describe what was happening in all the known parts of the world at once" (II.37). So he also deals with affairs in Greece, dominated at the time by two rival leagues of cities, the Achaean League and the Aetolian League (Polybius himself was an Achaean). Rome was drawn into Greek affairs through its invasion of Illyria, the latter being at the time in conflict with the Aetolians. With the ending of Book II Polybius has completed his preamble and is prepared for his grand theme: "how, when and why all the known parts of the world were brought under the domination of Rome."
One has to admire the ambition of Polybius' historical enterprise, and fundamentally he was right: the growth of Roman power was the salient fact of contemporary history. But it has to be admitted that Polybius' deliberate shifts of focus for brief periods, from Greece to Macedon, to Egypt to Spain, in order to fulfil his project of writing world history in an annalistic form, are disruptive and, for the non-expert, confusing. What really holds the work together, apart from the central narrative of the Punic Wars, is Polybius' authorial personality, marked by his recurring preoccupations with causality, comparison, constitutional factors, the lessons of experience, and the influence of fortune, which he groups together under the name of "pragmatic history." As historical narrative, one of its strengths—not surprising from the author of a lost book on tactics—is in its handling of military matters, which make up much of the whole. Here, unlike some other ancient historians, Polybius is precise and analytic, and inspires confidence. At the heart of the narrative of the Punic Wars is the protracted invasion of Italy by the Carthaginian general Hannibal, to whose extraordinary career and outstanding generalship Polybius pays full tribute.
As we have seen, Polybius follows Thucydides in his repudiation of history written merely to entertain. He is severe, for example, on Phylarchus for loading his narrative with picturesque imaginative detail:
In his eagerness to arouse the pity of his readers and enlist their sympathy through his story he introduces graphic scenes of women clinging to one another, tearing their hair and baring their breasts, and in addition he describes the tears and lamentations of men and women accompanied by their children and aged parents as they are led away into captivity. (II.56)
Polybius—prototype of the gruffness of the future professional historian—will follow only the austere path of historical truth. (There will presumably always be Phylarchans and Polybians.) We have an example of this austerity at a moment when it seems particularly repressive, in the account of Hannibal's army crossing the Alps, the most famous episode in the history (III.47–55). Picturesque historians, Polybius claims, contradict themselves by hailing Hannibal as a great and far-sighted general while being compelled by the requirements of dramatic narration to present the crossing as beset by appalling difficulties, and in order to resolve the contradiction they introduce the guidance of gods into "what is supposed to be a factual history." Polybius himself seems bent on vindicating Hannibal's calculation and on literally demythologizing his exploit. He has, he tells us, traversed the terrain to form his own impressions. But only a few paragraphs later we have tracks made impossible by an avalanche, pack animals falling over precipices, mules and horses stuck fast in snowdrifts—all the stage furniture of picturesque history. This is, perhaps, less a piece of backsliding in Polybius as a historian than testimony to the impossibility of presenting the passage of an army over the Alps, with or even without elephants, as a piece of ordinary military logistics: sometimes history just _is_ picturesque.
The same is perhaps true of another moment of high drama, the Carthaginian capture of the city of Tarentum, to which Polybius does full justice: the conspirators inside the walls sending fire signals from an empty part of the city reserved (unusually, in antiquity, within the walls) as a cemetery, deserted but for the tombs; the Roman officers carousing; the entrance, in another part of the city, of a young man, known to the guards, with a dead boar, as though from a hunt, but followed by Carthaginian soldiers; the terror of the Tarentines and the massacre of the Romans (VIII.24–30). (Livy's account (XXXV.8–10), which makes more of the hunting ruse, is for once less dramatic.) Polybius could no doubt justify this by an interest in the stratagem employed, but stratagem and excitement happily coincide.
Excelling in the description of tactics and weaponry and their influence on military outcomes, Polybius devotes a whole section of his history (VI.19–42) to a systematic and extended description of the organization, tactics and equipment of the Roman army—a digression much admired in the Renaissance. His attention to relevant technicalities is also evident elsewhere, for example in his account of the naval warfare in Sicily, where he describes in detail the Roman invention of an offensive contraption known as "the raven," attached to the ships' bows, and its decisive effects (I.22–3). He also praises the extraordinary ingenuity shown by the mathematician Archimedes in inventions designed to thwart the Roman besiegers of Syracuse (VIII.3–6). (It is, however, characteristically Livy's story which has Archimedes killed by a soldier when the city fell, while he was drawing diagrams in the dust.) Polybius' descriptions of the encounters of the Romans with the Celtic warriors of northern Italy offer a nice study in the diversity of weapons and tactics. The Roman javelin was effective against the Gauls partly because of the deficiencies of the Gallic shields:
The shield used by the Gauls does not cover the whole body, and so the tall stature of these naked troops made the missiles all the more likely to find their mark...The Roman shields, I should explain, were far better designed for defence, and so were their swords for attack, since the Gallic sword can only be used for cutting and not for thrusting. (II.30)
The Gallic swords had the further disadvantage of bending, and "unless the men are given time to straighten them with the foot against the ground, the second blow has virtually no effect" (II.33). For the connoisseur of military hardware and its qualities, Polybius is exceptional in detail and clarity. His accounts of the crucial battles of Trasimene (217 BC) and, above all, Cannae (216 BC), by which Hannibal established his army in Italy, are convincing, without apparent exaggeration or rhetorical embellishment. It is noteworthy that while Polybius gives, as one might expect, the gist of the generals' addresses to the troops, he presents them in indirect speech. However, the number of speeches given verbatim increases towards the end of the history, when Polybius was much closer to events and even a participant in them.
Polybius' weakness, at least in our eyes, is not rhetorical over-elaboration but sententiousness. It being his creed that history must above all be written to be useful, he is as ready to adorn his campaign and battle narratives with useful tips as he is to offer advice on political matters. Sometimes, it has to be admitted, the result is platitudinous or even tautological. Polybius can assure us that human nature is fallible(II.7) and that we should not rely on a continuation of Fortune's favours(I.35), while for a general to be a coward or a fool is productive of damaging consequences. The tendency to tautology is exemplified in a claim such as that "rashness, excessive audacity, blind impetuosity, vanity or foolish ambition" are exploitable weaknesses, where Polybius seems indifferent that the choice of nouns and adjectives, especially the latter, has already predetermined the outcome (III.81). More interesting and less trite are maxims which tempt one to apply the anachronistic adjective "Machiavellian." One such, for example, is the distinction Polybius makes between men who come to terms as a matter of yielding to circumstances and those who do so because their spirit is broken. The latter may be trusted, the former not (III.12). Another is the dictum that humanity after victory, as practised by Scipio, is good policy(X.36). More "Machiavellian" in the vulgar sense is the endorsement of Scipio's employment of superstition to encourage his troops (X.11). Polybius' approval of the politic use of religion was more than occasional: it was one of his reasons for admiring the Roman state.
We have already seen that for Polybius history must above all be truthful, because only on the basis of truth can we distil from it the lessons of experience—painless ones, unlike those from direct experience—which constitute history's utility and justification. There are many references to Polybius' understanding of the historian's task scattered throughout his work, but the fullest account occurs in Book XII, which is wholly and almost obsessively devoted to castigating the errors of other historians, and above all of Timaeus. The attack contains a good deal that seems petty and pedantic, but the book as a whole gives a conspectus of contemporary attitudes to history, seen from Polybius' distinctive point of view. Timaeus, according to Polybius, is a man of book-learning and documents: "he has quite neglected the business of making first-hand inquiries, which is the historian's most important duty." Since the historian cannot be an eyewitness to everything, what remains is to question as many people as possible, and to exercise judgement on what he hears. Timaeus, however, is prey to accounts of dreams, prodigies and other superstitions. In setting down the speeches of generals and statesmen, he has first made up his mind what ought to have been said and has then composed imaginary speeches as though writing a rhetorical exercise: "He has neither set down what was said nor the real sense of what was said."
Because, in Athens, he had access to the works of other historians, Timaeus assumed he had the material for writing history. He belongs to the class of historians who haunt libraries and dwell among memoirs and records. Timaeus, according to Polybius, is—to invoke a term from Carlyle—the historian as "Dryasdust." Documentary sources are of some value, but wholly inadequate to write a history of recent events(XII.25e). For this one needs—enter Polybius—"those who have played some part in affairs themselves." Those without such experience, civil and military, are disabled from understanding affairs and hence incapable of instructing others in their lessons. "It is in fact equally impossible for a man who has had no experience of action in the field to write well about military operations as it is for a man who has never engaged in political affairs and their attendant circumstances to write well on those topics" (XII.25h, g). The study of documents is therefore only third in importance to the historian, behind a knowledge of the relevant topography and practical experience. Timaeus, not an eyewitness to events, "has preferred to employ his ears," which is the inferior way—and even in this Timaeus' method is inferior, since the ears may learn "either by reading or by the examination of witnesses" and Timaeus has preferred the former (XII.27a). (It is interesting that Polybius regards reading as receiving information through the ears.) Polybius himself did not altogether disdain documents, such as the bronze tablet he had discovered which Hannibal himself had caused to be inscribed with the details of his forces: this was another form of eyewitness testimony, and "I considered this to be an absolutely trustworthy piece of evidence" (III.33).
For Polybius, the relationship between history and the experienced man of affairs is reciprocal. The latter makes the best historian, but in turn the best-instructed man of action is one versed in the lessons of history. Polybius is one of the first we know of explicitly to regard history as a training for a political career, though he presents it as a common claim. All historians, he says, have asserted that "the study of history is at once an education in the truest sense and a training for a political career, and that the most infallible, indeed the only, method of learning how to bear with dignity the vicissitudes of Fortune [ _Tyche_ ] is to be reminded of the disasters suffered by others" (I.1).
In being concerned above all with the study of causes and their consequences, and with the rise of Rome as the central event to be explained, the two aspects of Polybius' historical writing, "universal history" and "pragmatic history," are not in conflict but complementary. It is only large-scale comparative history which affords the parallels on which the study of causes can be based (what J. S. Mill was to call "the method of difference"): according to Polybius, "by far the most important part of historical writing lies in the consideration of the consequences of events, their accompanying circumstances, and above all their causes...All these tendencies can be recognized and understood from a general history," but not from monographic treatments(III.32).
Making a comparison with the method of observation in the practice of medicine, Polybius, in approaching the outbreak of the Second Punic War, draws distinctions between the beginning, the pretext and the cause. The cause is what shapes purposes and decisions, the beginning is what gives effect to them (III.6). In tracing causes, Polybius gives primacy, as one might expect him to, to the influence of laws and institutions, and above all, in the case of Rome, to its political constitution, to which a whole book is devoted (VI). In the long, hard struggle with the Carthaginians, the quality of the Romans—their patriotism, resolution and steadfastness—was tested to the utmost. It was moral qualities rather than material resources that, at critical moments, gave them ultimately the advantage. These qualities were fostered by Rome's laws and institutions. Roman customs were nurseries of public spirit—like, for example, the funeral rites and orations in honour of distinguished men: "It would be hard to imagine a more impressive scene for a young man who aspires to win fame and to practise virtue"(VI.53–4). "Virtue," as always in this way of thinking, means public virtue, the qualities proper to a man ( _vir_ ). Polybius here goes on to tell the now well-known story of the heroic self-sacrifice of Horatius at the bridge, holding back Rome's enemies. This episode, also told by Livy (who makes Horatius survive), was used by Macaulay in his _Lays of Ancient Rome._ (This Horatius, Horatius Cocles, is not to be confused with Publius Horatius, whose story inspired Jacques-Louis David's painting _The Oath of the Horatii._ ) As we have already seen, the manipulation of superstition through public religious rites was admired by Polybius: it was what "holds the Roman state together," for the masses can be controlled only by mysterious terrors. Religious matters are treated in Rome "with such solemnity and introduced so frequently both into public and into private life that nothing could exceed them in importance." For Polybius the introduction of such beliefs and rites was a wise and politic measure by "the ancients," and the moderns take great risks in rejecting them (VI.56).
The Roman state was tested almost to destruction by the great defeat by Hannibal at Cannae. It was, according to Polybius, only the "peculiar virtues" of the Roman constitution that allowed it to surmount this crisis (III.118). The constitution was itself then in its prime, and according to Polybius "a noble spectacle," while that of the Carthaginians was already declining (i.e. becoming more democratic). After Aristotle, and particularly with a strong revival from the Renaissance to the eighteenth century, Polybius was to become the foremost authority on the idea of the three types of constitution, and the cycle through which they pass as each pure form mutates into its corrupt counterpart (kingship into tyranny, aristocracy into oligarchy, democracy into mob rule) and then into its corrective opposite. Polybius subscribed to the view, particularly popular in England in the seventeenth and eighteenth centuries, that the cycle could be suspended, at least for a while, by a balance held between the three elements; for the English there was a particular resonance in the notion, proclaimed by Polybius, that the Romans arrived at their balance not by abstract reasoning but by trial and error (VI.10)—something the English Whig tradition never tired of ascribing to England, in contrast to the French.
Though, according to Polybius, the Roman constitution at the time of the Second Punic War was still at its optimum point, an inseparable feature of the concept of balance and its concomitant, corruption, was anxiety. Polybius sees no time limit to Rome's empire, but Rome's constitution "will pass through a natural evolution to its decay," for "each constitution possesses its own inherent and inseparable vice" (VI.9, 10). Before the lost battles of Trasimene and Cannae, says Polybius in one particularly pessimistic reference, the signs were already showing themselves in a demagogic tendency which led to the appointment of the wrong generals, a lurch toward an overbalance of the democratic element, and hence, "the first step in the demoralization of the Roman people" (II.21).
Demagogy as a symptom of corruption and decline was directly linked to the theory of the cycle. The other main indication of both ills was a more traditional blight: luxury, often associated with foreign influences. As the Greeks had blamed the Persians, so the Romans, notably that stern moralist and historian of early Rome Cato the Censor, came to blame the Greeks. Polybius endorses this view. Luxury and ostentation are the price of conquest and empire, as when the riches of Macedon were brought to Rome (XXXI.25). Too much prosperity is dangerous. It is as though the old conception of hubris, which in Herodotus brought down King Croesus, byword for wealth, has acquired the anxieties of a rude agricultural community waxing rich and transforming itself through commerce and conquest. It was an anxiety which, derived by Renaissance historians, notably Machiavelli, from the Roman historians Sallust and Livy, passed on from them into the eighteenth century, where it was to inform Edward Gibbon's account of the fall of the Roman empire, the vast counterpart to Polybius' work two thousand years earlier. The corrupting political consequences of luxury and venality became a staple of opposition rhetoric in eighteenth-century England. It is again no accident that the period from the Renaissance to the eighteenth century was the heyday of Polybian influence on European historical and political thought.
Polybius, had he been able to reflect on it, would have ascribed Rome's fall not only to luxury, corruption and the loss of the balanced constitution, as did his remote successors, but, as he also ascribed Rome's rise, to Fortune. We can be sure of this, because there was no large-scale historical development which, for all his interest in mediating causes, he was not prepared to ascribe to Fortune. It was Fortune which decided on and brought about Rome's world dominion. Fortune "steered almost all the affairs of the world in one direction and forced them to converge upon one and the same goal." Rome triumphant was a tour de force on the part of Fortune, which had never before "put on such a show-piece as that which we have witnessed in our own times," an achievement "the most excellent and profitable to contemplate"(I.4). This sounds like the language of religion, and certainly Polybius seems to personify Fortune as a goddess. It was as such that she was hailed in the Middle Ages and the Renaissance. It is natural to speak of Polybius' "Fortune" as a kind of divine providence. But elsewhere Polybius speaks of Fortune as one does of blind chance. Fortune disposes of events simply arbitrarily. Fortune as contingency is what is left as an explanation when human reason has exhausted its resources. It is reasonable, Polybius says, to attribute such unforeseeable contingencies "to the work of a god or of chance" (XXXVI.17). But this conception of Fortune as contingency does not seem one that could plan and bring about such a long-term and immense phenomenon as the rise of Rome, which does not appear to defy human reason, since Polybius can enunciate its causes, and does, while admitting that at times the Punic Wars were a close-run thing. We have to conclude that Polybius' conception of Fortune, though certainly of great importance to him, was essentially incoherent.
Polybius wanted to instruct, not to charm, his readers. He stands among the historians who have tried to use history as the basis for what would later be called historical, or more frequently political, science. For such a science the eighteenth century, in particular, was to turn to Polybius himself. In his desire to find a central theme in history, in his relative detachment and concern with accuracy, as well as in his interest in drawing usable lessons from history, he is closer to being our intellectual contemporary than are the great Roman historians who came after him, Sallust and Livy, whose focus was moral rather than what Polybius called pragmatic. This perhaps benefits him less than one might expect among modern readers, compared with the passionate, even agonized, involvement in their own times of other ancient historians, with whom we can empathize. We have our own, more recent, instructors in the lessons of history: Machiavelli, Montesquieu, Hume, Henry Adams. But Polybius is their precursor. In embodying his concerns in a massive account of the central theme of his time, the rise of Rome, he made enduringly for himself a prominent and distinctive role in the history of historiography.
**FIVE**
**Sallust: A City for Sale**
Gaius Sallustius Crispus (Sallust) wrote the works on which his reputation rests in the early 40s BC. The periods on which he wrote were recent. His fame rests on two short monographs of near-contemporary history; he also wrote a general history covering a decade of the 70s and 60s BC, but only fragments survive. In the ancient world and in the Middle Ages, Sallust was highly regarded both for the content of his histories and for his famously economical style, and he has been ever since. Tacitus admired him, as Sallust himself admired Thucydides, and St. Augustine quoted him. His is an extraordinary reputation to rest on two short monographs on events that were arguably not of the first importance.
One was a colonial war in North Africa against the Numidian king Jugurtha at the end of the second century BC. Sallust makes the usual claim for his subject's importance—it marked, he says, the beginning of civil strife in the Roman world—but this is special pleading: the fame of the Jurgurthine War owes more to Sallust than his to it; it seems likely he was drawn to the subject because he himself had been governor of Numidia half a century later. His other subject, the conspiracy of Catiline to overthrow the consular authority in Rome by force (65–62 BC) was certainly noteworthy and has been immortalized not only by Sallust but also in four celebrated orations by Cicero, who was a consul during the critical period. Sallust claims the conspiracy represented a threat without parallel to the Roman republic, and was an enterprise of unprecedented wickedness and criminality. Certainly had it succeeded the consequences would have been far-reaching; it was a symptom of the instability and lawlessness which several times plunged Rome into civil war in the first century BC.
Sallust was himself a politician in that insecure world. The Jugurthine War was still within easy living memory when he was a young man; Catiline's conspiracy was a contemporary event, which Sallust described twenty years afterwards. Sallust was a client of Julius Caesar, who was suspected, probably unjustly, of complicity in the plot and it has been claimed, though without much credence, that _The Conspiracy of Catiline_ was a propagandist account to exonerate Caesar. Though Sallust produced two case studies, rather than a continuous account, his works introduce the major figures in the political scene of the first half of the first century: the two political adventurers Marius and Sulla, who successively mastered Rome in its early decades, and conducted ruthless massacres and proscriptions (outlawing of political opponents). Both made their reputations in the Jugurthine War, and Sallust draws portraits of both, admiring their qualities and deploring their subsequent careers. There is no intimate portrait of Cicero, or of Pompey the Great, who is a peripheral figure in the _Catiline,_ but Caesar is given a notable speech and a eulogy.
At the heart of the problem of the Roman state in the first century, as its contemporary historians generally recognized, was the growth of a plutocracy, fattened on the proceeds of conquest, which engrossed much of the land of Italy and farmed it with slave labour. This, and the length of service far from Rome which the empire required, caused a breakdown of the old military system of a citizen militia whose members in theory returned to their farms at the end of their service. Marius had recognized the new reality by greatly reducing the property qualification for service, creating, in effect, a largely professionalized army, dependent on its commanders for procuring its pay and pensions. This was realistic, but had dire consequences as the provincial governors who commanded large armies became semi-independent warlords whose troops' loyalty was to them rather than to the republic. Marius, Sulla and subsequently Caesar conducted what were in effect military takeovers, and Pompey threatened to do the same. At the same time, a significant political fissure, much to the fore in Sallust's account of his times, became evident between the old senatorial elite, determined to cling to its privileges and monopolize the greater offices of state, and "new men"—Marius was the archetype—who won their way by ability and in some cases achieved great wealth, and who resented the arrogance of the old nobility.
Sallust, who in 52 BC became a tribune of the plebs, their official representative and a high authority, very clearly gave his sympathies to the new men and is eloquent in condemnations—in speeches, notably by Marius ( _Jugurthine War,_ 85.10–48), and in his own person—of the grasping selfishness of the traditional elite. His own official career seems to have been decidedly shady, which constitutes an ironic comment on his frequent and vehement denunciations of avarice and the low standard of public life: he was expelled from the Senate in 50 BC for alleged immorality, but Caesar supported his career. After serving Caesar as an officer in the civil war, in 45 BC Sallust was installed by him as governor of Numidia, where he became notorious for extortion. He retired from politics, after Caesar's murder, a very rich man. In his prefaces he extols the life of retirement and rejoices in his escape from the envy, backbiting and servility of public life. The true life of man—there is a Stoic or perhaps Platonic high-mindedness on display here—is the cultivation of one's intellectual powers and the pursuit of a worthy fame. The depiction of great deeds in writing history belongs to the best life, no less than the performance of them. His self-congratulation on his present tranquillity of mind, and on his impartial detachment and happiness at having escaped the sordid conditions of political advancement in a corrupted political state, may express a genuine relief—now he was rich—but at times one is driven to suspect the bitterness or at least ambivalence of the superseded politician in his denunciations.
Sallust shares with other contemporaries a deep pessimism: Roman public morality is in steep decline. He clearly admires some of the traditionalist attitudes associated with Cato the Censor (Chapter 5), and he gives an effective speech very much in that vein to Cato's great-grandson of the same name ( _Catiline,_ 52.8–53.4). Sallust became a highly influential critic of luxury and avarice as the vices which had undermined the old Roman virtues of austerity and integrity, speaking disapprovingly, for example, of Asiatic indulgences and a taste, corrupting to the Roman soldiery, for statues, pictures and embossed plate ( _Catiline,_ 10.6); the reference to embossed plate carries conviction. He often focuses on something he must have known very well: the characters and weaknesses, especially the vulnerability to bribery, of public men. He shares the typical idealization of ancestral Roman public virtue; its decline is a recent and a humiliating contrast. His characterizations of the new manners are generally pointed and precise, even if surely exaggerated. His two monographs are centred on two rather different, though related, kinds of political vice, which are represented as virtually omnipresent: venality in _The Jugurthine War,_ unprincipled ambition as a form of greed in the _Catiline._
_The Jugurthine War_ has two aspects: the overt and military, and the covert workings of political intrigue; it is the latter which is usually decisive, up to and including the capture, by treachery, of Jugurtha which brings the war to an end. Below the overt military operations, in which the Romans generally prevail, forcing the Numidians to adopt hit-and-run tactics in a manner characteristic of colonial wars, is a second, less visible, contest, nicely poised because of the venality of Roman senators and Numidian courtiers alike. In Sallust's account, the course of events is often dictated by who has successfully bribed whom: it is a world in which the corruption of individuals goes far towards controlling public affairs. Jugurtha, a client king whose methods of obtaining the throne have discredited him in Rome, devotes himself with considerable success to suborning the Senate by bribery and bringing a number of senators to serve his interests. Sallust's comment is "The public good, as so often happens, was sacrificed to private interests" ( _Jugurthine War,_ 25.5). Meanwhile the Roman commanders sent to bring Jugurtha to heel bribe his aides and allies to betray him. But they themselves are not all immune. Of the consul sent to command the army Sallust characteristically comments that his good qualities were made useless by his avarice. Bribed, he agrees to a truce (28.2).
Sallust seems to choose as his spokesman at this point a tribune-elect, Gaius Memmius, a well-known opponent of the nobility, to whom he attributes a speech which Sallust says, in his habitual formula, "ran somewhat as follows." Denouncing the abuses committed by the clique of nobles to secure their own position, Memmius makes their corruption the crux of his attack:
What manner of men are they who have made themselves rulers of the state? They are evildoers whose hands are red with blood. Covetous beyond measure, and stained with guilt, they are none the less swollen with pride, and there is nothing that they will not sell: honour, reputation, natural affection, every virtue indeed—as well as every vice—is to them a source of profit. (31.16)
Memmius' speech reflects the enduring bitterness left by the events of the 130s and 120s BC, when the tribunes of the plebs Tiberius and Gaius Gracchus, having introduced radical measures to benefit the poor, were successively murdered—events to which Memmius clearly refers. It is likely to strike the reader that the passion he expends on the ostensible issue, how Jugurtha was to be treated, is excessive, but it is explained by the climate of intense hostility to the nobles. Sallust himself attributes the mob's animosity towards Jugurtha to class hatred rather than patriotism. This mutual political hatred is, as it were, the subplot of _The Jugurthine War._ Sallust traces it to the defeat of Carthage (146 BC). With no external threat, a salutary restraint was removed. Civil strife was the product of peace and prosperity:
The division of the Roman state into warring factions, with all its attendant vices, had originated some years before, as a result of peace and of that material prosperity which men regard as the greatest blessing. Down to the destruction of Carthage the people and the Senate shared the government peaceably and with due restraint, and the citizens did not compete for glory or power; fear of its enemies preserved the good morals of the state. But when the people were relieved of this fear, the favourite vices of prosperity—licence and pride—appeared as a natural consequence. Thus the peace and quiet, which they had longed for in time of adversity proved, when they obtained it, to be even more grievous and bitter than the adversity. (41.5)
The nobles abused their power, the people their liberty, "every man snatching and seizing what he could for himself." But the nobility held the advantage: "The people were burdened with military service while the spoils of war were snatched by the generals." However, when the day came "when noblemen rose to power who preferred true glory to unjust dominion: then the state was shaken to its foundations by civil strife, as by an earthquake" (41.5); Sallust is referring to the Gracchi, murdered by the oligarchs. By their political murders and banishments, the intransigent nobles condemned themselves to live in fear. The resort to extremes, the reader is led to feel, is irreversible; one is reminded, perhaps rightly, of Thucydides' account of the breakdown of civil order in Corcyra.
As a sequel to Memmius' speech, Jugurtha—who has meanwhile been bribing the Roman officers left in charge in Numidia—is summoned to Rome. There he is protected from the fury of the people by one of the tribunes, whom he has bought. Ordered out of Rome by the Senate, he is made to exclaim apocalyptically, as he turns and looks back at the city, "Yonder is a city put up for sale and its days are numbered if it finds a buyer" (37.3).
The campaign against Jugurtha in Numidia is dominated, according to Sallust, by the same motive: cupidity. The Roman commander, Aulus Albinus, chooses to besiege the town where Jugurtha's treasury was kept—not an unreasonable military move, one would have thought. Jugurtha, however, succeeds in corrupting Roman officers, even down to centurions, and this, by promoting treachery in the army, gives him a victory. The new consul sent to command in Numidia, Metellus, is unbribable, to Jugurtha's concerned astonishment. (The reader shares his surprise.) Jugurtha sends envoys on a mission of corruption, but the biter is bit when Metellus turns them by his inducements. Metellus is superseded by Marius, elected consul in 107 BC; he is an able commander, lowly born and intensely ambitious. Sallust, who had earlier praised his good qualities (63.6), deplores the arrogance which grows on him as consul, but gives him, nonetheless, one of the most effective speeches in _The Jurgurthine War_ (85.10–48), contrasting his own merits and achievements with the inherited privileges of men of family, and presenting himself as a plain, blunt man of the people, without superficial elegancies:
I cannot, to justify your confidence in me, point to the portraits, triumphs and consulships of my ancestors. But if need be I can show spears, a banner, medals, and other military honours, to say nothing of the scars on my body—all of them in front. These are my family portraits, these my title of nobility, one not bequeathed to me, as theirs were to them, but won at the cost of countless trials and perils.
My words are not carefully chosen. I attach no importance to such artifices, of which true merit stands in no need, since it is plainly visible to all. It is my adversaries who require oratorical skill to help them cover up their turpitude. Nor have I studied Greek literature; I had no interest in a branch of learning which did nothing to improve the character of its professors...They call me vulgar and unpolished because I do not know how to put on an elegant dinner and do not have actors at my table or keep a cook who has cost me more than my farm overseer...
Marius was a successful general in Numidia, but the final act—appropriately—was a result of treachery and the scheming of Marius' deputy, Sulla. Jugurtha's ally, the Moorish king Bocchus, has second thoughts after two defeats. Sallust cannot resist adding "perhaps, too, he sought the advice of some friends whom Jugurtha had omitted to bribe" (103.1). Bocchus betrays Jugurtha, who is taken captive to Rome, appearing in the triumph awarded to Marius.
In the decades following the conclusion of the Jugurthine War in 105 BC, the political situation in Rome markedly deteriorated as Marius and Sulla fought their bloody political and military battle for supremacy. This accounted for the intensely nervous political climate in which Catiline, a ruined aristocrat and would-be demagogue, plotted to overthrow the consuls and make himself and his followers all-powerful. The projected coup was brewing over several years (66–62 BC)—Sallust has been accused of some chronological inaccuracy here—before matters came to a head with a rising in central Italy led by one of Catiline's supporters in 63 BC.
For Sallust _The Conspiracy of Catiline_ was another study in the corruption of Roman public life. Here the keynote is not venality, though Catiline and his friends are clearly greedy and instigated to revolt by financial ruin brought on by their extravagances (there is a practical role for "luxury" here). The central impulse (for, being out of office, they have nothing to sell) is inordinate, unprincipled and lawless ambition. Ambition, unlike avarice, and venality, is not seen as ignoble in itself, but perverted to pure selfishness and loosed from all restraint it represents the greatest possible danger to the state.
As so often in Roman polemic and diagnosis, Sallust's denunciation is thrown into relief by idealization of the past. The republic, in its early days, according to him, had advanced with such extraordinary rapidity because, relieved from the constraint represented by a jealous monarch, men burned to distinguish themselves and acquire glory in the service of the state ( _Catiline,_ 7.4). (This argument was to become popular in the Renaissance.) Avarice was almost unknown, and virtue was held in high esteem. Frugality and piety were the norms. Such virtue brought its rewards in the form of conquests, culminating in the destruction of Rome's great rival, Carthage. But then "fortune turned unkind" and converted Rome's success into disaster. Leisure and riches were fatal:
Growing love of money and the lust for power which followed it engendered every kind of evil. Avarice destroyed honour, integrity and every other virtue, and instead taught men to be proud and cruel, to neglect religion and to hold nothing too sacred to sell...Rome changed: her government, once so just and admirable, became harsh and unendurable. (10.6)
Sulla's dictatorship gave licence to robbery and pillage. Wealth became everything:
As soon as wealth came to be a mark of distinction and an easy way to renown, military commands, and political power, virtue began to decline. Poverty was now looked on as a disgrace and a blameless life as a sign of ill-nature. Riches made the younger generation a prey to luxury, avarice and pride. Squandering with one hand what they grabbed with the other, they set small value on their own property while they coveted that of others. Honour and modesty, all laws divine and human, were alike disregarded in a spirit of recklessness and intemperance. (13.5)
Avarice and ambition became blended, and the greed of Catiline and his followers, whetted by luxury, assumed a directly political form. Catiline himself is presented as a study in frustrated ambition perverted to political crime.
His associates are, according to Sallust, a rabble of cut-throats, perjurers, bankrupt gamblers and spendthrifts—Sallust gives a list of Catiline's supporters in the Senate, who are not excluded from these categories. Catiline's influence is fatal even to the innocent. He was said to have killed his own son to curry favour with his mistress and to be marked with the signs of a terrible remorse; one is tempted, anachronistically, to speak of the portrait as Miltonic or Byronic:
His unclean mind, hating god and fellow man alike, could find rest neither waking nor sleeping; so cruelly did remorse torture his frenzied soul. His complexion was pallid, his eyes hideous, his gait now hurried and now slow. Face and expression plainly marked him as a man distraught. (16.4)
The absence of troops from Italy, because they were serving under Pompey in the East, and the presence of many discontented veterans seem to give Catiline his chance. Aristocratic Roman youths are drawn to him and into his conspiracy. He makes a speech in which, by emphasizing the narrowness of the oligarchic clique which monopolizes power (20.12), he represents his followers as the poor and dispossessed, "whether our birth be noble or base," and promises his supporters everything from magistracies and priesthoods to cancellation of debts. Sallust makes amply clear the eclectic nature of Catiline's following: poor men and bankrupt aristocrats, Sulla's discontented veterans and his victims. In central Italy Catiline's lieutenant, Manlius, recruits the local brigands, while sons of noble families are detailed to murder their fathers (42.2). The city populace of Rome, Sallust suggests, were favourably disposed to anything promising a riot (39.5).
The last part of the _Catiline_ is concerned with discussion of the punishment of the plotters when they had been rounded up. It chiefly takes the form of two speeches, respectively urging banishment and death, the first from Julius Caesar and the second from Marcus Cato, great-grandson of the censor. This gives Sallust the opportunity for a final set of references to the loss of ancestral virtues. Interestingly, he then singles out among the factors which had made Rome great "the preeminent merit of a few citizens" (54.3). Rome was living now on the capital their actions had accumulated for it, and it was this alone which had so far saved it from the consequences of the poor quality of its modern generals and magistrates. In their contrasting but equally admirable qualities, the two speakers, Caesar and Cato, were notable exceptions to this. Sallust was a friend and supporter of Caesar, and owed him much; the encomium he pronounces is no surprise. In that of Cato, however, Sallust seems to pay his respects to old Roman austerity, of which the lavishly extravagant and manifestly ambitious Caesar was hardly an exemplar (53.5). It may be that in this way Sallust is hinting that Rome needs new men of splendid abilities, like Caesar, but also the ballast of old Roman virtues.
In focusing on the quality of public men, despite the disgust with the present and the seductions of the legends of republican virtue which may have led him to produce an overdrawn contrast, Sallust was displaying a characteristic hard-headedness, welcome compared with clichés (we shall find them in Livy) about the morality-sapping cultural miasma diffused by Greek bric-a-brac and effeminate Asiatic tastes. One does not always have to trust his judgement, of course, though it is not implausible that North African courtly intrigues, and for that matter those of the Roman Senate, could be notably corrupt. Sallust's attribution, in _The Jugurthine War,_ of almost every action to avarice has an air of obsessiveness. But there is more to it than that. It is interesting that two very different sets of people, the courtiers of a semi-barbarian North African monarch and (many) Roman senators, are under the sway of the same passion of avarice. This might seem an example of the idea of a more or less uniform human nature spoken of by Thucydides and Polybius, but it cannot be, because in Sallust, as later in Livy, the Romans of former times were not like that at all. Whence, then, the modern similarity? Sallust does not try to unpack the paradox, but it is easy to imagine an answer which, accepting Sallust's versions of the Romans, both ancient and modern, invokes a kind of inverse relationship between a sense of public duty and selfish individualism. If that is accepted, then the individualism (venality) of the barbarian and the senator, though contemporary, actually belong to different historical epochs. The former, the argument assumes, has never had a sense of the _res publica_ and its claims on him (though one might have thought personal loyalty might perform the same function). However, as Sallust says, the individualism of the corrupted Roman belongs to a society in which the hold of public duty and an idea of the republic has been eroded by an influx of new wealth and the unprincipled avarice it has fostered.
Sallust does not extend his analysis to barbarian corruptibility: barbarians are like that—unreliable—just as they fight or run away as seems expedient. (Sallust's geographical and ethnographical "digression" on Africa is brief and jejune) (17.4–19.1.) Barbarians, one might say—Sallust cannot—are like the rabble Livy would later describe as making up the early population of Rome before Romulus and Numa forged them into a polity and gave them laws and a collective idea of the sacred (Chapter 6). Corruption, then, is not just a perennial tendency of human nature but a cultural phenomenon and, when exemplified by Rome, a historical one. Sallust is here only a step away from something like a conception of cultural history—the conception which the great scholar Erich Auerbach claimed was precluded by the moralism of ancient historiography ( _Mimesis: The Representation of Reality in Western Literature,_ 1953). Of course a step can be a long one, and it is not altogether clear whether the example offered by Sallust's argument, or one might say his half-argument, tells for or against Auerbach's denial to the Romans of a sense of cultural change. Sallust goes halfway, but only halfway, towards it. However, it does not seem to be _moralism_ which is closing off the possibility; rather, this is what is opening it up.
Sallust is a master of economical, lucid and dramatic narrative, and of acid if exaggerated comment. He has perhaps some of the weaknesses of the disillusioned retired politician—though prolixity is not one of them—embittered, it may be, despite his protestations, at the ending of his career. But he has eminently the strengths of the breed, above all a robust understanding of the motives of political men and their often disingenuous manipulation of political rhetoric. His denunciations of modern Roman corruption are powerfully impressive, and must surely have influenced Livy. His diatribes are part of the background to—and sometimes emerge explicitly in—the copious political literature devoted to luxury and the loss of public spirit from the Renaissance to the eighteenth century.
**SIX**
**Livy: _From the Foundation of the City_**
It was a practice of the ancient Greeks and Romans to ascribe their distinctive institutions to the wisdom of a founder, perhaps with divine guidance: Lycurgus in Sparta, Solon in Athens, Romulus and his supposed successor Numa in Rome. At one time Alfred the Great was credited with this role in England, inventing, among other meritorious institutions, the jury and the University of Oxford. But the Romans added something more. It is the conception of an accretive constitution, ancient and continuous, but modified piecemeal over a long period, rather as the English, in the nineteenth century, prided themselves as their predecessors had done, on the antiquity and continuity of their constitution; one of its great merits was its gradual evolution, from precedent to precedent.
In Rome, a profound institutional conservatism came to be accompanied by a recognition of a certain desirable pragmatism and flexibility, mediating between the cherished inheritance bequeathed from the past and the needs and demands of the present. The Romans never developed this as a guiding idea to the pitch of self-consciousness and sophistication it acquired in England; in fact there was no word for it. It is implied, however, in the conjunction of two ideas they undoubtedly did have: respect for ancestral custom, the old Roman way, and moderation or conciliation, which implied flexibility—making necessary and timely concessions to preserve the unity, the concord, between the different orders of society. A particular example which clearly mattered to Livy was the erosion, over time and accompanied by much ill-feeling and resistance, of the rule that only members of the hereditary senatorial order, the patricians, were eligible for the supreme executive posts, the two annual consulships, and other offices. The Romans came to realize, and Livy illustrated it, that their constitution had to be understood not just as a single foundational act, like that of Lycurgus in Sparta, but historically. As Polybius had put it, the Romans owed their constitution not to theory but to experience. They had arrived at the result achieved by the power of reasoning by the great Spartan legislator Lycurgus, but the Romans "did not do so by means of abstract reasoning, but rather through the lessons learned from many struggles and difficulties; and finally, by always choosing the better course in the light of experience acquired from disasters" (Polybius, VI.10).
It is dangerous to generalize about ancient historiography, because so much of it is lost altogether, or known only by repute or fragmentarily, but it does seem that the Romans developed an attitude to their history which gave it, and particularly its earliest and most obscure years, a more profound significance than other peoples and cities attributed to their annals and foundation myths. With the growth of Rome's power, they came to dwell on the contrast between the humble beginnings of the city spoken of by the tradition—the huts of the shepherds clustering on the Roman hills in the eighth century BC and the rabble of runaways and vagabonds with whom Romulus augmented his foundation—and the world empire which Rome had won by the middle of the second century BC. Even Athens, the chief imperialist power of fifth-century Greece, had achieved no such dizzying eminence. Only the empire of Alexander did so (which drew Livy into an uncharacteristic digression: would the Romans have beaten him? [IX.17–10]); but Alexander's empire was an intensely personal achievement, and as rapid in its rise as in its dissolution. It was certainly not the product of qualities nurtured by a long tradition, or of a gradual, hard-won expansion, like that which, over some centuries, had made Rome master first of its immediate vicinity, then of central Italy and the whole Italian peninsula, and then, by victory in the long-drawn-out wars with Carthage, over the western Mediterranean, and finally over the western part of Alexander's empire, in Macedon, Greece and the Near East. This is the story which, with one hiatus, the extant books of Livy's history tell, up to the middle of the second century BC. By this time Rome's empire was a subject of awestruck contemplation, as it was still more in Livy's own time, the second half of the first century BC, the Age of Augustus.
Polybius, over a century earlier, had recognized the power of Rome as the central fact of the contemporary world, and other Greeks had written on Roman history, but for a Roman there was obviously in it also an invitation to patriotic pride, often accompanied by a nervous anxiety. Rome's achievement fostered a retrospective piety towards the ancient Roman virtues to which it was ascribed, but by the first century, when Livy was writing his history (from 30 BC onward), it had come to seem that there was a dreadful paradox in Rome's success: the qualities which had procured it had been subverted by the wealth and ease which it had brought.
This combination of pride and pessimism which is manifest in Livy is also evident in the work of his contemporaries the Augustan poets Virgil and Horace. The Romans were a great people, who had deserved their extraordinary fortune and perhaps the divine favour which had permitted it, but—the theme so strongly stressed by Sallust—the modern Romans were not the men their forebears had been. We have seen, in Polybius, what became commonplaces about the corruptive effect of luxury, but for a Roman a century and more later it was more than a sociological truth or warning; it had become a source of profound misgiving as well as of rhetorical denunciation.
A number of cultural streams originally fed the ancient tendency to exalt the past over the present and to think of change as degenerative. Polybius, as we have seen, manufactured out of Aristotle's typology of constitutions, in which each form had its corrupt antitype, a theory of constitutional change in which each successful form was essentially unstable, though the Roman constitution was best able to stave off eventual inevitable degeneration. But there was also a general tendency to exalt the past: Homer's heroes were stronger than four modern men. One of the chief impulses to the writing of history was, as Herodotus says (above, Chapter 1), to preserve the great deeds of earlier generations, which thereafter could be invoked for encouragement and emulation, as Xenophon's beleaguered army was reminded of the Greeks' victories over the Persians. Similar invocations of Roman achievements are, in Livy, a staple of the general's address of encouragement to his troops before battle. But the element of pessimism in Roman pride had been less specific among the Greeks, as a sense, as we find it in Herodotus and Thucydides, of the fragility of good fortune and greatness, expressed in the formula of hubris followed by nemesis. We have seen earlier, too, the self-congratulatory contrast of Greek freedom, frugality and hardihood and Persian luxury, effeminacy and servility. But, as the Roman imperium extended to the Near East, Alexander was sometimes instanced as an example of the contagious nature of Asiatic luxury, notably by Livy. In luxury lay the seeds of corruption, and in corruption the seeds of downfall. In Livy, and in Sallust before him, the question of moral fibre, nurtured by war, weakened by peace and ease, became the core of Roman history.
The preoccupation with Roman virtue and the piety towards long-standing Roman institutions and traditions, whose loss Livy deplores, expressed a sense of present malaise, but they also constituted a powerful impulse directing attention to the earliest centuries of Roman history, when Rome's institutional foundations were laid and the Roman qualities of character were in their original purity. This attention focused on traditions, on legends of patriotic deeds, on foundation myths, and on what survived of records and annals: in short, it led to historical research and writing in the form of antiquarianism. The Greek cities had had their myths of divine or heroic foundation and their legendary genealogies, and some of their citizens had compiled annals. Hellanicus, in the fifth century BC, had written a general account of the foundation of the greatest cities, and apparently provided the first history of early Athens. Polybius identified this as one of the types of history, but also distinguished such parochialism from his own practice of universal history. Only in the case of Rome, he realized, did city-state history and universal history in the long run coincide. By the second century BC nothing that could be known about Rome was merely parochial, and antiquarians who began to study the early history of the city could presumably think of this as no petty task.
Livy tells us that most of the early records were destroyed by a catastrophic fire at the time of the city's occupation by the Gauls in 386 BC. The importance of this has been seen as debatable, but early records certainly seem to have been sparse. Livy is to a considerable extent aware of the fragility of the evidential basis for his accounts of the earliest centuries, and unapologetic about it. They have, he says, "more of the charm of poetry than of sound historical record" (I.1), but he retells the legends with gusto, though with periodic expressions of scepticism and reluctance to choose between rival versions. In writing the early history of the city he had a number of precursors, on whom he drew, but most wrote near his own times. Livy's literary gifts and the survival of approximately a quarter of his immense history of Rome from the conventional date of the foundation of the city (753 BC in the Christian calendar) down to his own day, including the first ten books, have made his _the_ history of early Rome for subsequent generations. (Livy's work is conventionally divided into units of ten books, known as "decades.")
Greek historians besides Polybius had written on the history of Rome, including its earliest period, though mostly not exclusively. One of these was Polybius' whipping-boy Timaeus, and his work would have been available to Roman historians. A Greek contemporary of Livy, Diodorus of Sicily, included Rome in his universal history, which he said should include the earliest traditions of peoples. Another Greek contemporary, Dionysius of Halicarnassus, wrote specifically on Rome. His _Roman Antiquities,_ of which half survives, traced Rome's rise from its mythological origins, stressing the allegedly Greek character of the Romans.
Among Roman authors, the first prose historian of Rome was Fabius Pictor—frequently cited, not always uncritically, by Livy. Fabius Pictor wrote in Greek, though he came of a highly distinguished Roman family, the Fabii. Livy instanced him as an example of the distorting effect of family pride among historians. He flourished towards the end of the third century BC, and was a senator and perhaps a pontiff or chief priest. The Roman priesthood was not a caste set apart, but consisted of representatives of aristocratic families, qualified to perform ritual sacrifices. His history, now lost, traced the development of Rome from its supposed origins to his own day, and it is likely that a considerable part is preserved in substance in Livy and other historians. In his possible pontificate, and his apparently obtrusive family pride, we have two of the kinds of source in which facts and traditions from Rome's early history were preserved.
It was the custom each year for the pontiffs to have a record of events inscribed on a publicly displayed whitened board. When the board was wiped for another year, a transcription was made. At the end of the second century BC the pontifical annals were published as a chronicle, which seems to have run from around 400 BC, to which accounts purporting to cover the foundation of the city and its early history had been added. It is not known exactly what it contained. Another publication, of the proceedings of the Senate, was produced from the first century BC, as were records of notable speeches.
Another source of recording was the family pride of the most eminent Roman clans. Rome was a society much given to commemoration. We have seen (above, Chapter 4) Polybius' approval of the funeral orations in honour of distinguished men, as a nursery of civic virtue. Roman religion was domestic as well as civic, akin to ancestor worship; ancestral busts were preserved, and their features would have been familiar to their descendants. Some families like the Fabii and the Claudii had been long prominent in the republic. Other possible sources of this kind, so-called "banquet songs," have been thought to commemorate notable men and deeds, and to have been used by probably the first Roman historian to write in Latin, Marcus Cato—Cato the Censor—of whom Livy's history provides a vivid portrait, as also does Plutarch later. Cato, though a plebeian, was consul in 195 BC. His frugality and austerity, his intense conservatism and his hostility to foreign ways made him an archetypal figure in Roman public memory. His _Origins,_ which has survived only very fragmentarily, included a treatment of Roman history from its foundation legends to the Second Punic War.
Two other authors of histories of early Rome both wrote their narratives in verse: Ennius, who in the second century BC wrote eighteen books of annals from the foundation, and Varro, writing in the first century BC, whose history is completely lost. Both are sometimes cited as authorities. For an ancient historian, Livy is fairly profuse in naming his sources, because he wants to explain his difficulties in deciding between them, which often he does not. Apart from Fabius and Cato, he mentions an author of the second century from another distinguished family, Calpurnius Piso, and, nearer to his own time, Cornelius Nepos and Licinius Macer. The latter is mentioned particularly because of his discovery of a list of early magistrates on linen rolls held in the temple of Jupiter Moneta, running from around 445 BC.
Livy himself was essentially a literary artist applying himself to history, not an antiquarian scholar. The accounts of the first three centuries from the foundation, including the expulsion of the kings and the establishment of the republic, conventionally dated to 509 BC, with which Livy begins his second book, are essentially legendary, though they may have a substratum of fact preserved in tradition. As history, thanks to the work of his predecessors, we can begin to have more confidence in Livy's work from around the middle of the fifth century.
Livy's history, _From the Foundation of the City_ ( _Ab urbe condita_ ), was written in 142 books, of which 35 survive. Understandably, the early books cover much longer periods than those which follow, so that I–X cover almost four hundred years, down to the beginning of the third century, and deal, after the foundational and legendary period, with the establishment of Roman power in central Italy. The next books we have (XXI–XXX), dealing with the Second Punic War, cover only twenty-three years, and the remaining books extant, recounting the establishment of the Roman imperium in the eastern Mediterranean (XXXI–XLV), take fifteen books to narrate the events of only thirty-three years. The work as we have it ends in 167 BC, over a hundred years before Livy's own lifetime. The size of the whole work, including what is lost, is awesome. What survives is around two thousand pages.
It was the early books which made the greatest impression on subsequent generations. For the books on the Hannibalic War Livy seems to have followed Polybius quite extensively, without much acknowledgement, though Livy's account became better known. Livy was unashamedly a picturesque historian of the kind Polybius denigrated, though perhaps the most accomplished of them. He is, in fact, a very considerable prose artist. It is characteristic that we find some of the "human interest" stories from the Punic Wars in Livy, not Polybius: the young Hannibal taking an oath, at his father's bidding, to be an implacable enemy of Rome (XXI.1), and Archimedes, rapt in the contemplation of a mathematical problem, killed by a soldier in the storming of Syracuse (XXXV.31). Livy also enjoyed the posthumous advantage of writing in Latin; in the West, more boys have learned Latin than Greek even from the Renaissance onward, and many, like myself, must have had Livy among their early assignments.
Unlike most ancient historians, Livy (Titus Livius) had no official career. He was born in Padua around 60 BC, and spent most of his life, which he dedicated to writing his history, in Rome. According to Suetonius, the historian of the first twelve emperors, he helped the young Claudius, the future emperor, with his own early literary efforts. He was known to Augustus, though in no way an intimate, and he takes his place now with the other great celebrators and critics of imperial Rome in the Augustan Age, Virgil and Horace.
His history seems to have been an almost immediate success. Together with the biographies by Plutarch, the first ten books, in particular, have supplied Europe subsequently with a repertoire of legend and folklore only slightly less familiar and vital than that of the Bible and Greek mythology. Livy records the legends as worthy to be preserved and known. Like Polybius before him, and like Virgil, he regards the greatness of Rome as in some sense preordained. It was "written in the book of fate that this great city of ours should arise, and the first steps be taken to the founding of the mightiest empire the world has known—second only to God's" (I.3). Recognizing their frailty as history, he neither endorses nor rejects the old stories. There is no reason "to object when antiquity draws no hard line between the human and the supernatural: it adds dignity to the past, and if any nation deserves the privilege of claiming a divine ancestry, that nation is our own" (I.1). Livy's attitude would be unfairly described as credulous, but it is certainly pious.
So the early books introduce us to these stories, some of them still familiar: the she-wolf's suckling the abandoned twins Romulus and Remus; the flights of birds as an omen of the brothers' kingship (birds were always important in Roman divination); Romulus' murder of his brother in the ensuing dispute. Livy, as seems to have become customary, combines the story of Aeneas, the refugee from Troy, with the Romulus story. Aeneas settled in Latium, and united the indigenous Latins with his Trojans; Romulus was in his line of descent. Tradition dated the foundation of the city to 753 BC. A steadily more extensive inclusiveness is a continuous theme in Livy's history. Romulus makes Rome an asylum for runaways; the Sabine women are captured, so that henceforth, though there is enmity and a desire for revenge, Romans and Sabines are kin. Neighbouring peoples are steadily included within Rome's orbit, and the ultimate inclusion is the extension of eligibility for citizenship to the provinces (see VIII.14). With the extension of the empire there is the same pattern of assimilation, by conquest and adoption.
One of the most famous stories is that of the rape of Lucretia by the son of King Tarquin the Proud, leading to the expulsion of the Tarquinian dynasty and the ending of the monarchy ( _c._ 509 BC) (I.59). It was memorably used as a theme in painting, notably by Cranach (Lucretia's suicide) and by Titian (with a fiercely energetic rapist), and in Shakespeare's sensuous narrative poem. The expulsion of the Tarquins provoked war with the Etruscans, and in fact much of Livy's early history, after the dwellers on the seven Roman hills have been brought together, is concerned with wars with the adjacent cities and peoples, within a steadily expanding radius: Veii, only nine miles inland from Rome; Fidenae even closer; the Albans, Volscians, Latins, Etruscans, Aequians, Sabines; dealt with in later books are the Samnites and Campanians to the south. The Gauls, established in northern Italy, are the only people actually to enter the city of Rome. These conflicts, which can grow wearisomely repetitive, are diversified by what really mattered to subsequent generations—the legendary stories of Roman patriotic heroism and self-sacrifice.
Several such stories relate to the defence of the new republic against the attempt of the Etruscan king Lars Porsenna to restore the Tarquinian dynasty. Porsenna, in these stories, is, however, magnanimously disposed to honour Roman courage, even in the young Roman aristocrat, Gaius Mucius, who attempts to assassinate him. When foiled, Mucius boldly acknowledges his intention, and is not surprisingly condemned to be burnt to death. Declaiming "See how cheap men hold their bodies when they care only for honour," he holds his right hand in the flames as a demonstration of fortitude. Porsenna pardons him, and he is known thereafter as "Scaevola" or left-handed. Porsenna also pardons a girl, Cloelia, who makes a reputation as a heroine by escaping when detained as a hostage. The Romans honourably give her back, and Porsenna, moved by her courage, even more honourably returns her. She was subsequently honoured in Rome by a public equestrian statue(II.13).
Other stories are more sinister, one even having echoes of human sacrifice. A great chasm having opened in the Forum and been deemed an evil omen, the soothsayers call for a sacrifice if the Roman republic is to last for ever. A young soldier, Marcus Curtius, puts on full equipment and rides his horse fearlessly into the gulf (VII.6); the story was dramatically illustrated in a painting by Benjamin Haydon, the English Romantic artist and friend of Keats and Fuseli—the horse looks understandably reluctant. More emotionally complex is the story, also told by Plutarch (where Shakespeare found it), of Coriolanus, the proud oligarch who, exiled from Rome, is appointed general of an invading Volscian army. Moved by the entreaties of his mother, Volumnia, and his wife and children, he turns back and Rome is saved (II.40). The story is actually one of the triumph of affection over duty (at least to the Volscians), which seems to have been endorsed only because it benefited Rome. The reverse is much more common: Roman patriotic zeal begot what almost deserves to be called a heroic subgenre of the victories of public duty over private feelings, or, to put it another way, of the state over the family. This was Roman "firmness" or "sternness." The archetypal figure is Lucius Junius Brutus, the leader against the Tarquins and therefore the father of the republic. As consul, Brutus had to preside over the execution of his two sons for a conspiracy to restore the monarchy (II.5). A similar scene is enacted in Book VII (10), where Titus Manlius, son of the consul of the same name, wins a ritualized battle against the Latins by engaging in single combat, against the consuls' orders. He becomes a popular hero, but is nonetheless executed for his disobedience, at his father's orders. Least sympathetic of all such actions (and its outcome) is that of Publius Horatius, one of the three Horatii, brothers who fought the Alban champions, the Curatii, in what was again a highly ritualized though mortal combat. Horatius, the only survivor, meets his sister, who had been betrothed to one of the Curatii and is weeping for him, and in a rage kills her. At his subsequent trial the word of his, and the girl's, father exonerates him: a Roman daughter who could mourn an enemy deserved to die; Horatius is freed(I.26).
A different and more amiable kind of republican virtue, which also became exemplary, was embodied in the story of Cincinnatus, a former consul (460 BC), who fell into poverty and was reduced to working with his own hands on his three-acre farm. A crisis having arisen, it is decided that he is the man of the hour and must be made dictator, a temporary post of supreme power reserved for emergencies. Summoned by a delegation from the Senate, "he told his wife Rocilia to run to their cottage and fetch his toga. The toga was brought, and wiping the grimy sweat from his hands and face he put it on." Cincinnatus, having overcome the crisis, resigned fifteen days later. The story had a resonance in another new republic, in America, as a precedent for the president's reversion to the status of a private citizen after his term of office; there had been fears that Washington would make his presidency perpetual or even make himself king. Cincinnatus returned to his plough, or rather to his Virginia estate. Roman pseudonyms—Cato, Junius—were popular among authors of republican or at least oppositional pamphlets in eighteenth-century England as well as America. The collective pseudonym of the American Federalists was "Publius," though their writing was coolly sociological compared with the straining for republican virtue in emulation of Rome among the French revolutionaries a little later. The most extreme egalitarian among these, Babœuf, adopted "Gracchus" as his first name, after Tiberius and Gaius Gracchus, the second-century leaders of the plebs.
It is not surprising, in fact, that in late-eighteenth-century France inspiration was found in the legends of Roman public virtue and patriotism. Jacques-Louis David's painting _The Oath of the Horatii_ (1785), depicting the three brothers holding up their swords and dedicating themselves to patriotic self-sacrifice while the women weep, was found uplifting; collective oath-taking was to be a significant part of the public theatre of the Revolution, of which David was to be the celebrator and in a sense the stage manager. The cult of patriotic virtue, already evident in the 1780s, was in part a reaction against rococo erotic depictions of Greek mythology, so that in a sense they and scenes from Roman history were in competition. Others of David's works, like _The Lictors bringing Brutus the Bodies of his Sons_ (1789) and _The Intervention of the Sabine Women_ (1799)—Livy tells the story of their mediation between their new husbands and their enraged families in I.13—could have appeared earlier, but other paintings from Roman history had an immediate topicality in these years. David's _The Generosity of Roman Women_ (1791), in which the protagonists make a patriotic sacrifice of their adornments to the needs of the state, actually inspired a group of Parisian women to do the same when the new French republic was threatened by a foreign invasion to restore the Bourbons, like the intervention of the Etruscans on behalf of Tarquin. David was not the only painter to be inspired by Roman scenes. Equally topical, for example, was Jean-Baptiste Wicar's 1789 sketch for a subsequent painting of _Brutus Vowing to Expel the Tarquins._
Livy, perhaps more than any other ancient historian (unless Plutarch's _Lives_ are counted as histories), bequeathed to Europe, from the Renaissance onward, the conception of history as moral teaching by examples. The Roman character, rather than, as with Polybius, foresight and the unique virtues of the Roman constitution (which, by the time Livy was writing, was manifestly not working properly), was at the heart of Livy's history. Threats to Roman austerity were the importation of foreign cults, leading to the neglect of the native gods; luxury; the love of novelty; and the craze for extravagant entertainments—all of which provoke Livy to eloquent denunciation. His exaltation of the old Roman ways was of course not at all singular. Augustus, for example, fretted about the integrity of Roman family life and the fertility of Roman matrons. It is characteristic that in Rome there were long-established public officials, the censors, charged with keeping Roman manners and morals in order. The appraisal of character also provided a focus for the oratory of the Senate and the courts and the marking of public occasions: in the eulogy ( _laudatio_ ), often at funerals, and in the case for the prosecution, the invective, the formidable character assassination of which the supreme virtuoso had been Cicero.
Livy seems to have understood the relationship between national institutions (including religious rites) and national character as a reciprocal one, each supporting the other. The Spartans, in Livy's view, did themselves much harm when they abandoned the ancient institutions bequeathed to them by Lycurgus (XXXVIII.34, XXXIX.33–9). The Romans, in being true to their institutions, would be true to themselves. Reciting the details of an ancient custom, he adds that now "the memory of every practice, religious and secular, has been effaced by our preference for all that is new and foreign in place of what is native and traditional" (VIII.11). Livy is that not unusual kind of commentator, in the heyday of empire, who holds simultaneously that dominion is won by strength of character and that the country is going to the dogs: "Even the authority of parents over children is held cheap and of slight account" (XXVI.22). Symptoms of decay now abound, begun by the importation of the luxuries of Asia. Livy mentions bedspreads, tapestries, sideboards and female lutenists at banquets, while "the cook, who had been to the ancient Romans the least valuable of slaves and had been priced and treated accordingly, began to be highly valued, and what had been a mere service came to be regarded as an art"(XXXIX.6).
In the earliest books of his history, apart from its strikingly pessimistic preface, to which we shall return, Livy is naturally concerned with origins rather than decay. He itemizes the establishment of familiar Roman institutions and customs, many, though not all, being ascribed to Romulus or to his successor, Numa, the supposedly divinely guided lawgiver, who corresponds to the Athenian Solon. Among the creations of the founding period are the religious rites and festivals (these are described in Ovid's _Fasti_ ); the dedication of temples, including the temple of Vesta and the institution of the Vestal Virgins; the corps of pontiffs and augurs (who take the omens); the Senate, the Census and the organization of the army; and the main public works: the draining of the Forum, the building of the Circus, and the installation of the Cloaca Maxima, the great sewer of the city. With the republic came the two annual consulships (henceforth available as a method of dating), the tribunes and other public offices.
The details obviously have to be taken on trust, as Livy has taken them, but there is a factual basis. He does not gloss over the poverty and primitiveness of the early inhabitants: there is mention of cattle-raiding as an occasion for war. The lowlier Rome's origins, the more astounding its rise. Aeneas, in Virgil's poem, has a vision of future greatness, contrasting with the rusticity and wildness before him. Livy acknowledges the cultural borrowings from the higher civilization of the Etruscans, including motifs thought of as distinctively Roman such as the "curule" chair of the magistrate, the purple-bordered toga, and the twelve lictors, who subsequently became the bodyguard of the consuls. There is archaeological evidence for similar borrowings. Though the details of the early wars seem invented—the single combats, for example, and the ultra-patriotic sentiments uttered—Rome did establish its pre-eminence among the neighbouring peoples; the communities of shepherds on the Roman hills coalesced as a polity, with the Forum as the centre of civic life instead of a marsh.
Rome was subsequently able to assert a dominion over the peoples of central Italy more widely, and this certainly implies some combination of cohesion and toughness, of foresight and military organization and prowess, to which the self-esteem embodied in the early legends must have contributed. The Romans—so long as they were truly Roman, that is—could think of themselves as stern, steadfast, public-spirited, and devoted to their laws and traditions, for the legends told them so. Livy's brisk characterizations of the national traits of other peoples point the contrast: the Carthaginians are treacherous (XXI.4, XXII.7); Spaniards unreliable (like all foreigners) (XXII.22); Numidians over-sexed (XXIX.23); Athenians easily suggestible (XXXI.44); Thessalians restless and given to rioting (XXXIV.51); Syrians servile (XXXVI.13); Gauls initially fearsome, but easily discouraged (XXII.2, XXVII.48). All very un-Roman—though Livy is always suspicious of mobs, and applauds the religious institutions of Numa for giving the population something to think about in an interval of peace when "there was an obvious danger of a general relaxation of the nation's moral fibre." Numa is seen as inculcating fear of the gods, and instituting the rites to placate them, as a social discipline (I.19–20). Steady Roman patriotism was a product of time and institutions, and even the period of the monarchy was valuable in shielding the populace—"a rabble of vagrants, mostly runaways and refugees"—from the abuse of a premature liberty and ensuring that it would have to be hard-won. Patriotism, like the constitution, is a product of slow growth (II.1).
The two main threats to its continuance were of foreign origin: foreign luxuries and foreign cults. Luxury operated as a kind of sociological disease, affecting other peoples—the Carthaginians and the Gauls—as well as the Romans, and it had its traditional and noted plague-spots: Capua in southern Italy and Boeotia in Greece, as well as Asia as a whole. His army's stay in Capua was described as "Hannibal's Cannae" (i.e. the equivalent of his own defeat of the Romans)—the army was never the same again (VII.32, 38). Boeotia was similarly fatal to the army of King Antiochus of Syria (XXXVI.2). Like a plague bacillus, the infection was brought back to Rome by the city's victorious armies—particularly after the conquest of Macedon, when enriched with the spoils of Alexander's empire, and so ultimately from Asia.
Foreign cults, the second insidious threat, were periodically forbidden by the Roman authorities, with Livy's approval. Alien, unauthorized rites were apparently pernicious in themselves (VI.30, XXXIX.16), but a particularly lurid episode was the detection and punishment of a covert Bacchanalian cult in 186 BC. Initiated by a Greek soothsayer and hierophant of secret, orgiastic, nocturnal ceremonies, involving promiscuous mingling of the sexes and even murder, it had already spread widely. Its exposure caused panic. It was said that more than seven thousand men and women were involved. Some were punished; others fled or committed suicide. In hindsight this gives an interesting foretaste of future attitudes to Christianity (XXXIX.8–18), but Bacchanalianism, though perhaps ineradicable in some form, had less staying power.
It has to be remembered in considering Livy's mention of events like these that the fundamental form of his history was annalistic, though his command of smooth, continuous, well-paced narrative is such that it is easy to forget this. Some events were recorded just because they happened, just as each year was marked by the consulships and any notable omens or prodigies. We may be grateful for Livy's residual annalistic inclusiveness, because it gives us episodes which provide sidelights, even amusing ones, on "social history" which Polybius would have thought it beneath the dignity of history to record. Other ancient historians recorded exceptional natural events, such as earthquakes, partly as chronological markers, and noted supernatural events if their interpretation as portents and warnings influenced policy. Livy does too, but he seems to record ominous phenomena with particular relish and sometimes for their own sake: statues weeping or sweating, fallen or disfigured; showers of blood, stones or, on one more interesting occasion, meat (III.10); the untoward behaviour of birds and other creatures; monstrous births; a talking cow and a nodding Juno. Livy does not endorse the accounts of these events, but admits his weakness for them and regrets their neglect by modern historians: "My own outlook, as I write about events in times gone by, becomes in some way old-fashioned" (XLIII.13). We can read this as an endorsement of historical empathy.
Among the flashes of social history, it is impossible to omit the great flute-players' strike of 311 BC. Angry at being forbidden by the censors to hold their usual feast in the temple of Jupiter, they downed instruments, leaving sacrifices to be performed without music. They won. Their right to their feast was restored, and they were allowed "on three days a year to roam the city in fancy dress, making music and enjoying the licence which is now customary" (IX.30). More potentially serious was the case of the witty Vestal Virgin in 420 BC. She was exonerated of a charge of unchastity, but was ordered to amend her over-elegant costume and not to make any more jokes. Another overdressed Vestal, who in 337 was actually convicted of unchastity, was buried alive.
The question of women's dress brought women out on the streets in an unprecedented demonstration in 195 BC. The Lex Oppia, a sumptuary law imposed on women during the Punic Wars, was proposed for repeal, only for the repeal to be blocked by the tribunes. Women flooded the streets, not only from Rome but from outlying areas. This provoked Marcus Cato, byword for austerity and discipline, to an impressive speech given him by Livy (though his own, also extant, was different). It is a good example of a style of oratory: Livy's speeches, unlike Thucydides' speeches as situation reports, not only embodied a studied rhetoric but were supposed to reflect the characteristics and manners of the speaker. Cato's vehement speech is one in which a fearful misogyny leads on to a thin-end-of-the-wedge argument—what will they ask for next?—as well as characteristic praise of older and sterner times:
Citizens of Rome, if each one of us had set himself to retain the rights and dignity of a husband over his own wife, we should have less trouble with women as a whole sex. As things are, our liberty, overthrown in the home by female indiscipline, is now being crushed and trodden underfoot here too, in the Forum. It is because we have not kept them under control individually, that we are now terrorized by them collectively...
Our ancestors refused to allow any woman to transact even private business without a guardian to represent her; women had to be under the control of fathers, brothers, or husbands. But we (heaven preserve us) are now allowing them even to take part in politics, and actually to appear in the Forum and to be present at our meetings and assemblies! What are they now doing in the streets and at the street corners?...Give a free rein to their undisciplined nature, to this untamed animal, and then expect them to set a limit on their own licence! Unless you impose that limit, this is the last of the restraints imposed on women by custom or by law which they resent. What they are longing for is complete liberty, or rather—if we want to speak the truth—complete licence.
This leads Cato on to a more comprehensive denunciation of modern degeneracy as a result of luxury and foreign influences. The ironies of conquest are on full display:
As the fortune of our Commonwealth grows better and happier day by day, and as our empire increases—and already we have crossed into Greece and Asia (regions full of all kinds of sensual allurements) and are even laying hands on the treasures of kings—I am the more alarmed lest these things should capture us instead of us capturing them; these statues brought from Syracuse, believe me, were hostile standards brought against this city. And now I hear far too many people praising the ornaments of Corinth and Athens, and jeering at the terracotta antefixes [figures at the angles of the gables of temples] of the Roman gods. For my part, I prefer to have those gods propitious to us—as I trust they will be propitious if we allow them to remain in their own abodes. (XXXIV.1–4)
Cato's speech was followed by another, in rebuttal, and the law was repealed. The women had won.
An episode in 295 BC casts an unusual light on the long-standing friction between patricians and plebeians. Having married a plebeian and so become déclassée, the daughter of a patrician was barred by the patrician matrons from sacrifices at the shrine of Patrician Chastity "in the cattle market by the round temple of Hercules." The woman, showing spirit, set up an altar in her own home to Plebeian Chastity, and invited duly qualified matrons to the sacrifices. Unfortunately, Livy says, the new cult became discredited by the presence of women who, though plebeian, were apparently not matrons, or chaste (X.23).
The social discord between patricians and plebeians comes to the surface intermittently in Livy's history, even in the early books. The conflicts to which it gave rise were particularly menacing at moments of external crisis. There was as yet no professional army, and at such moments the plebeian citizen soldiers felt their power and sometimes exercised it, though—and this is Livy's main message—at the eleventh hour the Romans always learned to compromise and present a united front. As Livy shows, and at one point in particular acknowledges(VII.19), the conflict designated as being between patricians and plebeians was really two, more or less unrelated, conflicts, involving very different sections of the plebeian population. The problems of the poorer plebeians were chiefly those of debt and the difficulty of retaining their smallholdings as the rich became richer as a result of conquest (this was really the solid basis of the luxury argument) and, since patricians were debarred from commerce, sought to buy up land, which could be farmed by another product of conquest, slaves. Landholding and military matters were also intertwined. Soldiers returning home after service found their farms in disrepair and, struggling to pay the taxes the war had made necessary, fell into debt. Later the displacement of the smallholder, together with the huge extent of the empire, made the citizen soldier an unworkable idea and a professional army inevitable, but these problems were to come to a head in the first century BC, Livy's history of which is lost.
The problem of debt, however, is one to which he frequently returns. Huge resentment was caused by the fact that the debtor was bound to his creditors, being effectively enslaved, often after (or even arguably for) having served his city. The abolition in 326 BC of enslavement of debtors—long demanded—Livy calls "a second birth" of the liberty of the Roman people (VIII.28). The demands for an agrarian law—a redistribution of public land (again swollen by conquest) and restrictions on the size of landholdings—rumbled on, though the greatest crises, associated with the names of the two tribunes of the people Tiberius and Gaius Gracchus, occurring in the later second century, lie beyond the extant books of Livy's history. The establishment of tribunes of the people, however, recorded by Livy, was a significant step for the plebs (II.23, 32, 53, 58; III.55; VI.31; VII.29).
The issue of the hereditary privileges of the patricians was not, as Livy admits (VII.19), of much interest to the poor smallholders, but only to rich and prominent plebeians. Only patricians—members of the early families of senatorial rank, and therefore a kind of hereditary caste, the fathers or _patres_ (hence "patrician")—could hold the highest offices of state, or were qualified to perform the religious rites attached to them. The religious duties in fact were made much of by patrician speakers attempting to hold back the pressure of the more notable plebeians for access to office, it being held to be impious for anyone not from the qualified families to perform them. Eventually, with much bitterness and strife, patrician hereditary exclusiveness became eroded: the first non-patrician consul was elected, according to Livy, in 366 BC (VII.1). The religious duties are vividly presented in a speech which Livy gives to a spokesman for the plebs, but this time it is the thin-end-of-the-wedge argument in reverse. By 300 BC the plebs had the right to consulships, censorships and triumphs from which they were formerly excluded. So much has been conceded, the speaker says; how can the rest be withheld?
What god or man can think it unbecoming if men whom you have honoured with curule chairs, the purple-edged toga, the palmembroidered tunic, and with the decorated toga, the triumphal crown and laurel wreaths, whose houses you have marked out from the rest by the enemy's spoils fastened to their walls—if such men add the insignia of pontiffs and augurs? Shall the man resplendent in the robes of Jupiter Best and Highest, who has been carried through the City in a gilded chariot to ascend the Capitol, not be seen with sacrificial cup and augur's crook when with covered head he slaughters the victim or receives an augury from the citadel? (X.7)
Livy's attitude to these conflicts is to praise moderation and conciliation, and to disapprove of mob behaviour and aristocratic arrogance and intransigence. However, "true moderation in defence of political liberties" is difficult, because the latter can easily get out of hand. He takes pride in the fact that difficult changes were accomplished with a minimum of violence. Cohesion, harmony, is the great need. The supreme test of the solidarity of the Roman people is the aftermath of the disaster of Cannae (216 BC), when, with a Roman army destroyed and Hannibal at the gates of Rome— _Hannibal ad portas_ —the Romans, in all their consternation and even panic, essentially keep their nerve and surmount the crisis in united fashion, not thinking of suing for peace: "No other nation in the world could have suffered so tremendous a series of disasters, and not been overwhelmed" (XXII.54).
Livy, to say it again, was not a "research" historian, and his social thought seems commonplace—which is another way of saying representative. He is above all a superb narrator. The speeches he gives his characters are cogent and artfully wrought, according to the rules of ancient rhetoric. It is not fanciful to think that in them we catch something of the thunder-roll of senatorial eloquence in the golden age of Roman oratory. Livy's qualities are not easy to illustrate, especially in a foreign language. They depend on expansiveness, and so do not take kindly to brevity and selectivity. In the many campaign narratives, though he is notoriously weak on tactics and topography, he is good, it has to be said, with weather—as in the test, almost as severe as the Alps, imposed on Hannibal's men by the Apennines:
Heavy rain and a violent wind right in their faces made progress impossible; they could not hold their weapons, and if they tried to struggle on, the wind spun them round and flung them off their feet. The strength of it made it impossible to breathe, so all they could do was to turn their backs to it, crouching on the ground. Then the sky seemed to burst in a roar of sound, and between the horrific thunderclaps lightning flashed. They were blinded and deafened and benumbed with terror. (XII.58)
This may not be exactly history, but it is arresting imaginative writing.
Of all the stories of Roman "firmness," the one which most lodges in the mind, without the self-conscious declamatory style of such stories' usual protagonists or their uncongenial fanaticism, is the account of the Gauls' occupation of Rome in 386 BC. The story of the sacred geese, whose cackling alerted the sleeping guards on the Capitol to a night assault, has entered into European folklore, but Livy's main rhetorical effect is kept for the aged former magistrates who refused to place themselves as a burden on the garrison in the citadel on the Capitol (which held out) but chose to die in their houses. Putting on their ceremonial robes, they took their seats in their courtyards, in the ivory-inlaid curule chairs of the magistracy, and waited. When the Gauls entered, Livy says,
something akin to awe held them back at what met their gaze—those figures seated in the open courtyards, the robes and decoration august beyond reckoning, the majesty expressed in those grave, calm eyes like the majesty of gods. They might have been statues in some holy place, and for a while the Gallic warriors stood entranced; then, on an impulse, one of them touched the beard of a certain Marcus Papirius—it was long, as was the fashion in those days—and the Roman struck him on the head with his ivory staff.
The spell was broken, the barbarian killed him, and the others were butchered where they sat (V.41).
The aftermath of the Gallic invasion also allowed Livy the opportunity for one of his surely most deeply felt speeches, given to the (temporary) dictator Camillus. Rome itself lay devastated, but the neighbouring city of Veii had been captured and lay at Rome's mercy. A proposal was made to transfer the city there instead of laboriously rebuilding Rome. Camillus was opposed to this, and recalled the sacred sites and customs of Rome for his audience:
We have a city founded with all due rites of auspice and augury; not a stone of her streets but is permeated by our sense of the divine; for our annual sacrifices not the days only are fixed, but the places too, where they may be performed: men of Rome, would you desert your gods?...Think, for instance, of Jupiter's feast: how could his couch be decked anywhere but on the Capitol? What of Vesta's eternal fires, or of the image preserved in her shrine as a pledge of Rome's dominion...The Vestal Virgins have their place—their _own_ place, from which nothing but the capture of the City has ever moved them; the Flamen [priest] of Jupiter is forbidden by our religion to spend even one night outside the city walls—yet you would make them, one and all, go and live for ever in Veii. Ah, Vesta! Shall thy Virgins desert thee? Shall the Flamen of Jupiter live abroad night after night and stain himself and our country with so deep a sin?
The speech turns into a eulogy of the city, of its growth and situation and the future prophesied for it (V.51–2). One is reminded here both of Virgil and of Pericles' love letter to Athens in the Funeral Oration; it is a tribute to Livy that he is not diminished by the comparisons.
We do not know what Livy's account of his own century was like, but his preamble to the whole work is deeply pessimistic. He professes to expect that his readers will want to hurry on to the latest times and to read of how "the might of an imperial people is beginning to work its own ruin." Livy feels the opposite. Absorbed in antiquity, "I shall be able to turn my eyes from the troubles which for so long have tormented the modern world." He invites the reader to contemplate "the kinds of lives our ancestors lived" and Rome's rise, and then to
trace the progress of our moral decline, to watch, first, the sinking of the foundations of morality as the old teaching was allowed to lapse, then the rapidly increasing disintegration, then the final collapse of the whole edifice, and the dark dawning of our modern day when we can neither endure our vices nor face the remedies needed to cure them. The study of history is the best medicine for a sick mind.
In the past, poverty among the Romans went hand in hand with contentment.
Whatever the merits of its application here, the idea of long-term decline, the product of Livy's combination of patriotism and pessimism, is highly interesting as a conception of history, for it itself is an intrinsically historical one; in it the idea of a perennial human nature is superseded. The past was unlike the present not only in superficial, material terms, but morally and intellectually. They did things differently then, and thought and felt differently. Livy had a conception, though a negative one, of a distinctive culture of modernity, compounded of religious indifference, cosmopolitanism and a febrile love of novelty. He was in love with the past to an extent no historian before him seems to have been, despite the dedication of historians to preserving the memory of great deeds, because he was convinced that its virtues had been lost. He contemplated the past with yearning— _pace_ Macaulay, who said the exact opposite of him—not with a sense of confident possession.
Livy's work became famous almost at once, though its length doomed its chances of survival _in toto:_ copying was, understandably, of decades of books rather than of the whole work, which is why some complete decades survive and others do not. A new edition of the first ten books was made around AD 396, and it is to this that we owe their survival. Livy had understandably less to say to the early Christian centuries than to his contemporaries, but interest in his work never died altogether. Copying of the first ten books was carried out in the ninth century, and again in the early fourteenth century. Livy became a central figure of the Renaissance, with a famous commentary by Machiavelli in his _Discourses,_ and he remained a staple of education for five centuries beyond that. The doyen of French intellectual life in the later nineteenth century, Hippolyte Taine, wrote a thesis on Livy for his _agrégation,_ and the Victorian imperialist publicist Sir John Seeley began his career with an edition of Livy's first two books. Macaulay derived from him the inspiration for his _Lays of Ancient Rome,_ and in early-nineteenth-century Germany, following the work of Friedrich Wolf on Homer, Barthold Niebuhr drew from the early books of Livy a Romantic conception of the creative, collective self-consciousness of the _Volk._ This, reapplied by David Friedrich Strauss to the Hebrews and the Bible, became one of the seminal and most disturbing thoughts of the nineteenth century. If the whole of Livy's work had been lost, though Polybius, Dionysius and Plutarch would have filled some of the gaps, the subtraction from the future culture of Europe would have been incalculable.
Sir Richard Southern, in his _The Making of the Middle Ages_ (1953), recounts that in the monastery of Cluny around 1040 a list was made of the books chosen by the monks from their library for private study. Predictably, most chose works of the Christian Fathers, or biblical commentaries, lives of the saints, or books on ecclesiastical history or monastic discipline. The exception was one monk, out of the sixty-four, who chose Livy. One wishes it were possible to know more about him.
**SEVEN**
**Civil War and the Road to Autocracy: Plutarch, Appian and Cassius Dio**
Livy continued working on his history until his death in AD 17, but the later books which carried it down to his own times are lost. The chief surviving accounts of the period of the Civil Wars include Caesar's own narrative of his war with Pompey, while for the Second Civil War, between Caesar's murderers, Brutus and Cassius, and his avengers, Mark Antony and Octavian, which ended with the victory of the latter at the battle of Philippi in 41 BC, we have the accounts of three Greeks living in the first and second centuries AD: Plutarch, Appian and Cassius Dio.
Caesar's account is presented in his typically dry manner. Plutarch's, in his Lives of Brutus and Antony, is a very different matter, for his forte was to be vivid and inspiring. He has left a deep mark on subsequent European literature, particularly through the English translation (not from the original but from a French translation) by Sir Thomas North in the sixteenth century, which was used by Shakespeare for his _Coriolanus, Julius Caesar_ and _Antony and Cleopatra._ Plutarch was also, with Livy, a profound influence on later ideas of Roman republican virtue, most notoriously in late-eighteenth-century France. He wrote essentially to entertain and to provide moral lessons, to compare the heroes of the Greek political tradition—such figures as Solon, Themistocles, Pericles and Alcibiades—with Roman ones such as Coriolanus, Cato the Censor, the Gracchi, Brutus and Antony. This intention and the arrangement of his work makes it impossible to deal fully with it in the present context. It is not the intention here to deal in detail with biography as a genre, and Plutarch's _Parallel Lives_ as a whole is made unwieldy by being spread for essentially didactic purposes over the whole of antiquity, Greek and Roman, back to legendary times and down to the end of the Roman republican period. However, there are always times to make exceptions, and the Lives of Brutus and Antony are particularly useful here, since they deal with the crisis of the republic in a way that no surviving work of a major ancient historian does. Fortunately Plutarch's qualities as a writer can be illustrated from a sample of the _Lives._
In using these two brief biographies (the "Brutus" is some fifty pages, "Antony" longer), Shakespeare followed Plutarch so closely, sometimes quoting the translation almost verbatim, that even for new readers they have an air of familiarity. Our Brutus is Shakespeare's Brutus, who is Plutarch's, and the same is true of Antony—though of course the prose works are more detailed. Plutarch's "Brutus" begins with a contrast with Brutus' great ancestor (though his descent had been disputed), the father of the republic, Lucius Junius Brutus, who was hard and uncultivated. Marcus Brutus possessed an ideal temperament for the practice of virtue, being drawn to the inner life of self-cultivation but also able to respond to a call to action ("...the elements / So mix'd in him that Nature might stand up / And say to all the world, 'This was a man!'"— _Julius Caesar,_ V.v.72–4). We get more on his character, the school of philosophy to which he subscribed, the traits of his oratory in Latin and in Greek, his political principles, and his high-minded reason for attaching himself to the party of Pompey. Although of Pompey's faction, after Pompey's death he received Caesar's pardon and even his favour, including appointment to the governorship of Cisalpine Gaul, where he displayed a notable and unusual integrity and benevolence. However, he kept his distance from Caesar and was subject to the influence of Cassius, his brother-in-law and rival. Plutarch makes Caesar's suspicions of pale, lean men apply to Brutus as well as to Cassius.
Shakespeare was able to follow Plutarch almost step by step because the latter's work is so tautly constructed, dramatic and full of incident—almost, already, like a play, with, of course, the climax of Caesar's murder. Brutus is troubled by the solicitations of Cassius and of anonymous letters reminding him of his ancestry, and his wife is driven distracted to see him so troubled. Caesar, on the fatal day, reverses at the last moment his decision not to attend the Senate, and dies at the foot of Pompey's statue ("Et tu, Brute" is a post-classical invention). Cinna the poet suffers his poignant death by mistaken identity. The spectre that appears to Brutus and promises, correctly, to meet him again at Philippi was not invented for the Elizabethan groundlings, but provides Plutarch with a no doubt welcome macabre touch, though it is not explicitly identified with the ghost of Caesar, as in Shakespeare by Brutus and the stage direction. Cassius commits suicide as a result of a misunderstanding, and is eulogized by Brutus as the last of the Romans—"The sun of Rome is set. Our day is gone"—the line given to Titinius in _Julius Caesar_ (V.iii.64). Antony declares Brutus the only disinterested conspirator, though not necessarily over his body. The hangers-on are the familiar crew.
Plutarch's "Antony," for him a long biography, is more diffuse and, until the death scene in Cleopatra's mausoleum, less dramatic, so Shakespeare has to do a good deal with description, like the famous speech by Enobarbus (himself an invention) on Cleopatra in her barge, which is a close poetic rendering of a piece of Plutarch's prose. A fifth of Plutarch's work is devoted rather remarkably to Antony's campaign against the Parthians, which Shakespeare dismisses summarily. Plutarch's reading of Antony's character is interesting though familiar: boisterous, boyish, humorous, extravagant; he is a good commander, but unmanned by his passion for Cleopatra, who, in Plutarch, though not all ancient writers, is devoted also to him (in Shakespeare, "O Antony!" is almost her last word). Apart from the barge scene, the most striking direct transcription is the dying words given to her attendant, Charmian, when she is asked was Cleopatra's suicide well done: "Very well...and meet for a Princess descended from the race of so many noble kings" (North's translation; cf. _Antony and Cleopatra,_ V.ii.325–6). Charmian's very last words, the wonderfully human and enigmatic "Ah, soldier!," however, are pure Shakespeare.
Plutarch can be psychologically complex as well as vivid and dramatic, but one does not go to him for historical vision or explanation. When confronted with the general question of the transition from republic to empire, he is content, as was quite common, to leave matters to the higher powers: "But the day of the Republic was past, it would seem, and so the gods, wishing to remove from the scene the destined master of the world, kept from Brutus the knowledge of his success, even though it almost reached him in time" ("Brutus," 47). Personal motives and actions, and the contingencies which beset individuals and are controlled by the gods (or Fate) for their inscrutable purposes, do everything.
To turn from Plutarch to his younger contemporary, and fellow-Greek, Appian is to meet a genuine historian—though a somewhat colourless one—with a historian's typical concerns. In the part of his general history of Rome dealing with the first century BC (now published as _The Civil Wars_ ) Appian has a great theme, and knows it: the progressive decline of the republic into political violence, gangsterism, civil war and chaos. His starting point in these five books is the murders of the people's tribunes Tiberius and Gaius Gracchus, in 133 and 121 BC respectively, when they attempted a land redistribution on behalf of the poor. This, according to Appian, marked the end of traditional Roman political moderation: "No sword was ever brought into the assembly, and no Roman was ever killed by a Roman, until Tiberius Gracchus, while holding the office of tribune and in the act of proposing legislation, became the first man to die in civil unrest" ( _Civil Wars,_ I.2). The moral barrier once broken, violence escalates into the murderous strife between the factions of Marius and Sulla in the early first century, followed by the civil war between Caesar and Pompey in the mid-century. Appian considers the grounds of the popular discontent which underlies the outbreaks of class conflict—grounds which are manipulated by successful soldiers turned political adventurers and gangleaders: the monopolization of land in Italy by the rich, who create great ranches worked by slave labour, in place of the small freeholders on whom initially fell the burden of military service (I.7). Hence the critical issue of land redistribution, which brought violent civil strife to Rome itself. Under Gaius Gracchus, too, began what was to become an established practice, the distribution of free grain to each citizen from public funds (I.21). Appian later says that this drew the riff-raff of the provinces to Rome (II.20). Appian also deals with the vexed issue of the extension of Roman citizenship to the provinces (I.23, 34). Marius becomes the hope of the popular cause as the Gracchi had been, and Sulla's faction represents the interests of the senatorial class. Their rivalry degenerates into warfare even in the capital itself: "In this way the episodes of civil strife escalated from rivalry and contentiousness to murder, and from murder to full-scale war; and this [Sulla's] was the first army composed of Roman citizens to attack their own country as though it were a hostile power" (I.60).
The account of Caesar's murder, of which Appian greatly disapproves, has none of the attributes of Plutarch's in "Brutus," but is much fuller on the political context, though Appian too relies on Fate for a higher causality. Caesar went to the Senate on the fatal day despite the warning of the soothsayer, "for Caesar had to suffer Caesar's fate"(II.116). Elsewhere we are told explicitly that "it would seem that the divine will was interfering with public affairs to bring about change"(III.61).
There has been disagreement about how far Appian was a mere compiler of other men's work. It is possible to say with certainty, for example, that his account of Catiline's conspiracy derived heavily from Sallust, but in general the ancient habit of mentioning sources only in cases of disagreement makes it impossible to be sure. He is an uneven writer, sometimes very lucid, sometimes confusing, which encourages the idea that his sources may bear much of the responsibility for this. His authorial personality is also rather thin—an impression perhaps accentuated by the fact that we know very little about him other than that he was an advocate in Rome, practising in the times of Hadrian and Antoninus Pius in the second century AD.
The chief idiosyncrasy of Appian's final books is the enormously extended treatment he gives to the proscriptions of Brutus' and Cassius' alleged supporters, who were hunted down and murdered. "Some went down wells, some descended into the filth of the sewers, and others climbed up into the smoky rafters or sat in total silence under close-packed roof-tiles" (IV.13). The sequence of vivid anecdotes of sacrifice, treachery, suicide, escape, capture and miserable death is clearly intended by Appian, or his source, to evoke pity and terror, and the stories are indeed gripping (IV.5–51). Appian claims that he himself investigated the death of the most celebrated of the victims, Cicero, on the spot, but the inordinate length of this section (more than a third of Book IV), which has been called "Tales of the Proscriptions," may suggest that Appian, having found a good source, rather overdid it. He may also be copying Tacitus, who devotes time to the similar persecutions of the Roman aristocracy under Tiberius and Nero (Chapter 8).
Rome had become ungovernable, and Fate, by implication, sanctioned the transition to the autocracy of Augustus. For a subtle account of how this was effected, we have to turn to another Greek, Cassius Dio.
Dio, born around AD 163, migrated to Rome as a young man from the Greek province of Bithynia, south-west of the Black Sea. He entered the Senate under the emperor Commodus, and later held a consulship, which by then, of course, was not at all the chief executive office it had been under the republic. He held governorships including the proconsulship of Africa, and a second consulship in AD 229. His _Roman History,_ as was not uncommon, covered the whole period from Aeneas and Romulus to his own proconsulship. Of the eighty books of his _Roman History,_ those for the period 68 BC to AD 46 have survived intact. The books on the principate of Augustus are not only almost complete but are the only substantial ancient narrative of the period from 32 BC to AD 14.
At the beginning of the period, with Brutus and Cassius dead, Mark Antony is now based in Egypt, having renounced his Roman wife, Octavian's sister Octavia, to live in Egypt with Cleopatra and his children by her. In an elaborate ceremony in Alexandria in 34 BC, Antony issued a deliberate challenge. He proclaimed Caesarion, Cleopatra's supposed son by Caesar, to be Caesar's legitimate heir—the title claimed by Caesar's nephew and adopted son Octavian—and divided the territories of the East between the boy and himself. This division, known as the Donations of Alexandria, became the greatest of Octavian's grievances (50.1). Antony was suspected of intending to hand over Rome to Cleopatra and to transfer the government to Egypt. In the propaganda war from Rome which ensued, directed against Antony, and in Octavian's speech to his troops before the decisive battle of Actium, there was some choice ethnic and misogynist abuse of "the woman from Egypt," of Egyptian customs, and of Antony's adoption of them. For example, he follows the Queen on foot with her eunuchs, and wears an oriental dagger. The Egyptians are slaves to a woman (no woman ever did or could rule in Rome); it is a disgrace to Roman soldiers even to serve as her bodyguards. Antony adopts mythological names for himself, and the Egyptians worship reptiles and beasts as gods (50.5, 24–5).
After the battle of Actium (31 BC) and the retreat of Cleopatra and Antony to Egypt, the story Dio tells of their end is mainly the familiar one, focused on Cleopatra's mausoleum, partly because her treasure had been stored there. Cleopatra is seen by Dio as wishing for Antony's death in order to make terms with Octavian (51.8, 10). Antony, having tried to kill himself, is lifted up into the mausoleum to die in her arms. Finding Octavian impervious, Cleopatra prefers to die rather than be led in his triumph in Rome; she dies regally, though the asp is mentioned only as one possible cause of death, the other being a poisoned pin (51.12–14). She and Antony are placed in the same tomb. (Plutarch's [earlier] account in his Life of Antony, used by Shakespeare, who here again includes translated quotations verbatim, is essentially the same, though rather more favourable to Cleopatra and much fuller of pathetic detail.) Octavian shows a politic clemency to the Egyptians, and views the body of Alexander the Great, touching the nose and accidentally damaging it. Offered a sight of the embalmed Ptolemy rulers of Egypt, he says he had wished to see a king, not corpses, and he refuses to enter the shrine of the bull-god Apis, saying he worshipped gods not cattle (51.16).
In Rome, the crucial step is taken of offering Octavian the tribunician powers for life: tribunes had personal immunity and important powers of veto. In the provinces, he begins to receive the honours of a god, while those paid to him in Rome, though stopping short of this, are exceptional. Cleopatra's effigy is carried in his triumph, with a display of her treasures; her ornaments, Dio says, now adorn the Roman temples, while she herself is represented in gold in the temple of Venus. Roman hostility had clearly contained a good deal of fascination. Games were held, wild animals were slaughtered, and the hippopotamus and the rhinoceros made their debuts in Rome (51.20–22).
Octavian had behaved with modesty and, by Roman standards, moderation. Book 52 deals with the consolidation of his power, after a long preamble in which he is said to ask for advice on the form of his government from Agrippa and Maecenas, who respond with long speeches respectively in support of a republic and of autocratic rule. One is reminded of the (uncharacteristic) Persian debate in Herodotus when, the throne being vacant, the successful conspirators are presented as considering, in highly abstract form, the same issue (Herodotus, _Histories,_ III.80–82), and one wonders if Dio was remembering it. Agrippa's speech is highly abstract and platitudinous. Maecenas' advice relates more closely to Octavian's actual situation; he must, Maecenas says, retain power or perish. Maecenas makes a strong plea for ruling in association with "the best men in Rome" and suppressing the licence of the mob (52.14–15). His advice, which corresponds closely to the course Octavian was to take, is to retain the republican offices as positions of honour but to reduce their powers and make sure that Octavian controls who holds them (52.20).
He advocates a professional standing army, rather than a citizen militia, and puts the central dilemma in a nutshell: "We cannot survive without soldiers, and soldiers will not serve without pay," and the need for troops would arise whatever the form of government (52.28). He proposes that they should be paid and pensioned from the interest produced by the sale of public lands and by regular taxation (an innovation). Rome is to be beautified and entertained with spectacles (52.30). Personal disrespect to Octavian should be ignored, and sharply distinguished from conspiracy, which should be dealt with by the Senate. Excessive indications of his status are to be avoided: "No man ever became a god by vote" (52.35). Alien religions—an old theme this—should be suppressed; they are associated with secret societies and conspiracies. Sorcerers are to be banned, and—this was to be a later preoccupation—even philosophers are suspect (52.36). Octavian is warned against deceivers—another later concern, noteworthy in Tacitus, which Dio seems to make Maecenas anticipate. Power is best held in reserve and exercised with restraint, and external peace is preferable to war, though readiness for war must be kept up (52.37–38). Maecenas concludes prophetically:
If in reality you prefer the fact of monarchical rule yet fear the name of king as accursed, you need not accept that title but can still rule under the style of Caesar. If you need other titles besides, the people will give you that of Imperator, as they did your father Julius, and they will pay honour to your august status by yet another form of address. In this way you can enjoy to the full the reality of kingship without the stigma that attaches to that name. (52.40)
Dio, in writing this long speech, had of course the advantages of hindsight, and in addition to drawing on a knowledge of Octavian's subsequent policies (as Augustus) he must surely have had an eye on his own times. By comparison with the imperial monsters he had served, and survived (Commodus, Caracalla, Elagabalus), Augustus must have seemed a model monarch, as well as having the advantage of being a name of immense power and authority. Maecenas' speech falls into the long tradition of "Advice to Princes" (sometimes with the implication of criticism of those who do not follow it) as well as holding up the example of a revered forebear. In this case the speech is represented as having immediate effect. Once again, in ancient historiography, one sees a speech used both to give a survey of a situation, with its opportunities and dangers, and supposedly to reveal—in this case perfectly plausibly—the thinking behind policies pursued. Octavian, in Dio's account, inclines to Maecenas' advice but, in a final refinement, introduces the advocated changes over a period, to avoid the dangers in attempting "to change the natural proclivities of mankind at a single stroke" he even leaves some to be accomplished by his successors (52.41). Dio is not averse to recording examples of deviousness in Octavian's policy of autocracy by stealth. Recognizing the nervousness of Antony's former supporters, Octavian gives it out that Antony's papers have been burnt, but actually takes care to keep some for possible future use (52.42).
For the following year (28 BC), in which Octavian was consul for the sixth time and also used for the first time the title "Princeps" ("First"), Dio's fifty-third book gives a typically annalistic record of spectacles held and temples dedicated, including, at Agrippa's expense, the Pantheon, which survives. The centrepiece of the book, however, is Octavian's speech to the Senate renouncing all his offices and announcing his intention to retire into private life: all his actions, he claims, have merely been taken to save Rome from the dangers which threatened it (53.3–11). As intended, the effect when the Senate persuades him to change his mind is that he receives from it autocratic powers as though against his will. His first subsequent act is to double the pay of his bodyguard compared with that of the ordinary troops—"So genuine," Dio says sarcastically, "was his desire to lay down absolute power" (53.11). Octavian then engrosses for himself proconsular powers for ten years in all the provinces where more than one legion was stationed, giving him effective control of the army (53.12–13). Dio notes that, though the time limitation became a dead letter, it was subsequently the custom of emperors to celebrate each decade of their reigns, as though their sovereignty were thus renewed. At this time (27 BC) Octavian assumes the name Augustus, giving up, as too "royal," his desire to take that of Romulus.
Seemingly quoting Tacitus, Dio has no doubt that what has occurred is the institution of a monarchy:
The entire conduct and direction of affairs depend on the wishes of the one man who holds power at the time. And yet in order to maintain the impression that this authority is derived from the laws and not from their own supremacy, the emperors have arrogated to themselves all the functions, together with their actual titles, attached to those offices in which power resided.
As _imperator_ they have inherited the powers of the consuls; as censor (though the title eventually fell into disuse) they scrutinize morals; as Pontifex Maximus they are supreme in religious matters; the grant to them of the traditional powers of the tribunes (who continue to exist) gives them the right of the veto and of personal immunity, their persons being treated as sacred (53.17). In addition they have been set beyond the law ( _legibus solutus_ ) (53.18).
Although Dio allows himself some sarcasm at the expense of the facade thus created, his overall verdict is that it had become "impossible for the people to have lived in safety under a republic" (53.19). His only real complaint, interestingly, is as a historian. In the past, when all matters were brought before the Senate and the people, government was transparent: many people recorded what was done, so that the true version of events was publicly known. Now, crucial events are concealed from common knowledge: reports are misleading and cannot be properly investigated. In future he, Dio, will perforce follow the official versions, but will add his own opinion "as I have been able—relying on the many details I have gathered from my reading, or from hearsay, or from what I have seen—to form a judgement which tells us something more than the common report" (53.19). His complaint is an important observation, from a historian whose access, in his own time, to the very highest circles of power would have given him ample opportunity to register the differences between the reality of political action and the versions presented to the public, but it had been preceded, as we shall see, by a similar complaint by Tacitus.
The chief remaining events recorded of Augustus' reign are the loss of the three legions commanded by Varus in Germany in AD 9, which gave Augustus perhaps his greatest shock (56.18–24), and the successive deaths of the younger members of Augustus' family, which pave the way for the accession of his stepson Tiberius. Dio mentions ("some people allege") the rumours which had been retailed by Suetonius linking Augustus' wife Livia to these deaths, and indeed to Augustus' own, and he neither endorses nor rejects them. His verdict on Augustus draws a sharp distinction between his conduct as Octavian under the necessities of the period of civil war and his rise to power and his later exercise of it; Dio's judgement of the latter is wholly favourable. Under Augustus the Romans enjoyed the best of all worlds: "They were subjects of royalty without being slaves and citizens of a democracy without suffering discord" (56.43).
We know little about Dio's sources, though it is clear he knew the works of Tacitus and Suetonius and would have known Augustus' own autobiography. If we rely on impressions, he strikes the reader as a generally sober and trustworthy historian; he avoids excess of sentiment or partisanship, and is not unduly conventionally rhetorical, except perhaps in describing the battle of Actium. Unlike a Roman of the senatorial class like Tacitus, he has no perceptible nostalgia for the republic, but abhors, like Livy, the excesses of both tyranny and anarchy. His account of the way Augustus presents the facade of a republic while coverting it into an autocracy contributed to an image, particularly relevant to Opposition suspicions in eighteenth-century England, of the subversion of a free constitution by stealth. It was a double image, depicting both a sinisterly crafty ruler, undermining the constitution from within, whom an ill-intentioned monarch might emulate, and, equally plausibly, a wise and moderate statesman, bringing a golden age of peace and plenty to a city long distracted by civil strife.
**EIGHT**
**Tacitus: "Men fit to be slaves"**
Edward Gibbon, in his earliest work, an _Essay on the Study of Literature_ (1761), took Tacitus as his model for the "philosophic historian." Tacitus, he wrote, "employs the force of rhetoric only...to instruct the reader by sensible and powerful reflections." Gibbon's essay was in some respects consciously backward-looking. Tacitus' reputation was waning a little compared with the immense vogue he had enjoyed from the late sixteenth to the late seventeenth century, the European "Age of Absolutism." Gibbon's praise was part of a defence of humanist learning against the new "philosophic"—we should say "scientific"—spirit which condemned knowledge of the past as useless. The phrase "philosophic historian" laid claim to a foothold in both camps, the humanist by the noun and the modern by the adjective. Tacitus was a good candidate for the title because he had recently been much admired as a political thinker, compressing the wisdom gleaned from history into epigrammatic form. Montaigne, another admirer, had expressed the sense of a link between his own times and those of the Roman historian, which does much to explain the attention paid to the latter: "You could often believe that we were the subject of his narrating and berating" ("On the Art of Conversation," _Essays_ ).
It was, of course, commonplace to turn to the ancient world for parallels: Appian, for example, seems to have been widely read during the period of the French civil wars of the sixteenth century. It was natural for people in the Age of Absolutism to find much in Tacitus that seemed relevant to their own situations; drawing lessons from Tacitus seems also sometimes to have been a covert way of referring to the disreputable political wisdom purveyed by Machiavelli. Contemporary history and drama offered parallels which, to the well read, must often have seemed like reminiscences. Shakespeare's Richard III, for example, like Tiberius and Nero, wears a mask of humility and virtue until he achieves undisputed power, when he emerges as a bloodthirsty tyrant. Shakespeare's Iago is the ambitious, intriguing, tale-bearing lieutenant whose archetype could seem to be Tacitus' portrayal of Tiberius' henchman Sejanus. Shakespeare's contemporary Ben Jonson wrote a play on Sejanus (1603), while his _Catiline_ (1611) was based on Sallust and Cicero. Modern monarchs and princes and their intimates dabbled in astrology and magic, as their counterparts did in Tacitus. The French queen mother Catherine de' Medici, to whom the St. Bartholomew's Day Massacre was widely attributed, must readily have recalled to some the formidable and sinister imperial wives and mothers Livia and Agrippina; rumours of serial poisoning attached themselves to all three. Tacitus not only described such a world, but did so as a public man who clearly felt guilt for the compromises he had made to survive as a senator during the bloodstained reign of Domitian, as he explains particularly in the biography of his father-in-law, Agricola ( _Agricola,_ 3, 45).
Cornelius Tacitus was a boy in the time of Nero, and in the year of civil war in the Roman world, AD 69—"the year of the four emperors," which is covered by the extant part of his _Histories._ He seems to have produced these in the very early second century. They were followed by his other major work, the _Annals,_ which covers, now with lacunae, the earlier period from Tiberius to Nero. His public career progressed under Vespasian and Titus, and after the death of Domitian (AD 96) he became consul in 97 and later was made governor of the province of Asia. In the preface to the _Histories_ he celebrates the happy age into which he has lived following the death of Domitian—the age of Nerva and Trajan ( _Histories,_ I.i).
If the reign of Caligula and parts of the reigns of Claudius and Nero had not been missing from the _Annals,_ and also the period AD 71–96 from the _Histories,_ his works would have formed a continuous series covering almost the whole of the first century. The reigns of Claudius and Nero, as well as those of the other first twelve emperors, are of course extant in the Lives of the Caesars produced by his contemporary Suetonius. Suetonius' work is biographical rather than historical; he organizes his account around character traits rather than the sequence of events, according to the precedents set by the well-known genres of the panegyric and the invective, and also includes much that would have been thought too trivial for history. Tacitus' works also overlap with that of Cassius Dio, most notably on the principate of Tiberius. Dio's history continued until his own time, the early third century, but the later part now exists only as Byzantine summaries. Ignoring these, however, the line of major historians from Livy to Tacitus covers the last three centuries BC and the first century AD, from the consolidation of Roman power in Italy to the end of the first century of the empire. The gaps, in Livy and Tacitus in particular, are damaging, but we are fortunate to have so much. The _Annals_ survived only in two medieval manuscripts, one of one part and one of the other.
Tacitus, then, wrote contemporary and near-contemporary history. He cites the testimony of eyewitnesses, old men when he was young, whose memories stretched back well into the time of Tiberius ( _Annals,_ III.16). He certainly used the memoirs left by Tiberius and Claudius and the published transactions of the Senate, to which, as a senator, he would have had easy access. At the beginning of the _Histories,_ and also in his _Dialogue on Oratory,_ Tacitus shows himself highly self-conscious about the changing formative influences, cultural and political, on oratory and the writing of history, associated with the ending of a republican form of public life and the advent of the empire. In the _Dialogue_ we have, in contrasting speeches, the rudiments of a kind of literary or cultural history, with attempts at appraisal. At the beginning of the _Histories,_ speaking of historians, Tacitus remarks that "So long as republican history was their theme, they wrote with equal eloquence of style and independence of outlook," but when "the interests of peace required the concentration of power in the hands of one man, this great line of classical historians came to an end. Truth, too, suffered, in more ways than one. To an understandable ignorance of policy, which now lay outside public control, was in due course added a passion for flattery, or else a hatred of autocrats." Dio (53.19) was to make the same complaint of the inaccessibility of the real causes of events. Tacitus shows an acute, even perhaps excessive, awareness of the difference between public profession and private motive, and speculates freely about motives. For his own claim to impartiality—a standard one, and not always plausible—he found at the opening of the _Annals_ a formula which became classic: he had written, he said, without passion or partiality—" _sine ira et studio._ "
Montaigne, in noting Tacitus' biases, turns his perception of them into something like a compliment: one can see, he says, that Tacitus' judgements sometimes do not fit the evidence, because he has presented it without distortion ("On the Art of Conversation"). There is a good deal in this, especially in the presentation of Tiberius, the most striking of Tacitus' portraits. Tacitus does not discount the lurid rumours of sexual vices surrounding Tiberius' retirement on Capri, but he is less interested and circumstantial than Suetonius. Tiberius' public actions and sayings as reported by Tacitus seem almost always sensible, humane and even generous, and commendably if cynically down to earth and sane, indicating a positive dislike of imperial pretension and of flattery. These traits are, of course, not incompatible with bizarre sexual malpractices or, more importantly for the Senate, with lethal spasms of suspicion. The Senate, even while he is holding it up to contempt and derision, is almost always at the centre of Tacitus' picture; it was, after all, his own order. In his account, Tiberius is always seen as a darkly threatening, brooding presence, the enigma of whose character, guarded by his taciturnity and preference for seclusion, is always given the most unfavourable interpretation, in terms of hypocrisy and duplicity, even when his overt words and actions give no support to this; to the charge of hypocrisy these can offer no defence. Tiberius' reclusiveness, even his modesty, are proof for Tacitus of his moral turpitude and his need to conceal it; his silences, invisibility and sometimes enigmatic utterances are invested with malevolence and menace. How much justification there may have been for this interpretation it is now impossible to say, though it is certainly arresting and memorable, but the relentlessness of Tacitus' adverse judgements ends by arousing suspicion.
What survive with apparent certainty, as recorded by Tacitus, are Tiberius' actions and sayings. Tiberius, Tacitus does not dispute, lived frugally and unpretentiously. He was not miserly, and we are given numerous examples of his generosity in alleviating public and private misfortunes ( _Annals,_ I.75; II.37–48, 87; IV.64; VI.17). There are occasions when he shows clemency and compassion, or discountenances proceedings against individuals for disrespect to himself (I.74), as when he vetoes a prosecution of a man for having melted down a silver statue of him. It was said that on leaving the Senate he exclaimed in Greek, "Men fit to be slaves" (III.65). But Tacitus' view is that he was hostile to servility and freedom alike. He is seldom given full credit for his words and actions; they are always seen as the instruments of a deep, nefarious policy. His toleration of free speech, for example, was a way of discerning "the truth which servility hides" (VI.38). Tiberius' rejection of the proposal of a shrine to himself in Spain seems a model of sanity: "I am human, performing human tasks, and content to occupy the first place among men. That is what I want later generations to remember...Marble monuments, if the verdict of posterity is unfriendly, are mere neglected sepulchres" (IV.38). But even this was interpreted, Tacitus records without dissent, as attributable to a sense of guilt or to poverty of spirit.
In his interpretation of Tiberius, Tacitus seems to waver between seeing a real deterioration of character and attributing the change to Tiberius' growing authority and diminishing need for concealment of his true nature. It has been suggested that ancient writers—one sees the same pattern in Suetonius—were inclined to see characters as fixed and to attribute the lapsing of virtues and emergence of vices only to the disappearance of dissimulation. Tacitus admits the evidence of Tiberius' earlier virtues, but emphasizes the deterioration. Nonetheless, the summary he provides of the earlier, "good," part of Tiberius' rule amounts to an impressive tribute, indeed a description of a model government: freedom of discussion, the choice of worthy men for office, the due enforcement of law, and measures of public relief sponsored by the Emperor in cases of bad harvests. Officials were kept in check and not allowed to be extortionate. "His estates in Italy were few, his slaves unobtrusive, his household limited to a few freedmen. Any disputes he had with private citizens were settled in the law courts"(IV.6). The great change coincided with the advancement of Sejanus. It is tempting, but perhaps unwarranted, to think that Tacitus may have sought to reconcile discordant accounts under the general rubric of hypocrisy.
Most of the _Annals_ —in fact virtually all that is not concerned with the campaigns of Roman armies on the frontiers, in Parthia, Macedonia, Armenia, Germany and Britain—is focused on the relations between the Emperor and the Senate. Tacitus' attitude to the Senate is both scathing and solicitous. He is anxious to preserve whatever examples of merit, worthy of earlier generations, he can find. He subscribes to a standard view of history as the servant of morality: "to ensure that merit is recorded, and to confront evil deeds and words with the fear of posterity's denunciation" (III.65). Nevertheless, he is sometimes prepared to allow piety, concern for a family's good name, to draw a veil over abject behaviour: "They are dead, and I feel I owe it to their ancestors not to name them" (XIV.14). But in general the shifts to which servility is driven in its search for advantage, or at least for survival, are observed with the ironic contempt of a man who understands them perhaps only too well. Tiberius' well-known hostility to sycophancy adds a further twist: a show of independence was the only flattery left (I.8). Public conduct was conditioned by the anxious eye always kept on the ruler to judge its effect, so that, on the accession of Tiberius, the senators "must show neither satisfaction at the death of one emperor nor gloom at the accession of another: so their features were carefully arranged in a blend of tears and smiles, mourning and flattery" (I.7). "Political equality," as Tacitus almost redundantly observes, "was a thing of the past" (I.2). Despite his recording of occasional examples of virtue and independence, sometimes incurring death, the world of high politics described by Tacitus is predominantly one of slanderous accusations, malicious prosecutions, spying and tale-bearing, hypocrisy and subservience.
Informing became a deadly blight on the age:
Friends and relatives were as suspect as strangers, old stories as damaging as new. In the Forum, at a dinner party, a remark on any subject might mean prosecution. Everyone competed for priority in marking down the victim. Sometimes it was self-defence, but mostly it was a kind of contagion, like an epidemic. (VI.7)
The atmosphere of suspicion and fear in Tacitus' narrative is greatly intensified by the crooked ambition and arbitrary power and cruelty of Tiberius' chosen lieutenant, Sejanus, commander of the Praetorian Guard.
At Rome there was unprecedented agitation and terror. People behaved secretively even to their intimates, avoiding encounters and conversation, shunning the ears both of friends and strangers. Even voiceless, inanimate objects—ceilings and walls—were scanned suspiciously. (IV.69)
Surveillance has since been enhanced by technology, but clearly its essential nature remains the same. Elsewhere, in the _Agricola,_ speaking of his own experience of tyranny and the sense of shame it brought, Tacitus says, "The worst of our torments under Domitian was to see him with his eyes fixed upon us" (45), which reminds the modern reader of the court of Stalin, just as Sejanus is reminiscent of Stalin's chief of police, Beria.
Eventually Sejanus overreaches himself and is murdered. The appalling ruthlessness of Roman political atrocity is pitifully caught in the plea of his young daughter:
They were taken to prison. The boy understood what lay ahead of him. But the girl uncomprehendingly repeated: "What have I done? Where are you taking me? I will not do it again!" She could be punished with a beating, she said, like other children. Contemporary writers report that, because capital punishment of a virgin was unprecedented, she was violated by the executioner, with the noose beside her. Then both were strangled and their young bodies thrown on to the Gemonian steps. (V.6)
But the persecutions do not stop. For Tacitus, the underlying cause of Sejanus' rise to power was "heaven's anger against Rome" (IV.1). He laments not only the fate of Rome but his own as its historian:
I am aware that much of what I have described and shall describe may seem unimportant and trivial. But my chronicle is quite a different matter from histories of early Rome. Their subjects were great wars, cities stormed, kings routed and captured. Or, if home affairs were their choice, they could turn freely to conflicts of consuls with tribunes, to land-and corn-laws, feuds of conservatives and commons. Mine, on the other hand, is a circumscribed, inglorious field.
But, he goes on, even the apparently insignificant is worth examining, because it often causes major developments, whether the country is a democracy, an oligarchy or an autocracy (a mixture of these, he adds, never lasts long). When there was a democracy, it was necessary to understand the mind of the people in order to control them; when the Senate was powerful the wisest experts were those who best knew its mind. Similarly,
now that Rome has virtually been transformed into an autocracy, the investigation and record of these details concerning the autocrat may prove useful. Indeed, it is from such studies—from the experience of others—that most men learn to distinguish right and wrong, advantage and disadvantage...So these accounts have their uses. But they are distasteful. What interests and stimulates readers is a geographical description, the changing fortune of a battle, the glorious death of a commander. My themes, on the other hand, concern cruel orders, unremitting accusations, treacherous friendships, innocent men ruined—a conspicuously monotonous glut of downfalls and their monotonous causes. (IV.32–3)
One may also, Tacitus adds, excite animosity by mentioning the disgraced ancestors of the living, or the latter may see their own behaviour mirrored in that of others and be accordingly resentful, or be put to shame by glorious examples.
Tacitus' account of Caligula is missing, and that of Claudius severely truncated—what remains of it focuses chiefly on the treachery of Claudius' wives, Messalina and Agrippina. Tacitus invokes the authority of unnamed witnesses in recording this: "What I shall tell is the truth. Older men have heard and recorded it" (XI.27). The earlier part of Nero's reign is similarly dominated by his mother, Agrippina, though foreign affairs and campaigns, including those in Britain, claim the usual share of attention accorded them in annals. The burning of Rome, of course, forms another episode. Nero's own scandalous behaviour finds new forms of excess, and populace and senators, and their wives and daughters, are seen as grossly corrupted by sycophancy and debauchery, with the Emperor as leader of the revels. Even the spread of Christianity—"the deadly superstition"—to Rome itself is additional evidence of decadence: "All degraded and shameful practices flourish in the capital" (XV.44). The climax—Tacitus' extant account stops short of the fall of Nero—is the persecutions which follow an unsuccessful plot. Tacitus describes a number of cases in detail, dwelling particularly on dignified suicides. Ever conscious of the effect on his readers, he laments that they may find the otherwise unremitting ignominy tedious: "This slavish passivity, this torrent of wasted bloodshed far from active service, wearies, depresses and paralyses the mind"(XVI.14). But the ends of notable men deserve not to be ignored, and in any case their ignominy was not their fault but—again—heaven's anger with Rome. As he appreciates, it seems a far cry from the impulse, going back as far as Herodotus, to write history to preserve the memory of notable deeds.
It seems in Tacitus' account, in fact, that the last recourse open to a member of the senatorial class to show worth was in the manner of taking his own life. The moral antithesis here is the Stoic one, between the striving for a worthless power and the dignity of a well-conducted withdrawal from an intolerable life or from imminent execution. Freedom is found only in the liberation of death and, sometimes, in the belated frankness or defiance it releases. But Tacitus, while recognizing this, recognizes also the passivity contrasted with death on active service. Roman virtue, it seems, is no longer active, patriotic and political, but self-centred and, in Stoic terms, "philosophic." Its reward is not fame but merely escape; at best death takes place calmly, like the prototypical death of Socrates, in the company of friends, an intimate elite.
One motive to suicide, as Tacitus notes, is that it spares one's family the sequestration that followed conviction for treason: the concern for others was for family, not country. There is an ironic contrast here, though Tacitus does not make it explicit. Early examples of Roman "firmness," recorded, notably by Livy, from Lucius Junius Brutus onward, characteristically involved the sacrifice of family-feeling to the service of the state—a moral victory for public law over ties of kinship, which we may think of as a crucial step in the development of an idea of the state and its claims. In the world of tale-bearing and despotic terror described by Tacitus, the Roman state seemed a nostalgic memory; personal traits in the emperor, or the accusations covertly fed to him, or malicious or sycophantic prosecutions in the Senate, prevail everywhere; for the victim, the citizen _in extremis,_ personal and family interests are again predominant. Tacitus is surely well aware of this as marking the contrast between republic and empire—hence his lament over the task imposed on the modern historian.
The _Annals,_ as we have them, break off in mid-sentence in AD 66. The _Histories,_ which were actually written first, take up the account after Nero's death, in AD 68, and record, in much more detail than is common in the _Annals,_ the events of AD 69–70. They continued until almost the end of the century, but the rest is lost. The year AD 69 was one of exceptional eventfulness, comprising civil wars and usurpations, four changes of emperor, and the eventual establishment of a new dynasty by Vespasian: the century-long dominance of the Julio-Claudians, established by Augustus, was at an end. Hence AD 69 became known as "the year of the four emperors": Galba, Otho, Vitellius and Vespasian. Ultimate power rested for a while nakedly with the soldiery, and the conditions of the late republic were re-created in a highly condensed form. Three of the emperors had a power base in one of the military provinces, where each was proclaimed emperor by his troops: Galba in Spain, Vitellius in Germany, and Vespasian—who took longest to reach Rome—in the east, while Otho, an associate of Nero, was proclaimed by the Praetorian Guard in Rome in a kind of Neronian backlash against Galba. Hence Rome was involved in civil wars for the first time for a century.
Deploring this, Tacitus implies that the elevation of Galba, the first to be proclaimed, should have been accepted, though he recognizes that the eventual winner, Vespasian, was in many ways the worthiest of the contenders. For Otho, a partner in Nero's vices, and Vitellius, always spoken of as lazy and compulsively gluttonous, he has no use at all. Drawing the parallel with the late republic, he adds that Caesar and Pompey, Augustus and Brutus, had been honourable antagonists, while when Otho and Vitellius accused each other of wickedness and debauchery "here at least both were in the right" (I.74). The successive usurpers, the response of the Romans to them, and the conduct of the soldiers provoke Tacitus to savage ironies and dismissive epigrams. On Galba, for example: "So long as he was a subject he seemed too great a man to be one, and everyone would have judged him worthy to rule if only he had not ruled" (I.49). Vitellius is commemorated for "the excessive and imprudent generosity with which he squandered both what was his to give and what was not" (I.52). It is easy to understand the impression made on the young Edward Gibbon and to see the marks of Tacitus' influence on some of the stylistic traits as well as on the attitudes of _The Decline and Fall of the Roman Empire._
Tacitus registers the affronts to the dignity of Rome in the disturbances of AD 69 and the behaviour of the Praetorian Guard with an obvious sense of outrage which, even compared with the dreadful events he was to chronicle in the _Annals,_ testifies to the settled character that the empire had nonetheless assumed:
Roman troops made ready to murder an old defenceless man who was their emperor, just as if they were set on deposing a Vologaeses or Pacorus [rulers of Parthia]...Forcing their way through the crowd, trampling the Senate under foot, with weapons ready and horses spurred to a gallop, they burst upon the Forum. Such men were not deterred by the sight of the Capitol, the sanctity of the temples that looked down upon them, nor the thought of emperors past and emperors to come. (I.40)
Despite the delinquencies which had disfigured it, the imperial title had become, it seems, what Augustus' word had claimed for it: august. The injuries done to the city as well as its people, and worst of all the burning of the temple of Jupiter on the Capitoline hill, are clearly felt by Tacitus with indignation that they were the work of Roman armies, who treated Rome itself, as they had treated the peaceful provinces through which they marched, with all the harshness of conquering invaders.
As described by Tacitus, the world of AD 69 has in fact become a world turned upside down: the soldiers, insubordinate to their commanders as well as to the emperors, become mobs, while the Roman mob becomes homicidal, and—a particular outrage—senators are alleged to have taken swords into the Senate house. Peaceful Roman towns like Cremona—founded, Tacitus ironically recalls, as a bulwark against the Gauls—and later Rome itself, are sacked by Roman armies. The Gallic provinces try to placate the soldiers who pass through them "to secure peace in the absence of war" (I.63). When Vitellius enters Rome, he makes a speech "as if he were addressing the senate and people of a foreign state" (II.90). In the capital, Tacitus says, the populace were so corrupted that they applauded the fighting as though watching a gladiatorial combat:
Close by the fighting stood the people of Rome like the audience at a show, cheering and clapping this side or that in turns as if this were a mock battle in the arena. Whenever one side or the other gave way, men would hide in shops or take refuge in some great house. They were then dragged out and killed at the instance of the mob...The whole city presented a frightful caricature of its normal self; fighting and casualties at one point, baths and restaurants at another, here the spilling of blood and the litter of dead bodies, close by prostitutes and their like—all the vice associated with a life of idleness and pleasure, all the dreadful deeds typical of a pitiless sack. (III.83)
But Rome nursed a moral plague (Sallust's influence seems very evident here), infecting the armies with the taint of luxury, softness and indiscipline (as Capua and Boeotia, one could add, had once been blamed for doing; the centre of corruption was now the capital itself)(II.69). Tacitus describes the armies' excesses almost pathetically as "quite un-Roman" (II.73). The soldiers, with whom power lay, were, he makes clear, themselves fickle, disorderly and confused as well as grasping. At one point what begins as no more than a drunken misunderstanding is construed by the Emperor (Otho) and the senators as a full-scale mutiny, in what becomes an episode of comic and ignominious terror. The Emperor's dinner-party guests, in a state of acute anxiety, but trying to be nonchalant, look to him for a cue. "They eyed Otho's expression. As is the way of suspicious minds, although Otho felt alarm, he also inspired it." He tells them to leave. This is the signal for a general stampede. Magistrates throw away their badges of office. The troops panic in turn, and rush the banqueting hall demanding that the Emperor show himself. "The whole building was a hubbub of weapons and threats...Their blind and panic-stricken frenzy, finding no single target for its anger, clamoured for a clean sweep of everybody." Eventually Otho appears and cajoles them with "tears and entreaties," and they return to barracks—"but grudgingly and with bad consciences. Rome resembled a captured city the next day. The great houses were shuttered, the streets almost empty, the populace in mourning. The downcast glances of the troops displayed sullenness rather than regret" (I.81–2). In this confusion the successive emperors and claimants were less the directors of events than flotsam on their surface, bewildered, hesitant and passive. The abjectness of the Senate, frightened of the troops and anxious only to hedge against any possible outcome, is presented as pitiful. While Otho reigned and Vitellius advanced, some, in denouncing Vitellius, timed their invectives for moments of uproar when they could not be heard clearly or took care to be incoherent (I.85). Otho's own speech was unexpectedly restrained in its reference to Vitellius' supporters, which Tacitus is inclined to attribute to his speechwriter's anxiety to protect his own skin (1.90). (In the _Annals_ [XIII.3], Tacitus says that Nero was the first emperor to employ a speechwriter.)
In a world turned upside down, vice became virtue, poverty a blessing, and wealth a curse, and Tacitus' irony was fully equal to the opportunities. The Senate, in pardoning three of its members convicted under Claudius and Nero for extortion, changed the charge to the less grave one of treason, since the latter, by misuse, had lost (for the Senate) its gravity (I.77). Soldiers with the Othonian army, when captured, claimed credit with Vitellius for causing a debacle which was actually the result of inefficiency. Vitellius took them at their word and "acquitted them of good faith" (II.60). The wealth of one senatorial victim led to his will being set aside, while "Piso's last wishes were respected because he was poor" (I.48). The same Piso, adopted as his heir (Caesar) by Galba, procured from his elevation only one advantage: that of being murdered before the elder brother to whom he had been preferred.
With Vespasian installed as the victor, affairs return to a shaken normality, symbolized, it seems, by the description of the ceremonies for the rededication of the restored temple of Jupiter, though there are already ominous remarks about the new Emperor's son Domitian, who was to succeed his elder brother Titus in AD 81. Much of the remainder of the work which is left to us is concerned with debates in the Senate, apart from Book V, which is devoted to a garbled and decidedly hostile account of Jewish history and religion, made topical by the campaigns of Vespasian and Titus against the Jewish rebellion described in detail by Tacitus' contemporary the Jewish historian Josephus. The modern reader can for once enjoy the sense of thinking he knows better than Tacitus, though it is interesting to see what slanderous legends were current.
Tacitus clearly does not know the Bible, already available in a Greek translation. Thus, in the account of the flight from Egypt on which, Tacitus says, most authorities agree, it is the Israelites who are expelled, on the recommendation of the god Ammon (Tacitus called him Hammon) as an act of purification to cure Egypt of a plague they carry (generally identified as leprosy). Moses, boldly declaring to the Israelites that their own god has deserted them, exhorts them to rely only on their own resources. He follows his own precept by discovering, as a result of observing the behaviour of some wild asses, a source of water in the desert. On the seventh day of their emigration the Jews expel the Canaanites from their land, where they build a city and a temple, whose shrine, in gratitude for their deliverance, contains an image of a wild ass, though elsewhere they are said to erect no images and to practise a purely spiritual monotheism. They devote not only every seventh day, marking the end of their exodus, but also every seventh year to idleness. Their new religion was prescribed by Moses, and its practices are sinister and revolting: they ban exogamy, practise circumcision, and teach contempt of the gods and of patriotism. We are given fairly accurate descriptions of the Jordan and the Dead Sea, and also a reference to a desolate plain once supporting great and populous cities. The Jews are a degraded nation, and their religion, despite being a spiritual monotheism, is "superstitious." The description as we have it ends with a brief account of the siege and capture of Jerusalem by Titus.
Book V, then, was a classic digression of a conventional kind, and it is only the existence of another, fuller and more familiar, source which enables the non-specialist to be more critical of it than of other such ethnographic digressions in ancient historiography. Tacitus observed such conventions, though his manner—he seems to have admired Sallust most among his predecessors—in its famous terseness, its epigrams and ironies, is highly distinctive. In the _Annals,_ of course, he accepts the responsibilities of the annalist, though the _Histories,_ in devoting so much space to such a short period, has more the character of a monograph. In the _Annals,_ in particular, consulships, portents and notable deaths are recorded as they occur, and large amounts of attention are paid to campaigns on the borders of the empire; indeed, Tacitus seems to regret that conservative imperial attitudes to further extension of the empire do not allow him more opportunities for such descriptions ( _Annals,_ IV.32). Occasionally one can see Tacitus self-consciously testing the limits of the annalistic form and looking towards a more thematic treatment: "If I had not proposed to record each event under its own year, I should have liked to anticipate and recount immediately..." (IV.71). At another point he confesses to having merged the events of two summers and even to condensing, into one, campaigns conducted by two imperial governors over a period of years, "since piecemeal description would cast a strain on the memory" (as, reading from a scroll, it would indeed have done) (XII.40).
Affairs on the frontiers also provide the themes for two famous works, published not as digressions but as free-standing monographs. One is the biography—essentially a eulogy—of his father-in-law, Agricola, governor of Britain, of which it includes an account. The other, probably owing much to the elder Pliny, is widely regarded as the outstanding ethnographic monograph to have come down to us from the ancient world, the _Germania._ It has certainly had the most profound subsequent influence. From the Renaissance onward, this work of Tacitus, as we shall see later, was to be a key text in the scholarly rewriting, from ancient sources, of the early history of the European nations, replacing the legendary genealogies which had typically traced them to heroic refugees from Troy. In the seventeenth century it became a crucial text in constitutionalist opposition to absolutism. Much was made of Tacitus' declaration that the Germans had no hereditary kingships, so that the barbarians of German stock, for example the Franks and Anglo-Saxons, who established national kingdoms in Europe on the ruins of the Roman empire, presumably had originally had none either: freedom was older than absolutism. From the nineteenth century onward, as the study of European prehistory became more accented towards the uncovering of the racial origins and characteristics of the various nations, much stress came to be laid, with eventually ominous implications, on Tacitus' casual claim that the German tribes had always inhabited Germany and were racially unmixed. Tacitus' own text, divorced from these later preoccupations, is an accomplished and vivid work, whose depiction of German manners is not, in its effect, an unfriendly one: the account of the simplicity of German life, and particularly of its uncorrupted sexual mores, has been read as an implied criticism of the very different manners of Rome. Tacitus sets the Germans' drunkenness, idleness and quarrelsomeness against their sexual temperance and chastity, independence, courage and loyalty.
Germany also figures notably in the _Annals,_ as the chief external danger to Rome: the destruction of three Roman legions under Varus in the Teutoburger Wald by the German hero Arminius (to whom Tacitus pays generous tribute), just before Tacitus' work opens, was an immense shock. Tacitus picturesquely evokes the horror of the battlefield, with its debris of bones, as a later Roman army (under Germanicus) found it.
The scene lived up to its horrible associations. Varus' extensive first camp, with its broad extent and headquarters marked out, testified to the army's labours. Then the half-ruined breastwork and shallow ditch showed where the last pathetic remnant had gathered. On the open ground were whitening bones, scattered where men had fled, heaped up where they had stood and fought back. Fragments of spears and of horses' limbs lay there—also human heads, fastened to tree-trunks. In groves nearby were outlandish altars at which the Germans had massacred the Roman officers. (I.60)
The Germans were formidable adversaries, and the campaigns that Tacitus describes in the lower Rhine area and along the Danube were matters of vital concern, increased by the horror clearly evoked by the fearful terrain in which they were fought. In the _Germania,_ Tacitus speaks feelingly of how unappealing the lands of the Germans are to anyone not born there, being "covered either by bristling forests or by foul swamps" (5). Near the beginning of the _Annals,_ Roman anxiety about these sensitive frontiers is greatly augmented by dangerous mutinies in the armies stationed there. In reading Tacitus' account here one has to remember that he wrote it immediately after treating, in the _Histories,_ the civil wars of his own time, to whose outbreak the indiscipline and rapacity of the armies had largely contributed.
The situation half a century earlier was restored by the generalship of Augustus' grandson and Tiberius' nephew Germanicus, whose premature death soon afterwards plunged the empire into mourning amid strong rumours of poisoning. The peril represented by the mutinies and the difficulty in dealing with them are caught particularly by Tacitus in two notable speeches, one by a private soldier setting out the troops' grievances and the other by Germanicus, reproachful, scolding and placatory. Tacitus observes the convention of writing speeches, though his are often quite short; sometimes he puts them in indirect speech. Those of Germanicus and the soldier are given ostensibly verbatim. Under a despotism, a mutinous army becomes for a while a democracy, with all the volatility and instability of a democracy founded in grievances and without settled habits and traditions. Oratorical persuasiveness becomes again a vital political instrument.
Tacitus does not gloss over the hardships and injustices suffered by the soldiers, but Erich Auerbach warns us against interpreting the vividness of their recital as sympathy with the mutineers' demands. It is a product only of Tacitus' artistry, and the mutineers' leader, Percennius, who makes their case, is described as a former agitator in the theatre, which certainly conveys contempt. To Auerbach ( _Mimesis,_ ch. 2) Tacitus' speeches are "sheer display." One hardly suspects him, it is true, of endorsing Percennius' case: the mutineers represented a grave danger, and had to be suppressed. Yet the question of imaginative sympathy remains tantalizing. Auerbach admits that the rhetorical genre of writing speeches "allowed a certain sympathetic entering into the thoughts of the supposed speaker." The question remains how there could be both sympathetic understanding, for purely aesthetic purposes, and also none. Certainly for the modern reader, Percennius' speech is cogent as well as eloquent. Imaginatively, at least, Tacitus understands his case:
Old men, mutilated by wounds, are serving their thirtieth or fortieth year. And even after your official discharge your service is not finished; for you stay on with the colours as a reserve, still under canvas—the same drudgery under another name! And if you manage to survive all these hazards, even then you are dragged off to a remote country and "settled" in some waterlogged swamp or untilled mountainside. Truly the army is a harsh, unrewarding profession! Body and soul are reckoned at two and a half sesterces a day—and with this you have to find clothes, weapons, tents, and bribes for brutal centurions if you want to avoid chores. Heaven knows, lashes and wounds are always with us! So are hard winters and hardworking summers, grim war and unprofitable peace. There will never be improvement until service is based on a contract—pay, four sesterces a day; duration of service, sixteen years with no subsequent recall...( _Annals,_ I.17)
In response Germanicus stages a spectacular suicide attempt, which causes a revulsion of feeling, and when order has been restored the ringleaders are butchered. The soldiers, Tacitus says, not merely tolerated this but "revelled in the massacre as though it purged them of their offences" (I.44).
The soldiers' case has at least been allowed to be forcibly put. The same is true of Tacitus' treatment of barbarians. Indeed, the tribute to Arminius amounts to a memorable obituary: "He was unmistakably the liberator of Germany [the subsequent title "liberator Germaniae" comes from Tacitus]...To this day the tribes sing of him. Yet Greek historians ignore him...We Romans, too, underestimate him, since in our devotion to antiquity we neglect modern history" (II.88). There is a touch of the sentiment of Kipling's "The Ballad of East and West" about Tacitus' tribute, as Arminius is promoted into an international heroic Valhalla. Tacitus was alert to barbarians' courage and dignity, and even presents their reproaches to the Romans in memorable epigrams. The British leader Caratacus, taken captive to Rome, tells his conquerors, in a scene which became canonical in Victorian schoolroom history, "If you want to rule the world, does it follow that everyone else welcomes enslavement?" (XII.36). Tacitus often uses this last word for Roman conquest. He makes a German, denouncing Roman greed, memorably express his rejection of the terms offered: "We may have nowhere to live but we can find somewhere to die!" (XIII.56). Again, in a phrase which has become canonical, in the _Agricola_ the spokesman for the Britons before their decisive defeat is made to pass what, in reading it, can seem like a definitive judgement on Roman imperialism. Speaking of the Britons as "the last of the free," he identifies the Romans as "the only people on earth to whose covetousness both riches and poverty are equally tempting. To robbery, butchery, and rapine, they give the lying name of 'government' they create a desolation and call it peace" (30).
Tacitus' own judgements are often emphatic, but, taken overall, they are not simple. He sees, and regrets, a nobility of character in the republican period now largely lost, but he insists on recording modern exceptions and he deprecates, as in the case of Arminius, an exclusive adulation of antiquity. He recognizes that peace and security required the supersession of the republic by the empire, of whose aberrations and the servility they evoke he is the merciless chronicler. He satirically describes the sycophancy of the Senate, but admits he knows by experience the terror of being under a tyrant's eye and the near-impossibility of reconciling survival with the preservation of personal integrity. The price of survival in such circumstances is high and lasting, even when the tyranny is relaxed. Speaking of the terrible time of Domitian, he laments the lost years, the erosion of energy and integrity, suffered by himself and his generation:
Think of it. Fifteen whole years—no small part of a man's life—taken from us. Many have died by the chance happenings of fate; all the most energetic have fallen victims to the cruelty of the emperor. And the few of us that survive are no longer what we once were, since so many of our best years have been taken from us...( _Agricola,_ 3).
So far as Rome is concerned, Tacitus' conception of moral decline, though real, seems less deterministic than some; he also speculates that changes in manners may go in cycles ( _Annals,_ III.54). He certainly regrets aspects of the past, but he is not a natural idealizer, just as his endorsement of Roman imperialism is hardly bland. Heaven, certainly, has been angry with Rome and has employed tyrants as its scourges—an idea that resurfaces in St. Augustine. Rome's sufferings prove that "the gods are indifferent to our tranquillity, but are eager for our punishment" ( _Histories,_ I.3). Still, the present is at least much better than the recent past, and he is thankful to have lived to see it: "Modern times are indeed happy as few others have been, for we can think as we please, and speak as we think"—a tribute to the age of Nerva and Trajan, in which he was writing (I.1).
Tacitus is in one sense a stern moralist, denouncing the extremes of vice, which he attributes freely to individuals and to the Roman Senate and populace, as well as to emperors: sycophancy, malevolent suspicion, spying and false accusation, riot and cruelty. He subscribes emphatically to the view that the function of history is to foster virtue and to castigate vice by preserving examples of both ( _Annals,_ III.65; _Histories,_ III.51), though compared with Livy, who shared this attitude, his examples are more often minatory than inspiring. (We do not, of course, know what Livy's would have been as they approached his own times.) Tacitus regards the hypocrisy inherent in the Augustan system—the preservation of the outward forms of a republican egalitarianism—as actually enhancing rather than mitigating the horrors of servitude ( _Annals,_ I.81), but there is no alternative, and in taking the consulship (in AD 97) he himself played a notable part in the charade.
These complexities, and perhaps a training in outward conformity, are aptly expressed as irony, of which Tacitus is a master. We have noted the affinity with him felt by Edward Gibbon, who, in _The Decline and Fall,_ explored the same ambivalences and contradictions—including those of the apparently incompatible virtues of civilization and barbarism, as well as their inseparable accompanying vices. The moral and political world is neither pure nor simple. When Gibbon writes that the Greeks, in the heyday of the empire, had been "long since civilized and corrupted" (II) or—it is almost a precis of Tacitus—that the Caledonians preserved "their wild independence, for which they were not less indebted to their poverty than to their valour" (I), it is easy to fancy that Tacitus nods in appreciation, as Gibbon surely must often have done when reading Tacitus.
**NINE**
**A Provincial Perspective: Josephus on the Jewish Revolt**
When Vespasian, the last of the four emperors to be proclaimed in AD 69, was persuaded to stake his claim, he was commander of the Roman legions in Palestine, attempting to suppress the Jewish revolt which had begun three years earlier. He left his son Titus in command, and with him he left the former Jewish rebel commander in Galilee, Josephus, the future historian of the revolt. Taken prisoner by the Romans, Josephus had won Vespasian's favour by prophesying that he would become emperor; Josephus apparently had a reputation for accuracy in such matters. He was thus enabled to be present, with the Romans, at the siege of Jerusalem, undertaken by Vespasian and completed by Titus, of which he subsequently gave a detailed account. After the war, with his prophecy vindicated and Vespasian now emperor, he continued in favour and, with a pension and Roman citizenship, returned to a property in Rome, where he wrote four works, all extant.
The first, entitled _The Jewish War,_ was a piece of largely contemporary history of the war in which he had participated on both sides; Josephus wrote it, he said, to correct current errors and provide an authentic record. It was written first in his own language, Aramaic, and then translated into Greek. He had the benefit of access to the memoirs of Vespasian and Titus, who both apparently vetted his text. His later works, including a spare and self-serving autobiography vindicating his role in the war, were written in Greek. The other two concern not himself but his people. His _Jewish Antiquities_ is a long precis and paraphrase of the Hebrew Bible, with minor omissions and additions, to which is added the subsequent history of Palestine down to his own times, which repeats and greatly amplifies the first third of _The Jewish War,_ to which it is a historical introduction, covering approximately two centuries.
The variations from the biblical text in _Jewish Antiquities_ are sometimes the product of Josephus' own learning in Jewish law and traditions, but they also show anxiety to give no opportunities to foreign denigration. He omits, for example, the episode of the worship of the golden calf, presumably in order not to reinforce a current slander that the Jews worshipped animals ( _JA_ , III.99; see also Tacitus, above, Chapter 8).
Josephus was a Jewish scholar, a priest and a Pharisee. In view of his role in the war—which he ended, from a nationalist point of view, as a collaborator with the Romans—it is important to stress that, although he was no political radical, and deplored the rebellion and the behaviour of the nationalist leaders, whom he saw as fanatics and terrorists bringing misery to his people, he was nonetheless a proud and patriotic Jew, anxious that the Greek-speaking world should understand and respect Jewish law and customs. This comes out most clearly in his last, polemical, work, which acquired the title _Against Apion_ (Apion being one of the Greek critics he attacked), in which he set himself to refute Greek slanders against Jewish beliefs and practices and dismissal of the antiquity of the Hebrew sacred records.
Despite his brief period as a rebel general, in the Roman empire Josephus was a peace-loving provincial, who made it his task to explain the affairs of his troubled province and his unusual and often slandered people to the Greek-speaking part of the empire—which included, of course, the Roman educated class. In his narrative of the war, though the Roman governor is not excused, the heaviest criticisms are for the Jewish intransigents, and not for the Roman commanders, his patrons Vespasian and Titus. We have no reason to suppose these criticisms did not represent his real views. Nonetheless, his account is given chiefly from a Jewish perspective: it is the suffering of the ordinary population, especially the inhabitants of Jerusalem, and the faction-fighting and criminal behaviour of the insurgents that draw his attention, while his lamentation over the destroyed city and its Temple is clearly deeply felt.
When considering, earlier in this book, the originality of Herodotus' project of "inquiry," the obvious point was made that it was he who was the interrogator and describer of exotic peoples: such peoples neither interrogated Greeks for the same purpose, nor, except in their answers to Herodotus, portrayed themselves for a foreign audience. This now needs some elaboration, because Josephus, much later, is precisely such a self-referential historian and ethnographer, as in the digression in _The Jewish War_ in which he explains the beliefs of the Sadducees, Pharisees and Essenes. In fact, not long after Herodotus, non-Greek writers had appeared who adopted the Greek fashion for ethnographic description and the writing of history and practised it reflexively; they did so, it seems, in imitation of Herodotus, so they were practising a Greek genre, as indeed was Josephus. In the third century BC the Roman historian Fabius Pictor (see above, Chapter 6) wrote the history of Rome from Romulus in Greek, thus presenting it to the Greek world. The third-century Egyptian priest Manetho, whose history of Egypt in Greek was well known in the ancient world, is one of Josephus' targets. Attacking him, Josephus contradicts the old legend, which was purveyed much earlier by Hecataeus and which we have seen in Tacitus, that the Hebrews were ejected from Egypt as lepers, rather than, as the book of Exodus recounts, extorting their release from a reluctant Pharaoh ( _JA_ , III.265).
In the Greek world, the idea of the presence among the Jews of wise philosophers, analogous to the Persian magi, also seems to have been current. It may be the beginning of an image which much later made the Hebrew Kabbalah one of the key magical texts of the European occult tradition and gave King Solomon his role in Freemasonry. But for the next half dozen centuries the image stood, if it stood at all, instead of closer acquaintance. Though the Bible was available in a Greek translation, for the benefit of Greek-speaking Jewish exiles, before the end of the second century BC, and was, as we have seen, paraphrased in Greek by Josephus in the first century AD, it seems to have made little impression on the Gentile world except with the advent of Christianity (see below, for example Chapter 12), in which the relation to Judaic tradition was a serious issue. On the other hand, it is thanks mainly to Josephus' _The Jewish War_ that we know more of Palestine in the first century AD than of any other part of the empire: in Josephus, a Roman province found a voice and spoke of itself and of the experience of Roman rule. A writer neither Greek nor Roman had exhibited his people to a wider world, within the conventions of Graeco-Roman historiography. We should know incomparably more about Roman Britain if there had been a similarly able British historian.
Josephus begins his long historical preamble to _The Jewish War_ with the second century BC, when Palestine was contested territory between two of the successor kingdoms of Alexander's empire, Syria and Egypt. We are told of the Jews' successful assertion of their independence, led by the family of the Maccabees, after the Syrian king's desecration of the Temple had acted as a trigger for revolt. Josephus' narrative rapidly advances to the career of Herod the Great, who in the second half of the first century BC established himself in power by cultivating the favour of the Romans, now the dominant power in the region, and was recognized by them as king of the Jews in 40 BC, though Judaea itself was subsequently brought under direct Roman rule (in AD 6) under a Roman governor. Herod's successors were still local client rulers half a century later, and the most successful of them, Agrippa II, played a peripheral role (described by Josephus) on the side of Rome and the Jewish moderates in the war of AD 67–70. Herod the Great himself had been a consummate politician, managing brilliantly to negotiate the rapid and violent transitions in the Roman politics of his day, dealing successively with Pompey, Caesar, Cassius, Antony and Octavian (whom Josephus calls "Caesar," as he does later emperors), and remaining fast friends with all of them; if this was time-serving, he seems to have raised it to the level of an art.
Josephus introduces us to Herod's copious building projects, in the restoration and improvement of the Temple, in other notable buildings inside and outside Palestine, and in the creation, on a previously fairly insignificant site, of Caesarea as a great Hellenistic city on the coast of northern Palestine. Herod's lurid and tormented domestic life is also put under close scrutiny by Josephus, who had access to Herod's memoirs. The sections concerned, and the struggles for the succession, are a tax on the reader's concentration, perhaps largely because the royal dynasty, prolific in slander and treachery, was parsimonious with names, making do chiefly with permutations of Antipas, Antipater, Aristobulus, Agrippa and, of course, Herod. Royal families, it is true, often behave like this, but it seems sheer wantonness on Herod's part to have murdered his first wife, Mariamme, only to marry another also called Mariamme. Salome, the notorious daughter of Herodias (a variation enforced by gender), does not figure in Josephus, but she was clearly a chip off the old block. Her great-grandmother Salome (of course) not only incited Herod to kill his wife, but, according to her nephew, entered his bedroom and had intercourse with him against his will. It was that sort of family ( _JW_ , I.443, 499).
In the increasing friction between the Romans, with their client local rulers, and the Jewish population, the sanctity of the Temple is, in Josephus' account, a recurring source of tension. Pompey had violated the sanctuary to enter it with his officers, but had refrained from looting; Crassus, however, in 55 BC, had robbed the treasury. Herod, a Hellenized Edomite, not a Jew, provoked a riot in 5BC by adorning the gate with a golden eagle. Some young men zealous for the observance of the Law—the point was apparently debatable—tore it down, and they and the rabbis who had incited them were ordered by Herod to be burnt to death. The emperor Caligula, almost half a century later, posed a grave threat of violence by his determination to have a statue of himself in the guise of Zeus set up in the Temple; the situation was saved only by his timely murder. Josephus' description of the state of Palestine in the period leading up to the rebellion which broke out in AD 66–7 offers a vivid and plausible picture of a country sliding helplessly into anarchy under the influence of false prophets, bandit warlords, communal hatreds and an unsatisfactory governor: "The religious frauds and bandit chiefs joined forces and drove numbers to revolt—threatening with death those who submitted to Rome" (II.264). Some of the religious extremists regarded all secular authority as evil and illegitimate, and behaved accordingly. Inevitably, as well as murderous internecine conflict between various factions, there was much mutual provocation between the more firebrand nationalist and religious fanatics and the imperial power, compounded by inter-communal friction. In Caesarea a conflict arose between Jews and Greeks over a partially blocked access to a synagogue. When young Jews took action to clear it, some Greeks retaliated by placing a chamber pot in front of the synagogue and pretending to sacrifice over it; bloodshed inevitably ensued (II.289).
Florus, the Roman procurator, whose avarice is bitterly criticized by Josephus, made his contribution by raiding the Temple treasury for money, of which he may have been short. Confronted by civil disturbances in Jerusalem, he countered by encouraging what amounted to a Roman military riot, with looting and much loss of life, after which he withdrew to Caesarea. The hotheads bent on rebellion took possession of the Temple and made it their headquarters; at one point, before the beginning of the siege superseded internal conflict, one faction held the upper level of the Temple complex, another the lower, and they fought each other ferociously. A contingent of Roman soldiers, after being given terms of surrender, were slaughtered as soon as they disarmed, and respectable Jews fled the city if they could.
The siege of Jerusalem by the Romans is naturally the core of Josephus' history, but first—equally naturally perhaps, if egotistically—he devotes considerable space to the preliminary campaign in Galilee, where he himself had been appointed a Jewish commander; it was very short-lived, because the Romans were quickly victorious, and was hardly crucial. Josephus' version of his achievements there is not over-modest. He was, by his account, fearless, adroit and an excellent military organizer; he admired Roman drill, and writes an interesting digression describing it. He was, apparently, the hope of his people, vital to their morale, and his capture by the Romans was felt by them as a national disaster.
His account of his capture is ingenuous without being altogether convincing, and it has to be said that if he behaved worse than he said he did then he behaved very badly indeed. Making his escape from his defeated army, "helped by some divine providence," he finds himself in hiding with forty companions. Discovered and offered safe conduct by the Romans, he intends to give himself up, but his companions protest at being abandoned and recommend suicide. Josephus gives himself a decidedly well constructed, somewhat academic speech condemning suicide in general, which not surprisingly fails to grip his audience. ( Josephus' speeches given to others are also often more than usually implausible as products of the spur of the moment, under stress.) Josephus then suggests they draw lots to kill each other, the final survivor killing himself. This was the formula later adopted by the defenders of Masada, which Josephus knew of before he wrote, since he describes it. The others comply, killing and offering themselves to be killed according to the rules, until—"shall we put it down to divine providence or just to luck"?—only Josephus and one other are left alive. Surrounded presumably by the corpses of the compliant, Josephus apparently persuades his companion that, though his idea, the concept is fundamentally flawed. They give themselves up (III.341–91).
After a spell in captivity, Josephus is freed to begin his career of collaboration and the enjoyment of Vespasian's favour as a result of his prophecy of the latter's future greatness; Josephus seems genuinely to have believed in his ability to foretell the future and to interpret prophecies and his own prophetic dreams. He experiences the siege of Jerusalem from a ringside if not altogether safe position—he is struck once by a missile from the walls—shouting to the besieged in their own tongue that resistance is useless—which he believed—and that they should surrender; he also mentions the information on conditions inside the city that he received from deserters. His role would have been described in later military language as "propaganda and intelligence." In judging his conduct here we have to remember (and there is no reason not to take his word) that he had opposed the war, thought the rebellion doomed, deplored its leadership, and participated in it against his judgement. Earlier on, he gives King Agrippa a long speech in which he harangues the Jews on the might of Rome and the futility of their resistance, and this seems to express Josephus' own feelings(II.242–404). He had himself visited and lived in Rome for a while. He had every motive for presenting the leaders of the insurgents in the city as murderers, thieves and fanatics, which he repeatedly does, denouncing their faction-fighting, their desecration of the Temple, their ruthlessness to ordinary citizens, and the uselessness of it all—the immense suffering and the utter devastation inflicted on the Temple and the city.
Josephus is at pains to explain that the Jews brought the devastation on themselves, and largely caused it. He speaks of their leaders as zealots, and the only positive quality he concedes them—even, as a Jew, takes pride in—is their extraordinary and indomitable courage. The final cataclysm is apocalyptically described:
While the Sanctuary was burning, looting went on right and left and all who were caught were put to the sword...little children and old men, laymen and priests alike were butchered; every class was held in the iron embrace of war, whether they defended themselves or cried for mercy. Through the roar of the flames as they swept relentlessly on could be heard the groans of the falling: such were the height of the hill and the vastness of the blazing edifice that the entire city seemed to be on fire, while as for the noise, nothing could be imagined more shattering or more horrifying...Yet more terrible than the din were the sights that met the eye. The Temple Hill, enveloped in flames from top to bottom, appeared to be boiling up from its very roots...(VI.270–77)
In his summing up of the causes of the catastrophe, Josephus singles out particularly the irony that, while clear divine warnings were ignored, false prophecies and spurious messiahs were gullibly embraced: the Jews had, literally, misread the signs. Some Jews still holding out ask for a parley. Titus answers them in an implausibly long speech in justification of Roman policy and in condemnation of the wickedness of the Jews in taking up a hopeless cause. According to Josephus, Titus had earlier wished to spare the city and the Temple; now "he gave his men leave to burn and sack the city" (IV.253), which they very thoroughly do, while systematically looting the Temple treasury; the Roman soldiers carrying off the sacred vessels depicted over the arch of Titus in Rome is one of our abiding images from the ancient world.
The destruction of Jerusalem makes an unsurpassable climax to Josephus' history, but there was room for two more notable set pieces. One is the Roman triumph of Vespasian and Titus, described in all its state and splendour. Most interesting, and apparently unusual, are the travelling tableaux, three or four storeys high, showing stages of the war: men in flight and in captivity, siege engines battering down walls, an army streaming inside the ramparts, temples on fire, and a blazing countryside. The fall of Jerusalem had clearly caught the imagination of the Romans. Afterwards, one of the rebellion's ringleaders, Simon son of Gioras, a particular bête noire of Josephus, was executed in the Forum, and Vespasian deposited the treasures from the Temple—the golden table, the seven-branched candlestick and the rest—in his newly built temple of peace. The other set piece is the fall of the rock fortress of Masada, in AD 73, with the mass suicide of its defenders; no one seems to have played the role of Josephus and survived.
It was common for ancient historians to claim some superlative for their theme: "the greatest" in some relevant category. Disregarding arithmetic—always dubious in Josephus anyway—few who reach the conclusion of Josephus' story of the war will feel inclined, emotionally speaking, to challenge his own melancholy superlative: "No destruction ever wrought by God or man approached the wholesale carnage of this war" (IV.423–30). Ancient historians dwell much on the horrors of war, and Josephus' account is second to none in its manipulation of horror, with its descriptions of massacres, of suicidal intransigence prompted by nationalist and religious fanaticism, of ruthless terrorism directed towards peaceful populations, of fratricide between rival groups and their warlords, with sacred buildings used as strongpoints, and of the final, crushing intervention of the imperial power, driven beyond all patience and discrimination to its own atrocities and to wholesale destruction. It is a sobering reflection that if, among ancient historians' accounts of human savagery, that of Josephus is peculiarly harrowing, it is also peculiarly familiar.
**TEN**
**Ammianus Marcellinus: The Last Pagan Historian**
Ammianus Marcellinus has been spoken of as "the lonely historian." He was a pagan Greek, writing in Latin towards the end of the fourth century AD, in a now officially Christian world. From him back to Tacitus, three centuries earlier, stretches something like a historiographical desert, devoid of substantial surviving histories of the first rank. To some extent the appearance of a hiatus is a product of the accidents of survival, or rather of extinction, of manuscript sources. As we have seen, Cassius Dio continued his own Roman history down to his lifetime, the end of the third century, but the latter part survives only in a Byzantine summary. We have just over half of Ammianus' work, which itself survived only in a single ninth-century manuscript. The earlier part, which began at the end of the first century, more or less where Tacitus left off, is lost, so his history is exclusively contemporary by accident. The whole must have been an unbalanced work, since the previous two and a half centuries were covered in only thirteen books, while the remaining seventeen books cover only twenty-four years (AD 354–78); it was not uncommon for works of history to become denser as they reached their author's own times.
The historiographical desert was traversed fourteen hundred years later by Edward Gibbon, whose _Decline and Fall_ begins with the mid second century; from the notes to his twenty-third chapter we can get an idea of the difficulties it created. Gibbon used the extant summaries of Dio, and also other Greek historians: Herodian, Aurelius Victor (concise)—whom Ammianus may have known—and Zosimus (largely lost). He also used, with many complaints of its quality, but without realizing that it too was a contemporary work, though it purported to be otherwise, the Latin _Historia Augusta,_ a sensationalist compilation of imperial biographies. Gibbon's snorts of disgust with it are frequent: "this wretched biographer" "a most inaccurate writer." It is with evident relief that, though critical of his "coarse and undistinguishing pencil," Gibbon arrives at the firm ground of Ammianus' history (XVIII, n. 5).
Ammianus came from Antioch and served in the Roman army. Eventually he retired to Rome, where he produced his history in the early 390s. The period covered in the last and surviving part of it, which contains the reigns of Constantius, Julian and Valentinian, was one that he was in a position to observe at first hand, and on which he could interrogate eyewitnesses, which he did. Since the earlier books are lost, the work now begins abruptly, in 354, without preamble. Constantine the Great has been dead seventeen years, and his son Constantius, the survivor of a struggle for the succession, is emperor, though he has called on his younger cousin Gallus, elder brother of Ammianus' future hero Julian, to be his Caesar, the name now used for the junior imperial partner. "Fortune," Ammianus tells us—and for him this was no mere rhetorical flourish—has let loose imperial misdeeds on the empire. Gallus, on whom the spotlight falls, is violent and bloodthirsty and played on by rumour-mongers and spies; "Men began to fear even the walls." As often with Ammianus, we can suspect an echo, conscious or perhaps occasionally unconscious, of an admired author, in this case probably Tacitus. One of Gallus' exploits to keep himself informed is to wander in disguise in the streets of Rome at night (an imitation of Nero perhaps)—even, as Ammianus says interestingly, "in a city where the brightness of the street-lighting made night as clear as day." Gallus is incorrigible. He "raised high the hammer of self-will" and was carried away by his passions "like a rushing river to overturn the obstacles which lay in its path" (14.1). Violent passions are common in Ammianus, and so is a tumble of metaphors, either predictable or, sometimes, incongruous.
After a quick excursion to the Persian frontier and a digression on the habits of the Saracens (digressions are another frequent feature), Book 14 switches to the emperor Constantius, who is wintering in Arles and displaying the familiar imperial propensity to suspicion and arbitrary cruelty and susceptibility to flattery and slander. This traditional theme occurs three times, in fact, with Gallus, Constantius and, later, Valentinian, and is much the same in each case, forming a good deal of what might be called the "politics" of Ammianus' history, alongside military accounts. Tumults in Rome over a wine shortage, however, trigger an excursus on the city of Rome itself—"a city destined to endure as long as the human race survives"—and its inhabitants. The remarkable past and the immense idea of Rome are contrasted with, as Ammianus says rather apologetically, talk of riots and taverns. We get first a brief summary of Rome's rise, won by valour, and its maturity and age, categorized (following Seneca) like the ages of man. Now in old age the Roman people prefer peace and, like a wise parent, have handed over the government of the empire to its present rulers; this ingeniously manages to make Rome old and young and to incorporate a metaphorical defence of the transition from republic to empire. In any case, Rome is venerable not senile, "accepted in every region of the world as mistress and queen; everywhere the authority of its senators is paid the respect due to their grey hairs, and the name of the Roman people is an object of reverence and awe" (14.6).
However, in Rome there is a minority which discredits the majority by a frivolous competition in wealth and display, aspiring to gilded statues, and flaunting embroidered garments and high carriages. (Cato the Censor is called on for his usual growl of disapproval.) As usual, we are told that the ancestors were frugal, modest in dress and fortune. So far, so traditional; but then we get a sudden and fascinating swoop into thinly disguised autobiography, a comedy-of-manners vignette of the treatment of a stranger by fashionable Roman society, anticipating eighteenth-century satirical—we should probably add "neoclassical"—descriptions of the fickle, heartless politeness of "the Town" as in Fielding's _Tom Jones_ and Voltaire's _Candide._ The ingénu is first received as though he were a long-lost friend; next day he is not recognized and has to begin again. If, after years of association, he goes away for a while and returns, he will have to start yet again from the beginning. As someone to be invited he will rank behind racing drivers and gamblers and pretenders to esoteric knowledge, but invitations can be procured from the attendants by a bribe. Rich Romans race at full speed through the capital, the women in litters, attended by swarms of their slaves and eunuchs, for whom Ammianus has a particular abhorrence. It is the old theme of luxury set against Rome's ancient virtue and venerability, but presented with exceptional animation and circumstantiality and a strong suggestion of personal slights remembered; it clearly hurts particularly that on the occasion of a threatened famine, when foreigners were expelled from the city, no exception was made for professors of the liberal arts, but dancers and dancing masters were exempt. As always with Ammianus, too, one has to suspect the presence of literary models. He refers, without the name, to a "comic poet" (Terence, apparently) (14.6).
In a later book (28.4) we hear of the extravagances associated with dinner parties and even picnics. The houses of the great
are the resort of idle gossips, who greet every word uttered by the great man with various expressions of hypocritical applause, like the toady in the comedy who inflates the pride of the boastful soldier by attributing to him heroic exploits in sieges and in fights against overwhelming odds. In the same way our toadies admire the beauty of columns in a high façade or the brilliant sight presented by walls of coloured marble, and extol their noble owners as more than mortal. Sometimes too at their dinner parties scales are called for to weigh the fish, birds and dormice that are served. The guests are bored to death by repeated expressions of wonder at the unheard-of size of the creatures, especially when some thirty secretaries are in attendance with writing-cases and notebooks to take down the statistics, and all that is wanting to complete the appearance of a school is the schoolmaster...
A journey of fair length to visit their estates or be present at a hunt where all the work is done by others seems to some of them the equivalent of a march of Alexander the Great or Caesar. If they sail in their smart yachts from Lake Avenius or Puteoli, they might be going after the golden fleece, especially if they undertake the adventure in hot weather. If a fly settles on the silk fringes of their garments as they sit between their gilded fans, or if a tiny sunbeam finds its way through a hole in the awning over them, they wish that they had been born in the land of the Cimmerians...
Arriving at the public baths with an entourage of fifty, the aristocrats shout peremptorily, "What has become of our girls?" Then we get another descent into low life, with a fine list of proletarian nicknames—"Hogshead," "Sausage," "Pig's Belly" and so on—whose holders spend their time in gambling, discussing the rival merits of charioteers, and hanging about under the awnings of theatres. Some features of life in a big city are apparently perennial, and Harry the Horse, Ikey the Pig, Feet Samuels and Last-Card Louie would quickly have felt at home. The requirements of satire and censure could sometimes override the dignity of history, for which Ammianus seems half to apologize. He is very conscious of the decorum of history, and in fact ends his work with an injunction to historians to use the grand style, but his control of it is wobbly.
We have to return to Gallus in the East, and his erratic and savage behaviour, described in Book 14; Ammianus mentions his love of gladiatorial shows as an example of Gallus' taste for cruelty. The Emperor begins understandably to prepare the way for Gallus' removal and the latter, "like a snake wounded by a spear or stone" (14.7)—Ammianus' addiction to animal similes is another striking feature of his prose—begins a violent purge, after which Ammianus lowers the temperature with a digression on the eastern provinces. Back with Gallus again—who is now "a lion who has tasted human flesh" (14.9)—officials are prosecuted, tortured, executed. He has to be dealt with, and, full of foreboding and with his sleep tormented by the spectres of his victims, he is lured back to Italy, where he is executed. Ammianus concludes this book with a set of reflections on divine justice, invoking Adrastia (the daughter of Jupiter), who was also called Nemesis, "who punishes evil and rewards good deeds...Queen over all causation and arbiter and umpire of all events, she controls the urn from which men's lots are cast and regulates their vicissitudes of fortune." The myths of antiquity, he adds, endowed her with wings to symbolize speed and a rudder in her hand and a wheel beneath her feet "to signify that she runs through all the elements and governs the universe" (14.11). The book ends with famous examples from Greek and Roman history of sudden and extreme turns of fortune.
For us, since Ammianus' work as we have it begins abruptly, and Book 14 is an arbitrary starting point, there has been something to be said for illustrating the work's central characteristics by considering this book in some detail. There is much here to which the reader of the later books becomes accustomed: imperial suspiciousness and cruelty; the ethnographic and geographical digressions; the veneration for Rome's past and for the city itself, despite the satirical descriptions of its population; the piety towards the ancient gods; the literary self-consciousness and allusiveness and the parade of historical examples; the metaphorical excess in the writing and the addiction to wild-beast imagery.
One frequent interest, not brought out in this book, is Ammianus' belief in portents and divination, which he regards as a branch of knowledge, though an inexact one, open to abuse. Later (21.1) he gives a pious rationale for auguries: "We do not owe auguries and auspices to the will of birds; they have no knowledge of the future, and no one would be such a fool as to say that they have. The truth is that their flight is directed by God...By these means a gracious deity loves to reveal impending events to men." Like consulships, portents were often recorded by annalists as reference points. Ammianus follows the annalistic conventions, though not obtrusively, which accounts for some of the rapid switches in the narrative from East to West and back, though later he handles chronology more loosely and provides a semiapology for doing so (26.5; 28.1). But his interest in portents and prophetic dreams is far more than a matter of convention. They could be said to be one of the themes of his work, and he marks out divination and augury as "practices followed by worshippers of the old gods" (21.2). Julian was addicted to them and prided himself on his skill. The rise of the future emperor Julian is another note not sounded in the first extant book, though it was to be the core of Ammianus' work.
In Book 15 enters Ammianus himself—a transition he always marks by the use of the first person plural—on the staff of the general Ursicinus, who is sent by the Emperor to lure an imperial pretender into a trap. The pretender, Silvanus, is a victim of Constantius' suspiciousness and of false accusation, which has left him no alternative but flight or rebellion. It is characteristic of Ammianus that, confessing the fears of himself and his colleagues on their dangerous and not altogether honourable mission to deceive Silvanus, he should claim that they were consoled by a banal saying of Cicero's. Pretending to support the pseudo-Emperor, they suborn some of his soldiers and have him murdered. At this point Constantius, to pacify Gaul, advances Gallus' younger brother, Julian, to be Caesar and to handle a dangerous situation there—which he brilliantly does, by his talents and charm and the shining example of his character. Ammianus is sometimes critical of him, but he greatly praises Julian's military abilities, even though his final campaign, against the Persians, ended in disaster. Julian's appointment to Gaul leads Ammianus to an ethnographic digression, memorable for his account of the bellicosity of the Gaulish women: in a quarrel "the woman, with swollen neck and gnashing teeth [barbarians in Ammianus do a lot of teeth-gnashing], swings her great arms and begins to deliver a rain of punches mixed with kicks" (15.12).
In Book 16 Ammianus celebrates Julian's pacification of Gaul and his frugal, self-disciplined and consciously self-improving character, and mentions his still-concealed piety towards the old gods. Ammianus also returns the narrative to the neurotic atmosphere of Constantius' court, and the intrigues of his sinister henchmen—some of them picturesquely named, like "Paul the Chain" and "Count of Dreams." The chief diversification is the Emperor's state visit to Rome (AD 357) (16.10). This gives Ammianus the chance for a set-piece description and a tribute to the ancient city. Constantius in the procession holds himself rigidly and does not spit or rub his nose, which Ammianus finds remarkable and attributes to affectation. Constantius, on this, his first visit, is amazed by the sights: the Forum, "that sublime monument of pristine power," and the shrine of Tarpeian Jupiter, "beside which all else is like earth compared with heaven, or the buildings of the baths as big as provinces, or the solid mass of stone from Tibur that forms the amphitheatre, with its top almost beyond the reach of human sight, and the Pantheon, spread like a self-contained district under its high and lovely dome" and so on. Of the Forum of Trajan Ammianus says, "Its grandeur defies description and can never again be approached by mortal men." It is a remarkable picture of fourth-century Rome, almost on the eve of its fall, seen by a Greek from Antioch and a Roman emperor who was a stranger to it. Ammianus' Rome, though then intact, is of course more recognizable to us than earlier descriptions of the city.
Ammianus was in the East during Julian's campaigns in Gaul and Germany. His numerous battle scenes are an odd mixture of epic cliché, with rivers foaming with blood and arrows darkening the sky, and sharp and clearly first-hand observation, like the squalors of being besieged in the city of Amida by the Persians, his terror and concealment during the sack, and his fortunate if somewhat ignominious escape (19.8). He was also very scared of elephants. Clearly the result of personal observation too is the reference to the engineer, "whose name escapes me," who was mangled out of recognition by the backwards discharge of a carelessly loaded catapult (24.4). Ammianus scrupulously follows Julian's military career, to the point where, under pressure from the soldiers, he usurps the title of emperor and undertakes a long march from Gaul into the Balkans to make good his claim, only to be forestalled by the convenient death of Constantius from fever (20).
For posterity the greatest interest in Julian's brief reign (AD 361–3) is his abortive attempt to reinstate the worship of the ancient gods. Ammianus was essentially in sympathy, so his criticisms of Julian's excess of pagan zeal are striking. One of Julian's edicts which Ammianus censures most strongly banned Christians from practising as teachers of rhetoric or literature (22.10). Julian's worship was unduly ostentatious: "The victims with which he drenched the altars of the gods were all too numerous" (22.12), so that the troops were gorged with meat and demoralized, at huge cost. According to Ammianus, the number of the sacrifices Julian performed led him to be called "Axeman"—Ammianus was always alert to nicknames (22.14). Julian's enthusiasm embraced all non-Christian cults, so that he attempted unsuccessfully to rebuild the Temple in Jerusalem, and in Mesopotamia sacrificed to the moon according to the local rite (23.1, 3). Divination, on which he prided himself, became a kind of public craze (22.12); Ammianus disapproves of freelance soothsayers—as was traditional (Tacitus mentions it). It was a world, in any case, agog with suspicions of sorcery. Julian's intolerance also closed the great church at Antioch, in rage at the burning down, apparently by accident, of the temple of Apollo (22.13). Julian's piety, like Ammianus' own, contained, naturally, a marked strain of antiquarianism. He acted, just as Ammianus wrote, consciously in the shadow of precedents and past authorities: Ammianus cites Thucydides and Polybius, as well as quoting Virgil and Cicero and examples from Greek and Roman history. Typically, Julian, attempting to revive the famous Castalian spring as a source of prophecy, ordains that bodies buried there should be removed "with the same rite as that used by the Athenians when they purified the island of Delos" (22.12). The reference is to Thucydides, an attempted reading back across almost a thousand years, and a testimony, fostered by a deliberate antiquarianism, to the cultural unity of the ancient world.
Julian's death, from a wound received in the disastrous Persian campaign of AD 363, is another such conscious tribute, full of reminiscences of the relevant canonical event, the death of Socrates. Julian expires discoursing with philosophers on the sublimity of the soul and forbidding his followers to mourn "for a prince who was restored to heaven and numbered with the stars" (this was a Neoplatonic belief) (25.3). Paganism on its deathbed was pulling out all the stops. Ammianus' own paganism, though clearly devout, was of a more restrained and genial kind. In tacit contrast to Julian, he praises the toleration practised by the otherwise deplorable (Christian) emperor Valentinian, and speaks of "the plain and simple religion of the Christians," which "preaches only justice and mercy." He has none of the prejudice shown earlier by Tacitus, while noting the atrocities the Christians practise on each other: "no wild beasts are such dangerous enemies to man as Christians are to one another" (21.16, 22.11, 22.5). Ammianus gives the impression that men could differ about religion without harming or impeding each other, though in estimating the extent of his tolerance it must be remembered that he was writing under a Christian emperor.
Julian so dominates Ammianus' work that there is a natural tendency to regard the events recorded under his successors as an anticlimax, but in fact these included two of the most significant and ominous events in the later history of the empire: the permitted migration of the Goths across the Danube into Roman territory in AD 376 and the defeat and death of the emperor Valens in battle against the Persians at Adrianople in AD 378. Ammianus describes these, and is fully aware of the catastrophes they represent: the Goths immediately go on the rampage, and Adrianople, he says, is a disaster second in Roman military annals only to Cannae (31.8, 31.12–13). Of course he does not know these, as historians are now likely to see them, as stages in a rapid downward descent culminating in the sack of Rome itself in AD 410 and the loss of the western empire. For Ammianus, the empire, though passing through a time of great tribulations, was still the world's central fact, as Rome itself was the Eternal City. He can still speak without irony, referring to modern communications, of "our great and glorious latter days" (21.10). The empire is so widely beset that it is "as if the whole world were at the mercy of the Furies" (31.10), but we must not read more into this than was intended. The extent, if any, of Ammianus' historical pessimism has been a matter of debate, but it would be wrong to endow him, even obscurely, with foresight. He did not know that he was writing materials for a chapter in _The Decline and Fall of the Roman Empire._
**ELEVEN**
**General Characteristics of Ancient Historiography**
Compared with the centuries which followed, ancient historical writing, from the Greeks to Ammianus, formed a single genre, which the Renaissance attempted to revive. Before continuing, it is worth trying to summarize what the characteristics were which made it an entity, with common standards, preoccupations and assumptions, especially as these have been subject to some misrepresentation.
Throughout the classical period, historians were intensely aware of their predecessors, Greek and later Roman, and of many more of them than we now have access to or even knowledge of, thanks to the vicissitudes of manuscript survival. Ammianus, the last of the classical historians, was among the most self-conscious of all in this respect, and cites precedents freely from what we think of, as he did, as the classical canon, both literary and historiographical. Although, annoyingly to us, it was not the normal practice to name sources, ancient historians clearly treated their predecessors and contemporaries not only as sources but as models, rivals and, it is tempting to say, colleagues, across the two languages and many centuries. They use, occasionally quote, and cite in cases of disagreement or uncertainty; they echo, plagiarize (though it was not thought reprehensible to do so), emulate, criticize, denigrate—all of this, no doubt, more often than we can be aware of—in a manner characteristic of a scholarly and literary community, in this case exceptionally long-enduring, of co-workers consciously aware of it and of their heritage. We noted Polybius' confidence that if he died before finishing his work others would take up and complete it; such continuations are an occasional feature of ancient historiography.
In this sense ancient historians are an even more coherent group than the poets or philosophers, though the former were bound together by admiration for Homer and the latter characteristically belonged to "schools." Herodotus and Thucydides, though not all historians admired both, early achieved a pre-eminence comparable to that of Plato and Aristotle for philosophers. The transmission of their works, through centuries of transcription by hand, never hung by a thread, as did that of Polybius, Tacitus and others; nor was their work lost, like most of Livy's history. Because many authors have disappeared without trace, or exist only in fragments, many of the intellectual threads that bound this community together must be invisible to us or have to be the subject of guesswork. We can be reasonably confident that there has been a substantial correlation between merit, judged by both the copyists and ourselves, and survival, which is some consolation, but our sense of the collective life, as it were, of the ancient historians is irreparably damaged.
Some of the continuities, of course, were given by the continuities of Graeco-Roman culture and ancient public life more generally: the literary culture grounded in Homer and the Olympic pantheon, Greek philosophy, and the ethics of Stoicism. More directly relevant to historical writing are the rules for prose composition and oratory formulated by, and widely accepted from, the teachers of rhetoric, which have sometimes been thought to have exercised a baleful influence on history as accurate recording and which certainly exercised some constraint on what it was proper for historians to write about. Historical writing, as is made clear by the recipes for it propounded by Cicero, was above all a literary art, even if one with its own additional rules concerning truth-telling. Continuities in the framework of public life, which the historians chiefly attended to, and reflected, were provided, among other things, by a respect for oratory, indicated by the many invented speeches composed for their characters by the historians in the tradition established by Thucydides. Another continuity was the worship of the gods and a respect for their temples (paying attention to their foundation and occasional destruction), and for the venerated objects stored and displayed in them, which sometimes acted as mementos of significant events and were mentioned by historians accordingly. Associated with this is the recording of portents and auguries, prophetic dreams and oracles, and supernatural or monstrous occurrences generally. It is tempting to see a steady "secularization" in the historians' attitudes to these, but this is hard to do, though Livy, for example, deplores modern scepticism. The historians themselves range from pretty clear unbelief (Thucydides) to complete acceptance (Xenophon, Ammianus), but these examples do not suggest a steady trajectory.
Recording portents was part of the annalistic form which, from Thucydides onward, provided the frame for most works of history, though some historians treat it more freely than others: portents were recorded as part of the year's notable events, just as consulships continued to be recorded as chronological reference points long after they had ceased in Rome to be politically significant. Ethnographic and geographical digressions, pioneered by Herodotus, and the set-piece speeches which, as we have noted, were Thucydides' innovation (though he was criticized for making all his speakers sound alike) remained notable features of historical works. Both, especially the digressions, were freely used by Ammianus. The concentration on the events of public life, presented in an elevated style and with long, though sometimes conventional, accounts of military campaigns, is an enduring feature, but much that we would find of interest is excluded as beneath "the dignity of history," to use the phrase Lord Bolingbroke in the eighteenth century ( _Letters on the Study and Use of History_ [1752], Letter 5.2) applied approvingly to Thucydides and Xenophon. This inhibition did not apply in the related art of biography, nor in ethnographic digressions. In the motives for the writing of history, the one announced by Herodotus, the preservation of the memory of great deeds, is an abiding one. The emphasis on the "greatness" of one's theme—the greatest war, siege, conflict, achievement, city, empire—remains a constant from Herodotus onward, seconded, in some cases, by claims for one's immediate access to the events. Tacitus virtually apologizes for being unable to claim the former: it is a sign of the degeneration of Roman public life. Among the Roman writers, of course—and the view is shared by Polybius and Ammianus, though Greeks—the theme of the degeneracy of manners is also a constant, from Sallust to Tacitus.
The view, first articulated among the Romans, that the function of history is to inculcate virtue and castigate vice, through the presentation of examples of inspiring and ignominious behaviour, becomes standard. The rival, more intellectually formulated, Greek view that we find in Thucydides and Polybius, of the utility of history in offering examples of success and failure, practical wisdom and folly, takes second place to it. It was common throughout, of course, for history to be written after a distinguished public career, both political and military, which Polybius makes a _sine qua non_ ; the great exception is Livy. The interest in causality and in distinguishing different types of causes, again of keen interest to Thucydides and Polybius, is by modern standards low. What is of great interest, from Herodotus to Tacitus, as an explanation of success or failure and decline, is the moral fibre of a people, which forms a central theme in Livy and which, in the case of the Romans, is also acknowledged by Polybius. Typically, this interest and explanation are expressed in antitheses whose components vary but also display a certain congruity: primitive hardihood opposed to luxury; West to East (initially Greeks and Persians; later Romans and orientals or, chiefly, Greeks); freedom to servitude; past to present. The same basic opposition can be given an ethnic, military, political or historical dimension, often related to each other, as required. The contrast is often attended with adjectives suggesting masculinity and effeminacy: luxury and servility are unmanly. It was left to the Romans to convert them into something like a general conception of historical decline.
Of course the broad consensus about historical writing also accommodated disputes and, over time, notable shifts of perspective. Among the former, for which Polybius' polemics provide a good example, were the rival claims of truthfulness, whose importance for history was universally agreed in theory, and picturesque and appealing narration. Over time, the most obvious effect of the shift of focus from Greece to Rome is that the subject matter becomes more concentrated: a single city dominates the narrative, so that the historian does not have to switch, sometimes bewilderingly, from one city to another. The problems in establishing a common chronology were also greatly alleviated. On the debit side there is a perceptible loss of a characteristic Greek cosmopolitanism, a Homeric emotional even-handedness in dealing with the two sides of a conflict, between Greek cities or even between Greeks and Persians. Livy and Tacitus are hardly parochial writers—the theme of imperial expansion and the need constantly to return to affairs on the frontiers, as well as the ethnographic tradition to which Tacitus contributes (though Livy's barbarians are usually just stereotypes), ensured that. But, for Livy, Roman patriotism is overriding, and this issues, of course, in an antiquarian attention to the city's origins. Despite the existence of a genre of local histories (mostly lost), no Greek city seems to have paid its past such close and pious attention, leaving it generally to highly speculative genealogies and foundation myths. For this reason the contrast between past and present does not have the same function, while in Rome, from Sallust onward, a conception of long-term moral decline became established. Tacitus' writing under the empire (and the same may be true of Livy's lost books dealing with the last years of the republic) is clearly ambivalent, wistful for a lost past but not committed to the possibility of its recovery. Such wistfulness seems wholly un-Greek. The quality of Ammianus' nostalgia, if that is the right word, is harder to assess. His not always well-judged dippings into the bran tub of literary allusion, quotation and historical parallels can seem like gestures to a grander past, but he may just be showing off.
There are certain abiding popular misrepresentations—certainly misrepresentations if taken too generally and literally—which seem to recur in references by non-specialists to the general features of classical historiography; sometimes, it seems, these derive from taking Thucydides as being wholly representative, or from interpreting too crudely what he claims to be doing. In this way and by over-selective and uncontextualized quotation elsewhere, a number of myths, overstatements or half-truths about classical historiography can be too uncritically propagated. Though there is substance in some of them, all are false if stated as general truths without qualification. We can deal with them schematically:
1. _That all history was contemporary history._ This is up to a point true of early Greek historiography; as an account of Roman history, including that written by Greeks, it is not true at all. It is true of Thucydides, of Xenophon and of the earliest (though not the later) Alexander historians. Herodotus' history, disregarding the early references to legend and to Homer, stretched back some three-quarters of a century before his own time. We should also not deny the title of historian to the Greeks who wrote of the foundation of cities and of the history of the Mediterranean area, particularly the latter. It is certainly not true of Livy, nor of the Roman antiquaries who preceded him, nor of his Greek contemporary Dionysius of Halicarnassus, whose work also began with the foundation of Rome, nor of Dio, whose history covered the whole history of Rome down to his own times in the second century AD but a great proportion of which has not survived. It is not even true of Josephus, if one puts his two major works together—the _Jewish Antiquities,_ which begins with biblical history, with _The Jewish War,_ which itself has a substantial opening section beginning two centuries before the war itself. Ammianus, as we have seen, is a contemporary historian accidentally, since his first books, covering the century and a half before his own time, are lost. Tacitus can just be regarded as a contemporary historian only if one allows the memory of the oldest living eyewitnesses to count as contemporary.
2. _That ancient historiography is exclusively political and military._ This is broadly true as a prescription and declaration of intent, though it excludes biography, which moves between the official and the intimate; the earliest biographies (Greek) seem to have been of philosophers and literary men. In history proper, the most obvious exception is the often very extensive ethnographic digressions, from Herodotus onward, which deal with physical appearance, beliefs, clothing, diet, hygiene, habits, as well as marriage and funeral customs. Even in the accounts of the historians' own societies one would have to define religious matters as political—not unreasonably perhaps—to make the generalization hold, which for a modern readership is misleading. There are also, as we saw in Livy, the incidental passages of social history occasioned by the annalistic form or by the censure, in both Livy and Ammianus, of modern manners: the terrible stories of the proscriptions, in Appian and Tacitus, with Roman aristocrats driven to flight or into hiding, give us glimpses of the interiors of houses, domestic habits, and the relations of slaves, faithful or vindictive, with their masters. An admittedly jejune social and economic history is also inescapably part of Livy's and Appian's recording of the social conflicts in Rome, with their references to the depression and indebtedness of the small freeholder. Descriptions of aristocratic and proletarian misbehaviour, of rowdiness and rioting in the streets of Rome, and of plutocratic domestic luxury and lower-class fecklessness are common and often vivid in Sallust, Livy, Tacitus and Ammianus.
3. _That there was no conception of long-term historical change._ This is to some extent answered above in the discussion of "contemporary" history, but it is worth elaborating, for if we mean literally a conception it is certainly not true of Thucydides, the most ardent advocate of contemporary history. He thought, as others did not, that it was not possible to write long-term history to the required standards of accuracy, but that is another matter. Near the outset of his history he gives a sketch, of a kind that became fashionable during the eighteenth-century Enlightenment, of the development of early Greek society "from rudeness to refinement" (to use the eighteenth-century phrase) chiefly—another anticipation of the eighteenth century—as a result of commerce. No one would seriously claim this as one of Thucydides' major achievements as a historian, and it seems to have exercised no discernible influence, but as evidence that he was well aware that life in Greece had once been very different it is irrefutable. Polybius also gives a brief conjectural account of the origins of political society out of primitive beginnings in which human beings were little different from herd-animals (VI.5–7). Livy, similarly, knows very well that the Forum had once been a marsh, surrounded by the primitive huts of a shepherd population. Elsewhere he gives a thumbnail sketch of early human society as such, conceived in terms of what later came to be called "the state of nature." More subtly, as I have argued, the conception of a long-term deterioration in Roman character and manners is itself a conception of historical change, implying the emergence of fundamental differences that were more than merely material. Despite the long consensus on the conventions of historiography, and the near-consensus on its ends and methods, some writers even display an awareness that historiography itself is not unaffected by historical change. Polybius thought that with the rise of Rome a new kind of history, universal history, became possible. Tacitus, who wrote a dialogue in which styles of oratory are treated historically, was conscious that the tasks of the historian and the manner in which it was possible to write had been reconstituted by the change from republic to autocracy. In such views the possibility of "a history of histories" was at least logically present if not made conscious.
4. _That there was a notion that all change is cyclical._ This has in a sense been dealt with above, but it is still worth confronting it directly. It is true that among the Greeks, notably in Polybius, there is a conception, deriving from Aristotle's categorization of constitutions and their corrupt forms, of the dynamics of constitutional change which seems to imply a cyclical set of transformations. But not only was this view of constitutional change not universally subscribed to and deployed, the cyclical conception itself was not notably applied to other dimensions of life, though Tacitus, in passing, at one point wonders if it could be. (He speaks of it as though of a new thought, not a cliché.) Of course there was a widely diffused view, from Herodotus onward, that all prosperity and glory were held on a very uncertain tenure, but it is a long way from there to the more precise and technical conception of the omnipresence of cycles. Thucydides certainly did not expect Greek society to return to its earlier state, nor did Livy envisage a future for the Forum as a pasture for cattle (though if he had done so he would have been right). Sadly for him, there is no sign that he expected the Roman character to regenerate itself, or that Tacitus expected a return of the republic. On the other hand even the worst tyranny was ended by the death of the tyrant, and there is no suggestion of an institutionalized self-perpetuating tyranny such as Orwell envisages in his _Nineteen Eighty-four._
It is sometimes implied that a cyclical view and the absence of a conception of long-term change were entailed by a belief in an unchanging human nature, which is indeed proclaimed by Thucydides and accepted by Polybius as grounds for asserting the usefulness of history. We have already looked at the inadequacy of this considered as an interpretative master key to Thucydides' work. In Polybius it is counterbalanced in its effects by his claim to discern in history the emergence of a great new fact: Rome's rise, and Fortune's intention to promote this from now on as the central thrust of universal history. The relation between an idea of universal human nature and an awareness of social variability across space and time is conceptually a very complex one, which cannot be entered into here. It may be sufficient as an indication of that complexity to mention the case of Herodotus, who is at once broadmindedly cosmopolitan and fascinated by the cultural—and it is natural to say psychological—variety exhibited by the different peoples into which mankind is divided. Thucydides' abstention from ethnographic digressions and Polybius' circumspection with them seem to owe more to austerity towards history written to entertain than to any dogma about human nature. Herodotus, however, is an author who, with a touch of pathos or humour, can transcend in a second the gulfs created by cultural prejudice and by time, as in the description of the tears of Xerxes, yet can also dwell with a fascinated attention on the bizarre forms—to us and to the Greeks—that human nature can assume. The other distinguished ethnographer among the ancient historians, Tacitus, similarly, though to a lesser degree, sees the Germans both as comprehensibly and even sympathetically human and also as very unlike the modern Romans, just as he and Livy see the latter as mostly no longer like the Romans of old. The variety of human nature was in fact of the keenest interest to the ancient historians generally; Thucydides is the exception rather than the rule.
In this first section of this book much attention has been given to a handful of ancient historians, compared with the immensely larger number of historians to be considered (or, alas, ignored) in the sections to come. But the former remained unchallenged as historical models and authorities, and highly regarded as sources for moral and political wisdom, for approximately two thousand years, in a way that no other group of historians has been. This alone justifies extended attention, as does the fact, not universally true of even influential historians, that they are mostly highly rewarding to read.
**PART III**
> **CHRISTENDOM**
**TWELVE**
**The Bible and History: The People of God**
The differences between biblical and biblically inspired and classical historical writing are manifold. The historical books of the former are not written as literary exercises in the leisure following a public career, and they are not bound to the rules of classical rhetoric. Though the Bible incorporates the grandeur and sublimity of myth and epic, as well as ecstatic song and personal lamentation, it is also often homely and earthy: nothing human (or divine) seems alien to it. Despite the incidence of prominent priests and patriarchs, kings and prophets, the scriptures are concerned essentially with a people, the children of Israel, in their relationship with their God and its vicissitudes through time; religion and history are inextricably intertwined, because God is not primarily the god of a perennial nature, but the mover of history. He is omnipresent in a way utterly different from the pious references to fate, justice and the will of heaven found in classical historiography.
Hebrew history is insistently providentially guided; the writing of it, even before Christianity, was regarded as directly inspired, so there are no comparisons of authorities or attempts to reconcile conflicting ones, no expressions of doubt or boasts of inquiry: the authority is God himself. Despite the presence within it of a recurrent pattern of sin and retribution, which, with redemption, is also its overall pattern, it is essentially linear and directional—far more so than is even the theme of the rise (and decadence) of Rome. It had a beginning (Adam, Noah and the promise to Abraham); it will have an end in the Apocalypse, which is foreshadowed in the part of the Bible designated by Christians as the Old Testament. So there is a beginning and an end: sin and then the Last Judgement, with salvation for some. These are linked logically as well as historically by Christ's Incarnation as the middle term between them, which from the eighth century became the pivotal moment for dating: without Sin, no History.
The Pentateuch, the first five books of the Hebrew scriptures, probably referring to events towards the end of the second millennium BC, was written down in the first half of the first millennium BC; it was gathered, in the second half, into the canon of the biblical books much as we have it today, but it made little impression on Gentile culture until the coming of Christianity. As the Septuagint (from the seventy-two translators), it was translated into Greek in Alexandria in the third century BC, for the use of Jewish exiles, but remained largely of interest only to them. It was expounded for the pagan world early in the Christian era by Philo of Alexandria and Josephus in the first century AD, but their impact was on Christians, not pagans. With Christianity firmly established, the authoritative Latin translation throughout the Middle Ages, the Vulgate, was provided by St. Jerome (AD _c._ 340–420).
Considered as a whole, of course, the Hebrew scriptures, which Christianity adopted as the Old Testament, are immensely heterogeneous, comprising various genres of ancient literature: creation myth, national epic, wisdom literature, genealogies and king lists, songs and prayers, laws and detailed ritual prescriptions, prophecy and protracted warnings of divine wrath, often clothed in symbolism, though without the oracular sites prominent in the Hellenistic world. They also contain something like "political history," especially in the books of Samuel, Kings and Chronicles. Yet on a wider view they have too an extraordinary narrative coherence, presenting a view of the fate of mankind and subsequently of God's Chosen People from the Creation and the expulsion from Paradise as a result of Adam's sin to the second beginning of mankind with Noah, after the Flood; then to God's promise of land and favour to Abraham, vindicated in the deliverance from Egyptian captivity; then the journey of the people through the wilderness under their great leader Moses, by whom they are handed down God's law. They conquer and settle their Promised Land, the land of Canaan. All this can be described as foundation myth merging into epic. In the ensuing books comes an apparent transition from priestly to kingly rule, with conflicts with neighbouring peoples akin to the Romans' fight for survival and the conquest of central Italy, but with exogamy and assimilation as a menace, not an achievement, because they threaten their monotheistic religion. The people transgress again, rather as the world's inhabitants had done before the divine punishment of the Flood; they suffer a second exile, this time to Babylon, under Nebuchadnezzar; the Temple in Jerusalem, the centre of their national cult of Yahweh, is destroyed. Their return from this exile, or "captivity," and the rebuilding of the Temple are owed to the Persian Great King Cyrus, whose empire has succeeded that of Babylon as supposedly (though spuriously) prophetically described in the Book of Daniel.
This unity of theme, whose protagonist is the children of Israel and whose divine orchestrator is Yahweh, is no doubt the product of judicious editing and selectivity, with the part sometimes standing for the whole and significant events symbolically magnified and simplified. The Egyptian pharaoh of the first captivity, and Exodus, is unfortunately unidentifiable in the Egyptian records, but with the Assyrian and Persian monarchs Nebuchadnezzar, Cyrus and Darius, who appear from the end of 2 Chronicles and in Ezra, Nehemiah and Daniel, Hebrew history becomes integrated with the known history of the Mesopotamian world of the seventh and sixth centuries BC, and even datable. In Samuel, Kings and Chronicles we have Near Eastern kingly and priestly history on a small scale, given world-historical significance by the interventions and presiding will of Yahweh. In the narratives, constructed with great maturity for their period, there are vividly drawn political agents, court intrigues and dynastic struggles—as well as innumerable wars (described with less sophistication, because Yahweh's inclination is the deciding factor)—depicted sometimes in richly dramatic terms comparable with Homer. Together they represent a historiography that can stand comparison with the Persian sections of Herodotus, though crucially without the element of "inquiry" instead, in Ezra, there is an attention to documentation which modern historians would approve of. Ezra and Nehemiah, in the fifth century BC, are recognizably priestly historians and scribes writing a kind of contemporary history and quoting sources. With the subsequent history described in the uncanonical books of the Maccabees, we enter the world described later by Josephus, after his epitomization of the Bible in _Jewish Antiquities,_ and embark on the theme which would continue to the very end of the Jewish state in the first century AD, of the resistance of the Israelites to their Hellenizing rulers in defence of their religion and the purity of the Temple.
Judaism became a matter of vital interest in the Gentile world with the diffusion of Christianity. The classical heritage was too strong for the Bible ever entirely to monopolize the historical consciousness of Gentile Christians, but for something like a thousand years, and in many contexts for much longer, it was to become central to it. From the fifth century AD onward most writers were content to take their Roman history in the form of digests, with which they were duly provided, while the classical historians, with the notable exception of Sallust (who was both moralistic and brief ), became largely ignored or even unknown.
The impact of the Bible on Christian conceptions of history, from the earliest Christian centuries to the nineteenth, was radical and pervasive. It was not only that the sin of Adam, the Incarnation and the Last Judgement framed all history. The fact that biblical history presented the dealings of God with his Chosen People in something like a recurrent pattern of transgression, punishment and deliverance meant that the same pattern could be expected to be repeated so long as history lasted: history presented a recurring series of types and situations within the historical macrocosm of primal sin and final judgement. Christendom naturally took to itself the role of the Chosen People. Subsequently this role proved to be of almost infinite application, to any nation or sect which, abetted by its chroniclers, chose to assume it: the people's faith, greatness or sufferings, the divine favour or the chastisement visited on it, demonstrated the equation. Given the notion of repetition, the Bible made available as roles in contemporary history a gallery of memorable characters and deeds: warrior, rebel, judge, prophet, great king (or tyrant), ensnaring or patriotically homicidal women, all for recognition by the historian or adoption by the agents themselves and their eulogists or detractors. As Gibbon said, recording one set of such applications, which became at some times common currency of public abuse, "The characters of Eve, of the wife of Job, of Jezabel, of Herodias, were indecently applied to the mother of the emperor [Valentinian II]" ( _Decline and Fall,_ XXVII).
But above all the Bible offered an archetypal pattern, repeated many times, of covenant with God and entry to the Promised Land, of collective transgression and its punishment by devastation, exile, captivity, followed by deliverance and return, symbolized by the rebuilding of Jerusalem and the Temple. It is a pattern, it may be noted, which makes human beings the prime movers of history only through their transgressions: transgression is their role in the historical dynamic, though there is a subsidiary one for the instruments of punishment, whether tyrants or barbarians, and for the individual bringers of deliverance—types of Moses and the Messiah. It hardly needs saying that in secular history collective misbehaviour (particularly infidelity and fornication), destruction, oppression and occasional better times, or at least the promise of them, as well as successful migrations, have been common enough to make the pattern frequently recognizable. It was up to a point recognized in pagan history too: the pharaoh who restored order to troubled times was a stock figure in Egyptian inscriptions, as was the early lawgiver in Greek and Roman legend. Augustus was the Roman archetype of a prince of peace, whose advent could be mythically represented as the return of the goddess of justice, Astraea, to earth; she was all too often a _dea abscondita._ But in Judaeo-Christian thought Moses and Christ himself became the archetypal deliverers, as David was the successful warrior king, prophet and songmaker. In pious and courtly rhetoric such monarchs as the emperors Constantine and Charlemagne, Queen Elizabeth and William of Orange, and a good many others, were allowed by their adulators and commentators to awaken the resonances of providential salvation history.
The adoption of the Hebrew scriptures as the Old Testament by Gentile early Christians may not seem inescapable; after all, they claimed to have superseded the original Chosen People, as the children of a new covenant. In some respects the Old Testament was even potentially embarrassing, as was the behaviour of the Olympians in Homer, and in both cases one response was to allegorize the more disreputable episodes, after which the Old Testament could furnish suitable material for pious meditation. One very notable exception was Marcion, a Christian scholar of the second century, for whom the Jewish God was creator of the wicked world and was not the same as God the father of Jesus. He refused the allegorical option in treating the Old Testament, and so found Yahweh ethically unacceptable. He was excommunicated in AD 144. But Gentile Christians needed the Hebrew Bible. They needed its prophecies, alleged to be of Christ, to vindicate the claim that he was the Messiah; they also felt a need to rebut the charge that Christianity was a recent innovation, a parvenu religion (Eusebius, _History of the Church,_ 1.3–4). The Old Testament provided Christianity's ancestral title deeds. Once adopted, it offered a satisfying, comprehensive interpretation of human history and a rich repertoire of symbolic identifications applicable to subsequent history, now widely seen as a drama which Providence continued to inscribe, its characteristic ways of proceeding being starkly and awfully revealed by God's earlier dealings with his Chosen People.
Since the Hebrew scriptures were requisitioned to provide a prophetic series of figures or types of Christ as the Messiah—Moses, Joshua, David and others—as well as apparent actual prophecies, the habit of looking for such types, along with new identifications of God's elect, became readily transferred to modern history. We have to be a little careful. Not all analogies with biblical characters amount to what is specifically typological or figural thinking, nor are such identifications entirely unprecedented. Some are just analogies, applied often in flattery or contempt, and characters from pagan history could also be invoked as prototypes. Josephus had very aptly compared Moses to Solon and Romulus. More polemically, in the sixth century Gregory of Tours combined Roman and scriptural allusion when he called King Chilperic "the Nero and Herod of his time." In classical times, self-identification with figures from legend and history seems to have been common, as Alexander the Great identified and vied with, and may have imitated, Achilles. The emperor Caracalla was obsessed with Alexander himself, and had a picture produced, one half of whose face was Alexander's while the other half was Caracalla's. Sometimes, with the figural habit of mind ingrained, semi-identifications could be made on the basis of names. Henry VII's son Arthur was hailed as a revenant in this way.
But figural thinking proper, as it became current in patristic thinking from the second century AD, required the attribution of a specified providential role for which the Bible, and even Christ, provided the archetype. In this way the salient events of biblical and Jewish history could be recycled as interpretations of subsequent events, in polemic, eulogy or historical commentary. The fall of Jerusalem in AD 70 became such an archetypal event, and Josephus' account of it was much read by Christians. It powerfully affected Christian imaginations as early as the New Testament itself, where Jesus is made to prophesy it: "There shall not be left one stone upon another, that shall not be thrown down" (Mark 13:2, cf. Luke 21:6, 24). The catastrophe raised crucially the question why God had allowed it—and in a way that evoked a response in terms of sin, judgement, prophecy and deliverance.
Punishment entailed the need for deliverance and implied a deliverer. The counterparts, in the Christian era, to the figural anticipation of Christ in the Old Testament were the deliverer monarchs and leaders of later times, just as new Israels, chosen, sinful and chastised, also identified themselves through their prophets.
We shall see shortly how Eusebius gave a biblical and messianic aura to the emperor Constantine as a divine instrument and deliverer of God's people. The earliest British history—though it is more jeremiad than history—is a biblically phrased account of the sins and consequent sufferings of the Britons at the hands of the barbarians after the withdrawal of the Roman legions in AD 410. This was the _Ruin of Britain,_ by the sixth-century monk Gildas. He presents the Britons' disasters in biblical terms as a divine chastisement for transgression, while he hails the martyr St. Alban as a Joshua, himself—again the name ("Jesus" in Greek) helps—taken as one of the types of Christ. Other such parallels could be more encouraging. Bede, taking his cue from Gildas, saw his own people, the English, as the instruments of God's justified wrath against the Britons, and hence as a chosen people. As the greatest Catholic (i.e. non-heretical—most barbarians were Arian heretics) modern people, the Franks seem to have received encouragement from the papacy to see themselves as the new Israel—though the point has been argued. Protestant Bible-reading in the vernacular gave new impetus to such ways of thinking in the sixteenth and seventeenth centuries. In sixteenth-century England the preface of the _Acts and Monuments_ of John Foxe, better known as Foxe's _Book of Martyrs_ and a key Protestant text, spoke of the recently crowned Queen Elizabeth as the new Constantine; the latter had become a standard type of the Messiah as deliverer. In Foxe's modern martyrology the English were God's elect people, an idea powerfully present in the next century among the Puritan opponents of Charles I, encouraging their intractability. Many Protestant sects were to see themselves as seeking to build the New Jerusalem, an aspiration passed on to the secular utopians of the nineteenth century. The Puritan colonists of New England in their new land naturally drew on the same conception. John Winthrop, the first governor of Massachusetts, spoke of them as having made a covenant with God; the Spanish conquerors of Mexico, a century earlier, had on their first sight of Montezuma's capital hailed it as "the Promised Land," just as Martin Luther King's dream promised it to his people five and a half centuries later. The soldier historian of the Spanish conquest, Bernal Díaz (Chapter 24), was prompted by the siege and destruction of the Aztecs' capital to recall the fall of Jerusalem.
Of course such rhetoric tended to be the currency of historical agents, of sects and colonizers, rather than of reflective historians and chroniclers, but the latter were not immune. They generally held aloof from the prophetic, millenarian speculations prompted by the Hebrew and Christian eschatologies and keys to world history, most notably the spuriously prophetic dreams in the Book of Daniel and the Revelation of St. John the Divine. These influenced historical schemata like that of the twelfth-century prognostications of Joachim of Flores (1135–1202). Joachim's threefold division of the epochs of history, which included a strong prophetic element, was based on the Trinity: the epochs of the Father, the Son and the Holy Spirit, which would be the last. Eusebius, the first church historian, whom we must look at shortly, disapproved of millenarian speculation. But the biblical archetypes and imagery could colour secular descriptions, even without full commitment to the biblically grounded providential scheme. Ancient historiography, for example, records many catastrophic sieges and destructions of cities, but Josephus' account of the fall of Jerusalem reached a new pitch of intensity and lamentation, which seems to owe something to the Hebrew prophetic tradition. The prophets were the unsurpassed, as well as inspired, connoisseurs and experts in divine wrath, in the iniquities and faithlessness of peoples and their dire punishment:
For, behold, the Lord cometh forth out of his place, and will come down, and tread upon the high places of the earth.
And the mountains shall be molten under him, and the valleys shall be cleft, as wax before the fire, and as the waters that are poured down a steep place.
For the transgression of Jacob is all this, and for the sins of the house of Israel...
Therefore I will make Samaria as an heap of the field...
And all the graven images thereof shall be beaten to pieces, and all the hires thereof shall be burned with the fire, and all the idols thereof will I lay desolate.
(Micah 1:3–7)
Such admonitions, in the English of the Authorized Version, left an indelible impress on imaginations nurtured on the Bible; Josephus' _Jewish War_ also provided popular Protestant reading.
Hear Thomas Carlyle's peroration at the end of his _The French Revolution_ (1837), fusing in his unique fashion prophetic, iconoclastic fervour and satire:
Imposture is in flames, Imposture is burnt up: one Red-sea of Fire, wild-billowing, enwraps the World; with its fire-tongue, licks at the very Stars. Thrones are hurled into it, and Dubois Mitres, and Prebendal Stalls that drop fatness...Higher, higher yet flames the Fire-Sea; crackling with new dislocated timber; hissing with leather and prunella. The metal Images are molten; the marble Images become mortar-lime...
Carlyle was exceptional in the energy with which he could exploit the apocalyptic mode, but in secular history it was revolution which, more than any other theme, invited it. For the early Christians, however, the vital concomitant of judgement and condemnation was the promise of salvation brought by Christ. For the church historian Eusebius this promise was typified in contemporary history by the conversion to Christianity of the emperor Constantine and the deliverance of the Church from persecution.
**THIRTEEN**
**Eusebius: The Making of Orthodoxy and the Church Triumphant**
In the early fourth century the Christian Church endured intermittent periods of imperial persecution, with their crops of martyrdoms, the last and severest of which ended only in AD 312 with the conversion of the emperor Constantine and the transformation of the Church's position from persecuted sect to its eventual privileged status. Few great historical transitions have been more sudden. But the Church had been constructing itself since apostolic times, over almost three centuries, and, as expectations of an imminent apocalypse waned, it was beginning to become conscious not only of its present and future but also of its past.
The notion of the Church as the new community of the elect, and its commitment to teaching and evangelization, gave a crucial importance to the question of its identity and purity; as the generation of the Apostles slid further into the past, it became essential to distinguish and defend the authentic tradition. Questions of historical continuity were vital, especially before the earliest formulations of doctrine from the fourth century by councils of the Church, which provided reference points. It was essential to demonstrate continuity with the Old Testament prophecies, with the testimony of the Apostles and the lines of descent from them of Christian bishops, and with the teachings of those who had become established as authorities, as Fathers of the Church. These continuities were necessary to weed out heresies, spurious books, ecstatic false prophets and enthusiasts, and deviant interpretations of the sacred writings. Orthodoxy had to be forged in the face of competing zealotries and of the intellectual problems posed by the assimilation of Greek—above all Platonic—intellectual traditions to those of the Hebrew sacred writings and the Gospels. There were formidable difficulties in interpreting complex texts recognized to be at least in part symbolic. The question of the unity of the Church was essentially the same as that of its continuity; it has to be remembered that by the time of Constantine's conversion the ministry of Christ was as distant as the death of Louis XIV from ourselves. The Acts of the Apostles had in a sense laid the foundations for what was now explicitly attempted: a first history of the Church, and it was provided in the early fourth century by Eusebius, bishop of Caesarea.
Eusebius was a Greek, born probably in Caesarea, in Palestine, of which he became bishop in the early 260s; he lived to see the Church severely persecuted (AD 305–12)—his own teacher was one of its martyrs—and then delivered by the conversion of the emperor Constantine; he was a notable participant in the Council of Nicaea (AD 325), where he met and supported Constantine. Nicaea defined the nature of Christ so that the followers of Arius, for whom Christ was not of the same nature as the Father but subordinate to him, became heretics. Eusebius, who has been suspected of having Arian tendencies, seems to have prized the unity of the Church above all else; he treated the formula adopted at Nicaea as definitive. At Caesarea, Eusebius was heir to the great biblical commentator Origen ( _c._ 185– _c._ 254), who pioneered symbolic and typological readings of the Bible. Eusebius' own reading of even secular history was highly providentialist—understandably in someone who had witnessed what must have seemed the miracle of the Emperor's conversion immediately following the most severe and protracted persecution in the history of the Church. Constantine's conversion became for Eusebius almost like a second Incarnation, and Constantine was clearly God's representative on earth.
Besides his _History of the Church,_ which covers the whole period from the birth of Christ to the time of writing in the 320s, and which was very soon to be translated into Latin, Eusebius was well known in the Middle Ages as the author of a _Chronicle,_ epitomizing the history of the great peoples of antiquity down to the Romans, with an attempted interpretation, in parallel columns, of pagan and Hebrew chronologies. This was to be much drawn on for the preambles to medieval chronicles, being continued by St. Jerome and sometimes therefore known as "Eusebius–Jerome." Part of its purpose for Eusebius was to demonstrate the superior antiquity of the religion of the Hebrews—also a preoccupation of Josephus—over the others.
Eusebius wrote in another genre which was to achieve great popularity, Christian martyrology, an account of the recent great persecution in Caesarea. Martyrdom, prefigured in the crucifixion of Christ, with remoter precedents in the books of the Maccabees and the death of Socrates, became a central Christian preoccupation. Martyrs triumphed in the manner of their deaths, and were translated at once to heaven. Commemoration of them became an important feature of Christian liturgy, and accounts of their sufferings—often highly circumstantial, as they are in Eusebius—became, like the slightly later Lives of the saints, with which they overlapped, a form of popular Latin literature or what one might paradoxically call documented folklore. Martyrs corresponded in some respects to the heroes of pagan legend and history, but their deaths were self-chosen, they included a significant number of women, and their ends were not, ultimately, tragic but triumphant. The details of the physical agonies of martyrs, complementing those of Christ, were to be a rich source of Christian iconography. Martyrs and saints were influential citizens of heaven, and gifts and pilgrimages to their shrines would be a source of much wealth to the cathedrals and abbeys fortunate enough to possess their bodies or any fragments of their remains.
Eusebius' history was highly original in conception and became _the_ history of the early Church, in the sense that his successors sought only to continue, not to supplant it. It had an entirely new subject matter, and reads quite differently from any classical history. Church history was inevitably heavily involved in polemic, in establishing the undeviating line of orthodox tradition and distinguishing it from the snares of heresy, inauthentic books and deviant interpretations. It also recorded the more notable episcopacies and martyrdoms. Jews, pagans and, above all, heretics are Eusebius' polemical targets, and he writes his history to confound them, with no claim, like that which had become standard in classical histories, to write without passion or partisanship: he wrote avowedly to establish and vindicate Christian orthodoxy. Compared with the classical writers of history, this may seem to put a greater distance between him and modern notions of historical propriety, but it also had the ironic result of bringing his practice closer to that of modern historians, just as the religious controversies of the Reformation were later to do. Because he writes not to entertain but to prove and to vindicate, he needs the authority of sources, and the question of which are to be treated as authoritative (though of course this is not just or even primarily a question of reliable witnesses) is crucial for him. Hence he quotes them, copiously, and sometimes reproduces others to discount their deviant interpretations. Indeed, without the library of which he was so proud, his book could hardly have been written at all. Hence he may be less of a historian than his classical predecessors, but he is more of a scholar—though certainly not a disinterested one.
One feature of his book, before it becomes a record of contemporary events, is a succession list of the bishops of the main Christian sees from apostolic times. As we have seen, succession lists of pharaohs, kings, pontiffs and consuls are among the earliest forms of historical recording. In Eusebius, of course, they are crucial in establishing the apostolic succession, as the institutional spine of orthodoxy. Another vital interest, analogous to it in the realm of doctrine, is the establishment and justification of the accepted canon of scriptural books—there were a number of additional candidates—and, alongside these, the acceptance of approved biblical commentators and the rejection of their heretical rivals. One way, therefore, in which Eusebius' book seems hardly a history at all is that, after the Incarnation and until the conversion of Constantine, there are, apart from martyrdoms and the inevitable apostasies under persecution, virtually no events. As one of Eusebius' modern commentators puts it, "his history is peopled not by men or events but by those who wrote books." The precedent was not so much classical narratives of political and military affairs as the Greek histories of philosophers and their schools. Often one feels one is reading a kind of select bibliography, with a polemical commentary and extensive quotations. Here he is on Clement of Alexandria:
Of Clement's works, the _Miscellanies,_ all eight books are in my possession [there follows a list of titles]...In the _Miscellanies_ he has woven a tapestry combining Holy Writ with anything that he considered helpful in secular literature. He includes any view generally accepted, expounding those of Greeks and non-Greeks alike, and even correcting the false doctrines of the heresiarchs, and explains a great deal of history, providing us with a work of immense erudition. With all these strands he has blended the arguments of philosophers, so that the work fully justifies the title of _Miscellanies._ (6.13)
This gives quite a good idea of the creation, from diverse materials, Christian and pagan, of a body of relevant Christian erudition. Eusebius himself speaks of the condemnations of heretics and their opinions as giving "the names and dates of those who through a passion for innovation have wandered as far as possible from the truth" (1.1). The individuality of Eusebius' work lies therefore not in a manner of narration, but in what he chooses to record, and approve or condemn. He is, above all, making a record of the right and wrong paths, but he is content largely to let those who took them either speak for themselves or exist just as names and dates. This is not to say that his tone is dispassionate. The Devil is at work in the making of heresies, which are detestable impostures; heretical sects "crawled like poisonous reptiles over Asia and Phrygia" (5.14).
Merely scholarly error is sometimes more gently treated, but the anatomy of false prophecy in Eusebius and his sources—much is quotation—is often vivid and precise, like his description of a pseudoprophet in a state of unnatural ecstasy (5.17), and an account of the practices of a heresiarch in Antioch, who arranged "for women to sing hymns to himself in the middle of the church on the great day of the Easter festival: one would shudder to hear them" (7.30). On the aids to orthodox devotion the sources can be rhapsodic: the bones of a martyr are "more precious than stones of great price, more splendid than gold" (4.15).
It is in the recording of martyrdoms that Eusebius' work ceases to be a matter of disputes over words, and this gives a distinct character to the last three, contemporary, books, which are particularly devoted to the martyrs. Bookishness is put aside, and Eusebius, who in the establishment of orthodoxy can often seem a complacent writer, is provoked to a quasi-biblical eloquence—with again, of course, much quotation and allusion—by the contemplation of God's chastisement and of the Church's eventual victory:
The Lord in his anger [at the arrogance and quarrels of certain church leaders] covered the daughters of Zion with a cloud, and cast down from Heaven the glory of Israel...Everything indeed has been fulfilled in my time; I saw with my own eyes the places of worship thrown down from top to bottom, the inspired holy Scriptures committed to the flames in the middle of the public squares, and the pastors of the churches hiding disgracefully. (8.2)
Some behaved disgracefully, others heroically. The former are not identified, but the latter are named and glorified in a way that recalls one of the earliest impulses to historiography: that great deeds may not be uncommemorated.
Eusebius' writing also rises to the highest pitch of excitement when describing in detail the physical symptoms of decay suffered by the persecuting emperor Galerius:
[He was pursued by a divinely ordained punishment, which began with his flesh and went on to his soul. Without warning suppurative inflammation broke out round the middle of his genitals, then a deep-seated fistular ulcer; these ate their way incurably into his innermost bowels. From them came a teeming, indescribable mass of worms, and a sickening smell was given off; for the whole of his hulking body, thanks to overeating, had been transformed even before his illness into a huge lump of flabby fat, which then decomposed and presented those who came near with a revolting and horrifying sight. (8.16)
The doctors were unable to bear the extraordinary stench. It is no wonder that Gibbon said that Eusebius described "the symptoms and progress of [Galerius'] disorder with singular accuracy and apparent pleasure" ( _Decline and Fall,_ XIV, n. 37). The symptoms, in fact, bear a considerable resemblance to those ascribed by Josephus to Herod the Great. Those detailed by Eusebius with which God afflicted the last of the persecuting emperors, Maximin, are somewhat different but equally picturesque:
He was wasted by hunger, and the whole of his flesh was consumed by an invisible fire sent from God, so that all the contours of his former shape disintegrated and disappeared, and nothing but a collection of dry bones, like a phantom reduced by long years to a skeleton, was left, so that the onlookers could imagine nothing else than that his body had become the grave of his soul, which was interred in what was already a corpse and completely disintegrated. As the fever that consumed him blazed up ever more fiercely from the depths of his marrow, his eyes stood out of his head and fell from their sockets, leaving him blind. (9.1)
Maximin, perhaps not surprisingly, confessed his offences against Christ and was permitted to die.
God, having scourged his people, relents towards them. Constantine overcomes his co-emperor in battle in a manner compared by Eusebius to the destruction of the armies of Pharaoh in the Red Sea. In the account of Constantine's triumphant entry into Rome, the Emperor—"God's friend"—is welcomed by the people as their deliverer and saviour, to the accompaniment of much quoting by Eusebius of biblical rejoicings, especially from the Psalms. Constantine and his son Crispus, "holding out a saving hand to all who were perishing," are closely paralleled to God and his Son, while "his adversary thus finally thrown down" (actually Maximin) seems a type of the Devil (10.9). Constantine is compared to Christ as miraculous healer: "Taking the only Quickener of the dead as ally and co-worker he raised up the fallen Church, after first cleansing her and curing her sickness; and he clothed her with a garment" (10.4). Eusebius is referring to the rebuilt places of worship whose rededications he celebrates, and specifically to that of Tyre, for which he himself gave the oration, which he reproduces for us. Eusebius presents the deliverance of the Church virtually as a type of the final redemption. All dissensions between Christians fall away: "There was one power of the divine Spirit coursing through all the members, one soul in them all" (10.3).
The deliverance of the Church is conflated in Eusebius with the reunification of the empire, the peace of the Church being, it seems, identical with the Pax Romana. Constantine and his son "reunified the Roman Empire into a single whole, bringing it all under their peaceful sway, in a wide circle embracing north and south alike from the east to the furthest west." Freed from their fear, men "kept dazzling festival; light was everywhere, and men who once dared not look up greeted each other with smiling faces and shining eyes. They danced and sang in city and country alike..." (10.9). The empire and the Church are both healed, and there seems no significant difference between them. Eusebius not only draws on the Christian notion of redemption, but also stands at the end of a long tradition of the myth of Rome as the Eternal City and of the empire without end proclaimed by Virgil. It could never be a wholly Christianized idea, since for Christians history was to have an end in the Second Coming of Christ, but in Eusebius, who was notably cool towards speculations about the coming end (e.g. 3.39), it came as near as possible to being so.
Supplemented by St. Peter's alleged martyrdom there, the myth of Rome and its enduring authority was to be exploited in the high Middle Ages by the papacy, through the conceptions of the canon lawyers, and was to be competed for by the Holy Roman Emperors. A century after Eusebius, it had been contested by Augustine. Confronting the clamour of pagans claiming that the sack of Rome by the Goths (AD 410) was a consequence of the rejection of the old gods and hence the fault of the Christians, Augustine, in his _City of God_ (after AD 412), refused to identify God's people with any earthly community; the distinction between the earthly and the heavenly city was not an external one but referred to two different states of the soul, which would acquire their visible embodiments only with the Last Judgement. But Augustine cared enough about the argument advanced by the pagans to inspire his disciple Orosius to write his _Seven Books of History against the Pagans_ (AD 417), to show that disasters like the sack of Rome had occurred at all periods in the past. Orosius made much of the occupation of the city by the Gauls, as described by Livy; the modern disaster was therefore not attributable to the abandonment of paganism. Orosius discharged his task in an epitome of world history—with special attention to Roman history—which, together with a similar and later epitome by Isidore of Seville, became one of the key works of the Middle Ages, being constantly drawn on for the general preambles to chronicles and local histories of all kinds. But Orosius also did more than Augustine can have intended, in a way that draws him closer to Eusebius' view of history than to Augustine's. Orosius, in fact, succumbed, more explicitly than Eusebius, to the myth of eternal Rome, and, like Eusebius, identified the Church with the fortunes of the empire.
There were two schemata current at the time for world history: the sequence of four empires derived from the prophetic dream recounted in the biblical Book of Daniel (2.31–44), which became highly influential, and the classical six ages of man, followed, for Christians, by an eternal Sabbath. Augustine adopted the latter, but made no use of the former; Orosius subscribed to both. In both of them the Roman empire figured as the last of the series, the final act of human history, to be followed only by the last days. It would remain, therefore, without a human successor, and would endure to the end of secular time. The empire figured prominently and not negatively in Orosius as part of the providential scheme. In particular, the Augustan peace was linked to the pivotal moment of the Incarnation—not only in time, but as the necessary condition for the spreading of the Gospel. The rise of the Roman empire was therefore part of the divine plan. The conversion of the barbarians in modern times was concomitant with the extension of the empire, of which Orosius accordingly approved: "fall" is, despite the incursions of the Goths, not part of his conception of the empire's ordained role.
The fact that Christianity came to flourish under imperial patronage ensured that Eusebius' Rome-centred view of Christian history would prevail. The collapse of the empire in the West before the barbarian invaders did not displace the Eusebian–Orosian conception of universal history as one might have expected. The Pope in the city of Rome—where the tomb of St. Peter, to whom Christ was supposed to have confided his Church, was located—and later the Merovingian kings and the Carolingian emperors, all had a strong vested interest in this version of the relation of Rome to Christianity. The eventual adoption in the West of the phrase "Holy Roman Empire" only made explicit what had long been the dominant conception of Christian historiography.
**FOURTEEN**
**Gregory of Tours: Kings, Bishops and Others**
It was a characteristic of medieval historical writing, compared with classical historiography, to be at once all-encompassingly universal and highly and unapologetically particular and local. Typically, a chronological account, more or less full or episodic, of the chronicler's parent institution or locality would be prefaced by an outline of universal history—often a mixture of biblical events with those of secular ancient history, drawn from Orosius or St. Isidore of Seville (AD _c._ 560–636), and the biblical-secular chronologies begun by Eusebius and continued by Jerome (above, Chapter 13). After that, more or less abruptly, the contemporary chronicle could begin, mainly local in focus but with occasional interpolations, quite often garbled, from the wider world of great events. In the West, after the deposition of the last Roman emperor in AD 476 there was no longer, as there had been ever since Polybius, a single dominating secular protagonist for history, the Roman empire. The western medieval empire—never except briefly under Charlemagne anything like universal—and the consolidation of the national kingdoms were yet to come. The _Liber Pontificalis,_ the house chronicle of the popes, dealt initially with the local concerns of the bishop of Rome. In a politically fragmented world, whose chief binding agents were councils of bishops and the dynastic alliances of ruling families, almost all history was, on a greater or smaller scale, primarily local and was written accordingly; the most important form taken by record-keeping, as with the pagan Roman _fasti_ (books of ceremonies and festivals) and pontifical annals, was the successions of popes, bishops and abbots and the chief events of the liturgical year based on the lunar calendar, into which narratives of heterogeneous contemporary events, often very brief, could be interpolated.
The ten ample books that constitute the _Histories_ of Gregory, metropolitan bishop of Tours from AD 573 to 594, show the typical mixture of universal history in the first book, rapidly narrowing to the general affairs of Gaul after the Frankish invasions of the fifth century and to a localized contemporary chronicle in the last six books. However, the history of the previous century is unusually full, and the contemporary chronicle is exceptionally copious and vivid, strongly marked by the personality of its author, who displays some of the self-consciousness, though not the literary airs, of a classical historian. Gregory's life and the contemporary events he records were centred on, though not entirely confined to, the Loire valley. The Roman empire still existed, of course, in the East, and in its nearest embodiment in Italy, centred on Ravenna, but it was essentially peripheral. The Pope in Rome was more relevant but not particularly intrusive. The great monastic movement, particularly Irish, which criss-crossed Europe, transcending tribal and diocesan boundaries, belonged more to the seventh century than to Gregory's sixth. Generally the French metropolitans managed the affairs of their localities, meeting occasionally in conclave. Gregory, though he travelled widely in Gaul, never went south of the Alps or east of the Rhine. As a metropolitan bishop, he was at the centre of the world that concerned him.
He begins his work with a summary of world history based on the Bible, passing to the Incarnation of Christ and the general history of the Church, before proceeding quickly to Gaul, where he alights, still in his first book, at Clermont-Ferrand, his birthplace. For the modern reader the descent from the universal to the particular may seem precipitous, but it clearly seemed quite natural to Gregory, and is not untypical. His continuator—known, inauthentically, as "Fredegar"—mentions in his prologue his preparatory reading in the chronicles (he identified Jerome, Isidore of Seville and Gregory himself ) "from the beginning of the world to the decline of Guntramn's reign." What other chronological marker for the present could he have used? It is the reference also to the Creation—the mingling, in other words, of sacred and secular history—which makes his statement odd to us. Bede, writing in the same century (the eighth) in England and focusing on ecclesiastical history, which Fredegar largely ignores, solved the problem of chronological notation by dating forward and backward from the Incarnation, which historians have done ever since, despite the slight inconvenience of counting dates BC backwards. Gregory mingles secular and ecclesiastical history, "the wars waged by kings and the holy deeds of martyrs," and defends doing so by the precedents of Eusebius and Jerome (II, preface). He will, he says, set down events "in the muddled and confused order in which they occurred," and he does.
Gregory's concerns are misrepresented by the later entitling of his book _The History of the Franks._ Gregory was not a Frank, and his interest in them—chiefly their kings—seems largely incidental; it was because, as the rulers of sixth-century Gaul, they were inescapably there, but he has nothing like the interest in their destiny that, for example, Polybius had in the Romans who had conquered his people. His ethnic awareness and interest is generally minimal: he does not even mention that Gaul was a country of two utterly distinct languages, Latin and Frankish.
St. Gregory himself, like many of the bishops of Frankish Gaul, was a Gallo-Roman aristocrat, from a family productive of senators, bishops and saints. He was brought up in Clermont-Ferrand in the household of his uncle St. Gallus, bishop there from AD 525, and he eventually succeeded his cousin St. Euphronius as bishop of Tours in AD 573. Sainthood seems to have been an informal _Légion d'honneur_ of the sixth-century Gallic episcopate, and there was a marked dynastic element to it, despite the electoral process in the appointment of bishops; the process of papal canonization was not formalized until the twelfth century. The bishops were the defenders of Catholic orthodoxy and public morals, the leaders of their clergy and their people, and the administrators and trustees of the wealth of the Church, for the upkeep of buildings (in Tours especially the cathedral and shrine of St. Martin) and the relief of the poor; they were necessarily the familiars—remonstrating, mediating, conciliating—of the Frankish kings. All these concerns marked out the themes of Gregory's history.
The cities of Gaul, under their aristocratic native bishops, remained in large measure the heirs of Roman administrative traditions, just as the heritage of Rome persisted in the vernacular Latin of Gregory and his congregations; it is not clear that he knew any Frankish, and he never refers to the language. He was not well instructed in the classical literary heritage, and he knew nothing of that of Greece. His apologies for his unpolished Latin are a little more than the customary humility, but he says he writes to be understood—not for him the artificial archaizing of some late antique writers. It has been plausibly held that his history is a kind of continuation of his pastoral work, an anthology of tales with a homiletic purpose. Among his other writings, to which he often refers the reader, are martyrologies, Lives of the Fathers of the Church and of St. Martin of Tours, and a collection of miracles.
We have seen Eusebius recording martyrdoms. The prototypical saints' Lives, initiating a staple of medieval popular clerical reading, were Athanasius' _Life of Antony,_ the fourth-century hermit beleaguered by picturesque demons in the Egyptian desert, which was soon translated into Latin, and, nearer home and later, Sulpicius Severus' largely fictitious _Life of St. Martin_ ( _c._ 403), which effectively initiated the cult of the saint at Tours, where the inestimable treasure of his bones was housed in the cathedral. Cult centres of this kind were a great source of miraculous cures, posthumously effected by the saint, guaranteeing his sanctity and influence with God and drawing a lucrative stream of pilgrims and donations. Saints' Lives were formulaic, retailing the saint's birth and origins, and dwelling on the holy and triumphant death, but largely reducing the intervening life to a succession of miracles, often echoing those recounted in the Gospels, and apparently often parasitic on earlier models of the genre, but also sometimes including transmuted stories from classical antiquity and Christianized folk tales. Like St. Antony, the saint was characteristically beset by diabolic adversaries—though these were not necessarily temptations—and miracles were the armoury from which he drew to confound them. As moralizing entertainment we might compare them with Plutarch's _Lives_ in the pagan world, but by comparison they are naively presented and generally unindividualized.
After death, saints' bodies characteristically refused to putrefy and gave off a sweet perfume—Gregory is enthusiastic about this—and they wrought miracles by contact with the relics, which included objects touched by the saint. Competition for relics was understandably keen. Gregory's story of the arrival of St. Martin's corpse in Tours gives an example. The men of Tours and the men of Poitiers, assembled at St. Martin's deathbed, had disputed possession of the body:
The men of Poitiers said: "As a monk he is ours. He became an abbot in our town. We entrusted him to you, but we demand him back. It is sufficient for you that, while he was a Bishop on earth, you enjoyed his company, you shared his table, you were strengthened by his blessing and above all you were cheered by his miracles. Let these things suffice for you, and permit us at least to carry away his body." To this the men of Tours replied: "If you say that we should be satisfied with the miracles that he performed for us, then admit that while he was with you he did more than in our town. If all his other miracles are left out of the count, he raised two dead men for you and only one for us..." They went on with their argument until the sun went down and night began to fall. The body was placed in the middle of the room, the doors were locked and he was watched over by the two groups. The men of Poitiers planned to carry off the body as soon as morning came, but Almighty God would not allow the town of Tours to be deprived of its patron. In the end all the men of Poitiers fell asleep and there was not one who remained on guard. When the men of Tours saw that the Poitevins had fallen asleep, they took the mortal clay of that most holy body and some passed it through the window while others stood outside to receive it. They placed it in a boat and all those present rowed down the River Vienne. (I.48)
They carried the body back to Tours in triumph along the Loire, "praising God and chanting psalms." The detailed touches of the window, the boat and the singing are typical of Gregory's anecdotal methods.
The acquisition of St. Martin was a great moment, but for Gregory no circumstance was too humble or physically particular, no action too purely domestic, to be unworthy of narration if there was a moral or spiritual point to be made. There could be no question of his history being confined to political (or even political and ecclesiastical) events, because it is not clear that he had any particular concept of the political as a category. Apart from the combating of the Arian heresy, Gregory's world, even that of the ruling Merovingian dynasty and its rival rulers (perhaps "chieftains" would be a better word than "kings"), is a highly personal one, in the same way that the importance and wealth of the city of Tours derived largely from St. Martin's body. Motives and conduct are particular to the individual and are hardly ever generalized except as pleasing or displeasing to God, and, unlike in some Christian histories, such as those of Gildas and Bede in Britain, it is characteristically individuals, not peoples, whom God punishes or favours. The second sentence of Gregory's preface refers, in mentioning the animators of the contemporary events he proposes to chronicle, to "kings losing their temper." There is no place here for abstractions like "royal policy" or "aristocratic resistance." Great men behave much as small ones do—out of anger, cupidity, wilfulness, revenge. No such abstract concept as the blood feud among the Franks is mentioned: Gregory and perhaps his readers take it for granted. Even the saints, in Gregory's world, know how posthumously to avenge affronts.
Gregory begins his work with the memorable, and entirely accurate, reflection that "A great many things keep happening, some of them good, some of them bad." It is probably not misguided to read into this that we must not look to Gregory's history for an overall plot, though attempts have been made to impute one to him—such as a framework lifted from Old Testament history, the notion of the Franks as a people with a mission, or the idea of a steady decline of behaviour in Gaul from the time of King Clovis (d. AD 511) (e.g. IV.45). Though he goes on to speak of the quarrels of "the inhabitants of different countries," and occasionally describes the behaviour of mobs, Gregory's protagonists are individuals. It is true that he speculates briefly about the origins of the Franks (II.9), the invaders of Gaul in the previous century, quoting an earlier Latin historian, now lost, and—in a rare case of generalization—about the origins of the "long-haired kings" and the kind of kingship they possessed. (It was believed in Gregory's day that a Merovingian who had his hair shorn lost his eligibility for kingship.)
The most emphatically "historical" single event in the history of the Franks, because it was also an event in the history of the Church, was the conversion of King Clovis to the Catholic faith—according to Gregory, with his whole army—in AD 496; the other barbarian peoples who overran the West had been converted to the Arian version of Christianity. Gregory, rather uncharacteristically, makes the obvious and usual analogy in such circumstances: with Constantine. The current Roman emperor gave Clovis the title of consul. The baptism forged the link between the Franks and the papacy that would, three hundred years later, be crucial to the emergence of a Frankish, Carolingian, western empire. After Clovis, however, the Frankish kings (there were four simultaneously in Gregory's time, thanks to the division of the inheritance), and their nobles, appear in his history not as figures of historic destiny but simply because they were the secular powers, the great men, of their day. They are able to do good, chiefly in the form of donations to St. Martin's shrine, and, more often, harm, in their internecine wars, often indistinguishable from raids for plunder, with extortion and occasionally massacre visited on peaceful town and peasant populations. To Gregory, civil wars were a particular evil of the times, and he prays that they may cease.
The homicidal feuds within the dynasty, however, he records in deadpan fashion, shocking the reader by apparently treating them as normal, which they were. Merovingians, one comes to feel, just were like that. The reader is, of course, as Gregory presumably knows, sometimes registering the existence of the blood feud, but it does not occur to Gregory to categorize it as an institution of the Franks, so it looks like mere vengefulness or even motiveless ferocity. Gregory's account of and epitaph on Clovis—who has been spoken of as the hero of his history (he is hardly that) and who was certainly the greatest warrior king of the Merovingian line, establishing his rule over much of Gaul as well as embracing Catholic Christianity—are notable. Gregory drily recounts Clovis's behaviour, including his ogreish habit of unexpectedly bisecting people with his axe, and even seems to appreciate his gallows humour. After recounting Clovis's axe-work on two of his relatives, and his command to put a third to death, Gregory adds:
As soon as all three were slain Clovis took over their kingdom and their treasure [the kingdom was centred on Cambrai] In the same way he encompassed the death of many other kings and blood relations of his, whom he suspected of conspiring against his kingdom. By doing this he spread his dominion over the whole of Gaul. One day when he had called a general assembly of his subjects, he is said to have made the following remark about the relatives whom he had destroyed: "How sad a thing it is that I live among strangers like some solitary pilgrim, and that I have none of my own relations left to help me when disaster threatens!" He said this not because he grieved for their deaths but because in his cunning way he hoped to find some relative still in the land of the living whom he could kill. (II.42)
Yet Gregory's biblically phrased summary is that "Day in day out God submitted the enemies of Clovis to his dominion and increased his power, for he walked before Him with an upright heart and did what was pleasing in His sight" (II.40). Great kings, like the prototype, the biblical David, who enjoyed God's favour, were admittedly not expected to be perfect, but—well! It mattered, of course, that detestation of the Arian heresy provided the only "ideological" element in Gregory's history. He records, ostensibly verbatim, his own acrimonious debate with an Arian heretic (V.40) and also Catholic martyrdoms in Arian Visigothic Spain, which actually seem to have been very rare. He retails the sufferings of a Catholic girl who, forced to endure rebaptism as an Arian, cries out in defiance—and with a good recollection of a nice distinction bred from Greek philosophy—"I believe the Father with the Son and the Holy Ghost to be of one substance," before being consecrated to Christ by having her head cut off (II.2).
The Frankish kings are one element in Gregory's history, and at times it focuses on them and their often atrocious deeds. They do not, however, dominate the history. This can best be seen by a summary of his first four books; the later ones, when the history becomes contemporary and Gregory himself appears, are better dealt with thematically. In Book I, after a very short time with the Old Testament and the history of the Christian Church, we arrive at the conversion of Gaul, followed by the unedifying marital affairs of the first bishop of Clermont-Ferrand, and shortly afterwards by the death in AD 397 of St. Martin, third bishop of Tours, with which the book ends. The framework of Gregory's book is, above all, episcopal history in the area of the Loire valley. The Franks make their entry in Book II, but most of the earlier part of that too is episcopal. However, the latter part of the book is dominated by Clovis, and ends with his death in AD 511. The next two books interleave Frankish and episcopal themes as Gregory has promised. Endearingly, he greets the opening of Book V, with which the era of his personal knowledge opens, with the words, "Here, I am glad to say, begins Book V. Amen." His own first appearance is a dramatic instance of a confrontation in which the Church, embodied as St. Martin's shrine, with Gregory as its guardian, confounds the representative of royal Frankish power. The episode illustrates Gregory's way of telling a story, and so is worth quoting at length:
Next Roccolen marched on Tours, having received orders to do so from Chilperic...He pitched his camp on the further bank of the river Loire and sent messengers to me saying that I must expel Guntram from my church, for he was accused of having killed Theudebert. If I did not carry out his commands he would order the city and all its suburbs to be burnt to the ground.
Chilperic was one of the four Frankish kings, grandsons of Clovis, at that time. His half-brother Guntram had indeed killed Chilperic's son Theudebert in battle. Roccolen is presented to us without introduction. Gregory is defiant:
As soon as I heard this I sent a deputation to Roccolen to say that what he demanded had never been done down all the centuries from ancient times, and that it was quite unthinkable that the holy church could be violated; that if it were to happen it would bring small profit to him or to the King who had sent such orders; and that he would do better to shake in fear before Saint Martin the Bishop, whose miraculous power only the day before had made paralysed limbs straight [recorded by Gregory in his Life of St. Martin]. Roccolen was no whit abashed. He pulled to pieces the church-house on the opposite bank of the Loire, in which he had his quarters. The building was nailed together: the men of Maine, who formed Roccolen's army, put the nails in their pockets and sneaked off with them, destroying the harvest and ruining everything else as they went. When Roccolen was committing these outrages he was punished by God, for he fell ill with jaundice and turned bright yellow.
Seeking recovery, Roccolen is carried to the church he had threatened to violate, but he remains ill and personally unforgiven. He was clearly really unregenerate, for
We were then in the holy month of Lent: and he kept on eating baby rabbits. He had earlier drawn up for the first day of March certain ordinances by which he planned to mulct and ruin the people of Poitiers. Twenty-four hours before that he died; and with him died his overweening arrogance. (V.4)
The next section begins, "It was at this time that Felix, Bishop of the city of Nantes, wrote an abusive letter to me." The ostensible occasion was a complex affair involving the murder of Gregory's brother Peter, a deacon, who was accused of murdering his own bishop, whose office he coveted. The real reason, according to Gregory, was that Felix, a greedy and arrogant man, coveted some church land in Gregory's see. He retorts in kind, giving us the best bit of his letter and adding, "I will say no more for fear that you begin to think I am much the same myself."
Gregory's work is not particularly egocentric or self-justificatory, but after the beginning of Book V the history becomes still more focused on Tours and his own diocese. Within it his range of attention is wide, and reflects the multifarious interests and concerns of a Gallic bishop as pastor of his people and clergy; at times reading him reminds one of the recollections of a magistrate or judge (which in a sense he was) of broad human sympathies, with a taste for gossip. Gregory was clearly a born raconteur, and no level of human life (except perhaps, as individuals, that of the peasantry, for whom he has only a generalized concern) was inaccessible to him: domestic slaves, artisans, contumacious nuns, impostors (a particular worry), drunks—including several bishops (one also mentally unbalanced) and an ascetic who became an alcoholic—and fornicating priests and abbots, as well as ferocious Frankish nobles like Roccolen and fratricidal monarchs, all people his stories.
The story of Roccolen exemplifies the tension between bishops like Gregory, determined to maintain the Church's rights, and the Frankish kings and their representatives, with their incessant raiding and feuding, in which the ordinary populace, to whom the bishops are pastors, is caught up. Roccolen was acting on the King's orders, in pursuit of a blood feud within the royal family, and also acting as a tax-collector, to give a misleadingly bureaucratic name to something so random and sporadic. Such confrontations of royal power with the Church's privileges were, of course, common, and not only in Gaul. Perhaps the most famous, immortalized in a dramatic painting by Van Dyck, was the refusal of the Archbishop of Milan, St. Ambrose, to allow the sinful emperor Theodosius to enter his cathedral until he had sought forgiveness. What comes foremost in Gregory's account is that Roccolen is a bad, arrogant man, disrespectful to St. Martin's shrine, and a meat-eater in Lent (his illness might surely have earned a dispensation—Gregory makes it sound as though he devoured the rabbits whole), as well as an extortionist. St. Martin, however, can and does avenge affronts to his relics and his church; there are other instances. A not uncommon disease is attributed to divine intervention and punishment.
Miracles are a frequent preoccupation of Gregory's, but it has been plausibly argued that, though he can recognize them, he does not have anything like a precise modern definition of them as suspensions of a regular order of nature. Rather, the whole world witnesses all the time to God's power, but some aspects and events more strikingly and instructively than others, chiefly because of their obvious moral or spiritual import. Where a moral context is lacking, Gregory admits himself at a loss. Noting some exceptional astronomical behaviour, he assumes, as historians for the previous thousand years would have done, that it is a sign—but "I have no idea what it all meant" (V.22). Given a morally relevant situation, however, timely cases of cirrhosis, such as Roccolen's, are clearly miraculous and proper for pious contemplation. God looks after his own, at least when they are as influential or have such a powerful patron as St. Martin. Some miracles are more gratuitous than this, especially perhaps the fragrance associated with saints' relics, but these minister to faith and pious joy. Miracles confound the Devil, as well as the wicked men who are his agents. The constant metaphysical strife, waged partly by physical means, is not a walkover, however. The Devil also has exceptional powers, which is why impostors are so persuasive and dangerous. The Devil, it appears, afflicted a man of Bourges with a swarm of flies, which sent him mad. He became a wandering ascetic, dressed in animal skins, and "In order to encourage him in his deception the Devil gave him the power of prophesying the future." He set up as a prophet and eventually as Christ, and gathered a large following—part fanatics, part bandits. These were eventually attacked by some muscular servants of the Bishop of Le Puy, whom he had threatened; they killed the false Christ and dispersed his followers, some of whom never returned to full sanity. Such impostors, who acquire "great influence over the common people," are a big nuisance. Gregory says, "I saw quite a few of them myself. I did my best to argue with them" (X.25).
Gregory as raconteur can often remind the reader of later writers of (fictional) tales such as Boccaccio. The stories even tend to begin in the same ways, though more often as a preamble to murder or sometimes a miracle—and sometimes both—rather than to amorous intrigues, though these are not altogether absent. "A certain merchant called Christopher travelled to the city of Orleans..." "At this time there lived near the town of Nice a recluse called Hospicius..." "A Breton Count killed three of his brothers..." These stories as often involve people otherwise obscure as they do the great. Every soul and therefore every wicked or good deed matters. When Gregory gives in his preface an entirely traditional reason for writing history, "to keep alive the memory of those dead and gone, and to bring them to the notice of future generations," he may have been thinking chiefly of the "great," especially martyrs and saints. But the later, contemporary, part of his history, which attends to many others who are not great, is perhaps better covered by what he goes on to speak of: "the quarrels between the wicked and the righteous." Martyrdoms had become rare, but of such quarrels there was no end, and all were, by definition, significant.
One striking feature of Gregory's writing is his confident sense of rapport with his readers, expressed in homely, confiding terms (though these may be exaggerated by translation from the Latin). At one point—one can envisage him scratching his head and clicking his tongue—he even apologizes for forgetting his chronicler's manners: "I meant to tell you earlier on of a conversation I had with Saint Salvius, the Bishop. I forgot to mention it so perhaps you will not mind if I include it here" (V.50). His own feelings and judgements are not intrusive but are sometimes explicit—above all when, in a famous passage, he laments the death of young children from the plague: "And so we lost our little ones, who were so dear to us and sweet, whom we cherished in our bosoms and dandled in our arms, whom we had fed and nurtured with such loving care. As I write, I wipe away my tears..."(V.34). He goes on to quote Job. Gregory has the ability, like Herodotus, to annihilate historical time in contemplation of a common humanity. His anecdotes, typically several pages long, tend to be highly circumstantial and detailed, with invented or roughly recollected conversations, descriptions of what exactly people were doing at the moment of crisis, the prevailing weather, and the precise manner in which a murder or an assault was committed and the physical consequences. Only people's appearance is neglected. Apart from the required long hair, we have generally no idea what Gregory's Merovingian kings looked like, in contrast, for example, to Einhard's later description of Charlemagne; Einhard had learned from Suetonius. The larger context too, if there is one, is largely or wholly left to the reader to supply, while the agents, as we saw with Roccolen, are given little or no introduction. All this, of course, is consistent with a homiletic purpose. One can imagine some stories interpolated into a sermon for the moral, though Gregory is not generally an explicit moralizer: he is sometimes, in fact, remarkably abstemious in this respect.
This is episcopal history in a broad sense as well as at times a narrower one: Trollope with bloodshed. The more institutional concerns include disputed episcopal elections and a long account of a revolt in a nunnery, led by a daughter of King Charibert, who seems to exemplify the family traditions except that she does not actually murder anyone personally (though she does organize a gang of roughs who assault the prioress and fight the abbess's supporters in riots leading to deaths). The revolt is ended by violence, with rebellious nuns having their hair shorn—and, according to Gregory, in some cases also hands, ears and noses. It had been a vexatious affair, and he recounts it at considerable length (IX.38–43, X.15–17). This was exceptional, but disturbances of all kinds were the everyday currency of Gregory's life as a bishop. His tenth and last book ends with a list of the succession of the bishops of Tours, and brief descriptions of their deeds, before his final computation of the years (5,792) between the Creation and his own consecration as bishop in "the fifth year of Gregory, Pope of Rome, the thirty-third of King Guntram and the nineteenth of King Childebert II." The kings' names act as chronological markers, but it is the bishops who provide the list and the summarized deeds.
Medieval chronicles were frequently composite, one author, often anonymous, taking over from another to continue the record. Even where authors are identified, earlier ones are often epitomized (Orosius being a favourite, and later Isidore of Seville) as part of the compilation. Gregory, however, who seems to have an unusually acute sense of himself as author and of the integrity of his work, concludes with a plea that, while his successors are welcome to rewrite it in verse if they wish (so far as we know the offer was not taken up), they must above all leave the work in its entirety. He was fortunate: his work was highly popular, and was preserved and copied intact.
It was also, however, epitomized, together with a continuation of some seventy years after the end of Gregory's own work, in the chronicle which the Renaissance was to attribute to "Fredegar" the name has stuck. In all, albeit patchily, continuations took the chronicle into the Carolingian period, in the _Annals of the Realm of the Franks_ (AD 741–827). These continuations, including Fredegar, are quite properly described as chronicles of the Franks: there is, that is to say, a focus, above all, on the Frankish courts and their wars, as well as an increasing stress on the Franks as a people, with further embroideries of their mythological origins. Fredegar's is the first work in which we find the long-lived claim of a Trojan ancestry, which of course made them kin to the Romans. It is worth dwelling briefly on Fredegar's chronicle, because it is typical in a way that Gregory's work is not, and it brings out Gregory's distinctive characteristics.
After the usual world-historical preamble and chronology—he acknowledges Jerome and Isidore—and the résumé of Gregory's books, Fredegar shuttles between the various Merovingian courts, with news flashes of world events, sometimes obviously seen through a fog of rumour. He is much more thematically consistent than Gregory, though also generally terser and secular in his interests, with little on bishops or the supernatural. There is, however, one episode to which he gives particularly extended treatment, which is of exceptional interest. It concerns the expulsion from Burgundy in AD 610, by the Merovingian king Theuderic, of the Irish St. Columbanus, who, there and in Italy, exercised by his foundations an immense influence on the development of monasticism. Fredegar gives the domestic circumstances of the expulsion, instigated by the King's grandmother Brunechildis (Brunhild), who, not wishing to be outshone by a young queen, takes exception to Columbanus' admonition to the King to abandon his promiscuous ways and take a wife. The account of Columbanus hints warily at unused supernatural powers.
Brunechildis dominates part of Fredegar's narrative just as her sister-in-law and rival wicked queen, Fredegund, is ubiquitous in Gregory's last six books. Among the latter's numerous lurid crimes was the attempted assassination of Brunechildis, though it seems one of the more venial. Fredegund's exploits memorably include, in Gregory's account, slamming down the lid of a treasure chest on the neck of her daughter, who had annoyed her by referring to her low birth:
She leant on it with all her might and the edge of the chest pressed so hard against the girl's throat that her eyes were soon standing out of her head. One of the servant-girls who was in the room screamed out at the top of her voice: "Quick! Quick! Mistress is being choked to death by her mother." (IX.34)
After this, according to Gregory, relations between the two women deteriorated. Fredegund's son, King Chlotar, according to Fredegar, succeeded in catching Brunechildis, and after accusing her of the murder of no less than ten Frankish kings, and having her tortured, he had her led between the ranks of the soldiers on a camel and then tied to a wild horse, by which she was torn to pieces.
Fredegar is not often so colourful. Here he is, typically, in an example taken at random, at his most terse and annalistic:
In November this year [Fredegar dates by regnal years] Gundoald, supported by Mummolus and Desiderius, dared to invade part of Guntramn's kingdom and to destroy his cities. Guntramn sent to meet them an army under Leudegesil the constable and Aegyla the patrician. Gundoald took to flight and sought refuge in the city of Comminges, whence Duke Boso hurled him from the cliffs; and thus he died. (4.6)
Fredegar often breaks up narratives—of the pursuit of feuds, for example—for the requirements of strict chronology. As he tells us in his preface, "chronicle is a Greek word meaning in Latin the record of the years" (a definition from Isidore of Seville's _Etymologies_ ), and he takes the obligations it imposes seriously: "How this came about I shall set down in the right year under its proper sequence." His occasional brief interpolations or news flashes from contemporary world history—common in medieval chronicles—are sometimes startling, notably the report of the baptism of "the emperor of Persia" and sixty thousand of his subjects, followed by the conversion of all Persia to Christianity: it seems like an echo of the conversion of Clovis.
"Fredegar," whoever he was, has nothing like the anecdotal art and human curiosity of Gregory. Gregory himself is hardly a great historian: he is too episodic, too uninterested in generalization and context, and takes too much for granted. But, to use a metaphor understandably favoured by commentators, he opens a window on vivid, varied and animated scenes, at different social levels, in sixth-century Gaul—all domesticated and made personal. There is nothing else like it.
**FIFTEEN**
**Bede: The English Church and the English People**
Augustine, who was to be the first archbishop of Canterbury, was dispatched by Pope Gregory the Great on a mission to the English, and arrived in Kent in AD 597. On his way northward from Italy, he and his retinue of monks broke their journey at Tours, in 596. Gregory of Tours, who had died in 594, thus missed wishing them well by just two years. But for that, one would be able to speak literally of Augustine passing through Gregory's present into the past which was to engage the attention of the first English historian, Bede (who knew Gregory's work), over a century later. Bede's history begins with the dealings of the Romans with Britain, the early progress of Christianity there, and the establishment of the Saxon, Angle and Jutish invaders in the later fifth century. But for him the history of the English as a people really begins with Augustine's mission and the foundation of the English Church, and this is the true starting point of his _Ecclesiastical History of the English People_ ( _c._ 735).
Bede's history, like Gregory's—though it resembles it in little else—deals much with episcopacy, but whereas in Gaul the Christian episcopate was already established when the Franks arrived, and inherited much of the tradition of Roman rule in the towns, in England the kings of the various and shifting political entities into which the country was divided by the invaders were crucial to the success of the Christian missionary endeavour: their patronage almost guaranteed success; their hostility represented a severe setback. Bede is if anything even less interested than Gregory in secular history for its own sake, but kings play quite a different, and generally much more positive, role in his history. Bede also has a much more acute ethnic interest than Gregory: he distinguishes carefully between the various peoples who composed the population of Britain, and he has a clear conception, surprising perhaps in view of their political divisions, of the Anglo-Saxon people as a whole, as "the English."
Bede's work occupied an exceptionally distinguished place among the histories of the various barbarian successors to Roman rule in the West, which are not really exemplified in Gregory's subsequently mistitled _History of the Franks._ Apart from Gregory's work and Bede's, we have also the _History of the Deeds of the Goths_ by Jordanes (d. _c._ 554), a history of the Goths and Vandals by Isidore of Seville (d. 636), consisting mainly of extracts, and the _History of the Lombards_ by a Carolingian courtier and monk of Monte Cassino, Paul the Deacon (d. 799). The best account of the Huns comes from a Roman history by a fifth-century Greek, Priscus, whose work is largely lost but from which survives a fascinating description of a Roman embassy to Attila which Priscus accompanied. But the category of "barbarian" history proves on examination to be more complex and heterogeneous than it might seem. Apart from the question of quality, where Bede stands in a class of his own, the authors and their purposes were disparate; while the Goths and Lombards were Arian heretics, their historians were Catholics, believers in the Trinity, which set a distance between themselves and the peoples they considered and also gave them interests and concerns which transcended ethnicity. Jordanes, though apparently he had Gothic ancestry, is emphatically a man of the eastern Roman world (though he wrote in Latin) and an admirer of the contemporary emperor Justinian, while Paul belonged equally emphatically to the world of Charlemagne, conqueror of the Lombards.
Bede, however, is writing about his own people, though in Latin, and he has no sentiment on behalf of the Roman empire, only for the Roman Church. He is conscientious in retailing (not always accurately) the main episodes in Romano-British history, but then goes on to distinguish the origins of the invading tribes and their settlement as a patchwork of realms (he speaks of provinces). These were those of the South, East and West Saxons, of Kent (settled by the Jutes), of East Anglia, Northumbria and Mercia (the Midland kingdom), as well as those of minor or subordinate kinglets. But he is freed from any nostalgia for Roman Britain not only by his own race, but by his manifest contempt for the Britons, including the quality of their Christianity. Of the four peoples and five languages (the fifth being Latin) of Britain, he speaks warmly though not uncritically of the Irish monks established in northern Britain (he usually calls them Scots); the Picts are shadowy and the Britons disreputable; the English are a chosen people. The scholarship of the last couple of decades has impressively answered the obvious question why Bede is so confident that the English are one people, despite political multiplicity much greater than in Gregory's Gaul (which had a single ruling dynasty, though its sovereignty was often divided). One particular source of this idea, before Bede, seems to have been Pope Gregory, who in his letters speaks of the "gens Anglorum" (not "Saxonum") as an entity for conversion. Why Gregory preferred eponymous Angles to Saxons seems unclear: the story of the famous papal pun ("not Angles but Angels") provoked by the sight of boys, presumably slaves, in the marketplace, which Bede retails as "a story handed down to us by the tradition of our forebears" (II.1), embodies but does not explain the preference. What is absolutely clear, however, is that Gregory conceived from the outset that the Church to be established among the English was to be a single entity, with its primate at Canterbury, and that Bede was happy to follow him, not least in constant references to "the English people."
In a sense therefore, the English owe their existence as a people, or at least the recognition of it, to the papacy, and it was confirmed by Bede's history, which, in all Church matters—which form the core of his book—takes a wholly "Gregorian" point of view. The precondition of that existence is Catholic orthodoxy and unity in subordination to the see of Canterbury. Bede's book is the history of the achievement of that unity, through the manner of the Christian conversion of England and the establishment of its Church, over the century following Augustine's mission and prior to Bede's work. Bede is as hostile to heresy as Eusebius or Gregory—he mentions particularly Pelagianism, the British-born heresy that salvation was possible without grace, and Arianism—but they were not issues of his own time, any more than was paganism, at least among the educated and powerful. What exercises him is the dissidence of the Celtic, Irish strand of Christianity, established in much of Northumbria by the missionary efforts of Irish monks, over the date of the observance of Easter, and this is something to which he frequently reverts. Bede himself was an expert computator, so this was home territory for him, but what was most painful was the threat to unity.
Bede's history, then, is a history of the English people, but it is above all an ecclesiastical history. Kings, once converted, sometimes figure as model rulers with biblical overtones—partly, one assumes, _pour encourager les autres_ in Bede's own time. The situation of the English, as, following Bede, we may call them, was, after all, like that of the Israelites when, after defeating the original holders, they were settling their land. Bede had written a commentary on the books of Samuel and Kings, and was used to providing a providential as well as allegorical gloss on the events of apparently secular and tribal history; his Latin, which he wrote admirably, sometimes echoes the language of the Latin Vulgate Bible. The English conquest, as the chastisement of the unworthy Britons—a theme he gets from the British Jeremiah, Gildas, writing around the second quarter of the sixth century—is part of God's providential design. The English are therefore God's chosen instrument for punishment of sin.
Bede (born _c._ 673), who was entered into the recently founded monastery of Monkwearmouth, in Northumbria, at the age of seven, and spent his whole life there and in the nearby sister house of Jarrow, made himself into the most learned man in Europe, with an impressively varied list of authoritative writings to his credit, most copiously in biblical commentary, but also in the computation of time, and in hagiography, martyrology, hymnology and poetry. Besides the _Ecclesiastical History,_ he wrote a history of the abbots of his monastery. The precondition of his learning was his access to the library, remarkable for its time, collected mostly by his abbot and mentor Benedict Biscop on travels in Italy and Gaul. Bede lived and worked in remotest Northumbria, but it was in ecclesiastical terms a vibrant and sophisticated place; Monkwearmouth had over six hundred monks by AD 710. By that time England was itself sending missionaries to Germany.
The reader who for the first time encounters Book I of Bede's _History_ is likely to become aware of an authorial presence of great power and authority. Bede, one feels, is as reliable as he can be. This may be a slight exaggeration given the didactic moral purpose he has announced in his preface: to encourage good conduct by recording notable examples of goodness and wickedness. He is calm (except on heresy), measured, and apparently completely in control of his theme and his sources. He is generally chronologically lucid, being apparently the first author to date, as became normal, to and from the birth of Christ—an immense convenience. But he is also not slavishly tied to strict chronological sequence in the manner of an annalist, and he sees no objection to useful retrospect. He is scrupulous in giving his sources. Hagiography is prominent among them, and he records miracles with, presumably, a mixture of belief and homiletic purpose. As in Gregory, while some are genuinely "impossible," others could be put down to timely good fortune: seas calmed, winds opportunely changing, ailments recovered from. Bede's miracles always have a moral or spiritual point, most usually the triumph of Christianity over paganism; they are, then, weapons in the war of gods, and of course the pagan gods are demons. He never recounts the fantastic merely for its own sake. But Bede's _History_ is not only impressively controlled—at least until his own times, when it becomes more source-driven and less structured. It is also highly dramatic. The miraculous stories and the confrontations of Christian and pagan are brought vividly to life, as are the leading personalities, especially King Oswald, Bishop Aidan and St. Wilfrid. It is not surprising that Bede's work, now regarded as the masterpiece of early medieval historiography, soon obtained a very wide currency on the Continent as well as in England.
His preface is in the form of a letter to King Coelwulf of Northumbria, to whom he sends his work at the King's request. Coelwulf is said eagerly to desire "to know something of the doings and sayings of men of the past, and of famous men of our own race in particular." History, Bede tells us in a formula by now standard, as it had been from Roman times, gives us good and bad examples to emulate and avoid. Then, exceptionally, Bede lists his main sources, just as a modern historian might do. Above all there is Abbot Albinus, who has transmitted to him through Nothelm, "a priest of the church of London," ecclesiastical documents held at Canterbury. Nothelm, clearly a travelled man, has also himself been able to see and transcribe documents from the papal archives in Rome, including letters of Pope Gregory the Great, all of which he has made available to Bede, who also acknowledges help from bishops and monasteries in various parts of the country. For Northumbria itself he is not dependent on any particular source, but "on countless faithful witnesses," though he particularly acknowledges a debt to the Lindisfarne Life of St. Cuthbert, which he has used in his own life of the saint and partly reused in his history.
The work proper begins, as Gildas had done, with a brief but accurate geographical survey of the British Isles. It is enthusiastic, and has been thought to show deliberate resonances with the idea of Paradise, before the Fall. The narrative begins, as so many histories of Britain have done since, with the invasion of Julius Caesar, which Bede dates at 60 years before the Incarnation and 693 from the foundation of Rome, thereby correlating Christian and Roman chronology. Bede claims that stakes driven into the bed of the Thames as part of the Britons' defences are still (700 years later) to be seen, and he describes them. He knows about the later, Claudian, invasion, but makes a hash of dating and explaining the Roman wall (on which Gildas is also misled), which he must have known well, and relating it to the Roman earth-work linking the firths of Forth and Clyde. There is no Boudicca. Instead, apparently from the _Liber Pontificalis,_ there is a legendary British king, Lucius, who calls for Christian instruction in a letter to Pope Eleutherius (I.4). Persecutions of Christians in Britain under the empire are mentioned, particularly Diocletian's, and the death of the British martyr St. Alban (AD 301) is recounted in detail, with the miracles attending his execution (I.7). Though unsound on the wall, Bede makes interesting reference to other surviving physical evidence of the Roman occupation: cities, lighthouses, bridges and paved roads(I.11). The eventual arrival of St. Augustine's mission in Kent is the main event of Book I, but (citing Gildas by name as a source) Bede also devotes considerable attention to the cowardice, moral delinquency and spiritual inertia of the Britons: God justifiably punishes them, with the Saxons and other invaders as his chosen instruments (I.12–16). Bede is the first, but not the last, English historian to be anti-Briton.
In Bede's account of the invasions we first meet figures taken from Gildas and perhaps not entirely legendary, though it is impossible to tell. First of these is the British king Vortigern, who invites the incomers as military allies, which would have been common Roman practice; the "Angles or Saxons" arrive in three longships. (The Goths also, according to Jordanes, migrated in three ships.) There are the Saxon chiefs Hengist and Horsa, whose claimed descent from Woden Bede records without comment. (There is nothing odd about the claim, which is found also in Frankish royal genealogies, but one wonders if Bede knows who Woden was. It may seem implausible that he did not, though he takes little interest in the details of paganism; but since he regards the pagan deities as devils it seems equally unlikely that he would have made no comment if he did.) Then we have mention of the British war-leader Ambrosius Aurelianus, again from Gildas, and his victory over the invaders at the (unidentified) Badon Hill. (Subsequent commentators tried to link him with a much later arrival in literature, Arthur, king of the Britons.) Bede's discrimination of the tribes as Angles, Saxons and Jutes has stood up well to archaeological investigation, and some of their divisions are perpetuated, of course, in the names of English counties such as Sussex and Essex, though Wessex has slipped from administrative recognition into a literary conceit in the armoury of tourist boards.*1
For Bede, the chief hero of the conquest period, after St. Alban, is the Gallic saint St. Germanus of Auxerre (commemorated, like St. Alban, by a place name: St. Germans in Cornwall). Germanus comes to refute the Pelagian heresy rife among the Britons, and he does so—assisted by some notable miracles—despite the Devil's contriving a fall which breaks his leg; Bede compares him to Job (I.19). Germanus, undaunted by his injury, gamely insists on leading the British side to victory against a mixed pagan force of Picts and Saxons (I.20). But, despite this demonstration, the Britons are incorrigible, criminally failing in any missionary endeavour towards the newcomers. However, God "did not utterly abandon the people he had chosen," and thus instigated Pope Gregory to dispatch Augustine and his monks for their salvation (I.23). The foothold in Canterbury was allowed by the Kentish king Ethelbert, whose wife, Bertha, was a Christian Frank. But the fragility of a Christianizing enterprise dependent on dynastic politics is revealed when, in Kent, King Ethelbert, who had converted, is followed by apostate sons. Christianity was also insecurely established in Northumbria.
The initial conversion of Northumbria furnished Bede with several notable stories. One concerns the pagan high priest Coifi, who, after initially a somewhat pragmatic assessment of the Christian case ("If the gods had any power, they would surely have favoured myself, who have been...zealous in their service"), declares himself convinced and celebrates his laicization with the acquisition of arms, forbidden to a priest:
Thus equipped he set out to destroy the idols. Girded with a sword and with a spear in his hand, he mounted the king's stallion and rode up to the idols. When the crowd saw him they thought he had gone mad; but without hesitation, as soon as he reached the temple, he cast into it the spear that he carried and thus profaned it. Then, full of joy at his knowledge of the worship of the true God, he told his companions to set fire to the temple and its enclosures and destroy them. The site where these idols once stood is still shown, not far east of York, beyond the river Derwent, and is known today as Goodmanham. [It still is.] (II.14)
It has been pointed out that a spear was part of Woden's ritual, which Bede could scarcely have known, which seems to reinforce the story's authenticity.
A different style of argument, in the form of a parable, was provided by another of the Northumbrian chief men:
Your Majesty, when we compare the present life of man on earth with that time of which we have no knowledge, it seems to me like the flight of a single sparrow through the banqueting-hall where you are sitting at dinner on a winter's day with your thanes and counsellors. In the midst there is a comforting fire to warm the hall; outside the storms of winter rain or snow are raging. The sparrow flies swiftly in through one door of the hall, and out through another. While he is inside, he is safe from the winter storms, but after a few moments of comfort, he vanishes from sight into the wintry world from which he came. Even so, man appears on earth for a little while; but of what went before this life or of what follows, we know nothing. Therefore, if this new teaching has brought any more certain knowledge it seems only right that we should follow it. (II.13)
The vividness of the imagery of the warm and lighted hall, the centre of comfort and feasting, contrasted with the surrounding darkness and the uncharted wide world beyond, and the brevity of the sparrow's flight through it, has ensured this speech a deserved subsequent fame. It has the quality and the same kind of imagery and sensibility as that of Anglo-Saxon poetry in the vernacular:
_Bright were the buildings, halls where springs ran, high, horngabled, much throng noise; these many meadhalls men filled with loud cheerfulness: Wierd changed that._
("The Ruin")
_Where is the house of the feast? Where is the hall's uproar?_
_Alas, bright cup! Alas, burnished fighter!_
_Alas, proud prince! How that time has passed,_ _dark under night's helm, as though it had never been._
("The Wanderer")
Poetry was one of Bede's many interests and there is later (IV.24) a digression on Caedmon, a monk of Whitby, discovering his poetic gift, which Bede clearly greatly appreciates and regards as miraculous.
Perhaps a more typical insight into pagan mentalities, which Bede does not generally attempt to penetrate, is provided by the story of the three pagan sons of the Christian king of the East Saxons, who demand, though unbaptized, to be given the consecrated communion bread administered to their convert father. The bishop explains to them the indispensability of baptism, to no avail:
The bishop answered, "If you will be washed in the waters of salvation as your father was, you may share in the consecrated bread as he did: but so long as you reject the water of life, you are quite unfit to receive the bread of life." They retorted, "We refuse to enter that font and see no need for it; but we want to be strengthened with this bread."
The bishop stands firm, and is expelled from the kingdom (II.5).
Christianity flourishes in Northumbria, however, under King Oswald (604–42). He erects a cross which becomes a site of cult, and many miracles are performed there. Oswald is a Christian because, in exile, he has been instructed by the Scots (Irish), who at his request send him Bishop Aidan from Iona; Aidan establishes his see, under Oswald's patronage, on the island of Lindisfarne, off the coast of Northumbria. This leads Bede on to the missionary efforts of the Irish in North Britain, which have predated those from Canterbury and which he treats in a notable, retrospective chapter on the conversion of the Picts by St. Columba, in 565 (II.4). Bede speaks enthusiastically of the purity of life and love of God found in the Irish settlement of Iona, off the west coast of Scotland, but he deplores the mistaken view taken there on the date of Easter, which he attributes to the monks' isolation. Aidan, transferred to Northumbria but not being proficient in English, has King Oswald himself to translate for him. Bede is exceptionally warm in his commendation of Aidan's holy life and humility, which he contrasts with "the apathy of our own times" (III.5). We learn from Bede's account of him that monks in the Irish tradition preferred to walk rather than ride, keeping their pedestrian habits when becoming English bishops, who generally kept more state. King Oswald gives Aidan a fine horse, which Aidan gives to a beggar (III.14, also IV.3). Oswald reproaching him for this, Aidan reads him a lesson on the greater value of a human being compared with a horse. Oswald, moved by his humility, begs his forgiveness:
At the bishop's urgent request, the king sat down and began to be merry; but Aidan, on the contrary, grew so sad that he began to shed tears. His chaplain asked him, in his own language, which the king and his servants did not understand, why he wept. Aidan replied: "I know that the king will not live very long; for I have never before seen a humble king." (III.14)
There is elsewhere a story of a Christian king killed by his kinsmen for forgiving his enemies, i.e. renouncing the ethics of the feud (III.22). Oswald in fact dies in battle with the pagan King Penda of Mercia (642), who had previously put an end to Edwin, the first Christian king of Northumbria. Oswald's sanctity is vouched for by posthumous miracles.
Aidan too, while still alive, performs miracles. Bede's obituary for him shows again, touchingly, his mixed feelings about Celtic monasticism. Aidan cultivated peace and love, poverty and humility, and used his priestly authority to check the proud and powerful, comforting the sick and relieving the poor. "I greatly admire and love all these things about Aidan, because I have no doubt that they are pleasing to God: but I cannot approve or commend his failure to observe Easter at the proper time, whether he did it through ignorance of the canonical times or in deference to the customs of his own nation" (III.17). The wooden pillar against which Aidan leaned when he was dying was miraculously preserved in a fire; this motif of the wooden pillar has been seen as a folkloric one from the pagan cult of Thor.
Further successes for the Christian mission continue to be interleaved with descriptions of saintly lives in Bede's account. It is noticeable that Bede tends to prefer examples of the good for edifying purposes, while Gregory of Tours's preference—or perhaps just his experience—furnishes more of the opposite kind: it has been held more plausibly of Bede than of Gregory that he is an idealizer of earlier generations, in contrast with a less heroic and exalted present. The crux of his history is his account of the Synod of Whitby (AD 664), by which the vexatious question of the date of Easter is settled and the unity of the Church in Britain (except, of course, for the Britons) is secured. The synod was called and presided over by the Northumbrian king Oswy, another of Bede's model Christian kings, who followed the Irish method of computation, while his wife, instructed in Kentish (Roman) customs, observed the Roman one, so that one observed Lent while the other feasted—clearly an inconvenience, if no worse, in a royal household. At Whitby, for the first time, we meet the formidable figure of Wilfrid, a Northumbrian who has been to Rome and Gaul (we might be inclined to say Brussels) and has returned with strongly Roman (or Eurocratic) notions on Easter and the shape of the monastic tonsure; he emerges at Whitby as a spokesman for the Roman party.
_Pace_ Bede, the issue of the date of Easter was not really a matter of Irish barbarism and rusticity; the case of the basis for computation could be argued either way, but Wilfrid has a strong argument, which he exploits very effectively, in the isolation of the Irish compared with the breadth of consensus and the tradition of the Church arrayed against them.
Our Easter customs are those that we have seen universally observed in Rome, where the blessed Apostles Peter and Paul lived, taught, suffered, and are buried. We have also seen the same customs generally observed throughout Italy and Gaul when we travelled through these countries for study and prayer. Furthermore, we have learnt that Easter is observed by men of different nations and languages at one and the same time, in Africa, Asia, Egypt, Greece, and throughout the world wherever the Church of Christ has spread. The only people who stupidly contend against the whole world are these Scots [i.e. Irish] and their partners in obstinacy the Picts and Britons, who inhabit only a portion of these two uttermost islands of the ocean. (III.25)
Bede is scrupulous in summarizing the arguments, but a fuller account of the technical issue is provided later in a letter from Bede's abbot, Ceolfrid (V.21). King Oswy, in his summing-up, makes much of the authority of the See of St. Peter, and the power of the popes inherited from St. Peter to bind and loose, and he decides for the Roman position. Some of the Irish, unable to accept the new practice, leave Northumbria to return to Iona, with some of Aidan's bones. Bede again pays tribute to the austerity of life in the Irish monasteries and the devotion they receive from their people, contrasting it with the inferiority of his own time: "For in those days the sole concern of these teachers was to serve God, not the world, to satisfy the soul, not the belly" (III.26).
Bede's attitude to Wilfrid, with whose position he agreed, has been much discussed: the consensus seems to be that he respected but did not greatly like him, and gave him none of the affection he could not help feeling for Aidan. Wilfrid's stormy later career is alluded to by Bede: thanks to his dictatorial and uncompromising character, it included exile and even imprisonment before he was reinstated as a bishop, as well as missionary work in Sussex. Bede seems even to have glossed over some of its features—Wilfrid, for example, became extremely rich (IV.12–13, V.19).
After Whitby the Irish gradually fell into line everywhere (in the south of Ireland they had never been dissident, so the issue was not straightforwardly Celtic Christianity versus the rest). Bede cared deeply about the unity of the Church, and for him it was clearly a great moment when, in 716, even the intransigents in Iona accepted the Roman computation and tonsure:
This seemed to happen by a wonderful dispensation of God's grace, in order that the nation which had willingly and ungrudgingly laboured to communicate its own knowledge of God to the English nation might later, through the same English nation, arrive at a perfect way of life which they had not hitherto possessed. (V.22)
Earlier the high point of the history seems to have been reached at the opening of Book IV, when the new Archbishop of Canterbury, Theodore, dispatched by the Pope from Italy in 669, became "the first archbishop whom the whole Church of the English obeyed," thereby setting the seal on the period of conversion and the establishment of a united episcopacy in England in close communion with Rome. It is perhaps this sense of a mission accomplished, as well as the greater volume of recent and contemporary testimony reaching him from all over England within the period of living memory, which makes Bede's last two books markedly more hagiographical. There is, for example, no mention of secular affairs after around 690; there had no need to be.
It is in this section of the book (IV.27–32) that Bede incorporates material from his own _Life of St. Cuthbert,_ who became, reluctantly, bishop of Lindisfarne. Bede's accounts are artfully composed, and hagiography and its incidentals can tell historians of the period much about the mentality which produced it, but the ordinary modern reader of Bede's last two books is likely to be fairly easily sated with holy deaths, visions of heaven and hell, potent relics, uncorrupted and fragrant corpses, and even the setting to rights of a distressed horse (a posthumous miracle of St. Oswald) (III.9). The heroic age of conversion was over, but the individual heroism of the saints still did its work: the Church in England was doing well in its unceasing war with the demons.
Bede concludes, in 731, with a view of the present in Britain: a state of peace, secular and ecclesiastical (V.23). The Picts and Scots are Christian, docile and content. Only the Britons impotently nurse their resentments and preserve their own bad customs in respect to Easter. But they have—clearly for their own good—"been brought in part under the subjection of the English."
As was said earlier, Bede has throughout his history tended to prefer to recount good examples rather than bad, and he has none of Gregory of Tours's candour about ecclesiastical scandals—disciplinary matters were a bishop's business, hardly a monk's, except within his own monastery. If we looked only at the conclusion of the history we might think him complacent. In fact a letter written towards the end of his life shows him deeply disturbed by the lethargy, worldliness and wealth of the contemporary Church in England. Nonetheless, granted foresight as well as hindsight, he would have had to agree, as to some extent he clearly did, that he had lived in fortunate times: the battles with pagan idols and Celtic obstinacy had been won in the previous century; the end of his own was to see the first incursions of the Northmen.
**PART IV**
> **THE REVIVAL OF SECULAR HISTORY**
**SIXTEEN**
**Annals, Chronicles and History**
**Annals and Chronicles**
In 789 there is an ominous entry in the vernacular _Anglo-Saxon Chronicle:_
In this year King Brihtric married Offa's daughter Eadburh. And in his days there came for the first time three ships of Northmen and then the reeve rode to them and wished to force them to the king's residence, for he did not know what they were; and they slew him. Those were the first ships of Danish men which came to the land of the English.
Those three ships again! They were a portent of great damage to, among much else, the settled, scholarly monastic life that Bede had known, with its libraries and schools and riches. The Chronicle entries for the previous years record much internecine warfare; eventually the Danish invasions would play a part in the consolidation of a unitary kingdom, ruled, in the earlier eleventh century, by a Danish king (Cnut or Canute). Four years after the first entry on the Danes, the Chronicle records "the ravages of heathen men miserably destroyed God's church on Lindisfarne [Aidan's], with plunder and slaughter." The following year it was the turn of Bede's own monastery of Jarrow.
The bare entries of the vernacular Chronicle provide the record of these disastrous years. Bede's great achievement, popular and revered though it became—on the Continent as well as in England—had no remotely comparable successor for four hundred years. There was a hiatus, which _The Anglo-Saxon Chronicle,_ terse though its entries are, especially the earlier ones, was left to fill. It was a remarkable enterprise, and at this point one unique in the whole of Europe. The Chronicle seems to have been first compiled around the ninth century: the chronicler writing up 789 has hindsight. It was continued thereafter with copying and variations in a number of monasteries, so that the whole Chronicle runs from the landing of the Saxons in 494 to the end of the Anglo-Norman dynasty in 1154. Exactly when, where and why came the initiative for commencing it remains conjectural: Wessex, and a royal impetus, have been suggested; in its last years the centre of its production was Peterborough; King Alfred has been mentioned as its progenitor, and certainly with Alfred the entries become fuller and less purely annalistic, with campaign narratives extending over several paragraphs.
The differences between annals and chronicles, and between chronicles and histories, are a good example of quantitative change turning into qualitative. In individual cases, of course, and at the margins, the classification can become arguable, but the categories are pretty clear. _The Anglo-Saxon Chronicle_ offers, over time, an example of annals mutating into chronicle, just as some diaries are merely lists of engagements and others can be regarded as contributions to literature and history. If annals sometimes grew into chronicles, annals themselves seem often to have grown from calendars, kept, of necessity, in monasteries, chiefly for calculating each year the date of Easter. A diversity of matter—portents, weather, local, national and even international events—might then be integrated or else inscribed in the margin. Deaths and successions of popes, bishops, abbots and kings were of particular interest, as were battles, fires and other disasters. For a typical example, chosen at random, we may take a thirteenth-century entry in the long-sustained Chronicle of Bury St. Edmunds. It was, of course, originally in Latin:
1239. William Raleigh was elected Bishop of Norwich on 10th April. That horrible race of men known as the Tartars, which had once come swarming from remote fastnesses and overrun the face of the earth, laid waste Hungary and the neighbouring territories. On 18th June Eleanor, queen of England, gave birth to her eldest son Edward. His father was Henry...
The rest is Henry's genealogy back to Alfred, and we are given a cross-reference to an earlier genealogy for Alfred back to Adam. It is a rich entry: the Mongol (Tartar) invasions, the birth of Edward I, and the election of an East Anglian bishop. We would, of course, be much mistaken in expecting narrative; there are no thematic connections between the entries, except the birth and the genealogy. We should think instead of a newspaper whose timescale is the year, not the day. We are ourselves unperturbed by the most diverse news stories appearing in juxtaposition, with the conventional division into panels.
History as a genre, though often attending to the passage of the years as most classical historians had done, characteristically involves extended narrative, relevant circumstantial detail, and thematic coherence; the recording of facts is dictated by thematic, dramatic and explanatory considerations, rather than just chronological juxtaposition and convention. These distinctions were recognized in the twelfth century by Gervase of Canterbury. We may say that, as types, annals are disconnected, chronicles are episodic, history is ideally continuous; particular classifications can sometimes be disputable.
Though annals continued to be kept, chronicles and histories take us to the twelfth century. But before considering them we cannot ignore what flows in, given a chance, to supply the place of history for the next three hundred years: legend. The Britons, or Welsh, though they had a rich bardic literature, had so far as we know produced only two short and fragmentary histories between the fifth and the twelfth centuries, and certainly nothing like so coherent, authoritative, substantial and respected a work as Bede's. In the twelfth century an attempt was made to give them one: _The History of the Kings of Britain,_ by a secular clerk, apparently based in Oxford, Geoffrey of Monmouth. With a problematic relation to possible earlier writings and to oral tradition, Geoffrey is full and circumstantial, above all on the dark years of the fifth century when even Bede is largely terse or silent: the era of the Saxon invasions. We can, in fact, no longer postpone speaking of Arthur, the great British hero king, as Alfred, three centuries later, became the English one. But while the contemporary narrative sources for Alfred—chiefly the vernacular Chronicle and Asser's Life of Alfred, whose authorship and date have been much debated—are meagre (the story of the cakes is a twelfth-century addition), knowledge of Arthur kept getting fuller and fuller. Alfred was to remain of interest, but from the twelfth to the seventeenth century, despite dissentient voices, there is no doubt who was the hero king enshrined in the national collective memory: it was Arthur.
**Pseudo-History: Geoffrey of Monmouth**
So far as we know, Arthur first appears in historical writing in the ninth century, a hundred years before Alfred, but already four centuries after he was supposed to have lived—which is a problem. The fifth-century British monk Gildas, who wrote of the Saxon invasions and who lived close to the time of Arthur's supposed flourishing, does not mention him, though as we have seen he does mention a Romano-British leader or _dux_ called Ambrosius Aurelianus, who defeats the Saxons in a great battle, and Bede copies him. Arthur does not appear under his own name until the _History of the Britons_ ( _c._ 830) by Nennius, who wrote in Latin but clearly knew Welsh and Welsh genealogies and traditions. Yet in Nennius Arthur, although a war-leader who wins battles against the Saxons, is still scarcely more than a name. It was Geoffrey of Monmouth, three centuries later still, who, from whatever legendary scraps or nothing but the name, launched Arthur on his career as great king and national hero, holder of a renowned court to which knights flocked.
Arthur's historical credentials, in fact, are thin almost to vanishing point, but his legend, like the stories of the Greek heroes and the _Iliad,_ and Aeneas and Romulus in Rome, is a historical fact of a kind—and an important one, in the sense that it powerfully influenced or even dominated the picture many people in Britain, including the English, had of their past, particularly from the twelfth to the seventeenth century. Then, perhaps with more sensitivity to the difference between history and legend but with a willing suspension of disbelief, he figures notably in the nineteenth. After Geoffrey, the remains of Arthur and his queen, Guinevere, were "discovered" by the monks of Glastonbury in 1191, and Arthur and his knights became central to the emerging ethos of chivalry, with a number of sub-legends, worked up in France and Germany, to inspire the creation of medieval orders of chivalry which aped Arthur's Round Table, like the fourteenth-century Order of the Garter created by Edward III. Preceded by a number of medieval romances, Sir Thomas Malory's _Le Morte d'Arthur_ in the next century gave classic shape to the legends of Arthur for English readers; it was one of the first books selected for publication on Caxton's new printing press. In the sixteenth century Arthurian prototypes were a cult at the Tudor court; the Tudors were sometimes self-consciously Welsh, and Henry VII's first son was christened Arthur. Milton, in the next century, meditated an epic of Arthur before settling for Satan. Renaissance scholarship did some damage to Arthur's historicity, but, as we have said, from the seventeenth to the nineteenth century Alfred and the Saxons enjoyed a renewed vogue, largely for political reasons. The Victorian imagination divided its allegiances between the gentlemanly and romantic Arthur and the more populist image of Alfred; it is apt that the greatest Victorian poet to exploit the legends of Arthur was christened Alfred.
Without Geoffrey of Monmouth, the legends of Arthur—if they can be imagined—would have called for only a passing mention here at the most. Geoffrey's work, written in the 1130s, not only launched the Arthurian legend but did it in a manner which claimed, and in a number of respects plausibly claimed, to be history. The interest of Geoffrey's work is not exhausted by consideration of whether it has any factual basis. Geoffrey clearly knew what his contemporaries expected a history to be like, and was talented enough, free apparently from any danger of allowing his narrative to be dominated by its sources, amply to give it to them. Nennius' work alone could never have launched the legend: it is too genealogical, fragmented, incoherent and in many places bare—more a compilation than a history. It speaks more fully of Merlin than Arthur, and it also gives a more extensive account than Bede of the British king Vortigern and his uncontrollable Saxon allies Hengist and Horsa, all of which Geoffrey borrows and elaborates further. Geoffrey, though he understandably retails genealogies, remedies all Nennius' defects.
Geoffrey does not claim to be the author, but only to have translated into Latin from the Welsh "a certain very old book" shown him by Walter, archdeacon of Oxford. Whether such a book, for which there is virtually no independent evidence, existed and, if so, what it may have contained are interesting but not crucial questions. If Geoffrey was only the translator, the problem of the sources for his assertions is merely transferred to the old book, though if this did exist the question of how old it was, i.e. how much nearer the alleged time of Arthur, would become important. The consensus seems to be that, though there may be traces of Welsh legend and genealogy, partly from oral tradition, in Geoffrey's work, it is essentially his creation. If it were true, it would certainly be a very remarkable part of what has been called the twelfth-century historiographical renaissance—remarkable not only for the important gap it purports to fill in historical knowledge of Britain, but also for the accomplished and assured manner of Geoffrey's narration. In one sense it deserved the popularity it achieved. It has, however, no known sources apart from some well-known (and some garbled) fragments of Roman history, and items from Gildas, Bede and Nennius. Much more damagingly, some of what it describes at length, notably Arthur's great victorious battle with the Roman procurator Lucius Hiburnus in Gaul, left no trace in Roman records and historiography: so great an event could not have been overlooked in a highly literate and historically self-conscious society.
About a fifth of Geoffrey's _History_ concerns Arthur, though this proportion rises to a third if one includes Merlin, who, apart from arranging Arthur's conception, plays no part in the later story. The book begins with a foundation myth, along the lines of the _Aeneid._ Brutus, great-grandson of Aeneas (with a genealogy back to Noah), whom Gregory culled from Nennius, after somewhat Virgilian wanderings arrives with some Trojan followers and becomes the eponymous founder of Britain, at that time (as Bede says) called Albion, and inhabited only by giants. The most formidable of these, Gogmagog, is hurled into the sea, apparently off Plymouth, by the eponymous founder of Cornwall, Corineus. Geoffrey is fond of such etymologies; his greatest coup of this kind is the British king Lud (London, Ludgate). Geoffrey's work, though well constructed, is not evenly paced. What are sometimes little more than king lists are interspersed with extended narratives. Some of the more perfunctory kings have subsequently had niches created for them for reasons generally incidental to Geoffrey's purpose and chiefly on account of the memorability of their names: Gorbaduc (subject of one of the earliest English tragedies) and Hudibras (the subject of a seventeenth-century satirical poem) both have walk-on parts in Spenser's _Faerie Queene;_ there is King Coel—neither old nor merry; Cymbeline is unaccompanied by Shakespeare's Iachimo and Imogen; but King Leir appears with his whole family, with their usual names, and his story is as Shakespeare tells it, but with a happy ending—Cordelia becomes queen of Britain.
In Book VIII (of twelve) we arrive at familiar legendary territory with Merlin. His greatest feat, apart from contriving the conception of Arthur by some shape-changing work on Arthur's father, Uther Pendragon, is to transport the Giants' Ring (Stonehenge) from Ireland to Wiltshire as a burial place for British kings. Geoffrey, who is rather chary of the supernatural, makes the process sound more like engineering than magic: the King's men, understandably, are making a mess of it when Merlin arrives like a clerk of works who knows his business. Merlin's forte is actually prophecies, and Geoffrey provides them in Book VII, which was written independently and produced first as "The Prophecies of Merlin." It had a large vogue, not least because it hints at a future resurgence of the British or Welsh. Though there is hardly any room for doubt that it was seriously intended, it is so easily parodied that for the modern reader it is inescapably funny, not least in the bathos of the combination of high-flying animal symbolism with the homely names of the English counties: "A Wolf will act as standard-bearer and lead the troops and it will coil its tail around Cornwall. A soldier in a chariot will resist the Wolf and transform the Cornish people into a Boar." Significant beasts abound: the Ass of Wickedness, a Hedgehog loaded with apples, a bad-tempered Mountain Bull, the Dragon of Worcester, the Boar of Totnes, the Adder of London, and more. One has the impression Geoffrey could do this kind of thing in his sleep. Merlin tells Vortigern, perhaps redundantly for the reader, that the Red Dragon standing for Britain will be overrun by the White Dragon of the Saxons whom the King has unwisely welcomed, but "the race that is oppressed shall prevail in the end" it is a rather rare glimpse of history from a British perspective, the opposite of Bede's.
Geoffrey is rather more circumspect with the complementary later legend that Arthur is not dead but will return, as in the German myth of the Sleeping Kaiser. Geoffrey's account of Arthur's end is slightly ambiguous. After his last great battle with Mordred, Arthur is said to be "mortally wounded" he is not said to have died, but is reported to have been taken to the Isle of Avalon, mythically a place of healing, though later identified as Glastonbury, "so that his wounds might be attended to."
When Geoffrey was writing, the ideas of chivalry, which were to do so much to amplify and shape the Arthurian legend, and derived so much from it, had not assumed their later predominance. Apart from some exotica we will consider in a moment, his book is chiefly an apparently reliable account of Dark Age power politics. Arthur is a great king and commander, in the moulds of Caesar and Charlemagne, rather than the later knight errant. His wife is Guinevere, who is seduced by Mordred, Arthur's nephew, but there is no Lancelot and so no ideal of courtly love, which was soon to attach itself firmly to him. There is also no Holy Grail. Gawaine, Kay and Bedevere are there, and Arthur's sword is recognizable in its guise of Caliburn—though his spear, which, rather surprisingly to the modern reader, is called Ron, is not. There is no sword-in-the-stone. Arthur's court at Caerleon-on-Usk (Book IX) is of incomparable splendour, and sets the fashion for knights everywhere. Arthur also holds court in Paris after conquering Gaul. He additionally conquers the Picts, Norwegians and Danes (there seems to be some reminiscence of Canute ruling a great northern empire). Arthur is on the point of marching on Rome when he is recalled to deal with Mordred's treachery.
Arthur sometimes displays individual prowess, notably in giant-slaying (X.3, 4), and he measures the relative strength of the giants he has killed like a connoisseur. One, anthropophagous, giant has taken residence at the top of Mont Saint-Michel; another has a cloak made of the beards of the men he has killed. Such exploits read like intrusions from the world of _Beowulf_ into the mode of Roman history which Geoffrey seems to be adopting for Arthur's campaigns; they may stem from separate, pre-existing legends adapted by Geoffrey. But meeting a challenge from Lucius Hiburnus, the Roman procurator in Gaul, Arthur seems to slip easily into Roman habits himself. He raises an army of 180,300 men—excluding foot soldiers, whom Geoffrey says were not easy to count. Rome musters a world-representative army of 400,160 men. In the ensuing battle, preceded by generals' speeches in classical fashion, the armies stood face to face
with javelins raised...As soon as they heard the sound of battle-trumpets, the legion commanded by the King of Spain and Lucius Catellus charged boldly at the division led by the King of Scotland and the Duke of Cornwall, but the latter stood firm, shoulder to shoulder, and the Roman force was not able to break it. As the Roman legion persisted in its fierce attack, the division commanded by Gerin and Duke Boso moved up at the double. The Roman legion fought bravely, as has already been said, but this fresh division attacked it with a sudden cavalry charge, broke through and came into contact with the legion which the King of the Parthians was directing against the division of Aschil, King of the Danes...
and so on (X.9–12).
It is very convincing, apart from the numbers perhaps. If only it convinced. Making allowance for the fact that we are reading Geoffrey in a modern translation, it is instructive to compare his with the account of the same battle (Sassy) by Malory in his _Le Morte d'Arthur_ (1469), three centuries later:
Then the battles [divisions] approached and shove and shouted on both sides, and great strokes were smitten on both sides, many men overthrown, hurt and slain; and great valiances, prowesses and appertyces [conspicuous deeds] of war were that day showed, which were over long to recount and noble feats of every man, for they should contain an whole volume. But in especial, King Arthur rode in the battle...(V.viii)
Arthur kills Lucius with his own hand. Geoffrey, in neoclassical mode, has room for troop dispositions and tactics, despite his reluctance to count foot soldiers. Malory's account is of a chivalric rough-and-tumble, brought to an end by single combat.
Geoffrey has a number of devices for conveying authenticity, some of them subtle. For example, when Leir's grandson by Regan becomes king, "In his time it rained blood for three days and men died from the flies which swarmed"—a nice _faux-naïf_ chronicler's touch. Nennius had a rain of blood, like many historians before him, including Livy, but no flies. Geoffrey also produces presumably spurious charters, but this was a common skill. He also makes chronological cross-references between biblical, Roman and his own British history, in the manner pioneered by Eusebius. Isaiah was prophesying while the British flies were swarming; Cordelia was coeval with Romulus and Remus, and Brutus the Trojan with Eli the judge in Israel.
What is one to make of all this: of Geoffrey's intentions and the spirit in which he wrote? They are not perhaps quite the same thing. British patriotism and a drive for ecclesiastical preferment—he tried two dedicatees—seem sufficient motive, but how did Geoffrey regard his own creation, as we assume it to be? It is perhaps not wrong, and certainly tempting, to use the word hoax, but whether Geoffrey was in his own mind a hoaxer seems more problematic. He was a parodist of near-genius, and a considerable imaginative writer. How far he may have convinced himself that what he wrote is what the records would have said if they were fuller, or existed, seems impossible to say. It has been suggested that his reputation would stand higher if he had presented his work openly as a literary epic, like the _Aeneid._ Perhaps. The notion of Geoffrey delighting in his own mischief tends to appeal so much to the modern sensibility, with its taste for literary hoaxes, concealed jokes, parody and the confounding of the genres of fact and fiction, that we should no doubt be wary of it. Geoffrey's skill and success may even seem to reinforce a postmodern scepticism about the truth claims of history: it looks like history, it smells like history, so why isn't it history? In fact—blessed word!—it suggests the opposite of scepticism. In the Renaissance the exposure of the so-called Donation of Constantine, conferring on the Pope the succession to the sovereignty of the Roman emperors in the West, gave rise to, or reinforced, scepticism about the documentary grounds of all history. But this was the wrong inference: the point was that it had been possible to demonstrate that the Donation was a forgery. Similarly with Geoffrey's work. It is not the same as that of the chroniclers who were his contemporaries, however ungainly, gullible, careless or biased they may sometimes have been, and we know why it is not: the concept of fraudulence logically presupposes that of authenticity. William of Newburgh, half a century after Geoffrey, said that _The History of the Kings of Britain_ was all made up, "either from an inordinate love of lying, or for the sake of pleasing the Britons." Apart from Geoffrey's obvious quest for preferment, these are pretty much the alternatives. It is time to return to the terra firma of twelfth-century chronicle and history, even if it conceals pitfalls.
**Secular History and Chronicle: William of Malmesbury's _Modern History_ and the Scurrilities of Matthew Paris**
_The Anglo-Saxon Chronicle_ was unique in its time, and so, in his way, was Geoffrey of Monmouth. But chronicles were produced all over Europe in the Middle Ages. In France, as in England, a centralizing monarchy gave a focus for secular chronicles and created the possibility of a kind of national history. This was represented, above all, in France by the _Grandes Chroniques de France,_ issued in the vernacular with royal sponsorship from the thirteenth century. To avoid giving a lifeless and perfunctory survey, however, it is necessary here to narrow the perspective to a single country. Accessibility to an English-speaking readership, if nothing else, suggests that the focus shall be England, which fortunately had a rich chronicle literature, of which a proportion has been translated from the Latin.
By the twelfth century the island of Britain had undergone five major transformations, all of them the result of incursions from the European mainland: invasion and conquest by the Romans, the Saxons, the Danes and the Normans, and conversion to Catholic Christianity, from Ireland and from Rome. Each had left a residue in the records: in Caesar and Tacitus, in Gildas and Bede, in _The Anglo-Saxon Chronicle,_ and, most recently, in the late-eleventh-and twelfth-century accounts of the campaign of Hastings and the Norman Conquest. In the early twelfth century, when first-class chroniclers began to proliferate, the Conquest was only just receding from living memory, and the Anglo-Norman chroniclers included it as a matter of course. Some even seem to have been spurred to write by a divided inheritance and a sense of the massiveness of the transformation effected in the previous generation. For actual contemporary accounts we are less well off. The best prose account is that of the chronicler William of Poitiers, who is a eulogist of Duke William of Normandy. But in many respects the best record is a unique document, the Bayeux Tapestry.
Historically, the Tapestry covers much the same ground as William of Poitiers. It is still visible in its 230 foot by 91/2 inch glory and splendidly displayed in the town of Bayeux where it was in all probability made, at the order of its bishop, William the Conqueror's brother, Odo of Bayeux. It is also a text, and not even in the fashionably metaphorical sense, for it carries a verbal narrative in Latin running across it. As an object of beauty, it is enhanced by the narrow horizontal bands, top and bottom, like marginal annotations, which contain rather heraldic-looking birds and beasts, and scenes of ordinary lives and deaths, realistically depicted.
The narrative itself begins with the dispatch to Normandy of Earl Harold of Wessex, on an embassy to Duke William from King Edward the Confessor. Harold is shipwrecked and captured by a local magnate, but is freed by William. They go campaigning together, and then Harold swears on holy relics his fealty to the Duke. Back in England, Edward dies and Harold, ignoring his oath to William, takes the crown. William prepares an army, lands in Sussex, and defeats the Saxons in battle; Harold dies with apparently—it has been disputed—an arrow in his eye. There are vivid scenes of contemporary life throughout: sea voyages; shipbuilding; hunting; porters with their burdens; soldiers foraging, preparing food, building a wooden fortress, and setting fire to a house from which a woman and her child are seen emerging. In the margins, the birds and beasts give way to small archers shooting and soldiers mutilated, dead, unceremoniously stripped of their armour. It is a dazzling and humane document, comparable in historical importance to Trajan's Column, but easier to inspect.
_The Anglo-Saxon Chronicle,_ as one would expect, is woeful about the Conquest and treats it, as Gildas had done the Saxon one, as a punishment for sin. Then and later the Chronicle is more than usually sensitive to the sufferings of ordinary people. The slightly later Anglo-Norman chroniclers Orderic Vitalis, Robert of Jumièges, Henry of Huntingdon and William of Malmesbury borrow from the Chronicle (as well as from Geoffrey of Monmouth) and also, not improperly of course, from each other. Orderic and William were of mixed parentage, but all the chronicles treat Harold with respect. It is impossible to do justice to any of them by flitting between them. If a choice has to be made—and it has—it is the work of William of Malmesbury which suggests itself most strongly.
William, as his name implies, was a monk of Malmesbury Abbey in Wiltshire, of which he was librarian; like a number of monastic writers, he was the author of a history of his own abbey, as well as of Glastonbury. His _oeuvre,_ though varied in the manner to which we have become accustomed, shows an exceptional concern for orderliness and thematic choice. Hence in his two largest works he treats secular and ecclesiastical history separately, in the _Deeds of the Kings of the English_ ( _Gesta Regum Anglorum_ ) and the _Gesta Pontificum Anglorum._ The latter is a kind of continuation of Bede. In the _Deeds of the Kings_ he laments the long hiatus in the recording of the English past, except in the Chronicle(i.e. _The Anglo-Saxon Chronicle_ ) and declares it to be shameful. He also denounces what he calls the lies of the Britons. The most easily summarizable of his works, which here will have to stand, imperfectly, for the whole, is his work of contemporary history, the _Historia Novella_ (i.e. "Modern History"), which covers the years 1126–42, and is brought to an end by his death.
His preface to the _Deeds of the Kings,_ however, explains his approach to the historian's task and sources. He has, he says, no warrant for the truth of "long past transactions but the consonance of the times...Whatever I have recorded of later times, I have either myself seen, or heard from credible authority." More unusually, he is self-conscious about historical narrative and the need for thematic coherence. On Alfred, for example, he says, "To trace in detail the mazy labyrinth of his labours was never my design," for to give a recitation of his exploits "in their exact order of time would confuse the reader" William considerately summarizes. In his account of the Conquest—not very different from others—he takes on the responsibility of assessing its consequences, portraying the Saxon and Norman characters (respectively feckless and dissipated and calculating and frugal), and focusing particularly on the state of the Church at the time of the Conquest (bad). The point is not the adequacy of William's assessment, but his making the attempt. In the _Modern History_ too he is prepared to defy strict chronology and "to make into a parcel, as it were, the main points scattered through my text bearing on the conduct of Robert, earl of Gloucester, King Henry's son, and to present them in a recapitulation for the reader to evaluate." The Earl of Gloucester was William's dedicatee and patron and also one of the dedicatees of Geoffrey of Monmouth's _History._
As narrative, the _Modern History_ begins with the ending of Henry I's reign and the disputed succession which follows. Henry's only legitimate son had died in the fateful sinking in the Channel of the vessel called the _White Ship_ (1120). William laments this in the _Deeds of the Kings_ : "No ship was ever productive of so much misery for England, none was ever so widely celebrated throughout the world." The events of the _Modern History_ are those of the troubled reign (1135–54) of King Stephen and the civil war produced by the claim of Henry's daughter Matilda to the crown. Henry had made the nobles swear fealty to Matilda, but after his death many of them supported Stephen. It has to be admitted that William's promised recapitulation of the last part of King Henry's reign seems dispensable, concerning itself with the shamefulness of the fashion of long hair for men, a territorial dispute between the bishops of Llandaff and St. David's, a disputed papal election, and a cattle plague: William here seems to have stepped into chronicle territory. But the rest of the _Modern History_ is a fairly tautly constructed monograph on the civil war which follows the seizure of the throne by Stephen, count of Blois, aided by his brother the Bishop of Winchester, whom William knows. William addresses his monograph to posterity, in the preface to Part III, reverting to his point that the absence of a record is a disgrace: "In the year of the Lord's Incarnation 1142, I am undertaking to unravel the trackless maze of events and occurrences that befell in England, with the aim that posterity should not be ignorant of these matters through our lack of care" the chief lesson to be learned is the changeability of fortune, the mutability of the human lot.
For a comparable monograph on contemporary history one really has to go back to Roman times; William, like most medieval authors, was aware of Sallust and Livy, as well as knowing Virgil, Juvenal, Cicero and other classical writers. The story he has to tell is nothing less than that of the great crisis of the hitherto strong Anglo-Norman state; William points the contrast with the peace and security of Henry I's reign. _The Anglo-Saxon Chronicle,_ on which William draws in the _Deeds of the Kings,_ was eloquent on the sufferings of the populace as anarchy ensues and rival barons turn into no better than bandits:
1137. They oppressed the wretched people of the country severely with castle-building. When the castles were built they filled them with devils and wicked men. Then, both by night and day, they took those people they thought had any goods—men and women—and put them in prison and tortured them with indescribable torture to extort gold and silver.
Then the chronicler describes the torture. The same entry provides the famous remark that men "said openly that Christ and his saints were asleep." Then, in the manner of chroniclers, the subject switches to the building works of an abbot of Peterborough—the Chronicle was by now being produced there—and the alleged torture and blasphemous crucifixion by the Jews of Norwich of the child afterwards hailed as St. William of Norwich. William of Malmesbury echoes the Chronicle on the suffering created by the civil war, but his forte is really high politics. Castle-building figures in the _Modern History_ too, but chiefly as one of the stumbling blocks to peace. Bishops, presumably moved by the desire for security, have been building castles, contrary to canon law, and the King, violating the immunities of the clergy, has had some of them imprisoned. Both sides had a case, and William is good at presenting such dilemmas.
Although a monk, William was clearly also a man of the world, with acquaintances and sources among the great; he evidently went to a good deal of pains to inform himself of the inside of events. He is close to his patron, Matilda's illegitimate brother, the Earl of Gloucester, and also to the papal legate, Henry of Winchester. He also knows well Henry I's chancellor, a man of great power in the previous reign, Roger, bishop of Salisbury, and the local oppressor of William's abbey. Roger was, William says, "conscious of his power and abused God's indulgence rather more than was proper for such a man," while greatly glorifying his see of Salisbury. His attempt on the independence of Malmesbury Abbey is turned by William into an elegant epigram. The Bishop had also tried to elevate his own priory of Sherborne into an abbey: "Roger tried to turn abbeys into a bishopric, the property of a bishopric into an abbey." (It reads better in Latin: "abbatias in episcopatum, res episcopatus in abbatiam." One feels like applauding.) The Bishop, under the new regime, falls from his high estate. William says, "While to many he seemed pitiable, few pitied him," adding "which I myself am sorry for" (II.33).
The fortunes of the civil war fluctuate; Stephen, genial and kindly, is also weak; Matilda is headstrong and uncooperative (I.14–15). Allegiances are tested, repudiated, sway this way and that. William largely bypasses the military events and concentrates on the attempts at reconciliation, notably in the two councils called in 1139 and 1141 by the Bishop of Winchester, the papal legate, and the reasons for their failure. William attended the first of the councils. He clearly prefers Stephen personally, but blames him for his violation of the charter of liberties of the Church, to which he had agreed, and which William transcribes. Reckless men take advantage of Stephen's weakness, some coming as mercenaries from overseas to enjoy the pickings (I.18). In King Henry's reign, William says, foreigners came to England as refugees from strife at home; now they come for plunder (II.36). The unspoken threat behind the proceedings of the council is Stephen's possible excommunication for arresting bishops, but matters do not proceed so far—not only, William says, for lack of direct papal authorization, but also for fear of more violence (presumably against the Church) in England. William, one feels, understands the motives of public men biding their time, doing, he says, not what they would but what they could. The interests of the Church are clearly close to his heart, but he is not excessively judgmental, nor particularly cynical, though he clearly disapproves of Matilda—"this virago"—his patron Robert of Gloucester's half-sister.
William knows of the second legatine council only by report. He laments that nothing had been achieved: "So all this year [1141], whose tragedies I have briefly related, was ill-omened and almost mortal for England, which after thinking that it might now in some sort draw a breath of freedom, fell back again into misery, and thus, unless God's mercy sends a remedy, it will long remain" (III.59). The central event of this time was the escape of Matilda from Oxford, where she had been besieged. Other writers sensed the picturesque opportunity—she had been lowered from the walls before escaping across the snow. William does not. The outlines of the eventual solution to the conflict—Stephen's enjoyment of the crown for life and the succession of Matilda's son Henry, the future Henry II—are adumbrated in William's text, but he was prevented by death from seeing it implemented, so his history ends abruptly.
Since William mentions 1141 as ill-omened, it is perhaps worth repeating an event he records of the previous year: an eclipse of the sun on 20 March 1140, which, he says, "was thought and said by many to presage disaster for King Stephen [he lost a battle]." William adds that, at Malmesbury, "at first men sitting at table...feared the primeval chaos; then, hearing what it was, they went out and saw the stars around the sun" (II.38). "Then, hearing what it was"—an eclipse was still an omen, but it was also clearly reassuring to find it a recognized astronomical event.
William does not mute his criticisms of the great, but he speaks, generally, with decorum. Even the word "virago" applied to Matilda, subsequently having only unfavourable connotations, could at least in classical Latin sometimes mean "heroine" as well as "mannish woman." The great exception is made for Robert Fitz-Hubert, "the cruellest of all men and also a blasphemer against God." Robert has prisoners smeared with honey and then put in the glare of the sun to be tormented by insects (II.39). We are all glad when he is hanged. We may feel that William is sometimes discreet, but contemporary history, as he says, is particularly tricky. He is less inhibited in the _Deeds of the Kings,_ sometimes offering vivid physical descriptions of kings in the manner of Suetonius and Einhard. William Rufus, for example, is florid (of course), pot-bellied and powerful, with differently coloured eyes. He is also bad. William of Malmesbury is also perhaps something of a snob. Some Londoners—a portent—came to the second Winchester council to make the case for King Stephen. The Bishop of Winchester describes them as "magnates," because they have come from so great a city. William seems to regard them as gatecrashers; they claim to represent "what they call the commune of London" (III.49)—William's "quam vocant" has the effect of a raised eyebrow. The first London Chronicle would be produced half a century later, in 1188.
"Discreet" is not a word that could ever be applied to our next protagonist, Matthew Paris: he is populist, scathing, cynical, violently partisan, prejudiced and funny. Aware that his chronicle, of which his own manuscript copy survives, was full of indiscretions, he obviously tried subsequently to erase some of the most egregious. It was wasted labour. Not only were the erasures inefficient, the indiscretions were too many for him—bowdlerizing Matthew Paris was like trying to de-vein Gorgonzola. He is not a historian but a chronicler, but his episodic and heterogeneous _Greater Chronicle_ is vivid, entertaining, and held together by a highly personal view of the world.
Matthew was born soon after 1200, and became a monk of the Benedictine abbey of St. Albans. He compiled, as was common, a general history of the world from the Creation, incorporating the work of his predecessor in St. Albans, Roger of Wendover, narrowing down to 1259. Matthew's own chronicle covers something over two decades, but it is still very large, including European as well as English events. Recognizing this, he epitomized himself, twice over, into a shorter chronicle on English history alone, and then an abbreviation of that. He also produced an appendix of documents, the _Liber Additamentorum._ Matthew's material remained heterogeneous, but the new orderliness was at work in him also. He wrote hagiographies, including a life of St. Alban, in Anglo-Norman verse, and so obviously intended for the laity and women, but his _Greater Chronicle_ is almost entirely mundane, though not unecclesiastical, in its interests. The eminent medievalist V. H. Galbraith called it "the high-water mark of medieval historical writing in England" ( _Kings and Chroniclers,_ 1982).
Because the chronicle form, to which Matthew remained wedded, does not allow for extended narrative development, an attempt to give a flavour of the _Greater Chronicle_ is bound to be largely a recital of Matthew's prejudices and snorts of indignation and disgust. Fortunately they are entertaining. His sympathies are popular in so far as they are not purely Benedictine, and he is consistently against authority—papal, royal and episcopal—especially when it attempts to exact money and ignores chartered and customary rights, particularly those of monasteries. He is sensitive to public opinion, and treats it, perhaps for the first time, as something to be reckoned with. Hostility to Henry III's foreign favourites was a theme of the time, and Matthew joins in enthusiastically, referring to "hungry foreign nobles...with empty stomachs and open mouths gaping for the king's money" (27); the last is a favourite image with him, and is applied also to the Pope.
Periodically Matthew reports briefly but sympathetically on Parliament's remonstrations against royal exactions and foreigners. His reporting is always colloquial and down to earth: the king "lost his temper and said to his counsellors, 'It is your fault that the magnates have been alienated from me. Look! I am on the point of losing Gascony and have been robbed of Poitou. My treasury is empty! What can I do?'" (64). King Henry is not only rapacious, but also parsimonious and without dignity. He takes the cross (i.e. vows himself to a crusade) merely as a money-raising device. He attempts to establish his own temporary fair in London, for the sake of the tolls, prohibiting other retail sales for its duration to the detriment of the tradesmen; the site is plagued with wind, tearing at the awnings, and rain, so that the merchants were "cold, wet, hungry and thirsty...Their feet were dirtied by the mud and their merchandise spoilt by rain" (70). Matthew, a century after William of Malmesbury, writes with sympathy of the opposition of the mayor and commune of London to the King, though in monastic chronicles generally references to citizens and townspeople and their fractiousness seem to be disapproving. Matthew is not only hostile to foreigners and exacting great overlords: he has a Benedictine prejudice against the new mendicant orders, and is cynical about the motives of those who congregate in the new universities: "The world is now become elated with pride and despises the religion of the cloister," adding characteristically, "aiming at despoiling the monks of their property" (110).
Except when he quotes papal and episcopal letters, which are naturally formal, none of the great are allowed to retain their dignity. Matthew describes with relish the behaviour of the Archbishop of Canterbury, Boniface of Savoy, the King's uncle. In 1250, visiting the church of St. Bartholomew in London, and finding the canons showing a spirit of independence, Boniface falls into a rage, rushes at the sub-prior, and "vigorously [strikes] that holy man, a priest and a monk, with his fist, as he [stands] in the middle of the church, truculently repeating the blows, now on the aged face, now on his grey-haired head, and yelling 'This is how English traitors should be dealt with'" (148). He causes serious injuries, and the citizens of London, understandably enraged in their turn, cause a tumult. We seem to be back in the world of Gregory of Tours—or perhaps we have never really left it. Secular ecclesiastics get drunk and vomit; friars dress like lords and are used by the Pope as smooth-tongued extortionists (8)—Matthew disapproves of religious wandering about. Matthew has been called an early constitutionalist. This is not exactly wrong, but over-theoretical. He stands, certainly, for established rights of all kinds, though especially monastic ones; he is certainly sympathetic to the baronial "opposition" and to the citizens of London, and is unimpressed by great autocrats. He sometimes reminds one of a modern tabloid editor: disrespectful, populist, xenophobic and anti-intellectual.
He was also, uniquely, a talented illustrator of his own work. His marginal illustrations are not cartoons, though they have something of their quality, but elegant, attractive, sometimes quaint coloured drawings of relevant objects and scenes, as well as purely decorative heraldic shields. Of course they do not have the delicacy and richness of some illustrated missals, Gospels and books of hours. They are like Matthew's writing, lively and demotic and sometimes macabre: a pillory, a purse (relevantly), a Franciscan, a Jew, maltreated prisoners, hangings, maps (including the Holy Land, with camel), a galley, a battle, a shipwreck, a collapsing building illustrating that old standby of the chronicler an earthquake, and martyrdoms (St. Alban, Becket). Most unusual is the town band of Cremona, mounted, surprisingly, on an elephant; the city is relevant because of the ignominious part it played in the Italian resistance to the emperor Frederick Barbarossa, whose deeds seem to fascinate Matthew; they were the subject of a contemporary work by Otto of Freising. To read Matthew in an illustrated edition, with its birds, shields, satirical drawings and the bric-a-brac of contemporary life, adds considerably to the pleasure.
**Two Abbey Chronicles: St. Albans and Bury St. Edmunds**
Matthew Paris also wrote, in another common monastic genre, a "Deeds of the Abbots" for his own house, St. Albans. In his preamble, he provides for what was essentially conventual local history the same kind of rationale as was standard for national histories, with a twist at the end. It was done
so that neither their good deeds nor indeed their bad ones will perish in the future times through oblivion and by this means not only will people now and in the future be incited to do good, but evil people will be deterred by fear of scandal. Furthermore, if any secular or ecclesiastical person has piously conferred benefits on this church, not only his name but the benefit itself...will be perpetuated without a thread of falsity.
Keeping a record of benefactions, not only those in kind but those in land and privileges, was a motive to recording the history of the abbey, as it was to the forging of charters. It gave greater security of possession of donations, which had often been made in a more customary and less documentary age and therefore lacked what was now increasingly required in cases of dispute, a written record. It also acted as a kind of inventory, and hence a guard against misappropriation.
Matthew's writing, as his frank prefatory remarks suggest, loses nothing of its saltiness and suspicion of authority by being turned inward: he is no more a respecter of abbots per se than of kings and archbishops. Hagiographer though he also is, and staunch defender of conventual rights, he is here easily recognizable as the author of the _Greater Chronicle._ His interests are ecclesiastical and mundane. The abbey suffers, for example, from that perennial menace the dodgy builder, who gives "treacherous advice to add unnecessary, futile and excessively expensive ornaments," so that the abbot loses his nerve, the work is abandoned, and a wall falls down. To pay for the completion, the abbot sends "a certain cleric called Amphibalus, whom the Lord had raised from the dead on the fourth day through the good offices of St. Alban and St. Amphibalus," on a preaching tour with the saints' relics, to raise money (15). Matthew seems to record this simply as a fund-raising technique, with no other comment. But the work still languishes. Eventually, however, a refectory, a dormitory and privies are completed. St. Alban, as usual, can look after his own. When the abbey is threatened with losing a lawsuit, as a result of a charter forged by a treacherous monk who has been bribed by the other side, the traitor is discovered and exiled to the daughter house of Tynemouth, which seems, perhaps for climatic reasons, to have functioned as a sort of penitentiary. There, drunk and gorged, he meets his deserts by dying in the privy, calling out "Take him, Satan!" There is a similar death in Gregory of Tours, perhaps in imitation of that of the heresiarch Arius, who was said to have died voiding his bowels. Of course people do die there. In the St. Albans case, Matthew says that there were doubts whether the deceased merited Christian burial, but, to avoid a scandal, the brethren "kept silent about many things" (18).
An abbey was a polity in miniature, in which endless friction and disputes were produced by the often uncertain division of power and customary rights between kings, bishops—we have seen the Bishop of Salisbury oppressing Malmesbury Abbey—abbots, conventual officials and the assisting cloister monks, and the abbey's tenants. The abbot was an autocrat, but in times of vacancy the monks became (under restrictions) an electorate, with the right of proposing candidates for royal ratification. The monk they were electing would be a great man, a magnate of the kingdom. Matthew gives a detailed account of the proper procedures for elections, and also of the proper conduct of abbots-elect. Conflicts with ecclesiastical and lay neighbours on land and rights of jurisdiction, use and tolls were fertile in lawsuits and even, on occasion, violence—usually confined to the abbey's servants, though not always. The lay enemy of St. Albans, who had the charter forged, disputed the abbot's right to depose the prior of a church of which he himself was patron. He flourished the same or another charter, but also, less subtly, laid siege to the church as if it were a castle, threatening to castrate the new prior and the monks with him, who were reduced, according to Matthew, to the last stages of thirst and starvation. The abbey appealed to King John, who comes rather well out of the story, though we are told he hated the Earl. Exclaiming "Ho! By God's feet who has heard of such things!," he intimidates the Earl's men into abandoning the siege (20).
Matthew offers all kinds of information on the internal affairs of the monastery, from the commissioning of paintings for the church and the acquisition of books for the library to the withdrawal of a customary allowance which had become used for boozing. The death of Abbot John, who had panicked over the affair of the overcharging builder, is recounted by Matthew not only piously but touchingly. One of John's accomplishments had been that he was "an incomparable judge of urine" (30), but his failing eyesight meant that he was unable to use this diagnostic skill on himself. The deaths and solemn obsequies of abbots were great occasions. Abbot John's farewell to his monks is genuinely moving, though Matthew also characteristically records his transgressions as he had promised, naming his flatterers and deploring his tyrannical use of exile to remote priories or "cells" of the monastery as a punishment. The monks seek and obtain a charter prohibiting this from the next abbot, but he breaks it; one of the victims weeps and begs to remain at home among his brethren. Exile could in some cases apparently mean solitary confinement. It is another charge against the new abbot, William of Trumpington—a young man—that he prefers to dine with laymen rather than in "the friendly society of the cloister" (49). Matthew is also free with his criticisms of the abbey's officials.
Abbot William, in other respects, seems not a bad sort, and among other benefactions he performs a highly meritorious act in obtaining—we are not told how—a rib of St. Wulfstan from the Bishop of Worcester (49). Matthew, again according to his undertaking, is highly interested in recording acquisitions and embellishments, by commission or donation—images, paintings, the beautification of the altars, the strengthening of the church roof with lead. Abbots and monks, as well as laypeople, are among the donors. Matthew pays generous tribute to the painters and craftsmen responsible for the work, as well as to the quality of the gifts, though he also has to record the reprehensible and shocking theft, twice, of the Eucharist and its holy and jewelled gold and silver vessels from the church. At one point, recording a lay benefaction, Matthew gives what seems a premonitory glimpse of countless future entries in countless parish magazines, recording innumerable gifts of stoles, altar cloths and hassocks, piously fringed, crocheted and embroidered: "Mistress Alice, the daughter of Henry Cocus, bequeathed a red silk chasuble, nicely worked with gold, to this altar" (50).
St. Albans was a notable centre for the production of chronicles. We have already seen that Matthew continued the chronicle, from the Creation, of Roger of Wendover. The consecutive St. Albans Chronicle continues to the year 1440—more than two hundred years in the writing. Galbraith, in fact, speaks of "the St. Albans school of history"—Thomas of Walsingham, a St. Albans monk, was a notable chronicler of the Lancastrian period. Other monastic centres of chronicle production were Worcester, Canterbury, Durham and Peterborough. One such centre, however, Bury St. Edmunds, produced, like Matthew in the midst of the St. Albans Chronicle, a gem among the prevailing worthy mediocrity, though one, unlike Matthew's, focused only on the affairs of the monastery.
At the end of the eleventh century the abbey of St. Edmund had produced a Miracles of their saint, murdered by the Danes in 876, whose body it possessed. The document also traced the history of the abbey up to the dedication of its new church in 1094, as well as its running dispute over its autonomy with the Bishop of Thetford (a see later transferred to Norwich). As seems not uncommon, hagiography led to local history, just as the latter sometimes incorporated national history through outside pressure on the abbey's immunities, particularly from the King and particularly financial. Bury also produced an architectural history of the abbey and a continuation of its chronicle up to the second half of the thirteenth century, one of the notable events in which is a revolt by the young townsmen against the abbey in 1264. It also refers to the much more serious revolt in Norwich in 1272, in which the cathedral was set on fire and thirty or so servants of the monks were dragged out and murdered; the Bury chronicle estimates the number of citizens involved at over thirty thousand, including many women.
The gem of the Bury sequence, however, is the famous Chronicle of Jocelin of Brakelonde, which was given a permanent place in English literature when, soon after its publication by the Camden Society in 1840, Thomas Carlyle adopted it to stand for the past in admonitory contrast to the present in his tract for the times _Past and Present_ (1842). It is to Carlyle's credit that he so early recognized Jocelin's quality, though his condescension is at times irritating. Jocelin's chronicle covers the last decades of the twelfth century (Henry II, Richard I) and continues into the early thirteenth, in the reign of King John. It begins, domestically, in the time of Abbot Hugh, who is old and feeble and who has let things get out of hand. Hence, though it is called a chronicle, Jocelin's work exhibits one of the archetypal themes of historical narrative: the aged ruler, chaos, a deliverer, and renovation. The abbot, and under him his deputy the prior, should have been responsible for maintaining all St. Edmund's rights and liberties against its great neighbours, and against the King and the Pope, as well as ultimately for internal discipline. However, the abbey is going to ruin, piling up debts through its officers, chiefly the sacrist and the cellarer, who are out of control. The sacrist was responsible for the services of the church and its fabric, and for collecting dues from the townspeople, who were all tenants of the abbey. The cellarer was responsible for the purchase of provisions for the monks and guests, and for collecting revenues, in dues and kind, from the abbey's lands. The sacrist and cellarer were thus both responsible for raising and spending money (the distinction is therefore not the same as that between bursar and steward, though it has some similarities). There were ample opportunities for both to get into more debt, and they had done so, apparently from laxity and incompetence rather than peculation, each going his own way and becoming ever more hopelessly entangled with interest payments and further debts.
The old abbot dies, and there is an interregnum; the monks regularly pray for a new abbot—though, Jocelin ironically comments, if some had known who it was to be they "would not have prayed so devoutly" (11). Jocelin's description of the ensuing election is a superb piece of writing, retailing with humour and immense vitality, with much use of direct speech, the whispering and lobbying and the airing of old resentments and prejudices and new anxieties and calculations as the monks approach their great decision. The election opens up divisions between the old and the young, the learned and the unlettered, and through Jocelin we can see suddenly revealed the qualities sought in an abbot and the jealousies, resentments and apprehensions by which the monks are swayed, as well as their individual dispositions: timid, proud, even cynical. Intellectual snobbery is countered by anti-intellectualism. One says, "Abbot Ording was a good man and ruled the house wisely" an unlearned man might make a good abbot, "though he is not so perfect a philosopher as some others."
To this another made answer, "How may this be? How can he, a man who has no knowledge of letters, preach a sermon in Chapter, or on feast days to the people? How shall he who does not understand the Scriptures have knowledge how to bind and loose, seeing that 'the rule of souls is the art of arts and the science of sciences'? God forbid that a dumb image should be set up in the Church of St. Edmund, where it is known that there are many men of learning and of understanding." Again another said of yet another, "That brother is literate, eloquent and prudent, strict in his observance of the Rule; he has greatly loved the Convent, and has endured many ills for the possessions of the Church; he is fitting to be made Abbot." And another replied, "From all good clerks, O Lord deliver us; that it may please Thee to preserve us from all vexatious Norfolk disputants, we beseech Thee to hear us."
Jocelin himself supports the intellectual side, though in his subsequent account he seems to reproach himself for youthful priggishness:
And I indeed, being a young man, "understood as a child and spoke as a child," and I said that I would not agree to any man being made Abbot, unless he knew something of dialectic and could distinguish between false argument and true. And another, who thought himself wise, said, "May God Almighty give us for our shepherd one who is a fool and ignorant, so that he will have to ask us to help him!" (12)
Jocelin's inexperience in elections leads him into an embarrassing indiscretion:
On one occasion I could not contain my spirit, but blurted out what I thought, thinking that I spoke to faithful ears, and I said that a certain brother was unworthy to be Abbot, though he had loved me and conferred many benefits upon me; and said I thought another worthy to be Abbot, and named a man whom I loved less. I spake as my conscience bade me, considering the common good rather than my own advancement, and I spoke the truth, as the sequel proved. And behold! one of the sons of Belial revealed what I had said to my benefactor and friend; for which cause to this day I have never either by prayer or gift been able to recover his favour to the full...and, if I live long enough to see the abbacy vacant once again, I shall take care what I say.
The argument sways this way and that. Some take counsel of their fears: a man apparently humble "while he is in cloister, yet, if he chances to hold any office, he is apt to be disdainful, scorning monks and loving men of the world more than he should." (This was also a theme in Matthew Paris.) Another candidate simply had a speech impediment.
Abbot Samson is elected, and, after an anxious wait, the King's assent is given. Samson, former sub-sacrist, becomes, by the suffrage of his fellow monks, a great man in the kingdom; the abbey dominates not only the town but its surrounding area, and is a direct tenant of the Crown, like the great lay nobles. As in Matthew's chronicle, much solemn attention is given to the embalming and funeral of the new abbot's predecessor, as well as to the installation of the successful candidate, when over a thousand people dine afterwards. Jocelin becomes the abbot's chaplain, and hence gets to know him intimately, and he shares this intimacy with the reader. Samson is a strong man, autocratic in his ways and a firm ruler, but also attractively human. He causes resentment by not having favourites, even from his former supporters. It is also resented when, clearly not trusting the sacrist and cellarer to do their jobs properly, he installs his own nominees—secular clerks, not monks—at their elbows, to see that the accounts are kept in proper order.
There were, as ever, ample causes of dispute, both internal and external, and Jocelin's chronicle is largely a record of them, great and small: with neighbouring lay magnates, with the Archbishop of Canterbury over jurisdiction, with the Bishop of Ely over timber, and with the townspeople over customary rights and dues, over dung and buildings, over an unauthorized mill (which Samson has demolished) and over catches of eels. Samson is in dispute with King Richard over the disposal of a ward of St. Edmund whom the King wishes to marry to his own nominee. Samson stands firm, and it is the King who blinks first, subsequently behaving with some magnanimity. Samson has wrought a great renovation in the abbey's fortunes, but his relations with the monks become tense over such issues as the damage done by the abbot's fishpond to the monks' meadow, over dues from cows' pasture, and over the division of costs of hospitality—over whose guests were the abbot's charge and whose the monastery's. The cellarer was, as always, in the thick of things. Samson generally carries matters with a high hand, but he feels the cares of office and confides in Jocelin that he would have preferred to be a humble schoolmaster, which he had been before taking the habit, or a minor official of the monastery, rather than an abbot. There is a very touching scene when some of the opposition, in fear of the abbot's powers, capitulate, and Samson, victorious, dissolves publicly into tears at the acrimony which has invaded the house. The brethren, themselves now moved, weep also, and both sides exchange the kiss of peace.
The great set-piece description, over many pages, is of the opening of St. Edmund's shrine and the reverent unwrapping of the body, following a fire which required renovation work. It is clearly a huge privilege for some of the monks to see and even touch their patron, and it causes some envy. Twelve monks, including the highest officers, were chosen.
So, while the convent slept, these twelve put on albs and, drawing forth the coffin from the feretory [shrine], they carried it and setting it on a table near the ancient place of the feretory, they made ready to remove the lid, which was attached and fastened to the coffin by sixteen very long iron nails...The Abbot, therefore, standing close by, looked within and found first of all a silken cloth veiling the whole body, and after that a linen cloth of wondrous whiteness: and over the head was a small linen cloth, and beneath it a small cloth of silk, finely woven, like a nun's veil. And afterwards they found the body wrapped in a linen cloth, and then at last the lineaments of the holy body were revealed. Here the Abbot stopped, saying that he did not dare go further, to see the sacred flesh unclothed. Therefore taking the head between his hands, he said groaning, "Glorious Martyr, Saint Edmund, blessed be the hour when thou wast born! Glorious Martyr, turn not to my perdition this my boldness, that I, a miserable sinner, now touch thee; thou knowest my devotion, thou knowest my intent." And he proceeded to touch the eyes and the nose, which was very large and prominent, and afterwards he touched the breast and arms and, raising the left hand, he touched the fingers and placed his fingers between the fingers of the saint; and going further he found the feet turned stiffly upwards as of a man dead that self-same day, and he touched the toes of the feet and counted them as he touched them. (113–14)
After this high point there are further wrangles and reconciliations; the life of the monastery goes on. The last word, however, deserves to go to King John. After his coronation he visits the monastery, "led thither by devotion and a vow that he had made." Unfortunately, John's devotion comes cheap. "We indeed believed that he would make some great oblation; but he offered nothing save a single silken cloth which his servant had borrowed from our Sacrist—and they have not yet paid the price. And he had received the hospitality of St. Edmund at great cost to the Abbey, and when he departed he gave nothing at all to the honour or advantage of the Saint save some thirteen pence sterling..." (116–17).
St. Edmund deserves to become the patron saint of all those who entertain royalty.
**SEVENTEEN**
**Crusader History and Chivalric History: Villehardouin and Froissart**
**Villehardouin's _The Conquest of Constantinople_**
When, in 1187, Abbot Samson of Bury St. Edmunds heard that the Crusader kingdom (Outremer) had lost Jerusalem to the Muslims, after eighty-seven years of Christian occupation, he put on a hair shirt and drawers and abstained from meat. Jerusalem had been the great prize of the First Crusade. Inevitably the Crusades, as the foreign events most exciting to Christians, found a place in chronicles. William of Malmesbury, in the _Deeds of the Kings,_ gave considerable attention to Pope Urban II's proclamation of the First Crusade at the Council of Clermont in 1095, including documentation, and then took his readers on an excursus through Rome, Constantinople and Jerusalem, with descriptions. Matthew Paris, a century later, gave intermittent attention—which is the attention he gave everything—to the crusading of the German emperor Frederick II, whose career and dubious reputation (he was excommunicate when setting out on crusade) clearly fascinated him.
From 1096, when the first Christian forces assembled, the Crusades to the East covered a period of three centuries, and they not surprisingly affected European imaginations and sensibilities. As we shall see shortly, they also revived the earlier very common historical genre of the campaign monograph. The last example—by the Byzantine historian Procopius in the work entitled _The Wars_ —had dealt with a campaign in Europe, mounted from Byzantium by the emperor Justinian to reconquer the Gothic kingdom of Italy for the Roman empire in the first half of the sixth century. (Procopius' other well-known work, _The Secret History,_ is chiefly a hysterical and obsessive account of the sexual practices of the empress Theodora.) The Crusades were the first sustained effort of European expansion eastward, other than missionary endeavour (including Charlemagne's forcible conversion of the Saxons), since the time of Julian the Apostate's campaign in Persia in AD 363.
A crusade was, by definition, a pilgrimage, and a crusader acquired the merit and privileges of a pilgrim, including the remission of sins. Only a pope could grant this, so only a pope could proclaim a crusade, just as each individual could only become a crusader by taking a vow. There were to be many crusades, and not just in Asia Minor. They became a papal weapon in Europe too—against the Muslims in Spain, the pagans on the Baltic seaboard, the Albigensian heretics in southern France and northern Italy, and the popes' political opponents. They answered Stalin's later query "How many divisions has the Pope?" Given a crusade, a good many—but only intermittently and temporarily. They were also like missiles, once launched almost impossible for him to control, as was vividly exemplified by the Fourth Crusade, which set out ostensibly for Jerusalem and ended by conquering Constantinople and much of the Greek Christian empire.
The background to Pope Urban's innovation in preaching the First Crusade was the increasing harassment of Christian pilgrims to the Holy Land by its Muslim rulers and the crumbling of Byzantine military resistance to the Turks, which led to calls for help. The motives of some of the crusaders seem to have included land-grabbing by themselves from the outset, and the setting-out of a crusade also tended to be associated with pogroms against European Jews—classed with Muslims as common enemies of Christ. Crusades also tended, not without reason, to be seen by the Byzantines in the light of invasions rather than assistance. Clashes and bloodshed were not uncommon, partly because the First Crusade, in particular—it might be better to speak of a flow of bands and individuals eastward rather than of a single entity—was badly organized and provisioned. Jerusalem fell to the crusaders, however—in 1099, to the accompaniment of sack and massacre—and a Christian king of Jerusalem was elected, though little more than first among equals.
There are several contemporary eyewitness accounts of the First Crusade. One of the most interesting, because of its perspective more than its intrinsic merits, is that by the daughter of the Byzantine emperor Alexius I, the _Alexiad_ of Anna Comnena. It is a eulogy of her father, but it also provides an insight into Byzantine attitudes to the crusaders—numerous as the stars in heaven or the sands of the seashore—whom she calls Franks (the usual name), barbarians and also, most frequently, Kelts. Her animus against them, whom she calls treacherous and greedy, is understandable. One of the leaders, Bohemond of Taranto, had only recently been waging a war against the Greeks in Albania from his base in southern Italy. To Anna he was an unmitigated villain. Other leaders she refuses to name: the syllables of the names are too barbarous (10.x). The Pope, Urban's predecessor Gregory VII, is "the abominable pope" (1.xiii). These judgements are understandable, but, though her work is well organized, they largely rob her of the historian's first requisite, curiosity, just as her portrait of her father the Emperor is repetitively eulogistic. It is a pity that the very accomplished and perceptive court history of the Byzantine statesman Michael Psellus ends in 1078, a decade before the crusaders began to arrive. Anna does, however, note the crusaders' self-destructive impetuosity: "Kelts are indomitable in the opening cavalry charge, but afterwards, because of the weight of their armour and their own passionate nature and recklessness, it is actually very easy to beat them." "The Keltic race...combines an independent spirit and impudence, not to mention an absolute refusal to cultivate a disciplined art of war" (11.vi). These comments were to remain valid down to the last, fatal, charge of the Frankish knights at the battle of Nicopolis in 1396, where their hopes were finally extinguished. The crusader settlements, amid a sea of hostile Muslims and Greeks, never attracted the Latin, Catholic immigration which could alone have secured them. Crusading was at times impressive, and to Byzantines and Muslims threatening, but it was always too intermittent, the enthusiasm too short-lived (as in a sense it was intended to be: the crusader's vow did not include settlement) for long-term security.
The most accessible account of the Fourth Crusade, which had been proclaimed by Pope Innocent III in 1198, is the work unabashedly and accurately called _The Conquest of Constantinople,_ by Geoffroy de Villehardouin, who was a major participant in it. The eyewitness work produced by the Sixth, a generation later, the _Life of St. Louis_ by Jean de Joinville, has a vividness and humanity that Villehardouin lacks and is more enjoyable to read, but it is more personal memoir and biography than a general history, and the abortive campaign of the French king Louis IX in North Africa cannot rival in interest the extraordinary crusader attack on Constantinople. Villehardouin was in one sense superbly placed to write the history of the Crusade in which he played a prominent part. He accompanied it throughout, and was high in its councils, a number of times acting as envoy and negotiator—to the Venetians who carried the expedition in their ships, to the Greeks, and between warring lords among the crusaders themselves once they had carved out their territories. Only the Muslims lay outside his range, and indeed they play virtually no part in his book. (Alexius I had actually negotiated an alliance with them in fear of the crusaders.)
But his book suffers from the same handicaps as Anna Comnena's, whose prejudices, without knowing it, he echoes and reverses: Greeks are treacherous and not to be trusted. He is so identified with the strategic decision taken by the main body of the crusaders—essentially to support the Venetians in their war of acquisition of Christian territory across the Adriatic, and then to attack Constantinople—that opponents of it—like the Greeks who, essentially, are being invaded—are denied the exercise of historical understanding or even curiosity: they are simply saboteurs. He was, as one of the envoys sent by the intending crusaders to arrange transportation with the Venetians, actually an architect of the course they subsequently followed. Over-optimistic about the numbers to be given passage, the envoys contracted a huge debt to Venice, which the crusaders, who had to meet their own expenses, were unable or unwilling to pay. They therefore became subservient to Venetian designs on the Byzantine empire, the Venetians being bent at the very least on extracting trading privileges.
Some crusaders, understandably, wanted to keep clear of Venice, to cross the Adriatic from the Norman French base in southern Italy and to make their way directly to the coast of Syria. It was a legitimate point of view, but Villehardouin will have none of it, and even the predicament with which his book suddenly ends in 1207 (we do not know what happened to him), with the crusader lords fighting to defend their lands newly acquired from the Greek empire against the pressure of the Christian King Johanitza of Bulgaria and Wallachia, seems to give him no second thoughts. Nor, for that matter, does the crusaders' very thorough and destructive sack and devastation of Constantinople in 1204, and the division of its booty. He seems a classic example of the general—he was marshal of Champagne and later of "Romania," the Latin conquests—whose conviction that his course was the right one overrides any concession even of goodwill to his opponents in strategy. He continually lumps together the diversity of dissident views as belonging to "those who wished to break up the army," and he makes little distinction between those who went back to Europe, who may indeed have been seeking a way out, and those who went on independently to Syria, "where they could do no good." His self-serving account sometimes reminds one of Josephus, but the latter is incomparably the better narrator and analyst. For a military leader, Villehardouin can be unhelpful on tactics, with "as God willed it" functioning as a frequent all-purpose explanation of success or failure. He sometimes refers to knights having won honour and distinction, in the manner of heralds reporting on battles and tournaments, and he records important deaths, but shaming seems to be an important part of his motive for recording. His lists of named backsliders are much longer than those of the meritorious; the former are always, in his unwearied litany, "those who wished to break up the army."
Villehardouin's history is written in the vernacular and in prose, though the vernacular was more often used for verse. It is a history, not a chronicle: a controlled monograph, beginning at an appropriate place, the preaching of the Crusade, rather than with the creation of Adam, and presented as a coherent continuous narrative. In that respect it recalls a number of ancient historians, though Villehardouin falls a good way short of them in execution, and the recollection seems to be ours rather than his. He has apparently no awareness of predecessors; his book lacks the conventional preface and its promises, and he has no literary airs, which is rather a relief after the agonizingly literary self-consciousness of Anna Comnena. In fact it is doubtful whether he could read or write, so his work was probably dictated.
In spite of the absence of a preface, his motives in composing it, however unselfconscious, are not hard to make out. He knows that the fall of Constantinople to the crusaders, in which he has played a part, was an extraordinary event—he calls it an achievement. In the absence of a preface there are no customary claims for the greatness of the events recounted, but he is fully aware of them: of the splendour of the city and the enormity of the crusaders' undertaking, even though it was ostensibly initially to restore an heir who had been deposed from the imperial throne. He is several times overcome by the size and grandeur of the fleet which carries the crusaders (though it was smaller than it should have been). He is not generally a picturesque writer, but the disembarkation arouses him to a kind of celebration: "a most marvellous sight; knights and sergeants [men-at-arms] swarming out of the warships, numbers of sturdy war-horses taken out of the transports, countless fine tents and pavilions unloaded ready to pitch" (46). As for Constantinople itself, those who have not seen it cannot imagine that so fine a place exists. The crusaders
noted the high walls and lofty towers encircling it, and its rich palaces and tall churches, of which there were so many that no one would have believed it to be true if he had not seen it with his own eyes, and viewed the length and breadth of that city which reigns supreme over all others. There was indeed no man so brave and daring that his flesh did not shudder at the sight...Never before had so grand an enterprise been carried out by any people since the creation of the world. (59)
A description of the booty taken by the crusaders arouses him to similar wonder and hyperbole.
It is hard to say how devout Villehardouin is, despite the frequency of "as God willed it." Certainly his interest in ecclesiastical matters seems low. He mentions the Pope's disapproval of the attack on a Christian Adriatic city, but not the excommunication of the Venetians. He records dissension over its direction among the Cistercians with the crusade, but their leading dissident, the abbot of Vaux, who eventually makes his own way to Syria in disgust, is a deserter and "a disgrace." There is no mention of the installation of a Catholic patriarch in Constantinople and a Catholic chapter in the church of St. Sophia, which aroused fierce resentment among the Orthodox Greeks. More surprisingly, his gloating over the booty of the city makes no mention of the immense number of relics, which were much prized and indeed commanded high prices in the West. Villehardouin mentions the Pope's indulgence as a motive for the crusaders, but Jerusalem as an object disappears very rapidly from his pages and does not recur. He records an interesting declaration, presumably not untypical, by a man deserting a ship returning from Syria to join the crusader army, who calls to his former associates as he leaves them, "I'm going with these people, for it certainly seems to me that they'll win some land for themselves" (57). Villehardouin passes no comment on this not very pilgrim-like sentiment.
**Froissart: "Matters of great renown"**
According to Villehardouin, some of the French knights who took the cross in 1199 and sent him and others to negotiate their passage with the Venetians were attending a tournament at Ecry, in his own province of Champagne. Tournaments resembled crusades in that, though for even more restricted periods, they brought together members of the knightly and noble classes, sometimes with an international dimension. As mimic warfare, sometimes lethal, they had some of the characteristics of real conflict, though crucially without archers or sieges: there were heralds to record and proclaim feats performed, much glitter and display, courtesy to opponents, and even vows, though of more idiosyncratic and even eccentric kinds than the crusading vow which turned a warrior into a pilgrim. They even seem to have influenced the ways in which actual warfare was reported and conducted, with much emphasis on the exhibition of individual prowess and thereby the acquisition of distinction and honour; we find several examples in Villehardouin. Among the informants for his history used by Jean Froissart, nearly two centuries later, seem to have been heralds, and some of his descriptions have what might be called a heraldic perspective, as a kind of scorecard. As combats with rules, tournaments—though sometimes productive of bad blood as well as actual blood, and even tumult and disorder—fostered the solidarity of the knightly class and both exemplified and encouraged notions of chivalrous behaviour, the conduct proper to a knight: foul strokes were dishonourable.
The idea of chivalry, as a code of conduct for the aristocratic warrior class, fostered by the clerical authors of the vernacular romances centred on Arthur and his knights, had been evolving since the twelfth century and reached its height in the fourteenth. As the central rite of chivalry, apart from the ceremony of knighting itself, tournaments mutated from the chaotic free-for-all—the melee—of the twelfth century (Matthew Paris records a number in the early thirteenth being cancelled for fear of disorder, as well as on account of that perennial English enemy of sporting events, the weather) into something increasingly lavish and ceremonial. The early melees must have been rather like early football matches between villages, with few rules, no clearly demarcated areas, and some deaths. Later the circumscribed area of the town square was preferred, with ordered sequences of individual combats, and stands for spectators, including ladies, and festivities to follow.
Like warfare, again, the tournament had its commercial side, since the valuable horse and armour of the vanquished became the property of the victor; something like "professionals" existed among poor knights. In war, the prisoner, spared to be ransomed, was a negotiable commodity as well as a brother knight in misfortune; prisoners could be bought as a speculation on the ransom, rather like discounting bills of exchange. Chivalrous, that is courteous and fraternal, treatment of prisoners was another hallmark of class solidarity, which commoners did not benefit from.
Chivalry, as an articulated personal idea for the layman of knightly birth, stood at the confluence of several influences, of which we have just considered one. The ecclesiastical influence was seen chiefly in the ceremony of knighting, which came increasingly to resemble a religious rite, and in the oath sworn, to uphold justice and protect the weak. It was an oath-taking society: the central social bond was that between vassal and lord, established by an oath of fealty, and a disloyal vassal was held to be forsworn. It seems to have been common for crusaders to regard themselves as sworn vassals of Christ as their lord, whose honour was insulted by Muslim possession of the Holy Places. Vows also played a part in another source of chivalric ideas: the cult of courtly love, developed in the twelfth century chiefly in Aquitaine, which was both fostered by and reflected in the later elaborations of the Arthurian legends and in prose and verse romances. This represented an attempt, more female than ecclesiastical, to soften the manners of a warrior aristocracy. The devotion of the lover to his lady was a kind of vassalage. To be disloyal was to be dishonoured, while the knight's devotion was supposedly both an inspiration and a kind of purification, tested, like his courage, by ordeal. The vulgarized form of the ideal in the north saw the knight as lover exhibiting greater prowess, fighting better; we find examples of this idea, as we shall see, in Froissart.
Of course the gap, in both love and war, between ideal and reality could be very wide: sacking, looting, massacring were normal concomitants of fourteenth-century warfare, and Froissart's _Chronicles_ shows them to us. The knight was often, like the crusaders with Villehardouin dividing the loot of Constantinople, still recognizably kin to the Frankish chieftain as brigand known to Gregory of Tours, and the motives for warfare remained much the same, alongside the desire for honour and reputation through a full scorecard.
Froissart was the supreme historian of fourteenth-century warfare seen in the chivalric mode, as well as of a good deal besides. His work has a few chronicler's features, of which one is his incorporation of the work of a predecessor, Jean le Bel, which he acknowledges and which largely makes up his book from 1327 (the accession in England of Edward III) to the early 1360s, when it becomes wholly his own; it ends in 1400, after the deposition of Richard II. Froissart is also not averse to including interpolated stories for no better reason than that he has been told them and likes them. But in general he is a master of fluent, controlled and relevant narrative, in which, constantly, there seems some influence from the fictional prose romances, often Arthurian, popular in his time. Conversations are frequently used to carry forward the narrative. Froissart's work is history, but he does not make a fetish of the historian's sobriety: one has sometimes the sense that the general direction of the truth will do, and he is slapdash with details. His is a free and easy narrative manner, which is highly readable and not at all pedantic. He wrote in the vernacular, clearly for a lay audience, and was first translated into English in 1523–5. Centred on northern France, England and Flanders, with excursions to Scotland, Gascony and Spain, his account is a panoramic view, from a particular and largely—but not exclusively—chivalric perspective, on his century. It is given focus and direction by the protracted conflict between the English and French monarchies, though it is not confined to this. It also incorporates revolts in Paris, London and Ghent, and the French peasant risings known as jacqueries. The Peasants' Revolt in England (1381) is given extended if not very confidence-inspiring treatment. The Black Death is merely mentioned, though the sect of the Flagellants to which it gave rise is described. The Papal Schism (1378–1417), deriving from the election of rival popes, receives some attention.
Froissart was born in the county of Hainaut around 1337, into a bourgeois family with an apparent involvement in the unchivalrous trade of moneylending. He was early drawn to the court of Edward III of England in the wake of his countrywoman, Edward's queen, Philippa of Hainaut, for whom he wrote an account of the recent Anglo-French wars in verse, for which he says he was rewarded well. He eventually returned to the Continent, though he paid a later visit to England, and, having taken orders, he died a canon of Chimay, near Liège, in 1405. In the preface to his _Chronicles_ he tells us it was written at the request of his patron, Robert of Namur: "In order that the honourable enterprises, noble adventures and deeds of arms which took place during the wars waged between France and England should be fittingly related and preserved for posterity, so that brave men should be inspired thereby to follow such examples, I wish to place on record these matters of great renown." The references to posterity and providing examples were a standard formula, but the diction is the idiom of chivalry: "honourable enterprises," "noble adventures" and "great renown" through bravery. It is no surprise when he then goes on to tell us that he has sought the acquaintance of great lords in France, England, Scotland, Brittany and elsewhere, among whom his inquiries have been made.
To place Froissart's work in a category of "chivalric history," however, requires amplification, and even some qualification. Concepts of honour, prowess and courtesy are freely deployed, and we get plenty of examples of the class camaraderie of knighthood, as well as its commercial aspect. In the campaign that ended at Crécy (1346), some French knights, threatened with being slaughtered in a general massacre, call out to an English knight they recognize "because they had campaigned together in Granada and Prussia [both places of authorized crusades]...in the way knights do meet each other." They beg him to make them his prisoners, which would mean his protection. "When he heard this Sir Thomas was delighted, not only because their capture meant an excellent day's work and a fine haul of valuable prisoners" (75). Sir Thomas's shining catch represented simultaneously the solidarity of the crusaders' old-school tie, the knight's duty to display mercy to the vanquished, and what the eighteenth century, celebrating the civilizing effect (the civic successor to "courtesy") of international trade, would later call _le doux commerce._ In English, "courtesy," a key component of chivalry, is only etymologically a half-step from "courtly" ( _courtois, höflich_ ), related to the manners of the court, just as "chivalry," which largely faded away later, retains its original horsiness.
There is an amusing account, apparently given to Froissart at first hand, of an attempt to teach chivalry to the Irish, who were, it seems, recalcitrant to its values and customs. They do not fight in chivalrous fashion, taking no ransoms and running away when expedient (the inability of French knights, in particular, to distinguish between withdrawal and disgrace was a considerable tactical handicap). Eating their enemies' hearts, which the Irish are also said to do, is definitely not chivalrous. Froissart's informant is a pained English knight (named), who has undertaken to instruct four Irish kings in the knightly code—which includes table manners, though not the menu. They must sit at a separate table from their minstrels and dependants, and not share utensils with them. The kings become angry, apparently, when the hall furniture is rearranged to observe degree. It also has to be explained to them that they have been incorrectly knighted, with an unceremonious dubbing at the age of seven. Paedo-investiture will not do, and they will have to be re-knighted properly, rather as St. Wilfrid insisted on Irish monks being re-tonsured. They must also learn to wear breeches. It is very like missionary work, unaided by miracles, and no sparrow appears to point a moral, though chivalry would presumably have required something of gaudier plumage. The Irish, after being taught "the virtues and obligations of chivalry," are then, duly instructed and presumably breeched, knighted according to the proper rite by Richard II. Incidentally, though not bare-legged, the Germans too, according to Froissart, do not really understand chivalry as the English, French and Scottish do, and mistreat their prisoners (414–16).
Chivalry was, of course, a matter of honour and prowess as well as courtesy and humanity. Froissart is a connoisseur of knightly feats. In this respect the battle of Poitiers (1356) apparently excelled that of Crécy: Poitiers had more class—"There were incomparably more fine feats of arms than at Crécy" (138). In both, it is true, Froissart acknowledges the crucial role of archery. At the sea battle of Winchelsea (1340) Edward III adopts the tactics of the tournament, treating his ship as both horse and lance: "Steer at that ship straight ahead of us. I want to have a joust at it" (116). (The ships meet head-on, and Edward's is so badly damaged that it has to be abandoned.) As a royal command this ranks with that of the French king at Crécy, also in its way chivalric, when the Genoese crossbowmen in his army were retreating: "Kill all that rabble. They are only getting in our way" (89).
Sometimes in Froissart, as was mentioned earlier, one catches echoes of the idea, derived from the ethic of courtly love, that being in love was a combat advantage. Sir Eustace d'Aubricourt performed fine feats of arms and was irresistible, "for he was young and deeply in love and full of enterprise" (161). His is not the only example. Froissart adds that Eustace won great wealth by his deeds; presumably his love also prospered. On another occasion two knights, French and English, show personal animosity because they wear ladies' favours of exactly the same shade of blue; whether their suspicions are aroused is not specified.
In terms of the ethic and manners of chivalry, the centrepiece of Froissart's work is the conduct of Edward's son the Black Prince towards his captive the French king John the Good, made prisoner at Poitiers. The preamble is unseemly, the King being such a valuable prize that there is a fight over possession of him, with people of all sorts grabbing him and shouting "He's mine"—almost, one feels inclined to say, as though he were a relic, which would also have had a market value. The King is understandably relieved to be taken into custody by the Prince, who that evening, at a supper for King John and his captive nobles, stages what can only be described as a kind of chivalric ballet:
He himself served in all humility both at the King's table and at the others, steadfastly refusing to sit down with the King in spite of all his entreaties. He insisted that he was not yet worthy to sit at the table of so mighty a prince and so brave a soldier as he had proved himself to be on that day. He constantly kneeled before him, saying: "Beloved sire, do not make such a poor meal, even though God has not been willing to heed your prayers today. My noble father will certainly show you every mark of honour and friendship...In my opinion you have good cause to be cheerful, although the battle did not go in your favour, for today you have won the highest renown of a warrior, excelling the best of your knights. I do not say this to flatter you, for everyone on our side, having seen how each man fought, unanimously agrees with this and awards you the palm and crown, if you will consent to wear them." (144)
The French and English present all approve highly of the Prince's conduct and agree that "in him they would have a most chivalrous lord and master."
For the modern reader, at least, the pendant to this scene is Froissart's description of the massacre of the citizens of Limoges ordered by the Black Prince after its capture. For once Froissart forgets his aristocratic sympathies:
There were pitiful scenes. Men, women and children flung themselves on their knees before the Prince, crying: "Have mercy on us, gentle sir!" But he was so inflamed with anger that he would not listen. Neither man nor woman was heeded, but all who could be found were put to the sword, including many who were in no way to blame. I do not understand how they could have failed to take pity on people who were too unimportant to have committed treason. Yet they paid for it, and paid more dearly than the leaders who had committed it.
There is no man so hard-hearted that, if he had been in Limoges on that day, and had remembered God, he would not have wept bitterly at the fearful slaughter which took place. More than three thousand persons, men, women and children, were dragged out to have their throats cut. May God receive their souls, for they were true martyrs. (178)
Froissart, a bourgeois himself, is consistently more sympathetic to townspeople than to peasants. He has no sympathy for the participants in the French jacqueries, the peasant uprisings against the higher classes, or for those in the English Peasants' Revolt in 1381. He gives allegedly verbatim an egalitarian sermon by their clerical leader, John Ball, which six centuries later seems very cogent, but Ball is "crack-brained," the rebels "wicked men," and their actions "devilry" (212). He shows much more respect for the rebels in Ghent against the counts of Flanders; they are almost his own people, but their defeat, followed by their massacre at the battle of Coutrai (1302), is still "greatly to the honour and advantage of all Christendom and of all the nobility and gentry" (350). Anything which threatened order and degree was wicked per se. Yet it is not invariably only in his comments on burghers that Froissart's sympathies seem reversed. He loves, of course, the flourish and display associated with chivalry—the silken pavilions, elaborate tableaux, royal entries, festivities, knighting and marriage ceremonies. But in recounting the French preparations for an (aborted) invasion of England, after describing the orgy of painting and embellishing of the ships, including gold leaf, he catches himself and adds "and it was all paid for by poor people throughout France" (305–6). Froissart accepts, but it is not true to say that he endorses, the concomitants of glorious war, which he understands very well: looting, burning and devastation. "So was the good, fat land of Normandy ravaged and pillaged by the English"—so effectively, in fact, that "the very servants in the army turned up their noses at fur-lined gowns" (71).
There is in Froissart one striking example of bourgeois heroism, as well as the author's pity, in the famous final act of Edward III's siege of Calais, which is forced to capitulate by starvation. When it is announced that Edward will spare the inhabitants if six leading citizens will make their submission to him with halters around their necks and the keys of the town in their hands, six quickly volunteer. The pathos of the subsequent scene in which Queen Philippa pleads for their lives, eventually successfully, with her initially implacable husband greatly appealed to the Edwardians who set up a cast of Rodin's tableau of the wretched men's predicament, still visible on the Embankment in London. The townsfolk were expelled by Edward, who intended to plant the town with English, and Froissart again expresses pity, though it seems the expulsions may have been exaggerated. It is interesting that the knights captured were released on parole, Edward declaring that "they are gentlemen and I can trust them to keep their word" (109–10). The acceptability of the knight's parole was a function of the sanctity of the chivalric oath. The extreme of honourable and chivalrous behaviour in this respect was exhibited by the French King. Released to his own country after undertakings have been given, he finds that one of his followers has broken the conditions, whereupon he insists on returning voluntarily to his captivity in England, thereby, one assumes, having his tit-for-tat with the Prince. His arrival was predictably the occasion for great festivities. But parole applied only within the knightly class. When the young Count of Flanders was made captive by his bourgeois subjects, "They watched him so closely that he was hardly able to go and piss" (99).
Froissart, then, was a writer at once courtly and earthy, chivalric but sometimes more widely humane, with the bourgeois sometimes peeping past the associate of knights and nobles. But of course it is as the historian of the world of chivalry that he is above all remembered, a reputation he would have embraced. There is a painting by the American-born history painter Benjamin West, in the age of George III, of Edward III crossing the Somme during the Crécy campaign; it is a subject taken directly from Froissart, and conveys very well the impression made on later generations by his work. The French disputed the crossing. Here is Froissart's description:
The two Marshals of England sent their banner-bearers forward, in the name of God and St. George, and followed closely themselves. The bravest knights herded their horses into the river, with the best mounted in the lead. There were many jousts in the river, and many unhorsings on both sides, for Sir Godomar and his men defended the crossing bravely. A number of his knights, with others from Artois and Picardy, had decided not to wait on the bank but to ride into the ford and fight there in order to win greater distinction. So there was, as I have said, many a joust and many a skilled piece of fighting...The Genoese also did much damage with their crossbows, but the English archers shot so well together that it was an amazing sight to see...(80)
Froissart has no word of reproof here for the French knights who throw away their tactical advantage in possession of the bank and go down into the river in their search for "greater distinction," though later, in his account of Crécy itself, he blames the pride and vanity of the French for their fatal indiscipline, "for there were too many great lords among them, all determined to show their power" (86).
West's picture puts Edward himself centre stage and in midstream. There are some very small English archers doing their bit on an out-crop in the background. It is not realism; mud is unthinkable. It is Froissart's already partly idealized world, idealized still further by historical distance and the conventions of history painting, with none of Froissart's own occasional earthiness and realism. But the picture is in the highest degree animated and glittering. The focal point is the King himself, beneath his fluttering banner quartering the leopards of England and the fleurs-de-lis of France; the same quarters appear on his shield and surcoat, and his horse is similarly emblazoned. Heraldic devices served the purpose of identification: Edward seems quadruply insured against anonymity. But he makes a gallant and colourful figure, waving his battleaxe in a way that seems more debonair (another approved knightly quality), and encouraging to his followers, than menacing to the French already engaged, close on the further bank. The painting is a selective but not inappropriate evocation of the way Froissart often, though not invariably, sees his world. It is appropriate too, as George III observed, that it was to hang in Windsor Castle, where King Edward instituted the Order of the Garter, and which, according to Froissart, had been built by King Arthur (67).
**EIGHTEEN**
**From Civic Chronicle to Humanist History: Villani, Machiavelli and Guicciardini**
The gatecrashing Londoners at the Council of Winchester (above, Chapter 16) were a portent. As cities grew, in wealth and population, and developed corporate institutions for self-regulation, corporate consciousness grew also, and produced record-keeping, just as it did in monasteries. We have looked at conventual chronicles, at national histories presented as the deeds of kings, and at vernacular history as the representation of the deeds of a chivalric warrior aristocracy. Town chronicles, also in the vernacular, came to express a similar self-consciousness, in their case that of the largely unlearned but literate burgesses—the merchants, financiers and employers of labour, chiefly in the various aspects of the cloth industry—who dominated town life. Theirs was sometimes a contentious and divided form of self-assertion, with substantial citizens set against nobles and overlords and also against the lower grades of artisans with their own grievances. There were serious insurrections in the later 1370s and early 1380s in the major centres of population and urban prosperity and power: Paris, London, Ghent and other towns of Flanders, and Florence, which witnessed its first serious rising of the inferior artisans, the _ciompi._ Town life had flourished in central Italy, and manifested itself in Florence in city chronicles from the eleventh and twelfth centuries onward.
In central and northern Italy some cities became virtually autonomous, essentially city states; the conflicts of popes and emperors, and the primarily German concerns of the latter, made imperial suzerainty increasingly nominal. The result was in some cases, most notably in Venice and Florence, the emergence of a republican form of internal politics and the political fragmentation of Italy into a number of mutually suspicious and hostile states, both of which had implications for the writing of history. This was also to be shaped by the revival of classical learning known as humanism, which from the fifteenth century onward incorporated a close attention to the manner as well as the content of exemplary Roman historians, particularly Livy. But the local situation was not that of the empire, nor of a national kingdom: it was rather that faced by republican Rome, with its early conflicts with the rival states which were its close neighbours, and its growing internecine class conflicts. As the eminent historian of the Renaissance Denys Hay put it, "The Anglo-Saxon Chronicle, Malmesbury, the St. Albans writers down to Matthew Paris, were, whether they knew it or not, writing the history of England...And so, in a less marked fashion, with France. But no works like that can be found in Italy" ( _The Italian Renaissance,_ 1977).
But city chronicles proliferated, and from them, in conjunction with the classical revival, emerged in Florence in the fifteenth century a way of writing history much more closely modelled than before on Roman practice, and a new way of reflecting on the lessons of history. It incorporated not merely the conventions of Roman historiography—Greek historical writing was still much less well known, though translations were beginning to be made—but also the substance of Roman and modern Italian history as a source of republican inspiration and political lessons. The most favoured classical historians were still those admired in medieval times, Sallust and Livy, until the great vogue for Tacitus began in the late sixteenth century. But the image of Rome assumed a different form from that current in the Middle Ages, focused not on the imperial city, the _urbs aeterna,_ but on the struggling early republic to which the Romans of the first century BC had themselves looked back as a lost era of patriotic republican virtue.
Vernacular chronicle-writing implied not a certain detachment, as humanist history, with one foot in the classical world, was later to do, but the reverse: a close and intimate involvement in the life of the city, as the abbey chroniclers were involved in their own, smaller, communities. Monastic chronicles seem to have grown in some cases out of calendars; in the case of a lay, mercantile community such as Florence there were, instead, the family and business archives—they were the same thing—in which a literate laity recorded their affairs. These archives naturally came to incorporate public matters, with lists of office-holders and the interpolation of notices of general civic or even international events. In Italy, such public occurrences were events, such as wars, which involved the city with one or more neighbouring republic or principality; on a wider scale, the doings of popes and even of kings, crusaders, Turks and Mongols were of interest. In more local terms there was a kind of continuum between public life and the domestic and familial, with feuds between the greater families being a matter of acute public concern, since they were a frequent cause of disturbances, while family alliances were one of the keys to power. The tenure of public offices in Florence was very short, so rotation was rapid and participation wide; having held an office was an important mark of status for families as well as individuals.
There is a traceable evolution in Florence from private memoirs and family histories to vernacular chronicles and then to humanist histories of the Florentine polity. But the transition between the last two is more like a fracture, with the latter being superimposed on the former—as Machiavelli's history, written in the vernacular, incorporates sections from Villani's _Chronicle,_ composed two centuries earlier, into a work which, overall, has a wholly different character and set of interests and is written for a different and more sophisticated readership. This transition which represents a rupture as well as continuity is of more than local significance: it was pregnant for European historiography generally, and represents a vital moment in its development. The humanist historian, distancing himself from the colloquial manner and the largely local though promiscuous interests of the chronicler, was set apart by his education, of which he was proudly conscious, by his constant (rather than occasional) awareness of his classical models, and by a stern sense of relevance imposed by what Machiavelli's contemporary Guicciardini called "the laws of history." A leading modern authority on Guicciardini, Mark Phillips, has shown how his unpublished works exhibit the transition we have been considering here in a single literary career: he wrote, among other things, a family history, from which grew a Florentine history; this was later supplemented by another, which represented an attempt at an ambitiously polished humanist history, with, for example, formal invented speeches, but also incorporating much documentary research. This remained incomplete and in draft. Guicciardini finally went on to the monumental _History of Italy_ by which he is remembered.
To appreciate the change more generally, we first have to look more closely at what is by common agreement the most rewarding of the medieval Florentine chronicles, that by Giovanni Villani. It was written in the earlier part of the fourteenth century; Villani died in the Black Death, the European catastrophe which dominated the middle years of the century. He was a member of a minor but reputable commercial family with interests in banking and trading ventures. Within an overall framework of providential history, ultimately derived from Orosius (above, Chapter 18), which he took from a medieval papal chronicle, he traced the story of his city (he always uses the possessive: "our") from what he supposed to be its origins to his own times; for the usual obvious reasons, the _Chronicle_ becomes much more detailed the nearer it approaches the present. It is a mark of the communal character of his enterprise that it was continued after his death by his brother and nephew, as, in a monastery, it would have been continued by a younger monk. Initially prosperous, Villani was ruined by the establishment of a more democratic city regime after the overthrow of the brief foreign rule of the Duke of Athens (the title was a reminder of the Crusades), son of the King of Naples, in 1342–3. Before that he had served several times as one of the eight priors who sat, each for two months at a time, in the city's executive government, the Signoria.
Villani's _Chronicle_ has no explanatory preface, but later in the work he provides what amounts to one. Pope Boniface VIII had proclaimed 1300 as a jubilee year (always good for the prosperity of Rome), with indulgences for pilgrims to the city of St. Peter. Villani, as he tells us, was one such pilgrim; he pays tribute to the admirable organization of the provisions for the pilgrims' reception (VIII.36). Like Ammianus a thousand years earlier and Gibbon four and a half centuries later, he was greatly moved by the venerable grandeur of the city and was inspired, he tells us (like Gibbon), to the composition of the work to which his name is attached—though, again like Gibbon, he seems not to have begun writing for several decades. He was perhaps scarcely aware of the contrast, of which Gibbon makes so much in his circumstantial account of his work's conception, between the Rome of the republic and the Caesars and the clerical Rome of his own day. Viewing, Villani says,
the great and ancient things therein, and recalling the stories and the great days of the Romans [he mentions Virgil, Sallust, Livy and Orosius among others] and other masters of history, who write alike of small things and great [a good self-description, but hardly likely to appeal to some of the authors he cites], of the deeds and actions of the Romans, and also of foreign matters throughout the world, I myself, to preserve memorials and give examples to those which should come after, took up their style and design, although as a disciple I was not worthy of such a work. But considering that our city of Florence, the daughter and creature of Rome, was rising, and had great things before her, whilst Rome was declining, it seemed to me fitting to collect in this volume and new chronicle [the title his work was given] all the deeds and beginnings of the city of Florence...and in this year 1300, having returned from Rome, I began to compile this book, in reverence to God and the blessed John and in commendation of our city of Florence.
Florence was the daughter of Rome because, according to tradition, it had been founded by Julius Caesar. The humanists, with their surer grasp of Roman literature and history, were later to correct this, placing the foundation firmly in republican times—to Villani, Caesar was an emperor—some decades earlier, as a settlement for Sulla's veterans, as Sallust describes in his _Catiline._ But Villani also gives the foundation a wider context in the colonization of Italy. After a brief reference to the Tower of Babel and then the foundation of Florence's older rival Fiesole, on the hill above the river, by King Attat, Villani moves smoothly into familiar Virgilian territory with the emigration from Troy. Italian history is, as usual, overrun by vagrant Trojan émigré princes. After a short digression on the Roman eagle and the Florentine lilies as emblems—Villani was always interested in such insignia—we have the Romans building on the site of Florence a temple of Mars, their god, in black and white marble, which Villani clearly takes to be Florence's cathedral (which still stands). In an imperial persecution in AD 270 Florence acquires its first martyr, St. Miniato, a hermit and son of the King of Armenia, who had migrated to Italy. Beheaded, the saint has his head replaced, so that he is able to walk up the hill to where his church now stands, before expiring and being buried. In due course the temple of Mars was consecrated and dedicated to St. John, as Florence's cathedral, which, on account of the favourable astral situation at its foundation, was able to survive the devastation wrought by the Goths(I.32, 35). Villani is careful to tell us that heavenly influences do not absolutely bind human fate or human free will, but all the same...(III.1).
We are still in legendary times, for a little later we learn from Villani, or rather his sources, as news from abroad, of the birth of Merlin in Great Britain ("which is now called England") from a virgin. It was Merlin who ordained the Round Table of knights errant for King Uther Pendragon, descended from Brutus, grandson of Aeneas, and this was afterwards restored by his son Arthur, "as the Romances of the Britons make mention" (II.4). In other chapters (10, 12, 13) we get a brief and essentially historical account of the Franks and their emancipation of Italy and the Church from the Lombards. Siena, Florence's chief rival after Fiesole, was first inhabited, we are told, by invalids accompanying the Frankish king Charles Martel, for "Siena" derives from "non sana," unwell. This leads up to the coronation and empire of Charlemagne, by whom, it is alleged, Florence was rebuilt (III.1). Villani in fact gives a detailed description of the rebuilding, comparing the result with the modern city and again invoking a favourable disposition of the planetary bodies, though he sadly traces the subsequent dissensions of the city to the original mixture in its population of "the noble Romans and the cruel and fierce Fiesolans" the alleged reception of the Fiesolans into the city (IV.6) sounds rather like the assimilations in early Roman history as described by Livy. From the twelfth century—which is rapidly reached after a certain amount of attention to the quarrels of the empire and the papacy in the eleventh—the _Chronicle_ becomes much more substantial and circumstantial. The feuds and constitutional arrangements of the Florentines, and the defence of their liberties in the face of external threats, now take centre stage—though still with excursions elsewhere, and with an account of the origins of the strife between Guelfs and Ghibellines in Florence and of which great families embraced which cause (e.g. V.39). Florence was predominantly a Guelf (papalist) city, and Villani's own sympathies were Guelf. The Ghibellines were supporters of the emperors.
There is something very attractive about Villani's civic pride and his obvious responsiveness to the city's topography and the physical texture of its life. He accepts and sometimes deploys as an explanation Sallustian commonplaces about the dangers of too much tranquillity and prosperity, as nurseries of pride and factiousness, yet he also cannot help showing his delight at their manifestations. In 1300
our city of Florence was in the greatest and happiest state which had ever been since it was rebuilt, or before, alike in greatness and power and number of people [30,000 is his estimate, with 70,000 men of military age in the whole territory controlled by the city]...whereupon the sin of ingratitude, with the instigation of the enemy of the human race, brought forth from the said prosperity pride and corruption, which put an end to the feasts and joyaunce of the Florentines. For hitherto they had been living in many delights and dainties, and in tranquillity and with continual banquets; and every year throughout almost all the city on the first day of May there were bands and companies of men and women, with sports and dances. But now it came to pass that through envy there arose factions among the citizens...(VIII.39)
Around 1300, the factions began to be called the Whites and the Blacks, so that there could, for example, be Black Guelfs and White Guelfs. The change to the new, apparently empty, categories seems a recognition that the feuds were essentially those of great families (whom Villani, as usual, names) and their adherents: Florentine faction-fighting contained class dimensions, of nobles against citizens and rich against poor. Villani's later books take up the descriptions of these factions and the tumults and bloodshed to which they gave rise. (These are also dealt with in Machiavelli's later _Florentine History,_ with, as one would expect, a more probing political curiosity, though in no less detail but with less "surface.")
Villani's Florence is a city not only of factions and constitutional complexities but also of banners and emblems under which the citizens range themselves and to which, in moments of danger and tumult, they rally, summoned by bells to gather in militias and processions (e.g. VI.39). According to Villani, in the late twelfth century even the faction-fighting could at times be suspended in fellowship. Florentines allegedly became so used to their civil wars that "one day they would be fighting and the next day they would be eating and drinking together and telling tales of one another's valour and prowess in these battles" (V.2).
Villani gives a detailed account of the constitutional changes made by the temporarily dominant Ghibelline faction in 1266 (VII.13). Attempting to appear even-handed, the Ghibellines appointed two temporary rulers ( _podestà_ ) who were not imperial nominees but knights of an order called officially the Knights of St. Mary but known as "The Jovial Friars of Bologna." Villani describes the robes and insignia of the order, but adds that it was short-lived, "for the fact followed the name, to wit, they gave themselves more to joviality than to anything else." Colleges were set up from the various trades guilds (trades and districts partly coincided) which were a persistent feature of the city's governing structure, though their numbers varied over time. Villani lists them and the emblems and colours they bore as banners: judges and notaries, cloth merchants, money-changers (who bore a standard depicting gold florins on a red ground), wool merchants (a white sheep), physicians and apothecaries, silk merchants and furriers. Later, inferior trades were incorporated, including butchers (a black goat), stonemasons and carpenters (saw and axe) and smiths (pincers) (VII.13). Villani's most memorable description of this kind is that of the civic totem, the _carroccio,_ which the Florentines took with them in war against Siena in 1260:
The _carroccio,_ which was led by the commonwealth and people of Florence, was a chariot on four wheels, all painted red, and two tall red masts stood up together on it, on which was fastened and waved the great standard of the arms of the commune, divided white and red, and may still be seen today in S. Giovanni [the cathedral] And it was drawn by a great pair of oxen covered with red cloth, which were set apart solely for this...This _carroccio_ was used by our forefathers in triumphs and solemnities, and when they went out with the host, the neighbouring counts and knights [usually severely excluded from civic matters] brought it from the armoury of S. Giovanni and brought it to the piazza of the Mercato Nuovo, and having halted by a landmark, which is still there, in the form of a stone carved like a chariot, they committed it to the keeping of the people...
It was given a bodyguard of the best soldiers, and acted as a rallying point. A month before it was to set out a bell tolled from one of the city gates without ceasing:
And this thing they did in their pride, to give opportunity to the enemy...to prepare themselves. And some called it Martinella and some the Asses' Bell. And when the Florentine host went forth, they took down the bell from the arch and put it into a wooden tower upon a car, and the sound guided the host. By these two pomps of the _carroccio_ and the bell was maintained the lordly pride of the people of old and of our forefathers in their expeditions. (VI.75)
Machiavelli's view of the superiority of a citizens' militia to mercenary troops, presented as rooted in a nostalgic Roman republicanism, does not strike one as particularly visually imaginative. One can only speculate on the impression this passage might have made on his imagination and sensibility.
Another civic possession was the lions. Lions were also kept in the Tower of London, where they belonged to the king, but these were republican lions. One of them justifies his place in the _Chronicle_ in time-honoured fashion, as a portent. "A fine young lion," having been presented to the commonwealth by Pope Benedict VIII, was, remarkably, kicked to death by an ass, either in fear "or through a miracle." Those learned in matters of divination said this boded no good to the Pope, who shortly died (VII.62). The appearance of the other lion, in 1258, seems more gratuitous: a remarkable event is described by Villani because it happened. By the negligence of its keeper the lion escaped, terrorizing the citizens, and caught a boy, holding him between its paws. But it allowed the boy's mother to snatch him from it, "and the lion did no hurt either to the woman or to the child [later called 'Orlanduccio of the lion'], but only gazed steadfastly and kept still," which was to be attributed either to the innate nobility of the lion's nature or to fortune (VI.69). There was, however, a political footnote: when a member of the city's supreme government, which at that time had acquired a reputation for arrogance, "took and sent to his villa a grating which had belonged to the lion's den and was now lying about in the mud of the piazza of S. Giovanni, he was condemned therefore to a fine of 1,000 lire for embezzling the goods of the commonwealth"(VI.65). This is a good example of republican mores—it is difficult to imagine Lorenzo de' Medici being fined for nicking municipal debris from the Piazza, or the episode being thought worthy of the attention of history. The telling of the story, at some length, serves very effectively to mark off Villani's kind of chronicle-writing from the humanist historiography which had arrived by the age of Lorenzo. It was not that Villani did not have wider views, but they were theological, astrological and apocalyptic, rather than comprehensively political and historiographical.
It is seldom that one is able to see the transition of genres—indeed, one can speak of it as the transition from the Middle Ages to the Renaissance—in such sharp focus as one can in passing in a century from Villani's _Chronicle_ (though chronicles continued to be written) to the new humanist historiography written in Florence. This was presented in polished and learned Latin prose informed by a neoclassical political perspective and a sense of what historical writing in emulation of Sallust and Livy (which Villani had naively supposed himself to be doing) really involved. In the mid twentieth century Hans Baron hailed the work of the early-fifteenth-century Florentine humanist scholar Leonardo Bruni as seminal in this transition, speaking of "something like a Copernican revolution" in which Bruni's _History of the Florentine People_ was a major landmark ( _The Crisis of the Early Italian Renaissance,_ 1955). From this claim derives the term "civic humanism," which a number of scholars have adopted to express the kind of republican political turn and Ciceronian moral outlook which Bruni's work has been taken to represent. John Pocock, in _The Machiavellian Moment,_ has traced a long route for this attitude through Machiavelli and on to the polemics of opposition to the executive in eighteenth-century England and to the political thought of the American founding fathers and the United States Constitution. Certainly there came to be in Italy, and from the seventeenth century onward in other parts of Europe, a revival of interest in the concept of early Roman virtue and an identification with the Ciceronian idea of the active public life in the service of the state. More recent scholarly work has challenged Baron's claims for Bruni's unique originality and the importance of what Baron regarded as its political matrix, the threat to Florence's independence represented by the power and aggressive designs of the Duke of Milan, Giangaleazzo Visconti. Even if we do not take Bruni as a uniquely original Copernican figure, it is now accepted that in central Italy an articulated republican ideology established itself and found expression in humanist writings which changed the way that ancient Rome was seen and appealed to, most notably in Machiavelli.
Bruni himself was a notable scholar, being one of the earliest Italians to read Greek. He was highly successful in his attempt to re-create the manner of the ancient historians, and he read Thucydides and Polybius in the original when this was very rare. Machiavelli, a century later, though well educated, is not generally credited with any Greek at all; certainly, though he makes occasional references to Greek history (we know that he read Plutarch), Thucydides and Polybius are not the presence in his work that one might have expected in view of some of their affinities of temperament and interests. Bruni was the teacher of one of the greatest of all Italian humanist textual scholars, Lorenzo Valla, who made an unsatisfactory Latin translation of Thucydides. (The difficulties are great.)
Bruni was head of the Florentine chancery, the state secretariat, in which a century later Machiavelli would also serve. It had responsibility for correspondence with other states, so the humanist training in classical rhetoric was highly relevant. The term "humanist" derived from the "humanities" ( _umanista_ ) component of the educational curriculum: essentially an education in Latin rhetoric based on close imitation of the most highly regarded ancient models, notably Cicero and Seneca, but also including Sallust, Livy and Virgil. It contrasted, therefore, with the training in law, theology and dialectics which had formed the apex of medieval education, when grammar and rhetoric were regarded as more elementary and were often based only on excerpts from ancient authors; Bede, for example, quotes Virgil, but it has been doubted if he knew his work except in extracts.
The rise in prestige of rhetoric was essential to what is spoken of as humanism, and with it came the intensive study, imitation and attempted recovery in their pristine state of the approved classical authors. From the time of Petrarch in the later fourteenth century, humanist scholars aspired not merely to learn from classical writers, as their medieval predecessors had done, but to re-create the spirit and manner and moral world of Cicero and Seneca. In a sense they wished to _be_ classical authors, imitating their eloquence and adopting their values, and paying attention therefore to their letters and other forms of self-revelation as well as to more public compositions; Petrarch was said to have treated Cicero as an alter ego. As the art of persuasion, rhetoric was valued in the Roman world as the essential political skill. The value attached, in principle at least, throughout the Middle Ages, to the ascetic, contemplative life, institutionalized in monasticism and subsequently vigorously attacked by Machiavelli, was superseded among the humanists by a Ciceronian and republican endorsement of the active life in the service of the republic, with its necessary qualities of nerve and will as well as of eloquence and public spirit. The public life of the Italian republican city states in the later Middle Ages—open, contentious, sometimes dangerous, exhibiting sudden and violent reversals of fortune, which had to be mastered if possible—both required and gave scope to these qualities.
Using earlier chronicles—including Villani—Bruni and others rewrote the history of Florence, in Latin and in the humanist, Livian, manner and from a humanist perspective inspired by republican Roman history. It was not hard to see in the struggles of early Rome with its neighbours an analogue with the conditions of Florence. This emphasis on republican rather than imperial Roman history was in a sense reversing the historiographical coup effected by Eusebius and Orosius a thousand years earlier, when they made Augustus' and Constantine's empire part of the providential Christian story, by treating (and welcoming) the empire as the necessary condition for the spreading of the Christian message and the Christian Church. In the East, the Christianized empire had lasted, in theory at least, ever since, and it had been renewed, with papal sanction, by Charlemagne in the West. By resorting to the beleaguered early Roman republic as a model, humanism was not only reviving the ethics of public life as expressed in Cicero, Sallust and Livy, with honour and fame as the reward for service, but also raising the possibility of learning political lessons, as advocated by Polybius for example, from Rome's success in overcoming its rivals. The connection made explicit by Sallust (see above, Chapter 5) between republican emulation in the pursuit of honour, free institutions and the conquering energy exhibited by the Romans made Rome not the pivot of Christian history but an example and an inspiration. The message was particularly appropriate to Florence, whose civic freedom had been asserted _against_ the residual imperial power, so that anti-Ghibelline sentiments were, as we learn from Villani, predominant, though not universal.
It was to be Machiavelli who notoriously took the humanist ethic of activity in the pursuit of glory to an extreme, refusing to blink at the incompatibility, when taken to such an extreme, with Christian principles, and even seeming to relish exposing it. It was also he who attempted to make the history of republican Rome, together with examples from modern history, into a basis for lessons of permanent political usefulness—systematically in the _Discourses on the First Ten Books of Livy_ (completed in 1519), but also in places throughout his writings. His _Florentine History,_ published in 1532, frequently reveals the same aspiration. Several of his predecessors in the chancery, after Bruni, had also produced such histories; Machiavelli presented his own to Giulio de' Medici, who had become Pope Clement VII. It represented a partial rehabilitation for him with the Medici, but he was never able to resume the official career he had lost in 1512 when the Medici had returned to power in Florence, overthrowing the republican regime for which he had worked and in whose service he had undertaken ambassadorial missions to France, Rome and the emperor Maximilian. He had also worked to create a militia which was supposed to free Florence from its dependence on mercenaries. His friend the historian Guicciardini also had experience in diplomatic missions on the Florentine government's behalf. This practice of keeping close watch on one's neighbours by accredited ambassadors was a new development, born of the tensions and anxieties created by the existence of a number of independent states in close, mistrustful juxtaposition, with constantly shifting intentions and alliances.
One product of this was the articulation by Guicciardini of a consciously held conception of the balance ( _contrapeso_ ) of power between states, by which the independence of each might be preserved and none enabled to become too powerful. Among the important duties of an ambassador was to write reports (which subsequently became important sources for historians [Chapter 25])—appraisals of the power and preparedness, the intentions, intrigues and struggles for influence, in the host state. Together with an education in the Latin classics as a training in rhetoric, and the study of the ancient historians, this was a highly appropriate preparation for writing contemporary and recent Italian history. Both Machiavelli and Guicciardini were clearly fascinated by the experience of diplomacy, though Machiavelli was the more confident in framing political generalizations for future application; Guicciardini was more sceptical, and wrote a critique of Machiavelli's _Discourses_ along these lines. For the appraisal of the unique configurations of circumstances of which he saw politics as being made up, history was the ideal medium. Guicciardini's _History of Italy_ —which is essentially what came to be called "diplomatic history"—is accordingly immensely long and detailed. It was published posthumously in 1561, though written in the 1530s, and concludes in 1527.
Machiavelli's _Florentine History_ alternates the treatment of internal and external affairs. In its later books it too is highly detailed, but the political scientist is always struggling to break free into general maxims and widescale comparisons, which he gives at the opening of each book. They are tailored appropriately to the themes revealed by the main events to be described, which are grouped into the category of a particular political problem and are considered generally before being recounted. These themes include why the internal divisions of Rome could be bridged by conciliation—a notable theme in Livy—and even added to the warlike spirit of the Romans, while those of the Florentines only enfeeble them (III); why the Romans were able to maintain their institutions with little change, while the Florentines are constantly changing theirs (IV); why states oscillate between order and disorder, to which the answer, provided first by Sallust, is that "valour begets tranquillity, tranquillity ease, ease disorder, and disorder ruin. And conversely out of ruin springs order, from order valour, and thence glory and good fortune" (V). This leads on to a condemnation of the dangerously seductive and enervating effects of literary pursuits, much in the spirit of Cato the Censor, whose law expelling philosophers is favourably cited.
Book VI endorses the plundering habits of the states of antiquity; modern wars are, on the other hand, impoverishing, because the vanquished are spared and the spoils are taken by the soldiery—hence, in part, Machiavelli's campaigning against the use of mercenaries. Book VII opens with a warning against factions that was to echo down the centuries, particularly in eighteenth-century Britain and America, until in the nineteenth century constitutional opposition became respectable; Machiavelli makes the point that the more powerful the regime, the more opposition can be expressed only as conspiracy, which he acknowledges should form the preamble to Book VIII, which includes the famous (and recent) Pazzi conspiracy against the Medici in 1478, but he excuses himself on the grounds that he has dealt with the issue elsewhere, i.e. in the _Discourses._ In his subsequent account of the outcome of the plot, in which Lorenzo de' Medici himself was wounded and his brother Giuliano assassinated, but which nevertheless failed, Machiavelli the political connoisseur cannot refrain from itemizing the qualities required by the political assassin: coolness, courage and resolution, bred by long experience in affairs of life and death(VIII.5).
Connoisseurship and an irrepressible taste for maxims of advice, even to a cause we cannot imagine him in any way sympathetic to, is shown by Machiavelli also in his account earlier of the important insurrection of the poorer artisans, the _ciompi,_ in 1378. A leading rioter offers his fellows advice in a speech which, though there may be an element of parodic exaggeration, bears strongly the stamp of the author:
That our old outrages may be forgiven us we must needs in my judgement commit new, multiplying offences, redoubling our burnings and pillagings, and endeavouring to enlist as many as possible as companions in our crimes. For when the offenders are many, none are punished; petty offences are chastised, but great and grave offences are rewarded; and where many suffer wrong, few seek to avenge it. For wrongs that touch all equally are borne with more patience than those directed against individuals. (III.13)
This is Machiavellianism for the lower classes. The speaker winds up with a general reflection on virtues punished and vices rewarded which makes one suspect Machiavelli of indulging in self-parody, or perhaps the expression of his real opinion under the safe fictional cover of a reported speech by a malefactor. The routes to wealth and power in the world are force and fraud. Those who fail to use them are the losers: "The faithful servant is a servant always, and the good are always poor. None escape from bondage but the unfaithful and the bold, and none from poverty but the rapacious and dishonest."
A good deal of the earlier part of Machiavelli's history is familiar from Villani, but Machiavelli brings out more clearly the roles of the papacy and the emperors in keeping Italy divided by their rivalry and hence permitting the de facto independence of the greater cities. He pays tribute, in fact, to the diplomatic skill with which the popes, originally dependent on the emperors, raised themselves to parity (I.9–11). Machiavelli adopts the classical practice of inventing speeches, introducing them by "to this effect." One of the most notable is allegedly delivered during the resistance to the Duke of Athens in 1343, when the populace rise and crowd into the piazza with home-made standards (the Duke having confiscated the previous symbols of freedom and corporate identity). One of the priors makes a long, defiant speech on the traditional freedom of the city, whose mementos are "the seats of the magistracy, the standards of the free companies," which will not allow it to be forgotten (II.34). "What is it that you can do," the Duke is asked, "that will outweigh the delights of freedom and make men cease from longing to revert to their old condition?" One has to wonder in what mood of nostalgia, bitterness or hope Machiavelli wrote this celebration of civic freedom.
After the accounts of faction-fighting and constitutional changes in Florence, culminating in the rise of the Medici, Machiavelli turns (in Book V) to what is essentially skilfully told diplomatic history, which is given an ironic turn by his contempt for mercenary armies, whose leaders and troops alike are concerned only with their own interest and who change sides readily when it suits them. The result has been a state of neither peace nor war, without tranquillity but also without patriotism or valour: "For the wars of Italy were brought to such a degree of futility as to be entered on without fear, waged without danger and ended without loss" (V.1). After a protracted and detailed narrative, in the Roman manner, of an apparently hard-fought battle between the Florentines and the troops of the _condottiere_ Niccolò Piccinino, Machiavelli clearly enjoys giving the casualty list:
And in this great rout and protracted engagement, lasting from four in the afternoon to eight in the evening, one man only was slain, and even he perished not from wounds, or any blow dealt him in combat, but from being trampled on after falling from his horse. With such safety did men then fight; for all being mounted on horseback and sheathed in mail, and assured against death should he surrender, there was no reason why they should die, their armour protecting them while they fought, and surrender saving them when they could fight no longer. (V.33)
Though it deals much with relations between Florence and her neighbours, there is relatively little strictly military history in Machiavelli's _Florentine History,_ though he wrote a treatise on _The Art of War_ ; there is certainly less actual fighting than in Livy. Machiavelli's treatment of external relations traces the constantly shifting pattern of alliances and the motives behind it. The professional commanders of mercenary armies, the _condottieri,_ ostensibly employees, act sometimes like leaders of independent states, which several became. Machiavelli deals, with exemplary lucidity, with the calculations, threats, demonstrations, treacheries and covert dealings of the various relevant governments. One could reasonably call this "ambassador's history." In Book VI, for example, he gives a masterly account, too long to quote, of rival views in Florence on whether to favour the aspiration of the _condottiere_ Francesco Sforza to the dukedom of Milan, showing how they are conditioned by a complex interaction of domestic political considerations and estimations of Florence's interests as a state (VI.23). He speaks admiringly in this respect of the successes achieved by Venice: "It had been, as it were, the destiny of the Venetian republic to lose in war and gain in negotiating peace" (VI.19). Venice is cited by Machiavelli for the stability of its republican institutions. The Venetian republic as a political model was to be a legacy to the political discourse of the eighteenth century. In Machiavelli's brief digression on the early history of Venice, he speaks of it as a republic which, "both for its institutions and for its importance, deserves to be celebrated beyond every other Italian state," though now (that is, in the 1520s) the Venetians live, "like all the rest of the Italian princes, at the mercy of others"(I.28, 29).
Machiavelli can also admire the skill with which the popes, by adopting a kind of balance-of-power policy, have kept Italy divided, and he recognizes that this division is what has made the free development of the Italian states possible, though now it makes them hopelessly vulnerable to the intervention of the states beyond the Alps—"the barbarians," as he and Guicciardini sometimes call them (I.23, 28). He is similarly ambivalent about the internal divisions in Florence. In so far as they proceed from emulation and avidity for public distinction he speaks of them, following Sallust, as a source of energy and strength. He draws a distinction between this and the creation of a faction by the distribution of favours and the exercise of patronage, which is objectionable and which has been characteristic of Florence, to its harm. This is a preamble (VII) to a consideration of the rise to power of Cosimo de' Medici, and clearly points at him.
Considering that Machiavelli wrote his _Florentine History_ in a sense with Medicean encouragement, it is less sycophantic and more independent in its judgements than one feels entitled to expect, helped, of course, by the fact that the Medicean rulers were officially republicans. For example, in his concluding eulogy of Lorenzo de' Medici, with whose death the book ends, Machiavelli speaks favourably of Lorenzo's careful adherence to "the simplicity of republican manners," despite the munificence of his patronage, and of his restraint in not seeking princely foreign marriages for his children. There is a bitter irony, however, in his account of the conclusion of the Pazzi conspiracy against the Medici. When the cry of "Liberty" was raised, as in the past, there was no rallying to it, for "the ears of the people had been stopped by the prosperity and bounty of the Medici, and liberty was no longer known in Florence" (VIII.8).
For Machiavelli, Lorenzo's death in 1492 marked a turning point, because, in Machiavelli's concluding words, "Italy, remaining bereft of his counsel, found no resource in those who survived him either to satisfy or restrain the ambition of Ludovico Sforza, guardian of the Duke of Milan." It was Ludovico, as Machiavelli's readers knew, whose invitation had led to the invasion by Charles VIII of France in 1494, from which both Machiavelli and Guicciardini dated the present abject state of Italy, at the mercy of foreign powers. Machiavelli spoke of the invasion as "all these evil seeds, which...very soon ruined, and still continue to ruin, Italy." For the elaboration of these consequences, readers were able to turn, after its publication in 1561, to Guicciardini's _History of Italy,_ which begins where Machiavelli's ends.
Francesco Guicciardini, more socially distinguished than his friend Machiavelli, had served as an ambassador, to Spain in 1512–13, but he both came of an eminent Florentine family and remained on good terms with the Medici, so that under the two Medici popes, Leo X and Clement VII, he rose very high in the government of the papal possessions, as president of the Romagna and lieutenant general of the papal armies and later governor of Bologna. After his fall from power, when he devoted himself in his retirement, like many ancient historians, to writing his history, he spoke with some irony of the grandeur of his former state, referring to himself, as he does in his history, in the third person: his fellow citizens would not have recognized him "with his house full of tapestries, silver...surrounded by a guard of more than a hundred landsknechts, with halberdiers and other cavalry in attendance...never riding out with less than one hundred or one hundred and fifty horse; immersed in governing bodies, titles, 'Most illustrious lords...'." There was an irony in Guicciardini's eminence in the papal service, for, as he makes clear in the history, he was contemptuous of the abject state of the papacy and the Church in Italy, and came to deplore the role of the former in Italian affairs as pernicious. Privately he said that if "the position I have served under several popes [had not] obliged me to desire their greatness for my own self-interest...I would have loved Martin Luther as myself." His account of the policies and characters of the popes—written, of course, after his fall—is unsparing, though there is no approval in his references to Luther.
Guicciardini's history is immensely long and covers only forty-four years, from 1490 to 1534, so it is also immensely detailed—though for the first time, by focusing entirely on the relations of states, he was able to produce a work which considers Italy as a whole. Guicciardini's general characterization of historical change, in contrast to Machiavelli's famous reiteration of the notion of a cycle of prosperity followed by downfall, warns us of his commitment to the fine grain of historical events and hence to multiple explanations, which he indeed provides, though they mostly concern the intersection of many motives, intentions, calculations, misconceptions, irrational impulses and fleeting or enduring psychological dispositions; Guicciardini seldom offers one motive for an action when he can think of three or more. Human affairs, he says at the outset of his history, are mutable, "not unlike a sea whipped by winds," and the reader of his work comes to feel the force of the analogy.
Guicciardini's commitment to the particular, to the uniqueness of each situation, has two important consequences. One, of which he is fully aware, is a warning against overconfidence in commentators and, more crucially, in statesmen: arrogance is folly. The second, which informs the whole of Guicciardini's history, is a commitment to explanation through narrative, through recounting the dense particularity of each important historical moment. (On this see Phillips, _Guicciardini,_ Chapter 7ff.) Close narration is indispensable, and Guicciardini provides it. But his history also has a general theme. That of Machiavelli's history, despite its attention to foreign affairs, is above all factionalism and the loss of civic liberty, though this is perhaps not as clear as it might have been made if the methods of Cosimo and Lorenzo de' Medici in manipulating the government and effectively ruling with a republican facade, though acknowledged by Machiavelli, had been traced in more detail. Guicciardini is also concerned for his city's liberty, which he understandably approaches as a rich oligarch, but the central theme of his history is determined by the period it covers and by its focus on external affairs: it is the tragedy, felt also by Machiavelli, but not traced by him, of foreign dominance of Italian affairs and the reduction of the Italian states to a condition of subservience. With extraordinary powers of organization, and clearly much research, though it is hardly exhibited, Guicciardini narrates in one after another episode of complex negotiations how this outcome has been arrived at. The only comparable historiographical achievements are the studies in the nineteenth and twentieth centuries of what came to be called "the European states system."
But Guicciardini's work, though precise, controlled and dignified, is also in its totality a lament, an aspect which sometimes surfaces in the narrative, and there are villains. These are the Italian rulers, among whom the popes figure prominently, whose reckless and short-sighted ambitions allow the tragedy to occur. Fortune is sometimes invoked, but, while rulers should take account of it, Guicciardini's account is not deterministic: it traces the consequences of human folly, in many guises. Guicciardini's passionate indictment of the narrow selfishness of the leaders of the Italian states is prompted by a sense that the tragedy need not have happened. Only Lorenzo, who stands as a contrast, and a memory, is lavishly praised—for his understanding of the necessity for an intra-Italian balance of power. Potential strong men emerge, notably Cesare Borgia (the subject of Chapter VII of Machiavelli's _The Prince_ ) and Pope Julius II, but they overreach themselves and fail. No Italian measures up to the magnitude of the danger; instead they help to aggravate it, sometimes in rashness, sometimes in ignorance. As a political actor and former ambassador, Guicciardini understands very well the contrast between surface appearances and the considerations that lie behind them, as well as the multiplicity of factors and pressures to be calculated and responded to, and the misconceptions and rumours which cloud judgement.
The initial French invasion in 1494 was actually short-lived, but it opened the gates and revealed the appalling military superiority and ruthlessness of the transalpine states. Nothing, Guicciardini sees, is the same again. The French fight to win, burning houses and refusing to take prisoners, and they employ terrifying new weapons with irresistible power, above all field artillery. Guns as siege weapons were well known and ponderous, but the French cannon, firing bronze balls, were rapid and manoeuvrable, drawn by horses, not oxen: "They used this diabolical rather than human weapon not only in besieging cities but also in the field" (I). Such weapons "rendered ridiculous all former weapons of attack which had been used by the ancients." Guicciardini, in fact, is quick to note the developments of his own day which surpass the knowledge and techniques of the ancient world. He notes Venice's loss of the spice trade to the Portuguese, who sail around the southern coast of Africa direct to the Spice Islands, and the growth in the power of Spain. These voyages "have made it clear that the ancients were deceived in many ways regarding a knowledge of the earth" (VI).
It seems appropriate that Guicciardini's own history, though in a sense humanist-bred, was not at all slavishly imitative of current models. He conforms to the convention of confecting speeches, which review policies and alternatives, but his own accounts of the complex webs of diplomatic relations are highly original; the cutting from one power centre to another is exceptionally rapid and at times, it must be admitted, bewildering. Guicciardini's technique is jagged, where humanist history is typically smooth and even bland. The power of Guicciardini's history is moral and intellectual; it is a tragedy presented as a tour de force in the organization of masses of detailed material and complex narrative. It is a challenge rather than a pleasure to read: there is point to the legendary story which tells of the prisoner who, made to choose between reading Guicciardini's accounts of the wars of Florence with Pisa or the galleys, after a few pages chooses the galleys. Not since Thucydides had the interaction between a number of states—small, vulnerable, ambitious and fearful, sharing a common language and culture but varying in constitutions—been subjected to such close and coldly rational analysis. There is nothing of comparable complexity in Livy's accounts of the early Roman republic in its struggles for survival and expansion, though, it has to be remembered, Livy's descriptions, unlike Thucydides' and Guicciardini's, were not contemporary or even nearly so. In general, however, the difficulties in translating Thucydides and the humanist's requirement of models for Latin eloquence—together with Machiavelli's preoccupation (derived from Livy and Sallust) with republican morale, the successes it brought, and the dangers to it that those successes represented—ensured that it would be the Roman historians who would chiefly inspire the writing of history in the humanist tradition down to the eighteenth century.
**PART V**
> **STUDYING THE PAST**
**NINETEEN**
**Antiquarianism, Legal History and the Discovery of Feudalism**
"Studying the Past," the title of this last group of chapters, marks a new beginning: the use, from the sixteenth century onward, of the textual methods of Renaissance humanism to reveal and understand not only the works of ancient philosophers and poets but the European past, which from the later seventeenth century began to be called the Middle Ages. This technique was archival historical research, though that phrase did not become current until much later, and by this means, inquiries could be carried back beyond the memories of the historian, or of eyewitnesses, and freed from dependence on earlier historians and chroniclers. This was a great transformation, and we have to ask how it happened. Its importance for history was subsequently overlooked. Partly this was the result of focusing on what was, from the Renaissance, the dominant prescription for historical writing: imitation of the revered ancient models, which meant chiefly political narrative from which morals could be drawn. Partly also it was a matter of limited intentions and a relative lack of self-consciousness. In a good many cases, sixteenth-and seventeenth-century scholars, then spoken of as antiquaries, stumbled into archival research rather than adopted it as a programme for history, and their motives, as we shall see, were often political rather than what the nineteenth century, when an ideal of historical research was made explicit, approvingly came to call "scientific." In the eighteenth century such work was often denigrated and condescended to in comparison with elegant imitation of classical models, as uncouth, useless, pedantic and, given the state of many manuscripts, dirty and ungentlemanly.
The long-term result, even after attitudes to archival research had changed, was an undervaluing of the admittedly far from elegant or even readable results of the work of antiquaries and hence a distorted perspective on the history of historical studies. In particular, this encouraged an exaggerated view (with the full backing of its protagonists) of the acclaimed "revolution" in historical methods in the nineteenth century, particularly in Germany. Germany certainly then took the lead, with state sponsorship, in creating institutions for historical scholarship and in establishing an education for apprentice historians in the emerging historical profession—above all in the famous Berlin seminar of Leopold von Ranke—with the critical use of primary sources as its defining ethic. This was immensely influential, but it is doubtful whether it counts as an intellectual revolution (see Chapter 25). A different emphasis has been made possible by the work of scholars over the past half-century. Therefore in this chapter we need to consider the work and motives of sixteenth-and seventeenth-century scholars, though their work is of a very different kind from the histories discussed in other chapters, since it consists not of available and potentially enjoyable narratives, but typically of commentaries on highly specific philological and antiquarian topics. It is essential, however, to have some idea of its consequences.
But before this we have to take further account of the most approved genre of historical writing in the period: political narrative written in emulation of the ancient historians. Guicciardini's _History of Italy,_ though in the humanist tradition, is highly original. After this achievement, humanist narrative historiography for the next two centuries has been widely regarded as meagre in interest if not in quantity, leaving few if any monuments whose appeal transcends their period and the concerns of specialists. This may be a lazy judgement, but it is not one I am able to controvert. The humanist prescription for history was imitation of the best ancient models—most notably Livy—by a smooth eloquence and an exclusively political focus. History was a literary genre in which truth took second place to rhetorical effectiveness in the provision of inspiring examples of good and great conduct. This is, to a modern reader, not an appetizing set of prescriptions.
The fierce republican enthusiasm which had given early humanist history an edge, though it continued to inform European traditions of political thought down to the English, American and French revolutions, waned in historiography during the European Age of Absolutism. The office of historiographer royal (established in England in 1661) in France was held by some distinguished—and subversive—incumbents, from François Hotman in the sixteenth century to Voltaire in the eighteenth. William Robertson, whom we shall attend to shortly, was historiographer royal for Scotland in the mid eighteenth century. But the title is potentially a double-edged gift to Clio: patronage, but also the possibility of control. In Tudor England and in the France of Louis XIV, prudent historians got the approval of authority in advance when handling sensitive subjects. The reign of Richard II, who had been deposed, was a ticklish subject in the time of Elizabeth. A French historian, Nicolas Fréret, found himself in the Bastille in 1714 for indiscreet views on the Germanic—and hence Tacitean—character of Frankish society and a forthright dismissal of Trojan origins: Tacitus' _Germania,_ a text newly available from the beginning of the sixteenth century, notoriously supported the absence of hereditary monarchy among the Germans. A Cambridge professor, Isaac Dorislaus—suspiciously a Dutchman, and therefore tainted by republicanism—had his lectures suppressed for the same offence a century before Fréret's misfortune, in 1627.
It will not do, of course, to dismiss the whole genre of humanist historical narrative as bland literary exercises in pedagogy and sycophancy. The author of the most exhaustive survey of humanist historical writing in sixteenth-century Italy has found it relatively free from obvious political censorship and warns against taking the pieties endlessly reiterated in prefaces entirely seriously as a guide to what lay beyond them (Fryde, _Humanism and Renaissance Historiography_ ). The growing vogue, in the later sixteenth century and all through the seventeenth, for Tacitus' _Annals_ and _Histories_ as the most admired model, rather than Livy, introduced a welcome diversification, though the enthusiasm for Tacitus also expressed itself in collections of maxims of political conduct: he was safer to cite than Machiavelli. Some historians and commentators aspired to a Tacitean epigrammatic brevity and subtle psychological portraiture—though Guicciardini could also be a source.
There was a particular fascination with dissimulation, as is sometimes revealed in memoirs of the period, which, being more covert, are sometimes more interesting than the histories. The _Memoirs_ of Cardinal de Retz, one of the leaders of the Parisians in the disturbances during Louis XIV's minority in the 1640s, are wonderfully candid, for example, about the ways in which, as distributor of alms for the Archbishop of Paris, he had used his position to build up a political following. An interest in court motives and the contrasts between public profession and private intention were particularly apt to a modern age of absolutism, not altogether unlike the times in which Tacitus had lived, when, as he himself recognized, the conduct of government was curial rather than public and displayed in oratory; the _arcana imperii,_ the closet secrets of despotic policy, exercised a fascination precisely because they were closed. Even in parliamentary early-eighteenth-century England the Viscount Bolingbroke, who had himself headed a government, shared Polybius' snobbery towards mere scholars, who "have seldom the means of knowing those private passages on which all public transactions depend...They cannot see the working of the mine, but their industry collects the matter that is thrown out" ( _Letters on the Study and Use of History,_ Letter 5). It was also true, however, in France at least, that public records of the past were becoming better organized and inspectable by recognized scholars, who were sometimes their keepers. The Valois and Bourbon rulers recognized the value of the past as a potential propaganda arsenal, as well as that of archives as a necessary practical resource, even if they had to be handled with care.
Two of the most interesting "Tacitean" narrative histories of the period were both written in the early seventeenth century: Francis Bacon's _History of the Reign of King Henry VII_ (1622) and the Venetian Paolo Sarpi's _History of the Council of Trent_ (1619), the great church council which in 1545–63, in the wake of Luther's Reformation, had refurbished the weapons of the Catholic Church. Bacon's work broke no new ground in its employment of unpublished sources, though he used some; mostly he relied on the available chronicles. But his psychological portrayal of the King impressed itself, perhaps ineradicably, on subsequent generations: Henry is seen as avaricious, parsimonious, prudent, suspicious and secretive—there seem to be elements of Tacitus' Tiberius—though Bacon also allows for his better, humane, qualities. Sarpi is presented by his leading modern English commentator, David Wootton, as himself a lifelong arch-dissimulator, for he was a friar and, according to Wootton, a convinced atheist. In his account of the Council, which to Sarpi's dismay laid the foundations of the Catholic Counter-Reformation, Sarpi excels, in Tacitean–Guicciardinian fashion, in the dissection of psychology and motives and in describing the historical ironies by which all the agents find their hopes defeated. Wootton sees an additional originality in the depiction of how individuals are governed not merely by individual motives but by the collective interests they represent: bureaucratic necessity rules their actions.
Blandness was by no means always a feature of humanist historiography. The civil wars in later-sixteenth-century France produced a copious stream of contemporary histories of a highly partisan kind. But the greatest contribution of humanism to history in the sixteenth and seventeenth centuries was not neoclassical prescriptions for narrative historiography, but the application of the methods of humanist critical scholarship to the study of the past. Though its full effects, which are still with us, were not to be felt until the eighteenth century, this is such an important innovation that it requires closer attention, even though there is no single, much less readily accessible, work of the period in which the non-specialist reader can find its principles, applications and achievements epitomized. It is an expanding tradition of practice and its widening applications, rather than a corpus of enduring works, which is of interest.
As the early Christian centuries, culminating in the world-historical summary of Orosius, marked a new epoch in the conception of the past and the approach to it (above, Chapter 13)—an epoch which was to last for a thousand years and more—so we now have to consider the opening of the modern era in the study of the past. Though there have been changes since, particularly those associated with the establishment of a historical profession, which we shall have to look at later, there has been nothing of comparably transforming influence. The main novelty is expressed in the phrase we have already glanced at: "the study of the past." It is this phrase that we must now try to unpack.
The impulses to historiography to be found in Herodotus are investigative and commemorative, but the former is in the service of the latter. In a sense, all history beyond mere transcription must in some sense be investigative, more or less profoundly or superficially, but as we have seen, for a long period, and not least in humanist historical narratives, the commemorative aspect loomed larger. History was the retrieval and presentation of what deserved to be remembered, against the remorseless flow of time. For a long time there was relatively little sense that the past lay inert, but potentially revivable, until quickened by the researcher and historian, in documents and archives. Hence the recording and writing of history could seem like a contest with an otherwise all-enveloping oblivion. Now, however, though it would be another two centuries and more before its full potential was realized, the investigative impulse was becoming urgent, guided and controlled by the methods that humanists had been developing from the late fourteenth century onward.
Initially this essential characteristic of the historian's craft derived not from an impulse which it is altogether appropriate to call historical, but rather from the impulse to retrieve, admire, even to imitate. It is easier, initially, to call it literary and moral rather than historical, and it was applied chiefly to past documents—texts—which were in a broad sense literary, including the work of classical historians. The method was philological, and its chief purificatory agent in the search for the authentic text was the detection of anachronism. This was made possible only by a cultivated sense, the product of erudition, of what the grammar, style and diction of late-republican Latin (in particular) made it possible, or more importantly impossible, for an educated Roman or literary master to have written and meant. (The model prized above all was Cicero.) The most dramatic single demonstration of the power of critical scholarship to revise the past by the detection of anachronisms was its application not to a classical and literary text, but to a medieval and political one. It was the famous demonstration by the great early-fifteenth-century Italian humanist Lorenzo Valla that the document purporting to convey authority in the West from the emperor Constantine to Pope Sylvester, the "Donation of Constantine," could not possibly belong to the fourth century AD and was therefore a medieval forgery. In the long run, even more far-reaching in its effects was to be the application of humanist critical techniques to the foundational texts of Christianity in the New Testament, notably by Erasmus in the early sixteenth century. But in general historiography the most influential and profoundly transforming development—initially in Italy in the fifteenth century, and flowering in France in the sixteenth—was to be their application to the corpus of Roman jurisprudence, the _Corpus Juris Civilis,_ compiled in the reign of the emperor Justinian and handed down as authoritative ever since.
Humanist scholarship was a powerful solvent of received views of the past and versions of past texts. The texts the humanists prized from the ancient world, which included those of Roman law, were in their modern incarnations the products of centuries of transmission and transcription. They were therefore apt to be overlaid with well-intentioned but anachronistic commentaries, interpretations, interpolations and misunderstandings, as well as scribal errors and forgeries, which to the humanist scholar were the silt and encrustations beneath which could be glimpsed the pristine gold. The method of obtaining it was literary archaeology conducted with a moral and aesthetic passion for recovering and making available for imitation the literary manners and cultural and political values of the cherished period. The humanists therefore developed an acute sense of what might be called the pathologies of cultural transmission: the vicissitudes, misfortunes and well-intentioned ignorant maltreatment as well as deliberate manipulations (with no sense of guilt) to which the priceless wisdom and literary heritage of the ancients had for up to fifteen centuries been subjected.
The application of philological methods to the reconstruction, in due course, not merely of Roman literature but of Roman history and eventually of the more recent—though still often remote—European past generally was a process of extension and transference. The methods of humanist scholarship were naturally applied to the Roman law texts, and through them to a greater understanding of the Roman institutions to which the laws referred. Then outside Italy, and in particular in sixteenth-century France and half a century later in England, the practices of legal historical scholarship came to be applied to the barbarian and medieval antecedents of modern legal institutions and customs. How, with what motives, and with what eventual consequences for the future general understanding of the European past will be the questions to which the rest of this chapter is chiefly devoted.
To summarize at this point the first part of the argument: literary archaeology led to and provided the tools for legal archaeology, and legal humanism led in turn to the study not chiefly of events but of institutions, the most important of which in the Middle Ages, namely feudal institutions, were so pervasive as to constitute what the eighteenth century would come to call "the state of society." "The state of society in the feudal age(s)," for example, is a common piece of eighteenth-century diction, but two centuries of energetic scholarship, mostly legal, were necessary to make intelligible what came to be so casually referred to. Concomitant with that scholarship were also the vastly increased dissemination of information of all kinds, including scholarly, following the invention of printing in the later fifteenth century, and the better organization of and access to archives, generally speaking (though there were still many obstacles). In these developments was to lie the future of historical studies, and indeed the very possibility of systematically "studying the past." Later we shall look at the long-term consequences for what, until the eighteenth century, remained relatively unaffected in its own humanist mould, imitative rather than investigative, overawed by its classical precursors: narrative historiography.
We begin, however, with the notion of legal humanism. Later, legal scholarship was to become notably entangled with and to be both stimulated and warped by political conflicts, particularly in England, but initially the application of humanist techniques to legal texts seems to have been prompted by the same enthusiasm for recovering and purifying the heritage of the Roman world that inspired humanism generally. But the consequences of this application in France, particularly centred in the middle and later years of the sixteenth century in the University of Bourges, were to be ironic. Inquiries conducted to uncover and apply the legal wisdom of the Romans, purged of the dross of medieval commentaries which had in fact been part of the process of the adaptation and application of Roman law in more recent societies, ended by identifying a Roman law so pure that it was in fact manifestly alien, the law of a past and different society. To purify turned out, unintentionally, to be to historicize and to distance in a disconcerting fashion. Some jurists bluntly faced this. In his _Anti-Tribonian_ (1567), François Hotman proclaimed that Roman law, being the law of another time and place, was useless to French jurisprudence; he focused attention instead on the study of native, customary law.
In fact the study of Roman law in humanist fashion made it historical in a double sense: not only was it made remote, but it was historically relativized. It was not an ideal, frozen in time, but was itself the product of a long legal history, during which Rome had itself altered greatly; it had been collected only at the end by Justinian's legists, who themselves seemed sometimes uncomprehending of what they were dealing with. There was, then, no single jurisprudential moment, permanently available, once recovered and returned to its original condition, for use in the present.
There was a parallel here with the disputes among humanists over the authority to be accorded to the works of Cicero: was the Latin of Cicero mandatory for all time? That it should be was recognized as an indefensibly extreme view and was mocked accordingly, but the problems of jurists were greater. Working with an ideal but arguably unduly restrictive model, it was possible to write Latin prose which could pass as that of the first century BC, though it was at the expense of the flexibility of the colloquial Latin which in some circles was still a living tongue, now stigmatized as barbarous. But even if the ideal legal model could be identified, lawyers could not avoid questions of its applicability to an intractable contemporary reality. The alternatives were to relegate Roman law to the status of a purely academic study or to face the necessities of adaptation such as the medieval glossators had faced in their day.
A consequence of making Roman law a historical phenomenon rather than a timeless model, and of the growing perception of stubborn realities, presumably of more recent origin, to which it could not be fitted, was the recognition that in large part the characteristic laws, customs and institutions of the modern nations must derive from these nations' barbarian ancestry, or at least have evolved since the establishment of the barbarian kingdoms, owing perhaps nothing to Rome and to imperial edicts. One possible inference was an exaltation of local customary law, as suited to the people or peoples who lived by it. This was Hotman's solution. The charge of barbarism, in a sense true, had to be faced down, and historical scholarly attention had to be refocused from Rome and the empire to—as we should say—the early and high Middle Ages. In discerning the alienness of much of the late-Roman codes, the sixteenth-century French jurists were in a sense discovering medieval history—though not yet by that name—and recognizing medieval Europe not merely as ignorant, barbarous and uncomprehending, but as creative. For jurists who wished to discern the origins and guiding principles of their own laws, the Middle Ages were fundamental.
There was a patriotic and even sometimes a populist dimension in this. Roman civil law, in the late-antique compilations in which it had been chiefly inherited, tended naturally to be markedly authoritarian. Roman-law maxims included the notorious "Quod principi placuit legis vigorem habet" ("What is pleasing to the prince has the force of law") and the idea that the prince is "legibus solutus" (not bound by the law). Customary law, on the other hand, tended to endorse inherited private and corporate rights and privileges: to be, in fact, feudal in character. Imperial and customary were in a sense antitheses—as were the rival versions, Trojan and barbarian, of the origins of the European peoples and monarchies. The Trojan connection, following the example set by Virgil, embodied in Brutus, Francus and other eponymous émigré Trojan princes, was a flattering one, linking the people concerned, through cousinship to Aeneas and common descent from Priam, to the Romans. In the early modern period, as humanist scholarship promoted an ever-increasing scepticism and eventual dismissal of legendary Trojan princelings unknown alike to history and to ancient literature, and as Tacitus' _Germania_ became increasingly known and influential, the original "freedom" of the Germans became launched on a long career in European historiography.
We must not, however, produce an oversimplified picture of humanist scholars scouring the prehistory of Europe with the detergent of philological method: their own works could provide new sources of confusion, and even new forgeries. Moreover it was not only the ubiquitous Trojans who had given kings and legendary inhabitants to pre-Christian, pre-literate Europe. All of them must in theory have been traceable back to Noah, and some were explicitly so, for the Bible, as well as Virgil's fiction and classical legend, was a major source for early history, providing giants (who seem to have been common) and near-relatives of Noah like his grandson Gomer, who became regarded as progenitor of the Gauls. The sixteenth-century antiquary John Bale had pre-Trojan Britain founded by Samothes, son of Noah's son Japhet, one of whose descendants, Druys, founded the order of the Druids. Imaginary genealogies were only reluctantly abandoned, partly because an assured grasp of the authentic classical canon was necessarily a work of erudition and time, and partly because the alternative, apart from the discovery of the _Germania,_ was not so much new knowledge as a void. With the potential of archaeological delving still unrecognized—though the importance of artefacts (and particularly classical artefacts) as witnesses was beginning to be appreciated—virtually nothing could be known of the pre-Roman period in the past of the northern European nations. In the 1570s, in his history of Scotland, George Buchanan, an accomplished French-nurtured Scottish humanist, though a partisan and unreliable Protestant historian of his own time, the reign of Mary Stuart, was purveying with apparent confidence a list of forty legendary pre-Christian Scottish kings with such names as Thereus, Durstan, Mogallus and Athirco. He only reluctantly began to lose confidence in them after their rejection by the Welsh antiquary Humphrey Llwyd, who was scornful of Scottish pretensions, though himself happy with Geoffrey of Monmouth.
The world of sixteenth-century humanism was a copious and complicated one to find one's way around, even for the erudite, and particularly if we include works overlapping with narrative history—historical topographies, etymologies, "universal histories" and even the beginnings of annotated bibliographies. It has become customary among modern scholars to distinguish strongly in this period and down to the eighteenth century between neoclassically modelled narrative histories and humanist antiquarian learning with no pretensions to elegance. It was a distinction which contemporaries themselves remarked, and on the historians' side it was reinforced with a good deal of abuse of useless, pedantic and uncouth learning. In his _Letters on the Study and Use of History,_ Bolingbroke, whose historiographical models were the trinity revered by the humanists, Sallust, Livy and Tacitus, and who propounded the exiled statesman's view of history as made useful by the instructive examples it could yield, denounced scholars who made "fair copies of foul manuscripts, give the signification of hard words, and take a great deal of grammatical pains" as mere under-labourers. Even with the most eminent of them—he mentions Scaliger, one of the greatest of Renaissance scholars—he is prepared "to avow a thorough contempt for the whole business of these learned lives; for all the researches into antiquity, for all the systems of chronology and history" he would rather make mistakes in these "than sacrifice half my life to collect all the learned lumber that fills the head of an antiquary" (Letter 1). Now, however, we tend to find the humanist prescriptions for history depressingly imitative, while at least some of the admittedly often disorderly and miscellaneous work of the antiquaries is pregnant with reappraisals of the European past.
Ancestors mattered—even remote ones. Medieval Europe was a legalistic society in which, except specifically in contexts where Roman civil and canon law held sway, rights and therefore much of the structure of recognized obligations were characteristically justified by existing inherited title rather than philosophical principle. Sometimes the only witness to such entitlements was custom, though increasingly they came to seem insecurely grounded without documentary support—a powerful incentive to forgery, at least in the benign form of apparently guaranteeing what one did have rather than laying claim to what one did not.
At the highest political level, documents such as the spurious "Donation of Constantine" could be powerful support for or even be deemed essential to a vital claim. We are told of Edward I of England circulating the monasteries, where such documentary "memory" was most likely to be preserved, if only in the form of recorded custom, for support for his claim to the overlordship of Scotland; when he had the opportunity, he destroyed Scottish archives. In the sixteenth and seventeenth centuries, precedent-hunting and the search for documentary authority became, in some contexts, urgent, calling on the resources of modern interpretative skills and erudition, honed on the study of Roman law but increasingly applied to more recent national history. Given the powerful tendency to cast political claims and controversies into a legalistic mould, the appropriation and possession of an apparently authoritative version of the national past could be highly relevant. Some version of "ancient constitutionalism" as political argument can be found in a number of European countries, including Scotland, where it was purveyed by Buchanan, and the Netherlands.
Historical–legal arguments were not, of course, the only ones. Biblical precedents, theological dicta, and even references to Aristotle's _Politics,_ along with the lessons of experience and history, were also canvassed. The historical examples—particularly but not exclusively the Roman ones—could be treated in Machiavelli's fashion, as a storehouse of examples rather than of rights and precedents. But the sixteenth and seventeenth centuries, particularly in France and then, half a century later, in England in the period preceding the Civil War, were the great age of invocation of origins, precedents and long-established or immemorial rights in constitutional and political debate. Such arguments were naturally the province of lawyers, or at least of the legally trained, as many of the educated laity were. Although this mode of argument could sometimes produce only a fairly mindless sequence of citations, regardless of historical contexts, it became increasingly historically sophisticated. In some contexts, too, the legal-antiquarian passion, applied heterogeneously to various aspects of the past, bypassed political argument to become an investigation of origins and derivations conducted for its own sake, or from an impulse that was broadly patriotic rather than directly political.
Both patriotism and politics, in fact, were present in the legal—historical scholarship of sixteenth-century France, fusing in what came to be called "Gallicanism," the assertion of the historic distinctiveness of the realm of France, independent of both papacy and empire. But Gallicanism could accommodate also, though by no means always, a kind of anti-royalist populism, in references to Tacitean and Frankish "freedom"—which the word "Frank" itself was held to embody. It was harder, obviously, to incorporate the Gauls, later the Gallo-Romans, in the story of an independent France, though attempts were made and there were Celticists as well as Germanists. The circumstances of later sixteenth-century France—plunged into civil war, massacres, political assassinations and a succession crisis—gave such historical ideas a polemical relevance. Religious conflict threatened to tear France apart. A number of the most notable jurists were Huguenots or at any rate moderates drawn to the party of patriotism and order, the Politiques. The extreme Catholic party, the Holy League, under the leadership of the family of the Guises from Lorraine and, in collaboration with Spain, holding a papalist, "ultramontane" position, could be seen as representing the prospect of the de facto subjection of France to a foreign power. It was the great crisis of the French monarchy, as the first half of the seventeenth century was to be for the English one when supporters of parliamentary privileges and the royal prerogative searched the record offices, in Westminster and the Tower of London, for legal precedents—the earlier the better—for their respective positions. It was strongly disputed whether Parliament, including the House of Commons, which in fact owed its origin only to royal writ in the thirteenth century, was coeval with or even older than monarchy. The question of Saxon "freedom" and its alleged continuity with the present, unaffected by the Norman Conquest, was, like the equivalent for the Germans of which it was an offshoot, launched on a historical career which carried it well into the nineteenth century. Even in Elizabeth's reign, Anglo-Saxon studies were patronized by Archbishop Parker in the hope that they would demonstrate that the Church in England had always been independent of Rome. It was a preoccupation analogous to aspects of French Gallicanism.
The most politically embattled of the French sixteenth-century jurists was François Hotman. In his _Franco-Gallia_ (1573) Hotman unrolls precedent after precedent from the chroniclers, including Gregory of Tours and Fredegar, to show that the ancient kings of France were elected and could be deposed. For Hotman, the French Estates General were descended from the old Germanic assemblies. The evolution of Germanic custom also embraced the feudal bond. Some authors derived this ultimately from the institution of clientage in the Roman world, others from the Teutonic _comitatus_ of the chief's immediate entourage, whose position later became secured by hereditary tenure of land on condition of military service, establishing the full feudal relationship.
One feature of the enthusiasm for custom and the history of law was that, though it encouraged an interest in antecedents, it could also accommodate the idea of a subsequent evolution and even be accompanied by a scepticism about the possibility of tracing remote origins. Seen in this way, the history of law could provide the basis for a new kind of national and indeed comparative history, encompassing customs, manners and even ideas. For example, the _Recherches de la France_ (from 1560) of Etienne Pasquier (whose choice of the vernacular is striking) was professedly a search for the distinctive _ésprit_ of France and was highly patriotic in inspiration. It was not a narrative history but a collection of monographs on different aspects of French culture, essentially philological in method, with commentaries on the history of words for institutions and customs. French authors in the period, notably Jean Bodin, were also making attempts in the direction of secular universal history, tracing a history of civilization from primitive origins; Bodin specifically cites Thucydides and attacks the Christian and medieval periodization of the Four Empires, derived from the Book of Daniel, which made Rome the last empire before secular time would give way to eschatology. The word "civilization" itself was not coined (in France) until the later eighteenth century, so we have a construction like "le temps ou a commencé la civilité" (Loys le Roy, 1575). The need was proclaimed, if not satisfied, for secular general history of religion, laws, customs and manners ( _mœurs_ ), not provided by the classical historians.
There was no such rigorous reappraisal of the past in sixteenth-century England as was offered by legal humanism in France: that had to wait until the mid seventeenth century and the spur provided by the conflict of the Crown and Parliament. But antiquarian learning was vigorously pursued in Elizabethan England towards the end of the century, often under the influence of a patriotic enthusiasm, a kind of love affair with England and Englishness in which sixteenth-century English histories and chronicles played a notable part. After the accession of Elizabeth in 1558 on the death of her Catholic sister Mary, the story of English history was refurbished in a fashion that was Protestant, patriotic and providential. The greatest single influence seems to have been John Foxe's _Acts and Monuments,_ which went through successive and enlarged editions until an eight-hundred-page folio, with woodcut illustrations, was published in 1570. Because its second half set out the record of Protestant martyrdoms in Mary's reign, which was the task with which Foxe had begun, it became known as Foxe's _Book of Martyrs._ But Foxe set these martyrs in a wider context of Christian and English history as a whole, which also became influential. For the facts in this he relied heavily on the Italian humanist Polydore Virgil's _History of England_ (1535), which Foxe adapted to Protestant purposes in terms of a long struggle in which the Christian faith, grounded in the Bible and sometimes represented by a godly English monarch (Arthur, Alfred, Elizabeth), confronts first paganism (Saxons, Danes) and then the authority of the corrupt Church of Rome. After the sixteenth-century English translation of the Bible, Foxe's book has been spoken of as the greatest single influence on English Protestant thinking of the late Tudor and early Stuart period.
Following the Reformation, there was a strong desire to show that England had received the Christian revelation independently of Rome, and this involved a rehabilitation of the Christian Britons whom Bede had despised; the Britons were already fashionable owing to the cult of Arthur and the accession of the Welsh Tudor dynasty. The idea of an autonomous origin for English Christianity was fostered by Elizabeth's Archbishop Parker (1504–75), a notable patron of antiquarian studies. The supposed visit to England in the first century AD of Joseph of Arimathea, one of Christ's disciples, and possibly even of Christ himself, derived from a medieval legend (part of the Holy Grail cycle), was appealing, and still echoes in William Blake's _Jerusalem_ (1804–20). Protestant history required significant reappraisals. Henry II was the hero, not the villain, of his conflict with Archbishop Becket, defender of clerical privileges. Henry IV was not merely the usurper of his cousin Richard II's throne, but was condemned as persecutor of the Lollards, the English proto-Protestants of the fifteenth century, who were patriotically credited with a seminal role in the European Reformation.
True Christianity had triumphed in modern times. The Tudor dynasty, having reconciled the nation after the Wars of the Roses and put an end to the Lancastrian usurpation of the Crown, had produced in Henry VIII the great vindicator of the rights of an independent English Church against the spurious claims of the papacy. His daughter Mary's reign (1553–8), however, had seen not only the persecution of the true Protestant faith but also the dominance of foreigners, the Pope and Mary's husband, King Philip of Spain, in English affairs. Elizabeth's accession was therefore a providential deliverance not only of Protestantism but of the nation, which the scattering of the Spanish Armada in 1588 by "the winds of God" confirmed. Foxe compared Elizabeth to the emperor Constantine (born in England and the son of a British Christian queen, Helena) as liberator and protector of the Christian Church, and himself to the historian Eusebius. Religious truth and national independence were bound together, while England was the major Protestant power in Europe. It was obvious that the English were a Chosen People, and English history a providential story of national and religious independence. The Elizabethan chroniclers John Stow and Ralph Holinshed (drawn upon by Shakespeare) popularized this version of the nation's past and role, much assisted by cheap printings in convenient, readily portable octavo volumes. Stow's _Summary of English Chronicles_ (1565) and Holinshed's _The Chronicles of England, Scotland and Ireland_ (1577), which was influenced by Foxe, were highly successful publishing ventures.
Patriotic history was given solidity and depth by patriotic antiquarianism. Fine private libraries and manuscript collections were formed by Parker, Lord Burleigh, Elizabeth's chief minister, and Sir Robert Cotton, and these later came to form the bases of public, university and college libraries. The Elizabethan Society of Antiquaries, composed largely of lay gentry with, as was common, some legal training, was evidence of a cooperative spirit which made such holdings accessible by fellow scholars.
Another aspect of the patriotic antiquarian impulse was the appearance of the great topographical surveys of England, whose precursor was John Leland's _Itinerary_ (not published until 1710). The land itself and the evidence it bore of its history became objects of loving attention and investigation. The two strands, Protestant providential history and topographical antiquarianism, were not wide apart. Leland's disciple John Bale (above, Chapter 19) became a radical Protestant and Foxe's mentor. The most fully achieved topographical work was William Camden's _Britannia_ (1586). Other notable publications in this vein were William Lambarde's _Perambulation of Kent_ (1574), the first known county history, and John Stow's remarkable _Survey of London_ (1598–1603).
As a scholarly genre, topographical antiquarianism had had European predecessors—particularly Florio Biondo, whose _Italia Illustrata_ dated from the mid fifteenth century. A more immediate model was the map-making with in-depth scholarly historical surveys which is known as chorography, and whose sponsor was the great Flemish map-maker Ortelius, who encouraged Camden's work. After Camden, with the monasteries gone, the patrons of local history in England were inevitably the gentry, who had an overriding preoccupation with genealogies. But tenures and the names and histories of offices were of interest, and from this sprang the work on the history of tenures by the great antiquaries of early Stuart England John Selden and Sir Henry Spelman, which was also to throw light on the early history of Parliament.
Feudalism, as the possible critical break between Roman customs and those of the barbarian kingdoms, inassimilable to Roman civil law, was of peculiar interest in sixteenth-century France. Hotman wrote a treatise on feudal tenures ( _De Feudis,_ 1572). Such endeavours involved working back from more modern times, looking in particular at changes in legal terminology and trying to make out if they were superficial changes of nomenclature or pointed to real changes in practices. In his classic study of the beginnings of the historical study of law in seventeenth-century England, and its consequences, _The Ancient Constitution and the Feudal Law,_ John Pocock showed half a century ago how, although the early-seventeenth-century English lawyers focused attention on legal precedent and documentation, their addiction to notions of an immemorial common law and an immemorial constitution inhibited proper historical investigation of the kind the French jurists had pursued so effectively. In France the legal situation was complex, with no single common law, with a distinction between the areas of customary law and those of written law (the _pays du droit écrit_ ), and with a larger presence of Roman law. This complexity had acted as a spur to sophisticated historical investigation to trace the respective strands, and even to the beginnings of a comparative approach to law. England, by contrast, was parochial and self-satisfied, and the legal reverence for the authority of the past was essentially _un_ -historical: English law was what it had always been, a perennial and therefore in a sense timeless body of custom.
The breakthrough, in Pocock's highly persuasive account, was chiefly to be found in Sir Henry Spelman's work on the history of feudal tenures, beginning with his _Glossarium Archaeologicum_ (1626). The feudal tenures revealed by this were distant not only from the institutions recognized in Roman law, but also from those of recent times. The multifaceted character of feudalism, to use the modern term, made it particularly well adapted to stand for the character of a whole type of society. Its central relationship was at once legal and social, military and economic, accompanied in its maturity by an ethic of loyalty and honour, all perpetuated through inheritance. It was at once a way of organizing military force, a social hierarchy, an ethos and what Marx would later call a mode of production.
Some of this was, of course, as yet more implicit than explicitly conceptualized. Spelman's method was, as usual, philological. In form, as was common for such antiquarian inquiries, his work was a dictionary or glossary of terms for usages, ranks, offices, customs, tenures and so on, arranged alphabetically like an encyclopedia. It firmly rejected Brutus and legendary Trojan origins. English law, Germanic and Anglo-Saxon in origin, and the feudal relationship were the product of an evolution, an adaptation of barbarian custom to new circumstances of settlement on the land and its hereditary transmission. The _feud_ or _feudum_ had been, in fact, the key to English law as a whole; Parliament, as the House of Lords, was originally the council made up of the king's tenants-in-chief, i.e. those with no feudal superior between themselves and the Crown, attending as one of their obligations to the feudal lord; the House of Commons, not mentioned in Magna Carta (1215), could not therefore have been in existence before the thirteenth century. Spelman also knew that the feudal relationship had declined, though he had no account of this. As Pocock puts it, "The feudal revolution in English historiography was to impose on English history the division into pre-feudal, feudal and post-feudal periods which has ever since characterized it." The great public crisis of Spelman's lifetime, the Civil War between Parliament and King, culminating in regicide, confirmed the conception of a post-feudal, modern period, for the regicide was not an act of the barons but of the House of Commons, which essentially arrogated sovereignty to itself.
The most general inferences from this for an overall conception of history were drawn by James Harrington in his _Oceana_ (1656). Harrington was a republican, and he produced a kind of optimistically republican schematization of general history, whose most modern part rested on the view that the monarchy had been overthrown not by the feudal tenants-in-chief but by the independent gentry represented in the House of Commons, who were now the decisive power in the commonwealth. From the vantage this provided, Harrington produced a remarkable retrospective summary—not a narrative—of general European history, with Roman republicanism as its starting point, modern freeholder republicanism as its present state, and the feudal relations based on a distinctive form of land tenure, which Harrington called "the Gothic balance" (between king and barons, with the people as a makeweight), as the pivot. The key to his history is the distribution of landed property, which is the basis of all power. The engrossing of land by the rich, unchecked by the attempts of the Gracchi to impose redistribution by agrarian laws, had destroyed the Roman republic in destroying the basis of its citizen armies. The emperors depended instead on a hired professional soldiery who became increasingly "Gothicized." In due course the military benefices distributed as the rewards for service became hereditary and conditional on future service; this was the essence of the feudal tenure of land and military vassalage and hence of the Gothic balance, with the people as subordinate tenants of feudal overlords.
Harrington treats the decline of the balance, initially in favour of the king, as recent, citing, from Bacon's _History,_ Henry VII's legislation against private armies of retainers. Crucial also, in his account, however, is a diffusion of landownership among the gentry following the dissolution of the monasteries, creating thereby a powerful class of independent freeholders, outside the feudal hierarchy. The opportunity thus exists for a revival of a freeholder republic with a citizen militia. Pocock sums up Harrington's _Oceana_ as "a Machiavellian meditation on feudalism." Of course Harrington's account of the decline of feudalism is not acceptable to modern scholars; his terminology differs from theirs, and his monocausal scheme is crude. Nonetheless it is possible to make out here the general characterization of the course of European history, and some of its causes, which, with modifications and fits of restlessness, historians have mainly lived by ever since. Though Harrington's scheme is less comprehensively ambitious than the earlier tentative moves towards secular universal history (including social and cultural history) among the French, it is more specific and fully achieved. The key had initially been that of custom, and subsequently the conception of feudal tenure as an institution which had had a beginning and an end, even if both, and particularly the latter, were still obscure. Harrington provided an overview firmly based on what the eighteenth century would freely call "the state of society," the most significant changes in which—not the sequence of the Four Empires—would be the successive turning points of the historical story.
**TWENTY**
**Clarendon's _History of the Rebellion_ : The Wilfulness of Particular Men**
The exaltation of the Tudor monarchy and the cult of Queen Elizabeth among the chroniclers during her reign is so marked that the subject taken by England's greatest seventeenth-century historian, Edward Hyde, earl of Clarendon (1609–74), "the Great Rebellion," as he called it, has almost the appearance of a paradox. Yet there is a continuity of a kind. Though the breakdown of relations between King and Parliament from the 1620s to the 1640s was fed by fiscal and constitutional tensions, a powerful underlying as well as often overt theme in it was the idea of an "incomplete" religious revolution. Elizabethan historiography's incorporation of the idea of the English as a Chosen People was not a piece of complacency: it was implicitly a call to action, to the fulfilment of an exacting, divinely imposed role. Despite their discontent with the compromises of the Elizabethan church establishment, the more extreme Protestants, who were beginning to be called Puritans, had never turned against the Queen herself, whom they saw as their deliverer from Catholic persecution. The continuing fear of a Catholic succession would alone have held disloyalty in check. But radical Protestant aspirations, combined with what was seen as defence of the historic privileges of Parliament, became importunate in the reign of Charles I (1625–49). It was the breakdown of relations, the formation of two opposed sides in Parliament, and the ensuing civil war that provided Clarendon, who had been intimately involved, with the subject of his history.
Clarendon's history provided the bedrock for all subsequent accounts of the English revolution. _The History of the Rebellion and Civil Wars in England_ was begun in the thick of the events described, in the 1640s, with the encouragement and assistance of King Charles, and clearly with the advantage of many eyewitness accounts. Clarendon, whom until his ennoblement after the Restoration we should think of as Edward Hyde, was highly advantageously placed, first as a leading member of the Long Parliament and later as a member of the King's Privy Council and one of his chief advisers. The book was completed, as many ancient histories had been, by the author as an elder statesman in exile, having fallen from power as Charles II's Chancellor in 1667. He died in France in 1674, but the _History_ was not published until the beginning of the eighteenth century, when it became an immense success and probably the only work of history whose publication financed the erection of a great building, the Clarendon Building in Oxford.
Clarendon's history both is and is not a work in the classical mould of historiography; it is certainly not in the later Enlightenment one. Some of it was originally written in the form of an autobiography, and it retains some of the informality of a memoir, though it aspires to be comprehensive and to treat of events that Clarendon knew of only from a distance as well as those to which he was privy. It is undeniably from the latter, however, that the history chiefly derives its striking vitality. His manner is noticeably more relaxed and less stately than, say, Hume's in his history of the period published half a century later. Hume, for example, half apologizes for retailing from Clarendon some absurd behaviour of James I as "though minute...not undeserving of a place in history" (Chapter 14). Clarendon himself clearly felt no need for justification. His work is, of course, focused on public events, as an attempt to understand a catastrophe. It fastens on the characters and behaviour of the public men the author had known—including the King—diagnosing the weaknesses, misjudgements, ambitions and anxieties which had led to the catastrophe, in the classical and humanist manner. Clarendon frequently weighs up alternatives and discusses how things had been allowed to go wrong, and is free with personal reminiscence. Though he sometimes uses the first person and sometimes "Hyde," his references to himself, though they do not strike the reader as vain or obsessively self-justificatory, have none of the chilly austerity that Thucydides, for example, preserves towards his own role in events.
Clarendon's history is naturally and unashamedly partisan: he writes to judge as well as to recount. But his was the partisanship of a moderate who above all sought reconciliation, first in Parliament and later during the war. He had friends—and enemies—on both sides, and he is measured, not passionate, in his judgements. It was the King who had suggested the project to him, and it was intended as a vindication of most, though not all, of Charles's actions. Clarendon quotes and comments on many of the documents produced by the two sides in the early 1640s, some of which he had drafted for the King: this is, frankly, more than the general reader requires. Clarendon's view of events is naturally to some extent circumscribed after the outbreak of the war. He knows the royalist side better than the parliamentarian, and the campaigns in the south, from his residence in Oxford, better than those in the north. He is also, of course, conditioned by his own cast of mind, which strikes the reader as honourable, shrewd, conciliatory where possible, but also, as one would expect in an early-seventeenth-century parliamentarian, legalistic. He had originally been himself a leader among the parliamentary opposition, but his early associates, notably Pym and Hampden, came to take up positions he regarded as untenably extreme. He did not so much change sides as retain his original principles while events moved on around him, and as the parliamentary leadership became more radical and innovatory this inevitably propelled him into becoming spokesman for the King; it is possible that, after the disasters of the later 1640s, he wrote the history, with hindsight, as more of a royalist than he had been even after he entered the King's service. He was prejudiced against the Scots, disliking their Presbyterian obsessions and their treatment of both Charles I and his son when they were in their power. After having originally taken the parliamentary view of the need to curb the royal prerogative and recognized the parliamentary radicals' fears for their own safety and their need to ensure it, Clarendon came to regard the hard core of the parliamentarians—"the violent party," led by Pym—as having a long-term plan to subvert the existing constitution.
His general view of events was far from that of Hume later, though neither sympathized with Puritanism or arbitrary rule. Clarendon's was the outlook of a political tactician who saw problems, in so far as he did not view them legalistically, as a matter of getting the right men in the right jobs, and who attributed very wide consequences to the King's frequent failure to do so and to the misguided advice he too often received and acted on in consequence, as well as the harm done by the excluded who were made unnecessarily disaffected. A prime case of the latter was, according to Clarendon, the treatment of the Earl of Essex, who, having been dismissed from the Privy Council, remained in London when the King moved his court to Oxford (1642) and became general of the parliamentary armies. He could, in Clarendon's view, have been won over, with incalculable consequences. If he had been retained in office "he would never have been prevailed with to have taken command of that army which was afterwards raised against the King's...And there can be little doubt in any man who knew well the nature and temper of that time, that it had been utterly impossible for the two Houses of Parliament to have raised an army then if the Earl of Essex had not consented to be general of that army" (V.33). Clarendon may have been over-optimistic, but he had the invaluable advantage over later historians like Hume, writing consciously from a wider perspective, of remembering a time when nothing seemed inevitable.
Clarendon's history, following the unfolding of events, falls in fact though not formally into three sections: the approach to war, the war itself, and the years of exile after the royalist defeat. Initially he was in the thick of the parliamentary debates, though increasingly alienated from the radicals, participating in negotiations with the Court and later within the Privy Council, and taking a leading part in the war of words, of declarations, remonstrances and rebuttals. After the final rupture and the outbreak of war, he was in the royal capital of Oxford, as a member of the Council and adviser to the King. He was influential, but necessarily a spectator of the war itself, though a well-informed one. His accounts of the military operations are free from convention or attempts at the picturesque. They are down to earth, with an awareness, obviously derived from participants, of the misunderstandings and of plans gone awry in the fog of war. Above all, he dwells on the importance of logistics and the problem of discipline, on pay and supply, and on ordnance, weapons and ammunition and the value of capturing them. He meticulously notes and sometimes laments the deaths in action of prominent nobles and gentry. This was a classical and chivalric convention, but in a war in which recruitment and discipline depended so much on personal and local status and support it was also a highly practical matter. It was, as Clarendon notes, not so in the parliamentary army: "The officers of the enemy's side were never talked of, being for the most part of no better families than their common soldiers" (VIII.160).
Clarendon was eager throughout the war for any possibility of reconciliation. One moment in particular beckoned; it was delusive, but for a short while it gave Clarendon great personal prominence and the role to which he was suited—conducting a war of words, once again face to face and with the possibility of backstairs negotiation and the cultivation of personal relations with opponents. This was the Uxbridge Conference, held with a view to drawing up possible articles of peace, in February 1645. Clarendon was one of the King's commissioners there, and he describes the events of the next three weeks with great particularity, over thirty-five pages (VIII.215–50). Though he does the arguments full justice, his account of the negotiations also evokes a solid world, of carefully arranged chairs and meeting rooms, and men from both sides huddled around the fire on cold nights reviving old companionships. According to Clarendon, the parliamentary commissioners were made wary by the presence of their allies the Scots, which made them fearful to be seen alone with the royalists, "their old friends whom they loved better than their new." The three main subjects of contention, to each of which was allotted a certain number of days, were religion, the control of the militia, and the settlement of the Irish rebellion. In discussing the religious issues—essentially the demand for the adoption of a Presbyterian form of church government—the royal commissioners exploited the divisions among their opponents and the ambiguities in key terms. The Scottish Earl of Lauderdale, being put up to explain, made a hash of it, and the royalists demanded a clarification in writing, which "put the Scots' commissioners in great passion; for all the English sat still without speaking a word, as if they were not concerned. Various divines on each side were brought in to argue the points."
Days were thus consumed fruitlessly, and Clarendon's hostility to the Scots is apparent. One example of relaxed conversation out of the session is introduced by Clarendon as "a pleasant accident."
The commissioners of both sides, either before their sitting or after their rising, entertaining themselves together by the fire-side, as they sometimes did, it being extremely cold, in general and casual discourses, one of the King's commissioners asking one of the other, with whom he had familiarity, in a low voice, why there was not any mention of the Lord's Prayer, the Creed, or the Ten Commandments [in Parliament's religious proposals], (as indeed there is not), the Earl of Pembroke, overhearing the discourse, answered aloud, and with his usual passion, that he and many others were very sorry that they had been left out; that the putting them in had taken up many hours' debate in the House of Commons, and that at last the leaving them out had been carried by eight or nine voices...Which made many smile, to hear that the Lord's Prayer, the Creed and the Ten Commandments had been put to the question, and rejected...(VIII.232)
The other two topics were as intractable, control of the militia touching the parliamentarians' fears for their own safety after a treaty, and the King's failure to suppress the Irish rebellion being as much an embarrassment to his side as Presbyterianism to the English parliamentarians. Privately, relations warmed, and Clarendon was able to note the cracks appearing between the parliamentary peers, in particular, and the more radical members of the Commons and the army. But he says he placed no hopes in the former, for the earls of Pembroke and Salisbury feared the latter more than they hated them, and, though they would rather have seen the radicals destroyed than the King, "they had rather the King and his posterity should be destroyed than that Wilton should be taken from the one of them or Hatfield [their estates] from the other; the preservation of both of which from any danger they both believed to be the highest point of prudence and politic circumspection" (VIII.245). The conference collapsed without outcome. It had sat regularly, Clarendon tells, till one or two in the morning, apart from the preparing of papers, "so that if the treaty had continued longer it is very probable many of the commissioners would have fallen sick for want of sleep."
From 1645 onward Clarendon was condemned by the parliamentary victory to a wandering life in the entourage of the Prince of Wales, in which his chief endeavour was to prevent the latter from compromising himself in the eyes of English opinion by association with the Catholic and Francophile inclinations of his mother, the Queen. Chiefly, throughout the years of Charles II's exile, Clarendon says he counselled doing nothing but wait and endure, which was surely the correct tactic. It is not surprising that he saw the Restoration in 1660 as an act of Providence. Meanwhile, Clarendon's account, from a distance, of the indignities to which Charles I was subjected, though second-hand, has a pathos which clearly derived from the personal attachment Clarendon had formed to him. There is the description of his physical appearance, for example, which Hume took from him: grey, with his hair uncut and his clothes old, "so that his aspect and appearance was very different from what it had used to be," though he was unexpectedly cheerful (XI.157). Clarendon refuses to dwell on the King's last hours and his execution, on the grounds that they are both too distressing and too well known.
The main interlude in his account of the years of exile is the familiar story of Charles II's escape after the battle of Worcester. Second-hand but clearly from the best possible source—the King himself—it is highly circumstantial, taking twenty-two pages of text (XII.84–106). Clarendon takes an understandable pleasure in the "concurrence of good nature, charity, and generosity, in persons of the meanest and lowest extraction and condition" who assisted the King, not knowing him but knowing they could get a reward for revealing the fugitive. He also mentions the King's recounting "many particulars of the barbarous treatment" he had received in Scotland, where he had been subjected to sermons and obliged to sign the Presbyterian Covenant.
Inevitably, in general the years of exile and even the period of the Civil War do not have quite the day-to-day dramatic interest of the early years in the Long Parliament, when Clarendon, as a leading parliamentarian and later adviser to the King, was striving to avert the breach which led to war, and it is to these that one wants to turn in conclusion to take the measure of the quality and interest of the _History._ Clarendon's accounts of the sessions and the private conversations are extraordinarily vivid simply because of the way he situates his narrative, among the often tired, anxious, bewildered Members of Parliament, meeting sometimes heatedly in overcrowded, noisy, candlelit rooms. In the debate in 1641 on the proposed abolition of episcopacy
it was so late every day before the House was resumed, (the Speaker commonly leaving the chair about nine of the clock, and never resuming it till four in the afternoon), that it was very thin; they only who prosecuted the bill with impatience remaining in the House, and others who abhorred it, growing weary of so tiresome an attendance, left the House at dinner-time, and afterwards followed their pleasures: so that the Lord Falkland was wont to say that they who hated bishops hated them worse than the devil, and that they who loved them did not love them so well as their dinner. (III.241)
The public hubbub is interspersed with private confidences, informal attempts at persuasion, and the judgements of men's qualities and interests—the everyday currency of parliamentary life in a period of extraordinary tension and significance. In recording debates, Clarendon usually summarizes, with injections of verbatim quotation of what was presumably particularly memorable. Some of Clarendon's reports of private conversations passed into political folklore and have been quoted by generations of historians. At the time of the impeachment of the Earl of Strafford, the King's formidable lieutenant in Ireland,
Mr. Hyde going to a place called Pickadilly, (which was a fair house for entertainment and gaming, and handsome gravel walks with shade, and where were an upper and lower bowling-green, whither very many of the nobility and gentry of the best quality resorted, both for exercise and conversation) he was accosted by the Earl of Bedford, according to whom the King would make no difficulty if Strafford could be guaranteed his life. (I.161)
The obstacle was the Earl of Essex, to whom Bedford takes Hyde to try to persuade him to be less rigid. Essex is immovable, fearing that Strafford, pardoned, might yet be a danger to them all. "He shook his head, and avowed 'Stone-dead hath no fellow'" (I.164).
A particularly bitter battle was the debate on the Grand Remonstrance, the iteration of parliamentary grievances whose passage in November 1641 Clarendon saw as a decisive blow to prospects of conciliation:
The debate being entered on about nine of the clock in the morning, it continued all that day; and candles being called for when it grew dark (neither side being very desirous to adjourn it until the next day; though it was evident very many withdrew themselves out of pure faintness, and disability to attend the conclusion), the debate continued till after it was twelve of the clock, with much passion.
The Remonstrance was carried by just nine votes. As they at last emerged from the House, Oliver Cromwell, "(who at that time was little taken notice of )," whispered in the ear of Lord Falkland "that if the Remonstrance had been rejected he would have sold all he had the next morning, and never have seen England more." So near, sighs Clarendon, "was the poor kingdom at that time to its deliverance!"(IV.52).
With the final rupture, with some Members, including Hyde, following the King to what was to be his wartime capital in Oxford, the question of the allegiances of the members of the Privy Council also became an important issue. This gives Clarendon the chance for a number of character studies, in which he has a conscious virtuosity. His prose style is sometimes prolix and syntactically involved, especially when deploying an argument. It is the manner of an orator rather than a writer—not in the sense of being rhetorically ornate or impassioned, but in obviously having been written with the assumption that punctuation is a matter for the voice, sometimes injected with a telltale "so I say that" in the midst of a long sequence of clauses, to be followed by yet more. But the portraits of public men, measured and judicious though they often are, are also sometimes tartly economical and epigrammatic in a fashion that recalls Tacitus and is almost certainly meant to. Tacitus is the ancient historian most frequently quoted, though Clarendon also cites Livy, Plutarch and Thucydides. Such depictions, not only in the time-honoured fashion as obituaries, are scattered through the _History._
Among the most memorable is that of the Earl of Arundel, made general of the army designed to coerce the Scots. He "was thought to be made choice of only for his negative qualities: he did not love the Scots; he did not love the Puritans; which good qualities were alloyed by another negative; he did not love nobody else" (II.25). The disreputable Colonel (later Lord) Goring was so good at dissimulation "that men were not ordinarily ashamed, or out of countenance, with being deceived but twice by him" (VII.69). Of General Mon[c]k, the main author of the Restoration, Clarendon writes that "it is glory enough to his memory, that he was instrumental in bringing those mighty things to pass, which he had neither wisdom to foresee, nor courage to attempt, nor understanding to contrive" (XVI.115). Sir Arthur Aston, the governor of Oxford, preferred to the post by the Queen, "had the fortune to be very much esteemed where he was not known, and very much detested where he was" (VIII.121). In the middle of a longish appraisal of the Earl of Northumberland we have, "If he had thought the King as much above him as he thought himself above other considerable men, he would have been a good subject" (VI.398).
Other portraits need to be taken at nearer full length, like the characterization of the Earl of Warwick, commander of the fleet for Parliament:
He was a man of pleasant and companionable wit and conversation, of an universal jollity, and such licence in his words and actions that a man of less virtue could not be found out: so that a man might reasonably have believed that a man so qualified would not have been able to have contributed so much to the overthrow of a nation and kingdom. But with all these faults he had great authority and credit with that people who in the beginning of the troubles, did all the mischieve; and by opening his doors and making his house the rendezvous of all the silenced ministers [of religion] in the time when there was authority to silence them, and spending a good part of his estate, of which he was very prodigal, upon them, and by being present with them at their devotions, and making himself merry with them, and at them, which they dispensed with [excused], he became the head of that party, and got the style of _a godly man._ (VI.404)
The largest character study is the obituary for Clarendon's friend Lord Falkland, a casualty of the war (1643), who was
of that inimitable sweetness and delight in conversation, of so flowing and obliging a humanity and goodness to mankind, and of that primitive simplicity and integrity of life, that if there were no other brand upon their odious and accursed civil war than that single loss, it must be the most infamous and execrable to all posterity. (VII.217)
More interesting because more equivocal is Clarendon's estimate of the parliamentary leader John Hampden, killed in the same battle (Chalgrove):
He was of that rare affability and temper in debate, and of that seeming humility of judgement, as if he brought no opinions with him, but a desire of information and instruction; yet he had so subtle a way of interrogating, and under the notion of doubts insinuating his objections, that he left his opinions with those from whom he pretended to learn and receive them. (VII.83)
Clarendon came to attribute to him deep designs of subversion.
Clarendon had a clear sense of the ways in which men's differing qualities made them valuable in different ways. The Earl of Pembroke, for example, "had a choleric office [Lord Chamberlain], which entitled him to the exercise of some rudenesses, and the good order of the Court had some dependence upon his incivilities" (VII.399). It was a mistake, as the impatient and impolitic Prince Rupert was apt to do, to estimate a body like the Privy Council from the infirmities of its individual members—"the heaviness of this man, the levity of that, the weakness and simplicity of a third"—for "all great enterprises and designs that are to be executed have many parts, even in the projection, fit for the survey and disquisition of several faculties and abilities, and equally for the decision of sharper and more phlegmatic understandings" (VII.279–81).
In the fashion that was to become virtually universal until the nineteenth century, Clarendon saw Cromwell as a deep-designing hypocrite, moved always by the lust for power; but even his portrait is not wholly unfavourable. He was "not a man of blood," and had some virtues. He was also a man of immense ability and power of will, who made all Europe fear him; "he will be looked upon by posterity as a brave bad man" (XV.147–56). It was, for Clarendon, important to discriminate, lest, contemplating the wreck of the King's fortunes, one be led "to believe that a universal corruption of the hearts of the whole nation had brought forth these lamentable effects; which proceeded only from the folly and the forwardness, from the weakness and the wilfulness, the pride and the passion, of particular persons" (IX.1). Clarendon had none of the sophisticated general concepts of historical analysis that we find in Hume and Robertson a century later. For him it was axiomatic that, under Providence, the rebellion was made by individual men, not all of them even wicked. He is worth listening to, because he knew them.
**TWENTY-ONE**
**Philosophic History**
**Hume: Enthusiasm and Regicide**
Crucial to the emergence of the Enlightenment genre of the history of customs, manners and opinion was what was coming to be seen as an indisputable fact of European history: "the progress of society." The growth of commerce and the end of the "feudal anarchy," the "revival of learning and the surpassing of the ancients" in the discovery of the New World and the printing press and the improvements in the art of war (something that Guicciardini had been one of the first to note) all contributed to this perception. To this came to be added a conception of the improvement of "manners" over the previous two centuries, from the rough, pedantic, fiercely intolerant religious zeal and polemics of the time of the Reformation to the eighteenth-century cultivation of a polite, tolerant sociability as the mark of a refined society which was mild, humane and rational.
The ghost at the feast of reason and self-congratulation, in England at least, was the seventeenth-century revolution, the brief but unforgotten reign of the sectaries under the general characterization of "enthusiasm," and the menace of an egalitarian republicanism. It had apparently been exorcized, but events from the late eighteenth century onward could sometimes give it the appearance of a precedent as well as a warning. Revolution, in relation to progress, came to play something like the role earlier played by luxury and enervation in the classical and civic humanist paradigm: that of a nemesis. "The great cause of revolutions," Macaulay told the House of Commons in the debates before the first Reform Act in 1832, "is this, that while nations move onwards, constitutions stand still." Marx was to say much the same, though from a revolutionary stance, and Hume would say something similar (Chapter 21) of the government of Charles I. So salient was the revolutionary experience to become, and the fear and hope of it, that we have to attend to how historians rose to the challenge this represented to their comprehension and their art.
David Hume is now famous as a philosopher; as a historian he is scarcely known. In the eighteenth and nineteenth centuries it was the reverse. Then he was primarily the author of a monumental, authoritative, though much contested six-volume _History of England_ (1754–62). Hume was one of the key figures of the Scottish Enlightenment, and his history is a characteristic product of it, applying to a long tract of political, constitutional and social history some of its most basic ideas: the association of fundamental changes in manners and opinion with the decline of feudalism and the growth of commerce, and a consideration of the influence of religion on social and political life. William Robertson, his contemporary, applied these to the safer ground of sixteenth-century Europe, though his location of John Knox and the Scottish Reformers in a rude age whose characteristics they shared was not without polemical point. But Hume, in taking up the seventeenth century in England as his initial challenge (it was a peculiarity of his history that it was written and published backwards in terms of historical chronology), was confronting some of the most contentious issues in modern English political life. The political labels "Whig" and "Tory" derived from the two great factions of the seventeenth century, and even the more sophisticated modern labels "Court" and "Country" paralleled to a significant extent the seventeenth-century divisions. Hume's history of the seventeenth-century revolution, the first part of his English history, was published in 1754. In deference to James VI of Scotland's accession to the English throne in 1601 as James I, it was entitled _The History of Great Britain, Containing the Reigns of James I and Charles I._ For three-quarters of a century it came to dominate the field, though it was one of many histories of the English revolution, published from all points on the political spectrum, during that period. G. O. Trevelyan's life of Macaulay shows the latter clearly gratified as well as amused to find Hume's _History_ in a bookseller's window, a century after its publication, labelled "valuable as an introduction to Macaulay."
Hume's offences to contemporaries' pieties were several, but the chief was his refusal to accept the Whig notion of an enduring "ancient constitution" subverted by the Stuarts. To him, English constitutional precedents, by the early seventeenth century, were chaotically contradictory, reflecting the shifting balance between Crown and nobility over the preceding centuries: the constitution was "unintelligible"(Chapter 7). The early Stuarts could find precedents for most if not all of what they did, and the Tudors had ruled much more absolutely while avoiding theoretical claims. Essentially it was the parliamentary leaders in the 1640s who became the innovators, and there was for Hume in principle nothing wrong with that: the country needed a "regular system of liberty," which eventually became established. But, though Hume was prepared to welcome the outcome, he offended the more radical Whigs by his obvious distaste rather than admiration for most of the leading parliamentarians, who were unpolished and fanatical. Hume's terms for seventeenth-century English Puritanism and Scottish Presbyterianism were uncompromising. In them "the genius of fanaticism displayed itself in its full extent," and their inflamed imaginations poured themselves out "in wild, unpremeditated addresses to the Divinity" (Chapter 4). Abuse of the seventeenth-century sectaries was common enough in eighteenth-century Britain, but Hume's own notorious religious scepticism, which had prevented him from obtaining, like Robertson, an Edinburgh or Glasgow academic chair, was offensive to many. With so much weighted against it, the dominance exerted by Hume's _History_ and its publishing success—it made him rich—are striking.
With some exceptions we shall come to, it is not picturesque: its power is intellectual, in the quality of the reflection and the cogency of the narrative. For, despite the disquisitions with which it is interspersed—Adam Smith, a traditionalist in such matters, objected to the obstruction of the linear narrative flow—it is not an Enlightenment sociological essay but a detailed and full-bodied history, classical in its sense of decorum and annalistic in its arrangement. The convention of invented speeches was falling into disrepute, and Hume prefers to summarize, often representing the opinions of many rather than an individual. Otherwise, however, despite the disquisitions—the longest of which, on changes in society, was in subsequent editions relegated to an appendix—Hume's history is recognizably within a classical tradition of dealing with public affairs and public men. Of course the eighteenth-century genres of the essay on customs and the history of civil society form part of the intellectual inspiration.
One of the most controversial features of Hume's _History_ seems at first sight at odds with the sociological concerns of the Enlightenment and Hume's habitually detached and ironic authorial stance. It is the extent of the sympathy accorded to the victims, history's prominent losers—above all, of course, Charles I. There is in fact a conscious exploitation of the pathetic, the "sentimental" (a technical, rather than a pejorative, term), in the description of the King's captivity, trial and execution.
It is confessed that the King's behaviour, during this last period of his life, does great honour to his memory; and that in all appearances before his judges, he never forgot his part, either as a prince or as a man. Firm and intrepid, he maintained, in each reply, the utmost perspicuity and justness, both of thought and expression: Mild and equable, he rose into no passion at that unusual authority which was assumed over him. His soul, without effort or affectation, seemed only to remain in the situation familiar to it, and to look down with contempt on all the efforts of human malice and iniquity. (p. 678)
The cultivation of sentiment in eighteenth-century historical writing was taken up both by Hume and by his contemporary Robertson, notably in their treatments of the execution of Mary, Queen of Scots. (On this, see Phillips, _Society and Sentiment._ ) The pathos of Hume's account of Charles's end was often seen as a proof of Hume's "Toryism." Hume was not a Tory or a Jacobite, but he did sit loosely to traditional Whig pieties, and in his essays he revealed a position not common in eighteenth-century Britain. He had lived in France, and he set out a sharp distinction, based not only on the study of the ancient republics but also on the position of a subject of the absolute French monarchy, between public and private liberty. The antique republics had not understood private liberty at all, but, given an ordered monarchy, ruling through the law, it was not incompatible with the absence of public liberty. The life and interests of the individual could be as secure as under a representative system. Indeed, Hume, in Britain, feared the prospect of anarchy, as a result of unrestrained factionalism, more than absolute monarchy; he described the latter as the "easiest death, the true euthanasia of the British Constitution" he was predictably alarmed by the riots on behalf of the radical John Wilkes at the beginning of the reign of George III.
But the issue with "sentimental" history is not a matter of party labels or liberty or absolutism. One of the objects of Hume's _History_ was to abate the violent spirit of faction by the play of an enlightened reason over the contentious recent past. Sentimentalism, in that context, was not partisan but eirenic. Sentiment as an item in the historian's repertoire developed in relation to manifestations of the later-eighteenth-century exploration of sensibility generally, including an increasing value attached to immediacy of representation and empathy in historical narrative. (On this see Phillips's perceptive discussion in _Society and Sentiment._ ) Such exercises in empathy and concrete immediacy in historiography have more usually been placed in the nineteenth century (where they sometimes gave a pretext for denigrating the eighteenth for lacking them). Their earlier appearance was related not only to the cult of sensibility, but to attempts to encourage a wider readership for history, notably among women, of which Hume was certainly aware. Pathetic effects are not incongruous with the enlightened detachment to which Hume also aspires, but are actually in a sense part of it. It is _because_ Charles is to be seen not as a would-be despot seeking to subvert an established and inherited constitution but as a victim of historical changes, in manners, opinion and the balance of power and property, which he cannot understand or adjust to, that it is appropriate for the enlightened historian, who can understand them, to "shed a generous tear" for him.
This, indeed, though here turned to purposes of pathos, is the central message of Hume's early Stuart history. Charles could not be expected to have the long perspective, requiring hindsight and the appropriate conceptual equipment, of the philosophic historian. He was not, Hume says, "endowed with that masterly genius, which might enable him to perceive, in their infancy, the changes that arose in national manners, and know how to accommodate his conduct to them" (Chapter 23). He was a man lost between two worlds and two roles: "Had he been born an absolute prince, his humanity and good sense had rendered his reign happy and his memory precious: had the limitations on prerogative been, in his time, quite fixed and ascertained, his integrity had made him regard, as sacred, the boundaries of the constitution" (p. 684). His fatal flaw, humanly venial but politically disastrous, was a failure to read the signs of the times.
What those signs were is set out by Hume in a number of disquisitions, including a whole chapter (VI), subsequently made into an appendix, which strongly anticipates the famous Chapter III on "social" history in Macaulay's _History,_ which aroused criticism from the fastidious for the triumphalism of its insistent contrasts of "then" and "now." Hume's chapter also uses this device, though without the fanfare for modernity. It is not perfunctory, and, exceptionally, it incorporates some statistics. But the central contention is twofold: Hume, like Robertson, sees a new spirit of liberty and independence of thought developing from the sixteenth century especially in association with the rise of commerce (he is explicit about why the greater towns favoured the parliamentary side). And like Harrington and the "Country Party" polemicists of the earlier eighteenth century, particularly Bolingbroke, he sees power passing from the Lords to the Commons as feudalism ends and landed property is diffused:
The first rise of commerce and the arts had contributed, in preceding reigns, to scatter those immense fortunes of the barons, which rendered them so formidable to both king and people. The farther progress of these advantages began, during this reign, to ruin the small proprietors of land, and, by both events, the gentry, or that rank which composed the house of commons, enlarged their power and authority. The early improvements in luxury were seized by the greater nobles, whose fortunes, placing them above frugality or even calculation, were soon dissipated in expensive pleasures.
The Commons was discovering its power, assisted by the unwisdom of the first two Stuart monarchs in making a theoretical issue of their prerogatives. For Hume, all authority rested ultimately not on right but on opinion, which it was therefore essential to manage. The chief support of established authority was habit and tradition. It was fatal to encourage a disputatious inclination on the part of subjects by theoretical assertions, in religious or political matters.
The fiscal situation, too, favoured Parliament. The Crown was still expected to live on its traditional revenues, though expenses were increasing. Parliament was unable to grasp this, but was very ready to try to take advantage of the King's financial difficulties to exact concessions. Charles's government was driven to resort either to the revival of archaic sources of revenue or to the exploitation of new ones, both of which were bitterly resented. In one respect James and Charles were simply unfortunate, though they unwisely fanned the flames rather than seeking to dampen them. The spirit of religious fanaticism, stemming from the Reformation, had reached a high point. Hume drew on a concept which he had worked out in a well-known essay ("Of Superstition and Enthusiasm"). His name for it—not original, but turned from a pejorative term into a theoretical category—was "Enthusiasm." Enthusiasm could arise at any time, being essentially random (rather like Max Weber's "charisma"), defying all calculations of prudence and considerations of individual interests: "The fanatical spirit, let loose, confounded all regards to ease, safety, interest, and dissolved every moral and civil obligation" (Back Matter). For Hume the opposite of Enthusiasm in the dynamics of religious belief was Superstition. Superstition was irrational also, but arose from the impulse to propitiate and to curry favour. It was therefore servile, and a support of religious and civil establishments; its historical embodiment was Catholicism. But Enthusiasm, filled with antinomian intimations of unique, individual possession by the Spirit, was bold, aggressive, zealous and destructive; it was Puritanism which, with its remembrances of the terrible fanaticism of the seventeenth-century sectaries, could still bring a shiver of anxiety to the Age of Reason. It was encouraging to stress the historical distance between seventeenth-century fanaticism and what Hume called "the mildness and humanity of modern manners" (Chapter 6), but the historical examples were still admonitory. James I, according to Hume, was not wrong to see fanaticism as a threat to civil as well as religious authority, but was misguided in challenging it directly.
Hume's own accounts frequently resort to irony—for example on the zeal of the Scots for transplanting abroad their system of church government: "Never did refined Athens so exult in diffusing the sciences and liberal arts over a savage world...as the Scotch now rejoiced, in communicating their barbarous zeal and theological fervour, to the neighbouring nations" (Chapter 26). Hume's distaste was not reduced by his recognition that, in a manner to which the eighteenth century was becoming accustomed, Enthusiasm's historical function was something other than its intrinsic nature. Without the pernicious "epidemical frenzy" (Chapter 26), the defence of parliamentary freedoms could not have been sustained and the eventual "regular system of liberty" (Chapter 19) would have been aborted. The more "natural" outcome of the decline of the power of the feudal baronage, seen on the Continent as the rise of monarchical absolutism, would have been consummated. In the circumstances of the time, only a pious fanaticism, regardless of consequences, could have braced men to the necessary hazards and sacrifices to resist this. Behind "that singular and happy government, which at present we enjoy" (Chapter 15) lay a dark and fierce irrationality. In the case of the Scots, Hume is particularly explicit on the moral disjunction between the cause and the outcome: "The Scotch nation were first seized with that frenzy of reformation, which was so pernicious during the time, and which has since proved so salutary in the consequences" (Chapter 9). Historical causation is another thing than moral quality, and the connection between virtue and liberty is heavily qualified. It was a lesson which, applied by Hume to the Reformation and the political crisis of the seventeenth century, was, as we shall see, applied by his younger contemporary Gibbon to the history of Christianity as such and to the long-term relations of Barbarism and Civilization.
Hume's distaste for "Enthusiasm" was common in his century, and he expressed it with particular force. In the next century it was to become more muted, as the religious zealotry of the seventeenth century receded further into the past, and as a religious revival brought it greater respect. But Hume's other main preoccupation in relation to the English revolution, the long-term movement of society, was to remain one of the major themes of the history which was to be hailed as the great successor to and replacement for it, Macaulay's _History of England._
**Robertson: "The State of Society" and the Idea of Europe**
"The state of society" was to be a key conception for the most important historical writing of the eighteenth century. There were other innovations too. Some of these are well exemplified by _The History of the Reign of the Emperor Charles V_ (1769) by William Robertson (1721–93), which has a claim to be regarded as the first modern work of history. Of course the criteria of modernity are multiple; others would point to different examples, but the claim is defensible.
Modernity is in fact an idea relevant to Robertson's book, in a double sense: the book is about it—about, that is, the emergence of "modern" Europe in the sixteenth century—and it could not have been written but for other kinds of modernity. In Robertson's view, modernity was a secular cosmopolitanism, both cultural and political (in the latter sense expressed as the idea of "the balance of power"), which in the sixteenth century was making Europe modern and also making it, for the same reasons, an entity whose history could be written. But Robertson's book is also cosmopolitan—that is, European—not only in its viewpoint but in its genesis and its intention. He had made his name with a successful _History of Scotland,_ ten years earlier. _Charles V_ was a deliberate bid for wider, in fact for European-wide, attention in the literary market; Robertson arranged for its translation into French as soon as it was published.
It is difficult to think of a precedent for Robertson's book. Hitherto historians had been drawn chiefly to write the histories of their own nations or cities, which needed no special explanation, or to narratives of events in which they had been personally involved or of which they could at least claim some special knowledge and interest. Sarpi's history of the Council of Trent might be claimed as an exception, but he was nevertheless investigating the most significant recent episode in the history of the Church to whose clergy he belonged. Hume's _History of England,_ which Robertson considered emulating, before he thought better of it, is not really an exception to this, despite Hume's Scottish perspective. But Robertson had nothing at all to draw him to write of Charles V's empire—nothing, that is, except its centrality to early modern Europe, and the intrinsic interest and importance of the latter as a historical period (which is how Robertson saw it). Robertson's book in fact ranges considerably wider than even the extensive territories of Charles's empire, to encompass the whole history of Europe, and it was the possibility of doing this that made him choose his focal point.
This free range of choice of subject was new, and made possible only by the deluge from the printing presses over the previous two hundred years, including the printing of historical documents. Robertson maintained a network of foreign correspondents in writing his books, notably in Spain, and he lists them in the preface to his subsequent _History of America_ (1777), for which he adopted the highly modern expedient of a questionnaire. Even for the _History of Scotland_ he lists his main debts in his preface, as well as identifying important sources in appendices, as a modern scholar would characteristically do. Travel was not yet normally part of a historian's working practice: Gibbon, for example, never returned to Rome after his first, fateful, visit. Robertson never travelled on the Continent, but libraries, the book trade and helpful correspondents made it possible for him to write a detailed history of Charles V's empire from Edinburgh, where he was a minister of the Church of Scotland and for thirty years principal of Edinburgh University. There was nothing superficial about his way of writing history: he was a scholar as well as a literary historian.
But it was possible to write for the market, and Robertson did, receiving for _Charles V_ a sum from the publisher which awed his contemporaries. He had a patron, Lord Bute, George III's Scottish prime minister, who procured for him the revival of the office of historiographer royal for Scotland, but his clerical living was also important. Though he presided over an academic institution, he was not of course a "professional" historian in the modern sense of teaching history for a living. History was still not, never had been, and would not for another century be part of the academic curriculum, though there were a small number of endowed professorships. It was easy in the Scottish system to lecture on historical subjects from chairs of rhetoric and belles-lettres, moral philosophy, and law, as Adam Smith and John Millar did, though Robertson did not do so. But published history could pay well. The Earl of Clarendon's _History of the Rebellion_ had made unprecedented amounts; Hume's _History of England_ had brought him the rewards he had vainly sought from philosophy. The move from dependence on a patron to the independence achieved by production for the market was a theme in eighteenth-century Scottish thought; it was also being achieved in literary men's lives. Robertson was a tolerant, self-consciously modern Scottish cleric—one of the self-dubbed "Moderates"—and a modern literary man as well as an improving university administrator.
In using the word "modern" so freely, I am, of course, not travestying but at least semantically simplifying. The eighteenth century was more copious in its terminology: "polished," "polite," "refined" "civil," "civility" and "civilized" (quite common) and even "our enlightened age" (Gibbon). We get "Enlightenment" only toward the end of the century and in German, and "civilization" (first available in French) also in the second half of the century. But if our own utility word "modernity" was not in use, the concept certainly was, with its antitheses: "that rude age," "illiterate ages," "superstitious ages," "barbarism" and "the feudal anarchy." The contrast focused Robertson's attention, and it is impossible to explain what he was trying to do in _Charles V,_ and even in much of the _History of Scotland,_ without invoking it. The antitheses were contrasts of, in another key term, "manners," which included customs and conventions, values and characteristic conduct—in French, _mœurs._ The relation between an idea of modernity expressed in these terms and an idea of Europe is reciprocal. The history of Europe—and more widely of mankind—could be categorized in terms of manners, changing over time. Modernity could also be characterized in terms of what Robertson in _America,_ in a startlingly prophetic phrase, calls the "mode of subsistence": as the supersession of "the feudal system" by commerce, which is the hallmark of modernity, and by the associated softening and refinement of manners compared with the military spirit of feudalism. In politics—and this is what above all focuses attention in _Charles V_ —feudalism is replaced by "one great system" governing the relations of the European states, which is "the balance of power."
The eighteenth-century sense that the state of society and manners passed through successive stages gave rise over the course of the century to a characteristic genre: schematic abridgement of human history into "stages," with speculative accounts of the reasons for the transitions. In some cases, if the taxonomic element prevailed over the sequential, as in Montesquieu's _Spirit of the Laws_ (1748), the result was more a set of sociological categories, which was neither detailed narrative nor necessarily overtly historical scholarship, though evidence from different parts of the world and different historical eras was cited. History as a sequence of stages, of states of society or of the human mind, was in a sense the "enlightened" successor to the long-lived Christian universal history, derived from the Bible, Augustine, Orosius and the historical sequence of the Four Empires. The most obvious examples of such hostile rivalry between Christian and Enlightenment universal history were Voltaire's _Essay on Customs_ ( _Mœurs_ ) (1756) and, at the end of the century, Condorcet's anticlerical _Sketch for an Essay on the Progress of the Human Mind_ (1794). In France we have also Rousseau's _Discourse on the Origins of Inequality_ (1755). Scotland was a notable contributor, with some of David Hume's Essays, Adam Ferguson's _The History of Civil Society_ (1767), Adam Smith's _Lectures on Jurisprudence,_ John Millar's _Origin of the Distinction of Ranks_ (1771), Lord Kames's _Sketches of the History of Man_ (1774) and James Dunbar's _Essays on the History of Mankind in Rude and Cultivated Ages_ (1780). Robertson contributed to the genre with parts of his _History of America,_ on the manners and beliefs of the indigenous peoples, and also, in a more limited and strictly European fashion, with the "View of the Progress of Society in Europe from the Subversion of the Roman Empire to the Beginning of the Sixteenth Century" which he prefixed to _Charles V._
The motives for the creation of these schematic overviews were to some extent diverse. The most obvious was a general desire to make history "philosophic" that is, to uncover its underlying causes and make it a basis for useful generalizations. Montesquieu's enterprise was partly conditioned by what has been called a "feudal reaction" against the absolutism of Louis XIV in France. By identifying "despotism" (exemplified particularly by the Ottoman state) as a category distinct from monarchy, and based on the principle of fear, he left the way open to a characterization of monarchy as embedded in the rule of law and checked and supported by a vigorous aristocracy actuated by the principle of honour. Voltaire's and Condorcet's works were attacks on priestcraft and superstition. For the Scots the focus was on the forms assumed by civil society and manners, as in Montesquieu, and the forms of property-holding, as in Harrington. In Smith's _Lectures on Jurisprudence,_ in particular, we get a clear delineation of the so-called "four stages" of society: hunting-and-gathering savagery; pastoral nomadism, with the beginnings of property rights; agriculture (which in Europe after the barbarian invasions was held as property by the institution of the _feud_ ); and, the latest stage, commerce.
The Scottish focus on civil society, rather than, as was still to some extent true of Montesquieu, on forms of political constitution, was understandable. After the Act of Union (1707) Scotland was no longer a polity, and Edinburgh no longer a political capital. On the other hand the growth of Scotland's prosperity and the concomitant refinement of manners were very obvious. The concept of a polite and progressive civil society offered another form of self-assessment and of possible emulation. The half-jocular but competitive conversations between Johnson and Boswell on the relative merits of English and Scottish society are good informal examples of this.
It was also relevant, however, that within Scotland there were vast differences in characteristic forms of society, not only in the urban–rural contrasts which all Europeans knew, but geographically. North of the Highland line was, so to speak, Indian territory, which during the '45 rebellion had terrifyingly erupted into the civilized streets of Edinburgh. Highland clan society was assimilated by Robertson into his grim picture of Scottish feudalism as an additional source of strength and independence for the feudal warlords, enhancing their ungovernability. Clan society does not seem in general in the eighteenth century to have been carefully distinguished as a category, as it was to be in the nineteenth, when Marxists, in particular, adopted the term "gentile society" (from the Latin _gens_ ) as a stage distinct from and prior to feudalism. (Hume came nearer to recognizing this than Robertson, distinguishing sharply between the Scottish version, which, with primogeniture, reinforced feudalism, and the Irish one, with equal division of land among siblings, which was more barbarous.) Similarly, where anthropologists in the later nineteenth century found kinship systems to be the organizing principle of primitive societies, eighteenth-century commentators like Robertson tended to regard such societies as wholly sexually promiscuous, and so as mere hordes. Without property, why would they need to trace descent or identify kin?
On the other hand the perception of the ways in which feudalism was superseded had become much more sophisticated, compared for example with Harrington—particularly in Adam Smith. This marked a characteristic and important shift in the approach to historical causality more generally. The notion of a sequence of stages or forms of society (rather than polities) and of manners had implications for the understanding of the transitions from one to another. Despite the seventeenth-century English enthusiasm for customary law and the idea of an immemorial (and unchanging) constitution, laws and constitutions generally sounded like the kinds of thing that would, and certainly could, be deliberately enacted. Manners, however, seemed unlikely to be legislated into or out of existence. Their progress could be artificially retarded, but it seemed likely that when they changed for the better they did so gradually and, to use a favourite word of Gibbon's, also employed by Robertson, "insensibly." The mythic reputations of the great lawgivers—Lycurgus, Solon, Numa—at last began to be dismissed. As Adam Ferguson significantly wrote, "Nations stumble upon establishments which are indeed the result of human action, but not the execution of any human design." In _The Wealth of Nations_ (III.iii and IV) Smith gave his classic account of the gradual erosion of feudalism not by legislation but simply by human nature presented with the opportunities of the market. Given the increasing availability of goods as a result of the productivity of the towns and of commerce, the great feudal lords are drawn to expend their agricultural surplus on these rather than converting it into military and political power by keeping armed retainers and by imposing military obligations on their tenants; instead these obligations are increasingly commuted for money rents.
Robertson in his _Scotland_ gave an account of the ending of feudalism in France and England largely along Harringtonian lines: it was the result of the deliberate policy of Louis XI of France, implemented by cunning manipulation, and of deliberate legislative action by the English Henry VII, and his son's dissolution of the monasteries, which diffused the ownership of property. The failure of Scotland to emerge from the feudal era was a failure, for various reasons, of the Scottish kings. But in his "Progress of Society" essay Robertson proposes a more elaborate set of reasons, including the exposure of the European aristocracy to the refinements of Constantinople and the East in the Crusades, and the rise of the commercial city republics of Italy as a rival social and political model to the dominance of the feudal lords. His belief in the benign effects of the Crusades, born though they were in fanaticism and superstition, was an example of another sophistication in the notion of historical causality: the idea of "unintended consequences," of which Smith's chapters in Part III of _The Wealth of Nations_ provide a particularly vivid example. Vices could have desirable consequences; virtue was no guarantee of them. Robertson makes the point explicit in the case of Henry VIII's extravagance, vanity and wilfulness, which had the Dissolution and the English Reformation as its consequences. It was a mismatch of intention and outcome whose natural treatment in historical writing was as irony; Gibbon was to take profuse advantage of it.
Robertson's two introductory-survey chapters in his _Scotland,_ devoted to the enduring pernicious power of the Scottish feudal nobility and the weakness of the monarchy, made in a sense a more restricted and bleaker counterpart to the optimistic, European, version he provided ten years later in the "Progress of Society" preamble to _Charles V._ But whether entrenched or overcome, feudalism, intellectually speaking, would not go away. Scholarship continued to be assiduous in its attention to it. Only two years before Robertson's _Scotland,_ in which feudalism played such a leading part, Sir John Dalrymple, the Scottish antiquary to whom Robertson acknowledged debts in his preface, had published _An Essay towards a General History of Feudal Property in Great Britain._ In his highly negative view of the historical role of the Scottish feudal nobility, Robertson set his face against the "ancient constitutional" tradition fostered by Buchanan in the sixteenth century, in which the nobility were the guardians of liberty. For Robertson, liberty was modern not ancient, while about the ancient history of Scotland before the Romans nothing at all could be known. The introductory survey completed, the narrative part of Robertson's _Scotland_ begins with the reign of Mary, Queen of Scots, and the book ends with the accession of her son James VI to the throne of England on the death of Elizabeth in 1603. From then on, as king of England as well as Scotland, James disposed of force and resources which dwarfed those of the Scottish nobility; the history of Scotland as a modern, i.e. post-feudal, society could begin. Like Hume in an English context, Robertson was not an ancient-constitutionalist but a modern Whig. Feudalism had meant anarchy.
It was in _Charles V,_ dealing with a period he had chosen for this purpose, that Robertson was able to narrate the consequences of the decline of the feudal nobility on a European scale. Because of this decline, and because of the consequent concentration of power in the major European monarchies, the European states system was created; Robertson's debt to Guicciardini is very obvious. The history of Europe as a whole could be treated in narrative terms, and not merely in a survey or as the histories of individual states. One is reminded of Polybius on the consequences for historical writing of the emergence of Roman power as a unifying theme, but in Robertson the unification was not that of a European-wide empire, which would have been a disaster for mankind. (Robertson, like Polybius, invokes Providence—but not to explain the rise of empire but to give thanks for its failure.) What made Europe an entity, for Robertson, was the balance of power. The unity of Europe, in this paradoxical way, strikes one as rather similar to Smith's account of the creation of the market: each individual (seller, purchaser, statesman) seeks only his own profit/security, but the outcome is an order. Robertson enthusiastically calls it "one great family" and "that grand system" ( _Charles V,_ XI). Gibbon would later compare it to "one great republic" ( _Decline and Fall,_ "General Observations").
Robertson, of course, does not just generalize, but densely narrates the incessant shifts in relative power, the recalculations of advantage by kings and the German princes, the constant reshuffling of alliances, as well as the occasional irruptions of overconfidence (usually from Charles), and of vanity in pursuit of honour rather than interest (Francis I). Resentment and the desire for revenge also impair the operation of the sober estimates of interest and the policies they prescribe. The latter constitute the distinctive quality of modern statecraft, so that Francis's chivalric notions of honour represent a kind of cultural lag.
The greatest intrusion, however, because it involves zeal and intransigence, is the distinctively modern one of the Reformation, which Robertson, in a standard move, associates with the new "spirit of inquiry" and which wrecks Charles's chances of consolidating still-feudal Germany into a unified modern state. Robertson's treatment of the Reformation is notably political: the corruptions of the Church are condemned, but theological issues are marginalized. But if the Reformation is in a sense an aspect of modernity, it is also, as in the _History of Scotland,_ carefully culturally distanced from Robertson's own time. Luther and Knox, both performing a necessary task and fitted for it even by their defects on the scale of civility (another example of private vices, public benefits), are also figures firmly located in the past; they belong to a ruder, fiercer, more intransigent age than the present. In this they have some of the same traits as the Scottish nobles who butchered the royal secretary and favourite David Rizzio in the presence of their terrified, pregnant Queen, and who "fill us with horror at the manners...of that age" ( _Scotland,_ IV). But "in passing judgement on the characters of men we ought to try them by the principles and maxims of their own age, not those of another. For although virtue and vice are at all times the same, manners and customs vary continually" ( _Scotland,_ VIII). There is as yet for Robertson no established periodizing terminology, for example between "Early Modern" and "Enlightenment," but the way he holds the former period at arm's length is of more than academic significance. It has been pointed out that by his historical contextualizing of the Reformers, the ancestors of the rigid eighteenth-century Calvinist opponents of Robertson's kind of tolerant modern Presbyterianism, he was tacitly depriving them of their claim to be uniquely and authentically scriptural in all their characteristics. The concept of at least two periods since the revival of learning is already present. Even academically this was important. Part of the experience of reading history is constituted by such cultural distancing, just as much as its converse, the sense that what lies on the other side of the gulf can be made intelligible by historical imagination and scholarship. Robertson is as sensitive to such issues as any writer in the eighteenth century, and his commitment to studying a period as close as the sixteenth century makes this particularly evident.
The contrasting "politeness" of the eighteenth century is established not only by assertion but also by the deployment of what his age was beginning to term "sensibility"—imaginative sympathy or "feeling" indulged even to lachrymosity—and also by the polished, calm equability of the prose which enforces Robertson's interpretations and opinions. The former is chiefly in evidence in his treatment of Mary, Queen of Scots, who, morally, femininely, frail and politically misguided, is made pitiable in her misfortunes by her conjectured feelings, as in the scene of her enforced abdication: "Mary, when she subscribed these deeds, was bathed in tears; and while she gave away with her own hands, the sceptre which she had swayed so long, she felt a pang of grief and indignation, one of the severest, perhaps, which can sway the human heart" ( _Scotland,_ V). Implacable Buchanan, who knew her, merely says that she "reluctantly agreed to name guardians for her son, and that procurators were sent to arrange that the king should be crowned at Stirling."
For the moral and aesthetic effect of Robertson's prose one has to take an example, more or less at random. It is said that when Robertson read through one of his own letters he did so beating time, as though hearing a piece of music. It is quite credible. Consider this, from _Charles V_ :
Even Melancthon, whose merit of every kind entitled him to the first place among the Protestant divines, being now deprived of the manly counsels of Luther, which were wont to inspire him with fortitude and to preserve him steady amidst the storms and dangers that threatened the church, was seduced into unwarrantable concessions by the timidity of his temper, his fond desire of peace, and his excessive complaisance towards persons of high rank. (X)
Even such a small change as inverting "timidity of his temper" and "fond desire" would impair the calculated euphony of Robertson's sentence, but much more damaging—like the omission of a bar in a sonata—would be the semantically relatively harmless excision of "excessive," "of every kind," "manly" or "unwarrantable." Robertson's syntax would be rendered insistent and staccato instead of calmly controlled and seemingly inevitable—more like Guicciardini, in fact. Reading Robertson is smooth, easy and reassuring; Guicciardini is abrupt, disturbing and, as a literary experience, a rugged and challenging one. Gibbon spoke no more than the truth when he paid tribute in his _Memoir_ to "the perfect composition, the nervous language, the well-timed periods of Dr. Robertson."
One of Robertson's Edinburgh pupils was Walter Scott. To say this is not particularly to claim that Scott's view of history was influenced more by Robertson than by the historical ideas of the Scottish Enlightenment generally, though it is true that it was Robertson who most focused on Scottish history. But the point is a more general one, about what could and could not be thought and articulated at a given historical moment. Scott learned to be intensely self-conscious about this from the Scottish Enlightenment's understanding of national changes in ideas and manners. Distance in time was also distance in prevailing ideals and modes of conduct. Scott's novel _Waverley_ (1814), the first of the series that came to bear its name, set in the 1745 Jacobite Rebellion, originally bore the subtitle "'tis Fifty Years since," amended to "'tis Sixty Years since" a decade later to retain accuracy. As the novel's eponymous hero travels north, from England to Scotland and from Lowlands to Highlands, he also travels backwards in time, to a partially inadvertent participation in the rebellion. The English manor house in which he is reared is, as Scott makes clear, itself an anachronism: Cavalier, High Tory, Jacobite in sympathies. The uncle who brings him up is obsessed with family tradition and genealogy. When Waverley joins the army, his uncle regrets that it is no longer the fashion for him to take with him a retinue of followers from the estate. His tutor is a nonjuring clergyman (i.e. one who has refused to take the oath of allegiance to the Hanoverian sovereign), who writes unreadable High Church Jacobite tracts. His aunt is obsessed with the visit to the house by Charles II seeking shelter after the battle of Worcester. The house is a seventeenth-century time capsule with earlier feudal residues.
In the Scottish Lowlands, Waverley lodges with an even earlier version of the past. The Baron of Bradwardine seems to belong roughly to the fourteenth and fifteenth centuries. He is learned in feudal terms and in heraldry, and deeply attached to the symbols of feudalism. He has no thoughts of a warlord kind of semi-independence, but is loyal (to the Stuarts) even to subservience, being a pedantic antiquary who sets much store by his family's hereditary office, attached to his tenure, of removing the sovereign's boots after a battle, which he supports with copious quotation in barbarous Latin. But the genuine feudal article as a contemporary military reality is the Highland chieftain Fergus Mac-Ivor, whom Waverley meets next. He is a clan leader and warlord who engages in the rebellion not so much out of loyalty as out of self-interest, aspiring to become a great man at a restored Jacobite English court and prepared to use his clansmen's loyalty to himself to achieve this by augmenting Prince Charles Edward's army. But he is a divided figure, and knows it. In acting the part of chief at a clan gathering that Waverley attends, complete with a Gaelic bard, he remains detached, half-apologizing to Waverley for its barbarism. The clan world, it is hinted, is in a sense older than feudalism; Scott makes several Homeric parallels. The contrast with Bradwardine, who is mocked by Fergus as a pedant, reinforces this. But Fergus is also a polished modern gentleman, brought up at the French court. Scott, sensitive as always to period, characterizes him as possible only at that particular point: "Had Fergus Mac-Ivor lived Sixty Years sooner than he did, he would, in all probability, have lacked the polished manner and knowledge of the world which he now possessed; and had he lived Sixty Years later, his ambition and love of rule would have lacked the fuel which his situation now afforded." His anachronisms are calculated and manipulative: he maintains the lavish hospitality of a clan chief and crowds his estate with a largely redundant tenantry as a recruiting ground—all in pursuit of a peerage from a restored Stuart dynasty.
Fergus is half-modern, or inwardly almost wholly so; but modernity has two other main faces in the novel. One is Waverley's father, who has reconciled himself to the Hanoverians and become a parliamentary follower of Sir Robert Walpole—a byword for corruption—for the rewards of place and profit. Abandoning the family principles, he seems freed from all principle; in the novel he is only an offstage presence. But modernity has a more honourable face: a military one. Waverley's English prisoner, Colonel Talbot, though able, in the eighteenth-century phrase, to "make an interest" to procure Waverley's pardon, is wholly devoted to the service of his country. His sense of duty and clear-eyed (Whig) perception of the nation's best interests are contrasted with the unthinking loyalty of the humbler Highlanders and with Fergus's self-interested independence, half military, half political. Robertson, who approved of professional armies, would surely have approved of Talbot, as he would have recognized the characterization of Mac-Ivor.
But the general point here is that Scott's own delineation of his characters and his sense of the significance of their spiritually diverse historical locations could hardly have been presented in that way a century earlier. More than a century of historical awareness and reflection lay behind Scott's ability to see them as he did. Scott was thoroughly self-knowing about this. He was an antiquary who made fun of antiquaries; a modern Tory with a fundamentally Whig view of British history, who knew imaginatively what it was to be a Jacobite; a Romantic who knew how to domesticate his Romanticism, knowing that the point about the Sublime, as in Burke's famous essay, was that it was to be enjoyed from a distance. History, offering as it did, in modern representations of it, a gallery of manners or, as Scott's contemporary Sir James Mackintosh put it, a museum of mankind, now afforded a variety of vicarious experiences and even partial identifications which contemporary life could not match. Scott's own house, Abbotsford, was made into such a museum. Fergus's passionately Jacobite sister, with whom Waverley falls temporarily in love, draws an ironic picture of Waverley's future of domestic happiness and lettered indolence; he is not made, as she is, for the cruelty and heroism of a civil war: "And he will refit the old library in the most exquisite Gothic taste, and garnish its shelves with the most valuable volumes;—and he will draw plans and landscapes, and write verses, and rear temples and dig grottoes...—and he will be a happy man." This was at least a partial ironic self-portrait by Scott, with history transmuted into an epicurean, imaginative antiquarianism. The Victorians loved this trait in him, but remained oblivious that its enabling condition was the Scottish Enlightenment's conception of the history of manners.
**Gibbon: Rome, Barbarism and Civilization**
Gibbon fascinated scholars in the later twentieth century as he did not, in Britain at least, in the nineteenth, and the interest shows no sign of abating. The bicentenary of Gibbon's death in 1794 was impressively celebrated; the festivities, suitably inaugurated by a superb new edition of _The Decline and Fall of the Roman Empire,_ by David Womersley (1994), included a conference at Magdalen College, Oxford, so memorably denounced by Gibbon for its neglect of him as an undergraduate. One of Gibbon's predictions in his _Memoir_ of his life is clearly falsified: the University of Oxford "will as cheerfully renounce me for a son as I am willing to disclaim her for a mother." Our modern understanding of the rich archaeology of his mind has been enhanced not only by studies of him but by copious work on the Enlightenment, on the Machiavellian-humanist tradition, on seventeenth-and eighteenth-century scholarship and antiquarianism, and on the Scottish Enlightenment in particular, to which his debt is at various points apparent and which to the nineteenth century was not even a concept, much less an object of study. Arnaldo Momigliano, in his classic account of the division of seventeenth-and eighteenth-century historical writing into elegant but unscholarly narratives and miscellaneous antiquarian learning, made Gibbon a kind of synthesis, the climax and resolution of the story, as a historian who was also a scholar and a scholar who was an unsurpassed narrator.
Gibbon himself was aware of the situation that Momigliano analysed, and confronted it in his first published essay, originally in French, entitled an _Essay on the Study of Literature_ (1761). This was a defence of literature in the wide sense—humane learning, including history—against the contempt often and fashionably expressed in the eighteenth century for mere erudition. In the wake of Descartes's method of philosophical doubt, all facts about the past seemed dubious; the more antiquaries sought to ascertain such facts, the more disconnected and even trivial many of their findings seemed to be and the more uncouth, pedantic and otiose antiquarian learning appeared when judged by the "philosophic" standards of clarity, system and utility. In his preliminary discourse to the French _Encyclopédie_ the mathematician d'Alembert relegated history to the faculty of mere memory. Mathematics, not humanist erudition, was the esteemed paradigm; the ancients had been decisively surpassed, and there was little or nothing more to be learned from them.
Against this Gibbon, already conscious of a vocation as a historian, protested. He himself did not mind teasing the antiquarian "mere compiler" with charges of pedantry and obsessive, wasted effort, as the footnotes to _The Decline and Fall of the Roman Empire_ later made clear, but in his essay he invoked the possibility of a synthesis of the old and new models of intellectual achievement in the "philosophic historian"—someone who would be learned in the literature of antiquity, devoted to factual accuracy, but also capable of seeing in history a tissue of events connected by deeper causes than those most apparent, and able to present them coherently and perspicuously. Montesquieu was one model: learned but also intellectually probing and systematic. Among the ancients Gibbon above all invoked Tacitus, "who employs the force of rhetoric only to display the connection between the links which form the chain of historical events, and to instruct the reader by sensible and profound reflections" ( _Study of Literature_ ).
Gibbon did not himself work with manuscript sources, though he was indebted to and appreciative of those who did, particularly the French Benedictine scholars of the Congregation of St. Maur, who, from the later seventeenth century, had produced massive and deeply learned critical editions of the works of the Fathers of the Church—extensively used by Gibbon—and documents of early medieval history. But though Gibbon did not work in archives he was a natural scholar, to whom erudition was a pleasure and not merely an adjunct to literary composition. In his _Memoir_ he wrote gleefully of his acquisition for twenty pounds of the twenty volumes of the _Mémoires_ of the French Academy of Inscriptions, "nor would it have been easy, by any other expenditure, to have produced so large and lasting a fund of rational amusement." The work of the Academy was a testimony to the antiquarian passion devoted, particularly, to the study of ancient artefacts—coins, medals, funerary inscriptions and the like: the kind of material on which historians of the ancient world still heavily rely. The study of it was one way in which Gibbon was later able to go beyond his ancient literary sources, including the historians whom the humanists had been content to recover, imitate and follow. Superficially regarded, Gibbon might seem only a late example of them, with his own narrative depending in the first instance, as among the classical historians themselves, on following and sometimes criticizing or seeking to reconcile the work of earlier historians. But thanks to the scholarship of his own day and of the previous century (much of it denigrated as antiquarian), his own amazingly wide erudition, and his sharply critical approach to his sources, far exceeding the rather reluctant occasional criticisms by ancient historians of the accounts of their predecessors, Gibbon massively transcended them.
It is difficult now to recognize how innovative was Gibbon's choice of his life's work. It was not the practice to write histories of the ancient world (though Adam Ferguson had published a history of the Roman republic), because the ancient historians were thought unsurpassable, both through merit and through superior access. Instead one wrote commentaries—at which Gibbon, as exercises, tried out his apprentice hand on Sallust and Livy—or imitated them on other periods. Half a century later, in the early nineteenth century in England, the topicality of democracy as a contentious political issue gave rise to rival histories of Greece by William Mitford and George Grote. But in its time Gibbon's work was unique, and not only in its scale. To attempt it was undoubtedly a bold step, which Gibbon came to take only gradually. It is noticeable that even when committed to it he chose to begin with the period half a century after Tacitus' work, in the middle of the second century AD. He uses, and sometimes expresses gratitude to, or dissatisfaction with, Dio and Ammianus as well as a number of lesser or epitomized historians, but he had avoided any direct challenge to the author he regarded as the greatest of the ancient historians. Later, of course, as he emerged into the Christian, medieval western and Byzantine era—his history ends only in the fifteenth century, with the fall of Constantinople to the Turks—he was into another world, for the authorities for which he had often as much contempt as regard. But to go there was not originally his intention, for Gibbon's history far outgrew its origins.
Though it is well known, and for that matter somewhat imaginative, it is impossible not to quote Gibbon's account in the _Memoir_ of the moment of its conception, though it took many years to bring it to birth: "It was at Rome, on the fifteenth of October, 1764, as I sat musing amidst the ruins of the Capitol, while the barefooted fryars were singing Vespers in the temple of Jupiter, that the idea of writing the decline and fall of the City first started to my mind." Two points need to be made at once. One is that the original idea was to write the history of the city—a more antiquarian kind of enterprise—not the empire; it is to the ruins of the city that Gibbon returns only at the end of the vast work which came to range so far beyond it. The history expanded to include Russia, Persia, Mongolia and China, as well as to the limits of the empire in North Africa, Britain and the Near East. It includes the Christian theological controversies before and after the conversion of Constantine, the history of Byzantium to 1453, and the earlier history of its Turkish conquerors; before this Gibbon gives us the rise of Islam and the Crusades. The other point is that the mood of exaltation, clearly evident in his letters at the time, on this, his only, visit to Rome, was aroused in Gibbon chiefly by reminiscences of the _republic_ : "Each memorable spot where Romulus stood, or Tully [Cicero] spoke, or Caesar fell, was at once present to my eye." Gibbon did not lament the fall of the empire, though he deplored some of its aspects: the lamentation was, though only occasionally explicitly, for an earlier catastrophe—the supersession of the republic. Gibbon was as hostile to the idea of universal empire as Robertson, and as enthusiastic about the modern balance of power: empire drained vitality, which was promoted by independence and rivalry. Rome, for Gibbon, at the outset of his history was already on a gradient of decline, even before Christianity and barbarism began their work. In the first chapter we hear again, as it were, the chanting of the friars: the Campagna, Rome's hinterland, was "the theatre of her infant victories [an echo of Livy]. On that celebrated ground the first consuls deserved triumphs; their successors adorned villas, and _their_ posterity have erected convents."
Meditation on, and in, the ruins of former greatness was, in the wake of the Romans themselves, an eighteenth-century English classical taste, and one to which Gibbon was always susceptible. Nor was the elegiac note offered only to the Romans of the classical age: it could be sounded even in a context with which Gibbon had no sympathy. The great church of St. Sophia in Constantinople, after it has been, for the Byzantine Greeks, polluted by the introduction of the Latin rite, stands deserted "and a vast and gloomy silence prevailed in that venerable dome, which had so often smoked with a cloud of incense, blazed with innumerable lights, and resounded with the voice of prayer and thanksgiving" ( _Decline and Fall,_ LXVIII).
After three introductory survey chapters, and with frequent excursions to the provinces, almost all of Gibbon's first volume (of six), before its conclusion with two notorious chapters on early Christianity (XV, XVI), is focused on Rome itself and the failure to solve the problem of peaceful imperial succession, bedevilled by the indiscipline and venality of the standing army and especially the Praetorian Guard, which at one point puts the empire up for auction. In these dozen chapters Gibbon is very close to the moral world and understanding of historical dynamics of Sallust, Livy and Tacitus. It was a cliché of the nineteenth century that eighteenth-century historians had no ability to empathize with past times or recognize their distinctive moral characters. This is nonsense. But what is true is that, in speaking of Rome before the conversion of Constantine, Gibbon's work exists in an easy community of values with the historians of the late republic and with Tacitus, and this is sometimes a handicap in approaching the Christian Church (we remember Tacitus' contempt) and Byzantium. It matters much less in his dealings with the barbarian invaders (where the Tacitus of the _Germania_ was actually a sympathetic guide), which are not coloured by republican nostalgia and by Gibbon's regretful respect for the purely politic endorsement by the educated of a generally graceful and tolerant civic polytheism.
It is not surprising that in his presentation of Rome's decline in the third, fourth and fifth centuries we often hear echoes or explicit endorsements of the Machiavellian diagnosis of the nemesis of conquest in the form of corruption followed by loss of liberty—a diagnosis itself grounded in ancient Roman moralizing and satire. It had also been adopted by one of Gibbon's early models, Montesquieu, in his _Considerations on the Causes of the Greatness and Decadence of the Romans_ (1734). Elements of it, focusing on the threat to freedom represented by professional standing armies and the dangers of corruption, became embedded in the English rhetoric of political opposition to the executive from the early eighteenth century. Gibbon summarizes it in the appendix to his third volume (Ch. XXXVIII), which at one time looked like being the conclusion to the whole work (with the conversion of Clovis to Catholicism as the groundwork of Charlemagne's "new" western empire), and which he entitled "General Observations on the Fall of the Roman Empire in the West":
The decline of Rome was the natural and inevitable effect of immoderate greatness. Prosperity ripened the principle of decay; the causes of destruction multiplied with the extent of conquest...The victorious legions, who in distant wars acquired the vices of strangers and mercenaries, first oppressed the freedom of the republic, and afterwards violated the majesty of the Purple. The emperors, anxious for their personal safety and the public peace, were reduced to the base expedient of corrupting the discipline which rendered them alike formidable to their sovereign and to the enemy...
We should beware of taking the "General Observations" as a short cut to Gibbon's thinking. In fact he offers a number of causes of decline, though still within the neoclassical, civic-humanist frame of reference. In Chapter II he tells us that it was the long peace and uniform government of the Romans that undermined spirit and energy; in Chapter VII it is the mingling of the Romans with the servile provincials and in Chapter XXVII, inevitably, luxury and effeminacy. He also sees the transfer of its seat to Constantinople as marking the empire's definitive orientalization, epitomized as effeminacy and servility, for which he has a Catonic contempt going back, ironically, to the ancient Greeks—the court ritual of the emperors' "slaves" (there is a reminiscence here of Montesquieu's model of despotism ruling by fear and identified with the Orient) replacing the senatorial dignity which had been at least outwardly observed in Rome itself. All this, apart from the reference in Chapter II to the "long peace and the uniform government of the Romans, [which] introduced a slow and secret poison into the vitals of the empire," would have seemed common sense to the Romans themselves: that peace presented dangers was a thought which went back at least to Sallust; but the danger of "uniform government" signals modern ideas of a mixed constitution, national governments and the balance of power.
Gibbon also gave, as Montesquieu in the _Considerations_ did not, a role in decline to Christianity; the ancient pagans too, when they became aware of it, had blamed Christianity, though on account of its impiety towards the gods. For Gibbon, theological controversies—on which he had made himself expert—promoted civil strife, while Christian ethics, and particularly monastic asceticism, detracted from the martial virtues. Although it does not stand alone, the humanist formula of decay left deep marks on Gibbon's history and outlook: the fatal sequence of virtue, conquest, luxury, corruption, loss of freedom, and ultimate surrender to hardy barbarian conquerors is taken by Gibbon as something like a universal law, for the barbarian conquerors themselves are, inevitably, launched into the same sequence. It was something the Roman historians had at least hinted at, as in Livy's account of the devastating effect on Hannibal's victorious troops of the enervating luxury of Capua. In Gibbon, Goths, Vandals, Arabs, even the Mongol conquerors of China, succumb to the same civilized virus: "Alaric [the Goth conqueror of the city of Rome] would have blushed at the sight of his unworthy successor, sustaining on his head a diadem of pearls, incumbered with a flowing robe of gold and silver embroidery, and reclining on a litter or car of ivory, drawn by two white mules" (LI). Power and even civilization itself seem locked into a self-defeating cycle.
After the fractiousness and venality of the professional soldiery—Gibbon's account often reads like a sequel to Tacitus' _Histories,_ including the use of irony—the next major theme announced in _The Decline and Fall,_ in Chapters XV and XVI but recurring thereafter as a continuous and highly influential presence in both East and West, is Christianity. Gibbon, though he deprecated the fiercer attacks on religion by the French _philosophes_ —he calls Voltaire a fanatic—approached it, of course, with the rational and humane distaste of his age for blind superstition and "enthusiastic" zeal, which Hume had distinguished as the two poles between which the religious mentality oscillated. Much of the eighteenth century's hostility to religious fanaticism derived from a sense of gratitude that the era of religious wars, persecutions and massacres which had characterized the two previous centuries seemed effectively over, if perhaps only dormant. Gibbon's "our enlightened age" is, among other things, a sigh of relief, and at the outbreak of the French Revolution he immediately saw, in the French "patriotic" zealots, the swarms of monks whose fanaticism he had so often castigated in _The Decline and Fall._
His principal weapon against religion was, notoriously, irony. This takes a number of forms. Sometimes we have a phrase which, depending on the preconceptions of the reader, can be read either devoutly or sceptically, as in St. Augustine's "progress from reason to faith," or a sentence like "The laws of nature were frequently suspended for the benefit of the church." There were precedents for this in a scholarly work of the previous century that Gibbon admired, Pierre Bayle's _Philosophical Dictionary,_ but Gibbon's examples are much more polished and economical. Much of the mischief, too, is in the diction, especially the employment of an urbane, even prim, eighteenth-century vocabulary for the passionate and often eccentric zeal of early Christian asceticism and belief: "prudent," "singular," "experience." Origen, the Church Father who was alleged to have embraced chastity to the point of self-castration, had "judged it the most prudent to disarm the tempter." St. Simeon Stylites, living on top of a pillar as a gesture of withdrawal, had achieved fame "by the singular invention of an aerial pennance." Again, "In the primitive church, the influence of truth was very powerfully strengthened by an opinion which, however it may deserve respect for its usefulness and antiquity, has not been found agreeable to experience. It was universally believed, that the end of the world, and the kingdom of Heaven, were at hand" (XV, XXXVII, XV). It was not empathy, though it was magnificent.
There was nothing perfunctory, however, about Gibbon's treatment of Christian theology and practice or the Christological controversies of the Fathers and the great church councils of the age of Constantine. He is as magisterial, precise and detailed about these as he is about Byzantine court politics or the conquests, manners and new kingdoms of the barbarian invaders. For these last he had some contemporary monographs, chiefly French, such as the history of the Huns by des Guignes—with, of course, the late antique and medieval historians, Jordanes, Priscus, Paul the Deacon and Gregory of Tours. The garrulous charm and weak or non-existent concept of high politics of Gregory were met by Gibbon with stern Ciceronian disapproval: "in a prolix work...he has omitted almost everything that posterity desires to learn. I have tediously acquired, by a painful perusal, the right of pronouncing this unfavourable sentence" (XXXVIII).
Conceptually, however, Gibbon was indebted above all to the "histories of civil society" produced in the Scottish Enlightenment, for his characterization of what he calls (XXVI) the "Manners of the Pastoral Nations." Gibbon has derived from these a general concept of the way of life and accompanying manners of pastoral nomadism exemplified by the Huns, Arabs and Mongols and, a little doubtfully, the Germans. Gibbon shows these peoples' peculiar aptitude for war and conquest. His marshalling of the barbarian nations as they successively sweep over the Roman world may recall to us Herodotus' introduction in successive ethnographic and geographical digressions of the nations conquered by the Persian Great King, which are then mustered by him (and Herodotus) for the invasion of Greece. Gibbon clearly enjoys and takes pride in his historian's role of impresario of the nations: "I shall lead the Arabs to the conquest of Syria, Egypt, and Africa, the provinces of the Roman empire; nor can I check their victorious career till they have overthrown the monarchies of Persia and Spain"(XLVIII). But his own characterizations of barbarian manners offer much more than the lists of barbarian customs and beliefs given by ancient historians. Through the organizing concept of economic "stages," with their appropriate manners, unfolded in the history of civil society, he makes these traits intelligible as related aspects of a whole way of life. But he remains a historian, not just a sociological categorist, tracing the impact of the barbarians on the settled peoples and the interactions between them, and attending to their special characteristics, such as the formation of the creed of "Mahomet."
Gibbon's relations with leading members of the Scottish Enlightenment, as well as his intellectual debts, are worth attending to for a moment, for the influence was also literary and personal. As a pendant to his famous musing on the steps of the temple of Jupiter we may add another, much less well-known, quotation, for together they represent the two intellectual poles, neoclassical moralizing and modern sociological sophistication, between which his work stands. On 8 April 1776 David Hume wrote to the printer William Strachan saying that he was much taken with the new work by Mr. Gibbon which Strachan's firm had published, adding that "Dr. Smith's is another excellent work that has come from your press this year." To have published in the same year the first two volumes of _The Decline and Fall_ and Smith's _Wealth of Nations_ was indeed a publishing double-first deserving congratulation. The conjunction was apt, for aspects of Gibbon's work are as much an offshoot of the Scottish Enlightenment as Smith's work is its most famous literary monument. Gibbon was on cordial terms of intellectual comradeship with Smith, Ferguson, Robertson and Hume; with the last two, the great historians, even of discipleship. In recounting proudly in his _Memoir_ the successful reception of his first two volumes, it was to them that Gibbon gave pride of place: "The candour of Dr. Robertson embraced his disciple; a letter from Mr. Hume overpaid the labour of ten years." Contemplating their achievements as historians had at one time filled him, as he said, "with a mixed sensation of delight and despair," and this was, as he makes clear, mainly a matter of their mastery of narrative. In them he found what he had looked to in the _Essay on the Study of Literature_ : elegance, fluency and lucidity combined with learning devoid of pedantry or ungainliness. His own prose is manifestly related to, say, that of Robertson in _Charles V,_ which we have seen that he admired, but more mannered, pointed and memorable. Antithesis is a favourite device, and the syntax is often artfully balanced or cumulative, as when a triad of assertions forms a cumulative series, rounded off with a climax or the false climax of bathos. We have already seen several examples of this. An amusing inversion of bathos is his well-known description of the Roman conquest of Britain, where three ignominious adjectives are concluded with a genuine climax: "After a war of about forty years, undertaken by the most stupid, maintained by the most dissolute, and terminated by the most timid of all the emperors, the far greater part of the island submitted to the Roman yoke" (I).
Another notable feature of Gibbon's writing which has become famous or notorious is the footnotes, which do far more than just identify sources. Robertson and Hume had used footnotes as a way of bringing erudition to the support of the text without cluttering it with documents. Gibbon made them into an idiosyncratic art form, a commentary in which he gives rein to a relaxed, garrulous intimacy which acts in counterpoint with the tautly controlled formality of the text. They evoke the community of historians, scholars and antiquaries, ancient and modern, in a kind of camaraderie of admiration, scorn and sometimes smut. It is best just to quote a few random examples: "See an excellent dissertation on the origin and migrations of nations in the Mémoires de l'Académie des Inscriptions...it is seldom that the antiquarian and the philosopher are so happily blended" (IX). On the eternal damnation of the pagans: "The Jansenists who have so diligently studied the works of the Fathers, maintain this sentiment with distinguished zeal; and the learned M. de Tillemont never dismisses a virtuous emperor without pronouncing his damnation" (XV). "The modern Greeks [he means historians]...have displayed the love, rather than the talent, of fiction" (XXXII). "I have somewhere heard or read of the confession of a Benedictine abbot: 'My vow of poverty has given me a hundred crowns a year; my vow of obedience has raised me to the rank of a sovereign prince.' I forget the consequences of his vow of chastity"(XXXVII). On the execution of a heretical bishop: "The bishopric is now worth 20,000 ducats a year...and is therefore much less likely to produce the author of a new heresy" (XXVII).
Irony is, of course, a distancing device, as when Gibbon contrasts Roman imperial decadence with the virtues of the republic, or holds up with tweezers for enlightened contemplation the eccentricities of Christian ascetics. It can also be turned inward, with an effect of equivocation. This is a frequent feature of Augustan English writing generally, as well as common in Gibbon, as when a pair of contrasting motives is assigned as the cause of an action: "the credulity or prudence," "the avarice or humanity" and so on. Equivocation can also become unease and paradox. The sequence of virtue, conquest, luxury, corruption is a frequent generator of such paradoxes. Perhaps virtue and even civilization are self-doomed. The unease as well as the positive weight attached to the word "civilized" appears in the first paragraph of the history. The opening is proud: "In the second century of the Christian Æra, the empire of Rome comprehended the fairest part of the earth, and the most civilized portion of mankind." But what does it mean to be civilized? Three sentences later we find that "Their peaceful inhabitants enjoyed and abused the advantages of wealth and luxury." Perhaps to enjoy _was_ to abuse. Gibbon's irony feeds not only off the paradox of conquest turning apparently inexorably into servitude—the Roman and civic-humanist paradox—but also off the rather different paradoxes generated by the history of civil society, particularly its concept of "unintended consequences," which could be benign as well as disastrous but all the more morally equivocal for leaving ill-conduct unpunished or even rewarded. The extreme form of this paradox, which in its formulation was almost universally condemned as well as borrowed, occurred in Bernard de Mandeville's satire _The Fable of the Bees_ (1714), whose message was summarized as "private vices, public benefits." It was a thought highly familiar to Gibbon, and was even used by him to subvert the language of republican virtue: "The historian Sallust, who usefully practised the vices which he has so eloquently censured..." (XXXI, n. 105). In his presentation of Christianity Gibbon inverts the paradox and bends it to his ironic purpose. The Christian virtues such as piety and zeal must be acknowledged to be virtues, yet their consequences often seem highly regrettable: private virtues, public detriment.
Gibbon, of course, finds this unsurprising, but the double-edged character of civilization was disturbing. He finds hope in the "modern" thought, announced by Montesquieu and fostered by Hume, that it is only luxury procured by conquest that is enervating, but that the "opulence"—a neutral word—obtained by industry is innocuous, since it requires a sustained output of energy and discipline.
_The Decline and Fall_ in fact has in a sense two conclusions, one of which at least hints at optimism. The most obvious one is the fall of Byzantium to the Turks and the extinction of the last embers of the Roman empire, by the last of the barbarian invasions. But is it the last? Gibbon in his "General Observations" had invoked the notion of the revival of European civilization at the Renaissance. The other climax, in Chapter LXX, is the return, as promised, to the city of Rome, where we see the republican revival and the symbolic inauguration of the Renaissance by the coronation of Petrarch as poet laureate. It is a false dawn, however. Republicanism is crushed; the Renaissance itself was timidly imitative rather than a new beginning. Really to infuse his narrative with optimism Gibbon would have had to take it beyond the fifteenth century, to the eighteenth. Schematically and only in a survey, this is what he has done in the "General Observations." If it had stood, as originally intended, at the end of the entire work the optimistic trajectory would have been clear, for Gibbon's view of the Europe of his own day, until the outbreak of the French Revolution, is one of undiluted approval. The arts of life have been improved. The fatal uniformity of the Roman empire and the passivity it encouraged have been replaced by a beneficial fragmentation into a diversity of nation states and the energy generated by their rivalry. Under the auspices of the balance of power, Europe can be regarded as "one great republic" (a significant word): tense, energetic, diverse, but also a partnership in the arts of peace. Civilization has not merely survived the fall of the Roman empire which had for so long been its chief home, but has improved itself. At the end of _The Decline and Fall_ as we have it, something like this thought is at least obliquely alluded to. The ruins of the city, left by the decline and fall of the empire, which Gibbon referred to as "the greatest, perhaps, and most awful scene, in the history of mankind," are now "devoutly visited by a new race of pilgrims from the remote, and once savage, countries of the North" (LXXI). The pilgrims are civilized or they would not be devout, and their savage past is far behind them. Rome's fall can be contemplated as from a balcony, as an eighteenth-century connoisseur of the Sublime might enjoy its terrors from a secure distance. Gibbon's history is such a balcony, and is therefore part of the victory of civilization in surviving the fall of Rome.
It is useless to look, in Britain at least, for Gibbon's immediate successors. There are none. In France and the United States (Chapter 24) matters were different. The great Anglophile nineteenth-century French historian François Guizot annotated and translated Gibbon's work. In America, Gibbon was revered by the outstanding historians Prescott, Parkman and Henry Adams. But in England there was on the whole neglect and hostility. This is symptomatic of something wider. It is a fairly safe rule that any general Victorian pronouncement on the historical sense or the historiography of the eighteenth century will be dismissive, ignorant and distorting. The mentality of that century which, more than any other, has established the categories of the modern understanding of the history of Europe is habitually characterized in nineteenth-century England as "unhistorical." The reasons for this nonsense would make a good subject for the history of ideas. One has the impression that, having been so categorized, the eighteenth-century historians were little read. Denigrating clichés were passed unexamined from pen to pen, even among serious scholars. Leslie Stephen's pioneering _History of English Thought in the Eighteenth Century_ (1876) was notably deficient in appreciation of the period's historical writing. One can still hear echoes of this denigration; it is time they died away.
Partly it seems to derive from taking as representative of the eighteenth century the expressions of hostility to history that Gibbon tried to answer in his _Essay._ Partly it seems to derive from an unsubtle reading, as in the case of some ancient historians, of affirmations of the existence of a common human nature, ignoring the fact that these were often followed by reminders of the very different forms this nature could assume under different historical circumstances. One relevant consideration is surely the cultural fault line created by the French Revolution, which made some eighteenth-century pronouncements seem complacent and which retrospectively, in England, tainted the whole French Enlightenment. (The Scottish Enlightenment, as a concept, was of course established only in the later twentieth century.) Voltaire, rather than Montesquieu, though criticized by Robertson and Gibbon as unscholarly and superficial, was taken as epitomizing the age. In the nineteenth century, too, the categories of "race" and "nation" were reified and taken with extreme seriousness as bearers of precise and indelible qualities; eighteenth-century cosmopolitanism such as one finds in Hume and Gibbon could be regarded as more evidence of superficiality, whereas we would be inclined to see a sensible scepticism. Against Gibbon, too, as a notorious scoffer and sneerer, stood Christian earnestness. So, for that matter, did agnostic earnestness. The Victorians, as Nietzsche noted, expected critics of Christianity to go about their business with a proper solemnity. It was axiomatic for most that the Catholic Church was wrong. That it might also be funny was a thought not entertained. The literary appreciation of Gibbon in Britain had to wait for Lytton Strachey and Winston Churchill (an odd couple), both of whom admired and in different ways imitated him in the early twentieth century. In the world of scholarship Gibbon's revenge has been comprehensive, and he is now securely installed in the historians' pantheon. This is caught by the phrase used of himself and his colleagues by the outstanding contemporary historian of "Late Antiquity"—a concept we largely owe to him—Peter Brown: "In Gibbon's Shade."
**TWENTY-TWO**
**Revolutions: England and France**
**Macaulay: The Glorious Revolution**
Interest in England's seventeenth-century constitutional crisis, which had been central to English historiography in the eighteenth century, remained so in the nineteenth. It is true that the later period saw an awakened appreciation of the medieval in cultural contexts, and that the Saxons, partly owing to the sponsorship of Sir Walter Scott, enjoyed a renewed popularity. Towards the end of the century, too, imperialist enthusiasm prompted something of a cult of the age of Elizabeth. But the conflicts of Cavaliers and Roundheads continued to provide political reference points, and inspired a popular iconography. Foreign historians were also drawn to them. Two of the greatest nineteenth-century historians, François Guizot and Leopold von Ranke, wrote on the English Civil War period. In England, two of the leading proponents of a new professionalization in the study of history, Samuel Gardiner and Sir Charles Firth, devoted themselves to setting the seventeenth-century record straight and correcting the errors of their unprofessional predecessors. By far the most popular of these historical accounts, from the mid-century onward, when the _History of England from the Accession of James II_ was published (1848–61), was by a Whig Member of Parliament and minister and retired Anglo-Indian official, Thomas Babington (Lord) Macaulay (1800–59).
Macaulay's _History,_ which ends with the death of William III in 1702, is centred on the "second" seventeenth-century revolution, the "Glorious Revolution" of 1688, by which the Catholic and allegedly tyrannical James II had been overthrown and the crown conferred on William of Orange and his wife, James's daughter Mary. In the early nineteenth century three Whig politicians, including a subsequent prime minister, had written histories of this Revolution: Charles James Fox, Lord John Russell and Sir James Mackintosh (who bequeathed to Macaulay a valuable collection of documents, particularly strong in the pamphlet literature of the period). It was not a surprising interest. Parliamentary reform became the Whig policy in Opposition, leading to the Reform Act of 1832; the constitution to be reformed was that established by the Revolution in 1688 and its confirming legislation, which some regarded as definitive for all time. The Revolution had to be resituated in the Whig canon. It had to be a matter of separating the spirit from the letter. The best way of respecting the achievement of the men of 1688, according to reformers, was not to enshrine their work in constitutional marble, but to do as they had done and make the constitutional adjustments necessary to meet new circumstances. The necessary resituating was the task that Macaulay, drawing on the work of the Whig constitutional historian Henry Hallam, carried through to extraordinary popular acclaim, for where Hallam was dry Macaulay was vivid, dramatic, eloquent and exhilarating.
Two contemporary events helped to shape Macaulay's _History._ One was the 1832 Reform Act. In supporting it, Macaulay, as a Whig parliamentary spokesman, scored a spectacular oratorical success. Macaulay's speeches on the Reform Bill still resound on the page. They have an irresistible brio, and they are informed throughout by a sense of history. Macaulay compares the crisis of his times to that confronting Charles I and the Long Parliament, and his diagnosis is essentially the same as Hume's:
[Charles] would govern, I do not say ill, I do not say tyrannically; I only say this: he would govern the men of the seventeenth century as if they had been the men of the sixteenth century; and therefore it was, that all his talents and all his virtues did not save him from unpopularity, from civil war, from a prison, from a bar, from a scaffold. These things are written for our instruction. Another great intellectual revolution has taken place; our lot has been cast on a time analogous, in many respects, to the time which immediately preceded the meeting of the Long Parliament. There is a change in society. There must be a corresponding change in the government.
The progress of society had created a large, newly prosperous and politically self-conscious class which must be admitted to the franchise and included in the political nation by constitutional change, or revolution would follow, as it had in seventeenth-century England, as it had in France. Macaulay's political convictions were suffused by history and by political reminiscence. He had a strong feeling for the great parliamentary occasions, and they always recalled for him those of the past. He saw contemporary political events as though they were already historical: writing to his sister, he compared being present at the crucial vote on the Reform Bill to seeing Caesar stabbed in the Forum or Cromwell taking the mace from the table of the House of Commons when he expelled the Members and inaugurated his personal rule.
The second, recent, event by which Macaulay's mind and political ideas were decisively shaped was the French Revolution of 1848. The publication of his _History_ had been immediately preceded by the wave of revolutions which spread all over Europe in 1848, and he alluded to them as a dire warning, but also as a contrast to the English "Glorious Revolution" of 1688, which formed the centrepiece of his _History._ England had remained tranquil when, as he wrote in his preface, "the proudest capitals of Western Europe have streamed with civil blood." Macaulay had no doubt that it was the "preservative" Revolution of 1688 and the timely and peaceful passage of the Reform Act of 1832 which had saved England from the revolutions experienced on the Continent, above all in France. The connection in Macaulay's mind between 1688 and 1832, "between the Revolution which brought the crown into harmony with the Parliament, and the Revolution which brought the Parliament into harmony with the nation," would have been clearer had his _History_ run its planned course up to 1832, instead of finishing, because of its author's death, at the beginning of the eighteenth century. As it is, we have some inkling of what the eighteenth-century volumes might have been like in the published collections of his essays, including those on eighteenth-century statesmen.
Reconciliation is a recurring theme of the _History_ itself. Its hero is William of Orange, the Deliverer, who, James having fled, presides over the restoration of order and liberty. But the nation, though corrupted by the reigns of the last two Stuart kings, is also an accomplice in its own deliverance, not only through the lords who risk their necks in summoning William, but in the stiffening of constitutional and Protestant resolve in the later stages of James's reign. There is, in Macaulay's account, a rallying in defence of law and liberty and in resistance to James's arbitrariness which foreshadows the moment of (temporary) near-unity in the Revolution itself, when, the King having fled and the Prince of Orange not yet having taken control, Whigs and Tories momentarily sink their differences in defence of law and order and in an impressive show of unity by the respectable part of the nation.
It is natural to take the essay Macaulay wrote on "History" in 1828 as his definitive statement on the historian's task. This can be misleading as a guide to his achievement. It is sometimes taken as a call for "social history" (a promise fulfilled only in one chapter [III] of the _History_ ), as well as for historians to take lessons from the techniques developed in the novel, and especially by Scott. History should discard some of its dignity and descend into the common haunts of men, the exchange and the coffee house. How steep a descent this indicated is open to question: one can go a good deal lower and wider than either of these. Macaulay may have thought he was fulfilling his prescription by his frequent attempts, like Hume's, to summarize the state of public opinion. He certainly succeeded in imparting to historical narrative an unprecedented vividness and dramatic and emotional intensity, for which he saw Scott as a model, but he was not a "social historian." His _History_ remained chiefly focused on royal policies, parliamentary debates and state trials, and the appraisal of the intentions and qualities of public men.
Macaulay himself was born, not indeed into the Whig aristocracy where he found his early patrons, but into the comfortable middle class, and also in an atmosphere of great public causes. His father, Zachary, was a leading member of the Anti-Slavery League. At Cambridge in the early 1820s Macaulay shone as a debater, shedding the family Toryism and evangelical Christianity. He became an advanced Whig and reformer, but never a Radical. After winning a reputation as a reviewer and essayist and after his success in Parliament, he went in the 1830s to a lucrative post in India, from which he returned with an independent income. He later returned to Parliament and became a minister, but the last part of his life was really devoted to his _History._
He can, in fact, be seen as in many ways the last great neoclassical historian, writing history after a successful public career. His own culture was above all classical, broadened by a love of eighteenth-century novels and drama, as well as by his reading of Burke, whom he revered, and Scott. Where he was exceptional was not in being a social historian, which he certainly was not, but in the emotional range and depth, the almost pictorial vividness and concreteness, and the dramatic intensity of his historical writing. None of this was exactly new—it was foreshadowed in Robertson and Hume—but in no other historian was it so copiously and even extravagantly displayed. His own model historians were Tacitus and Thucydides, but he had an enduring distaste for Plutarch, on whom he wrote an early essay, and for the tradition of heroic—Macaulay would say posturing—classical republicanism which looked back to him, in late seventeenth-century England and above all in the French Revolution. He was not unmitigatedly hostile to the latter, thinking it productive of good in the long run, but he detested its strain of Plutarchian demogoguery.
Though the Scottish Enlightenment did much to shape his view of the progress of society, he was a modern English Whig who (unlike Hume) cherished the pieties of his political tradition with an eloquence unique except in the work of Burke. In the essay on Plutarch, noting the political rootlessness of his French admirers, he wrote that "Senate has not to our ears a sound so venerable as Parliament. We respect the Great Charter more than the laws of Solon. The Capitol and the Forum impress us with less awe than our own Westminster Hall and Westminster Abbey, the place where the great men of twenty generations have contended, the place where they sleep together." Macaulay's frequent invocations of earlier English history are eirenic more than partisan. A typical example (in Chapter V of the _History_ ) is the reference to the chapel in the Tower of London where the remains of the Duke of Monmouth were interred after his unsuccessful rebellion and his execution in 1685, which Macaulay has described in detail:
The head and body were placed in a coffin covered in black velvet, and were laid privately under the communion table of St. Peter's Chapel in the Tower. Within four years the pavement of the Chancel was again disturbed, and hard by the remains of Monmouth were laid the remains of [ Judge] Jeffreys. In truth there is no sadder spot on the earth than that little cemetery. Death is there associated not, as in Westminster Abbey and St. Paul's, with genius and virtue, with public veneration and with imperishable renown; not, as in our humbler churches and churchyards, with everything that is most endearing in social and domestic charities; but with whatever is darkest in human nature and in human destiny, with the savage triumph of implacable enemies, with the inconstancy, the ingratitude, the cowardice of friends, with all the miseries of fallen greatness and of blighted fame.
Examples follow of the disgraced and inglorious dead.
This was a melancholy set piece, but Macaulay was a high-spirited man who responded to glitter, energy and bustle, without qualms. One example occurs in James's reign with the King's attempt to overawe the capital with a military camp at Hounslow. Civil society is too much for his coercive intentions:
The Londoners saw this great force assembled in their neighbourhood with a terror which familiarity soon diminished. A visit to Hounslow became their favourite amusement on holidays. The camp presented the appearance of a vast fair. Mingled with the musketeers and dragoons, a multitude of fine gentlemen and ladies from Soho Square, sharpers and painted women from Whitefriars, invalids in sedans, monks in hoods and gowns, lacqueys in rich liveries, pedlars, orange girls, mischievous apprentices and gaping clowns, was constantly passing and repassing through the lower lines of tents. From some pavilions were heard the noises of drunken revelry, from others the curses of gamblers. In truth the place was merely a gay suburb of the capital. (VI)
It is a Hogarthian scene, or, if we look for a mid-Victorian parallel, one like Frith's _Derby Day,_ full of animation, variety and small vignettes. But there is a significance too, though Macaulay's conscious awareness of it is perhaps not certain. It is a kind of parody of a familiar topos from the stern classical republican tradition: the army suborned and corrupted by the luxury of the capital (Chapter 8). But Macaulay, though perhaps not used to hailing painted women as agents of the unintended benefit of constitutional liberty, approved of luxury, the fruit of peaceful industry and commerce. The army is not so much corrupted as reabsorbed by the civil society which James had hoped to use it to coerce, assimilated to the nation socially if not yet constitutionally.
The pictorial quality of Macaulay's narratives could also be given a more domestic or melodramatic tone, evoking pity and intimacy. There is, for example, his account of the flight of James's queen from Whitehall by river. James entrusts her and her baby to two French gentlemen:
Lauzun gave his hand to Mary; Saint Victor wrapped up in his warm cloak the ill-fated heir of so many kings. The party stole down the back stairs and embarked in an open skiff. It was a miserable voyage. The night was bleak; the rain fell; the wind roared; the waves were rough; at length the boat reached Lambeth; and the fugitives landed near an inn, where a coach and horses were waiting. Some time elapsed before the horses could be harnessed. Mary, afraid that her face might be known, would not enter the house. She remained with her child, cowering for shelter from the storm under the tower of Lambeth Church, and distracted by terror whenever the ostler approached her with his lanthorn. (IX)
It is a perfect Victorian genre scene of "Beauty in Distress," shaped by the heritage of eighteenth-century "sentimentalism": the wild night, the open boat and the heaving dark river; the young woman with her child, cowering for shelter beneath the dark tower of the church, shrinking from the ostler's lantern, the source of light in the picture.
The two rebellions—the abortive one by Monmouth and the successful invasion by William three years later—provide, of course, other pictorial and dramatic opportunities. For Macaulay the crushing of the Monmouth rebellion was the end of the ranting, fanatical aspects of radical Whiggism, republicanism and Puritanism, as well as a tragedy for the Protestant peasantry of the West Country who supported it. Its worst aspect was epitomized in the old intriguer Robert Ferguson, who drafted Monmouth's justificatory declaration. Macaulay's character study is as vehement as his often were:
Violent, malignant, regardless of truth, insensible to shame, insatiable of notoriety, delighting in intrigue, in tumult, in mischief for its own sake, he toiled during many years in the darkest mines of faction. He lived among libellers and false witnesses. He was the keeper of a secret purse from which agents too vile to be acknowledged received hire, and the director of a secret press whence pamphlets, bearing no name, were daily issued. (V)
In 1688 Ferguson attempts to join William's expedition, but is cold-shouldered. Macaulay uses him to make the contrast between William's enterprise and Monmouth's earlier one: "He had been a great man in the knot of ignorant and hotheaded outlaws who had urged the feeble Monmouth to destruction: but there was no place for a lowminded agitator, half maniac and half knave, among the grave statesmen and generals who partook the cares of the resolute and sagacious William" (IX).
Monmouth's rebellion had been an amateur affair. It was part of Macaulay's distaste for classical republicanism and its revival from the Renaissance onward that he despised its cult of the citizen army as anachronistic in an age of commerce and specialization. He sometimes refers to the armed West Country peasantry as "clowns" they were no match for James's professional army. As a follower of the Scottish Enlightenment and a Victorian patriot, Macaulay could be enthusiastic about professional armies—in the right cause and properly subjected to civilian authority. Robertson would have agreed (Chapter 21). Macaulay uses the parade of William's troops through Exeter, the first city to fall to him, further to underline the contrast with Monmouth's pathetic rabble. In its stressed exoticism William's army recalls—and is quite probably meant to—the multi-ethnic composition of Xerxes' great host as reviewed by Herodotus (Chapter 1). Potentially barbaric and terrible, it is made benign, as well as effective, by discipline and by William's firm, controlling hand.
From the West Gate to the Cathedral Close, the pressing and the shouting on each side was such as reminded Londoners of the crowds on the Lord Mayor's day. The houses were gaily decorated. Doors, windows, balconies, and roofs were thronged with gazers...the people of Devonshire, altogether unused to the splendour of well ordered camps, were overwhelmed with delight and awe. Descriptions of the martial pageant were circulated all over the kingdom. They contained much that was well fitted to gratify the vulgar appetite for the marvellous. For the Dutch army, composed of men who had been born in various climates, and had served under various standards, presented an aspect at once grotesque, gorgeous, and terrible to islanders who had, in general, a very indistinct notion of foreign countries. First rode Macclesfield at the head of two hundred gentlemen, mostly of English blood, glittering in helmets and cuirasses, and mounted on Flemish warhorses. Each was attended by a negro brought from the sugar plantations on the coast of Guinea. The citizens of Exeter, who had never seen so many specimens of the African race, gazed with wonder on those black faces set off by embroidered turbans and white feathers. Then, with drawn broadswords, came a squadron of Swedish horsemen in black armour and fur cloaks. They were regarded with a strange interest; for it was rumoured that they were natives of a land where the ocean was frozen and where the night lasted through half the year, and that they had themselves slain the huge bears whose skins they wore. (IX)
The last great scene of 1688 is provided by the debates on a constitutional settlement in the Convention Parliament, at which James was declared to have abdicated and, after much wrangling, the crown was offered to William and Mary jointly (X). Oratory is one of the great themes of Macaulay's history, though it is presented in summarized form, not ostensibly verbatim. He has a reverential sense for the great parliamentary occasion, and, as we have seen in his description of the passing of the Reform Act (Chapter 22), gives it additional solemnity by historical reminiscence and prospect. In describing the Convention Parliament he makes space to record the presence of William Sacheverell, "an orator whose great parliamentary abilities were, many years later, a favourite theme of old men who lived to see the conflicts of Walpole and Pulteney." House folklore is an aspect of enduring institutions, and it was especially cherished by Macaulay, who in this case was himself a member of the club.
The debates in the Convention Parliament are, of course, given full measure. Macaulay stresses how little they had to do with political philosophy, how much with English law and its stretching to an unprecedented situation: "If it were a legal maxim that the throne could never be vacant, it was also a legal maxim that a living man could have no heir. James was still living. How then could the Princess of Orange be his heir?" And so on. In 1828, in an essay on Hallam, Macaulay had deplored the petty-mindedness of the parliamentarians of 1688. Now, two French revolutions later, he endorsed their pragmatism. The declaration which gave the throne to William and Mary was illogical and contrary to fact, but it worked, and the settlement it created endured. By a benign fiction that no revolution had taken place, the work of the revolution, once done, continued to be ballasted by precedent. The account of the manner of the Parliament's deliberation is not just description for description's sake, as the subsequent words make clear. It is a lesson and an endorsement:
As our revolution was a vindication of ancient rights, so it was conducted with strict attention to ancient formalities. In almost every word and act may be discerned a profound reverence for the past. The Estates of the Realm deliberated in the old halls and according to the old rules...The sergeant with his mace brought up the messengers of the Lords to the table of the Commons; and the three obeisances were duly made. The conference was held with all the antique ceremonial. On the one side of the table, in the Painted Chamber, the managers of the Lords sat covered and robed in ermine and gold. The managers of the Commons stood bareheaded on the other side. The speeches presented an almost ludicrous contrast to the revolutionary oratory of every other country. Both the English parties agreed in treating with solemn respect the ancient constitutional traditions of the state. The only question was, in what sense those traditions were to be understood...When at length the dispute had been accommodated, the new sovereigns were proclaimed with the old pageantry...To us who have lived in the year 1848 it may seem almost an abuse of terms to call a proceeding, conducted...with such minute attention to prescriptive etiquette, by the terrible name of revolution. And yet this revolution, of all revolutions the least violent, has been of all revolutions the most beneficent...
One of Macaulay's contemporaries, Thomas Carlyle, had a puritanical contempt for all ceremonious shams and pretences, as well as a considerable contempt for Parliament. It is not surprising that he chose to become a historian of the French Revolution, which to him was the vengeance of history on outworn forms pretending still to be realities. His sensibility and Macaulay's, respectively puritanical and Burkean Whig, represent two facets of the early nineteenth-century mind confronting history.
**Carlyle's _French Revolution_ : History with a Hundred Tongues**
Thomas Carlyle was born in 1795, the year that signalled the end of the French Revolution, though not, of course, of its effects: the Parisian rising against the new, moderate and corrupt government of the Directory was put down by the cannon commanded by "Artillery Officer Buonaparte," as Carlyle calls him. It is the last episode in Carlyle's book on the Revolution, published over forty years later. To reach that point in the book leaves the assiduous reader with something of the same emotional exhaustion as must have been felt by the survivors of the Revolution, with all the ideals and most of the leading actors in the six years since 1789 dishonourably buried: "The Notabilities of France disappear, one after one, like lights in a Theatre, which you are snuffing out" (Volume III, Book VI.III). Carlyle inimitably recognizes the shared fatigue: "O Reader! Courage. I see land!"
Carlyle was born into a poor, Bible-reading, Calvinist family in south-west Scotland. He received a good education, both classical and scientific, at the University of Edinburgh: geology, chemistry and astronomy provided him with a stock of metaphors. Losing his Christianity while retaining a good deal of his Calvinism, he was unable to make a career as a minister of the Kirk, and had to establish himself as a man of letters, which led to his migration to London. He first displayed his highly individual literary idiom—biblical, Germanic, burlesque and hectoring—in his personal philosophical statement in _Sartor Resartus_ (1833–4). _The French Revolution_ (1837) was his great bid for literary recognition, and it succeeded. It was, among other things, like all his works in some way or other, an admonition. The Revolution was for him the advent of Democracy (always capitalized), terrible and sublime, in the modern world. It was a demonstration of divine justice, passed on a corrupt aristocracy which believed in nothing.
The Revolution had naturally encouraged apocalyptic speculation. Carlyle was not, of course, a literal believer in the Christian Apocalypse, though he was a close friend of the millenarian preacher Edward Irving, but the imagery of apocalypse as conflagration and destruction always came readily to him, as we see in the extraordinary concluding page of the book (see Chapter 12) and in the account of the taking of the Bastille. The book was written in the early years of the liberal "July Monarchy" in France, which was brought into being by a second revolution in 1830. In Britain there had been fears of revolution (not altogether extravagant) not only in the 1790s but in the period following the Napoleonic Wars and at the time of the campaign for the Reform Bill in the early 1830s. The decade in which Carlyle's history was written, the 1830s, saw the growth of Chartism, which aroused much apprehension and on which Carlyle wrote a long essay in 1839.
Carlyle responded to the French Revolution with a mixture of awe and complicity and grim enthusiasm, combined with horror and pity. He never really adjusted to the more tranquil period of the 1850s and '60s, which he viewed, with a kind of baffled impatience, as stagnation. The French Revolution, though he was condescending to its sentimentalities and appalled by its atrocities, appealed to his Old Testament and Calvinist sensibilities, as a divine scourging. Carlyle's ambivalence comes out most clearly in his treatment of what he calls "Sansculottism"—fuelled by hunger and desperation, capable of heroism and atrocious cruelty. Carlyle at his most abstract—and he is rarely abstract for long—presented the Revolution as the clash of such personified abstractions: "Sansculottism," "Patriotism," "Respectability," "Philosophism," "Clubbism." But his rhetoric is often highly concrete, and these abstractions are sometimes given physical characteristics reminiscent of allegory; those of "Sansculottism" (sometimes also personified as the poverty-stricken Parisian suburb of "Saint-Antoine") are almost tangible: "many-headed, fire-breathing." The Paris mob is sometimes "sooty Saint-Antoine," just as the courtiers are "the Œil-de-Bœuf" (a salon of assembly in the palace of Versailles). In reintroducing characters—in a useful mnemonic but also a deliberate evocation of epic convention, for Carlyle saw the Revolution as epic—he tags them with a repeated adjective or phrase in Homeric fashion: "Usher Maillard, Méry of the Thousand Orders" (said to have been written by him in the Hôtel de Ville on the day of the fall of the Bastille), "Old-Dragoon Drouet." Marshal de Broglie, whom the court relies on vainly for a royalist military coup, is "Mars," while the court usher, the Marquis de Brézé, is "Mercury" (he carries messages), and several times the source of excellent but also symbolic comedy. It is he who carries to the National Assembly the King's command to disperse, which the Assembly defies with the Tennis Court oath. De Brézé, who also acts as a kind of doorkeeper, and stands for the _Ancien Régime_ and etiquette, by which he lives, is trying to shut the door on world history (I.V.II). Brézé has his symbolic moment, but some of the most frequent, memorable and ominous reappearances are those of Jean-Paul Marat—to Carlyle "horseleech" or "sooty" Marat—who croaks his messages of hatred and at whose entrance, Carlyle says, the lamps in the room turn blue.
Marat, squalid and malevolent, is something like the evil genius of the Revolution, responsible for the massacres in the prisons in September 1792, but in embodying the hatreds and fanaticism of his Sanculottic followers, he has for Carlyle a kind of authenticity and power which the "Anglomaniac" National Assembly and later the "respectable" republican faction of the Girondins do not. The great men of the Revolution, Mirabeau and Danton, have this quality of authenticity or, in Carlyle's word, "Reality," and even Marat, without any finer qualities, has it. Carlyle respected fanaticism. "Reality," wherever found, is a kind of divine emanation, while frivolity and shams and theories are mere historical debris, to be shovelled away by men who are in earnest. He has no sympathy with middle-class, respectable constitutionalism—monarchist or republican—equating it with pedantry and legalism. The crowd scenes in the Revolution, which wind up his rhetoric to an extraordinary pitch of intensity and almost delirium, are for him sublime because spontaneous, whereas the 14 July 1790 commemoration of the fall of the Bastille, by an immense national festival and by oath-swearing on an "altar of the fatherland," is sentimental play-acting because prearranged. Mobs, however, _are_ authentic:
Your mob is a genuine outburst of Nature; issuing from, or communicating with, the deepest deep of Nature. When so much goes grinning and grimacing as a lifeless Formality, and under the stiff buckram no heart can be felt beating, here once more, if nowhere else, is a Sincerity and Reality. Shudder at it; or even shriek over it, if thou must; nevertheless consider it. Such a Complex of human Forces and Individualities hurled forth, in their transcendental mood, to act and react, on circumstances and on one another; to work out what it is in them to work. The thing they will do is known to no man; least of all to themselves. It is the inflammablest immeasurable Fire-work, generating, consuming itself. With what phases, to what extent, with what result it will burn off, Philosophy and Perspicacity conjecture in vain. (I.VII.IV)
The French revolutionary mob is a portent, and the Revolution itself is still active in the world, "the crowning Phenomenon of our Modern Time" and a lesson to Carlyle's contemporaries. Feudal aristocracy is everywhere being replaced by a moneybag aristocracy, but "That there be no second Sansculottism in our Earth for a thousand years, let us understand well what the first was; and let Rich and Poor of us go and do _otherwise_ " (III.VII.VI). The French Revolution is both the epic of Democracy and an admonition.
It must be apparent by now that Carlyle is no ordinary historian. John Stuart Mill called him a poet, though it is worth remembering that Lord Acton, not a man to underrate the historian's responsibilities, paid tribute to Carlyle's book as "the volumes that delivered our fathers from thraldom to Burke." The book is slow to start, and the reader has to use the early books, dealing with the _Ancien Régime,_ to get used to Carlyle's peculiar authorial manners and his experiments with diction and syntax, the frequent vocatives addressed to both the reader and the historical characters, and the occasional use of Gallicisms to convey the flavour of a translation (e.g. "there to consider himself"). A decadent society on the verge of extinction is depicted through a kind of collage of symbolic vignettes, held together by a rumbling, Calvinistic authorial sermon. There is an occasional welcome touch of grotesque humour or bathos. The colourful dresses at a kind of _fête champêtre_ to watch a balloon ascent (the chapter is significantly entitled "Windbags") are like banks of flowers, so, by a piece of surreal logic, the coaches in which their wearers sit are flowerpots:
Manifold, bright-tinted, glittering with gold; all through the Bois de Boulogne, in longdrawn variegated rows;—like longdrawn living flower-borders, tulips, dahlias, lilies of the valley; all in their moving flower-pots (of new-gilt carriages): pleasure of the eye, and pride of life! So rolls and dances the Procession: steady, of firm assurance, as if it rolled on adamant and the foundations of the world; not on mere heraldic parchment,—under which smoulders a lake of fire. Dance on, ye foolish ones; ye sought not wisdom, neither have ye found it. Ye and your fathers have sown the wind, ye shall reap the whirlwind. Was it not from of old written: _The wages of sin is death?_ (I.II.VI)
The narrative really picks up with the summoning of the Estates General to deal with the public bankruptcy (which Carlyle views with grim complacency), leading to the election of a National Assembly in which the nation's hopes become invested, and are, of course, doomed to disappointment. Carlyle, as a narrator of extraordinary idiosyncratic power, needs events, the more rapid and momentous the better, to give the reader his best.
One has to accept Carlyle as a historian, if at all, for what he is; it is no use expecting what he did not attempt to be: a lucid purveyor of linear narrative and careful analyses of cause and effect. These things can be found in the midst of Carlyle's accounts, but his stranger effects were entirely deliberate, made largely out of epic precedent, an Old Testament style of vision, a fierce pulpit manner, and an idiosyncratic cosmic view: a metaphysics made concrete through symbolism. The effect on narrative is a rapid cutting from individuals, often humble and seen only momentarily, and highly particular situations, rendered in full concrete circumstantiality, to cosmic and world-historical perspectives, with many intermediary points between.
Carlyle had meditated much on the writing of history, producing two essays on the subject, and to more penetrating effect than in Macaulay's parallel reflections. Carlyle's view and its effects on his chosen rhetorical strategies are summarized in two notable quotations: "History is the essence of innumerable biographies" and "Narrative is _linear,_ Action is _solid._ " The former warns us to expect no restrictions imposed by "the dignity of history" the latter not to expect straightforward chains of cause-and-effect explanation and accounts of the pursuit of considered, purposive politics. Everything in history is multiply determined; the actors scarcely see beyond their feet, and in every moment an immensity of different events is occurring, any of which may be significant. Our observations have to be successive though the things done were often simultaneous, and "shape after shape bodies itself forth from innumerable elements." Narrative, therefore, though it strive against its own linear nature, must try, as it were, to move sideways as well as forwards.
Carlyle's devices for bringing this about are essentially two: the selection of certain events, characters and actions as symbolic of larger realities, and extraordinarily innovative experiments in what may be called multi-voiced narrative, where the authorial voice, so often peremptory, intrusive and bullying, sometimes seems temporarily suspended in favour of a cacophony of other voices, of which he is the impresario, making a babble of catchphrases out of quotations from the newspapers, pamphlets, placards and memoirs he has consulted. This imagined babble in the midst of the revolutionary crowd, where Carlyle aims to place the reader—always, of course, in the present tense—is, to use his own term, combustible. A particular word or incident can ignite it into action and almost randomly determine its direction: to the Bastille, to Versailles, to the palace of the Tuileries, and hence to some of the central events of the Revolution. The combustibility is made up of hunger and hatred, suspicion and rumour. Suspicion, for example, of a royalist military coup, which leads to the storming of the Bastille. On 12 July
the streets are all placarded with an enormous-sized _De par le Roi,_ "inviting peaceable citizens to remain within doors," to feel no alarm, to gather in no crowd. Why so? What mean these "placards of enormous size"? Above all, what means this clatter of military; dragoons, hussars, rattling in from all points of the compass towards the Place Louis Quinze...
Have the destroyers descended on us then? From the Bridge of Sèvres to utmost Vincennes, from Saint-Denis to the Champs-de-Mars, we are begirt! Alarm, of the rogue unknown, is in every heart...Are these troops verily come out "against Brigands"? Where are the Brigands? What mystery is in the wind?—Hark! a human voice repeating articulately the Job's-news: _Necker, People's Minister, Saviour of France,_ is dismissed. Impossible; incredible! Treasonous to the public peace!...(I.V.IV)
Carlyle gives a similar preamble/explanation for the march to Versailles, involving the market women of Paris, which in the following October brings the royal family as virtual prisoners back to Paris and inaugurates the next act of the Revolution. There are rumours of a disloyal banquet held at Versailles to celebrate the arrival of a new regiment, where the nation has been insulted and its new tricolour cockade trampled underfoot:
Yes, here with us is famine; but yonder at Versailles is food; enough and to spare!...bloody-minded Aristocrats, heated with excess of high living, trample on the National Cockade. Can the atrocity be true? Nay, look: green uniforms faced with red; black cockades,—the colour of Night! Are we to have military onfall; and death also by starvation?...
In one of the Guardhouses of the Quartier Saint-Eustache, "a young woman" seizes a drum,—for how shall National Guards give fire on women, on a young woman? The young woman seizes the drum; sets forth beating it, "uttering cries relative to the dearth of grains." Descend, O mothers; descend ye Judiths, to food and revenge!—All women gather and go; crowds storm all stairs, force out all women: the female Insurrectionary Force, according to Camille [Desmoulins] resembles the English Naval one; there is a universal "Press of women." Robust Dames of the Halle, slim Mantua-makers, assiduous, risen with the dawn; ancient Virginity tripping to matins; the Housemaid, with early broom; all must go. Rouse ye, O women; the laggard men will not act; they say, we ourselves may act!
And so, like snowbreak from the mountains, for every staircase is a melted brook, it storms...(I.VII.III–IV)
The idiom here, of course, is that of epic, but the narrative technique is also an interpretation, right or wrong.
Interpretations of the French Revolution began almost as early as the Revolution itself. Conspiracy was the initially preferred explanation, attributed first to manipulation, using his vast wealth, by the radical head of the younger branch of the royal house, Philippe, duc d'Orléans, who later adopted the name Philippe Egalité and voted for the death of Louis XVI. In the later 1790s the French Catholic exiles, notably the Abbé Barruel, produced a rival conspiratorial version which attributed the Revolution to the Freemasons. In Britain, explanations in terms of "deeper" causes, particularly economic, soon replaced these early versions, but the conspiratorial view had life in it, at least at the popular level, for a long time. An example is Dickens's _A Tale of Two Cities_ (1859), which is sometimes cited as a case of Carlyle's influence. This may be true of the crowd scenes, but in their interpretations of the Revolution the two authors diverged widely. Dickens dwells on an entirely imaginary network of plotters, organized as a secret society, with mysterious signs and passwords for mutual recognition; this reflects conditions in Europe in the mid nineteenth century, particularly in Italy and France, but has nothing to do with either the actual French Revolution or Carlyle. Carlyle presents the French as stumbling from phase to phase of the Revolution, without clear intention, driven by events, by suspicion and fear, as well as by idealism and fanaticism, and arriving where no one had intended or foreseen. Carlyle's originality consisted most of all in the ways he dramatizes this view and enacts it before the reader's eyes—and ears, for one imagines Carlyle's narrator, as he invites one to do, as a heard voice.
The kind of voice it was can be sharply appreciated by the contrast with Macaulay, whose own mode of address was often oratorical. Take, for example, Macaulay's account of what he sees as the moment when the Revolution of 1688 might have descended into anarchy, with William not crowned, the army disbanded, James in flight, and the anti-Popish mob in full cry:
It was a terrible moment. The king was gone. The prince had not yet arrived. No regency had been appointed. The great seal, essential to the administration of ordinary justice, had disappeared. It was soon known that Feversham had, on receipt of the royal order, instantly disbanded his forces. What respect for law or property was likely to be found among soldiers, armed and congregated, emancipated from the restraints of discipline, and destitute of the necessaries of life? On the other hand, the populace of London had, during some weeks, shown a strong disposition to turbulence and rapine. The urgency of the crisis united for a short time all who had any interest in the peace of society. ( _History,_ X)
It is a situation report from the historian, but the manner, in diction and the measured ordering of the sentences, is also the parliamentary manner. By converting the verbs to the present tense, and inserting "Mr. Speaker" or "My Lords" at intervals, it is easy to imagine it as delivered by a minister to Parliament (not then sitting). "My Lords, the king is gone. The prince is not yet arrived. The Great Seal has disappeared. We understand that Lord Feversham, on receipt of an order from His Majesty, has..." etc. The voice, whosoever's it is, is in control, and so, the reader infers, is the institution for which it stands. There is nothing like Carlyle's tormented, fractured syntax, just as there is no voice with this kind of authority in Carlyle's _French Revolution._ In England we understand from Macaulay that Rationality and Respectability, however much under stress, are ultimately in control and will win. In Carlyle there is no such reassurance on behalf of either.
Both Macaulay and Carlyle stood at the apex of a long movement, from the eighteenth century, before austere professionalism spoiled the game, to render history for the reader in its full sensuous and emotional immediacy and circumstantiality. But their perspectives are different. The emotions in which Macaulay mainly involves the reader are parliamentary ones, or at least could have been expressed in Parliament—and none the worse for that. Carlyle said that the great element missing from our attempted entry into the past is Fear; he set himself to re-enact it, and succeeded extraordinarily well. His syntax is designed to embody a distracted groping for certainties in a fog of rumour and of events at best only half-understood, in moods of acute anxiety, rage and sometimes dangerous exaltation.
Carlyle's effects to this end were not to be exceeded until the twentieth century and in a different medium. To read Carlyle now is to be reminded of Sergei Eisenstein's cinematic technique in the handling of crowd scenes in another revolution, with the camera panning in and out from the most highly individualized close-up moments to the widest perspectives. Very Eisensteinian, as it were, is not only the intimacy Carlyle achieves but also the sudden shifts of perspective, so that we see the crowd as though from a camera high above, as a river to which each doorway and staircase is a tributary. Another such sudden shift, this time to a cosmic perspective, is achieved by a focusing, very cinematic in anticipation, on an indifferent non-human witness. As the siege rages, "the great Bastille clock ticks (inaudible) in its Inner Court there, at its ease, hour after hour; as if nothing special, for it or the world, were passing" (I.V.VI). It is part of the infinity of simultaneous events in history, which are also events in infinite space and time. On the evening the Bastille is taken the July sun falls "on reapers amid peaceful woody fields; on old women spinning in cottages; on ships far out in the silent main; on Balls at the Orangerie of Versailles, where high-rouged Dames of the Palace are even now dancing with double-jacketed Hussar-Officers" (I.V.VII). World history itself is dwarfed by the omnipresence and persistence of the repetitions of daily life and the enduring natural world. The peasant in Breughel's picture who ploughs his field while a distant Icarus falls from the sky is a very Carlylean emblem. At the height of the Terror in Paris twenty-three theatres play nightly and sixty assembly rooms for dancing are open: "In startling transitions, in colours all intensated, the sublime, the ludicrous, the horrible succeed one another; or rather, in crowding tumult, accompany one another." The historian, he says, would be glad of a hundred tongues. Lacking them, he must snatch for the reader "this or the other significant glimpse of things, in the fittest sequence we can" (III.V.I).
In contrast to Macaulay, Carlyle sometimes excuses himself from attending the tedious sessions of the National Convention, making its irrelevant constitution. He visits it, for preference, only in the moments when parliamentary decorum breaks down and the Convention becomes itself a kind of mob—as when the women from Paris, wet, bedraggled, famished and angry, break into its orderly session in Versailles and the President has to order in food for them from neighbouring cookshops, which Carlyle greets, after many allusions, with a genuine Homeric quotation: "nor did any soul lack a fair share of victual": loaves, wine and "great store of sausages" (I.VII.VIII). Sausages do not generally appear in national assemblies, nor in history. For Carlyle the Insurrection of Women was both burlesque and sublime. The confrontation of the desire for a constitution with the desire for sausages was an encapsulation of much that the Revolution was about. Always the mundane, the quotidian and the domestic reassert themselves, and Carlyle is as complicit with them as he is roused to celebration of the world-historical moment. Even then, in the epic of Democracy, the heroes are of course ordinary men: "On then, all Frenchmen that have hearts in their bodies. Roar with all your throats of cartilage and metal, ye Sons of Liberty; stir spasmodically whatever of utmost faculty is in you, soul, body or spirit; for it is the hour! Smite, thou Louis Tournay, cartwright of the Marais, old-soldier of the Regiment Dauphiné smite at that Outer Drawbridge chain..." (I.V.VI). The main inspiration, it is clear, is Homer; the rhapsodic attempt to render the multitudinous, and the vast energy of the mass, hint for the modern reader also at Walt Whitman.
Carlyle's use of bathos is not only funny but humanizes his narrative, which is not, of course, always at full stretch. When the Goddess of Reason, having been worshipped on the altar, returns home, "ungoddessed," to her husband, what do we imagine they talk about that evening over supper? It is a question one can imagine no other historian asking, then or perhaps since, and for it one can forgive him much pulpit rhetoric—and has to. History, we are vividly reminded, is the essence of innumerable biographies. Bathos occurs too in the account of the defence of the frontiers by the revolutionary armies, which was for Carlyle another epic of the Revolution. The threat of the invasion by the reactionary powers rouses him to the same helter-skelter mimetic narrative as the storming of the Bastille or the march to Versailles, the rapid concatenation of names and nouns matching the urgency of the hour:
Does not the Coalition, like a fire-tide, pour in; Prussia through the opened North-East; Austria, England through the North-West?...On Toulon Arsenal there flies a flag,—nay, not even the Fleur-de-lys of a Louis Pretender; there flies that accursed St. George's Cross of the English and Admiral Hood...Beleaguer it, bombard it, ye Commissioners Barras, Fréron, Robespierre Junior; thou General Cartaux, General Dugommier; above all, thou remarkable Artillery-Major, Napoleon Buonaparte! (III.IV.V)
But there is pathos as well as comic bathos in the expostulation of the municipality of Verdun, known for its patisserie, at finding itself reluctantly implicated in world history and expected to be heroic: "Resist him [the Duke of Brunswick] to the death? Every day of retardation precious? How, O General Beaurepaire (asks the amazed Municipality) shall we resist him?...Retardation, Patriotism is good; but so likewise is peaceable baking of pastry and sleeping in whole skin" (III.I.III). Verdun, ironically in view of its later status as an icon of resistance, capitulates tamely.
Carlyle, to the modern reader, is a paradox, an extraordinary compound of the archaic, or, worse, the deeply unfashionable, and elements we think of as distinctly modern. Swift and Rabelais before him suggest parallels, as does Whitman after. Carlyle's classmates nicknamed him "the Dean" in recognition of a Swiftian quality. Relations to earlier historiography are harder to see, though Carlyle greatly admired Schiller and came towards the end of a long period of erosion of the idea of the dignity of history. Carlyle has much time for the sublime, none for dignity. It is not surprising that the classical republican historians make no figure in his work, despite the French revolutionaries' devotion to them. The chief borrowing is, rather surprisingly, from Florence: Carlyle a number of times employs the _carroccio_ (Chapter 18) as an image of a symbolic rallying point. For the rest, no humanist he, or _philosophe,_ but an Old Testament-nurtured Puritan, at home with the mundane and the transcendental. Humorous, hectoring and at times almost frenzied, he found one guide to his monumental task in Homer's epic realism, and for the rest he went his own way. At his best—and he has more than one kind of best—his history and his prose have enormous imaginative energy, whose degenerations into bombast are the price the reader pays.
**Michelet and Taine: The People and the Mob**
In France itself, the serious study of the French Revolution began in the 1820s and has never ceased. Interpretation of the Revolution was central to current political attitudes and conflicts throughout the nineteenth century and beyond. The political factions which fought, politically, polemically and sometimes violently, for their different versions of France found the Revolution an inescapable point of reference. The principle of hereditary monarchy continued to be upheld by those spoken of as Legitimists or Ultras (i.e. ultra-royalist) and prevailed during the Restoration of the Bourbon monarchy from 1814 to 1830. The liberal, junior, Orléans branch of the family ascended the throne in 1830 after the July Revolution of that year in the person of King Louis-Philippe, whose father had voted for the execution of his cousin Louis XVI. After the Revolution of 1848, this "July Monarchy," which was supported by moderate liberal and generally Anglophile constitutionalists like the historian François Guizot (Chapter 23), who became minister of education and later leader of the government, was replaced by the Second Republic, in which class conflict soon produced a second, abortive, insurrection of the Parisian working class, whose leader, Louis Blanc, was himself a historian of the first French Revolution. As Karl Marx ironically noted, the overthrow of the Republic by Louis Napoleon, nephew of the Emperor, in 1851 recapitulated the supplanting of the first Republic by Napoleon. The Second Empire lasted until 1871, ending in military defeat by Prussia, followed by the violent suppression of the Paris Commune, whose methods had aroused vivid recollections of the revolutionary Terror of 1793–4. The Third Republic, liberal, anticlerical but mainly antisocialist, was led by another historian of the Revolution, Adolphe Thiers.
French nineteenth-century history often seemed engaged in recapitulating the different phases of the first Revolution: a partly revived _Ancien Régime,_ a liberal-constitutionalist phase echoed in the July Monarchy, the establishment of a Republic, a revolutionary Terror focused on Paris, with Bonaparte always waiting in the wings. The historiography of the Revolution could thus hardly be other than heavily politically charged. All political factions could find historical correlatives for their political allegiances and fears in the events of 1789–97. The continued strife over the status and role, particularly in education, of the Catholic Church, which had been one of the early targets of the Revolution, would alone have kept the divisions which had opened up then in the forefront of politics. Anticlericalism was an article of faith for most republicans.
The political prominence of the past ensured also the prominence of historians. Guizot was an eminent liberal historian, though he did not write on the Revolution. Thiers published one of the earliest histories of the Revolution, in the 1820s, before going on to hold high office under the July Monarchy, when he became Guizot's rival. He was eventually elected as the first president of the Third Republic, established in 1871. Another historian of the first Revolution, the poet Alphonse de Lamartine, was one of the leaders of the Second Republic. Louis Blanc's substantial history of the Revolution has already been mentioned. Under the Third Republic, France's leading socialist, Jean Jaurès, produced a _Socialist History of the French Revolution._
It was not uncommon in the nineteenth century for politicians to write history: as we have seen, in Britain, early in the century such prominent politicians as Charles James Fox, Sir James Mackintosh, Macaulay and a future prime minister, Lord John Russell, all wrote histories of the English revolution of 1688. In no case, however, was immediate political influence attributed to their histories. By the early nineteenth century, though there were nuances of interpretation, the 1688 Revolution had become in England a symbol of consensus. In the case of France, endorsement or denunciation of the leading actors of the Revolution—Mirabeau, the Girondins, Danton, Robespierre, the Commune of Paris and, for that matter, Bonaparte—immediately established one's contemporary political identity. The word "Jacobin" itself fell out of contemporary political usage, presumably for lack of overt claimants, though the Jacobins were not without admirers, but the Revolution still provided symbols of allegiance and a political vocabulary, including "Left" and "Right," referring to the seating of the factions in the Chamber.
By general consensus the outstanding history of France of the first half, perhaps of the whole, of the nineteenth century is that of Jules Michelet, which eventually grew to twenty-three volumes (1833–67). Into it, out of chronological sequence, was interpolated his two-volume _History of the French Revolution_ (1847–53). Though he did not take up a political career, Michelet's political commitment was unmistakable. His lectures at the Collège de France were suspended on the orders of Guizot, his patron and predecessor, now minister; this was a double irony, for Guizot's own lectures from the same chair had earlier been suspended by government order during the Restoration. Under the Second Empire Michelet lost his chair permanently, and the position in the National Archives that Guizot had procured for him, and retired into exile in Brittany, where he continued to work in local archives as well as writing his history and some other, highly idiosyncratic, works.
Before he came to the French Revolution he was known and admired for the medieval volumes of his _History of France._ All Michelet's work was deeply personal and emotional, but one early intellectual influence is noteworthy. In 1827 he translated the _New Science_ (1725) of the early-eighteenth-century Neapolitan thinker Giambattista Vico. The extent of Vico's influence on the thought of late-eighteenth-century Germans, most notably Herder, with whom he has a lot in common, and who also impressed Michelet, has never been easy to trace, but Vico's influence on Michelet is clear. To Vico, culture was a collective product of whole peoples. Mythology, in particular, gave a key to the mentalities of "the first peoples, who were everywhere naturally poets" ( _New Science,_ Chapter 22). Through it we can trace "a history of the ideas, the customs and the deeds of mankind" (Chapter 22), because they were "the manner of thinking of entire peoples" (p. 816). These, lacking the ability to form abstract concepts, expressed their ideas through personifications (Chapter 15). Lacking the faculty of abstraction, they apprehended and felt particular perceptions all the more vividly, which explains their sublimely poetic mentality (p. 819). In this way Vico establishes what were later to become almost a set of commonplaces, which were certainly subscribed to by Michelet, namely a set of antitheses not only between earlier and later times but between the popular and the educated mentality, in which the former is poetic and sensuous, the latter metaphysical and abstract. The populist Romanticism current in France in the 1830s and '40s, in which Michelet was immersed, was highly sympathetic to such ideas, but in his own case the influence of Vico was direct. _The History of France,_ with its emphasis on the life, experiences, thoughts and sentiments of the French people, reads at times like a massive embodiment of Vico's ideas. Michelet's manners as a writer were, as we shall see, declamatory and exclamatory in the Romantic fashion, carried, as in Carlyle, to idiosyncratic extremes, in a way that is now highly unfashionable. But the ways in which his attention as a historian is focused, and his passion for imaginative re-creation—he called writing history "resurrection"—find an echo in the interests of modern French historians.
In one respect Michelet's injunction to himself to bring about "resurrection," and his interest in the popular mind, overrode other personal predilections. He was heir to the anticlericalism of the French Enlightenment and the Revolution, but the medieval volumes of his history were admired by the Catholic Right. His treatment of Joan of Arc—whom he regarded as incarnating the self-consciousness of France as a nation—was particularly acclaimed. Michelet's history was thickly textured as well as dramatic; he worked assiduously in the National Archives, to which his post there gave him easy access. He has been called "the Victor Hugo of history" Hippolyte Taine compared him with the painter Delacroix. His identification with France and its people was so close that he regarded the _History_ as his spiritual autobiography: "It is through personal sorrows that the historian feels and reproduces the sorrows of nations." He rejoiced that in his vocation "God has given me in History the means of participating in everything."
It was his republican anticlericalism, however, particularly his hostility to the continuing influence of the Jesuits in French education, which turned him, prematurely in terms of his overall scheme for the _History,_ to the French Revolution. He had also been exploring what he saw as his roots and resurrecting his own early life in a short monograph called _The People_ (1846). In the dedicatory preface, to his friend Edgar Quinet, yet another historian of the Revolution, he celebrated his artisan origins. His father was an unsuccessful printer, exactly from the stratum from which the revolutionary crowds were chiefly recruited. Michelet had worked for him for a while, before the educational ladder created by the Revolution opened wider prospects to him. The paper wars of the Revolution were good for the business, but the Napoleonic censorship had been bad. Before writing on the recent and contemporary life of the People (we need to retain the upper case), Michelet had, as he explained, gone among them, talked, asked questions, and listened. But he also says he found his chief material in the reminiscences of his youth:
To know the life of the people, their toils and sufferings, I had but to interrogate my memory. For I too, my friend, have worked with my hands. The true name of modern man, that of _workman,_ I am entitled to it in more than one sense. Before I made books, I _composed_ them literally. I arranged letters before I grasped ideas; and I am not ignorant of the dismalness of the workshop, and the wearisomeness of long hours.
In turning in 1847 to the Revolution, Michelet in his own view was entering into the People's greatest historical achievement. The People was the collective hero; for him, as for Carlyle, the Revolution was the epic of Democracy; individual political leaders were secondary and sometimes highly culpable, but the People as such could do no wrong: in the Revolution's "benevolent period the whole people were the actors; in the period of cruelty only a few individuals." In the first crisis of the Revolution, among the populace "each one felt greatness in his breast from hour to hour."
There are considerable similarities between Michelet and Carlyle, which make the differences instructive. They worked entirely independently, but were subject to some of the same cultural influences, and possessed some similarities of temperament. Both conceived of the task of the historian as being to re-create and re-enact, and both threw their authorial personalities into the action, apostrophizing and exhorting, in moods often of exaltation, almost of frenzy. Michelet said, "I struggled physically with the clergy and the Terror." Carlyle complained loudly of the nervous cost of writing history: for a believer in silent toil he could be vigorously plaintive. In speaking of the actions of the crowd, the French language offered Michelet the convenience of an impersonal pronoun. He often spoke of it as " _on,_ " thus avoiding both "I" and "they" Carlyle sometimes used "we." Michelet, in speaking of himself as narrator, asserts his own presence: "I was at the foot of the Bastille. I was raising on its tower the immortal flag." Both authors were conscious of straining against the limitations of language. Carlyle, as we have seen, wished the historian had a hundred tongues, while Michelet yearned for "a new language, the language of a serious, loving Rabelais." They were, predictably, drawn to the same metaphors. Both spoke with abhorrence of the "mechanical," which stood for the abstract and soulless. Both invoked volcanoes, and spoke of the making of the new revolutionary world as "fermentation." Michelet calls on both images in his description of Danton—whom both he and Carlyle admired with qualifications—and of Danton's club, the Cordeliers, the rival to the Jacobin Club. We have to see the Cordeliers, Michelet says, "bubbling and fermenting together in their night sessions in the base of their Etna." Danton, his face ravaged by the pox, is hideous but sublime: "This almost eyeless face seems a volcano without a crater—volcano of mud and fire, within whose closed forge one hears the conflicts of nature." Mud and fire were also Carlyle's two most frequently invoked elements. Without precise recollection, and allowing for translation, the most expert reader could be deceived about which author was responsible for the above quotation.
Both, again, were susceptible to the apocalyptic, but here the differences begin to surface. Michelet's visions are sweeter; there may be a trace of his reading of the medieval mystical philosopher of history Joachim of Flores (Chapter 12), for whom the third age of the world was to be one of freedom, love and harmony. Carlyle's apocalypses were made of sterner, Hebraic stuff: scouring, retributive conflagrations. Carlyle's sensibility was Protestant and Judaic, while Michelet was touched by Indian mysticism. He would never have spoken of God, as Carlyle did, as "the almighty Taskmaster." Nature, for Michelet, came to acquire a pantheistic hue. Although, under the influence of German metaphysics, Carlyle was also fond of speaking of the interconnectedness of all things, he, like Karl Marx, conceived of physical nature as something for man to struggle with and wrest his subsistence from.
We have space for only two specific comparisons of Carlyle's and Michelet's respective treatments of key episodes of the Revolution. The first episode is the loyalist banquet at Versailles to welcome a new, royalist, regiment's officers, rumours of which fed revolutionary indignation and fears of counter-revolution in Paris, leading to the march, led by the market women, to Versailles to bring the royal family back to the capital. Carlyle's treatment is uncharacteristically bluff, almost amused. Young men get drunk, brag, say and do foolish things:
Suppose champagne flowing; with pot-valorous speech, with instrumental music, empty feathered heads growing ever the noisier, in their own emptiness, in each other's noise! Her Majesty, who looks unusually sad to-night (His Majesty sitting dulled with the day's hunting), is told that the sight of it would cheer her. Behold! She enters there, issuing from her State-rooms, like the Moon from clouds, this fairest, unhappy Queen of Hearts...Could featherheaded young ensigns do other than, by white Bourbon Cockades, handed them from fair fingers; by waving of swords, drawn to pledge the Queen's health; by trampling of National Cockades; by scaling the Boxes, whence intrusive murmurs may come; by vociferation, tripudiation [dancing in triumph], sound, fury and distraction, within doors and without,—testify what tempest-tost state of vacuity they are in? Till champagne and tripudiation do their work; and all lie silent, horizontal; passively slumbering. (I.VII.II)
Carlyle's comment is untypically indulgent: "It was so natural, yet so unwise."
Michelet takes the episode with deadly seriousness and a sense of outrage. The whole event, in his narration, is hectic, irrational and almost diabolic, as well as operatic. The officers are not only drunk but dazzled and disoriented when the King and Queen enter the royal theatre, "where the boxes, lined with looking-glasses, reflect a blaze of light in every direction." The officers tear off their revolutionary red-white-and-blue cockades, the new national emblem, and trample them underfoot. Michelet was always sensitive to symbolism, and employs it himself: in the _History of France_ the English, in destroying Joan the Maid, "thought they were deflowering France" (X). In Versailles
The music continued, ever more impassioned and ardent; it played the Marche des Hulans, and sounded the charge. They all leaped to their feet, looking around for the enemy to appear; for want of adversaries they scaled the boxes, rushed out into the _Cour de Marbre_...The frenzy of that moral orgy seemed to infect the whole court. (II.vii)
The differences appear most sharply, however, in Michelet's treatment of the first "Festival of the Federation," on the anniversary of the fall of the Bastille, 14 July 1790. Carlyle's version is above all ironic: the professions of universal goodwill are shortly to give way to massacre and the guillotine. But, in any case, humankind cannot sustain very much fraternity. Though he acknowledges that the Federation movement began spontaneously in the provinces and aroused popular enthusiasm all over France, he treats it as a kind of contagious intoxication and the greatest of the festivals, on the Champs de Mars in Paris, as manifestly artificial and stage-managed, which of course it was. To Carlyle its gospel was merely sentimental: he had, as we have seen, a Presbyterian sourness towards ritual, though he could be indulgent to spontaneous violence (as in the Scottish Reformation). But for Michelet the Federation is the high point of his history and in French national consciousness, pointing the way to a better future. He said that writing about it marked one of the great moments of his life. His description has an ominous element: at the sacramental moment, the swearing of the oath of fraternity, the surly demeanour of the royal family strikes a jarring note. But irony is almost absent, though he goes on to mourn over the contrasting future. The moment of the Federation was "the holy epoch in which the entire nation marched under one fraternal banner." He compares the marches to Paris by the participants from all over France to the Crusades: "What Jerusalem attracts thus a whole nation?...the Jerusalem of hearts, the holy unity of Fraternity, the great living city made of men..." its name is _patrie._ At the oath-swearing in the Champs de Mars
The plain is suddenly shaken by the report of forty pieces of cannon. At that clap of thunder, all rise and stretch forth their hands to heaven...O King! O People! pause...Heaven is listening and the sun is breaking expressly through the cloud...Attend to your oaths! Oh! how heartily the people swear! How credulous they still are!...But why does the King not grant them the happiness of seeing him swear at the altar? Why does he swear under cover, in the shade, and half-concealed from the people?...For God's sake, sire, raise your hand so that all may see it. (III.xii)
Hippolyte Taine, a quarter of a century later, offered his own version of the Federation oath, and the Revolution itself, in his _History of the French Revolution,_ which formed the second part of a longer work, _The Origins of Contemporary France_ (1875–95). His account of the Revolution is in almost every respect the opposite of Michelet's. The contrast between the two invites being treated as archetypal. Lord Acton coupled them as two works reading which formed an epoch in the reader's life: "No man feels the grandeur of the Revolution until he reads Michelet, or the horror of it without reading Taine." George Rudé, the chief modern analyst of the composition of the revolutionary crowd, spoke of historians, when referring to the crowd, as following the traditions established respectively by Michelet and Taine, and speaking of it accordingly as "the people" or "the mob." Rudé is highly critical of Taine's use of documents to characterize the make-up of the crowd, but it is a kind of tribute that Rudé expounds his own version, three-quarters of a century later, mainly in the form of an argument with Taine. Taine's research had been thorough, if insufficiently critical of documents which supported his case, and his rhetoric has an enduring power to shock and alarm. Where Michelet saw the essence of the Revolution, the role of the People, as benign, fraternal and inspiring, and blamed for its horrors only those who rejected the fraternal embrace, for Taine the Revolution was from the outset a pathological social phenomenon. The people, in the form of the mob, released from normal restraints, was irrational, uncontrolled and highly dangerous. He wrote in the shadow of the Paris Commune of 1871, which had revived memories of the excesses—and for some the heroism—of the first Revolution, and it shows.
Taine saw the revolutionary leaders as similarly out of control, intoxicated by general ideas which inspired an overconfidence exacerbated by political inexperience. Virtually the only point in common in the attitudes of Michelet and Taine was that neither was at all disposed to idealize, as some had done, the Anglophile constitutionalist leadership, above all that of Mirabeau, in the first stage of the Revolution. But their reasons for this rejection were characteristically different. Michelet, a strongly Anglophobe republican, had no sympathy with constitutional monarchy and the _juste milieu,_ while Taine denied that the Revolution had at any time been anything but recklessly utopian. In particular, Taine argued, from the moment, in July 1789, when the Assembly used the people as its shock troops and accepted the popular distribution of arms, the Revolution was set on a predetermined course.
Taine was himself a liberal constitutionalist, naturally drawn to the July Monarchy, which ended when he was twenty—the year he entered the Ecole Normale Superièure. He envied England's constitutional stability and responsible and experienced governing class, and read Macaulay's political essays as a source of political wisdom. He played no part in the Revolution of 1848, in which students were prominent, and wrapped himself self-consciously in the mantle of "science," far removed from political strife, as his vocation. He was a liberal for all that, believing strongly in freedom of thought and speech. Falling under suspicion by the clergy, who were influential in education in the Second Republic, he was forced into a kind of exile in the provinces for a while, despite his academic brilliance. He established his reputation, however, by his writings in the 1860s, on culture, art and psychology, and for several decades, from the '60s to the '80s, he became the dominant figure in French intellectual life.
Taine stood, above all, for a scientific approach—by which he often meant a psychological one—to questions of art, literature and (a particular interest) national character. In biology he was a follower of Lamarck among others, believing in the inheritance of acquired characteristics, and his slogan " _race, milieu, moment_ "—intended to provide a frame for the explanation of all cultural phenomena and collective psychologies—can be rendered roughly as "inheritance, circumstances and epoch." He believed that each cultural milieu and era had its master idea or disposition, which determined all its manifestations: thus the French mentality of the eighteenth century, which for him provided the motor of the Revolution, was characterized by an overriding confidence. This confidence was, for Taine, exemplified above all by Rousseau, and resulted in the application of simple abstract ideas of universal rationality—the _esprit classique_ expressed during the Revolution by the idea of popular sovereignty and embodied in the Declaration of the Rights of Man.
But until the 1870s, while Taine conducted himself with intellectual hauteur towards government and people alike, his interests were above all in psychology. What was intended as his masterwork, the result of years of study which included attendance at dissections and observation of the insane, was _On Intelligence_ (1870). This set out a conception of the mind which attempted, rather tentatively, to combine the philosophy of mind with neurology. In it the idea of the stable ego was dispensed with. Insanity was nearer the surface of the human mind than optimists would think; Taine's psychology had a distinctly pathological turn.
Taine explicitly regarded his work on culture and national character, and later on French history, as applied psychology, so it is important to grasp the outlines of his distinctive view. Taine regarded the Revolution as marking the onset of a disease from which France was still suffering; in a letter, he once compared it to the long-term effects of syphilis. With his psychological theory established, with the Third Republic inaugurated, and impelled by an appalled response to the Commune, Taine set out in _The Origins of Contemporary France,_ and in particular in the volumes on the Revolution, to trace its pathology.
Taine's psychological theory was a modification of that tradition in the philosophy of mind which is sometimes spoken of as empiricist, or, perhaps more helpfully, as sensationalist. Our knowledge of the world originates in sensations, which the mind combines as images. These remain in the mind after the original sensory input which caused them has ceased. They are therefore in a sense illusions, and in a state of sanity are known to be so. Taine calls them "true hallucinations." But since the mind harbours only its own images, the line between those which continue to convey useful information, and are confirmed by current sensations, and those which are simply, as it were, free-floating is a disturbingly indistinct one. Images jostle for attention in the mind—Taine uses an explicitly Darwinian analogy. Sometimes, prompted by some trigger of memory or emotion, particularly in states of reverie or high excitement, those which are no longer confirmed by sensation as real are activated and take over.
The surrender to unreality is of two opposed kinds (this line of thought was developed more fully by Taine's friend and follower Théodule Ribot). On the one side lies a jumble of images, none of them fixed or connected; this is mental confusion, advancing to instability and eventually delirium. On the other side lies the possibility that one, perhaps quite inappropriate, image becomes fixed, supplants all the others, and becomes incorrigible. This is the _idée fixe,_ and the state it produces is obsession or monomania. (There is an obvious analogy with the contrast between anarchy and despotism.) The mental poise which can correct and control the hallucinatory images and walk the tightrope between delirium and obsession is precarious; loss of grip on reality—insanity—is always ready to pounce. Taine regarded the French Revolution as a collective insanity; the distinction between the two types largely corresponds to the crowd and the leaders. He said privately that since 1789 France had been either infantile or mad.
Taine's account of the Revolution may have been decisively shaped by theory—or, if one prefers, prejudice—but it was also the product of extensive research, and is heavily documented, if not always critically. France in the 1790s is in the grip of an _idée fixe_ : the idea of the sovereignty of the people, expressed in Rousseau's _Social Contract_ and in the Declaration of the Rights of Man ( _FR_ I.IV.iii, VI.I.i). This idea is fanatically promulgated by some, who employ it to manipulate and coerce others, who have to defer to it on pain of exile or even death. The revolutionary leaders are intoxicated by their idea; the revolutionary mob, which has no critical powers, is similarly intoxicated by the contagion of mutual excitement which its numbers generate, and driven by need, fear and hatred (I.IV.v). (Taine's work was the forerunner of later studies in crowd psychology and behaviour, particularly of the classic _The Crowd_ (1895) by Gustave Le Bon, in which Taine's ideas are applied and amplified.) Elements of the mob, Taine insists, are also bribed.
The ostensible political sovereign, the Assembly and later the National Convention, is in fact at the mercy of the mob and of the political clubs like the Jacobins from which the revolutionary leaders promulgate their demands. The Assembly itself is virtually a mob (Le Bon says this too), in a constant state of hubbub and rowdy confusion, easily distracted, conducting its debates by means of slogans, which seek the applause of the galleries of spectators. These, since they represent the sovereign people, are incorrigible. They add to the din, intimidate those who dare utter unpopular opinions, and are in fact participants rather than onlookers (II.I.i). In the circumstances it is not surprising that the Assembly is given to sudden fits of ill-considered enthusiasm, resulting in hasty and confused legislation. Over-excitement becomes a kind of drug; the Assembly is not a conference for business but "a patriotic opera" (II.I.i). France's men of experience, the _intendants_ (governors) of the provinces, the members of the local _parlements,_ the ecclesiastical rulers of great dioceses, are in the main excluded (III.II.iii). The hubris begotten on inexperience by a-priori ideas is unchecked. Suspicion also thrives, and denunciation is encouraged. (Taine always speaks as though all supposed counter-revolutionary plots were fantasies.) The "bad counsellors" of the Assembly are fear and theory. Abstract ideas and conceit among the leaders, the urge to tumult and bloodshed among the people, feed off each other. Government makes way for "an intermittent despotism, for factions blindly impelled by enthusiasm, credulity, misery and fear" (I.II.vi). Henceforth, beyond the King, beyond the Assembly,
appears the real monarch, the people—that is to say the mob of a hundred, a thousand, a hundred thousand beings gathered together haphazard, on an impulse, on an alarm, suddenly and irresistibly made legislators, judges and executioners...who, with its mother, howling and misshapen Liberty, sits at the threshold of the Revolution like Milton's two spectres at the gates of hell. (I.II.viii)
Michelet, as we have seen, had a strong sympathy with what we may call the folkish elements of the early Revolution: the dancing, singing, street theatre and carnival. Taine, predictably, regards these as ominous as well as orgiastic and pathological. The oath of the Federation in Paris in July 1790 and the outbursts of federative enthusiasm all over France which preceded it are regarded by Taine as mass delusions: "Never was such an effort made to intoxicate the senses and strain the nerves beyond their powers of endurance...The difference between magniloquence and sincerity, between the false and the true, between show and substance is no longer distinguishable." A whole nation is losing its grip on reality in a kind of delirium which is taken for fraternity. But there is also manipulation, even if it does not recognize itself for what it is. Children, as young as nine years old, declaim patriotic orations: "it occurs to no one that they are puppets," with words put into their mouths. But people remain as they were: they avoid paying their debts, they lay their hands on public property if they get the chance—"everywhere there is philanthropy in words and symmetry in the laws; everywhere there is violence in acts and disorder in all things" (III.I.i). Taine does not draw the parallel, but the reader of Thucydides can hardly fail to be reminded of the anarchy and political fanaticism in Corcyra, where words lost or reversed their meanings, just as, elsewhere, Taine's account of the Terror reminds us of Tacitus' description of the omnipresent, suspicious eye of the despot and of the miseries of the Roman proscriptions (Chapter 5, Chapter 7). Where Michelet saw in 1790 a nation in the process of formation, Taine sees a society in a state of disintegration.
Taine does not often narrate events. Rather, he takes soundings of the state of French society, of the agents and the sufferers, across the social and institutional spectrum: in the Assembly and the Convention; in the psychology of the leaders, and in the political clubs which provide their power bases; in the Parisian mob and in the Paris Commune and sections; in the revolutionary tribunals and their victims; and in the provinces—including the representatives, the feared _representants en mission,_ sent from Paris with despotic powers to enforce the government's will and often their own. Rather than narration—though there are many interpolated anecdotes—Taine offers a steady accumulation of evidence, from documents, from observers, from the quoted remarks of leading political agents, and from memoirs. He is not writing a narrative but compiling an indictment, which he does with as much weight and skill as vehemence.
The time and energy required by "active citizenship" tends to bring forward, according to Taine, the worst elements in the population—those with plenty of both to spare and the impulse to agitate and dominate. They control the politics of the small-scale units, the sections: "Politics became a profession." The capital was more feverish than the provinces, the towns than the villages; Taine uses the simile of an abscess. The incoherence of the legislation produced by the Assembly, with no possibility of judicial review, plays into the hands of local leaders, who interpret it, and implement it or not, as they please (II.III.iv). The political club becomes "the champion, judge, interpreter and administrator of the rights of man" (II.III.v).
Book IV gives us Taine's analysis of the psychology and tactics of the Jacobins, which is essentially an elaboration of what he has set out already as the revolutionary mentality. He sometimes draws the analogy with the Puritans. The revolutionary is characteristically an antinomian, convinced of his righteousness, out of touch with reality. He sees himself as the authorized executor of the common will. "Marching along in the procession formed for him by this imaginary crowd, sustained by millions of metaphysical wills created by himself in his own image, he has their unanimous assent, and, like a chorus of triumphant shouts, he will fill the outer world with the inward echo of his own voice." He is therefore a pathological case: "Something which is not himself, a monstrous parasite, a foreign and disproportionate conception, lives within him" (IV.I.iii). The link to the argument of _On Intelligence_ is particularly apparent here.
In the next three books, Taine goes on to consider at length how the Jacobins' power is established and exercised. Book VIII is devoted to "the Governed": to the nobles, clergy, bourgeoisie and populace, and how they respectively fared during the Revolution. His history ends with the advent of Bonaparte. The Jacobin dictatorship could not last because it lacked the essential characteristic of a political society, mutual respect, particularly between governors and governed, and therefore was unable to establish mutual trust and confidence. In French civilian society by 1797 "there is not one among the three thousand legislators who have sat in the sovereign assemblies that can count on the deference and loyalty of a hundred Frenchmen." In the army, however—"military France"—it is otherwise (IX.I.x). Taine has largely ignored the wars on the frontiers: he speaks always of the revolutionary leadership as autonomous, driven on, deterministically, by its own vanity and obsession, ignoring the pressures to which the leaders were subjected. His indictment is immensely powerful, but it has the weaknesses of a deterministic demonstration.
Now, however, the frontiers, and the army which has seized and expanded them, become relevant. It is not just a matter of discipline. In the army have grown up mutual dependence, respect and sympathy. Hence the army is a society, and with its consent its commander can wield power, while "civil France" will welcome him as its liberator and restorer. The outcome is a despotism. The Revolution had left itself no alternative to the omnipotence of the state—towards which it had tended, though incoherently, from the outset. What was left from revolutionary chaos was the omnipresence of government as a result of "the absence of local and private initiative, the suspension of voluntary free associations, the gradual dispersion of small, spontaneous groupings, the preventative interdiction of prolonged hereditary works, the extinction of sentiments by which the individual lives beyond himself in the past or in the future" (IX.I.x). From this passage alone one could tell that Taine was an admirer of Burke and of Tocqueville. His metaphor for the state of France is a barracks: clean, well-built, symmetrical and "better adapted to the discipline of the average and lower elements of human nature...In this philosophical barracks we have lived for eighty years."
Taine's work was predictably subjected to heavy criticism for its apparent mono-causal determinism, its simplifications, its indulgence towards dubious sources which suited his case. One of the most prolonged attacks was the book entitled _Taine, Historian of the French Revolution,_ published in 1907 by Alphonse Aulard, who in 1886 had taken up the chair of the history of the French Revolution newly founded at the Sorbonne by the municipal council of Paris. Aulard, in announcing his credentials, identified himself as "a respectful and grateful son of the Revolution which has emancipated humanity and science." To understand the Revolution, he declared, one must love it. His attitudes were not far from Michelet's, though his manner was much more sober—"dull" is an adjective it has incurred. Like Michelet he regarded the people as the hero of the Revolution, though he insisted that the conduct of the revolutionary leaders should be judged in the light of their circumstances and the reasonable fears these engendered. He also seemed to go some way to justifying the Terror by arguing that it was necessary for survival and to preserve the gains of the Revolution; this was an argument which others, later, would apply to Stalin.
This is hindsight, but appreciation of Taine is likely to be enhanced by an awareness of the grimmer features of twentieth-century history. Admittedly Taine exaggerates. His work is far from, and does not aspire to, what was becoming an ideal of disinterested history (but so is that of all the other historians of the Revolution in the period). However, when one has allowed for exaggeration, for Taine's monocular vision and his own obsessions with Rousseau and the effects of the _esprit classique,_ his indictment remains formidable, as well as a rhetorical tour de force, a powerful psycho-drama. His easy acceptance of some dubious sources does not invalidate others. In the 1870s he described the emergence of the characteristics which, in the twentieth century, came to be spoken of as "totalitarian."
**TWENTY-THREE**
**History as the Story of Freedom: Constitutional Liberty and Individual Autonomy**
**Stubbs's _Constitutional History_ : From Township to Parliament**
"We have no thread through the enormous intricacy and complexity of modern politics, except the idea of progress towards more perfect and assured freedom," Lord Acton, newly appointed as Regius professor of history, told his Cambridge-undergraduate lecture audience in 1895. He could have added "history" to "politics" and many nineteenth-century historians in various European countries would have agreed with him. François Guizot (1787–1874), for example, France's greatest nineteenth-century constitutional historian, had taken this as the theme of his lectures at the Sorbonne in the 1820s (published as _The History of Civilization in Europe,_ English translation 1846). Other civilizations, he argued, had been theocratic, despotic, democratic or regulated by caste, but in Europe no one principle had ever dominated all the others. Europe's dynamism derived from liberty and the diversity which had preserved it—a consequence of the multiplicity of influences which had shaped European civilization: Roman, Christian and barbarian.
The importance of Rome and the heritage of civic republicanism which in the ancient world had stood for liberty was a debatable issue. It was common, above all in Britain, to draw a line under the ancient city state, as essentially archaic and hence, in the modern world, an inappropriate model, whose dangers had been revealed by the enthusiasm felt for it at the time of the French Revolution. It was from the German invaders of the Roman empire, bringing with them the Tacitean freedom of the German woods, that modern European liberty essentially descended (Chapter 19). Guizot, in insisting on recognition of the Romanized character of Gaul, especially in the south, from which he himself came, was more moderate. He stressed in his _History of Civilization in France_ (1829–32) that the municipal institutions of Roman Gaul had survived into the early Middle Ages, coalescing with the government of the urban Christian communities under their bishops, who were magistrates as well as pastors. But Guizot was obliged to admit that no direct continuity could be established between them and the city communes of the later Middle Ages; a French bourgeoisie was the product of the recovery of commercial life and the growth of the professions.
In any case, the semi-independence of the late-medieval towns had been crushed by the centralizing absolutism of Richelieu and Louis XIV. Much the same had occurred among the late-medieval city states of northern Italy, which, as Guicciardini had described, had succumbed, apart from Venice, to their own internal divisions and the military power of France and the Empire. The classic historical account of this for the nineteenth century was provided by the Swiss historian Charles de Sismondi, whose _History of the Italian Republics_ began to appear in 1807. Sismondi, a liberal, described the life of the free Italian communes with enthusiasm, but his story was one of failure, which he explained, in traditional humanist and indeed Roman republican fashion, chiefly by failures of character and public virtue. It seemed that the autonomous city republic, as a political form, had proved a blind alley.
But England, at least, seemed to prove that the liberty of the German woods had proved enduring, and that its future lay with a liberal, parliamentary constitutionalism and limited monarchy. For the French, however, there was a difficulty. Not only was monarchy in France a deeply divisive issue since the first French Revolution, but just as, above all for the British, the republican model had been discredited by that Revolution, so, in France, the notion of the Teutonic inheritance was tainted by its association with aristocratic domination. French aristocratic writers, from the late seventeenth century, had claimed descent for the French nobility from the Teutonic conquerors of Gaul, the Franks. The point was to rebut the claims of monarchical absolutism: it was an axiom, from Tacitus, that German kings were elected and their powers limited; we have seen this argument in Hotman in the sixteenth century (Chapter 19). It also followed, however, that the common people of France—the so-called Third Estate (i.e. neither noble nor clerical)—being the descendants of the Gauls, were a conquered people. There was, understandably, something like a plebeian reaction, most influentially, at the time of the Revolution, in the Abbé Sièyes's pamphlet _What is the Third Estate?_ (1789). If the nobility were essentially German invaders, then they were aliens whose titles derived from an act of violent usurpation. It was time for the descendants of the Gauls to recover their sovereign rights as a nation, of which the nobility formed no part.
No such divisiveness clouded the English understanding of the Teutonic invasion and settlement of post-Roman Britain as essentially free and even democratic. In the works of John Mitchell Kemble ( _The Saxons in England,_ 1849), Edward Freeman ( _The History of the Norman Conquest,_ 1867–79) and William Stubbs ( _The Constitutional History of England in its Origin and Development,_ 1873–8), England's Teutonic settlement was celebrated as the origin of the national tradition of liberty. It was above all Stubbs who stamped his authority for several generations on the constitutional history of medieval England and gave it a central place in the newly established history faculties in the universities, which from the 1860s onward permitted students to take examinations and to graduate in history (Chapter 25). For Stubbs, though not for Freeman, the heritage of Rome stood not for civilization but for tyranny, and it was gratifying to him that in England there was so little evidence of it. England's Teutonic invaders were settlers and pioneers. They had not, as in Gaul, established themselves as a landed aristocracy—as later did the Normans—served by an underclass of native Romano-British; these had been, it was confidently asserted, exterminated or driven to the margins in the Welsh mountains, not interbred with or enslaved.
For a while, on this basis, there were close relations between English and German historians. Kemble had been a pupil of the great German folklorist, linguist and legal antiquarian Jacob Grimm. As the French legal historians had done in the sixteenth century, from the late eighteenth century onward German scholars led Europe in the scholarly exploration of the legal customary roots of ancient Greek, Roman and Teutonic societies, notably with the work of Barthold Niebuhr on early Rome (Chapter 6), of Karl von Savigny, who was also a Roman legal historian, and of Grimm, whose work was in the field of early German language, myth and customs. German scholarship in this period was inspired and infused by a populist conception of the uniqueness and creativity of each primordial _Volk,_ expounded above all by Johann Gottfried Herder (1744–1803); the possible connection between his ideas and those of Vico (Chapter 22) is unclear. The spirit of the _Volk_ manifested itself in all aspects of life: language, myth and customary law and institutions. It was an understandable enthusiasm among a people increasingly conscious of national identity but without a nation state to stand as the protagonist of its history.
Most relevant of those investigations here is the attempted reconstruction by German scholars of early Teutonic customs, and particularly those of the supposed primitive nucleus of ancient Teutonic society, the village or _Mark_ community. English scholars followed them in this from the mid-century, though in Stubbs's case with reservations—not enough, according to his later admirers and critics. There were distinguished German experts on English constitutional history, to whom Stubbs paid tribute. Each country seemed to have something to offer the other. It was in Germany, particularly in Schleswig, that some customary evidence survived that might give a clue to the institutional ideas and practices carried with them by the Saxons. But in England the Teutonic traditions could, it seemed, show what Germany could not: an unbroken lineage from Germanic institutions to a fully developed national parliamentary constitution.
Stubbs (1825–1901) began his scholarly career as an editor of medieval manuscripts. Reaching behind these to make out the practices and conceptions they referred to and embodied became the bedrock of his work and his reputation, leading to his election in 1866 to the Oxford chair of medieval history. Through them one could register the tremors of slow-moving social and institutional changes which characteristically no legislation had brought into being but which were registered, for those who could detect them, in technical terms and procedural forms, in administrative, fiscal and judicial devices, in the growth and waning of privileges and exactions, in the augmentation and dwindling of functions. In Anglo-Saxon England, Stubbs wrote, "there are no constitutional revolutions, no violent reversals of legislation: custom is far more potent than law, and custom is modified infinitesimally every day. An alteration of law is often a mere registration of custom, when men have recognised its altered character. The names of offices and assemblies are permanent, while their character has imperceptibly undergone essential change" ( _CH_ , Chapter 4). A simple example of such changes is the mutation of the functions of royal officials such as the steward and the chamberlain: originally servants in the king's household, they become first great officers of state and then dwindle into mere honorific court titles.
Ever since the work of the legal historians of the sixteenth and seventeenth centuries, the possibility had existed of a kind of history that was impersonal, undramatic and technical, concerned with long-term changes rather than deliberately implemented policies. So far it had been chiefly exhibited in antiquarian monographs devoted to particular, often narrowly specific, institutions. The Enlightenment concept of "unintended consequences" had fostered that possibility from another angle: of tracing large-scale, unplanned social changes, which Gibbon called "insensible revolutions," but generally in the form of essays rather than monographs, as in Adam Smith's brilliant few pages in Book III of _The Wealth of Nations_ on the rise and decline of feudalism. Now the new constitutional history, in Stubbs's hands, partly prompted by the German idea of culture and custom as the productions of the anonymous _Volk,_ brought the two approaches together. Stubbs's _History_ was minutely investigative and scholarly but large-scale, with the notion (again stemming from the sixteenth century) of an inheritance of Teutonic liberty as its guiding theme. Stubbs brought the antiquarian, investigative tradition to a new level of dense scholarly rigour, and synthesized its results in a three-volume survey, ending in the fifteenth century, of English medieval history as the story of the preservation and growth of constitutional liberty, beginning in the smallest units of local self-government and culminating in a national parliament.
Stubbs was both self-conscious and confident about the kind of history which could do this. He knew it involved sacrificing the dramatic episodes and clash of personalities, the celebration of heroic deeds—in short the sympathetic identification with the figures of the past and the inspiration of noble characters—which probably drew most readers to history. He said so in his preface:
The History of Institutions affords little of the romantic interest, or of the picturesque groupings which constitute the charm of History in general, and holds out small temptation to the mind that requires to be tempted to the study of Truth. But it has a deep value and an abiding interest to those who have the courage to work upon it. It presents, in every branch, a regularly developed series of causes and consequences, and abounds in examples of that continuity of life, the realisation of which is necessary to give the reader a personal hold on the past and a right judgement of the present. For the roots of the present lie deep in the past, and nothing in the past is dead to the man who would learn how the present comes to be what it is...Constitutional History has a point of view, an insight, and a language of its own; it reads the exploits and characters of men by a different light from that shed by the false glare of arms, and interprets positions and facts in words that are voiceless to those who have only listened to the trumpet of fame.
Though kings and barons play their parts, the principal agents of Stubbs's history are multiple, obscure and even anonymous. They build up the fabric of English institutions over time, by countless imperceptible actions, as Stubbs's contemporary Charles Darwin saw coral reefs created by "myriads of tiny architects." The protagonist of Stubbs's _Constitutional History_ is the English people itself, obscurely operating and slowly transforming, in small spontaneous innovations and adaptations, the institutions by which it lived. Reading Stubbs often seems like witnessing the scholarly illustration and historical elaboration of the thought of Edmund Burke: such slow, accumulative growth and constant adaptation was a sign of life. Continuity in the history of English institutions was not merely the guardian of English liberty: it _was_ English liberty, in constant, spontaneous action. And, because it was spontaneous, anomalies abounded; method and substance were ultimately the same: "Complexity is a sign of growth; simplicity of detail signifies historically the extinction of earlier framework. That which springs up, as our whole system has done, on the principle of adapting present means to present ends, may be complex and inconvenient and empiric, but it is natural, spontaneous, and a crucial test of substantial freedom" ( _Lectures on Early English History,_ Chapter 21).
In Stubbs's account of English medieval history the old perceived tension and dialectic between civilization and progress and the survival of liberty was still alive. The threat to liberty was the necessary framework of law and order imposed by a powerful monarchy. "In general," Stubbs wrote in his _Lectures on Early English History_ (published in 1906) "the chances are greatly in favour of tyranny, resulting from the destruction of the old bases; England alone has a history in which ancient freedom made its way through, and utilized all that is good in feudalism, widening from precedent to precedent into perfect political liberty" (Chapter 18). The greatest threat to inherited English liberties had been the Norman Conquest. The survival of the basic Teutonic institutions on to which strong monarchy and feudal conceptions had been imposed by the Conquest had been a close-run thing. They had survived, according to Stubbs, in the smallest and most obscure forms of local self-government, the village moot and the jury:
In the preservation of the old forms,—the compurgation [exoneration by sworn oath] of the accused, the responsibility for the wergild [compensation], the representation of the township in the court of the hundred, and that of the hundred in the court of the shire; the choice of witnesses; the delegation to chosen committees of the common judicial rights of the suitors of the folk-moot; the need of witnesses for the transfer of chattels, and the evidence of the hundred and shire as to criminals and the duty of enforcing their production and punishment, and the countless diversity of customs in which the several communities went to work to fulfil the general injunction of the law—in those remained the seeds of future liberties...They were the humble discipline by which a downtrodden people were schooled to act together in small things until the time came when they could act together for greater ones. ( _CH_ , Chapter 5)
It was in these minuscule ways that the original Teutonic liberties had survived. Stubbs did not accept the full _Mark_ theory, as it was called, of an original village co-proprietorship of land. Some late-nineteenth-century polemicists saw in the co-proprietorship theory a justification for modern socialism and a demonstration that private property represented a usurpation of communal rights. Stubbs, a Tory in politics and eventually a bishop (of Oxford), was not one of them. For him it was on the free holding of land, known as allodialism, that the original Teutonic liberty had rested. Celtic peoples might have held land communally, as a group of kin—there was evidence to that effect—but the Teutonic evidence pointed to the individual freeholder, a member of the self-regulating village community in which each freeman had a voice: "In the allodial system is the germ of all the institutions of freedom" ( _EEH_ , Chapter 15). The history of English freedom, however, partly on account of the Norman Conquest, was not a simple, triumphant advance but a kind of pilgrim's progress, beset by many threats and vicissitudes until the trumpets could sound for the transformation into national parliamentary sovereignty. The lineage which linked the Teutonic freeholder to the modern voter and juryman was allegedly unbroken but was admitted to have been at times almost subterranean, invisible except to the eye of minute scholarship.
The English nineteenth-century vaunting of that heritage could seem more than a little smug. Acton—cosmopolitan and a Catholic—saw in it a regrettable parochialism and neglect of universal principle. Even the best institutions were valuable only as they provided security for a higher interest: the liberty of the individual and the use he made of it. For Acton there was a more important kind of history: intellectual, moral and religious. As he told the audience for his inaugural lecture as Regius professor of history in Cambridge in 1895, "A speech of Antigone, a simple sentence of Socrates, a few lines that were inscribed on an Indian rock before the Second Punic War, the footsteps of a silent yet prophetic people who dwelt by the Dead Sea and perished in the fall of Jerusalem, come nearer to our lives than the ancestral wisdom of barbarians who fed their swine on the Hercynian acorns." The lofty dismissal of barbarian swineherds is accentuated by the use of the classical name for the German forest. A general "History of Freedom" was the cherished project that was to have been the climax of Acton's scholarly life, but it was never written, and he left only fragmentary clues to its nature. It is clear that freedom as a moral principle, and its ultimate recognition as such, would have been the core of the story. By contrast, for Acton the English tradition was over-concerned with property rights and, latterly, with race. The Revolution of 1688 had been a half-hearted muddle, prompted by self-interest. It was only in the demand of the Puritan Independents in the 1650s that the English tradition reached the level of general principle, and only in America that it became fully self-conscious and came to understand, as it were, its higher self, in the formulation of the concept of universal human rights.
Guizot had distinguished two kinds of history of freedom, the institutional and the intellectual: "I do not propose to study with you the history of the interior of the human soul; it is the history of...the visible and social world that I shall occupy myself with." Acton's Cambridge lectures made similar gestures. As tradition and the syllabus suggested, they followed a predictable course of mainly political history: Philip II, the Thirty Years War, Louis XIV, before culminating in the American Revolution. (There was another series devoted wholly to the French Revolution.) But Acton gave plenty of indications that the interior of the soul was his true if postponed business: religion was for him virtually the same as the history of freedom, since both were concerned ultimately with conscience. He also gestured more than once to "the movement of ideas, which are not the effect but the cause of public events." The "History of Freedom," we can conjecture, would have been a wide-ranging history of ideas, with the development of free institutions as secondary to the development of full, self-conscious awareness of freedom and its moral responsibilities as the self-realization of humanity. But, possibly because of its fragmentary indications, Acton's kind of history of ideas seems alarmingly spasmodic—a series of under-explained epiphanies. The spirit seemed to blow where it listed among a historically motley collection of Catholic theologians, Puritan zealots and American squires. Acton was not shy of naming the day and the hour when revelation descended. Thus it was in America, as he said in a later Cambridge lecture, that the law of nature, which he sometimes called "the higher law," became incarnate: "On that evening of 16th December 1773 it became for the first time the reigning force in history."
It was the religious struggles of the Reformation which had—unwillingly, since the leading Reformers were not tolerant men—begotten religious liberty and the idea of freedom of conscience. But Acton's musings on the history of human self-consciousness also found a place for the Renaissance, though it had played no part in the inculcation of respect for political and human rights. In Acton's characterization of the Renaissance in his Cambridge lectures there seem to be traces of the outstanding nineteenth-century treatment of it, and indeed of the culture of any epoch, the Swiss Jacob Burckhardt's _The Civilization of the Renaissance in Italy_ (1860). Acton spoke of this, in a review of a book on the Borgias, as "the most penetrating and subtle treatise on the history of civilization that exists in literature." In his lectures he referred to "the finished individual of the Renaissance, ready for emergencies, equal to every fortune, relying on nothing inherited, but on his own resource...little recking rights of others, little caring for the sanctity of life." This is a portrait recognizable to any reader of Burckhardt, and it forms another, in some respects lurid, chapter in the nineteenth century's preoccupation with the history of freedom.
**Modernity's First-born Son: Burckhardt's Renaissance Man**
Burckhardt's book forms part of, yet also stands at an angle to, the nineteenth-century conception of the history of freedom. There is enough emotional complicity in Burckhardt's work to prevent us from regarding it as an ironic critique of that conception, but it takes it into wild and alarming moral territory. It would be wrong to think of contemporary ideas of modernity as simply congratulatory: many were highly critical, focusing, as the century wore on, on philistinism and conformity, against which Burckhardt reacted. There were other kinds of criticism dwelling on atomization and selfishness. Burckhardt's work broke new ground, however, in its way of presenting the origins of modernity and the newly emancipated European in Renaissance Italy, who was seen as fiercely amoral, but fascinating and capable of greatness. Out of the Machiavellian concept of _virtù_ (energy and prowess) and traits from European Romanticism, Burckhardt compounded an enduring archetype—as his younger colleague at the University of Basle, Friedrich Nietzsche, with help from the idea of the Dionysiac, was to create a new image of the ancient Greek. Both authors are exhilarating to read. Burckhardt's man of the Renaissance is the liberal hero without his surrounding decorum of principles: "the spirit of freedom" becomes antinomian, self-justifying and freed from all restraint.
Burckhardt's work is of enduring interest both in its method, which is highly original, and in the coherence and vividness of its interpretation, which, though recognizable as at best an exaggeration, lodges abidingly in the mind. Burckhardt spoke of "a" civilization, not just of civilization as such, though one which was a moment in a longer development, a civilization "which is the mother of our own." In Burckhardt, narrative has virtually disappeared, though illustrative anecdote abounds and is, in fact, a major part of the method: it is as though we are given the stories told by Gregory of Tours in a clearly envisaged, tenaciously realized conception of contemporary morals and mores. One precedent (though Burckhardt's task was wider and his evidence less tangible and therefore more difficult to handle) was the attempts made, since the art historian Johann Winckelmann's studies of the Greeks a century earlier, to regard the character of an artistic or architectural style as the product of a whole people, and to discern from this their inner character and aspirations. Ruskin, for example, immediately before Burckhardt, in _The Seven Lamps of Architecture_ (1849) and _The Stones of Venice_ (1851–3), had obsessively moralized (and in the case of the Renaissance immoralized) the formal characteristics of architecture. Art and life were inseparable, and art revealed the soul of a society, in its elevation or baseness. Burckhardt himself was highly visually sensitive: he lectured on art history at Basle, wrote a guide to Italian art (the _Cicerone_ ), and left art out of the _Renaissance_ only because he intended to treat it separately elsewhere.
In Germany the idea of art as the expression of spirit is associated particularly with Hegel, as it is later in France with Taine. Burckhardt shied away from the teleological meaning Hegel gave to his conception of the World Spirit whose moments form the moral and intellectual history of mankind, but there seems no doubt it helped him to see a unity in the particular moment of civilization represented by the Italian Renaissance. Rather than just categorizing this moment, he set out copiously to illustrate and in the process to explain it. The _Renaissance in Italy_ lives above all, apart from its anecdotes, in its adjectives. The origins of art history had lain in connoisseurship. Burckhardt was a connoisseur of the manifestations of the human personality, historically and geographically located. This, too, was a German preoccupation: since the time of Goethe and Schiller the aesthetics of personality was a matter of acute moral and cultural interest, and the categories borrowed from the appraisal of art were a major aspect of it.
The concept of the Renaissance as a period had been significantly elaborated since its self-consciousness about its achievements had expressed itself in the claim to be a rebirth of art and letters. This had sometimes explicitly consigned the preceding thousand years to the status of a barbarian interval or hiatus, a "middle" age. The late-seventeenth-century German scholar Christoph Keller (Cellarius) has been credited with being the first systematically to employ the familiar tripartite periodic division, Ancient, Medieval, Modern. But the concept of modernity itself increasingly needed differentiation, as the age of the humanists receded into the past. We have seen in Guicciardini a registration of the achievements—printing, the discovery of the New World, and modern artillery—which surpassed those of the ancients, and therefore made the notion of revival inadequate. In Hume and Robertson we have seen how the age of the Reformation too, uncouth and fanatical, was distanced from the current enlightened and polite one. But established periodic nomenclature was still somewhat meagre. Earlier, Voltaire in his _The Age of Louis XIV_ (1751) had enlarged his concept of the age beyond that of the reign, by including cultural as well as political achievements and by extending the scope of the age both laterally and chronologically, so that the cultural achievements became those of seventeenth-century Europe: Galileo, Bacon, Locke and Newton as well as Descartes and the great French literary figures of the _Grand Siècle._ But his method here remained essentially that of a catalogue or inventory of achievements. Arguably the seventeenth century still lacks a satisfactory period label; Carl J. Friedrich, in the middle of the twentieth century, understandably borrowed from art history and chose _The Age of the Baroque_ (1952) for his general study of European history and culture. It was, however, in the 1830s that the term "the Renaissance" was first employed in the modern comprehensive and periodic sense—by Michelet, when he gave it to a volume of his _History of France._ "Early Modern" is a twentieth-century coinage.
Acton was right to recognize Burckhardt's subtlety: his is a book which insinuates its themes and the thinking behind them. At first one can feel overwhelmed by the sequence of anecdotes, from chronicles, biographies and memoirs. But, as one reads on, the outlines come into focus and a coherent picture, explanatory but never merely reductionist, emerges.
The first part is entitled "The State as a Work of Art," a concept which obviously owes something to Machiavelli, though Burckhardt has plenty of other evidence. The key figure, it transpires, is that of the _condottiere._ Burckhardt, while recognizing the cruelty and treachery that were essential elements in the mercenary captains' trade, and their ruthless ambition to found dynasties of their own, treats them more respectfully than does Machiavelli, to whom they are sometimes risible. In Burckhardt the _condottiere_ typically displays a Machiavellian _virtù,_ but divorced from civic patriotism: boldness and resolution coexist with cool calculation and a ruthless determination to let no person, principle or loyalty stand in the way of the path to greatness. The _condottiere_ is a military entrepreneur and political adventurer, acting entirely for himself and relying only on his own skill, adroitness and foresight for his survival and success. The network of class, chivalric code (including loyalty) and social hierarchy based on birth, which surrounds the warrior nobles of northern Europe, means nothing to him. He is, literally, the freest of freelances, of no state, with no fixed social position, pitting his wits, courage and determination against fortune for high stakes: wealth, a dukedom, a principality even. Even when he becomes a ruler his power remains personal not dynastic, and essentially without legitimacy; only awe, splendour, popularity or fear can sustain it, and it is always insecure.
Burckhardt makes this role, and the type of personality which embraces and can sustain it, into the conceptual core of his book, because the other skills and roles he sees as characteristic of the Italian Renaissance are in a sense civil versions of the military adventurer, and those who possess and enact them are the bearers of Renaissance civilization: artists and architects, humanist scholars and men of letters. They too are the entrepreneurs of their own talents and personalities. Footloose, they typically move from patron to patron, in search of advancement or as the result of quarrels. Like Michelangelo, they sometimes proudly assert their own worth even to the most highly placed, like the Pope. Like the _condottieri_ and petty rulers, they benefit from the multiplicity of autonomous states that the unresolved conflict for domination of Italy between the papacy and the empire has allowed to grow up; the frontier is always helpfully near, beyond which another patron, with luck, will be accommodating. They too, in Burckhardt's picture, are typically without the supports and the constraints of traditional institutions—of guilds or universities—or are only very loosely attached to them. Their habitat is courts and cities, where the golden prizes are won—and as suddenly lost. They and the rulers are linked by mutual need: they for rewards and status, the despots for the legitimation conferred by the talents and skills of artists and humanists. The parvenu rulers needed to be surrounded by splendour; to be impressively represented at foreign courts and on public occasions, like the reception of ambassadors by elegant Latinists; to be celebrated, eulogized and elegized, and lauded as munificent patrons.
The hazards encountered by a soldier of fortune are self-evident, but Burckhardt takes pains to stress that they applied in a remarkable degree to what might otherwise seem to be in all probability the quiet lives of scholars and craftsmen:
For an ambitious youth, the fame and the brilliant position of the humanists were a perilous temptation...He was thus led to plunge into a life of excitement and vicissitude, in which exhausting studies, tutorships, secretaryships, professorships, offices in princely households, mortal enmities and perils, luxury and beggary, boundless admiration and boundless contempt, followed confusedly one upon the other. (Part III)
The humanist characteristically has no fixed home; Burckhardt draws a parallel with the Greek Sophists, "but the scholar of the Renaissance was forced to combine great learning with the power of resisting the influence of ever-changing pursuits and situations." An inordinate pride was a necessity for survival: the humanists are "the most striking examples and victims of an unbridled subjectivity." Burckhardt uses an early-sixteenth-century work, Piero Valeriano's _On the Infelicity of the Scholar._ The lives Valeriano describes are indeed lurid. We are introduced, Burckhardt says, to men who
in times of trouble lose, first their incomes, and then their places...to unsociable misers, who carry about their money sewn into their clothes, and die mad when they are robbed of it; to others, who accept well-paid offices, and then sicken with a melancholy longing for their lost freedom. We read how some died young of a plague or fever, and how the writings which had cost them so much toil were burnt with their bed and clothes; how others lived in terror of the murderous threats of their colleagues; how one was slain by a covetous servant, and another caught by highwaymen on a journey, and left to pine in a dungeon...(Part III)
Valeriano obviously had a ready ear for academic grumbles, but the picture, as a picture, has an extraordinary power. This was freedom, but it was a picaresque freedom of extreme hazard and an unrelenting demand for self-reliance. It was also compatible with extraordinary achievement, and even a spur to it.
Burckhardt makes much of the erosion of class barriers in the Italian towns, of the free intercourse between nobles, scholars, artists and rulers. Birth allegedly counts for little, ability for almost everything. To speak in modern terms of a "career open to talents" seems almost a mockery, implying regular hierarchies to ascend. The world conjured up by Burckhardt is one in which one stakes one's will and talents on the turn of fortune's wheel. At times Burckhardt's man of the Renaissance seems a kind of victim of modernity: he is the first to experience, in a radically unrestrained and unprotected form, the spiritual ordeal of the modern condition, alone and with only his own resources to sustain him—"The Italian of the Renaissance had to bear the first mighty surging of a new age" (Part VI).
The first three parts of the work, dominated as they are by the _condottiere_ as their archetypal image, are the most exhilarating in their brio and excitement. Part II, "The Development of the Individual," treating of the intense self-awareness and the interest, in biography and portraiture, in the qualities of the individual personality, is central. Burckhardt's contrast of the medieval condition in this respect and that of the Renaissance has inevitably drawn criticism for exaggeration:
In the Middle Ages both sides of human consciousness—that which was turned within as that which was turned without—lay dreaming or half awake beneath a common veil. The veil was woven of faith, illusion, and childish prepossession, through which the world and history were seen clad in strange hues. Man was conscious of himself only as a member of a race, people, party, family, or corporation—only through some general category. In Italy this veil first melted into air; an _objective_ treatment and consideration of the state and of all the things of this world became possible. The _subjective_ side at the same time asserted itself with corresponding emphasis; man became a spiritual _individual,_ and recognized himself as such.
The third part, "The Revival of Antiquity," though it contains the colourful account of the humanists' lives quoted above, is naturally a little more predictable. Then "The Discovery of the World and of Man" picks up the theme of objectivity, which is presented as another characteristic of the Renaissance individual, forced into the need for cool calculation and into an understanding of the world as it is and not as it might or should be. But the clear-sighted contemplation of the external world is also a source of delight to him.
Though Burckhardt's book stresses the pagan amorality of the Renaissance and hardly recognizes the quality of Renaissance piety, the final part, "Morality and Religion," shows awareness of the difficulties in striking the moral balance sheet of a whole society. Presenting the Italian Renaissance as an example of refined sensuality and cruelty, addicted to the elegantly macabre, especially in the pursuit of revenge, was no novelty. It rested on the Borgias' reputation for poisoning people; on Machiavelli and the English Jacobean revenge tragedies, habitually set in Italy; on recollections of stories from Dante—all these fostered this reputation, which was added to by the eighteenth-century Gothic novel, again often with an Italian setting, and by the poetry of Byron and Browning. In Germany, the _Sturm und Drang_ movement in literature sometimes went to Italy for scenes of violence and horror, while the German love affair with Italy epitomized in Goethe's _Italian Journey_ dwelt on the country's pagan seductiveness. Burckhardt, coming after all this, still managed to create an indelible impression, and, though he sometimes emphasizes distinctly Italian traits, it is above all the Renaissance as modern, as _proto_ -modernity, that takes his attention. The Renaissance Italian was "the firstborn among the sons of modern Europe."
Acton qualifies his praise of Burckhardt's book by saying that "its merit lies in the originality with which the author uses common books, rather than in actively new investigations." This was a piece of characteristic Actonian condescension: the author would be none the worse for a more protracted spell in the archives. That is more than doubtful, but there was a truth here, though condescension is not called for. Despite the range of Burckhardt's references, and though he does not particularly signal the fact, it is possible to see in his general conception of the Renaissance personality and its circumstances, and therefore in his original treatment of it, a synthesis of a handful of notable works and characters which dominate the book: Vasari's _Lives of the Artists_ (skill, technique, pride); Benvenuto Cellini's _Autobiography_ (a picaresque, amoral life, led by a master artist, sustained by a self-confidence amounting to impudence); Aretino (the pen as a weapon of self-assertion, cruelty and revenge); Machiavelli, of course; Francesco Sforza, the most successful of all the _condottieri,_ who made himself duke of Milan; Petrarch, the pioneer of an intense self-awareness as well as a passion for antiquity; Castiglione's _The Courtier,_ most celebrated of pattern books for the "all-round man," in any case a preoccupation of recent German, neo-Hellenic moral and philosophical cultural concerns. To say this is not to detract from Burckhardt's originality, but rather to admire the way he wove these themes together into a plausible and arresting synthesis which, however much qualified, has remained a challenge and point of reference for historians of the Renaissance, enshrined in a still very readable book.
It is hard, in fact, to imagine that Burckhardt will ever altogether lack readers, at least so long as a Nietszchean version of the liberated hero continues to fascinate. In general the nineteenth-century histories of freedom have themselves passed into history. G. M. Trevelyan's Italian Risorgimento trilogy (1907–11), focused on Garibaldi, was perhaps the last substantial work of history to be conceived as a liberal-democratic as well as nationalist epic. Trevelyan himself, who was Macaulay's great-nephew and fought a long campaign against the new professionalism's scorn for picturesque narrative, spoke in his preface to _Garibaldi's Defence of the Roman Republic_ (1907) of the "chord of poetry and romance" in Garibaldi's life, but wondered "whether his memory will now appeal to the English of a generation...said to be at once more sophisticated and less idealistic than the Victorian." He was too pessimistic at the time, but surely right in the long run. In general, the new intellectual climate of the last years of the nineteenth century, gripped by fears of socialism supported by mass electorates, often prompted glum reassessments of democracy and dampened enthusiasm for tracing the origins of parliamentary institutions. So too did the cult of "science" and "objectivity" in history, associated with the rise of a historical profession. Charles Petit-Dutaillis, in the preface to his 1908 _Supplement_ to Stubbs, spoke condescendingly of Stubbs's intellectual genesis in the liberal patriotic German scholarship from which he derived his "optimistic and patriotic conception of English history." Nowadays, he went on, "when so many illusions have been dissipated, when parliamentary institutions set up by almost every civilized nation have more openly revealed as they developed their inevitable littleness and when the formation of nationalities has turned Europe into an armed camp, history is written with less enthusiasm." He was right that nationalism, so long associated with Romantic, liberal versions of history, was now finding its watchword in _Realpolitik,_ the new term for "reason of state," the pursuit of success by any means, and often of domination rather than merely independence. The implications of the "new" nationalism for historiography will mainly concern us in a later chapter (Chapter 25). But first there is a new world to discover.
**TWENTY-FOUR**
**A New World: American Experiences**
**The Halls of Montezuma: Díaz, Prescott and the Conquest of New Spain**
The making of the United States of America could be seen as a story of freedom in the widest possible geographical frame. To the first historian of the United States, George Bancroft (1800–1891), American freedom was a democratic extension of the freedom born in the German forests. Others, notably Frederick Jackson Turner (1861–1934), saw it in more self-contained terms, as shaped by the land itself, by the open frontier which had ceased to exist only in his own lifetime. The Tudor Puritan conception of the English as an elect nation, brought by the seventeenth-century settlers of New England, also became woven into North American self-consciousness.
But the history of America can also be told as a story of dispossession, subjugation and enslavement, beginning with the earliest European penetration of the continent, the Spanish conquest of Mexico. The first historians of the New World were Spanish, and they told the story of the conquest, naturally, as an epic achievement, coloured by the conception of a crusade. When the New England settlers of Plymouth Plantation struggled, in the 1620s, for bare survival, under the eye of their God, it was already a hundred years since Cortés and the Spanish conquistadores had overthrown the Aztec empire of Montezuma and founded "New Spain" on its ruins. It was over fifty years since the Spaniards had begun to erect the vast baroque cathedral of Mexico City on the site of the great Aztec temple in which human sacrifices had been conducted on an almost industrial scale. The story of the conquest, which took only two years (1519–21), is one of the most extraordinary that history records. Xenophon's "Persian Expedition," which shares some of the characteristics of that of Cortés, led to no new foundation. The expeditions of Alexander were on an even greater scale, but considered just as a story they do not have quite the dramatic cohesion of the Spanish one, as Alexander and his army, after their conquest of the "barbarian" empire, move on eastward.
The Spanish historians who had been drawn to chronicle their countrymen's epic adventure and its aftermath can, to most English readers, be mainly only names. One exception is Fra Bartolomé de las Casas, whose championing of the cause of the conquered Indians (as they will be called here) has recommended him to posterity. Another is one of Cortés's companions on his march from the coast to the Aztec capital and in the hard fighting before the city was taken, Bernal Díaz. His account, written in old age, is a vivid and vigorous narrative, free of conventional tropes and rhetorical flourishes, from the landing of approximately four hundred Spanish soldiers, with sixteen horses and ten brass guns, on the coast of Yucatan, to the bloody siege, destruction and capture of Tenochtitlán (Mexico City) which marked the completion of the Spanish victory. Its merits as an eyewitness account and as a narrative have earned it a modern English translation (as _The Conquest of New Spain_ ).
The other classic narrative in English is _The Conquest of Mexico_ (1843) by William Hickling Prescott (1796–1859). Prescott, like all the notable American historians who wrote in the mid nineteenth century, was a Harvard-educated New Englander. His considerable body of work—a triumph over near-blindness—began with a history of Spain (1838) under the monarchs who presided over its unification in the late fifteenth and early sixteenth centuries and were the patrons of Columbus, Ferdinand and Isabella. Later he added a pendant to his _Mexico, The Conquest of Peru_ (1847), and an unfinished history of Philip II of Spain. For _The Conquest of Mexico_ he used Díaz's account extensively, referring to him in his now published _Notebooks_ as "my staple authority," though condescending to his humble literary merits and to the "homely texture" of his style ( _Mexico,_ V, endnote). Díaz claims no more for himself. Prescott was a considerable scholar, who had worked in the Spanish archives. He was therefore less dependent on Díaz than Robertson had been for the conquest section of his _History of America_ three-quarters of a century earlier. Prescott published a preface of acknowledgements, and provided substantial footnotes to a variety of sources.
Both Díaz and Prescott can still be read enjoyably, and their merits are contrasting and complementary. Their story begins essentially with the appointment of Cortés, an adventurer with a somewhat wild past, recounted by Prescott, to command an expedition sponsored by the governor of Cuba, Velasquez, to explore the Yucatan peninsula. This had been discovered by the Spaniards two years earlier, in 1517; Díaz had accompanied them, and recounts their experiences. The governor soon regretted Cortés's appointment and tried to detain him, but Cortés slipped away. Velasquez subsequently sent a larger force after him, but Cortés, with masterly tactics, first defeated and then recruited it. The story of the expedition is that of a military campaign in which Cortés both fights and wins over the satellite peoples of the Aztec empire, through whose territories the army passes. He is able to exploit both their restlessness under the rule of the Aztec emperor Montezuma and the awe produced by the Spaniards: bearded men in armour with firearms, a few riding horses, all wholly outside the Indians' experience. Cortés poses as their deliverer; Díaz does not comment on the duplicity involved in this, but Prescott is explicit:
Alas! they could not read the future, or they would have found no cause to rejoice in this harbinger of a revolution more tremendous than any predicted by their bards and prophets...The light of civilization would be poured on their land; but it would be the light of a consuming fire, before which their barbaric glory, their institutions, their very existence as a nation, would wither and become extinct! Their doom was sealed when the white man had set his foot on their soil. (II. vii)
The Spaniards were aided throughout, and not least by its influence on the mind of the emperor Montezuma, by the Mexicans' myth that their god Quezalcoatl had prophesied the arrival to rule them of a race of white-skinned demigods.
The strange terrain through which the Spaniards marched and their progressively deeper acquaintance with Mexican civilization make this one of the most fascinating of campaign narratives. The Spaniards wondered at the Mexicans' technological skills and distinctive craftsmanship in gold and silver, cotton and feathers, and at the extensive, well-planned cities and stone architecture, but were horrified by the frequent evidence of highly ritualized and copious human sacrifice. The culminating revelation was the city of Tenochtitlán, standing in a lake and intersected by canals running at right angles and crossed by causeways. Its huge population (by European standards), the vast marketplace, the great pyramidal towers for sacrifices facing each other down long avenues, and the palace of Montezuma, with its gardens, menagerie and elaborate court ritual, were like nothing previously seen in the new world discovered by Columbus almost three decades earlier. It is understandable that, as Bernal Díaz tells us, the Spanish soldiers felt as though they were living in the fantastic stories of romance, which were the staple popular tales of the early sixteenth century.
Díaz—who was, as he admits, "no scholar"—sometimes confesses to a sense of inadequacy in expressing his and his companions' reactions. He mentions their wonder, but knows that he cannot adequately convey it, though he tries. But his pedestrian honesty may seem an advantage to the modern reader, as it did to Prescott, preserving him from ready-made conventional metaphors and stylized diction. He refers, certainly, to chivalric romances, and aptly compares the siege and destruction of the great city to the fall of Jerusalem. He also mentions that Cortés, in his addresses to his troops, compared their feats with those of the Romans ( _New Spain,_ Chapter 24, Chapter 8). But his history in general is immune to literary self-consciousness. He tells his reader, in the modern phrase, how it was. Prescott, who called him an "untutored child of nature," also paid tribute to the qualities of his narrative. Prescott compares it to photography, or rather daguerrotype, which must be one of the earliest instances of such a literary comparison: "He introduces us into the heart of the camp, we huddle around the bivouac with the soldiers, loiter with them on their wearisome marches, listen to their stories, their murmurs of discontent, their plans of conquest, their hopes, their triumphs, their disappointments" ( _Mexico,_ V, endnote).
This is no more than just. Díaz not only tells us his extraordinary story well (though readers are indebted to the modern translator who has excised the digressions and repetitions); he humanizes the conquistadores, occasionally referring to them by name, or saying "I forget his name," and casually individualizes them. Cortés himself is an inspiring leader but tricky; Fra Olmedo, the humane priest with the expedition, sometimes exercises a wise restraint on the soldiers' iconoclastic zeal. But there are also humbler figures: the very ugly musketeer whom Cortés says the Mexicans will take for one of their idols (Chapter 7); the Spaniard found on the coast of Yucatan who has "gone native" and refuses to return to his own people, of whom Díaz speaks without censure (Chapter 3); the incompetent astrologer (Chapter 19), who leads Díaz into an uncharacteristically and surely unconsciously Homeric touch: "His astrology did not help him, for he too died there with his horse." Then there is the soldier, "a decent man, though difficult to understand," who, posted guard over the captive Aztec emperor, shouts, "To hell with this dog!...I'm sick to death of always guarding him!" (Chapter 17). There is Montezuma himself, of course, who is respectfully and sympathetically described, and among ordinary Mexicans there are, particularly memorably, the fat _cacique_ and his ugly niece (Mexican women are often described by Díaz as beautiful, and he takes one himself ), whom Cortés has to receive as a gift "with a show of pleasure" (Chapter 8). It is noteworthy that resulting children of mixed race who were born to the leaders, including Cortés's own, became recruited, as recorded by Prescott, into the Spanish aristocracy.
Díaz can strike the reader as both callous and humane. He gives the impression of a tolerant man who took human beings of all kinds as he found them, who pities Montezuma, and who is appalled, as all the Spaniards were, by human sacrifices to the Mexican "idols." It is a mark of the munificence and dignity of Montezuma, conveyed by Díaz, that the reader as well as the Emperor is shocked when Cortés has him put in chains. When he is killed—Díaz says accidentally, during an insurrection of the people against the foreigners—the Spaniards mourn apparently unaffectedly: "Cortés and all of us captains and soldiers wept for him, and there was no one among us that knew him and had dealings with him who did not mourn him as if he were our father, which was not surprising, since he was so good" (Chapter 19). On the other hand Díaz is entirely unrepentant, and rebuts the subsequent condemnation by Las Casas, over the massacre perpetrated on a large number of Indians suspected, perhaps correctly, by the Spaniards of treachery. Díaz's argument is twofold: the conquest was justified—a crusade in fact—and the massacre was essential to the army's survival, without which there would have been no conquest. Prescott consistently insists that this should be judged by the standards and ideas of the sixteenth century, not those of the nineteenth, while admitting that a more selective and restrained reprisal would probably have sufficed ( _Mexico,_ III.vii).
Though he inevitably offended the descendants of both peoples, Prescott himself strikes the reader as commendably judicious. He is no Puritan. He had learned on his travels in southern Europe to respect Catholicism. This saves him from the insistent and aggressive provincialism (despite an education partly in Germany and a diplomatic career) which disfigures the work of another Harvard historian on the sixteenth century, John Lothrop Motley's _The Rise and Fall of the Dutch Republic_ (1855). In Prescott's gentlemanly tolerance there is a distinct flavour of the eighteenth-century Enlightenment, without its iconoclastic zeal: he uses the significant word "philosophical" freely. His own stance towards the conquest is equally far from Christian triumphalism, reflex Protestant anti-Catholicism, nineteenth-century racial arrogance, and the modern principled opposition to imperialism. To those who would find it a fault that his work is not written with a fierce partisanship on behalf of any of these he had his own answer, though he addressed it only to himself, in his _Notebooks_ : "Never call hard names; it is unhistorical, unphilosophical, ungentlemanlike." The three adjectives, turned into their positive form, do well in characterizing Prescott. He views the conquest both as an extraordinary achievement and adventure—he applies the words "epic" and "romance" to it—and as the doom of a culture and a people, though he is anxious not to overrate the Mexican achievement on the scale of civilizations.
Prescott is not averse to judgement: the age of a rigorous cult of "objectivity," meaning not judgement but abstention from it, had not yet dawned in historiography. As we have seen, he sees the standards of the sixteenth century as relevant to appraising the conduct of sixteenth-century men. Nor, on the other hand, is he blinded by the horrors of human sacrifice to either the beliefs which prompted it or the fact that hundreds of thousands of Montezuma's subjects led apparently peaceful, confidently traditional lives which the Spanish conquest would utterly sweep away, and which had been greatly preferable to the future, for themselves and their descendants, as involuntary Catholics and subjects of Charles V and Philip II. The brave Mexican allies of Cortés in his campaign against the Aztec empire are pathetic dupes, unknowingly collaborating in the destruction of their relative autonomy and their culture.
Prescott's tolerant even-handedness is bound up with what has proved perhaps a still more controversial quality, much admired in the nineteenth century but sometimes harshly denigrated in the twentieth: the self-conscious, and it is not inappropriate to say "Augustan," literariness of his writing—precisely the quality from which Díaz's account was wholly free. It is clear from Prescott's notes to himself that the manner mattered to him as much as the substance, and that, though he was naturally drawn to the conquest as a subject by his work on earlier Spanish history, it also attracted him as a theme with high epic and picturesque possibilities. Comparing it with Washington Irving's life of Columbus, he spoke of his own project as "the most poetic subject ever offered to the pen of the historian." He had no axes to grind, but he had been inspired to become a historian by Gibbon's autobiography, and there are many Gibbonian echoes in his prose. We also find him brooding on Livy's manner of narrating Hannibal's crossing of the Alps, and this too seems to have echoes in _The Conquest of Mexico._ He reminds himself of the need to diversify the military history with
the description of the grand and picturesque scenery through which the Spaniards march; faithful descriptions of the various architectural remains, of the vegetable products, the mountains, the towns, etc, contrasted, moreover, with their present condition, affording an agreeable and instructive variety, and giving life and colouring to the picture; the description of Montezuma's palace, court ceremonies, way of life, his garden, collections of natural history, etc; the city of Mexico, its buildings, markets, manners of the people, etc...
The modern reader, knowing Prescott's history was produced almost halfway through the nineteenth century, is quite likely to register something old-fashioned, "eighteenth-century," in Prescott's writing(e.g. "agreeable and instructive"). There are some signs that he was aware of this, particularly in relation to his great eighteenth-century predecessor in recounting the conquest (though much more briefly), William Robertson. "Beware of Robertson," he admonished himself in his journal. "Never glance at him till after subject moulded in my mind and thrown into language—I must not get my manner from him." The admonition was perhaps not wholly successful. Prescott was born in the eighteenth century, and in the early decades of the nineteenth American literature was still highly derivative from English (and Scottish) eighteenth-century models. Twentieth-century commentators have objected strongly to this in Prescott. Quite why it should trouble us, however, that an author writing a century and a half ago should sometimes sound like one writing over two centuries ago seems unclear; perhaps it seems un-American.
Prescott makes very Gibbonian use of "or," both to point an antithesis and to insert a doubt or qualification: "The religion, or to speak correctly the superstition, of Montezuma..." ( _Mexico,_ II.vi). Italics too are used as Gibbon characteristically does to insinuate doubt: "Six thousand victims _are said_ to have been annually offered up in their sanguinary shrines" (II.vi). There is a Gibbonian loftiness in his comment on interpretations of Mexican religion. Originally it was seen by the Spaniards as devil-worship, but
before a century had elapsed the descendants of the same Spaniards discerned in the mysteries of the Aztec religion the features, obscured and defaced indeed, of the Jewish and Christian revelations. Such were the opposite conclusions of the unlettered soldier and of the scholar. A philosopher, untouched by superstition, might well doubt which of the two was the more extraordinary. (IV.iii)
This was "the philosophic historian" still at work: the raised eyebrow in the last word descends directly from the eighteenth century. Elsewhere Prescott seems to play variations on a famous antithesis in Gibbon ("To resist was fatal, and it was impossible to fly"— _Decline and Fall,_ III): Cortés's army seems trapped so that "To fight and to fly seemed equally difficult" ( _Mexico,_ III.vi).
The Augustan stateliness of Prescott's prose makes it an apt instrument for registering the grandeur of the Aztec capital and the refinement, luxury and ceremonial which surrounds Montezuma. Prescott does not miss, of course, the opportunity to draw the oriental analogies, including the standard reference to "effeminacy":
[Fishponds] afforded a retreat on their margins to various tribes of water-fowl, whose habits were so carefully consulted that some of these ponds were of salt water, as that which they most loved to frequent. A tessellated pavement of marble enclosed the ample basins, which were overhung by light and fanciful pavilions, that admitted the perfumed breezes of the gardens, and offered a grateful shelter to the monarch and his mistresses in the sultry heats of summer...
The place is now a tangled wilderness of wild shrubs, where the myrtle mingles its dark, glossy leaves with the red berries and delicate foliage of the pepper-tree. Surely there is no spot better suited to awaken meditation on the past; none where the traveller, as he sits under those stately cypresses gray with the moss of ages, can so fitly ponder on the sad destinies of the Indian races and the monarch who once held his courtly revels under the shadow of their branches. (IV.i)
To the Aztecs, of course, it was the Spaniards who were wondrous, and Prescott catches well the awe and astonishment the latter inspire, as when they build sailing ships for the siege of Tenochtitlán and these are seen for the first time gliding across the lake which surrounds the city: "It was a novel spectacle to the simple natives; and they gazed with wonder on the gallant ships, which, fluttering like sea-birds on their snowy pinions, bounded lightly over the waters..." (VI.iv).
Where the eighteenth-century flavour of Prescott's tropes and diction does become undeniably tiresome is when he has to use them to describe battle and bloodshed, where the weight of past conventions lies heavily on the narrative. Predictably, Díaz is good precisely at this—detailed, realistic and soldierly. He is particularly good at conveying the exhaustion produced by prolonged combat, as well as the terror beforehand. In other respects, of course, Prescott's range includes effects which are not within Díaz's reach or aspiration.
Historical perspective, as we have seen, is one of these effects, and with it a fuller sense of the pathos of the Mexicans' situation, which Díaz registers only when, under his own eyes, it affects an individual and a monarch, Montezuma. Sensitivity, too, to the sublime and picturesque qualities of the landscape which forms the setting is, as we have also seen, a quality deliberately cultivated by Prescott, but it also marks changes in European sensibility in the three hundred years between Díaz and himself. Prescott was working within aesthetic categories of the sublime and the picturesque developed over the previous century. Díaz responds to gardens and flowers, to ease and abundance, and in this he is not only a soldier, with an eye to comfort, but a man of his time. A sublime landscape is characteristically an uncomfortable and even dangerous one, and its appreciation, which began in the ancient world, is essentially hyper-civilized. In a way Díaz does not, Prescott revels in the landscapes progressively revealed to Cortés's men: the characteristic vegetation, or lack of it, of the diverse microclimates created by differences of altitude, and precipices which give glimpses of the profuse vegetation of the valley floors. (Prescott drew on the recent work of the geologist and geographer Alexander von Humboldt for some of his descriptions of landscape.) The army's march led it across the arid remains of volcanic activity:
Working their way across this scene of desolation, the path often led them along the borders of precipices, down whose sheer depths of two or three thousand feet the shrinking eye might behold another climate, and see all the glowing vegetation of the tropics, choking up the bottom of the ravines. (III. i)
Then, of course, there were the distant vistas—above all, that which gave the conquistadores their first sight of the Aztec capital. Their approach to it has Díaz expressing a sense of his inadequacy and reaching for an analogy with the popular romance of _Amadis de Gaul_ : it was "like an enchanted vision from the tale of Amadis. Indeed, some of our soldiers asked whether it was not all a dream...It was all so wonderful that I do not know how to describe this first glimpse of things never heard of, seen or dreamed of before" ( _New Spain,_ Chapter 15). For Prescott the wonder begins with the distant prospect of
the fair city of Mexico, with her white towers and pyramidal temples, reposing, as it were, on the bosom of the waters,—the far-famed "Venice of the Aztecs." High over all rose the royal hill of Chapoltepec, the residence of the Mexican monarchs, crowned with the same grove of gigantic cypresses which at this day fling their broad shadows over the land. In the distance beyond the blue waters of the lake, and nearly screened by intervening foliage, was seen a shining speck, the rival capital of Tezcuco, and, still further on, the dark belt of porphyry, girdling the valley around...(III.viii)
Prescott is prolific with such effects, and, as he intended, they give his history some of its character and appeal. Another landscape, the North American wilderness, knowledge of which in the seventeenth and eighteenth centuries was virtually bounded by the Mississippi river, was to be the chosen scene of the histories written by the second great nineteenth-century American historian.
**Outposts in the Wilderness: Parkman's History of the Great West**
The land itself, which he often called simply "the forest," was the central inspiration of Prescott's younger New England contemporary Francis Parkman. It was more dominant than in Prescott, because in the great northern wastes there was no unexpected indigenous civilization to focus the historian's and the reader's attention. Instead there were the forest-dwellers, both indigenous and European or half-European: hunters and trappers—in the French phrase (and most of the latter were French), _coureurs de bois._ Parkman was fascinated both by them and above all, as he made clear, from his earliest years by the forest itself; this included its Indian inhabitants and the French Catholic missionaries and traders who first explored it along its river systems, from the St. Lawrence and the Great Lakes to the Gulf of Mexico. Parkman recognized that the English-descended pioneer farmers in western Pennsylvania, New York and Virginia, as settlers, had more durability and represented the land's future. But Parkman was a Romantic, not a propagandist for Anglo-Saxon virtues, and he was frankly less interested in them. He knew that they spelt destruction to the forest and to the Indians' way of life, as surely as Cortés and his men did to the ancient civilization of Mexico, and his feelings about them were at best fatalistic, never enthusiastic.
Parkman's intense love affair with the American forest began early. On vacations from school in New England and as an undergraduate at Harvard, where he read the ancient historians, he spent as much time as possible in the backwoods of New Hampshire and Maine. His first, very distinguished, publication was a first-person account of his extended and physically demanding visit to the far west, _The California and Oregon Trail_ (1849), usually known as _The Oregon Trail_ (the reference to California was a topical touch in the year of the gold rush). It is one of the outstanding travel books of the century, alongside C. M. Doughty's _Travels in Arabia Deserta_ (1888), with which it bears some analogies. Parkman, despite the lifelong ill health which dogged him—like Prescott, he had among other afflictions severe eye trouble—lived for some weeks with the Indians, who still preserved their traditional way of life.
He makes clear in his historical works that he was often drawing on this experience, though the Indians he describes there were not the mounted buffalo-hunters of the prairies he had seen at first hand but the forest-dwellers in the area of the Great Lakes and the Ohio and Mississippi valleys. In _The Oregon Trail_ one learns a good deal about Parkman himself and his attitudes: his view of life as a struggle (before Darwinism, though one sometimes sees its influence later); his attraction to the picturesque though also squalid life of the Indians and their characteristics, combined with a determination not to idealize them; his own stoicism and admiration for bravery and self-reliance in others; his complete conviction of Indian untrustworthiness and lack of fixity of purpose.
Parkman's _oeuvre_ began back to front. His first historical work, _The Conspiracy of Pontiac_ (1851), treated the Indian rising against the British, after the latter's final victory over their French rivals in North America, in the 1760s. After an introductory section on the Indian way of life and the characters of the various tribes, _Pontiac_ (named after the exceptionally able Ottawa chief who led the rising) goes on to consider the two European powers which had confronted each other in North America for two centuries, and their relations with the Indians: good in the case of the French, who in the west were missionaries, trappers and traders and interbred freely with the Indians; bad in the case of the British settlers and the British government, which, after the defeat of the French, was tactless and arrogant—the immediate cause of the rising of the confederation of Indian tribes organized by Pontiac.
The three chapters in which Parkman surveys Anglo–French–Indian relations over the previous century are an initial statement of the interests which led to his subsequent seven volumes on the French and English (chiefly the former) in North America, from the sixteenth century to the final resolution of their conflict in the mid eighteenth century. These volumes, apart from _Pontiac_ and the final volume, appeared in chronological order up to 1892, shortly before Parkman's death. In between he published two novels and a book on roses, on which he was an expert. The nature of his historical _oeuvre_ is adequately conveyed by the separate titles: _Pioneers of France in the New World_ (1865), _The Jesuits in North America_ (1867), _La Salle and the Discovery of the Great West_ (1869), _The Old Regime in Canada_ (1874), _Count Frontenac and New France under Louis XIV_ (1877), _Montcalm and Wolfe_ (1884), and _A Half-Century of Conflict, 1700–1750_ (1892). The predominance of France in Parkman's interests is clear. For him the romance of American history, in so far as it did not dwell in the forest itself and its Indian inhabitants, pursuing their immemorial and doomed way of life, belonged to the French: to the _coureurs de bois,_ who sometimes became virtually Indians themselves, and to the heroic Jesuit missionaries, who established isolated outposts in the remotest wilderness, of which they were among the earliest European explorers and in which they often found the martyrdom they sought, and whose work proved in the long term as fragile as the traditional ways of the Indians. But Parkman does not neglect the struggle between the European powers, which he dwells on particularly from the perspective of the French governors of Quebec.
Perhaps the most interesting, as he was the most ambitious, of the explorers, Cavelier La Salle (1643–87), was not a priest but initially a trader. He was a French bourgeois who was the first to travel from the Great Lakes down the Mississippi to the Gulf of Mexico, and who dreamed of creating a vast French empire in the west, linked by the river and with its southern port on the Gulf, thus challenging both Britain and Spain. He is one of Parkman's heroes, along with the aristocratic—and autocratic—French governors, Champlain at the end of the sixteenth century and Frontenac in the late seventeenth, who established a chain of outposts along the Great Lakes, and the two great soldiers, Montcalm and Wolfe. One is tempted to add Pontiac, for, though he by no means draws Parkman's unqualified approval, Parkman admires Pontiac's fortitude and statesmanlike qualities and understands the desperation that drives him to revolt. The capture of Quebec by the British and the capitulation to them of the whole of New France was, for the Indians, Parkman says, "wholly disastrous."
His comments on the British and German settlers, ploddingly extending the borders of cultivation in western Pennsylvania, New York and Virginia, are respectful of their spirit of independence, in which they far surpass the docile French peasants of Quebec province under their feudal overlords, and of their dogged perseverance, in which they contrast with the footloose _coureurs de bois._ But they have no romance: "In every quality of efficiency and strength, the Canadian fell miserably below his rival; but in all that pleases the eye and interests the imagination he far surpassed him" ( _Pontiac,_ I.iii). Among the British there is no great vision like Champlain's, no imperial dreams for the continent like La Salle's. It is they and their kind, and not settlers, who stir Parkman's imagination, as in the climactic moment when, canoeing down the Mississippi, La Salle and his companions sense the approach not only of spring but of a warmer, southern climate and, still more, after over a thousand miles of dangerous travelling and harsh privations they (like Xenophon's men) feel the proximity of the sea:
With every stage of their adventurous progress, the mystery of this vast New World was more and more unveiled. More and more they entered the realms of spring. The hazy sunlight, the warm and drowsy air, the tender foliage, the opening flowers, betokened the reviving life of Nature. For several days more they followed the writhings of the great river, on its tortuous course through the wastes of swamp and canebreak, till on the thirteenth of March [1682] they found themselves wrapped in a thick fog. Neither shore was visible; but they heard on the right the booming of an Indian drum and the shrill outcries of the war-dance.
After meeting for the first time Indians who live in houses of baked mud and have a temple in which they worship the Sun with human sacrifices, they know that they are nearing their journey's end.
As he drifted down the turbid current, between the low and marshy shores, the brackish water changed to brine, and the breeze grew fresh with the salt breath of the sea. Then the broad bosom of the great Gulf opened on his sight, tossing its restless billows, limitless, voiceless, lonely as when born of chaos, without a sail, without a sign of life.
La Salle plants a column bearing the arms of France, and a cross, and proclaims the sovereignty of Louis XIV. Parkman's comment intimates the evanescence of the dream:
On that day, the realm of France received on parchment a stupendous accession. The fertile plains of Texas; the vast basin of the Mississippi, from its frozen northern springs to the sultry borders of the Gulf; from the woody ridges of the Alleghenies to the bare peaks of the Rocky Mountains—a region of savannahs and forests, suncracked deserts, and grassy prairies, watered by a thousand rivers, ranged by a thousand warlike tribes, passed beneath the sceptre of the Sultan of Versailles; and all by virtue of a feeble human voice, inaudible at half a mile. ( _La Salle,_ XX)
Even within its narrower boundaries, the state of Louisiana was not to remain in the long run French; the Mississippi would not become, as La Salle called it, after the great French statesman, the river Colbert.
Parkman clearly enjoys these references to the ancient world of Europe, to which La Salle returns to outfit an expedition to the Gulf by sea, which, missing the mouths of the Mississippi, perishes miserably on the inhospitable shore, where La Salle himself is murdered by mutineers. The frequent reminders of the strange conjunction of the western wilderness and the intrigues and formalities of the French court appealed to Parkman's imagination. "Many a gallant gentleman, many a nobleman of France, trod the black mould and oozy mosses of the forest with feet that had pressed the carpets of Versailles" ( _Pontiac,_ I.ii). He worked in archives in England and France as well as North America. Books like his had to be constructed very largely from primary sources: letters, memoirs, dispatches. He was himself a kind of pioneer, and he recognized the analogy. His estimation of the French, while it recognizes the incongruous conjunctions of old and new, emphasizes not only the rigidity of the social structure they carry with them to Quebec but also their flexibility in their relations with Indians, which goes right up to the governor. Count Frontenac (1620–98) lorded it among the Indians like a paternal and affable _grand seigneur,_ and "plumed and painted like an Indian chief, danced the war-dance and yelled the warsong at the camp fires of his delighted allies" ( _Pontiac,_ I.iii). Among the English, William Penn enjoyed exceptionally good relations with the Indians, but it is difficult to imagine him in a similar situation. The Jesuit missionaries even more strikingly straddled the two worlds:
We see them among the frozen forests of Acadia, struggling on snow-shoes, with some wandering Algonquin horde, or crouching in the crowded hunting-lodge, half stifled in the smoky den, and battling with troops of famished dogs for the morsel of sustenance. Again we see the black-robed priest wading among the white rapids of the Ottawa, toiling with his savage comrades to drag the canoe against the headlong water. Again, radiant in the vestments of his priestly office, he administers the sacramental bread to kneeling crowds of plumed and painted proselytes in the black forests of the Hurons...( _Pontiac,_ I.ii)
But the Quebec heartlands of New France had "no principle of increase," while in the forest "the savages did not become French, but the French became savages." Meanwhile, further south, the British settlements were slowly but steadily chipping away at the wilderness, where "forests crashing to the axe, dark spires of smoke ascending from autumnal fires, were heralds of the advancing host." In the valley of the St. Lawrence and along the coast of the Atlantic, "Feudalism stood arrayed against Democracy, Popery against Protestantism, the sword against the ploughshare" ( _Pontiac,_ I.ii). But the British settlers, by the mid eighteenth century, were still only beginning to pass the barrier of the Alleghenies, so that the campaigns of the British army in the west against the French and the Indians were fought out just beyond the borders of Pennsylvania and western Virginia. After the British victory it was the settlers there who, apart from the tiny garrisons in the remote outposts taken over from the French, bore the brunt of the Indian uprising instigated by Pontiac, and suffered its horrors.
The most complex situation was in Pennsylvania, and Parkman brings to it a kind of grim humour and not unlimited sympathy. The frontiersmen, goaded by the fate of their families and friends, turned their anger and hatred against the domesticated Indian Christian converts living peacefully in the state. When these were taken into custody in Philadelphia for their own protection, the hard, undisciplined men of the frontier caused a riot and tried to storm the jail. The protectors of the Indians were the Quakers. They figure less well, in Parkman's account, however, as sentimentalists who are sceptical about the atrocity stories from the frontier, and as pacifists who are recalcitrant in organizing a defence force against the rising. The exasperated frontiersmen were largely of Protestant Ulster stock, and their militant invocations of the Bible, calling for the punishment of evil-doers, are starkly at odds with the rival Quaker reading of the holy book. Parkman adopts the detached posture of "a student of human nature" (Prescott would have said "a philosophic mind") and deplores both "the enormities of white barbarism," which sometimes outdo those of the Indians, and the absurdity, as he sees it, of pacifism (which sometimes became compromised) in the face of Indian war parties ( _Pontiac,_ II.viii).
But Parkman's story of the rising to which he affixed the name of Pontiac (who remains a largely shadowy though impressive figure) is mainly the story of the small forts scattered along the Great Lakes which are immediately put under siege. All but that of Detroit, which, supplied across the lake, holds out for over a year, are overwhelmed and their garrisons are butchered or made prisoners; some of the prisoners later escape or are released, and it is from their accounts that Parkman gets much of his information, though the story of unconquered Detroit is naturally largest and fullest. Parkman brings his distinctive descriptive gift to bear on the situation of these small, isolated groups of beleaguered soldiers (mostly Scottish), utterly cut off from civilization, as well as on the ordeal suffered by the soldiers who became prisoners. The escapee faced loneliness, terror, hunger and bewilderment, and Parkman clearly used his own experience to evoke his plight, surrounded by "the thousand pitfalls and impediments of the forest," made more formidable in darkness:
At length, he can hear the gurgle of a neighbouring brook, and turning towards it, he wades along its pebbly channel, fearing lest the soft mould and rotten wood of the forest might retain traces enough to direct the bloodhound instinct of his pursuers. With the dawn of a misty and cloudy morning, he is still pushing on his way, when his attention is caught by the spectral figure of an ancient birch-tree, which, with its white bark hanging about it, seems woefully familiar to his eye. Among the neighbouring bushes, a blue smoke curls faintly upward, and, to his horror and amazement, he recognizes the very fire from which he had fled a few hours before.
Parkman continues reconstructing the typical experiences of the novice in the forest, with death and torture close behind him until by chance he reaches some frontier post or "perishes in despair, a meagre banquet for the wolves" ( _Pontiac,_ II.v).
To the Indian, of course, the forest is home and sustenance. "He will not learn the arts of civilization, and he and his forest must perish together. The stern, unchanging features of his mind excite our admiration from their very immutability" ( _Pontiac,_ I.i). Some of Parkman's references to the Indians' character and appearance will grate on post-colonial susceptibilities (he is fond of "snake-eyed," for example), but by the standards of his time his attitudes are complex. He would have said that they matched the inconsistencies of the Indian character, which have to be classed "with the other enigmas of the fathomless human heart." He admires the Indians' "haughty independence," dignity and fortitude. The Indian is not always centre stage, but it seems that for Parkman he is the cynosure: "to depict him is the aim of the ensuing history" ( _Pontiac,_ I.i).
One cannot take the measure of Parkman's attitude to the Indian apart from his attitude to civilization, and both seem equivocal, even tragic. All life is a mixture of good and evil; beauty and the picturesque are often at odds with utility; he speaks, in a significant phrase, of the cotton snake's "loathsome beauty." His cultural roots lie not in previous historiography but in the literature of European and American Romanticism: in Byron and Scott, who frequently provide the epigraphs for chapters in _The Oregon Trail,_ and in Fenimore Cooper, though he also consulted the scholarly authority on the Indians, Henry Schoolcraft. Equally powerful, however, is what one feels inclined to call a prospective Darwinism. It is a matter of sensibility as well as opinion. There is a vignette in _The Oregon Trail_ (i.e. ten years before the publication of _The Origin of Species_ ) which bears this out:
I went and lay down by the side of a deep, clear pool, formed by the water of the spring. A shoal of little fishes of about a pin's length were playing in it, sporting together, as it seemed, very amicably; but on closer observation I saw that they were engaged in a cannibal warfare among themselves. Now and then a small one would fall a victim, and immediately disappear down the maw of his voracious conqueror. Every moment, however, the tyrant of the pool, a monster about three inches long,...would slowly issue forth with quivering fins and tail from under the shelving bank..."Soft-hearted philanthropists," thought I, "may sigh long for their peaceful millennium; for from minnows up to men, life is an incessant battle." (XIX)
It is no surprise to find that Parkman had no sympathy with the Boston anti-slavery abolitionists and was notably hostile to the campaign for women's suffrage. His attitudes seem to have a good deal in common with those of the Mississippian Basil Ransom in Henry James's _The Bostonians._ The regrets, whose expression generally Parkman stoically repressed, found vent in irony. Describing a volcanic out-crop along the Mississippi once adorned with an Indian painting of what a Jesuit eyewitness described as a "monster," Parkman adds a footnote: "In 1867, when I passed this place, a part of the rock had crumbled away, and instead of Marquette's monster, it bore a huge advertisement of 'Plantation Bitters'" ( _La Salle,_ V). Elsewhere, commenting on the regrettable aesthetic effects of geological erosion in the vicinity of Minneapolis, he speaks of "other changes equally disastrous, in an artistic point of view" in the form of the city of Minneapolis, "which, in 1867, counted ten thousand inhabitants, two national banks, and an opera house," while the rival city on the opposite shore "boasted a gigantic water-cure and a State university" and so was no longer picturesque ( _La Salle,_ XVIII). Civilization and geological change are equally irresistible. The former, so often seen in the eighteenth century as a precious and even fragile inheritance, is now, it seems, to be seen as a kind of juggernaut, whose course is marked by banality and whose message is adapt or die.
But, though a stoic, Parkman is anything but a bleak writer. Some austerely fastidious twentieth-century commentators found him an excessively exuberant one. In fact he is a sensitive literary artist, a master of evocative, sensuous prose. He was naturally shaped by his period, as were Macaulay and Michelet, who also transgress later and more thin-blooded notions of literary restraint. The American wilderness had already been lauded and celebrated. In Parkman's histories it is omnipresent and central, and it provides, more than anything else, the powerful imaginative dynamics of his work.
**Henry Adams: From Republic to Nation**
American historical writing follows a familiar course: from history as a highly literary form of composition, written by amateur men of letters, to increasing professionalism and a commitment to objectivity sometimes spoken of as "scientific." From Prescott's work in the 1840s to that of Henry Adams in the 1890s marks the distance travelled in this direction. There is also another difference. Prescott wrote on the history of a continent as yet unsettled by Europeans, Parkman of one where the English colonies had not yet coalesced into a political union, while Adams was to write on the early history of the new republic.
History written in English about the experience of the New World had begun at the time of the earliest English settlement in the seventeenth century. William Bradford, the first historian of the New England colony, was also a leader of the colonists when they made their landing on Cape Cod in 1620. They were a group of separatist English Puritans who had left their country to live as a godly congregation, worshipping God only in the manner they believed he had prescribed, and uncontaminated by the corruptions of the world. America was incidental, a last resort: initially they had transplanted themselves as a body to Leiden, in Holland, but though free there from religious persecution they found it hard to sustain themselves and to keep their young people, in particular, from being drawn into the wider, ungodly, world. After their eventual landfall in North America, Bradford became their governor—though not, of course, as the name would later imply, appointed by the Crown. Unlearned but not uneducated, he was clearly a man of talent, and one of his talents was as a historian. The understandable self-consciousness of a community situated by its own collective will, as his was, naturally inspired a desire to record its fortunes and, above all, the mercies of God towards it despite its frailties. Bradford's _History of Plymouth Plantation, 1620–1647_ was consulted in manuscript by later historians, but was not actually published until 1912. The journal of John Winthrop, first governor of the settlement in Massachusetts Bay, also long remained in manuscript, but was published as _The History of New England from 1630 to 1649_ at the end of the eighteenth century. Captain John Smith wrote accounts of the older colony in Jamestown, Virginia: his concern was to advertise the attractions of American colonization.
But the _History of Plymouth Plantation_ is unmistakably history—or one might prefer to call it chronicle, for Bradford's work has, of necessity, some of the characteristics of medieval monastic chronicles, in its localism and its enforced concentration on the strictly bounded community, given a universal dimension by an overarching piety. Peter Gay has called it "an authentic masterpiece." After an account of the landing and the desolation which awaited the settlers, Bradford's book becomes the well-written annals of the affairs of the colony. Bradford is eloquent about the difficulties, whose starkness emphasizes the goodness of God's providence in preserving his saints: "Being thus passed the vast ocean...they had no friends to welcome them, nor inns to entertain or refresh their weatherbeaten bodys, no houses or much less townes to repaire to, to seek for succoure...Besides, what could they see but a hideous and desolate wilderness, full of wild beasts and wild men." Bradford wants the children of the settlers to know "what difficulties their fathers wrestled."
The settlement was to be that of a covenanted people: "By these presents solemnly and mutually in the presence of God, and one of another, [we] covenant and combine ourselves, together into a civil body politic." But these were not "Americans" but Englishmen, whose self-consciousness was that of a godly "gathered people." When Winthrop drew an image that was to become famous, this was the conception he was invoking: "For we must consider that we shall be as a city upon a hill. The eyes of all people are upon us." They were, that is, to be exemplary, but as a community of God's saints, voluntarily set apart, not yet primarily as inhabitants of a new land. Perry Miller showed more than half a century ago, in his pioneering study _The New England Mind,_ that it took several generations for that self-consciousness to become distinctively American. With inevitable disillusionments came also the adoption of a repeated historical dynamic, drawn from the Old Testament, and applied to the colony's history, in what Miller calls the "jeremiads." The people were constantly slipping from grace and as constantly visited with divine wrath. This was the pattern which, in some contrast to Bradford's matter-of-fact providentialism, informed the general history of New England produced by the most prominent Calvinist minister of the day, Cotton Mather, in 1702.
The best history of the political crises of the 1760s and '70s is generally agreed to be the _History of the Colony and Province of Massachusetts Bay,_ written by the state's loyalist governor Thomas Hutchinson, who completed his account, like many a public man turned historian before him, in exile, in Britain. It is a reminder that the War of Independence had some of the characteristics of a civil war. The most prominent of the historical celebrators of the Independence period was to be the Unitarian minister and teacher (later president) at Harvard Jared Sparks (1789–1866). He produced a twelve-volume Life of George Washington, which was essentially an edition of Washington's letters. Sparks held at Harvard, from 1839, the first chair of history in the United States. His work was necessarily rather in the field of the accumulation of documents and in the inspiration and patronage of historical writing generally than as a historian himself. Parkman, who was a pupil, dedicated _Pontiac_ to him.
The first historian of the United States, George Bancroft (1800–1891) was a protégé of Sparks. He was not a man of means like Prescott and Parkman, and his career included a spell teaching history at Harvard, after study in Germany, in Göttingen and Berlin, where he took a doctorate. He also, like Motley, became American minister in London, and later (1867–74) in Berlin. He had had a directly political career as Secretary to the Navy. His history eventually reached twelve volumes, published between 1834 and 1882, and was massively popular. He was a Jeffersonian Democrat, and his history was an uncritical celebration of America as the land of liberty and democracy. His dictum that "the organization of society must more and more conform to the principle of freedom" places him firmly among the liberal historians we considered in the last chapter. He was, however, also an enthusiastic democrat, which by no means all his European counterparts were: for him it was true that the voice of the people was the voice of God. Bancroft's somewhat naive, parochial and uncritical approach to American history, while it may have ensured his nineteenth-century popularity, brought an inevitable reaction, and his history, though not ill-written, does not have the literary eminence to raise it above the shifting currents of historiographical and ideological fashion. Twelve volumes—American nineteenth-century historians were nothing if not copious—are also rather a lot even for sympathizers. His history is undeniably a monument, but seems likely for the foreseeable future to remain a largely unvisited one.
Henry Adams's detailed study, originally in nine volumes, of the third and fourth presidencies, _The History of the United States of America during the Administrations of Jefferson and Madison_ (1889–91), is a very different matter, and is still highly regarded. Adams is a historians' historian. This is said neither in compliment nor in denigration, but simply as a statement of fact. He himself, commenting with characteristic irony on the small number of copies he had sold, remarked that history was made an aristocratic pursuit by the paucity of its material rewards. In fact Adams himself was the nearest America could get to a hereditary aristocrat: the Bostonian great-grandson of America's second president and the grandson of another. Adams's stance or pose of fastidious disillusionment, presented in his autobiographical _The Education of Henry Adams,_ confirmed his patrician image. He conformed to type in that his career included Harvard (studied and later taught), German universities, and a spell at the London embassy, that magnet for American historians, when his father was ambassador.
As a historian, Adams began as a medievalist, first collaborating on a book on Anglo-Saxon law and institutions. His choosing the 1800s for his American magnum opus has sometimes been thought to require explanation, but the suggestion that it was because the period contrasted badly with the presidency of his great-grandfather John Adams (1797–1801) has been strongly disputed. He had become expert in the period by editing the papers of a statesman he much admired, Albert Gallatin, who was Secretary of the Treasury under Jefferson and Madison and had played a part in negotiating the Treaty of Ghent (1814), which ended the war with Britain, and which forms a kind of climax to Adams's history. Adams's narration, as he explores the major episodes during the two presidencies, especially the purchase of Louisiana from the French and the onset of the war with Britain, is dense, intricate and long for its scope. In the main narrative part of the text he deals principally with the traditional subjects for history: legislation and the strife of parties, diplomacy and war. Like Prescott and Parkman, he worked in archives in Europe as well as the United States; his themes obliged him to estimate the motives and manoeuvres of Talleyrand, Napoleon and the British prime minister Spencer Perceval, as well as those of the American presidents and other politicians. He deals as expertly with the proceedings and dissensions in the British House of Commons as with those of Congress. The tone of his history is predominantly dispassionate—he is nothing like Bancroft—but undeniably patriotic: he sets up an antithesis between American energy and, increasingly, willingness to innovate and European conservatism and rigidity, personified in a simile of the British Life Guardsman in boots and cuirass, while the American is like a prizefighter stripped for the ring.
To the non-professional reader, however, the most attractive part of Adams's work is not the narrative, which frequently offers the experience of navigating an impressive but daunting complexity, but the long prologue, followed by a shorter epilogue of the same character, in which Adams surveys the state of American civilization in 1800 and which can still be read with profit and pleasure. It provides an analogy with Macaulay's famous chapter on the state of England in the reign of Charles II, and the comparison is to Adams's advantage. His survey is all the better for being without the declamatory triumphalism in which Macaulay incongruously clothed his essay in social history. It also sometimes recalls, not altogether accidentally, Taine's formula for cultural explanation: " _race, milieu, moment_ "—in Adams's case, inheritance, land, and the current historical challenge and opportunity. Adams shared Taine's desire to see history as a deterministic science of social and cultural development. The concluding section of the prologue, on "American Ideals," and the conclusion of the epilogue (which attempts to measure developments between 1800 and 1817), entitled "American Character," clearly represent for Adams the crown and a large part of the point of the whole exercise: once characterized, ideals and character represent what is to be explained by the survey and the history. This admittedly difficult attempt is not wholly successful, but the prologue as a whole—which is divided into a superb section on "Physical and Economical Conditions," one on "Popular Characteristics," and three on the "Mind," respectively, of New England, the South and "the Middle States"—is, however, a tour de force. Geography initially dominates the reader's impressions, as according to Adams it still not only shaped but interposed massive barriers to the progress of American civilization and national unity. He sees the land not, like Parkman, romantically, except in so far as there is a kind of romance in mastering it, but as presenting formidable obstacles, of which he gives many examples, to intercommunication within the recently founded republic.
Adams emphasizes the primitiveness and parochialism of much of American life. Drawing on his knowledge of medieval Europe, he compares the situation of the settlers in the western states (as they then were) with those of the Angles and Jutes of early England, while "Even in New England the ordinary farmhouse was hardly so well-built, so spacious or so warm as that of a well-to-do contemporary of Charlemagne." If to Parkman civilization sometimes presents itself as a kind of steamroller or juggernaut, to Adams it seems in 1800 more like a cart jolting over rough ground, with its wheels often sticking in the mud. The United States, politically unified, was a raw society whose scattered elements made its disparate character more evident than its unity. Even in the long-settled east the delays, hazards and hardships of travel between the major cities were formidable and the average journey times discouraging: the regular fast mail route from Maine to Georgia habitually took twenty days. The "hero" of this part of Adams's book, and even, symbolically, of the book as a whole, was the steamship designed by Robert Fulton, which had its maiden voyage in 1807. It brought the promise of easy and regular travel for passengers and freight through the great waterways, the rivers and lakes, of America. Thus it carried unification and the efficient exploitation of the continent's resources on its decks. Adams could not resist a triumphant jeer: compared with this "the medieval barbarisms of Napoleon and Spencer Perceval signified little more than the doings of Achilles and Agamemnon."
The only relative failures in these admirable surveys, at once detailed and central to the imaginative economy of Adams's book and the contrasts it draws, are the finale of the prologue and the corresponding one on "American Character" at the very end. As an essayist attempting to grasp this tantalizing but elusive phenomenon, Adams is decisively inferior to contemporaries like Walter Bagehot and Taine (in his _Notes on England,_ 1862). It seems surprising that he nowhere mentions Tocqueville's _Democracy in America_ (1835 and 1840). In an attempt to preserve a stance of objectivity, Adams for much of the time hides behind the positive and negative opinions of others, which he juxtaposes but hardly manages to compose into a satisfying picture. From him we get scattered judgements rather than anything like a synthesis or a novel and coherent interpretation.
In considering such a long and in many ways highly impressive achievement as Adams's history it may seem perverse and unfair to dwell on a particular weakness. This is in a sense true, but there is evidence that these sections, relatively short as they are, far from being peripheral are for Adams the most important of all. They are the testing ground (as the manoeuvrings of politicians to which he devoted so much more space could not be) for a scientific historical explanation of social development, to which he aspired but which he spoke of rather than practised. Adams's enthusiasm for this possibility, and for the concept of "science" generally, appears to have had a number of sources: Darwinism, of course, but also, it often seems, Herbert Spencer and Taine, as well as the attempts at a science of history presented by Auguste Comte and by H. T. Buckle. It results sometimes in portentous and not particularly lucid pronouncements not characteristic of him as a historian. He wrote to a friend, "History is simply social development along the line of least resistance" the "line of least resistance" is part of Herbert Spencer's general formulation of the idea of evolution, not Darwin's. In his _History,_ Adams wrote that the great men of an age helped "more or less unconsciously to reach the new level that society was about to seek." For Adams there was a conceptual connection between "scientific history" and the activities of "the average man" which made American history a particularly promising field. Despite his lack of enthusiasm for democracy as a political form, he wrote to Parkman that "the people" was the only subject for history; its "fixed and necessary development was to be revealed by psychology, physiology and history" (this, in a different order, seems to be Taine's " _race, milieu, moment_ " again).
Adams provided a fuller statement in the epilogue to his history: "Should history ever become a true science it must expect to establish its laws not from the complicated story of rival European nationalities, but from the economical evolution of a great democracy." North America provided the best prospect "for the spread of a society so large, uniform and isolated to answer the purposes of science." There the scientific historian would be able to study "a single homogeneous society...under conditions of undisturbed growth." Apart from registering that it is not much of a historical science that backs away from "the complicated story of rival European nationalities" (which makes up a good deal of its modern subject matter), we can recognize here a recurrent nineteenth-century inclination to establish a particular nationality as a norm—Max Weber's later concept of an "ideal type" would be useful here—whose history would constitute the central line of historical development, from which deviations could be identified. We find this in Guizot's _Civilization in Europe_ and in Buckle's _History of Civilization in England_ (1857–61), presented, in Buckle's case, specifically as a basis for scientific historical explanation; Buckle drew an analogy with the statistical mean. Adams was now proposing the United States, because it showed "the steady growth of a vast population without the social distinctions that confused other nations." ("Confused" seems rich.) The outcome was to be "to define national character," but it cannot be said that Adams got very far with this. This is not to say that his interpretations were particularly ill-judged, but that they are ill-matched with the proclaimed, wider, scientific purpose.
The rhetoric of "science" was becoming a historical commonplace from the 1880s onward. Not all historians went beyond a defining notion of "objectivity" to embrace that of "laws," as Adams clearly wished to do; Acton did not. But Adams's commitment to objectivity was becoming a convention. The historian, he said, "should study his own history in the same spirit and by the same methods with which he studied the formation of a crystal" he means, presumably, by the close and objective examination of documents, as practised by many of his contemporaries and juniors. Actually, though Adams generally adopted a dispassionate, analytic stance, and wrote a prose that was restrained and even austere compared with Prescott's or Parkman's, his _History_ is some way from the neutrality of crystallography, and we may be grateful for it. Adams has a vision of the new United States, its difficulties and potentialities, as Parkman has one of the American forest. The Louisiana Purchase and the war with Britain are not merely the salient events in the two presidencies he has undertaken to recount: they represent the future of the United States on the continent and a vital stage in its unification as a nation.
The Louisiana Purchase, the negotiation of which Adams presents as a tense drama with amusing aspects, finally settled the question whether Spain, France or the United States would dominate the continent in the territories beyond the Mississippi. The war with Britain—not so much the war itself as the willingness to accept war rather than knuckle under to British demands to stop and search American ships and confiscate cargoes—he sees as the first great test of American nationhood. The countermeasures against Britain, which severely but selectively damaged American commerce, with New England the chief loser, placed the Union under great strain. (Adams, of course, was writing after the much more severe test of the Civil War.) The negotiations with Britain were long and tortuous, and Adams approaches them from both sides. America's resistance and the war were for him the demonstration that the nation existed and that the Union could act in defence of its national honour and its vital interests; anything less would have been shameful and would have shown that the republic still fell short of nationhood. In his characterization of American attitudes in the approach to war, Adams's own neutrality drew the line at those who put sectional commercial interest before patriotism. He clearly took much pleasure in such successes as the war brought, particularly in naval actions, and was scathing about the conduct of military ones. For him the war was the culmination of the process of feeling not Englishmen in America, not members of a group of associated states, but Americans. Patriotism, not naive or blatant, a sense of continental destiny, and a strong feeling for the integrity of the Union are the guiding undercurrents of Adams's _History._
Adams's attraction to the notion of "scientific history," as we have seen, was an aspiration only, and one he shared with numerous others. Adams was born to be a mandarin, or rather Brahmin, but the world was changing, and his role with it. By the end of his teaching career at Harvard he was behaving like a "professional" historian. He was training, to use the fashionable phrase—"baking" he more colourfully called it—future historians (see Chapter 25). The number of history teachers in American universities was growing rapidly; the _American Historical Review_ began publication in 1895. Adams was president of the American Historical Association in 1893 when Frederick Jackson Turner pointed out that the open frontier was now closed and asserted its central importance in the shaping of American society. Harvard was losing its near-monopoly; Johns Hopkins was another nursery of the new professionalism. Turner himself grew up in Wisconsin, and his famous "frontier thesis" was a socio-economic challenge to the Anglocentric notion that American democracy had been shaped by its Germanic and English heritage. Another step in the same general direction was Charles A. Beard's heretical _An Economic Interpretation of the American Constitution_ (1913), which saw the move to independence as a struggle against British mercantilist economic policy, and the Founding Fathers as essentially concerned to secure private property against "levelling" tendencies. From the 1880s it was—as one of the marks of the emergence of a historical profession—an age of "theses." One of the most notable was A. T. Mahan's _The Influence of Sea Power on History, 1660–1783_ (1889), which, unusually for a work of history, was to exercise, inadvertently, a baleful influence on European power politics by fostering Kaiser Wilhelm II's determination to build Germany a navy to challenge Britain's.
The lifetimes of Prescott, Parkman and Adams span a century and a quarter (Prescott was born in 1796; Adams died in 1918). The sequence of their works has allowed us to consider in turn the conquest of Mexico by America's first European colonists, the Spaniards; then the virgin land of North America, its indigenous inhabitants and its earliest European, i.e. French, explorers; and finally the emergence—slow, partial and at times painful—of an American national consciousness, which Adams saw as still being created in the early nineteenth century. But there is another sequence, a cultural one, affecting the ways in which history was written. Prescott's work, closest to the eighteenth century, is aptly described as "neo-Gibbonian" (Adams also greatly admired Gibbon, but influence is indiscernible); Parkman's belongs unmistakably with the literature of Romanticism, with Wordsworth, Scott, Byron and Lermontov, and in America with Fenimore Cooper and Thoreau. Adams, the only university teacher of the three, is, in his pronouncements about history and in his later career, part of the rapid professionalization of history, in Europe as well as America, towards the end of the nineteenth century. It is this that we now have to consider more generally, as the chief influence on the writing of history from then on.
**TWENTY-FIVE**
**A Professional Consensus: The German Influence**
**Professionalization**
The multi-volumed, collaborative _Cambridge Modern History_ began publication in 1902. The prospectus, by its first editor, Lord Acton, has become famous for its commitment to objectivity and impersonality in historical writing: "Contributors will understand that...our Waterloo must satisfy French and English, German and Dutch alike; that nobody can tell, without examining the list of authors, where the Bishop of Oxford [Stubbs] laid down his pen, and whether Fairbairn or Gasquet [a Catholic cardinal], Liebermann or Harrison took it up." Readers here may enjoy an unintended ironical reminiscence of the anonymity and serial contributions of the medieval chroniclers, but to contemporaries this represented the acme of modernity in the writing of history and the ethics of the historical profession. (Acton, incidentally, was in the narrow sense no professional, but an aristocrat of private means, who taught in a university only at the end of his life.) Even more eloquent than the prospectus was the enterprise itself, a multi-authored compendium of international scholarship, testifying to a belief in the collaborative and cumulative character, appropriate to a science, which history had come to assume. In that sense the series was a product of several decades of growing European (and also now American) self-consciousness about the requirements of rigour and objectivity in the scholarly practice of history. The moment seemed ripe for such a project, and Acton its ideal editor. He was qualified by his cosmopolitanism as well as by the acknowledged range and profundity of his erudition and his frequently proclaimed emphasis on the critical treatment of sources as the defining characteristic of a historian.
The development of a historical profession in the most advanced countries in the later nineteenth and early twentieth centuries was part of a more general process of professionalization and specialization as middle-class education expanded, which of course also created increased opportunities for teaching careers. Specialism was an obvious and, though sometimes deplored, inevitable response to the rapid growth of knowledge, which itself was both a cause and a consequence of the research ideal. In the natural sciences it had in some cases an obvious utility, which other learned professions came to envy and claim a share in, as well as sometimes loftily to repudiate. History was an ancient intellectual practice, but not one, until the early nineteenth century—Göttingen was the forerunner—with a firm university teaching base, apart from a scattering of endowed professorships. It benefited, though it had sometimes to fight for its independence, from close association with the more ancient academic disciplines of classical scholarship and law, and was thus well placed, compared with modern literature, sociology or anthropology, to become a major player in the expanding academic world. The old tradition of "pragmatic history," as proclaimed by Polybius (Chapter 4), could be refurbished to support the idea that history was useful in the education of statesmen and civil servants. It might also be able to foster patriotism, a national consciousness and consensus, opposing ultra-radical and socialistic tendencies.
The character of the most advanced and industrialized states at the end of the nineteenth century was, with local variations, increasingly bureaucratic, and the organization of education, and even research, was part of this. Earlier in the nineteenth century it had been a patriotic act for governments to sponsor the serial printing and publication of medieval manuscripts of national importance (there was some commercial, private printing as well, such as the Camden Society edition of Jocelin's chronicle used by Carlyle). The acknowledged and sometimes envied precursor was the _Monumenta Germaniae Historica,_ which began publication in 1821, sponsored by the Prussian statesman Karl von Stein. Guizot, when he became a minister in the 1830s, was able to foster a similar enterprise in France. Such series became in a sense schools of research and (not always without difficulty) establishers of common scholarly standards. Stubbs acquired his scholarly training for himself by working as an editor of documents for the equivalent English enterprise, the so-called Rolls series.
The growth and prestige of history as a teaching and research profession was most advanced in Germany. The organization of historical faculties as miniature bureaucratic hierarchies was longest resisted in Oxford and Cambridge, particularly the former, where the older notion of the individual academic freehold, analogous to the clerical living, to which many Fellows of colleges succeeded, long retained a hold. Tensions ensued between professors who wished to organize "their" faculties, as on the Continent, and Fellows of the quasi-independent colleges. These tensions also tended to coincide, though not absolutely, with those between research and pedagogical models of the historical profession. On the Continent and increasingly in the USA and in the new English provincial universities, relations between professors and research students, i.e. the next generation of historians, tended to be closer and to be central to the self-image and practice of the profession, retaining an element of clientage in an otherwise bureaucratic structure.
Though there were also ancillary professionals such as archivists, the core of professionalization was the growth of paid posts in universities, combining teaching and research. More or less autonomous historical faculties grew up (in Oxford and Cambridge, as independent faculties, from the early 1870s)—sometimes, particularly initially, in association with other disciplines—with provision of training for research. In England, written examinations as a requirement for graduation, with classified lists of candidates, gave control over the content of the syllabus to those who set them. The existence of such university history schools both nurtured and in a sense presupposed a professional consensus about standards of research (as the qualification for appointments) and about its presentation, with the maintenance of sobriety—some said dullness—and a neutrality of tone becoming de rigueur. The results of research would characteristically be presented, at least initially, in the specialized academic journals recently founded for that purpose, whose editors were in a position to enforce such standards. Germany was again the precursor: the _Historische Zeitschrift_ began in 1859, and an earlier publication, the _Historisches Taschenbuch_ ran from 1830 to 1892. The _Revue Historique_ was founded in 1876—French historiography was heavily influenced by the example of Germany after 1870. The _English Historical Review_ began in 1886, and the American Historical Association was inaugurated in 1884. The _American Historical Review_ followed in 1895.
Consensus not only on how history should be written but on what history was about was not just a product of enthusiasm: for the aspiring professional it was enforceable. In the later nineteenth and early twentieth centuries history meant above all political history, including, notably, in modern history, a focus on the relations of states in the European inter-state system. It also included constitutional and legal origins and, with increasing prestige, economic history. The only German historian to whom Acton gave as much prominence as to Leopold von Ranke in his article on "German Schools of History" for the first number of the _English Historical Review_ was the economic historian Wilhelm Roscher; one of the handful of university lecturers in history appointed in Cambridge in the same decade to teach for the recently (1872) independent History Tripos (School) was the economic historian William Cunningham. Social and cultural history, despite the example of Burckhardt, were decidedly poor relations. To most historians these priorities no doubt seemed self-evident, as it also seemed that history, in acquiring professional recognition and organization, had found its identity. The long history of historical writing and inquiry from Herodotus onward had reached its terminus: Clio was unveiled and holding a chair, probably in Germany.
The rhetoric for expressing this in the period was, of course, that of "science": history, properly practised, was an objective and cumulative form of knowledge, the accumulation of results of the industry of many dedicated professionals. More would, of course, be learned, but modern historical practice was, it was sometimes claimed, itself the result of a "Copernican revolution," in the diffusion of obligatory standards for the critical examination of sources. It was reasonable to expect no subsequent revolutions: one cannot be more scientific than science. History was too innocent of overarching theory, other than the generalized exhortation to critical rigour, for the results of research to produce seismic theoretical consequences such as those beginning to be felt in physics from Clerk Maxwell to Einstein. How had this consensus on how to study history and, more problematically, what history was itself about become established? It was not, after all, much like the recent developments we considered in Chapter 22 above, with their dramatic narratives and increasing focus on "the people" as a historical protagonist. To begin to answer this we have to turn to Germany.
**German Historicism: Ranke, God and Machiavelli**
From the early years of the nineteenth century onward German scholarship, and particularly historical scholarship, enjoyed immense prestige. Its organization and productions were envied and to some extent imitated in France, Britain and the United States. Associated with that prestige was the reputation of Leopold von Ranke, widely regarded as the doyen of the German historical profession on account of his longevity (1795–1886), his strategic situation as professor of history in Berlin, the distinction of his pupils, and the sheer impressiveness and volume of his historical works (sixty, in the collected edition). His life almost spanned the century, and at its end his pupils, though some deviated up to a point from his interests and precepts, dominated the German historical profession.
German historiography had been shaped from the medieval period onward by the fact that, like Italy, Germany was seen as a nation but not, despite the German base of the Holy Roman Empire, a state: in comparison with Italy, however, the power of the Emperor and the princes was greater than that of the "free cities." Germany as a whole had never been subjected to Roman rule, so for both these reasons ancient Roman civic republicanism could not have the same resonances as in Italy. Tacitus' account of the Germanic tribes and their villages became the ancient sourcebook for German history. The results are understandable: on the one hand an imperial perspective, that of Christendom or universal history, and on the other a focus on the local, including but also other than the city—the principality, or else the basic institutions of local rural society in which were thought to reside the residues of the freedom described by Tacitus.
Then, in the mid nineteenth century, came a watershed, corresponding to the crisis in German liberalism after it became apparent from the failure of the all-German parliament in Frankfurt in 1848 that the unification of Germany, desired with increasing ardour, could be accomplished only by Prussian or Austrian power. First the idea and then, after 1871, the fact of a German nation state became mesmerizing. Local particularism was less cherished or was even denigrated; the universalism of the eighteenth century and the Enlightenment and the notion of an overarching natural law, binding even on sovereign states, were thrust aside. A new Machiavellianism, _Realpolitik_ —the term coined in 1851 originally to express the idea of unification only by power, but coming to legitimate the pursuit of national goals by any means—became current. In the work of the so-called "Prussian School," which included Ranke's pupils Heinrich von Sybel and Heinrich von Treitschke, German history came to be seen teleologically (as English liberals saw the history of their Parliament), with the Prussian ruling dynasty, the Hohenzollern, as the predestined instrument of unification. (Though his standards of scholarship were much patchier, one should probably annex Carlyle to the canon of the Prussian School; Goebbels read his life of Frederick the Great to the Führer, as an aid to morale, in the Chancellery bunker in Berlin in 1945.) Acton spoke of the Prussian School as "that garrison of distinguished historians that prepared the Prussian supremacy together with their own, and now [1886] hold Berlin like a fortress." But not all Ranke's pupils travelled this route. They also included the medieval constitutional historians of an older persuasion Georg Waitz, Rudolf von Gneist and Reinhold Pauli, while Burckhardt too was a pupil, though a deviant one.
In the nineteenth century, after the Napoleonic invasion of 1806, in which the Prussian state had crumbled, and then in the so-called "national awakening" of 1813 and the renewal of the war against Napoleon, the conception of the state moved centre stage, though without displacing that of the nation. _Völkisch_ scholars (see Chapter 23) had initially regarded it with aversion, as a lifeless machine clamped down on the spontaneous creativity of the _Volk._ The eighteenth-century German courts were French-oriented in their culture, and the model "Enlightened" states were despotic and highly rationalized. It is not surprising that the first analysts of bureaucracy were German. Even the state's admirers focused on its mechanical artifice and efficiency. But as, after the 1813–14 "war of liberation," the concept of the nation became increasingly politicized, the "Idea of the State" was gathered into the thought world of Romanticism; the State as conceived by Fichte and Hegel—as the embodiment of ethical life, a spiritual agent—became, in the metaphysical technical term derived from Herder, not just a machine but a historical "Individuality," the complement of Herder's concept of the Nation, also a unique Individuality. The two concepts achieved a kind of fusion in the idea of the nation state.
Participation in the nation state did not entail democracy, or anything like it, but an intensification of the consciousness of belonging to a higher spiritual entity, and identification with it. The State represented the individual's highest self. Hence the State was not a mere instrumentality but a spiritual reality, a self-realizing Idea, which, in the European states system, contended with other states, other Individualities, in its quest for survival and self-fulfilment, and was energized and schooled by these encounters. There is an echo here of the Roman and Machiavellian notion of the vitalizing effects of conflict and the dangers of tranquillity (Chapter 6, Chapter 18). For Hegel, in the last part of his _Philosophy of Right_ (1821) (Chapter 21), it was crucial that the State, in war, could call on the citizen to sacrifice his life. War was no longer, as in the eighteenth century, an affair merely for mercenaries. The State's right to the individual's life was, according to Hegel, the definitive demonstration that the State was not just an instrument for his protection (the contract theory), or for the production of welfare (Enlightened Despotism), but a higher spiritual entity than the individual. The requirement of his life was not tyranny but self-sacrifice, submission to one's own higher will and participation in the life of a higher entity.
German liberal intellectuals in the first half of the nineteenth century became increasingly irked by the political fragmentation of Germany into a multiplicity of states and were impatient for unification. Treitschke even at one point contemplated a Prussian invasion of the princely states to unify Germany by force. The conception of the nation state, striving to fulfil itself, as the supreme modern historical Idea became central to the nationalists' conception of modern European history; many German liberals had adopted the label "National Liberal," a significant addition to the noun. European history was the encounter and interaction of self-moving, autonomous spiritual Individualities. Germany, it was argued (not by Ranke), could not become a full player until the nation was politically embodied in a single state, whose need for fulfilment overrode both the interests of individual citizens and the dictates of a law of nations grounded in natural law.
From the mid nineteenth century onward, in the work of Ranke and his pupils, there was a renewed fascination with the idea, formulated in sixteenth-century Italy and echoed ever since, of the interaction of European states as an unstable balance of powers; this was seen as the keynote of modern European history. Much of Ranke's work was devoted to examining this, from the sixteenth century onward, beginning, in the early modern period, with his _History of the Popes_ (1834–7). States as Individualities in history were to Ranke thoughts in the mind of God, whose presence was continuously sensed. God's thoughts dwelt much in "the Great Powers" (the title of an essay by Ranke). Ranke was not a philosopher, and affirmed the superiority of history to philosophy. We do not find the metaphysics which clearly in some sense suffuses his writing laid out systematically: "the plans of the divine government" to which he refers ( _History of the Popes_ ) are implicit, not spelled out in detail, and are glimpsed by the historian, not abstractly summarized as in Hegel's philosophy of history. Nevertheless, they are sometimes unignorable, as in the guiding idea of the existence of unique historical Individualities. It was the faith in these points of access to the divine mind which underwrote the labour Ranke devoted to reconstructing, on a scrupulously investigated documentary basis, what to him were the central themes of European history. It was the historian, and he alone, who could discern the hand of God in unique historical configurations of events and forces. This is the core of what is sometimes called "Historicism" ( _Historismus_ ). Ranke repudiated Hegelian teleology, which embodied the idea of successive stages towards an eventual consummation in history, which philosophy could grasp. He did not dissent, however, from Hegel's rationale for the power relations between states; certainly he did not deprecate war.
The link between the commitment to exhaustive, exact scholarship and the underlying metaphysics was the notion that the historian's business is not with abstractions but with the unique spiritual entities, the Individualities concentrated as states, which are his protagonists and with the dense texture of their relations with each other. The understanding of history could not be teleological, as in Hegel, because every age, as a famous Rankean aphorism put it, was equally "immediate to God" and therefore required not to be placed in a sequence (though the idea of "universal history," to which he devoted his last years, was always a brooding presence in Ranke's work), but detailed, objective investigation. This repudiation of historical teleology brings Ranke closer to modern ideas than some of his contemporaries; his confidence in the possibility of objectivity, of seeing, in another famous aphorism, "wie es eigentlich gewesen" ("how it really happened"), which was then common, does not.
Ranke's _oeuvre_ was vast (most of it is untranslated into English), and it is hard to summarize a modern consensus on him because it is not clear that one exists. Almost everything seems contested: his almost exclusive devotion to political and, above all, "diplomatic" history; the extent of his influence; the importance to him of metaphysical or mystical notions; even the quality of his narrative, which was much praised for its Olympian detachment. Acton called it "colourless," and probably meant this as a compliment. He wrote that "Ranke is the representative of the age which instituted the modern study of history. He taught it to be critical, to be colourless and to be new. We meet him at every step, and he has done more for us than any other man." But it can hardly be complimentary when, in his review (1867) of Ranke's work on English history, Acton says that "scenes which Macaulay had made as vivid as anything in epic poetry, are described with elaborate dullness," or speaks of his later works as "tame and frigid."
The extent of Ranke's overriding preoccupation with the affairs of states is the most serious of the disagreements about him, because it affects the general argument being made here. It is not advisable to be dogmatic about sixty volumes: it can only be said, impressionistically, that it was so—counter-arguments seem to run the risk of making much of sightings of occasional swallows. Acton also said that Ranke had produced more excellent works of history than anyone else, but no single masterpiece. This is perhaps an assured route to esteem and influence among one's contemporaries; it is less recommendable if one wants to captivate readers a century and a half later. For most such readers, especially English and American ones, it seems most realistic to try to understand Ranke as an influence, a model and a portent rather than a still-living author. To do so is perhaps, in any case, an implication of the idea of a historical profession and its cumulative body of knowledge.
**Not Quite a Copernican Revolution**
In his Cambridge inaugural lecture, which we have already visited, Acton introduced his hearers to the new era, for which Ranke archetypally stood: "The accession of the critic in place of the indefatigable compiler, of the artist in coloured narrative, the skilled limner of character, the persuasive advocate of good, or other, causes, amounts to a change of dynasty, in the historic realm. For the critic is one who, when he lights on an interesting statement, begins by suspecting it." The new dynasty was the historical professionals, with their defining craft, the critical examination of sources, and Ranke (Acton had attended his lectures in Berlin) was its doyen.
The inaugural lecture was a rhetorical occasion calling for piety and exhortation rather than dissent. Nine years earlier, in his article in the new _English Historical Review_ on "German Schools of History," Acton's view of Ranke had been more qualified. Now he simplified and therefore somewhat exaggerated the novelty of the historiographical revolution in critical scholarship he attributed to the nineteenth century. He recognized that its principles had been well understood earlier; he was on safer ground in claiming that it was a novelty to apply them in writing narratives of modern history, which also depended on the opening of state archives, but this was a qualification easily forgotten. It was also important to apply them comprehensively and to seek out manuscript sources systematically, not sporadically, which he accused even the younger Ranke of doing. To consult _all_ the relevant sources is good advice, but it is hardly a Copernican revolution. That primary sources are to be preferred to secondary, and that misrepresentation, ignorance and even forgery are all always possibilities; that secondary sources are therefore to be rigorously tested for possible motives to distortion and for access to the truth—these were, as he admitted, not a mystery but more like common sense.
The distinction between primary and secondary sources is recognized as a crucial one for historians. Essentially it is that between a document by which something was done, like an order, a commission, a charter or a contract, and a commentary or narrative. In the first, truth and falsehood are not the issue (apart from forgery), though good faith and intention may be, and meaning certainly is. As Acton said, "The chronicle is a mixture of memory, imagination and design. The charter is reality itself." Another primary source is documents in which agents attempt to influence events, which therefore form part of what the historian investigates; the question whether the agent told the truth is probably secondary to questions of intention and efficacy. In secondary sources—typically narratives and commentaries—the reverse is the case. The most obvious and vital question is whether they are true. There is also, of course, in assessing them, a relevant distinction between eyewitness testimony and hearsay. There is an analogy between the practice of the critical historian and the passion of the fifteenth-century humanists for recovering the purest, least contaminated version of a classical text: the suspicion with which a historian like Ranke regarded chroniclers and previous historians parallels the humanists' scorn for the medieval commentators.
The distinction between primary and secondary is a flexible one, however. The same document may be both, as a commentary may be written as a political act. A document that is secondary in one kind of inquiry may be primary in another. Ranke apparently liked to form an impression of the personality of the source of a document, as part of an assessment of its trustworthiness. The biases of the chronicler stood as a kind of screen between the historian and the truth. But in Germany, in particular, in the study of the early periods it had been fully acknowledged that myth might be treated as forming a part of their history. This had inspired some highly influential scholarship and speculation, including, for example, one of the seminal works of the nineteenth century, David Friedrich Strauss's _Life of Jesus,_ in which the Gospels are dismissed as history but reinstated as evidence for the first-century Jewish mind. But in searching for the truth of mainly modern history it seems not to have occurred to Ranke (though Acton was aware of it in principle) that the lenses through which the chroniclers looked at the world were themselves material for history, part of the times in which they wrote, and in that sense potentially primary sources. This reflects the primacy assumed in the Rankean world by high politics—chroniclers were small beer, useful or unreliable servants—and of the assumption that facts and events were what history was essentially made of. Chroniclers were recorders of these, not part of history themselves. We find the same attitude in so distinguished a medievalist as V. H. Galbraith, writing in the 1950s on medieval English chroniclers as irritating colleagues. Any historian now old will probably have met this attitude—or shares it.
Much of Ranke's influence seems to have been personal, through the systematic way he trained his pupils in his famous Berlin seminar in the practice of the critical use of sources, with discussions of documentary examples. In this, as Acton knew, he had been anticipated by classical philologists, who also taught in this way. An American student recalling his days in Berlin at this time wrote an enthusiastic reminiscence of his seminar experience: "There the student appears, fortified by books and documents borrowed from the university library, and prepared with his brief of points and citations, like a lawyer about to plead a case in the court room...Authorities are discussed, parallel sources are cited; old opinions are exploded, standard histories are riddled by criticism." In this enthusiasm for a system now familiar—even if it does not always generate such breathless excitement—one can hear the doors of a new pedagogical experience opening. Henry Adams, who, like most notable nineteenth-and early-twentieth-century American historians, had trained in Germany, introduced such a seminar at Harvard in 1871 (there was another at Johns Hopkins) and supervised several of the earliest Harvard PhDs in the early 1870s, half a century before such seminars began to be introduced in England.
Much distinguished history was produced, of course, by Ranke and his pupils. In the process, the kind of history on which they chiefly focused became virtually definitive of modern history. The professional imperative to test particular kinds of source for their truth to events, joined with the contemporary political interests and pressures we considered earlier, promoted a focus on the uncovering and dissection of state papers, some only recently available, as the historian's central task. Other kinds of history, apart from constitutional, which had its own firm base among medievalists, were marginalized, though in any case the emphasis on political history and foreign affairs was a continuation of the long-standing ancient and humanist conception of what was proper to the dignity of history. There was also in the nineteenth century a reciprocal relation between the assumed role of the historian in ferreting out secrets and a history concerned with the intentions of statesmen: the closet politics of the sixteenth century and the _Ancien Régime_ ; the pursuit of covert diplomacy (the adjective is almost redundant). On all these Ranke wrote extensively. The more covert the policies, the greater the challenge to the professional historian to uncover them by investigating archives. The first book of documents on which Ranke worked closely was a collection of Venetian ambassadors' reports from the sixteenth to the eighteenth century, which seem to have become something of a fetish and a paradigm for him; even contemporaries sometimes complained that his history was excessively princely.
There were costs, as Peter Burke points out, claiming that Ranke took history away again from the broadened "civil history" of the eighteenth century, and returned it to the more restricted subject matter of humanist history. Burke also draws attention to the breadth of the interests of the Göttingen school of the later eighteenth century, "cut off in their prime by the historical revolution associated with Ranke," though he admits that Ranke's own interests were more eclectic than he is sometimes given credit for.
The point can be given further amplification. There is a marked contrast with the "sentimental" and dramatic treatments of events cultivated in the eighteenth century, as well as with Carlyle's and Michelet's demotic kind of history of the French Revolution. It was noted earlier (Chapter 21) that writing history for the market included writing for women. The emergent profession, however, was independent of the market, and was masculine in its ethos and largely in its composition; the smell of pipe smoke still clung to it in the 1950s. Its prejudices were to be long-lasting and restrictive, narrowing the illumination the study of the past could bring to a culture. In general, overly triumphant versions of the nineteenth-century "revolution" in historical studies were an aspect of, and helped to feed, what we have already noticed: the century's apparent ignorance of the scholarly innovations of the sixteenth and seventeenth centuries and its systematic denigration of eighteenth-century historiography. The notion of a nineteenth-century "Copernican revolution" reinforced an enduringly distorted version of the history of historiography, slanted towards the nineteenth century and Germany, which the present book has attempted to correct. The professional consensus of which this notion formed an important, and restrictive, part was to be a tough and lasting one, persisting well into the twentieth century. The methods of examining documents, of course, remain the same now as then, and as they were earlier still.
The rhetoric of "objectivity" and "science" did eventually become muted or abandoned in the twentieth century, with greater awareness of the complexity of the issues the terms raised and some awareness of their costs in the writing of history. The hierarchy of esteem for various kinds of history, however, proved more tenacious, with political history and international affairs sharing some of their prestige only with institutional history, chiefly medieval. Social and cultural history stood somewhat on the margins. Intellectual history, if it existed, was probably someone else's business. German Idealist metaphysics and English pragmatism ironically concurred in the primacy, in the modern period, of political history. The English took the centrality of Parliament—and earlier of their kings—for granted because it had occupied that position in English life for so long. The Germans exalted the state because, as the nation state, it had arrived for them so recently. (This occurred in an age when, in historical scholarship, France and the United States, whose intellectual traditions were different, became something like clients of the Germans.) How the professional consensus around the primacy of political history and the dedication to a "scientific" ideal of objectivity began to disintegrate we have to consider in the next chapter.
**TWENTY-SIX**
**The Twentieth Century**
**Professionalism and the Critique of "Whig History": History as a Science and History as an Art**
In this final chapter I have attempted a necessarily brief account of the main innovative developments in twentieth-century historical writing, of which there are many, and have tried to show what is interesting or valuable about them. There is an obvious danger, which I have consciously resisted, of any more comprehensive attempt growing disproportionately to the rest of the book, if only because so much more history was written during the twentieth century than in any previous one, and a general survey of even the most impressive historical writing the century produced could issue only in a kind of opinionated bibliography. A list of important and original twentieth-century historians who are not mentioned here would therefore be a very long and distinguished one. It would include almost all—even the most original—who contributed to genres which were established and familiar by the later nineteenth century, including the study of high politics and constitutional, diplomatic and military history. The attention given in this chapter to social and economic history, however, which assumed much importance in the middle and later twentieth century, is not meant to imply that there were no notable earlier examples, as there were for cultural history, which has also risen greatly in prominence in the twentieth century. Precedents, as noted in earlier chapters, have included, for example, the sixteenth-century French advocates of "perfect" (total or comprehensive) history; the Enlightenment interest in the "History of Civil Society" or civilization, and the flourishing study in Germany, particularly in Göttingen at the end of the eighteenth century, of "universal history" Vico's pioneering ideas about cultural history, and later the influential thought of Herder and the work of Michelet. To a greater extent therefore even than in earlier chapters, I can make no claim to completeness, but I am consoled by the thought that readers will probably be more familiar with twentieth-century historical writing than with that of previous eras.
It is important to recognize at the outset that what is intended in a distinction between new or relatively new and more traditional types of history is not a contrast in quality. Not all innovative work is good, and far from all the work in well-established genres is routine. If in (roughly) the last third of the century in particular a hundred historiographical flowers bloomed, some very soon also wilted. I have therefore tried in this chapter only to consider examples of novelty which were influential or representative, as well as some I find particularly impressive. I absolutely do not conceive it as my business to tell my fellow historians how to do their own, or to offer rebukes or prizes, issue prescriptions and make predictions, any of which would have given the book at the end a polemical character alien to its purposes. Meanwhile the older genres continue, sometimes in novel guises, and we should be grateful for it, though I have no space here to express proper appreciation of them.
In surveying the history of historical writing in the twentieth century it is tempting to try to relate the various forms it most typically assumed in the various European countries and in the United States to those countries' recent historical experiences. Any suggestions along these lines are bound to be subject to qualification, but it does seem possible to make out some broad outlines. As we shall see, the notably innovatory kinds of history produced in France have involved moves away from the focus on the nation state towards everyday life, as well as downwards to the regions and villages and outward to the history of the world. This seems to reflect the decline of France, earlier than that of Britain or Germany, as a great power, and disillusionment with the notoriously centralizing tendencies of the French state inherited from Richelieu, Louis XIV and the Revolution. Historians in the United States have naturally been drawn to study the creation of their nation and its testing in the ordeal of civil war, and have had to confront the steps by which it became the world's superpower; this has meant calling on the resources of traditional genres of political history, including the history of foreign policy, in which the United States has enjoyed an exceptional freedom of choice. The sense has been that America represents the world's future, so that to write of America is to engage with the history of the world.
In Germany, attention to the steps to the nation state, which riveted the attention of nineteenth-century historians, has necessarily been succeeded by attempts to understand its two great catastophes in the twentieth century, in 1914 and 1933, providing explanations in the long or the short term, again drawing predominantly on the well-established genres of political, diplomatic, military and to some extent national cultural history. Though it and the largely institutional reasons for it form too large a subject to be dealt with here, account has to be taken too of the fact that much internationally influential thinking about the major social and cultural transformations of Europe from the early modern period has been conducted in Germany under the label of sociology rather than history. It is enough to mention the names of Max Weber, Werner Sombart, Georg Simmel, Theodor Adorno, Norbert Elias and Jürgen Habermas.
Britain has been receptive, since the 1970s, to developments elsewhere, notably in France. It has also been remarkable for the range of its attention to the histories of other countries. An interest in Germany and Russia in this period may be taken for granted, as may attention to Asia and Africa, of which Britain once ruled so much. No historical imperative seems to have dictated that English historians should have become leading authorities, even in those countries themselves, on the histories of Spain, Italy, Poland and Sweden, but such has been the case. One of the outstanding English historians of his generation, Richard Cobb, who devoted his life to the history of France in the revolutionary period, earned from _Le Monde_ the accolade of "l'étonnant Cobb."
In the early twentieth century what I called the professional consensus was essentially unshaken, but it was becoming critical of what in Chapter 23 was spoken of as the liberal conception of history as the story of a continuous growth of freedom. The new professional temper was cooler, and we saw an example of this earlier in the prefatory remarks of Charles Petit-Dutaillis to his _Supplement to Stubbs' Constitutional History_ in 1908 (Chapter 23). This conveniently marks the difference in period between Stubbs's work, with its enthusiasm for the freedom of the German woods and the continuities of parliamentary history in England, and that of its revision and supplementation over a generation later. Secularization was one aspect of the transition. Stubbs was devout. Frederic William Maitland, the outstanding and sharply sceptical legal and institutional English historian of the early twentieth century, was not. Stubbs, a cleric who passed easily from an Oxford chair to the bishopric of Oxford in 1884, had declared in his inaugural lecture in 1866 that the study of history was essentially religious; even then this had raised a few eyebrows. Not all histories of liberty were overtly confident of finding in it the presence of divine providence, as Stubbs's was. But that God had willed the English constitution or the freedom of the United States was a thought by no means alien to the mid nineteenth century.
It is true that militant secularism had tended in France to drive a wedge between religion and liberty—though not for Guizot—while in Germany in the age of Ranke the God who presided over history had characteristically more statist concerns, and endorsed _Realpolitik._ But in the later nineteenth century liberalism as a political creed was entering choppier waters. Universal suffrage and the prospect it raised of "socialism"—a loose term at the time, meaning any kind of collectivism or welfare legislation—were widely perceived as threatening. Economic protection in the service of nationalism, which made a motive for imperial acquisitiveness, eroded the central liberal orthodoxy of Free Trade. These were powerful challenges to liberal optimism, to which the First World War administered an even graver shock.
Liberal historical triumphalism was essentially a narrative; professional zeal for exactitude, exhaustiveness and causal explanation naturally tended to tilt the balance from narrative to analysis. The tension between the two was to become particularly evident in both Britain and France from the later 1920s, though it was already apparent in the eighteenth century, with explanations presented in disquisitions or digressions. As we have seen (Chapter 21), Hume made use of these, to Adam Smith's disapproval. The professional historians of the middle years of the twentieth century became notably drawn to technical explication and the search for long-term causes in ways which naturally diverted attention to an important extent from narrating the events themselves. This tendency came to full fruition with an increasing input from sociology, anthropology and Marxism (which inevitably stressed "underlying" changes which were the business of economic as well as social historians). The narrative of events seemed superficial by comparison. The French called it _histoire événementielle,_ contrasting it with the _longue durée_ of structural change; Marxists saw it as tracing only changes at the level of the political "superstructure," compared to those of the "real" economic forces and the formation of classes. The new watchword, in both cases, was "structure." In 1929 Lewis Namier, in his _The Structure of Politics at the Accession of George III_ , struck a powerfully influential blow at key aspects of the liberal grand narrative of freedom—particularly at the idea of the historical continuity of the two great English political parties, Whigs (the defenders of freedom) and Tories, and at the notion that later-eighteenth-century English politics had been dominated by a Whig defence of constitutional principles against a monarch, George III, bent on restoring royal power.
Lewis Namier—the names were a drastic Anglicization—was a Polish immigrant and a Jew. He arrived in England in 1908, young enough to spend student years at the London School of Economics and Balliol College, Oxford. He was to devote himself to the politics of eighteenth-century England—almost, it seems, as a kind of solace for the ideological illusions of nineteenth-century liberalism and Europe's chaotic twentieth-century politics, finding relief in an adoptive English patriotism and a period of history which he chose to see as devoid of ideological pressures. By the time Namier left it, party principle had virtually been swept out of eighteenth-century English politics altogether, leaving only the motives and manoeuvrings of perennial political man in an ideological vacuum. Namier was said to have emptied the ideas out of history. In doing so, with immense erudition in his chosen field, he also emptied history out of it—i.e. much of what was historically specific.
Namier's devotion to the narrowest of time bands may have been exaggerated by his failure, like that of virtually all historians, to carry out all his projects, but it seems highly unlikely that these would have failed to embody his enduring detestations, his scholarly inclination towards miniaturism, and his contempt for evocative narrative and for the history of ideas. It was an axiom with him that the real considerations at work were to be found in the private correspondence of ministers and Members of Parliament—relatively unlikely repositories for inspiring declarations of shared principles. That the movement of events was essentially dictated by individual motives and calculations and that political ideas were merely rhetorical froth was thus more or less circularly guaranteed by the particular choice of sources. Public utterances were prima facie discredited because they were public. Namier is an extreme case of the tendency for the devotion above all to manuscript sources to predetermine what was to count as real history.
Namier's characteristic method— _The Oxford English Dictionary_ gave its imprimatur to the verb "Namierize" in 1976—was the close examination, exhaustively carried out, of the lives and political connections of individual Members of Parliament—what is known also, when more generally applied, as "prosopography," which was notably used by Sir Ronald Syme in 1939 for the study of the later Roman republic and by A. H. M. Jones for that of the empire. Namier's initiative resulted in the collectively produced _History of Parliament_ (actually of the House of Commons), undertaken by a group of historians, including Namier himself, with government sponsorship. "Namierizing" was what "The History of Parliament" apparently _meant_ : nothing like Stubbs's liberty "widening from precedent to precedent" (Chapter 23) here. Namier's influence on the profession in England—but virtually only in England—was very powerful and formidably narrowing. George III was rehabilitated, however, and the concept of the continuity of parties, which G. M. Trevelyan had declared only in 1926 to be "a great fact in English history," was left very battered, though the concept of party itself, in the eighteenth century, proved not beyond rehabilitation in the hands of younger scholars.
A parallel attack on the tradition of liberal history to Namier's was Herbert Butterfield's essay—it was no more than that— _The Whig Interpretation of History,_ published in 1931. The two historians were entirely independent of each other, and Butterfield's relations with Namier were always to be tense and sometimes antagonistic, so the fact that they should have pointed coincidentally in the same direction seems all the more significant. Butterfield's essay was an attack on liberal triumphalism in history, for which he chose rather unfortunately to use the parochial English name "Whig." He identified it as anachronistic and unhistorical or even anti-historical. Whig history, in Butterfield's definition—there was virtually no illustration—purveyed a conception of progress as the central theme of English history, dividing historical agents into canonized forefathers and mere obstacles, depriving the latter of any real benefit of historical understanding of their intentions and predicaments, and travestying even the "winners" by tacitly crediting them with a kind of obscure foreknowledge of later ideas, needs and practices. These anticipations were their significance for history. In these ways, the past, which had once been its own "present," with its own interests, concerns and urgencies, was sacrificed to modern concerns and turned into a bland and benign anticipation of the present. Believers in Providence like Stubbs, we may add in retrospect, were really on firmer ground than their successors in presuming a long-term significance to history of which historical figures were the agents. It is quite proper and in a sense not anachronistic to attribute historical foresight to Providence, if not to men. Butterfield actually had his own, not altogether explicit, conception of Providence in history, but Providence manifested itself, it seemed, chiefly in a capacity for irony: for playing games with unintended consequences and confounding mere men's anticipations and progressive notions.
As his use of "Whig" indicates, Butterfield's polemic was focused essentially on the deficiencies of a partisan version of English history, but the logic of his argument, as has subsequently come to be widely recognized, is capable of and in a sense requires much wider application. It has become common among historians to speak of "whig history"—the use of the lower case is a useful indicator of the wider sense—for any subjection of history to what is essentially a teleological view of the historical process. It is a description which will often be employed in this chapter. In this sense, because history has supposedly an anticipated terminus from which it derives its moral and political point, Marxist history is characteristically whig. So were the older, simpler, versions of the history of science. The advance of science could be seen as a series of victories over pre-scientific thinking, and only the views of the genuine and hence progressive scientists (or even only those parts of their views which were authentically scientific, for some were regrettably eclectic) were of much interest. This was an intelligible stance in a modern scientist, but in a historian it was a kind of self-contradiction. It was magnificent, but it was not history.
Butterfield's own later trajectory was to be more erratic and in some ways elusive than one might have supposed. During the Second World War he published an essay ( _The Englishman and His History_ ) in which the Whig (not "whig") interpretation, while still being held at a distance as history, was said to have had desirable political consequences: "Whatever it may have done to our history, it has had a wonderful effect on our politics." This essay can be regarded as a _pièce d'occasion,_ an effort of patriotism under stress, but Butterfield also recognizes an enduring emotional pull: "In every Englishman there is hidden something of a whig that seems to tug at the heart-strings." Butterfield was also to be one of Namier's strongest critics, denouncing the extrusion of principle from eighteenth-century politics, and of narrative and the broad sweep from historical writing. It can be argued that his own work pointed to no satisfactory reconciliation of historical writing with what he called "technical history," which needed to have room made for it but which ought not to be allowed to take over. An unease, however, was evident.
This is not surprising. A story is inherently whiggish, and the longer its timescale the more marked this will be, requiring an artificial protagonist enduring beyond the span of individual lives and therefore, generally, of individual purposes. A story is selective, looking forward to its later episodes or its eventual outcome for its criteria of relevance. A sense of its artifice has drawn authors of modernist fiction to disrupt or play with narrative expectations in order to draw attention to this. In the present book I have consciously resisted the narrative urge to impose a unitary story. To succumb to this would be to turn the book into a teleology (see Introduction). Here the warning in Butterfield's early essay, slight and in some ways unsatisfactory as it is, still casts its long shadow. The impulse to write history has nourished much effective narrative, and narrative—above all in Homer—was one of the sources of history as a genre. It would be a strange paradox if narrative and history turned out to be incompatible. But the example of Homer may teach us not to take the paradox too tragically. The _Iliad_ has a climax, the fall of Troy, but it has many perspectives, and it would be a drastically impoverished reading of Homer's epic that saw as its "point" an explanation of Troy's fall. The concept of a story is in essence a simple one, but that does not make all narrators either simple-minded or single-minded. Narrative can be capacious as well as directional. What is the "point" of _War and Peace_ or Thucydides?
The tension noted though not resolved by Butterfield between setting out the results of technical scholarship and literary narrative presentation was a long-standing one in England. Its recognition had provoked controversy, and was to continue to do so. We can see it in the determination of the founders of the _English Historical Review_ in 1886 to make their journal the preserve of "scientific" history and to keep out the men of letters. In Oxford and Cambridge in the last decades of the nineteenth century the creation of the undergraduate syllabuses and examinations had divided the new history faculties roughly between the proponents of a broad historical education (mostly the college tutors) and the advocates of "training in research" (mostly the professors). In Oxford, the inaugural lecture in 1904 of the new Regius professor, Charles Firth, who had made his criticisms locally specific, produced an indignant collective response from the tutors, who thought he had impugned their scholarship. In Cambridge in 1903 the inaugural lecture of J. B. Bury, entitled "History as a Science," had a similar tenor but remained at the level of generalities; it contained the by then usual hymn of praise to German scholarship, and was made controversial by Bury's rejection of "literary" history and his notorious conclusion that history is "simply a science, no less and no more." A rejoinder was published immediately by Macaulay's great-nephew G. M. Trevelyan, who repeated it with alterations in 1913 in his _Clio: A Muse._ There Trevelyan, as well as effectively attacking the analogy with physical science, made a plea for the educative function of history, and associated this with the effectiveness of its literary presentation. Lamenting the fact that "Two generations back, history was part of our national literature," but it had now become "merely the mutual conversation of scholars," he declared that "the art of history remains always the art of narrative." He himself was to achieve an enormous popular success with his _English Social History_ (1944).
In the next generation the leading proponent of Trevelyan's prescription for history as part of the national culture was his Cambridge pupil J. H. Plumb. Though Plumb, as a historian of the eighteenth century, had to show that he had mastered the "Namierizing" techniques for parliamentary history in vogue from the 1930s to the 1950s, his emotional allegiance was given to Trevelyan, and in his retrospective, semi-autobiographical reflections _The Making of an Historian_ (1988) it was natural for him to take Namier and Trevelyan as the two representative figures of rival traditions of scholarly introversion on the one hand and humane literature with responsibility to a public wider than the merely professional on the other. One might object that "traditions" is over-rigid. Though it may be difficult to combine the historian's dual responsibilities in a single work, many historians—including Plumb—have combined them in their careers. The contemporary representative, for Plumb, of a dedication to narrowly technical scholarship more or less chose himself: his Cambridge rival the Tudor historian Geoffrey Elton. That is, Elton at least partially acted up to the role assigned to him, even if it was to some extent an exaggeration. Elton did at times reach out to a wider readership, at least in schools, but his own views of the historian's responsibilities, as combatively expressed as Plumb's, were significantly different (cf. _The Practice of History_ [1967]). Elton had made his name as an administrative historian with _The Tudor Revolution in Government_ (1953). To him—it is tempting to say in the Rankean tradition—the training of research students was also a vital part of his role. He could speak of the Public Record Office as of a garden of delights.
Elton, like Namier, was an immigrant—in his case from Germany—and was similarly Anglophile, though much better assimilated. He was crustily resistant to foreign fads like Freudianism (unlike Namier), Marxism and sociology: the historian had to keep at arm's length not only literary fudging but also theory. Yet, looking at a career so dedicated to the study of bureaucracy as the engine room of the Tudor monarchy, it is hard not to see a German predilection for the state rather than the English shibboleth of Parliament. It was in Germany that, in the late eighteenth century, bureaucracy was first distinguished as a category, while the concepts for analysing its tendencies and effectiveness were given classic form at the beginning of the twentieth century by Max Weber. Elton, however, referred to Weber only in connection with his _The Protestant Ethic and the Spirit of Capitalism_ —and with characteristic trenchancy: "historical nonsense."
Plumb too, though drawn when young to Marxism, had no use for theory. He had wanted originally to write novels, like his friend and early patron C. P. Snow. His conversation had some of the qualities of Balzac's _Comédie humaine_ and of Thackeray's and Rowlandson's versions of Regency England. It was appropriate that, as he tells us, he should have begun his writing career as a historian (with _England in the Eighteenth Century,_ 1950) in a Brighton hotel. He delighted in the wider reputation his writing brought him, and encouraged his pupils to be adventurous. In a time and place whose preferred tone inclined to be dry—"grey" was Plumb's word—he had a liking for grandeur and excess, undercut by a sense of the comic and grotesque. Individual tuition from him was, as he intended, intimidating, exhilarating and never dull.
Coincidence or not, it is remarkable how prominent a part Plumb's pupils and younger colleagues at Christ's College, Cambridge, have played in bringing history before a wide public in the last couple of decades. One example was Roy Porter, prolific historian of medicine and madness, chiefly in the eighteenth century. David Cannadine, in addition to tracing elegantly the stratagems of the British aristocracy and monarchy in the modern world, is the author of a study of Trevelyan ( _G. M. Trevelyan: A Life in History,_ 1992), whose view that history should be written so as to form part of the national culture he has polemically restated. The widest audiences have, of course, been reached by television, as in Niall Ferguson's programmes subsequently published as _Empire: How Britain Made the Modern World_ (2003) and Simon Schama's engagingly written and presented outline _History of Britain_ (published in three volumes, 2000–2003). The success of Schama's series, in particular, surely owes something to the contrast its sweeping overview presents to the teaching of history in schools as a set of disconnected "projects." These seem to reflect a priority of "method" over comprehensive understanding, and an imitation at an early age of a feature of the historical profession: specialization. Trevelyan at his most pessimistic can hardly have expected Bury to triumph even among schoolchildren.
**"Structures": Cultural History and the _Annales_ School**
Namier had an interest not only in the working out of individual psychological drives and dispositions, which is evident in his studies of eighteenth-century politics, but also, almost inevitably in a Central European Jew of his generation, in Freudian psychology, which is not. In France, however, debts to psychology and claims for the help it might offer the historian were more overt, particularly in the work of Lucien Febvre (1878–1956). Febvre was, with the medievalist Marc Bloch, one of the two founders in 1929 (the same year as Namier's _Structure of Politics_ ) of the journal which became known simply as the _Annales_ ; its full original title was _Annales d'histoire économique et sociale,_ and it gave its name to a whole school of French historians, the most influential such school in twentieth-century historiography. "The _Annales_ school" is now an indispensable piece of historians' shorthand, though it covers a number of complexities.
Before coming to Febvre, however, and his contribution to cultural history in particular, we have to consider the work of the most notable cultural historian to publish in the years after the First World War, working essentially in the German tradition of cultural history, the eminent Dutch historian Johan Huizinga. Huizinga, whose importance was recognized by Febvre, is known chiefly by his classic _The Waning of the Middle Ages,_ first published in 1919. He too was interested in psychology; he had studied under the famous German psychologist Wilhelm Wundt. Huizinga had a conception of culture as a kind of inventive game that human beings play, involving the masks worn by different ages, by which they presented and recognized themselves, and established their sense of identity. "Representations," the term employed in the sociology and anthropology of Emile Durkheim, later became the fashionable term for the conceptual images through which people structured their sense of the world and of their society, and the self-images and beliefs they shared as part of a common cultural stock. These were clearly historically variable, and therefore invited treatment by historians, though they were just as unamenable as social and economic structures to traditional narrative treatment. The achievement of Burckhardt, who was the most obvious precursor to Huizinga's work, in _The Civilization_ [actually "Kultur"] _of the Renaissance in Italy,_ had been of an essentially non-narrative kind. Huizinga's work, which stood to the later Middle Ages as Burckhardt's to the Renaissance, and was clearly intended to, now bears some archaic features, not least the title and the periodic resort to organic metaphors. But it has retained an enduring popularity. It has clear claims to be regarded as the outstanding work of cultural history in the first half of the twentieth century, and a notable predecessor of much that was to come.
Huizinga was a highly self-conscious historian, who in some of his essays, published as _Men and Ideas_ (1960), set out to characterize cultural history, which he says must concentrate on "deeper general themes" ("The Task of Cultural History," 1926). Culture exists only as a "configuration," so that only "when the scholar has to determine the pattern of life, art and thought taken all together can there be a question of cultural history." Elsewhere he speaks of the objects of cultural history as "the forms and functions of civilization...as they consolidate into cultural figures, motifs, themes, symbols, concepts, ideals, styles and sentiments" "form and function" was a phrase current in the study of animal and plant morphology.
Huizinga is clearly touched (as was to become a feature of avant-garde historiography) by influences from contemporary psychology, sociology and anthropology, as well as from biology. He mentions as relevant, for example, the German sociologist Max Scheler, and the French anthropologist Marcel Mauss, a disciple of Durkheim. It is not surprising therefore to find him adumbrating what were to become aspirations for the _Annales_ school, or to find his work recognized by Febvre, who was to call for "a history of the emotions." Huizinga spoke of the possibility of a "history of vanity," and claimed that the seven deadly sins were seven chapters in the history of culture waiting for treatment. He also exhibits a marked anti-whiggism in Butterfield's more generalized sense. By making someone a precursor "one lifts him out of the framework of his own time, and in doing so one distorts history" ("Historical Ideals of Life"). He praises Burckhardt as being the first to view the Renaissance apart from the concern with enlightenment and progress, and to see it not just as a prelude but as _sui generis._
In _The Waning of the Middle Ages,_ Huizinga attempts to trace the configurations mainly of the courtly culture of the Burgundian lands in the later medieval period, focusing on the attitudes, often embodied in ceremonial and artistic aspects of life, to birth, love, marriage, death and knighthood. He is particularly interested in the rituals and symbols expressive of ideal forms of life, religious and chivalric. Like Burckhardt, he is visually sensitive, and aware of art as a source for uncovering ideas when its meanings are decoded. His book is highly sensuous in its evocations of town life at all levels, including everyday experiences and special fêtes, like those associated with royal entries, marriages and tournaments. He regards thinking in images and personifications as particularly characteristic of his period, developed and systematized to the point of decadence. In such thinking, natural objects are never merely themselves and seen as such (their perception is the subject of a whole chapter in Burckhardt), but are always the bearers of specific meanings, often multiple, within an overall symbolic code which is generally understood. The world is structured as a hierarchy of such symbolic meanings, in which colours, plants and flowers, animals, minerals, planets, biblical stories and the ordering of society all play parts and constantly evoke each other and the moral qualities with which they are associated. Resemblances, of any kind, constitute symbolic connections; it is a world not so much of causality as of correspondences, which relate objects in this way rather than, as a later world of thought would relate them, taxonomically and causally.
Huizinga recognizes that this is a kind of game, though a serious one, because for him man is a game-playing animal; one of his other works is entitled _Homo Ludens._ But for those who participate in it—which in some degree Huizinga thinks is everyone, since the distinction between high and popular culture is not as marked as it became later—it constitutes their world and conditions their responses to it. Huizinga lays particular stress on a pervasive mood of melancholy, which contrasts with the sense of renewal associated with the Renaissance. It is an autumnal world, and "long after nobility and feudalism had ceased to be the really essential factors in the state and in society, they continued to impress the mind as dominant forms of life." To understand an age, one has to understand not only the forces transforming it but its illusions.
Huizinga, though he wrote in Dutch, belonged to a German tradition of cultural historiography, _Kulturgeschichte._ As a result of the diaspora created by Nazism, a strand of that tradition was to be relocated to London with the library of the art historian and mythographer Aby Warburg, which formed the nucleus of the later Warburg Institute. This was particularly associated with iconography as a tool of both art history and the history of ideas, and with the later Middle Ages and the Renaissance. The affinities here with Huizinga are obvious, but, as was mentioned earlier, there were also points of overlap between him and Febvre in France and hence with the _Annales_ school, though the emphasis of _Annales_ became more emphatically social. Febvre's chief collaborator, Marc Bloch (1886–1944), was a medievalist, whose interest in the history of mentalities led him to produce a classic and eventually highly influential study of the healing powers attributed to the medieval kings, as an aspect of their sacred character (1924, translated into English as _The Royal Touch,_ 1973). Despite the _Annales_ ' proclaimed interest in economic history, the influences of psychology and anthropology were strong. Both Febvre and Bloch had been taught at the Ecole Normale Supérieure by the anthropologist Lucien Lévy-Bruhl, author of _Primitive Mentality_ (1922). Behind him in the Ecole tradition lay Durkheim, and beyond him his own teacher, its director and a distinguished historian of ideas and institutions, Numa Denis Fustel de Coulanges, author of a classic study of the rituals he claimed constituted the ancient household, which eventually became subsumed in the city ( _The Ancient City,_ 1864).
Febvre's 1938 _Annales_ article "History and Psychology," which begins with a reference to Huizinga, goes on to proclaim the need to reconstitute the whole physical, intellectual and moral universe of each successive generation; not surprisingly, Michelet was another revered predecessor. In a later article entitled "Sensibility and History" (1941) Febvre spoke as Huizinga had done of the need to recover the emotional life of the past. Febvre thus became associated with an aspect of the _Annales_ ' interests which came to be known as the study of historical "mentalities" ( _mentalités_ )—the ways, in particular societies at particular times, of structuring and representing views of the world and human life. The emphasis was on unconscious assumptions rather than articulated theories, expressed in current symbols, metaphors and categorical distinctions ritually expressed. The similarity to the way in which Durkheimian anthropology had developed in the early twentieth century, particularly from Durkheim's last work, _The Elementary Forms of the Religious Life_ (1917), was very apparent. There, identifying religion as an essentially public, social phenomenon, so that assertions of belief are to be seen only as the provision of subsequent rationales for established practices, Durkheim had gone on to speak of societies as recognizing, and therefore in a sense constituting, themselves through public rituals enacting "collective representations," through which, to use the term he had coined earlier, a "collective consciousness" ( _conscience collective_ ) was constructed. Durkheim's study was based empirically on descriptions of totemism among the Aborigines of Australia, but it was conceptually a short step to an interest in the mentalities and culture of the medieval and early modern European peasantry.
The _Annales_ approach was essentially anthropological. The study of "mentalities" as a way of doing cultural history came to incur drawbacks associated both with anthropology in the mid twentieth century and with the earlier, often organic, metaphors and models employed by cultural historians. Holistic, historical conceptions of culture stretched back to the early nineteenth century and beyond, with the German term _Zeitgeist,_ the spirit of the age, as one of the earliest signals of their presence. Other terms, apart from the Marxist "ideology," which came to the fore in the twentieth century, were the sociologist Karl Mannheim's _Weltanschauung,_ world view, drawn from astronomy, and Michel Foucault's _epistème._ The trouble with treating cultures as integrated wholes was that it seemed to imply a view of them as totally integrated: static, and impermeable to change. Both anthropologists and cultural historians came eventually to recognize the need to complicate things (though Marxists, with the _force majeure_ of materialist explanation, operating at the economic level, to account for change, felt the impulse less). But it took time. Febvre, for example, famously declared in his major substantive work _The Problem of Unbelief in the Sixteenth Century: The Religion of Rabelais_ (1942, trans. 1983) that atheism in the sixteenth century was impossible, which seems to have been simply untrue (he contradicted it elsewhere). "Popular culture," too, came to seem harder to circumscribe than the phrase itself gave warning of. The problems which global conceptions of thought-worlds had in explaining change were also indirectly acknowledged as intractable by Michel Foucault in _L'archéologie du savoir_ (1969, translated as _The Archeology of Knowledge,_ 1972), where he waved a dismissive hand at them and announced he was doing not intellectual history but the archaeology of knowledge: archaeologists, apparently, did not need to bother about such things. The dismissal of narrative, or what narrative existed to deal with, namely change, left a perceptible unfilled space when narrative was superseded by the analysis of structures and mentalities.
The next generation of scholars associated with the _Annales_ was dominated by the figure of Fernand Braudel (1902–85). With him the _Annales_ returned more to its roots in social and economic history, and also developed an interest in counting things—sometimes, it seemed, more or less anything. Braudel's own great classic study of the Mediterranean area in the sixteenth century is rooted in geography (more prominent in French education than elsewhere) and economics, though it also transcends them ( _The Mediterranean and the Mediterranean World in the Age of Philip II_ [1949, 2nd edn trans. 1972]). It came to be enlarged to two volumes, the first of which is an enormous panoramic study of the interaction of the physical and "human" geography of the Mediterranean area over what Braudel calls _la longue durée,_ brought to life chiefly through the use of sixteenth-century archives, which makes Braudel's work one of the great historical studies produced in the twentieth century. It is an imaginative and scholarly achievement of the first order; it was intially an afterthought to the sixteenth-century political history of the second volume, but Braudel came to see it as an essential preamble.
The last part of the first volume is fairly orthodox (though impressively thorough) economic history, with an emphasis on currency, prices, demography and transport. The first part is an immense survey of life on the Mediterranean seaboard and its hinterlands in the sixteenth century, as structured by distances, natural resources, and their opportunities and imperatives; by the configurations of land and sea and islands, the varieties of terrain and peoples, and the latter's interactions. Braudel considers the shaping influences of mountains, with their unyielding, rocky soils and their seasonal semi-nomadism between uplands and lowlands; the plains, often marshy and malarial as well as cultivatable; peninsulas, islands and harbours; cities and villages and the trade between them, large-and small-scale. Then he turns to the river systems, the trade routes mainly northward, to Rouen, to Hamburg and Danzig, and to the Black Sea and Russia.
Despite its diversity, Braudel insists, the Mediterranean area is also a unity, the sea criss-crossed by trading ships, from headland to headland and island to island. The floating population of sailors was mixed and polyglot; sailors transferred readily from one service to another, so a single crew might include almost every people around the Mediterranean, from Spain to Greece. In _la longue durée,_ regular patterns, the almost unchanging repetitions conditioned by topography and habitat, are constantly being re-enacted in different places, like "the eternal war between peasant and shepherd." Local economies expand and contract, cities rise and sink, but the needs they supplied remain, to be more amply satisfied elsewhere. It is an unforgiving world that Braudel depicts, with no idealization, but with a human sympathy that seems perceptible, though unuttered, through the detail of the archives. The reader is left with a powerful sense of the harshness and hazards of life: toil, epidemics (especially in the cities), the chances of trade, pirate raids and kidnappings for enslavement (just as in the ancient world), near-starvation at times on the subsistence afforded by poor, stony soil—"The clearing of stones from the fields of Minorca in the plain behind Mahon was not completed until the eighteenth century." The islands generally were "lands of hunger," dependent on imports and impoverished by monoculture for export. Sometimes they were besieged, while the mountains of their interiors were "the no-man's land of the Mediterranean, the refuge of the poor, bandits and outlaws."
In the generation after Braudel, from the 1970s onward, the influences from the _Annales_ group became wider and more diffuse, and began to acquire outliers in Britain and the United States. Area studies, much more modest in scale than Braudel's—essentially a way of doing French local history with a geographical preamble—were produced under his influence. As mentioned earlier, the _Annales_ school became known for embracing quantitative methods. It was inevitable that computers should have an impact on historical studies. In the United States, and Britain too, quantification became a recognized historical technique; the term ironically coined was "cliometrics." It became controversial in the States when applied to the question of the economic balance sheet of slavery; its practitioners seemed to be displaying the same insensitivity as "One that would peep and botanize/Upon his mother's grave" (Wordsworth)—or someone else's.
In Britain it was historical demography that responded most to the quantitative impulse, epitomized in the work of the Cambridge Group for the History of Population and Social Structure. Its godfather, Peter Laslett, presented its early results in popular form in his _The World We Have Lost_ (1965). Drawing on parish registers, Laslett's perhaps most startling conclusion, in refutation of received ideas, was that the nuclear family was as much the norm in pre-modern as in modern England. New families typically set up new households; people waited to marry until they could do so; widows moved out. The received picture of the pre-modern "traditional extended family," with three generations and some collaterals under the same roof, was apparently a myth.
In France, developments from the 1970s onward took up where Febvre had left off; in the Braudel years the study of mentalities and popular culture was somewhat in abeyance, and Braudel himself devoted no attention to the notion of a distinctive Mediterranean culture centred on honour, shame, chastity and vendetta. From the 1970s, studies in ancient, medieval and modern history began to be produced addressing the history of the emotions and the senses, and conceptions of and attitudes to the universals of life and death, whose historical dimensions were increasingly explored. Philippe Ariès, pioneer of the history of childhood, declared that childhood was not a concept until the seventeenth century. Dying, sex, the body, cleanliness and dirt, and even smells, have been given historical treatment.
In a way, some of this harks back to the ethnographic digressions in ancient historiography, from Herodotus onward, but it is transformed, as folklore was converted into "popular culture," by the historical and anthropological turn given to it. Perennial features of life and death could be historicized because all were perceived and conceptualized through "representations" which were historically variable, as was much else. As Huizinga had claimed, for example, the spaces assigned to public and private life varied from one society to another: in the Middle Ages the public sphere was much larger, the private virtually non-existent. The German sociologist Jürgen Habermas has before identified the history of "the public sphere" in ways which have been influential since.
But before pursuing some of these themes to the end of the century we have first to halt at the mid twentieth century, and even to some extent to reach back behind it, to take account of one of the most powerful influences on historiography in that period: Marxism. In particular, attempts were made, beginning in the 1920s and culminating in the 1950s, to fasten Marxist explanatory categories on to the two great revolutions, the English Civil War—henceforth "the English Revolution"—and the French Revolution.
**Marxism: The Last Grand Narrative?**
Marxism achieved a substantial embodiment in history only in the middle years of the twentieth century. Marx's own reputation when he died in 1883 was as an economic writer who had predicted the collapse of capitalism through successively more severe crises of underconsumption. In America, just before and after the First World War, it had looked for a while as though some historians, in turning towards social history, might also be on the verge of adopting a Marxist explanatory scheme for it, in the so-called "New History" of James Harvey Robinson and Charles Beard. But American Marxist history, like the American proletariat and the American socialist party, did not materialize. The Marxist historian of ancient Greece Moses Finley established himself in Britain from the 1950s. In Europe, however, the Russian Revolution, the 1929 crash and the Depression, which seemed as though it might be the capitalist collapse Marx had predicted, all helped to foster a Marxist view of contemporary history and, as a corollary, of the past. The rise of fascism, moreover, seemed in the 1930s to promise to bring nearer the final, perhaps military, confrontation of Communism and capitalism, after burying "bourgeois liberalism."
It was from the late 1930s onward, in particular, that attention focused on the English Civil War and the French Revolution as test cases for the Marxist conception of history in what became, in the twentieth century, the party line. If these conformed to the Marxist category of "bourgeois revolutions," sweeping up the debris of feudalism and paving the way for the advent of capitalism as the necessary preliminary to its own eventual collapse, there was all the more reason to expect that collapse and thus the final victory of the proletariat, already apparently presaged by the Bolshevik Revolution.
In England, the Marxist interpretation had roots, not yet systematically Marxist, earlier in the century and even in the later nineteenth. The turn, begun then, to economic history as the shaper of social change had a leftward slant from early on—though not by any means among all economic historians. This bias was understandable, because the origins of economic history in the later nineteenth century lay to a large extent in the impulse to turn from pure economic theory to measuring the social implications of the operation of its laws, and the social costs of industrial capitalism. Such an impulse significantly prompted the activities of the late-nineteenth-century German Verein für Sozialwissenschaft (Social Science Association) and the work in economic history of the German so-called "socialists of the [academic] chair," Wilhelm Roscher and Gustav Schmoller. Religion, as well as ethics, was in some cases a component; in England, the writings of John Ruskin are also often to be found among the intellectual antecedents. The latter applied to Arnold Toynbee, uncle of the author of the grandiose philosophy of history which made a stir in the 1940s and '50s. The elder Toynbee was credited with coining the phrase "the Industrial Revolution," in lectures delivered in Oxford in the 1880s. He introduced his lectures by saying that political economy had been too much divorced from history; it was not hard to decode this. Toynbee was followed, in the next generation, by R. H. Tawney (1880–1962), who fired one of the opening shots in what was to become a scholarly battle over the causes of the English Civil War, in an article ("The Rise of the Gentry, 1558–1640") in _Economic History Review_ (1941).
Tawney was at that point the author of two main publications, united by an ethical socialism and a deep detestation of capitalism. He was an Anglican Christian, and what is still perhaps his best-known work, _Religion and the Rise of Capitalism_ (1926), was first delivered in the series of Gifford theological lectures at Glasgow University. The title acknowledged the influence of Max Weber's _The Protestant Ethic and the Spirit of Capitalism_ (1904–5), from which it derived its central thesis, but in Tawney's hands this thesis became something notably different. Weber's book was a study in the aptitude of a particular religious psychology, that of sixteenth-century Calvinism, to foster the spirit of capitalist enterprise and a frugal self-discipline which encouraged reinvestment and the maximizing of profits. Weber approved of these, and thought the German bourgeoisie of his day had too little of them. His approach was coolly analytic, and his own ideological drive was nationalistic. Tawney, with a formidable capacity for Christian indignation, and no liking at all for entrepreneurial virtues, turned the Weberian thesis into a study of the relaxation, in the course of the English Reformation, of traditional clerical attempts to moderate economic competition.
Tawney's other major work, _The Agrarian Problem of the Sixteenth Century_ (1912), took up, as a matter for analytic economic history, the sixteenth-century protests against the enclosure of land and the conversion of arable to pasture, with consequences in dispossession and unemployment. This was the starting point of Tawney's interest in agrarian history, on which his "Gentry" article drew. The article revived the seventeenth-century thesis of Harrington (Chapter 19) which attributed the overthrow of the monarchy to the eclipse of the feudal baronage, pursued as a policy by Henry VII, and the economic rise of the gentry, accelerated by the dissolution of the monasteries and the dispersal of their lands. Politically the gentry were dominant in the House of Commons. Tawney attempted to demonstrate their rise economically. The Second World War then intervened, but the debate, which assumed an exceptional acrimony, was afterwards taken up in _Economic History Review,_ by Lawrence Stone (1948), supporting Tawney's thesis, and by Hugh Trevor-Roper (1950), attacking it. The controversy ended with Tawney's thesis no nearer being established and with no firmer grounds for asserting either an economic decline of the aristocracy or a relative rise of the gentry.
Tawney was an English socialist, not a Communist. He was too old to embrace the intellectual Communism current in the 1930s. Very much a product of that enthusiasm was the other historian who in the 1940s focused on the supposed socio-economic origins of the English Civil War, Christopher Hill, whose Marxist credentials were prominently displayed in his first book, _The English Revolution_ (1940). The war was boldly declared to have been a bourgeois revolution of the classic Marxist type. Hill, who later became Master of Balliol, the college which was also that of Toynbee and Tawney, had been, not unusually for the period, a student Communist before the war, and, more unusually, had gone to Soviet Russia for a year to sit at the feet of Soviet historians of the English seventeenth century. _The English Revolution_ was the outcome. Hill, who left the Communist Party after the invasion of Hungary in 1956, subsequently modified but never abandoned its central thesis. "The English Revolution" was the work of an entrepreneurial bourgeois class, which included gentry. Its main effect was to break through restraints on the further development of capitalism.
A generation later the main consensus of historians of the period had turned decisively against this interpretation. Some aristocrats, as well as some gentry, had behaved entrepreneurially, but there was no discernible correlation of this, or of more traditional economic attitudes, with political inclinations. Even the political allegiances of the leading citizens of the towns provided no simple picture of bourgeois class consciousness. In the work of the more extreme "revisionists," as they are called, even the Civil War itself seems to be dissolved into a series of conflicts between local elites, unleashed by the collapse of authority at the centre, where it was divided by religion.
Ever since the early nineteenth century, English radicalism, at least with the (very gradual) softening of the grievances of Dissenters against the Established Church, has tended to fasten on two features of English history. The first, as in Tawney's _Agrarian Problem,_ was the dispossession of the rural poor from their rights in the land, reaching a peak with the Enclosure movement of the eighteenth century. The second was the social consequences of industrialization from the late eighteenth to the mid nineteenth century. These preoccupations became combined, from the 1820s to the 1840s, in the debate called "The Condition of England" question, to which Macaulay and Carlyle notably contributed on opposite sides. Later historiography tended to echo the terms of this debate, particularly in Toynbee's and Tawney's socialist treatment of English economic and social history, promoted by the writings of the great Victorian social critics and prophets, Carlyle, Ruskin and William Morris.
A late product of their tradition was E. P. (Edward) Thompson, whose first major work was a massive study of William Morris. His highly influential _The Making of the English Working Class_ (1963) had noticeably Marxist features but sat loosely to Marxist notions of the historically correct, and particularly to economic determinism. Thompson's chosen period was from the English radicalism of the time of the French Revolution to the Chartism of the 1840s, but his book was much more than a study of the thought and activities of English radicals. It dealt also with rural discontent and resistance to mechanization in agriculture; with the experiences of the increasingly unemployed handicraft workers and various other categories of workers; and with ethical ideas and traditions of organization in working-class communities. Its guiding thread, however, was the notion of the coalescence and increasing self-articulation of the labouring poor in a national working-class consciousness. Marx himself had devoted much of the latter part of his life to the cause of educating the English workers in their class interest and promoting their organization in trade unions. But Marxists, both theoreticians and historians, had devoted little attention to _how_ class consciousness was actually formed: it tended to be assumed to follow almost automatically from the concentration of workers by the system of factory production; they became unified and disciplined in collective action by bargaining over wages and conditions, and perhaps helped to articulate their interests and historical role by middle-class adepts in Marxist theory. Thompson's account went deeper, and emphasized the pre-existence of ideas of justice and rights, as well as the multiple forms of working-class association. He accepted the importance of the changing "objective" conditions of existence, but repudiated economic determinism.
In so far, however, as it is a study in the formation of the sense of identity of a class (which therefore, presumably, is then poised to become a historical agent, though we are not really taken that far), Thompson's, like all Marxist histories, can hardly avoid being a kind of whig story, a sort of _Pilgrim's Progress,_ in which the protagonist, the working class, encounters various obstacles, which Thompson identifies, on its historical path to self-realization. The faith in the English constitution is one, which helps to foster political radicalism but also initially misdirects it. Methodism taught methods of self-organization, but its political quietism was pernicious and its emphasis on personal salvation a deviation. The promises of the ideology of the employers, of increased prosperity for all from entrepreneurship and industrial production, were seductively delusive. Thompson's indictments are formidable and at times are expressed, as he surely knew, with something of the vehemence of the radical early-nineteenth-century journalist William Cobbett: the unholy trinity of "magistrates, mill-owners and Methodists" is almost pastiche Cobbett. It is easy to think of the various snares threatening the working class's recognition of itself in allegorical capital letters. Accosted by "Introspective Guilt," "Odious Subservience" and "Emotional Onanism," Bunyan's Christian would have needed to think quickly. As in an inverted temperance tract, the educated Radical Francis Place is led into bad company by his inability to drink in taverns, leading to his consorting instead with "Utilitarian and Malthusian doctrines." The reader can only shake his head over a good man gone wrong. The metaphors, too, for states of emerging but still unachieved class consciousness are significant, suggesting malign enchantment: it is "transfixed" by the ideas of Tom Paine, which included respect for the profits of entrepreneurship, and "enmeshed" in constitutional argument.
The Marxist interpretation of the French Revolution may be said to have begun in the 1920s, in the work of Albert Mathiez, though he was not a Marxist but a pupil of Aulard, whom we have noted (Chapter 23) as a critic of Taine. It was continued by Albert Soboul, and, at a higher level of scholarship and sophistication, by Georges Lefebvre, in the mid-century. It was Lefebvre who declared that "the Revolution is only the crown of a long economic and social evolution which has made the bourgeoisie the masters of the world." Soboul explained the Revolution in bluntly Marxist terms, as the outcome of "a contradiction between the relations of production and the character of the productive forces" in other words, the contradictions at the level of the substructure were grinding out as usual their consequences for the arrangements of the political superstructure. At that level "the commercial bourgeoisie...guided, with a sure awareness of its interests, the Revolution to its objective."
The first really effective shots against this were fired by an English historian of France, Alfred Cobban, particularly in his _The Social Interpretation of the French Revolution_ (1964). He noted there that the demolition of the Marxist appropriation of "the English Revolution" had already begun. It had "blown up the supposed bourgeois revolution, leaving aristocracy and gentry, royal officials, lawyers, merchants, people, rising and falling classes, feudal and bourgeois society, labourers and peasants, scattered in fragments about monographs and textbooks." The society of the French _Ancien Régime_ had been no less complex and intractable; no single formula availed to express an individual's social position, or the "class" represented by an aggregate of such individuals. It was a society of multiple roles and various ways of estimating status: social, economic and legal criteria intersected in ways which defied simple class categorization. For Cobban, as for the English Civil War revisionists, the Revolution was essentially political and exemplified no single underlying determinism. It was also, to a significant degree, a revolt _against_ modernity.
The leading French critic of the Marxist interpretation, François Furet, particularly in his _Interpreting the French Revolution_ (1978, trans. 1981), argued to similar effect but with more attention to the motives and mindset of its proponents. He pointed out that the French Revolution had remained part of politics, rather than being recognized as history, where he made a plea for it finally to be relocated. Having shaped French political life for a century and a half, "The French Revolution is over." Instead of remaining an object of commemoration, it needed to be emancipated for open-minded historical inquiry. Noting that in the nineteenth century the Revolution was already thought of as a stage destined to be superseded, rather than as an achieved political event, Furet argued that its interpretation was bound to remain politicized so long as the Revolution was taken as presaging the proletarian revolution to come. The first act of that proletarian revolution had been identified as the Bolshevik Revolution of 1917, leading Mathiez to make a connection between the Jacobins and the Bolsheviks.
The hero of Furet's argument, in a sense, is Alexis de Tocqueville (1805–59), who, in _The Ancien Régime and the Revolution_ (1850), had established an administrative continuity between the centralizing drive of the monarchy from the seventeenth century onwards and that which also characterized the Revolution. According to Furet, we have to go behind the consciousness of the revolutionaries to see the work of the Revolution as the progressive seizure of power from the institutions of civil society and from local communities by the centralizing state. The Revolution was certainly an important event (Cobban once asked, "Was there a French Revolution?"), but it was a political and ideological one, centralizing power and diffusing and sanctifying a democratic ethos. The Marxist interpretation, however, embraces a number of categorical confusions in describing it, conflating monarchy and aristocracy, aristocracy and feudalism, bourgeoisie and capitalism. The attitudes of the bourgeoisie to entrepreneurialism were complex, and the aristocracy was by no means devoid of entrepreneurial interests.
It is the nature of historical revision to have no end: revisions of revisions are no doubt taking place even now. That is the norm for the historical profession. It does seem likely, however, that strictly Marxist interpretations of the two revolutions—interpretations whose collapse, incidentally, was prior to that of Soviet Communism—have become past episodes. A simple class terminology no longer sings its siren song to historians, and it is hard to imagine that an aspirant future Hill or Soboul, if there is one, could again see glad, confident morning while hailing the rising bourgeoisie.
**Anthropology and History: Languages and Paradigms**
In 1963 Keith Thomas published a striking article in the leftist journal of social history _Past and Present_ (founded 1952). Entitled "History and Anthropology," it was a protest against historical specialization by subject matter, contrasting this with the way in which anthropologists studied small-scale societies in their totality. He also pointed to substantive overlaps between the matters to which anthropologists characteristically gave attention in the preliterate societies they studied, such as witchcraft, the blood feud, myths and genealogies, and some of the phenomena of preliterate European societies studied by historians. The plausibility of Thomas's case, in pointing to anthropology as a possible source of guiding practice for historians, really depended, apart from the French example, on the revolution which had taken place in British anthropology since the 1920s, partly under the influence of Durkheimian ideas, away from evolutionism and towards the conceptual understanding of small, self-contained exotic societies through the key terms of structure and function.
Sir James Frazer, the most famous of all British anthropologists at the beginning of the century, had seen the possibilities in combining the study of exotic or "primitive" societies with that of pre-modern European folk beliefs and practices, then habitually described as "folklore." But he and his peers had worked under the aegis of a prevailing evolutionary perspective, which relegated folkloric superstitions, magical beliefs and the recurring rituals of rural societies to the category of "survivals," fragments of earlier stages in human society, left stranded in the wake of evolutionary progress. From the conception of them as survivals nothing could really be made except antiquarian recording, and confirmation that European societies had once, as evolutionism dictated, been savage. The adoption of "function" as a central explanatory concept offered an alternative, first in the study of non-European societies, but also in the study of the European past. The attention of anthropologists themselves remained for the most part fastened on the former, though the Oxford anthropologist E. E. Evans Pritchard, exceptionally, wrote an essay on "Anthropology and History" (reprinted in his _Essays in Social Anthropology,_ 1962).
The concept of function had the same capacity to transform folklore, and the study of the thought-worlds of pre-modern, mainly rural, European societies, as it had done in anthropology. As Thomas saw, by asking what thoughts hitherto conceptualized as survivals meant to those who thought them, and what these people's ritual and other practices did for them, it could be possible to see how these phenomena fitted into people's lives and conceptions of the world, opening up the study of pre-modern societies in a new way, alongside what Thomas described as specialization by subject—the familiar studies of political, constitutional and ecclesiastical history and the analysis of socio-economic structures and conditions. Folklore could be repositioned as the study of popular culture and mentalities, in the fashion advocated earlier in France.
It is perhaps significant that the most obvious counterpart in the United States to Thomas's article, later reinforced by his own classic study _Religion and the Decline of Magic_ (1971), was entitled _Sociology and History: Methods_ (1968), edited by a historian and a sociologist, Richard Hofstadter and Seymour Martin Lipset. Albeit looking in slightly different directions, it was a similar kind of call for history to be opened up to the influences of the social sciences. Sociology was becoming of interest to British historians too, but its position in the United States was more assured. The American tradition in anthropology focused on the distinctive traits of particular cultural circles (from the German _Kulturkreise_ ), or on individual psychological development, as in the Samoan work of Margaret Mead, rather than on the dynamics of small-scale societies. Hofstadter, in his preface to _Sociology and History,_ mentions, as themes to draw to the attention of historians, occupational structure, social stratification, quantitative methods (he cites opinion polls), "religious styles," immigration, and "the social role of ideologies." As is clear from this list, the freedom of the German woods being unfashionable, the Middle Ages was no longer expected to be part of the American story, which now began with Puritanism. In America, Max Weber was a much more potent external influence than Durkheim, and Weber is emphatically a Protestant individualist in his social thinking, while Durkheim's lies in the neo-Catholic tradition dominant in French sociological thought, with its interest in the social functions of religion and ritual, ever since Comte and Fustel de Coulanges in the mid nineteenth century.
In Europe, where Marc Bloch was the obvious pioneer, medieval and early modern historians from the 1970s onward became increasingly attentive to what they had always known but were now seeing new ways of approaching: that the world they studied was peopled with spirits and demons and exorcisms, and was infused with ideas of the sacred, with initiation rituals, purifications, miracles, cults and sacraments as points of access between the seen and unseen worlds. "Superstition" was a term from the Enlightenment, "faith" one from Christianity; a phrase like "system of belief" relativized and historicized both, and was implicitly comparative. The eminent historian of Late Antiquity (a category he has led us to see as a necessary one) Peter Brown, in his studies of early Christianity, has employed to telling effect an essentially anthropological conception of the sacred—in considering, for example, the Eastern Christian cult of the holy man, qualified for his role by ascetic practices and armed with powers of exorcism ( _Society and the Holy in Late Antiquity,_ 1982). Christianity in the fourth and fifth centuries had, like some modern historians, been much preoccupied with attitudes to the body.
It may seem a long step from the study of mentalities, largely through images, ritual practices, symbols and assumptions, to the intellectual history created by highly literate elites. It too, however, has been influenced since the 1960s and '70s by a turn, if not to anthropology, then to the shared web of language. The "history of ideas," at the highest level of abstraction, found a home initially, between the wars, chiefly in the United States. The _Journal of the History of Ideas_ (1941) was established under the influence of A. O. Lovejoy, whose remarkable work _The Great Chain of Being_ (1936) put forward the notion that there was a limited stock of what he called "unit ideas" (he drew an analogy with the supposedly finite number of fictional plots), which could be seen working themselves out in history. "The Great Chain of Being" was one of these; it proved difficult to find others. The "Great Chain" was the conception of a hierarchy of forms of being, from the most spiritual to the most material; ramifications of the idea, Lovejoy showed, spread out through philosophy, theology, science and literature from the Greeks to the eighteenth century. Lovejoy's book was a notable tour de force, but it did not provide the pattern for subsequent work that he had clearly hoped for. One thing this prescription for the "history of ideas" tended to generate was histories of a _word,_ for example "Gothic," which certainly ramified into different intellectual fields, but was not quite what had been envisaged.
From a historical point of view, Lovejoy's work was at least an improvement on focusing simply on the ideas of great thinkers, treated in isolation but arranged in chronological sequence. This conception of the history of ideas, if it can be called a conception, was thoroughly superseded between the late 1950s and the late 1970s, most notably by two scholars trained as historians, John Pocock and Quentin Skinner. The dominant genre here, chiefly for institutional reasons—it is hard to see others—was, and to a significant extent remains, the history of political thought. Courses in this subject were and are to be found in the syllabuses of history and politics departments; courses in intellectual history as such, at least as a practice that dares to speak its name, were generally not. (Whether we speak of intellectual history or the history of ideas here is of little importance.) The logic, however, in studying, say, Rousseau's _Social Contract_ but not his _Emile,_ or Locke on Civil Government but never Adam Smith on Moral Sentiments, is elusive. Pocock's chief scholarly contributions in these years were _The Ancient Constitution and the Feudal Law_ (1957) and _The Machiavellian Moment_ (1975), and Skinner's _The Foundations of Modern Political Thought_ (1978), all of which have been drawn on in this present book. Skinner also produced a number of sparkling polemical articles in the 1960s, attacking ahistorical and "whig" treatments of the canon of political thinkers (reprinted in _Visions of Politics,_ vol. 1).
Pocock's early methodological reflections testified to a kind of "linguistic turn." To see the web of intellectual life as one of socially transmitted meanings (which does not mean the reduction of its study to social history) has become common ground in intellectual history. Pocock spoke of the rival "languages" in which political and social debates were constructed. In his preface to his edition of Harrington's works he distinguished three such relevant languages in seventeenth-century England: the language of the common-law tradition, that of Puritan millenarianism, and that of republican virtue and corruption. Compared with others exhibiting the linguistic turn—Foucault, for example, who seems to regard language as a kind of prison—Pocock sees such languages both as multiple, and hence malleable, and as enabling. Skinner has emphasized the necessity of interpreting texts, mostly from the Renaissance (including the pictorial) and the seventeenth century, by setting "vocabularies" in the web of contemporary meanings on which they drew and without which they would have been unintelligible to their age. Though it is not usually done, it might seem appropriate even to speak of the "mentalities" of the learned, provided it is understood that the analysis of them is not psychological.
Recollections of the intellectual influences at work in this scholarly revolution vary among the participants. They were not so much precepts or models as converging indicators, more or less prominent for each individual, of a relevant intellectual climate in the 1950s and '60s. My own would include Michael Oakeshott's metaphor for intellectual life as "conversation," and the Wittgensteinian slogans, perhaps not very well understood, "A language is a form of life" and "The limits of my language are the limits of my world." Others might cite R. G. Collingwood's historicizing of philosophy and J. L. Austin's concept of "performative utterances," language as doing, which was effectively used by Skinner. It was also important that Duncan Forbes (my own Cambridge undergraduate supervisor in 1956) was then virtually single-handedly creating and expounding the conception, now indispensable, of an eighteenth-century "Scottish Enlightenment" I remember another Cambridge historian referring to it as "something Scottish, I believe." Pocock then spoke, at least in the exposition of his ideas on "languages," of Thomas Kuhn's use of the notion of "paradigms" in the history of science (T. Kuhn, _The Structure of Scientific Revolutions,_ 1962).
Kuhn's thesis was a bold extrapolation from the familiar concept of a Copernican revolution. A paradigm in Kuhn's sense establishes a particular way of explaining phenomena which is mandatory for a given scientific or philosophical community. It is wider than a theory: it specifies the criteria a successful theory must meet. Evolution, often spoken of as a theory, would be for Kuhn an example of a paradigm, since it establishes a whole mode of explanation, initially in opposition to creationism or the "argument from design," which was the previous one. From a paradigm, subordinate theories and explanations can be generated. To explain the structure of the eye by reference to its functions could be compatible with either the paradigm of design or that of evolution by natural selection. But a paradigm can also accommodate debates, which generate research. Rival theories, for example, of which line of hominids culminates in _Homo sapiens_ would both have to accept the evolutionary paradigm.
Paradigms are historical phenomena. A "paradigm shift," according to Kuhn, occurs when the whole explanatory thrust is changed or reversed: creationism (usually) speaks of the world as adapted to man; Darwinian evolution by natural selection speaks of man as fitted for survival in the world. The classic case of such a shift, literally a transposition, replaced the earth by the sun at the centre of its system in the Copernican revolution. Paradigm shifts occur when the accumulation of anomalies generated by the accepted paradigm requires so much "explaining away" that the simplicity achievable by a canvassed new paradigm becomes more attractive. Why, Darwin asked himself, did the beaks of Galapagos finches differ slightly from island to island? Why did God do that? (Most of these examples are not Kuhn's.) A paradigm, as an established consensus, is socially as well as intellectually sustained, and is enforced especially on apprentices, who thereafter acquire a vested interest in it. Hence a scientific community, which is the tribunal which determines what counts as an explanation, is essentially conservative, but subject to occasional drastic upheavals. The history of science is only in part a history of accumulation: it is also periodically revolutionary.
Kuhn's book had the effect of such an upheaval on a view of the history of science as the progressive accumulation of truth through the use of scientific method. Predictably, his thesis has been challenged, particularly for its apparent relativism in making historically variable paradigms determine what counts as explanation. But, whatever flaws it may have, its importance here is that it identified science as a human and collaborative practice, and characterized a mature scientific community partly in terms of the exercise of power. In doing so it opened up the possibility of a different kind of history of science, with more of a role for historians. A good deal of work, for example, has been done on the formation of an evolutionary consensus, and its varieties, in mid-nineteenth-century Britain; a paradigm shift can have resemblances to a political revolution.
Simultaneously but independently, emanating from the tradition of iconography and cultural history embodied in the Warburg tradition (Chapter 26), the work of the Renaissance scholar Frances Yates, in particular, rejected scientific whiggism in another fashion. Whiggish history of science had not only characteristically divided its cast into enlightened pioneers and their opponents. It had also used later and hence anachronistic criteria to discriminate what was interesting, even within the thinking of the pioneers of science, discarding the religious, astrological and alchemical rubbish with which it had sometimes been encumbered as of no concern to the history of _science._ Given a particular definition of the phrase this was consistent, but it is not what historians mean by history. Frances Yates, beginning with her _Giordano Bruno and the Hermetic Tradition_ (1964), shifted the focus of attention from identifying and appraising the portents of modernity to the total thought-world of her protagonists, much of it subsequently set aside as of only antiquarian interest. ("Antiquarian" used pejoratively, by historical whigs or non-historians, means "historical." Historians of political thought have sometimes been accused by political philosophers or political scientists of being antiquarian, meaning that they are historians.) Yates's work was in a sense the high-intellectual equivalent of taking the witchcraft beliefs and practices of a peasant society seriously, as a historical phenomenon.
**Suppressed Identities and Global Perspectives: World History and Micro-History**
Thompson's _Making of the English Working Class_ is a Janus-faced work, looking backward to its Marxist origins and also forward to the historiography of the last quarter of the twentieth century. Thompson's "Working Class" was still incipiently the revolutionary proletariat. But the formation of working-class consciousness, as Thompson richly describes it, is also the formation of what a more neutral sociological language would call a subculture, to which the historian needs to attend for its own sake. Thompson's famous statement of his intention, in his preface, identified his book as a precursor of new historical attitudes in the 1970s and '80s, while the concept of working-class consciousness traced its origin back to Marxist orthodoxy: "I am seeking to rescue the poor stockinger, the Luddite cropper, the 'obsolete' hand-loom weaver, the 'utopian' artisan, and even the deluded follower of Joanna Southcott [a millenarian prophet], from the enormous condescension of posterity." This chimed resonantly with the mood of the 1970s and after. Reconstructing the thought-world of the inarticulate or illiterate and semi-literate (though some of Thompson's radical cast were very articulate and literate indeed) came to seem a worthy task for the historian. Prospectively fashionable, too, was the way Thompson explained his title: " _Making,_ because it is a study in an active process, which owes as much to agency as to conditioning."
We find the same verb, and the same reasons for it, in one of the most notable American works of the 1970s, Eugene Genovese's _Roll, Jordan, Roll: The World the Slaves Made_ (1974). "Made" here has perhaps an even more striking active force than Thompson's "Making." Using letters extensively, and attempting to do justice to the variety of ways in which slavery was experienced, Genovese traced the culture the slaves created for themselves, under the most adverse of circumstances, out of the Bible, contact with white society, and African influences, as something new and unique, not merely traditional or imitative. Genovese's is a highly impressive book, which made an important general point. The notion of collective identities being culturally forged, as an autonomous creation, gripped imaginations in the latter part of the twentieth century. It meant seeing as agents and makers groups which had hitherto tended to figure in historical accounts only as sufferers of their grim fates or, for the administrative or governmental mind (reflected in some historians), as problems and even nuisances.
We can illustrate this by a contrast. It dates from the later nineteenth century, but the way of thinking it enunciates remained powerful, even orthodox, for another three-quarters of a century. The prefatory note to the first number of the _English Historical Review,_ in 1886, anonymous but actually by James Bryce, announced, more or less redundantly at the time one might have thought, that "States and nations will be the chief part of its [the _Review_ 's] subject." The reasons Bryce gives are rather more interesting than the declaration itself, and they are relevant here. Rejecting as too vague the view that history meant providing "a picture of the whole past," he defined it as "the record of human action," adding that "the acts of nations, and of the individuals who have played a great part in the affairs of nations, have usually been more important than the acts of private citizens." The use of "important," rather than the weaker "worth attending to," makes the claim tautological.
Bryce's argument was echoed seventy-five years later, in the course of an argument against Butterfield's _Whig Interpretation,_ in E. H. Carr's influential _What Is History?_ (1961). Carr's perspective was emphatically whiggish: "History properly so-called can be written only by those who feel and accept a sense of direction in history itself" ("history itself" seems to be the phrase in serious need of analysis here). Carr went on, "History is, by and large, a record of what people did, not of what they failed to do; to this extent it is inevitably a success story" (or an inevitable success story?). One recognizes the echo, if diminished a little, of Bryce's "played a great part." Under cover of making a general point about history, Bryce and Carr are issuing a prescription about what kind of history is worth doing—indeed, what counts as history. All historians choose what history interests them, but the choice need not be universalized and made mandatory; a confident consensus was speaking here.
In fact the arguments are fragile throughout. The antithesis that Bryce refers to, "a picture of the whole past," is vague only because of the level of generality at which it is presented, which makes it seem absurd as a prescription. To reformulate it less generally, as a claim that _any_ aspect of the past _may_ be made the subject of historical inquiry, makes it neither vague nor absurd. It is worth reflecting for a moment on what Bryce may have been thinking of, and repudiating for the _EHR_. It seems most likely to have been the publications of the local antiquarian and archaeological societies which were the provincial and established rivals of the new national journal of constitutional and political history. The contrast was not only between old and new, political and domestic, but between metropolitan and local. Local history was in a sense the predecessor of what we have soon to turn to, the modern genre of "micro-history," and therefore needs a moment's attention.
What we might have thought, and what Bryce probably thought, as others certainly did, was wrong with antiquarian and archaeological publications was not that they were vague but that they were all too specific, though miscellaneously so. John Richard Green, another of the _EHR_ 's founders, spoke loftily of such societies' archaeological interests as "ecclesiastical architecture slightly tempered by an enthusiasm for Roman camps and old helmets," and referred to their memberships as "country parsons and old maids." Antiquarian publications typically dealt in inventories of facts about costume, buildings, weapons and other artefacts, aspects of domestic life and locally notable families, local traditions, and random cullings from local archives. They were certainly eclectic, and in that sense took "the whole past" as their province. They were clearly not products of disciplined specialization, and were therefore in Bryce's sense "vague." One can convey their flavour best through a series of titles, which are sufficiently self-explanatory and even disarmingly frank. Here are some from the first volume (1875) of the _Transactions of the Royal Historical Society._ Despite its grand name and London base, the Society, founded in 1868, had been born in the world of antiquarian printing societies, and at this stage still bore its marks. The articles were predominantly if variously local: "The Personal Expenses of Charles II in the City of Worcester" "The Mounds at Dunblane and the Roman station at Alauna" "Tudor Prices in Kent, chiefly in 1577." Clearly men with access to and perhaps the care of a particular local archive bore a heavy responsibility for a list of this kind. The archive was consulted because it was there—though it has to be said that in the 1950s articles in the _Economic History Review_ (founded in 1929) were not invariably excitingly related to wider historical issues either.
By the mid twentieth century, however, as Keith Thomas saw, history other than "the affairs of nations" had the chance to reconcile comprehensiveness and coherence. Anthropologists had learned how to do so. There were indeed critics of the static worlds offered by British anthropology, and there were charges of triviality and references to anthropological monographs on particular societies as "stamp collecting," but in the 1960s and '70s small no longer meant trivial. Thomas had chosen his moment well. To the emerging historical sensibility, cultures and the collective identities they helped to constitute were "made" by their participants, mainly anonymously, in sustaining a particular collective way of life. Virtually all adults were in that sense agents, and even successful ones, active participants in social relations, mediated by participation in a language: passing on memories and precepts between generations; receiving, reshaping and transmitting, often confusedly, ideas about the world and human life. We are reminded of Carlyle's "History is the essence of innumerable biographies"—a thought which underlies the modern genre of "oral history." Oral historians record individual memories while their possessors, who would be unlikely to write them down, are still able to transmit them through the spoken word. It is a kind of revival of one of the oldest of historians' practices, the interrogation of eyewitnesses; largely superseded in the intervening period by the profusion of written documentation, it has been resuscitated by the invention of the tape recorder.
Modern conceptions of culture and its value are recognizably related to those of Vico, Herder and Michelet, which were marginalized in the age of Ranke and _Realpolitik._ The impulse is sometimes similar to Herder's desire to awaken in the Germans a sense of their then under-valued identity, through a cultivated awareness of the anonymous creation and transmission to the present of a common national culture. Though this might seem at odds with the modern concern with subcultures, it in fact also made room for focusing on the traditional ways of life of local communities. This left a significant trace in historiography, as we have seen (Chapter 23), through the once fashionable conception of the Teutonic village community of co-proprietors as the supposed bedrock of all Teutonic (including English and Scandinavian) society. This anticipates modern interest in the small-scale, but the latter is no longer necessarily thought of as germinative. All the same, _Volk,_ though virtually a proscribed word, is still an influential concept, though we squeamishly prefer "people's history."
The later twentieth-century revival of cultural history, of which some of the interests of the _Annales_ school were precursors, came to coincide and typically coalesce with the awakening of an articulated self-consciousness or awareness of identity among groups which thought of themselves as hitherto suppressed, ignored or marginalized. These were often not Herder's "nations," though some could be thought of as such, for example in "post-colonial" contexts; much less were they Bryce's. But the nineteenth-century conception of nationality could sometimes embrace something less comprehensive than "German culture." Augustin Thierry, author of _The History of the Conquest of England by the Normans_ (1825), which embraced the cause of the subjugated Saxons, had a particular interest in suppressed "nationalities"—Bretons, Aquitaineans, Provençals—which many contemporaries might have refused to recognize as such (and the French state certainly did). Thierry's ideas, rooted in the populist strain in European Romanticism, can also seem prophetic, now that the sense of identity is not monopolized by the nation state. Awareness of a multiplicity of possible historical protagonists has become much more acute, common and assertive.
To take an example near home, the United Kingdom seems a more problematic historical creation than it did in the heyday of Whiggism. (See Linda Colley—another Plumb pupil— _Britons: Forging the Nation,_ 1707–1837, 1992.) Ethnic minorities find voices; so do populations once colonized and incorporated into someone's imperial dream. Emancipated from this dream, the creation of identity, and winning assent to it, can be a matter of political and even literal life or death. (A West African student of mine once said, "What my country needs is a lot more whig history" I thought this profound as well as witty, recalling the later Butterfield's distinction between the historical demerits and the political advantages of whiggism.) For of course the characteristics of whig history can be not rejected but simply relocated, endowed with a different protagonist; manifest destiny is a game any people can play. So, from the 1970s onward, there has been a proliferation of identities recognized and invested in by historians: the history of formerly suppressed nationalities, women's history, black history, working-class history, ethnic history, peasant history, as well as the history of minority sexual inclinations, of bandits, of rebels and of unfashionable religious sects. Collectively, much of this has come to be known, not always accurately, as "history from below." The partial displacement of the nation state as the essential protagonist of history has encouraged not only an interest in categories which cut across it, like those of ethnicity and gender, but also moves outward and downward, to the history of the world in one direction and to that of small communities such as the workshop and, more often, the village or parish.
World history, which is still more an aspiration than an established body of historical writing, has many precedents. The older term, going back to Polybius, was "universal history" and medieval chronicles characteristically began with universal history before narrowing to often very local concerns. "Universal," of course, has to be understood relatively. Polybius made his focus the rise of Rome to domination of the Mediterranean and its hinterlands; even in his day it would have been recognized that this excluded the Persian empire. Medieval universal history, derived from Orosius, Jerome and Isidore of Seville, arose from attempts to blend Judaeo-Christian biblical history with the history of the Graeco-Roman world (Chapter 13). New versions appeared in the twelfth century and beyond. Germany and Italy, lacking the focus of a nation state, were particularly receptive to schematized and apocalyptic versions of universal history, including prophecy, as in the strongly Augustinian work of Otto of Freising (1114–58), who was a member of the imperial family, and the writings of the Franciscan mystic Joachim of Flores (Chapter 12). Joachim had an enduring influence on later apocalyptic ideas.
The best-known universal history of the Renaissance period, at least in England, Sir Walter Raleigh's _History of the World_ (1614), was emphatically biblical. The eighteenth-century Enlightenment, as we saw above (Chapter 21), produced what were in effect though not in name universal histories that were emphatically and even polemically secular. Voltaire's _Essay on Customs,_ which exalted the Egyptians as against the Jews, was written _against_ the _Universal History_ (1681) of the Catholic Bishop Bossuet. The new concept of civilization provided a kind of key, and the history of mankind was presented, highly schematically and conjecturally, in two kinds of story: the history of the human mind, in which the overcoming of superstition was crucial, and the history of the socio-economic stages of civil society. The writings of Marx and Engels, in their historical dimension, can be seen as a continuation of the "civil society" tradition, in which the economic organization of civil society was seen as determining the political order and ideas of each of its stages.
Germany continued to be receptive to the idea of universal history. In the later eighteenth century it was a major theme in the flourishing historical school based in Göttingen. Although this was supplanted in the nineteenth century, as described in the previous chapter, by the school of Ranke, with its major focus on the political and diplomatic history of early modern Europe, Ranke himself, in old age, reverted to the earlier interest in universal history by producing a seventeen-volume world history (1880–86).
By the more stringent standards of the later twentieth century, in the age of post-colonialism, all these forays into the universal were not nearly universal enough and were vitiated by an essentially European perspective. The same could be alleged, in different degrees, of the best-known world histories produced in the earlier twentieth century, between the wars: Oswald Spengler's _The Decline of the West_ (1918–22), H. G. Wells's popular _Outline of History_ (1920), infused with the idea of evolutionary progress, and, most massively, Arnold J. Toynbee's ten-volume _A Study of History_ (1934–54). Toynbee's reputation, always controversial, became greatly dimmed, and his particular preoccupation with the history of religion has had no perceptible long-term influence, but originally he blazed a trail. One of the most highly regarded modern world historians, the American scholar William McNeill, began his career as Toynbee's collaborator; the chief torch-bearer for world history among English historians, Geoffrey Barraclough, succeeded to Toynbee's chair at the Royal Institute of International Affairs (Chatham House) in London in 1956. More recent notable English contributions have been J. M. Roberts's _History of the World_ (1995) and in 2004, engaging with the modern period of "global history"—a phrase becoming increasingly fashionable—C. A. Bayly's _The Birth of the Modern World 1780–1914_.
One stimulus to a global perspective has been Marxism, in which, from the outset, capitalism has been seen as an international force and the motor of modern world history. Marx and Engels wrote in the _Communist Manifesto_ (1848) that "the bourgeoisie has through its exploitation of the world market given a cosmopolitan character to production and consumption in every country...In place of the old local and national seclusion and self-sufficiency, we have intercourse in every direction, universal inter-dependence of nations." A modern historian who has applied this thought over a series of four volumes covering the period from the French Revolution to the late twentieth century—a series which has become more and more a history of the world and not just of Europe as capitalism's birthplace—is Eric Hobsbawm. The last of the series, _The Age of Extremes: The Short Twentieth Century, 1914–1991_ (1994), is essentially a recent history of the world. Hobsbawm's Marxism has been comprehensively shorn of any lingering utopianism. One can sense, however, a wry pleasure in his observation that the collapse of the Soviet system exemplifies a Marxist truth: "Rarely has there been a clearer example of Marx's forces of production coming into conflict with the social, institutional and ideological superstructure which had transformed backward agrarian economies into advanced industrial ones, up to the point where they turn from forces into fetters of production" (Back Matter).
Britain has not, however, been particularly academically hospitable to world history compared with the United States, where it has found a more assured place in university curricula. But the most famous later-twentieth-century historian to be drawn towards world history has been Braudel. After his study of the sixteenth-century Mediterranean world he went on to produce works of even greater scope: in 1963 a _History of Civilizations_ (in the English translation), and then a three-volume study of material civilization and capitalism from the fifteenth to the eighteenth century. Braudel's geographical and economic interests and his interdisciplinary inclinations naturally pointed towards the perspective of world history. Characteristically, he was anxious to create a basis for empirical historical research on it, though in this respect his legacy in France has not been as vigorous as he must have hoped.
The connection of world history with social-science concepts and themes is not accidental. In its evolutionary days, in the later nineteenth century, sociology was often, as in Herbert Spencer, a highly abstract and schematized history of the social development of mankind, while the more empirically substantial and historically rooted work of Max Weber also pointed outward to non-European societies, most notably China, which could be drawn on for the application of such characteristically Weberian concepts as "bureaucracy," "traditional authority" and "charisma."
World or global history necessarily raises questions of definition. These are helpfully discussed in _Writing World History 1800–2000_ (eds. B. Stuchtey and E. Fuchs), published under the auspices of the German Historical Institute in London in 2003, which also provides an introduction to past and recent work. World history, being essentially comparative and concerned with, in Braudel's phrase, _la longue durée,_ needs large-scale organizing categories which are not distinctively national or specific to a narrowly defined period, and it is therefore likely that it will share these with other disciplines. Histories of societies recently emerging from alien rule, written from their own perspective rather than that of their former rulers, may make contributions to it, but in themselves they are candidates for the long-established genre of national histories, including, sometimes, references to emancipation from alien oppression or domination. Stubbs's _Constitutional History,_ in which the Normans are the colonial power, is describable in these terms, as is Thierry's _Conquest of England_ and a good deal of German history. But in dealing with wider themes and aiming at comprehensiveness, projected world histories necessarily focus on themes which are also of interest in other disciplines: cultural contact and exchange, economic take-off and the world economy, colonization and decolonization, slavery, migrations, urbanization, industrialization and other experiences loosely grouped as modernization. Archaeology, anthropology, geography, sociology and economics are all relevant.
Turning from the global to the village or other small community at a particular place and time brings us to micro-history. Its affinities are with anthropology, from which it has derived ideas about the life of small-scale societies, but also with the novel and with biography—though the biography of the obscure and even inarticulate. The three examples presented here can form not a conclusion or climax, but a suitable kind of coda to this book; they are respectively Italian, French and English. They are research-driven rather than theory-driven: that is to say they originate from a particular archive or text and a historian with the imagination to see its potentialities.
Characteristically, micro-history takes a small area, a narrow time band, perhaps a protagonist, though one with varying degrees of dominance, and a small community. It illuminates something more general than itself, but it is not necessarily to be thought of exactly as "evidence" of a given type, a brick to be added to an edifice of generalization built on the accumulation of cases, though in principle it could eventually be used in this way. Sometimes its subject is a single central event, or a sequence of them, understanding whose meaning requires painstaking scholarship, and perhaps some resort to available generalizations from elsewhere; sometimes its events are more diffuse, though focused in a certain way. In other words there may be unity of character and plot or something looser—an area, a particular time, a pattern of behaviour and of beliefs. There is typically a single main archival source, with Inquisition, court and parochial records prominent.
A popular and well-known early example was Emmanuel Le Roy Ladurie's _Montaillou_ (1975), a study in the mentality of peasants in a village in Languedoc in the early fourteenth century, filtered through the records of the interrogations conducted by the Inquisition investigating the Cathar heresy, locally rife. It grew in a sense out of the _Annales_ genre of area studies: an earlier, more general, work of Ladurie's was _The Peasants of Languedoc_ (1966). A later work, also in the _Annales_ tradition, has been _Carnival_ (trans. 1980). Natalie Zemon Davis's _The Return of Martin Guerre_ (1983), a vignette of French rural life in the sixteenth century, focused on marital relations, has been made into a film. _The Great Cat Massacre_ (1984), by Robert Darnton, a distinguished American historian of eighteenth-century France, was derived, he tells us, from a class in Princeton taught jointly with the anthropologist Clifford Geertz: the title essay interprets a bizarre act of hostility by the employees of a Parisian printer in the 1730s.
Carlo Ginzburg's _The Cheese and the Worms: The Cosmos of a Sixteenth-Century Miller_ (1976) is, like _Montaillou,_ based on Inquisition records. Ginzburg's protagonist is an autodidact miller known as Menocchio; he was born in northern Italy in 1533, and was burnt as a heretic, after being given, as was customary, a second chance by the church authorities, in 1599. He was denounced, not altogether surprisingly, by the parish priest, with whom he argued and whose authority he therefore challenged. He paid the penalty of an apparently irrepressible impulse to argue and to share his idiosyncratic ideas on religion with his uncomprehending fellow villagers. It was this inveterate disposition, exercised without associates and apparently for no other reason than his own satisfaction and because he could not help it, that eventually wore out the Inquisition's patience. He was sentenced, imprisoned, and in due course released, but his candour survived the ordeal and he was eventually burnt. There was nothing half-hearted about his heretical opinions, though they were peculiar to himself and associated with no sect. He was a solid man, apparently popular, who had begotten eleven children and been mayor of his village, but, though his views are fascinating when dissected by his historian, one begins to feel he was a man to avoid in a taverna.
His opinions were very unorthodox indeed. His conception of the world as a cheese inhabited by worms—a miller would be familiar with worms, which would appear to him to be self-generating—was only the beginning. The worms were angels, and God was created at the same time as they were. Moreover, and the Inquisition thought inconsistently, "Everything we see is God, and we are gods." Jesus was an ordinary man—the Virgin Birth was dismissed on common-sense grounds—and lower than the Holy Spirit, which was in all men, including infidels. The Scriptures were partly true and partly not, and clerical ordination was worthless. Almost Menocchio's only stock opinion seems to have been his disapproval of the gap between rich clerics and the poor. Ginzburg is very aware of the possibility of distortion of Menocchio's views, both as recorded by his scandalized but frequently baffled interrogators and as coming from a man whose life might depend on his answers. In general, however, Menocchio's honesty seems to have been as marked as his garrulity. What he thought, he said, and what he said was very odd indeed. It was not, however, entirely evolved out of his own mind.
For Menocchio was literate, and an avid reader of the books that randomly came his way; he and his opinions were made possible only by literacy and books. The case of Menocchio is explored by Ginzburg to show the inadequacy of studying the history of literacy simply through titles and statistics of book production. We have to try to understand _how_ books were read and what was made of that reading, by someone like Menocchio. Inferring ingeniously from Menocchio's garbled answers to his inquisitors, and drawing on his own erudition in the literary culture of the Renaissance, Ginzburg uncovers the archaeology of Menocchio's remarkable opinions and makes some sense of them.
It is clear that Menocchio was far from merely reproducing opinions he had read. He had read little, and he fastened with all the greater intensity on what he had haphazardly found, and reshaped it in terms of his own mental and social world. This was what produced the opinions which so startled and alarmed the court which tried him. He read actively, supplying his own emphases and interpretations. He read as an autodidact reads, ignoring much, fastening on what appealed to him and elaborating it. He was certainly no fanatic: it was reading and indiscretion, not religious zeal, that was his undoing. Menocchio had learned a kind of religious relativism and a general tolerance from reading accounts of other religions and beliefs, which he found in the travels of Sir John Mandeville. He even excavated heresy from Boccaccio. His authors would not have been so indiscreet as he was. It was all too much.
In Ginzburg's hands the case of Menocchio rebukes too simple an understanding of the idea of popular culture and also too simple a conception of literacy and the dissemination of books, and their implications. Menocchio died horribly partly because he was a man at the interface of two worlds, literate but uneducated.
Bryce had made room for one additional category as worthy of the _EHR_ : those who, like Luther, had changed the world's opinions. Menocchio, a heretic but not a heresiarch, had hardly done that, though his opinions are what make him interesting. Yet few if any readers would deny that Ginzburg's study of him is history "properly so called," as Carr would say, and a distinguished example of it. This gives a measure of an intellectual distance travelled, not only since the 1880s but since the 1960s. For it is not clear that Menocchio "did" anything but read and talk too much for his own good, but that is surely not the point.
Another case: Alain Corbin, in _The Village of Cannibals: Rage and Murder in France, 1870_ (1990), investigates the public torture and murder by a number of assailants of a young aristocrat in a village in the Dordogne during the dying days of the Bonapartist Second Empire. Corbin's work lies recognizably in the tradition of the _Annales_ ; he is also known as the author of a history of smells (1982, trans. as _The Foul and the Fragrant: Odour and the French Social Imagination_ ). _The Village of Cannibals_ is a study in rumour, class hatred and political attitudes. To the outsider they seem confused, but Corbin makes them intelligible. The murder was the immediate outcome of acute, misdirected collective anxiety seeking an outlet. French defeats on the frontiers by the Prussians were made ominous by recollections of earlier Prussian invasions, in 1793 and 1815, the first threatening counter-revolution, the second instrumental in restoring the Bourbon monarchy. Counter-revolution, to the peasants, meant the possible revival of feudal obligations. Therefore aristocrat = Prussian, and the unfortunate young man—who was personally popular and not a harsh landlord or politically active—was described as a Prussian by his murderers and tormentors. He was also, falsely, and one would have thought inconsistently, alleged to have called out "Long live the Republic." Hostility to republicans seems to have dated from unpopular taxation during the Second Republic twenty years earlier.
The Bonapartist murderers were not his immediate neighbours, who seem to have looked on or, in the case of the mayor, protested half-heartedly. They were gathered from the surrounding area for a village fair, and did not know him personally. In the afternoon, for several hours, they pounded him to death, almost ritually, each taking a share, with agricultural implements, as though threshing, and then burnt his body—an act which was referred to as like roasting a pig. There was no literal cannibalism: that was a term of horror adopted in wider, shocked comment on the deed, which appalled the general public, irrespective of political inclinations.
Corbin turns this into an illustration of changing attitudes to violence and mutilation of the body. To the peasants who perpetrated the deed, used to killing animals, what they did seemed natural as well as politically virtuous. They did it openly, as a patriotic act, and expected to be congratulated and rewarded. But there had been, fairly recently Corbin claims, a marked change in attitudes to the body and to pain. The ringleaders, who were subsequently tried and executed, were caught in a cultural time-lag between two historical types of sensibility. Corbin tells his story dramatically, and through it he reveals much in the politics and sentiments of the time and place that would otherwise be baffling.
Finally an English case: a remote Exmoor parish in the south-west of England in the age of the Reformation, described in Eamon Duffy's _The Voices of Morebath_ (2001). Morebath was in a conservative part of the country, but the parish seems to have been made unusual chiefly by the particularly full and vividly presented parish accounts, kept by the priest, Sir Christopher Trychay. ("Sir" was the normal prefix for the ordinary secular clergy.) Sir Christopher's incumbency spanned the whole course of the Reformation, from the solidly Catholic earlier years of Henry VIII, in 1520, to 1574 in the reign of Elizabeth. He was therefore parish priest not only when the first shocks were felt from Henry's breach with Rome, but through the vigorously reforming years of Edward VI's reign and the Catholic revival under Mary, and a further if rather milder re-Protestantizing of worship after the accession of Elizabeth. Duffy obviously likes Sir Christopher, and treats sympathetically his clearly reluctant bending to each wave of reforming zeal: loyal in his heart to the Catholic old regime, his strongest commitment was to the parish. Duffy does not milk the situation for pathos, but the poignancy of Sir Christopher's situation must be apparent to every reader, while, despite the author's restraint, even keenly Protestant ones may end by loathing the zealously reforming Dean of Exeter, the proximate source of enforced innovation, representative of the distant and fearsome Privy Council. Dean Heynes, in offending the conservative chapter of Exeter Cathedral by his contempt for tradition, may remind more than one reader of Trollope's Mr. Slope, though without his hypocrisy.
Duffy uses the priest's accounts, which he quotes at length and "translates," to reconstruct the rich and complex life of the parish centred on the church and its worship in the pre-Reformation period. It was focused largely on the cults of the various saints in the parish church, which were the responsibility of different groups of villagers, represented in rotation by various "wardens." Because of the wardens' duty to report on their financial stewardship to the parish, their reports figure largely in the priest's account books. It is clear that Sir Christopher wrote up all the reports on the wardens' behalf. Rotation meant that wardenships fell to poor men and minors, so that this was probably very necessary, and he seems to have presented the reports by reading them aloud. The entries have the characteristics of the spoken word, and the priest clearly used the public meetings at which they were read to encourage devotion and to conciliate in cases of dispute. It all presents a picture of very wide social involvement by the parishioners, and also the value attached to consensus as well as accountability. As Duffy says, "The accounts are saturated with a rhetoric of collective identity and shared responsibility."
The cults of the church's particular saints were central to all this, and their extinction, on the orders of distant authority, had a devastating effect on the social interactions of the parishioners: involvement perceptibly declined and narrowed as a result. Responsibility for the money used for the adornment and upkeep of the saints' images and tabernacles, and their lights, having been assigned to specific groups, the funds so established were called "stores" and were designated by the saint's name. The roles of the young men and young women, corporately organized and with their own wardens (with heads of households sometimes acting for them), are particularly striking. There was the Young Men's store and the Maidens' store. The Young Men were also responsible for the management of the "Church Ales," periodic festivities at which they were expected to make a profit from sales. This profit was an important resource, and the banning of Church Ales by Puritan zeal was financially grievous. Otherwise the parish's collective wealth was in its sheep, assigned to sustaining the various stores for the upkeep of their sainted patrons, so that one could speak of "Our Lady's sheep," "St. Antony's sheep," "St. Sidwell's sheep" (St. Sidwell—female—was the only Exeter saint).
The destruction of much of this is the English Reformation in miniature, and it tilted the balance from the community to the individual. Duffy shows, from an obscure and half-obliterated entry, misread previously, that five young men were equipped by the parish to join the 1549 West Country rebellion against the religious changes and several may have died. Duffy is a leading authority on the English Reformation; his other well-known work is a general study, _The Stripping of the Altars: Traditional Religion in England c. 1400–c.1580_ (1992). His study of Morebath could not have been made so illuminating without that general erudition, but it is also a remarkable tour de force to have reconstructed so much of the parish's life, and the personality of Sir Christopher Trychay, from a single, apparently unpromising, source. It required exceptional imagination and sensitivity, as well as erudition, to bring the latter to life.
There is some similarity, of course, to the medieval monastic chroniclers' recording of the lives of their communities, with special attention to benefactions, which needed to be remembered in prayers and kept accounted for. There is even a parallel with what was described above, in the Prologue, as one of the earliest impulses to historical recording and narrative: the ruler's gifts to the god, housed in the temple. It is fascinating, with Duffy's guidance, to see lists turning into narratives in a similar fashion, and the occasions are the same: recording and accounting for offerings to the god, or in this case Christ and the saints. The narratives are brief, but rhetorically vastly different from just the recitation of lists. They explain how the gifts came to be made, and paid for, and what is done with them: "Item as for the 6/8d for Roger Budd's grave with other bequests...it is assured that Agnes Budd will be here shortly and pay it [wylbe here schortly and pay hyt]" The past comes to life in a sentence, and more fully in
Item we received by the death of Agnes at Hayne [a surname] a gown and a ring at price of 12.s, in the which money Nicholas at Hayne is contented to send to London by William Hurley, when he goeth thither next, and to buy us a banner of silk and so to bring it into this church ["this" indicates where the account was presented]: and if the banner do not cost the full 12.s he saith ye shall have that is left when the banner cometh in to this church.
The Morebath accounts were first edited, with some errors, by a Devonshire antiquarian, in 1904. We should be grateful that they were preserved, and have to applaud the interest that prompted the task of editing. But if there is continuity here there is also a large intellectual distance. Questions of individual talent and imagination apart, the earlier period alone would surely have prevented the first editor from producing anything at all like _The Voices of Morebath._ But why? Duffy does not speculate, nor was he obliged to. It is not a responsibility of busy historians to make life easy for the historian of their craft, a marginal figure at best. But perhaps methodological reticence, not shared by Alain Corbin, who has a French attitude to methodological matters, is itself significant. There are no general references to conflict resolution in small-scale societies, or to the social functions of ritual—which, in general terms, a Catholic can assume anyway. But in seeing no need for these, Duffy, apart from personal inclination, is (especially if one compares this with Thomas's justified urgency in 1963) perhaps registering by omission a greatly widened imaginative consensus among historians about what it is possible for them to do.
This book can have no conclusion: the study and writing of history is still going on. Technological innovations may already be pointing to a new era. From the 1970s onwards, microfilm gave historians access at a distance to manuscripts as well as published works. The Internet now opens even greater possibilities for research, whose boundaries and implications we cannot yet see. For the moment, the greatest technological innovation in the writing of history remains the printing press, which immediately guaranteed the survival of works which might in the past have been lost. But we should remember, as we noted in chapter 21, how gradually the full implications of the printing press were absorbed, enabling for example Robertson, in the eighteenth century (with some help from Spain), to write an erudite history of the reign of the emperor Charles V without leaving Edinburgh. It is probably safe to say that the Internet will result in _more_ history being written, if only because it reduces research time; we may not know whether that history will be different or better for some while.
In the presentation of history, the chief new medium is obviously television. Access to film archives is an important aspect, but has been a mixed blessing, concentrating attention on the twentieth century and on what news agencies, film makers and government propaganda thought worth recording. Television programmes, apart from this, are often like extensions, to a vast audience, of the old formula of the lantern-slide lecture: a presenter or commentator supported by images. They can be well done, as some of the examples mentioned earlier in this chapter, or badly, of which there is now a plethora. But genuinely new possibilities are also being explored, as with great distinction in the remarkable series made by Ken Burns on the American Civil War. Restrained but informative commentary, sensitive editing, haunting photographs and music and readings from letters and diaries made this a deeply moving production, matching the scale of the events it recounted in a way no printed book could do. Considered as the presentation of an epic theme on a grand scale this has claims to be the outstanding work of history of the late twentieth century.
So there is no conclusion, only extensions of existing possibilities, and only the ending of a book. I hope it will be attributed more to modesty than to self-importance if I finish on a purely personal note. I have had two purposes. First, to convey the particular qualities of the histories discussed, and particularly what makes them readable with pleasure by non-specialists. In doing so I have also tried to convey the authors' intentions in writing them. Second, I have tried, as in this chapter for the most part, to trace and if possible to explain major shifts in the ways in which attention to the past was directed and applied. But I have always been conscious of what historians share. The notion that they formed a kind of community was registered as long ago as by Polybius, in his belief, probably correct, that if he died before he completed his history another historian would carry the subject on (Chapter 4). Medieval chroniclers were intrinsically collaborative, though sequentially. It hardly needs stressing that there is now a professional community—or perhaps it would be more realistic to say a set of overlapping ones—and, though not always recognized, past historiography is part of it. Historians are, after all, interested in the past. One can never tell when or how the work of a dead historian will suddenly seem relevant, or evoke fellow feeling or annoyance, as with a colleague. We have the advantage of hindsight, but historians have learned to be wary of overexploiting this. One has usually only to utter the dreaded word "whig" to induce a sudden modesty; one of the advantages of hindsight is to have learned not to abuse it.
I have, as I say, always been aware here of writing about a practice which stretches back two and a half thousand years, and of its practitioners as a kind of community of the dead and the living. To all of them the past mattered: it was worth investigating and recording and keeping alive for future generations. It is with the same respect and interest, and often with admiration, that I have written about them. To some I have done scant justice; to innumerable others, some of whom I am guiltily aware of, I have been able to do none at all.
**Select Bibliography**
Most translations from Greek, Latin, French and Spanish are those of editions of the texts published by Penguin Books. I have very occasionally provided an alternative version of the translation of a particular quotation. I am also indebted to the very helpful introductions to these editions. The place of publication of these is in all cases London.
The citations to the texts of ancient historians are to books and the numbered subsections into which they are conventionally divided, not to pages. Reference can therefore be made to any edition. This applies also to the section numbers given for Vico, Hegel and Stubbs. Dates given in this bibliography are of the editions used, not of first publication, which are normally provided in the main text. The list of secondary works does not claim to be comprehensive: I have selected the works I have found particularly useful, and I hope their authors will accept this general acknowledgement of my indebtedness to them. Occasionally I have cited a modern author in my own text, where I was conscious of following a particular interpretation especially closely and wished to acknowledge the fact.
**GENERAL**
Boyd, K. (ed.), _Encyclopaedia of Historians and Historical Writing_ (2 vols., London, 1999)
Cameron, J., et al. (eds.), _The Blackwell Dictionary of Historians_ (Oxford, 1988)
Hay, D., _Annalists and Historians: Western Historiography from the VIIIth to the XVIIIth Century_ (London, 1977) is more limited in scope but useful.
Kelley, D. R., _Faces of History: Historical Writing from Herodotus to Herder_ (New Haven and London, 1988)
Momigliano, A., _Essays in Ancient and Modern Historiography_ (Oxford, 1977)
Thompson, J. W., _A History of Historical Writing_ (2 vols., New York, 1942) is still useful for reference.
Woolf, D. R. (ed.), _A Global Encyclopaedia of Historical Writing_ (New York and London, 1998)
Two classics which I have often disagreed with but which are still a source of inspiration are E. Auerbach, _Mimesis: The Representation of Reality in Western Literature,_ trans. W. R. Trask (Princeton, 1953), and R. G. Collingwood, _The Idea of History_ (Oxford, 1946).
**PROLOGUE**
Butterfield, H., _The Origins of History_ (London, 1981)
Gardiner, A., _Egypt and the Pharaohs_ (Oxford, 1961)
Gurney, O. R., _The Hittites_ (rev. edn, London, 1990)
Staggs, H. W. F., _The Babylonians_ (2nd edn, London, 1998)
**GREECE**
**TEXTS**
Arrian, _The Campaigns of Alexander,_ trans. A. de Sélincourt, intro. and notes J. R. Hamilton (Penguin, 1971)
Herodotus, _The Histories,_ trans. A. de Sélincourt, rev. with intro. and notes J. Marincola (Penguin, 2003)
Quintus Curtius Rufus, _The History of Alexander,_ trans. J. Yardley, intro. and notes W. Heckel (Penguin, 1984)
Thucydides, _History of the Peloponnesian War,_ trans. R. Warner, intro. and notes M. I. Finley (Penguin, 1972)
Xenophon, _The Persian Expedition,_ trans. R. Warner, intro. and notes G. Cawkwell (Penguin, 1972)
**SECONDARY**
Adcock, F., _Thucydides and His History_ (Cambridge, 1963)
Bengtson, H., et al., _The Greeks and the Persians: From the Sixth to the Fourth Century_ (London, 1968)
Connor, W. R., _Thucydides_ (Princeton, 1984)
Fornara, C. W., _Herodotus: An Interpretative Essay_ (Oxford, 1971)
——— _The Nature of History in Ancient Greece and Rome_ (Berkeley and London, 1983)
Gould, J., _Herodotus_ (London, 1989)
Hornblower, S., _Thucydides_ (London, 1987)
Marincola, J., _Authority and Tradition in Ancient Historiography_ (Cambridge, 1997)
Moxon, I. S., et al. (eds.), _Past Perspectives: Studies in Greek and Roman Historical Writing_ (Cambridge, 1986)
Murray, O., _Early Greece_ (London and Cambridge, Mass., 1993)
Pearson, L., _Early Ionian Historians_ (Oxford, 1939)
Thomas, R., _Herodotus in Context: Ethnography, Science and the Art of Persuasion_ (Cambridge, 2000)
Usher, S., _The Historians of Greece and Rome_ (Bristol, 1985)
For Xenophon, Arrian and Curtius Rufus consult the introductions to the Penguin editions.
**ROME**
**TEXTS**
Ammianus Marcellinus, _The Later Roman Empire AD 354–378_, selected and trans. W. Hamilton, intro. and notes A. Wallace-Hadrill (Penguin 1976)
Appian, _The Civil Wars,_ trans. and intro. J. Carter (Penguin, 1996)
Cassius Dio, _The Roman History: The Reign of Augustus,_ trans. I. Scott-Kilvert, intro. J. Carter (Penguin, 1987)
Josephus, _The Jewish War,_ trans. G. A. Williamson, ed. and intro. E. M. Smallwood (Penguin, 1981)
Livy, _The History of Rome from its Foundation:_
(I–V) _The Early History of Rome,_ trans. A. de Sélincourt, intro. R. M. Ogilvie (Penguin, 1971)
(VI–X) _Rome and Italy,_ trans. B. Radice, intro. R. M. Ogilvie (Penguin, 1982)
(XXI–XXX) _The War with Hannibal,_ trans. A. de Sélincourt, ed. with intro. B. Radice (Penguin, 1965)
(XXXI–XLV) _Rome and the Mediterranean,_ trans. H. Bettenson, intro. A. H. McDonald (Penguin, 1976)
Plutarch, _Makers of Rome,_ trans. and intro. I. Scott-Kilvert (Penguin, 1985)
Polybius, _The Rise of the Roman Empire,_ trans. I. Scott-Kilvert, selected and intro. F. Walbank (Penguin, 1979)
Sallust, _The Jugurthine War. The Conspiracy of Catiline,_ trans. and intro. S. A. Handford (Penguin, 1963)
Suetonius, _The Twelve Caesars,_ trans. R. Graves, ed. and intro. J. B. Rives (Penguin, 2007)
Tacitus, _The Agricola and the Germania,_ trans. and intro. H. Mattingly (Penguin, 1970)
——— _The Annals of Imperial Rome,_ trans. and intro. M. Grant (Penguin, 1959)
—— _The Histories,_ trans. and intro. K. Wellesley (Penguin, 1975)
**SECONDARY**
Dorey, T. A. (ed.), _Latin Historians_ (London, 1966)
Grieve, E. S., _The Image of Rome_ (Princeton, 1969)
Wiseman, T. P., _Historiography and Imagination: Eight Essays on Roman Culture_ (Exeter, 1994)
Also, Fornara, _The Nature of History,_ Marincola, _Authority and Tradition,_ and Usher, _The Historians of Greece and Rome,_ all cited above.
Barnes, T. D., _Ammianus Marcellinus and the Representation of Historical Reality_ (Ithaca, NY, 1998)
Barrow, R. H., _Plutarch and his Times_ (London, 1967)
Dorey, T. A. (ed.), _Tacitus_ (London, 1969)
Earl, D. C., _The Political Thought of Sallust_ (Cambridge, 1961)
Jones, C. P., _Plutarch and Rome_ (Oxford, 1971)
Luce, T. J., _Livy: The Composition of His History_ (Princeton, 1977)
Martin, R., _Tacitus_ (London, 1981)
Millar, F., _A Study of Cassius Dio_ (Oxford, 1964)
Walbank, F., _Polybius_ (Berkeley and London, 1972)
Wallace-Hadrill, A., _Suetonius: The Scholar and His Caesars_ (London, 1983)
Walsh, P. G., _Livy: His Historical Aims and Methods_ (Cambridge, 1961)
Woodman, A. J., and Luce, T. J. (eds.), _Tacitus and the Tacitean Tradition_ (Princeton, 1993)
**CHRISTENDOM AND THE BARBARIANS**
**TEXTS**
Eusebius, _The History of the Church,_ trans. G. A. Williamson, rev. edn and intro. A. Louth (Penguin, 1989)
_The Fourth Book of the Chronicle of Fredegar with its Continuations,_ trans. J. M. Wallace-Hadrill (London, 1960)
Gregory of Tours, _The History of the Franks,_ trans. and intro. L. Thorpe (Penguin, 1974)
Munz, A., _From Roman to Merovingian Gaul: A Reader_ (Peterborough, Ont., 1999)
**SECONDARY**
Brown, P., _The Cult of the Saints: Its Rise and Function in Western Christianity_ (London, 1988)
—— _The Rise of Western Christendom: Triumph and Diversity AD 200–1000_ (Oxford, 1996)
Chadwick, H., _The Early Church_ (London, 1960)
Dahmus, J., _Seven Medieval Historians_ (Chicago, 1982)
Markus, R. A., _Saeculum: History and Society in the Theology of St. Augustine_ (Cambridge, 1970)
Smalley, B., _The Study of the Bible in the Early Middle Ages_ (London, 1974)
For figural interpretation of the Bible the classic source is E. Auerbach, _Scenes from the Drama of European Literature_ (Gloucester, 1973).
Barnes, A. T. D., _Constantine and Eusebius_ (London, 1981)
Wallace-Hadrill, D. S., _Eusebius of Caesarea_ (London, 1960)
Wallace-Hadrill, J. M., _The Barbarian West 400–1000_ (London, 1967)
—— _The Long-Haired Kings_ (London, 1962)
Wormald, P., et al. (eds.), _Ideal and Reality in Frankish and Anglo-Saxon Society_ (Oxford, 1983)
Young, F., _From Nicaea to Chalcedon_ (London, 1983)
**BEDE AND ANGLO-SAXON ENGLAND**
**TEXTS**
_The Anglo-Saxon Chronicle,_ ed. D. C. Douglas and D. Whitelock (Norwich, 1961)
Bede, _A History of the English Church and People,_ trans. and intro. L. Shirley-Price, rev. edn R. E. Latham (Penguin, 1968)
_English Historical Documents,_ vol. 1: _c.500_ – _1042_ , ed. D. Whitelock (London, 1968)
**SECONDARY WORKS**
Bonner, G. (ed.), _Famulus Christi: Essays in Commemoration of the Thirteenth Centenary of the Venerable Bede_ (London, 1976)
Campbell, J., _Essays in Anglo-Saxon History_ (London, 1986)
Mayr-Harting, H., _The Coming of Christianity to Anglo-Saxon England_ (3rd edn, London, 1991)
**BRITISH HISTORY**
**TEXTS**
Gildas, _The Ruin of Britain and other Works,_ ed. and trans. M. Winterbottom (London, 1978)
Nennius, _History of the Britons,_ ed. J. A. Giles in _Six Old English Chronicles_ (London, 1848)
**SECONDARY WORKS**
Hanning, R. W., _The Vision of History in Early Britain: From Gildas to Geoffrey of Monmouth_ (London, 1966)
Leckie, R. W., Jr., _The Passage of Dominion: Geoffrey of Monmouth and the Periodization of Insular History in the Twelfth Century_ (Toronto, Buffalo and London, 1981)
**MEDIEVAL ENGLAND**
**TEXTS**
_The Chronicle of Jocelin of Brakelonde,_ trans. H. E. Butler (London, 1949)
_Chronicles of Matthew Paris,_ ed. and trans. A. Vaughan (Gloucester, 1984); quotations from _Deeds of the Abbots_ are from this edition.
_The Ecclesiastical History of Ordericus Vitalis (IV_. _vii–viii_ ), ed. and trans. M. Chibnall (Oxford, 1975)
_English Historical Documents,_ vol. 2: _1042–1189_ , ed. D. C. Douglas and G. W. Greenaway (London, 1953)
Geoffrey of Monmouth, _The History of the Kings of Britain,_ trans. and intro. L. Thorpe (Penguin, 1966)
_The Illustrated Chronicles of Matthew Paris,_ ed. and trans. A. Vaughan (Stroud, 1993); quotations from the _Greater Chronicle_ are from this edition.
William of Malmesbury, _Historia Novella: The Contemporary History,_ ed. E. King, trans. K. R. Potter (Oxford, 1998)
——— _The Kings of England,_ trans. J. A. Giles (London, 1847)
**SECONDARY WORKS**
Galbraith, V. H., _Kings and Chroniclers: Essays in English Medieval History_ (London, 1982)
Gransden, A., _Historical Writing in England, c.550–1307_ (London, 1974)
Hay, _Annalists and Historians,_ cited above, is useful here.
**CRUSADER AND CHIVALRIC HISTORY**
**TEXTS**
_The Alexiad of Anna Comnena,_ trans. E. R. A. Sewtor (Penguin, 1969)
Jean Froissart, _Chronicles,_ selected, trans. and ed. G. Brereton (Penguin, 1968)
Joinville and Villehardouin, _Chronicles of the Crusades,_ trans. and intro. M. R. B. Shaw (Penguin, 1963)
**SECONDARY WORKS**
Barber, R., _The Knight and Chivalry_ (London, 1970)
Keen, M., _Chivalry_ (New Haven and London, 1984)
Riley-Smith, L. and J., _The Crusades: Ideas and Reality 1095–1292_ (London, 1981)
**CIVIC CHRONICLE AND HUMANIST HISTORY: THE RENAISSANCE**
**TEXTS**
Guicciardini, F., _History of Italy,_ trans. S. Alexander (New York and London, 1972)
Machiavelli, Niccolò, _The Florentine History,_ trans. N. H. Thomson (2 vols., London, 1906)
Villani, Giovanni, _Florentine Chronicle: Selections,_ ed. P. H. Wicksteed (London, 1906)
**SECONDARY WORKS**
Baron, H., _In Search of Florentine Civic Humanism_ (Princeton, 1988)
Fryde, E. B., _Humanism and Renaissance Historiography_ (London, 1983)
Gilbert, F., _Machiavelli and Guicciardini: Politics and History in Sixteenth-Century Florence_ (Princeton, 1965)
Grafton, A., and Blair, A., _The Transmission of Culture in Early Modern Europe_ (Philadelphia, 1990)
Green, L., _Chronicle into History: An Essay on the Interpretation of History in Florentine Fourteenth-Century Chronicles_ (Cambridge, 1972)
Phillips, M., _Francesco Guicciardini: The Historian's Craft_ (Toronto, 1977). I am greatly indebted to this book in the section on Guicciardini.
—— _Marco Parenti: A Life in Medici Florence_ (London, 1989). This is an original study of civic life and intellectual history based on the papers of a fifteenth-century Florentine citizen.
Reynolds, L. D., and Wilson, N. G., _Scribes and Scholars: A Guide to the Transmission of Greek and Latin Literature_ (Oxford, 1974). Also useful for the classical inheritance in the Middle Ages.
Skinner, Q., _The Foundations of Modern Political Thought_ (2 vols., Cambridge, 1978), vol. 1
—— _Machiavelli_ (Oxford, 1981)
—— _Visions of Politics_ (3 vols., Cambridge, 2002), vol. 2
**THE SIXTEENTH AND SEVENTEENTH CENTURIES**
**TEXTS**
Bacon, Francis, _The History of the Reign of King Henry VII_ (Cambridge, 1998)
Buchanan, George, _The Tyrannous Reign of Mary Stewart_ (Edinburgh, 1958)
Clarendon, Lord (Edward Hyde), _The History of the Rebellion and Civil Wars in England_ (6 vols., Oxford, 1961)
Hotman, F., _Franco-Gallia_ in _Constitutionalism and Resistance in the Sixteenth Century,_ ed. J. Franklin (New York, 1969)
_The Political Works of James Harrington,_ ed. J. Pocock (Cambridge, 1977)
**SECONDARY WORKS**
Douglas, D. C., _English Scholars 1660–1730_ (London, 1951)
Ford, F. L., _Robe and Sword: The Regrouping of the French Aristocracy after Louis XIV_ (Cambridge, Mass., 1962). Discusses the continuation of ancient constitutionalist arguments into the eighteenth century.
Grafton, A., _The Footnote: A Curious History_ (London, 1997)
Hale, J. R., _The Evolution of British Historiography from Bacon to Namier_ (London, 1967)
Haller, W., _Foxe's Book of Martyrs and the Elect Nation_ (London, 1963)
Huppert, G., _The Idea of Perfect History: Historical Erudition and Historical Philosophy in Renaissance France_ (Urbana, Ill., 1970)
Kelley, D. R., _Foundations of Modern Historical Scholarship in Language, Law and History in the French Renaissance_ (New York and London, 1970)
Kendrick, T. D., _British Antiquity_ (London, 1970)
Levy, F. J., _Tudor Historical Thought_ (San Marino, Cal., 1967)
McKisack, M., _Medieval History in the Tudor Age_ (Oxford, 1961)
Phillipson, N., and Skinner, Q. (eds.), _Political Discourse in Early Modern Britain_ (Cambridge, 1993)
Piggott, S., _Ruins in a Landscape_ (Edinburgh, 1976)
Pocock, J. G. A., _The Ancient Constitution and the Feudal Law: A Study in English Historical Thought in the Seventeenth Century_ (Cambridge, 1957)
—— _Politics, Language and Time: Essays in Political Thought and History_ (Cambridge, 1971)
Ranum, O., (ed.), _National Consciousness, History and Political Culture in Early Modern Europe_ (Baltimore, 1975)
Skinner, Q., _The Foundations of Modern Political Thought_ (2 vols., Cambridge, 1978), vol. 2
—— _Visions of Politics_ (3 vols., Cambridge, 2002), vol. 3
Smith Fussner, F., _The Historical Revolution: English Historical Writing and Thought, 1580–1640_ (London, 1962)
Wootton, D., _Paolo Sarpi: Between Renaissance and Enlightenment_ (Cambridge, 1983)
Worden, B., _Roundhead Reputations: The English Civil Wars and the Passions of Posterity_ (London, 2002). Indispensable for an understanding of the place of the Civil War period in English historical memory and scholarship.
Wormald, B. M. G., _Clarendon: Politics, Historiography and Religion 1640–1660_ (Cambridge, 1951)
**THE EIGHTEENTH CENTURY: THE ENLIGHTENMENT**
**TEXTS**
Bolingbroke, _Historical Writings,_ ed. I. Kramnick (Chicago, 1972)
Gibbon, Edward, _The History of the Decline and Fall of the Roman Empire,_ ed. D. Womersley (3 vols., Penguin, 1994)
Hume, David, _The History of Great Britain: The Reigns of James I and Charles I,_ ed. D. Forbes (London, 1970)
Lehmann, W. C., _John Millar of Glasgow_ (Cambridge, 1960) incorporates Millar's _Origin of the Distinction of Ranks_ (3rd edn, 1871)
Voltaire, _The Age of Louis XIV and Other Selected Writings_ , ed. and abridged J. H. Brumfitt (London, 1966)
_The Works of William Robertson_ (10 vols., London, 1821)
**SECONDARY WORKS**
Brown, S. J. (ed.), _William Robertson and the Experience of Empire_ (Cambridge, 1997)
Burrow, J. W., _Gibbon_ (Oxford, 1985)
Forbes, D., _Hume's Philosophical Politics_ (Cambridge, 1975)
Hont, I., and Ignatieff, M. (eds.), _Wealth and Virtue: The Shaping of Political Economy in the Scottish Enlightenment_ (Cambridge, 1983)
Kidd, C., _Subverting Scotland's Past: Scottish Whig Historians and the Creation of an Anglo-British Identity_ (Cambridge, 1993)
Kramnick, I., _Bolingbroke and His Circle: The Politics of Nostalgia in the Age of Walpole_ (Cambridge, Mass., 1968)
McKitterick, R., and Quinault, R. (eds.), _Edward Gibbon and Empire_ (Cambridge, 1997)
Meek, R. L., _Social Science and the Ignoble Savage_ (Cambridge, 1971)
Momigliano, A., _Studies in Historiography_ (London, 1966). The classic study of Gibbon's relation to the erudite tradition. For an aspect of the latter, see D. Knowles, _Great Historical Enterprises_ (London, 1963) on Benedictine historical scholarship.
O'Brien, K., _Narratives of Enlightenment_ (Cambridge, 1977)
Phillips, M., _Society and Sentiment: Genres of Historical Writing in Britain, 1740–1820_ (Princeton, 2000). I owe much to this pathbreaking book and to conversations with its author.
Phillipson, N., _Hume_ (London, 1989)
Pocock, J. G. A., _Barbarism and Religion_ (Cambridge, 1999–); four volumes of this monumental study of Gibbon and his intellectual context have so far appeared.
——— _The Machiavellian Moment: Florentine Political Thought and the Atlantic Republican Tradition_ (Princeton, 1975)
Womersley, D. (ed.), _Edward Gibbon: Bicentenary Essays_ (Oxford, 1997)
**REVOLUTIONS**
**TEXTS**
Carlyle, Thomas, _The French Revolution_ (London, 1902)
Macaulay, T. B., _The History of England from the Accession of James II_ (2 vols., London, 1903)
Michelet, Jules, _History of the French Revolution,_ abridged ed. G. Wright, trans. C. Cocks (Chicago and London, 1967)
Michelet's dedicatory epistle to _The People_ is translated in F. Stern, _The Varieties of History from Voltaire to the Present_ (Cleveland, 1956)
_The New Science of Giambattista Vico,_ trans. T. G. Bergin and M. H. Fisch (Ithaca, N.Y., and London, 1968) is considered here in connection with Michelet.
Taine, Hippolyte, _The French Revolution,_ trans. J. Durand (3 vols., Indianapolis, 2002)
**SECONDARY WORKS**
Burrow, J. W., _A Liberal Descent: Victorian Historians and the English Past_ (Cambridge, 1981). Part I elaborates some of the points made on Macaulay in the present book.
Clive, J., _Macaulay: The Shaping of the Historian_ (New York, 1973)
Mitzman, A., _Michelet, Historian: Rebirth and Romanticism in Nineteenth-Century France_ (New Haven, 1990). Useful on Michelet's biography, though the insistent Freudianism may put off some readers.
Rosenberg, J. D., _Carlyle and the Burden of History_ (Oxford, 1985)
Weinstein, L., _Hippolyte Taine_ (New York, 1972) is a useful introduction to Taine's thought generally.
**THE HISTORY OF FREEDOM**
**TEXTS**
Burckhardt, Jacob, _The Civilization of the Renaissance in Italy,_ trans. S. G. C. Middlemore (London, 1951)
Guizot, F., _History of Civilization in Europe,_ trans. W. Hazlitt, ed. L. Siedentop (London, 1997). The discussion of the context in the editor's introduction is more fully developed in Siedentop's _Tocqueville_ (Oxford, 1994), ch. 2.
_Selected Writings of Lord Acton,_ ed. Rufus J. Fears (3 vols., Indianapolis, 1985)
Stubbs, William, _The Constitutional History of England in its Origin and Development_ (3 vols., Oxford, 1873–8). For a vulgarized version see Edward Freeman, _The Growth of the English Constitution from the Earliest Times_ (London, 1872).
**SECONDARY WORKS**
Burrow, J. W., _A Liberal Descent_ (see above), Parts II and III for Stubbs and Freeman
McClelland, C. E., _The German Historians in England_ (Cambridge, 1971)
G. P. Gooch, _History and Historians in the Nineteenth Century_ (2nd edn, London, 1962) and J. W. Thompson, _A History of Historical Writing_ (cited above) are still useful for reference.
**AMERICA**
**TEXTS**
Adams, Henry, _The History of the United States of America during the Administrations of Jefferson and Madison,_ abridged edn E. Samuels (Chicago, 1967)
Bradford, William, _History of Plymouth Plantation, 1620–1647_ , ed. S. E. Morison (New York, 1966)
Díaz, Bernal, _The Conquest of New Spain,_ trans. J. M. Cohen (Penguin, 1963)
_The Literary Remains of William Hickling Prescott_ (Norman, Okla., 1961)
Parkman, Francis, _The Conspiracy of Pontiac and the Indian War after the Conquest of Caroda_ (2 vols., London, 1899)
——— _La Salle and the Discovery of the Great West_ (Bristol, 1962)
Prescott, William H., _History of the Conquest of Mexico_ (rev. edn, London, 1887)
**SECONDARY WORKS**
Bishop, F., _Henry Adams_ (Boston, 1979)
Gale, R. L., _Francis Parkman_ (New York, 1973)
Gay, P., _A Loss of Mastery: Puritan Historians of Colonial America_ (Berkeley and Los Angeles, 1966)
Kraus, M., _A History of American History_ (New York, 1937)
Levin, D., _History as Romantic Art: Bancroft, Prescott, Motley and Parkman_ (Stanford, Cal., 1954)
Miller, P., _The New England Mind_ (2 vols., Boston, 1953)
Vitzthum, R. C., _The American Compromise: Theme and Method in the Histories of Bancroft, Parkman and Adams_ (Norman, Okla., 1974)
Wills, G., _Henry Adams and the Making of America_ (Boston and New York, 2005)
**PROFESSIONALIZATION**
Heyck, T. W., _The Transformation of Intellectual Life in Britain_ (London, 1982)
Kenyon, J. P., _The History Men: The Historical Profession in England Since the Renaissance_ (London, 1983)
Levine, P., _The Amateur and the Professional: Antiquarians, Historians and Archaeologists in Victorian England 1838–1886_ (Cambridge, 1986)
Novik, P., _That Noble Dream: The "Objectivity" Question and the American Historical Profession_ (Cambridge, 1988)
Ringer, F., _The Decline of the German Mandarins: The German Academic Community 1890–1933_ (Cambridge, Mass., 1969)
Rothblatt, S., _The Revolution of the Dons: Cambridge and Society in Victorian England_ (Cambridge, 1981)
—— _Tradition and Change in English Liberal Education: An Essay in History and Culture_ (London, 1976)
Slee, P. R. H., _Learning and a Liberal Education: The Study of Modern History in the Universities of Oxford, Cambridge and Manchester, 1800–1914_ (Manchester, 1986)
Soffer, R., _Discipline and Power: The University, History and the Making of an English Elite 1870–1930_ (Stanford, Cal., 1994)
**GERMANY**
Barnard, F. M., _Herder's Social and Political Thought: From Enlightenment to Nationalism_ (Oxford, 1965)
Butterfield, H., _Man on His Past: The Study of the History of Historical Scholarship_ (Cambridge, 1969). On German historical scholarship before Ranke, the Göttingen school.
Gooch, _History and Historians in the Nineteenth Century_ (cited above), is useful on the "Prussian School."
Grafton, _The Footnote_ (cited above) gives a brilliant account of Ranke's methods in a perspective provided by Renaissance and eighteenth-century scholarship.
Iggers, G., _The German Conception of History: The National Trend of Historical Thought from Herder to the Present_ (Middletown, Conn., 1968)
Iggers, G., and Powell, J. M. (eds.), _Leopold von Ranke and the Shaping of the Historical Discipline_ (Syracuse, N.Y., 1990)
Krieger, L., _Ranke: The Meaning of History_ (Chicago, 1977)
Laue, H. von, _Leopold Ranke: The Formative Years_ (Princeton, 1950)
Meinecke, F., _Cosmopolitanism and the National State,_ trans. R. B. Kimber (Princeton, 1970)
—— _Historism: The Rise of a New Historical Outlook,_ trans. J. E. Anderson (London 1972). ("Historism" has not become current; "historicism" is the now usual translation of the German _Historismus._ ) Meinecke was both a commentator on and a participant in German historicism.
Ranum, O. (ed.), _National Consciousness, History and Political Culture in Early Modern Europe_ (Baltimore, 1975), ch. by L. Krieger
Reill, P. H., _The German Enlightenment and the Rise of Historicism_ (Berkeley, 1975)
Stuchtey, B., and Wende, P. (eds.), _British and German Historiography 1750–1950_ (Oxford, 2000)
**THE TWENTIETH CENTURY**
I have treated the citations of key texts in this chapter as self-explanatory. I have, however, given at the end the publication details of the three last works to be discussed in some detail.
The Critique of "Whig History": History as a Science and an Art
Bentley, M., _Modernising England's Past: English Historiography in the Age of Modernism 1870–1970_ (Cambridge, 2005). Balances the elements of persistence and of innovation. Particularly good on Butterfield and Namier.
—— _Modern Historiography: An Introduction_ (London, 1999)
——(ed.), _Companion to Historiography_ (London, 1997)
Cannadine, D., _G. M. Trevelyan: A Life in History_ (London, 1992)
Collini, S., _English Pasts: Essays in History and Culture_ (Oxford, 1999). Also relevant in the "Marxism" section, below, for Tawney.
Colley, L., _Namier_ (London, 1989)
Elton, G. R., _Modern Historians of Britain 1485–1945: A Critical Bibliography 1945–1969_ (London, 1970)
Lamont, W. (ed.), _Historical Controversies and Histories_ (London, 1998)
Huizinga and the _Annales_ School
**TEXTS**
Huizinga, J., _Men and Ideas: History, the Middle Ages, the Renaissance,_ essays trans. James S. Holmes and Hans van Marle (London, 1960)
—— _The Waning of the Middle Ages,_ trans. F. Hopman (Penguin, 1955)
_A New Kind of History from the Writings of Febvre,_ ed. P. Burke, trans. K. Folca (London, 1973)
**SECONDARY WORKS**
Burke, P., _The French Historical Revolution: The Annales School 1929–1989_ (London, 1990)
—— _Varieties of Cultural History_ (Ithaca, N.Y., 1997). Relevant also in the "Cultural History" section below.
Marxism
Coleman, D. C., _History and the Economic Past: The Rise and Decline of Economic History in Britain_ (London, 1987)
Eley, G., and Hunt, W. (eds.), _Revising the English Revolution: Reflections and Elaboration on the Work of Christopher Hill_ (London, 1988)
Furet, F., _Interpreting the French Revolution,_ trans. E. Foerster (Cambridge, 1981)
Hofstadter, R., _The Progressive Historians: Turner, Beard, Parrington_ (New York, 1968)
Kaye, H. J., _The British Marxist Historians_ (Cambridge, 1984)
Lucas, C. (ed.), _Rewriting the French Revolution_ (Oxford, 1991)
Cultural History, "History from Below" and World History
Bryce's anonymous "Prefatory Note" to the first _English Historical Review_ is reprinted in F. Stern, _The Varieties of History,_ cited above.
Burke, P. (ed.), _History and Social Theory_ (Cambridge, 1992)
——(ed.), _New Perspectives in English Historical Writing_ (2 vols., Cambridge, 2001)
Stuchtey, B., and Fuchs, E. (eds.), _Writing World History 1800–2000_ (Oxford University Press, 2003, for German Historical Institute, London)
Micro-History
**TEXTS**
Corbin, A., _The Village of Cannibals: Rage and Murder in France,_ 1870, trans. A. Goldhammer (Cambridge, 1992)
Duffy, E., _The Voices of Morebath: Reformation and Rebellion in an English Village_ (New Haven and London, 2001)
Ginzburg, C., _The Cheese and the Worms: The Cosmos of a Sixteenth-Century Miller,_ trans. J. and A. Tedeschi (London, 1980)
*1The author of the 1955 translation of Bede cited here spoke of his work as "a treasury of tales loved by every English child." That would clearly be over-optimistic now, and perhaps it was then, but the author of this book attended, sixty years ago, a school in which the "houses" were designated Angles, Saxons, Jutes and Vikings. It would now be fashionable to see in this an example of collective "memory" I have always suspected it was the headmasters's ironic comment on our level of literacy.
Return to text.
**A NOTE ABOUT THE AUTHOR**
John Burrow was for many years Professor of Intellectual History at the University of Sussex. From 1994 to 2000 he was the first Professor of Intellectual History at Oxford. He is author of _A Liberal Descent: Victorian Historians and the English Past; Gibbon; Whigs and Liberals: Continuity and Change in English Political Thought; The Crisis of Reason: European Thought 1848–1914_ ; and _That Noble Science of Politics_ , with Stefan Collini and Donald Winch. He will be Distinguished Visiting Professor at Williams College in Massachusetts in 2008.
ALSO BY JOHN BURROW
_Evolution and Society: A Study in Victorian Social Theory_
_A Liberal Descent: Victorian Historians and the English Past_
_Gibbon_
_Whigs and Liberals: Continuity and Change in English Political Thought_
_The Crisis of Reason: European Thought 1848–1914_
_That Noble Science of Politics_
(with Stefan Collini and Donald Winch)
THIS IS A BORZOI BOOK
PUBLISHED BY ALFRED A. KNOPF
Copyright © 2007 by John Burrow
All rights reserved. Published in the United States by Alfred A. Knopf, a division of Random House, Inc., New York.
www.aaknopf.com
Originally published in Great Britain by Allen Lane, an imprint of Penguin Books Ltd., London, in 2007.
Portions of this book previously appeared in
_The Yale Review_.
Knopf, Borzoi Books, and the colophon are registered trademarks of Random House, Inc.
Library of Congress Control Number: 2008920675
eISBN: 978-0-307-26852-5
v3.0
**A History of Histories?**
Why "A History of Histories," or, more explicitly, why not "The History of History"? History, even if we allow it to be in its broadest sense a single kind of activity, is nonetheless a highly diverse one. Plagues, invasions, emigrations; the foundation, working and development of constitutional arrangements and political systems; wars, external and civil, revolutions, changes in religion and culture, gradual or abrupt, the formation of various kinds of collective identity--confessional, national, ideological--providential history in the sense of the dealings of God with man: all these and much else are properly regarded as history. Some histories are virtually pure narrative; others are virtually pure, almost atemporal, analyses, being essentially structural or cultural surveys. History is contiguous with many other genres and lines of inquiry, from epic and myths of origin to various social sciences, and touching also biography, drama, political and moral polemic, ethnography, novels, inquests and judicial investigations. It was, so far as we know, Herodotus who first used the term _historia_ (inquiry) for what we call history. A _histor_ in Homer was someone who passed judgement based on the facts as a result of investigation, so the link between history and inquest is a very old one.
How can this variety be converted into a single historical narrative: "The History of History"? There is one answer which is obvious, being to some degree necessary to a narrative. This is by establishing a terminus, an end to which the episodes of the story are in some sense subordinate and contributing, so that they become moments in a progression. In the case of the history of historical writing--a genre which did not exist until the twentieth century--it was inevitable, given the period and its historiographical culture, that this was most commonly and easily done by taking the present state of the subject (or what it was assumed to be) as the terminus. By the early twentieth century this present state was characterized, variously but with a fair degree of consensus, as pure or "scientific" or (tacitly) as "professional" history, all identified perhaps with "the idea of history" or the study of the past "for its own sake." Professional history, in particular, was explicitly or simply by assumption associated with systematic archival research and the critical examination of sources, which had come to be thought of as constitutive of all serious history. Within this general consensus there could be differences of opinion, as for example between J. B. Bury and G. M. Trevelyan, over whether history was a "science" or an "art," and over how far, if at all, the historian in pursuit of his "science" should be concerned to establish laws (anathema to R. G. Collingwood in his classic book _The Idea of History,_ 1946). But despite these differences there was enough consensus to provide the basis for a selective grand narrative of the history of historiography, in which past historians were highlighted and assessed for their roles, necessarily partial, but helpful (or perhaps backsliding), in the general progression to the twentieth-century historian's views and approved practice. In this sense it was possible to write "The History of History."
I do not want to be understood as speaking simply in denigration of the assumptions underlying this possibility, as though of a past cultural episode. The central concerns--above all with history as truth-telling and, at least as an ideal, as free from bias--were already very old ones and, though shaken, are still in some sense with us, for those of us for whom a distinction between, say, history and imaginative fiction is still an important one. In this view Herodotus was taking an important step in distinguishing his own _Histories_ from the work of the poets, and Thucydides, though he may have judged unfairly, was invoking relevant criteria when he sneered by implication at Herodotus as belonging with authors less concerned to tell the truth than to entertain the public. Some distance between the search for historical understanding and merely emotionally or polemically effective writing is still part of the self-image and intentions of the historical profession. Of course, in the history of historiography zeal for truth has been a spectrum rather than an absolute--truth mattered, fairly obviously, more to Polybius than to Livy--but someone who wholly and perhaps wilfully falls off the negative end of the scale, like Geoffrey of Monmouth (this is not the place to argue the individual case), counts rather as a parodist or imitator of history.
All this may be true, but it remains true also that establishing a grand narrative of the history of historiography, by adopting a twentieth-century professional consensus as its terminus, was an impoverishingly narrowing strategy to adopt, eliminating or sidelining many interesting and potentially illuminating questions about approaches to historical writing, and indeed to the past as such, current in former times. There is, for example, the whole large and fascinating question of the surely very diverse motives for writing history. What did people in the past find interesting in _their_ past, and why did they? Which "pasts" did it lead them to focus attention on, as well as shaping how they chose to present them, and how and why did these change over time or how did different answers to these questions in a single period reflect and express differences within the culture? Why did new genres of historical writing emerge? Surely not only or invariably as a result of an extension of a pre-existing "scientific" curiosity, though that was sometimes a factor.
This book aims to provide answers to these questions. They have not been entirely neglected, and historians of historiography have been concerned to divide their subject in terms of genre as well as of method. But there is a balance to be redressed, as well as an allegiance to be proclaimed. I have tried to focus here on the question of the pasts that people have chosen for themselves and why, as well as on how they investigated and presented them. This may seem scarcely revolutionary, but like all chosen strategies it involves sacrifices. In particular, in grouping historians according to subject matter I have sometimes been cavalier with strict chronology--a lesson historians learned when they moved away from annals as the dominant form. So, for example, the "Alexander" historians are treated here as part of the story of the Greek encounters with the Persian empire, even though the historians whose work is still extant wrote much later, under the Roman empire. Perhaps most controversially, consideration of the Bible and its influence on historiography is postponed until its impact on the Gentile world in the early Christian centuries, rather than being placed, as chronology dictates, between Egyptian and Babylonian historians and Herodotus.
So, "A History of Histories" is intended to recognize the plurality of "histories" and the interests embodied in them, and to disclaim the ambition to construct a single grand narrative with the present as its terminus, which I see not only as implausibly prescriptive but also as narrowing the possibilities of exploration. There are, however, some exclusions which are intended also to be suggested in the title, with its implied retreat from comprehensiveness. No attempt has been made to deal with historiography outside the European cultural tradition (to which Egypt and Babylon are taken to contribute), notably Arabic and Chinese examples. Such exclusions are merely concessions to limitations of space, time and the author's knowledge. Another exclusion perhaps needs more apology, because it is at least in some measure arbitrary and the line of demarcation is a wavering one. I have taken the term "histories," though itself a generous one, as excluding biography and memoirs. In a book in danger of trying to include too much, I have thought this necessary, though the criteria are admittedly not always easy to apply: memoirs are clearly close kin to eyewitness history, and a "Life and Times" proclaims itself a mixed genre.
A word must be said about the treatment of the individual histories, which of course vary enormously in density and complexity, as well as in their accessibility to the modern reader. It is reasonable to assume that most readers of this book will not have read many or most of the historical works it discusses; indeed, that is part of the book's justification. One, primary, task therefore is to try to give a sense of the experience of reading these histories and what may be enjoyable about them. For many, perhaps most, historians, history has been a leisurely art, often requiring many volumes. It is not exclusively devoted to narrative, but narrative has long been at the core of it. It is therefore not enough just to convey the historian's intentions and views: some attempt must be made to convey not merely the structure of the narrative, but also its texture and qualities. In that respect, histories--which also often incorporate surveys, disquisitions, arguments and analyses--also resemble novels. I have made an attempt here, therefore, to give a sense, where appropriate, of what an eclectic, multi-layered, many-toned project a dense historical narrative may be. In attempting to render histories' special qualities I have not only resorted to a good deal of quotation but have attempted, sympathetically and with an awareness of the periods when they were written, to convey the literary qualities which form a large part of the experience of reading them. But these appraisals are also to be seen within, and perhaps as contributing to an overall understanding of, a more general context: the aims of historians in a particular period, the conventions which shaped their writing, and the ways these changed. I have also attended to historians' relations to the sources which made their work possible and partly conditioned it, and also briefly to the question of the particular writer's reliability. Awareness of this is part of an understanding of historians and therefore of the experience of reading them; history can never be, by definition, a purely literary endeavour. I have not, however, made any systematic attempt to focus on particular errors, for which, in any case, I should lack the necessary knowledge. That is the business of modern historians of the period, and they need no help from me. Such a checklist would, in any case, be intolerably tedious to read.
Historiography is not only a (wide) genre in itself, exhibiting continuities and revivals as well as shifting focuses of attention. It is also a part of Western culture as a whole--at times a highly influential and even central part--as well as being obviously a receptacle for the concerns of that culture and influenced by its fluctuations. European societies at different times and with varying emphases have attached immense importance to versions of their pasts and to notions of historical development, as well as plundering historical writings for legendary, heroic, tragic and pathetic motifs and topoi for poetry, drama and painting (in the eighteenth century, "historical" painting was regarded as the highest pictorial genre), and for exemplary, inspiring and minatory rhetoric. Ideas of history and of aspects of the past have intersected with and partly constituted ideas of religion, morals and politics. They have embodied authority, and provided means with which to challenge it. Above all, perhaps, they have provided focuses of allegiance, self-identification and "memory" for ethnic, national, religious, political, cultural and social collectivities and so helped to constitute them. Versions of the past have been offered, sometimes obliquely but often with visible anxiety, as diagnoses of contemporary predicaments or malaises.
We are accustomed to think of the intellectual history of Europe in terms of the history of philosophy, of science and religion, of art, literature and of ideas of social order and political authority. But the history of ideas about the past, as expressed in historical writing, and how the present stands in relation to it, is also part of that history; this book aims to contribute to an understanding of it. Among its chief components are conceptions of the distinctiveness of European civilization, contrasted chiefly with the empires of Asia; ideas of republican virtue, embodied in early Rome, supposedly corrupted by conquest and luxury; and the myth of Eternal Rome as mistress of the world, which became transmuted into the idea of a Christian empire. The Bible contributed ideas of collective transgression, punishment and redemption. From the sixteenth century onward we find the idea, largely derived from the Roman historian Tacitus, of an early freedom of the "Germanic" peoples, and of the existence of "ancient constitutions," with a continuing authority in the present, which European countries allegedly derive from their invasion by the "Gothic" barbarians. Eighteenth-century historical writing incorporated concepts of the progress of "civil society," chiefly in association with commerce, and of the end of "feudal anarchy" (or, in Marxist terms, the supersession of the feudal nobility by the hegemony of the bourgeoisie). The nineteenth century was the great age of the preoccupation with national identity, in association with ideas of national liberation and the creation of the nation state as the normal political form. This has passed on into the modern aspiration to give a voice to suppressed minorities. History, in other words, to name only the most prominent influences, has been republican, Christian, constitutionalist, sociological, Romantic, liberal, Marxist and nationalist. All these have left residues in subsequent historical writing; none at the moment dominates it.
I have therefore made a conscious effort not to treat the history of historiography in isolation, but to be aware of its place in the wider culture, of the cultural and political influences playing on it, and of the ways in which it fostered, transformed and transmitted them. "A History of Histories" can and should be more than a record of the achievements, strengths and weaknesses of historians and the schools and traditions to which they belonged. It is itself a historical enterprise, one of the ways in which we attempt to understand the past.
**Keeping Records and Making Accounts: Egypt and Babylon**
History--the elaborated, secular, prose narrative (all these qualifications are necessary) of public events, based on inquiry--was born, we can claim with confidence, in Greece between roughly 450 and 430 BC. If we want to add Thucydides' very different kind of history to that of Herodotus, who is sometimes spoken of as "the father of history," then we must speak instead of the second half of the fifth century BC. Even with this extension, and with the qualifications built into the description of the genre, it is extraordinary that we can speak of so short a period for its abrupt genesis, yet it appears to be justified. It is equally astonishing that we can plausibly claim that neither historian was to be excelled for over two thousand years subsequently--until, in fact, changes in methods and types of history begin to make comparison unrealistic.
To see what is meant by claiming that Herodotus and Thucydides were, so far as we know, the first historians, we need to recognize some basic distinctions which separate their work from examples of what we may call perhaps "proto-history" in the ancient civilizations of Egypt and Babylon. Herodotus himself paid tribute to the Egyptians for their preservation of knowledge of the past: "by their practice of keeping records of the past, [they] have made themselves much the most learned of any nation of which I have had experience" ( _Histories,_ II.77). In fact he seems to have believed too readily what he was told in the temples of Egypt when conducting the historical inquiries he incorporated into the panoramic survey of the known world with which he prefaced his account of the great Persian invasion of Greece in the early fifth century BC. His own references to Egyptian history are notably garbled, in contrast to what he has to say of the civilization of Babylon, on which he is much more reliable. Yet, whatever it may have pleased the Egyptian temple servants to tell him--it is not clear that his acquaintance ranged far up the priestly hierarchy--Herodotus' compliment to the Egyptians was not misplaced. Modern Egyptologists can know much more than Herodotus about ancient Egypt precisely because so much of it, thanks to the early establishment of a centralized bureaucratic state and the use of durable materials for inscriptions, has been preserved. To this we may add the effects of a dry climate and a traditional habit of mind: the records so preserved go back more than two thousand years before Herodotus' own time, the mid fifth century BC. The Egyptians were indeed then the world's premier record-keepers. The distinction between historiography and recording--between the sense in which Herodotus was, so far as we know, the first historian and the learning of his Egyptian informants--is therefore one to give us pause. It is valid enough, though of course, like all such distinctions, it becomes a little less rigid as we examine it.
Record-keeping is, in origin, commercial and bureaucratic good practice; it is not an art. Many of the factors which have preserved so much of the Egyptian past existed also in the ancient civilization of Mesopotamia, with its records inscribed on stones and clay tablets and, for most of the greatest matters, on the walls of temples, tombs and palaces. Every modern historian understands what we are speaking of here--namely archives--and regards them as in some form indispensable, as Herodotus, who worked by interrogation of "learned men," did not. The inscriptions were intended as records from the beginning: their endurance was deliberate, as that of casually collected documents is often not. It is only the humbler artefacts, such as inscribed clay tablets, which have survived inadvertently. Inscriptions being essentially records, this produces a kinship between their authors and Herodotus, who says, in his initial statement of intention at the outset of his _Histories,_ that he wrote to preserve the memory of great deeds (below, Part 1).
The key difference, of course, is the word that Herodotus used to describe his work: _historia,_ inquiry. His method of acquiring the information for his _Histories_ was chiefly interrogation. When he questioned the Egyptian temple servants and guardians, he, who seems to have known only Greek, was at a remove from the documents in a way that a modern historian would not care for. But, for all the deficiencies or secretiveness of his informants, we recognize something like an intelligible relationship: it is that between historian and archivist. It was he who, in the service of a systematic inquiry, history, was interrogating them; not vice versa. Similarly, when he interrogated "overseas" Greeks and perhaps "native informants" for knowledge of other parts of the world, as he must have done, it was he who was the anthropologist or ethnographer. There was, to state the obvious, no Scythian ethnographer; those whom the Greeks called Scythians, of whose customs Herodotus wrote an extended account, were illiterate nomads living north of the Black Sea. We have therefore argued ourselves into the position of saying that, as far as we know, there were no Egyptian or Babylonian historians earlier than Herodotus. The Egyptians were, as Herodotus says, learned; respect for Egyptian wisdom was high and even exaggerated in Greece, where, for example, the Greek gods, under other names, were thought to have originated in Egypt. But the Egyptians were recorders, not historians.
So far so simple. As we shall see in Chapter 1, Herodotus' notion of systematic inquiry was not entirely unique in Greece in his time. The inquiries closest to his own seem to have been mainly geographical (including of course "human geography"), a concern which is very evident also in Herodotus' work. But inquiry, systematic research, is not the only characteristic of historiography. Another is the rendering of the results of the inquiry into connected historical prose: narrative. There is in fact a route in the ancient world from recording to more or less extended historical narrative which blurs a little the distinction between recording and history that seems so firm when we attend only to the element of research.
The earliest writing seems, understandably, to have been concerned with practical transactions, and can be thought of either as part of the transaction or just as a subsequent record. Early public inscriptions, in the grandest contexts, recorded on the walls of temples, seem also to have taken on this transactional character, as the rendering of an account by the ruler of his stewardship on behalf of the god: his buildings erected, his gifts, and the toils and achievements, including victories, by which they were procured. Inventories are common. Other kinds of lists include the earliest materials for systematic chronology--king lists, for example--and are therefore crucial to the possibility of reliable large-scale history. Laws too, like the early so-called law codes of ancient Mesopotamia, are also essentially lists, in a way familiar, though much later, from the books of Leviticus and Numbers in the Old Testament from the end of the second millennium BC.
The connection between narrative and divine stewardship seems to lie in explanations. That is, to render an account, initially in the form of a list, can come to involve an explanation, which in turn may take the form of a more or less elaborated narrative. Historians are accustomed to insisting, perhaps rather combatively since it is their stock-in-trade, that an explanation may take the form of a narrative. They may be less keen on claiming an ancient kinship with accountants, and through them to the earliest uses of writing: making lists. The conceptual overlap here, however, is still caught in the related meanings, indicated by prepositions, of the English word "account," which itself recalls its origin in the act of counting. It makes good sense if bad style, with each repetition making a different point, to say, "Please account for the errors in the accounts you have presented to the board by giving your account of how they were compiled."
Narrative inscriptions which were also explanations and part of the rendering of accounts can be seen growing fuller and more circumstantial and far more "human" in their references to motives and action. Campaign narratives--with their natural climax in victories (or pretended victories) and hence conquests, subjection or capture of alien peoples, and the acquisition of spoils--are among the earliest extended narratives, and are also, of course, characteristically narratives of expeditions, like that of Xerxes in Herodotus, and, nearly two centuries later, the greatest of all campaign expeditions, that of Alexander, who took historians with him. This is an Egyptian account from the middle of the second millennium BC of a victory and its trophies:
His Majesty proceeded on his chariot to Kashabu, alone and without a companion, and returned thence in a short time bringing sixteen living Maryannu at the side of his chariot, twenty hands at the foreheads of his horses and sixty cattle driven in front of him. Submission was made to His Majesty by this town. Now as His Majesty was going south in the plain of Sharon, he found a messenger of the prince of Nahrin carrying a clay tablet at his neck, and took him as a living prisoner at the side of his chariot...Arrival of His Majesty at Memphis with joyful heart like a victorious bull. Amount of plunder...(Gardiner, _Egypt and the Pharaohs,_ p. 197)
Then follows an itemization, with numbers, of slaves, horses, chariots, weapons and musical instruments. Expeditions were also apparently conducted by the pharaohs in the quest for valuables such as minerals.
Another Egyptian form of narrative pattern is the onset of a period of confusion and disasters, brought to an end by the advent of a ruler who restores order. These narratives of deliverance--whose archetypal protagonist later, under Hebrew influence, became the Messiah figure--have had, like the campaign narrative, a long history in Western historiography, influencing the representation of such figures as the emperors Augustus and Constantine and, in English history, Queen Elizabeth I and William III. Thus some of the archetypal patterns of historical narrative were established as early as the inscriptions recording the deeds of the rulers of ancient Egypt, Mesopotamia and Asia Minor. In the Bible, the books of Exodus and Deuteronomy are essentially expedition and campaign narrative, which features also in the work of some of the greatest later historians.
Another way in which the rendering of accounts, in the form of vindication, elaborates itself into narrative seems to have been cultivated particularly among the Hittites. That is, a record was made to establish the justice of what had been done, and to refer the matter to divine arbitration. A leading authority on the Hittite empire, O. R. Gurney, describes a document of this kind as showing a highly developed political conscience. The more legalistic form of this kind of record-keeping, and the provision of a narrative on which judgement was to be based, concerned, naturally, the making and breaking of treaties. Treaties are the only documents to be transcribed at length by Thucydides, much of whose opening book is concerned with the rupturing of treaties which constituted the opening of the Peloponnesian War. The Hittites, a millennium earlier, had begun the practice of making the preambles to treaties into occasions for brief historical narratives explaining the treaties' origins; eventually the prefatory narratives became detached from the treaties and decrees and became free-standing annals, the chronicle being a presentation of the ruler's actions as an offering to the god.
The annalistic form of recording is found also among the Assyrians, as well as in early "historians" in Greek city states, along with accounts of supposed mythological origins. It is not employed by Herodotus--a work on such a large scale would have made it unusable--but it shapes Thucydides' history of the Peloponnesian War, with a division of each year into summer and winter for greater precision. It was to remain a fundamental historiographical form to the end of the Middle Ages.
Mention of annals brings to the surface a troublesome issue for early historiography: the need to find widely recognizable ways of marking chronology. This was less of a problem in centralized dynastic states like Egypt and Babylon--which is not to say that early king lists were always as helpfully drawn up as they might have been: attempting to eliminate all trace of a discredited ruler was a common Egyptian practice, while the lengths ascribed to reigns are clearly absurd. But in Greece, with no central political authority or record-keeping, the problem was acute. The only pan-Greek institutions were the oracle at Delphi and the four-yearly Olympic Games, with their famous winning athletes, and the Olympiads were eventually used as chronological markers. Thucydides dated events from the onset of the war he was recounting. For Athens it was possible to use the periods of office of the archons, the chief officials of the _polis,_ who held their posts for a year, though they may not have been easy to recall. The Romans from earliest times kept lists of the pontiffs, the chief priests, which served the same purposes as the names of the rulers of Egypt and Babylon. Some idea of the difficulties of establishing a common chronology can be seen in an example of Thucydides overloading:
The thirty years' truce which was entered into after the reconquest of Euboea lasted for fourteen years. In the fifteenth year, the forty-eighth year of the priestess-ship of Chrysis at Argos, the year when Aenesias was ephor at Sparta, and two months before the end of the archonship of Pythodorus at Athens, six months after the battle at Potidaea, just at the beginning of spring, a Theban force...(II.2)
One other form of proto-historiography which calls for attention is the counterpart, of wider application, of dynastic lists, namely the construction of genealogies, often with mythic alleged origins, with which the very early Greek city-state historians also seem to have concerned themselves. This interest in origins leaves occasional marks in Herodotus' _Histories,_ though as usual he does not always believe what he hears. Self-extolling histories of the great clans seem to have provided materials for the early history of Rome and to have been valuable sources despite their inevitably self-serving character. A particular cachet attached to descent from a god, Heracles being a favourite, or from a hero of the Trojan War. The Romans managed both, by being descended from Aeneas of Troy, who was the son of Venus. A well-attested genealogy could be useful or even indispensable, as we see in the Old Testament Book of Nehemiah, where it carried qualification for ritual office, and a flaw constituted disqualification (Nehemiah 7:5, 64). Alexander believed, or wished people to believe, that he was descended from the god Ammon.
Secular historical prose narratives (setting aside both epic poetry and the Bible, the latter being left for consideration later with its impact on Christian historiography) in the ancient empires were emerging gradually from more basic forms of bureaucratic record-keeping, though it is still a long step--a leap, in fact--from these to the richly humane and artfully controlled extended narratives of Herodotus. There are campaign and expedition narratives, and what may be called redemption and vindication narratives, accounts of disasters and rebellions and subsequent restorations, and of breaches of trust suffered and avenged. There were also the beginnings, among the Hittites, of annals as a form of recording.
The most elaborate, circumstantial and continuous of the early campaign narratives, though they exerted no direct influence on later ones, can at least claim a kind of kinship with them. One such is the account of the pharaoh Tuthmoses III (1490-1436 BC) in the campaign which culminated, in the twenty-third year of his reign, in the battle of Megiddo, recorded on the walls of his additions to the Temple of Amon-Re at Karnak. It contains an account of a council of war, with dialogue, and what has been called the earliest full description of a decisive battle. Before the battle, a debate arises over which route to take. The counsellors are notably sycophantic, but clearly have their own ideas when invited to express their opinions: "How can one go upon this road which is so narrow? It is reported that the enemy stand outside and have become numerous. Will not horse have to go behind horse, and soldiers and people likewise? Shall our own vanguard be fighting, while the rear stands here...?" The temple walls also exhibit, as was common, lists of defeated peoples, together with depictions. The narrative can also be checked against other accounts. Here then, as in comparable if less full or less vivid accounts, we have something which is certainly narrative, but which we may hesitate whether to call historical or proto-historical. Although fuller and less formulaically bombastic than other examples, it is an account of an episode, though a highly important one. It does not exhibit individualized character depiction, a sense of historical perspective (in this respect being rather like the two-dimensional side-face Egyptian conventions of representation) or the surrounding architecture of a large-scale historical design. Much the same could be said, of course, of many chronicles over the next two thousand years. It could not be said of Herodotus.
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 6,384
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Q: Push button is reducing voltage I'm making a little flashlight that'll turn on and off with a push button, but the problem is that my push button changes 3.2 V into 2.8 V.
I don't know if it's a normal issue or how to fix it. Any help would be appreciated.
A: Likely it has nothing to do with the switch. If you have a button cell battery and connect a LED directly, without any current limiting resistor, the battery output voltage will drop to the level of LED forward voltage, current being limited only by the internal resistance of the battery, LED, and wires. Which means the current can be too large for the battery or the LED.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,922
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Clients often request their agents to provide them a new policy to replace one they already have. Whatever their reasons are, it is important for you, the agent, to be sure the new policy provides the same coverage as the old policy did. Far too often, both the agent and client assume the coverage is the same when, in fact, it does not provide the exact same coverage as the original policy.
Occasionally, a client wants the new policy to have different coverage than the original one. It is imperative that you understand exactly what coverage your client requires. Claims against insurance agents E&O insurance policies are often based on a replacement policy not providing the exact same coverage as the original one. If your client has requested any change in coverage in the new policy, you must document the client's change request. Here is an example of a claim that was filed on the grounds that the agent made a mistake in replacing one policy with another one when the new policy did not provide the same coverage.
Review the example and the E&O insurance prevention tip that follows it.
Claim Description: A contractor, who did work for his clients in several different states, including California, had a policy that provided him coverage in any and all states where he worked with no exclusions under a Multi-State Commercial General Liability policy. Subsequently, a new policy was bound with a surplus lines carrier that attached an endorsement specifically excluding any work performed in California. Apparently, neither the agent nor the insured noticed the exclusion for work performed in California until the contractor filed a claim with his insurer. The contractor was sued for damages related to alleged defects in work he performed in California. When the contractor's claim was denied, he filed a claim against his agent. Estimated cost of the claim: $38,000.
Prevention Tip: When clients ask for a policy to replace their original policy, find out exactly what coverage the client expects from the replacement policy. Carefully compare both policies to be sure the new policy provides the coverage the client has requested. Have the client carefully review the new policy and agree that it provides the needed and requested coverage. If changes in coverage have been made in the replacement policy, have the client acknowledge in writing that the new policy conforms to the client's request. As insurance agents ourselves, the American Agents Alliance offers a very affordable, A-Rated, and admitted E&O insurance program for P&C insurance agents. If you are a P&C insurance agent, you can read about our E&O insurance program right here.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 5,060
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\section{Introduction}
Recently, the Event Horizon Telescope (EHT) Collaboration processed images of the supermassive black hole (BH) Sgr A* at the center of the galaxy \cite{SgrA_1_2022,SgrA_2_2022,SgrA_3_2022,SgrA_4_2022,SgrA_5_2022,SgrA_6_2022}. Similar to the images of M87* \cite{M87_1_2019,M87_2_2019,M87_3_2019,M87_4_2019,M87_5_2019,M87_6_2019}, these images are relatively dark in the middle and have a bright ring. The central dark region is known as the shadow of the BH, and the bright ring corresponds to the photon sphere \cite{Shoom:2017ril}. The black hole's gravity bends the trajectories of photons emitted by a distant light source behind the black hole, forming a shadow and photon sphere. Modern astrophysical observations suggest that the emissions from BHs are mainly disk-like accretions \cite{Remillard:2006fc,Yuan:2014gma}. The times of photons intersecting with the accretion disk plays an important role in the optical appearances of shadows. The photon ring consists of light rays that intersect the accretion disk three or more times \cite{Gralla:2019xty}.
The observation of BH shadows is considered to be an important way to detect BHs directly, which can enhance our understanding of their properties \cite{PRD_Tsupko_2017,ApJ_Kumar_2020,AAS_Broderick_2022}. Furthermore, the structures of shadows can be used to examine strong gravitational fields and general relativity \cite{NatAstron_Mizuno_2018}. As a result, analyzing the images of BH shadows and photon spheres has become one of the important research topics. Schwarzschild BH's shadow was first studied in \cite{MNRAS_Synge_1966}. Sooner, Bardeen et al. studied Kerr BH's shadow \cite{Astrophys_Bardeen_1972}. Through analytical calculation, the first figures of BH shadow with a rotating accretion disk are proposed in \cite{Astronomy_Luminet_1979}. In some modified gravity theories, the shadows of rotating BHs were also investigated \cite{Dastan:2016vhb,Long:2020wqj}. The research on BH shadows has been expanded to many aspects in recent years \cite{Qiao:2022jlu,Chakhchi:2022fls,Lin:2022ksb,Belhaj:2022kek,Sun:2022wya,Guo:2021bhr,Zhong:2021mty,Cunha:2016wzk,CPC_He_2022,Gan:2021xdl,Shaikh:2021yux,Heydari-Fard:2022jdu,Bambi:2019tjh,Vagnozzi:2019apd,Vagnozzi:2020quf,Roy:2021uye,Chen:2022nbb,Vagnozzi:2022moj}.
There are some pieces of evidence suggesting that magnetic fields may exist near BHs. After two years of in-depth research and data processing, the EHT Collaboration unveiled the shadow of M87* with polarized light \cite{M87_7_2021,M87_8_2021}. The image clearly showed that the shadow of the BH is affected by magnetic fields, implying there probably exists a strong magnetic field surrounding the BH. Just recently, the motion of the hot spot orbiting Sgr A* is predicted to be relevant to the magnetic fields surrounding the BH \cite{Wielgus:2022heh}.
To eliminate the singularity of point-like charge in Maxwell electrodynamics, Born and Infeld proposed the BI electrodynamics \cite{Nature_BI_1933}, which was inspired by special relativity and modified the Lagrangian of electromagnetic fields. In addition to BI electrodynamics, many other theories have been proposed to modify Maxwell electrodynamics to avoid singularity \cite{PRB_Kruglov_2012,PRD_Kruglov_2007,IJGMMP_Kruglov_2015}. These Maxwell's theory extensions are usually known as nonlinear electrodynamics (NLED). Among NLEDs, BI is distinguished for it is the only one that ensures no birefringence \cite{AIP_Kerner_2001}. Surprisingly, BI Lagrangian can be exactly derived from low energy string theory \cite{PLB_Fradkin_1985}. In addition, BI has been studied as coupling with general relativity \cite{PR_Hoffmann_1935}. In recent years, Einstein-Born-Infeld gravity has been studied in various papers \cite{Babar:2021nst,Wang:2020ohb,Jafarzade:2020ova,Ali:2022zox,Yang:2021bhv,Zhang:2021kha,Falciano:2021kdu,Hendi:2018cfr,Mazharimousavi:2014vza,Dehghani:2019noj,Jing:2020sdf,Gan:2019jac,Liang:2019dni,Tao:2017fsy,Bi:2020vcg}. Magnetically charged BHs, and their properties, have been discussed theoretically \cite{Lee_PRL_1992,Lee_PRD_1992,Lee_PRL_1994}. Considering two NLED models and comparing the shadows of these models to the observation of M87*, an upper bound on the magnetic charge of BHs was given \cite{Allahyari:2019jqz}. The potential astrophysical signatures of magnetically charged BHs are also studied \cite{Ghosh:2020tdu}. According to \cite{BenavidesGallego:2018odl}, the magnetic charge strongly affects the outer horizons and the ergospheres of rotating BHs, but the photon circular orbits are not explicitly depended on magnetic charge. Furthermore, the effects of strong magnetic fields on shadows are examined \cite{PRD_Haroldo_2021}. Theoretically, photons move along the null geodesics of the spacetime governed by the BH. Being affected by NLED, photons move along the null geodesics of the effective metric rather than that of the background metric \cite{IJMPD_Bergliaffa_2004}.
This paper investigates the observational appearances of magnetically charged BI BH. The shadows and photon spheres of BHs are derived using the backward ray tracing method \cite{Astronomy_Luminet_1979}. Additionally, we investigate the impact of the accretion's dynamics and shapes on the shadows. This paper is organized as follows. In section \ref{geo}, we investigate the metric and the trajectories of photons deflected by a magnetically charged BI BH. Additionally, we discuss the range of the magnetic charge and the BI parameter. Section \ref{spheric} studies the shadows and photon spheres with spherical accretion. In section \ref{ring}, the shadows produced by thin accretion disks are investigated. In section \ref{concl}, we discuss and conclude our works.
\section{Geodesics\label{geo}}
\subsection{The metric}
In this section, we derive the metric of BI BHs with magnetic charge in static spherically symmetric spacetime. The Einstein-Born-Infeld action is defined as follows \cite{PRD_Cai_2004}
\begin{align}
\mathcal{S}=\int\sqrt{-g}\left[ R+\mathcal{L}(F) \right]{\rm{d}}^{4}x,
\end{align}
where $R$ is the Ricci scalar, $F=F_{\mu\nu}F^{\mu\nu}$. The BI Lagrangian $\mathcal{L}(F)$ takes the form as
\begin{align}
\mathcal{L}(F)=4\beta^{2}\left( 1-\sqrt{1+\frac{F}{2\beta^{2}}} \right),
\label{LF}
\end{align}
where $\beta$ is called BI parameter with the dimension of mass. When $\beta\to\infty$, $\mathcal{L}(F)$ degenerates into the standard electromagnetic form.
In this paper, we consider static and spherically symmetric spacetime. The general form of metric is \cite{EPJC_He_2022}
\begin{align}
{\rm{d}}s^{2}=-f(r){\rm{d}}t^{2}+\frac{1}{f(r)}{\rm{d}}r^{2}+r^{2}({\rm{d}}\theta^{2}+\sin^{2}\theta{\rm{d}}\phi^{2}),
\label{ds^2_raw}
\end{align}
and the electromagnetic tensor generated by magnetically charged BH takes the form as
\begin{align}
F_{\theta\phi}=-F_{\phi\theta}=P\sin\theta,
\label{EMfield}
\end{align}
where $P$ is a parameter representing the magnetic charge.
The metric function $f(r)$ for magnetically charged BI BHs is given in \cite{EPJC_He_2022}
\begin{align}
f(r)=1-\frac{2M}{r}+\frac{2\beta^{2}r^{2}}{3}-\frac{2\beta^{2}r^{2}}{3}{_{2}F_{1}}\left(-\frac{3}{4},-\frac{1}{2};\frac{1}{4};-\frac{P^{2}}{r^{4}\beta^{2}}\right),
\end{align}
where $M$ is the mass of the BH, and $_{2}F_{1}$ is the hypergeometrical function. When $\beta\to\infty$, $f(r)$ degenerates into the form of magnetically charged Reissner-Nordström (RN) BH.
\begin{align}
f(r)\overset{\text{expand at }\beta=\infty}{=\!=\!=\!=\!=\!=\!=\!=\!=}1-\frac{2M}{r}+\frac{P^{2}}{r^{2}}+O\left(\frac{1}{\beta}\right)^{2}.
\end{align}
In addition, the equation $f(r)=0$ has two roots. The larger one $r_{\text{h}}$ represents the radius of the event horizon. Since it is not easy to obtain analytical solutions of $r_{\text{h}}$, its numerical solutions of different $\beta$ are listed in Table \ref{NumSol_changeBeta}. Setting the minimum of $f(r)$ to zero will yield the critical value of $P$ with respect to different $\beta$, as Fig. \ref{lgbeta_lgP} and Table \ref{NumSol_Pcrit} show.
\begin{figure}[htb]
\centering
\includegraphics[width=9cm]{lgbeta_lgP_crit.pdf}
\caption{Region plot in $\lg\beta-\lg P$ space for magnetically charged BI BH with fixed $M=1$. Black holes can only exist in the light yellow area. The critical value of the magnetic charge $P$ increases with the decrease of $\beta$. The black dashed line ($P=1$) represents extreme RN BHs. To display a wider plot range, we use logarithmic coordinates.}
\label{lgbeta_lgP}
\end{figure}
\begin{table}[!htbp]
\setlength{\tabcolsep}{5mm}
\begin{center}
\begin{tabular}[b]{c|cccccc}
\midrule[2pt]
$\beta$ & $0.05$ & $0.1$ & $0.15$ & $0.2$ & $0.3$ & $\infty$ \\
\toprule[1pt]
$P_{\text{crit}}$ & 2.35678 & 1.87057 & 1.63410 & 1.48467 & 1.29698 & 1.00000 \\
\midrule[2pt]
\end{tabular}
\end{center}
\centering
\caption{Data of critical value of magnetic charge $P_{\text{crit}}$ for respect to different $\beta$ with fixed $M=1$. The critical value of magnetic charge decreases as BI effect weakens until $\beta=\infty$, when $P_{\text{crit}}$ becomes the same as magnetically charged RN BH.}
\label{NumSol_Pcrit}
\end{table}
\subsection{The innermost stationary circular orbit\label{isco}}
Next, we derive the innermost stationary circular orbit (ISCO) for massive particles. The Lagrangian of a particle with unit mass, $\mathcal{L}_{m}$, has the form as
\begin{align}
\mathcal{L}_{m}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu},
\label{Lagrangian_massive}
\end{align}
where the dot on $x^{\mu}$ represents the derivative with respect to the proper time $\tau$. For massive particles, there are two conserved quantities: energy $E_{m}$ and angular momentum $L_{m}$, which are obtained by
\begin{align}
E_{m}=-\frac{\partial {\mathcal{L}}_{m}}{\partial \dot{t}}=f(r)\dot{t},\qquad L_{m}=\frac{\partial \mathcal{L}_{m}}{\partial \dot{\phi}}=r^{2}\dot{\phi}.
\label{conservations}
\end{align}
Substituting Eq. (\ref{conservations}) into Eq. (\ref{Lagrangian_massive}), noting that $\mathcal{L}_{m}=-1$, we derive the equation of radial motion
\begin{align}
\dot{r}^{2}+\frac{L^2}{r^{2}}f(r)+f(r)=E^{2}.
\label{rdot_massive}
\end{align}
We define the effective potential of massive particles to be
\begin{align}
V_{m}(r)=\frac{L^2}{r^{2}}f(r)+f(r).
\label{eff_potential_massive}
\end{align}
The radius of the ISCO satisfies
\begin{align}
\left.\frac{{\rm{d}}V_{m}}{{\rm{d}}r}\right|_{r=r_{\text{ISCO}}}=0,\qquad \left.\frac{{\rm{d}}^{2}V_{m}}{{\rm{d}}r^{2}}\right|_{r=r_{\text{ISCO}}}>0.
\label{ISCO_conditions}
\end{align}
Solving Eq. (\ref{ISCO_conditions}) numerically, we obtain the radii of the innermost stationary circular orbit $r_{\text{ISCO}}$ for different BI parameters $\beta$ and magnetic charges $P$ in Tables \ref{NumSol_changeBeta} and \ref{NumSol_changeP}.
We calculate the value of $r_{\text{ISCO}}-3r_{\text{h}}$ for different parameters $\beta$ and $P$ in Tables \ref{NumSol_changeBeta} and \ref{NumSol_changeP} to compare the spacetime of magnetically charged BI BH to that of Schwarzschild BH, for $r_{\text{ISCO}}=3r_{\text{h}}$ in Schwarzschild spacetime. The non-monotonicity of $r_{\text{ISCO}}-3r_{\text{h}}$ shown in Tables \ref{NumSol_changeBeta} and \ref{NumSol_changeP} implies that there is a critical value of $\beta$ that determines at which $r_{\text{ISCO}}-3r_{\text{h}}$ switches from negative to positive. Fig. \ref{risco3rh} is the contour plot of $r_{\text{ISCO}}-3r_{\text{h}}$ in $\beta$-$P$ space. We choose to start from $\beta=0.05$ rather than $\beta=0$ since numerical instability occurs at $\beta$ close to 0. Changes in $\beta$ and $P$ have contrary impacts on $r_{\text{ISCO}}-3r_{\text{h}}$ on each side of the critical values. In the red region, $r_{\text{ISCO}}-3r_{\text{h}}$ increases with $P$ and $\beta$. In the blue region, the impact of $P$ on $r_{\text{ISCO}}-3r_{\text{h}}$ is opposite to that of $\beta$. In other words, the impact of $P$ is contrary to that of $\beta$ if it is less than the critical value and vice versa. When $P=0$, a magnetically charged BI BH degenerates into a Schwarzschild BH, regardless of the BI effect, as the leftmost bolded line in Fig. \ref{risco3rh} shows.
\begin{figure}[htb]
\centering
\includegraphics[width=10cm]{risco3rh.pdf}
\caption{Contour plot in $\beta-P$ space with fixed $M=1$. The colors represent the value of $r_{\text{ISCO}}-3r_{\text{h}}$, and the contours are the isosurfaces of $r_{\text{ISCO}}-3r_{\text{h}}$. On the bolded curve, $r_{\text{ISCO}}=3r_{\text{h}}$. The impact of the magnetic charge $P$ is opposite to the BI parameter $\beta$ below the bolded curve, and vice versa.}
\label{risco3rh}
\end{figure}
\subsection{The trajectories of photons}
Next, we derive the trajectories of photons. The trajectories of photons are null geodesics in the effective metric rather than the background metric due to the effect of BI. The effective metric is \cite{EPJC_He_2022}
\begin{align}
{\rm{d}} s_{\text{eff}}^{2}=k(r)\left[-f(r){\rm{d}} t^{2}+\frac{1}{f(r)}{\rm{d}} r^{2}+h(r)({\rm{d}} \theta^{2}+\sin^{2}\theta {\rm{d}}\phi^{2})\right],
\label{eff_ds^2}
\end{align}
where
\begin{align}
k(r)=\sqrt{1+\frac{P^{2}}{\beta^{2}r^{4}}},\quad h(r)=r^{2}\left(1+\frac{P^{2}}{\beta^{2}r^{4}}\right).
\end{align}
Thus, we can derive the trajectories of photons. The Lagrangian of a photon is
\begin{align}
\mathcal{L}=\frac{1}{2}G_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu},
\label{Lagrangian}
\end{align}
where $G_{\mu\nu}$ are the covariant components of the effective metric, and the dot on $x^{\mu}$ represents the derivative with respect to the affine parameter $\lambda$. We simply focus on the equatorial plane ($\theta=\pi/2$ and $\dot{\theta}=0$) because the metric is spherically symmetric. Moreover, since the Lagrangian does not explicitly depend on coordinates $t$ and $\phi$, there are two conserved quantities along the geodesics
\begin{align}
E=-\frac{\partial\mathcal{L}}{\partial\dot{t}}=k(r)f(r)\dot{t},\qquad L=\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=k(r)h(r)\dot{\phi}.
\label{conservations_photons}
\end{align}
For photons $\mathcal{L}=0$, one can derive the following equations representing the motion of photons
\begin{align}
-\frac{E^{2}}{k(r)f(r)}+\frac{k(r)}{f(r)}\dot{r}^{2}+\frac{L^{2}}{k(r)h(r)}=0.
\label{rdot1}
\end{align}
The equations of photons' motion are derived by redefining the affine parameter $\lambda$ into $\lambda/L$ and setting the impact parameter $b=L/E$.
\begin{align}
\dot{t}=\frac{1}{bk(r)f(r)},
\label{tdot2}
\end{align}
\begin{align}
\dot{\phi}=\pm \frac{1}{k(r)h(r)},
\label{phidot2}
\end{align}
\begin{align}
k^{2}(r)\dot{r}^{2}=\frac{1}{b^{2}}-\frac{f(r)}{h(r)},
\label{rdot2}
\end{align}
where $\pm$ in Eq. (\ref{phidot2}) represents the counterclockwise and clockwise motion of the photon respectively. We define the effective potential $V(r)=f(r)/h(r)$. Specially, when $\dot{r}|_{r=r_{\text{ph}}}=0$ and ${\ddot{r}}|_{r=r_{\text{ph}}}=0$, it suggests that the photon can circle the BH for infinite times in an unstable orbit called photon sphere, which indicates
\begin{align}
V(r_{\text{ph}})=\frac{1}{b_{\text{ph}}^{2}},\quad \left.\frac{{\rm{d}}V}{{\rm{d}}r}\right|_{r=r_{\text{ph}}}=0,
\label{eff_potential}
\end{align}
where $r_{\text{ph}}$ is the radius of the photon sphere, $b_{\text{ph}}$ is the corresponding impact parameter. Since it is not easy to derive an analytical expression of $r_{\text{ph}}$ and $b_{\text{ph}}$, we list their numerical solutions in Tables \ref{NumSol_changeBeta} and \ref{NumSol_changeP}.
As Table \ref{NumSol_changeBeta} shows, $r_{\text{h}}$, $r_{\text{ISCO}}$, $r_{\text{ph}}$, and $b_{\text{ph}}$ decrease monotonically when $\beta$ increases. Despite that $r_{\text{ph}}$ drops steeper than $r_{\text{h}}$ as $\beta$ increases, $r_{\text{h}}<r_{\text{ph}}$ is always satisfied because a BI BH degenerates into an RN BH when $\beta\to\infty$.
\begin{table}[htbp]
\setlength{\tabcolsep}{3mm}
\begin{center}
\begin{tabular}[b]{c|cccccc}
\midrule[2pt]
$\beta$ & $0.05$ & $0.1$ & $0.15$ & $0.2$ & $0.3$ & $\infty$ \\
\toprule[1pt]
$r_{\text{h}}$ & 1.88896 & 1.87594 & 1.87134 & 1.86928 & 1.86758 & 1.86603 \\
$r_{\text{ISCO}}$ & 5.62174 & 5.61066 & 5.60845 & 5.60766 & 5.60710 & 5.60664 \\
$(r_{\text{ISCO}}-3r_{\text{h}})\times 10^{3}$ & -45.1400 & -17.1501 & -5.57996 & -0.17694 & 4.36839 & 8.56722 \\
$r_{\text{ph}}$ & 4.02971 & 3.52690 & 3.09423 & 3.06541 & 2.94078 & 2.82288 \\
$b_{\text{ph}}$ & 6.57031 & 5.86395 & 5.31786 & 5.27579 & 5.11771 & 4.96791 \\
\midrule[2pt]
\end{tabular}
\end{center}
\centering
\caption{Data of $r_{\text{h}}$, $r_{\text{ISCO}}$, $r_{\text{ph}}$, and $b_{\text{ph}}$ for different $\beta$ with fixed $M=1$, $P=0.5$.}
\label{NumSol_changeBeta}
\end{table}
Numerical solutions of different $P$ are listed in Table \ref{NumSol_changeP}. With the increase of $P$, $r_{\text{h}}$, and $r_{\text{ISCO}}$ decrease, while $r_{\text{ph}}$ and $b_{\text{ph}}$ increase monotonically.
\begin{table}[htbp]
\setlength{\tabcolsep}{3mm}
\begin{center}
\begin{tabular}[b]{c|cccccc}
\midrule[2pt]
$P$ & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.9 \\
\toprule[1pt]
$r_{\text{h}}$ & 1.95519 & 1.92050 & 1.87594 & 1.82139 & 1.75661 & 1.59483 \\
$r_{\text{ISCO}}$ & 5.86321 & 5.75424 & 5.61066 & 5.42955 & 5.20713 & 4.61999 \\
$(r_{\text{ISCO}}-3r_{\text{h}})\times 10^{3}$ & -2.37168 & -7.25105 & -17.1501 & -34.6012 & -62.6931 & -164.491 \\
$r_{\text{ph}}$ & 3.23266 & 3.37470 & 3.52690 & 3.68206 & 3.83617 & 4.13392 \\
$b_{\text{ph}}$ & 5.49393 & 5.67308 & 5.86395 & 6.05783 & 6.24991 & 6.61985 \\
\midrule[2pt]
\end{tabular}
\end{center}
\centering
\caption{Data of $r_{\text{h}}$, $r_{\text{ISCO}}$, $r_{\text{ph}}$, and $b_{\text{ph}}$ for different $P$ with fixed $M=1$, $\beta=0.1$.}
\label{NumSol_changeP}
\end{table}
The effective potentials with respect to different parameters $\beta$ and $P$ are shown respectively in Figs. \ref{geo_change_beta_V(r)} and \ref{geo_change_P_V(r)}. The trajectories of photons can be derived by combining Eqs. (\ref{phidot2}) and (\ref{rdot2}), yielding
\begin{align}
\frac{{\rm{d}}r}{{\rm{d}}\phi}=\pm \frac{1}{k(r)}\sqrt{\frac{1}{b^{2}}-\frac{f(r)}{h(r)}}\equiv F(r),
\label{geodesics}
\end{align}
where $\pm$ represents the photons moving radial inward and outward respectively. We can obtain the null geodesics by integrating Eq. (\ref{geodesics}). For $b>b_{\text{ph}}$, photons approach the BH before reaching the closest approach $r=r_{0}$, then escape towards infinity. The closest approach $r_{0}$ can be obtained by solving $F(r)=0$. In the backward ray tracing method \cite{Astronomy_Luminet_1979}, $r_{0}$ is significant for determining the photons' trajectories as it can be the lower integral limit. Meanwhile, for $b<b_{\text{ph}}$, photons approach the BH continuously before falling into it.
\begin{figure}[htb]
\begin{center}
\subfigure[$\ V(r)$]{\includegraphics[width=6cm]{V_P0.6_changebeta.pdf}\label{geo_change_beta_V(r)}}
\subfigure[$\ \beta=0.05$]{\includegraphics[width=6cm]{geobeta0.05P0.6.pdf}\label{geo_change_beta_beta=0.05}}
\subfigure[$\ \beta=0.1$]{\includegraphics[width=6cm]{geoP0.6beta0.1.pdf}\label{geo_change_beta_beta=0.1}}
\subfigure[$\ \beta=0.2$]{\includegraphics[width=6cm]{geobeta0.2P0.6.pdf}\label{geo_change_beta_beta=0.2}}
\end{center}
\caption{The effective potential $V(r)$ for different BI parameter $\beta$ are shown in the top-left panel. The other figures are trajectories of photons for different values of $\beta$ with fixed $P=0.6$ and $M=1$.}
\label{geo_change_beta}
\end{figure}
We set the photons incident from infinity on the x-axis. Firstly, we study the geodesics for fixed $P=0.6$ and $M=1$ as $\beta$ varies. The numerical solutions of $r_{\text{h}}$, $r_{\text{ph}}$, and $b_{\text{ph}}$ are shown in Table \ref{NumSol_changeBeta}. Interestingly, $r_{\text{ph}}$ and $b_{\text{ph}}$ decrease dramatically with the increase of $\beta$ when it is less than 0.2. When $\beta$ increases, $r_{\text{ph}}$ and $b_{\text{ph}}$ slowly decline and gradually converge to the case of RN BH. Therefore, we select $\beta=0.05,\ 0.1,\ 0.2$ to investigate the photons' trajectories in Fig. \ref{geo_change_beta}.
In Figs. \ref{geo_change_beta_beta=0.05}, \ref{geo_change_beta_beta=0.1}, and \ref{geo_change_beta_beta=0.2}, the black disk represents the BH. And the orange and grey trajectories represent photons originating with $b>b_{\text{ph}}$ and $b<b_{\text{ph}}$ respectively. In the case of photons coming from $b=b_{\text{ph}}$, they continually approach the BH and keep rotating around it in an unstable orbit located at $r=r_{\text{ph}}$, i.e., the photon sphere, which is shown by red dashed lines. The BI parameter significantly affects the radius of the photon sphere and the curvature of the geodesics. However, as $\beta$ increases, the decrease in $r_{\text{h}}$ becomes so insignificant that it is scarcely visible in those figures.
\begin{figure}[htb]
\begin{center}
\subfigure[$\ V(r)$]{\includegraphics[width=6cm]{V_beta0.1_changeP.pdf}\label{geo_change_P_V(r)}}
\subfigure[$\ P=0.3$]{\includegraphics[width=6cm]{geoP0.3beta0.1.pdf}\label{geo_change_P_P=0.3}}
\subfigure[$\ P=0.6$]{\includegraphics[width=6cm]{geoP0.6beta0.1.pdf}\label{geo_change_P_P=0.6}}
\subfigure[$\ P=0.9$]{\includegraphics[width=6cm]{geoP0.9beta0.1.pdf}\label{geo_change_P_P=0.9}}
\end{center}
\caption{The effective potential $V(r)$ for different magnetic charges $P$ are shown in the top-left panel. The other figures are trajectories of photons for different values of $P$ with fixed $\beta=0.1$ and $M=1$.}
\label{geo_change_P}
\end{figure}
Then, we study the influence of magnetic charge on geodesics with fixed $\beta=0.1$ and $M=1$. Using the same method, we list the numerical solutions of $r_{\text{h}}$, $r_{\text{ph}}$, and $b_{\text{ph}}$ in Table \ref{NumSol_changeP}. The geodesics of different magnetic charges are plotted in Fig. \ref{geo_change_P_P=0.3}, \ref{geo_change_P_P=0.6}, and \ref{geo_change_P_P=0.9}. As the BH gets more magnetically charged or the BI effect becomes stronger, the range between the photon spheres and the event horizon extends.
\section{Shadows and photon spheres with spherical accretions\label{spheric}}
\subsection{Stationary spherical accretions\label{static}}
In this subsection, we study the BH shadows of static and spherically symmetric accretions. We employ the backward ray tracing method to study the specific intensity received by a distant observer \cite{Astronomy_Luminet_1979}. The emissivity of photons can be expressed as \cite{CPC_He_2022}
\begin{align}
j(\nu_{\text{e}})\propto\rho(r)P(\nu_{\text{e}}),
\label{j}
\end{align}
where $\rho(r)$ is the density of photons at a given radius $r$, and $P(\nu_{\text{e}})$ is the probability of photons emitting at a given frequency $\nu_{\text{e}}$. For spherical accretions, we assume that the density of photons follows a logarithmic normal distribution \cite{ARXIV_2206.04430}
\begin{align}
\rho(r)=\frac{1}{r}\sqrt{\frac{\gamma}{\pi}}{\rm{e}}^{-\gamma\ln^{2}\frac{r}{r_{\text{m}}}},
\label{rho}
\end{align}
where $\gamma$ is a constant that affects how fast the density of photons decays with respect to the distance from the BH, $r_{\text{m}}$ is the median of the logarithmic normal distribution. Since the photon density reaches the maximum at the photon sphere, we select $r=r_{\text{ph}}$ as the highest value of the probability density of the lognormal distribution. Solving $\left.\frac{{\rm{d}}\rho(r)}{{\rm{d}}r}\right|_{r=r_{\text{ph}}}=0$, one can obtain
\begin{align}
r_{\text{m}}=r_{\text{ph}}{\rm{e}}^{\frac{1}{2\gamma}}.
\label{rph_and_rm}
\end{align}
We introduce a non-monochromatic light source satisfying a normal distribution with central frequency $\nu_{\text{c}}$, which takes the form as
\begin{align}
f(\nu)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(\nu-\nu_{\text{c}})^{2}}{2\sigma^{2}}\right).
\end{align}
In this paper, we only take the frequency within $\nu_{\text{c}}-\sqrt{2}\sigma<\nu<\nu_{\text{c}}+\sqrt{2}\sigma$ into account and ignore the rest. The probability of a photon whose frequency falls within the aforementioned range is
\begin{align}
P(\nu_{\text{c}})=\int_{\nu_{\text{c}}-\sqrt{2}\sigma}^{\nu_{\text{c}}+\sqrt{2}\sigma}f(\nu){\rm{d}}\nu\approx 0.84270.
\label{probability}
\end{align}
Combining Eqs. (\ref{j}), (\ref{rho}), (\ref{rph_and_rm}), and (\ref{probability}) together, one can derive
\begin{align}
j(\nu_{\text{e}})\propto \frac{1}{r}\sqrt{\frac{\gamma}{\pi}}{\rm{e}}^{-\gamma\ln^{2}\frac{r}{r_{\text{m}}}}.
\label{j_int}
\end{align}
Integrating Eq. (\ref{j_int}) along the geodesics and setting the coefficient of proportion to be unit, we obtain the specific intensity
\begin{align}
I(b)=\int_{\text{geo}} g^{3} j(\nu_{\text{e}}){\rm{d}}l_{\text{prop}},
\label{Ib}
\end{align}
where $g=\nu_{\text{o}}/\nu_{e}$ is the redshift factor, $\nu_{\text{o}}$ is the frequency received by the observer, ${\rm{d}}l_{\text{prop}}$ is the infinitesimal proper length observed in the instantaneous rest frame of the photon. The redshift factor $g$ is given by
\begin{align}
g=\frac{p_{\alpha}u_{\text{o}}^{\alpha}}{p_{\beta}u_{\text{e}}^{\beta}},
\label{g}
\end{align}
where $p_{\alpha}$ and $p_{\beta}$ are the 4-momenta of the photons received and those emitted, respectively. And $u_{\text{o}}^{\alpha}$ is the 4-velocity of the observer, $u_{\text{e}}^{\beta}$ is the 4-velocity of the static accretions. Setting the observer at $r=\infty$, from $G_{\mu\nu}u_{\text{e}}^{\mu}u_{\text{e}}^{\nu}=1$ we derive
\begin{align}
\begin{split}
u_{\text{o}}^{\alpha}&=\left(1,\ 0,\ 0,\ 0\right),\\
u_{\text{e}}^{\beta}&=\left( \sqrt{\frac{1}{k(r)f(r)}},\ 0,\ 0,\ 0 \right).
\end{split}
\label{4velocity_static}
\end{align}
Substituting $p_{t}=-E$ into the Hamitonian of photons $\mathcal{H}=G^{\mu\nu}p_{\mu}p_{\nu}=0$, and redefining the affine parameter $\lambda$ to $\lambda/L$, we derive
\begin{align}
p_{\mu}=\left( -1,\ \pm\sqrt{\frac{1}{f(r)}\left( \frac{1}{f(r)}-\frac{b^2}{h(r)} \right)},\ 0,\ \pm b \right),
\label{4momentum_static}
\end{align}
where the first positive and negative sign, respectively, represent photons moving radially inward and outward. And $\pm b$ represents photons moving counterclockwise and clockwise. Substituting Eqs. (\ref{4velocity_static}) and (\ref{4momentum_static}) into Eq. (\ref{g}), we get
\begin{align}
g=\sqrt{k(r)f(r)}.
\label{g_static_derived}
\end{align}
And the infinitesimal proper length can be obtained as follows
\begin{align}
{\rm{d}}l_{\text{prop}}=\pm \sqrt{G_{ij}{\rm{d}}x^{i}{\rm{d}}x^{j}}=\pm \sqrt{\frac{k(r)}{f(r)}+h(r)k(r)\left(\frac{{\rm{d}}\phi}{{\rm{d}}r}\right)^{2}}{\rm{d}}r,
\label{dl}
\end{align}
where ${\rm{d}}\phi/{\rm{d}}r=1/F(r)$, can be obtained from Eq. (\ref{geodesics}). Substituting Eqs. (\ref{j_int}) and (\ref{dl}) into Eq. (\ref{Ib}), we obtain the specific intensity $I$ of the impact parameter $b$. For different decay factors $\gamma$, we plot the specific intensity in Fig. \ref{Ib_static_change_gamma}.
\begin{figure}[htb]
\centering
\includegraphics[width=9cm]{Ib_static_changegamma.pdf}
\caption{The specific intensity with different $\gamma$ in static spherical accretion with fixed $\beta=0.1,\ P=0.6$, and $M=1$.}
\label{Ib_static_change_gamma}
\end{figure}
From Fig. \ref{Ib_static_change_gamma} we can see that the specific intensity grows slightly as $b<b_{\text{ph}}$. But then surges to the maximum as $b$ approaches $b_{\text{ph}}$, steadily decline when $b>b_{\text{ph}}$, and gradually approaches 0 until $b=\infty$. What is of importance in this figure is that the maximum of the specific intensity located at $b=b_{\text{ph}}$, as a result of the photons revolving around the BH multiple times. Therefore, $b_{\text{ph}}$ is an intrinsic property of the effective metric independent of $\gamma$.
For convenience, we set $\gamma=1$ in the following discussions. We set the inclination angle of the observer to be zero due to the spherically symmetric of the spacetime. The impact parameter $b$ determines the light ray's trajectory, and thus determines the observed intensity \cite{EPJC_Zeng_2020}. Thereon, we can obtain the specific intensity $I(b)$ from Eq. (\ref{Ib}) and then plot the simulated shadows in Figs. \ref{shadows_static_change_beta} and \ref{shadows_static_change_P}. The specific intensity is represented by several colors, which are brighter if the specific intensity is greater, and vice versa.
\begin{figure}[htb]
\begin{center}
\subfigure[$\ I(b)$]{\includegraphics[width=6cm]{Ib_static_P0.6_changebeta.pdf}\label{I(b)_static_change_beta}}
\subfigure[$\ \beta=0.05$]{\includegraphics[width=6cm]{shadow_static_beta0.05P0.6.pdf}\label{shadow_static_change_beta_beta=0.05}}
\subfigure[$\ \beta=0.1$]{\includegraphics[width=6cm]{shadow_static_P0.6beta0.1.pdf}\label{shadow_static_change_beta_beta=0.1}}
\subfigure[$\ \beta=0.2$]{\includegraphics[width=6cm]{shadow_static_beta0.2P0.6.pdf}\label{shadow_static_change_beta_beta=0.2}}
\end{center}
\caption{The specific intensity $I$ with respect to different impact parameters $b$ in the top-left panel. And the other figures are the BH shadows with static accretions for different BI parameters $\beta$ with $P=0.6$ and $M=1$.}
\label{shadows_static_change_beta}
\end{figure}
The specific intensity and the shadows with static accretions when $P=0.6$ and $M=1$ with respect to different $\beta$ are shown in Fig. \ref{shadows_static_change_beta}. The photon spheres are the brightest rings in Figs. \ref{shadow_static_change_beta_beta=0.05}, \ref{shadow_static_change_beta_beta=0.1}, and \ref{shadow_static_change_beta_beta=0.2}. And the radii of the photon spheres decrease continuously with the increase of $\beta$, but the decreasing speed gradually slows down, which is consistent with Table \ref{NumSol_changeBeta}. Also, we discuss the specific intensities and shadows in the case of $\beta=0.1,\ M=1$ as $P$ varies in Fig. \ref{shadows_static_change_P} by applying the same method above. The radii of the photon spheres obviously increase as the magnetic charge increases as described in Table. \ref{NumSol_changeP}. In summary, the average radial distance between the photons received by the observer and the BH would increase thus cause a rise in the specific intensity, if the BH gets more magnetically charged or the effect of BI becomes more significant.
\begin{figure}[htb]
\begin{center}
\subfigure[$\ I(b)$]{\includegraphics[width=6cm]{Ib_static_beta0.1_changeP.pdf}\label{I(b)_static_change_P}}
\subfigure[$\ P=0.3$]{\includegraphics[width=6cm]{shadow_static_P0.3beta0.1.pdf}\label{shadow_static_change_P_P=0.3}}
\subfigure[$\ P=0.6$]{\includegraphics[width=6cm]{shadow_static_P0.6beta0.1.pdf}\label{shadow_static_change_P_P=0.6}}
\subfigure[$\ P=0.9$]{\includegraphics[width=6cm]{shadow_static_P0.9beta0.1.pdf}\label{shadow_static_change_P_P=0.9}}
\end{center}
\caption{The specific intensity $I$ with respect to different impact parameters $b$ in the top-left panel. And the other figures are the BH shadows with static accretions for different magnetic charge with $\beta=0.1$ and $M=1$.}
\label{shadows_static_change_P}
\end{figure}
\subsection{Infalling spherical accretions\label{infalling}}
In this subsection, we study the shadows of free-falling spherically symmetric accretions in a static and spherically symmetric spacetime. The accretions have radial velocity pointing to the BH in this model. Similarly, we also employ Eq. (\ref{Ib}) to describe the intensities of the shadows. In infalling spherical accretions, the redshift factor $g$ can be calculated through Eq. (\ref{g}). The 4-velocity of a spherically symmetric infalling accretion influenced by the magnetic charged BI BH is \cite{NPB_ZENG_2022}
\begin{align}
u_{\text{e}}^{\beta}=\left( \frac{1}{A(r)},\ -\sqrt{\frac{1-A(r)}{A(r)B(r)}},\ 0,\ 0 \right).
\label{4velocity_infalling}
\end{align}
For simplicity, we set $A(r)=k(r)f(r)$ and $B(r)=1/[k(r)f(r)]$. The observer's position is fixed as the same as that in section \ref{static}, through the same method in obtaining Eqs. (\ref{4velocity_static}) and (\ref{4momentum_static}), we derive
\begin{align}
g=\frac{1}{\frac{1}{k(r)f(r)}\pm\sqrt{1-f(r)}\sqrt{\frac{1}{f(r)}\left(\frac{1}{f(r)}-\frac{b^{2}}{h(r)}\right)}},
\label{g_infalling_derived}
\end{align}
\begin{figure}[htb]
\begin{center}
\subfigure[$\ \beta=0.05.$]{
\includegraphics[width=6cm]{Ib_staticinfalling_beta0.05P0.6.pdf}\label{I(b)_staticinfalling_change_beta_beta=0.05}
\hspace{2mm}
\includegraphics[width=6cm]{shadow_infalling_beta0.05P0.6.pdf}\label{shadow_infalling_change_beta_beta=0.05}
}
\subfigure[$\ \beta=0.2.$]{
\includegraphics[width=6cm]{Ib_staticinfalling_beta0.2P0.6.pdf}\label{I(b)_staticinfalling_change_beta_beta=0.2}
\hspace{2mm}
\includegraphics[width=6cm]{shadow_infalling_beta0.2P0.6.pdf}\label{shadow_infalling_change_beta_beta=0.2}
}
\end{center}
\caption{The specific intensity $I$ with respect to different impact parameters $b$ in the left panels with static and infalling accretions for different $\beta$ with fixed $P=0.6$ and $M=1$. And the shadows with these parameters for infalling accretions are shown in the right panels.}
\label{staticinfalling_change_beta}
\end{figure}
where the $\pm$ sign represents the photons moving radial inward and outward respectively. Therefore, we obtain the specific intensity $I$ with the impact parameter $b$ in Fig. \ref{staticinfalling_change_beta} by combining Eqs. (\ref{Ib}) and (\ref{g_infalling_derived}). The specific intensity in infalling accretion decrease slower compared to the static one. We extend the plot range to $b\leq 25$ to display the properties of shadows more comprehensively.
\begin{figure}[htb]
\begin{center}
\subfigure[$\ P=0.3.$]{
\includegraphics[width=6cm]{Ib_staticinfalling_P0.3beta0.1.pdf}\label{I(b)_staticinfalling_change_P_P=0.3}
\hspace{2mm}
\includegraphics[width=6cm]{shadow_infalling_P0.3beta0.1.pdf}\label{shadow_infalling_change_P_P=0.3}
}
\subfigure[$\ P=0.9.$]{
\includegraphics[width=6cm]{Ib_staticinfalling_P0.9beta0.1.pdf}\label{I(b)_staticinfalling_change_P_P=0.9}
\hspace{2mm}
\includegraphics[width=6cm]{shadow_infalling_P0.9beta0.1.pdf}\label{shadow_infalling_change_P_P=0.9}
}
\end{center}
\caption{The specific intensity $I$ with respect to different impact parameters $b$ in the left panels with static and infalling accretions for different $P$ with fixed $\beta=0.1$ and $M=1$. And the shadows with these parameters for infalling accretions are shown in the right panels.}
\label{staticinfalling_change_P}
\end{figure}
As shown in Fig. \ref{staticinfalling_change_beta}, the specific intensities of infalling accretion are obviously darker than those of the static one. Compared to static accretion, the infalling one has a smaller redshift factor $g$. As a result, the impact of redshift on infalling accretion is more obvious than that on static accretion. If the photons' impact parameter $b$ is small, their trajectories would get close to the BH. As a consequence, the integration along the geodesics contributes less to the specific intensity, resulting in a significantly lower value. Interestingly, the specific intensity of infalling accretion decreases inapparently as $b\to 0$ when $b<b_{\text{ph}}$, compared to that of static accretion. That is because the average radial distance between photons and the BH increases as the impact parameter increases, and the impact of the redshift factor on photons attenuates with their distance from the BH.
Additionally, the specific intensity of either static or infalling accretion, converges to zero as $b$ approaches infinity. Firstly, the average distance between photons and the BH is positively correlated to the impact parameter $b$. And the photon density decreases with the distance from the BH. As a result, the density of photons is less where the impact parameter is greater. Secondly, as illustrated at the end of the previous paragraph, the effect of the redshift factor on photons becomes weaker with increasing distance from the BH.
It is also noteworthy that, in both static and infalling accretion, the impact parameter corresponding to the peak of the specific intensity is always the same. This is because, in this paper, we ignore the other factors that may affect the photons' equivalent metric, such as the electromagnetic interaction between the accretion matters and photons \cite{ARXIV_2206.04430}.
We also investigate the impact of different magnetic charges on the specific intensity with infalling spherical accretion with fixed $\beta=0.1$ in Fig. \ref{staticinfalling_change_P}. Due to the slow increase of $b_{\text{ph}}$ as the BH gets more magnetically charged (see Table \ref{NumSol_changeP}), the differences between Figs. \ref{shadow_infalling_change_P_P=0.3} and \ref{shadow_infalling_change_P_P=0.9} are not as obvious as that between Figs. \ref{shadow_infalling_change_beta_beta=0.05} and \ref{shadow_infalling_change_beta_beta=0.2}. The BH shadow in the observational appearance shrinks as the BH becomes more magnetically charged or the influence of BI becomes more significant. The reasons are the same as our discussions in the end of the previous subsection.
\section{Shadows produced by accretion disks\label{ring}}
\subsection{Direct emission, lens ring, and photon ring}
In this section, we study the shadows cast by a thin accretion disk. The accretion disk can be placed on the equatorial plane by defining that the z-axis is perpendicular to the disk plane, for spherically symmetric spacetime. Inspired by \cite{Gralla:2019xty}, the trajectories of photons can be classified into three categories according to the total laps $n$ that orbit the BH. The trajectories with $n<3/4$ are defined as the direct emission, where photons intersect the accretion disk only once. The lens ring, where photons pass through the accretion disk twice, corresponds to the orbit's laps within $3/4<n<5/4$. Photons on the photon ring travel across the accretion disk at least three times, satisfying $n>5/4$. The ranges of direct emission, lens ring, and photon ring with respect to the impact parameter $b$ are numerically solved in Table \ref{NumSol_dir_lens_phtring} and Fig. \ref{nb} with different parameters $\beta$ and $P$. Fig. \ref{axes} is a visualization for Table \ref{NumSol_dir_lens_phtring}.
\begin{table}[htbp]
\setlength{\tabcolsep}{3mm}
\begin{tabular}{@{}|c|c|c|c|@{}}
\toprule[1pt]
& $\beta=0.1$ and $P=0.6$ & $\beta=0.2$ and $P=0.6$ & $\beta=0.2$ and $P=0.9$ \\ \toprule[1pt]
\begin{tabular}[c]{@{}c@{}}Direct emission\\ $0\leq n<3/4$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$0\leq b<5.98186$\\ $b>6.62840$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$0\leq b<5.17489$\\ $b>6.08053$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$0\leq b<5.33738$\\ $b>6.05728$\end{tabular} \\ \toprule[1pt]
\begin{tabular}[c]{@{}c@{}}Lens ring\\ $3/4<n<5/4$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$5.98186<b<6.05358$\\ $6.06645<b<6.62840$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$5.17489<b<5.30104$\\ $5.32213<b<6.08053$\end{tabular} & \begin{tabular}[c]{@{}c@{}}$5.33738<b<5.42038$\\ $5.43528<b<6.05728$\end{tabular} \\ \toprule[1pt]
\begin{tabular}[c]{@{}c@{}}Photon ring\\ $n>5/4$\end{tabular} & $6.05358<b<6.06645$ & $5.30104<b<5.32213$ & $5.42038<b<5.43528$\\ \toprule[1pt]
\end{tabular}
\caption{The ranges of direct emission, lens ring, and photon ring with respect to the impact parameter $b$ in the left panels for different parameters $\beta$ and $P$ with fixed $M=1$.}
\label{NumSol_dir_lens_phtring}
\end{table}
\begin{figure}[htb]
\centering
\subfigure[$\ \beta=0.1,\ P=0.6.$]{\includegraphics[width=8cm]{regionP0.6beta0.1.pdf}}\\
\subfigure[$\ \beta=0.2,\ P=0.6.$]{\includegraphics[width=8cm]{regionbeta0.2P0.6.pdf}}\\
\subfigure[$\ \beta=0.2,\ P=0.9.$]{\includegraphics[width=8cm]{regionbeta0.2P0.9.pdf}}\\
\caption{Visualization of Table \ref{NumSol_dir_lens_phtring}. The grey, red, and yellow ranges are direct emission, lens ring, and photon ring, respectively.}
\label{axes}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=10cm]{n_b.pdf}
\caption{The relationship between the total laps of photons rotating the black hole $n$ and the impact parameter $b$.}
\label{nb}
\end{figure}
Table \ref{NumSol_dir_lens_phtring}, Figs. \ref{axes} and \ref{nb} present that the range of direct emission extends with the increase of $P$, but the ranges of the lens ring and photon ring both decrease. Interestingly, all these three ranges are shifted outwards with the increase of $P$ or the decrease of $\beta$. Hence, in the presence of magnetic charge, the increase of BI effect obviously strengthens the effect of magnetic charge, which is similar to the result of the blue region of Fig. \ref{risco3rh} in section \ref{isco}. Changes in the BI parameter and the magnetic charge thus have contrary impacts on these three regions.
\begin{figure}[htb]
\begin{center}
\subfigure[$\ \beta=0.1,\ P=0.6.$]{\includegraphics[width=5.4cm]{geo_DirLenPht_P0.6beta0.1.pdf}}
\subfigure[$\ \beta=0.2,\ P=0.6.$]{\includegraphics[width=5.4cm]{geo_DirLenPht_beta0.2P0.6.pdf}}
\subfigure[$\ \beta=0.2,\ P=0.9.$]{\includegraphics[width=5.4cm]{geo_DirLenPht_beta0.2P0.9.pdf}}
\end{center}
\caption{Trajectories of photons. The black, red, and yellow curves represent the light rays corresponding to direct emission, lens ring, and photon ring, respectively. And the photon sphere is depicted here by black dashed lines. The black disk represents the BH. The difference in impact parameters between two adjacent trajectories are $0.05$, $10^{-3}$, and $10^{-4}$ in direct emission, lens ring, and photon ring, respectively.}
\label{geo_DirLenPht}
\end{figure}
We plot the photons' trajectories close to the BH in Fig. \ref{geo_DirLenPht} to illustrate how the magnetic charge and BI effect affect these three ranges. From Fig. \ref{geo_DirLenPht} we note that when $\beta$ is fixed, the increase of $P$ significantly extends the photon ring, and slightly reduces the radius of the event horizon. Meanwhile the range of the lens ring becomes narrower. If the magnetic charge is fixed, the increase of the BI parameter would expand the range of the lens ring. Interestingly, the impact of weakening BI effect is similar to that of reducing the magnetic charge of BH, which is consistent with our previous analysis.
\subsection{Observed specific intensities and transfer function}
We assume that the thin accretion disk emits isotropically in the rest frame of static worldlines, and the accretion matter is set to be fixed. The accretion disk is placed at the equatorial plane (the y-axis in Fig. \ref{geo_DirLenPht}), and the observer is located at infinity above the north pole ($x=+\infty$ in Fig. \ref{geo_DirLenPht}). The specific intensity received by the observer with emission frequency $\nu_{\text{e}}$ is \cite{Gralla:2019xty}
\begin{align}
I_{\text{obs}}(r,\nu_{\text{o}})=g^{3}I_{\text{emi}}(r,\nu_{\text{e}}),\qquad g=\sqrt{k(r)f(r)},
\label{received_intensity_def}
\end{align}
where $I_{\text{emi}}(r,\nu_{\text{e}})$ is the emitted specific intensity by the accretion disk, $\nu_{\text{o}}$ is the observed frequency, and $g$ is the redshift factor derived in Eq. (\ref{g_static_derived}). The total specific intensity $I_{\text{o}}(r)$ can be obtained by integrating all
frequencies of $I_{\text{obs}}(r,\nu_{\text{o}})$, which is denoted as
\begin{align}
I_{\text{o}}(r)=\int_{0}^{+\infty}I_{\text{obs}}(r,\nu_{\text{o}}){\rm{d}}\nu_{\text{o}}=\int_{0}^{+\infty}g^{4}I_{\text{emi}}(r,\nu_{\text{e}}){\rm{d}}\nu_{\text{e}}=k^{2}(r)f^{2}(r)I_{\text{e}}(r),
\label{received_intensity_derived}
\end{align}
where we have defined $I_{\text{e}}(r)=\int_{0}^{+\infty}I_{\text{emi}}(r,\nu_{\text{e}}){\rm{d}}\nu_{\text{e}}$ as the total emitting specific intensity.
We assume the disk is optically thin, i.e., no photons are absorbed by the disk. When the photons are emitted from the accretion disk, the light rays are bent by the BH and may intersect with the accretion disk for several times. For the light rays in the regions of direct emission, they intersect with the accretion disk only once. The photons with $3/4<n<5/4$ cross the equatorial plane twice. The light rays intersecting the accretion disk three or more times form the photon ring.
The sum of the observed specific intensities from each intersection determines the total received intensity, for photons contribute the specific intensity received by the observer once every time they pass through the accretion disk \cite{Li:2021ypw}, yielding
\begin{align}
I_{\text{o}}(r)=\sum_{n=1}^{\infty}\left.k^{2}(r)f^{2}(r)I_{\text{e}}(r)\right|_{r=r_{n}(b)}.
\label{received_intensity_sum}
\end{align}
The $r_{n}(b)$ above is the transfer function, where $n$ is a positive integer. It is the radial position where the photons cross the accretion disk for the $n$-th time. The slope of a transfer function is called the demagnification factor \cite{Gralla:2019xty}. At given impact parameter $b$ it represents how much the corresponding part of the accretion disk is demagnified in the observer's vision at infinity. For different $\beta$ and $P$, the first three transfer functions $r_{n}(b)$ are plotted in Figure \ref{transfunc}. Here, we ignore the transfer functions of $n\geq 4$ due to limited numerical calculation precision.
\begin{figure}[htb]
\begin{center}
\subfigure[$\ \beta=0.1,\ P=0.6.$]{\includegraphics[width=5.4cm]{transfunc_P0.6beta0.1.pdf}}
\subfigure[$\ \beta=0.2,\ P=0.6.$]{\includegraphics[width=5.4cm]{transfunc_beta0.2P0.6.pdf}}
\subfigure[$\ \beta=0.2,\ P=0.9.$]{\includegraphics[width=5.4cm]{transfunc_beta0.2P0.9.pdf}}
\end{center}
\caption{The first three transfer functions of a magnetically charged BI BH with thin accretion disk. The black, red and yellow lines represent the first, second and third transfer functions, respectively.}
\label{transfunc}
\end{figure}
It is noteworthy that the demagnification factors ${\rm{d}}r/{\rm{d}}b$ of these transfer functions are quite different. The first transfer function corresponds to the direct emission, which looks like a straight line with a demagnification factor of 1 approximately. So $r_{1}(b)$ is related to the redshifted source profile. The second transfer function relates to the lens ring (including the photon ring). The slope of $r_{2}(b)$ is much greater than that of $r_{1}(b)$, which indicates the lens ring is a significantly demagnified image of the back side of the accretion disk. The third transfer function is a line almost perpendicular to the horizontal axis and only contains the photon ring. Hence, the image of the photon ring is an extremely demagnified image of the front side of the disk. Thereon, the demagnification factors of further transfer functions approach infinity and they contribute negligibly to the observed intensity. Ignoring the higher-order transfer functions, therefore, has little effect on the total observed intensity.
\subsection{Observational appearances}
In this subsection, we study the specific intensity with some numerical toy models based on Eq. (\ref{received_intensity_sum}). The first emissive function $I_{\text{e1}}$ is assumed to be an exponential distribution that starts from $r=r_{\text{ISCO}}$.
\begin{align}
I_{\text{e1}}(r)/I_{0}=\begin{cases}
\exp\left[-(r-r_{\text{ISCO}})\right],\ r\geq r_{\text{ISCO}},\\
0,\ r<r_{\text{ISCO}},
\end{cases}
\label{emitted_intensity1}
\end{align}
where $I_{0}$ is the maximum intensity, similarly hereinafter. It is plotted in the top-left figures in Figs. \ref{shadow_ring_beta0.1_P0.6}, \ref{shadow_ring_beta0.2_P0.6}, and \ref{shadow_ring_beta0.2_P0.9} for different parameters $\beta$ and $P$. It reaches its peak at $r=r_{\text{ISCO}}$, and plummets to zero as $r<r_{\text{ISCO}}$.
The second emissive function $I_{\text{e2}}$ is an inverse proportional function multiplied by an exponential distribution, which starts from $r=r_{\text{ph}}$. We call it power law exponential model for convenience in the following text.
\begin{align}
I_{\text{e2}}(r)/I_{0}=\begin{cases}
\frac{1}{r-r_{\text{ph}}+1}\exp\left[-(r-r_{\text{ph}})\right],\ r\geq r_{\text{ph}};\\
0,\ r<r_{\text{ph}}.
\end{cases}
\label{emitted_intensity2}
\end{align}
This emissive function is plotted in the top-middle figures in Figs. \ref{shadow_ring_beta0.1_P0.6}, \ref{shadow_ring_beta0.2_P0.6}, and \ref{shadow_ring_beta0.2_P0.9} for different parameters $\beta$ and $P$. Obviously, it has a peak at $r=r_{\text{ph}}$ and drops steeper than the first emissive function as $r>r_{\text{ph}}$.
\begin{figure}[htb]
\begin{center}
\subfigure[\ Exponential model.]{\begin{minipage}{.31\linewidth}
\includegraphics[width=\linewidth]{Ie1_P0.6beta0.1.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{Io1_P0.6beta0.1.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{shadow_ring_I1_P0.6beta0.1.pdf}
\end{minipage}
}
\subfigure[\ Power law exponential model.]{\begin{minipage}{.31\linewidth}
\includegraphics[width=\linewidth]{Ie2_P0.6beta0.1.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{Io2_P0.6beta0.1.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{shadow_ring_I2_P0.6beta0.1.pdf}
\end{minipage}
}
\subfigure[\ Bell-shaped model.]{\begin{minipage}{.31\linewidth}
\includegraphics[width=\linewidth]{Ie3_P0.6beta0.1.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{Io3_P0.6beta0.1.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{shadow_ring_I3_P0.6beta0.1.pdf}
\end{minipage}
}
\end{center}
\caption{Observational appearances of a thin disk for different models with fixed $\beta=0.1$ and $P=0.6$. The first row shows the emissive functions $I_{\text{e}}(r)$, and the figures in the second row are the observed intensities $I_{\text{o}}(b)$. In the third row, we plot these observed intensities into two-dimensional disks.}
\label{shadow_ring_beta0.1_P0.6}
\end{figure}
The third emissive function is assumed to be a bell-shaped function starting from $r=r_{\text{h}}$.
\begin{align}
I_{\text{e3}}(r)/I_{0}=\begin{cases}
\frac{1-\tanh \left[r-(r_{\text{ISCO}}-r_{\text{h}}+0.5)\right]}{1-\tanh \left[r_{\text{h}}-(r_{\text{ISCO}}-r_{\text{h}}+0.5)\right]},\ r\geq r_{\text{h}};\\
0,\ r<r_{\text{h}}.
\end{cases}
\label{emitted_intensity3}
\end{align}
The third emissive function is plotted in the top-right figures in Figs. \ref{shadow_ring_beta0.1_P0.6}, \ref{shadow_ring_beta0.2_P0.6}, and \ref{shadow_ring_beta0.2_P0.9} for different parameters $\beta$ and $P$. After reaching its maximum at $r=r_{\text{h}}$, it decreases slowly at first and then rapidly. By employing Eq. (\ref{received_intensity_sum}) to these emissive functions, we numerically calculate the observed intensities, which are then plotted in the middle row in those figures corresponding to these three emissive functions.
\begin{figure}[htb]
\begin{center}
\subfigure[\ Exponential model.]{\begin{minipage}{.31\linewidth}
\includegraphics[width=\linewidth]{Ie1_beta0.2P0.6.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{Io1_beta0.2P0.6.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{shadow_ring_I1_beta0.2P0.6.pdf}
\end{minipage}
}
\subfigure[\ Power law exponential model.]{\begin{minipage}{.31\linewidth}
\includegraphics[width=\linewidth]{Ie2_beta0.2P0.6.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{Io2_beta0.2P0.6.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{shadow_ring_I2_beta0.2P0.6.pdf}
\end{minipage}
}
\subfigure[\ Bell-shaped model.]{\begin{minipage}{.31\linewidth}
\includegraphics[width=\linewidth]{Ie3_beta0.2P0.6.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{Io3_beta0.2P0.6.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{shadow_ring_I3_beta0.2P0.6.pdf}
\end{minipage}
}
\end{center}
\caption{Observational appearances of a thin disk for different models with fixed $\beta=0.2$ and $P=0.6$. The first row shows the emissive functions $I_{\text{e}}(r)$, and the figures in the second row are the observed intensities $I_{\text{o}}(b)$. In the third row, we plot these observed intensities into two-dimensional disks.}
\label{shadow_ring_beta0.2_P0.6}
\end{figure}
\begin{figure}[htb]
\begin{center}
\subfigure[\ Exponential model.]{\begin{minipage}{.31\linewidth}
\includegraphics[width=\linewidth]{Ie1_beta0.2P0.9.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{Io1_beta0.2P0.9.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{shadow_ring_I1_beta0.2P0.9.pdf}
\end{minipage}
}
\subfigure[\ Power law exponential model.]{\begin{minipage}{.31\linewidth}
\includegraphics[width=\linewidth]{Ie2_beta0.2P0.9.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{Io2_beta0.2P0.9.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{shadow_ring_I2_beta0.2P0.9.pdf}
\end{minipage}
}
\subfigure[\ Bell-shaped model.]{\begin{minipage}{.31\linewidth}
\includegraphics[width=\linewidth]{Ie3_beta0.2P0.9.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{Io3_beta0.2P0.9.pdf}\vspace{2pt}
\includegraphics[width=\linewidth]{shadow_ring_I3_beta0.2P0.9.pdf}
\end{minipage}
}
\end{center}
\caption{Observational appearances of a thin disk for different models with fixed $\beta=0.2$ and $P=0.9$. The first row shows the emissive functions $I_{\text{e}}(r)$, and the figures in the second row are the observed intensities $I_{\text{o}}(b)$. In the third row, we plot these observed intensities into two-dimensional disks.}
\label{shadow_ring_beta0.2_P0.9}
\end{figure}
The middle-left panels of Figs. \ref{shadow_ring_beta0.1_P0.6}, \ref{shadow_ring_beta0.2_P0.6}, and \ref{shadow_ring_beta0.2_P0.9} correspond to the observed intensities of different parameters and different emissive functions. For the exponential model, the corresponding impact parameter for the first peak of the observed intensity is about $b\approx 6.1$ for $\beta=0.1$ and $P=0.6$ ($b\approx 5.3$ for $\beta=0.2$ and $P=0.6$; $b\approx 5.5$ for $\beta=0.2$ and $P=0.9$), which is located at the photon ring. The lens ring causes the second, considerably higher peak that occurs immediately after the first one. In the middle-left panels of Figs. \ref{shadow_ring_beta0.1_P0.6} and \ref{shadow_ring_beta0.2_P0.6}, the observed intensities reach their maximum after the second peak. For $\beta=0.1$ and $P=0.6$, the third peak of the observed intensity's corresponding impact parameter is approximately $b\approx 6.6$ ($b\approx 6.1$ for $\beta=0.2$ and $P=0.6$), representing the peak observed intensity produced by direct emission. In the case of $\beta=0.2$ and $P=0.9$, there are only two peaks in the observed intensity, but the second peak is higher than those in the other cases. The reason is that the lens ring is merged with the peak corresponding to the direct emission. In the bottom-left panels of Figs. \ref{shadow_ring_beta0.1_P0.6} and \ref{shadow_ring_beta0.2_P0.6} we can find a narrow ring just within the brightest ring, which is the lens ring. One may zoom in to find the extremely thin photon ring just within the lens ring. The brightness and width of the whole shadow are dominated by direct emission in the exponential model.
For the power law exponential model, all the observed intensities shoot up to their first peak at $b\approx 4.9$, $b\approx 4.1$, and $b\approx 4.3$ in the central panel of Figs. \ref{shadow_ring_beta0.1_P0.6}, \ref{shadow_ring_beta0.2_P0.6}, and \ref{shadow_ring_beta0.2_P0.9} respectively. These local maximums are related to the direct emission. The observed intensities then steadily decline until they reach the maximum produced by the combination of the photon ring and the lens ring. The bottom-center panels of those three figures show that the lens ring and photon ring are combined to generate a halo with a highly luminous but extremely narrow profile. And the direct emission has a low observed intensity but a wide range. Among these figures, the lens ring and the photon ring are indistinguishable. For this model, the brightness of the whole shadow is dominated by the combination of the photon ring and the lens ring, but the direct emission contributes the most to the width.
In the middle-right panels of Figs. \ref{shadow_ring_beta0.1_P0.6}, \ref{shadow_ring_beta0.2_P0.6}, and \ref{shadow_ring_beta0.2_P0.9}, three peaks appear from left to right. The luminosity climbs to the first maximum produced by the direct emission. After the first peak, the observed intensities drop gradually except for a dramatic rise to their maximum and fall immediately. The maximum is approximately $b\approx 6.1$ for $\beta=0.1$ and $P=0.6$ ($b\approx 5.3$ for $\beta=0.2$ and $P=0.6$; $b\approx 5.5$ for $\beta=0.2$ and $P=0.9$), which corresponds to the photon ring. The photon ring is a lower peak very close to the lens ring. Although we can identify the lens ring and the photon ring through $I_{\text{o3}}(b)$ curves, it is hard to distinguish them in the simulated shadow images. As a result, the lens ring and photon ring produce a thin and bright halo, and the direct emission contributes a relatively dark but wide ring. For the bell-shaped model, the lens ring, combining with the photon ring, dominate the maximum brightness of the whole shadow, while the direct emission makes the most contribution to the width.
Combining Figs. \ref{shadow_ring_beta0.1_P0.6}, \ref{shadow_ring_beta0.2_P0.6}, and \ref{shadow_ring_beta0.2_P0.9}, we arrive at some interesting conclusions. Firstly, it is easy to notice the differences in the shadows of magnetically charged BI BHs cast by the accretion disks starting from various radii. The radius of the central dark area has a strong relationship with the model. Secondly, the BI effect and the magnetic charge both have a considerable effect on the shadows. It is noticeable that the increase of the magnetic charge intensifies the luminosity produced by direct emission apparently, and the increase of the BI parameter weakens the effect of the magnetic charge. Lastly, the third is the most consistent with astronomical observations.
Different emission models have a large impact on the observed intensity of direct emission but have little impact on the lens ring and the photon ring. This is due to the small range of the impact parameter related to the lens ring and photon ring. The range of the radial distance from the BH of photons emitted within the aforementioned range of the impact parameter is also narrow. The shapes and locations of the lens ring and the photon ring are almost unchanged with the emission models for the emissivity of the accretion disk is almost constant in such a short range of the radial distance. The direct emission, however, relates to a wide range of the impact parameter. Photons with an impact parameter in this range have a wide range of radial distance from the BH when they are emitted from the accretion disk. The shapes and locations of observed intensity corresponding to the direct emission are also changing because the emission intensity of the accretion disk varies with the change in radial distance. Actually, the lens ring and the photon ring of the three models are difficult to identify for the ranges of the impact parameter corresponding to the transfer functions of $n=2$ and $n=3$ are very narrow. In this case, regardless of the accretion model, the emission intensity is nearly constant in such a narrow range of radial distance. Therefore, it is difficult to distinguish the photon ring and the lens ring only by changing the emission model.
\section{Conclusion and discussion\label{concl}}
In this paper, we investigated the observational appearances of magnetically charged Born-Infeld black hole. Firstly we presented the metric for BI BH with magnetic charge, and studied the existence of black hole with unit mass in the parameter space of the BI parameter $\beta$ and magnetic charge $P$. The critical value of the magnetic charge increases with the increasing effect of BI nonlinear electrodynamics. The radii of the event horizon, the innermost stationary circular orbit and the photon sphere, i.e., $r_{\text{h}}$, $r_{\text{ISCO}}$, and $r_{\text{ph}}$, and their corresponding impact parameter $b_{\text{ph}}$ decrease as $\beta$ increases. $r_{\text{ph}}$ and $r_{\text{ISCO}}$ monotonically decrease as $P$ increases, but $r_{\text{ph}}$ and $b_{\text{ph}}$ increase consistently.
To compare the spacetime of a magnetically charged BI BH with that of the Schwarzschild BH, we further investigated the values of $r_{\text{ISCO}}-3r_{\text{h}}$ and found that $\beta$ has critical values for each magnetic charge. The value $r_{\text{ISCO}}-3r_{\text{h}}$ increases monotonically as the BI effect weakens, i.e., the increase of $\beta$. But the increase of the magnetic charge decreases the aforementioned value if $\beta$ is below the critical value, and vice versa. The geodesics are then derived from the equivalent metric of photons. The photon trajectories are null geodesics in the effective metric rather than the background metric because of the effects of BI. We found that the photon geodesics are more bent as the BI parameter decreases or the magnetic charge increases.
BH shadows are produced by photons that are deflected in curved spacetime. We first consider static spherically symmetric accretion, assuming that the density of photons follows a lognormal distribution with a maximum at the radius of the photon sphere. Also, we supposed that the frequency of the photons emitted is not monochromatic but has a normal distribution with central frequency $\nu_{\text{c}}$. The specific intensity received by a distant observer was obtained by integrating along the geodesics using the backward ray tracing method. And we plotted the specific intensity on a two-dimensional plane with the observer located at infinity above the north pole to display the BH shadows. The intensity grows slightly as $b<b_{\text{ph}}$, then rises sharply to the maximum as $b$ approaches $b_{\text{ph}}$, and asymptotically drops to zero when $b>b_{\text{ph}}$. The specific intensity reaches the maximum when the impact parameter of photons is equal to the critical value, as a result of the photons orbiting the BH infinite times.
We also took infalling spherically symmetric accretion models into account. The shadows of infalling accretion are obviously darker than the static ones with the same BI parameter $\beta$ and magnetic charge $P$ because the impact of the redshift factor $g$ on specific intensity becomes significant when the photons get close to the BH. In both infalling and static accretion, the brightest ring in each shadow represents its photon sphere. The radii and intensities of the shadows vary depending on the BI parameter and the magnetic charge. It is also noteworthy that the impact parameter $b$ corresponding to the peak of the specific intensity is always the same in both static and infalling accretions, indicating that the critical value $b_{\text{ph}}$ is only related to the effective metric. Moreover, the radius of the photon sphere is independent of the accretion disks.
We then studied the shadow produced by accretion disks. The photons emitted by the disk are divided into direct emission, lens ring, and photon ring based on their numbers of intersections with the disk. The regions of direct emission, lens ring, and photon ring with parameter changes are similar to the previous analysis of $r_{\text{ISCO}}$. We introduced transfer functions to describe the relationship between the radial positions where the photons intersect the accretion disk and their impact parameters. From the demagnification factors of the transfer functions, we illustrated that the slope of direct emission is almost fixed. However, the lens ring and the photon ring are noticeably demagnified. Then the observational appearances are simulated using three simple emission models of the accretion disk. Among all the simulations, the direct emission dominates the width of the shadow. The photon ring and the lens ring can hardly be distinguished only by changing the emission model.
With increasing resolutions of astrophysical observations, the nuances between the inner radii of shadows and the theoretical solutions for $r_{\text{ISCO}}$ may be found, implying the existence of magnetic charge combined with the BI NLED is probable. We expect this paper to provide a reference for the investigation of shadows of BI BH and the effect of magnetic fields on BI BH.
\begin{acknowledgments}
The authors are grateful to Peng Wang, Aoyun He, and Yadong Xue for useful discussions. This work is supported by NSFC Grant No. 12147207 and 12175212.
\end{acknowledgments}
\bibliographystyle{unsrt}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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**BASKETBALL
AND ITS GREATEST PLAYERS**
**inside _sports_**
# **BASKETBALL
AND ITS GREATEST PLAYERS**
**E DITED BY SHERMAN HOLLAR**
Published in 2012 by Britannica Educational Publishing
(a trademark of Encyclopædia Britannica, Inc.)
in association with Rosen Educational Services, LLC
29 East 21st Street, New York, NY 10010.
Copyright © 2012 Encyclopædia Britannica, Inc. Britannica, Encyclopædia Britannica, and the Thistle logo are registered trademarks of Encyclopædia Britannica, Inc. All rights reserved.
Rosen Educational Services materials copyright © 2012 Rosen Educational Services, LLC.
All rights reserved.
Distributed exclusively by Rosen Educational Services.
For a listing of additional Britannica Educational Publishing titles, call toll free (800) 237-9932.
First Edition
Britannica Educational Publishing
Michael I. Levy: Executive Editor, Encyclopædia Britannica
J.E. Luebering: Director, Core Reference Group, Encyclopædia Britannica
Adam Augustyn: Assistant Manager, Encyclopædia Britannica
Anthony L. Green: Editor, Compton's by Britannica
Michael Anderson: Senior Editor, Compton's by Britannica
Sherman Hollar: Associate Editor, Compton's by Britannica
Marilyn L. Barton: Senior Coordinator, Production Control
Steven Bosco: Director, Editorial Technologies
Lisa S. Braucher: Senior Producer and Data Editor
Yvette Charboneau: Senior Copy Editor
Kathy Nakamura: Manager, Media Acquisition
Rosen Educational Services
Shalini Saxena: Editor
Nelson Sá: Art Director
Cindy Reiman: Photography Manager
Matthew Cauli: Designer, Cover Design
Introduction by Shalini Saxena
**Library of Congress Cataloging-in-Publication Data**
Basketball and its greatest players / edited by Sherman Hollar.
p. cm. —(Inside sports)
"In association with Britannica Educational Publishing, Rosen Educational Services."
Includes bibliographical references and index.
ISBN 978-1-61530-558-2 (eBook)
1. Basketball players—Juvenile literature. I. Hollar, Sherman.
GV885.1.B36 2011
796.323—dc22
2010052666
**On the cover, page**: NBA superstar LeBron James of the Miami Heat plays against the Detroit Pistons. _Miami Herald/MCT via Getty Images_
Pp. –7, , , , © www.istockphoto.com/Jeff Milner; pp. , , , , © www.istockphoto.com/Thomas Levack; back cover, remaining interior background image Shutterstock.com
## **C ONTENTS**
**I NTRODUCTION**
**C HAPTER 1 THE GAME OF BASKETBALL**
**C HAPTER 2 HISTORY OF THE GAME**
**C HAPTER 3 WOMEN'S BASKETBALL AND INTERNATIONAL COMPETITION**
**C HAPTER 4 NOTABLE PLAYERS**
**C ONCLUSION**
**G LOSSARY**
**F OR MORE INFORMATION**
**B IBLIOGRAPHY**
**I NDEX**
## **INTRODUCTION**
While baseball has long been considered the American national pastime and football the most watched televised sport in the country, basketball—conceived entirely by Massachusetts physical education instructor James Naismith—is also a truly American institution. From casual one-on-one games in a gym to international Olympic-level competitions, "shooting hoops" has come to occupy a unique place on the both the American and international cultural landscapes. This volume examines the history of this much-loved form of recreation and introduces some of the game-changing professional players whose abilities have elevated the artistry of the game.
At first glance, basketball does not seem to be a complicated sport. As its name suggests, the key components of a basketball game are a ball and a basket into which the ball must be thrown in order to score points (the actual peach baskets used in early games were later replaced with the open hoops in use today). However, a number of elements add a certain amount of drama to each game. Within a short amount of time after gaining possession of the ball, each team must attempt to sink a shot into the basket that the opposing team is defending. Various offensive and defensive strategies may be employed throughout the course of a game. The press, for example, is a defensive strategy often used to force the opposition to hurry its movements and to commit errors that result in turnovers.
Originally designed as an indoor game for Naismith's students to play in the winter, basketball has since evolved in many respects. Although it is still enjoyed by students and casual athletes both indoors and out, its national and international appeal has escalated dramatically. The National Basketball Association (NBA) can be credited with much of this change. While it competed with other professional leagues in its early years, it eventually emerged as the primary professional league for male players in the United States and transformed the game. With increased visibility and popularity both at home and abroad, the NBA's teams and players have sold out arenas and attracted new generations of fans.
With basketball quickly gaining traction around the country after its introduction, it was only a matter of time before women and international players took to the court. Women's teams emerged soon after men's teams, and basketball has thus retained popularity among women nearly as long as it has men. Like the men's game, women's basketball has undergone a number of modifications over the years. The establishment of the Women's National Basketball Association (WNBA) in 1997 as the primary professional league for female players in the United States cemented the presence of women's games on national television and made the game accessible to wider audiences. International teams and players have similarly adopted the game and helped make basketball a global phenomenon.
**_The first basketball court was set up in this Springfield, Mass., gymnasium in 1891 by James Naismith_. Hulton Archive/Getty Images**
Although basketball is a team sport that in many ways demands that the whole be greater than the sum of its parts, individual players have been critical to advancing the sport and introducing new moves and dynamics to the game. After all, one can hardly mention basketball without recalling the acrobatics of Michael Jordan, or Air Jordan, as he aptly came to be known. The icons of today—Shaquille O'Neal, Kobe Bryant, LeBron James, to name a few—owe much to the generations of athletes who preceded them and have continued their legacy of innovation in exciting fashion.
From its humble beginnings in a school gym to selling out Madison Square Garden, basketball has endured and evolved immensely in the years since its invention. Still, it retains much of its original character and stands as a testament to the imagination of one teacher as much as it does to the dedication of the remarkable players that have reinvented the idea of skill.
## **CHAPTER 1
THE GAME OF BASKETBALL**
It could have been called boxball. In the winter of 1891, James Naismith, an instructor at a YMCA training school in Springfield, Mass., asked the janitor to hang a couple of boxes from the gymnasium balcony for an experimental indoor ball game. The game became known as basketball because the janitor, unable to find boxes to make the elevated goals, nailed up two half-bushel peach baskets. Naismith came up with the game in hopes of curing the winter doldrums of his students who had grown bored with the routine of gymnastics and calisthenics. Naismith first experimented with indoor versions of rugby, lacrosse, and other sports, but they proved too violent. The former divinity student eventually struck upon the idea of upright goals that would minimize the force on the ball and keep some distance between the players and the actual scoring. Thus the internationally popular game of basketball was born.
**_James Naismith, the creator of basketball_. Hulton Archive/Getty Images**
### **_H OW THE GAME IS PLAYED_**
The first basketball games played by James Naismith's students had two nine-man teams and were played with a soccer ball. In the first years of basketball competition, the game was played under a wide range of rules. The number of players varied from 10 to 18, the courts were often irregularly shaped, and baskets were attached to balconies that allowed fans to swat shots from the basket. Within the next decade the familiar rules and equipment of the modern game were established and continued to be modified to enhance the excitement of the competition.
#### **_R ULES AND EQUIPMENT_**
The dimensions of the indoor playing floor are 50 feet (15.2 meters) by 94 feet (28.7 meters) for professional and college play. High school courts may be smaller. The international court is 49.2 feet (15 meters) by 92 feet (28 meters). Every court includes a line that divides the court in half, and at the midpoint of that line is the center circle. Other court markings include the free throw line and lane markings, and a three-point line that is 19 feet 9 inches (6 meters) from the basket in college and high school play and 23 feet 9 inches (7.2 meters) in the National Basketball Association (NBA). At each end of the court is a goal, or basket, 10 feet (3 meters) above the floor. The first goals were literally baskets, and a ladder was kept nearby to clear out the basket after each successful goal. When metal baskets were substituted, a pole was needed to poke the ball out of a hole in the bottom. The metal rim of the hoop was not invented until 1906. A bag of braided cord netting was attached to the hoop, and after a score the ball was released by pulling a cord. The current basket is a metal orange ring, 18 inches (46 centimeters) in diameter, with an open net suspended from the rim.
**_For NBA competition a basketball court is laid out on a hardwood floor in a gymnasium or other indoor arena. The area at each end of the court marked by the free throw lane and its restraining circle, popularly called the key, is named for its resemblance to a keyhole. The standard dimensions of 94 feet (28.7 meters) in length and 50 feet (15.2 meters) in width may vary in high school and international play_. © Merriam-Webster Inc**.
The backboard was introduced in 1895 in order to prevent fans sitting in balconies from interfering with the baskets. The backboard is now an important factor in the game and is used by skilled players to make shooting easier. Backboards are either rectangular or fan-shaped and may be made of any rigid material—usually glass. The soccer balls used in early basketball games were replaced by laced leather balls, then laceless ones, and finally the molded leather-or composition-covered balls in use today. The inflated ball has a grippable, pebbled surface. It measures 29.5 to 30 inches (74.9 to 76 centimeters) in circumference and weighs 20 to 22 ounces (567 to 624 grams).
**_A modern basketball_. © www.istockphoto.com/Jason Lugo**
In 1897 the number of players per side for a basketball game was set at five. Typically the five players are a center, two forwards (one power forward and one small forward), and two guards (one point guard and one shooting guard). While there are five players per team on the court at any given time, the total number of players on a team may be higher (no more than 15 in the NBA), and substitutions may occur during the game. For collegiate, professional, and international games there are three referees, who are assisted by two timekeepers and two scorers. (A single timekeeper and a single scorer may be used if acceptable to the referee.)
The length of the game is regulated according to the age and physical strength of the players. College basketball teams play two 20-minute halves, with a 15-minute rest between halves. If the score is tied at the end of the two halves, play continues for as many five-minute overtime periods as it takes for one team to break the tie. High school teams play four eight-minute periods, with a ten-minute intermission at halftime, plus three-minute overtime periods if needed. NBA games consist of four 12-minute periods with a 20-minute rest between halves. The tie-breaking overtime periods are five minutes long.
The visiting team has the choice of baskets at the start of the game, and the teams change baskets for the second half. Play begins when the referee tosses up the ball between two opposing players, who stand inside the center circle and jump to tap the ball to a teammate. Other players must stay outside the circle until the ball is tapped.
**_A player dribbles a ball down a court_. Darryl Bautista © The Rosen Publishing Group and Darryl Bautista**
Each team strives to outscore the other by making as many baskets as possible. Any player may shoot at the basket. A successful shot is a field goal, which counts for two points. If the shot is attempted beyond the three-point line, then the successful shot counts for three points. Successful free throws (unhindered shots at the basket from the free throw line) count for one point each.
A player with possession of the ball must pass or shoot before taking two steps or must start dribbling (bouncing the ball) before taking a second step. The dribble continues until the player touches the ball with both hands at once, permits it to come to rest, or loses control of it. The ball may be batted with the hands, passed, bounced, or rolled in any direction.
When a field goal or free throw attempt fails, play continues. After each successful basket, the team that did not score puts the ball in play from behind the end line at the basket it is defending.
**_The shot clock is displayed above the backboard in a game of college basketball. College teams are required to sink or attempt a shot that makes contact with the rim within 35 seconds of taking control of the ball_. Jonathan Daniel/Getty Images**
In 1954 the shot clock was introduced in the NBA in order to increase scoring and the pace of the game. The shot clock begins when a team takes possession of the ball. Teams are required to take a shot that is either successful or makes contact with the rim within a set amount of time. The NBA and international leagues have a 24-second clock; college has a 35-second clock. There is no shot clock in high school basketball.
#### **_V IOLATIONS AND FOULS_**
Penalties are given for two general types of offenses—violations and fouls. The most common violations include: running or walking with the ball without dribbling; double dribbling (using both hands at the same time to dribble or stopping and restarting the dribble); and goaltending (interfering with a shot on the ball's downward path into the basket). Other violations include kicking the ball intentionally and, in the case of most American competition, touching the ball while it is on the rim of the basket. Most violations are punished by awarding the ball out-of-bounds to the other team.
**S HOTS FROM THE FIELD**
**One of the main field shots in the game of basketball is the layup, in which the shooter, while close to the basket, jumps and lays the ball against the backboard so it will rebound into the basket or just lays it over the rim. Away from the basket, players use a one-hand push shot from a stride, jump, or standing position and a hook shot, which is overhead. The jump shot is a particularly effective offensive weapon, since it is released at the top of the jump, making it difficult to defend. Some players can dunk or slam-dunk the ball, jamming the ball down into the basket**.
**_A player slam-dunks a ball through the hoop_. Shutterstock.com**
Fouls may be either personal or technical. Since basketball is theoretically not a body contact sport, a personal foul can result from any physical involvement with an opposing player. Pushing, pulling, bumping, holding, tripping, and charging are all infractions of the rules. A personal foul may be a common foul (neither obvious nor intentional), double foul (in which two opponents commit personal fouls against each other at about the same time), or multiple foul (in which two players on one team commit personal fouls against an opponent at about the same time). A player control foul is a common foul committed by a player while controlling the ball.
The officials call the fouls, penalizing the offending team either by awarding the ball out-of-bounds to its opponents or by awarding free throws. If a player is fouled in the act of shooting and misses the basket, he receives two free throws (three, if fouled attempting a three-point shot). If he is fouled in the act of shooting and makes the basket, the score counts and he receives an additional free throw. A high school or college player who commits five personal fouls must leave the game. NBA players are allowed six fouls. When a player is called for a personal foul, the foul is also registered against the team. When a team exceeds its limit—four team fouls per quarter in professional play and six per half in college and high school play—the opposing team receives extra free throws.
**_A defending player_ (right) _fouls an offensive player handling the ball_. Darryl Bautista © The Rosen Publishing Group and Darryl Bautista**
A technical foul may be committed by either a player or a nonplayer, such as coaches or spectators. Technical fouls are called against a team for delaying the game, unsportsman-like tactics, illegal substitutions, or illegal timeouts. The penalty throw for a technical foul may be made by any member of the opposing team, which then is usually given possession of the ball out-of-bounds at midcourt.
### **_S TYLES OF PLAY_**
Teams employ different styles of play on both the offensive and defensive ends of the court. One of the best-known styles of offensive play is the motion offense. In the motion offense, all the players weave in and out of the area in front of the basket, trying to block defenders and free their teammates for open shots. A common play within the motion offense is the pick-and-roll, in which a player sets a "pick" for a teammate with the ball by getting in the way of the defender guarding him. If the defender continues to follow the ball handler, the player setting the pick moves, or "rolls," toward the basket and is often able to get open for a pass and shot. Some teams prefer the fast-break style of play that emphasizes pushing the ball quickly toward the offensive end and taking shots before the defense is set.
**_Ryan Kelly (#34) and Kyle Singler (#12) of the Duke Blue Devils block Jay-R Strowbridge (#55) of the Oregon Ducks_. Jonathan Ferrey/Getty Images**
The two primary styles of defense are zone and man-to-man. In a zone defense, each defensive player guards a certain area on the court. In man-to-man defense, each player guards a specific player on the opposing team. In a combination defense, two or three players guard certain zones while the others play man-to-man defense. The press, which forces the offensive team to move the ball quickly, is used often by a team that is losing toward the end of the game. In employing a press, a team goes all out to try to block and steal the ball from the other team. Most teams use a man-to-man press, a full-court zone press, or a half-court zone press.
## **CHAPTER 2
HISTORY OF THE GAME**
Basketball is the only major sport that is completely American in origin. Following its invention by Naismith in 1891, the game quickly spread through the United States, and the first professional teams emerged in the Northeast at the turn of the 20th century. These teams were instrumental in developing the game, introducing key elements such as the bounce pass and the free throw following a foul. The first professional league was the National Basketball League, which formed in 1898. The early professional games were very physical and often stirred the emotion of fans to such a point that the atmosphere at games was frequently hostile. Chicken wire cages (and later rope netting) was placed around the court to separate players and fans. Even decades after this practice disappeared, basketball players were referred to as "cagers."
**T HE BARNSTORMING ERA**
**Three barnstorming professional teams had a major impact on the sport—the Original Celtics, the New York Renaissance (often called the Rens), and the Harlem Globetrotters. Led by 6-foot, 5-inch (1.96-meter) Joe Lapchick, the Original Celtics were formed in New York City and dominated professional play in the 1920s with their switching man-to-man defense and superior passing game. The Celtics, an all-white team, had a memorable clash with the all-black New York Rens in 1925–26 when the two teams split 6 games. Formed by Robert Douglas in 1922, the New York Rens were one of the finest teams of the barnstorming era. In the 1932–33 season, the Rens won 88 consecutive games, and in 1939 they won the World Professional Tournament. The Harlem Globetrotters, an all-black squad founded in 1927 by promoter and coach Abe Saperstein, originally played exhibitions in a Chicago ballroom. By the 1930s the Globetrotters had attracted top black players and were dominating other teams with their superior ball-handling ability. As the games became increasingly lopsided, the team developed its trademark comic style—players spinning the ball on their fingers, head-bouncing it into the basket, dribbling behind the back, and making blind passes—in order to make the games entertaining**.
**_A five-photograph composite shows members of the Harlem Globetrotters team in 1931_. Chicago History Museum/Archive Photos/Getty Images**
### **_T HE EARLY YEARS OF THE NBA_**
Basketball's premiere league, the NBA, was formed in 1949 when the National Basketball League merged with the Basketball Association of America. The NBA flourished in its early years due in large part to the brilliant play of Minneapolis Lakers center George Mikan. Standing 6 feet 10 inches (2.08 meters), Mikan was the game's first dominant center, and his playing style had an influence on many subsequent big men. In 1950 the NBA was integrated by African American players Earl Lloyd, Chuck Cooper, and Nathaniel "Sweetwater" Clifton. The NBA was unchallenged as the top basketball league until the American Basketball Association (ABA) emerged in 1967. The ABA introduced the three-point basket and played a more open style of basketball that soon attracted large crowds. The ABA competed with the NBA for top players but eventually collapsed under financial burdens, though four ABA teams were absorbed into the NBA.
**_When Earl Lloyd was selected to play with the Washington Capitols in 1950, he became one of the first black players in the NBA. Here he poses for a photograph after joining the Syracuse Nationals_. NBA Photos/National Basketball Association/Getty Images**
### **_A N EW ERA OF SUPERSTARS_**
By the early 1980s the NBA was plagued by money-losing franchises, low attendance, and declining television ratings. The league soon rebounded under the leadership of David Stern, NBA commissioner from 1984, who helped transform it into an international entertainment company. Aggressive marketing highlighted star players such as Magic Johnson, Larry Bird, and, especially, Michael Jordan. Other innovations included league limits on player salaries, lucrative broadcast rights for network and cable television, and expanded All-Star Game festivities.
**_NBA commissioner David Stern_ (right) _gives a ring to Larry Bird of the Boston Celtics during the 1984 Championship ring ceremony_. Dick Raphael/NBAE via Getty Images**
### **_T HE NBA TODAY_**
As of 2011, the NBA had a total of 30 teams organized into Eastern and Western conferences and further divided into six divisions. In the Eastern Conference the Atlantic Division comprises the Boston Celtics, the New Jersey Nets (in Newark), the New York Knicks, the Philadelphia 76ers, and the Toronto Raptors; the Central Division is made up of the Chicago Bulls, the Cleveland Cavaliers, the Detroit Pistons, the Indiana Pacers (in Indianapolis), and the Milwaukee (Wisconsin) Bucks; the Southeast Division comprises the Atlanta Hawks, the Charlotte (North Carolina) Bobcats, the Miami Heat, the Orlando (Florida) Magic, and the Washington (D.C.) Wizards. In the Western Conference the Southwest Division comprises the Texas-based Dallas Mavericks, Houston Rockets, and San Antonio Spurs, the Memphis (Tennessee) Grizzlies, and the New Orleans Hornets; the Northwest Division is made up of the Denver Nuggets, the Minnesota Timberwolves (in Minneapolis), the Oklahoma City Thunder, the Portland (Oregon) Trail Blazers, and the Utah Jazz (in Salt Lake City); the Pacific Division comprises the Phoenix Suns and the California-based Golden State Warriors (in Oakland), Los Angeles Clippers, Los Angeles Lakers, and Sacramento Kings.
**_The Boston Celtics in 2009. The Celtics are one of 15 teams in the NBA's Eastern Conference_. Brian Babineau/NBAE via Getty Images**
**_The Los Angeles Lakers in 2010. The Lakers are one of 15 teams in the NBA's Western Conference_. Andrew D. Bernstein/NBAE via Getty Images**
**_New York City's Madison Square Garden, home of the New York Knicks, is often packed with fans during games, such as this one between the Knicks and the Miami Heat in 2005_. Jim McIsaac/Getty Images**
**T HE NBDL**
**In 2001 the NBA launched the National Basketball Development League (NBDL). The NBDL served as a kind of "farm system" for the NBA. Through its first 50 years the NBA did not have an official system of player development or a true minor league system for bringing up young and inexperienced players such as the one that exists in major league baseball. College basketball has been the area from which the NBA did the vast majority of its recruiting. By 2000 this had begun to change somewhat, as players began to be drafted straight out of high school with increasing frequency. In 2005 the NBA instituted a rule stipulating that domestic players must be at least age 19 and have been out of high school for one year to be eligible for the draft, which in effect required players to spend at least one year in college or on an international professional team before coming to the NBA**.
The play-offs follow the traditional 82-game schedule, involving 16 teams and beginning in late April. Played as a best-of-seven series, the final pairings stretch into late June. Although basketball is traditionally a winter game, the NBA still fills its arenas and attracts a national television audience in late spring and early summer.
As the popularity of the league grew, player salaries rose to an annual average of more than $5 million by mid-2000s, and some superstars earned more than $20 million yearly. The NBA has a salary cap that limits (at least theoretically, as loopholes allow many teams to exceed the cap) the total amount a team can spend on salaries in any given season.
## **CHAPTER 3
WOMEN'S BASKETBALL AND INTERNATIONAL COMPETITION**
Basketball has always enjoyed great popularity. Within a decade of its invention there were women's teams, professional circuits, and an intercollegiate conference. The game quickly spread to Canada, Europe, and Australia. Today, young people play basketball in grade school, high school, college, and at athletic clubs. In addition to amateur teams, there are professional leagues and independent professional clubs throughout the world.
### **_D EVELOPMENT OF THE WOMEN'S GAME_**
Women have been playing basketball for almost as long as men. Senda Berenson, a physical education teacher at all-female Smith College in Northampton, Mass., observed men playing the game at a nearby institution in the early 1890s and decided to try it with her students. Other colleges followed suit, and teams from Stanford University and the University of California at Berkeley met in 1896 for the first women's intercollegiate basketball game.
Although the women's rules at the high school, collegiate, and professional levels today are similar to those for the men's game, this was not the case when the sport began. Clara Baer, who introduced basketball at the H. Sophie Newcombe College for women in New Orleans, misread a diagram of the court sent to her by James Naismith and restricted the movement of offensive and defensive players to one end of the court. The mistake, however, was quickly embraced as many people considered it unladylike for women to play competitive sports and preferred the limitations placed on movement. Others worried about the health of female athletes and tried to compensate by shortening the time in each play period. Modifications occurred through the years, but the full-court, five-player team did not become standard until the early 1970s.
**_The U.S. women's national team won the gold medal in basketball at the 2000 Olympic Games held in Sydney, Australia. Members of the team are pictured during the playing of the national anthem_. Andrew D. Bernstein/NBAE via Getty Images**
The International Women's Sports Federation included basketball in its version of the Olympics in 1924. In 1926 the Amateur Athletic Union started sponsoring an annual basketball tournament for women. The World Amateur Basketball Championship for women began in 1953, with the United States team taking the gold medal. A squad from the Soviet Union won the first Olympic gold medal for women's basketball at the 1976 Summer Games in Montreal. U.S. women's teams were crowned Olympic champions in 1984, 1988, 1996, 2000, and 2004.
Women's basketball, and women's sports in general, received a major boost in 1972 with the introduction in the United States of Title IX, a law banning sexual discrimination at educational institutions receiving federal funds. As a result, more high schools fielded women's teams, and opportunities increased for female athletes to receive college scholarships.
Collegiate women's basketball continued to change when control switched from the Association for Intercollegiate Athletics for Women to the National Collegiate Athletic Association (NCAA) in the early 1980s. The NCAA ran the sport more efficiently and increased its visibility, resulting in the attendance figures for women's collegiate basketball games skyrocketing to more than five times their 1982 levels by 2005.
### **_T HE RISE OF THE WNBA_**
Unlike their male counterparts, female hoopsters had little opportunity to play professionally in the United States for some time. An attempt to change this situation came in 1978 with the formation of the Women's Professional Basketball League, but financial losses caused the eight-team league to disband after three seasons. Ann Meyers, a former star at the University of California at Los Angeles, signed a one-year contract with the Indiana Pacers of the NBA in 1979, but she did not make the team. In 1985 Lynette Woodard became the first female member of the Harlem Globetrotters, and Olympian Nancy Lieberman became the first woman to play in a men's professional basketball league when she joined the now-defunct United States Basketball League in 1986. Most American women who wanted to play professionally, however, resorted to traveling overseas to join flourishing leagues in Europe, South America, Asia, and Australia. Some chose to stay in the United States to work in such sports-related fields as coaching and broadcasting.
**_Lynette Woodard, the first female member of the Harlem Globetrotters, joins her teammates on the court_. Focus On Sport/Getty Images**
**_The Los Angeles Sparks, one of 14 teams in the WNBA, in 2010_. Johnny Vy/NBAE via Getty Images**
With the popularity of women's basketball at an all-time high in the United States following the 1996 Atlanta Olympics, new efforts were made to establish professional leagues for women. The American Basketball League (ABL), organized with start-up money from private businesspeople, debuted in October 1996 with eight teams. Backed by the powerful NBA, the Women's National Basketball Association (WNBA) tipped off in June 1997. Eight NBA cities were chosen to host WNBA franchises. Unlike the ABL, whose teams played during the traditional basketball season, WNBA teams played during the summer, with postseason semifinal and championship games held at the end of August. This enabled the women to use the same arenas as the men and helped the teams receive television exposure by not having to compete against collegiate and professional men's basketball. Cable stations chose to show many weekday games, and the league received network coverage on the weekends. The success of the WNBA allowed it to draw better talent and it forced the ABL to fold in 1998. By 2006 the WNBA had grown to 14 teams.
### **_I NTERNATIONAL COMPETITION_**
Despite its American origins, basketball has become a sport without boundaries. The Fédération Internationale de Basketball, the world governing body of basketball, was established in 1932, and the sport made its official Olympic debut in 1936. World championships were established in 1950 for men and in 1953 for women. Professional leagues developed in Europe, Asia, and Australia, and in the second half of the 20th century, many American players, unable to land a spot on an NBA team, traveled overseas to play professionally. The United States dominated international play for decades, but the quality of play outside the United States improved dramatically in the 1970s, and international competition was no longer one-sided. In the 1980s U.S. colleges began to actively recruit players from overseas, and during that time players from Europe—such as Detlef Schrempf of Germany and Drazen Petrovic of Yugoslavia—became key contributors on NBA teams. Today, players from Africa, Asia, Europe, and Australia are on NBA rosters. In 2002 Yao Ming of China became the first non-American player to be selected as the top pick in the NBA draft.
**_D'or Fischer_ (left), _an American playing for Spain's Real Madrid, attempts to defend against Theodoros Papaloukas of Greece's Olympiakos_. Pierre-Philippe Marcou/AFP/Getty Images**
**_Yao Ming of the Houston Rockets slam-dunks in a 2003 game against the Los Angeles Lakers_. Brian Bahr/Getty Images**
## **CHAPTER 4
NOTABLE PLAYERS**
While most of the original rules of basketball posted by James Naismith a century ago have endured, style and performance give the modern game a wholly different look. Over the course of the game's development, superstars of each generation perfected moves that earlier players could never have imagined—Wilt Chamberlain's finger roll, Kareem Abdul-Jabbar's sky hook, Julius Erving's slam dunk, Magic Johnson's no-look pass, Michael Jordan's incredible hang time. The careers of these and other influential players are discussed on the following pages.
### **_E ARLY NBA PLAYERS_**
#### **_G EORGE MIKAN_**
In a 1950 Associated Press poll, George Mikan was selected as the greatest basketball player of the first half of the 20th century. He was the first of the outstanding big men in the post-World War II professional game. Born in Joliet, Ill., on June 18, 1924, he played center for the Minneapolis Lakers from 1947 through 1954 and for a brief stint in the 1955–56 season, scoring 11,764 points in 520 regular-season games for an average of 22.6 points a game. In 91 playoff games, he scored 2,141 points for a 23.5 average. In an era when the professional game was known for its rough play, the lanky and nearsighted Mikan, wearing thick protective goggles, hardly looked the part of a basketball star. However, despite numerous broken bones and countless cuts and bruises, Mikan thrived in the sport. With Mikan at center, the Lakers won six championships from 1947–48 through 1953–54 (1950–51 season excepted). After retiring as a player, he coached the Lakers for part of the 1957–58 season and later served as commissioner of the ABA. Mikan was inducted into the Basketball Hall of Fame in 1959. He died on June 1, 2005, in Scottsdale, Ariz.
**_George Mikan of the Minneapolis Lakers, wearing his signature goggles, shows off his hook shot_. NBA Photos/NBAE via Getty Images**
#### **_B OB COUSY_**
Known as "Houdini of the Hardwood," Bob Cousy dazzled fans with his dribbling skill and behind-the-back passes. One of the game's great ball-handling guards, he was adept both at scoring and at playmaking. Born in New York, N.Y., on Aug. 9, 1928, he starred at the College of the Holy Cross before breaking into the NBA. During his career with the Boston Celtics from 1950 to 1963, he led the NBA in assists for eight consecutive years, his one-game record of 28 assists (1959) standing until 1978. Teamed with talented players such as Bill Russell, Bill Sharman, and K.C. Jones, Cousy adopted the competitive spirit of his coach Red Auerbach and directed the Celtics' play in six championship seasons (1957, 1959–63). After he left the Celtics, Cousy coached the Boston College Eagles from 1963 to 1969 and the NBA's Cincinnati/Kansas City Royals from 1969 to 1974. From 1975 to 1979 he served as commissioner of the American Soccer League and later became a marketing consultant and part-time television commentator for the Celtics. He was elected to the Basketball Hall of Fame in 1970.
#### **_B ILL RUSSELL_**
Standing 6 feet 10 inches (2.08 meters) tall, Bill Russell set standards by which other exceptionally tall players were judged. On five occasions he was voted the Most Valuable Player in the NBA, and he was selected by the Associated Press as the outstanding professional basketball player of the 1960s. Today many regard him as the best defensive center ever to play the game. Born in Monroe, La., on Feb. 12, 1934, Russell played on the gold medal-winning U.S. Olympic team in 1956 and for the Boston Celtics from 1956 through 1969. With Russell turning shot-blocking into an art form, Boston dominated the NBA for more than a decade. He helped lead the Celtics to 9 championships (1957, 1959–66) in 10 seasons. The team won two more championships (1968–69) with Russell as player and coach. Russell was the first African American to coach a major professional U.S. sports team. He later coached the Seattle SuperSonics from 1973 to 1977 and the Sacramento Kings from 1987 to 1988 and served the Kings briefly as vice-president of operations from 1988 to 1989. Russell was enshrined in the Basketball Hall of Fame in 1975.
#### **_W ILT CHAMBERLAIN_**
The press nicknamed him Wilt the Stilt, but he preferred to be called the Big Dipper. Playing center, Wilt Chamberlain was the first outstanding 7-foot (2.13-meter) player in basketball and is still considered by some the greatest offensive player in the history of the game.
Wilton Norman Chamberlain was born in Philadelphia on Aug. 21, 1936. He played two years at the University of Kansas and one season with the Harlem Globetrotters before joining the Philadelphia (later, San Francisco) Warriors of the NBA in 1959. After only one season with the Warriors, he was named the league's Most Valuable Player. During the 1961–62 season he claimed three NBA records when he scored 4,029 points in 80 regular-season games, reached an average of 50.4 points a game, and scored 100 points in a single game. Among Chamberlain's signature moves were the finger roll and fadeaway jump shot.
In the middle of the 1964–65 season Chamberlain was traded to the Philadelphia 76ers. In 1966 he became the second NBA player to attain a career total of 20,000 points, and the next season he led his team to the NBA title and reached a record 68.3 shooting average. Before the 1968–69 season, Chamberlain was traded to the Los Angeles Lakers. In 1968 he reached a record 25,000 career points and became the first center to lead the league in assists. Chamberlain's powerful presence drove the Lakers to the 1972 NBA title and helped them win a record 33 consecutive victories. When he retired in 1973, he had a record total of 23,924 rebounds and 31,419 points. He was named to the Basketball Hall of Fame in 1978. Chamberlain died on Oct. 12, 1999, in Los Angeles.
**_Wilt Chamberlain_ (center) _of the Los Angeles Lakers attempts a shot in a game against the Seattle SuperSonics_. Wen Roberts/AFP/Getty Images**
#### **_O SCAR ROBERTSON_**
Known as the Big O, Oscar Robertson was long considered the best all-around player in NBA history. Born on Nov. 24, 1938, he grew up in Indianapolis, Ind., where he led Crispus Attucks High School to two state championships. In 1956 he received an athletic scholarship to the University of Cincinnati and became the first African American to play basketball there. In three seasons of collegiate basketball, he averaged 33.8 points per game and helped the Cincinnati Bearcats twice reach the final four of the NCAA tournament. In 1960 he won a gold medal in Rome as a member of the U.S. Olympic team. That same year he began his professional career as a guard with the Cincinnati Royals. A superior ball handler, he led the league in assists six times. He was named the NBA Most Valuable Player for the 1963–64 season, in which he averaged 31.4 points, 9.9 rebounds, and 11 assists per game. He was traded to the Milwaukee Bucks in 1970 and, with Kareem Abdul-Jabbar, won four Midwest division titles for the Bucks, as well as the NBA championship in 1971. Robertson retired from the NBA in 1974 with 26,710 career points (25.7 per game), 7,804 rebounds (7.5 average), and 9,887 assists (an NBA record at the time). He was elected to the Basketball Hall of Fame in 1979.
### **_R ECENT NBA LEGENDS_**
#### **_K AREEM ABDUL-JABBAR_**
His extraordinary height of 7 feet 2 inches (2.18 meters) combined with extraordinary skills enabled Kareem Abdul-Jabbar to dominate the game of basketball throughout the 1970s and early '80s. He was known for his excellent shooting touch and wide range of graceful post moves, including his sweeping, nearly indefensible sky hook.
**_Kareem Abdul-Jabbar of the Los Angeles Lakers shoots a graceful sky hook in a game against the Dallas Mavericks_. Mike Powell/Getty Images**
Abdul-Jabbar was born Ferdinand Lewis Alcindor, Jr., on April 16, 1947, in New York City. He adopted the better-known Arabic name in 1971, six years after he joined the Black Muslim movement. As a high school basketball player, Lew Alcindor led his team to a 95–6 record and scored 2,067 points, a New York City record for three years. He received more than 100 offers of college scholarships and chose the University of California at Los Angeles (UCLA), which he led to three consecutive NCAA championships.
Upon graduation from UCLA, he was drafted by the Milwaukee Bucks. He continued to play spectacularly at the center position, earning rookie of the year honors in 1970. In 1975 Abdul-Jabbar was traded to the Los Angeles Lakers. He helped the Lakers win the NBA championship in 1980, 1982, 1985, 1987, and 1988. In 1989, the year of his retirement, he became the first player to score more than 38,000 points.
Away from the basketball court, Abdul-Jabbar has earned recognition as a writer. His autobiography, _Giant Steps_ , was published in 1983. In addition to his own experiences, he has written on the African American experience, including _Black Profiles in Courage: A Legacy of African American Achievement_ (1996; with Alan Steinberg).
#### **_J ULIUS ERVING_**
Basketball great Julius Erving, better known as Dr. J., once said of his amazing airborne moves: "It's easy once you learn how to fly." His flights quickly made him one of basketball's all-time top scorers.
Julius Winfield Erving was born in Hempstead, N.Y., on Feb. 22, 1950. In his first year at the University of Massachusetts, he broke records for scoring and rebounding. He left the university in 1971 after his junior year to sign a contract as a forward with the Virginia Squires of the ABA.
In 1973 Erving moved to the New York Nets and, with an average of 27.4 points and 10.7 rebounds per game, led them to the ABA title. Sports enthusiasts believe that when the NBA absorbed the ABA in 1976, it was primarily to get Erving, who joined the Philadelphia 76ers. His first years in Philadelphia were frustrating because the 76er style of play and tendinitis in his knees limited him. But by the 1979–80 season conditions had changed, and Erving was averaging 26.9 points a game. Erving led the 76ers to the NBA finals four times, including their 1983 championship win.
Erving changed the idea of the way the game of basketball should be played. His airborne feats popularized the slam dunk. His ability to block shots and rebound, as well as pass and handle the ball skillfully, encouraged faster and flashier playing.
Erving's honors included being named ABA Most Valuable Player in 1974, 1975, and 1976, NBA Most Valuable Player in 1981, and numerous elections to ABA and NBA all-star teams. Before his retirement after the 1986–87 season, Erving became only the third player to score 30,000 points in a professional career.
#### **_L ARRY BIRD_**
The superstar of the Boston Celtics during the 1980s, Larry Bird had a talent for breathing life into tired basketball organizations. In addition to transforming the Celtics into a championship team, he also helped rejuvenate the sport's popularity worldwide. Bird is considered one of the greatest pure shooters of all time.
**_Larry Bird of the Boston Celtics attempts a shot during a game_. Getty Images**
Larry Joe Bird was born on Dec. 7, 1956, in a small rural community near French Lick, Ind. The self-described Hick from French Lick, Bird almost did not play college basketball. Although he was recruited by coach Bobby Knight of Indiana University in 1975, the homesick teenager, overwhelmed by the demands of university life, hitchhiked home after only 3½ weeks at school. A year later, he was recruited to play basketball for Indiana State University in Terre Haute, where he became an immediate phenomenon. In his senior year, Bird led the team into the 1978–79 NCAA championship game with a 33–0 season record. The contest between Bird's Indiana State and Magic Johnson's Michigan State foreshadowed the professional rivalry that would electrify the NBA in the 1980s. Indiana State lost the championship game to Michigan State, but Bird defeated Johnson in the vote for college player of the year.
In 1979 Bird signed a contract with the ailing Boston Celtics, the team that he would stay with for his entire professional career. Leading the team in scoring, rebounding, steals, and minutes played, Bird sparked one of the greatest single-season turnarounds in NBA history. The Celtics improved from a record of 29–53 in 1978–79 to a division-leading record of 61–21 in 1979–80. Although Johnson's Los Angeles Lakers won the 1980 championship, Bird was voted NBA rookie of the year. He went on to lead the Celtics to three NBA championships (1981, 1984, and 1986). Bird also won three consecutive league Most Valuable Player (MVP) awards (1984–86), the first noncenter to do so.
Bird retired in 1993 with career averages of 24.3 points per game and 10 rebounds per game. In 1998 he was elected to the Basketball Hall of Fame. He became the head coach of the Indiana Pacers in 1997 and was named coach of the year after his first season. Bird resigned in 2000 and became the Pacers' president of basketball operations in 2003.
#### **_M AGIC JOHNSON_**
Magic Johnson led the Los Angeles Lakers to five NBA championships in the 1980s. Standing 6 feet 9 inches (2.06 meters) tall, he was exceptionally tall for a point guard and was able to use his size to rebound and score inside. However, he was best known for his creative passing and expert floor leadership.
**_Magic Johnson of the Los Angeles Lakers dribbles a ball down the court during a 1987 game_. Rick Stewart/Getty Images**
Earvin Johnson, Jr., was born in Lansing, Mich., on Aug. 14, 1959, the sixth of ten children. He got his start playing basketball on the playgrounds of Lansing. As a senior he led the Everett High School team to the Michigan Class A championship. After a particularly amazing display of basketball skill, during which he scored 36 points, grabbed 18 rebounds, and had 16 assists, a Lansing sportswriter christened him Magic. The hometown Michigan State University wooed the All-Stater in 1977. During his sophomore year at Michigan State, he led the Spartans to the NCAA championship—handing Larry Bird and Indiana State its only defeat of that season.
Johnson was drafted by the Los Angeles Lakers in 1979. With Magic leading the team, the Lakers won the NBA championship in 1980, 1982, 1985, 1987, and 1988. He was chosen play-off MVP three times (1980, 1982, and 1987) and was the first rookie to be so named. He was the league's MVP three times (1987, 1989, and 1990), and he helped lead the U.S. team to a basketball gold medal at the 1992 Olympics in Barcelona, Spain. His 9,921 career assists set an NBA record (broken by John Stockton in 1995).
With his dazzling smile, style, and blind passes, Johnson was among those credited with professional basketball's surge in popularity in the 1980s. The sports world was stunned on Nov. 7, 1991, when Johnson announced his immediate retirement from professional basketball due to HIV infection. Later he served briefly as head coach of the Lakers (1994), and he returned as a player for a portion of the 1995–96 season. Johnson went on to become a successful entrepreneur and a noted HIV/AIDS activist. He was elected to the Basketball Hall of Fame in 2002.
#### **_M ICHAEL JORDAN_**
Both literally and figuratively, Michael Jordan soared higher than any NBA guard before him. His extraordinary leaping ability and acrobatic maneuvers inspired his nickname, "Air Jordan." He was the NBA's top scorer for a record-breaking 10 seasons. Also outstanding at defense, Jordan was one of the greatest all-around players in the history of the game.
**_Michael Jordan of the Chicago Bulls demonstrates one of his storied leaping maneuvers as he seeks to score against the Charlotte Hornets in a 1995 game_. Jonathan Daniel/Getty Images**
Michael Jeffrey Jordan was born on Feb. 17, 1963, in Brooklyn, N.Y., but he grew up in Wilmington, N.C. Although he was cut from the varsity basketball team in his sophomore year of high school, he later became one of the team's star players. Jordan earned a scholarship to the University of North Carolina at Chapel Hill, where he helped lead the school's basketball team to the NCAA championship during his freshman year. In both his sophomore and junior years, he was named the NCAA college player of the year. After his junior year he left to join the NBA and was drafted by the Chicago Bulls.
In his first season with the Bulls, the 6-foot-6-inch (1.98-meter) Jordan averaged 28.2 points per game and was named rookie of the year. Jordan won seven consecutive scoring titles from the 1986–87 season through the 1992–93 season. He led the Bulls to three consecutive NBA championships, in 1991, 1992, and 1993. He also led the U.S. basketball team to a gold medal in both the 1984 and 1992 Olympic Games.
Saying that he did not have "anything else to prove," Jordan retired from professional basketball in October 1993. In 1994 he signed to play for a minor league baseball team, but after one season he decided to return to basketball. He rejoined the Bulls late in their 1994–95 season. After leading the team to three more championships in 1996, 1997, and 1998, Jordan retired for the second time in January 1999.
In 2000 Jordan bought a share of the Washington Wizards, but he soon wanted to return to the court. He gave up his ownership position with the Wizards in 2001 in order to play on the team. In the 2002–03 season he became the first player in NBA history age 40 years or older to score more than 40 points in a game. Jordan's final retirement from basketball came in May 2003.
Jordan was named the NBA's Most Valuable Player in 1988, 1991, 1992, 1996, and 1998. At the time of his retirement in 2003, Jordan ranked third in career scoring, with a total of 32,292 points, behind Kareem Abdul-Jabbar and Karl Malone. Jordan's scoring average of 30.12 points per game was the highest in league history. Jordan became part owner of the NBA's Charlotte Bobcats in 2006 and took over control of the team as its majority owner in 2010; he was the first former NBA player to become a majority owner of one of the league's teams. Jordan was inducted into the Basketball Hall of Fame in 2009.
### **_C URRENT NBA PLAYERS_**
#### **_S HAQUILLE O'NEAL_**
With his intimidating size and outstanding skills, Shaquille O'Neal overwhelmed the competition in the NBA. A 7-foot-1-inch (2.16-meter) center who weighed at least 315 pounds (142.9 kilograms), "Shaq" was nevertheless an agile athlete. Along with teammate Kobe Bryant and coach Phil Jackson, O'Neal led the Los Angeles Lakers to three consecutive NBA championships (2000–02).
Shaquille Rashaun O'Neal was born on March 6, 1972, in Newark, N.J. At age 13 O'Neal, already 6 feet 6 inches (1.98 meters) tall, met Louisiana State University (LSU) head basketball coach Dale Brown, who made an early pitch for O'Neal to join his team. O'Neal ultimately chose LSU, where he became a hot prospect for the pros. As the first pick of the 1992 NBA draft, he was signed by the Orlando Magic to a seven-year, 40-million-dollar contract, becoming the highest-paid rookie in the NBA. O'Neal went on to be chosen NBA rookie of the year.
**_Shaquille O'Neal of the Cleveland Cavaliers controls the ball while Brad Miller of the Chicago Bulls attempts to defend against him_. Gregory Shamus/Getty Images**
After four years with the Magic, O'Neal was lured to join the Lakers in 1996. Under coach Phil Jackson's guidance, O'Neal developed into more of a team player, paying greater attention to his defense and rebounding. In 2000 the Lakers won the NBA championship, and O'Neal was named MVP of the regular season, the All-Star Game, and the NBA finals. The Lakers captured the NBA title again in 2001 and 2002. O'Neal was named the MVP of the finals for both years. The Lakers returned to the NBA finals in 2004 but were defeated. Soon thereafter the team underwent major changes, chief among them a trade that sent O'Neal to the Miami Heat. O'Neal helped lead that team to its first NBA championship, in 2006.
In February 2008 O'Neal was traded to the Phoenix Suns. His playing style did not mix well with the Suns' up-tempo game, however, and—despite having had a very solid 2008–09 season—he was traded to the Cleveland Cavaliers in June 2009. Following the completion of the 2009–10 season, O'Neal signed a two-year contract to play with the Boston Celtics.
#### **_K OBE BRYANT_**
One of the most prolific scorers in the history of the NBA, Kobe Bryant helped lead the Los Angeles Lakers to five NBA championships (2000–02, 2009–10).
Kobe Bean Bryant was born on Aug. 23, 1978, in Philadelphia, Pa. Bryant—whose father, Joe Bryant, spent eight seasons in the NBA—played basketball at Lower Merion High School in Ardmore, Pa., where he earned national player of the year honors before opting to forgo college and enter the 1996 NBA draft. Chosen by the Charlotte Hornets with the 13th overall pick in the draft, Bryant was traded to the Lakers and soon proved his merit with the team. In just his second season, he was selected for the NBA All-Star Game, becoming the league's youngest-ever all-star.
Under the leadership of Phil Jackson, who became coach of the Lakers in 1999, Bryant, a shooting guard, and his all-star teammate Shaquille O'Neal, a center, meshed into a remarkably effective combination, and by the time Bryant was 23, the Lakers had won three consecutive NBA championships. The Lakers returned to the NBA finals in 2004 but were upset by the Detroit Pistons. O'Neal subsequently was traded to the Miami Heat, and Bryant emerged as the team's sole leader. He led the league in scoring during the 2005–06 and 2006–07 seasons. In 2008 he was named the league's MVP for the first time in his career; he also starred on the U.S. men's basketball team that captured the gold medal at the Olympic Games in Beijing. In 2009 he won his fourth NBA title as the Lakers decisively defeated the Orlando Magic 4 games to 1 in the finals. Bryant averaged 32.4 points per game in the series and was named the finals MVP. In 2009–10 he was once more named NBA finals MVP after the Lakers defeated the Boston Celtics in a seven-game series.
**_Kobe Bryant of the Los Angeles Lakers celebrates a 2000 Lakers victory against the Seattle SuperSonics_. Dan Levine/AFP/Getty Images**
#### **_L EBRON JAMES_**
After entering the NBA directly from high school in 2003, LeBron James quickly established himself as one of the league's superstars. An extraordinarily versatile small forward who was capable of playing multiple positions, James was selected as the NBA's MVP in 2009 and 2010, becoming only the 10th player in NBA history to have earned that honor in consecutive seasons.
LeBron Raymone James was born on Dec. 30, 1984, in Akron, Ohio. In high school, he was named Ohio's Mr. Basketball three times. In his senior season, he was the consensus national high school player of the year before being chosen by the Cleveland Cavaliers as the first overall pick of the 2003 NBA draft.
**_LeBron James_ (right) _of the Cleveland Cavaliers and NBA MVP in 2009 receives his trophy from NBA commissioner David Stern_. Gregory Shamus/Getty Images**
James made an immediate impact on the Cavaliers, leading the team in scoring, steals, and minutes played during the 2003–04 season en route to claiming the NBA's rookie of the year award. He was named to the NBA All-Star team for the first time in 2005, and in 2007 he guided Cleveland to the franchise's first berth in the NBA finals. Although James posted a spectacular average of 25 points, 8 rebounds, and 8 assists per game throughout the play-offs that year, the Cavaliers were swept in the finals by the San Antonio Spurs.
During the 2007–08 season, James led the NBA in scoring with an average of 30 points per game and became the youngest player in league history to tally 10,000 career points. The following season he piloted the Cavaliers to a team-record 66 regular-season wins. During both the 2008–09 and 2009–10 seasons, James continued to display his prolific scoring ability, averaging 28.4 and 29.7 points per game, respectively. In addition to his achievements in the NBA, James was a member of the U.S. men's Olympic basketball teams that won the bronze medal at the 2004 Games and the gold medal at the 2008 Games.
At the end of the 2009–10 season James became arguably the most sought-after free agent in NBA history when his contract with the Cavaliers expired. In an unprecedented hour-long television special on the ESPN sports cable network, James announced his decision to sign with the Miami Heat.
**_Nancy Lieberman of the U.S. women's national team_. Tim DeFrisco/Getty Images**
### **_WNBA P LAYERS_**
#### **_N ANCY LIEBERMAN_**
A pioneer in women's basketball, Nancy Lieberman recorded several unprecedented accomplishments in a playing career that spanned three decades.
Born July 1, 1958, in Brooklyn, N.Y., Lieberman garnered attention in the male-dominated New York basketball scene with her natural ability and court savvy. She entered Old Dominion University in Virginia in 1976 and led the school to consecutive Association for Intercollegiate Athletics for Women championships in 1978–79 and 1979–80. An extraordinarily quick point guard, she became known for her precision passing and accurate shooting touch, which enabled her to average 18.1 points per game over her four-year collegiate career. At the international level, she helped lead the United States to a gold medal in the 1975 Pan American Games. She was also a member of the silver-medal-winning 1976 U.S. Olympic team; she made the 1980 team as well, but the squad did not compete because of an American boycott of the Games.
In 1980 Lieberman was the number-one draft pick of the Dallas Diamonds of the Women's Basketball League (WBL), a fledgling women's professional league that folded in 1982. In 1984 Lieberman was again the first draft pick of a newly created professional circuit, the Women's American Basketball Association (WABA). Because fan interest for a women's professional league still was not strong enough to generate financial success, however, the WABA was also short-lived.
Reluctant to leave the United States for Europe, where she had several offers to play professionally, Lieberman continued to look for new opportunities at home. In 1986 she became the first woman to play in a men's professional league, the United States Basketball League, with the Springfield Fame. In 1988 Lieberman was chosen by the Washington Generals to play against the Harlem Globetrotters, making her the first woman to participate in a Harlem Globetrotters world tour. Approaching the age of 40 but still a talented player, she joined the Phoenix Mercury of the newly formed WNBA in 1996.
Outside of her basketball career, Lieberman established her own sports marketing company and became an accomplished broadcaster and a well-known public speaker. She was inducted into the Basketball Hall of Fame in 1996.
#### **_C YNTHIA COOPER_**
The first MVP of the WNBA was Cynthia Cooper of the Houston Comets. In the WNBA's inaugural 1996–97 season, Cooper led the league in scoring while guiding the Comets to the championship. She was named MVP of both the regular season and the playoffs that year.
Born in Chicago, Ill., on April 14, 1963, Cooper was raised in the Watts section of Los Angeles. She began playing organized basketball at age 16 and quickly took to the sport. She earned a scholarship to the University of Southern California, where she played in the shadow of Cheryl Miller while helping the team to national championships in 1983 and 1984. After college Cooper played professionally in Europe, primarily for a team in Parma, Italy, where she blossomed into a potent scorer and a tenacious defender. She was a member of the 1988 U.S. national team that won the gold medal at the Olympic Games in Seoul, South Korea.
**_Cynthia Cooper of the Houston Comets shoots a layup in a 2000 game against the New York Liberty_. Ronald Martinez/Getty Images**
By the end of the WNBA's inaugural season, Cooper had established herself as the league's first great player. Along with star teammates Sheryl Swoopes and Tina Thompson, Cooper led the Comets to titles in 1998, 1999, and 2000, each time being recognized as the MVP of the play-offs. She was named the league MVP for the second time in 1998. Cooper retired in 2000 and became the head coach of the WNBA's Phoenix Mercury the following year. She returned to playing basketball in 2003, again with the Comets, and permanently retired from the game in 2004 with WNBA career per-game averages of 21 points, 4.9 assists, 3.3 rebounds, and 1.56 steals. Cooper was named the women's basketball head coach at Prairie View (Texas) A&M University in 2005. She was enshrined in the Basketball Hall of Fame in 2010.
## **C ONCLUSION**
By the early 21st century, basketball had developed into a truly global game. In the United States, it was firmly established at the forefront of the sporting scene, alongside such traditional leaders as baseball and football, and in numerous other countries around the world, the game continued to grow steadily both in popularity and importance. Interest in the game has deepened over the years as a result of increased exposure via network and cable television coverage. More importantly, however, it is the nature of the game of itself—its fast and exciting pace, creative and intricate styles of play, frequently dramatic and high-scoring contests, and the skills and athleticism demanded of players at the highest levels—that ensures that the sport of basketball will always have a broad appeal.
## **G LOSSARY**
**assist** The action (as a throw or pass) of a player who enables a teammate to make a basket.
**barnstorm** To travel from place to place staging games.
**berth** Placement in an athletic tournament or contest.
**calisthenics** Free body exercises performed with varying degrees of intensity and rhythm that employ such motions as bending, stretching, and jumping, as well as such specialized movements as push-ups, sit-ups, and chin-ups.
**championship** A contest held to determine the winning team of a season; the final game of the play-offs.
**center** Player on a basketball team who usually plays near the basket and is typically the tallest person on the team.
**draft** A system whereby exclusive rights to selected new players are apportioned among professional teams.
**dribble** To bounce a ball by hand.
**forward** A player in basketball who plays at the front of his team's formation near the basket at which his team is attempting to score.
**free throw** An unhindered shot in basketball made from behind a set line and awarded because of a foul by an opponent.
**guard** A player stationed in the back court in basketball.
**hook shot** A shot in basketball made usually while standing sideways to the basket by swinging the ball up in an arc with the far hand.
**jump shot** A shot in basketball made by jumping into the air and releasing the ball with one or both hands at the peak of the jump.
**layup** A shot in basketball made from near the basket usually by playing the ball off the backboard.
**pick-and-roll** A basketball play in which a player sets a screen between a ball-handling teammate and a defender from the opposing team and then cuts toward the basket for a pass.
**play-off games** A series of contests played after the end of the regular season to determine the teams that will compete in the championship game.
**point guard** A guard in basketball who is chiefly responsible for running the offense.
**power forward** A basketball forward whose size and strength are used primarily in controlling play near the basket.
**press** An aggressive pressuring defense in basketball employed against the opposing team often over the entire court area.
**prolific** Marked by abundant inventiveness or productivity.
**shooting guard** A guard in basketball whose chief role is as an outside shooter.
**slam dunk** A shot in basketball made by jumping high into the air and throwing the ball down through the basket.
**small forward** A basketball forward who is usually smaller than a power forward and whose play is characterized by quickness and scoring ability.
## **F OR MORE INFORMATION**
Canada Basketball (CB)
1 Westside Drive, Suite 11
Toronto, ON M9C 1B2
Canada
(416) 614-8037
Web site: http://www.basketball.ca
CB is the national governing body of amateur basketball in Canada. The site provides information on teams and players throughout the country as well as the Long-Term Athlete Development Model.
Canadian Colleges Athletic Association (CCAA)
St. Lawrence College
2 Belmont Street, Windmill Point
Cornwall, ON K6H 4Z1
Canada
(613) 937-1508
Web site: http://www.ccaa.ca
The CCAA provides information on college athletics throughout Canada as well as rankings, scores, and regulations.
The National Basketball Association (NBA)
645 Fifth Avenue
New York, NY 10022
(212) 407-8000
Web site: http://www.nba.com
The NBA site provides information on everything from player statistics to game schedules to the history and rules of the NBA. Information on the NBDL and the WNBA is also available.
The National Collegiate Athletic Association (NCAA)
700 W. Washington Street
P.O. Box 6222
Indianapolis, IN 46206
(317) 917-6222
Web site: http://www.ncaa.org
The NCAA site provides information on college athletics in the United States and addresses issues facing student athletes. It also offers information on NCAA-sponsored scholarships for students and post-graduates.
USA Basketball
5465 Mark Dabling Boulevard
Colorado Springs, CO 80918
(719) 590-4800
Web site: http://www.usabasketball.com
USA Basketball is the national governing body of basketball in the United States. The site provides information on men's and women's teams throughout the country as well as the various international competitions in which they compete.
The Women's National Basketball Association (WNBA)
645 Fifth Avenue
New York, NY 10022
(212) 407-8000
Web site: http://www.wnba.com
The WNBA site provides information on player stats, game schedules, and the official history and rules of the WNBA. Information on the WNBA's outreach program and a glossary of relevant terms are also available.
### _W EB SITES_
Due to the changing nature of Internet links, Rosen Educational Services has developed an online list of Web sites related to the subject of this book. This site is updated regularly. Please use this link to access the list:
http://www.rosenlinks.com/spor/bskt
## **B IBLIOGRAPHY**
Anderson, Dave. _The Story of Basketball_ , rev. ed. (Morrow, 1997).
Bird, Larry, and MacMullan, Jackie. _Bird Watching: On Playing and Coaching the Game I Love_ (Warner Books, 2000).
Fleder, Rob. _The Basketball Book_ (Time Books, 2007).
Gifford, Clive. _Basketball_ (PowerKids Press, 2009).
Lannin, Joanne. _A History of Basketball for Girls and Women_ (Lerner Sports, 2000).
Mullin, Chris and Coleman, Brian. _Basketball_ (DK, 2000).
Ramen, Fred. _Basketball: Rules, Tips, Strategy, and Safety_ (Rosen Central, 2007).
Steen, Sandra, and Steen, Susan. _Take It to the Hoop: 100 Years of Women's Basketball_ (Twenty-First Century Books, 2003).
Thomas, Keltie. _How Basketball Works_ (Maple Tree Press, 2005).
Thomas, Ron, and Herran, Joe. _Getting Into Basketball_ (Chelsea House Publishers, 2006).
## **I NDEX**
**A**
Abdul-Jabbar, Kareem, , –59,
All-Star Game, , ,
American Basketball Association (ABA), , , ,
American Basketball League (ABL),
Atlanta Hawks,
Auerbach, Red,
**B**
backboard, about the,
Baer, Clara,
ball, about the,
barnstorming teams, –29
basket, about the, –14
basketball
court/equipment for, –14
history of, –12, –37
integration of, –31
rules of, –20
styles of play, –26
types of shots, –22
violations and fouls, –24
Basketball Association of America,
Basketball Hall of Fame, , , , , , , , , ,
Berenson, Senda, –39
Bird, Larry, , –63,
Boston Celtics, , , , , –63, ,
Brown, Dale,
Bryant, Kobe, , –75
**C**
Chamberlain, Wilt, , –55
Charlotte Bobcats, , –70
Chicago Bulls, , ,
Cincinnati Royals, ,
Cleveland Cavaliers, , , –77
Clifton, Nathaniel "Sweetwater,"
college basketball
court for, ,
fouls and,
international players and,
recruiting for NBA and,
rules for, ,
women's, –40,
Cooper, Chuck,
Cooper, Cynthia, –83
court size and markings, –13
Cousy, Bob, –51
**D**
Dallas Mavericks,
defensive play, –25,
Denver Nuggets,
Detroit Pistons, ,
Douglas, Robert,
draft, basketball,
dribbling, ,
**E**
Eastern Conference, –34
Atlantic Division,
Central Division, –34
Southeast Division,
Erving, Julius, , –60
**F**
fast-break style,
Fédération Internationale de Basketball,
fouls, , –24,
personal, ,
technical,
free throw line, ,
free throws, , ,
**G**
goaltending,
Golden State Warriors,
**H**
Harlem Globetrotters, , , , ,
high school basketball
court for, ,
fouls and,
recruiting for NBA and,
rules for, ,
women's, ,
hook shot,
Houston Comets, ,
Houston Rockets,
**I**
Indiana Pacers, , ,
international basketball, , , , , –47
International Women's Sports Federation,
**J**
Jackson, Phil, , ,
James, LeBron, –77
Johnson, Magic, –32, , , –66
Jones, K.C.,
Jordan, Michael, , , –70
jump shot,
**K**
Knight, Bobby,
**L**
Lapchick, Joe,
layup,
Lieberman, Nancy, –43, –81
Lloyd, Earl,
Los Angeles Clippers,
Los Angeles Lakers, , , , , , , , , , –75
**M**
Malone, Karl,
man-to-man defense, ,
Memphis Grizzlies,
Meyers, Ann,
Miami Heat, , , ,
Mikan, George, , –50
Miller, Cheryl,
Milwaukee Bucks, , ,
Minneapolis Lakers, ,
Minnesota Timberwolves,
motion offense, –26
**N**
Naismith, James, –12, , ,
National Basketball Association (NBA)
court for,
early years of, –31
era of superstars in, –32
fouls and,
international players in,
notable players, –83
rules for, , ,
today, –37
WNBA and,
National Basketball Development League (NBDL),
National Basketball League, ,
National Collegiate Athletic Association (NCAA), , , , , ,
New Jersey Nets,
New Orleans Hornets,
New York Knicks,
New York Renaissance (Rens), –29
**O**
offensive play, –26
Oklahoma City Thunder,
Olympics basketball in the, , , , –66, , ,
women's basketball in, , , ,
O'Neal, Shaquille, –72, –75
one-hand push shot,
Original Celtics,
Orlando Magic, , –72,
out-of-bounds, awarding ball, ,
**P**
Philadelphia 76ers, , ,
Philadelphia Warriors,
Phoenix Mercury, ,
Phoenix Suns, ,
pick-and-roll, –26
play-offs,
Portland Trail Blazers,
press defense,
**R**
referees,
Robertson, Oscar, –56
Russell, Bill, –52
**S**
Sacramento Kings, ,
salaries, players',
San Antonio Spurs, ,
Saperstein, Abe,
Seattle SuperSonics,
Sharman, Bill,
shot clock,
slam dunk, ,
Stern, David,
Stockton, John,
Swoopes, Sheryl,
**T**
Thompson, Tina,
three-point line, ,
three-point shot, , ,
Title IX,
Toronto Raptors,
**U**
United States Basketball League, ,
Utah Jazz,
**V**
violations, –22
**W**
Washington Wizards, ,
Western Conference, –36
Northwest Division,
Pacific Division, –36
Southwest Division, –35
Women's American Basketball Association (WABA),
women's basketball, –44, , –83
development of, –39
rise of the WNBA, –44
rules for,
Women's Basketball League (WBL),
Women's National Basketball Association (WNBA), , –83
Women's Professional Basketball League, –42
Woodard, Lynette,
World Amateur Basketball championship,
**Z**
zone defense,
|
{
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}
| 144
|
The European Physical Journal - Applied Physics
Sputtered-deposited thin brass ...
August 2006 , pp. 93-105
Sputtered-deposited thin brass films in a modified glow discharge Grimm-type source
K. I. Grais (a1), M. A. Eid (a1), N. L. Tawfik (a1), M. S. Abd-El-Aal (a1) and A. A. Shaltout (a1)...
1 Spectroscopy Department, Physics Division, National Research Centre, Dokki, Giza, Egypt
DOI: https://doi.org/10.1051/epjap:2006071
Published online by Cambridge University Press: 19 July 2006
Modification of the non-assisted gas flow-line across the target surface in a Grimm-type glow discharge source is described. The new flow–line permits the gas to flow through a cylindrical annular space ending with a disc-space annular gap, facing the target surface. This configuration would cause directed jet assisted gas flow rays to impinge on infinite points across the cathode surface. Improvement has been achieved in the V-I characteristics where $\Delta V$ / $\Delta I$ increases from 1.8 to 3.5 V/mA. The sputtering as well as simultaneous deposition rates, have been increased by a factor of 16 and 17 respectively. These roll over with increasing sputtering time, their maximum values at a characteristic time, t oc of 21 min. The t oc value was constant for different operating parameters provided that the source geometry assembly is kept fixed. The presence of a glass substrate in the anode cavity has, apparently, no effect on the obtained data. Improvements have also been achieved in the crater profile, characterized by an approximately flat crater bottom with nearly vertical walls, and less re-deposited particles on the crater depth and edge. Fixing the distance Z of the substrate from target surface, along the cell axis, and varying the deposition time from 1 to 30 min, a sequence of changes in the deposited film were observed by X-ray diffraction and energy dispersion X-ray (EDX). These changes start with an amorphous structure, followed by the appearance of Cu and Zn crystallites and a probable deposition of Cu5Zn8 clusters. The profile of the number of sputtered particles at different Z values is characterized by a number of peaks and troughs. This behavior has been explained by the occurrence of local cluster-dissociation and formation, by different collision processes. The improvements achieved by the application of the present jet assisted gas flow can be of value in the analytical application of this type of glow discharge.
khairiiggrais@hotmail.com
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URL: /core/journals/the-european-physical-journal-applied-physics
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,480
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Q: Jmeter OAuth 1.0 Request I have a Java code for OAuth 1.0 Authentication and each time i have to execute it before testing each request. I am currently working on JSR223 sampler.
But happen to see this OAuth 1.0 Plugin, can someone explain me how this OAuth 1.0 based Authentication in the screenshot works.
I have some parameter value to be passed in the Authorization header output example given below for each request.
For the first URL, I have two parameter
*
*Key
*Secret Key
Which give the Access token and secret access token for the Second URL and for the third it will add
*
*request Payload
*Email
*Status
without these the request will fail.
My output code looks like the below in Eclipse:-
Authorization :OAuth oauth_signature="Dynamic Value",scope="Parameter Value",oauth_version="Dynamic Value",oauth_nonce="Dynamic Value",oauth_signature_method=HMAC-SHA1,oauth_consumer_key="Parameter Value",oauth_token="Dynamic Value",oauth_timestamp="Dynamic Value"
Is there a way that i run the piece of Java code directly with all the Jar files in Jmeter [without doing any changes] is that possible?
OAuth 1.0 Plugin
A: I would suggest adding OAuth client libraries to JMeter's classpath (just drop them to /lib folder of your JMeter installation)
Also download the latest version of groovy-all.jar and put it to the same /lib folder. Restart JMeter to pick the jars up.
Then add JSR223 Sampler to your Test Plan, choose "groovy" in "Language" drop-down and put your Java code to "Script" area. Valid Java code in 99% of cases will be valid Groovy code so you can run it this way.
See How to Run Performance Tests on OAuth Secured Apps with JMeter guide for more detailed information on the domain.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,677
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\section{Introduction}
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth]{figures/overview.png}
\end{center}
\caption{(Top) Two pairs of positive samples are fed to the coupled-encoder ($f_f(.)$ and $f_p(.)$). Each color in the contrastive space presents a distinct identity. Black arrows demonstrate the attraction between two representations, and the red arrows increase the distance. Solid and dashed circles represent profile and frontal representations, respectively.
Due to the memory buffer, the number of instances in the contrastive space is more than the mini-batch size. (Bottom) Illustrating the distance distributions between positive (shown in blue) and negative (shown in orange) pairs. Our optimization improves the similarity between the different views of the same identity while increasing the distance between different identities.}
\label{fig:overview}
\vspace{-2mm}
\end{figure}
With the advancement of technology and increasing demand for security, biometrics are among the most essential and surfed applications of computer vision \cite{meden2021privacy}. Among biometric traits, the face has received particular attention since it is naturally exposed, offers better hygiene in the acquisition, and can be acquired in an unconstrained setting without direct participation of the user \cite{masi2018deep}. Face Recognition (FR) has been a major interest in computer vision for many years, and FR methods have advanced significantly over the years \cite{masi2018deep}.
Classical FR techniques are mainly based on extracting hand-crafted features, and the primary concern is extracting features with high intra-class compactness, and inter-class separability \cite{ahonen2006face}.
Modern approaches address this issue by incorporating learning models based on the Convolutional Neural Networks (CNNs) \cite{schroff2015facenet}.
CNNs have demonstrated extraordinary performance in FR; however, their performance drastically degrades for profile views \cite{zhao2018towards}.
There are four primary issues with profile face images as compared to frontal images: 1) self-occlusion, 2) background distraction, 3) shift in the distribution of data, and 4) inaccuracy in the alignment due to the lack of accurate landmarks \cite{cao2018pose,taherkhani2020pf}.
There are two mainstream approaches for profile-to-frontal FR. The first approach handles pose variation by extracting pose-invariant features \cite{liu2017sphereface,wang2017normface}.
The second approach estimates the frontal view of a given profile face and then utilizes it for the recognition task \cite{tran2017disentangled, ju2022complete}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{figures/SupConMemory.png}
\end{center}
\caption{With memory, for each sample in mini-batch B, we can have different contrastive pairs and calculate the loss with the samples which are not in the current mini-batch. Different colors refer to representation of different identity in the dataset}
\label{fig:long}
\label{fig:SupConMemory}
\vspace{-2mm}
\end{figure}
In the first approach, Softmax with a cross-entropy loss is mainly adopted to supervise a deep classifier. Although Softmax promotes the separability of representations, provided features are insufficiently discriminative in practical FR problems \cite{wang2017normface}. To address this issue, pioneering works of \cite{schroff2015facenet,wen2016discriminative,sun2014deep} employ sample-to-sample comparison as their loss functions to reduce the intra-class variations. However, most of the recent FR methods mainly focus on the sample-to-prototype comparison, and they improve the discriminative power of representations by applying several margin penalties on the Softmax loss function \cite{liu2017sphereface,wang2017normface}.
Despite the remarkable performance, a common issue of these approaches is that FR datasets contain a large number of identities, and only a few identities are presented in each mini-batch, which complicates finding an optimal decision boundary \cite{hou2019learning}. Increasing the mini-batch size may alleviate the problem. However, it does not guarantee performance improvement \cite{keskar2016large,you2017scaling}, and it may not be possible (due to the memory constraints). Moreover, shortcomings such as sensitivity to noisy labels \cite{zhang2018generalized}, likelihood of poor margin \cite{elsayed2018large},
and convergence difficulty on the networks with small embedding feature size \cite{li2019airface} have led to diminishing generalization.
In the second approach, FR process is separated into two tasks: identity-preserving face generation and frontal face recognition.
Among face generation modules, the Generative Adversarial Networks (GANs) have received special attention \cite{tran2017disentangled, ju2022complete}. Despite the remarkable results concerning image quality and human perception, GANs add high-frequency components to the synthesized images, which negatively affects the recognition process \cite{wang2018orthogonal}.
Besides, from the optimization perspective, profile-to-frontal face generation is an intrinsically ill-posed problem, and multiple frontal faces exist for each profile face \cite{huang2017beyond}. Also, there are several other nuisance factors in face images, including expression, illumination, and the quality of images. These factors result in a large gap between features of real and synthesized frontal faces in the identity metric space, which significantly deteriorates the final performance \cite{wang2018orthogonal}.
In this paper, we hypothesize that profile and frontal faces have latent connection in an abstract embedding space. We exploit this hidden connection in the embedding domain using a deep coupled model consisting of dedicated networks for the profile and frontal views of the face. These two networks share the same discriminative latent embedding, see Fig.~\ref{fig:overview}.
Using the proposed Pose-Aware Contrastive learning (PAC), we enforce the agreement of the features by maximizing the lower bound of the mutual information between the representations of the same identity \cite{tian2019contrastive}.
In this manner, the model aims to pull closer the representations from pairs of the same identity compared to representations of different identities \cite{khosla2020supervised}. PAC also helps the model to implicitly benefit from hard negative/positive instances \cite{khosla2020supervised}. Hard samples are close to the decision boundary in the embedding space, and emphasizing them in training leads to faster convergence, and better generalization \cite{schroff2015facenet}. Also, we leverage Pose-Aware Contrastive with Memory buffer (PACM), a simple yet effective way to help the loss utilize a massive number of identities' representations without increasing the mini-batch size.
Aiming to further reduce the gap between the profile and frontal images in the embedding space, we employ our proposed Pose-Aware Adversarial Domain Adaptation (PADA) learning approach to enforce the model to learn an asymmetric mapping from profile to frontal representation. Our experiments, evaluations, and ablations studies show that the proposed framework achieves notable performance in learning pose-invariant discriminative representations. Contributions of this paper can be summarized as follows:
\begin{itemize}
\item A novel profile-to-frontal face recognition model is developed, which utilizes a pose-aware contrastive learning to maximize the mutual information between the profile and frontal representations from the same identity in an embedded space.
\item A novel pose-aware domain adaptation approach is developed to enforce the agreement of features from different poses.
\item A novel approach is proposed to learn pose-agnostic representations from a larger number of instances than the mini-batch size in a multiview setting
\end{itemize}
\section{Related Works}
Deep learning has been applied to various applications since its advent \cite{deng2019arcface,nourelahi2022machine,nourelahi2022explainable,aghdaie2022morph,saffari2021robust,mosharafian2022deep,mosharafian2021gaussian}. Biometric has been one the most surfed area due to the availability of large-scale datasets which can be either in the form of signals or images. Among biometric traits, the face has received special attention. In this section, we briefly summarize recent attempts in FR.
\subsection{Deep Face Recognition}\label{section:DFR}
The availability of computing power has made CNN the primary tool in computer vision \cite{masi2018deep}. During the past decade, the introduction of new network architectures, accessibility to large-scale datasets, and modifications of the loss functions have led to significant achievements in deep FR \cite{masi2018deep,cao2018vggface2}.
Along with other supervised deep learning frameworks, Softmax with a cross-entropy loss is of the most popular criterion for FR \cite{khosla2020supervised}.
Intrinsically, features provided by the Softmax have angular distribution \cite{deng2019arcface, wen2016discriminative}. Studies have shown that considering angular distance instead of Euclidean distance significantly improves the FR performance \cite{wang2017normface,wen2016discriminative}. Based on this characteristics, multiple training paradigms have been proposed to adapt various kinds of margins to the Cosine based embedding space \cite{wang2017normface,wen2016discriminative,liu2017sphereface}.
Sample-to-sample loss functions are also well stablished in deep FR \cite{sun2014deep,wen2016discriminative,schroff2015facenet}.
Sun {\it et al.} \cite{sun2014deep} combined the identification and Margin Contrastive Loss (MCL) loss for having a more powerful supervisory signal.
In \cite{schroff2015facenet}, Schroff {\it et al.} presented the Triplet loss, which forces the representations of a triplet to be discriminative. To increase intra-class compactness, Wen {\it et al.} \cite{wen2016discriminative} introduced the Center loss in which the model learns a center for each class. They used it in combination with the Softmax loss to keep representations from collapsing to zero.
\begin{figure*}
\begin{center}
\includegraphics[width=1.0\linewidth]{figures/proposed.png}
\end{center}
\caption{A mini-batch (B) of frontal and profile faces is fed into the coupled-encoder. This model provides representations with high mutual information (high similarity in the embedding space) for genuine pairs and far distant representations for the imposter pairs. Here, positive pairs are shown in the same color. Without memory buffer, the contrastive loss is calculated within the mini-batch (two negative samples for each image). However, memory buffer provide \(\small{K}\) negative samples, \(\small{K>B}\), for each sample. Red-dashed arrows show the gradient back-propagation. The pose-aware adversarial domain adaptation (PADA) loss only affects the profile encoder.}
\label{fig:propose_architecture}
\end{figure*}
\subsection{Pose Robust Face Recognition}
Although near-frontal FR is considered a solved problem in common cases, FR in extreme poses, where enrolled faces in the gallery and the probe images have large pose disparity, has still remained a difficult effort \cite{tu2021joint}. There are two main approaches to cope with profile-to-frontal FR \cite{huang2017beyond,hu2018pose,masi2018learning,meng2021poseface}. One major research avenue is based on synthesizing a frontal face from its profile input and then utilizing the synthesized frontal face for recognition \cite{huang2017beyond,hu2018pose,qian2019unsupervised,yin2017towards,tran2017disentangled}.
Despite the satisfactory results, this approach has a handful of intrinsic drawbacks.
First, the problem of recovering a face's canonical view from its profile pair is under-defined \cite{zhao2018towards, huang2017beyond}. Second, since the frontal faces should be generated first in order to train the classifier, end-to-end training of the generator and classifier is unattainable \cite{tu2021joint}.
Another main line of inquiry for profile-to-frontal FR is to learn pose-agnostic mapping. For instance, Masi {\it et al.} \cite{masi2018learning} used 3D rendering to synthesize multiple views of a profile image. These images were used to train pose-specific deep feature extractors. Then, features from all networks are fused to construct the final prediction.
DREAM \cite{cao2018pose} is based on finding a mapping between profile and frontal embeddings. The authors hypothesized a gradual connection from profile to frontal representation. They utilized a residual mapping that adds pose-adaptive residuals to the features extracted from profile faces.
Meng {\it et al.} \cite{meng2021poseface} disentangled pose and identity representations by mapping face to identity and landmark subspaces.
To disentangle identity from the pose, Yin {\it et al.} adopted a multi-tasking framework \cite{yin2017multi}.
PF-cpGAN \cite{taherkhani2020pf} seeks to learn pose-agnostic representation by employing face frontalization as a sub-task.
Although these methods have shown promising results, there is still a significant performance gap for faces with extreme poses in unconstrained conditions \cite{zhao2018towards}.
In comparison, our proposed coupled-encoder benefits from PACM and PADA losses. They encourage coupled-encoder to map the faces with the same identity to close representations and faces with different identities to representations far from each other. In addition, the memory buffer elevates the efficiency of proposed contrastive learning by looking at a relatively much larger number of identities than the mini-batch size.
\section{Proposed Method}
We introduce a coupled-encoder architecture to map the profile and frontal face images to a shared embedding space.
PACM and PADA losses force the coupled-encoder to learn pose-agnostic representations.
Our model incorporates a massive number of instances to PACM loss function. For example, on a single NVIDIA TITAN X GPU and the mini-batch size of 32, the coupled-encoder calculates the loss between more than 6000 distinct instances, which is beneficial for contrastive learning to maximize the mutual information between two different views of the face images of an identity \cite{van2018representation}. Furthermore, PADA loss improves the compatibility of representations by forcing the profile encoder to map the off-angle faces close to its frontal pairs.
\subsection{Pose-Aware Contrastive Learning}
Our proposed profile-to-frontal FR framework is based on recent information-theoretic techniques for contrastive representation learning \cite{van2018representation,tian2019contrastive}. During training, we aim to maximize the mutual information between profile and frontal face images from the same identity (positive pair) and minimize the mutual information between face images with different identities (negative pairs). During testing, we make the decision considering the distance between representations of a pair of face images in the embedding space. The first step during training is to select face images that represent the same identity but distinct views. Then, each image in positive pair is employed to choose instances that represent distinct identities and views. For example, given a profile face image, we should pick negative frontal instances and vice versa. Then, these pairs of frontal and profile face images are used to train two deep encoders.
Given a training set \( \small{ D=D_f \bigcup D_p}\), \(\small{D_f:{\{(x_{f,i},y_{f,i}) \}_{i=0}^{N_f}}}\) and \(\small{D_p:\{(x_{p,i},y_{p,i}) \}_{i=0}^{N_p}}\) represent profile and frontal subsets of dataset, respectively.
\(N_p\) is the number of profile samples and \(N_f\) frontal samples.
As presented in Fig.~\ref{fig:propose_architecture}, the coupled-encoder consists of \(\small{f_f(.)}\) and \(\small{f_p(.)}\), which are the frontal and profile dedicated embedding sub-networks. These two sub-networks map the frontal and profile faces to a \(\small{d}\)-dimensional embedding space: \(\small{f_f(.) : \mathbb{R}^{3 \times w \times h} \rightarrow \mathbb{R}^d}\) and \(\small{f_p(.) : \mathbb{R}^{3 \times w \times h} \rightarrow \mathbb{R}^d}\), respectively. For convenience, we ignore the index \(\small{i}\) that reflects the index of the sample in their corresponding subset. \(\small{z_{f} = f_f(x_{f})}\) and \(\small{z_{p} = f_p(x_{p})}\) are the representations of the frontal, $x_f$, and profile, $x_p$, images generated by their corresponding encoders.
The first step toward our training paradigm is to select pair of images with the same identity and pose disparity. To construct our training samples, we define a genuine (positive) pairs as:
\begin{equation}\label{GenuinePair}
\small
\begin{aligned}
&P=\{(z_{f},z_{p})|y_{f}=y_{p}\},
\end{aligned}
\end{equation}
where \(\small{y_{f}}\) and \(\small{y_{p}}\) represent labeled identities of \(\small{z_{f}}\) and \(\small{z_{p}}\), respectively. In contrast, we define an imposter (negative) pair as:
\begin{equation}\label{ImposterPair}
\small
\begin{aligned}
&N=\{(z_{f},z_{p})|y_{f}\neq y_{p}\},
\end{aligned}
\end{equation}
for positive pairs, we first choose a random identity \(\small{y_0}\) as an anchor. Then, sampling two images independently from \(\small{D_f}\) and \(\small{D_p}\) given the selected identity. Consequently, we define the joint distribution of genuine pair as \cite{van2018representation}:
\begin{equation}\label{JointGenuine}
\small
\begin{aligned}
p(z_{f},z_{p})=&\sum_{y_0 \in C}{p(y_{f}=y_0)p(z_{f}|y_{f}=y_0)}&\\
&\times{p(y_{p}=y_0)p(z_{p}|y_{p}=y_0)}\\
&=\sum_{y_0 \in C}{p(y_{f}=y_0,y_{p}=y_0)}\\
&\times{p(z_{f},z_{p}|y_{f}=y_{p}=y_0)}\\
&=\sum_{y_0 \in C}{p(z_{f},z_{p},y_{f}=y_0,y_{p}=y_0)},
\end{aligned}
\end{equation}
where \(\small{C}\) reflects the total identities presented in the dataset. Assuming a high entropy of negative pairs and the large number of identities,
which is the case for the FR, we approximate the sampling of the negative pairs with sampling from product of marginals \cite{van2018representation}:
\begin{equation}\label{JointImposter}
\small
\begin{aligned}
p(z_{f})p(z_{p})\approx & \sum_{y_0 \in C}{\sum_{\substack{y_0^{'} \in C \\ y_0^{'} \neq k}}{p(y_{f}=y_0)}p(z_{f}|y_{f}=y_0)} & \\
& \times p(y_{p}=y_0^{'})p(z_{p}|y_{p}=y_0^{'}).&
\end{aligned}
\end{equation}
We aim to train $f_f(.)$ and $f_p(.)$ such that face images of an identity in different views are mapped closely in the embedding space. To this end, we maximize the mutual information between positive pair representations by maximizing the KL divergence between Eqs.~\ref{JointGenuine} and \ref{JointImposter} \cite{tian2019contrastive}. Hence, we aim to learn a function, \(\small{h(.)}\), which provides a low value for negative pairs and high value for positive pairs \cite{van2018representation}.
\begin{equation}\label{cosineh}
\small
\begin{aligned}
&h(z_f,z_p) = \exp({\frac{1}{\tau}\frac{z_f \cdot z_p}{||z_f|| \cdot ||z_p||}}),
\end{aligned}
\end{equation}
\(\small{h(.)}\) reflects the cosine similarity between latent representations and \(\tau\) is the temperature \cite{hinton2015distilling}, which plays an important role in concentration of representations in the hypersphere \cite{wu2018unsupervised,wang2017normface}.
The contrastive learning aims to pull an anchor \(z_{f,i_0}\) and positive samples \(z_{p,i_0}\) close in the embedding space while pushing the anchor away from many negative samples \cite{tian2019contrastive}
\begin{equation}\label{NCELoss1}
\small
\begin{aligned}
&L_{cont} = - \mathop{\mathbb{E}}_{S}\left[\log{\frac{{h(z_{f,i_0},z_{p,i_0})}}{\sum_{i=0}^{k}{{h(z_{f,i_0},z_{p,i})}}}}\right],
\end{aligned}
\end{equation}
where \(\small{S:\{(z_{f,i_0},z_{p,i})\}_{i=0}^{k}}\) is a set of \(k\) negative pairs and one positive pair \cite{van2018representation}.
In \cite{van2018representation}, it is proven that optimal \(\small{h(.)}\) is in direct proportion with the ratio of joint distribution and product of marginals distributions: \(\small{h \propto \frac{p(z_f,z_p)}{p(z_f)p(z_p)}}\). Replacing \(\small{h(.)}\) with the density ratio results \cite{van2018representation}:
\begin{equation}\label{proof}
\small
\begin{aligned}
L_{cont}^{optim}
&\geq \log(k) - \mathop{\mathbb{E}}_{(z_f,z_p)\sim p_{z_f,z_p}}\log{\left[\frac{p(z_{f},z_{p})}{p(z_{f})p(z_{p})}\right]},
\end{aligned}
\end{equation}
recalling the mutual information between two random variable \(z_f\) and \(z_p\): \(I(z_f;z_p)=\mathop{\mathbb{E}}_{p_{z_f,z_p}}\left[{\frac{p(z_f,z_p)}{p(z_f)p(z_p)}}\right]\). Consequently, for any positive pair of frontal and profile faces: \(\small{I(z_f,z_p) \geq \log{(k)}-l_{cont}^{optim}}\) \cite{van2018representation}.
Hence, minimizing the contrastive loss results in maximizing the lower bound to \(\small{I(z_f;z_p)}\).
Without loss of generality, we consider the features are normalized: \(\small{||z_{f}||=||z_{p}||=1}\). Consequently, given a mini-batch, the PAC loss for a frontal anchors is:
\begin{equation}\label{SupConLoss_frontalAnchor}
\small
\begin{aligned}
&L_{PAC}^{f}=-\sum_{i=1}^{|B|}{{\log{\frac{\exp(\frac{1}{\tau}{z_{f,i} \boldsymbol{\cdot} z_{p,a_i}} )}{\sum_{j\in N_p(i)}{\exp(\frac{1}{\tau}{z_{f,i} \boldsymbol{\cdot} z_{p,j} }})}}}},
\end{aligned}
\end{equation}
where \(\small{B}\) is the mini-batch and \(\small{N_p(i)}\) is a set of one positive and many negative profile samples corresponding to \(\small{z_{f,i}}\). Symmetrically, considering profile samples as anchors:
\begin{equation}\label{SupConLoss_profileAnchor}
\small
\begin{aligned}
&L_{PAC}^{p}=-\sum_{i=1}^{|B|}{{\log{\frac{\exp(\frac{1}{\tau}{z_{p,i} \boldsymbol{\cdot} z_{f,a_i}} )}{\sum_{j\in N_f(i)}{\exp({ \frac{1}{\tau}z_{p,i} \boldsymbol{\cdot} z_{f,j} }})}}}},
\end{aligned}
\end{equation}
there are four main advantages in using this loss function. 1) Comparison with every negative sample within mini-batch at the same time, denominator in Eqs.~\ref{SupConLoss_frontalAnchor} and \ref{SupConLoss_profileAnchor}, 2) rather than optimizing the angle between the representations and their corresponding prototypes, the model directly learns the angle between representations \cite{liu2017sphereface}, 3) PAC provides the implicit hard negative/positive mining to the model \cite{khosla2020supervised}, and 4) PAC maximizes the mutual information between representations from different views of the shared context \cite{bachman2019learning}.
\begin{table}[]
\addtolength{\tabcolsep}{-2pt}
\small
\caption{Verification accuracy (\%) and standard deviation for CFP-FP over standard 10-folds. Results of the \cite{cao2018pose,chen2016unconstrained} are copied from \cite{taherkhani2020pf}.}
\vspace{2mm}
\begin{tabular}{l|cccc}
\hline
\multirow{2}{*}{{Method}} & \multicolumn{2}{c}{{Frontal-Profile}} & \multicolumn{2}{c}{{Frontal-Frontal}} \\
& {Accuracy} & {EER} & {Accuracy} & {EER} \\ \hline \hline
\footnotesize{PR-REM} \cite{cao2018pose} & \footnotesize{{93.25(2.23)}} & \footnotesize{7.92(0.98)} & \footnotesize{98.1(2.19)} & \footnotesize{1.1(0.22)} \\
\footnotesize{DCNN} \cite{chen2016unconstrained} & \footnotesize{84.91(1.82)} & \footnotesize{14.97(1.98)} & \footnotesize{96.4(0.69)} & \footnotesize{3.48(0.67)} \\
\footnotesize{p-CNN} \cite{yin2017multi} & \footnotesize{94.39(1.17)} & \footnotesize{5.94(0.11)} & \footnotesize{97.79(0.40)} & \footnotesize{2.48(0.07)} \\
\footnotesize{FRN-TI} \cite{tu2021joint} & \footnotesize{95.62} &-&-&- \\
\footnotesize{DR-GAN} \cite{tran2017disentangled} & \footnotesize{93.41(1.17)} & - &\footnotesize{97.84(0.79)}& - \\
\footnotesize{PF-cpGAN} \cite{taherkhani2020pf} & \footnotesize{93.78(2.46)} & \footnotesize{7.21(0.65)} & \footnotesize{98.88(1.56)} & \footnotesize{0.93(0.14)} \\
\footnotesize{PIM} \cite{zhao2018towards} & \footnotesize{93.1(1.01)} & \footnotesize{7.69(1.29)} & \textbf{\footnotesize{99.44(0.36)}} & \footnotesize{0.86(0.49)} \\\hline
ours & \textbf{\footnotesize{95.85(1.07)}} & \textbf{\footnotesize{4.22(0.15)}} & \footnotesize{99.37(0.4)} & \textbf{\footnotesize{0.63(0.05)}} \\ \hline
\end{tabular}
\label{table:cfp}
\vspace{-2mm}
\end{table}
\subsection{Pose-Aware Contrastive Learning with Memory Buffer}
Due to the large number of classes in FR datasets and having a small number of identities within a mini-batch, the conventional FR methods could not cover a large number of identities at each step of loss calculation \cite{hou2019learning}. Increasing the mini-batch size may alleviate the issue, but it does not ensure the improvement in the performance, and in many cases, due to the memory constraint, it is impractical \cite{keskar2016large,you2017scaling}.
We mitigate this issue by adopting the memory buffer framework \cite{wu2018unsupervised} to the loss function to benefit from more negative instances. The memory buffer consists of the latent representations of profile and frontal faces from past iterations. During each learning iteration, representations \(\small{z_f}\) and \(\small{z_p}\) are updated to the memory at the corresponding instance entry:
\begin{equation}\label{memory_update}
\small
\begin{aligned}
&r_{f,t+1}= m * r_{f,t} + (1-m) * z_f\\
&r_{p,t+1}= m * r_{p,t} + (1-m) * z_p,
\end{aligned}
\end{equation}
where \(\small{r_f}\) and \(\small{r_p}\) stand for features saved in frontal and profile memory buffer and \(\small{m}\) is the momentum coefficient in updating \cite{wu2018unsupervised}.
Therefore, Eq.~\ref{SupConLoss_frontalAnchor} can be rewritten as:
\begin{equation}\label{SupConLoss_frontalAnchor_memory}
\small
\begin{aligned}
&L_{PACM}^{f}=-\sum_{i=1}^{|B|}{{\log{\frac{\exp({\frac{1}{\tau} z_{f,i} \boldsymbol{\cdot} r_{p,a_i}})}{\sum_{j\in N_p^M(i)}{\exp({\frac{1}{\tau} z_{f,i} \boldsymbol{\cdot} r_{p,j} }})}}}},
\end{aligned}
\end{equation}
where \(\small{N_p^M(i)}\) represent a set of one positive and multiple negative profile representations drawn from the memory. Similarly:
\begin{equation}\label{SupConLoss_profileAnchor_memory}
\small
\begin{aligned}
&L_{PACM}^{p}=-\sum_{i=1}^{|B|}{{\log{\frac{\exp({\frac{1}{\tau} z_{p,i} \boldsymbol{\cdot} r_{f,a_i}} )}{\sum_{j\in N_f^M(i)}{\exp({\frac{1}{\tau} z_{p,i} \boldsymbol{\cdot} r_{f,j} }})}}}}.
\end{aligned}
\end{equation}
Instances within a mini-batch form the \(\small{N_f(i)}\) and \(\small{N_p(i)}\); however, \(\small{N_f^M(i)}\) and \(\small{N_p^M(i)}\) are drawn from the memory and are not constrained to the mini-batch size.
Therefore, we can chose the number of negative pairs such that: \(\small{|N_p^M(i)|\gg |N_p(i)|}\) and \(\small{|N_f^M(i)|\gg |N_f(i)|}\).
Memory allows us to choose different samples for every instance in each mini-batch and not be limited to the mini-batch size \cite{trosten2021reconsidering}, see Fig.~\ref{fig:SupConMemory}.
Finally, the overall loss for the Pose-Aware Contrastive with Memory buffer (PACM) is:
\begin{equation}\label{MCM}
\small
\begin{aligned}
&L_{PACM}=L_{PACM}^f + L_{PACM}^p.
\end{aligned}
\end{equation}
\addtolength{\tabcolsep}{0pt}
\begin{table*}[t]
\small
\caption{Performance (\%) comparison of our framework on Setting1 and Setting2 of Multi-PIE dataset. Last three rows represent our results with single (first two) and couple network (last row) with 200 negative samples. }
\begin{center}
\begin{tabular}{l|cccccc|cccccc}
\hline
& \multicolumn{6}{c|}{{Setting 1}} & \multicolumn{6}{c}{{Setting 2}} \\
\multirow{-2}{*}{{Method}} & \(\pm\){15\(\boldsymbol{^{\circ}}\)} & \(\pm\){30\(\boldsymbol{^{\circ}}\)} & \(\pm\){45\(\boldsymbol{^{\circ}}\)} & \(\pm\){60} & \(\pm\){75\(\boldsymbol{^{\circ}}\)} & \(\pm\){90\(\boldsymbol{^{\circ}}\)} & \(\pm\){15\(\boldsymbol{^{\circ}}\)} & \(\pm\){30\(\boldsymbol{^{\circ}}\)} & \(\pm\){45\(\boldsymbol{^{\circ}}\)} & \(\pm\){60} & \(\pm\)\textbf{75\(\boldsymbol{^{\circ}}\)} & \(\pm\){90\(\boldsymbol{^{\circ}}\)} \\ \hline \hline
\multicolumn{1}{l|}{FF-GAN \cite{yin2017towards}} & - & - & - & - & - & - & 94.6 & 92.5 & 89.7 & 85.2 & 77.2 & 61.2 \\
\multicolumn{1}{l|}{DR-GAN \cite{tran2017disentangled}} & - & - & - & - & - & - & 94.0 & 90.1 & 86.2 & 83.2 & - & - \\
\multicolumn{1}{l|}{FNM+Light CNN \cite{qian2019unsupervised}} & 99.9 & 99.5 & 98.2 & 93.7 & 81.3 & 55.8 & - &- & - &- & - & - \\
\multicolumn{1}{l|}{CAPG-GAN \cite{hu2018pose}} & 99.95 & 99.37 & 98.28 & 93.74 & 87.40 & 77.10 & 99.82& 99.56 & 97.33 & 90.63 & 83.05 & 66.05 \\
\multicolumn{1}{l|}{TP-GAN \cite{huang2017beyond}} & 99.78 & 99.85 & 98.58 & 92.93 & 84.10 & 64.03 & 98.68 & 98.06 & 95.38 & 87.72 & 77.43 & 64.64\\
\multicolumn{1}{l|}{PIM \cite{zhao2018towards}} & 99.80 & 99.40 & 98.30 & 97.70 & 91.20 & 75.00 & 99.30 & 99.00 & 98.50 & 98.10 & 95.00 &86.50\\
\multicolumn{1}{l|}{PoseFace \cite{meng2021poseface}} & \textbf{100} & 99.97 & 99.62 & 98.55 & 96.07 & 90.58 & - &- & - &- & - &-\\
\multicolumn{1}{l|}{FFWM \cite{wei2020learning}} & \textbf{100} & \textbf{100} & \textbf{100} & 98.86 & 96.54 & 88.55 & 99.86 & 99.80 & 99.37 & 98.85 &97.20 &93.17\\
\multicolumn{1}{l|}{PF-cpGAN \cite{taherkhani2020pf}} & 99.9 & 99.9 & 98.9 & 97.6 & 94.2 & 88.1 & - & - & - & - & - &-\\ \hline
\multicolumn{1}{l|}{Ours-single-wo PADA} & \textbf{100} & \textbf{100} & 99.97 & {99.53} & {97.26} & {91.70} & 99.98 & 99.97& 99.82& 99.47& 97.19& 91.51\\
\multicolumn{1}{l|}{Ours-single} & \textbf{100} & \textbf{100} & 99.97 & {99.63} & {97.50} & {92.65} & 99.95 & 99.83& 99.70& 98.97& 96.64& 91.79\\
\multicolumn{1}{l|}{Ours-couple} & \textbf{100} & \textbf{100} & 99.97 & \textbf{99.74} & \textbf{98.04} & \textbf{94.64} & \textbf{100} & \textbf{100}& \textbf{99.98}& \textbf{99.76}& \textbf{98.21}& \textbf{94.49}\\ \hline
\end{tabular}
\end{center}
\label{table:results_cmu}
\vspace{-2mm}
\end{table*}
\subsection{Pose-Aware Adversarial Domain Adaptation Learning}
For each identity, the ideal scenario is to have an identical profile and frontal feature representations. To further improve the similarity of these features, we adapt the idea of adversarial adaptation \cite{tzeng2017adversarial}, which aims at making the representations of profile and frontal images as similar as possible. To this end, we aim to fool a binary classifier (view discriminator), which is going to be trained to distinguish between profile and frontal representations. This mimics the policy which is used in the GANs in which the generator tries to produce samples that are indistinguishable from real samples \cite{tzeng2017adversarial}.
The view discriminator \(\small{D}\) classifies whether a feature vector comes from a profile or frontal image. Therefore, \(\small{D}\) should maximize cross-entropy loss function:
\begin{equation}\label{CE_D}
\small
\begin{aligned}
L_{D}(x_{p},x_{f},f_p(.),f_f(.))=&\mathbb{E} \left[ \log{D(f_f(x_f)}\right] \\
&+ \mathbb{E} \left[ \log{(1-D(f_p(x_p))}\right].
\end{aligned}
\end{equation}
It is important to remember that for adversarial domain adaptation learning, positive samples are used, \(y_f=y_p\). The profile encoder's objective is to fool the view discriminator by maximizing the following:
\begin{equation}\label{adversarial_profile}
\small
\begin{aligned}
&L_{encoder}(x_{p},f_p(.))=\mathbb{E} \left[ \log{D(f_p(x_p)}\right],
\end{aligned}
\end{equation}
considering Eqs.~\ref{CE_D} and \ref{adversarial_profile}, we conclude that the profile encoder and the view discriminator play a minmax game:
\begin{equation}\label{adv}
\small
\begin{aligned}
L_{PADA} =\min_{f_p(.)}\max_{D}&\{\mathbb{E} \left[ \log{D(f_f(x_f)}\right]\\
&+ \mathbb{E} \left[ \log{(1-D(f_p(x_p))}\right]\},
\end{aligned}
\end{equation}
where \(f_f(.)\) is fixed during the adversarial training and \(f_p(.)\) is trained by Eq.~\ref{adv}. Consequently, the model learns an asymmetric mapping from profile to frontal representation that modifies the profile encoder to learn representations that match the representations of frontal faces \cite{tzeng2017adversarial}.
Based on the above discussion, we formulate the total training loss function as:
\begin{equation}\label{lossOveral}
\small
\begin{aligned}
&L_{total} = \lambda_1 l_{PADA} + \lambda_2 l_{PACM},
\end{aligned}
\end{equation}
where \(\small{\lambda_1}\) and \(\small{\lambda_2}\) are the training regularization parameters.
\section{Experiments}
We study the performance of the coupled-encoder on four FR datasets. We report the results of the proposed framework for verification and identification setup and compare them with the state-of-the-art (SOTA) methods. Furthermore, we investigate the impact of the number of negative samples and effect of different terms in Eq.~\ref{lossOveral}.
\subsection{Training Setup}
For all datasets, MTCNN \cite{zhang2016joint} is considered to detect and align faces. All the images are resized to 112\(\times\)112, and pixel values are normalized to \([-1,1]\). Most of the FR datasets are imbalanced in two aspects: 1) the number of per identity samples, and 2) the number of profile and frontal samples for each identity. Consequently, there is a good chance that one of the networks learns a degenerate solution \cite{du2020semi}.
We initialize the encoders sub-networks with pre-trained weights on the VggFace2 dataset \cite{cao2018vggface2} with the Softmax loss to mitigate this problem.
For selecting the frontal and profile pairs, we apply \cite{ruiz2018fine} to the datasets to create frontal and profile subsets based on the yaw angle. Then, face images with an absolute yaw value less than 15$^\circ$ are considered frontal. For training Eq.~\ref{lossOveral}, the initial learning rate is set to 0.001 and is multiplied by 0.1 every ten epochs, and weight decay and momentum are 0.00001 and 0.9, respectively. The model is trained for 20 epochs. During the training, \(\lambda_1=0.1\) and \(\lambda_2=1.0\), and the number of negative samples is 6,000.
We adopt ResNet50 \cite{deng2019arcface} as the encoder networks for the profile and frontal views. The final feature is of size 512$\times$7$\times$7. Feature maps are reshaped to form a vector of size 25,088 and passed to a fully-connected layer of size 512 to construct the final representation. The last feature vectors of encoders are enqueued to the frontal and profile memory buffer, see Fig.~\ref{fig:propose_architecture}. The view discriminator is an MLP with two hidden layers of size 256, each followed by batch normalization and leaky relu activation function. At the top of these hidden layers, there is a single neuron with a sigmoid activation function.
The model is trained using Stochastic Gradient Descent (SGD) with a mini-batch size of 32 on an NVIDIA TITAN X GPU using Pytorch \cite{paszke2019pytorch}.
\subsection{Results}
\textbf{CMU Multi-PIE} dataset \cite{gross2010multi} includes 750,000 images of 337 identities with variations in pose, illumination, and expression in the controlled environment. It contains images of 15 different views from 20 illuminations in different expressions. For fair comparison, we use neutral images in 13 views of $\{0^\circ,\pm15^\circ,\pm30^\circ,\pm45^\circ,\pm60^\circ,\pm75^\circ,\pm90^\circ\}$, and all the variations in illumination are included as \cite{huang2017beyond}. More specifically, there are two main settings for this dataset in the literature: {Setting1} and {Setting2}. In both settings, face images with neutral expression are used. In the Setting1, 250 identities from session 01 of the dataset are employed. The first 150 identities are selected for training and the rest for testing. The test set consists of probe and gallery sets. The galley set consists of one frontal image per identity in neutral illumination, and the probe set contains images with a yaw angle other than zero. There is no overlap between training and testing identities. In Setting2, face images of the 200 identities from four sessions are used for training. The probe and gallery sets for testing are constructed as Setting1. In this dataset, images with zero yaw degree are considered frontal, and all the other views are considered profile. Following \cite{tran2017disentangled,qian2019unsupervised,
huang2017beyond,zhao2018towards,meng2021poseface,wei2020learning,taherkhani2020pf}, we fine-tune the coupled-encoder on the training sets of Setting1 and Setting2, separately.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth]{figures/MI.png}
\end{center}
\caption{The distance distributions between positive (shown in blue) and negative (shown in orange) pairs. In near frontal poses, the normalized Softmax (top in each block) and the PACM (bottom in each block) separate positive from negative pairs. In near-profile, \(\small{\pm60^{\circ}}\) and \(\small{\pm75^{\circ}}\), and complete profile, \(\small{\pm90^{\circ}}\), only PACM perfectly separates these two distributions (High mutual information between positive pairs).}
\label{fig:distribution}
\vspace{-2mm}
\end{figure}
Table~\ref{table:results_cmu} demonstrates the performance of our model in comparison with SOTA models on the Multi-PIE dataset and we investigate the effect of face angle on the FR performance. Almost all methods perform above 92\% for pose variation in \([-60^{\circ},+60^{\circ}]\). However, beyond this range, their performance drastically decreases.
Table~\ref{table:results_cmu} shows that the coupled-encoder outperforms both face frontalization and pose-invariant feature learning algorithms for the face with \(\pm90^{\circ}\) pose. Our framework improves \cite{taherkhani2020pf} by almost 6\%. Even without the pose-aware adversarial domain adaptation, the presented model achieves better results compared to other methods (in Setting1). Considering PoseFace \cite{meng2021poseface}, coupled-encoder performs better for every pose, and it outperforms PoseFace for face images with \(\pm90^{\circ}\) pose by almost four percentage points. Same as Setting1, the coupled-encoder surpasses other algorithms in Setting2. The improvement is more noticeable for the extreme pose of \(\pm90^{\circ}\).
\begin{table}[]
\small
\caption{Verification accuracy (\%) for IJB-B and IJB-C dataset. Results of the \cite{schroff2015facenet,qian2019unsupervised,cao2018pose} are copied from \cite{taherkhani2020pf}}
\label{table:ijb}
\begin{center}
\begin{tabular}{l|cc|cc}
\hline
\multirow{2}{*}{{Metohd}} & \multicolumn{2}{c|}{\scalebox{.8}{{IJB-C(TAR@FAR)}}} & \multicolumn{2}{c}{\scalebox{.8}{{IJB-B(TAR@FAR)}}} \\
& {0.001} & {0.01} & {0.001} & {0.01} \\ \hline \hline
{GOTs} \cite{whitelam2017iarpa,maze2018iarpa} & 36.3 & 62.1 & {33.0} & 60.0 \\
{VGG-CNN} \cite{whitelam2017iarpa,maze2018iarpa} & 74.3 & 87.2 & {72.0} & 86.0 \\
{CFR-GAN} \cite{ju2022complete} & 74.81 & 86.46 & {73.54} & 85.34 \\
{FaceNet} \cite{schroff2015facenet} & 66.3 & 82.3 & - &- \\
{FNM} \cite{qian2019unsupervised} & 80.4 & 91.2 & - &- \\
{PR-REM} \cite{cao2018pose} & 83.4 & 92.1 & - &- \\
{PF-cpGAN} \cite{taherkhani2020pf} & 86.1 & 93.8 & 84.21 & 90.02 \\
{Lin {\it et al.}} \cite{lin2022mitigating} &89.85&\textbf{95.99}&87.55& \textbf{95.08} \\
{SSA} \cite{lin2021domain} &\textbf{90.91}&95.90&88.27&94.88 \\
\hline
{ours} & {90.05} & {95.70} & {\textbf{88.35}} & {94.87} \\ \hline
\end{tabular}
\end{center}
\vspace{-2mm}
\end{table}
The \textbf{Celebrities in Frontal-Profile in the Wild (CFP)} dataset \cite{sengupta2016frontal} includes facial images of 500 different identities with 10 frontal and 4 profile samples per identity. Following \cite{sengupta2016frontal}, we evaluate our method on 10-fold protocol, and each fold consists of 350 genuine and 350 imposter pairs.
Considering results on the CFP-FP dataset in Table~\ref{table:cfp}, our framework performs better than the other SOTA methods with at least a margin of 0.23\% in terms of accuracy and 1.72\% improvements in terms of EER.
The \textbf{IJB-B} \cite{whitelam2017iarpa} is challenging in the wild dataset, which was further extended to \textbf{IJB-C} \cite{maze2018iarpa}. These datasets are of the most challenging benchmarks for FR, containing large pose variation and diversity in resolutions.
For evaluating on these datasets, we follow the protocol in \cite{whitelam2017iarpa,maze2018iarpa}. Baseline for comparison on IJB-B and IJB-C datasets are PR-REM \cite{cao2018pose}, FNM \cite{qian2019unsupervised}, FaceNet \cite{schroff2015facenet}, GOTs \cite{whitelam2017iarpa,maze2018iarpa}, VGG-CNN \cite{whitelam2017iarpa,maze2018iarpa}, PF-cpGAN \cite{taherkhani2020pf}, CFR-GAN \cite{ju2022complete}, SSA \cite{lin2021domain}, and Lin {\it et al.} \cite{lin2022mitigating}. Table~\ref{table:ijb} shows that coupled-encoder improves the True Acceptance Rate (TAR) at False Acceptance Rate (FAR) of 0.001. Coupled-encoder performs in pare with SSA \cite{lin2021domain} and \cite{lin2022mitigating}.
\begin{table}[]
\addtolength{\tabcolsep}{-3pt}
\small
\caption{Recognition accuracy (\%) of the proposed framework on setting1 of Multi-PIE dataset.}
\begin{center}
\begin{tabular}{l|llllll}
\hline
& \multicolumn{6}{c}{{view}}\\ \cline{2-7}
\multicolumn{1}{l|}{{loss}} & \(\pm\){15\(\boldsymbol{^{\circ}}\)} & \(\pm\){30\(\boldsymbol{^{\circ}}\)} & \(\pm\){45\(\boldsymbol{^{\circ}}\)} & \(\pm\){60\(\boldsymbol{^{\circ}}\)} & \(\pm\){75\(\boldsymbol{^{\circ}}\)} & \(\pm\){90\(\boldsymbol{^{\circ}}\)} \\ \hline \hline
\multicolumn{1}{l|}{{Joint Softmax}} & 99.89 & 99.71 & 98.64 & 94.47 & 86.92 & 75.18 \\
\multicolumn{1}{l|}{{MCL}} & 99.82 & 99.80 & 98.84 & 96.17 & 89.04 & 77.71 \\
\multicolumn{1}{l|}{{PAC}} & 100 & 99.87 & 99.28 & 98.65 & 96.49 & 91.06 \\
\multicolumn{1}{l|}{{PAC+PADA}} & 100 & 100 & 99.94 & 99.71 & 97.66 & 93.50 \\
\multicolumn{1}{l|}{{PACM+PADA}} & 100 & 100 & 99.97 & 99.74 & 98.04 & 94.64 \\ \hline
\end{tabular}
\label{table:abloss}
\end{center}
\vspace{-4mm}
\end{table}
\subsection{Ablation studies}
In this section, we analyze the effects of different terms of our proposed framework. First, we study the impact of the loss function's components, {\it i.e.,} pose-aware contrastive, memory buffer, and pose-aware adversarial domain adaptation on the performance. Then we report the results of a single encoder. We report on the Multi-PIE dataset to better illustrate the influence of shift in distribution caused by variation in view angle, see Fig.~\ref{fig:distribution}.
The encoders are pre-trained ResNet50 with Softmax on the VggFace2 dataset. We finetune them with the corresponding loss for the equal number of iterations on the training set of Setting1. The optimizer is SGD for all the experiments. We consider multiple learning rates for training MCL and softmax loss. The learning rate for other experiments is chosen to be 0.001. Also, we have to choose an appropriate margin for training with MCL; therefore, we conduct experiments with different margin values and best results are reported in Table~\ref{table:abloss}.
From Table~\ref{table:abloss}, all the losses perform similarly for the absolute view angles less than 60$^{\circ}$.
\begin{table}[]
\addtolength{\tabcolsep}{-3pt}
\small
\caption{\small{Recognition accuracy (\%) of the proposed framework on the Setting1 for Multi-PIE dataset with varying number of negative samples.}}
\begin{center}
\begin{tabular}{l|llllll}
\hline
& \multicolumn{6}{c}{{view}}\\ \cline{2-7}
\multicolumn{1}{l|}{{Num Negative}} & \(\pm\){15\(\boldsymbol{^{\circ}}\)} & \(\pm\){30\(\boldsymbol{^{\circ}}\)} & \(\pm\){45\(\boldsymbol{^{\circ}}\)} & \(\pm\){60\(\boldsymbol{^{\circ}}\)} & \(\pm\){75\(\boldsymbol{^{\circ}}\)} & \(\pm\){90\(\boldsymbol{^{\circ}}\)} \\ \hline \hline
\multicolumn{1}{l|}{{32}} & 100 & 100 & 99.91 & 99.69 & 97.67 & 93.19 \\
\multicolumn{1}{l|}{{100}} & 100 & 100 & 99.97 & 99.56 & 97.97 & 94.12 \\
\multicolumn{1}{l|}{{150}} & 100 & 100 & 99.95 & 99.94 & 97.98 & 94.39 \\
\multicolumn{1}{l|}{{200}} & 100 & 100 & 99.97 & 99.74 & 98.04 & 94.64 \\
\multicolumn{1}{l|}{{500}} & 100 & 100 & 100 & 99.61 & 98.01 & 94.61 \\ \hline
\end{tabular}
\end{center}
\label{table:ablation_nce}
\vspace{-4mm}
\end{table}
In our experiments, utilizing PAC outperforms Softmax and MCL by almost 15\% in identification accuracy for the extreme poses of \(\pm90^{\circ}\). At the same time, it improves the accuracy in near frontal views, except for the \(\pm45^{\circ}\). In the cases of \(\pm30^{\circ}\) and \(\pm45^{\circ}\), we observe that when adding the adversarial domain adaptation loss, the performance is improved by 0.13\% and 0.66\%, respectively. This emphasizes the role of asymmetric mapping in aligning profile and frontal representations.
Contrastive learning frameworks benefit from a larger number of negative samples \cite{van2018representation}. Thus, we expect improvement by integrating the memory bank into the loss function, consistent with our experiment.
In Table~\ref{table:ablation_nce}, we further study the effect of varying the number of negative samples on the Setting1 of Multi-PIE dataset. The performance constantly improves until 200 negative samples and then saturates. This saturation is due to the fact that, in the Setting1 of the Multi-PIE, only 150 identities are used for training, and most of the 500 negative samples represent repetitive identities. Moreover, Table~\ref{table:results_cmu} shows the performance of the proposed framework with single and coupled-encoder. As the pose disagreement between probe and gallery images increases, coupled encoder presents more improvement, which emphasizes that we need a dedicated frontal encoder to have more flexible mapping for profile faces.
\section{Conclusion}
In this paper, we focused on solving FR in extreme pose scenario.We proposed a new coupled-encoder framework with two distinct encoders that maximize the mutual information between the embeddings of profile and frontal face images. For this goal, we adopt a pose-aware contrastive loss and pose-aware asymmetric training. They force the coupled-encoder to map faces with the same identity to close representations and faces with different identities to the far representations. Furthermore, the memory buffer improves the effectiveness of suggested contrastive learning, by looking at a massive number of identities compared to the mini-batch size. We conducted experiments on multiple benchmarks, showing the capability of our approach to outperform SOTA methods. These performance improvements illustrate the effect of a domain-dedicated feature extractor and employing PACM loss on projecting images to an embedding space where all the images of the same person are close together and far from other individuals, regardless of the view angle. Moreover, the role of each part of our loss function is investigated in the ablation study.
\section{Acknowledgement}
This research is based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA R\&D Contract No. 2022-21102100001. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.
{\small
\bibliographystyle{ieee}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 131
|
Tonight's X Factor will see Nile Rodgers replaced Robbie on the judging panel once again.
Last week when Robbie was absent for his tour in South America, his group United Vibe were eliminated from the show.
Many fans were furious they were dropped while their mentor was away.
But, contestant Brendan Murray has insisted Robbie's absence didn't have an impact on United Vibe's exit.
Speaking exclusively to Express.co.uk, he said: "I think, obviously there are circumstances with sound and everything on Saturday night.
"Maybe it was the wrong song or something else, it's hard to know really.
Brendan went on to comment on if he thought it was unfair some performances were replayed following the sound issues.
"Everyone had different opinions of it," he noted.
"Some people were happy, some people weren't happy.
"But, you know, you just had to think of the two lads.
"If that was you in that position, then obviously you would want the same performance, be given a chance. There was difference in opinions.
"I think there was going to be disappointment either way with the way it went because it was such a short window.
Brendan is currently the favourite to be eliminated from the show this week.
A spokesman for bookmakers OddsMonkey said: "Dalton is just getting more and more popular, every week he impresses and that means more and more money, we are being told that that vast majority of all bets is for Dalton.
Brendan is set to perform Everybody Hurts from Bewitched.
Will he be able to impress the judges and viewers enough to be saved?
X Factor airs tonight on ITV at 8.30pm.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,344
|
Ryggsträng (även notokord eller korda; vetenskapligt: chorda dorsalis) är en stavformig bildning i embryon eller larvformer hos alla ryggsträngsdjur. Ryggsträngen innehåller en elastisk stödjevävnad (axialt mesoderm), och kan ge kroppen en viss skelettlik stadga. Den kvarstår hos lansettfiskar, men tillbakabildas helt hos manteldjuren. Hos de flesta ryggradsdjur omvandlas eller ersätts den i huvudsak av ryggraden. Den finns hos vuxna däggdjur kvar som en liten rest i diskarna mellan ryggkotorna.
Se även
Svalgsträngsdjur
ryggen
Embryologi
Evolution
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 416
|
template < T> class Capsule
{
private:
vector<T> cdata;
SecByteBlock ckey;
SecByteBlock iv;
unsigned long times;
unsigned long base;
public:
bool save(string filepath);
bool load(string filepath);
vector<T> getCryptedData();
SecByteBlock getCryptedKey();
SecByteBlock getIV();
unsigned long getNumberOfOperations();
unsigned long getBase();
};
#endif //_CAPSULE_H
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,475
|
As anticipated, we both slept in this morning, but it was a little chilly so the heater went on and soon warmed us up – have we mentioned that we love our diesel heater? Apart from the chill, it seemed to be an alright start to the day so we went for a tour around Jugiong. Within the grounds of the free camp area there are some tennis courts, and right next door is the local swimming pool, a playground, and public toilets. There's a great old pub that is currently undergoing some quite extensive renovations, and an obviously very popular store and cafe. We stopped there for coffee but unfortunately it's closed on Tuesdays. Further along the road we spotted an interesting horse-drawn cart and there is also a small yard full of sculptures of mainly horses made from pieces of scrap metal. There is a small motel with what appears to be a nice dining room. The locals at Jugiong have done a great job in attracting visitors and we would recommend vanners and motorhomers should definitely add Jugiong to their list of places to visit.
As we headed back to the Hume Highway, it started to rain. It soon became very heavy rain and driving became very unpleasant with a good number of trucks in a hurry, and various other vehicles in even more of a hurry.
Not great weather for driving.
Eventually we turned off the highway into Yass, where we stopped for coffee and a hot snack, and a respite from the weather and traffic.
Refreshed and more relaxed we returned to the highway and eventually turned off onto the Barton Highway into Canberra. The weather wasn't much better at this stage.
The Barton Highway into Canberra usually looks much better than this!
We often stop at Murrumbateman but it didn't look too inviting today so we drove straight through.
Not even Murrumbateman looked inviting.
On the Canberra side of Murrumbateman there is a stretch of road that was under construction for quite a long time a couple of years ago. In our opinion it is a wasted opportunity – they could/should have straightened out the bend and made it twin lanes both ways, and much safer given the weather conditions in the area.
Fortunately the weather did improve a little as we entered Canberra and arrived at the Alivio Tourist Park. We came with a little trepidation, as we stayed here several times some years ago but it was quite run-down so we found somewhere else to stay when we visited Canberra. Katie told us that there had been some changes here so we thought that we'd give it another try.
After setting up and enjoying a nice hot lunch in Bertha, we ventured out for a walk around the park, then back home for afternoon coffee.
This place is under new management, has a new name and the changes are significant. There are now a number of ensuite sites and the main amenities block has had an amazing upgrade, including heating. What used to be a very old secondary amenities block has now become a magnificent camp kitchen. There is a large cooking area and another large room with colourful tables and chairs, both rooms with large TV screens, plus a great outside deck area. They even provide a herb garden for the use of park guests. If cooking isn't your thing, there is a very nice looking cafe/restaurant here which serves breakfast and dinner 7 days per week, plus they have take-away, all at reasonable prices.
Newly created Camp Kitchen – luxury camping at its best.
Alivio also has plenty of playgrounds and activities for kids, several bbq areas, modern cabins and views to the Black Mountain communication tower.
Views of Alivio Tourist Park.
We will be going to Katie's for dinner tonight and will be spending the next few days with her, so we will probably be off the air until we hit the road again.
We got off to a ridiculously early start this morning and were out of the gate in Bertha about 6.45am in order to miss the early morning traffic. In fact we slept in Bertha last night as part of our preparations – it's amazing how many things you think about that you might not have packed etc when you're actually "on the premises". Plus it was a good chance to make sure that the diesel heater worked. It did!
One of our concerns about leaving that early was that it might be foggy, but thankfully it was a nice clear morning and the traffic was flowing well so we made great progress. That was until about Wallan, when we hit dense fog and very limited visibility. We took advantage of the situation to stop at the servo and top Bertha up with diesel and to treat ourselves to a hot breakfast and a hot coffee.
Foggy and wet at Wallan.
The weather improved somewhat as we went further north along the Hume Freeway and we settled for just constant drizzle.
Weather is a bit better.
Next stop was at a rest stop near Euroa for a cup of coffee, then it was back on the road until we both felt that we needed a stretch and a walk so we turned off the Hume into Glenrowan. It was still drizzling lightly so we put on our coats and went for a wander. In a "small world" moment we bumped into our long-time friend Simon who I had worked with many moons ago and who we hadn't seen for several years. Simon and his wife were on their way home to Melbourne from Canberra where they had been visiting relatives and after a brief chat we headed off in our respective directions.
Ann's mother's name was Kelly!
We continued our slightly damp stroll around Glenrowan before stopping off at the Glenrowan Bakehouse for something hot for lunch. .
Back on the highway the weather improved and the sun made an appearance and by Wodonga I actually needed my sunnies. Travelling in sunny and clear weather is a lot more pleasant than driving in rain and drizzle so we bypassed Holbrook, where we would normally have stayed and continued towards Gundagai. It was also great to see green pastures and some healthy looking livestock. We stopped just north of Holbrook for another stretch before continuing on to our final stop at Jugiong free camp.
Healthy looking pastures and livestock.
We have stayed at Jugiong before and it is a beautiful spot. It's a bit cold at the moment and there aren't many people here but we have our heater and the music on and we're quite cosy.
Jugiong is a great place to stay.
Today has been an exceptional day and certainly not typical, as we very rarely travel anything like 556 kms in a day. However, an early start and quite decent weather, most of the time, have combined to create a long but successful day with the end result that we don't have nearly as far to travel tomorrow to visit our daughter Katie in Canberra.
We will sleep well tonight though!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,564
|
{"url":"https:\/\/www.metlink.org\/maths-for-planet-earth\/city-emission-levels\/","text":"Home \u00bb\u00a0Maths for Planet Earth \u00bb City Emission Levels\n\n# City Emission Levels\n\nThe emissions of a city from 2000 to 2012 are modelled by the equation\n\n$$p\\left( t \\right) = \\frac{1}{10}\\ln\\left( t + 1 \\right) \u2013 \\cos\\frac{t}{2} + \\frac{1}{10}t^{\\frac{3}{2}} + 199.3$$\n\n$0 \\leq t \\leq 12$\n\na) Show that the emissions reach a local maximum in the interval $$8.5 \\leq t \\leq 8.6$$\n\n[5 marks]\n\nThe emissions reach a local minimum between 9 and 11 years after the measurements began.\n\nb) Using the Newton-Raphson procedure once and taking $$t_{0} = 9.9$$ as a first approximation, find a second approximation of when the emissions reach a local minimum.\n\n[6 marks]\n\n## More Maths for Planet Earth\n\nA Level\n##### A Solar Sine Curve\nYou are given the equation [fleft( x right) = 5costheta \u2013 8sintheta] a) Express f(x) in the form (Rcos{(theta + alpha})) where (R >\nGCSE\n##### Climate Striking Students\nThe pie chart shows information about students going to a \u2018Fridays for Future\u2019 climate strike. 3360 more female students went to the strike than\nGCSE\n##### Surviving Species\nClimate change affects the habitats and environments of many species, some of which won\u2019t be able to adapt fast enough to survive in their\nA Level\n##### A Climate Aware Citizen\nA person decides in 2020 that they want to completely eradicate their carbon footprint in 20 months. Following this decision, they begin to use","date":"2023-02-08 00:12:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5030423402786255, \"perplexity\": 4168.363297981513}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500664.85\/warc\/CC-MAIN-20230207233330-20230208023330-00050.warc.gz\"}"}
| null | null |
Q: Display element if DOMs' scrollposition is greater then xyz I want to display a fixed element if the users scroll position is greater then a certain height. So I tried to do this:
$(document).scroll(function() {
if ($(document).scrollTop() > 700px) {
$('#trigger_lisa').css("display", "block");
}
})
To explain it, if the users focus is greater then 700px I want to display the element #trigger_lisa. The way I did the "700px" is weird, but I have no clue on how to do better.
Thanks for helping
A: 700px is a syntax error. "700px" would be valid but wrong. Just use 700.
However, if you want to hide it again when the user scrolls back, use toggle:
$(document).scroll(function() {
$('#trigger_lisa').toggle($(document).scrollTop() > 700);
});
body {
height: 10000px;
}
#trigger_lisa {
position: fixed;
top: 0;
background: blue;
height: 200px;
width: 200px;
display: none;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div id="trigger_lisa"></div>
A: Your code has incorrect notation, an "px" is not need for this case, "700" must be a number. In general case your code works:
$(document).scroll(function() {
if($(document).scrollTop()>700)
{
$('#trigger_lisa').css("display","block");
}
});
.wrapper {
height: 3000px;
}
#trigger_lisa {
display: none;
position: fixed;
top: 50%;
left: 50%;
margin-top: -10px;
margin-left: -50px;
width: 100px;
height: 20px;
background-color: #323232;
}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<div class="wrapper">
<div id="trigger_lisa"></div>
</div>
https://jsfiddle.net/Romanzhivo/p7bcjvm4/1/
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,060
|
'use strict';
var list = require('postcss').list;
var cssList = require('css-list');
var balancedMatch = require('balanced-match');
var functions = [
'calc',
'cubic-bezier',
'gradient',
'hsl',
'rect',
'rgb',
'rotate3d',
'scale',
'scale3d',
'transform3d',
'translate3d',
'url',
'var'
];
function optimise (decl) {
decl.value = cssList.map(decl.value, function (value, type) {
if (type !== 'func') {
return value;
}
var match = balancedMatch('(', ')', value);
if (!~functions.indexOf(match.pre)) {
return value;
}
return [
match.pre,
'(',
list.comma(match.body).map(function (value) {
return list.space(value).join(' ');
}).join(','),
')',
match.post
].join('');
});
}
module.exports = function () {
return function (css) {
functions.forEach(function (fn) {
css.eachDecl(optimise);
});
};
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 567
|
package Paws::ElastiCache::CreateCacheSecurityGroupResult;
use Moose;
has CacheSecurityGroup => (is => 'ro', isa => 'Paws::ElastiCache::CacheSecurityGroup');
has _request_id => (is => 'ro', isa => 'Str');
1;
### main pod documentation begin ###
=head1 NAME
Paws::ElastiCache::CreateCacheSecurityGroupResult
=head1 ATTRIBUTES
=head2 CacheSecurityGroup => L<Paws::ElastiCache::CacheSecurityGroup>
=head2 _request_id => Str
=cut
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,807
|
Q: one real analysis problem to prove the existence of a sequence and limit of a sequence Question: Suppose that S is a nonempty subset of R that is bounded above and put s = sup S.Show that there is a sequence (xn) such that xn ∈ S for all n and lim as n apporaches infinity xn = s.
Hi, I am a beginner for real analysis, the question about is the one that I am trying to do on my own, but I don't know how to analyze or solve it after I write the definitions out.
What I am thinking is that,
Given: S is a non-empty subset of R that is bounded above and s equals to the sup A
Prove: <1> there is a sequence (xn) such that xn belongs to s for all n,
<2> limit of xn as n approaches to infinity is s
Thoughts: since S is a nonempty subset of R that is bounded above, then there exists a number t such that t is greater or equal to s for all s belongs to S, so we know t is an upper bound of s.
Since s = sup S, then s is the least upper bound of S. since t is an upper bound for S, and s is the least upper bound, then t is also greater or equal to sup S. For the limit part, according to the definition of limit, we can write that |an-s|< epsilon.
I stop here because I don't how to do the following steps, I don't know how to use the information that I have so far to explain this questions.
Any help will be super appreciated
Thanks a lot
A: Let $s= \sup S$. We know that:
$$
\text{ for any $\varepsilon>0$, there is $x\in S$ such that $s-\varepsilon < x$}.
$$
Thus for each $n>0$ you can choose $x_n\in S$ such that
$$
s - \frac 1 n < x_n.
$$
Clearly, then, $s - \frac 1 n < x_n\le s$ for all $n>0$, so $\lim_n x_n$ exists and equals $s$.
A: Hint: Put $s=\sup S$. If $s\in S$, define $x_n\equiv s$ and we are done.
Otherwise $s\notin S$. For each positive integer $n$, select $x_n\in S$ with $x_n\in (s-\frac1n,s)$ (why is this possible?). Then by construction $x_n \to s$ and each $x_n\in S$.
A: There is an alternating definition for $\sup$ will be useful here, $s=\sup S$ iff 1) $s$ is an upper bound of $S$ and 2) for any $\epsilon>0$, $s-\epsilon$ is NOT an upper bound of $S$. (Try prove it using your definition)
Using above statement, for all $n\in\mathbb{N}$, $s-\frac{1}{n}$ is not an upper bound of $s$, which means $\exists x_n\in S$, such that $x_n>s-\frac{1}{n}$. Also since $s$ is an upper bound of $S$, $x_n\le s$. Hence we have $$s-\frac{1}{n}<x_n\le s,\forall n\in\mathbb{N}$$
Can you continue here to justify $x_n\to s$?
A: Let $s=\sup S$.
We know that:
$\forall ε>0 \;\exists x∈S/\;s−ε<x \;$(By definition of supremum)
$\Rightarrow$ Given $ε_{1} = 1\gt0,\,\exists x_{1}\in S/ s-1\lt x_{1}\le s \;$(By def. of sup)
And
$\Rightarrow$ Given $ε_{2} = \frac{1}{2}\gt0,\,\exists x_{2}\in S/ s-\frac{1}{2}\lt x_{2}\le s \;$(By def. of sup)
And
$\Rightarrow$ Given $ε_{3} = \frac{1}{3}\gt0,\,\exists x_{3}\in S/ s-\frac{1}{3}\lt x_{3}\le s \;$(By def. of sup)
In an iterative way:
$\Rightarrow$ Given $ε_{m} = \frac{1}{m}\gt0,\,\exists x_{m}\in S/ s-\frac{1}{m}\lt x_{m}\le s,\, m\in \mathbb{N},\;$(By def. of sup)
Then, choosing all $x_{m}$ and forming the sequence $(x_{n})_{n}$, we have:
$$s-\frac{1}{n}\lt x_{n}\le s,\forall n\in \mathbb{N},$$
$\Rightarrow \lim\limits_{n \to \infty} x_{n} = s$
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,336
|
\section{Introduction}
The Internet of Things (IoT) is the extension of the Internet to include connected embedded computing devices with the intention of transferring data over such a network without human-to-machine interaction \cite{madakam2015internet}. In a typical IoT set-up, data from various IoT sensors and edge devices can be produced and transmitted through gateways to the Cloud \cite{6192937}. The collected IoT data can be processed, analyzed, and stored by different cloud instances using Artificial Intelligence (AI) on cloud infrastructures. The conventional cloud-dominant centralized architecture has tremendous benefits for many IoT applications \cite{2018Brokering}, such as optimizing the cost and energy in pricing problems \cite{2018Economic, St2016Cloud} and maximizing the quality of service \cite{QUARATI2016403}. However, it also comes with inevitable shortcomings especially those involving local users' autonomous control over their data, control that usually requires the decision making process to be conducted at the device side or edge side for better security \cite{9301390} and privacy protection \cite{9163078}.
Specifically, we now consider a typical IoT scenario where data streams from various IoT devices can be transmitted to the Cloud and stored on a cloud database. Our initial observation is that most IoT devices start to transmit data at a fixed transmission frequency, and such a transmission frequency is typically set by default or pre-defined by the device manufacturer with limited options made available to users. However, some advanced IoT devices with edge intelligence, e.g. Raspberry Pis and the Jetson series toolkit from Nvidia, can now be programmed to promptly respond to changes in the external environment \cite{10.1117/12.2571307, 8230004}, and can also be deployed with deep learning algorithms to satisfy stringent low-latency transmission requirements for time-sensitive IoT applications \cite{9287960, 9289509}. This approach does not sufficiently cater for a practical situation where groups of IoT devices may work collaboratively with limited operational resources enforced by the external environment. In fact, implementing IoT devices in a resource-constrained environment may impose two types of design problems that are of particular interest to us in this paper: 1) how to determine an adaptive transmission frequency for each IoT device so that an overall utility of the group of devices can be maximized in response to the dynamic changes of the environment; and 2) how to ensure that different kinds of network resources can be better managed in a way that heterogeneous IoT devices can be engaged with the network in a secure, privacy-aware and plug-and-play manner.
To be specific, privacy-awareness in our context refers to the fact that the mapping between the utility and the transmission dynamics of a given IoT device should not be revealed to any unrelated devices, third-party gateway and untrusted cloud units or instances. We highlight that this design consideration is important in practice because if this information is revealed publicly it may be possible for an attacker to identify which IoT device is more vulnerable in a given system \cite{2020A}. In terms of network resources, we observe that the capacity of a cloud-based database instance is typically limited in storage space and it often comes with a capped time-based throughput for a given user. For instance, an IBM Cloudant database instance allows 1 GB of data storage with 10 writes/sec for its Lite Plan users, and 20 GB of data storage with 50 writes/sec for its Standard Plan users \cite{bienko2015ibm}. Given this scenario, it can be envisioned that a writing congestion event, e.g. a REST-API writing failure, can be triggered for a group of IoT devices if the Maximum Writing Frequency (MWF) of the data is not managed properly.
To solve this challenge, in this paper we propose a transmission frequency management system for IoT edge devices in a decentralized architecture with an anomaly detection mechanism. Thus the MWF can be managed optimally by a group of IoT devices and any abnormal writing frequency occurrences can be detected by the gateway. To carry out optimization, we assume that each IoT device is associated with a utility function with some concavity \cite{7106504, 2019Utility}, in a way that only the user of the device can specify. Here, the utility refers to how a user can practically benefit from a given Data Flow Writing Frequency (DFWF). For instance, a utility function can easily describe the accuracy of a trained model with respect to DFWF of a given IoT device for an Edge AI type of IoT applications \cite{LV202190}. Furthermore, as previously mentioned, such a utility function may also potentially reflect the significance or vulnerability of an IoT device in a specific scenario. For instance, a faster transmission frequency of a webcam in a bank system may be more desirable, i.e., have higher utility, especially in an emergency, than that of a $CO_2$ detector.
With this idea in mind, our main objective in this paper is to maximize the overall utility of the group of IoT devices given the predefined and limited MWF and storage capacity of the database. We will show that the presented challenge can be formulated as a concave optimization problem with constraints. This problem will then be solved using the well-known Alternating Direction Method of Multipliers (ADMM) algorithm \cite{boyd2011distributed} in a decentralized optimization framework where each utility function is locally defined on the edge device and will not be revealed to any unrelated devices and untrusted management platforms, such as other smart gateways and cloud units/instances. The proposed solution aims to provide flexibility in data transmission for IoT systems and applications especially in resource-constrained environments. As we shall see, the designed system is fully autonomous and can be easily deployed to optimally manage various IoT transmission frequencies with anomaly detection capabilities.
The remainder of this paper is organized as follows. In Section II, the architecture of the proposed system is presented. The optimization problem is formulated in Section III and its implementation is discussed in Section IV. Establishing and configuring real-world simulations and their results are discussed in Section V. The anomaly detection mechanism is demonstrated in Section VI. Finally, a conclusion from the current research and potential future research directions are provided in Section VII.
\section{System Architecture}\label{sec:System_model}
Our proposed system architecture is illustrated in Fig. \ref{fig:architecture1}. The system consists of four main components, including IoT edge devices, gateways, a cloud platform and users. The main functionalities of each component are described as follows:
\begin{itemize}
\item[1.] \textbf{IoT devices}: sensors/devices connected to a gateway, having the capabilities of defining utility functions and the ability to solve a local optimization problem in a decentralized manner.
\item[2.] \textbf{Gateway}: collects data from IoT devices/sensors, passing data to the Cloud, and conducts basic data processing tasks including anomaly detection to protect and inform users.
\item[3.] \textbf{Cloud platform}: a central hub for data analysis, monitoring and storage.
\item[4.] \textbf{Users}: the owner of the IoT devices who wishes to use the IoT devices in some collaborative application scenarios.
\end{itemize}
In the proposed system, a gateway starts by waiting for connection from IoT devices. When an IoT device initially connects to the gateway, the decentralized optimization algorithm is activated to calculate the optimal transmission frequencies for all connected devices whilst taking account of the resource constraints of the system. After that, the gateway starts to collect data streams from all IoT devices after the transmission frequencies are established. Finally, data collected by the gateway is transmitted to the cloud platform for the purpose of data storage and further analysis of the IoT devices if specifically requested by the users.
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.5\textwidth, height=3in]{fig1_new.png}
\caption{Schematic diagram of the system architecture.}
\label{fig:architecture1}
\vspace{-0.1in}
\end{figure}
\section{Problem Statement}
We now present the specific problem statement to be solved in this paper. A user wishes to determine the optimal DFWF of every IoT edge device so that the overall utility of the whole group can be maximised, given $N$ number of devices connected to the gateway, the utility $f_i(x_i)$ of the $i^{th}$ device with current DFWF $x_i$, MWF $c$, total data storage available per received data packet, $d$, and $a_i$ the data size required for the $i$'th device per writing request.
Mathematically, this problem can be formulated as follows:
\begin{equation} \label{eq}
\begin{gathered}
\underset{x_1, x_2, \ldots, x_{N}}{\max} \quad
\sum\limits_{i = 1}^{N} f_{i}\left(x_i \right),\\
{\text{such that}} ~
\sum\limits_{i = 1}^{N} x_i \leq c, ~ \sum\limits_{i = 1}^{N} a_i x_i \leq d, ~ x_i \geq 0 \\
\end{gathered}
\end{equation}
We shall only require that each utility function $f_i(x_i)$ can be modelled as a continuously differentiable, non-decreasing, strictly concave function, which is a common assumption for modelling the utility of internet data traffic \cite{srikant2004mathematics}. For example, utility functions may be modelled as a cluster of negative quadratic functions.
\section{System Implementation}
The classic ADMM algorithm proposed in \cite{boyd2011distributed} is particularly suited to solving the formulated optimization problem \eqref{eq} as the problem can be converted to a convex optimization problem with convex constraints. Here we briefly recall the ADMM algorithm for solving \eqref{eq}, which is shown in Algorithm \ref{ADMM}, where $x$ and $z$ are
updated in an alternating fashion and $u$ is a dual update variable.
\begin{algorithm}[htbp]
\caption{ADMM Algorithm}
\begin{algorithmic}[1]
\State $\textbf{x}^{k+1}:= \argmaxB_\textbf{x} ( \sum\limits_{i = 1}^{N} f_i(x_i) + (\rho/2) || \textbf{x} - \textbf{z}^{k} + \textbf{u}^{k} ||_{2}^{2})$
\State $\textbf{z}^{k+1}:= \Pi_{\mathcal{C}} (\textbf{x}^{k+1} + \textbf{u}^{k})$
\State $\textbf{u}^{k+1}:= \textbf{u}^{k} + \textbf{x}^{k+1} - \textbf{z}^{k+1}$
\end{algorithmic}
\label{ADMM}
\end{algorithm}
Note that the above ADMM algorithm can be implemented in a decentralized manner as our objective function is separable which implies that both $\textbf{x}$ and $\textbf{u}$ vector updates in the algorithm can be implemented in parallel. Finally, the $\textbf{z}$ update depends on inputs from both $\textbf{x}$ and $\textbf{u}$. Given these inputs, the projection operator $\Pi_{\mathcal{C}}$ projects the resulting vector to the constrained convex space $\mathcal{C}$. Thus, the $\textbf{z}$ update needs to be implemented on a gateway. Note that $\rho$ is the augmented Lagrangian parameter and we take $\rho = 1.0$, being equivalent to a $\rho/2$ step size in $x$ update. The ADMM algorithm in its decentralized format is shown in Algorithm \ref{DecentralisedADMM}.
\begin{algorithm}[htbp]
\caption{Decentralized ADMM Algorithm}
\begin{algorithmic}[1]
\State ${x_i}^{k+1}:= \argmaxB_{x_i} ( f_i(x_i) + (\rho/2) || {x_i}^{k} - {z_i}^{k} + {u_i}^{k} ||_{2}^{2})$
\State $\textbf{z}^{k+1}:= \Pi_{\mathcal{C}} (\textbf{x}^{k+1} + \textbf{u}^{k})$
\State ${u_i}^{k+1}:= {u_i}^{k} + {x_i}^{k+1} - {z_i}^{k+1}$
\end{algorithmic}
\label{DecentralisedADMM}
\end{algorithm}
With this algorithm in mind, the proposed system can be implemented in the following steps, which are illustrated in Fig. \ref{fig:flow_chart}.
\begin{itemize}
\item[S1:] During the initialization stage, a user needs to specify some parameters before running the
algorithm. This includes $N$, $c$, $d$, $a_i$ and the utility function $f_i(x_i)$ of each device.
\item[S2:] When the initialization step finishes, the ADMM algorithm will be implemented in an iterative manner on the edge IoT devices to determine the optimal DFWF by computing the optimal ${x_i}^{k+1}$ as per Algorithm \ref{DecentralisedADMM}.
\item[S3:] During each iteration, the gateway gathers all the optimal ${x_i}^{k+1}$ from all devices, calculates and broadcasts the updated $\textbf{z}$ value to local edge devices. Upon receiving the $\textbf{z}$ value, each edge device updates ${u_i}^{k+1}$ correspondingly.
\item [S4:] If there are any resource changes during runtime, the algorithm can dynamically capture the changes to recalculate the optimal solution given the new context.
\item[S5:] When the algorithm converges, the optimal DFWF will be set by each device, and these devices
can then start pushing data to the cloud accordingly.
\item[S6:] The gateway keeps monitoring the data injection and detects if an anomaly happens on any of the transmission frequencies. If so, the user will be alerted and the optimal solution will be recalculated and reset after the anomaly has been remedied.
\item[S7:] Finally, all transmitted data streams will be stored on the cloud and an authorised user can leverage the stored data for visualization and analysis by making a request.
\end{itemize}
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.4\textwidth]{fig1_flow.png}
\caption{System implementation flowchart. }
\label{fig:flow_chart}
\vspace{-0.3in}
\end{figure}
\begin{figure}[ht]
\vspace{0.1in}
\centering
\includegraphics[width=0.4\textwidth]{fig_devices.jpg}
\caption{System simulation device setup.}
\label{fig:devices}
\vspace{-0.1in}
\end{figure}
\section{Simulation}
This section presents simulation results to evaluate the performance of the proposed system. As shown in Fig. \ref{fig:devices}, the system consists of a laptop as the central node (i.e., as a smart gateway in this work), three IoT devices (Jetson AGX Xavier, Nvidia), and a router for the communication between the gateway and the IoT devices. Typically, IoT s connect to router using wireless. However, in our setup, since the IoT devices do not have the capability of wireless transmission, they transmit data to the router via cables, and the laptop communicates with router wirelessly. Decentralized ADMM optimization and data transmission are implemented on both the gateway the and devices via socket programming. System parameters for the simulations are set as $N = 3$, $c = 10$, $d = 15$, $a_1 = 2$, $a_2 = 3$, and $a_3 = 5$. The utility functions in this simulation are presented in Table \ref{UtilityF} and have the characteristics previously specified to successfully apply the ADMM algorithm. We simulate the system in two scenarios: a) resources are sufficient for the data transmission request, and b) resources are insufficient for the data transmission request from all devices. For each device $i$, its transmission frequency $x_i$ is defined as data is transmitted $x_i$ times per second. In particular, $x_i = 0$ implies that the $i^{th}$ device is not transmitting data. Thus, for each device, an extra constraint, $x_i >= \gamma_i$ applies to indicate the minimum transmission frequency. For simplicity, we set $\gamma_1 = \gamma_2 = \gamma_3 = 1$ in our simulation.
It is worth noting that the gateway is not able to access the utility function of each device in order to cater for privacy concerns, and also that the transmission frequency of each device is calculated locally and not explicitly exposed to the gateway. However, a DFWF may be estimated by the gateway by evaluating the time intervals of the consecutively received data packets and an averaged DFWF is calculated over 300 data packets after the optimal DFWF is assigned.
\renewcommand\arraystretch{1.5}
\begin{table}[ht!]
\centering
\caption{Utility Functions}\label{UtilityF}
\vspace{-0.2cm}
\begin{tabular}{|c|c|}
\hline
\textbf{Device index} &
\textbf{Utility Functions} \\ \hline
1 & $f_1(x_1) = -(x_1+9)^2 - x_1^3 + 900$ \\ \hline
2 & $f_2(x_2) = -(x_2-4)^2 + 500$ \\ \hline
3 & $f_3(x_3) = -(2x_3+3)^2 - x_3^3 + 110$ \\ \hline
\end{tabular
\end{table}
\subsection{System with sufficient resources}
In this scenario, only device $1$ and device $2$ are connected to the gateway (i.e., parameter $N = 2$) and all other system parameters are kept by default, i.e., $c = 10$, $d = 15$, $a_1 = 2$, $a_2 = 3$ with the associated utility functions $f_1(x_1)$ and $f_2(x_2)$ shown in Table \ref{UtilityF}. With these parameters, the theoretical optimal results of the ADMM implementation are $x_1^{*} = 1$ and $x_2^{*} = 4$ for optimization problem (1). This result implies that the gateway expects to receive $1$ and $4$ data packet(s) per second from device $1$ and $2$ on average. In this setup, the capacity provided by the system is sufficient since $x_1^{*} + x_2^{*} < c$ and $a_1 * x_1^{*} + a_2 * x_2^{*} <d$. With the decentralized ADMM implemented using the simulation setup, the optimization results and resource consumption of the system are illustrated in Fig. \ref{fig2a} and Fig. \ref{fig2b}, respectively. In particular, Fig. \ref{fig2a} shows the evolution of the calculated DFWF for both devices as estimated by the gateway. The DFWFs are estimated along with the number of received data packets, indicated by the red and green lines for device $1$ and device $2$, respectively. Concretely, our results show that the estimated DFWFs are $0.9984\ Hz$ and $3.9318\ Hz$ for device $1$ and $2$, respectively, as shown in Table \ref{simulationA}. The estimated DFWFs are just slightly below the the theoretical optimal DFWFs, indicated by the dotted-line in Fig. \ref{fig2a}. The decay of the DFWF is accounted for by the internet speed, while the communication between the gateway and the devices is based on a router, resulting in a $1.6\ ms$ and a $4.3\ ms$ delay for device $1$ and device $2$, respectively. Meanwhile, we find that the fluctuation of the estimated DFWFs is caused by the data jamming when the gateway is receiving data packets with high writing frequency. Fig. \ref{fig2b} shows the sum of DFWFs as well as the size of total data packets of all connected devices per second transmitted to the gateway. The dotted-line indicates the maximum total DFWF (in red) and received data size (in green) for each data packet. Since the system can provide sufficient resources, the total DFWF and the writing data size has not reached the resource boundary after the transmission frequencies are optimized, indicating that the proposed system is robust as long as the system resources are sufficient for this specific data transmission task.
\begin{table}[!h]
\centering
\caption{Simulation results (average)}\label{simulationA}
\vspace{-0.2cm}
\begin{tabular}{|c|c|c|}
\hline
\textbf{ } & \textbf{DFWF (Hz)} & \textbf{DFWF (Hz)} \\ \hline
\textbf{ } & \textbf{Device 1} & \textbf{Device 2} \\ \hline
\textbf{Theoretical} & 1.0000 & 4.0000 \\ \hline
\textbf{Actual} & 0.9984 & 3.9318 \\ \hline
\textbf{Absolute Error} & 0.0016 & 0.0682 \\ \hline
\end{tabular
\end{table}
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.42\textwidth]{fig2a.png}
\caption{Decentralized optimization process of transmission frequency for Device 1 and Device 2.}
\label{fig2a}
\vspace{-0.1in}
\end{figure}
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.42\textwidth]{fig2b.png}
\caption{System resources consumption.}
\label{fig2b}
\vspace{-0.1in}
\end{figure}
\subsection{System with insufficient resources}
In this scenario, after device $1$ and device $2$ have connected to the gateway and the optimized transmission frequencies have been calculated, a new device, device $3$, connects to the gateway and the timing of connection is recorded. Given $N = 3$, $c = 10$, $d = 15$, $a_1 = 2$, $a_2 = 3$, $a_3 = 5$ and the corresponding utility functions $f_1(x_1)$, $f_2(x_2)$, $f_3(x_3)$ reported in Table \ref{UtilityF}, the theoretical optimal results of the ADMM implementation are calculated as $x_1^{*} = 1.00$, $x_2^{*} = 2.66$ and $x_3^{*} = 1.00$ for optimization problem (1). This result implies that, on average, the gateway expects to receive $1$, $2.66$ and $1$ data packet(s) per second from devices $1$, $2$, and $3$ respectively.
Based on the simulation platform, the decentralized optimization process and system resource usage are shown in Fig. \ref{fig4} and Fig. \ref{fig5} in the scenario of insufficient resources. We note that before the connection of device $3$, device $1$ and device $2$ transmit their data packets under the corresponding optimized transmission frequencies exactly as described in the first scenario with sufficient resources. As shown in Fig. \ref{fig4}, after the device $3$ connects to the system (indicated by the red arrow), the DFWF of device $2$ is readjusted and converges to a new optimal value. The DFWF of device $1$ remains unchanged since the recalculated optimal result equals the previously assigned DFWF before the connection of device $3$. After the decentralized ADMM solution is found for device $3$ (indicated by the magenta circle), device $3$ pushes data packets to the gateway using its optimal DFWF. After all three devices are transmitting data steadily (i.e., after the magenta circle), our results show that the estimated DFWFs are $0.9984\ Hz$, $2.6410\ Hz$ and $0.9984\ Hz$ for device $1$, $2$, and $3$, respectively, which are reported in Table \ref{simulationB}. Again, these estimated DFWFs are slightly below the theoretical optimal DFWFs, indicated by dotted-lines, reflecting time delays of $1.6\ ms$, $3.7\ ms$ and $1.6\ ms$ for devices $1$, $2$, $3$, respectively during their transmissions.
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.41\textwidth]{fig4.png}
\caption{Decentralized optimization process of transmission frequencies for Device 1, Device 2 and Device 3.}
\label{fig4}
\vspace{-0.1in}
\end{figure}
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.41\textwidth]{fig5.png}
\caption{System resource consumption in boundary conditions. The magenta cycle shows the connection point of device 3.}
\label{fig5}
\vspace{-0.1in}
\end{figure}
After the optimal transmission frequencies are established, as shown in Fig. \ref{fig5}, device $3$ starts to push data (marked by the magenta circle) and the total writing data size reaches the level of the system resource boundary immediately. This indicates that the proposed system is able to reallocate the system resources to finish the data transmission task effectively using the ADMM approach. Finally, for comparison purposes, we evaluate the overall utility under the ADMM-optimized DFWFs, with non-optimized average distributed DFWFs (i.e., $x_i = c/N$), and non-optimized proportionally distributed DFWFs (i.e., $x_i = (a_i*c)/\sum a_i$) as two baselines given the same MWF $c$. The results shown in Table \ref{UtilityValue} find that the utility under ADMM-optimized DFWFs achieves the largest value, which demonstrates that the proposed system obtains the best result compared to other trivial system setups that have not undergone any optimization process.
\begin{table}[!h]
\centering
\caption{Simulation results (average)}\label{simulationB}
\vspace{-0.2cm}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{ } & \textbf{DFWF (Hz)} & \textbf{DFWF (Hz)} & \textbf{DFWF (Hz)} \\ \hline
\textbf{ } & \textbf{Device 1} & \textbf{Device 2} & \textbf{Device 3} \\ \hline
\textbf{Theoretical} & 1.0000 & 2.6667 & 1.0000 \\ \hline
\textbf{Actual} & 0.9984 & 2.6410 & 0.9984 \\ \hline
\textbf{Absolute Error} & 0.0016 & 0.0257 & 0.0016 \\ \hline
\end{tabular
\vspace{-0.2in}
\end{table}
\begin{table}[ht!]
\centering
\caption{Utility Evaluation}\label{UtilityValue}
\vspace{-0.2cm}
\begin{tabular}{|c|c|}
\hline
\textbf{DFWFs} &
\textbf{Utility Value} \\ \hline
ADMM optimized & 1381.22 \\ \hline
Average distributed & 1190.35 \\ \hline
Proportionably distributed & 1086.00 \\ \hline
\end{tabular
\vspace{-0.1in}
\end{table}
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.41\textwidth]{fig6.png}
\caption{Monitoring of $z_{2}$ and $x_{2}$ during the decentralized optimization process for Device 2.}
\label{fig6}
\vspace{-0.2in}
\end{figure}
\section{Abnormal transmission frequency detection}
While the transmission frequencies are determined and allocated by the system, all the devices push data steadily with their specified DTWF. However, an IoT device may transmit data with an unexpected transmission frequency when an edge device intensively manipulates its converged DFWF to a different, more desired, state. Alternatively, a malicious node in the network may tamper with the converged DFWF of an edge device intentionally. In this section, manipulation of transmission frequencies is briefly discussed for the examination of abnormal transmission frequency detection at the side of gateway.
According to the fundamental mechanism of the ADMM algorithm, the gateway only has access to $\textbf{z}$. Since $\textbf{x}$ achieves convergence to $\textbf{z}$ eventually, as a specific example (i.e., $z_{2}$ and $x_{2}$) shown in Fig. \ref{fig6}, we argue that the gateway is able to detect the anomaly of $\textbf{x}$ during the whole transmission process based on its knowledge of the latest value of $\textbf{z}$. Specifically, this detection process can be described in the following three steps:
\begin{itemize}
\item[S1:] Gateway accesses the value of $z_{i}$ for each device.
\item[S2:] Gateway estimates the DFWF (i.e., the converged value of $x_{i}$) for each device according to the received time-stamped data flow.
\item[S3:] If the estimated DFWF is significantly different to the reference value of $z_{i}$ (i.e., $|z_{i}-x_{i}| \geq \delta$, where $\delta$ is a threshold depending on the network delay), the optimal transmission frequency can regarded as anomalous and as being manipulated.
\end{itemize}
\section{Conclusion}\label{sec:Conclusion}
In this paper, we propose a novel transmission frequency management system for IoT edge devices. This innovative system is able to find the optimal transmission frequency for each IoT device in a resource-constrained, privacy-aware environment. In addition, it is able to detect the connection of a new device and determine and reassign the new optimal transmission frequencies automatically. Our simulation results show that the proposed system is effective in real-world scenarios, with a high accuracy for estimation of transmission frequency in a low-latency ($5\ ms$) router-based experimental IoT network. Finally, we have introduced an abnormal frequency detection mechanism for simple scenarios where the converged DFWF may have been manipulated. Our results show that the ADMM-based algorithm can successfully identify this type of undesirable anomaly during real-time IoT data transmission.
As part of our future work, different kinds of abnormal frequency detection mechanisms will be further investigated by taking account of more complicated application scenarios where malicious manipulations of utility functions of devices are considered. We will also investigate the dynamics of system behaviours when the utility functions are non-smooth and non-convex.
\section*{Acknowledgement}
This work was support in part by Science Foundation Ireland grant SFI/12/RC/2289\_P2, and in part by Postgraduate Research Scholarship from the Faculty of Engineering and Computing at Dublin City University.
\bibliographystyle{ieeetran}
\section{Introduction}
The Internet of Things (IoT) is the extension of the Internet to include connected embedded computing devices with the intention of transferring data over such a network without human-to-machine interaction \cite{madakam2015internet}. In a typical IoT set-up, data from various IoT sensors and edge devices can be produced and transmitted through gateways to the Cloud \cite{6192937}. The collected IoT data can be processed, analyzed, and stored by different cloud instances using Artificial Intelligence (AI) on cloud infrastructures. The conventional cloud-dominant centralized architecture has tremendous benefits for many IoT applications \cite{2018Brokering}, such as optimizing the cost and energy in pricing problems \cite{2018Economic, St2016Cloud} and maximizing the quality of service \cite{QUARATI2016403}. However, it also comes with inevitable shortcomings especially those involving local users' autonomous control over their data, control that usually requires the decision making process to be conducted at the device side or edge side for better security \cite{9301390} and privacy protection \cite{9163078}.
Specifically, we now consider a typical IoT scenario where data streams from various IoT devices can be transmitted to the Cloud and stored on a cloud database. Our initial observation is that most IoT devices start to transmit data at a fixed transmission frequency, and such a transmission frequency is typically set by default or pre-defined by the device manufacturer with limited options made available to users. However, some advanced IoT devices with edge intelligence, e.g. Raspberry Pis and the Jetson series toolkit from Nvidia, can now be programmed to promptly respond to changes in the external environment \cite{10.1117/12.2571307, 8230004}, and can also be deployed with deep learning algorithms to satisfy stringent low-latency transmission requirements for time-sensitive IoT applications \cite{9287960, 9289509}. This approach does not sufficiently cater for a practical situation where groups of IoT devices may work collaboratively with limited operational resources enforced by the external environment. In fact, implementing IoT devices in a resource-constrained environment may impose two types of design problems that are of particular interest to us in this paper: 1) how to determine an adaptive transmission frequency for each IoT device so that an overall utility of the group of devices can be maximized in response to the dynamic changes of the environment; and 2) how to ensure that different kinds of network resources can be better managed in a way that heterogeneous IoT devices can be engaged with the network in a secure, privacy-aware and plug-and-play manner.
To be specific, privacy-awareness in our context refers to the fact that the mapping between the utility and the transmission dynamics of a given IoT device should not be revealed to any unrelated devices, third-party gateway and untrusted cloud units or instances. We highlight that this design consideration is important in practice because if this information is revealed publicly it may be possible for an attacker to identify which IoT device is more vulnerable in a given system \cite{2020A}. In terms of network resources, we observe that the capacity of a cloud-based database instance is typically limited in storage space and it often comes with a capped time-based throughput for a given user. For instance, an IBM Cloudant database instance allows 1 GB of data storage with 10 writes/sec for its Lite Plan users, and 20 GB of data storage with 50 writes/sec for its Standard Plan users \cite{bienko2015ibm}. Given this scenario, it can be envisioned that a writing congestion event, e.g. a REST-API writing failure, can be triggered for a group of IoT devices if the Maximum Writing Frequency (MWF) of the data is not managed properly.
To solve this challenge, in this paper we propose a transmission frequency management system for IoT edge devices in a decentralized architecture with an anomaly detection mechanism. Thus the MWF can be managed optimally by a group of IoT devices and any abnormal writing frequency occurrences can be detected by the gateway. To carry out optimization, we assume that each IoT device is associated with a utility function with some concavity \cite{7106504, 2019Utility}, in a way that only the user of the device can specify. Here, the utility refers to how a user can practically benefit from a given Data Flow Writing Frequency (DFWF). For instance, a utility function can easily describe the accuracy of a trained model with respect to DFWF of a given IoT device for an Edge AI type of IoT applications \cite{LV202190}. Furthermore, as previously mentioned, such a utility function may also potentially reflect the significance or vulnerability of an IoT device in a specific scenario. For instance, a faster transmission frequency of a webcam in a bank system may be more desirable, i.e., have higher utility, especially in an emergency, than that of a $CO_2$ detector.
With this idea in mind, our main objective in this paper is to maximize the overall utility of the group of IoT devices given the predefined and limited MWF and storage capacity of the database. We will show that the presented challenge can be formulated as a concave optimization problem with constraints. This problem will then be solved using the well-known Alternating Direction Method of Multipliers (ADMM) algorithm \cite{boyd2011distributed} in a decentralized optimization framework where each utility function is locally defined on the edge device and will not be revealed to any unrelated devices and untrusted management platforms, such as other smart gateways and cloud units/instances. The proposed solution aims to provide flexibility in data transmission for IoT systems and applications especially in resource-constrained environments. As we shall see, the designed system is fully autonomous and can be easily deployed to optimally manage various IoT transmission frequencies with anomaly detection capabilities.
The remainder of this paper is organized as follows. In Section II, the architecture of the proposed system is presented. The optimization problem is formulated in Section III and its implementation is discussed in Section IV. Establishing and configuring real-world simulations and their results are discussed in Section V. The anomaly detection mechanism is demonstrated in Section VI. Finally, a conclusion from the current research and potential future research directions are provided in Section VII.
\section{System Architecture}\label{sec:System_model}
Our proposed system architecture is illustrated in Fig. \ref{fig:architecture1}. The system consists of four main components, including IoT edge devices, gateways, a cloud platform and users. The main functionalities of each component are described as follows:
\begin{itemize}
\item[1.] \textbf{IoT devices}: sensors/devices connected to a gateway, having the capabilities of defining utility functions and the ability to solve a local optimization problem in a decentralized manner.
\item[2.] \textbf{Gateway}: collects data from IoT devices/sensors, passing data to the Cloud, and conducts basic data processing tasks including anomaly detection to protect and inform users.
\item[3.] \textbf{Cloud platform}: a central hub for data analysis, monitoring and storage.
\item[4.] \textbf{Users}: the owner of the IoT devices who wishes to use the IoT devices in some collaborative application scenarios.
\end{itemize}
In the proposed system, a gateway starts by waiting for connection from IoT devices. When an IoT device initially connects to the gateway, the decentralized optimization algorithm is activated to calculate the optimal transmission frequencies for all connected devices whilst taking account of the resource constraints of the system. After that, the gateway starts to collect data streams from all IoT devices after the transmission frequencies are established. Finally, data collected by the gateway is transmitted to the cloud platform for the purpose of data storage and further analysis of the IoT devices if specifically requested by the users.
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.5\textwidth, height=3in]{fig1_new.png}
\caption{Schematic diagram of the system architecture.}
\label{fig:architecture1}
\vspace{-0.1in}
\end{figure}
\section{Problem Statement}
We now present the specific problem statement to be solved in this paper. A user wishes to determine the optimal DFWF of every IoT edge device so that the overall utility of the whole group can be maximised, given $N$ number of devices connected to the gateway, the utility $f_i(x_i)$ of the $i^{th}$ device with current DFWF $x_i$, MWF $c$, total data storage available per received data packet, $d$, and $a_i$ the data size required for the $i$'th device per writing request.
Mathematically, this problem can be formulated as follows:
\begin{equation} \label{eq}
\begin{gathered}
\underset{x_1, x_2, \ldots, x_{N}}{\max} \quad
\sum\limits_{i = 1}^{N} f_{i}\left(x_i \right),\\
{\text{such that}} ~
\sum\limits_{i = 1}^{N} x_i \leq c, ~ \sum\limits_{i = 1}^{N} a_i x_i \leq d, ~ x_i \geq 0 \\
\end{gathered}
\end{equation}
We shall only require that each utility function $f_i(x_i)$ can be modelled as a continuously differentiable, non-decreasing, strictly concave function, which is a common assumption for modelling the utility of internet data traffic \cite{srikant2004mathematics}. For example, utility functions may be modelled as a cluster of negative quadratic functions.
\section{System Implementation}
The classic ADMM algorithm proposed in \cite{boyd2011distributed} is particularly suited to solving the formulated optimization problem \eqref{eq} as the problem can be converted to a convex optimization problem with convex constraints. Here we briefly recall the ADMM algorithm for solving \eqref{eq}, which is shown in Algorithm \ref{ADMM}, where $x$ and $z$ are
updated in an alternating fashion and $u$ is a dual update variable.
\begin{algorithm}[htbp]
\caption{ADMM Algorithm}
\begin{algorithmic}[1]
\State $\textbf{x}^{k+1}:= \argmaxB_\textbf{x} ( \sum\limits_{i = 1}^{N} f_i(x_i) + (\rho/2) || \textbf{x} - \textbf{z}^{k} + \textbf{u}^{k} ||_{2}^{2})$
\State $\textbf{z}^{k+1}:= \Pi_{\mathcal{C}} (\textbf{x}^{k+1} + \textbf{u}^{k})$
\State $\textbf{u}^{k+1}:= \textbf{u}^{k} + \textbf{x}^{k+1} - \textbf{z}^{k+1}$
\end{algorithmic}
\label{ADMM}
\end{algorithm}
Note that the above ADMM algorithm can be implemented in a decentralized manner as our objective function is separable which implies that both $\textbf{x}$ and $\textbf{u}$ vector updates in the algorithm can be implemented in parallel. Finally, the $\textbf{z}$ update depends on inputs from both $\textbf{x}$ and $\textbf{u}$. Given these inputs, the projection operator $\Pi_{\mathcal{C}}$ projects the resulting vector to the constrained convex space $\mathcal{C}$. Thus, the $\textbf{z}$ update needs to be implemented on a gateway. Note that $\rho$ is the augmented Lagrangian parameter and we take $\rho = 1.0$, being equivalent to a $\rho/2$ step size in $x$ update. The ADMM algorithm in its decentralized format is shown in Algorithm \ref{DecentralisedADMM}.
\begin{algorithm}[htbp]
\caption{Decentralized ADMM Algorithm}
\begin{algorithmic}[1]
\State ${x_i}^{k+1}:= \argmaxB_{x_i} ( f_i(x_i) + (\rho/2) || {x_i}^{k} - {z_i}^{k} + {u_i}^{k} ||_{2}^{2})$
\State $\textbf{z}^{k+1}:= \Pi_{\mathcal{C}} (\textbf{x}^{k+1} + \textbf{u}^{k})$
\State ${u_i}^{k+1}:= {u_i}^{k} + {x_i}^{k+1} - {z_i}^{k+1}$
\end{algorithmic}
\label{DecentralisedADMM}
\end{algorithm}
With this algorithm in mind, the proposed system can be implemented in the following steps, which are illustrated in Fig. \ref{fig:flow_chart}.
\begin{itemize}
\item[S1:] During the initialization stage, a user needs to specify some parameters before running the
algorithm. This includes $N$, $c$, $d$, $a_i$ and the utility function $f_i(x_i)$ of each device.
\item[S2:] When the initialization step finishes, the ADMM algorithm will be implemented in an iterative manner on the edge IoT devices to determine the optimal DFWF by computing the optimal ${x_i}^{k+1}$ as per Algorithm \ref{DecentralisedADMM}.
\item[S3:] During each iteration, the gateway gathers all the optimal ${x_i}^{k+1}$ from all devices, calculates and broadcasts the updated $\textbf{z}$ value to local edge devices. Upon receiving the $\textbf{z}$ value, each edge device updates ${u_i}^{k+1}$ correspondingly.
\item [S4:] If there are any resource changes during runtime, the algorithm can dynamically capture the changes to recalculate the optimal solution given the new context.
\item[S5:] When the algorithm converges, the optimal DFWF will be set by each device, and these devices
can then start pushing data to the cloud accordingly.
\item[S6:] The gateway keeps monitoring the data injection and detects if an anomaly happens on any of the transmission frequencies. If so, the user will be alerted and the optimal solution will be recalculated and reset after the anomaly has been remedied.
\item[S7:] Finally, all transmitted data streams will be stored on the cloud and an authorised user can leverage the stored data for visualization and analysis by making a request.
\end{itemize}
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.4\textwidth]{fig1_flow.png}
\caption{System implementation flowchart. }
\label{fig:flow_chart}
\vspace{-0.3in}
\end{figure}
\begin{figure}[ht]
\vspace{0.1in}
\centering
\includegraphics[width=0.4\textwidth]{fig_devices.jpg}
\caption{System simulation device setup.}
\label{fig:devices}
\vspace{-0.1in}
\end{figure}
\section{Simulation}
This section presents simulation results to evaluate the performance of the proposed system. As shown in Fig. \ref{fig:devices}, the system consists of a laptop as the central node (i.e., as a smart gateway in this work), three IoT devices (Jetson AGX Xavier, Nvidia), and a router for the communication between the gateway and the IoT devices. Typically, IoT s connect to router using wireless. However, in our setup, since the IoT devices do not have the capability of wireless transmission, they transmit data to the router via cables, and the laptop communicates with router wirelessly. Decentralized ADMM optimization and data transmission are implemented on both the gateway the and devices via socket programming. System parameters for the simulations are set as $N = 3$, $c = 10$, $d = 15$, $a_1 = 2$, $a_2 = 3$, and $a_3 = 5$. The utility functions in this simulation are presented in Table \ref{UtilityF} and have the characteristics previously specified to successfully apply the ADMM algorithm. We simulate the system in two scenarios: a) resources are sufficient for the data transmission request, and b) resources are insufficient for the data transmission request from all devices. For each device $i$, its transmission frequency $x_i$ is defined as data is transmitted $x_i$ times per second. In particular, $x_i = 0$ implies that the $i^{th}$ device is not transmitting data. Thus, for each device, an extra constraint, $x_i >= \gamma_i$ applies to indicate the minimum transmission frequency. For simplicity, we set $\gamma_1 = \gamma_2 = \gamma_3 = 1$ in our simulation.
It is worth noting that the gateway is not able to access the utility function of each device in order to cater for privacy concerns, and also that the transmission frequency of each device is calculated locally and not explicitly exposed to the gateway. However, a DFWF may be estimated by the gateway by evaluating the time intervals of the consecutively received data packets and an averaged DFWF is calculated over 300 data packets after the optimal DFWF is assigned.
\renewcommand\arraystretch{1.5}
\begin{table}[ht!]
\centering
\caption{Utility Functions}\label{UtilityF}
\vspace{-0.2cm}
\begin{tabular}{|c|c|}
\hline
\textbf{Device index} &
\textbf{Utility Functions} \\ \hline
1 & $f_1(x_1) = -(x_1+9)^2 - x_1^3 + 900$ \\ \hline
2 & $f_2(x_2) = -(x_2-4)^2 + 500$ \\ \hline
3 & $f_3(x_3) = -(2x_3+3)^2 - x_3^3 + 110$ \\ \hline
\end{tabular
\end{table}
\subsection{System with sufficient resources}
In this scenario, only device $1$ and device $2$ are connected to the gateway (i.e., parameter $N = 2$) and all other system parameters are kept by default, i.e., $c = 10$, $d = 15$, $a_1 = 2$, $a_2 = 3$ with the associated utility functions $f_1(x_1)$ and $f_2(x_2)$ shown in Table \ref{UtilityF}. With these parameters, the theoretical optimal results of the ADMM implementation are $x_1^{*} = 1$ and $x_2^{*} = 4$ for optimization problem (1). This result implies that the gateway expects to receive $1$ and $4$ data packet(s) per second from device $1$ and $2$ on average. In this setup, the capacity provided by the system is sufficient since $x_1^{*} + x_2^{*} < c$ and $a_1 * x_1^{*} + a_2 * x_2^{*} <d$. With the decentralized ADMM implemented using the simulation setup, the optimization results and resource consumption of the system are illustrated in Fig. \ref{fig2a} and Fig. \ref{fig2b}, respectively. In particular, Fig. \ref{fig2a} shows the evolution of the calculated DFWF for both devices as estimated by the gateway. The DFWFs are estimated along with the number of received data packets, indicated by the red and green lines for device $1$ and device $2$, respectively. Concretely, our results show that the estimated DFWFs are $0.9984\ Hz$ and $3.9318\ Hz$ for device $1$ and $2$, respectively, as shown in Table \ref{simulationA}. The estimated DFWFs are just slightly below the the theoretical optimal DFWFs, indicated by the dotted-line in Fig. \ref{fig2a}. The decay of the DFWF is accounted for by the internet speed, while the communication between the gateway and the devices is based on a router, resulting in a $1.6\ ms$ and a $4.3\ ms$ delay for device $1$ and device $2$, respectively. Meanwhile, we find that the fluctuation of the estimated DFWFs is caused by the data jamming when the gateway is receiving data packets with high writing frequency. Fig. \ref{fig2b} shows the sum of DFWFs as well as the size of total data packets of all connected devices per second transmitted to the gateway. The dotted-line indicates the maximum total DFWF (in red) and received data size (in green) for each data packet. Since the system can provide sufficient resources, the total DFWF and the writing data size has not reached the resource boundary after the transmission frequencies are optimized, indicating that the proposed system is robust as long as the system resources are sufficient for this specific data transmission task.
\begin{table}[!h]
\centering
\caption{Simulation results (average)}\label{simulationA}
\vspace{-0.2cm}
\begin{tabular}{|c|c|c|}
\hline
\textbf{ } & \textbf{DFWF (Hz)} & \textbf{DFWF (Hz)} \\ \hline
\textbf{ } & \textbf{Device 1} & \textbf{Device 2} \\ \hline
\textbf{Theoretical} & 1.0000 & 4.0000 \\ \hline
\textbf{Actual} & 0.9984 & 3.9318 \\ \hline
\textbf{Absolute Error} & 0.0016 & 0.0682 \\ \hline
\end{tabular
\end{table}
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.42\textwidth]{fig2a.png}
\caption{Decentralized optimization process of transmission frequency for Device 1 and Device 2.}
\label{fig2a}
\vspace{-0.1in}
\end{figure}
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.42\textwidth]{fig2b.png}
\caption{System resources consumption.}
\label{fig2b}
\vspace{-0.1in}
\end{figure}
\subsection{System with insufficient resources}
In this scenario, after device $1$ and device $2$ have connected to the gateway and the optimized transmission frequencies have been calculated, a new device, device $3$, connects to the gateway and the timing of connection is recorded. Given $N = 3$, $c = 10$, $d = 15$, $a_1 = 2$, $a_2 = 3$, $a_3 = 5$ and the corresponding utility functions $f_1(x_1)$, $f_2(x_2)$, $f_3(x_3)$ reported in Table \ref{UtilityF}, the theoretical optimal results of the ADMM implementation are calculated as $x_1^{*} = 1.00$, $x_2^{*} = 2.66$ and $x_3^{*} = 1.00$ for optimization problem (1). This result implies that, on average, the gateway expects to receive $1$, $2.66$ and $1$ data packet(s) per second from devices $1$, $2$, and $3$ respectively.
Based on the simulation platform, the decentralized optimization process and system resource usage are shown in Fig. \ref{fig4} and Fig. \ref{fig5} in the scenario of insufficient resources. We note that before the connection of device $3$, device $1$ and device $2$ transmit their data packets under the corresponding optimized transmission frequencies exactly as described in the first scenario with sufficient resources. As shown in Fig. \ref{fig4}, after the device $3$ connects to the system (indicated by the red arrow), the DFWF of device $2$ is readjusted and converges to a new optimal value. The DFWF of device $1$ remains unchanged since the recalculated optimal result equals the previously assigned DFWF before the connection of device $3$. After the decentralized ADMM solution is found for device $3$ (indicated by the magenta circle), device $3$ pushes data packets to the gateway using its optimal DFWF. After all three devices are transmitting data steadily (i.e., after the magenta circle), our results show that the estimated DFWFs are $0.9984\ Hz$, $2.6410\ Hz$ and $0.9984\ Hz$ for device $1$, $2$, and $3$, respectively, which are reported in Table \ref{simulationB}. Again, these estimated DFWFs are slightly below the theoretical optimal DFWFs, indicated by dotted-lines, reflecting time delays of $1.6\ ms$, $3.7\ ms$ and $1.6\ ms$ for devices $1$, $2$, $3$, respectively during their transmissions.
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.41\textwidth]{fig4.png}
\caption{Decentralized optimization process of transmission frequencies for Device 1, Device 2 and Device 3.}
\label{fig4}
\vspace{-0.1in}
\end{figure}
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.41\textwidth]{fig5.png}
\caption{System resource consumption in boundary conditions. The magenta cycle shows the connection point of device 3.}
\label{fig5}
\vspace{-0.1in}
\end{figure}
After the optimal transmission frequencies are established, as shown in Fig. \ref{fig5}, device $3$ starts to push data (marked by the magenta circle) and the total writing data size reaches the level of the system resource boundary immediately. This indicates that the proposed system is able to reallocate the system resources to finish the data transmission task effectively using the ADMM approach. Finally, for comparison purposes, we evaluate the overall utility under the ADMM-optimized DFWFs, with non-optimized average distributed DFWFs (i.e., $x_i = c/N$), and non-optimized proportionally distributed DFWFs (i.e., $x_i = (a_i*c)/\sum a_i$) as two baselines given the same MWF $c$. The results shown in Table \ref{UtilityValue} find that the utility under ADMM-optimized DFWFs achieves the largest value, which demonstrates that the proposed system obtains the best result compared to other trivial system setups that have not undergone any optimization process.
\begin{table}[!h]
\centering
\caption{Simulation results (average)}\label{simulationB}
\vspace{-0.2cm}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{ } & \textbf{DFWF (Hz)} & \textbf{DFWF (Hz)} & \textbf{DFWF (Hz)} \\ \hline
\textbf{ } & \textbf{Device 1} & \textbf{Device 2} & \textbf{Device 3} \\ \hline
\textbf{Theoretical} & 1.0000 & 2.6667 & 1.0000 \\ \hline
\textbf{Actual} & 0.9984 & 2.6410 & 0.9984 \\ \hline
\textbf{Absolute Error} & 0.0016 & 0.0257 & 0.0016 \\ \hline
\end{tabular
\vspace{-0.2in}
\end{table}
\begin{table}[ht!]
\centering
\caption{Utility Evaluation}\label{UtilityValue}
\vspace{-0.2cm}
\begin{tabular}{|c|c|}
\hline
\textbf{DFWFs} &
\textbf{Utility Value} \\ \hline
ADMM optimized & 1381.22 \\ \hline
Average distributed & 1190.35 \\ \hline
Proportionably distributed & 1086.00 \\ \hline
\end{tabular
\vspace{-0.1in}
\end{table}
\begin{figure}[ht]
\vspace{-0.1in}
\centering
\includegraphics[width=0.41\textwidth]{fig6.png}
\caption{Monitoring of $z_{2}$ and $x_{2}$ during the decentralized optimization process for Device 2.}
\label{fig6}
\vspace{-0.2in}
\end{figure}
\section{Abnormal transmission frequency detection}
While the transmission frequencies are determined and allocated by the system, all the devices push data steadily with their specified DTWF. However, an IoT device may transmit data with an unexpected transmission frequency when an edge device intensively manipulates its converged DFWF to a different, more desired, state. Alternatively, a malicious node in the network may tamper with the converged DFWF of an edge device intentionally. In this section, manipulation of transmission frequencies is briefly discussed for the examination of abnormal transmission frequency detection at the side of gateway.
According to the fundamental mechanism of the ADMM algorithm, the gateway only has access to $\textbf{z}$. Since $\textbf{x}$ achieves convergence to $\textbf{z}$ eventually, as a specific example (i.e., $z_{2}$ and $x_{2}$) shown in Fig. \ref{fig6}, we argue that the gateway is able to detect the anomaly of $\textbf{x}$ during the whole transmission process based on its knowledge of the latest value of $\textbf{z}$. Specifically, this detection process can be described in the following three steps:
\begin{itemize}
\item[S1:] Gateway accesses the value of $z_{i}$ for each device.
\item[S2:] Gateway estimates the DFWF (i.e., the converged value of $x_{i}$) for each device according to the received time-stamped data flow.
\item[S3:] If the estimated DFWF is significantly different to the reference value of $z_{i}$ (i.e., $|z_{i}-x_{i}| \geq \delta$, where $\delta$ is a threshold depending on the network delay), the optimal transmission frequency can regarded as anomalous and as being manipulated.
\end{itemize}
\section{Conclusion}\label{sec:Conclusion}
In this paper, we propose a novel transmission frequency management system for IoT edge devices. This innovative system is able to find the optimal transmission frequency for each IoT device in a resource-constrained, privacy-aware environment. In addition, it is able to detect the connection of a new device and determine and reassign the new optimal transmission frequencies automatically. Our simulation results show that the proposed system is effective in real-world scenarios, with a high accuracy for estimation of transmission frequency in a low-latency ($5\ ms$) router-based experimental IoT network. Finally, we have introduced an abnormal frequency detection mechanism for simple scenarios where the converged DFWF may have been manipulated. Our results show that the ADMM-based algorithm can successfully identify this type of undesirable anomaly during real-time IoT data transmission.
As part of our future work, different kinds of abnormal frequency detection mechanisms will be further investigated by taking account of more complicated application scenarios where malicious manipulations of utility functions of devices are considered. We will also investigate the dynamics of system behaviours when the utility functions are non-smooth and non-convex.
\section*{Acknowledgement}
This work was support in part by Science Foundation Ireland grant SFI/12/RC/2289\_P2, and in part by Postgraduate Research Scholarship from the Faculty of Engineering and Computing at Dublin City University.
\bibliographystyle{ieeetran}
|
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"redpajama_set_name": "RedPajamaArXiv"
}
| 7,844
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FCMB may refer to:
First City Monument Bank, Nigerian bank
FC Montceau Bourgogne, French football club
|
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"redpajama_set_name": "RedPajamaWikipedia"
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| 1,180
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\section{Introduction}
Barkhausen noise in soft magnets \cite{Bohn2018,BohnCorreaCararaPapanikolaouDurinSommer2014,BohnCorreaViegasPapanikolaouDurinSommer2013} originates from complex microscopic magnetization processes through the jerky motion of magnetic domain walls. First discovered by H.~Barkhausen \cite{Barkhausen1919} in 1919, it is the oldest example of depinning and avalanche motion \cite{SethnaDahmenMyers2001,DurinZapperi2006b,Wiese2021}.
Observables readily accessible are the avalanche size and duration \cite{PerkovicDahmenSethna1995,DurinZapperi2000}, as well as the avalanche shape \cite{Zapperi2005a, PapanikolaouBohnSommerDurinZapperiSethna2011,LaursonIllaSantucciTallakstadyAlava2013,DurinBohnCorreaSommerDoussalWiese2016}.
On the theoretical side, mean-field models pioneered in 1990 by Alessandro, Beatrice, Bertotti and Montorsi (ABBM) \cite{AlessandroBeatriceBertottiMontorsi1990,AlessandroBeatriceBertottiMontorsi1990b} have shaped our thinking. In these models, the domain wall is represented by a single degree of freedom, its centre of mass, a.k.a.\ {\em mean field} (MF). The question then arises how good this description is. A partial answer was given by experiments, which show the existence of two universality classes differing in the kind and range of interactions governing the domain-wall dynamics~\cite{DurinZapperi2000,DurinZapperi2006b}; they separate amorphous materials with short-range (SR) interactions from polycrystals with long-range (LR) interactions, the latter attributable to strong dipolar interactions. Another distinction is whether eddy currents (EC) play a noticeable effect for the wall motion~\cite{DurinZapperi2006b,DobrinevskiLeDoussalWiese2011b,PapanikolaouBohnSommerDurinZapperiSethna2011, Bohn2018, Zapperi2005a}, an aspect experimentally tunable by varying the sample thickness~\cite{Zapperi2005a, PapanikolaouBohnSommerDurinZapperiSethna2011, Bohn2018}.
For the first class, key observables such as the avalanche-size exponent $\tau\simeq 1.27$ differ from their MF prediction $\tau_{\rm MF}=3/2$, while they are correctly accounted for by field-theoretic models \cite{DSFisher1998,DobrinevskiLeDoussalWiese2014a,DobrinevskiPhD}. For LR magnets the experimentally observed exponents agree with their MF predictions, which led to the belief that MF theory is valid there.
In view of the solid evidence for critical exponents, a key question is whether an experiment can be designed which contradicts the ABBM model in one of its key predictions for LR magnets. We show below that this is indeed the case for the correlator of forces acting on the
domain wall, or equivalently the correlator of its positions.
In order to understand this, consider the equation of motion of an interface with SR interactions,
\begin{eqnarray}\label{eom}
\eta \partial_t u(x, t) &=& \nabla^2 u(x, t) + m^2 [w - u(x, t)] + F(x, u(x,t)), \nonumber\\
w &=& vt.
\end{eqnarray}
Here $w$ is proportional to the external applied field, and we suppose that the latter increases very slowly.
The term proportional to $m^2$, the {\em demagnetization factor}, is always present in magnetic systems, usually denoted $k\equiv m^2$ \cite{DurinZapperi2006b}.
Under slow driving conditions, most of the time the l.h.s.\ of \Eq{eom} vanishes. Averaging \Eq{eom} over $x$, we arrive a
\footnote{We denote by $u_w(x):=u(x,t)$ the interface position given $w=vt$, and
\begin{equation}\nonumber
u_w : = \frac 1{L^d} \int_x u_w(x), \quad F_w := \frac1{L^d} \int_x F(x,u_w(x)).
\end{equation}
}
\begin{equation}
\eta \dot u_w = m^2 \left[w-u_w \right] + F_w .
\end{equation}
This is Newton's law: At rest, the forces exerted by the confining potential are balanced by the forces from the disorder.
The key observable we study here are the interface correlations,
\begin{equation}\label{obs}
\hat \Delta_v(w-w') := \overline{\left[w-u_w \right] \left[w'-u_{w'} \right]}^{\rm c} \simeq \frac1 {m^4}
\overline{F_w F_{w'}}^{\rm c} .
\end{equation}
The index $v$ reminds that $\hat\Delta_v$ depends on the driving velocity. The normalization is different from the one used in the field theory \cite{LeDoussalWiese2006a}, which contains an additional factor of $m^4L^d$, with $L$ the system size, on the r.h.s. Our choice is motivated by a lack in the knowledge of $m^2$ and $L$, and by the reduction of scales in $\hat\Delta_v(w)$ to a single one, namely the correlation length $\rho$ in $w$-direction.
We are particularly interested in its zero-velocity limit
\begin{align}
\hat \Delta (w ) = \lim_{v \to 0} \hat \Delta_{v }(w ). \label{DeltaDef}
\end{align}
The observable $\hat\Delta(w)$ is the central object of our work.
It is also the central object of the field theory, necessary for all quantitative predictions \cite{NattermannStepanowTangLeschhorn1992,NarayanDSFisher1993a,LeDoussal2006b,LeDoussalWiese2006a,Wiese2021}.
In an experiment, it is impossible to take the limit of $v\to 0$.
The effect of the finite driving velocity $v$ is to round the cusp $|\hat\Delta^\prime (0^+)|=\sigma $ (see \Eq{DeltaABBM}) in a boundary layer of size $\delta_{w} \sim v \tau$, where {\begin{figure*}[t]
{\setlength{\unitlength}{1cm}\begin{picture}(17.8,6.9)
\put(0,0){ \includegraphics[width=.49\textwidth]{SRNEC200nmUdotBark.png}}
\put(8.9,0){ \includegraphics[width=.49\textwidth]{SRNEC200nmForceDropsv2.png}}
\put(4.4,6.6){(a)}
\put(13.2,6.6){(b)}
\end{picture}}
\caption{Barkhausen noise in an amorphous FeSiB film with thickness of $200$~nm, a bulk magnet with SR elasticity and without eddy-current effects. The signal in $\textbf{(a)}$, depicted in terms of $\dot u_w$ and $w$ in our notation, is characterised by sudden bursts of activity which are recorded as a voltage signal in the pickup coil, due to changes in the magnetic flux. Background instrumental white noise is visible, leading to negative $\dot u_w$. $\textbf{(b)}$ The connected part of the interface position, $w-u_w$, obtained after integration. The linearly increasing parts have slope one by construction, and correspond to an increasing magnetic field, followed by sudden jumps in force when the domain wall moves forward.}
\label{FigSRNECForcedrops}
\end{figure*}}$\tau$ is the timescale set by the response function $R(t)\simeq \frac1\tau \mathrm{e}^{-t/\tau}$ (see Fig.~\ref{Fig2}(c) for an example).
One can show \cite{terBurgWiese2020} that
\begin{align}
\hat \Delta_v(w) = \int_0^\infty {\rm d} t \int_0^\infty {\rm d} t^\prime R(t) R(t^\prime ) \hat\Delta(w - v(t - t^\prime)) . \label{DeltaDefRounded-text}
\end{align}
In Appendix~\ref{s:unfolding} we summarize the method of Ref.~\cite{terBurgWiese2020} to {\em unfold} \Eq{DeltaDefRounded-text} and reconstruct $\hat\Delta(w)$ from the measured $\hat\Delta_v(w)$. The result is
\begin{align}
\hat\Delta(w) = \hat \Delta_v(w) + \tau^2 \hat\Delta_{\dot{u}}(w) \label{OperatorUnfolding1-text},
\end{align}
where $\hat\Delta_{\dot u}(w)$ is the auto-correlation function of $\dot u_w$, readily accessible in our experiment.
This
allows us to extract the correlation function $\hat\Delta(w)$ in \Eq{DeltaDef}, by plotting the r.h.s.\ and finding the time scale $\tau$ that best eliminates the boundary layer. As we demonstrate below, Eq.\ \eqref{OperatorUnfolding1-text} allows us to remove a relatively large boundary layer of size $\delta_w = v \tau$, but it creates a smaller one of size $\delta^\prime_{w} = v \tau^\prime$. This we believe is due to additional fast modes contributing to the response function in Eq.\ \eqref{DeltaDefRounded-text}.
This is further discussed in Appendix~\ref{s:Higher-order unfolding}.
In the small-$v$ limit \Eq{DeltaDefRounded-text} can be approximated by a boundary-layer ansatz \cite{terBurgWiese2020}. Whereas this method may be more robust for noisy data, it is less precise. We discuss this in Appendix~\ref{s:unfolding}.
The ABBM model assumes the forces $F_w$ to perform a random walk, and as a consequence%
\footnote{Since $F_w$ performs a random walk, $\overline{F_w}$ is not well defined, whereas \Eq{DeltaABBM} is.}
\begin{equation}
\frac1{m^{4}}\frac{1}2 \overline{[F_w-F_{w'}]^2} = \hat \Delta(0)- \hat \Delta(w-w')\simeq \sigma |w-w'|. \label{DeltaABBM}
\end{equation}
Field theory \cite{ChauveLeDoussalWiese2000a,LeDoussalWieseChauve2002,Wiese2021} predicts a combination $\hat\Delta(0)-\hat\Delta(w)$ which as \Eq{DeltaABBM} grows linearly for small $w$ but saturates for large $w$. Its shape is distinct in SR and LR systems; analytical expressions are given in Appendix~\ref{s:theory-Delta}.
The above framework has been tested in simulations \cite{MiddletonLeDoussalWiese2006,RossoLeDoussalWiese2006a,terBurgWiese2020}, experiments on wetting \cite{LeDoussalWieseMoulinetRolley2009}, as well as RNA/DNA peeling \cite{WieseBercyMelkonyanBizebard2019}. Compared to these systems, magnets have important advantages: Firstly, they permit easy repetition, increasing the statistics; secondly, they are the only system where both LR and SR elasticity are realized.
Our measurement procedure is as follows:
\Eq{obs} above uses $u_w $, the position of the center of mass of the interface.
Our experimental data consist of the Barkhausen-noise time series, itself proportional to the center-of-mass velocity $\dot u_w$. As can be seen in Fig.~\ref{FigSRNECForcedrops}(a), this signal is characterized by bursts when the domain wall moves forward, and a vanishing signal when the wall is at rest, combined with background white noise from the measurement device\footnote{Since the interface only moves forward \cite{Middleton1992} $\dot u_w\ge 0$, incidences on Fig.~\ref{FigSRNECForcedrops} violating this condition are due to noise.}. This allows us to reconstruct $u_w$, as depicted in Fig.~\ref{FigSRNECForcedrops}(b).
The domain-wall position $u_w$ is characterized by linearly increasing parts corresponding to an increasing magnetic field (i.e.\ $w$), followed by sudden drops in $w-u_w$ when the domain wall moves forward. Since the linear increase is due to $w$, its slope is one.
This allows us to reconstruct the a priori unknown scale between $\dot u_w$ and the induced current, reducing the unknown scales in the experiment to a single one. {\begin{figure*}[t]
{\setlength{\unitlength}{1cm}\begin{picture}(17.8,7.2)
\put(0,0){ \includegraphics[width=.49\textwidth]{SRNEC200nmUnfoldingAspectRatiov4.png}}
\put(8.9,0){ \includegraphics[width=.49\textwidth]{SRNEC200nmfixedPointsv4.png}}
\put(4.4,7){(a)}
\put(13.2,7){(b)}
\put(1,1){\color{red} raw data}
\put(1.3,1.3){\color{red}\vector(0,5){1.47}}
\put(3.2,5.3){\color{blue} unfolded data}
\put(3.1,5.3){\color{blue}\vector(-3,-1){2.3}}
\put(3.1,4.){\color{grey} extrapolation}
\put(3,4.03){\color{grey}\vector(-3,-1){1.9}}
\put(11.,2.85){\color{red} exponential}
\put(11.4,2.7){\color{red}\vector(-0,-1){0.69}}
\put(13.1,1.5){\color{blue} $d=0$}
\put(13,1.5){\color{blue}\vector(-1,-1){0.45}}
\put(9.85,1.76){\color{black} 1-loop}
\put(9.9,1.4){\color{black} $d=d_{\rm c}$}
\put(10.75,1.5){\color{black}\vector(2,1){0.45}}
\put(10.17,1){\color{orange} 2-loop}
\put(10.2,0.73){\bfseries\color{orange} $ {\it d}=2$}
\put(11.07,0.9){\color{orange}\vector(2,1){0.70}}
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\end{picture}}
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\put(0,0){ \includegraphics[width=.49\textwidth]{SRWEC25gr002timeDeltaAllVelInsetv2.png}}
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\put(4.4,7){(c)}
\put(13.2,7){(d)}
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\put(9.7,1.7){\color{black} 1-loop}
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\put(10.5,1.5){\color{black}\vector(2,1){0.4}}
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\put(10.03,0.65){\color{orange} $d=2$}
\put(10.9,0.85){\color{orange}\vector(2,1){0.6}}
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\put(14.0,1.32){\color{grey}\vector(-4,-1){1.75}}
\end{picture}}
\caption{\textbf{(a)} Construction of the fixed-point function $\hat\Delta(w)$ for the FeSiB film (SR elasticity, no eddy-current effects). In red the raw data.
In blue dashed, the result from Eq.~\eqref{OperatorUnfolding1-text} using $\tau =0.17$.
In dotted gray the extrapolation to $w=0$. The small remaining peak (blue dashed)
is at a much smaller time scale.
\textbf{(b)}
Comparison of the fixed-point function using the dotted gray curve of {(a)}, to theoretical candidates, fixing scales by $\hat\Delta(0)$ and $\hat\Delta'(0^+)$. The theory candidates from top to bottom are: exponential function (red, dotted), solution in $d = 0$ \cite{LeDoussalWiese2008a,terBurgWiese2020} (blue, dashed), 2-loop FRG for $d = 2$ obtained by Pade resummation (orange, dotted), 1-loop FRG solution, valid for $d=d_{\rm c}$ (black, dot-dashed). Error bars in green represent 1-$\sigma$ confidence intervals. The inset shows theory minus data in the corresponding color code, favouring the $d = 2$ fixed point at two loops (with error bars for this curve only).
{\bf (c)} Check of the unfolding procedure, \Eq{OperatorUnfolding1-text}, for the FeCoB ribbon (SR elasticity, noticeable eddy currents), at different driving velocities $v$, using the same time scale $\tau=0.025$; magnified in the inset.
Apart from a small deviation for $v = 3$ they extrapolate to the same function. $\textbf{(d)}$ Comparison of $\hat\Delta(w)$ from (c) to the theoretical candidates, using the same color code as in Fig.~(b).
The data is consistent with the 2-loop FRG fixed point at $\epsilon = 2 $.
}
\label{Fig2}
\end{figure*}}
Details of the construction of $u_w$ from $\dot{u}_w$ are discussed in Appendix~\ref{s:Numerical methods}.
We analyse $\hat\Delta(w)$ in different classes of materials, including polycrystals and amorphous materials. They are respectively characterized by long-range or short-range elasticity, both for thick and thin samples, i.e., with and without eddy currents.
Table~\ref{tab1} summarises our set of samples and their properties.
Details for the samples and experiments are given in Appendix~\ref{s:Samples}, and for the data analysis in Appendix~\ref{s:Analysisdetails}. There we also included the relevant information to convert our units of $w$ to physical time and space.
\begin{table}[b]
\begin{tabular}{lcc}
\hline\hline
{\bf \footnotesize Sample} & ~ {\bf \footnotesize \makecell[c]{Interactions /\\ Eddy currents}} ~ &~ {\bf \footnotesize Correlation Length}~\\
\hline
\footnotesize \makecell[l]{Amorphous $200$-nm-thick \\ FeSiB film} &\footnotesize SR / No & \footnotesize 7.5 {\rm ms} $\sim$ 495 ${\rm \mu m }$ \\
\footnotesize Amorphous FeCoB ribbon~ &\footnotesize SR / Yes & \footnotesize 0.1 {\rm s} $\sim$ 67.5 ${\rm \mu m }$ \\
\footnotesize \makecell[l]{Polycrystalline $200$-nm-thick \\ NiFe film} &\footnotesize LR / No& \footnotesize 12.5 {\rm ms} $\sim$ 500 ${\rm \mu m }$ \\
\footnotesize Polycrystalline FeSi ribbon & \footnotesize LR / Yes & \footnotesize 35 {\rm ms} $\sim$ 0.9695 ${\rm \mu m }$ \\
\hline \hline
\end{tabular}
\caption{The set of samples investigated in this work, belonging to two universality classes ascribed to the kind of interaction governing the domain-wall dynamics, i.e.\ short-range and long-range interactions. In addition, two of them are thin films with negligible eddy-current effects, while the other two are ribbons, known for the retarding effect of eddy currents. The correlation length $\rho$ is given in physical units using the information in Appendix~\ref{s:Analysisdetails}. }
\label{tab1}
\end{table}
\smallskip
\noindent
{\bf{Short-range interactions without eddy currents.}}
Our first sample is an amorphous FeSiB film with a thickness of 200~nm.
Fig.\ \ref{Fig2}(a) shows that the raw data for $\hat\Delta(w)$ are
rounded in a boundary layer of size $\delta_w \approx 0.6$, due to the finite driving velocity.
To obtain the zero-driving-velocity limit $\hat\Delta(w)$,
we use \Eq{OperatorUnfolding1-text} with $\tau=0.17$.
\begin{figure*}[t]
{\setlength{\unitlength}{1cm}\begin{picture}(17.8,7.2)
\put(8.9,0){ \includegraphics[width=.49\textwidth]{LRWECfixedPointsV2.png}}
\put(0,0){ \includegraphics[width=.49\textwidth]{LRNEC200nmFixedPointInsetv4.png}}
\put(4.4,7){(a)}
\put(13.2,7){(b)}
\put(10.7,3.4){\color{red} exponential}
\put(11.4,3.35){\color{red}\vector(-0,-1){1.4}}
\put(12.9,1.4){\color{blue} $d=0$}
\put(12.8,1.45){\color{blue}\vector(-1,-1){0.47}}
\put(9.75,1.66){\color{black} 1-loop}
\put(9.75,1.3){\color{black} $d=d_{\rm c}$}
\put(10.7,1.43){\color{black}\vector(2,1){0.53}}
\put(10,0.9){\color{orange} 2-loop}
\put(10.03,0.58){\color{orange} $d=2$}
\put(10.9,0.8){\color{orange}\vector(2,1){0.85}}
\put(13.4,1.1){\color{grey} measurement}
\put(13.9,1.1){\color{grey}\vector(0,-1){0.65}}
\put(2.4,3.1){\color{red} exponential}
\put(2.63,3){\color{red}\vector(-0,-1){0.55}}
\put(4.3,1.95){\color{blue} $d=0$}
\put(4.2,1.95){\color{blue}\vector(-1,-1){0.55}}
\put(0.95,1.66){\color{black} 1-loop}
\put(0.95,1.3){\color{black} $d=d_{\rm c}$}
\put(1.9,1.4){\color{black}\vector(2,1){0.85}}
\put(2.2,0.9){\color{orange} 2-loop}
\put(2.23,0.58){\color{orange} $d=2$}
\put(3.1,0.75){\color{orange}\vector(2,1){0.75}}
\put(4.75,1.2){\color{grey} measurement}
\put(5.38,1.1){\color{grey}\vector(0,-1){0.56}}
\end{picture}
}
\caption{The measured function $\hat\Delta(w)$ for our two LR samples, i.e., (a) a polycrystalline $200$-nm-thick NiFe film (with negligible eddy-current effects), and (b) a polycrystalline FeSi ribbon (with eddy currents). Both measurements are in agreement with the 1-loop FRG fixed point. }
\label{Fig3}
\end{figure*}
This reduces the boundary layer (non-straight part) from $\delta_w \approx 0.6$ (in red, solid) to $\delta_w \approx 0.1$ (in blue, dashed), allowing us to extrapolate to $w=0$ (shown in dotted grey). It is this curve we report as our final result on Fig.\ \ref{Fig2}(b) (in solid grey).
The measured values for $\hat\Delta(0)$ and $\hat\Delta'(0^+)$ are then used to fix all scales in the fixed-point functions we wish to compare to on Fig.~\ref{Fig2}(b).
These are from bottom to top (analytic expressions are given in Appendix~\ref{s:theory-Delta}): 1-loop FRG (black, dot-dashed, relevant for $d=d_{\rm c}$, i.e.\ LR elasticity), $2$-loop FRG in $d= 2$ (relevant for SR elasticity, in orange, dotted) \cite{ChauveLeDoussalWiese2000a,LeDoussalWieseChauve2002}, the $d = 0$ solution \cite{LeDoussalWiese2008a,terBurgWiese2020} (blue, dashed) and a pure exponential (red, dotted), the latter, not realized in magnets, given as reference. The data agree best, and within error bars, with the 2-loop FRG fixed point predicted by the theory for $d = 2$. We note that from Fig.~\ref{Fig2}(b) we extract a correlation length $\rho := \hat\Delta(0)/\hat\Delta^\prime(0) \approx 3 $. This agrees with the scale on which $\hat\Delta_{\dot{u}}(w)$ decays to 0 (see Fig.~\ref{Fig8Suppl}(a) in Appendix~\ref{s:Velocity correlations}).
\smallskip
\noindent
{\bf{Short-range interactions with eddy currents.}} Our second sample with SR elasticity is an amorphous FeCoB ribbon where eddy currents are non-negligible. Here a range of different driving velocities is at our disposal. As eddy-current effects and non-linearities become more relevant as $v$ increases, we focus on the small-$v$ limit of $v = 1,2,3$. Whereas for the previous sample, finding $\hat\Delta_v(0)$ was sufficient, here there is additional (white) noise contributing to $\dot{u}$. After integration this contributes a linear function to $\hat\Delta(w)$, and what we measure is
\begin{align}
\hat\Delta_v^{\rm raw}(0)-\hat\Delta_v^{\rm raw}(w) = \hat\Delta_v(0) - \hat\Delta_v(w) + \sigma_{\rm noise} |w|, \label{DeltaNoisecontr}
\end{align}
necessitating to subtract a linear term $\sigma_{\rm noise} |w|$ (see Fig.~\ref{Fig9} in Appendix~\ref{s:SRWoutECdetails}). Fig.~\ref{Fig2}(c) shows $\hat\Delta_v(w)$ after this subtraction, for driving velocities $v = 1,2,3$ in blue, red and green.
The inset shows a zoom into the boundary layer with unfolding by \Eq{OperatorUnfolding1-text} in the same colour code. Having data at different $v$ allows us to test that
\begin{enumerate}
\item[(i)] the boundary layer scales linearly in $v$, i.e. $\delta_w\sim v \tau$.
\item [(ii)] $\hat\Delta_v(w)$ for $v = 1,2, 3$ unfold to the same $\hat\Delta(w)$.
\end{enumerate}
Both conditions are satisfied
using $ \tau= 0.025$.
Comparison to the fixed-point candidates proceeds as before and is shown in Fig.~\ref{Fig2}(d), using both $v=1$ and $v=2$ to improve the statistics. Although the error bars are non-negligible, the data is in agreement with the predicted 2-loop fixed point in $d= 2$, as for the FeSiB film with SR elasticity without eddy currents shown in Fig.~\ref{Fig2}(b).
Note though that for $w > 0.7$ we observe a slower decay and the data slightly deviates from the 2-loop result, albeit well within error bars.
We do not know whether this is a statistical fluctuation, or
due to the eddy currents.
\smallskip
\noindent
{\bf{Long-range interactions without eddy currents.}}
Long-range elasticity arises in materials, here a polycrystalline NiFe film with a thickness of $200$~nm, due to strong dipolar interactions between parts of the magnetic domain wall. For long-range elasticity the upper critical dimension $d_{\rm c} = 2$ coincides with the dimension of the domain wall.
The common belief is that then MF theory, i.e.\ the ABBM model, is sufficient to describe the system.
A glance at Fig.~\ref{Fig3}(a) shows that the experimental result is in contradiction to the prediction \eq{DeltaABBM} of the ABBM model. While the latter holds at small $w$, at larger $w$ the correlator $\hat\Delta(w)$ decays to zero.
Field theory predicts \cite{ChauveLeDoussalWiese2000a,LeDoussalWieseChauve2002,FedorenkoStepanow2002,LeDoussalWiese2003a} that fluctuations are relevant at the upper critical dimension, and that the correlator $\hat\Delta(w)$ is given by the 1-loop FRG fixed point. Fig.~\ref{Fig3}(a) shows that this is indeed the case.
\smallskip
\noindent
{\bf{Long-range interactions with eddy currents.}}
Our fourth sample is a polycrystalline FeSi ribbon where the elasticity is long-range and eddy currents non-negligible. Fig.~\ref{Fig3}(b) shows a comparison of the unfolded data to the four theoretical candidates.
As for the NiFe film with LR elasticity without eddy-current effects, the agreement is excellent with the 1-loop FRG solution, and inconsistent with ABBM. We refer to Appendix~\ref{s:LRWECdetails} and Fig.~\ref{Fig10} for details on the data analysis for this sample.
\begin{figure}[t]
\centerline{(a)\hspace{.24\textwidth}(b)}
\vspace*{-0.2cm}
\centerline{
\includegraphics[width=.25\textwidth]{SRWECanticorrelations
\hfill
\includegraphics[width=.25\textwidth]{LRWECanticorrelations}}
\caption{Anticorrelation of avalanches as a function of $w$, for the two samples with eddy-current effects, (a) the FeCoB with SR elasticity and (b) the FeSi one with LR elasticity. The solid line is the prediction $-\hat\Delta''(w)$ from \Eq{e:anticorrelations}, as obtained from the experiment. The dashed lines are bounds on the maximally achievable reduction from the $\epsilon$-expansion \eq{bound}, with error bars in cyan, for the SR elasticity case. There are no fitting parameters.}
\label{Fig4}
\end{figure}
\smallskip
Curiously, the question of the effective force-force or center-of-mass correlations, the correlations of velocities or avalanche sizes have never been addressed experimentally.
Here we aim at closing this gap. We show that these correlations have a universal form, predicted by the functional renormalization group \cite{NarayanDSFisher1993a,ChauveLeDoussalWiese2000a,LeDoussalWieseChauve2002,LeDoussal2006b,LeDoussalWiese2006a,Wiese2021}, both for short-range and long-range elasticity and mostly independent of eddy currents. An important property of the experiments is that force-force correlations are always bounded, and do not grow indefinitely as in mean-field models such as the ABBM model \cite{AlessandroBeatriceBertottiMontorsi1990,AlessandroBeatriceBertottiMontorsi1990b}, see \Eq{DeltaABBM}. It has been shown (see Ref.~\cite{Wiese2021}, section 4.20, or Eq.~(8) of Ref.~\cite{ThieryLeDoussalWiese2016}) that
as a consequence, avalanches are anti-correlated
\begin{equation}\label{e:anticorrelations}
\frac {\left< S_{w_1} S_{w_2} \right>}{\left< S\right> ^2}-1 = - {\hat\Delta''(w_1-w_2)}.
\end{equation}
Here $S_w$ is the size of an avalanche at $w$, and by definition $\left<S_w\right> = \left< S\right>$. In the numerator is $\left< S_{w_1} S_{w_2} \right> $,
the expectation of the product of avalanche sizes, given that one was triggered at $w=w_1$, and a second at $w=w_2$.
The experimental verification of this relation is shown on Fig.~\ref{Fig4}. Despite large statistical fluctuations, both the functional form as the amplitude agree.
Since $\hat\Delta(w)$ is convex, $\hat\Delta''(w)\ge 0$. On the other hand, $ {\left< S_{w_1} S_{w_2} \right>\ge 0}$, thus $\hat\Delta''(w)\le 1$. This bound is impossible to reach, as the toy-model \eq{Delta-DPM} in dimension $d=0$ has $\hat\Delta''(0^+)=0.5$.
The $\epsilon$-expansion \cite{Wiese2021} gives
\begin{equation}\label{bound}
\hat\Delta''(0^+)\le \frac29 + 0.107533 \epsilon + \ca O(\epsilon^2),
\end{equation}
which evaluates to $0.437$ for SR, and $0.222$ for LR correlations.
A glance at Fig.~\ref{Fig4} shows that this bound is saturated, both for the SR and LR sample. This is surprising as both systems have multiple domain walls, estimated to be around five for the samples on Fig.~\ref{Fig4}. So either all but one domain wall are completely stuck, or these multiple walls are so highly correlated that they effectively behave as a single wall.
We hope this work inspires the experimental community to look beyond the commonly studied observables and beyond mean field. Further experimental systems which are interesting to explore are sheered colloids or foams, DNA unzipping, and earthquakes.
\medskip
\acknowledgements
We thank A.~Douin, F.~Lechenault, G.~Mukerjee and A.~Rosso for discussions.
F.B.\ and R.L.S.\ acknowledge financial support from the Brazilian agencies CNPq and CAPES.
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\else
\renewcommand{\doi}[2]{\href{http://dx.doi.org/#1}{#2}}
\fi
\providecommand{\link}[2]{\href{#1}{#2}}
\providecommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{#1}}
\providecommand{\hal}[1]{\href{https://hal.archives-ouvertes.fr/hal-#1}{hal-#1}}
\providecommand{\mrnumber}[1]{\href{https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=#1&loc=fromreflist}{MR#1}}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 106
|
34 Hilariously Cute Photos Of Animals Taking A Bath…. #12 Might Be Too Relaxed.
No one would be surprised if you told them that bath time is the best part of your day. Baths are places to contemplate, relax, and unwind from the day - and they're even better if you've got some bubbles in there.
As it turns out, humans aren't the only ones who love a good soak. These 34 animals are making the most out of their evening wash. #22 might never get out of the water.
#1. Doesn't know how she got here.
#2. Contemplating his life choices.
#3. Giving herself the bath of a lifetime.
#4. Next step: Light a candle.
#5. This is the best day of his life.
#6. Prefers to be alone and outdoors.
#7. Loves reconnecting with the water.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,846
|
Q: Jquery function will only work once, but I need to use it multiple times I have been working on a simple Frogger game as an assignment and have run into an issue with one of my functions.
function collision($frogger, $car1) {
var x1 = $frogger.offset().left; var y1 = $frogger.offset().top;
var h1 = $frogger.outerHeight(true); var w1 = $frogger.outerWidth(true);
var b1 = y1 + h1; var r1 = x1 + w1;
var x2 = $car1.offset().left; var y2 = $car1.offset().top;
var h2 = $car1.outerHeight(true); var w2 = $car1.outerWidth(true);
var b2 = y2 + h2; var r2 = x2 + w2;
if (b1 < y2 || y1 > b2 || r1 < x2 || x1 > r2) {
document.onkeydown = function() {
document.getElementById('jump').play();
}
}
else {
$('#frogger').hide();
}
}
I am using this to detect collisions between the first car and the frog, however I need 8 instances of this function as there are 8 lanes on my map.This is my function for the second car frogger will have to cross
// ** 2nd Lane ** //
function collision2($frogger, $car2) {
var x1 = $frogger.offset().left; var y1 = $frogger.offset().top;
var h1 = $frogger.outerHeight(true); var w1 = $frogger.outerWidth(true);
var b1 = y1 + h1; var r1 = x1 + w1;
var x2 = $car2.offset().left; var y2 = $car2.offset().top;
var h2 = $car2.outerHeight(true); var w2 = $car2.outerWidth(true);
var b2 = y2 + h2; var r2 = x2 + w2;
if (b1 < y2 || y1 > b2 || r1 < x2 || x1 > r2) {
console.log("false");
}
else {
$('#frogger').hide();
}
}
Is there a simpler way to write this function for all 8 instances? Otherwise, why is this function only running once in its first instance?
A: Something like this ?
// Put cars into an array
var cars = [$car1, $car2];
// Modify the function so that car is a function param
function collision($frogger, car) {
var x1 = $frogger.offset().left; var y1 = $frogger.offset().top;
var h1 = $frogger.outerHeight(true); var w1 = $frogger.outerWidth(true);
var b1 = y1 + h1; var r1 = x1 + w1;
var x2 = car.offset().left; var y2 = car.offset().top;
var h2 = car.outerHeight(true); var w2 = $car.outerWidth(true);
var b2 = y2 + h2; var r2 = x2 + w2;
if (b1 < y2 || y1 > b2 || r1 < x2 || x1 > r2) {
document.onkeydown = function() {
document.getElementById('jump').play();
}
}
else {
$('#frogger').hide();
}
}
// Run the function across all cars from array
cars.map(function(item){
collision($frogger, item);
});
A: I see you again.the problem of you is you want to share the code for others.you can use jQuery Deferred.then you can customize your behavior out of the function.for more details you can see $.when and $.Deferred
function collision($frogger, $car) {
var x1 = $frogger.offset().left; var y1 = $frogger.offset().top;
var h1 = $frogger.outerHeight(true); var w1 = $frogger.outerWidth(true);
var b1 = y1 + h1; var r1 = x1 + w1;
var x2 = $car.offset().left; var y2 = $car.offset().top;
var h2 = $car.outerHeight(true); var w2 = $car.outerWidth(true);
var b2 = y2 + h2; var r2 = x2 + w2;
var dfd=$.Deferred();
if (b1 < y2 || y1 > b2 || r1 < x2 || x1 > r2) {
dfd.reject();
}
else {
dfd.resolve();
}
return dfd;
}
then,you can use the different behavior out of the collision.for example:
var hidden=function(){
$('#frogger').hide();
};
$.when(collision($frogger,$car1)).then(hidden).fail(function(){
document.onkeydown = function() {
document.getElementById('jump').play();
}
});
$.when(collision($frogger,$car2)).then(hidden).fail(function(){
console.log("false");
});
if you process collision are all the same when hitted you can by introduce another function to share the code likes the code below:
function test($frogger, $car){
return $.when(collision($frogger, $car)).then(function () {
$('#frogger').hide();
});
}
test($frogger, $car1).fail(function () {
document.onkeydown = function () {
document.getElementById('jump').play();
}
});
test($frogger, $car2).fail(function () {
console.log("false");
});
this methodology be called as Refactor,make your code more clean & simple.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,929
|
Q: Using cin to insert elements to a map in C++ I have no idea on how to use cin in order to insert data to a map.
I believe it should look something like this:
for(int i=0; i<=N;i++){
mapas.insert ( '/*word from cin*/',i );
/*or*/
mapas[/*word from cin*/]=i;
}
Here is my full code:
#include <iostream>
#include<algorithm>
#include <string>
#include <map>
using namespace std;
int main() {
map<string,int>mapas;
map<string,int>::iterator it;
int N;//how many words there will be
string zodis;//the word I will be looking for later
cin>>N;
cin>>zodis;
for(it=mapas.begin();it!=mapas.end();it++){
cout<<it->first<<endl;}
}
Any ideas would be appreciated because Im about to kill myself
A: you need
mapas[<string>] = <int>;
e.g.,
mapas[zodis] = N;
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,772
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Woods is pleased to announce that its partner Patrick Ouellet has become a Fellow of the American College of Trial Lawyers, one of the premier legal associations in North America. The induction ceremony at which Patrick became a Fellow took place recently before an audience of 700 persons during the recent Induction Ceremony at the 2018 Spring Meeting of the College in Phoenix, Arizona. The meeting had a total attendance of 850.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,748
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\section{Introduction}\label{intro}
In the last decades, the importance of copulas in mathematical modeling was recognized by many researchers; see e.g.\ \cite{tank,pr1,packham}. Many applications come from actuarial and financial mathematics, where the joint distribution of a vector of random variables is studied frequently. Typical problems are the pricing of basket options or the derivation of the Value at Risk of a portfolio. In this context, an interesting question concerns the best or worst case when the marginal distributions are given but the dependence structure of the underlying random vector is unknown or only partially known. Such situations appear frequently since dependence structures are in general more difficult to calibrate from empirical data than marginal distributions. Thus we are interested in maximizing the value of an integral by considering all possible copulas as integrators.\\
The underlying problem is in general open, however there exist solutions for some particular classes of integrand functions $f$. For instance, Rapuch and Roncalli \cite{rap} consider basket option pricing when no information on the dependence of the underlying random variables is available. They derive bounds for the prices of several options of European type in the Black-Scholes model, where the integrand function has a mixed second derivative with constant sign on the unit square. Tankov \cite{tank} extends these results to the greater class of two-increasing (or supermodular) functions $f$, a definition will be given in the next section. Furthermore the author gives an extension to option pricing problems under partial information on the dependence of the underlying random variables. Note that the above results are based on classical findings due to Tchen \cite{tchen}.\\
Similar results and applications in number theory are presented by Fialov\'a and Strauch \cite{fial}. They consider bounds for functionals which depend on two uniformly distributed point sequences. Under similar conditions as in \cite{rap}, they show that the Fr\'echet-Hoeffding bounds $W$ and $M$ are the copulas for which the extremal values are obtained. We remark that the underlying problem was formulated as an open problem in the unsolved problem collection of \emph{Uniform Distribution Theory} \footnote{Problem 1.29 in the open problem collection as of 19. January 2013 (http://www.boku.ac.at/MATH/udt/unsolvedproblems.pdf)}. A more detailed introduction to applications in uniform distribution theory is given in Section \ref{udt} of this article.\\
A list of results for a different class of functions $f$ exists in the context of financial risk theory; see e.g.\ Puccetti and R\"uschendorf \cite{pr1} or Albrecher et al.\ \cite{aak}. In \cite{pr1} the authors derive sharp bounds for quantiles of the loss of a portfolio, represented by a finite sum of dependent random variables, when no or only partial information on the dependence structure within the portfolio is available. Such quantities play an important role in actuarial and financial mathematics, for instance in the computation of the Value at Risk. Recently this approach has been generalized to derive optimal bounds for the expected shortfall of a portfolio; see Puccetti \cite{pr3}. Many of these results rely on the so-called rearrangement method due to R\"uschendorf \cite{ru}. Note that the application of the rearrangement method requires a rather strong regularity of the integrand function; see e.g.\ \cite{pr3}. The optimal bounds in the articles mentioned above are attained by using so-called shuffles of $M$-class of copulas, which we define in the next section.\\
The structure of our paper is the following: in the next section, after a short introduction to copulas, we present our main results, which are bounds on integrals of piecewise constant functions. Furthermore, we formulate an approximation technique for a very general class of integrand functions. In the third section, we apply our results to problems in uniform distribution theory and financial mathematics.
\section{Main Results}\label{mainsec}
In the sequel we consider expectations,
\begin{equation}\label{start}
\mathbb{E}[f(X,Y)],
\end{equation}
where $f$ is a function on $[0,1[^2$ and $X,Y$ are uniformly distributed random variables on the unit interval. In this situation the joint distribution function $C$ of $X$ and $Y$ is a copula.
\begin{definition}[Copula]\label{cop}
Let $C$ be a positive function on the unit square. Then $C$ is called (two)-copula iff for every $x,y \in [0,1[$
\begin{align*}
C(x,0) &= C(0,y) = 0,\\
C(x,1) &= x \text{ and } C(1,y) = y,
\end{align*}
and for every $x_1,x_2,y_1,y_2 \in [0,1[$ with $x_2 \geq x_1$ and $y_2 \geq y_1$
\begin{equation}\label{two-inc}
C(x_2,y_2) - C(x_2, y_1) - C(x_1, y_2) + C(x_1,y_1) \geq 0.
\end{equation}
A function which satisfies \eqref{two-inc} is called two-increasing or supermodular. In the sequel we denote by $\mathcal{C}$ the set of all two-copulas.
\end{definition}
Note that the restriction to uniformly distributed marginals is insignificant since by Sklar's Theorem, see e.g.\ \cite[Theorem 2.3.3]{nelsen}, we can write every continuous two-dimensional distribution function $H$ as
\begin{equation*}
H(x,y) = C(F(x),G(y)),
\end{equation*}
where $F,G$ denote the marginal distributions of $H$ and $C$ is a copula. Moreover if $F$ and $G$ are continuous, then $C$ is unique and we have
\begin{equation*}
\int_{[0,1[^2} f(x,y) dH(x,y) = \int_{[0,1[^2} f(F^{-1}(x),G^{-1}(y)) dC(x,y),
\end{equation*}
where $F^{-1},G^{-1}$ denote the inverse distribution functions of the marginals.\\
Copulas can be ordered stochastically, where the upper and lower bounds are called Fr\'echet-Hoeffding bounds (see e.g.\ \cite[Theorem 2.2.3]{nelsen}). More precisely, for every two-copula $C$ we have
\begin{equation}\label{frechet}
\max(x+y-1,0) \leq C(x,y) \leq \min(x,y), \quad \text{ for all } (x,y) \in [0,1[^2.
\end{equation}
It is also well-known that the Fr\'echet-Hoeffding lower and upper bounds $W(x,y) = \max(x+y-1,0)$ and $M(x,y) = \min(x,y)$ are copulas in the two dimensional setting. For higher dimensions an analogon of \eqref{frechet} exists, however the lower bound is in general not a copula, see \cite[Theorem 3.2 and 3.3]{joe}. For a detailed introduction to copulas see \cite{nelsen,joe}.\\
Thus, according to the discussion in the beginning of Section \ref{intro}, we are interested in bounds of the form
\begin{equation}\label{bounds}
\int_{[0,1[^2} f(x,y) dC_{\min}(x,y) \leq \int_{[0,1[^2} f(x,y) dC(x,y) \leq \int_{[0,1[^2} f(x,y) dC_{\max}(x,y),
\end{equation}
for all $C \in \mathcal{C}$, where $C_{\min}, C_{\max}$ are copulas. As mentioned above a particularly interesting subclass of copulas for our problems are so-called shuffles of $M$, see \cite[Section 3.2.3]{nelsen}.\\
\begin{definition}[Shuffles of $M$]\label{shuf}
Let $n \geq 1$, $s = (s_0, \ldots, s_n)$ be a partition of the unit interval with $0 = s_0 < s_1 < \ldots < s_n = 1$, $\pi$ be a permutation of $S_n = \{1,\ldots,n\}$ and $\omega \colon S_n \rightarrow \{-1, 1\}$. We define the partition $t= (t_0, \ldots, t_n), ~0 = t_0 < t_1 < \ldots < t_n = 1$ such that each $[s_{i-1}, s_i[ \times [t_{\pi(i)-1}, t_{\pi(i)}[$ is a square. A copula $C$ is called shuffle of $M$ with parameters $\{n, s, \pi, \omega\}$ if it is defined in the following way: for all $i \in \{1, \ldots, n\}$ if $\omega(i) = 1$, then $C$ distributes a mass of $s_i - s_{i-1}$ uniformly spread along the diagonal of $[s_{i-1}, s_i[ \times [t_{\pi(i)-1}, t_{\pi(i)}[$ and if $\omega(i) = -1$ then $C$ distributes a mass of $s_i - s_{i-1}$ uniformly spread along the antidiagonal of $[s_{i-1}, s_i[ \times [t_{\pi(i)-1}, t_{\pi(i)}[$.
\end{definition}
Note that the two Fr\'echet-Hoeffding bounds $W, M$ are trivial shuffles of $M$ with parameters $\{1, (0,1), (1), -1\}$ and $\{1, (0,1), (1), 1\}$, respectively. Furthermore, it is well-known that every copula can be approximated arbitrarily close with respect to the supremum norm by a shuffle of $M$; see e.g.\ \cite[Theorem 3.2.2]{nelsen}. In the sequel we denote by $I_n$ the partition of the unit interval which consists of $n$ intervals of equal length.\\
In next theorem we illustrate the close relation of \eqref{bounds} to problems in optimization theory, namely linear assignment problems of the form
\begin{equation}\label{lin}
\max_{\pi \in \mathcal{P}} \sum_{i = 1}^n a_{i, \pi(i)},
\end{equation}
where $\mathcal{P}$ is the set of all permutations of $\{1, \ldots, n\}$. Such problems are well understood and can be solved efficiently, for example by using the celebrated Hungarian Algorithm due to Kuhn \cite{kuhn}. For a detailed description of assignment problems and related solution algorithms we refer to \cite{burk}.
\begin{theorem}\label{main1}
Let $n \geq 1$, $A = \{a_{i,j}\}_{i,j=1, \ldots,n}$ be a real-valued $n \times n$ matrix and let the function $f$ be defined as
\begin{equation*}
f(x,y) := a_{i,j}, \quad (x,y) \in \left[ \frac{i-1}{n}, \frac{i}{n} \right [ \times \left[ \frac{j-1}{n}, \frac{j}{n} \right[.
\end{equation*}
Then the copula $C_{\max}$ which maximizes
\begin{equation}\label{int}
\max_{C \in \mathcal{C}} \int_{[0,1[^2} f(x,y) dC(x,y)
\end{equation}
is given as a shuffle of $M$ with parameters $\{n, I_n, \pi^*, 1\}$, where $\pi^*$ is the permutation which solves the assignment problem
\begin{equation*}
\max_{\pi \in \mathcal{P}} \sum_{i = 1}^n a_{i, \pi(i)}.
\end{equation*}
Moreover, the maximal value of \eqref{int} is given as
\begin{equation}\label{maxval}
\int_{[0,1[^2} f(x,y) dC_{\max}(x,y) = \frac{1}{n} \sum_{i = 1}^n a_{i, \pi^*(i)}.
\end{equation}
\end{theorem}
\begin{proof}
Let $\{C_k(x,y), k = 1,\ldots, n! = N\}$ be the set of all shuffles of $M$ with parameters of the form $\{n, I_n, \pi_k, 1\}$ and let $t_k \geq 0, ~k = 1, \ldots, N$, where $\sum_{k = 1}^N t_k = 1$. Then $C'(x,y) = \sum_{k = 1}^N t_k C_k(x,y)$ is always a copula satisfying
\begin{equation*}
\int_{[0,1[^2} f(x,y) dC'(x,y) \leq \frac{1}{n} \sum_{i = 1}^n a_{i, \pi^*(i)},
\end{equation*}
where $\pi^*$ is given in the statement of the theorem.\\
For an arbitrary copula $C \in \mathcal{C}$ we define the matrix $B_C$ as
\begin{equation*}
B_C(i,j) = n \int_{\left [ \frac{i-1}{n}, \frac{i}{n} \right [ \times \left [ \frac{j-1}{n}, \frac{j}{n} \right [} dC(x,y).
\end{equation*}
It follows by Definition \ref{cop} that $B_C$ is doubly stochastic and by Definition \ref{shuf} that $B_{C_k}$ is a permutation matrix. Furthermore it follows by the Birkhoff-von Neumann Theorem that the set of doubly stochastic matrices coincides with the convex hull of the set of permutation matrices, see e.g. \cite{Mirsky}. Thus for every $C$ there exist $t_k \geq 0, ~k = 1,\ldots,N $ with $\sum_{k = 1}^N t_k = 1$ such that
\begin{equation*}
B_C(i,j) = \sum_{k = 1}^N t_k B_{C_k}(i,j), \quad \text{ for every } i,j,
\end{equation*}
and hence
\begin{equation*}
\int_{[0,1[^2} f(x,y) dC(x,y) = \sum_{k = 1}^N t_k \int_{[0,1[^2} f(x,y) dC_k(x,y) \leq \frac{1}{n} \sum_{i = 1}^n a_{i, \pi^*(i)}\ .\qquad \qed
\end{equation*}
\end{proof}
Note that the maximal copula in Theorem \ref{main1} is by no means unique, since for instance the value of the integral in \eqref{int} is independent of the choice of $\omega$.\\
Obviously, we can derive a lower bound in Theorem \ref{main1} by considering \linebreak$g(x,y) = -f(x,y)$. Furthermore, it is easy to see that Theorem \ref{main1} applies to all functions $f$ which are constant on sets of the form
\begin{equation*}
I_{i,j} = \left[ s_i, s_{i+1} \right[ \times \left[ t_j, t_{j+1} \right[, \quad i = 0, \ldots, n-1, ~ j = 0, \ldots, m-1,
\end{equation*}
where $0 = s_0 < s_1 < \ldots < s_n = 1$ and $0 = t_0 < t_1 < \ldots < t_m = 1$ are rational numbers.\\
The following generalization of our approach applies to a wide class of functions on the unit square.
\begin{theorem}\label{main2}
Let $f$ be a continuous function on $[0,1]^2$, let the sets $I^n_{i,j}$ be given as
\begin{equation*}
I^n_{i,j} = \left[ \frac{i-1}{2^n}, \frac{i}{2^n} \right [ \times \left[ \frac{j-1}{2^n}, \frac{j}{2^n} \right[ \quad \text{ for } i,j = 1, \ldots, 2^n,
\end{equation*}
for every $n > 1$ and define the functions $\underline{f}_n, \overline{f}_n$ as
\begin{align}
\underline{f}_n(x,y) &= \min_{(x,y) \in I^n_{i,j}} f\left( x, y \right), \quad \text{ for all } (x,y) \in I^n_{i,j},\notag\\
\overline{f}_n(x,y) &= \max_{(x,y) \in I^n_{i,j}} f\left( x, y \right), \quad \text{ for all } (x,y) \in I^n_{i,j}.\label{mini}
\end{align}
Furthermore, let $\underline{C}^n_{\max}, \overline{C}^n_{\max}$ be the copulas which maximize
\begin{equation*}
\max_{C \in \mathcal{C}} \int_{[0,1[^2} \underline{f}_n(x,y) dC(x,y) \text{ and } \max_{C \in \mathcal{C}} \int_{[0,1[^2} \overline{f}_n(x,y) dC(x,y),
\end{equation*}
respectively. Then
\begin{align}
\int_{[0,1[^2} \underline{f}_n(x,y) d\underline{C}^n_{\max}(x,y) &\leq \sup_{C \in \mathcal{C}} \int_{[0,1[^2} f(x,y) dC(x,y) \notag\\
&\leq \int_{[0,1[^2} \overline{f}_n(x,y) d\overline{C}^n_{\max}(x,y),\label{bound}
\end{align}
for every $n$, and
\begin{align}
\lim_{n \rightarrow \infty} \int_{[0,1[^2} \underline{f}_n(x,y) d\underline{C}^n_{\max}(x,y) &= \lim_{n \rightarrow \infty} \int_{[0,1[^2} \overline{f}_n(x,y) d\overline{C}^n_{\max}(x,y) \notag\\
&= \sup_{C \in \mathcal{C}} \int_{[0,1[^2} f(x,y) dC(x,y).\label{conv}
\end{align}
\end{theorem}
\begin{proof}
The inequalities in \eqref{bound} follow immediately from the construction of $\underline{f}_n,\overline{f}_n$ and Theorem \ref{main1}. Furthermore since $f$ is continuous on $[0,1]^2$ we have that for every $\epsilon > 0$ there exists an integer $n$ such that
\begin{equation}\label{eps}
| \overline{f}_n(x,y) - \underline{f}_n(x,y) | < \epsilon, \quad \text{ for all } (x,y) \in [0,1]^2.
\end{equation}
Moreover, by Theorem \ref{main1}, for every $n$ we can write
\begin{equation*}
\int_{[0,1[^2} \underline{f}_n(x,y) d\underline{C}^n(x,y) = \frac{1}{2^n} \sum_{i = 1}^{2^n} a_{i, \pi^*(i)},
\end{equation*}
for a permutation $\pi^*$ and a real valued matrix $A = \{a_{i,j}\}_{i,j=1, \ldots,n}$ with
\begin{equation*}
a_{i,j} = \min_{(x,y) \in I^n_{i,j}} f\left( x, y \right), \quad \text{ for } i,j = 1, \ldots, 2^n.
\end{equation*}
Using \eqref{eps}, we get that
\begin{equation*}
\int_{[0,1[^2} \overline{f}_n(x,y) d\overline{C}^n(x,y) \leq \int_{[0,1[^2} (\underline{f}_n(x,y) + \epsilon) d\underline{C}^n(x,y) = \frac{1}{2^n} \sum_{i = 1}^{2^n} (a_{i, \pi^*(i)} + \epsilon)
\end{equation*}
and thus
\begin{equation*}
\left | \int_{[0,1[^2} \overline{f}_n(x,y) d\overline{C}^n(x,y) - \int_{[0,1[^2} \underline{f}_n(x,y) d\underline{C}^n(x,y) \right | < \epsilon.
\end{equation*}
Combining this with \eqref{bound}, we get \eqref{conv}. \qquad \qed
\end{proof}
The assumption that $f$ is continuous can, perhaps, be relaxed to the case that $f$ is $C$-continuous a.e.\ for all $C \in \mathcal{C}$. This is required to make sure that
\begin{equation*}
\int_{[0,1[^2} f(x,y) dC(x,y)
\end{equation*}
exists for all $C \in \mathcal{C}$.\\
By defining the function families $\underline{f}_n, \overline{f}_n$ differently, we might get an approximation technique which converges faster to the optimal value, for instance we could use
\begin{equation*}
f_n(x,y) = f\left(\frac{i}{2^n}, \frac{j}{2^n} \right), \quad \text{ for all } (x,y) \in I^n_{i,j}.
\end{equation*}
Furthermore, the mini- and maximization steps in \eqref{mini} can be time-consuming, for instance when these problems are not explicitly solvable. However the advantage of the present approach lies in the fact that we get an upper and lower bound of the optimal value for every $n$, which is obviously useful for numerical applications.\\
In numerical investigations, where \eqref{mini} could not be solved explicitly, we used mini- and maximization over a fixed grid in each $I^n_{i,j}$. This results in a fast computation, however we obviously lose the property of upper and lower bounds for every $n$.\\
By assuming Lipschitz-continuity of $f$, we can describe the rate of convergence of our method.
\begin{corollary}
Let the assumptions of Theorem \ref{main2} hold and, in addition assume that $f$ is Lipschitz-continuous on $[0,1]^2$ with parameter $L$. Then
\begin{equation*}
\left | \int_{[0,1[^2} \overline{f}_n(x,y) d\overline{C}^n(x,y) - \int_{[0,1[^2} \underline{f}(x,y) d\underline{C}(x,y) \right| \leq L \frac{\sqrt{2}}{2^n}.
\end{equation*}
\end{corollary}
\begin{proof}
Following the proof of Theorem \ref{main2} and using the Lipschitz-continuity of $f$ we get
\begin{equation*}
| \overline{f}_n(x,y) - \underline{f}_n(x,y) | \leq L \frac{\sqrt{2}}{2^n}, \quad \text{ for all } (x,y) \in [0,1]^2,
\end{equation*}
and thus
\begin{equation*}
\left | \int_{[0,1[^2} \overline{f}_n(x,y) d\overline{C}^n(x,y) - \int_{[0,1[^2} \underline{f}_n(x,y) d\underline{C}^n(x,y) \right | \leq L \frac{\sqrt{2}}{2^n}.
\end{equation*}
\qed
\end{proof}
\section{Applications}
In this section we present two numerical examples in which we apply the approximation technique presented in Theorem \ref{main2}. We use an implementation of the Hungarian Algorithm in MatLab, which makes it possible to derive the solution of the linear assignment problem \eqref{lin} for a given matrix $A$ of size $2^{10} \times 2^{10}$ within seconds. The involved mini- or maximization of the integrand function on a given grid can be done efficiently, since the integrand functions are piecewise smooth.
\subsection{Uniform Distribution Theory}\label{udt}
A deterministic sequence $(x_n)_{n > 1}$ of points in $[0,1[$ is called uniformly distributed (u.d.) iff
\begin{equation*}
\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n = 1}^N \mathbf{1}_{[a, b[} (x_n) = b-a
\end{equation*}
for all intervals $[a, b[ \subseteq [0,1[$. Furthermore, we call $g$ the asymptotic distribution function (a.d.f.) of a point sequence $(x_n,y_n)_{n > 1}$ in $[0,1[^2$ if
\begin{equation*}
g(x,y) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n = 1}^{N} \mathbf{1}_{[0,x[ \times [0,y[}(x_n,y_n),
\end{equation*}
holds in every point $(x,y)$ of continuity of $g$, for a survey of classical results in this field see \cite{sp}. In \cite{fial}, Fialov\'a and Strauch consider
\begin{equation*}
\limsup_{N \rightarrow \infty} \frac{1}{N} \sum_{n = 1}^N f(x_n, y_n),
\end{equation*}
where $(x_n)_{n > 1}, (y_n)_{n > 1}$ are u.d.\ sequences in the unit interval and $f$ is a continuous function on $[0,1[^2$, see also \cite{ps}. In this case the a.d.f.\ $g$ of $(x_n,y_n)_{n > 1}$ is always a copula and we can write
\begin{equation}\label{inte}
\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n = 1}^N f(x_n, y_n) = \int_0^1 \int_0^1 f(x,y) dg(x,y).
\end{equation}
Now we can derive upper bounds for \eqref{inte} by maximizing $g$ over the set of all copulas. This has already been done in \cite{fial} for functions $f$ where $\frac{\partial^2 f}{\partial_x \partial_y}(x,y)$ has constant sign for all $(x,y) \in [0,1[^2$. Note that this condition is equivalent to the two-increasing property of $f$ provided that $\frac{\partial^2 f}{\partial_x \partial_y}(x,y)$ exists on the unit square.\\
As a numerical example, we consider
\begin{equation*}
\limsup_{N \rightarrow \infty} \frac{1}{N} \sum_{n = 1}^N \sin(\pi (x_n + y_n)).
\end{equation*}
The numerical results are illustrated in Table \ref{tab:t2}. Note that the approximations of the lower bound can be easily computed using the symmetry of the sine function.\\
A further interesting question concerns the sequences $(x_n)_{n > 1}, (y_n)_{n > 1}$ which maximize \eqref{inte}. Let $(x_n)_{n > 1}$ be a u.d.\ sequence and $C(x,y)$ a shuffle of $M$, then it is easy to see that $(f(x_n))_{n > 1}$ is u.d., where $f$ is the support of $C$. Thus if $C$ is the shuffle of $M$ which attains the maximum in \eqref{inte}, an optimal two-dimensional sequence is given as $(x_n, f(x_n))_{n > 1}$, where $(x_n)_{n > 1}$ is an arbitrary u.d.\ sequence. In Figure \ref{fig: max1}, we present the copula which attains the upper bound for the maximum in our approximation when $n = 7$.\\
Although we can not give a rigorous proof, by increasing $n$ it seems that the copula $C'$ which attains the maximum is the shuffle of $M$ with parameters $\{2, (0,0.75,1), (1), \{\omega(1) = -1, \omega(2) = 1 \}\}$. In this case we have
\begin{align*}
\int_0^1 \int_0^1 \sin(\pi (x + y)) dC'(x,y) &= \int_0^1 \sin(\pi (x + f'(x))) dx\\
&= \int_0^{\frac{3}{4}} \sin(\pi (x + 0.75 - x)) dx + \int_{\frac{3}{4}}^1 \sin(\pi 2 x) dx\\
&= \frac{3}{4 \sqrt{2}} - \frac{1}{2 \pi} \approx 0.371175,
\end{align*}
where $f'$ denotes the support of $C'$.
\begin{table}[!ht]
\centering
\begin{tabular}[h]{|r|r|r|r|r|r|r|}
\toprule
$n$ & 5 & 6 & 7 & 8& 9 & 10\\
UB & 0.3933 & 0.3824 & 0.377 & 0.3741 & 0.3727 & 0.3712\\
LB & 0.3482 & 0.3598 & 0.3655 & 0.3684 & 0.3698 & 0.3711\\
\bottomrule
\end{tabular}
\caption{Upper and lower bounds for the maximum in \eqref{inte} with respect to $n$.}
\label{tab:t2}
\end{table}
\begin{figure}[H]
\centering
\includegraphics[width=7cm]{fig1.pdf}
\caption{Support of copula which attains upper bound for $\sin( \pi (X + Y))$ and $n = 7$.}
\label{fig: max1}
\end{figure}
\subsection{First-to-default Swaps}
A first-to-default swap (FTD) is a contract in which a protection seller (PS) insures a protection buyer (PB) against the loss caused by the first default event in a portfolio of risky assets. The PB pays regularly a fixed constant premium to the PS, the so-called spread, until the maturity $T$ of the contract or the first default event, whichever occurs first. In exchange, the PS compensates the loss caused by the default at the time of default.\\
We assume that the underlying portfolio consists of two risky assets, for which the marginal default distributions are known, but the joint distribution is unknown. We want to derive a worst case bound in this setting. For the valuation of the FTD we follow the paper of Schmidt and Ward \cite{schmidt}. Note that Monte Carlo methods for the evaluation of first-to-default swaps, where the dependences within the portfolio is modeled by a copula, are e.g.\ presented in Aistleitner et al.\ \cite{aht} and Packham and Schmidt \cite{packham}.\\
Let $\tau_1, \tau_2$ denote the random default times of the two risky assets, let the notional be equal to one for both assets and $R_i, i = 1,2,$ be the so-called recovery rates, which are the percental amounts of money that can be liquidized in case of the default of an asset. We assume that the distribution of $\tau_i$ is given as
\begin{equation*}
\mathbb{P}(\tau_i \leq t) = 1 - e^{-\lambda_i t}, \quad t > 0,
\end{equation*}
where the intensity $\lambda_i$ can be derived from the credit default swap market as
\begin{equation*}
\lambda_i = \frac{s_i}{1 - R_i},
\end{equation*}
and $s_i$ is the premium of an insurance against the default of asset $i$.\\
Now denote by $\tau = \min(\tau_1, \tau_2)$ the first default time in the portfolio, let $0 = t_0 < t_1 < \ldots < t_n = T$ be the payment times of the constant spread and assume that there exists a risk free interest rate $r \geq 0$. Then, to guarantee a fair spread $s$, we obtain that the expected, discounted premium and default payments are equal, i.e.\
\begin{equation*}
s \sum_{i = 0}^n e^{- r t_i} \mathbb{P}(\tau > t_i) = \sum_{i = 1}^2 \mathbb{E}\left[ (1- R_i) e^{- r \tau} \mathbf{1}_{\{ \tau < T \wedge \tau = \tau_i\}} \right].
\end{equation*}
By the above assumptions we obtain that
\begin{align*}
&\mathbb{P}(\tau > t_i) = \int_{[0,1[^2} \mathbf{1}_{\left\{f(x, \lambda_1) > t_i ~\wedge ~f(y, \lambda_2) > t_i \right\}} dC(x,y),\\
&\sum_{i = 1}^2 \mathbb{E}\left[ (1- R_i) e^{- r \tau} \mathbf{1}_{\{ \tau < T ~\wedge ~\tau = \tau_i\}} \right] = \\
&\int_{[0,1[^2} e^{-r \min\left( f(x, \lambda_1), f(y, \lambda_2) \right)} \biggl(\mathbf{1}_{\left\{ f(x, \lambda_1) \leq \min\left(f(y, \lambda_2), T\right) \right\}} (1 - R_1) \\
&+ \mathbf{1}_{\left \{f(y, \lambda_2) \leq \min\left(f(x, \lambda_1), T\right) \right\}} (1 - R_2) \biggr) dC(x,y),
\end{align*}
where $f(x, \lambda) = \frac{- \log(1-x)}{\lambda}$ is the inverse distribution function of an exponential distribution with parameter $\lambda$ and $\mathbf{1}_{\{(x,y) \in B\}}$ denotes the characteristic function of set $B \subseteq [0,1[^2$.\\
Now we want to calculate the maximal spread $s$ by maximizing over all copulas. We obtain for the spread that
\begin{align}
s &= \int_{[0,1[^2} \frac{e^{-r \min\left( f(x, \lambda_1), f(y, \lambda_2) \right)}}{\sum_{i = 0}^n e^{- r t_i} \mathbf{1}_{\left\{f(x, \lambda_1) > t_i ~\wedge ~f(y, \lambda_2) > t_i \right\}} }\notag \\
&\cdot \biggl(\mathbf{1}_{\left\{ f(x, \lambda_1) \leq \min\left(f(y, \lambda_2), T\right) \right\}} (1 - R_1) \notag\\
&+ \mathbf{1}_{\left \{f(y, \lambda_2) \leq \min\left(f(x, \lambda_1), T\right) \right\}} (1 - R_2)\biggr) dC(x,y). \label{ftd}
\end{align}
Note that the value of the integral is finite since the first payment is made at $t_0 = 0$. Furthermore, the integrand function in this example is not continuous, thus Theorem \ref{main2} cannot be applied. Nevertheless, it is clear that our technique provides upper and lower bounds for the optimal values, and since these bounds converge to each other our approach still works.\\
In Table \ref{tab:t1} we present numerical results for a concrete example with three payment times, $t_i = 0,1,2$. One can observe that the resulting copulas (given in Figures \ref{fig: max2} and \ref{fig: max3} for $n = 7,8$, respectively) are highly irregular in left upper quarter of the unit square. Nevertheless for $n = 10$ the upper and lower bounds for the optimal values are almost equal.
\begin{table}[!ht]
\centering
\begin{tabular}[h]{|r|r|r|r|r|r|r|r|}
\toprule
$\lambda_1$ & $\lambda_2$ & $R_1$ & $R_2$ & $T$ & $r$ & $t_i$ &\\
$\frac{1}{3}$ & $\frac{1}{2}$ & 0.5 & 0.7 & 2 & 0.05 & (0, 1, 2)&\\
\midrule
$n$ & 3 & 4 & 5 & 6 & 7 & 8& 10\\
\midrule
$\overline{UB}$ & 0.3601 & 0.3355 &0.3301 &0.326 & 0.322 & 0.3202 &0.3195\\
$\overline{LB}$ & 0.2956 & 0.3031 & 0.314 & 0.318 & 0.3183 & 0.3189 & 0.3195\\
\midrule
$\underline{UB}$ & 0.1714 & 0.1674 & 0.1567 & 0.1535 & 0.1519 & 0.1505 & 0.1498\\
$\underline{LB}$ & 0.1453 & 0.1456 & 0.1458 & 0.1480 & 0.1492 & 0.1492 & 0.1495 \\
\bottomrule
\end{tabular}
\caption{Approximation of the maximal spread of a FTD, where $\overline{UB}$ and $\overline{LB}$ and $\underline{UB}$ and $\underline{LB}$ denote the values of the upper and the lower bounds of the maximal and minimal value of the integral, and $n$ the fineness of the approximation according to Theorem \ref{main2}.}
\label{tab:t1}
\end{table}
\begin{figure}[H]
\centering
\includegraphics[width=7cm]{fig2.pdf}
\caption{Copula which attains the upper bound for the maximal value with $n = 7$.}
\label{fig: max2}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=7cm]{fig3.pdf}
\caption{Copula which attains the upper bound for the maximal value with $n = 8$.}
\label{fig: max3}
\end{figure}
\section{Conclusions}
The method presented in this paper can be used to derive sharp bounds for integrals of piecewise constant functions with respect to copulas. This extends the scientific literature on this topic, that is in general still open. The numerical effectiveness of our method was illustrated in two numerical examples from different branches of applied mathematics.\\
A starting point for further research is an extension of the presented technique to higher dimensional problems, since founding bounds for multidimensional integrals with respect to copulas has several applications in fields of mathematics such as number theory, financial and actuarial mathematics. Of course our aim is to study and investigate general problems and try to find a link between different branches of mathematics. Nevertheless, since the resulting so-called multi-index assignment are in general NP-hard, we plan to investigate heuristics; see e.g.\ \cite{burk}.
\section*{Acknowledgements}
The authors would like to thank Prof.\ Robert Tichy from TU Graz and Prof.\ Oto Strauch from the Slovak Academy of Science for helpful remarks and suggestions. Furthermore the authors are indebted to two anonymous referees who helped to improve the paper.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,906
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Zaključёnnye (Заключённые) è un film del 1936 diretto da Evgenij Veniaminovič Červjakov.
Trama
Note
Collegamenti esterni
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,493
|
Polystachya suaveolens är en orkidéart som beskrevs av Phillip James Cribb. Polystachya suaveolens ingår i släktet Polystachya och familjen orkidéer. Inga underarter finns listade i Catalogue of Life.
Källor
Orkidéer
suaveolens
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 263
|
La figlia di un soldato non piange mai (A Soldier's Daughter Never Cries) è un film del 1998 diretto da James Ivory.
Trama
Una famiglia americana a Parigi tra anni '60 e '70: il padre Bill, scrittore di successo, la madre Marcella, espansiva ed affettuosa e la figlia Channe. Un giorno i genitori decidono di adottare Benoit, sei anni. Quando sente crescere intorno a sé affetto e tenerezza, Benoit chiede di essere chiamato Billy. Quando Channe è ormai adolescente Bill annuncia che la famiglia tornerà negli Stati Uniti a causa delle sue cattive condizioni di salute.
Trasferitisi sulla costa orientale, Bill, malato, cerca di finire il suo nuovo romanzo mentre Channe comincia ad uscire con un ragazzo, che presenta anche a casa. Il padre tenta di approfondire con lei l'argomento dei rapporti sessuali, ma la malattia avanza: un giorno invita Channe a leggere il diario che la madre di Billy scrisse mentre lo aspettava. Dopo la morte di Bill, Marcella consegna il diario a Billy, ma lui rifiuta di leggerlo e lo affida alla sorella.
Ora che il suo padre americano è morto, Billy si sente veramente legato a lui e alla sua seconda famiglia.
Curiosità
Il libro da cui è tratto il film è stato scritto da Kaylie Jones figlia del più famoso James Jones (Da qui all'eternità, La sottile linea rossa) ed è semi-autobiografico.
Collegamenti esterni
Film drammatici
Film diretti da James Ivory
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,851
|
Q: Assigned form value into var in JavaScript I have a html form below:
<label class="sr-only" for="phone">Phone Number </label>
<input type="text" name="phone" placeholder="eg. 123456XXXXXXXX" class="phone form-control" id="phone">
</div>
<div class="f1-buttons">
<button type="button" class="btn btn-next">Next</button>
</div>
When the user fills in the form, the form value will be auto assigned in the script below:
<script>var data = "{ \"phone\":\"--this will get value from phone ID field\", \"tenantID\":\"0\" }"; .....</script>
Tried using the document.getElementById("phone").value; but it is not working. Any have suggestions how to achieve this?
var data = "{ \"MSISDN\":\"document.getElementById("phone").value\", \"tenantID\":\"0\" }";
var xhr = new XMLHttpRequest();
xhr.withCredentials = true;
xhr.addEventListener("readystatechange", function () {
if (this.readyState === 4) {
console.log(this.responseText);
}
});
xhr.open("POST", "https://api/v1/uat/user");
xhr.setRequestHeader("accept", "application/json;charset=UTF8");
xhr.setRequestHeader("content-type", "application/json");
xhr.setRequestHeader("api-key", "12345565");
xhr.setRequestHeader("cache-control", "no-cache");
xhr.send(data);
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/1.2.3/jquery.min.js"></script>
<fieldset>
<h4>Please fill phone Number:</h4>
<div class="form-group">
<label class="sr-only" for="phone">Phone Number (123456789xxx) </label>
<input type="text" name="phone" placeholder="eg. 123456789xxx" class="MSISDN form-control" id="phone">
</div>
<div class="f1-buttons">
<button type="button" class="btn btn-next">Next</button>
</div>
</fieldset>
I edited my question, the phone number field is to call an API response using post method in body request. What I want to do is, when user filled up the phone number will POST to the API URL to get a response if valid the response will print in the next form
Thanks
A:
var data = "{ \"MSISDN\":\"document.getElementById("phone").value\", \"tenantID\":\"0\" }";
Putting some JavaScript source code inside a string literal is just going to make a string literal containing that source code and not the result of executing it.
Generating JSON by hand is a poor idea anyway. The value might contain special characters that would break it.
Create an object. Add a value to that object. Then use a library function to encode it.
var data = {
MSISDN: document.getElementById("phone").value,
tenantID: "0"
};
var json = JSON.stringify(data);
A: You are assigning the JSON into a variable var data = "{ \"phone\":\"--this will get value from phone ID field\", \"tenantID\":\"0\" }". So, you can access directly to data as the following snippet shows:
console.log(data)
<script>var data = "{ \"phone\":\"--this will get value from phone ID field\", \"tenantID\":\"0\" }"</script>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,522
|
//
// WDECPayPalPayment.h
// WDeCom
//
// Created by Sedlak, Stefan on 10/12/15.
// Copyright © 2018 Wirecard Technologies GmbH. All rights reserved.
//
#import <WDeCom/WDeCom.h>
/** @addtogroup ios_sdk
* @{
*/
/**
* @brief Defines PayPal payment method.
*/
@interface WDECPayPalPayment : WDECPayment
/**
@brief It describes recurring transactions.
*/
@property (strong, nonatomic, nullable) WDECPeriodic *periodic;
/**
@brief It is used by risk library. The data collected by the Magnes library is used to complement the data that is obtained from the PayPal hosted pages.
*/
@property (strong, nonatomic, nullable) NSString *riskReferenceId;
/**
@brief It is used by risk library. For merchants using an Advertising ID (IDFA), this parameter should be populated with the IDFA.
*/
@property (strong, nonatomic, nullable) NSString *advertisingIdentifier;
@end
/** @} */
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,021
|
package azkaban.db;
import java.sql.Connection;
import java.sql.SQLException;
import org.apache.commons.dbcp2.BasicDataSource;
import org.apache.log4j.Logger;
public class MySQLDataSource extends AzkabanDataSource {
private static final Logger logger = Logger.getLogger(MySQLDataSource.class);
public MySQLDataSource(final String host, final int port, final String dbName,
final String user, final String password, final int numConnections) {
super();
final String url = "jdbc:mysql://" + (host + ":" + port + "/" + dbName);
addConnectionProperty("useUnicode", "yes");
addConnectionProperty("characterEncoding", "UTF-8");
setDriverClassName("com.mysql.jdbc.Driver");
setUsername(user);
setPassword(password);
setUrl(url);
setMaxTotal(numConnections);
setValidationQuery("/* ping */ select 1");
setTestOnBorrow(true);
}
/**
* This method overrides {@link BasicDataSource#getConnection()}, in order to have retry logics.
*/
@Override
public synchronized Connection getConnection() throws SQLException {
Connection connection = null;
int retryAttempt = 0;
while (retryAttempt < AzDBUtil.MAX_DB_RETRY_COUNT) {
try {
/**
* when DB connection could not be fetched (e.g., network issue), or connection can not be validated,
* {@link BasicDataSource} throws a SQL Exception. {@link BasicDataSource#dataSource} will be reset to null.
* createDataSource() will create a new dataSource.
* Every Attempt generates a thread-hanging-time, about 75 seconds, which is hard coded, and can not be changed.
*/
connection = createDataSource().getConnection();
if (connection != null) {
return connection;
}
} catch (final SQLException ex) {
logger.error(
"Failed to find DB connection. waits 1 minutes and retry. No.Attempt = " + retryAttempt,
ex);
} finally {
retryAttempt++;
}
}
return null;
}
@Override
public String getDBType() {
return "mysql";
}
@Override
public boolean allowsOnDuplicateKey() {
return true;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,947
|
Be it your intention to imprint the shape of your favorite animal or an effort to make the image of a loved pet a part of your body, find out more about animal tattoos below. This section houses articles that not only give you myriad ideas about a whole range of animal tattoos, they also unveil the hidden meanings behind these body images that will help you gain greater insight of what you wish to make a part of your personality for eternity.
One of the most popular tattoo ideas for women, a hummingbird represents a variety meaning. Read this Buzzle post to find the various symbolism you can associate your tattoo with.
From geometric styles to a simple black ink technique, getting a hummingbird tattoo requires a lot of research and patience at your end. But with Buzzle, you can trust to find impressive tattoo designs that will blow you away.
Seahorse tattoos make a style statement and are a cool tattoo design to have on your body. They are very popular among women. Seahorse tattoos are also said to be a symbol of fatherhood.
The most popular meaning of a dove tattoo is peace and love. But, did you know that a dove represents many other meanings and symbolism?
Starfish tattoos not only look great, the symbolism of starfish make it an interesting choice for a tattoo. You can try inking them with other sea creatures to make an unusual and attractive tattoo.
If you're not the one to shy away from doing what's extreme, an octopus tattoo will surely mesmerize you. Buzzle gives you 5 amazing octopus tattoo design ideas that will leave you speechless.
Cranes hold a deep symbolism in many cultures, which sure makes them one of the best options in tattoo designs. This Buzzle piece has amassed some very beautiful crane tattoo designs along with their meanings.
Animal themes have always been a vogue among tattoo enthusiasts, and if the animal is a fox, the tattoo becomes all the more foxy! A fox is associated with many traits, both good and bad.
If you are looking for the cute tattoo designs, then penguin tattoos are just for you. Check some awesome penguin designs and their meanings here.
Looking for a unique tattoo design? How about using the modest dragonfly as an inspiration? Here is more information on designs and meanings associated with dragonfly tattoos.
Insect tattoos have become very popular these days. In them, bumble bee designs are a big hit with tattoo lovers. The following passages elucidate more information on the same.
Getting a tattoo of a lion indicates strength, drive, and confidence in life. Read the Buzzle article as it illustrates which type of design you should go for.
Ladybug tattoos are small, but they can look quite sensational. Find out the symbolic meaning of a ladybug tattoo, along with some simple and cool designs to create an unique tattoo.
The bull has always been a symbol of fierce, masculine strength. If you wish to get this creature as a tattoo, use some of these design ideas, with an explanation of its symbolism.
Commonly associated with death and evil, raven tattoos also symbolize intelligence and wit. This Buzzle article gives you some amazing raven tattoo designs and their meanings.
Tattoos interest many, and each one wants to be different with the design. If aggression and strength is what you want to convey, a bull tattoo is worth considering. This is because, bulls are known for their aggression and strength.
Like the idea of getting a dove tattoo but don't know much about it and what to look for exactly? In this article we will give you a brief look into what a dove tattoo represents and what you can do with it.
Animal tattoo ideas are depictions of human desire to be as free as them in thought and being. Read on to know what animal tattoo designs mean.
Eagle tattoos designs are very popular, and this Buzzle gives you some interesting ideas you can use if you are contemplating getting this tattoo done.
Ever thought of getting rooster tattoos? This Buzzle article features some really interesting designs to get you started.
Considering getting a fish tattoo inked? Perhaps the following article can help you finalize a design and make it your own.
The turtle tattoo is one of the unique designs found today. This article will give you more information on meanings of these tattoos with different designs ideas.
A sparrow tattoo symbolizes love, freedom, loyalty, etc. This write-up provides information on its meaning and design ideas.
Cats will always be a subject of scary tales and myths. Whether you love them or hate them, you cannot deny their presence in our day-to-day lives. Same goes for cat tattoo designs.
Raging bull or bull head tattoos are the most popular masculine tattoo designs today. Here is more information on it.
A peacock feather tattoo is a design, in which there is no exaggeration, no artificial coloring, and no unusual theme attached to it. Here is more information on this wonderful tattoo design.
Dolphin tattoos look elegant and beautiful, on both men and women. These designs are a part of the traditional Hawaiian tattoo designs.
Eagle tattoos are able to capture the majesty and beauty of this regal bird of prey. Of all the various tattoo designs, ones with the bald eagle are the most popular among body art aficionados.
When it comes to tattoo designs, there's no paucity to the array of motifs available. Let us take a quick look at the popularity of tiger designs today.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,369
|
This painting is part of a series of 13 pieces inspired by the philosophies and techniques of European winemakers. The piece is on a thick gallery wrapped canvas with finished white edges. It comes ready to hang.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,447
|
Нидерэбларн () — община () в Австрии, в федеральной земле Штирия.
Входит в состав округа Лицен. Население составляет 560 человек (на 31 декабря 2005 года). Занимает площадь 20,95 км². Официальный код — 61227.
Политическая ситуация
Бургомистр общины — Вальтер Граймайстер (АНП) по результатам выборов 2005 года.
Совет представителей общины () состоит из 9 мест.
Распределение мест:
АНП занимает 7 мест;
СДПА занимает 2 места.
Ссылки
Официальная страница
Города Штирии
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,111
|
\section{\label{sec:Intro} Introduction}
The study of hadron structure in terms of quantum chromodynamics and the
underlying quarks and gluons is a major focus of modern physics. Due to the
nature of confinement and the complex internal dynamics involved, however, QCD
calculations of hadron properties have proved challenging. The recent proton
radius ``puzzle''~\cite{Pohl:2013,Gao:2021} and the many measurements and theoretical
developments it has spurred, have emphasized that, while the proton is one of
the basic building blocks of matter and the most familiar of all hadrons, we
still do not fully understand its properties and structure. Advances in effective field
theories~\cite{Lensky:2010,Griesshammer:2012,Lensky:2014,Lensky:2015}, dispersion relation
analyses~\cite{Drechsel:1999,Holstein:1999,Drechsel:2003,Pasquini:2007}, and lattice QCD~\cite{Detmold:2006}
have added impetus to obtain more accurate measurement of hadron structure observables, such as
polarizabilities and charge radii.
An object's polarizabilities characterize its internal response to applied external
electric ($\vec{E}$) and magnetic ($\vec{H}$) fields; they are fundamental properties such as mass and the charge and at the microscopic level, they can be accessed via Compton scattering.
In the expansion of the effective Hamiltonian in incident photon energy $\omega_\gamma$, the electric ($\alpha_{E1}$) and magnetic ($\beta_{M1}$) polarizabilities enter at $\mathcal{O}(\omega_{\gamma}^2)$~\cite{Babusci:1998}:
\begin{equation}
H^{(2)}_{eff} = -4\pi \Bigl[\frac{1}{2} \alpha_{E1} \vec{E}^2 + \frac{1}{2} \beta_{M1} \vec{H}^2 \Bigr];
\label{eq:H2}
\end{equation}
while the four spin polarizabilities ($\gamma_{EiMj}$) are included at $\mathcal{O}(\omega_{\gamma}^3)$:
\begin{align*}
H^{(3)}_{eff} &= -4\pi \Bigl[ \frac{1}{2} \gamma_{E1E1} \vec{\sigma} \cdot (\vec{E} \times \dot{\vec{E}}) + \frac{1}{2} \gamma_{M1M1} \vec{\sigma} \cdot (\vec{H} \times \dot{\vec{H}}) \\
&\quad- \gamma_{M1E2} E_{ij} \sigma_{i}H_{j} + \gamma_{E1M2} H_{ij} \sigma_{i}E_{j} \Bigr],
\stepcounter{equation}\tag{\theequation}\label{eq:H3}
\end{align*}
where $\vec{\sigma}$ are the proton's Pauli spin matrices, $\dot{\vec{E}} = \partial_{t} \vec{E}$ and $E_{ij} = \frac{1}{2}(\nabla_i E_j + \nabla_j E_i$) are partial derivatives with respect to time and space, respectively.
Considerable experimental effort has been expended over the
last half century to obtain the scalar polarizabilities of the
proton~\cite{Baranov:1974,Federspiel:1991,MacGibbon:1995,Olmos:2001}, and recent measurements have
resulted in extractions of the proton's heretofore unknown individual spin
polarizabilities~\cite{Martel:2015,Paudyal:2020}. Work has also been carried out on the
neutron's scalar polarizabilities, but due to the absence of a free-neutron
target their unambiguous extraction has proved more
challenging~\cite{Hornidge:2001,Kossert:2002,Schumacher:2005,Myers:2014}.
Polarizabilities are of interest, not only in the study of hadron structure
where they can provide input to the QCD puzzle, but also in other fields including precision atomic physics and astrophysics.
They yield an appreciable correction to the proton charge radius via the Lamb shift and
hyperfine structure~\cite{Drell:1967,Bernabeu:1983,Faustov:2000,Carlson:2011} and
influence neutron star properties~\cite{Bernabeu:1974}. Moreover, a precise
determination of the proton scalar polarizabilities is integral to the
extraction of the proton spin polarizabilities, as the latter appear at higher
order in the expansion of the Compton scattering
Hamiltonian~\cite{Low:1954,GellMann:1954,Klein:1955,Petrunkin:1961,Babusci:1998}.
Furthermore, based solely on improved chiral effective field theory analyses and attempts to curate a statistically consistent database~\cite{Lensky:2010,Griesshammer:2012,Lensky:2014,Lensky:2015}, the Particle Data Group have recently adjusted their values of the proton scalar
polarizabilities~\cite{PDG:2014} without using new experimental data. This clearly demonstrates the necessity to obtain a new Compton scattering dataset with high statistical accuracy and low systematic errors in order to constrain the extraction
of the scalar polarizabilities.
The measurement reported here represents the highest statistics Compton
scattering data ever obtained on the proton --- roughly one million Compton
events below pion photoproduction threshold --- resulting in an improvement of a factor of
approximately five over the world's previous best measurement~\cite{Olmos:2001}.
It builds on our recently reported pilot experiment where the photon beam asymmetry below pion
threshold was obtained for the very first time~\cite{Sokhoyan:2017}.
\section{\label{sec:Exp} Experimental Setup and Data Handling}
The experimental data were obtained in two beamtimes in 2018 using tagged photons at the MAMI electron microtron facility~\cite{Kaiser:2008,Jankowiak:2006}.
The electron beam, with an energy of $883$~MeV, impinged on a $10~\mu$m thin diamond radiator, producing a linearly polarized photon beam via coherent Bremsstrahlung~\cite{Lohmann:1994}, with a degree of polarization up to $78\%$.
The recoiling electrons from the Bremsstrahlung process were momentum analyzed using the Glasgow photon tagging spectrometer~\cite{McGeorge:2007}.
Only photons with energies in the range $\omega_{\gamma} = 85 - 140$~MeV were considered in this analysis.
The resulting photon beam passed a $3$-mm-diameter lead collimator and was incident on a $10$-cm-long liquid hydrogen target.
The final state particles were detected using the same Crystal Ball (CB)/TAPS detector system as in the pilot experiment~\cite{Sokhoyan:2017} with a nearly complete solid angle coverage. Additional information on the apparatus used for these measurements can be found in Refs.~\cite{Unverzagt:2008,Prakhov:2008}.%
\begin{figure}[]%
\begin{center}%
\includegraphics[width=0.45\textwidth,angle=0]{Fig1.pdf}%
\end{center}%
\caption{Example of missing mass distribution for $\theta_{\gamma'} = 120-130^{\circ}$ and $\omega_{\gamma} = 118-130~\text{MeV}$. The measured missing mass distribution is shown together with the simulated Monte Carlo one, in black and orange, respectively. The gray dotted lines show the selection applied in the analysis.}
\label{fig:missingmass}
\end{figure}
\begin{figure*}%
\begin{center}%
\subfloat[\label{fig:Results:a}Unpolarized differential cross-section.]{\includegraphics[width = 0.33\linewidth]{Fig2a_1.pdf} \includegraphics[width = 0.33\linewidth]{Fig2a_2.pdf}}%
\subfloat[\label{fig:Results:b}Beam asymmetry.]{\includegraphics[width = 0.33\linewidth]{Fig2b.pdf}}%
\end{center}%
\caption{Examples of our new data on the proton Compton scattering. Panel (a) shows 24 of the 60 total points of unpolarized differential cross-section. The azure triangles show previous measurement from the TAPS collaboration~\cite{Olmos:2001} for the closest energy bin ($\omega_{\gamma}$ = 98.9 and 134.7 MeV from left to right, respectively). Panel (b) shows 12 of the 36 total points of beam asymmetry $\Sigma_3$. In all the plots, the errors are statistical only. The systematic uncertainties are depicted as gray and azure bars for the new points and the TAPS results, respectively. The solid blue curves represent the Born contribution only. The long-dashed orange, short-dashed purple and dotted-dashed green curves represent our fit results reported in \cref{tab:results}, obtained within DR~\cite{Drechsel:1999,Holstein:1999,Pasquini:2007}, B$\chi$PT~\cite{Lensky:2010} and HB$\chi$PT~\cite{McGovern:2012} frameworks, respectively. All the measured points are reported and plotted in the Supplemental Material~\cite{SuppMat}}%
\label{fig:Results}%
\end{figure*}%
The experiment was performed with two different orthogonal orientations of the photon polarization vector (formed by the momentum of the incoming photon and its electric field vector) and to minimize the systematic uncertainty the polarization vector was flipped approximately every two hours.
The degree of linear polarization, that depends on the photon energy and crystal orientation, was directly extracted from experimental data following the procedure described in Ref.~\cite{Livingston:2008}.
The Compton scattering events, $\vec{\gamma} p \to \gamma p$, were selected by requiring exactly one photon in the CB (for the incident photon energies of interest the recoil protons are typically not energetic enough to reach the detector apparatus).
The photon is identified as a cluster in the calorimeter without any associated hit in the Particle Identification Detector nor in any of the two Multiwire Proportional Chambers.
Due to a significant beam-related electromagnetic background in the forward region, only events with outgoing photon scattering angle $\theta_{\gamma'} > 30^{\circ}$ were considered.
Moreover, due to the relatively high photon flux, a time coincidence within $3$~ns was required between the scattered photon and the hits in the tagger focal plane.
To remove the random coincidences in the selected time window, a side-band subtraction was also performed by selecting a background sample on each side of the prompt peak.
Furthermore, the background generated from the the non-hydrogenic components of the target (Kapton cell, insulating material, etc.) was sampled during dedicated empty-target runs, and subtracted from the full-target sample.
\cref{fig:missingmass} shows a sample of the missing mass distribution calculated as
\begin{equation}
M_{miss} = \sqrt{(\omega_{\gamma} + m_p - \omega_{\gamma'})^2 - (\vec{k} - \vec{k}')^2 },
\label{eq:missingmass}
\end{equation}
where $k = (\omega_{\gamma}, \vec{k})$ and $k' = (\omega_{\gamma'}, \vec{k'})$ are the incoming and scattered photon four-momenta, respectively, and $m_p$ is the target proton mass at rest.
For a Compton scattering event from a proton in the target cell we expect $M_{miss}$ to be in agreement with the proton mass.
The distribution is plotted together with the one obtained from a Monte Carlo simulation of the full experimental apparatus, based on the Geant4 package~\cite{GEANT4:2002}.
The good agreement between the data and the simulated distribution indicates a low background contamination in the final sample, and to remove any remaining background a cut on missing mass is applied (\cref{fig:missingmass}), which was optimized using the Monte Carlo simulation.
From the final dataset, the unpolarized differential cross-section was extracted for five approximately $10$~MeV-wide photon energy bins spanning the range $\omega_{\gamma} = 86$ to $140$~MeV and twelve $10^{\circ}$-wide polar angular bins from $\theta_{\gamma'} = 30^{\circ}$ to $150^{\circ}$, for a total of 60 points.
A sample of the obtained results is plotted in \cref{fig:Results:a} as the statistical-error-weighted average between the two beamtimes.
The error bars are statistical only with the systematic uncertainty given by gray bars. They include both correlated and point-to-point uncorrelated errors.
The former comes from three independent sources that give in total a systematic uncertainty of $3\%$: target density ($1\%$), photon flux normalization ($2\%$), and analysis cuts and Monte Carlo simulation ($2\%$).
The latter comes from the remaining background contamination in the final sample and it was estimated from the small yield evident to the right of the proton peak before the missing mass cut (\cref{fig:missingmass}).
This background yield was found to be energy- and polar-angle-dependent and ranges from $10\%$ at low beam energy and scattering angle to $0.2\%$ at high energy and central angular bins.
The consistency of the two beamtimes was checked looking at the distribution of the normalized residuals of the two cross-sections calculated using the two different periods.
The results were found to be in perfect agreement for all energy bins, but the lowest one ($\omega_{\gamma} = 86.3 - 98.2~\text{MeV}$) in which one of the two beamtimes was about $5\%$ higher than the other one.
To account for this, a $3\%$ systematic uncertainty was included in the uncorrelated systematic uncertainties of this lowest energy bin.
The numerical values are tabulated and plotted in the Supplemental Material~\cite{SuppMat}.
Our results in \cref{fig:Results:a} are compared with the previously published data from the TAPS collaboration~\cite{Olmos:2001}.
The Born term, describing the proton as a point-like particle without internal structure aside from the anomalous magnetic moment, is shown along with the fit results reported in \cref{tab:results}, obtained within different theoretical frameworks: Dispersion relation (DR)~\cite{Drechsel:1999,Holstein:1999,Pasquini:2007}, Baryon Chiral Perturbation theory (B$\chi$PT)~\cite{Lensky:2010}, and Heavy Baryon Chiral Perturbation theory (HB$\chi$PT)~\cite{McGovern:2012}.
The main difference between the two $\chi$PT variants is whether the nucleon is treated relativistically (B$\chi$PT) or an amplitude expansion in powers of $1/M_{N}$ is performed (HB$\chi$PT) (see Ref.~\cite{Lensky:2012} for a comprehensive comparison).
The three theories fit our data equally well, as confirmed also from the $\chi^2$ values in \cref{tab:results}.
The experimental data show divergence from a calculation based on only the Born term, showing the sensitivity of the data to the proton internal structure, which at this energy is mainly, but not exclusively, described by the scalar polarizabilities.
The improvement in the statistical quality of the new data compared to the previous TAPS measurement is clear from \cref{fig:Results:a}. The new measurement also provides a wider angular coverage and smaller systematic errors.
The combination of the linearly polarized photon beam with the unpolarized LH$_2$ target results in azimuthal dependence to the Compton scattering cross-section, allowing for the determination of the single polarization observable $\Sigma_3$ defined as~\cite{Babusci:1998}
\begin{equation}
\Sigma_3 = \dfrac{d\sigma_{\parallel} - d\sigma_{\perp}}{d\sigma_{\parallel} + d\sigma_{\perp}},
\label{eq:sigma3}
\end{equation}
where $d\sigma_{\parallel(\perp)}$ is the polarized cross-section obtained with one of the two orthogonal orientations of the photon polarization vector, usually named ``parallel'' and ``perpendicular''.
The beam asymmetry was extracted in the same energy and angular range as the unpolarized cross-section, using a procedure similar to the one in Ref.~\cite{Sokhoyan:2017}. Due to the larger statistical uncertainties, to achieve adequate statistics the current data were binned in three photon energy regions, for a total of 36 new points.
A sample of our results is shown in \cref{fig:Results:b}.
The error bars shown are statistical only. The systematic uncertainties are depicted as gray bars.
The main contribution to the systematic uncertainty comes from the procedure to extract the linear polarization degree from the data, and an upper limit to this is estimated to be $5\%$, uniformly distributed.
An additional uncorrelated point-to-point contribution coming from the background contamination was also estimated, using a method similar to that employed for the unpolarized cross-section.
The numerical values are tabulated and plotted in the Supplemental Material~\cite{SuppMat}.
Our asymmetry results in \cref{fig:Results:b} are plotted together with the fit results obtained using the same theoretical frameworks as for the differential cross-sections shown in \cref{fig:Results:a}.
Also in this case, the three models can fit our results equally well.
However, in all three photon energy bins the fit results are distinct from the leading Born term, indicating that the asymmetry is sensitive to the proton structure constants.
A more comprehensive description of this work can be found in Ref.~\cite{Mornacchi:2021}.
\section{\label{sec:Results} Results and Discussion}
\begin{table*}[]
\caption{Scalar polarizabilities extracted by fitting the new unpolarized cross-section and beam asymmetry data using HDPV DR~\cite{Drechsel:1999,Holstein:1999,Pasquini:2007}, B$\chi$PT~\cite{Lensky:2010}, and HB$\chi$PT~\cite{McGovern:2012} code. The errors are statistical, systematic, and from the spin polarizabilities, respectively. The spin polarizabilities were fixed to the last experimental values available~\cite{Paudyal:2020}. $s_{\sigma}$ and $s_{\Sigma}$ are the normalization factors for the unpolarized cross-section and the beam asymmetry, respectively. The scalar polarizability values are given in units of $10^{-4}~\text{fm}^3$.} \label{tab:results}
\resizebox{\linewidth}{!}{%
\begin{tabular}{|l|c|c|c|}
\hline
& HDPV & B$\chi$PT & HB$\chi$PT \\ \hline
$\alpha_{E1}$ & $11.23 \pm 0.16 \pm 0.46 \pm 0.02$ & $10.65 \pm 0.16 \pm 0.47 \pm 0.04$ & $11.10 \pm 0.16 \pm 0.47 \pm 0.17$ \\
$\beta_{M1}$ & $2.79 \pm 0.20 \pm 0.23 \pm 0.11$ & $3.28 \pm 0.21 \pm 0.24 \pm 0.09$ & $3.36 \pm 0.21 \pm 0.24 \pm 0.20$ \\ \hline
$s_{\sigma}$ & $1.011 \pm 0.015$ & $1.013 \pm 0.015$ & $1.043 \pm 0.016$ \\
$s_{\Sigma}$ & $0.994 \pm 0.015$ & $0.996 \pm 0.015$ & $1.001 \pm 0.015$ \\ \hline
$\chi^2$/DOF & $82.10/93 = 0.89$ & $82.96/93 = 0.89$ & $83.16/93= 0.89$ \\ \hline
\end{tabular}%
}
\end{table*}
A fit to extract the proton scalar polarizabilities from all of the new data was performed using three different models: fixed-t DRs~\cite{Drechsel:1999,Holstein:1999,Pasquini:2007}, B$\chi$PT~\cite{Lensky:2010}, and HB$\chi$PT~\cite{McGovern:2012}.
The unpolarized cross-section and the beam asymmetry were given as input to the fitter as two independent datasets.
The uncorrelated point-to-point systematic errors were added in quadrature to the statistical ones.
The correlated systematic uncertainties were included in the fit as common normalization factors, one for each dataset, and treated as additional fit parameters.
Their deviations from the expected value of 1 were also accounted for in the $\chi^2$ function to be minimized~\cite{DAgostini:1993}.
The minimization was performed by using
the MINUIT minimization routine~\cite{James:1975}.
In order to emphasize the sensitivity of the new data to $\alpha_{E1}$ and $\beta_{M1}$ as much as possible, the spin polarizabilities were kept fixed to the most recent experimental values~\cite{Paudyal:2020}.
Moreover, to minimize the statistical uncertainty, the well-known Baldin Sum rule was included as an additional data point to be fitted at $\alpha_{E1} + \beta_{M1} = 13.8 \pm 0.4$ (in the usual units)~\cite{Olmos:2001}.
\begin{figure}[]
\begin{center}
\includegraphics[width=0.45\textwidth,angle=0]{Fig3.pdf}
\end{center}
\caption{Results of $\alpha_{E1}$ vs $\beta_{M1}$ for the proton, obtained from different experiments and theories. The extraction from our data is depicted as blue full ellipse. The loosely dotted azure ellipse shows the result from the TAPS collaboration~\cite{Olmos:2001}. The dotted purple circle is the B$\chi$PT prediction~\cite{Lensky:2010}, the green dashed-dotted curve is the extraction within HB$\chi$PT~\cite{McGovern:2012}, and the orange dashed curve is the bootstrap-based fit using DR~\cite{Pasquini:2019, Pedroni:2019}. The black circle shows the values quoted by the PDG~\cite{Zyla:2020}. The Baldin sum rule constraint was used in the present extraction, as well as in those from TAPS, HB$\chi$PT, and HDPV. All contours correspond to 1$\sigma$ level.}
\label{fig:alphabeta2D}
\end{figure}
The fit results are summarized in \cref{tab:results}.
The errors quoted in the central values of $\alpha_{E1}$ and $\beta_{M1}$ are statistical, systematic and from the spin polarizabilities, respectively.
The first one was obtained by performing the fit without the normalization factors.
The second error is given by how much the errors on the parameters changed by the inclusion of the systematic errors.
The third error is given by the variation in the best value of $\alpha_{E1}$ and $\beta_{M1}$ when the spin polarizabilities are not fixed, but rather free to vary within their experimental errors.
The small systematic error for the latter term indicates the new dataset has only a limited dependency on the spin polarizabilities, and thus making it well suited for a precise study of the two scalar terms.
The extractions of the scalar polarizabilities reported in \cref{tab:results} --- in particular of $\beta_{M1}$ --- exhibits a moderate model dependence.
To provide a best estimate of the central values for the two parameters, the results from the three theories were combined using weighted average, taking the quadratic sum of the statistical and systematic uncertainties as weights. For each error the largest contributions among the different theories was assigned.
Additionally, the largest of the differences between each theory and the average was used to estimate an additional error due to the model dependence for both $\alpha_{E1}$ and $\beta_{M1}$.
The best values for the extraction of the scalar polarizabilities from the new data using the Baldin sum rule constraint are
\begin{align*}
&{}\alpha_{E1} = 10.99 \pm 0.16 \pm 0.47 \pm 0.17 \pm 0.34 \\
&{}\beta_{M1} = 3.14 \pm 0.21 \pm 0.24 \pm 0.20 \pm 0.35 \stepcounter{equation}\tag{\theequation}\label{eq:finalresults}
\end{align*}
where the errors are statistical, systematic, spin polarizability dependent and model dependent.
A correlation coefficient between the two scalar polarizabilities of $\rho_{\alpha_{E1}-\beta_{M1}} = -0.75$ was also reported by the fitter.
The effect of the constraint was checked by repeating the fits without the additional point at $\alpha_{E1} + \beta_{M1} = 13.8 \pm 0.4$.
The obtained values for $\alpha_{E1}$ and $\beta_{M1}$ are in agreement with the ones of \cref{eq:finalresults} within $1.5\sigma$ and $0.5\sigma$, respectively, indicating the limited effect of the constraint on the final results.
\Cref{fig:alphabeta2D} shows the scalar polarizability extraction from this work as the blue full ellipse. Also shown are various previously published global extractions and predictions of these two parameters.
The azure dotted circle shows in particular the results from the TAPS collaboration~\cite{Olmos:2001}, the highest statistics dataset published previously.
The improvement in the uncertainty of the scalar polarizabilities extracted from the new data is clearly visible.
\section{\label{sec:final} Summary}
In summary, a new precision measurement of the proton Compton scattering unpolarized cross-section and
beam asymmetry is presented.
A fit to the new data using different theoretical models resulted in an extraction of the scalar polarizabilities $\alpha_{E1}$ and $\beta_{M1}$ from one consistent dataset with an unprecedented precision. The new results will be important for resolving the current ambiguities in the extraction of these fundamental quantities.
Moreover, these new experimental data can be used in combination with the already published ones on single and double polarization observables from the A2 collaboration~\cite{Martel:2015,Sokhoyan:2017,Paudyal:2020}, to obtain the first combined extraction of all the six proton polarizabilities from experimental data measured at a single facility, achieving an important new milestone in the MAMI program.
\begin{acknowledgments}
The authors wish to acknowledge the outstanding support of the accelerator group and operators of MAMI. We also wish to acknowledge and thank J.~McGovern, V.~Pascalutsa, and B.~Pasquini for providing us with their theory codes, together with H.~Grie{\ss}hammer and M.~Vanderhaegen for the theoretical contributions and support.
This project has received funding from the European Union's Horizon 2020 research and
innovation program under grant agreement No 824093.
This work has been supported by the U.K. STFC (ST/L005719/1, ST/P004458/1, ST/T002077/1,ST/P004385/2, ST/V002570/1, ST/P004008/1 and ST/L00478X/2) grants, the Deutsche Forschungsgemeinschaft (SFB443, SFB/TR16, and SFB1044), DFG-RFBR (Grant No. 09-02-91330), Schweizerischer Nationalfonds (Contracts No. 200020-175807, No. 200020-156983, No. 132799, No. 121781, No. 117601), the U.S. Department of Energy (Offices of Science and Nuclear Physics, Awards No. DE-SC0014323, DEFG02-99-ER41110, No. DE-FG02-88ER40415, No. DEFG02-01-ER41194) and National Science Foundation (Grants NSF OISE-1358175; PHY-1039130, PHY-1714833, PHY-2012940 No. IIA-1358175), INFN (Italy), and NSERC of Canada (Grant No. FRN-SAPPJ2015-00023).
\end{acknowledgments}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Breakthrough research offers alternative to battery power
Ground-breaking research from the University of Surrey and Augmented Optics Ltd, in collaboration with the University of Bristol, has developed potentially transformational technology which could revolutionise the capabilities of appliances that have previously relied on battery power to work.
This development by Augmented Optics Ltd, could translate into very high energy density super-capacitors making it possible to recharge your mobile phone, laptop or other mobile devices in just a few seconds.
It could also revolutionise the use of electric cars, allowing the possibility for them to recharge as quickly as it takes for a regular non-electric car to refuel with petrol and to travel from London to Edinburgh, for example, without the need to recharge.
The technology was adapted from the principles used to make soft contact lenses, which Dr Donald Highgate (of Augmented Optics, and an alumnus of the University) developed following his postgraduate studies at Surrey 40 years ago.
Supercapacitors, an alternative power source to batteries, store energy using electrodes and electrolytes and both charge and deliver energy quickly, unlike conventional batteries which do so in a much slower, more sustained way. Supercapacitors have the ability to charge and discharge rapidly over very large numbers of cycles. However, because of their poor energy density per kilogramme (approximately just one twentieth of existing battery technology), they have, until now, been unable to compete with conventional battery energy storage in many applications.
The technology uses materials based on large organic molecules composed of many repeated sub-units and bonded together to form a three-dimensional network.
Dr Brendan Howlin, of the University of Surrey, explained: "There is a global search for new energy storage technology and this new ultra capacity supercapacitor has the potential to open the door to unimaginably exciting developments."
Co-Principal Investigator Dr Ian Hamerton at University of Bristol said: "While this research has potentially opened the route to very high density supercapacitors, the technology has many other possible uses in which tough, flexible conducting materials are desirable, including bioelectronics, sensors, wearable electronics, and advanced optics. We believe that this is an extremely exciting and potentially game changing development."
For more details see www.Supercapacitormaterials.com
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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College welcomes Class of 2021
As they arrived on campus Wednesday and Thursday, Connecticut College's newest camels were greeted by cheering student leaders, colorful signs and a simple message: Welcome home.
"This is where you will live for the next four years—a place of self-development, relationship and social responsibility; a place of intimate face-to-face interaction; a place of connections," Dean of the College Jefferson Singer told the new students at a welcome assembly.
The 447 members of the Class of 2021 and 24 transfer students are an impressive group. They hail from 25 countries, including Belgium, China, Egypt, Jamaica, Jordan, Kenya, Mauritius, Morocco, Pakistan, South Korea, Turkey and Vietnam, and they speak 28 languages, including Armenian, Bengali, Burmese, Haitian Creole, Hindi, Swahili, Tagalog and Urdu. Sixty-seven members of the class are the first in their families to go to college, while 18 are at least the second members of their families to attend Conn, and 73 percent graduated in the top 20 percent of their high school class.
The incoming students are early beneficiaries of Connections, which officially launched last year. Connections is the College's reinvention of the liberal arts; a new kind of curriculum that lets students integrate their interests into a meaningful educational pathway to carry them through college and into a fulfilling, effective career and life.
With Connections, the students will orchestrate their educations with the support of a team of advisers; learn to collaborate, innovate and solve new problems in a complex world; and put the world together in new ways with interdisciplinary study, a relevant internship, a world language and an interconnected outlook.
The new students will learn all about Connections, meet with their team of advisers, register for classes, and get to know each other and the faculty and staff of the College over the next several days. The five-day orientation, "Welcome Home," will culminate with the 103rd Convocation, a celebration of the opening of the academic year, on Monday, Aug. 28, at 4:30 p.m. on Jean C. Tempel '65 Green. Anne Bernhard, professor of biology, will deliver the keynote address, "Humpback Whales, Bacteria and the Liberal Arts."
Check out scenes from Arrival Day below. To see more photos, visit Connecticut College on Facebook.
Arrival Day 2017
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Team Advising
New arboretum director named
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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