text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Q: Laravel Mailables - Send Collection to View I am using Laravel 5.4 Mailables and cannot pass data from my collection to the view.
I have a collection of $users:
[
0 => {
"name": "Justin",
"email": "justin@test.com",
"passwordExpires": "7 days"
}
1 => {
"name": "Max",
"email": "max@test.com",
"passwordExpires": "2 days"
}
]
Call mailable with a collection of users:
Mail::to($users)->send(new PasswordExpiring());
But PasswordExpiring() does not have access to the current user being emailed. I could pass all users in the constructor but I have no way of knowing which one Mail is trying to send to.
How would I pass the current user data into my mailable view?
A: I just added a for loop and send the emails one at a time.
$users = $this->repo->getExpiring();
foreach ($users as $user)
{
$mail_user = $this->transform($user);
Mail::to($mail_user)->send(new PasswordExpiring($mail_user));
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,914 |
Wayne Levere Hays (* 13. Mai 1911 im Richland Township, Ohio; † 13. Februar 1989 in Wheeling, West Virginia) war ein US-amerikanischer Politiker der Demokratischen Partei. Vom 3. Januar 1949 bis zum 1. September 1976 war er Mitglied des Repräsentantenhauses der Vereinigten Staaten für den 18. Kongressdistrikt des Bundesstaates Ohio.
Biografie
Hays wurde im Osten Ohios geboren. 1933 graduierte er an der Ohio State University. Von 1939 bis 1945 sammelte er erste kommunalpolitische Erfahrungen. In dieser Zeit diente er als Bürgermeister von Flushing. Gleichzeitig zu diesem Amt saß Hays von 1914 bis 1942 im Senat von Ohio. Von 1945 bis 1949 war er vier Jahre lang Commissioner im Belmont County. 1941 wurde Hays zum Militärdienst bei der US Army einberufen. 1942 wurde er aufgrund medizinischer Probleme aus dem Militärdienst entlassen.
Bei den Kongresswahlen 1948 wurde Hays als Kandidat der Demokraten im 18. Kongressdistrikt von Ohio ins US-Repräsentantenhaus gewählt. Insgesamt gelangen ihm 13 Wiederwahlen. Während dieser Zeit war er von 1971 bis zu seinem Rücktritt Vorsitzender des einflussreichen United States House Committee on House Administration. Zu den Präsidentschaftswahlen 1972 erhielt Hays 5 Stimmen bei der Democratic National Convention, ohne jedoch eine Kandidatur angekündigt zu haben. Am 1. September 1976 trat Hays von seinem Mandat aufgrund eines Sex-Skandals zurück. Von 1979 bis 1981 war Hays nochmals politisch aktiv, er saß in dieser Zeit im Repräsentantenhaus von Ohio.
Hays starb 1989 in Wheeling in West Virginia. Er wurde auf dem Saint Clairsville Union Cemetery in St. Clairsville beigesetzt.
Weblinks
Mitglied des Repräsentantenhauses der Vereinigten Staaten für Ohio
Mitglied des Senats von Ohio
Mitglied des Repräsentantenhauses von Ohio
Bürgermeister (Flushing, Ohio)
Mitglied der Demokratischen Partei (Vereinigte Staaten)
Militärperson (United States Army)
US-Amerikaner
Geboren 1911
Gestorben 1989
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,607 |
\section{Introduction}
Random matrix theory (RMT) was introduced in the 1950s to describe the highly-excited energy levels of heavy nuclei \cite{wigner_1}-\cite{bohigas}. It has subsequently found numerous and important applications in many different branches of physics. A review of various research directions in RMT is given, for example, in \cite{nina, RMTbook}. Of all these directions, one is particularly unexpected: the use of RMT in number theory. Though statistical methods have many applications in number theory (see e.g. \cite{kac}) the importance of RMT has attracted particular attention in recent years. This line of investigation started with a theorem of Montgomery \cite{montgomery} relating to the pair correlation of the non-trivial zeros of the Riemann zeta function, and Dyson's remark that Montgomery's formula can be interpreted as saying that statistical properties of the zeros are the same as those of eigenvalues of large hermitian matrices with independent random elements. Odlyzko's extensive numerical computations \cite{Odlyzko} provided compelling support for this conjecture, but also drew attention to the slow approach to the limit where RMT is expected to hold. In the case of pair correlation, the approach to the limit was shown to be arithmetical in origin by Berry \cite{Berry} and a precise formula was derived in \cite{bogomolny_keating} which matches the numerical data extremely well \cite{SIAM}. An analogous formula for the zeros of Dirichlet $L$-functions was recently derived in \cite{bogomolny_keating2}.
Further developments in this direction came from a conjecture that the moments of the Riemann zeta function (as well other $L$-functions) can be calculated using the moments of characteristic polynomials in RMT \cite{keating, keating2}. Today there exists a large collection of conjectures which predict (in a good agreement with existing numerics) mean values of many different quantities related to number-theoretic zeta and $L$-functions using ideas from RMT. For a review of the background to this area, see \cite{KS, KS2}.
Unfortunately, only a limited number of rigorous results have been obtained in this field. Most conjectures are based on heuristic arguments which are very difficult (if not impossible) to justify mathematically. As different heuristics stress different points, it is of interest and importance to compare different methods of calculation. It is also often the case that these heuristic methods lead to new insights into similar problems concerning the spectral statistics of quantum chaotic systems (see, e.g., \cite{bogomolny_keating, KM}).
The purpose of this paper is to calculate a formula, like that derived in \cite{bogomolny_keating}, for the three-point correlation function for the Riemann zeros using a method proposed in \cite{varenna}. This function has already been obtained in \cite{triple} by Conrey and Snaith using the ratio conjecture \cite{ratio}, which follows from heuristic manipulations of the approximate functional equation of the zeta function developed in \cite{CFKRS}. Our calculation was carried out independently of, and at the same time as \cite{triple} but we hesitated to publish the result mainly because both methods are general, permit to calculate, in principle, all correlation functions, and should lead to the same formulae.
We decided to present our calculations now because there has recently been renewed interest in these kinds of formulae (see, e.g.~\cite{Zeev, CS}) and we believe that the approach we take here sheds significant new light on their structure. We emphasize that this approach is completely different to the one involving the ratios conjecture. It is based on the idea of exploiting random matrix universality, and it incorporates universal and non-universal (arithmetic) components in a novel way from the outset. In particular, it combines the determinental structure of random matrix theory with the arithmetical terms in a way that appears more natural than in other approaches. We see this as a significant advantage, because it is often a major difficulty to identify the combinatorial identities underpinning this structure \cite{nonlinearity, RudSar, Zeev, CS}. It is also worth remarking that the formulae that emerge from this kind of approach are useful for applications (cf.~\cite{nearest, DHKMS}). And finally, there has recently been considerable focus on random matrix universality (see, e.g., \cite{RMTbook}), and our approach is likely to be of interest in that context too.
The plan of the paper is the following. Section~\ref{rmt_universality} is devoted to a short discussion of random matrix universality. It is well known that standard random matrix ensembles with different one-body potentials lead to different (non-universal) densities of eigenvalues. Nevertheless, it is widely accepted that after unfolding, local statistical properties of eigenvalues for all 'reasonable' ensembles are the same. This universality leads to an explicit expression for the random matrix kernel, conjecturally valid for any mean density. In Section~\ref{riemann} an expression for the density of Riemann zeros calculated by taking into account a large but finite number of prime numbers is presented. The primes entering this formula are chosen in such a way that they can be considered as independent under an average over a large window of heights on the critical axis. Inserting the finite expression for the Riemann zeros density into the random matrix kernel gives us formal correlation functions of Riemann zeros. But the result inevitably has oscillations related with short primes. Averaging such oscillations over a large window leads to our main conjecture for correlation functions presented in Section~\ref{conjecture}. In Section~\ref{two_point} it is demonstrated how it is possible to derive from this conjecture the two-point correlation function of Riemann zeros. Section~\ref{three_point} contains an explicit calculation of the tree-point correlation function. The final expression agrees with the result of \cite{triple} obtained by a completely different method. Section~\ref{summary} is a brief summary of important formulae.
\section{Random matrix universality}\label{rmt_universality}
It is well known (see e.g. \cite{mehta}) that the standard Gaussian Unitary Ensemble of $N\times N$ random matrices (GUE) is determined as an ensemble of Hermitian matrices ($M_{mn}=M_{mn}^{\dag}$) whose elements are random variables with the joint probability
\begin{equation}
P(M_{mn})=C_N \mathrm{e}^{-\mathrm{Tr}\, M^2}\prod_{j=1}^N \mathrm{d}M_{nn}
\prod_{1\leq m<n\leq N}\mathrm{d} \mathrm{Re}\,M_{mn}\ \mathrm{d} \mathrm{Im}\,M_{mn}
\label{gaussian}
\end{equation}
where $C_N$ is a normalization constant.
A natural generalization of this ensemble consists in choosing instead of $\mathrm{Tr}\, M^2$ in the exponent an 'arbitrary' function $\mathrm{Tr}\, f(M)$ (called often the one-body potential)
\begin{equation}
P(M_{mn})=C_N \mathrm{e}^{-\mathrm{Tr}\,f( M)}\prod_{j=1}^N \mathrm{d}M_{nn}
\prod_{1\leq m<n\leq N}\mathrm{d} \mathrm{Re}\,M_{mn}\ \mathrm{d} \mathrm{Im}\,M_{mn}\ .
\label{ensemble_f}
\end{equation}
All such ensembles permit one to integrate over angle-type variables and get the joint probability density for the eigenvalues $\lambda_j$ of the matrices $M_{mn}$ \cite{mehta}
\begin{equation}
P(\lambda_1,\lambda, \ldots, \lambda_N)=C^{\prime}_N \prod_{1\leq i<j\leq N}(\lambda_j-\lambda_i)^2 \exp \left (-\sum_{k=1}^Nf(\lambda_k) \right )\ .
\end{equation}
To calculate the $n$-point correlation function one has to fix $n$ eigenvalues $x_j=\lambda_j$ with $j=1,\ldots, n$ and to integrate over the remaining $N-n$ variables. For the ensemble considered, this can be done by the method of orthogonal polynomials \cite{mehta}. One first introduces polynomials, $p_k(x)$ orthogonal with respect to the measure $\exp \left (-f(x) \right )$
\begin{equation}
\int \mathrm{e}^{-f(x) }p_k(x)p_r(x)\mathrm{d}x=\delta_{kr}\ .
\end{equation}
Then the $n$-point correlation function for the ensemble \eqref{ensemble_f} takes the form of the $n\times n$ determinant \cite{mehta}
\begin{equation}
R(x_1,\ldots,x_n)=\det( K_N(x,y) )_{x,y=x_1,\ldots, x_n}
\label{n_corr_functions}
\end{equation}
where the kernel $K_N(x,y)$ is expressed through the orthogonal polynomials as follow
\begin{equation}
K_N(x,y)=\sum_{j=0}^{N-1} p_j(x)p_j(y)\mathrm{e}^{-f(x)/2-f(y)/2}\ .
\end{equation}
In particular, the mean level density, $\bar{\rho}(x)$, (i.e. one-point correlation function) is
\begin{equation}
\bar{\rho}(x)=K_N(x,x)\ .
\label{density}
\end{equation}
Usually one is interested in the limit $N\to\infty$ and the main question is what is then the limiting behaviour of this kernel. For the Gaussian ensemble \eqref{gaussian} the answer is well known \cite{mehta} since the orthogonal polynomials in this case are the usual Hermite polynomials
\begin{equation}
p_j(x)=\frac{1}{\sqrt{2^jj!\sqrt{\pi}}}H_j(x)\ .
\end{equation}
The mean level density \eqref{density} in this case is given by the famous "semicircle law",
\begin{equation}
\bar{\rho}(x)=\left \{ \begin{array}{cc}\dfrac{1}{\pi}\sqrt{2N-x^2},& |x|<\sqrt{2N}\\0,&|x|>\sqrt{2N}\end{array}\right . \ ,
\end{equation}
and there exists an explicit formula for the kernel $K_N(x,y)$ when $N$ is large. This formula takes an especially simple form in the bulk of the spectrum when $|x|,|y|\ll \sqrt{N}$
\begin{equation}
K_N(x,y)=\frac{\sin \pi \bar{\rho}(0)(x-y)}{\pi (x-y)}
\label{kernel_gue}
\end{equation}
where $\bar{\rho}(0)=\sqrt{2N}/\pi$ is the level density \eqref{density} at small $x$.
For a general one-body potential $f(x)$ the situation is more difficult. The mean level density can be calculated for large $N$ from the Dyson equation \cite{mehta}
\begin{equation}
\mathrm{P}\int_{-\infty}^{\infty}\frac{\bar{\rho}(z)}{x-z}\mathrm{d}z=\tfrac{1}{2}f^{\prime}(x)
\label{dyson}
\end{equation}
where P indicates the principal value integral, but correlation functions are more difficult to obtain though
there exists a vast literature on this subject (see e.g. \cite{orthogonal} and references therein).
Instead of using rigorous asymptotic formulae, we shall argue as follows. It is well known that the mean level density is not a universal quantity. Different one-body potentials, $f(x)$, lead to different densities (cf. \eqref{dyson}). On the other hand, it is widely accepted that after unfolding all correlation functions in the local scale should be universal. The unfolding here means that one calculates statistical properties not of the true levels, $\lambda_n$, (which, in general, have a non-universal mean density $\bar{\rho}(x)$) but of new quantities
\begin{equation}
e_n=\bar{N}(\lambda_n)
\end{equation}
where $\bar{N}(x)$ is the mean number of levels with $\lambda_j<x$, $\bar{N}(x)=\int^x \bar{\rho}(y) \mathrm{d}y$.
By construction, the new levels, $e_n$, have unit mean density, and random matrix universality asserts that for these quantities the kernel has exactly the same form as in \eqref{kernel_gue} with $\bar{\rho}(0)=1$
\begin{equation}
K(x,y)=\frac{\sin \pi (\bar{N}(x)-\bar{N}(y) )}{\pi (x-y)}\ .
\label{k_general}
\end{equation}
This formula is assumed to be valid when (i) points $x$ and $y$ are far from the ends of the spectrum, and (ii) the mean number of levels $\bar{N}(x)$ is a smooth function of $x$ i.e. it changes slowly in the scale of the nearest levels.
This expression is the concise manifestation of random matrix universality but we are not aware that it has been proved in full generality. Nevertheless, it agrees with all we know (or conjecture) about universal behaviour of random matrix ensembles (at least for GUE types) and we shall apply it below for the Riemann zeta function.
\section{Riemann zeta function}\label{riemann}
The Riemann zeta function is defined when Re$s>1$ as the sum over all integers (see e.g. \cite{riemann})
\begin{equation}
\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}
\end{equation}
or as the Euler product over prime numbers
\begin{equation}
\zeta(s)=\prod_{p}\left (1-\frac{1}{p^s}\right )^{-1}\ .
\label{euler}
\end{equation}
It has an analytic continuation to the rest of the complex $s$-plane, except for a pole at $s=1$. The celebrated Riemann Hypothesis states that all non-trivial zeros of this function, $\zeta(s_j)=0$, have the form
\begin{equation}
s_j=\tfrac{1}{2}+\mathrm{i}E_j
\end{equation}
with real $E_j$ (which may be thought of as analogous to quantum energies).
Assuming the Riemann Hypothesis, it is easy to write down a formal expression for the density of these zeros (called in the mathematical literature Weil's explicit formula \cite{weil}). Indeed if one writes $\zeta(1/2+\mathrm{i}E)\sim \prod_j(E-E_j)$ then the density of $E_j$ is
\begin{equation}
d(E)=-\frac{1}{\pi }\mathrm{Im}\,\frac{\partial }{\partial E}\ln \zeta \left (\tfrac{1}{2}+\mathrm{i}(E+\mathrm{i}\varepsilon)\right )_{\varepsilon\to +0}\ .
\end{equation}
Using the functional relation for the zeta function and \eqref{euler} one gets that, as usual, the density of zeros is a sum of two terms
\begin{equation}
d(E)=\overline{d(E)} +d^{(\mathrm{osc})}(E) \ ,
\end{equation}
where $ \overline{d(E)}$ is the smooth part of the density, which, as $E\rightarrow\infty$ is given by
\begin{equation}
\overline{d(E)}\approx\frac{1}{2\pi}\ln \frac{E}{2\pi} \ ,
\end{equation}
and $d^{(\mathrm{osc})}(E)$ is the oscillating part of the density
\begin{equation}
d^{(\mathrm{osc})}(E)=-\frac{1}{2\pi}\sum_{p}\sum_{n=1}^{\infty}
\frac{\ln p}{ p^{n/2}}\left (\mathrm{e}^{\mathrm{i} n E\ln p}+\mathrm{e}^{-\mathrm{i} n E\ln p}\right )\ .
\label{osc}
\end{equation}
Of course, the sum over all primes $p$ diverges at real $E$ and this expression has no (clear) mathematical meaning (similar to all "physical" trace formulae). It gains such a meaning when integrated against a sufficiently smooth test function.
Nevertheless, it is legitimate to write the 'true' density as a finite sum over primes with $p<p^*$ and an unknown remainder related with large primes satisfying $p>p^*$
\begin{equation}
d(E)=\bar{\rho}(E)+\mathrm{large\;primes},\qquad \bar{\rho}(E)=\overline{d(E)} +\widetilde{d(E,p^*)}
\label{full_density}
\end{equation}
where $\widetilde{d(E,p^*)}$ is the same sum as in \eqref{osc} but taken over a finite set of primes with $p<p^*$ (the value of $p^*$ will be chosen below)
\begin{equation}
\widetilde{d(E,p*)}=-\frac{1}{2\pi}\sum_{p<p*}\sum_{n=1}^{\infty}
\frac{\ln p}{ p^{n/2}}\left (\mathrm{e}^{\mathrm{i} n E\ln p}+\mathrm{e}^{-\mathrm{i} n E\ln p}\right )=
\frac{1}{2\pi \mathrm{i}}\frac{\partial}{\partial E}\sum_{p<p^*}
\ln\frac{1-A_p\mathrm{e}^{\mathrm{i}\Phi_p(E)}}{1-A_p\mathrm{e}^{-\mathrm{i}\Phi_p(E)}}\ .
\label{d}
\end{equation}
Here for further convenience we introduce the notation
\begin{equation}
A_p=\frac{1}{\sqrt{p}},\qquad \Phi_p(E)=E\ln p \ .
\end{equation}
The knowledge of $\bar{\rho}(E)$ permits easily to calculate the mean number of levels corresponding to this density
\begin{equation}
\bar{N}(x,p^*)\equiv \int^x_0 \bar{\rho}(E)\mathrm{d}E= \int^x_0 ( \overline{d(E)}+\widetilde{d(E,p^*)} )\mathrm{d}E\ .
\end{equation}
It is plain that
\begin{equation}
\mathrm{e}^{2\pi \mathrm{i} \bar{N}(E,p^*)}=\mathrm{e}^{2\pi \mathrm{i} \overline{N(E)}}\prod_{p<p^*}
\frac{1-A_p \mathrm{e}^{\mathrm{i}\Phi_p(E)} }{1-A_p\mathrm{e}^{-\mathrm{i}\Phi_p(E)} }
\label{N}
\end{equation}
where
\begin{equation}
\overline{N(E)}=\frac{E}{2\pi}\ln\frac{E}{2\pi \mathrm{e}}+\mathrm{const}\ .
\end{equation}
For the Riemann zeta function the constant is known ($7/8$) but is irrelevant for our purpose.
\section{Main conjecture}\label{conjecture}
The principal point in the approach to statistical properties of Riemann zeros advocated here consists in the assumption that the large primes indicated in \eqref{full_density} give rise to GUE correlations \eqref{n_corr_functions} with random matrix kernel \eqref{k_general} where $\bar{N}(E,p*)$ is determined by small primes \eqref{N}. Precisely,
\begin{equation}
K(E_i,E_j)=\frac{\sin(\pi (\bar{N}(E_i,p*)-\bar{N}(E_j,p*)))}{\pi (E_i-E_j)}\ .
\label{main_kernel}
\end{equation}
Of course, in such an approach the exact mechanism by which large primes conspire to give this kernel is completely ignored. But as we shall show below this assumption permits us to calculate all low order terms for correlation functions of Riemann zeros in agreement with ones calculated by different mathods.
When $n$-point correlation functions are calculated from \eqref{n_corr_functions} using the kernel \eqref{main_kernel}, the result necessarily has oscillations related with oscillations in the "mean" density of zeros \eqref{d} produced by short primes.
Usually one is looking for statistical properties of a set of zeros close to a large value of $E$. In this case it is natural to write $E_j=E+e_j$ and then to average around $E$. It means that we propose to calculate correlation functions of the Riemann zeros from the following expression
\begin{equation}
R_n(e_1,e_2,\ldots, e_n)=\left \langle \left \langle \det \Big ( K(E+e_i,E+e_j )\Big )_{i,j=1,\ldots,n}\right \rangle \right \rangle_{\Delta E} \ .
\label{R_n_definition}
\end{equation}
Here the average indicated by $\langle \langle \ldots \rangle \rangle_{\Delta E}$ is to be carried out over a large window of heights $E$
\begin{equation}
\langle \langle F(E) \rangle \rangle_{\Delta E} \equiv \frac{1}{\Delta E}\int_{E-\Delta E/2}^{E+\Delta E/2} F(E^{\prime})\mathrm{d}E^{\prime} \ .
\end{equation}
Let us choose the cut-off prime, $p^*$, and the window, $\Delta E$, to fulfil the inequalities
\begin{equation}
1\ll p^*\ll\Delta E\ll E\ .
\label{inequalities}
\end{equation}
This choice permits one, at least formally, to calculate all necessary mean values. In particular, one has
\begin{equation}
\langle \langle \mathrm{e}^{\mathrm{i} n E\ln p } \rangle \rangle_{\Delta E} =0,\qquad \mathrm{for}\; p<p^*,\qquad n\in Z^*\ ,
\end{equation}
and
\begin{equation}
\langle \langle \mathrm{e}^{ \mathrm{i} E(n_1\ln p_1 -n_2\ln p_2)} \rangle \rangle_{\Delta E} =\delta_{n_1,n_2} \delta_{p_1,p_2}\qquad \mathrm{for}\; p_1,p_2<p^*,\qquad n_1,n_2\in Z^* \ .
\label{orthogonality}
\end{equation}
Therefore
\begin{equation}
\langle \langle \tilde{d}(E,p^*) \rangle \rangle_{\Delta E} =0,\qquad \langle \langle \overline{d(E)} \rangle \rangle_{\Delta E} = \overline{d(E)}\ .
\label{rho_average}
\end{equation}
More generally, phases $E\ln p$ associated with different primes $p<p^*$ can be considered as independent random phases and the procedure of averaging a quasi-periodic function of these phases is reduced to the integration over them
\begin{equation}
\left \langle \left \langle F\Big (\mathrm{e}^{\mathrm{i}E\ln p_1},\ldots, \mathrm{e}^{\mathrm{i}E\ln p_n} \Big ) \right \rangle \right\rangle_{\Delta E}
=\int_0^{2\pi}\frac{\mathrm{d}\phi_1}{2\pi}\cdots \int_0^{2\pi}\frac{\mathrm{d}\phi_n}{2\pi}
F\Big (\mathrm{e}^{\mathrm{i}\phi_1},\ldots, \mathrm{e}^{\mathrm{i}\phi_n} \Big ) \ .
\label{good_average}
\end{equation}
Any averaging procedure for which this relation is fulfilled is suitable for our purposes and it can serve as the definition of 'good' averaging.
Eq.~\eqref{R_n_definition} together with \eqref{main_kernel} and \eqref{N} are our main formulae for correlation functions of zeros of the Riemann zeta function. In the next Sections we show how such formula can be used to calculate explicitly the two and three-point correlation functions. When performing the calculations we shall see that, after averaging, the remaining terms can be divided into two groups. The first contains various sums over primes such that they have well defined values in the formal limit $p^*\to\infty$. The second, which includes divergent contributions, can be transformed to the formally divergent product (with imaginary $s$)
\begin{equation}
f(s,p^*)= \prod_{p<p^*}\dfrac{1-p^{-1}}{1-p^{-1-s}}\ .
\end{equation}
Under the assumption that
\begin{equation}
1 \ll \ln(p^*)\ll 1/|s|\;.
\label{ll}
\end{equation}
This can be done as follows
\begin{eqnarray}
& &f(s,p^*)=\lim_{t\to 0}\prod_{p<p^*}\frac{1-1/p^{1+t}}{1-1/p^{1+s}}
=\lim_{t\to 0}\frac{\zeta(1+s)}{\zeta(1+t)}
\prod_{p>p^*}\frac{1-1/p^{1+s}}{1-1/p^{1+t}}\nonumber \\
&\approx &\lim_{t\to 0}\frac{\zeta(1+s)}{\zeta(1+t)}
\exp (\int_{\ln(p^*)}^{\infty}\frac{\mathrm{d}u}{u}(\mathrm{e}^{-tu}-\mathrm{e}^{-su}))
=\lim_{t\to 0}\frac{\zeta(1+s)}{\zeta(1+t)}\exp \ln(s/t)=s\zeta(1+s)\ .
\end{eqnarray}
In the last step we use that according to our assumption $s\ln (p^*)\ll 1$ (with, of course, $t\ln(p^*)\ll 1$), and
\begin{equation}
\int_{0}^{\infty}\frac{\mathrm{d} u}{u}(\mathrm{e}^{-tu}-\mathrm{e}^{-su})=\ln s -\ln t\ .
\end{equation}
(A more careful derivation can be given by using Eq.~3.14.1 of \cite{riemann}.)
This leads to the conclusion that under \eqref{ll}
\begin{equation}
\prod_{p<p^*}\frac{1-p^{-1}}{1-p^{-1-s}}\underset{p^*\to\infty}{\longrightarrow} s\zeta(1+s)
\label{zeta}
\end{equation}
and we shall use this expression throughout the paper.
\section{Two-point correlation function of Riemann zeros}\label{two_point}
The simplest non-trivial example of a correlation function of Riemann zeros is the two-point correlation function. It was calculated in \cite{bogomolny_keating} by using the explicit form of the Hardy-Littlewood conjecture concerning the distribution of prime pairs. See \cite{bogomolny_keating2} for an extension to Dirichlet $L$-functions. A formula identical to that obtained in \cite{bogomolny_keating} was shown also to follow from the ratios conjecture \cite{ratiosCS}. Here we show that exactly the same result is obtained form Eq.~\eqref{R_n_definition} based on completely different assumptions.
From \eqref{R_n_definition} one gets
\begin{eqnarray}
R_2(e_1,e_2)&=& \left \langle \left \langle \det \left ( \begin{array}{cc} K(E+e_1,E+e_1 ) & K(E+e_1,E+e_2 \\ K(E+e_2,E+e_1 )& K(E+e_2,E+e_2) \end{array} \right )\right \rangle \right \rangle_{\Delta E} \nonumber\\
& =& \langle \langle \bar{\rho}(E+e_1)\bar{\rho}(E+e_2)\rangle \rangle_{\Delta E} -
\langle \langle K^2(E+e_1,E+e_2) \rangle \rangle_{\Delta E} \ .
\end{eqnarray}
Here we use $K(E,E)=\bar{\rho}(E)$ and $K(E_1,E_2)=K(E_2,E_1)$. It is worth remarking that in this approach we start with a determinant - in other approaches the main difficultly lies in identifying the combinatorial identities that match with a determinental form. Noting that
\begin{eqnarray}
K(E_1,E_2)&=&\frac{\sin^2(\pi (\bar{N}(E_1,p*)-\bar{N}(E_2,p*)))}{\pi^2 (E_1-E_2)^2}\\
&=&-\frac{1}{4\pi^2(E_1-E_2)^2}\left (\mathrm{e}^{2\pi \mathrm{i} (\bar{N}(E_1,p^*)-\bar{N}(E_2,p^*)}-2+ \mathrm{e}^{-2\pi \mathrm{i} (\bar{N}(E_1,p^*)-\bar{N}(E_2,p^*)} \right ) \nonumber
\end{eqnarray}
and assuming the validity of \eqref{rho_average}, one gets
\begin{equation}
R_{2}(e_1,e_2)=\overline{d(E)}^{\,2}+R_{2}^{\mbox{c}}(e_{1},e_2)\ .
\end{equation}
$R_{2}^{\mbox{c}}(e_{1},e_2)$ is the connected part of the two-point correlation function equals to the sum of two terms, the smooth term, $R_{2}^{\mbox{diag}}(e_1, e_2)$, and the oscillatory term, $R_{2}^{\mbox{osc}}(e_1,e_2)$,
\begin{eqnarray}
R_2^{\mbox{c}}(e_1,e_2)&\equiv & \langle \langle K^2(E+e_1,E+e_2) \rangle \rangle_{\Delta E} =
R_{2}^{\mbox{diag}}(e_1, e_2)+R_{2}^{\mbox{osc}}(e_1,e_2) \ ,
\label{rtwo}
\end{eqnarray}
where
\begin{equation}
R_2^{\mbox{diag}}(e_1,e_2)= \langle \langle d_1\ d_2 \rangle \rangle_{\Delta E} -\frac{1}{2\pi^2\epsilon^2}
\label{R_diag}
\end{equation}
and
\begin{equation}
R_{2}^{\mbox{osc}}(e_1,e_2)=\frac{1}{4\pi^2 \epsilon^2} \langle \langle \mathrm{e}^{2\pi \mathrm{i} (N_1-N_2)}+\mathrm{e}^{-2\pi \mathrm{i} (N_1-N_2)}\rangle \rangle_{\Delta E}\ .
\label{R_osc}
\end{equation}
Here and below we use the following notations: $ d_j=\widetilde{d(E+e_j, p*)}$, $N_j=\bar{N}(E+e_j,p*)$, and $\epsilon=e_1-e_2$.
\subsection{Smooth terms}
The calculation of $\langle \langle d_1\ d_2 \rangle \rangle_{\Delta E}$ is straightforward and corresponds to the so-called diagonal approximation. From \eqref{d} one has
\begin{eqnarray}
&4\pi^2 & \langle \langle d_1\ d_2 \rangle \rangle_{\Delta E} =\sum_{p_{1,2}<p*}\sum_{n_{1,2}=1}^{\infty}
\frac{\ln p_1 \ln p_2}{p_1^{n_1/2}p_2^{n_2/2}} \Big \langle \Big \langle \left (\mathrm{e}^{\mathrm{i} n_1 (E+e_1)\ln p_1}+\mathrm{e}^{-\mathrm{i} n_1 (E+e_1)\ln p_1}\right ) \nonumber \\
&\times & \left (\mathrm{e}^{\mathrm{i} n_2 (E+e_2)\ln p_2}+\mathrm{e}^{-\mathrm{i} n_2 (E+e_2)\ln p_2}\right )\Big \rangle \Big \rangle_{\Delta E}
\nonumber\\
&=&\sum_{p<p*}\sum_{n=1}^{\infty}\frac{\ln^2 p}{p^{n} }
\Big (\mathrm{e}^{\mathrm{i} n \epsilon \ln p}+\mathrm{e}^{-\mathrm{i} n \epsilon\ln p}\Big )
= - \frac{\partial^2}{\partial \epsilon^2}\sum_{p<p*}\sum_{n=1}^{\infty}\frac{1}{n^2 p^{n} }
\Big(\mathrm{e}^{\mathrm{i} n \epsilon \ln p}+\mathrm{e}^{-\mathrm{i} n \epsilon\ln p}\Big )\nonumber\\
&=& - \frac{\partial^2}{\partial \epsilon^2}\sum_{p<p*}\sum_{n=2}^{\infty}\frac{1}{n^2 p^{n} }
\left (\mathrm{e}^{\mathrm{i} n \epsilon\ln p}+\mathrm{e}^{-\mathrm{i} n \epsilon\ln p}\right )
- \frac{\partial^2}{\partial \epsilon^2}\sum_{p<p*}\frac{1}{ p }
\left (\mathrm{e}^{\mathrm{i} \epsilon \ln p}+\mathrm{e}^{-\mathrm{i} \epsilon\ln p}\right )
\label{d_1_d_2}
\end{eqnarray}
When $p^*\to\infty$ only the last term diverges. To calculate this it is convenient to use Eq.~\eqref{zeta}. By taking the logarithm of the both parts of this relation and of its complex conjugate one obtains
\begin{equation}
\sum_{p<p^*}\sum_{n=1}^{\infty}\frac{1}{n p^n}\left (\mathrm{e}^{\mathrm{i} n s \ln p}+\mathrm{e}^{-\mathrm{i}n s\ln p}\right )=
2\ln s +\ln |\zeta(1+\mathrm{i}s)|^2 + C
\end{equation}
with a constant $C=-2\ln \prod_{p<p^*}(1-1/p)$. Consequently,
\begin{equation}
\sum_{p<p*}\frac{1}{ p }
\left (\mathrm{e}^{\mathrm{i} s\ln p}+\mathrm{e}^{-\mathrm{i} s\ln p}\right )=2\ln s +\ln |\zeta(1+\mathrm{i}s)|^2 -\sum_{p<p^*} \sum_{n=2}^{\infty}\frac{1}{n p^n}\left (\mathrm{e}^{\mathrm{i} n s \ln p}+\mathrm{e}^{-\mathrm{i}n s\ln p}\right )+ C\ .
\end{equation}
Substituting this relation (with $s=\epsilon$) into \eqref{d_1_d_2} and taking into account the fact that the sums with $n\geq 2$ converge and $C$ is independent on $\epsilon$, we conclude that
\begin{equation}
\langle \langle d_1\ d_2 \rangle \rangle_{\Delta E}=\frac{1}{2\pi^2 \epsilon^2}-\frac{1}{4\pi^2} \frac{\partial^2}{\partial \epsilon^2}\ln |\zeta(1+\mathrm{i} \epsilon )|^2
-\frac{1}{4\pi^2} \frac{\partial^2}{\partial \epsilon^2}\sum_{p<p*}\sum_{n=2}^{\infty}\frac{1-n}{n^2 p^{n} }
\left (\mathrm{e}^{\mathrm{i} n \epsilon\ln p}+\mathrm{e}^{-\mathrm{i} n\epsilon \ln p}\right )\ .
\end{equation}
Combining this expression and Eq.~\eqref{R_diag}, and using $\sum_{n=2}^{\infty}(1-n)x^n=-x^2/(1-x)^2$ gives
\begin{equation}
R_{2}^{\mbox{diag}}(\epsilon)=-\frac{1}{4\pi^2}\frac{\partial^2}{\partial \epsilon^2}
\ln |\zeta(1+\mathrm{i}\epsilon)|^2-
\frac{1}{4\pi^2}\sum_p \ln^2 p\left( \frac{1}{(p^{1+\mathrm{i}\epsilon }-1)^2}+
\frac{1}{(p^{1-\mathrm{i} \epsilon}-1)^2} \right )\ .
\label{r_2diag}
\end{equation}
\subsection{Oscillatory terms}
The next step consists in calculating the oscillatory part of the two-point correlation function given by \eqref{R_osc}. Substituting Eq.~\eqref{N} into \eqref{R_osc}, one gets
\begin{equation}
\left \langle \left \langle \mathrm{e}^{2\pi \mathrm{i} (N_1-N_2)} \right \rangle \right \rangle_{\Delta E} =
\mathrm{e}^{2\pi \mathrm{i} \overline{d(E)}(e_1-e_2)} \langle \langle R_p(E;e_1,e_2) \rangle \rangle_{\Delta E}
\label{mean_exp_N}
\end{equation}
where
\begin{equation}
R_p(E;e_1,e_2)= \prod_{p<p^*} \frac{[1-A_p \mathrm{e}^{-\mathrm{i}(\Phi_p(E)+e_2\ln p )}][1-A_p \mathrm{e}^{\mathrm{i}(\Phi_p(E)+e_1\ln p)}] }
{[1-A_p\mathrm{e}^{-\mathrm{i}(\Phi_p(E)+e_1\ln p)}][1-A_p \mathrm{e}^{\mathrm{i}(\Phi_p(E)+e_2\ln p)}] }\ .
\end{equation}
The averaging of $R_p(E,e_1,e_2)$ over $E$ can be done by using Eq.~\ref{good_average}. Therefore, the average over $E$ corresponds to the independent integration over phases $\Phi_p(E)=E\ln p_j$
\begin{equation}
\langle \langle R_p(E;e_1,e_2) \rangle \rangle_{\Delta E} =\prod_{p<p^*} \langle \mathrm{R}_p (\Phi_p; e_1,e_2) \rangle_{\Phi_p}
\end{equation}
where the average $\langle \mathrm{R}_p (\Phi_p; e_1,e_2) \rangle_{\Phi_p}$ is simply the mean value over all $\Phi_p$
\begin{equation}
\langle \mathrm{R}_p (\Phi_p; e_1,e_2) \rangle_{\Phi_p}=\frac{1}{2\pi}\int_0^{2\pi}\mathrm{R}_p (\Phi_p; e_1,e_2)\mathrm{d}\Phi_p
\ ,
\label{phi_average}
\end{equation}
and
\begin{equation}
\mathrm{R}_p (\Phi_p; e_1,e_2)=
\frac{[1-A_p \mathrm{e}^{-\mathrm{i}(\Phi_p +e_2\ln p)}][1-A_p \mathrm{e}^{\mathrm{i}(\Phi_p +e_1\ln p)} ] }
{[1-A_p\mathrm{e}^{-\mathrm{i}(\Phi_p +e_1\ln p)}][1-A_p \mathrm{e}^{\mathrm{i}(\Phi_p +e_2\ln p)}] }\ .
\label{R_p}
\end{equation}
The calculation of the integral can conveniently be performed by complex integration. Putting $z=\mathrm{e}^{\mathrm{i}\Phi_p}$, one gets
\begin{equation}
\langle \mathrm{R}_p (\Phi_p;e_1,e_2) \rangle_{\Phi_p} =
\frac{1}{2\pi} \oint \frac{[1-A_p z^{-1}\mathrm{e}^{-\mathrm{i}e_2\ln p }][1-A_p z \mathrm{e}^{\mathrm{i}e_1\ln p}] }
{[1-A_p z^{-1} \mathrm{e}^{-\mathrm{i}e_1\ln p}][1-A_p z \mathrm{e}^{\mathrm{i}e_2\ln p}] }\frac{\mathrm{d}z}{z}
\end{equation}
where the integral is taken over the unit circle in the complex plane.
As $A_p=p^{-1}<1$, inside the integration contour there are two poles. The first is at $z=0$ and the second at $z=A_p \mathrm{e}^{-\mathrm{i}e_1\ln p}$. Straightforward calculation gives
\begin{eqnarray}
\langle \mathrm{R}_p (\Phi_p;e_1,e_2) \rangle_{\Phi_p}&=&\mathrm{e}^{\mathrm{i}\epsilon \ln p}
+\frac{(1-A_p^2)(1-\mathrm{e}^{\mathrm{i}\epsilon \ln p })}{1-A_p^2\mathrm{e}^{-\mathrm{i}\epsilon \ln p }}=
\frac{1-2A_p^2+A_p^2\mathrm{e}^{\mathrm{i} \epsilon \ln p}}{1-A_p^2 \mathrm{e}^{-\mathrm{i}\epsilon \ln p }} \label{ww}\\
&=& \frac{(1-A_p^2)^2}{|1-A_p^2\, \mathrm{e}^{\mathrm{i}\epsilon \ln p }|^2}
\left ( 1-\frac{A_p^4(1-\mathrm{e}^{\mathrm{i}\epsilon \ln p })^2}{(1-A_p^2)^2} \right ) \nonumber \ ,
\label{average_R_p}
\end{eqnarray}
where as above $\epsilon=e_1-e_2$.
To calculate the product over $p<p^*$ we use Eq.~\eqref{zeta}; the final answer is
\begin{equation}
R_{2}^{\mbox{osc}}(\epsilon)=\frac{1}{4\pi^2}\mathrm{e}^{2\pi \mathrm{i} \overline{d(E)}\epsilon}|\zeta(1+\mathrm{i}\epsilon)|^2
\prod_p \left (1-\frac{(1-p^{\mathrm{i}\epsilon })^2}{(p-1)^2}\right ) + \mbox{ c.c. }\ .
\label{r_2osc}
\end{equation}
Expressions \eqref{r_2diag} and \eqref{r_2osc} constitute two parts of the connected two-point correlation function of zeros of the Riemann zeta function. It is important to stress that the same expressions were obtained in \cite{bogomolny_keating} by a completely different method based on the Hardy-Littlewood conjecture concerning the distribution of prime pairs. Exactly the same formulae are also derived from the ratio conjecture \cite{ratiosCS}. Of course, all these methods are heuristic and cannot be considered as a true mathematical proof. Nevertheless, their mutual agreement means that if such a formula exists, it is likely to be given by the above expressions.
It is plain that when $\epsilon\to 0$, Eq.~\ref{r_2osc} reproduces the two-point correlation function for the GUE ensemble of random matrices
\begin{equation}
R_2^{\mbox{c}}(\epsilon)\underset{\epsilon\to 0}{\longrightarrow} -\frac{\sin^2(\pi \bar{d}\epsilon)}{\pi^2 \epsilon^2}\ .
\end{equation}
In \cite{nonlinearity} using an averaged version of the Hardy-Littlewood conjecture it was shown that at small separations all correlation functions of the Riemann zeros agree with random matrix predictions.
\section{Three-point correlation function}\label{three_point}
The purpose of this Section is to calculate explicitly the three-point correlation function of zeros of the Riemann zeta function using the method discussed in the previous Sections.
By definition
\begin{equation}
R_3(e_1,e_2,e_3)=\left \langle \left \langle \left |
\begin{array}{ccc}
\bar{\rho}_1&K_{12}&K_{13}\\
K_{21}&\bar{\rho}_2&K_{23}\\
K_{31}&K_{32}&\bar{\rho}_3
\end{array}\right | \right \rangle \right \rangle_{\Delta E}
\end{equation}
where $\bar{\rho}_j=\rho(E+e_j)$ and $K_{ij}=K(E+e_i, E+e_j)$.
Expanding the determinant one gets
\begin{equation}
R_3(e_1,e_2,e_3)=\bar{\rho}_1\, \bar{\rho}_2\, \bar{\rho}_3+K_{12}\, K_{23}\, K_{31}+K_{21}\, K_{13}\, K_{32}-
\bar{\rho}_1\, K_{23}K_{32}-\bar{\rho}_2\, K_{13}\, K_{31}-\bar{\rho}_3 \, K_{12}\, K_{21} \ .
\end{equation}
Each $K_{ij}$ is given by \eqref{main_kernel} and it is straightforward to check that
\begin{equation}
K_{12}\, K_{23}\, K_{31}=\frac{1}{(2\pi \mathrm{i})^3 e_{12} e_{23 }e_{31}}
\left (
\mathrm{e}^{2\pi \mathrm{i} (N_1-N_2)}+\mathrm{e}^{2\pi \mathrm{i} (N_2-N_3)}+\mathrm{e}^{2\pi \mathrm{i} (N_3-N_1)}-
\mathrm{ c.c. }\right )
\end{equation}
with $e_{ij}=e_i-e_j$.
From \eqref{full_density} $\bar{\rho}_j\equiv \bar{\rho}(E+e_j)$ is the sum of two terms, $\bar{\rho}_j=\overline{d(E+e_j)} +\widetilde{d(E+e_j,p^*)}$. Using $\overline{d(E+e_j)}=\overline{d(E)}+ \mathcal{O}(e_j/E)$, ignoring the last correction terms, and, as above, denoting $d_j=\widetilde{d(E+e_j,p^*)}$ one obtains
\begin{equation}
R_3(e_1,e_2,e_3)=\overline{d(E)}^{\ 3}+\overline{d(E)} \, R_2^{\mbox{c}}(e_{12})+
\overline{d(E)} \, R_2^{\mbox{c}}(e_{23})+
\overline{d(E)} \, R_2^{\mbox{c}}(e_{31}) +R_{3}^{\mbox{c}}(e_1,e_2,e_3)\ .
\end{equation}
Here $R_2^{\mbox{c}}(e_{ij})$ is the connected two-point correlation function \eqref{rtwo} calculated in the previous Section, and $R_{3}^{\mbox{c}}(e_1,e_2,e_3)$ is the connected three-point function conveniently written as the sum of two terms
\begin{equation}
R_{3}^{\mbox{c}}(e_1,e_2,e_3)= R_{3}^{\mbox{diag}}(e_1,e_2,e_3)+R_{3}^{\mbox{osc}}(e_1,e_2,e_3) .
\label{rthree}
\end{equation}
Here $R_{3}^{\mbox{diag}}(e_1,e_2,e_3)$ is a smooth part
\begin{equation}
R_{3}^{\mbox{diag}}(e_1,e_2,e_3)=\langle \langle d_1 d_2 d_3 \rangle \rangle_{\Delta E}\ ,
\label{diag}
\end{equation}
and $R_{3}^{\mbox{osc}}(e_1,e_2,e_3)$ is an oscillatory part
\begin{equation}
R_{3}^{\mbox{osc}}(e_1,e_2,e_3)=-\frac{1}{(2\pi i)^3}\left (
\frac{r_{12 }}{e_{12}^2}+
\frac{r_{23}}{e_{23}^2}+
\frac{r_{31}}{e_{31}^2}+\mbox{ c.c. }\right )
\end{equation}
where
\begin{eqnarray}
r_{12}&=&\left \langle \left \langle \left [ 2\pi \mathrm{i}
d_3-\frac{2 e_{12}}{e_{23}e_{31}}\right ]\mathrm{e}^{2\pi \mathrm{i} (N_1-N_2)}\right \rangle \right \rangle_{\Delta E} \ ,\\
r_{23}&=&\left \langle \left \langle \left [ 2\pi \mathrm{i}
d_1-\frac{2 e_{23}}{e_{12}e_{31}}\right ]\mathrm{e}^{2\pi \mathrm{i} (N_2-N_3)} \right \rangle \right \rangle_{\Delta E}\ ,\\
r_{31}&=&\left \langle \left \langle \left [ 2\pi \mathrm{i}
d_2 -\frac{2e_{31}}{e_{12}e_{23}}\right ]\mathrm{e}^{2\pi \mathrm{i} (N_3-N_1)}\right \rangle \right \rangle_{\Delta E}\ .
\end{eqnarray}
\subsection{Smooth terms}
The calculation of smooth (diagonal) contributions \eqref{diag} for the three-point correlation function of Riemann zeros is simplified by the fact that after averaging terms divergent when $p^*\to \infty$ disappear and only convergent sums remain.
Indeed, the average over $E$ removes all products with different primes (cf. \eqref{orthogonality}). Therefore
\begin{eqnarray}
&&R_{3}^{\mbox{diag}}(e_1,e_2,e_3)\equiv \langle \langle d_1\, d_2\, d_3\rangle \rangle_{\Delta E} =-\frac{1}{(2\pi)^3}\sum_p \ln^3 p \sum_{n_1,n_2,n_3=1}^{\infty}A_p^{n_1+n_2+n_3} \nonumber\\
&\times &
\Big \langle
\left (\mathrm{e}^{\mathrm{i} n_1(\Phi_p(E)+e_1\ln p)}+\mathrm{e}^{-\mathrm{i} n_1(\Phi_p(E)+e_1\ln p)}\right)
\left (\mathrm{e}^{\mathrm{i} n_2(\Phi_p(E)+e_2\ln p)}+\mathrm{e}^{-\mathrm{i} n_2(\Phi_p(E)+e_2\ln p)}\right)\nonumber\\
&\times &\left(\mathrm{e}^{\mathrm{i} n_3(\Phi_p(E)+e_3\ln p)}+\mathrm{e}^{-\mathrm{i} n_3(\Phi_p(E)+e_3\ln p)}\right )
\Big \rangle_{\Phi_p}\ .
\end{eqnarray}
After averaging over $\Phi_p$, non-zero result gives only diagonal terms with $n_1=n_2+n_3$, $n_2=n_1+n_3$, and $n_3=n_1+n_2$. So
\begin{eqnarray}
&&R_{3}^{\mbox{diag}}(e_1,e_2,e_3)= -\frac{1}{(2\pi)^3}\sum_p \ln^3 p \left [
\frac{1}{(p^{1-\mathrm{i} e_{12}}-1)(p^{1-\mathrm{i} e_{13}}-1)}+\right . \nonumber\\
&&+\left .\frac{1}{(p^{1-\mathrm{i} e_{21}}-1)(p^{1-\mathrm{i} e_{23}}-1)}
+ \frac{1}{(p^{1-\mathrm{i} e_{32}}-1)(p^{1-\mathrm{i} e_{31}}-1)}\right ]+\mathrm{c.c.}\ .
\label{rdiag}
\end{eqnarray}
\subsection{Oscillatory terms}
Using Eqs.~(\ref{N}) and (\ref{d}) we obtain
\begin{eqnarray}
&& \left \langle \left \langle 2\pi \mathrm{i} d_3 \mathrm{e}^{2\pi \mathrm{i} (N_1-N_2)}\right \rangle \right \rangle_{\Delta E} \nonumber\\
&&=\mathrm{e}^{2\pi \mathrm{i} \overline{d(E)} e_{12}}\frac{\partial}{\partial e_3}
\left \langle \left \langle \left [ \sum_{p<p^*}
\ln \frac{1-A_p\mathrm{e}^{\mathrm{i}\Phi_p(3)}}{1-A_p\mathrm{e}^{-\mathrm{i}\Phi_p(3)}} \right ]
\prod_{p<p^*} \frac{(1-A_p\mathrm{e}^{-\mathrm{i}\Phi_p(2)})(1-A_p \mathrm{e}^{\mathrm{i}\Phi_p(1)})}
{(1-A_p\mathrm{e}^{-\mathrm{i}\Phi_p(1)})(1-A_p\mathrm{e}^{\mathrm{i}\Phi_p(2)})} \right \rangle \right \rangle_{\Delta E}
\label{sum_product}
\end{eqnarray}
where $\Phi_p(j)\equiv \Phi_p(E+e_j)=\Phi_p(E)+e_p\ln p$.
According to Eq.~\eqref{good_average} the average over $E$ is equivalent to the mean value over all phases. In Eq~\eqref{sum_product} the sum over primes is multiplied by the product over the same primes. Therefore the average of each term in the sum, say with $p=q$, reduces to the following product of the averages
\begin{eqnarray}
& &\left \langle \left \langle \frac{\partial}{\partial e_3}
\ln \left [ \frac{1-A_q\mathrm{e}^{\mathrm{i}\Phi_q(3)}}{1-A_q\mathrm{e}^{-\mathrm{i}\Phi_q(3)}} \right ]
\prod_{p<p^*} \frac{(1-A_p\mathrm{e}^{-\mathrm{i}\Phi_p(2)})(1-A_p \mathrm{e}^{\mathrm{i}\Phi_p(1)})}
{(1-A_p\mathrm{e}^{-\mathrm{i}\Phi_p(1)})(1-A_p\mathrm{e}^{\mathrm{i}\Phi_p(2)})} \right \rangle \right \rangle_{\Delta E}\nonumber \\
&=& T_q(e_1,e_2,e_3)\ \prod_{p\neq q} \langle R_p(\Phi_p ; e_1,e_2)\rangle_{\Phi_p}
\end{eqnarray}
where
\begin{eqnarray}
T_q( e_1,e_2,e_3)&=&\int_0^{2\pi} \frac{\mathrm{d}\Phi_q}{2\pi} \frac{\partial}{\partial e_3}
\ln \left [\frac{1-A_q\mathrm{e}^{\mathrm{i}(\Phi_q+e_3\ln q) }}{1-A_q\mathrm{e}^{-\mathrm{i}(\Phi_q+e_3\ln p)}}\right ]\nonumber\\
&\times &
\frac{(1-A_q\mathrm{e}^{-\mathrm{i}(\Phi_q+e_2\ln q)})(1-A_q \mathrm{e}^{\mathrm{i}(\Phi_q+e_1\ln q)})}
{(1-A_q\mathrm{e}^{-\mathrm{i}(\Phi_q+e_1\ln q)})(1-A_q\mathrm{e}^{\mathrm{i}(\Phi_q+e_2\ln q)})}\ ,
\end{eqnarray}
with $R_p(\Phi_p ; e_1,e_2)$ given by Eq.~\eqref{R_p}, and its average by Eq.~\eqref{average_R_p}.
As in the previous Section it is convenient to calculate $T_q(\Phi_q; e_1,e_2,e_3)$ by complex integration. Denoting $z=\mathrm{e}^{\mathrm{i}(\Phi_q}$, $A_q=A$, and $f_j=e_j\ln q$ one has
\begin{equation}
T_q( e_1,e_2,e_3) = - \frac{\ln q}{2\pi}\oint \left [\frac{Az\mathrm{e}^{\mathrm{i}f_3} }{1-Az\mathrm{e}^{\mathrm{i}f_3}} + \frac{A\mathrm{e}^{-\mathrm{i}f_3}}{z-A\mathrm{e}^{-\mathrm{i}f_3}} \right ] \frac{(1-A z \mathrm{e}^{\mathrm{i} f_1})(z-A \mathrm{e}^{-\mathrm{i}f_2})}{(z-A\mathrm{e}^{-\mathrm{i}f_1})(1-Az\mathrm{e}^{\mathrm{i}f_2})}\frac{\mathrm{d}z}{z}\ .
\end{equation}
Inside the unit circle the integrand has 3 poles, $z=0$, $z=A\mathrm{e}^{-\mathrm{i}f_1}$, and $z=A\mathrm{e}^{-\mathrm{i}f_3}$.
Direct calculations give ($\phi_{ij}=f_i-f_j$)
\begin{eqnarray}
&&T_q( e_1,e_2,e_3) = -\mathrm{i}\ln q \left [- \mathrm{e}^{\mathrm{i}\phi_{12}} +\Big (
\frac{A^2 \mathrm{e}^{\mathrm{i}\phi_{31}} }{1-A^2 \mathrm{e}^{\mathrm{i}\phi_{31}}}+
\frac{1}{\mathrm{e}^{\mathrm{i}\phi_{31}}-1} \Big )
\frac{(1-A^2)(1-\mathrm{e}^{\mathrm{i}\phi_{12}})}{1-A^2\mathrm{e}^{\mathrm{i}\phi_{21}}} \right .\nonumber \\
&&+ \left . \frac{(1-A^2\mathrm{e}^{\mathrm{i}\phi_{13}})(1-\mathrm{e}^{\mathrm{i}\phi_{32}})}{(1-\mathrm{e}^{\mathrm{i}\phi_{31}})(1-A^2\mathrm{e}^{\mathrm{i}\phi_{23}})}\right ]
= -\mathrm{i}\ln q \frac{(1-A^2)(1-\mathrm{e}^{\mathrm{i}\phi_{12}})}{1-A^2\mathrm{e}^{\mathrm{i}\phi_{21}}}\left [
\frac{A^2 \mathrm{e}^{\mathrm{i}\phi_{31}}}{1-A^2 \mathrm{e}^{\mathrm{i}\phi_{31}}}+\frac{A^2 \mathrm{e}^{\mathrm{i}\phi_{23}}}{1-A^2 \mathrm{e}^{\mathrm{i}\phi_{23}}}\right ] \ .
\end{eqnarray}
Re-introducing the full notation it follows that ($e_{ij}=e_i-e_j$)
\begin{equation}
T_q( e_1,e_2,e_3)=\frac{\partial}{\partial e_3} \ln \left [ \frac{1-A_q^2 \mathrm{e}^{\mathrm{i}e_{31}\ln q}}{1-A_q^2 \mathrm{e}^{\mathrm{i}e_{23}\ln q}} \right ] \frac{(1-A_q^2)(1-\mathrm{e}^{\mathrm{i}e_{12}\ln q})}{1-A_q^2\mathrm{e}^{\mathrm{i}e_{21}\ln q}} \ .
\end{equation}
Using Eq.~\eqref{ww} one gets
\begin{eqnarray}
r_{12}&=&\mathrm{e}^{2\pi \mathrm{i} \overline{d(E)}e_{12}}\prod_{p<p^*}
\frac{(1-A_p^2)^2}{|1-A_p^2 \mathrm{e}^{\mathrm{i}e_{12}\ln p}|^2}
\left (1-\frac{A_p^4}{(1-A_p^2)^2}(1-\mathrm{e}^{\mathrm{i}e_{12}\ln p} )^2\right )\times \\
&\times &\frac{\partial}{\partial e_3}\left [\sum_{q<p^*}
\frac{(1-A_q^2)(1-\mathrm{e}^{\mathrm{i}e_{12}\ln q})}{1-2A_q^2+A_q^2 \mathrm{e}^{\mathrm{i}e_{12}\ln q}}
\ln \frac{1-A_q^2\mathrm{e}^{\mathrm{i}e_{31}\ln q}}{1-A_q^2\mathrm{e}^{\mathrm{i}e_{23}\ln q}}+ \ln \frac{e_{31}^2}{e_{32}^2} \right ] \ .
\end{eqnarray}
The summand in the square brackets can be transformed as follows
\begin{eqnarray}
&&\frac{(1-A^2)(1-\mathrm{e}^{\mathrm{i}\phi_{12}})}{1-2A^2+A^2\mathrm{e}^{\mathrm{i}\phi_{12}}}
\ln \frac{1-A^2\mathrm{e}^{\mathrm{i}\phi_{31}}}{1-A^2\mathrm{e}^{\mathrm{i}\phi_{23}}}=
\ln \frac{1-A^2\mathrm{e}^{\mathrm{i}\phi_{31}}}{1-A^2\mathrm{e}^{\mathrm{i}\phi_{23}}}+\ln \frac{1-A^2\mathrm{e}^{-\mathrm{i}\phi_{31}}}{1-A^2\mathrm{e}^{-\mathrm{i}\phi_{23}}} +\nonumber \\
&&+\frac{A^2-\mathrm{e}^{\mathrm{i}\phi_{12}}}{1-2A^2+A^2\mathrm{e}^{\mathrm{i}\phi_{12}}}\ln \frac{1-A^2\mathrm{e}^{\mathrm{i}\phi_{31}}}{1-A^2\mathrm{e}^{\mathrm{i}\phi_{23}}}-\ln \frac{1-A^2\mathrm{e}^{-\mathrm{i}\phi_{31}}}{1-A^2\mathrm{e}^{-\mathrm{i}\phi_{23}}}=\nonumber \\
&&=\ln \left |\frac{1-A^2\mathrm{e}^{\mathrm{i}\phi_{31}}}{1-A^2\mathrm{e}^{\mathrm{i}\phi_{23}}}\right |^2 + A^2\frac{(1-\mathrm{e}^{\mathrm{i}\phi_{12}})^2}{1-2A^2+A^2\mathrm{e}^{\mathrm{i}\phi_{12}}}\ln \frac{1-A^2\mathrm{e}^{\mathrm{i}\phi_{31}}}{1-A^2\mathrm{e}^{\mathrm{i}\phi_{23}}}-\nonumber \\
&&-\left [\mathrm{e}^{\mathrm{i}\phi_{12}}\ln \frac{1-A^2\mathrm{e}^{\mathrm{i}\phi_{31}}}{1-A^2\mathrm{e}^{\mathrm{i}\phi_{23}}}+ \ln \frac{1-A^2\mathrm{e}^{-\mathrm{i}\phi_{31}}}{1-A^2\mathrm{e}^{-\mathrm{i}\phi_{23}}}\right ]\ .
\label{terms}
\end{eqnarray}
The expansions of all terms except the first one starts with $A^4\equiv A_q^4$ and, consequently, their sum over $q$ converge for large primes. In the divergent part, as above, we use \eqref{zeta}, and
\begin{equation}
\sum_{q<p^*} \ln \left |\frac{1-A_q^2\mathrm{e}^{\mathrm{i}e_{31}\ln q}}{1-A_q^2\mathrm{e}^{\mathrm{i}e_{23}\ln q}}\right |^2=
\ln \left |\prod_{q<p^*} \frac{1-A^2\mathrm{e}^{\mathrm{i}e_{31}\ln q}}{1-A^2\mathrm{e}^{\mathrm{i}\phi_{23}\ln q}}\right |^2
\underset{p^*\to\infty}{\longrightarrow} \ln \left |\frac{e_{23}\zeta(1+\mathrm{i} e_{23})}{e_{31}\zeta(1+\mathrm{i} e_{31})}\right |^2\ .
\end{equation}
The other terms in \eqref{terms} can be transform as follows
\begin{eqnarray}
&&\frac{\partial}{\partial e_3}\left [\mathrm{e}^{\mathrm{i}\phi_{12}}\ln\frac{1-A^2\mathrm{e}^{\mathrm{i}\phi_{31}}}{
1-A^2\mathrm{e}^{\mathrm{i}\phi_{23}}}+ \ln\frac{1-A^2\mathrm{e}^{-\mathrm{i}\phi_{31}}}{1-A^2\mathrm{e}^{-\mathrm{i}\phi_{23}}}\right ]\nonumber \\
&&=\mathrm{i}\ln q\ A^4\left [ \frac{\mathrm{e}^{-\mathrm{i}\phi_{23}}(\mathrm{e}^{\mathrm{i}\phi_{32}}-\mathrm{e}^{\mathrm{i}\phi_{31}})}
{(1-A^2\mathrm{e}^{\mathrm{i}\phi_{32}})(1-A^2\mathrm{e}^{\mathrm{i}\phi_{31}})} +\frac{\mathrm{e}^{-\mathrm{i}\phi_{31}}(\mathrm{e}^{-\mathrm{i}\phi_{31}}-\mathrm{e}^{-\mathrm{i}\phi_{32}})}{(1-A^2\mathrm{e}^{-\mathrm{i}\phi_{31}})(1-A^2\mathrm{e}^{-\mathrm{i}\phi_{32}})}\right ]\ .
\end{eqnarray}
Similarly, the other terms in Eq.~(\ref{terms}) takes the form
\begin{eqnarray}
&&\frac{\partial}{\partial e_3}\left [
A^2\frac{(1-\mathrm{e}^{\mathrm{i}\phi_{12}})^2}{1-2A^2+A^2\mathrm{e}^{\mathrm{i}\phi_{12}}}\ln\frac{1-A^2\mathrm{e}^{\mathrm{i}\phi_{31}}}{1-A^2\mathrm{e}^{\mathrm{i}\phi_{23}}}\right ]=\nonumber \\
&&=-\mathrm{i}\ln q \ A^4\frac{(1-\mathrm{e}^{\mathrm{i}\phi_{12}})^2}{1-2A^2+A^2\mathrm{e}^{\mathrm{i}\phi_{12}}}\left (\frac{\mathrm{e}^{\mathrm{i}\phi_{31}}}{1-A^2\mathrm{e}^{\mathrm{i}\phi_{31}}} + \frac{\mathrm{e}^{-\mathrm{i}\phi_{32}}}{1-A^2\mathrm{e}^{-\mathrm{i}\phi_{32}}} \right )\ .
\end{eqnarray}
Combining all terms together one finds
\begin{eqnarray}
&&R_3^{\mbox{osc}}(e_1,e_2,e_3)=
-\frac{\mathrm{e}^{2\pi \mathrm{i}\overline{d(E)} e_{12}}}{(2\pi \mathrm{i})^3}
|\zeta(1+\mathrm{i}e_{12})|^2
\prod_p\left (1 -\frac{(1-p^{\mathrm{i}e_{12}})^2}{(p-1)^2}\right )
\left [\frac{\partial}{\partial e_3}
\ln\left |\frac{\zeta(1+\mathrm{i}e_{32})}{\zeta(1+\mathrm{i}e_{31})}\right |^2 \right . \nonumber\\
&&\left . -\mathrm{i} \sum_q \ln q \left ( \frac{ q^{\mathrm{i} e_{12}}-1}
{(q^{1+\mathrm{i} e_{23}}-1)(q^{1+\mathrm{i} e_{13}} -1)}\right . \right . +\frac{q^{\mathrm{i} e_{12}}-1}{(q^{1+\mathrm{i} e_{31}}-1)(q^{1+\mathrm{i} e_{32}}-1)}+\label{rosc}\\
&&+ \left . \left .\frac{(1-q^{\mathrm{i} e_{12}})^2}{q-2+q^{\mathrm{i} e_{12}}}
(\frac{1}{q^{1-\mathrm{i}e_{31}}-1}+\frac{1}{q^{1-\mathrm{i}e_{23}}-1}) \right )\right ]+\mathrm{cyclic\; permutations}
+\mathrm{c.c.}\ .
\nonumber
\end{eqnarray}
Here "cyclic permutations" means that one has to add 2 other terms corresponding to cyclic permutations of indices $(1,2,3)$, i.e.
terms with substitutions: $1\to 3,\; 2\to 1,\; 3\to 2$ and $1\to 2,\; 2\to 3,\; 3\to 1$.
\section{Summary}\label{summary}
The principal ingredients of the proposed method for calculating correlation functions for the Riemann zeros are the following:
\begin{itemize}
\item A 'universal' formula for the kernel of a GUE-type ensemble of random matrices with a given mean eigenvalue density, $\bar{\rho}(E)$,
\begin{equation}
K(x,y)=\frac{\sin \pi \int_{x}^y \bar{\rho}(E)\mathrm{d}E }{\pi (x-y)}\ .
\end{equation}
\item The relation with the Riemann zeta function is established by fixing $\bar{\rho}(E)$ in the above expression as the finite part of the density of the zeros
\begin{equation}
\bar{\rho}(E)=\frac{1}{2\pi}\ln \frac{E}{2\pi} -\frac{1}{2\pi}\sum_{p<p*}\sum_{n=1}^{\infty}
\frac{\ln p}{ p^{n/2}}\left (\mathrm{e}^{\mathrm{i} n E\ln p}+\mathrm{e}^{-\mathrm{i} n E\ln p}\right )
\end{equation}
where the summation is performed over all prime numbers up to a certain cut-off value of $p^*$.
\item Correlation functions are calculated by the averaging the GUE determinantal formula over a large window of $E$
\begin{equation}
R(e_1,\ldots, e_n)=\left \langle \left \langle \det( K (x,y) )_{x,y=E+e_1,\ldots, E+e_n}\right \rangle \right\rangle_{\Delta E}
\ .
\end{equation}
\item The average is assumed to be such that phases $E\ln p$ with different primes $p<p^*$ can be considered as independent random phases and the procedure of averaging is carried out by integration over these phases
\begin{equation}
\left \langle \left \langle F\Big (\mathrm{e}^{\mathrm{i}E\ln p_1},\ldots, \mathrm{e}^{\mathrm{i}E\ln p_n} \Big ) \right \rangle \right\rangle_{\Delta E}
=\int_0^{2\pi}\frac{\mathrm{d}\phi_1}{2\pi}\cdots \int_0^{2\pi}\frac{\mathrm{d}\phi_n}{2\pi}
F\Big (\mathrm{e}^{\mathrm{i}\phi_1},\mathrm{e}^{\mathrm{i}\phi_2},\ldots, \mathrm{e}^{\mathrm{i}\phi_n} \Big )\ .
\end{equation}
\item After the averaging, the result consists of different sums over prime less than $p^*$. Those sums which converge when $p^*\to\infty$ are substituted by the sums over all primes. Sums divergent at large $p^*$ can be transformed to one particular product (or its logarithm or its derivative) whose limiting value is
\begin{equation}
\prod_{p<p^*}\dfrac{1-p^{-1} }{1-p^{-1-s}}\underset{p^*\to\infty}{\longrightarrow} s\zeta(1+s)\ .
\end{equation}
\end{itemize}
When the above rules are accepted, the calculation of correlation functions of Riemann zeros at a large height $E$ on the critical line reduces to purly algebraic manipulations (cf. Sections \ref{two_point} and \ref{three_point}). Our purpose here was to explain this new method, which, as we have emphasized, has the advantage over other heuristic approaches that it incorporates the determinanetal structure of RMT at the beginning (usually, it requires delicate combinatorial manipulations to establish this \cite{nonlinearity, RudSar, Zeev, CS}). The method also extends straightforwardly to similar quantum chaotic problems.
For convenience we rewrite the obtained expressions given by Eqs.~(\ref{rthree}), (\ref{rosc}), and (\ref{rdiag}) together with Eqs.~\eqref{rtwo}, \eqref{r_2diag}, \eqref{r_2osc} for the two-point and three-point correlation functions.
\begin{center} \textbf{Two-point correlation function} \end{center}
\begin{equation}
R_2(\epsilon)=\overline{d(E)}^{\, 2}+R_2^{\mbox{c}}(\epsilon),\qquad
R_2^{\mbox{c}}(\epsilon)=R_{2}^{\mbox{diag}}(\epsilon)+R_{2}^{\mbox{osc}}(\epsilon)\ .
\end{equation}
where
\begin{eqnarray}
R_{2}^{\mbox{diag}}(\epsilon)&=&-\frac{1}{4\pi^2}\frac{\partial^2}{\partial \epsilon^2}
\ln|\zeta(1+\mathrm{i}\epsilon)|^2 -
\frac{1}{4\pi^2}\sum_p \ln^2 p \left (\frac{1}{(p^{1+\mathrm{i}\epsilon}-1)^2}
+\frac{1}{(p^{1-\mathrm{i}\epsilon}-1)^2}\right )\ ,
\label{twodiag}\\
R_{2}^{\mbox{osc}}(\epsilon)&=&\frac{\mathrm{e}^{2\pi i \overline{d(E)}\epsilon}}{4\pi^2}|\zeta(1+\mathrm{i}\epsilon)|^2
\prod_p\left (1-\frac{(1-p^{\mathrm{i}\epsilon})^2}{(p-1)^2}\right ) + \mbox{ c.c.}\ .
\label{twoosc}
\end{eqnarray}
\begin{center} \textbf{Three-point correlation function} \end{center}
\begin{eqnarray}
R_3(e_1,e_2,e_3)&=&\overline{d(E)}^{\, 3}+\overline{d(E)}R_2^{\mbox{c}}(e_{12})+
\overline{d(E)} R_2^{\mbox{c}}(e_{23})+
\overline{d(E)} R_2^{\mbox{c}}(e_{31})+R_3^{\mbox{c}}(e_1,e_2,e_3)\ , \nonumber\\
R_3^{\mbox{c}}(e_1,e_2,e_3)&=& R_{3}^{\mbox{diag}}(e_1,e_2,e_3)+R_{3}^{\mbox{osc}}(e_1,e_2,e_3)\ ,
\end{eqnarray}
where
\begin{eqnarray}
&&R_{3}^{\mbox{diag}}(e_1,e_2,e_3)= -\frac{1}{(2\pi)^3}
\sum_p \ln^3 p\left (
\frac{1}{(p^{1-\mathrm{i} e_{12}}-1)(p^{1-\mathrm{i} e_{13}}-1)}\right . +
\frac{1}{(p^{1-\mathrm{i} e_{21}}-1)(p^{1-\mathrm{i} e_{23}}-1)}\nonumber\\
&& \left . +\frac{1}{(p^{1-\mathrm{i} e_{32}}-1)(p^{1-\mathrm{i} e_{31}}-1)}\right ) +\mbox{c.c.}
\end{eqnarray}
and
\begin{eqnarray}
&&R_3^{\mbox{osc}}(e_1,e_2,e_3)=-\frac{\mathrm{e}^{2\pi \mathrm{i}\overline{d(E)}e_{12}}}{(2\pi \mathrm{i})^3}
|\zeta(1+\mathrm{i}e_{12})|^2
\prod_p \left (1 -\frac{(1-p^{\mathrm{i}e_{12}})^2}{(p-1)^2}\right ) \left [
\frac{\partial}{\partial e_3}\ln
\left |\frac{\zeta(1+\mathrm{i}e_{32})}{\zeta(1+\mathrm{i}e_{31})}\right |^2 \right .\nonumber\\
&&\left .-\mathrm{i} \sum_p \ln p \left ( \frac{ p^{\mathrm{i} e_{12}}-1}
{(p^{1+\mathrm{i} e_{23}}-1)(p^{1+\mathrm{i} e_{13}}-1)}\right .\right .+\frac{p^{\mathrm{i} e_{12}}-1}{(p^{1-\mathrm{i} e_{13}}-1)(p^{1-\mathrm{i} e_{22}}-1)}+
\label{threeosc}\\
&&+ \left . \left .\frac{(1-p^{\mathrm{i} e_{12}})^2}{p-2+p^{\mathrm{i} e_{12}}}
\Big (\frac{1}{p^{1-\mathrm{i}e_{31}}-1}+\frac{1}{p^{1-\mathrm{i}e_{23}}-1}\Big ) \right )\right ]+
\mathrm{ cyclic \;permutations}+\mathrm{c.c.}\ .
\nonumber
\end{eqnarray}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,111 |
import * as React from "react";
import { CarbonIconProps } from "../../";
declare const Chemistry24: React.ForwardRefExoticComponent<
CarbonIconProps & React.RefAttributes<SVGSVGElement>
>;
export default Chemistry24;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,180 |
\section{Introduction}
Nuclear dimension for $C^*$-algebras was introduced by Winter and Zacharias in \cite{winter-zacharias}, as a noncommutative generalization of covering dimension. Since then, it has come to play a crucial role in structure and classification of $C^*$-algebras: indeed, it is now known that simple unital separable $C^*$-algebras with finite nuclear dimension and which satisfy the UCT are classified via the Elliott invariant (\cite{TWW,EGLN}). It was shown in \cite{winter-zacharias} that if $Y$ is a locally compact metrizable space, then $\dimnuc(C_0(Y))$ coincides with the covering dimension of $Y$, and the property of having finite nuclear dimension is preserved under various constructions: forming direct sums and tensor products, passing to quotients and hereditary subalgebras, and forming extensions. There has been considerable interest in seeing to what extent finite nuclear dimension is preserved under forming crossed products, which in particular led to the development of the notion of Rokhlin dimension for various group actions; see \cite{HWZ,hirshberg-phillips,szabo,SWZ, gardella} for actions of finite groups, $\Z$, $\Z^m$, and compact group actions.
The case of flows, that is, actions of $\R$, was addressed in a joint paper \cite{HSWW16} by the authors and Szab{\'o} and Winter. We developed a theory of Rokhlin dimension for flows which to a great extent parallels the theory for actions of $\Z$ (though the technicalities worked out to be rather different). This generalized Kishimoto's Rokhlin property for flows (\cite{kishimoto96}). In particular, we showed that if $\alpha$ is a flow on a $C^*$-algebra $A$ with finite Rokhlin dimension, and $A$ has finite nuclear dimension, then the crossed product $A \rtimes_{\alpha} \R$ has finite nuclear dimension. We furthermore showed that if $Y$ is a locally compact metrizable space with finite covering dimension,
then any flow $\alpha$ on $C_0(Y)$ induced from a \emph{free} topological flow on $Y$ has finite Rokhlin dimension. Thus, $C_0(Y) \rtimes_{\alpha} \R$ has finite nuclear dimension.
This leaves the case of non-free flows. Those cannot have finite Rokhlin dimension.
We addressed the parallel case of non-free $\Z$-actions on topological spaces in \cite{Hirshberg-Wu16}, where we showed that if $Y$ is a locally compact metrizable space with finite covering dimension and $\alpha$ is \emph{any} automorphism of $C_0(Y)$ then $C_0(Y) \rtimes_{\alpha} \Z$ has finite nuclear dimension. As in the case of actions of $\Z$, non-free actions of $\R$ are quite prevalent. Indeed, if $Y$ is a compact smooth manifold then any vector field on $Y$ gives rise to a flow, but typically such flows may have fixed points or periodic orbits.
The purpose of this paper is to provide a parallel to \cite{Hirshberg-Wu16} for possibly non-free flows. We show in Theorem \ref{thm-main} that
if $Y$ is a locally compact, metrizable with finite covering dimension, and $\alpha$ is a flow on $C_0(Y)$, then
\[
\operatorname{dim}_\mathrm{nuc}({C}_{0}(Y) \rtimes {\mathbb{R}^{}} ) \leq 5 \big( \dim(Y) \big) ^2 + 12 \dim(Y) + 6 \; .
\]
In fact, this estimate for flows implies a similar estimate for $\Z$-actions, by applying the mapping torus construction to obtain a flow from a $\Z$-action (see Corollary~\ref{cor:mapping-torus}). Thus we also recover the main theorem of \cite{Hirshberg-Wu16}, albeit with a less sharp bound. It is worth pointing out that despite the similarity between these results, there is some significant difference in the technical tools used in them (see the comments after Corollary~\ref{cor:mapping-torus}).
As an application, we give an estimate to the nuclear dimension of $C^*$-algebras associated to orientable line foliations. It was shown by Whitney (\cite{Whitney}) that any orientable line foliation on a locally compact metrizable space arises from a flow. A crucial object in the study of index theorems for foliations (\cite{Connes-Skandalis}) is the construction of the holonomy groupoid $G_{\mathcal{F}}$ of a foliation $\mathcal{F}$ and the $C^*$-algebra $C^*(G_{\mathcal{F}})$ thereof (see also \cite{moore-schochet} for a discussion of $C^*$-algebras associated to foliations). However, for an orientable line foliation $\mathcal{F}$, this foliation $C^*$-algebra $C^*(G_{\mathcal{F}})$ typically differs from the crossed product by the flow that gives rise to the foliation.
To make a link between foliation $C^*$-algebras and crossed product $C^*$-algebras, we develop a general theory for a kind of groupoid homomorphisms between topological groupoids that we call \emph{fiberwise groupoid covering maps} (see Definition~\ref{def:fiberwise-groupoid-covering}). They have the ability to induce quotient maps between the corresponding maximal groupoid $C^*$-algebras (see Theorem~\ref{thm:fiberwise-groupoid-covering-locally-compact} and~\ref{thm:fiberwise-groupoid-covering-quotient}). A canonical quotient map from the transformation groupoid of a flow to the holonomy groupoid $G_{\mathcal{F}}$ of the induced foliation is shown to be a fiberwise groupoid covering map (see Proposition~\ref{prop:foliation-covering}).
As a consequence, $C^*(G_{\mathcal{F}})$ can be obtained as a quotient of the crossed product associated to the flow (see Corollary~\ref{cor:foliation-covering}). Thus we have
\[
\operatorname{dim}_\mathrm{nuc}(C^*(G_{\mathcal{F}}) ) \leq 5 \big( \dim(Y) \big) ^2 + 12 \dim(Y) + 6 \; .
\]
where $Y$ is the underlying space of the foliation (see Corollary~\ref{cor:foliation-algebra-dimnuc}).
We now briefly sketch the idea of our proof of the main theorem. The basic idea is similar to the case of integer actions, though the techniques are different. The case in which the flow has uniformly compact orbits, that is, when all points are fixed or periodic with period bounded by some constant $R$, was already done in \cite[Section 3]{Hirshberg-Wu16} (indeed, we included it in that generality for use in the present paper), except for an estimate of the covering dimension of the quotient space $Y / \R$, which we complete in Section~\ref{section: bounded periods}. We showed there that one can find in such a case a bound on the nuclear dimension of $C_0(Y) \rtimes_{\alpha} \R$, which, crucially, does not depend on the maximal period length $R$.
Now, if we fix a some $R>0$, we can consider the set of points which are periodic with orbit length $\leq R$, which we denote by $Y_{\leq R}$, and we let $Y_{>R} = Y \setminus Y_{\leq R}$. Then one shows that $Y_{\leq R}$ is an invariant closed subset, and we have an equivariant extension
$$
0 \to C_0(Y_{> R}) \to C_0(Y) \to C_0(Y_{\leq R}) \to 0
.
$$
Although the restriction of $\alpha$ to $C_0(Y_{>R})$ does not necessarily have finite Rokhlin dimension, we can use a quantitative version of the techniques we used in \cite{HSWW16}, based on \cite{BarLRei081465306017991882} and \cite{Kasprowski-Rueping}, to construct dimensionally controlled long and thin covers of $C_0(Y_{>R})$, provided ``long'' means ``not too long compared to $R$''. This technique, which was used in \cite{HSWW16} to construct Rokhlin elements, is then used to construct decomposable approximations for $C_0(Y_{>R}) \rtimes_{\alpha} \R$, for certain finite subsets and a level of precision which improves as $R$ increases. (One could also use those covers to construct Rokhlin-type elements and then use them to construct decomposable approximations; however in the abelian setting, constructing suitable flow-wise Lipschitz partitions of unity, which in the setting of \cite{HSWW16} is a step towards constructing Rokhlin elements, is enough to obtain the required decomposable approximations, with an improved bound, so we do not need those Rokhlin elements for our result.) Having constructed those decomposable approximations for both $C_0(Y_{>R}) \rtimes_{\alpha} \R$ and for $C_0(Y_{\leq R}) \rtimes_{\alpha} \R$, we patch them together to get decomposable approximations for $C_0(Y) \rtimes_{\alpha} \R$ in a manner similar to the one done in \cite{Hirshberg-Wu16} for the case of $\Z$-actions, where the required precision of the final approximation predetermines how large $R$ has to be at the beginning of this analysis.
We note that in previous proofs of finite nuclear dimension for uniform Roe algebras and, more generally, groupoid $C^*$-algebras, such as those in \cite{winter-zacharias} and \cite{GWY}, Arveson's extension theorem is used in the construction of the downward completely positive map in the nuclear approximation. In our case, we give instead an explicit description of this map as a sum of compositions of compressions by a partition of unity of the unit space and conditional expectations associated to the clopen inclusion of relatively compact subgroupoids into larger ones. In the course of doing this, we prove a general result that any clopen inclusion of locally compact groupoids induces a conditional expectation between their reduced groupoid $C^*$-algebras (see Theorem~\ref{thm:clopen-subgroupoid}). Aside from simplifying the proof somewhat, the fact that it keeps track of the Cartan subalgebras could make it useful for future work.
\paragraph{\textbf{Acknowledgements:}} The authors would like to thank George Elliott, Alexander Kumjian, and Claude Schochet for some helpful discussions. Part of the research underlying this paper was carried out during the authors' visits to the Mittag-Leffler Institute, University of M\"{u}nster, the Penn State University, Centre de Recerca Matem\`{a}tica in Barcelona, the Fields Institute, and the Banff International Research Station.
\section{Preliminaries}
\label{section:prelim}
Throughout the paper, we use the following conventions. To simplify formulas, we may use the notations $\dimnucone(A) = \dimnuc(A)+1$, $\dim^{+1}(Y) = \dim(Y)+1$ and $\dr^{+1}(A) = \dr(A)+1$. If $A$ is a $C^*$-algebra, we denote by $A_+$ the positive part, and by $A_{+, \leq 1}$ the set of positive elements of norm at most $1$. If $G$ is a locally compact Hausdorff group and $A$ is a $C^*$-algebra, we denote by $\alpha \colon G \curvearrowright A$ an action, that is, a continuous homomorphism $\alpha \colon G \to \aut(A)$, where $\aut(A)$ is topologized by pointwise convergence. If $X$ is a metric space, $S \subseteq X$ and $r>0$, we denote $N_r(S) = \left\{x \in X \middlebar \operatorname{dist}(x,S)<r \right\}$.
We are interested here in the case in which $A$ is commutative, that is, $A \cong C_0(Y)$ for some locally compact Hausdorff space $Y$ (namely the spectrum $\widehat{A}$). By Gel'fand's theorem, an action $\alpha \colon G \curvearrowright C_0(Y)$ is completely determined by a continuous action $\calpha \colon G \curvearrowright Y$ on the spectrum, and vice versa. They are related by the identity $\alpha_g (f) = f \circ \calpha_{g^{-1}}$ for any $f \in C_0(Y)$ and any $g \in G$. Thus taking a $C^*$-algebraic point of view, we will denote by $\calpha \colon G \curvearrowright Y$ an action on a locally compact Hausdorff space by homeomorphisms, and save the notation $\alpha$ for the corresponding action on $C_0(Y)$.
When $G = \R$, the additive group of the real numbers, we often call a topological dynamical system $(Y, \R, \calpha)$ a \emph{flow}. We also make the following definitions. For any $y \in Y$, we let $\mathrm{per}_{\calpha}(y) = \inf\left\{t > 0 \middlebar \calpha_t(y) = y \right\}$ be the minimal period of the flow at $y$, or in other words, the length of the orbit of $y$ (with the convention $\mathrm{per}_{\calpha}(y) = \infty$ if $\calpha_t(y) \neq y$ for any $t \neq 0$). Thus this number is constant within each orbit. For a positive real number $R$, we decompose $Y$ as $Y_{\leq R} \sqcup Y_{> R}$, where
\begin{align*}
Y_{\leq R} & = \left\{ y \in Y \middlebar \mathrm{per}_{\calpha}(y) \leq R \right\} = \left\{ y \in Y \middlebar \calpha_{\R}(y) = \calpha_{[0,R]} (y) \right\} \; , \\
Y_{> R} & = Y \setminus Y_{\leq R} = \left\{ y \in Y \middlebar \mathrm{per}_{\calpha}(y) > R \right\} \; .
\end{align*}
This decomposition is then invariant under the flow. Intuitively, $Y_{\leq R}$ and $Y_{> R}$ are the short-period part and the long-period part of the flow, respectively.
We also recall some basic facts about (possibly non-Hausdorff) locally compact groupoids and their $C^*$-algebras. A \emph{topological groupoid} is a small category whose morphisms are all invertible and whose set of morphisms is equipped with a topology such that the multiplication map $G^2 = \left\{(x,y) \in G \times G \middlebar d(x) = r(y) \right\} \to G$ given by $(x,y) \to x \cdot y$ and the inversion map $G \to G$ given by $x \to x ^{-1}$ are both continuous, where $d, r \colon G \to G^0$ (the subset of $G$ consisting of all units, called the \emph{unit space}) are the domain and range maps, and $G^2$ (called the set of composable pairs) is given the subset topology inherited from the product topology on $G \times G$. We also write $G^u = r^{-1}(\left\{u\right\})$ and $G_u = d^{-1}(\left\{u\right\})$ for $u \in G^0$.
\begin{defn}
Let $Y$ be a topological space. We say that $Y$ is \emph{locally Hausdorff} if any point has a closed neighborhood which is compact Hausdorff.
\end{defn}
\begin{notn}\label{nota:C_c_0}
Let $Y$ be a topological space. We denote by $C_c(Y)_0$ the set of complex-valued functions $f$ for which there exists a Hausdorff open set $U$ so that $f$ vanishes outside of $U$, and $f|_U$ is continuous and compactly supported. Note that such functions may not be continuous when $Y$ is not Hausdorff. We then define $C_c(Y)$ to be the linear span of $C_c(Y)_0$ in the linear space of all complex-valued functions on $Y$.
\end{notn}
Observe that if $X$ is an open subset of $Y$, then we have canonical embeddings $C_c(X)_0 \hookrightarrow C_c(Y)_0$ and $C_c(X) \hookrightarrow C_c(Y)$. Also, if $Y$ is locally compact and Hausdorff, then both $C_c(Y)_0$ and $C_c(Y)$ are simply the set of compactly-supported continuous functions on $Y$.
\begin{defn}(\cite[Definition~2.2.1 and~2.2.2]{paterson}])\label{def:locally-compact-groupoids}
A \emph{locally compact locally Hausdorff groupoid} is a topological groupoid that satisfies the following axioms:
\begin{enumerate}
\item\label{def:locally-compact-groupoids:unit-space} $G^0$ is locally compact Hausdorff in the relative topology inherited from $G$;
\item\label{def:locally-compact-groupoids:locally-Hausdorff} there is a countable family $\mathcal{C}$ of compact Hausdorff subsets of $G$ such that the family $\left\{\mathcal{C}^0 \middlebar C \in \mathcal{C} \right\}$ of interiors of members of $\mathcal{C}$ is a basis for the topology of $G$. (In particular, $G$ is locally Hausdorff;)
\item\label{def:locally-compact-groupoids:sections} for any $u \in G^0$, $G^u$ is locally compact Hausdorff in the relative topology inherited from $G$;
\item\label{def:locally-compact-groupoids:Haar-system} $G$ admits a \emph{(left) Haar system} $\left\{\lambda^u \right\}_{u \in G^0}$, in the sense that each $\lambda^u$ is a positive regular Borel measure on the locally compact Hausdorff space $G^u$, such that
\begin{enumerate}
\item\label{def:locally-compact-groupoids:Haar-system:full-support} the support of each $\lambda^u$ is the whole of $G^u$,
\item\label{def:locally-compact-groupoids:Haar-system:continuous} for any $g \in C_c(G)$, the function $g^0$ given by the integral $\int_{G^u} g \, d \lambda^u$, belongs to $C_c(G^0)$,
\item\label{def:locally-compact-groupoids:Haar-system:invariance} for any $x \in G$ and $f \in C_c(G)$, we have
\[
\int_{G^{d(x)}} f(xz) \, d \lambda^{d(x)} (z) = \int_{G^{r(x)}} f(y) \, d \lambda^{r(x)} (y) \; .
\]
\end{enumerate}
\end{enumerate}
\end{defn}
Note that most of the technicality in this definition is due to our need to deal with the non-Hausdorffness of certain groupoids coming from holonomies of line foliations. If a locally compact groupoid is indeed Hausdorff, then
the existence of a Haar system is known to be a consequence of the first three axioms, Nevertheless, the uniqueness may still fail (a counterexample is given by the pair groupoid construction below); hence we always fix a left Haar measure as part of the data of a locally compact groupoid.
\begin{eg}\label{example:groupoids}
Basic examples of locally compact (Hausdorff) groupoids include:
\begin{enumerate}
\item the \emph{pair groupoid}\footnote{It is also known as the \emph{trivial groupoid} in \cite{paterson}.} $X \times X$ for a locally compact metrizable space $X$, where $\left( X \times X \right)^0$ is the diagonal, $d(x,y) = (y, y)$, $r(x,y) = (x , x)$ and $(x,y) \cdot (y,z) = (x, z)$, for any $x, y, z \in X$, and the positive regular Borel measures on $X$ with full support are in one-to-one correspondence with the left Haar measures of $X \times X$;
\item the \emph{transformation groupoid} $X \rtimes_\calpha G$ for an action $\calpha$ of a locally compact group $G$ on a locally compact metrizable space $X$, which is defined to be $X \times G$ with $\left( X \rtimes G \right)^0 = X \times \{1\}$, $d(x,g) = \left(\calpha^{-1}_g (x), 1 \right)$, $r(x,g) = \left( x, 1 \right)$ and $(x, g) \cdot (\calpha^{-1}_g (x), h) = (x, gh)$, where the left Haar measure of $G$ induces a left Haar system for $X \rtimes_\calpha G$;
\item if $G$ is a locally compact groupoid and $U$ is an open subset of $G^0$, then $G_U$, the \emph{reduction} of $G$ to $U$, as defined by $\{ x \in G \mid d(x) \in U, \, r(x) \in U \}$, is an open subset of $G$ that inherits the structure of a locally compact groupoid from $G$, where the left Haar system is induced by the inclusion $C_c(G_U) \subset C_c(G)$.
\end{enumerate}
\end{eg}
The main motivation for considering Haar systems lies in the representation theory for a groupoid $G$, which is often better studied through representations of the convolution $*$-algebra $C_c(G)$, where the convolution product is defined by the formula (c.f., \cite[(2.20) and (2.21)]{paterson})
\begin{equation}\label{eq:groupoid-multiplication-r}
f \ast g (x) = \int_{G^{r(x)}} f(y) g(y^{-1}x) \, d \lambda^{r(x)} (y)
\end{equation}
or equivalently
\begin{equation}\label{eq:groupoid-multiplication-d}
f \ast g (x) = \int_{G^{d(x)}} f(xt) g(t^{-1}) \, d \lambda^{d(x)} (t)
\end{equation}
for any $f, g \in C_c(G)$ and any $x \in G$, and the $*$-operation is given by
\begin{equation}\label{eq:groupoid-star-operation}
f^* (x) = \overline{f(x^{-1})}
\end{equation}
for any $f \in C_c(G)$ and any $x \in G$ (c.f., \cite[(2.22)]{paterson}).
Important for us is the reduced $C^*$-algebra of a locally compact groupoid $G$. To this end, we write $\left\{\lambda_u\right\}_{u \in G^0}$ for the \emph{right Haar measure} corresponding to the left Haar measure $\lambda = \left\{\lambda^u\right\}_{u \in G^0}$, where $\lambda_v(E) = \lambda^v(E^{-1})$ for any Borel set $E \subset G_v$. Now any $v \in G^0$ determines a $*$-representation $\operatorname{Ind}v$, the \emph{left regular representation at $v$}, of $C_c(G)$ on the Hilbert space $L^2(G_v, \lambda_v)$, defined by the formula (c.f., \cite[(3.41)]{paterson})
\[
\left( \operatorname{Ind}v (f) \cdot \xi \right) (x) = \int_{t \in G_v} f(xt^{-1}) \xi(t) \, d\lambda_v(t)
\]
for any $f \in C_c(G)$, any $\xi \in L^2(G_v, \lambda_v)$ and any $x \in G_v$; thus in terms of the inner product, we have
\[
\left\langle \operatorname{Ind}v (f) \cdot \xi, \eta \right\rangle = \int_{x \in G_v} \int_{t \in G_v} f(xt^{-1}) \xi(t) \overline{\eta(x)} \, d\lambda_v(t) \, d\lambda_v(x)
\]
for any $\eta \in L^2(G_v, \lambda_v)$. It is known that the $C^*$-seminorm induced by $\operatorname{Ind}v$ is bounded by the \emph{$I$-norm} (c.f., \cite[(2.25)]{paterson}) and thus $\sup_{v \in G^0} \| \operatorname{Ind} v (f) \| < \infty$. In fact, this estimate works for any $*$-representation.
\begin{defn}\label{def:reduced-groupoid-algebra}
Let $G$ be a locally compact groupoid with a left Haar system $\lambda$. The \emph{maximal groupoid $C^*$-algebra} $C^*(G, \lambda)$ of $G$ is the $C^*$-envelop of $C_c(G)$. The \emph{reduced groupoid $C^*$-algebra} $C^*_{r}(G, \lambda)$ of $G$ is the $C^*$-completion of $C_c(G)$ under the norm $\| f \|_{\operatorname{red}} = \sup_{v \in G^0} \| \operatorname{Ind} v (f) \|$. Equivalently, $C^*_{r}(G, \lambda)$ is the completion of the image of the $*$-representation $\bigoplus_{v \in G^0} \operatorname{Ind}v$ of $C_c(G)$ on $\bigoplus_{v \in G^0} L^2(G_v, \lambda_v)$.
\end{defn}
We may write $C^*(G)$ and $C^*_{r}(G)$ instead of $C^*(G, \lambda)$ and $C^*_{r}(G, \lambda)$ if the Haar system $\lambda$ is clear from the context. Also useful for us is the canonical embedding
\begin{equation} \label{eq:canonical-embedding-unit-space}
C_0(G^0) \hookrightarrow M(C^*_r(G)) \; ,
\end{equation}
where $M(C^*_r(G))$ is the multiplier algebra of $C^*_r(G)$, so that for any $ f \in C_0(G^0) $ and for any $ g $ in the dense subalgebra $ C_c(G) $ in $ C^*_r(G) $, the convolution product of $ g $ by $ f $ from the left and right are also in $C_c(G)$ and are given by
\[
(f \ast g) \ (x) = f(r(x)) \cdot g (x) \ \text{and}\ (g \ast f ) \ (x) = g(x) \cdot f \left(d (y)\right)
\]
for any $x \in G$.
\begin{eg}\label{example:reduced-groupoid-algebra}
For the groupoids in Example~\ref{example:groupoids}, we have:
\begin{enumerate}
\item for any locally compact Hausdorff space $X$ and a positive regular Borel measure $\mu$ on $X$ with full support, the reduced $C^*$-algebra of its pair groupoid $X \times X$ together with the left Haar measure associated to $\mu$ is isomorphic to the algebra of compact operators on $L^2(X, \mu)$ (c.f., for example, \cite[Theorem~3.1.2]{paterson});
\item for any action $\calpha$ of a locally compact group $G$ on a locally compact metrizable space $X$, the reduced $C^*$-algebra of its transformation groupoid is isomorphic to the reduced crossed product $C_0(X) \rtimes_r G$, and in fact, when $G$ is unimodular with a Haar measure $m$, the canonical identification between $C_c(X \rtimes G)$ and $C_c(G, C_c(X))$, where a function on $X \times G$ is identified with an iterated function on $G$ and $X$, intertwines the convolution products and the $*$-operation. Here the convolution product and the $*$-operation on $C_c(G, C_c(X))$ are defined by
\[
(f \ast f') (g)(x) = \int_{G} f(h)(x) \cdot f'(h^{-1}g)(\calpha_{h^{-1}}(x)) \, d m (h)
\]
and
\[
f^* (g)(x) = \overline{f(g^{-1})( \calpha_{g^{-1}}(x) )}
\]
for any $g \in G$ and $x \in X$, so that it forms a dense subalgebra of $C_0(X) \rtimes_r G$;
\item if $G_U$ is the reduction of a locally compact groupoid $G$ to an open subset $U$ of the unit space $G^0$, then the inclusion $C_c(G_U) \subset C_c(G)$ induces an embedding $C^*_r(G_U) \hookrightarrow C^*_r(G)$, whose image coincides with the hereditary subalgebra $C_0(U) \cdot C^*_r(G) \cdot C_0(U)$, where we used the canonical embeddings $C_0(U) \subset C_0(G^0) \hookrightarrow M(C^*_r(G))$.
\end{enumerate}
\end{eg}
Next we review some facts about order zero maps and nuclear dimension. If $A = A_0 \oplus A_1 \oplus \ldots \oplus A_d$ is a $C^*$-algebra, and $\varphi^{(k)} \colon A_k \to B$ are order zero contractions into some $C^*$-algebra $B$ for $k=0,1,\ldots,d$, we say that the map $\varphi = \sum_{k=0}^d \varphi^{(k)}$ is a \emph{piecewise contractive} $(d+1)$-decomposable completely positive map.
The following fact concerning order zero maps is standard and used often in the literature. It follows immediately from the fact that cones over finite dimensional $C^*$-algebras are projective. See \cite[Proposition 1.2.4]{winter-covering-II} and the proof of \cite[Proposition 2.9]{winter-zacharias}. We record it here for further reference.
\begin{Lemma}
\label{Lemma:lifting-decoposable-maps}
Let $A$ be a finite dimensional $C^*$-algebra, let $B$ be a $C^*$-algebra and let $I \lhd B$ be an ideal. Then any piecewise contractive $(d+1)$-decomposable completely positive map $\varphi \colon A \to B/I$ lifts to a piecewise contractive $(d+1)$-decomposable completely positive map $\widetilde{\varphi} \colon A \to B$. \qed
\end{Lemma}
We record for the reader's convenience a few lemmas from \cite{Hirshberg-Wu16} which we will re-use in this paper.
\begin{Lemma}[{\cite[Lemma 1.2]{Hirshberg-Wu16}}]
\label{Lemma:finite-dimnuc}
Let $B$ be a separable and nuclear $C^*$-algebra and $B_0$ a dense subset of the unit ball of $B$. Then $\dimnuc(B) \leq d$ if and only if for any finite subset $F \subseteq B_0$ and for any $\eps>0$ there exists a $C^*$-algebra $A_{\eps} = A_{\eps}^{(0)} \oplus \cdots \oplus A_{\eps}^{(m)}$ and completely positive maps
\[
\xymatrix{
B \ar[dr]_{\psi = \bigoplus_{l=0}^m \psi^{(l)} \quad } \ar@{.>}[rr]^{\id} & & B \\
& A_{\eps} = \bigoplus_{l=0}^m A_{\eps}^{(l)} \ar[ur]_{\quad\varphi = \sum_{l=0}^m \varphi^{(l)}} &
}
\]
so that
\begin{enumerate}
\item $\psi$ is contractive,
\item each $\varphi^{(l)}$ is a sum $\varphi^{(l)} = \sum_{k=0}^{ d^{(l)} } \varphi^{(l,k)}$ of $(d^{(l)} + 1)$-many order zero contractions,
\item $\|\varphi(\psi(x)) - x\| < \eps$ for all $x \in F$, and
\item $\displaystyle \sum_{l=0}^m ( \dimnuc(A_{\eps}^{(l)}) + 1) (d^{(l)}+1) \leq d+1$.
\end{enumerate}
\qed
\end{Lemma}
\begin{lem}[{\cite[Lemma 1.3]{Hirshberg-Wu16}}]\label{lem:separable-dimnuc}
Let $G$ be a locally compact Hausdorff and second countable group, and let $A$ be a $G$-$C^*$-algebra. Then any countable subset $S \subset A$ is contained in a $G$-invariant separable $C^*$-subalgebra $B \subset A$ with $\dimnuc(B) \leq \dimnuc(A)$. In particular, $A$ can be written as a direct limit of separable $G$-$C^*$-algebras with nuclear dimension no more than $\dimnuc(A)$.
\qed
\end{lem}
\begin{lem}[{\cite[Lemma 1.4]{Hirshberg-Wu16}}] \label{lem:quasicentral-approximate-unit}
Let $X$ be a locally compact Hausdorff space, let $G$ be a locally compact Hausdorff group, and let $\calpha: G \curvearrowright X$ be a continuous action. Suppose $U$ is a $G$-invariant open subset of $X$. Then there is a quasicentral approximate unit for $C_0(U) + C_0(U) \rtimes_\alpha G \subset M(C_0(X) \rtimes_\alpha G)$ which is contained in $C_c(U)_{+, \leq 1}$.
\qed
\end{lem}
We record the following two results from classical dimension theory, which are used later in the paper.
Those two results apply to the case of metrizable spaces, since any metrizable space is paracompact, Hausdorff and totally normal. For a discussion of different variants of paracompactness and normality, we refer the reader to \cite[Chapter 1, section 4]{Pears75}.
\begin{thm}[{\cite[Chapter 3, Theorem 6.4]{Pears75}}]\label{thm:Pears75}
If $M$ is a subspace of a totally normal space $X$, then $\dim (M) \leq \dim (X)$.
\qed
\end{thm}
\begin{prop}[{\cite[Chapter 9, Proposition 2.16]{Pears75}}] \label{prop:Pears75}
If $X$ and $Y$ are weakly paracompact $T_4$-spaces and $f\colon X \to Y$ is a continuous open surjection such that $f^{-1}(y)$ is finite for each point of $Y$, then $\dim(X) = \dim (Y) $. \qed
\end{prop}
\section{Subtubular covers for flows without short periods}
\label{Section:Tube}
In this section, we study the case where there is a nontrivial lower bound on the periods, i.e.\ $Y = Y_{> R}$ for some $R>0$.
Our aim is to show that when this lower bound $R$ is large enough, we can obtain covers of the space by long enough tubes (or subsets of tubes) with controlled dimensions, and these in turn yield partitions of unity made up of almost invariant functions. The content of this section parallels and extends \cite[Section 6]{HSWW16}, where the case of free flows is treated. Thus unsurprisingly, this section also makes uses of technical results from \cite{Kasprowski-Rueping} (which is a generalization of results from \cite{BarLRei081465306017991882}).
Before we introduce the necessary terminology borrowed from \cite{BarLRei081465306017991882} and \cite{Kasprowski-Rueping}, we point out that we are not using the full power of the constructions and results in those papers. Instead we are looking at a somewhat simplified situation: while in those papers, they need to consider a proper action by a discrete group $G$ which commutes with the flow, this is of no particular interest to us in the present paper, and thus we consider only the flow itself. In other words, we take $G$ to be the trivial group.
\begin{defn}[cf.~{\cite[Definition 2.2]{BarLRei081465306017991882}}]
\label{defn:tubes}
A \emph{box} (or a \emph{tube}) is a compact subset $B \subset Y$ such that there exists a real number $ l = l_B $ with the property that for every $y \in B$, there exist real numbers $ a_-(y) \le 0 \le a_+(y)$ and $\varepsilon(y) > 0 $ satisfying
\begin{align*}
l = &\ a_+(y) - a_-(y) ;\\
\calpha_t(y) \in &\ B \ \text{ for\ } t \in [a_-(y), a_+(y) ] ;\\
\calpha_t(y) \not\in &\ B \ \text{ for\ } t \in ( a_-(y) - \varepsilon(y), a_-(y) ) \cup (a_+(y), a_+(y) + \varepsilon(y) ).
\end{align*}
Moreover, for any tube $B$, the following data are associated to it:
\begin{enumerate}
\item The \emph{length} $ l_B $;
\item The topological interior $B^o$ is called an \emph{open tube};
\item The subset $ S_B = \left\{ y \in B \middlebar a_-(y) + a_+(y) =0 \right\}$ is called the \emph{central slice} of $B$;
\item The subsets $ \partial_+ B $ and $ \partial_- B $, respectively called the \emph{top} and the \emph{bottom} of $B$, are defined by
$$ \partial_\pm B = \left\{ y \in B \middlebar a_\pm (y) =0 \right\} = \left\{ \calpha_{a_\pm(y)} (y) \middlebar y \in S_B \right\} ; $$
\item Similarly, the \emph{open top} $ \partial_+ B ^o $ and the \emph{open bottom} $ \partial_- B ^o $ are defined by
$$ \partial_\pm B^o = \left\{ \calpha_{a_\pm(y)} (y) \middlebar y \in S_B \cap B^o \right\} \; , $$
and the \emph{open central slice} is defined by
\[
S_{B^o} = S_B \cap B^o \; .
\]
\end{enumerate}
\end{defn}
Intuitively speaking, what a tube is to a dynamical system of ${\mathbb{R}^{}}$ is what a Rokhlin tower is to a dynamical system of ${\mathbb{Z}^{}}$, in that they function as a local trivialization of the action.
\begin{lem}[cf.~{\cite[Lemma 2.6]{BarLRei081465306017991882}}]
\label{lem:tubes-basics}
Let $B \subset Y$ be a tube of length $l = l_B$. Then
\begin{enumerate}
\item The maps
$$ a_\pm: B \to {\mathbb{R}^{}}, \ y \mapsto a_\pm(y) $$
are continuous;
\item There exists $\varepsilon_B > 0 $ depending only on $B$ such that the numbers $\varepsilon(y)$ appearing in the definition of a tube can be chosen so that $ \varepsilon(y) \ge \varepsilon_B$ holds for all $y \in B$;
\item The map
$$ S_B \times \left[ - \frac{l}{2}, \frac{l}{2} \right] \to B \; , \quad (y, t) \mapsto \calpha_t(y) $$
is a homeomorphism, whose inverse is given by the map
\[
B \to S_B \times \left[ - \frac{l}{2}, \frac{l}{2} \right] \; , \quad y \to \left( \calpha_{\frac{a_-(y) + a_+(y)}{2}} (y), \frac{l_B}{2} - a_+(y) \right) \; .
\]
\end{enumerate}
\end{lem}
\begin{rmk}\label{rmk:tube-basics}
The last statement in the previous lemma can be turned into an alternative definition for tubes: a tube is a pair $(S, l)$, where $S \subset Y$ is compact, $l >0$, and the map
\[
S \times \left[ - \frac{l}{2}, \frac{l}{2} \right] \to Y , \ (y, t) \mapsto \calpha_t(y)
\]
is an embedding. To relate to Definition~\ref{defn:tubes}, we can establish, for any $y \in S$ and $t \in \left[ - \frac{l}{2}, \frac{l}{2} \right]$, the identities
\[
a_- \left( \calpha_t(y) \right) = - t - \frac{l}{2} \text{ and } a_+ \left( \calpha_t(y) \right) = - t + \frac{l}{2} \; .
\]
\end{rmk}
Recall that $\mathrm{per}_{\calpha}(y) = \inf\left\{t > 0 \middlebar \calpha_t(y) = y \right\}$ is the minimal period of the orbit of $y$. Observe that the existence of a tube $B$ around a point $y \in Y$ requires the necessary condition $l_B < \mathrm{per}_{\calpha}(y)$. In fact, this is also sufficient:
\begin{lem}[{\cite[Lemma 2.11 and Lemma 2.16]{BarLRei081465306017991882}}]\label{lem:existence-tubes}
For any $y \in Y$ not fixed by $\calpha$, and for any $l \in (0, \mathrm{per}_{\calpha}(y))$, there exists a tube $B$ with $l_B = l$ and $y \in S_B \cap B^o$.
\end{lem}
\begin{cor}\label{cor:long-period-open}
The subset $Y_{> R}$ is open and $\calpha$-invariant, while $Y_{\leq R}$ is closed and $\calpha$-invariant.
\end{cor}
\begin{proof}
The $\calpha$-invariance is obvious. To show $Y_{> R}$ is open, we see that given any $y \in Y_{> R}$, by Lemma~\ref{lem:existence-tubes}, there is a tube $B_y$ such that $l_{B_y} = R$ and $y \in S_{B_y} \cap B_y^o$. This implies that for any point $y'$ in the open neighborhood $B_y^o$ around $y$, we have $\mathrm{per}_{\calpha}(y') > R$, that is, the neighborhood $B_y^o \subset Y_{> R}$. Therefore $Y_{> R}$ is open while $Y_{\leq R}$ is closed.
\end{proof}
Occasionally we will need to \emph{stretch} a tube, as formalized in the following lemma, whose proof is immediate from the definition.
\begin{lem}\label{lem:stretching-tube}
Let $B$ be a tube and $\displaystyle 0 < L < \frac{\varepsilon_B}{2} $. Then the set $ \calpha_{[-L, L]} (B) $ is also a tube with the same central slice and a new length $ l_B + 2 L $.
\end{lem}
In order to study the nuclear dimension of the crossed product ${C}_{0}(Y) \rtimes {\mathbb{R}^{}} $, we would like to decompose $Y$ in a dimensionally controlled fashion into open subsets such that the flow is trivialized when restricted to each open subset. This motivation leads to the definition of \emph{tube dimension} in \cite[Definition~7.6]{HSWW16}, which works well for free flows. In our current situation, we need a quantitative generalization of the notion of tube dimension.
\begin{Notation}
\label{def:mult}
Given a topological space $X$ with a collection $\mathcal{U}$ of subsets of $X$, its \emph{multiplicity} $\operatorname{mult}(\mathcal{U})$ is the infimum of natural numbers $d$ satisfying that the intersection of any $d+1$ pairwise distinct elements in $\mathcal{U}$ is empty, while its \emph{chromatic number} $\operatorname{chrom}(\mathcal{U})$ is the infimum of natural numbers $d$ satisfying that $\mathcal{U}$ can be written as a union of $d+1$ many families of disjoint subsets of $X$.
\end{Notation}
\begin{rmk}\label{rmk:multiplicity_vs_chromatic_number}
It is clear from the definition that $\operatorname{mult}(\mathcal{U}) \leq \operatorname{chrom}(\mathcal{U})$ for any collection $\mathcal{U}$. The opposite direction does not hold in general.
\end{rmk}
\begin{defn}\label{defn:subtubular-cover}
Let $ (Y, {\mathbb{R}^{}}, \calpha) $ be a topological flow and let $K \subset Y$ be a subset. Let $L \in [0,\infty)$ and $d \in \Z^{\geq 0}$. A \emph{subtubular cover of $K$ with width $\geq L$ and multiplicity (respectively, chromatic number) $\leq d+1$} is a finite collection $ \mathcal{U} $ of open subsets of $Y$ satisfying:
\begin{enumerate}
\item\label{defn:subtubular-cover:width} for any $ y\in K $, there is $U \in \mathcal{U}$ such that $ \calpha_{[-L, L]}(y) \subset U $;
\item\label{defn:subtubular-cover:subtubular} each $ U \in \mathcal{U} $ is contained in a tube $B_U$;
\item\label{defn:subtubular-cover:mult} $ \operatorname{mult} ( \mathcal{U} ) \leq d+1$ (respectively, $ \operatorname{chrom} ( \mathcal{U} \le d + 1 $).
\end{enumerate}
\end{defn}
\begin{rmk}
The relation with tube dimension given in \cite[Definition~7.6]{HSWW16} is that $ \operatorname{dim}_\mathrm{tube} (\calpha) \leq d$ if and only if for any $L> 0 $ and any compact subset $K \subset Y$, there exists a subtubular cover of $K$ with width $\geq L$ and multiplicity $\leq d+1$. In fact, one may also use the chromatic number in place of the multiplicity in the above statement, thanks to \cite[Proposition~7.23]{HSWW16}.
\end{rmk}
We shall prove that for a flow $\calpha$ on a locally connected, locally compact and metrizable space $ Y $ with finite topological dimension, when the lower bound on the periods is not too small compared to $L$, any compact subset $K \subset Y$ possesses a subtubular cover with width $\geq L$ and multiplicity $\leq d+1$. For this, we will need to invoke a result by Kasprowski and R\"{u}ping (\cite[Theorem~5.3]{Kasprowski-Rueping}), which itself is an improvement of a construction of the so-called ``long thin covers" by Bartels, L\"{u}ck and Reich (\cite[Theorem~1.2, Proposition~4.1]{BarLRei081465306017991882}). Since we are dealing with a simplified situation, we shall give a somewhat different formulation of their theorem that is sufficient for our purposes, cf. Remark~\ref{rmk:KRcoverbytubes-differences}.
\begin{thm}[cf.~{\cite[Theorem~5.3]{Kasprowski-Rueping}}]
\label{thm:KRcoverbytubes}
Let $Y$ be a locally compact and metrizable space with a continuous action $\Phi$ by $\mathbb{R}$. Let $L$ be a positive number. Then there is a collection of open tubes of multiplicity at most $5 (\operatorname{dim}(Y) + 1)$ satisfying the property that for any point $y \in Y_{>20L}$, there is an open tube in this collection containing $\Phi_{[-L, L]}(y)$.
\qed
\end{thm}
\begin{rmk}\label{rmk:KRcoverbytubes-differences}
We explain some deviation from the original formulation of the above theorem in \cite{Kasprowski-Rueping}:
\begin{enumerate}
\item In the original version, the authors consider not only a flow $\Phi$ on $Y$, but also a proper action of a discrete group $G$ that commutes with $\Phi$, and the cover they produce is required to be a so-called \emph{$\mathcal{F}in$}-cover with regard to the second action: it is invariant, and for each open set $U$ in the cover, only finitely many elements of $G$ fix $U$, while any other element carry $U$ to a set disjoint from $U$ ({\cite[Notation~1.3(4)]{Kasprowski-Rueping}}). Since this is not needed for proving our main result, we drop this assumption, or equivalently, we assume this extra group $G$ that appears in their theorem to be the trivial group, in which case the {$\mathcal{F}in$}-cover condition is automatic.
\item The dimension estimate in the original version is in terms of the \emph{small inductive dimension} $\mathrm{ind}(Y)$, but as they remarked in \cite[Theorem~3.5]{Kasprowski-Rueping}, in the context their Theorem~5.3 applies to, where $Y$ is locally compact and metrizable, the small inductive dimension is equal to the covering dimension $\operatorname{dim}(Y)$.
\item It is not made explicit in the original statement of their theorem that the cover consists of open tubes, but this is evident from their proof: the cover is made up of the sets $\Phi_{(-4L, 4L)} (B_i^k)$ for $i \in \mathbb{N}$ and $k \in \left\{0, \ldots, \operatorname{dim}(Y)\right\}$, and each of them is the interior of a tube $\displaystyle \Phi_{[-4L, 4L]} (\overline{B_i^k} )$, which is restricted from the larger tube $\Phi_{[-10L, 10L]} (S_i)$ constructed in \cite[Lemma~4.6]{Kasprowski-Rueping}.
\item It is also clear here that although in the original statement of the theorem, it is only claimed that the cover has dimension at most $5 \operatorname{dim}^{\!+1}(Y)$, but in fact the cover they obtained has multiplicity at most $5 \operatorname{dim}^{\!+1}(Y)$.
\end{enumerate}
\end{rmk}
\begin{cor}\label{cor:estimate-subtubular-cover}
Let $Y$ be a locally compact and metrizable space and $\Phi$ a flow on $Y$. Then for any $L \in [0,\infty)$, any $d \in \Z^{\geq 0}$ and any compact subset $K \subset Y_{> 20 L}$, there is a subtubular cover of $K$ with width $\geq L$ and multiplicity $\leq 5 \cdot \operatorname{dim}^{\!+1}(Y)$.
\end{cor}
\begin{proof}
This is almost a direct consequence of Theorem~\ref{thm:KRcoverbytubes}. The only point to be clarified is the finiteness of the cover. Let $L \in [0,\infty)$, $d \in \Z^{\geq 0}$ and a compact subset $K \subset Y_{> 20 L}$ be given. By Theorem~\ref{thm:KRcoverbytubes}, there is a collection $\mathcal{U}$ of open tubes of multiplicity at most $5 (\operatorname{dim}(Y) + 1)$ satisfying the property that for any point $y \in Y_{>20L}$, there is an open tube $U_y$ in this collection containing $\Phi_{[-L, L]}(y)$. For any $y \in K$,
we may write $y = \calpha_{t_y}(z_y)$ for $z_y$ in the central slice $S_{\overline{U_y}}$ and $t_y \in \left[ -\frac{l_{\overline{U_y}}}{2} , \frac{l_{\overline{U_y}}}{2}\right]$. By the local trivialization associated to the tube $\overline{U_y}$ as in Lemma~\ref{lem:tubes-basics}, we have $[t_y - L , t_y + L] \subset \left( -\frac{l_{\overline{U_y}}}{2} , \frac{l_{\overline{U_y}}}{2}\right)$. Thus there exists $\delta_y > 0$ such that $t_y - L - \delta_y > -\frac{l_{\overline{U_y}}}{2}$ and $t_y + L + \delta_y < \frac{l_{\overline{U_y}}}{2}$. Define the set $V_y = \Phi_{(t_y - \delta_y , t_y + \delta_y)}(S) \cap U_y$. Then using the local trivialization, we see that $V_y$ is an open neighborhood of $y$ such that $\Phi_{[-L, L]}(V_y) \subset U_y$.
By the compactness of $K$, there is a finite subset $\left\{y_1 , \ldots, y_n\right\} \subset K$ such that $K \subset V_{y_1} \cup \ldots \cup V_{y_n}$. Define a finite subcollection $\mathcal{U}'$ of $\mathcal{U}$ to be $\left\{U_{y_1} , \ldots , U_{y_n}\right\}$. By our construction above, $\mathcal{U}'$ satisfies Condition~\ref{defn:subtubular-cover:width} in Definition~\ref{defn:subtubular-cover}, while it inherits Conditions~\ref{defn:subtubular-cover:subtubular} and~\ref{defn:subtubular-cover:mult} from $\mathcal{U}$. Therefore $\mathcal{U}'$ is a subtubular cover of $K$ with width $\geq L$ and multiplicity $\leq d+1$.
\end{proof}
\section{From subtubular covers to partitions of unity}
In order to apply Definition~\ref{defn:subtubular-cover} and Corollary~\ref{cor:estimate-subtubular-cover} to the context of $C^*$-algebras, our next step is to show that because the subtubular covers we obtained have large overlaps along flow lines, they give rise to partitions of unity that are \emph{almost flat} along the flow lines. What follows is a refinement of the argument presented in \cite[Section~8]{HSWW16}.
\begin{defn}[{\cite[Definition~7.11]{HSWW16}}]\label{definitionofflowwiseLipschitz}
Let $ \Phi: {\mathbb{R}^{}} \curvearrowright Y $ be a flow and $ F : Y \to X $ a map to a metric space $(X, d)$. The map $F$ is called \emph{$\Phi$-Lipschitz with constant $\delta$}, if for every $y \in Y$, the map $ t \mapsto F(\Phi_t (y) ) $ is Lipschitz with constant $\delta$. In other words, we have
\[
d( F(\Phi_t (y) ), F(y) ) \le \delta \cdot |t|
\]
for all $y \in Y$ and $t \in {\mathbb{R}^{}}$.
\end{defn}
As in \cite[Remark~8.12]{HSWW16}, one convenient way to produce flow-wise Lipschitz functions from any given function is to \emph{smear} it along the flow. More precisely, for any bounded Borel function $f$ on $Y$ and any $\lambda_+ , \lambda_- \in {\mathbb{R}^{}} $ such that $ \lambda_+ > \lambda_- $, we define $\mathbb{E} (\Phi_*)_{[\lambda_-, \lambda_+]} (f) : Y \to {\mathbb{C}^{}}$ by
\begin{equation}\label{definitionofsmearing}
\mathbb{E} (\Phi_*)_{[\lambda_-, \lambda_+]} (f) (y) = \frac{1}{\lambda_+ - \lambda_-} \int_{\lambda_-} ^{\lambda_+} f (\Phi_{-t} (y) ) \: d t .
\end{equation}
This has the advantage that it preserves, if applied to a family of functions, the property of being a partition of unity. This works for the following generalization of partitions of unity.
\begin{defn}[{\cite[Definition~7.14]{HSWW16}}]\label{def:relative-pou}
Let $ X $ be a topological space and $ A \subset X $ a subset. Let $ \mathcal{U} = \left\{ U_i \right\}_{i\in I} $ be a locally finite collection of open sets in $X$ such that $A \subset \bigcup \mathcal{U}$. Then a \emph{partition of unity for $ A \subset X $ subordinate to $ \mathcal{U} $} is a collection of continuous functions $ \left\{ f_i \middlebar X \to [0,\infty) \right\}_{i\in I} $ such that
\begin{enumerate}
\item for each $i\in I$, the support of $f_U$ is contained in $ U_i\in I $;
\item for all $x \in A$, one has $\displaystyle \sum_{i\in I} f_i (x) =1 $.
\end{enumerate}
\end{defn}
Also useful for us are certain simplicial techniques common in dimension theory, which allows us to pass from the weaker notion of multiplicity to the stronger one of chromatic number.
\begin{defn}[{\cite[Definition~7.17]{HSWW16}}]\label{def:simplicial-complex}
For us, an \emph{abstract simplicial complex} $Z$ consists of:
\begin{itemize}
\item a set $Z_0$, called the set of \emph{vertices}, and
\item a collection of its finite subsets closed under taking subsets, called the collection of \emph{simplices}.
\end{itemize}
We often write $\sigma \in Z$ to denote that $\sigma$ is a simplex of $Z$. We also associate the following structures to $Z$:
\begin{enumerate}
\item\label{def:simplicial-complex:dimension} The dimension of a simplex is the cardinality of the corresponding finite subset minus $1$, and the (simplicial) dimension of the abstract simplicial complex is the supremum of the dimensions of its simplices.
\item\label{def:simplicial-complex:realization} The \emph{geometric realization} of an abstract simplicial complex $Z$, denoted as $|Z|$, is the set of tuples
\[
\bigcup_{\sigma \in Z} \left\{ (z_v)_{v} \in [0,1]^{Z_0} ~\middle|~ \sum_{v\in\sigma} z_v = 1 \,, ~\text{and}~ z_v = 0 ~\text{for~any}~ v \in Z_0 \setminus \sigma \right\}.
\]
Similarly for a simplex $\sigma$ of $Z$, we define its \emph{closed} (respectively, \emph{open}) \emph{geometric realization} $\overline{|\sigma|}$ (respectively, $|\sigma|$) as follows:
\begin{align*}
\overline{|\sigma|} & = \left\{ (z_v)_v \in |Z| \middlebar \sum_{v\in\sigma} z_v = 1 \right\} \\
|\sigma| & = \left\{ (z_v)_v \in |Z| \middlebar \sum_{v\in\sigma} z_v = 1 ~ \text{with}~ z_v >0 ~\text{for~any~} v\in\sigma \right\}\; .
\end{align*}
\item\label{def:simplicial-complex:metric} Although usually $|Z|$ is equipped with the weak topology, for our purposes we consider the $\ell^1$-topology, induced by the $\ell^1$-metric $d^1: |Z| \times |Z| \to [0, 2]$ defined by
\[
d^1\Bigl( (z_v)_v , (z'_v)_v \Bigl) = \sum_{v \in Z_0} |z_v - z'_v | \; .
\]
\item\label{def:simplicial-complex:star} For any vertex $v_0 \in Z_0$, the \emph{(simplicial) star} around $v_0$ is the set of simplices of $Z$ that contain $v_0$, and the \emph{open star} around $v_0$ is the union of the open geometric realizations of such simplices in $|Z|$, that is, the set
\[
\left\{ (z_v)_v \in |Z| \middlebar z_{v_0} > 0 \right\} \; .
\]
\item\label{def:simplicial-complex:cone} The \emph{simplicial cone} $CZ$ is the abstract simplicical complex
\[
\left\{ \sigma, \sigma \sqcup \left\{\infty\right\} \middlebar \sigma \in Z \right\} \; ,
\]
where $\infty$ is an additional vertex. More concretely, we have $(CZ)_0 = Z_0 \sqcup \left\{\infty\right\}$, each simplex $\sigma$ in $Z$ spawns two simplices $\sigma$ and $\sigma \sqcup \left\{\infty\right\}$ in $CZ$, and all simplices of $CZ$ arise this way.
\item\label{def:simplicial-complex:subcomplex} A \emph{subcomplex} of $Z$ is an abstract simplicial complex $Z'$ with $Z'_0 \subset Z_0$ and $Z' \subset Z$. It is clear that there is a canonical embedding $|Z'| \subset |Z|$ preserving the $\ell^1$-metric.
\end{enumerate}
\end{defn}
As in \cite[Section~8]{HSWW16}, by making use of the smearing technique and certain simplicial techniques, we can obtain the passage from a subtubular cover as in Corollary~\ref{cor:estimate-subtubular-cover} to a certain partition of unity that we can later use to estimate the nuclear dimension of the crossed product $C^*$-algebra. The idea is:
\begin{enumerate}
\item shrink the cover along the flow lines \textendash\ the width of the cover tells us how long we can do this without destroying the covering ability of the shrunken sets;
\item use \cite[Lemma~8.15]{HSWW16} pick a partition of unity subordinate to the shrunken cover;
\item apply \cite[Lemma~8.13]{HSWW16} to smear this partition of unity to get a new one that is flow-wise Lipschitz, in which process the supports of the functions are allowed to grow back to the original unshrunken cover;
\item in order to obtain a control on the chromatic number (instead of just the multiplicity), we make use of the nerve complex of the original cover: viewing the above flow-wise Lipschitz partition of unity as a flow-wise Lipschitz map from the space to the nerve complex of the original cover, we can use it to pull back a canonical partition of unity on this finite dimensional simplicial complex subordinate to a canonical cover with controlled chromatic number.
\end{enumerate}
\begin{prop}\label{prop:get-simplicial-complex}
For any $d \in \Z^{\geq 0}$ and $\delta > 0$, there exists $Q = Q(\delta, d) > 0$ such that for any flow $\calpha$ on a locally connected, locally compact and metrizable space $Y$ with topological dimension $\leq d$, and for any compact subset $ K \subset Y_{> Q} $, there exists a finite simplicial complex $ Z $ of dimension at most $5(d+1)-1$, along with a map $ F: Y \to | CZ | $ satisfying:
\begin{enumerate}
\item
\label{prop:get-simplicial-complex-a}
$ F $ is $\calpha$-Lipschitz with constant $\delta$;
\item
\label{prop:get-simplicial-complex-b}
for any vertex $ v \in Z_0 $ (the vertex set of $Z$), the preimage of the open star around $v$ is contained in a tube $B_v$;
\item
\label{prop:get-simplicial-complex-c}
$ F(K) \subset Z $.
\end{enumerate}
\end{prop}
\begin{proof}
Given $d \in \Z^{\geq 0}$ and $\delta > 0$, set $ L = \frac{ 5(d+1) +1 }{\delta} $ and $Q = 20L$. Now given any flow $\calpha$ on a locally connected, locally compact and metrizable space $Y$ with topological dimension $\leq d$ and any compact subset $ K \subset Y_{> Q} $, we define $ \widehat{K} = \calpha_{ \left[ - \frac{L}{4}, \frac{L}{4} \right] } (K)$, which is also a compact subset of $ Y_{> Q} $, and then apply Corollary~\ref{cor:estimate-subtubular-cover} with $\widehat{K}$ in place of $K$, to obtain a subtubular cover of $ \widehat{K} $ with width $\geq L$ and multiplicity $ \leq 5(d+1)$.
For each $U \in \mathcal{U}$, define $ U' = \left\{ y \in Y \middlebar \calpha_{[-L, L]}(y) \subset U \right\} $, which is open by \cite[Lemma~8.22]{HSWW16}. By our construction, we have $ \calpha_{[-L, L]}(U') \subset U $. Because of the first condition in Definition~\ref{defn:subtubular-cover}, $ \mathcal{U}' = \left\{ U' \right\}_{U\in\mathcal{U}} $ covers $ \widehat{K} $. Now fix a partition of unity $ \left\{ f_U \right\}_{U\in\mathcal{U}} $ for $ \widehat{K} \subset Y $ subordinate to $ \mathcal{U}' $.
For any $U \in \mathcal{U} $, set $\widehat{f}_U = \mathbb{E} (\alpha)_{ \left[ - L, L \right] } (f_U)$.
Then by \cite[Lemma~8.16]{HSWW16}, the collection $ \big\{ \widehat{f}_U \big\}_{U \in \mathcal{U}} $ is a partition of unity for the inclusion $ K \subset X$ subordinate to $ \mathcal{U} $, whose members are $\calpha$-Lipschitz with the constant $ \frac{1 }{L} $, and so is the function $ \displaystyle \mathrm{1}_X - \sum_{U \in \mathcal{U}} \mathbb{E} (\alpha)_{ \left[ - L, L \right] } (f_U) $.
Hence by \cite[formula~(8.2) in Lemma~8.21]{HSWW16}, if we let $ Z = \mathcal{N}(\mathcal{U}) $, the nerve complex of $\mathcal{U}$ (see \cite[Definition~8.19]{HSWW16}), we have a map
$$ F= \left( \bigoplus_{U\in\mathcal{U}} \widehat{f}_U \right) \oplus \left( \mathrm{1}_X - \sum_{U \in \mathcal{U}} \mathbb{E} (\alpha)_{ \left[ - L, L \right] } (f_U) \right) : Y \to |\mathcal{N}(\mathcal{U}^+) | = | C Z |, $$
which is $\calpha$-Lipschitz with regard to the $ \mathit{l}^{1} $-metric with constant $ \frac{ 5(d+1) +1 }{L} = \delta $, as at most $ (5(d+1) +1) $ summands of $F$ are non-zero at each point. It also maps $ K $ into $ | \mathcal{N}(\mathcal{U}) | $ as $ \left( \mathrm{1}_X - \sum_{U \in \mathcal{U}} \mathbb{E} (\alpha)_{ \left[ - L, L \right] } (f_U) \right) $ vanishes on $K$. Finally, the preimage of the open star around each vertex $ U \in \mathcal{U} $ is contained in $ \mathrm{supp}(\widehat{f}_U) \subset U $, which is in turn contained in a tube $B_U$.
\end{proof}
The following lemma allows us to shrink the support of a partition of unity in a way that respects the Lipschitz constants.
\begin{lem}\label{lem:shrink-pou}
Let $(X,\operatorname{dist})$ be a metric space and let $d \in \mathbb{N}$ and $\eta > 0$. Suppose $\left\{f_i\right\}_{i \in I}$ is a partition of unity for $X$ such that
\begin{enumerate}
\item\label{lem:shrink-pou:1-mult} for any subset $F \subset I$ with cardinality greater than $d+1$, we have $\prod_{i \in F} f_i = 0$;
\item\label{lem:shrink-pou:2-Lipschitz} for any $i \in I$, $f_i$ is Lipschitz with constant $\eta$.
\end{enumerate}
Then for any $r \in \left[ 0, \frac{1}{\eta (d+1)} \right)$, there is a partition of unity $\left\{g_i\right\}_{i \in I}$ for $X$ such that
\begin{enumerate}\setcounter{enumi}{2}
\item\label{lem:shrink-pou:3-supp} for any $i \in I$, $N_{r} \big( \supp(g_i) \big) \subset \supp(f_i) ^o $;
\item\label{lem:shrink-pou:4-Lipschitz} for any $i \in I$, $g_i$ is Lipschitz with constant $\displaystyle \frac{ (d+2) \eta }{\big( 1 - (d+1) \eta r \big)^2}$.
\end{enumerate}
\end{lem}
\begin{proof}
Choose $\varepsilon \in \left( \eta r, \frac{1}{d+1} \right)$ such that
\begin{equation}\label{eq:lem:shrink-pou:Lipschitz}
\frac{ \big( 1 - (d+1) \varepsilon \big) \eta + ( 1 - \varepsilon ) (d+1) \eta }{\big( 1 - (d+1) \varepsilon \big)^2} < \frac{ (d+2) \eta }{\big( 1 - (d+1) \eta r \big)^2} \; ,
\end{equation}
which is possible because $\eta r < \frac{1}{d+1}$ and the strict inequality~\eqref{eq:lem:shrink-pou:Lipschitz} holds when we replace $\varepsilon$ with $\eta r$. Now define
\[
h (t) = \max\left\{t-\varepsilon, 0\right\} \, .
\]
So $h$ is Lipschitz with constant $1$. Hence for each $i \in I$, $h \circ f_i : X \to [ 0, \infty )$ is Lipschitz with constant $\eta$, while $f_i \leq 1$ and $\varepsilon < \frac{1}{\eta (d+1)} \leq 1$ together also imply the image of $h \circ f_i$ is within $[0, 1 - \varepsilon ]$. Moreover, for any $x \in \supp(h \circ f_i)$, we have $f_i(x) \geq \varepsilon$. Since $f_i$ is Lipschitz with constant $\eta$, any $x'$ with $\operatorname{dist}(x, x') < r$ satisfies $f_i(x') > \varepsilon - \eta r > 0$, and thus we have $B_r(x) \subset (f_i)^{-1} \big( \mathbb{R} \setminus (\eta r - \varepsilon, \varepsilon - \eta r) \big) $. Consequently, we have $N_{r} \big( \supp(h \circ f_i) \big) \subset \supp(f_i) ^o $.
Since for any $x \in X$, we have $\sum_{i \in I} f_i (x) = 1 $ and our assumption on $\left\{f_i\right\}_{i \in I}$ implies that the number of $i \in I$ with $f_i (x) > 0$ is between $1$ and $d+1$, we have
\[
0 < 1 - (d + 1) \varepsilon \leq \sum_{i \in I} ( h \circ f_i) (x) \leq 1 - \varepsilon \; .
\]
Similarly, one checks that the function $\displaystyle \sum_{i \in I} ( h \circ f_i)$ is Lipschitz with constant~$(d+1) \eta$.
Consequently, we may define a new partition of unity $\left\{g_i\right\}_{i \in I}$ for $X$ by
\[
g_i(x) = \frac{( h \circ f_i) (x)}{ \displaystyle \sum_{ j \in I} ( h \circ f_j) (x) } \quad \text{for~any~} x \in X \; .
\]
Then for any $i \in I$, we have $\supp(g_i) = \supp(h \circ f_i)$ and thus $N_{r} \big( \supp(g_i) \big) \subset \supp(f_i) ^o $. A direct computation using the ranges and the Lipschitz constants of $\displaystyle \sum_{i \in I} ( h \circ f_i)$ and $h \circ f_i$, for $i \in I$, shows that each $g_i$ is Lipschitz with a constant equal to the left-hand side of \eqref{eq:lem:shrink-pou:Lipschitz}, and thus also the right-hand side, as desired.
\end{proof}
\begin{prop}\label{prop:get-partition-of-unity}
For any $d \in \Z^{\geq 0}$, any $\eta > 0$ and any $L> 0$, there exists $R = R(L, \eta, d) > 0$ such that for any flow $\calpha$ on a locally connected, locally compact and metrizable space $Y$ with topological dimension $\leq d$, and for any compact subset $ K \subset Y_{> R} $, there exist a finite partition of unity $ \left\{ f_i \right\}_{i \in I} $ for the inclusion $ K \subset Y$ satisfying:
\begin{enumerate}
\item
\label{prop:get-partition-of-unity-a}
for any $ i\in I $, $ f_i $ is $\calpha$-Lipschitz with constant $\eta$;
\item
\label{prop:get-partition-of-unity-b}
for any $ i\in I $, $ \calpha_{[-L, L]} \big( \mathrm{supp}( f_i ) \big) $ is contained in a tube $B_i$;
\item
\label{prop:get-partition-of-unity-c}
there is a decomposition $I = I^{(0)} \cup \cdots \cup I^{(5(d+1)-1)} $ such that for any $ l \in \left\{ 0, \dots, 5(d+1)-1 \right\} $ and any two different $i, j \in I^{(l)} $, we have
$$ \calpha_{[-L, L]} \big( \mathrm{supp}( f_i ) \big) \cap \calpha_{[-L, L]} \big( \mathrm{supp}( f_j ) \big) = \varnothing . $$
\end{enumerate}
\end{prop}
\begin{proof}
Given $d \in \Z^{\geq 0}$ and $L, \eta > 0$, choose $\delta \in \left( 0, \big( 2(d+2)^2 (d+3 ) (2 d + 5) L \big) ^{-1} \right)$ such that
\begin{equation}\label{eq:prop:get-partition-of-unity:eta}
\delta \cdot \frac{ 2(d+2) (d+3 )^2 (2 d + 5) }{\big( 1 - 2(d+2)^2 (d+3 ) (2 d + 5) \delta L \big)^2} < \eta \; .
\end{equation}
Set $R = Q(\delta,d)$ as in Proposition~\ref{prop:get-simplicial-complex}. Now given any flow $\calpha$ on a locally connected, locally compact and metrizable space $Y$ with topological dimension $\leq d$ and any compact subset $ K \subset Y_{> R} $, we apply Proposition~\ref{prop:get-simplicial-complex} to obtain a finite simplicial complex $ Z $ of dimension at most $5(d+1)-1$ and a map $ F: Y \to | CZ | $ satisfying:
{
\renewcommand{\theenumi}{\theequation}
\renewcommand{\labelenumi}{(\theenumi)}
\begin{enumerate}
\stepcounter{equation}\item\label{eq:prop:get-partition-of-unity:F-a}
$ F $ is $\calpha$-Lipschitz with constant $\delta$;
\stepcounter{equation}\item\label{eq:prop:get-partition-of-unity:F-b}
for any vertex $ v \in Z_0 $ (the vertex set of $Z$), the preimage of the open star around $v$ is contained in a tube $B_v$;
\stepcounter{equation}\item\label{eq:prop:get-partition-of-unity:F-c}
$ F(K) \subset Z $.
\end{enumerate}
}
By \cite[Lemma~8.18]{HSWW16}, there is a partition of unity $ \left\{ \nu_\sigma \right\}_{\sigma \in CZ } $ for $|CZ|$ indexed by the simplices of $CZ$, such that each $\nu_\sigma$ is $2(\dim(CZ) + 1) (\dim(CZ) + 2 ) (2\dim(CZ) + 3)$-Lipschitz and for any different simplices $\sigma$ and $\sigma'$ with the same dimension, we have $\nu_\sigma\nu_{\sigma'} = 0$. In particular, $ \left\{ \nu_\sigma \right\}_{\sigma \in CZ } $ satisfies Conditions~\eqref{lem:shrink-pou:1-mult} and~\eqref{lem:shrink-pou:2-Lipschitz} of Lemma~\ref{lem:shrink-pou}, with $d+1$ in place of $d$ and $2(d+2) (d+3 ) (2 d + 5)$ in place of $\eta$.
Set $r = \delta L$. Since $2(d+2) (d+3 ) (2 d + 5) \cdot (d+2) \cdot r < 1$, we can apply Lemma~\ref{lem:shrink-pou} to $ \left\{ \nu_\sigma \right\}_{\sigma \in CZ } $ and $r$ to obtain a partition of unity $\left\{\mu_\sigma\right\}_{\sigma \in CZ } $ for $|CZ|$ satisfying
{
\renewcommand{\theenumi}{\theequation}
\renewcommand{\labelenumi}{(\theenumi)}
\begin{enumerate}
\stepcounter{equation}\item\label{eq:prop:get-partition-of-unity:mu-containment} for any $\sigma \in CZ$, $N_{r} \big( \supp(\mu_\sigma) \big) \subset \supp(\nu_\sigma) ^o $;
\stepcounter{equation}\item\label{eq:prop:get-partition-of-unity:mu-Lipschitz} for any $\sigma \in CZ$, $\mu_\sigma$ is Lipschitz with constant \[
\displaystyle \frac{ (d+3) \cdot 2(d+2) (d+3 ) (2 d + 5) }{\big( 1 - (d+2) \cdot 2(d+2) (d+3 ) (2 d + 5) r \big)^2} \;. \]
\end{enumerate}
}
Define $I = Z$, that is, the collection of all simplices in $Z$. Then there is a decomposition $ I = \bigsqcup_{l=0}^{d} I^{(l)} $, where $I^{(l)}$ consists of all $l$-dimensional simplices in $Z$, for $l = 0, \ldots, d$. For any $\sigma \in I$, define $ f_\sigma = \mu_\sigma \circ F : Y \to [0,1] $. We claim that $ \left\{f_\sigma\right\}_{\sigma\in I} $ together with the decomposition $ I = \bigcup_{l=0}^{d} I^{(l)} $ is a partition of unity for the inclusion $ K \subset Y $ satisfying the required conditions.
First, by \eqref{eq:prop:get-partition-of-unity:F-c}, we know that $F(y) \in |Z|$ for any $y \in K$, which then implies $\sum_{\sigma \in Z} \nu_\sigma ( F(y) ) = 1 $ and $\nu_\sigma ( F(y) ) = 0$ for any $\sigma \in CZ \setminus Z$. By \eqref{eq:prop:get-partition-of-unity:mu-containment}, this implies $\mu_\sigma ( F(y) ) = 0$ for any $\sigma \in CZ \setminus Z$ and thus $\sum_{\sigma \in Z} \mu_\sigma ( F(y) ) = 1 $ because $\left\{\mu_\sigma\right\}_{\sigma \in CZ } $ is a partition of unity for $|CZ|$. This shows that $ \left\{f_\sigma\right\}_{\sigma\in I} $ is a partition of unity for the inclusion $ K \subset Y $. It remains to verify the three conditions in the statement of the proposition.
\begin{enumerate}
\item For any $\sigma \in I$, since we know the Lipschitz constants of $\mu_\sigma$ and $F$, we see that $f_\sigma$ is Lipschitz with a constant equal to
\[
\delta \cdot \frac{ (d+3) \cdot 2(d+2) (d+3 ) (2 d + 5) }{\big( 1 - (d+2) \cdot 2(d+2) (d+3 ) (2 d + 5) \delta L \big)^2} \;,
\]
and thus it is also Lipschitz with constant $\eta$, because of \eqref{eq:prop:get-partition-of-unity:eta}.
\item For any $\sigma \in I$, since $F$ is Lipschitz with constant $\delta$ and $\delta L = r$, we have
\begin{equation}\label{eq:prop:get-partition-of-unity:containments}
F \left( \calpha_{[-L, L]} \big( \mathrm{supp}( \mu_\sigma \circ F ) \big) \right) \subset N_r \big( \mathrm{supp}( \mu_\sigma ) \big) \subset \mathrm{supp}( \nu_\sigma ) ^o \; .
\end{equation}
By \cite[Lemma~8.18]{HSWW16}, $\mathrm{supp}( \nu_\sigma ) ^o$ is contained in the open set $V_\sigma$, which, by its definition, is contained in the open star of any vertex in $\sigma$. Hence the preimage of $\calpha_{[-L, L]} \big( \mathrm{supp}( \mu_\sigma \circ F ) \big) $ under $F$ is contained in an open star in $|Z|$. Since $F$ was chosen to satisfy \eqref{prop:get-simplicial-complex-b} of Proposition~\ref{prop:get-simplicial-complex}, we see that $\calpha_{[-L, L]} \big( \mathrm{supp}( \mu_\sigma \circ F ) \big) $ is contained in a tube.
\item For any $ l \in \left\{ 0, \dots, 5(d+1)-1 \right\} $ and any two different $\sigma, \sigma' \in I^{(l)} $, by \cite[Lemma~8.18]{HSWW16}, we have $V_\sigma \cap V_{\sigma'} = \varnothing$, and thus $\mathrm{supp}( \nu_\sigma ) ^o \cap \mathrm{supp}( \nu_{\sigma'} ) ^o = \varnothing$. It then follows from \eqref{eq:prop:get-partition-of-unity:containments} that
\[
\calpha_{[-L, L]} \big( \mathrm{supp}( f_i ) \big) \cap \calpha_{[-L, L]} \big( \mathrm{supp}( f_j ) \big) = \varnothing \; .
\]
\end{enumerate}
\end{proof}
\section{Actions with bounded periods}
\label{section: bounded periods}
In this section, we shift our attention to the case where a flow $(Y, \R, \calpha)$ has bounded periods, i.e.\ $Y = Y_{\leq R}$ for some $R > 0$.
Notice that this is equivalent to saying that there is $R > 0$, such that for any $y \in Y$, the orbit of $y$ is equal to $\left\{ \calpha_t (y) \middlebar t \in [0,R] \right\}$. In particular, this implies (and in fact is equivalent to) the following condition defined more generally for a locally compact Hausdorff second countable group $G$ and a locally compact Hausdorff space $Y$:
\begin{defn}[{\cite[Definition~3.1]{Hirshberg-Wu16}}]
A continuous action $\calpha: G \curvearrowright Y$ is said to have \emph{uniformly compact orbits} if there exists a compact subset $K \subset G$ such that for any $y \in Y$, we have $G \cdot y = K \cdot y$.
\end{defn}
Hence by \cite[Lemma~3.2]{Hirshberg-Wu16}, we know that the quotient space $Y / \mathbb{R}$ is Hausdorff and locally compact and the quotient map $\pi \colon Y \to Y / \mathbb{R}$ is proper.
\begin{lem}\label{lem:quotient-is-metrizable}
Let $Y$ be a locally compact metrizable space and let $\calpha: \R \curvearrowright Y$ be a continuous action with bounded periods. Then $Y / \mathbb{R}$ is also metrizable.
\end{lem}
\begin{proof}
Observe that the metrizability of a locally compact Hausdorff space is equivalent to the separability of its $C^*$-algebra. Since the quotient map $\pi \colon Y \to Y/ \mathbb{R}$ is a proper map between two locally compact Hausdorff space, by the Gelfand duality, we see that $C(Y/\mathbb{R})$ embeds into $C(Y)$ and thus the separability of the former follows from that of the latter.
\end{proof}
We would like to estimate the dimension of $Y / \mathbb{R}$, for which it is helpful to recall the notion of local dimension.
\begin{defn}[{\cite[Chapter 5, Definition~1.1]{Pears75}}]
The \emph{local dimension}, $\operatorname{locdim} (X)$, of a topological space $X$ is defined as follows. If $X$ is empty, then $\operatorname{locdim}(X) = -1$. Otherwise $\operatorname{locdim}(X)$ is the least integer $n$ such that for any $x \in X$, there is some open set $U$ containing $x$ such that $\operatorname{dim}(\overline{U}) \leq n$, or $\infty$ if there is no such integer.
\end{defn}
The local nature of this notion makes it useful when studying the behavior of dimensions with regard to continuous maps.
\begin{prop}[{\cite[Chapter 5, Proposition~3.4]{Pears75}}]\label{prop:locdim-dim}
If $X$ is a weakly paracompact normal space (e.g., if $X$ is metrizable), then $\operatorname{locdim}(X) = \dim(X)$.
\end{prop}
Returning to the case of $\R$-actions, we have:
\begin{prop}\label{prop:dim-quotient-space-no-fixed-points}
Let $Y$ be a locally compact metrizable space and let $\calpha: \R \curvearrowright Y$ be a continuous action with bounded periods but no fixed points. Then $\dim (Y / \mathbb{R}) \leq \dim(Y) $.
\end{prop}
\begin{proof}
By \ref{prop:locdim-dim}, it suffices to show $\operatorname{locdim} (Y / \mathbb{R}) \leq \dim(Y) $. Now given any $x_0 \in Y / \mathbb{R}$, we write $x_0 = \pi(y_0)$ for some $y_0 \in Y$. Since $\calpha$ has no fixed points, by Lemma~\ref{lem:existence-tubes}, there is a tube $B$ with $y_0 \in S_B \cap B^o$. Observe that $\pi(B^o)$ is an open neighborhood of $x_0$ since $\pi^{-1}(\pi(B^o)) = \mathbb{R} \cdot B^o$ is open, and thus the compact set $\pi(B)$ is a closed neighborhood of $x_0$. Moreover, we have $\pi(S_B) = \pi(B)$ by the definition of tubes.
Let $l_B$ be the length of the tube $B$ and let $R$ be such that $Y = Y_{\leq R}$. We claim that for any $x \in Y / \mathbb{R}$, we have
\begin{equation}\label{eq:bounding-preimage}
| \pi^{-1}(x) \cap S_B | < \frac{R}{l_B} + 1 \; .
\end{equation}
Indeed, for any $y \in \pi^{-1}(x) \cap S_B$, we know that the orbit $\pi^{-1}(x) = \left\{ \calpha_t (y) \middlebar t \in [0,R] \right\}$. Hence if we define $D = \left\{ t \in [0, R] \middlebar \calpha_t (y) \in S_B \right\}$, then the map $\calpha_{-} (y) \colon t \in D \mapsto \calpha_t (y) \in \pi^{-1}(x) \cap S_B$ is surjective. Now for any $t \in D$, by Lemma~\ref{lem:tubes-basics}, we know that the map $\calpha_{-} (y) \colon s \in \left[ t - \frac{l_B}{2}, t + \frac{l_B}{2} \right] \mapsto \calpha_t (y) \in \pi^{-1}(x)$ is injective, which implies by translation that the map $\calpha_{-} (y) \colon s \in \left[ t , t + l_B \right] \mapsto \calpha_t (y) \in \pi^{-1}(x)$ is injective, and hence $\left( t , t + l_B \right] \cap D = \varnothing$. It follows that $D$ is discrete in $[0, R]$ and is of the form $\left\{d_0, d_1, \ldots, d_n\right\}$ with $d_0 = 0$ and $d_i > d_{i - 1} + l_B$ for $i = 1, \ldots, n$. Hence $n l_B < R$, which implies
\[
| \pi^{-1}(x) \cap S_B | \leq |D| < \frac{R}{l_B} + 1 \; .
\]
To finish the proof, we observe that since both $S_B$ and $\pi(S_B)$ are compact Hausdorff, the topology of the latter coincides with the quotient topology induced by the continuous surjection $\pi|_{S_B} \colon S_B \to \pi(S_B)$. Furthermore, it follows from \eqref{eq:bounding-preimage} that $\pi|_{S_B}$ is a finite-to-one map. Hence by Theorem~\ref{thm:Pears75} and Proposition~\ref{prop:Pears75}, we have $\dim(\pi(S_B)) = \dim(S_B) \leq \dim(Y)$. To summarize, we have found a closed neighborhood $\dim(\pi(S_B))$ of $x_0$ of dimension at most $\dim(Y)$. Since $x_0$ is arbitrary, we have $\operatorname{locdim} (Y / \mathbb{R}) \leq \dim(Y) $.
\end{proof}
\begin{prop}\label{prop:dim-quotient-space}
Let $Y$ be a locally compact metrizable space and let $\calpha: \R \curvearrowright Y$ be a continuous action with bounded periods. Then $\dim (Y / \mathbb{R}) \leq \dim(Y) $.
\end{prop}
\begin{proof}
Observe that the fixed-point set $Y^{\R}$ is a closed subset of $Y$. Thus $\dim(Y^{\R} / \R) = \dim(Y^{\R}) \leq \dim(Y)$. By Proposition~\ref{prop:dim-quotient-space-no-fixed-points}, we have $\dim((Y \setminus Y^{\R}) / \R) \leq \dim(Y)$. Since $Y \setminus Y^{\R}$ is an open subset of $Y$, its image $(Y \setminus Y^{\R}) / \R$ is an open subset of $Y / \R$. Now the desired inequality follows by applying \cite[Chapter 3, Corollary~5.8]{Pears75} to the decomposition $Y / \R = Y^{\R} / \R \sqcup (Y \setminus Y^{\R}) / \R$.
\end{proof}
Recall the following theorem:
\begin{thm}\label{thm:dimnuc-uniformly-compact-orbits} \cite[Theorem 3.4]{Hirshberg-Wu16}
Let $\calpha: G \curvearrowright Y$ be a continuous action with uniformly compact orbits. Then
\[
\dimnuc^{+1} ( C_0(Y) \rtimes G ) \leq \dim^{+1} (Y / G) \cdot \sup_{y \in Y} \dimnuc^{+1} ( C^*(G_y) )
\]
and
\[
\dr^{+1} ( C_0(Y) \rtimes G ) \leq \dim^{+1} (Y / G) \cdot \sup_{y \in Y} \dr^{+1} ( C^*(G_y) ) \; .
\]
\end{thm}
\begin{cor}\label{cor:dimnuc-bounded-periods}
Let $Y$ be a locally compact metrizable space and let $\calpha: \R \curvearrowright Y$ be a continuous action with bounded periods. Then
\[
\dr^{+1} ( C_0(Y) \rtimes \R ) \leq 2 \dim^{+1}(Y) \; .
\]
\end{cor}
\begin{proof}
Since for any $y \in Y$, the stabilizer $\R_y$, being a closed subgroup of $\R$, must be either $0$, $\R$, or $p\Z$ for some $p>0$, thus $\dr ( C^*(\R_y) )$ is either $0$ or $1$. The result then follows from Theorem~\ref{thm:dimnuc-uniformly-compact-orbits} and Proposition~\ref{prop:dim-quotient-space}.
\end{proof}
\section{Clopen subgroupoids and conditional expectations}
In this section, we take on a digression to prove a general result about the reduce groupoid $C^*$-algebra of a locally compact groupoid $G$: a clopen subgroupoid $H$ of $G$ induces an inclusion $C^*_r(H) \subset C^*_r(G)$ which admits a natural \emph{conditional expectation}, that is, a contractive, completely positive map $e \colon C^*_r(G) \to C^*_r(H)$ such that
\begin{equation} \label{eq:conditional-expectation}
e(abc) = a \cdot e(b) \cdot c \text{~and~} e(a) = a
\end{equation}
for any $b \in C^*_r(G)$ and $a, c \in C^*_r(H)$ (see \cite{tomiyama} or \cite[II.6.10]{blackadar-operator-algebras} for other equivalent definitions). Such conditional expectations will play a role in the nuclear dimension estimate in our main result.
The following lemma is straightforward from Definition~\ref{def:locally-compact-groupoids}.
\begin{lem}\label{lem:open-subgroupoid}
Let $G$ be a locally compact groupoid with a left Haar system $\lambda = \left\{\lambda^u \right\}_{u \in G^0}$. Let $H$ be an open subset of $G$ which forms a subgroupoid (i.e., $H^0 = H \cap G^0$ and the functions $d$ and $r$ as well as the multiplication are inherited from $G$). Then $H$ is also a locally compact groupoid, the linear space $C_c(H)$ embeds into $C_c(G)$ by extending functions trivially from $H$ to $G$, and $\lambda$ restricts to a left Haar system $\lambda_H$ on $H$ via the embedding $C_c(H) \subset C_c(G)$, such that this inclusion extends to an inclusion $C^*_r(H, \lambda_H) \subset C^*_r(G, \lambda)$ of $C^*$-algebras. \qed
\end{lem}
Our main result of this section is the following:
\begin{thm}\label{thm:clopen-subgroupoid}
Let $G$ be a locally compact groupoid with a left Haar system $\lambda = \left\{\lambda^u \right\}_{u \in G^0}$. Let $H$ be a \emph{clopen} subset of $G$ which forms a subgroupoid and let $\lambda_H$ be the restriction of $\lambda$ to $H$ by Lemma~\ref{lem:open-subgroupoid}. Then the natural inclusion $C^*_r(H, \lambda_H) \subset C^*_r(G, \lambda)$ admits a conditional expectation $e \colon C^*_r(G, \lambda) \to C^*_r(H, \lambda_H)$.
\end{thm}
\begin{proof}
For any open Hausdorff subset $U$ of $G$, the subset $H \cap U$ is clopen in $U$, and thus for any compactly supported continuous function $f$ on $U$, the pointwise product $f \cdot \chi_{H \cap U}$, where $\chi_{H \cap U}$ is the characteristic function of $H \cap U$ in $U$, is also continuous and compactly supported in $U$. This allows us to define a map \[\rho_H \colon C_c(G) \to C_c(H) \subset C_c(G)\] given by $f \mapsto f \cdot \chi_H$, where $\chi_H$ is the characteristic function of $H$ in $G$ and the multiplication ``$\cdot$'' is pointwise. Similarly define \[\rho_{G \setminus H} \colon C_c(G) \to C_c(G \setminus H) \subset C_c(G)\] by $f \mapsto f \cdot \chi_{G \setminus H}$, where $C_c(G \setminus H)$ is defined in the same way as $C_c(G)$ and $C_c(H)$. It is clear that
\begin{enumerate}
\item both $\rho_H$ and $\rho_{G \setminus H}$ are idempotents, that is, $\rho_H \circ \rho_H = \rho_H$ and $\rho_{G \setminus H} \circ \rho_{G \setminus H} = \rho_{G \setminus H}$,
\item the image of $\rho_H$ is $C_c(H)$ and that of $\rho_{G \setminus H}$ is $C_c(G \setminus H)$, and
\item $\rho_H + \rho_{G \setminus H} = \operatorname{id}_{C_c(G)}$.
\end{enumerate}
Moreover, since the set $H \cdot (G \setminus H) \cdot H$ given by
\[
\{ x_1 \cdot y \cdot x_2 \mid x_1, x_2 \in H ,\, y \in G \setminus H, \, d(x_1) = r(y), \, d(y) = r(x_2) \}
\]
has no intersection with $H$, by the way the convolution in $C_c(G)$ is defined, we have
\[
C_c(H) \ast C_c(G \setminus H) \ast C_c(H) \subset C_c(G \setminus H) \; ,
\]
that is, the kernel of $\rho_H$. It follows that for any $g \in C_c(G)$ and $f_1, f_2 \in C_c(H)$,
\begin{equation}\label{eq:conditional-expectation-proof}
\rho_H(f_1 g f_2) = \rho_H \big(f_1 \rho_H(g) f_2 \big) + \rho_H \big(f_1 \rho_{G \setminus H}(g) f_2 \big) = f_1 \rho_H(g) f_2 \; .
\end{equation}
Following Definition~\ref{def:reduced-groupoid-algebra}, for any $v \in G^0$, we write $\operatorname{Ind}^G v \colon C_c(G) \to B(L^2(G_v, \lambda_v))$ for the left regular representation at $v$. Similarly, if $v \in H^0$, we have the left regular representation $\operatorname{Ind}^H v \colon C_c(H) \to B(L^2(H_v, (\lambda_H)_v))$. Observe that since for any $v \in H^0$, the measure $(\lambda_H)_v$ is restricted from $\lambda_v$, we have an isometry
\[
V_v \colon L^2(H_v, (\lambda_H)_v) \to L^2(G_v, \lambda_v)
\]
given by trivially extending $L^2$-functions on $H_v$ to $G_v$, which induces, via compression, the completely positive contraction
\[
\varphi_v \colon B(L^2(G_v, \lambda_v)) \to B(L^2(H_v, (\lambda_H)_v)), \; T \mapsto V_v^* T V_v \; .
\]
For any $v \in H^0$, we have the following:
\begin{enumerate}
\item For any $f \in C_c(H)$, we have $\varphi_v ( \operatorname{Ind}^G v (f) ) = \operatorname{Ind}^H v (f)$, because for any $\xi, \eta \in L^2(H_v, (\lambda_H)_v)$, we can calculate the inner product
\begin{align*}
& \left\langle \varphi_v ( \operatorname{Ind}^G v (f)) \cdot \xi \;,\; \eta \right\rangle_{L^2(H_v, (\lambda_H)_v)} \\
= & \left\langle V_v^* \cdot \operatorname{Ind}^G v (f) \cdot V_v \cdot \xi \;,\; \eta \right\rangle_{L^2(H_v, (\lambda_H)_v)} \\
= & \left\langle \operatorname{Ind}^G v (f) \cdot (V_v \xi) \;,\; V_v \eta \right\rangle_{L^2(G_v, \lambda_v)} \\
= & \int_{x \in G_v} \int_{t \in G_v} f(xt^{-1}) \cdot (V_v \xi)(t) \cdot \overline{(V_v \eta)(x)} \, d\lambda_v(t) \, d\lambda_v(x) \\
= & \int_{x \in H_v} \int_{t \in H_v} f(xt^{-1}) \cdot \xi(t) \cdot \overline{ \eta(x)} \, d\lambda_v(t) \, d\lambda_v(x) \\
= & \left\langle \operatorname{Ind}^H v (f) \cdot \xi \;,\; \eta \right\rangle_{L^2(H_v, (\lambda_H)_v)} \; .
\end{align*}
\item For any $f \in C_c(G \setminus H)$, we have $\varphi_v ( \operatorname{Ind}^G v (f) ) = 0$, because for any $\xi, \eta \in L^2(H_v, (\lambda_H)_v)$, we can similarly calculate the inner product
\begin{align*}
& \left\langle \varphi_v ( \operatorname{Ind}^G v (f)) \cdot \xi \;,\; \eta \right\rangle_{L^2(H_v, (\lambda_H)_v)} \\
= & \int_{x \in H_v} \int_{t \in H_v} f(xt^{-1}) \cdot \xi(t) \cdot \overline{ \eta(x)} \, d\lambda_v(t) \, d\lambda_v(x) \\
= & 0 \;
\end{align*}
as $xt^{-1} \in H$ for any $x, t \in H_v$ but $f(H) = \{0\}$.
\end{enumerate}
Hence for any $f \in C_c(G)$, we have
\[
\varphi_v \left( \operatorname{Ind}^G v (f) \right) = \varphi_v \left( \operatorname{Ind}^G v \left(\rho_H(f) + \rho_{G \setminus H}(f) \right) \right) = \operatorname{Ind}^H v \left(\rho_H(f)\right) + 0 \; ,
\]
and thus we have shown $\varphi_v \circ \operatorname{Ind}^G v = \operatorname{Ind}^H v \circ \rho_H$ for any $v \in H^0$. Writing $\varphi_v = 0$ for any $v \in G \setminus H$, we define a completely positive contraction
\[
\varphi = \bigoplus_{v \in G^0} \varphi_v \colon B\left(\bigoplus_{v \in G^0} L^2(G_v, \lambda_v)\right) \to B\left(\bigoplus_{v \in H^0} L^2(H_v, (\lambda_H)_v)\right) \; .
\]
It follows that
\[
\varphi \circ \left(\bigoplus_{v \in G^0} \operatorname{Ind}^G v\right) = \left(\bigoplus_{v \in H^0} \operatorname{Ind}^H v\right) \circ \rho_H \; .
\]
Since by Definition~\ref{def:reduced-groupoid-algebra}, $C^*_r(G, \lambda)$ (respectively, $C^*_r(H, \lambda_H)$) is the norm completion of the image of $\left(\bigoplus_{v \in G^0} \operatorname{Ind}^G v\right)$ (respectively, $\left(\bigoplus_{v \in H^0} \operatorname{Ind}^H v\right)$), we see that $\varphi$ restricts to a completely positive contraction from $C^*_r(G, \lambda)$ to $C^*_r(H, \lambda_H)$ that extends the map $\rho_H \colon C_c(G) \to C_c(H)$. Since $\rho_H|_{C_c(H)} = \operatorname{id}_{C_c(H)}$, by a density argument, we have $\varphi(a) = a$ for any $a \in C^*_r(H, \lambda_H)$. Similarly, \eqref{eq:conditional-expectation-proof} implies $\varphi(abc) = a \varphi(b) c$ for any $b \in C^*_r(G, \lambda)$ and $a, c \in C^*_r(H, \lambda_H)$.
\end{proof}
\begin{rmk}
By applying a theorem of Tomiyama (\cite{tomiyama}), which says every contractive projection from a $C^*$-algebra to a $C^*$-subalgebra of it is a conditional expectation, we may shorten the proof of Theorem~\ref{thm:clopen-subgroupoid}, e.g., by leaving out the part that gives us \eqref{eq:conditional-expectation}. However, we choose to present a more down-to-earth proof for the sake of completeness.
\end{rmk}
\begin{eg}
If $G$ is an \'{e}tale groupoid and $H$ is taken to be its unit space $G^0$, which is a clopen subgroupoid with trivial operations, then our construction recovers the standard conditional expectation from $C^*_r(G, \lambda)$ onto its Cartan subalgebra $C_0(G^0)$.
\end{eg}
\begin{prop}\label{prop:clopen-subgroupoid-hereditary}
Let $G$, $H$ and $e \colon C^*_r(G, \lambda) \to C^*_r(H, \lambda_H)$ be as in Theorem~\ref{thm:clopen-subgroupoid}. Then under the canonical embeddings
\[
C_0(H^0) \hookrightarrow C_0(G^0) \hookrightarrow M(C^*(G, \lambda))
\]
as in \eqref{eq:canonical-embedding-unit-space}, we have
\[
e(abc) = a \cdot e(b) \cdot c
\]
for any any $b \in C^*_r(G, \lambda)$ and any $a,c \in C_0(H^0)$. In particular, for any $a \in C_0(H^0)$, the map $e$ restricts to a conditional expectation
\[
e|_{a C^*_r(G, \lambda) a^*} \colon a C^*_r(G, \lambda) a^* \to a C^*_r(H, \lambda_H) a^* \; .
\]
\end{prop}
\begin{proof}
Since $C_c(G)$ is dense in $C^*_r(G, \lambda)$, it suffices to show $e(abc) = a \cdot e(b) \cdot c$ for any any $b \in C_c(G)$ and any $a,c \in C_0(H^0)$. To this end, we notice that the set $H^0 \cdot (G \setminus H) \cdot H^0$ given by
\[
\{ x_1 \cdot y \cdot x_2 \mid x_1, x_2 \in H^0 ,\, y \in G \setminus H, \, d(x_1) = r(y), \, d(y) = r(x_2) \}
\]
has no intersection with $H$. By the way the embedding $C_0(G^0) \hookrightarrow M(C^*(G, \lambda)) $ is defined, we have
\[
C_0(H^0) \ast C_c(G \setminus H) \ast C_0(H^0) \subset C_c(G \setminus H) \; ,
\]
where $\ast$ denotes the convolution product in $M(C^*(G, \lambda))$. Let $\rho_H \colon C_c(G) \to C_c(H)$ and $\rho_{G \setminus H} \colon C_c(G) \to C_c(G \setminus H)$ be defined as in the proof of Theorem~\ref{thm:clopen-subgroupoid}. Then for any $b \in C_c(G)$ and $a, c \in C_0(H^0)$, we have
\begin{equation*}
\rho_H(a b c) = \rho_H \big(a \rho_H(b) c \big) + \rho_H \big(a \rho_{G \setminus H}(b) c \big) = a \rho_H(b) c \; ,
\end{equation*}
because $\rho_H + \rho_{G \setminus H} = \operatorname{id}_{C_c(G)}$ and $\operatorname{ker} \rho_{H} = C_c(G \setminus H)$.
\end{proof}
\begin{rmk}
By a standard approximation trick, we can also show that
\begin{equation*}
e(abc) = a \cdot e(b) \cdot c
\end{equation*}
for any $b \in C^*_r(G)$ and any $a$ and $c$ in the closure of $C^*_r(H)$ in the multiplier algebra $M(C^*_r(G))$ under the strict topology.
\end{rmk}
\section{Subgroupoids associated to a tube}
Recall from Example~\ref{example:groupoids} that a topological flow $(Y, {\mathbb{R}^{}}, \calpha)$ gives rise to the transformation groupoid $Y \rtimes_{\calpha} \mathbb{R}$. We often drop the subscript and simply write $Y \rtimes \mathbb{R}$ when there is no confusion. Here we associate two open subgroupoid $\mathcal{G}_B$ and $\mathcal{H}_B$ to any tube $B$ in the space.
\begin{defn}\label{def:tube-groupoids}
Let $(Y, {\mathbb{R}^{}}, \calpha)$ be a topological flow and let $B$ be a tube in $Y$ with length $l_B$. Let
\[
a_+, a_- \colon B \to \mathbb{R}
\]
be as in Definition~\ref{defn:tubes}, which are continuous functions by Lemma~\ref{lem:tubes-basics}. Let $S_{B^o}$ be the open central slice of $B$ as in Definition~\ref{defn:tubes}. Let $Y \rtimes \mathbb{R}$ be the transformation groupoid as in Example~\ref{example:groupoids}: as a set, we have $Y \rtimes \mathbb{R} = Y \times \mathbb{R}$. When there is no confusion, we also identify the unit space $Y \times \{0\}$ with $Y$. We define subsets of $Y \rtimes \mathbb{R}$ as follows:
\[
\mathcal{G}_B = \{ x \in Y \rtimes \mathbb{R} \mid d(x) \text{ and } r(x) \in B^o \} = \{ (y, t) \in Y \times \mathbb{R} \mid \calpha_{-t}(y) \text{ and } y \in B^o \}
\]
and
\[
\mathcal{H}_B = \left\{ (y, t) \in Y \times \mathbb{R} \middlebar y \in B^o \text{ and } - t \in \big(a_-(y), a_+(y)\big) \right\} \; .
\]
We also define an auxiliary groupoid
\[
\mathcal{K}_B = S_{B^o} \times \left( \left( - \frac{l_B}{2} , \frac{l_B}{2} \right) \times \left( - \frac{l_B}{2} , \frac{l_B}{2} \right) \right) \;,
\]
i.e., the product of the open central slice $S_{B^o}$, as a space, with the pair groupoid $\left( \left( - \frac{l_B}{2} , \frac{l_B}{2} \right) \times \left( - \frac{l_B}{2} , \frac{l_B}{2} \right) \right)$, equipped with the product topology, such that the unit space $\left(\mathcal{K}_B\right)^0$ is given by $\{ (y, t, s) \in \mathcal{K}_B \mid t = s \}$, we have $d(y,t,s) = (y,s,s)$, $r(y,t,s) = (y,t,t)$, and $(y,t,s) \cdot (y, s, s') = (y, t, s')$ for any $y \in S_{B^o}$ and $t,s,s' \in \left( - \frac{l_B}{2} , \frac{l_B}{2} \right)$, and the Haar system is given by the Lebesgue measure on $\left( - \frac{l_B}{2} , \frac{l_B}{2} \right)$, which is identified with each $\left( \mathcal{K}_B \right)^u$.
\end{defn}
\begin{lem}\label{lem:groupoids-G-H}
Let $(Y, {\mathbb{R}^{}}, \calpha)$ be a topological flow and let $B$ be a tube in $Y$. We have
\begin{enumerate}
\item $ \mathcal{G}_B $ is an open subgroupoid of $Y \rtimes \mathbb{R}$, and
\item $ \mathcal{H}_B $ is a clopen subset of $\mathcal{G}_B$.
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}
\item This is clear because $\mathcal{G}_B$ is the reduction of $Y \rtimes \mathbb{R}$ to the open subset $B^o$ of the unit space $Y$ (see Example~\ref{example:groupoids}).
In particular, it is open since it can be written as $d^{-1}(B^o) \cap r^{-1}(B^o)$, the intersection of two open subsets.
\item We first note that $\mathcal{H}_B \subset \mathcal{G}_B$ because for any $(y,t) \in \mathcal{H}_B$, we have $y \in B^o$ and $\calpha_{-t}(y) \in B^o$ by Definition~\ref{defn:tubes}.
To see $\mathcal{H}_B$ is clopen in $\mathcal{G}_B$, we first note that $\mathcal{H}_B$ is open because it is the preimage of the open subset $\{ (t_1, t_2, t_3) \in \mathbb{R}^3 \colon t_1 < t_2 < t_3 \} \subset \mathbb{R}^3$ under the continuous map $(y, t) \in \mathcal{G}_B \to \left(a_-(y) , - t , a_+(y)\right) \in \mathbb{R}^3$.
On the other hand, by Definition~\ref{defn:tubes} and the continuity of $\calpha$, for any $y \in B^o$, we have $\calpha_{a_-(y)} (y) \notin B^o$, that is, $r\left( y, - a_-(y) \right) \notin B^o$, and thus $\left( y, - a_-(y) \right) \notin \mathcal{G}_B$. Similarly, $\left( y, - a_+(y) \right) \notin \mathcal{G}_B$. Hence the image of the (same) continuous map $(y, t) \in \mathcal{G}_B \to \left(a_-(y) , - t , a_+(y)\right) \in \mathbb{R}^3$ does not intersect the subset $\{ (t_1, t_2, t_3) \in \mathbb{R}^3 \colon t_1 = t_2 \text{~or~} t_2 = t_3 \}$. It follows that $\mathcal{H}_B$ is also the preimage of the closed subset $\{ (t_1, t_2, t_3) \in \mathbb{R}^3 \colon t_1 \leq t_2 \leq t_3 \}$ under this continuous map, which shows it is also closed.
\end{enumerate}
\end{proof}
\begin{lem}\label{lem:groupoids-H-K}
Let $(Y, {\mathbb{R}^{}}, \calpha)$ be a topological flow and let $B$ be a tube in $Y$ with length $l_B$ and let a map
\[
\widetilde{\tau}_B \colon \mathcal{K}_B \to Y \rtimes \mathbb{R} , \quad (y, t, s) \mapsto \left( \calpha_t(y) , t - s \right)
\]
be defined. Then we have
\begin{enumerate}
\item $ \widetilde{\tau}_B $ is a groupoid homomorphism (i.e., it intertwines the unit space, the maps $d$ and $r$, and the multiplication),
\item the image of $ \widetilde{\tau}_B $ is the open subset $\mathcal{H}_B$,
\item $ \widetilde{\tau}_B $ is a homeomorphism onto its image (and thus $\mathcal{H}_B$ is a locally compact Hausdorff groupoid which is isomorphic to $\mathcal{K}_B$ as a topological groupoid), and
\item $ \widetilde{\tau}_B $ intertwines $\lambda_{\mathcal{K}}$, the left Haar system on $\mathcal{K}_B$, and $\lambda_{\mathcal{H}}$, the restriction of the left Haar system on $Y \rtimes \mathbb{R}$ to $\mathcal{H}_B$.
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}
\item For any $(y, t, t) \in \left(\mathcal{K}_B\right)^0$, we have $\widetilde{\tau}_B (y, t, t) = \left( \calpha_{t} (y), 0 \right) \in \left( Y \rtimes \mathbb{R} \right)^0$. This shows $\widetilde{\tau}_B \left(\left(\mathcal{K}_B\right)^0\right) \subset \left( Y \rtimes \mathbb{R} \right)^0$. Moreover, for any $y \in S_{B^o}$ and $t,s,s' \in \left( - \frac{l_B}{2} , \frac{l_B}{2} \right)$, we use Example~\ref{example:groupoids} and Definition~\ref{def:tube-groupoids} to check that
\[
d \left(\widetilde{\tau}_B (y,t,s)\right) = d \left( \calpha_{t} (y), t - s \right) = \left( \calpha_{s} (y), 0 \right) = \widetilde{\tau}_B (y,s,s) = \widetilde{\tau}_B \left( d(y,t,s)\right) \; ,
\]
\[
r \left(\widetilde{\tau}_B (y,t,s)\right) = r \left( \calpha_{t} (y), t - s \right) = \left( \calpha_{t} (y), 0 \right) = \widetilde{\tau}_B (y,t,t) = \widetilde{\tau}_B \left( r(y,t,s)\right) \; ,
\]
and
\begin{align*}
\left(\widetilde{\tau}_B (y,t,s)\right) \cdot \left(\widetilde{\tau}_B (y,s,s')\right) & = \left( \calpha_{t} (y), t - s \right) \cdot \left( \calpha_{s} (y), s - s' \right) \\
&= \left( \calpha_{t} (y), t - s' \right) \\
&= \widetilde{\tau}_B (y,t,s') \; .
\end{align*}
Together these show that $ \widetilde{\tau}_B $ is a groupoid homomorphism.
\item We first show $ \widetilde{\tau}_B \left(\mathcal{K}_B\right) \subset \mathcal{H}_B$. For any $(y,t,s) \in \mathcal{K}_B$, where $y \in S_{B^o}$ and $t, s \in \left( - \frac{l_B}{2}, \frac{l_B}{2} \right)$, it follows from Lemma~\ref{lem:tubes-basics} that $\calpha_{t}(y)$ falls inside $B^o$, and
\[
a_-(\calpha_{t}(y)) = - \frac{l_B}{2} - t \text{~and~} a_+(\calpha_{t}(y)) = \frac{l_B}{2} - t \; .
\]
Hence we have
\[
a_-(\calpha_{t}(y)) = - \frac{l_B}{2} - t < s-t < \frac{l_B}{2} - t = a_+(\calpha_{t}(y)) \; .
\]
Thus by Definition~\ref{def:tube-groupoids}, we have $\widetilde{\tau}_B (y,t,s) \in \mathcal{H}_B$. This shows $ \widetilde{\tau}_B \left(\mathcal{K}_B\right) \subset \mathcal{H}_B$.
To prove the other direction, recall from Lemma~\ref{lem:tubes-basics} that there are a pair of homeomorphisms
\[
\tau_B \colon S_B \times \left[ - \frac{l_B}{2}, \frac{l_B}{2} \right] \to B \; , \quad (y, t) \mapsto \calpha_t(y)
\]
and
\[
\theta_B \colon B \to S_B \times \left[ - \frac{l_B}{2}, \frac{l_B}{2} \right] \; , \quad y \to \left( \calpha_{\frac{a_-(y) + a_+(y)}{2}} (y), \frac{l_B}{2} - a_+(y) \right) \; .
\]
that implement a local trivialization for $B$. We write $\theta_{B,1} \colon B \to S_B$ and $\theta_{B,2} \colon B \to \left[ - \frac{l_B}{2}, \frac{l_B}{2} \right]$ for the two components of $\theta_B$ and define a map
\[
\widetilde{\theta}_B \colon \mathcal{H}_B \to \mathcal{K}_B \, , \quad (y, t) \mapsto \left( \theta_{B,1}(y), \theta_{B,2}(y), \theta_{B,2}(y) - t \right) \, ,
\]
which is well defined since $\theta_{B,1} \left(B^o\right) \subset S_{B^o}$, $\theta_{B,2} \left(B^o\right) \subset \left( - \frac{l_B}{2}, \frac{l_B}{2} \right)$, and $\theta_{B,2} (y) - t \in \left( - \frac{l_B}{2}, \frac{l_B}{2} \right)$ because
\[
- t \in \left(a_-(y) , a_+(y) \right) = \left( - \frac{l_B}{2} - \theta_{B,2} (y) , \frac{l_B}{2} - \theta_{B,2} (y) \right) \; .
\]
Then we check that for any $(y,t) \in \mathcal{H}_B$, we have
\begin{align*}
\widetilde{\tau}_B \left( \widetilde{\theta}_B (y,t) \right) &= \widetilde{\tau}_B \left( \theta_{B,1}(y), \theta_{B,2}(y), \theta_{B,2}(y) - t \right) \\
&= \left( \tau \left( \theta (y) \right) , \theta_{B,2}(y) - \left( \theta_{B,2}(y) - t \right) \right) \\
&= (y,t) \; .
\end{align*}
This shows that $\widetilde{\tau}_B \cdot \widetilde{\theta}_B = \operatorname{id}_{\mathcal{H}_B}$ and thus $\widetilde{\tau}_B\left(\mathcal{K}_B\right) = \mathcal{H}_B$.
\item We have just checked $\widetilde{\tau}_B \cdot \widetilde{\theta}_B = \operatorname{id}_{\mathcal{H}_B}$ above. Now for any $(y,t,s) \in \mathcal{K}_B$, we have
\begin{align*}
\widetilde{\theta}_B \left( \widetilde{\tau}_B (y,t,s) \right) &= \widetilde{\theta}_B \left( \tau (y,t), t - s \right) \\
&= \left( \theta_{B,1}(\tau (y,t)), \theta_{B,2}(\tau (y,t)), \theta_{B,2}(\tau (y,t)) - (t-s) \right) \\
&= (y,t,s) \; .
\end{align*}
This shows $\widetilde{\theta}_B$ and $\widetilde{\tau}_B$ are mutual inverses. Both are continuous, by the continuity of $\tau$ and $\theta$. Therefore $ \widetilde{\tau}_B $ is a homeomorphism onto its image.
\item For any $u = (y, t, t) \in \left( \mathcal{K}_B \right)^0$, we have $\widetilde{\tau}_B(u) = (\calpha_t (y), 0)$ and $\widetilde{\theta}_B$ restrict to a map
\[
\widetilde{\theta}_B |_{ \lambda_{\mathcal{K}}^{u} } \colon \left(\mathcal{K}_B \right)^{u} \to \left(\mathcal{H}_B \right)^{\widetilde{\tau}_B(u)} \, , \quad (y, t, s) \mapsto \left( \calpha_t (y), t - s \right) \; ,
\]
which is a linear map in $s$ with coefficient $-1$. Since the left Haar measures $\lambda_{\mathcal{K}}^{u}$ on $\left( \mathcal{K}_B \right)^{u}$ and $\lambda_{\mathcal{H}}^{\widetilde{\tau}_B(u)}$ on $\left( \mathcal{H}_B \right)^{\widetilde{\tau}_B(u)}$ are given, respectively, by the Lebesgue measures on the third coordinate and the second coordinate, we see that the push-forward measure $\left( \widetilde{\theta}_B \middle|_{ \lambda_{\mathcal{K}}^{u} } \right)_* \left( \lambda_{\mathcal{K}}^{u} \right)$ agrees with $\lambda_{\mathcal{H}}^{\widetilde{\tau}_B(u)}$.
\end{enumerate}
\end{proof}
Intuitively speaking, if we think of elements in the groupoid $Y \rtimes \mathbb{R}$ as directed paths following the orbits of the action $\calpha$, then $\mathcal{G}_B$ contains all the paths that both start and end in $B^o$, while $\mathcal{H}_B$ contains all the paths that lie entirely inside $B^o$.
Next, we look at the $C^*$-algebras of the groupoids $\mathcal{G}_B$ and $\mathcal{H}_B$.
\begin{lem}\label{lem:groupoid-algebra-H}
Let $(Y, {\mathbb{R}^{}}, \calpha)$ be a topological flow and let $B$ be a tube in $Y$. Then
\begin{enumerate}
\item $C^*_r(\mathcal{H}_B)$ is isomorphic to the stabilization of $C_0(S_{B^o})$, and
\item The embedding $\mathcal{H}_B \hookrightarrow \mathcal{G}_B$ induces an inclusion $C^*_r(\mathcal{H}_B) \subset C^*_r(\mathcal{G}_B)$ that admits a conditional expection $e_B \colon C^*_r(\mathcal{G}_B) \to C^*_r(\mathcal{H}_B)$.
\end{enumerate}
\end{lem}
\begin{proof}
The first statement follows from Lemma~\ref{lem:groupoids-H-K}, Example~\ref{example:reduced-groupoid-algebra} and the fact that $C^*_r(\mathcal{G}_1 \times \mathcal{G}_2) \cong C^*_r(\mathcal{G}_1) \otimes_{\text{min}} C^*_r(\mathcal{G}_2)$ for locally compact groupoids $\mathcal{G}_1$ and $\mathcal{G}_2$.
Since $\mathcal{H}_B$ embeds into $\mathcal{G}_B$ as a clopen subgroupoid by Lemma~\ref{lem:groupoids-G-H}, the second statement follows from Theorem~\ref{thm:clopen-subgroupoid}.
\end{proof}
Adopting the intuitive picture as above, we see that the effect of the conditional expectation $e$ is that it annihilates all the paths in $\mathcal{G}_B$ that roam outside of $B$. On the other hand, we see that if a path is very short, then having both ends inside $B^o$ implies that its entirety falls inside $B^o$. To make this idea precise, we make use of the notion of propagation.
\begin{defn}\label{def:propagation}
For any element $(y, t)$ in the transformation groupoid $Y \rtimes \mathbb{R}$, we define its \emph{propagation} $\operatorname{prop}(y, t)$ to be the absolute value of $t$.
For any $f \in C_c(Y \rtimes \mathbb{R})$, we define its \emph{propagation} $\operatorname{prop}(f)$ to be the supremum of $\operatorname{prop}(x)$, where $x$ ranges in the support of $f$.
\end{defn}
\begin{rmk}\label{rmk:def:propagation}
Note that since $f$ has compact support and taking propagation constitutes a continuous function from $Y \rtimes \mathbb{R} \to [0, \infty)$, the aforementioned supremum is achieved by some element in the support of $f$ and is thus always finite. Also note that under the standard identification of the convolution algebras $C_c(Y \rtimes \mathbb{R})$ and $C_c(\mathbb{R}, C_c(Y))$ as given in Example~\ref{example:reduced-groupoid-algebra}, the propagation of $f \in C_c(\mathbb{R}, C_c(Y))$ is equal to the largest possible $|t|$ for $t \in \mathbb{R}$ with $f(t) \not= 0$.
\end{rmk}
Recall that there is a canonical embedding
\begin{equation} \label{eq:canonical-embedding-Y}
{C}_{0}(Y) \hookrightarrow M({C}_{0}(Y) \rtimes {\mathbb{R}^{}}) \cong M(C^*_r(Y \rtimes \mathbb{R}))
\end{equation}
as given in \eqref{eq:canonical-embedding-unit-space}, so that for any $ f \in {C}_{0}(Y) $ and for any $ g $ in the dense subalgebra $ {C}_{c}(Y \rtimes \mathbb{R}) $ in $ C^*_r(Y \rtimes \mathbb{R}) $, the multiplication of $ g $ by $ f $ from the left and right are also in ${C}_{c}(Y \rtimes \mathbb{R})$ and are given by
\[
(f \cdot g) \ (y, t) = f(y) \cdot g (y, t) \ \text{and}\ (g \cdot f ) \ (y, t) = g(y, t) \cdot \alpha_{t} ( f ) (y) = g(y, t) \cdot f \left( \calpha_{-t} (y)\right)
\]
for any $(y,t) \in Y \rtimes \mathbb{R}$.
\begin{lem}\label{lem:groupoid-algebra-compression}
Let $(Y, {\mathbb{R}^{}}, \calpha)$ be a topological flow and let $B$ be a tube in $Y$. Let $f \in C_0(Y)$ have support inside $B$. Then for any $g$ in the dense subalgebra $C_c(Y \rtimes \mathbb{R})$, we have $f g f^* \in C_c(\mathcal{G}_B)$ and $\operatorname{prop}(f g f^*) \leq \operatorname{prop}(g)$. Moreover, the hereditary subalgebra $f C^*_r(Y \rtimes \mathbb{R}) f^*$ is contained in the subalgebra $C^*_r(\mathcal{G}_B)$ and is in fact equal to $f C^*_r(\mathcal{G}_B) f^*$.
\end{lem}
\begin{proof}
It is straightforward to see that for any $g \in C_c(Y \rtimes \mathbb{R})$, the support of $f g f^* \in C_c(Y \rtimes \mathbb{R})$ is
\[
\overline{\left\{ (y,t) \in Y \rtimes \mathbb{R} \mid g(y,t) \not= 0 , \, f(y) \not= 0 ,\, f(\calpha_{-t}(y)) \not= 0 \right\}} \; ,
\]
which is contained in the support of $g$ and also in $\mathcal{G}_B$ because $\{ y \in Y \mid f(y) \not= 0 \}$ is contained in $B^o$. The first containment gives us $\operatorname{prop}(f g f^*) \leq \operatorname{prop}(g)$, while the second tells us $f g f^* \in C_c(\mathcal{G}_B)$. Thus we have $f C^*_r(Y \rtimes \mathbb{R}) f^* \subset C^*_r(\mathcal{G}_B)$ by a density argument. Hence
\[
f C^*_r(Y \rtimes \mathbb{R}) f^* = f^2 C^*_r(Y \rtimes \mathbb{R}) (f^2)^* \subset f C^*_r(\mathcal{G}_B) f^* \subset f C^*_r(Y \rtimes \mathbb{R}) f^* \; ,
\]
which implies $f C^*_r(Y \rtimes \mathbb{R}) f^* = f C^*_r(\mathcal{G}_B) f^*$.
\end{proof}
\begin{lem}\label{lem:groupoid-propagation-G-H}
Let $(Y, {\mathbb{R}^{}}, \calpha)$ be a topological flow and let $B$ be a tube in $Y$. Let $\epsilon$ be a positive number as in Definition~\ref{defn:tubes}. Then any $x \in \mathcal{G}_B$ with $\operatorname{prop}(x) \leq \epsilon$ is contained in $\mathcal{H}_B$.
\end{lem}
\begin{proof}
Given $(y,t) \in \mathcal{G}_B$ with $\operatorname{prop}(x) \leq \epsilon$, we have $y \in B^o$, $\calpha_{-t}(y) \in B^o$ and $|t| \leq \epsilon$. By Definition~\ref{defn:tubes} and by the continuity of $\calpha$, the first two conditions above imply that $-t \notin \left[ a_-(y) - \epsilon, a_-(y) \right] \cup \left[ a_+(y), a_+(y) + \epsilon \right]$. Since $a_-(y) < 0 < a_+(y)$ and $-t \in [-\epsilon, \epsilon]$, we have $-t \in \left( a_-(y), a_+(y) \right)$. Therefore $(y,t) \in \mathcal{H}_B$.
\end{proof}
\begin{lem}\label{lem:groupoid-algebra-propagation-G-H}
Let $Y$, $\calpha$, $B$ and $\epsilon$ be as in Lemma~\ref{lem:groupoid-propagation-G-H}. Then any $g \in C_c(\mathcal{G}_B)$ with $\operatorname{prop}(g) \leq \epsilon$ is contained in the subalgebra $C_c(\mathcal{H}_B)$. In particular, if we let $e_B \colon C^*_r(\mathcal{G}_B) \to C^*_r(\mathcal{H}_B)$ be the conditional expectation as in Lemma~\ref{lem:groupoid-algebra-H}, we have $e_B(g) = g$.
\end{lem}
\begin{proof}
This follows immediately from Lemma~\ref{lem:groupoid-propagation-G-H}.
\end{proof}
\section{The main result}
In this section, we bound the nuclear dimension of the crossed product algebra $ {C}_{0}(Y) \rtimes {\mathbb{R}^{}} $ in terms of the covering dimension of $Y$.
\begin{thm}\label{thm-main}
Let $(Y, {\mathbb{R}^{}}, \calpha)$ be a topological flow. Assume $Y$ is a locally compact metrizable space with finite covering dimension. Then
\[
\operatorname{dim}_\mathrm{nuc}({C}_{0}(Y) \rtimes {\mathbb{R}^{}} ) \leq (\dim(Y)+1)(5\dim(Y)+7) - 1 \; .
\]
\end{thm}
\begin{proof}
\setcounter{clminproof}{\value{Thm}}
Set $d=\dim (Y)$. We need to show $\dimnuc ( C_0(Y) \rtimes_\alpha \R ) + 1 \leq (d+1)(5d+7)$. Since $ {C}_{c}({\mathbb{R}^{}}, {C}_{c}(Y)) $ is dense in $ {C}_{0}(Y) \rtimes {\mathbb{R}^{}} $, it suffices to show that for any finite subset $ F \subset {C}_{c}({\mathbb{R}^{}}, {C}_{c}(Y)) \cap ( {C}_{0}(Y) \rtimes {\mathbb{R}^{}})_{\leq 1} $ and $ \varepsilon > 0 $, the condition of Lemma~\ref{Lemma:finite-dimnuc} holds for $F$ and $(7d+10) \varepsilon$. Given such $ F $ and $\varepsilon$, we start by finding $ L > 0 $ such that $ F \subset {C}_{c}( (-L, L), {C}_{c}(Y)) \subset {C}_{c}({\mathbb{R}^{}}, {C}_{c}(Y)) $.
Denote by $ \left\| b \right\|_1 $ the $L^{1}$-norm of $ b \in F $ as a function over $\mathbb{R}$.
Set
\begin{equation}\label{eq:defn-eta}
\eta = \frac{\varepsilon ^2 }{ 4 L ^3 \left( \sup_{b \in F} \left\| b \right\|_1 \right) ^2 } \, .
\end{equation}
We define $R = R(L, \eta, d) > 0$ as in Proposition~\ref{prop:get-partition-of-unity}. Recall that
\begin{align*}
Y_{\leq R} & = \left\{ y \in Y \middlebar \mathrm{per}_{\calpha}(y) \leq R \right\} = \left\{ y \in Y \middlebar \calpha_{\R}(y) = \calpha_{[0,R]} (y) \right\} \; , \\
Y_{> R} & = Y \setminus Y_{\leq R} = \left\{ y \in Y \middlebar \mathrm{per}_{\calpha}(y) > R \right\} \; .
\end{align*}
By Corollary~\ref{cor:long-period-open}, $Y_{> R}$ is an open $\calpha$-invariant subset, and $Y_{\leq R}$ is a closed $\calpha$-invariant subset. Thus we have the following exact sequence
\begin{equation}\label{eq:exact-sequence}
0 \to C_0(Y_{> R}) \rtimes \R \overset{\theta}{\longrightarrow} C_0(Y ) \rtimes \R \overset{\pi}{\longrightarrow} C_0(Y_{\leq R}) \rtimes \R \to 0 \; .
\end{equation}
Let us first focus on the quotient algebra in this exact sequence. Since the restricted action $\R \curvearrowright Y_{\leq R}$ has bounded periods, by Corollary~\ref{cor:dimnuc-bounded-periods}, we have
\[
\dimnuc ( C_0(Y_{\leq R}) \rtimes \R ) \leq \dr ( C_0(Y_{\leq R}) \rtimes \R ) \leq 2 (d + 1) - 1 \; .
\]
Hence by definition, we can find a piecewise contractive $2 (d + 1)$-decomposable approximation for $(\pi (F), \varepsilon)$:
\begin{equation}\label{eq:decomposable-approximation-for-quotient}
\xymatrix{\displaystyle
C_0(Y_{\leq R}) \rtimes \R \ar[dr]_{\psi_{\leq R} = \bigoplus_{l=0} ^{2d+1} \psi_{\leq R}^{(l)} \ \ \ } \ar@{.>}[rr]^{\id} & & C_0(Y_{\leq R}) \rtimes \R \\
\displaystyle & A_{\leq R} = \bigoplus_{l=0} ^{2d+1} A_{\leq R}^{(l)} \ar[ur]_{\ \ \ \ \varphi_{\leq R} = \sum_{l=0} ^{2d+1} \varphi_{\leq R}^{(l)} } &
}
\end{equation}
By Lemma~\ref{Lemma:lifting-decoposable-maps}, we may lift $\varphi_{\leq R}$ (against $\pi$) to a piecewise contractive $2 (d + 1)$-decomposable completely positive map
\[
\widetilde{\varphi}_{\leq R} = \sum_{l=0} ^{ 2d+1} \widetilde{\varphi}_{\leq R}^{(l)} : A_{\leq R} \to C_0(Y) \rtimes \R \; .
\]
By \cite[Proposition 1.4]{winter-zacharias}, there exists $\delta > 0$ such that for any positive contraction $e \in C_0(Y) \rtimes \R$, if $\left\| [ (1 - e), \widetilde{\varphi}_{\leq R}^{(l)} (a) ] \right\| \leq \delta \|a\| $ for any $a \in A_{\leq R}$ and for any $l \in \left\{0, \ldots, 2d+1\right\}$, then there are completely positive contractive order zero maps $\widehat{\varphi}_{\leq R}^{(l)} : A_{\leq R} \to C_0(Y ) \rtimes \R $ such that
\[
\left\| \widehat{\varphi}_{\leq R}^{(l)} (a) - (1 - e)^\frac{1}{2} \widetilde{\varphi}_{\leq R}^{(l)} (a) (1 - e)^\frac{1}{2} \right\| \leq \varepsilon \| a \|
\]
for all $a \in A_{\leq R}$ and for all $l \in \left\{0, \ldots, 2d+1\right\}$.
By Lemma~\ref{lem:quasicentral-approximate-unit}, there is a quasicentral approximate unit for
\[
C_0(Y_{> R}) + C_0(Y_{> R}) \rtimes \R \subseteq M(C_0(Y ) \rtimes \R)
\]
which is contained in $C_c(Y_{> R})_{+, \leq 1}$. Thus, we may choose an element $e \in C_c(Y_{> R})_{+, \leq 1}$ which satisfies:
\begin{enumerate}
\item $\left\| [ (1 - e), \widetilde{\varphi}_{\leq R}^{(l)} (a) ] \right\| \leq \delta \|a\| $ for any $a \in A_{\leq R}$ and for any $l \in \left\{0, \ldots, 2d+1\right\}$;
\item $\left\| e^\frac{1}{2} \: b \: e^\frac{1}{2} + (1-e)^\frac{1}{2} \: b \: (1-e)^\frac{1}{2} - b \right\| \leq \varepsilon $ for any $b \in F$;
\item $\left\| (1-e)^\frac{1}{2} ( (\widetilde{\varphi}_{\leq R} \circ \psi_{\leq R} \circ \pi) (b) - b ) (1-e)^\frac{1}{2} \right\| \leq 2 \varepsilon $ for any $b \in F$.
\end{enumerate}
Therefore, we can find maps $\widehat{\varphi}_{\leq R}^{(l)}$ as described in the previous paragraph, and sum them up to obtain a piecewise contractive $ 2(d+1)$-decomposable completely positive map
\[
\widehat{\varphi}_{\leq R} = \sum_{l=0} ^{ 2d+1} \widehat{\varphi}_{\leq R}^{(l)} : A_{\leq R} \to C_0(Y ) \rtimes \R
\]
such that for any $a \in A_{\leq R}$,
\[
\left\| \widehat{\varphi}_{\leq R} (a) - (1 - e)^\frac{1}{2} \widetilde{\varphi}_{\leq R} (a) (1 - e)^\frac{1}{2} \right\| \leq 2(d+1) \varepsilon \| a \| \; .
\]
\begin{clminproof}\label{clm:estimate-1:thm:estimate-dimnuc-Z}
The diagram
\[
\xymatrix{\displaystyle
C_0(Y) \rtimes \R \ar[dr]_{\psi_{\leq R} \circ \pi \ \ } \ar@{.>}[rr]^{\id} & & C_0(Y) \rtimes \R \\
\displaystyle & A_{\leq R} \ar[ur]_{\ \widehat{\varphi}_{\leq R} } &
}
\]
commutes on $(1 - e)^\frac{1}{2} F (1 - e)^\frac{1}{2}$ up to errors bounded by $(2d+4) \varepsilon$.
\end{clminproof}
\begin{proof}
Observe that for any $b \in F$,
\begin{align*}
& \left\| (\widehat{\varphi}_{\leq R} \circ \psi_{\leq R} \circ \pi ) \left( (1 - e)^\frac{1}{2} b (1 - e)^\frac{1}{2} \right) - (1 - e)^\frac{1}{2} b (1 - e)^\frac{1}{2} \right\| & \\
= & \ \left\| (\widehat{\varphi}_{\leq R} \circ \psi_{\leq R} \circ \pi ) \left( b \right) - (1 - e)^\frac{1}{2} b (1 - e)^\frac{1}{2} \right\| & \Big[\scriptscriptstyle{ \pi(e) = 0 }\Big] \\
\leq & \ \left\| (\widehat{\varphi}_{\leq R} \circ \psi_{\leq R} \circ \pi ) ( b ) - (1 - e)^\frac{1}{2} \cdot ( \widetilde{\varphi}_{\leq R} \circ \psi_{\leq R} \circ \pi ) ( b ) \cdot (1 - e)^\frac{1}{2} \right\| \\
& + \left\| (1 - e)^\frac{1}{2} \big( (\widetilde{\varphi}_{\leq R} \circ \psi_{\leq R} \circ \pi ) ( b ) - b \big) (1 - e)^\frac{1}{2} \right\| \\
\leq & \ 2(d+1) \varepsilon \left\| (\psi_{\leq R} \circ \pi ) ( b ) \right\| + 2 \varepsilon \\
\leq & \ (2d+4) \varepsilon . & \Big[\scriptstyle{\|b\| \leq 1 }\Big]
\end{align*}
This proves the claim.
\end{proof}
Our next step is to find an approximation for $ e^\frac{1}{2} F e^\frac{1}{2} $, which concerns the kernel algebra in the exact sequence in~\eqref{eq:exact-sequence}. Since $e$ is compactly supported and $\mathrm{supp}(e) \subset Y_{> R}$, we may pick a compact subset $K \subset Y_{> R}$ such that $ \mathrm{supp}(e) \subset K^o $. Our choice of $R$ allows us to apply Proposition~\ref{prop:get-partition-of-unity} to obtain a finite partition of unity $\left\{f_i\right\}_{i \in I}$ for the inclusion $K \subset Y_{>R}$ satisfying
\begin{enumerate}
\item \label{prop:get-partition-of-unity-a-re1} for any $ i\in I $, $ f_i $ is $\calpha$-Lipschitz with constant $\eta$;
\item \label{prop:get-partition-of-unity-b-re1} for any $ i\in I $, $ \calpha_{[-L, L]} \big( \mathrm{supp}( f_i ) \big) $ is contained in a tube $B_i$ in $Y_{>R}$;
\item \label{prop:get-partition-of-unity-c-re1} there is a decomposition $I = I^{(0)} \cup \cdots \cup I^{(5(d+1)-1)} $ such that for any $ l \in \left\{ 0, \dots, 5(d+1)-1 \right\} $ and any two different $i, j \in I^{(l)} $, we have
$$ \calpha_{[-L, L]} \big( \mathrm{supp}( f_i ) \big) \cap \calpha_{[-L, L]} \big( \mathrm{supp}( f_j ) \big) = \varnothing . $$
By eliminating redundancies, we may assume without loss of generality that $I^{(0)} , \cdots , I^{(5(d+1)-1)}$ are disjoint.
\end{enumerate}
Now for each $ i\in I $, let us shrink the tube $B_i$ to
\[
\widehat{B}_i = \left\{ y \in Y_{>R} \middlebar \calpha_{[-L,L]}(y) \subset B_i \right\} \; ,
\]
which is a tube with the same central slice but a shorter length $l_{\widehat{B}_i} = l_{{B}_i} - 2 L$, while
\begin{equation}\label{eq:varepsilon_B-hat}
\varepsilon_{\widehat{B}_i} \geq \varepsilon_{{B}_i} + 2L > 2 L \; ,
\end{equation}
which follows immediately from the definition of the value $\varepsilon_{{B}_i}$ in Lemma~\ref{lem:tubes-basics}. Additionally, condition (\ref{prop:get-partition-of-unity-b-re1}) above implies that $ \mathrm{supp}( f_i ) \subset \widehat{B}_i$. These properties of the shrunken tubes will be used in Claim~\ref{clm:estimate-2:thm:estimate-dimnuc-Z}.
For each tube $\widehat{B}_i$, we define $\mathcal{G}_{\widehat{B}_i}$ and $\mathcal{H}_{\widehat{B}_i}$ as in Definition~\ref{def:tube-groupoids}. Then by Lemma~\ref{lem:groupoids-G-H} and~\ref{lem:groupoids-H-K}, $\mathcal{G}_{\widehat{B}_i}$ is an open subgroupoid of $Y_{>R} \rtimes \mathbb{R}$ (itself an open subgroupoid of $Y \rtimes \mathbb{R}$) and $\mathcal{H}_{\widehat{B}_i}$ is a clopen subgroupoid of $\mathcal{G}_{\widehat{B}_i}$. There are, by Lemma~\ref{lem:open-subgroupoid}, embeddings
\begin{equation}\label{eq:groupoid-algebra-embeddings}
C^*_r(\mathcal{H}_{\widehat{B}_i}) \subset C^*_r(\mathcal{G}_{\widehat{B}_i}) \subset C^*_r( Y_{>R} \rtimes \mathbb{R} ) \subset C^*_r( Y \rtimes \mathbb{R} ) \; ,
\end{equation}
by Lemma~\ref{lem:groupoid-algebra-H}, a conditional expectation
\[
e_i = e_{\widehat{B}_i} \colon C^*_r(\mathcal{G}_{\widehat{B}_i}) \to C^*_r(\mathcal{H}_{\widehat{B}_i})
\]
and, by Lemma~\ref{lem:groupoid-algebra-compression}, a compression map
\[
\kappa_i \colon C^*_r( Y \rtimes \mathbb{R} ) \to f_i^{\frac{1}{2}} C^*_r(\mathcal{G}_{\widehat{B}_i}) f_i^{\frac{1}{2}} \, , \quad b \mapsto f_i^{\frac{1}{2}} b f_i^{\frac{1}{2}} \; ,
\]
which is completely positive and contractive because $\left\| f_i \right\| \leq 1$.
Since $e \in C_c(Y_{> R})$ and $L$ is chosen such that $ F \subset {C}_{c}( (-L, L), {C}_{c}(Y))$, we have $ e^\frac{1}{2} F e^\frac{1}{2} \subset {C}_{c}( (-L, L), {C}_{c}(Y_{>R}))$. By Example~\ref{example:reduced-groupoid-algebra} and Remark~\ref{rmk:def:propagation}, under the canonical identification between $C_0(Y_{>R}) \rtimes \mathbb{R}$ and $C^*_r( Y_{>R} \rtimes \mathbb{R} )$, we have, for any $b \in F$, that $e^\frac{1}{2} b e^\frac{1}{2} \in C_c( Y_{>R} \rtimes \mathbb{R} )$ with propagation $\operatorname{prop}\left( e^\frac{1}{2} b e^\frac{1}{2} \right) < L$. It follows by Lemma~\ref{lem:groupoid-algebra-compression} that for each $i \in I$ and any $b \in F$, we have
\[
\operatorname{prop}\left(\kappa_i\left(e^\frac{1}{2} b e^\frac{1}{2}\right)\right) \leq \operatorname{prop}\left( e^\frac{1}{2} b e^\frac{1}{2} \right) < L
\]
and thus
\begin{equation}\label{eq:e-kappa}
e_i \left(\kappa_i\left(e^\frac{1}{2} b e^\frac{1}{2}\right)\right) = \kappa_i\left(e^\frac{1}{2} b e^\frac{1}{2}\right)
\end{equation}
by Lemma~\ref{lem:groupoid-algebra-propagation-G-H} and \eqref{eq:varepsilon_B-hat}.
By Proposition~\ref{prop:clopen-subgroupoid-hereditary}, $e_i$ restricts to a conditional expectation
\[
e_i \left|_{f_i^{\frac{1}{2}} C^*_r(\mathcal{G}_{\widehat{B}_i}) f_i^{\frac{1}{2}}} \right. \colon f_i^{\frac{1}{2}} C^*_r(\mathcal{G}_{\widehat{B}_i}) f_i^{\frac{1}{2}} \to f_i^{\frac{1}{2}} C^*_r(\mathcal{H}_{\widehat{B}_i}) f_i^{\frac{1}{2}} \;,
\]
and thus we may define, for each $i \in I$, a completely positive contraction
\[
\psi_i = e_i \left|_{f_i^{\frac{1}{2}} C^*_r(\mathcal{G}_{\widehat{B}_i}) f_i^{\frac{1}{2}}} \right. \circ \kappa_i \colon C^*_r( Y \rtimes \mathbb{R} ) \to f_i^{\frac{1}{2}} C^*_r(\mathcal{H}_{\widehat{B}_i}) f_i^{\frac{1}{2}}
\]
and a $*$-homomorphism
\[
\varphi_i \colon f_i^{\frac{1}{2}} C^*_r(\mathcal{H}_{\widehat{B}_i}) f_i^{\frac{1}{2}} \to C^*_r( Y \rtimes \mathbb{R} )
\]
given by the embeddings induced from \eqref{eq:groupoid-algebra-embeddings}.
For $ l = 0, \dots, 5(d+1)-1 $, we define algebras
\begin{align*}
A_{> R}^{(l)} &= \bigoplus_{i\in I^{(l)}} f_i^{\frac{1}{2}} C^*_r(\mathcal{H}_{\widehat{B}_i}) f_i^{\frac{1}{2}} \\
A_{> R} &= \bigoplus_{l = 0} ^{5(d+1)-1} A_{> R}^{(l)} = \bigoplus_{i\in I} f_i^{\frac{1}{2}} C^*_r(\mathcal{H}_{\widehat{B}_i}) f_i^{\frac{1}{2}}
\end{align*}
and maps
\begin{align*}
\psi_{> R}^{(l)} &= \bigoplus_{i\in I^{(l)}} \psi_i \colon C^*_r( Y \rtimes \mathbb{R} ) \to A_{> R}^{(l)} \\
\varphi_{> R}^{(l)} &= \sum_{i\in I^{(l)}} \varphi_i \colon A_{> R}^{(l)} \to C^*_r( Y \rtimes \mathbb{R} ) \; .
\end{align*}
Then each $\psi_{> R}^{(l)} $ is again a completely positive contraction, and since the family $ \left\{ f_i ^{\frac{1}{2}} \right\}_{i\in I^{(l)}} $ is orthogonal, the collection of subalgebras $\left\{ f_i ^{\frac{1}{2}} C^*_r(\mathcal{H}_{\widehat{B}_i}) f_i^{\frac{1}{2}} \right\}_{i\in I^{(l)}}$ is also orthogonal, which implies that each $ \varphi_{> R}^{(l)} $ is again a homomorphism, in particular, a completely positive order zero contraction. Hence we have the following diagram:
\begin{displaymath}
\xymatrix{
{C}_{0}(Y) \rtimes {\mathbb{R}^{}} \ar@{.>}[rr]^{\mathrm{Id}} \ar[rd]^{\bigoplus_{i\in I} \kappa_i} \ar@/_/[rdd]_{\bigoplus_{l = 0} ^{5d+4} \psi_{> R}^{(l)} = \: \bigoplus_{i\in I} \psi_i} & & {C}_{0}(Y) \rtimes {\mathbb{R}^{}} \\
& { \bigoplus_{i\in I} f_i^{\frac{1}{2}} C^*_r(\mathcal{G}_{\widehat{B}_i}) f_i^{\frac{1}{2}} } \ar@/_/[d]_{\bigoplus_{i\in I} e_i} \ar[ru]^{\sum_{i\in I} \mathrm{Incl}_i} & \\
& A_{> R} = \bigoplus_{i\in I} f_i^{\frac{1}{2}} C^*_r(\mathcal{H}_{\widehat{B}_i}) f_i^{\frac{1}{2}} \ar@/_/@{^{(}->}[ruu]_{\sum_{l = 0} ^{5d+4} \varphi_{> R}^{(l)} = \: \sum_{i\in I} \varphi_i} \ar@/_/@{^{(}->}[u] &
}
\end{displaymath}
where $\displaystyle \operatorname{Incl}_i \colon f_i^{\frac{1}{2}} C^*_r(\mathcal{G}_{\widehat{B}_i}) f_i^{\frac{1}{2}} \hookrightarrow {C}_{0}(Y) \rtimes {\mathbb{R}^{}}$ denotes the canonical embedding induced from \eqref{eq:groupoid-algebra-embeddings}.
\begin{clminproof}\label{clm:estimate-2:thm:estimate-dimnuc-Z}
The large triangle in the above diagram commutes on $e^\frac{1}{2} F e^\frac{1}{2}$ up to errors bounded by $5(d+1) \varepsilon$.
\end{clminproof}
\begin{proof}
First we observe that it suffices to show the upper triangle commutes on $e^\frac{1}{2} F e^\frac{1}{2}$ up to errors bounded by $5(d+1) \varepsilon $. Indeed, for any $ b \in F$, we have
\begin{equation}\label{eq:large-triangle-upper-triangle}
\left( \sum_{i \in I} \varphi_i \right) \circ \left( \bigoplus_{i \in I} \psi_i \right) \left( e^\frac{1}{2} b e^\frac{1}{2} \right)
= \sum_{i \in I} e_i \circ \kappa_i \left( e^\frac{1}{2} b e^\frac{1}{2} \right)
= \sum_{i \in I} \kappa_i \left( e^\frac{1}{2} b e^\frac{1}{2} \right)
\end{equation}
by \eqref{eq:e-kappa}, and the right-hand side can also be written as
\[
\left( \sum_{i \in I} \operatorname{Incl}_i \right) \circ \left( \bigoplus_{i \in I} \kappa_i \right) \left( e^\frac{1}{2} b e^\frac{1}{2} \right) \; ,
\]
which is the composition of the two lower arrows of the upper triangle.
Thus we just need to bound the norm of
\[
\left( e^\frac{1}{2} b e^\frac{1}{2} \right) - \sum_{i \in I} \kappa_i \left( e^\frac{1}{2} b e^\frac{1}{2} \right) \; ,
\]
which, by the definition of $\kappa_i$, is equal to
\[
\left( e^\frac{1}{2} b e^\frac{1}{2} \right) - \sum_{i \in I} f_i^{\frac{1}{2}} \cdot \left( e^\frac{1}{2} b e^\frac{1}{2} \right) \cdot f_i^{\frac{1}{2}} \; .
\]
To do this, we compute, for any $b \in F$, for any $i \in I$ and for $ t \in (- L, L) $,
\begin{align*}
& \left\| \left( f_i^{\frac{1}{2}} \cdot { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } - { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } \cdot f_i^{\frac{1}{2}} \right) (t) \right\|_{{C}_{0}(Y)} \\
= &\ \left\| f_i^{\frac{1}{2}} \cdot { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } (t) - { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } (t) \cdot \alpha_{t} \left(f_i^{\frac{1}{2}} \right) \right\|_{{C}_{0}(Y)} \\
= &\ \left\| \left( f_i^{\frac{1}{2}} - \alpha_{t} \left( f_i^{\frac{1}{2}} \right) \right) \cdot e^\frac{1}{2} \cdot \alpha_{t}(e^\frac{1}{2}) \cdot b(t) \right\|_{{C}_{0}(Y)} \\
\le &\ \left\| f_i^{\frac{1}{2}} - \alpha_{t} \left( f_i^{\frac{1}{2}} \right) \right\|_{{C}_{0}(Y)} \cdot \left\| e^\frac{1}{2} \right\|_{{C}_{0}(Y)} \cdot \left\| \alpha_{t}(e^\frac{1}{2}) \right\|_{{C}_{0}(Y)} \cdot \left\| b (t) \right\|_{{C}_{0}(Y)} \\
\leq &\ \sup_{y\in Y} \left| f_i^{\frac{1}{2}} (y) - f_i^{\frac{1}{2}} \left( \alpha_{-t} (y) \right) \right| \cdot \left\| b (t) \right\|_{{C}_{0}(Y)} \; .
\end{align*}
Since $f_i$ was taken to be $\calpha$-Lipschitz with constant $\eta$, the last line above is bounded above by
\begin{align*}
\sup_{s \in {\mathbb{R}^{\ge 0}} , \Delta s \in [0, \eta L ] } \left| \sqrt{ s + \Delta s } - \sqrt{s} \right| \cdot \left\| b (t) \right\|_{{C}_{0}(Y)} = &\ \sqrt{ \eta L } \cdot \left\| b (t) \right\|_{{C}_{0}(Y)} \; .
\end{align*}
Combined with fact that $b$ is supported inside $(-L, L)$ and our definition of $\eta$ in~\eqref{eq:defn-eta}, this gives the estimate
\[
\left\| f_i^{\frac{1}{2}} \cdot { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } - { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } \cdot f_i^{\frac{1}{2}} \right\|_1 \le 2 L \cdot \sqrt{ \eta L } \cdot \left\| b \right\|_1 \leq \varepsilon
\]
in the function space $L^1(\R, {C}_{0}(Y))$, and thus
\[
\left\| f_i^{\frac{1}{2}} \left[ f_i^{\frac{1}{2}}, { e^\frac{1}{2} b e^\frac{1}{2} } \right] \right\|_{{C}_{0}(Y)\rtimes {\mathbb{R}^{}}} \le \left\| f_i^{\frac{1}{2}} \right\|_{{C}_{0}(Y)} \cdot \left\| \left[ f_i^{\frac{1}{2}}, { e^\frac{1}{2} b e^\frac{1}{2} } \right] \right\|_1 \le \varepsilon \; .
\]
Moreover, for each $ l \in \left\{0,\dots ,5d+4 \right\} $, since for any different $ i, j \in I^{(l)} $, we have
$$ \calpha_{[-L, L]}(\mathrm{supp}(f_i)) \cap \calpha_{[-L, L]}(\mathrm{supp}(f_j)) = \varnothing , $$
thus for any $ b \in F \subset {C}_{c}( (-L, L), {C}_{c}(Y)) $,
$$ \left( f_i^{\frac{1}{2}} \cdot { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } \cdot f_j^{\frac{1}{2}} \right) (t) = \left( f_i^{\frac{1}{2}} \cdot \calpha_{t} \left(f_j^{\frac{1}{2}} \right) \right) \cdot \left( e^\frac{1}{2} \cdot \calpha_{t}(e^\frac{1}{2}) \right) \cdot b(t) = 0 , $$
both when $ |t| \ge L $, as $b(t) = 0$, and when $ |t| < L $, as $f_i^{\frac{1}{2}} \cdot \calpha_{t} \left(f_j^{\frac{1}{2}} \right) = 0$. Therefore $ f_i^{\frac{1}{2}} \cdot { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } \cdot f_j^{\frac{1}{2}} = 0 $, and it follows that $ f_i^{\frac{1}{2}} \left[ f_i^{\frac{1}{2}}, { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } \right] $ and $ f_j^{\frac{1}{2}} \left[ f_j^{\frac{1}{2}}, { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } \right] $ are orthogonal to each other. Consequently, using the fact that $ \sum_{i\in I} f_i (y) = 1$ for any $ y \in K$, we see that for any $ b \in F $,
\begin{align*}
& \left\| { e^\frac{1}{2} b e^\frac{1}{2} } - \sum_{i\in I} f_i^{\frac{1}{2}} \cdot { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } \cdot f_i^{\frac{1}{2}} \right\| \\
= &\ \left\| \left( \sum_{i\in I} f_i \right) \cdot { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } - \sum_{i\in I} f_i^{\frac{1}{2}} \cdot { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } \cdot f_i^{\frac{1}{2}} \right\| \\
= &\ \left\| \sum_{i\in I} f_i^{\frac{1}{2}} \left[ f_i^{\frac{1}{2}}, { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } \right] \right\| \\
\leq &\ \sum_{l=0} ^{5d+4} \left\| \sum_{i\in I^{(l)}} f_i^{\frac{1}{2}} \left[ f_i^{\frac{1}{2}}, { \left( e^\frac{1}{2} b e^\frac{1}{2} \right) } \right] \right\| \\
\le &\ 5(d+1) \varepsilon \; ,
\end{align*}
where we used the orthogonality in the norm estimate of the sum. Combined with \eqref{eq:large-triangle-upper-triangle}, we obtain $\displaystyle \left\| { e^\frac{1}{2} b e^\frac{1}{2} } - \left( \sum_{i \in I} \varphi_i \right) \circ \left( \bigoplus_{i \in I} \psi_i \right) \left( e^\frac{1}{2} b e^\frac{1}{2} \right) \right\| \le 5(d+1) \varepsilon$.
\end{proof}
To finish the proof of Theorem~\ref{thm-main}, we define $\psi \colon C_0(Y) \rtimes \R \to A_{\leq R} \oplus A_{> R}$ so that
$$\psi (b) = ( \psi_{\leq R} \circ \pi ) \left( (1 - e)^\frac{1}{2} b (1 - e)^\frac{1}{2} \right) \oplus \psi_{> R} \left( e^\frac{1}{2} b e^\frac{1}{2} \right)
$$
for any $b \in C_0(Y) \rtimes \R$ and consider the diagram
\[
\xymatrix{\displaystyle
C_0(Y) \rtimes \R \ar[dr]_{\psi \ } \ar@{.>}[rr]^{\id} & & C_0(Y) \rtimes \R \\
\displaystyle & A_{\leq R} \oplus A_{> R} \ar[ur]_{\ \ \ \varphi = \widehat{\varphi}_{\leq R} + \varphi_{> R} } &
}
\]
Observe that:
\begin{enumerate}
\item $\psi$ is completely positive and contractive.
\item $\varphi_{\leq R}$ is a sum of $2(d + 1)$-many order zero contractions, and $\varphi_{> R}$ is a sum of $5(d+1)$-many $*$-homomorphisms (which, in particular, are order zero contractions).
\item For all $b \in F$, we compute, using the bounds given by Claims~\ref{clm:estimate-1:thm:estimate-dimnuc-Z} and~\ref{clm:estimate-2:thm:estimate-dimnuc-Z} and using the properties of $e$:
\begin{align*}
& \left\|\varphi(\psi(b)) - b \right\| \\
\leq & \ \left\| ( \widehat{\varphi}_{\leq R} \circ \psi_{\leq R} \circ \pi ) \left( (1 - e)^\frac{1}{2} b (1 - e)^\frac{1}{2} \right) - (1 - e)^\frac{1}{2} b (1 - e)^\frac{1}{2} \right\| \\
& + \left\| ( \varphi_{> R} \circ \psi_{>R} ) \left( e^\frac{1}{2} b e^\frac{1}{2} \right) - e^\frac{1}{2} b e^\frac{1}{2} \right\| \\
& + \left\| (1 - e)^\frac{1}{2} b (1 - e)^\frac{1}{2} + e^\frac{1}{2} b e^\frac{1}{2} - b \right\| \\
\leq & \ (2d+4) \varepsilon + 5(d+1) \varepsilon + \varepsilon \\
= & \ (7d+10) \varepsilon \, .
\end{align*}
\item Since $A_{\leq R}$ is a multi-matrix algebra (see~\eqref{eq:decomposable-approximation-for-quotient}), we have $\dimnuc(A_{\leq R}) = 0$. On the other hand, by the permanence properties of nuclear dimension with regard to direct sums and hereditary subalgebras (see \cite[Proposition~2.3 and~2.5]{winter-zacharias}), we have
\[
\dimnuc(A_{>R}) = \max_{i \in I } \operatorname{dim}_\mathrm{nuc} \left( f_i^{\frac{1}{2}} C^*_r(\mathcal{H}_{\widehat{B}_i}) f_i^{\frac{1}{2}} \right) \leq \max_{i \in I } \operatorname{dim}_\mathrm{nuc} \left( C^*_r(\mathcal{H}_{\widehat{B}_i}) \right) \;
\]
and by Lemma~\ref{lem:groupoid-algebra-H}, the permanence property of nuclear dimension with regard to stabilization (see \cite[Corollary~2.8]{winter-zacharias}), and the subset permanence property of covering dimension, it follows that
\[
\dimnuc(A_{>R}) \leq \max_{i \in I } \operatorname{dim}_\mathrm{nuc} \left( C_0(S_{\widehat{B}_i^o}) \right) = \max_{i \in I } \operatorname{dim} \left( S_{\widehat{B}_i^o} \right) \leq \operatorname{dim} (Y) = d \;
\]
\end{enumerate}
With these we can tally up the dimensions:
\begin{align*}
& (\dimnuc(A_{\leq R}) + 1) 2(d+1) + ( \dimnuc(A_{>R}) + 1) 5(d+1) \\
\leq & \ 2(d+1) + (d + 1) \cdot 5(d+1) \\
= & \ (d+1)(5d+7) \; .
\end{align*}
Therefore the result follows from Lemma~\ref{Lemma:finite-dimnuc}.
\setcounter{Thm}{\value{clminproof}}
\end{proof}
The following corollary shows that the metrizability requirement in Theorem~\ref{thm-main} may be removed.
\begin{cor}\label{cor:thm-main-nonmetrizable}
Let $(Y, {\mathbb{R}^{}}, \calpha)$ be a topological flow. Assume $Y$ is a locally compact Hausdorff space with finite covering dimension. Then
\[
\operatorname{dim}_\mathrm{nuc}({C}_{0}(Y) \rtimes {\mathbb{R}} ) \leq (\dim(Y)+1)(5\dim(Y)+7) - 1 \; .
\]
\end{cor}
\begin{proof}
We reduce the situation to the metrizable case as follows. By Lemma \ref{lem:separable-dimnuc}, $C_0(Y)$ is the union of $\R$-invariant separable $C^*$-subalgebras $C_0(X)$ with $\dim(X) \leq \dimnuc (C_0(Y))$. Therefore if we let $I$ be the net of all $\R$-invariant separable $C^*$-subalgebras of $C_0(Y)$ with nuclear dimension no more than $\dimnuc(C_0(Y))$, ordered by inclusion, then since the spectrum of a commutative separable $C^*$-algebra is metrizable, we have
\begin{align*}
\dimnuc^{+1}(C_0(Y) \rtimes \R) \leq & \liminf_{C_0(X) \in I} \dimnuc^{+1}(C_0(X) \rtimes \R)
\end{align*}
and hence the statement follows immediately from Theorem~\ref{thm-main}.
\end{proof}
We are grateful to George Elliott for pointing out a connection to the main result of \cite{Hirshberg-Wu16}: applying the mapping torus construction, we can apply Theorem~\ref{thm-main} to obtain a nuclear dimension bound for crossed products associated to topological $\mathbb{Z}$-actions on finite dimensional space.
\begin{cor}\label{cor:mapping-torus}
Let $X$ be a locally compact Hausdorff space and $\calpha \in \operatorname{Homeo}(X)$. Then
\[
\operatorname{dim}_\mathrm{nuc}({C}_{0}(X) \rtimes_{\calpha} {\mathbb{Z}} ) \leq (\dim(X)+2)(5\dim(X)+12) - 1 \; .
\]
\end{cor}
\begin{proof}
Let $Y$ be the (topological) mapping torus of $\calpha$, that is, the quotient of $X \times \R$ by the diagonal action of $\Z$, where $\Z$ acts on $X$ by $\calpha$ and on $\R$ by translation (in the reverse direction). Since this $\Z$-action commutes with the translation action by $\R$ on the second factor, we obtain a topological flow $(Y, \R, \widehat{\beta})$. It follows that the embedding $X \times (0,1) \subset X \times \R$ induces an embedding $X \times (0,1) \subset Y$, which is invariant under the action of the subgroup $\Z < \R$, and the restriction of $\widehat{\beta}$ to $X \times (0,1) \subset Y$ and $\Z < \R$ agrees with $\calpha \times \operatorname{id}$. Moreover, $(Y \rtimes_{\widehat{\beta}} \R)_{X \times (0,1)}$, the reduction of the transformation groupoid $Y \rtimes_{\widehat{\beta}} \R$ to $X \times (0,1)$ (as a subset of the unit space), decomposes as a product of the transformation groupoid $X \rtimes_{\calpha} \Z$ and the pair groupoid $(0,1) \times (0,1)$. Hence $C^*_r\left((Y \rtimes_{\widehat{\beta}} \R)_{X \times (0,1)}\right)$, a hereditary subalgebra of $C^*_r(Y \rtimes_{\widehat{\beta}} \R)$ by Example~\ref{example:reduced-groupoid-algebra}, is isomorphic to the stabilization of $C^*_r(X \rtimes_{\calpha} \Z)$. It follows from the permanence properties of nuclear dimension with regard to stabilizations and hereditary subalgebras that
\begin{equation}\label{eq:mapping-torus-dimnuc}
\dimnuc \left(C_0(X) \rtimes_{\calpha} \Z\right) = \dimnuc \left(C^*_r\left((Y \rtimes_{\widehat{\beta}} \R)_{X \times (0,1)}\right)\right) \leq \dimnuc \left(C_0(Y) \rtimes_{\widehat{\beta}} \R\right) \; .
\end{equation}
Since the complement $Y \setminus (X \times (0,1))$ is homeomorphic to $X$, we see that
\begin{equation}\label{eq:mapping-torus-covdim}
\dim (Y) \leq \max \{ \dim(X \times (0,1)) , \dim (X) \} \leq \dim(X) + 1 \; .
\end{equation}
Our claim then follows by combining \eqref{eq:mapping-torus-dimnuc}, \eqref{eq:mapping-torus-covdim}, and Theorem~\ref{thm-main}.
\end{proof}
Corollary~\ref{cor:mapping-torus} essentially recovers \cite[Theorem~5.1]{Hirshberg-Wu16}, with a less sharp bound. Despite the similarity of the end results, the underlying technical machinery in this paper, namely the long thin covers on flow spaces, is different from the marker property techniques powering the result of \cite{Hirshberg-Wu16}. The former was developed in the study of the Farrell-Jones conjecture by Bartels, L\"{u}ck and Reich \cite{BarLRei081465306017991882} (and improved by Kasprowski and R\"{u}ping \cite{Kasprowski-Rueping}), while the latter was introduced in the context of topological dynamics by Gutman \cite{Gutman-marker} based on \cite{Lindenstrauss95} and later adapted by Szab\'{o} \cite{szabo}.
\section{Fiberwise groupoid coverings and surjective $*$-homomorphisms}\label{section:groupoid-coverings}
We take on another digression to discuss locally compact groupoids in general: we are going to define what we call \emph{fiberwise groupoid covering maps} and show that these maps induce quotient maps between maxiaml groupoid $C^*$-algebras. The motivating examples are constructed from orientable line foliations, which will be the topic of Section~\ref{section:foliation}.
\begin{defn}\label{def:fiberwise-groupoid-covering}
Let $G$ and $H$ be two topological groupoids. A map $\pi \colon G \to H$ is called a \emph{fiberwise groupoid covering (map)} if
\begin{enumerate}
\item \label{def:fiberwise-groupoid-covering-homomorphism} $\pi$ is a groupoid homomorphism (i.e., it intertwines the unit space, the maps $d$ and $r$, and the multiplication),
\item \label{def:fiberwise-groupoid-covering-unit-space} $\pi$ restricts to a homeomorphism between the unit spaces $G^0$ and $H^0$
\item \label{def:fiberwise-groupoid-covering-surjective} $\pi$ is surjective, and
\item \label{def:fiberwise-groupoid-covering-local-homeo} $\pi$ is a local homeomorphism.
\end{enumerate}
Suppose $G$ and $H$ are locally compact groupoids with Haar systems $\lambda_G$ and $\lambda_H$. Then a fiberwise groupoid covering $\pi \colon G \to H$ is said to \emph{intertwine the Haar systems $\lambda_G$ and $\lambda_H$} if for any $u \in G^0$ and any open subset $U \subset G$ for which $\pi|_U$ is a homeomorphism onto its image, the push-forward of the measure $\left.\lambda_{G}^u \middle| _{U \cap G^u} \right.$ under $\left. \pi \middle| _{U \cap G^u} \right.$ is equal to $\left.\lambda_{H}^{\pi(u)}\middle| _{\pi(U) \cap H^{\pi(u)}} \right.$.
\end{defn}
Some immediate algebraic consequences are listed below.
\begin{lem}\label{lem:fiberwise-groupoid-covering-algebraic}
Let $\pi \colon G \to H$ be a fiberwise groupoid covering map. Then the following are true:
\begin{enumerate}
\item \label{lem:fiberwise-groupoid-covering-algebraic-r-s} For any $y \in H$, there are unique $u,v \in G^0$ with $\pi(u) = r(y)$ and $\pi(v) = d(y)$. Moreoever, we have $r(\pi^{-1}(y)) = \left\{ u \right\}$ and $d(\pi^{-1}(y)) = \left\{ v \right\}$.
\item \label{lem:fiberwise-groupoid-covering-algebraic-group} For every $v \in H^0$, $\pi^{-1}(v)$ is a group.
\item \label{lem:fiberwise-groupoid-covering-algebraic-preimage} For any $x \in G$, writing $u = \pi(r(x))$ and $v = \pi(d(x))$, we have
\[
\pi^{-1}(\pi(x)) = \pi^{-1}(u) \cdot x = x \cdot \pi^{-1}(v)
\]
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}
\item The first claim follows from the fact that $\pi$ restricts to a bijection from $G^0$ to $H^0$. For the second claim, we notice that $\pi(r(\pi^{-1}(y))) = r(\pi(\pi^{-1}(y))) = r(\{y\}) = \{\pi(u)\}$, which implies $r(\pi^{-1}(y)) = \left\{ u \right\}$, again by the bijectivity of $\pi$ on $G^0$. Similarly, we have $d(\pi^{-1}(y)) = \left\{ v \right\}$.
\item This follows immediately from part~\eqref{lem:fiberwise-groupoid-covering-algebraic-r-s} since all the elements in $\pi^{-1}(v)$ have the same range and domain, both being equal to the unique preimage of $v$.
\item We first observe that $\pi\left( \pi^{-1}(u) \cdot x \right) = \pi\left( \pi^{-1}(u) \right) \pi(x) = u \pi(x) = \pi(x)$, which implies $\pi^{-1}(u) \cdot x \subset \pi^{-1}(\pi(x))$. Conversely, for any $y \in \pi^{-1}(\pi(x))$, we notice that $d(y) = d(x)$ and $r(y) = r(x)$ by part~\eqref{lem:fiberwise-groupoid-covering-algebraic-r-s}, and that $\pi(y x^{-1}) = \pi(y) \pi(x)^{-1} = \pi(x) \pi(x)^{-1} = \pi\left(x x^{-1}\right) = \pi(r(x)) = u$, which implies $y \in \pi^{-1}(u) \cdot x$. Hence $\pi^{-1}(\pi(x)) \subset \pi^{-1}(u) \cdot x$. Combining the two direction, we get $\pi^{-1}(\pi(x)) = \pi^{-1}(u) \cdot x$. Similarly, $\pi^{-1}(\pi(x)) = x \cdot \pi^{-1}(v)$.
\end{enumerate}
\end{proof}
Note that a fiberwise groupoid covering map $\pi \colon G \to H$ need not be a topological covering map from $G$ to $H$ (i.e., for any $y \in H$, there is an open neighborhood $V$ of $y$ in $H$ such that $\pi^{-1}(V)$ is a disjoint union of open subsets of $G$, each of which is homeomorphic to $V$ through $\pi$). An example is given by taking $G$ to be the transformation groupoid $[0,1] \rtimes \mathbb{Z}/2\mathbb{Z}$ associated to the trivial action, and $\pi$ to be the map that collapses every fiber to a single unit element except for the fiber at $0 \in [0,1]$. However, the following observation justifies our use of the term ``fiberwise groupoid covering''.
\begin{lem}\label{lem:fiberwise-groupoid-covering-topological-covering}
Let $\pi \colon G \to H$ be a fiberwise groupoid covering map. Then for every $u \in G^0$, the restriction $\left. \pi \middle| _{G^u} \right. \colon G^u \to H^{\pi(u)}$ (respectively, $\left. \pi \middle| _{G_u} \right. \colon G_u \to H_{\pi(u)}$) is a regular covering map that is implemented by the action of the group $\pi^{-1}(u)$ on $G^u$ by left multiplication (respectively, on $G_u$ by right multiplication).
\end{lem}
\begin{proof}
We only prove the lemma for $\left. \pi \middle| _{G^u} \right. \colon G^u \to H^{\pi(u)}$. The case for $\left. \pi \middle| _{G_u} \right. \colon G_u \to H_{\pi(u)}$ follows by noticing that $\pi$ also serves as a fiberwise groupoid covering from the opposite groupoid $G^{\operatorname{op}}$ to $H^{\operatorname{op}}$.
Given $y \in H^{\pi(u)}$, we are going to find an open neighborhood $V$ of $y$ in $H^{\pi(u)}$ and an open set $W$ in $G^u$ such that $\pi^{-1}(V)$ is the disjoint union of $z \cdot W$, where $w$ ranges over the group $\pi^{-1}(u)$, and $\pi$ maps each $z \cdot W$ homeomorphically onto $\pi(W)$. Since $\pi$ is surjective, we can choose a certain $x \in \pi^{-1}(y)$. Lemma~\ref{lem:fiberwise-groupoid-covering-algebraic} implies that $\pi^{-1}(y) \subset G^u$, $\pi^{-1}(v)$ is a group, and $\pi^{-1}(y) = \pi^{-1}(v) \cdot x$.
Since $\pi$ is a local homeomorphism, there is an open neighborhood $U$ of $x$ in $G$ such that $\left. \pi \middle| _{U} \right.$ is a homeomorphism onto its image. Let $W = U \cap G^u$ and $V = \pi(W)$, which is contained in $H^{\pi(u)}$. Note that $\pi^{-1}(V) = \pi^{-1}(v) \cdot W$ by Lemma~\ref{lem:fiberwise-groupoid-covering-algebraic}\eqref{lem:fiberwise-groupoid-covering-algebraic-preimage}. Also notice that for any $z \in \pi^{-1}(v)$, the restriction of $\pi$ to $z W$ is equal to the composition of $\left. \pi \middle| _{W} \right.$ and the left multiplication map by $z^{-1}$, and is thus a homeomorphism onto its image $V$. Lastly, we show $\left\{ z \cdot W \colon z \in \pi^{-1}(v) \right\}$ is a disjoint family of subsets. Suppose we could find $w \in (z \cdot W) \cap (z' \cap W)$ for different $z , z' \in \pi^{-1}(v)$, we would get two different elements $z^{-1}w$ and $(z')^{-1}w$ in $W$ having the same image under $\pi$, thus contradicting our assumption that $\left. \pi \middle| _{U} \right.$ is a homeomorphism onto its image.
In summary, we have shown that $\pi^{-1}(V)$ is a disjoint union of open subsets $z \cdot W$, for $z \in \pi^{-1}(v)$, each of which is homeomorphic to $V$ through $\left. \pi \middle| _{G^u} \right.$.
\end{proof}
\begin{nota}\label{nota:fiberwise-groupoid-covering-topological-C_c_pi}
Let $\pi \colon G \to H$ be a fiberwise groupoid covering map. Then
\begin{enumerate}
\item we write $C_c(G)_\pi$ for the set of complex-valued functions $f$ on $G$ for which there exists a Hausdorff open set $U \subset G$ so that $\left.\pi\middle|_U\right.$ is a homeomorphism onto its image, $f$ vanishes outside of $U$, and $f|_U$ is continuous and compactly supported;
\item we write $C_c(H)_\pi$ for the set of complex-valued functions $g$ on $H$ for which there exists a Hausdorff open set $U \subset G$ so that $\left.\pi\middle|_U\right.$ is a homeomorphism onto its image, $f$ vanishes outside of $\pi(U)$, and $f|_U$ is continuous and compactly supported.
\end{enumerate}
\end{nota}
\begin{lem}\label{lem:fiberwise-groupoid-covering-topological-C_c_pi}
Let $\pi \colon G \to H$ be a fiberwise groupoid covering map. Then $C_c(G)$ (respectively, $C_c(H)$), as given in Definition~\ref{nota:C_c_0}, is linearly spanned by $C_c(G)_\pi$ (respectively, $C_c(H)_\pi$).
\end{lem}
\begin{proof}
It is clear that $C_c(G)_\pi \subset C_c(G)_0 \subset C_c(G)$. Thus by Definition~\ref{nota:C_c_0}, one just needs to show each $f \in C_c(G)_0$ lies in the linear span of $C_c(G)_\pi$, but this is a standard argument using a partition of unity. The case for $H$ is similar.
\end{proof}
\begin{lem}\label{lem:fiberwise-groupoid-covering-topological-finite-preimage}
Let $\pi \colon G \to H$ be a fiberwise groupoid covering map. Then for any compact set $K \subset G$ and $y \in H$, the set $K \cap \pi^{-1}(y)$ is finite. Therefore, for any complex function $f$ on $G$ with compact support and $y \in H$, there are only finitely many $x \in \pi^{-1}(y)$ such that $f(x) \not= 0$.
\end{lem}
\begin{proof}
Let $u \in G^0$ be such that $\pi(u) = r(y)$. Then $K \cap G^u$, being a closed subset of the compact set $K$, is also compact. By Lemma~\ref{lem:fiberwise-groupoid-covering-topological-covering}, the restriction $\left. \pi \middle| _{G^u} \right. \colon G^u \to H^{\pi(u)}$ is a covering map, which entails that $\pi^{-1}(y)$ is a closed discrete subset of $G^u$, and thus $K \cap \pi^{-1}(y)$, being a compact discrete set, must be finite.
\end{proof}
\begin{lem}\label{lem:fiberwise-groupoid-covering-topological-pi-star}
Let $\pi \colon G \to H$ be a fiberwise groupoid covering map. Then the formula
\begin{equation}\label{eq:sum-preimage}
\pi_*(f)(y) = \sum_{x \in \pi^{-1}(y)} f(x) \quad \text{for~} f \in C_c(G) \text{~and~} y \in H
\end{equation}
defines a surjective linear map $\pi_* \colon C_c(G) \to C_c(H)$. Similarly, for any $u \in G^0$, the formula
\begin{equation}\label{eq:sum-preimage-u}
\pi_*^u(f)(y) = \sum_{x \in \pi^{-1}(y)} f(x) \quad \text{for~} f \in C_c(G^u) \text{~and~} y \in H^{\pi(u)}
\end{equation}
defines a surjective linear map $\pi_*^u \colon C_c(G^u) \to C_c\left(H^{\pi(u)}\right)$. Moreover, we have, for any $x \in G$ and $f \in C_c \left(G^{d(x)}\right)$,
\begin{equation}\label{eq:sum-preimage-equivariant}
\pi_*^{r(x)}(x \cdot f) = \pi(x) \cdot \pi_*^{d(x)}(f) \; ,
\end{equation}
where $x \cdot f (z) = f(x^{-1} z)$ and $\pi(x) \cdot \pi_*^{d(x)}(f) (y) = \pi_*^{d(x)}(f) \left( \left(\pi(x) \right)^{-1} y\right)$ for any $z \in G^{r(x)}$ and $y \in H^{\pi(r(x))}$.
\end{lem}
\begin{proof}
It follows from Lemma~\ref{lem:fiberwise-groupoid-covering-topological-finite-preimage} that for any $f \in C_c(G)_0$ and $y \in H$, the sum in \eqref{eq:sum-preimage} is always finite. The same is thus also true for all $f \in C_c(G)$ by linearity. Then the assignment $f \mapsto \pi_*(f)$ defines a linear map from $C_c(G)$ to the space of all functions on $H$. For any Hausdorff open subset $U \subset G$ where $\pi$ is restricted to a homeomorphism onto its image, we observe that if $f \in C_c(U)$, then $\pi_*(f)$ is supported in $\pi(U)$ and $\left. \pi_*(f)\middle|_{\pi(U)} \right. = f \circ \left(\pi\middle|_U\right)^{-1}$, and every $g \in C_c(\pi(U))$ can be realized this way. Hence $\pi_*$ maps $C_c(G)_\pi$ surjectively onto $C_c(H)_\pi$. It follows by linearity that $\pi_* \left(C_c(G)\right) = C_c(H)$.
The case for $\pi_*^u$ is entirely analogous.
To prove the last equation, we observe that for any $x \in G$ and $y \in H^{\pi(r(x))}$, if we write $\widetilde{y} $ for an arbitrary element in $\pi^{-1}(y)$, we have by Lemma~\ref{lem:fiberwise-groupoid-covering-algebraic} that
\[
\pi^{-1} \left( \left(\pi(x) \right)^{-1} y\right) = \pi^{-1} \left( \pi\left(x^{-1} \widetilde{y} \right) \right) = x^{-1} \widetilde{y} \cdot \pi^{-1} (d(y)) = x^{-1} \cdot \pi^{-1} \left( y \right) \; .
\]
Hence for any $f \in C_c \left(G^{d(x)}\right)$, we have
\begin{align*}
\pi(x) \cdot \pi_*^{d(x)}(f) (y) &= \pi_*^{d(x)}(f) \left( \left(\pi(x) \right)^{-1} y\right) \\
& = \sum_{z \in \pi^{-1} \left(\left(\pi(x) \right)^{-1} y\right)} f(z) \\
& = \sum_{z \in x^{-1} \cdot \pi^{-1} \left( y \right)} f(z) \\
& = \sum_{z' \in \pi^{-1} \left( y \right)} f(x^{-1} z') \\
& = \pi_*^{r(x)}(x \cdot f) (y)
\end{align*}
as desired.
\end{proof}
The following lemma is a special case of a general fact about regular covers. We include a proof for completeness.
\begin{lem}\label{lem:fiberwise-groupoid-covering-topological-pi-star-kernel}
Let $\pi \colon G \to H$ be a fiberwise groupoid covering map. Then for any $u \in G^0$ such that $G^u$ and $H^{\pi(u)}$ are locally compact and Hausdorff, the kernel of $\pi_*^u \colon C_c(G^u) \to C_c\left(H^{\pi(u)}\right)$ is linearly spanned by functions of the form $f - z \cdot f$, for $f \in C_c(G^u)$ and $z \in \pi^{-1}(u)$, where $z \cdot f$ denotes the function $x \mapsto f(z^{-1} \cdot x)$.
\end{lem}
\begin{proof}
We first observe that for any $f \in C_c(G^u)$ and $z \in \pi^{-1}(u)$, since $\pi_*^u(z \cdot f) = \pi_*^u(f)$, the difference $f - z \cdot f$ is indeed in $\ker \left( \pi_*^u \right)$. Thus it suffices to show that any $g \in \ker \left( \pi_*^u \right)$ can be written as a finite sum of such elements.
We first assume there is an open subset $U \subset G^u$ such that $\pi$ maps $U$ homeomorphically to its image and $g$ is supported in $\pi^{-1}(\pi(U))$, which, by Lemma~\ref{lem:fiberwise-groupoid-covering-topological-covering}, is equal to $\bigsqcup_{z \in \pi^{-1}(u)} z \cdot U$. Thus in this case, $g$ can be written as $\sum_{z \in \pi^{-1}(u)} g_z$, where each $f_z$ is supported in $z \cdot U$. Since $g$ is compactly supported, this is a finite sum, i.e., there are $z_1, \ldots, z_n \in \pi^{-1}(u)$ such that $g = \sum_{i = 1}^n g_{z_i}$. Observe that
\[
\pi_*^u \left( \sum_{i = 1}^n z_i^{-1} \cdot g_{z_i} \right) = \sum_{i = 1}^n \pi_*^u \left( z_i^{-1} \cdot g_{z_i} \right) = \sum_{i = 1}^n \pi_*^u \left( g_{z_i} \right) = \pi_*^u(g) = 0 \; .
\]
On the other hand, $\sum_{i = 1}^n z_i^{-1} \cdot g_{z_i}$ is supported inside $\bigcup_{i=1}^n z_i^{-1} \cdot z_i \cdot U$, which is just $U$. It follows that its image under $\pi_*^u$ is supported in $\pi(U)$, and
\[
\left. \pi_*^u \left( \sum_{i = 1}^n z_i^{-1} \cdot g_{z_i} \right) \middle|_{\pi(U)} \right. = \left( \sum_{i = 1}^n z_i^{-1} \cdot g_{z_i} \right) \circ \left( \pi \middle|_U \right)^{-1} \; .
\]
Combining these equations, we see that $\sum_{i = 1}^n z_i^{-1} \cdot g_{z_i} = 0$. Therefore we have
\[
g = g - 0 = \sum_{i = 1}^n \left( g_{z_i} - z_i^{-1} \cdot g_{z_i} \right)
\]
as desired.
For a general $g \in \ker \left( \pi_*^u \right)$, we cover the support of $\pi_*^u(g)$ by finitely many open sets $W_i$, for $i = 1, \ldots, n$, where each $W_i$ is the homeomorphic image of some open subset $U_i \subset G^u$ under $\pi$. Let $\{ f_i \colon i = 1, \ldots, n \}$ be a set of functions on $H^{\pi(U)}$ such that each $f_i$ is supported in $W_i$ and $\sum_{i=1}^{n} f_i$ is equal to $1$ on the support of $\pi_*^u(g)$. For $i = 1, \ldots, n$, writing $\widetilde{f}_i = f_i \circ \pi$, we observe that $\pi_*^u(\widetilde{f}_i g) = f_i \pi_*^u(g) = 0$ and $\widetilde{f}_i g$ is supported in $\pi^{-1}(W_i)$; hence by the previous paragraph $\widetilde{f}_i g$ is in the linear span of $\{ f - z \cdot f \in C_c(G^u), z \in \pi^{-1}(u) \}$. Since $\sum_{i=1}^{n} \widetilde{f}_i$ is equal to $1$ on the support of $g$, we have $g = \sum_{i=1}^{n} \widetilde{f}_i g$. It follows that $g$ is also in the linear span of $\{ f - z \cdot f \in C_c(G^u), z \in \pi^{-1}(u) \}$.
\end{proof}
\begin{lem}\label{lem:fiberwise-groupoid-covering-topological-intertwine}
Let $G$ and $H$ be locally compact groupoids with Haar systems $\lambda_G$ and $\lambda_H$. Let $\pi \colon G \to H$ be a fiberwise groupoid covering map. Then $\pi$ intertwines $\lambda_G$ and $\lambda_H$ if and only if for any $u \in G^0$ and $f \in C_c(G^u)$, we have
\[
\int_{G^u} f \, d \lambda_G^u = \int_{H^{\pi(u)}} \pi_*^u(f) \, d \lambda_H^{\pi(u)} \; .
\]
\end{lem}
\begin{proof}
By linearity, it suffices to show, for any $u \in G^0$ and any open subset $U \subset G$ for which $\pi|_U$ is a homeomorphism onto its image, that $\left.\lambda_{H}^{\pi(u)}\middle| _{\pi(U) \cap H^{\pi(u)}} \right.$ is the pushforward of $\left. \lambda_{G}^u \middle| _{U \cap G^u}\right.$ under $\left. \pi \middle| _{U \cap G^u} \right.$
if and only if for any $f \in C_c(U \cap G^u)$, we have
\[
\int_{G^{u}} f \, d \lambda_G^{u} = \int_{H^{\pi(u)}} \pi_*^u(f) \, d \lambda_H^{\pi(u)} \; .
\]
But this is true since the former statement is equivalent to that for any $f \in C_c(U \cap G^u)$, we have
\[
\int_{U \cap G^u} f \, d \lambda_G^{u} = \int_{\pi(U) \cap H^{\pi(u)}} f \circ \left( \pi\middle|_{U \cap G^u} \right)^{-1} \, d \lambda_H^{\pi(u)} \; ,
\]
while the left-hand side of this equation is equal to $\int_{G^{u}} f \, d \lambda_G^{u}$ and the right-hand side is equal to $\int_{H^{\pi(u)}} \pi_*^u(f) \, d \lambda_H^{\pi(u)}$ because $\pi_*^u(f)$ is supported in $\pi(U) \cap H^{\pi(u)}$ and $\left. \pi_*^u(f) \middle|_{\pi(U) \cap H^{\pi(u)}} \right. = f \circ \left( \pi\middle|_{U \cap G^u} \right)^{-1}$. This shows the two statements are equivalent, as desired.
\end{proof}
\begin{thm}\label{thm:fiberwise-groupoid-covering-locally-compact}
Let $\pi \colon G \to H$ be a fiberwise groupoid covering map. Then the following are true:
\begin{enumerate}
\item If $H$ is a locally compact groupoid with Haar system $\lambda_H$, then $G$ is also locally compact and can be equipped with a Haar system $\lambda_G$ such that $\pi$ intertwines $\lambda_G$ and $\lambda_H$.
\item If $G$ is a locally compact groupoid with Haar system $\lambda_G$, then $H$ is also locally compact and can be equipped with a Haar system $\lambda_H$ such that $\pi$ intertwines $\lambda_G$ and $\lambda_H$.
\end{enumerate}
\end{thm}
\begin{proof}
For each of the two statements, to prove a topological groupoid is actually a locally compact groupoid, we need to check the four conditions in Definition~\ref{def:locally-compact-groupoids}. In fact, the first three conditions are straightforward \emph{for both statements}:
\begin{itemize}
\item Condition~\eqref{def:locally-compact-groupoids:unit-space} is a direct consequence of Definition~\ref{def:fiberwise-groupoid-covering}\eqref{def:fiberwise-groupoid-covering-unit-space}.
\item Condition~\eqref{def:locally-compact-groupoids:locally-Hausdorff} can be easily verified using the fact that $\pi$ is a local homeomorphism.
\item Condition~\eqref{def:locally-compact-groupoids:sections} follows from Lemma~\ref{lem:fiberwise-groupoid-covering-topological-covering} and the fact in elementary topology that given a topological covering map, if either the domain or the codomain is locally compact (respectively, Hausdorff), then so is the other.
\end{itemize}
Therefore, all that is left is to construct Haar systems. For this we use the construction in Lemma~\ref{lem:fiberwise-groupoid-covering-topological-pi-star} and treat the two statements separately.
\begin{enumerate}
\item We first prove the statement where we assume $H$ is a locally compact groupoid with Haar system $\lambda_H$.
In this case, we define, for each $u \in G^0$, a linear map
\[
\Lambda_G^u \colon C_c(G^u) \to \mathbb{C}
\]
such that for any $f \in C_c(G^u)$ and $u \in G^0$, we have
\[
\Lambda_G^u (f) = \int_{H^{\pi(u)}} \pi_*^u(f) \, d \lambda_H^{\pi(u)} \;.
\]
Clearly $\Lambda_G^u$ maps non-negative function to non-negative functions, and thus, by the Riesz representation theorem, determines a positive regular Borel measure $\lambda_G^u$ on $G^u$.
We claim that $\left\{ \lambda_G^u \right\}_{u \in G^0}$ is a Haar system we look for. To this end, we verify conditions~\eqref{def:locally-compact-groupoids:Haar-system:full-support}-\eqref{def:locally-compact-groupoids:Haar-system:invariance} in Definition~\ref{def:locally-compact-groupoids}:
\begin{itemize}
\item Condition~\eqref{def:locally-compact-groupoids:Haar-system:full-support} follows from the observation that any nonzero non-negative function $f \in C_c(G^u)$ is taken by $\pi_*$ to a nonzero non-negative function in $C_c\left(H^{\pi(u)}\right)$, and thus $\Lambda_G^u (f) > 0$ since $\lambda_H^{\pi(u)}$ has full support.
\item To prove condition~\eqref{def:locally-compact-groupoids:Haar-system:continuous}, we define a linear map
\[
\Lambda_G \colon C_c(G) \to C_c(G^0)
\]
such that for any $f \in C_c(G)$ and $u \in G^0$, we have
\[
\Lambda_G (f) (u) = \int_{H^{\pi(u)}} \pi_*(f) \, d \lambda_H^{\pi(u)} \; .
\]
Here $\Lambda_G (f)$ is indeed in $C_c(G^0)$ thanks to Definition~\ref{def:locally-compact-groupoids}\eqref{def:locally-compact-groupoids:Haar-system:continuous} and Definition~\ref{def:fiberwise-groupoid-covering}\eqref{def:fiberwise-groupoid-covering-unit-space}. Comparing the constructions, we see that for any $f \in C_c(G)$ and $u \in G^0$, we have $\int_{G^u} f \, d \lambda_G^u = \Lambda_G^u \left(\left. f \middle|_{G^u} \right.\right) = \Lambda_G (f) (u)$. Hence the function $u \mapsto \int_{G^u} f \, d \lambda_G^u$ is in $C_c(G^0)$.
\item To prove condition~\eqref{def:locally-compact-groupoids:Haar-system:invariance}, we apply Equation~\eqref{eq:sum-preimage-equivariant} to see that for any $x \in G$ and $f \in C_c(G)$, we have
\[
\pi_*^{d(x)} \left(x^{-1} \cdot f\middle|_{G^{r(x)}} \right) = \pi(x^{-1}) \cdot \pi_*^{r(x)} \left( f \middle|_{G^{r(x)}} \right) \; ,
\]
which, together with condition~\eqref{def:locally-compact-groupoids:Haar-system:invariance} for $\{\lambda_H^u\}_{u \in H^0}$, implies
\begin{align*}
& \ \int_{H^{\pi(d(x))}} \pi_*^{d(x)} \left(x^{-1} \cdot f\middle|_{G^{r(x)}} \right) \, d \lambda_H^{\pi(d(x))} \\
& = \int_{H^{\pi(d(x))}} \pi(x^{-1}) \cdot \pi_*^{r(x)} \left( f \middle|_{G^{r(x)}} \right) \, d \lambda_H^{\pi(d(x))} \\
& = \int_{H^{\pi(r(x))}} \pi_*^{r(x)} \left( f \middle|_{G^{r(x)}} \right) \, d \lambda_H^{\pi(r(x))} \;.
\end{align*}
Unwrapping the definitions, we see that
\[
\int_{G^{d(x)}} f(xz) \, d \lambda_G^{d(x)} (z) = \int_{G^{r(x)}} f(y) \, d \lambda_G^{r(x)} (y) \; .
\]
\end{itemize}
Therefore we have shown that $G$ is a locally compact groupoid with a Haar system $\lambda_G$. By Lemma~\ref{lem:fiberwise-groupoid-covering-topological-intertwine}, it is clear from our construction that $\pi$ intertwines $\lambda_G$ and $\lambda_H$.
\item Then we turn to the second statement where we assume $G$ is a locally compact groupoid with Haar system $\lambda_G$. For each $u \in G^0$, consider the linear functional
\[
\Lambda_G^u \colon C_c(G^u) \to \mathbb{C} \, , \quad f \mapsto \int_{G^u} f \, d \lambda_G^u \; .
\]
By condition~\eqref{def:locally-compact-groupoids:Haar-system:invariance}, for any $z \in \pi^{-1}(\pi(u))$, since $r(z) = d(z) = u$, we have
\[
\Lambda_G^u(z \cdot f) = \int_{G^u} z \cdot f \, d \lambda_G^u = \int_{G^u} f \, d \lambda_G^u = \Lambda_G^u(f) \; .
\]
Applying Lemma~\ref{lem:fiberwise-groupoid-covering-topological-pi-star-kernel}, we have
\[
\ker \Lambda_G^u \supset \operatorname{span} \left\{ f - z \cdot f \colon f \in C_c(G^u), z \in \pi^{-1}(\pi(u)) \right\} = \ker \pi_*^u \; .
\]
Since $\pi_*^u \colon C_c(G^u) \to C_c\left( H^{\pi(u)} \right)$ is a surjection, we see that $\Lambda_G^u$ factors through a linear functional on $C_c(H^{\pi(u)})$, which we may denote by $\Lambda_H^{\pi(u)}$ without any ambiguity because $\pi$ maps $G^0$ bijectively onto $H^0$.
The positivity of $\Lambda_G^u$ implies that of $\Lambda_H^{\pi(u)}$ because by a partition-of-unity argument, any non-negative function in $C_c\left( H^{\pi(u)} \right)$ has a non-negative preimage in $C_c(G^u)$. Hence by the Riesz representation theorem, $\Lambda_H^{\pi(u)}$ determines a positive regular Borel measure $\lambda_H^{\pi(u)}$ on $H^{\pi(u)}$. It satisfies
\[
\int_{G^u} f \, d \lambda_G^u = \int_{H^{\pi(u)}} \pi_*^u(f) \, d \lambda_H^{\pi(u)}
\]
for any $f \in C_c(G^u)$.
We claim that $\{\lambda_H^u\}_{u \in H^0}$ is a Haar system we look for. To this end, we verify conditions~\eqref{def:locally-compact-groupoids:Haar-system:full-support}-\eqref{def:locally-compact-groupoids:Haar-system:invariance} in Definition~\ref{def:locally-compact-groupoids}:
\begin{itemize}
\item Condition~\eqref{def:locally-compact-groupoids:Haar-system:full-support} follows from the observation that by a partition-of-unity argument, any nonzero non-negative function $f \in C_c\left( H^{\pi(u)} \right)$ has a nonzero non-negative preimage in $C_c(G^u)$, and thus $\Lambda_H^{\pi(u)} (f) > 0$ since $\lambda_G^u$ has full support.
\item Condition~\eqref{def:locally-compact-groupoids:Haar-system:continuous} for $\lambda_H^{\pi(u)}$ follows directly from that for $\lambda_G^u$.
\item Condition~\eqref{def:locally-compact-groupoids:Haar-system:invariance} for $\lambda_H^{\pi(u)}$ also follows from that for $\lambda_G^u$. Indeed, for any $x \in H$ and $f \in C_c(H)$, choosing $\widetilde{x} \in \pi^{-1}$ and $\widetilde{f} \in \left(\pi_*\right)^{-1} (f)$ by surjectivity, we apply Equation~\eqref{eq:sum-preimage-equivariant} to get
\[
\pi_*^{d(\widetilde{x})} \left(\widetilde{x}^{-1} \cdot \widetilde{f} \middle|_{G^{r(\widetilde{x})}} \right) = x^{-1} \cdot \pi_*^{r(\widetilde{x})} \left( \widetilde{f} \middle|_{G^{r(\widetilde{x})}} \right) = x^{-1} \cdot \left. {f} \middle|_{H^{r({x})}} \right. \; ,
\]
which implies
\begin{align*}
& \ \int_{H^{d(x)}} f(xz) \, d \lambda_H^{d(x)} (z) \\
& = \int_{G^{d(\widetilde{x})}} \widetilde{f}(\widetilde{x}y) \, d \lambda_G^{d(\widetilde{x})} (y) \\
& = \int_{G^{r(\widetilde{x})}} \widetilde{f}(y') \, d \lambda_G^{r(\widetilde{x})} (y') \\
& = \int_{H^{r(x)}} f(z') \, d \lambda_H^{r(x)} (z') \; ,
\end{align*}
as desired.
\end{itemize}
Therefore we have shown that $H$ is a locally compact groupoid with a Haar system $\lambda_H$. By Lemma~\ref{lem:fiberwise-groupoid-covering-topological-intertwine}, it is clear from our construction that $\pi$ intertwines $\lambda_G$ and $\lambda_H$.
\end{enumerate}
\end{proof}
\begin{thm}\label{thm:fiberwise-groupoid-covering-quotient}
Let $G$ and $H$ be locally compact groupoids with Haar systems $\lambda_G$ and $\lambda_H$. Let $\pi \colon G \to H$ be a fiberwise groupoid covering map that intertwines $\lambda_G$ and $\lambda_H$. Then the surjection $\pi_* \colon C_c(G) \to C_c(H)$ given in Lemma~\ref{lem:fiberwise-groupoid-covering-topological-pi-star} is a $*$-homomorphism between the convolution $*$-algebras. Therefore, the $C^*$-algebra $C^*(H)$ is a quotient of $C^*(G)$.
\end{thm}
\begin{proof}
Lemma~\ref{lem:fiberwise-groupoid-covering-topological-intertwine} tells us that for any $f \in C_c(G)$, we have
\begin{equation}\label{eq:push-forward-integrals}
\int_{G^u} f \, d\lambda^u = \int_{ H^{\pi(u)}} \pi_*(f) \, d\lambda^{\pi(u)} \; ,
\end{equation}
which then entails, by Equation~\eqref{eq:groupoid-multiplication-r},
\[
\pi_*( f \ast g) = \pi_*( f) \ast \pi_*(g) \;
\]
for any $g \in C_c(G)$. It is also straightforward to check
\[
\pi_*(f^*) = (\pi_*(f))^*
\]
by the defining Equation~\eqref{eq:groupoid-star-operation}. This shows $\pi_* \colon C_c(G) \to C_c(H)$ is a $*$-homomorphism.
The last statement about $C^*$-algebras follows from the surjectivity of $\pi_*$ by a standard density argument.
\end{proof}
\begin{eg}
The quotient homomorphism from $\mathbb{R}$ to $\mathbb{R} / \mathbb{Z}$, as locally compact groups, is a fiberwise groupoid covering. It thus induces a surjection from $C^*(\mathbb{R})$ to $C^*(\mathbb{R} / \mathbb{Z})$. Using Pontryagin duality, this corresponds to the restriction $*$-homomorphism $C_0(\widehat{\mathbb{R}})$ to $C_0(\widehat{\mathbb{R} / \mathbb{Z}})$ induced by the inclusion $\widehat{\mathbb{R} / \mathbb{Z}} \cong \mathbb{Z} \hookrightarrow \mathbb{R} \cong \widehat{\mathbb{R}}$.
\end{eg}
A more interesting class of examples is given by orientable line foliations. We will discuss this in the next section.
\section{Orientable line foliations} \label{section:foliation}
In this section, we show that if $\mathcal{F}$ is a one-dimensional orientable foliation of a locally compact metrizable space with finite covering dimension, then the associated foliation $C^*$-algebra has finite nuclear dimension. We refer the reader to \cite{moore-schochet} for a discussion of the groupoid and $C^*$-algebra associated to a foliation (though in our case we do not need to assume that $Y$ is a manifold), and to \cite{paterson} for a discussion of the $C^*$-algebra associated to a locally compact groupoid, which we summarized in Section~\ref{section:prelim}.
\begin{defn}[\cite{moore-schochet}]\label{def:line-foliation}
Recall that a \emph{line foliation chart} or a \emph{one-dimensional foliation chart} on a locally compact metrizable space $Y$ is a triple $(U, T, \varphi)$, where $U$ is an open subset of $Y$, $T$ is some locally compact metrizable space and $\varphi \colon U \to \R \times T$. The preimages under $\varphi$ of sets of the form $\R \times \{x\}$ are called \emph{plaques}. Two line foliation charts $(U, T, \varphi)$ and $(U', T', \varphi')$ are coherent if for any plaque $P$ in $(U, T, \varphi)$ and $P'$ in $(U', T', \varphi')$, we have $P \cap P'$ is open in both $P$ and $P'$. A \emph{line foliation} or a \emph{one-dimensional foliation} on $Y$ is a foliated atlas, i.e., a collection of coherent foliation charts that cover $Y$, where two foliated atlases are consider to produce the same foliation if there union is also a foliated atlas. The relation of being in the same plaque in a chart generates an equivalence relation, whose equivalence classes are called the \emph{leaves} of the foliation. For each $y \in Y$, if $(U, T, \varphi)$ is a chart such that $y \in U$, then $T$, together with the base point given by the second coordinate of $\varphi(y)$, is called a \emph{transversal} of $\mathcal{F}$ at $y$. Although this depends on the choice of the chart, since any two charts $(U, T, \varphi)$ and $(U', T', \varphi')$ that cover $y$ agree on their intersection, there is thus a canonical homeomorphism between neighborhoods of the base points in $T$ and $T'$.
\end{defn}
\begin{defn}[\cite{moore-schochet}]\label{def:orientable-line-foliation}
A line foliation is \emph{orientable} if it has a foliated atlas such that for any two line foliation charts $(U, T, \varphi)$ and $(U', T', \varphi')$ in this atlas, the transition map
\[
\left. \varphi \middle|_{U\cap U'} \right. \circ \left( \varphi' \middle|_{U\cap U'} \right)^{-1} \colon \varphi'(U\cap U') \to \R \times T
\]
is increasing in the first coordinate in the sense that for any $(s, x)$ and $(t, x)$ in $\varphi'(U\cap U')$, whenever $s \leq t$, we have the first coordinate of $\left. \varphi \middle|_{U\cap U'} \right. \circ \left( \varphi' \middle|_{U\cap U'} \right)^{-1} (s, x)$ is less than or equal to that of $\left. \varphi \middle|_{U\cap U'} \right. \circ \left( \varphi' \middle|_{U\cap U'} \right)^{-1} (t, x)$.
\end{defn}
It is clear that a flow without fixed points gives rise to an orientable line foliation, where the collection of all tubes provides an atlas of local charts, by Lemma~\ref{lem:tubes-basics}. It is a remarkable theorem that the converse is also true.
\begin{thm}[{\cite[Theorem~27A]{Whitney}}]\label{thm:Whitney}
If $\mathcal{F}$ is an orientable line foliation of a locally compact metrizable space $Y$, then there exists a flow $\calpha$ on $Y$ with no fixed points such that the leaves of $\mathcal{F}$ are exactly the orbits of $\calpha$.
\end{thm}
\begin{cnstrct}[\cite{moore-schochet}]\label{construction:holonomy-groupoid}
Let us also recall the construction of the holonomy groupoid of a line foliation, which is in fact defined for general foliations and plays an important role in the index theory for the longitudinal elliptic operators (\cite{Connes-Skandalis}). Suppose $\mathcal{F}$ is a line foliation of a locally compact Hausdorff space $Y$. We first observe that for any line foliation chart $(U, T, \varphi)$ and any $y, y' \in U$, if $y$ and $y'$ are on the same plaque, then $T$ can be taken to be a transversal at both $y$ and $y'$, with the same base point. It follows that in this situation, for any transversals $T'$ at $y$ and $T''$ at $y'$, there is a canonical homeomorphism between neighborhoods of the base points of $T'$ and $T''$. For any path $\gamma \colon [0,1] \to Y$ whose image is contained in one of the leaves of the foliation and any transversals $T$ at $\gamma(0)$ and $T'$ at $\gamma(1)$, if we cover $\gamma$ by a sequence of foliation charts $\{(U_i, X_i, \varphi_i)\}_{i \in \{1, \ldots, n\}}$ such that $U_i \cap U_{i+1} \not= \varnothing$ for each $i \in \{0, \ldots, n-1 \}$ and choose $t_0, \ldots, t_{n} \in [0,1]$ such that $t_0 = 0$, $t_{n} = 1$, $t_0 < \ldots < t_{n}$, $\gamma(t_i) \in U_i$ for each $i \in \{1, \ldots, n \}$ and $\gamma(t_i) \in U_{i+1}$ for each $i \in \{0, \ldots, n - 1\}$, then we may define a homeomorphism $H_\gamma$ between neighborhoods of the base points of $T$ and $T'$ by composing the homeomorphisms between neighborhoods of the base points of some chosen transversals at $x_i$ and $x_{i+1}$. Although this homeomorphism depends on the choice of the cover $\{(U_i, X_i, \varphi_i)\}_{i \in \{1, \ldots, n\}}$, the points $t_0, \ldots, t_{n}$ and the transversals thereat, the \emph{germ} of this homeomorphism, denoted by $[H_\gamma]$, is independent of all these choices, in the sense that for two different choices of the above items, the resulting homeomorphisms will coincide on neighborhoods of the base points. We define the \emph{holonomy class} $[\gamma]_{\hol}$ of $\gamma$ to be the set of all paths $\gamma' \colon [0,1] \to Y$ such that
\begin{enumerate}
\item the image of $\gamma'$ is contained in one of the leaves of $\mathcal{F}$;
\item $\gamma(0) = \gamma'(0)$ and $\gamma(1) = \gamma'(1)$;
\item $[H_\gamma] = [H_{\gamma'}]$.
\end{enumerate}
It can be checked that the equivalence relation of being in the same holonomy class is preserved under concatenation and reversal of paths, and is weaker than homotopy. We define the \emph{holonomy groupoid} $G_{\mathcal{F}}$ to be the set of all holonomy classes of paths along the leaves of $\mathcal{F}$, where the unit space consists of (classes of) the constant paths and is identified with $Y$, taking inverse amounts to reversing (class of) a path, and the range and source of $[\gamma]_{\hol}$ are $\gamma(0)$ and $\gamma(1)$, respectively. This groupoid is topologized as follows: for any $[\gamma] \in G_{\mathcal{F}}$ where $\gamma \colon [0,1] \to Y$ is a path whose image is contained in one of the leaves of the foliation, we choose a sequence of foliation charts $\{(U_i, X_i, \varphi_i)\}_{i \in \{1, \ldots, n\}}$ as above, and define a set $V(\gamma, \{(U_i, X_i, \varphi_i)\}_{i \in \{1, \ldots, n\}})$ consisting of the holonomy classes of all paths $\gamma' \colon [0,1] \to Y$ whose image is contained in one of the leaves of the foliation and also in the union $\displaystyle \bigcup_{i=1}^{n} U_i$. We let these sets generate the topology on $G_{\mathcal{F}}$ as a subbase (in fact, it suffices to pick one representative for each holonomy class to carry out the construction). This topology is locally Hausdorff and locally compact. When restricted to the unit space $G_{\mathcal{F}}^0$, it coincides with the topology on $Y$, and when restricted to $G_{\mathcal{F}}^u$ for any $u \in G_{\mathcal{F}}^0$, it makes $G_{\mathcal{F}}^u$ into a topological one-dimensional manifold.
In the prominent case where $Y$ is a smooth manifold and $\mathcal{F}$ is smooth (i.e., given by an atlas whose the transition functions are smooth), the holonomy groupoid $G_{\mathcal{F}}$ becomes a Lie groupoid. Its Haar systems thus are in one-to-one correspondence with $1$-densities on the dual of its Lie algebroid.
\end{cnstrct}
\begin{prop}\label{prop:foliation-covering}
Let $Y$ be a locally compact metrizable space and let $\calpha$ be a topological flow on $Y$ without fixed points. Let $\mathcal{F}$ be the line foliation induced from $\calpha$. Then there is a fiberwise groupoid covering map $\pi$ from the transformation groupoid $Y \rtimes \R$ to $G_{\mathcal{F}}$.
\end{prop}
\begin{proof}
Given any $y \in Y$ and any $t \in \R$, we define a curve $\gamma_{y,t}:[0,1] \to Y$ by $\gamma_{y,t}(s) = \calpha_{-ts}(y)$. Noting that the leaves of the foliation $\mathcal{F}$ are all either copies of $\R$ or of the circle $S^1$, it is evident that if $y$ and $y'$ are two points on the same leaf, then any curve from $y$ to $y'$ along the leaf is homotopic inside the leaf via a homotopy which fixes end-points to a curve of the form $\gamma_{y,t}$ for some $t$ such that $\calpha_{-t}(y) = y'$. Thus, the map $\pi$ from $Y \rtimes \R$ to $G_{\mathcal{F}}$ given by
\[
\pi(y,t) = [\gamma_{y,t}]_{\hol}
\]
is a surjective map. It is straightforward to see that $\pi$ is also a groupoid homomorphism, and when restricted to the unit spaces, $\pi$ coincides with the identity map on $Y$.
It remains to show that $\pi$ is a local homeomorphism. Fix $(y,t) \in Y \rtimes \R$. Then there exist tubes $B_1, \cdots,B_n \subseteq Y$ such that $y \in S_{B_1^o}$, $\calpha_{-t}(y) \in S_{B_n^o}$ and for any $z \in S_{B_1^0}$, we have $\calpha_{-st}(z) \in \bigcup_{k=1}^n B_k^o$ for all $s \in [0,1]$, and $\calpha_{-t}(z) \in B_n^o$. While there is no reason to expect that $\calpha_t(z)$ is in the central slice of $B_n$, there is a unique $s(z) \in \left( - \frac{l_{B_n}}{2} , \frac{l_{B_n}}{2} \right)$ for which $\calpha_{-t - s(z)}(z) \in S_{B_n^o}$, and we note that the map $z \mapsto r(z)$ is continuous. For any $w \in B_1^o$, pick $r(w) \in \left( - \frac{l_{B_1}}{2} , \frac{l_{B_1}}{2} \right)$ such that $\calpha_{-r(w)}(w) \in S_{B_1^o}$, and note that $w \mapsto r(w)$ is continuous. Set $T(w) = t+r(w)+s(\calpha_{-s(w)}(w))$, so that $\calpha_{-T(w)}(w) \in S_{B_n^o}$. Since $T$ is continuous and $T(y) = t$, the set
\[
W(y,t, \{B_1, \cdots,B_n\}) = \left\{(w, T(w) + q) \in Y \rtimes \R \middlebar w \in B_1^o , ~ q \in \left( - \frac{l_{B_n}}{2} , \frac{l_{B_n}}{2} \right) \right\}
\]
is an open neighborhood of $(y,t)$ in $Y \rtimes \R$. Since tubes can be made arbitrarily small, the collection of all such $W(y,t, \{B_1, \cdots,B_n\})$ will form a local base of the topology at $(y,t)$. On the other hand, $\pi$ is injective on each $W(y,t, \{B_1, \cdots,B_n\})$ because if the paths $\gamma_{w, T(w) + q}$ and $\gamma_{w', T(w') + q'}$ are holonomy equivalent for $w, w' \in B_1^o$ and $ q , q' \in \left( - \frac{l_{B_n}}{2} , \frac{l_{B_n}}{2} \right)$, then the endpoint condition yields $w = w'$ and $\calpha_{-(T(w) + q)}(w) = \calpha_{-(T(w') + q')}(w')$, which implies $\calpha_{-q}\left(\calpha_{-T(w)}(w)\right) = \calpha_{-q'}\left(\calpha_{-T(w)}(w)\right)$ and thus $q = q'$. Moreover, it follows from the definition of the subsets $V(\gamma_{y,t}, \{(U_i, X_i, \varphi_i)\}_{i \in \{1, \ldots, m\}})$ in $G_{\mathcal{F}}$ earlier in this section, we can see that if $\bigcup_{k=1}^n B_k^o \subset \bigcup_{l=1}^m U_l$ then we have $\pi\left( W(y,t, \{B_1, \cdots,B_n\}) \right) \subset V(\gamma_{y,t}, \{(U_i, X_i, \varphi_i)\}_{i \in \{1, \ldots, m\}})$ and vice versa. Hence the collection of all possible $\pi\left( W(y,t, \{B_1, \cdots,B_n\}) \right)$ forms a local basis of $G_{\mathcal{F}}$ at $\pi(y,t)$. This shows that $\pi$ is a local homeomorphism.
\end{proof}
\begin{cor}\label{cor:foliation-covering}
Let $Y$, $\calpha$, $\mathcal{F}$ and $\pi$ be as in Proposition~\ref{prop:foliation-covering}. Let $\mu$ be the Haar system on $Y \rtimes \R$ determined by the Lebesgue measure on $\R$ as in Example~\ref{example:groupoids}. Then there is a Haar system $\lambda$ on $G_{\mathcal{F}}$ such that $\pi$ intertwines $\mu$ and $\lambda$. In this case, $C^*(G_{\mathcal{F}}, \lambda)$ is a quotient of $C_0(Y) \rtimes \R$.
\end{cor}
\begin{proof}
This follows from Theorem~\ref{thm:fiberwise-groupoid-covering-locally-compact} and~\ref{thm:fiberwise-groupoid-covering-quotient}.
\end{proof}
Intuitively speaking, the Haar system $\lambda$ is obtained by identifying each leave, i.e., each orbit, with the quotient of $\R$ by the stabilizer group of the orbit, and then taking the Lebesgue measure on this quotient, which is either $\R$ or a circle.
We return to the discussion of nuclear dimension. By \cite{MRW87}, two different Haar systems on a locally compact groupoid will yield groupoid $C^*$-algebras that are Morita equivalent. Since the nuclear dimension is invariant under Morita equivalent (\cite[Corollary~2.8]{winter-zacharias}), there is no ambiguity in talking about the nuclear dimension of the $C^*$-algebra associated to a locally compact groupoid.
\begin{Cor}\label{cor:foliation-algebra-dimnuc}
If $Y$ is a locally compact metrizable space with covering dimension $d$, and $\mathcal{F}$ is an orientable one-dimensional foliation of $Y$, then the nuclear dimension of the $C^*$-algebra associated to the holonomy groupoid is at most $5d^2+12d+6$.
\end{Cor}
\begin{proof}
By Theorem~\ref{thm:Whitney}, $\mathcal{F}$ arises from a flow $\calpha$ on $Y$. Thus for the holonomy groupoid $G_{\mathcal{F}}$, we may use, without loss of generality, the Haar system $\lambda_{G_{\mathcal{F}}}$ produced in Proposition~\ref{prop:foliation-covering}. Corollary~\ref{cor:foliation-covering} guarantees $C^*(G_{\mathcal{F}}, \lambda_{G_{\mathcal{F}}})$ is a quotient of $C_0(Y) \rtimes \R$. By \cite[Proposition 2.7]{winter-zacharias}, we have $\dimnuc(C_0(Y)\rtimes \R ) \geq \dimnuc(C^*(G_{\mathcal{F}}, \lambda_{G_{\mathcal{F}}}))$. The result then follows Theorem~\ref{thm-main}.
\end{proof}
\bibliographystyle{alpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,048 |
\section{\label{}}
\section{Introduction}
The time evolution of the $B^0_q$-$\Bqb$ system is described by the
Schr\"{o}dinger equation
\begin{equation}
i\frac{\partial}{\partial
t}\left(\begin{array}{c}{|B^0_q}(t)\rangle\\{|\Bqb}(t)\rangle\end{array}\right)=
\left(\mbox{\boldmath $\rm M$}^q-\frac{i}{2}\mbox{\boldmath $\rm
\Gamma$}^q\right)
\left(\begin{array}{c}{|B^0_q}(t)\rangle\\{|\Bqb}(t)\rangle\end{array}\right),
\end{equation}
where the \mbox{\boldmath $\rm M$}$^q$ and \mbox{\boldmath $\rm \Gamma$}$^q$
matrices are Hermitian, and $CPT$ invariance requires
$M^q_{11}=M^q_{22}$ and $\Gamma^q_{11}=\Gamma^q_{22}$.
The off-diagonal elements, $M^q_{12}$ and $\Gamma^q_{12}$, of these matrices describe the off-shell (dispersive)
and on-shell (absorptive) contributions
to $B^0_q$-$\Bqb$ mixing, respectively.
The mixing can be described by three physical quantities: $|M^q_{12}|$, $|\Gamma^q_{12}|$ and the relative phase $\phi^q_{12} = \arg\left(-\frac{M^q_{12}}{\Gamma^q_{12}}\right)$. They are related to the experimental observables \citep{Lenz:2012mb} as: $\Delta M_q \simeq 2|M^q_{12}|$, $\Delta \Gamma_q \simeq 2|\Gamma^q_{12}|\cos\phi^q_{12}$, and $a^q_{\rm sl} \simeq \frac{\Delta \Gamma_q}{\Delta m_q}\tan \phi^q_{12}$, where $\Delta M_q$ is the mass difference between the heavy and the light mass eigenstates, $-\Delta \Gamma_q$ the width difference, and $a^q_{\rm sl}$ the semileptonic (or flavor specific) $CP$ asymmetry. The phase difference, denoted as $\phi_q$, between $\arg(M_{12}^q)$ and the decay phase in $b\to c\bar{c}s$ transitions is also an experimental observable. In the $B_s$ case, $\phi_s = -2\arg[V_{ts}V_{tb}^*/V_{cs}V_{cb}^*]$, is small and accurately predicted as $-0.0363^{+0.0016}_{-0.0015}$ rad~\citep{Charles:2011va}. New Physics \citep{Bobeth:2011st} in mixing could add new phases to $M_{12}^q$ or/and $\Gamma_{12}^q$ and modify $\phi^q_{12}$ or/and $\phi_q$ from their Standard Model (SM) predictions.
\section{Formalism}
The time dependent decay rates for initial $B^0_q$ or $\Bqb$ decays to a $CP$ eigenstate ($f_{\rm CP}$) are:
\begin{eqnarray}\label{Eq-t}
\Gamma(B_q^0(t)\to f_{\rm CP})= {\cal N} e^{-\G_q t}\left\{\frac{1+|\lf|^2}{2}\ch + \frac{1-|\lf|^2}{2}\cs
-\RE(\lf)\sh - \IM(\lf)\sn\right\},\nonumber\\
\Gamma(\overline{B}_q^0(t)\to f_{\rm CP})= {\cal N} e^{-\G_q t} \left\{\frac{1+|\lf|^2}{2}\ch - \frac{1-|\lf|^2}{2}\cs
-\RE(\lf)\sh + \IM(\lf)\sn\right\},
\end{eqnarray}
where $\lambda = (q/p)(\bar{A}_f/A_f)$, $\G_q$ is average width of two mass eigenstates, $A_f$ ($\bar{A}_f$) is the amplitude of $B^0_q$ ($\Bqb$) decay, and $|p/q|=1$ is used assuming no $CP$ violation in the mixing.
The $CP$ asymmetry is\begin{equation}
A_{f_{CP}}(t)\equiv \frac{\Gamma(\overline{B}_q^0(t)\to f_{CP})-\Gamma(B_q^0(t)\to f_{CP})}{\Gamma(\overline{B}_q^0(t)\to f_{CP})+\Gamma(B_q^0(t)\to f_{CP})}
=\frac{\displaystyle {A^{\rm dir}}\cs + {A^{\rm mix}} \sn}{\displaystyle \ch + {A^{\Delta \G}}\sh}.
\end{equation}
There are three $CP$ variables but only two are independent: direct $CP$ asymmetry $A^{\rm dir}=\frac{\displaystyle |\lf|^2-1}{\displaystyle |\lf|^2+1}$,
mixing induced $CP$ asymmetry $A^{\rm mix}=\frac{\displaystyle 2\IM \lf}{\displaystyle |\lf|^2+1}$ and
$A^{\Delta \G}=-\frac{\displaystyle 2\RE \lf}{\displaystyle |\lf|^2+1}$ satisfying $(A^{\rm dir})^2+(A^{\rm mix})^2+(A^{\Delta \G})^2=1$.
\section{Resonant components and $\phi_s$ determination in $\Bs \to J/\psi \pi^+ \pi^-$}
Motivated by a predication in Ref. \citep{Stone:2008ak}, the LHCb collaboration made the first observation of $\Bs \to J/\psi f_0(980), f_0(980)\to \pi^+\pi^-$ \citep{Aaij:2011fx}, which was subsequently confirmed by others \citep{Li:2011pg,Aaltonen:2011nk,Abazov:2011hv}. This mode is a $CP$-odd eigenstate and can be used to determine $\phi_s$ without the need for an angular analysis, as is required in the $J/\psi \phi$ final state \citep{LHCb:2011aa,Abazov:2011ry,CDF:2011af}. With 0.4~fb$^{-1}$ data, LHCb used the candidates within $\pm90$ MeV of $f_0(980)$ mass and measured $\phi_s=-0.44\pm0.44\pm0.02$ rad \citep{LHCb:2011ab}. Whenever two uncertainties are given, the first is statistical and the second systematic. However, the used events are only about half of $J/\psi \pi^+ \pi^-$ signal.
To optimize $J/\psi \pi^+ \pi^-$ usefulness, we need to understand the $CP$ content of this final state. With 1.0~fb$^{-1}$ data, LHCb \citep{LHCb:2012ae} preformed a modified Dalitz-plot analysis which fits to the $m^2(\pi^+\pi^-)$, $m^2(J/\psi \pi^+)$ and $J/\psi\to \mu^+\mu^-$ helicity angle ($\theta_{J/\psi}$) distributions after integrating the angle between $J/\psi$ and $\pi^+\pi^-$ decay planes. Different from the classical ``Dalitz-plot'' analysis, the vector $J/\psi$ particle has 3 helicity amplitudes that must be considered. The projection of $m^2(\pi^+\pi^-)$ overlayed with the best fit is shown in Fig. \ref{DP1}. The components and fractions of the best fit are given in Table~\ref{tab:ff1}. The fraction of $J/\psi \rho$ is measured less than 1.6\% at 95\% confidence level (CL); the $J/\psi \rho$ final state only can be present in higher order processes. The final states listed in Table \ref{tab:ff1} are all $CP$-odd states, expect for $f_2(1270)$ with helicity $|\Lambda| = 1$ which has mixed-$CP$. Combining with this and $\rho$, the fraction of $CP$-even states is less than 2.3\% at 95\% CL. So the whole mode is dominated by $CP$-odd state, and can be used for $\phi_s$ measurement without need of angular analysis. The relative branching ratio between $\Bs\to J/\psi \pi^+\pi^-$ and $J/\psi\phi$ is measured as (19.79$\pm$0.47$\pm0.52$)\% \citep{LHCb:2012ae}.
\begin{figure*}[h!t!]
\centering
\includegraphics*[width=0.7\textwidth, bb= 0 539 530 843]{Fig12.eps}
\caption{Dalitz fit project of $m^2(\pi^+\pi^-)$ from the best fit. The points with error bars are data, the signal fit is shown with a (red) dashed line, the background with a (black) dotted line, and the (blue) solid line represents the total.} \label{DP1}
\end{figure*}
\begin{table}[h!]
\begin{center}
\caption{Resonance fractions in $\Bs\rightarrow J/\psi \pi^+\pi^-$ over the full mass range \citep{LHCb:2012ae}. The final-state helicity of the D-wave is denoted by $\Lambda$. Only statistical uncertainties are quoted.}\label{tab:ff1}
\begin{tabular}{lcc}
\hline
~~~Resonance & Normalized fraction (\%)\\
\hline
$f_0(980)$ &$69.7\pm2.3$ \\
$f_0(1370)$ &$21.2\pm2.7$\\
non-resonant $\pi^+\pi^-$ & ~\,$8.4\pm1.5$\\
$f_2(1270)$, $\Lambda=0$ &~\,$0.49\pm0.16$\\
$f_2(1270)$, $|\Lambda| = 1$ &~\,$0.21\pm0.65$\\
\hline
\end{tabular}
\end{center}
\end{table}
The $\Bs\to J/\psi \pi^+\pi^-$ decay complements $J/\psi \phi$ in determining $\phi_s$. The event yield is about 40\% of $J/\psi \phi (\phi \to K^+ K^-)$. In $\Bs$ decay to a $CP$-odd final state via $b\to c\bar{c}s$ transition, the time dependent decay rates are given in Eq.~(\ref{Eq-t}) with $\lambda = -e^{-\phi_s}$. We also need to take into account experimental effects, such as acceptance in decay time, time resolution, and dilution due to wrong flavor tagging. The decay time resolution in LHCb is about 40 fs thanks to the excellent vertex detector and large momentum of $\Bs$.
LHCb updated the $\phi_s$ measurement using an sample of 1 fb$^{-1}$ data \citep{LHCb:2012ad}. The $m(J/\psi \pi^+\pi^-)$ and $m(\pi^+\pi^-)$ distributions are shown in Fig. \ref{jpsipipi} after applying a Boosted Decision Tree selection rather than a cut-based selection used in Fig. \ref{DP1} . With about 7400 $\Bs$ signal candidates, LHCb finds $\phi_s = -0.019^{+0.173+0.004}_{-0.174-0.003}$ rad, consistent with the SM expectation~\citep{Charles:2011va}. The largest systematic uncertainty arises from allowing direct $CP$ violation. Here $\phi_s$ changes by $-0.0020$ from the default value and $|\lambda|=0.89\pm0.13$ consistent with no direct $CP$ violation. The systematic uncertainty due to a possible $CP$-even component is $-0.0008$. The measured $\phi_s$ is consistent with the LHCb preliminary result $-0.001\pm0.101\pm0.027$ rad using $\Bs\to J/\psi \phi$ \citep{LHCb-CONF-2012-002}. Combining the two LHCb measurements gives $\phi_s=-0.002\pm0.083\pm0.027$ rad \citep{LHCb-CONF-2012-002}.
\begin{figure*}[h!t!]
\centering
\includegraphics[width=0.44\textwidth]{Fig2.eps
\includegraphics[width=0.46\textwidth]{Fig3.eps}
\caption{Left: The $m(J/\psi \pi^+\pi^-)$ distribution from the candidates with $m(\pi^+\pi^-)\in [775,1550]$ MeV, shown also the $\Bs$ signal as the red solid line and other various background components. Right: The $m(\pi^+\pi^-)$ from the candidates with $\pm20$ MeV around $\Bs$ mass peak. The arrows indicate the region used for $\phi_s$ measurement and the red dash line shows the background.} \label{jpsipipi}
\end{figure*}
Other possible modes for measuring $\phi_s$ are
\begin{itemize}
\item $\Bs\to J/\psi f_2^{\prime}(1525)$ which was first observed by LHCb \citep{Aaij:2011ac}. This mode could be useful but it needs to include additional D-wave in transversity amplitudes. Sizeable S-wave over the entire $m(K^+K^-)$ in $J/\psi K^+K^-$ is also seen \citep{Aaij:2011ac}.
\item $\Bs\to J/\psi \eta^{(\prime)}$ which were first observed by Belle \citep{Belle:2012aa}. They are $CP$-even states with large branching fraction, but neutral detection is difficult for experiments in a hadron collider.
\item $\Bs\to \psi(2S) \phi, \psi(2S)\to \mu^+\mu^-$ and $\Bs \to D_s^+ D_s^-$. The event yields are of order of 5-10\% of $(J/\psi \to \mu^+\mu^-)(\phi\to K^+ K^-)$.
\end{itemize}
\section{$CP$ asymmetries in $\Bs \to h^+ h^{(\prime)-}$ decays}
Two-body charmless $\Bs$ decays have significant contribution of penguin diagrams, providing an entry point for New Physics.
Under U-spin symmetry, where $s$ quarks are changed to $d$ quarks, $\Bs \to K^+ K^-$ is analogous to $B^0 \to K^+\pi^-$ and $\Bs \to K^-\pi^+$ to $B^0\to \pi^+ \pi^-$. Specifically it turns out that $A_{KK}^{\rm dir}$ in $\Bs$ should be equal to the direct $CP$ asymmetry in $B^0 \to K^+\pi^-$, in the limit of exact U-spin symmetry and neglecting certain diagrams with annihilation topologies contributing to the decay amplitude \citep{Fleischer:1999pa}.
The direct $CP$ asymmetry in the flavor-specific decay $\Bs \to (f = K^- \pi^+)$,
\begin{equation}
A_{\rm CP}(\Bs \to K\pi) = \frac{\displaystyle |\overline{A}_{\bar{f}}|^2-|A_f|^2}{\displaystyle |\overline{A}_{\bar{f}}|^2+|A_f|^2},
\end{equation}
is measured by LHCb using 0.35 fb$^{-1}$ data, where an untagged time-integrated method is used. The measured charge asymmetry after correcting detection and production biases is equal to $A_{\rm CP} + {\cal O}(a^s_{\rm sl})$. The semileptonic asymmetry is measured as $a^s_{\rm sl}=(-1.81\pm1.06)\%$ by the D0 Collaboration~\citep{Abazov:2011yk}, where the uncertainty is sum of statistical and systematic. The LHCb Collaboration recently measured $a^s_{\rm sl}=(-0.24\pm0.54\pm0.33)\%$ \citep{LHCb-CONF-2012-022}. Thus ${\cal O}(a^s_{\rm sl})$ is negligible compared to the measured $CP$ asymmetry shown later. The $K^+\pi^-$ and $K^-\pi^+$ mass spectra for the candidates are shown in Fig. \ref{acp}. A Clear difference in the yields of two peaks at the $\Bs$ mass is seen.
LHCb finds $A_{\rm CP}(\Bs \to K\pi) = 0.27\pm0.08\pm0.02$ \citep{Aaij:2012qe}. It is 3.3$\sigma$ from zero, providing the first evidence for $CP$ violation in the decays of $\Bs$ mesons. The result for $A_{\rm CP}(\Bs \to K\pi)$ is consistent with the only previous measurement from CDF of $0.39\pm0.15\pm0.08$ \citep{Aaltonen:2011qt}.
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\textwidth]{ACP.eps}
\caption{Invariant $K\pi$ mass spectra for (a) $K^+\pi^-$ and (b) $K^-\pi^+$ combinations.} \label{acp}
\end{figure*}
LHCb \citep{LHCb-CONF-2012-007} made the first measurement of $CP$ asymmetry in $\Bs \to K^+K^-$. It's a time-dependent flavor-tagged analysis. The large statistical sample $\Bd \to K^- \pi^+$ is used to measure flavor tagging efficiency and mistag probability that are then constrained in the fit to $\Bs \to K^+K^-$. The mass distribution of selected candidates is shown in Fig. \ref{acpkk} (left); fit obtains $7155\pm97$ signal candidates. The measured $CP$ asymmetries (preliminary) are
\begin{eqnarray}
A_{KK}^{\rm dir}&=&0.02\pm0.18\pm0.04,\nonumber\\
A_{KK}^{\rm mix}&=&0.17\pm0.18\pm0.05.
\end{eqnarray}
The raw asymmetry is shown in Fig. \ref{acpkk} (right).
\begin{figure*}[h!t!]
\centering
\includegraphics[width=0.45\textwidth]{fig6top.eps
\includegraphics[width=0.44\textwidth]{fig7.eps}
\caption{Left: Invariant mass projection for the $\Bs\to K^+ K^-$. The data are described by the overall fit result (solid blue line) which is the sum
of: $\Bs \to K^+ K^-$ signal (dotted dark yellow line); $\Bd\to K\pi$ cross-feed background
(dashed double-dotted red line); partially reconstructed three-body (dashed orange line)
and combinatorial (dashed dotted grey line) backgrounds. Right: Time-dependent raw asymmetry in the $\Bs\to K^+K^-$ signal mass region with the
result of the fit superimposed. The offset $t_0 = 0.95$ ps corresponds to the selection requirement
where the decay time acceptance starts.} \label{acpkk}
\end{figure*}
\section{Effective lifetimes of $\Bs$ from $CP$-eigenstate final states}
The untagged decay time distribution summing $\Bs$ and $\Bsb$ is
\begin{equation}
\Gamma(t) \propto (1-A^{\Delta \Gamma_s})e^{-\Gamma_Lt}+(1+A^{\Delta \Gamma_s})e^{-\Gamma_Ht},
\end{equation}
where $A^{\Delta \Gamma_s}=-\frac{\displaystyle 2\RE \lf}{\displaystyle |\lf|^2+1}$, as in Eq.~(\ref{Eq-t}), and $\tau_H$ and $\tau_L$ are the lifetimes of heavy and light mass eigenstates. The $CP$-odd state $J/\psi f_0(980)$ has $A^{\Delta \Gamma_s}=\cos\phi_s=1$ and $CP$-even state $K^+K^-$ has $A^{\Delta \Gamma_s}\approx-1$ (the small difference from unity is discussed in Ref. \citep{Fleischer:2011cw}). Therefore, their effective lifetimes are approximately equal to the lifetimes of heavy and light mass eigenstates, respectively.
The effective lifetime in $J/\psi f_0(980)$ final state is measured to be $1.700 \pm 0.040 \pm 0.026$ ps by LHCb \citep{Aaij:2012nt} and $1.70^{+0.12}_{-0.11}$$\pm0.03$ ps by CDF \citep{Aaltonen:2011nk}. The most precise measurement on the effective lifetime in $K^+K^-$ final state comes from LHCb. The measured value is $1.455\pm0.046 \pm0.006$ ps \citep{:2012ns}. Those measurements can be used to improve the determination of $\Gamma_s$ and $\Delta \Gamma_s$ using the method carried out by HFAG ~\citep{Amhis:2012bh}.
\section{Summary}
In summary, the latest world best measurements on $CP$ violation in other $\Bs$ decays are discussed. They are mostly provided by LHCb collaboration thanks to the larger statics and better detector performance.
\bigskip
\begin{acknowledgments}
Work supported by U.S. National Science Foundation.
\end{acknowledgments}
\bigskip
\newpage
\input{Liming.bbl}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,882 |
Deborah W. Post - Professor Emeritus
Deborah W. Post
B.A., cum laude, 1971, Hofstra University
J.D., 1978, Harvard Law School
Email: dpost@tourolaw.edu
Room: 411E
Faculty Assistant: Sue Mori
smori@tourolaw.edu
Professor Deborah W. Post began her legal career working in the corporate section of a law firm in Houston, Texas, Bracewell & Patterson, now renamed Bracewell & Guiliani. She left practice to teach at the University of Houston Law School and moved to New York to Touro Law Center in 1987. She has been a visiting professor at Syracuse Law School, DePaul Law School, and State University of New Jersey Rutgers School of Law Newark. She also has taught as an adjunct at Hofstra Law School, UMass Dartmouth and St. Johns University School of Law. Professor Post has written for and about legal education. Among her most notable publications are a book on legal education, Cultivating Intelligence: Power, Law and the Politics of Teaching written with a colleague, Louise Harmon and a casebook in Contract, Contracting Law, with co-authors Amy Kastely and Nancy Ota. She has been a member of the Society of American Law Teachers Board of Governors for ten years and was co-president of that organization with Professor Margaret Barry from 2008-2010.
Visit Personal Web Site
Cultivating Intelligence: Power, Law and the Politics of Teaching, written with a colleague, Louise Harmon and published by New York University Press.
Casebook on contracts: Contracting Law with co-authors Amy Kastely and Nancy Ota.
"Continuity and Change: Partnership Formation Under the Common Law," Villanova Law Review (1987)
"Reflections on Identity, Diversity, and Morality," Berkeley Women's Law Journal (1990-91).
"Race, Riots and the Rule of Law," Denver Law Review (1993)
"Profit, Progress and Moral Imperatives," Touro Law Review (1993)
"Critical Thoughts About Race, Exclusion, Oppression and Tenure," Pace Law Review (1994)
"Power and Morality of Grading: A Case Study and a Few Critical Thoughts on Grade Normalization," University of Missouri at Kansas City Law Review (1997). | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,485 |
Ландшафтна архитектура е проектирането на обществени места на открито, паркове и градини, както и други структури за постигане на екологични, социално-поведенчески, или естетични резултати. Това включва системното изучаване на съществуващите социални, екологични и геоложки условия и процеси в ландшафта, и проектирането на интервенции, чрез които ще се достигне до желания резултат. Обхватът на професията включва: дизайн на градска среда (ландшафтен дизайн); планиране за местата, управление на дъждовни води; жилищно или градско планиране; екологично възстановяване; планиране на паркове и места за отдих, управление на визуален ресурс; планиране на зелена инфраструктура и озеленяване на частни и други имоти, в които специалистите планират и проектират в различни мащаби ландшафтния дизайн, изграждането и управлението му. Специалистите в професията на ландшафтната архитектура се наричат ландшафтни архитекти.
Ландшафтни архитекти работят върху всички видове структури и външно пространство – големи или малки, градски, крайградски и селски обекти, и с "твърди" (вградени) и "меки" (засадени) материали, за постигането на екологична устойчивост. Най-ценният принос може да бъде направено най-първия етап на проект за генериране на идеи с техническо разбиране и творчески усет за проектирането, организацията и използването на пространства. Архитектурният пейзаж може да обхване цялостната концепция и да се състави генерален план, от който се изготвят подробни проекти, чертежи и технически спецификации. Спрямо него се разглеждат предложенията, разрешават се и се контролират договорите за строителни работи. Други дейности могат да включват изготвянето на оценки на въздействието върху околната среда и екологичен контрол върху състоянието на ландшафта.
История
Ландшафтна архитектура и нейните естетически аспекти включват форма, растения, цвят, аромат, размер, климат и функция. Градинските площи се нуждаят от непрекъсната поддръжка, за да запазят плевелите и други нежелани природни явления далеч от себе си. Градините се променят със сезоните и климата, както и с цикъла на живот на растенията. Исторически градините намират приложение повече в частни имения отколкото публично.
Известно ни е, че градините са били използвани от дълбоки времена, но имаме малко знания за градини и ландшафтна архитектура през античните времена, първите писмени документи за тях идват доста по-късно, под формата на наръчници за изграждане на градини, след което като критики и културни коментари. Историята на градинарството отбелязва значително развитие през късния 20 век, чрез историци на изкуството в САЩ и литературни историци във Великобритания начело. Колкото повече вървим напред към модерната литература, толкова по богата е библиографията, но има и много важни разработки от античността.
Древни народи като Египтяни, Римляни и Гърци развиват собствените си особености за изграждане на градини. По времето на Италианския ренесанс се развиват външните градини, които се смятат за продължение на сградата. Един чудесен пример в Тиволи, Италия е Вила д`Есте, която е известна с това, че има един от най-красивите паркове в Европа. Построена, когато кардинал Иполито II д'Есте взима решение да превърне досегашния манастир във вила. Тя се намира в непосредствена близост до столицата- Рим, което я прави често предпочитана дестинация за дневни пътешествия на туристи, но не само близостта и до великия град, а най-вече и изящните и пищни градини наслоени от зеленина, изготвени с помощта на това, което днес ние наричаме ландшафтна архитектура.
Вижте също
Озеленяване
Ландшафтна архитектура (ЛТУ)
Външни препратки
Ландшафтна архитектура в Start.bg
Ландшафтна архитектура в Links.bg
Ландшафтна архитектура и история
Източници | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,175 |
PITTSBURGH, PA. Carol H. Ware, of Pittsburgh, Pa., formerly of Kinnairds Point, Worton, died peacefully Sunday, Dec. 25, 2011, after a brief illness. She was born in 1928. Carol was the wife of the late Robert L. Ware; mother of four children, Frank (Debi) Ware of Virginia, Peter Ware of Arkansas, Wendy Ware of California and Priscilla (Richard) Kwiatkowski of Pennsylvania; and had seven grandchildren. She was preceded in death by her parents, Wilson and Alice Hoag, and her sister, Audrey (Harlan) Strader. A memorial Mass of Christian Burial was held Dec. 30, 2011, in West Mifflin, Pa. The burial will be held at a later date in Kent County. Memorial contributions may be made to a charity of one's choice or to ovarian cancer research. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,045 |
Q: Force Azure Re-Authentication for each request in C# I have a requirement where I want to fetch the Authorization Code for Azure login for each request in web API. As of now once the user signs in to Azure, after that I am not getting the authorization code as the user is already signed in.
How can I force the user to sign in again? This is the code I have been using as of now in the owin_startup file in web API?
app.SetDefaultSignInAsAuthenticationType(CookieAuthenticationDefaults.AuthenticationType);
app.UseCookieAuthentication(new CookieAuthenticationOptions()
{
CookieSecure = (CookieSecureOption)Convert.ToInt32(cookieSecure), // CookieSecureOption.NeverAlways
CookieManager = new SystemWebCookieManager(),
CookieHttpOnly = false,
});
app.UseOpenIdConnectAuthentication(new OpenIdConnectAuthenticationOptions
{
ClientId = clientId,
Authority = Authority,
RedirectUri = RedirectUri,
});
A: According to the code and the cases you post before, I think it it not about Azure ad b2c, so here I will give a reply for azure ad.
When you request an authorization code, there is a prompt=login property which is indicate the user should be prompted to reauthenticate.
Also here is an article about Forcing reauthentication with Azure AD which suggest use Token Max Age to achieve it.
You can append max_age= to the authorization URL (or just put 0 to force password authentication at all times). So once user gets redirected to the URL, they will be presented with an information to login again.
public class RequireReauthenticationAttribute : Attribute, IAsyncResourceFilter
{
private int _timeElapsedSinceLast;
public RequireReauthenticationAttribute(int timeElapsedSinceLast)
{
_timeElapsedSinceLast = timeElapsedSinceLast;
}
public async Task OnResourceExecutionAsync(ResourceExecutingContext context, ResourceExecutionDelegate next)
{
var foundAuthTime = int.TryParse(context.HttpContext.User.FindFirst(AppClaimTypes.AuthTime)?.Value, out int authTime);
if (foundAuthTime && DateTime.UtcNow.ToUnixTimestamp() - authTime < _timeElapsedSinceLast)
{
await next();
}
else
{
var state = new Dictionary<string, string> { { "reauthenticate", "true" } };
await context.HttpContext.Authentication.ChallengeAsync(OpenIdConnectDefaults.AuthenticationScheme, new AuthenticationProperties(state)
{
RedirectUri = context.HttpContext.Request.Path
}, ChallengeBehavior.Unauthorized);
}
}
}
A: It appears that the reason you wish to re-authenticate the user is to get/refresh the token in memory even when the auth cookie is present. Simplest way to achieve this would be decorate your controller with the AuthorizeForScopes attribute. You can check the sample project on Github here
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,811 |
__author__ = 'cosmin'
class Loan:
'''
Entity to represent a loan, which is basically a pair of Client, Book. Every book can have only one Client at a given moment.
Also, every Client can have one or more books rented.
So the Loan has:
- Renter (Client)
- Rented Book
'''
def __init__(self, client, book):
self._client = client
self._book = book
def __repr__(self):
'''
Function to print the Object in a nice way
'''
return "Client %s has the book #%d with the Title: %s" % (self._client.getName(), self._book.getId(), self._book.getTitle())
def __eq__(self, other):
return (isinstance(other, self.__class__)
and self.__dict__ == other.__dict__)
def getClient(self):
'''
Function to get the Client from a specific Loan
:return: a Client object
'''
return self._client
def getBook(self):
'''
Function to get the Book from a specific Loan
:return: a book object
'''
return self._book | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,021 |
Las Cabras may refer to:
Las Cabras, Chile
Las Cabras, Herrera, Panama | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,518 |
{"url":"http:\/\/www.mathworks.com\/help\/control\/ref\/getcompsensitivity.html?nocookie=true","text":"Accelerating the pace of engineering and science\n\n# getCompSensitivity\n\nComplementary sensitivity function from generalized model of control system\n\n## Syntax\n\n\u2022 T = getCompSensitivity(CL,location) example\n\u2022 T = getSensitivity(CL,location,opening) example\n\n## Description\n\nexample\n\nT = getCompSensitivity(CL,location) returns the complementary sensitivity measured at the specified location for a generalized model of a control system.\n\nexample\n\nT = getSensitivity(CL,location,opening) specifies additional loop openings for the complementary sensitivity function calculation. Use an opening, for example, to calculate the complementary sensitivity function of an inner loop, with the outer loop open.\n\nIf opening and location list the same point, the software opens the loop after adding the disturbance signal at the point.\n\n## Examples\n\nexpand all\n\n### Complementary Sensitivity Function at a Location\n\nCompute the complementary sensitivity at the plant output, X.\n\nCreate a model of the system by specifying and connecting a numeric LTI plant model G, a tunable controller C, and the AnalysisPoint block X. Use the AnalysisPoint block to mark the location where you assess the complementary sensitivity (plant output in this example).\n\n```G = tf([1],[1 5]);\nC = ltiblock.pid('C','p');\nC.Kp.Value = 3;\nX = AnalysisPoint('X');\nCL = feedback(G*C,X);```\n\nCL is a genss model that represents the closed-loop response of the control system from r to y. The model contains the AnalysisPoint block, X, that identifies the analysis-point location.\n\nCalculate the complementary sensitivity, T, at X.\n\n```T = getCompSensitivity(CL,'X');\ntf(T)```\n```ans =\n\nFrom input \"X\" to output \"X\":\n-3\n-----\ns + 8\n\nContinuous-time transfer function.\n```\n\n### Specify Additional Loop Opening for Complementary Sensitivity Function Calculation\n\nCalculate the inner-loop sensitivity at the output of G2, with the outer loop open.\n\nCreate a model of the system by specifying and connecting the numeric plant models, tunable controllers, and AnalysisPoint blocks. G1 and G2 are plant models, C1 and C2 are tunable controllers, and X1 and X2 are AnalysisPoint blocks that mark potential loop-opening locations.\n\n```G1 = tf(10,[1 10]);\nG2 = tf([1 2],[1 0.2 10]);\nC1 = ltiblock.pid('C','pi');\nC2 = ltiblock.gain('G',1);\nX1 = AnalysisPoint('X1');\nX2 = AnalysisPoint('X2');\nCL = feedback(G1*feedback(G2*C2,X2)*C1,X1);```\n\nCalculate the complementary sensitivity, T, at X2, with the outer loop open at X1.\n\n```T = getCompSensitivity(CL,'X2','X1');\ntf(T)```\n```ans =\n\nFrom input \"X2\" to output \"X2\":\n-s - 2\n----------------\ns^2 + 1.2 s + 12\n\nContinuous-time transfer function.```\n\n## Input Arguments\n\nexpand all\n\n### CL \u2014 Model of control systemgeneralized state-space model\n\nModel of a control system, specified as a Generalized State-Space Model (genss).\n\nLocations at which you can perform sensitivity analysis or open loops are marked by AnalysisPoint blocks in CL. Use getPoints(CL) to get the list of such locations.\n\n### location \u2014 Locationstring | cell array of strings\n\nLocation at which you calculate the complementary sensitivity function, specified as a string or cell array of strings. To extract the complementary sensitivity function at multiple locations, use a cell array of strings.\n\nEach string in location must match an analysis point in CL. Analysis points are marked using AnalysisPoint blocks. Use getPoints(CL) to get the list of available analysis points in CL.\n\nExample: 'u' or {'u','y'}\n\n### opening \u2014 Additional loop openingstring | cell array of strings\n\nAdditional loop opening used to calculate the complementary sensitivity function, specified as a string or cell array of strings. To open the loop at multiple locations, use a cell array of strings.\n\nEach string in opening must match an analysis point in CL. Analysis points are marked using AnalysisPoint blocks. Use getPoints(CL) to get the list of available analysis points in CL.\n\nUse an opening, for example, to calculate the complementary sensitivity function of an inner loop, with the outer loop open.\n\nIf opening and location list the same point, the software opens the loop after adding the disturbance signal at the point.\n\nExample: 'y_outer' or {'y_outer','y_outer2'}\n\n## Output Arguments\n\nexpand all\n\n### T \u2014 Complementary sensitivity functiongeneralized state-space model\n\nComplementary sensitivity function of the control system, T, measured at location, returned as a Generalized State-Space Model (genss).\n\n\u2022 If location specifies a single analysis point, then T is a SISO genss model.\n\n\u2022 If location is a string specifying a vector signal, or a cell array identifying multiple analysis points, then T is a MIMO genss model.\n\nexpand all\n\n### Complementary Sensitivity\n\nThe complementary sensitivity function, T, at a point is the closed-loop transfer function around the feedback loop measured at the specified location. It is related to the open-loop transfer function, L, and the sensitivity function, S, at the same point as follows:\n\n$T=\\frac{L}{1-L}=S-1.$\n\nUse getLoopTransfer and getSensitivity to compute L and S.\n\nConsider the following model:\n\nThe complementary sensitivity, T, at y is defined as the transfer function from dy to y.\n\nObserve that, in contrast to the sensitivity function, the disturbance, dy, is added after the measurement, y.\n\n$\\begin{array}{l}y=-GK\\left(y+dy\\right)\\\\ \\to y=-GKy-GKdy\\\\ \\to \\left(I+GK\\right)y=-GKdy\\\\ \\to y=\\underset{T}{\\underbrace{-{\\left(I+GK\\right)}^{-1}GK}}dy.\\end{array}$\n\nHere, I is an identity matrix of the same size as GK. The complementary sensitivity transfer function at y is equal to -1 times the closed-loop transfer function from r to y.\n\nComplementary sensitivity at multiple locations, for example, u and y, is defined as the MIMO transfer function from the disturbances to measurements:\n\n$T=\\left[\\begin{array}{cc}{T}_{du\\to u}& {T}_{dy\\to u}\\\\ {T}_{du\\to y}& {T}_{dy\\to y}\\end{array}\\right].$","date":"2014-12-25 16:44:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 3, \"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\": 0.5674380660057068, \"perplexity\": 2687.1974815702156}, \"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-2014-52\/segments\/1419447547854.96\/warc\/CC-MAIN-20141224185907-00020-ip-10-231-17-201.ec2.internal.warc.gz\"}"} | null | null |
Anyone even mildly familiar with bread and the science of bread baking would probably take one look at the title of this article and assume that I had lost more than a few of my marbles.
"Sourdough in three hours?! Nope. Can't be done," they would boldly exclaim. To a certain extent, they are right. A true sourdough cannot be rushed, accelerated, or otherwise prodded along; you could, of course, go out and purchase a sourdough starter, but where's the challenge in that?
For the uninitiated, the big divider between an average loaf of bread you can slap together in around three hours and a sourdough lies in a simple mixture of flour and water. This "starter" is allowed to sit out for weeks, months, even years on end, and as it lies around, it creates a home for wild yeast. As time passes, these feisty yeast cultures produce carbon dioxide and alcohol, which impart distinct sour flavor into what was at first a flavorless blob of dough.
So how do we shortcut such an excruciatingly long process into a few hours? Quite simply, you don't. Science does not enjoy being cheated, so we'll still give our bread rise using the power of yeast, but look elsewhere for that distinct sour flavor. This month, we're staring a seemingly impossible task in the face to create: the three-hour sourdough.
A few weeks ago in my breads class at Le Cordon Bleu, our chef challenged us to create our own original breads (okay, it was actually an exam, but I totally took it as a challenge). Only a few parameters were put in place: it couldn't be too heavily enriched with things like eggs and butter, and had to maintain a fairly basic shape. Other than that, we had a good deal of freedom.
As always, finding inspiration was as easy as cracking open the fridge to see what was sitting around. It just so happened that I had prepped one of my favorite snacks the day before: pickled sweet onions in apple cider vinegar, brown sugar, and low-country blackening spice. My plan for them was fairly simple: in a bread that only contained flour, water, yeast, and salt, I would replace a portion of the water with my intensely flavored pickling liquid (as well as add the onions) to up the flavor of the dough.
So, dutifully following bread baking procedure, I whipped together my recipe with that slight amendment and handed the result to my chef. When he sliced into my loaf, he revealed a dense, under-cooked dough; horrified, I couldn't help but wonder what could have gone wrong.
As it turns out, I hadn't accounted for the harsh conditions that acid creates for yeast. In addition to weakening the gluten structure, acid inhibits the yeast's ability to produce essential gases and alcohol that give bread its rise and flavor. Because of this, a dough has a ceiling to the amount of acid it can handle, and will need almost double the amount of time to rise (which I hadn't counted on).
Aside from the doughy, raw nature of my failed loaf, it did have a pleasant sour hint to it. So was there anything I could do to this recipe to develop that sour character while still making a well-made bread? It was time to hit the kitchen and find out.
Earlier in this article, I knocked a little bit on purchasing a pre-grown sourdough starter. There is absolutely nothing wrong with doing this, as it is a guaranteed route to great flavor; the only problem is that you are essentially growing a live yeast culture that demands constant, daily care. Perhaps not the best route fora casual home cook. Travis suggested something much more low-maintenance: simply take already a baked sourdough loaf (especially a staled one that would otherwise be tossed), and soak a portion of it in the water called for in the recipe.
Such an idea stems from the classic French "old dough" technique, in which an older portion of unbaked dough is added to a recipe. With the older dough that's already been fermented it automatically boosts the flavor of the new dough that it is added to. The yeast in a baked loaf of sourdough has dies in the high heat of an oven, but the goal is to carry the sour flavor from the baked loaf to the one being mixed, not the yeast.
What resulted was pleasing, if not ideal; tasters could not identify any carryover in sourness, but using the old, well-developed bread bolstered the flavor profile of an otherwise basic bread recipe. But was it so drastic a change that it will forever change the way you bake bread? Probably not. On to the next idea, then.
When we think about acids in cooking, we tend to work with two major categories: acetic and lactic. To over simplify it, acetic acids are the harsher type that we find in such products as vinegar, while lactic acids are the gentler ones that give a slight tang to some of our favorite dairy products.
One of the main contributors of flavor within a developed sourdough is lactic acid, so Travis suggested using whey protein runoff from a dairy product like yogurt. Using this alone, or even in conjunction with the harsher pickling liquid, would in theory contribute a tangy note to the bread, without being so acidulated that the bread couldn't form. Taking time to hang the yogurt would push me outside of my three-hour window, but at this point I was hungry for results. One container of yogurt, some cheesecloth, and a little time later, I had accumulated more than enough whey to give this interesting idea a go.
Fortunately and unfortunately, the bread did form nicely, but without any pronounced sour flavor. I had a nice loaf of sandwich bread, but sourdough this was not.
Several loaves of bread and zero satisfying results later, frustration was setting in. What had been the difference between my raw and sour dough and the well-baked but neutrally flavored breads I produced? Surely there was a middle ground somewhere that I could hit.
After some deep thought, it occurred to me that the problem with my original underdone batch of bread might have been the pickled onions. I had calculated the maximum amount of pickling liquid I could add to the recipe, but wasn't taking into account the liquid that the onions held onto after coming out of the jar. This small amount of extra acidic liquid was a likely candidate in the downfall of my bread.
So I would bank everything on two last batches: both would have high (but not the maximum) levels of acid, and one would contain onions, but pickled in the exact amount of liquid the recipe called for. Finally! A positive result emerged, and I was so happy I could have (and maybe did) kiss the damn things.
Both fully-baked loaves did indeed produce a slight twang, but it was undeniable in a taste test that the loaf with the pickled onions exhibited concentrated blasts of tangy brightness. As an added bonus, the acid creates a looser "crumb" on the bread, helping the inside look less like a plain Jane sandwich bread and more like the artisanal sourdough it aspires to be.
I wouldn't say that it matches the flavor profile of a sourdough whose starter has had years to develop, but it is on par with one given a few weeks to grow. Not bad!
I realize that a lot of people would rather run to their local bakery and grab a loaf, but bread baking is a relatively simple and rewarding process that I encourage anyone to try. As usual, I'll leave you with the recipe, as well as a few general bread-making tips for the home cook who still feels a bit shy about the process.
Ingredients in a bread recipe are meticulously measured to produce exacting results. My recipe happens to translate fairly well to cups and measuring spoons, but this is rarely the case, and can really throw off your bread consistency.
Also known as bread machine yeast, instant yeast can be mixed directly into your dry ingredients without being dissolved in water first like active dry yeast. It will shave about five minutes off of your cook time, which is invaluable if you are doing large-batch baking.
A large portion of your bake time in breads comes in two large chunks of time in which you have to allow the yeast to feed on the starches and sugars in a dough to produce gases. One of these byproducts is carbon dioxide, which does contribute to a dough's rise as it expands, but actually prevents further yeast activity.
A common technique for regulating this CO2 development is to "punch" or press the gases from the dough. Rather than completely deflating the dough, fold it onto itself by gently lifting it from one end and pulling the dough up and over. Doing so will maximize rise and flavor, which is particularly useful when making a dough such as the Three-Hour Sourdough, which needs as much help as it can get.
No Proofing Box? No problem.
Professional bakers use large, temperature-controlled boxes to create a warm, moist environment that yeast loves, and causes breads to rise faster, thus shaving more time off the cooking process. Home chefs try to recreate this using a low oven and a pan of hot water, but here's what I propose: fill a coffee mug halfway with water, microwave it to near boiling (about a minute should do), then place your covered bread in the now warm, moist "proofing box." Boom.
A few last tips for finishing your bread. Simple French breads such as the one in this article are in part characterized by a crisp, golden crust. Professional bakers achieve this by having hot steam-injected ovens. Recreate this effect by leaving an empty baking sheet on the bottom rack of the oven while it preheats. By the time your bread goes in, a little water poured onto the hot sheet pan will create billows of steam to crisp up your bread's crust and give it deep, golden brown color.
1. Preheat the oven to 500 degrees. Place an empty baking sheet onto the bottom rack of the oven.
2. Mix together the flour, salt, and yeast in a medium bowl. Add the pickled onions, pickling liquid, and water. Stir with a wooden spoon until combined.
3. Once the dough has absorbed the liquid, turn the dough out onto a floured work surface and knead until smooth, about 8-10 minutes.
4. Place the kneaded dough into a lightly oiled bowl, cover with a damp towel, and let rise in a warm, moist environment for an hour, folding the dough after 30 minutes.
5. Fold the dough once more, shape into a round, and place on a cutting board generously dusted with cornmeal. Cover with a damp towel and let rise for one hour.
6. Uncover the dough, brush with a small amount of water, and sprinkle with black sea salt and a bit of flour. Using the blade of a sharp paring knife, score an "x" in the top of the loaf.
7. Immediately place the scored loaf directly onto a preheated pizza stone or onto a parchment-lined baking sheet and place in the oven. Carefully pour a small amount of water onto the empty baking sheet, and immediately close the oven door.
8. Allow the loaf until deep golden brown, about 18-20 minutes. Remove from the oven and let cool on a rack for at least 20 minutes before slicing it open. Serve with bacon butter.
1. Combine ingredients in a Ziploc bag, close tightly, and allow to sit in the refrigerator for at least one hour, or up to two weeks.
1. Place bacon strips in an even layer in an unheated sauté pan. Cook over medium low heat until bacon is crispy on one side, about 8-10 minutes. Flip and allow to crisp on the other side.
2. Drain cooked bacon on a paper towel-line plate. Once cooled, crumble into pieces and mix in a small bowl with the softened butter. Serve immediately or store in an airtight container in the fridge for later use. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,465 |
Q: While loop is repeating one record? I have created while loop which is used to fetch record matching the word getting from URL. The variable is getting the word but while loop is repeating only one record don't know why? here is my code.
include_once('dbconnect.php');
$GetChR=isset($_REQUEST['authchar'])?$_REQUEST['authchar']:'';
//get the function
include_once ('function.php');[![enter image description here][1]][1]
$page = (int) (!isset($_GET["page"]) ? 1 : $_GET["page"]);
$limit = 25;
$startpoint = ($page * $limit) - $limit;
//to make pagination
$statement = " table2, table1 WHERE table1.col2 like '%$GetChR%'";
echo '<div class="records round">';
//show records
$query = mysqli_query($con,"SELECT table2.col2 AS a,table1.col2 AS b, table1.col1 AS c, table1.q_url AS d, table1.tags AS e FROM {$statement} LIMIT {$startpoint} , {$limit}");
while ($row = mysqli_fetch_array($query))
{
$input='';
//$Authorname =$row['a'];
$url = "http://$_SERVER[HTTP_HOST]$_SERVER[REQUEST_URI]";
$url=explode('/',$url);
echo '<div class="record round">'; $input .='<a href="http://localhost/quotes/'.$url[5].'/'.$row['d'].'.html">';
echo $input .=$row['b'];echo'</a>';echo '</div>';
// $count++;
}
echo'</div>';
echo pagination($statement,$limit,$page);
A: These are the main issues with your code:
*
*You're not joining your tables. There's no condition or join statement that describes how table1 and table2 are to be linked. Results are unpredictable, but would need to see your data to be sure if this is what causes the repeated record.
*You're creating a massive SQL injection risk by dumping a request parameter straight into your query. Use prepared statements instead.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,114 |
Why has T.J. Dillashaw's doping case instantly incurred our anger, but others haven't?
At Friday's UFC news conference, there was audible dismay and disappointment from the fans in attendance when they heard UFC President Dana White say that Brock Lesnar might not be returning any time soon to fight for the heavyweight title. You know, now that his doping suspension is over.
And UFC light heavyweight champion Jon Jones, whose drug tests continue to pulse with picograms of turinabol? We were very sorry that he was too sick to make it to the stage Friday, but we're still very much looking forward to his next title defense in July.
Meanwhile, in New York this week, Bellator had a press conference of its own to hype a bout between Lyoto Machida and Chael Sonnen, a man who was run out of the UFC after multiple doping infractions. This required Sonnen to take a brief break from his other job as a talking head on ESPN, where his expertise as a former EPO user came in handy this week.
Part of it is probably personality. Dillashaw wasn't exactly beloved before this. He had that public breakup with Team Alpha Male, then gleefully announced his intention to kill the flyweight division. He didn't have a ton of goodwill built up when he needed it, and maybe now he's feeling it.
There's also the issue of how this substance meshes with Dillashaw's style and achievements. He's a high-paced fighter who used constant pressure to break opponents and claim a UFC title. We were wondering if he might be one of the best bantamweights to ever do it. Then we found out there might have been a reason he was able to do all that, and it went beyond simple hard work in the gym. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,671 |
{"url":"http:\/\/cohomeworkvfum.voteforthegirls.us\/an-analysis-of-vector.html","text":"# An analysis of vector\n\nPage 3 vector analysis operations vector analysis operations a variety of tntmips processes are involved in topological vector analysis the processes consid. Performance and integrity analysis of the vector tracking architecture of gnss receivers a dissertation submitted to the faculty of the graduate school. This prize-winning study traces the rise of the vector concept from the discovery of complex numbers through the systems of hypercomplex numbers created by hamilton. \u2022 two dimensional vector analyses is accomplished by the use of geometric methods \u2022 graphic analysis uses scale drawings to find the sum or difference of two or. Instead of using a simple lifetime average, udemy calculates a course's star rating by considering a number of different factors such as the number of ratings, the. A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied (scaled) by numbers, called scalars.\n\nFunctional and structured tensor analysis for engineers a casual (intuition-based) introduction to vector and tensor analysis with reviews of popular. An introduction to tensors for students of physics and engineering an introduction to tensors for students and similar higher-order vector products. Data analysis royalty free stock illustrations big collection of cliparts, vectors, illustration and vector arts. Download analysis images and photos over 62,932 analysis pictures to choose from, with no signup needed download in under 30 seconds. A good application for vector analysis is the calculation of an airplane's flight path this requires knowing the magnitude and direction of both the plane's thrust. Cloning vectors the molecular analysis of dna has been made possible by the cloning of cloning vector - a dna molecule that carries foreign dna into a host.\n\n1 vector analysis a1 vectors a11 introduction some physical quantities like the mass or the temperature at some point only have magnitude we can represent these. Appendix d vector analysis 1 appendix d vector analysis the following conventions are used in this appendix and throughout the book: fg`\u02c6are scalar functions of xt.\n\nIn multivariate analysis a vector is a matrix with either only one column or only one row a column vector has only one column a row vector has only one row. Harvey mudd college math tutorial: elementary vector analysis in order to measure many physical quantities, such as force or velocity, we need to determine.\n\n## An analysis of vector\n\nAn analysis of vector measurement accuracy enhancement techniques 2 agenda \u2022 accuracy enl-lancementf:or one -port networks \u2022 one-portcalibration techniques. To evaluate the refractive outcomes for the correction of low to moderate astigmatism up to 1 year following small incision lenticule extraction (smile) surgery this.\n\nVector calculus, or vector analysis, is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional euclidean. 3 analytical kinematics in this course we first discuss analytical kinematic analysis analytical kinematics is based on projecting the vector loop equation(s. Preface this book covers calculus in two and three variables it is suitable for a one-semester course, normally known as \u201cvector calculus\u201d, \u201cmultivariable. To better understand the science of propulsion it is necessary to use some mathematical ideas from vector analysis most people are introduced to vectors. Analysis of i-vector length normalization in speaker recognition systems daniel garcia-romero and carol y espy-wilson department of electrical and computer. Vector in c++, a vector is a sequence of elements that can be accessed by an index, but - unlike an array - it does not have a \ufb01xed size vector = v \/\/ start.\n\nPower vector analysis of refractive, corneal, and internal astigmatism in an elderly chinese population: the shihpai eye study. Called a vector algebra or vector analysis def\u2014as distinguished from vectors the real (positive or negative) quantities of ordinary algebra are called scalars1. Definition of vector analysis in the financial dictionary - by free online english dictionary and encyclopedia what is vector analysis meaning of vector analysis as. Signalvu-pc is the foundation of rf and vector signal analysis software that helps you easily validate rf designs it is based on the signal analysis engine of the. The direction of a vector $\\vecb{v}$ in 3-space is specified by its components in the $x$, $y$, and $z$ directions, respectively: (x,y,z) \\quad {\\small\\textrm{or. The physics classroom \u00bb physics tutorial \u00bb vectors - motion and forces in two dimensions vectors vector components vector resolution component addition.\n\nAn analysis of vector\nRated 4\/5 based on 17 review","date":"2018-05-26 23:11:26","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\": 0.5536657571792603, \"perplexity\": 1479.9584103037532}, \"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-2018-22\/segments\/1526794867949.10\/warc\/CC-MAIN-20180526225551-20180527005551-00032.warc.gz\"}"} | null | null |
\section{Introduction}\label{sec:intro}
Time series forecasting plays an important role in both research and industries. Correctly forecast time series can greatly benefit various business sectors such as traffic, weather, and electricity forecasting~\citep{hyndman2018forecasting}.
As a result, tremendous efforts have been devoted to developing better forecasting models~\citep{petropoulos2020forecasting,bhatnagar2021merlion,triebe2021neuralprophet}, with a recent focus on deep neural networks~\citep{li2019enhancing,xu2021autoformer,yue2021ts2vec,zhou2021informer} thanks to their impressive capabilities to discover hierarchical latent representations and complex dependencies.
However, real-world deployments are often interested in the online time series forecasting scenario~\citep{anava2013online,liu2016online} where the deployed model needs to adapt to the non-stationary environments quickly.
Despite the promising results of deep models on batch-learning of time series forecasting~\cite{oreshkin2019n,zhou2021informer}, training deep forecasters online remains challenging for two major reasons.
First, naively train deep neural networks on data streams often converges slowly~\citep{sahoo2018online,aljundi2019online} because the benefits of offline training such as mini-batches or training for multiple epochs are not available.
Second, learning in non-stationary environments requires dealing with concept drifts in which the underlying data generation process changes over time~\citep{gama2014survey}. Together with the catastrophic forgetting phenomenon~\citep{mccloskey1989catastrophic}, the model loses the ability to forecast the observed patterns when they reappear, which further hinders the performance.
With such difficulties, online time series forecasting presents a promising yet challenging problem.
To address the above limitations, we innovatively formulate online time series forecasting problem as the \emph{online, task-free continual learning} problem~\citep{aljundi2019online,aljundi2019task}.
Particularly, continual learning requires balancing two conflicting objectives: (i) utilizing past knowledge to facilitate fast learning of current patterns; and (ii) maintaining and updating the already acquired knowledge. These two objectives closely match the aforementioned challenges and are usually referred to as the \emph{stability-plasticity} dilemma~\citep{grossberg1982does}.
With this connection, we develop an efficient online time series forecasting framework motivated by the \emph{Complementary Learning Systems (CLS) theory}~\citep{mcclelland1995there,kumaran2016learning}, a neuroscience framework for continual learning.
Specifically, the CLS theory suggests that humans can continually learn thanks to the interactions between the \emph{hippocampus} and the \emph{neocortex}. Moreover, the hippocampus is responsible for modeling pattern-specific representations of recent experiences to facilitate fast learning. The hippocampus interacts with the neocortex to consolidate, retrieve, and update such experiences to form a more general representation, which supports generalization to new experiences, as illustrated in Figure~\ref{fig:cls}.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{figs/CLS.pdf}
\caption{An illustration of the fast and slow learning according to the CLS theory. Figure is inspired from~\citet{parisi2019continual}.} \label{fig:cls}
\vspace*{-0.2in}
\end{figure}
Motivated from the CLS theory, we propose a novel framework of Fast and Slow learning Networks (FSNet) to train deep neural forecasters on the fly for online time-series forecasting. FSNet augments a standard deep neural network backbone (slow learner) with two complementary components to facilitate fast adaptation to both new and recurrent concepts.
To this end, FSNet employs a per-layer adapter to monitor each layer's contribution to the forecasting loss via its partial derivative. The adapter transforms each layer's weight and feature at each step based on its recent gradient, allowing a fine-grain, per-layer fast adaptation to optimize the current loss.
In addition, FSNet employs a second and complementary associative memory component~\citep{kaiser2017learning} to store important, recurring patterns observed during training. When encountering such events, the adapter interacts with the memory to store, update, and retrieve the previous transformations, facilitating fast learning of such patterns.
As a result, the whole model can adapt to the fast-changing and the long, recurring patterns in time series. As an analogy, in FSNet, the deep neural network plays the role of neocortex while the adapter and its memory play acts as a hippocampus component.
In summary, our work makes the following contributions. First, we propose FSNet, a novel framework to forecast time series on the fly with deep models. Second, motivated from the CLS theory for continual learning, we propose a fast-and-slow learning paradigm of FSNet to handle both the fast changing and long-term knowledge in time series. Lastly, we conduct extensive experiments on both real and synthetic datasets to demonstrate FSNet's efficacy and robustness to the concept drift, recurring patterns.
\section{Preliminary and Related Work} \label{sec:related}
This section provides the necessary background of time series forecasting and continual learning.
\subsection{Time Series Forecasting Settings}\label{sec:ts-setting}
Let ${\mathcal{X}} = ({\bm{x}}_1,\ldots,{\bm{x}}_T) \in \mathbb{R}^{T\times n}$ be a time series of $T$ observations, each has $n$ dimensions.
The goal of time series forecasting is that given a look-back window of length $e$ starting at time $i$: ${\mathcal{X}}_{i,e} = ({\bm{x}}_i,\ldots,{\bm{x}}_{i+e})$, predict the next $H$-steps of the time series as $f_{\omega}({\mathcal{X}}_{i,H}) = ({\bm{x}}_{i+e+1} \ldots {\bm{x}}_{i+e+H})$, where $\omega$ denotes the parameter of the forecasting model.
We refer to a pair of look-back and forecast windows as a sample.
For multiple-steps forecasting ($H>1$) we follow the standard approach of employing a linear regressor of dimension to forecast all $H$ steps in the horizons~\citep{zhou2021informer} simultaneously.
\subsubsection{Online Time Series Forecasting}\label{sec:online-ts}
Online time series forecasting is relevant is many real-world scenarios~\citep{anava2013online,liu2016online,gultekin2018online,aydore2019dynamic} due to the sequential nature of time series.
In this setting, there is no separation of training and evaluation. Instead, learning occurs over a sequence of rounds. At each round, the model receives a look-back window and predicts the forecast window. Then, the true answer is revealed to improve the model's predictions of the incoming rounds.
Due to its challenging nature, existing online time series forecasting studies only focus on linear models~\citep{anava2013online,liu2016online}, while there is limited progress in building deep forecasting model online~\citep{aydore2019dynamic}. However, with the expressive capability of deep neural networks, we argue that training deep neural networks forecaster on the fly presents a challenging yet promising research problem.
In addition, we note that hyper-parameter cross-validation and data normalization are important for training deep neural networks but are not available in the traditional online learning setting~\citep{hazan2019introduction}. Therefore, we follow the setting in~\citet{chaudhry2019tiny} in reserve the first 25\% data of the time series as a warm-up phase to cross-validate all hyper-parameters and calculate the mean, standard deviations to normalize the samples during online learning.
\subsection{Continual Learning}
Continual learning~\citep{lopez2017gradient} is an emerging topic aiming to build intelligent agents that can learn to perform a series of tasks sequentially, with only limited access to past experiences. A general online continual learning problem can be formulated as a problem of learning over a sequence of samples: ${\mathcal{S}} = \{{\bm{s}}_1, \ldots, {\bm{s}}_T\}$ where each sample ${\bm{s}}_t = \{{\bm{x}}_t, y_t\}$ consists of an input observation ${\bm{x}}_t$ and its corresponding label $y_t$. In addition, each sample is assumed to be drawn from an underlying task-distribution $P^t$, which can change from time $t$ to another distribution $P^{t+1}$ at the next step, indicating a task switch.
A continual learner must achieve a good trade-off between maintaining the acquired knowledge of previous tasks and facilitating the learning of future tasks, which is also known as the \emph{stability-plasticity} dilemma~\citep{grossberg1982does,grossberg2013adaptive}.
Due to its connections to how humans learn, several neuroscience frameworks have motivated the development of various continual learning algorithms. One popular framework is the complementary learning systems theory for a dual learning system~\citep{mcclelland1995there,kumaran2016learning}.
Continual learning methods inspired from the CLS theory augments the slow, deep networks with the ability to quickly learn on data streams, either via the experience replay mechanism~\citep{lin1992self,riemer2018learning,rolnick2019experience,aljundi2019online,buzzega2020dark} or via explicit modeling of each of the fast and slow learning components~\citep{pham2021dualnet,anonymous2022learning}. Such methods have demonstrated promising results on more controlled benchmarks constructed from image data.
Our study further extends this research direction to the real-world scenario of online time series forecasting. Our FSNet designs a specific fast learning module comprising an adapter and an associative memory to enhance the deep neural networks with the ability to forecast on the fly.
\begin{figure*}[!h]
\centering
\includegraphics[width=0.95\textwidth]{figs/framework.pdf}
\caption{An overview of FSNet where its interactions with the backbone are denoted as blue arrows. Best view in colors.}
\label{fig:framework}
\vspace*{-0.1in}
\end{figure*}
\section{Proposed Framework}
This section formulates the online time series forecasting as a task-free online continual learning problem and details the proposed FSNet framework.
\subsection{Online Time Series Forecasting as a Continual Learning Problem}\label{sec:ts-cl}
Our formulation is motivated by the locally stationary stochastic processes observation, where a time series can be splitted into a sequence of stationary segments~\citep{vogt2012nonparametric,dahlhaus2012locally,das2016measuring}. Since the same underlying process generates samples from a stationary segment, we refer to forecasting each stationary segment as a learning task for continual learning.
We note that this formulation is general and encompasses existing learning paradigms. For example, splitting into only one segment indicates no concept drifts, and learning reduces to online learning in stationary environments. Online continual learning~\citep{aljundi2019online} corresponds to the case of there are at least two segments. Moreover, we also do not assume that the time points of task switch are given to the model, which is a common setting in many continual learning studies~\citep{kirkpatrick2017overcoming,lopez2017gradient}. However, obtaining such information in real-world time series can be expensive because of the missing or irregularly sampled data~\citep{li2020learning,farnoosh2021deep}.
Therefore, while we assume that the data comprises several tasks, the task-changing points are not provided by the environment, which corresponds to the online, task-free continual learning formulation~\citep{aljundi2019online,aljundi2019task}.
We now highlight a key difference in our formulation compared to the existing task-free continual learning studies~\citep{aljundi2019online,aljundi2019task}, which mainly focus on image domains. Such benchmarks are commonly constructed from existing image datasets such as ImageNet~\citep{deng2009imagenet} or CIFAR~\citep{krizhevsky2009learning}, allowing for full control over the task boundaries from the researchers' perspective~\citep{aljundi2019task}. As a result, each task has a separate test set to evaluate the model's generalization. In contrast, time series evolves and old patterns may not reappear exactly in the future. Therefore, we are not interested in predicting old patterns precisely but predicting how they will evolve. \textit{For example, we do not need to predict the electricity consumption over the last winter. But it is more important to predict the electricity consumption this winter, assuming that it is likely to have the same patterns as the last one.} In summary, online time series forecasting does not require a separate test set, and the model is evaluated by the error accumulated throughout learning. Such a formulation is more realistic and is gaining interest in recent continual learning studies~\citep{hu2020drinking,cai2021online}.
\subsection{Fast and Slow Learning Networks (FSNet)}
By reconciling online time series forecasting with continual learning, we motivate from the CLS theory to develop a fast and slow learning framework for online time series forecasting. Particularly, we propose FSNet to jointly address the challenges of quickly adapting to abrupt changes and facilitating learning of recurrent patterns. While the former challenge arises from the time series's nature, tackling the latter one requires the model to selectively retain and accumulate past knowledge over a long time horizon (beyond the current look-back window) without catastrophic forgetting~\citep{mccloskey1989catastrophic,ratcliff1990connectionist}.
Naively training a neural network online converges \emph{slowly} and cannot handle the aforementioned challenges~\citep{sahoo2018online,aljundi2019online}.
FSNet addresses the fast adaptation to abrupt changes by modeling the partial derivatives (Section~\ref{sec:fast-adapter}) and facilitates learning of recurring patterns via a sparse memory interaction (Section~\ref{sec:ctrl-mem}).
Both mechanisms are implemented in FSNet via two complementary components of the adapter and associative memory.
We consider Temporal Convolutional Network (TCN)~\citep{bai2018empirical} as the backbone deep neural network (that learns slowly online) to extract a time-series feature representation due to the simple forward architecture and promising results~\citep{yue2021ts2vec}. The backbone has $L$ layer with parameters $\bm \theta = \{ \bm \theta_l\}_{l=1}^L$.
FSNet improves the TCN backbone with \emph{two} complementary components: a per-layer adapter $\bm \phi_l$ and a per-layer associate memory $\mathcal M_l$.
The total \emph{trainable} parameters is $\bm \omega = \{ \bm \theta_l, \bm \phi_l \}_{l=1}^L$ and the total associate memory is $\mathcal M = \{ \mathcal M_l\}_{l=1}^L$. Figure~\ref{fig:framework} provides an illustration of FSNet.
\vspace*{-0.1in}
\subsubsection{Fast-adaptation Mechanism}\label{sec:fast-adapter}
Recent works have demonstrated a \emph{shallow-to-deep} principle in that shallower networks can quickly adapt to the changes in data streams or learn more efficiently with limited data~\citep{sahoo2018online,phuong2019distillation}. Therefore, it is more beneficial to learn in such scenarios with a shallow network first and then gradually increase its depth via a multi-exit architecture~\citep{phuong2019distillation}.
Our fast adaptation mechanism generalizes such approaches by allowing each layer to adapt independently rather than restricting to the network's depth. In the following, we omit the superscript indicating time $t$ for brevity.
The key observation allowing for a faster adaptation is that \emph{the partial derivative $\nabla_{\bm \theta_l}\ell$ characterizes the contribution of layer $\bm \theta_l$ to the forecasting loss $\ell$}. Since shallow networks can learn more efficiently~\citep{phuong2019distillation}, we propose to monitor and modify each layer independently to learn the current loss better.
As a result, we implement an adapter to map the layer's recent gradients to a set of smaller, more compact transformation parameters to adapt the backbone.
In online training, because of the noise and non-stationary of time series data, a gradient of a single sample can highly fluctuate~\citep{bottou1998online} and introduces noises to the adaptation parameters. Therefore, we use the exponential moving average (EMA) of the backbone's gradients to smooth out online training's noises as:
\begin{equation}\label{eqn:ema}
\hat{{\bm{g}}}_{l} \gets \gamma \hat{{\bm{g}}}_{l} + (1-\gamma) {\bm{g}}_l^t,
\end{equation}
where ${\bm{g}}_l^t$ denotes the gradient of the $l$-th layer at time $t$ and $\hat{{\bm{g}}}_{l}$ denotes the EMA gradient.
The adapter takes $\hat{{\bm{g}}}_{l}$ as input maps it to the adaptation coefficients ${\bm{u}}_l$.
In this work, we adopt the element-wise transformation~\citep{dumoulin2018feature-wise} as the adaptation process thanks to its simplicity and promising results in continual learning~\citep{pham2021contextual,yin2021mitigating}. Particularly, the adaptation parameter ${\bm{u}}_l$ consists of two components: (i) a weight adaptation parameter $\bm \alpha_l$; and (ii) a feature adaptation parameter $\bm \beta_l$, concatenated together as ${\bm{u}}_l = [\bm \alpha_l ; \bm \beta_l]$. We absorb the bias transformation parameter into $\alpha_l$ for brevity.
The adaptation process for a layer $\bm \theta_l$ involves two steps: a weight and a feature adaptation.
First, the weight adaptation parameter $\bm \alpha_l$ acts on the corresponding weight of the backbone network via an element-wise multiplication as
\begin{align}\label{eqn:adapter}
\tilde{\bm \theta_l} =& \mathrm{tile}(\bm \alpha_l) \odot \bm \theta_l, \quad \bm \alpha_l \in \mathbb{R}^{C},\; \bm \theta_l \in \mathbb{R}^{I\times C \times L}.
\end{align}
In Equation~\ref{eqn:adapter}, $\bm \theta$ is a stack of $I$ features maps of $C$ channels and length $L$, $\bm \theta_l$ denotes the adapted weight, tile($\bm \alpha_l$) denotes that the weight adaptor is applied per-channel on all filters via a tile function, and $\odot$ denotes the element-wise multiplication. Similarly, the feature adaptation $\beta_l$ also interacts with the output feature map $\bm h_l$ as
\begin{align}
\tilde{\bm h_l} =& \mathrm{tile}(\bm \beta_l) \odot \bm h_l, \quad \bm \beta_l \in \mathbb{R}^{C},\; \bm h_l \in \mathbb{R}^{I\times C \times L}.
\end{align}
A naive adapter implementation directly maps the model's gradient to the per-element adaptation parameter and results in a very high dimensional mapping. Therefore, we implement the \emph{chunking} operation~\citep{ha2016hypernetworks} to split the gradient into equal size chunks and then maps each chunk to an element of the adaptation parameters. We denote this chunking operator as $\Omega(\cdot; \bm \phi_l)$ and provide the detailed description in Appendix~\ref{app:FSNet}.
Moreover, adaptation is applied \emph{per-channel}, which greatly reduces the memory overhead, offers compression and generalization~\citep{ha2016hypernetworks,von2019continual}.
In summary, let $\circledast$ denotes the convolution operation, at step $t$, the FSNet's fast adaptation procedure for the $l$-th layer is summarized as
\begin{align}\label{eqn:adaptation}
[\bm \alpha_l, \bm \beta_l] =& {\bm{u}}_l,\; \mathrm{where}\; {\bm{u}}_l = \bm \Omega(\hat{{\bm{g}}}_l; \bm \phi_l) \\
\tilde{\bm \theta_l} =& \mathrm{tile}(\bm \alpha_l) \odot \bm \theta_l \notag \\
\tilde{\bm h_l} =& \mathrm{tile}(\bm \beta_l) \odot \bm h_l ,\; \mathrm{where}\; \bm h_l = \tilde{\bm \theta_l} \circledast \tilde{h}_{l-1}. \notag
\end{align}
It is interesting to note that FSNet's adaptation is a general case of the NCCL design~\citep{yin2021mitigating}, which directly trains the adaptation parameter. However, such an approach lacks the ability to facilitate fast adaptation via monitoring the gradient. In contrast, FSNet offers fast adaptation with compression and generalization by using an adapter to map the gradient's EMA to the adaptation parameters. We will empirically compare in Section~\ref{sec:exp:ablation}.
\subsubsection{Remembering Recurring Events with an Associative Memory}\label{sec:ctrl-mem}
In time series, old patterns may reappear in the future, and it is beneficial to recall similar knowledge in the past to facilitate learning further. While storing the \emph{original data} can alleviate this problem, it might not be applicable in many domains due to privacy concerns. Therefore, as the second key element in FSNet, we implement an associative memory to store the adaptation coefficients of repeating events encountered during learning. While the adapter can handle fast recent changes over a short time scale, recurrent patterns are stored in the memory and then retrieved when they reappear in the future.
For this purpose, we equip each adapter with an associate memory $\mathcal M_l \in \mathbb{R}^{N \times d}$ where $d$ denotes the dimensionality of ${\bm{u}}_l$, and $N$ denotes the number of elements.
The associate memory only \emph{sparsely} interacts with the adapter to store, retrieve, and update such important events, as we discuss below.
\begin{table*}[th!]
\centering
\setlength\tabcolsep{3pt}
\caption{Final cumulative MSE and MAE of algorithms at the end of learning. Best results are in bold.}
\label{tab:main}
\begin{tabular}{lccccccccccccc}
\toprule
\multicolumn{2}{c}{Method} & \multicolumn{2}{c}{FSNet} & \multicolumn{2}{c}{DER++} & \multicolumn{2}{c}{ER} & \multicolumn{2}{c}{OGD} & \multicolumn{2}{c}{OGD (L)} & \multicolumn{2}{c}{TFCL} \\ \midrule
Data & H & MSE & MAE & MSE & MAE & MSE & MAE & MSE & MAE & MSE & MAE & MSE & MAE \\ \midrule
\multirow{3}{*}{ETTh2} & 1 & {\bf 0.466} & {\bf 0.368} & 0.508 & 0.375 & 0.508 & 0.376 & 0.527 & 0.384 & 0.486 & 0.410 & 0.557 & 0.472 \\
& 24 & {\bf 0.687} & {\bf 0.467} & 0.804 & 0.534 & 0.818 & 0.541 & 0.996 & 0.587 & 0.827 & 0.572 & 0.866 & 0.563 \\
& 48 & {\bf 0.846} & {\bf 0.515} & 1.149 & 0.576 & 1.152 & 0.578 & 1.202 & 0.604 & 1.083 & 0.571 & 0.904 & 0.583 \\ \midrule
\multirow{3}{*}{ETTm1} & 1 & 0.085 & {\bf 0.191} & {\bf 0.083} & 0.192 & 0.086 & 0.197 & 0.105 & 0.229 & 0.111 & 0.237 & 0.110 & 0.237 \\
& 24 & {\bf 0.115} & {\bf 0.249} & 0.196 & 0.326 & 0.211 & 0.341 & 0.283 & 0.400 & 0.254 & 0.378 & 0.195 & 0.326 \\
& 48 & {\bf 0.127} & {\bf 0.263} & 0.216 & 0.346 & 0.229 & 0.358 & 0.283 & 0.401 & 0.263 & 0.389 & 0.198 & 0.331 \\ \midrule
\multirow{3}{*}{ECL} & 1 & {3.143} & {0.472} & {\bf 2.718} & {\bf 0.424} & 2.790 & 0.458 & 2.945 & 0.487 & 2.858 & 0.499 & 3.740 & 0.677 \\
& 24 & {\bf 10.467} & {\bf 1.022} & 11.167 & 1.077 & 11.765 & 1.127 & 13.217 & 1.117 & 12.749 & 1.134 & 14.999 & 1.611 \\
& 48 & {\bf 11.996} & {\bf 0.972} & 12.199 & 1.124 & 12.616 & 1.165 & 14.133 & 1.182 & 14.007 & 1.186 & 15.602 & 1.635 \\ \midrule
\multirow{2}{*}{Traffic} & 1 & {\bf 0.321} & {\bf 0.260} & 0.412 & 0.295 & 0.412 & 0.298 & 0.422 & 0.306 & 0.419 & 0.306 & 0.332 & {0.268} \\
& 24 & {\bf 0.421} & {\bf 0.312} & 0.431 & 0.304 & 0.430 & 0.306 & 0.442 & 0.316 & 0.435 & 0.314 & 0.532 & {0.360} \\ \midrule
\multirow{3}{*}{WTH} & 1 & {\bf 0.162} & {\bf 0.216} & 0.171 & 0.230 & 0.178 & 0.239 & 0.199 & 0.266 & 0.206 & 0.278 & 0.221 & 0.293 \\
& 24 & {\bf 0.188} & {\bf 0.274} & 0.287 & 0.350 & 0.295 & 0.358 & 0.316 & 0.373 & 0.308 & 0.367 & 0.257 & 0.339 \\
& 48 & {\bf 0.223} & {\bf 0.301} & 0.305 & 0.365 & 0.308 & 0.368 & 0.317 & 0.373 & 0.301 & 0.364 & 0.268 & 0.349 \\ \midrule
\multirow{2}{*}{S-Abrupt} & 1 & {\bf 1.391} & {\bf 0.929} & 2.291 & 1.169 & 2.440 & 1.202 & 2.638 & 1.213 & 2.767 & 1.238 & 2.359 & 1.156 \\
& 24 & {\bf 1.299} & {\bf 0.904} & 3.362 & 1.358 & 3.826 & 1.478 & 3.904 & 1.491 & 3.427 & 1.370 & 3.414 & 1.366 \\ \midrule
\multirow{2}{*}{S-Gradual} & 1 & {\bf 1.760} & {\bf 1.038} & 2.291 & 1.169 & 2.440 & 1.202 & 2.849 & 1.286 & 2.811 & 1.284 & 2.697 & 1.263 \\
& 24 & {\bf 1.299} & {\bf 0.904} & 3.771 & 1.469 & 3.826 & 1.478 & 3.904 & 1.491 & 3.897 & 1.490 & 3.827 & 1.478 \\ \bottomrule
\end{tabular}
\vspace*{-0.1in}
\end{table*}
\textbf{Sparse Adapter-Memory Interactions }
Interacting with the memory at every step is expensive and susceptible to noises. Therefore, we propose to trigger this interaction only when a substantial change in the representation is detected.
Interference between the current and past representations can be characterized in terms of a dot product between the gradients~\citep{lopez2017gradient,riemer2018learning}.
Therefore, we propose to monitor the cosine similarity between the recent and longer term gradients and trigger the memory interaction when their interference fails below a threshold, which could indicate the pattern has changed significantly.
To this end, in addition to the gradient EMA in Equation~\ref{eqn:adaptation}, we deploy a second gradient EMA $\hat{{\bm{g}}}'_{l}$ with a smaller coefficient $\gamma' < \gamma$ and measure their cosine similarity to trigger the memory interaction as:
\begin{equation} \label{eqn:trigger}
\mathrm{Trigger\;if:}\; \mathrm{cos}(\hat{{\bm{g}}}_l, \hat{{\bm{g}}}'_l) = \frac{\hat{{\bm{g}}}_l \cdot \hat{{\bm{g}}}_l'}{||\hat{{\bm{g}}}_l||\, ||\hat{{\bm{g}}}_l||} < -\tau,
\end{equation}
where $\tau>0$ is a hyper-parameter determining the significant degree of interference.
Moreover, we want to set $\tau$ to a relatively high value (e.g. 0.7) so that the memory only remembers significant changing patterns, which could be important and may reappear. We will discuss the hyper-parameter settings in Appendix~\ref{app:hyper}.
\textbf{The Adapter-Memory Interacting Mechanism }
Since the current adaptation parameter may not capture the whole event, which could span over a few samples, we
perform the memory read and write operations using the adaptation parameter's EMA (with coefficient $\gamma'$) to fully capture the current pattern. The EMA of ${\bm{u}}_l$ is calculated in the same manner as Equation~\ref{eqn:ema}.
When a memory interaction is triggered, the adapter queries and retrieves the most similar transformations in the past via an attention read operation, which is a weighted sum over the memory items:
\begin{enumerate}
\item Attention calculation: ${\bm{r}}_l = \mathrm{softmax}(\mathcal M_l \hat{{\bm{u}}}_l)$;
\item Top-k selection: ${\bm{r}}^{(k)}_l = \mathrm{TopK}({\bm{r}}_l)$;
\item Retrieval: $\tilde{{\bm{u}}_l} = \sum_{i=1}^K {\bm{r}}^{(k)}_l[i]\mathcal M_l[i]$,
\end{enumerate}
where ${\bm{r}}^{(k)}[i]$ denotes the $i$-th element of ${\bm{r}}^{(k)}_l$ and $\mathcal M_l[i]$ denotes the $i$-th row of $\mathcal M_l$.
Since the memory could store conflicting patterns, we employ a sparse attention by retrieving the top-k most relevant memory items, which we fix as $k=2$. The retrieved adaptation parameter characterizes old experiences in adapting to the current pattern in the past and can improve learning at the present time by weighted summing with the current parameters as
\begin{equation}
{\bm{u}}_l \gets \tau {\bm{u}}_l + (1-\tau) \tilde{{\bm{u}}}_t,
\end{equation}
where we use the same threshold value $\tau$ to determine the sparse memory interaction and the weighted sum of the adaptation parameter. Then we perform a write operation to update and accumulate the knowledge stored in $\mathcal M_l$:
\begin{align}\label{eqn:mem-update}
\mathcal M_l \gets& \tau \mathcal M_l + (1-\tau) \hat{{\bm{u}}}_l \otimes {\bm{r}}^{(k)}_l, \notag \\
\mathcal M_l \gets& \frac{\mathcal M_l}{\max(1, ||\mathcal M_l||_2)},
\end{align}
where $\otimes$ denotes the outer-product operator, which allows us to efficiently write the new knowledge to the most relevant locations indicated by ${\bm{r}}^{(k)}_l$~\citep{rae2016scaling,kaiser2017learning}.
The memory is then normalized to avoid its values scaling exponentially.
Lastly, we note that FSNet is suitable for the task-free, online continual learning scenario because we do not need to detect when tasks switch explicitly. Instead, we relax the task boundaries definition and allow the model to improve its learning on current samples continuously.
\section{Experiments}
Our experiments aim at investigating the following hypotheses: (i) FSNet facilitates faster adaptation to both new and recurring concepts compared to existing strategies; (ii) FSNet achieves faster and better convergence than other methods; and (iii) modeling the partial derivative is the key ingredients for fast adaptation. Due to space constraints, we provide the key information of the experimental setting in the main paper and discuss the full details in the Appendix.
\subsection{Experimental Settings}\label{sec:exp-settings}
\begin{figure*}
\captionsetup[subfigure]{justification=Centering}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[height=0.35in]{figs/legend.pdf}
\end{subfigure}
\subcaptionbox{ETTh2\label{fig3:etth2}}{\includegraphics[width=2.2in]{figs/ETTh2.pdf}}\hspace{0.1em}
\subcaptionbox{ETTm1\label{fig3:etth2}}{\includegraphics[width=2.2in]{figs/ETTm1.pdf}}\hspace{0.1em}
\subcaptionbox{ECL\label{fig3:ecl}}{\includegraphics[width=2.2in]{figs/ECL.pdf}}
\subcaptionbox{Traffic\label{fig3:traffic}}{\includegraphics[width=2.2in]{figs/Traffic.pdf}}\hspace{0.1em}
\subcaptionbox{WTH\label{fig3:traffic}}{\includegraphics[width=2.2in]{figs/WTH.pdf}}\hspace{0.1em}
\subcaptionbox{S-Abrupt\label{fig3:traffic}}{\includegraphics[width=2.2in]{figs/Toy.pdf}}
\caption{Evolution of the cumulative loss of different methods during training with forecasting horizon $H=24$.}
\label{fig:curve}
\vspace*{-0.1in}
\end{figure*}
\textbf{Datasets } We explore a wide range of time series forecasting datasets. {\bf ETT}\footnote{\url{https://github.com/zhouhaoyi/ETDataset}}~\citep{zhou2021informer} records the target value of ``oil temperature" and 6 power load features over a period of two years. We consider the ETTh2 and ETTm1 benchmarks where the observations are recorded hourly and in 15-minutes intervals respectively.
{\bf ECL} (Electricty Consuming Load)\footnote{\url{https://archive.ics.uci.edu/ml/datasets/ElectricityLoadDiagrams20112014}} collects the electricity consumption of 321 clients from 2012 to 2014.
{\bf Traffic}\footnote{\url{https://pems.dot.ca.gov/}} records the road occupancy rates at San Francisco Bay area freeways.
{\bf Weather}\footnote{\url{https://www.ncei.noaa.gov/data/local-climatological-data/}} records 11 climate features from nearly 1,600 locations in the U.S in an hour intervals from 2010 to 2013.
We also construct two synthetic datasets to explicitly test the model's ability to deal with new and recurring concept drifts. We synthesize a task by sampling $1,000$ samples from a first-order autoregressive process with coefficient $\varphi$ AR$_{\varphi}$(1), where different tasks correspond to different $\varphi$ values.
The first synthetic data, {\bf S-Abrupt} contains abrupt, and recurrent concepts where the samples abruptly switch from one AR process to another by the following order: AR$_{0.1}$(1), AR$_{0.4}$(1), AR$_{0.6}$(1), AR$_{0.1}$(1), AR$_{0.3}$(1), AR$_{0.6}$(1). The second data, {\bf S-Gradual} contains gradual, incremental shifts, where the shift starts at the last 20\% of each task. In this scenario, the last 20\% samples of a task is an averaged from two AR process with the order as above. Detailed information is provided in Appendix~\ref{app:synthetic}.
\textbf{Implementation Details } We split the data into warm-up and online training phases by the ratio of 25:75 and consider the TCN backbone~\citep{yue2021ts2vec} for experiments. In the online training phase, all methods are trained by the L2 loss and AdamW optimizer~\citep{loshchilov2017decoupled}. Both the epoch and batch size are set to {\bf one}, and the memory for ER and DER++ is implemented as a reservoir sampling buffer~\citep{vitter1985random} with a total of 500 samples (around 2\%-5\% of data).
In the validation phase, we calculate the mean and standard deviation to normalize online training samples and perform hyper-parameter cross-validation. For all benchmarks, we set the look-back window length to $w=60$ and the forecast horizon of $H=1$. We also explore the model's ability to forecast longer horizons by varying $H\in\{24,48\}$. We cross-validate and discuss the the hyper-parameter setting in Appendix~\ref{app:hyper}.
\textbf{Baselines } We consider four online training under non-stationary environment methods for comparison. First, the Online Gradient Descent \emph({OGD})~\citep{zinkevich2003online} strategy that simply trains continuously. We also include a \emph{OGD (L)}, a large variant of OGD with \emph{twice} the TCN's filters per layer, resulting in a roughly twice number of parameters\footnote{Our FSNet only increases the model size by a factor of 1.5.}. Then, we implement the Experiment Replay~\citep{lin1992self,chaudhry2019tiny} strategy where a buffer is employed to store previous data and interleave old samples during the learning of newer ones. The next competitor is DER++~\citep{buzzega2020dark} which further adds a knowledge distillation~\citep{hinton2015distilling} loss to ER. Finally, we compare TFCL~\citep{aljundi2019task}, which augments the standard ER with a task-free variant of MAS~\citep{aljundi2018memory} to only update important parameters to past patterns.
We emphasize that ER, DER++, and TFCL are strong baselines in the online setting since they enjoy the benefits of training on mini-batches, which greatly reduce noises from singe samples and offers faster, better convergence~\citep{bottou1998online}.
\subsection{Online Forecasting Results}
\textbf{Cumulative Performance } Table~\ref{tab:main} reports cumulative mean-squared errors (MSE) and mean-absolute errors (MAE) at the end of training. We observe that ER and DER++ are strong competitors and can significantly improve over the OGD strategies. However, such methods still cannot work well under multiple task switches (S-Abrupt). Moreover, no clear task boundaries (S-Gradual) presents an even more challenging problem and increases most models' errors. On the other hand, our FSNet shows promising results on all datasets and outperforms most competing baselines across different forecasting horizons. Moreover, the improvements are significant on the synthetic benchmarks, indicating that LSFNet can quickly adapt to the non-stationary environment and recall previous knowledge, even without clear task boundaries.
\begin{figure*}[t]
\captionsetup[subfigure]{justification=Centering}
\begin{subfigure}{\textwidth}
\centering
\includegraphics[height=0.35in]{figs/legend1.pdf}
\end{subfigure}
\vspace*{-0.1in}
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=3in]{figs/900.pdf}
\vspace*{-0.05in}
\caption{Prediction from time $t=900$}
\end{subfigure}
~
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=3in]{figs/5900.pdf}
\vspace*{-0.05in}
\caption{Prediction from time $t=5900$}
\end{subfigure}
\vspace*{-0.05in}
\caption{Visualizations of models' prediction at the early and final stages of online learning process.}
\label{fig:quant-vis}
\vspace*{-0.1in}
\end{figure*}
\textbf{Convergent behaviors of Different Learning Strategies}
Figure~\ref{fig:curve} reports the convergent behaviors on the considered methods. We omit the S-Gradual dataset for spaces because we observe the same behavior as S-Abrupt.
The results show the benefits of ER by offering faster convergence during learning compared to OGD. However, it is important to note that storing the original data may not apply in many domains. On S-Abrupt, most baselines demonstrate the inability to quickly recover from concept drifts, indicated by the increasing error curves.
We also observe promising results of FSNet on most datasets, with significant improvements over the baselines on the ETT, WTH, and S-Abrupt datasets. The ECL dataset is more challenging with missing values~\citep{li2019enhancing} and large magnitude varying \emph{within and across} dimensions, which may require calculating a better data normalization. While FSNet achieved encouraging results on ECL, handling the above challenges can further improve its performance.
Overall, the results shed light on the challenges of online time series forecasting and demonstrate promising results of FSNet.
\textbf{Visualization }
We explore the model's prediction quality on the S-Abrupt since it is a univariate time series. The remaining real-world datasets are multivariate are challenging to visualize.
Particularly, we plot the model's forecasting at two-time points: at $t=900$ and the end of learning, $t=5900$ in Figure~\ref{fig:quant-vis}. We can see the difficulties of training deep neural networks online in that the model struggles to learn at the early stages, where it only observed a few samples. With the limited samples per task and the presence of multiple concept drifts, the standard online optimization collapsed to a naive solution of predicting random noises around zero. However, FSNet can successfully capture the time series' patterns and provide better predictions.
\begin{table}[t]
\centering
\setlength\tabcolsep{3pt}
\caption{Final comulative MSE and MAE of different FSNet variants. Best results are in bold.}
\label{tab:ablation}
\begin{tabular}{lccccccc}
\toprule
\multirow{2}{*}{Variant} & & \multicolumn{6}{c}{FSNet} \\ \cmidrule{3-8}
& & \multicolumn{2}{c}{Full} & \multicolumn{2}{c}{No Memory} & \multicolumn{2}{c}{Naive}\\
\midrule
Data & H & MSE & MAE & MSE & MAE & MSE & MAE \\ \midrule
\multirow{3}{*}{ETTh2} & 1 & \textbf{0.466} & \textbf{0.368} & 0.539 & 0.385 & 0.534 & 0.418 \\
& 24 & \textbf{0.687} & \textbf{0.467} & 0.689 & 0.468 & 0.860 & 0.555 \\
& 48 & \textbf{0.846} & \textbf{0.515} & {0.924} & {0.526} & {0.973} & {0.570} \\ \midrule
\multirow{2}{*}{Traffic} & 1 & \textbf{0.321} & \textbf{0.260} & {0.325} & {0.263} & 0.337 & 0.275 \\
& 24 & \textbf{0.421} & \textbf{0.312} & {0.425} & {0.315} & {0.480} & {0.349} \\ \midrule
\multicolumn{1}{l}{\multirow{2}{*}{S-Abrupt}} & 1 & \textbf{1.391} & \textbf{0.929} & {1.734} & {1.024} & 3.318 & 1.413 \\
\multicolumn{1}{l}{} & 24 & \textbf{1.299} & \textbf{0.904} & 1.390 & 0.933 & 3.727 & 1.467 \\ \midrule
\multicolumn{1}{l}{\multirow{2}{*}{S-Gradual}} & 1 & {1.760} & {1.038} & \textbf{1.734} & \textbf{1.024} & 3.318 & 1.414 \\
\multicolumn{1}{l}{} & 24 & \textbf{1.299} & \textbf{0.904} & 1.415 & 0.940 & 3.748 & 1.478 \\ \bottomrule
\end{tabular}
\vspace*{-0.2in}
\end{table}
\vspace*{-0.1in}
\subsection{Ablation Studies of FSNet's Design}\label{sec:exp:ablation}
We now analyze the contribution of each FSNet component. First, we explore the benefits of using the associate memory in Section~\ref{sec:ctrl-mem} by constructing a \emph{No Memory} variant of FSNet that only uses an adapter and does not interact with the memory. Second, we further remove the adapter, which results in the \emph{Naive} variant that deploys the adaptation coefficients $\bm \alpha_t, \bm \beta_t$ as trainable parameters with the backbone, which also corresponds to the NCCL design~\citep{yin2021mitigating}. The Naive variant demonstrates the benefits of monitoring the layer's gradients, which is our key idea for fast adaptation (Section~\ref{sec:fast-adapter}). Note that we cannot directly compare with NCCL because its implementation is not available and it requires the task identifier, storing the original data, which are not available in our problem.
We report the results in Table~\ref{tab:ablation}.
We first observe that FSNet achieves almost the same results with the No Memory variant on the Traffic and S-Gradual datasets. One possible reason is the insignificant representation interference in the Traffic dataset and the slowly changing representations in the S-Gradual dataset. In such cases, the representation changes can be easily captured by the adapter alone.
On the other hand, ETTh2 and S-Abrupt may have challenging and sudden drifts, we observe the associative memory's benefits and more significant improvements compared to not using a memory.
Second, the Naive variant does not achieve satisfactory results, indicating the benefits of modeling the recent gradients for adaptation. It is also worth noting that FSNet without the memory can still out performs many baselines reported in Table~\ref{tab:main}.
Overall, these results demonstrated the complementary of each FSNet's components to deal with different type of concept drifts in time series.
\subsection{Memory Comparison}
We now explore each method's performance under a fixed memory budget. For FSNet, the memory is allocated to store the transformation parameters of previous patterns. We consider ER and DER++ as two replay-based baselines which use the memory to store the original data.
In this experiment, we consider the Traffic and ETTh2 datasets and gradually increase the memory budget in Megabyte (Mb), which correspond to the number of items in the memory of each method. Table~\ref{tab:mem} reports the result of this experiment. First, we observe that all methods performs better as the budget increases, which allows storing more items. Second, we observe that the three memory thresholds are $4.52$Mb, $2.26$Mb, and $1.13$Mb correspond to storing 1280, 640, and 320 items in FSNet, which is equivalent to storing 128, 64, and 32 items per-layer. On the other hand, storing the original data is more expensive in both cases, especially in the Traffic dataset where the time series is high dimensional and has many covariates. Overall, FSNet shows promising scalability with the memory size by consistently outperforms both ER and DER++ in all cases.
\begin{table}[t]
\setlength\tabcolsep{3pt}
\caption{Final cummulative MSE and MAE of different methods as the memory size increases. Best results are in bold.}
\label{tab:mem}
\begin{tabular}{lccccccc}
\toprule
\multirow{2}{*}{Method} & \multicolumn{3}{c}{Traffic} & \multicolumn{3}{c}{ETTh2} & \multirow{2}{*}{Mb} \\ \cmidrule{2-7}
& $|\mathcal M|$ & MSE & MAE & $|\mathcal M|$ & MSE & MAE & \\ \midrule
FSNet & 1280 & {\bf 0.413} & {\bf 0.306} & 1280 & {\bf 0.616} & {\bf 0.456} & \multirow{3}{*}{4.52} \\
ER & 14 & {0.440} & {0.313} & 840 & {0.806} & {0.537} & \\
DER++ & 13 & 0.439 & 0.310 & 720 & 0.776 & 0.532 & \\ \midrule
FSNet & 640 & {\bf 0.417} & {\bf 0.309} & 640 & {\bf 0.711} & {\bf 0.469} & \multirow{3}{*}{2.26} \\
ER & 7 & 0.443 & 0.316 & 560 & 0.821 & 0.544 & \\
DER++ & 6 & 0.440 & 0.313 & 520 & 0.803 & 0.537 & \\ \midrule
FSNet & 320 & {\bf 0.421} & {\bf 0.312} & 320 & {\bf 0.687} & {\bf 0.467} & \multirow{3}{*}{1.13} \\
ER & 4 & 0.445 & 0.315 & 280 & 0.814 & 0.538 & \\
DER++ & 3 & 0.441 & 0.312 & 240 & 0.767 & 0.53 & \\ \bottomrule
\end{tabular}
\end{table}
\section{Conclusion}
We have investigated the limitations of training deep neural networks for online time series forecasting in non-stationary environments, where they lack the capability to adapt to new or recurring patterns quickly. We then propose Fast and Slow learning Networks (FSNet) by extending the CLS theory for continual learning to online time series forecasting. FSNet augments a neural network backbone with two key components: (i) an adapter for adapting to the recent changes; and (ii) an associative memory to handle recurrent patterns. Moreover, the adapter sparsely interacts with its memory to store, update, and retrieve important recurring patterns to facilitate learning of such events in the future. Extensive experiments demonstrate the FSNet's capability to deal with various types of concept drifts to achieve promising results in both real-world and synthetic time-series data.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,496 |
Q: Controlar velocidade de animação utilizando o componente JSlider Bom galera, estou desenvolvendo um projetinho que visa simular o ambiente controlado de trilhos de trem, onde tenho 3 trens circulando em sentido horário, onde os três passam pelo mesmo local em determinados trechos. Estou com dificuldade em implementar a velocidade dos trens dinamicamente utilizando o componente JSlider. Consegui criar os 3 componentes, porém ainda não consegui associá-los aos meus trens.
Segue o código:
public class Trens {
private static final double QUADROS_POR_SEGUNDO = 20.0;
public static void preparar() {
JFrame t = new JFrame();
t.setLayout(null);
t.setSize(1200, 900);
t.setTitle("Semáforo");
t.setLocationRelativeTo(null);
t.setResizable(false);
t.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
Trilho t1 = new Trilho(300, 100, 300, 170);
Trilho t2 = new Trilho(600, 100, 300, 170);
Trilho t3 = new Trilho(450, 270, 300, 170);
Jslider slider = new Jslider(30, 150, 300, 170);
Jslider slider2 = new Jslider(900, 150, 300, 170);
Jslider slider3 = new Jslider(150, 330, 300, 170);
Trem a = new Trem(t1, Color.BLUE, 350, 100, 100.0);
Trem b = new Trem(t2, Color.GREEN, 650, 100, 100.0);
Trem c = new Trem(t3, Color.RED, 450, 335, 100.0);
t.add(a);
t.add(b);
t.add(c);
t.add(t1);
t.add(t2);
t.add(t3);
t.add(slider);
t.add(slider2);
t.add(slider3);
Runnable moverTudo = () -> {
EventQueue.invokeLater(() -> {
a.mover(1 / QUADROS_POR_SEGUNDO);
b.mover(1 / QUADROS_POR_SEGUNDO);
c.mover(1 / QUADROS_POR_SEGUNDO);
});
};
Executors.newSingleThreadScheduledExecutor().scheduleAtFixedRate(moverTudo, 0, (int) (1000 / QUADROS_POR_SEGUNDO), TimeUnit.MILLISECONDS);
t.setVisible(true);
}
public static void main(String[] args) {
EventQueue.invokeLater(Trens::preparar);
}
public static class Trilho extends JComponent {
public Trilho(int x, int y, int width, int height) {
this.setBounds(x, y, width, height);
this.setBorder(BorderFactory.createLineBorder(Color.BLACK));
}
}
public static class Trem extends JComponent {
private Color cor;
private Trilho trilho;
private int x;
private int y;
private double velocidade; // pixels por segundo
private double restante; // Frações de pixels que faltou andar.
public Trem(Trilho trilho, Color cor, int x, int y, double velocidade) {
this.trilho = trilho;
this.cor = cor;
this.x = x;
this.y = y;
this.velocidade = velocidade;
this.setBounds(x - 5, y - 5, 10, 10);
}
@Override
public void paintComponent(Graphics g) {
g.setColor(cor);
g.fillRect(0, 0, getWidth(), getHeight());
}
public void mover(double deltaT) {
if (velocidade == 0) return;
boolean sentidoHorario = velocidade > 0;
double distancia = Math.abs(restante + velocidade * deltaT);
int tLeft = trilho.getX();
int tTop = trilho.getY();
int tRight = tLeft + trilho.getWidth();
int tBottom = tTop + trilho.getHeight();
for (int i = 0; i < (int) distancia; i++) {
// Se deve ir à esquerda:
if (x > tLeft && y == (sentidoHorario ? tBottom : tTop)) {
x--;
// Se deve ir à direita:
} else if (x < tRight && y == (sentidoHorario ? tTop : tBottom)) {
x++;
// Se deve ir para cima:
} else if (y > tTop && x == (sentidoHorario ? tLeft : tRight)) {
y--;
// Se deve ir para baixo:
} else if (y < tBottom && x == (sentidoHorario ? tRight : tLeft)) {
y++;
// Se não for nenhum dos anteriores, o trem está descarrilhado. Coloca de novo no trilho.
} else {
x = tLeft;
y = tTop;
}
}
restante = distancia % 1;
setLocation(x - 5, y - 5);
}
}
public static class Jslider extends JPanel{
private Jslider(int x, int y, int width, int height){
this.setBounds(x, y, width, height);
JSlider control = new JSlider(0,100,25);
control.setMajorTickSpacing(50);
control.setMinorTickSpacing(10);
control.setPaintTicks(true);
control.setFont(new Font("Serfi", Font.ITALIC, 12));
control.setPaintLabels(true);
control.setSnapToTicks(true);
add(control);
}
}
public class EventoSlider implements ChangeListener{
@Override
public void stateChanged(ChangeEvent e) {
JSlider source = (JSlider) e.getSource();
int fps = (int) source.getValue();
System.out.println(fps);
}
}
}
A: Consegui, até que foi simples:
package trens;
import java.awt.Color;
import java.awt.EventQueue;
import java.awt.Font;
import java.awt.Graphics;
import java.util.concurrent.Executors;
import java.util.concurrent.TimeUnit;
import javax.swing.BorderFactory;
import javax.swing.JComponent;
import javax.swing.JFrame;
import javax.swing.JPanel;
import javax.swing.JSlider;
public class Trens {
private static final double QUADROS_POR_SEGUNDO = 20.0;
public static void preparar() {
JFrame t = new JFrame();
t.setLayout(null);
t.setSize(1200, 900);
t.setTitle("Semáforo");
t.setLocationRelativeTo(null);
t.setResizable(false);
t.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
Trilho t1 = new Trilho(300, 100, 300, 170);
Trilho t2 = new Trilho(600, 100, 300, 170);
Trilho t3 = new Trilho(450, 270, 300, 170);
Trem a = new Trem(t1, Color.BLUE, 350, 100, 100.0);
Trem b = new Trem(t2, Color.GREEN, 650, 100, 100.0);
Trem c = new Trem(t3, Color.RED, 450, 335, 100.0);
VelocityControl slider = new VelocityControl(a, 30, 150, 300, 170);
VelocityControl slider2 = new VelocityControl(b, 900, 150, 300, 170);
VelocityControl slider3 = new VelocityControl(c, 150, 330, 300, 170);
t.add(a);
t.add(b);
t.add(c);
t.add(t1);
t.add(t2);
t.add(t3);
t.add(slider);
t.add(slider2);
t.add(slider3);
Runnable moverTudo = () -> {
EventQueue.invokeLater(() -> {
a.mover(1 / QUADROS_POR_SEGUNDO);
b.mover(1 / QUADROS_POR_SEGUNDO);
c.mover(1 / QUADROS_POR_SEGUNDO);
});
};
Executors.newSingleThreadScheduledExecutor().scheduleAtFixedRate(moverTudo, 0, (int) (1000 / QUADROS_POR_SEGUNDO), TimeUnit.MILLISECONDS);
t.setVisible(true);
}
public static void main(String[] args) {
EventQueue.invokeLater(Trens::preparar);
}
public static class Trilho extends JComponent {
public Trilho(int x, int y, int width, int height) {
this.setBounds(x, y, width, height);
this.setBorder(BorderFactory.createLineBorder(Color.BLACK));
}
}
public static class Trem extends JComponent {
private Color cor;
private Trilho trilho;
private int x;
private int y;
private double velocidade; // pixels por segundo
private double restante; // Frações de pixels que faltou andar.
public Trem(Trilho trilho, Color cor, int x, int y, double velocidade) {
this.trilho = trilho;
this.cor = cor;
this.x = x;
this.y = y;
this.velocidade = velocidade;
this.setBounds(x - 5, y - 5, 10, 10);
}
@Override
public void paintComponent(Graphics g) {
g.setColor(cor);
g.fillRect(0, 0, getWidth(), getHeight());
}
public void setVelocidade(double novaVelocidade) {
this.velocidade = novaVelocidade;
}
public void mover(double deltaT) {
if (velocidade == 0) return;
boolean sentidoHorario = velocidade > 0;
double distancia = Math.abs(restante + velocidade * deltaT);
int tLeft = trilho.getX();
int tTop = trilho.getY();
int tRight = tLeft + trilho.getWidth();
int tBottom = tTop + trilho.getHeight();
for (int i = 0; i < (int) distancia; i++) {
// Se deve ir à esquerda:
if (x > tLeft && y == (sentidoHorario ? tBottom : tTop)) {
x--;
// Se deve ir à direita:
} else if (x < tRight && y == (sentidoHorario ? tTop : tBottom)) {
x++;
// Se deve ir para cima:
} else if (y > tTop && x == (sentidoHorario ? tLeft : tRight)) {
y--;
// Se deve ir para baixo:
} else if (y < tBottom && x == (sentidoHorario ? tRight : tLeft)) {
y++;
// Se não for nenhum dos anteriores, o trem está descarrilhado. Coloca de novo no trilho.
} else {
x = tLeft;
y = tTop;
}
}
restante = distancia % 1;
setLocation(x - 5, y - 5);
}
}
public static class VelocityControl extends JPanel {
private VelocityControl(Trem trem, int x, int y, int width, int height) {
this.setBounds(x, y, width, height);
JSlider control = new JSlider(0, 100, 25);
control.setMajorTickSpacing(50);
control.setMinorTickSpacing(10);
control.setPaintTicks(true);
control.setFont(new Font("Serif", Font.ITALIC, 12));
control.setPaintLabels(true);
control.setSnapToTicks(true);
this.add(control);
control.addChangeListener(e -> trem.setVelocidade(control.getValue()));
}
}
}
Mudei o nome da classe Jslider para VelocityControl para não confundir com a JSlider (note que uma tem o S maiúsculo e a outra minúsculo).
O truque para fazer o valor escolhido refletir na velocidade é o control.addChangeListener(e -> trem.setVelocidade(control.getValue())); com a sintaxe do lambda, isso fica bem simples e limpo.
Para ligar os trens aos controles, bastou passar o trem como um parâmetro mais para o construtor do VelocityControl.
Um último detalhe é que você tinha usado "Serfi" ao invés de "Serif" no nome da fonte, um erro de digitação.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,831 |
package brooklyn.util.text;
import java.util.ArrayList;
import java.util.LinkedHashMap;
import java.util.List;
import java.util.Map;
/** utility which takes a bunch of segments and applies shortening rules to them */
public class StringShortener {
protected Map<String,String> wordsByIdInOrder = new LinkedHashMap<String,String>();
protected String separator = null;
protected interface ShorteningRule {
/** returns the new list, with the relevant items in the list replaced */
public int apply(LinkedHashMap<String, String> words, int maxlen, int length);
}
protected class TruncationRule implements ShorteningRule {
public TruncationRule(String id, int len) {
this.id = id;
this.len = len;
}
String id;
int len;
public int apply(LinkedHashMap<String, String> words, int maxlen, int length) {
String v = words.get(id);
if (v!=null && v.length()>len) {
int charsToRemove = v.length() - len;
if (length-charsToRemove < maxlen) charsToRemove = length-maxlen;
words.put(id, v.substring(0, v.length() - charsToRemove));
length -= charsToRemove;
if (charsToRemove==v.length() && separator!=null && length>0)
length -= separator.length();
}
return length;
}
}
protected class RemovalRule implements ShorteningRule {
public RemovalRule(String id) {
this.id = id;
}
String id;
public int apply(LinkedHashMap<String, String> words, int maxlen, int length) {
String v = words.get(id);
if (v!=null) {
words.remove(id);
length -= v.length();
if (separator!=null && length>0)
length -= separator.length();
}
return length;
}
}
private List<ShorteningRule> rules = new ArrayList<StringShortener.ShorteningRule>();
public StringShortener separator(String separator) {
this.separator = separator;
return this;
}
public StringShortener append(String id, String text) {
String old = wordsByIdInOrder.put(id, text);
if (old!=null) {
throw new IllegalStateException("Cannot append with id '"+id+"' when id already present");
}
// TODO expose a replace or update
return this;
}
public StringShortener truncate(String id, int len) {
String v = wordsByIdInOrder.get(id);
if (v!=null && v.length()>len) {
wordsByIdInOrder.put(id, v.substring(0, len));
}
return this;
}
public StringShortener canTruncate(String id, int len) {
rules.add(new TruncationRule(id, len));
return this;
}
public StringShortener canRemove(String id) {
rules.add(new RemovalRule(id));
return this;
}
public String getStringOfMaxLength(int maxlen) {
LinkedHashMap<String, String> words = new LinkedHashMap<String,String>();
words.putAll(wordsByIdInOrder);
int length = 0;
for (String w: words.values()) {
if (!Strings.isBlank(w)) {
length += w.length();
if (separator!=null)
length += separator.length();
}
}
if (separator!=null && length>0)
// remove trailing separator if one had been added
length -= separator.length();
List<ShorteningRule> rulesLeft = new ArrayList<ShorteningRule>();
rulesLeft.addAll(rules);
while (length > maxlen && !rulesLeft.isEmpty()) {
ShorteningRule r = rulesLeft.remove(0);
length = r.apply(words, maxlen, length);
}
StringBuilder sb = new StringBuilder();
for (String w: words.values()) {
if (!Strings.isBlank(w)) {
if (separator!=null && sb.length()>0)
sb.append(separator);
sb.append(w);
}
}
String result = sb.toString();
if (result.length() > maxlen) result = result.substring(0, maxlen);
return result;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 96 |
#include "OgreStableHeaders.h"
#include "Animation/OgreSkeletonTrack.h"
#include "Math/Array/OgreMathlib.h"
#include "Math/Array/OgreBoneTransform.h"
#include "Math/Array/OgreKfTransformArrayMemoryManager.h"
#include "OgreException.h"
namespace Ogre
{
SkeletonTrack::SkeletonTrack( uint32 boneBlockIdx,
KfTransformArrayMemoryManager *kfTransformMemoryManager ) :
mKeyFrameRigs( 0 ),
mNumFrames( 0 ),
mBoneBlockIdx( boneBlockIdx ),
mUsedSlots( 0 ),
mLocalMemoryManager( kfTransformMemoryManager )
{
}
//-----------------------------------------------------------------------------------
SkeletonTrack::~SkeletonTrack()
{
}
//-----------------------------------------------------------------------------------
void SkeletonTrack::setNumKeyFrame( size_t numKeyFrames )
{
mKeyFrameRigs.reserve( numKeyFrames );
}
//-----------------------------------------------------------------------------------
void SkeletonTrack::addKeyFrame( Real timestamp, Real frameRate )
{
assert( mKeyFrameRigs.empty() || timestamp > mKeyFrameRigs.back().mFrame );
mKeyFrameRigs.push_back( KeyFrameRig() );
KeyFrameRig &keyFrame = mKeyFrameRigs.back();
keyFrame.mFrame = timestamp * frameRate;
keyFrame.mInvNextFrameDistance = 1.0f;
if( mKeyFrameRigs.size() > 1 )
{
KeyFrameRig &prevKeyFrame = mKeyFrameRigs[mKeyFrameRigs.size()-2];
prevKeyFrame.mInvNextFrameDistance = 1.0f / (keyFrame.mFrame - prevKeyFrame.mFrame);
}
mLocalMemoryManager->createNewNode( (KfTransform**)(&keyFrame.mBoneTransform) );
}
//-----------------------------------------------------------------------------------
void SkeletonTrack::setKeyFrameTransform( Real frame, uint32 slot, const Vector3 &vPos,
const Quaternion &qRot, const Vector3 vScale )
{
KeyFrameRigVec::iterator itor = mKeyFrameRigs.begin();
KeyFrameRigVec::iterator end = mKeyFrameRigs.end();
while( itor != end && Math::Abs( itor->mFrame - frame ) < 1e-6f )
++itor;
if( itor == mKeyFrameRigs.end() )
{
OGRE_EXCEPT( Exception::ERR_ITEM_NOT_FOUND, "Frame not found.",
"SkeletonTrack::setKeyFrameTransform" );
}
itor->mBoneTransform->mPosition.setFromVector3( vPos, slot );
itor->mBoneTransform->mOrientation.setFromQuaternion( qRot, slot );
itor->mBoneTransform->mScale.setFromVector3( vScale, slot );
mUsedSlots = std::max( slot+1, mUsedSlots );
}
//-----------------------------------------------------------------------------------
inline void SkeletonTrack::getKeyFrameRigAt( KeyFrameRigVec::const_iterator &inOutPrevFrame,
KeyFrameRigVec::const_iterator &outNextFrame,
Real frame ) const
{
KeyFrameRigVec::const_iterator prevFrame = inOutPrevFrame;
KeyFrameRigVec::const_iterator nextFrame = inOutPrevFrame;
if( frame >= nextFrame->mFrame )
{
while( nextFrame != (mKeyFrameRigs.end() - 1) && nextFrame->mFrame <= frame )
prevFrame = nextFrame++;
}
else
{
while( prevFrame != mKeyFrameRigs.begin() && prevFrame->mFrame > frame )
nextFrame = prevFrame--;
}
inOutPrevFrame = prevFrame;
outNextFrame = nextFrame;
}
//-----------------------------------------------------------------------------------
void SkeletonTrack::applyKeyFrameRigAt( KeyFrameRigVec::const_iterator &inOutLastKnownKeyFrameRig,
float frame, ArrayReal animWeight,
const ArrayReal * RESTRICT_ALIAS perBoneWeights,
const TransformArray &boneTransforms ) const
{
KeyFrameRigVec::const_iterator prevFrame = inOutLastKnownKeyFrameRig;
KeyFrameRigVec::const_iterator nextFrame;
getKeyFrameRigAt( prevFrame, nextFrame, frame );
const Real scalarW = (frame - prevFrame->mFrame) * prevFrame->mInvNextFrameDistance;
ArrayReal fTimeW = Mathlib::SetAll( scalarW );
size_t level = mBoneBlockIdx >> 24;
size_t offset = mBoneBlockIdx & 0x00FFFFFF;
ArrayVector3 * RESTRICT_ALIAS finalPos = boneTransforms[level].mPosition + offset;
ArrayVector3 * RESTRICT_ALIAS finalScale = boneTransforms[level].mScale + offset;
ArrayQuaternion * RESTRICT_ALIAS finalRot = boneTransforms[level].mOrientation + offset;
KfTransform * RESTRICT_ALIAS prevTransf = prevFrame->mBoneTransform;
KfTransform * RESTRICT_ALIAS nextTransf = nextFrame->mBoneTransform;
ArrayVector3 interpPos, interpScale;
ArrayQuaternion interpRot;
//Interpolate keyframes' rotation not using shortestPath to respect the original animation
interpPos = Math::lerp( prevTransf->mPosition, nextTransf->mPosition, fTimeW );
interpRot = ArrayQuaternion::nlerp( fTimeW, prevTransf->mOrientation,
nextTransf->mOrientation );
interpScale = Math::lerp( prevTransf->mScale, nextTransf->mScale, fTimeW );
//Combine our internal flag (that prevents blending
//unanimated bones) with user's custom weights
ArrayReal fW = (*perBoneWeights) * animWeight;
//When mixing, also interpolate rotation not using shortest path; as this is usually desired
*finalPos += interpPos * fW;
*finalScale *= Math::lerp( ArrayVector3::UNIT_SCALE, interpScale, fW );
*finalRot = ArrayQuaternion::nlerp( fW, ArrayQuaternion::IDENTITY, interpRot ) *
(*finalRot);
inOutLastKnownKeyFrameRig = prevFrame;
}
//-----------------------------------------------------------------------------------
void SkeletonTrack::_bakeUnusedSlots(void)
{
assert( mUsedSlots <= ARRAY_PACKED_REALS );
if( mUsedSlots <= (ARRAY_PACKED_REALS >> 1) )
{
KeyFrameRigVec::const_iterator itor = mKeyFrameRigs.begin();
KeyFrameRigVec::const_iterator end = mKeyFrameRigs.end();
while( itor != end )
{
size_t j=0;
for( size_t i=mUsedSlots; i<ARRAY_PACKED_REALS; ++i )
{
Vector3 vTmp;
Quaternion qTmp;
itor->mBoneTransform->mPosition.getAsVector3( vTmp, j );
itor->mBoneTransform->mPosition.setFromVector3( vTmp, i );
itor->mBoneTransform->mOrientation.getAsQuaternion( qTmp, j );
itor->mBoneTransform->mOrientation.setFromQuaternion( qTmp, i );
itor->mBoneTransform->mScale.getAsVector3( vTmp, j );
itor->mBoneTransform->mScale.setFromVector3( vTmp, i );
j = (j+1) % mUsedSlots;
}
++itor;
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,851 |
Oscar Arvid Antonsson (* 31. Januar 1898 in Lund; † 23. Februar 1960 in Stockholm) war ein schwedischer Kunsthistoriker, Zeichner und Bildhauer.
Leben
Oscar Antonsson studierte bis 1923 Kunstgeschichte an der Universität Lund und wurde 1937 an der Universität Uppsala mit einer Dissertation über den Hermes des Praxiteles promoviert wurde. Seit 1930 war er am Nationalmuseum in Stockholm tätig, zunächst als Assistent, ab 1934 als Kurator, ab 1944 als stellvertretender Direktor und ab 1946 als Direktor.
Er unternahm Studienreisen in ganz Europa und widmete sich schon als Student auch der Bildenden Kunst, insbesondere als Zeichner, dann aber vor allem als Bildhauer mit Bronzeskulpturen.
Veröffentlichungen (Auswahl)
The Praxiteles marble group in Olympia. Petterson, Stockholm 1937 (Dissertation).
Sergels ungdom och Romtid. Norstedt, Stockholm 1942.
Antik konst. En konstbok från Nationalmuseum. Ehlins, Stockholm 1958.
mit Boo von Malmborg: Gustaf III:s Antikmuseum på Stockholms Slott. Stockholm 1958.
Literatur
Weblinks
Nationalmuseum Stockholm
Werke bei artnet.de
Kunsthistoriker
Bildhauer (Schweden)
Zeichner (Schweden)
Wissenschaftlicher Mitarbeiter eines Museums in Schweden
Schwede
Geboren 1898
Gestorben 1960
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,147 |
D.C. Circuit Refuses To …
Mike Scarcella, Blog of the Legal Times
Sicherheitsfirmen
Tote Sicherheitsthemen & Konfliktzonen: Allgemein Gerichtsprozesse & Regulierung: Allgemein
The full U.S. Court of Appeals for the D.C. Circuit...said it will not review a panel's decision to revive the prosecution of former Blackwater private security guards charged in the shooting deaths of Iraqi civilians in 2007. The nine active judges of the court issued an order today, without comment, unanimously rejecting the request from the guards that the full court review and overturn a three-judge panel decision in April. The panel decision remanded the case back to Washington federal district court for further proceedings. The panel ruling reversed U.S. District Judge Ricardo Urbina's decision in December 2009 to dismiss the indictment. Urbina concluded the government, in building its case, improperly used compelled statements the guards made in the hours and days after the fatal shooting in Baghdad. The judge said the indictment itself was tainted...The guards were indicted in Washington federal district court in December 2008. The guards claim they acted in self-defense in the shooting, which left 34 civilians dead or wounded. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 311 |
{"url":"https:\/\/brilliant.org\/problems\/i-learnt-something-new-today\/","text":"# I learnt something new today!\n\nCalculus Level 4\n\n$\\large \\displaystyle \\int^{4}_{0}\\dfrac{x^3}{\\sqrt{16-x^2}} \\, dx$\n\nIf the value of above expression is in the form $$\\dfrac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers, find $$a+b$$.\n\n\u00d7","date":"2018-01-16 11:52:30","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\": 0.8306172490119934, \"perplexity\": 298.41661381443396}, \"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-2018-05\/segments\/1516084886416.17\/warc\/CC-MAIN-20180116105522-20180116125522-00574.warc.gz\"}"} | null | null |
\section*{Author Declarations}
\subsection*{Conflict of Interest}
The authors have no conflicts to disclose.
\subsection*{Author Contributions}
V.U. and M.B. analyzed the data and wrote the paper, M.B., C.A.F.V., B.R., D.F., G.S., F.D., F.K., R.M., H.T.L., C.S., U.S. performed XFEL and PEEM experiments, E.R and C.A.R. synthesized the sample, C.A.F.V., M.B. and E.A.M. prepared the sample for PEEM and characterized it with scanning electron microscopy, C.D. prepared gratings, S.G. performed micromagnetic simulations, M.B., C.D., C.S. and U.S. jointly conceived the project.
\section*{Data Availability Statement}
The data is available from the PSI repository.
\nocite{*}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,832 |
@class DMResource;
typedef void (^DMChainedCallback)(DMResource* resource, DMCallback next);
@interface DMCallbackChain : NSObject
- (id)initWith:(DMResource*)resource callbacks:(NSArray*)callbacks;
- (void)next;
- (void)done;
@end | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,968 |
HomeBikesNaked Sports BikeNew Suzuki Gixxer Price In Bangladesh
New Suzuki Gixxer Price In Bangladesh
Add to wishlistAdded to wishlistRemoved from wishlist 19
Product is rated as #2 in category Naked Sports Bike
৳ 241,950.00 – ৳ 264,950.00
Choose an optionCarburetorFuel Injection
Choose an optionNormal Braking SystemSingle Channel ABS Clear
New Suzuki Gixxer Price In Bangladesh quantity
New Suzuki Gixxer is one of the most popular bikes produced by Suzuki Motorcycle. It's a stylish bike. This bike design is very Unique.
New Suzuki Gixxer is one of the most popular bikes produced by Suzuki Motorcycle. It's a stylish bike. This bike design is very Unique. To know more about the bike's Specification please click on the Specification tab.
Unless you're living under a rock or something, you're probably familiar with the New Suzuki Gixxer. This is the latest Suzuki bike, and it's available in Bangladesh. It was a bike that was filled with hope, aspirations. Since its launch, this bike has attracted a lot of attention. It is easily one of the most anticipated bikes for 2022.
Suzuki Gixxer has been present in South Asian markets since 2014. Although it is not the most popular bike series, the Suzuki Gixxer has earned the respect of all riders. The Suzuki Gixxer has been the most popular competitor to the Yamaha FZ and has always been the best in terms of features and preferences. The New Suzuki Gixxer, which was launched in Bangladesh in July 2020, is the latest instalment.
The New Suzuki Gixxer's most appealing feature is its new design. Everyone fell in love with its new sporty and elegant design. The bike has a big vibe due to the large air scoops that are located beside it. It's a completely new tank, with cutouts that give more room for the legs. The tail section is longer with a pillion seat that is more generously sized. It is also refreshing to see the new design of the taillight as well as the headlight. The retro look of the headlight and the curved tail light gives it an almost comical look.
The New Suzuki Gixxer has LED treatments available. Both the low and high beams can be used by the LED-equipped headlight. The LEDs will ensure you are adequately covered in darkness and don't require any extra powerful LEDs. A dual-tone LED tail light makes it visible from a distance of over a mile. While it wouldn't have much impact, LED indicators could be an option to this bike. They would have added style to the minimalistic bike.
The New Suzuki Gixxer is available in three colors. They come in:
Grey.
All colors listed above are glossy. Except for brandings, the bikes do not have any decals. The bike's glossy finish combined with minimal decals make it look mature.
The Suzuki Gixxer instrument cluster is one of its most notable advantages. The new Suzuki Gixxer also has an instrument cluster that is as informative as the old versions. The instrument cluster has a clock, a clock, and a gear indicator. It doesn't really matter if it's new or old, anyone will appreciate an informative cluster.
The New Suzuki Gixxer has a height of 800mm and is quite tall. Although it can accommodate riders up to 5'7″, the bike is not suitable for younger riders. A 12liter average-sized fuel tank is included on the bike, which is sufficient for city commutes but could be problematic on the highway.
The New Suzuki Gixxer is 1200mm high, 3000mm long, wide, and weighs 136kg. The bike is not heavy but it's not the lightest. This will make aerodynamics easy. The average wheelbase is 1335mm. This allows the bike to balance perfectly.
Engine and Transmission:
The New Suzuki Gixxer features a single-cylinder, 4-stroke single-cylinder engine of 154.9cc. The engine is air-cooled and has two valves, SOHC. There are two types of engine options: fuel-injected and carbureted. Both versions produce 14.1BHP power at 8000rpm, and 14Nm torque at 6000rpm. The controls of the bike are the same as for the previous version. The bike is expected have a fuel consumption of 35/40 kmpl (carb/FI span)
The New Suzuki Gixxer features an essential wet multi-plate clutch. The 5-speed gearbox on the bike is reasonable considering its power. This bike would have performed better with a 6-speed transmission.
Brakes, Suspensions, and Wheels:
The New Suzuki Gixxer has a unique feature: the brakes. It features dual-disc brakes. The ABS is on the FI version, while the primary brake for the carb model of the bike is provided by the primary front brake. The bike's front brake is suitable for both models, as is its rear brake. Because the rear brakes have discs, they are perfectly suited.
The New Suzuki Gixxer features telescopic forks at its front. The front forks have oil-dampened coil springs. The front suspension is slightly stiff which could be problematic for Bangladesh's rough city commutes. The rear is equipped with a mono-shock, swingarm-based mono-shock suspension. This should perform well.
The New Suzuki Gixxer comes with cast alloy wheels. Both the front and rear tires have 100/80 and 130/60 section tires, respectively. The large rear tire allows for easier and more confident cornering. The front wheel and rear wheel are perfectly balanced.
The New Suzuki Gixxer has all the right features for novice riders. These features allow riders to quickly learn how to ride the bike and provide a great learning experience. This bike is more forgiving than the carburetor. This motorcycle is great for anyone looking for a nice-looking naked bike.
The competitors of the New Suzuki Gixxer are named below:
FZs FI series.
Bajaj Pulsar NS160 ABS FI.
Suzuki Gixxer 155 (older versions)
TVS Apache RTR 4V Dual Disc.
FAQ- Frequently Ask Question
1. Is New Suzuki Gixxer a good bike?
Ans- Suzuki is one of the largest motorcycle manufacturers in the world, it can be trusted and is reliable with its products.
2. What is the mileage of the New Suzuki Gixxer?
Ans- The New Suzuki Gixxer has a mileage of around 40-45kmpl.
3. Is there any naked version of the New Suzuki Gixxer?
Ans- There is a naked version of the New Suzuki Gixxer, known as the New Suzuki Gixxer.
4. Is there any carbureted version of the New Suzuki Gixxer?
Ans- There is a carbureted version of the New Suzuki Gixxer, but it does not have any ABS.
Specification: New Suzuki Gixxer Price In Bangladesh
Rancon Motorbike Bangladesh
CC Category
2. Engine & Transmission
Wet Multiplate
Carburetor, Fuel Injection
13.6ps@8000 rpm, 14.1 PS @ 8000 rpm
13.8Nm@6000rpm, 14 NM @ 6000 rpm
3. Dimensions
4. Brakes, Wheels & Suspensions
Normal Braking System, Single Channel ABS
single downtube
Single Disc
100/80 – 17
Swing arm, Mono Suspension
140/60R – 17
5. Top Speed and Mileage
Mileage (Average)
6. Electricals
12V, 3Ah
Pass Switch
RPM Meter
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Lifan KPS 150 Price In Bangladesh | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 31 |
Hey there. I'm Aubrey, managing editor of the very blog you're reading right now!
Let me tell you a secret. Sure, I'm a marketer by day, but my true passion?
Now, it's probably no surprise that a marketer loves words. Whoop-di-doo, Aubrey.
What is important is that my words brought you here. And that's the trademark of a good blog.
And, that's it! You've got a blog. Right? No?
Alright, I confess: it's not just about those two things—but once you have those defined, your blog is off to a pretty good start.
From there, you're not alone. In fact, I was just on a panel of fellow marketing bloggers to share our learnings on what exactly makes a good blog.
I love this question because I can't answer it with a number. I can't give you a magic quantity or time of day to post to ensure that your B2B blog takes off.
A question like this was meant to have a simple answer, but the reality is so much more interesting than that!
How do you measure your blog post success?
How do you convert a press release into a blog post most efficiently?
How do you attract readers?
Check it out for yourself here. And, of course, don't forget to subscribe to our weekly blog digest to see for yourself what our blog strategy looks like. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,347 |
{"url":"https:\/\/economics.stackexchange.com\/questions\/15939\/making-mathematical-sense-of-the-expression-for-realized-bond-return","text":"# Making mathematical sense of the expression for realized bond return\n\nI came across the following statement regarding the realized 10-year maturity bond's return over a year:\n\nThe realized bond return (H) over a year has two components: the yield income earned over time and the capital gain or loss due to yield changes: $$H_{10} \\approx Y_{10}-\\text{Duration}_{10} \\times \\Delta Y_{10}.$$\n\nI am a complete economics rookie and I'm trying to understand what's going on here from the mathematical point of view, which I'm gonna present here, but my calculations don't seem to add up.\n\nIf we denote the bond's coupon with $C$, and the bond's time $t=0$ yield to maturity with $y_0$, then the bond's value at time $t=0$ equals: $$V_0=\\frac{C}{1+y_0}+\\frac{C}{(1+y_0)^2}+\\dots+\\frac{C}{(1+y_0)^9}+\\frac{F+C}{(1+y_0)^{10}}.$$\n\nAt time $t=1$, we can express the bond's value as the function of the time $t=1$ yield to maturity $y$, so we have $$V_1(y)=\\frac{C}{1+y}+\\frac{C}{(1+y)^2}+\\dots+\\frac{C}{(1+y)^8}+\\frac{F+C}{(1+y)^{9}}.$$ Derivative of $V_1$ with respect to $y$ is equal to: $$\\frac{dV_1}{dy}=-1\\cdot\\frac{C}{(1+y)^2}-2\\cdot\\frac{C}{(1+y)^3}-\\dots-8 \\cdot \\frac{C}{(1+y)^9}-9 \\cdot\\frac{F+C}{(1+y)^{10}} .$$ Now, we can apply some basic calculus here and state that for $\\Delta y$ small \"enough\", we have that $$V_1(y_0+\\Delta y)\\approx V_1(y_0)+\\frac{d V_1}{dy}(y_0)\\cdot \\Delta y.$$ So now, if we consider the absolute return on our position (buying this bond at time $t=0$, selling it at $t=1$) from the time $t=0$ perspective, under the assumption that the time $t=1$ bond's yield to maturity is $y_1=y_0+\\Delta y$, we have that: $$\\text{AbsReturn} \\approx-V_0+\\frac{C}{1+y_0}+\\frac{V_1(y_0)+\\frac{d V_1}{dy}(y_0)\\cdot \\Delta y}{1+y_0}.$$ That is - we buy the bond for $V_0$, at the end of the first year we are paid the coupon which discounted value is $\\frac{C}{1+y_0}$, and the approximation of the time $t=1$ bond's value taking into the account the YTM change is $V_1(y_0)+\\frac{d V_1}{dy}(y_0)\\cdot \\Delta y$ and we also discount it to time $t=0$.\n\nNow, we can simplify the expression for AbsReturn since $-V_0+\\frac{C}{1+y_0}+\\frac{V_1(y_0)}{1+y_0}=0$ and we get: $$\\text{AbsReturn}= \\frac{\\frac{d V_1}{dy}(y_0)\\cdot \\Delta y}{1+y_0} ,$$ which I guess we can also divide with our initial investment of $V_0$ to get the rate of return so we get: $$\\text{RateOfReturn}= \\frac{\\frac{d V_1}{dy}(y_0)\\cdot \\Delta y}{V_0(1+y_0)} ,$$ aaand this is where I completely lose it. I can't seem to understand the connection between the original expression and the thing I end up with. What does the term $\\text{Duration}_{10}$ in the original formula even stand for - I guess it is the derivation of bond's value with respect to yield - but bond's value at what time: $t=0$ or $t=1$? Does it even make any difference? If it is at time $t=0$, how can we be using linear approximation of that function for approximating bond's value change at time $t=1$? I'm completely puzzled over this. Am I doing something completely wrong in this derivation? I appreciate any insights on this. Thanks!\n\n\u2022 The expression you try to understand is very vaguely and badly expressed. For example, the \"yield income\" is nothing but the coupon, because the holder of the bond gets the coupon and nothing else at the end of the year. And there is no change in the coupon, as the $\\Delta Y_{10}$ symbol appears to suggest. \u2013\u00a0Alecos Papadopoulos Mar 28 '17 at 0:15","date":"2020-01-21 23:03:47","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\": 0.8660343885421753, \"perplexity\": 307.37904754564113}, \"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-2020-05\/segments\/1579250606226.29\/warc\/CC-MAIN-20200121222429-20200122011429-00491.warc.gz\"}"} | null | null |
package fi.veikkaus.dcontext;
/**
* Created by arau on 28.6.2016.
*/
public interface HelpfulContextTask extends ContextTask {
String help();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,402 |
{"url":"http:\/\/math.stackexchange.com\/questions\/106912\/applied-odes-in-trajectory-problem","text":"# Applied ODEs in trajectory problem\n\nI'm having a hard time solving this problem:\n\nLet there be a town $A$ in a shore of a river. Let $x=0$ be the shore. Let $(0,0)$ be the location of the town. Let $B$ be another town, in the oppossite shore, $x=b$, and let the town be in $(b,0)$. Suppose a person from town $A$ goes in a boat with velocity $v$ to $B$, always poiting at $B$ (see the picture), and let the river flow in the positive direction of $y$ with velocity $u$. Find the curve that gives the person's trayectory over time.\n\nIn an arbitrary point of the curve there will always be a system of three vectors, $v$, $u$ and $w=v+u$ in the following manner, where $w$, the tangent vector, is $u+v$. Their modulus is constant, thus since $v+u$ varies, the variables here are the angles and the modulus. Remember that $v$ always points at $B$. IN the image, the lengths of $v$ and $v_1$ and of $u$ and $u_1$ should be the same (since they're the same vector).\n\nYou can see that $v$ is \"pushing\" to get to $B$ but the vector $u$ will always modify $v$'s direction (and thus the man's), making the tangent actually be $w = u+v$. The dotted parallel lines are a reference to the tangent angle which is that between the dotted line and $w$. My approach is:\n\n$$\\tan \\theta = \\frac{dy}{dx}$$\n\nand the modulus of $v+u$ will satisfy, being a function of time.\n\n$$|| v+u||(t) = \\frac{ds}{dt}$$\n\nI just need then to find a way of relating the modulus to the angle to find the solution.\n\nAnother thing that comes to mind is thinking as the solution in a parametric way, thus first finding $$\\frac{dy}{dt} = f(t)$$ and $$\\frac{dx}{dt} = g(t)$$ then taking the quotient and integrating.\n\n-\nJust to clarify, does $v$ have constant magnitude in the problem statement, or does $u + v$ have constant magnitude? If $u + v$ has constant magnitude, then I would expect both the magnitude and direction of $v$ to vary with the position of the boat, because $u$ is a constant vector. \u2013\u00a0Geoff Oxberry Feb 8 '12 at 5:01\n$v$ and $u$ are constant. The magnitude of $v+u$ which I understand is the instat velocity of the boat varies depending on how we combine the vectors. \u2013\u00a0Pedro Tamaroff Feb 8 '12 at 5:24\nan answer that stays in the scope of the th[e]ory of (...) persecution problems... Never heard about such a theory. Sounds cool, though. \u2013\u00a0Did Feb 13 '12 at 10:02\nNo, thanks... My point is to recommend some more restraint in the use of the word theory (which in this case is, I think, inappropriate). \u2013\u00a0Did Feb 13 '12 at 12:06\n@DidierPiau Well, in ODE courses persecution problems is a common topic to see when studying isogonal families of curves. I don't really appreaciate the ironical approach, you can be straightforward and I won't mind. \u2013\u00a0Pedro Tamaroff Feb 13 '12 at 12:10\n\nI assume that $|u|$ and $|v|$ are constant. Since the boat is always pointing towards $(b.0)$, we know that $$\\frac{v_y}{v_x}=\\frac{-y}{b-x}\\tag{1}$$ Since $|v|^2=v_x^2+v_y^2$, we can square $(1)$ and add $1$, or square the reciprocal of $(1)$ and add $1$ to get $$\\begin{array}{} v_x=|v|\\frac{b-x}{\\sqrt{(b-x)^2+y^2}}&\\text{and}&v_y=|v|\\frac{-y}{\\sqrt{(b-x)^2+y^2}} \\end{array}\\tag{2}$$ Since the river flow is in the positive $y$ direction, $u_y=|u|$ and $u_x=0$. The slope of the trajectory is the ratio of the total $y$-velocity to the total $x$-velocity: \\begin{align} \\frac{\\mathrm{d}y}{\\mathrm{d}x} &=\\frac{v_y+u_y}{v_x+u_x}\\\\ &=\\frac{v_y+|u|}{v_x}\\\\ &=\\frac{|u|\\sqrt{(b-x)^2+y^2}-|v|y}{|v|(b-x)}\\tag{3} \\end{align} which becomes $$|v|((b-x)\\;\\mathrm{d}y-y\\;\\mathrm{d}(b-x))=-|u|\\sqrt{(b-x)^2+y^2}\\;\\mathrm{d}(b-x)\\tag{4}$$ Dividing $(4)$ by $(b-x)^2$ and rearranging yields $$0=|v|\\;\\mathrm{d}\\frac{y}{b-x}+|u|\\sqrt{1+\\left(\\frac{y}{b-x}\\right)^2}\\;\\mathrm{d}\\log\\left(b-x\\right)\\tag{5}$$ Dividing $(5)$ by $\\sqrt{1+\\left(\\frac{y}{b-x}\\right)^2}$ yields $$|v|\\;\\mathrm{d}\\operatorname{arcsinh}\\left(\\frac{y}{b-x}\\right)+|u|\\;\\mathrm{d}\\log(b-x)=0\\tag{6}$$ which means $$|v|\\operatorname{arcsinh}\\left(\\frac{y}{b-x}\\right)+|u|\\log(b-x)=C|v|\\tag{7}$$ which leads to \\begin{align} y &=(b-x)\\sinh\\left(C-\\frac{|u|}{|v|}\\log(b-x)\\right)\\\\ &=(b-x)\\sinh\\left(\\frac{|u|}{|v|}\\log\\left(\\frac{b}{b-x}\\right)\\right)\\\\ &=\\frac{b-x}{2}\\left[\\left(\\frac{b}{b-x}\\right)^{|u|\/|v|}-\\left(\\frac{b-x}{b}\\right)^{|u|\/|v|}\\right]\\tag{8} \\end{align} Note that $$\\lim_{x\\to b^-}y=\\left\\{\\begin{array}{}0&\\text{if }|u|<|v|\\\\b\/2&\\text{if }|u|=|v|\\\\\\infty&\\text{if }|u|>|v|\\end{array}\\right.\\tag{9}$$\n\nPath of Crossing:\n\nAll paths below, except $\\dfrac{|u|}{|v|}=1$, end directly across the river from the starting point, but most are very steep at the end. For $\\dfrac{|u|}{|v|}=1$, the boat never reaches the other side, but it limits to the point $\\frac12$ the river width downstream.\n\nTime to Cross:\n\nAs the relative speed of the stream gets closer to the speed of the boat, the time to cross the stream increases. Combining $(2)$ and $(8)$ yields \\begin{align} T &=\\int_0^b\\frac{\\mathrm{d}x}{v_x}\\\\ &=\\frac{1}{|v|}\\int_0^b\\frac{\\sqrt{(b-x)^2+y^2}}{b-x}\\;\\mathrm{d}x\\\\ &=\\frac{1}{|v|}\\int_0^b\\sqrt{1+\\left(\\frac{y}{b-x}\\right)^2}\\;\\mathrm{d}x\\\\ &=\\frac{1}{|v|}\\int_0^b\\sqrt{1+\\frac14\\left[\\left(\\frac{b}{b-x}\\right)^{|u|\/|v|}-\\left(\\frac{b-x}{b}\\right)^{|u|\/|v|}\\right]^2}\\;\\mathrm{d}x\\\\ &=\\frac{1}{2|v|}\\int_0^b\\left[\\left(\\frac{b}{b-x}\\right)^{|u|\/|v|}+\\left(\\frac{b-x}{b}\\right)^{|u|\/|v|}\\right]\\;\\mathrm{d}x\\\\ &=\\frac{1}{2|v|}\\left[\\frac{b}{1-|u|\/|v|}+\\frac{b}{1+|u|\/|v|}\\right]\\\\ &=\\frac{b}{|v|}\\frac{1}{1-(|u|\/|v|)^2}\\tag{10} \\end{align} Thus, with no cross-current ($|u|=0$), $T=\\dfrac{b}{|v|}$, and as $|u|\\to|v|$, $T\\to\\infty$; both as expected.\n\n-\neverything seems ok except you're missing a factor of $(b-x)$ in your final answer. i.e, the solution should be $$\\frac{{b - x}}{2}\\left[ {{{\\left( {\\frac{b}{{b - x}}} \\right)}^c} - {{\\left( {\\frac{{b - x}}{b}} \\right)}^c}} \\right]$$ where $c$ is the quotient. Could you expand a little on (2) and (3)? \u2013\u00a0Pedro Tamaroff Feb 13 '12 at 15:54\nA got it all except $$\\frac{{dy}}{{dx}} = \\frac{{{v_y} + \\left| u \\right|}}{{{v_x}}}$$ \u2013\u00a0Pedro Tamaroff Feb 13 '12 at 16:18\nare you around? \u2013\u00a0Pedro Tamaroff Feb 13 '12 at 16:28\n@Peter: I had already caught the $(b-x)$, but I missed the $\\frac12$. Thanks for catching that. I will add a bit in the answer about $(2)$ and $(3)$. \u2013\u00a0robjohn Feb 13 '12 at 16:33\nI really appreciate your work. Thanks! \u2013\u00a0Pedro Tamaroff Feb 13 '12 at 16:35\n\n## Edit\n\nThis gives a solution to what I understood was the problem before the OP edited it.\n\nWe have a river of width $w>0$. City $A$ is in the shore $y=0$ at the point $(0,0)$. City $B$ is in the opposite shore, which I assume to be $y=w$, at point $(b,w)$. A person rows from $A$ to $B$, pointing always in the direction of $B$ and with speed $v$ and has to fight against a current going from $b$ to $A$ of speed $u$. Let $(x(t),y(t))$ be the position of the boat at time $t$. The velocity that the rower gives to the boat at this time is in the direction of $(x,y)$ to $B$, that is, it is proportional to $(b-x,w-y)$. Since its modulus is equal to $v$, it must be $$\\frac{v}{r}\\,(b-x,w-y),\\quad r=\\sqrt{(b-x)^2+(w-y)^2}.$$ Let $V=(V_x,V_y)$ be the real velocity of the boat. Then $$V_x=\\frac{v}{r}\\,(b-x)-u,\\quad V_y=\\frac{v}{r}\\,(w-y).$$ This leads to the following system of differential equations: \\begin{align*} x'&=\\frac{v}{r}\\,(b-x)-u,\\\\ y'&=\\frac{v}{r}\\,(w-y), \\end{align*}\\tag1 together with the initial conditions $x(0)=y(0)=0$. The system is equivalent to the equation $$\\frac{dy}{dx}=\\frac{w-y}{b-x-\\dfrac{u}{v}\\,\\sqrt{(b-x)^2+(w-y)^2}},\\quad y(0)=0.\\tag2$$ The change $z=w-y$, $t=b-x$ transforms it into an homogeneous equation, which can be solved explicitly. Observe that $$v\\,\\frac{b}{\\sqrt{b^2+w^2}}<u\\implies \\frac{dx}{dt}(t=0)<0,$$ and the boat will start going away from $B$. If $v$ is not too small, after a while the boat will turn and go towards $B$. In this case, system (1) and equation (2) are not equivalent.\n\nThis is how the trajectories of the boat look like\n\nIn the problem such as you stated it, the current seems to go perpendicular from one shore to the other. If this is what you really want, all you have to do is take the term $-u$ from the first equation in (1) to the second.\n\n-\nyou are placing $B$ in $(0,b)$ instead of $(b,0)$ \u2013\u00a0Pedro Tamaroff Feb 13 '12 at 15:57\n@Peter But if $B$ is in $(b,0)$ and $A$ is in $(0,0)$ and the shore is $y=0$, they are on the same shore. \u2013\u00a0Juli\u00e1n Aguirre Feb 13 '12 at 16:03\nThe shore is the $y$ axis, not $y=0$. I'll correct it. \u2013\u00a0Pedro Tamaroff Feb 13 '12 at 16:04","date":"2016-04-30 01:35:15","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\": 3, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9213086366653442, \"perplexity\": 308.29638802610935}, \"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-2016-18\/segments\/1461860111581.11\/warc\/CC-MAIN-20160428161511-00031-ip-10-239-7-51.ec2.internal.warc.gz\"}"} | null | null |
{"url":"http:\/\/tex.stackexchange.com\/questions\/30943\/nomenclatura-how-to-display-one-symbol-used-in-two-differents-contexts-or-more","text":"# Nomenclatura-How to display one symbol used in two differents contexts or more\n\nThe following example builds one nomenclatura (the list of some importa symbols).\n\nThe problem I met is the following one : if one symbol is used in two contexts, this symbol appears two times (see the letter a in my example).\n\nIs it possible to change this so as to obtain for example the letter a appears only one time in the nomenclatura following by list of the different uses of this letter ?\n\n\\documentclass{article}\n\\usepackage{nomencl}\n\\makenomenclature\n\n\\begin{document}\n\n\\section*{Main equations}\n\n$$a=\\frac{N}{A}$$%\n\n\\nomenclature{$a$}{The number of angels per unit area}%\n\\nomenclature{$N$}{The number of angels per needle point}%\n\\nomenclature{$A$}{The area of the needle point}%\n\nThe equation $\\sigma = m a$%\n\n\\nomenclature{$\\sigma$}{The total mass of angels per unit area}%\n\\nomenclature{$m$}{The mass of one angel} follows easily.\n\nLet's try with another $a$... %\n\\nomenclature{$a$}{The area of the needle point}%\n\n\\printnomenclature\n\n\\end{document}\n\n\nThe solution given just after by Werner does the job but there is still one problem from my point of view.\n\nInstead of...\n\na The number of angels per unit area\nThe area of the needle point\n\n\nit would be better to have something like...\n\na - The number of angels per unit area\n- The area of the needle point\n\n\nIs it possible to do that ?\n\n-\n\n## 1 Answer\n\nIn the following example I played around with the sort order of nomencl by using the <prefix> command:\n\n\\nomencl[<prefix>]{<symbol>}{<description>}\n\n\nHowever, it required to transform all the nomenclatures to the same type (string in this case). That is, using the prefix for each item (in text\/non-math) so that they are sorted in the same group.\n\n\\documentclass{article}\n\\usepackage{nomencl}% http:\/\/ctan.org\/pkg\/nomencl\n\\makenomenclature\n\n\\begin{document}\n\n\\section*{Main equations}\n\n$$a=\\frac{N}{A}$$%\n\n\\nomenclature[a]{$a$}{The number of angels per unit area}%\n\\nomenclature[N]{$N$}{The number of angels per needle point}%\n\\nomenclature[A]{$A$}{The area of the needle point}%\n\nThe equation $\\sigma = m a$%\n\n\\nomenclature[s]{$\\sigma$}{The total mass of angels per unit area}%\n\\nomenclature[m]{$m$}{The mass of one angel} follows easily.\n\nLet's try with another $a$... %\n\\nomenclature[a1]{}{The area of the needle point}%\n\\nomenclature[a2]{}{The distance to the moon}%\n\\nomenclature[a3]{}{The number of needles in a haystack}%\n\\nomenclature[a4]{}{The value of $pi$}%\n\\printnomenclature\n\n\\end{document}\n\n\nIn the above example, all items under the a nomenclature that should be printed without a, has been given a prefix a? where ? is a numeric to force some sort order.\n\nText-only nomenclatures can be used in the traditional form without the prefix <prefix>, like (say) \\nomenclature{C}{This is a C, see.}; included in the above MWE. You'll notice that the \\sigma entry has been sorted based on the text prefix s. This can be changed if needed.\n\nSince there is no connection between the items in the List of Nomenclatures and the grouped sorting has been manually altered (via specifying a1, a2, ... for items using the same symbol as a), there is no way to easily accommodate the following request automatically:\n\nManually, however, it is no problem. Just add the respective dashed (-- or en-dash) before each item. This should not be a major drawback, since the nomenclature items are formatted and inserted manually in the document anyway:\n\n\\documentclass{article}\n\\usepackage{nomencl}% http:\/\/ctan.org\/pkg\/nomencl\n\\makenomenclature\n\n\\begin{document}\n\n\\section*{Main equations}\n\n$$a=\\frac{N}{A}$$%\n\n\\nomenclature[a]{$a$}{--\\ The number of angels per unit area}%\n\\nomenclature[N]{$N$}{The number of angels per needle point}%\n\\nomenclature[A]{$A$}{The area of the needle point}%\n\nThe equation $\\sigma = m a$%\n\n\\nomenclature[s]{$\\sigma$}{The total mass of angels per unit area}%\n\\nomenclature[m]{$m$}{The mass of one angel} follows easily.\n\nLet's try with another $a$... %\n\\nomenclature[a1]{}{--\\ The area of the needle point}%\n\\nomenclature[a2]{}{--\\ The distance to the moon}%\n\\nomenclature[a3]{}{--\\ The number of needles in a haystack}%\n\\nomenclature[a4]{}{--\\ The value of $pi$}%\n\n\\printnomenclature\n\n\\end{document}\n\n-\nThanks for this explanation. Is there a way to have display some symbol before each item of the list ? \u2013\u00a0projetmbc Oct 9 '11 at 9:51\n@projetmbc: I thought you were interested in removing the symbol before each item in the duplicated symbol. Do you want some symbol printed before each item in the list? Please provide more detail; perhaps even by editing your original question to include this information. \u2013\u00a0Werner Oct 9 '11 at 15:06\nI've edited my first message so as to point what I'm looking for. \u2013\u00a0projetmbc Oct 9 '11 at 18:53\n@projetmbc: See my revised answer. Hope this helps. \u2013\u00a0Werner Oct 9 '11 at 21:07\nThanks for this. \u2013\u00a0projetmbc Oct 10 '11 at 18:06","date":"2015-11-27 12:16:10","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\": 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\": 3, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6895249485969543, \"perplexity\": 1744.7002258542032}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"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-2015-48\/segments\/1448398449145.52\/warc\/CC-MAIN-20151124205409-00086-ip-10-71-132-137.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
\label{S:1}
Recently topological insulators (TIs), that are insulating in the bulk, but have topologically protected conducting edge states with dissipationless charge current at the surface, have attracted a lot of attention in the field of condensed matter physics \cite{Zhang2010,Kane2010,Moore2010,Qi2011}. Several TIs from the V-VI group, e.g. Bi$_{2}$Se$_{3}$, Bi$_{2}$Te$_{3}$, Sb$_{2}$Te$_{3}$ are at the same time promising thermoelectric (TE) materials \cite{Zhang2009,Hsieh2009,Chen2009,Zhang2011} that can convert heat into electricity. The connection and common characteristics between TI, in particular $Z_2$ TIs which preserve time-reversal symmetry (TRS), and TE such as heavy elements, narrow band gaps have been recently pointed out \cite{Muechler2013,Xu2017}. In contrast to our knowledge Chern insulators -- the TRS broken analogue of TIs -- have received little attention concerning TE applications. Chern insulators are characterized by a non-zero Chern number in the absence of an external magnetic field \cite{Weng2015,Ren2016}. It is a significant challenge to realise robust Chern insulators. One strategy to achieve breaking of TRS is via doping of magnetic impurities into known topological insulators, such e.g. Mn-doped HgTe-, Cr-, Fe-doped Bi$_{2}$Te$_{3}$, Bi$_{2}$Se$_{3}$, Sb$_{2}$Te$_{3}$ \cite{Liu_Zhang,Yu_Zhang,Fang_Bernevig}. Further realization possibilities are $5d$ transition metal atoms on graphene \cite{Zhang2012,Zhou_Liu} as well as OsCl$_3$ \cite{Sheng2017}.
\begin{figure}[htbp!]
\centering
\includegraphics[width=8.8cm,keepaspectratio]{Fig1.pdf}
\caption{a) Schematic view of the (EuO)$_{n}$/(MgO)$_{m}$(001) superlattice with $n=2$ and b) the corresponding Brillouin zone.}
\label{fig:schematic_view}
\end{figure}
Transition metal oxides (TMO) with their rich functionality, resulting from the intricate interplay of spin, orbital and lattice degrees of freedom, have a greater tendency towards TRS breaking and larger band gaps compared to conventional $sp$ bonded systems and are thus an attractive class of materials to search for topologically non-trivial states. Intensive efforts have been directed at finding Chern insulators in TMO that host a honeycomb lattice, for which Haldane predicted a quantized anomalous Hall effect in his seminal work for spinless fermions \cite{Haldane}. A buckled honeycomb lattice can be formed in (111)-oriented A$X$O$_3$ perovskite superlattices (SL) by two consecutive triangular $X$-layers as proposed by Xiao et al. \cite{Xiao2011}. Several realizations have been proposed, e.g. SrIrO$_3$ and LaAuO$_3$ bilayers, however, considering correlation effects results in an AFM ground state for SrIrO$_3$ \cite{Lado2013,Okamoto2014}. In the $3d$ series, e.g. in (La$X$O$_3$)$_2$/(LaAlO$_3$)$_4$(111) SLs \cite{Doennig2016}, LaMnO$_3$ was identified as a Chern insulator with a band gap of ~150 meV when P321 symmetry is preserved, but the ground state is a trivial Mott insulator with Jahn-Teller (JT) distortion. Further candidates for quantum anomalous Hall insulators (QAHI) are LaRuO$_3$ and LaOsO$_3$ \cite{HongliNQM},
as well as the metastable symmetric ferromagnetic cases of LaPdO$_3$, LaPtO$_3$ and LaTcO$_3$ \cite{Guo2019,OKRP2019} honeycomb bilayers encased in LaAlO$_3$(111). Corundum-derived SL provide another realization of the honeycomb lattice. A systematic study of the $3d$ \cite{OKRP2018}, $4d$ and $5d$ \cite{JPCS2019} series of corundum-based honeycomb layers ($X_2$O$_3$)$_1$/(Al$_2$O$_3$)$_5$(0001) identified the metastable cases of $X$\,=\,Tc,\,Pt as Chern insulators with $C$\,=\,--2 and --1 and band gaps of 54 and 59 meV, respectively.
Other lattice types proposed as candidate Chern insulators are e.g. rutile-derived heterostructures \cite{Huang_Vanderbilt,Cai_Gong,Lado2016}, pyrochlore \cite{Fiete2015}, as well as rocksalt-derived superlattices as EuO/CdO \cite{Zhang2014} and EuO/GdN SLs\cite{Garrity2014}. The aim in the latter is to combine heavy elements with large SOC in the two initially topologically trivial components in a quantum well (QW) structure and induce a SOC-dirven band inversion, analogous to the HgTe/CdTe QW, where band inversion was originally predicted and observed\cite{Bernevig2006,Zhang2006,Konig2007}. EuO is one of the few ferromagnetically (FM) ordered semiconductors \cite{Mauger1986} with a Curie temperature ($T_c$) of 69 K. Similarly, in both EuO/CdO and EuO/GdN the CI state is achieved under considerable strain\cite{Zhang2014,Garrity2014}. In EuO/GdN the band inversion involves Eu $4f$ and the Gd $5d$\cite{Garrity2014}, whereas in EuO/CdO QW it takes place between the occupied Eu $4f$ and the Cd $5s$ states. The resulting band gap is in the meV range owing to the large $\Delta l\,=\,3$\cite{Zhang2014}.
Here we follow a different strategy: by combining EuO with the large band gap insulator MgO in a (EuO)$_{1}$/(MgO)$_{3}$(001) superlattice at the lattice constant of MgO (4.21 \AA) we achieve a band inversion in SL between Eu $4f$ and $5d$ states within the same component, while MgO merely plays the role of a spacer. We note that EuO/MgO(001) QW heterostructures have already been realized experimentally\cite{Tjeng2009,Tjeng2011,Mueller2013,Mueller2016} using lattice-matched yttria-stabilized zirconia (YSZ).
Already Hicks and Dresselhaus \cite{Hicks1993} proposed that the TE properties of materials can be improved in reduced dimensions as e.g. in QW. Experimentally, a giant Seebeck coefficient was reported in $\delta$-doped SrTiO$_3$ SLs\cite{Ohta2007}. The confinement- and strain-induced enhancement of TE properties was recently addressed based on first principles calculations in LaNiO$_3$/LaAlO$_3$(001) SLs \cite{Benjamin2018,Benjamin2019,Viewpoint} and Sr$X$O$_3$/SrTiO$_3$(001) quantum wells \cite{Verma2019}, as well as other SrTiO$_3$-based SLs \cite{Pallecchi2015,Filippetti2012,Delugas2013,Bilc2016}. For example in (LaNiO$_3$)$_1$/(LaAlO$_3$)$_1$(001) \cite{Benjamin2018} the confinement- and strain-induced metal-to-insulator transition (MIT) at $a_{\rm STO}$ leads to enhanced in-plane power factor and high Seebeck coefficient. In the following we discuss the electronic properties of (EuO)$_{n}$/(MgO)$_{m}$(001) SL and provide topological analysis for the non-trivial cases. Moreover, we address in particular the implications of the topological Chern state in (EuO)$_{1}$/(MgO)$_{3}$(001) QWs on the thermoelectric properties using Boltzmann transport theory and compare to the (EuO)$_{2}$/(MgO)$_{2}$(001) case.
\section{Theoretical methods}
\begin{figure} [htp!]
\includegraphics[width=8.6cm,keepaspectratio]{Fig2.pdf}
\caption{a-b) Element- and orbital-projected band structure of ferromagnetic (EuO)$_{n}$/(MgO)$_{m}$(001) with an out-of-plane lattice parameter $c=2a_{\rm MgO}$ and optimized $c$ for $n=2$, $m=2$. The corresponding projected density of states (DOS) are displayed in c-d). The orbital character is color coded, highlighting that the states below/above the Fermi level have predominantly Eu-$4f$ (blue)/Eu-$5d$ (green) as well as O-$2p$ (red) character, respectively.}
\label{fig:EuO1_MgO3}
\end{figure}
\begin{figure*} [htbp!]
\includegraphics[width=18cm,keepaspectratio]{Fig3.pdf}
\caption{GGA+$U$ band structures of ferromagnetic (EuO)$_{n}$/(MgO)$_{m}$(001) at fixed lateral lattice constant of MgO and a) constrained or b) optimized out-of-plane lattice constant $c$ for $n=1$, $m=3$, as well as c) optimized $c$ for $n=2$, $m=2$. Majority and minority channels are shown in dark blue/light orange. The corresponding GGA+$U$+SOC band structures are displayed in the bottom panels d-f).}
\label{fig:EuO_MgO_BS}
\end{figure*}
\begin{figure*} [htp!]
\centering
\includegraphics[width=12cm,keepaspectratio]{Fig4.pdf}
\caption{In a-b) GGA+$U$+SOC band structures for FM (EuO)$_{1}$/(MgO)$_{3}$(001) and (EuO)$_{2}$/(MgO)$_{2}$(001) with magnetization along the [001]-direction as well as c-d) Berry curvatures along the same $k$-path. The corresponding anomalous Hall conductivities (AHC) $\sigma_{xy}$ in units of $e^{2}/h$ as a function of the chemical potential are displayed in e-f).}
\label{fig:EuO1_MgO3_SOC}
\end{figure*}
\begin{figure*} [htp!]
\includegraphics[width=18cm,keepaspectratio]{Fig5.pdf}
\caption{a) The band-decomposed spin-texture in $k$-space from the GGA+$U$+SOC calculation with magnetization along [001] for the topmost occupied and lowest unoccupied bands (cf. Fig. \ref{fig:EuO1_MgO3_SOC}a) of the FM (EuO)$_{1}$/(MgO)$_{3}$(001) superlattice. The color scale denotes the projection on the $\hat{z}$-axis with red (blue) indicating parallel (antiparallel) orientation. The calculated edge states for (EuO)$_{1}$/(MgO)$_{3}$(001) and (EuO)$_{2}$/(MgO)$_{2}$(001) are shown in b-c) . Warmer colors (red/white) represent higher local DOS, blue regions denote the bulk energy gap and solid red lines are the edge states connecting the valence and conduction bands.}
\label{fig:EuO1_MgO3_ST_and_SS}
\end{figure*}
Density functional calculations were performed for (EuO)$_{n}$/(MgO)$_{m}$(001) SLs with the projector augmented wave (PAW) method\cite{PAW} as implemented in the VASP \cite{VASP} code. The cutoff energy of the plane-waves was set to 500 eV. For the exchange-correlation functional we used the generalized gradient approximation (GGA) by Perdew, Burke and Enzerhof \cite{GGA_PBE}. A $\Gamma$-centered $k$-point grid of 16$\times$16$\times$8 were adopted in the self-consistent calculations employing the tetrahedron method \cite{Bloechl1994}. Static electronic correlation effects were taken into account within the GGA\,+$U$ approach in the formulation of Liechtenstein et al. \cite{Liechtenstein}. Consistent with previous studies \cite{Larson2006,Larson2006PRB,Schlipf2013,Tong2014}, an on-site Coulomb repulsion parameter of $U$\,=\,7.4 eV and an exchange interaction parameter $J$\,=\,1.1 eV were considered for the Eu $4f$ states. With these values we obtain a band gap of 1.13 eV for antiferromagnetic coupling which is in very good agreement with the experimentally reported band gap of 1.12 eV \cite{Guentherodt1971} of room-temperature bulk EuO, whereas for the ferromagnetic ground state the band gap amounts to 0.65 eV. The optimized bulk lattice constant of EuO with ferro- and antiferromagnetic arrangement within GGA+$U$ is $a$\,=\,5.184\,\AA\, and $a$\,=\,5.193\,\AA, respectively, slightly higher than the experimental value of $a$\,=\,5.141\,\AA\ \cite{Vleck1961}. Similarly, for bulk MgO GGA yields a bulk lattice parameter of $a$\,=\,4.24\,\AA, somewhat larger than the experimental lattice constant $a$\,=\,4.21\,\AA \cite{Roessler1967,Boer1998,Fei1999}. We note that within GGA the band gap of MgO is significantly underestimated (4.28 eV), compared to the experimental value of 7.83 eV \cite{Whited1973} and can be improved only by considering many body effects \cite{Fuchs2008}. Still the GGA MgO band gap is much larger than the one of the active material EuO, thus the phenomena in the heterostructure are determined by the confined EuO and not affected by the size of the band gap of MgO. The heterostructures were modeled at the experimental lateral lattice constant of MgO and internal parameters were relaxed until the Hellman-Feynman forces are less than 1 meV/\AA, while the $c$ lattice constant was either fixed at the value of bulk MgO or relaxed. In the case of (EuO)$_{1}$/(MgO)$_{3}$(001) the topologically non-trivial case was also explored using the all-electron full-potential linearized augmented plane wave (LAPW) method as implemented in the Wien2k code \cite{wien2k}. In particular, the anomalous Hall conductivity (AHC) calculations were performed on a dense $k$-point mesh of 144$\times$144$\times$12 using the wannier90 code \cite{wannier90}. The transport coefficients based on input from the DFT calculations are obtained within the constant relaxation time approximation using the BoltzTraP code \cite{Boltztrap}.
\section{Results and discussion:}
\subsection{GGA+$U$ results for ({\rm EuO})$_{n}$/({\rm MgO})$_{m}$(001) quantum wells}
\label{sec:EuO_MgO_electronic_properties}
In this Section we discuss the electronic properties of the (EuO)$_{n}$/(MgO)$_{m}$(001) superlattices, referred to as ($n$,$m$) in the following. According to Hund's rule, Eu$^{2+}$ exhibits a formal $4f^7$ high-spin configuration with a closed shell and a large magnetic moment of $\sim 7.0$ $\mu_{\rm B}$. The ferromagnetic state is the ground state. The GGA+$U$ element- and orbitally resolved band structure and the spin-dependent projected density of states (DOS) of (EuO)$_{1}$/(MgO)$_{3}$(001) with $c_{\rm MgO}$ are shown in Fig. \ref{fig:EuO1_MgO3}a and c. Just below the Fermi level the band structure is dominated by the narrow (bandwidth $\sim$ 2 eV) half-filled Eu $4f$ bands, whereas the conduction bands are strongly dispersive, e.g. along M-$\Gamma$-X and of prevailing Eu $5d$ character (see Fig. \ref{fig:EuO1_MgO3}a), respectively. Moreover, the top of the valence and bottom of the conduction band of this quantum well touch along $\Gamma$-Z, rendering the system semimetallic with predominant contribution of majority spin bands. The main effect of the $c$ relaxation is the enhanced dispersion and overlap of conduction and valence bands along $\Gamma$-Z (cf. Fig. \ref{fig:EuO_MgO_BS}b). On the other hand, the band structure of (EuO)$_{2}$/(MgO)$_{2}$(001) with relaxed $c$ (cf. Figs. \ref{fig:EuO1_MgO3}b and \ref{fig:EuO_MgO_BS}c) bears some similarities to $n=1$, $m=3$ at the MgO $c$ lattice constant, in particular, the touching flat conduction and valence bands along $\Gamma$-Z, however exhibits a much stronger overlap and hybridization between the Eu $4f$ and O $2p$ bands and a pronounced O $2p$ contribution along $\Gamma$-Z just below the Fermi level, visible also in the orbitally projected DOS in Figs. \ref{fig:EuO1_MgO3}c and d.
\begin{figure*} [htp!]
\includegraphics[width=16.5cm,keepaspectratio]{Fig6.pdf}
\caption{The in- and cross-plane components of the electrical conductivity tensor divided by $\tau$, the Seebeck coefficient, $PF/\tau$ and the electronic contribution to the
figure of merit ZT$|_{el}$ are shown as a function of the chemical potential in a-b) at 300 K and 600 K for FM (EuO)$_{1}$/(MgO)$_{3}$(001) superlattice at $c$\,=\,8.4~\AA, whereas c-d) show the thermoelectric properties with the optimized out-of-plane parameter of $c_{opt}$\,=\,9.6~\AA, e-f) thermoelectric performance of (EuO)$_{2}$/(MgO)$_{2}$(001) SL.}
\label{fig:EuO1_MgO3_TE}
\end{figure*}
\subsection{Effect of spin-orbit coupling and topological analysis}
\label{sec:CI_phase}
Despite the similar features in the bandstructure, the effect of spin-orbit coupling is very distinct for the three systems. The corresponding band structures are displayed in Fig. \ref{fig:EuO_MgO_BS}d-f. While (EuO)$_{1}$/(MgO)$_{3}$(001) with relaxed $c$ remains metallic with no pronounced rearrangement of bands, for (EuO)$_{1}$/(MgO)$_{3}$(001) at $c=2a_{\rm MgO}$ a significant band gap of 0.51 eV is opened for SOC with out-of-plane magnetization direction.
Apparently, the degeneracy of the touching bands at the Fermi level is lifted giving rise to a band inversion along $\Gamma$-Z. The band inversion is present but the band gap is nearly vanishing for (EuO)$_{2}$/(MgO)$_{2}$(001) with relaxed $c$ (see inset in Fig. \ref{fig:EuO_MgO_BS}f). In order to analyze the origin of the band rearrangement and inversion we plot in Fig. \ref{fig:EuO1_MgO3_SOC}a and b the element and orbital projections on the band structure with SOC for (1,3) and (2,2).
In contrast to the previously reported band inversion between Eu $4f$ and Cd $5s$ bands in EuO/CdO(001) \cite{Zhang2014} or Eu $4f$ and Gd $5d$ states in EuO/GdN SL \cite{Garrity2014}, for(EuO)$_{1}$/(MgO)$_{3}$(001) at $c=2a_{\rm MgO}$ the band inversion takes place between the $4f$ and $5d$ states of Eu itself. The strong interaction of these bands of opposite parity and $\Delta l\,=\,1$ lead to a substantial band gap opening. Interestingly this bears analogies with previous reports of bulk EuO under pressure, where fluctuations between $(4f)^{7}(5d)^{0}$ and $(4f)^{6}(5d)^{1}$ configurations were suggested in experimental \cite{Zimmer1984} and theoretical studies \cite{Eyert1986PB,Eyert1986SS}. Upon inclusion of SOC the band structure of (EuO)$_{2}$/(MgO)$_{2}$(001) with relaxed $c$ shows a reduced contribution of O $2p$ along $\Gamma$-Z and a similar inversion of the topmost Eu $4f$ and lowest $5d$ band around $E_{\rm F}$, though with a vanishing band gap.
Having identified the origin of the band rearrangement and inversion for the two systems, we proceed to analyze the topological properties. The non-trivial nature of the (1,3) system is underpinned by the Berry curvature (cf. Fig. \ref{fig:EuO1_MgO3_SOC}c) which exhibits pronounced negative peaks along $\Gamma$-Z as well as M-$\Gamma$ paths, and a flat region along $\Gamma$-Z. This leads to the emergence of a broad plateau in the anomalous Hall conductivity $\sigma_{xy}$ at $E_{\rm F}$\ in Fig. \ref{fig:EuO1_MgO3_SOC}e, rendering (EuO)$_{1}$/(MgO)$_{3}$(001) a Chern insulator with a quantized $C$\,=\,--1. As shown in Fig. \ref{fig:EuO1_MgO3_SOC}d, sharp peaks arise in the Berry curvature $\Omega(k)$ of (EuO)$_{2}$/(MgO)$_{2}$(001) at the avoided crossing of Eu $4f$ and $5d$ bands along the M-$\Gamma$ and Z-R paths with values of 3000 and 8000 bohr$^{2}$.
For (EuO)$_{2}$/(MgO)$_{2}$(001) the Hall conductivity in Fig. \ref{fig:EuO1_MgO3_SOC}f shows substantial, nearly quantized values (--1.04 $e^{2}/h$) caused by the non-trivial bands with the plateau being just below $E_{\rm F}$\ and a finite value of 0.8\,$e^{2}/h$ at $E_{\rm F}$.
In Fig. \ref{fig:EuO1_MgO3_ST_and_SS}a we plot the spin-texture of the relevant bands of (1,3) (see Fig. \ref{fig:EuO1_MgO3_SOC}a). The occupied band exhibits only positive $s_z$ values throughout the whole BZ. In contrast, the out-of-plane spin component $s_z$ of the lower part of the unoccupied parabolic band is negative around $\Gamma$ but reverses sign further away from the BZ center. This switching of spin orientation can be ascribed to the SOC-induced band inversion between the occupied majority $4f$ states and unoccupied minority $5d$ states band along $\Gamma$-Z, occurring just above $E_{\rm F}$\ in Fig.~\ref{fig:EuO_MgO_BS}a. The surface states shown in Fig.~\ref{fig:EuO1_MgO3_ST_and_SS}b using Wanniertools \cite{wanniertools} based on the Maximally Localized Wannier functions (MLWF) method presents one topologically protected chiral edge state, connecting the valence and conduction band. In contrast, the edge state for (EuO)$_{2}$/(MgO)$_{2}$(001) in Fig.~\ref{fig:EuO1_MgO3_ST_and_SS}c is obscured due to the overlap of valence and conduction bands along $\Gamma$-Z.
\subsection{Thermoelectric properties}
\label{sec:thermoelectric}
In the following, we investigate the thermoelectric properties of the (EuO)$_{n}$/(MgO)$_{m}$(001) SLs with and without optimized out-of-plane lattice constants $c$. A central quantity related to the TE efficiency is the figure of merit:
\begin{equation}
\label{eq:1}
ZT = \frac{S^{2} \sigma}{\kappa} T
\end{equation}
where $S$ is the Seebeck coefficient, $\sigma$ the conductivity and $\kappa$ the thermal conductivity. Another related quantity is the power factor $PF=S^2\sigma$. In Fig. \ref{fig:EuO1_MgO3_TE} we plot $\sigma/\tau$, the Seebeck coefficient and $PF/\tau$ for the systems studied in Fig. \ref{fig:EuO_MgO_BS} at two different temperatures, 300 and 600 K. While (EuO)$_{1}$/(MgO)$_{3}$(001) with $c$\,=\,8.4~\AA\ is a Chern insulator upon inclusion of SOC (cf. Fig. \ref{fig:EuO1_MgO3_TE}), the other two systems remain metallic or semimetallic. This is reflected in the vanishing out-of-plane conductivity for the former, whereas the more dispersive bands of (EuO)$_{1}$/(MgO)$_{3}$(001) with relaxed $c$ leads to higher $\sigma/\tau$. On the other hand the flat touching bands along $\Gamma$-Z in (EuO)$_{2}$/(MgO)$_{2}$(001) result in a vanishing out-of-plane conductivity.
The Chern insulating system (EuO)$_{1}$/(MgO)$_{3}$(001) exhibits a much higher Seebeck coefficient, reaching values between 400-800~$\mu$VK$^{-1}$ and $PF/\tau$ of 0.8$\cdot$10$^{11}$ (300 K) and 1.2$\cdot$10$^{11}$ W/K$^{2}$ms (600 K). The electronic figure of merit ZT$|_{el}$ attains in- and out-of-plane values of $\sim 0.8$ and $\sim 0.5$ at 300 K and $\sim 0.8$ and $\sim 0.7$ at 600 K at the valence band edge while values of $\sim 0.4$ and $\sim 0.6$ at 300 K and $\sim 0.9$ and $\sim 0.7$ at 600 K at the conduction band edge are obtained, respectively. In contrast $S$, $PF/\tau$ and electronic figure of merit ZT$|_{el}$ are much lower for the metallic case with optimized $c$. On the other hand, (EuO)$_{2}$/(MgO)$_{2}$(001) SL exhibits significant values for $S$ ($\sim 200$ $\mu$VK$^{-1}$), $PF/\tau$ (0.8$\cdot$10$^{11}$ W/K$^{2}$ms at 600 K) and ZT$|_{el}$ ($\sim 0.6$) may be reached upon doping.
Thus, both systems with topologically non-trivial bands and in particular the Chern insulating phase show promising TE properties. The improved performance is associated with the mixture of flat bands around $\Gamma-$Z due to the SOC-induced band inversion that leads to a steep increase of DOS at the band edges, thereby enhancing $S$ and dispersive bands that contribute to the electrical conductivity. Moreover, while we consider here only the electronic contributions to the thermal conductivity, we expect that the high atomic number of Eu (63) and the phonon scattering at interfaces in this layered structure will be beneficial to reduce the lattice contribution to $\kappa$.
\section{Summary}
\label{sec:fin}
In summary, the effect of confinement and strain on the topological and thermoelectric properties of (EuO)$_{n}$/(MgO)$_{m}$(001) superlattices has been studied by DFT\,+\,\textit{U}\,+\,SOC calculations in conjunction with the semi-classical Boltzmann transport theory. Combining two topologically trivial materials EuO and MgO in a QW structure results in a Chern insulating phase. Particularly, (EuO)$_{1}$/(MgO)$_{3}$(001) SL with lattice parameters constrained to the ones of MgO exhibits semimetallic behavior. The inclusion of SOC opens a large band gap of 0.51 eV due to a band inversion between Eu $5d$ and $4f$ bands. This mechanism is distinct to the one in EuO/CdO \cite{Zhang2014} and EuO/GdN SLs\cite{Garrity2014}, where the band inversion takes place between bands of different elements in the two constituents: Eu $4f$ and Cd $5s$ or Gd $5d$, respectively. The resulting Chern insulating phase with $C$\,=\,--1 shows a sign reversal of the out-of-plane spin components $s_z$ along the loop of band inversion around $\Gamma$ and a single chiral edge state. A similar band inversion occurs also in (EuO)$_{2}$/(MgO)$_{2}$(001) SL but with a vanishing band gap. The resulting band rearrangement close to $E_{\rm F}$\ leads to sharp peaks of the Berry curvature at the avoided band-crossing along Z-R and a plateau in the anomalous Hall conductivity above $E_{\rm F}$\ with a notable value of --1.04 $e^{2}/h$.
Moreover, the (EuO)$_{1}$/(MgO)$_{3}$(001) SL exhibits enhanced thermoelectric performance in terms of Seebeck coefficient of 400-800~$\mu$VK$^{-1}$ and $PF/\tau$ of 0.8-1.2$\cdot$10$^{11}$W/K$^{2}$ms and an electronic figure of merit of 0.4-0.9 at the band edges depending on temperature, driven by both the confinement and topological nature of the system. Similarly, an out-of-plane electronic $ZT$ of 0.6 is achievable in (EuO)$_{2}$/(MgO)$_{2}$(001) SL.
The opening of a gap due to SOC-driven band inversion with steep increase of DOS at the band edges due to the flatness of bands along the $\Gamma$-Z direction promotes high values of the Seebeck coefficient and PF, whereas concomitant dispersive bands contribute to the electrical conductivity. Similar effect of systems at the verge of a metal-to insulator transition (though not topological) has been found in other oxide heterostructures \cite{Benjamin2018, Verma2019}. Furthermore, the results establish a link between topological and thermoelectric properties, in particular for systems with broken inversion symmetry.
\clearpage
\begin{acknowledgments}
We acknowledge useful discussions with M. M\"uller and A. Lorke on EuO/MgO quantum wells and funding by the German Science Foundation (Deutsche Forschungsgemeinschaft, DFG) within the CRC/TRR80, project G3 and computational time at the Leibniz Rechenzentrum Garching, project pr87ro. We also would like to thank G\'{e}rald K\"ammerer for performing initial calculations with Wien2k.
\end{acknowledgments}
| {
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} | 1,897 |
/* ------------------------------------------------------------------------- */
//
// Copyright (c) 2010 CubeSoft, Inc.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
//
/* ------------------------------------------------------------------------- */
namespace Cube.DataContract;
using System;
using System.Collections;
using Cube.DataContract.Internal;
using Cube.Text.Extensions;
using Microsoft.Win32;
/* ------------------------------------------------------------------------- */
///
/// RegistrySserializer
///
/// <summary>
/// Provides functionality to serialize to the registry.
/// </summary>
///
/* ------------------------------------------------------------------------- */
internal class RegistrySerializer
{
#region Methods
/* --------------------------------------------------------------------- */
///
/// Invoke
///
/// <summary>
/// Invokes the serialization to the specified registry key.
/// </summary>
///
/// <param name="dest">Root registry key.</param>
/// <param name="src">Object to be serialized.</param>
///
/* --------------------------------------------------------------------- */
public void Invoke<T>(RegistryKey dest, T src) => Set(typeof(T), dest, src);
#endregion
#region Implementations
/* --------------------------------------------------------------------- */
///
/// Set
///
/// <summary>
/// Sets the specified object to the specified registry key.
/// </summary>
///
/* --------------------------------------------------------------------- */
private void Set(Type type, RegistryKey dest, object src)
{
if (dest is null || src is null) return;
foreach (var pi in type.GetProperties())
{
var key = pi.GetDataMemberName();
if (!key.HasValue()) continue;
var value = pi.GetValue(src, null);
if (value is null) continue;
var pt = pi.GetPropertyType();
if (pt.IsEnum) dest.SetValue(key, (int)value);
else if (pt.IsGenericList()) Create(dest, key, e => SetList(pt, e, (IList)value));
else if (pt.IsArray) Create(dest, key, e => SetArray(pt, e, (Array)value));
else if (pt.IsObject()) Create(dest, key, e => Set(pt, e, value));
else Set(pt, dest, key, value);
}
}
/* --------------------------------------------------------------------- */
///
/// Set
///
/// <summary>
/// Sets the specified object to the specified registry key.
/// </summary>
///
/* --------------------------------------------------------------------- */
private void Set(Type type, RegistryKey dest, string key, object value)
{
switch (Type.GetTypeCode(type))
{
case TypeCode.Boolean:
dest.SetValue(key, ((bool)value) ? 1 : 0);
break;
case TypeCode.DateTime:
dest.SetValue(key, ((DateTime)value).ToUniversalTime().ToString("o"));
break;
default:
dest.SetValue(key, value);
break;
}
}
/* --------------------------------------------------------------------- */
///
/// SetArray
///
/// <summary>
/// Sets the specified array object to the specified registry key.
/// </summary>
///
/* --------------------------------------------------------------------- */
private void SetArray(Type type, RegistryKey dest, Array src)
{
if (src.Rank != 1) return;
var t = type.GetElementType();
var n = Digit(src.Length);
foreach (var name in dest.GetSubKeyNames()) dest.DeleteSubKeyTree(name);
for (var i = 0; i < src.Length; ++i)
{
Create(dest, i.ToString($"D{n}"), e => SetListElement(t, e, src.GetValue(i)));
}
}
/* --------------------------------------------------------------------- */
///
/// SetList
///
/// <summary>
/// Sets the specified list object to the specified registry key.
/// </summary>
///
/* --------------------------------------------------------------------- */
private void SetList(Type type, RegistryKey dest, IList src)
{
var ga = type.GetGenericArguments();
if (ga is null || ga.Length != 1) return;
var n = Digit(src.Count);
foreach (var name in dest.GetSubKeyNames()) dest.DeleteSubKeyTree(name);
for (var i = 0; i < src.Count; ++i)
{
Create(dest, i.ToString($"D{n}"), e => SetListElement(ga[0], e, src[i]));
}
}
/* --------------------------------------------------------------------- */
///
/// SetListElement
///
/// <summary>
/// Sets the specified list object to the specified registry key.
/// </summary>
///
/* --------------------------------------------------------------------- */
private void SetListElement(Type type, RegistryKey dest, object src)
{
if (type.IsObject()) Set(type, dest, src);
else Set(type, dest, "", src);
}
/* --------------------------------------------------------------------- */
///
/// Create
///
/// <summary>
/// Creates a new registry key with the specified registry key
/// and name, and invokes the specified action.
/// </summary>
///
/* --------------------------------------------------------------------- */
private void Create(RegistryKey dest, string name, Action<RegistryKey> callback)
{
using var e = dest.CreateSubKey(name);
if (e is not null) callback(e);
}
/* --------------------------------------------------------------------- */
///
/// Digit
///
/// <summary>
/// Gets the digit number of the specified value.
/// </summary>
///
/* --------------------------------------------------------------------- */
private int Digit(int n) => n.ToString().Length;
#endregion
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,195 |
Embedded Computer│Industrial computer │Low cost SBC │single board computer│industrial automation
← Ian Wright is Turning Fedex and Garbage Trucks Into High Performance EVs
Create a "Wheel of Excuses" With BASIC and the New Raspberry Pi single board computer →
Vulnerable "Smart" Devices Make an Internet of Insecure Things among network appliance
According to recent research, 70 percent of Americans plan to own network appliance in the next five years, at least one smart appliance like an internet-connected refrigerator or thermostat. That's a skyrocketing adoption rate considering the number of smart appliance owners in the United States today is just four percent.
Yet backdoors and other insecure channels have been found in many such network appliance devices, opening them to possible hacks, botnets, and other cyber mischief. Although the widely touted hack of smart refrigerators earlier this year has since been debunked, there's still no shortage of vulnerabilities in the emerging, so-called Internet of Things.
Enter, then, one of the world's top research centers devoted to IT security, boasting 700 students in this growing field, the Horst Gortz Institute for IT Security at Ruhr-University Bochum in Germany. A research group at HGI, led by Christof Paar—professor and networking aplliance chair for embedded security at the Institute—has been discovering and helping manufacturers patch security holes in Internet-of-Things devices like appliances, cars, and the wireless routers they connect with.
Paar, who is also adjunct professor of electrical and computer engineering at the University of Massachusetts at Amherst, says there are good engineering, technological, and even cultural reasons why security of the Internet of Things is a very hard problem.
For starters, it's hard enough to get people to update their laptops and smartphones with the latest security patches. Imagine, then, a world where everything from your garage door opener, your coffeemaker, your eyeglasses, and even your running shoes have possible network appliance vulnerabilities. And the onus is entirely on you to download and install firmware updates—if there are any.
Furthermore, most Internet-connected "things" are net-savvier iterations of designs that have long pre-Internet legacies—legacies in which digital security had previously never been a major concern. But, Paar says, security is not just another new feature to be added onto an networking aplliance device. Internet security requires designers and engineers embrace a different culture altogether.
"There's essentially no tolerance for error in security engineering."
"There's essentially no tolerance for error in security engineering," Paar says. "If you write software, and the software is not quite optimum, you might be ten percent slower. You're ten percent worse, but you still have pretty decent results. If you make one little mistake in security engineering, and the attacker gets in, the whole system collapses immediately. That's kind of unique to security and crypto-security in general."
Paar's research team, which published some of its latest findings in Internet-of-Things security this summer, spends a lot of time on physical and electrical engineering-based attacks on networking aplliance, also called side-channel attacks.
For instance, in 2013 Paar and six colleagues discovered rackmount in an Internet-connected digital lock made by Simons Voss. It involved a predictable, non-random number the lock's algorithm used when challenging a user for the passcode. And the flaws in the security algorithm were discoverable, they found, via the wireless link between the lock and its remote control.
The way they handled the network box discovery was how they handle all security rackmount exploit discoveries at the Institute, Paar says. They first revealed the weakness to the manufacturers and offered to help patch the error before they publicized the exploit.
"They fixed the network box system, and the new generation of their rackmount is better," he says. "They had homegrown crypto, which failed. And they had side-channel [security], which failed. So we had two or three vulnerabilities which we could exploit. And we could repair all of them."
Of the scores of papers and research reports the Embedded Security group publishes, Paar says one of the most often overlooked factors behind hacking is not technological vulnerabilities but economic ones.
"There's a reason that a lot of this hacking happens in countries that are economically not that well off," Paar says. "I think most people would way prefer having a good job in Silicon Valley or in a well-paying European company—rather than doing illegal stuff and trying to sell their services."
But as long as there are hackers, whatever their circumstances and countries of origin, Paar says smart engineering and present-day technology can stop most of them in their network box tracks.
"Our premise is that it's not that easy to do embedded security right, and that essentially has been confirmed," he says. "There are very few systems we looked at that we couldn't break. The shocking thing is the technology is there to get the security right. If you use state of the art technology, you can build systems that are very secure for practical rackmount applications."
refer to:
http://spectrum.ieee.org/riskfactor/computing/networks/vulnerable-smart-devices-make-an-internet-of-insecure-things
This entry was posted in Uncategorized and tagged automobile computer, Embedded and PC-Based, Embedded Computer, Embedded pc, Embedded SBC, Networking. Bookmark the permalink.
All-in-One Gaming Board
Embedded fanless
Embedded SBC
Gaming Platform
Fanless Computer
fanless pc
gaming embedded pc
Gaming I/O
gaming I/O controller | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,754 |
{"url":"https:\/\/math.stackexchange.com\/questions\/1443493\/mapping-class-group-and-normal-subgroups","text":"Mapping Class Group and normal subgroups\n\nI study braid groups and algebraic geometry, and I learned about the mapping class group (MCG).\n\nSpecifically, I encountered the MCG of the punctured unit disk (assume n punctures). I understand that the MCG of the punctured disk is a subgroup of the bijective diffeomorphisms group from the punctured disk to itself, and that diffeomorphisms are homotopic to the identity diffeomorphism on the boundary. How can I show that this subgroup is a normal subgroup of the bijective diffeomorphisms group?\n\nIf I name the diffeomorphisms group K, and the normal subgroup S, is the MCG K\/S? If this is true, how do I prove this?\n\nThanks!\n\n1 Answer\n\nEvery topological group $G$ fits into a short exact sequence\n\n$$1 \\to G_0 \\to G \\to \\pi_0(G) \\to 1$$\n\nwhere $\\pi_0(G)$ is the group of connected components and $G_0$ is the connected component of the identity. In particular, the fact that $G \\to \\pi_0(G)$ is a surjective homomorphism with kernel $G_0$ means that $G_0$ is normal.\n\nIf $G$ is taken to be a diffeomorphism group, then $\\pi_0(G)$ is the corresponding mapping class group. In particular, it is a quotient, not a subgroup, of the diffeomorphism group.\n\n\u2022 It's probably worth noting that the mapping class group usually can't be realized by diffeomorphisms; that is, there is usually not a section $\\text{MCG}(M) \\to \\text{Diff}(M)$. This is true for surfaces of genus at least 3, for instance. \u2013\u00a0user98602 Sep 21 '15 at 0:46","date":"2019-10-23 04:55:55","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\": 0.8679934740066528, \"perplexity\": 124.26030707836603}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"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-2019-43\/segments\/1570987829458.93\/warc\/CC-MAIN-20191023043257-20191023070757-00338.warc.gz\"}"} | null | null |
package org.apache.pdfbox.tools;
import java.awt.GraphicsEnvironment;
import org.apache.pdfbox.debugger.PDFDebugger;
import picocli.CommandLine;
import picocli.CommandLine.Command;
import picocli.CommandLine.ParameterException;
import picocli.CommandLine.Spec;
/**
* Simple wrapper around all the command line utilities included in PDFBox.
* Used as the main class in the runnable standalone PDFBox jar.
*/
@Command(name="pdfbox",
customSynopsis = "pdfbox [COMMAND] [OPTIONS]",
footer = {
"See 'pdfbox help <command>' to read about a specific subcommand"
},
versionProvider = Version.class)
public final class PDFBox implements Runnable
{
@Spec CommandLine.Model.CommandSpec spec;
/**
* Main method.
*
* @param args command line arguments
*/
public static void main(String[] args)
{
// suppress the Dock icon on OS X
System.setProperty("apple.awt.UIElement", "true");
CommandLine commandLine = new CommandLine(new PDFBox()).setSubcommandsCaseInsensitive(true);
if (!GraphicsEnvironment.isHeadless())
{
commandLine.addSubcommand("debug", PDFDebugger.class);
}
commandLine.addSubcommand("decrypt", Decrypt.class);
commandLine.addSubcommand("encrypt", Encrypt.class);
commandLine.addSubcommand("decode", WriteDecodedDoc.class);
commandLine.addSubcommand("export:images", ExtractImages.class);
commandLine.addSubcommand("export:text", ExtractText.class);
commandLine.addSubcommand("export:fdf", ExportFDF.class);
commandLine.addSubcommand("export:xfdf", ExportXFDF.class);
commandLine.addSubcommand("import:fdf", ImportFDF.class);
commandLine.addSubcommand("import:xfdf", ImportXFDF.class);
commandLine.addSubcommand("overlay", OverlayPDF.class);
commandLine.addSubcommand("print", PrintPDF.class);
commandLine.addSubcommand("render", PDFToImage.class);
commandLine.addSubcommand("merge", PDFMerger.class);
commandLine.addSubcommand("split", PDFSplit.class);
commandLine.addSubcommand("fromimage", ImageToPDF.class);
commandLine.addSubcommand("fromtext", TextToPDF.class);
commandLine.addSubcommand("version", Version.class);
commandLine.addSubcommand("help", CommandLine.HelpCommand.class);
commandLine.execute(args);
}
@Override
public void run()
{
throw new ParameterException(spec.commandLine(), "Error: Subcommand required");
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,518 |
require 'wired_for_change/version'
require 'wired_for_change/salsa_base'
require 'wired_for_change/salsa_connection'
require 'wired_for_change/salsa_supporter'
require 'wired_for_change/salsa_donation'
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,884 |
AAU Member Breaks Record
Original article found at milesplit.com
Brandon Miller had set over a handful of records at the AAU level entering his first year of high school at St. Louis John Burroughs (MO) High in 2017.
So it was only a matter of time before the freshman mid-distance wonder was at it for his high school track and field team.
Over the weekend, the 14-year-old not only claimed his first state title at the Missouri Track and Field Championships in Jefferson City, but he also secured a freshmen class record in the 800m with his time of 1:50.84 in the Class 3A race.
Miller broke the previous national freshmen record of 1:51.03 held by Michael Granville of Bell Gardens, California, back in 1993. Granville eventually went on to set the national record of 1:46.45 in 1996.
It followed an age group world record for 14-year-olds last July at the AAU Junior Olympics in Houston, where Miller ran a winning time of 1:51.23.
Miller's Class 3A time stood as the state championship's best 800m performance across all classifications and stands as a US No. 15 mark and US No. 1 for all freshmen.
Miller also contributed to wins in the 4x200, 4x400 and 4x800 as John Burroughs earned a Class 3A team title over Reeds Springs by two points. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,030 |
Q: How to create a mapping function that will map the Image pixel data to actual WGS84 coordinate data of the original Point Cloud Data Currently faced with this challenge of mapping the pixel data I have gotten from an image to the actual X,Y coordinate in a WGS84 system
Currently using Python script to do this - but I am relatively new with it so I may be missing some stuff here >.<
I have a black-white image created using Canny edge detection till it's only white lines outlining the image.
Now I need to convert this black-white 'outline' image
into a txt file and convert those pixels into the equivalent WGS84 of the actual outdoors.
The original image is a snapshot of the top down view of point cloud data in X,Y,Z
I am able to get the original mincoord(X1,Y2) and maxcoord (X2,Y2) of the WGS84 of the original image.
Before:
After:
Save pixel data in txt format in PIL
from PIL import Image
im = Image.open("outdoor.jpg")
fil = open('file', 'w')
pixel = im.load()
row, column = im.size
for y in range(column):
for x in range(row):
pixel = pix[x, y]
fil.write(str(pixel) + '\n')
fil.close()
How could I do this mapping from the 'local' pixel data to the 'global' WGS84 coordinate equivalent and write it to a txt file with the X Y coordinates?
Thank you and all help is always appreciated :)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,694 |
import os
import io
import re
from setuptools import setup, find_packages
cwd = os.path.abspath(os.path.dirname(__file__))
def read(*names, **kwargs):
with io.open(
os.path.join(cwd, *names), encoding=kwargs.get('encoding', 'utf8')
) as fp:
return fp.read()
def find_version(*file_paths):
version_file = read(*file_paths)
version_match = re.search(r'^__version__ = [\'"]([^\'"]*)[\'"]',
version_file, re.M)
if version_match:
return version_match.group(1)
raise RuntimeError('Unable to find version string.')
classifiers = [
# 3 - Alpha
# 4 - Beta
# 5 - Production/Stable
'Development Status :: 4 - Beta',
'Intended Audience :: Developers',
'Topic :: Software Development :: Build Tools',
'License :: OSI Approved :: MIT License',
'Programming Language :: Python :: 3',
'Programming Language :: Python :: 3.5',
'Programming Language :: Python :: 3.6',
'Programming Language :: Python :: 3.7',
]
setup(
name='wt',
version=find_version('wt', '__init__.py'),
description='Static blog generator',
long_description=read('README.md'),
long_description_content_type='text/markdown',
url='https://github.com/ysegorov/wt',
author='Yuri Egorov',
author_email='ysegorov@gmail.com',
license='MIT',
classifiers=classifiers,
keywords='blog static site generator',
packages=find_packages(exclude=['tests']),
package_data={
'wt': ['templates/*.html', 'templates/*.xml',
'fixtures/*.yaml', 'fixtures/*.md',
'fixtures/*.css', 'fixtures/*.png']},
install_requires=[
'markdown>=3.0.1',
'jinja2>=2.10',
'pyyaml>=3.13',
'cached-property>=1.3.0',
],
extras_require={
'dev': [
'twine>=1.8.1',
'coverage>=4.2',
'pytest>=3.7.0',
'pytest-cov>=2.3.1',
'pytest-describe>=0.11.0',
]
},
setup_requires=['pytest-runner>=2.0,<3dev'],
tests_require=[
'coverage>=4.2',
'pytest>=3.7.0',
'pytest-cov>=2.3.1',
'pytest-describe>=0.11.0'],
entry_points={
'console_scripts': [
'wt=wt.cli:main'
]
}
)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,902 |
HomeFAMOUS PEOPLEJim Dolan WABC, Bio, Wiki, Age, Height, Family, Wife, Married, Salary, Net Worth
Jim Dolan WABC, Bio, Wiki, Age, Height, Family, Wife, Married, Salary, Net Worth
Jim Dolan Biography and Wiki
Jim Dolan Age | How old is NBC Jim Dolan?
Jim Dolan Birthday
Jim Dolan Height and Weight
Jim Dolan Family
Jim Dolan Education
Jim Dolan Wife | Is NBC Jim Dolan Married?
Jim Dolan Salary
Jim Dolan Net Worth | How Rich Is Jim Dolan?
Jim Dolan Body Measurements
Jim Dolan KCEN Wikipedia
How Old is Jim Dolan?
Is Jim Dolan Married?
Jim Dolan Social Media Handles
Jim Dolan is a well-known and talented American news journalist currently working for WABC News as a news reporter.
Dolan is 67 years old as of 2022, she was born on February 17, 1955, in New York, United States.
Jim celebrates his birthday on the 17th of February every year.
Dolan stands at an average height of 5 feet 6 inches with a moderate weight.
Dolan was born to parents Helen Dolan and Chuck Dolan. It is known that his a cable industry icon and was the first to wire Manhattan for cable TV. Likewise, he was the founder of HBO. He grew up along with his two siblings whose identities are still not disclosed.
Jim graduated with a Bachelor's degree in Business and Communications from Boston University.
Dolan was previously married to her loving wife, meteorologist Linda Church. In their marriage, the couple was blessed with two children named Colleen Dolan, and Paul Dolan. Unfortunately, they divorced and parted ways due to irreconcilable differences.
Also Read About: John Del Giorno WABC, Bio, Wiki, Age, Height, Wife, Married, Salary, Net Worth
NBC News' Jim receives an annual salary of $85,000-$95,000 from his career as a news journalist. The normal pay of a news journalist ranges from between $ 24,292 and $ 72,507 which translates to an hourly average wage of between $ 10.15 and $ 31.32.
Dolan has an estimated net worth of $1 Million – $5 Million (Approx.) as of 2022, from his successful career. His primary source of income is his career as a journalist. Through his various sources of income, he has been able to accumulate good fortune but prefers to lead a modest lifestyle.
Jim Dolan Photo
Full Names: Jim Dolan.
Age: 67 years old.
Birthday: February 17.
Height: 5 feet 6 inches.
Wife: Linda Church.
Salary: $85,000 – $95,000.
Net Worth: $1 Million – $5 Million (Approx.).
Dolan is a well-known broadcast journalist who is currently working for ABC7 News. Here, he serves as the station's news reporter. Likewise, he provides report coverages for Eyewitness News on WABC-TV in New York City. He led the report coverage of The LA Riots.
Most noteworthy, Jim was among the first reporters that covered The Persian Gulf War. He was also on the ground during The War in Afghanistan live coverage. In addition, he was appointed to cover The death of Princess Diana and the Intifada in the Middle East.
It is known that Dolan started his career in broadcasting at WABC-TV in New York City in 1986. During this time, he provided live report coverages of The Oklahoma City Bombing as well as The LA Riots. Throughout his career, he has done reports in more than thirty countries on five continents.
Also Read About: Kris Wu Bio, Wiki, Age, Height, Parents, Nationality, Wife, Arrested, Rape, Net Worth and Songs
Jim was previously married to her loving wife, meteorologist Linda Church. In their marriage, the couple was blessed with two children named Colleen Dolan, and Paul Dolan. Unfortunately, they divorced and parted ways due to irreconcilable differences.
Famous Journalists in USA
Jeff Smith WABC, Bio, Wiki, Age, Height, Family, Wife, Married, Salary, Net Worth
Margaret Brennan (CBS) Bio, Wiki, Age, Height, Parents, Husband, Baby, Salary and Net Worth
USA JOURNALISTS
Alex Perez (ABC Journalist) Biography, Wiki, Age, Height, Family, Wife, Partner, ABC News, Salary and Net Worth
September 9, 2022 Jimmy Mutua USA JOURNALISTS
Alex Perez is an American news anchor and reporter who has worked in the broadcasting channels such as NBC Chicago. Perez is currently working for ABC News. […]
Bryant Reed KDKA, Bio, Wiki, Age, Height, Family, Wife, Married, Salary, Net Worth
Bryant Reed is a professional and talented American news journalist currently working for KDKA-TV. He has also worked for WECT News before coming to Channel 2 News. […]
Boris Sanchez (CNN) Bio, Wiki, Height, Wife, Married, Salary, Net Worth, CNN
November 30, 2022 Jimmy Mutua USA JOURNALISTS
Boris Sanchez is an American journalist who was born in Havana, Cuba, USA. Currently, he works as the White House correspondent for CNN covering the […] | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,754 |
Erato ("Umiłowana", "Ukochana", gr. Eratṓ, łac. Erato 'Pożądana', 'Namiętna') – w mitologii greckiej muza poezji miłosnej.
Uchodziła za córkę boga Zeusa i tytanidy Mnemosyne oraz za siostrę: Euterpe, Kalliope, Klio, Melpomene, Polihymnii, Talii, Terpsychory i Uranii.
Była jedną spośród dziewięciu muz olimpijskich (przebywały na Olimpie), które należały do orszaku Apollina (Apollon Musagetes), ich przewodnika. Wraz ze swoimi siostrami uświetniała śpiewem biesiady bosko-ludzkie (m.in. zaślubiny Tetydy i Peleusa oraz Harmonii i Kadmosa), a także uczty olimpijskie samych bogów.
W sztuce przedstawiana jest zwykle jako kobieta z kitarą (lub cytrą) – atrybutem symbolizującym dziedzinę sztuki, której patronowała.
Imieniem muzy nazwano jedną z planetoid – (62) Erato oraz rodzaj roślin z rodziny astrowatych – Erato.
Zobacz też
kameny
Pierydy
Uwagi
Przypisy
Bibliografia
Muzy
Bóstwa poezji | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,656 |
These regions are not to be confused with the former districts of the Northwest Territories.
Prior to the creation of Nunavut in 1999, there were five census division in the NWT. Their boundaries were altered somewhat as part of the adjustment. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,837 |
<?php
namespace Czim\CmsModels\ModelConfig;
class ModelList
{
/**
* @var array
*/
protected $main = [];
/**
* @var ListFilter[]
*/
protected $filters = [];
public function toArray(): array
{
return $this->buildListArray();
}
/**
* Disable filters even if they're specified
*
* @return ModelList|$this
*/
public function disableFilters(): ModelList
{
$this->main['disable_filters'] = true;
return $this;
}
/**
* Sets a list of filter definitions
*
* In addition to any Field-defined filters.
*
* @param ListFilter[] $filters
* @return ModelList
*/
public function filters(array $filters): ModelList
{
$this->filters = $filters;
return $this;
}
/**
* Set a sorting strategy to enable by default
*
* Stacks strategies if called more than once
*
* @param string $strategy alias/name or <FQN>@<method> for custom ordering
* @return ModelList|$this
*/
public function defaultSortingStrategy(string $strategy): ModelList
{
if (is_string($this->main['default_sort'])) {
$this->main['default_sort'] = [ $this->main['default_sort'] ];
$this->main['default_sort'][] = $strategy;
return $this;
}
$this->main['default_sort'] = $strategy;
return $this;
}
/**
* Set a list of sorting strategies to enable by default
*
* @param string[] $strategies list of alias/name or <FQN>@<method> for custom ordering
* @return ModelList|$this
*/
public function defaultSortingStrategies(array $strategies): ModelList
{
array_map([$this, 'defaultSortingStrategy'], $strategies);
return $this;
}
/**
* Set the (default) page size
*
* @param int $size
* @return ModelList|$this
*/
public function pageSize(int $size): ModelList
{
$this->main['page_size'] = $size;
return $this;
}
/**
* Set the available page size choices that users can switch to manually
*
* @param int[] $sizes first value will be default
* @return ModelList|$this
*/
public function pageSizes(array $sizes): ModelList
{
$this->main['page_size'] = $sizes;
return $this;
}
/**
* Disable the use and display of scopes, even if specified
*
* @return ModelList|$this
*/
public function disableScopes(): ModelList
{
$this->main['disable_scopes'] = true;
return $this;
}
/**
* Sets scopes for the listing
*
* @param string[] $scopes scopes or scoping strategies, keyed by scope name
* @return ModelList|$this
*/
public function scopes(array $scopes): ModelList
{
$this->main['scopes'] = $scopes;
return $this;
}
/**
* Adds an action to be performed when clicking a row.
*
* Will stack if called multiple times, in order.
* The first action that is permitted, is performed.
*
* @param string $action action alias
* @return ModelList|$this
*/
public function rowClickAction(string $action): ModelList
{
if ( ! array_has($this->main, 'default_action')) {
$this->main['default_action'] = [];
}
$this->main['default_action'][] = $action;
return $this;
}
/**
* Set a view to include before the listing
*
* @param string $view
* @param array $variables optional list of variable names to pass through to the included view
* @return ModelList|$this
*/
public function viewBefore(string $view, array $variables = []): ModelList
{
$this->main['before'] = compact('view', 'variables');
return $this;
}
/**
* Set a view to include after the listing
*
* @param string $view
* @param array $variables optional list of variable names to pass through to the included view
* @return ModelList|$this
*/
public function viewAfter(string $view, array $variables = []): ModelList
{
$this->main['after'] = compact('view', 'variables');
return $this;
}
// ------------------------------------------------------------------------------
// List parent relations
// ------------------------------------------------------------------------------
/**
* Hide everything but top-level list parents by default to this relation
*
* Useful to remove clutter for nested content with a click-through-to-children setup.
* Set to relation method name that must be present in 'parents'.
*
* @param string $relation
* @return ModelList|$this
*/
public function parentRelationActiveByDefault(string $relation): ModelList
{
$this->main['default_top_relation'] = $relation;
if ( ! array_get($this->main, 'parents')) {
// Assume the relation (= the field) if no parents set yet.
$this->main['parents'] = [
[
'relation' => $relation,
'field' => $relation,
],
];
}
return $this;
}
/**
* List parents for list hierarchy handling
*
* The relations by which this listing may be 'filtered'.
*
* @param string[] $relations either relation names, or key-value pairs: relation => field
* @return ModelList|$this
*/
public function parentRelations(array $relations): ModelList
{
$this->main['parents'] = [];
foreach ($relations as $key => $value) {
if (is_string($key)) {
$parent = [
'relation' => $key,
'field' => $value,
];
} else {
$parent = [
'relation' => $value,
'field' => $value,
];
}
$this->main['parents'][] = $parent;
}
return $this;
}
// ------------------------------------------------------------------------------
// Orderable
// ------------------------------------------------------------------------------
/**
* Allow manual ordering by dragging and dropping records
*
* This may also be enabled by calling orderable() on a corresponding Field entry.
*
* @param bool $orderable
* @return ModelList|$this
*/
public function orderable(bool $orderable = true): ModelList
{
$this->main['orderable'] = $orderable;
return $this;
}
/**
* The strategy by which the model can be ordered
*
* For now, this should always be 'listify'.
*
* @param string $strategy
* @return ModelList|$this
*/
public function orderableStrategy(string $strategy): ModelList
{
$this->main['orderable_strategy'] = $strategy ?: 'listify';
return $this;
}
/**
* The column used for the order strategy ('position' for listify)
*
* This may also be set by calling orderable() on a corresponding Field entry.
* This wil also automatically enable orderable.
*
* @param string $column
* @return ModelList|$this
*/
public function orderableColumn(string $column): ModelList
{
$this->main['orderable_column'] = $column;
return $this;
}
/**
* If listify is scoped in a way to restrict it for a relation's foreign key,
* set the relation name method
*
* @param string $relation
* @return ModelList|$this
*/
public function orderableRelationScope(string $relation): ModelList
{
$this->main['order_scope_relation'] = $relation;
return $this;
}
// ------------------------------------------------------------------------------
// Activatable
// ------------------------------------------------------------------------------
/**
* Allow manual activation
*
* Whether the model may be activated/deactived through the listing;
* ie. whether it has a manipulable 'active' flag.
*
* This may also be enabled by calling activatable() on a corresponding Field entry.
*
* @param bool $activatable
* @return ModelList|$this
*/
public function activatable(bool $activatable = true): ModelList
{
$this->main['activatable'] = $activatable;
return $this;
}
/**
* The column used for the activate strategy ('active' by default), will be toggled true/false
*
*
* This may also be set by calling activatable() on a corresponding Field entry.
* This wil also automatically enable activatable.
*
* @param string $column
* @return ModelList|$this
*/
public function activatableColumn(string $column): ModelList
{
$this->main['activatable_column'] = $column;
return $this;
}
protected function buildListArray(): array
{
$array = $this->main;
$this->applyFiltersDataToArray($array);
return $array;
}
protected function applyFiltersDataToArray(array &$array): void
{
if ( ! count($this->filters)) {
return;
}
foreach ($this->filters as $filter) {
$array['filters'][ $filter->getKey() ] = $filter->toArray();
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,872 |
{"url":"https:\/\/brilliant.org\/discussions\/thread\/simpler-solution-of-mechanics-warmups-level-4\/","text":"# Simpler solution of Mechanics Warmups: Level 4 Challenges, problem 3\n\nSince my friends and I are taking non-calculus based Physics classes, we were shocked by the solutions given for their complexity. We found out a much simpler way to solve the problem--even people who have just learnt AP Physics 1 can understand.\n\nFirst, assume that at the initial conditions (before oscillations start), both of the springs are at their initial lengths. At the instance when the pulley-spring-rope-mass system reaches equilibrium, Free-body Diagram of the movable pulley is shown below, (image 1) where tension in the rope with springs is T1 while tension in the rope to hang the mass is T2. The ropes are massless, so tension should have the same magnitude in a rope.\n\nAt that point, the movable pulley is in state of equilibrium so the net force acting on it is zero, which means taking upward to be positive)\nFnet=2T1-Mg= 0,\nT1=Mg\/2.\n\nAssume that the springs are uniform. By Hooke\u2019s Law, displacements of the springs have magnitude of\n\u0394X1=F1\/K1=T\/K1=Mg\/2K1 and\n\u0394X2= F2\/K2=T\/K2=Mg\/2K2, respectively.\nThe lengths of ropes are unchanged, so the pulley\u2019s displacement from initial position has a magnitude of\n\u0394Xpulley=total increased length of springs(and ropes, which equals 0)=(\u0394X1+\u0394X2)\/2=( Mg\/2K1+ Mg\/2K2)\/2.\n\nThen, consider the springs to be one equivalent spring, whose spring constant can\u2019t be obtained directly from method of connecting springs in series or in parallel (the two springs\u2019 amplitudes are DIFFERENT in the oscillation). Forces acting on the movable pulley can be now considered as: (image 2)\n\nThe net force again is zero, so\nF equivalent spring=T2=Mg.\nApply Hooke\u2019s Law for a second time,\nK equivalent spring=F\/\u0394X\n=F equivalent spring\/\u0394Xpulley=4K1K2\/(K1+K2).\nUsing quantities above, we can get period of the system\u2019s oscillation is\nT=2\u03c0\u221a(M\/K)= 2\u03c0\u221a(M\/K equivalent spring)= 2\u03c0\u221a(M(K1+K2)\/4K1K2)\n\u22480.7695s.\n\nNote by Veronica Luo\n1\u00a0week, 2\u00a0days ago\n\nThis discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution \u2014 they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.\n\nWhen posting on Brilliant:\n\n\u2022 Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .\n\u2022 Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting \"I don't understand!\" doesn't help anyone.\n\u2022 Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.\n\nMarkdownAppears as\n*italics* or _italics_ italics\n**bold** or __bold__ bold\n- bulleted- list\n\u2022 bulleted\n\u2022 list\n1. numbered2. list\n1. numbered\n2. list\nNote: you must add a full line of space before and after lists for them to show up correctly\nparagraph 1paragraph 2\n\nparagraph 1\n\nparagraph 2\n\n[example link](https:\/\/brilliant.org)example link\n> This is a quote\nThis is a quote\n # I indented these lines\n# 4 spaces, and now they show\n# up as a code block.\n\nprint \"hello world\"\n# I indented these lines\n# 4 spaces, and now they show\n# up as a code block.\n\nprint \"hello world\"\nMathAppears as\nRemember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.\n2 \\times 3 $2 \\times 3$\n2^{34} $2^{34}$\na_{i-1} $a_{i-1}$\n\\frac{2}{3} $\\frac{2}{3}$\n\\sqrt{2} $\\sqrt{2}$\n\\sum_{i=1}^3 $\\sum_{i=1}^3$\n\\sin \\theta $\\sin \\theta$\n\\boxed{123} $\\boxed{123}$\n\nSort by:\n\n- 6\u00a0days, 6\u00a0hours ago\n\nSure, I'll try to fix it. Thanks! :)\n\n- 6\u00a0days ago","date":"2021-05-06 10:30:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 8, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"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\": 0.9583323001861572, \"perplexity\": 3137.367213446144}, \"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\/1620243988753.91\/warc\/CC-MAIN-20210506083716-20210506113716-00250.warc.gz\"}"} | null | null |
Mahdi Mohammed Gulaid "Khadar" (en somalí: Mahdi Maxamed Guleed "Khadar", en árabe: مهدي محمد جوليد خضر) es un político somalí, primer ministro interino de la República Federal de Somalia desde el 25 de julio y el 23 de septiembre de 2020, y viceprimer ministro del 21 de marzo de 2017 al 25 de julio de 2020.
Biografía
Antes de entrar en política, ejerció la abogacía en Somalilandia y trabajó con la segunda Comisión Electoral de Somalilandia como asesor legal.
Gulaid fue nombrado viceprimer ministro de Somalia en marzo de 2017 por el entonces primer ministro Hassan Ali Khaire. Desde que ocupó este cargo, el Viceprimer Ministro ayudó a establecer las agendas nacionales y de desarrollo de Somalia y gestionó una amplia cartera de prioridades gubernamentales. Semanalmente, Gulaid supervisó el gabinete y presidió una serie de comités a nivel ministerial, particularmente en los sectores económico y social. También presidió las sesiones del Fondo para el Desarrollo y la Reconstrucción de Somalia, una plataforma vital que permite al gobierno y los socios para el desarrollo proporcionar orientación estratégica y supervisión para las actividades de desarrollo en Somalia.
Como viceprimer ministro, Gulaid también presidió el capítulo 4 de la Arquitectura de Seguridad Nacional de Somalia, incluida la prevención y la lucha contra el extremismo violento. Más recientemente, copresidió el Foro de Asociación de Somalia en Bruselas, que concluyó con un éxito rotundo.
El Presidente Abdullahi Mohamed nombró a Mahdi Mohammed Gulaid como el primer ministro en funciones el 25 de julio de 2020 a raíz de la moción de censura parlamentaria contra el gobierno del primer ministro, Hassan Ali Khaire.
Referencias
Enlaces externos
Primeros ministros de Somalia
Políticos de Somalia del siglo XXI | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,879 |
using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Windows;
using System.Windows.Controls;
using System.Windows.Data;
using System.Windows.Documents;
using System.Windows.Input;
using System.Windows.Media;
using System.Windows.Media.Imaging;
using System.Windows.Navigation;
using System.Windows.Shapes;
namespace EnumerableTest.Runner.Wpf
{
/// <summary>
/// KeyValueGrid.xaml の相互作用ロジック
/// </summary>
public partial class KeyValueGrid : UserControl
{
#region KeyItemTemplate
public static DependencyProperty KeyItemTemplateProperty { get; } =
DependencyProperty.Register(
"KeyItemTemplate",
typeof(DataTemplate),
typeof(KeyValueGrid),
new FrameworkPropertyMetadata()
{
PropertyChangedCallback = OnKeyItemTemplateChanged,
}
);
public DataTemplate KeyItemTemplate
{
get { return (DataTemplate)GetValue(KeyItemTemplateProperty); }
set { SetValue(KeyItemTemplateProperty, value); }
}
public static void OnKeyItemTemplateChanged(DependencyObject sender, DependencyPropertyChangedEventArgs e)
{
var @this = (KeyValueGrid)sender;
@this.Reset();
}
#endregion
#region ValueItemTemplate
public static DependencyProperty ValueItemTemplateProperty { get; } =
DependencyProperty.Register(
"ValueItemTemplate",
typeof(DataTemplate),
typeof(KeyValueGrid),
new FrameworkPropertyMetadata()
{
PropertyChangedCallback = OnValueItemTemplateChanged,
}
);
public DataTemplate ValueItemTemplate
{
get { return (DataTemplate)GetValue(ValueItemTemplateProperty); }
set { SetValue(ValueItemTemplateProperty, value); }
}
public static void OnValueItemTemplateChanged(DependencyObject sender, DependencyPropertyChangedEventArgs e)
{
var @this = (KeyValueGrid)sender;
@this.Reset();
}
#endregion
#region ItemsSource
public static DependencyProperty ItemsSourceProperty { get; } =
DependencyProperty.Register(
"ItemsSource",
typeof(IEnumerable),
typeof(KeyValueGrid),
new FrameworkPropertyMetadata()
{
PropertyChangedCallback = OnItemsSourceChanged,
}
);
public IEnumerable ItemsSource
{
get { return (IEnumerable)GetValue(ItemsSourceProperty); }
set { SetValue(ItemsSourceProperty, value); }
}
public static void OnItemsSourceChanged(DependencyObject sender, DependencyPropertyChangedEventArgs e)
{
var @this = (KeyValueGrid)sender;
@this.Reset();
}
#endregion
void Reset()
{
var itemsSource = ItemsSource;
var keyItemTemplate = KeyItemTemplate;
var valueItemTemplate = ValueItemTemplate;
if (itemsSource == null || keyItemTemplate == null || valueItemTemplate == null)
{
return;
}
grid.Children.Clear();
var rowIndex = 0;
foreach (var kv in itemsSource)
{
grid.RowDefinitions.Add(new RowDefinition() { Height = GridLength.Auto });
var type = kv.GetType();
var key = type.GetProperty("Key").GetValue(kv);
var value = type.GetProperty("Value").GetValue(kv);
var keyItem = (FrameworkElement)keyItemTemplate.LoadContent();
keyItem.DataContext = key;
Grid.SetRow(keyItem, rowIndex);
Grid.SetColumn(keyItem, 0);
grid.Children.Add(keyItem);
var valueItem = (FrameworkElement)valueItemTemplate.LoadContent();
valueItem.DataContext = value;
Grid.SetRow(valueItem, rowIndex);
Grid.SetColumn(valueItem, 1);
grid.Children.Add(valueItem);
rowIndex++;
}
}
public KeyValueGrid()
{
InitializeComponent();
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,351 |
Q: Returning a string array containing string forms of integers, replacing multiples of certain numbers with strings of text I need to have a function accept two arguments, int start, and int end. I need the two integers to give an integer series. For example, start = 1, end = 10. It would return 1,2,3,4,5,6,7,8,9. I need the function to return a string array containing the string forms of these numbers, except for multiples of 3, use "FirstName" instead of the number, for multiples of 5 use "LastName", and for multiples of both 3 and 5 use "FullName".
public class StringHandler() {
public static void function(int start, int end) {
int size = end - start;
System.out.println("Name - ");
for(int i = 1; i <= size; i++) {
if (i % 3 == 0 && i % 5 == 0) {
System.out.print("FullName" + " ");
} else if (i % 3 == 0) {
System.out.print("FirstName" + " ");
} else if (i % 5 == 0) {
System.out.print("LastName" + " ");
} else {
System.out.print(Integer.toString(i) + " ");
}
}
}
public static void main(String[] args) {
function(1, 30);
}
}
Just having a hard coding block right now and cannot figure out what I am doing wrong.
A: Your for-loop should be like this:
for(int i = start; i < end; i++)
based on:
For example, start = 1, end = 10. It would return 1,2,3,4,5,6,7,8,9.
EDIT:
The full source code would be something like this:
public static String[] function(int start, int end)
{
int size = end - start;
String[] array = new String[size];
int counter = 0;
for(int i = start; i < end; i++)
{
if(i % 3 == 0 && i % 5 == 0) array[counter] = "FullName";
else if(i % 3 == 0) array[counter] = "FirstName";
else if(i % 5 == 0) array[counter] = "LastName";
else array[counter] = Integer.toString(i);
if(i < end - 1) array[counter] += " ";
counter++;
}
return array;
}
A: This should do the trick:
public static String[] function(int start, int end) {
int size = end - start;
String[] result = new String[size];
System.out.println("Name - ");
for(int i = start; i <= end; i++) {
if (i % 3 == 0 && i % 5 == 0) {
result[i-start] = "FullName ";
} else if (i % 3 == 0) {
result[i-start] = "FirstName ";
} else if (i % 5 == 0) {
result[i-start] = "LastName ";
} else {
result[i-start] = Integer.toString(i) + " ";
}
}
System.out.println(Arrays.toString(result));
return result;
}
public static void main(String[] args) {
function(1, 30);
}
The println isn't necessary, just so you can see it in the console. Notice that the return type is no longer void but now String[], so you can do something along the lines of:
String[] stringArray = function(1,30);
when you call function in order to get hold of the resulting array.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,107 |
\section{Introduction}
Class-incremental event detection is a challenging setting in lifelong learning, where the model is incrementally updated on a continual stream of data for new event types while retaining the event detection capability for all the previously learned types. The main challenge of class-incremental event detection lies in the \emph{catastrophic forgetting} problem, where the model's performance on previously learned types significantly drops after it is trained on new data. Recent studies~\cite{wang2019sentence,lopez2017gradient} have revealed that replaying stored samples of old classes can effectively alleviate the catastrophic forgetting issue. However, simply fine-tuning the entire model on the limited stored samples may result in overfitting, especially when the model has a huge set of parameters. How to effectively leverage the stored examples still remains an important question.
Prompt learning, which is to simply tune a template-based or continuous prompt appended to the input text while keeping all the other parameters freezed, has recently shown comparable or even better performance than fine-tuning the entire model in many NLP tasks~\cite{gpt3,jiang2020how,fewshotlearner,prefixtuning,ptuning,warp}. It is especially flavored by the lifelong learning setting since it only tunes a small amount of parameters, thus has the potential to alleviate the catastrophic forgetting and exemplar memory overfitting issues. Moreover, the prompts can also be used to store task-specific knowledge.
In this work, we propose a simple but effective incremental prompting framework that introduces \textbf{E}psodic \textbf{M}emory \textbf{P}rompts (\textbf{EMP}) to store the learned type-specific knowledge. At each training stage, we adopt a learnable prompt for each new event type added from the current task. The prompts are initialized with event type names and fine-tuned with the annotations from each task. To encourage the prompts to always carry and reflect type-specific information, we entangle the feature representation of each event mention with the type-specific prompts by optimizing its type distribution over them. After each training stage, we keep the learned prompts in the model and incorporate new prompts for next task. In this way, the acquired task-specific knowledge can be carried into subsequent tasks. Therefore, our EMP can be considered as a soft episodic memory that preserves the old knowledge and transfers it to new tasks. Our contributions can be summarized as follows:
\begin{itemize}[noitemsep,nolistsep,wide]
\item We propose \textbf{E}psodic \textbf{M}emory \textbf{P}rompts (\textbf{EMP}) which can explicitly carry previously learned knowledge to subsequent tasks for class-incremental event detection. Extensive experiments validate the effectiveness of our method.
\item To the best of our knowledge, we are the first to adopt prompting methods for class-incremental event detection. Our framework has the potential to be applied to other incremental learning tasks.
\end{itemize}
\section{Problem Formulation}
Given an input text $x_{1:L}$ and a set of target spans $\{(x_i, x_j)\}$ from it, an event detection model needs to assign each target span with an event type in the ontology or label it as \textit{Other} if the span is not an event trigger. For class-incremental event detection, we aim to train a single model $f_{\theta}$ on a sequence of $T$ tasks $\{\mathcal{D}_1, ..., \mathcal{D}_T\}$ that consist of non-overlapping event type sets $\{\mathcal{C}_1, ..., \mathcal{C}_T\}$\footnote{Though the type sets from all tasks contain \textit{Other}, they have distinct meanings given different seen types.}. In each $t$-th task, the model needs to classify each mention to any of the types that have seen so far $\mathcal{O}_t=\mathcal{C}_1\bigcup...\bigcup\mathcal{C}_t$. The training instances in each task $\mathcal{D}_t$ consist of tuples of an input text $x_{1:L}^{t}$, a target span $\bar{x}^{t}$, and its corresponding label $\mathbf{y}^t$ where $\mathbf{y}^t\in\mathcal{C}_t$. For convenience, the notations are for the $t$-th training stage by default unless denoted explicitly in the following parts of the paper.
\section{Approach}
\subsection{Span-based Event Detection}
\label{sec:baseline}
Given an input sentence $x_{1:L}^t$ from task $\mathcal{D}_t$, we first encode it with BERT~\cite{bert} to obtain the contextual representations $\mathbf{x}_{1:L}^t = \text{BERT}(x_{1:L}^t)$. Note that we freeze BERT's parameters in our method and all baselines. For each span $\bar{x}^{t}$, we concatenate its starting and ending token representations and feed them into a multilayer perceptron (MLP) to get the span representation $\mathbf{h}_{span}^{t}$. Then, we apply a linear layer on $\mathbf{h}_{span}^{t}$ to predict the type distribution of the span $p^{t}=linear(\mathbf{h}_{span}^{t})$. We use cross-entropy loss to train the model on $\mathcal{D}_t$:
\begin{equation}
\mathcal{L}_{C} = -\sum_{(\bar{x}^{t},y^t)\in \mathcal{D}_t} \text{log}\ p^{t}.
\end{equation}
\subsection{Episodic Memory Prompting}
\label{sec_emp}
To overcome the catastrophic forgetting and exemplar memory overfitting issues, we design a continuous prompting approach with Episodic Memory Prompts (EMPs) to preserve the knowledge learned from each task and transfer to new tasks.
Given an incoming task $\mathcal{D}_t$ and its corresponding new event type set $\mathcal{C}_t = \{c_1^t,...,c_{n_t}^t\}$, we first initialize a sequence of new \emph{prompts} $\mathbf{C}^t=[\mathbf{c}_1^t,...,\mathbf{c}_{n_t}^t]$ where $\mathbf{c}_i^t\in\mathbb{R}^{1\times e}$ is a type-specific prompt for type $c_i^t$, $n_t$ is the number of event types in the $t$-th task. $e$ is the embedding dimension size. In our experiments, we use the event type name to initialize each event type prompt $\mathbf{c}_i^t$ (see Appendix~\ref{sec:implement} for details). Note that we always preserve the prompts learned from previous tasks, thus the accumulated prompts until the $t$-th task are represented as $\mathbf{I}^t=[\mathbf{C}^1,...,\mathbf{C}^t]$. Given a particular sentence $x_{1:L}^t$ from $\mathcal{D}_t$, we concatenate it with the accumulated prompts $\mathbf{I}^t$, encode the whole sequence with BERT, and obtain the sequence of contextual representations $[\tilde{\mathbf{x}}_{1:L}^t; \tilde{\mathbf{I}}^t]$, where $\tilde{\mathbf{x}}_{1:L}^t$ and $\tilde{\mathbf{I}}^t$ denote the sequence of contextual embeddings of $x_{1:L}^t$ and $\mathbf{I}^t$ respectively. $[;]$ is concatenation operation. Then, similar as Section~\ref{sec:baseline}, we obtain a representation $\tilde{\mathbf{h}}_{span}^{t}$ for each span based on $\tilde{\mathbf{x}}^t_i$, and predict the logits over all target event types $\tilde{p}^{t}=linear(\tilde{\mathbf{h}}_{span}^{t})$.
We expect the EMPs to be specific to the corresponding event types and preserve the knowledge of each event type from previous tasks. So we design an entangled prompt optimization strategy to entangle the feature representation of each span with the event type-specific prompts by computing an event type probability distribution over them. Specifically, given a span representation $\tilde{\mathbf{h}}_{span}^{t}$ and EMP representations $\tilde{\mathbf{I}}^t$, we compute the probability distribution over all prompts as $\tilde{p}^t_c = \text{MLP}(\tilde{\mathbf{I}}^t)\cdot\tilde{\mathbf{h}}_{span}^{t}$, where $\cdot$ is the dot product. Finally, we combine the original logits $\tilde{p}^{t}$ and $\tilde{p}^t_c$ to predict the event type label for each span:
\begin{equation}
\label{equa:emp}
\tilde{\mathcal{L}}_{C} = -\sum_{(\bar{x}^t,y^t)\in \mathcal{D}_t} \text{log}\ (\tilde{p}^{t} + \tilde{p}^t_c).
\end{equation}
At the end of each training stage, we keep the learned prompts from the current task $\mathbf{C}^t$ in the model, and then initialize a new prompt $\mathbf{C}^{t+1}$ for the next task incrementally: $\mathbf{I}^{t+1} = [\mathbf{I}^{t};\mathbf{C}^{t+1}]$.
\subsection{Lifelong Learning with Experience Replay and Knowledge Distillation}
To alleviate the catastrophic forgetting issue, a common strategy is to store a limited amount of data from old tasks in a memory buffer and pass them to later tasks. We follow this strategy and adopt two popularly used methods: (1) Experience Replay which is to repeatedly optimize the model on the stored data in subsequent tasks; and (2) Knowledge Distillation (KD) that is to ensure the output probabilities and features from the current and previous models to be matched, respectively.
Specifically, after training on $\mathcal{D}_t$, we apply the herding algorithm~\cite{herding} to select 20 training samples for each type into the memory buffer, denoted as $\mathcal{M}$. Similar as Equation~\ref{equa:emp}, the objective for experience repaly is:
\begin{equation}
\mathcal{L}_{ER} = -\sum_{(\bar{x}^r,y^r)\in \mathcal{M}} \text{log}\ (\tilde{p}^{t} + \tilde{p}^t_c).
\end{equation}
For knowledge distillation, following~\cite{cao2020increment}, we apply both \emph{prediction-level} and \emph{feature-level} distillation, and use a temperature parameter to rescale the probabilities of prediction-level KD.
The objectives for prediction-level KD and feature-level KD are computed as:
\begin{equation*}
\mathcal{L}_{PD} = - \sum_{(\bar{x}^r, y^r)\in\mathcal{M}} (\tilde{p}^{t-1} + \tilde{p}^{t-1}_c)\ \text{log}\ ((\tilde{p}^{t} + \tilde{p}^t_c)).
\end{equation*}
\begin{equation*}
\mathcal{L}_{FD} = \sum_{(x^r, (x_i^r, x_j^r),y^r)\in\mathcal{M}} 1 - g(\Bar{\mathbf{h}}_{span}^{t-1}, \Bar{\mathbf{h}}_{span}^{t}),
\end{equation*}
where $g$ is the cosine similarity function. $\Bar{\mathbf{h}}_{span}^{t-1}$ and $\Bar{\mathbf{h}}_{span}^{t}$ are $l_2$-normalized features from the model at $t-1$ and $t$ stages, respectively.
\paragraph{Optimization}
We apply a weighting factor $\lambda$ to control how much loss from experience replay and knowledge distillation to use in each batch. The final loss is computed as:
\begin{equation*}
\mathcal{L} = \tilde{\mathcal{L}}_C + \lambda(\mathcal{L}_{ER}+\mathcal{L}_{PD}+\mathcal{L}_{FD}).
\label{equa:baseline_train}
\end{equation*}
\section{Approach}
\subsection{Lifelong Event Detection Baseline}
\label{sec:baseline}
\paragraph{Span-based Event Detection}
In this paper, we develop our approach based on a span-based event detection model.
We first encode the sentence with BERT to obtain the contextual representations $\mathbf{x}_{1:L} = \text{BERT}(x_{1:L})$. Note that we freeze BERT's parameters in our method and all baselines.
For each span, we concatenate its starting and ending token representations and feed them into a multilayer perceptron (MLP) to get the span representation $\mathbf{h}_{span}$. Then, we apply a linear layer on $\mathbf{h}_{span}$ to predict the type distribution of the span $p^{t}=linear(\mathbf{h}_{span})$, where $t$ denotes the $t$-th task. We use cross-entropy loss to train the model on $\mathcal{D}_t$:
\begin{equation}
\mathcal{L}_{C} = -\sum_{(x^t, (x_i^t, x_j^t),y^t)\in \mathcal{D}_t} \text{log}\ p^{t}.
\end{equation}
\paragraph{Experience Replay} To alleviate the catastrophic forgetting issue, a common strategy is to store a limited amount of data from old types in a memory buffer and replay them in subsequent tasks. Thus, after training on $t$-th task, we apply the herding algorithm~\cite{}\lifu{add reference} to select 20\lifu{how many? for each type or in total}\minqian{solved} training samples for each type into the memory buffer, denoted as $\mathcal{M}$. We use $p^{t}$ to denote the logits predicted by the model at $t$-th stage, so the loss for experience repaly is:
\begin{equation}
\mathcal{L}_{ER} = -\sum_{(x^r, (x_i^r, x_j^r),y^r)\in \mathcal{M}} \text{log}\ p^{t}.
\end{equation}
\paragraph{Knowledge Distillation} As another popular strategy to alleviate the forgetting problem, knowledge distillation (KD) is to ensure the output probabilities from the current and previous model to be matched, and the features encoded by the current model not distorted too much from those learned from the previous model. Following~\cite{cao2020increment}, we apply both \emph{prediction-level} and \emph{feature-level} distillation, and use a temperature parameter to rescale the probabilities of prediction-level KD. Denote the scaled output probability in $t$-th task as $\hat{p}^{t}$. The prediction-level KD loss is computed as:
\begin{equation}
\mathcal{L}_{PD} = - \sum_{(x^r, (x_i^r, x_j^r),y^r)\in\mathcal{M}} \hat{p}^{t-1}\ \text{log}\ \hat{p}^{t}.
\end{equation}
The feature-level KD loss is:
\begin{equation}
\mathcal{L}_{FD} = \sum_{(x^r, (x_i^r, x_j^r),y^r)\in\mathcal{M}} 1 - g(\Bar{\mathbf{h}}_{span}^{t-1}, \Bar{\mathbf{h}}_{span}^{t}),
\end{equation}
where $g$ is the cosine similarity function. $\Bar{\mathbf{h}}_{span}^{t-1}$ and $\Bar{\mathbf{h}}_{span}^{t}$ are $l_2$-normalized features from the model at $t-1$ and $t$ stages, respectively.
\paragraph{Optimization}
We apply a weighting factor $\lambda$ to control how much loss from experience replay and knowledge distillation to use in each batch. The final loss is computed as:
\begin{equation}
\mathcal{L} = \mathcal{L}_C + \lambda(\mathcal{L}_{ER}+\mathcal{L}_{PD}+\mathcal{L}_{FD}).
\label{equa:baseline_train}
\end{equation}
\subsection{Episodic Memory Prompting}
To better overcome the catastrophic forgetting issue and improve the generalizability of event detection method under class-incremental setting, we further design a continuous prompting approach that relies on Episodic Memory Prompts to preserve the knowledge learned from each task and transfer to new tasks.
Given an income task $\mathcal{D}_t$ and its corresponding new event type set $\mathcal{C}_t = \{c_1^t,...,c_{n_t}^t\}$, we first initialize a sequence of new \emph{prompts} $\mathbf{C}^t=[\mathbf{c}_1^t,...,\mathbf{c}_{n_t}^t]$ where $\mathbf{c}_i^t\in\mathbb{R}^{1\times e}$ is a type-specific prompt for type $c_i^t$. $e$ is the embedding dimension size. In our experiments, we use the event type names as the prior knowledge to initialize the parameter of each event type prompt $\mathbf{c}_i^t$ (see Section~\ref{sec:implement} for details). Note that we always preserve the prompts learned from previous tasks, thus the accumulated prompts until the $t$-th task are represented as $\mathbf{I}^t=[\mathbf{C}^1,...,\mathbf{C}^t]$. Given a particular sentence $x_i^t$ from $D^t$, we concatenate the it with the accumulated prompts $\mathbf{I}^t$, encode the whole sequence with the pre-trained language model, e.g., BERT~\cite{}\lifu{cite}, and obtain the sequence of contextual representations $[\tilde{\mathbf{X}}^t_i; \tilde{\mathbf{I}}^t]$, where $\tilde{\mathbf{X}}^t_i$ and $\tilde{\mathbf{I}}^t_i$ denote the sequence of contextual embeddings of $x_i^t$ and $\mathbf{I}^t$ respectively. $[;]$ is concatenation operation. Then, similar as the baseline approach in Section~\ref{sec:baseline}, we will use $\tilde{\mathbf{X}}^t_i$ as the contextual representations to detect events and optimize the model with Equation~\ref{equa:baseline_train}.
At the end of each training stage, we keep the learned prompts from the current task $\mathbf{C}^t$ in the model, and then initialize and incorporate a new prompt $\mathbf{C}^{t+1}$ for the next task incrementally: $\mathbf{I}^{t+1} = [\mathbf{I}^{t};\mathbf{C}^{t+1}]$.
In other words, our EMP can serve as a parameterized episodic memory module that stores the knowledge for each learned type, and transfer the capability acquired from previous tasks to new tasks.
\paragraph{Entangled Prompt Optimization} We expect the EMPs to be specific to the corresponding event types and preserve the knowledge of each event type from previous tasks. To achieve this goal, we design an entangled prompt optimization strategy. Specifically, after obtaining the contextual representations of the sentence as $\tilde{\mathbf{X}}^t_i$ and EMPs as $\tilde{\mathbf{I}}^t_i$, for each span we get the span representation $\tilde{\mathbf{h}}_{span}$. Then we entangle the span features with event type specific prompts by computing an event type probability distribution over all prompts: $p_c = \text{MLP}(\tilde{\mathbf{I}}^t_i)\cdot\tilde{\mathbf{h}}_{span}$.
after the EMP is encoded from the language model, we first apply a MLP layer to further encode the prompt.
Then, we apply dot product between the encoded prompts $\mathbf{H}_{prom}$ and the span representation $\mathbf{h}_{span}$ to obtain the \emph{prompt logits}: $p_c = \mathbf{H}_{prom}^{T}\mathbf{h}_{span}$.
Finally, we combine the original logits and the \emph{prompt logits} to get the final logits for prediction in our framework: $p_{sum} = p + p_c$.
\section{Experiments and Discussion}
\paragraph{Experiment Settings} We conduct experiments on two benchmark datasets: ACE05-EN~\cite{ace} and MAVEN~\cite{maven}, and construct the class-incremental datasets following the \emph{oracle negative} setting in~\cite{yu2021lifelong}. We divided the ontology into 5 subsets with distinct event types, and then use them to constitute a sequence of 5 tasks denoted as $\mathcal{D}_{1:5}$. We use the same partition and task order permutations in \cite{yu2021lifelong}. During the learning process from $\mathcal{D}_1$ to $\mathcal{D}_5$, we constantly test the model on the entire test set (which contains the whole ontology) and take the mentions of unseen event types as negative instances. More implementation details, including parameters, initialization of prompts as well as baselines are shown in Appendix~\ref{appendix_exp}.
\begin{table*}[!htbp]
\centering
\resizebox{0.9\textwidth}{!}
{
\begin{tabular}{l | ccccc | ccccc}
\toprule
& \multicolumn{5}{c}{MAVEN} & \multicolumn{5}{c}{ACE05-EN} \\
\midrule
Task &1 &2 & 3 & 4 & 5 & 1 &2 & 3 & 4 & 5 \\
\midrule
BERT-ED & 63.51 & 39.99 & 33.36 & 23.83 & 22.69 & 58.30 & 43.96 & 38.02 & 21.53 & 25.71 \\
iCaRL*~\cite{icarl} & 18.08 & 27.03 & 30.78 & 31.26 & 29.77 & 4.05 & 5.41 & 7.25 & 6.94 & 8.94 \\
EEIL~\cite{eeil} & 63.51 & 50.62 & 45.16 & 41.39 & 38.34 & 58.30 & 54.93 & 52.72 & 45.18 & 41.95 \\
BIC~\cite{bic} & 63.51 & 46.69 & 39.15 & 31.69 & 30.47 & 58.30 & 45.73 & 43.28 & 35.70 & 30.80 \\
KCN~\cite{cao2020increment} & 63.51 & 51.17 & 46.80 & 38.72 & 38.58 & 58.30 & 54.71 & 52.88 & 44.93 & 41.10 \\
KT~\cite{yu2021lifelong} & 63.51 & 52.36 & 47.24 & 39.51 & 39.34 & 58.30 & \textbf{55.41} & 53.95 & 45.00 & 42.62 \\
\midrule
EMP (Ours) &\textbf{67.62} & \textbf{58.33} & \textbf{54.53} & \textbf{47.70} & \textbf{44.30} & \textbf{59.60} & 53.19 & \textbf{55.20} & \textbf{45.64} & \textbf{43.28} \\
\midrule
Upperbound (Ours) & / & / & / & / & 66.68 & / & / & / & / & 68.22 \\
\bottomrule
\end{tabular}
}
\caption{Comparison between our approach and baselines in terms of micro F-1 (\%) on 5 class-incremental tasks. We report the averaged results on 5 permutations of tasks to alleviate the affect of task order.}
\vspace{-1.2em}
\label{tab:main}
\end{table*}
\paragraph{Results}
We present the main results in Table~\ref{tab:main}. We have following observations:
(1) by comparing the performance of various approaches on Task 1 which are not affected by any catastrophic forgetting, our prompting based approach improves 4.1\% F-score on MAVEN and 1.3\% F-score on ACE05, demonstrating that by incorporating task-specific prompts, event detection itself can be significantly improved. EMPs even provide more improvement on MAVEN which contains a lot more event types than ACE05, suggesting the potential of incorporating EMPs for fine-grained event detection; (2) \textbf{KCN} can be viewed as an ablated version of our approach without EMPs. Our approach consistently outperforms \textbf{KCN} on almost all tasks on both datasets, demonstrating the effectiveness of EMPs on improving class-incremental event detection; (3) Comparing with \textbf{BERT-ED}, \textbf{KCN} adopts experience replay and knowledge distillation. Their performance gap verifies that these two strategies can dramatically alleviate the catastrophic forgetting problem. (4) There is still a large gap between the current lifelong learning approaches and the upperbound, indicating that catastrophic forgetting still remains a very challenging problem. Note that for fair comparison, for all approaches, we set the exemplar buffer size as 20, and allow one exemplar instance to be used in each training batch instead of the whole memory set, thus most results in our paper cannot be directly compared with the results reported in~\cite{yu2021lifelong}. We also analyze the effect of exemplar buffer size in Appendix~\ref{appendix_effect}.
\section{Experiments}
\subsection{Experiment Settings}
\paragraph{Class-incremental Dataset Construction} We use two datasets for the experiments. (1) ACE05-EN contains 33 event types with 597 documents and 5,311 mentions. (2) MAVEN contains 168 event types in the general domain with 3,623 documents and 96,987 mentions. For both ACE and MAVEN, we follow the \emph{oracle negative} setting in~\cite{yu2021lifelong} and construct the class-incremental dataset as follows. We divided the ontology into 5 subsets, with no overlapping event types between any two of them. These 5 subsets then constitute a sequence of 5 tasks denoted as $\mathcal{D}_{1:5}$. We use the same partition and task order permutations in \cite{yu2021lifelong}\footnote{\lifu{add one sentence to talk about the motivation of permutation}}.
\paragraph{Evaluation}
We use micro F1 score as our primary evaluation metric. During the learning process from $\mathcal{D}_1$ to $\mathcal{D}_5$, we constantly test the model on the entire test set (which contains the whole ontology) and take the mentions of unseen event types as negative instances.
\paragraph{Implementation Details}
\label{sec:implement}
We use AdamW~\cite{adamw} optimizer during training with learning rate set to $1e-4$ and weight decay set to $1e-2$. Different from previous work~\cite{yu2021lifelong}, we set the batch size to 1 as we encode each sentence once and consider all target spans in the sentence at the same time. We adopt gradient accumulation with the step set to 8. As the number of batches is large, we apply a periodic replay strategy with the interval set to 10 to reduce computational cost. For each class-incremental task $\mathcal{D}_{t}$, we set the maximum number of training epochs to 20. We adopt the early stopping strategy with patience 5, i.e., the training stops if the performance on the development set does not increase for 5 epochs. We set the weighting factor $\lambda=k/(s+k)$, where $s$ is the number of predicted spans and $k$ is set to 50.
The parameters of each prompt in EMPs are initialized with the corresponding event type name. Specifically, there are three cases in the initialization: (1) If the type name is \emph{single-token} and it is contained in BERT's vocabulary, we directly use the pre-trained embedding of this token to initialize the prompt; (2) If the type name is \emph{multiple-token} and the tokens are contained in BERT's vocabulary, we take the average of the pre-trained embeddings of these tokens to initialize the prompt; (3) If the type name contains \emph{Out-of-Vocabulary (OOV)} tokens, we replace the OOV tokens with the synonyms that are contained in BERT's vocabulary.
\begin{table*}[!htbp]
\centering
\resizebox{0.95\textwidth}{!}
{
\begin{tabular}{l | ccccc | ccccc}
\toprule
& \multicolumn{5}{c}{MAVEN} & \multicolumn{5}{c}{ACE05-EN} \\
\midrule
Task &1 &2 & 3 & 4 & 5 & 1 &2 & 3 & 4 & 5 \\
\midrule
BERT-ED & 63.51 & 39.99 & 33.36 & 23.83 & 22.69 & 58.30 & 43.96 & 38.02 & 21.53 & 25.71 \\
iCaRL*~\cite{icarl} & 18.08 & 27.03 & 30.78 & 31.26 & 29.77 & 4.05 & 5.41 & 7.25 & 6.94 & 8.94 \\
EEIL~\cite{eeil} & 63.51 & 50.62 & 45.16 & 41.39 & 38.34 & 58.30 & 54.93 & 52.72 & 45.18 & 41.95 \\
BIC~\cite{bic} & 63.51 & 46.69 & 39.15 & 31.69 & 30.47 & 58.30 & 45.73 & 43.28 & 35.70 & 30.80 \\
KCN~\cite{cao2020increment} & 63.51 & 51.17 & 46.80 & 38.72 & 38.58 & 58.30 & 54.71 & 52.88 & 44.93 & 41.10 \\
KT~\cite{yu2021lifelong} & 63.51 & 52.36 & 47.24 & 39.51 & 39.34 & 58.30 & \textbf{55.41} & 53.95 & 45.00 & 42.62 \\
\midrule
EMP (Ours) &\textbf{67.62} & \textbf{58.33} & \textbf{54.53} & \textbf{47.70} & \textbf{44.30} & \textbf{59.60} & 53.19 & \textbf{55.20} & \textbf{45.64} & \textbf{43.28} \\
\midrule
Upperbound (Ours) & / & / & / & / & 66.68 & / & / & / & / & 68.22 \\
\bottomrule
\end{tabular}
}
\caption{Comparison between our approach and baselines in terms of micro F-1 (\%) on 5 class-incremental tasks. We report the averaged results on 5 permutations of tasks to alleviate the affect of task order.}
\label{tab:main}
\end{table*}
\paragraph{Baselines} We consider the following baselines for comparison: (1) \textbf{BERT-ED}: simply trains the BERT based event detection model on new tasks without prompts, experience replay or knowledge distillation. It's the same as the span-based event detection baseline in Section~\ref{sec:baseline}. (2) \textbf{KCN}~\cite{cao2020increment}: as the original approach studied a different setting, we adapt their hierarchical distillation as the baseline\lifu{use one sentence to describe this approach}. (3) \textbf{KT}~\cite{yu2021lifelong}: transfer knowledge between old types and new types in two directions. (4) \textbf{Upperbound}: trains the same model on all types in the datasets jointly. We present the details of \textbf{iCaRL}~\cite{icarl}, \textbf{EEIL}~\cite{eeil}, and \textbf{BIC}~\cite{bic} in Appendix.\minqian{TODO}
For fair comparison, our approach and all baselines (except for the Upperbound baseline) are built upon \textbf{KCN} and use the same experience replay and knowledge distillation strategies described in Section~\ref{sec_emp}.
\subsection{Results}
We present the main results in Table~\ref{tab:main}. Note that as iCaRL does not use experience replay strategy and is not affected by the modification, we report the result in~\cite{yu2021lifelong} for reference. Based on Table~\ref{tab:main}, we have following observations: (1) by comparing the performance of various approaches on Task 1 which are not affected by any catastrophic forgetting, our prompting based approach improves 4.1\% F-score on MAVEN and 1.3\% F-score on ACE05, demonstrating that by incorporating task-specific prompts, event detection itself can be significantly improved. EMPs even provide more improvement on MAVEN which contains a lot more event types than ACE05, which shows the potential of incorporating EMPs for fine-grained event detection; (2) \textbf{KCN} can be viewed as an ablated version of our approach without the EMPs. Our approach consistently outperforms \textbf{KCN} on almost all tasks on both datasets, demonstrating the benefit of EMPs on improving the generalizability of event detection under class-incremental setting and addressing the catastrophic forgetting problem; (3) By comparing XX and XX, we see XX, \lifu{complete it} which verifies that experience replay and knowledge distillation can dramatically alleviate the catastrophic forgetting problem. (4) Comparing with the upperbound performance, there is still a large gap between the current lifelong learning approaches and the upperbound, indicating that catastrophic forgetting still remains a very challenging problem.
It is worth noting that either storing too many data or replaying the examples too frequently will increase the computational burden, and more critically, deteriorate the value of lifelong learning research. We thus constrain that only one exemplar instance is allowed to use in each training batch, compared with~\newcite{yu2021lifelong} that appends the whole memory set to each training batch. Therefore, most results in our paper cannot be directly compared with the results reported in~\cite{yu2021lifelong}.
\section{Discussion}
\paragraph{Analysis of New and Old Types in Lifelong Learning} Figure~\ref{fig:overall} shows the F-score on old and new event types in each training stage for both our approach and \textbf{KT}~\cite{yu2021lifelong} on the MAVEN dataset. Our approach consistently outperforms \textbf{KT} by a large margin on both old types and new types, demonstrating that our EMPs effectively preserve learned knowledge from old event types and significantly improve event detection when the annotations are sufficient. Interestingly, comparing the F-score on new types in Task 1 and old types in Task 2, both methods improve the performance on the types of Task 1, indicating that both methods have the potential of leveraging indirect supervision to improve event detection.
\begin{figure}[tbp]
\centering
\includegraphics[width=0.7\linewidth]{figure/macro_f1.png}
\caption{Performance on old types and new types in each lifelong task on one random permutation of MAVEN (best viewed in color). }
\vspace{-1.2em}
\label{fig:overall}
\end{figure}
\paragraph{Ablation Study}
For ablation study, we consider three ablated models based on our EMPs: (1) change the prompt initialization from using event type name representations\footnote{Details of using event type name to initialize prompts are shown in Appendix~\ref{appendix_exp}} to using random distribution; (2) remove the knowledge distillation loss $\mathcal{L}_{PD}$ and $\mathcal{L}_{FD}$; (3) use completely fixed prompts to replace the trainable soft prompts. From Table~\ref{tab:ablation}, we observes that: (1) using event type names to initialize the prompts is helpful in most tasks. We leave how to incorporate more effective prior knowledge into prompts for future work; (2) switching the continuous prompts to discrete prompts degrades the performance significantly, suggesting that the continuous prompts is generally more promising than the discrete prompts.
\begin{table}[!htbp]
\centering
\resizebox{0.45\textwidth}{!}
{
\begin{tabular}{l | ccccc}
\toprule
Task &1 &2 & 3 & 4 & 5 \\
\midrule
EMP (Ours) & \textbf{70.57} & \textbf{57.87} & \textbf{54.33} & 48.39 & \textbf{45.82} \\
- w/o EInit & 70.26 & 54.78 & 50.56 & \textbf{48.42} & 42.28 \\
- w/o KD & 70.57 & 54.82 & 53.24 & 45.37 & 41.22 \\
- Discrete & 67.57 & 54.86 & 49.99 & 45.51 & 39.08 \\
\bottomrule
\end{tabular}
}
\caption{Ablation study on event-specific prompt initialization (EInit), knowledge distillation (KD), and switching to discrete prompts (Discrete) on one random permutation of MAVEN}.
\vspace{-1em}
\label{tab:ablation}
\end{table}
\section{Related Work}
\paragraph{Lifelong Event Detection} Recent deep neural networks have shown state-of-the-art performance on conventional supervised event detection~\cite{chenEE2015,jointEE,feng2016,huang2020semi,wang2021query}. However, when moving to lifelong learning setting, the performance significantly drops~\cite{kirkpatrick2017overcoming,lwf,aljundi2019gradient,li2021refining,ke-etal-2021-adapting,madotto-etal-2021-continual}. Episodic memory replay (EMR)~\cite{lopez2017gradient,guo2020improved,dautume2019episodic,wang2019sentence,han2020continual} and knowledge distillation~\cite{llkd,cao2020increment,yu2021lifelong} have been the two most effective techniques to overcome the catastrophic forgetting problem. However, they highly rely on the stored data from old tasks, which is not the most realistic setting for lifelong learning.
\paragraph{Prompt Learning}
Conditioning on large-scale pre-trained language models, prompt learning~\cite{gpt3,ptuning,knowprompt,ptuningv2,transprompt} have shown comparable performance as language model fine-tuning. Several recent studies explore prompt learning in lifelong learning setting. \citet{lept5} use prompt tuning to train the model as a task solver and data generator in their proposed Lifelong Few-shot Language Learning problem. \citet{l2p} propose L2P for continual learning in the vision area. To the best of our knowledge, we are the first work to adopt prompt learning for class-incremental event detection.
\section{Conclusion}
In this paper, we propose a novel prompting framework, namely Episodic Memory Prompts (EMP), for class-incremental event detection. During each training stage, EMP learns type-specific knowledge via a continuous prompt for each event type. The EMPs trained in previous tasks are kept in the model, such that the acquired task-specific knowledge can be transferred into the following new tasks. Experimental results validate the effectiveness of our method comparing with competitive baselines. In addition, our extensive analysis shows that by employing EMPs, both event detection itself and the incremental learning capability of our approach are significantly improved.
\section{Baseline Details}
\section{Experimental Details}
\label{appendix_exp}
\paragraph{Baselines} We consider the following baselines for comparison: (1) \textbf{BERT-ED}: simply trains the BERT based event detection model on new tasks without prompts, experience replay or knowledge distillation. It's the same as the span-based event detection baseline in Section~\ref{sec:baseline}. (2) \textbf{KCN}~\cite{cao2020increment}: use a prototype-based example sampling strategy and hierarchical distillation. As the original approach studied a different setting, we adapt their prediction-level and feature-level distillation as the baseline. (3) \textbf{KT}~\cite{yu2021lifelong}: transfer knowledge between old types and new types in two directions. (4) \textbf{iCaRL*}~\cite{icarl}: use nearest-mean-of-exemplars rules to perform classification combined with knowledge distillation. iCaRL adopts different strategies for classification, experience replay, and distillation. We directly report the result in~\cite{yu2021lifelong} for reference. (5) \textbf{EEIL}~\cite{eeil}: use an additional finetuning stage on the balanced dataset. (6) \textbf{BIC}~\cite{bic}: use a bias correction layer after the classification layer. (7) \textbf{Upperbound}: trains the same model on all types in the datasets jointly.
For \textbf{iCaRL}, \textbf{EEIL}, and \textbf{BIC}, we use the same implementation in~\cite{yu2021lifelong}.
For fair comparison, our approach and all baselines (except for the Upperbound baseline) are built upon \textbf{KCN} and use the same experience replay and knowledge distillation strategies described in Section~\ref{sec_emp}.
\paragraph{Implementation Details}
\label{sec:implement}
During training, we use AdamW~\cite{adamw} optimizer with the learning rate set to $1e-4$ and weight decay set to $1e-2$. Different from previous work~\cite{yu2021lifelong}, we set the batch size to 1 as we encode each sentence once and consider all target spans in the sentence at the same time. We adopt gradient accumulation with the step set to 8. As the number of batches is large, we apply a periodic replay strategy with the interval set to 10 to reduce computational cost. For each lifelong task $\mathcal{D}_{t}$, we set the maximum number of training epochs to 20. We adopt the early stopping strategy with patience 5, i.e., the training stops if the performance on the development set does not increase for 5 epochs. We set the weighting factor $\lambda=k/(s+k)$, where $s$ is the number of predicted spans and $k$ is set to 50. The temperature parameter used in prediction-level distillation is set to 2.
The parameters of each prompt in EMPs are initialized with the corresponding event type name. Specifically, there are three cases in the initialization: (1) If the type name is \emph{single-token} and it is contained in BERT's vocabulary, we directly use the pre-trained embedding of this token to initialize the prompt; (2) If the type name is \emph{multiple-token} and the tokens are contained in BERT's vocabulary, we take the average of the pre-trained embeddings of these tokens to initialize the prompt; (3) If the type name contains \emph{Out-of-Vocabulary (OOV)} tokens, we replace the OOV tokens with the synonyms that are contained in BERT's vocabulary.
\begin{figure}[tbp]
\centering
\includegraphics[width=0.7\linewidth]{figure/buffer_size.png}
\caption{Performance with different buffer size in each task on MAVEN (best viewed in color).}
\label{fig:buffer}
\end{figure}
\section{Effect of Exemplar Buffer Size}
\label{appendix_effect}
We conduct an analysis on the effect of exemplar buffer size. We explore the buffer size for each type in \{0, 10, 20\}. Note that although we reduced the buffer size, we did not modify the replay frequency, as we want to investigate the effect of data diversity in memory buffer. We use \textbf{KT} as the baseline when buffer size is 20 and 10. Note that when buffer size is 0, we do not adopt either experience replay or knowledge distillation and thus use \textbf{BERT-ED} as the baseline. We plot the results on Figure~\ref{fig:buffer}.
We observed that: (1) Decreasing the buffer size for each type from 20 to 10 degrades the performance of both models. This indicates that reducing data diversity may result in the overfitting on example data, and thus deteriorates the performance; (2) The performance of our method is not affected as much as the baseline, demonstrating our prompting framework is more tolerant to smaller buffer size and remains very competitive performance when less data are available; (3) When the buffer size decreases to 0, the performance of both methods drops significantly. This shows that both approaches highly rely on the stored data to overcome the catastrophic forgetting problem. This calls for developing more advance techniques to reduce the dependence on stored examples, as storing past data could result in data leakage in real-world applications.
\section{Appendix}
\end{document}
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"redpajama_set_name": "RedPajamaArXiv"
} | 2,664 |
{"url":"https:\/\/math.stackexchange.com\/questions\/941948\/sum-of-sequence-of-cubes-and-summation-on-the-upper-index","text":"# Sum of sequence of cubes and summation on the upper index\n\nExpress the sum of the sequence of cubes as a polynomial in n using the summation on the upper index formula: $$\\sum\\limits_{k=0}^n\\binom{k}{m} = \\binom{n+1}{m+1}$$ It has been proven that the sum of sequence of cubes can be expressed as the following fraction: $$\\sum\\limits_{k=1}^n(k^3) = \\frac{n^2(n+1)^2}{4}$$ However, I am stuck and can't find a way to apply the said formula to get the fraction. I guess I haven't grasped yet the connection between binomial coefficients and powers. Can I get a hint?\n\n\u2022 You want to prove the middle formula via the binomial sum equality? Sep 22 '14 at 19:06\n\u2022 @user144248 Yes, exactly! Sep 22 '14 at 19:16\n\u2022 $k^3 = 6{k \\choose 3} + 3 k^2 -2k$ (I got this by expanding out ${k \\choose 3}$). Now eliminate the $k^2$ using ${k \\choose 2}$. Then again for the leftover $k$ term. Now, if you sum the LHS, you sum the RHS. But by the stated thm, you can calculate the sum of the RHS. Sep 25 '14 at 15:55\n\n## 2 Answers\n\nHint\n\nWith the formula we find:\n\n$$\\binom{n+1}{4}=\\sum_{k=0}^{n}\\binom{k}{3}=\\frac{1}{3!}\\sum_{k=0}^{n}k\\left(k-1\\right)\\left(k-2\\right)$$\n\nThis enables you to express $\\sum_{k=0}^{n}k^{3}$ in formulas of $\\sum_{k=0}^{n}k^{2}$ and $\\sum_{k=0}^{n}k$.\n\nYou can use a version of the formula again ($m=2$) to find likewise a formula for $\\sum_{k=0}^{n}k^{2}$ in $\\sum_{k=0}^{n}k$.\n\nedit:\n\n\u2022 $\\binom{n+1}{2}=\\sum_{k=0}^{n}\\binom{k}{1}=\\sum_{k=0}^{n}k$\n\n\u2022 $\\binom{n+1}{3}=\\sum_{k=0}^{n}\\binom{k}{2}=\\frac{1}{2}\\sum_{k=0}^{n}k\\left(k-1\\right)=\\frac{1}{2}\\sum_{k=0}^{n}k^{2}-\\frac{1}{2}\\sum_{k=0}^{n}k$\n\n\u2022 $\\binom{n+1}{4}=\\sum_{k=0}^{n}\\binom{k}{3}=\\frac{1}{6}\\sum_{k=0}^{n}k\\left(k-1\\right)\\left(k-2\\right)=\\frac{1}{6}\\sum_{k=0}^{n}k^{3}-\\frac{3}{6}\\sum_{k=0}^{n}k^{2}+\\frac{2}{6}\\sum_{k=0}^{n}k$\n\n\u2022 How do we get to $\\sum\\limits_{k=0}^n\\binom{k}{3}$ from $\\sum\\limits_{k=0}^n(k^3)$ ? Sep 22 '14 at 19:34\n\u2022 You are asked to find a formula for $\\sum k^{3}$, aren't you? Then you are supposed to do the opposite: get to $\\sum k^{3}$ from $\\sum\\binom{k}{3}$. My answer gives a hint how to do that. Sep 22 '14 at 20:28\n\nUsing a difference table for $(n^3)$, we get\n\n${\\color{red}0}\\;\\;\\;\\;1\\;\\;\\;\\;8\\;\\;\\;\\;27\\;\\;\\;\\;64$\n\n$\\;\\;\\;{\\color{red}1}\\;\\;\\;\\;7\\;\\;\\;19\\;\\;\\;\\;37$\n\n$\\;\\;\\;\\;\\;\\;{\\color{red}6}\\;\\;\\;12\\;\\;\\;18$\n\n$\\;\\;\\;\\;\\;\\;\\;\\;\\;{\\color{red}6}\\;\\;\\;\\;6$\n\n$\\hspace{.59in}0$\n\nThen $n^3=0\\binom{n}{0}+1\\binom{n}{1}+6\\binom{n}{2}+6\\binom{n}{3}$,\n\nso $\\displaystyle\\sum_{k=0}^{n}k^3=\\sum_{k=0}^{n}\\left[1\\binom{k}{1}+6\\binom{k}{2}+6\\binom{k}{3}\\right]=1\\binom{n+1}{2}+6\\binom{n+1}{3}+6\\binom{n+1}{4}$\n\nusing the given summation formula.\n\nNow show that this gives the stated formula for $\\displaystyle\\sum_{k=1}^nk^3$.\n\n\u2022 Thank you! I do understand the second equality, but I do not understand how we express $n^3$ as a sum of binomial coefficients. Can you clarify? Sep 22 '14 at 20:09\n\u2022 The easiest way I know to do this is to construct the difference table (by writing the terms of the sequence in the first line, and then continuing to take differences to get the succeeding lines), and then using the left diagonal to get the coefficients. You could also do this by setting $n^3=a\\binom{n}{0}+b\\binom{n}{1}+c\\binom{n}{2}+d\\binom{n}{3}$ and then substituting $n=0,1,2,3$ to solve for the coefficients. Sep 22 '14 at 20:14","date":"2021-12-05 05:41:23","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\": 0.9464845061302185, \"perplexity\": 175.71489422854026}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"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-2021-49\/segments\/1637964363135.71\/warc\/CC-MAIN-20211205035505-20211205065505-00027.warc.gz\"}"} | null | null |
// Copyright 2016 The Chromium Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.
package org.chromium.chrome.browser.ntp.cards;
import static org.junit.Assert.assertEquals;
import static org.junit.Assert.assertFalse;
import static org.junit.Assert.assertTrue;
import android.graphics.Bitmap;
import org.chromium.base.Callback;
import org.chromium.base.metrics.RecordHistogram;
import org.chromium.base.test.util.Feature;
import org.chromium.chrome.browser.ntp.snippets.CategoryStatus;
import org.chromium.chrome.browser.ntp.snippets.CategoryStatus.CategoryStatusEnum;
import org.chromium.chrome.browser.ntp.snippets.ContentSuggestionsCardLayout;
import org.chromium.chrome.browser.ntp.snippets.KnownCategories;
import org.chromium.chrome.browser.ntp.snippets.SnippetArticleListItem;
import org.chromium.chrome.browser.ntp.snippets.SuggestionsSource;
import org.chromium.testing.local.LocalRobolectricTestRunner;
import org.junit.Before;
import org.junit.Test;
import org.junit.runner.RunWith;
import org.robolectric.annotation.Config;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.Set;
/**
* Unit tests for {@link NewTabPageAdapter}.
*/
@RunWith(LocalRobolectricTestRunner.class)
@Config(manifest = Config.NONE)
public class NewTabPageAdapterTest {
/**
* Number of elements, not including content suggestions that are loaded
* in a populated recycler view.
* The 3 elements are: above-the-fold, header, bottom spacer
* TODO(dgn): Make this depend on the category info of the loaded sections
* instead of being a constant, as it needs to know if the MORE button is
* present for example.
*/
private static final int PERMANENT_ELEMENTS_COUNT = 3;
/**
* Number of elements that are loaded in an empty recycler view
* The 5 elements are: above-the-fold, header, status card, progress
* indicator, bottom spacer.
*/
private static final int EMPTY_STATE_ELEMENTS_COUNT = 5;
private static class FakeSnippetsSource implements SuggestionsSource {
private SuggestionsSource.Observer mObserver;
private final Map<Integer, List<SnippetArticleListItem>> mSuggestions = new HashMap<>();
private final Map<Integer, Integer> mCategoryStatus = new HashMap<>();
private final Map<Integer, SuggestionsCategoryInfo> mCategoryInfo = new HashMap<>();
public void setStatusForCategory(int category, @CategoryStatusEnum int status) {
mCategoryStatus.put(category, status);
if (mObserver != null) mObserver.onCategoryStatusChanged(category, status);
}
public void setSuggestionsForCategory(
int category, List<SnippetArticleListItem> suggestions) {
mSuggestions.put(category, suggestions);
if (mObserver != null) mObserver.onNewSuggestions(category);
}
public void setInfoForCategory(int category, SuggestionsCategoryInfo info) {
mCategoryInfo.put(category, info);
}
@Override
public void dismissSuggestion(SnippetArticleListItem suggestion) {
throw new UnsupportedOperationException();
}
@Override
public void fetchSuggestionImage(
SnippetArticleListItem suggestion, Callback<Bitmap> callback) {
throw new UnsupportedOperationException();
}
@Override
public void getSuggestionVisited(
SnippetArticleListItem suggestion, Callback<Boolean> callback) {
throw new UnsupportedOperationException();
}
@Override
public void setObserver(Observer observer) {
mObserver = observer;
}
@Override
public int[] getCategories() {
Set<Integer> ids = mCategoryStatus.keySet();
int[] result = new int[ids.size()];
int index = 0;
for (int id : ids) result[index++] = id;
return result;
}
@CategoryStatusEnum
@Override
public int getCategoryStatus(int category) {
return mCategoryStatus.get(category);
}
@Override
public SuggestionsCategoryInfo getCategoryInfo(int category) {
return mCategoryInfo.get(category);
}
@Override
public List<SnippetArticleListItem> getSuggestionsForCategory(int category) {
List<SnippetArticleListItem> result = mSuggestions.get(category);
return result == null ? Collections.<SnippetArticleListItem>emptyList() : result;
}
}
private FakeSnippetsSource mSnippetsSource = new FakeSnippetsSource();
private NewTabPageAdapter mNtpAdapter;
@Before
public void setUp() {
RecordHistogram.disableForTests();
mSnippetsSource = new FakeSnippetsSource();
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES, CategoryStatus.INITIALIZING);
mSnippetsSource.setInfoForCategory(
KnownCategories.ARTICLES, new SuggestionsCategoryInfo("Articles for you",
ContentSuggestionsCardLayout.FULL_CARD));
mNtpAdapter = new NewTabPageAdapter(null, null, mSnippetsSource, null);
}
/**
* Tests the content of the adapter under standard conditions: on start and after a snippet
* fetch.
*/
@Test
@Feature({"Ntp"})
public void testSnippetLoading() {
assertEquals(EMPTY_STATE_ELEMENTS_COUNT, mNtpAdapter.getItemCount());
assertEquals(NewTabPageListItem.VIEW_TYPE_ABOVE_THE_FOLD, mNtpAdapter.getItemViewType(0));
assertEquals(NewTabPageListItem.VIEW_TYPE_HEADER, mNtpAdapter.getItemViewType(1));
assertEquals(NewTabPageListItem.VIEW_TYPE_STATUS, mNtpAdapter.getItemViewType(2));
assertEquals(NewTabPageListItem.VIEW_TYPE_PROGRESS, mNtpAdapter.getItemViewType(3));
assertEquals(NewTabPageListItem.VIEW_TYPE_SPACING, mNtpAdapter.getItemViewType(4));
List<SnippetArticleListItem> snippets = createDummySnippets();
mSnippetsSource.setSuggestionsForCategory(KnownCategories.ARTICLES, snippets);
List<NewTabPageListItem> loadedItems = new ArrayList<>(mNtpAdapter.getItems());
assertEquals(NewTabPageListItem.VIEW_TYPE_ABOVE_THE_FOLD, mNtpAdapter.getItemViewType(0));
assertEquals(NewTabPageListItem.VIEW_TYPE_HEADER, mNtpAdapter.getItemViewType(1));
assertEquals(snippets, loadedItems.subList(2, loadedItems.size() - 1));
assertEquals(NewTabPageListItem.VIEW_TYPE_SPACING,
mNtpAdapter.getItemViewType(loadedItems.size() - 1));
// The adapter should ignore any new incoming data.
mSnippetsSource.setSuggestionsForCategory(KnownCategories.ARTICLES,
Arrays.asList(new SnippetArticleListItem[] {new SnippetArticleListItem(
"foo", "title1", "pub1", "txt1", "foo", "bar", 0, 0, 0)}));
assertEquals(loadedItems, mNtpAdapter.getItems());
}
/**
* Tests that the adapter keeps listening for snippet updates if it didn't get anything from
* a previous fetch.
*/
@Test
@Feature({"Ntp"})
public void testSnippetLoadingInitiallyEmpty() {
// If we don't get anything, we should be in the same situation as the initial one.
mSnippetsSource.setSuggestionsForCategory(
KnownCategories.ARTICLES, new ArrayList<SnippetArticleListItem>());
assertEquals(EMPTY_STATE_ELEMENTS_COUNT, mNtpAdapter.getItemCount());
assertEquals(NewTabPageListItem.VIEW_TYPE_ABOVE_THE_FOLD, mNtpAdapter.getItemViewType(0));
assertEquals(NewTabPageListItem.VIEW_TYPE_HEADER, mNtpAdapter.getItemViewType(1));
assertEquals(NewTabPageListItem.VIEW_TYPE_STATUS, mNtpAdapter.getItemViewType(2));
assertEquals(NewTabPageListItem.VIEW_TYPE_PROGRESS, mNtpAdapter.getItemViewType(3));
assertEquals(NewTabPageListItem.VIEW_TYPE_SPACING, mNtpAdapter.getItemViewType(4));
// We should load new snippets when we get notified about them.
List<SnippetArticleListItem> snippets = createDummySnippets();
mSnippetsSource.setSuggestionsForCategory(KnownCategories.ARTICLES, snippets);
List<NewTabPageListItem> loadedItems = new ArrayList<>(mNtpAdapter.getItems());
assertEquals(NewTabPageListItem.VIEW_TYPE_ABOVE_THE_FOLD, mNtpAdapter.getItemViewType(0));
assertEquals(NewTabPageListItem.VIEW_TYPE_HEADER, mNtpAdapter.getItemViewType(1));
assertEquals(snippets, loadedItems.subList(2, loadedItems.size() - 1));
assertEquals(NewTabPageListItem.VIEW_TYPE_SPACING,
mNtpAdapter.getItemViewType(loadedItems.size() - 1));
// The adapter should ignore any new incoming data.
mSnippetsSource.setSuggestionsForCategory(KnownCategories.ARTICLES,
Arrays.asList(new SnippetArticleListItem[] {new SnippetArticleListItem(
"foo", "title1", "pub1", "txt1", "foo", "bar", 0, 0, 0)}));
assertEquals(loadedItems, mNtpAdapter.getItems());
}
/**
* Tests that the adapter clears the snippets when asked to.
*/
@Test
@Feature({"Ntp"})
public void testSnippetClearing() {
List<SnippetArticleListItem> snippets = createDummySnippets();
mSnippetsSource.setSuggestionsForCategory(KnownCategories.ARTICLES, snippets);
assertEquals(PERMANENT_ELEMENTS_COUNT + snippets.size(), mNtpAdapter.getItemCount());
// If we get told that snippets are enabled, we just leave the current
// ones there and not clear.
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.AVAILABLE);
assertEquals(PERMANENT_ELEMENTS_COUNT + snippets.size(), mNtpAdapter.getItemCount());
// When snippets are disabled, we clear them and we should go back to
// the situation with the status card.
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.SIGNED_OUT);
assertEquals(EMPTY_STATE_ELEMENTS_COUNT, mNtpAdapter.getItemCount());
// The adapter should now be waiting for new snippets.
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.AVAILABLE);
mSnippetsSource.setSuggestionsForCategory(KnownCategories.ARTICLES, snippets);
assertEquals(PERMANENT_ELEMENTS_COUNT + snippets.size(), mNtpAdapter.getItemCount());
}
/**
* Tests that the adapter loads snippets only when the status is favorable.
*/
@Test
@Feature({"Ntp"})
public void testSnippetLoadingBlock() {
List<SnippetArticleListItem> snippets = createDummySnippets();
// By default, status is INITIALIZING, so we can load snippets
mSnippetsSource.setSuggestionsForCategory(KnownCategories.ARTICLES, snippets);
assertEquals(PERMANENT_ELEMENTS_COUNT + snippets.size(), mNtpAdapter.getItemCount());
// If we have snippets, we should not load the new list.
snippets.add(new SnippetArticleListItem("https://site.com/url1", "title1", "pub1", "txt1",
"https://site.com/url1", "https://amp.site.com/url1", 0, 0, 0));
mSnippetsSource.setSuggestionsForCategory(KnownCategories.ARTICLES, snippets);
assertEquals(PERMANENT_ELEMENTS_COUNT + snippets.size() - 1, mNtpAdapter.getItemCount());
// When snippets are disabled, we should not be able to load them
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.SIGNED_OUT);
mSnippetsSource.setSuggestionsForCategory(KnownCategories.ARTICLES, snippets);
assertEquals(EMPTY_STATE_ELEMENTS_COUNT, mNtpAdapter.getItemCount());
// INITIALIZING lets us load snippets still.
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.INITIALIZING);
mSnippetsSource.setSuggestionsForCategory(KnownCategories.ARTICLES, snippets);
assertEquals(PERMANENT_ELEMENTS_COUNT + snippets.size(), mNtpAdapter.getItemCount());
// The adapter should now be waiting for new snippets.
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.AVAILABLE);
mSnippetsSource.setSuggestionsForCategory(KnownCategories.ARTICLES, snippets);
assertEquals(PERMANENT_ELEMENTS_COUNT + snippets.size(), mNtpAdapter.getItemCount());
}
/**
* Tests how the loading indicator reacts to status changes.
*/
@Test
@Feature({"Ntp"})
public void testProgressIndicatorDisplay() {
int progressPos = mNtpAdapter.getBottomSpacerPosition() - 1;
ProgressListItem progress = (ProgressListItem) mNtpAdapter.getItems().get(progressPos);
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.INITIALIZING);
assertTrue(progress.isVisible());
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.AVAILABLE);
assertFalse(progress.isVisible());
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.AVAILABLE_LOADING);
assertTrue(progress.isVisible());
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.NOT_PROVIDED);
assertFalse(progress.isVisible());
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.CATEGORY_EXPLICITLY_DISABLED);
assertFalse(progress.isVisible());
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.SIGNED_OUT);
assertFalse(progress.isVisible());
mSnippetsSource.setStatusForCategory(KnownCategories.ARTICLES,
CategoryStatus.LOADING_ERROR);
assertFalse(progress.isVisible());
}
private List<SnippetArticleListItem> createDummySnippets() {
List<SnippetArticleListItem> snippets = new ArrayList<>();
snippets.add(new SnippetArticleListItem("https://site.com/url1", "title1", "pub1", "txt1",
"https://site.com/url1", "https://amp.site.com/url1", 0, 0, 0));
snippets.add(new SnippetArticleListItem("https://site.com/url2", "title2", "pub2", "txt2",
"https://site.com/url2", "https://amp.site.com/url2", 0, 0, 0));
snippets.add(new SnippetArticleListItem("https://site.com/url3", "title3", "pub3", "txt3",
"https://site.com/url3", "https://amp.site.com/url3", 0, 0, 0));
return snippets;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,603 |
{"url":"https:\/\/tex.stackexchange.com\/questions\/211034\/font-size-of-figures-and-table-captions","text":"# Font size of figures and table captions\n\nI need my figure and table captions to be of font size 8pt, how do I achieve this?\n\n\u2022 Welcome to TeX.SX! Please help us to help you and add a minimal working example (MWE) that illustrates your problem. It will be much easier for us to reproduce your situation and find out what the issue is when we see compilable code, starting with \\documentclass{...} and ending with \\end{document}. \u2013\u00a0yo' Nov 7 '14 at 18:13\n\nIf you are using one of the standard classes (book, report, article) or a class compatible with the caption package, then you can declare your default font specification using \\DeclareCaptionFont and then use this new declaration in \\captionsetup:\n\n\\documentclass{article}\n\\usepackage{caption}\n\n\\DeclareCaptionFont{mysize}{\\fontsize{8}{9.6}\\selectfont}\n\\captionsetup{font=mysize}\n\n\\begin{document}\n\n\\begin{figure}\n\\centering\nA\n\\caption{The caption of the figure will be typeset using \\texttt{8pt} font size}\n\\end{figure}\nRegular text.\n\n\\end{document}\n\n\nThe result:\n\n\u2022 what is the 9.6 that you have next to 8? \u2013\u00a0tilaprimera Nov 7 '14 at 18:37\n\u2022 @tilaprimera the value for the baselineskip (the vertical separation between two consecutive lines). \u2013\u00a0Gonzalo Medina Nov 7 '14 at 18:38","date":"2020-02-17 00:08:46","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\": 0.9358114004135132, \"perplexity\": 2114.41351972357}, \"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-2020-10\/segments\/1581875141460.64\/warc\/CC-MAIN-20200217000519-20200217030519-00210.warc.gz\"}"} | null | null |
Produced by David Edwards, Cindy Beyer and the online
Distributed Proofreaders Canada team at
http://www.pgdpcanada.net.
CITIES OF BELGIUM
=G r a n t A l l e n ' s H i s t o r i c a l G u i d e s=
Fcap. 8vo, green cloth, with rounded corners to
slip in the pocket, price 3s. 6d. net each.
I. P A R I S. By GRANT ALLEN
(_Second Edition_).
II. F L O R E N C E. By GRANT ALLEN
(_Second Edition_).
III. T H E C I T I E S O F B E L G I U M.
By GRANT ALLEN (_Second Edition_).
IV. V E N I C E. By GRANT ALLEN.
V. T H E C I T I E S O F N O R T H E R N
I T A L Y. By GEO. C. WILLIAMSON, Litt.D.
VI. T H E U M B R I A N T O W N S. By
Mr. and Mrs. J. W. CRUICKSHANK.
LONDON: GRANT RICHARDS
48 LEICESTER SQUARE
GRANT ALLEN'S HISTORICAL
GUIDES
=CITIES OF BELGIUM=
BY GRANT ALLEN
[Illustration]
LONDON: GRANT RICHARDS
48 LEICESTER SQUARE
BUTLER & TANNER,
THE SELWOOD PRINTING WORKS,
FROME, AND LONDON.
PREFACE TO THE SECOND EDITION
RECENT alterations, especially in the Brussels Gallery, make a new
edition of this book imperative, and, as I had been with my father
during its inception, I have undertaken such revision as is necessary.
In the main, however, my work has been merely mechanical, and the guide
remains substantially identical in detail with that originally published
in 1897.
Since that date it has been remarked in more than one quarter that many
interesting towns and objects have been omitted. I can only reply that
it would be impossible to deal exhaustively with a country so rich in
historical and artistic interest as Belgium in a single volume of this
size, and that my father only professed to point out such sights in the
chief towns as seemed to him most worthy of interest.
To alter even slightly the work of an author (especially when, as in
this case, that author is powerless to object) is a task to be
approached with the utmost diffidence, and I can only trust that those
who use this book will impute all blame for any errors or omissions
wholly to me, rather than to one who is beyond the reach of criticism.
JERRARD GRANT ALLEN.
July, 1902.
INTRODUCTION
THE object and plan of these Historical Handbooks is somewhat
different from that of any other guides at present before the public.
They do not compete or clash with such existing works; they are rather
intended to supplement than to supplant them. My purpose is not to
direct the stranger through the streets and squares of an unknown town
towards the buildings or sights which he may desire to visit; still less
is it my design to give him practical information about hotels, cab
fares, omnibuses, tramways, and other every-day material conveniences.
For such details, the traveller must still have recourse to the trusty
pages of his Baedeker, his Joanne, or his Murray. I desire rather to
supply the tourist who wishes to use his travel as a means of culture
with such historical and antiquarian information as will enable him to
understand, and therefore to enjoy, the architecture, sculpture,
painting, and minor arts of the towns he visits. In one word, it is my
object to give the reader in a very compendious form the result of all
those inquiries which have naturally suggested themselves to my own mind
during thirty-five years of foreign travel, the solution of which has
cost myself a good deal of research, thought, and labour, beyond the
facts which I could find in the ordinary handbooks.
For several years past I have devoted myself to collecting and arranging
material for a set of books to embody the idea I had thus entertained. I
earnestly hope they may meet a want on the part of tourists, especially
Americans, who, so far as my experience goes, usually come to Europe
with an honest and reverent desire to learn from the Old World whatever
of value it has to teach them, and who are prepared to take an amount of
pains in turning their trip to good account which is both rare and
praiseworthy. For such readers I shall call attention at times to other
sources of information.
These guide-books will deal more particularly with the Great Towns where
objects of art and antiquity are numerous. In every one of them, the
general plan pursued will be somewhat as follows. First will come the
inquiry why a town ever gathered together at all at that particular
spot—what induced the aggregation of human beings rather there than
elsewhere. Next, we shall consider why that town grew to social or
political importance and what were the stages by which it assumed its
present shape. Thirdly, we shall ask why it gave rise to that higher
form of handicraft which we know as Art, and towards what particular
arts it especially gravitated. After that, we shall take in detail the
various strata of its growth or development, examining the buildings and
works of art which they contain in historical order, and, as far as
possible, tracing the causes which led to their evolution. In
particular, we shall lay stress upon the origin and meaning of each
structure as an organic whole, and upon the allusions or symbols which
its fabric embodies.
A single instance will show the method upon which I intend to proceed
better than any amount of general description. A church, as a rule, is
built over the body or relics of a particular saint, in whose special
honour it was originally erected. That saint was usually one of great
local importance at the moment of its erection, or was peculiarly
implored against plague, foreign enemies, or some other pressing and
dreaded misfortune. In dealing with such a church, then, I endeavour to
show what were the circumstances which led to its erection, and what
memorials of these circumstances it still retains. In other cases it may
derive its origin from some special monastic body—Benedictine,
Dominican, Franciscan—and may therefore be full of the peculiar
symbolism and historical allusion of the order who founded it. Wherever
I have to deal with such a church, I try as far as possible to exhibit
the effect which its origin had upon its architecture and decoration; to
trace the image of the patron saint in sculpture or stained glass
throughout the fabric; and to set forth the connection of the whole
design with time and place, with order and purpose. In short, instead of
looking upon monuments of the sort mainly as the product of this or that
architect, I look upon them rather as material embodiments of the spirit
of the age—crystallizations, as it were, in stone and bronze, in form
and colour, of great popular enthusiasms.
By thus concentrating attention on what is essential and important in a
town, I hope to give in a comparatively short space, though with
inevitable conciseness, a fuller account than is usually given of the
chief architectural and monumental works of the principal art-cities. In
dealing with Paris, for example, I shall have little to say about such
modern constructions as the Champs Élysées or the Eiffel Tower; still
less, of course, about the Morgue, the Catacombs, the waxworks of the
Musée Grévin, and the celebrated Excursion in the Paris Sewers. The
space thus saved from vulgar wonders I shall hope to devote to fuller
explanation of Notre-Dame and the Sainte Chapelle, of the mediæval
carvings or tapestries of Cluny, and of the pictures or sculptures in
the galleries of the Louvre. Similarly in Florence, whatever I save from
description of the Cascine and even of the beautiful Viale dei Colli
(where explanation is needless and word-painting superfluous), I shall
give up to the Bargello, the Uffizi, and the Pitti Palace. The passing
life of the moment does not enter into my plan; I regard each town I
endeavour to illustrate mainly as a museum of its own history.
For this reason, too, I shall devote most attention in every case to
what is locally illustrative, and less to what is merely adventitious
and foreign. In Paris, for instance, I shall have more to say about
truly Parisian art and history, as embodied in St. Denis, the Île de la
Cité, and the shrine of Ste. Geneviève, than about the Egyptian and
Assyrian collections of the Louvre. In Florence, again, I shall deal
rather with the Etruscan remains, with Giotto and Fra Angelico, with the
Duomo and the Campanile, than with the admirable Memlincks and Rubenses
of the Uffizi and the Pitti, or with the beautiful Van der Goes of the
Hospital of Santa Maria. In Bruges and Brussels, once more, I shall be
especially Flemish; in the Rhine towns, Rhenish; in Venice, Venetian. I
shall assign a due amount of space, indeed, to the foreign collections,
but I shall call attention chiefly to those monuments or objects which
are of entirely local and typical value.
As regards the character of the information given, it will be mainly
historical, antiquarian, and, above all, explanatory. I am not a
connoisseur—an adept in the difficult modern science of distinguishing
the handicraft of various masters, in painting or sculpture, by minute
signs and delicate inferential processes. In such matters, I shall be
well content to follow the lead of the most authoritative experts. Nor
am I an art-critic—a student versed in the technique of the studios and
the dialect of the modelling-room. In such matters, again, I shall
attempt little more than to accept the general opinion of the most
discriminative judges. What I aim at rather is to expound the history
and meaning of each work—to put the intelligent reader in such a
position that he may judge for himself of the æsthetic beauty and
success of the object before him. To recognise the fact that this is a
Perseus and Andromeda, that a St. Barbara enthroned, the other an
obscure episode in the legend of St. Philip, is not art-criticism, but
it is often an almost indispensable prelude to the formation of a right
and sound judgment. We must know what the artist was trying to represent
before we can feel sure what measure of success he has attained in his
representation.
For the general study of Christian art, alike in architecture,
sculpture, and painting, no treatises are more useful for the tourist to
carry with him for constant reference than Mrs. Jameson's _Sacred and
Legendary Art_, and _Legends of the Madonna_ (London, Longmans). For
works of Italian art, both in Italy and elsewhere, Kugler's _Italian
Schools of Painting_ is an invaluable _vade-mecum_. These books should
be carried about by everybody everywhere. Other works of special and
local importance will occasionally be noticed under each particular
city, church, or museum.
I cannot venture to hope that handbooks containing such a mass of facts
as these will be wholly free from errors and misstatements, above all in
early editions. I can only beg those who may detect any such to point
them out, without unnecessary harshness, to the author, care of the
publisher, and if possible to assign reasons for any dissentient
opinion.
GRANT ALLEN
C O N T E N T S
PAGE
PREFACE TO THE SECOND EDITION 5
INTRODUCTION 6
HOW TO USE THESE GUIDE-BOOKS 12
ORIGINS OF THE BELGIAN TOWNS 13
ORDER OF THE TOUR 20
I BRUGES—
_A._ Origins of Bruges 22
_B._ The Heart of the City 25
_C._ The Hospital of St. John 35
_D._ The Town in General 45
_E._ The Churches 49
_F._ The Academy 59
II GHENT—
_A._ Origins of Ghent 66
_B._ The Core of Ghent 69
_C._ The Cathedral 77
_D._ The Outskirts 90
III BRUSSELS—
_A._ Origins of Brussels 98
_B._ The Heart of Brussels 100
_C._ The Picture Gallery 105
_D._ The Cathedral 138
_E._ The Upper Town 145
_F._ Surroundings 156
IV ANTWERP—
_A._ Origins of Antwerp 164
_B._ The Cathedral 168
_C._ The Picture Gallery 176
_D._ The Town in General 205
V HISTORICAL NOTES 217
INDEX 229
HOW TO USE THESE GUIDEBOOKS
_T__HE portions of this book intended to be read at leisure_ =at
home=, _before proceeding to explore each town or monument, are enclosed
in brackets [thus]. The portion relating to each_ =principal object=
_should be quietly read and digested_ =before= _a visit, and referred to
again afterwards. The portion to be read_ =on the spot= _is made as
brief as possible, and is printed in large legible type, so as to be
easily read in the dim light of churches, chapels, and galleries. The_
=key-note words= _are printed in_ =bold type=, _to catch the eye. Where
objects are numbered, the numbers used are always those of the latest
official catalogues._
_Baedeker's Guides are so printed that each principal portion can be
detached entire from the volume. The traveller who uses Baedeker is
advised to carry in his pocket one such portion, referring to the place
he is then visiting, together with the plan of the town, while carrying
this book in his hand. These Guides do_ =not= _profess to supply
practical information_.
_Individual works of merit are distinguished by an asterisk (*); those
of very exceptional interest and merit have two asterisks._ =Nothing=
_is noticed in this book which does not seem to the writer worthy of
attention_.
_See little at a time, and see it thoroughly._ =Never= _attempt to "do"
any place or any monument. By following strictly the order in which
objects are noticed in this book, you will gain a conception of the_
=historical evolution= _of the town which you cannot obtain if you go
about looking at churches and palaces hap-hazard. The order is arranged,
not quite chronologically, but on a definite_ =plan=, _which greatly
facilitates comprehension of the subject_.
ORIGINS OF THE BELGIAN TOWNS
THE somewhat heterogeneous country which we now call =Belgium= formed
=part of Gaul= under the Roman Empire. But though rich and commercial
even then, it seems to have been relatively little Romanised; and in the
beginning of the 5th century it was overrun by the =Salic Franks=, on
their way towards Laon, Soissons, and Paris. When civilization began to
creep northward again in the 9th century through the districts
barbarised by the Teutonic invasion, it was the Frankish =Charlemagne=
(Karl the Great) who introduced Roman arts afresh into the Upper and
Lower Rhinelands. The Rhine from Basle to Cologne was naturally the
region most influenced by this new Roman revival; but as Charlemagne had
his chief seat at Aix-la-Chapelle (Aachen), near the modern Belgian
frontier, the western Frankish provinces were also included in the
sphere of his improvements. When the kingdom of the Franks began to
divide more or less definitely into the Empire and France, the Flemish
region formed nominally part of the Neustrian and, later, of the French
dominions. From a very early date, however, it was practically almost
independent, and it became so even in name during its later stages. But
Brabant (with Brussels) remained a portion of the Empire.
The =Rhine= constituted the great central waterway of mediæval Europe;
the =Flemish towns= were its =ports= and its manufacturing centres. They
filled in the 13th and 14th centuries much the same place that
Liverpool, Glasgow, Manchester, and Birmingham fill in the 20th. Many
causes contributed to this result. Flanders, half independent under its
own Counts, occupied a middle position, geographically and politically,
between France and the Empire; it was comparatively free from the
disastrous wars which desolated both these countries, and in particular
(_see_ under Ghent) it largely escaped the long smouldering quarrel
between French and English which so long retarded the development of the
former. Its commercial towns, again, were not exposed on the open sea to
the attacks of pirates or hostile fleets, but were safely ensconced in
inland flats, reached by rivers or canals, almost inaccessible to
maritime enemies. Similar conditions elsewhere early ensured peace and
prosperity for Venice. The canal system of Holland and Belgium began to
be developed as early as the 12th century (at first for drainage), and
was one leading cause of the commercial importance of the Flemish cities
in the 14th. In so flat a country, locks are all but unnecessary. The
two towns which earliest rose to greatness in the Belgian area were thus
=Bruges= and =Ghent=; they possessed in the highest degree the combined
advantages of easy access to the sea and comparative inland security.
Bruges, in particular, was one of the chief stations of the Hanseatic
League, which formed an essentially commercial alliance for the mutual
protection of the northern trading centres. By the 14th century Bruges
had thus become in the north what Venice was in the south, the capital
of commerce. Trading companies from all the surrounding countries had
their "factories" in the town, and every European king or prince of
importance kept a resident minister accredited to the merchant Republic.
Some comprehension of the =mercantile condition of Europe= in general
during the Middle Ages is necessary in order to understand the early
importance and wealth of the Flemish cities. Southern Europe, and in
particular Italy, was then still the seat of all higher civilization,
more especially of the trade in manufactured articles and objects of
luxury. Florence, Venice, and Genoa ranked as the polished and learned
cities of the world. Further east, again, Constantinople still remained
in the hands of the Greek emperors, or, during the Crusades, of their
Latin rivals. A brisk trade existed _viâ_ the Mediterranean between
Europe and India or the nearer East. This double stream of traffic ran
along two main routes—one, by the Rhine, from Lombardy and Rome; the
other, by sea, from Venice, Genoa, Florence, Constantinople, the Levant,
and India. On the other hand, France was still but a half civilized
country, with few manufactures and little external trade; while England
was an exporter of raw produce, chiefly wool, like Australia in our own
time. The Hanseatic merchants of Cologne held the trade of London; those
of Wisby and Lübeck governed that of the Baltic; Bruges, as head of the
Hansa, was in close connection with all of these, as well as with Hull,
York, Novgorod, and Bergen. The position of the Flemish towns in the
14th century was thus not wholly unlike that of New York, Philadelphia,
and Boston at the present day; they stood as intermediaries between the
older civilized countries, like Italy or the Greek empire, and the newer
producers of raw material, like England, North Germany, and the Baltic
towns.
The local =manufactures= of Flanders consisted chiefly of woollen goods
and linens; the imports included Italian luxuries, Spanish figs and
raisins, Egyptian dates, Oriental silks, English wool, cattle, and
metals, Rhenish wines, and Baltic furs, skins, and walrus tusks.
In the early 16th century, when navigation had assumed new conditions,
and trade was largely diverted to the Atlantic, Antwerp, the port of the
Schelde, superseded the towns on the inland network. As Venice sank,
Antwerp rose.
The =art= that grew up in the Flemish cities during their epoch of
continuous commercial development bears on its very face the visible
impress of its mercantile origin. France is essentially a monarchical
country, and it is centralized in Paris; everything in old French art is
therefore regal and lordly. The Italian towns were oligarchies of
nobles; so the principal buildings of Florence and Venice are the
castles or palaces of the princely families, while their pictures
represent the type of art that belongs in its nature to a cultivated
aristocracy. But in Flanders, everything is in essence =commercial=. The
architecture consists mainly, not of private palaces, but of guilds,
town halls, exchanges, belfries: the pictures are the portraits of solid
and successful merchants, or the devotional works which a merchant donor
presented to the patron saint of his town or business. They are almost
overloaded with details of fur, brocade, jewellery, lace, gold, silver,
polished brass, glasswork, Oriental carpets, and richly carved
furniture. In order to understand Flemish art, therefore, it is
necessary to bear in mind at every step that it is =the art of a purely
commercial people=.
Another point which differentiates Flemish painting from the painting of
Italy during the same period is the complete absence of any opportunity
for the display of frescoes. In the Italian churches, where the walls
serve largely for support, and the full southern light makes the size of
the windows of less importance, great surfaces were left bare in the
nave and aisles, or in the lower part of the choir, crying aloud for
decoration at the hands of the fresco-painter. But in the northern
Gothic, which aimed above all things at height and the soaring effect,
and which almost annihilated the wall, by making its churches consist of
rows of vast windows with intervening piers or buttresses, the
opportunity for mural decoration occurred but seldom. The climate also
destroyed frescoes. Hence the works of pictorial art in Flemish
buildings are almost confined to =altar-pieces and votive tablets=.
Again, the great school of painting in early Italy (from Giotto to
Perugino) was a school of fresco-painters; but in Flanders no high type
of art arose till the discovery of oil-painting. Pictures were usually
imported from the Rhine towns. Hence, pictorial art in the Low Countries
seems to spring almost full-fledged, instead of being traceable through
gradual stages of evolution as in Italy. Most of the best early
paintings are small and highly finished; it was only at a comparatively
late date, when Antwerp became the leading town, that Italian influence
began to produce the larger and coarser canvases of Rubens and his
followers.
=Very early Flemish art= greatly resembles the art of the School of
Cologne. Only with Hubert and Jan van Eyck (about 1360-1440) does the
distinctively Flemish taste begin to show itself—the taste for delicate
and minute workmanship, linked with a peculiar realistic idealism, more
dainty than German work, more literal than Italian. It is an art that
bases itself upon truth of imitation and perfection of finish: its chief
æsthetic beauty is its jewel-like colour and its wealth of decorative
adjuncts. The subsequent development of Flemish painting—the painting
that pleased a clique of opulent commercial patrons—we shall trace in
detail in the various cities.
Whoever wishes to gain a deeper insight into Flemish painting should
take in his portmanteau Sir Martin =Conway's "Early Flemish Artists,"= a
brilliant and masterly work of the first importance, to which this Guide
is deeply indebted.
The =political history= of the country during this flourishing period of
the Middle Ages has also stamped itself, though somewhat less deeply, on
the character of the towns and of the art evolved in them. The =Counts
of Flanders=, originally mere lords of Bruges and its district, held
their dominions of the Kings of France. Their territory included, not
only Arras (at first the capital, now included in France) with Bruges,
Ghent, Courtrai, Tournay, and Ypres, but also the towns and districts of
Valenciennes, Lille, and St. Omer, which are now French. From the time
of Baldwin VIII. (1191), however, Arras became a part of France, and
=Ghent= was erected into the capital of Flanders. In the beginning of
the 13th century, two women sovereigns ruled in succession; under them,
and during the absence of the elective Counts on crusades, the towns
rose to be practically burgher republics. Bruges, Ypres, Ghent, and
Lille were said to possess each 40,000 looms; and though this is
certainly a mediæval exaggeration, yet the Flemish cities at this epoch
were at any rate the chief manufacturing and trading centres of northern
Europe, while London was still a mere local emporium.
In the 14th century, the cities acquired still greater freedom. The
citizens had always claimed the right to elect their Count; and the
people of Ghent now made treaties without him on their own account with
Edward III. of England. To this age belongs the heroic period of the Van
Arteveldes at Ghent, when the burghers became the real rulers of
Flanders, as will be more fully described hereafter. In 1384, however,
Count Louis III. died, leaving an only daughter, who was married to
=Philip the Bold of Burgundy=; and the wealthy Flemish towns thus passed
under the sway of the powerful princes of Dijon. Brabant fell later by
inheritance to Philip the Good. It was under the =Burgundian dynasty=,
who often held their court at Ghent, that the arts of the Netherlands
attained their first great development. Philip the Good (1419-1467)
employed Jan van Eyck as his court painter; and during his reign or just
after it the chief works of Flemish art were produced in Bruges, Ghent,
Brussels, and Tournai.
=Charles the Bold=, the last Duke of Burgundy, left one daughter, Mary,
who was married to Maximilian, afterwards Emperor. From that date
forward the history of the Flemish towns is practically merged in that
of the dynasty of Charles V., and finally becomes the story of an
unwilling and ever justly rebellious Spanish province. The subsequent
vicissitudes of Belgium as an Austrian appanage, a part of Holland, and
an independent kingdom, belong to the domain of European history. For
the visitor, it is the period of the Burgundian supremacy that really
counts in the cities of Belgium.
Yet the one great point for the tourist to bear in mind is really
this—that the art of the Flemish towns is essentially =the art of a
group of burgher communities=. It is frankly commercial, neither royal
nor aristocratic. In its beginnings it develops a strictly municipal
architecture, with a school of painters who aimed at portraiture and
sacred panel pictures. After the Reformation had destroyed sacred art in
Holland, painting in that part of the Netherlands confined itself to
portraits and to somewhat vulgar popular scenes: while in Belgium it was
Italianised, or rather Titianised and Veronesed, by Rubens and his
followers. But in its best days it was national, local, and sacred or
personal.
Take =Conway's "Early Flemish Artists"= with you in your portmanteau,
and read over in the evening his account of the works you have seen
during the day.
ORDER OF THE TOUR
IF possible, visit the cities of Belgium in =the order in which they
are treated in this Guide=:—Bruges first; then Ghent, Brussels, and
Antwerp. For this order you will find very good reasons. Bruges is the
most antique in tone and the least spoiled of all the Flemish towns; it
best exhibits the local peculiarities we have here specially to
consider; and it leads up naturally to the other cities. It is true,
Memling, the great painter whom we have chiefly to study at Bruges, is
later in date than Jan van Eyck, whose principal work (with that of his
brother Hubert) is to be seen at Ghent. But historical sequence in this
minor matter is somewhat less important than a due apprehension of the
general air of an old Flemish town such as those in which the art of the
Van Eycks arose; and besides, there is at least one characteristic Van
Eyck at Bruges, while there are many Memlings for comparison in other
cities.
As a rule, too little time is given by tourists to Bruges and Ghent, and
too much to Brussels. I should advise three or four days each to the
first-named towns, and a week to the capital.
Those who intend to combine a visit to =Holland= in the same tour should
certainly see Belgium in the order here given first, and then proceed to
Rotterdam, the Hague, Haarlem, and Amsterdam. For such a sequence, which
is geographically the easiest, is also chronologically natural. Bruges
is the most mediæval of all the towns, and has for its principal great
artist Memling. Ghent comes next, with the Van Eycks and a few later
painters. Brussels represents the end of the Middle Ages, and contains a
general metropolitan collection of early and middle Flemish art. Antwerp
gives us in particular Quentin Matsys and his contemporaries, as well as
Rubens and Van Dyck. And the Dutch towns lead us on through Van Dyck and
the later transitionals to Rembrandt, Van der Helst, Frans Hals, and the
other mighty masters of Holland. I may add that as the arrangement of
this Guide is roughly chronological, the tourist will use it best if he
follows its order.
The =Ostend route= takes the towns naturally in the sequence I suggest.
Visitors arriving by =Harwich= or =Calais= should not stop first at
Antwerp or Brussels, but go straight to Bruges, and then double back
again.
I
BRUGES
_A._ ORIGINS OF BRUGES
IN a lost corner of the great lowland flat of Flanders, defended from
the sea by an artificial <DW18>, and at the point of intersection of an
intricate network of canals and waterways, there arose in the early
Middle Ages a trading town, known in Flemish as =Brugge=, in French as
=Bruges= (that is to say, =The Bridge=), from a primitive structure that
here crossed the river. (A number of bridges now span the sluggish
streams. All of them open in the middle to admit the passage of
shipping.) Bruges stood originally on a little river, the =Reye=, once
navigable, now swallowed by canals: and the Reye flowed into the =Zwin=,
long silted up, but then the safest harbour in the Low Countries. At
first the capital of a petty Count, this land-locked internal harbour
grew in time to be the Venice of the North, and to gather round its
quays, or at its haven of Damme, the ships and merchandise of all
neighbouring peoples. Already in 1200 it ranked as the central mart of
the Hanseatic League. It was the port of entry for English wool and
Russian furs: the port of departure for Flemish broadcloths, laces,
tapestries, and linens. Canals soon connected it with Ghent, Dunkirk,
Sluys, Furnes, and Ypres. Its nucleus lay in a little knot of buildings
about the Grand' Place and the Hôtel de Ville, stretching out to the
Cathedral and the Dyver; thence it spread on all sides till in 1362 it
filled the whole space within the existing ramparts, now largely
abandoned or given over to fields and gardens. It was the wealthiest
town of Europe, outside Italy. In the 14th century, Bruges was
frequently the residence of the Counts of Flanders; and in the 15th it
became the seat of the brilliant court of the Dukes of Burgundy. Under
their rule, the opulent burghers and foreign merchants began to employ a
group of famous artists who have made the city a place of pilgrimage for
Europe and America, and to adorn the town with most of those buildings
which now beautify its decay.
The foreign traders in Bruges lived in "factories" or guilds, resembling
monasteries or colleges, and were governed by their own commercial laws.
The Bardi of Florence were among its famous merchants: the Medici had
agents here: so had the millionaire Fuggers of Augsburg.
Bruges is the best place in which to make =a first acquaintance= with
the towns and art of Flanders, because here almost all the principal
buildings are mediæval, and comparatively little that is modern comes in
to mar the completeness of the picture. We see in it the architecture
and the painting of Flanders, in the midst of the houses, the land, and
the folk that gave them origin. Brussels is largely modernised, and even
Ghent has great living manufactures; but Bruges is =a fossil of the 15th
century=. It was the first to flourish and the first to decay of the
towns of Belgium.
The =decline= of the town was due partly to the break-up of the
Hanseatic system; partly to the rise of English ports and manufacturing
towns; but still more (and especially as compared with other Flemish
cities) to the silting of the Zwin, and the want of adaptation in its
waterways to the needs of great ships and modern navigation. The old sea
entrance to Bruges was through the Zwin, by way of Sluys and Kadzand; up
that channel came the Venetian merchant fleet and the Flemish galleys,
to the port of Damme. By 1470, it ceased to be navigable for large
vessels. The later canal is still open, but as it passes through what is
now Dutch territory, it is little used; nor is it adapted to any save
ships of comparatively small burden. Another canal, suitable for craft
of 500 tons, leads through Belgian territory to Ostend; but few vessels
now navigate it, and those for the most part only for local trade. The
town has shrunk to half its former size, and has only a quarter of its
mediæval population. The commercial decay of Bruges, however, has
preserved its charm for the artist, the archæologist, and the tourist;
its sleepy streets and unfrequented quays are among the most picturesque
sights of bustling and industrial modern Belgium. The great private
palaces, indeed, are almost all destroyed: but many public buildings
remain, and the domestic architecture is quaint and pretty.
Bruges was the mother of the arts in Flanders. Jan van Eyck lived here
from 1428 to 1440: Memling, probably, from 1477 till 1494. Caxton, the
first English printer, lived as a merchant at Bruges (in the Domus
Anglorum or English factory) from 1446 to 1476, and probably put in the
press here the earliest English printed book (though strong grounds have
been adduced in favour of Cologne). Colard Mansion, the great printer of
Bruges at that date, was one of the leaders in the art of typography.
Those who desire further information on this most interesting town will
find it in James Weale's _Bruges et ses Environs_, an admirable work, to
which I desire to acknowledge my obligations.
* * * * *
At least two whole days should be devoted to Bruges: more if possible.
But the hasty traveller, who has but time for a glimpse, should neglect
the churches, and walk round the Grand' Place and the Place du Bourg to
the Dyver: spending most of his time at the =**Hôpital de St. Jean=,
which contains the glorious works of =Memling=. These are by far the
most important objects to be seen in the city. The description in this
Guide is written from the point of view of the more leisurely traveller.
Expect the frequent recurrence of the following =symbols= on houses or
pictures: (1) the Lion of Flanders, heraldic or otherwise, crowned, and
bearing a collar with a pendant cross; (2) the Bear of Bruges; (3) the
Golden Fleece (_Toison d'or_), the device of the Order founded by
Philippe le Bon in 1430, and appropriate to a country which owed its
wealth to wool; it consists of a sheep's skin suspended from a collar.
The Flemish emblem of the Swan is also common as a relief or decoration.
St. Donatian, Archbishop of Rheims, is the patron saint. His mark is a
wheel with five lighted candles.
_B._ THE HEART OF THE CITY
[The original =nucleus of Bruges= is formed by the =Bourg=,
which stands near the centre of the modern city. In 865, Baldwin
Bras-de-Fer, Count of Flanders, built a _château_ or _burg_ by
the Reye, in a corner of land still marked by the modern canal
of the Dyver, and near it a chapel, into which he transported
the relics of St. Donatian. This _burg_ grew in time into the
chief palace of the Counts of Flanders, now replaced by the
Palais de Justice; while the chapel by its side developed into
the first cathedral of Bruges, St. Donatian, now wholly
demolished. A bridge hard by crossed the little river Reye; and
from this bridge the town ultimately derives its name. The
_burg_ was built as a _tête-du-pont_ to protect the passage. A
town of traders gradually sprang up under the protection of the
castle, and developed at last into the great trading port of
Bruges. To this centre, then, we will first direct ourselves.]
Go from your hotel, down the Rue St. Amand, or the Rue St. Jacques, to
the =Grand' Place= or market-place of Bruges, noticing on your way the
numerous handsome old houses, with high-pitched roofs and gable-ends
arranged like steps, mostly of the 16th and 17th centuries. (Bruges is a
Flemish-speaking town: note the true names of the streets in Flemish.)
The very tall square tower which faces you as you enter the Grand' Place
is the =*Belfry=, the centre and visible embodiment of the town of
Bruges. The Grand' Place itself was the forum and meeting-place of the
soldier-citizens, who were called to arms by the chimes in the Belfry.
The centre of the Place is therefore appropriately occupied by a
colossal =statue group= (modern) of Pieter de Coninck and Jan Breidel,
the leaders of the citizens of Bruges at the Battle of the Spurs before
the walls of Courtrai in 1302, a conflict which secured the freedom of
Flanders from the interference of the Kings of France. The group is by
Devigne. The _reliefs_ on the pedestal represent scenes from the battle
and its antecedents.
The majestic =Belfry= itself represents the first beginnings of freedom
in Bruges. Leave to erect such a bell-tower, both as a mark of
independence and to summon the citizens to arms, was one of the first
privileges which every Teutonic trading town desired to wring from its
feudal lord. This (brick) tower, the pledge of municipal rights, was
begun in 1291 (to replace an earlier one of wood), and finished about a
hundred years later, the octagon (in stone) at the summit (which holds
the bells) having been erected in 1393-96. It consists of three stories,
the two lower of which are square and flanked by balconies with turrets;
the windows below are of the simple Early Gothic style, but show a later
type of architecture in the octagon. The _niche_ in the centre contains
the Virgin and Child (restored, after being destroyed by the French
revolutionists). Below it on either side are smaller figures holding
escutcheons. From the balcony between these last, the laws and the
rescripts of the Counts were read aloud to the people assembled in the
square.
The Belfry can be =ascended= by steps. Apply to the _concierge_; 25 c.
per person. Owing to the force of the wind, it leans slightly to the
S.E. The =*view= from the top is very extensive and striking; it
embraces the greater part of the Plain of Flanders, with its towns and
villages: the country, though quite flat, looks beautiful when thus
seen. In early times, however, the look-out from the summit was of
practical use for purposes of observation, military or maritime. It
commanded the river, the Zwin, and the sea approach by Sluys and Damme;
the course of the various canals; and the roads to Ghent, Antwerp,
Tournai, and Courtrai. The Belfry contains a famous set of =chimes=, the
mechanism of which may be inspected by the visitor. He will have
frequent opportunities of hearing the beautiful and mellow carillon,
perhaps to excess. The existing bells date only from 1680: the mechanism
from 1784.
The square building on either side of the Belfry, known as =Les Halles=,
was erected in or about 1248, and is a fine but sombre specimen of Early
Gothic civic architecture. The wing to the left was originally the
=Cloth Hall=, for the display and sale of the woollen manufactures of
Ghent and Bruges. It is now used as municipal offices. A door to the L.
gives access to a small Museum of Antiquities on the ground floor, which
may be safely neglected by all save specialist archæologists. (Admission
50 c.) The wing to the right is the meat market.
Now, stand with your back to the Belfry to survey the =Square=. The
brick building on your right is the Post Office (modern); the stone one
beyond it (also modern) is the Palace of the Provincial Government of
Flanders. Both have been erected in a style suitable to the town. In the
Middle Ages, ships could come up to this part of the Grand' Place to
discharge their cargo. The quaint houses that face you, with
high-pitched gable-ends, are partly modern, but mostly old, though
restored. On the left (W.) side of the Place, at the corner of the Rue
St. Amand, stands the square castle-like building known as _Au Lion de
Flandre_ and marked by its gold lion. It is one of the best brick
mediæval buildings in Bruges. According to a doubtful tradition, it was
occupied by Charles II. of England during his exile, when he was created
by the Brugeois King of the Crossbowmen of St. Sebastian (see later). In
the house beside it, known as the Craenenburg, the citizens of Bruges
imprisoned Maximilian, King of the Romans, from the 5th to the 17th of
February, 1488, because he would not grant the care of his son Philip,
heir to the crown of the Netherlands, to the King of France. They only
released him after he had sworn before an altar erected at the spot, on
the Host, the true Cross, and the Relics of St. Donatian, to renounce
his claim to the guardianship of his son, and to grant a general
amnesty. However, he was treacherously released from his oath by a
congress of Princes convened a little later by his father, the Emperor
Frederic IV.
* * * * *
From the corner of the Post Office, take the short Rue Breydel to the
=Place du Bourg=, the still more intimate centre and focus of the early
life of Bruges. This Place contained the old Palace of the Counts of
Flanders, and the original Cathedral, both now destroyed, as well as the
Town Hall and other important buildings still preserved for us.
The tallest of the three handsome edifices on the S. side of the Square
(profusely adorned with sculpture) is the =**Hôtel de Ville=, a
beautiful gem of Middle Gothic architecture, begun about 1376, and
finished about 1387. This is one of the finest pieces of civic
architecture in Belgium. The _façade_, though over-restored, and the six
beautiful turrets and chimneys, are in the main of the original design.
The sculpture in the niches, destroyed during the French Revolution, has
been only tolerably replaced by modern Belgian sculptors in our own day.
The lower tier contains the Annunciation, R. and L. of the doorway, with
figures of various saints and prophets. In the tiers above this are
statues of the Counts of Flanders of various ages. The _reliefs_ just
below the windows of the first floor represent episodes from Biblical
history:—David before Saul, David dancing before the Ark, the Judgment
of Solomon, the Building of Solomon's Temple, and other scenes which the
visitor can easily identify. The Great Hall in the interior is
interesting only for its fine pendant Gothic wooden roof.
The somewhat lower building, to the right of the Hôtel de Ville, is the
=**Chapelle du Saint Sang=. The decorated portal round the corner also
forms part of the same building.
[In the 12th and 13th centuries (age of the Crusades) the
chivalrous and credulous knights of the North and West who
repaired to the Holy Land, whether as pilgrims or as soldiers of
the Faith, were anxious to bring back with them relics of the
saints or of still more holy personages. The astute Greeks and
Syrians with whom they had to deal rose to the occasion, and
sold the simple Westerns various sacred objects of more or less
doubtful authenticity at fabulous prices. Over these treasured
deposits stately churches were often raised; for example, St.
Louis of France constructed the Sainte Chapelle in Paris, to
contain the Crown of Thorns and part of the True Cross, which he
had purchased at an immense cost from Baldwin, Emperor of
Constantinople. Among the earlier visitors to the Holy Land who
thus signalised their journey was Theodoric of Alsace, elected
Count of Flanders in 1128; he brought back with him in 1149 some
drops of the =Holy Blood= of the Saviour, said to have been
preserved by Joseph of Arimathea, which he presented to his
faithful city of Bruges. Fitly to enshrine them, Theodoric
erected a chapel in the succeeding year, 1150; and this early
church forms the lower floor of the existing building. Above it,
in the 15th century, when Bruges grew richer, was raised a
second and more gorgeous chapel (as at the Sainte Chapelle), in
which the holy relic is now preserved. Almost all the works of
art in the dainty little oratory accordingly bear special
reference to the Holy Blood, its preservation, and its transport
to Bruges. The dedication is to =St. Basil=, the founder of
eastern monasticism—a Greek Father little known in the West,
whose fame Theodoric must have learned in Syria. The nobles of
Flanders, it must be remembered, were particularly active in
organising the Crusades.]
The =exterior= has a fine figure of St. Leonard (holding the fetters
which are his symbol) under a Gothic niche. He was the patron of
Christian slaves held in duress by the Saracens. The beautiful
flamboyant =portal= and =staircase=, round the corner, erected in
1529-1533, in the ornate decorative style of the period, have (restored)
figures of Crusaders and their Queens in niches, with incongruous
Renaissance busts below.
To visit the =interior=, ring the bell in the corner: admission, 50 c.
per person.
The =Museum= of the Brotherhood of the Holy Blood, on the first floor,
which we first visit, contains by the =left wall= the handsome
silver-gilt Reliquary (of 1617), studded with jewels, which encloses the
drops of the Holy Blood. The figures on it represent Christ (the source
of the Blood), the Blessed Virgin, St. Basil (patron of the church), and
St. Donatian (patron of the town). The Blood is exhibited in a simpler
_châsse_ in the chapel every Friday; that is to say, on the day of the
Crucifixion. The great Reliquary itself is carried in procession only,
on the Monday after the 3rd of May. Right and left of the _châsse_ are
portraits of the members of the Confraternity of the Holy Blood by P.
Pourbus, 1556: unusually good works of this painter. A triptych to the
right, by an unknown master of the early 16th century, figures the
Crucifixion, with special reference to the Holy Blood, representing St.
Longinus in the act of piercing the side of Christ (thus drawing the
Blood), with the Holy Women and St. John in attendance; on the wings,
the Way to Calvary, and the Resurrection.
=Between the windows= is a curious chronological picture of the late
15th century, representing the History of Our Lady in the usual stages,
with other episodes. To the R. of it, a painting of the 15th century
shows Count Theodoric receiving the Holy Blood from his brother-in-law,
Baldwin, King of Jerusalem, and the bringing of the Holy Blood to
Bruges.
On the =right wall= there is a famous =*triptych= by Gerard David (the
finest work here), representing the Deposition in the Tomb, with the
Maries, St. John, Nicodemus, and an attendant holding a dish to contain
the Holy Blood, which is also seen conspicuously flowing from the
wounds; the left wing shows the Magdalen with Cleophas; the right wing,
the preservation of the Crown of Thorns by Joseph of Arimathea. The
portrait character of the faces is admirable: stand long and study this
fine work.
The original designs for the windows of the Chapel are preserved in a
=glass case= by the window; behind which are fragments of early
glass; conspicuous among them, St. Barbara with her tower.
On the =exit wall= is a fine piece of late Flemish tapestry,
representing the bringing of the body of St. Augustine to Pavia, with
side figures of San Frediano of Lucca and Sant' Ercolano of
Perugia—executed, no doubt, for an Italian patron.
The =Chapel= itself, which we next enter, is gorgeously decorated in
polychrome, recently restored. The stained glass windows, containing
portraits of the Burgundian Princes from the beginning of the dynasty
down to Maria Theresa and Francis I., were executed in 1845 from earlier
designs. The large window facing the High Altar is modern. It represents
appropriately the history of the Passion, the origin of the Sacred
Blood, its Transference to Bruges, and the figures of the Flemish
Crusaders engaged in its transport. At the summit of the window, notice
the frequent and fitting symbol of the pelican feeding its young with
its own blood.
In the little =side chapel= to the R., separated from the main building
by an arcade of three arches, is the _tabernacle_ or canopy from which
the Sacred Blood is exhibited weekly. To the right is hung a Crown of
Thorns. Notice, also, the Crown of Thorns held by the angel at the top
of the steps. The window to the L. (modern) represents St. Longinus, the
centurion who pierced the side of Christ, and St. Veronica, displaying
her napkin which she gave to the Saviour to wipe his face on the way to
Calvary, and which retained ever after the impress of the Divine
Countenance. Almost all the other objects in the chapel bear reference,
more or less direct, to the Holy Blood. Observe particularly in the main
chapel the handsome modern High Altar with its reliefs of
scenes of the Passion. Such scenes as the Paschal Lamb on its base, with
the Hebrew smearing the lintel of the door, are of course symbolical.
The =Lower Chapel=, to which we are next conducted, is a fine specimen
of late Romanesque architecture. It was built by Theodoric in 1150. Its
solid short pillars and round arches contrast with the lighter and later
Gothic of the upper building. The space above the door of the eastern of
the two chapels which face the entrance, is occupied by an interesting
mediæval relief representing a baptism with a dove descending. Notice as
you pass out, from the Place outside, the two beautiful _turrets_ at the
west end of the main chapel.
* * * * *
To the left of the Hôtel de Ville stands the ornate and much gilded
Renaissance building known as the =*Maison de l'Ancien Greffe=,
originally the municipal record office, but now employed as a
police-court. It bears the date 1537, and has been recently restored and
profusely covered with gold decoration. Over the main doorway is the
Lion of Flanders; on the architrave of the first floor are heads of
Counts and Countesses; and the building is surmounted by a figure of
Justice, with Moses and Aaron and emblematical statues. Note the Golden
Fleece and other symbols. The interior is uninteresting.
The E. side of the square is formed by the =Palais de Justice=, which
stands on the site of an old =palace of the Counts of Flanders=,
presented by Philippe le Beau to the Liberty of Bruges, and employed by
them as their town hall of the _Buitenpoorters_, or inhabitants of the
district outside the gate, known as the _Franc de Bruges_. The
Renaissance building, erected between 1520 and 1608, was burnt down and
replaced in the 18th century by the very uninteresting existing
building. Parts of the old palace, however, were preserved, one room in
which should be visited for the sake of its magnificent
=**chimney-piece=. In order to see it, enter the quadrangle: the
porter's room faces you as you enter; inquire there for the key;
admission, 50 c. per person. The _concierge_ conducts you to the
Court-Room, belonging to the original building. Almost the entire side
of the room is occupied by a splendid Renaissance chimney-piece,
executed in 1529, after designs by Lancelot Blondeel of Bruges (a
painter whose works are frequent in the town), and Guyot de Beaugrant of
Malines, for the Council of the Liberty of Bruges, in honour of Charles
V., as a memorial of the Treaty of Cambrai, in 1526. (This was the
treaty concluded after the battle of Pavia, by which François I^{er} of
France was compelled to acknowledge the independence of Flanders. Some
of the figures in the background are allusive to the victory.) The lower
part, or chimney-piece proper, is of black marble. The upper portion is
of carved oak. The marble part has four bas-reliefs in white alabaster
by Guyot de Beaugrant, representing the History of Susannah, a mere
excuse for the nude: (1) Susannah and the Elders at the Bath; (2)
Susannah dragged by the Elders before the Judge; (3) Daniel before the
Judge exculpating Susannah; (4) The Stoning of the Elders. The genii at
the corners are also by Beaugrant. The whole is in the pagan taste of
the Renaissance. The upper portion in oak contains in the centre a
statue of Charles V., represented in his capacity as Count of Flanders
(as shown by the arms on his cuirass): the other figures represent his
descent and the cumulation of sovereignties in his person. On the throne
behind Charles (ill seen) are busts of Philippe le Beau, his father,
through whom he inherited the Burgundian dominions, and Johanna (the
Mad) of Spain, his mother, through whom he inherited the united
Peninsula. The statues L. and R. are those of his actual royal
predecessors. The figures to the L. are his paternal grandfather, the
Emperor Maximilian, from whom he derived his German territories, and his
paternal grandmother, *Mary of Burgundy, who brought into the family
Flanders, Burgundy, etc. Mary is represented with a hawk on her wrist,
as she was killed at twenty-five by a fall from her horse while out
hawking. (We shall see her tomb later at Notre-Dame.) The figures on the
R. are those of Ferdinand of Aragon and Isabella of Castile, the
maternal grandfather and grandmother of Charles, from whom he inherited
the two portions of his Spanish dominions. The medallions at the back
represent the personages most concerned in the Treaty of Cambrai, and
the Victory of Pavia which rendered it possible. (De Lannoy, the
conqueror, to whom François gave up his sword, and Margaret of Austria.)
The tapestry which surrounds the hall is modern; it was manufactured at
Ingelmünster after the pattern of a few old fragments found in the
cellars of the ancient building. The mediocre painting on the wall
depicts a sitting of the court of the Liberty of Bruges in this room
(1659).
* * * * *
The N. side of the square is now occupied by a small Place planted with
trees. Originally, however, the =old cathedral= of Bruges occupied this
site. It was dedicated to St. Donatian, the patron of the city, whose
relics were preserved in it; but it was barbarously destroyed by the
French Revolutionary army in 1799, and the works of art which it
contained were dispersed or ruined. Figures of St. Donatian occur
accordingly in many paintings at Bruges. Jan van Eyck was buried in this
cathedral, and a =statue= has been erected to him under the trees in the
little Place. In order, therefore, mentally to complete the picture of
the Place du Bourg in the 16th century, we must imagine not only the
Hôtel de Ville, the Chapelle du Saint Sang, and the Ancien Greffe in
something approaching their existing condition, but also the stately
cathedral and the original Renaissance building of the _Franc de Bruges_
filling in the remainder.
An archway spans the space between the Ancien Greffe and the Hôtel de
Ville. Take the narrow street which dives beneath it, looking back as
you pass at the archway with its inscription of S.P.Q.B. (for Senatus
Populusque Brugensis). The street then leads across a bridge over the
river Reye or principal canal, and affords a good view of the back of
the earlier portion of the Palais de Justice, with its picturesque brick
turrets, and a few early arches belonging to the primitive palace. I
recommend the visitor to turn to the R. after crossing the bridge,
traverse the little square, and make his way home by the bank of the
Dyver and the church of Notre-Dame. The view towards the Hôtel de Ville
and the Belfry, from the part of the Dyver a little to the east behind
the Belfry, is one of the most picturesque and striking in Bruges.
_C._ THE HOSPITAL OF ST. JOHN
[The =Hospital of St. John=, one of the most ancient
institutions in Bruges, or of its kind in Europe, was founded
not later than 1188, and still retains, within and without, its
mediæval arrangement. =Its Augustinian brothers and nuns= tend
the sick in the primitive building, now largely added to. It
derives its chief interest for the tourist, however, from its
small =Picture Gallery=, the one object in Bruges which must
above everything else be visited. This is the only place for
studying in full the exquisite art of =Memling=, whose charming
and poetical work is here more fully represented than elsewhere.
In this respect the Hospital of St. John may be fitly compared
with the two other famous "one-man shows" of Europe—the Fra
Angelicos at San Marco in Florence, and the Giottos in the
Madonna dell' Arena at Padua. Many of the pictures were painted
for the institution which they still adorn; so that we have here
the opportunity of seeing works of mediæval art in the precise
surroundings which first produced them.
=Hans Memling=, whose name is also written _Memlinc_ and
_Memlin_, etc. (long erroneously cited as _Hemling_; through a
mistaken reading of the initial in his signature) is a painter
of whom little is known, save his work; but the work is the man,
and therefore amply sufficient. He was born about 1430, perhaps
in Germany, and is believed to have been a pupil of Roger van
der Weyden, the Brussels painter, whose work we shall see later
at Antwerp and elsewhere. Mr. Weale has shown that he was a
person of some wealth, settled at Bruges in his own house (about
1478), and in a position to lend money to the town. He died in
1495. His period of activity as a painter is thus coincident
with the earlier work of Carpaccio and Perugino in Italy; and he
died while Raphael was still a boy. In relation to the artists
of his own country, whose works we have still to see, Memling
was junior by more than a generation to Jan van Eyck, having
been born about ten years before Van Eyck died; he was also
younger by thirty years than Roger van der Weyden; and by twenty
or thirty years than Dierick Bouts; but older by at least twenty
than Gerard David. Memling has been called the Fra Angelico of
Flanders; but this is only true so far as regards Fra Angelico's
panel works; the saintly Frate, when he worked in fresco,
adopted a style wholly different from that which he displays in
his miniature-like altar-pieces. It would be truer to say that
Memling is the Benozzo Gozzoli of the North: he has the same
love of decorative adjuncts, and the same naïve delight in the
beauty of external nature.
Before visiting the Hospital, it is also well to be acquainted
in outline with =the history of St. Ursula=, whose _châsse_ or
=shrine= forms one of its greatest treasures. The Hospital
possessed an important relic of the saint—her holy arm—and
about 1480-1489 commissioned Memling to paint scenes from her
life on the shrine destined to contain this precious deposit.
The chest or reliquary which he adorned for the purpose forms
the very best work of Memling's lifetime.
=St. Ursula= was a British (or Bretonne) princess, brought up as
a Christian by her pious parents. She was sought in marriage by
a pagan prince, Conon, said to be the son of a king of England.
The English king, called Agrippinus in the legend, sent
ambassadors to the king of Britain (or Brittany) asking for the
hand of Ursula for his heir. But Ursula made three conditions:
first, that she should be given as companions ten noble virgins,
and that she herself and each of the virgins should be
accompanied by a thousand maiden attendants; second, that they
should all together visit the shrines of the saints; and third,
that the prince Conon and all his court should receive baptism.
These conditions were complied with; the king of England
collected 11,000 virgins; and Ursula, with her companions,
sailed for Cologne, where she arrived miraculously without the
assistance of sailors (but Memling adds them). Here, she had a
vision of an angel bidding her to repair to Rome, the threshold
of the apostles. From Cologne, the pilgrims went up the Rhine by
boat, till they arrived at Basle, where they disembarked and
continued their journey on foot over the Alps to Italy. At
length they reached the Tiber, which they descended till they
approached the walls of Rome. There, the Pope, St. Cyriacus,
went forth with all his clergy in procession to meet them. He
gave them his blessing, and lest the maidens should come to harm
in so wicked a city, he had tents pitched for them outside the
walls on the side towards Tivoli. Meanwhile, prince Conon had
come on pilgrimage by a different route, and arrived at Rome on
the same day as his betrothed. He knelt with Ursula at the feet
of the Pope, and, being baptized, received in exchange the name
of Ethereus.
After a certain time spent in Rome, the holy maidens bethought
them to return home again. Thereupon, Pope Cyriacus decided to
accompany them, together with his cardinals, archbishops,
bishops, patriarchs, and many others of his prelates. They
crossed the Alps, embarked again at Basle, and made their way
northward as far as Cologne. Now it happened that the army of
the Huns was at that time besieging the Roman colony; and the
pagans fell upon the 11,000 virgins, with the Pope and their
other saintly companions. Prince Ethereus was one of the first
to die; then Cyriacus, the bishops, and the cardinals perished.
Last of all, the pagans turned upon the virgins, all of whom
they slew, save only St. Ursula. Her they carried before their
king, who, beholding her beauty, would fain have wedded her. But
Ursula sternly refused the offer of this son of Satan; whereupon
the king, seizing his bow, transfixed her breast with three
arrows. Hence her symbol is an arrow; also, she is the patroness
of young girls and of virgins, so that her shrine is
particularly appropriate in a nunnery.
Most of the bones of St. Ursula and her 11,000 virgins are
preserved at Cologne, the city of her martyrdom, where they are
ranged in cases round the walls of a church dedicated in her
honour; but her arm is here, and a few other relics are
distributed elsewhere.
The Hospital is open daily from 9 to 6; Sundays, 3 to 6. 1 franc
per person. If you have Conway, take it with you.]
From the Grand' Place, turn down the Rue des Pierres, the principal
shopping street of Bruges, with several fine old _façades_, many of them
dated. At the Place Simon Stévin turn to the L., and go straight on as
far as the church of Notre-Dame. The long brick building with Gothic
arches, on your right, is the =**Hospital of St. John= the Evangelist.
First, examine the brick Gothic =exterior=. Over the outer doorway is
the figure of a bishop with a flaming heart, the emblem of St.
Augustine, this being an Augustinian hospital. Continue on to the
original main portal (now bricked up) with a broken pillar, and two 13th
century reliefs in the tympanum. That to the _right_ represents the
Death of the Virgin, with the Apostles grouped around, and the figure of
the Christ receiving her naked new-born soul as usual. Above is the
Coronation of Our Lady. That to the _left_ seems like a reversed and
altered _replica_ of the same subject, with perhaps the Last Judgment
above it. It is, however, so much dilapidated that identification is
difficult. (Perhaps the top is a Glory of St. Ursula.) Go on as far as
the little bridge over the canal, to inspect the picturesque river front
of the Hospital.
Return to the main portal and ring the inner bell. Admission, see above.
The =pictures= are collected in the former Chapter-house of the
Hospital, above the door of which is another figure of St. Augustine.
The centre of the room is occupied by the famous _châsse_ or =**shrine=
containing the arm of =St. Ursula=, a dainty little Gothic chapel in
miniature. It is painted with exquisite scenes from the legend, by
Memling, with all the charm of a fairy tale. He treats it as a poetical
romance. Begin the story on the side towards the window. (For a
penetrating criticism of these works, see Conway.)
_1st panel_, on the left: St. Ursula and her maidens, in the rich dress
of the Burgundian court of the 15th century, arrive at Cologne, the
buildings of which are seen in the background, correctly represented,
but not in their true relations. In a window in the background to the
R., the angel appears to St. Ursula in a vision.
_2nd panel_: the Virgins arrive at Basle and disembark from the ships.
In the background, they are seen preparing to make their way, one by
one, across the Alps, which rise from low hills at the base to snowy
mountains. From another ship Conon and his knights are disembarking.
**_3rd panel_: (the most beautiful:) the Maidens arrive at Rome. In the
distance they are seen entering the city through a triumphal arch; in
the foreground, St. Ursula kneels before St. Cyriacus and his bishops,
with their attendant deacons, all the faces having the character of
portraits. (Note especially the fat and jolly ecclesiastic just under
the arch.) At the same time, her betrothed, Conon, with his knights,
arrives at Rome by a different road, and is seen kneeling in a red robe
trimmed with rich fur beside St. Ursula. (Fine portrait faces of Conon
and an old courtier behind him.) The Pope and his priests are gathered
under the portals of a beautiful round-arched building, whose exquisite
architecture should be closely examined. To the extreme R., the new
converts and Conon receive baptism naked in fonts after the early
fashion. In the background of this scene, St. Ursula receives the
Sacrament. (She may be recognised throughout by her peculiar
blue-and-white dress, with its open sleeves.) To the left of her, Conon
makes confession. In this, as in the other scenes, several successive
moments of the same episode are contemporaneously represented. Look long
at it.
Now, =turn round= the shrine, which swings freely on a pivot, to see the
scenes of the return journey.
_1st panel_: (beginning again at the left:) the Pope and his bishops and
cardinals embark with St. Ursula in the boat at Basle on their way to
Cologne. Three episodes are here conjoined: the Pope cautiously stepping
into a ship; the Pope seated; the ship sailing down the Rhine. All the
faces here, and especially the timid old Pope stepping into the boat,
deserve careful examination. In the background, the return over the
Alps.
_*2nd panel_: the Maidens and the Pope arrive at Cologne, where they are
instantly set upon by the armed Huns. Conon is slain by the thrust of a
sword, and falls back dying in the arms of St. Ursula. Many of the
maidens are also slaughtered.
_*3rd panel_: continuous with the last, but representing a subsequent
moment: the Martyrdom of St. Ursula. The King of the Huns, in full
armour, at the door of his tent, bends his bow to shoot the blessed
martyr, who has refused his advances. Around are grouped his knights in
admirably painted armour. (Note the reflections.) All the scenes have
the character of a mediæval romance. For their open-air tone and
make-believe martyrdom, see Conway.
At the =ends= of the shrine are two other pictures, (1) *St. Ursula with
her arrow, as the protectress of young girls, sheltering a number of
them under her cloak (not, as is commonly said, the 11,000 Virgins).
Similar protecting figures of the saint are common elsewhere (Cluny,
Bologna, etc.). At the opposite end, (2) the Madonna and Child with an
apple, and at her feet two Augustinian nuns of this Hospital, kneeling,
to represent the devotion of the order.
The =roof= of the shrine is also decorated with pictures. (1) St. Ursula
receiving the crown of martyrdom from God the Father, with the Son and
the Holy Ghost; at the sides, two angels playing the mandoline and the
regal or portable organ; (2) St. Ursula in Paradise, bearing her arrow,
and surrounded by her maidens, who shared her martyrdom, together with
the Pope and other ecclesiastics in the background. (This picture is
largely borrowed from the famous one by Stephan Lochner on the High
Altar of Cologne Cathedral, known as the _Dombild_. If you are going on
to Cologne, buy a photograph of this now, to compare with Meister
Stephan later. His altar-piece is engraved in Conway. If you have it
with you, compare them.) At the sides are two angels playing the zither
and the violin. (The angels are possibly by a pupil.)
I have given a brief description only of these pictures, but every one
of them ought to be carefully examined, and the character of the figures
and of the landscape or architectural background noted. You will see
nothing lovelier in all Flanders.
Near the window by the entrance is a =**Triptych=, also by Memling,
commissioned by Brother Jan Floreins of this Hospital. The _central
panel_ represents the =Adoration of the Magi=, which takes place, as
usual, under a ruined temple fitted up as a manger. The Eldest of the
Three Kings (according to precedent) is kneeling and has presented his
gift; Joseph, recognisable (in all three panels) by his red-and-black
robe, stands erect behind him, with the presented gift in his hands. The
Middle-aged King, arrayed in cloth of gold, with a white tippet, kneels
with his gift to the L. of the picture. The Young King, a black man, as
always, is entering with his gift to the right. The three thus typify
the Three Ages of Man, and also the three known continents, Europe,
Asia, Africa. On the L. side of this central panel are figured the
donor, Jan Floreins, and his brother Jacob. (Members of the same family
are grouped in the well-known "Duchâtel Madonna," also by Memling, in
the Louvre.) To the right is a figure looking in at a window and wearing
the yellow cap still used by convalescents of the Hospital, (arbitrarily
said to be a portrait of Memling.) The _left panel_ represents the
Nativity, with our Lady, St. Joseph, and two adoring angels. The _right
panel_ shows the Presentation in the Temple, with Simeon and Anna, and
St. Joseph (in red and black) in the background. (The whole thus
typifies the Epiphany of Christ; left, to the Blessed Virgin; centre, to
the Gentiles; right, to the Jews.) The _outer panels_, in pursuance of
the same idea, have figures, right, of St. John Baptist with the lamb
(he pointed out Christ to the Jews), with the Baptism of Christ in the
background; and left, St. Veronica, who preserved for us the features of
our Lord, displaying his divine face on her napkin. The architectural
frame shows the First Sin and the Expulsion from Paradise. Note
everywhere the strong character in the men's faces, and the exquisite
landscape or architectural backgrounds. Dated 1479. This is Memling's
finest altar-piece: its glow of colour is glorious.
By the centre window, a =*triptych=, doubtfully attributed to Memling,
represents, in the centre, =the Deposition from the Cross=, with the
Holy Blood conspicuous, as might be expected in a Bruges work. In the
foreground are St. John, the Madonna, and St. Mary Magdalene; in the
background, the preparations for the Deposition in the Tomb. On the
=wings=: _left_, Brother Adrian Reins, the donor, with his patron saint,
Adrian, bearing his symbol, the anvil, on which his limbs were struck
off, and with his lion at his feet; _right_, St. Barbara with her tower,
perhaps as patroness of armourers. On the =exterior wings=, _left_, St.
Wilgefortis with her tau-shaped cross; _right_, St. Mary of Egypt, with
the three loaves which sustained her in the desert.
On the same stand is the beautiful =*diptych= by Memling, representing
=Martin van Nieuwenhoven= adoring the Madonna. The _left_ panel
represents Our Lady and the Child, with an apple, poised on a
beautifully painted cushion. A convex mirror in the background reflects
the backs of the figures (as in the Van Eyck of the National Gallery).
Through the open window is seen a charming distant prospect. The _right_
panel has the fine portrait of the donor, in a velvet dress painted with
extreme realism. Note the admirable prayer-book and joined hands. At his
back, a stained glass window shows his patron, St. Martin, dividing his
cloak for the beggar. Below, a lovely glimpse of landscape. This is
probably Memling's most successful portrait. Dated 1487: brought here
from the _Hospice_ of St. Julian, of which Martin was Master.
In all Flemish art, observe now the =wooden face of the
Madonna=—ultimately derived, I believe, from imitation of painted
wooden figures, and then hardened into a type. As a rule, the Madonna is
the least interesting part of all Flemish painting; and after her, the
women, especially the young ones. The men's faces are best, and better
when old: character, not beauty, is what the painter cares for. This is
most noticeable in Van Eyck, but is true in part even of Memling.
At the end of the room is the magnificent =*triptych= painted by Memling
for the =High Altar= of the Church of this Hospital. This is the largest
of his works, and it is dedicated to the honour of the two saints (John
the Evangelist and John the Baptist) who are patrons of the Hospital.
The =central panel= represents Our Lady, seated in an exquisite
cloister, on a throne backed with cloth of gold. To the right and left
are two exquisite angels, one of whom plays a regal, while the other, in
a delicious pale blue robe, holds a book for Our Lady. Two smaller
angels, poised in air, support her crown. To the left, St. Catherine of
Alexandria kneels as princess, with the broken wheel and the sword of
her martyrdom at her feet. The Child Christ places a ring on her finger;
whence the whole composition is often absurdly called "The Marriage of
St. Catherine." It should be styled "The Altar-piece of the St. Johns."
To the right is St. Barbara, calmly reading, with her tower behind her.
When these two saints are thus combined, they represent the meditative
and the active life (as St. Barbara was the patroness of arms:) or, more
definitely, the clergy and the knighthood. Hence their appropriateness
to an institution, half monastic, half secular. In the background stand
the two patron saints; St. John Baptist with the lamb (Memling's
personal patron), to the left, and St. John the Evangelist with the cup
and serpent, to the right. (For these symbols, see Mrs. Jameson.) Behind
the Baptist are scenes from his life and preaching. He is led to prison,
and his body is burned by order of Julian the Apostate. Behind the
Evangelist, he is seen in the cauldron of boiling oil. The small figure
in black to the right is the chief donor, Brother Jan Floreins, who is
seen further back in his secular capacity as public gauger of wine, near
a great crane, which affords a fine picture of mercantile life in old
Bruges. The _left wing_ represents the life of St. John the Baptist. In
the distance is seen the Baptism of Christ. In a room to the left, the
daughter of Herodias dances before Herod. The foreground is occupied by
the episode of the Decollation, treated in a courtly manner, very
redolent of the Burgundian splendour. Figures and attitudes are
charming: only, the martyrdom sinks into insignificance beside the
princess's collar. Other minor episodes may be discovered by inspection.
(The episodes on either wing overflow into the main pictures.) The
_right wing_ shows St. John the Evangelist in Patmos, writing the
Apocalypse, various scenes from which are realistically and too solidly
represented above him, without poetical insight. Memling here attempts
to transcend his powers. He has no sublimity. On the _exterior of the
wings_ are seen the four other members of the society who were donors of
the altar-piece; Anthony Zeghers, master of the Hospital, with his
patron, St. Anthony, known by his pig and tau-shaped crutch and bell:
Jacob de Cueninc, treasurer, accompanied by his patron, St. James the
Greater, with his pilgrim's staff and scallop-shell: Agnes Casembrood,
mistress of the Hospital, with her patron, St. Agnes, known by her lamb:
and Claire van Hulsen, a sister, with her patron, St. Clara. Dated,
1479.
By the entrance door is a =Portrait of Marie Moreel=, represented as a
Sibyl. She was a daughter of Willem Moreel or Morelli, a patron of
Memling, whom we shall meet again at the Museum. This is a fine portrait
of a solid, plain body, a good deal spoiled by attempted cleaning. It
comes from the _Hospice_ of St. Julian.
As you go out, cast a glance at the fine old brick buildings, and note
the cleanliness of all the arrangements.
Return more than once: do not be satisfied with a single visit.
The other pictures and objects formerly exhibited in this Hospital have
been transferred to the Potterie and another building. They need only be
visited by those whose time is ample.
After leaving the Hospital, I do not advise an immediate visit to the
Academy. Let the Memlings first sink into your mind. But the walk may be
prolonged by crossing the canal, and taking the second turning to the
R., which leads (over a pretty bridge of three arches) to the
=Béguinage=, a lay-nunnery for ladies who take no vows, but who live in
monastic fashion under the charge of a Superior. Above the gateway is a
figure of St. Elizabeth of Hungary, (to whom the church within is
dedicated) giving alms to a beggar. She wears her crown, and carries in
her hand the crown and book which are her symbol. Remember these,—they
will recur later. Pass under the gateway and into the grass-grown
precincts for an external glimpse of the quiet old-world close, with its
calm white-washed houses. The church, dedicated to St. Elizabeth, is
uninteresting. This walk may be further prolonged by the pretty bank of
the _Lac d'amour_ or _Minnewater_ as far as the external canal,
returning by the ramparts and the picturesque =Porte de Gand=.
_D._ THE TOWN IN GENERAL.
[The =town of Bruges itself= is more interesting, after all,
than almost any one thing in it. Vary your day by giving up the
morning to definite sight-seeing, and devoting the afternoon to
=strolls= through the town and neighbourhood, in search of
picturesqueness. I subjoin a few stray hints for such casual
rambles.]
(1) Set out from the Grand' Place, and turn down the Rue Breydel to the
Place du Bourg. Cross the Place by the statue of Jan van Eyck; traverse
the Rue Philippe Stock; turn up the Rue des Armuriers a little to the
R., and continue on to the Place St. Jean, with a few interesting
houses. Note here and elsewhere, at every turn, the little statues of
the Virgin and Child in niches, and the old signs on the fronts or
gables. The interesting =Gothic turret= which faces you as you go
belongs to the old 14th Century building called =De Poorters Loodge=, or
the Assembly Hall of the Noble Citizens Within the Gate, as opposed to
those of the =Franc de Bruges=. Continue on in the same direction to the
Place Jan van Eyck, where you open up one of the most charming views in
Bruges over the canal and quays. The Place is "adorned" by a modern
statue of Jan van Eyck. The dilapidated building to your L. is that of
the =Académie des Beaux-Arts= which occupies the site of the Citizens'
Assembly Hall: the ancient edifice was wholly rebuilt and spoilt in
1755, with the exception of the picturesque tower, best viewed from the
base of the statue. Opposite you, as you emerge into the Place, is the
charming =Tonlieu= or Custom House, whose decorated _façade_ and portal
(restored) bear the date 1477, with the arms of Pieter van Luxemburg,
and the collar of the Golden Fleece. The dainty little neighbouring
house to the L., now practically united with it, has a coquettish
_façade_: the saints in the niches are St. George, St. John Baptist, St.
Thomas à Becket, (or Augustine?) and St. John the Evangelist.
The Tonlieu is now fitted up as the Municipal Library. (Open daily,
free, 10 to 1, and 3 to 5, Saturday and Sunday excepted.) It contains
illuminated manuscripts and examples of editions printed by Colard
Mansion. All round the Place are other picturesque mediæval or
Renaissance houses.
The little street to the R. (diagonally) of the Tonlieu leads on to the
Marché du Mercedi, now called Place de Memling, embellished by a statue
of the great painter. Cross the Place diagonally to the Quai des
Espagnoles (Madonna and Child in front of you) and continue along the
quay, to the L., to the first bridge; there cross and go along the
picturesque Quai des Augustins to the Rue Flamande. (Quaint little
window to the left, as you cross the bridge.) Follow the Rue Flamande as
far as the Theatre, just before reaching which you pass, right, a
handsome mediæval stone mansion, (formerly the Guild of the Genoese
Merchants,) with a relief over the door, representing St. George killing
the Dragon, and the Princess Cleodolind looking on. At the Theatre, turn
to the R., following the tram-line, and making your way back to the
Grand' Place by the Rue des Tonneliers.
(2) As early as 1362, Bruges acquired its existing size, and was
surrounded by =ramparts=, which still in part remain. A continuous canal
runs round these ramparts, and beyond it again lies an outer moat. Most
of the =old gates= have unhappily been destroyed, but four still exist.
These may be made the objects of interesting rambles.
Go from your hotel, or from the Grand' Place, by the Rue Flamande, as
far as the Rue de l'Académie. Turn along this to the R., into the Place
Jan van Eyck, noting as you pass the Bear of Bruges at the corner of the
building of the old Academy. Follow the quay straight on till you reach
a second canal, near the corner of which, by the Rue des Carmes, is an
interesting shop with good beaten brasswork. Take the long squalid Rue
des Carmes to the right, past the ugly convent of the English Ladies,
with its domed church in the most painful taste of the later Renaissance
(1736). The mediæval brick building on your right, at the end of the
street, is the late Gothic =Guild-house of the Archers of St.
Sebastian=. Its slender octagonal tower has a certain picturesqueness.
(St. Sebastian was of course the patron of archery.) Charles II. of
England (see under the Grand' Place) was a member of this society during
his exile: his bust is preserved here. So also was the Emperor
Maximilian. Continue to the =ramparts=, and mount the first hill,
crowned by a windmill,—a scene of a type familiar to us in many later
Dutch and Flemish pictures. A picturesque view of Bruges is obtained
from this point: the octagonal Belfry, the square tower of St. Sauveur,
(the Cathedral), the tapering brick spire of Notre-Dame, with its
projecting gallery and the steeple of the new church of the Madeleine
are all conspicuous in views from this side. Follow the ramparts to the
R., to the picturesque =Porte de Ste. Croix=, and on past the barracks
and the little garden to the Quai des Dominicains, returning by the Park
and the Place du Bourg or the Dyver.
(3) Set out by the Grand' Place and the Place du Bourg; then follow the
Rue Haute, with its interesting old houses, as far as the canal. Do not
cross it, but skirt the quay on the further side, with the towers of St.
Walburge and St. Gilles in front of you. At the bridge, diverge to the
right, round the church of St. Anne, and the quaint little =Church of
Jerusalem=, which contains an unimportant imitation of the Holy
Sepulchre at Jerusalem, founded by a burgomaster of Bruges in the 15th
century. It is just worth looking at. Return to the bridge, and follow
the quay straight on to the modern Episcopal Seminary and the
picturesque old =Hospice de la Potterie=, which now harbours the Museum
of Antiquities belonging to the Hospital of St. John. I do not advise a
visit. (It contains third-rate early Flemish pictures, inferior
tapestry, and a few pieces of carved oak furniture. Admission, 50 c.:
entrance by the door just beyond the church, No. F, 79. The church
itself is worth a minute's visit.) This walk passes many interesting old
houses, which it is not necessary now to specify. Return by the =Porte
de Damme=, and the opposite side of the same canal, to the Pont des
Carmes, whence follow the pretty canal on the right to the Rue Flamande.
(4) Take the Rue St. Jacques, and go straight out to the =Porte d'
Ostende=, which forms an interesting picture. Cross the canal and outer
moat, and traverse the long avenue, past the gasometers, as far as the
navigable canal from Bruges to Ostend. Then retrace your steps to the
gateway, and return by the ramparts and the Railway Station to the Rue
Nord du Sablon.
These four walks will show you almost all that is externally interesting
in the streets and canals of the city.
* * * * *
The original Palace of the Counts of Flanders, we saw, occupied the site
of the Palais de Justice. Their later residence, the =Cour des Princes=,
in a street behind the Hôtel du Commerce, has now entirely disappeared.
Its site is filled by a large ornate modern building, belonging to the
Sisters of the Sacred Heart, who use it as a school for girls.
The water-system of Bruges is also interesting. The original river Reye
enters the town at the Minnewater, flows past the Hospital and the
Dyver, and turns northward at the Bourg, running under arches till it
emerges on the Place Jan van Eyck. This accounts for the apparently
meaningless way this branch seems to stop short close to the statue of
Van Eyck: also, for the mediæval ships unloading at the Grand' Place.
The water is now mostly diverted along the canals and the moat by the
ramparts.
_E._ THE CHURCHES
[The =original Cathedral= of Bruges (St. Donatian) was
destroyed, as we saw, by the French, in 1799; but the town still
possesses two fine =mediæval churches= of considerable
pretensions, as well as several others of lesser importance.
Though of very ancient foundation, the two principal churches in
their existing form date only from the most flourishing period
of Bruges, the 13th, 14th, and 15th centuries.
=St. Salvator= or =St. Sauveur=, the larger, was erected into
the =Cathedral= after the destruction of St. Donatian, whose
relics were transferred to it. To this, therefore, we will first
direct ourselves.]
Go down the Rue des Pierres as far as
THE CATHEDRAL,
which replaces a very ancient church built by St. Eligius (St. Éloy) in
646.
Externally, the edifice, which is built of brick, has rather a heavy and
cumbrous effect, its chief good features being the handsome square tower
and the large decorated windows of the N. and S. Transepts. The Choir
and its chapels have the characteristic French form of a _chevêt_. The
main portal of the N. Transept has been robbed of its sculpture. The
Choir is of the late 13th century: the Nave and Transept are mainly in
the decorated style of the 14th.
The best =entrance= is near the tower on the N. side. Walk straight on
into the body of the Nave, by the archway in the heavy tower, so as to
view the internal architecture as a whole. The =Nave= and _single_
=Aisles= are handsome and imposing, though the windows on the S. side
have been despoiled of their tracery. Notice the curious high-pointed
=Triforium= (1362), between the arches of the Nave and the windows of
the Clerestory. The =Choir= is closed by a strikingly ugly debased
Renaissance or rococo Rood-Screen, (1682), in black-and-white marble,
supporting the organ. It has a statue of God the Father by the younger
Quellin. The whole of the interior has been decorated afresh in somewhat
gaudy polychrome by Jean Béthune. The effect is on the whole not
unpleasing.
The Cathedral contains few works of art of high merit, but a preliminary
walk round the Aisles, Transept, and Ambulatory behind the Choir will
give a good idea of its general arrangement. Then return to view the
paintings. The sacristan takes you round and unlocks the pictures. Do
not let him hurry you.
Begin with the =Left Aisle=.
The =Baptistery=, on your L., contains a handsome font. R. and L. of the
entry to it are admirable brasses. In the Baptistery itself, _L. wall_,
are two wings of a rather quaint triptych, representing St. Martin
dividing his cloak with the beggar; St. Nicholas raising to life the
three boys who had been salted for meat; St. Mary Magdalen with the pot
of ointment (in the distance, as Penitent in the Desert); and St.
Barbara with her tower; dated 1613. Also a rude Flemish picture (16th
century) of the lives of St. Joachim and St. Anna, and their daughter
the Blessed Virgin:—the main episodes are the Marriage of the Virgin,
Birth of the Virgin, and Rejection of St. Joachim from the Temple, with
other scenes in the background.
The _end wall_ of the Baptistery has Peter Pourbus's masterpiece, a
*triptych painted for the =Guild of the Holy Sacrament=, attached to the
church of St. Sauveur, and allusive to their functions. The _outer
wings_, when closed, represent the Miracle of the Mass of St. Gregory,
when the Host, as he consecrated it, was changed into the bodily
Presence of the Saviour, to silence a doubter. It thus shows in a
visible form the tremendous mystery of Transubstantiation, in honour of
which the Guild was founded. Behind, the Brothers of the Confraternity
are represented (on the right wing) in attendance on the Pope, as
spectators of the miracle. One of them holds his triple crown. These may
rank among the finest portraits by the elder Pourbus. They show the last
stage in the evolution of native Flemish art before it was
revolutionized by Rubens. The _inner picture_ represents, in the centre,
the Last Supper, or rather, the Institution of the Eucharist, to
commemorate which fact the Guild was founded. The arrangement of the
figures is in the old conventional order, round three sides of a table,
with Judas in the foreground to the left. The _wings_ contain Old
Testament subjects of typical import, as foreshadowing the Eucharist.
_Left_, Melchisedec giving bread and wine to Abraham; _right_, Elijah
fed by the angel in the Wilderness. All the faces have still much of the
old Flemish portrait character.
On the _R. wall_ are the wings of a picture, by F. Pourbus (the son),
painted for the Guild of Shoemakers, whose chapel is adjacent. The
_inside_ contains portraits of the members. On the _outside_ are their
patrons, St. Crispinus and St. Crispianus, with their shoemakers'
knives. Also, an early Crucifixion, of the school of Cologne (about
1400), with St. Catherine holding her wheel and trampling on the tyrant
Maximin, by whose orders she was executed, and St. Barbara with her
tower. (These two also occur together in Memling's great triptych.) The
picture is interesting as the only specimen in Bruges of the precursors
of Van Eyck on the lower Rhine. The Baptistery contains, besides, a fine
old candlestick, and a quaint ciborium (for the Holy Oil) with
reliefs of the Seven Joys of Mary (1536).
The vistas from the =North Transept= are impressive. It terminates in
the =Chapel of the Shoemakers' Guild=, with a fine carved wooden door of
about 1470, and good brasses, as well as an early crucifix. It is
dedicated to the patron saints of the craft, and bears their arms, a
boot.
The first two chapels in the =Ambulatory= (behind the Choir) have good
screens.
The =third Chapel= encloses the tomb of Archbishop Carondelet, in
alabaster, (1544,) a fine work of the Italian Renaissance. The Descent
from the Cross by Claeissens, with the Crown of Thorns and the Holy
Blood in the foreground: on the wings, St. Philip, and the donor, under
the protection of (the canonized) Charlemagne. Near this is a
=*triptych= by Dierick Bouts, (falsely ascribed to Memling)
representing, in the _centre_, St. Hippolytus torn to pieces by four
horses. (He was the jailor of St. Lawrence, who converted him: _see_
Mrs. Jameson). The faces show well the remarkable power of this
bourgeois painter of Louvain. On the _left wing_ are the donors; on the
_right wing_ Hippolytus confesses himself a Christian, and is condemned
to martyrdom. Over the =altar=, _retable_, a Tree of Jesse, in carved
woodwork, with the family of Our Lady: on the wings, (painted,) the
legend of St. Hubert and the stag, and the legend of St. Lucy.
In the =Apse= is the Chapel of the Host.
The next =chapel=, of the Seven Sorrows, has a Mater Dolorosa of 1460
(copy of one at Rome); a fine _*brass_; and the *portrait of Philippe le
Beau, known as Philippus Stok (father of Charles V), and bearing the
collar of the Golden Fleece.
The =Choir=, (admirable architecturally,) contains the *stalls and arms
of the Knights of the Golden Fleece, with good carved Misereres.
The Cathedral contains many other pictures of interest, which, however,
do not fall within the scope of these Guides.
The =Chambre des Marguilliers=, or Churchwardens' Vestry, contains
manuscripts and church furniture, sufficiently described by the
sacristan.
In the =Sacristy= are still preserved the relics of St. Donatian.
Give the sacristan a franc, and then go round alone again, to inspect
the unlocked pictures at your leisure.
* * * * *
On leaving the Cathedral, go round the south side, which affords an
excellent view of the chapels built out from the apse. Then take the
little Rue du St. Esprit as far as the Church of
NOTRE-DAME,
which replaces a chapel, built by St. Boniface, the Apostle of Germany,
in 744, and enclosed in the town in 909.
Stand opposite it, in the small Place on the N. side, to observe the
somewhat shapeless architecture, the handsome brick tower crowned by a
tall brick steeple, and the beautiful little =*porch= or "Paradise,"
built out from the main structure in flamboyant Gothic of the 15th
century. The portal of this porch has been walled up, and the area is
now used as a chapel, approached from the interior. Notice the delicate
tracery of the windows, the fine finials and niches, and the charming
gable-end.
The picturesque building with turrets to the L. of the church was
originally the =mansion= of the family =Van der Gruuthuus=, one of the
principal mediæval stocks of Bruges. It had a passage communicating with
the family gallery in the church of Notre-Dame. The building, recently
restored, is now in course of being fitted up for the Town Museum of
Antiquities. A Museum of Lace is already installed in it; the entrance
is by a doorway over the bridge to the left (50 c. per person).
=Enter the church=, and walk straight into the Nave, below the great
West Window, a spot which affords a good view of the centre of the
church, the vaulted _double_ Aisles, and the angular Apse. The Choir is
shut off from the body of the church by a very ugly marble Rood-Screen
(1722), still bearing its crucifix, and with a figure of Our Lady,
patroness of the church, enshrined above its central arch. Rococo
statues of the Twelve Apostles, with their well-known symbols (1618),
are attached to the pillars. (Note these symbols: they recur in similar
situations everywhere.) In spite of hideous disfigurements, the main
portion of the interior is still a fine specimen of good middle Gothic
architecture, mainly of the 14th century.
Walk up the =outer left Aisle=. The last bay is formed by the
=Baptistery=, originally the =porch=, whose beautiful exterior we have
already viewed. Its interior architecture is also very charming. It
contains the Font, and the usual figure of the patron, St. John the
Baptist. This Aisle terminates in an apsidal chapel (of the Holy Cross)
containing inferior pictures of the 17th century, representing the
history of a relic of the True Cross preserved here.
The =inner left Aisle= leads to the =Ambulatory= or passage at the back
of the Choir. The Confessionals to the R. have fairly good rococo carved
woodwork, 1689. On the L. is the handsome mediæval woodwork =gallery=
(1474), belonging to the Van der Gruuthuus family, originally approached
by a passage from their mansion behind. Beneath it, is a screen of
delicate early Gothic architecture, with family escutcheons above the
door.
The windows of the =Apse= have good modern stained glass.
On the L., at the entrance to the Apse, Pourbus's Adoration of the
Shepherds, a winged picture, closed. The sacristan will open it. On the
wings are, _left_, the donor, Sire Josse de Damhoudere, with his patron,
St. Josse, and his four sons; _right_, his wife, Louise, with her five
daughters, and her patron St. Louis of France, wearing his crown and
robe of fleurs-de-lis, and holding the _main de justice_. He is
represented older than is usual, or indeed historical, and in features
somewhat resembles Henri IV. This is a fine picture for its master. On
the outer wings are the cognate subjects, the Circumcision and the
Adoration of the Magi, in grisaille.
The chapel in the Apse, formerly the Lady Chapel, now contains the Host.
It has a gaudy modern altar for the monstrance.
In the =South Ambulatory=, over a doorway, Foundation of the Church of
Santa Maria Maggiore at Rome, by Claeissens.
A =chapel= to the =L=., just beyond, locked, but opened by the sacristan
(1 franc; or, for a party, according to notice displayed at entrance),
contains the celebrated =**tombs of Mary of Burgundy and Charles the
Bold=, her father. Mary was the wife of Maximilian, and died by a fall
from her horse in 1482, when only twenty-five. Her =**monument= was
designed and executed by Peter Beckere of Brussels, by order of her son
Philippe le Beau, in 1502. The sarcophagus is of black marble: the
statue of the Princess, in gilt bronze, lies recumbent upon it. The
style is intermediate between that of the later Middle Ages and of the
full Renaissance. Beside it is the *tomb of Charles the Bold, of far
less artistic value. Charles was buried at Nancy, after the fatal
battle, but his body was transported to St. Donatian in this town by his
descendant Charles V, and finally laid here beside his daughter by
Philip II, who had this tomb constructed for his ancestor in imitation
of that of Mary.
(I advise the visitor after seeing these tombs and the great
chimney-piece of the _Franc de Bruges_ to read up the history of Charles
the Bold and his descendants, down to Charles V.)
The _east wall_ of this chapel, beyond the tomb of Charles the Bold, has
a fine picture of Our Lady of Sorrows, enthroned, surrounded by smaller
subjects of the Seven Sorrows. Beginning at the left, the Circumcision,
the Flight into Egypt, Christ lost by his parents in the Temple, the Way
to Calvary, (with St. Veronica holding out her napkin,) the Crucifixion,
(with Our Lady, St. John, and Mary Magdalen,) the Descent from the
Cross, and the Deposition in the Tomb. A fine work of its sort,
attributed to Mostart (or to Maubeuge). On the _west wall_ are two wings
from a triptych by Pourbus, with tolerable portraits, (centre-piece
destroyed,) and an early Flemish painting of the Deposition from the
Cross (interesting for comparison with Roger van der Weyden and Gerard
David). In the foreground lies the vessel containing the Holy Blood. On
the wings are the Crucifixion and the Resurrection. The whole is very
rudely painted. Outside are portraits of the donor and his wife and
children, with their patrons St. James (staff and scallop) and St.
Margaret (whose dragon just appears in the background).
On an =arcade=, a little further on, is a very early fresco (1350?) of a
saint (St. Louis of France?), and also a dainty small relief (about
1500) of a donor, introduced by his patron, St. Peter, adoring Our Lady.
The =end chapel= of the right aisle, that of the Holy Sacrament,
contains a celebrated and noble white marble =**Madonna and Child, by
Michael Angelo=, enshrined in a black marble niche. The pensive, grave,
and graceful face, the exquisite modelling of the dainty naked Child,
and the beautiful infantile pose of its left hand, all betray a design
of Michael Angelo, though the execution may possibly have been left to
pupils. But the modelling is softer and more feminine than is usual with
this great sculptor, except in his early period. In this respect, it
resembles most the unfinished Madonna in the Bargello at Florence.
Condivi mentions that Peter Mouscron of Bruges ordered of Michael Angelo
a Madonna and Child in bronze: he was probably mistaken as to the
material: and we have here doubtless the work in question. Apart from
its great artistic value, this exquisite group is interesting as
affording another link between Flanders and Italy.
The same chapel also contains some good 17th century pictures.
Near the confessional, as we return towards the West End of the church,
we find a good diptych of Herri met de Bles, of 1520, containing, _left
panel_, an Annunciation, with all the conventional elements; to the
left, as usual, is the angel Gabriel; to the right, Our Lady. These
relative positions are never altered. The lilies in the pot, the desk
and book, the bed with its furniture, the arcade in the background, and
the rich brocade, are all constant features in pictures of this subject.
Look out for them elsewhere. The _right panel_ has the Adoration of the
Magi, with the Old, Middle-aged, and Young Kings, the last-named a Moor.
This quaint and interesting work of a Flemish painter, with its archaic
background, and its early Italian reminiscences, also betrays the
influence of Dürer. Among the other pictures may be mentioned a triptych
in an adjacent small chapel: the _central panel_ shows the
Transfiguration, with the three apostles below, Moses, Elias, and the
Eternal Father above (perhaps by Jan Mostart). On the _wings_ (much
later, by P. Pourbus), are the portraits of the donor, his wife, and
their patron saints.
The =West Wall= of the church has several large pictures of the later
Renaissance, which can be sufficiently inspected on their merits by
those who care for them. The best of them are the Adoration of the Magi
by Seghers, and De Crayer's Adoration of the Infant Jesus. I do not
propose to deal at length with later Flemish art till we reach Brussels
and Antwerp: at Bruges, it is best to confine oneself to the
introductory period of Flemish painting—that of the Burgundian princes.
I will therefore only call attention here to the meaningless way in
which huge pictures like B. van Orley's Crucifixion, with subsidiary
scenes from the Passion, reproduce the form of earlier winged pictures,
which becomes absurd on this gigantic scale.
* * * * *
The =Church of St. Jacques= stands in the street of the same name,
conveniently near the Hôtel du Commerce. It is a good old mediæval
building (12th century, rebuilt 1457-1518), but hopelessly ruined by
alterations in the 17th century, and now, as a fabric, externally and
internally uninteresting. Its architecture is in the churchwarden style:
its decoration in the upholsterer's. The carved wooden pulpit is a
miracle of bad taste (17th century), surpassed only by the
parti- marble rood-screen. A few good pictures and decorative
objects, however, occur among the mass of paintings ranged round its
walls as in a gallery. The best is a =panel= of the old Flemish School
(by Dierick Bouts, or more probably a pupil), in the _left aisle_, just
beyond the second doorway. It tells very naïvely the History of St. Lucy
(see Mrs. Jameson). _Left_, she informs her mother that she is about to
distribute her goods to the poor, who are visibly represented in a
compact body asking alms behind her. _Centre_, she is hailed before the
consul Paschasius by her betrothed, whom she refuses to marry. She
confesses herself a Christian, and is condemned to a life of shame.
_Right_, she is dragged away to a house of ill-fame, the consul
Paschasius accompanying; but two very stumpy oxen fail to move her. The
Holy Ghost flits above her head. The details are good, but the figures
very wooden. Dated, 1480.
Beside it is an extravagant Lancelot Blondeel of St. Cosmo and St.
Damian, the doctor saints, with surgical instruments and pots of
ointment. The central picture shows their martyrdom.
Further on hangs a good Flemish triptych (according to Waagen, by Jan
Mostart), representing, the prophecies of Christ's coming: _centre_, the
Madonna and Child; with King Solomon below, from whom a genealogical
tree rises to bear St. Joachim and St. Anna, parents of Our Lady. R. and
L. of him, Balaam and Isaiah, who prophesied of the Virgin and Christ:
with two Sibyls, universally believed in the Middle Ages to have also
foretold the advent of the Saviour. The stem ends in the Virgin and
Child. _Left_, the Tiburtine Sibyl showing the Emperor Augustus the
vision of the glorious Virgin in the sky: _right_, St. John the
Evangelist in Patmos beholding the Apocalyptic vision of the Woman
clothed with the Sun. This is a fine work of its kind, and full of the
prophetic ideas of the Middle Ages.
Pass round the =Ambulatory= and =Choir= to the _first chapel_ at the
_east end_ of the _right Aisle_. It contains an altar with the Madonna
and Child in Della Robbia ware, probably by Luca. Also, a fine tomb of
Ferry de Gros and his two wives, the first of whom reposes by his side
and the second beneath him. This is a good piece of early Renaissance
workmanship (about 1530). The church also contains a few excellent later
works by Pourbus and others, which need not be specified. This was the
church of the Florentine merchants at Bruges (whence perhaps the Della
Robbia) and particularly of the Portinari, who commissioned the great
altar-piece by Van der Goes now in the Hospital of Santa Maria Nuova at
Florence.
The other churches of Bruges need not detain the tourist, though all
contain a few objects of interest for the visitor who has a week or two
at his disposition.
_F._ THE ACADEMY
[The =Académie des Beaux-Arts=, which formerly occupied the
_Poorters Loodge_ (or Guild Hall of the citizens within the
gates) has a small but valuable =collection of pictures=,
removed from the destroyed cathedral of St. Donatian and other
churches of Bruges, which well repays a visit. You will here
have an excellent opportunity for studying _Jan van Eyck_, whose
work I shall more particularly notice when we arrive at Ghent.
It is interesting, however, here to compare him with his great
successor, _Memling_, who is represented at the Academy by a
fine triptych. The little gallery also contains some admirable
works by _Gerard David_, one of the latest of the old School of
Flemish painters, whose work may thus be observed here side by
side with those of his two chief predecessors. Owing to the
ruinous state of the original building the collection has been
transferred to a temporary apartment, beyond the Hospital
bridge, near the Church of Notre-Dame. No tourist should leave
Bruges without visiting this interesting collection.]
The =Museum= is situated (at present) in a house on the right-hand side
of the Rue Ste. Catherine, nearly opposite a new church. Go to it past
the Hospital of St. John. Admission daily, 50 c. per person.
Begin in the centre of the =wall opposite the entrance=.
(1.) Jan van Eyck. =**Altar-piece=, ordered by George van der Palen, for
the High Altar of the original Cathedral of =St. Donatian=, of which he
was a canon. The centre of the picture is occupied by the Madonna and
Child, the face of Our Lady somewhat recalling German models. She sits
in the apse of a church, probably St. Donatian. The Child, whom it is
the fashion to describe as "aged-looking," fondles a parrot and grasps a
bunch of flowers. To the left stands St. Donatian the Archbishop, patron
saint of the church for which this altar-piece was painted. He bears his
usual symbol, the wheel with five lighted candles (as in the beautiful
panel by Gerard David in the National Gallery at London). This is a fine
and finely-painted figure. To the right, St. George, in full armour,
admirably represented, but in an affected attitude, lifts his casque
somewhat jauntily as he presents his namesake the Canon George to Our
Lady. In all this we get a touch of Burgundian courtliness: the event is
represented as a state ceremonial. With his left hand the Saint supports
his Red Cross banner. The portrait of the kneeling Canon
himself,—asthmatic, pudding-faced—is very admirable and life-like, but
by no means flattered. He grips his prayer-book with an old man's
tremulous hand. (For a profound criticism of this fine picture, see
Conway.) The insipid Madonna, the rather foolish St. George, the fine
portrait of the Canon, are all typical of Van Eyck's manner. The
accessories of architecture, decoration, and background, should also be
carefully noted. The capitals of the columns and the knobs of glass in
the window, as well as St. George's costume, are elaborated in Van
Eyck's finest fashion.
(2.) Jan van Eyck. *Portrait of his wife, painted for presentation to
the Bruges Guild of Painters, together with one of the artist himself,
now undiscoverable. This is a fine though evidently unflattered portrait
of a capable housewife, very stiffly arrayed in her best church-going
costume. It deserves close inspection.
Above it, (3.) Head of Christ, ascribed to Jan van Eyck, but in reality
a poor and reduced copy of the picture at Berlin.
(4.) Memling. **Triptych painted for =Willem Moreel= or =Morelli=, a
member of a wealthy Savoyard family settled at Bruges. Like Jan van
Eyck's portrait of the two Arnolfini in London, and Hugo van der Goes's
triptych of the Portinari at Florence, this picture marks well the
cosmopolitan character of old Bruges. In the _central panel_, St.
Christopher (whose altar in the church of St. Jacques it adorned) wades
with his staff through the water, feeling as he goes the increasing
burden of the Christ-Child on his shoulder. (For the legend, see Mrs.
Jameson.) To the left, above, is the diminutive figure of the hermit
with his lantern, which always accompanies St. Christopher. The left
foreground of the picture is occupied by St. Maurus, in his Benedictine
costume; to the right is St. Giles (St. Egidius) the hermit, with the
wounded doe, the arrow piercing the arm of the saint. The _left wing_
represents the donor, Willem Moreel, under the care of his patron, St.
William, who wears a hermit's dress above his coat of armour. (When a
saint places his hand on a votary's shoulder it usually implies that the
votary is a namesake.) Behind are Moreel's five sons. All these
portraits, but particularly that of the donor and his eldest son, who
closely resembles him, are admirable. The _right wing_ represents the
donor's wife, Barbara, under the protection of her patron, St. Barbara,
with her tower, showing as usual three windows (emblematic of the Holy
Trinity). Behind the lady are her two daughters, one of whom is habited
as a Benedictine nun, whence, doubtless, the introduction of St. Maurus
into the main altar-piece. This fine triptych originally decorated an
altar of St. Christopher in Moreel's private chapel in the church of St.
Jacques. One of his daughters is the "Sibylla Sambetha" represented at
the Hospital. The =wings= at the =back= represent in grisaille St. John
the Baptist with the lamb, and St. George with the dragon. It was usual
to paint the outer wings in grisaille or in low tones of colour, so that
the splendour of the interior hues might burst upon the spectator as the
triptych was opened.
(12) Attributed to Schoreel: really, by a master of the Brabant School.
Death of the Virgin. Our Lady is represented on her deathbed,
surrounded, as always, by the surviving apostles, who were miraculously
collected together to her chamber. The faces are those of Flemish
peasants or artisans. Above, Christ appears in glory, surrounded by a
halo of cherubs, to receive her new-born soul. Two angels support his
outer garment. This picture well shows the beginning of the later
Flemish tendency.
Now return to No. 5, by Gerard David, on the other side of the great Van
Eyck. This is a *triptych, painted for Jean des Trompes, for the High
Altar of the lower chapel of the Holy Blood. The _central panel_
represents the =Baptism of Christ=. In the middle, the Saviour wades in
the water of a diminutive Jordan, where the concentric circles show the
increased careful study of nature. On the right-hand side of the
picture, St. John-Baptist, patron saint of the donor, pours water on his
head. The relative positions of these two figures, and of the angel to
the left holding a robe, are conventional: they have descended from a
very early period of art. (In the Ravenna mosaics, the place of the
angel is filled by the river-god of the Jordan with his urn, afterwards
transformed and Christianized into an angel with a towel. Look out in
future for similar arrangements.) The central figures are weak; but the
robe of the angel is painted with Flemish minuteness. So are the flowers
and leaves of the foreground. Above, the dove descends upon the head of
the Saviour, while the Eternal Father pronounces from the skies the
words, "Behold my Beloved Son in whom I am well pleased."
In the background are two other episodes: L., the preaching of St.
John-Baptist (where Oriental costumes indicate the heathen); R., St.
John-Baptist pointing out Christ to his disciples with the words,
"Behold the Lamb of God." The distance shows two towns and a fine
landscape. Observe the admirable painting of the trees, with their good
shadows; also the ivy climbing up the trunk of one to the right. This
picture is among the earliest in which the gloom of a wood is accurately
represented: in many other respects it well illustrates =the rise of
landscape-painting=. (For an exhaustive criticism, see Conway.) The
_left wing_ has a portrait of the donor, with his other patron, St. John
the Evangelist, holding the cup. Beside the donor kneels his little son
Philip. This portrait, the face and foot of the Evangelist, the fur of
the donor's robe, the crane in the background, and many other
accessories deserve close attention. Two figures in the background dimly
foreshadow Teniers. The _right wing_ has a portrait of the donor's wife,
Elizabeth, with her four daughters. Behind her stands her patroness, St.
Elizabeth of Hungary, in Franciscan robes, with the crown on her head
and the double crown and book in her hands, as on the statuette at the
door of the Béguinage. The painting of a rosary here is excellent. The
_outer wings_ (turn them back) show, on the _left_, the Madonna and
Child with a bunch of grapes; on the _right_, the donor's second wife
Madeleine, introduced by her patroness, St. Mary Madeleine, who holds
the alabaster pot of ointment. By the lady's side kneels her daughter.
The background consists of a view, probably in the Bruges of that
period. Painted about 1507.
6 and 7. Gerard David. =*The Punishment of the Unjust Judge.= These two
panels are of a type commonly set up in courts of justice as a warning
to evil-doers. They were ordered by the Bruges magistracy. You will see
a similar pair by Dierick Bouts in Brussels. The story, a horrid one, is
taken from Herodotus. Sisamnes was a judge in Persia whom King Cambyses
detected receiving a bribe and ordered to be flayed alive. The king then
stretched his skin on the seat of judgment, and appointed the son of
Sisamnes to sit in his father's place, that he might remember to avoid a
like fate. The _first picture_ represents, in the background, the
bribery. In the foreground, King Cambyses, in a rich embroidered robe,
demonstrates on his fingers the guilt of the unjust judge. Sisamnes is
seized on his tribunal by a man of the people; courtiers, lawyers, and
burgesses looking on. The expression on his face and the painting of all
the accessories is admirable. In the _second picture_ we have the
flaying of the unjust judge, a horrible scene, powerfully rendered.
Cambyses stands by, holding his sceptre, surrounded by courtiers who
recall the last age of the Burgundian dominion. In the background (as a
subsequent episode) the son of Sisamnes is seen sitting in his father's
place: behind him hangs the skin of the father. Architecture, landscape,
ropes, and all other accessories of this painful picture should be
carefully noted.
15. J. Prévost. Last Judgment. Below, the dead are rising, half naked,
from the tomb, girt only with their shrouds; the good receiving garments
from angels, and the bad hurried away to a very Flemish and unimpressive
Hell. Above, Christ as Judge holds the sword. Two angels blow out the
words of blessing or malediction. On the spectator's left, Our Lady
shows the breast that suckled the Redeemer. Behind her are St. Peter
with the key, St. Paul with the sword, St. Bartholomew with the knife,
and other saints. On the right are St. John-Baptist with the lamb, King
David with the harp, Moses, horned (as always), with the tables of the
law, and a confused group of saints. This picture is rather curious than
beautiful. Above it is a later treatment of the same subject by Van
Coornhuuse, interesting for comparison as showing the usual persistence
of types and the conventional grouping of the individual figures.
Compare especially the corresponding personages in the lower left-hand
corners.
A few other pictures skied on this wall deserve passing notice. 29 is a
Death and the Miser, of the School of Quentin Matsys. 17, by Lancelot
Blondeel, the architect of the great chimney-piece of the _Franc de
Bruges_, represents St. Luke painting Our Lady, in one of the fantastic
frames in which this painter delighted. 18, by the same, has a St.
George and the Dragon, with the Princess Cleodolind looking on. Around
it are four smaller scenes of his martyrdom: (he was boiled, burnt with
torches, dragged by a horse, and finally decapitated). 11, is a good
diptych of the Flemish school, by an unknown contemporary of Gerard
David. It represents, left, a donor, with his patron St. John the
Almoner, holding his symbol, a sheaf of corn. On the right, his wife
with her patroness, St. Godeliva. 28, is an Adoration of the Magi, where
the Three Kings again illustrate the three ages of man and the three
continents. Beside it is a Nativity which exhibits all the traditional
features already noted.
The =end wall= has in its centre a tolerably good Adoration of the Magi,
of the German School, 15th century. Note once more the Three Kings, of
whom the youngest is a Moor. Left of this, a *drawing, by Jan van Eyck,
of St. Barbara, which should be closely inspected. She holds a palm of
martyrdom. In the background, workmen build her tower. It is interesting
as a scene of real life at this period. This is a replica of the
well-known picture at Antwerp. To the right, two drawings by
Gerard David from the life of St. John-Baptist. Above these hangs a
tolerable P. Pourbus of the Last Judgment, valuable for comparison with
the two previous treatments of the same subject on the principal wall.
Go from one to the other once or twice. Later painters of the
Renaissance use this solemn theme as a mere excuse for obtruding the
nude—and often the vulgar nude—into churches. On the same wall are a
good triptych in grisaille by P. Pourbus (Way to Calvary, Descent from
the Cross, Resurrection: from Notre-Dame at Damme), and other pictures.
The remaining walls have portraits and other works, from the 17th
century downwards, most of which need no explanation. A few of them,
indeed, are not without merit. But, as I have before observed, it is
best in mediæval Bruges to confine oneself to the 13th, 14th, 15th, and
early 16th centuries, leaving the rise of the Renaissance, and the later
Flemish School of painting, to occupy us at Antwerp, where they can be
studied to far greater advantage.
II
GHENT
_A._ ORIGINS OF GHENT
FLANDERS owes everything to its water communications. At the junction
of the =Schelde= with the =Lys= or Lei, there grew up in the very early
Middle Ages a trading town, named =Gent= in Flemish, and =Gand= in
French, but commonly Anglicised as =Ghent=. It lay on a close network of
rivers and canals, formed partly by these two main streams, and partly
by the minor channels of the Lieve and the Moere, which together
intersect it into several islands. Such a tangle of inland waterways,
giving access both to the sea and to Bruges, Courtrai, and Tournai, as
well as less directly to Antwerp and Brussels, ensured the rising town
in early times considerable importance. It formed the centre of a
radiating commerce. Westward, its main relations were with London and
the English wool ports; eastward with Cologne, Maastricht, the Rhine
towns, and Italy. Ghent was always =the capital of East Flanders=, as
Bruges or Ypres were of the Western province; and after the Counts lost
possession of Arras and Artois, it became in the 13th century their
principal residence and the metropolis of the country. The trade in
=weaving= grew rapidly in importance, and the Ghenters received from
their Count a charter of liberties of the usual mediæval burgher type.
As time went on, and the city advanced in wealth, its subjection to its
sovereigns became purely nominal. Ghent equipped large bodies of citizen
soldiers, and repulsed a considerable English army under Edward I. The
Ghenters were also determined opponents of the claims of the French
kings to interfere in the internal affairs of Flanders; thus they were
mainly instrumental in winning the famous =Battle of the Spurs= in 1302,
when the citizens of Bruges and Ghent put to flight the army of France
under the Count of Artois before the walls of Tournai, and dedicated as
trophies 700 golden spurs, worn by the French knights whom they had
routed. This battle, memorable as one of the chief triumphs of nascent
industrial freedom over the chivalry and royalty of mediævalism, secured
the liberties of the Flemish towns against French aggression.
Early in the 14th century, the burghers of Ghent, under their democratic
chief, Jacob or =Jacques Van Artevelde=, attained =practical
independence=. Till 1322, the Counts and people of Flanders had been
united in their resistance to the claims of France; but with the
accession of Count Louis of Nevers, the aspect of affairs changed. Louis
was French by education, sympathies, and interests, and aristocratic by
nature; he sought to curtail the liberties of the Flemish towns, and to
make himself despotic. The wealthy and populous burgher republics
resisted, and in 1337 Van Artevelde was appointed =Captain of Ghent=.
Louis fled to France, and asked the aid of Philip of Valois. Thereupon,
Van Artevelde made himself the =ally of Edward III. of England=, then
beginning his war with France; but as the Flemings did not like entirely
to cast off their allegiance—a thing repugnant to mediæval
sentiment—Van Artevelde persuaded Edward to put forward his trumped-up
claim to the crown of France, and thus induced the towns to transfer
their fealty from Philip to his English rival. It was therefore in his
character as King of France that Edward came to Flanders. The alliance
thus formed between the great producer of raw wool, England, and the
great manufacturer of woollen goods, Ghent, proved of immense commercial
importance to both parties. But as Count Louis sided with Philip of
Valois, the breach between the democracy of Ghent and its nominal
sovereign now became impassable. Van Artevelde held supreme power in
Ghent and Flanders for nine years—the golden age of Flemish
commerce—and was treated on equal terms by Edward, who stopped at Ghent
as his guest for considerable periods. But he was opposed by a portion
of the citizens, and his suggestion that the Black Prince, son of Edward
III., should be elected Count of Flanders, proved so unpopular with his
enemies that he was assassinated by one of them, Gerard Denys. The town
and states immediately repudiated the murder; and the alliance which Van
Artevelde had brought about still continued. It had far-reaching
results; the woollen industry was introduced by Edward into the Eastern
Counties of England, and Ghent had risen meanwhile to be the =chief
manufacturing city of Europe=.
The quarrel between the democratic weavers and their exiled Counts was
still carried on by =Philip van Artevelde=, the son of Jacques, and
godson of Queen Philippa of England, herself a Hainaulter. Under his
rule, the town continued to increase in wealth and population. But the
general tendency of later mediæval Europe towards centralised despotisms
as against urban republics was too strong in the end for free Ghent. In
1381, Philip was appointed dictator by the democratic party, in the war
against the Count, son of his father's old opponent, whom he repelled
with great slaughter in a battle near Bruges. He then made himself
Regent of Flanders. But Count Louis obtained the aid of Charles VI. of
France, and defeated and killed Philip Van Artevelde at the disastrous
battle of Roosebeke in 1382. That was practically the end of local
freedom in Flanders. Though the cities continued to revolt against their
sovereigns from time to time, they were obliged to submit for the most
part to their Count and to the Burgundian princes who inherited from him
by marriage.
The subsequent history of Ghent is that of the =capital of the
Burgundian Dukes=, and of the House of Austria. Here the German king,
Maximilian, afterwards Emperor, married Mary of Burgundy, the heiress of
the Netherlands; and here Charles V. was born in the palace of the
Counts. It was his principal residence, and he was essentially a
Fleming. Other historical reminiscences will be pointed out in the
course of our peregrinations.
The =old waterways=, partially artificial, between Ghent and the sea,
other than the circuitous route by the shallow Schelde, had silted up by
1827, when a ship canal was constructed to Terneuzen. This canal has
since been widened and deepened so as to admit vessels of 1,700 tons; it
has thus helped to some small degree to save the town from the fate of
Bruges. But as its mouth lies in what is now Dutch territory, and as
heavy tolls are levied, it is comparatively little used. Another and
somewhat frequented canal leads to Bruges; but Ghent owes most of its
existing prosperity to its =manufactures= (cotton, linen, engines,
leather) and to its central position on the =railway system=.
The important points for the tourist to bear in mind are these, however.
Ghent during the Middle Ages was =a merchant republic=, practically
independent, with its guilds and its belfry, the last of which was used
to summon the citizens to arms in case of danger. It was also =the chief
manufacturing town in Europe=, as Bruges was the chief commercial
centre. By treaty with Edward III., Bruges was made the "staple" or sole
port of entry for English wool: and this wool was woven into cloth for
the most part at Ghent.
Further details of the vicissitudes of Ghent can be found in Van Duyse,
_Gand, Monumental et Pittoresque_.
* * * * *
The chief objects of interest at Ghent are the Cathedral, with its great
=Van Eyck=; and the =Town Hall= and =Belfry=. These can be tolerably
seen in one day: but a stay of three or four days will not be too much
to explore the curious nooks of the early city.
_B._ THE CORE OF GHENT
[The =old town= of Ghent lies on the island formed by the
junction of the Lys and the Schelde, with their various
backwaters (all now largely artificial). Near this point, but
beyond the Lys, the Counts of Flanders early erected a strong
=castle=, the _Gravensteen_ or _Oudeburg_, beneath whose
protection, aided by the two navigable rivers, merchants and
weavers gradually settled. As at Bruges, the heart of the town,
however, is purely =municipal and mercantile= in its
architecture. The _Town Hall_, which was the meeting-place of
the citizens, and the _Belfry_, which summoned them to arms or
council, are the chief points of interest in the city. The
Schelde is still tidal to its very centre.
As most visitors will probably stop in one of the hotels on the
=Place d'Armes=, near the S. end of older Ghent, I shall frankly
take that square as our starting-point. It may facilitate
recognition at first sight to add that the large square tower,
visible to the R. from the Place d'Armes, is that of the
Cathedral, while the tapering spire, crowned by a gilt dragon,
belongs to the Belfry.]
Go first on a =tour of orientation= through early Ghent. If you follow
these directions implicitly, you can see everything important in one
short walk. Cross the Place d'Armes diagonally to the N.E. corner, and
follow the small and narrow streets which run due N. to the front of the
=Cathedral=. Walk round the S. side of this, to form a first general
impression, but do not enter it at present.
Then, from the West Front of the Cathedral, take the Rue St. Jean
straight before you. The tower with the gilded dragon which faces you as
you walk is that of the =Belfry=. It was designed in 1183, about a
century earlier than that of Bruges, but only erected between 1321 and
1339; it is a fine work in the Early Gothic style. Its windows have been
walled up. The tapering turret which crowns the tower is unfortunately
modern, and of iron. On the very summit stands a huge gilded =dragon=,
which universal tradition represents as having been brought from St.
Sophia at Constantinople to Bruges by the Crusader Baldwin of Flanders,
(1204), and removed as a trophy by the people of Ghent (under Philip van
Artevelde) in 1382. It certainly appears to be of Oriental origin, but
is stated on documentary evidence (discovered by M. Vuylsteke) to have
been made in Ghent itself in 1380. If so, it would seem at least to be
based on an Oriental model.
The small building at the foot of the Belfry, now in course of (over)
restoration, is the =Cloth Hall=, erected in 1424, a graceful but not
very important Gothic edifice (of the Decorated period), with niches
vacant of their statues. The _concierge_ of the Belfry now has a room in
it. Application must be made here to mount to the summit. (1 franc, or 2
for a party.) Dark and steep.
The =view= is extensive and beautiful, but not quite so striking as that
at Bruges. The principal buildings of the city lie just below you:
beyond, all Flanders. The =chimes= are celebrated. The chief bell is
known as Roelandt.
Now turn round into the _Botermarkt_ or _Marché au Beurre_ to the right,
and inspect the =Belfry= again from the little bay in the corner
opposite. This is the best near view of the tower. The portal to the R.
was formerly the entry to the =town prison=, beneath the Belfry, now in
course of complete restoration. In its gable is a too-famous 18th
century relief (the _Mammelokker_) representing the Roman daughter
feeding her father from her breast at the window of a prison, and
doubtless intended to excite the charity of passers-by. It certainly
serves no other function, for it is neither beautiful nor decorative.
Cross over to the R. side of the Butter-market. The building on the L.,
in two totally distinct portions, is the =Hôtel-de-Ville=. The part at
which you first arrive, (latest in point of time,) was rebuilt in the
early Renaissance style in 1595-1628. It is one of the earliest and in
many ways the best example of Renaissance architecture in Belgium, in
part because it retains certain good features of local domestic
building, such as the pointed gable-ends (round the corner to the L.)
and the projecting windows with dormers on the main _façade_. (Look out
for their origin elsewhere.) It has three storeys, with projecting half
colonnades, the columns being Doric on the ground floor, Ionic on the
first floor, and Corinthian on the second. Recollect the gable-ends and
dormers for comparison with others in old houses in Ghent hereafter.
Now, continue on to the corner, where we arrive at the earlier =Gothic
portion= of the Hôtel-de-Ville, erected in 1518-1535 by Dominic de
Waghemakere, who also built in part the cathedral at Antwerp. The
projecting polygonal corner, with its handsome balcony, is very
noticeable. The work is of the latest and most florid Gothic, somewhat
lacking in grace and dignity, but ornate in its splendour. Observe the
depressed arches, the noble cornice, the rich decoration of garlands. A
few of the niches have now been filled with modern statues of saints.
From the corner opposite, a good view is obtained of both parts of the
Hôtel-de-Ville and also of the Belfry.
Turn to the left into the Rue Haut-Port, to observe the =main front= of
this earlier Gothic building, with its fine projecting windows above,
its empty niches, its handsome entrance staircase and main portal, its
beautiful little balcony for addressing the people below, and the large
projecting window of its ancient chapel near the centre. Note how well
the _façade_ is thus broken up and diversified. This is the finest
specimen of florid Gothic in Belgium. Beyond it comes another
Renaissance portion, and then a handsome Renaissance dwelling-house. The
street also contains several fine early houses, the best of which (a
Gothic guildhall, known as the Cour St. Georges) stands at the corner to
the left, facing the Hôtel-de-Ville.
The =interior= of the Hôtel-de-Ville need not be visited, though it has
a handsome Gothic staircase (demolished, sold, built into a private
house, re-erected) and some fine halls and internal courts, interesting
to those who have plenty of time at their disposal.
Now, return to the Belfry and continue straight down the left-hand side
of the Rue de la Catalogne. The church on the right, round the base of
which houses have been allowed to cluster, is =St. Nicolas=—the oldest
in the town. This is one of the most solid pieces of architecture at
Ghent. It has a fine decorated tower, which has happily escaped
restoration, besides small turrets to the Transepts, and two, rather
larger, to the gable of the Nave. Go on into the _Koornmarkt_ or _Marché
aux Blés_, to the R.; stand there for a moment, at the end of the Rue de
la Catalogne, to observe the fine _coup d'œil_, which takes in St.
Nicolas, the Belfry, and the tower of the Cathedral. The main _façade_
of St. Nicolas faces the Koornmarkt. Over the door is a modern figure of
the Saint himself, raising the three boys who were salted down for meat.
Nicolas was the popular saint, the patron of the merchants and
burgesses; and the prominent position of his church on the Corn Market
is very characteristic of the burgher spirit of Ghent.
A hasty glance will suffice for the =interior=, which is a
characteristic specimen of the unrestored Belgian church, with figures
of the Twelve Apostles (as always) against the pillars of the Nave; an
ugly carved pulpit; short Transepts; an Apse with bad glass; and the
vaulting of Nave, Aisles, and Choir concealed by plaster. The tawdry
decorations render what might be a fine interior wholly unimpressive.
The High Altar has an altar-piece by Liemakere, representing, in the
confused style of the School of Rubens, the election of St. Nicholas as
Bishop of Myra. Above is an 18th century figure of the Saint, raising
the three boys from the tub. The early pillars of the Choir are really
handsome.
On emerging from the front of the church, continue straight on to the
bridge which crosses the Lys, affording a good view to the L. of the
Apse of =St. Michel=. Then, go along the side of this handsome church,
with late Gothic windows resembling English Perpendicular. It has a
solid but unfinished tower, and a good West Portal, robbed of its
sculpture and cruelly mutilated. A glimpse at the =interior=, which has
been scraped and renovated, will show at once the fine architecture. The
Nave has impressive round pillars, windows in the clerestory, and
excellent brick vaulting. The vaulted Aisles are surrounded by chapels.
The Choir is very handsome. In the N. Transept is a famous but overrated
*Crucifixion by Van Dyck, not without beauty of conception and
composition, but spoiled by restorations. Walk round the Transepts and
Ambulatory. There are some good works of the School of Rubens.
Now, continue along the =quay=, on the same side as St. Michel,
(observing as you go that the early town extended to _both_ banks of the
river), in order to view the _façade_ of the handsome =Maison des
Bateliers=, or Guild House of the Skippers, erected in 1531 for the
masters of the shipping of Ghent, in somewhat the same florid
late-Gothic style as the Hôtel-de-Ville. This is the finest existing
specimen of old Flemish houses. Over the doorway is an appropriate
relief of a ship, somewhat antiquated and heraldic in character. By the
side of this Guild-house are two others, less interesting: the first,
the Guild House of the Grain Measurers; the next, very old and
dilapidated, the Staple House of Corn, Romanesque, said to be the
earliest civil building in Belgium. Several fine gable-ends are seen to
the L., including one with Renaissance architecture, on this side of the
Lys. At the moment of writing, the houses next to the Skippers' Guild
are in course of demolition, exposing a bare side of the old Hall most
unpicturesquely.
Now, retrace your steps over the Bridge, and through the Corn Market,
almost wholly modernized, with the exception of a few gabled houses.
The next little square at which we arrive is the =Marché aux Herbes=.
Its W. side is occupied by the ancient but uninteresting Grande
Boucherie. Turn to the L. by the corner of the Boucherie, with Our Lady
and Child in a niche, and cross the bridge to the other side of the Lys.
On the left are two handsome old houses. In front rise the gateway and
bastions of the OUDEBURG, or Castle of the Princes. This was the
primitive palace of the Counts of Flanders in Ghent. The irregular
little square in front of it is known as the =Place Ste. Pharailde=. The
=castle= has recently been cleared from the numerous modern houses which
encumbered and hid it. The first stronghold on this site was erected in
868. The existing ruins of the =gateway=, with round Romanesque arches,
date back to 1180; the square =keep= behind is of the 10th century. In
this palace Jacob van Artevelde entertained Edward III. When Edward
returned to England, he left Queen Philippa here, and during his absence
she bore (in the Monastery of St. Bavon) her third son, John of Gaunt,
who took his well-known surname from the place of his birth. It was on
Edward's return to Flanders, accompanied by the ladies of Philippa's
suite, that he found the French fleet drawn up near Sluys to prevent his
entry into the port of Bruges, on which occasion he gained the first
great English naval victory. The Castle, which is now in course of
partial restoration, is closely bound up with the greatness of Van
Artevelde and the heroic period in the history of Ghent.
Walk round it to note its extent and its commanding position at the
point where the bridge crosses the Lys to the main part of the town.
The opposite corner of the Place Ste. Pharailde has a _Renaissance
gateway_, re-erected in imitation of the original by Arthus Quellin, and
adorned with sculptures of Neptune, the Schelde, and the Lys, the
sources of Ghent's greatness. It leads to the Fish-market. Around are
several good old houses.
Continue along the quay on the same side of the river as the Oudeburg,
as far as the Pont du Laitage, just before reaching which you pass on
your left two 17th century houses with reliefs, (the Works of Charity, a
Flying Hart, etc.). Cross the bridge and turn to the R. as far as the
=big cannon=, known as _Dulle Griete_ or Mad Margaret, dating back to
the 14th century. By the touch-hole are the Cross of St. Andrew and the
Arms of Phillipe le Bon of Burgundy.
Turn into the large square in front of you. The building which faces you
at the end of the street as you advance (with a tower at the corner and
high gables) is one of the best old mediæval houses in Ghent, the
Collacie-Zolder, or Municipal Council-Room, of the 13th or 14th century.
It has an interesting little pulpit or balcony at its corner, with a
bell, from which addresses could be made to the people. The towers that
face you a little to the L. are those of St. Jacques, to be visited
presently.
Continue into the square, at the corner of which is the Municipal
Council-Room. This is the _Vrydagmarkt_ or =Marché du Vendredi=, in
which a strikingly picturesque market is still held every Friday
morning. If possible, visit it. The square was the forum of old Ghent
and the meeting-place of the citizens. A few fine old buildings in the
native local style still surround it. The centre is appropriately
occupied by a modern colossal statue of Jacob van Artevelde, addressing
the citizens in his famous speech when he excited them to opposition to
the Count of Flanders with his Gallicising policy. At the base are
allegorical figures of Flanders, and of the Belgian towns, wearing mural
crowns. The reliefs represent Van Artevelde's three chief diplomatic
triumphs,—the League of Ghent with Bruges and Ypres; the League of
Flanders and England; the League of Flanders, Brabant, and Hainault. In
this square the most important events in the history of early Flanders
took place. Here the citizens of Ghent took the oath of allegiance to
each new Count on his accession, after they had compelled him to swear
in good old Teutonic style "to uphold and see upheld all the standing
wits (laws), fore-rights (regulations), freehoods, and wonts of the
Countship and town of Ghent." The guilds which had their halls around
met here to oppose arbitrary action on the part of their sovereign.
Here, too, the parties within the town itself frequently joined issue in
civil contest. In later times, the Duke of Alva perpetrated most of his
shameful executions on this spot. The site of the statue of Van
Artevelde was originally occupied by one of Charles V., who was born in
Ghent, in a palace now destroyed, and whose history (see later) is
intimately connected with this town, always one of his principal
residences. The statue was destroyed in 1794 by the French invaders:
(picture in the Museum).
Turn up at the corner by the Municipal Council-Room and take the first
street to the L., which leads you into the Place St. Jacques, occupied
by the Church of =St. Jacques=. The _façade_, with the two towers, was
Romanesque, but has been restored in such a wholesale way as to destroy
its interest. The remainder of the church is Gothic. Walk round it so as
to observe its features, noticing in particular the quaint stone spire
of the right-hand tower. The =interior= might be good, were it not
spoiled by tawdry decorations. The pulpit has a marble figure of the
patron, St. James, with the pilgrim's staff and gourd, emblematic of his
connection with the great place of pilgrimage of Santiago de
Compostella. The vaulting has been freed from excrescences, and is
excellent of its kind. The High Altar has a figure of St. James above,
and a painting of his martyrdom beneath.
This walk will have led you through the principal part of early Ghent.
Hence you may return either by the Cathedral, or by the chief line of
business streets which runs direct from the Pont du Laitage to the
modern Palais de Justice and the Place d'Armes.
_C._ THE CATHEDRAL
[The local =patron saint= of Ghent is =St. Bavon=, a somewhat
dubious personage, belonging to the first age of Christianity in
Flanders, of whom little is known. Legend describes him as a
"Duke of Brabant" in the 7th century (of course an anachronism).
He seems to have been a nobleman of Hesbaie who spent his life
as a soldier "and in worldly pleasures"; but when he was 50, his
wife died, and, overwhelmed with grief, he gave up all his
possessions to be distributed among the poor, and entered a cell
or monastery in Ghent, of which St. Amand (see later) was the
founder. Of this he became abbot. At last, finding the monastic
life not sufficiently austere, the new saint took refuge in a
hollow tree in a forest, and there spent the remainder of his
days. His emblem is a falcon. The =monastery of St. Bavon= long
existed at Ghent; some of its ruins still remain, and will be
described hereafter. To this local saint, accordingly, it might
seem fitting that the Cathedral of Ghent should be dedicated.
But in reality the building was at first a parish church under
the invocation of =St. John the Baptist=, and only received the
relics and name of St. Bavon after 1540, when Charles V.
destroyed the monastery, as will be described hereafter.
The real interest of the Cathedral centres, however, not in St.
Bavon, nor in his picture by Rubens, but in the great polyptych
of =the Adoration of the Lamb=, the masterpiece of =Jan van
Eyck= and his brother =Hubert=, which forms in a certain sense
the point of departure for the native art of the Netherlands.
This is therefore a convenient place in which to consider the
position of these two great painters. They were born at Maaseyck
or Eyck-sur-Meuse near Maastricht; Hubert, the elder, about 1360
or 1370; Jan, the younger, about 1390. The only undoubted work
of Hubert is the altar-piece in St. Bavon, and even this is only
his in part, having been completed after his death by his
brother Jan. =Hubert= probably derived his teaching from the
School of the Lower Rhine, which first in the North attained any
importance, and which had its chief exponents at Maastricht and
Cologne. Of this School, he was the final flower. Though not, as
commonly said, the inventor of =oil-painting=, he was the first
artist who employed the process in its developed form, and he
also made immense advances in naturalness of drawing and truth
of spirit. =Jan= was probably a pupil of Hubert; he lived at
Ghent while the great picture of the Adoration of the Lamb was
still being completed; later, he was painter by appointment to
the court of the Dukes of Burgundy, and had a house at Bruges,
where he died in 1440. He was also employed on various missions
abroad, accompanying embassies as far as to Portugal. His
painting, though less ideal and beautiful than that of his great
successor Memling, is marvellous in its truth: it has an
extraordinary charm of purity of colour, vividness of
delineation, and fine portrayal of character. Indeed, all the
early Flemish artists were essentially =portrait painters=; they
copied with fidelity whatever was set before them, whether it
were fabrics, furniture, jewellery, flowers, or the literal
faces and figures of men and women.
Hubert and Jan van Eyck, however, were not so much in strictness
the _founders_ of a school as the culminating point of early
German art, to which they gave a new Flemish direction. Their
work was almost perfect in its own kind. Their successors did
not surpass them: in some respects they even fell short of them.
The Adoration of the Lamb is =by far the most important thing=
to be seen at Ghent. But it is viewed at some disadvantage in
the church, and is so full of figures and meaning that it cannot
be taken in without long study. I strongly advise you,
therefore, to buy a =photograph= of the _entire composition_
=beforehand=, and try to understand as much as possible of the
picture by comparing it with the account here given, the
=evening before= you visit the picture. You will then be able
more readily to grasp the actual work, in form and colour, when
you see it.
The Cathedral is open daily (for viewing the pictures, etc.)
from 5 to 12, and from 3.30 to 6. Between 12 and 3.30 you can
also get in by knocking loudly on the door in the West Front.]
Go straight from your hotel to the Cathedral,—built as the parish
church of St. John about 1250-1300; re-dedicated to St. Bavon, 1540;
erected into a Bishop's see, 1599. Stand before the =West Front= at a
little distance, to examine the simple but massive architecture of the
tower and _façade_.
The =great portal= has been robbed of the statues which once adorned its
niches. Three have been "restored": they represent, centre, the Saviour;
L., the patron, St. Bavon, recognisable by his falcon, his sword as
duke, and his book as monk; he wears armour, with a ducal robe and cap
above it; R., St. John the Baptist, the earlier patron.
Then, walk, to the right, round the south side, to observe the external
architecture of the Nave, Aisles, and Choir. The latter has the
characteristic rounded or apsidal termination of Continental Gothic,
whereas English Gothic has usually a square end. Enter by the S. portal.
The =interior=, with single Aisles and short Transepts, (Early Gothic)
is striking for its simple dignity, its massive pillars, and its high
arches, though the undeniably noble effect of the whole is somewhat
marred to English eyes by the unusual appearance of the unadorned brick
walls and vaulting. The _pulpit_, by Delvaux (1745), partly in oak,
partly in marble, represents Truth revealing the Christian Faith to
astonished Paganism (figured as an old and outworn man:) it is a model
of all that should be avoided in plastic or religious art. The _screen_
which separates the Choir from the Transepts is equally unfortunate. The
apsidal end of the Choir, however, with its fine modern stained glass,
forms a very pleasing feature in the general _coup d'œil_.
Begin the examination in detail with the left or =N. Aisle=. The _1st
chapel_, that of the Holy Cross, contains a Pietà by Janssens and a
Descent from the Cross by Rombouts, good works of the school of Rubens.
The _3rd chapel_, that of St. Macarius or St. Macaire (an object of
local worship whom we shall meet again elsewhere at Ghent), has a modern
statue of the saint, and a pleasing decoration in polychrome. The right
or =S. Aisle= has nothing of importance.
A short flight of steps leads to the =Ambulatory=, whose black-and-white
marble screen, on the side toward the Choir, is not without dignity.
The =sacristan= opens the locked =Chapels in the Ambulatory=
(flamboyant), beginning at the steps on the R. or S. side of the Choir.
You will find him in the Sacristy, in the N. Transept. Do not let him
hurry you.
The _1st chapel_ contains a tolerable triptych by F. Pourbus (son of
Peter), with the Finding of Christ in the Temple for its central
subject, and the Circumcision and Baptism on the inner wings. Notice in
the last the conventional attitudes of the Baptist, the Saviour, and the
angel with the towel, as in the Gerard David and all old examples of
this subject: but the semi-nude figure undressing in the foreground is
an unhappy innovation of the Renaissance. Many of the heads in the
central picture are portraits: Alva, Charles V., Philip II., and Pourbus
himself. On the outer wings is a good *portrait of the donor (Viglius)
adoring the Saviour (1571).
_3rd chapel._ Crucifixion, by Gerard van der Meire, of Ghent. On the
left wing, Moses striking the Rock, symbolical of the fountain of living
water, Christ. On the right wing, the Elevation of the Brazen Serpent,
symbolical of the Crucifixion. This is a mystic "typical" picture,
interesting only for its symbolism. Note the Flemish love of such
subjects.
The _4th chapel_ contains a good tomb of Cornelius Jansen and Willem
Lindau, the two first bishops of Ghent (bishopric founded only in 1599)
with fair recumbent figures of the early 17th century.
_5th chapel._ Coxcie. Lazarus and Dives: a mediocre picture.
Mount the steps to the =Upper Ambulatory=.
The _6th chapel_ (of the Vydts family) contains the famous altar-piece
of the =**Adoration of the Lamb=, by HUBERT and JAN VAN EYCK, to study
which is the chief object of a visit to Ghent. See it more than once,
and examine it carefully. Ask the Sacristan to let you sit before it for
some time in quiet, or he will hurry you on. You must observe it in
close detail.
As a whole, the work before you is _not_ entirely by the two Van Eycks.
The Adam and Eve on the outer upper shutters of the interior (originally
by Hubert) have been altogether removed, and are now in the Museum at
Brussels, where we shall see them in due course. Their place has been
filled, not by copies (for the originals were nude), but by skin-clad
representations of the same figures, whose nudity seemed to the Emperor
Joseph II. unsuitable for a church. The lower wings, which were
principally (it is believed) by Jan van Eyck, have also been removed,
and sold to Berlin. They are replaced by very tolerable copies, made in
the early 16th century by Michael Coxcie. Thus, to form an idea of the
_detail_ of the original in its full totality it is necessary to visit,
not only Ghent, but also Brussels and Berlin. Nevertheless, I describe
the whole picture here _as it stands_, as this is the best place to
observe its general composition. I shall say a few words later as to
variations of this work from the original. There is a good copy of the
whole picture in the Museum at Antwerp, where you will be able to
inspect it at greater length and under easier conditions. The remaining
portions of the original still left here are believed to be for the most
part the work of Hubert van Eyck. Jan must rather be studied in many
scattered places,—Bruges, Brussels, Berlin, Paris, Madrid, and London.
The altar-piece was commissioned from Hubert van Eyck by Josse Vydts
(Latinised as Jodocus), a gentleman of Ghent, and his wife, Isabella,
about the year 1420. Hubert died while the polyptych was still
unfinished, and Jan completed it in 1432. Too much importance has been
attached by critics, I fancy, to the rhyming hexameter inscribed upon
it, (with the words "De Eyck" unmetrically introduced:) "Pictor Hubertus
major quo nemo repertus," etc. They have been twisted into a deliberate
expression of belief on the part of Jan that Hubert was a greater
painter than himself. If so, it seems to me, Jan was a worse critic than
painter. They are probably due, however, to a somewhat affected modesty,
or more probably still, to a priestly poet who was in straits to find a
rhyme for Hubertus.
I proceed to a =detailed explanation= of the picture.
The =subject=, in its entirety, is the Adoration of the Lamb that was
Slain, and it is mainly based on the passage in the Apocalypse: "I
looked, and lo, a Lamb stood on the Mount Zion, and with Him an hundred
and forty and four thousand, having His Father's name written in their
foreheads. . . . And I heard the voice of harpers harping with their
harps." Elsewhere we read: "I beheld, and, lo, a great multitude, which
no man could number, clothed with white robes, and palms in their
hands. . . . These are they which came out of great tribulation, and
have washed their robes, and made them white in the blood of the Lamb.
Therefore are they before the throne of God; and He shall feed them, and
shall lead them to living fountains of waters, and shall wipe away all
tears from their eyes." Much of the imagery, however, I believe, is also
taken from the Te Deum.
=Lower Tier.=
The _central panel_ (original: attributed to Hubert) represents in its
middle the altar, hung with red damask, and covered with a white cloth,
on which the Lamb of God is standing. His blood flows into a crystal
chalice. (This part is clearly symbolical of the Eucharist.) Upon Him,
from above, descends the Holy Ghost, in the form of a dove, sent out by
the Eternal Father, who occupies the central panel on top. Around the
altar are grouped adoring angels, with many- wings, holding the
instruments of the Passion—the Cross, the Spear, the Sponge, and the
Column to which Christ was fastened for flagellation. In front of it,
two angels swing censers. The flowery foreground is occupied by the
Fountain of Life, from which pure water flows limpid, to irrigate the
smiling fields of Paradise. Four bands of worshippers converge towards
this centre. On the left-hand side, stand, kneel, or ride, a group of
worshippers representing, as a whole, the _secular aspect_ of the
Christian Church—the laity. The foreground of this group is occupied by
the precursors of Christ. Conspicuous among them are the Jewish prophets
in front, and then the Greek poets and philosophers,—Homer, Plato,
Aristotle—whom mediæval charity regarded as inspired in a secondary
degree by the Spirit of Wisdom. Homer, in white, is crowned with laurel.
The group also includes kings and other important secular personages.
The right-hand side, opposite, is occupied by representatives of the
Church, showing the religious as opposed to the secular half of the
Christian world. In the front rank kneel 14 persons, the Twelve Apostles
(with Paul and Matthias) in simple robes, barefooted; behind them are
ranged all the orders of the hierarchy—canonized popes, with their
attendant deacons; archbishops, bishops, and other dignitaries.
The background shows two other groups, one of which (to the L.) consists
of the Martyrs, bearing their palms of martyrdom, and including in their
number popes, cardinals, bishops, and other ecclesiastics. The inner
meaning of this group is further emphasized by the symbolical presence
of a palm tree behind them. To balance them on the R. advance the
Virgins conspicuous among whom are St. Agnes with her lamb, St. Barbara
with her tower, St. Catherine, and St. Dorothy with her roses: many of
them carry palms of martyrdom. These various groups thus illustrate the
words of the Te Deum, representing "the glorious company of the
apostles," "the goodly fellowship of the prophets," "the noble army of
martyrs," "the Holy Church throughout all the world," etc., in adoration
of the Lamb that was Slain. (A chorus of Apostles, of Prophets, of
Martyrs, of Virgins is common in art.)
The more distant background is occupied by towered cities, typifying
perhaps the New Jerusalem, but adorned with Flemish or Rhenish turrets
and domes, and painted with Flemish minuteness and exactitude.
On the front of the altar are written in Latin the words, "Behold the
Lamb of God that taketh away the sins of the world."
The _Left Wings_ (inferior copy by Coxcie: originals, probably by Jan,
now at Berlin) form a continuation of the scene of the Prophets and the
secular side of Christendom in the central panel, and represent, in the
First or Inner Half, the Orders of Chivalry and the mediæval knighthood
riding, as on a crusade or pilgrimage, towards the Lamb that was Slain.
At their head go the soldier saints, St. George, St. Adrian, St.
Maurice, and St. Charlemagne (for the great emperor Karl is also a
canonized person). The action of the horses throughout is admirable. The
Second or Outer Half (ill described as "the Just Judges") represents the
Merchants and Burgesses, among whom two portraits in the foreground are
pointed out by tradition as those of Hubert and Jan van Eyck: (Hubert in
front, on a white horse: Jan behind, in a dark brown dress, trimmed with
fur). But this detail is unimportant: what matters is the colour and
composition on one hand, the idea on the other. These two panels,
therefore, with the group in front of them, are to be taken as
representing the Secular World—learned, noble, knightly, or
mercantile—in adoration of the central truth of Christianity as
manifested in the Holy Eucharist.
The corresponding _Right Wings_ (copy by Coxcie: originals, probably by
Jan, at Berlin) show respectively the Hermits and Pilgrims—the
contemplative and ascetic complement of the ecclesiastical group in
front of them: the monastic as opposed to the beneficed clerics. The
First or Inner Half shows the Eremites, amongst whom are notable St.
Anthony with his crutch, and, in the background, St. Mary Magdalen with
her box of ointment, emerging from her cave, (the Sainte Baume) in
Provence, in her character as the Penitent in the Desert. On the Second
or Outer Half, the body of Pilgrims is led by the gigantic form of St.
Christopher, with his staff and bare legs for wading; behind whom is a
pilgrim with a scallop-shell, and many other figures, not all of them
(to me) identifiable. Here again the presence of palms in the background
marks the esoteric idea of martyrdom.
I need not call attention throughout to the limpid sky, the fleecy
clouds, the lovely trees, the exquisite detail of architecture and
landscape.
=Upper Tier.=
The _three Central Panels_ (original) are attributed to Hubert. That in
the _middle_ represents, not (I feel sure) as is commonly said, Christ,
but God the Father ("Therefore they are before the throne of God")
wearing the triple crown (like the Pope), holding the sceptre, and with
his right hand raised in the attitude of benediction. His face is
majestic, grave, passionless: his dress kingly: a gorgeous morse fastens
his jewelled robe of regal red. At his feet lies the crown of earthly
sovereignty. He seems to discharge the Holy Ghost on the Lamb beneath
him. The word Sabaoth, embroidered on his garments, marks him, I think,
as the Father: indeed, the Son could hardly preside at the sacrifice of
the Lamb, even in the Eucharist.
On the right of the Father, in the _panel to the_ (spectator's) _left_
(Hubert: original), Our Lady, crowned, as Queen of Heaven, sits reading
in her blue robe. Her face is far more graceful than is usual in Flemish
art: indeed, she is the most charming of Flemish Madonnas. Behind her is
stretched a hanging of fine brocade.
The _panel to the right_ (Hubert: original) shows St. John the Baptist,
with his camel-hair garment, covered by a flowing green mantle. The
folds of all these draperies in Hubert's three figures, though simple,
have great grandeur.
The _Outer Wing to the left_ (substituted clothed figure, not a copy:
original, by Hubert, at Brussels) has Adam, as typical (with Eve) of
unregenerate humanity: a sense further marked by the Offerings of Cain
and Abel above it.
The _Outer Wing to the right_ has an Eve with the apple, (similarly
clad, not copied from the original, by Hubert, now at Brussels:) above
it, the First Murder.
The _Inner Left Wing_ (copy: the original, attributed to Jan, is at
Berlin) has a beautiful group of singing angels.
The _Inner Right Wing_ (copy: the original, likewise attributed to Jan,
is also at Berlin) has an angel (_not_ St. Cecilia) playing an organ,
with other angels accompanying on various musical instruments.
Taking it in its entirety, then, the altar-piece, when opened, is a
great =mystical poem= of the Eucharist and the Sacrifice of the Lamb,
with the Christian folk, both Church and World, adoring. It was in order
to prepare your mind for recognition of this marked strain of mysticism
in the otherwise prosaic and practical Flemish temperament, that I
called your attention at Bruges to several mystic or type-emphasising
pictures, in themselves of comparatively small æsthetic value.
The composition contains over 200 figures. Many of them, which I have
not here identified, can be detected by a closer inspection, which,
however, I will leave to the reader.
Now, ask the sacristan to =shut the wings=. They are painted on the
_outer side_ (all a copy) mainly in grisaille, or in very low tones of
colour, as is usual in such cases, so as to allow the jewel-like
brilliancy of the internal picture to burst upon the observer the moment
the altar-piece is opened.
The _lower wings_ have (in this copy) representations of the Four
Evangelists, in niches, in imitation of statuary. Observe the
half-classical pose and costume of Luke, the Beloved Physician. These
figures, however, were not so arranged in the original, as I shall
afterwards explain.
The _upper wings_ represent on their first or lowest tier, the
Annunciation, a frequent subject for such divided shutters. In the
centre is the usual arcade, giving a glimpse of the town of Ghent where
Hubert painted it. (The scene is said to be Hubert's own studio, which
stood on the site of the Café des Arcades in the Place d'Armes: the view
is that which he saw from his own windows.) To the L., as always, is the
angel Gabriel, with the Annunciation lily; to the R. is Our Lady,
reading. The Dove descends upon her head. The ordinary accessories of
furniture are present—prie-dieu, curtain, bed-chamber, etc. Note this
arrangement of the personages of the Annunciation, with the empty space
between Our Lady and the angel: it will recur in many other pictures.
Observe also the Flemish realism of the painter, who places the scene in
his own town at his own period: and contrast it with the mysticism of
the entire conception.
The _uppermost tier_ of all is occupied by figures of two Sibyls
(universally believed in the Middle Ages to have prophesied of Christ)
as well as two half-length figures of the prophets Zachariah and Micah,
(also as foretellers of the Virgin birth).
In several details the outer shutters in this copy =differ markedly from
the originals= at Berlin. Jan's picture had, below, outer panels (when
shut), portraits of Josse Vydts and his wife: inner panels, imitated
statues (in grisaille) of St. John the Baptist and St. John the
Evangelist, patrons at that time of this church. If you are going on to
Berlin, you will see them: if back to London, then go to the Basement
Floor of the National Gallery, where you will find the water-colour copy
done for the Arundel Society, which will give you an excellent idea of
the work in its original condition.
A few words must be given to the =external history= of this great
altar-piece. It was begun by Hubert about 1420. His death in 1426
interrupted the work. Jan probably continued to paint at it till 1428,
when he went to Portugal. On his return, he must have carried it to
Bruges, where he next lived, and there completed it in 1432. It was then
placed in this the family chapel of Josse Vydts. During the troubles of
the Reformation it was carried to the Hôtel-de-Ville, but after the
capitulation to the Duke of Parma it was restored to the chapel of the
Vydts family. Philip II. wished to carry it off, but had to content
himself with a copy by Coxcie, the wings of which are now in this
chapel. The panels with Adam and Eve were removed in 1784, after Joseph
II. had disapproved of them, and hidden in the sacristy. In 1794, the
remaining panels were carried to Paris: after the peace, they were
returned, but _only the central portions_ were replaced in the chapel.
The wings, save Adam and Eve, were sold to a Brussels dealer, and
finally bought by the King of Prussia, which accounts for their presence
at Berlin. As for Adam and Eve, the church exchanged them with the
Brussels Museum for the wings of Coxcie's copy. These various
vicissitudes will explain the existing condition of the compound
picture.
Do not be content with seeing it once. Go home, re-read this
description, and come again to study it afresh to-morrow.
* * * * *
The _chapel of the Holy Sacrament_, in the Apse, has very ugly rococo
monuments to bishops of the 18th century, in the worst style of the
debased Renaissance, and other monstrosities.
The _10th chapel_ has a famous *altar-piece by Rubens, St. Bavon
renouncing his worldly goods to embrace the monastic life. The Saint is
seen, attired as a Duke of Brabant of the 17th century, in his armour
and ducal robes, attended by his pages, making his profession at the
door of a stately Renaissance church, such as certainly did not exist in
the North in his time, and received with acclamation by a dignified body
of nobly-robed ecclesiastics, including St. Amand (see later, under the
Monastery of St. Bavon). The features of the patron saint are said to be
those of Rubens; they certainly resemble his portrait of himself at
Florence. The foreground is occupied by a group of poor, to whom St.
Bavon's worldly goods are being profusely scattered. On the L. are two
ladies, in somewhat extravagant courtly costumes, who are apparently
moved to follow the Saint's example. They are said to be the painter's
two wives, but the resemblance to their known portraits is feeble. This
is a fine specimen of Rubens's grandiose and princely manner, of his
feeling for space, and of his large sense of colour; but it is certainly
_not_ a sacred picture. It was appropriately painted for the High Altar
in the Choir (1624), after the church was dedicated to St. Bavon and
erected into a cathedral, but was removed from that place of honour in
the 18th century to make room for a vulgar abomination by Verbruggen. (I
defer consideration of Rubens and his school till we reach Brussels and
Antwerp.) Fair monument of a 17th century bishop.
Descend the steps again. Enter the =Choir=, a very fine piece of
architecture, cleared of the monstrosities of the last century: it has
beautiful grey stone arches (about 1300), a handsome Triforium, and
excellent brick vaulting. The lower portion, however, is still
disfigured by black-and-white marble screens and several incongruous
rococo tombs, some of which have individual merit. (That to the left,
Bishop Triest by Duquesnoy, is excellent in its own _genre_.) Over the
_High Altar_ flutters a peculiarly annoying and flyaway 17th century
figure of the Apotheosis of St. Bavon, the patron saint of the
Cathedral, who of course thus occupies the place of honour. It is by
Verbruggen. The huge copper candlesticks, bearing the royal arms of
England, as used by Charles I, belonged to his private oratory in Old
St. Paul's in London, and were sold by order of Cromwell. Impressive
view down the Nave from this point.
=Tip the Sacristan= at the rate of 1 franc per head of your party.
_D._ THE OUTSKIRTS
=Old Ghent= occupied for the most part the island which extends from the
Palais de Justice on one side to the Botanical Gardens on the other.
This island, bounded by the Lys, the Schelde, and an ancient canal,
includes almost all the principal buildings of the town, such as the
Cathedral, St. Nicolas, the Hôtel-de-Ville, the Belfry, and St. Jacques,
as well as the chief Places, such as the Marché aux Grains, the Marché
aux Herbes, and the Marché du Vendredi. It also extended beyond the Lys
to the little island on which is situated the church of St. Michel, and
again to the islet formed between the Lieve and the Lys, which contains
the _château_ of the Counts and the Place Ste. Pharailde.
In the later middle ages, however, the town had spread to nearly its
existing extreme dimensions, and was probably more populous than at the
present moment. But its ancient =fortifications= have been destroyed,
and their place has been taken by boulevards and canals. The line may
still be traced on the map, or walked round through a series of shipping
suburbs; but it is uninteresting to follow, a great part of its course
lying through the more squalid portions of the town. The only remaining
=gate= is that known as the =Rabot= (1489), a very interesting and
picturesque object, situated in a particularly slummy quarter. It can
best be reached by crossing the bridge near the church of St. Michel,
and continuing along the Rue Haute to the Boulevard du Béguinage, (where
stood originally the Grand Béguinage, whose place is now occupied by
modern streets.) Turn there along the boulevard to the R., till you
reach the gate, which consists of two curious round towers, enclosing a
high and picturesque gable-end. Owing to the unpleasant nature of the
walk, I do _not_ recommend this excursion.
* * * * *
The =S. quarter= of the town, beyond the Cathedral and St. Nicolas, has
been much modernized during the last two centuries. Its only interesting
points are the recent =Palais de Justice= and the =Kouter= or =Place
d'Armes=, (once the archery ground) in which a pretty flower-market is
held on Friday and Sunday mornings. The Café des Arcades, at its E. end,
occupies the site of Hubert van Eyck's studio.
The rest of the inner town contains little that throws light on its
origin or history.
* * * * *
There is, however, one small excursion which it would be well for those
to take who have a morning to spare, and who desire to understand the
development of Ghent—I mean to the =Monastery of St. Bavon=, which
alone recalls the first age of the city. Every early mediæval town had
outside its walls a ring of abbeys and monasteries, and Ghent was
particularly rich in this respect.
[=St. Amand= was the =apostle of Flanders= and the surrounding
countries. He was sent by the pious king Dagobert to convert the
Flemings _en bloc_, and is said to have built, about 630, a
little cell by the bank of the Lys, N.E. of the modern city. In
651, =St. Bavon= entered this infant monastery, which henceforth
took his name. The =abbey= grew to be one of the most important
in Flanders, and occupied a large area on the N.E. of the town,
near the Antwerp Gate. Eginhard, the biographer and son-in-law
of Charlemagne, was abbot in the 9th century. The Counts of
Flanders had rights of hospitality at St. Bavon's; hence it was
here, and not in the Oudeburg as usually stated, that Queen
Philippa gave birth to John of Gaunt. In 1539, however, Charles
V. that headstrong despot, angry at the continual resistance of
his native town to his arbitrary wishes, dissolved the monastery
in the high-handed fashion of the 16th century, in order to
build a citadel on the spot. As compensation for disturbance to
the injured saint, he transported the relics of St. Bavon to
what was then the parish church of St. John, which has ever
since borne the name of the local patron. Around the dismantled
ruins, the Emperor erected a great fort, afterwards known as the
Spaniards' Castle, (_Château des Espagnols_, or Het Spanjaards
Kasteel.) This gigantic citadel occupied a vast square space,
still traceable in the shape of the modern streets; but no other
relic of it now remains. The =ruins of the Abbey= are in
themselves inconsiderable, but they are certainly picturesque
and well worth a visit from those who are spending some days in
Ghent. The hurried tourist may safely neglect them.]
The direct route from the Place d'Armes to the =Abbey= is by the Quai du
Bas Escaut, and the Rue Van Eyck. A pleasanter route, however, is by the
Rue de Brabant and the Rue Digue de Brabant to the Place d'Artevelde,
passing through the handsomest part of the modern town. (In the Place
itself stands the fine modern Romanesque Church of St. Anne, the
interior of which is sumptuously decorated in imitation of mosaic.)
Thence, follow the Quai Porte aux Vaches to the Place Van Eyck. Cross
the bridges over the Upper and Lower Schelde, and the Abbey lies
straight in front of you.
Walk past the ivy-clad outer wall of the =ruins= to the white house at
the corner of the street beyond it, where you will find the _concierge_
(notice above the door). One franc is sufficient tip for a party. The
_concierge_ conducts you over the building, which has a picturesque
cloister, partly Romanesque, but mainly 15th century. The centre of the
quadrangle is occupied by a pretty and neatly-kept garden of the old
sweet-scented peasant flowers of Flanders. The most interesting part of
the ruins, however, is the octagonal Romanesque =Baptistery= or "Chapel
of St. Macaire," a fine piece of early vaulting, with round arches, very
Byzantine in aspect. The chapel rests on massive piers, and its
Romanesque arches contrast prettily with the transitional Gothic work of
the cloister in the neighbourhood. Within are several fragments of
Romanesque sculpture, particularly some *capitals of columns, with
grotesque and naïve representations of Adam and Eve with the Lord in the
Garden, and other similar biblical subjects. (Examine closely.) There is
likewise an interesting relief of St. Amand preaching the Gospel in
Flanders, and a man-at-arms in stone, of Artevelde's period, removed
from the old coping of the Belfry.
We next go on to the Crypt, the tombs of the monks, the monastery
cellars, etc., where are collected many pieces of ancient sculpture,
some found in the ruins and others brought from elsewhere. The Refectory
at the end, which for some time served as the Church of St. Macaire, is
now in course of transformation into a local Museum of Monumental Art.
It contains some good old tombs, and an early fresco (of St. Louis?)
almost obliterated. But the garden and cloister are the best of the
place, and make together a very pretty picture. You can return by the
Quai and the Rue St. Georges, or by the Place St. Bavon and the
Archiepiscopal Palace. (The castellated building to the L., much
restored, near the Cathedral, known as the =Steen of Gérard le Diable=,
is the sole remaining example of the mediæval fortified houses in
Ghent.)
* * * * *
Another monastery, a visit to which will lead you through the extensive
southern portion of the city, is the (wholly modernized) Benedictine
=Abbey of St. Pierre= (I do not recommend it). To reach it, you take the
Rue Courte du Jour and the Rue Neuve St. Pierre, to the large square
known as the Plaine St. Pierre, partly obtained by demolition of the
monastery buildings. It is situated on rising ground, which may pass for
a hill in Flanders. This is, in its origin, the oldest monastery in
Ghent, having been founded, according to tradition, by St. Amand
himself, in 630, on the site of an ancient temple of Mercury. The
existing buildings, however, hardly date in any part beyond the 17th
century. =The Church of Notre-Dame de St. Pierre= was erected between
1629 and 1720, in the grandiose style of the period. It is vast, and not
unimposing. The interior has a certain cold dignity. The pictures are
mostly of the school of Rubens, many of them dealing with St. Peter and
St. Benedict; among them are good specimens. The best, by De Crayer,
shows the favourite Benedictine subject of St. Benedict recognising the
envoy of King Totila, who personated the king.
The =Plaine de St. Pierre= is used for the amusing yearly =fair=, from
Mi-Carême to Easter.
* * * * *
The =Museum of Painting= (a small and unimportant gallery) is situated
in part of an old _Augustinian monastery_, which is reached by the
Oudeburg and the Rue Ste. Marguerite. (Church by the side, full of
Augustinian symbols.) Open daily from 9 to 12, and 2 to 5, free. (I do
not advise a visit, unless you have plenty of time to spare.) The
Picture Gallery is on the second floor.
The _Rooms to the L._ contain modern Belgian and French pictures, many
of them possessing considerable merit, but not of a sort which enters
into the scheme of these Guide-books.
The _Rooms to the R._ of the staircase contain the early pictures.
=1st Room=. F. Pourbus: A votive triptych for recovery from sickness. In
the centre, Isaiah prophesying to Hezekiah his recovery. On the wings,
the Crucifixion, and the donor with his patron St. James. Outside the
wings, in grisaille, the Raising of Lazarus (in two panels), giving a
symbolical meaning to this votive offering. On the wall beside it,
several tolerable pictures of the old Flemish School: a good Ex Voto of
a donor, with the Madonna and Child, by an unknown artist; a writhing
Calvary, by Van Heemskerk; a Holy Family, by De Vos; and a quaint
triptych of St. Anne and her family, with her daughter, the Madonna, and
her grand-child, the Saviour, at her feet. Around are grouped Joseph,
Mary Cleophas, Zebedee, Alpheus, Joachim, the husband of Anna, and Mary
Salome, with her children, James and John. This queer old work, by an
unknown artist, is interesting for comparison with the great Quentin
Matsys, which you will see at Brussels. St. Joseph holds in his hand the
rod that has flowered. (See _Legends of the Madonna_.)
Beneath this triptych are three interesting portrait groups of husbands
and wives, 16th century. On the wings, a "Noli Me Tangere"—Christ and
the Magdalen in the garden.
The _2nd Room_ has Dutch and Flemish works of the 17th century, mostly
self-explanatory. The Last Judgment, by R. Coxcie, shows a late stage of
a subject which we have already seen at Bruges, now reduced to an
opportunity for the display of exaggerated anatomical knowledge. There
are also several tolerable works of the School of Rubens, many of which
are interesting mainly as showing the superiority of the Master to all
his followers. Rombouts, The Five Senses, is, however, an excellent work
of its own class. The centre of the further wall is occupied by a
worthless picture of Duchastel's, representing the Inauguration of
Charles II. of Spain as Count of Flanders, in 1666, interesting mainly
as a view of old Ghent. The action takes place in the Marché du
Vendredi, the centre of which is occupied by the statue of Charles V.,
destroyed at the French Revolution. All round are the original
picturesque houses, with their high Flemish gable-ends. On the R. is the
Church of St. Jacques, much as at the present day. In front of the
Municipal Council Chamber a platform is erected for the inauguration.
The picture gives a good idea of the splendour of Ghent, even at the
period of the Spanish domination.
Near it, Rubens's St. Francis receiving the Stigmata, where the
conventional elements of the crucified six-winged seraph, the rays
proceeding from the five wounds to the saint's hands, feet, and side,
and the astonished brother, Leo, in the distance, are all preserved,
though enormously transfigured. The colour is unpleasing. This is almost
a replica of the work in the Cologne Museum. Rombouts—tolerable Holy
Family. Close by, some of Hondekoeter's favourite birds, and Zeghers's
flowers. Over the door, a fine De Crayer. In the centre of the room are
a series of pictures from the Gospel History, by F. Pourbus, with the
Last Supper and donor at the back of one, formerly a triptych.
The _3rd Room_ has pictures of the School of Rubens, many of them of
considerable merit, particularly De Crayer's Coronation of St. Rosalie
and Vision of St. Augustine, in both of which he approaches within a
measureable distance of the great master. His Judgment of Solomon is
also excellent. Some other pictures in the room, however, exhibit the
theatrical tendency of the 17th century in its worst form.
* * * * *
On the way back from the Picture Gallery, you pass on your L. the Rue
Longue des Pierres, down which, a little way on the R., is a small
=Museum of Antiquities=. I do not advise a visit to this. It contains
one good brass, and some silver badges worn by ambassadors of Ghent, but
otherwise consists, for the most part, of third-rate _bric-à-brac_.
* * * * *
Most visitors to Ghent go to see the =Grand Béguinage=. This was
originally situated in a little district by itself, close to the gate of
the Rabot, where its church, uninteresting, (dedicated, like that of
Bruges, to St. Elizabeth of Hungary), still stands; but the site has
been occupied by the town for new streets. The present Grand Béguinage
lies on the road to Antwerp. It is a little town in miniature, enclosed
by wall and moat, with streets and houses all very neat and clean, but
of no archæological interest. Yet it forms a pleasant enough end for a
short drive. And you can buy lace there. The description in Baedeker is
amply sufficient.
* * * * *
Bruges is full of memories of the Burgundian Princes. At Ghent it is the
personality of =Charles V.=, the great emperor who cumulated in his own
person the sovereignties of Germany, the Low Countries, Spain and
Burgundy, that meets us afresh at every turn. He was born here in 1500,
and baptized in a font (otherwise uninteresting) which still stands in
the N. Transept of the Cathedral. Ghent was really, for the greater part
of his life, his practical capital, and he never ceased to be at heart a
Ghenter. That did not prevent the citizens from justly rebelling against
him in 1540, after the suppression of which revolt Charles is said to
have ascended the Cathedral tower, while the executioner was putting to
death the ringleaders in the rebellion, in order to choose with his
brother Ferdinand the site for the citadel he intended to erect, to
overawe the freedom-loving city. He chose the Monastery of St. Bavon as
its site, and, as we have seen, built there his colossal fortress, now
wholly demolished. The Palace in which he was born and which he
inhabited frequently during life, was known as the Cour du Prince. It
stood near the Ancien Grand Béguinage, but only its name now survives in
that of a street.
The Spaniard's Castle was long the standing menace to freedom in the Low
Countries. Within its precincts Egmont and Hoorn were imprisoned in 1568
for several months before their execution.
During the early Middle Ages, the Oudeburg was the residence of the
Counts of Flanders in Ghent. Later on, that castellated building grew
out of keeping with the splendour of the Burgundian Princes, and its
place as a royal residence was taken by the Cour du Prince, already
mentioned, which was inhabited by Maximilian and his wife, Mary of
Burgundy, as well as by Philippe le Beau and Johanna of Spain, the
parents of Charles V. No direct memorials of the great Emperor now exist
in Ghent, his statue in the Marché du Vendredi having been destroyed;
but a modern street commemorates his name, and mementoes of him crop up
at every point in the city.
Though the Ghenters were rebellious subjects, Charles V. was proud of
his capital, and several of his very bad _bon mots_, punning on the
words _Gand_ and _gant_, have been preserved for us. As Baedeker repeats
these imperial jests, however, I need not detail them.
III
BRUSSELS
_A._ ORIGINS OF BRUSSELS
B=RUSSELS= was in a certain sense the ancient =capital= of =Brabant=,
as Bruges and Ghent were the ancient capitals of West and East Flanders.
It grew up (as early as the 8th century) on the banks of the little
river =Senne=, whose course through its midst is now masked by the
modern Inner Boulevards, built on arches above the unseen stream. The
Senne is one of the numerous rivers which flow into the Schelde, and the
original town clustered close round its banks, its centre being marked
by the Grand' Place and the church of St. Nicolas. Unlike Bruges and
Ghent, however, Brussels has always been rather an administrative than a
commercial centre. It is true, it had considerable trade in the Middle
Ages, as its fine Hôtel-de-Ville and Guild Houses still attest; but it
seems to have sprung up round a villa of the Frankish kings, and it owed
at least as much to its later feudal lords, the Counts of Louvain,
afterwards Dukes of Brabant, and to their Burgundian successors, as to
its mercantile position.
The Senne was never a very important river for navigation, though, like
most of the Belgian waterways, it was ascended by light craft, while a
canal connected the town with the Schelde and Antwerp: but the situation
of Brussels on the =great inland trade route= between Bruges or Ghent
and Cologne gave it a certain mercantile value. Bruges, Ghent, Brussels,
Louvain, Maastricht, and Aix-la-Chapelle all formed stations on this
important route, and all owed to it a portion of their commercial
prestige.
The burgher town which was thus engaged in trade and manufactures was
Flemish in speech and feeling, and lay in the hollow by the river and
the Grand' Place. But =a lordly suburb= began to arise at an early date
on the hill to eastward, where the Counts of Louvain built themselves a
mansion, surrounded by those of the lesser nobility. After 1380, the
Counts migrated here from too democratic Louvain. Later on, in the
fifteenth century, the Dukes of Burgundy (who united the sovereignty of
Brabant with that of Flanders) often held their court here, as the
population was less turbulent and less set upon freedom than that of
purely commercial and industrial Bruges and Ghent. Thus the distinctive
position of Brussels as the aristocratic centre and the =seat of the
court= grew fixed. Again, the Dukes of Burgundy were French in speech,
and surrounded themselves with French knights and courtiers; to suit the
sovereigns, the local nobility also acquired the habit of speaking
French, which has gradually become the language of one-half of Belgium.
But the people of the Old Town in the valley were, and are still,
largely Flemish in tongue, in customs, in sympathies, and in aspect;
while the inhabitants of the Montagne de la Cour and the court quarter
generally are French in speech, in taste, and in manners. We will trace
in the sequel the gradual growth of Brussels from its nucleus by the
river (the Lower Town), up the side of the eastern hill to the Palace
district (the Upper Town), and thence through the new Quartier Léopold
and the surrounding region to its modern extension far beyond the limits
of the mediæval ramparts.
Choose an hotel in the airy and wholesome Upper Town, as near as
possible to the =Park= or the Place Royale.
=St. Michael the Archangel= is the =patron saint= of Brussels: he will
meet you everywhere, even on the lampposts. For the patroness, St.
Gudula, see under the Cathedral.
_B._ THE HEART OF BRUSSELS
[The =nucleus of Brussels=, as of Paris, was formed by =an
island=, now no longer existing. Round this islet ran two
branches of the little =river Senne=, at present obliterated by
the Inner Boulevards. Brussels, in short, has denied its
parentage; the Senne, which is visible N. and S. of the Outer
Boulevards, being covered over by arches within the whole of the
Inner City.
The =centre of the island= is marked by the little _Place St.
Géry_, which the reader need not trouble to visit. Here, at the
end of the 6th century, St. Géry, Bishop of Cambrai and apostle
of Brabant, built a small chapel, succeeded by a church, now
demolished. The true centre of Brussels, however, may be
conveniently taken as the existing Bourse. Close by, as the town
grew, the =Grand' Place= or market-place was surrounded by noble
mediæval and Renaissance buildings. To this centre then, the
real heart of Brussels in the Middle Ages, we first direct
ourselves.]
Go from your hotel to the =Grand' Place=. It may be reached by either of
two convenient roads; from the _Place Royale_ by the Montagne de la Cour
and the Rue de la Madeleine, or from the _Park_ by the Montagne du Parc
(which takes various names as it descends), and the Galérie St. Hubert.
Either route brings you out at the end of the Galérie, whence a short
street to the L. will land you at once in the Grand' Place, undoubtedly
the finest square in Europe, and the only one which now enables us to
reconstruct in imagination the other Grand's Places of Belgium and the
Rhine country.
The most conspicuous building in the Place, with the tall tower and open
spire, is the HÔTEL-DE-VILLE, with one possible exception (Louvain) the
handsomest in Belgium. It consists of a tapering central tower, flanked
by two wings, their high-pitched roof covered with projecting windows.
The ground floor is arcaded. The first and second floors have Gothic
windows, altered into square frames in a portion of the building. The
edifice is of different dates. The original Hôtel-de-Ville consisted
only of the wing to your L., as you face it, erected in 1402. The R.
wing, shorter in façade, and architecturally somewhat different, was
added in 1443. The style of the whole, save where altered, is Middle
Gothic "Decorated"). The beautiful open =spire= should be specially
noticed. On its summit stands a colossal gilt metal figure (1454) of the
Archangel Michael, patron of the city. The _statues_ in the niches are
modern, and not quite in keeping with the character of the building.
Observe, over the main portal, St. Michael, patron saint of the town,
with St. Sebastian, St. Christopher, St. George, and St. Géry. Below are
the Cardinal Virtues. The figures above are Dukes of Brabant. Inspect
the whole _façade_ carefully. You will hardly find a nobler piece of
civic architecture in Europe. The carved wooden door has also a figure
of St. Michael. The gargoyles and the bosses near the staircase entrance
to the L. are likewise interesting.
Now, go round the corners to the L. and R., to inspect the equally fine
_façades_, facing the Rues de l'Hôtel-de-Ville and de la Tête-d'Or. The
_back_ of the building is 18th century and uninteresting. You may also
pass rapidly through the _courtyard_, which however has very little
character. But you need not trouble to inspect the _interior_, unless
you are an abandoned sight-seer.
The other important and beautiful building which faces the
Hôtel-de-Ville is the MAISON DU ROI, formerly used as the HALLE AU PAIN
or _Broodhuis_. It is of late florid Gothic, verging towards Renaissance
(1514, restored:) and is in three storeys, two of them arcaded. The
first floor has an open gallery, like the loggia of a Venetian palace,
whence ladies could view processions and ceremonies in the square below.
The building terminates in a high roof, with projecting windows, and a
handsome open tower and lantern. The whole has been recently rebuilt and
profusely gilded. Within, is a small Communal Museum (open free daily,
10 to 4). Come again often to view these two noble halls.
The third principal building (on the E. side of the Square) known as the
MAISON DES DUCS was the Public Weighing House, constructed in a debased
Renaissance style, and also profusely gilded. It bears the date 1698,
but is now unworthily occupied by sale rooms and shops.
The whole of the remaining space in this glorious square is surrounded
by magnificent =Guild Halls= of the various corporations.
Beginning on the =S. side= (that occupied by the Hôtel-de-Ville), we
have, first, L., two high-gabled houses of good 17th century domestic
architecture. Next to them, R., comes the =Hôtel des Brasseurs=, dated
1752, and lately surmounted by a bronze equestrian statue of Charles of
Lorraine. This was originally the _Guild Hall of the Brewers_. After
that, again, rises the house known as "The Swan," belonging to the
_Corporation of Butchers_. The small building at the corner, next the
Hôtel-de-Ville, with an open _loggia_, now in course of restoration, is
known as the Maison de l'Étoile: a gilt star surmounts its gable.
The finest group of houses, however, is that to the =W. side= of the
Square (R. of the Hôtel-de-Ville), unoccupied by any one prominent
building. Beginning on the L., we have, first, the house known as "The
Fox" (=Le Renard=), dated 1699: it is surmounted by a figure of St.
Nicholas resuscitating the three boys, and is adorned with statues of
Justice and the Four Continents on its first floor. Then comes the
=Guild Hall of the Skippers=, or _Maison des Bateliers_, its gable
constructed somewhat like the poop of a ship. The symbolism here is all
marine—sailors above; then Neptune and his horses, etc. R. of this, we
see the house known as "=La Louve=," bearing as a sign Romulus and Remus
with the wolf. This was originally the =Guild Hall of the Archers=. It
shows an inscription stating that it was restored, after being burnt
down, by the _Confraternity of St. Sebastian_ (patron of archers). Its
relief of the Saint with a bow is appropriate. The two remaining houses
are "_La Brouette_," dated 1697, and "_Le Sac_," bearing on its gable a
medallion with three faces.
The houses on the =N. side=, (that occupied by the Maison du Roi,) are
less interesting, except those on the extreme R. Next to the Maison du
Roi itself come two pretty little decorated houses, beyond which is the
=Guild Hall of the Painters=, known as "The Pigeon," and that called "La
Taupe," the =Hall of the Tailors=. The two last at the corner of the
street have been recently restored. Several other fine houses of the
same period close the vista of the streets round the corner.
This imposing group of Guild Halls dates, however, only from the end of
the 17th century, mostly about 1697. The reason is that in 1695 the
greater part of the Grand' Place was destroyed by Marshal de Villeroi
during the siege. Two years later, the Guild Houses were rebuilt in the
ornate and somewhat debased style of the Louis XIV. period. Fortunately,
the two great mediæval buildings, which stood almost isolated, did not
share the general destruction.
Continue your stroll through the =Lower Town=.
From the Grand' Place, take the Rue au Beurre, which leads W. towards
the Bourse. On your R. you will pass the now uninteresting and entirely
modernized =Church of St. Nicolas=. In its origin, however, this is one
of the oldest churches in Brussels, and though it has long lost almost
every mark of antiquity, it is instructive to recognise here again (as
at Ghent) the democratic patron saint of the merchants and burgesses in
close proximity to their Town Hall and their Guild Houses. The =Bourse=
itself, which faces you, is a handsome and imposing modern building. Go
past its side till you reach the line of the =Inner Boulevards=, which
lead N. and S. between the Gare du Nord and the Gare du Midi.
This superb line of streets, one of the finest set of modern boulevards
in Europe, has been driven straight through the heart of the =Old Town=,
and the authorities offered large money prizes for the best _façades_
erected along the route. Content yourself for the moment with a glance
up and down, to observe the general effect, and then continue on to your
L. along the Boulevard, where the first street on the R. will lead you
to the little =Place St. Géry=, now occupied by a market, but originally
the centre of Old Brussels. A stroll through the neighbouring streets is
interesting, past the =Halles Centrales=, and the modern Church of =St.
Catherine=, close by which stands the old Tower of St. Catherine, built
into a modern block of houses. A little further on is the picturesque
=Tour Noire=, the only remaining relic of the first fortifications of
the city. You may prolong this walk to the Place du Béguinage, with a
tolerable church. The quarter has no special interest, but it will serve
to give you a passing idea of the primitive nucleus of mediæval
Brussels.
I will interpolate here a few remarks about the more modern portion of
the Old Town. The best way to see it is to take the tram along the Inner
Boulevards from the Gare du Midi to the Gare du Nord. You will then
pass, first, the Outer Boulevards (see later): next, R., the _Palais du
Midi_; L., the _Place d'Anneessens_, with a statue of Anneessens, the
intrepid and public-spirited magistrate of Brussels who was put to death
in 1719 for venturing to defend the privileges of the city against the
Austrian authorities. Just opposite this, you get a glimpse, R., of the
_Place Rouppe_, to be noticed later. Passing the _Place Fontainas_,
where many streets radiate, you arrive at the _Bourse_, already noticed.
The handsome corner building (with dome) in front of you, which forms so
conspicuous an element in the prospect as you approach, is the _Hôtel
Continental_. Just in front of it expands a small new square (_Place de
Brouckere_) still unfinished, on which a monument is now being erected
to a late burgomaster (De Brouckere.) At this point, the Boulevard
divides, the western branch following the course of the Senne (which
emerges to light just beyond the Outer Boulevards,) while the eastern
branch goes straight on to the Gare du Nord, passing at the first corner
a handsome narrow house with gilt summit, which won the first prize in
the competition instituted by the Municipality for the best _façades_ on
the new line of streets.
After reaching the Gare du Nord, you can return to the Gare du Midi by
an alternative line of main streets, which also cuts through the heart
of the Old Town, a little to the E. of the Inner Boulevards. It begins
with the _Rue Neuve_, where a short street to the L. conducts you
straight to the =Place des Martyrs=, a white and somewhat desolate
square of the 18th century, (1775) adorned later with a Monument to the
Belgians who were killed during the War of Independence in 1830. Shortly
after this (continuing the main line) you pass two covered galleries, R.
and L., and then arrive at the =Place de la Monnaie=. On your R. is the
handsome building of the new =Post Office=; on your L., the white
Ionic-pillared =Grand Théâtre= or =Théâtre de la Monnaie=, (opera,
etc.). You then pass between _St. Nicolas_ on the L., and the _Bourse_
on the R., and continue on to the =Place Rouppe=, (ornamented with a
fountain and a statue of Brussels personified): whence the Avenue du
Midi leads you straight to the _Place de la Constitution_, in front of
the South Station.
The remainder of the =Western Half of the town= is, for the most part,
poor and devoid of interest, though it contains the principal markets,
hospitals, and barracks, as well as the basins for the canals which have
superseded the Senne.
_C._ THE PICTURE GALLERY
[I interpolate here the account of the =Brussels Picture
Gallery=, because it is the most important object to be seen in
the town, after the Grand' Place and its neighbourhood. You must
pay it several visits—three at the very least—and you may as
well begin early. Follow the roughly chronological order here
indicated, and you will understand it very much better. Begin
again next time where you left off last: but also, revisit the
rooms you have already seen, to let the pictures sink into your
memory. Intersperse these visits with general sight-seeing in
the town and neighbourhood.
The Brussels Gallery forms an excellent continuation to the
works of art we have already studied at Bruges and Ghent. In the
_first place_, it gives us some further examples of the =Old
Flemish masters=, of the Van Eycks and of Memling, as well as
several altar-pieces belonging to the mystical religious School
of the Brussels town-painter, =Roger Van der Weyden=, who was
Memling's master. These have been removed from churches at
various times, and gradually collected by the present
Government. It also affords us an admirable opportunity of
becoming well acquainted with the masterpieces of =Dierick
Bouts=, or Dierick of Haarlem, an early painter, Dutch by birth
but Flemish by training, who was town painter in democratic
Louvain, (which town may afterwards be made the object of an
excursion from Brussels).
But, in the _second place_, besides these painters of the early
school, the Brussels Gallery is rich in works of the
=transitional period=, and possesses in particular a magnificent
altar-piece by =Quentin Matsys=, the last of the old Flemish
School, and the first great precursor of the Renaissance in the
Low Countries. He was practically an Antwerp man (though born at
Louvain), and his place in art may more fitly be considered in
the Antwerp Museum.
From his time on we are enabled to trace, in this Gallery, the
evolution of Flemish art to its =third period=, the time of
=Rubens= (also better seen at Antwerp) and his successors, the
great =Dutch painters=, here fairly represented by Rembrandt,
Frans Hals, Van der Helst, Gerard Dou, and Teniers.
In the following list of the most noteworthy works of each
School, I have adhered, roughly speaking, to a chronological
order, but without compelling the reader unnecessarily to dance
up and down the various rooms of the collection from one work to
another. The Gallery itself is one of the most splendid in
Europe, and it has been recently re-arranged in a most
satisfactory manner.]
The national collection of pictures by Old Masters occupies the very
handsome modern building known as the =Palais des Beaux-Arts= in the Rue
de la Régence, immediately after passing through the Place Royale. (Four
large granite columns in front: bronze sculpture groups to R. and L.)
See plan on p. 108.
Enter by the big door with the four large granite columns. In the
vestibule, turn to the R., and mount the staircase. Then pass through
Room III. and Corridor A, to Room V. on the right, and on into
ROOM I.
=Hall of the Old Flemish Masters.=
This contains the most interesting works in the Gallery.
(You may also, if you like, pass through the collection of Sculpture in
the Hall below, entering by Corridor D; in which case, turn to the L.
into Rooms VIII. and II., and then to the R. into Room I., as above.
This is the handsomer entrance. Much of the sculpture has great merit:
but being purely modern, it does not fall within the scope of these
Historical Guides.)
Begin in the middle of the wall, with No. 170, =**Hubert van Eyck=: the
two outer upper shutters from the =Adoration of the Lamb= at Ghent,
representing Adam and Eve, whose nudity so shocked Joseph II. that he
objected to their presence in a church. These fine examples of the
unidealized northern nude are highly characteristic of the Van Eycks'
craftsmanship. The Adam is an extremely conscientious and able rendering
of an ordinary and ill-chosen model, surprisingly and almost painfully
true in its fidelity to nature. The foreshortening of the foot, the
minute rendering of the separate small hairs on the legs, the
large-veined, every-day hands, the frank exhibition of the bones and
sinews of the neck, all show the extreme northern love of realism, and
the singular northern inattention to beauty. Compare this figure with
the large German panels on a gold ground in the corners diagonally
opposite (No. 624), if you wish to see how great an advance in truth of
portraiture was made by the Van Eycks. The Eve is an equally faithful
rendering of an uninteresting model, with protruding body and spindle
legs. Above, in the lunettes, are the Offerings of Cain and Abel, and
the Death of Abel, in grisaille. The _backs of the shutters_ will be
opened for you by the attendant. They exhibit, above, two Sibyls, with
scrolls from their prophecies; below, the central portion of the
Annunciation in the total picture, with a view through the window over
the town of Ghent, and the last words of the angelic message, truncated
from their context. This portion of the picture is, of course, only
comprehensible by a study of the original altar-piece at Ghent.
[Illustration: THE PICTURE GALLERY AT BRUSSELS.]
Continue now along this wall to the R. of the Adam and Eve.
24. J. Gossart, called Mabuse (1470-1541), triptych with a
=Glorification of the Magdalen=, given by a special votary. The _central
panel_ contains the chief event in her history—the Supper at the House
of Simon the Pharisee. The host and one guest are admirably represented
by Flemish portraits, exquisitely robed, and reproduced in marvellous
detail. The figure of the Christ is, as usual, insipid. Beneath the
table, the Magdalen, as central figure, with her alabaster box of
ointment, kisses the feet of Christ. To the right, Judas, with his
traditional red hair, and bearing the purse, asks, with a contemptuous
gesture, Why this was not sold and given to the poor? In the background
are the Apostles. Conspicuous amongst them is the conventional round
face of St. Peter. The whole scene takes place in a richly decorated
interior, with charming colouring and a finely rendered clock, curtain,
and other accessories. Gossart visited Italy, and was one of the
earliest Flemings to be influenced by the Italian Renaissance. You will
not overlook the half-Gothic, half-Renaissance architecture, nor the
chained squirrel, nor the semi-grotesque episodes in the background,
very domestic and Flemish. (Moses above the Pharisee's head marks his
devotion.)
The _left panel_ has another principal event in the Magdalen's life, the
Resurrection of Lazarus. Here also the Christ is insipid, but the Peter
behind him, in a green robe, is finely characterized; and the John,
affected. Beside are the Magdalen (same dress as before) and Martha,
with a group of women and bystanders in singular head-dresses. In the
background rises a very ideal Bethany. The _right panel_ represents the
kneeling donor (an unknown Premonstratensian abbot); on his book is
written, "Mary Magdalen, pray for us." Above him is seen the floating
figure of the Magdalen, clad only in her own luxuriant hair, and raised
aloft by angels from her cave, the Sainte Baume, in Provence, to behold
the Beatific Vision. The background has Stations of the Cross, actually
copied (with the rest of the landscape) from those at the Sainte Baume,
which Gossart must have visited at his patron's instance. On the _backs
of the wings_, yet another scene in the life of the Saint, Christ and
the Magdalen in the Garden. All this triptych is finely modelled and
well-.
552. Three panels attributed to =Roger van der Weyden=, of Tournai, town
painter of Brussels, and teacher of Memling—a highly symbolical and
religious master. Scenes from the Life of the Virgin. In the _centre_,
the Presentation of the Virgin in the Temple. The foreground is occupied
by St. Joachim and St. Anna, parents of the little Virgin, who is seen
mounting the regulation fifteen steps of the Temple, assisted by a
somewhat unusual angel. At the head of the steps stands the High Priest.
Within, the Virgins of the Lord are seen reading. To the right, still in
the same panel, is the Annunciation, with the usual features, angel L.,
Madonna R., _prie-dieu_, bed, Annunciation lily, and arcade in the
foreground. The _left panel_ has the Circumcision; and the _right_,
Christ among the Doctors in the Temple, with some excellent portraits in
the background. (For Van der Weyden's place in art, see Conway; for the
Madonna ascending the steps, _Legends of the Madonna_.)
554. Also attributed to =Roger van der Weyden=: parts of the same
series, Way to Calvary and the Crucifixion. The _first_ has the usual
brutal soldiers and a suffering but not very dignified Christ. (Study
for comparison with others.) Beside the Virgin kneels the donor. The
_second_ has the conventional figures of the fainting Madonna, St. John,
the Magdalen, and the other Maries: sun and moon darkened. In the
distance of both, Flemish towns. (Good trees and landscape.)
105A. Fine Madonna with Child, and an apple by Van Clìve.
159. A crowded Calvary of the German School (late 15th century) with an
emaciated Saviour, writhing and distorted thieves, and rather wooden
spectators. Observe the St. Longinus in armour on the bay horse,
piercing the side of Christ, for comparison hereafter with such later
conceptions as Rubens's at Antwerp. To the L. is the group of the
Madonna, St. John, and the two Maries. The red eyes of St. John are
characteristic of this scene, and descend to Van Dyck. The Maries are
unmitigated German housewives. The Magdalen embraces the foot of the
Cross. On the right are spectators and a brawl between soldiers. The
background is full of characteristic German devils and horrors: also St.
Veronica, Peter and Malchus, Judas hanging himself, etc.
Above it, 523, German School, attributed to Wolgemut, Christ and the
Apostles: gold background. Very flavourless: shows the tendencies from
which the Van Eycks revolted.
517. Roger van der Weyden: head of a Woman Weeping. Perhaps a portion of
a large composition, or a study for one. More likely, a copy by a pupil.
Much damaged.
Now, return along the same wall, beyond the great Van Eyck in the
centre.
335. =Bernard van Orley= (transitional). Triptych (sawn in two), with
the Patience of Job inside, and Lazarus and Dives outside. In the
_centre panel_, the house falling upon the sons of Job. In the
background, Job and his comforters: his house in flames, etc. _Left
panel_, the flocks and herds of Job driven off by the Sabeans, with
Satan before the Almighty at the summit. _Right panel_, Job in his last
state more blessed than formerly: his comforters ask him to intercede
for them. Beyond this again, the outer shutters (the panels having been
sawn through): _extreme left_, Lazarus at the Rich Man's gate; above,
his new-born soul borne aloft to Heaven. Below, cooks, servants, etc.
_Extreme right_, the Rich Man dying, attended by his physician (compare
the Dropsical Woman by Gerard Dou in the Louvre). Below, Dives in
Torments (in a very Flemish Hell) calling to Lazarus. Above, Lazarus in
Abraham's bosom. This is a good characteristic example of the
=transitional period= between the early and later Flemish art, greatly
influenced by the Italian Renaissance. Van Orley travelled in Italy, and
imitated Raphael in composition and drawing.
Beyond it, attributed to Roger van der Weyden, 553 (3 panels arbitrarily
placed together). In the _centre panel_, two subjects. Left, the
Nativity, elements all conventional: ruined temple, shed, ox and ass
(extremely wooden), and St. Joseph in background. (He frequently bears a
candle in this scene in order to indicate that the time is night.)
Right, the Adoration of the Three Kings, old, middle-aged, young, the
last a Moor. St. Joseph examines, as often, the Old King's gift. Note
his costume; it recurs in Flemish art. _Left panel_, Joseph of Arimathea
with the Crown of Thorns, Nicodemus with the three nails, St. John, and
the three Maries at the sepulchre. _Right panel_, Entombment, with the
same figures: the Crown of Thorns and nails in the foreground. Great
importance is always attached to these relics, preserved in the Sainte
Chapelle, and at Monza, near Milan.
350. =Patinier.= Repose on the Flight into Egypt, with fine landscape
background.
122. =Cranach= the Elder. Hard portrait of a very Scotch looking and
Calvinistic Elder.
10. Amberger; German School, 16th century; excellent portrait of a
gentleman; good beard.
569. Fine portrait of a man, by unknown artist. Flemish School, 16th
century.
Above these, 618, Flagellation and Ascension, German School, with gilt
backgrounds.
The _skied pictures_ on this wall are only interesting as specimens of
the later transitional period; when Flemish Art was aiming ill at
effects unnatural to it.
Continue along the wall in the same direction.
By the door, 557, *portrait of Johanna of Spain (the Mad), mother of
Charles V.: fine 15th century work.
531 and 532. Excellent old Flemish portraits.
Above these, 541, Scenes from the Life of the Virgin, with a donor. L.,
Nativity. Note the conventional elements. R., Circumcision. Above, Angel
and patron saints.
557. *Portrait of Philippe le Beau, father of Charles V., companion to
his wife opposite. Observe the collar of the Golden Fleece, and the
united arms of Spain, Burgundy, etc., on his doublet. These portraits
were originally the wings of a triptych.
544. A Holy Family and St. Anne, with the donor, a Franciscan monk, by a
feeble imitator of Memling.
334. Tolerable portrait of a doctor, by Bernard Van Orley.
Next it, unnumbered, *Virgin and child. Gérard David. Our Lady feeds
Christ with a characteristic Flemish wooden spoon.
348. Patinier, a painter chiefly memorable for his landscapes (of which
this is a poor example). St. Jerome in the Desert, beating his breast
with a stone before a crucifix. Beside him, his cardinal's hat and lion.
Not a good example of the master.
641. Holbein the Younger. Portrait of *Sir Thomas More.
Above, 575. Triptych, Flemish School, early 16th century. _Centre
panel_, Miracle of St. Anthony of Padua and the Mule. (The Saint,
carrying the Host, met a scoffer's mule, which knelt till it passed.)
Above, St. Bonaventura, attired as bishop, praying. These must be the
chief objects of the donor's devotion: they are also represented on the
outer wings. _Right_ and _left_, the donor (whose name was Tobias), with
his personal patron, St. Raphael the Archangel (accompanying the young
Tobias), and his wife, with St. Margaret and the Dragon. (For Tobias and
the Fish, see Book of Tobit.)
174, skied, is a Last Judgment by Floris, also transitional and useful
for comparison with others elsewhere. To R. and L., the Fall of the
Damned and the Just Ascending recall early examples at Bruges.
534. Triptych of the Flemish School (Hugo van der Goes?); _Centre
panel_, Assumption of Our Lady. Round the empty tomb are gathered the
apostles; conspicuous among them, St. Peter with a censer, and St.
James. Above, Our Lady taken up in a glory by Christ and the Holy Ghost,
represented as like Him. In the background, her Funeral, St. Peter, as
Pope, accompanying. Note the papal dress of St. Peter; St. James holds
the cross as Bishop of Jerusalem. _Left wing_, the chief donor,
accompanied by his guardian angel and two of the apostles, one of whom
holds St. Peter's tiara, as if part of the main picture. In the
background, St. Thomas receiving the Holy Girdle from an Angel, a common
treatment in Flemish art, though Italians make him receive it from Our
Lady in person. (See my _Guide to Florence_.) _Right wing_, donor's son
and wife, with guardian angel. This triptych closely resembles No. 535
(which see later), except that that picture is in one panel, instead of
three. I think 535 must have been painted first, and this taken from it,
but made into a triptych; which would account for the unusual flowing
over of the main subject into the wings.
419. Martin Schongauer (of Colmar, a German largely influenced by Roger
van der Weyden), *Ecce <DW25>, painted like a miniature.
349. Patinier: another Repose on the Flight into Egypt. Observe
persistence of the main elements. Notice in particular, as compared with
the similar picture (350) close by, the staff, basket, etc., in the R.
foreground.
546. School of Memling, perhaps by the master: a Bishop preaching: M.
Fétis thinks, exhorting the Crusade in which Pope Nicholas V. wished to
interest the princes of Europe after the fall of Constantinople.
621. School of Dürer: Fine and thoughtful portrait of a man, perhaps
Erasmus.
Above it, 577, Flemish triptych (school of Van der Weyden) of the
Adoration of the Magi, the elements in which will by this time be
familiar to you. Right and left, Adoration of the Shepherds and
Circumcision. The exceptional frequency of the subject of the Adoration
of the Magi in the Low Countries and the Rhine district is to be
accounted for by the fact that the relics of the Three Kings are
preserved in Cologne Cathedral, and are there the chief object of local
cult.
At the corner,
84 and 255, two good portraits by the German de Bruyn (early 16th
cent.). Transitional: show Italian influence.
Between them, 619, unknown German, Wedding Feast at Cana. That you may
have no doubt as to the reality of the miracle, a servant is pouring
water into the jars in the foreground. He is much the best portion of
the picture. Behind are Christ, St. John, and Our Lady. Next to them,
the bride and bridegroom. (Compare the Gerard David in the Louvre.)
Above it, 624, a very quaint St. George and St. Catherine, early German
School, with gold background. St. George is stiffly clad in armour, and
painfully conscious of his spindle legs, with a transfixed dragon and
broken lance at his feet. St. Catherine looks extremely peevish, with a
Byzantine down-drawn mouth: she holds the sword of her martyrdom, and
has a fragment of her wheel showing behind her. Her face is highly
characteristic of the severity and austerity of early German art.
Companion piece (624) at opposite corner.
Now proceed to the next wall.
539. Fine portraits of a donor and his wife, with their patron saints,
Peter and Paul. The tops of both have been sawn off.
549 and 643. Two Madonnas. Not very important.
545. Between them, **unknown German Master (Lafenestre says, Flemish).
Panel with Our Lady and Virgin Saints, what is called a "=Paradise
Picture=," apparently painted for a church or nunnery in Cologne, and
with the chief patronesses of the city churches or chapels grouped
around in adoration. Our Lady, with her typical German features, sits in
front, in a robe of blue, before a crimson damask curtain upheld by
angels. Her face is sweetly and insipidly charming. She holds a regal
court among her ladies. In front of her kneels the Magdalen, with her
long hair and pot of ointment. To the L., St. Catherine of Alexandria,
crowned as princess, and with her wheel embroidered in pearls on her red
robe as a symbol. The Infant Christ places the ring on her finger.
Further L., St. Cecilia with a bell, substituted in northern art (where
the chimes in the belfry were so important) for the organ which she
holds in Italy. Then, St. Lucy, with her eyes in a dish, and St.
Apollonia, holding her tooth in a pair of pincers. In front of these
two, in a richly brocaded dress, and beautiful crown, St. Ursula, the
great martyr of Cologne, with the arrows of her martyrdom lying at her
feet. To Our Lady's right, St. Barbara, in a purple robe trimmed with
ermine and embroidered with her tower (of three windows), offers a rose
to the Infant. Her necklet is of towers. As usual in northern art, she
balances St. Catherine. Beside her kneels St. Agnes, in red, with her
lamb, and her ruby ring: beyond whom are St. Helena with the cross
(wearing a simple Roman circlet), St. Agatha, holding her own severed
breast in the pincers, and St. Cunera with the cradle and arrow, one of
the martyred companions of St. Ursula. In the background, the True Vine
on a trellis, the garden of roses ("is my sister, my spouse"), and a
landscape of the Rhine, in which St. George kills the dragon. This is a
particularly fine composition of the old German School.
65 and 66. **Dierick Bouts: Two companion panels, life-size figures,
known as =the Justice of the Emperor Otho=, and painted for the
Council-Room of the Hôtel-de-Ville at Louvain, as a warning to
evil-doers, perjurers, or unjust magistrates. (Compare the Gerard David
of the Flaying of Sisamnes in the Academy at Bruges.) It is first
necessary to understand _the story_. During the absence of the Emperor
Otho in Italy (according to tradition), his empress made advances to a
gentleman of the court, who rejected her offers. Piqued by this rebuff,
the empress denounced him to Otho on his return as having attempted to
betray her honour. Otho, without further testimony, had the nobleman
beheaded. His widow appeared before the Emperor's judgment-seat, bearing
her husband's head in her hands, and offered to prove his innocence by
the ordeal of fire. She therefore held a red-hot iron in her hand
unhurt. Otho, convinced of his wife's treachery by this miraculous
evidence, had the perjured empress burned alive. The _first panel_, to
the R., represents the scene in two separate moments. Behind, the
nobleman, in his shirt and with his hands tied, walks towards the place
of execution, accompanied by his wife in a red dress and black hood, as
well as by a Franciscan friar. In the foreground, the executioner
(looking grimly stern) has just decapitated the victim, and is giving
the head to the wife in a towel. The headless corpse lies on the ground
before him. The neck originally spurted blood; flowers have been painted
in to conceal this painful element. All round stand spectators, probably
portraits of the Louvain magistrates, admirably rendered in Bouts's dry
and stiff but life-like manner. Behind them, within a walled garden
belonging to a castle in the background, stand the Emperor with his
sceptre and crown, and the faithless Empress. Good town and landscape to
the L. The _second panel_, to the L., separated from this by a large
triptych, represents the nobleman's wife appearing before the enthroned
Otho. In her right hand she holds her husband's head; with her left she
grasps the red-hot iron, unmoved. The brazier of charcoal in which it
has been heated stands on the parti- marble floor in the
foreground. Around are several portraits of courtiers. Behind is
represented the scene of the empress burning, which closes the episode.
I need not call attention to the admirable painting of the fur, the
green coat, Otho's flowered red robe, the dog, the throne, and all the
other accessories. This is considered Dierick Bouts's masterpiece. (Go
later to Louvain to complete your idea of him.)
Between these two pictures are arranged =five of the finest works= in
the collection.
292 and 293. =Memling=: **Portraits of William Moreel (or Morelli),
Burgomaster of Bruges, and his wife Barbara, the same persons
(Savoyards) who are represented in the St. Christopher triptych in the
Academy at Bruges. Their daughter is the Sibyl Sambetha of the St.
John's Hospital. Both portraits, but especially the Burgomaster's, are
good, hard, dry pictures.
515. Memling: **Triptych: perhaps painted in Italy (if I permitted
myself an opinion, I would say, doubtfully by Memling). At any rate, it
is for the Sforza family of Milan. _Central panel_, the Crucifixion,
with Our Lady and St. John. Beautiful background of a fanciful
Jerusalem. Sun and moon darkened. In the foreground kneel Francesco
Sforza in armour, his wife Bianca Visconti, and his son Galeazzo-Maria.
Behind the duke, his coat of arms. _Left panel_: the Nativity. In the
foreground St. Francis with the Stigmata, as patron saint of Francesco,
and St. Bavon with his falcon. _Right panel_: St. John the Baptist, as
patron saint of Giovanni Galeazzo. Below, St. Catherine with her sword
and wheel, and St. Barbara with her tower, two charming figures. I do
not know the reason of their introduction, but they are common pendants
of one another in northern art. You can get an attendant to unfasten the
_outer wings_ of the triptych for you, but they are not important. They
contain, in grisaille, L., St. Jerome and the lion; R., St. George and
the dragon. (The presence of St. Bavon in this enigmatic picture leads
me to suppose it was painted for a church at Ghent. But what were the
Sforza family doing there? Perhaps it has reference to some local
business of the Sforzas in Flanders.)
190. =**Roger van der Weyden=: Portrait of Charles the Bold of Burgundy,
wearing the Golden Fleece. An excellent and characteristic piece of
workmanship. The arrow has a meaning: it is the symbol of St. Sebastian,
to whom (as plague-saint) Charles made a vow in illness, and whom ever
after he specially reverenced.
294. Memling: **Portrait of an unknown man, which may be contrasted for
its comparative softness of execution with the harder work of his master
beside it. Above these:—
211. Triptych, by Heemskerck (early Dutch School), representing,
_Centre_, the Entombment, Christ borne, as usual, by Nicodemus and
Joseph of Arimathea. In front, the crown of thorns. Behind, the
Magdalen; then the Madonna and St. John, the two Maries, and an unknown
man holding a vase of ointment. _L. and R._ the donor and his wife, with
their patron saints Peter and Mary Magdalen (keys, box of ointment).
542. =*Dierick Bouts= of Louvain: The Last Supper. A fine and
characteristic example of the town painter of Louvain. The faces are
those of peasants or small bourgeois. To the right are the donors,
entering as spectators: their faces are excellent. Judas sits in front
of the table. The Christ is insipid. Note the admirable work of the
pavement and background. The servant is a good feature. If you have
Conway with you, compare this picture with the engraving of the very
similar one by Bouts at Louvain, (p. 277:) only, the architecture there
is Gothic, here Renaissance.
139. *Descent from the Cross (Van der Weyden or his school). Notice the
white sheet on which the body is laid, as later in the great Rubens.
Nicodemus and Joseph of Arimathea support the body, St. John and one of
the Maries holds the fainting Madonna. Left, the Magdalen, with her long
hair. By her feet, her box of ointment. Close beside it, the nails,
hammer and pincers. (M. Lafenestre, following Bode, attributes this
picture to Petrus Christus, but with a query.)
29. Pleasing transitional Madonna, School of Van Orley, somewhat Italian
in feeling, in a pretty arcade, with nice landscape background.
Above these, 109, a triptych, by Coninxloo. _Centre_, Family of St.
Anne. Interesting for comparison with the great Quentin Matsys in the
centre of the room. L., Joachim's offering rejected in the Temple (small
episodes behind). R., the death of St. Anne. Come back to the central
panel after you have viewed the Quentin Matsys. (The component
personages are explained there.)
543. Tolerable triptych, Flemish school, representing the events of the
Infancy. _Centre_, Adoration of the Shepherds, with the usual
conventional features (ruined temple, shed, ox and ass, etc.), and St.
Joseph holding his candle, as often, to indicate night time. _Left_,
Annunciation, with the usual position of the angel reversed. Otherwise,
the portico and other features persist. Compare the great Van Eyck at
Ghent, from which some elements here are borrowed. _Right_, the
Circumcision. Symbolical figure of Moses on altar, full of the symbolism
of Van der Weyden's school. (Outer shutters, uninteresting, SS.
Catharine and Barbara.)
585. Above it, good family group of a donor and his sons, with St.
George; and his wife and daughters, with St. Barbara. (The crucifixes
mark monks and nuns.)
At the corner, 624, German school. St. Mary Magdalen and St. Thomas, on
gold background. Companion piece to 624. At opposite end.
626. School of Martin Schongauer: Christ and the Magdalen in the house
of the Pharisee. Very contorted. Compare with the Gossart.
Above it, 106, Flemish school. Mass of St. Gregory, with the Crucified
Christ appearing on the altar. (Recall the Pourbus at Bruges.) A most
unpleasant picture. Behind, are the elements of the Passion. L., the
donors; R., Souls in Purgatory, relieved by masses. Many minor episodes
occupy the area.
On either side of it, *27, 28, beautiful soft-toned German portraits
(? by Beham) of two children, Maximilian II. and his sister Anne of
Austria.
563. Lombard: Unimportant picture, meaninglessly described as Human
Misfortunes. It seems to commemorate an escape from shipwreck and from
plagues by the same person. _L. panel_: A ship sinking; a man saved on
the shore. In the background, under divine direction of an angel, he
finds his lost gold in a fish's body. _R. panel_, He lies ill of plague,
while above is seen the miracle of St. Gregory and the Angel of the
Plague (Michael) sheathing his sword on the Castle of St. Angelo.
540. Virgin and Child. Attributed without much certainty to Petrus
Christus.
535. Good unknown Flemish picture of the *Assumption of Our Lady
(closely resembling No. 534, which see again). The empty tomb stands in
the midst, with lilies; around, St. Peter and St. James, and the other
apostles; above, Our Lady ascending, borne by a duplicated figure of
Christ (one standing for the Holy Ghost), in an almond-shaped glory. R.,
Her Funeral, with St. Peter wearing the triple crown; L., St. Thomas
receiving the girdle from an angel. Compare with 534, which Lafenestre
judges to be the work of a different artist.
567. Good portrait attributed to Bernard Van Orley.
596. Six panels: Flemish School. Ornate, but not interesting. (1) The
Lord creating Eve; in the background the Temptation. (2) Abraham, Sarah,
and Isaac; in the background in three successive scenes, Abraham's
Sacrifice. (3) Noah and his Family with the Ark. (4) Esau asks the
Blessing of Isaac. (5) Meeting of Jacob and Esau. Note the grotesquely
urban conception of the Semitic nomads. (6) The Nativity.
559. Attributed to Van Orley. Pietà, with the usual group, and family of
donors. Interesting as a work of transition.
Above it, 580. Triptych, with Descent from the Cross, Flemish school.
Usual figures; identify them. On the wings, L., Agony in the Garden,
Kiss of Judas, Peter and Malchus: R., The Resurrection. Noli Me Tangere,
Disciples at Emmaus, etc.
107. P. Coecke, 16th century: A Last Supper. Only interesting as showing
transition. Compare with Dierick Bouts.
Above it, 300. Patinier: Dead Christ on the knees of the Virgin (Our
Lady of the Seven Sorrows), painfully emaciated. A sword pierces Our
Lady's breast (and will recur often). Around it, the rest of the Seven
Sorrows. Note the landscape, characteristic of the painter.
12. (Old number; no new number.) Coninxloo: Joachim and Anna, with the
rejected offering. From them, a genealogical tree bears the Madonna and
Child. L. and R., the Angel appearing to Joachim, and Joachim and Anna
at the Golden Gate. (Read up the legend.) Curious architectural setting.
301. Good portrait by an unknown (transitional) Fleming (Van Orley?),
probably of a lawyer; the charters seem to indicate a secretary of
Maximilian and Charles V.
The place of honour in the centre of the room is occupied by 299, a
magnificent =**triptych= by =Quentin Matsys=, one of the noblest works
of the _transitional school_, strangely luminous, with very
characteristic and curious colouring. It represents the favourite
Flemish subject of =the Family of St. Anne=. (It was painted for the
Confraternity of St. Anne at Louvain, and stood as an altar-piece in the
church of St. Pierre.) _Central panel_: An arcade, in the middle arch of
which appears St. Anne, in red and purple (throughout), offering grapes
to the Divine Child, who holds a bullfinch, and is seated on the lap of
Our Lady. R., Mary Salome, with her two sons James and John. L., Mary
Cleophas, with her sons James the Less, Simon, Thaddæus, and Joseph the
Just. Behind the parapet, beside St. Anne, her husband Joachim; and
beside Mary Salome, her husband Zebedee. Beside Our Lady, her husband
Joseph; beside Mary Cleophas, her husband Alphæus. Beautiful blue
mountain landscape. _L. panel_: The angel appearing to Joachim, in a
magnificent blue landscape. Joachim's dress is constant. The angel's
robe is most delicious in colour. _R. panel_: The Death of St. Anne,
with Our Lady and the other Maries in attendance. Behind, their
husbands. The young Christ gives the benediction.
Now, go round to the _back of the picture_, to observe the _outer
wings_. L., St. Joachim driven from the Temple by the High Priest R.
(chronologically the first), Joachim and Anna (much younger), making
their offerings (on marriage) to the High Priest in the Temple. (Same
High Priest, younger; same dresses.) The portrait behind recalls the
earlier Flemish manner; otherwise, the work is full of incipient
transition to the Renaissance. Little episode of Joachim and Anna
distributing alms in the background. (When the triptych is closed, this
wing comes in its proper place as first of the series.)
191. =Jan van Eyck=: (attribution doubtful; probably a later artist,
perhaps Gerard David): The Adoration of the Magi. Another good example
of this favourite Flemish subject. In the foreground, the Madonna and
Child: one of Van Eyck's most pleasing faces (if his). Then, the Old
King, kneeling; the Middle-aged King, half-kneeling; and the Young King,
a Moor, with his gift, behind. (The Old King in such pictures has almost
always deposited his gift.) In the background, Joseph, and the retinue
of the Magi. Ruined temple, shed, ox, ass, etc., as usual.
291. Dierick Bouts: *Martyrdom of St. Sebastian. Characteristic peasant
face; admirable cloak and background.
Now go into the next hall, marked
ROOM II.
on the plan. This contains mainly German and Flemish pictures of the
=transition=.
Right of the door, 338. Very Raphaelesque Holy Family by Bernard Van
Orley, showing in the highest degree the Italian influence on this
originally quite Flemish painter.
Above it, 92 and 92A. Portraits of the Micaul family.
105. J. Joest: St. Anne enthroned, Joseph, Our Lady, the Infant. Early
transitional.
193. Jean Gossaert, Adam and Eve. Good later Flemish nude.
50. J. Bosch: Appalling Flemish Temptation of St. Anthony, with perhaps
the silliest and most grotesquely repulsive devils ever painted.
2. Aertsen: *The Dutch Cook. A famous picture, showing well the earlier
stages of Dutch _genre_ development.
217. Van Hemessen: _Genre_ piece, absurdly given the name of The
Prodigal Son, by a sort of prescription, but really a Flemish tavern
scene of the sort which afterwards appealed to Dutch artists. A
characteristic work: transitional, but with good humorous faces,
especially to the right. Painters still thought all pictures must
pretend to be sacred.
591. German Adoration of the Magi. A fragment only.
603. Herri met de Bles: The Temptation of St. Anthony. Figures and
landscape show Italian influence.
336. Transitional Adoration of the Shepherds. Observe the growing
Renaissance feeling and Italian influence.
247, 248. Excellent portraits by Adrien Key.
41. Lancelot Blondeel: St. Peter enthroned as Pope: in one of his usual
extravagant architectural frameworks. In circles above, his Imprisonment
and Crucifixion.
Close by, unnumbered, two excellent portraits.
81. P. Brueghel the Younger: absurdly called The Census at Bethlehem. In
reality a Flemish winter scene.
318. Sir Anthony More: *Portrait of the =Duke of Alva=, with the firm
lips and cruel eyes of the ruthless Spaniard. One understands him.
359. Good portrait by Pourbus of a plump and well-fed Flemish gentleman.
80. P. Brueghel the Younger: Described as the Massacre of the Innocents.
Flemish winter. The beginning of _genre_ painting.
Most of the pictures skied above these are of some interest for
comparison with earlier examples of the same subjects.
565. Unknown French portrait of Edward VI. of England. Hard and dry and
of little artistic value.
566. Tolerable Flemish portrait of Guillaume de Croy (Golden Fleece).
573. Good Flemish portrait of a woman, dated 1504.
555. Flemish school: Annunciation. Chiefly interesting for its
conventional features, and its very quaint figure of St. Mary of Egypt,
with her three loaves, in the R. panel.
124B. (Old number; no new number given.) Unusual combined picture of St.
Jerome, uniting the subjects usually known as St. Jerome in the Desert
and St. Jerome in his Study.
622. Fine German portrait of the early 17th century.
316. Good strong portrait, by Sir Anthony More, of Hubert Goltzius.
123. =Cranach= the Elder (German 16th cent.): *Adam and Eve. Fine
specimens of the later northern nude of the early Renaissance,
interesting for comparison with the cruder realism of Van Eyck. As yet,
however, even the figure of Eve has relatively little idealism or
beauty. Excellent stag in the background.
361. A good Pourbus. Beyond the door, 536, Flemish school: (Hugo Van der
Goes?). Donor, a lady in a nun's dress (?), with her name-saint, St.
Barbara, bearing her palm as a martyr: in the background, her tower with
the three windows. To balance it, 536, Her brother (?) or husband, with
his patron, St. James. (Staff and scallop-shell.)
Above, 84, Triptych by Jan Coninxloo of the History of St. Nicholas.
(The wings are misplaced.) _R. wing_ (it should be L.), St. Nicholas,
three days old, stands up in his bath to thank God for having brought
him into the world. _Central panel_, the young St. Nicholas enthroned as
Bishop of Myra. _L. wing_ (should be R.), The Death of St. Nicholas,
with angels standing by to convey his soul to Heaven. A good
transitional Flemish picture. Beneath, tolerable portraits.
361A. A late Flemish Virgin, with portrait of the donor and St. Francis
receiving the stigmata.
572. Sir Anthony More. Portrait. Above it, 595, an Entombment, where
note again the conventional grouping.
337. Wings of a triptych by Bernard van Orley. The centre is missing.
L., Martyrdom of St. Matthias. R., The Doubting Thomas. In the
background, Lazarus and Dives, and other episodes. Renaissance
architecture.
4. Van Alsloot: The Procession of the Body of St. Gudula at Brussels: of
the Spanish period, with the guilds named. Interest purely
archæological. Each guild carries its mace and symbol. (The second part
comes later.)
586. Above this, skied, are four good female saints, transitional, named
on labels.
574. Portrait, of the school of Van Orley: lady with a pink, pleasing.
Italian influence is obvious.
505. Portrait of a lady, by M. De Vos. Early 17th century, marking the
latest transitional period. It belongs to a destroyed triptych.
79. Breughel the Elder: St. Michael the Archangel conquering the devils.
A hideous nightmare of a morbid and disordered imagination.
627. Crucifixion, by an unknown German, with small figures of donors,
and Rhine background.
597. German school. Tree of Jesse, of purely symbolical interest.
504. *Portrait by M. De Vos. Probably husband of (and pendant to) the
previous one. It was the other wing of the same triptych.
5. Van Alsloot: Remainder of the Procession of St. Gudula, with a quaint
dragon, and the Maison du Roi in the background. Observe, near the
centre, the personification of the patron, St. Michael: elsewhere are
St. Christopher, Ste. Gudule, etc.
561. Two panels from a triptych attributed to Van Orley. Centre,
missing. L., The Birth of the Virgin. Note this for the conventional
features: St. Anne in bed; attendant feeding her: bath for infant. In
the background presentation of the Virgin in the Temple: Joachim and
Anna below: the Virgin ascending: the High Priest welcoming her: the
Virgins of the Lord by the side. R., Joachim's offering rejected. In the
background, the meeting of Joachim and Anna at the Golden Gate, and the
angel foretelling the Birth of the Virgin. Compare this with the great
Quentin Matsys, observing especially the money falling from the table.
76. Another example of a later Last Judgment.
584. Mostart: Two stories from the life of St. Benedict. (1) The miracle
of his dinner. (2) As a youth, he mends by prayer the dish broken by his
nurse. (See Mrs. Jameson's _Monastic Orders_.)
Now pass through Room [VIII.], containing chiefly late Italian and
French pictures (which neglect for the present) and go on into
CORRIDOR A,
to the L., overlooking the Sculpture Gallery. This takes us at once into
the =Later Flemish school= of Rubens and his followers, whose works fill
all these large corridors, which are admirably adapted for them. Begin
to the R. of the door. (Remember that I do not attempt criticism, but
confine myself to historical indications.)
230. Jordaens: Fine landscape, with city to the right. As yet, however,
landscape dare not stand entirely on its own merits. Therefore, we have
here in the foreground figures of Eleazar and Rebecca at the well, which
retain the tradition that pictures must have some sort of sacred
purpose.
194A. Unknown. Interior of a picture-gallery, with well-known pictures.
L. of the door. 465. Van Thulden: Flemish Wedding Feast. Landscape is
beginning to triumph now; it gets rid of all pretence of sacredness, but
still retains small figures in the foreground. Landscape for landscape's
sake is hardly yet dreamed of.
135. De Crayer, one of the best imitators of Rubens: *Adoration of the
Shepherds, in the master's manner.
179A. Good Still Life by Fyt.
379. =Rubens=: **Coronation of the Virgin by God the Father and the Son,
the Holy Ghost hovering above in a glory. This altar-piece, for an altar
of Our Lady, is a magnificent specimen of the master's rich and luminous
colouring. The crimson robe of the Christ, the blue and lilac harmony on
the Madonna, and the faint yellow of the Father's robe, are admirably
contrasted. So are the darkness of the lower clouds and the luminosity
of the upper region, recalling Titian's famous Assumption at Venice. The
little boy angels are sweet and characteristic. Here you may begin to
appreciate the force, the dash, the lavish wealth of Rubens. (According
to Rooses, however, the work of a pupil, touched up by the master.)
Then, unnumbered, Jordaens: *Susannah and the Elders: a very Flemish and
matronly Susannah. The nude of Rubens, without the glorious touch of the
master: but a good picture.
392. Study by the same for the ceiling in Whitehall.
386. Fine *portrait by Rubens of a fair man (J. C. de Cordes). "Inferior
work." (Fromentin.)
390. Rubens: **Charming little Madonna and Child (called "Our Lady of
the Forget-me-not"), in a garden of roses (the landscape by J.
Brueghel). One of his best small pictures, in a careful style.
387. Rubens: *Portrait: Wife of the last: in his finest and richest
portrait manner, which contrasts in many ways with his larger and freer
allegorical style. (Fromentin thinks poorly of it.)
389. In the corner, four Fine *Heads of <DW64>s, a study for the Magi,
by Rubens. Not caricatured, but full of genuine <DW64> character.
Above it, 219. An Adoration of the Magi by Herreyns: Interesting only as
showing the persistence of the school into the 18th century.
235. Jordaens: *An Allegory of Abundance. Studies from the nude in the
style of the school: meritorious.
Pass the door of the Dutch school. Beyond it, more Still Life,
excellently painted.
240. Next door, Jordaens: *Nymph and Satyr. (This corridor is largely
given up to works by Jordaens, who was a Protestant, and preferred
heathen mythological subjects to Catholic Christian ones.
434. Snyders: 17th century: *Still Life, which now begins to be painted
on its own merits. This last is by the great animal painter of the
Flemish school.
238. Jordaens: Very Flemish *family group, with a somewhat superfluous
satyr. (Subject nominally taken from the fable of the Satyr and the
Wayfarer.)
302. Vandermeulen: View of Tournai and landscape, with the siege by
Louis XIV. introduced for the sake of figures in the foreground.
Above it, 240 (? old number). De Crayer: St. Anthony and St. Paul the
Hermit. Interesting for persistence of the typical figures.
The other pictures in this corridor are, I think, self-explanatory.
Now enter
ROOM III.
Right hand, further corner from door, Still Life by Snyders.
100. Good portrait by Philippe de Champaigne.
387A. Splendid *portrait by Rubens (according to Rooses, by Van Dyck).
This room also contains several fine pictures by Teniers (father or son)
and other late Flemish painters, deserving of attention, but needing no
explanation. (Portraits, etc.) Do not imagine because I pass them by
that you need not look at them.
Now enter
CORRIDOR B.
L. of the door, good works by De Crayer and others.
166. =Van Dyck= (the greatest pupil of Rubens, leading us on to the
later Dutch school). *St. Francis in ecstasy before the Crucifix. From
the Franciscan Capuchin Church in Brussels.
288. P. Meert, good portraits.
165. Companion to 166. Another Franciscan picture by Van Dyck. *St.
Anthony of Padua holding the Infant Jesus. (In neither is he seen to
great advantage.)
In the centre, 378, Rubens: **Assumption, high altar-piece from the
Carmelite Church in Brussels. A fine picture, of Ruben's early period,
smooth of surface and relatively careful, with the apostles looking into
the empty tomb, whence women are picking roses (See _Legends of the
Madonna_). To the R., the youthful figure of St. Thomas, stretching his
hands. Observe the fine contrast of colour between the lower and upper
portions. This is a noble specimen of the master's bold and dramatic
treatment, but without his later ease of execution.
503. *Good portraits, by C. De Vos, of himself and his family.
127. De Crayer: St. Anthony, with his pig and staff, and St. Paul the
Hermit, in his robe of palm-leaves, fed by a raven. In the background,
the Death of St. Paul; two lions dig his grave. R., below, late figure
of donor, seldom so introduced at this period. Jay in the background.
Good landscape.
Enter
ROOM IV.
with landscapes and still life of the later period. Left of the door,
381, Rubens: The Woman Taken in Adultery. 605. Good *family group of the
Van Vilsteren household, attributed to Van Dyck.
466. Interior of a picture-gallery, Teniers.
The room also contains several pictures worthy of note, but too modern
in tone to need explanation. Observe that landscape has now almost
vindicated its right to be heard alone, though figures in the foreground
are still considered more or less necessary.
Now enter
CORRIDOR C.
which contains good pictures of the =later Flemish school=. R. of door,
435, A. Van Utrecht: one of the favourite Dutch kitchen scenes, well
painted.
L. of the door.
95. Philippe de Champaigne: Presentation in the Temple, with
characteristic crude French colouring.
437. Stag Hunt by Snyders.
375. Rubens: *Martyrdom of the local Bishop, St. Lieven. His tongue is
torn out and given to dogs. Very savage pagans; rearing horse; and
characteristic angels, with celestial scene. In Rubens' less pleasing
"allegorical" manner. Plenty of force, but too fiercely bustling.
383. Rubens: Fine portrait of the Archduke Albert.
384. Rubens: companion portrait of the Infanta Isabella, wife of 383.
376. Rubens: *Painfully un-Christian subject: mainly by a pupil,
re-touched by the master: The Saviour about to destroy the World, which
is protected by St. Francis and Our Lady. A strange method by which a
votary seeks to impress his devotion to his own patrons. Behind, burning
towns, murder, etc.
374. Rubens: *The Way to Calvary. (Almost all these large Rubenses are
from High Altars.) In the foreground, the two thieves; then Christ
falling, and a very Flemish and high-born St. Veronica unconcernedly
wiping his forehead. Our Lady faints close by, supported by St. John.
St. Longinus mounted, and Roman soldiers. The composition somewhat
sketchy, but immensely vigorous. A gorgeous pageant, it wholly lacks
pathos.
377. Rubens: **Adoration of the Magi (Altar-piece of the Capuchin Church
at Tournai). One of his noblest works, magnificently and opulently
. The subject was one he often painted. Note still the Three
Kings, representing the three ages and continents, but, oh, how
transfigured! In their suite are Moors and other Orientals. Behind, St.
Joseph with flambeaux, representing the earlier candle. This is a
painting in Rubens's best Grand Seigneur manner—vast, throbbing,
concentrated. He thinks of a Nativity as taking place with all the pomp
and ceremony of the courts which he frequented. Charming pages in the
foreground.
382. Rubens: Venus in the Forge of Vulcan. A made-up picture. Splendid
studies of the exuberant nude by Rubens; with effects of light and shade
in a smithy, added in the late 17th century to make up for a lost
portion.
380. Rubens (much restored): *Christ on the knees of Our Lady. A noble
composition, greatly injured. In the foreground kneels the Magdalen (her
hair falling ungracefully), with the nails and Crown of Thorns. Notice
always her abundant locks. To the R., St. Francis, with the Stigmata,
bends over in adoration (a Franciscan picture). To the L., very fleshy
angels (Antwerp models) hold the instruments of the Passion. White sheet
and dead flesh in their usual strong combination. (Painted for the
Franciscan Capuchins of Brussels.)
The De Crayers, close by, contrast in the comparative crudity of their
colour with the splendid harmonies of the master.
507. Paul de Vos: Horse and Wolves. Full of spirit.
129. De Crayer. The Martyrdom of St. Blaise. Shows him combed with a
wool-carder.
Then flowers, hunting scenes, etc., requiring no comment.
Now pass through Room VII. (with Italian pictures to be considered
later) and enter
CORRIDOR D.
Right of door, nothing that requires comment.
Left of door, 231, Janssens: Our Lady appearing to St. Bruno.
125. De Crayer: *Assumption of St. Catherine, with her wheel and sword.
A fine picture, in which De Crayer approaches very near Rubens. In the
foreground are St. Augustine with the flaming heart; St. Gregory,
habited as Pope; St. Ambrose, and St. Jerome,—the four Doctors of the
Church, with other saints, contemplating devoutly the glory of St.
Catherine.
R. and L. of central door, 136, 137, good saints, by De Crayer. Beneath
them, excellent landscapes.
The remaining pictures in this room can be inspected by the visitor
without need for explanation.
It is interesting to stand by the =balustrade=, here, above the
sculpture gallery, not only for the general outlook upon the handsome
hall, but also to note how the colour of the Rubenses stands out at a
distance among the other pictures.
Now, go on through Room VIII. to Corridor A., reaching on the L.,
ROOM V.
containing the =Dutch Masters=. On these, for the most part, I shall
have little to say. Their landscapes, flower-pieces, and portraits are
admirable, indeed, but they are of the sort which explain themselves at
sight, and need rather for their appreciation critical faculty than
external knowledge. Begin on the L. =of the door=.
282. Nicolas Maes: Good portrait of a 17th century lady.
263. Leerman's Crucifixion, finely executed.
Near it, good landscape or flower-pieces, etc., by Cuyp, De Heem, and
Isaac van Ostade.
448. St. Pierre at Louvain.
525. Good hunting scene by Wouwerman.
279. **Admirable figure of an old woman fallen asleep over her reading,
by Nicolas Maes.
Above it, 284. *Good portrait by Nicolas Maes.
278. *Fine portrait by Luttichuys.
304. Van Mieris: Susannah and the Elders. Frankly anachronistic.
233 is a fine *landscape with cattle, by Karel du Jardin.
147. Van Delen: Excellent architectural piece, with good *portraits in
the foreground, painted in later by Emmanuel Biset.
=End Wall.=
44. Admirable *portrait by Bol.
45. Bol: *Portrait of a lady, probably wife of the last. On either side
310, 311, characteristic tavern scenes by Molinaer.
=Right Wall.=
398. One of Jacob Ruysdael's finest landscapes, with ruined tower.
399. Excellent *sea piece by Jacob Ruysdael, representing the Lake of
Haarlem in a storm. Good foam.
I pass by, on the same wall, many meritorious Dutch works which cannot
fail to strike the observer.
141. Albert Cuyp: *Cows. Excellent.
232. *Delicately luminous piece by Karel du Jardin, "L'Avant-garde du
Convoi."
153. Gerard Dou: **The artist drawing a Cupid by lamplight. One of his
finest studies in light and shade. It should be looked at long and
carefully.
On either side of it, delicate small pieces by Steen, A. van Ostade and
Dietrich.
47. *Good portrait by Bol.
281. *Portrait by Maes. Fine and audacious in colouring. Above it, 403,
good Ruysdael.
Do not imagine because I give little space to the pictures in this room
that they are not therefore important. As works of art, many of them are
of the first value; but they do not require that kind of explanation
which it is the particular province of these Guides to afford.
Now, pass through the small passage to
ROOM VI.
containing works also by the =Dutch masters=, the finest of which are
here exhibited.
=Left of the door.=
364. Van Ravestein, capital portrait.
280. Maes: **Old woman reading.
206. Fruit piece by De Heem. One of his finest.
490. Van der Velde, junior: Shipping on the Zuyder Zee. The Dutch
interest in the sea begins to make itself felt.
345. Portraits by Palamedes, arranged as a musical party.
352 (old number, no new number given). Molyn the Elder: Town fête by
night. Good effect of light.
333. Admirable *portrait by Nicolas Maes.
251 and 250. De Keyser: Two fine portraits of women.
46. Bol: **Excellent portrait of Saskia, wife of Rembrandt.
463. **Exquisite miniature portrait by Ter Burg, which should be
inspected closely.
328. Van der Neer: The Burning of Dordrecht. A lurid small piece.
501. A. de Voys: The Jolly Drinker. Highly characteristic of Dutch
sentiment.
528. Landscape by Wynants.
The other still life and fruit or flower pieces on this wall need no
comment.
=End wall.=
616. Weenix: Dutch lady dressing, with good effects of light and colour.
203. Frans Hals: **Portrait of W. van Heythuysen. One of his finest
works. Broadly executed, and full of dash and bravado.
496. Excellent still life by Jan Weenix.
522. De Witte: Fine architectural church interior.
222. Above it, *peacock and other birds by Hondecoeter, who painted
almost exclusively similar subjects. The solitary feather in the
foreground recalls his famous masterpiece at the Hague.
On each side of this, tolerable portraits by Van der Helst.
36. Fruit and still life by Van Beyeren.
444. *One of Jan Steen's most characteristic pieces of Batavian humour.
A Dutch lover offering affection's gift, in the shape of a herring and
two leeks, to a lady no longer in her first youth. Behind, her
unconscious husband. The painting of every detail is full of the best
Dutch merits, and the tone of the whole frankly repulsive.
=Right wall.=
221. Hobbema: *The Wood at Haarlem. Characteristic Dutch landscape.
202. Frans Hals: **Splendid portrait of Professor Hoornebeck of Leyden.
Extremely vivacious and rapidly handled.
220. *One of Hobbema's most famous mills.
Above this, 19, Storm at Sea, by Backhuysen.
368. Excellent **portrait by Rembrandt.
216. Portrait by Van der Helst. Not in his best manner.
445. A very characteristic and excellent Jan Steen, known as *The
Rhetoricians—that is to say, members of a Literary Club or Debating
Society, one of whom is engaged in reading his prize verses to a not too
appreciative audience outside. Even here, however, Jan cannot omit his
favourite touch of coarse Dutch love-making, with a tavern-girl
introduced out of pure perversity.
357. Paul Potter: *Pigs. Admirably piggy.
=End wall.=
223. More of Hondecoeter's unimpeachable *poultry.
48. Bol: *Portrait of a Mathematician and Anatomist. One of the
painter's masterpieces.
367. **Splendid portrait by Rembrandt ("L'Homme au grand chapeau"). An
excellent and characteristic example of his art. The light and shade,
the painting of the hair, and the masterly handling of the robe are all
in the great painter's noblest manner.
402. Capital *water scene by S. van Ruysdael: a ferry on the Meuse.
224. Hondecoeter. More poultry, this time dead, with realistic nails,
and other little tricks to catch the great public.
=Left wall= (R. of door).
88. Van der Capelle: Calm sea, with excellent fishing boats.
Now, return through Corridors A. and D. to
ROOM VII.
containing the =early Italian pictures=. Few of these are of much value,
and as they are not connected with Flanders or Brabant, I will not
enlarge upon them. Right of door,
631. An early Italian Adoration of the Magi, where you may compare the
Three Kings, Joseph with the gift, the ox and ass, etc., with Flemish
examples.
631 (left) is a characteristic example of St. Francis receiving the
stigmata. Study it for comparison with the Rubens at Ghent, and others.
628. Above is a set of panels containing events in the History of Our
Lady. I give the subjects, running along the top row first, with
necessary brevity: Joachim expelled from the Temple: Warned by the
Angel: Anna warned by the Angel: Meeting of Joachim and Anna at the
Golden Gate: Birth of the Virgin: Presentation of the Virgin in the
Temple: The Nativity: Adoration of the Magi: Christ found in the Temple:
Miracle at Cana: Raising of Lazarus: Death of the Virgin, with Christ
receiving her soul as a new-born baby. All these may be studied as early
examples of the subjects they represent. Above them, 629 and 630: two
Crucifixions of various ages.
140. Good characteristic Carlo Crivelli of *St. Francis with the
stigmata.
3. Adam and Eve. Albani.
Above it, a tolerable Veronese of *Juno scattering wealth into the lap
of Venice, St. Mark's lion beside her.
140. Beautiful Carlo Crivelli of *Our Lady and Child. This picture and
the corresponding one opposite are parts of a large altar-piece, the
main portion of which, a Pietà, is in the National Gallery in London.
415. Vannuchi (_not_ Perugino): Leda and the Swan.
412 is a good portrait of Mary of Austria.
634. A tolerable Marriage of the Virgin.
473. Tintoretto: Portrait of a Venetian gentleman.
474. Another by the same.
ROOM VIII.
opposite, also contains later =Italian= pictures, with a few French.
472 is a Martyrdom of St. Mark, by Tintoretto.
497 is a Holy Family by Paolo Veronese, with St. Theresa and St.
Catherine.
The other works in the room do not call for notice.
PASSAGE leading into ROOM VI.
Skied. 477. Perugino: Madonna and Child, with the infant St. John of
Florence, in a frame of Delia Robbia work. This is one of the best
Italian pictures in this Gallery, but not a good example. Near it,
School of Mantegna, Christ and St. Thomas with St. John the Baptist.
If you want further information about the pictures in the Brussels
Gallery, you will find it in Lafenestre and Richtenberger's _La
Belgique_, in the series of _La Peinture en Europe_.
[Illustration: CATHEDRAL OF BRUSSELS]
_D_. THE CATHEDRAL
[The =Cathedral= of Brussels is dedicated to St. Gudula or Ste.
Gudule, and to St. Michael the Archangel. Ste. Gudule is a holy
person who takes us back to the earlier ages of Christianity
among the Middle Franks. She was a member of the family of Pepin
d'Heristal, the kinsman of Charlemagne, and she died about 712.
She became a nun at Nivelles under her aunt, St. Gertrude. The
only fact of importance known as to her life is that she used to
rise early, in order to pay her devotions at a distant church,
whither she guided her steps by the aid of a lantern. Satan
frequently extinguished this light, desiring to lead her feet
astray, but the prayers of the saint as often rekindled it.
Hence she is usually represented carrying a lantern, with the
devil beside her, who endeavours to blow it out.
In the 10th century, the body of Ste. Gudule was brought to
Brussels from Morseel; and in the 11th (1047), Lambert, Count of
Louvain, built a church on this site above it: but the existing
building, still containing the body of the saint, was not begun
till 1220.
More important, however, than Ste. Gudule, in the later history
of Brussels Cathedral, is the painful mediæval incident of the
=Stolen Hosts=. The Jew-baiting of the 14th century led to a
story that on Good Friday, 1370, certain impious Jews had stolen
16 consecrated Hosts from the Cathedral, and sacrilegiously
transfixed them with knives in their synagogue. The Hosts
miraculously bled, which so alarmed the Jews that they restored
them to the altar. Their sacrilege was discovered by the aid of
an accomplice, and on this evidence several Jews were burned
alive, and the rest banished from Brabant for ever. A chapel on
the site of the synagogue still commemorates the event, and the
Miracle of the Hosts (as it is called) gives rise to several
works of art now remaining in the Cathedral. An annual ceremony
(on the Sunday after the 15th of July) keeps green the memory of
the miraculous bleeding: the identical wafers are then
exhibited. Visit the =interior= between 12 and 4, when the doors
are closed, but will be opened for you by a sacristan in the
South Portal, at a charge of 50 c. per head. You will then be
able to inspect the whole place peaceably at your leisure. Take
your opera-glasses.]
Approach the =Cathedral=, if possible, from the direction of the Grand'
Place. It is built so as to be first seen from this side, and naturally
turns its main West Front towards the older city. Go to it, therefore,
by the street known as the Rue de la Montagne and the short (modern) Rue
Ste. Gudule, which lead straight up to the handsome (recent) staircase
and platform. The building loses much by being approached sideways, as
is usually the case, from the Upper Town, which did not exist at all in
this direction when the Cathedral was built. Consider it in relation to
the nucleus in the valley.
First examine the =exterior=. The accompanying rough plan will
sufficiently explain its various portions.
The =façade= has two tall towers, and a rather low gable-end, with large
West Window. In style, it approaches rather to German than to French
Gothic. Over the _Principal Entrance_ are (restored) figures of the
Trinity, surrounded by angels, with the Twelve Apostles, each bearing
his symbol or the instruments of his martyrdom. Below, on the central
pillar, the Three Magi, the middle one a Moor. High up on the gable-end
is the figure of Ste. Gudule, the human patron, with the Devil
endeavouring to extinguish her lantern. Above her is the other and
angelic patron, St. Michael. (These two figures also occur on the middle
of the carved wooden doors.) At the sides, two bishops, probably St.
Géry and St. Amand. Though the sculpture is modern, it is of interest
from the point of view of symbolism. The _left portal_ has St. Joachim,
St. Anne, and the education of the Virgin. The _right portal_ has St.
Joseph and Our Lady with the Divine Infant.
* * * * *
Now, =go round the building= to the R., to observe its arrangement. You
pass first the _chapels_ or _bays_ of the =S. Aisle=, with
weather-beaten sculpture, and then reach the slightly projecting =South
Transept=. Beyond the South Portal, the =Choir= is hidden by the
addition of a large projecting =chapel= (that of
Notre-Dame-de-Délivrance), whose architecture will be better understood
from the interior. At the East End, you get a good view of the Gothic
=Choir= and =Apse=, with its external chapels and flying buttresses. The
extreme East point is occupied by the ugly little hexagonal rococo
_Chapel of the Magdalen_, a hideous addition of the 18th century. Still
passing round in the same direction, you arrive at a second projecting
=Chapel= (du Saint Sacremént), which balances the first. The best
=general view= is obtained from the North Side, taking in the beautiful
porch of the =North Transept=. (The handsome Louis XVI. building
opposite is the _Banque Nationale_.)
* * * * *
The Cathedral as an interior is disappointing. It contains no pictures
of any importance, and its architecture is less striking within than
without. =The stained glass=, indeed, is famous; none of it, however, is
mediæval. The best windows date only from the High Renaissance; the
remainder are 17th century or modern.
* * * * *
Walk first into the =centre of the Church=, where you can gain a good
idea of the high Choir, with its Apse and Triforium of graceful Early
Gothic architecture, as well as of the short Transepts, the two
additional chapels, R. and L., the Nave and single Aisles, and the great
west window.
* * * * *
Now, begin the tour of the church with the =South Aisle=, to the L. as
you enter. The glass here is modern. It represents the story of the
Stolen Hosts, some of the subjects being difficult to decipher. We see
the Jew bribing a Christian, who removes the Hosts in a monstrance: then
the Christian departing from the Jewish Synagogue with his ill-gotten
gains. The third window I do not understand. After that, we see the Jews
betrayed by one of their number; the Miracle of the Blood, with their
horror and astonishment; the Recovery of the Hosts; and in the =North
Aisle=, their Return to the Church in procession, and the various
miracles afterward wrought by them. I cannot pretend to have deciphered
all these accurately. The =Nave= has the usual Flemish figures of the
Twelve Apostles set against the piers, most of them of the 17th century.
The great =west window= has the Last Judgment, by Floris, a poor
composition, overcrowded with indistinguishable figures.
* * * * *
The =pulpit=, by Verbruggen, is one of the usual unspeakable
abominations of seventeenth century wood-carving. Below are Adam and Eve
driven from Paradise: above, on the canopy, the Virgin and Infant
Saviour wound the serpent's head with the cross: the Tree of Life,
supporting the actual platform, gives shelter to incredible birds and
animals. This ugly object was made for the Jesuits' Church at Louvain,
and given to the Cathedral by Maria Theresa on the suppression of the
Society of Jesus.
* * * * *
Return to the =Transepts=. The window in the _North Transept_ represents
Charles V., kneeling, attended by his patron, Charlemagne, who was a
canonized saint, but who bears the sword and orb of empire. Behind him,
Charles's wife Isabella, with her patroness, St. Elizabeth of Hungary,
holding the crown. This window, erected in 1538, from designs by Bernard
van Orley, was the gift of the Emperor. That in the _South Transept_
represents the Holy Trinity, with King Louis of Hungary kneeling in
adoration, attended by his patron, St. Louis of France. Behind him is
his Queen, Marie (sister of Charles V.), with her patron, the Blessed
Virgin. This window also is by Van Orley.
* * * * *
Now, enter the =Chapel= by the North Transept, that of =the Holy
Sacrament=, erected in 1535-39, in honour of the Miraculous (Stolen)
Hosts, which are still preserved here, and which are carried in
procession annually on the Sunday following the 15th of July. The
windows in this chapel, each of which bears its date above, were placed
in it immediately after its erection, and are the best in the Cathedral.
They exhibit the style of the Transitional Renaissance. Each window
shows, above, the story of the Stolen Hosts, with, below, the various
donors and their patrons. _First window_ as you enter: Above, the
Bribery: below, King John III. of Portugal with his patron, St.
John-Baptist; and Queen Catherine, his wife (sister of Charles V.), with
her patron, St. Catherine, holding her sword of martyrdom and trampling
on the tyrant Maximin: (all by Michael Coxcie). _Second window_: above,
the Hosts insulted in the Synagogue: below, Louis of Hungary, with his
patron, St. Louis; and Marie, his wife (sister of Charles V.), with her
patroness, Our Lady (Coxcie). _Third window_: above, same subject as in
the 3rd of the S. Aisle—perhaps the attack on the Jews: below, Francis
I. of France, with his patron, St. Francis, receiving the Stigmata;
behind him, Eleonora, his wife (sister of Charles V.), with her
patroness, St. Helena (Bernard van Orley). _Fourth window_: above,
Denunciation of the Jews: below, Ferdinand, brother of Charles V., with
his patron, St. Ferdinand; and his wife, Anne, with her patron, St. Anna
(Bernard van Orley). The _end window_ represents the Adoration of the
Holy Sacrament, and of the Lamb that was slain, in a composition
suggested by the Van Eyck at Ghent. Below, to the L. are an Emperor and
Empress (Charles V. and Isabella), a king and queen, and other
representatives of the world secular: to the R. are a pope, a cardinal,
bishops, prophets, and other representatives of the church or the world
ecclesiastical.
* * * * *
Now, proceed to the =opposite chapel=, by the S. Transept, that of =Our
Lady of Deliverance= (Notre-Dame de Délivrance). This chapel was erected
in 1649-53, to balance that in the N. Transept. Its windows, made after
designs by Van Thulden, in 1656, represent the continued decadence of
the art of glass-painting. The subjects are taken from the History of
Our Lady, above, with the donors and their patrons, princes of the House
of Austria, below. Unlike the last, the subjects here begin at the
_inner end_, near the altar. _First window_: the Presentation of Our
Lady in the Temple. She mounts the steps to the High Priest: below are
St. Joachim and St. Anna. _Second window_: The Marriage of the Virgin.
_Third window_: The Annunciation, with the Angel Gabriel and the Dove
descending in a glory. _Fourth window_: The Visitation of Mary to
Elizabeth: the figure of Mary, in its odd hat, taken from the Rubens in
Antwerp Cathedral. The Austrian Princes and Princesses below, in the
insipid taste of the 17th century, have commemorated their own names so
legibly on the bases that I need not enumerate them.
* * * * *
Now, return to the N. Transept, to make the tour of the =Ambulatory=. At
the entrance to the Apse, L., is a colossal statue of the patroness,
Ste. Gudule, with the Devil under her feet. The stained glass of the
Apse is good modern. Notice the fine pillars to your right. The
hexagonal rococo =Chapel of St. Mary Magdalen=, at the end of the Apse,
has modern windows of, L. and R., the two patrons, St. Michael and Ste.
Gudule, the latter with the lantern and Devil: and, Centre, the Trinity.
Exit from the Apse: L., gilded statue of the other patron, St. Michael,
to balance the Ste. Gudule. Beside it, curious wooden Easter Sepulchre,
with Nicodemus, Joseph of Arimathea, the Mater Dolorosa, and the Maries.
Above it, the Risen Christ, with Roman soldiers on the pediment. Fine
view from near this point of the Choir and Transepts.
* * * * *
The high =Choir= has in its Apse stained-glass windows (use your
opera-glass), representing Our Lady, and the patron saints, with various
kings and queens in adoration (middle of the 16th century). The
portraits are (1) Maximilian and Mary of Burgundy: (2) Philippe le Beau,
their son, with his wife, Johanna the Mad, of Castille: (3) Charles V.
and his brother Ferdinand, sons of Philippe: (4) Philip II. of Spain,
son of Charles V., with his second wife. The architecture here is Early
Gothic and interesting.
_E_. THE UPPER TOWN
From the Grand' Place, two main lines of streets lead towards the =Upper
Town=. The first, which we have already followed, runs straight to the
Cathedral; the second, known as the Rue de la Madeleine and then as the
=Montagne de la Cour=, mounts the hill to the =Place Royale=.
The city of the merchants lay about the Hôtel-de-Ville, the Senne, and
the old navigation. The town and court of the =Counts of Louvain= and
=Dukes of Brabant= clustered about the Castle on the high ground
overlooking the Lower City. On this hill, the =Caudenberg=, the Counts
of Louvain built their first palace, close to what is now the Place
Royale. Their castle was burnt down in 1731, but the neighbourhood has
ever since been the seat of the Belgian court for the time
being—Burgundian, Austrian, Dutch, or Coburger. All this quarter,
however, has been so greatly altered by modern "improvements" that
scarcely a relic of antiquity is now left in it, with the exception of a
few mediæval churches.
In spite of the competition of the Central or Inner Boulevards, the
=Montagne de la Cour=, which mounts directly from the Grand' Place to
the Cour (the residence of the Dukes or afterwards of the Emperors and
the Austrian Viceroys), still remains the principal street for shopping
in Brussels. It takes one straight into the =Place Royale=, one of the
finest modern squares in Europe, occupying in part the site of the old
Castle. Its centre is filled by the famous *statue of =Godfrey de
Bouillon= by Simonis: the great Crusader is represented on horseback,
waving his banner, and crying his celebrated cry of "Dieu le veut!" The
unimpressive =Church=, with Corinthian pillars, a crude fresco in the
pediment, and a green cupola, which faces you as you enter, is =St.
Jacques sur Caudenberg=. To R. and L. you open up vistas of the =Rue de
la Régence= and the =Rue Royale=. The former is closed by the huge mass
of the new =Palais de Justice=. The latter ends in the great domed
church of =Ste. Marie de Schaerbeck=.
In order to gain a proper conception of the Upper Town, one of the
best-arranged in Europe, you must take the Place Royale and the Ancienne
Cour (just below it) as your starting-point. The Place, the Park, and
the streets about them were all laid out, under Austrian rule, at the
end of the 18th century (1774) by the architect Guimard, who thus made
Brussels into the handsome town we now see it. Turning to the R. from
the Place Royale, towards the Rue de la Régence, you come first to the
gateway of a courtyard, guarded by sentinels. Disregarding these, push
past them into the court as if the place belonged to you. The quadrangle
you have entered is the site of the old Palace of the Dukes of Brabant,
for which the present building, known as the =Ancienne Cour=, was
substituted by the Austrian Stadtholders in 1731 after the great fire.
The first building to your L. is occupied by the Royal Museum and
Library. The portion of the building at the end of the court, in a
semi-circular recess, contains the MODERN PICTURE GALLERY (open daily
from 10 to 4, free). In this gallery are collected the chief works of
the modern Belgian School of Painters, which the tourist should not omit
to study, but a full description of which lies wholly outside the scope
of these Guide Books.
[This =modern Belgian School= was started in Antwerp, after the
Revolution of 1830. It answered at first to the romantic
movement in France (headed by Delaroche, Géricault, and others:)
but the Belgian painters dealt mainly in historical pictures
drawn from the struggles for liberty in their own country. The
most distinguished of these "romantic" Belgian artists were
Louis Gallait and Edouard de Bièfve, whose chief national works
are to be seen in this gallery. Though they belong to a type
which now strikes us as mannered and artificial, not to say
insipid, they may help to impress historical facts on the
spectator's memory. A very different side of the national
movement will meet us at Antwerp. The later Belgian School has
been gradually swamped by Parisian tendencies.]
Returning to the Place Royale, and continuing along the =Rue de la
Régence=, the first building on the L. closed with a grille, is the
Palace of the Comté de Flandre. Nearly opposite it (with four granite
pillars) is the Palais des Beaux-Arts, containing the Ancient Pictures
(already noticed). Further on to the R. we arrive at the church of
=Notre-Dame-des-Victoires= ("Église du Sablon"), to be described in
detail hereafter. The pretty and coquettish little garden on the L. is
the =Square= or =Place du Petit Sablon=. It contains a modern monument
to Counts Egmont and Hoorn, the martyrs of Belgian freedom, by Fraikin,
and is worth a visit. The little statuettes on the parapet of the square
represent artisans of the old Guilds of Brussels. The building at the
back of the Place is the Palace of the Duke d'Arenberg: its central part
was Count Egmont's mansion (erected 1548). Further on, to the L., come
the handsome building of the Conservatoire de Musique and then the
Jewish Synagogue. The end of the street is blocked by the gigantic and
massive _façade_ of the new =Palais de Justice=, one of the hugest
buildings of our period, imposing by its mere colossal size and its
almost Egyptian solidity, but not architecturally pleasing. The interior
need not trouble you.
* * * * *
=Northward= from the Place Royale, again, stretches the =Rue Royale=,
along which, as we walk, we have ever before us the immense gilt dome of
=Ste. Marie de Schaerbeck=. This fine street was admirably laid out in
1774 by the architect Guimard, who was the founder of the modern plan of
Brussels. It is a fine promenade, along the very edge of the hill,
beautifully varied, and affording several attractive glimpses over the
earlier town by means of breaks in the line of houses, left on purpose
by Guimard, some of which have, however, been unfortunately built up.
Starting from the Place Royale, we have first, on our R., the Hôtel
Bellevue; beyond which, round the corner, facing the Park, extends the
unprepossessing white _façade_ of the =King's Palace= (18th century,
rebuilt). Then, again on the R., we arrive at the pretty little =Park=,
laid out by Guimard in 1774, on the site of the old garden of the Dukes
of Brabant. This is a pleasant lounging-place, animated in the
afternoon, when the band plays. It contains ponds, sculpture,
nursemaids, children, and one of the principal theatres.
Continuing still northward, we pass the Statue of Belliard, in the first
break, and then the Montagne du Parc, L., leading direct to the Lower
Town. At the end of the Park, the Rue de la Loi runs R., eastward,
towards the Exhibition Buildings. The great block of public offices in
this street, facing the Park, includes the Chamber of Representatives
(=Palais de la Nation=) and the principal Ministries. Beyond these we
get, on the L., a glimpse of the Cathedral, and on the R. a number of
radiating streets which open out towards the fashionable =Quartier
Léopold=. Then, on the L., we arrive at the =Place du Congrès= with its
Doric =column=, commemorating the Congress which ratified the
Independence in 1831. It can be ascended (193 steps, spiral) for the
sake of its admirable =*view=, the best general outlook to be obtained
over Brussels. (A few sous should be given to the guardian.) The
prospect from the summit (morning light best) will enable you to
identify every principal building in the city (good map by Kiessling,
72, Montagne de la Cour).
Continuing our route, the street to the R. leads to the little Place de
la Liberté. Beyond this, the Rue Royale goes on to the Outer Boulevards,
and finally ends at =Ste. Marie de Schaerbeck=, a gigantic modern
Byzantine church, more splendid than beautiful, but a good termination
for an afternoon ramble.
* * * * *
The =Outer Boulevards= of Brussels, which ring round the original 14th
century city, have now been converted into magnificent =promenades=,
planted with trees, and supplied with special lanes for riders. These
Boulevards, perhaps the handsomest in the world, replace the =ancient
walls=, erected in 1357-1379, when the town had already reached such
considerable limits. Most of what is interesting or important in
Brussels is still to be found within the irregular pentagonal ring of
the Boulevards. A pleasant way of seeing the whole round is to take the
=electric tram=, from the Gare du Nord, by the =Upper Boulevards=, to
the Gare du Midi. You first mount the steep hill, with the Botanical
Gardens on your L., backed by the extensive hot-houses. The line then
crosses the Rue Royale, looking L. towards Ste. Marie de Schaerbeck, and
R. towards the Place Royale. As you turn the corner, you have on your L.
a small triangular garden, and on your R. the circular Place des
Barricades, with a statue of the great anatomist Vesalius, physician to
Charles V., and an indirect victim of the Inquisition. The rail then
bends round the Boulevard du Régent, with glimpses (to the R.) of the
Park, and (to the L.) of the Squares in the Quartier Léopold. You next
pass, R., the =Palais des Académies=, in its neatly kept garden, beyond
which you arrive at the private gardens of the Royal Palace and the
Place du Trône. Hence you continue to the Place de Namur and the
Fontaine de Brouckere, and continue on to the Place Louise, at which
point the open =Avenue Louise= leads direct to the pleasant =Bois de la
Cambre=. The Boulevard de Waterloo carries you on to the =Porte de Hal=,
the only one of the old gateways still standing. This is a massive
fortress of irregular shape, built in 1381, and it was used by the
Spanish authorities in the time of Alva as the Bastille of Brussels. The
=interior= (open free, daily) contains a fine winding staircase and a
small =collection of arms= and armour, with a little Ethnographical
Museum, which is worth ten minutes' visit in passing. Hence, the
Boulevard du Midi conducts you straight to the Gare du Midi, from which
point you can return, on foot or by tram, through the Inner Boulevards
or diagonally through the old town, to your hôtel.
* * * * *
The remainder of the =Outer Boulevards=, leading from the Gare du Midi
to the Gare du Nord by the _western_ half of the town, is commonly known
as the =Lower Boulevards=, (Note the distinction of Upper, Lower, and
Inner.) It passes through a comparatively poor quarter, and is much less
interesting than the other half. The only objects of note on its circuit
are the slaughter houses and the basins of the canal. Nevertheless, a
complete tour of the Boulevards, Upper, Lower, and Inner, will serve to
give you a better general conception of Brussels within the old walls
than you can otherwise obtain.
* * * * *
I cannot pretend in this Guide to point out all the objects of interest
in Modern Brussels, within this great ring. Speaking generally, the
reader will find pleasant walks for spare moments in the quarter between
the Rue Royale or the Rue de la Régence and the Upper Boulevards. This
district is high, healthy, and airy, and is chiefly given over to
official buildings. On the other hand, the quarter between these two
streets and the Inner Boulevards, especially southward about the Place
St. Jean and the Rue de l'Étuve, leads through some interesting portions
of 17th century and 18th century Brussels, with occasional good domestic
architecture. The district lying W. of the Inner Boulevards is of little
interest, save in its central portion already indicated. It is the
quarter of docks, entrepôts, and the more squalid side of wholesale
business.
* * * * *
The immense area of Brussels =outside the Outer Boulevards= I cannot
pretend to deal with. Pleasant walks may be taken at the E. end of the
town about the Chaussée de Louvain, the Square Marie-Louise, the
Exhibition Grounds, the Parc Léopold (near which is the too famous Musée
Wiertz), and the elevated land in the eastern quarter generally. The
=Bois de la Cambre=, the true park of Brussels, makes a delightful place
to walk or drive in the afternoon, especially on Sundays. It somewhat
resembles the Bois de Boulogne, but is wilder and prettier. Perhaps the
most satisfactory way of visiting it is to take the tram to the gate of
the wood, and then walk through it.
* * * * *
There are =three other churches=, beside the Cathedral, in the
neighbourhood of the Place Royale, which you may go to see, _if_ you
have plenty of time left, but which you need not otherwise trouble
about. The three can be easily combined in a single short round.
Go down the Montagne du Parc, and take the first turning to the L., Rue
des Douze Apôtres, which will bring you direct to the little =Chapelle
de l'Expiation=, erected in 1436, on the site of the synagogue where the
Stolen Hosts were profaned, and in expiation of the supposed crime. The
exterior of the building has been modernized, and indeed the whole is of
little interest, save in connection with the Cathedral and the Stolen
Hosts; but a glance inside is not undesirable. The interior, flamboyant
Gothic, is thoroughly well decorated throughout, in modern polychrome,
with scenes from the Gospel History. The Apse has good modern
stained-glass windows, and frescoes of angels holding the instruments of
the Passion. It is separated from the Nave by a high Rood-Loft, without
a screen. Modern taste has here almost entirely ignored the painful and
malicious story of the Stolen Wafers.
Now, continue down the Rue des Sols as far as the Rue de l'Impératrice
(where a slight _détour_ to the R. takes you in front of the Université
Libre, a large and somewhat imposing, but uninteresting building).
Continue rather to the L. down the Rue de l'Impératrice, crossing the
Montagne de la Cour, into the Rue de l'Empereur and the Rue d'Or, till
you arrive at the Place de la Chapelle, containing the church of
=Notre-Dame de la Chapelle=—after the Cathedral, the finest mediæval
church of Brussels. The =exterior= has lately (alas!) been quite too
much restored. It shows a fine Nave and Aisles of the 15th century, and
a much lower and very beautiful Choir of the 13th century, with some
Romanesque details of an earlier building (10th century?) Walk once
round the church, to observe the exterior architecture. The West Front
is massive rather than beautiful. The sculpture over the door (the
Trinity with angels, and Our Lady) is modern. Over the southern portal
is a modern relief, in a Romanesque tympanum, representing the
Coronation of Our Lady by God the Father and the Son. The Romanesque and
transitional work of the beautiful low Choir and Apse has unfortunately
been over-restored.
The =interior=, with its fine Nave and Aisles, is impressive, especially
as you look from the centre down towards the West end. The round pillars
of the Nave are handsome, and have the usual figures of the Twelve
Apostles. The _pulpit_ is one of the familiar 17th century
monstrosities, with palms, and Elijah in the Wilderness. The interior of
the pretty little Apse has been so completely modernized as to leave it
little interest. There are a few good pictures of the School of Rubens
(De Crayer, Van Thulden, etc.).
On emerging from the church, follow the tramway line up the hill to the
market-place of the =Grand Sablon=. Good views in every direction as you
enter the Place. The square is animated on Fridays and Sundays, when
markets are held here. Pass through the market-place, which contains an
absurd 18th century monument, erected by a Marquis of Ailesbury of the
period, in gratitude for the hospitality he had received from the
citizens of Brussels, and continue on to the Rue de la Régence, passing
on your R. the beautiful Apse of the church of Notre-Dame-des-Victoires,
now unhappily in course of restoration. The entrance is in the Rue de la
Régence, and the church is _not_ oriented.
=Notre-Dame-des-Victoires=, or Notre-Dame du Sablon, was founded in 1304
by the Guild of Crossbowmen; but the existing late-Gothic building is
almost entirely of the 15th and 16th centuries. It has been
over-restored in parts, and the beautiful crumbling exterior of the Apse
is now threatened with disfigurement.
The =interior= is pleasing. Over the Main Entrance, within, is a curious
_ex voto_ of a ship, in commemoration of the arrival of a sacred image,
said to have floated miraculously by sea.
The _first chapel_ to your L. as you enter has a *tomb of Count Flaminio
Gamier, secretary to the Duke of Parma, partly restored, but with fine
original alabaster reliefs of the early Renaissance, representing the
History of the Virgin. The series begins below: (1) Meeting of Joachim
and Anna at the Golden Gate; (2) The Birth of the Virgin; (3) The
Presentation of the Virgin in the Temple. Then, above: (4) Annunciation;
note the relative positions of the angel and Our Lady, the lily, the
_prie-dieu_, and the loggia in the background; (5) the Visitation, with
the usual arch; and (6) the Presentation of Christ in the Temple.
The =Apse= has restored figures of saints (named) in imitation of those
which were discovered in ruined fresco during the restoration. They are
a good typical collection of the saints most venerated in the Low
Countries in the Middle Ages.
The =Nave= has the usual figures of Apostles, named, and a small open
Triforium just below the Clerestory. The Pulpit has on its face a
medallion of Our Lady; R. and L., Moses and St. Augustine. Below, the
four beasts of the Evangelists.
You need not trouble about any other special building in Brussels; but
you may occupy yourself pleasantly with many walks through all parts of
the city.
* * * * *
You are now in a position to understand the =growth and spread of
Brussels=. From the very beginning, the merchant town occupied the
_valley_, while the capital of the Counts, Dukes, or Sovereigns spread
over the _hill_, in the neighbourhood of what are still significantly
called the Montagne de la Cour and the Place Royale. To this day the two
contrasted parts of the city are broadly distinct. The valley speaks
Flemish; the mountain, French. In the valley stand all the municipal and
mercantile buildings—the Hôtel-de-Ville, the Bourse, the Post-Office,
the markets, and the principal places of wholesale business. On the hill
stand the Royal Palace, the Government Offices, the Legislative Body,
the Ministries, the Palais de Justice, and the whole of the National
Museums and collections. From this point of view again, in our own day,
the valley is _municipal_, and the hill _national_. The contrasted
aspects of the Inner Boulevards and the Rue de la Régence well mark the
difference. In the valley, you will find, once more, the hotels of
commerce and of the passing traveller; on the hill, those frequented by
ambassadors and the wealthier class of foreign tourists. Near the Place
Royale were situated the houses of the old Brabant nobility, the Egmonts
and the Cuylenburgs; as at the present day are situated those of the
Arenbergs and the De Chimays.
=Historically=, the spread of the town from its centre began towards the
Castle of the =Counts of Louvain= and Dukes of Brabant, in the Ancienne
Cour, now occupied by the Royal Library and the Modern Picture Gallery,
as well as towards the ecclesiastical quarter of the Cathedral and the
Chancellerie. The antiquity of this portion of the Upper Town is well
marked by the continued existence of the mediæval churches of Notre-Dame
de la Chapelle, Notre-Dame-des-Victoires, and the Chapelle de
l'Expiation. Under the =Burgundian princes=, Brussels ranked second to
Ghent and Bruges; but after the =Hapsburgs= obtained possession of the
Low Countries, it was made the principal residence of the sovereigns in
their western domains. Charles V. inhabited it as one of his chief
capitals. Under Philip II. of Spain, it became the official residence of
the Stadtholder of the Netherlands; and Margaret of Parma, who bore that
office, held her court in the old Palace. From that time forth Brussels
was recognised as the common capital of the southern Low Countries. The
Austrian Stadtholders habitually lived here; and when, after the
Napoleonic upheaval, Belgium and Holland were united into a single
kingdom, Brussels was made the alternative capital with Amsterdam. By
the time that Belgium asserted her independence in 1830, Brussels had
thus obtained the prescriptive right to become the seat of government of
the new nation.
The old Palace had been burnt down in 1731, and the outer wings of the
existing Palace were built by the Austrians shortly after. It was they,
too, who laid out the Rue Royale and Place Royale, with the Park and its
surroundings, as we still see them at the present day. To the Austrian
rulers are also due the Parliamentary Buildings: but the Palais des
Académies was built under Dutch rule in 1829. Since 1830 the town has
been greatly beautified and improved. The Inner Boulevards have been
opened through the labyrinth of streets in the old centre: the Palais de
Justice has been built, the Quartier Léopold has grown up, and great
edifices have been erected at Schaerbeck and elsewhere on the outskirts.
At the present day, of Brussels =within the Boulevards=, the Hill
District is _governmental and fashionable_; the Central District,
_municipal and commercial_: the Western District contains the markets,
basins, canals, and _wholesale business_ side of the city. =Without the
Boulevards=, Fashion has spread eastward towards the Bois de la Cambre
and the Parc Léopold. The poorer districts run southward and westward.
But every part of the city is amply provided with wide thoroughfares and
open breathing-spaces. In this respect, Brussels is one of the
best-arranged cities in Europe.
_F_. SURROUNDINGS
The only =excursion= of interest in the immediate neighbourhood of
Brussels is that to =Laeken= (recommended), which may be taken by tram
from the Inner Boulevards, the Gare du Nord, the Gare du Midi, Bourse,
etc. Cars run every 10 minutes. The modern =Church of St. Mary= at
Laeken is a handsome unfinished building. A little to the R. lie the
Park and the =Royal Château=, inaccessible and unimportant. The road
behind the church ascends the Montagne du Tonnerre, a little hill with a
=Monument to Léopold I.=, not unlike the Albert Memorial in London. A
good =*view= of Brussels is obtained from the summit of the monument,
ascended by a winding staircase. (No fee.) The easiest way to make this
excursion is by carriage _in the afternoon_.
Unless you are a military man or a student of tactics, I do not advise
you to undertake the dull and wearisome excursion to =Waterloo=. The
battle-field is hot and shade-less in summer, cold and draughty in
spring and autumn. The points of interest, such as they are, lie at
considerable distances. Waterloo is country, and ugly country—no more.
The general traveller who desires to be conducted round the various
strategic landmarks of the field will find his wants amply catered for
by Baedeker. But I advise him to forego that foregone disappointment.
The time saved by _not_ visiting Waterloo may, however, be well devoted
to a morning excursion to
LOUVAIN.
[This ancient and important town, which should be visited both
on account of its magnificent =Hôtel-de-Ville=, and in order to
make a better acquaintance with =Dierick Bouts=, the town
painter, can be conveniently reached by train from the Gare du
Nord. The best trains take little more than half an hour to do
the journey. A single morning is sufficient for the excursion,
especially if you start early. Wednesday is the most convenient
day, as a quick train then returns about 1.30. (Consult
Bradshaw.) Lunch can be obtained (good) in the large white
building on the left-hand side of the Hôtel-de-Ville. (It is a
private club, but contains a public restaurant, on the R.,
within, to which, push through boldly.) If you have Conway, take
him with you on this excursion, to compare the doubtful Roger
van der Weyden at St. Pierre with the woodcut he gives of its
supposed original at Madrid. Read before you start (or on the
way) his admirable accounts of Roger van der Weyden and Dierick
Bouts.
=Louvain= is, in a certain sense, =the mother city of Brussels=.
Standing on its own little navigable river, the Dyle, it was,
till the end of the 14th century, the capital of the Counts and
of the Duchy of Brabant. It had a large population of weavers,
engaged in the cloth trade. Here, as elsewhere, the weavers
formed the chief bulwark of freedom in the population. In 1378,
however, after a popular rising, Duke Wenseslaus besieged and
conquered the city; and the tyrannical sway of the nobles, whom
he re-introduced, aided by the rise of Ghent or, later, of
Antwerp, drove away trade from the city. Many of the weavers
emigrated to Holland and England, where they helped to establish
the woollen industry.
During the early Middle Ages, Louvain was also celebrated for
its =University=, founded in 1426, and suppressed by the French
in 1797. It was re-established by the Dutch in 1817, but
abandoned by the Belgian Government in 1834, and then started
afresh in the next year as a free private Roman Catholic
University. Charles V. was educated here.
The modern town has shrunk far away within its ancient ramparts,
whose site is now for the most part occupied by empty
Boulevards. It is still the stronghold of Roman Catholic
theology in Belgium.]
As you emerge from the station, you come upon a small Place, adorned
with a statue (by Geefs) of Sylvain van de Weyer, a revolutionary of
1830, and long Belgian Minister in England. Take the long straight
street up which the statue looks. This leads direct to the Grand' Place,
the centre of the town, whence the chief streets radiate in every
direction, the ground-plan recalling that of a Roman city.
The principal building in the Grand' Place is the =Hôtel-de-Ville=,
standing out with three sides visible from the Place, and probably the
finest civic building in Belgium. It is of very florid late-Gothic
architecture, between 1448 and 1463. Begin first with the left _façade_,
exhibiting three main storeys, with handsome Gothic windows. Above come
a gallery and then a gable-end, flanked by octagonal turrets, and
bearing a similar turret on its summit. In the centre of the gable is a
little projecting balcony of the kind so common on Belgian civic
buildings. The architecture of the niches and turrets is of very fine
florid Gothic, in better taste than that at Ghent of nearly the same
period. The statues which fill the niches are modern. Those of the first
storey represent personages of importance in the local history of the
city: those of the second, the various mediæval guilds or trades: those
of the third, the Counts of Louvain and Dukes of Brabant of all ages.
The bosses or corbels which support the statues are carved with
scriptural scenes in high relief. I give the subjects of a few
(beginning L.): the reader must decipher the remainder for himself. The
Court of Heaven: The Fall of the Angels into the visible Jaws of Hell:
Adam and Eve in the Garden: The Expulsion from Paradise: The Death of
Abel, with quaint rabbits escaping: The Drunkenness of Noah: Abraham and
Lot: etc.
The main _façade_ has an entrance staircase, and two portals in the
centre, above which are figures of St. Peter, L., and Our Lady and
Child, R., the former in compliment to the patron of the church
opposite. This _façade_ has three storeys, decorated with Gothic
windows, and capped by a gallery parapet, above which rises the
high-pitched roof, broken by several quaint small windows. At either end
are the turrets of the gable, with steps to ascend them. The rows of
statues represent as before (in 4 tiers) persons of local distinction,
mediæval guilds, and the Princes who have ruled Brabant and Louvain.
Here again the sculptures beneath the bosses should be closely
inspected. Among the most conspicuous are the Golden Calf, the
Institution of Sacrifices in the Tabernacle, Balaam's Ass, Susannah and
the Elders, etc.
The gable-end to the R., ill seen from the narrow street, resembles in
its features the one opposite it, but this _façade_, which was even
finer than the others, is at present in course of wholesale restoration.
The best =general view= is obtained from the door of St. Pierre, or near
either corner of the Place, diagonally opposite.
Do not trouble about the =interior=.
Opposite the Hôtel-de-Ville stands the =Church of St. Pierre=,
originally erected in 1040, but entirely rebuilt in 1430, to which date
the whole existing edifice belongs. It is a handsome late-Gothic
building, with a fine West Front, never completed, and a truncated
tower. The central West Window is imposing, but the ruined portal has a
depressing effect. Walk round the church once outside to observe its
exterior architecture, obscured towards the Grand' Place by the usual
agglomeration of small Renaissance houses. The =main entrance= is in the
South Transept; above it stands a poor modern statue of the patron, St.
Peter. The High Choir, with its flying buttresses, would form a fine
element if the houses were cleared away, so as to afford a view of the
chapels below.
Now view the =interior=. Go at once into the body of the church. The
general effect is handsome, but the walls are cold and white-washed. The
church has a fine Nave, with single Aisles, short Transepts, high Choir,
and Ambulatory. The Nave, Transepts, and Choir, have all an exactly
similar clerestory, with an unusual Triforium of open latticework, and
tracery in the same style in the spandrils of the arches.
Go down to the W. end of the Nave. The entrance doors at this end have
good but not beautiful carved woodwork of the Renaissance.
=Left Aisle.= _First chapel._ Late Gothic copper font, with large crane,
to support a heavy iron cover, now removed. The other chapels on this
side contain nothing of interest.
=Right Aisle.= _First chapel_ (of San Carlo Borromeo), has an
altar-piece, copied from one by De Crayer, carried off by the French and
now at Nancy. It represents San Carlo ministering to the plague-stricken
at Milan. Also, a triptych, by Van de Baeren, 1594. Centre, St. Dorothea
beheaded. Her head praising God. L., Her trial before the governor,
Fabricius. R., Her torture in enduring the sight of her sister's
martyrdom. Statue of San Carlo by Geefs.
_Second chapel_, of the Armourers, has a railing with arms and cannon,
and contains an old blackened crucifix, much venerated because it is
said to have caught a thief who had entered the church to steal the
treasures.
The =pulpit= is a carved wooden monstrosity of the 18th century,
representing, behind, the Repentance of Peter, with the cock crowing, a
maladroit subject for a church dedicated to the saint. In front, the
Conversion of St. Paul, with his horse overthrown. Above are two palm
trees.
The =Choir= is separated from the Transepts and Nave by a very handsome
and elaborate *Rood-Loft, in the finest flamboyant late-Gothic style
(1450), one of the best still remaining examples in Europe. It supports
a Crucifixion, with St. John and Our Lady. Its arcade of three handsome
arches is surmounted by a sculptured balustrade, containing figures of
saints (the Saviour, Our Lady and Child, the Twelve Apostles with the
instruments of their martyrdom, the Doctors of the Church, and a few
others). Examine carefully.
Now, pass behind the Choir, into the =Ambulatory=, beginning on the N.,
or left side. The _first recess_ has a fine mediæval tomb of Mathilde de
Flandre. On your R., in the Choir, a little further on, is a beautiful
late-Gothic tabernacle or canopy of 1450, gilded, and containing scenes
from the Passion. Just behind the High Altar is a curious little 15th
century relief: _Centre_, the Crucifixion with St. John and Our Lady:
_R._, The Resurrection, with sleeping Roman soldiers: _L._, The donor,
with his patron St. John the Baptist.
The _first chapel_ beyond the High Altar contains =**The Last Supper=,
by Dierick Bouts. This picture forms the central piece of a =triptych=,
painted for the Confraternity of the Holy Sacrament. The _L. wing_ of it
is now at Munich, and the _R._ at Berlin. It represented, when entire,
the same mystical series of the Institution of the Eucharist which we
have already seen in the Pourbus of the Cathedral at Bruges. The
_central panel_ represented the Institution of the Eucharist; the L.
(Munich) has Melchizedeck offering bread and wine to Abraham; the R.
(Berlin) Elijah fed by ravens in the wilderness. On the outer sides of
the panels are two similar typical subjects: L. (Munich), the Gathering
of the Manna or food from Heaven; and R. (Berlin), the Feast of the
Passover, the Paschal Lamb being regarded as a type of the Christian
sacrifice. The picture as it stands in this chapel has of course lost
its mystical significance. It closely resembles the smaller Last Supper
in the Brussels Gallery; but the architecture here is Gothic, not
Renaissance. Study well, especially the figures of the donor (by the
door) and the servant. The floor is characteristic.
Also a =**triptych=, by Dierick Bouts, the Martyrdom of St. Erasmus,
patron against intestinal diseases: a bishop, martyred at Formia in the
persecution of Diocletian. It represents the hideous episode of the
unwinding of the saint's bowels. The executioner on the L. is a good
specimen of Dierick Bouts's rude artisan figures; he looks like a
cobbler. In the background is the Emperor Diocletian, richly attired,
with a courtier, whose attitude recalls more than one of those in the
Justice of Otho. The landscape is characteristic of Bouts's manner. This
is a good, hard, dry picture. The _L. panel_ has St. Jerome, robed as
cardinal, with his lion; the R. has St. Anthony, accompanied by a
vanquished demon. This, however, is a St. Anthony as the abbot, not as
the hermit in the desert.
On the roof of the _fourth chapel_ have recently been discovered some
frescoes, from which the plaster and whitewash is now being removed.
In the chapel next it to the R. is a =triptych=, the Descent from the
Cross (covered, the Sacristan will open it: 1 franc); usually attributed
to Roger van der Weyden, but much disputed. It is probably a smaller
(altered) copy of the famous composition in the Escurial at Madrid (see
Conway). The _central picture_ has Christ supported by Joseph of
Arimathea and Nicodemus, with the fainting Madonna, St. John, and the
other Maries. The singularly unpleasing fat cook-like Magdalen, in a
rich robe, is a constant feature in the group of Descents from the Cross
by Roger and his pupils. Study this picture. The _L. panel_ has a good
portrait of the donor, with his two sons, accompanied by his patron St.
James the Greater (or St. William?). The _R. panel_ has his wife, with
her two daughters and her patroness, St. Adelaide (or St. Elizabeth of
Hungary, holding the crown which she gave up for the Franciscan
profession?).
In the _sixth chapel_ is a fine Renaissance tomb, representing Adolf van
Baussede in adoration before the Trinity, introduced by his patron St.
Adolphus, with allegorical figures of Faith, Hope, and Charity. The work
is almost Italian in character.
Over the High Altar is a modern figure of the patron, St. Peter,
enthroned as pope, and with papal symbols behind him. Left of it is the
fine canopy we have already observed from the outside, with scenes from
the Passion. The architecture here is striking.
The great Quentin Matsys of the Family of St. Anne in the Brussels
Picture Gallery was formerly an altar-piece in this church.
There is nothing else at Louvain that need detain you. If you like, you
can stroll a little way down the Rue de Namur, just to the R. of the
Hôtel-de-Ville. It contains some good old houses. The desolate building
on your R. was originally the Halles, but is now the University. It was
built for the Guild of Clothmakers in 1317, and has been wholly
modernised; but there are some good Gothic arches on the basement floor
within (approach down the side street to the R.). Further on is the
Collège du St. Esprit on the R., and the Church of St. Michel
(uninteresting) on the L. The street which here runs off obliquely
conducts to the Collège Marie-Thérèse, and the Collège Adrien VI.,
uninteresting, and all used as hostelries for the students. The only
other objects to look at in Louvain are the =choir-stalls= in carved
wood, early Renaissance, at the Church of =St. Gertrude=, dedicated to
the Abbess of Nivelles and aunt of St. Gudula. It lies down the Rue de
Malines, in the opposite direction from the Rue de Namur. You have then
seen Louvain.
* * * * *
On your way from Brussels to Antwerp, you ought to visit Malines
Cathedral. The easiest way is to book your luggage through, and then
stop for an hour or two at Malines, going on by a later train.
IV
ANTWERP
_A_. ORIGINS OF ANTWERP
ANTWERP, the seaport of the Schelde estuary, is practically =the
youngest= (and the least interesting) of the great Belgian towns. It
should therefore be visited _last_ by the historically-minded tourist. A
small town, known in Flemish as =Antwerpen= ("at the Wharf"),—a name
altered in French and English into Anvers and Antwerp,—existed here, it
is true, as early as the 7th century, and suffered heavily in the 9th
from the ubiquitous Northmen. But its situation at the open mouth of the
great estuary of the Schelde, exposed to every passing piratical
invader, rendered it unfit for the purposes of early commerce. The trade
of Flanders, in its first beginnings, accordingly concentrated itself in
the more protected inland ports like Bruges and Ghent; while that of
Brabant, of which province Antwerp itself formed a part, found a safer
home in Brussels or Louvain, far up some minor internal river. Hence
=the rise of Antwerp= dates no further back than the end of the 15th and
beginning of =the 16th century=.
Its rise, that is to say, as =a great commercial port=, for from an
early period it was the capital of a petty margrave, under the Duke of
Brabant. As northern Europe grew gradually quieter during the 11th and
12th centuries, Antwerp rose somewhat in importance; and the
magnificence of its cathedral, the earliest part of which dates from
1352, sufficiently shows that the town was increasing in wealth and
population during the palmy period when Bruges and Ghent governed the
trade of the Continent. But when, in the 15th century and the beginning
of the 16th, Bruges began to decline (partly from political causes, but
more still from changes in navigation and trade routes), Antwerp rose
suddenly to the first position in the Low Countries and perhaps in
Europe. Its large, deep, and open port was better adapted to the
increasing shipping of the new epoch than were the shallow and narrow
canals or rivers of Ghent, Bruges, and Brussels. The discovery of
America, and of the route to India by the Cape of Good Hope, had
revolutionized both commerce and navigation; vessels were built larger
and of deeper draught; and the Schelde became for a time what the
Thames, the Clyde, and the Mersey have become in our own period. Antwerp
under Charles V. was probably even more prosperous and wealthier than
Venice. =The centre of traffic was shifting= from the _Mediterranean_ to
the _Atlantic seaboard_. The city reached its highest point of
prosperity about 1568, when it is said that thousands of vessels lay at
anchor in the Schelde, and that more than a hundred craft sailed and
arrived daily. Even allowing for the smaller burden of those days,
however, this is probably an exaggeration. The great =fairs= of Antwerp,
of which those of Leipzig and Nijni Novgorod are now the only modern
representatives, also drew thousands of merchants from all parts of the
world. The chief imports were wool and other agricultural produce from
England, grain from the Baltic, wines from France and Germany, spices
and sugar from Portuguese territory, and silks and Oriental luxuries
from Venice and other parts of Italy. The exports were the manufactured
goods of Flanders and Brabant, countries which still took the lead in
textile fabrics, tapestries, carpets, and many other important
industries.
It is to this late period of wealth and prosperity that Antwerp owes
most of the great buildings and works of art which still adorn it. Its
=cathedral=, indeed, varies in date in different parts from the middle
of the 14th to the beginning of the 16th century, and some portions were
not quite completed till the 17th; but the general aspect of the core of
the town is of the =Renaissance= epoch. It contains in its modern
gallery not a few Flemish paintings of the earlier period, produced by
the artists of Ghent, Bruges, and Brussels; but its own native art dates
no further back than =Quentin Matsys= (1466-1531), the last of the
painters of the Netherlands who adhered to the national type of art;
while it reached its highest point in =Rubens= (1577-1640), who
introduced into the Low Countries the developed style of the Italian
Renaissance, adapted and strained through an essentially robust Flemish
nature. It is only at Antwerp that these two great masters can be
studied to the highest advantage; they illustrate, one the rise, the
other the culmination and after-glow, of the greatness of their native
city. I say native advisedly, for though Rubens most probably was born
at Siegen (in Nassau), he was an Antwerper by descent, by blood, by
nature, and by residence.
The =decline= of the city in later times was due to a variety of
concurrent causes, some of them strangely artificial, which long
distracted trade from one of its most natural outlets in Europe. The
Spanish troops began the devastation, during the abortive attempt of the
southern provinces to shake off the yoke of Spain; in 1576, the Town
Hall and nearly a thousand noble buildings were burnt, while 8,000
people were ruthlessly massacred. In 1585, the Duke of Parma completed
the destruction of the local prosperity: the population was largely
scattered, and the trade of Antwerp completely ruined. The long and
unsuccessful rebellion, the division which it unhappily caused between
Holland and Belgium, and the rapid commercial rise, first of Amsterdam
and then of England, all contributed to annihilate the mercantile
importance of Antwerp. The Dutch erected forts on their own territory at
the mouth of the Schelde, and refused to allow shipping to proceed up
the river. Finally by the Treaty of Munster in 1648 it was agreed that
no sea-going vessel should be allowed to ascend the estuary to Antwerp,
but that all ships should unload at a Dutch port, goods being forwarded
by river craft to the former capital of European commerce. From that
date forward to the French occupation in 1794, Antwerp sank to the
position of a mere local centre, while Rotterdam and Amsterdam took its
place as commercial cities. In the latter year, however, the French
reopened the navigation of the Schelde, and destroyed the iniquitous
Dutch forts at the entrance to the river. Napoleon, in whose empire the
town was included, constructed a harbour and built new quays; but after
his fall, Antwerp was made over to Holland, and began to trade as a
Dutch seaport. The erection of Belgium into a separate kingdom in 1830
again told against it, as the Dutch maintained their unjust power of
levying tolls on the shipping; in addition to which drawback, Antwerp
had suffered heavily from siege during the War of Independence. In 1863,
however, the Dutch extortioners were bought off by a heavy money
payment, and Antwerp, the natural outlet of the Schelde, and to a great
extent of the German empire, once more regained its natural place as a
main commercial port of Europe. Since that date, its rise has been
extraordinarily rapid, in correspondence with the large development of
Belgian manufactures and still more with the new position of Germany as
a world-trading power. Indeed, nothing but the artificial restrictions
placed upon its commerce by the selfishness and injustice of the Dutch
could ever have prevented the seaport of the Schelde from ranking as one
of the chief harbours of the world, as soon as ocean-going ships
demanded ports of that size, and as commerce had no longer anything to
fear from marauding pirates.
As a consequence of these conditions, we have to expect in Antwerp
mainly =a central town of the 15th and 16th centuries=, with an immense
=modern outgrowth= of very recent origin. Save its fine Cathedral, and
its imported pictures, it has little or nothing of mediæval interest.
The =population= of Antwerp is almost entirely =Flemish=, though French
is the language of the higher commerce; and the town is the stronghold
of the old Flemish feeling in Belgium, as opposed to the Parisian tone
of Brussels.
Concurrently with the rise of its renewed commercial importance, Antwerp
has become once more =a centre of Belgian art=, and especially of the
pure Flemish school of _archaists_, who have chosen their subjects from
Flemish history, and followed to some extent the precedents of the early
Flemish painters. Examples of these will meet us later.
Choose an =hotel= on the =Place Verte=, if possible, or at least very
near it. You cannot gain a first impression of Antwerp in less than four
or five days.
Antwerp is a confused town, a maze without a plan: till you have learnt
your way about, I advise you to =follow the tram-lines=: you will thus
avoid the slummy streets which abound even in the best quarter.
_B_. THE CATHEDRAL
[The first thing to see at Antwerp is the High Church of Our
Lady, once the =Cathedral=, and still commonly so called, though
it is not now a bishop's see, but part of the diocese of
Malines. It is a fine early and middle Gothic church, with a
late Gothic or flamboyant tower; but, relatively to its fame, it
is externally disappointing. This is partly because mean houses
have been allowed to gather round it, but partly also because
its somewhat meretricious spire has been unduly praised by
earlier generations. Modern taste, which admires the simpler and
severer early forms of Gothic, finds it fantastic and
over-elaborate.]
The =Place Verte= opposite the Cathedral (once the churchyard), is
planted with trees, and has its centre occupied by a modern statue of
Rubens. This is one of the few points from which you can view (more or
less) the =exterior= of the Cathedral, the greater part of which is
obstructed by shabby shops clustered round its base. The only really
good views, however, are obtained from the second-floor windows of the
houses on the E. side of the Square, such as the Hôtel de l'Europe.
Nevertheless, it will be well to walk round the building outside, in
order to inspect as much of it as is visible.
The chief =portal= (practically), recently restored, and the =S.
Transept= are seen from the Place Verte. There is little sculpture on
them, save a small late figure of the patroness, Our Lady, with the
Child, high up between the angels of the gable-end.
Now, go round to the L., into the little triangular Place known as the
Marché aux Gants, to view the main =West Front=, best seen from the apex
of the triangle opposite. It has a fine central Portal and West Window,
flanked by two great towers, the southern incomplete. Its niches have
statues of six only out of the Twelve Apostles. The =northern tower=, up
to the first gallery, is middle Gothic of 1352-1449. The upper portion,
with the octagonal lantern of very open work, flanked by projecting
pinnacles, tied by small buttresses, is in later flamboyant Gothic, and
was erected in 1502-1518, by Dominic de Waghamakere, the architect of
the Gothic portion of the Town Hall at Ghent. This florid spire has been
excessively praised above its merits, but will hardly satisfy a modern
taste. It can be ascended (75 c.), but is dark and steep: the view,
though fine, hardly repays the trouble.
The =well= in the Marché aux Gants, near the front of the Cathedral, has
a beautiful wrought-iron canopy, to support its lid, said to have been
made by Quentin Matsys when he was a blacksmith, or rather a
metal-worker, before he took to painting. (But the legend is doubtful.)
It consists of a trellis of vine, supporting wild men and women with
clubs, and capped by a figure of Brabo, the eponymous hero of Brabant,
flinging the hand of the giant Antigonus (see later, under the
Hôtel-de-Ville).
Now, continue on round the =N. Side= of the Cathedral. A few glimpses of
the =N. Transept= and Aisles, as well as of the Nave and Choir, may be
obtained as we proceed, much of it unfortunately now marred by excessive
restoration. The beautiful Choir and Apse, with their flying buttresses,
are almost entirely concealed by neighbouring houses. If these were
cleared away, a fine view would be obtained of a noble piece of
architecture, now only visible by occasional glimpses from the upper
floors of surrounding houses. This portion of the church is further
disfigured by the abrupt terminations to the roofs of the Transepts, and
by the ridiculous pepper-caster top which replaces the central spire or
_flèche_ of the original conception. Continue on through the narrow
streets till you have made a complete tour of the Cathedral and returned
to the Place Verte and the door of the =S. Transept=. The best general
view, however, is not obtainable from any of these points, but from the
Grand' Place, and especially the upper windows of the Hôtel-de-Ville, to
be visited later.
Now, enter the Cathedral, by the door in the S. Transept. (Open, free,
from 8 to 12 on Sundays and Thursdays: or, every day, 12 to 4, on
payment of 1 franc per person. But if you wish really to inspect the
works of art it contains, pay your franc like a man, and see them at
your leisure when there are no services in progress. Fine music at High
Mass at 10 on Sundays.)
The =interior= is impressive and solemn, with its high Nave, Transepts,
and Choir, of good simple Gothic, and its _three_ rows of Aisles, the
perspective of which, with their many pillars, is extremely striking.
The Aisles, however, are unusually low in proportion to the height of
the central cruciform building. First walk down the Nave to the West
End, to form a general conception of the fine and impressive interior,
grand in its colossal simplicity, and commendably free from 18th century
disfigurements.
Now, begin at the =R. or S. Aisle=, which contains admirable modern
Stations of the Cross by Vinck and Hendrickx, excellently painted in the
archaic spirit. I do not describe these, as they need no explanation,
but each is worthy of individual attention. Do not hurry.
The _Chapel of the Sacrament_, at the end of this Aisle, has good
polychrome decoration, and fine stained glass windows (Last Supper,
1503: St. Amand converting Antwerp; St. Norbert preaching against the
heresy of Tanquelin at Antwerp, etc.): also, a reliquary of St. Roch,
and an interesting modern statue of that great plague-saint.
The =S. Transept= has a good modern stained glass window, and affords
fine views of the central Dome and Aisles.
On the _R. wall_ are the Marriage at Cana in Galilee, appropriately
painted for the Altar of the Wine-merchants, by M. de Vos (excellent for
comparison with others of the same subject), and a Last Supper by Otto
van Veen, the master of Rubens, formerly the Altar-piece of the Chapel
of the Sacrament.
The _L. wall_ of the S. Transept is occupied by =Rubens's= great
=triptych of St. Christopher=, commonly called (from its central
portion) =**The Descent from the Cross=. This is a splendid work,
conceived (as to idea) in the mystical spirit of old Flemish art, though
carried out, of course, in the utterly different and incongruous style
of Rubens. In order to understand it we must remember that _triptychs
were usually kept closed on the altar_, and that the picture which first
met the eye was that which occupies the outer shutters. It struck the
key-note. Now, the outer shutters of this work (seldom seen, unless you
ask the Sacristan to close it) are occupied by a figure of St.
Christopher, with the hermit who directed him to Christ, accompanied by
his lantern and owl, as in the earlier St. Christopher triptych by
Memling in the Academy at Bruges. This painting was ordered from Rubens
by the Guild of Arquebusiers, whose patron is St. Christopher. On the
outside, therefore, Rubens painted the saint himself, whose name (of
course) means the Christ-Bearer. But on the inner portion he painted
three other symbolical or allusive scenes of the Bearing of Christ:
_L._, The Visitation; the unborn Christ borne by His mother: _R._, The
Presentation in the Temple; the living Christ borne by Simeon: _Centre_,
The Descent from the Cross; the dead Christ borne by Joseph of Arimathea
and the Disciples.
The _L. wing_ shows us Our Lady, in a big Flemish hat, approaching St.
Elizabeth. Behind, Joseph and Zacharias, the two husbands, shake hands.
(This composition has been copied in the stained glass window of the
Cathedral at Antwerp.) In order to impress the mystical meaning of the
picture, the fact of Our Lady's pregnancy has been strongly insisted
upon.
The _central panel_ shows us the Descent from the Cross. Nicodemus holds
the body by one shoulder, while St. John, below, receives it in his
arms, and the Magdalen at the feet expresses her tenderness. Joseph of
Arimathea descends the ladder. The actual corpse forms the salient point
in the picture. It is usual to say that the contrast of the dead body
and white sheet is borrowed from the famous treatment of the same
subject by Daniele da Volterra in Santa Trinità de' Monti at Rome; and
indeed, the composition in this work has probably been suggested by the
Italian example; but a similar white sheet, with the dead body seen
against it, is found in all early Flemish art, and especially in works
of the School of Roger van der Weyden. (It is known as the Holy
Sudarium.) In this splendid and gorgeous conception, Rubens has given
the greatest importance to the body of the Saviour; but he is so
intensely occupied with the mechanical difficulties of its support, the
strain and stress of the dead weight, that he forgets feeling; in spite
of the agonised attitude of the Mater Dolorosa, the picture is sadly
lacking in pathos. He realizes the scene as to its material facts; he
fails to realize its spiritual significance. (For an opposite opinion,
see M. Max Rooses, who speaks of "the profound expression of a tender
and respectful love.") To my mind, the man who holds the Sudarium in his
teeth is a fault of taste of the most flagrant character. We think of
the whole work rather as a wonderful piece of art than as the fitting
delineation of a sacred subject. But as art it is triumphant. The faces
of the St. John and the Magdalen are also charming.
The _R. wing_, with the Presentation, and the aged Simeon receiving
Christ in his arms, is of less interest.
Next, enter the =Ambulatory=, behind the Choir.
_1st Chapel._ Good modern stained glass window of the Pietà.
_2nd Chapel._ Tomb of =John Moretus=, the son-in-law of Plantin, the
famous printer (see after, under Musée Plantin-Moretus) erected by
Martina Plantin, his widow, and with pictures by =Rubens=. Above, in an
oval, portrait of John Moretus (by a pupil, re-touched by Rubens).
Below, triptych; _centre_, *The Resurrection, emblematic of hope for his
glorious future. _L. wing_, his patron, St. John-Baptist; _R. wing_, his
widow's patroness, St. Martina. This triptych, too, loses by not being
first seen _closed_: on the outside are two angels, about to open a
door; as the wings unfold, you behold the luminous figure of the risen
Christ, grasping the red Resurrection banner. This figure is celebrated.
The dismay of the Roman soldiers is conceived in the thorough Rubens
spirit. Observe the arrangement of this triptych on the tomb: it will
help you to understand others in the Museum.
Opposite this, Tomb of a Premonstratensian Friar, with St. Norbert,
founder of the Order, in adoration, by Pepyn.
This chapel is also one of the best points of view for Rubens's famous
=**Assumption=, above the High Altar. We here see one of these great
altar-pieces (of which we shall meet many examples in the Museum) placed
in the situation for which it was originally designed. This Assumption
ranks as one of Rubens's masterpieces. Above, Our Lady is caught up into
the air by a circle of little cherubs, dimly recalling the earlier
Italian mandorla. Below, stand the Apostles, looking into the empty
tomb, with the youthful figure of St. Thomas stretching out his hands in
an attitude derived from the Italian subject of the Sacra Cintola. In
the centre of the foreground, the Holy Women, about to pick roses from
the empty tomb. (See a similar work in the Brussels Museum. This
composition can only be understood by the light of earlier Italian
examples.)
On the pier between this and the next chapel, Crucifixion, with Scenes
from the Passion.
_3rd Chapel._ Master of the School of Cologne, 14th century. A Glory of
the Angels. In the centre, St. Michael the Archangel slaying a dragon,
whose double tongue divides into many heads of kings. R. and L., the
insignificant donor and donatrix. On either side, choirs of angels in
hierarchies. Above, Christ enthroned in a mandorla (almond-shaped halo)
worshipped by angels. Beneath, in the predella, St. Stephen with his
stone; St. Ursula with bow and arrow; St. Peter (keys); a Pietà; St.
John the Evangelist; St. Agnes with her ruby ring; and St. Anthony the
Abbot with his staff and bell. A good picture of the school from which
Van Eyck was a reaction. Opposite it, Tomb of Bishop Ambrosio Capello,
by Arthus Quellin, the only one remaining tomb of a bishop in the
Cathedral.
_4th Chapel._ Good 16th century figure of Our Lady and Child. Tomb of
Plantin, with Last Judgment by De Backer.
_5th Chapel._ Beautiful modern archaic altar-piece of St. Barbara.
_6th Chapel._ Nothing of special interest, though in all these chapels
the stained-glass windows and polychromatic decorations are worthy of
notice.
Opposite it, on the back of the High Altar, painted imitations of
reliefs by Van Bree: an extraordinary illusion; Annunciation, Marriage
of the Virgin, Visitation. In front of these, Tomb of Isabella of
Bourbon, wife of Charles the Bold, and mother of Mary of Burgundy.
Altar-back, Death of the Virgin, 17th cent.
_7th Chapel._ Good modern archaic altar-piece, with a miracle of St.
John Berchman. The saints are named on it.
_8th Chapel._ Tolerable modern archaic altar-piece of Our Lady and
Child, with donors and saints.
On the pier, between this and the next chapel, School of Roger van der
Weyden, Selection of Joseph as the husband of the Virgin, and Marriage
of the Virgin; a good picture.
_9th Chapel_, of St. Joseph, patron saint of Belgium, and therefore
honoured with this larger shrine. On the Altar, modern carved and gilt
altar-piece, St. Joseph bearing the Infant Christ. Around it, Scenes
from his Life. L. (beginning below), Marriage of the Virgin and Joseph,
Nativity, Presentation in the Temple; R. (beginning above), Flight into
Egypt, Finding of Christ in the Temple, the Holy Carpenter's Shop.
Centre, Death of St. Joseph. On the wings, R., Philip IV. dedicating
Belgium to St. Joseph; L., Pius IX., accompanied by St. Peter,
appointing Joseph patron saint of Belgium.
Now enter the =N. Transept=.
_R. wall._ Rubens's famous *Elevation of the Cross. In form a triptych,
but with the same subject continued through its three members. _Centre_,
The Elevation: _L._, St. John, the Mater Dolorosa, and the Holy Women:
_R._, Longinus and the soldiers, with the two thieves. This is one of
Rubens's most bustling pictures, where the mere muscular effort almost
wholly chokes the sense of pathos. The dog in the foreground is an
exceptionally unhappy later addition by the master. The tone of colour
is brown and cold; the work is mainly painted for light and shade. It
was formerly the altar-piece in the Church of St. Walburga, who appears
with other saints on the outer shutters.
This Transept also contains stained glass of the 17th century.
On the _L. wall_ is a triptych by Francken: _Centre_, Christ among the
Doctors, said to be portraits of the Reformers. _L. wing_, St. Ambrose
baptizing Augustine, with the donor, kneeling. _R. wing_, Elijah causing
the widow's cruse of oil to be replenished.
The _Chapel_ in the N. Transept has nothing of interest.
Now, enter the =Choir=, with good modern carved stalls, and a different
but less impressive view of Rubens's Assumption.
The =N. Aisle= has little of interest, save its stained-glass windows,
and a Head of Christ, painted on marble, ascribed to Leonardo, but
really of Flemish origin. This is affixed to the first pillar of the
_Lady Chapel_. Further on in the Aisle, confessionals with tolerable
wood-carvings.
The =Nave= has the usual overloaded 17th century _pulpit_, with Europe,
Asia, Africa, and America.
I have only briefly enumerated the principal contents; but you will find
much more that is interesting for yourself if you spend an hour or two
longer in examining the Cathedral.
_C_. THE PICTURE GALLERY
[The chief object of interest at Antwerp, even more important
than the Cathedral itself, is the =Picture Gallery=, regally
housed in a magnificent Museum at the S. end of the town. The
building alone might make Trafalgar Square blush, if Trafalgar
Square had a blush left in it. To this collection you should
devote at least two or three mornings.
[Illustration: THE PICTURE GALLERY, ANTWERP.
Modern Pictures in the Rooms marked with an Italic capital.]
The Antwerp Gallery contains in its palatial rooms a large
number of =Flemish pictures=, many of them collected from the
suppressed Churches and Monasteries of the city. (Remember that
they were painted for such situations, not to be seen in
Museums.) You will here have an opportunity of observing a few
good pictures of the =early Flemish School=, and especially of
improving your slight acquaintance with =Roger van der Weyden=,
one of whose loveliest works is preserved in the gallery. You
will also see at least one admirable example of =Quentin
Matsys=, as well as several fine works of the Transitional
School between the early and the later Flemish periods.
But the special glory of the Antwerp Museum is its great
collection of =Rubenses=. It is at Antwerp alone, indeed, that
you can begin to grasp the greatness of Rubens, as you may grasp
it afterwards at Munich and Vienna. I do not say you will love
him: I will not pretend to love him myself: but you may at least
understand him. This, then, is the proper place in which to
consider briefly the position of Rubens in Flemish Art.
From the days of the Van Eycks to those of Gerard David,
painting in the Low Countries had followed a strictly _national_
line of development. Its growth was organic and internal. With
Quentin Matsys, and still more with Bernard van Orley, Pourbus,
and the rest, the influence of the _Italian Renaissance_ had
begun to interfere with the native current of art in the Low
Countries. It was Rubens who finally transformed Flemish
painting by adopting to a certain extent the grandiose style of
the later Italian and especially the Venetian Masters, at the
same time that he transfused it with local feeling and with the
private mark of his own superabundant and vigorous
individuality.
=Rubens= was =an Antwerp man=, by descent and education, though
accidentally born at Siegen in Nassau. His father was an Antwerp
justice of an important family, exiled for supposed Calvinistic
leanings, and disgraced for an intrigue with a royal lady, Anna
of Saxony, the eccentric wife of William of Orange. A gentleman
by birth and breeding, Peter Paul Rubens painted throughout life
in the spirit of a generous, luxurious aristocrat. His master
was Otto van Veen, Court Painter to the Dukes of Parma, and
himself an Italianised Flemish artist, whose work is amply
represented in the Museum. Early in life, Rubens =travelled in
Italy=, where he imbibed to a great extent the prevailing tone
of Italian art, as represented by Titian, Veronese, and to a
less extent, Tintoretto, as well as by Domenichino and the later
Roman School of painters. To these influences we must add the
subtler effect of the general spirit of the late 16th and early
17th centuries, the age when voyages to America and to India,
and the sudden opening of the Atlantic seaboard, had caused in
men's minds a great ferment of opinion and given rise to a new
outburst of activity and struggle. Romance was rife. The world
was turned upside down. It was the day of Spanish supremacy, the
day when the gold and silver of the Indies poured in vast sums
into Madrid and the Low Countries. The Mediterranean had given
way to the Atlantic, Venice to Antwerp. In England, this age
gave us the rich and varied Elizabethan literature; in the Low
Countries, it gave us the highly analogous and profusely lavish
art of the School of Rubens.
Rubens lived his life throughout on =a big scale=. He travelled
much. He was statesman and diplomatist as well as painter. He
moved from Paris to London, from Madrid to Mantua. All these
things give a tone to his art. He is large, spacious, airy,
voluptuous. He has a bold self-confidence, a prodigal freedom,
an easy opulence. He delights in colossal figures, in regal
costume, in court dresses and feathers,—the romance and
pageantry of the royal world he lived in. Space seems to swell
and soar on his canvas. Vast marble halls with huge pillars and
lofty steps are the architectural background in which his soul
delights. His outlines are too flowing to be curbed into stiff
correctness. His sturdy Flemish nature, again, comes out in the
full and fleshy figures, the florid cheeks, and the abundant
fair hair of his female characters. All scenes alike, however
sacred, are for him just opportunities for the display of
sensuous personal charm, enlivened by rich costume or wealthy
accessories. Yet in his large romantic way he is doing for
cosmopolitan mercantile Antwerp in the 17th century what Van
Eyck and Memling did for cosmopolitan Ghent and Bruges in the
15th.
One more peculiarity of his art must be mentioned. The early
painters, as we saw in the St. Ursula casket, had little sense
of real dramatic life and movement. Rubens had learned to admire
this quality in his Venetian masters, and he bettered their
instruction with Flemish force and with the =stir and bustle= of
a big seaport town in an epoch of development. His pictures are
full, not merely of life, but of strain, stress, turmoil. It is
more than animation—it is noise, it is tumult. He often forgets
the sacredness of a scene by emphasizing too much the muscular
action and the violent movement of those who participate in it.
This is particularly noticeable in the Descent from the Cross in
the Cathedral, and still more in the famous Coup de Lance at the
Museum.
The astonishing number of pictures which Rubens has left may be
accounted for in part by his incredible rapidity of
execution—he dashed off a huge picture in a fortnight,—but in
part also by the fact that he was largely assisted by a numerous
body of pupils. Of these, =Van Dyck= was by far the most
individual, the tenderest, the most refined: and not a few of
his stately and touching masterpieces may here be studied.
The =Dutch School= is also represented by several excellent
small pictures.
Of alien art, there are a few fine pieces by =Early Italian=
artists.]
The =entrance door= is under the great portico on the west front, facing
the river. Open daily, 9 or 10 to 4 or 5, 1 fr. per person: free on
Sundays. (Inquire hours of hotel porter.)
You pass from the Vestibule (sticks and umbrellas left) into a Hall and
Staircase of palatial dimensions, admirably decorated with fine modern
paintings by N. De Keyser, of Antwerp, representing the Arts and Artists
of the city, the influence upon them of Italian masters, and the
recognition extended to their work in London, Paris, Rome, Bologna,
Amsterdam, and Vienna. I do not describe these excellent pictures, as
the inscriptions upon them sufficiently indicate their meaning, but they
are well worth your careful attention.
The =rooms= are lettered (A. B. C. etc.) over the doorways. On reaching
the top of the staircase, pass at once through Rooms J. and I., and go
straight into
ROOM C.
=Hall of the Ancient Masters=, Flemish or foreign.
=Right of the door=,
224. Justus of Ghent: a bland old pope, probably St. Gregory, holding a
monstrance, between two angels. In the background, a curious
altar-piece, with the Annunciation, Nativity, Adoration of the Magi,
Flight into Egypt, Presentation in the Temple, and Finding of Christ in
the Temple. Above it, two female saints (or figures of Our Lady?). A
good work, in an early dry manner.
Above it, 463. Madonna and Child, by Van Orley: the landscape by
Patinier. From a tomb in the Cathedral.
383. Van der Meire. Triptych from an altar; Centre, Way to Calvary, with
St. Veronica offering her napkin, and brutal, stolid Flemish soldiers
bearing the hammer, etc. In the background, the Flight into Egypt. The
_wings_ have been transposed. _L._, (should be R.), the Finding of
Christ in the Temple. _R._ (should be L.), the Presentation in the
Temple.
Above it, 380. Van den Broeck (1530-1601): a Last Judgment. Interesting
for comparison with previous examples. Renaissance nude.
557. Unknown. Dutch School of the early 16th century. The Tiburtine
Sibyl showing the Emperor Augustus the apparition of the Virgin and
Child on the Aventine. A page, his robe embroidered with his master's
initial A., holds the Emperor's crown. Very Dutch architecture. (The
Catalogue, I think erroneously, makes it the Madonna appearing to
Constantine.)
560. Good hard early Dutch portrait.
527. Unknown. Resurrection, the Saviour, bearing the white pennant, with
red cross, and sleeping Roman soldiers.
42. An Adam and Eve, attributed to Cranach the Elder. Harsh northern
nude.
341. Good portrait by Susterman, _alias_ Lambert Lombard.
Above these, Madonna, in the Byzantine style, with the usual Greek
inscriptions.
521. School of Albert Dürer: Mater Dolorosa, with the Seven Sorrows
around her.
549. Good Flemish portrait of William I., Prince of Orange.
Above, 387, Van der Meire: an Entombment, with the usual figures,
Nicodemus and Joseph of Arimathea; the Magdalen in the foreground with
the box of ointment; the Mater Dolorosa supported by St. John (in red);
and, behind, the two Maries. In the background, a _Pietà_—that is to
say, the same group mourning over the Dead Saviour.
425. Van Hemessen: The Calling of Matthew from the receipt of custom.
Harsh and uninteresting.
568. School of Quentin Matsys: Christ and St. Veronica. Probably part
only of a Way to Calvary. The spiked club is frequent.
241. Quentin Matsys: a fine and celebrated *Head of the Saviour
Blessing, with more expression than is usual in the Flemish type of this
subject. Notice even here, however, close adhesion to the original
typical features.
242. Quentin Matsys: Companion *Head of Our Lady, as Queen of Heaven.
Full of charm and simplicity.
Between these, 4, =*Antonello da Messina= (an Italian profoundly
influenced by the School of Van Eyck, and the first to introduce the
Flemish improvements in oil-painting into Italy). Crucifixion, with St.
John and Our Lady. This work should be carefully studied, as a
connecting link between the art of Flanders and Italy. It is painted
with the greatest precision and care, and bears marks everywhere of its
double origin—Flemish minuteness, Italian nobility.
254. Memling: **admirable cold-toned portrait of a member of the De Croy
family. The hands, face, and robe, are all exquisitely painted.
Centre of the wall, 412, good early copy of Jan van Eyck's altar-piece
for Canon George van der Paelen, in the Academy at Bruges. If you have
not been there, see page 59 for particulars. Better preserved than the
original: perhaps a replica by the master himself.
519. Crucifixion, with Our Lady and St. John, on a gold background.
Interesting only as a specimen of the very wooden Dutch painting of the
14th century. Contrast it with the Van Eyck beneath it, if you wish to
see the strides which that great painter took in his art.
397. Good hard *portrait of Philippe le Bon of Burgundy, an
uninteresting, narrow-souled personage, wearing the collar of the Golden
Fleece, by Roger van der Weyden.
43. Cranach the Elder: Charity. A study of the nude, somewhat more
graceful than is the wont of this painter.
264. Mostaert (Jan, the Dutchman), tolerable hard portrait: same person
reappears in 262.
179. Gossaert: *a beautiful panel representing the Return from Calvary.
The Mater Dolorosa is supported by St. John. On the L., the Magdalen
with her pot of ointment; R., the other Maries. Very touching. Notice
the Flemish love for these scenes of the Passion and Entombment.
198. Hans Holbein the Younger: **admirable portrait of Erasmus. It
lives. Full of vivacity and scholarly keenness, with the quick face of a
bright intelligence, and the expressive hands of a thinker. The fur is
masterly.
180. Gossaert: group of figures somewhat strangely known as "The Just
Judges." Probably a single surviving panel from an extensive work of the
same character as the Adoration of the Lamb at Ghent.
263. Jan Mostaert: *very fine portrait of a man in a large black hat and
yellow doublet. Pendant to 264.
558. Holy Family. Dutch School. Early 16th century.
202. Lucas van Leyden: *portraits. Characteristic, and well thrown out
against the background.
566. School of Quentin Matsys: a genre piece; an unpleasant
representation of a young girl attempting to cut the purse strings of an
old man. Probably a companion picture to one now in the possession of
the Countess of Pourtalés, Paris.
Above these, 168, Triptych by Fyol, German School. _Centre_, the
Adoration of the Magi. The Old King has removed his crown, as usual, and
presented his gift. He is evidently a portrait: he wears a collar of the
Golden Fleece, and is probably Philippe le Bon. Behind him, the
Middle-aged King, kneeling; then the Young King, a Moor, with his
offering. (The story of the Three Kings—Gaspar, Melchior,
Balthasar—was largely evolved in the Cologne district, where their
relics formed the main object of pious pilgrimage.) To the R., an
undignified Joseph, with his staff, and the peculiar robe with which you
are now, I hope, familiar. In the background, the family of the donor,
looking in through a window. The _wings_ have, I think, been misplaced.
L., The Circumcision; R., Nativity: notice the ox and ass, and the
costume of Joseph.
325. Schoreel: Crucifixion, with Our Lady, St. John, the Magdalen, and
angels catching the Holy Blood. (A frequent episode.)
Above it, 570, School of Gossaert: Our Lady.
262. Jan Mostaert: The Prophecies of Our Lady. Above, she is represented
as Queen of Heaven, in an oval glory of angels, recalling the Italian
mandorla. Below, those who have prophesied of her: in the centre,
Isaiah, with scroll, "Behold, a Virgin shall conceive," etc.: R. and L.,
Micah and Zechariah. Further R. and L., two Sibyls. The one to the R. is
the same person as 264.
567. School of Quentin Matsys: Favourite subject of the Miser.
25. More monstrosities by Bosch.
=Beyond the door=,
534. Unknown: Flemish School: Assumption of Our Lady. Above, the Trinity
waiting to crown her.
123. Dunwege: German School. The Family of St. Anne, resembling in
subject the Quentin Matsys at Brussels. Centre, St. Anne enthroned.
Below her, Our Lady and the Divine Child. (Often Our Lady sits on St.
Anne's lap.) L., Joachim offers St. Anne and Our Lady cherries. (See
_Legends of the Madonna_.) R., St. Joseph, with his staff and robe. On
either side, the Maries, with their children, here legibly named, and
their husbands. (From a church at Calcar.)
Above this, 523. Triptych: Madonna and Child, with donors and patron
saints (Sebastian and Mary Magdalen). Note their symbols. On either
side,
Van der Meire: 388: Mater Dolorosa; her breast pierced with a sword: and
on the other side of the triptych 389 (attribution doubtful, according
to Lafenestre), a donatrix with St. Catherine, holding the sword of her
martyrdom.
569. School of Gossaert, Way to Calvary, with the usual brutal soldiers.
47. Herri Met de Bles: Repose on the Flight into Egypt. Notice the
sleeping St. Joseph, and the staff, basket, and gourd, which mark this
subject.
539. Good unknown Flemish portrait.
Beyond this, a frame containing five excellent small pictures.
243. Quentin Matsys: *St. Mary Magdalen with her alabaster box. Sweet
and simple. In reality, portrait of an amiable round-faced Flemish young
lady, in the character of her patron saint. Her home forms the
background.
526 and 538. Fine unknown portraits.
199. *Exquisite and delicate miniature by Hans Holbein the Younger.
(Lafenestre doubts the attribution.)
132. Fouquet, the old French painter, 1415-1485. Hard old French picture
of a Madonna and Child, of the regal French type, with solid-looking red
and blue cherubs. Said to be a portrait of Agnes Sorel, mistress of
Charles VII. From the Cathedral of Melun.
Then, another case, containing six delicate works of the first
importance.
396. *Roger van der Weyden (more probably, School of Van Eyck):
Annunciation. The angel Gabriel, in an exquisitely painted bluish-white
robe, has just entered. Our Lady kneels at her prie-dieu with her book.
In the foreground, the Annunciation lily; behind, the bed-chamber. The
Dove descends upon her head. This is one of the loveliest works in the
collection.
253. Memling: **Exquisite portrait of a Premonstratensian Canon.
28. Dierick Bouts: The Madonna and Child. An excellent specimen of his
hard, careful manner.
203. Lucas of Leyden: David playing before Saul.
30. Bril, 1556-1626. Fine miniature specimen of later Flemish landscape,
with the Prodigal Son in the foreground.
559. Unknown but admirable portrait of a man.
223. Justus van Ghent: Nativity, with Adoration of the Shepherds. A good
picture, full of interesting episodes.
Beyond these, another case, containing fine small works. A beautiful
little *Madonna with the Fountain of Life (411) by Jan van Eyck, closely
resembling a large one by Meister Wilhelm, in the Museum at Cologne. Two
good unknown portraits. A splendid **portrait of a medallist (5) by
Antonello da Messina (sometimes attributed to Memling). A portrait (33)
of François II. of France as a child, by Clouet, of the old French
School. A characteristic *Albert Dürer (124), portrait of Frederick III.
of Saxony: and a good Gossaert (182). These do not need description, but
should be closely studied.
The place of honour on this wall is occupied by 393, a magnificent
=**Seven Sacraments=, usually attributed to =Roger van der Weyden=,
though believed by some to be a work of his master, Robert Campin of
Tournai. At any rate, it is a work full of Roger's mystic spirit. In
form, it is a triptych, but the main subjects are continued through on
to the wings. The _central panel_ represents the Sacrament of the Mass,
typified in the foreground by a Crucifixion, taking place in the nave of
an unknown Gothic church. At the foot of the cross are the fainting
Madonna, supported by St. John (in red as usual) and a touching group of
the three Maries. The robe of one to the left overflows into the next
panel. In the background, the actual Mass is represented as being
celebrated at the High Altar. The architecture of the church (with its
triforium, clerestory, and apse, and its fine reredos and screen) is
well worth notice. So are the figures of Our Lady, St. Peter and St.
John, on the decorative work of the screen and reredos. I believe the
kneeling figure behind the officiating priest to be a portrait of the
donor. The _side panels_ represent the other sacraments, taking place in
the aisles and lateral chapels of the same church. L., Baptism,
Confirmation, Confession: in the Confirmation, the children go away
wearing the sacred bandage. R., Holy Orders, Matrimony, Extreme Unction.
Each of these groups should be carefully noted. The _colours_ of the
angels above are all symbolical:—white (innocence) for Baptism: yellow
(initiation) for Confirmation: red (love or sin) for confession and
absolution: green (hope) for the Eucharist: purple (self-sacrifice) for
Holy Orders: blue (fidelity) for Marriage: violet, almost black, (death)
for Extreme Unction. The picture is full of other episodes and mystical
touches. In all this beautiful and touching composition, the Mary to the
right of the Cross is perhaps the most lovely portion. For a fine
criticism, see Conway.
Beyond this, another frame with exquisite small works.
250. Quentin Matsys: Head of Christ, with the Crown of Thorns and Holy
Blood; painful.
540. Admirable unknown miniature portrait.
544. Excellent little St. Helena.
542. A little donor, with his patron, St. John.
204, 205, 206. Good Lucas of Leyden, of the Four Evangelists (John
missing). Luke, with the bull, painting; Matthew, with the angel, and
Mark, with the lion, writing.
537. Admirable unknown portrait. These little works again need no
description, but close study.
Above them, 244. Quentin Matsys (?). The Misers, one of the best known
of this favourite subject.
Then, another frame of miniatures.
517, 518. Unknown Flemish 14th century Madonna and Child, with donor and
wife.
541, 522. Tolerable portraits.
545. Fine portrait, of the Spanish period.
410. =**Van Eyck's= celebrated unfinished =St. Barbara=, holding her
palm of martyrdom, and with her tower in the background. It should be
closely studied, both as an indication of the master's method, and as a
contemporary drawing illustrating the modes of mediæval building. For a
careful criticism, see Conway.
Above these, Engelbrechtsen, 130. St. Hubert, attired as bishop, bearing
his crozier and hunting horn, and with the stag beside him, with the
crucifix between its horns.
127. The same. St. Leonard releasing prisoners.
Then, another case of good small pictures.
3. A Fra Angelico. Interesting in the midst of these Flemish pictures.
St. Romuald reproaching the Emperor Otho III. for the murder of
Crescentius.
32. Petrus Christus (?). A donor and his patron, St. Jerome.
64. A landscape by Patinir.
536. A Baptism of Christ, where note the conventional arrangement and
the angel with the robe.
561. Triptych. Madonna and Child. St. Christopher, and St. George. Harsh
and angular.
548. Mater Dolorosa, transpierced by the sword.
535. Good Flemish Madonna with angels.
207. Lucas of Leyden: Adoration of the Magi. You can now note for
yourself the ox, ass, Joseph, position, age, and complexion of Kings,
etc.
29. Attributed (doubtfully) to Dierick Bouts: St. Christopher wading,
with the Infant Christ. In the background, the hermit and lantern. (See
Mrs. Jameson.)
176. Giotto: A St. Paul with the sword. Characteristic of early
Florentine work.
257, 258, 259, 260. Simone Martini of Siena: Four panels. _Extreme
ends_, **Annunciation, closely resembling the figures in the Ufizzi at
Florence: Annunciations are often thus divided into two portions.
_Centre_, Crucifixion and Descent from the Cross. These exquisitely
finished little works are full of the tender and delicate spirit of the
early Sienese School. In the Crucifixion, notice particularly the
Magdalen, and St. Longinus piercing the side of Christ. Our Lady in the
Annunciation has the fretful down-drawn mouth inherited by early Italian
art from its Byzantine teachers.
177. Giotto: St. Nicolas of Myra with the three golden balls, protecting
a donor.
Above are three good portraits by Van Orley, and other works which need
no description.
On easels at the end, 255. =Attributed to Memling=: **Exquisite =Madonna
and Child= in a church. Our Lady, arrayed as Queen of Heaven, with a pot
of lilies before her, stands in the nave of a lovely early Gothic
cathedral, with a later Decorated apse, and admirable rood-screen. Every
detail of the tiles, the crown, the screen, and the robe, as well as Our
Lady's hair and hands, should be closely looked into. This is one of the
loveliest pictures here. It is a very reduced copy from one by Jan van
Eyck at Berlin: the church is that of the Abbey of the Dunes near
Furnes. Its attribution to Memling has been disputed: Conway believes it
to be by a follower. In any case, it is lovely.
256. **Companion panel, of the donor, a Cistercian Abbot of the Dunes,
in a sumptuous room, half bed-chamber, half study, with a beautiful
fireplace and fire. He kneels at his prayers, having deposited his mitre
on a cushion beside him, and laid his crozier comfortably by the
fireplace. Creature comforts are not neglected on the side-board. Here
also every decorative detail should be closely examined. These are two
of the very finest works of the School of Memling. Probably the Abbot
admired Jan van Eyck's Madonna, painted for a predecessor, and asked for
a copy, with himself in adoration on the other wing of the diptych.
At the back, on a revolving pivot,
530, 531. Christ blessing, and a Cistercian Canon in adoration. As
usual, the outer panels are less brilliant in colouring than the inner.
Notice the Alpha and Omega and the P. and F. (for Pater and Filius) on
the curtain behind the Saviour. These works are by an inferior hand.
The other easel has a fine (208, 209, 210) *Lucas van Leyden: Adoration
of the Magi, with fantastic elongated figures. Note the ruined temple.
The other features will now be familiar. Lucas's treatment is peculiar.
_L._, St. George and the Dragon. The saint has broken his lance and
attacks the fearsome beast with his sword. In the background, the
Princess Cleodolind and landscape. _R._, The donor, in a rich furred
robe, and behind him, St. Margaret with her dragon. At the back, 181,
_wings_, by the same, with a peculiar Annunciation (the wings being
open, reversed in order). Between them has been unwisely inserted an
Ecce <DW25> by Gossaert.
Now, go straight through Rooms H, F, and E, to three rooms _en suite_,
the last of which is
ROOM A,
containing the Transitional Pictures. (It is usual to skip these insipid
works of the intermediate age, and to jump at once from the School of
Van Eyck to the School of Rubens—I think unwisely—for Rubens himself
can only properly be appreciated as _the product of an evolution_, by
the light of the two main influences which affected him—his Flemish
masters, and his Italian models, Veronese and Giulio Romano.) Begin at
the =far end=, near the lettered doorway, and note throughout the effort
to imitate Italian art; the endeavour at classical knowledge; and the
curious jumble of Flemish and Tuscan ideas. But the Flemish skill in
portraiture still continues.
698. Good portrait of Giles van Schoonbeke, by P. Pourbus.
Next to it, 103, Martin De Vos, the Elder: St. Anthony the Abbot,
accompanied by his pig and bell, and his usual tempters, burying the
body of St. Paul the Hermit, whose grave two lions are digging. To the
R., hideous Flemish devils, grotesquely horrible. Above, phases of the
Temptation of St. Anthony.
372. Michael Coxcie: Martyrdom of St. George—one of his tortures. Good
transitional work, inspired by Italian feeling.
72. M. De Vos: Triptych, painted for the altar of the Guild of
Crossbowmen in the Cathedral. _Centre_, Triumph of the risen Christ. In
the foreground, St. Peter (keys), and St. Paul (sword), with open pages
of their writings. L., St. George, patron of the Crossbowmen, with his
banner and armour; R., St. Agnes with her lamb. _Left panel_, Baptism of
Constantine by St. Sylvester. _Right panel_, Constantine ordering the
erection of the Church of St. George at Constantinople. In the sky, the
apparition of Our Lady to the Emperor. A gigantic work, recalling the
later Italian Renaissance, especially the Schools of Bronzino and Giulio
Romano.
374. Michael Coxcie: Martyrdom of St. George; the other wing of the same
triptych in honour of St. George as 372; central portion lost.
89. M. De Vos: St. Conrad of Ascoli, a Franciscan friar, in devout
contemplation of the founder of his Order, St. Francis, receiving the
Stigmata. Around it, small scenes from the life of St. Conrad,
unimportant. Below, Devotion at the tomb of St. Conrad: royal personages
praying, offerings of rich images, and the sick healed by his relics. A
curious picture of frank corpse-worship.
699. Good portrait by Pourbus.
576. Triptych, unknown. St. Eligius of Noyon (St. Éloy), one of the
apostles of Brabant, preaching to a congregation really composed of good
local portraits. (A pious way of having oneself painted.) R. and L., St.
Eligius feeding prisoners, and St. Eligius healing the sick.
741. Another of Bernard van Orley's General Resurrections, the type of
which will now be familiar to you. In the centre, strangely introduced
group of portraits of the donors, engaged in burying a friend, whose
memory this triptych was doubtless intended to commemorate. On either
wing, the six works of Mercy (the seventh, burial, is in the main
picture).
77. Good transitional triptych, by M. De Vos, for the Guild of
Leather-dressers. _Centre_, The Incredulity of St. Thomas. On the
_wings_, Scenes from the life of the Baptist. L., Baptism of Christ;
where note the persistence of the little symbolical Jordan, with angels
almost inconspicuous. R., The Decollation of St. John. Salome receiving
his head in a charger. In the background, Herodias.
371. Coxcie the Younger: Martyrdom of St. Sebastian, patron saint of
Bowmen, from their altar in the Cathedral. An attempt to be very
Italian. The _wings_ of this triptych are by Francken. L., St. Sebastian
exhorting Marcus and Marcellinus to go to martyrdom. R., St. Sebastian
miraculously healing the dumb woman, with portrait spectators, in dress
of the period, deeply interested.
Now go on into
ROOM B,
(unlettered, the centre of the three). It contains works of an earlier
period.
The =left wall= is entirely occupied by three large panels of a fine old
Flemish 15th century picture, attributed to Memling (and apparently
accepted as his by Lafenestre), representing *Christ Enthroned, with orb
and cross, surrounded by choirs of angels; those in the central panel
singing; the others, playing various musical instruments. This is a
beautiful work, but less pleasing than those of the same school on a
smaller scale. It has been recently bought from the monastery of Najera
in Spain. It was intended, I think, to be seen at a height, probably on
an organ-loft, and loses by being placed so near the eye of the
spectator.
The opposite wall, R., is occupied by 245, =Quentin Matsys's
masterpiece=, the triptych of =**the Entombment=, painted for the altar
of the Guild of Cabinet-makers. The colouring is much more pleasing than
in the Family of St. Anne at Brussels. _Central panel_, The Entombment.
Nicodemus supports the emaciated body of the dead Saviour, while Joseph
of Arimathea wipes the marks of the crown of thorns from his head. The
worn body itself, with a face of pathetic suffering, lies on the usual
white sheet in the foreground. At the foot, Mary Magdalen, with her pot
of ointment and long fair hair, strokes the body tenderly. In the centre
is the fainting Madonna, supported, as always, by St. John, in his red
robe. Behind are the three Maries. The usual attendant (a ruffianly
Fleming, in a queer turban-like cap) holds the crown of thorns. At the
back, preparations for the actual placing in the sepulchre. In the
background, Calvary.
The _wings_ have scenes from the lives of the two St. Johns. L., The
daughter of Herodias, a very mincing young lady, in a gorgeous dress,
brings the head of St. John the Baptist on a charger to her mother and a
fiercely-bearded Herod. The queen appears to be about to carve it.
Above, a gallery of minstrels. Admirable drapery and accessories. The R.
wing has the so-called Martyrdom of St. John the Evangelist, in the
cauldron of boiling oil, with a delightful boy spectator looking on in a
tree. The Emperor Domitian (older than history), on a white horse,
behind. Flemish varlets stir the fire lustily. This noble work
originally decorated the altar in the Chapel of the _Menuisiers_ of
Antwerp in the Cathedral.
On easels, 649, Claeissens: Triptych of the Crucifixion, with the Way to
Calvary and the Resurrection. Elongated, attenuated figures.
680. Giles Mostaert (the elder): Singular complex picture, painted for
the Hospital of Antwerp; representing, above, The General Resurrection:
Christ enthroned between Our Lady and St. John-Baptist. Beneath, naked
souls rising from the tomb. To the L., St. Peter welcomes the just at
the gate of the Celestial City. To the R., devils drive the wicked into
the gaping jaws of Hell. Beneath, the courses that lead to either end:
the Seven Works of Mercy, inspired by the Redeemer, and the Seven Deadly
Sins, suggested by devils. I will leave you to identify them (it is
easy).
Go on into
ROOM D,
containing more works of the Transition. These large altar-pieces of the
early 17th century, the period of the greatest wealth in Antwerp, though
often frigid, as works of art, are at least interesting as showing the
opulence and the tastes of the Antwerp guilds during the epoch of the
Spanish domination. They are adapted to the huge Renaissance churches
then erected, as the smaller triptychs of the 15th century were adapted
to the smaller Gothic altars.
529. Feast of Archers, with the King of the Archers enthroned in the
background.
696, 697. Tolerable portraits by Pourbus.
183. A Madonna by Gossaert.
114. Frans Floris: St. Luke painting, with his bull most realistically
assisting, and his workman grinding his colours. From the old Academy of
Painters, whose patron was St. Luke. Italian influence.
135. Ambrose Francken: Loaves and fishes.
148. The same. Decollation of St. Cosmo and St. Damian: painted for the
Guild of Physicians, of whom these were the patron saints.
357. A splendid and luminous =Titian=, in the curious courtly ceremonial
manner of the Venetian painters. **Pope Alexander VI. (Borgia), in a
beautiful green dalmatic, introducing to the enthroned St. Peter his
friend, Giovanni Sforza da Pesaro, Bishop of Paphos, and admiral of the
Pope's fleet. At the bishop's feet lies his helmet, to show his double
character as priest and warrior. He grasps the banner of the Borgias and
of the Holy Church. In the background (to show who he is), the sea and
fleet. St. Peter's red robe is splendid. The Venetians frequently paint
similar subjects,—"Allow me to introduce to your Sainthood," etc. This
is a fine work in Titian's early harder manner, still somewhat
reminiscent of the School of Bellini. Its glorious but delicate colour
comes out all the better for the crudity of the works around it.
146. Ambrose Francken: St. Cosmo and St. Damian, the Doctor Saints,
amputating an injured leg, and replacing it by the leg of a dead Moor.
In the background, other episodes of their profession. (Wing of the
triptych for the Guild of Physicians.)
83. M. De Vos: Triptych, painted for the Guild of the Mint, and allusive
to their functions. _Centre_, The Tribute Money. "Render unto Cæsar,"
etc., with tempting Pharisees and Sadducees, and Roman soldiers. In the
foreground, St. Peter in blue and yellow, with his daughter Petronilla.
_Left wing_: Peter, similarly habited, finds the tribute money in the
fish's mouth. _Right wing_: The Widow's Mite. (The French titles, "Le
Denier de César," "Le Denier du Tribut," "Le Denier de la Veuve," bring
out the allusion better.)
88. M. De Vos: St. Luke painting Our Lady, with his bull, as ever, in
attendance. The wings by others. L., St. Luke preaching. R., St. Paul
before Felix. From the altar of the (painters') Confraternity of St.
Luke in the Cathedral.
113. Frans Floris: Adoration of the Shepherds. Note persistence of
formal elements from old School, with complete transformation of spirit.
663. Floris: Judgement of Solomon.
112. Frans Floris's horrible St. Michael conquering the devils; the most
repulsive picture by this repulsive and exaggerated master.
Right and left of it, good late Flemish portraits of donors.
483. Portrait by Van Veen, Rubens's master.
ROOM E
contains chiefly works of the school of Rubens, most of which can now be
satisfactorily comprehended by the reader without much explanation. I
will therefore treat them briefly.
=Left of the door=,
82. A Nativity, by De Vos. Can be instructively compared with earlier
examples.
57. Good 17th century landscape.
646. Attributed to Brueghel: Paying tithes.
644. P. Brueghel the Younger: A village merry-making ("Kermesse
Flamande"). With more than the usual vulgarity of episode.
722 and 724. Capital portraits.
Good Still life, etc.
ROOM F
contains nothing which the reader cannot adequately understand for
himself. Omit Room G for the present (it contains the Dutch Masters),
and turn instead into
ROOM H,
mostly devoted to works of the =School of Rubens=.
_End Wall_, 305. =Rubens=: *The Last Communion of the dying St. Francis
of Assisi. A famous work, in unusually low tones of colour—scarcely
more than chiaroscuro. St. Francis, almost nude, is supported by his
friars. Above, angels, now reduced to cherubs, wait to convey his soul
to Heaven. Painted for the altar of St. Francis in the Franciscan Church
of the Récollets. See it from the far end of the room, where it becomes
much more luminous.
On either side, 662, good portrait by S. De Vos (himself, dashing and
vigorous: every inch an artist): and
104. C. De Vos: Admirable and life-like **portrait of the messenger or
porter of the Guild of St. Luke, the Society of Painters of Antwerp,
exhibiting the plate belonging to his confraternity. He is covered with
medals, which are the property of the Society, and has the air of a
shrewd and faithful servant. This living presentment of a real man is
deservedly popular.
661. Tolerable portrait by C. De Vos.
403. Van Dyck's *Entombment (or Pietà), often called Descent from the
Cross. This is one of his noblest pictures, but badly restored.
335. Angry swans disturbed by dogs. Snyders.
215. Jordaens: Last Supper. The effect of gloom somewhat foreshadows
Rembrandt.
401. Van Dyck: **A Dominican picture (Guiffrey calls it "cold and
empty"), painted at his father's dying wish for the Dominican Nunnery at
Antwerp. The two great saints of the Order, St. Dominic, the founder,
and St. Catherine of Assisi, the originator of the female branch, stand
at the foot of the Cross, which is itself a secondary object in the
picture. St. Dominic looks up in adoration; St. Catherine, wearing the
crown of thorns, fervently embraces the feet of the Saviour. On the
base, a child angel, in a high unearthly light, with a half extinguished
torch, points with hope to the figure of the crucified Lord. The whole
is emblematic of belief in a glorious Resurrection, through the aid of
the Dominican prayers. Interesting inscription on the rock: "Lest earth
should weigh too heavily on his father's soul, A. van Dyck rolled this
stone to the foot of the Cross, and placed it in this spot."
381. Van Hoeck: Madonna and Child, with St. Francis, from the Franciscan
Church of the Récollets.
660. Tolerable portrait by C. De Vos.
406. Van Dyck's noble **Crucifixion, with the sun and moon darkened. One
of his most admirable pictures.
677. Jordaens: **Charming family scene, known by the title of "As sing
the Old, so pipe the Young." Three generations—grandparents, parents,
and children—engaged in music together. Very catching: a most popular
picture.
734. Good *portrait of a priest, by Van Dyck.
402. Fine *portrait of a bishop of Antwerp, by Van Dyck.
404. Van Dyck: **Pietà, altar-piece for a chapel of Our Lady of the
Seven Sorrows. Our Lady holds on her lap the dead Christ, while St. John
points out with his finger the wound in His hand to pitying angels. All
the formal elements in this scene—Our Lady, St. John, the angels,
etc.—belong to the earlier conception of the Pietà, but all have been
entirely transfigured by Van Dyck in accordance partly with the
conceptions of the School of Rubens, though still more with his own
peculiar imagination. It is interesting, however, to note in this
touching and beautiful picture, full of deep feeling, how far the type
of the St. John has been inherited, remotely, from the School of Van der
Weyden. Even the red robe and long hair persist. The features, too, are
those with which we are familiar. This is one of the gems of the
collection. It shows the direct influence of Italian travel modifying
Van Dyck's style, acquired from Rubens.
This room also contains several other excellent works of the School of
Rubens or his more or less remote followers, which I need not
particularize.
Now continue into
ROOM I,
containing what are considered to be the =gems= among the =Rubenses= and
the =later pictures=.
Right of the door, Rubens and Brueghel, 319: Small copy of the Dead
Christ. Schut, 327: The Beheading of St. George. A pagan priest, behind,
endeavours to make him worship an image of Apollo. Above, angels wait to
convey his soul to Heaven. This is a somewhat confused picture, with a
spacious composition and a fine luminous foreground: it is considered
its painter's masterpiece. Intended for the altar of the Archers (whose
patron was St. George), in Antwerp Cathedral.
673. Good still life by Gysels.
669. F. Francken: Portraits of a wealthy family in their own picture
gallery.
107. C. De Vos: *Portraits of the Snoek family, in devotion to St.
Norbert. This picture requires a little explanation. St. Norbert was the
Catholic antagonist of the heretic Tankelin at Antwerp in the 12th
century. In this frankly anachronistic picture the Snoek family of the
17th century, portly, well-fed burghers, are represented restoring to
the mediæval saint the monstrance and other church vessels removed from
his church during the Calvinist troubles. The Snoeks are _living
personages_; the Saint is envisaged as a _heavenly character_. It is, in
short, a highly allegorical picture of the family showing their devotion
to true Catholicism, and their detestation of current heresy. In the
background stands the town of Antwerp, with the Cathedral and St.
Michael. (From the burial chapel of the Snoek family at St. Michael.)
There is a Brueghel in Brussels Museum, representing St. Norbert
preaching against Tankelin.
307. Beyond the door, Rubens: **Triptych, to adorn a tomb, for the
funerary chapel of his friend Rockox. Compare, for size and purpose, the
Moretus tomb in the Cathedral. It shows the painter's early careful
manner, and represents in its _central piece_ the Incredulity of St.
Thomas. On the _wings_, the Burgomaster Nicolas Rockox, and his wife,
for whose tomb it was painted. The wings are finer than the central
portion. This early work, still recalling Van Veen's academic tone,
should be compared with the Van Veens and also with Rubens's fine
portrait of himself and his brother, with Lipsius and Grotius, in the
Pitti at Florence. It marks the earliest age, when he was still content
with comparatively small sizes, and gave greater elaboration to his
work, but without his later dash and vigour. M. Rooses thinks ill of it.
781. *Fine farmyard scene by Rubens, with the story of the Prodigal Son
in the foreground. One of the many signs of his extraordinary
versatility.
Beyond, on either side of the great Rubens, to be noticed presently, are
two pictures by his master, Otto van Veen: 480, The Calling of Matthew,
and 479, Zacchæus in the Fig-Tree. These two careful works recall the
later Italian Schools, more particularly Titian, and are good examples
of that careful academic transitional Flemish art which Rubens was to
transform and revivify by the strength of his own exuberant and powerful
personality. They are admirably placed here for comparison with
297. Rubens's famous altar-piece of the Crucifixion, for the Church of
the Franciscans, commonly known as the =**Coup de Lance=. In this
splendid work Rubens is seen in one of his finest embodiments. The
figure of Christ has fine virility. St. Longinus, to the L., on a white
horse, is in the very act of piercing his side. The Magdalen, embracing
the foot of the Cross, as ever, throws up her arms with supplicating
gesture. To the R. is the Madonna. Behind, a soldier is engaged in
breaking the limbs of the Impenitent Thief (always on Christ's L.) who
writhes in his torture. The whole work is full of Rubens's life and
bustle, well contrasted with the academic calm of the Van Veens beside
it. Even those who do not love Rubens (and I confess I am of them) must
see in such a work as this how his great powers _succeeded_ in effects
at which his contemporaries aimed ineffectually. Boldly dramatic, but
not sacred.
300. **Triptych by Rubens, commonly known as the =Christ à la Paille=,
painted for a tomb in the Cathedral (compare the Moretus one). In the
centre is a Pietà: Joseph of Arimathea supporting the dead body of the
Christ on the edge of a stone covered with straw. Behind, Our Lady and
another Mary, with the face of St. John just appearing in the
background. This "too famous" work is rather a study of the dead nude
than a really sacred picture. Some of its details overstep the
justifiable limits of horror. The _wings_ are occupied by, L., a
so-called Madonna and Child, really a portrait of a lady and boy—(his
wife and son?): R., St. John the Evangelist (patron of the person for
whose tomb it was painted), accompanied by his eagle.
706. Admirable *portrait by Rubens of Gaspard Groaerts, town secretary.
The bust is Marcus Aurelius.
171. J. Fyt: Excellent screaming eagles, with a dead duck. One of the
earliest and best presentations of wild life at home.
315. Rubens: Small copy (with variations) of the Descent from the Cross
in the Cathedral (by a pupil).
708. One of the best *portraits by Rubens in the Gallery, subject
unknown: lacks personal dignity, but Rubens has made the most of him.
The rest of this wall is occupied by some tolerable gigantic
altar-pieces and other good works of the School of Rubens. Most of them
derive their chief interest from their evident inferiority in design and
colour to the handicraft of the Master. They are the very same
thing—with the genius omitted.
_End wall_, 314, Rubens: called the =*Holy Trinity=. The Almighty
supports on His knees the figure of the dead Christ. Behind, hovers the
Holy Ghost. On either side, boy angels hold the crown of thorns, the
three nails, and the other implements of the Passion. This is really a
study in the science of foreshortening, and in the painting of the dead
nude, largely suggested, I believe, by a still more unpleasing Mantegna
in the Brera at Milan.
719. Above. Excellent fishmongery by Snyders.
212. Janssens: The Schelde bringing wealth to Antwerp, in the
allegorical taste of the period.
712. Rubens: St. Dominic.
172. Fyt: Excellent dogs and game.
299. Rubens: An **allegorical picture to enforce the efficacy of the
prayers of =St. Theresa=. The foundress of the Scalzi, dressed in the
sober robe of her Carmelite Order, is interceding with Christ for the
soul of Bernardino de Mendoza, the founder of a Carmelite convent at
Valladolid. Below, souls in Purgatory. In the left-hand corner stands
Bernardino, whom, at St. Theresa's prayer, angels are helping to escape
from torment. A fine luminous picture of a most unpleasing subject.
Painted for the altar of St. Theresa in the church of her own barefooted
Carmelites.
405. Van Dyck: Magnificent portrait of Cesare Alessandro Scaglia, in
black ecclesiastical robes, with lace cuffs and collar, and the almost
womanish delicate hands of a diplomatic, astute, courtier-like
ecclesiastic. The thoughtful eyes and resolute face might belong to a
Richelieu.
306. Rubens: =**The Education of the Virgin=, painted for a chapel of
St. Anne. A charming domestic picture of a wealthy young lady of
Flanders, pretending to be Our Lady, in a beautifully painted white silk
gown. Beside her, her mother, a well-preserved St. Anne, of aristocratic
matronly dignity. Behind is St. Joachim, and above, two light little
baby angels. The feeling of the whole is graceful courtly-domestic.
481, 482. Two scenes from the life of St. Nicholas, by Van Veen, the
master of Rubens. R., he throws through a window three purses of gold as
dowries for the three starving daughters of a poor nobleman. (This
ornate treatment contrasts wonderfully with the simpler early Italian
pictures of the same subject.) L., he brings corn for the starving poor
of Myra. Both pictures represent the _bourgeois_ saint in his favourite
character of the benefactor of the poor. They are here well placed for
contrast with
298. Rubens: =**Adoration of the Magi=, considered to be his finest
embodiment of this favourite subject, and one of his masterpieces. R.,
Our Lady and Child, with the ox in the foreground, and St. Joseph behind
her. L., two kings make their offerings. Behind them, the third, a Moor,
in an Algerian costume, leering horribly. Above, the ruined temple, the
shed, and the camels. M. Max Rooses calls this work "the _chef d'œuvre_
by which Rubens inaugurated his third manner," and other critics praise
loudly its gorgeous colouring, its audacious composition, its marvellous
certainty. To me, the great canvas, with its hideous ogling Moor, is
simply unendurable; but I give the gist of authoritative opinion.
312. Rubens: *The Holy Family, known as =La Vierge au Perroquet=. It is
chiefly remarkable as a rich and gorgeous piece of colouring, with a
charming nude boy of delicious innocence.
313. Rubens: *Crucifixion. One of his best embodiments of this subject.
Opposite wall.
709. Rubens, partly made-up: Jupiter and Antiope. A mythological
subject, treated in a somewhat Italian style, with a quaint little
huddling Cupid in the foreground.
Beyond this, three designs by Rubens for Triumphal Cars and Arches, on
the occasion of the entry of Ferdinand of Austria in 1635.
The whole of this room contains several other excellent altar-pieces,
many of which are Franciscan.
ROOM J.
R. and L. of the door, 105, C. De Vos: Portraits of a husband and wife,
with their sons and daughters.
370. Van Cortbemde: The Good Samaritan, pouring in oil and wine in a
most literal sense. In the background, the priest and the Levite.
109. Fine portrait of a well-fed Flemish merchant, William Van Meerbeck,
by C. De Vos. Behind him his patron, St. William.
748. Van Thulden: Continence of Scipio.
265. Murillo (Spanish School). St. Francis. A reminiscence of the older
subject of his receiving the Stigmata. It has the showy and affected
pietism of the Spaniards. A mere study.
214. Jordaens: Pharaoh in the Red Sea.
=Room N= contains several good portraits and views of the town and other
places, of the 17th and 18th centuries, many of them excellent as
studies of Old Antwerp, enabling us to appreciate the greatness of the
architectural losses which the city has sustained. These, however, are
essentially works for the visitor to inspect at his leisure. They need
little or no explanation. Notice especially 728, 348, 726.
775. Good unknown Flemish portrait.
22. Portraits by Boeyermans.
Room O, beyond, is filled for the most part with canvases of the school
of Rubens, mainly interesting for comparison with the works of the
master, and needing little comment.
Now return to
ROOM G,
containing the =Dutch Pictures=. Many of these are masterpieces of their
sort, but need here little save enumeration. The Reformation turned
Dutch art entirely upon portraiture, landscape, and domestic scenes.
Dutch art is =frankly modern=.
338. Jan Steen: Samson and the Philistines, as Jan Steen imaged it.
767. Admirable calm sea-piece, by Van der Capelle.
752. Weenix poaching on Hondecoeter's preserves.
502. A beautiful little Wynants.
399. W. van de Velde the younger: Calm sea, with ships.
398. Admirable cows, by A. van de Velde.
293. =Rembrandt=: **Admirable portrait of his wife, Saskia; almost a
replica of the one at Cassel, perhaps either painted by a pupil, or else
from memory after her death, and badly restored. It breathes Dutch
modesty.
349. Terburg: *Girl playing a mandoline.
705. Excellent *portrait of a Burgomaster, by Rembrandt.
324. A charming Schalken.
628. Unknown: perhaps =Frans Hals=: Excellent portrait of a calm old
lady.
668. Karel du Jardin: Admirable landscape, with cows.
188. Celebrated and vigorous **Fisher-boy of Haarlem, with a basket, by
=Frans Hals=, rapidly touched with the hand of a master.
339. One of Jan Steen's village merry-makings.
26. Delicate soft landscape, by J. and A. Both.
675. A mill, by Hobbema.
768. Van der Velde: Fine landscape, with cows.
427. Flowers by Van Huysum.
674. Admirable *portrait, by Frans Hals, of a round-faced, full-blooded,
sensuous Dutch gentleman. Full of dash and vigour.
738. Venus and Cupid, by W. van Mieris.
437. Excellent fishmonger, by W. van Mieris.
466. *The Smoker, by A. van Ostade.
682. Arch and charming portrait, by Mytens.
773. A fine Wynants.
382. B. van der Helst: Child with a dog.
679. Some of Molenaer's peasant folk.
713. Ruysdael: *Waterfall in Norway.
The room is full of other fine and delicately-finished pictures of the
Dutch School, of which I say nothing, only because they are of the kind
which are to be appreciated by careful examination, and which do not
need explanation or description.
Room K. contains Flemish works of the later School of Rubens and the
beginning of the decadence.
The remaining rooms of the Gallery have =modern pictures=, belonging to
the _historical_ and to the _archaic_ Schools of Antwerp. These works
lie without the scope of the present Guides, but many of them are of the
highest order of merit, and they well deserve attention both for their
own intrinsic excellence and for comparison with the works of the 15th
and early 16th centuries on which they are based. The paintings of Leys
and his followers, mainly in Room T, are especially worth consideration
in this connection. These painters have faithfully endeavoured to revert
to the principles and methods of the great early Flemish Masters, and
though their work has often the almost inevitable faults and failings of
a revival, it cannot fail to interest those who have drunk in the spirit
of Van Eyck and Memling.
On the _ground floor_, a good copy, 413, etc., of the Adoration of the
Lamb at Ghent, useful for filling up the gaps in your knowledge, and
more readily inspected at leisure and from a nearer point of view than
the original. The _portraits_ and _battle scenes_ on the remaining walls
need little comment.
_D_. THE TOWN IN GENERAL
[=Mediæval Antwerp=, now no more, lay within a narrow ring of
walls in the neighbourhood of the Cathedral. Its circumference
formed a rough semi-circle, whose base-line was the Schelde,
while its outer walls may still be traced on a good map about
the Rempart Ste. Catherine and the Rempart du Lombard. This
oldest district still remains on the whole an intricate tangle
of narrow and tortuous streets, with a few ancient buildings.
Later =Renaissance Antwerp= stretched to the limit of the
existing Avenues in their _northern_ part, though the _southern_
portion (from the Place Léopold on) extends beyond the boundary
of the 17th century city, and occupies the site of the huge
demolished Old Citadel, built by Alva. Antwerp, however, has
undergone so many changes, and so few relics of the mediæval age
now survive, that I can hardly apply to its growth the
historical method I have employed in other Belgian towns. It
will be necessary here merely to point out the principal
existing objects of interest, without connecting them into
definite excursions.]
The =centre of mediæval Antwerp= was the =Grand' Place=, which may be
reached from the Place Verte, through the little triangular Marché aux
Gants, in front of the main _façade_ of the Cathedral. It was, however,
so entirely modernized under the Spanish _régime_ that it now possesses
very little interest. The W. side of the square is entirely occupied by
the =Hôtel-de-Ville=, a poor Renaissance building, which looks very weak
after the magnificent Gothic Town Halls of Bruges, Ghent, Brussels, and
Louvain. The _façade_ is extremely plain, not to say domestic. The
ground floor has an arcade in imitation of Italian _rustica_ work, above
which come two stories with Doric and Ionic columns (and Corinthian in
the centre); the top floor being occupied by an open _loggia_,
supporting the roof. In the centre, where we might expect a spire, rises
a false gable-end, architecturally meaningless. The niche in the gable
is occupied by a statue of Our Lady with the Child (1585), the patroness
of the city, flanked by allegorical figures of Wisdom and Justice.
The =interior= has been modernized: but it contains one fine hall, the
_Salle Leys_, decorated with noble archaistic paintings by Baron Leys.
It may be visited before 9, or after 4 in the evening (1 franc to the
_concierge_). In the Burgomaster's Room is also a good Renaissance
chimney-piece, from the Abbey of Tongerloo, with reliefs of the Marriage
at Cana, the Brazen Serpent, and Abraham's Sacrifice.
The square contains a few =Guild Houses= of the 17th century, the best
of which is the _Hall of the Archers_, to the R. of the Hôtel-de-Ville,
a handsome and conspicuous building, lately surmounted by a gilt figure
of St. George slaying the Dragon, in honour of the patron saint of the
Archers. The older Guild Houses, however, were mostly destroyed by the
Spaniards. The square, as it stands, being Renaissance or modern, cannot
compare with the Grand' Place in most other Belgian cities.
The centre of the Place is occupied by a bronze =fountain=, with a
statue of Silvius Brabo, a mythical hero of mediæval invention, intended
to account for the name Brabant. He is said to have cut off the hand of
the giant Antigonus, who exacted a toll from all vessels entering the
Schelde, under penalty of cutting off the hand of the skipper,—a myth
equally suggested by a false etymology of Antwerp from _Hand Werpen_
(Hand throwing). The Hand of Antwerp, indeed, forms part of the city
arms, and will meet you on the lamp-posts and elsewhere. It is, however,
the ordinary Hand of Authority (Main de Justice), or of good luck, so
common in the East, and recurring all over Europe, as on the shields of
our own baronets. Such a hand, as an emblem of authority, was erected
over the gate of many mediæval Teutonic cities.
* * * * *
One of the objects best worth visiting in Antwerp, after the Cathedral
and the Picture Gallery, is the =Plantin-Moretus Museum= containing many
memorials of a famous family of =Renaissance printers=, whose monuments
we have already seen in the Cathedral. To reach it you turn from the
Place Verte into the Rue des Peignes, almost opposite the S. door of the
Cathedral. The second turning to the R. will lead you into the small
_Place du Vendredi_, the most conspicuous building in which is the
Museum.
Beyond advising a visit, it is difficult to say much about this
interesting old house and its contents. Those who are lovers of
_typography_ or of _old engravings_ will find enough in it to occupy
them for more than one morning. Such had better buy the admirable work,
_Le Musée Plantin-Moretus_, by M. Max Rooses, the conservator. On the
other hand, the general sight-seer will at least be pleased with the
picturesque courtyard, draped in summer by the mantling foliage and
abundant clusters of a magnificent old vine, as well as with the
spacious rooms, the carved oak doorways, balustrades, and staircases,
the delicious galleries, the tiles and fireplaces, and the many
admirable portraits by Rubens or others. Were it merely as a striking
example of a =Flemish domestic interior= of the upper class during the
Spanish period, this Museum would well deserve attention. Read the
following notes before starting.
The =house of Plantin= was established by Christopher Plantin of Tours
(born 1514), who came to Antwerp in 1549, and established himself as a
printer in 1555. He was made Archetypographer to the King by Philip II.,
and the business was carried on in this building by himself, his
son-in-law, Moretus, and his descendants, from 1579 till 1875. It was
Plantin's daughter, Martina, who married John Moretus (see the
Cathedral), and under the name of Plantin-Moretus the business was
continued through many generations to our own day. The firm were
essentially =learned printers=, setting up works in Latin, Greek, and
Hebrew, or even in Oriental types, and issuing editions of many
important classical authors. I will not describe the various rooms,
about which the reader can wander for himself at his own sweet will, but
will merely mention that they contain admirable portraits of the Plantin
and Moretus families, and of their famous editor, Justus Lipsius, by
Rubens, and others. (The Lipsius is particularly interesting for
comparison with the one at Florence in the Pitti.) The dwelling-rooms
and reception-rooms, of the family, with their fine early furniture, are
now open to the visitor. So is the quaint little =shop=, facing the
street, the composing room and proof-readers' room, the =study= occupied
by Lipsius, and the =library=, with examples of many of the books
printed by the firm. The original =blocks= of their woodcuts and of
their capital letters, with the plates of their engravings, are likewise
shown, together with old and modern impressions. Do not suppose from
this, however, that the place is only interesting to book-hunters or
lovers of engravings. The =pictures and decorations= alone,—nay, the
house itself—will amply repay a visit.
* * * * *
A walk should be taken from the Place Verte, by the Vieux Marché au Blé,
or through the Marché aux Gants, to the river front and =Port= of the
Schelde. (Follow the tram-line.) Here two handsome raised =promenoirs=
or esplanades, open to the public, afford an excellent *view over the
river, the old town, and the shipping in the harbour.
The _southernmost_ (and pleasantest) of these _promenoirs_ ends (S.)
near the =Porte de l'Escaut=, a somewhat insignificant gateway, designed
by Rubens, and adorned with feeble sculpture by Arthus Quellin. It stood
originally a little lower down the river, but has been removed, stone by
stone, to its present situation. The quaint red building, with hexagonal
turrets at the angles, visible from both esplanades, is the _Vieille
Boucherie_, or Butchers' Guild Hall, of 1503. It stands in a squalid
quarter, but was once a fine edifice. Near the N. end of this
_promenoir_, a =ferry-boat= runs at frequent intervals to the
=Tête-de-Flandre= on the opposite shore of the river. Here there is a
Kursaal and a strong fort. It is worth while crossing on a fine day in
order to gain a general view of the quays and the town. The
_northernmost promenoir_ is approached by an archway under the
castellated building known as the =Steen=. This is a portion of the old
Castle of Antwerp, originally belonging to the Margraves and the Dukes
of Brabant, but made over by Charles V. to the burghers of Antwerp. The
Inquisition held its sittings in this castle. It is now, though much
restored and quite modern-looking (except the portal), almost the only
remaining relic of Mediæval Antwerp, outside the Cathedral. It contains
a small Museum of Antiquities (unimportant; open daily, 10 to 4: 1 fr.:
Sunday and Thursday free). Unless you have plenty of time you need not
visit it.
A little way beyond the N. end of the _northern promenoir_ a tangled
street leads to the =Church of St. Paul=, which will be described
hereafter. Continuing along the _Quays_ in this direction you arrive at
last at the =Docks=. The large modern castellated building in front of
you is the =Pilotage=, round which sea-captains congregate in clusters.
Turning along the dirty quay to the R., you reach shortly on the L. the
site of the =Maison Hanséatique=, which was the _entrepôt_ in Antwerp of
the Hanseatic League. But it was burnt down a few years since, and its
place is now occupied by mean sheds and warehouses. All this quarter is
given over to the most unsightly and malodorous realities of modern
seafaring life and commerce.
* * * * *
Antwerp is somewhat ill provided with =drives= or country walks. The
prettiest of its =public gardens= is the little =Park=, which may be
reached from the Avenue des Arts by either of the three main Avenues
eastward, adorned respectively with statues of Quentin Matsys, Leys, and
Jordaens. The Park is a small but ingeniously laid out triangular area,
occupying the site of an old bastion, with a pleasing sheet of
ornamental water (originally the moat), crossed by a bridge, and backed
up by the twin spires of the modern Church of St. Joseph. Around it lies
the chief residential quarter of 19th century Antwerp. This is a cool
stroll in the afternoon, for one tired of sight-seeing. (Ask your hotel
porter when and where the band plays daily.) Further on in the same
direction is the pretty little public garden known as the =Pépinière=,
and lying in a pleasant open quarter. (Band here also.)
The =Zoological Garden=, just behind the Gare de l'Est, (admission 1
fr.) is well worth a visit if you are making a stay. It is particularly
well stocked with birds and animals, and has a rather pretty alpine
rock-garden. On Sunday afternoons, a good band plays here from 3 to 6,
and all Antwerp goes to listen to it.
* * * * *
A round of the =Avenues= may best be made in an open tram. The northern
portion, leading from the _Entrepôt_ and the _Goods Station_ as far as
the Place de la Commune, has few objects of interest. In the Place de la
Commune you pass, R., the handsome and ornate _Flemish Theatre_; while,
L., the Rue Carnot leads to the Zoological Garden, and to the
uninteresting industrial suburb of Borgerhout. Beyond this comes a
_Covered Market_, L., and then the Place Teniers, with a statue of
Teniers. Here the Avenue de Keyser leads, L., to the main Railway
Station (Gare de l'Est). Further on, L., the Avenue Marie-Thérèse, with
a statue of Matsys, runs to the Park. So, a little later, do the Avenue
Louise-Marie, with a statue of Leys, and the Avenue Marie-Henriette,
with a statue of Jordaens. The handsome building, with domed and rounded
turrets, on your R., just beyond the last-named Avenue, is the _Banque
Nationiale_, intended to contain the public treasure of Belgium in case
of war. Here the Chaussée de Malines leads off, S.E., to the
uninteresting suburb of Berchem. The heavy new building on the L., a
little further S., looking like a French mediæval _château_, is the
_Palais de Justice_. From this point the Avenue du Sud runs through an
unfinished district, occupying the site of the old _Citadel_ (Alva's)
past the _Museum_ and the _Palais de l'Industrie_, to the desolate Place
du Sud, with the _South Railway Station_. You can return by tram along
the Quays to the Hôtel-de-Ville and the Cathedral.
If you have plenty of time to spare, you may devote a day to
THE ROCOCO CHURCHES.
Most of the Antwerp churches, other than the Cathedral, are late Gothic
or Renaissance buildings, disfigured by all the flyaway marble
decorations so strangely admired during the 17th and 18th centuries. Few
of them deserve a visit, save for a picture or two of Rubens still
preserved on their altars. There are one or two, however, usually gone
through by tourists, and of these I shall give some brief account, for
the benefit of those who care for such things, though I do not think you
need trouble about them, unless you have plenty of time, and are
specially attracted by the later School of Antwerp.
The most important of these rococo churches is =St. Jacques=, the
principal doorway of which opens into the Longue Rue Neuve. The
pleasantest way to reach it, however, is to go from the Place Verte
through the Marché aux Souliers, following the tramway to the Place de
Meir. This broad street (one of the few open ones in Antwerp), lined by
baroque Renaissance mansions of some pretensions, has been formed by
filling up an old canal. The most imposing building on the R., marked by
two angels holding an oval with the letter L. (the king's initial), is
the =Royal Palace=. A little further on, upon the same side of the
street, is the =House of Rubens's Parents=, with his bust above, and an
inscription on its pediment signifying the fact in the Latin tongue. To
reach St. Jacques you need not go quite as far down the street as these
two buildings. Turn to your L. at the =Bourse=, a handsome modern
edifice, standing at the end of what looks like a blind alley. The road
runs through it, and it is practically used as a public thoroughfare.
The building itself is recent—1869-72—but it occupies the site of a
late-Gothic Exchange of 1531, erected by Dominic van Waghemakere. The
present Bourse resembles its predecessor somewhat in style, but is much
larger, has an incongruous Moorish tinge, and is provided with a
nondescript glass-and-iron roof. Turn to the R. at the end of the lane,
and continue down the Longue Rue Neuve, which leads you towards St.
Jacques, a late-Gothic church, never quite completed. The entrance is
not by the _façade_, but on the S. side, in the Longue Rue Neuve.
(Visitors admitted from 12 till 4 p.m., 1 fr. per person. Knock at the
door, and the sacristan will open.)
The =interior= is of good late-Gothic architecture, terribly over-loaded
with Renaissance tombs and sprawling baroque marble decorations. The
church was used as the Pantheon (or Westminster Abbey) for burials of
distinguished Antwerp families under the Spanish domination; and they
have left in every part of it their ugly and tasteless memorials.
Begin in the =S. Aisle=.
_1st chapel._ Van Dyck: St. George and the Dragon: mediocre. Above,
statue of St. George, to whom angels offer crowns of martyrdom. Good
modern marble reliefs of Scenes from the Passion, continued in
subsequent chapels.
At the end, Baptistery, with good font.
_2nd chapel_, of St. Antony. Temptation of St. Antony, by M. De Vos.
Italian 17th century Madonna.
_3rd chapel_, of St. Roch, the great plague-saint. It contains an
altar-piece by E. Quellin, angels tending St. Roch when stricken with
the plague. Above, the saint with his staff and gourd, in marble,
accompanied by the angel who visited him in the desert. On the window
wall, relics of St. Roch, patron against the plague. Round this chapel
and the succeeding ones are a series of pictures from the Life of St.
Roch, by an unknown Flemish master, dated 1517. They represent St. Roch
in prison; relieved by the dog; resting in the forest; visited by the
angel; etc. (See Mrs. Jameson.) A tomb here has a good Virgin and Child.
_4th chapel._ Fine old tomb; also, continuation of the History of St.
Roch.
_5th chapel._ More History of St. Roch. On the wall, relics of St.
Catherine, who stands on the altar-piece with her sword and wheel;
balanced, as usual, by St. Barbara. The chapel is dedicated to St. Anna,
who is seen above the altar, with Our Lady and the Infant.
_6th chapel._ Baptism of Christ, by Michael Coxcie, on the altar. Window
wall, M. De Vos: Triptych: Centre, Martyrdom of St. James; L., the
daughter of the Canaanite; R., the daughter of Jairus. (The wings are by
Francken.)
The =S. Transept= has Renaissance figures of the Apostles (continued in
the N. transept).
The =Choir= is separated from the Nave and Transepts by an ugly
Renaissance _rood-screen_.
The =Chapel of the Host=, in the S. transept, is full of twisting and
twirling Renaissance marble-work, well seconded by equally obtrusive
modern works in the same spirit.
The =Ambulatory= has a marble screen, separating it from the Choir, in
the worst taste of the Renaissance, with many rococo tombs and
sculptures of that period plastered against it.
_1st chapel_, of the Trinity, has a Holy Trinity for altar-piece, by Van
Balen.
The door to the L. gives access to the =Choir=, with an atrocious
sculptured High Altar, and carved choir-stalls.
_2nd_ and _3rd_ chapels, uninteresting.
The _end chapel_, behind the High Altar, is the =burial chapel of the
Rubens family=. The =altar-piece=, painted by Rubens for his family
chapel, represents the Madonna and Child adored by St. Bonaventura;
close by stands the Magdalen; to the L. a hurrying St. George
(reminiscent of the St. Sebastian by Veronese at Venice), and to the R.,
a very brown St. Jerome. The calm of the central picture, with its group
of women, is interfered with by these two incongruous male figures. It
is like parts of two compositions, joined meaninglessly together. Above
are infant cherubs scattering flowers. One would say, Rubens had here
thrown together a number of separate studies for which he had no
particular use elsewhere. But the colour is most mellow.
_5th chapel_, of San Carlo Borromeo (who practically replaced St. Roch
in later cosmopolitan Catholicism as the chief plague-saint). The
altar-piece, by Jordaens, represents the saint invoking the protection
of Christ and Our Lady for the plague-stricken in the foreground.
Painted for the town almoner.
_6th chapel._ Three good portraits.
_7th chapel._ Visitation, by Victor Wolfvoet.
The =N. Transept= has the continuation of the Twelve Apostles, with two
of the four Latin Fathers by the portal (the other two being at the
opposite doorway). The =Chapel= (=of Our Lady=) resembles that in the S.
Transept, and is equally terrible.
=N. Aisle=: The _2nd chapel_ has a fine triptych by M. De Vos, of the
Glory of Our Lady. _Centre_, the Court of Heaven, where the prominent
position of Our Lady is unusual, and marks an advanced phase of her
cult. In the assemblage of saints below, St. Peter, St. John-Baptist,
and many others, may be recognised by their symbols. The _L. wing_ has
the Calling of Matthew; the _R. wing_, St. Hubert, with the apparition
of the crucifix between the horns of the stag. Beneath are good
portraits of donors. The fine _stained glass window_ of this chapel is
noteworthy. It represents the Last Supper, with donors (1538).
The _3rd chapel_, of the Rockox family, has a good triptych, by Van
Orley, of the Last Judgment. On the _wings_ are portraits of the donor
and family. L., Adrian Rockox and sons, with his patron St. Adrian
(sword, anvil). R., his wife, Catherine, with her daughters, and her
patroness, St. Catherine.
_4th chapel._ Good triptych by Balen. Centre, Adoration of the Magi; R.
and L., Annunciation and Visitation. On a tomb opposite, good portraits
by Ryckaert.
_5th chapel._ Triptych, by M. De Vos: Presentation of Our Lady in the
Temple. L., The Pagans attempt in vain to burn the body of St. Mark; R.,
Martyrdom of St. Lucy.
Another church frequently visited by tourists is =St. Paul=, formerly
belonging to a Dominican Monastery by its side, and situated in a dirty
and malodorous district. Do not attempt to go to it direct. Reach it by
the Quays, turning to the R. near the end of the Northern Promenoir.
Over the outer doorway of the court is a rococo relief of St. Dominic
receiving the rosary from Our Lady. To the R., as you enter, is an
astonishing and tawdry Calvary, built up with rock and slag against the
wall of the Transept. It has, above, a Crucifixion; below, Entombment
and Holy Sepulchre. All round are subsidiary scenes: St. Peter, with the
crowing cock; Christ and the Magdalen in the Garden; Angels to lead the
way, etc. The church itself is an imposing late-Gothic building,
uglified by unspeakable rococo additions. (Admission, from 12 till 4.
Knock at the door: 1 fr. per person. But unless you are a great admirer
of Rubens, the sum is ill-bestowed for seeing one or two of his less
important pictures.) In the N. Transept is Rubens's *Scourging of
Christ, covered: the only thing here really worth seeing. In the N.
Aisle, one of his weakest Adorations of the Magi. On the altar of the
Sacrament, a so-called "Dispute on the Sacrament," by Rubens: really,
the Fathers and Doctors of the Church, especially the Dominicans,
represented by St. Thomas Aquinas, in devout contemplation of the
Mystery of the Eucharist. The other pictures in the church are
relatively uninteresting works of the School of Rubens; the best is a
Way to Calvary by Van Dyck.
If you want more Rubenses, you will find a Madonna, with a great group
of Augustinian and primitive saints, in the Church of =St. Augustine=
(Rue des Peignes), where there is also a good Ecstasy of St. Augustine
by Van Dyck; and in the Church of =St. Anthony of Padua= (Marché aux
Chevaux), a picture, partly by Rubens, representing St. Anthony
receiving the Child Jesus from the hands of the Virgin: but I do not
recommend either excursion.
* * * * *
Antwerp is strongly =fortified=, and a moat, filled with water, runs
round its existing _enceinte_. The Old Citadel to the S. has been
demolished (its site being now occupied by the Museum and the unfinished
quarter in that direction), and a New Citadel erected in the N. The
defensive works are among the finest in Europe.
If you wish to see =whither Flemish art went=, you must go on to
=Holland=. But if you wish to know =whence Flemish art came=, you must
visit the =Rhine Towns=.
If you are returning to England, _viâ_ Calais, stop on the way to see
the noble Romanesque and Transitional Cathedral at =Tournai=. You can
easily do this without loss of time by taking the _first_ boat train
from Brussels in the morning, stopping an hour or two at Tournai (break
permitted with through tickets), and going on by the _second_ train. You
can register your luggage through to London, and have no more bother
with it. You will then have seen everything of the first importance in
Belgium, except Ypres. And Ypres is so inaccessible that I advise you to
neglect it.
V
HISTORICAL NOTES
IN the separate Introductions to the various towns, dealing rather
with Origins than with History, I have laid stress chiefly on the
=industrial and municipal facts=, which in Belgium, indeed, are
all-important. I add here, however, a few =general notes= on the
=political history= of the country as a whole, chiefly =dynastic=. These
may serve for reference, or at least as reminders; and in particular
they should be useful as giving some information about the originals of
portraits in the various galleries.
The two portions of the modern kingdom of Belgium with which we are most
concerned in this Guide are the =County of Flanders= and the =Duchy of
Brabant=. The first was originally a fief of France; the second, a
component member of the Empire. They were commercially wealthier than
the other portions of the Gallo-German borderland which is now Belgium;
they were also the parts most affected by the Burgundian princes; on
both which accounts, they are still by far the richest in works of art,
alike in architecture, in painting, and in sculpture.
The vast Frankish dominions of the Merovingians and of the descendants
of Charlemagne—of the Merwings and Karlings, to be more strictly
Teutonic—showed at all times a tendency to break up into two distinct
realms, known as the =Eastern and Western Kingdoms= (Austria—not, of
course, in the modern sense—and Neustria). These kingdoms were not
artificial, but based on a real difference of race and speech. The
Eastern Kingdom (Franken or Franconia) where the Frankish and Teutonic
blood was purest, became first the Empire, in the restricted sense, and
later Germany and Austria (in part). The Western Kingdom (Neustria)
where Celtic or Gallic blood predominated, and where the speech was
Latin, or (later) French, became in time the Kingdom of France. But
between these two Francias, and especially during the period of unrest,
there existed a certain number of middle provinces, sometimes even a
middle kingdom, known from its first possessor, Lothar, son of
Charlemagne, as Lotharingia or Lorraine. Of these middle provinces, the
chief northern members were Flanders, Brabant, Hainault, and Liège.
=Flanders= in the early Middle Ages was =a fief of France=; it included
not only the modern Belgian provinces of _East_ and _West Flanders_, but
also _French Flanders_, that is to say the Department of the _Nord_ and
part of the _Pas de Calais_. As early as the Treaty of Verdun (843), the
land of Flanders was assigned to Neustria. But the county, as we know
it, really grew up from the possessions of a noble family at Bruges and
Sluys, the head of which was originally known as Forester or Ranger. In
862, the King of France, as suzerain, changed this title to that of
Count, in the person of Baldwin Bras-de-Fer (Baldwin I.). Baldwin was
also invested with the charge of the neighbouring coast of France
proper, on tenure of defending it against the Norman pirates. In 1006,
his descendant, Baldwin IV., seized the Emperor's town of Valenciennes;
and having shown his ability to keep his booty, he was invested by the
Franconian Henry II. with this district as a fief, so that he thus
became a feudatory both of France and of the Empire. He was also
presented with Ghent and the Isles of Zealand. Baldwin V. (1036) added
to the growing principality the districts of Alost, Tournai, and
Hainault. The petty dynastic quarrels of the 11th century are far too
intricate for record here; in the end, the domains of the Counts were
approximately restricted to what we now know as Flanders proper. A bare
list of names and dates must suffice for this epoch:—Baldwin V.
(1036-1067); Baldwin VI. (1067-1070); Robert II. (1093-1111); and
Baldwin VII. (1111-1119).
After this date, the native line having become extinct, the county was
held by =foreign elective princes=, under whom the power of the towns
increased greatly. Among these alien Counts, the most distinguished was
Theodoric (in French, Thierry; in German, Dietrich; or in Dutch,
Dierick) of Alsace, who was a distinguished Crusader, and the founder of
the Chapel of the Holy Blood at Bruges (which see).
Under Baldwin of Hainault (1191-1194) Artois was ceded to France,
together with St. Omer and Hesdin. Henceforth, =Ghent= superseded Arras
as =the capital=. Baldwin IX. (1194-1206) became a mighty Crusader, and
founded the Latin Empire of Constantinople. Indeed, the Crusades were
largely manned and managed by Flemings. He was followed in Flanders by
his two daughters, Johanna and Margaret, under whose rule the cities
gained still greater privileges. Margaret's son, Guy de Dampierre, was
the creature of Philippe IV. of France, who endeavoured to rule Flanders
through his minister, Châtillon. The Flemings answered by just revolt,
and fought the famous Battle of the Spurs near Courtrai, already
described, against the French interlopers (see Bruges). In 1322, Louis
de Nevers (Louis I.) became Count, and provoked by his Gallicising and
despotic tendencies the formidable rebellion under Van Artevelde (see
Ghent). The quarrel between the league of burghers and their lord
continued more or less during the reigns of Count Louis II. (1346) and
Louis III., who died in 1385, leaving one daughter, Margaret, married to
Philip the Bold (Philippe-le-Hardi) of Burgundy.
The political revolution caused in Flanders and Brabant by the accession
of the =Burgundian dynasty= was so deep-reaching that a few words must
be devoted to the origin and rise of this powerful family, a branch of
the royal Valois of France. The old Kingdom of Burgundy had of course
been long extinct; but its name was inherited by two distinct
principalities, the _Duchy of Burgundy_, which formed part of France,
and the _County of Burgundy_ (Franche Comté), which was a fief of the
Empire. In the 14th century, a new middle kingdom, like the earlier
Lotharingia, seemed likely to arise by the sudden growth of a
practically independent power in this debateable land between France and
Germany. In 1361, the _Duchy_ of Burgundy fell in to the crown of
France; and in order, as he thought, to secure its union with the
central authority, John the Good of France (Jean-le-Bon), during the
troublous times after the Treaty of Bretigny, conferred it as a fief
upon his son, Philippe de Valois (=Philip the Bold=, or
Philippe-le-Hardi) who married Margaret of Flanders, thus uniting two of
the greatest vassal principalities of the French crown. In 1385, on the
death of Louis III., Philip succeeded to the County of Flanders, now
practically almost an independent state. After him reigned three other
princes of his family. John the Fearless (Jean-sans-Peur, 1404-1419)
will be remembered by visitors to Paris as the builder of the Porte
Rouge at Notre-Dame de Paris. Philip the Good (Philippe-le-Bon,
1419-1467) was the patron of Van Eyck and Memling. (His portrait by
Roger van der Weyden is in the Antwerp Gallery.) Charles the Bold
(Charles-le-Téméraire, 1467-1477) raised the power of the house to its
utmost pitch, and then destroyed it. (His portrait by Memling is in the
Brussels Gallery.) Contrary, however, to the belief of John the Good,
the princes of the Valois dynasty in Burgundy, instead of remaining
loyal to the crown of France, became some of its most dangerous and
dreaded rivals.
All these Dukes, as French princes, played at the same time an important
part in the affairs of France. They also won, by marriage, by purchase,
by treaty, or by conquest, large territories within the Empire,
including most of modern Belgium and Holland, together with much that is
now part of France. They were thus, like their Flemish predecessors,
vassals at once of the Emperor and the French king; but they were really
=more powerful than either of their nominal over-lords=; for their
central position between the two jealous neighbours gave them great
advantages, while their possession of the wealthy cities of the Low
Countries made them into the richest princes in mediæval Europe. It was
at their opulent and ostentatious court that Van Eyck and Memling
painted the gorgeous pictures which still preserve for us some vague
memory of this old-world splendour. At the same time, the increased
power of the princes, who could draw upon their other dominions to
suppress risings in Flanders, told unfavourably upon the liberties of
the cities. The Burgundian dominion thus sowed the seeds of the Spanish
despotism.
Jean-sans-Peur was murdered by the Dauphin, afterwards Charles VII.; and
this cousinly crime threw his son, Philippe-le-Bon, into the arms of the
English. It was the policy of Burgundy and Flanders, indeed, to weaken
the royal power by all possible means. Philip supported the English
cause in France for many years; and it was his defection, after the
Treaty of Arras in 1435, that destroyed the chances of Henry VI. on the
Continent. The reign of Philippe-le-Bon, we saw, was the Augustan age of
the Burgundian dynasty. (Fully to understand Burgundian art, however,
you must visit Dijon as well as Brabant and Flanders.) Under =Charles
the Bold=, the most ambitious prince of the Burgundian house, the power
of the Dukes was raised for a time to its highest pitch, and then began
to collapse suddenly. A constant rivalry existed between Charles and his
nominal suzerain, Louis XI. It was Charles's dream to restore or
re-create the old Burgundian kingdom by annexing Lorraine, with its
capital, Nancy, and conquering the rising Swiss Confederacy. He would
thus have consolidated his dominions in the Netherlands with his
discontinuous Duchy and County of Burgundy. He had even designs upon
Provence, then as yet an independent county. Louis XI. met these
attempts to create a rival state by a policy of stirring up enemies
against his too powerful feudatory. In his war with the Swiss, Charles
was signally defeated in the decisive battles at Granson and Morat, in
1476. In the succeeding year, he was routed and killed at Nancy, whither
the Swiss had gone to help René, Duke of Lorraine, in his effort to win
back his Duchy from Charles. The conquered Duke was buried at Nancy, but
his body was afterwards brought to Bruges by his descendant, the Emperor
Charles V., and now reposes in the splendid tomb which we have seen at
Notre-Dame in that city.
This war had important results. It largely broke down the power of
Burgundy. Charles's daughter, =Mary=, kept the Low Countries and the
_County_ of Burgundy (Imperial); but the _Duchy_ (French) reverted to
the crown of France, with which it was ever after associated. The scheme
of a great Middle Kingdom thus came to an end; and the destinies of the
Low Countries were entirely altered.
We have next to consider the dynastic events by which the Low Countries
passed under the rule of the =House of Hapsburg=. In 1477, Mary of
Burgundy succeeded her father Charles as Countess of Flanders, Duchess
of Brabant, etc. In the same year she was married to =Maximilian of
Austria=, King of the Romans, son of the Emperor Frederic III. (or IV.).
Maximilian was afterwards elected Emperor on his father's death. The
children of this marriage were Philip the Handsome (Philippe-le-Beau, or
le-Bel; Philippus Stok), who died in 1506, and Margaret of Austria.
Philip, again, married Johanna (Juana) the Mad, of Castile, and thus
became King of Castile, in right of his wife. The various steps by which
these different sovereignties were cumulated in the person of Philip's
son, Charles V., are so important to a proper comprehension of the
subject that I venture to tabulate them.
Frederic III. (or IV.) Charles the Bold.
ǁ ǁ
Ferdinand = Isabella Maximilian = Mary
(of Aragon) ǁ (of Castile) (of Austria) ǁ (of Burgundy)
ǁ ǁ
Johanna the Mad = Philippe-le-Beau
(of Spain) ǁ (of Burgundy and Austria)
ǁ
Charles V.
During the lifetime of Maximilian, who was afterwards Emperor, Mary, and
her son =Philippe-le-Beau=, ruled at first in the Low Countries (for the
quarrel between Maximilian and Bruges over the tutorship of Philippe,
see p. 27). After the death of Isabella of Castile, Ferdinand retired to
Aragon, and Philippe ruled Castile on behalf of his insane wife, Juana.
Philippe died in 1506, and his sister, Margaret of Austria, then ruled
as Regent in the Netherlands (for Charles) till her death in 1530.
Charles V., born at Ghent in 1500, was elected to the Empire after his
grandfather, Maximilian I., and thus became at once Emperor, King of
Spain, Duke of Austria, and ruler of the Low Countries. (In 1516 he
succeeded Ferdinand in the Kingdom of Spain, and in 1519 was elected
Emperor.)
The same series of events carried the Netherlands, quite accidentally,
under =Spanish rule=. For Charles was an absolutist, who governed on
essentially despotic principles. His conduct towards Ghent in 1539
brought affairs to a crisis. The Emperor, in pursuance of his plans
against France, had demanded an enormous subsidy from the city, which
the burgesses constitutionally refused to grant, meeting the unjust
extortion by open rebellion. They even entered into negotiations with
François I^{er}; who, however, with the base instinct of a brother
absolutist, betrayed their secret to his enemy the Emperor. Charles
actually obtained leave from François to march a Spanish army through
France to punish the Flemings, and arrived with a powerful force before
the rebellious city. The Ghenters demanded pardon; but Charles, deeply
incensed, entered the town under arms, and took up his abode there in
triumph. Alva, his ruthless Spanish commander (portrait in the Brussels
Gallery), suggested that the town should be utterly destroyed; but the
Emperor could not afford to part with his richest and most populous
city, nor could even he endure to destroy his birth-place. He contented
himself with a terrible vengeance, beheading the ringleaders, banishing
the minor patriots, and forfeiting the goods of all suspected persons.
The city was declared guilty of _lèse-majesté_, and the town
magistrates, with the chiefs of the Guilds, were compelled to appear
before Charles with halters round their necks, and to beg for pardon.
The Emperor also ordered that no magistrate of Ghent should ever
thenceforth appear in public without a halter, a badge which became with
time a mere silken decoration. The privileges of the city were at the
same time abolished, and the famous old bell, Roland, was removed from
the Belfry.
Thenceforth Charles treated the Netherlands as =a conquered Spanish
territory=. He dissolved the monastery of St. Bavon, and erected on its
site the great Citadel, which he garrisoned with Spaniards, to repress
the native love of liberty of the Flemings (see Ghent). In subsequent
risings of the Low Countries, the Spaniards' Castle, the stronghold of
the alien force, was the first point to be attacked; and on it depended
the issue of freedom or slavery in the Netherlands. Charles also
established the Inquisition, which is said to have put to death no fewer
than 100,000 persons.
In 1555, the Emperor abdicated in favour of his son Philip, known as
=Philip II. of Spain=. But his brother Ferdinand, to whom he had
resigned his Austrian dominions, was elected Emperor (having been
already King of the Romans) as Ferdinand I. From his time forth, the
Empire became more exclusively German, so that its connexion with Rome
was almost forgotten save as a historic myth, degenerating into the mere
legal fiction of a Holy Roman Empire, with nothing Roman in it. Thus,
the Netherlands alone of the earlier Burgundian heritage remained in the
holding of the Austrian kings of Spain, who ruled them nominally as
native sovereigns, but practically as Spaniards and aliens by means of
imported military garrisons.
Philip II.—austere, narrow, domineering, fanatical—remained only four
years in the Netherlands, and then retired to Spain, appointing his
half-sister, =Margaret of Parma= (illegitimate daughter of Charles V.),
regent of the Low Countries (1559-1567). She resided in the Ancienne
Cour at Brussels. Her minister, Granvella, Bishop of Arras, made himself
so unpopular, and the measures taken against the Protestants were so
severe, that the cities, ever the strongholds of liberty, showed signs
of revolution. They objected to the illegal maintenance of a Spanish
standing army, and also to the Inquisition. In April, 1567, as a
consequence of the discontents, the =Duke of Alva= was sent with 10,000
men as lieutenant-general to the Netherlands, to suppress what was known
as the Beggars' League (Les Gueux), now practically headed by the Prince
of Orange (William the Silent). Alva entered Brussels with his Spanish
and Italian mercenaries and treacherously seized his two suspected
antagonists, Count Egmont and Count Hoorn. The patriotic noblemen were
imprisoned at Ghent, in the Spaniards' Castle, were condemned to death,
and finally beheaded in the Grand' Place at Brussels. (For fuller
details of the great revolutionary movement thus inaugurated, see
Motley's _Rise of the Dutch Republic_, and Juste's _Le Comté d'Egmont et
le Comté de Hornes_.) Alva also established in Brussels his infamous
"Council of Troubles," which put to death in cold blood no less than
20,000 inoffensive burghers. His cold and impassive cruelty led to the
=Revolt of the United Provinces= in 1568—a general movement of all the
Spanish Netherlands (as they now began to be called) to throw off the
hateful yoke of Spain. Under the able leadership of William of Orange,
the Flemings besieged and reduced the Spaniards' Castle at Ghent. In the
deadly struggle for freedom which ensued, the Northern Provinces
(Holland), aided by their great natural advantages for defence among the
flooded marshes of the Rhine delta, succeeded in casting off their
allegiance to Philip. They were then known as the United Netherlands.
The long and heroic contest of the Southern Provinces (Belgium) against
the Spanish oppressor was not equally successful. A desperate struggle
for liberty met with little result, and the Spanish sovereigns continued
to govern their Belgian dominions like a conquered country. In 1578,
Alessandro Farnese, Duke of Parma (son of Margaret), was sent as
Governor to the Netherlands, where he remained in power till 1596. In
the prosecution of the war against the Northern Provinces (Holland) he
besieged Antwerp, and took it after fourteen months in 1585. In the
"Spanish Fury" which followed, Antwerp was almost destroyed, and all its
noblest buildings ruined. Nevertheless, under Parma's rule, the other
cities recovered to a certain extent their municipal freedom; though the
country as a whole was still treated as a vanquished province.
The next great landmark of Belgian history is the passage of the Spanish
Netherlands under =Austrian rule=. The first indefinite steps towards
this revolution were taken in 1598, when Philip II. ceded the country as
a fief to his daughter the Infanta Isabella (Clara Isabella Eugenia) on
her marriage with Albert, Archduke of Austria, who held the provinces as
the Spanish Governor. (Portraits of Albert and Isabella by Rubens in the
Brussels Gallery.) The new rulers made the country feel to a certain
extent that it was no longer treated as a mere disobedient Spanish
appanage. After the troubles of the Revolt, and the cruel destruction of
Antwerp by Parma, trade and manufactures began to revive. Albert and
Isabella were strongly Catholic in sentiment; and it was under their
_régime_ that the greater part of the rococo churches of Antwerp and
other cities were built, in the showy but debased taste of the period,
and decorated with large and brilliantly- altar-pieces. They
also induced Rubens to settle in the Netherlands, appointed him Court
painter, and allowed him to live at Antwerp, where the trade of the Low
Countries was still largely concentrated. During their vice-royalty,
however, Brussels became more than ever the recognised capital of the
country, and the seat of the aristocracy.
After Albert's death in 1621, the Netherlands =reverted to Spain=, and a
dull period, without either art or real local history, supervened,
though the wars of the 17th and 18th centuries were in great part fought
out over these unfortunate provinces, "the cockpit of Europe." The
campaigns of Marlborough and Prince Eugene are too well-known as part of
English and European history to need recapitulation here. At the end of
the =War of the Spanish Succession=, the Peace of Rastadt, in 1714,
assigned the Spanish Netherlands to =Austria=, thus entailing upon the
unhappy country another hundred years of foreign domination.
Nevertheless, the =Austrian Netherlands=, as they were thenceforth
called (in contradistinction to the "United Netherlands" or Holland),
were on the whole tolerably well governed by the Austrian Stadtholders,
who held their court at Brussels, and who were usually relations of the
Imperial family. Few memorials, however, of Maria Theresa, of Joseph
II., or of Léopold II. now exist in Belgium, and those few are not
remarkable for beauty. It was during this relatively peaceful and
law-abiding time, on the other hand, that the Upper Town of Brussels was
laid out in its existing form by Guimard. As a whole, the Belgian
provinces were probably better governed under Austrian rule than under
any other _régime_ up to the period of the existing independent and
national monarchy.
The =French Revolutionists= invaded Belgium in 1794, and committed great
havoc among historical buildings at Bruges and elsewhere. Indeed, they
did more harm to the arts of the Netherlands than anybody else, except
the Spaniards and the modern "restorers." They also divided Belgium into
nine departments; and Napoleon half sneeringly, half cynically,
justified the annexation on the ground that the Low Countries were the
alluvial deposit of French rivers. The Belgian States formed part of
Napoleon's composite empire till 1814, when these Southern Provinces
were assigned by the Treaty of London to Holland. In 1815, during the
Hundred Days, the Allied Armies had their headquarters at Brussels, and
the decisive battle against Napoleon was fought at Waterloo. The
Congress of Vienna once more affirmed the union of Belgium with Holland;
they remained as one kingdom till the first revolutionary period in
1830. The Southern Provinces then successfully seceded from the Dutch
monarchy: indeed, the attempted fusion of semi-French and Catholic
Belgium with purely Teutonic and Protestant Holland was one of those
foredoomed failures so dear to diplomacy. A National Congress elected
Léopold of Saxe-Coburg as _King of the Belgians_ (Roi des Belges), and
the crown is now held by his son, Léopold II. For nearly seventy years
Belgium has thus enjoyed, for the first time in its history, an
independent and relatively popular government of its own choosing. The
development of its iron and coal industries during this epoch has vastly
increased its wealth and importance; while the rise of Antwerp as a
great European port has also done much to develop its resources. At the
present day Belgium ranks as one of the most thickly populated, richest,
and on the whole most liberal-minded countries of Europe. Its neutrality
is assured by the Treaty of London, and its army exists only to repel
invasion in case that neutrality should ever be violated.
I N D E X
Académie des Beaux-Arts, 46.
Adoration of the Lamb, The, 81-88.
Adoration of the Magi, The, 41, 201.
Alva, Duke of, 124, 225.
Antiquities, 48, 96.
Antwerp, 164-215.
Armour, Collection of, 150.
Assumption, The, 173.
Austrian Netherlands, The, 227.
Battle of the Spurs, 67.
Béguinages, 45, 32.
Belfries, 25, 70.
Bol, 134, 137.
Boulevards, 149.
Bouts, Dierick, 52, 116, 120, 161, 162.
Brabo, Legend of, 206.
Bruges, 22-64.
Brussels, 98-163.
Burgundy, Dukes of, 18, 219-222.
Cathedrals, 49-53, 77-89, 138-145, 163, 168-176.
Caxton, 214.
Charles I., of England, 89.
Charles II., of England, 27, 47.
Charles V., Emperor, 96, 97, 223.
Charles the Bold, 55, 221.
Chimney-piece, 32.
Christ à la Paille, The, 199.
Colard Mansion, 24, 46.
Cologne, School of, 17, 173.
Counts of Flanders, 17.
Coup de Lance, The, 198.
Cranach, 123.
Crucifixion, The, 202.
De Crayer, 133.
Descent from the Cross, The, 171.
De Vos, 197, 199.
Education of the Virgin, The, 200.
Edward III., of England, 67.
Egmont, Count, 147, 225.
Elevation of the Cross, The, 175.
Entombment, The, 192.
Ethnographical Museum, 150.
Flanders, History of, 217-219.
Fountain, 206.
French Revolution in Belgium, 227.
Gateways, 47, 74, 75, 150.
Gerard David, 30, 62, 63.
Gerard Dou, 134.
Ghent, 66-97.
Godfrey de Bouillon, 146.
Gossaert, 108.
Guild Halls, 47, 74, 102, 206.
Guimard, 148.
Hals, Frans, 135, 136, 203.
Hand of Antwerp, 206.
Hanseatic League, 14, 15, 22, 209.
Hobbema, 134.
Holbein the Younger, 112, 182.
Hondecoeter, 136.
Hoorn, Count, 147, 225.
Hôtels-de-Ville, 28, 71, 100, 158, 205.
Italian Pictures, 137, 138.
John of Gaunt, 75, 91.
Jordaens, 127.
Laeken, 156.
Lamb, The Adoration of the, 81-88.
Leys, 204, 206.
Louvain, 156-163.
Lucas van Leyden, 189.
Mabuse, 108, 182.
Maes, Nicolas, 133, 134.
Magi, Adoration of the, 41, 201.
Malines, 163.
Mary of Burgundy, 18, 33, 55.
Matsys, Quentin, 122, 169, 181, 184, 192.
Memling, 35, 38-44, 69, 117, 118, 182, 185, 188, 191.
Michael Angelo, 56.
Modern Belgian Pictures, 147, 204.
Moretus, 173, 207.
Orange, William of, 225.
Palais de Justice, The, 32, 148.
Parma, Duke of, 225.
Parma, Margaret of, 224.
Perugino, 138.
Philip II., 224.
Plantin-Moretus Museum, 207.
Pourbus, Peter, 50.
Rembrandt, 136, 203.
Rood-Loft, 161.
Rubens, 88, 128, 129, 131, 171, 173, 175, 177, 178, 195, 198-202, 213,
215.
Ruysdael, 134, 204.
St. Bavon, Legend of, 77.
Ste. Gudule, Legend of, 138.
Ste. Ursula, Legend of, 36, 37.
Spaniards' Castle, The, 91.
Spanish Rule in Flanders, 222-226.
Steen, Jan, 136, 137.
Teniers, 129.
Ter Burg, 135.
Titian, 193.
Tombs, 53, 93, 153, 173, 213.
Tournai, 216.
Town Halls, 28, 71, 100, 158, 205.
Universities, 157, 163.
Van Arteveldes, The, 67, 68, 76.
Van der Weyden, Roger, 110, 118, 160, 182, 186.
Van Dyck, 130, 196, 198, 202.
Van Eyck, Hubert, 78, 81-88, 167.
Van Eyck, Jan, 34, 59, 60, 78, 81-88, 182, 185, 187.
Van Orley, Bernard, 111, 126.
Van Ostade, 204.
Van Veen, Otto, 198, 201.
Veronese, 138.
Waghemakere, 72, 169.
Waterloo, 156.
Well, with canopy, 169.
Wiertz, 151.
Zoological Gardens, 210.
Butler & Tanner. The Selwood Printing Works, Frome and London.
TRANSCRIBER NOTES
Misspelled words and printer errors have been corrected.
Inconsistencies in punctuation have been maintained.
Some illustrations were moved to facilitate page layout.
[The end of _Cities of Belgium_, by Grant Allen.]
End of the Project Gutenberg EBook of Cities of Belgium, by Grant Allen
*** | {
"redpajama_set_name": "RedPajamaBook"
} | 1,785 |
\section{Introduction}\label{sec:intro}
The interstellar medium is hierarchically structured. The densest
entities are individual prestellar cores, which generally are found
within filaments or clumps in giant molecular clouds (GMCs) \citep{AndrePPVI,DobbsPPVI}; the GMCs may be part of molecular/atomic
complexes, and are typically found within spiral arms, arm
spurs/feathers, or sheared flocculent features \citep{Elmegreen1980,LaVigne2006}. At any given time within a galaxy, a distribution of
GMCs with various properties exists, and each forms stars according to
the distribution of clumps and cores within it. To understand the
intermediate scale between parsecs and kiloparsecs, the properties of
GMCs must be understood, and it is of particular interest to
investigate whether the characteristics of a GMC may be used to
predict its star formation rate.
There is a long history of characterizing ISM structures in
observations. Molecular lines, dust extinction, and dust emission maps
are used to identify regions with high column density or number
density. These density proxies are a convenient and readily available
way to identify structures, and from these measurements distributions
of cloud sizes and masses can be obtained.
In addition to measuring column densities from molecular or dust
emission, line emission is used to trace velocities of gas, and from
this the kinetic energy content of the structures can be estimated.
For example, based on CO surveys, GMCs in the Milky Way have masses $10^{4}-10^{6} \Msun$, radii between $10-50 \pc$, velocity dispersion between $1-7\kms$, and a linewidth-size relationship of $\sigma_{1D} = 0.9 \kms (R/\pc)^{1/2}$
\citep{Solomon1987,Blitz1993,Heyer2015},
and properties of resolved GMCs in nearby
galaxies are similar \citep{Bolatto2008}.
By combining an estimate of the mass, size, and velocity dispersion,
an estimate of the virial parameter $\alpha_v \equiv 2 \Esubt{k}/|\Esubt{g}|$ (for kinetic energy $\Esubt{k}$ and gravitational energy $\Esubt{g}$) can
be obtained \citep[e.g.][]{Roman-Duval2010,HernandezTan2015}.
Virial parameter estimates from observations typically adopt
$|\Esubt{g}|=3 GM^2/(5R)$
for the gravitational energy, as would apply for an isolated,
uniform-density sphere, where the effective radius is empirically
computed from projected area as $R=(A/\pi)^{1/2}$. Although the case
of ellipsoidal structures has been considered
\citep{Bertoldi1992},
more general effects from nonspherical cloud geometry are not
generally taken into account (even though the filamentary nature of
the ISM makes many clouds quite elongated); non-sphericity tends to reduce
gravitational binding. Internal stratification is sometimes taken
into account by assuming a power-law density profile, which can
increase the estimated $|\Esubt{g}|$ by up to a factor $\sim 2$ \citep{HernandezTan2015}.
Based on the simplest spherical estimate, clouds are considered
``bound'' if the estimated virial parameter $\alpha_v \equiv
5 \sigma^2 R/(GM)$ is less than or equal to 2, where $\sigma$ is the
line-of-sight velocity dispersion. However, traditional estimates of gravitational
binding energy are problematic even beyond the assumptions of
homogeneity and spherical geometry because the ``isolated cloud''
estimate of $|\Esubt{g}|$ does not properly take into account neighboring
structures. For a given local gravitational potential minimum at the
center of a cloud, tidal forces set the effective zero of the
gravitational potential not at infinite distance but along the first potential
contour that has a saddle point -- equivalent to the Roche
lobe for the case of two spherical bodies. As a result, tidal forces
effectively decrease gravitational binding energy $|\Esubt{g}|$ of dense
regions in close proximity to other dense regions, which is common
because of the hierarchical structure of density variations. In
addition, simple virial parameter estimates neglect magnetic
contributions to support, which can significantly add to the numerator
\citep{McKee1993PPIII,Heiles1993PPIII}.
Although
simple virial parameter estimates are inexact, they are often used to
assess whether a structure is a likely candidate for star formation.
Star formation is observed to take place within the densest structures at the
smallest scale within the ISM hierarchy, and it is important to
understand what dynamical processes lead to the onset of gravitational
collapse, and what controls the rate of star formation within a given
level of the hierarchy. More generally, it is of interest to
understand how star formation timescales are related to the properties
and corresponding timescales of gaseous structures. Because star
formation involves gravity, the most commonly invoked reference
timescale is the free-fall collapse time,
\begin{equation}\label{eq:tff}
\tff = \left(\frac{3\pi}{32G\rho}\right)^{1/2},
\end{equation}
where $\rho$ is the gas density. Perhaps the simplest way to
characterize the relationship between star formation and gas
properties is via the star formation efficiency per free-fall time
\citep{Krumholz2005,Krumholz2007},
defined as
\begin{equation}\label{eq:eff}
\eff \equiv \frac{\dot M_*}{M/\tff},
\end{equation}
where $\tff$ is the free-fall time at the mean density of the gas
contributing to $M$, and $\dot M_*$ is the star formation rate (SFR).
Other relevant timescales include the flow crossing time across a
structure that is supported by turbulent stresses, and the sound
crossing time for a structure that is supported by thermal pressure.
A class of theoretical models for star formation suggests that in
turbulent clouds, there is a critical density $\rho_\mathrm{crit}$
above which collapse occurs
within a free-fall time, with $\rho_\mathrm{crit}$ depending on the
the ratios of kinetic to gravitational energy (virial parameter),
turbulent to thermal velocity (Mach number),
and thermal to magnetic pressure (plasma beta parameter)
\citep{Krumholz2005,Padoan2011,Hennebelle2011,Federrath2012,Padoan2014}.
The underlying physical concept behind the idea of a
critical density is that the density must be high enough that thermal
pressure and magnetic stresses cannot support against collapse, and
that the collapse time is shorter than the timescale for shear to
tear apart a structure.
In addition to theoretical models, direct numerical simulations have
been used to characterize the dependence of SFRs on
gas properties. One idealized type of setup employs simulations with
isothermal, self-gravitating gas, in which turbulence is driven in
Fourier space. From a large set of driven-turbulence
simulations, \cite{2012ApJ...759L..27P} suggested that $\eff$ depends
primarily on the ratio of flow crossing time to free-fall time as
\begin{equation}
\eff \propto \exp(-1.6\tff/\tdyn).
\end{equation}
where $t_{\rm dyn} = R/\sigma_{3D} = R/(\sqrt{3}\sigma_{1D})$ is the flow crossing time for system size $2R$ ($=L$, the simulation box size for \cite{2012ApJ...759L..27P}). For a uniform spherical
cloud, the timescale ratio can be related to the virial parameter by
\begin{equation}\label{eq:tav}
\left(\frac{\tff}{\tdyn}\right)^{2}
= \frac{3\pi^{2}}{40}\alpha_{v};
\end{equation}
thus, these simulations suggest a strong suppression of star formation
at high $\alpha_v$.
Idealized simulations have the advantage of carefully controlled
conditions, but the disadvantages that the turbulence is driven in an
artificially prescribed manner to maintain a fixed overall turbulent
amplitude, and that the processes leading to cloud formation and
destruction are not followed. In reality, GMCs form due to a
combination of large-scale ISM flows (including turbulence, shear,
and epicyclic motion) and gravity (both stellar gravity and
self-gravity) that lead to collection of material from a large volume,
as mediated by thermal and magnetic pressure, and a change from atomic
to molecular phase as gas cools. Turbulence on scales less than
the scale height of the warm-cold ISM likely originates
primarily due to the feedback from young stars \citep{Maclow2004,McKee2007,Elmegreen2004}\footnote{Gravitational instabilities in the
combined gas-stellar system \citep[e.g.][]{1984ApJ...276..114J,1992MNRAS.256..307R,2001MNRAS.323..445R,2007ApJ...660.1232K} can drive horizontal
motions at very large scales,
as seen in numerical simulations
\citep[e.g.][and citations within]{2007ApJ...660.1232K,2008ApJ...684..978S,2009MNRAS.392..294A,2011MNRAS.417.1318D,2012MNRAS.421.3488H,2015ApJ...804...18A}, but generally
these motions do not reach supersonic amplitudes unless they are
associated with gravitational collapse. In addition, turbulence at scales
less than the disk scale height can be driven by spiral shocks
and by the magnetorotational instability, but numerical simulations show
that the corresponding amplitudes are relatively modest in cold gas
\citep[e.g.][and citations within]{2004MNRAS.349..270W,2006ApJ...649L..13K,2007MNRAS.374.1115D,2013MNRAS.430.1790B,2010ApJ...720.1454K,2005ApJ...629..849P,2007ApJ...663..183P}.},
whether
inherited from a GMC's formation stage or produced
internally.
Considering that GMCs live for at most a few turbulent crossing times
or free-fall times \citep{Kawamura2009,Kruijssen2019},
it is not clear that internal GMC conditions can control star formation in a
way that is entirely divorced from their formation and destruction
processes.
In recent years, (magneto)-hydrodynamic simulations have been used to
follow the star-forming multiphase ISM in kpc-size regions at high resolution.
In these simulations,
massive self-gravitating clouds naturally condense out of the diffuse
gas, and within these clouds localized collapse occurs that represents
star cluster formation \citep{Gatto2017,Iffrig2017,KimOstriker2017,Kannan2018,2018A&A...620A..21C}.
By modeling the return
of energy (representing radiative heating and supernova explosions)
from these cluster particles to their surroundings, a self-consistent,
self-regulated state can be reached in which all thermal phases of the
ISM are represented, and a hierarchy of structures is naturally
created. While the large-scale time-averaged SFR adjusts such that
feedback provides the energy and momentum needed to maintain overall
equilibrium in the ISM as a whole
\citep{Ostriker2010,Ostriker2011,Kim2011,Kim2013},
the collapse to make individual star clusters depends on local
conditions in overdense clouds. Simulations of this kind present an
opportunity to evaluate the role of gravity in binding ISM structures
that are part of a complex environment, and to assess common practices
for estimating gravitational boundedness. In addition, simulations of
this kind afford a realistic setting to test theoretical ideas
regarding the role of gravitational boundedness in controlling SFRs.
In this paper, we use a large-scale ISM simulation produced in the
TIGRESS framework \citep{KimOstriker2017}
to characterize the properties
of dense structures and their relationship to star formation. Our
structural decomposition analysis includes methods that are similar to
typical observational practices, in which objects are defined based on
density or column density. For sets of objects defined by different
density thresholds, we compute statistics of mass, size, and velocity
dispersion, which allows us to compute ``empirical'' virial parameters
and linewidth-size relations. We compute both traditional virial
parameters (only kinetic energy) and virial parameters including
thermal and magnetic energy. In addition, we apply another method of
defining structures based on contours of the gravitational potential
(rather than density contours). In this method we identify bound
objects as regions where the kinetic, thermal, and magnetic energy are
sufficiently low compared to the gravitational energy (computed
relative to a tidally-defined potential contour). These two analyses
allow us to relate traditional virial parameter estimates for objects to
measurements of gravitational binding that directly take into account
nonspherical geometry, internal stratification, and tidal forces. We
shall show that traditional virial parameter estimates can
significantly under- or over-state the true boundedness of ISM
structures.
To study the relationship between gas and star formation, we use
correlations between the temporal history of the SFR and the mass of
gas in different categories of objects, including objects defined by
density thresholds and objects defined by being gravitationally bound.
In this way, we are able to measure how $\eff$ varies as a function of
density and what $\eff$ is for objects that are gravitationally bound
(also allowing for different treatments of surface terms). We are
also able to measure time delays between the availability of a mass
reservoir and the star formation burst that it produces. We use
correlation analysis to quantify the relative predictive power of
different star formation models that depend on the traditional virial
parameter, and on our more sophisticated assessment of gravitational
binding.
The plan of this paper is as follows. In \autoref{sec:methods} we
describe our analysis methods, including how we identify bound objects
(\autoref{sec:mhbr}), the properties we measure for bound objects and
for density-defined objects (\autoref{sec:mprop}), and how we conduct
time-series correlation analyses (\autoref{sec:mtime});
\autoref{sec:analysissum} summarizes our methods and
\autoref{sec:tigress} describes the
primary TIGRESS simulation that we analyze.
\autoref{sec:results} presents an overview of structure (\autoref{sec:resultmaps})
and results of our analyses, including
statistics of object
properties (\autoref{sec:prop}) and time series correlation studies
(\autoref{sec:rtime}), with a summary of trends in the values of
$\eff$ and levels of correlations for various ways of selecting gas
in \autoref{sec:eff}.
In \autoref{sec:conc}
we summarize our results and discuss connections with other
current theory and observations.
\section{Methods}\label{sec:methods}
In this paper, we analyze the properties of dense and bound gas
structures, and investigate the relationship between the material in
these structures and the star formation rate, as applied to the
fiducial TIGRESS model described in \citet{KimOstriker2017}. The
methods we develop, described in some detail here, are quite
general and can be applied to other numerical simulation data. With
some modifications to allow for projected rather than fully
three-dimensional information, our methods can also be applied to
observed data sets.
We begin by describing methods for identifying objects based on
density isosurfaces or on the gravitational potential in comparison to
the kinetic, thermal, and magnetic energy densities (Section
\ref{sec:mhbr}); additional technical details of the algorithm are described
in Appendix \ref{sec:algorithm}.
We also describe how we quantify object properties
including mass, size, velocity dispersion, and virial parameter
(Section \ref{sec:mprop}). We then describe our use of time series to
compare the simulated SFR to the history of mass per free-fall time
for different categories of objects (Section \ref{sec:mtime}); this
involves fitting for optimal time delay and efficiency and Bayesian inference
to test models for the dependence on virial parameter. Finally, in
Section \ref{sec:tigress} we briefly summarize the numerical
implementation and parameters of the TIGRESS model to which we have
applied our analysis.
\subsection{Bound Objects}\label{sec:mhbr}
To motivate our procedure for identifying gravitationally bound structures,
consider what is bound in the Solar system. For example, a pebble on Earth
is bound to the Earth, the Earth-Moon system, the Sun, and the Galaxy,
but is not bound gravitationally to a nearby pebble. To make this
determination, both the gravitational potential contours and the relative
velocities of the structures involved are needed; a pebble as well as its
neighbors mutually lie within the Earth's gravitational potential as limited
by the Moon's tidal force, and do not have high enough velocities that
they could find themselves on the Moon or escape entirely from the
Solar system. Thus, we consider the pebble as part of the Earth.
For application to the ISM and star formation, boundedness can also
have a characteristic scale dependence. Matter on larger scales tends
to have a higher internal kinetic energy ($\sigma^{2} \propto L$) from
the scaling properties of turbulence, but also tends to be
increasingly bound by the gravitational potential ($GM/R \sim
\rho{}L^{2}$). Depending on the scale dependence of the density,
there may be a hierarchy of boundedness, with bound structures nested within
other bound structures.
With the above motivation, we identify a hierarchy of structures in
our ISM simulations based on contours of the gravitational potential,
and bound structures based on the energy of fluid elements relative to
the structure tree. The first level of the gravitational tree is
comprised of structures enclosed by isocontours that surround a single
minimum. Each successive level is comprised of material within
isocontours enclosing distinct sets of minima whose largest enclosing
isocontours are in contact. That is, branches merge into a new object
when their isocontours are in contact. This means that each object in
the tree can be uniquely identified with a critical point in the
gravitational potential, where isocontours come into contact.
\autoref{fig:well} provides a schematic illustration of this
procedure.
\begin{figure}
\includegraphics[width=\columnwidth]{\figformat{well}}
\caption{Schematic of HBPs (upper case) and associated HBRs (lower
case) as level sets within gravitational wells, plotting
gravitational energy against spatial coordinate. Each HBR is bound
relative to its associated HBP. For example, on both left and
right ``a'' represents material interior to an isocontour of the
gravitational well such that the HBR has 0 total energy, bound
relative to the gravitational contour ``A.'' Similarly, ``b'' is
bound relative to ``B.'' The regions ``a'' and ``b'' are
therefore both HBRs. On the left, we show an example of a region
``ab'' which is invalid as an HBR because it consists of two
non-contiguous parts. On the right, we show an example of a
contiguous HBR ``ab'' which merges ``a'' and ``b.'' We are generally
interested in only the largest HBRs in any hierarchy, so on the right
we would remove the objects ``a'' and ``b'' from further consideration.
This schematic also illustrates that each HBP object can be
identified by a critical point of the potential: ``A'' and ``B'' are
associated with their respective local minima, and ``AB'' is associated with
the central local maximum.
}
\label{fig:well}
\end{figure}
At each level in the gravitational potential contour tree, we denote the
object enclosed within a closed contour as a {\it hierarchical binding parent}
(HBP). Within each of these objects, we denote some subset of the gas
as a {\it hierarchical bound region} (HBR). The HBR is the set of cells for
which the total energy (kinetic, thermal, magnetic, gravitational) of the
region is 0. In this calculation, we assign a gravitational binding energy
to each cell based on the difference between its gravitational potential ($\Phi$) and the isocontour surface of the HBP ($\Phi_{0}$); i.e. the contribution
to the gravitational energy of the HBR is $(\Phi - \Phi_0)\rho dx^3$.
Hence, the HBP is responsible for binding these HBR cells.
Cells are added to a candidate HBR in order of gravitational potential depth
(deepest first). For the contribution to kinetic energy, the center of mass
(COM) velocity of the subset of cells is subtracted out first. Within a given
HBP, the most massive bound subset of cells is taken as the HBR; if no
cells are bound, there is no HBR.
In the above definition, we have not considered any effects from thermal
or turbulent stresses on the surface (defined by an equipotential) of objects.
Because the dynamics of turbulent systems are complex, surface stresses
could in principle either act to compress and help bind
structures (e.g. for a converging
flow) or act to disperse and unbind structures (e.g. for a shear flow).
While the complex dynamics makes it impossible to decide
between these alternatives in a general sense, we can still
investigate the potential magnitude of the effects that
surface stresses may have.
To do this, we begin by averaging the kinetic ($\Esubd{k}$), thermal
($\Esubd{th}$), and magnetic ($\Esubd{B}$) energy density over the
surface $\Omega$ of the HBP ($N$ cells) to compute the mean surface energy density
\begin{equation}
\Esubd{\Omega} = \frac{1}{N} \sum\limits_{i \in \Omega}
(\Esubd{k,i} + \Esubd{th,i} + \Esubd{B,i}).
\end{equation}
Here, $\Esubd{k,i}$ is the kinetic energy density computed relative to the
center of mass velocity of the surface cells.
We then define ``HBR+1'' and ``HBR-1'' objects, where
the object HBR$\pm$1 is the set of cells satisfying
\begin{equation}
\sum\limits_{i \in \mathrm{HBR\pm1}} \Esubd{k,i} + \Esubd{th,i} + \Esubd{B,i}
< \sum\limits_{i \in \mathrm{HBR\pm1}} (\Phi_{0} - \Phi_{i})\rho_{i} \pm {\Esubd{\Omega}},
\end{equation}
and now $\Esubd{k,i}$ is the kinetic energy density computed relative to
the center of mass velocity of the HBR$\pm1$ cells. Clearly, HBR+1
will be more massive than the corresponding HBR identified without the
surface energy terms, because the criterion for including cells
becomes less restrictive by adding $\Esubd{\Omega}$
on the right-hand
side. Similarly, HBR-1 will be less massive than the corresponding
HBR. This procedure could be generalized by adding or subtracting
arbitrary multiples of $\Esubd{\Omega}$, but for comparison purposes
we have found HBR-1, HBR, HBR+1 suffice. We can think of HBR+1
objects as structures in which surface stresses are treated as helping
to bind material; HBR-1 objects are those where surface stresses are
treated as reducing binding. Physically, addition of $\Esubd{\Omega}$
on the right-hand side in HBR+1 is equivalent to only
considering the excess of $\Esubd{k}$, $\Esubd{th}$, and $\Esubd{B}$
over ``ambient'' values when computing total
energy.\footnote{We note that when using the Virial Theorem
\citep[e.g.][]{1992ApJ...399..551M}, in the case of isotropic
magnetic fields and a spherical cloud the surface terms would enter in
exactly the same way as in the HBR+1 definition. That is, the mean surface
values of kinetic, thermal, and magnetic energy density would be
subtracted from the mean values within the volume.}
Subsequently, we will test the correlation of HBR and HBR$\pm$1 objects
with respect to the star formation rate; if surface terms play an important
physical role, we might expect this to be reflected in the relative
correlations with SFRs that we measure.
In the hierarchical contour tree, a nested sequence of HBPs
is uniquely defined by critical points in the equipotential, e.g.
in Figure~\ref{fig:well} ``A'' and ``B'' are nested within ``AB''.
At each level of the tree, HBRs can be identified with respect to the
corresponding HBPs.
An additional requirement for HBR (and HBR$\pm1$) objects to
be considered valid is spatial compactness. Physically, we impose
this requirement because a ``divided'' HBR within a single HBP could
not be trusted to form a contiguous object.
The case of a non-contiguous HBR occurs when the COM velocity of the
HBP is significantly different from that of its HBP branches, while
the surface potential of the HBP is not significantly higher than that
of its branches (this difference is equivalent to the difference
between the HBP surface and the HBP originating critical point). Then,
in this scenario, considering the HBP as a whole increases the kinetic
energy without sufficiently increasing the depth of the binding
gravitational well, resulting in separate regions that are unbound
relative to each other but may be individually
self-bound. \autoref{fig:well} shows an example of a contiguous
vs. non-contiguous HBR.
Both HBPs and HBRs are grown contiguously from cells of decreasing
potential depth, so a non-contiguous HBR would only form from an HBP
with multiple local minima and correspondingly multiple HBP
branches. The separate HBR subregions are each a subset of cells from
one of those separate HBP branches, which meet at the critical point
identified with the origin of the HBP. Hence, in our tree construction it
suffices to check that the HBR of an HBP contains the critical point
identifying that HBP, which is a convenient guarantee that the HBR is
contiguous.
We have so far described a process of building a contour tree of
gravitational potential isocontours. Each isocontour defines an HBP hosting
an HBR (which may be empty, or may be non-contiguous).
Because HBPs and HBRs are nested, one can consider levels in the hierarchy
separately (in which case given fluid elements are counted at each level
they appear), or one may apply a merging or pruning criterion to objects
to ``flatten'' the hierarchy, such that each fluid element appears in at
most a single object.
We are interested in regions self-bound on each scale. An HBP may bind a contiguous region of mass but have children binding non-contiguous regions.
This is analogous to a larger scale self-bound system (like a galaxy) containing subregions that are not self-bound (an unbound GMC, with separate bound subregions, would be a non-contiguous HBR). To enforce that every level of the hierarchy is self-bound, we build the HBR tree from the HBP tree bottom-up, starting with leaf HBPs around individual local minima. A parent HBP is only evaluated (computing its HBR) if all its children HBPs were evaluated and host contiguous HBR. If an HBR is evaluated and contiguous, it replaces its branch HBRs, thus becoming a leaf node of the subset of the full HBR tree. The leaf nodes of the subset of the full HBR tree are considered to be star-forming regions.
This method naturally selects the largest scale candidates for contiguous
collapse and hence star formation,
and are robust to small-scale fluctuations in
the gravitational potential (for example, the point-potentials of star
particles). Furthermore, we note that changes over time in the
gravitational potential structure can cause rapid changes in the
population of leaf nodes of the contour tree. As objects disperse,
merge, or evolve, critical points are created, destroyed, and
relocated. Leaf nodes can be sensitive to such changes in the
gravitational structure,
but contiguous HBRs
are more robust. For example,
a dispersing or merging object smoothly transitions to or from being
considered as multiple HBRs vs. a single HBR, because the relevant
parameter is the total energy content, which (roughly) continuously
changes.
We also checked the approach of building the HBR top-down, allowing a given HBR to contain non-contiguous HBRs, essentially allowing gas to be bound under any viable isocontour. This method produces contiguous HBRs that are as massive as possible.
Even with the above definitions, additional
choices can be made in computing contiguous HBRs.
We varied a single choice at a time to study their effects.
For example, the star particle potential contribution to the potential
could either be included or excluded.
However, we found that including the star particle potential did not
produce a significant quantitative effect on our results.
The most significant difference we found was when building the HBR tree
top-down: we found that in certain simulation snapshots (a
few per cent of the time) where the mass
happened to coalesce in a single region, the fraction of ``bound mass''
spiked by an order of magnitude.
For the rest of this paper, when we refer to ``HBR'' the choices adopted
are: building the HBR tree from the bottom-up, excluding the star particle potential, and ignoring surface
stresses.
We have found that that considering surface
stresses can have a large effect, and we report results separately for
objects identified as HBR$\pm1$, as above. Inclusion of surface stresses
as HBR+1 can lead to an order of
magnitude more mass being considered ``bound.'' However, as we shall show,
this does not have a
strong effect on the correlation between star formation and ``bound'' mass
over time.
\subsection{Object Definition and Properties}\label{sec:mprop}
After computing HBRs from the gravitational potential structure and
fluid properties (density, kinetic energy, thermal energy, magnetic
energy), we have a collection of objects, each of which contains a
number of simulation cells.
There are also other means of identifying potentially star-forming
objects that are closer to traditional observational methods that use
molecular tracers that may have a characteristic threshold density, or
dust emission/extinction maps with a minimum column. Here, we shall
apply number density thresholds ($\nhmin=$ 10, 30, and 100 $\pcc$) to
identify contiguous regions where the number density $\nh > \nhmin$, referring to these regions as ``$\nhmin$ objects.''\footnote{For this, we use Python package scipy, specifically the function scipy.ndimage.label, with a boundary correction for the shearing periodic box}
We also treat the set of HBPs for HBRs as objects.
We shall analyze HBRs, HBPs, and $\nhmin$ objects
in similar ways, both in terms of their properties and
their relation to star formation. For each set of object categories
in any simulation snapshot, we use member cells to calculate each
object's mass, volume, free-fall time from its mean (volume-weighted)
density, and a mass per free-fall time.
For HBR and $\nhmin$ objects we compute individual virial
parameters $\alpha_{v}$. We compute the thermal energy density
$\Esubd{th} = {\cal P}/(\gamma-1)$ for pressure ${\cal P}$ using
$\gamma = 5/3$.
With momentum density $\vect{p}=\rho \vect{v}$
and center-of momentum velocity
${\bf v}_{\rm COM}$, the kinetic energy density is $\Esubd{k} =(1/2)(
|\vect{p}|^{2}/\rho{} - \rho{}\vect{v}_{\rm COM}^{2})$, while magnetic
energy density is $\Esubd{B} = |\vect{B}|^{2}/8\pi$ for
magnetic field $\vect{B}$. These are summed over cells for each object to
define the total kinetic, thermal, and magnetic energy $\Esubt{k}$,
$\Esubt{th}$, and $\Esubt{B}$, respectively.
We define an effective object radius $R$ from
each object's volume via $V = (4\pi{}/3)R^{3}$, and then define an
estimated gravitational self-binding energy as
\begin{equation}\label{eq:Egdef}
\Esubt{g} \equiv
\frac{(3/5)GM^{2}}{R}
\end{equation}
using the total object mass $M$. We note that this is
{\it not} the true gravitational binding energy, but we adopt this
definition for the purpose of comparison with standard practices in
the field which assume isolated objects. With the above definitions, we set
\begin{align}\label{eq:virialdef}
\alpha_{v} &\equiv 2\frac{\Esubt{k}}{\Esubt{g}} \\
\label{eq:virialtotdef}
\alpha_{\rm v,total} &\equiv 2\frac{\Esubt{k} + \Esubt{th} + \Esubt{B}}{\Esubt{g}};
\end{align}
while the former
is used most often in the literature under the assumption that kinetic energy
dominates over both thermal and magnetic, the latter is more general.
We also examine the separate energy components of objects.
For each $\nhmin$ object, we find the mass fraction of
cells that are also within HBRs. This allows us to examine the overlap
between a method of identifying ISM structures (and possible
star-forming regions) that is simple but easily applied, and a method
that is sophisticated and physically motivated, but not directly
applicable in observations. This mass
fraction is also the probability of gas being bound given the observation
that it is high density ($P(\rm bound | \rm dense)$). We colloquially
refer to this as the ``bound fraction.''
\subsection{Time Series Analysis}\label{sec:mtime}
A question of significant interest is the detailed correlation in time
between the mass in identifiable star-forming structures and the
actual star formation rate (SFR). To investigate this question, for each
simulation snapshot and object type defined as described in
Sections~\ref{sec:mhbr} and \ref{sec:mprop}, we sum the mass, volume,
and mass per free-fall time of all objects of that type within the
snapshot. This procedure provides a set of time series (representing the
ratio of mass per free-fall time for selected gas subsets) that we can
use to test the connection to time-dependent star formation rate
$\SFR(t)$.
To create time series for comparison to $\SFR(t)$, we also consider the
collective material above minimum gas surface density thresholds
$\Sigma_0=$ 10, 30, and 100 $\surfunit$. For each threshold and for
each simulation snapshot, we compute the mass above the threshold,
the volume ($\nhmin$ objects only), and
mass per free-fall time. For the free-fall time, we use a
mass-weighted average density. For logarithmic bins of number density
of half-decade width, we also compute the snapshot mass, volume, and
mass per free-fall time, using the volume-weighted average
density. This average density tends to be the lower edge of the bin
when looking at the high density side of the distribution.
We compute the SFR at any given time $t$ by taking the total mass of
all star particles whose age $t_{*}$ is less than some maximum age
$t_{*,\rm max}$, and dividing by that age:
\begin{equation}\label{eq:sfrage}
\rm{SFR}(t) = \sum\limits_{t_{*} < t_{*,\rm max}}M_{*}/t_{*,\rm max}
\end{equation}
This is observationally motivated but also naturally smooths the SFR
time series. This also introduces a delay shift of $t_{*,\rm max}/2$
in the time series relative to the simulation because mass that has
formed stars at a given time $t$ contributes equally to the SFR at
later times until $t+t_{*,\rm max}$, with midpoint centered on
$t+t_{*,\rm max}/2$. As long as only young stars are considered and
$t_{*,\rm max}$ is small, these effects are not problematic.
We use time series comparisons to
compute the star formation efficiency per free-fall time
(\autoref{eq:eff}) for each subset of the gas. For comparison to $\mathrm{SFR}(t)$,
we use the individual time series $\mff$ from each defined gas
subset (e.g. HBR, HBP, number density thresholds,
number density bins, surface density threshold). Note that the typical
density and free-fall time of a given definition do not significantly
change over time, so correlating $\mathrm{SFR}$ with total mass per free-fall
time $\mff$ is similar to correlating $\mathrm{SFR}(t)$ with the total mass
$M(t)$ in a defined subset.
We treat our time series of simulation snapshots as a set of 2-D
samples in $\mathrm{SFR}$ and $\mff$, and apply a simple linear regression to
estimate $\eff$ using the model $\mathrm{SFR} = \eff\mff$. Hence, our
inferred $\eff$ is simply
\begin{equation}\label{eq:effinfer}
\eff = \frac{\sum_{N} \SFR_{i} (\mff)_{i}}{\sum_{N} (\mff)_{i}^{2}} = \frac{\langle(\SFR)\mff\rangle}{\langle (\mff)^{2} \rangle}
\end{equation}
with standard error in the fitted coefficient
\begin{equation}\label{eq:effstderr}
\Delta\eff{}^{2} = \frac{1}{N-2}\frac{\sum_{N} \Delta{}\SFR_{i}^{2}}{\sum_{N} \left(\mff_{i} - \langle\mff{}\rangle\right)^{2}}
\end{equation}
where
\begin{equation}\label{eq:deltaSFR}
\Delta\SFR_{i} = \SFR_{i} - \eff{}\mff_{i}
\end{equation}
and
$\langle\mff\rangle = (1/N)\sum_{N}(\mff)_{i}$.
We note that the uncertainty in the inferred $\eff$ from
\autoref{eq:effstderr} will tend to decrease with increasing sample size.
The normalized variance in the error of the ``data'' SFR compared to the
``model'' $\eff M/\tff$ is then
\begin{equation}\label{eq:SFRerr}
\sigma_{\Delta \SFR/\langle \SFR \rangle}^{2} = \frac{1}{N-1}\frac{\sum_{N}
\Delta\SFR_{i}^{2}}{\langle \SFR \rangle^{2}}.
\end{equation}
We interpret a smaller normalized error
$\sigma_{\Delta \SFR/\langle \SFR \rangle}$
as a stronger relationship between SFR and
$\mff$. Note that the covariance and Pearson correlation coefficient
between two variables $X$ and $Y$ both increase with a term
$E[XY]$. The standard error increases with
$\Delta\SFR_{i}^{2} = \SFR_{i}^{2} + \eff^{2}(\mff)_{i}^{2} - 2\eff\SFR_{i}(\mff)_{i}$, hence
decreasing with $E[(\SFR)\mff]$. Thus, qualitatively, a smaller standard
error corresponds to a larger covariance or correlation coefficient,
and demonstrates a stronger dependence of SFR on $\mff$.
The above approach has the strength of being a statistically
uncontroversial way to consistently estimate both $\eff$ and the
strength of the relationship between SFR and $\mff$. It also benefits
from giving more weight to simulation snapshots with more $\mff$ or
equivalently more total mass. This is similar to treating each unit of
mass as a single sample and averaging these
samples. Conceptually, it is best to measure $\eff$ in snapshots where
both SFR and $\mff$ are large.
We note that we also experimented with other methods of
estimating $\eff$ and quantifying the connection between SFR and
$\mff$, but use \autoref{eq:effinfer} and \autoref{eq:SFRerr} because these methods
have the most statistical simplicity, physical motivation, and
consistent results of our candidate methods. Other ways of estimating
$\eff{}$ included $\mean{\SFR}/\mean{\mff}$ and
$\mean{\SFR/(\mff)}$. Other ways of quantifying the connection included
the covariance, the Pearson correlation coefficient, the standard
deviation of $[\SFR/(\mff)]_{i}$, and the root mean square of $\SFR -
\eff{}\mff$.
In practice, we modify the above to consider the effect of time
delays. First, as already alluded to, our definition and observable
definitions of SFR are already shifted. Furthermore, given a gaseous
object it is reasonable to expect that it may not presently be forming
stars but rather will form stars after a delay which scales with the
free-fall time. In detail, we might expect that temporal peaks in the
mass of low density gas would lead to temporal peaks in the mass of high
density gas after a delay comparable to the low density free fall
time. Correspondingly, temporal peaks in the mass of gas at yet
higher density might be expected after a subsequent shorter delay,
comparable to the high density free fall time.
To allow for temporal delays, we apply the analysis described by
\autoref{eq:effinfer} and \autoref{eq:SFRerr} to time-shifted sets of SFR and
$\mff$, interpolating when necessary.
For any time series, we identify the delay time $t_{d}$ which
minimizes
$\sigma_{\Delta \SFR/\langle \SFR \rangle}$,
assuming that SFR lags behind $\mff$
by $t_{d}$. We present results for $\eff{}$ and
$\sigma_{\Delta \SFR/\langle \SFR \rangle}$
for this choice of $t_{d}$. This allows for the maximum correlation
between SFR and $\mff$, under the assumption that $\mff$
causes future SFR.
\subsubsection{Dependence on virial parameter}\label{sec:effmodels}
In varying galactic environments, gas at a given density may be in
different dynamical states, in ways that would affect future star
formation. For example, increasing contrast of the density in a cloud
relative to its environment may reflect a more bound state, and clouds
that are more bound might be more susceptible to forming stars.
Following typical practice in the field, we can characterize the ``boundedness''
of individual structures based on their virial parameter.
We test
the effect of the virial parameter on susceptibility to star formation
using our time series, comparing the actual $\SFR(t)$ (from star particles)
with model predictions
\begin{equation}\label{eq:modelsfr}
\SFR_{m}(t) = \sum\limits_{\mathrm{object\ i}} \frac{\eff(\alpha_{v,i}) M_{i}}{\tffi{i}}.
\end{equation}
For each temporal snapshot, the right-hand side is a sum over objects in
a given category at that time (with ``objects'' being HBRs or density-defined objects), and
$\eff(\alpha)$ is a specified model. For each object,
$\alpha_{\rm v,i}$, $M_i$, and $t_{\rm ff,i}$ are the virial parameter, mass, and
free-fall time.
Our simplest model is to take constant $\eff$, that is
\begin{equation}
\eff(\alpha_{v}) = \effi{0}.
\end{equation}
where $\effi{0}$ defines the normalization of this model and models to follow.
Our second model is a generalization of the dependence proposed by
\citet{2012ApJ...759L..27P},
\begin{equation}\label{eq:padoanmodel}
\eff(\alpha_{v}) = \effi{0} \exp(-\beta(3\pi^{2}/40)^{1/2}\alpha_{v}^{1/2}).
\end{equation}
In their simulations of self-gravitating driven-turbulence periodic
boxes for models with a range of global ratios of $\tff/t_{\rm dyn}$,
they found that $ \eff =\effi{0} \exp(-\beta \tff/t_{\rm dyn})$ with
$\beta = 1.6$ followed the overall trend for the dependence of
SFR on the value of $\tff/t_{\rm dyn}$ (see Section\ref{sec:intro}).
Another possible model is a simple $\alpha_{v}$ cutoff,
\begin{equation}\label{eq:cutoffmodel}
\eff(\alpha_{v}) = \effi{0} H(\alpha_{v,\mathrm{cutoff}} - \alpha_{v}),
\end{equation}
where H is the Heaviside step function, thus taking only objects with $\alpha_{v} < \alpha_{v,\mathrm{cutoff}}$ but weighting them equally.
Since we are interested in comparing model $\SFR_{m}(t)$ to simulation
$\SFR(t)$, we apply Bayes's theorem,
\begin{equation}\label{eq:Bayes}
P(A|B) = \frac{P(B|A)P(A)}{P(B)}
\end{equation}
where A represents the model $\SFR_{m}(t)$ given by \autoref{eq:modelsfr} and B represents the simulated SFR(t) given by \autoref{eq:sfrage}.
For the likelihood $P(B|A)$ we assume
\begin{equation}\label{eq:likely}
P(B|A) = \prod_{i} \frac{1}{\sqrt{2\pi\sigma^{2}}} e^{-\frac{(\Delta\SFR(t_{i})/\langle\SFR\rangle)^{2}}{2\sigma^{2}}},
\end{equation}
taking the product over discrete time samples $t_{i}$, and where
\begin{equation}
\Delta\SFR(t) = \SFR(t) - \SFR_{m}(t-t_{d}).
\end{equation}
We select subsets of $\{t_{i}\}$ for each delay time $t_{d}$ so that the
likelihood $P(B|A)$ is always computed using the same number of
samples/snapshots regardless of $t_{d}$. We normalize by the time
averaged global star formation rate $\langle\SFR\rangle$ so that
$\sigma$ is dimensionless.
For a given object class and model $\eff(\alpha_{v})$, we evaluate the
likelihood $P(B|A)$ over the parameter vector $\theta$ that includes
time delay $t_d$, $\effi{0}$, additional model parameters ($\beta$ or
$\alpha_{v,\mathrm{cutoff}}$ as appropriate), and $\sigma$.
Since $A$ represents SFR$_m$ and depends only on the parameter vector
$\theta$, the posterior in \autoref{eq:Bayes} is
\begin{equation}
P(\theta|\SFR) = \frac{P(\SFR|\theta)P(\theta)}{P(\SFR)}.
\end{equation}
Note that $P(\theta) = \prod_{i} P(\theta_{i})$, the product of
priors, which we briefly describe. We use uniform linear priors for
time delay $t_{d}$ and slope $\beta$ (allowing negative values), and
uniform logarithmic priors for $\effi{0}$, $\sigma$, and
$\alpha_{v,\mathrm{cutoff}}$. Using uniform linear priors instead of logarithmic does
not substantially change our results.
Since $P(\SFR)$ does not vary with $\theta$, we estimate the
marginalized distribution for parameter $x$ by integrating over other
parameters $\Theta = \{y \in \theta | y \neq x\}$
\begin{equation}
P(x|\SFR) = \frac{\int P(\theta|\SFR) d\Theta}{\int P(\theta|\SFR) d\theta} = \frac{\int P(\SFR|\theta) P(\theta) d\Theta}{\int P(\SFR|\theta) P(\theta)d\theta}
\end{equation}
thus inferring mean values of each parameter
\begin{equation}\label{eq:Bayesmean}
\hat{x} = \int x P(x|\SFR) dx
\end{equation}
and variance from
\begin{equation}\label{eq:Bayesvariance}
\mathrm{Var}(x) = \hat{x^{2}} - \hat{x}^{2}
\end{equation}
For the constant $\eff$ model, inferring $\eff$ is equivalent to obtaining
the simple linear regression described above in Equation~\ref{eq:effinfer}.
From the definition of $\sigma$ in \autoref{eq:likely}, the inferred
value of $\sigma$ is equivalent to $\sigma_{\Delta \SFR/\langle \SFR
\rangle}$, and is a measure of the goodness of fit of each model to
the data for the inferred parameter values. Beyond the single value
of $\sigma$, it is also interesting to compare the distributions in
$\Delta \SFR_i/\langle \SFR\rangle$ for different models and different
gas subsets.
\subsection{Analysis Methods Summary}\label{sec:analysissum}
To summarize, we compute two categories of properties from a set of
simulation snapshots: object-by-object properties of all objects from
all snapshots, and time series of simulation snapshot totals.
Object-by-object properties include mass, volume, mean density,
free-fall time, mass per free-fall time, and virial parameters, and
only apply to number density threshold, HBR, and HBP objects. The
bound fraction is an object-by-object property only applying to $\nhmin$
objects. Simulation snapshot total properties
include mass, volume, and mass per free-fall time, which we calculate
for all definitions: surface density threshold, number density
threshold, number density bins, HBRs, and HBPs. We use the
object-by-object properties to study the population of star-forming
objects. We use the time series to study temporal
correlations between the star formation rate and various subsets of
the simulation gas.
\subsection{TIGRESS Simulations}\label{sec:tigress}
Although our methods can be applied more broadly, we focus our tests
on the TIGRESS simulations. TIGRESS is a suite of MHD simulations
which self-consistently models star formation and effects of feedback
in the three-phase ISM at parsec scales. Details of the TIGRESS numerical
algorithms are presented in \citet{KimOstriker2017}, along with
results on the basic properties (and a convergence study) of
a model with parameters representative of the Solar neighborhood. We
use two versions of this model for the tests in the present paper, one
with 4 pc resolution and one with 2 pc resolution. Data dumps from these
models that we use have a cadence of 1 Myr, with different minimum and
maximum times as indicated in Table \ref{tab:simparam}. While the
surface density declines over time, the typical value is $\sim 10 \surfunit$.
The features in TIGRESS include self-gravity, sink particles,
supernova rates and FUV luminosity from a population synthesis model,
resolved supernova remnant evolution prior to cooling,
FUV-dependent photoelectric heating, optically thin cooling, and galactic shear.
TIGRESS uses shearing periodic boundaries in the galactic plane and
outflow in the vertical direction. The shearing periodic boundaries
affect the computation of gravitational potential isocontours. We use
an algorithm wherein each cell only needs to know which cells are its
immediate neighbors, so to correct for shearing periodic boundaries
(or any other boundary) we simply correct the neighbor list of cells
on the boundary. The shear velocity is included in computing the
kinetic energy of objects, but is a small effect. However, a
correction is necessary across a shearing periodic boundary, since an
object lying across the boundary contains cells with extra velocity
$qL\Omega = 28.7 \kms$.
\begin{deluxetable*}{l l l l l l l}\label{tab:simparam}
\tablecaption{Simulation Parameters \label{tab:table1}}
\tablehead{\colhead{Name} & \colhead{Resolution} & \colhead{Cadence} & \colhead{$t_{\mathrm{min}}$ (Myr)} & \colhead{$t_{\mathrm{max}}$ (Myr)} & \colhead{$\Sigma_{\mathrm{min}} (M_{\odot}/\pc^{2})$} & \colhead{$\Sigma_{\mathrm{max}} (M_{\odot}/\pc^{2})$}}
\startdata
MHD\_4pc & 4 pc & 1 & 300 & 700 & 8 & 13\\
MHD\_2pc & 2 pc & 1 & 351 & 421 & 9 & 10\\
\enddata
\end{deluxetable*}
\subsection{Dendrograms}
We can use a dendrogram\footnote{See \cite{2008ApJ...679.1338R,2009Natur.457...63G,2013ApJ...770..141B} for previous dendrogram analyses} as a graphical representation of the
gravitational potential contour
tree. Simultaneously, a dendrogram represents the structure of the
gravitational potential and shows where HBRs are relative to that
structure. We compute a dendrogram so that local minima in the
gravitational potential are spaced evenly (``Tree Index'') and ordered
so that two objects that intersect are nearby.
Then, the intersections can be represented by
non-overlapping horizontal lines, and distances in ``Tree Index''
roughly encode 3-D spatial distances, since intersecting isocontours
are obviously in contact with each other. We start with a list of all
isocontours on top of the tree with no parents. A complete contour
tree should only have one such isocontour containing all points, but
it may be desired to terminate the evaluation of the contour tree
early. Then, each member of the list is replaced with itself followed
by its immediate children, the isocontours that merged to form
it. This repeats for each new member of the list and can be performed
recursively. Forming this list top-down keeps children together, with
only their descendents between them, which ensures that intersections
do not overlap. Then, the tree is plotted in reverse, since deeper
descendents appear later in the list and need to be plotted first, as
the average of their ``Tree Index'' determines the ``Tree Index'' of
their parents. Local minima are plotted first and given integer ``Tree
Index,'' which evenly spaces them, as desired.
\begin{figure}
\includegraphics[height=0.95\textheight,width=\columnwidth,keepaspectratio]{\figformat{progression}}
\caption{A progression of surface density snapshots from the TIGRESS
2pc (MHD\_2pc) simulation at 370, 390, and 410 Myr (from top to bottom)
comparing
energy-identified objects (left) to density-identified
objects (right). Red contours show projections of
HBR (left) and $\nhmin = 100\pcc$ objects
(right). Black contours show HBP (left) and $\nh > 10\pcc$ objects
(right).}
\label{fig:progression}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{\figformat{tree411}}
\caption{A representation of the contour tree, or dendrogram,
showing objects according to their gravitational potential value
($\Phi$, in units of $(\kms)^{2}$) and relative position in the
tree. This corresponds to the bottom panels of
\autoref{fig:progression}, from the 2pc resolution simulation at
410 Myr. Downward carets show local minima of the gravitational
potential, with red carets showing minima hosting HBR. The bases
of upward green carets show the maximum $\Phi$ isocontour of each
HBR, bound relative to a horizontal black line delineating the
maximum $\Phi$ of its HBP. Regions between critical points are
represented as vertical black lines, and critical points are
horizontal black lines where those regions intersect and merge in
this tree diagram.}
\label{fig:tree}
\end{figure}
\section{Results}\label{sec:results}
\subsection{Structure geography and object dendograms}\label{sec:resultmaps}
Sample surface density snapshots from the MHD\_2pc TIGRESS model can be seen in
\autoref{fig:progression}. Also shown, in left-hand panels,
is a comparison between HBR and HBP objects, projected onto the horizontal
plane. In the right-hand panels, we similarly show projections of objects
defined by density thresholds $\nhmin = 10\pcc$ and $\nhmin = 100\pcc$.
This comparison highlights the smoother and more selective nature of energy-identified objects.
A sample dendogram of the HBP and HBR objects identified in
\autoref{fig:progression}e is shown in \autoref{fig:tree}. The
dendrogram reveals several qualitative properties. For example, ISM
turbulence is of the order $v \sim 1-10 \kms$, so it is expected that
bound material must be found in wells with depths of $\Delta\Phi \sim
1-100 \mathrm{km^{2}/s^{2}}$. At a glance, this is apparent in
\autoref{fig:tree}. Most local minima, and most of the regions
represented by the tree do not host bound regions. The differences
between the tops of HBRs and the tops of HBPs, roughly represents the
total energy and corresponds to $v \sim 1 \kms$.
Furthermore, we can see that the merging criterion described in
\autoref{sec:mhbr} usually prefers the smallest scale isocontours at
this resolution, corresponding to critical isocontours containing only
1 local minimum. That is, no merging occurs to produce HBR in the
snapshot represented by \autoref{fig:tree}, and in general this is rare. A merged
HBR would appear as a green upward caret on a vertical line
(representing a volume) stemming above (containing) a horizontal line
(representing an intersection between isocontours of multiple local
minima). Qualitatively, this is because merging adds very little
$\Delta\Phi$ for each merge, as evidenced by short vertical lines in
\autoref{fig:tree} corresponding to $v \sim \kms$, but quickly moves
to larger length scales and higher velocity dispersion.
Note that \autoref{fig:progression} (panels e and f) shows that the
gas is mostly contained in a single large-scale region, which should
result in an overall potential well of the simulation. This is represented in
the dendrogram as the overall well shape, except for the large
isocontour at (-200 pc, 300 pc) corresponding to index 44. The densest
gas and the bound gas in the hierarchy tends to be near the bottom of
the overall well of the simulation.
\subsection{Gas Distribution and Object Properties}\label{sec:prop}
First, we summarize some of the basic properties of the gas in the
simulations. In \autoref{fig:hist_bin} we show the number density
distribution in the 4pc and 2pc simulations, taken over all times. We
show mass fractions of half-decade bins in number density and
normalize the continuous distribution accordingly. In both simulations
the mass pdfs are centered near $\nh = 1~\pcc$
(mass-weighted mean densities are $\nh=4.84$ and $10.1\pcc$
for 4pc simulation and 2pc simulation, respectively)
with maximum density of
$10^{2.5}\pcc$ in the 4pc simulation and $10^{3}\pcc$ in the 2pc
simulation. The mode of the distribution is at density of $\nh=0.7\pcc$ and
$\nh=0.8\pcc$
for 4pc simulation and 2pc simulation, respectively.
The distribution depends on resolution at high number
density due to the criterion for introducing sink particles when
collapsing objects become unresolved.
The density distribution is dominated by a roughly log-normal
distribution, with a secondary cold dense component. In
\autoref{fig:hist_thresh} we show the mass fractions above number
density thresholds. In the 2pc simulation, roughly half of the mass is
denser than $\nh=1 \pcc$, roughly a tenth of the mass is denser than
$30\pcc$, and a few per cent of the mass is denser than $100\pcc$.
Compared to the lower-resolution model, the 2 pc model
has slightly larger mass fractions at higher density.
\begin{figure}
\includegraphics[width=\columnwidth,keepaspectratio]{\figformat{histbin}}
\caption{The hydrogen number density distributions of TIGRESS Solar
neighborhood simulations with 2 pc (red) and 4 pc (black)
resolution, taken at late times ($t > 300$ Myr). Half decade bins
are shown for the 4 pc simulation case, showing the fraction of
the total mass in each bin. }
\label{fig:hist_bin}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth,keepaspectratio]{\figformat{histthresh}}
\caption{As in Figure~\ref{fig:hist_bin} except with cumulative
distributions of the hydrogen number density.}
\label{fig:hist_thresh}
\end{figure}
Next we describe properties of $\nhmin$ objects, using
thresholds of $\nhmin=$ 10, 30, and $100\pcc$, and compare them to HBR
objects. For the following examination of object properties we only
use the 2pc simulation so that objects are better resolved.
\begin{figure*}
\includegraphics[width=\textwidth]{\figformat{histx}}
\caption{Number-weighted (blue, right axis) and mass-weighted
(black, left axis) distributions of mass, radius, number density,
and free-fall time, for objects defined in
the 2 pc resolution simulation. The top
two rows show distributions for HBP (Parent) and HBR (Bound)
objects, and the bottom two rows show distributions for objects
defined by density threshold of $\nhmin=$ 10 and $100\pcc$. The
radii are computed from the volume as $R = \sqrt[3]{3V/4\pi{}}$.}
\label{fig:hist_x}
\end{figure*}
In \autoref{fig:hist_x} we present number-weighted and mass-weighted
distributions of mass, radius, density, and free-fall time for HBR and
HBP objects as well as objects defined by number density thresholds
$\nhmin=$ 10 and $100\pcc$. For HBR objects, the typical mass is
$10^{3}-10^{4} \Msun$, with very few above $10^{4}\Msun$.
The HBR objects are mostly dense,
with $\nh$ a few $100 \pcc$. Hence, it is useful to compare HBR to
density objects (both threshold and bin) around $100\pcc$.
The characteristic free-fall times of HBR objects are short ($\tff\sim 3$ Myr).
Radii of HBR objects are typically several pc, which demonstrates that they are
well-resolved with 2pc resolution.
We note that a barely resolved 4x4x4 region of cells would have a
volume of $8^3=512\pc^{3}$. For number densities of 10, 30, and
$100\pcc$, such a region would respectively have masses of 170, 500,
and $1700\Msun$ assuming $\mu = 1.4$. We find very few HBR objects below this
lower mass limit set by resolution.
HBP objects have larger sizes and masses than HBR objects, with lower
characteristic densities (a few tenths) with free-fall times nearly 10
Myr.
There are a large number of $\nhmin$ objects at small
masses and radii, since we place no lower cutoff on their
size. However, most of the mass is in objects of large mass and
radius. For $\nhmin = 10\pcc$, typical (in a mass-weighted sense) objects
are $10^{5}-10^{6}\Msun$ and $\lesssim 100 \pc$. For $\nhmin = 100\pcc$,
typical objects are $10^{4}-10^{5}\Msun$ and $\lesssim 10 \pc$.
\begin{figure*}
\includegraphics[width=\textwidth]{\figformat{avc}}
\caption{Distributions in virial parameters and mass of various
defined objects in the 2pc simulation. Left panels show HBRs, and
central and right panels show $\nhmin = 10, 30$, and $100\pcc$
objects. In the top row (panels a, b, c, and d), only kinetic
energy is considered for $\alpha_{v}$ (Equation~\ref{eq:virialdef}).
In the bottom row (panels
e, f, g, h) $\alpha_{v,\mathrm{total}}$ considers kinetic,
thermal, and magnetic energy (Equation~\ref{eq:virialtotdef}).
Dashed horizontal lines delineate
$\alpha_{v}=2$ or $\alpha_{\rm v,total} = 2$, corresponding to
$\Esubt{k} = \Esubt{g}$ in top panels and
$\Esubt{k} + \Esubt{B} + \Esubt{th} = \Esubt{g}$ in bottom
panels. Vertical lines represent the minimum mass estimate from
\autoref{eq:wellmass}. Contours show the distribution of
$\nhmin$-objects whose mass has less than one per cent overlap with
HBR. Scatter points are individual $\nhmin$-objects whose color reflects
their mass fraction overlap with HBR. Truly bound objects (red points)
with order unity overlap with HBR tend to be low mass (around
$10^{3}\Msun$) with virial parameters $\alpha_v \lesssim
2$. Especially at high masses, many apparently ``bound'' objects
based on $\alpha_{v} < 2$ are not in fact HBR-bound (i.e. they are colored
blue-green-yellow). Additionally, many $\alpha_{v} > 2$ and
$\alpha_{v,total} > 2$ objects at low and moderate density
have significant HBR overlap (red). These results show
that ``observed'' virial
parameter is not a good indication of true gravitational binding.}
\label{fig:av}
\end{figure*}
\autoref{fig:av} shows the distribution of virial parameters and
masses for HBR and $\nhmin$ objects, via contours and scatter plots.
For scatter plots of individual $\nhmin$ objects, the color of each
point indicates the fraction of its mass that is bound, based on
overlap with HBR objects. We find that very few $\nhmin$ objects at
the low mass end ($< 10^{3}\Msun$) have overlap with HBR, even if
their kinetic virial parameter $\alpha_{v} < 2$. $\nhmin$ objects with
negligible (less than one per cent) overlap with HBR are represented
by contours enclosing $20\%$, $40\%$, $60\%$, $80\%$, and $90\%$ of
the objects.
The depth of a well behaves as $GM/R \sim G\rho{}R^{2}$. At constant
density, only a sufficiently large object will have a well deep enough
to bind material. A rough estimate comparing $GM/R > v^{2}$ with $M =
(4\pi{}/3)\rho{}R^{3}$ yields a minimum mass that follows
\begin{equation}\label{eq:wellmass}
M^{2} > \frac{v^{6}}{\frac{4\pi}{3}G^{3}\rho{}}.
\end{equation}
For $v = 1\kms$ and $\nhmin = 30\pcc$, this minimum mass is
$2 \times 10^{3}\Msun$. The $\nhmin = 10\pcc$ and $\nhmin = 30\pcc$ objects
overlapping HBR lie above their respective minimum mass
(for $v=1\kms$), whereas the
$\nhmin = 100\pcc$ objects all lie above the resolution minimum mass
(which is greater than the well minimum mass).
For objects of mass $\sim 10^{3.5}-10^4\Msun$, those at the lower range of
$\alpha_v$ and $\alpha_{v,total}$ have the largest fraction of bound gas (i.e.
red colored points), which is consistent with general expectations. However,
the actual values of $\alpha_v$ and $\alpha_{v,total}$ in the objects with
$>50\%$ HBR-overlap cover a wide range from
$\alpha_v \sim 0.6-6$, generally decreasing at higher density. This shows that
the ``observed'' virial parameter is not a very accurate
quantitative measure of gravitational boundedness.
Furthermore, Figure~\ref{fig:av} shows that unlike low-mass
($\sim 10^{3.5}-10^4\Msun$) $\nhmin$ objects with
$\alpha_v \lesssim 2$ which are generally bound, $\nhmin$ objects at the
high mass end ($ > 10^{4.5}\Msun$) have very little overlap with HBRs
even at low virial parameter ($\alpha_{v} < 2$). That is, high mass
objects ($10^{6}\Msun$) can appear bound based on simple criteria
using their mass, size, and velocity dispersion (Equation~\ref{eq:virialdef}),
but in reality this
is not consistent with full gravitational potential structure. This
is in part due to tidal fields preferentially unbinding larger-scale
objects, and in part due to substructure. Substructure within a
$\nhmin$ object manifests itself as multiple separate HBR objects which
comprise a small fraction of the mass because most of the mass lies in
between HBRs.
In summary, we find that overdense objects with $\alpha_v \lesssim 2$ are
truly bound only if their masses are low; high-mass objects are generally
unbound even when a simple estimate suggests otherwise.
\begin{figure}
\includegraphics[width=\columnwidth,keepaspectratio]{\figformat{pbound}}
\caption{The fraction of mass in objects defined by density
thresholds that is also within HBR objects; this is equivalent to
the conditional probability $P(\rm bound | \rm dense)$, in the 2pc
simulation. This increases with density $\nhmin$, but even at high
densities (100 $\pcc$) only $\sim 10\%$ of the mass is bound.}
\label{fig:pbound}
\end{figure}
In \autoref{fig:pbound} we show the HBR fraction of the mass above
number density thresholds $\nhmin = 10$, $30$, and $100
\pcc$, representing the probability of gas being bound given that it
is dense. This fraction is only a few per cent for $\nhmin = 10\pcc$ and
$30\pcc$, increasing to $10\%$ for $\nhmin = 100\pcc$. The fraction
roughly follows the power law
$\nhmin^{0.80}$. Since HBR mass tends to be
dense, each density threshold should contain nearly all the HBR
mass. Thus, the overlap fraction should roughly follow the reciprocal
of the threshold distribution as shown in \autoref{fig:hist_thresh}
(for the 2pc simulation). The threshold distribution at these number
densities follows $\nhmin^{-0.87}$, whose reciprocal has a similar slope
as expected. Although the $\nhmin = 100 \pcc$ threshold is within an
order of magnitude of the maximum density in the simulation, and the
Larson-Penston density is one of the criteria for star particle
formation, we do not find strong evidence for a critical density for
boundedness within the range $\nh = 10-100\pcc$.
\begin{figure*}
\includegraphics[width=\textwidth]{\figformat{lws}}
\caption{The linewidth-size ($\sigma_{3D}$ and radius from
$V = (4\pi/3)R^{3}$ for object volume $V$)
relationship of various objects (number
density threshold, HBR, HBP) from the 2 pc TIGRESS solar
neighborhood simulation. In panels a-e, contours show the full
distribution and the blue line shows the median value of radius
bins. The contours contain $20\%$, $40\%$, $60\%$, $80\%,$ and
$90\%$ of the objects. For reference, the black dashed line
represents $\sigma \propto R^{1/2}$ with normalization set by the
measured velocity dispersion of $T < 2\times{}10^{4} K$ gas for
$R$ equal to the measured scale height in the simulation. In
panels a-c, the black dotted line represents spheres at the threshold
density with equal kinetic and potential energy
($\sigma \propto R \nhmin^{1/2}$).
The solid black line is $\sigma \propto R^{1/2}$ with normalization similar
to that of Milky Way GMCs
$\sigma_{3D} = \sqrt{3} \times 0.9\kms (R/\pc)^{1/2}$ \citep{Heyer2015}.
The median relations of all object
types are stacked in the bottom-right panel.}
\label{fig:lws}
\end{figure*}
We show in \autoref{fig:lws} the linewidth-size relationships of
$\nh$-threshold, HBR, and HBP objects.
Here, the linewidth for an object is defined as $\sqrt{2\Esubt{k}/M}$ where $M$ is the mass and $\Esubt{k}$ is the kinetic energy in the object's center of mass frame.
The size is computed from object volume $V$ taking $V = (4\pi/3)R^{3}$.
Panels a-e of the figure
include both contours of each distribution and median relationships, with
bins for the latter chosen to each contain 100 objects, except for the
final bin. We also include several reference lines with slopes
$\sigma \propto R^{1/2}$ and $R$ for comparison.
$\nhmin$ objects and HBRs have linewidth-size relationships that are
somewhat steeper than $\sigma \propto R^{1/2}$, which is what would
be expected if all the objects simply sampled from the same power
spectrum of highly compressible ISM turbulence with an outer scale
much larger than typical object sizes. The objects with the closest
linewidth-size relation to $\sigma \propto R^{1/2}$ are HBPs, which
are formed by isocontours of the gravitational potential. While
there are detailed differences among the median linewidth-size
relationships for different object categories,
\autoref{fig:lws}f shows that the median relationships are in fact quite
similar across all categories.
For $\nhmin$-objects, in each panel of \autoref{fig:lws} we also include
the line corresponding to a spherical object with density equal to the
threshold density and kinetic energy equal to the potential energy,
which has slope $\sigma \propto R \nhmin^{1/2}$. Moving to higher $\nhmin$ shifts the
$\sigma \propto R$ line from self-gravitation upward since
$\sigma_{3D} = (\rho G 8\pi/5)^{1/2} R$.
For density threshold $\nhmin=10, 30 \pcc$, most
objects lie above the marginally self-gravitating locus. For the
$\nhmin=100 \pcc$ threshold, the median relationship follows the
marginally-bound relation quite well. These results are consistent with the
results for the distributions of $\alpha_v$ shown in \autoref{fig:av},
which shows that the typical $\alpha_v$ decreases with $\nh$ threshold.
We again emphasize that even with order-unity $\alpha_v$, most of the material
in objects above a density threshold $\nhmin=100 \pcc$ is not part of bound
regions.
\subsection{Time Series}\label{sec:rtime}
We now turn to the results of our time series analysis, based on methods
described in
\autoref{sec:mtime}. Since a large number of snapshots is necessary for this
analysis,
we primarily use the 4pc simulation, which was run for a longer time.
\begin{figure}
\includegraphics[width=\columnwidth]{\figformat{timemass}}
\caption{Time series of mass fractions for various categories of
objects and density bins, as labeled, for the 4 pc resolution
Solar neighborhood simulation. HBRs are bound objects and HBPs are
their parents; see text in \autoref{sec:mhbr} for explanation of
treatment of surface terms in HBR$\pm$1. The quasi-periodic
variations in the $\nh$ bin time series reflect the natural 50 Myr
vertical oscillation timescale in the galactic potential. }
\label{fig:time_mass}
\end{figure}
In \autoref{fig:time_mass} we show how the mass fractions of different
categories of material evolve over time. The top panel shows HBR and HBP
material as well as HBR$\pm$1, where the latter allows for surface terms
to help confine or else disperse material, and HBP$\pm$1 are the ``parents''
of HBR$\pm$1 objects (see \autoref{sec:mhbr} for details of definitions).
HBP+1 and HBR+1 include more mass and HBP-1 and HBR-1 include less mass,
as described in \autoref{sec:mhbr}.
The bottom panel shows the mass fractions of material
in half-decade number
density bins.
Overall, the amplitude of fluctuations increases for categories with lower
mean mass fractions. In addition, upward fluctuations are successively
delayed in time for increasing $\nh$ bins; we discuss this effect
further below.
Roughly $10\%$ of the mass
is in low density bins and in HBR+1/HBP+1 objects. HBR+1 and HBP+1 objects
effectively subtract the surface value of kinetic, thermal, and
magnetic energy from each cell in the object, such that
low density material in the
shallower regions of a potential well is considered bound and the object
mass is higher than if surface terms were neglected.
The HBR+1 mass is close to the
HBP+1 mass because nearly all the material within an isocontour is
considered bound.
A few per cent of the mass is within HBP-1 and HBP objects,
which have relatively similar mass histories.
Only of order $10^{-3}$ of all the material in the simulation is in
HBR and HBR-1 objects. Relative to HBR, HBR-1 objects have slightly less
mass because the surface energy terms are added to each cell, reducing the
amount of material that is considered bound.
\begin{figure}
\includegraphics[width=\columnwidth]{\figformat{mbtsfr}}
\caption{Comparison between star formation rate (SFR) and mass
divided by free-fall time $M/\tff{}$ of various gas populations from
$t=300-700$ Myr in the 4 pc resolution Solar neighborhood
simulation. The SFR shown is smoothed over 5 Myr and normalized to
its time-average value. The $M/\tff{}$ time series are also
normalized relative to their time-averaged values. Individual
panels compare (a) surface density thresholds, (b) number density
thresholds, (c) number density bins, and (d) HBP and HBR. In
all cases the denser, more restrictive definition leads to a
qualitatively better match between SFR and $M/\tff{}$.}
\label{fig:mbtsfr}
\end{figure}
It is interesting to compare the SFR history with the evolution of
$M/\tff$ at lower and higher gas surface density $\Sigma$, lower and
higher density $\nh$, and for less and more bound objects. By
comparing time series in \autoref{fig:mbtsfr}, it is evident that
more restrictive definitions have an improved correlation with $\SFR$.
This holds
for increasing threshold $\Sigma$, increasing $\nh$ threshold,
increasing $\nh$ bins, and increasing boundedness from HBP to HBR.
Intriguingly, \autoref{fig:mbtsfr}b,c demonstrate that simple density
criteria (high density threshold or bin) yield {\it better}
correlation with SFR than the more complex criteria based on total
energy in the full gravitational potential landscape that go into the
definition of HBR, shown in \autoref{fig:mbtsfr}d. The visual
impressions of these histories already suggest that gravitational
binding is not a guarantee that star formation will be successful; we
return to this quantitatively below.
\begin{figure*}
\includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{\figformat{timedensity}}
\caption{Comparison between SFR and $M/\tff{}$ as in
\autoref{fig:mbtsfr} for half-decade number density bins
Here
$M/\tff{}$ is normalized by a factor of $\eff{}/\mean{\SFR}$,
where $\eff{}$ is the result of the simple linear regression
(\autoref{eq:effinfer}). The inferred delay time $t_{d}$ is used
to offset the time series, which is labeled as ``delayed'' for
each number density bin. It is clear that the correlation between
SFR and $M/\tff{}$ improves and $t_{d}$ decreases as density
increases.}
\label{fig:time_density}
\end{figure*}
\begin{figure}
\includegraphics[width=\columnwidth,keepaspectratio]{\figformat{timelag}}
\caption{The delay time $t_{d}$ vs. lower density edge of some of
the half-decade density bins shown in
\autoref{fig:time_density}. Lines for
$\tff{} = [3\pi/(32G\mu n_{\mathrm{H}})]^{1/2}$
are computed using the
lower and upper number density edge, respectively resulting in a
maximum and minimum free-fall time. For denser bins, the delay
time roughly follows the minimum free-fall time.}
\label{fig:time_lag}
\end{figure}
\begin{figure*}
\includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{\figformat{timeothers}}
\caption{Comparison of $\SFR$ and $\epsilon_{\rm ff} M/\tff$ as
in \autoref{fig:time_density} for surface density thresholds
($\Sigma=$ 10 and $30\Msun\pc^{-2}$), number density thresholds ($\nhmin=$ 10, 30, and $100\pcc$), and HBR (bound) objects, showing only the delayed time
series.}
\label{fig:time_others}
\end{figure*}
As described in Section \ref{sec:mtime}, we can apply linear
regression to obtain both the optimal time delay to match the shape of
each $M/\tff$ time series to the SFR, and the corresponding
normalization amplitude $\eff$ that measures the best-fit star
formation efficiency per free-fall time. \autoref{fig:time_density}
shows the result of applying this linear regression, demonstrating
that higher $\nh$ bins correlate more strongly to SFR, with a smaller
time delay, compared to lower $\nh$ bins. Even at low
$\nh = 10^{0.5}\pcc$ some correlation is apparent, but not for lower
densities. Thus, while the amount of gas in low density bins
($\nh=10^{-0.5}-10^{0.5}$) comprising the bulk of ISM mass (see
\autoref{fig:hist_bin}) varies in time due to large-scale vertical and
horizontal oscillations that produce ISM compressions and
rarefactions, these variations do not appear to directly {\it induce}
star formation.
In \autoref{fig:time_lag} we show that the time delay inferred from
the fit is comparable to the free-fall time associated with the $\nh$
bin upper edge,
$t_{\mathrm{ff,min}} = [3\pi/(32G\mu n_{\mathrm{H,max}})]^{1/2}$, with
$n_{\mathrm{H,max}} = 10^{0.5}n_{\mathrm{H,min}}$. This is consistent
with the idea that a variation in mass or $M/\tff$ at a given density
can only lead to a variation in SFR after the gas is able to dynamically
respond; the minimum response time is the free-fall collapse time at that
density, and this indeed appears to be the defining timescale, even
though the efficiency of collapse is less than unity.
In \autoref{fig:time_others} we compare histories of star formation
with the time series of $\epsilon_{\rm ff} M/\tff$ for best-fit time
delay and $\epsilon_{\rm ff}$ for material defined by surface density
threshold, number density threshold, and HBR objects. For surface
density and number density, correlation with SFR improves with a
higher threshold. This correlation is visually similar for $\Sigma >
30 \surfunit$, $\nh > 10, 30\pcc$. Although HBR and $\nh > 100\pcc$ both
follow $\SFR$ quite closely (and in particular follow lows much better than predictions based on lower density thresholds),
in fact the mass of HBR gas provides a slightly worse prediction of $\SFR$
than the mass of gas at $\nh > 100\pcc$. We quantify this in the next
subsection.
\subsection{Star Formation Efficiency}\label{sec:eff}
\begin{figure*}
\includegraphics[width=\textwidth,keepaspectratio]{\figformat{effsigma}}
\caption{Inferred $\eff{}$ (a, b, e, f) and RMS error $\seff$ (c, d
, g, h) based on time series of SFR compared to time series of
$M/\tff$ for selected subsets of the gas, as labeled. Gas
selection criteria include density ($\nh$) thresholds and bins,
surface density ($\Sigma$) thresholds, bound objects (HBR) and
their parents (HBP), overlaps between density and HBR objects
($j$), combined density and $\alpha_v<2$ criteria ($v$); see text
for details. In each panel, results from 4pc are shown as darker
points with 2pc ($\eff$ only)shown as lighter points. Errorbars from \autoref{eq:Bayesvariance} are not shown but would lie within markers, decreasing with the number of time snapshots used.}
\label{fig:eff}
\end{figure*}
In this section, we present results on our inference for efficiency
per free-fall time $\eff$ for various subsets of gas, based on
application of the methods of Section \ref{sec:mtime}. We are interested
in both the measures of $\eff$ and how they depend on the criteria for
defining a subset of the gas, and quantitative assessment of the
relative performance for predicting SFR. We include results from both
the 4pc simulation time series, which based on its larger number of
snapshots is advantageous in terms of sample size, and the 2pc
simulation, which allows us to test whether our results are converged
with respect to numerical resolution.
\autoref{fig:eff} shows for several different categories of objects
the values for $\eff$ and for $\sigma_{\Delta \SFR/\langle \SFR \rangle}$ based on linear
regression. The top two rows show results for gas subsets defined by
$\Sigma$ and $\nh$ thresholds and by $\nh$ bins, also comparing to HBR
results. For both $\Sigma$ and $\nh$ thresholds
(\autoref{fig:eff}a,c), the total mass decreases faster than the
free-fall time as the threshold increases. As a result, $\mff$ decreases at
increasing density,
leading to an increase in $\eff$ with threshold level. At the same time,
$\sigma_{\Delta \SFR/\langle \SFR \rangle}$
mostly decreases with increasing threshold, implying
better correlation of denser gas with SFR; this is consistent with the
visual impression from previous plots.
When we consider subsets of gas in density bins (\autoref{fig:eff}b,d),
$\eff$ increases and $\sigma_{\Delta \SFR/\langle \SFR \rangle}$
decreases at higher densities $\nh$.
However, at $\nh < 10^{1.5}\pcc$, $\eff$ is roughly flat at roughly $0.06$.
The value $\eff \sim 0.01$ for $\Sigma > 10 \surfunit$ represents the mean
efficiency for the bulk of the material in the simulation.
Interestingly, while the
values of $\eff$ for high density thresholds are similar to the value
of $\eff$ for gas in HBR (a few tenths), the value of
$\sigma_{\Delta \SFR/\langle \SFR \rangle}$
is {\it lower} for density thresholds than for HBR gas. While HBR gas is
mostly quite dense, this says that the additional criteria of
requiring that every parcel of gas is bound within an HBR does
{\it not} lead to better agreement in the histories.
Looking at HBP and HBR variants in \autoref{fig:eff}f,g, higher mass
variants (all HBP, and HBR+1) have lower $\eff$ ($\sim 0.1$).
HBR has $\eff \approx 0.4$ and HBR-1 has
$\eff \approx 0.6$: in both cases the efficiency is nearly order
unity, as might be expected of truly collapsing objects. However,
we cannot distinguish between the case that each bound object takes a total time
$\tff/\eff$ to collapse, vs. the case that the probability
of each bound object being dispersed is $1-\eff$ and the collapse
timescale for surviving objects is $\tff$.
The variants of HBR and HBP have a similar correlation to SFR, but HBR
and HBP have slightly lower $\sigma_{\Delta \SFR/\langle \SFR \rangle}$.
It is interesting that isocontours alone (HBP) provide a
reasonable correlation to SFR, and that varying treatment of surface
terms has little effect.
We can also consider combined criteria and test the correlation with
SFR. In \autoref{fig:eff}e,g, we show results for ``$j$'' time
series, consisting of material that exceeds certain density
thresholds and also overlaps with HBRs, and for ``$v$'' time series,
material in $\nhmin$ objects that also satisfy $\alpha_{v} < 2$
(kinetic energy only, excluding thermal and magnetic, in the virial
parameter).
The $v$ series
has similar results to $\nhmin$ objects themselves, but with
slightly greater $\eff$ and comparable
$\sigma_{\Delta \SFR/\langle \SFR \rangle}$.
Whereas $\eff$
for $\nh > 10\pcc$ doubles when considering $\alpha_{v}$ in the $v$ series,
it is only higher by $30\%$ in the $\nh > 100\pc$ case.
Considering the virial parameter mainly affects lower density gas.
The ``$j$'' series over material overlapping between HBR and $\nhmin$
objects is mostly similar to HBR because most HBR mass also
satisfies $\nh \sim 100\pcc$. The exception is $\nh > 100\pc$ in the
4pc case because $100\pcc$ is close to the maximum density of the
simulation.
Note that although $\eff$ for surface and $\nhmin$ objects are
resolution dependent (4pc and 2pc points differ), the values of $\eff$
for low- and moderate-$\nh$ bins are resolution independent. Thus,
differences in $\nhmin$ objects are due to the lack of gas in high
density bins for lower resolution simulations. Unlike the high-$\nh$
series, the HBR and HBP series (and their variants) have essentially
the same values of $\eff$ at 4pc and 2pc resolution. The difference
in $\sigma_{\Delta \SFR/\langle \SFR \rangle}$ with resolution primarily reflects the different number of snapshots available at each resolution.
Whereas multiple cycles of star formation are available in the 4pc simulation, only 70 Myr of snapshots are available from the 2pc simulation. Therefore, we do not draw any conclusions from the 2pc values of $\sigma_{\Delta \SFR/\langle \SFR \rangle}$ and do not show those values in \autoref{fig:eff}.
\begin{figure}
\includegraphics[width=\columnwidth,height=0.9\textheight,keepaspectratio]{\figformat{errorhist}}
\caption{A comparison of the distribution of $\Delta\mathrm{SFR}/\langle\mathrm{SFR}\rangle$ (see Equation~\ref{eq:deltaSFR}), the difference between actual and predicted SFR using different categories of gas, based on (a)-(b) gas surface density threshold, (c)-(e) number density threshold, and (f) HBR objects,
corresponding to the time series shown in \autoref{fig:time_others}.
The mean error is shown as a blue vertical line, and quartiles are shown in black lines (median in solid, 25 and 75 dashed).}
\label{fig:error_hist}
\end{figure}
In \autoref{fig:error_hist} we illustrate how correlation changes for
different gas selection criterion
in further detail by providing histograms of the
error $\Delta\mathrm{SFR}/\langle\mathrm{SFR}\rangle$
(Equation~\ref{eq:deltaSFR}), where a
positive error means that the simulated star formation rate is higher
than the model star formation rate based on $M/\tff$ for a single snapshot.
For less restrictive selection criteria, such as lower $\nh$ or $\Sigma$,
the mean and median in the distributions shift to the left, indicating that the
predicted $\eff M/\tff$ exceeds the actual SFR. This is clearly evident in
\autoref{fig:time_others}: for low density thresholds, there is little
predicted variation in the SFR about the mean, whereas the true SF history is
mostly below the mean level, with some sharp peaks.
\autoref{fig:error_hist} also shows that worse correlation at lower $\nh$ or
$\Sigma$ threshold is associated with
a larger number of snapshots wherein $\Delta\mathrm{SFR}/\langle\mathrm{SFR}\rangle \sim -0.5-1$
and $\Delta\mathrm{SFR}/\langle\mathrm{SFR}\rangle > 1$.
Again, this is evident in the missed
``long valleys'' and ``sharp peaks''
for the prediction based on $\eff M/\tff$ in the
$\Sigma > 10 \surfunit$, $\nh > 10 \pcc$ cases in
\autoref{fig:time_others}.
Since missing sharp peaks occurs during periods of high SFR, considering an alternative version of \autoref{eq:SFRerr} by weighting by $\SFR(t_{i})$ would amplify the improvement of correlation with increasing density.
\autoref{fig:error_hist}e,f also quantifies the visual impression from \autoref{fig:time_others} that
the restriction that gas be bound (HBR) does not offer better predictive
power for $\SFR$ compared to a simple high density threshold. In particular,
the HBR prediction misses a peak at $t\sim 400$Myr, which accounts for the
positive error tail in $\Delta \SFR$ compared to $\nh > 100 \pcc$.
Counterintuitive to the immediate
visual impression from \autoref{fig:time_others},
HBR has larger $\sigma_{\Delta \SFR/\langle \SFR \rangle}$ than even the $\nh >10 \pcc$ and
$\nh > 30 \pcc$ for the 4pc model, although for the 2pc model HBR performs
better. The primary reason for this is the overall much larger range of
predicted $\SFR$ from HBR; since this has high peaks that can be slightly
offset from the peaks in the true $\SFR$, this leads to a broader distribution
of errors in \autoref{fig:error_hist}.
Another reason is that HBR can be too selective, where there are snapshots with high SFR but insufficient corresponding HBR gas mass.
\subsubsection{Dependence on virial parameter}
\begin{figure}
\includegraphics[width=\columnwidth,height=0.9\textheight,keepaspectratio]{\figformat{cphne}}
\caption{Comparison of inferred model parameters and goodness of fit for three
models of the dependence of SFR on $\alpha_v$ as described in
Section \ref{sec:effmodels}. Results are shown for model with
no $\alpha_v$ dependence (blue points), model with exponential
dependence on $\alpha_v^{1/2}$ (orange points;
\autoref{eq:padoanmodel}), and model with an $\alpha_v$ cutoff
(green points; \autoref{eq:cutoffmodel}). Points and bars represent the mean
(\autoref{eq:Bayesmean}) and standard
deviation (from \autoref{eq:Bayesvariance})
of marginalized distributions for time delay
$t_{\mathrm{delay}}$ (in Myr), efficiency $\eff$, slope $\beta$ for the
exponential model, and cutoff $\alpha_{v}$.
Standard deviations of normalized $\SFR$ errors
$\sigma_{\Delta\SFR/\langle\SFR\rangle}$
(inferred $\sigma$ in \autoref{eq:likely}) are shown for all models.
Reference values $\beta = 1.6$ and $\alpha_{v} = 2$ are shown with
horizontal dashed lines. Columns left to right use thresholds
$\nh>10$, $30$, and $100 \pcc$, and energy-based
criteria (HBR) to define objects.}
\label{fig:cphne}
\end{figure}
As discussed in Section \ref{sec:effmodels}, we can apply Bayesian
inference to our time series to evaluate parameters and explore the
relative goodness of fit for different models that have been proposed
for the dependence of star formation on the virial parameter. In our
tests, we separately examine objects defined by number density
thresholds with $n_{\mathrm{H,min}} = 10, 30$, and $100\pcc$, as well
as HBR objects.
\autoref{fig:cphne} presents the results of our analyses. From left
to right, panels show results for objects defined by different density
thresholds and by the HBR criterion. Each row gives values of
parameters obtained for the three models under consideration: constant
$\eff$ (blue points, equivalent to the results presented in
\autoref{fig:eff}), an exponential dependence on $\alpha_v^{1/2}$
(orange points, generalizing \citealt{2012ApJ...759L..27P}), and a
cutoff in $\alpha_v$ (green points).
The plotted value of $\eff$ represents
$\epsilon_{\rm ff,0}$ for the exponential and cutoff models.
For all models, from lower density to higher density thresholds the
inferred time delay decreases, consistent with
\autoref{fig:time_density}. At the same time, the inferred $\eff$
increases with $\nh$ for both the constant-$\eff$ and the
virial-cutoff models.
The inferred $\epsilon_{\rm ff,0}$ does not
monotonically vary with $\nh$ for the exponential model.
Going from $\nh >30$ to $\nh >100$, the RMS error $\sigma_{\Delta\SFR/\langle\SFR\rangle}$
decreases for both the exponential and cutoff model, similar to what was
shown previously for the model with no $\alpha_v$ dependence. Based on
RMS error levels for any given gas selection criterion, there is generally
no significant preference for the $\alpha_v$-dependent models for $\SFR$ compared to the
$\alpha_v$-independent model. This can be understood considering the
inexact correspondence between apparent $\alpha_v$ and true boundedness,
and the previously-discussed limitations of boundedness as a detailed
predictor of the $\SFR$.
The exception is the lower-density
$\nh>10$ object class, in which both models that
account for the virial parameter perform better than the
constant-$\eff$ model (see further discussion below).
For the exponential model, only the low density
threshold case shows a similar slope $\beta \approx 1.6$ to that found by
\citealt{2012ApJ...759L..27P}; at high density
$\beta \sim 0$ is preferred, which is equivalent to constant $\eff$.
This is consistent with expectations, considering the differences between
the types of simulations of \citealt{2012ApJ...759L..27P} vs.
the current models and analysis. In \citealt{2012ApJ...759L..27P}, the
comparison was between global SFRs for small-box simulations of cold gas
in which turbulence was driven to different levels. This is most similar
to our selection of objects with $\nh > 10 \pcc$, and then assigning
a relative probability of SF depending on level of the virial parameter
(which has a large variation at low density, with only the lower $\alpha_v$
objects being massive enough to host star formation,
as shown in \autoref{fig:av}).
When we instead select objects that are already quite overdense compared
to the average, the range of virial parameters is smaller
(as seen in \autoref{fig:av}) so there is little ``leverage.''
For a low density threshold $\nh > 10\pcc$, the $\alpha_{v}$ cutoff
model prefers $\alpha_{v} \approx 2$, for similar reasons to the higher
preferred $\beta$ in the exponential model. However, at $\nh>30$ and
$\nh>100$, the inferred cutoff $\alpha_v$ values are larger, demonstrating
that the density threshold itself provides a good correlation with the
SFR and that removing high $\alpha_{v}$ material is not
preferred. Energy-selected HBR objects also do not benefit from a
virial parameter cutoff.
\section{Conclusion}\label{sec:conc}
\subsection{Summary}\label{sec:summary}
In this work, we have applied structure-finding techniques to
TIGRESS simulations of the star-forming ISM, and characterized the
properties of the objects we identify. In addition, we have
investigated families of relationships between the SFR and material
that could be considered ``eligible'' for star formation, by being
part of a subset of the gas with defined properties. For the latter,
we consider both collections of objects and more general gas
subsets.
Our primary comparison of structures is between those
defined based on density or surface density (bins or thresholds)
and those that are defined based on the gravitational potential (also
considering kinetic, thermal, and magnetic energy). The former is
more analogous to the definitions of ISM structure typically used in
observations (where boundaries are often defined by observed
intensity), whereas the latter more directly connects to dynamics.
The definitions and techniques used to identify structures are
described in Section \ref{sec:mhbr} and \ref{sec:mprop}.
For both material defined by density selection criteria and material
defined by energy selection criteria, we compare time series of $M/\tff$
to the SFR history. We use these comparisons to fit for time delays ($t_d$)
and
efficiencies per free-fall time ($\eff$). In addition, we apply Bayesian
inference to compare three different models for star formation with different
dependence on the virial parameter. Techniques are described in
Section \ref{sec:mtime} and \ref{sec:effmodels}.
Key results are as follows:
1. {\it Object properties}:
Basic statistics
(mass, size, density, free-fall time) of HBRs (bound
objects) and HBPs (their parents) are compared to statistics of $\nhmin$
objects defined by density contours in \autoref{fig:hist_x}. Typical masses
of HBRs are $\sim 10^3-10^4 M_\odot$, with $\nh \sim 100 \pcc$. HBPs have
slightly lower density and masses that extend up to $\sim 10^5 M_\odot$. Thus,
bound objects are dense. Most of the mass of $\nhmin$ objects is in
large structures: typical values are
$R\sim 30-100 \pc$ and $M\sim 10^{5}-10^{6}\Msun$ for
$\nhmin = 10\pcc$, and $R\sim 10 \pc$ and $M\sim 10^{4}\Msun$ for $\nhmin = 100\pcc$.
Not all dense objects are bound (see below).
2. {\it Virial parameters and boundedness}: Figure \ref{fig:av} shows
the distribution of values of the virial parameter $\alpha_v$ most
commonly used in observations, which compares kinetic energy with
gravitational energy, assuming an isolated sphere with the same
mass and volume to compute $\Esubt{g}$
(\autoref{eq:virialdef}). Figure \ref{fig:av} also shows results for
a variant $\alpha_{\rm v,tot}$ that includes thermal and
magnetic energy (\autoref{eq:virialtotdef}). Because magnetic and
kinetic energy are comparable, we find that neglect of magnetic energy
in estimating the virial parameter is not justified. Interestingly, while HBR
objects are defined based on bound material, they have a range of
values for ``observed''
$\alpha_v \sim 0.5-5$ and $\alpha_{\rm v,total}\sim 2-7$.
\autoref{fig:av} also depicts the fraction of gas in each
$\nh$-defined object that is truly bound when considering the full
gravitational potential. Many objects that appear bound based on
$\alpha_v$ in fact contain only a small fraction of bound gas; this is
especially an issue at high mass $M\sim 10^4-10^6 M_\odot$. Massive,
moderate-density objects exist but they are not bound by gravitational
wells even when $\alpha_v <2$ (\autoref{fig:av}b,c). The probability
of gas being bound increases with $\nhmin$ (\autoref{fig:pbound}).
3. {\it Linewidth-size relations}: \autoref{fig:lws} shows that the
median linewidth-size relation for low-$\nhmin$ structures is shallower
than and lies above the $\sigma \propto R$ relation for bound objects
with fixed density,
but is fairly close to the mean $\sigma \propto R^{1/2}$
relation expected for supersonic turbulent gas with
outer scale exceeding the cloud scale.
At high $\nhmin$, the median linewidth does follow the $\alpha_{v} = 2$
linewidth-size relation
$\sigma_{3D} \approx (\rho G 8\pi/5)^{1/2} R$
for $\rho = \mu \nhmin$.
HBPs follow the same
$\sigma \propto R^{1/2}$ relation as objects selected with a low
density threshold, and in both cases the
normalization is consistent with the large-scale velocity dispersion
and overall scale height of the ISM in the simulation.
4. {\it Temporal histories}:
From the time series, we find that on average only a few tenths of
percent of the simulation mass is in bound structures (HBRs), while
$\sim 10\%$ is at densities at least an order of magnitude
above the median density ($\nh \approx 1 \pcc$) in the simulation
(\autoref{fig:time_mass}). The time series of the bound mass also has
high variability on short (few Myr)
timescales and large-amplitude fluctuations.
Fluctuations in the mass of gas at high densities $\nh > 100 \pcc$
exceed an order of magnitude, and the same is true for the gas mass at
high $\Sigma > 100 \surfunit$ (\autoref{fig:mbtsfr}). In contrast,
the mass of moderate-density gas fluctuates only over a factor $\sim
3$ with a timescale comparable to large-scale galactic vertical and
horizontal oscillation times in the galactic potential. Generally,
upward fluctuations in any mass bin are delayed relative to those in
lower-density mass bins, and star formation fluctuations are delayed
by $\sim \tff(\nh)$ relative to the mass of gas with density $\sim
\nh$ (\autoref{fig:time_density}, \autoref{fig:time_lag}).
5. {\it Star formation efficiency per free-fall time}: By correlating
the time history of $M/\tff$ in different gas subsets with the time
history of the SFR, we measure $\eff$; results are reported in
\autoref{fig:eff}. While $\eff$ is fairly flat in density bins at
$\nh \lesssim 30\pcc$, it increases to a few tenths when $\nh >
100\pcc$. This is close to the value for bound objects ($\eff =0.4$
for HBR gas). The degree of correlation between the detailed temporal history of
$\eff M/\tff$ and SFR($t$) secularly increases with increasing
density, as shown in \autoref{fig:time_others} and \autoref{fig:eff}d.
Even though the time series of $\eff M/\tff$ for HBR gas mostly tracks
SFR($t$) quite closely (\autoref{fig:time_others}), the RMS
error (defining $\Delta \SFR = \SFR - \eff M/\tff $) is worse than for
moderate-density gas because the large-amplitude variations in the
mass of HBR gas imply any ``miss'' is strongly penalized.
6. {\it Dependence of star formation on virial parameter}: In addition
to considering the simplest star formation model prescription in which
$\eff$ is constant for all gas in a given density bin, we test two
models in which $\eff$ depends on the virial parameter $\alpha_v$ of
defined density structures. For one model, $\eff$ has an exponential
dependence on $\alpha_v^{1/2}$, and for the other, $\eff$ is zero
above some cutoff in $\alpha_v$. We use Bayesian inference to obtain
marginalized model parameters and RMS errors, as shown in
\autoref{fig:cphne}. We find that allowing for a dependence on
$\alpha_v$ improves the correlation with SFR for moderate-density gas
($\nhmin = 10 \pcc$), but does not alter the strength of the correlation
for high-density gas ($\nhmin = 30,\ 100 \pcc$) or for energy-selected
HBR objects.
Overall, we conclude that the performance and parameters for
$\alpha_v$-dependent models of star formation, when applied to the
full multiphase ISM, may depend on how objects are defined (e.g. a
dependence on density contrast relative to ambient), and/or on global
aspects of ISM dynamics and star formation (including the space-time
correlations of feedback with gas structures).
\subsection{Discussion}\label{sec:discussion}
\subsubsection{Quantifying the role of self-gravity: are GMCs bound?}
There are a number of reasons why apparent virial parameters disagree
with detailed measurements of boundedness.
For example, $\alpha_{v}$ or $\alpha_{\rm v, total}$ could
underestimate boundedness because a uniform cloud is assumed, but the
actual gravitational potential can more strongly bind material in the
center of an object if it is stratified. Also, our HBR definition
considers gravitational energy relative to a surrounding potential
isocontour, where the potential considers all material rather than just
an isolated structure.
Material in and beyond the HBP surrounding an HBR contributes to
defining the bounding equipotential and to determining how
deep the potential well is.
Thus, an HBR can be more bound than it would appear from using just
an object's own mass in
$\alpha_{v}$ or $\alpha_{\rm v, total}$ (as in e.g. \autoref{fig:av}e)
because mass outside of itself contributes to defining the
equipotentials and containing the gas in a local region.
At the same time, objects can also be less bound than would be
predicted based on the traditional virial ratio of \autoref{eq:virialdef},
because the assumption of an isolated object
with vacuum boundary conditions overestimates $|\Esubt{g}|$ compared to the
real case in which tidal forces limit the region that can be bound to
a given center. Considering the gravitational potential computed
globally, including tidal forces, means that dense objects that are
near other dense objects will be less bound than the naive estimates
used in $\alpha_v$ or $\alpha_{\rm v, total}$. This explains why many
of the moderate-$\nhmin$ objects with low apparent virial parameter in
\autoref{fig:av}b,c,f,g mostly consist of unbound gas.
Due to all these effects, both HBR bound and unbound objects can
appear bound or unbound according to $\alpha_{v}$ and total
$\alpha_{v}$.
All of the above effects will be an issue for real clouds as well as
the structures in our simulations. Thus, we caution that simple
estimates of gravitational energy relative to kinetic energy are
generally inadequate for assessing whether observed GMCs are genuinely
bound structures.
To determine whether observed GMCs are genuinely
bound, a similar procedure to what we have applied in this paper would
be required. That is, the first step would be to compute the
gravitational potential from all relevant material. While three-dimensional
structure is not in general known, previous tests have shown that
projected surface density combined with an estimated line-of-sight depth
is sufficient when clouds mutually lie in a planar configuration
\citep{Gong2011}. Inclusion of the gravitational potential from all
surrounding material is particularly important for GMCs that are found in
spiral arms, where the close proximity of clouds leads to significant
tidal effects.
Our finding that the traditional virial parameter (\autoref{eq:virialdef}
with \autoref{eq:Egdef})
is at best an approximate
measure of boundedness has implications for interpretations of $\alpha_v$ in
observations that are otherwise quite puzzling. For example,
\citet{Roman-Duval2010} found that GMCs identified from $^{13}$CO
Galactic Ring Survey observations have median $\alpha_v \sim 0.5$, with mode
$\sim 0.3$. Because a low level of kinetic energy would rapidly
lead to collapse,
it is difficult to understand how this situation could arise
unless GMCs are strongly magnetically supported, which empirically does
not seem to be the case \citep[e.g.][]{Crutcher2012,Thompson2019}.
Indeed, in purely hydrodynamic simulations, isolated clouds
that are initiated with $\alpha_v$ significantly below 1 go through a stage rapid of
contraction, such that $\alpha_v \approx 1$ by the time star formation
commences \citep{Raskutti2016}. The low median
$\alpha_v$ in the \citet{Roman-Duval2010} observations could be understood
if $|\Esubt{g}|$ has been overestimated by, for example, neglecting tidal effects.
Observational surveys of nearby galaxies at $\sim 50-100$pc resolution find
values of the traditional $\alpha_v \sim 1.5-3$ for gas in resolved
structures \citep{Sun2018}. This
suggests that most clouds are bound, which combined with the estimated
completeness of $> 50\%$ would suggest that most molecular material is
in bound clouds.
However, in this case the low observed $\eff \sim 0.01$ for molecular gas
\citep{Utomo2018} would
be in significant tension with our finding that bound objects (HBRs) have
$\eff \sim 0.4$. The driven-turbulence simulations of
\citet{2012ApJ...759L..27P} have similarly found $\eff \sim 0.2-0.5$ when
$\alpha_v \sim 1$. A possible resolution is again that the traditional
observed $\alpha_v$ may overestimate boundedness by treating each cloud as
isolated.
\subsubsection{Star formation efficiency: variations and correlations}
Our results regarding the low value $\eff \sim 0.01$ of the efficiency
per free-fall time at ``average'' gas conditions is consistent with
previous observational work across a range of galaxies
\citep[e.g.][and citations within]{Evans2009,Krumholz2012,Evans2014,Lee2016,Ochsendorf2017,Utomo2018} as well as previous
numerical simulations \citep[][]{Kim2013}. In addition, some
observations have indicated an increase of $\eff$ with density of
individual structures within given galaxies
\citep[e.g.][]{Krumholz2007,Vutisalchavakul2016}, consistent with the
trend we have identified. Since star formation is only occurring in
the very densest regions, the variations of $\eff$ with density
threshold in a given environment, both in observations and in our
simulations, reflects the relative abundances of gas at different
densities, i.e. the density probability density function (pdf).
Analyses of the power-law portion of pdfs in Milky-Way molecular clouds
\citep[e.g.][]{Schneider2015a,Schneider2015b}
imply a decrease of $M/\tff$ at higher density, which is compatible
with the increase of $\eff$ with density that we have found
(\autoref{fig:eff}a,b).
The density pdf in turn reflects a ``nested'' dynamical evolution:
successively denser structures form in a hierarchical fashion, with
only a fraction of the gas at a given density experiencing net
compression by gravity and by thermal, turbulent, and magnetic
pressure to attain a higher density. Our temporal analysis provides
evidence for hierarchical dynamics at work, in that mass histories at
varying density are offset by time delays that scale with the
gravitational free-fall time.
Recent observations across varying galactic environments have suggested
that $\eff$ is not a function of absolute density, but of density
contrast relative to ambient levels
\citep[e.g.][]{Garcia2012,Longmore2013,Usero2015,Gallagher2018,Querejeta2019}, although this
interpretation is complicated by uncertainties in environmental
variation of conversion factors for dense gas tracers
\citep{Shimajiri2017}. While our present analysis considers only a
single galactic environment, we will be able to test the extent to
which $\eff$ depends on relative vs. absolute density via analysis of
additional TIGRESS simulations which have been completed for
inner-galaxy and galactic-center environments.
In addition to systematically larger $\eff$ at higher density, our
analysis shows systematically better correlations of the temporal
histories of $\SFR$ and (time-offset) histories of $\eff M/\tff$ at
higher density (\autoref{fig:time_density} and
\autoref{fig:time_others}). This can be quantified by the systematic
decrease in $\sigma_{\Delta\SFR/\langle\SFR\rangle}$ for higher
density gas as shown in \autoref{fig:eff}. A simulation
provides the benefit of being able to correct for the time delay between
the formation of a given defined structure and the resulting star
formation. Since the $\SFR$ is highly variable, this time delay
produces deviations between the simultaneous $\eff M/\tff$ and $\SFR$
on the order of $t_{\mathrm{delay}}{d(\SFR)/dt}$.
For lower density gas in which
$t_{\rm delay} \sim \tff$ is long, time delays inherently
make SFRs in observations appear less correlated with the ``simultaneous''
gas mass than they really should be (as in
\autoref{fig:mbtsfr}). The combination of the stronger inherent
correlation in amplitude variations and smaller time delays implies
that there should be less scatter in the observed statistical
correlations between $\SFR$ and mass of high density tracers in
comparison to low density tracers (assuming that the measurement of
the $\SFR$ is based on a tracer with a short timescale that does not
itself wash out the signal).
With a sufficiently large sample of environments such that galactic
conditions can be controlled (e.g. specifying limited ranges of both
total gas and stellar surface density), and such that all phases of the star
formation cycle are well sampled for given conditions, increasingly
quantitative measures of the relationship between gas and star
formation become possible. For example, full sampling over temporal
history can minimize effects of time delays when evaluating the
overall $\eff$ for low-density gas. In addition, it will be possible
to quantify increases in the correlation of SFR and $M/\tff$ with
density (we measure this by a reduction in
$\sigma_{\Delta\SFR/\langle\SFR\rangle}$) while controlling for environment;
steps towards this have
already been taken \citep[e.g.][]{Gallagher2018,Jimenez2019}.
Given sufficiently
high resolution observations, it may also be possible to use analysis
of spatial correlations between high density tracers and star
formation \citep[e.g.~as in][]{Kruijssen2019} as a proxy to measure
temporal correlations between $\SFR$ and dense gas mass that we have
identified using simulations, thereby characterizing the bursty nature of
$\SFR$.
Finally, we remark on the relation between our work and other
theoretical/computational studies that address the relationship between
gas and star formation. Many studies have focused exclusively
on the cold and dense ISM, because this is the material most proximate
to star formation. With a narrower focus it is also possible to define
an idealized system with a reduced number of parameters; a minimal set
of parameters to describe gas in molecular clouds would include the
turbulent Mach number, the ratio of the mean Alfv\'en speed to
the sound speed, and the ratio of the Jeans length to cloud size (or
equivalently free-fall time to turbulent crossing time) \citep{Ostriker1999}.
Based on a set of idealized simulations of
this kind, with turbulence driven to maintain a fixed level,
\citep{2012ApJ...759L..27P} proposed that $\eff$ exponentially declines
with increasing virial parameter. As noted above,
for moderate density threshold ($\nhmin = 10$) our fitted
coefficients are consistent with their results. However, this is not the
case when we consider gas at higher density thresholds. This may be because
of limited resolution at higher density thresholds in our simulations, or
because physical feedback in our simulations differs from idealized
turbulent driving, which (together with the multiphase nature) means that
all scales are not equivalent.
A class of simple theoretical models for star formation rates in
turbulent systems is predicated on the notion that there is a critical
density $\rho_{\rm crit}$, with structures at density above $\rho_{\rm
crit}$ collapsing before they can be torn apart by ambient
turbulence \citep[e.g.][]{Krumholz2005,Padoan2011,Hennebelle2011,Federrath2012}.
These theoretical models are intended to represent idealized GMC conditions,
with gas effectively isothermal and turbulence highly supersonic; they are
therefore not immediately applicable to the present multiphase ISM simulations.
Still, it is interesting to note that our analysis does not provide evidence
that there is a ``point of no return'' at any particular density. Rather,
there is an order of magnitude variation in the density of bound clouds
(\autoref{fig:hist_x}g), with the probability of gas being bound and $\eff$
both increasing with density (\autoref{fig:pbound}, \autoref{fig:error_hist}).
The present analysis does not provide information about individual cloud
lifetimes, however. Both for large-scale multiphase ISM simulations and for
smaller-scale simulations of star-forming clouds,
numerical measurements of the lifetimes of individual structures are needed
in order to test theoretical concepts of gravoturbulent fragmentation,
and to assess whether simulations agree with observational constraints \citep[e.g.][]{Murray2011,2016ApJ...833..229L,2019MNRAS.488.1501G}.
While some estimates of object lifetimes can be obtained via
frame-to-frame differences in structural decompositions, the most direct
way to follow evolution is via Lagrangian tracer particles. Tracers are
commonly implemented to follow baryon cycles of gravitational
collapse and dispersal by feedback in cosmological simulations of
galaxy formation \citep[e.g.][]{Genel2013,Cadiou2019}, and for the same
reasons would be a valuable tool for future numerical studies
of the star-forming interstellar medium.
\acknowledgments
The work was supported by the National Science Foundation under grant
AST-1713949 and NASA under grant NNX17AG26G to ECO, and grant
DGE-1148900 providing a Graduate Research Fellowship to SAM.
| {
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{"url":"https:\/\/www.scipedia.com\/wd\/index.php?title=Nadunkandi_et_al_2011a&diff=99612&oldid=prev","text":"## Abstract\n\nWe propose a fourth\u2010order compact scheme on structured meshes for the Helmholtz equation given by ${\\displaystyle R(\\phi ):=f(x)+\\Delta \\phi +\\chi 2\\phi =0}$. The scheme consists of taking the alpha\u2010interpolation of the Galerkin finite element method and the classical central finite difference method. In 1D, this scheme is identical to the alpha\u2010interpolation method (J. Comput. Appl. Math. 1982; 8(1):15\u201319) and in 2D making the choice ${\\displaystyle \\alpha =0.5}$ we recover the generalized fourth\u2010order compact Pad\u00e9 approximation (J. Comput. Phys. 1995; 119:252\u2013270; Comput. Meth. Appl. Mech. Engrg 1998; 163:343\u2013358) (therein using the parameter ${\\displaystyle \\gamma =2}$). We follow (SIAM Rev. 2000; 42(3):451\u2013484; Comput. Meth. Appl. Mech. Engrg 1995; 128:325\u2013359) for the analysis of this scheme and its performance on square meshes is compared with that of the quasi\u2010stabilized FEM (Comput. Meth. Appl. Mech. Engrg 1995; 128:325\u2013359). In particular, we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate ${\\displaystyle O((\\chi l)^{4})}$, where ${\\displaystyle \\chi ,l}$ represent the wavenumber and the mesh size, respectively. An expression for the parameter \u03b1 is given that minimizes the maximum relative phase error in a sense that will be explained in Section 4.5. Convergence studies of the error in the ${\\displaystyle L^{2}}$ norm, the ${\\displaystyle H^{1}}$ semi\u2010norm and the ${\\displaystyle l^{\\infty }}$ Euclidean norm are done and the pollution effect is found to be small. Copyright \u00a9 2010 John Wiley & Sons, Ltd.\n\nBack to Top\n\n### Document information\n\nPublished on 19\/12\/18\nSubmitted on 19\/12\/18\n\nDOI: 10.1002\/nme.3043\nLicence: CC BY-NC-SA license\n\n### Document Score\n\n0\n\nTimes cited: 3\nViews 28\nRecommendations 0","date":"2019-08-20 18:20:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 8, \"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\": 0.3617574870586395, \"perplexity\": 1657.4689469559726}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"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-2019-35\/segments\/1566027315558.25\/warc\/CC-MAIN-20190820180442-20190820202442-00357.warc.gz\"}"} | null | null |
OCDA Rackauckas defends himself against allegations of prosecutorial misconduct
Posted on September 14, 2017 by Editor Posted in OCDA, Orange County, Tony Rackauckas
ORANGE COUNTY DISTRICT ATTORNEY NEWS RELEASE
OP-ED BY DISTRICT ATTORNEY TONY RACKAUCKAS DEBUNKING RECENT HARVARD STUDY ON PROSECUTORIAL MISCONDUCT
The numbers were startling.
According to media reports, a newly formed group affiliated with Harvard Law School called the "Fair Punishment Project" (FPP) supposedly reviewed court decisions from 2010 to 2015 involving the Orange County District Attorney's Office. The "study" found that the OCDA had seven convictions reversed due to prosecutorial misconduct and 24 overall findings of prosecutorial misconduct, and these numbers were ranked amongst the highest in the state. Curiously, the authors of the report chose not to "show their work" behind their claims, or list any of the qualifying cases they had deemed to be prosecutorial misconduct. The media accused our prosecutors of "cheating" to win, and wrote an editorial stating "these are not cases of errors, but of willful and serious misconduct," while a local politician who covets the Office of the District Attorney proclaimed we had "the worst record in the entire state."
There was just one problem: The numbers in the study were completely wrong.
In their rush to criticize the men and women of the OCDA who handle over 60,000 new cases each year, the media, politicians, and other opportunists all neglected to take the time to confirm whether the report's information was actually correct.
The truth is the claims do not match the evidence.
What's even more puzzling, the FPP got wind that they may be exposed and they "corrected" their study, yet the numbers were still wrong.
Publicly available records show that in five of the seven alleged reversals, the appellate decision contained no finding of prosecutorial misconduct. In fact in one decision, the Court of Appeal explicitly stated, "We do not find fault with the prosecution," yet the report included it anyway as a reversal due to prosecutorial misconduct. Two of the seven cases were not even reversals, but rather legal motions that did not involve misconduct by the prosecutors. In reality, of the seven cases there were more reversals due to decisions by trial judges (three), than due to decisions by prosecutors (two). Interestingly enough, nobody is suggesting there is a "systemic" effort by our trial judges to cheat defendants. Such a suggestion would be wrong, just as it is wrong to suggest that there is a systemic effort by OCDA prosecutors to do so.
Further, the appellate justices deciding these cases understood what the media did not: that as a legal term, "prosecutorial misconduct" can include unintentional errors and the slightest of misstatements. As a result, the Court of Appeal did not find the vast majority of misconduct cited in the report to be willful or serious, and instead used descriptions like "a rather minor digression," "a few stray words," or in one instance, "a single improper question in the course of a lengthy trial."
Putting the correct numbers of the FPP in perspective, they cite over 150 cases between 2010 and 2015 where a convicted defendant claimed he/she didn't get a fair trial due to prosecutorial misconduct, and in all but two cases, over 98 percent of the time, the Court of Appeal disagreed. While a single reversal due to misconduct is unacceptable, these numbers in context display what I have long known to be true: that the OCDA has a consistent superb record of fairly convicting guilty defendants to keep our community safe, and give victims and their families a sense of justice.
It's unfortunate Harvard University has sullied itself by publishing such an unscholarly study. It is irresponsible for the group, no matter its anti-prosecutorial and public safety agenda, to publish such a "study" and double down when they are about to be exposed, and for the media to parrot it even when they had the raw data. Finally, it is pathetic for a politician to try to cheer on case reversals and seek political gain from pain to crime victims.
The OCDA is aware of the huge responsibility that the prosecution has and our number one duty is to do justice while seeking public safety. Academics, reporters, and politicians also have their respective responsibilities to state the truth. Blindly repeating claims of misconduct is reckless at best, and promotes an unwarranted and dangerous distrust within our community, even when those claims are proven false.
Let us hope the next time such claims are made, the response is to search for the truth, rather than a headline.
Tony Rackauckas
TONY RACKAUCKAS, District Attorney
Susan Kang Schroeder, Chief of Staff
Michelle Van Der Linden,Spokesperson
OCDA Tony Rackauckas
« The Lincoln Club of Orange County endorses Tyler Diep for the 72nd Assembly District
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,001 |
// Copyright (c) Microsoft. All rights reserved.
// Licensed under the MIT license. See LICENSE file in the project root for full license information.
using System;
using System.Collections.Generic;
using System.Globalization;
using System.Linq;
using Microsoft.Spark.CSharp.Core;
using NUnit.Framework;
namespace Microsoft.Spark.CSharp.Samples
{
class PairRDDSamples
{
[Sample]
internal static void PairRDDCollectAsMapSample()
{
var map = SparkCLRSamples.SparkContext.Parallelize(new[] { new KeyValuePair<int, int>(1, 2), new KeyValuePair<int, int>(3, 4) }, 1).CollectAsMap();
foreach (var kv in map)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(map.ContainsKey(1) && map[1] == 2);
Assert.IsTrue(map.ContainsKey(1) && map[3] == 4);
}
}
[Sample]
internal static void PairRDDKeysSample()
{
var keys = SparkCLRSamples.SparkContext.Parallelize(new[] { new KeyValuePair<int, int>(1, 2), new KeyValuePair<int, int>(3, 4) }, 1).Keys().Collect();
Console.WriteLine(keys[0]);
Console.WriteLine(keys[1]);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.AreEqual(1, keys[0]);
Assert.AreEqual(3, keys[1]);
}
}
[Sample]
internal static void PairRDDValuesSample()
{
var values = SparkCLRSamples.SparkContext.Parallelize(new[] { new KeyValuePair<int, int>(1, 2), new KeyValuePair<int, int>(3, 4) }, 1).Values().Collect();
Console.WriteLine(values[0]);
Console.WriteLine(values[1]);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.AreEqual(2, values[0]);
Assert.AreEqual(4, values[1]);
}
}
[Sample]
internal static void PairRDDReduceByKeySample()
{
var reduced = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 1),
new KeyValuePair<string, int>("b", 1),
new KeyValuePair<string, int>("a", 1)
}, 2)
.ReduceByKey((x, y) => x + y).Collect();
foreach (var kv in reduced)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(reduced.Contains(new KeyValuePair<string, int>("a", 2)));
Assert.IsTrue(reduced.Contains(new KeyValuePair<string, int>("b", 1)));
}
}
[Sample]
internal static void PairRDDReduceByKeyLocallySample()
{
var reduced = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 1),
new KeyValuePair<string, int>("b", 1),
new KeyValuePair<string, int>("a", 1)
}, 2)
.ReduceByKeyLocally((x, y) => x + y);
foreach (var kv in reduced)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(reduced.Contains(new KeyValuePair<string, int>("a", 2)));
Assert.IsTrue(reduced.Contains(new KeyValuePair<string, int>("b", 1)));
}
}
[Sample]
internal static void PairRDDCountByKeySample()
{
var countByKey = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 1),
new KeyValuePair<string, int>("b", 1),
new KeyValuePair<string, int>("a", 1)
}, 2)
.CountByKey();
foreach (var kv in countByKey)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.AreEqual(2, countByKey["a"]);
Assert.AreEqual(1, countByKey["b"]);
}
}
[Sample]
internal static void PairRDDJoinSample()
{
var l = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 1),
new KeyValuePair<string, int>("b", 4),
}, 1);
var r = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 2),
new KeyValuePair<string, int>("a", 3),
}, 1);
var joined = l.Join(r, 2).Collect();
foreach (var kv in joined)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(joined.Contains(new KeyValuePair<string, Tuple<int, int>>("a", new Tuple<int, int>(1, 2))));
Assert.IsTrue(joined.Contains(new KeyValuePair<string, Tuple<int, int>>("a", new Tuple<int, int>(1, 3))));
}
}
[Sample]
internal static void PairRDDLeftOuterJoinSample()
{
var l = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 1),
new KeyValuePair<string, int>("b", 4),
}, 2);
var r = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 2),
}, 1);
var joined = l.LeftOuterJoin(r).Collect();
foreach (var kv in joined)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(joined.Any(kv => kv.Key == "a" && kv.Value.Item1 == 1 && kv.Value.Item2.IsDefined && kv.Value.Item2.GetValue() == 2));
Assert.IsTrue(joined.Any(kv => kv.Key == "b" && kv.Value.Item1 == 4 && !kv.Value.Item2.IsDefined));
}
}
[Sample]
internal static void PairRDDRightOuterJoinSample()
{
var l = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 2),
}, 1);
var r = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 1),
new KeyValuePair<string, int>("b", 4),
}, 2);
var joined = l.RightOuterJoin(r).Collect();
foreach (var kv in joined)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(joined.Any(kv => kv.Key == "a" && kv.Value.Item1.IsDefined && kv.Value.Item1.GetValue() == 2 && kv.Value.Item2 == 1));
Assert.IsTrue(joined.Any(kv => kv.Key == "b" && !kv.Value.Item1.IsDefined && kv.Value.Item2 == 4));
}
}
[Sample]
internal static void PairRDDFullOuterJoinSample()
{
var l = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 1),
new KeyValuePair<string, int>("b", 4),
}, 2);
var r = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 2),
new KeyValuePair<string, int>("c", 8),
}, 2);
var joined = l.FullOuterJoin(r).Collect();
foreach (var kv in joined)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(joined.Any(kv => kv.Key == "a" && kv.Value.Item1.IsDefined && kv.Value.Item1.GetValue() == 1 &&
kv.Value.Item2.IsDefined && kv.Value.Item2.GetValue() == 2));
Assert.IsTrue(joined.Any(kv => kv.Key == "b" && kv.Value.Item1.IsDefined && kv.Value.Item1.GetValue() == 4 &&
!kv.Value.Item2.IsDefined));
Assert.IsTrue(joined.Any(kv => kv.Key == "c" && !kv.Value.Item1.IsDefined &&
kv.Value.Item2.IsDefined && kv.Value.Item2.GetValue() == 8));
}
}
[Sample]
internal static void PairRDDPartitionBySample()
{
Func<dynamic, int> partitionFunc = key =>
{
if (key < 3) return 1;
if (key >= 3 && key < 6) return 2;
else return 3;
};
var partitioned = SparkCLRSamples.SparkContext.Parallelize(new[] { 1, 2, 3, 4, 5, 6, 1 }, 3)
.Map(x => new KeyValuePair<int, int>(x, x + 100))
.PartitionBy(3, partitionFunc)
.Glom()
.Collect();
foreach (var partition in partitioned)
{
foreach (var kv in partition)
{
Console.Write(kv + " ");
}
Console.WriteLine();
}
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.AreEqual(3, partitioned.Length);
// Assert that the partition distribution is correct with partitionFunc
Assert.IsTrue(partitioned.Count(p => p.All(key => key.Key < 3)) == 1);
Assert.IsTrue(partitioned.Count(p => p.All(key => key.Key >= 3 && key.Key < 6)) == 1);
Assert.IsTrue(partitioned.Count(p => p.All(key => key.Key >= 6)) == 1);
}
}
[Sample]
internal static void PairRDDCombineByKeySample()
{
var combineByKey = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 1),
new KeyValuePair<string, int>("b", 1),
new KeyValuePair<string, int>("a", 1)
}, 2)
.CombineByKey(() => string.Empty, (x, y) => x + y.ToString(CultureInfo.InvariantCulture), (x, y) => x + y).Collect();
foreach (var kv in combineByKey)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(combineByKey.Contains(new KeyValuePair<string, string>("a", "11")));
Assert.IsTrue(combineByKey.Contains(new KeyValuePair<string, string>("b", "1")));
}
}
[Sample]
internal static void PairRDDAggregateByKeySample()
{
var aggregateByKey = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 1),
new KeyValuePair<string, int>("b", 1),
new KeyValuePair<string, int>("a", 1)
}, 2)
.AggregateByKey(() => 0, (x, y) => x + y, (x, y) => x + y).Collect();
foreach (var kv in aggregateByKey)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(aggregateByKey.Contains(new KeyValuePair<string, int>("a", 2)));
Assert.IsTrue(aggregateByKey.Contains(new KeyValuePair<string, int>("b", 1)));
}
}
[Sample]
internal static void PairRDDFoldByKeySample()
{
var FoldByKey = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 1),
new KeyValuePair<string, int>("b", 1),
new KeyValuePair<string, int>("a", 1)
}, 2)
.FoldByKey(() => 0, (x, y) => x + y).Collect();
foreach (var kv in FoldByKey)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(FoldByKey.Contains(new KeyValuePair<string, int>("a", 2)));
Assert.IsTrue(FoldByKey.Contains(new KeyValuePair<string, int>("b", 1)));
}
}
[Sample]
internal static void PairRDDGroupByKeySample()
{
var groupByKey = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, int>("a", 1),
new KeyValuePair<string, int>("b", 1),
new KeyValuePair<string, int>("a", 1)
}, 2)
.GroupByKey().Collect();
foreach (var kv in groupByKey)
Console.WriteLine(kv.Key + ", " + "(" + string.Join(",", kv.Value) + ")");
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(groupByKey.Any(kv => kv.Key == "a" && kv.Value.Count == 2 && kv.Value[0] == 1 && kv.Value[1] == 1));
Assert.IsTrue(groupByKey.Any(kv => kv.Key == "b" && kv.Value.Count == 1 && kv.Value[0] == 1));
}
}
[Sample]
internal static void PairRDDMapValuesSample()
{
var mapValues = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, string[]>("a", new[]{"apple", "banana", "lemon"}),
new KeyValuePair<string, string[]>("b", new[]{"grapes"})
}, 2)
.MapValues(x => x.Length).Collect();
foreach (var kv in mapValues)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(mapValues.Any(kv => kv.Key == "a" && kv.Value == 3));
Assert.IsTrue(mapValues.Any(kv => kv.Key == "b" && kv.Value == 1));
}
}
[Sample]
internal static void PairRDDFlatMapValuesSample()
{
var flatMapValues = SparkCLRSamples.SparkContext.Parallelize(
new[]
{
new KeyValuePair<string, string[]>("a", new[]{"x", "y", "z"}),
new KeyValuePair<string, string[]>("b", new[]{"p", "r"})
}, 2)
.FlatMapValues(x => x).Collect();
foreach (var kv in flatMapValues)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(flatMapValues.Any(kv => kv.Key == "a" && kv.Value == "x"));
Assert.IsTrue(flatMapValues.Any(kv => kv.Key == "a" && kv.Value == "y"));
Assert.IsTrue(flatMapValues.Any(kv => kv.Key == "a" && kv.Value == "z"));
Assert.IsTrue(flatMapValues.Any(kv => kv.Key == "b" && kv.Value == "p"));
Assert.IsTrue(flatMapValues.Any(kv => kv.Key == "b" && kv.Value == "r"));
}
}
[Sample]
internal static void PairRDDGroupWithSample()
{
var x = SparkCLRSamples.SparkContext.Parallelize(new[] { new KeyValuePair<string, int>("a", 1), new KeyValuePair<string, int>("b", 4)}, 2);
var y = SparkCLRSamples.SparkContext.Parallelize(new[] { new KeyValuePair<string, int>("a", 2)}, 1);
var groupWith = x.GroupWith(y).Collect();
foreach (var kv in groupWith)
Console.WriteLine(kv.Key + ", " + "(" + string.Join(",", kv.Value) + ")");
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(groupWith.Any(kv => kv.Key == "a" && kv.Value.Item1[0] == 1 && kv.Value.Item2[0] == 2));
Assert.IsTrue(groupWith.Any(kv => kv.Key == "b" && kv.Value.Item1[0] == 4 && !kv.Value.Item2.Any()));
}
}
[Sample]
internal static void PairRDDGroupWithSample2()
{
var x = SparkCLRSamples.SparkContext.Parallelize(new[] { new KeyValuePair<string, int>("a", 5), new KeyValuePair<string, int>("b", 6) }, 2);
var y = SparkCLRSamples.SparkContext.Parallelize(new[] { new KeyValuePair<string, int>("a", 1), new KeyValuePair<string, int>("b", 4) }, 2);
var z = SparkCLRSamples.SparkContext.Parallelize(new[] { new KeyValuePair<string, int>("a", 2) }, 1);
var groupWith = x.GroupWith(y, z).Collect();
foreach (var kv in groupWith)
Console.WriteLine(kv.Key + ", " + "(" + string.Join(",", kv.Value) + ")");
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.IsTrue(groupWith.Any(kv => kv.Key == "a" && kv.Value.Item1[0] == 5 && kv.Value.Item2[0] == 1 && kv.Value.Item3[0] == 2));
Assert.IsTrue(groupWith.Any(kv => kv.Key == "b" && kv.Value.Item1[0] == 6 && kv.Value.Item2[0] == 4 && !kv.Value.Item3.Any()));
}
}
//TO DO: implement PairRDDFunctions.SampleByKey
//[Sample]
//internal static void PairRDDSampleByKeySample()
//{
// var fractions = new Dictionary<string, double> { { "a", 0.2 }, { "b", 0.1 } };
// var rdd = SparkCLRSamples.SparkContext.Parallelize(fractions.Keys.ToArray(), 2).Cartesian(SparkCLRSamples.SparkContext.Parallelize(Enumerable.Range(0, 1000), 2));
// var sample = rdd.Map(t => new KeyValuePair<string, int>(t.Item1, t.Item2)).SampleByKey(false, fractions, 2).GroupByKey().Collect();
// Console.WriteLine(sample);
//}
[Sample]
internal static void PairRDDSubtractByKeySample()
{
var x = SparkCLRSamples.SparkContext.Parallelize(new[] { new KeyValuePair<string, int?>("a", 1), new KeyValuePair<string, int?>("b", 4), new KeyValuePair<string, int?>("b", 5), new KeyValuePair<string, int?>("a", 2) }, 2);
var y = SparkCLRSamples.SparkContext.Parallelize(new[] { new KeyValuePair<string, int?>("a", 3), new KeyValuePair<string, int?>("c", null) }, 2);
var subtractByKey = x.SubtractByKey(y).Collect();
foreach (var kv in subtractByKey)
Console.WriteLine(kv);
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.AreEqual(2, subtractByKey.Length);
subtractByKey.Contains(new KeyValuePair<string, int?>("b", 4));
subtractByKey.Contains(new KeyValuePair<string, int?>("b", 5));
}
}
[Sample]
internal static void PairRDDLookupSample()
{
var rdd = SparkCLRSamples.SparkContext.Parallelize(Enumerable.Range(0, 1000).Zip(Enumerable.Range(0, 1000), (x, y) => new KeyValuePair<int, int>(x, y)), 10);
var lookup42 = rdd.Lookup(42);
var lookup1024 = rdd.Lookup(1024);
Console.WriteLine(string.Join(",", lookup42));
Console.WriteLine(string.Join(",", lookup1024));
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
CollectionAssert.AreEqual(new[] { 42 }, lookup42);
Assert.AreEqual(0, lookup1024.Length);
}
}
[Sample]
internal static void PairRDDSortByKeySample()
{
var rdd = SparkCLRSamples.SparkContext.Parallelize(new[] { new KeyValuePair<string, int>("B", 2),
new KeyValuePair<string, int>("a", 1), new KeyValuePair<string, int>("c", 3),
new KeyValuePair<string, int>("E", 5), new KeyValuePair<string, int>("D", 4)}, 3);
var sortedRdd = rdd.SortByKey(true, 2);
var sortedInTotal = sortedRdd.Collect();
var sortedPartitions = sortedRdd.Glom().Collect();
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.AreEqual(2, sortedPartitions.Length);
// by default SortByKey is case sensitive
CollectionAssert.AreEqual(new[] { "B", "D", "E", "a", "c" }, sortedInTotal.Select(kv => kv.Key).ToArray());
}
// convert the keys to lower case in order to sort with case insensitive
sortedRdd = rdd.SortByKey(true, 2, key => key.ToLowerInvariant());
sortedInTotal = sortedRdd.Collect();
sortedPartitions = sortedRdd.Glom().Collect();
if (SparkCLRSamples.Configuration.IsValidationEnabled)
{
Assert.AreEqual(2, sortedPartitions.Length);
CollectionAssert.AreEqual(new[] { "a", "B", "c", "D", "E" }, sortedInTotal.Select(kv => kv.Key).ToArray());
}
}
}
}
| {
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Нісе́ко (, ) — містечко в Японії, в повіті Абута округу Сірібесі префектури Хоккайдо. Станом на площа містечка становила км². Станом на населення містечка становило осіб.
Примітки
Джерела та література
Посилання
Офіційна сторінка Нісеко
Містечка префектури Хоккайдо | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,450 |
The Fortean
Posted on August 24, 2019 August 26, 2019 by Cropster
Ghost Writer: The Humpty Doo Poltergeist
In an earlier post I covered my own experiences with the Humpty Doo poltergeist in Australia in 1998.
Two Australian journalists who also visited the Humpty Doo house wrote positive pieces about the case; Frank Robson in the Sydney Morning Herald 'Good Weekend' magazine and freelance writer Max Anderson for The Australian Weekend magazine.
Journalist Max Anderson at the Humpty Doo house in 1998. Copyright: Max Anderson.
Max's article appeared in May 1998 and he has graciously allowed me to reprint it here. Check out his great blog.
Enjoy a different perspective on probably the best documented poltergeist case of all time.
When writer Max Anderson hitched a ride with Today Tonight on the recent 'Humpty Doo' ghost scoop he knew he had a scorcher of a yarn. Camera-toting TV journos chasing ghosts in Australia's most haunted house? He couldn't lose. Only he reckoned without one thing…
I couldn't believe it. I'd found someone who cheerfully admitted to being a professional ghost buster. True story: Stephen Bishop will rid your home or office of unwanted entities for up to $50 a room. He even teaches apprentice psychics in his 150-strong Chiara College of Metaphysics in Sydney.
"It's a big industry," he said over beers one evening in Balmain's Exchange Pub, "Sydney's huge for it. I ghost-busted a Woolhara brothel last week. It was affecting business."
Intrigued, amused, deeply sceptical, I scribbled a pad-full of notes while his conversation went merrily bump in the night. Then I secured a promise he'd take me on his next bust.
Two weeks later, events took a bizarre twist. On 3 April I learned that Channel Seven's current affairs program Today Tonight had stitched up a deal with residents of a 'haunted house' in a Northern Territory town called Humpty Doo. Local media had been reporting 'a ghost' going berserk, until Seven signed a cheque for an exclusive report. I called Today Tonight reporter Greg Quail and did some horse trading: he could have my ghost buster if he'd take both of us to Humpty Doo – I, of course, being free to scoop my own story.
Sixteen hours later, I was on a plane to Darwin with Bishop, Quail and a film crew. Angry ghosts and ratings-hungry current affairs? Truly, Christmas had come early.
Saturday 4 April
Up close, TV is scary. Ten minutes out of Darwin airport and already there was stuff flying around – namely money. Hotel rooms, hire cars, extra video tape, even a thermal camera flown up from Brisbane. And Quail had quickly quarantined my ghost-buster in the Darwin hotel lest he "scare any spirits away".
Speeding south, story details were materialising on 'Australia's most haunted house'. There'd been six weeks of supernatural aggro inflicted on five residents including 'flying objects' and words appearing on floors in gravel and Scrabble letters — FIRE, SKIN, CAR, HELP, and the name TROY. Troy and his friend had been incinerated when their Ute, loaded with thinners, pancaked into a nearby tree two months ago. He was best mates with a resident at the house called Murphy.
But there'd also been talk by the Seven team of an elaborate hoax. The Humpty Doo (Humpty Don't?) story had broken on April Fool's Day. Most chilling of all was the spectre of 'The Great Carlos', a hoax psychic set up by 60 Minutes in 19xx. Competing shows had swallowed Carlos like mullet.
Certainly, driving through the torpid NT flatlands with its spiky pandanus trees and creaky cottage industry ("4 SALE, STUFFED CROCKS") it was hard to imagine a setting less like the moonlit imaginings of Stoker and Shelley.
The Humpty Doo house in McMinns Drive.
Number 90, McMinns Drive sat behind a high cyclone fence, down a long gravel driveway. It was surrounded by five flat acres, studded with mango trees and wrecked vehicles.
We parked beside the single storey house painted a curious eggshell blue, and walked to the back where an extension of the baking roof formed an outdoor 'breezeway'. Under this stood a long galvanised steel table, chairs, a fridge and a Harley Davidson.
The Harley's owner Dave stood to meet us. "Gidday, how are ya?" He was heavy and bearded, wore t-shirt and stubbies, and spoke quietly. "Yeah, we've had stuff moved, thrown, broken, smashed…"
"But we're hoping it's gone," said his girlfriend Jill, a thin woman with tousled blonde hair and lit cigarette. "A clairvoyant rang from Brisbane today, said she'd got rid of it. And it's been quiet all day."
Inside, the house was sparsely furnished. The dining area was empty save for a large cabinet, its three windows held together with starbursts of orange tape. The violence of the image surprised me. Beneath the cabinet, a crucifix and Bibles with pages 'torn during a priest's visit'. The kitchen window was smashed, 'by a flying beer mug'. CDs and stereo had been 'toppled in the lounge'.
And on the bathroom floor, Scrabble letters read 'NO TV'. Someone in our group sniggered.
The man Murphy arrived home. He was short and powerfully built, tatts on his mahogany shoulders. "What do all youse fuckn vultures want?" The guy's hostility caught me off guard and I stood feeling awkward in my long pants, already wet with sweat. Quail tried to placate, assuring him we were there to take a look and hopefully expel any spooks.
"How y'gonna get rid of it?" he snapped. We talked about Steve. "Well where is he? And when can he come? I'm sick of the fuckn thing."
The remaining residents showed up, Burnie a muscular driller, and his wife Kirstie [Cropster: It was actually Kirsty!] with their baby, Jasmine. Kirstie was thin, dark haired, tired and scowling.
For the fourth time, Jill said, "I reckon it's gone with that clairvoyant woman. It's been quiet all day. I'm sure it's gone."
Jill cried out around 4pm, came running from a bedroom, hugging her arms through her thin dress. "It's happened again," she said, "Murph, your bedroom – yer mattress is up in the bedroom."
Cameras, people, all there in a flash, peering into Murph's small room. A foam mattress stood upended, thrown with bedding against a dressing table. We murmured, unsure. An operator carried his thermal camera into the lounge, looking for unusual heat activity, finding nothing. I followed him, Jill talking all the time: "That's how it starts, nothing for hours, then…"
Standing in the lounge, there was a smart 'CRACK!' on a cabinet — and I saw an AA battery land on the floor, just half a meter away.
Jill chattered: "There? See? See that? It's happening!"
I glanced to where it might have been thrown from. No-one. I heard myself yell — "GUYS! IT'S HAPPENED!" — only it was a silly voice, over-dramatic, stoked. Striding figures came into the room to see me pointing dumbly.
"I don't know where it came from," I said.
Three rational men were sitting at midnight in a hotel bar, trying to explain what we'd seen.
And not only the battery's sudden appearance. I couldn't account for the steak knife which bounced off the steel table onto the floor at Kirstie's feet. Or the heavy glass lid which fell into view while Kirstie was looking inside the fridge. The sound recordist couldn't account for the spanner he saw crash into a kitchen cupboard, hurled with such force from the empty lounge that it shook a video camera mounted in the kitchen. Both cameramen were baffled by another knife that struck the hire car while they were stowing their cameras, no-one behind them.
Steve Bishop urged us to recount events in detail, amazed at so much activity. "But I don't think it's connected with Troy," he said. I sat up, listened to him. "It's too soon after his death. And if Troy was a friend, why's he causing trouble? No, it's bigger than that."
I decided that descriptions of objects being 'thrown' or 'flying around' were inapt. No, our objects had appeared in our peripheral vision; we only saw them upon or after impact, followed by any movement on the rebound which gave us clues of origin and trajectory.
Still, since these objects were obeying physics (and the angle of incidence does indeed equal the angle of reflection) then maybe we could determine who was throwing them.
"And God help whichever one it is. Burnie and Murph are at the end of their rope. Burnie said if they caught someone throwing stuff, they'd kill 'em."
That night, I slept with the light on.
The Humpty Doo house in 1998. Copyright: Max Anderson.
Sunday 5 April
Palm Sunday. The NO TV message in the bathroom had changed to NO CAMERA.
I watched with Kirstie while a Seven man recorded the Scrabble letters in Hi 8. In an instant we heard a scratching noise at ceiling height, all spun around to see a piece of glass the size of a playing card fall at Kirstie's heel. Again the rush of adrenalins, the excitement of seeing something utterly, inexplicably fantastic.
Minutes later, in the dining room with Kirstie there was a sharp SMACK! on the wall, a meter above my head. A small chunk of glass rebounded at my feet.
"Jesus, it's going off!" I walked into a bedroom empty except for piles of clothes and toys and for some reason the skin shrank on the back of my neck. I retreated and announced, "That room gives me the shivers!"
"What room? What fuckn room?" Murph was quickly in my face, eyes wide, yelling, pointing. "Look you don't know nuthn! You're guessin' like all the fuckn rest of 'em!"
Then Kirstie started up: "Leave the bloke alone! He's got an opinion! No-one really knows, Murph! Leave the bloke alone!"
Maybe it was being confronted by very real human stress, but I began listening to the residents and that morning, (while Scrabble letters hit the roof, Murph's mattress up-ended twice and gravel fell from nowhere), I tried to ascertain exactly what was bugging them.
Safety wasn't a concern. It was rare that anyone had been touched by an object, let alone hurt, and for the most part, damage had been inconvenient rather than costly. They were jumpy but only occasionally terrified, and they liked the place. "Buggered if we're gonna be shifted by a ghost!" they said.
No, I decided they were freaked by not knowing whether it was connected with the death of Troy; they were tired from mentally wrestling with their reality of esoteria; and they were utterly frustrated and upset to the point of fury that Humpty Doo, Darwin and soon the rest of Australia thought they were liars, druggies or mentally unstable. From their perspective, they were experiencing what US Vietnam veterans went through: no understanding and no sympathy while shit rained down all around them.
The afternoon turned into a waiting game. Quail, yet to witness anything himself, suffering from flu and desperate for pictures, had set five video cameras rolling inside.
So of course the action shifted outside. When the local priest, Father Tom, crunched up the driveway in his car, a .44 magnum bullet flew onto the steel table with a resounding CLANG. I felt my gut hollow: knives, glass and now bullets. The dilemma of live ammunition slamming into steel wasn't lost on me, either. The Priest, who'd seen it all before while blessing the house, and now quite media-shy, departed minutes later.
An hour passed, the sweat thickening to waxy grime, the talk punctuated by the 'crack-crush' of chilled beer tins and scrit-scrit-scratch of disposable lighters. Then a cameraman yelled in frustration: "Come ON!" At which a stone flew at great speed from the empty driveway, CRACK! onto the table. Quail's eyes widened. "I've seen it! My God!"
The cameraman called out again: "At least show us on camera so we can go!" Within minutes, the crew had an incident on three cameras: a baby's bottle being toppled from a microwave in the kitchen. But conclusive evidence? On tape, Dave's leg occludes a clear-shot camera the split second the bottle leaves the microwave, throwing the whole incident open to question. The timing is not just excellent: it's work of genius.
Monday 6 April
On the morning of the third day, the NT News tabloid reported the builder of the house as saying his energy was stalking the property because the banks had forced him out. Elsewhere a friend had accused Murphy of "only being in it for Channel Seven's money". And the ABC had reported a five-figure payout from Seven. (I can tell you the five residents came away with nothing like it.)
But by now my cynical 'ghosts and media' reportage story was an irrelevance.
In the quiet of the day I found myself stalking the hot blue walls of the house, my t-shirt hanging in the humidity. I was hungry for more incidents, not least so I could assuage an angry crowd of sceptics gathered in my head. I wanted – demanded – 100 per cent proof or disproof, or even a 50:50 doubt:certainty for a conclusive "Who knows?". But the persistent 95 per cent "I can't explain" accompanied by a five per cent "just maybe" drove me nuts.
By the end of the day, despite seeing scissors appearing in the pool, a coin hitting the roof and another bullet landing behind me, I was plagued with sensory denial. I dug up grounds for doubt on every incident, convinced that Kirstie was somehow culpable. The appearance of the Scrabble letters 'GO' on top of the sound recordist's fluffy microphone made me laugh. "Only three points?"
That night, a cameraman and I slept over. As I lay melting on the living room floor, I reasoned that our witnessed 'unexplainable' events were as reliable as reports filed on accidents. Cloudy re-tellings of fleeting moments poorly perceived.
I slept fitfully as the house lay quiet.
Tuesday 7 April
4pm, on the fourth day; and as ghost buster Stephen Bishop came through the gates of Number 90, I was confident of two things: (1) if anyone was pulling stunts it was Kirstie and (2) Kirstie wasn't pulling stunts.
My sceptic bubble had been burst that morning.
Kirsty from the Humpty Doo house.
The other four residents had left for work. Jasmine was crying in the bedroom. I went to quieten her with Kirstie. The mother hoisted her child into the crook of her left arm, then turned and walked with me, smoking a cigarette with her right hand. I passed Murph's bedroom door on my right with Kirstie on my left. I spontaneously checked the room. I opened the door, saw it was empty and undisturbed, began to close it and heard a sudden BANG. I was confused, heard Kirstie say, "That came from in there."
Opened the door again to see a piece of broken glass against the far wall. The fact is she could not have thrown that glass. After re-enacting the event on camera, I quietly declared for the record: "Poltergeist".
But now the cameras were rolling for Steve who'd emerged from his car at the gates. "How're you feeling?" I asked him.
"I was pretty nervous this morning," he said. "That's partly the cameras, but my energy is being disturbed. I can tell you this is the most extreme case I've ever come across. I'm not even sure I can deal with it."
Dressed in white t-shirt, baggy shorts and cap, he began moving among the mango trees in the front acreage, closing his eyes behind his spectacles, putting palms out, taking breaths. "The land's dead, lost its soul," he pronounced to the camera.
I swallowed, suddenly feeling horribly responsible for setting him up as a 'ghost buster' in front of TV land. He answered Quail's questions with confidence– "Steve, tell us what you're doing now," – but I feared he sounded preposterous.
As darkness fell, he met the residents in the breezeway (the house was quiet) and asked them about their feelings, dreams, the baby's reactions, like a doctor probing for symptoms.
"It seems to be connected with the energy of the earth and the land around here. If that's the case then there's nothing I can do. It also seems to have intelligence which means it could be very dangerous."
Inside, he found the house "oppressive" while in some rooms he perceived a 'residual', like grey slime. He teamed with Dave, the Harley rider, who professed to feel it strongly as Steve mentally cleaned and re-set each room. The entity was squeezed using 'psychic seals' into a single room, where Steve, in faith healer's voice, suggested it was going away, and all would be right. Everyone was running with sweat, cooking under the camera lights.
Later that night, I watched the baby Jasmine collecting up a bunch of fibre pens once arranged on the floor to spell 'FIRE'. I thanked Steve for his help and patience, for his good faith, which seemed to give some of the residents strength. "I look at it like psychic science," he said. "If you're religious, then it would work just as well because you're focussing your energies into that."
"And you think it's gone?"
He looked over his glasses. "I don't know."
Sunday, 12 April.
Writing these events, I'm still sure of what I saw. What's more, I realise that like it or not I now belong to a minority group which wears a number of labels including Fraud, Pratt, Dreamer and Liar. None of them good for a writer/journalist.
"Tell me about it," said Greg Quail, returned to Sydney after a full seven days in Darwin. And guess what?"
"It's back. Came back Friday. And worse than before." Quail described a vase of flowers being smashed in the empty lounge room. "Shit everywhere."
"I believe it," I said.
Friday, 24 April
Quail's show rates well on Monday night after a media blitz. Today Tonight allocates a huge 15 minutes to the story. More witnesses are scheduled for the "Story of the year" on Tuesday, then Bishop will bust on Wednesday.
In the interim, the residents are threatened by the owner of Number 90 with eviction for damage caused to property. Court action proceeds.
On Sunday, a freelance cameraman based in NT was commissioned by Seven for extra shots, who captured a flying object on tape. Quail rejected the shot for Monday's show because it could not be cross checked with other camera angles. On Tuesday morning a tape editor notices a glimpsed reflection of a figure apparently tossing the object over the head of the cameraman.
Quail is in shock. Two seconds of tape will trash two weeks of investigation involving some 30 incidents and 18 witnesses. We think the figure is Kirstie, and after the fanfare of Monday, her timing is terrifyingly bad.
"I was called into the bosses office, this morning" groans Quail over a beer. "He played me the tape, demanded an explanation. I just wanted to run out of the building."
"What are you going to do?" I ask.
"You know there's something in there, I know there's something in there, but what can I do? With that one incident, she's blown their whole story."
"But what about – " Once again, we painfully pick over the incidents of the four days, dividing them into 'suspect' and 'sound'. But none of it is of consequence. Quail has to make a decision quickly: TV management doesn't believe in ghosts and TV Land will ultimately judge haunted Humpty Doo on this single clumsy piece of fraud. "The story's a turd," concludes one TV veteran, "and you can't polish a turd."
The ghost busters of Today Tonight must become hoax busters. Bishop's story never gets made; sceptics are interviewed; furious phone calls are exchanged between Quail and the residents. Kirsty admits she threw the object for the sake of being believed but on Wednesday denies everything. The residents circle the wagons and say "Enough".
On Friday, anchor Peter Luck can barely keep the contempt off his face as he says they'll be revisiting Humpty Doo, "Hopefully for the last time". A two minute end-piece on TT tells why.
Part of me too, is relieved. I'm back in the land of the rational-thinking, albeit wearing my 'Pratt' label for falling victim to the thin woman with the baby. But if I sit down and really think through those four days?
Then I'm certain a poltergeist somewhere is laughing its head off.
Posted in Poltergeist
Published by Cropster
View all posts by Cropster
Prev Mean Monkey of the Megalong
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Angela and the Stones
The Faces In the Window
Who Let The Djinn In?
Curious Rain of Fish in Sri Lanka.
A Truly Abominable Snowman
Larry Jordan on Stones Fly in San Jose
Ralphie on Stones Fly in San Jose
Larry Arnold on A Poltergeist In Vastareina, Chechnya.
Larry Arnold on Señora Nunez and the Burning Stones
Malcolm Smith on Tracks of the Kempsey Creature
Entombed Animals
Fish Fall
Religious Mystery
Strange Falls
Strange Images
Strange Rain
© Copyright 2021 – The Fortean | {
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Q: Textarea not storing the newline in update with ajax I have some text stored in mysql table in below format:
*
*This is the demo line 1
*This is the demo line 2
*This is the demo line 3
*This is the demo line 4
*This is the demo line 5
but, when I update any line in the text area, it displays the below output.Actually, I'm trying to make a text file with this if I don't update it to output me the above content in a text file but, When I update the content to give me the below output in the text file.
Here is my code:
This is the demo line 1
This is the demo line 2This is the demo line 3This is the demo line 4This is the demo line 5
<table>
<tr class="head">
<th style="display: none">Lable Name</th>
<th style="display: none">Tag Details</th>
</tr>
<?php
$sql1 = "SELECT * from tagto_print where print_tag_id='8'";
$reslabel=mysql_query($sql1);
$i=1;
while($row=mysql_fetch_array($reslabel))
{
$printid=$row['print_tag_id'];
$tagdetailstoprint=$row['print_tag_details'];
?>
<tr id="<?php echo $printid; ?>" class="edit_tr">
<td width="10%"><?php echo $row['print_tag_labelname'];?></td>
<td width="50%" class="edit_td">
<span id="details_<?php echo $printid; ?>" class="text"><pre><?php echo $tagdetailstoprint; ?></pre></span>
<textarea class="form-control" id="details_input_<?php echo $printid; ?>" rows="8"> <?php echo $tagdetailstoprint; ?></textarea></td>
</tr>
<?php
$i++;
}
?>
</table>
<style>
.form-control
{
display:none;
}
</style>
<script type="text/javascript">
$(document).ready(function()
{
$(".edit_tr").click(function()
{
var ID=$(this).attr('id');
$("#details_"+ID).hide();
$("#details_input_"+ID).show();
}).change(function()
{
var ID=$(this).attr('id');
var tagdetailsarea=$("#details_input_"+ID).val();
var dataString = 'printid='+ ID +'&printdetails='+tagdetailsarea;
$("#details_"+ID).html('<img src="load.gif" />');
if(tagdetailsarea.length>0)
{
$.ajax({
type: "POST",
url: "print-tags-edit.php",
data: dataString,
cache: false,
success: function(html)
{
$("#details_"+ID).html(tagdetailsarea);
}
});
}
else
{
alert('You can not Print Blank Data');
}
});
$(".form-control").mouseup(function()
{
return false
});
$(document).mouseup(function()
{
$(".form-control").hide();
$(".text").show();
});
});
</script>
File to update print-tags-edit.php:
<?php
if($_POST['printid'])
{
$id=$_POST['printid'];
$details=$_POST['printdetails'];
$sql = "update tagto_print set print_tag_details='$details' where print_tag_id='$id'";
mysql_query($sql);
}
?>
A: When you print it just put it inside this function , this to print in php
nl2br($txt)
if thing is inverted , just view source before writing data to file
if you found br then replace it with new line \n
A: I sorted my problem. This solution is mix of many references. I fetched the record from the table and put the (br) in content
$tagdetailstoprint=$row['print_tag_details'];
$breaks = nl2br($tagdetailstoprint);
$text = str_replace($breaks, "\r\n", $tagdetailstoprint);
In my textarea, I did this:
<textarea class="form-control" id="details_input_<?php echo $printid; ?>" rows="8"> <?php echo htmlspecialchars($text); ?></textarea>
and in style, I added the style sheet suggested by the AND
<style>
.text{
white-space: pre-line;
}
</style>
In my file to update print-tags-edit.php, I did this:
$details=nl2br($_POST['printdetails']); // i stored the nl2br in my mysql table
This code is to create the text file:
$today = date("Y-m-d-Hi");
header("Content-type: application/octet-stream");
header("Content-Disposition: attachment; filename=".$today."_Backup.txt");
$query = "SELECT * FROM tagto_print where print_tag_id='8'";
$result = mysql_query($query);
$data = "";
$breaks = array("<br />","<br>","<br/>"); // i put the br in array
while ($row=mysql_fetch_object($result)) {
$text = str_replace($breaks, "\r\n", $row->print_tag_details); // and repalced it with the "\r\n"
$data .= "{$text}";
$data .="\r\n";
}
echo $data;
exit;
}
| {
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Xenoblade Chronicles 2 Rare Blades are an important part of the game. They give you different kinds of Rare Blade perks, called Blade Arts. You also give various bonuses and boosts through Field Skills and Battle Skills. To get Rare Blades in Xenoblade Chronicles 2, you need to collect Core Crystals. Our guide on Rare Blades in Xenoblade Chronicles 2 will break down how to get them and Core Crystals, how perks work, swapping Rare Blades among characters, and more.
How to get Rare & Legendary Blades in Xenoblade Chronicles 2?
To get Xenoblade Chronicles 2 Rare Blades, you have to bond with Core Crystals. There are several different types of Core Crystals: Common, Rare, and Legendary. Once you have them, you can then bond one of them to your character of choice. It appears that which Blade you get is a bit random. You usually get Rare Blades with Rare Core Crystals, but sometimes, you'll get a Common Blade. Then again, sometimes a Rare Blade will pop out of a Common Core Crystal. You can expect to get your first Rare Blade about 15 or so hours in.
As for Legendary Blades, you'll normally get them by progressing through the story of the game. You might come across side quests that reward you with a Legendary Blade, though.
For more detailed info, check out our Xenoblade Chronicles 2 How to Get New Blades – Bond Blade Guide.
How to Obtain Rare & Legendary Core Crystals?
You obtain Legendary & Rare Core Crystals in Xenoblade Chronicles 2 as rewards for various activities. For example, you can get them as a reward for completing various quests. They can also drop from enemies, or you can find them in chests. Since drops in Xenoblade Chronicles 2 can get a bit chaotic sometimes, keep your eyes peeled for those Core Crystals.
Different Blades can give your characters different perks, which are called Blade Arts. Let's take Pyra as an example. She gives a boost to your accuracy and also increases you next Driver Art's power. That's not the only thing Blades can give you. Every single one of them offers different Field Skills and Battle Skills. You can also equip your Blades with Aux Cores to further boost them.
As we've noted before, every blade gives you different kinds of bonuses and boosts. Among other things, they offer various Field Skills. These can give you different element mastery, focuses, and skills that will help you when you're out and about, such as Lockpicking, Botany, Ichthyology, and the like. In the list below, we'll show you what the game's Main Blades will give you.
How to Swap Rare Blades Between Characters?
The first step to changing Rare Blades among your drivers is to go into the Manage Blades menu. From there, you can select Use Overdrive. This will allow you to take your Rare Blades and bond them to a different driver. That way, you can grant characters different perks, depending on what you need at the time.
Xenoblade Chronicles 2 – How Many Rare Blades?
There are almost forty rare blades in the game, and most of them are obtained through random bonding. Luckily, some of them are quest rewards you'll get either during the main story, or by completing side missions. Our list is divided by role – attack, tank and healer.
To get the Lucky Core Crystal you'll need to buy the Greedy Monster Item from the Informant in Alba Cavanich – Empire of Mor Ardain. Now you can spawn a rare monster Gluttonous Marrin from the salvage point near Chansagh Wastes, Empire of Mor Ardain – Lower Level. This creature drops the crystal upon defeat.
Kassandra is actually found in an Unlucky Core from killing a unique monster in Mor Ardain. You find the location of the monster by talking to the informant in Alba Cavinac. | {
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Taphozous perforatus est une espèce de chauve-souris de la famille des Emballonuridae.
Distribution
Cette espèce se rencontre en Afrique, au Moyen-Orient et dans l'ouest de l'Asie du Sud.
Liste des sous-espèces
Selon :
Taphozous perforatus haedinus Thomas, 1915
Taphozous perforatus perforatus Geoffroy, 1818
Taphozous perforatus senegalensis Desmarest, 1820
Taphozous perforatus sudani Thomas, 1915
Publication originale
É. Geoffroy, 1818 : Description de l'Égypte.
Notes et références
Voir aussi
Faune d'Oman
Bibliographie
Harrison D. L., 1968 : On three mammals new to the fauna of Oman, Arabia, with the description of a new subspecies of bat. Mammalia, , , .
Liens externes
Chiroptère (nom scientifique)
Emballonuridae
Faune d'Afrique de l'Ouest
Faune d'Afrique de l'Est
Faune d'Asie de l'Ouest
Faune d'Asie du Sud
Chauve-souris du sous-continent indien | {
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\section{Proofs}
\begin{proof}[Detailed proof of Lemma~\ref{lemmaEstimatorUnbiased}]
The expected value of the exponential of the estimator defined in (\ref{eqnUnbiasedEst}) is
\begin{align*}
\Exv{\exp(\epsilon_x)}
=
\Exv{\exp\left( \sum_{\phi \in S} s_{\phi} \log\left( 1 + \frac{\Psi}{\lambda M_{\phi}} \phi(x) \right) \right)}.
\end{align*}
Since the $s_{\phi}$ are independent, this becomes
\begin{align*}
\Exv{\exp(\epsilon_x)}
=
\prod_{\phi \in \Phi} \Exv{\exp\left( s_{\phi} \log\left( 1 + \frac{\Psi}{\lambda M_{\phi}} \phi(x) \right) \right)}.
\end{align*}
Each of these constituent expected values is an evaluation of the moment generating function of a Poisson random variable. Applying the known expression for this MGF,
\begin{align*}
\Exv{\exp(\epsilon_x)}
=&
\prod_{\phi \in \Phi}
\exp\left(
\frac{ \lambda M_{\phi} }{ \Psi }
\left(
\exp\left(
\log\left( 1 + \frac{\Psi}{\lambda M_{\phi}} \phi(x) \right)
\right)
-
1
\right)
\right) \\
=&
\prod_{\phi \in \Phi}
\exp\left(
\frac{ \lambda M_{\phi} }{ \Psi }
\left(
\frac{\Psi}{\lambda M_{\phi}} \phi(x)
\right)
\right) \\
=&
\prod_{\phi \in \Phi}
\exp\left(
\phi(x)
\right) \\
=&
\exp(\zeta(x)).
\end{align*}
This is what we wanted to show.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thmMINTreversible}]
The probability of transitioning from $(x, \epsilon_{x(i)})$ to $(y, \epsilon_{y(i)})$ for two distinct states $x$ and $y$ which differ only in variable $i$ will be the probability that: we decide to re-sample variable $i$, we choose this particular value $\epsilon_{y(i)}$ when we sample it at random from $\mu_y$, and we sample $v = y(i)$ from $\rho$ when we sample the new value of variable $i$.
So the transition probability will be
\[
T((x, \epsilon_{x(i)}), (y, \epsilon_{y(i)}))
=
\frac{1}{n} \cdot \mu_y(\epsilon_{y(i)}) \cdot \Exv{ \frac{\exp(\epsilon_{y(i)})}{\sum_{w=1}^D \exp(\epsilon_w)} }
\]
where this expected value is taken over the randomly sampled $\epsilon_w$ (except when $w \in \{ x(i), y(i) \}$ where the value of $\epsilon_w$ is already determined).
If we multiply both sides by $\bar \pi(x, \epsilon)$ which by definition for some fixed $Z$ is equal to
\[
\bar \pi(x, \epsilon) = \frac{1}{Z} \mu_x(\epsilon) \cdot \exp(\epsilon)
\]
then we get
\begin{align*}
&\bar \pi(x, \epsilon) T((x, \epsilon_{x(i)}), (y, \epsilon_{y(i)})) \\
&\hspace{2em}=
\frac{1}{n Z} \cdot \Exv{ \frac{\mu_x(\epsilon) \cdot \exp(\epsilon) \cdot \mu_y(\epsilon_{y(i)}) \cdot \exp(\epsilon_{y(i)}) }{\sum_{w=1}^D \exp(\epsilon_w)} }
\end{align*}
and this expression is clearly symmetric in $x$ and $y$, so the chain is reversible and $\bar \pi$ is indeed its stationary distribution.
The second result, about the marginal distribution in $x$, follows directly from the definition of expected value.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thmMINTgap}]
This proof will use the technique of \emph{Dirichlet forms}~\cite{levin2009markov}.
The Dirichlet form of a Markov chain with transition matrix $T$ is defined for function argument $f: \Omega \rightarrow \R$ as
\[
\mathcal{E}(f) = \frac{1}{2} \sum_{x,y \in \Omega} (f(x) - f(y))^2 \pi(x) T(x, y).
\]
Similarly, the variance of the function $f$ under the distribution $\pi$ is defined as
\[
\Var[\pi]{f} = \frac{1}{2} \sum_{x,y \in \Omega} (f(x) - f(y))^2 \pi(x) \pi(y);
\]
this is equivalent to the standard definition of variance.
It is a standard result (from \citet{levin2009markov}) that the spectral gap can be written as
\[
\gamma = \min_f \frac{\mathcal{E}(f)}{\Var[\pi]{f}}.
\]
Now, consider the Dirichlet form of the MIN-Gibbs chain.
Let this form be $\bar{\mathcal{E}}(f)$, where for $f: \Omega \times \R \rightarrow \R$,
\begin{dmath*}
\bar{\mathcal{E}}(f)
=
\frac{1}{2} \sum_{x,y \in \Omega}
\sum_{\epsilon_x \in \dom(\mu_x)}
\sum_{\epsilon_y \in \dom(\mu_y)}
(f(x, \epsilon_x) - f(y, \epsilon_y))^2 \bar \pi(x, \epsilon_x) \bar T((x, \epsilon_x), (y, \epsilon_y)).
\end{dmath*}
From the result of Theorem~\ref{thmMINTreversible}, we know that for some $\bar Z$,
\[
\bar \pi(x, \epsilon) = \frac{1}{\bar Z} \mu_x(\epsilon) \cdot \exp(\epsilon).
\]
We also recall from the proof of that theorem that, for $x$ and $y$ which differ only in variable $i$,
\[
\bar T((x, \epsilon_{x}), (y, \epsilon_{y}))
=
\frac{1}{n} \cdot \mu_y(\epsilon_{y}) \cdot \Exv{ \frac{\exp(\epsilon_{y})}{\sum_{w=1}^D \exp(\epsilon_w)} },
\]
where the $\epsilon_w$ are each sampled independently from $\mu_w$.
Otherwise, the transition probability is zero.
As a result, if we let $Q \subset \Omega \times \Omega$ denote the pairs of states which differ only in a single variable, we can rewrite our Dirichlet form as
\begin{dmath*}
\bar{\mathcal{E}}(f)
=
\frac{1}{2 n \bar Z} \sum_{(x,y) \in Q}
\sum_{\epsilon_x \in \dom(\mu_x)}
\sum_{\epsilon_y \in \dom(\mu_y)}
(f(x, \epsilon_x) - f(y, \epsilon_y))^2
\mu_x(\epsilon_x) \cdot \exp(\epsilon_x) \cdot
\mu_y(\epsilon_y) \cdot \Exv{ \frac{\exp(\epsilon_y)}{\sum_{w=1}^D \exp(\epsilon_w)} }.
\end{dmath*}
Now, by the definition of expected value, if we suppose that $\epsilon_x$ and $\epsilon_y$ are random variables sampled from $\mu_x$ and $\mu_y$ respectively, then
\begin{dmath*}
\bar{\mathcal{E}}(f)
=
\frac{1}{2 n \bar Z} \sum_{(x,y) \in Q}
\Exv{
(f(x, \epsilon_x) - f(y, \epsilon_y))^2
\cdot
\frac{\exp(\epsilon_x) \exp(\epsilon_y)}{\sum_{w=1}^D \exp(\epsilon_w)}
}.
\end{dmath*}
Using a similar argument, we can also analyze the original vanilla Gibbs chain.
For simplicity, define
\[
\zeta(x) = \sum_{\phi \in \Phi} \phi(x),
\]
and suppose that for some $Z$
\[
\pi(x) = \frac{1}{Z} \exp(\zeta(x)).
\]
Now we can define the Dirichlet form of the original Gibbs chain as $\mathcal{E}(g)$, where $g: \Omega \rightarrow \R$ and
\begin{dmath*}
\mathcal{E}(g)
=
\frac{1}{2 n Z} \sum_{(x,y) \in Q}
(g(x) - g(y))^2
\cdot
\frac{\exp(\zeta(x)) \exp(\zeta(y))}{\sum_{w=1}^D \exp(\zeta(w))}.
\end{dmath*}
Now, we know by the condition of the theorem that
\[
\Abs{\epsilon_x - \zeta(x)} \le \delta.
\]
As a result, we can bound the Dirichlet form of our MIN-Gibbs chain with
\begin{dmath*}
\bar{\mathcal{E}}(f)
\ge
\frac{1}{2 n \bar Z} \sum_{(x,y) \in Q}
\Exv{
(f(x, \epsilon_x) - f(y, \epsilon_y))^2
\cdot
\frac{\exp(\zeta(x) - \delta) \exp(\zeta(y) - \delta)}{\sum_{w=1}^D \exp(\zeta(w) + \delta)}
}
=
\frac{1}{2 n \bar Z} \sum_{(x,y) \in Q}
\frac{\exp(\zeta(x) - \delta) \exp(\zeta(y) - \delta)}{\sum_{w=1}^D \exp(\zeta(w) + \delta)}
\Exv{ (f(x, \epsilon_x) - f(y, \epsilon_y))^2 }.
\end{dmath*}
On the other hand, the variance form associated with the MIN-Gibbs chain is
\begin{dmath*}
\Var[\bar \pi]{f}
=
\frac{1}{2} \sum_{x,y \in \Omega}
\sum_{\epsilon_x \in \dom(\mu_x)}
\sum_{\epsilon_y \in \dom(\mu_y)}
(f(x, \epsilon_x) - f(y, \epsilon_y))^2 \bar \pi(x) \bar \pi(y)
=
\frac{1}{2 \bar Z^2} \sum_{x,y \in \Omega}
\sum_{\epsilon_x \in \dom(\mu_x)}
\sum_{\epsilon_y \in \dom(\mu_y)}
(f(x, \epsilon_x) - f(y, \epsilon_y))^2 \mu_x(\epsilon_x) \cdot \exp(\epsilon_x) \cdot \mu_y(\epsilon_y) \cdot \exp(\epsilon_y)
\le
\frac{1}{2 \bar Z^2} \sum_{x,y \in \Omega}
\exp(\zeta(x) + \delta) \cdot \exp(\zeta(y) + \delta) \cdot
\Exv{ (f(x, \epsilon_x) - f(y, \epsilon_y))^2 }.
\end{dmath*}
Now, we need some way to get rid of these expected values.
One way to do it is to recall that for positive numbers $a_1, a_2, \ldots, a_N$ and $b_1, b_2, \ldots, b_N$,
\[
\frac{\sum_{i=1}^n a_i}{\sum_{i=1}^n b_i} \ge \min_i \frac{a_i}{b_i}
\]
or, in terms of expected value, if $a$ and be are nonnegative functions and $Z$ is a random variable,
\[
\frac{\Exv{a(Z)}}{\Exv{b(Z)}}
=
\frac{1}{\Exv{b(Z)}} \Exv{b(Z) \frac{a(Z)}{b(Z)}}
\ge
\frac{1}{\Exv{b(Z)}} \Exv{b(Z) \min_z \frac{a(z)}{b(z)}}
=
\min_z \frac{a(z)}{b(z)}.
\]
Equivalently, we can say that there exists a fixed $z$ (the $z$ that minimizes the expression on the right) such that
\begin{equation}
\label{eqnABZ}
\frac{\Exv{a(Z)}}{\Exv{b(Z)}} \ge \frac{a(z)}{b(z)}.
\end{equation}
It follows that
\begin{dmath*}
\bar \gamma
=
\min_f \frac{\bar{\mathcal{E}}(f)}{\Var[\bar \pi]{f}}
\ge
\min_f \frac{
\frac{1}{2 n \bar Z} \sum_{(x,y) \in Q}
\frac{\exp(\zeta(x) - \delta) \exp(\zeta(y) - \delta)}{\sum_{w=1}^D \exp(\zeta(w) + \delta)}
\Exv{ (f(x, \epsilon_x) - f(y, \epsilon_y))^2 }
}{
\frac{1}{2 \bar Z^2} \sum_{x,y \in \Omega}
\exp(\zeta(x) + \delta) \cdot \exp(\zeta(y) + \delta) \cdot
\Exv{ (f(x, \epsilon_x) - f(y, \epsilon_y))^2 }
}
=
\min_f \frac{
\Exv{
\frac{1}{2 n \bar Z} \sum_{(x,y) \in Q}
\frac{\exp(\zeta(x) - \delta) \exp(\zeta(y) - \delta)}{\sum_{w=1}^D \exp(\zeta(w) + \delta)}
(f(x, \epsilon_x) - f(y, \epsilon_y))^2
}
}{
\Exv{
\frac{1}{2 \bar Z^2} \sum_{x,y \in \Omega}
\exp(\zeta(x) + \delta) \cdot \exp(\zeta(y) + \delta) \cdot
(f(x, \epsilon_x) - f(y, \epsilon_y))^2
}
},
\end{dmath*}
where the randomness in this expression is over the random variables $\epsilon_x \sim \mu_x$ for each $x \in \Omega$.
We can think of this as a ratio of huge sums over all possible assignments of $\epsilon_x$ for all $x$,
which will be lower bounded by the ratio for some fixed assignment of $\epsilon_x$, as per (\ref{eqnABZ}).
That is, for some assignment of values to each $\epsilon_x$ for $x \in \Omega$,
\begin{dmath*}
\bar \gamma
=
\min_f \frac{\bar{\mathcal{E}}(f)}{\Var[\bar \pi]{f}}
\ge
\min_f \frac{
\frac{1}{2 n \bar Z} \sum_{(x,y) \in Q}
\frac{\exp(\zeta(x) - \delta) \exp(\zeta(y) - \delta)}{\sum_{w=1}^D \exp(\zeta(w) + \delta)}
(f(x, \epsilon_x) - f(y, \epsilon_y))^2
}{
\frac{1}{2 \bar Z^2} \sum_{x,y \in \Omega}
\exp(\zeta(x) + \delta) \cdot \exp(\zeta(y) + \delta) \cdot
(f(x, \epsilon_x) - f(y, \epsilon_y))^2
}.
\end{dmath*}
If we now define
\[
g(x) = f(x, \epsilon_x)
\]
then we can simplify this ratio to
\begin{dmath*}
\bar \gamma
\ge
\min_g \frac{
\frac{1}{2 n \bar Z} \sum_{(x,y) \in Q}
\frac{\exp(\zeta(x) - \delta) \exp(\zeta(y) - \delta)}{\sum_{w=1}^D \exp(\zeta(w) + \delta)}
(g(x) - g(y))^2
}{
\frac{1}{2 \bar Z^2} \sum_{x,y \in \Omega}
\exp(\zeta(x) + \delta) \cdot \exp(\zeta(y) + \delta) \cdot
(g(x) - g(y))^2
}
=
\exp(-5 \delta)
\min_g \frac{
\frac{1}{2 n} \sum_{(x,y) \in Q}
\frac{\exp(\zeta(x)) \exp(\zeta(y))}{\sum_{w=1}^D \exp(\zeta(w))}
(g(x) - g(y))^2
}{
\frac{1}{2 \bar Z} \sum_{x,y \in \Omega}
\exp(\zeta(x)) \cdot \exp(\zeta(y)) \cdot
(g(x) - g(y))^2
}.
\end{dmath*}
Now, we notice that these are the same as the expressions for the original chain!
In fact,
\begin{dmath*}
\bar \gamma
\ge
\exp(-5 \delta)
\frac{\bar Z}{Z}
\min_g \frac{
\mathcal{E}(g)
}{
\Var[\pi]{g}
}.
\end{dmath*}
All that remains is to bound this ratio of the $Z$ and $\bar Z$. We can do this with
\begin{dmath*}
\frac{\bar Z}{Z}
=
\frac{
\sum_{x \in \Omega} \sum_{\epsilon \in \dom(\mu_x)}
\mu_x(\epsilon) \exp(\epsilon)
}{
\sum_{x \in \Omega} \exp(\zeta(x))
}
\ge
\frac{
\sum_{x \in \Omega} \sum_{\epsilon \in \dom(\mu_x)}
\mu_x(\epsilon) \exp(\zeta(x) - \delta)
}{
\sum_{x \in \Omega} \exp(\zeta(x))
}
=
\exp(-\delta)
\frac{
\sum_{x \in \Omega} \exp(\zeta(x))
}{
\sum_{x \in \Omega} \exp(\zeta(x))
}
=
\exp(-\delta),
\end{dmath*}
and so
\begin{dmath*}
\bar \gamma
\ge
\exp(-6 \delta)
\min_g \frac{
\mathcal{E}(g)
}{
\Var[\pi]{g}
}
=
\exp(-5 \delta) \gamma.
\end{dmath*}
This is what we wanted to show.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lemmaUnbiasedConcentration}]
The estimator we want to bound is
\[
\epsilon_x = \sum_{\phi \in S} s_{\phi} \log\left( 1 + \frac{\Psi}{\lambda M_{\phi}} \phi(x) \right).
\]
where the parameter of the Poisson-distributed random variable $s_{\phi}$ is
\[
\Exv{s_{\phi}} = \frac{\lambda M_{\phi}}{\Psi}.
\]
First, we note that since $0 \le \phi(x)$ and the logarithm is concave,
\[
0
\le
\log\left( 1 + \frac{\Psi}{\lambda M_{\phi}} \phi(x) \right)
\le
\frac{\Psi}{\lambda M_{\phi}} \phi(x)
\le
\frac{\Psi}{\lambda}.
\]
Second, note that since the variance of a Poisson random variable is equal to its expected value, the variance of the estimator is
\begin{dmath*}
\Var{\epsilon_x}
=
\sum_{\phi \in \Phi} \Var{s_{\phi}}
\left( \log\left( 1 + \frac{\Psi}{\lambda M_{\phi}} \phi(x) \right) \right)^2
\le
\frac{\Psi^2}{\lambda^2} \sum_{\phi \in \Phi} \Var{s_{\phi}}
=
\frac{\Psi^2}{\lambda^2} \sum_{\phi \in \Phi} \frac{\lambda M_{\phi}}{\Psi}
=
\frac{\Psi^2}{\lambda}.
\end{dmath*}
Thus, by the Bernstein inequality, if we can write $\epsilon_x$ as a sum, then
\begin{dmath*}
\Prob{ \Abs{\epsilon_x - \Exv{\epsilon_x}} \ge t }
\le
2 \exp\left(
-\frac{
\frac{1}{2} t^2
}{
\Var{\epsilon_x}
+
\frac{1}{3} C t
}
\right)
=
2 \exp\left(
-\frac{
\frac{1}{2} t^2
}{
\frac{\Psi^2}{\lambda}
+
\frac{1}{3} C t
}
\right)
\end{dmath*}
where $C$ is the maximum magnitude of any component of the sum.
But, since the Poisson distribution is infinitely divisible, we can make arbitrarily many components of the sum, so we can push $C$ arbitrarily close to $0$.
By a continuity argument we can set it equal to $0$, and so
\begin{dmath*}
\Prob{ \Abs{\epsilon_x - \Exv{\epsilon_x}} \ge t }
\le
2 \exp\left(
-\frac{
\lambda t^2
}{
2 \Psi^2
}
\right).
\end{dmath*}
Next, we need to bound $\Exv{\epsilon_x}$.
First, note that by Jensen's inequality,
\[
\exp(\Exv{\epsilon_x})
\le
\Exv{\exp(\epsilon_x)}
=
\exp(\zeta(x)),
\]
so $\epsilon_x$ is an underestimator of $\zeta(x)$ in expectation.
We thus need to bound it from below.
Since the expected value of $s_{\phi}$ is known, we can write the expected value explicitly as
\begin{dmath*}
\Exv{\epsilon_x}
=
\sum_{\phi \in \Phi}
\frac{\lambda M_{\phi}}{\Psi}
\log\left( 1 + \frac{\Psi}{\lambda M_{\phi}} \phi(x) \right).
\end{dmath*}
We know for positive $z$ that $\log(1+z) \ge x - x^2/2$, so
\begin{dmath*}
\Exv{\epsilon_x}
\ge
\sum_{\phi \in \Phi}
\frac{\lambda M_{\phi}}{\Psi}
\left(
\frac{\Psi}{\lambda M_{\phi}} \phi(x)
-
\left( \frac{\Psi}{\lambda M_{\phi}} \phi(x) \right)^2
\right)
\ge
\sum_{\phi \in \Phi}
\frac{\lambda M_{\phi}}{\Psi}
\left(
\frac{\Psi}{\lambda M_{\phi}} \phi(x)
-
\frac{\Psi^2}{\lambda^2}
\right)
=
\sum_{\phi \in \Phi}
\left(
\phi(x)
-
\frac{\Psi M_{\phi}}{\lambda}
\right)
=
\zeta(x)
-
\frac{\Psi^2}{\lambda}.
\end{dmath*}
Thus,
\begin{dmath*}
\Prob{ \Abs{\epsilon_x - \zeta(x) } \ge t + \frac{\Psi^2}{\lambda} }
\le
2 \exp\left(
-\frac{
\lambda t^2
}{
2 \Psi^2
}
\right).
\end{dmath*}
Now, suppose we want $\Prob{ \Abs{\epsilon_x - \zeta(x) } \ge \delta} \le a$.
If we assign $t = \delta / 2$, and require that
\[
\lambda \ge \frac{2 \Psi^2}{\delta},
\]
then we just need
\begin{dmath*}
a
\ge
2 \exp\left(
-\frac{
\lambda \delta^2
}{
8 \Psi^2
}
\right).
\end{dmath*}
So it suffices to set
\[
\lambda
\ge
\frac{
8 \Psi^2
}{
\delta^2
}
\log\left( \frac{2}{a} \right).
\]
This proves the lemma.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thmMGPMHreversible}]
Statistically, the sampling procedure in Algorithm~\ref{algMGPMH} is equivalent to sampling a variable $i$, and then sampling a Poisson random variable $s_{\phi}$ for each $\phi \in \Phi$.
Let $T_{i,s}(x, y)$ denote the probability of transitioning from state $x$ to $y$ given that we have already chosen to sample variable $i$ with minibatch coefficients $s$.
Then, the overall transition matrix will be
\[
T(x, y) = \Exv{ T_{i,s}(x, y) }
\]
where this expected value is taken with respect to the random variables $i$ and $s$.
If we look at $T_{i,s}(x,y)$ for two states that differ only at variable $i$, it will be equal to
\[
T_{i,s}(x,y) = \rho(y(i)) \cdot \min(a, 1),
\]
which is the probability of proposing $y$ times the probability of accepting that proposal.
(Otherwise, if $x$ and $y$ differ at a variable other than $i$, this transition probability will be $T_{i,s}(x,y) = 0$.)
We can expand this expression to
\begin{dmath*}
T_{i,s}(x,y)
=
\frac{\exp(\epsilon_{y(i)})}{\sum_{w=1}^D \exp(\epsilon_w)}
\cdot
\min\left(
\frac{
\exp\left(
\sum_{\varphi \in A[i]} \varphi(y)
\right)
}{
\exp\left(
\sum_{\varphi \in A[i]} \varphi(x)
\right)
}
\cdot
\frac{
\exp\left(
\epsilon_x
\right)
}{
\exp\left(
\epsilon_y
\right)
}, 1 \right)
=
\frac{\exp(\epsilon_{y(i)})}{\sum_{w=1}^D \exp(\epsilon_w)}
\cdot
\min\left(
\frac{
\exp\left(
\sum_{\varphi \in \Phi} \varphi(y)
\right)
}{
\exp\left(
\sum_{\varphi \in \Phi} \varphi(x)
\right)
}
\cdot
\frac{
\exp\left(
\epsilon_x
\right)
}{
\exp\left(
\epsilon_y
\right)
}, 1 \right)
=
\frac{\exp(\epsilon_{y(i)})}{\sum_{w=1}^D \exp(\epsilon_w)}
\cdot
\min\left(
\frac{
\exp\left( \zeta(y) \right)
}{
\exp\left( \zeta(x) \right)
}
\cdot
\frac{
\exp\left(
\epsilon_{x(i)}
\right)
}{
\exp\left(
\epsilon_{y(i)}
\right)
}, 1 \right).
\end{dmath*}
Multiplying this by $\pi(x)$,
\begin{dmath*}
\pi(x) T_{i,s}(x,y)
=
\frac{1}{Z} \exp(\zeta(x))
\cdot
\frac{\exp(\epsilon_{y(i)})}{\sum_{w=1}^D \exp(\epsilon_w)}
\cdot
\min\left(
\frac{
\exp\left( \zeta(y) \right)
}{
\exp\left( \zeta(x) \right)
}
\cdot
\frac{
\exp\left(
\epsilon_{x(i)}
\right)
}{
\exp\left(
\epsilon_{y(i)}
\right)
}, 1 \right)
=
\frac{1}{Z \sum_{w=1}^D \exp(\epsilon_w)}
\cdot
\min\left(
\exp\left( \zeta(y) \right)
\cdot
\exp\left( \epsilon_{x(i)} \right)
,
\exp\left( \zeta(x) \right)
\cdot
\exp\left( \epsilon_{y(i)} \right)
\right).
\end{dmath*}
This last expression is clearly symmetric in $x$ and $y$. So,
\[
\pi(x) T_{i,s}(x,y)
=
\pi(y) T_{i,s}(y,x).
\]
But this implies that
\begin{dmath*}
\pi(x) T(x,y)
=
\pi(x) \Exv{ T_{i,s}(x,y) }
=
\Exv{ \pi(x) T_{i,s}(x,y) }
=
\Exv{ \pi(y) T_{i,s}(y,x) }
=
\pi(y) \Exv{ T_{i,s}(y,x) }
=
\pi(y) T(y,x),
\end{dmath*}
so the whole chain is reversible.
This is what we wanted to prove.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thmMGPMHgap}]
As before, we will accomplish this proof via the technique of Dirichlet forms.
First, recall that the transition probability matrix $T$ of vanilla Gibbs sampling is, for any $x$ and $y$ which differ in only variable $i$,
\begin{dmath*}
T(x, y)
=
\frac{1}{n} \frac{
\exp(\zeta(y))
}{
\sum_{w = 1}^D \exp\left(\zeta(y_{i \rightarrow w}) \right)
},
\end{dmath*}
where here $y_{i \rightarrow w}$ denotes the state $y$ with variable $i$ assigned to $w$; that is,
\[
y_{i \rightarrow w}
=
(x \cap y) \cup \{(i, w)\}.
\]
Multiplying by $\pi(x)$,
\begin{dmath*}
\pi(x) T(x, y)
=
\frac{1}{n Z} \frac{
\exp(\zeta(x)) \exp(\zeta(y))
}{
\sum_{w = 1}^D \exp\left(\zeta(y_{i \rightarrow w}) \right)
},
\end{dmath*}
From the proof of Theorem~\ref{thmMGPMHreversible}, we have that the transition probability matrix of MGPMH (which we denote with $\bar T$) satisfies
\begin{dmath*}
\pi(x) \bar T(x,y)
=
\pi(x) \Exv[j,s]{ \bar T_{j,s}(x,y) }
=
\frac{1}{n} \pi(x) \Exv[s]{ \bar T_{i,s}(x,y) }
=
\frac{1}{n Z}
\Exv{
\frac{
\min\left(
\exp\left( \zeta(y) \right)
\cdot
\exp\left( \epsilon_{x(i)} \right)
,
\exp\left( \zeta(x) \right)
\cdot
\exp\left( \epsilon_{y(i)} \right)
\right)
}{
\sum_{w=1}^D \exp(\epsilon_w)
}
}.
\end{dmath*}
It follows that
\begin{dmath*}
\frac{
\pi(x) \bar T(x,y)
}{
\pi(x) T(x, y)
}
=
\frac{
\frac{1}{n Z}
\Exv{
\frac{
\min\left(
\exp\left( \zeta(y) \right)
\cdot
\exp\left( \epsilon_{x(i)} \right)
,
\exp\left( \zeta(x) \right)
\cdot
\exp\left( \epsilon_{y(i)} \right)
\right)
}{
\sum_{w=1}^D \exp(\epsilon_w)
}
}
}{
\frac{1}{n Z} \frac{
\exp(\zeta(x)) \exp(\zeta(y))
}{
\sum_{w = 1}^D \exp\left(\zeta(y_{i \rightarrow w}) \right)
}
}
=
\Exv{
\frac{
\sum_{w = 1}^D \exp\left(\zeta(y_{i \rightarrow w}) \right)
}{
\sum_{w=1}^D \exp(\epsilon_w)
}
\min\left(
\exp\left( \epsilon_{x(i)} - \zeta(x) \right)
,
\exp\left( \epsilon_{y(i)} - \zeta(y) \right)
\right)
}
=
\Exv{
\frac{
\sum_{w = 1}^D \exp\left(\zeta(y_{i \rightarrow w}) \right)
}{
\sum_{w=1}^D \exp(\epsilon_w)
}
\cdot
\frac{1}{
\max\left(
\exp\left( \zeta(x) - \epsilon_{x(i)} \right)
,
\exp\left( \zeta(y) - \epsilon_{y(i)} \right)
\right)
}
}
=
\Exv{
\frac{
\sum_{w = 1}^D \exp\left(\zeta(y_{i \rightarrow w}) \right)
}{
\sum_{w=1}^D
\max\left(
\exp\left( \epsilon_w + \zeta(x) - \epsilon_{x(i)} \right)
,
\exp\left( \epsilon_w + \zeta(y) - \epsilon_{y(i)} \right)
\right)
}
}.
\end{dmath*}
By Jensen's inequality, since $f(z) = 1/z$ is convex, we can bound this from below by
\begin{dmath*}
\frac{
\pi(x) \bar T(x,y)
}{
\pi(x) T(x, y)
}
\ge
\frac{
\sum_{w = 1}^D \exp\left(\zeta(y_{i \rightarrow w}) \right)
}{
\Exv{
\sum_{w=1}^D
\max\left(
\exp\left( \epsilon_w + \zeta(x) - \epsilon_{x(i)} \right)
,
\exp\left( \epsilon_w + \zeta(y) - \epsilon_{y(i)} \right)
\right)
}
}
=
\frac{
\sum_{w = 1}^D \exp\left(\zeta(y_{i \rightarrow w}) \right)
}{
\sum_{w=1}^D
\exp\left(\zeta(y_{i \rightarrow w}) \right)
\Exv{
\max\left(
\exp\left( \epsilon_w - \zeta(y_{i \rightarrow w}) + \zeta(x) - \epsilon_{x(i)} \right)
,
\exp\left( \epsilon_w - \zeta(y_{i \rightarrow w}) + \zeta(y) - \epsilon_{y(i)} \right)
\right)
}
}.
\end{dmath*}
Next, we notice that if we define $z = y_{i \rightarrow w}$ for a particular $w$, we can write
\begin{dmath*}
\Exv{
\max\left(
\exp\left( \epsilon_w - \zeta(y_{i \rightarrow w}) + \zeta(x) - \epsilon_{x(i)} \right)
,
\exp\left( \epsilon_w - \zeta(y_{i \rightarrow w}) + \zeta(y) - \epsilon_{y(i)} \right)
\right)
}
=
\Exv{
\max\left(
\exp\left( \epsilon_{z(i)} - \zeta(z) + \zeta(x) - \epsilon_{x(i)} \right)
,
\exp\left( \epsilon_{z(i)} - \zeta(z) + \zeta(y) - \epsilon_{y(i)} \right)
\right)
}
\end{dmath*}
Recall that the random variables in this expected value are the $\epsilon_w$, which are functions of the minibatch coefficients $s_{\phi}$ where
\[
\epsilon_{z(i)} = \sum_{\phi \in A[i]} \frac{s_{\phi} L}{\lambda M_{\phi}} \phi(z).
\]
So,
\begin{dmath*}
\Exv{
\max\left(
\exp\left( \epsilon_{z(i)} - \zeta(z) + \zeta(x) - \epsilon_{x(i)} \right)
,
\exp\left( \epsilon_{z(i)} - \zeta(z) + \zeta(y) - \epsilon_{y(i)} \right)
\right)
}
=
\Exv{
\max\left(
\exp\left(
\left( \sum_{\phi \in A[i]} \frac{s_{\phi} L}{\lambda M_{\phi}} \phi(z) \right)
-
\zeta(z)
+
\zeta(x)
-
\left( \sum_{\phi \in A[i]} \frac{s_{\phi} L}{\lambda M_{\phi}} \phi(x) \right)
\right)
,\\
\exp\left(
\left( \sum_{\phi \in A[i]} \frac{s_{\phi} L}{\lambda M_{\phi}} \phi(z) \right)
-
\zeta(z)
+
\zeta(y)
-
\left( \sum_{\phi \in A[i]} \frac{s_{\phi} L}{\lambda M_{\phi}} \phi(y) \right)
\right)
\right)
}
=
\Exv{
\max\left(
\exp\left(
\sum_{\phi \in A[i]} \left( \frac{s_{\phi} L}{\lambda M_{\phi}} - 1 \right) (\phi(z) - \phi(x))
\right)
,
\exp\left(
\sum_{\phi \in A[i]} \left( \frac{s_{\phi} L}{\lambda M_{\phi}} - 1 \right) (\phi(z) - \phi(y))
\right)
\right)
}.
\end{dmath*}
Since the maximum of two sums is less than the sum of the maximums, it follows that
\begin{dmath*}
\Exv{
\max\left(
\exp\left( \epsilon_{z(i)} - \zeta(z) + \zeta(x) - \epsilon_{x(i)} \right)
,
\exp\left( \epsilon_{z(i)} - \zeta(z) + \zeta(y) - \epsilon_{y(i)} \right)
\right)
}
=
\Exv{
\exp\left(
\max\left(
\sum_{\phi \in A[i]} \left( \frac{s_{\phi} L}{\lambda M_{\phi}} - 1 \right) (\phi(z) - \phi(x))
,
\sum_{\phi \in A[i]} \left( \frac{s_{\phi} L}{\lambda M_{\phi}} - 1 \right) (\phi(z) - \phi(y))
\right)
\right)
}
\le
\Exv{
\exp\left(
\sum_{\phi \in A[i]}
\max\left(
\left( \frac{s_{\phi} L}{\lambda M_{\phi}} - 1 \right) (\phi(z) - \phi(x))
,
\left( \frac{s_{\phi} L}{\lambda M_{\phi}} - 1 \right) (\phi(z) - \phi(y))
\right)
\right)
}
=
\Exv{
\exp\left(
\sum_{\phi \in A[i]}
\left( \frac{s_{\phi} L}{\lambda M_{\phi}} - 1 \right)
\max\left(
\phi(z) - \phi(x)
,
\phi(z) - \phi(y)
\right)
\right)
}
\le
\Exv{
\exp\left(
\sum_{\phi \in A[i]}
\left( \frac{s_{\phi} L}{\lambda M_{\phi}} - 1 \right) M_{\phi}
\right)
},
\end{dmath*}
where this last line follows from the fact that $\phi(z) - \phi(x) \le M_{\phi}$.
So, by independence of the $s_{\phi}$,
\begin{dmath*}
\Exv{
\max\left(
\exp\left( \epsilon_{z(i)} - \zeta(z) + \zeta(x) - \epsilon_{x(i)} \right)
,
\exp\left( \epsilon_{z(i)} - \zeta(z) + \zeta(y) - \epsilon_{y(i)} \right)
\right)
}
\le
\prod_{\phi \in A[i]}
\Exv{
\exp\left(
\left( \frac{s_{\phi} L}{\lambda M_{\phi}} - 1 \right) M_{\phi}
\right)
}
=
\prod_{\phi \in A[i]}
\exp(-M_\phi)
\Exv{
\exp\left(
\frac{s_{\phi} L}{\lambda}
\right)
}.
\end{dmath*}
This last expression is just the moment generating function of the Poisson random variable, evaluated at $t = \frac{L}{\lambda}$.
Since $s_{\phi}$ has parameter $\frac{\lambda M_{\phi}}{L}$, it follows from known properties of the Poisson distribution that
\[
\Exv{
\exp\left(
\frac{s_{\phi} L}{\lambda}
\right)
}
=
\exp\left(
\frac{\lambda M_{\phi}}{L}
\left(
\exp\left( \frac{L}{\lambda} \right) - 1
\right)
\right).
\]
Multiplying both sides by $\exp(-M_{\phi})$,
\[
\exp(-M_\phi)
\Exv{
\exp\left(
\frac{s_{\phi} L}{\lambda}
\right)
}
=
\exp\left(
M_{\phi}
\left(
\frac{\lambda}{L}
\left(
\exp\left( \frac{L}{\lambda} \right) - 1
\right)
-
1
\right)
\right).
\]
And substituing this back into our previous expression,
\begin{dmath*}
\Exv{
\max\left(
\exp\left( \epsilon_{z(i)} - \zeta(z) + \zeta(x) - \epsilon_{x(i)} \right)
,
\exp\left( \epsilon_{z(i)} - \zeta(z) + \zeta(y) - \epsilon_{y(i)} \right)
\right)
}
\le
\prod_{\phi \in A[i]}
\exp\left(
M_{\phi}
\left(
\frac{\lambda}{L}
\left(
\exp\left( \frac{L}{\lambda} \right) - 1
\right)
-
1
\right)
\right)
=
\exp\left(
\sum_{\phi \in A[i]}
M_{\phi}
\left(
\frac{\lambda}{L}
\left(
\exp\left( \frac{L}{\lambda} \right) - 1
\right)
-
1
\right)
\right)
=
\exp\left(
L
\left(
\frac{\lambda}{L}
\left(
\exp\left( \frac{L}{\lambda} \right) - 1
\right)
-
1
\right)
\right)
=
\exp\left(
\lambda
\left(
\exp\left( \frac{L}{\lambda} \right) - 1
\right)
-
L
\right).
\end{dmath*}
As long as $z \le 1$, we know that $\exp(z) - 1 \le z + z^2$.
So, as long as $L \le \lambda$, it follows that
\begin{dmath*}
\Exv{
\max\left(
\exp\left( \epsilon_{z(i)} - \zeta(z) + \zeta(x) - \epsilon_{x(i)} \right)
,
\exp\left( \epsilon_{z(i)} - \zeta(z) + \zeta(y) - \epsilon_{y(i)} \right)
\right)
}
\le
\exp\left(
\lambda
\left(
\frac{L}{\lambda} + \left( \frac{L}{\lambda} \right)^2
\right)
-
L
\right)
=
\exp\left(
\frac{L^2}{\lambda}
\right).
\end{dmath*}
Substituting this back into our previous expression above, we have that
\begin{dmath*}
\frac{
\pi(x) \bar T(x,y)
}{
\pi(x) T(x, y)
}
\ge
\frac{
\sum_{w = 1}^D \exp\left(\zeta(y_{i \rightarrow w}) \right)
}{
\sum_{w=1}^D
\exp\left(\zeta(y_{i \rightarrow w}) \right)
\Exv{
\max\left(
\exp\left( \epsilon_w - \zeta(y_{i \rightarrow w}) + \zeta(x) - \epsilon_{x(i)} \right)
,
\exp\left( \epsilon_w - \zeta(y_{i \rightarrow w}) + \zeta(y) - \epsilon_{y(i)} \right)
\right)
}
}
\ge
\frac{
\sum_{w = 1}^D \exp\left(\zeta(y_{i \rightarrow w}) \right)
}{
\sum_{w=1}^D
\exp\left(\zeta(y_{i \rightarrow w}) \right)
\cdot
\exp\left(
\frac{L^2}{\lambda}
\right)
}
=
\exp\left(
-\frac{L^2}{\lambda}
\right).
\end{dmath*}
Thus, it follows that
\[
\frac{
\pi(x) \bar T(x,y)
}{
\pi(x) T(x, y)
}
\ge
\exp\left(
-\frac{L^2}{\lambda}
\right).
\]
Now, we finally use the Dirichlet forms.
By definition,
\begin{dmath*}
\bar \gamma
=
\min_f
\frac{
\bar{\mathcal{E}}(f)
}{
\Var[\pi]{f}
}
=
\min_f
\frac{
1
}{
\Var[\pi]{f}
}
\sum_{x,y} \pi(x) \bar T(x, y)
\ge
\exp\left(
-\frac{L^2}{\lambda}
\right)
\cdot
\min_f
\frac{
1
}{
\Var[\pi]{f}
}
\sum_{x,y} \pi(x) T(x, y)
=
\exp\left(
-\frac{L^2}{\lambda}
\right)
\cdot
\min_f
\frac{
\mathcal{E}(f)
}{
\Var[\pi]{f}
}
=
\exp\left(
-\frac{L^2}{\lambda}
\right)
\cdot
\gamma.
\end{dmath*}
This is what we wanted to prove.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thmDoubleMINTreversible}]
The probability of transitioning from $(x, \xi_x)$ to $(y, \xi_y)$ for two distinct states $x$ and $y$ which differ only in variable $i$ will be the probability that: we decide to re-sample variable $i$, we choose $y$ as our proposal, we sample $\xi_y$ as the energy for state $y$, and we accept the proposed change.
We can write this transition probability as
\begin{dmath*}
T((x, \xi_x), (y, \xi_y))
=
\frac{1}{n}
\Exv{
\psi(y)
\cdot
\mu_y(\xi_y)
\cdot
\min(a, 1)
},
\end{dmath*}
where here the expected value is taken over the random minibatch coefficients $s_\phi$.
We can expand this out to
\begin{dmath*}
T((x, \xi_x), (y, \xi_y))
=
\frac{1}{n}
\Exv{
\frac{\exp(\epsilon_{y(i)})}{\sum_{u=1}^D \exp(\epsilon_u)}
\cdot
\mu_y(\xi_y)
\cdot
\min\left(
\frac{
\exp(\xi_y)
}{
\exp(\xi_x)
}
\cdot
\frac{
\exp\left(
\epsilon_{x(i)}
\right)
}{
\exp\left(
\epsilon_{y(i)}
\right)
}
, 1 \right)
}.
\end{dmath*}
If we define stationary distribution
\[
\pi(x, \xi)
=
\frac{1}{Z}
\mu_x(\xi) \cdot \exp(\xi),
\]
then
\begin{dmath*}
\pi(x, \xi_x)
\cdot
T((x, \xi_x), (y, \xi_y))
=
\frac{1}{n Z}
\Exv{
\mu_x(\xi_x) \cdot \exp(\xi_x)
\cdot
\frac{\exp(\epsilon_{y(i)})}{\sum_{u=1}^D \exp(\epsilon_u)}
\cdot
\mu_y(\xi_y)
\cdot
\min\left(
\frac{
\exp(\xi_y)
}{
\exp(\xi_x)
}
\cdot
\frac{
\exp\left(
\epsilon_{x(i)}
\right)
}{
\exp\left(
\epsilon_{y(i)}
\right)
}
, 1 \right)
}
=
\frac{1}{n Z}
\Exv{
\frac{
\mu_x(\xi_x)
\cdot
\mu_y(\xi_y)
}{\sum_{u=1}^D \exp(\epsilon_u)}
\cdot
\min\left(
\exp(\xi_y)
\cdot
\exp(\epsilon_{x(i)})
,
\exp(\xi_x)
\cdot
\exp(\epsilon_{y(i)})
\right)
}.
\end{dmath*}
This expression is clearly symmetric in $(x, \xi_x)$ and $(y, \xi_y)$, so it follows that the chain is reversible with stationary distribution $\pi$ as defined above.
This is what we wanted to prove.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thmDoubleMINTgap}]
As with the analysis of MIN-Gibbs, we will prove this result using the technique of Dirichlet forms.
From the result of Theorem~\ref{thmDoubleMINTreversible}, we know that for some $\bar Z$, the stationary distribution of DoubleMIN-Gibbs is
\[
\bar \pi(x, \xi) = \frac{1}{\bar Z} \mu_x(\xi) \cdot \exp(\xi).
\]
We also know from that same theorem that the transition probability matrix can be written as
\begin{dmath*}
\bar \pi(x, \xi_x)
\cdot
\bar T((x, \xi_x), (y, \xi_y))
=
\frac{1}{n \bar Z}
\Exv{
\frac{
\mu_x(\xi_x)
\cdot
\mu_y(\xi_y)
}{\sum_{u=1}^D \exp(\epsilon_u)}
\cdot
\min\left(
\exp(\xi_y)
\cdot
\exp(\epsilon_{x(i)})
,
\exp(\xi_x)
\cdot
\exp(\epsilon_{y(i)})
\right)
}.
\end{dmath*}
It follows from the same analysis as in the proof of Theorem~\ref{thmMINTreversible} that the Dirichlet form of DoubleMIN-Gibbs is
\begin{dmath*}
\bar{\mathcal{E}}(f)
=
\frac{1}{2 n \bar Z}
\sum_{(x, y) \in Q}
\Exv{
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
\cdot
\frac{
\min\left(
\exp(\xi_y)
\cdot
\exp(\epsilon_{x(i)})
,
\exp(\xi_x)
\cdot
\exp(\epsilon_{y(i)})
\right)
}{
\sum_{u=1}^D \exp(\epsilon_u)
}
},
\end{dmath*}
where $Q \subset \Omega \times \Omega$ as before is the set of pairs of stats which differ only in a single variable.
Here the expected value is also taken over random variables $\xi_x$ and $\xi_y$, which we suppose are sampled independently from $\mu_x$ and $\mu_y$, respectively.
By a similar argument, we can also determine that the MGPMH chain will have Dirichlet form
\begin{dmath*}
\bar{\mathcal{E}}(g)
=
\frac{1}{2 n \bar Z}
\sum_{(x, y) \in Q}
\Exv{
\left(g(x) - g(y) \right)^2
\cdot
\frac{
\min\left(
\exp(\zeta(y))
\cdot
\exp(\epsilon_{x(i)})
,
\exp(\zeta(x))
\cdot
\exp(\epsilon_{y(i)})
\right)
}{
\sum_{u=1}^D \exp(\epsilon_u)
}
}.
\end{dmath*}
Now, we know by the condition of the theorem that
\[
\Abs{\xi_x - \zeta(x)} \le \delta.
\]
Therefore, we can bound the Dirichlet form of DoubleMIN-Gibbs from below with
\begin{dmath*}
\bar{\mathcal{E}}(f)
\ge
\frac{1}{2 n \bar Z}
\sum_{(x, y) \in Q}
\Exv{
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
\cdot
\frac{
\min\left(
\exp(\zeta(y) - \delta)
\cdot
\exp(\epsilon_{x(i)})
,
\exp(\zeta(x) - \delta)
\cdot
\exp(\epsilon_{y(i)})
\right)
}{
\sum_{u=1}^D \exp(\epsilon_u)
}
}
=
\frac{\exp(-\delta)}{2 n \bar Z}
\sum_{(x, y) \in Q}
\Exv{
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
\cdot
\frac{
\min\left(
\exp(\zeta(y))
\cdot
\exp(\epsilon_{x(i)})
,
\exp(\zeta(x))
\cdot
\exp(\epsilon_{y(i)})
\right)
}{
\sum_{u=1}^D \exp(\epsilon_u)
}
}
=
\frac{\exp(-\delta)}{2 n \bar Z}
\sum_{(x, y) \in Q}
\Exv{
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
}
\Exv{
\frac{
\min\left(
\exp(\zeta(y))
\cdot
\exp(\epsilon_{x(i)})
,
\exp(\zeta(x))
\cdot
\exp(\epsilon_{y(i)})
\right)
}{
\sum_{u=1}^D \exp(\epsilon_u)
}
},
\end{dmath*}
where we can separate out the expected values like this because $\xi_x$ and $\xi_y$ are sampled independently from the other random variables $s_\phi$.
On the other hand, the variance form associated with the DoubleMIN-Gibbs chain is
\begin{dmath*}
\Var[\bar \pi]{f}
=
\frac{1}{2 \bar Z^2} \sum_{x \in \Omega} \sum_{y \in \Omega}
\Exv{
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
\cdot
\exp(\xi_x) \cdot \exp(\xi_y)
}
\le
\frac{1}{2 \bar Z^2} \sum_{x \in \Omega} \sum_{y \in \Omega}
\Exv{
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
\cdot
\exp(\zeta(x) + \delta) \cdot \exp(\zeta(y) + \delta)
}
=
\frac{\exp(2 \delta)}{2 \bar Z^2} \sum_{x \in \Omega} \sum_{y \in \Omega}
\exp(\zeta(x)) \cdot \exp(\zeta(y))
\Exv{
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
}.
\end{dmath*}
From here, we can write
\begin{dmath*}
\bar \gamma
=
\min_f
\frac{
\bar{\mathcal{E}}(f)
}{
\Var[\bar \pi]{f}
}
\ge
\min_f
\frac{
\frac{\exp(-\delta)}{2 n \bar Z}
\sum_{(x, y) \in Q}
\Exv{
\frac{
\min\left(
\exp(\zeta(y))
\cdot
\exp(\epsilon_{x(i)})
,
\exp(\zeta(x))
\cdot
\exp(\epsilon_{y(i)})
\right)
}{
\sum_{u=1}^D \exp(\epsilon_u)
}
}
\Exv{
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
}
}{
\frac{\exp(2 \delta)}{2 \bar Z^2} \sum_{x \in \Omega} \sum_{y \in \Omega}
\exp(\zeta(x)) \cdot \exp(\zeta(y))
\Exv{
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
}
}
=
\frac{\exp(-3 \delta) \bar Z}{n}
\min_f
\frac{
\sum_{(x, y) \in Q}
\Exv{
\frac{
\min\left(
\exp(\zeta(y))
\cdot
\exp(\epsilon_{x(i)})
,
\exp(\zeta(x))
\cdot
\exp(\epsilon_{y(i)})
\right)
}{
\sum_{u=1}^D \exp(\epsilon_u)
}
}
\Exv{
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
}
}{
\sum_{x \in \Omega} \sum_{y \in \Omega}
\exp(\zeta(x)) \cdot \exp(\zeta(y))
\Exv{
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
}
}.
\end{dmath*}
As before, using the fact that for positive numbers $a_1, a_2, \ldots, a_N$ and $b_1, b_2, \ldots, b_N$,
\[
\frac{\sum_{i=1}^n a_i}{\sum_{i=1}^n b_i} \ge \min_i \frac{a_i}{b_i}.
\]
we can bound this from below for some fixed $\xi_x$ and $\xi_y$ (which may still be a function of $f$) with
\begin{dmath*}
\bar \gamma
\ge
\frac{\exp(-3 \delta) \bar Z}{n}
\min_f
\frac{
\sum_{(x, y) \in Q}
\Exv{
\frac{
\min\left(
\exp(\zeta(y))
\cdot
\exp(\epsilon_{x(i)})
,
\exp(\zeta(x))
\cdot
\exp(\epsilon_{y(i)})
\right)
}{
\sum_{u=1}^D \exp(\epsilon_u)
}
}
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
}{
\sum_{x \in \Omega} \sum_{y \in \Omega}
\exp(\zeta(x)) \cdot \exp(\zeta(y))
\left(f(x, \xi_x) - f(y, \xi_y) \right)^2
}
\ge
\frac{\exp(-3 \delta) \bar Z}{n}
\min_g
\frac{
\sum_{(x, y) \in Q}
\Exv{
\frac{
\min\left(
\exp(\zeta(y))
\cdot
\exp(\epsilon_{x(i)})
,
\exp(\zeta(x))
\cdot
\exp(\epsilon_{y(i)})
\right)
}{
\sum_{u=1}^D \exp(\epsilon_u)
}
}
\left(g(x) - g(y) \right)^2
}{
\sum_{x \in \Omega} \sum_{y \in \Omega}
\exp(\zeta(x)) \cdot \exp(\zeta(y))
\left(g(x) - g(y) \right)^2
},
\end{dmath*}
where this last inequality holds because we can always set
\[
g(x) = f(x, \xi_x).
\]
Finally, we can rewrite this as
\begin{dmath*}
\bar \gamma
\ge
\frac{\exp(-3 \delta) \bar Z}{Z}
\min_g
\frac{
\frac{1}{2 n Z}
\sum_{(x, y) \in Q}
\Exv{
\frac{
\min\left(
\exp(\zeta(y))
\cdot
\exp(\epsilon_{x(i)})
,
\exp(\zeta(x))
\cdot
\exp(\epsilon_{y(i)})
\right)
}{
\sum_{u=1}^D \exp(\epsilon_u)
}
}
\left(g(x) - g(y) \right)^2
}{
\frac{1}{2 Z^2}
\sum_{x \in \Omega} \sum_{y \in \Omega}
\exp(\zeta(x)) \cdot \exp(\zeta(y))
\left(g(x) - g(y) \right)^2
}
=
\frac{\exp(-3 \delta) \bar Z}{Z}
\min_g
\frac{
\mathcal{E}(g)
}{
\Var[\pi]{g}
}
=
\frac{\exp(-3 \delta) \bar Z}{Z}
\gamma.
\end{dmath*}
All that remains is to bound the ratio of the $Z$ and $\bar Z$.
But this ratio is the same as it is in the proof of Theorem~\ref{thmMINTgap}, so we can conclude that
\[
\frac{\bar Z}{Z}
\ge
\exp(-\delta).
\]
So,
\[
\bar \gamma \ge \exp(-4 \delta) \gamma.
\]
This is what we wanted to show.
\end{proof}
\section{Experiments}
In this section, we describe the methodology of our experiments in more detail.
Our experiments are run on a synthetic Ising model with energy
\[
\zeta_{\text{Ising}}(x) = \sum_{i = 1}^N \sum_{j = 1}^N \beta \cdot A_{ij} \cdot (x(i) x(j) + 1),
\]
and a synthetic Potts model with energy
\[
\zeta_{\text{Potts}} = \sum_{i = 1}^N \sum_{j = 1}^N \beta \cdot A_{ij} \cdot \delta(x(i), x(j)).
\]
As is usually the case with these models, we laid out our variables on a grid, a $20 \times 20$ grid to be precise.
This resulted in $n = 400$ variables for both models.
Our goal here was to construct a dense synthetic model with a nontrivial interaction matrix (i.e. non-constant $A_{ij}$).
To do this, we assigned $A_{ij}$ (for $i \ne j$) according to the Gaussian kernel
\[
A_{ij} = \exp\left(-\gamma d_{ij}^2 \right)
\]
where $d_{ij}$ is the distance between variables $i$ and $j$ in the $20 \times 20$ grid.
Equivalently, if $x_i \in \{1, \ldots, 20\} \times \{1, \ldots, 20\}$ is the position of variable $i$ in the grid, then
\[
A_{ij} = \exp\left(-\gamma \|\mathbf x_i - \mathbf x_j\|^2 \right).
\]
This form may be more easily recognizable as a Gaussian RBF kernel.
We chose to set $\gamma = 1.5$ for both the Ising and Potts models.
For all experiments, we ran $1000000 = 10^6$ iterations of sampling.
For the Ising model, we set the inverse temperature $\beta = 1.0$, and for the Potts model, we set $\beta = 4.6$.
In both cases, we hand-tuned $\beta$ so that it was small enough that the marginal error of vanilla Gibbs sampling would clearly be observed to converge within $10^6$ iterations, but large enough that its convergence trajectory would be clearly distinct from the trivial convergence trajectory when $\beta = 0$.
In other words, we set $\beta$ so that, for plain Gibbs sampling, we would both observe non-trivial behavior and have a chain that converged fast enough to efficiently simulate.
We then used this value of $\beta$ to evaluate our algorithms.
\section{Introduction}
\label{secIntro}
Gibbs sampling is a Markov chain Monte Carlo method that is one of the most widespread techniques used with graphical models~\cite{koller2009probabilistic}.
Gibbs sampling is an iterative method that repeatedly resamples a variable in the model from its conditional distribution, a process that is guaranteed to converge asymptotically to the desired distribution.
Since these updates are typically simple and fast to run, Gibbs sampling can be applied to a variety of problems, and has been used for inference on large-scale graphical models in many systems~\cite{newman2007distributed,lunn2009bugs,
mccallum2009factorie,smola2010architecture,NIPS2012_4832,zhang2014dimmwitted}.
Unfortunately, for large graphical models with many factors, the computational cost of running an iteration of Gibbs sampling can become prohibitive.
Even though Gibbs sampling is a \emph{graph-local} algorithm, in the sense that each update only needs to reference data associated with a local neighborhood of the factor graph, as graphs become large and highly connected, even these local neighborhoods can become huge.
To make matters worse, complicated models tend to have categorical variables with large domains: for example, the variables can represent US States or ZIP codes, where there are dozens to thousands of possible choices.
The cost of computing a single iteration of Gibbs sampling is proportional to the \emph{product} of the size of the local neighborhood and the size of the domain of the categorical random variables.
\cmd{More explicitly, if the \emph{maximum degree} of a variable in the factor graph is $\Delta$, and each variable can take on $D$ possible values, then a single Gibbs sampling update takes $O(D \Delta)$ time to compute in the worst case.}
\begin{table*}[t]
\caption{Single-iteration computational complexity bounds of our minibatching algorithms compared with ordinary Gibbs sampling, for parameter settings which can slow down convergence, as measured by the spectral gap, by no more than a $O(1)$ factor.}
\label{tabSummary}
\vskip 0.15in
\begin{center}
\begin{tabular}{lll}
\toprule
\textbf{Algorithm} & \textbf{Compute Cost/Iteration} & \textbf{Notes} \\
\midrule
Gibbs sampling & $O(D \Delta)$ & \\
\midrule
MIN-Gibbs: Minibatch Gibbs & $O(D \Psi^2)$ & with high probability \\
MGPMH: Minibatch-Gibbs-Proposal Metropolis Hastings & $O(D L^2 + \Delta)$ & \\
DoubleMIN-Gibbs: Doubly Minibatch Gibbs & $O(D L^2 + \Psi^2)$ & with high probability \\
\bottomrule
\end{tabular}
\end{center}
\vskip -0.1in
\end{table*}
This effect limits the scalability of Gibbs sampling.
What started as a fast, efficient system can become dramatically slower over the course of development as models become more complicated, new factors are added, and new values are included in the variables' domains.
To address this, practitioners have developed techniques such as \emph{pruning}~\cite{rekatsinas2017holoclean} which improve scalability by making the factor graphs simpler.
While this does make Gibbs sampling faster, it comes at the cost of introducing bias which reduces fidelity to the original model.
In this paper, we explore a more principled approach to making Gibbs sampling scalable.
Our approach is inspired by \emph{minibatching} in stochastic gradient descent, which has been used with great success to scale up machine learning training.
The main idea is that, when a Gibbs sampling update depends on a large local graph neighborhood, we can instead \emph{randomly subsample} the neighborhood.
Supposing the random sample is representative of the rest of the neighborhood, we can proceed to perform the update using just the random sample, rather than the (much larger) entire neighborhood.
In this paper, we study minibatching for Gibbs Sampling, and we make the following contributions:
\begin{itemize}
\itemsep2pt
\item We introduce techniques for \emph{minibatching} which can dramatically reduce the cost of running Gibbs sampling, while provably adding \emph{no bias} to the samples.
\item We prove bounds on the convergence rates of our minibatch Gibbs algorithms, as measured by the \emph{spectral gap} of the Markov chain. We give a recipe for how to set the algorithms' parameters so that the spectral gap of minibatch Gibbs can be bounded to be arbitrarily close to the spectral gap of the original Gibbs chain.
\item For a class of graphs with bounded energy, we show doing this results in an asymptotic computation cost of $O(D + \Delta)$, a substantial speedup from the cost of Gibbs sampling which is $O(D \Delta)$.
\end{itemize}
\subsection{Background and Definitions}
\label{secSetup}
First, to make our claims rigorous, we describe our setting by defining a factor graph and the various conditions and conventions we will be using in the paper.
A factor graph~\cite{koller2009probabilistic} with $n$ variables determines a probability distribution $\pi$ over a state space $\Omega$.
In general, a state $x \in \Omega$ is an assignment of a value to each of the $n$ variables, and each variable has its own domain over which it can take on values.
For simplicity, in this paper we will assume all variables take on values in the same domain $\{1, \ldots, D\}$ for some constant $D$, which would make $\Omega$ the set of functions $x: \{1,\ldots,n\} \rightarrow \{1,\ldots,D\}$.\footnote{\cmd{This means that the state space $\Omega$ is discrete, and we will be focusing on discrete-valued models in this paper. Continuous-valued models could be approximated with our algorithms by discretizing their domains to any desired level of accuracy.}}
The probability distribution $\pi$ is determined as the product of some set of factors $\Phi$, such that for any $x \in \Omega$,
\[
\textstyle
\pi(x) \propto \exp\left( \sum_{\phi \in \Phi} \phi(x) \right);
\]
this is sometimes called the \emph{Gibbs measure}. In this document, we use the notation $\rho(v) \propto \exp(\epsilon_v)$ to mean the unique distribution that is proportional to $\exp(\epsilon_v)$, that is,
\[
\rho(v) = \frac{\exp(\epsilon_v)}{\sum_{w = 1}^D \exp(\epsilon_w)}.
\]
Equivalently, we can define an \emph{energy function} $\zeta$, where
\[
\textstyle
\zeta(x) = \sum_{\phi \in \Phi} \phi(x),
\]
and let $\pi(x) \propto \exp(\zeta(x))$.
\cmd{(Note that this definition implicitly excludes models with hard constraints or zero-probability states.)}
This is called a factor graph because the factors $\phi$ and variables $i$ have a dependency relation in that each $\phi$ depends only on the values of some of the variables, but typically not all $n$.
Formally, a factor $\phi$ depends on a variable if changing the value of that variable could change the value of $\phi$---meaning, if there exists states $x$ and $y$ which differ only in variable $i$ and for which $\phi(x) \ne \phi(y)$.
Using this, we define the relation
\[
A = \left\{ (i, \phi) \middle| \text{factor } \phi \text{ depends on variable } i \right\}.
\]
Equivalently, this is a bipartite graph on the variables and factors; this edge relation, together with the factor function descriptions, defines a factor graph.
Using this, we formally describe Gibbs sampling, which we write in Algorithm~\ref{algGibbs}.
Note that we use the notation $x(i) \leftarrow u$ to denote variable $i$ within state $x$ being reassigned the value $u$, and following standard relation notation we let $A[i] = \{ \phi | (i, \phi) \in A \}$ be the set of factors that depend on variable $i$.
\begin{algorithm}[t]
\caption{Gibbs sampling}
\begin{algorithmic}
\label{algGibbs}
\STATE \textbf{given:} initial state $x \in \Omega$
\LOOP
\STATE \textbf{sample} variable index $i$ uniformly from $\{1,\ldots,n\}$
\FORALL{$u$ \textbf{in} $\{1, \ldots, D\}$}
\STATE $x(i) \leftarrow u$
\STATE $\epsilon_u \leftarrow \sum_{\gamma \in A[i]} \phi(x)$
\ENDFOR
\STATE construct distribution $\rho$ over $\{1, \ldots, D\}$ where
\[
\textstyle
\rho(v) \propto \exp(\epsilon_v)
\]
\STATE \textbf{sample} $v$ from $\rho$.
\STATE $x(i) \leftarrow v$
\STATE \textbf{output sample} $x$
\ENDLOOP
\end{algorithmic}
\end{algorithm}
Next, we describe several conditions on the factor functions that we will be using throughout this paper.
For all that follows, we will suppose that $\phi(x) \ge 0$ for all $\phi$ and for all $x$; this holds without \cmd{loss of generality (for models without hard constraints)} because adding a constant to a factor function $\phi$ does not change the resulting distribution.
\begin{definition}[Factor graph conditions]
\label{defnFGC}
The \emph{maximum energy} of a factor $\phi$ is the smallest $M_{\phi} \in \R$ such that,
\[
0 \le \phi(x) \le M_{\phi} \text{ for all $x \in \Omega$}.
\]
The \emph{total maximum energy} $\Psi$ of a graph is the sum of all the maximum energies
\[
\textstyle
\Psi = \sum_{\phi \in \Phi} M_{\phi}.
\]
The \emph{local maximum energy} $L$ of a graph is the largest sum of all the maximum energies of factors that depend on a single variable $i$,
\[
\textstyle
L = \max_{i \in \{1, \ldots, n\}} \sum_{\phi \in A[i]} M_{\phi}.
\]
The \emph{maximum degree} $\Delta$ of a factor graph is the largest number of factors that are connected to a variable,
\[
\textstyle
\Delta = \max_{i \in \{1, \ldots, n\}} \Abs{ A[i] }.
\]
\end{definition}
For large models with many low-energy factors, the total maximum energy can be much smaller than the number of factors, and the local maximum energy can be much smaller than the maximum degree.
We will introduce several minibatch Gibbs variants in this paper, and their relative computation costs, in terms of the quantities defined in Definition~\ref{defnFGC}, are summarized in Table~\ref{tabSummary}.
\subsection{Related Work}
\label{secRelatedWork}
Several recent papers have explored applying minibatching to speed up large-scale Markov chain Monte Carlo methods.
Our work is inspired by \citet{li2017mini}, which shows how minibatching can be applied to the Metropolis-Hastings algorithm~\cite{hastings1970monte}.
They show that their method, MINT-MCMC, can outperform existing techniques that use gradient information, such as stochastic gradient descent and stochastic gradient Langevin dynamics, on large-scale Bayesian neural network tasks.
This is one among several papers that have also showed how to use minibatching to speed up Metropolis-Hastings under various conditions \cite{korattikara2014austerity,bardenet2014towards,chen2016efficient}.
\cmd{More general related approaches include Firefly MC, which implements a minibatching-like computational pattern using auxiliary variables \cite{maclaurin2014firefly}, and block-Poisson estimation, which constructs unbiased log-likelihood estimates using Poisson variables~\cite{quiroz2018mcmc}.}
Our results are distinct from this line of work in that we are the first to study minibatch Gibbs sampling in depth\footnote{\citet{
johndrow2015approximations} does evaluate a version of a minibatched Gibbs sampler on a particular application, but does not study Gibbs in depth.}, and we are the first to address the effect of categorical random variables with large domains that can limit computational tractability.
Additionally, none of these papers provides any theoretical bounds on the convergence rates of their minibatched algorithm compared to the original MCMC chain.
\section{MIN-Gibbs}
\begin{algorithm}[t]
\caption{MIN-Gibbs: Minibatch Gibbs sampling}
\begin{algorithmic}
\label{algMINTGibbs}
\STATE \textbf{given:} minibatch estimator distributions $\mu_x$ for $x \in \Omega$
\STATE \textbf{given:} initial state $(x, \epsilon) \in \Omega \times \R$
\LOOP
\STATE \textbf{sample} variable index $i$ uniformly from $\{1,\ldots,n\}$
\STATE $\epsilon_{x(i)} \leftarrow \epsilon$
\FORALL{$u$ \textbf{in} $\{1, \ldots, D\} \setminus \{x(i)\}$}
\STATE $x(i) \leftarrow u$
\STATE \textbf{sample} energy $\epsilon_u$ from $\mu_x$
\ENDFOR
\STATE construct distribution $\rho$ over $\{1, \ldots, D\}$ where
\[
\rho(v) \propto \exp(\epsilon_v)
\]
\STATE \textbf{sample} $v$ from $\rho$.
\STATE $x(i) \leftarrow v$
\STATE $\epsilon \leftarrow \epsilon_v$
\STATE \textbf{output sample} $(x, \epsilon)$
\ENDLOOP
\end{algorithmic}
\end{algorithm}
In this section, we will present our first algorithm for applying minibatching to Gibbs sampling.
As there are many options when doing minibatching\footnote{These options include: what the batch size is, whether the batch size is fixed or random, whether to do weighted sampling, whether to sample with replacement, etc.} we will present our algorithm in terms of a general framework for Gibbs sampling in which we use an estimate for the energy rather than computing the energy exactly.
Specifically, we imagine the result of minibatching on some state $x \in \Omega$ is a random variable $\epsilon_x$ such that
\[
\epsilon_x \approx \sum_{\phi \in \Phi} \phi(x).
\]
For example, we could assign $\epsilon_x$ as
\[
\epsilon_x = \frac{|\Phi|}{B} \sum_{\phi \in S} \phi(x)
\]
where $S$ is a randomly chosen subset of $\Phi$ of size $B$.
More formally, we let $\mu_x$ be the distribution of $\epsilon_x$, and we assume our algorithm will have access to $\mu_x$ for every state $x$ and can draw samples from it.
For simplicity, we assume $\mu_x$ has finite support, which is true for any minibatch estimator.
We can now replace the exact sums in the Gibbs sampling algorithm with our approximations $\epsilon_x$.
If sampling from $\mu_x$ is easier than computing this sum, this will result in a faster algorithm than vanilla Gibbs sampling.
There is one further optimization: rather than re-estimating the energy of the current state $x$ at each iteration of our algorithm, instead we can \emph{cache} its value as it was computed in the previous step.
Doing this results in Algorithm~\ref{algMINTGibbs}, MIN-Gibbs.
We call our algorithm MIN-Gibbs because it was inspired by the Mini-batch Tempered MCMC (MINT-MCMC) algorithm presented by \citet{li2017mini} for applying minibatching to Metropolis-Hastings, another popular MCMC algorithm.
MINT-MCMC also used the idea of caching the energy to form an augmented system with state space $\Omega \times \R$.
Compared with their algorithm, our contributions are that: (1) we modify the technique to apply to Gibbs sampling; (2) we show that \emph{tempering} (which is sampling from a modified distribution at a higher temperature) and other sources of bias can be circumvented by using a bias-adjusted minibatching scheme; and (3) we prove bounds on the convergence rate of our technique.
Because it uses an energy estimate rather than the true energy, this algorithm is not equivalent to standard Gibbs sampling.
This raises the question: will it converge to the same distribution $\pi$, and if not, what will it converge to?
This question can be answered by using the property of \emph{reversibility} also known as \emph{detailed balance}.
\begin{definition}
A Markov chain with transition matrix $T$ is \emph{reversible} if for some distribution $\bar \pi$ and all states $x, y \in \Omega$,
\[
\bar \pi(x) T(x, y) = \bar \pi(y) T(y,x).
\]
\end{definition}
It is a well known result that if a chain $T$ is reversible, the distribution $\bar \pi$ will be a stationary distribution of $T$, that is, $\bar \pi T = \bar \pi$.
Thus we can use reversibility to determine the stationary distribution of a chain, and we do so for Algorithm~\ref{algMINTGibbs} in the following theorem.
\begin{theorem}
\label{thmMINTreversible}
The Markov chain described in Algorithm~\ref{algMINTGibbs} is reversible and has stationary distribution
\[
\bar \pi(x, \epsilon) \propto \mu_x(\epsilon) \cdot \exp(\epsilon)
\]
and its marginal stationary distribution in $x$ will be
\[
\bar \pi(x) \propto \Exv[\epsilon \sim \mu_x]{\exp(\epsilon)}.
\]
\end{theorem}
Interestingly, this theorem shows that if we choose our estimation scheme such that for all $x \in \Omega$,
\begin{equation}
\textstyle
\label{eqnUnbiasedCondition}
\Exv[\epsilon \sim \mu_x]{\exp(\epsilon)} = \exp( \zeta(x) ) = \prod_{\phi \in \Phi} \exp\left( \phi(x) \right)
\end{equation}
then MIN-Gibbs will actually be \emph{unbiased}: we will have $\pi(x) = \bar \pi(x)$.
Next, we will show how we can take advantage of this by constructing estimators that satisfy (\ref{eqnUnbiasedCondition}) using minibatching.
Suppose without loss of generality that each $\phi$ is non-negative, $\phi(x) \ge 0$.
Let $\lambda$ be a desired average minibatch size, and for each factor $\phi$, let $s_{\phi}$ be an independent Poisson-distributed random variable with mean $\lambda M_{\phi} / \Psi$.
Let $S \subset \Phi$ denote the set of factors $\phi$ for which $s_{\phi} > 0$.
Then for any state $x$, define the estimator
\begin{equation}
\label{eqnUnbiasedEst}
\epsilon_x = \sum_{\phi \in S} s_{\phi} \log\left( 1 + \frac{\Psi}{\lambda M_{\phi}} \phi(x) \right).
\end{equation}
\begin{lemma}
\label{lemmaEstimatorUnbiased}
The estimator $\epsilon_x$ defined in (\ref{eqnUnbiasedEst}) satisfies the unbiasedness condition in (\ref{eqnUnbiasedCondition}).
\end{lemma}
\begin{proof}
The expected value of the exponential of the estimator defined in (\ref{eqnUnbiasedEst}) is
\begin{align*}
\textstyle
\Exv{\exp(\epsilon_x)}
=
\Exv{\exp\left( \sum_{\phi \in S} s_{\phi} \log\left( 1 + \frac{\Psi}{\lambda M_{\phi}} \phi(x) \right) \right)}.
\end{align*}
Since the $s_{\phi}$ are independent, this becomes
\begin{align*}
\textstyle
\Exv{\exp(\epsilon_x)}
=
\prod_{\phi \in \Phi} \Exv{\exp\left( s_{\phi} \log\left( 1 + \frac{\Psi}{\lambda M_{\phi}} \phi(x) \right) \right)}.
\end{align*}
Each of these constituent expected values is an evaluation of the moment generating function of a Poisson random variable. Applying the known expression for this MGF and simplifying gives us the expression in (\ref{eqnUnbiasedCondition}), which is what we wanted to prove.
\end{proof}
From the result of this lemma, we can see that using this \emph{bias-adjusted} estimator with MIN-Gibbs will result in an unbiased chain with stationary distribution $\pi$.
However, the chain being unbiased does not, by itself, mean that this method will be more effective than standard Gibbs.
For that to be true, it must also be the case that approximating the energy did not affect the convergence rate of the algorithm---at least not too much.
The convergence times of Gibbs samplers can vary dramatically, from time linear in the number of variables to exponential time.
As a result, it is plausible that approximating the energy as we do in Algorithm~\ref{algMINTGibbs} could switch us over from a fast-converging chain to a slow-converging one, and as a result even though the individual iterations would be computationally faster, the overall computation would be slow---or worse, would silently give incorrect answers when the chain fails to converge.
Recently, other methods that have been used to speed up Gibbs sampling, such as running asynchronously \cite{desa2016hoggibbs} and changing the order in which the variables are sampled \cite{he2016scan}, have been shown in some situations to result in algorithms that converge significantly slower than plain Gibbs sampling.
Because of this, if we want to use algorithms like MIN-Gibbs with confidence, we need to show that this will not happen for minibatching.
To show that minibatching does not have a disastrous effect on convergence, we need to bound the convergence rate of our algorithms.
To measure the convergence rate, we use a metric called the \emph{spectral gap}~\cite{levin2009markov}.
\begin{definition}
Let $T$ be the transition matrix of a reversible Markov chain.
Since it is reversible, its eigenvalues must all be real, and they can be ordered as
\[
1 = \lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_{\Abs{\Omega}}.
\]
The spectral gap $\gamma$ is defined as $\gamma = \lambda_1 - \lambda_2$.
\end{definition}
The spectral gap measures the convergence rate of a Markov chain in terms of the $\ell_2$-distance, in that the larger the spectral gap, the faster the chain converges.
The spectral gap is related to several other metrics of convergence for Markov chains.
For example, it is a standard result that for a lazy Markov chain (a Markov chain is called lazy if at each iteration, it stays in its current state with probability at least $1/2$) the \emph{mixing time} of the chain, which measures the number of steps required to converge to within total-variation distance $\epsilon$ of the stationary distribution $\pi$, is bounded by
\[
t_{\text{mix}}(\epsilon) \le \frac{1}{\gamma} \log\left( \frac{1}{\epsilon \cdot \min_{x \in \Omega} \pi(x)} \right).
\]
In order to determine the effect of using minibatching on convergence, we would like to bound the spectral gap of MIN-Gibbs in terms of the spectral gap of the original Gibbs sampling chain.
In the following theorem, we show that if the energy estimator $\epsilon_u$ is always sufficiently close to the true energy, then we can bound the spectral gap.
\begin{theorem}
\label{thmMINTgap}
Let $\bar \gamma$ be the spectral gap of MIN-Gibbs running with an energy estimator $\mu_x$ that has finite support and that satisfies, for some constant $\delta > 0$ and every $x \in \Omega$,
\[
\Prob[\epsilon_x \sim \mu_x]{\Abs{ \epsilon_x - \zeta(x) } \le \delta} = 1.
\]
Let $\gamma$ be the spectral gap of a vanilla Gibbs sampling chain running using the exact energy.
Then,
\[
\bar \gamma
\ge
\exp(-6 \delta) \cdot \gamma.
\]
\end{theorem}
That is, the convergence is slowed down by at most a constant factor of $\exp(-5 \delta)$---which is independent of the size of the problem.
This theorem guarantees that if we can restrict our estimates to being within a distance of $O(1)$ of the exact energy, then the convergence rate will not be slowed down by more than an $O(1)$ constant factor.
Unfortunately, the estimator presented in (\ref{eqnUnbiasedEst}) is not going to be always bounded by a constant distance from the exact energy. Still, \cmd{since it is the sum of independent terms with bounded variance}, we \emph{can} bound it with high probability.
\begin{lemma}
\label{lemmaUnbiasedConcentration}
For any constants $0 < \delta$ and $0 < a < 1$, if we assign an expected batch size
\[
\lambda
\ge
\max\left(
\frac{
8 \Psi^2
}{
\delta^2
}
\log\left( \frac{2}{a} \right)
,
\frac{2 \Psi^2}{\delta}
\right),
\]
then the estimator in $(\ref{eqnUnbiasedEst})$ satisfies
$\Prob{ \Abs{\epsilon_x - \zeta(x) } \ge \delta} \le a$.
\end{lemma}
This lemma lets us construct minibatch estimators that remain arbitrarily close to the true energy with arbitrarily high probability.
Furthermore, we can do this with a minibatch size independent of the number of variables or factors, depending instead on the total energy.
This means that if we have a very large number of low-energy factors, we can get significant speedup from MIN-Gibbs with high-probability theoretical guarantees on the convergence rate.
In particular, since the computational cost of running a single epoch of MIN-Gibbs is $D$ times the minibatch size, and this lemma suggests we need to set $\lambda = \Omega(\Psi^2)$, it follows that the total computational cost of MIN-Gibbs will be $O(\Psi^2 D)$.
\paragraph{Validation of MIN-Gibbs.} Having characterized the effects of minibatching on Gibbs sampling, we present a synthetic scenario where Algorithm~\ref{algMINTGibbs} can be applied.
The Ising model \cite{ising1925beitrag} is a probabilistic model over an $N \times N$ lattice, with domain $x(i) \in \{-1,1\}$.
The Ising model has a physical interpretation in which each $x(i)$ represent the magnetic \emph{spin} at each site.
The energy of a configuration is given by:
\[
\textstyle
\zeta_{\text{Ising}}(x) = \sum_{i = 1}^N \sum_{j = 1}^N \beta \cdot A_{ij} \cdot (x(i) x(j) + 1),
\]
where $A_{ij}$ is called the \emph{interaction} between variable $i$ and $j$, and $\beta$ is the \emph{inverse temperature}.\footnote{In some settings, the Ising model is used with an additional bias term (which physically represents a static magnetic field), but here for simplicity we do not include this term.}
We chose to use the Ising model to validate MIN-Gibbs because it is a simple model that nevertheless has non-trivial statistical behavior.
We chose an Ising model in which each site is fully connected (i.e. $\Delta = N^2 - 1$),
and the strength of interaction $A_{ij}$ between any two sites is determined based on their distance by a Gaussian kernel.
We simulated Algorithm~\ref{algMINTGibbs} on a graph with $n = N^2 = 400$ and inverse temperature $\beta = 1$. \cmd{This $\beta$ parameter was hand-tuned such that the Gibbs sampler seemed to mix in about the number of iterations we wanted to run. To have a fair comparison, the same setup was then used to evaluate MIN-Gibbs.
For this model, $L = 2.21$ and $\Psi = 416.1$.}\footnote{Note that since we chose $\beta$ large enough that $\Psi^2 > \Delta$ for this model, we do not expect MIN-Gibbs to be faster than Gibbs sampling for this particular synthetic example.}
We started the algorithm with a unmixed configuration where each site takes on the same state ($x(i) = 1$ for all $i$).
As the algorithm ran, we used the output samples to compute a running average of the marginal distributions of each variable.
By symmetry (since negating a state will not change its probability), the marginal distribution of each variable in the stationary distribution $\pi$ should be uniform, so we can use the distance between the estimated marginals and the uniform distribution as a proxy to evaluate the convergence of the Markov chain.
Figure~\ref{fig:MINT} shows the average $\ell_2$-distance error in the estimated marginals compared with the fully-mixed state.
Notice that as the batch size increases, MIN-Gibbs approaches vanilla Gibbs sampling.
\begin{figure}[!tb]
\centering
\includegraphics[width=\myfcwidth]{mingibbs.pdf}
\caption{Convergence of marginal estimates for MIN-Gibbs compared with vanilla Gibbs sampling.}
\vspace{-2ex}
\label{fig:MINT}
\end{figure}
\textbf{Using local structure.} Although MIN-Gibbs has a computational cost that is independent of the size of the graph, it still depends on the total maximum energy $\Psi$.
That is, unlike vanilla Gibbs sampling, which has a computational cost that depends only on the number of factors \emph{local} to the variable that is being sampled, MIN-Gibbs depends on \emph{global} properties of the graph.
This is because the estimators used by MIN-Gibbs are approximating the whole energy sum $\zeta(x)$, rather than approximating the sum over just those factors which depend on the variable that is being resampled.
How can we modify MIN-Gibbs to take advantage of this local structure?
One straightforward way to do it is to allow the estimators $\epsilon_x$ to be \emph{dependent}, rather than independent.
That is, instead of choosing a unique minibatch every time we estimate the energy, we choose one fixed minibatch for each iteration.
Once our minibatch is fixed, the conditional distribution computation will exhibit the same cancellation of irrelevant factors that vanilla Gibbs has, and we will be able to compute the transition probabilities using only local information.
One side-effect of this is that we can no longer use the energy-caching technique that MIN-Gibbs uses, as we will no longer ever be estimating the total energy, but rather only the local component of the energy (the part that is dependent on the variable we are re-sampling).
This would result in something like Algorithm~\ref{algLocalMinibatchGibbs}.
(We could also consider variants of this algorithm that use different minibatching schemes, such as (\ref{eqnUnbiasedEst}), but here for simplicity we present the version that uses standard minibatching.)
Note that Algorithm~\ref{algLocalMinibatchGibbs} is nearly identical to vanilla Gibbs sampling, except that it uses a single minibatch to estimate all the energies in each iteration.
\begin{algorithm}[t]
\caption{Local Minibatch Gibbs}
\begin{algorithmic}
\label{algLocalMinibatchGibbs}
\STATE \textbf{given:} initial state $x \in \Omega \times \R$, minibatch size $B$
\LOOP
\STATE \textbf{sample} variable index $i$ uniformly from $\{1,\ldots,n\}$
\STATE \textbf{sample} minibatch $S \subset A[i]$ uniformly s.t. $\Abs{S} = B$
\FORALL{$u$ \textbf{in} $\{1, \ldots, D\}$}
\STATE $x(i) \leftarrow u$
\STATE $\epsilon_u \leftarrow \frac{\Abs{A[i]}}{\Abs{S}} \sum_{\gamma \in S} \phi(x)$
\ENDFOR
\STATE construct distribution $\rho$ over $\{1, \ldots, D\}$ where
\[
\rho(v) \propto \exp(\epsilon_v)
\]
\STATE \textbf{sample} $v$ from $\rho$.
\STATE $x(i) \leftarrow v$
\STATE \textbf{output sample} $x$
\ENDLOOP
\end{algorithmic}
\end{algorithm}
Algorithm~\ref{algLocalMinibatchGibbs} will run iterations in time $O(B D)$, which can be substantially faster than plain Gibbs, which runs in $O(\Delta D)$.
We evaluate Algorithm~\ref{algLocalMinibatchGibbs} empirically, and demonstrate in Figure~\ref{fig:local} that it converges, with almost the same trajectory as plain Gibbs, for various values of the batch size $B$. The simulation here is on the same Ising model as in Algorithm~\ref{algMINTGibbs}, with the same parameters.
Unfortunately, it is unclear if there is anything useful we can say about its convergence rate or even what it converges to.
Unlike MIN-Gibbs, because of the lack of energy-caching there is no obvious reversibility argument to be made here, and so we can not prove bounds on the spectral gap, since those bounds require reversibility.
\begin{figure*}[t]
\centering
\subfigure[]
{
\includegraphics[width=0.30\linewidth]{localmingibbs.pdf}
\label{fig:local}
}
\subfigure[]
{
\includegraphics[width=0.30\linewidth]{mgpmh.pdf}
\label{fig:MGPMH}
}
\subfigure[]
{
\includegraphics[width=0.30\linewidth]{doublemingibbs.pdf}
\label{fig:double}
}
\vspace{-1em}
\caption{Convergence of (a) Local Minibatch Gibbs, (b) MGPMH, and (c) DoubleMIN-Gibbs, compared with vanilla Gibbs sampling.}
\label{fig:convergence}
\end{figure*}
\section{Minibatch-Gibbs-Proposal MH}
We left off in the previous section by presenting Algorithm~\ref{algLocalMinibatchGibbs}, which we showed has promising computational cost but gives us no guarantees on accuracy or convergence rate.
There is a general technique that we can use to transform such chains into ones that \emph{do} have accuracy guarantees: Metropolis-Hastings~\cite{hastings1970monte}.
This well-known technique uses updates from an arbitrary Markov chain, called the \emph{proposal distribution}, but then chooses whether or not to \emph{reject} the update based on the true target distribution $\pi$.
This rejection step reshapes the proposal chain into one that is reversible with stationary distribution $\pi$.
A natural next step for us is to use Metropolis-Hastings with Algorithm~\ref{algLocalMinibatchGibbs} as a proposal distribution.
Doing this results in Algorithm~\ref{algMGPMH}.\footnote{Note that Algorithm~\ref{algMGPMH} is not \emph{precisely} Metropolis-Hastings, because its acceptance probability differs somewhat from what Metropolis-Hastings would have as it is dependent on prior randomness (the selected variable $i$ and minibatch weights $s_{\phi}$).}
\begin{algorithm}[t]
\caption{MGPMH: Minibatch-Gibbs-Proposal MH}
\label{algMGPMH}
\begin{algorithmic}
\REQUIRE Initial model $x \in \Omega$ and average batch size $\lambda$
\LOOP
\STATE Sample $i$ uniformly from $\{1, \ldots, n \}$.
\FORALL{$\phi$ \textbf{in} $A[i]$}
\STATE Sample $s_{\phi} \sim \text{Poisson}\left( \frac{ \lambda M_{\phi} }{L} \right)$
\ENDFOR
\STATE $S \leftarrow \{ \phi | s_{\phi} > 0 \}$
\FORALL{$u$ \textbf{in} $\{1, \ldots, D\}$}
\STATE $\epsilon_u \leftarrow \sum_{\phi \in S} \frac{ s_{\phi} L }{ \lambda M_{\phi} } \phi(x)$
\ENDFOR
\STATE construct distribution $\psi(v) \propto \exp\left( \epsilon_v \right)$
\STATE \textbf{sample} $v$ from $\psi$.
\STATE construct update candidate $y \leftarrow x$; $y(i) \leftarrow v$
\STATE compute the update probability
\[
a
=
\frac{
\exp\left(
\sum_{\phi \in A[i]} \phi(y)
\right)
}{
\exp\left(
\sum_{\phi \in A[i]} \phi(x)
\right)
}
\cdot
\frac{
\exp\left(
\epsilon_{x(i)}
\right)
}{
\exp\left(
\epsilon_{y(i)}
\right)
}.
\]
\STATE Update $x \leftarrow y$ with probability $\min(a, 1)$.
\ENDLOOP
\end{algorithmic}
\end{algorithm}
Because Algorithm~\ref{algMGPMH} is based on Metropolis-Hastings, it is natural for it to be reversible and have $\pi$ as its stationary distribution.
We prove that this must be the case.
We also prove a bound on the spectral gap of the chain.
\begin{theorem}
\label{thmMGPMHreversible}
Algorithm~\ref{algMGPMH} is reversible, and it has stationary distribution $\pi$.
\end{theorem}
\begin{theorem}
\label{thmMGPMHgap}
Let $\bar \gamma$ be the spectral gap of MGPMH, and let $\gamma$ be the spectral gap of a vanilla Gibbs sampling chain running on the same factor graph.
Suppose the expected minibatch size is large enough that $L \le \lambda$.
Then,
\[
\bar \gamma
\ge
\exp\left(
-L^2 / \lambda
\right)
\cdot
\gamma.
\]
\end{theorem}
These theorems mean that MGPMH will converge to the correct stationary distribution $\pi$, and will do so with a convergence rate that is at most a factor of $\exp(L^2/\lambda)$ slower than the vanilla Gibbs chain.
By making $\lambda \approx L^2$, this difference in convergence rates can be made $O(1)$.
On the other hand, when we look at the computational cost of an iteration of Algorithm~\ref{algMGPMH}, we notice it will be $O(\Delta + D \Abs{S})$, since we spend $O(\Delta)$ time computing the minibatch coefficients $s_{\phi}$ and the acceptance probability $a$, and we spend $O(D \Abs{S})$ time computing the energy estimates $\epsilon_u$.
In expectation, the minibatch size $\Abs{S}$ will be
\[
\textstyle
\Exv{\Abs{S}} \le \sum_\phi s_{\phi} = \sum_\phi \frac{\lambda M_{\phi}}{L} = \lambda,
\]
so our average runtime for an iteration will be $O(\lambda D + \Delta)$.
If we want a guaranteed $O(1)$-factor-difference on the spectral gap, we need to set $\lambda = O(L^2)$.
Doing this produces a final computation cost of $O(L^2 D + \Delta)$ for MGPMH.
This can represent a significant speedup over the $O(D \Delta)$ complexity of vanilla Gibbs sampling.
\paragraph{Validation of MGPMH.} We again turn to a synthetic example to validate the theoretical guarantees in Algorithm~\ref{algMGPMH}. We simulate a generalization of the Ising model known as the Potts model ~\cite{potts1952some} with domain $\{1, \ldots, D\}$. The energy of a configuration is the following:
\[
\textstyle
\zeta_{\text{Potts}} = \sum_{i = 1}^N \sum_{j = 1}^N \beta \cdot A_{ij} \cdot \delta(x(i), x(j))
\]
where the the $\delta$ function equals one whenever $x(i) = x(j)$ and zero otherwise.
We simulate a graph with $n = N^2 = \Delta + 1 = 400$, $\beta = 4.6$, $D = 10$, and a fully connected configuration $A_{ij}$ that depends on site distance in the same way as before.
\cmd{This model has $L = 5.09$ and $\Psi = 957.1$: note that $L^2 \ll \Delta$ for this model.}
We run the simulation for one million iterations and illustrate the error in the marginals of MGPMH and that of vanilla Gibbs sampling in Figure~\ref{fig:MGPMH}.
We evaluate MGPMH for three average batch sizes $\lambda$, written in Figure~\ref{fig:MGPMH} as multiples of the local maximum energy squared ($L^2$). MGPMH approaches vanilla Gibbs sampling as batch size increases, which validates Theorem~\ref{thmMGPMHgap}.
\textbf{Combining methods. } Still, under some conditions, even MGPMH might be too slow.
For example, $\Delta$ could be very large relative to $L^2 D$.
In that setting, even though MGPMH would be faster than Gibbs sampling because it decouples the dependence on $D$ and $\Delta$, it still might be intractably slow.
Can the decoupling of $D$ and $\Delta$ from MGPMH be combined with the $\Delta$-independence of MIN-Gibbs?
The most straightforward way to address this question is to replace the exact energy computation in the acceptance probability of MGPMH with a \emph{second} minibatch approximation, like in MIN-Gibbs.
Doing this combines the effects of MIN-Gibbs, which conceptually replaces $\Delta$ with $\Psi^2$ in the computational cost, and MGPMH, which conceptually replaces $D \Delta$ with $D L^2 + \Delta$.
We call this algorithm \emph{DoubleMIN-Gibbs} because of its doubly minibatched nature.
It turns out DoubleMIN-Gibbs, Algorithm~\ref{algDoubleMINTGibbs}, has the same stationary distribution as MIN-Gibbs, and we can also prove a similar bound on its spectral gap.
\begin{algorithm}[t]
\caption{DoubleMIN-Gibbs: Doubly-Minibatched}
\label{algDoubleMINTGibbs}
\begin{algorithmic}
\REQUIRE Initial state $(x, \xi_x) \in \Omega \times R$
\REQUIRE Average first batch size $\lambda_1$
\REQUIRE Second minibatch estimators $\mu_x$ for $x \in \Omega$
\LOOP
\STATE Sample $i$ uniformly from $\{1, \ldots, n \}$.
\FORALL{$\phi$ \textbf{in} $A[i]$}
\STATE Sample $s_{\phi} \sim \text{Poisson}\left( \frac{ \lambda M_{\phi} }{L} \right)$
\ENDFOR
\STATE $S \leftarrow \{ \phi | s_{\phi} > 0 \}$
\FORALL{$u$ \textbf{in} $\{1, \ldots, D\}$}
\STATE $\epsilon_u \leftarrow \sum_{\phi \in S} \frac{ s_{\phi} L }{ \lambda M_{\phi} } \phi(x)$
\ENDFOR
\STATE construct distribution $\psi(v) \propto \exp\left( \epsilon_v \right)$
\STATE \textbf{sample} $v$ from $\psi$.
\STATE construct update candidate $y \leftarrow x$; $y(i) \leftarrow v$
\STATE \textbf{sample} $\xi_y \sim \mu_y$
\STATE compute the update probability
\[
a
=
\exp\left(\xi_y - \xi_x + \epsilon_{x(i)} - \epsilon_{y(i)}\right).
\]
\STATE Update $(x, \xi_x) \leftarrow (y, \xi_y)$ with probability $\min(a, 1)$.
\ENDLOOP
\end{algorithmic}
\end{algorithm}
\begin{theorem}
\label{thmDoubleMINTreversible}
The Markov chain described in Algorithm~\ref{algDoubleMINTGibbs} is reversible and has the same stationary distribution (and therefore the same marginal stationary distribution) as MIN-Gibbs with the same estimator (as described in Theorem~\ref{thmMINTreversible}).
\end{theorem}
\begin{theorem}
\label{thmDoubleMINTgap}
Let $\bar \gamma$ be the spectral gap of DoubleMIN-Gibbs running with an energy estimator $\mu_x$ that has finite support and that satisfies, for some constant $\delta > 0$ and every $x \in \Omega$,
\[
\Prob[\xi_x \sim \mu_x]{\Abs{ \xi_x - \zeta(x) } \le \delta} = 1.
\]
Let $\gamma$ be the spectral gap of a MGPMH sampling chain running using the same graph and average batch size.
Then,
\[
\bar \gamma
\ge
\exp(-4 \delta) \cdot \gamma.
\]
\end{theorem}
Essentially, this theorem tells us the same thing Theorem~\ref{thmMINTgap} told us about MIN-Gibbs.
As long as we can bound our estimator to be at most $O(1)$ away from the true energy $\zeta(x)$, convergence is slowed down by only at most a constant factor from MGPMH.
By Lemma~\ref{lemmaUnbiasedConcentration}, this happens with high probability when we use an estimator of the same average batch size (for the second minibatch) as we used for MIN-Gibbs: $B = O(\Psi^2)$.
Of course, this will only ensure an $O(1)$ difference from MGPMH; to ensure an $O(1)$ difference from vanilla Gibbs, we also need to invoke Theorem~\ref{thmMGPMHgap}, which tells us we need to use an average batch size of $O(L^2)$ for the first minibatch.
Along with the fact that the first minibatch sum needs to be computed $D$ times, this results in an overall computational complexity of $O(L^2 D + \Psi^2)$.
One potential obstacle to an implementation actually achieving this asymptotic rate is the sampling of the Poisson random variables to select $s_{\phi}$.
Naively, since there are up to $\Delta$ potential values of $\phi$, this could take up to $O(\Delta)$ time to compute.
However, it is known to be possible to sample a (sparse) vector of Poisson random variables in time proportional to \emph{the sum of their parameters} instead of the length of the vector.
To illustrate, suppose we want to sample $x_1, \ldots, x_m$ independently where $x_i \sim \textrm{Poisson}(\lambda_i)$.
To do this fast, first we notice if we let $B = \sum_{i=1}^m x_i$, then $B$ will \emph{also} be Poisson distributed, with parameter $\Lambda = \sum_{i=1}^m \lambda_i$.
Conditioned on their sum $B$, the variables $x_1, \ldots, x_m$ have a \emph{multinomial distribution} with trial count $B$ and event probabilities $p_i = \lambda_i / \Lambda$.
It is straightforward to sample from a multinomial distribution in time proportional to its trial count (ignoring log factors).
Thus, we can sample $x_1, \ldots, x_m$ by first sampling $B \sim \textrm{Poisson}(\lambda)$ and then sampling $(x_1, \ldots, x_m) \sim \textrm{Multinomial}(B, (p_1, \ldots, p_m))$, and this entire process will only take on average $O(\Lambda)$ time.\footnote{This still requires $O(m)$ time to compute $\Lambda$ and the probabilities $p_i$, but this can be computed once at the start of Algorithm~\ref{algDoubleMINTGibbs}, thereby not affecting the per-iteration computational complexity.}
For Algorithm~\ref{algDoubleMINTGibbs}, this means we can sample all the $s_{\phi}$ in average time $O(\lambda)$.\footnote{This technique can also be used to speed up the other algorithms in this paper, yet it is not necessary to establish their asymptotic computational complexity.}
This is enough to confirm that its overall computational complexity is in fact $O(L^2 D + \Psi^2)$.
\textbf{Validation of DoubleMIN-Gibbs.} We evaluated the DoubleMIN-Gibbs algorithm on the same synthetic Potts model that we used for MGPMH.
Figure~\ref{fig:double} illustrates the performance of DoubleMIN-Gibbs with a batch size of $L^2$ for the first (MGPMH) minibatch, while the batch size of the second (MIN-Gibbs) minibatch, which we denote $\lambda_2$ in Figure~\ref{fig:double}, is adjusted to multiples of $\Psi^2$. As the second minibatch size increases, DoubleMIN Gibbs approaches the trajectory of MGPMH and vanilla Gibbs sampling, which is what we would expect from the result of Theorem~\ref{thmDoubleMINTgap}.
\section{Conclusion}
We studied applying minibatching to Gibbs sampling.
First, we introduced MIN-Gibbs, which improves the asymptotic per-iteration computational cost of Gibbs sampling from $O(D \Delta)$ to $O(D \Psi^2)$ in the setting where the total maximum energy $\Psi^2$ is small compared to the maximum degree $\Delta$.
Second, we introduced MGPMH, which has an asymptotic cost of $O(D L^2 + \Delta)$ and is an improvement in the setting where $L^2 \ll \Delta$.
Finally, we combined the two techniques to produce DoubleMIN-Gibbs, which achieves a computational cost of $O(D L^2 + \Psi^2)$ and further improves over the other algorithms when $L^2 \le \Psi^2 \ll \Delta$.
We proved that all these algorithms can be made to be unbiased, and that these computation costs can be achieved with parameter settings that are guaranteed to have the same asymptotic convergence rate as plain Gibbs sampling, up to a constant-factor slowdown that can be made arbitrarily small.
Our techniques will potentially enable graphical model inference engines that use Gibbs sampling to scale up to much larger and more complicated graphs than were previously possible, and could become a part of a future of highly scalable probabilistic inference on big data.
\section*{Acknowledgments}
The authors acknowledge the support of NSF DMS-1721550 and 1407557.
The views and conclusions contained herein are those of
the authors and should not be interpreted as necessarily rep-
resenting the official policies or endorsements, either ex-
pressed or implied, of the NSF or the U.S. Government.
\bibliographystyle{plainnat}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,132 |
\section{Introduction}\label{sec:1}
Let $\T:=\{z\in\C :\vert z\vert=1\}$ be the unit circle in $\C$.
We write $\sigma$ for the normalized Lebesgue measure $d\theta/(2\pi)$ on
$([-\pi,\pi), \mcB([-\pi,\pi)))$, where $\mcB([-\pi,\pi))$ is the Borel $\sigma$-algebra
on $[-\pi,\pi)$; thus we have $\sigma([-\pi,\pi))=1$.
For $p\in [1,\infty)$, we write $L_p(\T)$ for the Lebesgue space of measurable functions $f:\T\to\C$
such that $\Vert f\Vert_p<\infty$, where
$\Vert f\Vert_p:=\{\int_{-\pi}^{\pi}\vert f(e^{i\theta})\vert^p \sigma(d\theta)\}^{1/p}$.
Let $L_p^{m\times n}(\T)$ be the space of $\C^{m\times n}$-valued functions on
$\T$ whose entries belong to $L_p(\T)$.
Let $d\in\N$. For $n\in\N$, we consider the block Toeplitz matrix
\[
T_n(w)
:=\left(
\begin{matrix}
\gamma(0) & \gamma(-1) & \cdots & \gamma(-n+1)\cr
\gamma(1) & \gamma(0) & \cdots & \gamma(-n+2)\cr
\vdots & \vdots & \ddots & \vdots \cr
\gamma(n-1) & \gamma(n-2) & \cdots & \gamma(0)
\end{matrix}
\right)
\in \C^{dn\times dn},
\]
where
\begin{equation}
\gamma(k):=\int_{-\pi}^{\pi}e^{-ik\theta}w(e^{i\theta})\frac{d\theta}{2\pi} \in \C^{d\times d},
\qquad k\in\Z,
\label{eq:gamma123}
\end{equation}
and the symbol $w$ satisfies the following two conditions:
\begin{align}
&\mbox{$w\in L^{d\times d}_1(\T)$ and $w(e^{i\theta})$ is a positive Hermitian matrix $\sigma$-a.e.}
\label{eq:A}\\
&w^{-1}\in L^{d\times d}_1(\T).
\label{eq:M}
\end{align}
In this paper,
we show novel explicit formulas for $T_n(w)^{-1}$
(Theorem \ref{thm:TforXwithM123}), which are especially useful for large $n$ (see \cite{BS}).
The formulas are new even for $d=1$.
The main ingredients of the formulas are the
Fourier coefficients of $h^*h_{\sharp}^{-1}=h^{-1}h_{\sharp}^*$, where
$h$ and $h_{\sharp}$ are $\C^{d\times d}$-valued outer functions on
$\T$ such that
\begin{equation}
w(e^{i\theta})
= h(e^{i\theta}) h(e^{i\theta})^*
= h_{\sharp}(e^{i\theta})^*h_{\sharp}(e^{i\theta}), \qquad
\qquad \mbox{$\sigma$-a.e.}
\label{eq:decomp888}
\end{equation}
(see \cite{HL}; see also Section \ref{sec:2}).
The unitary matrix valued function $h^*h_{\sharp}^{-1}=h^{-1}h_{\sharp}^*$ on $\T$
attached to $w$
is called the {\it phase function\/} of $w$
(see page 428 in \cite{Pe}).
In the proof of the above explicit formulas for $T_n(w)^{-1}$,
the dual process of
a stationary process that has $w$ as its spectral density plays an important role.
Thus, let $\{X_k:k\in\Z\}$ be a $\C^d$-valued, centered, weakly stationary process
that has spectral density $w$, hence autocovariance function $\gamma$ in (\ref{eq:gamma123}).
We write $\{X^{\prime}_k: k\in\Z\}$ for the
dual process of $\{X_k\}$ (see \cite{M60}; see also Section \ref{sec:2} below).
The key to the proof of the explicit formulas for $T_n(w)^{-1}$
is the following equality (Theorem \ref{thm:Tdual123}):
\begin{equation}
\left(T_n(w)^{-1}\right)^{s,t} = \langle X^{\prime}_s, P_{[1,n]}X^{\prime}_t\rangle,
\qquad s, t \in \{1,\dots,n\}.
\label{eq:Tdual123}
\end{equation}
Here, $\langle \cdot, \cdot \rangle$ stands for the Gram matrix
(see Section \ref{sec:3}) and
$P_{[1,n]}X^{\prime}_t$ denotes the best
linear predictor of $X^{\prime}_t$ based on the observations $X_{1},\dots,X_{n}$
(see Section \ref{sec:2} for the precise definition).
Moreover, for $n\in\N$, $M \in \C^{dn \times dn}$ and $s, t\in\{1,\dots,n\}$,
we write $M^{s,t}\in \C^{d\times d}$ for the $(s,t)$ block of $M$; thus
$M = (M^{s,t})_{1\le s, t\le n}$.
The equality (\ref{eq:Tdual123}) enables us to apply the $P_{[1,n]}$-related methods
developed in \cite{I00, I08, IK06, IKP1, IKP2} and others to derive
the explicit formulas for $T_n(w)^{-1}$.
We illustrate the usefulness of the explicit formulas for $T_n(w)^{-1}$ by two applications.
The first one is a strong convergence result for solutions of block Toeplitz systems.
For this application, we assume (\ref{eq:A}) as well as the following condition:
\begin{equation}
\mbox{$\sum_{k=-\infty}^{\infty} \Vert \gamma(k)\Vert <\infty$ and
$\min_{z\in\T}\det w(z)>0$.}
\label{eq:S}
\end{equation}
Here, for $a\in\C^{d\times d}$, $\Vert a\Vert$ denotes the
operator norm of $a$.
We note that the conditions (\ref{eq:A}) and (\ref{eq:S}) imply (\ref{eq:M})
(see Section \ref{sec:4}).
Under (\ref{eq:A}) and (\ref{eq:S}), for $n\in\N$ and a $\C^{d\times d}$-valued sequence
$\{y_k\}_{k=1}^{\infty}$ such that $\sum_{k=1}^{\infty} \Vert y_k\Vert < \infty$,
let
\begin{equation}
Z_n=(z_{n,1}^{\top},\dots,z_{n,n}^{\top})^{\top}\in \C^{dn\times d}\ \ \mbox{with}\ \
z_{n,k}\in\C^{d\times d},\ k\in\{1,\dots,n\},
\label{eq:Zz-314}
\end{equation}
be the solution to the block Toeplitz system
\begin{equation}
T_n(w)Z_n = Y_n,
\label{eq:TS183}
\end{equation}
where
\begin{equation}
Y_n := (y_1^{\top},\dots,y_n^{\top})^{\top}\in\C^{dn\times d}.
\label{eq:Rr-314}
\end{equation}
Also, let
\begin{equation}
Z_{\infty}=(z_{1}^{\top},z_{2}^{\top},\dots)^{\top}\ \ \mbox{with}\ \
z_{k}\in\C^{d\times d},\ k \in \N,
\label{eq:Zzinfty-314}
\end{equation}
be the solution to the corresponding infinite block Toeplitz system
\begin{equation}
T_{\infty}(w)Z_{\infty} = Y_{\infty},
\label{eq:TSinfty183}
\end{equation}
where
\begin{equation}
T_{\infty}(w) :=
\left(
\begin{matrix}
\gamma(0) & \gamma(-1) & \gamma(-2) & \cdots \cr
\gamma(1) & \gamma(0) & \gamma(-1) & \cdots \cr
\gamma(2) & \gamma(1) & \gamma(0) & \cdots \cr
\vdots & \vdots & \vdots & \ddots
\end{matrix}
\right)
\label{eq:Tinfty123}
\end{equation}
and
\begin{equation}
Y_{\infty} := (y_1^{\top},y_2^{\top},\dots)^{\top}.
\label{eq:Rrinfty-314}
\end{equation}
Then, our result reads as follows (Theorem \ref{thm:Bax-conv123}):
\begin{equation}
\lim_{n\to\infty} \sum_{k=1}^n \Vert z_{n,k} - z_k\Vert = 0.
\label{eq:conv-z-234}
\end{equation}
We explain the background of the result (\ref{eq:conv-z-234}).
Let $\{X_k:k\in\Z\}$ be as above.
The condition (\ref{eq:S}) implies that $\{X_k\}$ is a short-memory process.
For $n\in\N$, the finite and infinite predictor coefficients $\phi_{n,k}\in\C^{d\times d}$,
$k\in\{1,\dots,n\}$, and $\phi_k$, $k\in\N$, of $\{X_k\}$ are defined by
\[
P_{[1,n]} X_{n+1} = \sum_{k=1}^n \phi_{n,k} X_{n+1-k}
\qquad \mbox{and} \qquad
P_{(-\infty,n]} X_{n+1} = \sum_{k=1}^{\infty} \phi_{k} X_{n+1-k},
\]
respectively; see Section \ref{sec:3} for the precise definitions of
$P_{[1,n]}$ and $P_{(-\infty,n]}$.
We note that $\sum_{k=1}^{\infty} \Vert \phi_k\Vert < \infty$ holds under (\ref{eq:A}) and (\ref{eq:S})
(see Section \ref{sec:4} below and (2.16) in \cite{IKP2}).
Baxter's inequality in \cite{Bax, CP, HD} states that, under (\ref{eq:A}) and (\ref{eq:S}),
there exists $K\in (0,\infty)$ such that
\begin{equation}
\sum_{j=k}^{n}\Vert \phi_{n,k} - \phi_k \Vert
\le K\sum_{k=n+1}^{\infty}\Vert \phi_k\Vert,\qquad
n\in\N.
\label{eq:Baxter597}
\end{equation}
In particular, we have
\begin{equation}
\lim_{n\to\infty} \sum_{k=1}^n \Vert \phi_{n,k} - \phi_k\Vert = 0.
\label{eq:pred-234}
\end{equation}
If we put $\tilde{w}(e^{i\theta}):=w(e^{-i\theta})$,
then, $(\phi_{n,1},\dots,\phi_{n,n})$ is the solution to the block Toeplitz system
\[
T_n(\tilde{w})(\phi_{n,1},\dots,\phi_{n,n})^* = (\gamma(1),\dots,\gamma(n))^*,
\]
called the {\it Yule--Walker equation}, while $(\phi_{1},\phi_{2}, \dots)$
is the solution to the corresponding
infinite block Toeplitz system
\[
T_{\infty}(\tilde{w})(\phi_{1},\phi_{2},\dots)^* = (\gamma(1),\gamma(2),\dots)^*.
\]
Clearly, $\tilde{w}$ satisfies (\ref{eq:A}) and (\ref{eq:S}) since so does $w$.
Therefore, our result (\ref{eq:conv-z-234}) can be viewed as an extension to (\ref{eq:pred-234}).
It should be noted, however, that we prove (\ref{eq:conv-z-234}) directly,
without proving an analogue of (\ref{eq:Baxter597}).
The convergence result (\ref{eq:pred-234}) has various applications in time series analysis, such as
the autoregressive sieve bootstrap; see, e.g., \cite{IKP2} and the references therein.
We note that (\ref{eq:Baxter597}), hence (\ref{eq:pred-234}), is also proved for
univariate and multivariate long-memory processes in \cite{IK06} and \cite{IKP2}, respectively.
These long-memory processes have singular spectral densities $w$ but
our explicit formulas for $T_n(w)^{-1}$ above also cover such $w$.
Naturally, there arises the problem of proving results of the type (\ref{eq:conv-z-234}) for
such singular $w$. This problem is open.
The second application of the explicit formulas for $T_n(w)^{-1}$ is
closed-form formulas for $T_n(w)^{-1}$ with rational $w$ (Theorem \ref{thm:TforR123}).
More precisely, we assume that
$w$ is of the form
\begin{equation}
w(e^{i\theta})=h(e^{i\theta})h(e^{i\theta})^*,\qquad \theta\in [-\pi,\pi),
\label{eq:farima529}
\end{equation}
where $h:\T\to \C^{d\times d}$ satisfies the following condition:
\begin{equation}
\begin{aligned}
&\mbox{the entries of $h(z)$ are rational functions in $z$ that have}\\
&\mbox{no poles in $\overline{\D}$, and $\det h(z)$ has no zeros in $\overline{\D}$.}
\end{aligned}
\label{eq:C}
\end{equation}
Here $\overline{\D}:=\{z\in\C :\vert z\vert\le 1\}$ is the closed unit disk in $\C$.
The closed-form formulas for $T_n(w)^{-1}$ consist of several building block matrices
that are of fixed sizes independent of $n$.
The significance of the formulas for $T_n(w)^{-1}$ is that they provide us with
a linear-time algorithm to compute the solution
$Z\in \C^{dn\times d}$ to the block Toeplitz system
\begin{equation}
T_n(w)Z = Y
\label{eq:TS234}
\end{equation}
for $Y\in\C^{dn\times d}$ (see Section \ref{sec:6}).
The famous Durbin--Levinson algorithm solves the equation (\ref{eq:TS234}) for
more general $w$ in $O(n^2)$ time.
Algorithms for Toeplitz linear systems that run faster than $O(n^2)$
are called {\it superfast}. While our algorithm is restricted to the class of $w$
satisfying (\ref{eq:farima529}) and (\ref{eq:C}), the class is important in applications,
and the linear-time algorithm is ideally superfast in the sense that there is no algorithm faster than $O(n)$.
This paper is organized as follows.
We state the explicit formulas for $T_n(w)^{-1}$
in Section \ref{sec:2}.
In Section \ref{sec:3}, we first prove (\ref{eq:Tdual123}) and then
use it to prove the explicit formulas for $T_n(w)^{-1}$.
In Section \ref{sec:4}, we prove (\ref{eq:conv-z-234}) for $w$ satisfying
(\ref{eq:A}) and (\ref{eq:S}), using the explicit formulas for $T_n(w)^{-1}$.
In Section \ref{sec:5}, we prove the closed-form formulas for $T_n(w)^{-1}$ with
$w$ satisfying (\ref{eq:C}), using the explicit formulas for $T_n(w)^{-1}$.
In Section \ref{sec:6}, we explain how the results in Section \ref{sec:5}
give a linear-time algorithm to compute the solution to (\ref{eq:TS234}).
Finally, the Appendix contains the omitted proofs of two lemmas.
\section{Explicit formulas}\label{sec:2}
Let $\C^{m\times n}$ be the set of all complex $m\times n$ matrices;
we write $\C^d$ for $\C^{d\times 1}$.
Let $I_n$ be the $n\times n$ unit matrix.
For $a\in \C^{m\times n}$, $a^{\top}$ denotes the transpose of $a$, and
$\overline{a}$ and
$a^*$ the complex and Hermitian conjugates of $a$, respectively;
thus, in particular, $a^*:=\overline{a}^{\top}$.
For $a\in\C^{d\times d}$, we write $\Vert a\Vert$ for the operator norm of $a$:
\[
\Vert a\Vert:=\sup_{u\in \C^d, \vert u\vert \le 1}\vert au\vert.
\]
Here
$\vert u\vert:=(\sum_{i=1}^d\vert u^i\vert^2)^{1/2}$ denotes the Euclidean norm of
$u=(u^1,\dots,u^d)^{\top}\in \C^d$.
For $p\in [1,\infty)$ and $K\subset \Z$, $\ell_p^{d\times d}(K)$ denotes the space of
$\C^{d\times d}$-valued sequences
$\{a_k\}_{k\in K}$ such that $\sum_{k\in K}\Vert a_k\Vert^p<\infty$.
We write $\ell_{p+}^{d\times d}$ for $\ell_p^{d\times d}(\N\cup\{0\})$
and $\ell_{p+}$ for $\ell_{p+}^{1\times 1}=\ell_p^{1\times 1}(\N\cup\{0\})$.
Recall $\sigma$ from Section \ref{sec:1}.
The Hardy class $H_2(\T)$ on $\T$ is the closed subspace of
$L_2(\T)$ consisting of $f\in L_2(\T)$ such that
$\int_{-\pi}^{\pi}e^{im\theta}f(e^{i\theta})\sigma(d\theta)=0$ for $m=1,2,\dots$.
Let $H_2^{m\times n}(\T)$ be the space of $\C^{m\times n}$-valued functions on
$\T$ whose entries belong to $H_2(\T)$.
Let $\D:=\{z\in\C : \vert z\vert<1\}$ be the open unit disk in $\C$.
We write $H_2(\D)$ for the Hardy class on $\D$, consisting of
holomorphic functions $f$ on $\D$ such that
$\sup_{r\in [0,1)}\int_{-\pi}^{\pi}\vert f(re^{i\theta})\vert^2\sigma(d\theta)<\infty$.
As usual, we identify each function $f$ in $H_2(\D)$ with its boundary function
$f(e^{i\theta}):=\lim_{r\uparrow 1}f(re^{i\theta})$, $\sigma$-a.e.,
in $H_2(\T)$.
A function $h$ in $H_2^{d\times d}(\T)$ is called \textit{outer}\/ if $\det h$ is a
$\C$-valued outer function, that is, $\det h$ satisfies
$\log\vert \det h(0)\vert
=\int_{-\pi}^{\pi}\log\vert \det h(e^{i\theta})\vert \sigma(d\theta)$
(see Definition 3.1 in \cite{KK}).
We assume that $w$ satisfies (\ref{eq:A}) and (\ref{eq:M}).
Then $\log \det w$ is in $L_1(\T)$ (see Section 3 in \cite{IKP2}). Therefore
$w$ has the decompositions (\ref{eq:decomp888})
for two outer functions $h$ and $h_{\sharp}$ belonging to $H_2^{d\times d}(\T)$, and
$h$ and $h_{\sharp}$ are unique up to constant unitary factors
(see Chapter II in \cite{Roz} and Theorem 11 in \cite{HL}; see also Section 3 in \cite{IKP2}).
We may take $h_{\sharp}=h$ for the case $d=1$ but there is no such simple
relation between $h$ and $h_{\sharp}$ for $d\ge 2$.
We define the outer function $\tilde{h}$ in $H_2^{d\times d}(\T)$ by
\begin{equation}
\tilde{h}(z) := \{h_{\sharp}(\overline{z})\}^*.
\label{eq:outft628}
\end{equation}
All of $h^{-1}$, $h_{\sharp}^{-1}$ and $\tilde{h}^{-1}$ also belong to $H_2^{d\times d}(\T)$ since
we have assumed (\ref{eq:M}).
We define four $\C^{d\times d}$-valued sequences $\{c_k\}$, $\{a_k\}$, $\{\tilde{c}_k\}$ and $\{\tilde{a}_k\}$ by
\begin{align}
h(z)&=\sum_{k=0}^{\infty}z^kc_k,\qquad z\in\D,
\label{eq:MA111}\\
-h(z)^{-1}&=\sum_{k=0}^{\infty}z^ka_k,\qquad z\in\D,
\label{eq:AR111}\\
\tilde{h}(z)&=\sum_{k=0}^{\infty}z^k\tilde{c}_k,\qquad z\in\D,
\label{eq:MA222}
\end{align}
and
\begin{equation}
-\tilde{h}(z)^{-1}=\sum_{k=0}^{\infty}z^k\tilde{a}_k,\qquad z\in\D,
\label{eq:AR222}
\end{equation}
respectively. By (\ref{eq:M}),
all of $\{c_k\}$, $\{a_k\}$, $\{\tilde{c}_k\}$ and $\{\tilde{a}_k\}$ belong to $\ell_{2+}^{d\times d}$.
We define a $\C^{d\times d}$-valued sequence $\{\beta_k\}_{k=-\infty}^{\infty}$ as the
(minus of the) Fourier coefficients of the phase function $h^*h_{\sharp}^{-1}=h^{-1}h_{\sharp}^*$:
for $k \in \Z$,
\begin{equation}
\beta_k
=
-\int_{-\pi}^{\pi}e^{-ik\theta} h(e^{i\theta})^* h_{\sharp}(e^{i\theta})^{-1} \frac{d\theta}{2\pi}
=
-\int_{-\pi}^{\pi}e^{-ik\theta} h(e^{i\theta})^{-1} h_{\sharp}(e^{i\theta})^* \frac{d\theta}{2\pi}.
\label{eq:beta-def667}
\end{equation}
For $n\in\mathbb{N}$, $u \in \{1,\dots,n\}$ and $k\in\mathbb{N}$, we can define
the sequences $\{b_{n,u,\ell}^k\}_{\ell=0}^{\infty}\in \ell_{2+}^{d\times d}$
by the recursion
\begin{equation}
\left\{
\begin{aligned}
b_{n,u,\ell}^1&=\beta_{u+\ell},\\
b_{n,u,\ell}^{2k}
&=\sum_{m=0}^{\infty} b_{n,u,m}^{2k-1} \beta_{n+1+m+\ell}^*,\qquad
b_{n,u,\ell}^{2k+1}
=\sum_{m=0}^{\infty} b_{n,u,m}^{2k} \beta_{n+1+m+\ell}
\end{aligned}
\right.
\label{eq:recurs123}
\end{equation}
(see Section \ref{sec:3} below).
Similarly, for $n\in\mathbb{N}$, $u \in \{1,\dots,n\}$ and $k\in\mathbb{N}$, we can define
the sequences $\{\tilde{b}_{n,u,\ell}^k\}_{\ell=0}^{\infty}\in \ell_{2+}^{d\times d}$
by the recursion
\begin{equation}
\left\{
\begin{aligned}
\tilde{b}_{n,u,\ell}^1&=\beta_{n+1-u+\ell}^*,\\
\tilde{b}_{n,u,\ell}^{2k}
&=\sum_{m=0}^{\infty} \tilde{b}_{n,u,m}^{2k-1} \beta_{n+1+m+\ell},\qquad
\tilde{b}_{n,u,\ell}^{2k+1}
=\sum_{m=0}^{\infty} \tilde{b}_{n,u,m}^{2k} \beta_{n+1+m+\ell}^*.
\end{aligned}
\right.
\label{eq:til-recurs123}
\end{equation}
Since $T_n(w)$, hence $T_n(w)^{-1}$, is self-adjoint, we have
\begin{equation}
(T_n(w)^{-1})^{s,t} = ((T_n(w)^{-1})^{t,s})^*, \qquad s, t \in \{1,\dots,n\},
\label{eq:SA-123}
\end{equation}
where recall from Section \ref{sec:1} that $(T_n(w)^{-1})^{s,t}$ denotes
the $(s,t)$ block of $T_n(w)^{-1}$.
We use the following notation:
\[
s\vee t:=\max(s, t), \qquad s\wedge t:=\min(s, t).
\]
We are ready to state the explicit formulas for $(T_n(w))^{-1}$.
\begin{theorem}\label{thm:TforXwithM123}
We assume (\ref{eq:A}) and (\ref{eq:M}). Then the following two assertions hold.
\begin{enumerate}
\item For $n\in\N$ and $s, t\in\{1,\dots,n\}$, we have
\begin{equation}
\begin{aligned}
\left(T_n(w)^{-1}\right)^{s,t}
&= \sum_{\ell = 1}^{s\wedge t} \tilde{a}_{s - \ell}^* \tilde{a}_{t - \ell} \\
& + \sum_{u=1}^t \sum_{k=1}^{\infty}
\left\{
\sum_{\ell = 0}^{\infty} \tilde{b}_{n,u,\ell}^{2k-1} a_{n + 1 - s + \ell}
+ \sum_{\ell = 0}^{\infty} \tilde{b}_{n,u,\ell}^{2k} \tilde{a}_{s + \ell}
\right\}^* \tilde{a}_{t-u}.
\end{aligned}
\label{eq:TforXwithM222}
\end{equation}
\item
For $n\in\N$ and $s, t\in\{1,\dots,n\}$, we have
\begin{equation}
\begin{aligned}
\left(T_n(w)^{-1}\right)^{s,t}
&= \sum_{\ell=s\vee t}^n a_{\ell - s}^* a_{\ell - t} \\
& + \sum_{u=t}^n \sum_{k=1}^{\infty}
\left\{
\sum_{\ell = 0}^{\infty} b_{n,u,\ell}^{2k-1} \tilde{a}_{s + \ell}
+ \sum_{\ell = 0}^{\infty} b_{n,u,\ell}^{2k} a_{n + 1 - s + \ell}
\right\}^* a_{u-t}.
\end{aligned}
\label{eq:TforXwithM111}
\end{equation}
\end{enumerate}
\end{theorem}
The proof of Theorem \ref{thm:TforXwithM123} will be given in Section \ref{sec:3}.
\begin{corollary}\label{cor:TforXwithM456}
We assume (\ref{eq:A}) and (\ref{eq:M}). Then the following two assertions hold.
\begin{enumerate}
\item For $n\in\N$ and $s, t\in\{1,\dots,n\}$, we have
\begin{equation}
\begin{aligned}
\left(T_n(w)^{-1}\right)^{s,t}
&= \sum_{\ell = 1}^{s\wedge t} \tilde{a}_{s - \ell}^* \tilde{a}_{t - \ell} \\
& + \sum_{u=1}^s \tilde{a}_{s-u}^* \sum_{k=1}^{\infty}
\left\{
\sum_{\ell = 0}^{\infty} \tilde{b}_{n,u,\ell}^{2k-1} a_{n + 1 - t + \ell}
+ \sum_{\ell = 0}^{\infty} \tilde{b}_{n,u,\ell}^{2k} \tilde{a}_{t + \ell}
\right\}.
\end{aligned}
\label{eq:TforXwithM444}
\end{equation}
\item For $n\in\N$ and $s, t\in\{1,\dots,n\}$, we have
\begin{equation}
\begin{aligned}
\left(T_n(w)^{-1}\right)^{s,t}
&= \sum_{\ell=s\vee t}^n a_{\ell - s}^* a_{\ell - t} \\
& + \sum_{u=s}^n a_{u-s}^* \sum_{k=1}^{\infty}
\left\{
\sum_{\ell = 0}^{\infty} b_{n,u,\ell}^{2k-1} \tilde{a}_{t + \ell}
+ \sum_{\ell = 0}^{\infty} b_{n,u,\ell}^{2k} a_{n + 1 - t + \ell}
\right\}.
\end{aligned}
\label{eq:TforXwithM333}
\end{equation}
\end{enumerate}
\end{corollary}
\begin{proof}
Thanks to (\ref{eq:SA-123}), we obtain
(\ref{eq:TforXwithM444}) and (\ref{eq:TforXwithM333})
from
(\ref{eq:TforXwithM222}) and (\ref{eq:TforXwithM111}),
respectively.
\end{proof}
\begin{remark}\label{rem:1st-term-123}
Recall $T_{\infty}(w)$ from (\ref{eq:Tinfty123}).
For $n\in\N\cup\{0\}$, we have $\gamma(n)=\sum_{k=0}^{\infty} \tilde{c}_k \tilde{c}_{n+k}^*$ and
$\gamma(-n)=\sum_{k=0}^{\infty} \tilde{c}_{n+k} \tilde{c}_{k}^*$ (see (2.13) in \cite{IKP2}),
hence $T_{\infty}(w) = \tilde{C}_{\infty} (\tilde{C}_{\infty})^*$,
where
\[
\tilde{C}_{\infty} :=
\left(
\begin{matrix}
\tilde{c}_0 & \tilde{c}_1 & \tilde{c}_2 & \cdots \cr
& \tilde{c}_0 & \tilde{c}_1 & \cdots \cr
& & \tilde{c}_0 & \cdots \cr
\mbox{\huge 0} & & & \ddots
\end{matrix}
\right).
\]
On the other hand, it follows from $\tilde{h}(z)\tilde{h}(z)^{-1}=I_d$ that
$\sum_{k=0}^n \tilde{c}_k \tilde{a}_{n-k} = -\delta_{n0}I_d$ for $n\in\N\cup\{0\}$,
hence $\tilde{C}_{\infty} \tilde{A}_{\infty} = -I_{\infty}$, where
\[
\tilde{A}_{\infty} :=
\left(
\begin{matrix}
\tilde{a}_0 & \tilde{a}_1 & \tilde{a}_2 & \cdots \cr
& \tilde{a}_0 & \tilde{a}_1 & \cdots \cr
& & \tilde{a}_0 & \cdots \cr
\mbox{\huge 0} & & & \ddots
\end{matrix}
\right),
\qquad
I_{\infty} :=
\left(
\begin{matrix}
1 & 0 & 0 & \cdots \cr
0 & 1 & 0 & \cdots \cr
0 & 0 & 1 & \cdots \cr
\vdots & \vdots & \vdots & \ddots
\end{matrix}
\right).
\]
Combining, we have $T_{\infty}(w)^{-1} = (\tilde{A}_{\infty})^* \tilde{A}_{\infty}$.
Thus, we find that the first term $\sum_{\ell = 1}^{s\wedge t} \tilde{a}_{s - \ell}^* \tilde{a}_{t - \ell}$
in (\ref{eq:TforXwithM222}) or (\ref{eq:TforXwithM444}) coincides with the $(s,t)$ block of
$T_{\infty}(w)^{-1}$.
\end{remark}
For $n\in\N$, we define
\begin{equation}
\tilde{A}_n :=
\left(
\begin{matrix}
\tilde{a}_0 & \tilde{a}_1 & \tilde{a}_2 & \cdots & \tilde{a}_{n-1} \cr
& \tilde{a}_0 & \tilde{a}_1 & \cdots & \tilde{a}_{n-2} \cr
& & \ddots & \ddots & \vdots \cr
& & & \ddots & \tilde{a}_1 \cr
\mbox{\huge 0} & & & & \tilde{a}_0
\end{matrix}
\right) \in \C^{dn\times dn}
\label{eq:tildeAnU-234}
\end{equation}
and
\begin{equation}
A_n :=
\left(
\begin{matrix}
a_0 & & & & \mbox{\huge 0} \cr
a_1 & a_0 & & & \cr
a_2 & a_1 & \ddots & & \cr
\vdots & \vdots & \ddots & \ddots & \cr
a_{n-1} & a_{n-2} & \cdots & a_1 & a_0
\end{matrix}
\right) \in \C^{dn\times dn}.
\label{eq:AnL-234}
\end{equation}
The next lemma will turn out to be useful in Section \ref{sec:6}.
\begin{lemma}\label{lem:decomp234}
For $n\in\N$ and $s, t \in \{1,\dots,n\}$, we have the following two equalities:
\begin{align*}
\left(\tilde{A}_n^* \tilde{A}_n\right)^{s,t}
&= \sum_{\ell = 1}^{s\wedge t} \tilde{a}_{s - \ell}^* \tilde{a}_{t - \ell},
\\
\left(A_n^* A_n\right)^{s,t}
&= \sum_{\ell=s\vee t}^n a_{\ell - s}^* a_{\ell - t}.
\end{align*}
\end{lemma}
The proof of Lemma \ref{lem:decomp234} is straightforward and will be omitted.
\section{Proof of Theorem \ref{thm:TforXwithM123}}\label{sec:3}
In this section, we prove Theorem \ref{thm:TforXwithM123}.
We assume (\ref{eq:A}) and (\ref{eq:M}).
Let $\{X_k\}=\{X_k:k\in\Z\}$ be a $\C^d$-valued, centered,
weakly stationary process, defined on a probability space $(\Omega, \mcF, P)$,
that has spectral density $w$, hence autocovariance function $\gamma$.
Thus we have
$E[X_k X_0^*] = \gamma(k) = \int_{-\pi}^{\pi}e^{-ik\theta}w(e^{i\theta})(d\theta/(2\pi))$ for
$k\in\Z$.
Write $X_k=(X^1_k,\dots,X^d_k)^{\top}$, and
let $V$ be the complex Hilbert space
spanned by all the entries $\{X^j_k: k\in\Z,\ j=1,\dots,d\}$ in $L^2(\Omega, \mcF, P)$,
which has inner product $(x, y)_{V}:=E[x\overline{y}]$ and
norm $\Vert x\Vert_{V}:=(x,x)_{V}^{1/2}$.
For $J\subset \Z$ such as $\{n\}$,
$(-\infty,n]:=\{n,n-1,\dots\}$, $[n,\infty):=\{n,n+1,\dots\}$,
and $[m,n]:=\{m,\dots,n\}$ with $m\le n$,
we define the closed subspace $V_J^X$ of $V$ by
\[
V_J^X:=\cspn \{X^j_k: j=1,\dots,d,\ k\in J\}.
\]
Let $P_J$ and $P_J^{\perp}$ be the orthogonal projection operators of $V$ onto
$V_J^X$ and $(V_J^X)^{\perp}$, respectively, where
$(V_J^X)^{\bot}$ denotes the orthogonal complement of $V_J^X$ in $V$.
By Theorem 3.1 in \cite{I00} for $d=1$ and Corollary 3.6 in \cite{IKP1} for general $d\ge 1$, the conditions
(\ref{eq:A}) and (\ref{eq:M}) imply the following \textit{intersection of past and future}
property:
\begin{equation}
V_{(-\infty,n]}^X\cap V_{[1,\infty)}^X=V_{[1,n]}^X,\qquad n \in \N.
\label{eq:IPF}
\end{equation}
Let $V^d$ be the space of $\C^d$-valued random variables on
$(\Omega, \mcF, P)$ whose entries belong to $V$.
The norm $\Vert x\Vert_{V^d}$ of $x=(x^1,\dots,x^d)^{\top}\in V^d$ is given by
$\Vert x\Vert_{V^d}:=(\sum_{i=1}^d \Vert x^i\Vert_V^2)^{1/2}$.
For $J\subset\mathbb{Z}$ and
$x=(x^1,\dots,x^d)^{\top}\in V^d$, we write $P_Jx$ for $(P_Jx^1, \dots, P_Jx^d)^{\top}$.
We define $P_J^{\perp}x$ in a similar way.
For $x=(x^1,\dots,x^d)^{\top}$ and $y=(y^1,\dots,y^d)^{\top}$ in $V^d$,
\[
\langle x,y\rangle:=E[xy^*]=((x^k, y^{\ell})_V)_{1\le k, \ell\le d}
\in \C^{d\times d}
\]
stands for the Gram matrix of $x$ and $y$.
Let
\[
X_k=\int_{-\pi}^{\pi}e^{-ik\theta}\eta(d\theta),\qquad k\in\Z,
\]
be the spectral representation of $\{X_k\}$, where $\eta$ is a $\C^d$-valued
random spectral measure.
We define a $d$-variate stationary process $\{\varepsilon_k:k\in\Z\}$,
called the \textit{forward innovation
process}\/ of $\{X_k\}$, by
\[
\varepsilon_k:=\int_{-\pi}^{\pi}e^{-ik\theta}h(e^{i\theta})^{-1}\eta(d\theta),\qquad k\in\Z.
\]
Then, $\{\varepsilon_k\}$ satisfies
$\langle \varepsilon_n, \varepsilon_m\rangle = \delta_{n m}I_d$ and
$V_{(-\infty,n]}^X=V_{(-\infty,n]}^{\varepsilon}$ for $n\in\Z$,
hence
\[
(V_{(-\infty,n]}^X)^{\bot} = V_{[n+1, \infty)}^{\varepsilon},\qquad n\in\Z.
\]
Recall the outer function $h_{\sharp}$ in $H_2^{d\times d}(\T)$ from (\ref{eq:decomp888}).
We define the \textit{backward innovation process}\/ $\{\tilde{\varepsilon}_k: k\in\Z\}$ of $\{X_k\}$ by
\[
\tilde{\varepsilon}_k:=\int_{-\pi}^{\pi}e^{ik\theta}\{h_{\sharp}(e^{i\theta})^*\}^{-1}
\eta(d\theta),\qquad k\in\Z.
\]
Then, $\{\tilde{\varepsilon}_k\}$ satisfies
$\langle \tilde{\varepsilon}_n, \tilde{\varepsilon}_m\rangle=\delta_{n m}I_d$ and
$V_{[-n,\infty)}^X=V_{(-\infty, n]}^{\tilde{\varepsilon}}$ for $n\in\Z$,
hence
\[
(V_{[-n,\infty)}^X)^{\bot} = V_{[n+1,\infty)}^{\tilde{\varepsilon}},\qquad n\in\Z
\]
(see Section 2 in \cite{IKP2}). Moreover, by Lemma 4.1 in \cite{IKP2}, we have
\begin{equation}
\langle\varepsilon_{\ell}, \tilde{\varepsilon}_{m}\rangle = -\beta_{\ell+m},\qquad
\langle\tilde{\varepsilon}_{m}, \varepsilon_{\ell}\rangle = -\beta_{\ell+m}^*,
\qquad \ell, m \in \Z.
\label{eq:beta234}
\end{equation}
By (\ref{eq:beta234}), for $\{s_{\ell}\}\in \ell_{2+}^{d\times d}$ and $n\in\N$,
\begin{align}
P_{[1,\infty)}^{\perp}
\left(\sum_{\ell=0}^{\infty} s_{\ell} \varepsilon_{n+1+\ell}\right)
&=-\sum_{\ell=0}^\infty
\left(\sum_{m=0}^\infty s_m \beta_{n+1+\ell+m}\right)
\tilde{\varepsilon}_{\ell},
\label{eq:proj555}\\
P_{(-\infty,n]}^{\perp}
\left(\sum_{\ell=0}^\infty s_{\ell} \tilde{\varepsilon}_{\ell}\right)
&=-\sum_{\ell=0}^\infty
\left(\sum_{m=0}^\infty s_m \beta_{n+1+\ell+m}^*\right)
\varepsilon_{n+1+\ell}.
\label{eq:proj666}
\end{align}
Therefore,
\[
\left\{\sum_{m=0}^\infty s_m \beta_{n+1+\ell+m}\right\}_{\ell=0}^{\infty}, \
\left\{\sum_{m=0}^\infty s_m \beta_{n+1+\ell+m}^*\right\}_{\ell=0}^{\infty}\ \in \
\ell_{2+}^{d\times d}.
\]
See Lemma 4.2 in \cite{IKP2}. In particular,
for $n\in\mathbb{N}$, $u \in \{1,\dots,n\}$ and $k\in\mathbb{N}$, we can define
the sequences $\{b_{n,u,\ell}^k\}_{\ell=0}^{\infty}\in \ell_{2+}^{d\times d}$
and $\{\tilde{b}_{n,u,\ell}^k\}_{\ell=0}^{\infty}\in \ell_{2+}^{d\times d}$
by the recursions (\ref{eq:recurs123}) and (\ref{eq:til-recurs123}), respectively.
By (\ref{eq:A}) and (\ref{eq:M}), $\{X_k\}$ has the
dual process $\{X^{\prime}_k: k\in\Z\}$, which is a $\C^d$-valued, centered,
weakly stationary process
characterized by the biorthogonality relation
\[
\langle X_s,X^{\prime}_t\rangle=\delta_{st}I_d, \qquad s, t\in\Z
\]
(see \cite{M60}).
Recall $\{a_k\} \in \ell_{2+}^{d\times d}$ and $\{\tilde{a}_k\} \in \ell_{2+}^{d\times d}$
from (\ref{eq:AR111}) and (\ref{eq:AR222}), respectively.
The dual process $\{X^{\prime}_k\}$ admits the following two MA representations (see Section 5 in \cite{IKP2}):
\begin{align}
X^{\prime}_n&=-\sum_{\ell=n}^\infty a_{\ell-n}^* \varepsilon_{\ell}, \qquad n\in\Z,
\label{eq:MAdual111}\\
X^{\prime}_n&=-\sum_{\ell=-n}^{\infty} \tilde{a}_{\ell+n}^* \tilde{\varepsilon}_{\ell},\qquad n\in\Z.
\label{eq:MAdual222}
\end{align}
The next theorem is the key to the proof of Theorem \ref{thm:TforXwithM123}.
\begin{theorem}\label{thm:Tdual123}
We assume (\ref{eq:A}) and (\ref{eq:M}). Then, for $n\in\N$ and $s,t \in \{1,\dots,n\}$, we have (\ref{eq:Tdual123}).
\end{theorem}
\begin{proof}
Fix $n\in\N$. For $s\in \{1,\dots,n\}$, we can write
$P_{[1,n]}X^{\prime}_s = \sum_{k=1}^n q_{s,k} X_k$
for some $q_{s,k}\in\C^{d\times d}$, $k\in\{1,\dots,n\}$.
For $s,t \in \{1,\dots,n\}$, we have
\[
\begin{aligned}
\delta_{st}I_d
&= \langle X^{\prime}_s, X_t\rangle
= \langle X^{\prime}_s, P_{[1,n]}X_t\rangle
= \langle P_{[1,n]}X^{\prime}_s, X_t \rangle
= \left\langle \sum_{k=1}^n q_{s,k} X_k, X_t\right\rangle\\
&=\sum_{k=1}^n q_{s,k} \left\langle X_k, X_t \right\rangle
=\sum_{k=1}^n q_{s,k} \gamma(k-t),
\end{aligned}
\]
or $Q_n T_n(w) = I_{dn}$, where
$Q_n := (q_{s,k})_{1\le s, k\le n}\in \C^{dn\times dn}$. Therefore, we have
$Q_n = T_n(w)^{-1}$.
However,
\[
\langle X^{\prime}_s, P_{[1,n]}X^{\prime}_t\rangle
= \langle P_{[1,n]}X^{\prime}_s, X^{\prime}_t\rangle
= \left\langle \sum_{k=1}^n q_{s,k} X_k, X^{\prime}_t \right\rangle
= \sum_{k=1}^n q_{s,k} \langle X_k, X^{\prime}_t\rangle = q_{s,t}.
\]
Thus, the theorem follows.
\end{proof}
\begin{lemma}\label{lem:Tdual456}
We assume (\ref{eq:A}) and (\ref{eq:M}). Then,
for $n\in\mathbb{N}$ and $s,t \in \{1,\dots,n\}$, the following two equalities hold:
\begin{align}
\langle X^{\prime}_s, P_{[1,n]}X^{\prime}_t\rangle
&= \sum_{\ell=s\vee t}^n a_{\ell-s}^* a_{\ell-t}
+ \sum_{u=t}^n \langle X^{\prime}_s, P_{[1,n]}^{\bot} \varepsilon_u\rangle a_{u-t},
\label{eq:dual777}\\
\langle X^{\prime}_s, P_{[1,n]}X^{\prime}_t\rangle
&= \sum_{\ell=1}^{s\wedge t} \tilde{a}_{s-\ell}^* \tilde{a}_{t-\ell}
+ \sum_{u=1}^t \langle X^{\prime}_s, P_{[1,n]}^{\bot}\tilde{\varepsilon}_{-u}\rangle \tilde{a}_{t-u}.
\label{eq:dual888}
\end{align}
\end{lemma}
\begin{proof}
First, we prove (\ref{eq:dual777}). Since
$V_{[1,n]}^{X} \subset V_{(-\infty,n]}^{X}$, we have
\[
\langle X^{\prime}_s, P_{[1,n]} X^{\prime}_t\rangle
=\langle X^{\prime}_s, P_{[1,n]} P_{(-\infty,n]} X^{\prime}_t\rangle
=\langle X^{\prime}_s, P_{(-\infty,n]} X^{\prime}_t\rangle
- \langle X^{\prime}_s, P_{[1,n]}^{\bot} P_{(-\infty,n]} X^{\prime}_t\rangle.
\]
On the other hand, from (\ref{eq:MAdual111}), we have
$P_{(-\infty,n]} X^{\prime}_t = -\sum_{m=t}^n a_{m-t}^* \varepsilon_{m}$,
hence
\[
\langle X^{\prime}_s, P_{(-\infty,n]} X^{\prime}_t\rangle
= \left\langle \sum_{\ell=s}^{\infty} a_{\ell-s}^* \varepsilon_{\ell},
\sum_{m=t}^n a_{m-t}^* \varepsilon_{m}\right\rangle
=\sum_{\ell=s\vee t}^n a_{\ell-s}^* a_{\ell-t},
\]
and
\[
\langle X^{\prime}_s, P_{[1,n]}^{\bot} P_{(-\infty,n]} X^{\prime}_t\rangle
=-\left\langle X^{\prime}_s, P_{[1,n]}^{\bot}\left(\sum_{u=t}^n a_{u-t}^* \varepsilon_{u}\right)\right\rangle
=-\sum_{u=t}^n \langle X^{\prime}_s, P_{[1,n]}^{\bot}\varepsilon_{u}\rangle a_{u-t}.
\]
Combining, we obtain (\ref{eq:dual777}).
Next, we prove (\ref{eq:dual888}). Since
$V_{[1,n]}^{X} \subset V_{[1,\infty)}^{X}$, we have
\[
\langle X^{\prime}_s, P_{[1,n]} X^{\prime}_t\rangle
=\langle X^{\prime}_s, P_{[1,n]} P_{[1,\infty)} X^{\prime}_t\rangle
=\langle X^{\prime}_s, P_{[1,\infty)} X^{\prime}_t\rangle
- \langle X^{\prime}_s, P_{[1,n]}^{\bot} P_{[1,\infty)} X^{\prime}_t\rangle.
\]
On the other hand, from (\ref{eq:MAdual222}), we have
$P_{[1, \infty)} X^{\prime}_t
= -\sum_{m=1}^{t} \tilde{a}_{t-m}^* \tilde{\varepsilon}_{-m}$,
hence
\[
\langle X^{\prime}_s, P_{[1,\infty)} X^{\prime}_t\rangle
= \left\langle \sum_{\ell=-\infty}^{s} \tilde{a}_{s-\ell}^* \tilde{\varepsilon}_{-\ell},
\sum_{m=1}^{t} \tilde{a}_{t-m}^* \tilde{\varepsilon}_{-m} \right\rangle
=\sum_{\ell=1}^{s\wedge t} \tilde{a}_{s-\ell}^* \tilde{a}_{t-\ell},
\]
and
\[
\langle X^{\prime}_s, P_{[1,n]}^{\bot} P_{[1,\infty)} X^{\prime}_t\rangle
=-\left\langle X^{\prime}_s, P_{[1,n]}^{\bot}\left( \sum_{u=1}^{t} \tilde{a}_{t-u}^* \tilde{\varepsilon}_{-u} \right)\right\rangle
=-\sum_{u=1}^t \langle X^{\prime}_s, P_{[1,n]}^{\bot}\tilde{\varepsilon}_{-u}\rangle \tilde{a}_{t-u}.
\]
Combining, we obtain (\ref{eq:dual888}).
\end{proof}
For $n\in\mathbb{N}$ and $u \in \{1,\dots,n\}$,
we define the sequence $\{W_{n,u}^k\}_{k=1}^{\infty}$ in $V^d$ by
\begin{align*}
W_{n,u}^{2k-1}
&= - P_{[1, \infty)}^{\perp}
(P_{(-\infty, n]}^\perp P_{[1, \infty)}^{\perp})^{k-1} \varepsilon_u,\qquad k \in \N,
\\
W_{n,u}^{2k}
&=
(P_{(-\infty,n]}^{\perp} P_{[1, \infty)}^{\perp})^{k} \varepsilon_u,\qquad k \in \N.
\end{align*}
\begin{lemma}\label{lem:basic-rep123}
We assume (\ref{eq:A}) and (\ref{eq:M}). Then, for $n\in\mathbb{N}$ and $u\in\{1,\dots,n\}$, we have
\begin{equation}
P_{[1,n]}^{\perp} \varepsilon_u = -\sum_{k=1}^{\infty} W_{n,u}^{k},
\label{eq:basic111}
\end{equation}
the sum converging strongly in $V^d$.
\end{lemma}
\begin{proof}
Since $\varepsilon_u$ is in $V_{(-\infty,n]}^X$, (\ref{eq:basic111}) follows
from (\ref{eq:IPF}) and Theorem 3.2 in \cite{IKP2}.
\end{proof}
\begin{proposition}\label{prop:forward638}
We assume (\ref{eq:A}) and (\ref{eq:M}). Then, for $n\in\mathbb{N}$, $u\in\{1,\dots,n\}$ and $k\in\mathbb{N}$, we have
\begin{align}
W_{n,u}^{2k-1}
&=\sum_{\ell=0}^\infty b_{n,u,\ell}^{2k-1} \tilde{\varepsilon}_{\ell},
\label{eq:expan111}\\
W_{n,u}^{2k}
&=\sum_{\ell=0}^\infty b_{n,u,\ell}^{2k} \varepsilon_{n+1+\ell}.
\label{eq:expan222}
\end{align}
\end{proposition}
\begin{proof}
Note that, from the definition of $W_{n,u}^k$,
\[
W_{n,u}^{2k+1}=-P_{[1,\infty)}^\perp W_{n,u}^{2k},
\qquad
W_{n,u}^{2k+2}=-P_{(-\infty,n]}^\perp W_{n,u}^{2k+1}.
\]
We prove (\ref{eq:expan111}) and (\ref{eq:expan222}) by induction.
First, by (\ref{eq:beta234}), we have
\[
W_{n,u}^1=-P_{[1, \infty)}^{\perp} \varepsilon_u
= -\sum_{\ell=0}^{\infty} \langle \varepsilon_u, \tilde{\varepsilon}_{\ell}\rangle \tilde{\varepsilon}_{\ell}
=\sum_{\ell=0}^\infty \beta_{u+\ell} \tilde{\varepsilon}_{\ell}
=
\sum_{\ell=0}^\infty b_{n,u,\ell}^1 \tilde{\varepsilon}_{\ell}.
\]
For $k \in \N$, assume that
$W_{n,u}^{2k-1}=\sum_{\ell=0}^\infty b_{n,u,\ell}^{2k-1} \tilde{\varepsilon}_{\ell}$.
Then, by (\ref{eq:proj666}),
\[
\begin{aligned}
W_{n,u}^{2k}
&=-P_{(-\infty,n]}^{\perp}
\left(\sum_{\ell=0}^{\infty} b_{n,u,\ell}^{2k-1} \tilde{\varepsilon}_{\ell}\right)
=\sum_{\ell=0}^{\infty}
\left(\sum_{m=0}^\infty b_{n,u,m}^{2k-1} \beta_{n+1+m+\ell}^*
\right) \varepsilon_{n+1+\ell}\\
&=\sum_{\ell=0}^\infty b_{n,u,\ell}^{2k} \varepsilon_{n+1+\ell},
\end{aligned}
\]
and, by (\ref{eq:proj555}),
\[
\begin{aligned}
W_{n,u}^{2k+1}
&=-P_{[1, \infty)}^{\perp}
\left(\sum_{\ell=0}^\infty b_{n,u,\ell}^{2k} \varepsilon_{n+1+\ell}\right)
=\sum_{\ell=0}^{\infty}
\left(\sum_{m=0}^{\infty} b_{n,u,m}^{2k} \beta_{n+1+m+\ell}
\right) \tilde{\varepsilon}_{\ell}\\
&=\sum_{\ell=0}^{\infty} b_{n,u,\ell}^{2k+1} \tilde{\varepsilon}_{\ell}.
\end{aligned}
\]
Thus (\ref{eq:expan111}) and (\ref{eq:expan222}) follow.
\end{proof}
For $n\in\mathbb{N}$ and $u \in \{1,\dots,n\}$,
we define the sequence $\{\tilde{W}_{n,u}^k\}_{k=1}^{\infty}$ in
$V^d$ by
\begin{align*}
\tilde{W}_{n,u}^{2k-1}
&= - P_{(-\infty, n]}^\perp
(P_{[1, \infty)}^{\perp} P_{(-\infty, n]}^\perp)^{k-1} \tilde{\varepsilon}_{-u},
\qquad k \in \N,
\\
\tilde{W}_{n,u}^{2k}
&=
(P_{[1, \infty)}^{\perp} P_{(-\infty,n]}^{\perp})^{k} \tilde{\varepsilon}_{-u},
\qquad k \in \N.
\end{align*}
\begin{lemma}\label{lem:basic-rep456}
We assume (\ref{eq:A}) and (\ref{eq:M}). Then, for $n\in\mathbb{N}$ and $u\in\{1,\dots,n\}$, we have
\begin{equation}
P_{[1,n]}^{\perp} \tilde{\varepsilon}_{-u} = -\sum_{k=1}^{\infty} \tilde{W}_{n,u}^{k},
\label{eq:basic222}
\end{equation}
the sum converging strongly in $V^d$.
\end{lemma}
\begin{proof}
Since $\tilde{\varepsilon}_{-u}$ is in $V_{[1,\infty)}^X$, (\ref{eq:basic222}) follows
from (\ref{eq:IPF}) and Theorem 3.2 in \cite{IKP2}.
\end{proof}
\begin{proposition}\label{prop:til-forward638}
We assume (\ref{eq:A}) and (\ref{eq:M}).
Then, for $n\in\mathbb{N}$, $u\in\{1,\dots,n\}$ and $k\in\mathbb{N}$, we have
\begin{align}
\tilde{W}_{n,u}^{2k-1}
&=\sum_{\ell=0}^\infty \tilde{b}_{n,u,\ell}^{2k-1} \varepsilon_{n+1+\ell},
\label{eq:til-expan111}\\
\tilde{W}_{n,u}^{2k}
&=\sum_{\ell=0}^\infty \tilde{b}_{n,u,\ell}^{2k} \tilde{\varepsilon}_{\ell}.
\label{eq:til-expan222}
\end{align}
\end{proposition}
\begin{proof}
Note that, from the definition of $\tilde{W}_{n,u}^k$,
\[
\tilde{W}_{n,u}^{2k+1}=-P_{(-\infty,n]}^\perp \tilde{W}_{n,u}^{2k},
\qquad
\tilde{W}_{n,u}^{2k+2}=- P_{[1,\infty)}^\perp \tilde{W}_{n,u}^{2k+1}.
\]
We prove (\ref{eq:til-expan111}) and (\ref{eq:til-expan222}) by induction.
First, by (\ref{eq:beta234}), we have
\[
\begin{aligned}
\tilde{W}_{n,u}^1&= - P_{(-\infty,n]}^\perp \tilde{\varepsilon}_{-u}
= -\sum_{\ell=0}^{\infty} \langle \tilde{\varepsilon}_{-u}, \varepsilon_{n+1+\ell}\rangle \varepsilon_{n+1+\ell}\\
&=\sum_{\ell=0}^\infty \beta_{n+1-u+\ell}^* \varepsilon_{n+1+\ell}
=
\sum_{\ell=0}^\infty \tilde{b}_{n,u,\ell}^1 \varepsilon_{n+1+\ell}.
\end{aligned}
\]
For $k \in \N$, assume that
$\tilde{W}_{n,u}^{2k-1}=\sum_{\ell=0}^\infty \tilde{b}_{n,u,\ell}^{2k-1} \varepsilon_{n+1+\ell}$.
Then, by (\ref{eq:proj555}),
\[
\begin{aligned}
\tilde{W}_{n,u}^{2k}
&=-P_{[1,\infty)}^\perp
\left(\sum_{\ell=0}^{\infty} \tilde{b}_{n,u,\ell}^{2k-1} \varepsilon_{n+1+\ell}\right)
=\sum_{\ell=0}^{\infty}
\left(\sum_{m=0}^\infty \tilde{b}_{n,u,m}^{2k-1} \beta_{n+1+m+\ell}
\right) \tilde{\varepsilon}_l\\
&=\sum_{\ell=0}^\infty \tilde{b}_{n,u,\ell}^{2k} \tilde{\varepsilon}_{\ell},
\end{aligned}
\]
and, by (\ref{eq:proj666}),
\[
\begin{aligned}
\tilde{W}_{n,u}^{2k+1}
&=-P_{(\infty,n]}^{\perp}
\left(\sum_{\ell=0}^\infty \tilde{b}_{n,u,\ell}^{2k} \tilde{\varepsilon}_{\ell}\right)
=\sum_{\ell=0}^{\infty}
\left(\sum_{m=0}^{\infty} \tilde{b}_{n,u,m}^{2k} \beta_{n+1+m+\ell}^*
\right) \varepsilon_{n+1+\ell}\\
&=\sum_{\ell=0}^{\infty} \tilde{b}_{n,u,\ell}^{2k+1} \varepsilon_{n+1+\ell}.
\end{aligned}
\]
Thus (\ref{eq:til-expan111}) and (\ref{eq:til-expan222}) follow.
\end{proof}
We are ready to prove Theorem \ref{thm:TforXwithM123}.
\begin{proof}[Proof of Theorem \ref{thm:TforXwithM123}]
(i)\ For $n\in\N$, $s, u \in \{1,\dots,n\}$ and $k\in\N$,
we see from (\ref{eq:MAdual111}) and (\ref{eq:til-expan111}) that
\[
\langle X^{\prime}_s, \tilde{W}_{n,u}^{2k-1} \rangle
= - \sum_{\ell=0}^{\infty} a_{n+1-s+\ell}^* (\tilde{b}_{n,u,\ell}^{2k-1})^*,
\]
and from (\ref{eq:MAdual222}) and (\ref{eq:til-expan222}) that
\[
\langle X^{\prime}_s, \tilde{W}_{n,u}^{2k} \rangle
= - \sum_{\ell=0}^{\infty} \tilde{a}_{s+\ell}^* (\tilde{b}_{n,u,\ell}^{2k})^*.
\]
Therefore, by Lemma \ref{lem:basic-rep456},
\[
\langle X^{\prime}_s, P_{[1,n]}^{\perp} \tilde{\varepsilon}_{-u} \rangle
= - \sum_{k=1}^{\infty} \langle X^{\prime}_s, \tilde{W}_{n,u}^{k} \rangle
= \sum_{k=1}^{\infty}
\left\{
\sum_{\ell=0}^{\infty} \tilde{b}_{n,u,\ell}^{2k-1} a_{n+1-s+\ell}
+
\sum_{\ell=0}^{\infty} \tilde{b}_{n,u,\ell}^{2k} \tilde{a}_{s+\ell}
\right\}^*.
\]
The assertion (i) follows from this, Theorem \ref{thm:Tdual123} and Lemma \ref{lem:Tdual456}.
(ii)\ For $n\in\N$, $s, u \in \{1,\dots,n\}$ and $k\in\N$,
we see from (\ref{eq:MAdual222}) and (\ref{eq:expan111}) that
\[
\langle X^{\prime}_s, W_{n,u}^{2k-1} \rangle
= - \sum_{\ell=0}^{\infty} \tilde{a}_{s+\ell}^* (b_{n,u,\ell}^{2k-1})^*,
\]
and from (\ref{eq:MAdual111}) and (\ref{eq:expan222}) that
\[
\langle X^{\prime}_s, W_{n,u}^{2k} \rangle
= - \sum_{\ell=0}^{\infty} a_{n+1-s+\ell}^* (b_{n,u,\ell}^{2k})^*.
\]
Therefore, by Lemma \ref{lem:basic-rep123},
\[
\langle X^{\prime}_s, P_{[1,n]}^{\perp} \varepsilon_u \rangle
= - \sum_{k=1}^{\infty} \langle X^{\prime}_s, W_{n,u}^{k} \rangle
= \sum_{k=1}^{\infty}
\left\{
\sum_{\ell=0}^{\infty} b_{n,u,\ell}^{2k-1} \tilde{a}_{s+\ell}
+
\sum_{\ell=0}^{\infty} b_{n,u,\ell}^{2k} a_{n+1-s+\ell}
\right\}^*.
\]
The assertion (ii) follows from this, Theorem \ref{thm:Tdual123} and Lemma \ref{lem:Tdual456}.
\end{proof}
\section{Strong convergence result for Toeplitz systems}\label{sec:4}
In this section, we use Theorem \ref{thm:TforXwithM123} to show a strong convergence
result for solutions of block Toeplitz systems.
We assume (\ref{eq:A}) and (\ref{eq:S}).
Then $w$ is continuous on $\T$ since
$w(e^{i\theta})=(2\pi)^{-1}\sum_{k\in\Z} e^{ik\theta} \gamma(k)$.
In particular, (\ref{eq:M}) is also satisfied.
The conditions (\ref{eq:A}) and (\ref{eq:S}) also imply
that all of $\{a_k\}$, $\{c_k\}$, $\{\tilde{a}_k\}$ and
$\{\tilde{c}_k\}$ belong to $\ell_{1+}^{d\times d}$.
See Theorem 3.3 and (3.3) in \cite{KB18}; see also Theorem 4.1 in \cite{I08}.
In particular, we have
$h(e^{i\theta})^{-1} = - \sum_{k=0}^{\infty} e^{ik\theta} a_k$ and
$h_{\sharp}(e^{i\theta}) = \tilde{h}(e^{-i\theta})^* = \sum_{k=0}^{\infty} e^{ik\theta} \tilde{c}_k^*$,
hence, by (\ref{eq:beta-def667}),
\begin{equation}
\beta_k = \sum_{j=0}^{\infty} a_{j+k} \tilde{c}_j,\qquad k \in \N\cup\{0\}.
\label{eq:beta-a-c111}
\end{equation}
Under (\ref{eq:A}) and (\ref{eq:S}), we define
\[
F(n):=\left(\sum_{j=0}^{\infty}\Vert \tilde{c}_j\Vert\right)
\sum_{\ell=n}^{\infty}\Vert a_{\ell}\Vert, \qquad
n \in \N\cup\{0\}.
\]
Then $F(n)$ decreases to zero as $n\to\infty$.
We need the next lemma in the proof of Theorem \ref{thm:Bax-conv123} below.
\begin{lemma}\label{lem:estim4.2}
We assume (\ref{eq:A}) and (\ref{eq:S}). Then, for $n, k\in\N$ and $u\in\{1,\dots,n\}$, we have
\begin{equation}
\sum_{\ell=0}^{\infty}
\Vert \tilde{b}^{k}_{n,u,\ell}\Vert\le F(n+1)^{k-1}F(n+1-u).
\label{eq:ac-ineq985}
\end{equation}
\end{lemma}
\begin{proof}
For $m\in\N$,
we see from (\ref{eq:beta-a-c111}) that
\[
\sum_{\ell=0}^{\infty} \Vert \beta_{m+\ell}\Vert
\le \sum_{j=0}^{\infty} \Vert \tilde{c}_j\Vert \sum_{\ell=0}^{\infty} \Vert a_{m+j+\ell}\Vert
\le \sum_{j=0}^{\infty} \Vert \tilde{c}_j\Vert \sum_{\ell=m}^{\infty} \Vert a_{\ell}\Vert,
\]
hence
\begin{equation}
\sum_{\ell=0}^{\infty} \Vert \beta_{m+\ell}\Vert \le F(m).
\label{eq:beta-ineq519}
\end{equation}
Let $n\in\N$ and $u\in\{1,\dots,n\}$. We use induction on $k$ to prove (\ref{eq:ac-ineq985}).
Since $\tilde{b}^1_{n,u,\ell}=\beta_{n+1-u+\ell}^*$, we see from (\ref{eq:beta-ineq519}) that
\[
\sum_{\ell=0}^{\infty} \Vert \tilde{b}^1_{n,u,\ell} \Vert
=\sum_{\ell=0}^{\infty}\Vert \beta_{n+1-u+\ell} \Vert
\le F(n+1-u).
\]
We assume (\ref{eq:ac-ineq985}) for $k\in\N$.
Then, again by (\ref{eq:beta-ineq519}),
\[
\begin{aligned}
\sum_{\ell=0}^{\infty}
\Vert \tilde{b}^{k+1}_{n,u,\ell} \Vert
&\le \sum_{m=0}^{\infty} \Vert \tilde{b}^{k}_{n,u,m} \Vert
\sum_{\ell=0}^{\infty} \Vert \beta_{n + 1 + m + \ell}\Vert\\
&\le F(n+1) \sum_{m=0}^{\infty} \Vert \tilde{b}^k_{n,u,m} \Vert \le F(n+1)^{k}F(n+1-u).
\end{aligned}
\]
Thus (\ref{eq:ac-ineq985}) with $k$ replaced by $k+1$ also holds.
\end{proof}
For $\{y_k\}_{k=1}^{\infty} \in \ell_1^{d\times d}(\N)$, the solution
$Z_{\infty}$ to (\ref{eq:TSinfty183}) with (\ref{eq:Tinfty123}) and (\ref{eq:Rrinfty-314}) is
given by (\ref{eq:Zzinfty-314}) with
\begin{equation}
z_s = \sum_{t=1}^{\infty} \sum_{\ell = 1}^{s\wedge t}
\tilde{a}_{s - \ell}^* \tilde{a}_{t - \ell} y_t \in \C^{d\times d}, \qquad s\in\N
\label{eq:z-sequence123}
\end{equation}
(see Remark \ref{rem:1st-term-123} in Section \ref{sec:2}).
Notice that the sum in (\ref{eq:z-sequence123}) converges absolutely.
\begin{theorem}\label{thm:Bax-conv123}
We assume (\ref{eq:A}) and (\ref{eq:S}). Let $\{y_k\}_{k=1}^{\infty} \in \ell_1^{d\times d}(\N)$.
Then, for
$Z_n$ in (\ref{eq:Zz-314})--(\ref{eq:Rr-314}) and
$Z_{\infty}$ in (\ref{eq:Zzinfty-314})--(\ref{eq:Rrinfty-314}),
we have (\ref{eq:conv-z-234}).
\end{theorem}
\begin{proof}
By Theorem \ref{thm:TforXwithM123} (i), we have
\[
\begin{aligned}
z_{n,s} &= \sum_{t=1}^{n} \sum_{\ell = 1}^{s\wedge t}
\tilde{a}_{s - \ell}^* \tilde{a}_{t - \ell} y_t
+ \sum_{t=1}^{n} \sum_{u=1}^t
\sum_{\ell = 0}^{\infty} a_{n + 1 - s + \ell}^* \beta_{n+1-u+\ell} \tilde{a}_{t-u} y_t\\
&\quad\quad + \sum_{t=1}^{n} \sum_{u=1}^t \sum_{k=1}^{\infty}
\left\{
\sum_{\ell = 0}^{\infty} \tilde{b}_{n,u,\ell}^{2k+1} a_{n + 1 - s + \ell}
+ \sum_{\ell = 0}^{\infty} \tilde{b}_{n,u,\ell}^{2k} \tilde{a}_{s + \ell}
\right\}^* \tilde{a}_{t-u} y_t,
\end{aligned}
\]
hence, by (\ref{eq:z-sequence123}),
$\sum_{s=1}^n \Vert z_{n,s} - z_s\Vert
\le S_1(n) + S_2(n) + S_3(n) + S_4(n)$,
where
\begin{align*}
S_1(n) &:= \sum_{t=n+1}^{\infty} \sum_{s=1}^n \sum_{\ell = 1}^{s}
\Vert \tilde{a}_{s - \ell}\Vert \Vert \tilde{a}_{t - \ell}\Vert \Vert y_t\Vert,\\
S_2(n) &:= \sum_{s=1}^n \sum_{t=1}^{n} \sum_{u=1}^t \sum_{\ell = 0}^{\infty}
\Vert a_{n + 1 - s + \ell}\Vert \Vert \beta_{n+1-u+\ell}\Vert \Vert \tilde{a}_{t-u}\Vert
\Vert y_t\Vert,\\
S_3(n) &:= \sum_{s=1}^n \sum_{t=1}^{n} \sum_{u=1}^t \sum_{k=1}^{\infty} \sum_{\ell = 0}^{\infty}
\Vert \tilde{b}_{n,u,\ell}^{2k+1}\Vert \Vert a_{n + 1 - s + \ell}\Vert
\Vert \tilde{a}_{t-u}\Vert \Vert y_t\Vert
\end{align*}
and
\[
S_4(n) = \sum_{s=1}^n \sum_{t=1}^{n} \sum_{u=1}^t \sum_{k=1}^{\infty} \sum_{\ell = 0}^{\infty}
\Vert \tilde{b}_{n,u,\ell}^{2k}\Vert \Vert \tilde{a}_{s + \ell}\Vert
\Vert \tilde{a}_{t-u}\Vert \Vert y_t\Vert.
\]
By the change of variables $m=s-\ell+1$, we have
\[
\begin{aligned}
S_1(n)
&= \sum_{t=n+1}^{\infty} \sum_{s=1}^n \sum_{m = 1}^{s}
\Vert \tilde{a}_{m - 1}\Vert \Vert \tilde{a}_{t + m -s - 1}\Vert \Vert y_t\Vert\\
&=\sum_{t=n+1}^{\infty} \Vert y_t\Vert \sum_{m = 1}^{n} \Vert \tilde{a}_{m - 1}\Vert \sum_{s=m}^n
\Vert \tilde{a}_{t + m -s - 1}\Vert\\
&\le \left( \sum_{k = 0}^{\infty} \Vert \tilde{a}_{k}\Vert\right)^2 \sum_{t=n+1}^{\infty} \Vert y_t\Vert
\quad \to \quad 0,\qquad n\to\infty.
\end{aligned}
\]
By (\ref{eq:ac-ineq985}) with $k=1$ or (\ref{eq:beta-ineq519}), we have
\[
\begin{aligned}
S_2(n) &= \sum_{t=1}^{n} \sum_{u=1}^t
\Vert \tilde{a}_{t-u}\Vert \Vert y_t\Vert
\sum_{\ell = 0}^{\infty} \Vert \beta_{n+1-u+\ell}\Vert
\sum_{s=1}^n \Vert a_{n + 1 - s + \ell} \Vert\\
&\le \left( \sum_{s=1}^{\infty} \Vert a_{s}\Vert\right)
\sum_{t=1}^{n} \sum_{u=1}^t
\Vert \tilde{a}_{t-u}\Vert \Vert y_t\Vert F(n+1-u).
\end{aligned}
\]
Furthermore, by the change of variables $v=t-u+1$, we obtain
\[
\begin{aligned}
\sum_{t=1}^{n} \sum_{u=1}^t
\Vert \tilde{a}_{t-u}\Vert \Vert y_t\Vert F(n+1-u)
&=\sum_{t=1}^{\infty} \sum_{u=1}^t
\Vert \tilde{a}_{t-u}\Vert \Vert y_t\Vert 1_{[0,n]}(t)F(n+1-u)\\
&=\sum_{t=1}^{\infty} \sum_{v=1}^t
\Vert \tilde{a}_{v-1}\Vert \Vert y_t\Vert 1_{[0,n]}(t)F(n-t+v)\\
&\le \sum_{t=1}^{\infty} \sum_{v=1}^{\infty}
\Vert \tilde{a}_{v-1}\Vert \Vert y_t\Vert 1_{[0,n]}(t)F(n-t+v).
\end{aligned}
\]
Since
\begin{gather*}
\lim_{n\to\infty} \Vert \tilde{a}_{v-1}\Vert \Vert y_t\Vert 1_{[0,n]}(t)F(n-t+v)
=0,\qquad t, v\in\N,\\
\Vert \tilde{a}_{v-1}\Vert \Vert y_t\Vert 1_{[0,n]}(t)F(n-t+v)
\le F(1)\Vert \tilde{a}_{v-1}\Vert \Vert y_t\Vert,\qquad t, v\in\N,\\
\sum_{t=1}^{\infty} \sum_{v=1}^{\infty}
\Vert \tilde{a}_{v-1}\Vert \Vert y_t\Vert <\infty,
\end{gather*}
the dominated convergence theorem yields
\[
\lim_{n\to\infty} \sum_{t=1}^{\infty} \sum_{v=1}^{\infty}
\Vert \tilde{a}_{v-1}\Vert \Vert y_t\Vert 1_{[0,n]}(t)F(n-t+v) = 0,
\]
hence $\lim_{n\to\infty} S_2(n) = 0$.
Choose $N\in\N$ such that $F(N+1)<1$. Then, by Lemma \ref{lem:estim4.2}, we have,
for $n\ge N$,
\[
\begin{aligned}
S_3(n)&=\sum_{t=1}^{n} \sum_{u=1}^t
\Vert \tilde{a}_{t-u}\Vert \Vert y_t\Vert
\sum_{k=1}^{\infty} \sum_{\ell = 0}^{\infty} \Vert \tilde{b}_{n,u,\ell}^{2k+1}\Vert
\sum_{s=1}^n \Vert a_{n + 1 - s + \ell}\Vert\\
&\le F(1)\left( \sum_{s=1}^{\infty} \Vert a_{s}\Vert\right)
\sum_{t=1}^{n} \Vert y_t\Vert \sum_{u=1}^t
\Vert \tilde{a}_{t-u}\Vert
\sum_{k=1}^{\infty} F(n+1)^{2k}\\
&\le F(1)\left( \sum_{s=1}^{\infty} \Vert a_{s}\Vert\right)
\left( \sum_{u=0}^{\infty} \Vert \tilde{a}_{u}\Vert\right)
\left(\sum_{t=1}^{\infty} \Vert y_t\Vert\right)
\frac{F(n+1)^2}{1-F(n+1)^2}.
\end{aligned}
\]
Thus $\lim_{n\to\infty}S_3(n)=0$. Similarly, we have, for $n\ge N$,
\[
S_4(n)
\le F(1)
\left( \sum_{s=0}^{\infty} \Vert \tilde{a}_{s}\Vert\right)^2
\left(\sum_{t=1}^{\infty} \Vert y_t\Vert\right)
\frac{F(n+1)}{1-F(n+1)^2},
\]
hence $\lim_{n\to\infty}S_4(n)=0$.
Combining, we obtain (\ref{eq:conv-z-234}).
\end{proof}
\section{Closed-form formulas}\label{sec:5}
In this section, we use Theorem \ref{thm:TforXwithM123} to derive
closed-form formulas for $T_n(w)^{-1}$ with rational symbol $w$.
We assume that the symbol $w$ of $T_n(w)$ is of the form (\ref{eq:farima529})
with $h:\T\to \C^{d\times d}$ satisfying (\ref{eq:C}).
Then $h$ is an outer function in $H_2^{d\times d}(\T)$, and
another outer function $h_{\sharp}\in H_2^{d\times d}(\T)$ that appears in (\ref{eq:decomp888})
also satisfies (\ref{eq:C}); see Section 6.2 in \cite{IKP2}.
Notice that (\ref{eq:farima529}) with (\ref{eq:C}) implies (\ref{eq:A}) and (\ref{eq:M}).
We can write $h(z)^{-1}$ in the form
\begin{equation}
h(z)^{-1} = - \rho_{0,0} - \sum_{{\mu}=1}^{K} \sum_{j=1}^{m_{\mu}} \frac{1}{(1-\barp_{\mu}z)^j}\rho_{{\mu}, j}
- \sum_{j=1}^{m_0} z^j \rho_{0,j},
\label{eq:hinverse162}
\end{equation}
where
\begin{equation}
\left\{
\begin{aligned}
&K\in\N\cup\{0\}, \qquad m_{\mu}\in\N, \quad \mu \in \{1,\dots,K\}, \qquad m_0\in\N\cup\{0\},\\
&p_{\mu}\in\D\setminus \{0\}, \quad \mu \in \{1,\dots,K\}, \qquad p_{\mu}\neq p_{\nu}, \quad \mu\neq \nu,\\
&\rho_{{\mu}, j}\in\C^{d\times d}, \quad \mu \in \{0,\dots,K\},\ j \in \{1,\dots,m_{\mu}\},
\qquad \rho_{0,0} \in\C^{d\times d},\\
&\rho_{{\mu},m_{\mu}}\neq 0, \quad \mu \in \{1,\dots,K\},\\
&\rho_{0,m_0}\neq 0\quad \mbox{if}\ \ m_0\ge 1.
\end{aligned}
\right.
\label{eq:hinverse163}
\end{equation}
Here the convention $\sum_{k=1}^0=0$ is adopted in the sums on the right-hand side of (\ref{eq:hinverse162}).
For example, if $m_0=0$, then
\[
h(z)^{-1} = - \rho_{0,0} - \sum_{{\mu}=1}^{K} \sum_{j=1}^{m_{\mu}} \frac{1}{(1-\barp_{\mu}z)^j}\rho_{{\mu}, j},
\]
while, if $K=0$, then
\begin{equation}
h(z)^{-1} = - \rho_{0,0} - \sum_{j=1}^{m_0} z^j \rho_{0,j}.
\label{eq:K000}
\end{equation}
By Theorem 2 in \cite{I20},
$h_{\sharp}^{-1}$ has the same $m_0$ and the same poles
with the same multiplicities as $h^{-1}$,
that is,
for $m_0$, $K$ and $(p_1, m_1), \dots, (p_K, m_K)$ in
$(\ref{eq:hinverse162})$ with $(\ref{eq:hinverse163})$,
$h_{\sharp}^{-1}$ has the form
\begin{equation}
h_{\sharp}(z)^{-1} = - \rho_{0,0}^{\sharp} -
\sum_{{\mu}=1}^{K} \sum_{j=1}^{m_{\mu}} \frac{1}{(1-\barp_{\mu}z)^j}\rho_{{\mu}, j}^{\sharp}
- \sum_{j=1}^{m_0} z^j \rho_{0,j}^{\sharp},
\label{eq:hsharpinv162}
\end{equation}
where
\[
\left\{
\begin{aligned}
&\rho_{{\mu}, j}^{\sharp}\in\C^{d\times d}, \quad
\mu \in \{0,\dots,K\},\ j \in \{1,\dots,m_{\mu}\}, \qquad
\rho_{0,0}^{\sharp} \in\C^{d\times d},\\
&\rho_{{\mu},m_{\mu}}^{\sharp}\neq 0, \quad \mu \in \{1,\dots,K\},\\
&\rho_{0,m_0}^{\sharp}\neq 0\quad \mbox{if}\ \ m_0\ge 1.
\end{aligned}
\right.
\]
Notice that if $d=1$, then we can take $h_{\sharp}=h$, hence $\rho_{0,0}=\rho_{0,0}^{\sharp}$
and $\rho_{{\mu}, j} = \rho_{{\mu}, j}^{\sharp}$ for $\mu\in\{1,\dots,K\}$ and $j\in\{1,\dots,m_{\mu}\}$.
Recall $\tilde{h}$ from (\ref{eq:outft628}). From (\ref{eq:hsharpinv162}),
we have
\[
\tilde{h}(z)^{-1} = - \tilde{\rho}_{0,0} -
\sum_{{\mu}=1}^{K} \sum_{j=1}^{m_{\mu}} \frac{1}{(1 - p_{\mu} z)^j} \tilde{\rho}_{{\mu}, j}
- \sum_{j=1}^{m_0} z^j \tilde{\rho}_{0,j},
\]
where
\[
\tilde{\rho}_{0,0} := (\rho_{0,0}^{\sharp})^*, \qquad
\tilde{\rho}_{\mu,j} := (\rho_{\mu, j}^{\sharp})^*, \quad
\mu \in \{0,\dots,K\},\ j \in \{1,\dots,m_{\mu}\}.
\]
Recall the sequences $\{a_k\}$ and $\{\tilde{a}_k\}$ from (\ref{eq:AR111}) and (\ref{eq:AR222}), respectively.
We have
\begin{align}
a_n &= \sum_{{\mu}=1}^{K} \sum_{j=1}^{m_{\mu}} \binom{n+j-1}{j-1} \barp_{\mu}^n
\rho_{{\mu}, j}, \qquad n\ge m_0 + 1,
\label{eq:a363}\\
\tilde{a}_n &= \sum_{{\mu}=1}^{K} \sum_{j=1}^{m_{\mu}} \binom{n+j-1}{j-1} p_{\mu}^n
\tilde{\rho}_{{\mu}, j}, \qquad n\ge m_0 + 1
\label{eq:atilde361}
\end{align}
and
\begin{align}
a_n &= \rho_{0,n} + \sum_{{\mu}=1}^{K} \sum_{j=1}^{m_{\mu}} \binom{n+j-1}{j-1} \barp_{\mu}^n
\rho_{{\mu}, j}, \qquad n \in \{0,\dots,m_0\},
\label{eq:a364}\\
\tilde{a}_n &= \tilde{\rho}_{0,n} + \sum_{{\mu}=1}^{K} \sum_{j=1}^{m_{\mu}} \binom{n+j-1}{j-1} p_{\mu}^n
\tilde{\rho}_{{\mu}, j}, \qquad n \in \{0,\dots,m_0\},
\label{eq:atilde362}
\end{align}
where the convention $\binom{0}{0}=1$ is adopted; see Proposition 4 in \cite{I20}.
We first consider the case of $K = 0$. As can be seen from the following theorem,
in this case, we have simple closed-form formulas for $T_n(w)^{-1}$.
\begin{theorem}\label{thm:TforR777}
We assume (\ref{eq:farima529}), (\ref{eq:C}) and $K = 0$
for $K$ in (\ref{eq:hinverse162}).
Thus we assume (\ref{eq:K000}).
Then the following four assertions hold.
\begin{enumerate}
\item For $n \ge m_0 + 1$, $s \in \{1,\dots,n\}$ and $t \in \{1, \dots, n-m_0\}$, we have
\begin{equation}
\left(T_n(w)^{-1}\right)^{s,t}
= \sum_{\lambda=1}^{s\wedge t} \tilde{a}_{s-\lambda}^* \tilde{a}_{t-\lambda}.
\label{eq:TforR777}
\end{equation}
\item For $n \ge m_0 + 1$, $s \in \{1, \dots, n-m_0\}$ and $t \in \{1,\dots,n\}$, we have
\begin{equation}
\left(T_n(w)^{-1}\right)^{s,t}
= \sum_{\lambda=1}^{s\wedge t} \tilde{a}_{s-\lambda}^* \tilde{a}_{t-\lambda}.
\label{eq:TforR999}
\end{equation}
\item
For $n \ge m_0 + 1$, $s \in \{1,\dots,n\}$ and $t \in \{m_0+1, \dots, n\}$, we have
\begin{equation}
\left(T_n(w)^{-1}\right)^{s,t}
= \sum_{\lambda=s\vee t}^n a_{\lambda-s}^* a_{\lambda-t}.
\label{eq:TforR666}
\end{equation}
\item
For $n \ge m_0 + 1$, $s \in \{m_0+1, \dots, n\}$ and $t \in \{1,\dots,n\}$, we have
\begin{equation}
\left(T_n(w)^{-1}\right)^{s,t}
= \sum_{\lambda=s\vee t}^n a_{\lambda-s}^* a_{\lambda-t}.
\label{eq:TforR888}
\end{equation}
\end{enumerate}
\end{theorem}
\begin{proof}
For $w$ satisfying (\ref{eq:farima529}), (\ref{eq:C}) and $K = 0$,
let $\{X_k\}$, $\{X^{\prime}_k\}$, $\{\varepsilon_k\}$ and $\{\tilde{\varepsilon}_k\}$
be as in Section \ref{sec:3}.
(i)\quad By (\ref{eq:hsharpinv162}) with $K=0$, we have $\tilde{a}_0=\tilde{\rho}_0$,
$\tilde{a}_k=\tilde{\rho}_{0,k}$ for $k\in\{1,\dots,m_0\}$ and $\tilde{a}_k=0$ for
$k \ge m_0 + 1$. In particular, we have
$\sum_{k=0}^{m_0}\tilde{a}_{k}X_{u+k} + \tilde{\varepsilon}_{-u} = 0$ for $u\in\Z$; see (2.15) in \cite{IKP2}.
This implies $\tilde{\varepsilon}_{-u} \in V_{[1,n]}^X$, or $P_{[1,n]}^{\bot} \tilde{\varepsilon}_{-u} = 0$,
for $u\in\{1,\dots,n-m_0\}$.
Therefore, (\ref{eq:TforR777}) follows from Theorem \ref{thm:Tdual123} and (\ref{eq:dual888}).
(iii)\quad By (\ref{eq:K000}), we have $a_0=\rho_{0,0}$,
$a_k=\rho_{0,k}$ for $k\in\{1,\dots,m_0\}$ and $a_k=0$ for $k \ge m_0 + 1$. In particular,
$\sum_{k=0}^{m_0}a_{k}X_{u-k} + \varepsilon_u = 0$ for $u\in\Z$; see (2.15) in \cite{IKP2}.
This implies $\varepsilon_u \in V_{[1,n]}^X$, or $P_{[1,n]}^{\bot} \varepsilon_u = 0$,
for $u\in\{m_0+1, \dots,n\}$.
Therefore, (\ref{eq:TforR666}) follows from Theorem \ref{thm:Tdual123} and (\ref{eq:dual777}).
(ii), (iv)\quad By (\ref{eq:SA-123}), (ii) and (iv) follow from (i) and (iii), respectively.
\end{proof}
We turn to the case of $K \ge 1$.
In what follows in this section, for $K$ in (\ref{eq:hinverse162}), we assume
\[
K\ge 1.
\]
For $m_1,\dots,m_K$ in (\ref{eq:hinverse162}), we define $M\in \N$ by
\begin{equation}
M:=\sum_{\mu=1}^K m_{\mu}.
\label{eq:sum-mmu123}
\end{equation}
For $\mu\in\{1,\dots,K\}$, $p_{\mu}$ in (\ref{eq:hinverse162}) and $i\in\N$,
we define $p_{{\mu}, i}: \Z \to \C^{d\times d}$ by
\begin{equation}
p_{{\mu},i}(k):=
\binom{k}{i-1} p_{\mu}^{k - i+1} I_d, \qquad k\in\Z.
\label{eq:p362}
\end{equation}
Notice that
\[
p_{\mu,i}(0)=\binom{0}{i-1}p_{\mu}^{-i+1} I_d = \delta_{i, 1} I_d.
\]
For $n\in \Z$, we also define $\p_n \in \C^{dM\times d}$ by the
following block representation:
\[
\begin{aligned}
\p_n
&:=(p_{1, 1}(n), \dots, p_{1, m_1}(n)\ \vert\ p_{2, 1}(n), \dots, p_{2, m_2}(n)\ \vert \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\cdots \vert\
p_{K, 1}(n), \dots, p_{K, m_{K}}(n))^{\top}.
\end{aligned}
\]
Notice that
\[
\p_0 = (I_d, 0, \dots, 0\ \vert\ I_d, 0, \dots, 0 \vert
\cdots \vert\
I_d, 0, \dots, 0)^{\top} \in \C^{dM\times d}.
\]
We define $\Lambda\in\C^{dM\times dM}$ by
\[
\Lambda := \sum_{\ell=0}^{\infty} \p_{\ell} \p_{\ell}^*.
\]
For ${\mu}, {\nu}\in\{1,2,\dots,K\}$, we define $\Lambda^{{\mu},{\nu}}\in \C^{dm_{\mu}\times dm_{\nu}}$ by the
block representation
\[
\Lambda^{{\mu},{\nu}} := \left(
\begin{matrix}
\lambda^{{\mu}, {\nu}}(1, 1) & \lambda^{{\mu}, {\nu}}(1, 2) & \cdots & \lambda^{{\mu}, {\nu}}(1, m_{\nu}) \cr
\lambda^{{\mu}, {\nu}}(2, 1) & \lambda^{{\mu}, {\nu}}(2, 2) & \cdots & \lambda^{{\mu}, {\nu}}(2, m_{\nu}) \cr
\vdots & \vdots & & \vdots \cr
\lambda^{\mu,\nu}(m_{\mu},1) & \lambda^{\mu,\nu}(m_{\mu},2) & \cdots & \lambda^{\mu,\nu}(m_{\mu},m_{\nu}) \cr
\end{matrix}
\right),
\]
where, for $i \in \{1,\dots,m_{\mu}\}$ and $j \in \{1,\dots,m_{\nu}\}$,
\[
\lambda^{{\mu},{\nu}}(i, j)
:= \sum_{r=0}^{j-1}
\binom{i-1}{r} \binom{i+j-r-2}{i-1}
\frac{p_{\mu}^{j - r -1}\barp_{\nu}^{i-r-1}}{(1-p_{\mu}\barp_{\nu})^{i+j-r-1}} I_d
\in \C^{d\times d}.
\]
Then, by Lemma 3 in \cite{I20}, the matrix $\Lambda$ has the following block representation:
\[
\Lambda=\left(
\begin{matrix}
\Lambda^{1, 1} & \Lambda^{1, 2} & \cdots & \Lambda^{1, K} \cr
\Lambda^{2, 1} & \Lambda^{2, 2} & \cdots & \Lambda^{2, K} \cr
\vdots & \vdots & \ddots & \vdots \cr
\Lambda^{K, 1} & \Lambda^{K, 2} & \cdots & \Lambda^{K, K} \cr
\end{matrix}
\right).
\]
We define, for $\mu \in \{1,\dots,K\}$ and $j \in \{1,\dots,m_{\mu}\}$,
\begin{equation}
\theta_{{\mu}, j} := - \lim_{z\to p_{\mu}} \frac{1}{(m_{\mu} - j)!}
\frac{d^{m_{\mu} - j}}{dz^{m_{\mu} - j}} \left\{(z-p_{\mu})^{m_{\mu}} h_{\sharp}(z) h^{\dagger}(z)^{-1}\right\}
\in \C^{d\times d},
\label{eq:theta456}
\end{equation}
where
\begin{equation}
h^{\dagger}(z) := h(1/\overline{z})^*.
\label{eq:hdagger456}
\end{equation}
We define
$\Theta\in \C^{dM\times dM}$ by the block representation
\[
\Theta:=
\left(
\begin{matrix}
\Theta_1 & 0 & \cdots & 0\cr
0 & \Theta_2 & \cdots & 0\cr
\vdots & \vdots & \ddots & \vdots \cr
0 & 0 & \cdots & \Theta_{K}
\end{matrix}
\right),
\]
where, for $\mu \in \{1,\dots,K\}$, $\Theta_{\mu} \in \C^{dm_{\mu}\times dm_{\mu}}$ is defined by
\[
\Theta_{\mu} :=
\left(
\begin{matrix}
\theta_{{\mu}, 1} & \theta_{{\mu}, 2} & \cdots & \theta_{{\mu}, m_{\mu}-1} & \theta_{{\mu}, m_{\mu}} \cr
\theta_{{\mu}, 2} & \theta_{{\mu}, 3} & \cdots & \theta_{{\mu}, m_{\mu}} & \cr
\vdots & \vdots & & & \cr
\theta_{{\mu}, m_{\mu}-1} & \theta_{{\mu}, m_{\mu}} & & & \cr
\theta_{{\mu}, m_{\mu}} & & & & \mbox{\huge 0}
\end{matrix}
\right)
\]
using $\theta_{\mu,j}$ in (\ref{eq:theta456}) with (\ref{eq:hdagger456}).
For $n\in \Z$, we define
$\Pi_n\in \C^{dM\times dM}$ by the block representation
\[
\Pi_n:=
\left(
\begin{matrix}
\Pi_{1, n} & 0 & \cdots & 0\cr
0 & \Pi_{2, n} & \cdots & 0\cr
\vdots & \vdots & \ddots & \vdots \cr
0 & 0 & \cdots & \Pi_{K, n}
\end{matrix}
\right),
\]
where, for $\mu \in \{1,\dots,K\}$ and $n\in\Z$, $\Pi_{\mu, n} \in \C^{dm_{\mu}\times dm_{\mu}}$ is defined by
\[
\Pi_{{\mu},n} :=
\left(
\begin{matrix}
p_{{\mu}, 1}(n) & p_{{\mu}, 2}(n) & p_{{\mu}, 3}(n) & \cdots & p_{{\mu}, m_{\mu}}(n) \cr
& p_{{\mu}, 1}(n) & p_{{\mu}, 2}(n) & \cdots & p_{{\mu}, m_{\mu}-1}(n) \cr
& & \ddots & \ddots & \vdots \cr
& & & \ddots & p_{{\mu}, 2}(n) \cr
\mbox{\huge 0} & & & & p_{{\mu}, 1}(n)
\end{matrix}
\right)
\]
using $p_{{\mu}, i}(n)$ in (\ref{eq:p362}).
The next lemma slightly extends Lemma 17 in \cite{I20}.
\begin{lemma}\label{lem:beta162}
We assume (\ref{eq:farima529}), (\ref{eq:C}) and $K \ge 1$ for $K$ in (\ref{eq:hinverse162}).
Then,
for $n, k, \ell\in\Z$ such that $n+k+\ell\ge m_0$, we have
\[
\beta_{n+k+\ell+1}^* = \p_{\ell}^{\top} \Pi_n \Theta \p_k,
\]
hence
\[
\beta_{n+k+\ell+1} = \p_k^* (\Pi_n \Theta)^* \overline{\p}_{\ell}.
\]
\end{lemma}
The proof of Lemma \ref{lem:beta162} is almost the same as that of Lemma 17 in \cite{I20}, hence we omit it.
For $n\in \Z$, we define $G_n, \tilde{G}_n\in \C^{dM\times dM}$ by
\[
G_n := \Pi_n \Theta \Lambda,
\qquad
\tilde{G}_n := (\Pi_n \Theta)^* \Lambda^{\top}.
\]
\begin{lemma}\label{lem:bk123}
We assume (\ref{eq:farima529}), (\ref{eq:C}) and $K \ge 1$
for $K$ in (\ref{eq:hinverse162}). Then the following two
assertions hold.
\begin{enumerate}
\item
We assume $n\ge u\ge m_0+1$. Then, for $k\in\N$ and $\ell\in\mathbb{N}\cup\{0\}$, we have
\begin{align}
b_{n,u,\ell}^{2k-1} &= \p_{u-n-1}^* ( \tilde{G}_n G_n )^{k-1} (\Pi_n \Theta)^* \overline{\p}_{\ell},
\label{eq:bodd123}\\
b_{n,u,\ell}^{2k} &= \p_{u-n-1}^* (\tilde{G}_n G_n )^{k-1} \tilde{G}_n \Pi_n \Theta \p_{\ell}.
\label{eq:beven123}
\end{align}
\item
We assume $1\le u\le n-m_0$. Then, for $k\in\N$ and $\ell\in\mathbb{N}\cup\{0\}$, we have
\begin{align}
\tilde{b}_{n,u,\ell}^{2k-1} &= \p_{-u}^{\top} ( G_n \tilde{G}_n )^{k-1} \Pi_n \Theta \p_{\ell},
\label{eq:til-bodd123}\\
\tilde{b}_{n,u,\ell}^{2k} &= \p_{-u}^{\top} (G_n \tilde{G}_n )^{k-1} G_n (\Pi_n \Theta)^* \overline{\p}_{\ell}.
\label{eq:til-beven123}
\end{align}
\end{enumerate}
\end{lemma}
The proof of Lemma \ref{lem:bk123} will be given in the Appendix.
For $n\in\N$ and ${\mu}, {\nu}\in\{1,2,\dots,K\}$,
we define $\Xi_n^{{\mu},{\nu}}\in \C^{dm_{\mu}\times dm_{\nu}}$ by
the block representation
\[
\Xi_n^{{\mu},{\nu}} := \left(
\begin{matrix}
\xi_n^{{\mu}, {\nu}}(1, 1) & \xi_n^{{\mu}, {\nu}}(1, 2) & \cdots & \xi_n^{{\mu}, {\nu}}(1, m_{\nu}) \cr
\xi_n^{{\mu}, {\nu}}(2, 1) & \xi_n^{{\mu}, {\nu}}(2, 2) & \cdots & \xi_n^{{\mu}, {\nu}}(2, m_{\nu}) \cr
\vdots & \vdots & & \vdots \cr
\xi_n^{\mu,\nu}(m_{\mu},1) & \xi_n^{\mu,\nu}(m_{\mu},2) & \cdots & \xi_n^{\mu,\nu}(m_{\mu},m_{\nu}) \cr
\end{matrix}
\right),
\]
where, for $n\in\N$, $i \in \{1,\dots,m_{\mu}\}$ and $j \in \{1,\dots,m_{\nu}\}$,
$\xi_n^{{\mu},{\nu}}(i, j) \in \C^{d\times d}$ is defined by
\[
\xi_n^{{\mu},{\nu}}(i, j)
:= \sum_{r=0}^{j-1}
\binom{n+i+j-2}{r} \binom{i+j-r-2}{i-1}
\frac{p_{\mu}^{j - r -1}\barp_{\nu}^{n+i+j-r-2}}{(1-p_{\mu}\barp_{\nu})^{i+j-r-1}} I_d.
\]
For $n\in\N$, we define $\Xi_n\in\C^{dM\times dM}$ by
\[
\Xi_n:=\left(
\begin{matrix}
\Xi_n^{1, 1} & \Xi_n^{1, 2} & \cdots & \Xi_n^{1, K} \cr
\Xi_n^{2, 1} & \Xi_n^{2, 2} & \cdots & \Xi_n^{2, K} \cr
\vdots & \vdots & \ddots & \vdots \cr
\Xi_n^{K, 1} & \Xi_n^{K, 2} & \cdots & \Xi_n^{K, K} \cr
\end{matrix}
\right).
\]
We also define $\rho \in \C^{dM\times d}$ and $\tilde{\rho} \in \C^{dM\times d}$
by the block representations
\[
\rho
:=(\rho_{1, 1}^{\top}, \dots, \rho_{1, m_1}^{\top} \ \vert \
\rho_{2, 1}^{\top}, \dots, \rho_{2, m_2}^{\top} \ \vert \
\cdots \ \vert \
\rho_{K, 1}^{\top}, \dots, \rho_{K, m_{K}}^{\top})^{\top}
\]
and
\[
\begin{aligned}
\tilde{\rho}
&:=(\tilde{\rho}_{1, 1}^{\top}, \dots, \tilde{\rho}_{1, m_1}^{\top}
\ \vert \ \tilde{\rho}_{2, 1}^{\top}, \dots, \tilde{\rho}_{2, m_2}^{\top} \ \vert \
\cdots \ \vert \
\tilde{\rho}_{K, 1}^{\top}, \dots, \tilde{\rho}_{K, m_{K}}^{\top})^{\top}\\
&=\left(\overline{\rho_{1, 1}^{\sharp}}, \dots, \overline{\rho_{1, m_1}^{\sharp}}
\ \vert \ \overline{\rho_{2, 1}^{\sharp}}, \dots, \overline{\rho_{2, m_2}^{\sharp}} \ \vert \
\cdots \ \vert \
\overline{\rho_{K, 1}^{\sharp}}, \dots, \overline{\rho_{K, m_{K}}^{\sharp}}\right)^{\top},
\end{aligned}
\]
respectively.
For $n\in\N$, we define $v_n, \tilde{v}_n \in\C^{dM\times d}$ by
\[
v_n := \sum_{\ell=0}^{\infty} \p_{\ell} a_{n+\ell},
\qquad
\tilde{v}_n := \sum_{\ell=0}^{\infty} \overline{\p}_{\ell} \tilde{a}_{n+\ell}.
\]
Then, by Lemma 5 in \cite{I20}, we have
\[
v_n = \Xi_n \rho,
\qquad
\tilde{v}_n = \overline{\Xi}_n \tilde{\rho},
\qquad
n\ge m_0 + 1.
\]
Moreover, if $m_0\ge 1$, then we have
\[
v_n = \Xi_n \rho + \sum_{\ell=0}^{m_0-n} \p_{\ell} \rho_{0,n+\ell},
\qquad
\tilde{v}_n = \overline{\Xi}_n \tilde{\rho} + \sum_{\ell=0}^{m_0-n} \overline{\p}_{\ell} \tilde{\rho}_{0,n+\ell},
\qquad n \in \{1,\dots,m_0\}.
\]
For $n\in\Z$, we define $w_n, \tilde{w}_n \in\C^{dM\times d}$ by
\[
w_n := \sum_{\ell=0}^{\infty} \p_{\ell - n} a_{\ell},
\qquad
\tilde{w}_n := \sum_{\ell=0}^{\infty} \overline{\p}_{\ell - n} \tilde{a}_{\ell}.
\]
To give closed-form expressions for $w_n$ and $\tilde{w}_n$, we introduce some matrices.
For $n\in\Z$ and ${\mu}, {\nu}\in\{1,2,\dots,K\}$,
we define $\Phi_n^{{\mu},{\nu}}\in \C^{dm_{\mu}\times dm_{\nu}}$ by
the block representation
\[
\Phi_n^{{\mu},{\nu}} := \left(
\begin{matrix}
\varphi_n^{{\mu}, {\nu}}(1, 1) & \varphi_n^{{\mu}, {\nu}}(1, 2) & \cdots & \varphi_n^{{\mu}, {\nu}}(1, m_{\nu}) \cr
\varphi_n^{{\mu}, {\nu}}(2, 1) & \varphi_n^{{\mu}, {\nu}}(2, 2) & \cdots & \varphi_n^{{\mu}, {\nu}}(2, m_{\nu}) \cr
\vdots & \vdots & & \vdots \cr
\varphi_n^{\mu,\nu}(m_{\mu},1) & \varphi_n^{\mu,\nu}(m_{\mu},2) & \cdots & \varphi_n^{\mu,\nu}(m_{\mu},m_{\nu}) \cr
\end{matrix}
\right),
\]
where, for $n\in\Z$, $i=1,\dots,m_{\mu}$ and $j = 1,\dots,m_{\nu}$,
$\varphi_n^{{\mu},{\nu}}(i, j) \in \C^{d\times d}$ is defined by
\[
\varphi_n^{{\mu},{\nu}}(i, j)
:= \sum_{q=0}^{i-1} \sum_{r=0}^{j-1}
\binom{j-1}{r} \binom{r+q}{q} \binom{r - n}{i-q-1}
\frac{p_{\mu}^{r+q+1-i - n} \barp_{\nu}^{r+q}}{(1-p_{\mu}\barp_{\nu})^{r+q+1}} I_d.
\]
For $n\in\Z$, we define $\Phi_n\in\C^{dM\times dM}$ by
\[
\Phi_n:=\left(
\begin{matrix}
\Phi_n^{1, 1} & \Phi_n^{1, 2} & \cdots & \Phi_n^{1, K} \cr
\Phi_n^{2, 1} & \Phi_n^{2, 2} & \cdots & \Phi_n^{2, K} \cr
\vdots & \vdots & \ddots & \vdots \cr
\Phi_n^{K, 1} & \Phi_n^{K, 2} & \cdots & \Phi_n^{K, K} \cr
\end{matrix}
\right).
\]
Here are closed-form expressions for $w_n$ and $\tilde{w}_n$.
\begin{lemma}\label{lem:w516}
We have
\begin{align*}
w_n &= \Phi_n \rho + \sum_{\ell=0}^{m_0} \p_{\ell - n} \rho_{0,\ell}, \qquad n \in \Z,
\\
\tilde{w}_n &= \overline{\Phi}_n \tilde{\rho}
+ \sum_{\ell=0}^{m_0} \overline{\p}_{\ell - n} \tilde{\rho}_{0,\ell}, \qquad n \in \Z.
\end{align*}
\end{lemma}
The proof of Lemma \ref{lem:w516} will be given in the Appendix.
Recall $M$ from (\ref{eq:sum-mmu123}).
For $n \in \N$ and $s\in \{1,\dots,n\}$,
we define
\begin{align*}
\ell_{n,s} &:= \{w_{n+1-s} - v_{n+1-s}\}^* (I_{dM} - \tilde{G}_n G_n )^{-1} \in \C^{d \times dM},
\\
\tilde{\ell}_{n,s} &:= \{\tilde{w}_{s} - \tilde{v}_s\}^* (I_{dM} - G_n \tilde{G}_n )^{-1} \in \C^{d \times dM},
\\\
r_{n,s} &:= (\Pi_n \Theta)^* \tilde{v}_s + \tilde{G}_n \Pi_n \Theta v_{n+1-s} \in \C^{dM \times d}
\end{align*}
and
\[
\tilde{r}_{n,s} := \Pi_n \Theta v_{n+1-s} + G_n (\Pi_n \Theta)^* \tilde{v}_s \in \C^{dM \times d}.
\]
Here are closed-form formulas for $(T_n(w))^{-1}$ with $w$
satisfying (\ref{eq:C}) and $K \ge 1$.
\begin{theorem}\label{thm:TforR123}
We assume (\ref{eq:farima529}), (\ref{eq:C}) and $K \ge 1$ for $K$ in (\ref{eq:hinverse162}).
Then the following four assertions hold.
\begin{enumerate}
\item For $n \ge m_0 + 1$, $s \in \{1,\dots,n\}$ and $t \in \{1, \dots, n-m_0\}$, we have
\[
\left(T_n(w)^{-1}\right)^{s,t}
= \tilde{r}_{n,s}^* \tilde{\ell}_{n,t}^*
+ \sum_{\lambda=1}^{s\wedge t} \tilde{a}_{s-\lambda}^* \tilde{a}_{t-\lambda}.
\]
\item For $n \ge m_0 + 1$, $s \in \{1, \dots, n-m_0\}$ and $t \in \{1,\dots,n\}$, we have
\[
\left(T_n(w)^{-1}\right)^{s,t}
= \tilde{\ell}_{n,s} \tilde{r}_{n,t}
+ \sum_{\lambda=1}^{s\wedge t} \tilde{a}_{s-\lambda}^* \tilde{a}_{t-\lambda}.
\]
\item
For $n \ge m_0 + 1$, $s \in \{1,\dots,n\}$ and $t \in \{m_0+1, \dots, n\}$, we have
\[
\left(T_n(w)^{-1}\right)^{s,t}
= r_{n,s}^* \ell_{n,t}^*
+ \sum_{\lambda=s\vee t}^n a_{\lambda-s}^* a_{\lambda-t}.
\]
\item
For $n \ge m_0 + 1$, $s \in \{m_0+1, \dots, n\}$ and $t \in \{1,\dots,n\}$, we have
\[
\left(T_n(w)^{-1}\right)^{s,t}
= \ell_{n,s} r_{n,t}
+ \sum_{\lambda=s\vee t}^n a_{\lambda-s}^* a_{\lambda-t}.
\]
\end{enumerate}
\end{theorem}
\begin{proof}
(i)\quad We assume $n \ge m_0 + 1$, $s \in \{1,\dots,n\}$ and $t \in \{1, \dots, n-m_0\}$.
Then, by Lemma \ref{lem:bk123} (ii) above and Lemma 19 in \cite{I20}, we have
\[
\begin{aligned}
&\sum_{u=1}^t \sum_{k=1}^{\infty}
\left\{
\sum_{\lambda=0}^{\infty} \tilde{b}_{n,u,\lambda}^{2k-1} a_{n+1-s+\lambda}
\right\}^* \tilde{a}_{t-u}\\
&\qquad =\sum_{u=1}^t \sum_{k=1}^{\infty}
\left\{
\sum_{\lambda=0}^{\infty} \p_{-u}^{\top} ( G_n \tilde{G}_n )^{k-1} \Pi_n \Theta
\p_{\lambda} a_{n+1-s+\lambda}
\right\}^* \tilde{a}_{t-u}\\
&\qquad =\sum_{u=1}^t \sum_{k=1}^{\infty}
\left\{
\p_{-u}^{\top} ( G_n \tilde{G}_n )^{k-1} \Pi_n \Theta v_{n+1-s}
\right\}^* \tilde{a}_{t-u}\\
&\qquad =\sum_{u=1}^t
\left\{
\p_{-u}^{\top} ( I_{dM} - G_n \tilde{G}_n )^{-1} \Pi_n \Theta v_{n+1-s}
\right\}^* \tilde{a}_{t-u}\\
&\qquad =v_{n+1-s}^* (\Pi_n \Theta)^* (I_{dM} - \tilde{G}_n^* G_n^*)^{-1}
\sum_{u=1}^t \overline{\p}_{-u} \tilde{a}_{t-u}.
\end{aligned}
\]
Similarly, by Lemma \ref{lem:bk123} (ii) above and Lemma 19 in \cite{I20},
\[
\sum_{u=1}^t \sum_{k=1}^{\infty}
\left\{
\sum_{\lambda=0}^{\infty} \tilde{b}_{n,u,\lambda}^{2k} \tilde{a}_{s+\lambda}
\right\}^* \tilde{a}_{t-u}
= \tilde{v}_s^* \Pi_n \Theta G_n^* (I_{dM} - \tilde{G}_n^* G_n^*)^{-1}
\sum_{u=1}^t \overline{\p}_{-u} \tilde{a}_{t-u}.
\]
However,
$\sum_{u=1}^t \overline{\p}_{-u} \tilde{a}_{t-u}
=\sum_{\lambda=0}^{\infty} \overline{\p}_{\lambda-t} \tilde{a}_{\lambda}
-\sum_{\lambda=0}^{\infty} \overline{\p}_{\lambda} \tilde{a}_{t+\lambda}
=\tilde{w}_{t} - \tilde{v}_t$.
Therefore, the assertion (i) follows from Theorem \ref{thm:TforXwithM123} (i).
(iii)\quad We assume $n \ge m_0 + 1$, $s \in \{1,\dots,n\}$ and $t \in \{m_0+1, \dots, n\}$.
Then, by Lemma \ref{lem:bk123} (i) above and Lemma 19 in \cite{I20}, we have
\[
\begin{aligned}
&\sum_{u=t}^n \sum_{k=1}^{\infty}
\left\{
\sum_{\lambda=0}^{\infty} b_{n,u,\lambda}^{2k-1} \tilde{a}_{s+\lambda}
\right\}^* a_{u-t}\\
&\qquad =\sum_{u=t}^n \sum_{k=1}^{\infty}
\left\{
\sum_{\lambda=0}^{\infty} \p_{u-n-1}^* ( \tilde{G}_n G_n )^{k-1} (\Pi_n \Theta)^* \overline{\p}_{\lambda}
\tilde{a}_{s+\lambda}
\right\}^* a_{u-t}\\
&\qquad =\sum_{u=t}^n \sum_{k=1}^{\infty}
\left\{
\p_{u-n-1}^* ( \tilde{G}_n G_n )^{k-1} (\Pi_n \Theta)^* \tilde{v}_{s}
\right\}^* a_{u-t}\\
&\qquad =\sum_{u=t}^n
\left\{
\p_{u-n-1}^* ( I_{dM} - \tilde{G}_n G_n )^{-1} (\Pi_n \Theta)^* \tilde{v}_{s}
\right\}^* a_{u-t}\\
&\qquad = \tilde{v}_{s}^* \Pi_n \Theta (I_{dM} - G_n^* \tilde{G}_n^* )^{-1}\sum_{u=t}^n \p_{u-n-1} a_{u-t}.
\end{aligned}
\]
Similarly, by Lemma \ref{lem:bk123} (i) above and Lemma 19 in \cite{I20}, we have
\[
\begin{aligned}
&\sum_{u=t}^n \sum_{k=1}^{\infty}
\left\{
\sum_{\lambda=0}^{\infty} b_{n,u,\lambda}^{2k} a_{n+1-s+\lambda}
\right\}^* a_{u-t}\\
&\qquad
= v_{n+1-s}^* (\Pi_n \Theta)^* \tilde{G}_n^* (I_{dM} - G_n^* \tilde{G}_n^* )^{-1}
\sum_{u=t}^n \p_{u-n-1} a_{u-t}.
\end{aligned}
\]
However, $\sum_{u=t}^n \p_{u-n-1} a_{u-t} = w_{n+1-t} - v_{n+1-t}$.
Therefore, the assertion (ii) follows from Theorem \ref{thm:TforXwithM123} (ii).
(ii), (iv)\quad By (\ref{eq:SA-123}), (ii) and (iv) follow from (i) and (iii), respectively.
\end{proof}
\begin{exmp}\label{exmp:simple333}
Suppose that $K\ge 1$, $m_{\mu}=1$ for $\mu \in \{1,\dots,K\}$ and $m_0=0$. Then,
\[
h(z)^{-1} = - \rho_{0,0} - \sum_{{\mu}=1}^{K} \frac{1}{1-\barp_{\mu}z}\rho_{{\mu}, 1}, \qquad
h_{\sharp}(z)^{-1} = - \rho^{\sharp}_{0,0} -
\sum_{{\mu}=1}^{K} \frac{1}{1 - \barp_{\mu} z} \rho^{\sharp}_{{\mu}, 1}.
\]
We have
\begin{align*}
&\p_n^{\top} = (p_1^n I_d, \dots, p_K^n I_d)\in \C^{d \times dK}, \qquad n \in \Z,\\
&\rho^{\top}
=(\rho_{1, 1}^{\top}, \rho_{2, 1}^{\top}, \dots, \rho_{K, 1}^{\top}) \in \C^{dK\times d},
\qquad
\tilde{\rho}^{\top}
=\left(\overline{\rho^{\sharp}_{1, 1}}, \overline{\rho^{\sharp}_{2, 1}},
\dots, \overline{\rho^{\sharp}_{K, 1}}\right)
\in \C^{dK\times d}.
\end{align*}
We also have
\begin{align*}
\Theta &= \left(
\begin{matrix}
p_1 h_{\sharp}(p_1) \rho_{1, 1}^* & 0 & \cdots & 0 \cr
0 & p_2 h_{\sharp}(p_2) \rho_{2, 1}^* & \cdots & 0 \cr
\vdots & \vdots & \ddots & \vdots \cr
0 & 0 & \cdots & p_K h_{\sharp}(p_K) \rho_{K, 1}^*
\end{matrix}
\right)
\in \C^{dK\times dK},\\
\Lambda &= \left(
\begin{matrix}
\frac{1}{1-p_{1}\barp_{1}} I_d & \frac{1}{1-p_{1}\barp_{2}} I_d
& \cdots & \frac{1}{1-p_{1}\barp_{K}} I_d \cr
\frac{1}{1-p_{2}\barp_{1}} I_d & \frac{1}{1-p_{2}\barp_{2}} I_d
& \cdots & \frac{1}{1-p_{2}\barp_{K}} I_d \cr
\vdots & \vdots & \ddots & \vdots \cr
\frac{1}{1-p_{K}\barp_{1}} I_d & \frac{1}{1-p_{K}\barp_{2}} I_d
& \cdots & \frac{1}{1-p_{K}\barp_{K}} I_d \cr
\end{matrix}
\right)
\in \C^{dK\times dK},\\
\Pi_n &= \left(
\begin{matrix}
p_1^n I_d & 0 & \cdots & 0 \cr
0 & p_2^n I_d & \cdots & 0 \cr
\vdots & \vdots & \ddots & \vdots \cr
0 & 0 & \cdots & p_K^n I_d
\end{matrix}
\right)
\in \C^{dK\times dK},\qquad n \in \Z,\\
\Xi_n &= \left(
\begin{matrix}
\frac{\barp_{1}^{n}}{1-p_{1}\barp_{1}} I_d & \frac{\barp_{2}^{n}}{1-p_{1}\barp_{2}} I_d
& \cdots & \frac{\barp_{K}^{n}}{1-p_{1}\barp_{K}} I_d \cr
\frac{\barp_{1}^{n}}{1-p_{2}\barp_{1}} I_d & \frac{\barp_{2}^{n}}{1-p_{2}\barp_{2}} I_d
& \cdots & \frac{\barp_{K}^{n}}{1-p_{2}\barp_{K}} I_d \cr
\vdots & \vdots & \ddots & \vdots \cr
\frac{\barp_{1}^{n}}{1-p_{K}\barp_{1}} I_d & \frac{\barp_{2}^{n}}{1-p_{K}\barp_{2}} I_d
& \cdots & \frac{\barp_{K}^{n}}{1-p_{K}\barp_{K}} I_d \cr
\end{matrix}
\right)
\in \C^{dK\times dK}, \qquad n \in \N,\\
\Phi_n &= \left(
\begin{matrix}
\frac{p_{1}^{-n}}{1-p_{1}\barp_{1}} I_d & \frac{p_{1}^{-n}}{1-p_{1}\barp_{2}} I_d
& \cdots & \frac{p_{1}^{-n}}{1-p_{1}\barp_{K}} I_d \cr
\frac{p_{2}^{-n}}{1-p_{2}\barp_{1}} I_d & \frac{p_{2}^{-n}}{1-p_{2}\barp_{2}} I_d
& \cdots & \frac{p_{2}^{-n}}{1-p_{2}\barp_{K}} I_d \cr
\vdots & \vdots & \ddots & \vdots \cr
\frac{p_{K}^{-n}}{1-p_{K}\barp_{1}} I_d & \frac{p_{K}^{-n}}{1-p_{K}\barp_{2}} I_d
& \cdots & \frac{p_{K}^{-n}}{1-p_{K}\barp_{K}} I_d \cr
\end{matrix}
\right)
\in \C^{dK\times dK}, \qquad n \in \Z,\\
G_n &= \Pi_n \Theta \Lambda \in \C^{dK\times dK},
\qquad
\tilde{G}_n = (\Pi_n \Theta)^* \Lambda^{\top} \in \C^{dK\times dK},
\qquad
n \in \Z,\\
v_n &= \Xi_n\rho \in \C^{dK\times d}, \qquad \tilde{v}_n = \overline{\Xi}_n \tilde{\rho} \in \C^{dK\times d},
\qquad
n\in\N,\\
w_n &= \Phi_n\rho + \p_{-n}\rho_{0,0}\in \C^{dK\times d},
\qquad \tilde{w}_n = \overline{\Phi}_n \tilde{\rho} + \overline{\p}_{-n} \tilde{\rho}_{0,0} \in \C^{dK\times d},
\qquad
n\in\Z.
\end{align*}
\end{exmp}
\begin{exmp}\label{exmp:simple444}
In Example \ref{exmp:simple333}, we further assume $d=K=1$ and $\rho_{0,0}=0$. Then, we can write
$h(z) = h_{\sharp}(z) = -(1 - \barp z)/\rho$, where $\rho\in \C\setminus\{0\}$
and $p \in \D\setminus\{0\}$. It follows that
\begin{align*}
&c_0=-1/\rho, \qquad c_1 = \barp/\rho, \qquad c_k = 0\quad (k\ge 2),\\
&a_k = \rho (\barp)^k, \qquad \tilde{a}_k = \overline{a}_k, \qquad k \in \N\cup\{0\}.
\end{align*}
Since $\gamma(k) = \sum_{\ell=0}^{\infty} c_{k+\ell}\overline{c}_{\ell}$
and $\gamma(-k)=\overline{\gamma(k)}$ for $k\in\N\cup\{0\}$, we have
\[
T_2(w) =
\frac{1}{\vert \rho\vert^2}
\left(
\begin{matrix}
1 + \vert p \vert^2 & -p \cr
-\barp & 1 + \vert p \vert^2
\end{matrix}
\right),
\]
hence
\[
T_2(w)^{-1} =
\frac{\vert \rho\vert^2}{1 + \vert p \vert^2 + \vert p \vert^4}
\left(
\begin{matrix}
1 + \vert p \vert^2 & p \cr
\barp & 1 + \vert p \vert^2
\end{matrix}
\right).
\]
We also have
\[
\tilde{A}_2 = \overline{\rho}
\left(
\begin{matrix}
1 & p \cr
0 & 1
\end{matrix}
\right)
\quad
\mbox{and}
\quad
A_2 = \rho
\left(
\begin{matrix}
1 & 0 \cr
\barp & 1
\end{matrix}
\right)
\]
for $\tilde{A}_2$ and $A_2$ in (\ref{eq:tildeAnU-234}) and (\ref{eq:AnL-234}), respectively.
By simple calculations, we have
\begin{align*}
(\ell_{2,1}, \ell_{2,2})
&=
\frac{\overline{\rho}}{(\barp)^2(1 - \vert p\vert^6)}
(1 + \vert p\vert^2, \barp), \qquad
\left(\tilde{\ell}_{2,1}, \tilde{\ell}_{2,2}\right)
=
\left(\overline{\ell_{2,2}}, \overline{\ell_{2,1}} \right),\\
(r_{2,1}, r_{2,2})
&=
- \rho \barp \vert p\vert^2 (1 - \vert p\vert^2)
(\barp(1+\vert p\vert^2), \vert p\vert^2), \qquad
\left(\tilde{r}_{2,1}, \tilde{r}_{2,2}\right)
=
\left(\overline{r_{2,2}}, \overline{r_{2,1}} \right)
\end{align*}
hence
\[
T_2(w)^{-1}
=
\tilde{A}_2^* \tilde{A}_2
+
\left(
\begin{matrix}
\tilde{\ell}_{2,1} \cr
\tilde{\ell}_{2,2}
\end{matrix}
\right)
\left(
\tilde{r}_{2,1}, \tilde{r}_{2,2}
\right)
=
A_2^* A_2
+
\left(
\begin{matrix}
\ell_{2,1} \cr
\ell_{2,2}
\end{matrix}
\right)
\left(
r_{2,1}, r_{2,2}
\right)
\]
which agrees with equalities in Theorem \ref{thm:TforR123}.
\end{exmp}
\section{Linear-time algorithm}\label{sec:6}
As in Section \ref{sec:5}, we assume (\ref{eq:farima529}) and (\ref{eq:C}).
Let $K$ be as in (\ref{eq:hinverse162}) with (\ref{eq:hinverse163}).
In this section, we explain how Theorems \ref{thm:TforR777} and \ref{thm:TforR123}
above provide us with a linear-time algorithm to compute the solution $Z$
to the block Toeplitz system (\ref{eq:TS234}).
For
\begin{equation}
Y = (y_1^{\top},\dots,y_n^{\top})^{\top}\in\C^{dn\times d} \quad \mbox{with} \quad
y_s \in \C^{d \times d}, \quad s \in \{1,\dots,n\},
\label{eq:Y-999}
\end{equation}
let
\[
Z = (z_1^{\top},\dots,z_n^{\top})^{\top}\in\C^{dn\times d} \quad \mbox{with} \quad
z_s \in \C^{d \times d}, \quad s \in \{1,\dots,n\},
\]
be the solution to (\ref{eq:TS234}), that is, $Z = T_n(w)^{-1}Y$.
For $m_0$ in (\ref{eq:hinverse162}), let $n \ge 2m_0 + 1$ so that $n-m_0 \ge m_0+1$ holds.
Recall $\tilde{A}_n$ and $A_n$ from (\ref{eq:tildeAnU-234}) and (\ref{eq:AnL-234}), respectively.
If $K = 0$, then it follows from Lemma \ref{lem:decomp234} and Theorem \ref{thm:TforR777} (ii), (iv) that
\begin{align*}
z_s &= \tilde{\alpha}_{n,s}, \qquad s \in \{1,\dots,n-m_0\},
\\
z_s &= \alpha_{n,s}, \qquad s \in \{m_0+1,\dots,n\},
\end{align*}
where
\begin{align*}
(\tilde{\alpha}_{n,1}^{\top}, \dots, \tilde{\alpha}_{n,n}^{\top})^{\top}
&:= \tilde{A}_n^* \tilde{A}_n Y
\quad \mbox{with} \quad \tilde{\alpha}_{n,s} \in \C^{d \times d}, \quad s \in \{1,\dots,n\},
\\
(\alpha_{n,1}^{\top}, \dots, \alpha_{n,n}^{\top})^{\top}
&:= A_n^* A_n Y
\quad \mbox{with} \quad \alpha_{n,s} \in \C^{d \times d}, \quad s \in \{1,\dots,n\}.
\end{align*}
On the other hand, if $K \ge 1$, then we see from Lemma \ref{lem:decomp234} and Theorem \ref{thm:TforR123} (ii), (iv) that
\begin{align*}
z_s &= \tilde{\ell}_{n,s} \tilde{R}_n
+ \tilde{\alpha}_{n,s}, \qquad s \in \{1,\dots,n-m_0\},
\\
z_s &= \ell_{n,s} R_n
+ \alpha_{n,s}, \qquad s \in \{m_0+1,\dots,n\},
\end{align*}
where
\[
\tilde{R}_n := \sum_{t=1}^n \tilde{r}_{n,t} y_t \in \C^{d \times d},
\qquad
R_n := \sum_{t=1}^n r_{n,t} y_t \in \C^{d \times d}.
\]
Therefore, algorithms to compute $\tilde{A}_n^* \tilde{A}_n Y$ and $A_n^* A_n Y$
in $O(n)$ operations imply that of $Z$.
We present the former ones below.
For $n\in\N\cup\{0\}$, $\mu\in\{1,\dots,K\}$ and $j\in\{1,\dots,m_{\mu}\}$, we define
$q_{\mu,j}(n) \in \C^{d\times d}$ by $q_{\mu,j}(n):=p_{\mu,j}(n+j-1)$, that is,
\begin{equation}
q_{\mu,j}(n) = \binom{n+j-1}{j-1} p_{\mu}^nI_d.
\label{eq:a-224}
\end{equation}
For $n\in\N$, $\mu\in\{1,\dots,K\}$ and $j\in\{1,\dots,m_{\mu}\}$, we define
the upper triangular block Toeplitz matrix
$Q_{\mu,j,n} \in \C^{dn\times dn}$ by
\[
Q_{\mu,j,n} :=
\left(
\begin{matrix}
q_{\mu,j}(0) & q_{\mu,j}(1) & q_{\mu,j}(2) & \cdots & q_{\mu,j}(n-1) \cr
& q_{\mu,j}(0) & q_{\mu,j}(1) & \cdots & q_{\mu,j}(n-2) \cr
& & \ddots & \ddots & \vdots \cr
& & & \ddots & q_{\mu,j}(1) \cr
\mbox{\huge 0} & & & & q_{\mu,j}(0)
\end{matrix}
\right).
\]
Notice that
\[
Q^*_{\mu,j,n} :=
\left(
\begin{matrix}
q^*_{\mu,j}(0) & & & & \mbox{\huge 0} \cr
q^*_{\mu,j}(1) & q^*_{\mu,j}(0) & & & \cr
q^*_{\mu,j}(2) & q^*_{\mu,j}(1) & \ddots & & \cr
\vdots & \vdots & \ddots & \ddots & \cr
q^*_{\mu,j}(n-1) & q^*_{\mu,j}(n-2) & \cdots & q^*_{\mu,j}(1) & q^*_{\mu,j}(0)
\end{matrix}
\right)
\]
with $q^*_{\mu,j}(n) = \binom{n+j-1}{j-1} \barp_{\mu}^nI_d$.
For $n\in\N$, $\mu\in\{1,\dots,K\}$ and $j\in\{1,\dots,m_{\mu}\}$, we define
the block diagonal matrices
$\tilde{D}_{\mu,j,n} \in \C^{dn\times dn}$
and
$D_{\mu,j,n} \in \C^{dn\times dn}$ by
\[
\tilde{D}_{\mu,j,n} :=
\left(
\begin{matrix}
\tilde{\rho}_{{\mu},j} & 0 & \cdots & 0 \cr
0 & \tilde{\rho}_{{\mu},j} & \cdots & 0 \cr
\vdots & \vdots & \ddots & \vdots \cr
0 & 0 & \cdots & \tilde{\rho}_{{\mu},j}
\end{matrix}
\right)
\quad \mbox{and} \quad
D_{\mu,j,n} :=
\left(
\begin{matrix}
\rho_{{\mu},j} & 0 & \cdots & 0 \cr
0 & \rho_{{\mu},j} & \cdots & 0 \cr
\vdots & \vdots & \ddots & \vdots \cr
0 & 0 & \cdots & \rho_{{\mu},j}
\end{matrix}
\right),
\]
respectively.
Moreover,
for $n\ge m_0+1$, we define the upper and lower triangular block Toeplitz matrices
$\tilde{\Delta}_{n} \in \C^{dn\times dn}$
and
$\Delta_{n} \in \C^{dn\times dn}$ by
\[
\tilde{\Delta}_{n} :=
\left(
\begin{matrix}
\tilde{\rho}_{0,0} & \tilde{\rho}_{0,1} & \cdots & \tilde{\rho}_{0,m_0} & & \mbox{\huge 0} \cr
& \tilde{\rho}_{0,0} & \tilde{\rho}_{0,1} & & \ddots & \cr
& & \ddots & \ddots & & \tilde{\rho}_{0,m_0} \cr
& & & \ddots & \ddots & \vdots \cr
& & & & \tilde{\rho}_{0,0} & \tilde{\rho}_{0,1} \cr
\mbox{\huge 0} & & & & & \tilde{\rho}_{0,0}
\end{matrix}
\right)
\]
and
\[
\Delta_{n} :=
\left(
\begin{matrix}
\rho_{0,0} & & & & & \mbox{\huge 0} \cr
\rho_{0,1} & \rho_{0,0} & & & & \cr
\vdots & \rho_{0,1} & \ddots & & & \cr
\rho_{0,m_0} & & \ddots & \ddots & & \cr
& \ddots & & \ddots & \rho_{0,0} & \cr
\mbox{\huge 0} & & \rho_{0,m_0} & \cdots & \rho_{0,1} & \rho_{0,0}
\end{matrix}
\right),
\]
respectively. Note that both $\tilde{\Delta}_{n}$ and $\Delta_{n}$ are
sparse matrices in the sense that they have only
$O(n)$ nonzero elements.
By (\ref{eq:a363})--(\ref{eq:atilde362}), we have
\begin{align*}
\tilde{A}_n &= \tilde{\Delta}_{n} + \sum_{\mu=1}^K \sum_{j=1}^{m_{\mu}} Q_{\mu,j,n} \tilde{D}_{\mu,j,n},
\qquad n\ge m_0 + 1,
\\
A_n &= \Delta_{n} + \sum_{\mu=1}^K \sum_{j=1}^{m_{\mu}} Q^*_{\mu,j,n} D_{\mu,j,n},
\qquad n\ge m_0 + 1.
\end{align*}
Therefore, it is enough to give linear-time algorithms to compute
$Q_{\mu,i,n} Y$ and $Q^*_{\mu,i,n} Y$ for $Y \in \C^{dn\times d}$ in $O(n)$ operations.
The following two propositions provide such linear-time algorithms.
\begin{proposition}\label{prop:calcQ111}
Let $n\in\N$, $\mu\in\{1,\dots,K\}$ and $Y$ be as in (\ref{eq:Y-999}).
We put
$Z_{\mu,i}=Q_{\mu,i,n} Y$ for $i\in\{1,\dots,m_{\mu}\}$.
Then the component blocks $z_{\mu,i}(s)$ of
$Z_{\mu,i}=(z_{\mu,i}^{\top}(1),\dots,z_{\mu,i}^{\top}(n))^{\top}$
satisfy the following equalities:
\begin{align}
&z_{\mu,i}(n) = q_{\mu,i}(0) y_n, \qquad i \in \{1,\dots,m_{\mu}\},
\label{eq:z-recurs111}\\
&z_{\mu,1}(s) = p_{\mu} z_{\mu,1}(s+1) +q_{\mu,1}(0) y_s, \qquad s\in\{1,\dots,n-1\}
\label{eq:z-recurs222}\\
&\begin{aligned}
z_{\mu,i}(s) &= p_{\mu} z_{\mu,i}(s+1) + z_{\mu,i-1}(s) + \{q_{\mu,i}(0) -q_{\mu,i-1}(0)\} y_s,\\
&\qquad\qquad\qquad\qquad\qquad i\in\{2,\dots,m_{\mu}\}, \ s\in\{1,\dots,n-1\}.
\end{aligned}
\label{eq:z-recurs333}
\end{align}
\end{proposition}
\begin{proof}
From the definition of $Q_{\mu,i,n}$, (\ref{eq:z-recurs111}) is trivial.
For $q_{\mu,i}(k)$ in (\ref{eq:a-224}), Pascal's rule yields
the following recursions:
\begin{align}
&q_{\mu,1}(k+1) = p_{\mu}q_{\mu,1}(k), \qquad k \in \N\cup\{0\},
\label{eq:a-recurs111}\\
&q_{\mu,i}(k+1) = p_{\mu}q_{\mu,i}(k) +q_{\mu,i-1}(k+1),
\qquad i \in \{2,\dots,j\}, \ k \in \N\cup\{0\}.
\label{eq:a-recurs222}
\end{align}
For $s\in\{1,\dots,n-1\}$, we see, from (\ref{eq:a-recurs111}),
\[
\begin{aligned}
z_{\mu,1}(s)
&=q_{\mu,1}(0) y_s + \sum_{t=0}^{n-s-1}q_{\mu,1}(t+1) y_{s+t+1}
=q_{\mu,1}(0) y_s + p_{\mu} \sum_{t=0}^{n-s-1}q_{\mu,1}(t) y_{s+t+1}\\
&=q_{\mu,1}(0) y_s + p_{\mu} z_{\mu,1}(s+1),
\end{aligned}
\]
and, from (\ref{eq:a-recurs222}),
\[
\begin{aligned}
z_{\mu,i}(s)
&=q_{\mu,i}(0) y_s + \sum_{t=0}^{n-s-1}q_{\mu,i}(t+1) y_{s+t+1}\\
&= \{q_{\mu,i}(0) -q_{\mu,i-1}(0)\} y_s
+ p_{\mu} \sum_{t=0}^{n-s-1}q_{\mu,1}(t) y_{s+t+1}
+ \sum_{t=0}^{n-s}q_{\mu,i-1}(t) y_{s+t}\\
&=\{q_{\mu,i}(0) -q_{\mu,i-1}(0)\} y_s
+ p_{\mu} z_{\mu,i}(s+1) + z_{\mu,i-1}(s)
\end{aligned}
\]
for $i\in\{2,\dots,j\}$.
Thus, (\ref{eq:z-recurs222}) and (\ref{eq:z-recurs333}) follow.
\end{proof}
By Proposition \ref{prop:calcQ111}, we can compute $z_{\mu,i}(s)$, $i \in \{1,\dots,m_{\mu}\}$,
$s\in \{1,\dots,n\}$, in the following order in $O(n)$ operations:
\[
\begin{aligned}
&z_{\mu,1}(n) \ \to \ \cdots \to \ z_{\mu,1}(1) \ \to \ z_{\mu,2}(n) \ \to \cdots \to \ z_{\mu,2}(1)\\
&\qquad\qquad\qquad\qquad\qquad\qquad \to \cdots \to \ z_{\mu,m_{\mu}}(n) \ \to \cdots \to z_{\mu,m_{\mu}}(1).
\end{aligned}
\]
\begin{proposition}\label{prop:calcQ222}
Let $n\in\N$, $\mu\in\{1,\dots,K\}$ and $Y$ be as in (\ref{eq:Y-999}).
We put
$W_{\mu,i}=Q^*_{\mu,i,n} Y$ for $i\in\{1,\dots,m_{\mu}\}$.
Then the component blocks $w_{\mu,i}(s)$ of
$W_{\mu,i}=(w_{\mu,i}^{\top}(1),\dots,w_{\mu,i}^{\top}(n))^{\top}$
satisfy the following equalities:
\begin{align*}
&w_{\mu,i}(1) = q_{\mu,i}^*(0) y_1, \qquad i \in \{1,\dots,m_{\mu}\},
\\
&w_{\mu,1}(s+1) = \barp_{\mu} w_{\mu,1}(s) +q_{\mu,1}^*(0) y_{s+1}, \qquad s\in\{1,\dots,n-1\}
\\
&\begin{aligned}
w_{\mu,i}(s+1) &= \barp_{\mu} w_{\mu,i}(s) + w_{\mu,i-1}(s+1) + \{q_{\mu,i}^*(0) -q_{\mu,i-1}^*(0)\} y_{s+1},\\
&\qquad\qquad\qquad\qquad\qquad i\in\{2,\dots,m_{\mu}\}, \ s\in\{1,\dots,n-1\}.
\end{aligned}
\end{align*}
\end{proposition}
The proof of Proposition \ref{prop:calcQ222} is similar to that of
Proposition \ref{prop:calcQ111}; we omit it.
By Proposition \ref{prop:calcQ222}, we can compute $w_{\mu,i}(s)$, $i \in \{1,\dots,m_{\mu}\}$,
$s\in \{1,\dots,n\}$, in the following order in $O(n)$ operations:
\[
\begin{aligned}
&w_{\mu,1}(1) \ \to \ \cdots \to \ w_{\mu,1}(n) \ \to \ w_{\mu,2}(1) \ \to \cdots \to \ w_{\mu,2}(n)\\
&\qquad\qquad\qquad\qquad\qquad\qquad \to \cdots \to \ w_{\mu,m_{\mu}}(1) \ \to \cdots \to w_{\mu,m_{\mu}}(n).
\end{aligned}
\]
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 35 |
News and Programs
Feeding the 5000: Updating the Rules for Eating?
News and Programs Programs and Projects Slideshow
31 August 2013 2 September 2013 Ecojes Admin
Jaime Tatay, SJ
Is it legitimate to relate a contemporary advocacy campaign on food waste and a few marginal notes on a 16th century prayer book? In my opinion, I think it is.
Eating has often been considered a minor spiritual issue, especially compared to the centrality of prayer, silence, or biblical meditation. Yet it has always drawn the attention of religious founders and practitioners throughout history. Half a millennium ago, Ignatius of Loyola wrote eight short notes or "rules" on his Spiritual Exercises. They are known as the Rules for Eating. Setting these rules in a context of prayer, silence and contemplation was an attempt to introduce the question of eating into the spiritual matrix of the Exercises. Eating, for Ignatius, was an important habit in need of transformation.
As David Townsend puts it in his commentary on Digesting the Rules for Eating, the person "is asked to make conscious and deliberate decisions on his need and use of food and drink." The rules aim at transforming the habits and the sensibility of the person. In their effort to shape eating habits, religious ascetic traditions and contemporary advocacy groups find a common ground and a common end.
Feeding the 5000 is an advocacy campaign that asks each one of us to, precisely, "make conscious and deliberate decisions on his need and use of food and drink." Tristram Stuart, author of the book Waste: Uncovering the Global Food Scandal and leader of the Feeding the 5000 campaign in the UK, is a social activist trying to influence policy and public opinion on this complex issue.
Ignatius could never have imaged that the approximately 40 million tons of food wasted by US households, retailers, and food services each year would be enough to satisfy the hunger of the nearly one billion malnourished people in the world. He could never have imagined that 10% of rich countries' greenhouse gas emissions come from growing food that is never eaten. He could have never imagined that the UK, US, and Europe have nearly twice as much food as is required by the nutritional needs of their populations. He could have never imagined that up to half the entire food supply is wasted between the farm and the fork.
In short, Ignatius could never have imagined the scandalous "food waste facts" Feeding the 5000 campaign is trying to denounce and the complex connections between eating habits, agribusiness, agricultural subsidies, environmental degradation, and social justice.
However, in spite of the temporal anachronism, Ignatius and Tristram share a common interest. The Rules for Eating and the Feeding the 5000 campaign are not ultimately about food. They are about the importance of reflecting on the consequences of our daily actions. Reflecting on food could transform our daily eating habits into a movement willing to promote prudence, frugality, environmental stability and social justice.
"So, whether you eat or drink, or whatever you do, do everything for the glory of God." (1 Corinthians 10:31)
Digesting the Rules for Eating by David Townsend is available in pdf at The Way, an international journal of contemporary Christian spirituality published by the British Jesuits.
Feeding the 5000, food waste, Jaime Tatay SJ, Saint Ignatius, Spiritual Exercises
Teaching poverty, teaching transparency in our business schools
GMOs and Zambia | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,429 |
// Copyright 2016 Google Inc.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
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// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
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////////////////////////////////////////////////////////////////////////////////
package sample;
import tiger.GenerationTriggerAnnotation;
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import tiger.ScopeDependency;
import tiger.ScopedComponentNames;
import javax.inject.Singleton;
/**
* Injection information for the sample.
*/
@GenerationTriggerAnnotation
@PackageForGenerated("sample")
public class Scopes {
@ScopedComponentNames(scope = Singleton.class, name = "Application" )
public static class ForApplication {}
@ScopedComponentNames(scope = sample.ActivityScoped.class, name = "Activity" )
@ScopeDependency(scope = sample.ActivityScoped.class, parent = Singleton.class)
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| {
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} | 5,990 |
Home › Journal › Hannah Fox, Melbourne
Now reading: Hannah Fox, Melbourne
three hundred and two
Hannah Fox, Melbourne
Images by Tasha Tylee
Hannah is a painter, amongst other artistic mediums, and lives here with her husband David and their three boys Hugo 9, Isaac 7 and Jude 5.
"Our home is a Californian bungalow in the Melbourne inner city suburb of Northcote. When we bought it 9 years ago and it was a sad old house surrounded by a concrete garden and definitely ripe for demolition! But we loved the suburb and wanted to renovate. With a 3 month old baby we took on the front of the house and garden and over time have repaired and renewed the house...and had two more children! More recently we completed an extension to accommodate the family into the future. We did most of the work ourselves as David is a pro on the tools and I have always loved a bit of DIY! Having said that.... its been a long 9 years!"
"Now the house is well and truly our home and it serves our family perfectly. It is relaxed, yet refined and I think that helps to subdue the chaos of family life! Our boys are loud and busy little people but we all love being home together. I'm most content when we are all home and hosting family and friends. It's the people that give the spaces life and meaning."
"Our main environmental practice at home is the RRR. 'Reduce, Reuse, Recycle.' And we also look forward to having fresh eggs again and chook poo for the garden...sadly our three chickens were taken by a fox after three years of being part of our family! I'm still grieving! When I recover we'll get some more chooks! We plant all our own herbs and tomatoes too. I am often found down by our local creek with my son Hugo picking up litter on a Saturday morning...every little bit helps i hope."
Table dressed using an IN BED linen tablecloth in stone.__
Hannah's bed dressed in linen pillow slips in tobacco, flat & fitted sheets in dove grey, and a duvet cover in dove grey by IN BED.__
"I read a lot about the environment, education, urbanism, parenting and a little bit of 'light philosophy' ... I'm interested in ideas and peoples different approaches to life. I particularly love a good magazine like Dumbo Feather and also listening to a good podcast or radio interview. Being an artist can be isolating at times as it is such a solo pursuit...so I try to feed my brain and continue to learn and be engaged."
I studied Fine Art at university and majored in painting. After that I went on to study further and pursue a career in visual merchandising, styling events and graphic design...and then ended up back together with my true calling...painting.
"My practice has evolved over time and still continues to change. My main practice is painting, yet I like to explore other areas such as collage and ceramics... photography and drawing have also always helped to inform my paintings. I work from my home studio which is simply a dream! I can get lost in there and indulge in my work forgetting about the mess in the kitchen and the 5 loads of washing that need doing. There is always washing...ALWAYS!
"I exhibit in solo and group exhibitions and also host open studio days. I love working with other artists and creatives too and have collaborated with fashion designers, photographers and furniture designers. I think keeping my practice diverse keeps things interesting and keeps me motivated. "I've just completed a new body of work for an open studio event. I am also preparing for an art residency in Italy where I hope to create new work for another exhibition."
Most of my inspiration come from landscape and memory. I am most responsive to simple aesthetics...light, shape, texture and form.
"My art library has been growing steadily since I first inherited some Australian Landscape art books and old auction house catalogues from my Nana when I was a young teenager. I even had art books on our wedding gift registry! Forget the dinner set! I treasure my books and refer to them always...its a quick way into another world when I need to switch off or think outside the square a bit. My other treasured belongings are the pieces of art from other artists who I love and admire. Either by saving up to purchase them, being gifted a piece or by trading with other artists...I love having others creativity around me."
"I am definitely in the right neighbourhood! I love the people and the sense of community that my area has. From hipsters on skateboards to pram pushing parents, everyone looks happy here. There is a great mix of music venues, restaurants, community gardens, pop up shops etc... I can retreat to my suburban house and yet walk up to the main street where there is always a humming of activity and I love that." @hannahfoxart
three hundred and three
Cath & Jeremy Brown, Devon, UK
An incredible farmhouse in rural Devon.
three hundred and one
Caitlin Garcia-Ahern, Oaxaca, Mexico
A warm house in Oaxaca, home to Caitlin and the dogs she cares for. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,503 |
Никола́й Евге́ньевич Ларио́нов (19 февраля 1957, Волхов, Ленинградская область, РСФСР, СССР) — советский и российский футболист, российский футбольный тренер. Мастер спорта СССР.
Биография
Бо́льшую часть карьеры провёл в ленинградском «Зените», в составе которого стал чемпионом СССР в 1984 году. В дубль «Зенита» он перешёл в 17 лет, но заиграть сразу в команде мастеров ему не удалось, и на несколько сезонов Ларионов отправился в ленинградское «Динамо». В 1979 он вернулся в «Зенит», после чего более 10 сезонов защищал цвета клуба, в 1983—1985 годах был капитаном.
Один из немногих игроков «Зенита», постоянно выступавших за сборную СССР (больше него из зенитовцев за сборную сыграл только Василий Данилов). В 1983 году получил тяжелейшую травму, но через год, когда «Зенит» боролся за золото, вернулся на поле и в «золотом» матче против «Металлиста» открыл счет.
Бронзовый призёр чемпионата СССР 1980 года, обладатель Суперкубка СССР 1985 года. Трижды включался в список 33 лучших футболистов сезона в СССР (№ 2 — 1983, 1985, № 3 — 1984). Являлся игроком сборной СССР (19 игр, 2 гола), в её составе участвовал в ЧМ-1986.
Завершив карьеру игрока, работал тренером в СДЮШОР «Зенит». Был членом тренерского штаба «Зенита», одно время возглавлял дублирующий состав. В 2004—2006 годах работал в «Зените» начальником команды. В сентябре 2006 года был уволен с последней должности в связи с её сокращением.
С 2008 года работал в тренерском штабе молодёжной команды «Зенита», в 2009 году исполнял обязанности главного тренера. Под его руководством молодёжная команда стала чемпионом России 2009 года.
По состоянию на февраль 2017 года — селекционер в Академии ФК «Зенит».
Примечания
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{"url":"https:\/\/www.physicsforums.com\/threads\/geometric-series.245348\/","text":"Geometric Series\n\n1. Jul 16, 2008\n\nRossinole\n\n1. The problem statement, all variables and given\/known data\n\nDoes the series from n=1 to infinity of (2)\/(n^2-1) converge or diverge? If it converges, find the sum.\n\n2. Relevant equations\n\n3. The attempt at a solution\n\nI can see right away that the series converges by a limit comparison test by looking at the series. However, to find the sum I have re-write that as a geometric series. There is nothing, at least to me, that gives away how to re-write that as a geometric series. That's where I'm stuck.\n\nThanks for any help.\n\n2. Jul 16, 2008\n\nrock.freak667\n\nWhy don'y you write $\\frac{2}{n^2-1}$ in partial fractions and see if it is a telescoping series?\n\nFind\n\n$$\\sum_{n=1} ^{N} \\frac{2}{n^2-1}$$\n\nand then check what happens as $N \\rightarrow \\infty$\n\n3. Jul 16, 2008\n\nRossinole\n\nI see it now. Thank you.\n\n4. Aug 25, 2008\n\njainal36\n\nwhat's the summation of the series?\n\nwhat's the summation of the series?\n1.A^1 + 2.A^2 + 3.A^3 + .............+n.A^n.\n\nplease reply\n\n5. Aug 25, 2008\n\nrock.freak667\n\nRe: what's the summation of the series?\n\nWhat have you tried so far on it?\n\n6. Aug 25, 2008\n\nDefennder\n\nRe: what's the summation of the series?\n\nIs this related to the original post or something separate?\n\n7. Aug 26, 2008\n\nstatdad\n\nAre you sure you don't mean\n\n$$\\sum_{n=2}^\\infty \\frac{2}{n^2-1}$$\n\ni.e.- the sum starting at $$n = 2$$? If you try to start at $$n = 1$$ the very first term is undefined (can't divide by zero) so the series would not converge. As I mentioned - this makes the difference between the series converging and not converging, and will influence your value for the sum.\n\n8. Aug 26, 2008\n\nGib Z\n\nYes, he probably meant the sum to start at n=2.\n\nJainal36- you should really start new threads rather than hijacking others, but ill help anyway - thats just A times the derivative of a series you know how to sum.\n\nKnow someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook","date":"2017-09-26 11:12:49","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\": 0.8619726300239563, \"perplexity\": 952.8097440587898}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"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-2017-39\/segments\/1505818695439.96\/warc\/CC-MAIN-20170926103944-20170926123944-00192.warc.gz\"}"} | null | null |
Unemployed Greeks Look To Create Their Own Jobs
By Joanna Kakissis
Published April 10, 2012 at 12:39 AM PDT
Panos Papadopoulos, 28, is the co-founder of BugSense, which makes a service to track bugs in mobile phone applications. He also mentors other young entrepreneurs at CoLab, a business incubator in Greece.
In Greece, more than 21 percent of the working-age population is jobless. For Greeks under age 25, the rate is more than double that.
Some young Greeks are frightened that the economy, now in free fall, will take years to recover, so they're leaving for jobs abroad. A few entrepreneurs, however, are trying to start businesses during the worst recession in decades.
A magnet for these young entrepreneurs is CoLab, a business incubator in a weathered building near the Athens Cathedral in the city center. CoLab opened in 2009, with just one occupant — a Spanish travel writer.
Now, it has 45 occupants including software and programming whizzes, Web developers, corporate responsibility consultants and even a couple of yogis.
One of the stars here is Panos Papadopoulos, a 28-year-old computer scientist. He's the co-founder of BugSense, which tracks bugs in mobile phone applications. BugSense is already making a profit and has Silicon Valley investors.
When you don't have a job that means you have plenty of time. You should do something with that time.
"We have 5,000 developers from Japan to Argentina," Papadopoulos says. "Some of our customers include Samsung, Skype [and] VMware."
He mentors younger entrepreneurs like 25-year-old John Katsiotis, who co-founded a new company called Parking Defenders. It's developing a smartphone application that helps people find parking spots in Athens.
The app allows users to notify each other when they're about to vacate a spot. "I see a list of all the users that are interested, and I choose one," he says. "So that person can come and take my spot."
Katsiotis thinks the idea has potential, since parking is so scarce in Athens that people sometimes leave their cars on sidewalks.
A Need To Be More Creative
Dimitris Tsigos, who leads a young entrepreneurs association in Greece, also likes the idea. He says he especially likes that Katsiotis is exploring his idea creatively, something young Greeks don't do enough of, according to Tsigos.
Many of his own relatives thought he was nuts when he started his successful e-learning company, Virtual Trip, 12 years ago, when he was still in college. An aunt told him a real job meant working for the government.
"The Greek dream was that you are hired in the public sector," Tsigos says. "You go to work at 8, you leave at 12, and you get 1,200 euros."
That's about $1,600 a month. Not everyone worked these easy hours, but at least public sector jobs used to be safe. The constitution protected public workers from getting fired.
That changed in 2010, when Greece took billions of dollars in bailout loans to keep from defaulting on more than $400 billion in debt. International lenders — who include the European Union, the International Monetary Fund and the European Central Bank — are demanding that Greece fire 150,000 public workers over the next three years to cut costs. Austerity measures also forced more than 100,000 businesses to close last year.
That's left young Greeks with virtually no options, Tsigos says.
"That's why they are frightened," he says. "That's why you see all these demonstrations with people expressing anger, because they're frightened. Because they see all [these plans] they had in their mind is now destroyed."
He says he hears from about 10 young entrepreneurs a week, but says that's not enough to call it a trend.
Huge Job Losses
Most young Greeks say they feel adrift in an economy that shed more than 300,000 jobs last year. Venetia Kogkou, 31, lost her job as a librarian two years ago. Many of her friends are also unemployed.
Kogkou says she's sent out hundreds of resumes, with no response.
"When I was a little girl, my mom used to tell me that if I didn't study, I'd end up working at the supermarket," Kogkou says. "But you know what? I can't even get a job there."
She and her friends say they see no option but to leave Greece.
John Katsiotis, the young computer scientist, says he's going to stay. His smartphone app on parking hasn't made any money yet, and he's not sure it ever will.
"When you don't have a job, that means you have plenty of time," he says. "You should do something with that time."
He says he wants to use this time to create a job, even in this morbid economy. The Greek economy is now in its fifth year of recession, and economists have predicted more years of stagnation.
Joanna Kakissis
See stories by Joanna Kakissis | {
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\subsubsection*{\bibname}}
\begin{document}
\twocolumn[
\aistatstitle{Instructions for Paper Submissions to AISTATS 2023}
\aistatsauthor{ Author 1 \And Author 2 \And Author 3 }
\aistatsaddress{ Institution 1 \And Institution 2 \And Institution 3 } ]
\begin{abstract}
The Abstract paragraph should be indented 0.25 inch (1.5 picas) on
both left and right-hand margins. Use 10~point type, with a vertical
spacing of 11~points. The \textbf{Abstract} heading must be centered,
bold, and in point size 12. Two line spaces precede the
Abstract. The Abstract must be limited to one paragraph.
\end{abstract}
\section{GENERAL FORMATTING INSTRUCTIONS}
The camera-ready versions of the accepted papers are 8 pages,
plus any additional pages needed for references.
Papers are in 2 columns with the overall line width of 6.75~inches (41~picas).
Each column is 3.25~inches wide (19.5~picas). The space
between the columns is .25~inches wide (1.5~picas). The left margin is 0.88~inches (5.28~picas).
Use 10~point type with a vertical spacing of
11~points. Please use US Letter size paper instead of A4.
Paper title is 16~point, caps/lc, bold, centered between 2~horizontal rules.
Top rule is 4~points thick and bottom rule is 1~point thick.
Allow 1/4~inch space above and below title to rules.
Author descriptions are center-justified, initial caps. The lead
author is to be listed first (left-most), and the Co-authors are set
to follow. If up to three authors, use a single row of author
descriptions, each one center-justified, and all set side by side;
with more authors or unusually long names or institutions, use more
rows.
Use one-half line space between paragraphs, with no indent.
\section{FIRST LEVEL HEADINGS}
First level headings are all caps, flush left, bold, and in point size
12. Use one line space before the first level heading and one-half line space
after the first level heading.
\subsection{Second Level Heading}
Second level headings are initial caps, flush left, bold, and in point
size 10. Use one line space before the second level heading and one-half line
space after the second level heading.
\subsubsection{Third Level Heading}
Third level headings are flush left, initial caps, bold, and in point
size 10. Use one line space before the third level heading and one-half line
space after the third level heading.
\paragraph{Fourth Level Heading}
Fourth level headings must be flush left, initial caps, bold, and
Roman type. Use one line space before the fourth level heading, and
place the section text immediately after the heading with no line
break, but an 11 point horizontal space.
\subsection{Citations, Figure, References}
\subsubsection{Citations in Text}
Citations within the text should include the author's last name and
year, e.g., (Cheesman, 1985).
Be sure that the sentence reads
correctly if the citation is deleted: e.g., instead of ``As described
by (Cheesman, 1985), we first frobulate the widgets,'' write ``As
described by Cheesman (1985), we first frobulate the widgets.''
The references listed at the end of the paper can follow any style as long as it is used consistently.
\subsubsection{Footnotes}
Indicate footnotes with a number\footnote{Sample of the first
footnote.} in the text. Use 8 point type for footnotes. Place the
footnotes at the bottom of the column in which their markers appear,
continuing to the next column if required. Precede the footnote
section of a column with a 0.5 point horizontal rule 1~inch (6~picas)
long.\footnote{Sample of the second footnote.}
\subsubsection{Figures}
All artwork must be centered, neat, clean, and legible. All lines
should be very dark for purposes of reproduction, and art work should
not be hand-drawn. Figures may appear at the top of a column, at the
top of a page spanning multiple columns, inline within a column, or
with text wrapped around them, but the figure number and caption
always appear immediately below the figure. Leave 2 line spaces
between the figure and the caption. The figure caption is initial caps
and each figure should be numbered consecutively.
Make sure that the figure caption does not get separated from the
figure. Leave extra white space at the bottom of the page rather than
splitting the figure and figure caption.
\begin{figure}[h]
\vspace{.3in}
\centerline{\fbox{This figure intentionally left non-blank}}
\vspace{.3in}
\caption{Sample Figure Caption}
\end{figure}
\subsubsection{Tables}
All tables must be centered, neat, clean, and legible. Do not use hand-drawn tables.
Table number and title always appear above the table.
See Table~\ref{sample-table}.
Use one line space before the table title, one line space after the table title,
and one line space after the table. The table title must be
initial caps and each table numbered consecutively.
\begin{table}[h]
\caption{Sample Table Title} \label{sample-table}
\begin{center}
\begin{tabular}{ll}
\textbf{PART} &\textbf{DESCRIPTION} \\
\hline \\
Dendrite &Input terminal \\
Axon &Output terminal \\
Soma &Cell body (contains cell nucleus) \\
\end{tabular}
\end{center}
\end{table}
\section{SUPPLEMENTARY MATERIAL}
If you need to include additional appendices during submission, you can include them in the supplementary material file.
You can submit a single file of additional supplementary material which may be either a pdf file (such as proof details) or a zip file for other formats/more files (such as code or videos).
Note that reviewers are under no obligation to examine your supplementary material.
If you have only one supplementary pdf file, please upload it as is; otherwise gather everything to the single zip file.
You must use \texttt{aistats2023.sty} as a style file for your supplementary pdf file and follow the same formatting instructions as in the main paper.
The only difference is that it must be in a \emph{single-column} format.
You can use \texttt{supplement.tex} in our starter pack as a starting point.
Alternatively, you may append the supplementary content to the main paper and split the final PDF into two separate files.
\section{SUBMISSION INSTRUCTIONS}
To submit your paper to AISTATS 2023, please follow these instructions.
\begin{enumerate}
\item Download \texttt{aistats2023.sty}, \texttt{fancyhdr.sty}, and \texttt{sample\_paper.tex} provided in our starter pack.
Please, do not modify the style files as this might result in a formatting violation.
\item Use \texttt{sample\_paper.tex} as a starting point.
\item Begin your document with
\begin{flushleft}
\texttt{\textbackslash documentclass[twoside]\{article\}}\\
\texttt{\textbackslash usepackage\{aistats2023\}}
\end{flushleft}
The \texttt{twoside} option for the class article allows the
package \texttt{fancyhdr.sty} to include headings for even and odd
numbered pages.
\item When you are ready to submit the manuscript, compile the latex file to obtain the pdf file.
\item Check that the content of your submission, \emph{excluding} references, is limited to \textbf{8 pages}. The number of pages containing references alone is not limited.
\item Upload the PDF file along with other supplementary material files to the CMT website.
\end{enumerate}
\subsection{Camera-ready Papers}
If your papers are accepted, you will need to submit the camera-ready version. Please make sure that you follow these instructions:
\begin{enumerate}
\item Change the beginning of your document to
\begin{flushleft}
\texttt{\textbackslash documentclass[twoside]\{article\}}\\
\texttt{\textbackslash usepackage[accepted]\{aistats2023\}}
\end{flushleft}
The option \texttt{accepted} for the package
\texttt{aistats2023.sty} will write a copyright notice at the end of
the first column of the first page. This option will also print
headings for the paper. For the \emph{even} pages, the title of
the paper will be used as heading and for \emph{odd} pages the
author names will be used as heading. If the title of the paper
is too long or the number of authors is too large, the style will
print a warning message as heading. If this happens additional
commands can be used to place as headings shorter versions of the
title and the author names. This is explained in the next point.
\item If you get warning messages as described above, then
immediately after $\texttt{\textbackslash
begin\{document\}}$, write
\begin{flushleft}
\texttt{\textbackslash runningtitle\{Provide here an alternative
shorter version of the title of your paper\}}\\
\texttt{\textbackslash runningauthor\{Provide here the surnames of
the authors of your paper, all separated by commas\}}
\end{flushleft}
Note that the text that appears as argument in \texttt{\textbackslash
runningtitle} will be printed as a heading in the \emph{even}
pages. The text that appears as argument in \texttt{\textbackslash
runningauthor} will be printed as a heading in the \emph{odd}
pages. If even the author surnames do not fit, it is acceptable
to give a subset of author names followed by ``et al.''
\item The camera-ready versions of the accepted papers are 8
pages, plus any additional pages needed for references.
\item If you need to include additional appendices,
you can include them in the supplementary
material file.
\item Please, do not change the layout given by the above
instructions and by the style file.
\end{enumerate}
\subsubsection*{Acknowledgements}
All acknowledgments go at the end of the paper, including thanks to reviewers who gave useful comments, to colleagues who contributed to the ideas, and to funding agencies and corporate sponsors that provided financial support.
To preserve the anonymity, please include acknowledgments \emph{only} in the camera-ready papers.
\subsubsection*{References}
References follow the acknowledgements. Use an unnumbered third level
heading for the references section. Please use the same font
size for references as for the body of the paper---remember that
references do not count against your page length total.
\subsubsection*{\bibname}}
\begin{document}
\onecolumn
\aistatstitle{Instructions for Paper Submissions to AISTATS 2022: \\
Supplementary Materials}
\section{FORMATTING INSTRUCTIONS}
To prepare a supplementary pdf file, we ask the authors to use \texttt{aistats2022.sty} as a style file and to follow the same formatting instructions as in the main paper.
The only difference is that the supplementary material must be in a \emph{single-column} format.
You can use \texttt{supplement.tex} in our starter pack as a starting point, or append the supplementary content to the main paper and split the final PDF into two separate files.
Note that reviewers are under no obligation to examine your supplementary material.
\section{MISSING PROOFS}
The supplementary materials may contain detailed proofs of the results that are missing in the main paper.
\subsection{Proof of Lemma 3}
\textit{In this section, we present the detailed proof of Lemma 3 and then [ ... ]}
\section{ADDITIONAL EXPERIMENTS}
If you have additional experimental results, you may include them in the supplementary materials.
\subsection{The Effect of Regularization Parameter}
\textit{Our algorithm depends on the regularization parameter $\lambda$. Figure 1 below illustrates the effect of this parameter on the performance of our algorithm. As we can see, [ ... ]}
\vfill
\end{document}
\section{Introduction}
Quantifying the dependence between variables is a fundamental problem in science. Mutual information (MI) is one such descriptor of dependence and is widely utilized in information theory to measure the dependence between two random variables $ X$ and $ Y$.
MI has been employed in numerous fields, such as statistics \cite{zhang2022normal}, manufacturing \cite{zhao2022adversarial}, health \cite{huntington2021cytokine}, and machine learning \cite{sanchez2020learning, tschannen2019mutual, hjelm2018learning, belghazi2018mine}.
Intuitively, mutual information measures the reduction in uncertainty (the amount of information in common) about $ X$ by observing $ Y$ and vice-versa. Let $\mathcal{X}$ and $\mathcal{Y}$ be the respective ranges of $ X$ and $Y$ and $\mathbb{P}_{X,Y}$ their joint distribution defined on $\mathcal{X} \times \mathcal{Y}$. The MI between $X$ and $Y$ is defined as:
\begin{align*}
I(X, Y) &= H(X) - H(X\vert Y)\\
& = H(X) + H(Y) - H(X, Y),
\end{align*}
wher
$H(\cdot)$ is the entropy of a given random variable, and $H(X \vert Y)$ the conditional entropy of $X$ given $Y$.
Estimating MI for continuous multidimensional random variables is not a trivial task as it requires knowledge of $\mathbb{P}_{X,Y}$ or the corresponding joint probability density function. Several approaches have been proposed to estimate MI from data. A natural way to address this problem is to estimate the underlying probability density functions from data samples \cite{orlitsky2003always,orlitsky2015competitive} to later approximate the MI; however, estimating probability densities is not data efficient \cite{majdara2022efficient}. Other works, such as the one proposed in \cite{kraskov2004estimating}, are based on entropy estimates from k-nearest neighbor distances, although this method requires making assumptions on the underlying distributions that might not be true. While some of these approaches work well in an information-theoretic sense, many are often inefficient and have difficulty scaling to very large high-dimensional datasets. Given the notorious difficulties of measuring MI from finite and high dimensional data, popular alternative methods have emerged for maximizing (or minimizing) a lower (upper) bound on MI \cite{poole2019variational, mcallester2020formal} that is more tractable and scalable.\par
Modern approaches based on representation learning for estimating MI or its bounds have led to plethora of methods using MI maximization to approach unsupervised learning problems \cite{sordoni2021decomposed,tian2020contrastive,bachman2019learning,hjelm2018learning, oord2018representation}. Most of these methods are inspired by the InfoMax principle introduced by \cite{linsker1988self} to learn a representation that maximizes its MI with the input. However, instead of maximizing the MI of the representation and the input, a more tractable approach is to maximize the MI between the representations of two ``views'' of the input which has been shown to be a lower bound on the InfoMax cost function \cite{tschannen2019mutual}. This approach is convenient since the learned representations are typically encoded in a low-dimensional space.\par
There are several factors involved in MI estimation that are crucial for these methods to perform well, such as: the way the views are chosen, the MI estimator, and the architectures of the networks used to encode features. Specifically, DeepInfoMax \cite{hjelm2018learning} is a well known methodology that maximizes MI between global features extracted from the original image and local features from patches of the same image. Contrastive Multiview Coding (CMC) \cite{tian2020contrastive} estimates the MI between the encoded representations of two views representing different modalities of the same image (e.g. different image channels). Contrastive Predictive Coding (CPC) \cite{oord2018representation} considers data in a sequential fashion and computes a context latent representation that summarizes the features from previous image patches and estimates the MI between this context representation and the features from the patch at the current position. \par
The choice of MI estimator plays a significant role in performance as well. Among the common estimators are InfoNCE ($I_{\textrm{NCE}}$) \cite{oord2018representation}, Nguyen, Wainwright, and Jordan ($I_{\textrm{NWJ}}$) \cite{nguyen2010estimating}, Mutual Information Neural Estimation (MINE) \cite{belghazi2018mine} and difference-of-entropies (DoE) \cite{mcallester2020formal}. $I_{\textrm{NCE}}$ aims to maximize a lower bound on MI between a pair of random variables by discriminating positive pairs from negative ones \cite{wu2021rethinking}. $I_{\textrm{NWJ}}$ trains a log density ratio estimator to maximize a variational lower bound on the Kullback-Leibler (KL) divergence. A similar bound, denoted as $I_{\textrm{JS}}$, does the equivalent although using the Jensen-Shannon divergence instead. Similarly, MINE is based on a dual
representation of the KL-divergence and relies on a neural network to approximate the MI. Finally, DoE estimates MI as a difference of entropies, bounding the entropies by cross-entropy terms. This results in a quantity that is neither an upper bound nor a lower bound, but exhibits evidence that it can estimate large MI values. \par
Despite the popular use of MI maximization for learning representations, it has been observed that maximizing very tight bounds on MI can lead to inferior representations \cite{tschannen2019mutual}. The MI itself may be not enough to learn relevant representations and the quality of the representations is determined more by the architecture employed than the MI estimator used. Indeed, maximizing MI between the representation of paired views of the same instances does not guarantee lower MI between the representations of unpaired instances. Therefore, we propose a formulation that explicitly captures both maximization of mutual information between paired instances and minimization of mutual information between unpaired instances by using an information-theoretic quantity known as matrix-based entropy \cite{sanchezgiraldo2015measures}. The formulation, \textbf{Di}fference of \textbf{M}atrix-based \textbf{E}ntropies (DiME), compares the joint entropy of the two paired random variables (for example, representations of two different views) to the expected joint entropy between unpaired samples. The latter is obtained by randomly permuting the paired samples within a batch.\par
Our main contributions are:
\begin{itemize}
\item We introduce and motivate DiME, a quantity inspired by matrix-based entropy which serves as an objective function for problems seeking to maximize mutual information.
\item We demonstrate DiME's effectiveness in several tasks, namely: learning a shared representations between multiple views, disentangling latent factors, and training GANs.
\end{itemize}
\section{Background}
Before introducing DiME, we must first provide a description of the matrix-based entropy because it is the basic building block of DiME.
\subsection{Matrix-Based Entropy and Related Quantities}
Let $\mathbf{X} = \{x_i\}_{i=1}^{n}$ be a set of $n$ data points $x \in \mathcal{X}$ sampled from an unknown distribution $\mathbb{P}_X$ defined on $\mathcal{X}$.
Let $\kappa : \mathcal{X} \times \mathcal{X} \mapsto \mathbb{R}_{\geq 0}$ be a positive definite kernel such that $\kappa(x, x) = 1$ for all $x \in \mathcal{X}$. We can construct a Gram matrix $\mathbf{K}_{\mathbf{X}}$ consisting on all pairwise evaluation of the points in $\mathbf{X}$. Given $\mathbf{K}_{\mathbf{X}}$, the matrix-based entropy of order $\alpha>0$ is defined as:
\begin{equation}\label{eq:matrix_based_entropy}
S_{\alpha}\left(\mathbf{K}_{\mathbf{X}}\right) = \frac{1}{1-\alpha}\log{\left[\Tr{\left( \left( \frac{\mathbf{K}_{\mathbf{X}}}{n} \right)^{\alpha}\right) }\right]},
\end{equation}
where $\Tr$ denotes the trace operator and $\mathbf{K}^{\alpha}$ is an arbitrary matrix power, which is defined based on its eigenvalue decomposition \cite{bhatia1997}.
The matrix-based entropy $S_{\alpha}$ is an information-theoretic quantity that behaves similarly to Rényi's $\alpha$-order entropy, but it can be estimated directly from data without making strong assumptions about the underlying distribution \cite{sanchezgiraldo2015measures, sanchezgiraldo2013iclr}.
Equation~\ref{eq:matrix_based_entropy} is the main building block, from which other information-theoretic quantities can be defined.
\subsubsection{Matrix-Based Joint entropy}
Let $\mathcal{X}$ and $\mathcal{Y}$ be two nonempty sets for which there is a joint probability measure space $\left(\mathcal{X} \times \mathcal{Y}, \mathbf{B}_{\mathcal{X} \times \mathcal{Y}}, \mathbb{P}_{X,Y}\right)$ for a set $\left\{(x_i,y_i)\right\}_{i=1}^n$ of $n$ pairs sampled from a joint distribution $\mathbb{P}_{X,Y}$. By defining a kernel $\kappa : \left(\mathcal{X}\times\mathcal{Y}\right) \times \left(\mathcal{X}\times\mathcal{Y}\right) \mapsto \mathbb{R}$ we can extend \eqref{eq:matrix_based_entropy} to pairs of variables. A choice consistent with kernels on $\mathcal{X}$ and $\mathcal{Y}$ is the product kernel:
\begin{equation}\label{eq:product_kernel}
\kappa_{\mathcal{X} \times \mathcal{Y}}((x,y), (x', y')) = \kappa_{\mathcal{X}}(x, x')\kappa_{\mathcal{Y}}(y, y').
\end{equation}
This corresponds to the tensor product between all dimensions of the representations of $\mathcal{X}$ and $\mathcal{Y}$. For kernels $\kappa_{\mathcal{X}}$ and $\kappa_{\mathcal{Y}}$, such that $\kappa_{\mathcal{X}}(x,x) = 1$, the product kernel is equivalent to concatenating the dimensions of the Hilbert spaces resulting from taking the $\log$ of the kernel. For instance, if $\mathcal{X} \subset \mathbb{R}^{d_\mathcal{X}}$ and $\mathcal{Y} \subset \mathbb{R}^{d_\mathcal{Y}}$ and we use the Gaussian kernel for both $\kappa_{\mathcal{X}}$ and $\kappa_{\mathcal{Y}}$, the product kernel is a kernel on $\mathbb{R}^{d_\mathcal{X} + d_\mathcal{Y}}.$, which corresponds to direct concatenation of features in the input space.
This concatenation leads to the notion of matrix-based joint entropy, which can be expressed in terms of \eqref{eq:matrix_based_entropy} using the Hadamard product:
\begin{equation}\label{eq:represenation_joint_entropy_product}
S_{\alpha}(\mathbf{K}_\mathbf{X} \circ \mathbf{K}_\mathbf{Y}).
\end{equation}
For $\alpha > 0$ and non-negative normalized kernels $\mathbf{K}_\mathbf{X}$ and $\mathbf{K}_\mathbf{Y}$, we can verify that:
\begin{eqnarray}\label{eq:joint_entropy_as_upper_bound}
S_{\alpha}(\mathbf{K}_\mathbf{X} \circ \mathbf{K}_\mathbf{Y}) & \geq & S_{\alpha}(\mathbf{K}_\mathbf{X}) \\
S_{\alpha}(\mathbf{K}_\mathbf{X} \circ \mathbf{K}_\mathbf{Y}) & \geq & S_{\alpha}(\mathbf{K}_\mathbf{Y}),
\end{eqnarray}
which gives rise to the following property.
\begin{property}\label{prop:entropy_exponent} Let $\mathbf{K}$ be a normalized Gram matrix and $\mathbf{K}^{\circ \gamma}$ denote the matrix of entry-wise $\gamma$ power. If $\mathbf{K}$ is an infinitely divisible matrix, that is $\mathbf{K}^{\circ \gamma}$ is positive semidefinite for any nonnegative $\gamma$, $S_{\alpha}(\mathbf{K}^{\circ \gamma})$ is a monotonically increasing function of $\gamma$.
\begin{equation}
\mathrm{S}_{\alpha}(\mathbf{K}^{\circ \gamma_1}) \leq \mathrm{S}_{\alpha}(\mathbf{K}^{\circ \gamma_2}),
\end{equation}
for $0 < \gamma_1 \leq \gamma_2$.
\end{property}
For the Gaussian kernel, where taking the entry-wise exponent of the Gram matrix is equivalent to changing the width of the kernel, larger values of $\gamma$ are related to smaller width $\sigma$. Similarly to differential entropy, as we scale the random variable relative to $\sigma$, we can have larger or smaller matrix-based entropies for the same set of points.
\subsubsection{Matrix-Based Conditional Entropy and Mutual Information}
Based on \eqref{eq:joint_entropy_as_upper_bound}, the matrix-based conditional entropy can be posed as
\begin{equation}\label{eq:represenation_conditional_entroopy}
S_{\alpha}(\mathbf{K}_\mathbf{X} \vert \mathbf{K}_\mathbf{Y}) = S_{\alpha}(\mathbf{K}_\mathbf{X} \circ \mathbf{K}_\mathbf{Y}) - S_{\alpha}(\mathbf{K}_\mathbf{Y}).
\end{equation}
Note that for $\alpha \neq 1$, $S_{\alpha}(\mathbf{K}_\mathbf{X} \vert \mathbf{K}_\mathbf{Y}) \leq S_{\alpha}(\mathbf{K}_\mathbf{X})$ is not always true \cite{teixeira2012conditional}. Nevertheless, being able to adjust $\alpha$ can still be useful to emphasize high density regions of the data. Based on \eqref{eq:represenation_conditional_entroopy}, we can define matrix-based mutual information is defined as:
\begin{equation}\label{eq:representation_mutual_information}
I_{\alpha}(\mathbf{K}_\mathbf{X} ; \mathbf{K}_\mathbf{Y}) = S_{\alpha}(\mathbf{K}_\mathbf{X}) + S_{\alpha}(\mathbf{K}_\mathbf{Y}) - S_{\alpha}(\mathbf{K}_\mathbf{X} \circ \mathbf{K}_\mathbf{Y}).
\end{equation}
Note again that this quantity is well behaved for $\alpha = 1$. However, it has been shown that for certain density operators subadditivity, $S_{\alpha}(\mathbf{K}_\mathbf{X} \circ \mathbf{K}_\mathbf{Y}) \leq S_{\alpha}(\mathbf{K}_\mathbf{X}) + S_{\alpha}(\mathbf{K}_\mathbf{Y})$, still holds for Rényi's quantum entropy with $\alpha \neq 1$ \cite{camilo2019physrev}, and thus we don't restrict to $\alpha=1$ in our exposition.\par
It must be emphasized that these quantities are not estimators of Shannon's entropy or mutual information. For instance, it is possible for the differential entropy of continuous random variables to be negative, whereas matrix-based entropy is always nonnegative. Also, if two continuous random variables are equal, such as when $X=Y$ or $X$ and $Y$ are related through an invertible mapping, Shannon's mutual information $I(X, Y)$ is infinite. In contrast, $I_{\alpha}(\mathbf{K}_\mathbf{X} ; \mathbf{K}_\mathbf{X}) \leq S_{\alpha}(\mathbf{K}_\mathbf{X})$, exhibiting properties more similar to those of discrete random variables. However, as we mentioned above, scaling does affect matrix-based entropy, which is a property of continuous random variables. \section{Difference of Matrix-Based Entropies (DiME)}
The matrix-based mutual information between paired samples from random variables $X$ and $Y$ introduced in \eqref{eq:representation_mutual_information} works well in practice \cite{sanchezgiraldo2015measures, zhang2022icassp}, but it does require proper selection of kernel parameters. For instance, one needs to choose the appropriate bandwidth $\sigma$ (kernel size) when using the Gaussian kernel. Because of property \ref{prop:entropy_exponent}, decreasing the bandwidth parameter leads to a trivial maximization of \eqref{eq:representation_mutual_information} where $I_{\alpha}(\mathbf{K}_{\mathbf{X}} ; \mathbf{K}_{\mathbf{Y}}) = \log{n}$. On the other hand, a large kernel size makes $I_{\alpha}(\mathbf{K}_{\mathbf{X}} ; \mathbf{K}_{\mathbf{Y}})$ very small and unable to capture dependencies between $X$ and $Y$. Inspired by hypothesis testing for independence, where the matrix-based MI between paired samples $\mathbf{K}_{\mathbf{X}}$ and $\mathbf{K}_{\mathbf{Y}}$ is compared to a surrogate for the null distribution by sampling values of matrix-based MI between $\mathbf{K}_{\mathbf{X}}$ and $\bm{\Pi}\mathbf{K}_{\mathbf{Y}}\bm{\Pi}^{T}$, where $\bm{\Pi}$ is a random permutation, we propose using the following difference as our statistic:
\begin{equation}\label{eq:difference_of_mutual_information}
I_{\alpha}(\mathbf{K}_{\mathbf{X}} ; \mathbf{K}_{\mathbf{Y}}) - \mathbb{E}_{\bm{\Pi}}\left[ I_{\alpha}(\mathbf{K}_{\mathbf{X}} ; \bm{\Pi}\mathbf{K}_{\mathbf{Y}}\bm{\Pi}^{T})\right]
\end{equation}
Since the matrix-based entropy for $\mathbf{K}_{\mathbf{Y}}$ is invariant to permutations, \eqref{eq:difference_of_mutual_information} can be reduced to a difference of joint entropies:
\begin{equation}\label{eq:difference_of_matrix_entropies}
\mathbb{E}_{\bm{\Pi}}\left[ S_{\alpha}(\mathbf{K}_{\mathbf{X}} \circ \bm{\Pi}\mathbf{K}_{\mathbf{Y}}\bm{\Pi}^{T})\right] - S_{\alpha}(\mathbf{K}_{\mathbf{X}} \circ \mathbf{K}_{\mathbf{Y}}).
\end{equation}
Notice that unlike \eqref{eq:representation_mutual_information} the difference of matrix-based entropies \eqref{eq:difference_of_matrix_entropies} does not monotonically increase as the kernel bandwidth parameter goes to zero. Instead, this difference moves from small to large and back to small as we decrease the kernel size. In other words, by checking the difference of matrix based entropies, we can select the kernel parameter for which the matrix-based joint entropy of a set of points drawn from the joint distribution $\mathbb{P}_{X,Y}$ can be distinguished from the entropy of a surrogate of the product of marginals $\mathbb{P}_{X}\otimes \mathbb{P}_{Y}$ obtained by the random permutations. This difference of matrix-based entropies constitutes a lower bound on the matrix-based mutual information \eqref{eq:representation_mutual_information}. If we parameterize our kernel function, such that $\mathbf{K}_{\mathbf{X}}(\theta_X)$ and $\mathbf{K}_{\mathbf{X}}(\theta_Y)$ are now functions of the parameter $\theta = \{\theta_X, \theta_Y\}$, we can maximize the difference of matrix-based entropies \eqref{eq:difference_of_matrix_entropies} with respect to $\theta$, which yields DiME:
\begin{multline}
\label{eq:contrastive_representation_mutual_information}
\operatorname{DiME}_{\alpha}(\mathbf{X}; \mathbf{Y}) = \max_{\theta} \mathbb{E}_{\bm{\Pi}}\left[ S_{\alpha}(\mathbf{K}_{\mathbf{X}}(\theta_X) \circ \bm{\Pi}\mathbf{K}_{\mathbf{Y}}(\theta_Y) \bm{\Pi}^{T})\right] - S_{\alpha}(\mathbf{K}_{\mathbf{X}}(\theta_X) \circ \mathbf{K}_{\mathbf{Y}}(\theta_Y))
\end{multline}
To approximate the expectation in the first term of \eqref{eq:contrastive_representation_mutual_information}, we compute an average over a small number $p$ of random permutations. We have experimentally observed that even as single permutation works well. However, we generally use $p=5$ for most of the experiments.
\subsection{Time Complexity}
A key operation to calculate \eqref{eq:matrix_based_entropy} is eigendecomposition, which is used to compute the trace operator. It is well known that eigendecomposition for a square matrix has a time complexity of $O(n^3)$, where $n$ is the matrix size. For a very large $n$ this operation can be prohibitive. In our case, $n$ is the size of a mini-batch which is typically small. Due to this, we are able to compute DiME in a reasonable time without resorting to potentially faster, but less accurate, eigendecomposition approximations.
\section{Comparison to Mutual Information Variational Bounds}
We apply DiME to a toy problem described in \cite{poole2019variational}. We use their same specifications to construct the dataset. Specifically, we sample $(X, Y)$ from a 20-dimensional Gaussian with zero mean and correlation $\rho$, where $\rho$ increases after a given number of iterations. With this setup, the true MI can be computed as $I(X; Y) = -\frac{d}{2} \log (1 - \rho ^2) = -10 \log (1 - \rho ^2)$. The purpose of this experiment is to compute estimations of DiME and compare how it behaves in relation to the true MI. We also compute three variational methods for MI estimation: $I_{\textrm{JS}}$ \cite{hjelm2018learning}, $I_{\textrm{NWJ}}$ \cite{nguyen2010estimating,nowozin2016f}, and $I_{\textrm{NCE}}$ \cite{oord2018representation}, for reference. For each of the variational methods we maximize the bound by training an MLP critic with two hidden layers of 256 units, ReLU activations, and an embedding dimensionality of 32. These methods are in contrast to DiME, where we simply (and optionally) optimize two parameters: the kernel bandwidths in \eqref{eq:contrastive_representation_mutual_information} where $K$ is a Gaussian kernel. We initialize both bandwidths to $\sqrt{d}=\sqrt{20} \approx 4.5$ and use Adam \cite{kingma2014adam} as the optimizer for all estimators. \par
Results for this experiment are shown in Figure~\ref{fig:variationalbounds}. We show the performance of DiME both with and without optimization of kernel bandwidths. The variance of DiME seems to be less affected by the underlying true MI than $I_{\textrm{JS}}$ and $I_{\textrm{NWJ}}$, albeit with a higher variance for low MI and small batch size. As opposed to $I_{\textrm{NCE}}$, DiME continues to scale well when MI exceeds the log of the batch size. Furthermore, even without any training of kernel bandwidth, DiME functions as a reasonable estimator of MI. In this scenario, we are computing our measure of dependence directly from the data at each step. However, we observe that training kernel bandwidths using DiME is often helpful. DiME grows more consistently as the true MI increases. For this experimental setup, we discuss how DiME is affected by batch size and dimensionality in the Appendix. \par
Unlike estimators using variational bounds of MI, DiME can be directly calculated from batches of data. In the variational bound approaches, a critic network is used to provide a score for encoded data. Here, no additional critic network is required and our estimator of MI uses only the data.
\begin{figure*}[hbt!]
\centering
\begin{subfigure}[b]{0.9\textwidth}
\centering
\includegraphics[width=\textwidth]{imgs/64_joint_critic.png}
\label{fig:variationalbounds_64}
\end{subfigure}
\centering
\begin{subfigure}[b]{0.9\textwidth}
\centering
\includegraphics[width=\textwidth]{imgs/1024_separable_critic.png}
\label{fig:variationalbounds_256}
\end{subfigure}
\caption{Performance of different estimators of MI for a toy correlated gaussian dataset with increasing correlation over time. The black staircase line denotes the true MI of the dataset. Note that the DiME is on a different scale than other methods, but still behaves as an estimator of MI. \textbf{(top)} Trained using batch sizes of 64 and using joint critics for variational estimators. \textbf{(bottom)} Trained using batch sizes of 1024 and using separable critics for variational estimators. }
\label{fig:variationalbounds}
\end{figure*}
\section{Experiments}
In this section, we present several experiments showcasing potential applications for DiME. The main purpose of these experiments is to highlight the versatility of DiME when combined with other matrix-based information-theoretic quantities.
\subsection{Multiview representation learning}
\label{section:multiview}
As discussed previously, DiME is well suited for tasks involving the maximization of MI between random variables assigning more value to simpler configurations that exhibit high dependence. This can be used in contrastive learning and representation learning when trying to maximize dependence between two views of data, e.g. \cite{wang2015deep, oord2018representation, hjelm2018learning, zbontar2021barlow}. \par
Let $\mathbf{X}^{(1)} = \left\{x_i^{(1)}\right\}_{i=1}^N$ and $\mathbf{X}^{(2)} = \left\{x_i^{(2)}\right\}_{i=1}^N$ be paired sets of instances from two different views of some latent variable. The goal is to train encoders $f_1\in \mathcal{F}_1$ and $f_2 \in \mathcal{F}_2$ (potentially with shared weights) in order to maximize the MI between the encoded representations by using DiME as follows:
\begin{equation}
\underset{f_1 \in \mathcal{F}_1, f_2 \in \mathcal{F}_2}{\operatorname{maximize}} \operatorname{DiME}_{\alpha}\left(f_1(\mathbf{X}^{(1)}),f_2(\mathbf{X}^{(2)})\right)
\label{eq:dime_multiview}
\end{equation}
Next, we apply our objective function in two well-known multiview datasets and assess the results on downstream tasks such as classification, latent factor disentanglement, and style transfer.
\subsubsection{Multiview MNIST}
\label{subsection:multiview MNIST}
We follow the work of \cite{wang2015deep} in constructing a multiview MNIST dataset with two views. The first view contains digits rotated randomly between $-45$ and $45$ degrees. The second view contains digits with added noise sampled uniformly from $[0, 1]$, clamping to one all values in excess of one. The second view is then shuffled so that paired images are different instances from the same class. With this setup, the only information shared between views is the class label. We thus expect a maximization of dependence between the shared representation between views to carry only the class label information.\par
We encode each view using separate encoders which consist of the architecture described in Table \ref{tab:multiview_architecture}. DiME is optimized between the views as described in~\eqref{eq:dime_multiview}. We use a Gaussian kernel with fixed bandwidth $\sqrt{D}/2$. A batch size of 3000 is used.
\begin{figure}[!t]
\centering
\includegraphics[scale=0.44]{imgs/mean_stratos_test_accuracies.png}
\caption{Classification accuracy across dimensionality obtained via firstly training encoders with a dependence loss and then training a classifier consisting of one hidden layer after the frozen encoder. The accuracy of equivalently sized networks trained using direct supervision is also included.}
\label{fig:downstream}
\end{figure}
We compare to several baselines on the task of downstream classification accuracy. The baselines are: DCCA \cite{wang2015deep}, CKA \cite{cortes2012algorithms, kornblith2019similarity}, HSIC \cite{gretton2005measuring}, infoNCE \cite{oord2018representation}, TUBA \cite{poole2019variational}, MINE \cite{belghazi2018mine}, and Stratos' difference-of-entropies (DOE) \cite{mcallester2020formal}.
We define downstream classification accuracy as training an encoder using a measure of dependence followed by the training of a supervised classifier with one hidden layer acting on top of the encoder's output. Note that the encoder parameters are frozen after its initial training. While the encoder never has access to labels during training, we don't consider this approach unsupervised because label information is inherent in the paired dataset.\par
The downstream classification accuracies for varying latent dimensionalities, averaged over 10 runs, are shown in Figure \ref{fig:downstream}. From this, we observe that DiME outperforms both baselines for smaller latent dimensionalities. DiME requires fewer latent dimensions to be competitive with fully supervised accuracies. However, DiME does have higher variance for certain dimensionalities. Possible explanations for this are discussed in Section \ref{section:conclusion}.
\subsubsection{Disentanglement of Latent Factors}
\label{subsection:disentanglement}
So far we have extracted information that is common to two views. It becomes natural to ask if you can instead extract information that is exclusive to views. With the multiview MNIST dataset, exclusive information to View 1 would be rotations and for View 2 it would be noise. Because View 2 is shuffled in a way so that paired datapoints are different instances of the same class, exclusive information could also constitute latent factors such as stroke width, boldness, height, and width.\par
In order to separate shared and exclusive information, we use the matrix-based conditional entropy defined in \eqref{eq:represenation_conditional_entroopy}. Let $\bm{S}^{(i)}$ and $\bm{E}^{(i)}$ define the shared and exclusive information in view $i$ captured by the encoder $f_{i}$ for data $\bm{X}^{(i)}$. Then $f_i(\bm{X}^{(i)}) = [\bm{S}^{(i)}, \bm{E}^{(i)}]$, where the right side is a concatenation of dimensions over the batch. The matrix-based conditional entropy $S_{\alpha}(\bm{S}^{(i)} | \bm{E}^{(i)})$ quantifies the amount of uncertainty remaining for $\bm{S}^{(i)}$ (class label) after observing $\bm{E}^{(i)}$ (i.e. latent factors). Ideally this quantity should be maximized so that observing exclusive information gleans nothing about shared information.\par
\begin{figure*}[t]
\centering
\includegraphics[scale=0.5]{imgs/ica_nowhitening.png}
\caption{Walking on disentangled exclusive independent components. The walk is performed by encoding a digit prototype (center column) and modifying the exclusive dimensions. The leftmost and rightmost columns are moving -2 and +2 units, respectively, in the direction of the independent component, with evenly spaced steps in-between. }
\label{fig:ica_walk}
\end{figure*}
Here, we use the same encoder setup as described in Section~\ref{section:multiview}. To encourage the usage of exclusive dimensions by the encoder $f_i$, we also minimize reconstruction error by passing the full latent code through a decoder $g_{i}$. This is in contrast to Section \ref{section:multiview} where only an encoder is needed. The reconstruction of $\bm{X}^{(i)}$ is denoted as $\hat{\bm{X}}^{(i)} = g_i(f_i(\bm{X}^{(i)}))$. A separate encoder/decoder pair is used for each view. The full loss function used here is:
\begin{align*}
\mathcal{L} = & \textrm{DiME}(\bm{S}^{(1)}, \bm{S}^{(2)}) + S_{\alpha}(\bm{S}^{(1)}| \bm{E}^{(1)}) + S_{\alpha}(\bm{S}^{(2)}| \bm{E}^{(2)}) \\& - \textrm{MSE}(\bm{X}^{(1)}, \hat{\bm{X}}^{(1)}) - \textrm{MSE}(\bm{X}^{(2)}, \hat{\bm{X}}^{(2)}),
\end{align*}
which is maximized with respect to $f_1, f_2, g_1,$ and $g_2$. We use this loss to learn 10 shared dimensions and 5 exclusive dimensions. Unlike variational autoencoders, which also learn latent factors in an information-theoretic fashion, here we are not enforcing that each exclusive dimension is independent from the others \cite{kingma2019introduction}. In order to visualize what exclusive factors are being learned, we calculate the independent components of the exclusive dimensions with parallel FastICA~\cite{hyvarinen2000independent}. Using these independent components, we can start with a prototype and perform walks in the latent space in independent directions. By walking in a single independent direction, we intend to change only a single latent factor. In Figure \ref{fig:ica_walk}, we walk on the five exclusive independent components of View 1. Qualitatively, it seems that they encode height, width, stroke width, roundness, and rotation.
\subsubsection{Colored MNIST}
Similarly to \cite{sanchez2020learning} we performed experiments for learning disentangled representations by using DiME on the colored MNIST dataset. Given a pair of images belonging to the same class but from different views (colored background and colored foreground digits, see Figure \ref{fig:colored dataset}), we seek to learn a set of shared features that represent the commonalities between the images and disentangle the exclusive features of each view. It has been shown that these kinds of representations are convenient to perform downstream tasks, such as image retrieval by finding similar images to a query image by looking at either the shared or the exclusive features. We maximize the MI between the shared representations captured by a single encoder $f_{\textrm{sh}}$ and minimize the MI of the shared and exclusive features captured by a separate encoder $f_{\textrm{ex}}$ to encourage the disentanglement of the two components.\par
In particular, $f_{\textrm{sh}}(\mathbf{X}) = \mathbf{S}$ encodes the shared features of the two views (class-dependent features), and $f_{\textrm{ex}}(\mathbf{X}) = \mathbf{E} = [\mathbf{V}, \mathbf{Z}]$ extracts the class-independent attributes so that $\mathbf{V}$ contains the exclusive features of each view (view-dependent features). $\mathbf{Z}$ is used to encode residual latent factors needed for reconstruction.\par
Because we want the model to learn a shared space that is invariant to the view, we first maximize the MI between the shared representations of both views $ \mathbf{S}^{(1)}$ and $ \mathbf{S}^{(2)}$ via DiME as in equation \ref{eq:dime_multiview}. Here, the same encoder is used for both views. Once the shared representation is learned, we freeze the shared encoder and train the exclusive encoder $f_{\textrm{ex}}$. To disentangle all three latent sub-spaces, we minimize the MI between $\mathbf{S}$ and $\mathbf{E}$ and minimize the MI between $\mathbf{V}$ and $\mathbf{Z}$. This procedure encourages the exclusive encoder not to learn features that were already learned by the shared encoder. Additionally, we have a conditional prior on $\mathbf{V} = [\mathbf{V}_1,\mathbf{V}_2] $ such that $\mathbf{V}^{(1)} = [\mathbf{V}_1, \mathbf{0}] $ and $\mathbf{V}^{(2)} = [\mathbf{0},\mathbf{V}_2] $ intending different subspaces to independently encode the view-exclusive generative factors (see Figure \ref{fig:exclusive space}). We achieve this by minimizing the Jensen-Rényi divergence (JRD) \cite{osorio2022representation} of the view-exclusive features to samples from two mutually exclusive uniform random variables from 0 to 1, that we denote as $\mathbf{P}$. To ensure that features have high MI with the images, we also train a decoder $g \in \mathcal{G}$ that will reconstruct the original images using all three latent subspaces.\par
We train the exclusive encoder by searching $f_{\textrm{ex}}\in \mathcal{F}_{\textrm{ex}},g \in \mathcal{G}$ that minimize the following objective:
\begin{equation*}
\mathcal{L}_{\textrm{ex}} = I_\alpha(\mathbf{S},\mathbf{E}) + I_\alpha(\mathbf{V},\mathbf{Z}) + D_{\alpha} (\mathbf{V}||\mathbf{P}) + \textrm{MSE}(\mathbf{X},\hat{\mathbf{X}})
\end{equation*}
\begin{figure*}[!t]
\begin{subfigure}[b]{0.08\linewidth}
\centering
\includegraphics[width=\textwidth]{imgs/dataset.pdf}
\caption{}
\label{fig:colored dataset}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.25\linewidth}
\centering
\includegraphics[width=\textwidth]{imgs/sharedSpace.pdf}
\caption{}
\label{fig: shared space}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.18\linewidth}
\centering
\includegraphics[width=\textwidth]{imgs/nnShared.pdf}
\caption{}
\label{fig:nearest shared}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.26\linewidth}
\centering
\includegraphics[width=\textwidth]{imgs/exclusiveSpace.pdf}
\caption{}
\label{fig:exclusive space}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.19\linewidth}
\centering
\includegraphics[width=\textwidth]{imgs/nnExclusive.pdf}
\caption{}
\label{fig:nearest exclusive}
\end{subfigure}
\hfill
\caption{(a) Example of a pair of images from the multiview colored MNIST: view 1 corresponds to background colored digits and view 2 to foreground colored digits. (b) t-SNE of the shared features learned via DiME. Colors represent different digit labels. (c) Image retrieval by finding nearest neighbors on the shared features. (d) view-exclusive features learned by minimizing JRD to a desired prior. (e) Image retrieval by finding nearest neighbors on the view-exclusive features}
\label{fig:Results colored mnist}
\end{figure*}
\begin{figure}
\begin{subfigure}[b]{0.4\linewidth}
\centering
\includegraphics[width=\textwidth]{imgs/background_disentangled.pdf}
\caption{}
\label{fig:Background disentangled}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.4\linewidth}
\centering
\includegraphics[width=\textwidth]{imgs/foreground_disentangled.pdf}
\caption{}
\label{fig:foreground_disentangled}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.125\linewidth}
\centering
\includegraphics[width=\textwidth]{imgs/transferstyle.pdf}
\caption{}
\label{fig:style_transfer}
\end{subfigure}
\hfill
\caption{Disentangling of generative factors in the multiview colored MNIST dataset. (a) Walking on the exclusive dimension of view 1. (b) Walking on the exclusive dimension of view 2. (c) Style transfer results by swapping exclusive dimensions.}
\label{fig:colored mnist results}
\end{figure}
Both encoders have a similar architecture to the one used
in section 5.1.1 but batch-norm is omitted. Exact model details are provided in Table \ref{tab:coloredmultiview_architecture} in the appendix. The dimensionality of S, V and Z are 10, 2 and 6 respectively. To train
the shared encoder we use a batch size of 1500, whereas
for the exclusive encoder batch size of 64 is used.
A t-SNE visualization of the learned shared features can be seen in Figure \ref{fig: shared space} where it can be observed how digits from the same class are grouped into the same clusters regardless of the view. This is corroborated with image retrieval by finding the nearest neighbors to the shared representation of a query image as it can be seen in Figure~\ref{fig:nearest shared}. This substantiates the fact that DiME is well capturing the common attributes between the two views, which in this case is the class information. \par
On the other hand, the learned view-exclusive features (see Figure \ref{fig:exclusive space}) are class-agnostic and for a given query instance, the exclusive space nearest neighbors correspond to the same background/foreground color. This is independent
of the digit class as can be seen in Figure \ref{fig:nearest exclusive}. To further support the quality of the shared features learned by DiME, we generate samples by keeping the shared representation fixed and move along the view-exclusive dimensions. In Figures \ref{fig:Background disentangled} and \ref{fig:foreground_disentangled} we can see how the digit content is well preserved as the view generative factor is changing. \par
This approach is well suited for style transfer, by exchanging the style (exclusive $E$) between a query and a reference image while keeping the content (shared $S$) intact. Without any additional training required, we display style transfer in Figure \ref{fig:style_transfer}.
\paragraph{Representation disentanglement evaluation Multicolored MNIST}
Similarly to \cite{sanchez2020learning}, we evaluate the disentanglement of the shared and exclusive features by performing some classification experiments. Ideally, the shared representation only encodes common information between the views, that is, features relevant to the digit class. Therefore, a digit classifier trained on the shared features should be able to accurately classify the digit label. Conversely, it should not be able to classify the background or foreground colors since that information is not common across the views. Classifying colors based on shared features should perform at the level of a random classifier, that is 8.33\% since there are 12 different colors.
To evaluate this, we train a simple classifier composed of 2 fully connected layers with 1024 hidden units and compare our results to \cite{sanchez2020learning} and \cite{gonzalez2018image}. Shown in Table \ref{tab:accuracyShared}, DiME achieves the highest digit number classification accuracy on shared features. We only encode 10 shared features while \cite{sanchez2020learning} uses 64 and \cite{gonzalez2018image} 512 features. Additionally, to corroborate that the shared features encoded via DiME are disentangled from the exclusive ones, the foreground and background color accuracy are close to a random guess, achieving the lowest foreground color accuracy and a similar background accuracy to \cite{sanchez2020learning}.
We perform a similar experiment by using the exclusive features to train a digit or color classifier. The results of this analysis is shown in Table \ref{tab:accuracyExclusive}. Our method reaches the lowest digit number accuracy, being close to the ideal 10\%. This supports the fact that the exclusive features are truly class-agnostic. Furthermore, the size of our exclusive features is smaller (2 view exclusive features, 1 per view) compared to \cite{sanchez2020learning} (8 exclusive features) and \cite{gonzalez2018image} (8 exclusive features)
\begin{table}[]
\centering
\caption{Digit number, background and foreground color accuracy using the shared representation}
\label{tab:accuracyShared}
\begin{tabular}{l|l|l|l}
Method & Digit Number & Background Color & Foreground color \\ \hline
Ideal & 100\% & 8.33\% & 8.33\% \\
\cite{sanchez2020learning} & 94.48\% & \textbf{8.22\%} & 8.83\% \\
\cite{gonzalez2018image} & 95.42\% & 99.56\% & 29.81\% \\
DiME & \textbf{98.93\%} & 8.79\% & \textbf{8.41\%}
\end{tabular}
\end{table}
\begin{table}[]
\centering
\caption{Digit number, background and foreground color accuracy using the exclusive representation}
\label{tab:accuracyExclusive}
\begin{tabular}{l|l|l|l}
Method & Digit Number & Background Color & Foreground color \\ \hline
Ideal & 10.00\% & 100\% & 100\% \\
\cite{sanchez2020learning} & 13.20\% & \textbf{99.99\%} & \textbf{99.92}\% \\
\cite{gonzalez2018image} & 99.99\% & 71.63\% & 29.81\% \\
DiME & \textbf{10.30\%} & 98.84\% & 84.3\%
\end{tabular}
\end{table}
We also compared DiME to another mutual information estimator as MINE assess the impact of the mutual information estimator. For this experiment, we replaced the mutual information estimator while keeping the rest of the losses. The results are shown in \ref{tab:accuracy}. We can observe that while MINE learns a useful shared representation, a problem occurs in how MINE interacts with other matrix-based quantities used in the loss function leading to an inaccurate disentanglement. This is partly because when using MINE or similar mutual information estimators, it is necessary to properly tune the kernel bandwidth for matrix-based quantities to achieve satisfactory results. On the contrary, DiME naturally adjusts the data with a predefined sigma ($\sqrt{D/2}$ was used in the paper) to make the data suitable for matrix-based quantities. For the MINE experiment, we tested 5 different values of $\sigma$ for the kernels involved in the matrix-based quantities in the loss, selected in a log space around $\sqrt{D/2}$. However, none of them were able to perform adequately.
\begin{table}[]
\caption{Digit number (DN), Background (B) and Foreground (F) color accuracy using the shared (left) representation and exclusive (right) representations}
\label{tab:accuracy}
\begin{subtable}{.5\textwidth}
\begin{tabular}{l|l|l|l}
Method & DN & B & F \\ \hline
Ideal & 100\% & 8.33\% & 8.33\% \\
MINE & 98.92\% & 11.24 \% & 12.11\% \\
DiME & \textbf{98.93\%} & \textbf{8.79\% } & \textbf{8.41\%}
\end{tabular}
\end{subtable}%
\begin{subtable}{.5\textwidth}
\begin{tabular}{l|l|l|l}
Method & DN & B & F \\ \hline
Ideal & 10.00\% & 100\% & 100\% \\
MINE $\sigma_0$ & 10.77\% & 58.08\% & 48.1\% \\
MINE $\sigma_1$ & 11.30\% & 64.49\% & 58.51\% \\
MINE $\sigma_2$ & 10.85\% & 69.38\% & 70.24\% \\
MINE $\sigma_3$ & 10.99\% & 59.04\% & 63.64\% \\
MINE $\sigma_4$ & 11.01\% & 40.51\% & 42.28\% \\
DiME & \textbf{10.30\%} & \textbf{98.84}\% & \textbf{84.3}\%
\end{tabular}
\end{subtable}%
\end{table}
\subsection{DiME-GANs}
Motivated by the relation between MI and Jensen-Shannon divergence, we propose training of GANs as an alternating optimization of between two competing objectives. Similar to MMD-GAN \cite{Li2017mmdgan}, our discriminator network $f_\theta$ maps samples from data space $\mathcal{X}$ to a representation space $\mathcal{Y}$, and the generator network $g_\psi$ maps samples from the noise space $\mathcal{Z}$ to data space $\mathcal{X}$. \par
To train the discriminator, we use DiME to maximize the MI between samples of a mixture distribution and an indicator variable. The mixture distribution has two components: the distribution of real samples and the distribution of samples from a generator. The indicator variable is paired with each sample drawn from the mixture and indicates the component (real or fake) from which the sample came from. To train the generator, we maximize the matrix-based conditional entropy of the indicator variable given a sample from the mixture. This is equivalent to minimizing the matrix-based MI between the mixture and the indicator variable. Since the matrix-based MI is an upper bound of DiME, each generator update tries to reduce DiME by pushing down this upper bound. Conversely, each discriminator step tunes the representation space such that the upper bound given by the matrix-based MI is tight.
\begin{figure}[!t]
\centering
\includegraphics[width=0.5\linewidth]{imgs/samples_dcgan_laplacian64_101022.jpeg}
\caption{Samples generated using DiME-GAN}
\label{fig:GAN_cifar10}
\vspace*{-5pt}
\end{figure}
Figure \ref{fig:GAN_cifar10} show samples from DiME-GAN trained on CIFAR10 dataset. For the kernel we use the Laplacian kernel defined as $\kappa_{\sigma}(x, y) = e^{-\frac{\sum_{i=1}^D \vert (x)_i - (y)_i\vert}{\sigma}}$ with sigma fixed $\sigma = \sqrt{D/2}$. The discriminator objective tunes the parameters of $f_\theta$. For each iteration, we use 64 images from the true distribution and 64 images sampled from the generator network. Both optimizers use the Adam with $lr = 0.00005$. The architecture is similar to DCGAN, but without batch-normalization and gradient clippling during training. Details of the employed architecture and hyperparameters are provided in the Appendix.
\section{Related Work}
This work is not the first time that matrix-based entropy has been used in a multiview setting. In \cite{zhang2022icassp}, a supervised information-bottleneck approach is used to learn joint representations for views and their labels. Matrix-based entropy is utilized to simultaneously minimize MI between each view and its encodings and to maximize MI between fused encodings and labels. However, our proposal solves the issue of selecting bandwidth parameters and forcing the output of the networks to specified ranges which are necessary to avoid trivial solutions when using matrix-based entropy alone.\par
Barlow Twins is a method for self-supervised learning with the objective of reducing cross-correlation in projections of embedded data \cite{zbontar2021barlow}. Their method makes clear distinctions between the embedded space (used for downstream tasks) and the projected space where cross-correlation is calculated. The similarity between this method and ours lies in the projector. In our work, the ``projector'' is the implicit mapping related to the positive definite kernel function which projects data to the RKHS. It is in the RKHS where matrix-based entropy and related quantites are calculated. A similar method to Barlow Twins is the SSL-HSIC, which infuses the power of kernel methods with modern self-supervised learning problems \cite{li2021self}.\par
Our work uses permutations to decouple random variables. This is not a novel idea, and has been used before in, for example, FactorVAE \cite{kim2018disentangling} and MINE \cite{belghazi2018mine}. There, permutations are used to efficiently compute a product of marginals. However, the self regulating property of DiME, which uses permutations to provide the reference entropy, is novel and open for further exploration.
\section{Conclusions}
\label{section:conclusion}
We proposed DiME, a quantity which behaves like mutual information and is able to be estimated directly from data. DiME is built using the matrix-based entropy as its main component, which uses kernels to measure uncertainty in the dataset. We compared DiME's behavior as a mutual information estimator to variational estimators, and showed it was well behaved for sufficiently large batch sizes. Furthermore, DiME does not require an additional network and is calculated in the same space as the data. We applied DiME to the problems of multiview representation learning, disentanglement of latent factors, and as a GAN training objective.\par
To handle the variance of DiME, which can be very large, we are exploring alternatives to the Gram matrix formulation. For instance, by working directly in the feature space, we can aggregate results from small batches and reduce the variance of the estimators. There are several methods to provide an explicit feature space, such as using Random Fourier Feature approximations~\cite{rahimi2007random} to kernels.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,371 |
\section{Introduction}\label{sec:intro}
The spectral norm of a matrix, i.e., the maximal singular value, has numerous applications in pure and applied mathematics.
One of the fundamental reasons for the tremendous use of this norm is that it is polynomial-time computable and the software for its computation is easily available
on MAPLE, MATHEMATICA, MATLAB and other platforms.
Multiarrays, or $d$-mode tensors, i.e. $d\ge 3$, are starting to gain popularity due to data explosion and other applications. Usually, these problems deal with real valued tensors.
Since the creation of quantum physics, $d$-mode tensors over complex numbers became the basic tool in treating the $d$-partite states. Furthermore,
the special case of $d$-partite symmetric qubits, called bosons, is the basic ingredient in construction the boson sampling devices \cite{AA13, Netall}.
The ($\mathbb{F}$-)spectral norm of a tensor is a well defined quantity over the real ($\mathbb{F}=\mathbb{R}$) or complex numbers ($\mathbb{F}=\mathbb{C}$).
Unlike in the matrix case, the computation of the spectral norm in general can be NP-hard \cite{FL18,HL13}. Furthermore, the complex spectral norm of a real tensor can be bigger than its real spectral norm. In spite of these numerical difficulties, there is a need to compute these norms in special cases of interesting applications, as the geometric measure of entanglement. (See later in the Introduction and \S\ref{sec: spwcnrm}.) Even the simplest case of $d$-partite qubits poses theoretical and numerical challenges \cite{GFE09}. This can be partly explained by the fact that
the space $\otimes^d\mathbb{C}^2$ has dimension $2^d$.
In this paper we restrict ourselves to $d$-mode symmetric tensors over $\mathbb{F}^n$, denoted as $\rS^d\mathbb{F}^n$. The dimension of this space is ${n+d- 1\choose d}={n+d-1\choose n-1}$. Hence for a fixed $n$ this dimension is $O(d^{n-1})$. In particular, the dimension of $\rS^d\mathbb{C}^2$ is $d+1$.
A symmetric tensor $\cS\in\rS^d\mathbb{F}^n$ can be identified with a homogeneous
polynomial $f=f_{\cS}$ of degree $d$ in $n$ variables over $\mathbb{F}$, denoted here as $\rP(d,n,\mathbb{F})$. It was already observed by J. J. Sylvester \cite{Syl51} that binary forms, i.e., $n=2$, posses very special properties related to polynomials of one complex variable.
The spectral norm of $\cS\in\rS^d\mathbb{F}^n$ is denoted by $\|\cS\|_{\sigma,\mathbb{F}}$. Its value is equal to the following maximum of $f\in\rP(d,n,\mathbb{F})$ on the unit sphere in $\mathbb{F}^n$:
\[\|f\|_{\sigma,\mathbb{F}}=\max\{|f(\mathbf{x})|,\;\mathbf{x}\in\mathbb{F}^n, \|\mathbf{x}\|=1\}.\]
The spectral norm of a complex valued symmetric tensor is given as the maximum of the real part $\Re f$ over the set of complex valued vectors of norm one. The critical vector of $\Re f$ is an anti-eigenvector of $\bF=\frac{1}{d}\nabla f$. The critical value is the eigenvalue of this system. In this paper we first convert this eigenvalue problem to the anti-fixed point $\bF(\mathbf{x})=\bar \mathbf{x}$. Next we show that this anti-fixed point is a fixed point of another polynomial map $\bH$, where $\bH(\mathbf{x})=\overline{\bF(\overline{\bF(\mathbf{x})})}$. In the generic case, where the hypersurface $f(\mathbf{x})=0$ is a smooth hypersurface in the complex projective space, the number of fixed points is $(d-1)^{2n}$, counted with multiplicities.
In this case we find all fixed of $\bH$ and compute the $\|f\|_{\sigma,\mathbb{C}}$. In the real case $\mathbb{F}=\mathbb{R}$, it is enough to consider the fixed points of $\bF$.
Our approach is completely different from the standard optimization methods that are used now in the literature \cite{CHLZ12,LMV00,FMPS13,FT15,Nie14,Tong}.
We now highlight the new and most important results of our paper. Associate with $\cS\in\rS^d\mathbb{C}^n$ the polynomial $f(\mathbf{x})=f_{\cS}(\mathbf{x})=\cS\times\otimes^d\mathbf{x},\mathbf{x}\in\mathbb{C}^n$.
(See the beginning of \S\ref{sec: spwcnrm} for tensor notations.) Let $\mathbf{F}$ and $\bH$ be defined as above.
Each component of $\mathbf{F}$ and $\bH$ is a homogeneous polynomial of degree $(d-1)$ and $(d-1)^2$ respectively. Theorem \ref{theofoundthm} shows that the complex spectral norm of $\cS\in\rS^d\mathbb{C}^n$ can be computed by finding the fixed points of the homogeneous polynomial map $\mathbf{H}:\mathbb{C}^n\to\mathbb{C}^n$, which is the polynomial system of equations $\bH(\mathbf{x})-\mathbf{x}=\mathbf{0}$ . Let $\mathrm{H}(d,n)$ be the classical hyperdeterminant variety in the space $\rP(d,n,\mathbb{C})$ \cite{GKZ}. Assume that $f\in\rP(d,n,\mathbb{C})\setminus \mathrm{H}(d,n)$. Then the number of fixed points of $\bF$ and $\bH$ is $(d-1)^n$ and $(d-1)^{2n}$ respectively, (counted with multiplicities).
If in addition $\cS$ is real then its real spectral norm can be found by considering only the real fixed points of $\mathbf{F}$ in projective space $\mathbb{P}\mathbb{C}^n$.
Recall the elimination method for finding the roots of a system of $n$ polynomial equations in $n$ variables, each of degree at most $d$, with only isolated roots \cite{vdW, Mor}. Its complexity upper bound $d^{2^{n}}$ follows from Kronecker's work \cite{Hei}. If the system does not have roots at infinity then the solutions of this system have rational univariate representation. The arithmetic complexity of finding these roots is $O(d^{c\,n})$ for some $c>0$ \cite{Rou99,MST17}.
Using the above result, and the fact that the maps $\textbf{F}$ or $\bH$ corresponding to $f_{\cS}\in\rP(d,n,\mathbb{C})\setminus \mathrm{H}(d,n)$ have a finite number of fixed points with a polynomial univariate representation, we give an algorithm to find the $\mathbb{F}$-spectral norm of $\cS\in\rS^d\mathbb{F}^n$ with an arbitrary relative precision with respect to the Hilbert-Schmidt norm of $\|\cS\|$. See Theorems \ref{cesnsingcten}, \ref{cessingcten} and \ref{cesrealten}. The arithmetic complexity of this algorithm is $O(d^{c\,n})$.
We remark that in all papers on complexity of finding the roots of zero dimensional zero set of polynomial equations in more than one variable cited in this paper we did not see any results on approximation of these roots within $\delta>0$ precision.
We study in detail the case of $d$-mode symmetric qubits, which are tensors in $\rS^d\mathbb{C}^2$ of Hilbert-Schmidt norm one. We show that the nonzero fixed points of $\mathbf{H}$ can be computed by finding the roots of the corresponding
polynomial of one complex variable of degree at most $(d-1)^2+1$, provided that this symmetric qubit is not in the exceptional family, (Theorem \ref{computdqubspecnrm}). We give a polynomial time algorithm for a relative approximation for all symmetric qubits, (Theorem \ref{complexaynqub}). If $\cS$ is real valued then its real spectral norm depends only on the real roots of this polynomial,
or actually, on the real root of another polynomial of degree at most $d+1$.
Our results have an important application to the notion of the geometric measure of entanglement of $d$-partite symmetric states, (bosons), in quantum physics and its computation. Recall that a $d$-partite state is represented by a $d$-mode tensor $\cT$ of Hilbert-Schmidt norm one: $\|\cT\|=1$. One of the most important notion in quantum physics is the entanglement of $d$-partite systems \cite{EPR35, Sch35,Sch36,Bel64}. A state $\cT$ is called entangled if it is not a product state, (rank one tensor). The distance of a state $\cT$ to the product states is called the geometric measure of entanglement. It is given by $\sqrt{2(1-\|\cT\|_{\sigma,\mathbb{C}})}$, where $\|\cT\|_{\sigma,\mathbb{C}}$ is the $\mathbb{C}$-spectral norm of $\cT$. (See \S\ref{sec: spwcnrm}).
In particular, we deduce that the geometric measure of entanglement of a $d$-partite symmetric state $\cS\in\rS^d\mathbb{C}^n$, called a symmetric $d$-\emph{qunit}, is polynomial time computable in $d$ for a fixed $n$. For symmetric qubits our results have much better complexity than in the case $n>2$.
We now survey briefly the contents of our paper. In \S\ref{sec: spwcnrm} we state our notations for tensors. We recall the definition of the spectral norm of a tensor
$\cT$. We state the well known connection between the notion of the geometric measure of entanglement and the spectral norm of the $d$-partite state \eqref{defgment}.
In \S\ref{sec:symten} we first discuss the identification of $d$-symmetric tensors with the homogeneous polynomials of degree $d$. Then we study the spectral norm of $d$-symmetric tensors on $\mathbb{F}^n$. We recall the remarkable theorem of Banach \cite{Ban38} \eqref{Banchar} that characterizes the spectral norm of a symmetric tensor, which was rediscovered a number of times in the mathematical and physical literature \cite{CHLZ12,Fri13,Hubetall09}.
In \S\ref{sec:critpts} we study the critical points of the homogeneous polynomial $f$ of degree $d$ on the unit sphere in $\mathbb{F}^n$. We call a symmetric tensor $\cS$, where $f=f_{\cS}$, \emph{singular} if the system $\nabla f(\mathbf{x})=\mathbf{0}$ has a nontrivial solution in $\mathbb{C}^n$.
Equivalently, if the corresponding hypersurface $f(\mathbf{x})=0$ in the projective space $\mathbb{P}\mathbb{C}^n$ is singular.
We show that the critical points of the real part of $f(\mathbf{x})$ correspond to anti-fixed points of $\bF$ for $\mathbb{F}=\mathbb{C}$: $\bF(\mathbf{x})=\bar\mathbf{x}$ and to fixed points of $\pm \bF$ for $\mathbb{F}=\mathbb{R}$.
Theorem \ref{theofoundthm}, explained above, outlines our theoretical approach for numerical computation of the spectral norm of symmetric tensors. Using the degree theory we give in Theorem \ref{estnumbanteig} lower and upper bounds on the number of anti-fixed points of $\bF$ corresponding to nonsingular $\cS\in\rS^d\mathbb{C}^n$.
In \S\ref{sec:dnqudit} we study the available algorithms and their complexities to approximate the spectral norms of symmetric tensors in $\rS^d\mathbb{F}^n$ for a fixed $n$.
A symmetric tensor $\cS$, and the corresponding $f_{\cS}\in\rP(d,n,\mathbb{C})$, are called strongly nonsingular if they are nonsingular and the coordinates $x_1$ of the $(d-1)^{2n}$ fixed points of $\bH$ are pairwise distinct. We show that most of the symmetric tensors in $\rS^d\mathbb{C}^n$ are strongly nonsingular. The fixed points of $\bH$ corresponding to a strongly nonsingular tensor satisfy the conditions of the shape Lemma \cite{Rou99}.
In Theorem \ref{cesnsingcten} we consider the case of a strongly nonsingular $\cT\in\rS^d\mathbb{Z}[\bi]^n$,where $\mathbb{Z}[\bi]$ are Gaussian integers. Assume that the coordinates of $\cT$ are bounded in absolute value by $2^{\tau}$ for some $\tau\in\mathbb{N}$. Using recent results in \cite{BFS15} and \cite{MST17} we show that the bit complexity of the computation of an approximation $L(\cT)$ to the norm $\|\cT\|_{\sigma,\mathbb{C}}$ with a relative precision $2^{-e}, e\in\mathbb{N}$ is $\tilde O\big((\tau+e) d^{8n}\big)$.
In Theorem \ref{cessingcten} we discuss the computational complexity of an approximation $L(\cT)$ to a given $\cT\in\rS^d\mathbb{Z}[\bi]^n$, without assuming that $\cT$ is strongly nonsingular. We give a probabilistic algorithm to compute $L(\cT)$ with a similar bit complexity.
Similar results are obtained in Theorem \ref{cesrealten} for an approximation of the spectral norm of $\cT\in\rS^d\mathbb{Z}^n$ with slightly better complexity estimations.
Theorems \ref{theofoundthm}, \ref{cesnsingcten}, \ref{cessingcten}, and \ref{cesrealten} constitute the first major contribution of this paper.
An obvious question is what is the complexity of finding an approximation $L(\cT)$ to $\cT\in\rS^d\mathbb{Z}^n$ if we do not keep $n$ fixed.
Theorem \ref{NPhardquart} shows that an approximation of the spectral norm of homogeneous quartic polynomials with an arbitrary precision is NP-hard. This result follows from the old result of Motzkin-Straus \cite{MS65} relating the clique number of a graph to a certain maximum problem for the adjacency matrix of the graph, and its tensor interpretation to the spectral norm of tensors \cite[(8.2)]{FL18}. Unfortunately,
it was not observed in \cite{FL18} that the corresponding tensor is symmetric.
Note that the approximation algorithm outlined in Theorem \ref{cesrealten} has at least complexity $O(3^{n^2})$, while the brute force method looking over all possible subsets of $n$ vertices of the given graph is $O(2^n)$.
In \S\ref{sec:dqubit} we discuss in detail theoretical and numerical aspects of the computation of the spectral norm of $\cS\in\rS^d\mathbb{F}^2$. In Theorem \ref{computdqubspecnrm} we show that
the fixed points of the corresponding $\bH$ in this case can be reduced
to one polynomial equation of degree at most $(d-1)^2+1$, unless we are in the exceptional family. In the nonexceptional case we give a simple formula to compute the spectral norm. In Theorem \ref{complexaynqub} we give an approximation algorithm for the spectral norm of $\cT\in\rS^d\mathbb{Z}[\bi]^2 $ with a relative error $2^{-e}$ of bit complexity $\tilde\cO(d^2(d^4\max(d^2,\tau)+e))$.
For $\cT\in\rS^d\mathbb{Z}^2$ we have better complexity results. Theorems \ref{computdqubspecnrm} and \ref{complexaynqub} constitute the second main contribution of this paper.
In \S\ref{sec:excepcase} we analyze completely the exceptional family. We show how to obtain a relative approximation for symmetric tensors in this family. The complexity of this approximation is the same as for the nonexceptional family.
In \S\ref{sec:numerex} we give numerical examples of our method for calculating the complex and the real spectral norm of some of $\cS\in\rS^d\mathbb{C}^n$ for $n=2,3,4$. Many of our examples correspond to polynomials that are sum of two monomials. Lemma \ref{sum2mon} shows that in this case one can assume that the coefficients of these two monomials are nonnegative, if we compute the complex spectral norm. That is, the corresponding symmetric tensors have nonnegative entries. (This result is false if we consider the real spectral norm of real symmetric tensor which corresponds to a sum of two monomials.) Most of the examples of $d$-qubits considered in \cite{AMM10} are sums of two monomials. The authors believe that their examples for $d=4,5, \dots,12$ are the most entangled $d$-qubits. Our software confirms the values of the spectral norms of the examples in \cite{AMM10}. We also consider five one complex parameter families of these examples, and we compute a number of values of the spectral norms in these families. As expected, in all these computed examples the spectral norms are higher than in the examples in \cite{AMM10}.
In Appendix 1 we consider a standard orthonormal basis in $\rS^d\mathbb{C}^n$, the analog of Dicke states in $\rS^d\mathbb{C}^2$ \cite{Dic}, and the entanglement of each element in the basis. We give an upper bound on the entanglement of symmetric states in $\rS^d\mathbb{C}^n$.
In Appendix 2 we discuss the complexity results associated with a system of $m$ polynomial equations in $m$ variables with isolated roots and no roots at infinity.
We recall simple necessary and sufficient conditions on such systems. We define an $x_1$-simple system which has only simple solutions with pairwise distinct $x_1$ coordinates. For $x_1$-simple systems the reduced Gr\"obner basis with respect to the order $x_1 \prec\cdots \prec x_m$ satisfies the shape lemma.
We recall the known complexity results of finding the reduced Gr\"obner basis
in this case \cite{BFS15}. We also recall the complexity results for finding the roots of
a polynomial in one complex variable \cite{NR96}. Lemma 5 summarize the complexity results of finding all the roots of $x_1$-simple system with a given precision.
In Appendix 3 we discuss briefly the Majorana representation. This representation is used in physics literature \cite{AMM10,MGBB10}. We explain how Majorana representation suggests two kinds of the most entangled $d$-symmetric qubits, which solve either T\'oth's or Thomson's problems \cite{Why52,Tho04}. Most of the example in \cite{AMM10} are based on these two problems.
However, in certain cases as shown in \cite{AMM10} the most entagled symmetric states do not solve neither of the above problems.
\section{Spectral norm and entanglement}\label{sec: spwcnrm}
For a positive integer $d$, i.e., $d\in\mathbb{N}$, we denote by $[d]$ the set of consecutive integers $\{1,\ldots,d\}$.
Let $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$, $\mathbf{n}=(n_1,\ldots,n_d)\in\mathbb{N}^d$. We will identify the tensor product space
$\otimes_{i=1}^d \mathbb{F}^{n_i}$ with the space of $d$-arrays $\mathbb{F}^{\mathbf{n}}$.
The entries of $\cT\in\mathbb{F}^{\mathbf{n}}$
are denoted as $\cT_{i_1,\ldots,i_d}$. We also will use the notation $\cT=[\cT_{i_1,\ldots,i_d}]$.
So $\cT$ is called a vector for $d=1$, a matrix for $d=2$ and a tensor for $d\ge 3$. Note that the dimension of $\mathbb{F}^{\mathbf{n}}$ is $N(\mathbf{n})=n_1\cdots n_d$.
Assume that $d\ge 2$ is an integer and $k\in \{0,1\}$. For $\mathbf{n}=(n_1,\ldots,n_d)\in\mathbb{N}^d$ let $\mathbf{m}=(n_{k+1},\ldots,n_d)\in \mathbb{N}^{d-k}$. Assume that $\cT\in \mathbb{F}^\mathbf{n}$ and $\cS\in\mathbb{F}^{\mathbf{m}}$. Then $\cT\times \cS$ is the scalar $\sum_{i_1=\cdots=i_d}^{n_1,\ldots,n_d}\cT_{i_1,\ldots,i_d}\cS_{i_1,\ldots,i_{d}}$ for $k=0$ and a vector in $\mathbb{F}^{n_1}$, whose $i$-th coordinate is given by
$(\cT\times \cS)_{i}=\sum_{i_{2}=\cdots=i_d=1}^{n_2,\ldots,n_d} \cT_{i,i_2,\ldots,i_d}\cS_{i_{2},\ldots,i_{d}}$, for $k=1$.
The inner product on $\mathbb{F}^{\mathbf{n}}$ is given as $\langle\cS,\cT\rangle:=\cS\times \overline{\cT}$, where $\overline{\cT}=[\overline{\cT_{i_1,\ldots,i_d}}]$.
Furthermore, $\|\cS\|=\sqrt{\langle\cS,\cS\rangle}$ is the Hilbert-Schmidt norm of $\cS$. Assume that $\mathbf{x}_i=(x_{1,i},\ldots,x_{n_i,i})^\top\in\mathbb{F}^{n_i}$ for $i\in[d]$.
Then $\otimes_{i=1}^d \mathbf{x}_i$ is a tensor in $\mathbb{F}^{\mathbf{n}}$, with the entries $(\otimes_{i=1}^d \mathbf{x}_i)_{i_1,\ldots,i_d}=x_{i_1,1}\cdots x_{i_d,d}$.
($\otimes_{i=1}^d \mathbf{x}_i$ is called a rank one tensor if all $\mathbf{x}_i\ne \mathbf{0}$.) Assume that $\mathbf{x}_1=\cdots=\mathbf{x}_d=\mathbf{x}$. Then $\otimes^d\mathbf{x}=\otimes_{i=1}^d \mathbf{x}_i$.
Denote the unit sphere in $\mathbb{F}^n$ by $\rS(n,\mathbb{F})=\{\mathbf{x}\in\mathbb{F}^n, \|\mathbf{x}\|=1\}$. Recall that the spectral norm of $\cT\in \mathbb{F}^{\mathbf{n}}$ is given as
\begin{eqnarray*}\label{specnrmdef}
\|\cT\|_{\sigma,\mathbb{F}}=\max\{|\cT\times \otimes_{i=1}^d \mathbf{x}_i|, \; \mathbf{x}_i\in \rS(n_i,\mathbb{F}) \textrm{ for } i\in[d]\}.
\end{eqnarray*}
Assume that $d=2$. Then $\cT$ is a matrix $T\in\mathbb{F}^{n_1\times n_2}$. In that case
$\|T\|_{\sigma,\mathbb{F}}$ is the spectral norm of $T$, and is equal to its maximum singular value $\sigma_1(T)$. In particular, for $T\in \mathbb{R}^{n_1\times n_2}$ one has equality $\|T\|_{\sigma,\mathbb{R}}=\|T\|_{\sigma,\mathbb{F}}$. Furthermore, as $\sigma_1(T)^2$ is the maximum eigenvalue of hermitian matrix $T T^*$ or $T^*T$. It is well known that $\sigma_1(T)$ can be computed in polynomial time in the entries of $T$ and $\max(n_1,n_2)$. See for example \cite{GV13} for a general rectangular matrix, or \cite{KW92} for direct of Lancos algorithm for $T T^*$. Another method for $T$, with Gaussian integer entries, is as follows: First compute the characteristic polynomial $p(z)$ of the hermitian matrix $H(T)=\left[\begin{array}{cc}0 &T\\T^*&0\end{array}\right]$. The complexity of such an algorithm is described in \cite{DPW05}. Next recall that the eigenvalues of $H(T)$ are $\pm \sigma_i(T)$ and $0$ \cite[Theorem 4.11.1]{Frb16}. Now use well known algorithms as \cite{NR96} to approximate the roots $p(z)$.
In the rest of this paper we assume that $d\ge 3$, i.e., $\cT$ is a tensor, unless stated otherwise.
Unlike in the matrix case, for a real tensor $\cT\in \mathbb{R}^{\mathbf{n}}$ it is possible that $\|\cT\|_{\sigma,\mathbb{R}}<\|\cT\|_{\sigma,\mathbb{C}}$ \cite{FL18}.
For simplicity of notation we will let $\|\cT\|_{\sigma}$ denote $\|\cT\|_{\sigma,\mathbb{C}}$, and no ambiguity will arise.
A standard way to compute $\|\cT\|_{\sigma,\mathbb{F}}$ is an alternating maximization with respect to one variable, while other variables are fixed, see \cite{LMV00}. Other variants of this method is maximization on two variables using the SVD algorithms \cite{FMPS13}, or the
Newton method \cite{FT15,Tong}. These methods in the best case yield a convergence to a local maximum, which provide a lower bound
to $\|\cT\|_{\sigma,\mathbb{F}}$. Semidefinite relaxation methods, as in \cite{Nie14}, will yield an upper bound to $\|\cT\|_{\sigma,\mathbb{F}}$, which will converge in some cases to $\|\cT\|_{\sigma,\mathbb{F}}$.
Recall that in quantum physics $\cT\in\mathbb{C}^{\mathbf{n}}$ is called a state if $\|\cT\|=1$. (Furthermore, all tensors of the form $\zeta\cT$, where $\|\cT\|=1$ and
$\zeta\in\mathbb{C},|\zeta|=1$ are identified as the same state.
That is, the space of the states in $\mathbb{C}^{\mathbf{n}}$ is the quotient space $\rS(N(\mathbf{n}),\mathbb{C})/\rS(1,\mathbb{C})$. For simplicity of our exposition will ignore this identification.)
Denote by $\Pi^{\mathbf{n}}$ the product states in $\mathbb{C}^{\mathbf{n}}$:
\[\Pi^{\mathbf{n}}=\{\otimes_{i=1}^d \mathbf{x}_i,\; \mathbf{x}_i\in\rS(n_i,\mathbb{C}),i\in[d]\}.\]
The geometric measure of entanglement of a state $\cT\in\mathbb{C}^{\mathbf{n}}$ is
\begin{eqnarray}\label{defgment}
\mathrm{dist}(\cT,\Pi^{\mathbf{n}})=\min_{\cY\in\Pi^{\mathbf{n}}} \|\cT-\cY\|.
\end{eqnarray}
As $\|\cT\|=\|\cY\|=1$ it follows that $\mathrm{dist}(\cT,\Pi^{\mathbf{n}})=\sqrt{2(1-\|\cT\|_{\sigma})}$. Hence an equivalent measurement of entanglement is \cite{GFE09}
\begin{equation}\label{defetaT}
\eta(\cT)=-\log_2 \|\cT\|_\sigma^2.
\end{equation}
The maximal entanglement is
\begin{equation}\label{maxentn}
\eta(\mathbf{n})=\max_{\cT\in\mathbb{C}^{\mathbf{n}}, \|\cT\|=1} -\log_2\|\cT\|_\sigma^2.
\end{equation}
See \cite{DFLW17} for other measurements of entanglement using the nuclear norm of $\cT$. Lemma 9.1 in \cite{FL18} implies
\[\eta(\mathbf{n})\le \log_2 N(\mathbf{n}).\]
Let $n^{\times d}=(n,\ldots,n)\in\mathbb{N}^d$.
For $n=2$ we get that $\eta(2^{\times d})\le d$. In \cite{Jungetall08} it is shown that $\eta(2^{\times d})\le d-1$. A complementary result is given in \cite{GFE09}: For the set of states of Haar measure at least $1-e^{-d^2}$ on the sphere $\|\cT\|=1$ in $\otimes^d\mathbb{C}^{2}$ the inequality $\eta(\cT)\ge d - 2\log_2 d-2$ holds. A generalization of this result to $\otimes^d\mathbb{C}^{n}$ is given in \cite{DM18}.
\section{Symmetric tensors}\label{sec:symten}
A tensor $\cS=[\cS_{i_1,\ldots,i_d}]\in\otimes^d\mathbb{F}^n$ is called symmetric if $\cS_{i_1,\ldots,i_d}=
\cS_{i_{\omega(1)},\ldots,i_{\omega(d)}}$ for every permutation $\omega:[d]\to[d]$.
Denote by $\rS^d\mathbb{F}^n\subset \otimes^d\mathbb{F}^n$ the vector space of $d$-mode symmetric tensors on $\mathbb{F}^n$.
In what follows we assume that $\cS$ is a symmetric tensor and $d\ge 2$, unless stated otherwise. A tensor
$\cS\in\rS^d\mathbb{F}^n$ defines a unique homogeneous polynomial of degree $d$ in $n$ variables $f(\mathbf{x})=\cS\times\otimes^d\mathbf{x}$, where
\begin{eqnarray*}\label{defpolfx}
f(\mathbf{x})=\sum_{ j_k+1\in [d+1],k\in[n], j_1+\cdots +j_n=d} \frac{d!}{j_1!\cdots j_n!} f_{j_1,\ldots,j_n} x_1^{j_1}\cdots x_n^{j_n}.
\end{eqnarray*}
Conversely, a homogeneous polynomial $f(\mathbf{x})$ of degree $d$ in $n$ variables defines a unique symmetric $\cS\in\rS^d\mathbb{F}^n$ as given in part (4) of Lemma \ref{isolem}.
Hence it is advantageous to replace $\rS^d\mathbb{F}^n$ by the isomorphic space of all homogeneous polynomials of degree $d$ in $n$ variables over $\mathbb{F}$, denoted as $\rP(d,n,\mathbb{F})$. We now introduce the standard multinomial notation as in \cite{Rez92}.
Let $\mathbb{Z}_+$ be the set of all nonnegative integers. Denote by $J(d,n)$ be the set of all $\bj=(j_1,\ldots,j_n)\in\mathbb{Z}_+^n$ appearing in the above definition of $f(\mathbf{x})$:
\begin{eqnarray*}\label{defJdn}
J(d,n)= \{\bj=(j_1,\ldots,j_n)\in\mathbb{Z}_+^n,\; j_1+\cdots+j_n=d\}.
\end{eqnarray*}
It is well known that $|J(d,n)|={n+d-1\choose n}={n+d-1\choose d-1}$, see for example \cite{Rez92}.
Define
\begin{eqnarray*}\label{defcbj}
c(\bj)=\frac{d!}{j_1!\cdots j_n!}
\end{eqnarray*}
For $\mathbf{x}=(x_1,\ldots,x_n)^\top\in\mathbb{F}^n$ and $\bj=(j_1,\ldots,j_n)\in J(d,n)$ let $\mathbf{x}^{\bj}$ be the monomial $x_1^{j_1}\cdots x_n^{j_n}$. Then the above definition of $f(\mathbf{x})$ is equivalent to
\begin{equation}\label{defpolx1}
f(\mathbf{x})=\sum_{\bj\in J(d,n)} c(\bj)f_{\bj}\mathbf{x}^{\bj}.
\end{equation}
Let $\bU$ and $\bV$ be finite dimensional vector spaces over $\mathbb{F}$. Then $\bU$ and $\bV$ are isomorphic if and only if $\bU$ and $\bV$ have the same dimension.
Assume that $\bU$ and $\bV$ are two inner product vector spaces over $\mathbb{F}$ of the same dimension $N$. Then $L:\bU\to \bV$ is called an isometry if $L$ preserves the inner product. Assume that $\bu_1,\ldots,\bu_N$ in $\bU$ is an orthonormal basis in $\bU$. Then a linear transformation $L:\bU\to \bV$ is an isometry if and only if $\bv_1=L(\bu_1),\ldots,\bv_N=L(\bu_N)$ is an orthonormal basis in $\bV$.
The following lemma summarizes the properties of the isomorphisms of $\rS^d\mathbb{F}^n$ to $\rP(d,n,\mathbb{F})$ and to the auxiliary vector space $\mathbb{F}^{J(d,n)}$, and recalls Banach's characterization of the spectral norm of $\cS\in\rS^d\mathbb{F}^n$ as the maximum of the absolute value of the polynomial $\cS\times(\otimes^d\mathbf{x})$ on the unit sphere $\rS(n,\mathbb{F})$ \cite{Ban38}:
\begin{lemma}\label{isolem}
Let $\mathbb{F}^{J(d,n)}$ be the space of all vectors $\mathbf{f}=(f_{\bj}), \bj\in J(d,n)$. Assume that the inner product and the Hilbert norm on $\mathbb{F}^{J(d,n)}$ are given by
\begin{eqnarray*}\label{inprodFJ}
\langle \mathbf{f},\bg\rangle=\sum_{\bj\in J(d,n)} c(\bj)f_{\bj}\overline{g_{\bj}},\quad \|\mathbf{f}\|=\sqrt{\langle \mathbf{f},\mathbf{f}\rangle},\quad \mathbf{f}=(f_{\bj}),\bg=(g_{\bj})\in \mathbb{F}^{J(d,n)}.
\end{eqnarray*}
Then
\begin{enumerate}
\item Let $\mathbf{e}_{\bj}=(\delta_{\bj,\bk})_{\bk\in J(d,n)}, \bj\in J(d,n)$, where $\delta_{\bj,\bk}$ is Kronecker's delta function, be the standard basis in $\mathbb{F}^{J(d,n)}$. Then $\frac{1}{\sqrt{c(\bj})}\mathbf{e}_{\bj}, \bj\in J(d,n)$ is an orthonormal basis in $\mathbb{F}^{J(d,n)}$.
\item $\mathbb{F}^{J(d,n)}$ is isomorphic to $\mathbb{F}^{n+d-1\choose d-1}$. There is an isometry $L:\mathbb{F}^{J(d,n)}\to \mathbb{F}^{n+d-1\choose d-1}$ which maps the orthonormal basis $\frac{1}{\sqrt{c(\bj})}\mathbf{e}_{\bj}, \bj\in J(d,n)$ to the standard orthonormal basis in $\mathbb{F}^{n+d-1\choose d-1}$.
\item $\mathbb{F}^{J(d,n)}$ is isomorphic to $\rP(n,d,\mathbb{F})$, where $\mathbf{f}$ corresponds to $f(\mathbf{x})$ given by \eqref{defpolx1}.
\item The map $L:\rS^d\mathbb{F}^n\to\rP(d,n,\mathbb{F})$ which is given by $L(\cS)=f$, where $f(\mathbf{x})=S\times(\otimes ^d\mathbf{x})$, is an isomorphism and an isometry.
\item
Assume that $\cS\in\rS^d\mathbb{F}^n$ and let $\bF(\mathbf{x})=\cS\times(\otimes^{d-1}\mathbf{x})$. Then
\begin{eqnarray}\label{Fformulas}
&&\bF(\mathbf{x})=(F_1(x),\ldots,F_n(\mathbf{x}))=\frac{1}{d}\nabla f(\mathbf{x})=\frac{1}{d}(\frac{\partial f}{\partial x_1}(\mathbf{x}),\ldots,\frac{\partial f}{\partial x_n}(\mathbf{x})),\\
\label{Eulerform}
&&\sum_{i=1}^n x_iF_i(\mathbf{x})=f(\mathbf{x}).
\end{eqnarray}
\item The spectral norm of a symmetric tensor is given by Banach's characterization \cite{Ban38}:
\begin{equation}\label{Banchar}
\|\cS\|_{\sigma,\mathbb{F}}=\max\{\frac{|\cS\times(\otimes^d\mathbf{x})|}{\|\mathbf{x}\|^d}, \;\mathbf{x}\in\mathbb{F}^n\setminus\{\mathbf{0}\}\}=\max\{|\cS\times(\otimes^d\mathbf{x})|, \;\mathbf{x}\in\rS(n,\mathbb{F})\}, \quad \cS\in\rS^d\mathbb{F}^n.
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof} Parts (1)-(3) are straightforward.
\noindent Part (4). Let $\cS\in\rS^d\mathbb{F}^n$. Define $L(\cS)=f$, where $f(\mathbf{x})=\cS\times (\otimes^d \mathbf{x})$. Clearly $f\in\rP(d,n,\mathbb{F})$. Assume that $f(\bx)$ is given by \eqref{defpolx1}. For a given $i_1,\ldots,i_d\in[n]$ and $k\in[n]$ let $j_k$ be the number of times $k$ appears in the multiset $\{i_1,\ldots,i_d\}$. Set $\bj=(j_1,\ldots,j_n)$. Then $L(\cS)_{\bj}=f_{\bj}$.
It is straightforward to show that $L$ is an isomorphism. Furthermore, $\langle \cS,\cT\rangle=\langle L(\cS),L(\cT)\rangle$. Hence $L$ is an isometry.
\noindent
Part (5).
Observe that $\frac{\partial\;\;\;\;}{\partial x_m} (\cS_{i_1,\ldots,i_d}\mathbf{x}^{\bj})$ is not zero if and only if $i_l=m$ for some $l\in[d]$. So assume that $i_l=m$. Since $\cS$ is symmetric we can choose $l\in[d]$. Hence $d F_m=\frac{\partial f\;\;\;}{\partial x_m}$, and \eqref{Fformulas} holds. Equality \eqref{Eulerform} is Euler's formula for $f\in\rP(d,n,\mathbb{F})$ and $\bF=\frac{1}{d}\nabla f$.
Part (6) is Banach's theorem \cite{Ban38}, see \cite{FL18} for details.
\end{proof}
Banach's theorem \eqref{Banchar} was rediscovered several times since 1938. In quantum physics literature it appeared in \cite{Hubetall09} for the case $\mathbb{F}=\mathbb{C}$. In mathematical literature, for the case $\mathbb{F}=\mathbb{R}$, it appeared in \cite{CHLZ12,Fri13}. (Observe that a natural generalization of Banach's theorem to partially symmetric tensors is given in \cite{Fri13}.)
In view of Lemma \ref{isolem} it makes sense to introduce the spectral norm on $\mathbb{F}^{J(d,n)}$ and $\rP(d,n)$:
\begin{equation}\label{polspecnorm}
\|\ff\|_{\sigma,\mathbb{F}}=\|f\|_{\sigma,\mathbb{F}}=\max\{\frac{|f(\mathbf{x})|}{\|\mathbf{x}\|^d},\mathbf{x}\in\mathbb{F}\setminus\{\mathbf{0}\}\}=\max\{|f(\mathbf{x})|,\mathbf{x}\in\rS(n,\mathbb{F})\},
\end{equation}
where $f(\mathbf{x})$ is given by \eqref{defpolx1}.
We denote by $\cS(\bj)\in\rS^d\mathbb{C}^n$ the symmetric state corresponding to $\frac{1}{\sqrt{c(\bj})}\mathbf{e}_{\bj}, \bj\in J(d,n)$. Note that $\cS(\bj)$ corresponds to the monomial $\sqrt{c(\bj)} \mathbf{x}^{\bj}$, i.e., $\cS(\bj)\times\otimes^d\mathbf{x}=\sqrt{c(\bj)} \mathbf{x}^{\bj}$. We also let $\|f\|=\|\ff\|$.
\section{Critical points of $\Re (\cS\times\otimes^d \mathbf{x})$ on $\rS(n,\mathbb{F})$ }\label{sec:critpts}
Recall that the projective space $\mathbb{P}\mathbb{C}^N$ is the space of lines through the origin in $\mathbb{C}^N$. Thus a point in $\mathbb{P}\mathbb{C}^N$ is the equivalence class $[\by]=\{t\by, t\in\mathbb{C}\setminus\{0\}, \by\in\mathbb{C}^N\setminus\{\mathbf{0}\}\}$. Note that for each $[\by]\in\mathbb{P}\mathbb{C}^N$, the hyperplane $\{\bx\in\mathbb{C}^N,\;\langle \bx,\by\rangle=0\}$ is independent of the representative corresponding to $[\by]$.
Recall that $\cS\in\rS^d\mathbb{C}^n$ is called nonsingular \cite{FO14} if
\[\cS\times\otimes^{d-1}\mathbf{x}=\mathbf{0}\Rightarrow \mathbf{x}=\mathbf{0}.\]
Otherwise $\cS$ is called singular.
A nonzero homogeneous polynomial $f(\mathbf{x})$ defines a hypersurface $H(f):=\{\mathbf{x}\in\mathbb{C}^{n}\setminus\{\mathbf{0}\},f(\mathbf{x})=0\}$ in $\mathbb{P}\mathbb{C}^n$.
$H(f)$ is called a smooth hypersurface if $\nabla f(\mathbf{x})\ne \mathbf{0}$ for each $\mathbf{x}\ne \mathbf{0}$ that satisfies $f(\mathbf{x})=0$.
\begin{proposition}\label{relatfxS} Assume that $\cS\in\rS^d\mathbb{C}^n$.
Let $f(\mathbf{x})=\cS\times\otimes^d\mathbf{x}$. Then
$\cS$ is nonsingular if and only if $H(f)$ is a smooth hypersurface in $\mathbb{P}\mathbb{C}^n$.
\end{proposition}
\begin{proof} Let $\bF=\frac{1}{d}\nabla f$. Assume that $\bF(\mathbf{x})=\mathbf{0}$ for some $\mathbf{x}\ne \mathbf{0}$. Euler's identity yields that $f(\mathbf{x})=0$. Use part (5) of Lemma \ref{isolem} to deduce the proposition.
\end{proof}
The following result is well known \cite{GKZ}:
\begin{proposition}\label{hypdetvar} Denote by $\mathbb{P}\mathbb{C}^{J(d,n)}$ the complex projective space corresponding to the affine space $\mathbb{C}^{J(d,n)}$. With each $[\ff]\in \mathbb{P}\mathbb{C}^{J(d,n)}$ associate the hypersurface $f(\mathbf{x})=0$ in $\mathbb{P}\mathbb{C}^n$, where $f(\mathbf{x})$ is given by \eqref{defpolx1}. Then the set of singular hypersurfaces is the hyperdeterminant variety $\mathrm{H}(d,n)\subset \mathbb{P}\mathbb{C}^{J(d,n)}$, which is the zero set of the hyperdeterminant polynomial on $\mathbb{C}^{J(d,n)}$.
\end{proposition}
\begin{corollary}\label{hypdetvar} The set of singular symmetric tensors in $\rS^d\mathbb{C}^n$ is the zero set $V(d,n)\subset \rS^d\mathbb{C}^n$ of the polynomial on $\rS^d\mathbb{C}^n$ which is induced by the hyperdeterminant polynomial on $\mathbb{C}^{J(d,n)}$.
\end{corollary}
For a complex number $z=x+\bi y\in\mathbb{C}, x,y\in\mathbb{R}$ we denote $x=\Re z$ and $y=\Im z$.
Fix $\mathbf{x}\in\mathbb{C}^n$ and let $\zeta\in \mathbb{C}$. Then $\cS\times\otimes^d\left(\zeta\mathbf{x}\right)=\zeta^d\left(\cS\times\otimes^d\mathbf{x}\right)$. Hence there exists $\zeta\in\mathbb{C}, |\zeta|=1$ such that
$|\cS\times\otimes^d\mathbf{x}|=\Re\left(\cS\times\otimes^d\left(\zeta\mathbf{x}\right)\right)$.
Therefore for $\mathbb{F}=\mathbb{C}$ we can replace the characterization \eqref{Banchar} with:
\begin{eqnarray*}\label{maxcharspecnrmc}
\|\cS\|_{\sigma}=\max_{\mathbf{x}\in\rS(n,\mathbb{C})} \Re(\cS\times\otimes^d\mathbf{x}), \textrm{ for } \cS\in\rS^d\mathbb{C}^n.
\end{eqnarray*}
Let $f\in \rP(d,n,\mathbb{F})$. Consider $\Re f|\rS(n,\mathbb{F})$, the restriction of $\Re f$ to the sphere $\rS(n,\mathbb{F})$. Suppose first that
$\mathbb{F}=\mathbb{R}$. Then $\Re f|\rS(n,\mathbb{R})=f|\rS(n,\mathbb{R})$. A point $\mathbf{x}\in\rS(n,\mathbb{R})$ is called a critical point of $f|\rS(n,\mathbb{R})$ if the directional derivative of $f$ at $\mathbf{x}$ in direction of each $\mathbf{y}$, where $\langle \mathbf{y},\mathbf{x}\rangle =0$, is $0$. Assume that $\mathbf{x}$ is a critical point of $f|\rS(n,\mathbb{R})$. Then $f(\mathbf{x})$ is called a critical value of $f|\rS(n.\mathbb{R})$.
Assume now that $\mathbb{F}=\mathbb{C}$. View $\mathbb{C}^n$ as $\mathbb{R}^{2n}$ by writing $\mathbf{z}\in\mathbb{C}^n$ as $\mathbf{z}=\mathbf{x}+\bi\mathbf{y}$, where $\mathbf{x},\mathbf{y}\in\mathbb{R}^n$. Clearly $\Re f(\mathbf{z}), \mathbf{z}\in\mathbb{C}^n$ can be viewed as a homogeneous polynomial $g(\mathbf{x},\mathbf{y})$ of degree $d$ in $2n$ variables $(\mathbf{x},\mathbf{y})$.
We then identify $\rS(n,\mathbb{C})$ with $\rS(2n,\mathbb{R})$. Then $\Re f|\rS(n,\mathbb{C})$ is $g|\rS(2n,\mathbb{R})$. Thus a critical point of $\Re f|\rS(n,\mathbb{C})$ is the critical point of $g|\rS(2n,\mathbb{R})$, and the critical value of $\Re f|\rS(n,\mathbb{C})$ is the critical value of $g|\rS(2n,\mathbb{R})$.
Note that the points $\mathbf{x}\in\rS(n,\mathbb{F})$ where $\Re f|\rS(n,\mathbb{F})$ is maximum or minimum
are critical points, and the maximum and minimum values are critical values.
\begin{lemma}\label{critptlem} Assume that $\cS\in\rS^d\mathbb{F}^n, d\ge 2$.
A point $\mathbf{x}\in\rS(n,\mathbb{F})$ is a critical point of $\Re f(\mathbf{x})$ on $\rS(n,\mathbb{F})$
if and only if
\begin{equation}\label{critpt}
\cS\times \otimes^{d-1}\mathbf{x}=\lambda\overline{\mathbf{x}}, \quad \mathbf{x}\in\rS(n,\mathbb{F}), \lambda\in\mathbb{R},
\end{equation}
where $\overline{\mathbf{x}}$ denote the complex conjugate of $\mathbf{x}$. The number of critical values $\lambda$ satisfying (\ref{critpt}) is finite.
\end{lemma}
\begin{proof}
First assume that $\mathbb{F}=\mathbb{R}$. Let $\mathbf{x}\in\rS(n,\mathbb{R})$. First assume that $\mathbf{x}$ is a critical point of $f(\mathbf{z})=\cS\times\otimes^d\mathbf{z}$ for $\mathbf{z}\in\rS(n,\mathbb{R})$. Let $\mathbf{y}\in\mathbb{R}^n$ be orthogonal to $\mathbf{x}$: $\mathbf{y}^\top \mathbf{x}=0$. Then $\|\mathbf{x}+t\mathbf{y}\|=\sqrt{1+t^2\|\mathbf{y}\|^2}=1+O(t^2)$ for $t\in\mathbb{R}$. Clearly
\[\cS\times \otimes ^d (\mathbf{x}+t\mathbf{y})=\cS\times \otimes^d \mathbf{x}+td\mathbf{y}^\top (\cS\times \otimes^{d-1}\mathbf{x}) +O(t^2).\]
As $\mathbf{x}$ is a critical point of $\cS\times\otimes^d\mathbf{z}$ for $\mathbf{z}\in\rS(n,\mathbb{R})$ it follows that $\mathbf{y}^\top (\cS\times \otimes^{d-1}\mathbf{x})=0$ for each $\mathbf{y}$ orthogonal to $\mathbf{x}$.
Hence $\cS\times \otimes^{d-1}\mathbf{x}$ is colinear with $\mathbf{x}$. As $\bar\mathbf{x}=\mathbf{x}$ for each $\mathbf{x}\in\mathbb{R}^n$ we deduce (\ref{critpt}). Similar arguments show that if (\ref{critpt}) holds for
$\mathbf{x}\in\rS(n,\mathbb{R})$ then $\mathbf{x}$ is a critical point.
As $f(\mathbf{x})$ is a polynomial on $\mathbb{R}^n$ it follows that
the set of critical points of $f|\rS(n,\mathbb{R})$ is a real algebraic set. This algebraic set is a finite union of connected algebraic sets \cite[Proposition 1.6]{Cos05}. On each connected algebraic set of critical points $f$ is a constant function, whose value on this set is a critical value. This proves that the number of critical values for $\mathbb{F}=\mathbb{R}$ is finite.
Second assume that $\mathbb{F}=\mathbb{C}$. View $\mathbb{C}^{n}$ as $2n$-dimensional real vector space $\mathbb{R}^{2n}$ with the standard inner product $\Re (\mathbf{y}^*\mathbf{x})$, where $\mathbf{y}^*=\overline{\mathbf{y}}^\top$. Hence $\|\mathbf{x}\|=\sqrt{\Re (\mathbf{x}^*\mathbf{x})}$. Assume that $\mathbf{x}\in\rS(n,\mathbb{C})$
is a critical point of $\Re (\cS\times \otimes^d\mathbf{z})$ on $\rS(n,\mathbb{C})$. Let $\mathbf{y}\in\mathbb{C}^n$ be orthogonal to $\mathbf{x}$: $\Re (\mathbf{y}^*\mathbf{x})=\Re(\overline{ \mathbf{y}}^\top \mathbf{x})=0$. Then $\|\mathbf{x}+t\mathbf{y}\|=\sqrt{1+t^2\|\mathbf{y}\|^2}
=1+O(t^2)$ for $t\in\mathbb{R}$. Hence
\begin{eqnarray}\label{varforRSx}
\Re(\cS\times \otimes ^d (\mathbf{x}+t\mathbf{y}))=\Re(\cS\times \otimes^d \mathbf{x})+td\Re(\mathbf{y}^\top (\cS\times \otimes^{d-1}\mathbf{x})) +O(t^2).
\end{eqnarray}
As $\mathbf{x}$ is a critical point we deduce that
\[0=\Re(\mathbf{y}^\top (\cS\times \otimes^{d-1}\mathbf{x}))=\Re(\mathbf{y}^* (\overline{\cS\times \otimes^{d-1}\mathbf{x}})).\]
Hence $\overline{\cS\times \otimes^{d-1}\mathbf{x}}$ is $\mathbb{R}$-colinear with $\mathbf{x}$. Thus (\ref{critpt}) holds. Vice versa, suppose that \eqref{critpt} holds. As $\lambda\in\mathbb{R}$ and $0=\Re (\mathbf{y}^*\mathbf{x})=\Re (\mathbf{y}^\top \bar\mathbf{x})=0$ the equality \eqref{varforRSx} yields that $\mathbf{x}$ is a critical point.
Since $q(\mathbf{x}):=\Re(\cS\times \otimes^d\mathbf{x})$ is a polynomial on $\mathbb{C}^n\sim\mathbb{R}^{2n}$ it follows from the above arguments for $\mathbb{F}=\mathbb{R}$ that
$q|\rS(n,\mathbb{C})$ has a finite number of critical values.
\end{proof}
Clearly, a maximum point of $|\cS\times \otimes^d\mathbf{x}|$ on $\rS(n,\mathbb{F})$ is a critical point of $\Re\left(\cS\times\otimes^d\mathbf{x}\right)$ on $\rS(n,\mathbb{F})$. Hence:
\begin{corollary}\label{maxeig} Let $d,n\ge 2$ be integers.
\begin{enumerate}
\item Assume that $\cS\in\rS^d\mathbb{R}^n$. Then there exists $\mathbf{x}\in\rS(n,\mathbb{R})$ satisfying (\ref{critpt}) such that $|\lambda|=\|\cS\|_{\sigma,\mathbb{R}}$. Furthermore, $\|\cS\|_{\sigma,\mathbb{R}}$ is the maximum of all $|\lambda|$ satisfying (\ref{critpt}).
\item Assume that $\cS\in\rS^d\mathbb{C}^n$. Then there exists $\mathbf{x}\in\rS(n,\mathbb{C})$ satisfying (\ref{critpt}) such that $\lambda=\|\cS\|_{\sigma}$. Furthermore, $\|\cS\|_{\sigma}$ is the maximum of all $|\lambda|$ satisfying (\ref{critpt}).
\end{enumerate}
\end{corollary}
We call $\mathbf{x}\in\rS(n,\mathbb{F})$ and $\lambda\in\mathbb{F}$ an eigenvector and an eigenvalue of $\cS\in\rS^d\mathbb{F}^n$ if the following conditions hold \cite{CS}:
\begin{eqnarray}\label{defeigvl}
\cS\times\otimes^{d-1}\mathbf{x}=\lambda\mathbf{x}, \quad \mathbf{x}\in\rS(n,\mathbb{F}),\;\lambda\in \mathbb{F}, \quad \cS\in\rS^d\mathbb{F}^n.
\end{eqnarray}
Assume that $\mathbb{F}=\mathbb{R}$. Then the above equality is equivalent to (\ref{critpt}). First assume that $d$ is odd and $\mathbf{x}$ is an eigenvector of $\cS$. Then $-\mathbf{x}$ is an eigenvector of $\cS$ corresponding to $-\lambda$. Hence without loss of generality we can consider only nonnegative eigenvalues of $\cS$. Second assume that $d$ is even and $\mathbf{x}$ is an eigenvector of $\cS$. Then $-\mathbf{x}$ is also eigenvector of $\cS$ corresponding to $\lambda$.
A vector $\mathbf{x}\in\rS(n,\mathbb{C})$ and a scalar $\lambda\in\mathbb{R}$ that satisfy (\ref{critpt}) are called the \emph{anti-eigenvector} and \emph{anti-eigenvalue} of $\cS\in\rS^d\mathbb{C}^n$.
Note that if $\mathbf{x}$ is an anti-eigenvector and $\lambda$ a corresponding anti-eigenvalue then $\zeta\mathbf{x}$ is also anti-eigenvector with a corresponding anti-eigenvalue $\varepsilon\lambda$, where $\varepsilon=\pm 1$ and $\zeta^d=\varepsilon$. Hence, we can always assume that each nonzero anti-eigenvalue is positive, and there are $d$ different choices of $\zeta$ such that $\zeta\mathbf{x}\in\mathrm{span}(\mathbf{x})$ is an anti-eigenvector corresponding to a given positive anti-eigenvalue $\lambda$.
We now state the first main result of this paper, which gives the theoretical foundation for the computational methods of our paper.
\begin{theorem}\label{theofoundthm} Let $\cS\in\rS^d\mathbb{F}^n\setminus\{0\}$ and $d\ge 3$. Associate with $\cS$ the polynomial $f(\mathbf{x})=\cS\times \otimes^d\mathbf{x}\in\rP(d,n,\mathbb{F})\setminus\{0\}$. Let $\bF=\frac{1}{d}\nabla f$. Denote
\begin{eqnarray*}
\mathrm{fix}(\bF)=\{\mathbf{x}\in\mathbb{C}^n, \;\bF(\mathbf{x})=\mathbf{x}\}, \quad
\mathrm{afix}(\bF)=\{\mathbf{x}\in\mathbb{C}^n, \;\bF(\mathbf{x})=\bar \mathbf{x}\},
\end{eqnarray*}
the set of fixed and antifixed points of $\bF$ in $\mathbb{C}^n$ respectively.
Let $\omega_{d-2}=e^{\pi\bi/(d-2)}$, ( $\omega_{d-2}^{d-2}=-1$).
Assume that $f\in\rP(d,n,\mathbb{R})$. Denote
\begin{eqnarray*}
\mathrm{fix}_{\mathbb{R}}(\bF)=
\begin{cases}
\mathrm{fix}(\bF)\cap\mathbb{R}^n \textrm{ if } d \textrm{ is odd},\\
(\mathrm{fix}(\bF)\cup \bar\omega_{d-2}\mathrm{fix}(\bF))\cap \mathbb{R}^n \textrm{ if } d \textrm{ is even}.
\end{cases}
\end{eqnarray*}
\begin{enumerate}
\item Assume that $\mathbb{F}=\mathbb{R}$. Then
\begin{eqnarray}\label{Sinftynrmformr}
\|\cS\|_{\sigma,\mathbb{R}}=\|f\|_{\sigma,\mathbb{R}}=\max\{\frac{|f(\mathbf{x})|}{\|\mathbf{x}\|^d}, \mathbf{x}\in\mathrm{fix}_{\mathbb{R}}(\bF)\setminus\{\mathbf{0}\}\}.
\end{eqnarray}
\item Let $\mathbb{F}=\mathbb{C}$. Denote by $\overline{\bF}:\mathbb{C}^n\to\mathbb{C}^n$ the polynomial mapping given by the equality $\overline{\bF}(\mathbf{x})=\overline{\bF(\bar\mathbf{x})}$. Let $\bH=\overline{\bF}\circ \bF$.
Then the set of fixed points of $\mathrm{fix}(\bH)=\{\mathbf{x}\in\mathbb{C}^n, \bH(\mathbf{x})=\mathbf{x}\}$ contains $\mathrm{afix}(\bF)$. Furthermore, $\mathbf{x}$ is a fixed point of $\bH$ if and only if $(\mathbf{x},\mathbf{y})\in \mathbb{C}^n\times\mathbb{C}^n$ is a solution to the system:
\begin{eqnarray}\label{sysfixpH}
\bF(\mathbf{x})-\mathbf{y}=0, \quad \bar{\bF}(\mathbf{y})-\mathbf{x}=0.
\end{eqnarray}
Moreover
\begin{eqnarray}\label{Sinftynrmformc}
\|\cS\|_{\sigma}=\|f\|_{\sigma}=\max\{\frac{|f(\mathbf{x})|}{\|\mathbf{x}\|^d}, \mathbf{x}\in\mathrm{fix}(\bH)\setminus\{\mathbf{0}\}\}.
\end{eqnarray}
\item Assume that $\cS\in\rS^d\mathbb{C}^n$ is nonsingular. Then $\mathrm{fix}(\bF)$ and $\mathrm{fix}(\bH)$ have cardinalities $(d-1)^{n}$ and $(d-1)^{2n}$ respectively, counted with multiplicities. The origin $\mathbf{x}=\mathbf{0}$ is a fixed point of $\bF$ and $\bH$ of multiplicity one. Let $\mathbf{x}\in \mathrm{fix}(\bF)\setminus\{\mathbf{0}\}$ and $\mathbf{y}\in \mathrm{fix}(\bH)\setminus\{\mathbf{0}\}$.
Then $\phi\mathbf{x}\in \mathrm{fix}(\bF)\setminus\{\mathbf{0}\}$ and $\psi\mathbf{y}\in \mathrm{fix}(\bH)\setminus\{\mathbf{0}\}$ if and only if $\phi^{d-2}=1$ and $\psi^{(d-1)^2-1}=1$.
\end{enumerate}
\end{theorem}
\begin{proof} (1-2) Let
\begin{eqnarray*}
\begin{cases}
\alpha_{\mathbb{R}}=\sup\{\frac{|f(\mathbf{z})|}{\|\mathbf{z}\|^d}, \mathbf{z}\in\mathrm{fix}_{\mathbb{R}}(\bF)\setminus\{\mathbf{0}\}\},\\
\alpha_{\mathbb{C}}=\sup\{\frac{|f(\mathbf{z})|}{\|\mathbf{z}\|^d}, \mathbf{z}\in\mathrm{afix}(\bF)\setminus\{\mathbf{0}\}\}.
\end{cases}
\end{eqnarray*}
The characterization \eqref{polspecnorm} yields that $\alpha_{\mathbb{F}}\le \|f\|_{\sigma,\mathbb{F}}$.
Corollary \ref{maxeig} claims that $\|\cS\|_{\sigma,\mathbb{F}}=|\lambda^\star|$, where $| \lambda^\star|$ is the maximum of all $|\lambda|$ satisfying (\ref{critpt}). As $f\ne 0$ it follows that $\lambda^\star\in\mathbb{R}\setminus\{0\}$. From the discussion before this theorem it follows that we can assume that $\lambda^*>0$ unless $d$ is even and $\mathbb{F}=\mathbb{R}$.
Assume that $\bu\in\rS(n,\mathbb{F})$ satisfies $\bF(\bu)=\lambda^\star \bar\bu$. (I.e. $\bu$ satisfies (\ref{critpt}) with $\lambda=\lambda^\star$). Then there exists a positive $t$ such that $|\lambda^\star| t^{d-1}=t$. Let $\mathbf{x}=t\bu$. Then $\bF(\mathbf{x})=\frac{\lambda^\star}{|\lambda^\star|}\bar\mathbf{x}$. First assume that $\lambda^\star>0$. Then
$\mathbf{x}\in\mathrm{afix}(\bF)$.
Furthermore, if $\mathbb{F}=\mathbb{R}$ then
$\mathbf{x}\in\mathrm{fix}(\bF)\cap\mathbb{R}^n\subseteq \mathrm{fix}_{\mathbb{R}}(\bF)$.
Clearly, $\|\cS\|_{\sigma,\mathbb{F}}=|\lambda^\star|=\frac{|f(\mathbf{x})|}{\|\mathbf{x}\|^d}$. Hence $\|\cS\|_{\sigma,\mathbb{F}}=\alpha_{\mathbb{F}}$ in this case.
Second assume that $\mathbb{F}=\mathbb{R}$, $d$ is even and $\lambda^\star<0$. Then $\bF(\mathbf{x})=-\mathbf{x}$ and $\mathbf{x}\in\mathbb{R}^n$. Define $\mathbf{y}=\omega_{d-2}\mathbf{x}$. Then $\bF(\mathbf{y})=\mathbf{y}$.
Clearly $\bar\omega_{d-2}\mathbf{y}=\mathbf{x}\in\mathbb{R}^n$. Hence $\mathbf{x}\in \mathrm{fix}_{\mathbb{R}}(\bF)$. As
$\|\cS\|_{\sigma,\mathbb{R}}=|\lambda^\star|=\frac{|f(\mathbf{x})|}{\|\mathbf{x}\|^d}$ we deduce that $\alpha_{\mathbb{R}}=\|\cS\|_{\sigma,\mathbb{R}}$. This show the characterization \eqref{Sinftynrmformr}.
Assume that $\bF(\mathbf{x})=\bar \mathbf{x}$. Then
$$\bH(\mathbf{x})=\overline{\bF}(\bF(\mathbf{x}))=\overline{\bF}(\bar \mathbf{x})=\overline{\bF(\mathbf{x})}=\mathbf{x}.$$
Hence $\mathrm{afix}(\bF)\subseteq \mathrm{fix}(\bH)$. Similarly, $\mathbf{x}\in \mathrm{fix}(\bH)$ if and only if \eqref{sysfixpH} holds. (Set $\mathbf{y}=\bF(\mathbf{x})$.)
Let $\beta$ be the right hand side of \eqref{Sinftynrmformc}. By definition $\|f\|_{\sigma}\ge \beta$. As $\mathrm{afix}(\bF)\subseteq \mathrm{fix}(\bH)$ it follows that $\alpha_{\mathbb{C}}\le \beta$. Hence $\|f\|_{\sigma}=\beta$ and \eqref{Sinftynrmformc} holds.
\noindent
(3) Recall that $\mathrm{fix}(\bF)$ is the set of solutions of the system $\bF(\mathbf{x})-\mathbf{x}=\mathbf{0}$.
As $d\ge 3$ the highest homogeneous part of this system is $\bF(\mathbf{x})=\cS\times \otimes^{d-1}\mathbf{x}$. As $\cS$ is nonsingular we deduce that $\bF(\mathbf{x})=\mathbf{0}$ has the only solution $\mathbf{x}=\mathbf{0}$. That is, the system $\bF(\mathbf{x})-\mathbf{x}=\mathbf{0}$ does not have solutions at infinity. Therefore the Bezout theorem yields that the number of solutions counting with multiplicities is $(d-1)^n=\prod_{i=1}^n \deg \bF_i$. (See \cite{Fri77} for a proof using degree theory.)
Observe next that $\mathbf{0}\in \mathrm{fix}(\bF)$. As the Jacobian $D(\bF(\mathbf{x})-\mathbf{x})$ is $-I$ at $\mathbf{x}=\mathbf{0}$ it follows that $\mathbf{0}$ is a fixed point of multiplicity one. Assume that $\mathbf{x}\in\mathrm{fix}(\bF)\setminus\{\mathbf{0}\}$. Then $\bF(\phi\mathbf{x})=\phi^{d-1}\mathbf{x}=\phi^{d-2}(\phi \mathbf{x})$. Hence $\phi\mathbf{x}$ is a nonzero fixed point of $\bF$ if and only if $\phi^{d-2}=1$.
To show similar results for $\bH$ we first have to show that $\bH(\mathbf{x})=\mathbf{0}$ has the only solution $\mathbf{x}=\mathbf{0}$. As $\cS$ is nonsingular it follows that $\bar\cS$ is nonsingular.
Thus
$$\bH(\mathbf{x})=\bar{\bF}(\bF(\mathbf{x}))=\mathbf{0}\Rightarrow\bF(\mathbf{x})=\mathbf{0}\Rightarrow \mathbf{x}=\mathbf{0}.$$
As $\deg \bH_i=(d-1)^2$ for $i\in[n]$ we deduce the similar results for $\bH$.
\end{proof}
Thus our approach to compute the spectral norm of $\cS$ is to compute the fixed points of $\bF$ and $\bH$ using the available software as Bertini \cite{BHSW06} for polynomial system of equations, and then use \eqref{Sinftynrmformr} or \eqref{Sinftynrmformc}.
Note that to compute the fixed points of $\bH$ we can use also the system \eqref{sysfixpH}.
In \cite{HS14} the authors consider the dynamics of a special anti-holomorphic map of $\mathbb{C}$ of the form $z\mapsto \bar z^d +c$. They also note that the dynamics of the ``squared" map is given by the holomorphic map $z\mapsto (z^d +\bar c)^d +c$.
Thus the dynamics of the maps $\mathbf{x}\mapsto \overline{\bF(\mathbf{x})}$ and its square - $\bH$ are generalizations of the dynamics studied in \cite{HS14}.
Suppose that $\mathbb{F}=\mathbb{C}$. Assume that $\mathbf{x}\in\rS(n,\mathbb{C})$ and $\lambda\in\mathbb{C}$ are an eigenvector and the corresponding eigenvalue of $\cS\in\rS^d\mathbb{C}^n$, i.e., \eqref{defeigvl} holds. Let $\zeta\in\mathbb{C},|\zeta|=1$.
Then $\zeta\mathbf{x}$ is an eigenvector of $\cS$ with the corresponding eigenvalue $\zeta^{d-2}\lambda$. Assume that $\lambda\ne 0$. For $d>2$ we can choose $\zeta$ of modulus $1$ such that $\zeta^{d-2}\lambda=|\lambda|>0$.
Furthermore, the number of such choices of $\zeta$ is $d-2$. In this context it is natural to consider the eigenspace $\mathrm{span}(\mathbf{x})$, to which correspond a unique eigenvalue
$\lambda\ge 0$.
It is shown in \cite{CS} that the number of different
eigenspaces of generic $\cS\in\rS^d\mathbb{C}^n$ is
\begin{eqnarray*}\label{CSnumbereigv}
c(2,n) =n, \quad c(d,n)=\frac{(d-1)^n -1}{d-2} \textrm{ for } d\ge 3.
\end{eqnarray*}
For $d\ge 3$ and a nonsingular $\cS$ this result follows from part (3) of Theorem \ref{theofoundthm}, where we count the number of nonzero fixed points of $\bF$.
That is, each $\cS\in\rS^d\mathbb{C}^n\setminus V(d,n)$ has the above number of eigenspaces $\mathrm{span}(\mathbf{x}), \mathbf{x}\in\rS(n,\mathbb{C})$. The obvious question is: what is the maximal number
of eigenspaces $\mathrm{span}(\mathbf{x})$ corresponding to $\mathbf{x}\in\rS(n,\mathbb{R})$ for $\cS\in\rS^d\mathbb{R}^n\setminus V(d,n)$. Since $\cS\times \otimes^d\mathbf{x}$ has at least two critical points on
$\rS(n,\mathbb{R})$ for $\cS\ne 0$, corresponding to the maximum and minimum values, it follows that $\cS\ne 0$ has at least one real eigenspace.
In \cite{ABC13} the authors study the average number of critical points of a random homogeneous function $f(\mathbf{x})$ of degree $d$, where its coefficients are independent Gaussian random variables.
Assume that $d=2$. Then $\rP(2,n,\mathbb{F})$ is the space of quadratic forms in $n$ variables on $\mathbb{F}$, which correspond to the space of symmetric matrices $\rS^2\mathbb{F}^n$.
That is $f(\mathbf{x})=\mathbf{x}^\top S\mathbf{x}$, where $S\in \mathbb{F}^{n\times n}$ is symmetric.
For $\mathbb{F}=\mathbb{R}$ the critical points of $f(\mathbf{x})$ correspond to the eigenvalues of $S$. For $\mathbb{F}=\mathbb{C}$ recall Schur's theorem:
There exists a unitary matrix $U\in \mathbb{C}^{n\times n}$ such that
$U^\top SU=\operatorname{diag}(a_1,\ldots,a_n), a_1\ge \cdots\ge a_n\ge 0 $, where $\operatorname{diag}(a_1,\ldots,a_n)\in\rS^2\mathbb{C}^n$ is a diagonal matrix with the diagonal entries $a_1,\ldots,a_n$. As $\bar U$ is unitary it follows that $a_i=\sigma_i(S), i\in [n]$ are the singular values of $S$.
Let $U=[\bu_1,\cdots,\bu_n] $. Then $SU=\bar U \operatorname{diag}(a_1,\ldots,a_n)$ which is equivalent to $S\bu_i=a_i \bar\bu_i, i\in[n]$, which is a special case of (\ref{critpt}).
We now give an estimate of the number of different positive anti-eigenvalues for a nonsingular $\cS\in\rS^d\mathbb{C}^n$.
\begin{theorem}\label{estnumbanteig} Assume that $\cS\in\rS^d\mathbb{C}^n$ is nonsingular. Then the number of positive anti-eigenvalues with corresponding anti-eigenspaces
is finite. This number $\mu(\cS)$, counting with multiplicities, satisfies the inequalities
\begin{eqnarray*}\label{estnumbanteig1}
\frac{(d-1)^n-1}{d}\le \mu(\cS)\le \frac{(d-1)^{2n} -1}{(d-1)^2-1}=\sum_{k=0}^{n-1} (d-1)^{2k}.
\end{eqnarray*}
\end{theorem}
\begin{proof} Assume that $\cS\in\rS^d\mathbb{C}^n$ is nonsingular. First suppose that $d=2$. Schur's theorem implies that the number of different positive anti-eigenvalues of a complex symmetric
matrix, which are the singular values of $\cS$, is at most $n$. Hence our theorem holds.
Second suppose that $d>2$.
Assume that $\mathbf{x}\in\rS(n,\mathbb{C})$ is an anti-eigenvector with corresponding anti-eigenvalue $\lambda> 0$. As in the proof of Theorem \ref{theofoundthm} we can assume that $\mathbf{y}\in\mathrm{afix}(\bF)\setminus\{\mathbf{0}\}$. Recall that $\mathrm{afix}(\bF)\setminus\{\mathbf{0}\}\subset\mathrm{fix}(\bH)$. Theorem \ref{theofoundthm} yields that
$\mathrm{fix}(\bH)\setminus\{\mathbf{0}\}$ has cardinality $(d-1)^{2n}-1$. The subspace spanned by $\mathbf{y}\in \mathrm{fix}(\bH)\setminus\{\mathbf{0}\}$ contains $(d-1)^2-1$ fixed points corresponding to $\psi \mathbf{y}$, where $\psi^{(d-1)^2-1}=1$. This shows the upper bound in \eqref{estnumbanteig1}.
We now show the lower bound using the degree theory as in \cite{Fri77}.
Let $\mu=\min\{\|\bF\|, \|\mathbf{x}\|=1\}$. As $\bF(\mathbf{x})=\mathbf{0}\Rightarrow \mathbf{x}=\mathbf{0}$ it follows that $\mu>0$.
Let $\bG_t(\mathbf{x})=\bF(\mathbf{x})-t\bar(\mathbf{x})$ for $t\in[0,1]$. Then $\|\bG_t(\mathbf{x})\|\ge \mu\|\mathbf{x}\|^d -t\|\mathbf{x}\|$.In particular, $\lim_{\|\mathbf{x}\|\to\infty}\|\bG_t(\mathbf{x})\|=\infty$. Hence
$\bG_t: \mathbb{C}^n\to\mathbb{C}^n$ is a proper map. Let $\mathbb{C}^n\cup\{\infty\}$ be one point compactification of $\mathbb{C}^n$. So $\mathbb{C}^n\cup\{\infty\}$ is homeomorphic to the $2n$ dimensional sphere $\rS^{2n}$. Thus $\bG_t$ extends to a continuous map $\hat\bG_t:\rS^{2n}\to\rS^{2n}$. Let $\deg \hat\bG_t$ be the topological degree of $\hat\bG_t$. The above arguments so that this topological degree is constant for $t\in[0,1]$. Hence $\deg \hat\bG_1=\deg \hat\bG_0=\deg \hat\bF$. The topological degree $\hat\bF$ is just the covering degree of the proper complex polynomial map $\bF$, which is $(d-1)^n$. Hence the number of the antifixed points of $\bF$ is at least $\deg \hat\bG=(d-1)^n$.
As $\mathbf{0}$ is a simple fixed point of $\bH$, $\mathbf{0}$ is a simple zero of $ \bG$. Therefore $|\mathrm{afix}(\bF)\setminus\{\mathbf{0}\}|\ge (d-1)^n-1$. Recall that if $\mathbf{x}\in \mathrm{afix}(\bF)\setminus\{\mathbf{0}\}$ then $\zeta\mathbf{x}\in\mathrm{afix}(\bF)\setminus\{\mathbf{0}\}$ for $\zeta^d=1$. This establishes the lower bound in \eqref{estnumbanteig1}.
\end{proof}
Consider the following example: $f(\mathbf{x})=\sum_{i=1}^n x_i^d$. Then $\bF(\mathbf{x})=(x_1^{d-1},\ldots,x_n^{d-1})^\top$. It is straightforward to show that $|\mathrm{afix}(\bF)|=(d+1)^n$. Furthermore the number of positive eigenvalues of the corresponding $\cS$ is $\frac{(d+1)^{n-1}-1}{d}$.
In what follows we will need the following observation:
\begin{lemma}\label{FHest} Assume that $\cS\in\rS^d\mathbb{F}^n\setminus\{0\}$. Let $\bF(\mathbf{z})=\cS\times(\otimes^{d-1}\mathbf{z})$ and $\mathbf{H}=\bar \bF\circ \bF$.
Then
\begin{eqnarray*}\label{FHest1}
\|\bF(\mathbf{z})\|\le \|\cS\|_{\sigma,\mathbb{F}}\|\mathbf{z}\|^{d-1}, \quad \|\bH(\mathbf{z})\|\le \|\cS\|_{\sigma,\mathbb{F}}^d \|\mathbf{z}\|^{(d-1)^2}, \quad \mathbf{z}\in\mathbb{F}^n.
\end{eqnarray*}
For $\mathbf{z}\in\rS(n,\mathbb{F})$ satisfying $|\cS\times \otimes^d\mathbf{z}|=\|\cS\|_{\sigma,\mathbb{F}}$ equality holds in the above inequalities. Suppose furthermore that $d>2$ and $\mathbf{x}\in \mathbb{F}^n\setminus\{\mathbf{0}\}$ is a fixed point of $\bH$. Then
\begin{eqnarray*}\label{FHest2}
\|\mathbf{x}\|^{-(d-2)}\le \|\cS\|_{\sigma,\mathbb{F}},
\end{eqnarray*}
and this inequality is sharp.
\end{lemma}
\begin{proof} Since $\bF$ and $\mathbf{H}$ are homogeneous maps of degree $d-1$ and $(d-1)^2$ respectively, it is enough to prove the first two inequalities of our lemma for $\mathbf{z}\in\rS(n,\mathbb{F})$.
Assume that $\mathbf{z}\in\rS(n,\mathbb{F})$.
Let $\bw=\cS\times \otimes^{d-1}\mathbf{z}$. First assume that $\bw=\mathbf{0}$. Then $\bF(\mathbf{z})=\bH(\mathbf{z})=\mathbf{0}$ and the first two inequalities of our lemma trivially hold.
Second assume that $\bw\ne \mathbf{0}$.
Let $\bu=\frac{1}{\|\bw\|}\overline {\bw}$. Hence
\[\|\bF(\mathbf{z})\|=|\cS\times (\bu\otimes (\otimes^{d-1}\mathbf{z}))|\le \|\cS\|_{\sigma,\mathbb{F}}.\]
This establishes the first inequality of our lemma. Clearly, $\|\bar\cS\|_{\sigma,\mathbb{F}}=\|\cS\|_{\sigma,\mathbb{F}}$. Hence
\[\|\bH(\mathbf{z})\|=\|\bar\bF(\bF(\mathbf{z}))\|\le \|\cS\|_{\sigma,\mathbb{F}}(\|\bF(\mathbf{z})\|)^{d-1}\le \|\cS\|_{\sigma,\mathbb{F}} ( \|\cS\|_{\sigma,\mathbb{F}})^{d-1}= \|\cS\|_{\sigma,\mathbb{F}}^d.\]
This establishes the second inequality of our lemma.
Suppose that $|\cS\times \otimes^d\mathbf{z}|=\|\cS\|_{\sigma,\mathbb{F}}$ for $\mathbf{z}\in\rS(n,\mathbb{F})$.
First assume that $\mathbb{F}=\mathbb{C}$.
Hence there exists $\zeta\in\mathbb{C},|\zeta|=1$ such that $\mathbf{x}=\zeta\mathbf{z}$ satisfies (\ref{critpt}) with $\lambda=\|\cS\|_{\sigma}$. Clearly $\|\bF(\mathbf{z})\|=\|\bF(\mathbf{x})\|=\lambda=\|\cS\|_{\sigma}$.
Moreover
\[\bH(\mathbf{x})=\bar \bF(\lambda\bar\mathbf{x})=\lambda^{d-1}\bar \bF(\bar\mathbf{x})=\lambda^d\mathbf{x}=
\|\cS\|^{d}_{\sigma}\mathbf{x}.\]
Hence $\|\bH(\mathbf{z})\|=\|\bH(\mathbf{x})\|=\|\cS\|^{d}_{\sigma}$.
Second assume that $\mathbb{F}=\mathbb{R}$. Then $\mathbf{z}\in\rS(n,\mathbb{R})$ is a critical point of $\cS\times \otimes^d\mathbf{x}$ on $\rS(n,\mathbb{R})$. Corollary \ref{maxeig} yields that $\cS\times \otimes^{d-1}\mathbf{z}=\pm \|\cS\|_{\sigma,\mathbb{R}}\mathbf{z}$. Hence $\|\bF(\mathbf{z})\|=\|\cS\|_{\sigma,\mathbb{R}}$ and $\|\bH(\mathbf{z})\|=\|\cS\|_{\sigma,\mathbb{R}}^d$.
Assume finally that $\bH(\mathbf{x})=\mathbf{x}, \mathbf{x}\ne \mathbb{F}^n\setminus\{\mathbf{0}\}$. The second inequality of our lemma yields $\|\mathbf{x}\|= \|\bH(\mathbf{x})\|\le \|\cS\|^d_{\sigma,\mathbb{F}}\|\mathbf{x}\|^{(d-1)^2}$. Hence
$ \|\cS\|^d_{\sigma,\mathbb{F}}\ge \|\mathbf{x}\|^{-(d-1)^2+1}=\|\mathbf{x}\|^{-d(d-2)}$ which yields the third inequality of our lemma. If $\mathbf{x}$ corresponds to the critical vector $\mathbf{z}\in\rS(n,\mathbb{F})$ with the eigenvalue $\lambda$ satisfying $|\lambda|=\|\cS\|_{\sigma,\mathbb{F}}$ then $\|\mathbf{x}\|
^{-(d-2)}= \|\cS\|_{\sigma,\mathbb{F}}$.
\end{proof}
\section{Polynomial-time computability of the spectral norm of $\cS\in\rS^d\mathbb{F}^n$ for a fixed $n$}\label{sec:dnqudit}
In this section we assume that $d\ge 3$ and $\cS\ne 0$. (For $d=2$, (matrices), the spectral norm is the maximal singular value of the matrix, which is polynomially computable.)
{Furthermore we are going to use the results of Appendix 2.
\begin{definition}\label{dersnsing}A symmetric tensor $\cS\in \rS^d\mathbb{C}^n$, and the corresponding polynomial $f(\mathbf{x})=f_{\cS}(\mathbf{x})=\cS\times\otimes^d\mathbf{x}$, are called strongly nonsingular if the following conditions hold: First, $d\ge 3$ and $\cS$ is nonsingular, i.e. the hypersurface $f=0$ is smooth in $\mathbb{P}\mathbb{C}^n$. Second, the $x_1$ coordinates of $(d-1)^{2n}$ solutions $(\mathbf{x},\mathbf{y})$ of the system \eqref{sysfixpH} are distinct.
\end{definition}
\begin{lemma}\label{varsnons} The set of $f\in \rP(d,n,\mathbb{C})$ which are not strongly nonsingular, denoted as $\rP_{ss}(d,n,\mathbb{C})$, has the following structure: Identify $\rP(d,n,\mathbb{C})$ with $\mathbb{C}^{J(d,n)}$ and $\rP_{ss}(d,n,\mathbb{C})$ with $\rV_{ss}(d,n)$. Then $\rV_{ss}(d,n)$ is a disjoint union of hyperdeterminant variety $\mathrm{H}(d,n)$ and the set $\rV'_{ss}(d,n)\subset\mathbb{C}^{J(d,n)}$ which is characterized as follows. There exists a bihomogeneous polynomial $p(\bu, \bv), (\bu,\bv)\in C^{J(d,n)}\times C^{J(d,n)}$ of total degree $2((d-1)^{2n}-1)$ and of degree $(d-1)^{2n}-1$ in $\bu$ and $\bv$ such that $\bu\in \rV_{ss}'(d,n)$ if and only if $\bu\not\in \mathrm{H}(d,n)$ and $p(\bu,\bar\bu)=0$.
\end{lemma}
}
\begin{proof} Let $f,g\in\rP(d,n,\mathbb{C})$. Define $\bF=\frac{1}{d}\nabla f, \bG=\frac{1}{d}\nabla g$. Consider a generalization of the system \eqref{sysfixpH}:
\begin{eqnarray}\label{FGsys} \bF(\mathbf{x})-\mathbf{y}=\mathbf{0}, \quad \bG(\mathbf{y})-\mathbf{x}=\mathbf{0}.
\end{eqnarray}
As $d\ge 3$, the homogeneous part of the above system, i.e., the system \eqref{trivsolcond}, $\bF(\mathbf{x})=\mathbf{0},\bG(\mathbf{y})=\mathbf{0}$ has a unique solution $\mathbf{x}=\mathbf{y}=\mathbf{0}$ if and only if $f=0$ and $g=0$f are smooth hypersurfaces. That is, $f,g\not\in \mathrm{H}(d,n)$. In this case the system \eqref{FGsys} has $D=(d-1)^{2n}$ isolated solutions, counting with multiplicities, and no solution at infinity. (See Appendix 2).
Thus if we find the reduced Gr$\mathrm{\ddot{o}}$bner basis with respect to the order
\begin{eqnarray}\label{lexordxy}
x_1\prec \cdots\prec x_n\prec y_1\prec \cdots \prec y_n
\end{eqnarray}
then the last polynomial is a monic polynomial $p_1(x_1)$ of degree $(d-1)^{2n}$.
We now show an example where $p_1(x_1)$ has $(d-1)^{2n}$ distinct roots.
Let $f=g=h$, where $h=\sum_{i=1}^n x_i^d$. Then the system \eqref{FGsys} is the system \eqref{sysfixpH}. It splits to $n$ systems in $(x_k,y_k)$ for $k\in[n]$:
\begin{eqnarray*} x_k^{d-1}=y_k, \quad y_k^{d-1}=x_k, \quad k\in[n].
\end{eqnarray*}
Note that this system has $D$ distinct solutions. However $x_1$ in these solutions has only $(d-1)^2$ distinct values: $0$, and $(d-1)^{2}-1$ roots of unity. We now show how perturb $h$ so that the $x_1$-coordinates of the solutions of \eqref{FGsys} for $f=g=h$ are distinct. Let $\ba=(a_2,\ldots,a_n)^\top\in\mathbb{C}^{n-1}$ and denote
$h_{\ba}(\mathbf{x})=(x_1+\sum_{j=2}a_jx_j)^d+\sum_{i=2}^n x_i^d$. It is straightforward to check that $h_{\ba}$ is nonsingular. Assume that $\ba$ is close to $\mathbf{0}$. Then the $D$ solutions of \eqref{FGsys} corresponding to $h_{\ba}$ are close the $D$ distinct solutions of \eqref{FGsys} corresponding to $h_{\mathbf{0}}$. In particular each solution of of \eqref{FGsys} corresponding to $h_{\ba}$ is analytic in $\ba$ in the neighborhood of $\ba=\mathbf{0}$. It is left to show that we can choose $\ba$ close to $\mathbf{0}$ such that $x_1(\ba)$ are all distinct. Let $(\mathbf{x}_l(\ba),\mathbf{y}_l(\ba))=(x_{1,l}(\ba),\ldots, x_{n.l}(\ba),y_{1,l}(\ba),\ldots,y_{n,l}(\ba))^\top$ be the $l$-analytic solution of the system \eqref{FGsys} corresponding to $h_{\ba}$ for $l\in[D]$. Clearly if $x_{1,p}(\mathbf{0})\ne x_{1,q}(\mathbf{0})$ then $x_{1,p}(\ba)\ne x_{1,q}(\ba)$ for a small $\ba$. Thus we need to study the case where $x_{1,p}(\mathbf{0})=x_{1,q}(\mathbf{0})=\zeta$. That is, $\zeta$ is a root of $z^{(d-1)^2} -z=0$.
The two equations of the system \eqref{FGsys} corresponding $F_1(\mathbf{x})-y_1=0$ and $G_1(\mathbf{y})-x_1=0$ is
\begin{eqnarray*}
(x_1+\sum_{j=2}a_jx_j)^{d-1}-y_1=0, \quad (y_1+\sum_{j=2}a_jy_j)^{d-1}-x_1=0.
\end{eqnarray*}
To find the $\nabla x_{1,p}(\mathbf{0})$ we can assume that $x_{i,l}(\ba)=\mathbf{x}_{i,l}(\mathbf{0})$ and $y_{i,l}(\ba)=y_{i,l}(\mathbf{0})$ for $i\ge 2$. Observe that $y_{i,l}(\mathbf{0})=x_{i,l}^{d-1}(\mathbf{0})$ for $i\in[n]$ and $l\in[D]$.
First assume that $\mathbf{x}_{1,p}(\mathbf{0})=\zeta\ne 0$. Hence $\zeta^{(d-1)^2-1}=1$. Thus
\begin{eqnarray*}
&&x_{1,p}(\ba)=(y_{1,p}(\ba)+\sum_{j=2}^n a_jy_{p,j}(\ba))^{d-1}=y^{d-1}_{1,p}(\ba) +(d-1)\zeta^{d-2}\sum_{j=2}^n a_jy_{j,p}(\mathbf{0})+ \cO(\|\ba\|^2)=\\
&&(x_{1,p}(\ba)+\sum_{i=2}^n a_i x_{j,p}(\ba))^{(d-1)^2}+(d-1)\zeta^{d-2}\sum_{j=2}^n a_jx_{j,p}^{d-1}(\mathbf{0})+ \cO(\|\ba\|^2)=\\
&&x_{1,p}^{(d-1)^2}(\ba)+(d-1)^2\zeta^{(d-1)^2-1}\sum_{j=2}^n a_jx_{j,p}(\mathbf{0})
+(d-1)\zeta^{d-2}\sum_{j=2}^n a_jx_{j,p}^{d-1}(\mathbf{0})+ \cO(\|\ba\|^2).
\end{eqnarray*}
As $\zeta^{(d-1)^2-1}=1$ we obtain the equation
\begin{eqnarray*}
x_{1,p}(\ba)-x_{1,p}(\ba)^{(d-1)^2}=(d-1)\sum_{j=2}^n ((d-1)x_{j,p}(\mathbf{0})+\zeta^{d-2}x_{j,p}^{d-1}(\mathbf{0}))a_j + \cO(\|\ba\|^2).
\end{eqnarray*}
Hence
\begin{eqnarray*}
\nabla x_{1,p}(\mathbf{0})=-\frac{d-1}{d(d-2)}\sum_{j=2}^n ((d-1)x_{j,p}(\mathbf{0})+\zeta^{d-2}x_{j,p}^{d-1}(\mathbf{0}))a_j.
\end{eqnarray*}
Recall that $x_{j,p}(\mathbf{0})^{(d-1)^2}=x_{j,p}(\mathbf{0})$ for $j\ge 2$ and $p\in[D]$. Hence, for generic real $a_2,\ldots,a_n$ the values of $\nabla x_{1,p}(\mathbf{0})$ are disrinct for all solution $\mathbf{x}_{p}(\ba)$ such that $x_{1,p}(\mathbf{0})=\zeta\ne 0$.
Assume $\zeta=0$. The above arguments yield that $\nabla x_{1,p}(\ba)=\mathbf{0}$. Similarly, $\nabla y_{1,p}(\ba)=\mathbf{0}$. Hence the power series of $x_{1,p}(\ba)$ and $x_{1,p}(\ba)$ start from at least a quadratic polynomial. As $x_{1,p}(\ba)=(y_{1,p}(\ba)+\sum_{j=1}^n a_jy_{j,p}(\ba))^{d-1}$ we deduce that the power series of $x_{1,p}(\ba)$ start with a homogeneous polynomial of degree $d-1$ of the form
$(\sum_{j=2}^n a_j y_{j,p}(\mathbf{0}))^{d-1}$. Recall that $y_{j,p}^{(d-1)^2}=y_{j,p}(\mathbf{0})$ for $j\ge 2$ and $p\in[D]$. Hence for generic real $a_2,\ldots,a_n$ all polynomials $(\sum_{j=2}^n a_j y_{j,p}(\mathbf{0}))^{d-1}$ are different. Hence for a small real generic values of $a_2,\ldots,a_n$ we have that $p_1(x_1)$ have $D$ distinct roots. (Note that $x_1=0$ is always a root of $p_1(x_1)$.)
Assume that $f,g\in\rP(n,d,\mathbb{C})$ are represented by $\bu,\bv\in\mathbb{C}^{J(d,n)}$. Suppose that $\bu,\bv\not\in \mathrm{H}(d,n)$. Then $p_1(x_1)$ is a polynomial of degree $D$. Its coefficients are rational functions in $\bu,\bv$. By multiplying $p_1(x_1)$ by a corresponding polynomial in $\bu,\bv$ we obtain a polynomial $P_1(x_1)$ of degree $D$ whose coefficients are polynomials in $\bu,\bv$ each one of degree $D$.
$P_1(x_1)$ will not have $D$ distinct roots if and only if the discriminant of $P_1(x_1)$ is zero. This discriminant is a polynomial $p(\bu,\bv)$ of degree $2(D-1)$.
It is not hard to see that $p(\bu,\bv)$ is a bihomogeneous polynomial in $(\bu,\bv)$ of degree $D-1$ in $\bu$ and $\bv$ respectively.
Observe next that the system \eqref{sysfixpH} corresponds to a point $(\bu,\bar\bu)$.
For a real $\bu$ the system \eqref{FGsys} corresponds to $f=g$. We showed that for a generic choice
of $\bu$ $p(\bu,\bu)\ne 0$. Thus $V_{ss}(d,n)=\mathrm{H}(d,n)\cup \rV'_{ss}(d,n)$, where
$\rV'_{ss}(d,n)=\{\bu\in\mathbb{C}^{J(d,n)}, \bu\notin \mathrm{H}(d,n), \;p(\bu,\bar\bu)=0\}$.
\end{proof}
Let $\bi=\sqrt{-1}\in\mathbb{C}$, i.e., $\bi^2=-1$. Denote by $\mathbb{Z}[\bi]=\mathbb{Z}+\bi\mathbb{Z}\subset\mathbb{C}$ the integral domain of Gaussian integers, and by $\mathbb{Q}[\bi]$ the field of Gaussian rationals. Let $\mathbb{Z}^n\subset \mathbb{R}^n$ and $\mathbb{Z}[\bi]^n\subset \mathbb{C}^n$ be the $\mathbb{Z}$ and $\mathbb{Z}[\bi]$ modules of vectors with integer and Gaussian integer coordinates respectively.
We now give an upper bound on the complexity of finding the spectral norm $\|\cS\|_{\sigma}$, assuming first that $\cS$ is strongly nonsingular and the entries of $\cS$ are Gaussian rationals, i.e., $\cS\in\rS^d \mathbb{Q}[\bi]^n$. (Note that the assumption that $\cS$ is strongly nonsingular yields that $\cS\ne 0$.) Equivalently, we can assume that $\cS=\frac{1}{N}\cT$, where $\cT$ is a symmetric tensor with Gaussian integers entries $\cT\in\rS^d\mathbb{Z}[\bi]^n$ and $N\in\mathbb{N}$. Thus it is enough to estimate the spectral norm of $\cT$. We identify $\cT$ with $f(\mathbf{x})=\cT\times(\otimes^d \mathbf{x})$. We assume that each coefficient $f_{\bj}$ in \eqref{defpolx1} is $a_{\bj}+\bi b_{\bj}$, where $a_{\bj},b_{\bj}\in\mathbb{Z}$ and $|a_{\bj}|,|b_{\bj}|\le 2^{\tau}$ for each $\bj\in J(d,n)$ for a given integer $M\in\mathbb{N}$.
Next we compute $\bF=(F_1,\ldots,F_n)=\frac{1}{d}\nabla f$. As $\bF=\cT\times \otimes^{d-1}\mathbf{x}$, it follows that the coefficient of each monomial in the coordinates of $\bF$ is a Gaussian integer. Hence the bit length of an integer coefficient is $\tau$.
As $\cT$ is strictly nonsingular we deduce that the system \eqref{sysfixpH} has exactly $D=(d-1)^{2n}$ simple solutions. Furthermore, after finding the reduced Gr$\mathrm{\ddot{o}}$bner basis with respect to the lexicographical order \eqref{lexordxy} we have the conditions of Lemma \ref{shapelem}, the Shape Lemma, in Appendix 2. That is, the reduced Gr$\mathrm{\ddot{o}}$bner basis is of the form
\begin{eqnarray*}
p_1(x_1), x_2-p_2(x_1), \ldots,x_n-p_n(x_1), y_1-p_{n+1}(x_1),\ldots,y_n-p_{2n}(x_1).
\end{eqnarray*}
The degree of $p_1(x_1)$ is $D$, and $p_1(x_1)$ has simple zeros. The degree of each $p_i(x_1)$ is less than $D$ for $i>1$.
The solutions of the system \eqref{sysfixpH} are parametrized by the roots $x_1$ of $p_1(x_1)$. They are of the form $(\mathbf{x},\mathbf{y})$, where $\mathbf{x}=(x_1,p_2(x_1),\ldots,p_n(x_1)^\top$ and $\mathbf{y}=(p_{n+1}(x_1),\ldots,p_{2n}(x_1))^\top$. Recall that the fixed points of $\bH$ are the $\mathbf{x}$ part of the solutions $(\mathbf{x},\mathbf{y})$ of the solutions of \eqref{sysfixpH}. Let
\begin{eqnarray}\label{solfixH}
X=\{(x_1,p_2(x_1),\ldots,p_n(x_1))^\top\in \mathbb{C}^n, \;p_1(x_1)=0\}.
\end{eqnarray}
Thus $X$ is a parametrization of all fixed points of $\bH$ for strongly nonsingular $\cT\in\rS^d\mathbb{C}^n$. In particular, $|X|=D$.
Lemma \ref{eransol} of Appendix 2 gives the bit complexity of computing the coordinate of each solution $(\mathbf{x},\mathbf{y})$ of \eqref{sysfixpH} with precision $2^{-\ell}$, for a given $\ell\in\mathbb{N}$.
Recall that for a nonzero fixed point $\mathbf{x}$ of $\bH$ corresponding to $\cS$ the third inequality of Lemma \ref{FHest} holds. As $\|\cT\|_{\sigma}\le \|\cT\|$ we obtain that a nonzero fixed point of $\bH$ corresponding to $\cT$ satisfies the inequality $\|\mathbf{x}\|\ge \|\cT\|^{-\frac{1}{d-2}}$. Hence Lemma \ref{FHest} yields
\begin{equation}\label{charcTspecnrm}
\|\cT\|_{\sigma}=\max\{\|\mathbf{x}\|^{-(d-2)}, \; \mathbf{x}\in X, \; \|\mathbf{x}\|\ge \|\cT\|^{-\frac{1}{d-2}}\}, \quad f(\by)=\cT\times(\otimes^d \mathbf{x}).
\end{equation}
\begin{theorem}\label{cesnsingcten} Let $d\ge 3$ be an integer. Assume that $\cT\in\rS^d\mathbb{Z}[\bi]^n$, and each coordinate of $\cT$ is bounded above by $2^{\tau}$ for some $\tau\in\mathbb{N}$. If
$\cT$ is strongly nonsingular then for a given $e\in\mathbb{N}$ we can compute rational $L(\cT)$ satisfying
\begin{eqnarray*}\label{Lctineq}
|\|\cT\|_{\sigma}-L(\cT)|\le 2^{-e} \|\cT\|_{\sigma}.
\end{eqnarray*}
The bit complexity of computing $L(\cT)$ is $\tilde O\big((\tau+e) d^{8n}\big)$
\end{theorem}
\begin{proof} Recall that $\mathbf{x}\in X\setminus\{\mathbf{0}\}$ satisfies the inequality $\|\mathbf{x}\|\ge \|\cT\|^{-\frac{1}{d-2}}$. As $\cT\in \rS^d \mathbb{Z}[\bi]^n\setminus\{0\}$ it follows that
\begin{eqnarray*}
1\le\|\cT\|\le \sqrt{n+d-1\choose d}2^{\tau}\le (d+1)^{(n-1)/2} 2^{\tau}.
\end{eqnarray*}
We compute the coordinates of $\mathbf{x}\in X\setminus\{\mathbf{0}\}$ with precision $2^{-(e+k)}$, where $k\in\mathbb{N}$ is specified below. This will give an approximation $\hat \mathbf{x}(\mathbf{x})=(\hat x_1,\ldots,\hat x_n)^\top$ of $\mathbf{x}$. Observe that
$\|\mathbf{x}-\hat\mathbf{x}(\mathbf{x})\|\le \sqrt{n}2^{-(e+k)}$. We assume that
\begin{eqnarray*}
\sqrt{n}2^{-k}\le 2^{-d}(d+1)^{-(n-1)/2(d-2)}2^{-\tau/(d-2)}.
\end{eqnarray*}
To satisfy the above inequality we choose
\begin{eqnarray}\label{defkLTest}
k=\lceil d+(1/2)\log_2 n+\frac{n-1}{2(d-2)}\log_2(d+1)+\tau/(d-2)\rceil.
\end{eqnarray}
Hence for any nonzero fixed point one has the inequality
\begin{eqnarray*}
\|\mathbf{x}-\hat \mathbf{x}(\mathbf{x})\|\le \sqrt{n}2^{-(e+k)}\le 2^{-(e+d)}\|\cT\|^{-\frac{1}{d-2}}\le 2^{-(e+d)}\|\mathbf{x}\|.
\end{eqnarray*}
In particular $(1-2^{-(e+d)})\|\mathbf{x}\|\le \|\hat\mathbf{x}(\mathbf{x})\|\le (1-2^{-(e+d)})\|\mathbf{x}\|$.
Let
\begin{eqnarray*}
L(\cT)=\max\{\|\hat\mathbf{x}(\mathbf{x})\|^{-(d-2)}, \mathbf{x}\in X\setminus\{\mathbf{0}\}\}.
\end{eqnarray*}
We claim that the inequality \eqref{Lctineq} holds. For a nonzero fixed point of $\bH$ we estimate
$|\|\mathbf{x}\|^{-(d-2)}-\|\hat\mathbf{x}(\mathbf{x})\|^{-(d-2)}|$. First observe that
\begin{eqnarray*}
&&|\|\mathbf{x}\|^{d-2}-\|\hat\mathbf{x}(\mathbf{x})\|^{d-2}|\le |\|\mathbf{x}\|-\|\hat\mathbf{x}(\mathbf{x})\||(d-2)\max(\|\mathbf{x}\|^{d-3}, \|\hat\mathbf{x}(\mathbf{x})\|^{d-3})\le \\
&&2^{-(e+d)}(d-2)(1+2^{-(e+d)})^{d-2}\|\mathbf{x}\|^{d-2}.
\end{eqnarray*}
Hence
\begin{eqnarray*}
&&|\|\mathbf{x}\|^{-(d-2)}-\|\hat\mathbf{x}(\mathbf{x})\|^{-(d-2)}|=|\|\mathbf{x}\|^{d-2}-\|\hat\mathbf{x}(\mathbf{x})\|^{d-2}|\|\mathbf{x}\|^{-(d-2)}\|\hat\mathbf{x}(\mathbf{x})\|^{-(d-2)}\le\\
&&2^{-(e+d)}(d-2)(1+2^{-(e+d)})^{d-2}(1-2^{-(e+d)})^{-(d-2)}\|\mathbf{x}\|^{-(d-2)}.
\end{eqnarray*}
As $e\ge 1$and $d\ge 3$ it follows that $(1+2^{-(e+d)})/(1-2^{-(e+d)})\le 17/15$. It is straightforward to show that $(d-2)(1+2^{-(e+d)})^{d-2}(1-2^{-(e+d)})^{-(d-2)}\le 2^d$ for an integer $d\ge 3$. Hence$|\|\mathbf{x}\|^{-(d-2)}-\|\hat\mathbf{x}(\mathbf{x})\|^{-(d-2)}|\le 2^{-e}\|\mathbf{x}\|^{-(d-2)}$.
First choose a fixed point $\bx$ of satisfying $\|\mathbf{x}\|^{-(d-2)}=\|\cT\|_{\sigma}$. Therefore $L(\cT)\ge (1-2^{-e})\|\cT\|_{\sigma}$. Assume that $L(\cT)=\|\hat\mathbf{x}(\mathbf{x})\|^{-(d-2)}$ for some $\mathbf{x}\in X\setminus\{\mathbf{0}\}$. Then
$L(\cT)\le (1+2^{-e})\|\mathbf{x}\|^{-(d-2)}\le (1+2^{-e})\|\cT\|_{\sigma}$. Thus $ |\|\cT\|_{\sigma}-L(\cT)|\le 2^{-e} \|\cT\|_{\sigma}$.
It is left to show that the bit complexity of computing $L(\cT)$ is $\tilde\cO\big((\tau+e)d^{8n}\big)$. This follows from the proof of Lemma \ref{eransol} in Appendix 2.
First note that in Lemma \ref{eransol} $m=2n$. Next observe that the value of $\ell$ in
Lemma \ref{eransol} is $e+k$, where $k$ is given by \eqref{defkLTest}.
\end{proof}
We now present the main result of this section:
\begin{theorem}\label{cessingcten} Let $d\ge 3$ be an integer. Assume that $\cJ\in \rS^d\mathbb{Z}[\bi]^n$ is a strongly nonsingular tensor that satisfies $\|\cJ\|\le 2^{c}, c\in\mathbb{N}$. Suppose that $\cT\in\rS^d\mathbb{Z}[\bi]^n$, and each coordinate of $\cT$ is bounded above by $2^{\tau},\tau\in\mathbb{N}$.
For a given $b,e\in\mathbb{N}$ we can compute $L(\cT)$ satisfying
\begin{eqnarray*}\label{Lctineqm}
|\|\cT\|_{\sigma}-L(\cT)|\le 2^{-e} \|\cT\|,
\end{eqnarray*}
with probability greater than $1-(d-1)^{-2nb}$.
The bit complexity of computing $L(\cT)$ is $\tilde O\big((\tau+c+2nb\log_2 (d-1)+e) d^{8n}\big)$.
\end{theorem}
\begin{proof}
Clearly, it is enough to assume that $\cT\ne 0$.
Lemma \ref{varsnons} yields that the set of strongly singular symmetric tensors is the zero set of $\Delta(\bu)p(\bu,\bar\bu)$, where $\bu\in \mathbb{C}^{J(d,n)}$ represents the polynomial $f(\mathbf{x})=\cS\times \otimes^d\mathbf{x}$, for $\cS\in\rS^d\mathbb{C}^n$. Here $\Delta(\bu)$ is a polynomial of degree $n(d-1)^{n-1}$, and $\Delta(\bu)=0$ is the hyperdeterminant variety.
The polynomial $p(\bu,\bar\bu)$ is a polynomial of degree $2((d-1)^{2n}-1)$.
Let $g(\mathbf{x})=\cJ\times\otimes^d\mathbf{x}$. Denote by $\bg\in\mathbb{C}^{J(d,n)}$ the vector corresponding to $g(\mathbf{x})$. We now consider the affine line of polynomials $f_t(\mathbf{x})=t f(\mathbf{x})+g(\mathbf{x})$ for $t\in\mathbb{R}$. The value of the polynomial $\Delta p$ on $t\ff+\bg$ is $q(t)=\Delta(t\ff+\bg)p(t\ff+\bg,t\bar\ff+\bar\bg)$. As $\cJ$ is strongly nonsingular $q(0)\ne 0$. Hence $q(t)$ is a nonzero polynomial of degree at most $n(d-1)^{n-1}+2((d-1)^{2n}-1)< 3(d-1)^{2n}$.
Let $A=\{M,M+2,\ldots, M+N-1\}\subset \mathbb{Z}$. We assume that $M=2^{c+e+2}$ and $N=3(d-1)^{2n(b+1)}$.
The cardinality of $A$ is $N$.
For each $a\in A$, let us consider the tensor $\cT(a)=a\cT+\cJ$. Note $q(t)$ vanishes at most at $n(d-1)^{n-1}+2((d-1)^{2n}-1)$ points of $A$. Choose a random $a\in A$ from the uniform distribution on A. Then with probability greater than $1-(d-1)^{-2nb}$ the tensor $\cT(a)$ is strongly nonsingular.
Let $L(\cT(a))$ be the approximation given by Theorem \ref{cesnsingcten}, where we replace $e$ by $e+2$. We now choose $L(\cT)=\frac{1}{a} L(\cT(a))$. We claim that
$ |\|\cT\|_{\sigma}-L(\cT)|\le 2^{-e} \|\cT\|$. Indeed, the inequality of Theorem \ref{cesnsingcten} yields
\[|\|\cT+\frac{1}{a}\cJ\|_{\sigma}- L(\cT)|\le \frac{2^{-e}}{4} \|\cT+\frac{1}{a}\cJ\|_{\sigma}.\]
As $\cT\in\rS^d\mathbb{Z}[\bi]\setminus\{0\}$ it follows that $\|\cT\|\ge 1$. Since $a\ge 2^{c+e+2}$ and $\|\cJ\|\le 2^{c}$ we deduce
\begin{eqnarray*}
&&|\|\cT\|_{\sigma}-L(\cT)|\le |\|\cT+\frac{1}{a}\cJ\|_{\sigma}-L(\cT)\|+|\|\cT\|_{\sigma}-\|\cT+\frac{1}{a}\cJ\|_{\sigma}\|\le\\
&&2^{-(e+2)}\|\cT+\frac{1}{a}\cJ\|_{\sigma}+\frac{1}{a}\|\cJ\|_{\sigma}\le
2^{-(e+2)}\|\cT\|_{\sigma}+(2^{-(e+2)}+1)\frac{1}{a}\|\cJ\|_{\sigma}\le\\
&&2^{-(e+2)}(\|\cT\|_{\sigma}+
2^{-(e+2)}+1)\le
2^{-(e+2)}(\|\cT\|+(2^{-(e+2)}+1)\|\cT\|)< 2^{-e}\|\cT\|.
\end{eqnarray*}
It is left to show that the bit complexity of finding $L(\cT)$ is $\tilde O\big((\tau+c+2nb\log_2(d-1)+c) d^{8n}\big)$. This follows from Theorem \ref{cesnsingcten}. Indeed observe that each entry by $\cT(a)$ is bounded by $2^{\tau+c+2}(d-1)^{2n(b+1)}$.
\end{proof}
We remark that we can find a strongly nonsingular $\cJ\in \rS^d\mathbb{Z}^n$ as follows: We choose a tensor of the form given in the proof of Lemma \ref{varsnons} by choosing $a_2,\ldots,a_n\in\mathbb{Z}$ at random.
We now give a similar complexity result for an approximation of $\|\cT\|_{\sigma,\mathbb{R}}$:
\begin{theorem}\label{cesrealten} Let $d\ge 3$ be an integer. Suppose that $\cT\in\rS^d\mathbb{Z}^n$, and each coordinate of $\cT$ is bounded above by $2^{\tau},\tau\in\mathbb{N}$.
For a given $e\in\mathbb{N}$ we can compute $L(\cT)$ that satisfies the following conditions:
\begin{enumerate}
\item Assume that $\cT$ is strongly nonsingular. Then $|\|\cT\|_{\sigma,\mathbb{R}}-L(\cT)|\le 2^{-e} \|\cT\|_{\sigma,\mathbb{R}}$.
The bit complexity of computing $L(\cT)$ is $\tilde\cO\big((\tau+e)d^{4n}\big)$.
\item Assume that $\cJ\in \rS^d\mathbb{Z}^n$ is a strongly nonsingular tensor that satisfies $\|\cJ\|\le 2^{c}, c\in\mathbb{N}$. For a given $b\in\mathbb{N}$ we can compute $L(\cT)$ with probability
greater than $1-(d-1)^{-2nb}$ that satisfies $ |\|\cT\|_{\sigma,\mathbb{R}}-L(\cT)|\le 2^{-e} \|\cT\|$. The bit complexity of computing $L(\cT)$ is $\tilde O\big((\tau+c+2nb\log_2 (d-1)+e) d^{4n}\big)$. With probability
\end{enumerate}
\end{theorem}
\begin{proof} We point out briefly the corresponding modifications of the proofs of Theorems \ref{cesnsingcten} and \ref{cessingcten} respectively. Let $f(\mathbf{x})=\cT\times\otimes^d\mathbf{x}$. First observe that $\bar\bF=\bF$. Hence any fixed point of $\bF$ is a fixed point $\bH$.
First assume that $\cT$ is nonsingular. Then $x_1$ coordinates of $(d-1)^n$ fixed points of $\bF$ are pairwise distinct. We now find the Gr\"obner basis of the system $\bF(\mathbf{x})-\mathbf{x}=\mathbf{0}$ with respect to the order $x_1\prec\cdots\prec x_n$.
It is of the form $p_1(x_1), x_2-p_2(x_1),\ldots,x_n-p_n(x_1)$, where $p_1,\ldots,p_n\in\mathbb{R}[x_1]$. Here $\deg p_1=D$, where $D=(d-1)^n$. Furthermore $p_1(x_1)$ has $D$ distinct roots. Recall that $\deg p_i<D$ for $i>1$. The set of the fixed points of $\bF$ is given by \eqref{solfixH}.
Recall that $\|\cT\|_{\sigma,\mathbb{R}}$ can be computed using \eqref{Sinftynrmformr}.
Thus we need to determine $\mathrm{fix}_{\mathbb{R}}(\bF)$ with precision $2^{e+k}$, as in the proof of Theorem \ref{cesnsingcten}.
This can be done as follows. First consider $X_{\mathbb{R}}=X\cap\mathbb{R}^n$. For each approximation of a root of $x_1$ with precision $2^{e+k}$, we check if the disk $|z-x_1|<2^{e+k}$ contains a real root of $p_1(x_1)$. If yes, we replace $x_1$ by $\tilde x_1\in\mathbb{R}$. Then a real approximation of the real fixed point is $(\tilde x_1,p_2(\tilde x_1),\ldots,p_n(\tilde x_n))^\top$. Similarly we can find a real approximation of $\bar\omega_{d-2}X\cap \mathbb{R}^n$ for an even $n$. (See Theorem \ref{theofoundthm}.)
Now we proceed as in the proof of Theorem \ref{cesnsingcten}.
For a general $\cT\in\rS^d\mathbb{Z}^n$ we repeat the arguments of the proof of Theorem \ref{cessingcten} taking into account the above results for a strongly nonsingular $\cT\in\rS^d\mathbb{Z}^n$.
\end{proof}
We conclude this section with the following NP-hardness result for an arbitrary approximation of the spectral norm of a real or complex valued homogeneous quartic polynomial in $n$ variables.
\begin{theorem}\label{NPhardquart} Let $A=[A_{i,j}]$ be an $n\times n$ nonzero symmetric matrix with $\{0,1\}$ entries and zero diagonal. Set
\[f_A=\sum_{i=j=1}^n A_{i,j}x_i^2x_j^2.\]
\begin{enumerate}
\item Let $2e$ be the number of nonzero elements in $A$. Then
\[\|\mathbf{f}_A\|=\frac{\sqrt{2e}}{\sqrt{3}}, \quad \frac{\sqrt{2}}{\sqrt{3}}\le \|\mathbf{f}_A\|\le \frac{\sqrt{n(n-1)}}{\sqrt{3}}<n,\]
where $\|\mathbf{f}_A\|$ is the norm defined in Lemma \ref{isolem}.
\item Equality
\[\|f_A\|_{\sigma,\mathbb{R}}=\|f_A\|_{\sigma}=1-1/\kappa(A)\]
holds, where $\kappa(A)$ is an integer in the set $[n]$.
\item It is NP-hard to compute an approximation of $\|f_A\|_{\sigma}$ within relative precision $\delta<\frac{1}{2n^2(n+1)}$ with respect to $\|\mathbf{f}_A\|$.
\end{enumerate}
\end{theorem}
\begin{proof} (1) Assume that $f_A$ corresponds to $\cS=[\cS_{k_1,k_2,k_3,k_4}]\in \rS^4\mathbb{C}^n$. Then $\cS_{k_1,k_2,k_3,k_4}=1/3$ if the multiset $\{k_1,k_2,k_3,k_4\}$ is of the form $\{i,i,j,j\}$, $i<j$ and $A_{i,j}=A_{j,i}=1$. (The number of nonzero $\cS_{k_1,k_2,k_3,k_4}$ corresponding to $\{i,i,j,j\}$ is $4!/(2!)^2=6$.) Otherwise $\cS_{k_1,k_2,k_3,k_4}=0$. Hence $\|\mathbf{f}_A\|=\|\cS\|=\sqrt{2e}/\sqrt{3}$.
The second inequality follows straightforward.
\noindent
(2) The matrix $A$ is the adjacency matrix of the following simple undirected graph $G=(V,E)$:
Here $V=[n]$ and the edge $(i,j)$ is in $E$ if and only if $A_{i,j}=1$. Let $\kappa(A)$ be the cardinality of the maximal clique in $G$, (the clique number of $G$). The equality $\|f_A\|_{\sigma,\mathbb{R}}=1-1/\kappa(A)$ is \cite[(8.2)]{FL18}. As $\cS$ has nonnegative entries it follows that $\|f_A\|_{\sigma,\mathbb{R}}=\|f_A\|_{\sigma}$ \cite{FL18}.
\noindent
(3) Since $\|f_A\|_{\sigma}=1-1/\kappa(A)$, an approximation of $\|f_A\|_{\sigma}$ within relative precision $\varepsilon<\frac{1}{2n^2(n+1)}$ with respect to $\|\mathbf{f}_A\|$ determines the clique number. However, it is an NP-complete problem to determine the clique number of a simple graph \cite{Ka}.
\end{proof}
\section{Polynomial-time computability of spectral norm of symmetric $d$-qubits}\label{sec:dqubit}
In this section we improve the results of the previous section for the case $n=2$.
We parametrize
$\cS=[\cS_{i_1,\ldots,i_d}]\in \rS^d\mathbb{F}^2$ by
$\mathbf{s}=(s_0,\ldots,s_d)$ as follows: $\cS_{i_1,\ldots,i_d}=s_k$ if exactly $k$ indices from the multiset
$\{i_1,\ldots,i_d\}$ are equal to $2$. Note that exactly $d \choose k$ entries of $\cS$ are equal to $s_k$. Hence
$\|\cS\|=\sqrt{\sum_{k=0}^d {d\choose k}|s_k|^2}$.
The following lemma is a restatement of some results of Lemma \ref{isolem}.
\begin{lemma}\label{qubspecnrmlem} Let $\cS\in\rS^d\mathbb{C}^2$ and associate with $\cS$ the vector $\mathbf{s}=(s_0,\ldots,s_d)^\top\in \mathbb{C}^{d+1}$. Denote
\begin{equation}\label{defphiz}
\phi(z)=\sum_{j=0}^d {d\choose j} s_j z^j.
\end{equation}
Then
\begin{enumerate}
\item
Let $f(\mathbf{x})=\cS\times\otimes ^d\mathbf{x}$ and $\cS\times\otimes^{d-1} \mathbf{x}=\mathbf{F}(\mathbf{x})=(F_1(\mathbf{x}),F_2(\mathbf{x}))^\top$, where $\mathbf{x}=(x_1,x_2)^\top$. Then
\begin{eqnarray*}\label{deffx}
&&f(\mathbf{x})=\sum_{j=0}^d {d\choose j} s_j x_1^{d-j}x_2^j=x_1^d \phi(\frac{x_2}{x_1}),\\
\label{formF0}
&&F_1(\mathbf{x})=\sum_{j=0}^{d-1} {d-1\choose j} s_jx_1^{d-1-j}x_2^j =\frac{1}{d}\frac{\partial f}{\partial x_1},\;
F_2(\mathbf{x})=\sum_{j=0}^{d-1} {d-1\choose j}s_{j+1} x_1^{d-j-1}x_2^j=\frac{1}{d}\frac{\partial f}{\partial x_2}.\label{formF1}
\end{eqnarray*}
\item For $\mathbf{s}=(0,\ldots,0,s_d)^\top$ we have $\|\cS\|_{\sigma,\mathbb{F}}=|s_d|$.
\end{enumerate}
\end{lemma}
\begin{proof} (1). Use equalities \eqref{defpolx1} and \eqref{Fformulas}.
\noindent
(2). Assume that $\mathbf{s}=(0,\ldots,0,s_d)^\top$. Then $\cS\times \otimes^d\mathbf{x}=s_d x_2^d$. Hence $\|\cS\|_{\sigma,\mathbb{F}}=|s_d|$.
\end{proof}
The next proposition studies the fixed points of $\bF$ and $\bH$ for the case $n=2$.
\begin{proposition}\label{fixptHFn=2} Suppose that the assumptions and notations of Lemma \ref{qubspecnrmlem} hold.
Assume that $d\ge 3$ and
\begin{equation}\label{Sassump}
\mathbf{s}\ne (0,\ldots,0,s_d)^\top.
\end{equation}
\begin{enumerate}
\item
Define polynomials $p(z), q(z)$ and the rational function $r(z)$ as follows
\begin{eqnarray*}\label{fzfor}
p(z)=\sum_{j=0}^{d-1} {d-1\choose j}s_{j+1} z^j,\; q(z)=\sum_{j=0}^{d-1} {d-1\choose j} s_jz^j,\; r(z)=\frac{p(z)}{q(z)}.\;
\end{eqnarray*}
Then
\begin{equation}\label{relpqrphi}
p(z)=\frac{1}{d}\phi'(z),\; q(z)=\phi(z)-\frac{1}{d}z\phi'(z), \;r(z)=\frac{\phi'(z)}{d\phi(z)-z\phi'(z)}.
\end{equation}
\item
Suppose that $\bF(\mathbf{x})=(x_1,x_2)^\top\ne \mathbf{0}$. First assume that $x_1\ne 0$. Let $z=x_2/x_1$. Then
\begin{equation}\label{fixqub1}
z q(z)-p(z)=0 \textrm{ and } q(z)\ne 0.
\end{equation}
Vice versa, for each $z\in\mathbb{C}$ satisfying the above conditions there are exactly $d-2$ fixed points of $\bF$ of the form $(x_1,x_1 z)$, where $x_1$ satisfies $x_1^{d-2}=\frac{1}{q(z)}$.
Second assume that $x_1=0$. Then $s_{d-1}=0$ and $s_d\ne 0$.
Vice versa, if $s_{d-1}=0$ and $s_d\ne 0$ then there are exactly $d-2$ nonzero fixed points of the form $(0,x_2)$, where $x_2$ satisfies $x_2^{d-2}=1/s_d$.
\item Suppose that $\bF(\mathbf{x})=(\bar x_1, \bar x_2)\ne \mathbf{0}$. First assume that $x_1\ne 0$. Let $z=x_2/x_1$. Then
\begin{equation}\label{antifixqub1}
\bar z q(z)-p(z)=0
\end{equation}
and $q(z)\ne 0$.
Vice versa, for each $z\in\mathbb{C}$ satisfying the above conditions there are exactly $d$ antifixed points of $\bF$ of the form $(x_1,x_1 z)$, where $x_1$ satisfies $x_1=\frac{\zeta}{|q(z)|^{1/(d-2)}}$ and $\zeta^d=1$.
Second assume that $x_1=0$. Then $s_{d-1}=0$ and $s_d\ne 0$.
Vice versa, if $s_{d-1}=0$ and $s_d\ne 0$ then there are $d$ nonzero fixed points of the form $(0,\frac{\zeta}{|s_d|^{1/(d-2)}})$ and $\zeta^d=1$.
\item Let
\begin{eqnarray*}\label{defbarpqf}
\bar p(z)=\sum_{j=0}^{d-1} {d-1\choose j}\bar s_{j+1} z^j, \;\bar q(z)=\sum_{j=0}^{d-1} {d-1\choose j} \bar s_jz^j,\;\bar r(z)=\frac{\bar p(z)}{\bar q(z)},\;g(z)=\bar r (r(z)).
\end{eqnarray*}
Then $g(z)=\frac{u(z)}{v(z)}$, where
\begin{eqnarray}\label{defuz}
u(z)= \sum_{j=0}^{d-1} {d-1\choose j}\bar s_{j+1}
\big(\sum_{k=0}^{d-1} {d-1\choose k} s_{k+1} z^k\big)^{j}\big(\sum_{k=0}^{d-1} {d-1\choose k} s_kz^k\big)^{d-1-j},\\
\label{defvz}
v(z)= \sum_{j=0}^{d-1} {d-1\choose j}\bar s_{j}
\big(\sum_{k=0}^{d-1} {d-1\choose k} s_{k+1} z^k\big)^{j}\big(\sum_{k=0}^{d-1} {d-1\choose k} s_kz^k\big)^{d-1-j}.\quad
\end{eqnarray}
Suppose that $\bF(\mathbf{x})=(\bar x_1, \bar x_2)^\top \ne \mathbf{0}$ as in (3). Assume that $x_1\ne 0$. Then each solution of \eqref{antifixqub1} satisfies
\begin{equation}\label{poleqfixpoint}
zv(z)-u(z)=0
\end{equation}
and $v(z)\ne 0$.
This is a polynomial equation of degree at most $(d-1)^2+1$.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) The assumption $\mathbf{s}\ne (0,\ldots,0,s_d)^\top$ is equivalent to the assumption that the polynomial $q(z)$ is not zero identically. Assume that $x_1\ne 0$. Let $z=\frac{x_2}{x_1}$. Lemma \ref{qubspecnrmlem} yields:
$\frac{F_2(x_1,x_1)}{x_1^{d-1}}=p(z)$ and $\frac{F_1(x_1,x_1)}{x_1^{d-1}}=q(z)$.
Recall that
\begin{eqnarray*}
dF_2=\frac{\partial f}{\partial x_2}=\frac{\partial (x_1^{d}\phi(\frac{x_2}{x_1}))}{\partial x_2}=x_1^{d-1}\phi'(\frac{x_2}{x_1}),\\
dF_1=\frac{\partial f}{\partial x_1}=\frac{\partial (x_1^{d}\phi(\frac{x_2}{x_1}))}{\partial x_1}=dx_1^{d-1}\phi(\frac{x_2}{x_1})-x_2x_1^{d-2}\phi'(\frac{x_2}{x_1}).
\end{eqnarray*}
These equalities yield (\ref{relpqrphi}).
\noindent
(2) Assume that $\bF(\mathbf{x})=(x_1,x_2)^\top \ne \mathbf{0}$.
Suppose first that $x_1\ne 0$. Then
$x_1^{d-1} q(z)=F_1(\mathbf{x})=x_1$ and $x_1^{d-1}p(z)=F_2(\mathbf{x})=x_2$. As $x_1\ne 0$ it follows that $q(z)\ne 0$.
Divide the second equality by the first one to deduce \eqref{fixqub1}. Note that $x_1^{d-2}=\frac{1}{q(z)}$.
Assume that $z\in\mathbb{C}$ satisfies \eqref{fixqub1}. Suppose furthermore that $x_1^{d-2}=\frac{1}{q(z)}$. Let $\mathbf{x}=(x_1,x_1 z)^\top$. Then
\[F_1(\mathbf{x})=x_1^{d-1} q(z)=x_1, \quad F_2(\mathbf{x})=x_1^{d-1}p(z)=x_1^{d-1} zq(z)=x_1^{d-1} (x_2/x_1)q(z)=x_1^{d-2}q(z)x_2=x_2.\]
Hence each $z$ that satisfies \eqref{fixqub1} gives rise to exactly $d-2$ distinct nonzero fixed points of $\bF$.
Assume that $x_1=0$. Hence $x_2\ne 0$. Note that $F_1(\mathbf{x})=s_{d-1}x_2^{d-1}$.
As $F_1(\mathbf{x})=x_0=0$ we deduce that $s_{d-1}=0$. Clearly, $F_2(\mathbf{x})=s_d x_2^{d-1}=x_2$. As $x_2\ne 0$ it follows that $s_d\ne 0$.
Suppose that $s_{d-1}=0$ and $s_d\ne 0$. Then all nonzero fixed points of $\bF$ of the form $(0,x_2)^\top$ are exactly those $x_2$ satisfying $x_2^{d-2}=1/s_d$.
\noindent
(3) The proof of this part is very similar to the part (2) and we leave it to the reader.
\noindent
(4) From the definitions of $p(z),q(z),\bar p(z), \bar q(z), \bar r(z) $ and $g(z)$ we deduce straightforward the identities (\ref{defuz}) and (\ref{defvz}).
Let
\[u(z)=\sum_{k=0}^{(d-1)^2} u_kz^k, \quad v(z)=\sum_{k=0}^{(d-1)^2} v_kz^k.\]
Then
\begin{eqnarray*}\label{uvdfor}
u_{(d-1)^2}=\sum_{j=0}^{d-1} {d-1\choose j}\bar s_{j+1}s_{d}^j s_{d-1}^{d-1-j},\;
v_{(d-1)^2}= \sum_{j=0}^{d-1} {d-1\choose j}\bar s_{j}s_{d}^j s_{d-1}^{d-1-j}.
\end{eqnarray*}
Hence the polynomial $zv(z)-u(z)$ is of degree at most $(d-1)^2+1$. Suppose $\mathbf{x}=(x_1,x_2)^\top\in\mathrm{fix}(\bH), x_1\ne 0$:
$\overline{\mathbf{F}}(\bF(\mathbf{x}))=\mathbf{x}$.
Since we assumed that $x_1\ne 0$ it follows that
\[u(z)=\frac{\bar\bF_2(\bF(\mathbf{x}))}{x_1^{(d-1)^2}}, \quad v(z)=\frac{\bar\bF_1(\bF(\mathbf{x}))}{x_1^{(d-1)^2}}=x_1^{-(d-1)^2 +1}\ne 0.\]
Therefore
\[\bar r( r(z))=\frac{\bar\bF_2(\bF(\mathbf{x}))x_1^{-(d-1)^2}}{\bar\bF_1(\bF(\mathbf{x}))x_1^{-(d-1)^2}}=\frac{x_2}{x_1}=z.\]
This yields (\ref{poleqfixpoint}), which is a polynomial equation of degree at most $(d-1)^2 +1$.
\end{proof}
The following theorem gives much more efficient way to compute the spectral norm of symmetric qubits than the general methods suggested in \S\ref{sec:dnqudit}.
\begin{theorem}\label{computdqubspecnrm} Let $\cS\in\rS^d\mathbb{C}^2$, $d>2$ and associate with $\cS$ the vector $\mathbf{s}=(s_0,\ldots,s_d)^\top\in \mathbb{C}^{d+1}$. Let $\phi(z)$ be given by \eqref{defphiz}. Assume the notations of Proposition \ref{fixptHFn=2}. Then
\begin{enumerate}
\item The polynomial $zv(z)-u(z)$ is a zero polynomial if and only if one of the following conditions hold. Either
\begin{eqnarray*}\label{geqiden}
\mathbf{s}=A(\delta_{1(k+1)},\ldots,\delta_{(d+1)(k+1)})^\top \textrm{ for } k\in[d-1], (f(\mathbf{x})=A{d\choose k}x_1^{d-k}x_2^k),
\end{eqnarray*}
where $A$ is a nonzero scalar constant and $\delta_{ij}$ is Kronecker's delta function. For this $\cS\in\rS^d\mathbb{F}^2$ we have
\begin{equation}\label{geqidenSform}
\|\cS\|_{\sigma,\mathbb{F}}=|A|{d \choose k} \big(1-\frac{k}{d}\big)^{\frac{d-k}{2}}\big(\frac{k}{d}\big)^{\frac{k}{2}}.
\end{equation}
Or $\cS$ has corresponding $\phi$ given by
\begin{eqnarray}\label{geqiden1}
&&\phi(z)=A (z+a)^p(z+b)^{d-p},\quad A\ne 0,\\
&&a=e^{-\theta\bi}c,\; b=-e^{-\theta\bi}c^{-1},\;c\in\mathbb{R}\setminus\{0\}, \; \theta\in\mathbb{R},\;p\in[d-1].\notag
\end{eqnarray}
Assume that $\cS\in\rS^d\mathbb{C}^2$ corresponds to $\phi$ is of the form (\ref{geqiden1}). Then $\|\cS\|_{\sigma}$ can be computed to an arbitrary precision as explained in \S\ref{sec:excepcase}.
\item Let
\begin{eqnarray*}
R_1=\{z\in\mathbb{C},\;zv(z)-u(z)=0\}\label{defR1}, \quad R_1'=R_1\cap \mathbb{R}.
\end{eqnarray*}
Suppose that $\cS\in\rS^d\mathbb{C}^2$ is not of the form given in (1). (Hence the set $R_1$ is finite.) Then $\|\cS\|_{\sigma}$ has the following maximum characterization:
\begin{eqnarray}
\label{specnrmSform1}
\|\cS\|_{\sigma}= \max\left\{|s_d|, \max\{\frac{|\phi(z)|}{(1+|z|^2)^{\frac{d}{2}}}, z\in R_1\}\right\}.
\end{eqnarray}
\item Assume that $\cS\in\rS^d\mathbb{R}^2$. Let
\begin{eqnarray*}\label{defR}
R=\{z\in\mathbb{C}, \;z q(z)-p(z)=0\},\quad R'=R\cap\mathbb{R}
\end{eqnarray*}
Then $zq(z)-p(z)$ is a zero polynomial if and only if $d$ is even and $\phi(z)=A(z^2+1)^{d/2}$. In this case $\|\cS\|_{\sigma,\mathbb{R}}=|A|$.
Assume that $\phi(z)$ is not of the form $A(z^2+1)^{d/2}$. Then $\|\cS\|_{\sigma,\mathbb{R}}$ has the following maximum characterizations:
\begin{eqnarray}\label{realforspecnrm}
\|\cS\|_{\sigma,\mathbb{R}}=\max\left\{|s_d|, \max\{\frac{|\phi(z)|}{(1+|z|^2)^{\frac{d}{2}}}, z\in R'\}\right\}.
\end{eqnarray}
\end{enumerate}
\end{theorem}
\begin{proof}
\noindent
(1) The analysis of the exceptional cases is given in \S\ref{sec:excepcase}.
\noindent
(2) Assume that $\mathbf{s}=(s_0,\ldots,s_d)^\top\in\mathbb{C}^{d+1}$ is the vector corresponding to $\cS=[\cS_{i_1,\ldots,i_d}]\in\rS^d\mathbb{C}^2$. Let $\omega$ be the righthand side of \eqref{specnrmSform1}. We claim that Banach's characterization \eqref{Banchar} yields that $\omega \le \|\cS\|_{\sigma}$. Indeed, $\|\cS\|_{\sigma}\ge |\cS\times(\otimes^d(0,1)^\top)|=|\cS_{2,\ldots,2}|=|s_d|$. Assume now that $\mathbf{x}=(x_1,x_2)^\top$ and $x_1\ne 0$. Let $z=x_2/x_1$. Then $|\cS\times(\otimes^d \mathbf{x})|/\|\mathbf{x}\|^{d}=|\phi(z)|/(1+|z|^2)^{d/2}$. Hence $\omega \le \|\cS\|_{\sigma}$. As any maximal point of $|\cS\times(\otimes^d \mathbf{x})|$ on $\rS(2,\mathbb{C})$ is either
$\zeta(0,1)^\top, |\zeta|=1$, or of the form $\zeta(1,z),\zeta\in\mathbb{C}\setminus\{0\}$, where $z\in R_1$ we deduce that $\omega =\|\cS\|_{\sigma}$.
\noindent
(3) Assume that $zq(z)-p(z)$ is identically zero. The equalities \eqref{relpqrphi} yield that
\[z\phi=(1/d)(z^2+1)\phi'(z)\Rightarrow (d/2) (\ln (z^2 +1))'=(\ln \phi)'\Rightarrow
\phi = A(z^2+1)^{d/2}.\]
As $\phi$ is a polynomial, it follows that $d$ is even. Note that $\cS\times (\otimes ^d\mathbf{x})=A(x_0^2+x_1^2)^{d/2}$. Hence $\|\cS\|_{\sigma,\mathbb{R}}=|A|$.
Assume now $\phi$ is not of the form $A(z^2+1)^{d/2}$. Then the arguments of the proof of part (2) yield the equality \eqref{realforspecnrm}.
\end{proof}
We now give the complexity of finding the spectral norm of $\cS\in\rS^d\mathbb{Z}[\bi]^2$ or $\cS\in \rS^d\mathbb{Z}^2$.
\begin{theorem}\label{complexaynqub} Let $\cS\in\rS^d\mathbb{Z}[\bi]^2$, $d>2$ and associate with $\cS$ the vector $\mathbf{s}=(s_0,\ldots,s_d)^\top\in \mathbb{Z}[\bi]^{d+1}$.
Assume that $|s_{i-1}|\le2^{\tau}, i\in[d+1]$ for some $\tau\in\mathbb{N}$. Let $\phi(z)$ be given by \eqref{defphiz}.
Assume the notations of Proposition \ref{fixptHFn=2}. For a given $e\in\mathbb{N}$ we can compute a rational $L(\cS)$ satisfying
\begin{eqnarray*}\label{LcSineqm}
|\|\cS\|_{\sigma,\mathbb{F}}-L(\cS)|\le 2^{-e} \|\cS\|_{\sigma,\mathbb{F}}
\end{eqnarray*}
under the following conditions:
\begin{enumerate}
\item
The tensor $\cS\in \rS^d\mathbb{Z}[\bi]^2$ does not satisfy condition (1) of Theorem \ref{computdqubspecnrm}.
The bit complexity of computation of $L(\cS)$ for $\mathbb{F}=\mathbb{C}$ is $\tilde\cO(d^2(d^4\max(d^2,\tau)+e))$.
\item
The tensor $\cS$ has real integer entries. The bit complexity of computation of $L(\cS)$ for $\mathbb{F}=\mathbb{R}$ is $\hat\cO(d(d^2\max(d,\tau)+e))$.
\end{enumerate}
\end{theorem}
\begin{proof} (1) The assumption that condition (1) of Theorem \ref{computdqubspecnrm} does not hold is equivalent to the assumption that the polynomial $h(z)=zv(z)-u(z)$ is a nontrivial polynomial, whose degree $D$ satisfies $D\le (d-1)^2+1<d^2$. (See part (4) of Proposition \ref{fixptHFn=2}.) Observe that $h(z)=zv(z)-u(z)\in (\mathbb{Z}[\bi])[z]$, and its coefficients are bounded by $2^{2\tau+2d}$.
As in the proof of Theorem \ref{cesnsingcten},
we approximate the roots of $h(z)$ with precision $2^{-(e+k)}$, where $k\in\mathbb{N}$ is specified later. Now we repeat the arguments of Theorem \ref{cesnsingcten} by replacing the characterization \eqref{charcTspecnrm} with \eqref{specnrmSform1}.
This yields the estimate $ |\|\cS\|_{\sigma}-L(\cS)|\le 2^{-e} \|\cS\|_{\sigma}$. The bit complexity estimate $\tilde\cO(d^2(d^4\max(d^2,\tau)+e))$ follows from \cite{NR96}.
(See Appendix 2.)
\noindent (2) We now repeat the arguments of the proof of (1) with the following modifications. Assume that $\phi(z)$ corresponds to $\cS\in \rS^d \mathbb{Z}^n$. Suppose first $\phi(z)=A(z^2+1)^{d/2}$, where $d$ is even. Then $\|\cS\|_{\sigma,\mathbb{R}}=|A|$ and let $L(\cS)=|A|$. Assume now that $\phi(z)\ne A(z^2+1)^{d/2}$. Then $h(z)=zq(z)-p(z)$ is a nonzero polynomial of degree $D\le d$. We approximate the roots of $h(z)$ to approximate the set $R'$ as given in part (3) of Theorem \ref{computdqubspecnrm}.
We use characterization \eqref{realforspecnrm} to compute $L(\cS)$.
\end{proof}
\section{The exceptional cases}\label{sec:excepcase}
\subsection{Analysis of the exceptional cases}\label{sec:analexcas}
In this subsection we discuss part (1) of Theorem \ref{computdqubspecnrm}.
Assume that $g(z)=\bar r(r(z))=z$ identically. Recall that $r(z)$ can be viewed as a holomorphic map of the Riemann sphere. The degree of this map is $\delta\in\mathbb{N}$ since $g$ is not a constant map. Hence the degree of the map $g$ is $\delta^2$. Since $g$ is the identity map, its degree is $1$, it follows that $\delta=1$ and $r(z)$ is a M\"obius map:
\[r(z)=\frac{az+b}{cz+d}, \quad ad-bc\ne 0.\]
Use the formula for $r(z)$ in (\ref{relpqrphi}) to deduce
\[\frac{1}{d}(\log \phi(z))'=\frac{1}{d}\frac{\phi'(z)}{\phi(z)}=\frac{az+b}{cz+d +z(az+b)}.\]
Let $l$ be the number of distinct roots of $\phi(z)$. Then the logarithmic derivative of $\phi(z)$ has exactly $l$ distinct poles. Comparing that with the above formula of the logarithmic derivative of $\phi$, we deduce that $\phi$ has either one (possibly) multiple root or two distinct roots.
First assume that $\phi(z)$ has one root of multiplicity $k$: $\phi(z)=A(z+a)^k$ for $k\in[d]$. Then
\[r(z)=\frac{k}{(d-k)z+d a}, \quad \bar r(z)=\frac{k}{(d-k)z+d \bar a}.\]
In this case $g(z)\equiv z$ if and only if $k\in[d-1]$ and $a=0$.
Clearly, if $\phi(z)=Az^k$, where $A\ne 0$, then $g(z)\equiv z$.
In this case $\cS\times\otimes^d \mathbf{x}=A{d\choose k} x_1^{d-k}x_2^k$. To find $\|\cS\|_{\sigma,\mathbb{F}}$ we need to maximize $|A|{d\choose k}|x_1|^{d-k}|x_2|^k$ subject to
$|x_1|^2+|x_2|^2=1$. The maximum is obtained for
$|x_1|^2=1-\frac{k}{d}, |x_2|^2=\frac{k}{d}$. This proves (\ref{geqidenSform}).
Second assume that $\phi(z)$ has two distinct zeros: $\phi(z)=A(z+a)^p(z+b)^q$, where $a\ne b$, $p,q\in\mathbb{N}$ and $p+q\le d$. Then
\[r(z)=\frac{(z+a)^{p-1}(z+b)^{q-1}\big((p+q)z+pb+qa\big)}{(z+a)^{p-1}(z+b)^{q-1}\big(d(z+a)(z+b)-z\big((p+q)z+pb+qa\big)\big)}.\]
Suppose first that $p+q<d$. In order that $r(z)$ will be a M\"obius transformation
we need to assume that $(p+q)z+pb+qa$ divides $(z+a)(z+b)$. This is impossible,
since $\phi'$ has exactly $p+q-2$ common roots with $\phi$. Hence we are left with the case $p+q=d$. In this case
\begin{eqnarray*}\label{rxfroexcase}
r(z)=\frac{dz+\alpha}{\beta z+dab}, \quad \alpha=pb+qa, \beta=d(a+b)-\alpha, p+q=d.
\end{eqnarray*}
The assumption that $g(z)\equiv z$ is equivalent to the following matrix equality
\begin{eqnarray*}\label{defmatA}
\bar B B=\gamma^2 I_2, \quad B=\left[\begin{array}{cc}d&\alpha\\\beta&dab\end{array}\right] \quad \gamma\ne 0.
\end{eqnarray*}
Taking the determinant of the above identity we deduce that $\gamma^4=|\det B|^2>0$. So $\gamma^2=\pm \tau^{-2}$ for some $\tau>0$.
Let $C=\tau B$. Suppose first that $\gamma^2=\tau^{-2}$. Then $\bar C$ is the inverse of $C$. So $\det C=\delta, |\delta|=1$. Then
\begin{eqnarray*}\label{analeq}
dab=\delta d,\; d=\delta d\bar a\bar b,\;-\alpha=\delta\bar \alpha, -\beta=\delta \bar\beta.
\end{eqnarray*}
Hence $ab=\delta$.
We next observe that if we replace $\phi(z)$ with $\phi_{\theta}(z):=\phi(e^{\theta\bi }z)$ for any $\theta\in\mathbb{R}$ we will obtain the following relations
\[p_{\theta}(z)=e^{\theta \bi}p(e^{\theta \bi}z),\; q_{\theta}(z)=q(e^{\theta \bi}z),\; \overline{p_{\theta}}(z)=e^{-\theta \bi}p(e^{-\theta \bi}z),\; \overline{q_{\theta}}(z)=\bar q(e^{-\theta \bi}z).\]
Let
\[r_{\theta}(z)=\frac{p_\theta(z)}{q_\theta(z)}, \;\overline{r_\theta}(z)=\frac{\overline{p_\theta}(z)}{\overline{q_\theta}(z)},\; g_\theta(z)=\overline{r_\theta}(r_\theta(z)).\]
A straightforward calculation shows that $g_\theta(z)=z$ for all $\theta\in\mathbb{R}$. Note the two roots of $\phi_\theta(z)$ are $-ae^{-\theta\bi}, -be^{-\theta\bi}$. Hence we can choose $\theta$ such that $\delta=-1$. Assume for simplicity of the exposition this condition holds for $\theta=0$, i.e., for $\phi$. So $\alpha$ and $\beta$ are real. In particular $a+b$ is real. So $b=-a^{-1}$ and $a-a^{-1}$ is real. Hence $a$ is real and also $b$ is real.
Suppose now that $\gamma^2=-\tau^2$. Then $\bar B=-B^{-1}$. So $\det B=\delta, |\delta|=1$. Then
\begin{equation}\label{analeq1}
dab=-\delta d,\; d=-\delta d\bar a\bar b,\;\alpha=\delta\bar \alpha, \beta=\delta \bar\beta.
\end{equation}
By considering $\phi_\theta$ as above we may assume that $\delta=1$ which gives again that $a\in\mathbb{R}\setminus\{0\}$ and $b=-a^{-1}$. This proves (\ref{geqiden1}).
Vice versa, assume that $\phi(z)$ is of the form (\ref{geqiden1}), where $a\in\mathbb{R}\setminus\{0\}$ and $b=-\frac{1}{a}$. We claim that $\bar r(r(z))\equiv z$.
The above arguments show that $r(z)=(dz+\alpha)/(\beta z+dab)$. Furthermore
\begin{eqnarray*}\label{excforalphbet}
ab=-1,\;\alpha=(d-p)a -\frac{p}{a}, \beta=pa -\frac{d-p}{a}, \; p\in[d-1].
\end{eqnarray*}
Consider the matrix $B$ as given above. Note that the trace of this matrix is zero. We claim that $B$ is not singular. Indeed
\begin{eqnarray*}\notag
&&\det B=-(d^2 +\alpha\beta)=-(d^2 - (d-p)^2 -p^2 +p(d-p)(a^2+a^{-2}))=\\
&&-p(d-p)(a+a^{-1})^2.\label{formdetA}
\end{eqnarray*}
Hence $B$ has two distinct eigenvalues $\{\gamma,-\gamma\}$ and is diagonalizable.
As $B$ is a real matrix we get that $\bar B B= B^2=\gamma^2 I_2$. Therefore
$\bar r(r(z))\equiv z$. As $\bar r_\theta(r_\theta(z))\equiv z$ for each $s\in\mathbb{R}$ we deduce that for each $\phi$ of the form (\ref{geqiden1}) $zq(z)-p(z)$ is a zero polynomial.
\subsection{Computation of $\|\cS\|_{\sigma}$ in the exceptional cases}\label{sec:compexcas}
Assume now that $\rS_\theta\in\rS^d\mathbb{C}^2$ is induced by $\phi$ of the form (\ref{geqiden1}). Then
\begin{eqnarray*}\label{formFxcom}
\cS_{\theta}\times\otimes^d\mathbf{x}=A(x_2+ce^{-\theta\bi}x_1)^p(x_2-c^{-1}e^{-\theta\bi}x_1)^{d-p}
, \; c\in\mathbb{R}\setminus\{0\},\theta\in\mathbb{R},p\in[d-1],
\end{eqnarray*}
Clearly, $\|{\cS}_{\theta}\|_{\sigma}=\|\cS_0\|_{\sigma}$. Thus it is enough to find $\|\cS_0\|_{\sigma}$. For simplicity of notation we let $\cS=\cS_0$.
We now suggest the following simple approximations to find $\|\cS\|_{\sigma}$ using the case (2) of Theorem \ref{computdqubspecnrm}. It is enough to assume that $A=1$, i.e., $\cS$ corresponds to $\phi(z)=(z+c)^p(z-c^{-1})^{d-p}$.
Let $\omega$ be a rational in the interval $(0,1)$. Set $\phi(z,\omega)=\phi(z)+\omega$.
Let $\cS(\omega)\in\rS^d\mathbb{R}^2$ be the symmetric tensor corresponding to $\phi(z,\omega)$. Then $S(\omega)\times(\otimes^d \mathbf{x})=(x_2+cx_1)^p(x_2-c^{-1}x_1)^{p-d}+\omega x_1^d$.
By choosing $\mathbf{x}=(0,1)^\top$ we deduce that $\|\cS(\omega)\|_{\sigma}\ge 1$. It is straightforward to show that
$\|\cS(\omega)-\cS\|_{\sigma}= \omega$. As $\omega\le \omega\|\cS(\tau)\|_{\sigma}$ we obtain
\[\|\cS\|_{\sigma}\in [\|\cS(\omega)\|_{\sigma}(1-\omega),\|\cS(\omega)\|_{\sigma}(1+\omega)].\]
Observe next that $\phi'(z,\omega)=\phi'(z)$. Hence a common root of $\phi(z,\omega)$ and
$\phi'(z,\omega)$ can not be a common root of $\phi(z)$ and $\phi'(z)$. Therefore
$\phi(z,\omega)$ and $\phi'(z,\omega)$ can have at most one common root. As $d\ge 3$ we deduce that
$\cS(\omega)$ does not satisfy the assumptions of part (2) of Theorem \ref{computdqubspecnrm}.
Assume that $c\in \mathbb{Q}$.
Let $\delta$ be rational in the interval $(0,1)$ and assume that $\omega=\delta/4$. Use the case (1) of Theorem \ref{complexaynqub} to find an approximation $L(\cS(\omega))$ that satisfies $|\|\cS(\omega)\|_{\sigma}-L(\cS(\omega))|\le (\delta/4)\|\cS(\omega)\|$. Let us take $L(\cS(\omega))$ to be an approximation for $\|\cS\|$. Then
\begin{eqnarray*}
|\|\cS\|_{\sigma} - L(\cS(\omega))|\le| \|\cS\|_{\sigma} - \|\cS(\omega)\|_{\sigma}|+|\|\cS(\omega)\|_{\sigma} - L(\cS(\omega))|\le (\omega +\delta/4)\|\cS(\omega)\|_{\sigma}\le \frac{\delta}{2}\frac{1}{1-\delta/4}\|\cS\|_{\sigma}<\delta \|\cS\|_{\sigma}.
\end{eqnarray*}
Thus we found a rational approximation of $\|\cS\|_{\sigma}$ that satisfies the inequality of Theorem \ref{complexaynqub} for $\mathbb{F}=\mathbb{C}$. The complexity of $L(\cS(\omega))$ is given in case (1) of
Theorem \ref{complexaynqub}.
\section{Numerical examples}\label{sec:numerex}
In this section we give some numerical examples of applications of
Theorem \ref{computdqubspecnrm} for the qubit cases, and of Theorem \ref{theofoundthm} for $n>2$.
In many examples here $f(\mathbf{x})$ is a sum of two monomials. In this case we show that to compute the complex spectral norm of $f$ one can assume that the coefficients of the two monomials
are nonnegative. Equivalently, we can assume the symmetric tensor $\cS$ has nonnegative entries.
Hence $\|\cS\|_{\sigma}=\|\cS\|_{\sigma,\mathbb{R}}=\|f\|_{\sigma}$ \cite{FL18}.
However, if $f$ has real coefficients then it may happen that $\|f\|_{\sigma,\mathbb{R}}<\|f\|_{\sigma}$. See Example 2 below.
\begin{lemma}\label{sum2mon} Let $f\in\rP(d,n,\mathbb{C})$, where $d,n\ge 2$, and assume that $f=a\mathbf{x}^{\bj}+b\mathbf{x}^{\bk}$, where $\bj\ne \bk$. Denote $g(\mathbf{x})=|a|\mathbf{x}^{\bj}+|b|\mathbf{x}^{\bk}$.
Then $\|f\|_{\sigma}=\|g\|_{\sigma}=\|g\|_{\sigma,\mathbb{R}}=g(\mathbf{y})$ for some $\mathbf{y}\ge \mathbf{0}, \|\mathbf{y}\|=1$.
\end{lemma}
\begin{proof} Clearly, it is enough to assume that $a\ne 0$ and $b\ne 0$. Let $\mathbf{x}=(x_1,\ldots,x_n)^\top$ and denote $\mathbf{x}_+=(|x_1|,\ldots,|x_n|)^\top$. Note that $\|\mathbf{x}\|=\|\mathbf{x}_+\|$.
Assume that $\|\mathbf{x}\|=1$. As $|f(\mathbf{x})|,|g(\mathbf{x})|\le g(\mathbf{x}_+)$ it follows that $\|f\|_{\sigma}\le \|g\|_{\sigma}=g(\mathbf{y})$, for some $\mathbf{y}\ge 0$ and $\|\mathbf{y}\|=1$. Hence $\|g\|_{\sigma}=\|g\|_{\sigma,\mathbb{R}}$.
Observe next that $f(\mathbf{x})=\mathbf{x}^{\bl}(a\mathbf{x}^{\bj'}+b\mathbf{x}^{\bk'})$. We can choose $\bl\ge 0$ so that the monomials $\mathbf{x}^{\bj'}$ and $\mathbf{x}^{\bk'}$ do not have a common variable.
Assume that $\mathbf{z}=(z_1,\ldots,z_n)^\top$ such that $\mathbf{z}_+=\mathbf{y}$. Then it is possible to choose the arguments of $z_1,\ldots,z_n$ such that $|a\mathbf{z}^{\bj'}+b\mathbf{z}^{\bk'}|=|a|\mathbf{y}^{\bj'}+|b|\mathbf{y}^{\bk'}$. Hence $|f(\mathbf{z})|=g(\mathbf{y})=\|g\|_{\sigma}$. Therefore $\|f\|_{\sigma}=\|g\|_{\sigma}$.
\end{proof}
In some of the examples we discuss, we modified the examples of $f$, that satisfy conditions of Lemma \ref{sum2mon}, by considering $f_e$:
\begin{eqnarray}\label{ftform}
&&f_e(\mathbf{x})=\sqrt{1-|e|^2}f(\mathbf{x}) + eh(\mathbf{x}), \quad e=t\omega,\\
&&t=0,1/5,1/4/,1/3,1/2,1,\quad \omega=1,-1,i, 1/2 +i\frac{\sqrt{3}}{2},-1/2 +i\frac{\sqrt{3}}{2},\notag\\
&&f=a\mathbf{x}^{\bj}+b\mathbf{x}^{\bk}, \; h=c\mathbf{x}^{\bl},\; \bj\ne \bk, \bj\ne \bl,\bk\ne \bl,\; a,b,c>0, \|f\|=\|h\|=1.\notag
\end{eqnarray}
Note that $\|f_e\|=1$, and $f_0=f, f_1=h$. Hence, when we give our results for $\|f_e\|_{\sigma}$ we give separately the values of $\|f\|_{\sigma},\|h\|_{\sigma}$, and $\|f_e\|_{\sigma}$ for $t=1/5,1/4/,1/3,1/2$ and the above values of $\omega$.
First we discuss the symmetric $d$-qubits. If the
$d$-qubits are real we compute both the complex and real spectral norms of a given tensor $\cS$.
In Theorem \ref{computdqubspecnrm}, we showed that the spectral norm of a given $d$-qubit can be found by solving the polynomial equation $zv(z)-u(z) =0$ (see (\ref{poleqfixpoint})), which is of degree of at most $(d-1)^2+1$. In what follows that we assume that the polynomial $zv(z)-u(z)$ is not a zero polynomial. (That is, we are not dealing with the exceptional cases that are discussed in \S\ref{sec:excepcase}.) We use formula (\ref{specnrmSform1}) to find $\|\cS\|_{\sigma}$, and formula (\ref{realforspecnrm}) to find $\|\cS\|_{\sigma,\mathbb{R}}$.
For $n>2$ we use Theorem \ref{theofoundthm} to compute $\|f\|_{\sigma}$. Thus we need to assume that $\mathrm{fix}(\bH)$ is finite, or if all the monomials of $f$ have nonnegative coefficents then the set of the real fixed points of $\bF$ is finite.
In this paper we use Bertini \cite{BHSW06} (version 1.5, released in 2015), which is a well developed software to find all the complex solutions of a given polynomail system. It is worth noting that, in contrast to the theoretical results of this paper, the output of Bertini is not certified and may be incorrect by an unbounded error if the software jumps between solutions when tracking the homotopy paths.
All the computation are implemented with Matlab R2018b on a MacBook Pro 64-bit OS X (10.12.6) system with 16GB memory and 2.9 GHz Intel Core i7 CPU. In the display of the computational results, only four decimal digits are shown. The default parameters in Bertini are used to solve the polynomial equation $zv(z)-u(z) =0$.
Since all examples only take few seconds we will not show the computing time.
In our examples all polynomials correspond to the symmetric tensors $\cS\in \rS^d\mathbb{C}^n$ of Hilbert-Schmidt norm one. Furthermore, if $\cS$ has nonnegative entries then $\|\cS\|_{\sigma}=\|\cS\|_{\sigma,\mathbb{R}}$ \cite{FL18}.
Assume that $d=2$. Then $|s_d|=|\cS_{2,\ldots,2}|$, and if $\cS$ has nonnegative entries then one can use part (2) of Theorem \ref{computdqubspecnrm}.
In these examples we find the degree of the polynomial $zv(z)-u(z)$, the number of real and complex roots, and the number of roots that fail to satisfy (\ref{antifixqub1}).
\subsection{Three examples of symmetric $3$-qubits}\label{subsec:first3q}
These examples are interesting to us, and some of them are discussed in other papers.
\setcounter{theorem}{0}
\begin{example}\emph{\cite{Nie14}} \label{exm:1:d:qubit} Let
$f=0.3104 x_1^3 - 1.4598x_1^2x_2 - 0.6558 x_1 x_2^2+0.2235 x_2^3$.
The polynomial $zv(z)-u(z)$ has degree $5=2^2+1$. It has 5 roots, 3 of them are real and the other 2 are complex. We have $R=R_1$, $R'=R_1'$. Then
\[ \|f\|_{\sigma~~} \approx 0.7027,\quad
\|f\|_{\sigma,\mathbb{R}} \approx 0.6205.\]
\end{example}
\begin{example} \emph{\cite{FL18}} \label{exm:1:B:d:qubit} Let $f=\frac{3}{2}x_1^2x_2- \frac{1}{2} x_2^3$.
The polynomial $zv(z)-u(z)$ has degree 4. It has 4 roots, 2 of them are real and the other 2 are complex. We have $R=R_1$, $R'=R_1'$. Then
\[\|f\|_{\sigma~~}\approx 0.7071, \quad \|f\|_{\sigma,\mathbb{R}}=0.5.\]
\end{example}
\begin{example}\label{exm:2:d:qubit}
Let $f=\frac{1}{\sqrt{5}}x_1^3-\frac{3}{2\sqrt{5}}x_1^2x_3-\frac{3}{\sqrt{5}}x_1 x_2^3+ \frac{1}{2\sqrt{5}}x_2^3$.
The polynomial $zv(z)-u(z)$ has degree 5. It has 5 roots, 3 of them are real and the other 2 are complex. We have $R=R_1$, $R'=R_1'$. Then
\[ \|f\|_{\sigma~~} \approx 0.7071, \quad \|f\|_{\sigma,\mathbb{R}} \approx 0.5000.\]
\end{example}
\subsection{Five examples from \cite{AMM10} and their variations}\label{subsec:AMM10}
In \cite{AMM10} the authors give examples of $d$-symmetric qubits for $d=4,\ldots,12$, which they assume to have the lowest complex spectral norm. Their examples are motivated by the Majorana model, see Appendix 3. (Note that some examples have at least two versions (a) and (b).) Our software gave the same values of the spectral norms for the examples in \cite{AMM10}. We could not find with our software examples of symmetric $d$-qubits with lower complex spectral norm.
In the following examples we find the spectral norm of $f_e$ of the form \eqref{ftform}, where $f$ is the polynomial given in \cite{AMM10}. The polynomial $h$ corresponds to a monomial given by Dicke basis with the lowest spectral norm. (See Appendix 1.)
\begin{example}\emph{\cite[Corresponds to example 6.1]{AMM10}} \label{exm:9:d:qubit} Let $f= \frac{1}{\sqrt{3}}x_1^4+\frac{\sqrt{8}}{\sqrt{3}}x_1x_2^3$.
The polynomial $zv(z)-u(z)$ has degree $10=3^2+1$. It has 10 roots, 4 of them are real and the other 6 are complex. We have $R= R_1$, $R'=R_1'$. Then
$\|f\|_{\sigma~~} \approx 0.5774$. According to \cite{AMM10}, $\|f\|_{\sigma}=\frac{1}{\sqrt{3}}$.
Let $h= \sqrt{6}x_1^2x_2^2$. Then $\|h\|_{\sigma}=\frac{\sqrt{3}}{\sqrt{8}}\approx 0.6124$. (See \eqref{specnrmSj1jn}.) Table 1 gives the results for $\|f_e\|_{\sigma}$:
\begin{table*}[htb]
\centering
\begin{scriptsize}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
$\|f_e\|_{\sigma}$ & $t= \frac{1}{5}$ & $t= \frac{1}{4}$ & $t= \frac{1}{3}$ & $t= \frac{1}{2}$ \\ \hline
$\omega = 1$ & 0.6787 & 0.7012 & 0.7358 & 0.7918 \\ \hline
$\omega = -1$ & 0.6314 & 0.6442 & 0.6645& 0.6989 \\ \hline
$\omega = i$ & 0.6662 & 0.6863 & 0.7172 & 0.7676 \\ \hline
$\omega = \frac{1}{2}+\frac{\sqrt{3}}{2}i$ & 0.6314 & 0.6442 &0.6645 & 0.6989 \\ \hline
$\omega = -\frac{1}{2}+\frac{\sqrt{3}}{2}i$ & 0.6787 &0.7012 & 0.7358 & 0.7918 \\ \hline
\end{tabular}
\end{scriptsize}\caption{Computational Results of $\|f_e\|_{\sigma}$ for Example \ref{exm:9:d:qubit} with different $t$ and $\omega$.} \label{different:ab:results:qubit:order4}
\end{table*}
\end{example}
\begin{example}\emph{\cite[Corresponds to example 6.2(b), figure 5(b)]{AMM10}} \label{exm:10:d:qubit} Let $f=\frac{1}{\sqrt{1+A^2}} x_1^5 +\frac{\sqrt{5}A}{\sqrt{(1+A^2)}}x_1x_2^4$, where $A \approx 1.53154$.
The polynomial $zv(z)-u(z)$ has degree $17=4^2+1$. It has 17 roots, 5 of them are real and the other 12 are complex. Four roots do not satisfy (\ref{antifixqub1}). Furthermore $R'=R_1'$. Then $\|f\|_{\sigma~~} =0.5467$.
According to \cite{AMM10}, $\|f\|_{\sigma}=\frac{1}{\sqrt{1+A^2}}$.
Let $h=\sqrt{10}x_1^3x_2^2$. Then $\|h\|_{\sigma}=\frac{6\sqrt{6}}{25}\approx 0.5879$. (See \eqref{specnrmSj1jn}.) Here is the table for $\|f_e\|_{\sigma}$:
\begin{table*}[htb]
\centering
\begin{scriptsize}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
$\|f_e\|_{\sigma}$ & $t= \frac{1}{5}$ & $t= \frac{1}{4}$ & $t= \frac{1}{3}$ & $t= \frac{1}{2}$ \\ \hline
$\omega = 1$ & 0.5930&0.6038 &0.6214 & 0.6573 \\ \hline
$\omega = -1$ & 0.5930 &0.6038 &0.6214& 0.6573 \\ \hline
$\omega = i$ & 0.5622 &0.5692 &0.5793 & 0.5941 \\ \hline
$\omega = \frac{1}{2}+\frac{\sqrt{3}}{2}i$ & 0.5759 &0.5830 & 0.5941 & 0.6133 \\ \hline
$\omega = -\frac{1}{2}+\frac{\sqrt{3}}{2}i$ & 0.5759 &0.5830 &0.5941 & 0.6133 \\ \hline
\end{tabular}
\end{scriptsize}\caption{Computational Results of $\|f_e\|_{\sigma}$ for Example \ref{exm:10:d:qubit} with different $t$ and $\omega$.} \label{different:ab:results:qubit:order5}
\end{table*}
\end{example}
\begin{example}\emph{\cite[Corresponds to example 6.3]{AMM10}} \label{exm:11:d:qubit} Let $f=\sqrt{3}(x_1^5 x_2 + x_1 x_2^5)$.
The polynomial $zv(z)-u(z)$ has degree $25<5^2+1$. It has 25 roots, 7 of them are real and the other 18 are complex. For the 25 roots $z$, (\ref{antifixqub1}) fails to hold for 5 roots. Six of the seven real roots satisfy $zq(z)-p(z)=0$. Then
$\|f\|_{\sigma~~} =0.4714$.
According to \cite{AMM10}, $\|f\|_{\sigma}=\frac{\sqrt{2}}{3}$. Let $h=\sqrt{20}x_1^3x_2^3$. Then $\|h\|_{\sigma}=\frac{\sqrt{5}}{4}\approx 0.5590$. (See \eqref{specnrmSj1jn}.) Here is the table for $\|f_e\|_{\sigma}$:
\begin{table*}[htb]
\centering
\begin{scriptsize}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
$\|f_e\|_{\sigma}$ & $t= \frac{1}{5}$ & $t= \frac{1}{4}$ & $t= \frac{1}{3}$ & $t= \frac{1}{2}$ \\ \hline
$\omega = 1$ & 0.5382 &0.5590 &0.5946 &0.6545 \\ \hline
$\omega = -1$ & 0.5382 &0.5590 & 0.5946 &0.6545 \\ \hline
$\omega = i$ &0.4777 &0.4811 &0.4886 & 0.5076 \\ \hline
$\omega = \frac{1}{2}+\frac{\sqrt{3}}{2}i$ & 0.5054 &0.5148 &0.5312 & 0.5688\\ \hline
$\omega = -\frac{1}{2}+\frac{\sqrt{3}}{2}i$ & 0.5054 &0.5148 & 0.5312 & 0.5688\\ \hline
\end{tabular}
\end{scriptsize}\caption{Computational Results of $\|f_e\|_{\sigma}$ for Example \ref{exm:11:d:qubit} with different $t$ and $\omega$.} \label{different:ab:results:qubit:order6}
\end{table*}
\end{example}
\begin{example}\emph{\cite[Corresponds to example 6.4]{AMM10}} \label{exm:12:d:qubit} Let $f=\frac{\sqrt{7}}{\sqrt{2}}(x_1^6x_2+x_1x_2^6)$.
The polynomial $zv(z)-u(z)$ has degree $36<6^2+1$. It has 36 roots, 6 of them are real and the other 30 are complex. For the 36 roots $z$, (\ref{antifixqub1}) fails to hold for 11 roots. Five of the six real roots satisfy $zq(z)-p(z)=0$. Then
$\|f\|_{\sigma~~} =0.4508$. Let $h=\sqrt{35}x_1^4x_2^3$.
Then $\|h\|_{\sigma}=\frac{48\sqrt{15}}{7^3}\approx 0.5420$. (See \eqref{specnrmSj1jn}.)
Here is the table for $\|f_e\|_{\sigma}$:
\begin{table*}[htb]
\centering
\begin{scriptsize}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
$\|f_e\|_{\sigma}$ & $t= \frac{1}{5}$ & $t= \frac{1}{4}$ & $t= \frac{1}{3}$ & $t= \frac{1}{2}$ \\ \hline
$\omega = 1$ & 0.5006 & 0.5131 &0.5346 & 0.5796\\ \hline
$\omega = -1$ & 0.4939 &0.5048 & 0.5229 & 0.5597 \\ \hline
$\omega = i$ & 0.4988 &0.5109 & 0.5314 & 0.5742 \\ \hline
$\omega = \frac{1}{2}+\frac{\sqrt{3}}{2}i$ & 0.4998 & 0.5121 &0.5332 & 0.5772 \\ \hline
$\omega = -\frac{1}{2}+\frac{\sqrt{3}}{2}i$ &0.4975 &0.5092 &0.5291 & 0.5703 \\ \hline
\end{tabular}
\end{scriptsize}\caption{Computational Results of $\|f_e\|_{\sigma}$ for Example \ref{exm:12:d:qubit} with different $t$ and $\omega$.} \label{different:ab:results:qubit:order7}
\end{table*}
\end{example}
\begin{example} \emph{\cite[Corresponds to example 6.5]{AMM10}}
\label{exm:13:d:qubit} Let $f=4(0.336\sqrt{2}x_1^7x_2+0.3705\sqrt{7}x_1^2x_2^6)$.
The polynomial $zv(z)-u(z)$ has degree $42<7^2+1$, which has 41 roots, 7 of them are real and the other 34 are complex. One of the real roots $z=0$ has multiplicity 2. For the 41 roots $z$, (\ref{antifixqub1}) fails to hold for 10 roots. Then
$\|f\|_{\sigma~~} =0.4288$. Let $h=\sqrt{70}x_1^4x_2^4$.
Then $\|h\|_{\sigma}=\frac{\sqrt{70}}{16}\approx 0.5229$. (See \eqref{specnrmSj1jn}.)
Here is the table for $\|f_e\|_{\sigma}$:
\begin{table*}[htb]
\centering
\begin{scriptsize}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
$\|f_e\|_{\sigma}$ & $t= \frac{1}{5}$ & $t= \frac{1}{4}$ & $t= \frac{1}{3}$ & $t= \frac{1}{2}$ \\ \hline
$\omega = 1$ & 0.4946 & 0.5108&0.5374& 0.5867\\ \hline
$\omega = -1$ & 0.4841 &0.4979 &0.5206 & 0.5630 \\ \hline
$\omega = i$ & 0.4919 & 0.5075 & 0.5330& 0.5806 \\ \hline
$\omega = \frac{1}{2}+\frac{\sqrt{3}}{2}i$ & 0.4934 &0.5093 &0.5354 & 0.5840\\ \hline
$\omega = -\frac{1}{2}+\frac{\sqrt{3}}{2}i$ & 0.4898 &0.5049 & 0.5297 & 0.5759\\ \hline
\end{tabular}
\end{scriptsize}\caption{Computational Results of $\|f_e\|_{\sigma}$ for Example \ref{exm:13:d:qubit} with different $t$ and $\omega$.} \label{different:ab:results:qubit:order8}
\end{table*}
\end{example}
More details on the above examples and additional examples for $n=2$ are given in \cite{FW16}.
\subsection{A family of symmetric $3$-qutrits}\label{exsym3qtr} Let us consider the following family of symmetric $3$-qutrits.
\begin{eqnarray*}
f_{a,b}(x_1,x_2,x_3)=a(x_1^3+x_2^3+x_3^3)+bx_1x_2x_3, \quad 3|a|^2+|b|^2/6=1.
\end{eqnarray*}
This example is inspired by \cite[Example (1)]{CGL99}. It is straightforward to show
that
$$\|f_{1/\sqrt{3},0}\|_{\sigma}=\|f_{1/\sqrt{3},0}\|_{\sigma,\mathbb{R}}=\frac{1}{\sqrt{3}}\approx 0.5774, \quad \|f_{0,\sqrt{6}}\|_{\sigma}=\|f_{0,\sqrt{6}}\|_{\sigma,\mathbb{R}}=\frac{\sqrt{2}}{3}=0.4714.$$
Note that $f_{0,\sqrt{6}}$ corresponds to the most entangled Dicke basis $\cS(3,3)$ \eqref{defSdn}. Observe that $\|f_{a,b}\|_{\sigma}=\|f_{\bar a,\bar b}\|_{\sigma}=\|f_{\zeta a,\zeta b}\|_{\sigma}$ for $|\zeta|=1$. Here is the table for $\|f_{a,b}\|_{\sigma}$:
\begin{table*}[htb]
\centering
\begin{scriptsize}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
$a$ & $b$ & No. of real fixed points & No. of complex fixed points & $\|f_{a,b}\|_{\sigma,\mathbb{R}}$ & $\|f_{a,b}\|_{\sigma}$ \\ \hline
$\frac{1}{3}$ & 2 & 8& 56& 0.5774 & 0.5774 \\ \hline
$\frac{1}{2}$ & $\sqrt{\frac{3}{2}}$& 8& 56&0.5244&0.5244\\ \hline
$\frac{1}{3}$ & -2 & 8& 56 &0.4975& 0.5092\\ \hline
$\frac{1}{2}$ & -$\sqrt{\frac{3}{2}}$& 8& 56 &0.5000 &0.5000\\ \hline
$0$ & $\sqrt{6}$ & 5& 50& 0.4714 & 0.4714 \\ \hline
$\frac{1}{\sqrt{3}}$ & 0& 8& 56 &0.5774& 0.5774\\ \hline
$\frac{1}{6}+\frac{\sqrt{3}}{6}i$& $\sqrt{2}-\sqrt{2}i$ & 1 &63 &-&0.5730 \\ \hline
$\frac{1}{4}+\frac{\sqrt{3}}{4}i$& $\frac{\sqrt{6}}{4}+\frac{3\sqrt{2}}{4}i$ & 1 &63 & -& 0.5244\\ \hline
\end{tabular}
\end{scriptsize}\caption{Computational Results for \ref{exsym3qtr} with different $a$ and $b$.} \label{tensor:computation:results:exm:ab:real}
\end{table*}
Let us consider the real case. For different real $a$ and $b$, we use Bertini to solve the equation $F(y) = y$, and get the real spectral norm of tensor $f_{a,b}$. Results are shown in the Table \ref{tensor:computation:results:exm:ab:real}. As we see, the number of real fixed point is $8=(3-1)^3$ the maximum possible as given by part (3) of Theorem \ref{theofoundthm}, except for $f_{0,\sqrt{6}}$ corresponding to $\cS(3,3)$.
Thus, among all these examples the Dicke state $\cS(3,3)$ is the most entangled one.
\subsection{A family of symmetric $4$-ququadrits}\label{exsym4qtr}
Recall that most entangled $3$-qubit is $\cW\in\rS^3\mathbb{C}^2$, which corresponds to $f=\sqrt{3}x_1^2x_2$ \cite{TWP09,CXZ10}. (It is the Dicke basis element $\cS(3,2)$.)
It is of interest to consider the tensor product state $\cW\otimes \cW\in \otimes^6\mathbb{C}^2$ \cite{CCDW,CF18}. Note that this state is not symmetric. Lemma 2.3 in \cite{DFLW17} yields
$$\|\cW\otimes\cW\|_{\sigma}=\|\cW\|_{\sigma}^2=\|\cS(3,2)\|^2=(\frac{2}{3})^2=\frac{4}{9}\approx 0.4444.$$
It is possible to represent $\cW\otimes\cW$ as a symmetric tensor $\cS\in \rS^3\mathbb{C}^4$.
It is represented by polynomial $f=x_1^2 x_4+2x_1x_2x_3$ \cite{CCDW}.
Let us consider $f_{a,b}=ax_1^2 x_4+2bx_1x_2x_3$,
where $|a|^2+2|b|^2 = 3$. (So $\|f_{a,b}\|=1$.) Note that $f=f_{1,1}$. In Table \ref{tensor:order3:n4:exm:results:ab}, we show $\|f_{a,b}\|_{\sigma}$ for some values of $a,b$.
\begin{table*}[htb]
\centering
\begin{scriptsize}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline
$a$ & $b$ & $\|f_{a,b}\|_{\sigma}$ \\ \hline
1 & 1 & 0.4444 \\ \hline
$\frac{\sqrt{2}}{2}$& $\frac{\sqrt{5}}{2}$ & 0.4536\\ \hline
$\frac{\sqrt{5}}{2}$ & $\sqrt{\frac{7}{8}}$& 0.4491 \\ \hline
$\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$ & $\frac{1}{2}+\frac{\sqrt{3}}{2}i$ & 0.4444 \\ \hline
$\frac{\sqrt{2}}{2}(\frac{1}{2}-\frac{\sqrt{3}}{2}i)$ &$\frac{\sqrt{5}}{2}(\frac{\sqrt{3}}{4}+\frac{\sqrt{13}}{4}i)$ & 0.4536\\ \hline
$\frac{\sqrt{6}}{2}(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)$ & $\frac{\sqrt{3}}{2}(\frac{\sqrt{3}}{2}+\frac{1}{2}i)$ & 0.4714\\ \hline
\end{tabular}
\end{scriptsize}\caption{Computational Results for Example \ref{exsym4qtr} with different $a$ and $b$.} \label{tensor:order3:n4:exm:results:ab}
\end{table*}
These computations point out that probably $\|\cS\|=\|f\|_{\sigma}$ is equal to $\|\cW\otimes\cW\|_{\sigma}$. This contrast with the results in \cite{CCDW,CF18} that $\mathrm{rank }\; \cS=7<\mathrm{rank }\;\cW\otimes\cW=8 <(\mathrm{rank }\;\cW)^2=9$. (Recall that the rank of a tensor $\cT$ is the minimal number of summands in a representation of
$\cT$ as a sum of rank one tensors.) Next observe that $f$ does not have the minimal complex spectral norm in the above examples. Finally, $\|f\|_{\sigma}<\|\cS(3,4)\|=\frac{\sqrt{2}}{3}$.\\
\emph{Acknowledgement}: We thank the two referees for their useful remarks and comments.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,434 |
It's a Love Cult è il decimo album in studio della band norvegese Motorpsycho.
Tracce
#1, #2, #4, #7, #9 di Sæther.
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#6 di Gebhardt.
Arrangiamenti orchestrali sulla traccia #5, #7 di B. Slagsvold.
Arrangiamenti dei fiati sulla traccia #6 di L. Horntveth, H. Gebhardt.
Formazione
Bent Sæther: basso, voce, chitarra, piano, harmonium, Mellotron, percussioni, organo Viscount
Hans Magnus Ryan: chitarre, voce, piano Rhodes, ARP, sidstation, harmonium elettrico, organo Viscount, percussioni, lap-steel
Håkon Gebhardt: batteria, voce, banjo, percussioni, zither, chitarre, glockenspiel, lap-steel
con:
Helge Sten (Deathprod): audio virus, Echoplex, filtri, Theremin, percussioni
Baard Slagsvold: piano, voce, Mellotron, Clavinette, organo Hammond
e:
Archi suonati da: Øyvind Fossheim, Vegard Johnsen, André Orvik, Hans Morten Stensland, Jon W. Sønstebø e Anne Britt Søvig Årdal
Corni e fiati suonati da: Ketil Vestrum Einarsen, Lars Horntveth, Anne-Grethe Orvik, Øyvind Brække, Mathias Eick.
Collegamenti esterni | {
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Back Lift in Troy | Body Lift Costs near Bloomfield Hills & Rochester Hills Ellen A. Janetzke, M.D.
1.3 Why Is A Back Lift Necessary?
1.4 How Is A Back Lift Performed?
1.5 How Will I Care For My Body After A Back Lift If Driving In From Detroit, MI ?
1.6 Will I Have Scars From My Back Lift Surgery In Birmingham, MI?
Candidates are healthy men or women who have back fat associated with aging, sun damage, or weight loss. If you are dealing with pockets of sagging fat on the back or around the "bra line", a back lift may be able to help.
Why Is A Back Lift Necessary?
Though you may first notice loose skin on the front of your body especially after losing a large amount of weight, your back may exhibit similar issues. Just like your thighs and abdomen, your back has experienced weight loss and if gravity has also played a role, it may now appear loose. This can result in skin bunching around your bra area and above your waistline—the much-despised "back rolls." For men and women, a back lift can help to offer a more proportionate body silhouette.
How Is A Back Lift Performed?
A back lift surgery in Birmingham, MI is an outpatient procedure and is completed in a similar fashion to a tummy tuck. The procedure usually takes 1-2 hours to complete, and general anesthesia will be used. Most often, Dr. Ellen will make two careful incisions across the center of your back, near the bra line. The loose skin between these two incisions will be removed. When the intended result is achieved, Dr. Ellen will close the incisions with sutures and cover them with a surgical dressing. In some cases, Dr. Ellen may also use liposuction to remove fat from the area.
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When you leave her practice, Dr. Ellen will ask you to wear a compression garment, which will work to reduce swelling and maintain the new contours of your back. Your activity will need to be limited after surgery, but you will be able to walk. Any swelling you experience will subside and discomfort can be managed with medication. Recovery time will differ from patient to patient, though typically lasts about two weeks.
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As with any surgery, some scarring will be present. The visibility of scarring will vary by patient, however. In the case of a back lift, Dr. Ellen will take care to hide the incisions within the natural contours of your body and under your bra strap area. Your scars will fade over time and you should be able to conceal them easily. After you heal from your back lift, you should be able to proudly show off your body in clothing you previously struggled with. Many men and women also experience a boost in overall confidence following their back lift procedure.
If you are looking for a board certified plastic surgeon serving the Detroit, Rochester Hills, Sterling Heights and Troy areas, give Dr. Ellen a call to schedule your consultation. Birmingham plastic surgeon Dr. Ellen is one of the best Detroit plastic surgeons to help you tone and slenderize your physique. Dr. Ellen and her staff are eager to meet with you to discuss your body contouring options. Conveniently located just 6 miles North of Royal Oak and 10 miles East of West Bloomfield, Dr. Ellen serves South East Michigan. We encourage you to email us or call our office in Birmingham at 248-220-6760. | {
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{"url":"https:\/\/socratic.org\/questions\/what-is-the-molarity-of-a-solution-produced-by-dissolving-4-moles-of-sugar-in-2-","text":"# What is the molarity of a solution produced by dissolving 4 moles of sugar in 2 L of water?\n\n$M = \\frac{{n}_{2}}{{V}_{1}} = \\frac{4}{2} = 2 M = 2 m o l {L}^{-} 1$\n$M = \\frac{{n}_{2}}{{V}_{1}} =$ (number of moles of the solute{mol} )$\/$(volume of the solvent {L})\n$M = \\frac{{n}_{2}}{{V}_{1}} = \\frac{4}{2} = 2 M = 2 m o l {L}^{-} 1$","date":"2019-09-21 17:13:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 4, \"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\": 0.43038642406463623, \"perplexity\": 4324.360059067247}, \"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-2019-39\/segments\/1568514574588.96\/warc\/CC-MAIN-20190921170434-20190921192434-00019.warc.gz\"}"} | null | null |
The Emperor's New Clothes announced as next BSL integrated and Captioned show for Easter 2022
Derby Theatre announces its next fully accessible production perfect for the whole family
We are delighted to announce our next BSL integrated and Captioned production, The Emperor's New Clothes, on from Friday 1 – Saturday 16 April 2022.
The Emperor's New Clothes will be brought vividly to life with a wonderful cast of actors including musicians, hearing, Deaf and interpreting actors.
It will follow on from our BSL integrated and Captioned family productions, the highly acclaimed The Jungle Book in 2019 and our current, smash-hit Christmas production of Treasure Island.
The show will be a co-production between Derby Theatre, Hiccup Theatre and Polka Theatre.
BSL Intro
The Emperor likes nothing more than a fashion parade and you are all invited to ohh and ahh at his talent and eye-popping outfit. Meet the vain Emperor, the pandering advisors who offer uncritical support, and the crowd who fail to tell the truth, preferring lies to be told instead.
Hilarious and heart-warming, magical and musical, The Emperor's New Clothes is a stylish theatrical treat for all ages. Hiccup Theatre spin music, puppetry, BSL and storytelling into a very special yarn.
"Hiccup Theatre's family shows always take you to unexpected places in such imaginative ways" The Guardian
About the Co-Producers
Hiccup Theatre have been behind previous, phenomenally popular Christmas shows for young audiences in the Derby Theatre Studio over the past few years – including Jack, Little Red Riding Hood, Goldilocks and The Three Bears and The Gingerbread Man.
Hiccup is a theatre company who make work for children, young people and their families and encourage them to discover, question, explore, create, express, laugh, inspire and be inspired.
Polka Theatre is one of the leading theatres in the UK dedicated exclusively to children who produce Inspirational Theatre and Creative Experiences for Children.
Accessible Performances
All performances will have fully integrated British Sign Language and Captioning.
Relaxed Performance: Saturday 9 April at 2:30pm
Specifically aimed to appeal to audience members who are on the autistic spectrum or have sensory sensitivities, or those who would benefit from a more relaxed atmosphere as regards to making noise or moving around– this performance will be available at reduced capacity.
Live Audio Described Performance: Thursday 14 April,1pm (with Touch Tour available)
Best and recommended view for integrated captioning: seats on rows D-H
Closed Captions: we have a small number of handheld tablets that the captions for The Emperor's New Clothes can be viewed on (a large screen on stage displaying the captions will also be visible throughout the entire performance). To book one of these tablets please email DerbyTheatreTickets@derby.ac.uk
Priority Booking Open from Mon 13 Dec, 12pm
Priority booking is open to Friend members, Assisted Access Scheme members and users with an access requirement tag on their theatre account from Monday 13 December at 12pm.
Members must login to their theatre account to unlock early access to booking. Alternatively, please email DerbyTheatreTickets@derby.ac.uk, text 07717 346 964 or call the Box Office on 01332 59 39 (available Tue - Sat from 10am to 4pm).
General Sale from Fri 15 Dec
Tickets on general sale from Friday 17 December at 10am. | {
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\section{Introduction}
\label{sec:intro}
CMC surfaces in space forms are given by their associated $\mathbb C^*$-family of flat $\SL(2,\mathbb{C})$-connections
$\nabla^\lambda$.
For simply connected surfaces, all flat connections are trivial, and the associated family of gauge equivalence classes $[\nabla^\lambda]$
does not contain any information. This is in stark contrast to compact surfaces of genus $g\geq1.$
For CMC tori, it was shown by Hitchin \cite{H} that the conformal type of the torus (as well as its energy) are determined by the asymptotic behavior of $[\nabla^\lambda]$ as $\lambda\to0.$ Similarly, it was shown in \cite{He3} that
the energy of Lawson symmetric CMC surfaces of genus $2$ can also be recovered from the asymptotic behavior of $[\nabla^\lambda].$
In this paper, we prove that also the conformal type of a compact CMC surface of genus $g\geq2$ with simple umbilics is determined by the asymptotic behavior of the associated family of gauge equivalence classes $[\nabla^\lambda]$ locally in the Teichm\"uller space:
The trace of the monodromy of $\nabla^\lambda$ along curves $\gamma$ (satisfying a certain condition)
give the period of the square root of the Hopf differential along $\gamma,$ see formula \eqref{asymptotic_trace}.
Then, we reprove that the periods of the square root of a holomorphic quadratic differential with simple zeros (defined on the so-called Hitchin covering)
determine the Riemann surface structure locally in the Teichm\"uller space. We then apply this observation to
prove our main theorem.
\section{The family of flat connections and its asymptotic monodromy}\label{family}
Let $f\colon M \to S^3$ be a conformal immersion of constant mean curvature $H$ from a compact Riemann surface to the round $S^3.$ The immersion $f$ is determined by its associated
$\mathbb{C}^*$-family $\nabla^{\lambda}$ of flat $\SL(2, \mathbb{C})$ connections on the trivial $\mathbb{C}^2$ bundle over $M$, see \cite{H, B}:
\begin{The}[\cite{H,B}]
\label{The1}
For a conformal CMC immersion $f\colon M\to S^3$ there exists
an associated family of flat $\SL(2,\mathbb{C})$-connections
\begin{equation}\label{associated_family}
\lambda\in\mathbb{C}^*\mapsto \nabla^\lambda=\nabla+\lambda^{-1}\Phi-\lambda\Phi^*
\end{equation}
on a complex bundle $V\to M$ of rank $2,$ where $\Phi$ is a nilpotent and nowhere vanishing
complex linear $1$-form and $\Phi^*$ is its
adjoint with respect to a unitary metric. The connections $\nabla^\lambda$
are unitary for $ S^1\subset\mathbb{C}^*$ and trivial for
$\lambda_1\neq\lambda_2\in S^1$. \\
Conversely, for every $\mathbb{C}^*$-family of flat connections satisfying the properties listed above, the $SU(2)$-valued gauge transformation between the trivial connections
$\nabla^{\lambda_1}$ and $\nabla^{\lambda_2}$ is a CMC immersion into $S^3=SU(2)$ with mean curvature $H=i\frac{\lambda_1+\lambda_2}{\lambda_1-\lambda_2}.$ \end{The}
Because of the following theorem we cannot apply the methods developed by Hitchin \cite{H} in order to study
CMC surfaces of higher genus.
\begin{The}[\cite{He1}]
For compact CMC surfaces of genus $g\geq2$ the flat $SL(2, \mathbb{C})$-connections $\nabla^{\lambda}$ are irreducible for generic spectral parameter $\lambda\in\mathbb{C}^*.$
\end{The}
\subsection{Monodromy}\label{mon}
Because $\Phi$ in Theorem \ref{The1} is nilpotent and nowhere vanishing, there exists a holomorphic line bundle $S^*=\ker \Phi\subset V$
which automatically satisfies $(S^*)^2=K^*,$ i.e., $S^*$ is the dual of a holomorphic spin bundle $S.$ With respect to
the unitary decomposition
$V=S^*\oplus S$
the connections $\nabla^\lambda$ are given by
\begin{equation}\label{gceqn}
\nabla^\lambda=\dvector{&\nabla^{spin^*} &
-\frac{i}{2} Q^*\\ & -\frac{i}{2} Q & \nabla^{spin}}+\lambda^{-1}\dvector{0 & \tfrac{1}{2}\\0&0} -\lambda\dvector{0&0\\i\vol &0},\end{equation}
see \cite{He1,He3}.
In \eqref{gceqn} $Q$ is a holomorphic quadratic differential -- the Hopf differential of the CMC surface -- and $\nabla^S$ is the spin connection of the induced Levi-Civita connection.
Let $\zeta^2=\lambda$ and consider the $\zeta$-dependent gauge transformations
\[g=\dvector{\tfrac{1}{\sqrt{\zeta}} &0\\0& \sqrt{\zeta}}\]
with respect to the decomposition $V=S^*\oplus S.$
Then,
\begin{equation}\label{twisted}
\nabla^\zeta:=\nabla^\lambda.g=\tilde\nabla+\zeta^{-1}\dvector{0 & \tfrac{1}{2}\\-\frac{i}{2} Q&0} -\zeta\dvector{0&-\frac{i}{2} Q^*\\i\vol &0},
\end{equation}
is the twisted family of flat connections. For $\zeta^2=\lambda$ the traces of the monodromies of $\nabla^\zeta$ and $\nabla^\lambda$
along any closed curve are the same.
Instead of working with $\nabla^\lambda$ or $\nabla^\zeta$ we will work with another family of flat connections:
Consider the Hopf differential $Q$ of the CMC surface of genus $g\geq2$ and assume that it
has only simple zeros. Then there exists a double covering $\hat M\to M$ (the Hitchin covering \cite{H1}) branched over the umbilics, i.e., the zeros of $Q$.
$\hat M$ has genus $4g-3$ and there exists a holomorphic 1-form $\omega$ on $\hat M$ such that $\tfrac{i}{4}Q=-\omega^2$. Note that in \cite{H1} $Q$ is pulled back as a section and not as a quadratic differential. Nevertheless, it follows easily
that the pullback of $Q$ as a quadratic differential has a globally well-defined square root which is a holomorphic 1-form on the Hitchin covering.
The holomorphic Higgs field $\Phi=\dvector{0 & \tfrac{1}{2}\\-\frac{i}{2} Q&0}$ (with respect to $\dbar^{\tilde\nabla}$) diagonalizes on $\hat M$ \cite{H1},
i.e., there is a decomposition $V=L\oplus L^*\to\hat M$ such that
\[\Phi=\dvector{\omega&0\\0&-\omega},\]
where $\Phi$ is the pull-back as an endomorphism-valued 1-form.
In the following, we will work with the pull-back connections $\hat\nabla^\zeta$
of $\nabla^\zeta$ on $V=L\oplus L^*\to\hat M.$
In order to obtain
informations from the asymptotic behavior of the monodromy of $\hat\nabla^\zeta$ (respectively $\nabla^\lambda$)
we need to recall basic knowledge from the asymptotic analysis of linear ordinary differential equations. For a more extensive treatment of this topic we refer to \cite{Was,Fed}. A short treatment appropriate for our purposes is given in Appendix C in \cite{GMN}:
Let $I=(\varphi_0;\varphi_1)\subset\mathbb{R}$
and consider the sector \[S:=\{te^{i\varphi}\mid t\in\mathbb{R}^{>0},\varphi\in I\}.\] Let $\gamma\colon [0;1]\to\hat M$ be a closed curve such that
\begin{equation}\label{direction_condition}
\Re(e^{i\varphi}\omega(\gamma'(t)))>0
\end{equation}
for all directions $e^{i\varphi}\in S^1$ with $\varphi\in I$ and all
$t\in [0;1].$ Then the trace of the monodromy along $\gamma$ satisfies
\begin{equation}\label{asymptotic_trace}
\Tr(M(\hat\nabla^\zeta,\gamma))\exp{(-\tfrac{1}{\zeta}\int_\gamma\omega)}\xrightarrow[\tfrac{1}{\zeta}\in S,\,\zeta\to 0]{} const\neq0\end{equation}
for some $const\in\mathbb{C}\setminus\{0\}.$ Hence, we obtain:
\begin{Pro}\label{as-be}
For closed curves $\gamma$ satisfying \eqref{direction_condition} the value of
the period
$\int_\gamma\omega$
is determined by the asymptotic behavior for $\lambda\to0$ of the trace of the monodromy representation of $\nabla^\lambda.$
\end{Pro}
\section{The Riemann surface structure}\label{RSS}
Let $M_t,$ $t\in(-\delta,\delta),$ be a (smooth) family of compact Riemann surfaces of genus $g,$ and
let $Q_t$ be a family of holomorphic quadratic differentials with respect to $M_t.$
Assume that the differentials $Q_t$ have only simple zeros, and consider
the family $\pi_t\colon\hat M_t\to M_t$ of Hitchin coverings as well as the corresponding (holomorphic) involutions
$\sigma_t$ interchanging the sheets.
We can identify the smooth surfaces $\hat M=\hat M_t$ in such a way that all $\sigma_t$ are the same (smooth map $\sigma\colon \hat M\to\hat M$). In this situation we get:
\begin{Lem}\label{torelli}
Assume that
on the Hitchin covering $\hat M\to M$ the periods of $\omega_t=\sqrt Q_t\in H^0(\hat M_t,K_{\hat M_t})$
are constant in $t.$ Then the Riemann surfaces $M_t$ are equivalent. Moreover, the Hopf differentials
$Q_t$ are also equivalent.
\end{Lem}
\begin{Rem}
It is a well-known fact (see \cite{KH} or $\S 1.1.$ in \cite{KZ}) that the space of Riemann surfaces equipped with
holomorphic differentials $\omega$ having a prescribed number of zeros of prescribed order has local coordinates given
by the so-called relative periods $\int_{\gamma_k}\omega,$ where the curves $\gamma_k,\, k=1,..,$ generate the first homology
group relative to the set of zeros of $\omega.$
We briefly reprove this fact in the situation of this paper along with
the necessary computation of the ``half-periods'' $\int_\gamma\omega$ for curves connecting
zeros of $\omega.$
\end{Rem}
\begin{proof}
Consider the smooth surface $\hat M$ together with the family of closed complex valued 1-forms $\omega_t.$
They are holomorphic with respect to a $t$-dependent Riemann surface structure $J_t,$ and their zeros
are all of order $2.$ Hence, we can assume that the zeros coincide on the smooth surface $\hat M.$ Moreover,
we can also assume without loss of generality that in a local neighborhood $U\subset\hat M$ of its common zeros, the 1-forms coincide $(\omega_t)_{\mid U}=(\omega_0)_{\mid U}.$
As the periods of $\omega_t$ are independent of $t$ we get
\[\omega_t=\omega_0+ df_t\]
for some complex-valued function $f_t\colon\hat M\to\mathbb{C}$ depending smoothly on $t.$
Moreover, the differential $df_t\in\Omega^1(\hat M,\mathbb{C})$ vanishes in the neighborhood $U$ of the zeros of $\omega.$
In order to apply the Moser trick to deform the forms $\omega_t$ to $\omega_0$, we first need to establish that the values of $\frac{\partial{f_t}}{\partial t}$ near the zeros of
$\omega_t$ are $0$.
Recall that we have already identified the involutions $\sigma_t=\sigma\colon \hat M\to\hat M.$
Let $\gamma$ be a curve connecting
two zeros of $\omega_t$ and consider the closed curve $\Gamma=\sigma(\gamma^{-1})\circ\gamma.$
Because $\sigma^*\omega_t=-\omega_t$ for all $t$ we
obtain
\[2\int_\gamma\omega_t=\int_{\Gamma}\omega_t=\int_{\Gamma}\omega_0=2\int_\gamma\omega_0.\]
Therefore, the values of $f_t$ at any zero of $\omega_t$ are the same, and without loss of generality we have that
they are $0$ independently of $t.$
Next we consider the closed $1$-form
\[\Omega=\omega_0+df_t+\frac{\partial{f_t}}{\partial t} dt=\omega_t+\frac{\partial{f_t}}{\partial t} dt\]
on $\hat M\times(-\delta,\delta).$ We want to show that there exists a (real) vector field $X$ on
$\hat M\times(-\delta,\delta)$ such that
$\Omega(X)=0$ and $dt(X)=1.$ We already know that $f_t$ is $0$ in the neighborhood $U$ of the zeros of $\omega_t$
for all $t\in(-\delta,\delta).$ Therefore, the existence and uniqueness of $X$ follows from the fact that
for all $p\in \hat M\setminus \{\text{zeros of } \omega_t\}$ the linear map $(\omega_t)_p\colon T_p\hat M\to\mathbb{C}$ is an isomorphism.
Clearly, $X$ is complete.
Then
\begin{equation}\label{lieder}\mathcal L_X\Omega=d i_X\Omega+i_X d\Omega=0,\end{equation}
and the flow at time $s$ \[\Phi_s\colon \hat M\times(-\delta,\delta)\to \hat M\times(-\delta,\delta)\]
is of the form
\[\Phi_s(p,t)=(\phi_s(p),t+s).\]
By construction and \eqref{lieder}, $\phi_s\colon\hat M\to\hat M$ is a diffeomorphism which satisfies
$\phi_s^*\omega_s=\omega_0.$
This also shows that $\phi_s$ is a holomorphic diffeomorphism with respect to the Riemann surface structures $M_0$ and $M_s.$
\end{proof}
\subsection{Deformations of CMC surfaces}
In the abelian case of tori, it is easy to show that the monodromy representation (based at some fixed point) already determines the conformal type of the immersed CMC torus $f\colon M\to S^3$:
The asymptotic behavior of the traces of the monodromies determines the periods of a (non-zero) holomorphic differential on the
torus, see \cite{H} for details. Clearly, these periods determine the conformal type of the torus.
The main result of this note is the following generalization to compact CMC surfaces of genus $g\geq2:$
\begin{The}\label{Main}
Let $M$ be a compact oriented surface.
Let $f_t\colon M\to S^3$ be a family of CMC surfaces
such that
the gauge equivalence classes of associated family of flat connections ${^t}{\nabla}^\lambda$
are equal for all $t$ and generic $\lambda\in\mathbb{C}^*.$ If the CMC surfaces have only simple umbilics (for generic $t$).
Then the induced Riemann surface structures $\Sigma_t$ are all equivalent and the Hopf differentials coincide.
\end{The}
\begin{Rem}
The theorem remains true for CMC surfaces with periods, i.e., for CMC surfaces into a flat $S^3$ bundle over $M$
(in the sense of Hitchin's \cite{H} gauge-theoretic harmonic map equations).
These are given by solutions $(g,Q)$ of the Gauss-Codazzi equations which are globally defined on the surface $M.$
Here $g$ is a Riemannian metric which induces the spin connection in\eqref{gceqn}, and $Q$ is the Hopf differential. The Gauss-Codazzi equations are then equivalent
to the flatness of all connections $\nabla^\lambda$ in \eqref{gceqn}. The theorem also remains true for CMC surfaces (with periods) in the space forms $\mathbb R^3$ and $\mathbb H^3.$
\end{Rem}
\begin{proof}
We show that there exists closed curves $\gamma_i,$ $i=1,...,$ on $\hat M$
which satisfy \eqref{direction_condition} for non-empty sectors and which generate the first homology over $\mathbb Q.$
Then the proof follows from Lemma \ref{torelli} and Proposition \ref{as-be}.
We apply Lemma 4.4 in \cite{For}. This Lemma and its proof tell us, that (the pull-back of) $Q$ on $\hat M$ is the limit of holomorphic quadratic differentials $Q_i$ with respect to suitable Riemann
surface structures $\hat M_i$ (note that we identify $\hat M_i$ and $\hat M$ as smooth surfaces) and which satisfy the following properties:
\begin{enumerate}
\item
all $Q_i$ have the same number of zeros of the same order as $Q;$
\item for all $i\in\mathbb{N}$ the horizontal distribution $\Re(\sqrt{Q_i}))=0$
contains closed curves which do not hit any zeros of $Q_i$ and which generate a Lagrangian subspace $\mathcal L$ in the symplectic space $H_1(\hat M,\mathbb{R})$;
\item for any Lagrangian subspace $\Lambda\subset H_1(\hat M,\mathbb{R})$ the differentials $Q_i$ can be chosen in such a way that $\mathcal L\cap \Lambda=\{0\};$
\item there is an open neighborhood $U$ of the zeros of $Q$ such that $(Q_i)_{\mid U}=Q_{\mid U}$ for all $i\in\mathbb{N}.$
\end{enumerate}
We claim that for $Q_i$ close to $Q$ and a closed curve $\gamma\colon S^1\to\hat M\setminus\{\text{zeros of } Q_i\}$ satisfying
\[\Re(\sqrt{Q_i}(\gamma'))=0\]
condition \eqref{direction_condition} is satisfied for a non-empty sector $S$. This follows from the fact that $\sqrt{Q_i}_{\mid U}=\sqrt{Q}_{\mid U}$ on $U$ and that point-wise on the compact set $\hat M\setminus U$ the norms of $\sqrt{Q}$ and $\sqrt{Q_i}$ are bounded
from below uniformly. Note that because of (iii) we obtain enough curves $\gamma$ which generate the first homology $H_1(\hat M;\mathbb Q).$
\begin{Rem}
The statement of the theorem remains true for some classes of CMC surface with higher order umbilics. In particular, the conformal type of a $(k,l)$-symmetric CMC surface (as defined in \cite{HHSch}) is determined
uniquely by the asymptotic behavior \eqref{asymptotic_trace} of the monodromy representation.
\end{Rem}
\begin{Rem}
Combining Theorem \ref{Main} with Theorem 7 of \cite{He3} shows that deformations of compact CMC surfaces (with simple umbilics) which preserve the conjugacy classes of the monodromy representations of the associated families
of flat connections must be induced by a family of dressing transformations. Dressing is a transformation of the family of flat connections induced by a $\lambda$-dependent gauge which becomes singular
at certain spectral parameters $\lambda_0\notin S^1\cup\{0,\infty\}$ where parallel eigenlines with respect to $\nabla^{\lambda_0}$ exist. In the case of higher genus $g\geq2,$ families of dressing transformations
can only exists if there is a spectral parameter $\lambda_0$ for which $\nabla^{\lambda_0}$ is trivial (up to an diagonal $\mathbb{Z}_2$-gauge). We expect that this would occur only in very exceptional situations.
\end{Rem}
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,882 |
package com.coinkite.api;
import com.fasterxml.jackson.annotation.JsonIgnoreProperties;
import com.fasterxml.jackson.annotation.JsonInclude;
import com.fasterxml.jackson.annotation.JsonProperty;
import java.math.BigDecimal;
/**
* {
"help_msg": "Responses are being rate-limited: min gap time 2s",
"message": "Too Many Requests",
"status": 429,
"wait_time": 1.759
}
*/
@JsonIgnoreProperties(ignoreUnknown = true)
@JsonInclude(value = JsonInclude.Include.NON_NULL)
public class RestError {
@JsonProperty("help_msg")
private String helpMsg;
@JsonProperty("message")
private String message;
@JsonProperty("status")
private int status;
@JsonProperty("wait_time")
private BigDecimal waitTime;
// @JsonProperty("so_far")
// private
public String getHelpMsg() {
return helpMsg;
}
public void setHelpMsg(String helpMsg) {
this.helpMsg = helpMsg;
}
public String getMessage() {
return message;
}
public void setMessage(String message) {
this.message = message;
}
public int getStatus() {
return status;
}
public void setStatus(int status) {
this.status = status;
}
public BigDecimal getWaitTime() {
return waitTime;
}
public void setWaitTime(BigDecimal waitTime) {
this.waitTime = waitTime;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 795 |
Few words look better together in the English language than "happy hour," especially in Reykjavík, where liquor comes at a premium. We know alcohol is a necessity for life, and trust us, it's increasingly possible to sip something light and sour or dark and sweet without having to first mortgage your pancreas.
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If that's not delectable enough for you—you snob—they also have sparkling Thursdays where flutes of bubbly are 1,000 ISK and bottles are 5,000 ISK all day.
Check out more info about happy hours on our amazing app, Appy Hour, available on iOS and Android. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,673 |
\section{Introduction}
The aim of this note is to describe a field theoretical model, motivated by an attempt to extend the understanding of certain aspects of 2+1 dimensional gauge theories to 3+1 dimensions.
In 2+1 dimensions one has a very simple and straightforward relation between confinement and spontaneous breaking of a discrete magnetic symmetry\cite{thooft},\cite{dualreview}. Additionally, in 2+1 dimensions, nonabelian gauge theories, exhibiting the phenomenon of confinement, are related to abelian theories on the effective field theory level, by a simple simmetry breaking deformation. This deformation breaks the continuous $U(1)$ magnetic symmetry of an abelian theory down to a discrete group $Z_N$ for $SU(N)$ gauge theories with adjoint matter. The mere fact of the presence of this deformation, coupled with the spontaneous breaking of the residual discreet group leads with certainty to a confining long distance behavior\cite{dualreview},\cite{chris}.
In this note we discuss a 3+1 dimensional model that displays similar features. The model is designed to implement certain aspects of abelian-nonabelian transition, similar to 2+1 dimensional case. Although it clearly cannot be taken literally as the effective theory of QCD, curiously enough it does have some similarity with Fadeev-Niemi model, that has been proposed as an effective theory of glueballs from a completely different perspective\cite{niemi}.
Before discussing the model itself, we will recall briefly the story of 2+1 dimensional gauge theories. As a prototypical Abelian gauge theory consider scalar QED. It posesses a continuous $U_\mu(1)$ global symmetry generated by the total magnetic flux through the plane of the system, $\Phi=\int d^2x B(x)$. The order parameter
for this symmetry is one complex field $V$, which creates pointlike magnetic vortices. In the Coulomb phase $\langle V\rangle=v \ne 0$ and $U_\mu(1)$ is spontaneously broken. The low energy dynamics is qualitatively described by the effective "`dual"' Lagrangian
\begin{equation}\label{l}
L=-\partial^\mu V\partial_\mu V^*-\lambda (V^*V-\frac{e^2}{8\pi})^2
\end{equation}
The Goldstone boson of the $U_\mu(1)$ symmetry breaking is identified with the massless photon, while the electric charge in the dual formulation is the topological charge of the field $V$
\begin{equation}
J_\mu=\frac{1}{e}\epsilon_{\mu\nu\lambda}\partial_\nu V^*\partial_\lambda V
\end{equation}
A charged state of QED in the effective description appears as a hedgehog - like soliton of $V$: $V(x)=ve^{i\theta(x)}$, with $\theta=\tan^{-1} y/x$.
This effective formulation is also a good basis for description of confinement in nonabelian theories. In particular the effective theory of a weakly interacting $SU(N)$ model is essentially the same as eq.(\ref{l}), except with a potential which breaks the magnetic $U_\mu(1)$ symmetry down to $Z_N$
\begin{equation}\label{lN}
L=-\partial^\mu V\partial_\mu V^*-\lambda (V^*V-\frac{e^2}{8\pi})^2+\mu(V^N+V^{*N})
\end{equation}
The perturbation reduces the infinite degeneracy of vacua of the Abelian theory to a finite number of degenerate vacuum states connected by the $Z_N$ transformations. As a result, a charged state does not have a rotational symmetry anymore, but the winding is concentrated within a quasi one dimensional "`flux tube"'\cite{dualreview}.
This is a very simple picture, and a very appealing one inasmuch as it identifies charged objects with topological defects which inherently have long range interactions due to their topological nature. It also identified photons with Goldstone bosons, providing a natural symmetry based explanation for masslessness of the photon.
It is natural to ask, whether in 3+1 dimensions one can have a similar description, which encompasses the massless nature of photons in QED as well as topological mechanism of confinement in nonabelian theories. The situation here of course, is much more complicated. First of all, in 3+1 dimensions photons are vector particles and so it is not clear at all whether they can be understood as Goldstone bosons. Even if such a case can be made for photons, it is not easy to identify the relevant conserved current that breaks spontaneously. It is clear that the current has to be related to the dual field strength $\tilde F_{\mu\nu}$ consistent with the fact that photons have spin one\cite{photongoldstone}. The dual field strength, however has no local order parameter, and thus is an object of a very different kind than ordinary vector currents, which we are used to deal with. Another complication is, that classical effective description assumes weakly interacting theory, while QCD is of course strongly interacting.
All these are difficult questions, to which we do not attempt to provide answers here. Instead we will be content to construct a model that encompasses the following basic features:
1. The model should describe dynamics of scalar fields, and contain no fundamental gauge fields.
2. The model should have the limit (putative "Abelian regime") in which it has two massless degrees of freedom, which are identified as Goldstone bosons. These massless Goldstone bosons in our model are intended to play the role of photons.
3. In the Abelian regime the model must provide for existence of classical topological solitons, which play the role of electrically charged particles. We require the topological charge that is carried by these solitons to reflect the mapping of the spatial infinity onto the manifold of vacua, and thus be given by $\Pi_2(M)$. The energy of these solitons has to be finite in the infrared. The energy density of a soliton solution should decrease as $1/r^4$ far from the soliton core. This is nontrivial in 3+1 dimensions, since our model has no gauge fields, while scalar fields that contribute to $\Pi_2$ have to be long range.
4. Soliton must become confined in the "Nonabelian regime", when a symmetry breaking perturbation is added. This same perturbation must eliminate massless Goldstones by expicitly breaking the (previously) spontaneously broken symmetry group down to a discreet subgroup. Confinement should be accompanied by formation of string between the solitons.
In this note we present a model which has all the above features and discuss its properties, which are somewhat unusual. In particular, the requirement of the finiteness of the energy of a topological soliton in the Abelian regime is very restrictive. It leads to rather unusual properties of the confining strings in the Nonabelian regime such as existance of an infinite number of zero modes. This degeneracy can be lifted, however that requires the addition of another perturbation which is not clearly related with breaking of symmetries of the theory.
We stress, that the model we discuss does not provide perfect emulation of many properties of gauge theories. In particular, even in the Abelian regime, it contains classical solutions with magnetic charge density, and thus the effective dual field strength tensor is not conserved. Related to that, although we are able to construct solutions of equations of motion that behave as single photons, the model has no solutions that correspond to a two photon state with arbitrary polarization vectors. Nevertheless, we think that the model has many similarities with gauge theories, and thus is sufficiently interesting to be considered perhaps as a simplified prototype for future work.
\section{The Abelian Model.}
\subsection{The Field Space and the Lagrangian.}
As explained in the introduction, we wish to construct a model of scalar fields which contains two massless degrees of freedom and solitons of finite energy. The simplest option that we adopt is a theory with two scalar degrees of freedom endowed with $SU(2)$ symmetry. Spontaneous breaking of this symmetry must lead to two massless modes. Thus we choose as the configuration space the $O(3)$ nonlinear $\sigma$-model.
\begin{equation}
\phi^a, \ a=1,2,3; \ \ \ \ \phi^2=1
\end{equation}
The moduli space allows for the requisit topology $\Pi_2(S^2)$. The topological charge associated with it, is identified with the electric charge of QED
\begin{equation}\label{topch}
Q=\frac{e}{4\pi^2}\int d^3x\epsilon_{abc}\epsilon_{ijk}\partial^i\phi^a\partial_j\phi^b\partial_k\phi^c
\end{equation}
As the first task, we have to contend with the following potential problem. In a theory with the standard kinetic term, the energy of a state with nonvanishing topological charge diverges in the infrared. A typical topologically nontrivial field configuration is a rotationally symmetric hedgehog
\begin{equation}\label{hedge}
\phi_h^a(x)=\frac{r^a}{|r|}f(|r|); \ \ \ \ \ \ \ \ \ f(|r|)\rightarrow_{r\rightarrow\infty}1
\end{equation}
The standard two derivative kinetic energy diverges quadratically on such a configuration. In order to make the energy of the soliton finite, we need to introduce a kinetic term with more than two derivatives.
In fact, there exists a unique four derivative term which is a natural choice for a kinetic term for our model. The identification of the electric charge with the topological charge eq.(\ref{topch}) also naturally leads to the identification of the electric current as
\begin{equation}
J^\mu=\frac{e}{4\pi^2}\epsilon_{abc}\epsilon^{\mu\nu\lambda\sigma}\partial_\nu\phi^a\partial_\lambda\phi^b\partial_\sigma\phi^c
\end{equation}
and therefore electromagnetic field tensor as
\begin{equation}\label{field}
F^{\mu\nu}=\epsilon^{abc}\epsilon^{\mu\nu\lambda\sigma}\phi^a\partial_\lambda\phi^b\partial_\sigma\phi^c
\end{equation}
Since our goal is to construct a model that resembles QED as close as possible, the natural choice for the kinetic term is the square of the field strength tensor, which is just the well known Skyrme term.
Hence we consider the model of a triplet of scalar fields defined by the following Lagrangian:
\begin{equation}\label{ourl}
L=\frac{1}{16e^2}F^{\mu\nu}F_{\mu\nu}+\lambda(\phi^2-1)^2
\end{equation}
We note, that the sign of the $F^2$ term in the Lagrangian is opposite to that in QED. In the framework of eq.(\ref{ourl}) the sign is determined so that the Hamiltonian is positive, rather than negative definite. This feature is common to models related by duality. For example the same is true in the 2+1 dimensional models described in the introduction, where the kinetic term in the Lagrangian of the effective theory when written in terms of the field strength tensor has the opposite sign to that of Electrodynamics. The reason for this inversion, is that while in QED the electric field is proportional to the time derivative of the basic field (in this case $A_\mu$), in the effective dual description it is the magnetic field that contains time derivative of the vertex field $V$. Thus in order for the Hamiltonian of the two models in terms of $E$ and $B$ to be the same, the Lagrangians have to have opposite sign. This inversion of sign also takes place in our model, and is the natural consequence of the relation between the field strength tensor and the basic scalar degrees of freedom eq.(\ref{field}).
In the strongly coupled limit $\lambda\rightarrow \infty$, the isovector $\phi$ has unit length, and the field strength is trivially conserved
\begin{equation}
\partial_\nu F^{\mu\nu}=0
\end{equation}
This limit therefore corresponds to QED without charges. In this limit the energy of the soliton eq.(\ref{hedge}) diverges linearly in the ultraviolet. At finite coupling $\lambda$ the variation of the radial component of the field $\phi^a$ softens the UV behavior, and the soliton energy is UV finite. It is also IR finite thanks to our choice of the four derivative action.
In fact on the hedgehog configuration eq.(\ref{hedge}) the " electric field" decreases as $E_i(x)\propto \frac{r^1}{|r|^3}$, and the energy density away from the soliton core decreases as $1/r^4$, just like the Coulomb energy of a static electric charge in the electrodynamics.
\subsection{The Equations of Motion}
We now derive the equations of motion for the model. For conveniene we define in the strong coupling limit
\begin{equation}
\phi_3 = z, \ \ \ \ \ \ \psi = \phi_1 + i \phi_2 = \sqrt{1-z^2} e^{i \chi}
\end{equation}
With this parametrization one has
\begin{equation} F^{\mu \nu} = \epsilon^{\mu \nu \alpha \beta} \epsilon_{abc} \phi^a \partial_ \alpha \phi^b \partial_\beta \phi^c = - 2 \epsilon^{\mu\nu\alpha\beta}\partial_\alpha z \partial_\beta \chi\end{equation}
The Lagrangian can be written as
\begin{equation} L=\frac{1}{4e^2}(\partial_\mu z\partial_\nu \chi-\partial_\mu\chi\partial_\nu z)^2\end{equation}
The equations of motion read
\begin{eqnarray}
&&\partial^\mu\Big[\frac{1}{e^2}\partial^\nu\chi\left(\partial_\mu z\partial_\nu \chi-\partial_\nu z\partial_\mu \chi\right)\Big]=0\nonumber\\
&& \partial^\mu\Big[\frac{1}{e^2}\partial^\nu z\left(\partial_\mu z\partial_\nu \chi-\partial_\nu z\partial_\mu \chi\right)\Big]=0
\end{eqnarray}
This can be combined into
\begin{equation}\label{eqmo}
\frac{1}{e^2}\partial_\nu G(z,\chi)\partial_\mu\left(\partial_\mu z\partial_\nu \chi-\partial_\nu z\partial_\mu \chi\right)= \frac{1}{e^2}\partial_\nu\Big[ G(z,\chi)\partial_\mu\left(\partial_\mu z\partial_\nu \chi-\partial_\nu z\partial_\mu \chi\right)\Big]=0
\end{equation}
where $G(z,\chi)$ is an arbitrary function of two variables. These equations have a form of conservation equations for currents defined as
\begin{equation}\label{current}
J_\nu^G=G(z,\chi)\partial^\mu\left(\partial_\mu z\partial_\nu \chi-\partial_\nu z\partial_\mu \chi\right)
\end{equation}
\subsection{The Symmetries and the Correspondence to Electrodynamics.}
The conserved currents of eq.(\ref{current}) can indeed be identified with conserved Noether currents. An unexpected consequence of the choice of the Skyrme term as the kinetic term in the Lagrangian, is that the global symmetry group of the model is much larger than the $SO(3)$ group we have started with.
To see this note, that the field strength as defined in eq.(\ref{field}) is related to an infinitesimal area on a configuration space. Let us be more precise here. A given field configuration $\phi^a(x)$ defines a map from space-time to a sphere $S^2$. Consider a given component the field strength tensor, say $F_{12}$ at some point $x$. To calculate it in terms of $\phi$ we consider three infinitesimally close points $A\equiv x^\mu$, $B\equiv x^\mu+\delta^{\mu 1}a$ and $C\equiv x^\mu+\delta^{\mu 2}a$. These three points in space-time, map into three infinitesimally close points on the sphere $\phi^a(A),\ \phi^a(B),\ \phi^a(C)$. The field strength $F_{12}$ is proportional (up to the factor $a^{-2}$ to the area of the inifnitesimal triangle on $S^2$ defined by these three points.
Since the action of our toy model depends only on $F^{\mu\nu}$, it is clear that any reparametrization of the sphere which preserves area is an invariance of our action.
Thus the $SO(3)$ global symmetry we started with, is a {\it small} subgroup of the area preserving diffeomorfisms of $S^2$, which we denote $Sdiff(2)$\cite{sdiff}. This is the group of canonical transformations of a classical mechanics of one degree of freedom.
The infinitesimal symmetry transformation in terms of $z$ and $\chi$ is
\begin{equation}\label{sdiff}
z\rightarrow z+\frac{\partial G}{\partial \chi}; \ \ \chi\rightarrow \chi-\frac{\partial G}{\partial z}
\end{equation}
with arbitrary $G(z,\chi)$.
The appropriate Noether currents are precisely those of eq.(\ref{current}) and the equations of motion are indeed equivalent to conservation equations of these currents.
It is amusing to note that this symmetry is similar to the world sheet diffeomorphism invariance of the Nambu-Gotto string. Indeed, if one thinks of the fields $z$ and $\chi$ as the world sheet string coordinates, the world sheet diffeomorphism invariance is precisely eq.(\ref{sdiff})\cite{misha}. Although our setup looks very different from a string theory, there may be more to this analogy than meets the eye, as the basic "`order parameters"' of the magnetic symmetry in $QED_4$ are indeed magnetic vortex strings\cite{photongoldstone}. The $S^2$ topology of the world sheet then implies closed string loops. We will not develop this analogy any further here, and instead will return to the field theoretical approach.
The enhanced symmetry means that the moduli space is much larger than $S^2$ as would be naively the case for symmetry breaking pattern $SO(3)\rightarrow SO(2)$. Any configuration $\phi^a(x)$ that maps the configuration space into an arbitrary {\bf one dimensional} curve on $S^2$ has vanishing action and is thus a point on the moduli space. The moduli space is therefore the union of maps $\phi^a(x)$ that map $R^4$ to $L$, where $L$ is an arbitrary point or a one dimensional curve on $S^2$.
Nevertheless, even though the moduli space is not a simple sphere, the topological charge $Q$ is quantized for any smooth classical configuration of fields $\phi(x)$. A twist in the tale is that there are many more degenerate soliton configurations than just the rotationally invariant hedgehog of eq.(\ref{hedge}). Any $Sdiff(2)$ transformation coresponding to an arbitrary regular function $G$ of eq.(\ref{sdiff}) applied to the configuration eq.(\ref{hedge}) generates a soliton configuration $\phi^{aG}_h(x)$ which is degenerate in energy with $\phi^a_h(x)$. Note, that although these are different field configurations, they all correspond to the same electric field $E_i=\epsilon_{ijk}\epsilon^{abc}\phi^a\partial_j\phi^b\partial_k\phi^c$, since the electric (as well as magnetic) field is invariant under the action of $Sdiff(2)$ transformations.
\subsection{Plane waves - photon states.}
Returning to the Lagrangian eq.(\ref{ourl}), the natural question to ask is how much of a relation does it have with electrodynamics. With the identification eq.(\ref{field}), we know that the field strength $F^{\mu\nu}$ satisfies half of Maxwell's equations. The equations of motion eq.(\ref{eqmo}) are quite reminsicent of the other half of Maxwell's equations. They can be rewritten in terms of $F^{\mu\nu}$ as
\begin{equation}\label{max}[\partial_\nu G(z,\chi)]\partial_\mu \tilde F^{\mu\nu}=0\end{equation}
Thus, any configuration of the fields $z,\ \chi$ that satisfies $\partial_\mu \tilde F^{\mu\nu}=0$, also satisfies the equations of motion of our model. The converse is not true: there are solutions of the equations of motion eq.(\ref{eqmo}) which do not satisfy the equations of motion of electrodynamics. We give an example of such a solution in the Appendix.
The model eq.(\ref{ourl}) is therefore not equivalent to electrodynamics. Nevertheless, it is interesting to ask whether the spectrum of solutions of eq.(\ref{ourl}) contains basic excitations of QED, in particular the photons. This is a slight abuse of language, since we are dealing with a classical theory. We will nevertheless refer to plane wave configurations of $F^{\mu\nu}$ with lightlike momenum as photons.
Our aim in this section is to show that the free wave excitations are indeed solutions of equations eq.(\ref{eqmo}). To this end consider the configuration
\begin{equation} \label{photons} \chi(x)=A\epsilon^\mu x_\mu; \ \ \ \ \ \ z(x)=\sin k^\mu x_\mu\end{equation}
where the vector $\epsilon^\mu$ is normalized as $\epsilon^\mu\epsilon_\mu=-1$.
On this configuration
\begin{equation} \label{photons1}\tilde F^{\mu\nu}=A(\epsilon^\mu k^\nu-\epsilon^\nu k^\mu)\cos k\cdot x\end{equation}
Thus
\begin{equation} \partial_\mu \tilde F^{\mu\nu}=-A\Big[(\epsilon\cdot k)k^\nu-k^2\epsilon^\nu\Big]\sin k\cdot x\end{equation}
This vanishes, provided the momentum vector is lightlike and the polarization vector $\epsilon$ is perpendicular to $k$:
\begin{equation} \label{photons2} k^2=0; \ \ \epsilon\cdot k=0\end{equation}
For a given lightlike momentum $k_\mu$ this equation has three independent solutions for $\epsilon^\mu$. One of them, however is proportional to $k_\mu$ itself. With this polarization vector, the field strength tensor vanishes. Thus there are two independent polarization vectors $\epsilon^\mu_\lambda,\ \ \lambda=1,2$ that correspond to plane wave solutions for $F^{\mu\nu}$. Just like in QED, it is convenient to choose the polarization vectors so that their zeroth component vanishes $\epsilon_\lambda^\mu=(0,\epsilon^i_\lambda)$.
The constant $A$ is the overall amplitude of the electromagnetic wave, whose square is porportional to the number of photons with a given momentum and a given polarization in the wave.
The arbitrariness in the choice of the polarization vectors is precisely the same as in the choice of polarization vectors in Electrodynamics
\begin{equation} \epsilon^\mu\rightarrow \epsilon^\mu+ak^\mu\end{equation}
Note that this shift of polarization vector is affected by the transformation
\begin{equation} \chi\rightarrow \chi+a\arcsin z\end{equation}
which is one of the $Sdiff(2)$ transformations eq.(\ref{sdiff}). In fact the two field configurations eq.(\ref{photons}) can be transformed by any element of $Sdiff(2)$ without changing $F^{\mu\nu}$.
The solution eqs.(\ref{photons},\ref{photons1},\ref{photons2}) describes a state in all respects equivalent to the freely propagating photon, and we will refer to it as such. The setup here is dual to the usual free QED. Normally one introduces the vector potential $A_\mu$ via $F^{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. This relation potentiates the homogeneous Maxwell's equation $\partial_\mu\tilde F^{\mu\nu}=0$. However, in the free chargeless QED entirely analogously one can potentiate the other Maxwell equation by introducing the dual vector potential via $\tilde F^{\mu\nu}=\partial_\mu \tilde A_\nu-\partial_\nu \tilde A_\mu$. The dynamics of the dual vector potential $\tilde A_\mu$ is identical to that of $A_\mu$, and it can be expanded in exactly the same polarization basis as $A_\mu$. In this formulation QED pocesses a dual gauge symmetry $\tilde A_\mu\rightarrow \tilde A_\mu+\partial_\mu\lambda(x)$.
To make the correspondence between our model and the Electrodynamics more transparent, we can define a dual vector potential
\begin{equation}\label{dualv} \tilde A_\mu=z\partial_\mu \chi\end{equation}
Under the $Sdiff(2)$ transformation eq.(\ref{sdiff}) it transforms as
\begin{equation} \label{transf}\tilde A_\mu\rightarrow \tilde A_\mu +\partial_\mu[G-z\frac{\partial G}{\partial z}]\end{equation}
which has a form reminiscent of the dual gauge transformation in Electrodynamics with the gauge function $\lambda(x)=G-z\frac{\partial G}{\partial z}$.
The analogy of eq.(\ref{dualv}) with the dual vector potential of QED is suggestive, but one has to be aware that this is only an analogy rather than equivalence. First, the tranformation eq.(\ref{transf}) is not a gauge transformation, but rather the action of a global symmetry transformation of the Lagrangian on $\tilde A_\mu$ of eq.(\ref{dualv}). More importnantly, the vector field defined in eq.(\ref{dualv}) in terms of two scalar fields is not the most general form of a vector field, even allowing for a possible gauge ambiguity. For that reason the variation of the Lagrangian with respect to such a constrained vector potential does not lead to the homogeneous Maxwell's equation directly, but instead to eq.(\ref{max}), which allows additional solutions.
Finally we note, that the solution eq.(\ref{photons}) gives a nice and simple interpretation for the properties of the photon states in terms of the effective theory. The photon momentum is the momentum associated with the variation of the third component of the isovector $\phi^a$, while the direction of the photon polarization vector is determined by the spatial variation of the phase $\chi$.
Although a plane wave $\tilde F_{\mu\nu}$ solves the equations of motion, the equations eq.(\ref{max}) are not linear in the basic field variables, and thus an arbitrary superposition of two such solutions, itself is not a solution. We may try to construct a two photon state by slightly extending the ansatz eq.(\ref{photons}).
\begin{equation}\label{photons3} \chi = \lambda_\mu x_\mu; \ \ \ z= a \sin{k^\mu x_\mu}+ b\sin{p^\mu x_\mu}\end{equation}
with $k^\mu$ and $p^\mu$ - both lightlike vectors, $\lambda^\mu k_\mu=\lambda^\mu p_\mu=0$ and $\lambda^\mu \lambda_\mu=-1$.
The latter two conditions can be satisfied by taking
\begin{equation}\lambda^\mu=\alpha\Big[\epsilon^\mu- \frac{\epsilon \cdot k}{k \cdot p} p_\mu - \frac{\epsilon \cdot p}{k \cdot p} k_\mu \Big]\label{lambda}\end{equation}
with an arbitrary vector $\epsilon^\mu$ and an appropriate normalization constant $\alpha$.
The dual field strength tensor can be written as:
\begin{equation} \tilde{F}_{\mu \nu} = a (k_{\mu}\epsilon^k_\nu-k_\nu\epsilon^k_\mu)
\cos{k\cdot x} +b (p_{\mu}\epsilon^p_\nu-p_\nu\epsilon^p_\mu)
\cos{p\cdot x} \end{equation}
with the polarization vectors
\begin{equation} \epsilon^k_\mu=\lambda_\mu- \frac{\lambda_0}{k_0}k_\mu; \ \ \ \epsilon^p_\mu=\lambda_\mu- \frac{\lambda_0}{p_0}p_\mu;\end{equation}
This is a bona fide two photon state. Unfortunately by varying $\lambda_\mu$ at fixed $p$ and $k$ one cannot obtain two most general polarization vectors for photons with momenta $k$ and $p$. This is obvious since both polarization vectors $\epsilon^k$ and $\epsilon^p$ have equal component in the direction perpenducular to the plane spanned by $p^i; \ k^i$. Thus we are lacking one degree of freedom to be able to construct a two photon state with arbitrary polarizations of both photons. In the Appendix we show that this is problem is not restricted to our ansatz eq.(\ref{photons3}), but is a genuine limitation of our Lagrangian.
\section{The Nonabelian perturbation and the string solution.}
In analogy with 2+1 dimensions we now perturb the theory with the simplest perturbation which breaks the $O(3)$ global symmetry. This perturbation should eliminate the vacuum degeneracy associated with the spontaneous symetry breaking. We find it convenient to choose a potential that fixes the vacuum expectation value of the field $z$ to be equal to unity. We thus consider the Lagrangian
\begin{equation} L=\frac{1}{16e^2}F^{\mu\nu}F_{\mu\nu} - \frac{2}{e^2}\Lambda^2 (z-1)^2\end{equation}
The perturbation breaks not only the $SO(3)$ symmetry, but also generic $Sdiff(2)$ transformations. Nevertheless, the subgroup generated by
\begin{equation}\label{subgroup}
\chi\rightarrow\chi -\frac {dG(z)}{dz}
\end{equation}
remains unbroken. We keep this in mind throughout the discussion of this section.
The equations of motion now are
\begin{eqnarray}
&&\partial^\mu\Big[\frac{1}{e^2}\partial^\nu\chi\left(\partial_\mu z\partial_\nu \chi-\partial_\nu z\partial_\mu \chi\right)\Big]=\frac{4}{e^2}\Lambda^2(z-1)\nonumber\\
&& \partial^\mu\Big[\frac{1}{e^2}\partial^\nu z\left(\partial_\mu z\partial_\nu \chi-\partial_\nu z\partial_\mu \chi\right)\Big]=0
\end{eqnarray}
With this perturbation there are no finite energy solutions with nonvanishing topological charge $Q$. Instead, we expect to find translationally invariant stringlike solution. In the presense of a soliton and antisoliton such strings will connect the two and will provide for a linear confining potential. To find such a solution consider a static field configuration translationally invariant in the third direction. For such a configuration the only non-vanishing components of $F_{\mu\nu}$ are:
\begin{equation} F^{0 3 } = 2\epsilon^{ij}\partial_i z \partial_j \chi\end{equation}
We look for a solution invariant under rotations in the plain perpendicular to the string
\begin{equation}
\chi(x)=\theta(x); \ \ \ \ \ \ \ \ z(x)=z(r)
\end{equation}
where $r$ and $\theta$ are the polar coordinates in the $x_1,x_2$ plain.
Such a configuration has a unit winding in the $x_1,x_2$ plain which is precisely what one expects from the string connecting the soliton and antisoliton. The soliton, resides at some very large negative value of $x_3$. far to the left of the soliton the field configuration must be vacuum $\phi^1=\phi^2=0; \ z=1$. Thus the topological charge calculated on a surface enclosing such a soliton is equal to the two dimensional topological charge - the winding of the phase $\chi$ on any plain pierced by the string. An identical argument applies for the antisoliton, which resides at large positive value of $x_3$. Thus indeed our ansatz is appropriate for the string connecting a soliton and an antisoliton residing far apart. For our ansatz the equation of motion for the field $\chi$ is trivially satisfied. The equation for $z$ becomes
\begin{equation} 4z''=4\Lambda^2(z-1)\end{equation}
where $z'\equiv \frac{dz}{dr^2}$
For the solution to be well defined in the middle of the string and approach vacuum far away from it, $z$ must have the asymptotic behavior:
\begin{equation} z(0)=-1, \mbox{ } z(\infty)=1\end{equation}
The solution is easy to find
\begin{equation} z(r^2) = 1 - 2 \exp\{-\Lambda r^2\} \label{zetsifir}\end{equation}
The solution has some intuitively expected properties: it has a finite width, outside which the fields approach vacuum, while inside the string the field values are different from the vacuum and thus it carries a finite energy density. The string tension can be calculated exactly
\begin{equation} \sigma=8\pi\frac{\Lambda}{e^2}\end{equation}
Nevertheless, the solution is rather peculiar, since it does not approach the vacuum exponentially, but rather as a Gaussian. the string therefore has a very sharply defined width, outside of which the vacuum is reached very quickly. In a theory with a finite mass gap and a finite number of massive excitations such behavior is impossible.
This strange behavior can be traced back to our noncanonical kinetic term. Indeed, for simple dimensional reasons, the kinetic energy for a rotaionally invariant configuration is a second derivative with respect to $r^2$ rather than $r$, which results in a Gaussian rather than exponential decay of the solution.
\section{The $Z_N$ preserving perturbation.}
The perturbation considered in the last section was the sumplest potential that breaks the $SO(3)$ as well as the $Sdiff(2)$ symmetries but leaves an $O(2)$ subgroup of $SO(3)$ and large subgroup $Sdiff(2)$, eq.(\ref{subgroup}) intact. Following the parallel with the 2+1 dimensional discussion, we expect the global symmetries to be broken down to $Z_N$ if our effective theory has a chance of mimicking some properties of $SU(N)$ gauge theories. In this section therefore we consider an additional perturbation, which breaks the remaining $O(2)$ symmetry down to the $Z_N$ subgroup.
We modify the Lagrangian to
\begin{equation} L =\frac{1}{16e^2}F^2 - \frac{2}{e^2}\Lambda^2 (z - 1)^2 \Big[1- \mu (\psi^N + \psi^{\star N})\Big] = \frac{1}{16e^2}F^2 - \frac{2}{e^2}\Lambda^2 (z - 1)^2 \Big[1- 2 \mu (1-z^2)^{N/2} \cos{N \chi}\Big] \label{tamlang} \end{equation}
We will only consider regime where the minimum of the potential is at $z=1$. It is easy to see that this is the case as long as
\begin{equation} \mu<\frac{1}{2}\end{equation}
For field configurations which do not depend on the longitudinal coordinate $x_3$, the energy per unit length is given by
\begin{equation}\label{energy}
E=\int d^2x \frac{1}{2e^2}(\epsilon_{ij}\partial_iz\partial_j\chi)^2+\frac{2}{e^2}\Lambda^2 (z - 1)^2 \Big[1- 2 \mu (1-z^2)^{N/2} \cos{N \chi}\Big]
\end{equation}
\subsection{Perturbative solution}
Let us first consider the perturbation to be small, $\mu \ll 1$, and find the first order corrections to the string solution of the previous section.
We take the following ansatz for the perturbative solution:
\begin{equation}
z(r, \theta) = z(r); \chi = \theta + \chi_1(r, \theta)=\theta+f(r)\sin N\theta\end{equation}
where $z(r)$ is given by eq.(\ref{zetsifir}). This is not the most general form of the perturbation, but which nevertheless yields a solution to first order in $\mu$, as we now show.
To first order in $\mu$ the equation for $\chi_1$ is
\begin{equation}
\frac{1}{e^2}8N^2(z')^2f\sin N\theta=\frac{1}{e^2}N\mu\Lambda^2(z-1)^2(1-z^2)^{N/2}\sin N\theta\end{equation}
solved by
\begin{equation}\label{pertsol} f(r^2)=\frac{\mu}{N}\Big[2e^{-\Lambda r^2}(1-e^{-\Lambda r^2})\Big]^{N/2}
\end{equation}
The second minimization equation reads
\begin{equation} \frac{1}{e^2}8N\Big[2z''f+z'f'\Big]=\frac{1}{e^2}4\mu\Lambda\Big[2(z-1)(1-z^2)^{N/2}-Nz(z-1)^2(1-z^2)^{N/2-1}\Big]\end{equation}
It is straightforward to check that this equation is indeed satisfied by the perturbative solution eq.(\ref{pertsol}) and $z(r)$ given by eq.(\ref{zetsifir}).
Calculating the longitudinal electric field corresponding to this solution we find
\begin{equation} F^{03}=-4\Lambda e^{-\Lambda r^2}\Big[1+\mu\Big(2e^{-\Lambda r^2}(1-e^{-\Lambda r^2})\Big)^{N/2}\cos N\theta\Big]\end{equation}
The electric field is concentrated within the radius $\Lambda^{1/2}$ in the transverse plane, with the $Z_N$ invariant perturbation providing a slight angular modulation, as expected.
\subsection{The General Solution}
Let us now consider the minimization equations beyond perturbation theory and beyond the simple ansatz of the previos subsection.
Minimization fo the energy functional eq.(\ref{energy}) gives the following equations:
\begin{eqnarray}
&&\frac{1}{e^2}\epsilon_{ij}\partial_j\chi\partial_iF=\frac{\partial U}{\partial z}\nonumber\\
&&\frac{1}{e^2}\epsilon_{ij}\partial_jz\partial_iF=-\frac{\partial U}{\partial \chi}
\end{eqnarray}
where
\begin{equation} F\equiv \frac{1}{2}F^{03}=\epsilon_{ij}\partial_iz\partial_j\chi\end{equation}
and $U$ is the potential energy in eq.(\ref{energy}).
Multiplying the first equation by $\partial_k z$, the second by $\partial_k \chi$ and subtracting, we find:
\begin{equation}\label{relation} \frac{1}{2e^2}\partial_k (F^2)=\partial_k U\end{equation}
For any finite energy configuration the electric field vanishes at infinity. Since the potential $U$ appearing in eq.(\ref{energy}) also vanishes at infinity, the integration constant needed to integrate eq.(\ref{relation}) is trivial and we find
\begin{equation} F^2=2e^2 U; \ \ \ \ \ F=\sqrt{2e^2U}\end{equation}
To solve the equation it is convenient to use the coordinates $(\tau=r^2,\theta)$:
\begin{equation}\label{equation} \partial_\tau z \partial_\theta \chi - \partial_\theta z \partial_\tau \chi = \sqrt{\frac{1}{2}e^2U}\end{equation}
This equation obviously has many solutions. The infinite degeneracy follows from a symmetry of the energy functional eq.(\ref{energy}). Consider a transformation
\begin{equation}\label{sdiffr} (z(x),\chi(x))\rightarrow (z(x'),\chi(x')); \ \ \ \ \ \ \frac{\partial(x'^{1}, x'^{2})}{\partial(x^1, x^2)}=1\end{equation}
Such transformations form the group of area preserving diffeomorphisms on a plain $SDiff(R^2)$. Note that it is a diffeomorphism transformation on the coordinate space rather than on the field space, and thus is very different from $Sdiff(2)$, which we discussed in the previous section.
This transformation is clearly a symmetry of the energy functional eq.(\ref{energy}). Thus, starting from any string solution one can generates an infinite number of degenerate solutions with the help of $SDiff(R^2)$ transformations. Note, that since the longitudinal electric field is itself invariant under eq.(\ref{sdiffr}), all these solutions have the same electric field profile.
We will discuss here two such solutions. Let us look for solution with a prescribed and simple dependence of $\chi$ on the angle : $\chi=\theta$.
Eq.(\ref{equation}) then becomes an equation for $z$:
\begin{equation} \partial_\tau z = \sqrt{\frac{1}{2}e^2U} = \sqrt{ \Lambda^2 (z - 1)^2 \Big[1- 2 \mu (1-z^2)^{N/2} \cos{N \theta}\Big] }\end{equation}
The dependence on $\theta$ here is parametric, and so for every value of $\theta$ it is solved by
\begin{equation}\tau=\int_{-1}^{z(\tau)} dz\frac{1}{ \sqrt{ \Lambda^2 (z - 1)^2 \Big[1- 2 \mu (1-z^2)^{N/2} \cos{N \theta}\Big] }}\end{equation}
The solution has the correct asymptotics, since as $\tau\rightarrow\infty$ the function $z$ has to approach unity for the RHS to diverge. In fact it is easy to find the large distance asymptotics of the solution. When $z$ is close to unity, we can neglect the term proportional to $\mu$ in the denominator, and for the IR asymptotics we have
\begin{equation}\tau=\int_{-1}^{z(\tau)} dz\frac{1}{ \sqrt{ \Lambda^2 (z - 1)^2 }}\end{equation}
which is solved by
\begin{equation} z(\tau\rightarrow\infty)=1-2e^{-\Lambda\tau}\end{equation}
This is the same as eq.(\ref{zetsifir}), showing that the IR asymptotics of the string solution is unaffected by the $Z_N$ perturbation.
Let us now consider a solution where $z$ only has radial dependence. In this case, we have:
\begin{equation} \partial_\tau z \partial_\theta \chi = \sqrt{ \Lambda^2 (z - 1)^2 \Big[1- 2 \mu (1-z^2)^{N/2} \cos{N \chi}\Big] }\end{equation}
This can be formally solved for $\theta$ at fixed $r$:
\begin{equation} \theta = \int_0^{\chi{(r,\theta)}} \frac{z' d \chi}{\sqrt{ \Lambda^2 (z - 1)^2 \Big[1- 2 \mu (1-z^2)^{N/2} \cos{N \chi}\Big] }}\end{equation}
The right hand side can be expressed in terms of elliptic integrals:
\begin{equation} \theta = \frac{2}{N}\frac{z'}{\sqrt{\Lambda^2 (z-1)^2 (1-2\mu (1-z^2)^{N/2})}}F(\frac{N \chi}{2}, \frac{4\mu (1-z^2)^{N/2}}{2\mu(1-z^2)^{N/2} -1})\end{equation}
where $F(\phi,m)$ is the incomplete elliptic integral of the first kind:
\begin{equation} F(\phi,m) = \int_0^\phi (1-m\sin{\theta}^2)^{-1/2} d\theta\end{equation}
The solution has to satisfy the boundary condition
\begin{equation}\chi(\theta+2\pi) = \chi(\theta)+ 2 \pi\end{equation}
Imposing this condition gives the equation for the radial dependence of $z$. Using the relation $F(\frac{k \pi}{2},m) = k K (m)$, where $K(m)$ is the complete elliptic integral of the first kind, we have:
\begin{equation} 2\pi = \frac{4z'}{\sqrt{\Lambda^2 (z-1)^2 (1-2\mu (1-z^2)^{N/2})}}K(\frac{4\mu (1-z^2)^{N/2}}{2\mu(1-z^2)^{N/2} -1})\end{equation}
One can easily check, that in the infrared for $z\rightarrow 1$ the equation reduces to
\begin{equation} z'=\Lambda (1-z)\end{equation}
and thus has the same asymptotics as in eq.(\ref{zetsifir}).
\section{Discussion}
In this paper we tried to follow the template of 2+1 dimensions and, based on a couple of simple requirements "`guess"' a scalar theory which could be a candidate of the effective theory of 3+1 dimensional gauge theories. The theory we ended up with is not entirely satisfactory, but it does have several interesting and intriguing features.
In the Abelian limit it has a large global symmetry group, which is spontaneously broken by lowest energy classical solutions. This symmetry is not reflected in the observables which we tentatively identified with the observables of QED. This is similar to 2+1 dimensions, where the electromagnetic field was invariant under the magnetic $U(1)$ symmetry which acted nontrivially on the vortex field. In our 3+1 dimensional model the electromagnetic field is also invariant under the action of the (large) global symmetry group $Sdiff(2)$ which is nontrivially represented on the effective scalar fields.
Just like in 2+1 dimensions, this global symmetry is broken by the lowest energy configurations. However, the situation here is more complicated. Whereas in 2+1 dimensions the symmetry breaking pattern is the standard one, in our 3+1 dimensional model the space of vacuum configurations is very large. It includes field configurations that have nontrivial spatial dependence, and thus breaks translational invariance in addition to the global $Sdiff(2)$ symmetry. In fact, it could well be that classical analysis fails in this model quite badly. Many of the classical vacua differ from each other only in finite region of space. Generically in such a case one expects that upon quantization these configurations become connected by tunneling transitions of finite probability. Thus it is natural to expect that the quantum portrait of moduli space is significantly different from the classical one. This is a very interesting question, but it is far beyond the scope of the present work.
Upon introduction of the symmetry breaking perturbation, the model provides a simple classical description of confinement similarly to the 2+1 dimensional case. However here also there are some peculiarity. In particular, string solutions are infinitely degenerate, as static energy for configurations which depend only on two coordinates has an additional diffeomorphism invariance. This is a different invariance than in the Abelian limit, as it involves diffeomorphism transformations in coordinate space rather than in field space. Nevertheless it results in degeneracy of the solutions, although all such solutions yield the same electric field. In the sense of electric field distributions, the solution seems to be unique. This again begs the question about the behavior of such a string in a quantum theory, since it carries a large entropy associated with large degeneracy.
We note that the string degeneracy is lifted if one adds the standard kinetic term for the $O(3)$ sigma model fields, $\partial^\mu\phi^a\partial_\mu\phi^a$. Addition of such a term also makes our model identical with the one proposed by Faddeev and Niemi in \cite{niemi} as an effective theory of QCD. Interestingly, the picture we suggest is quite distinct from and complementary to that of \cite{niemi}. Whereas the authors of \cite{niemi} concentrated on closed string solutions supposedly representing glueballs, our picture is that of open strings, with the endpoints corresponding to "`constituent gluons"' like in 2+1 dimensions\cite{dualreview},\cite{baruch}. The stability of closed strings in the Faddeev-Niemi model is achieved due to nontrivial twisting of the phase of the scalar field along the string. Open strings on the other hand, do not require any twist and in principle can break into shorter strings similarly to QCD. The approximate stability of relatively long strings must be due to dynamical properties of the theory which should make the endpoints sufficiently heavy\cite{greensite}.
Finally we note that with the standard kinetic term our model becomes very similar to $CP^1$ model, which has been recently discussed in the literature in relation to effective models of confinement\cite{cpn}.
\section{Appendix.}
In this appendix we show that the model considered in this paper does not admit two photon solutions with arbitrary polarizations.
We are looking for two photon solutions for which the electromagnetic tensor is of the form:
\begin{equation} \tilde{F}_{\mu\nu}= \partial_{[\mu}z\partial_{\nu]}\chi=A(k_{\mu}\epsilon^1_{\nu}-k_{\nu}\epsilon^1_{\mu}) \cos{kx} + B(p_{\mu}\epsilon^2_{\nu}-p_{\nu}\epsilon^2_{\mu}) \cos{px}\end{equation}
Foer simplicity we choose the case when the first photon has momentum $k$ in $x$-direction and polarization $a$ in $y$-direction, while the second photon has momentum $p$ in $y$-direction and polarization $b$ in $z$ direction. Note that this case is not covered by our construction of two photon states in the body of the paper.
Now, for components of $\tilde{F}_{\mu\nu}$, we have:
\begin{equation}\partial_{[0}z \partial_{1]}\chi = 0 = \partial_{[1}z \partial_{3]}\chi = 0\end{equation}
\begin{equation} \partial_{[0}z \partial_{2]}\chi = ka \cos{kx} = - \partial_{[1}z \partial_{2]}\chi\end{equation}
\begin{equation} \partial_{[0}z \partial_{3]}\chi = pb \cos{px} = -\partial_{[2}z \partial_{3]}\chi \end{equation}
Introducing new coordinates $(x,y,z,t) \rightarrow (\bar{x}=t-x, \bar{y} = t-y, \bar{t} = t, \bar{z} = z)$, and using unbarred symbols for notational simplicity, we have:
\begin{equation} \label{first}\partial_{[t}z \partial_{y]}\chi = \partial_{[t}z \partial_{z]}\chi = \partial_{[x}z \partial_{z]}\chi =0\end{equation}
\begin{equation} \partial_{[t}z \partial_{x]}\chi = \partial_{[x}z \partial_{y]}\chi= -ka \cos{kx}\end{equation}
\begin{equation}\label{last} \partial_{[y}z \partial_{z]}\chi = pb \cos{py}\end{equation}
These equations have no solutions. Assuming $\partial_tz\ne 0$, the first two equations in eq.(\ref{first}) imply $ \partial_{y}z \partial_{z}\chi- \partial_{z}z \partial_{y}\chi = 0$, which contradicts eq.(\ref{last})the last equation. Alternatively, assuming $\partial_tz= 0$, implies vanishing of either $\partial_t\chi$, or two other partial derivatives of $z$ . It is then easy to see that both these options are in conflict with the rest of the equations. The result is that a two photon state with this polarization pattern cannot be constructed in this model.
The model also contains solutions which do not satisfy the homogeneous Maxwell equation. As an example of such a solution consider the configuration
\begin{equation} \chi=\sin p\cdot x; \ \ \ z=\sin k\cdot x\end{equation}
It is easy to see that this configuration satisfies equations of motion, provided
\begin{equation} (p\cdot k)^2-p^2k^2=0\end{equation}
A simple example is a lightlike momentum $k^\mu$ and a spacelike momentum $p^\mu$ satisfying $p\cdot k=0$. This yields the dual field strength
\begin{equation} \tilde F_{\mu\nu}\propto (k_\mu p_\nu-k_\nu p_\mu)[\cos (p+k)\cdot x+\cos (p-k)\cdot x]\end{equation}
which is not conserved
\begin{equation} \partial^\mu\tilde F_{\mu\nu}\propto p^2 k_\nu [\sin (p+k)\cdot x+\sin(p-k)\cdot x]
\end{equation}
In fact, both momenta $k+p$ and $k-p$ are spacelike, and thus $\tilde F_{\mu\nu}$ looks tachyonic. However, as mentioned in the Discussion, since the model classically has many degenerate vacua with broken translational invariance, interpretaton of classical solutions as excitations is not so clear.
\section*{Acknowledgments}
The research was supported by the DOE grant DE-FG02-92ER40716. We thank Alexei Yung and Michael Lublinsky for useful conversations.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,147 |
I am coding a tool in which we can add edit or delete from a file but when i read the file and try to write one record at the end of file, m getting an error.
"ALLOC SHR REU FILE(HI) DATASET('S2.REXX.FILE.BKUP')"
"ALLOC SHR REU FILE(HI1) DATASET('S2.REXX.FILE.BKUP')"
System abend code 913, reason code 00000056.
Abend in host command EXECIO or address environment routine TSO.
EXECIO error while trying to GET or PUT a record.
ICH* messages are RACF messages and it clearly says that you don't have edit/update access to the dataset.
It says that the ID is intending to UPDATE/EDIT the dataset, but the ID has only READ access.
The ID doesn't have access to write to 'this' rexx logfile but Nitika doesn't know why there is a write to this file.
I suspect you are not showing the entire rexx program and the problem is in another section of the program.
Add a TRACE('R') to the beginning of the program.
This seems to be a part of the process and the error may be coming from a different part of code. You may need to check what is the high level qualifier GFST with your colleagues.
Please analyze/let us know the whole process to see where it is going to write that log file.
*Pedro submitted the post before I did But we are trying to say the same thing.
Now that I have coded it I can read your code better. Please code it yourself in future.
I see that with the first ALLOC you allocate to FILE(HI) but your DISKR references HI1. | {
"redpajama_set_name": "RedPajamaC4"
} | 637 |
Q: Can the Bessel functions tend to a plane wave? Can the Bessel functions tend to a plane wave?
If I have this function:
$$
y(u)= c_1J_{-\sqrt{b}/2}(e^{2u}/6)+c_2J_{\sqrt{b}/2}(e^{2u}/6)+c_1J_{-i\sqrt{b}/2}(e^{2u}/6)+c_2J_{i\sqrt{b}/2}(e^{2u}/6)
$$
with b a positive number, for $u \rightarrow - \infty$ I find an exponential behaviour using Mathematica with "Series[]". But I don't know how to move on and how to collect together the same kind of exponential. May you help me?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,913 |
Bibliographical Note
This Dover edition, first published in 1966 and republished in 2003, is an unabridged republication of the work originally published by Macmillan and Company, London, in 1870. As such, it is the complete text of the third revised edition.
Library of Congress Cataloging-in-Publication Data
Abbott, Edwin Abbott, 1838–1926.
A Shakespearian grammar : an attempt to illustrate some of the differences between Elizabethan and modern English / E.A. Abbott.
p. cm.
Originally published: London : Macmillan, 1870.
Includes index.
9780486148731
1. Shakespeare, William, 1564–1616—Language. 2. English language—Early modern, 1500–1700—Grammar. I. Title.
PR3075.A4 2003
822.3'3—dc21
2003053173
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
# Table of Contents
Title Page
Copyright Page
PREFACE TO THIRD EDITION.
PREFACE TO FIRST EDITION.
INTRODUCTION.
GRAMMAR.
PROSODY.
SIMILE AND METAPHOR.
NOTES AND QUESTIONS. \- MACBETH, ACT III.
INDEX TO THE QUOTATIONS \- FROM SHAKESPEARE'S PLAYS.
VERBAL INDEX.
DOVER BOOKS
# PREFACE TO THIRD EDITION.
THE success which has attended the First and Second Editions of the "SHAKESPEARIAN GRAMMAR," and the demand for a Third Edition within a year of the publication of the First, has encouraged the Author to endeavour to make the work somewhat more useful, and to render it, as far as possible, a complete book of reference for all difficulties of Shakespearian syntax or prosody. For this purpose the whole of Shakespeare has been re-read, and an attempt has been made to include within this Edition the explanation of every idiomatic difficulty (where the text is not confessedly corrupt) that comes within the province of a grammar as distinct from a glossary.
The great object being to make a useful book of reference for students, and especially for classes in schools, several Plays have been indexed so fully that with the aid of a glossary and historical notes the references will serve for a complete commentary. These Plays are, As You Like It, Coriolanus, Hamlet, Henry V., Julius Cœsar, Lear, Macbeth, Merchant of Venice, Midsummer Night's Dream, Richard II., Richard III., Tempest, Twelfth Night. It is hoped that these copious indexes will meet a want, by giving some definite work to be prepared by the class, whether as a holiday task or in the work of the term. The want of some such distinct work, to give thoroughness and definiteness to an English lesson, has been felt by many teachers of experience. A complete table of the contents of each paragraph has been prefixed, together with a Verbal Index at the end. The indexes may be of use to students of a more advanced stage, and perhaps may occasionally be found useful to the general reader of Shakespeare.
A second perusal of Shakespeare, with a special reference to idiom and prosody, has brought to light several laws which regulate many apparent irregularities. The interesting distinction between thou and you (Pars. 231—235), for example, has not hitherto attracted the attention of readers, or, as far as I am aware, of commentators on Shakespeare. The use of the relative with plural antecedent and singular verb (Par. 246) ; the prevalence of the third person plural in -s (Par. 333), which does not appear in modern editions of Shakespeare ; the "confusion of proximity" (Par. 412) ; the distinction between an adjective before and after a noun ; these and many other points which were at first either briefly or not at all discussed, have increased the present to more than thrice the size of the original book. I propose now to stereotype this edition, so that no further changes need be anticipated.
It may be thought that the amplification of the Prosody is unnecessary, at all events, for the purpose of a school-book. My own experience, however, leads me to think that the Prosody of Shakespeare has peculiar interest for boys, and that some training in it is absolutely necessary if they are to read Shakespeare critically. The additions which have been made to this part of the book have sprung naturally out of the lessons in English which I have been in the habit of giving ; and as they are the results of practical experience, I am confident they will be found useful for school purposes. A conjectural character, more apparent however than real, has perhaps been given to this part of the book from the necessity that I felt of setting down every difficult verse of Shakespeare. where the text was not acknowledged as corrupt, or where the difficulty was more than slight. Practically, I think, it will be found that the rules of the Prosody will be found to solve most of the difficulties that will present themselves to boys—at least, in the thirteen Plays above mentioned.
Besides obligations mentioned in the First Edition, I must acknowledge the great assistance I have received from MÄTZNER'S Englische Grammatik (3 vols., Berlin, 1865), whose enormous collection of examples deserves notice. I am indebted to the same author for some points illustrating the connection between Early and Elizabethan English. Here, however, I have received ample assistance from Mr. F. J. Furnivall, Mr. R. Morris, and others, whose kindness I am glad to have an opportunity of mentioning. In particular, I must here acknowledge my very great obligation to the Rev. W. W. Skeat, late Fellow of Christ's College, Cambridge, whose excellent edition of William of Palerne (Early English Text Society, 1867), and whose Mœso-Gothic Dictionary (Asher, London, 1866), have been of great service to me. Mr. Skeat also revised the whole of the proof-sheets, and many of his suggestions are incorporated in the present work. I may add here, that in discussing the difference between "thou" and "you" (231–5), and the "monosyllabic foot" (480–6), I was not aware that I had been anticipated by Mr. Skeat, who has illustrated the former point (with reference to Early English) in William of Palerne, p. xlii., and the latter in his Essay on the Metres or Chaucer (vol. i., Aldine Edition, London, 1866). The copious Index to Layamon, edited by Sir Frederick Madden, has also been of great service. I trust that, though care has been taken to avoid any unnecessary parade of Anglo-Saxon, or Early English, that might interfere with the distinct object of the work, the information on these points will be found trustworthy and useful. The Prosody has been revised throughout by Mr. A. J. Ellis, whose work on Early English Pronunciation is well known. Mr. Ellis's method of scansion and notation is not in all respects the same as my own, but I have made several modifications in consequence of his suggestive criticisms.
I have now only to express my hope that this little book may do something to forward the development of English instruction in English schools. Taking the very lowest ground, I believe that an intelligent study of English is the shortest and safest way to attain to an intelligent and successful study of Latin and Greek, and that it is idle to expect a boy to grapple with a sentence of Plato or Thucydides if he cannot master a passage of Shakespeare or a couplet of Pope. Looking, therefore, at the study of English from the old point of view adopted by those who advocate a purely classical instruction, I am emphatically of opinion that it is a positive gain to classical studies to deduct from them an hour or two every week for the study of English. But I need scarcely say that the time seems not far off when every English boy who continues his studies to the age of fifteen will study English for the sake of English; and where English is studied Shakespeare is not likely to be forgotten.
E. A. A.
30th May, 1870.
# PREFACE TO FIRST EDITION.
THE object of this work is to furnish students of Shakespeare and Bacon with a short systematic account of some points of difference between Elizabethan syntax and our own. The words of these authors present but little difficulty. They can be understood from glossaries, and, even without such aid, a little reflection and attention to the context will generally enable us to hit the meaning. But the differences of idiom are more perplexing. They are more frequent than mere verbal difficulties, and they are less obvious and noticeable. But it need hardly be said, that if we allow ourselves to fancy we are studying Shakespeare critically, when we have not noticed and cannot explain the simplest Shakespearian idiom, we are in danger of seriously lowering our standard of accurate study, and so far from training we are untraining our understanding. Nor is it enough to enumerate unusual idioms without explaining them. Such is not the course we pursue in Latin and Greek, and our native tongue should either not be studied critically at all, or be studied as thoroughly as the languages of antiquity.
The difficulty which the author has experienced in teaching pupils to read Shakespearian verse correctly, and to analyse a metaphorical expression, has induced him to add a few pages on Shakespeare's prosody and on the use of simile and metaphor.
A very important question in the study of English is, what should be the amount and nature of the assistance given to students in the shape of notes. It is clear that the mere getting up and reproducing a commentator's opinions, though the process may fill a boy with useful information, can in no sense be called a training. In the Notes and Questions at the end of this volume I have tried to give no more help than is absolutely necessary. The questions may be of use as a holiday-task, or in showing the student how to work the Grammar. They have been for the most part answered by a class of boys from fourteen to sixteen years old, and some by boys much younger.
In some of the sections of the Prosody I must acknowledge my obligations to Mr. W. S. Walker's work on Shakespeare's Versification. Other obligations are acknowledged in the course of the work; but the great mass of the examples have been collected in the course of several years' close study of Shakespeare and contemporaneous authors. I am aware that there will be found both inaccuracies and incompleteness in this attempt to apply the rules of classical scholarship to the criticism of Elizabethan English, but it is perhaps from a number of such imperfect contributions that there will at last arise a perfect English Grammar.
# REFERENCES.
The following works are referred to by the pages :—
Ascham's Scholemaster . (Mayor) . | London, | 1863.
---|---|---
The Advancement of Learning | Oxford, | 1640.
Bacon's Essays (Wright) | London, | 1868.
Ben Jonson's Works (Gifford) | London, | 1838.
North's Plutarch | London, | 1656,
Florio's Montaigne | London, | 1603.
Wager, Heywood, Ingelend, &c., and sometimes Beaumont and Fletcher, are quoted from "The Songs of the Dramatists," J. W. Parker, 1855.
# WORKS REFERRED TO BY ABBREVIATIONS.
Some of the plays of Shakespeare are indicated by the initials of the titles, as follow:
A. W. | All's Well that Ends Well.
---|---
A. and C. | Antony and Cleopatra.
A. Y. L. | As You Like It.
C. of E. | Comedy of Errors.
J. C. | Julius Caesar.
L. L. L. | Love's Labour Lost.
M. for M. | Measure for Measure.
M. of V. | Merchant of Venice.
M. W. of W. | Merry Wives of Windsor.
M. N. D. | Midsummer Night's Dream.
M. Ado . | Much Ado about Nothing.
P. of T. | Pericles of Tyre.
R. and J. | Romeo and Juliet.
T. of Sh. | Taming of the Shrew.
T. of A. | Timon of Athens.
T. A. | Titus Andronicus.
Tr. and Cr. | Troilus and Cressida.
T. N. | Twelfth Night.
T. G. of V. | Two Gentlemen of Verona.
W. T. | Winter's Tale.
(The quotations are from the Globe edition unless otherwise specified.)
Asch. | Ascham's Scholemaster.
---|---
B. E. | Bacon's Essays.
B. and F. | Beaumont and Fletcher
B. J. | Ben Jonson.
B, J. E. in & c. | Every Man in his Humour.
E. out & c. | Every Man out of his Humour.
Sil. Wom. | Cynthia's Revels.
Sil. Wom. | Silent Woman.
Sejan. | Sejanus.
Sad Sh. | Sad Shepherd.
L. C. | Lover's Complaint.
N. P. | North's Plutarch.
P. P. | Passionate Pilgrim.
R. of L. | Rape of Lucrece.
Sonn. | Shakespeare's Sonnets.
V. and A. | Venus and Adonis.
Numbers in parentheses thus (81) refer to the paragraphs of the Grammar.
# INTRODUCTION.
ELIZABETHAN English, on a superficial view, appears to present this great point of difference from the English of modern times, that in the former any irregularities whatever, whether in the formation of words or in the combination of words into sentences, are allowable. In the first place, almost any part of speech can be used as any other part of speech. An adverb can be used as a verb, " They askance their eyes " (R, of L.) ; as a noun, "the backward and abysm of time" (Sonn.) ; or as an adjective, "a seldom pleasure" (Sonn.). Any noun, adjective, or neuter verb can be used as an active verb. You can "happy" your friend, " malice " or "foot" your enemy, or "fall" an axe on his neck. An adjective can be used as an adverb ; and you can speak and act "easy," "free," "excellent :" or as a noun, and you can talk of "fair" instead of "beauty," and "a pale" instead of "a paleness." Even the pronouns are not exempt from these metamorphoses. A "he" is used for a man, and a lady is described by a gentleman as "the fairest she he has yet beheld." Spenser asks us to
"Come down and learne the little what
That Thomalin can sayne."—Calend. Jul. v. 31 (Nares).
And Heywood, after dividing human diners into three classes thus—
"Some with small fare they be not pleased,
Some with much fare they be diseased,
Some with mean fare be scant appeased,"
adds with truly Elizabethan freedom—
" But of all somes none is displeased
To be welcome."
In the second place, every variety of apparent grammatical inaccuracy meets us. He for him, him for he; spoke and took, for spoken and taken; plural nominatives with singular verbs; relatives omitted where they are now considered necessary ; unnecessary antecedents inserted ; shall for will, should for would, would for wish ; to omitted after "I ought," inserted after " I durst ;" double negatives ; double comparatives ("more better," &c.) and superlatives ; such followed by which, that by as, as used for as if ; that for so that ; and lastly, some verbs apparently with two nominatives, and others without any nominative at all. To this long list of irregularities it may be added that many words, and particularly prepositions and the infinitives of verbs, are used in a different sense from the modern. Thus—
" To fright you thus methinks I am too savage,"—
Macb. iv. 2. 70.
does not mean "I am too savage to fright you." "Received of the most pious Edward" (170) does not mean "from Edward," but "by Edward;" and when Shakespeare says that "the rich" will not every hour survey his treasure, "for blunting the fine point of seldom pleasure," he does not mean "for the sake of," but "for fear of" blunting pleasure.
On a more careful examination, however, these apparently disorderly and inexplicable anomalies will arrange themselves under certain heads. It must be remembered that the Elizabethan was a transitional period in the history of the English language. On the one hand, there was the influx of new discoveries and new thoughts requiring as their equivalent the coinage of new words (especially words expressive of abstract ideas); on the other hand, the revival of classical studies and the popularity of translations from Latin and Greek authors suggested Latin and Greek words (but principally Latin) as the readiest and most malleable metal, or rather as so many ready-made coins requiring only a slight national stamp to prepare them for the proposed augmentation of the currency of the language. Moreover, the long and rounded periods of the ancients commended themselves to the ear of the Elizabethan authors. In the attempt to conform English to the Latin frame, the constructive power of the former language was severely strained.
The necessity of avoiding ambiguity and the difficulty of connecting the end of a long sentence with the beginning, gave rise to some irregularities, to the redundant pronoun (242), the redundant 'that' (285), and the irregular 'to' (416).
But, for the most part, the influence of the classical languages was confined to single words, and to the rhythm of the sentence. The syntax was mostly English both in its origin and its development, and several constructions that are now called anomalous (such as the double negative [406] and the double comparative [409]) have, and had from the earliest period, an independent existence in English, and are merely the natural results of a spirit which preferred clearness and vigour of expression to logical symmetry. Many of the anomalies above mentioned may be traced back to some peculiarities of Early English, modified by the transitional Elizabethan period. Above all, it must be remembered that Early English was far richer than Elizabethan English in inflections. As far as English inflections are concerned the Elizabethan period was destructive rather than constructive. Naturally, therefore, while inflections were being discarded, all sorts of tentative experiments were made: some inflections were discarded that we have restored, others retained that we have discarded. Again, sometimes where inflections were retained the sense of their meaning and power had been lost, and at other times the memory of inflections that were no longer visibly expressed in writing still influenced the manner of expression. Thus Ben Jonson writes :—
"The persons plural keep the termination of the first person singular. In former times, till about the reign of King Henry VIII. they were wont to be formed by adding en thus :—Loven, sayen, complainen. But now (whatsoever is the cause) it is quite grown out of use, and that other so generally prevailed that I dare not presume to set this on foot again."
He appears to be aware of the Midland plural in en (332) which is found only very rarely in Spenser and in Pericles of Tyre, but not of the Northern plural in es (333), which is very frequently found in Shakespeare, and which presents the apparent anomaly of a plural noun combined with a singular verb. And the same author does not seem to be aware of the existence of the subjunctive mood in English. He ignores it in his "Etymology of a Verb," and, in the chapter on " Syntax of a Verb with a Noun," writes as follows :—
" Nouns signifying a multitude, though they be of the singular number, require a verb plural :
" 'And wise men rehearsen in sentence,
Where folk be drunken there is no resistance.'"
—LYDGATE, lib.ii.
And he continues thus :—" This exception is in other nouns also very common, especially when the verb is joined to an adverb or conjunction : 'It is preposterous to execute a man before he have been condemned.'" It would appear hence that the dramatist was ignorant of the force of the inflection of the subjunctive, though he frequently uses it. Among the results of inflectional changes we may set down the following anomalies:—
I. Inflections discarded but their power retained. Hence (a) "spoke" (343) for "spoken," "rid" for "ridden." (b) "You ought not walk" for " You ought not walken " (the old infinitive). (c) The new infinitive (357) "to walk" used in its new meaning and also sometimes retaining its old gerundive signification. (d) To "glad" (act.), to "mad" (act.), &c. (290) for to "gladden," "madden," &c. (e) The adverbial e (1) being discarded, an adjective appears to be used as an adverb : "He raged more fierce," &c. (ƒ) "Other" is used for "other(e)," pl. "other men," &c. (g) The ellipsis of the pronoun (399) as a nominative may also be in part thus explained.
II. Inflections retained with their old power.
(a) The subjunctive inflection frequently used to express a condition—" Go not my horse," for "If my horse go not." Hence (b) as with the subj. appears to be used for as if, and for and if, but (in the sense of except) for except if, &c. (c) The plural in en; very rarely. (d) The plural in es or s ; far more commonly. (e) His used as the old genitive of he for of him. Me, him, &c. used to represent other cases beside the objective and the modern dative : "I am appointed him to murder you."
III. Inflections retained but their power diminished or lost.
(a) Thus 'he' for 'him,' 'him' for 'he ;' 'I' for 'me,' 'me' for 'I,' &c. (b) In the same way the s which was the sign of the possessive case had so far lost its meaning that, though frequently retained, it was sometimes replaced (in mistake) by his and her.
IV. Other anomalies may be explained by reference to the derivations of words and the idioms of Early English.
Hence can be explained (a) so followed by as; (b) such followed by which (found in E. E. sometimes in the form whuch or wuch) ; (c) that followed by as; (d) who followed by he; (e) the which put for which; (ƒ) shall for will, should for would, and would for wish.
The four above-mentioned causes are not sufficient to explain all the anomalies of Elizabethan style. There are several redundancies, and still more ellipses, which can only be explained as follows.
V. (a) Clearness was preferred to grammatical correctness, and (b) brevity both to correctness and clearness. Hence it was common to place words in the order in which they came uppermost in the mind without much regard to syntax, and the result was a forcible and perfectly unambiguous but ungrammatical sentence, such as :
* (a)"The prince that feeds great natures they will sway him."
B. J. Sejanus.
* (b)As instances of brevity :—
" Be guilty of my death since of my crime.—R. of L.
" It cost more to get than to lose in a day."—B. J. Poetaster.
VI. One great cause of the difference between Elizabethan and Victorian English is, that the latter has introduced or developed what may be called the division of labour. A few examples will illustrate this.
The Elizabethan subjunctive (see VERBS, SUBJUIVCTIVE) could be used (1) optatively, or (2) to express a condition or (3) a consequence of a condition, (4) or to signify purpose after "that." Now, all these different meanings are expressed by different auxiliaries—"would that !" "should he come," "he would find," "that he may see,"—and the subjunctive inflection is restricted to a few phrases with "if." " To walk" is now either (1) a noun, or (2) denotes a purpose, "in order to walk." In Elizabethan English, "to walk" might also denote "by walking," "as regards walking," "for walking;" a licence now discarded, except in one or two common phrases, such as "I am happy to say," &c. Similarly, Shakespeare could write "of vantage" for "from vantage-ground," "of charity" for "for charity's sake," "of mine honour" for "on my honour," "of purpose" for "on purpose," "of the city's cost" for "at the city's cost," "of his body" for "as regards his life," "made peace of enmity" for "peace instead of enmity," "we shall find a shrewd contriver of him" for "in him," "did I never speak of all that time" for "during all that time." Similarly "by" has been despoiled of many of its powers, which have been divided among "near," "in accordance with," "by reason of," "owing to." "But" has been forced to cede some of its provinces to "unless" and "except." Lastly, "that," in Early English the only relative, had been already, before the Elizabethan times, supplanted in many idioms by "who" and "which;" but it still retained its meanings of "because," "inasmuch as," and "when;" sometimes under the forms "for that," "in that;" sometimes without the prepositions. These it has now lost, except in a few colloquial phrases.
As a rule, then, the tendency of the English language has been to divide the labour of expression as far as possible by diminishing the task assigned to overburdened words and imposing it upon others. There are, of course, exceptions to this rule—notably "who" and "which ;" but this has been the general tendency. And in most cases it will be found that the Victorian idiom is clearer but less terse than the corresponding Elizabethan idiom which it has supplanted.
VII. The character of Elizabethan English is impressed upon its pronunciation, as well as upon its idioms and words. As a rule their pronunciation seems to have been more rapid than ours. Probably the greater influence of spoken as compared with written English, sanctioned many contractions which would now be judged intolerable if for the first time introduced. (See 461.) This, however, does not explain the singular variation of accent upon the same words in the same author. Why should "exile," "aspect," "confessor," and many other words, be accented now on the first, now on the second syllable? The answer is, that during the unsettled Elizabethan period the foreign influence was contending with varying success against the native rules of English pronunciation. The English rule, as given by Ben Jonson, is definite enough. "In dissyllabic simple nouns " (by which it is to be supposed he means un-compounded), "the accent is on the first, as 'bélief,' 'hónour,' &c." But he goes on to say, that " all verbs coming from the Latin, either of the supine or otherwise, hold the accent as it is found in the first person present of those Latin verbs." Hence a continual strife over every noun derived from Latin participles : the English language claiming the new comer as her naturalized subject, bound by English laws; the Latin, on the other hand, asserting a partial jurisdiction over her emigrants. Hence accéss and áccess, precépt and précept, contráct (noun) and cóntract, instinct and instinct, relápse and rélapse. The same battle raged over other Latin words not derived from participles : commérce and cómmerce, obdúrate and ódbdurate, sepúlchre and sépulchre, contráry and cóntrary, authórize and aúthorize, perséver and persevére, cónfessor and conféssor. The battle terminated in a thoroughly English manner. An arbitrary compromise has been effected between the combatants. Respéct, relápse, succéss, succéssor, were ceded to the Latin: áspect, cóllapse, áccess, sépulchre, were appropriated by the English. But while the contest was pending, and prisoners being taken and retaken on either side, we must not be surprised at finding the same word ranged now under native, now under foreign colours.
VIII. Words then used literally are now used metaphorically, and vice versâ.
The effect of this is most apparent in the altered use of prepositions. For instance, "by," originally meaning "near," has supplanted "of" in the metaphorical sense of agency, as it may in its turn be supplanted by "with" or some other preposition. This is discussed more fully under the head of prepositions (138). Here a few illustrations will be given from other words. It is not easy to discover a defined law regulating changes of metaphor. There is no reason why we should not, with Beaumont and Fletcher, talk of living at a "deep rate" as well as a "high rate." But it will be found with respect to many words derived from Latin and Greek, that the Elizabethans used them literally and generally; we, metaphorically. and particularly. Thus "metaphysical" was used by Shakespeare in the broader meaning of " supernatural ; " and " fantastical " could be applied even to a murder, in the wide sense of "imagined." So "exorbitant was " out of the path," " uncommon ;" now only applied to that which is uncommonly "expensive." So extravagant (" The extravagant and erring spirit," Hamlet, i. 1) has been restricted to " wandering beyond the bounds of economy." "To aggravate" now means, except when applied to disease, "to add to the mental burdens of any one," hence "to vex ;" but in Sonn. 146 we find "to aggravate thy store" in the literal sense of "to add to the weight of" or "increase." So "journall" meant "diurnal" or " daily ; " now it is restricted to a "daily" newspaper or memoir. The fact is that, in the influx of Greek and Latin words into the English language, many were introduced to express ideas that either could be, or were already, expressed in the existing vocabulary. Thus we do not require "metaphysical" to express that which is supernatural, nor "fantastical" to express that which is imagined; "exorbitant" is unnecessary in the sense of "uncommon ;" "extravagant" (though it has a special force in "the extravagant and erring spirit," Hamlet, i. 1) is not in most cases so obvious as "wandering ;" "increase" is simpler than "aggravate," and "daily" more English than "diurnal." Similarly "speculation is unnecessary to express the power of seeing, "advertised" useless in the sense of "warned" or " informed" (Lear, iv. 6. 214), "vulgar" in the sense of common. Such words, once introduced into the language, finding the broader room which they had been intended to fill already occupied, were forced to take narrower meanings. They did this, for the most part, by confining themselves to one out of many meanings which they had formerly represented, or by adopting metaphorical and philosophical instead of literal and material significations ; and as the sense of their derivation and original meaning became weaker, the transition became easier. This is not merely true of words derived from Latin and Greek. " Travail," for example, finding itself supplanted in its original sense by "work" or "labour," has narrowed itself to a special meaning : the same is true of "beef," "pork," &c.
On the other hand, some Latin and Greek words that express technicalities have, as the sense of their exact meaning was weakened, gradually become more loosely and generally used. Thus, "influence" means now more than the mere influence of the stars on men; "triumph," "preposterous," "pomp," " civil," "ovation," and "decimate," have lost much of their technical meaning. Of these words it may be said, that Shakespeare uses them more literally and particularly than we do. Thus, "triumph" is used for a show at a festival; "civil" is used for peaceful; "preposterous ass" (T. of Sh. iii. 1. 9) is applied to a man who put music before philosophy; "decimation " (T. of A. v. 1. 31) is used in its technical sense for " a tithed death."
One cause that has affected the meaning of Latin-derived words has been the preference with which they have been selected in order to express depreciation. This has narrowed some words to an unfavourable signification which they did not originally possess. Thus, "impertinent" in Elizabethan authors meant "not to the point;" "officious" could then mean " obliging," and a clever person could be described as "an admirable conceited fellow" (W. T. iv. 4. 203).
A classical termination (446) may sometimes be treated as active or as passive. Hence "plausibly" is used for "with applause" actively.
"The Romans plausibly did give consent."
—R. of L.
"A very inconsiderate (inconsiderable) handful of English."
N. P. Appendix 31.
Thus, on the one hand, we have "fluxive eyes" (eyes flowing with tears: L. C. 8), and on the other the more common passive sense, as "the inexpressive she" (the woman whose praises cannot be expressed).
With respect to words of English or French origin, it is more difficult to establish any rule. All that can be said is that the Elizabethan, as well as the Victorian meaning, may be traced to the derivation of the word. Why, for instance, should not Ben Jonson write—
"Frost fearing myrtle shall impale my head."
—Poetast. i. 1.
i.e. "take in within its pale, surround," as justifiably as we use the word in its modern sense of " transfixing ? " Why should not sirens "train" (draw or decoy—trahere) their victims to destruction, as well as educators "train" their pupils onward on the path of knowledge ? We talk of "a world of trouble" to signify an infinity ; why should not Bacon (E. 38) talk of "a globe of precepts?" Owing to the deficiency of their vocabulary, and their habit of combining prepositions with verbs, to make distinct words almost like the Germans, the Elizabethans used to employ many common English words, such as "pass," " hold," " take," in many various significations. Thus we find "take" in the sense of (1) "bewitch;" (2) "interrupt" ("You take him too quickly, Marcius," B. J. Poetast.) ; (3) "consider" (" The whole court shall take itself abused," B. J. Cy.'s Rev. v. 1) ; (4) "understand " (" You'll take him presently," E. out & c. i. 1) ; and (5) "resort to" ("He was driven by foule weather to take a poor man's cottage," N. P. 597). With prepositions the word has many more meanings. "Take out"=" copy;" " take in"=" subdue;" "take up "= "borrow;" "take in with " (Bacon)=" side with;" "take up "="pull up" of a horse. And these meanings are additional to the many other meanings which the word still retains. To enter further into the subject of the formation and meaning of words is not the purpose of this treatise. The glossaries of Nares and Halliwell supply the materials for a detailed study of the subject. One remark may be of use to the student before referring him to the following pages. The enumeration of the points of difference between Shakespearian and modern English may seem to have been a mere list of irregularities and proofs of the inferiority of the former to the latter. And it is true that the former period presents the English language in a transitional and undeveloped condition, rejecting and inventing much that the verdict of posterity has retained and discarded. It was an age of experiments, and the experiments were not always successful. While we have accepted copious, ingenious , disloyal, we have rejected as useless copy (in the sense of "plenty"), ingin, and disnoble. But for freedom, for brevity and for vigour, Elizabethan is superior to modern English. Many of the words employed by Shakespeare and his contemporaries were the recent inventions of the age; hence they were used with a freshness and exactness to which we are strangers. Again, the spoken English so far predominated over the grammatical English that it materially influenced the rhythm of the verse (see Prosody), the construction of the sentence, and even sometimes (460) the spelling of words. Hence sprung an artless and unlaboured harmony which seems the natural heritage of Elizabethan poets, whereas such harmony as is attained by modern authors frequently betrays a painful excess of art. Lastly, the use of some few still remaining inflections (the subjunctive in particular), the lingering sense of many other inflections that had passed away leaving behind something of the old versatility and audacity in the arrangement of the sentence, the stern subordination of grammar to terseness and clearness, and the consequent directness and naturalness of expression, all conspire to give a liveliness and wakefulness to Shakespearian English which are wanting in the grammatical monotony of the present day. We may perhaps claim some superiority in completeness and perspicuity for modern English, but if we were to appeal on this ground to the shade of Shakespeare in the words of Antonio in the Tempest,—
"Do you not hear us speak?"
we might fairly be crushed with the reply of Sebastian—
" I do; and surely
It is a sleepy language."
# GRAMMAR.
# ADJECTIVES.
1. Adjectives are freely used as Adverbs.
In Early English, many adverbs were formed from adjectives by adding e (dative) to the positive degree: as bright, adj. ; brighte, adv. In time the e was dropped, but the adverbial use was kept. Hence, from a false analogy, many adjectives (such as excellent) which could never form adverbs in e, were used as adverbs. We still say colloquially, "come quick;" "the moon shines bright," &c. But Shakespeare could say:
"Which the false man does easy."—Macb. ii. 3. 143.
"Some will dear abide it."—J. C. iii. 2. 119.
"Thou didst it excellent."—T. of Sh. i. 1. 89.
"Which else should free have wrought."—Macb. ii. 1. 19.
"Raged more fierce."—Rich. II. ii. 1. 173.
"Grow not instant old."—Ham. i. 5. 94.
"'Tis noble spoken."—A. and C. ii. 2. 99.
"Did I expose myself pure for his love."—T. N. v. 1. 86.
" Equal ravenous as he is subtle."—Hen. VIII. i. 1. 159.
We find the two forms of the adverb side by side in:
" She was new lodged and newly deified."—L. C. 84.
The position of the article shows that mere is an adverb in:
"Ay, surely, mere the truth. "—A. W. iii. 5. 58.
So
" It shall safe be kept."—Cymb. i. 6. 209.
"Heaven and our Lady gracious has it pleas'd."
1 Hen. VI. i. 2. 74.
"(I know) when the blood bums how prodigal the soul
Lends the tongue vows."—Hamlet, i. 3. 116.
Such transpositions as "our lady gracious," (adj.) where "gracious" is a mere epithet, are not common in Shakespeare. (see 419.) In
"My lady sweet, arise,"—Cymb. ii. 3. 29.
"My-lady" is more like one word than "our lady," and is also an appellative. In appellations such transpositions are allowed. (See 13.)
Sometimes the two forms occur together:
"And she will speak most bitterly and strange."
M. for M. v. 1. 90.
2. Adjectives compounded. Hence two adjectives were freely combined together, the first being a kind of adverb qualifying the second. Thus:
"I am too sudden-bold."—L. L. L. ii. 1. 107.
"Fertile-fresh"—M. W. of W. v. 5. 72.
"More active-valiant or more valiant-young."
1 Hen. IV. v. 1. 90.
"Daring-hardy."—Rich. II. i. 3. 43.
"Honourable-dangerous."—J. C. i. 3. 124. See ib. v. 1. 60.
"He lies crafty-sick."—2 Hen. IV. Prol. 37.
"I am too childish-foolish for this world."—R. III. i. 3. 142.
"You are too senseless-obstinate, my lord."—R. III. iii. 1. 44.
"That fools should be so deep-contemplative."—A. Y. ii. 7. 31.
" Glouc. Methinks the ground is even.
Edg.
Horrible-steep."—Lear, iv. 6. 3.
In the last example it is hard to decide whether the two adjectives are compounded, or (which is much more probable) "horrible" is a separate word used as in (1) for "horribly," as in T. N. iii. 4. 196. In the West of England "terrible" is still used in this adverbial sense.
There are some passages which are only fully intelligible when this combination is remembered :
" A strange tongue makes my cause more strange-suspicious."
Hen. VIII. iii. 1. 45.
Erase the usual comma after "strange."
" Here is a silly-stately style indeed."—I Hen. VI. iv. 7. 72.
Perhaps " He only in a general-honest thought."—J. C v. 5. 71.
3. Adjectives, especially those ending in ful, less, ble, and ive, have both an active and a passive meaning ; just as we still say, " a fearful (pass.) coward," and " a fearful (act.) danger."
" To throw away the dearest thing he owed,
As 'twere a careless trifle."—Macbeth, i. 4. 11,
" Such helpless harmes yt's better hidden keep."—SPEN. F. Q. i. 5. 42.
"Even as poor birds deceived with painted grapes,
Like those poor birds that helpless berries saw."
V. and A. 604 ; Rich. III. i. 2. 13.
"Upon the sightless couriers of the air. "—Macbeth, i. 7. 23.
"How dare thy joints forget
To pay their awful duty to our presence?"—Rich. II. iii. 3. 76.
"Terrible" is "frightened" in Lear, i. 2. 32; "dreadful," "awe-struck," Hamlet, i. 2. 207; "thankful" is "thankworthy," P. of T. v. 1. 285. So "unmeritable" (act. Rich. III. iii. 7. 155; J. C. iv. 1. 12); "medicinable" (act. Tr. and Cr. iii. 3. 44); "sensible" (pass. Macb. ii. 1. 36; Hamlet, i. 1. 57); "insuppressive" (pass. J. C. ii. 1. 134) ; "plausive" (pass. Hamlet, i. 4. 30) ; "incomprehensive" (pass. Tr. and Cr. iii. 3. 198); "respective" (act. R. and J. iii. 1. 128; pass. T. G. of V. iv. 4. 200); "unexpressive" (pass. A. Y. L. iii. 2. 10); "comfortable" (act. Lear, i. 4, 328); "deceivable" (act. R. II. ii. 3. 84; T. N. iv. 3. 21).
"Probable," "contemptible," and "artificial," are active in—
"The least of all these signs were probable."—2 Hen. VI. iii. 2. 178.
"'Tis very probable that the man will scorn it, for he hath a very contemptible spirit."—M. Ado, ii. 3. 188.
"We, Hermia, like two artificial gods
Have with our needles created both one flower."
M. N. D. iii. 2. 204.
Hence even " The intrenchant air."—Macbeth, v. 8. 9.
"Unprizable" (T. N. v. 1. 58) means "not able to be made a prize of, captured."
"Effect" (Rich. III. i. 2. 120) seems used for "effecter" or "agent " if the text is correct.
4. Adjectives signifying effect were often used to signify the cause. This is a difference of thought. We still say " pale death," "gaunt famine," where the personification is obvious ; but we do not say—
"Oppress'd with two weak evils, age and hunger."
A. Y. L. ii. 7. 132.
" Like as a sort of hungry dogs ymet
Doe fall together, stryving each to get
The greatest portion of the greedie pray."
SPENS. F. Q. vi. 11. 17.
"And barren rage of death's eternal cold."—Sonn. 13.
Nor should we say of the Caduceus—
"His sleepy yerde in hond he bare upright."—CHAUC. C. T. 1390. Compare also
" Sixth part of each!
A trembling contribution !"—Hen. VIII. i. 2. 95.
Here "trembling" is used for "fear-inspiring."
So other Elizabethan authors (Walker): "idle agues," "rotten showers," "barren curses."
5. Adjectives are frequently used for Nouns, even in the singular.
" A sudden pale usurps her cheek."—V. and A.
"Every Roman's private (privacy or private interest)."
B. J. Sejan. iii. 1.
"'Twas caviare to the general."—Hamlet, ii. 2. 458.
"Truth lies open to all. It is no man's several."—B. J. Disc. 742 b. "Before these bastard signs of fair (beauty) were born."—Sonn. 68.
So "fair befal," Rich. II. ii. 1. 129 ; Rich. III. i. 3. 282. But see 297.
" Till fortune, tired with doing bad,
Threw him ashore to give him glad."—P. of T. ii. Gower, 37.
"That termless (indescribable) hand
Whose bare outbragg'd the web it seem'd to wear."—L. C. 95.
" In few" = "in short."—Hamlet, i. 3. 126; Temp. i. 2. 144.
" Small (little) have continual plodders ever won."
L. L. L. i. 1. 86.
"By small and small."—Rich. II. iii. 7. 198; Rich. III. i. 3.111.
" Say what you can, my false o'erweighs your true."
M. for M. ii. 4. 170.
"I'll make division of my present (money) with you."
T. N. iii. 4. 380.
If the text were correct, the following would be an instance of an adjective inflected like a noun:
" Have added feathers to the learned's wing."—Sonn. 78.
But probably the right reading is "learned'st."
"Wont," the noun (Hamlet, i. 4. 6), is a corruption from "woned," from the verb "wonye" E. E., "wunian" A.-S., "to dwell." Compare θος.
6. Adjectives comparative. The inflection er instead of more is found before "than."
" Sir, your company is fairer than honest."—M. for M. iv. 3. 185.
The comparative "more wonderful" seems to be used, as in Latin, for "more wonderful than usual," if the following line is to be attributed to Cicero as in the editions:
" Why, saw you anything more wonderful ?"—J. C. i. 3. 14.
In Hamlet iv. 7. 49, "my sudden and more strange return," means "sudden, and even more strange than sudden."
7. The comparative inflection-er was sometimes used even when the positive ended in-ing,-ed,-id,-ain,-st,-ect. These terminations (perhaps because they assimilate the adjective to a participle by their sound) generally now take " more."
" Horrider," Cymb. iv. 2. 331; "curster," T. of Sh. iii. 2. 156; "perfecter," Coriol. ii. 1. 91; "certainer," M. Ado, v. 3. 62.
8. Superlative. The superlative inflection est, like the Latin superlative, is sometimes used to signify "very," with little or no idea of excess.
"A little ere the mightiest Julius fell."—Hamlet, i. 1. 114.
" My mutest conscience" (Cymb. i. 6. 116) may perhaps mean "the mutest part or corner of my conscience," like "summus mons."
9. The superlative inflection est is found after-ent,-ing,-ed, -ect. Thus, "violentest" (Coriol. iv. 6. 73); "cursedst" (M. of V. ii. 1. 46); "lyingest" (T. of Sh. i. 2. 25); "perfectest," (Macb. i. 5. 2).
This use of -est and -er (see 7) is a remnant of the indiscriminate application of these inflections to all adjectives which is found in Early English. Thus, in Piers Plowman, we have "avarousere" (B. i. 189), "merveillousest" (B. viii. 68).
10. The superlative was sometimes used (as it is still, but with recognized incorrectness) where only two objects are compared.
" Between two dogs which hath the deeper mouth,
Between two blades which bears the better temper,
Between two horses which doth bear him best,
Between two girls which has the merriest eye."
1 Hen. IV. ii. 4. 15.
"Not to bestow my youngest daughter
Before I have a husband for the elder."—T. of Sh. i. 1. 50.
"Of two usuries, the merriest was put down, and the worser allowed."—M. for M. iii. 2. 7.
Here it seems used for variety to avoid the repetition of the comparative.
11. Comparative and superlative doubled.—The inflections -er and -est, which represent the comparative and superlative degrees of adjectives, though retained, yet lost some of their force, and sometimes received the addition of more, most, for the purpose of greater emphasis.
"A more larger list of sceptres."—A. and C. iii. 6. 76.
"More elder."—M. of V. iv. 1. 251.
"More better."—Temp. i. 2. 19.
"More nearer."—Hamlet, ii. 1. 11.
"Thy most worst."—W. T. iii. 2. 180.
"More braver."—Temp. i. 2. 439.
"With the most boldest."—J. C. iii. 1. 121.
"Most unkindest."—J. C. iii. 2. 187.
"To some more fitter place."—M. for M. ii. 2. 16.
" I would have been much more a fresher man."
Tr. and Cr. v. 6. 21.
Ben Jonson speaks of this as "a certain kind of English atticism, imitating the manner of the most ancientest and finest Grecians."—B. J. 786. But there is no ground for thinking that this idiom was the result of imitating Greek. We find Bottom saying :
"The more better assurance."—M. N. D. iii. 1. 4.
Note the anomaly: "Less happier lands."—R. II. ii. 1. 49.
12. The Adjectives all, each, both, every, other, are sometimes interchanged and used as Pronouns in a manner different from modern usage.
All for any :
"They were slaine without all mercie."—HOLINSHED.
"Without all bail."—Sonn. 74.
"Without all reason."—ASCH. 48.
(Comp. in Latin "sine omni, &c.") Heb. vii. 7 : Wickliffe, "with-outen ony agenseiyinge ;" Rheims, Geneva, and A. V. "without all contradiction."
This construction, which is common in Ascham and Andrewes, is probably a Latinism in those authors. It may be, however, that in "things without all remedy," Macb. iii. 2. 11, "without" is used in the sense of "outside," "beyond." See Without (197).
All for every :
" Good order in all thyng."—ASCH. 62.
"And all thing unbecoming."—Macb iii. 1. 14.
We still use "all" for "all men." But Ascham (p. 54) wrote : "Ill commonlie have over much wit," and (p. 65) "Infinite shall be made cold by your example, that were never hurt by reading of bookes." This is perhaps an attempt to introduce a Latin idiom. Shakespeare, however, writes :
" What ever have been thought on."—Coriol. i. 2. 4.
Each for "all" or "each one of :"
"At each his needless heavings."—W. T. ii. 3. 35.
So every (i.e. "ever-ich," "ever-each") :
"Of every these happen'd accidents."—Temp. v. 1. 249.
And "none :" "None our parts."—A. and C. i. 3. 36.
Each for "both :"
"And each though enemies to either's reign
Do in consent shake hands to torture me."—Sonn. 28.
"Each in her sleep themselves so beautify."—R. of L. 404.
"Tell me
In peace what each of them by the other lose."—Coriol. iii. 2. 44.
This confusion is even now a common mistake. Compare
"How pale each worshipful and rev'rend guest
Rise from a Clergy or a City feast."—POPE, Imit. Hor. ii. 75.
Each for "each other :"
" But being both from me, both to each friend."—Sonn. 144.
(i.e. both friends each to the other.)
Both seems put for "each," or either used for "each other," in
"They are both in either's powers."—Temp. i. 2. 450.
There may, however, be an ellipsis of each after both :
"They are both (each) in either's powers."
Compare "A thousand groans. . . . .
Came (one) on another's neck."—Sonn. 131.
It is natural to conjecture that this is a misprint for "one or other's." But compare
"I think there is not half a kiss to choose
Who loves another best."—W. T. iv. 4. 176. (See 88)
Every one, Other, Neither, are used as plural pronouns:
"And every one to rest themselves betake."—R. of L.
"Every one of these considerations, syr, move me."—ASCH. Dedic.
"Everything
In readiness for Hymenæus stand."—T. A. i. 1. 325.
"Smooth every passion
That in the nature of their lord rebel."—Lear, ii. 2. 82.
"Every" is a pronoun in
" If every of your wishes had a womb."
A. and C. i. 2. 38 ; A. Y. L. v. 4. 178.
"Thersites' body is as good as Ajax'
When neither are alive."—Cymb. iv. 2. 252.
"Other have authoritie."—ASCH. 46.
"And therefore is the glorious planet Sol
In noble eminence enthron'd and spher'd
Amidst the other."—Tr. and C. i. 3. 89.
Other is also used as a singular pronoun (even when not preceded by "each") :
" Every time gentler than other."—J. C. i. 2. 231.
"With greedy force each other doth assail."—SPENS. F. Q. i. 5. 6. i.e. "each doth assail the other."—Rich. II. i. 1. 22.
"We learn no other but the confident tyrant.
Keeps still in Dunsinane."—Macb. v. 4. 8.
" He hopes it is no other
But, for your health and your digestion's sake,
An after-dinner's breath."—Tr. and Cr. ii. 3. 120.
"If you think other."—Othello, iv. 2. 13.
"Suppose no other."—A. W. iii. 6. 27.
In the two last passages "other" may be used adverbially for "otherwise," as in Macbeth, i. 7. 77, which may explain
"They can be meek that have no other cause."—C. of E. ii. 1. 33. i.e. "no cause otherwise than for meekness."
The use of all(e) and other(e) as plural pronouns is consistent with ancient usage. It was as correct as "omnes" and "alii" in Latin, as "alle" and "andere" in German. Our modern "others said" is only justified by a custom which might have compelled us to say "manys" or "alls said," and which has induced us to say "our betters," though not (with Heywood) "our biggers." The plural use of neither, "not both," depends on the plural use of either for "both," which is still retained in "on either side," used for "on both sides." This is justified by the original meaning of ei-ther, i.e. "every one of two," just as whe-ther means "which of two." "Either" in O.E. is found for "both." Similarly we say "none were taken" instead of "none (no one) was taken." We still retain the use of other as a pronoun without the in such phrases as "they saw each other," for "they saw each the other." Many is also used as a noun. (See 5.) Hence we have :
"In many's looks."—Sonn. 93.
Beside the adjective "mani," "moni" (many), there was also in Early English the noun "manie" or "meine" (multitude, from Fr. "maisgnée," Lat. "minores natu"). But it is doubtful whether this influenced the use just mentioned.
13. The possessive Adjectives, when unemphatic, are sometimes transposed, being really combined with nouns (like the French monsieur, milord).
"Dear my lord."—J. C. ii. 1. 255.
" Good my brother."—Hamlet, i. 3. 46.
"Sweet my mother."—R. and J. iii. 5. 200.
"Oh! poor our sex."—Tr. and Cr. v. 2. 109.
" Art thou that my lord Elijah ? "—1 Kings xviii. 7.
"Come, our queen."—Cymb. ii. 3. 68.
So probably, vocatively :
"Tongue-tied our queen speak thou."—W. T. i. 1. 27.
Compare "Come on, our queen."—Rich. II. i. 2. 222.
"Good my knave. "—L. L. L. iii. 1. 153.
"Good my friends."—Coriol. v. 2. 8.
"Good your highness, patience."—A. and C. ii. 5. 106.
"Good my girl."—1 Hen. VI. v. 4. 25.
Hence, by analogy, even
"Good my mouse of virtue."—T. N. i. 5. 69.
The emphatic nature of this appellative "good" is illustrated by
"Good now, sit down."—Hamlet, i. 1. 70:
where the noun is omitted. So W. T. v. 1. 19 ; Tempest, i. 1. 16. "Gunnow" (good now) is still an appellative in Dorsetshire.
Sometimes, but very rarely, the possessive adjective used vocatively is allowed to stand first in the sentence:
" Our very loving sister, well be met."—Lear, v. 1. 20.
It is possible that this use of "my," "our," &c. may be in part explained from their derivation, since they were originally not adjectives, but the possessive cases of pronouns. Thus, "sweet my mother," = "sweet mother of me," or "sweet mother mine."
Similar vocatives are
" The last of all the Romans, fare thee well."—J. C. v. 3. 99.
" The jewels of our father, with wash'd eyes,
Cordelia leaves you."—Lear, i. 1. 271.
So Folio, "Take that, the likeness of this railer here."
3 Hen. VI. v. 5. 38 (Globe "thou").
14. The Adjectives just, mere, proper, and very were sometimes used as in Latin.
Just = exact. "A just seven-night."—M. Ado, ii. 1. 375.
"A jus! pound."—M. of V. iv. 1. 327.
Whereas we retain this sense only in the adverbial use, "just a week." Compare "justum iter."
15. Mere = "unmixed with anything else :" hence, by inference, "intact," "complete."
"The mere perdition of the Turkish fleet."—O. ii. 2. 3.
i.e. the "complete destruction."
" Strangely-visited people,
The mere despair of surgery. "—Macbeth, iv. 3. 132.
i.e. "the utter despair." So Rich. III. iii. 7. 263.
The word now means "unmixed," and therefore, by inference, "nothing but," "bare," "insignificant." But, in accordance with its original meaning, "not merely," in Bacon, is used for "not entirely." So Hamlet, i. 2. 137.
16. Proper = "peculiar," "own."
"Their proper selves."—Temp. iii. 3. 60.
" With my proper hand."—Cymb. iv. 2. 97; T. N. v. 1. 327.
i.e. "with my own hand," as in French. So J. C. i. 2. 41, v. 3. 96.
Very = "true." " My very friends."—M. of V. iii. 2. 226.
17. More (mo-re) and most (mo-st) (comp. E. E. ma or mo ; mar or mor ; maest, mast, or most) are frequently used as the comparative and superlative of the adjective "great." [Moe, or mo, as a comparative (Rich. II. ii. 1. 239 ; Rich. III. iv. 4. 199), is contracted from more or mo-er. Compare "bet" for "bett-er," "leng" for "leng-er," and "streng" for "streng-er," in O. E. See also "sith," 62.]
"At our more leisure."—M. for M. i. 3. 49.
"A more requital."—K. J. ii. 1. 34.
"With most gladness."—A. and C. ii. 2. 169.
"Our most quiet " (our very great quiet).—2 Hen. IV. iv. 1. 71.
" So grace and mercy at your most need help you."
Hamlet, i. 5. 180.
Hence we understand :
"Not fearing death nor shrinking for distress,
But always resolute in most extremes."—I Hen. VI. iv. 1. 38.
i.e. not "in the majority of extremities," as it would mean with us, but "in the greatest extremes."
Hence:
" More (instead of greater) and less came in with cap and knee."
1 Hen. IV. iv. 3. 68.
"And more and less do flock to follow him."
2 Hen. IV. i. 1. 209.
"Both more and less have given him the revolt."
Macbeth, v. 4. 12.
That "less" refers here to rank, and not to number, is illustrated by
" What great ones do the less will prattle of."—T. N. i. 2. 33.
So Chaucer :
" The grete giftes to the most and leste."—C. T. 2227.
18. One is used for "above all," or "alone," i.e. "all-one," in Elizabethan English with superlatives.
"He is one the truest manner'd."—Cymb. i. 6. 164.
"One the wisest prince."—Hen. VIII. ii. 4. 49.
"Have I spake one the least word."—Ib. 153.
But in Early English one is thus used without a superlative :
"He one is to be praised."
"I had no brother but him one."
"He was king one."
(Here Mr. Morris conjectures that the O. E. "ane" stands for A.-S. dative "an-um.")
So in Latin "justissimus unus;" and in Greek μóνος is similarly used. So "alone" = "above all things."
"That must needs be sport alone."—M. N. D. iii. 2. 119.
"I am alone the villain of the earth."—A. and C. iv. 6. 30.
" So full of shapes is fancy
That it alone is high fantastical."—T. N. i. 1. 15.
None. See 53.
19. Right (which is now seldom used as an adjective, except with the definite article, as the opposite of "the wrong," e.g. "the right way," not "a right way"), was used by Shakespeare, with the indefinite article, to mean "real," " down-right."
"I am a right maid for my cowardice."—M. N. D. iii. 2. 302. Compare A. and C. iv. 12. 28, "a right gipsy." It means "true" in "A right description of our sport, my lord."—L. L. L. v. 2. 522.
20. Self (se = swa [so]; -lf. = Germ. leib, "body:" Wedgewood, however, suggests the reciprocal pronoun, Lat. se, Germ. sich, and he quotes, "Et il ses cors ira," i.e. "and he him self will go," Old French, and still retained in Creole patois) was still used in its old adjectival meaning "same," especially in "one self," i.e. "one and the same," and "that self." Compare the German "selbe."
"That self chain."—C. of E. v. 1. 1.0.
"That self mould."—Rich. II. i. 2. 23.
"One self king."—T. N. i. 1. 39.
Compare 3 Hen. VI. iii. 1. 11; A. and C. v. 1. 21; M. of V. i. 1. 148.
Hence we can trace the use of himself, &c. The early English did not always use "self," except for emphasis; their use was often the same as our modern poetic use :
"They sat them down upon the yellow sand."—TENNYSON.
In order to define the him, and to identify it with the previous he, the word self (meaning "the same," "the aforesaid") was added : "He bends himself." Thyself and myself are for thee-self, me-self. "One self king" may be illustrated by "one same house."—MONTAIGNE, 228. We also find the adjectival use of "self" retained in
"The territories of Attica selfe."—N. P. 175.
"The city selfe of Athens."—N. P. 183.
"Itself" is generally, if not always, written in the Folio "it selfe."
There is a difficulty, however, in such a phrase as "I myself saw it." Why do we not find "I-self," "he-self," in such cases? Why, even in A.-S., do we find the rule that, when self agrees with the subject of the sentence, the pronoun has to be repeated in the dative before self: "he (him) self did it," but when the noun is in an oblique case self is declined like any other adjective, and agrees with its noun: "he hine seolfne band," i.e. "he bound himself?" The fact is, that in the second case "self" is an ordinary adjective used as an adjective: "he bound the same or aforesaid him." But in the former case "himself" is often an abridgment of a prepositional expression used as an adverb: "he did it by himself," "of himself," "for himself," and, being a quasi-adverb, does not receive the adjectival inflection. It follows that "my," "thy," in "myself" and "thyself," are not pronominal adjectives, but represent inflected cases of the pronouns. Thus "ourself" for "ourselves" is strictly in accordance with the A.-S. usage in
"We will ourself in person to this war,"—Rich. II. i. 4. 42. though of course Shakespeare only uses it for "myself" in the mouth of a dignified personage. Similarly in Piers Plowman (B. viii. 62) we have "myn one " ( = "of me one," i.e. "of me alone" [see One]) used for "by myself," and "him one" (William of Palerne, 17) for "by himself;" and here "myn" is the genitive of "I," and "him" the dative of "he," and "one" is an adjective. This is also illustrated by the Scottish "my lane," i.e. "my, or by me, alone." Hence, instead of "ourselves" we have in Wickliffe, 2 Cor. x. 2, "but we mesuren us in us silf and comparisownen us silf to us," and, a line above, "hem silf " for "themselves."
Very early, however, the notion became prevalent that the inflected pronoun was a pronominal adjective, and that "self" was a noun. Hence we find in Chaucer, "myself hath been the whip," "and to prove their selfes" in Berners' Froissart; and in Shakespeare, Temp. i. 2. 132, "thy crying self." Hence the modern "ourselves," "yourselves."
The use of "self" as a noun is common in Shakespeare: "Tarquin's self," Coriol. ii. 2. 98; "my woeful self," L. C. 143. Hence the reading of the Folio may be correct in the first of the following lines :
"Even so myself bewails good Gloucester's case,
With sad unhelpful tears and with dimm'd eyes
Look after him."—2 Hen. VI. iii. 1. 217.
But the change to the first person is more in accordance with Shakespeare's -usage, as:
"This love of theirs myself have often seen."
T. G. of V. iii. 1. 23.
So T. G. iii. 1. 147; ib. iv. 2. 110.
So "himself" is used as a pronoun, without "he," in
"Direct not him whose way himself will choose."
Rich. II. ii. 1. 29.
"Self-born arms" (Rich. II. ii. 3. 80) seems to mean "divided against themselves," "civil war."
21. Some, being frequently used with numeral adjectives qualifying nouns of time, as "some sixteen months" (T. G. of V. iv. 1. 21), is also found, by association, with a singular noun of time.
"Some hour before you took me."—T. N. ii. 1. 22.
"I would detain you here some month or two."—M. of V. iii. 2. 9.
"Some day or two."—R. III. iii. 1. 64.
It would seem that in such expressions "some" has acquired an adverbial usage, as in the provincialisms, "It is some late," "Five mile or some" (MÄTZNER, ii. 253). Compare
"I think 'tis now some seven o'clock."—T. of Sh. iv. 3. 189.
"Sum" is, however, found in Early English and Anglo-Saxon in the sense of "a certain." Compare A.-S. "Sum jungling hym fyligde," Mark xiv. 51. So Wickliffe, where A. V. has "A certain young man followed him." "Other-some" (M.N.D. i. 1. 226), see p. 6.
22. The licence of converting one part of speech into another may be illustrated by the following words used as adjectives:
"The fine point of seldom (rare) pleasure."—Sonn. 52.
"Each under (inferior) eye."—Sonn. 7.
"This beneath (lower) world."—T. of A. i. 1. 44. "The orb below
As hush (silent) as death."—Hamlet, ii. 2. 508.
See also still, below (22).
"Most felt (palpable) and open this."—B. J. Sejan. i. 2.
"Most laid (plotted) impudence."—B. J. Fox.
As still with us, any noun could be prefixed to another with the force of an adjective: "water-drops," "water-thieves," "water-fly," &c.
This licence, however, was sometimes used where we should prefer the genitive or an adjective. Thus, "the region kites" (Hamlet, ii. 2. 607,) for "the kites of the region;" and "the region cloud," Sonn. 33. So perhaps, " a moment leisure," Hamlet, i. 3. 133. We say "heart's ease," but Shakespeare, Hen. V. ii. 2. 27, says "heart-grief;" "heart-blood," Rich. II. i. 1. 172, &c.; "faction -traitors," ib. ii. 2. 57. Again, a word like "music" is not commonly used by us as a prefix unless the suffix is habitually connected with "music:" thus "music-book," "music-master," &c., but not "music " for "musical " as in
"The honey of his music vows."—Hamlet, iii. 1. 164.
Compare "venom mud," R. of L. 561; "venom clamours," C. of E. v. i. 69, for "venomous;" "venom sound," Rich. II. ii. 1. 19; "venom tooth," Rich. III. i. 3. 291.
This licence is very frequent with proper names.
"Here in Philippi fields."—J. C. v. 5. 19.
"Draw them to Tiber banks."—J. C. i. 1. 63.
"There is no world without Verona walls."—R. and J. iii. 3. 17,
"Within rich Pisa walls."—T. of Sh. ii. 1. 369.
"To the Cyprus wars."—O. i. 1. 151.
"Turkey cushions."—T, of Sh. ii. 1. 355, as we still say.
"From Leonati seat."—Cymb. v. 4. 60.
" Venice gold."—T. of Sh. ii. 1. 366.
The reason for this licence is to be found in an increasing dislike and disuse of the inflection in 's. Thus we find, "sake" frequently preceded in 1 Hen. IV. by an uninflected noun: "for recreation sake," 1 Hen. IV. i. 2. 174; ib. ii. 1. 80; ib. v. 1. 65; "for fashion sake," A. Y. L. iii. 2. 271.
# ADVERBS.
23. It is characteristic of the unsettled nature of the Elizabethan language that, while (see 1) adjectives were freely used as adverbs without the termination ly, on the other hand ly was occasionally added to words from which we have rejected it. Thus: "fastly" (L. C. 9); "youngly" (Coriol. ii. 3. 244).
24. Adverbs with prefix a-: (1) Before nouns. In these adverbs the a- represents some preposition, as "in," "on," "of," &c. contracted by rapidity of pronunciation. As might be expected, the contraction is mostly found in the prepositional phrases that are in most common use, and therefore most likely to be rapidly pronounced. Thus (Coriol. iii. 1. 261-2) Menenius says: "I would they were in Tiber," while the Patrician, " I would they were a-bed." Here a- means "in," as in the following:
"3d Fisherman. Master, I marvel how the fishes live in the sea.
1st Fisherman. Why, as men do a-land."—P. of T. ii. 1. 31.
A\- is also used where we should now use "at." Compare, however, O. E. "on work."
" Sets him new a-work."—Hamlet, ii. 2. 51; Lear iii. 5. 8.
So R. of L. 1496. And compare Hamlet, ii. 1. 58, "There (he) was a' gaming," with
"When he is drunk, asleep, or in his rage
At gaming."—Hamlet, iii. 3. 91.
Sometimes "of" and "a-" are interchanged. Compare "a-kin" and "of kind," "of burst" and "a-thirst," "of buve" and "above." Most frequently, however, "a-" represents our modern "on" or "in." Compare "a-live" and "on live."
"Bite the holy cords a-twain."—Lear, ii. 2. 80; L. C. 6.
Compare "That his spere brast a-five," i.e. "burst in five pieces." (HALLIWELL.) So
"A-front."—1 Hen. IV. ii.4. 222. "A-fire."—Temp. i. 2. 212.
"Look up a-height" (perhaps).—Lear, iv. 6. 58.
"Beaten the maids a-row."—C. of E. v. 1. 170.
"And keep in a-door."—Lear, i. 4. 138.
Thus, probably, we must explain
" Thy angel becomes a fear."—A. and C. ii. 3. 22.
i.e. "a-fear." The word "a-fere" is found in A.-S. in the sense of "fearful" (M tzner, i. 394). And in the expressions "What a plague?" (1 Hen. IV. iv. 2. 56,) "What a devil?" (1 Hen. IV. ii. 2. 30,) "A God's name" (Rich. II. ii. 1. 251,) and the like, we must suppose a to mean "in," "on," or "of." There is some difficulty in
"I love a ballad in print a life" (so Folio, Globe, "o' life"). W. T. iv. 4. 264.
It might be considered as a kind of oath, "on my life." Nares explains it "as my life," but the passages which he quotes could be equally well explained on the supposition that a is a preposition. The expression "all amort" in 1 Hen. VI. iii. 2. 124, and T. of Sh. iv. 3. 36, is said to be an English corruption of " à la mort."
"To heal the sick, to cheer the alamort."—NARES.
The a (E. E. an or on) in these adverbial words sometimes for euphony retains the n:
"And each particular hair to stand an end."—Ham. i. 4. 19.
So Hamlet, iii. 4. 122, Rich. III. i. 3. 304; and compare "an hungry," "an hungered" below, where the an is shown not to be the article. So
" A slave that still an end turns me to shame,"—T. G. of V. iv. 4. 67. where "an end" (like "run on head" (Homilies), i.e. "run a-head") signifies motion "on to the end."
These adverbial forms were extremely common in earlier English, even where the nouns were of French origin. Thus we find: "a-grief," "a-fyn" for "en-fin," "a-bone" excellently, "a-cas" by chance. Indeed the corruption of en\- into a\- in Old French itself is very common, and we still retain from this source "a-round" for "en rond " and "a-front " for "en front."
(2) Before adjectives and participles, used as nouns.
When an adjective may easily be used as a noun, it is intelligible that it may be preceded by a-. Compare "a-height," quoted above, with our modern "on high," and with
"One heaved a-high to be hurled down below."
Rich. III. iv. 4. 86.
It is easy also to understand a-before verbal nouns and before adjectives used as nouns, where it represents on:
"I would have him nine years a-killing."—O. iv. 1. 188.
i.e. "on, or in the act of killing." So
"Whither were you a-going?"—Hen. VIII. i. 3. 50.
i.e. "in the act of going."
"The slave that was a-hanging there."—Lear, v. 3. 274.
"Tom's a-cold."—Lear, iii. 4. 59.
i.e. "a-kale," E. E. "in a chill."
Some remarkable instances of this form are subjoined, in which nouns are probably concealed.
" I made her weep a-good."—T. G. of V. iv. 4. 170.
i.e. "in good earnest;" but "good" may be a noun. Compare "a-bone" above.
"The secret mischiefs that I set abroach."—R. III. i. 3. 325;
R. and J. i. 1. 111.
where a is prefixed to "broach," now used only as a verb. " On broach" and "abroach" are found in E. E. Compare
" O'er which his melancholy sits on brood."
Hamlet, iii. 1. 173.
Compare "That sets them all agape."—MILTON, P. L. v. ; which is to be explained by the existence of an old noun, "gape."
(3) As the prefix of participles and adjectives.
In this case a- represents a corruption of the A.-S. intensive of. Thus from E. E. "of feren," we have "afered" or "afeared;" from A.-S. "of-gán," "a-gone." The of before a vowel or h is sometimes changed into on or an. See On, 182. And indeed the prefixes an-, on-, of-, a-, were all nearly convertible. Hence "of-hungred" appears not only as "afingred," but also "an-hungered," as in St. Matthew xxv. 44, A. V.: "When saw we thee an hungered or athirst?" It would be a natural mistake to treat an here as the article: but compare
" They were an hungry,"—Coriol. i. 1. 209.
where the plural "they" renders it impossible to suppose that an is the article.
Perhaps, by analogy, a\- is also sometimes placed before adjectives that are formed from verbs. It can scarcely be said that weary is a noun in
"For Cassius is a-weary of the world."
J. C. iv. 3. 95; 1 Hen. IV. iii. 2. 88.
Rather "a-weary," like "of-walked," means "of-wery," i.e. "tired out."
25. Adverbs ending in "s" formed from the possessive inflection of Nouns. Some adverbs thus formed are still in common use, such as "needs" = "of necessity."
"Needs must I like it well."—Rich. II. iii. 2. 4.
"There must be needs a like proportion."—M. of V. iii. 4. 14.
But we find also in Shakespeare:
"He would have tickled you other gates than he did."
T. N. v. 1. 198.
i.e. "in another gate or fashion."
In this way (compare "sideways," "lengthways," &c.) we must probably explain
" Come a little nearer this ways."—M. W. of W. ii. 2. 50.
And "Come thy ways"—T. N. v. 2. 1.
Compare also the expression in our Prayer-book:
"Any ways afflicted, or distressed."
Others explain this as a corruption of "wise."
"Days" is similarly used:
"'Tis but early days."—Tr. and Cr. iv. 5. 12.
i.e. "in the day," as the Germans use "morgens." Compare "now-a-days," and N. P. 179, "at noondaies."
A similar explanation might suggest itself for
"Is Warwick friends with Margaret?"
3 Hen. VI. iv. 1. 115; A. and C. ii. 5. 44.
But "I am friends" is not found in E. E., and therefore probably it is simply a confusion of two constructions, "I am friend to him" and "we are friends."
26. After was used adverbially of time:
"If you know
That I do fawn on men, and hug them hard,
And after scandal them."—J. C. i. 2. 76.
Now we use afterwards in this sense, using after rarely as an adverb and only with verbs of motion, to signify an interval of space, as "he followed after."
27. The use of the following adverbs should be noted:
Again (radical meaning " opposite") is now only used in the local sense of returning, as in "He came back again, home again," &c.; and metaphorically only in the sense of repeating, as in "Again we find many other instances," &c. It is used by Shakespeare metaphorically. in the sense of "on the other hand." Thus—
"Have you
Ere now denied the asker, and now again (on the other hand)
Of him that did not ask but mock, bestow
Your sued-for tongues? "—Coriol. ii. 3. 214.
"Where (whereas) Nicias did turne the Athenians from their purpose, Alcibiades againe (on the other hand) had a further reach," &c.—N. P. 172. So Rich. II. ii. 9. 27.
It is also used literally for "back again." "Haste you again," A. W. ii. 2. 73, does not mean "haste a second time," but "hasten back."
Again is used for " again and again," i.e. repeatedly (a previous action being naturally implied by again), and hence intensively almost like "amain."
"For wooing here until I sweat(ed) again."—M. of V. iii. 2. 205.
"Weeping again the king my father's wreck."
Tempest, i. 2. 390.
For omission of -ed in "sweat" (common in E. E.), see 341.
28. All (altogether) used adverbially:
"I will dispossess her all."—T. of A. i. 1. 139.
"For us to levy power is all unpossible. "—Rich. II. ii. 2. 126.
In compounds all is freely thus used, "All-worthy lord;" "all-watched night;" "her all-disgraced friend," A. and C. iii. 12. 22. Sometimes it seems to mean "by all persons," as in "all-shunned." So, "this all-hating world," Rich. II. v. 5. 66, does not mean "hating all," but "hating (me) universally."
All used intensively was frequently prefixed to other adverbs of degree, as "so."
"What occasion of import
Hath all so long detain'd you from your wife?"
T. of Sh. iii. 1. 105.
The connection of all and "so" is perpetuated in the modern "also." Still more commonly is all prefixed to "too."
"In thy heart-blood, though being all too base
To stain the temper of my knightly sword."
Rich. II. iv. 1. 28.
"Our argument
Is all too heavy to admit much talk."—2 Hen. IV. v. 2. 24.
So Cymb. v. 5. 169; T. G. of V. iii. 1. 162; Sonn. 18, 61, 86; R. of L. 44, 1686.
There are two passages in Shakespeare where all-to requires explanation:
"It was not she that called him all to nought."—V. and A. 993.
"The very principals (principal posts of the house) did seem to rend And all to topple."—P. of T. iii. 2. 17.
(1) In the first passage all-to is probably an intensive form of "to," which in Early English (see Too, below) had of itself an intensive meaning. Originally "to" belonged to the verb. Thus "to-breke" meant "break in pieces." When "all" was added, as in "all to-breke," it at first had no connection with "to," but intensified "to-breke." But "to" and "too" are written indifferently for one another by Elizabethan and earlier writers, and hence sprang a corrupt use of "all-to," caused probably by the frequent connection of all and too illustrated above. It means here "altogether."
(2) In the second passage some (a) connect "to-topple," believing that here and in M. W. of W. iv. 4. 57, "to-pinch," "to" is an intensive prefix, as in Early English. But neither of the two passages necessitates the supposition that Shakespeare used this archaism. (See M. W. of W. iv. 4. 5 below, To omitted and inserted, 350.) We can, therefore, either (b) write "all-to" (as in the Globe), and treat it as meaning "altogether," or (c) suppose that "all" means "quite," and that "to topple," like "to rend," depends upon "seem." This last is the more obvious and probable construction.
From this use of "all too" or "all to," closely connected in the sense of " altogether," it was corruptly employed as an intensive prefix, more especially before verbs beginning with be-: "all-to-be qualify," B. J.; "all-to-bekist," ib.; and later, "he all-to be-Gullivers me," SWIFT; "all-to-be-traytor'd," NARES.
29. Almost, used for mostly, generally:
"Neither is it almost seen that very beautiful persons are of
great virtue. "—B. E. 163.
Our modern meaning nearly is traceable to the fact that anything is nearly done when the most of it is done.
Almost (see also Transpositions) frequently follows the word which it qualifies.
" I swoon Almost with fear."—M. N. D. ii. 3. 154.
"As like almost to Claudio as himself."—M. for M. v. 1. 494.
Hence in negative sentences we find " not-almost " where we should use "almost not," or, in one word, "scarcely," " hardly."
"You cannot reason (almost) with a man."—Rich. III. ii. 2. 39. The Globe omits the parenthesis of the Folio.
"And yet his trespass, in our common reason,
Is not almost a fault . . . to incur a private check."—O. iii. 3. 66. i.e. "is not (I may Almost say) fault enough to," &c. or "is scarcely fault enough to," &c. So
"I have not breath'd almost since I did see it."—C. of E. v. 1. 181.
It was natural for the Elizabethans to dislike putting the qualifying "almost" before the word qualified by it. But there was an ambiguity in their idiom. "Not almost-a-fault" would mean " not approaching to a fault;" "not-almost a fault," "very nearly not a fault." We have, therefore, done well in avoiding the ambiguity by disusing " almost" in negative sentences. The same ambiguity and peculiarity attaches to interrogative, comparative, and other conjunctional sentences.
"Would you imagine or almost believe ?"—Rich. III. iii. 5. 35. i.e. "Would you suppose without evidence, or (I may almost say) believe upon evidence?" &c.
" Our aim, which was
To take in many towns ere almost Rome
Should know we were afoot."—Coriol. i. 2. 24.
Alone, see One, 18.
30. Along is frequently joined to " with" and transposed, as:
"With him is Gratiano gone along."—M. of V. ii. 8. 2.
Hence the "with me" being omitted, "along" is often used foi "along with me."
"Demetrius and Egeus, go along,
I must employ you in some business. "—M. N. D. i. 1. 123.
Note, that here, as in T. of Sh. iv. 5. 7; 2 Hen. IV. ii. 1. 191 ; O. i. 1. 180 ; "go" is used where we should say "come." The word is used simply to express the motion of walking by WICKLIFFE: Acts xiv. 8. MONTAIGNE, Florio, 230.
Sometimes the verb of motion is omitted, as in
"Will you along (with us) ?"—Coriol. ii. 3. 157.
"Let's along" is still a common Americanism.
Sometimes the ellipsis refers to the third person.
"Go you along (with him)."—A. and C. v. 1. 69.
Perhaps we ought (to the advantage of the rhythm) to place a comma after along; in
"Therefore have I entreated him along,
With us to watch the minutes of this night."—Ham. i. 1. 26.
30 a. Anon. The derivative meaning of anon (an-ane) is "at one instant," or " in an instant," and this is its ordinary use. But in
"Still and anon."—K. J. iv. 1. 47.
"Which ever and anon he gave his nose."
1 Hen. IV. i. 3. 38.
anon seems to mean "the moment after," a previous moment being implied by "still," "ever." Compare our "now and then."
31. Anything, like Any ways, is adverbially used:
"Do you think they can take any pleasure in it, or be anything delighted?"—MONTAIGNE, 31.
"Any ways afflicted, or distressed."—Prayer-book.
"Ways" is, perhaps, genitive. See 25.
32. Away.
"She could never away with me."—2 Hen. IV. iii. 2. 231.
i.e. "she could not endure me." A verb of motion is probably omitted. Compare our " I cannot get on with him," "put up with him," and the provincial " I cannot do with him."
"I could not do withal. "—M. of V. iii. 2. 72.
So "she could never away with me" = "she could not go on her way," i.e. "get on with me." For the omission of the verb of motion compare
"Will you along !"—Coriol. ii. 3. 157.
33. Back, for "backward."
"Goes to and back lackeying the varying tide."
A. and C. i. 4. 46.
Where we should say " to and fro."
34. Besides = "by the side of the main question," i.e. "in other respects," "for the rest."
"This Timæus was a man not so well knowne as he, but besides (for the rest) a wise man and very hardy."—N. P. 174.
Similarly besides is used as a preposition in the sense "out of."
"How fell you besides your five wits?"—T. N. iv. 2. 92.
35. Briefly = " a short time ago," instead of (as with us) "in a short space of time."
" Briefly we heard their drums.
How couldst thou . . . bring thy news so late?"
Coriol. i. 6. 16.
Similarly we use the Saxon equivalent "shortly" to signify futurity.
36. By (original meaning " near the side." Hence "by and by" = "very near," which can be used either of time or, as in Early English, also of place) is used for "aside," "on one side," "away," in the phrase
"Stand by, or I shall gall you."—K. J. iv. 3. 94.
Whereas, on the other hand, "to stand by a person" means "to stand near any one."
37. Chance appears to be used as an adverb :
" How chance thou art returned so soon ? "—C. of E. i. 2. 42.
But the order of the words "thou art," indicates that Shakespeare treated chance as a verb. " How may it chance or chances that," as Hamlet, ii. 2. 343, "How chances it they travel?" Compare—
"How chance the roses there do fade so fast?"
M. N. D. i. 1. 129.
So Tr. and Cr. iii. 1. 151; 2 Hen. IV. iv. 4. 20 ; Rich. III. iv. 2. 103 ; M. W. if W. v. 5. 231; P. of T. iv. 1. 23.
Compare, however, also—
"If case some one of you would fly from us."—3 Hen. VI. v. 4. 34. where "case" is for the Old French "per-case."
This use of chance as an apparent adverb is illustrated by
"Perchance his boast of Lucrece' sovereignty
Suggested this proud issue of a king:
Perchance that envy of so rich a thing
Braving compare, disdainfully did sting."—R. of L. 39.
Here "perchance" seems used first as an adverb, then as a verb, "it may chance that." So Shakespeare, perhaps, used chance as an adverb, but unconsciously retained the order of words which shows that, strictly speaking, it is to be considered as a verb.
38. Even. "Even now" with us is applied to an action that has been going on for some long time and still continues, the emphasis being laid on "now." In Shakespeare the emphasis is often to be laid on "even," and "even now" means "exactly or only now," i.e. "scarcely longer ago than the present:" hence "but now."
"There was an old fat woman even now with me."
M. W.of W. iv. 5. 26.
Often "but even now" is used in this sense: M. of V. i. 1. 35. On the other hand, both "even now" and "but now" can signify "just at this moment," as in
" But now I was the lord
Of this fair mansion ; . . . and even now, but now,
This house, these servants, and this same myself
Are yours."—M. of V. iii. 2. 171.
We use "just now" for the Shakespearian "even now," laying the emphasis on "just." Even is used for "even now," in the sense of "at this moment," in
"A certain convocation of politic worms are even at him."
Hamlet, iv. 3. 22.
So "even when" means "just when" in
"(Roses) die, even when they to perfection grow."
T. N. ii. 4. 42.
39. Ever (at every time) freq. :
"For slander's mark was ever yet the fair."—Sonn. 70.
The latter use is still retained in poetry. But in prose we confine "ever" (like the Latin "unquam") to negative, comparative, and interrogative sentences.
Ever seems contrary to modern usage in
"Would I might
But ever see that man."—Temp. i. 2. 168.
"But," however, implies a kind of negative, and "ever" means "at any time."
40. Far, used metaphorically for "very."
"But far unfit to be a sovereign."—3 Hen. VI. iii. 2. 92.
So 2 Hen. VI. iii. 2. 286.
41. Forth, hence, and hither are used without verbs of motion (motion being implied) :
"I have no mind of feasting forth to-night."—M. of V. ii. 2.37.
"Her husband will be forth."—M. W. of W. ii. 2. 278.
" By praising him here who doth hence remain."—Sonn. 39.
"From thence the sauce to meat is ceremony. "—Macb. iii. 4. 36.
"Methinks I hear hither your husband's drum."—Coriol. i. 3. 32.
"Prepare thee hence for France. "—Rich. II v. 1. 31.
Forth, "to the end:"
"To hear this matter forth."—M. for M. v. 1. 255.
Forth, as a preposition: see Prepositions.
42. Happily, which now means "by good hap," was sometimes used for "haply," i.e. "by hasp," just as " success " was sometimes "good," at other times " ill."
"Hamlet. That great baby you see there is not yet out of his swaddling-clouts.
Ros. Happily he's the second time come to them."—Hamlet, ii. 2. 402.
"And these our ships, you happily may think,
Are like the Trojan horse (which) was stuffed within
With bloody veins."—P. of T. i. 4. 29.
" Though I may fear
Her will recoiling to her better judgment
May fall to match you with her country forms,
And happily repent."—Othello, iii. 3. 238.
It means "gladly" in Macbeth, i. 3. 89.
43. Here is used very freely in compounds: "they here approach" (Macb. iv. 3. 133); "here-remain" (ib. 148). Perhaps here may be considered as much an adjective, when thus used, as "then" in "our then dictator" (Coriol. ii. 2. 93). So in Greek.
44. Hitherto, which is now used of time, is used by Shakespeare of space:
"England from Trent and Severn hitherto."
1 Hen. IV. iii. 1. 74.
45. Home. We still say "to come home," "to strike home," using the word adverbially with verbs of motion, but not
" I cannot speak him home," i.e. completely.
Coriol. ii. 2. 107.
"Satisfy me home."—Cymb. iii. 5. 83.
"(Your son) lack'd the sense to know her estimation home."
"That trusted home
A. W. v. 3. 4.
Might yet enkindle you unto the crown."—Macbeth, i. 3. 121.
46. How (adverbial derivative from hwa = hwu, O. E.) used for "however:"
" I never yet saw man
How wise, how noble, young, how rarely featured,
But she would spell him backward."—M. Ado, iii. 1. 60.
"Or whether his fall enraged him or how 'twas."
Coriol. i. 3. 69.
How is perhaps used for "as" in V. and A. 815:
"Look, how a bright star shooteth from the sky,
So glides he in the night from Venus' eye."
This, which is the punctuation of the Globe, is perhaps correct, and illustrated by
"Look, as the fair and fiery-pointed sun
Rushing from forth a cloud bereaves our sight,
Even so," &c.—R. of L. 372.
So V. and A. 67; M. of V. iii. 2. 127.
Similarly, GASCOIGNE (Mätzner) has :
"How many men, so many minds."
47. Howsoe'er for " owsoe'er it be," "in any case."
"Howsoe'er, my brother hath done well."—Cymb. iv. 2. 146.
So However. See 403.
48. Last. Such phrases as "at the last," "at the first," are common, but not
"The last (time) that e'er I took her leave at court." A. W. v. 3. 79.
Merely, completely. See Adjectives, Mere, 15.
More, Most. See Adjectives, 18.
49. Moreabove = "moreover."—Hamlet, ii. 2. 126.
50. Moreover precedes "that," like our "beside that."
"Moreover that we much did long to see you."
Hamlet, i. 2. 2.
51. Much, More, is frequently used as an ordinary adjective, after a pronominal adjective, like the Scotch mickle, and the E. E. muchel. (So in A.-S.)
"Thy much goodness."—M. for M. v. 1. 534.
"Yet so much (great) is my poverty of spirit."
Rich. III. iii. 7. 159.
Much was frequently used as an adverb even with positive adjectives.
"I am much ill."—2 Hen. IV. iv. 4. 111.
So Tr. and Cr. ii. 3. 115; J. C. iv. 3. 255.
"Our too much memorable shame."—Hen. V. ii. 4. 53.
So Rich. II. ii. 2. 1.
More is frequently used as a noun and adverb in juxtaposition.
" The slave's report is seconded and more
More fearful is deliver'd."—Coriol. iv. 6. 63. Comp. K. J. iv. 2. 42.
"More than that tongue that more hath more express'd."—Sonn. 23. "If there be more, more woeful, hold it in."—Lear, v. 3. 202.
We sometimes say "the many" (see 12), but not "the most," in the sense of "most men." Heywood, however, writes—
"Yes, since the most censures, believes and saith
By an implicit faith."—Commendatory Verses on B. J.
Needs. See 25.
52. Never is used where we now more commonly use "ever" in phrases as :
" And creep time ne'er so slow,
Yet it shall come for me to do thee good."—K. J. iii. 3. 31.
So 1 Hen. VI. v. 3. 98 ; Rich. II. v. 1. 64.
There is probably here a confusion of two constructions, (1) "And though time creep so slow as it never crept before," and (2)
"And though time never crept so slow as in the case I a supposing." These two are combined into, "And though time creep- (how shall I describe it? though it crept) never so slow." Construction (2) is illustrated by
"Never so weary, never so in woe,
I can no further crawl, no further go."—M. N. D. iii. 2. 442.
Here, strictly speaking, the ellipsis is " I have been," or "having been;" "I have never been so weary." But it is easy to see that "never so weary" being habitually used in this sense, Hermia might say, "I am never-so-weary," or still more easily, "though I were never-so-weary."
In such phrases as "never the nearer," never seems to man "nought." So Wickliffe, John xix. 21 :
" But how he now seeth we wite nere," i.e. "we know not."
53. None seems to be the emphatic form of "no," like "mine" of "my" in the modern idiom :
"Satisfaction (there) can be none but by pangs of death."
T. N. iii. 4. 261.
For we could not say "there can be none satisfaction." This emphatic use of the pronoun at the end of a sentence is found very early. None seems loosely used for "not at all," like "nothing" (55), "no-whit," i.e. "not." And this may, perhaps, explain :
"None a stranger there
So merry and so gamesome."—Cymb. i. 6. 59.
Here either none means "not," "ne'er," or a comma must be placed after none: "none, being a stranger," which is a very harsh construction.
The adverbial use of " none " may be traced to Early English and Anglo-Saxon. Under the form "nan," i.e. "ne-an" (compare German "nein"), we find "nan more," and also "none longer," "whether he wolde or noon" (CHAUCER, Mätzner). " Nan " was used as an adverbial accusative for "by no means" even in A.-S. (Mätzner, iii. 131.) In Rich. II. v. 2. 99, "He shall be none," the meaning is, "he shall not be one of their number." "None" is still used by us for "nothing," followed by a partitive genitive, "I had none of it ; " and this explains the Elizabethan phrase
"She will none of me."—T. N. i. 3. 113.
i.e. "She desires to have (321) nothing from, as regards to do with, me." So
"You can say none of this."—T. N. v. 1. 342.
54. Not is apparently put for "not only" in the two following passages:
" Speak fair ; you may salve so
Not what is dangerous present, but the loss
Of what is past."—Coriol. iii. 2. 71.
" For that he has
Given hostile strokes, and that not in the presence
Of dreaded justice, but on the ministers
That do distribute it."—Coriol. iii. 3. 97.
55. Nothing, like "no-way," "naught," "not," (A.-S. náht, i.e. "no whit,") is often used adverbially.
" And that would set my teeth nothing on edge."
1 Hen. IV. iii. 1. 133.
" I fear nothing what may be said against me."
Hen. VIII. i. 2. 212.
where "what" is not put for "which."
56. Off (away from the point) :
"That's off: that's off. I would you had rather been silent." Coriol. ii. 2. 64.
To be off = to take off one's hat:
"I will practise the insinuating nod and be off to them most counterfeitly."—Coriol. ii. 3. 107.
57. Once ("once for all," "above all "):
"Once, if he require our voices, we ought not to deny him."
Coriol. ii. 3. 1.
"'Tis once thou lovest,
And I will fit thee with the remedy."—M. Ado, i. 1. 320.
Hence "positively."
"Nay, an you be a cursing hypocrite, once you must be looked to."—M. Ado, V. i. 212.
"Nay, an you begin to rail on society, once I am sworn not to give regard to you."—Timon, i. 2. 251.
The Folio and Globe place the comma after once
Once is sometimes omitted:
"This is (once) for all."—Hamlet, i. 3. 131.
Once sometimes "in a word:"
"Once this—your long experience of her wisdom,
Her sober virtue, years, and modesty,
Plead on her part some cause to you unknown."
C. of E. iii. 1. 90.
At once is found in this or a similar sense:
"My lords, at once; the cause why we are met
Is to determine of the coronation."—Rich. III. iii. 4. 1.
" My lords, at once; the care you have of us
Is worthy praise."—2 Hen. VI. iii. 1. 66.
Once seems to mean "at some time (future)" in
" I thank thee, and I pray thee, once to-night
Give my sweet Nan this ring."—M. W. of W. iii. 4. 103.
But the word may be taken as above.
58. Only, i.e. on(e)ly, is used as an adjective. See But (130), and Transpositions (420).
"The only (mere) breath."—SPENS. F. Q. i. 7. 13.
"It was for her love and only pleasure."—INGELEND.
"By her only aspect she turned men into stones."—BACON, Adv. of L. 274.
We have lost this adjectival use of only, except in the sense of "single," in such phrases as " an only child."
Only, like "alone" (18), is used nearly in the sense of "above all," "surpassing."
'Oph. You are merry, my lord.
Ham. Who? I?
Oph. Ay, my lord.
Ham. O God, your only jig-maker."—Hamlet, iii. 2. 131.
"Your worm is your only emperor for diet."—Ib. iv. 3. 22.
58 a. Over means "over again" in
"Trebles thee o'er."—Tempest, ii. 1. 221.
i.e. "repeats thy former self thrice." Compare
"I would be trebled twenty times myself."—M. of V. iii. 2. 154.
59. Presently = "at the present time," "at once," instead of, as now, "soon, but not at once."
"Desd. Yes, but not yet to die.
Othello. O yes, presently."—Othello, v. 2. 52.
So Rich. II. iii. 1. 3; 2. 179.
60. Round, used adverbially in the sense of " straightforwardly." "Round," like "square" with us, from its connection with "regular," "symmetrical," and "complete," was used to signify "plain and honest." Hence
" I went round to work."—Hamlet, ii. 2. 139.
means just the opposite of "circuitously."
61. Severally ("sever," Lat. separo), used for "separately." So
"When severally we hear them rendered."—J. C. iii. 2. 10.
And "Contemplation doth withdraw our soule from us, and severally employ it from the body."—MONTAIGNE, 30.
Thus, "a several plot" (Sonn. 137) is a "separate" or "private plot" opposed to "a common."
62. Since (A.-S. sith = "time," also adv. "late," "later;" "sith-than" = "after that") adverbially for "ago."
"I told your lordship a year since."—M. Ado, ii. 2. 13.
This must be explained by an ellipsis :
" I told your lordship (it is) a year since (I told you)."
Compare a transitional use of "since" between an adverb and conjunction in " Waverley ; or, 'tis Sixty Years since." Omit "'tis," and since becomes an adverb.
So since is used for "since then," like our "ever since" in
'And since, methinks, I would not (do not wish to) grow so
fast."—Rich. III. ii. 4. 14.
Since, when used adverbially as well as conjunctionally, frequently takes the verb in the simple past where we use the complete present:
"I did not see him since."—A. and C. i. 3. 1.
This is in accordance with an original meaning of the word, "later," ("sith.") We should still say, "I never saw him after that;" and since has the meaning of "after."
We also find the present after "since," to denote an action that is and has been going on since a certain time. (So in Latin with "jampridem.")
" My desires e'er since pursue me. "—T. N. i. 1. 23.
See Conjunctions, 132.
63. So (original meaning "in that way") is frequently inserted in replies where we should omit it:
"Trib. Repair to the Capitol.
Peop. We will so."—Coriol. ii. 3. 262.
"T. Fortitude doth consist, &c.
D. It doth so indeed, sir."—B. J. Sil. Wom. iv. 2.
Here so means "as you direct, assert." "As" is, by derivation, only an emphatic form of so. See 106.
64. So is sometimes omitted after " I think," "if," &c.
"G. What, in metre?
Luc. In any proportion or language.
G. I think, or in any religion."—M. for M. i. 2. 24.
"Will the time serve to tell? I do not think (so)."
Coriol. i. 6. 46.
"Haply you shall not see me more; or if,
A mangled shadow."—A. and C. iv. 2. 27.
"Not like a corse; or if, not to be buried."—W. T. iv. 4. 131.
"Do not plunge thyself too far in anger, lest thou hasten thy trial, which if, Lord have mercy on thee for a hen. "—A. W. ii. 3. 223.
Compare
"What though; yet I live like a poor gentleman born."
M. W. of W. i. 1. 287; Hen. V. ii. 1. 9; A. Y. L. iii. 3. 51.
"O, if it prove,
Tempests are kind and salt waves fresh in love."
T. N. iii. 4. 418.
65. So is put for the more emphatic form, al-so.
"Demetrius, thou dost over-ween in all,
And so in this, to bear me down with braves."
T. A. ii. 1. 30.
"It is a cold and heat that does outgo
All sense of winters and of summers so."—B.J. Sad Sh. ii. 1.
"Mad in pursuit, and in possession so. "—Sonn. 129.
"Good morrow, Antony.
Ant. So to most noble Cæsar."—J. C. ii. 2. 117.
So approaches " also" in
"Cousin, farewell; and, uncle, bid him so."
Rich. II. i. 3. 247.
So that; so as. (See Pronouns, Relative, 275, 276.)
66. So (like the Greek o τω δ ) is often used where we should use " then." " In this way" naturally leads to " thus," " on this," " tltereupon," "then."
"And when this hail some heat from Hermia felt
So he dissolved."—M. N. D. i. 1. 245.
So is, therefore, sometimes more emphatic than with us, as in (arrange thus, not as Globe)—
" Olivia. To one of your receiving enough is shown ;
A cypress, not a bosom, hides (Fol.) my heart——(pauses)
So (i.e. after this confession) let me hear you speak.
Vio.
I pity you."
T. N. iii. 1. 133.
So in conditional clauses. See Conjunctions, 133.
67. So was often, and correctly, used (where we use the adverbial "such" or " so" with "a") before an adjective, e.g. "so great faith" where we say "such great faith," "so long time" where we say "so long a time." We seem to feel that "so" (being an adverb, and therefore more liable to transposition than the adjective "such") requires to be attached to the word which it qualifies, either (1) by introducing the article which necessarily links together the words thus : "so-great a-loss;" or else (2) by placing "so" in a position where its effect is equally unmistakeable : "a-loss so-great."
When the noun is in the plural we cannot use the former method ; we are, therefore, driven to the latter, and instead of saying
" So hard termes."—N. P. 176.
we say "terms so hard."
" In so profound abysm I throw all care."—Sonn 112.
" My particular grief
Is of so flood-gate and o'erbearing nature."—O. i. 3. 55.
" And I will call him to so strict account."—1 Hen,IV. iii. 2. 149.
" With so full soul."—Temp. iii. 1. 44.
" Of so quick condition."—M. for M. i. 1. 54.
But note that in these instances the "so" follows a preposition. After prepositions the article (see Article, 90) is frequently omitted. Shakespeare could have written
" My grief is of nature so floodgate," &c.
" I will call him to account so strict that," &c.
Our modern usage was already introduced side by side with the other as early as Wickliffe. Compare
"So long time. "—St. John xiv. 9.
with "So long a time. "—Hebrews iv. 7.
68. Something used adverbially, like "somewhat."
" A white head and something a round belly."
2 Hen. IV. i. 2. 212.
We should say "a somewhat round," placing the adverb between the article and the adjective so as to show unmistakeably that the adverb qualifies the adjective. " Something" may possibly be so taken (though "somehow" would make better sense) in
"This something-settled matter in his breast."—Ham. iii. 1. 181.
68 a. Sometimes, like "sometime," is used by Shakespeare for "formerly" in
"Thy sometimes brother's wife."—Rich. II. i. 2. 54.
So probably
"Sometimes from her eyes
I did receive fair speechless messages."—M. of V.i. 1. 163.
Compare "olim" in Latin.
69. Still used for constantly, in accordance with the derivation of the word, "quiet," "unmoved." It is now used only in the sense of "even now," "even then." The connection between "during all time up to the present " and "even at the present " is natural, and both meanings are easily derived from the radical meaning, "without moving from its place." Comp. the different meanings of dum, donec, ως, &c.
" Thou still hast been the author of good tidings."
Hamlet, ii. 2. 42.
"But this thy countenance still lock'd in steel
I never saw till now."—T. and C. iv. 5. 195.
i.e. "because it was constantly lock'd in steel."
And this is the best, though not the most obvious, interpretation of
"But still the house affairs would draw her hence."
Othello, i. 3. 147.
It is used as an adjective for constant (though some suggest "silent") in
"But I of thee will wrest an alphabet,
And by still practice learn to know the meaning."
T. A. iii. 2. 44.
This interpretation is corroborated by
"But that still use of grief makes wild grief tame,
My tongue should to thy ears not name my boys."
Rich. III. iv. 3. 229.
70. Than is used for then :
" And their ranks began
To break upon the galled shore and than
Retire again."—R. of L. 456.
Then for than, freq. in North's Plutarch, Ascham, &c.
In O. E. the commonest forms are "thanne" = then; "then" = than.
Then and than (like tum and tam, quum and quam in Latin) are closely connected, and, indeed, mere varieties of the same word. They were originally inflections of the demonstrative, and meant "at that (time)," "in that (way)." As "that" is used as a relative, "than" has the signification of "in the way in which" (quam), just as then (71) is used for "at the time at which" (quum). It is usual to explain "He is taller than I" thus : "He is taller; then I am tall." This explanation does not so well explain "He is not taller than I." On the whole, it is more in analogy with the German als, Latin quam, Greek , to explain it thus: "In the way in which I am tall he is taller." The close connection between "in that way," "at that time," "in that place," &c., is illustrated by the use of there for thereupon, or then.
"Even there resolved my reason into tears."—L. C. 42.
71. Then apparently used for "when." So in E. E. See That ,284.
"And more more strong, then lesser is my fear,
I shall endue you with; meantime but ask," &c.
K. J. iv. 2. 42.
72. To-fore, which was as common in E. E. as "be-fore" and " a-fore," is found in
" O would thou wert as thou to-fore hast been."
T. A. iii. 2. 294.
73. Too, which is only an emphatic form of "to" (compare πρóς in Greek, used adverbially), is often spelt "to" by Elizabethan writers (Sonn. 38, 86); and conversely, "too" is found for "to" (Sonn. 56, 135).
Too seems used, like the E. E. "to," for "excessively" in Spenser, Shephaeard's Calendar, May:
"Thilke same kidde (as I can well devise)
Was too very foolish and unwise."
Perhaps, also, in
"Lest that your goods too soon be confiscate."—C. of E. i. 2. 2.
though the meaning may be "the goods of you also. "
"Tempt him not so too far."—A. and C. i. 3. 11.
And there is, perhaps, an allusion to the E. E. meaning in "too-too," which is often found in Elizabethan English.
Too is often used in the phrase, "I am too blame" (Folio)
" I am much too blame."
O. iii. 3. 211, 282; M. of V. v. 1. 166; Rich. III. ii. 2. 13.
This is so common in other Elizabethan authors, that it seems to require more explanation than the confusion between "to" and "too" mentioned above. Perhaps "blame" was considered an adjective, as in
"In faith, my lord, you are too wilful-blame."
1 Hen. IV. iii. 1. 177.
and " too" may have been, as in E. E., used for " excessively."
Too seems used for "very much," or "too much," in
"Tell him that gave me this (wound), who lov'd him too,
He struck my soul and not my body through."
B. and F. F. Sh. iii. 1.
The context will hardly admit of the interpretation, "Me who also lov'd him."
The transition from the meaning of progressive motion to that of "increasingly" or "excessively," and from "excessively" to the modern " to excess," is too natural to require more than mention.
73 a. What, when. What and when are often used as exclamations of impatience:
" What, Lucius, ho !"—J. C. ii. 1. 1.
" When, Lucius, when?"—Ib. 5.
Some ellipsis is to be supplied, "What (is the matter)?" "When (are you coming) ?" So in
" Gaunt. Throw down, my son, the duke of Norfolk's gage.
K. Rich. And, Norfolk, throw down his.
Gaunt. When, Harry, when ?"—Rich. II. i. 1. 162.
See also What, 253.
74. Whilst. "The while" is often used in accordance with the derivation of the word for "(in) the (mean) time." The inflected forms whiles and whilest are generally used as conjunctions. But we have
" If you'll go fetch him
We'll say our song the whilst."—Cymb. iv. 2. 254.
75. Why (instrumental case of E.E. hwa, "who"), used after "for," instead of "wherefore." Like the Latin "quid enim ?" it came after a time to mean "for indeed," as
"And send the hearers weeping to their beds ;
For why, the senseless brands will sympathise."
Rich. II. v. 1. 40.
i.e. "wherefore? (because) the senseless," &c. The provincialism "whyfore" still exists. "For" does not correspond to "enim," but is a preposition by derivation. Later writers, however, and possibly Shakespeare, may have used "for" in "for why" as a conjunction. Some, however, maintain that the comma should be removed after "for why," and that "for why" (like νθ ν) means "for this that," "because," the relative containing an implied antecedent.
A distinction seems drawn between "why" and "for what" in
" Why, or for what these nobles were committed
Is all unknown to me, my gracious lady."—Rich. III. ii. 4. 48.
Why, perhaps, refers to the past cause, for what to the future object.
" Ant. S. Shall I tell you why?
Drom. S. Ay, sir, and wherefore ; for they say every why hath a wherefore."—C. of E. ii. 2. 43-45.
i.e. "every deed said to be done owing to a certain cause is really done for a certain object."
Compare
"Say, why is this? Wherefore? What shall we do ? "
Hamlet, i. 4. 57.
"Why" and "how" are both derivatives of the relative, and are sometimes interchanged in A.-S. "Why" seems to have been the ablative of instrument, and " how" the adverbial derivative of manner, from "who."
76. Yet (up to this time) is only used now after a negative, "not yet," "never yet," &c. Then it was also used before a negative.
"For (as) yet his honour never heard a play."—T. of Sh. Ind. 1. 96.
" Yet I have not seen
So likely an ambassador of love."—M. of V. ii. 9. 92.
" Yet (up to this time) they are not joined."—A. and C. iv. 12. 1.
" I will make one of her women lawyer to me, for I yet not understand the case myself."—Cymb. ii. 3. 80.
The following is a remarkable passage:
" Hel. You, Diana,
Under my poor instructions yet (still) must suffer
Something in my behalf.
Diana. Let death and honesty
Go with your impositions, I am yours
Upon your will to suffer.
Hel. Yet (i.e. for the present) I pray you ;
But with the word the time will bring on summer," &c.
A. W. iv. 4. 30.
i.e. "a little longer I entreat your patience, but," &c.
Yet is also used in this sense without a distinct negative:
"Solan. What news on the Rialto?
Salar. Why yet it lives there uncheck'd that Antonio," &c.
M. of V. iii. 1. 1.
77. The adverbs backward and inward are used as nouns.
"In the dark backward and abysm of time. "—Temp. i. 2. 50.
" I was an inward of his."—M. for M. iii. 2. 138.
So " Thou losest here a better where to find."—Lear, i. 1. 264.
"Nor can there be that deity in my nature
Of here-and-everywhere."—T. N. v. 1. 235.
i.e. "the divine attribute of ubiquity."
Then, as with us, was used as an adjective.
"Our then dictator."—Coriol. ii. 2. 93.
So
"Good sometime queen."—Rich. II. v. 1. 37.
"Our here approach."—Macb. iv. 3. 133.
See Compounds.
78. Adverbs after "is." We still say "that is well;" but, perhaps, no other adverb (except "soon") is now thus used. Shakespeare, however, has
"That's verily."—Tempest, ii. 1. 321.
"That's worthily."—Coriol. iv. 1. 53.
"Lucius' banishment was wrongfully."—T. A. iv. 4. 16.
Some verb, as "said" or "done," is easily understood. "In harbour" has the force of a verb in
"Safely in harbour
Is the king's ship."—Tempest, i. 2. 226.
# ARTICLES.
79. An, A, (Early Eng. An, Ane, On, One, a, o,) our indefinite Article, is now distinguished from our Numeral "one." In Early English, as in modern French and German, there was no such distinction. Hence, even in Elizabethan English, a (since it still represented, or had only recently ceased to represent, "one") was more emphatic than with us, a fact which will explain its omission where we insert it, and its insertion where we should use some more emphatic word, "some," "any," "one," &c.
80. An and one, pronunciation of. The connection between "an " and "one" appears more obvious when it is remembered that "one" was probably pronounced by Shakespeare, not as now "won," but "un." This is made probable by the constant elision of " the " before " one " in " th' one " as in " th' other: " compare " th' one" in
" Th' one sweetly flatters, t' other feareth harm."—R. of L. 172. So Rich. II. v. 2. 18. Ben Jonson (783) mentions as authorized contractions, " y'once " for " ye once" along with "y'utter." Compare also the pun in T. G. of V. ii. 1. 3:
"Speed. Sir, your glove.
Val. Not mine; my gloves are on.
Speed. Why, then, this may be yours, for this is but one."
This will explain the rhyme:
" So thanks to all at once and to each one
Whom we invite to see us crowned at Scone."
Macbeth, v. 8. 74–5.
In the dialect of the North of England and of Scotland, the " w " is still not sounded.
" An " was always used in A. -S. and mostly in E. E. before consonants as well as vowels: "ane kinges ... dohter" (STRATMANN). I have not found an instance in Shakespeare of "an" before an ordinary consonant, but it occurs before "w":
" Have an wish but for't. "—P. of T. iv. 4. 2.
81. A was used for one in such expressions as "He came with never a friend," &c.
" He and his physicians are of a mind."—A. W. i. 3. 244.
"'Fore God, they are both in a tale."—M. Ado, iv. 2. 33.
" An two men ride of a horse one must ride behind."
Ib. iii- 5. 44.
" For in a night the best part of my power
Were in the Washes . . . devoured."—K. J. v. 7. 64.
So
" The Images were found in a night all hacked and hewed."
N. P. 172.
" We still have slept together,
Rose at an instant, learn'd, play'd, eat together."
A. Y. L. i. 3. 76.
" Myself and a sister both born in an hour."—T.N. ii. 1. 20.
" You, or any living man, may be drunk at a time, man."
Othello, ii. 3. 319.
i.e. " at one time," " for once."
" These foils have all a length."—Hamlet, v. 2. 277.
We find " one" and "a" interchanged in
" Hear me one word :
Beseech you, tribunes, hear me but a word."
Coriol. iii. 1. 266.
" But shall we wear these honours for a day?
Or shall they last "—Rich. III. iv. 2. 5.
We never use the possessive inflection of the unemphatic one as an antecedent ; but Shakespeare writes :
"For taking one's part that is out of favour. "—Lear, i. 4. 111.
We also find in Early English :
" Thre persones in a Godhede. "—HALLIWELL.
where a is for one. Compare Scotch "ae" for "one."
It seems used for "any," i.e. ane-y, or one-y, in
"There's not a one of them."—Macb. iii. 4. 131.
"Ne'er a one to be found."—B. J. E. in &c. iii. 2.
So Cymb. i. 1. 24.
And emphatically for "some," "a certain," in
"There is a thing within my bosom tells me."
2 Hen. IV. iv. 1. 183.
'I should impart a thing to you from his majesty."
Hamlet, v. 2. 92.
"Shall I tell you a thing?"—L. L. L. v. 1. 152.
" I told you a thing yesterday."—Tr. and Cr. i. 2. 185.
"And I came to acquaint you with a matter."
A. Y. L. i. 1. 129.
82. A and The omitted in archaic poetry. In the infancy of thought nouns are regarded as names, denoting not classes but individuals. Hence the absence of any article before nouns. Besides, as the articles interfere with the metre, and often supply what may be well left to the imagination, there was additional reason for omitting them. Hence Spenser, the archaic poet, writes
"Fayre Una—whom salvage nation does adore."
F. Q. i. 6. Title.
" And seizing cruell clawes on trembling brest."—Ib. i. 3. 19.
"Faire virgin, to redeem her deare, brings Arthure to the fight."—Ib. i. 8. Title.
" From raging spoil of lawlesse victors will. "—Ib. i. 3. 43.
" With thrilling point of deadly yron brand."—Ib. i. 3. 42.
Shakespeare rarely indulges in this archaism except to ridicule it:
"Whereat with blade, with bloody blameful blade,
He bravely broached his boiling bloody breast ;
And Thisby, tarrying in mulberry shade,
His dagger drew and died."—M. N. D. v. 1. 147.
Somewhat similar is
"In glorious Christian field. "—Rich. II. iv. 1. 93.
" When lion rough in wildest rage doth roar."
M. N. D. v. 1. 224.
" Ah ! Richard with the eyes of (my or the) heavy mind."
Rich. II. ii. 4. 18.
" So, longest way shall have the longest moans."
Ib. v. 1. 90.
In antitheses, as
" And with no less nobility of love
Than that which dearest father bears his son,"
Hamlet, i. 2. 111.
the omission of the is intelligible, since the whole class is expressed. But it appears not uncommon to omit the article before superlatives :
"Best safety lies in fear. "—Hamlet, i. 3. 41.
This is, perhaps, explained by the double meaning of the superlative, which means not only "the best of the class," but also "very good." See 8.
83. A and The are also sometimes omitted after as, like, and than in comparative sentences :
"As falcon to the lure away she flies."—V. and A. 1027.
"The why is plain as way to parish church."
A. Y. L. ii. 7. 52.
"More tuneable than lark to shepherd's ear."
M. N. D. i. 1. 184.
This is, however, common both in early and modern English. In such sentences the whole class is expressed, and therefore the article omitted. It might be asked, however, why "the lure" on this hypothesis? The is put for its. So in E. E. (MÄTZNER, iii. 195) "ase hound doth (chase) the hare," i.e. "its prey the hare."
A is still omitted by us in adverbial compounds, such as "snail-like," "clerk-like," &c. Then it was omitted as being unnecessarily emphatic in such expressions as :
"Creeping like snail."—A. Y. L. ii. 7. 146.
" Sighing like furnace."—Ib. 148.
"And like unletter'd clerk."—Sonn. 85.
"Like snail " is an adverb in process of formation. It is intermediate between "like a snail" and "snail-like."
84. A being more emphatic than with us, was sometimes omitted where the noun stands for the class, and might almost be replaced by the corresponding adjective. "If ever I were traitor," Rich. II. i. 3. 201 = traitorous. Similarly
'And having now shown himself open enemy to Alcibiades."
N. P. 176.
So, though we find "never a master" in the sense of "not one master," yet where the "never " is emphasized and has its proper meaning, "at no time," the a is omitted :
" Those eyes which never shed remorseful tear."
Rich. III. i. 2. 156.
" In war was never lion rag'd so fierce."—Rich. II. ii. 1. 173.
" Never master had a page so kind. "—Cymb. v. 5. 85.
"Was ever king that joy'd an earthly throne."
2 Hen. VI. iv. 9. 1.
"'Twas never merry world since," &c.—T. N. iii. 1. 109.
On the other hand, in contrast to the example first quoted, when the "never" is omitted and an is emphatic, almost like one, it is inserted :
'My manly eyes did scorn an humble tear."
Rich. III. i. 2. 165.
A is also omitted before collective nouns, such as "plenty," "abundance," &c., and therefore before "great number" in
"Belike you slew great number of his people."—T. N. iii. 3. 29.
85. A inserted after some adjectives used as adverbs:
"It was upon this fashion bequeathed me by will but poor a thousand pounds. "—A. Y. L. i. 1. 2.
This usage is found in the earlier text of LAYAMON (A.D. 1200): "Long a time (longe ane stunde)," ii. 290, &c., where the adjective appears merely to be emphasized, and not used adverbially. In the later text the adjective is placed, here and in other passages, in its ordinary position. The adjectives "each," "such," "which," (used for "of what kind,") and "many" were especially often thus used. "At ich a mel" = "at each meal," Piers Plough. Crede. 109. (So in Scotch "ilka.") "Whiche a wife was Alceste," CHAUCER, C. T. 11754 = "what a wife." "On moni are (later text, mani ane) wisen," LAYAMON, i. 24 ; "monianes cunnes," ib. 39 ; "of many a kind (l. t. of manian erthe)," "of many an earth."
The last-quoted passages render untenable the theory (Archbishop Trench, English Past and Present) which explains "many a man" as a corruption of "many of men." In these passages, e.g. "moni anes cunnes" ("of many a race "), the article or numeral adjective "an " is declined like an adjective, while "moni " is not. The inference is, that "moni" is used adverbially. In the same way the Germans say "mancher (adj.) mann," but "manch (adv.) ein mann," "ein solcher (adj.) mann," but "solch (adv.) ein mann." In A.-S. the idiom was "many man," not "many a man." The termination in y, causing "many" to be considered as adverbially used, may not perhaps account for the introduction of the a into E. E., but it may account for its retention in Elizabethan and modern English. Nor can it escape notice that most of the adjectives which take a after them end in ch, or lic ("like"), an adverbial termination. So beside the adjectives enumerated above, "thellich" (modern Dorsetshire, "thilk" or "thick"), "the like," answering to "whilk" ("which"), is followed by a. So after the adverb "ofte," we have "a day" in
"Ful ofte a day he swelde and seyde alas!"
CHAUCER, Knighte's Tale, 498.
It is perhaps some such feeling, that "many" means "often," which justifies the separation of "many" and "a" in the following:
" I have in vain said many
A prayer upon her grave."—W. T. v. 3. 144.
Perhaps in this way (as an adjective used adverbially) we must explain (compare " none (adj.) inheritance," Acts vii. 5):
"Exceeding pleasant; none (adv.) a stranger there
So merry and so gamesome."—Cymb. i. 6. 59.
like "ne'er a stranger," unless after "none" we supply "who was."
A is pleonastically used in
" I would not spend another such a night. "—R. III. i. 4. 5.
In "What poor an instrument" (A. and C. v. 2. 236), "what" is used for "how."
86. A was sometimes omitted after "what," in the sense of "what kind of."
"Cassius, what night is this? "—J. C. i. 3. 42.
(A has been unnecessarily inserted by some commentators.)
"I'll tell the world
Aloud what man thou art."—M. for M. ii. 4. 153.
"Jove knows what man thou mightst have made."
Cymb. iv. 2. 207.
" What dreadful noise of waters in mine ears."
Rich. III. i. 4. 22.
" What case stand I in?" (W. T. i. 2. 352) = In what a
position am I ?
" What thing it is that I never
Did see man die!"—Cymb. iv. 4. 35.
We omit the article after "what" before nouns signifying a collective class, saying "what wickedness!" but "what a crime !" " what fruit !" but "what an apple !" Hence the distinction in the following : "What a merit were it in death to take this poor maid from the world ! What corruption in this life that it will let this man live!"—M. for M. iii. 1. 240.
A is omitted after "such:"
" Showers of blood,
The which, how far off from the mind of Bolingbroke
It is such crimson tempest should bedrench," &c.
Rich. II. iii. 3. 46.
Here "such" probably means "the aforesaid," referring to the "showers of blood."
After "such" in this sense the indefinite article is still omitted; naturally, since "such " is used in a defining sense.
A is omitted after "many" in "Many time and oft" (2 Hen. VI. ii. 1. 93). Here "many-time," like "some-time," "often-times," "many-times" (MONTAIGNE, Introduction), seems used as one word adverbially.
A is omitted before "little," where we commonly place it in the sense of "some:"
"O, do not swear ;
Hold (a) little faith, though thou hast too much fear."
T. N. v. 1. 174.
It is perhaps caused by the antithesis which assimilates the use of "little" to the use of "much." "In (a) little time "(V. and A. 132) is to be explained as a prepositional phrase approximating to an adverb : see 89.
87. A was frequently inserted before a numeral adjective, for the purpose of indicating that the objects enumerated are regarded collectively as one. We still say "a score," "a fo(u)rt(een)-night." But we also find :
"An eight days after these sayings."—Luke ix. 28.
"A two shilling or so."—B. J. E. in &c. i. 4 ad fin.
"'Tis now a nineteen years agone at least."—B. J. Case is altered.
Also in E. E. :
"An five mile."—HALLIWELL.
This usage is not common in Shakespeare, except after "one."
"But one seven years."—Coriol. iv. 1. 55.
The a is omitted in
"But this our purpose now is twelve-month old."
1 Hen. IV. i. 1. 28.
Compare " This three mile."—Macbeth, v. 5. 37.
The a in "a many men," "a few men," is perhaps thus to be explained. Compare " This nineteen years" (M. for M. i. 3. 21), with "This many summers" (Hen. VIII. iii. 2. 360). So
"A many merry men."—A. Y. L. i. 1. 121.
"A many thousand warlike French."—K. J. iv. 2. 199.
So Hen. V. iv. 1. 127 ; iv. 3. 95. And still more curiously :
"But many a many foot of land the worse."—K. J. i. 1. 183.
Some explain "a many" by reference to the old noun "many," "a many men," for "a many (of) men." And the word is thus used:
"A many of our bodies."—Hen. V. iv. 3. 95.
"O thou fond many, with what loud applause
Didst thou beat heaven."—2 Hen. IV. i. 3. 91.
"In many's looks."—Sonn. 93.
So perhaps A. W. iv. 5. 55. Add "their meiny," Lear, ii. 4. 35.
Nor can it be denied that in E. E. "of" is often omitted in such phrases as "many manner (of) men," "a pair (of) gloves," &c. just as in German we have "diese Art Mensch." But we also say "a few men" (an expression that occurs as early as Robert of Brunne), and "few" seems to have been an adjective.
It is probable that both the constructions above-mentioned are required to explain this use of a. Thus "a hundred men" is for "a hundred (of) men," but in "a twelvemonth," "a fortnight," "twelve" and "fourteen" are not regarded as simple nouns, but as compound nouns used adjectively. Compare the double use of "mille," "millia," in Latin.
88. An-other. A is apparently put for the in
"There is not half a kiss to choose who loves an other best."
W. T. iv. 4. 176.
This is, however, in accordance with our common idiom: "they love one an other," which ought strictly to be either "they love, the one the other," or "they love, one other." The latter form is still retained in "they love each other;" but as in "one other" there is great ambiguity, it was avoided by the insertion of a second "one" or "an," thus, "they love one an-other." This is illustrated by Matt. xxiv. 10 (TYNDALE) : "And shall betraye one another and shall hate one the other;" whereas WICKLIFFE has, "ech other." So I Cor. xii. 25 : WICKLIFFE, "ech for other;" the rest "for one another." "One another" is now treated almost like a single noun in prepositional phrases, such as, "We speak to one another." But Shakespeare retains a trace of the original idiom in
"What we speak one to an other."—A. W. iv. 1. 20.
89. The was frequently omitted before a noun already defined by another noun, especially in prepositional phrases.
"In number of our friends."—J. C. iii. 1. 216.
"Since death of my dearest mother."—Cymb. iv. 2. 190.
"At heel of that defy him."—A. and C. ii. 2. 160.
"In absence of thy friend."—T. G. of V. i. 1. 59.
"To sternage of their navy."—Hen. V. iii. Prol. 18.
"To relief of lazars."—Ib. i. 1. 15.
"For honour of our land."—Ib. iii. 5. 22.
"Thy beauty's form in table of my heart."—Sonn. 24.
"Some beauty peep'd through lattice of sear'd age." L. C. st. ii.
"Forage in blood of French nobility."—Hen. V. i. 2. 110.
"In cradle of the rude imperious surge."—2Hen. IV. iii. 1. 20.
"Proving from world's minority their right."—R. of L.
"On most part of their fleet."—Othello, ii. 1. 24.
So 1 Hen. VI. i. 2. 77; 2 Hen. VI. i. 2. 36, 79; Rich. II. i. 3. 136. We could say "in season," but not
"We at (the right) time of (the) year
Do wound the bark."—Rich. II. iii. 4. 57.
So even in Pope :
"Alas, young man, your days can ne'er be long;
In flower of age you perish for a song."
POPE, Imit. Hor. i. 102.
90. The is also omitted after prepositions in adverbial phrases.
" At door."—W. T. iv. 4. 352 ; T. of Sh. iv. 1. 125.
" At palace."—W. T. iv. 4. 731.
"At height."—Hamlet, i. 4. 21.
" Ere I went to wars."—M. Ado, i. 1. 307.
" To cabin."—Tempest, i. 1. 17.
" The grace 'fore meat and the thanks at end."
Coriol. iv. 7. 4.
" You were in presence then."—Rich. II. iv. 1. 62.
i.e. "in the presence-chamber."
" And milk comes frozen home in pail."—L. L. L. v. 2. 925.
" With spectacles on nose and pouch on side."
A. Y. L. ii. 7. 159.
" This day was viewed in open as his queen."
Hen. VIII. iii. 2. 405.
" He foam'd at mouth."—J. C. i. 2. 256.
" Sticks me at heart."—A. Y. L. i. 2. 254.
" Exeunt in manner as they entered."—Ib. ii. 4. 242.
" Than pard or cat-o'-mountain."—Tempest, iv. 1. 262.
And with adjectives :
" In humblest manner."—Tempest. ii. 4. 144.
"In first rank."—Tr. and Cr. iii. 3. 161.
" In pail" is as justifiable as "in bed," except that the former, not being so common as the latter, has not the same claim to the adverbial brevity which dispensed with the article. Both are adverbial phrases, one of which has been accepted, the other rejected. Thus in
" Stealing unseen to west with this disgrace."—Sonn. 33.
" to-west" is as much an adverb as "west-ward."
Sometimes a possessive adjective is thus omitted:
" Not Priamus and Hecuba on knees."—Tr. and Cr. v. 3. 53.
So in E. E. "a-knee."
Compare our "I have at hard."
Perhaps this may explain the omission of "the" after "at" in
" We are familiar at first."—Cymb. i. 4. 112.
where " at first " is not opposed to " afterwards " (as it is with us), but means "at the first," or rather "from the first," "at once."
The omission of "the" in
"On one and other side Trojan and Greek
Sets all on hazard."—Tr. and Cr. i. 1. 21.
is in accordance with our idiom, "one another" and "each other."
On the other hand, where "the" is emphatic, meaning "that" or "the right," it is sometimes inserted before "one."
"Morocco. How shall I know if I do choose the right ?"
Portia. The one of them contains my picture, prince."
M. of V. ii. 7. 11.
91. The was inserted in a few phrases which had not, though they now have, become adverbial. "At the length" (N. P. 592), "At the first," "At the last," &c.
"There in the full convive we."—Tr. and Cr. iv. 5. 272.
" In the favour of the Athenians. "—N. P. 177.
92. The used to denote notoriety, &c. Any word when referred to as being defined and well known may of course be preceded by the article. Thus we frequently speak of "the air." Bacon (E. 231) however wrote, " The matter (the substance called matter) is in a perpetual flux."
The is sometimes used (compare Latin "ille") for "the celebrated," "the one above all others," occasionally with "alone," as
" I am alone the villain of the earth. "—A. and C. iv. 6. 30.
Or with a superlative :
" He was the wretched'st thing when he was young."
Rich. III. ii. 4. 18.
" The last (prayer) is for my men : they are the poorest ;
But poverty could never draw 'em from me."
Hen. VIII. iv. 2. 148.
But also without these :
"Am I the man yet ?"—A. Y. L. iii. 3. 3.
"Smacks it not something of the policy ?"—K. J. ii. 1. 396.
" For their dear causes
Would to the bleeding and the grim alarm
Excite the mortified man."—Macbeth, v. 2. 4.
The ellipsis to be supplied is added in
"Are you the courtiers and the travell'd gallants?
The spritely fellows that the people talk of?"
B. and F. Elder Brother, iv. 1.
The seems to mean "the same as ever" in
"Live you the marble-breasted tyrant still."—T. N. v. 1. 127.
It is not often that "the" is used in this sense before English proper names. In
"The Douglas and the Percy both together."
1 Hen. IV. v. 1. 116.
the second the may be caused by the first, which, of course, is still used, "the Bruce," "the Douglas," being frequent, and explicable as referring to the chief of the Douglases and Bruces. But we also have
"To leave the Talbot and to follow us."—I Hen. VI. iii. 3. 20, 31. and so in Early English "the Brute," "the Herod."
The is seldom used, like the article in French, for the possessive adjective :
'The king is angry : see, he bites the lip."
Rich. III. iv. 2. 27.
The word "better" is used as a noun, and opposed to "the worse," (compare the French proverb, "le mieux est l'ennemi du bien,") in
" Bad news, by'r lady; seldom comes the better." Rich. III. ii. 3. 4.
" Death," the ender of life, seems more liable to retain the mark of notoriety than "life." Hence
" Where they feared the death, they have borne life away."
Hen. V. iv. 1. 81 ; Rich. III. i. 2. 179 ; ii. 3. 55.
So " Dar'd to the combat."—Hamlet, i. 1. 84. i.e. "the combat that ends all dispute." French influence is perceptible in these two last instances, and in
" To shake the head."—M. of V. iii. 2. 15.
The which (see Relative), 270.
93. The frequently precedes a verbal that is followed by an object :
" Whose state so many had the managing."—Hen. V. Epilog.
" You need not fear the having any of these lords."
M. of V. i. 2. 109.
"The seeing these effects will be
Both noisome and infectious."—Cymb. i. 5. 25.
"P. Pray, sir, in what?
D. In the delaying death."—M. for M. iv. 2. 172.
"Nothing in his life
Became him like the leaving it."—Macb. i. 4. 8.
"The locking up the spirits."—Cymb. i. 5. 41.
So Lear, iv. 4. 9 ; Hen. VIII. iii. 2. 347 ; M. for M. iii. 2. 126 ; M. of V. iv. 1. 309 ; M. Ado, ii. 2. 53 ; O. iii. 4. 22 ; T. N. i. 5. 84.
The question naturally arises, are these verbals, "locking," &c. nouns? and, if so, why are they not followed by "of,"—e.g. "the locking of the spirits"? Or are they parts of verbs? and in that case, why are they preceded by the article? The fact that a verb in E. E. had an abstract noun in -ing (A.-S. -ung)—e.g. "slaeten," to hunt; "slaeting," hunting—renders it a priori probable that these words in -ing are nouns. Very early, however, the termination -ng was confused with, and finally supplanted, the present participle termination in -nde. Thus in the earlier text of Layamon (iii. 72) we have "heo riden singinge," i.e. "they rode singing;" and in the later text the proper participial form "singende." An additional element of confusion was introduced by the gerundial inflection enne, e.g. "singenne," used after the preposition "to." As early as the twelfth century "to singenne" (Morris, E. E. Specimens, p. 53) became "to singende," and hence (by the corruption above mentioned) "to singinge." Hence, when Layamon writes that the king went out "an-slaeting" (ii. 88), or "a-slatinge" (iii. 168), it is not easy to prove that the verbal noun is here used : for the form may represent the corruption of the gerund used with the preposition "an" instead of with "to." And as early as Layamon we find the infinitive "to kumen" side by side with the present participle "to comende" (i. 49); and the gerund "cumene" side by side with the verbal "coming" (iii. 231) ; and the noun "tiding(s)" spelt in the earlier text "tidind" or "tidinde," the present participle (i. 59). The conclusion is, that although "locking" is a noun, and therefore preceded by "the," yet it is so far confused with the gerund as to be allowed the privilege of governing a direct object. The "of" was omitted partly for shortness, as well as owing to the confusion above mentioned.
It is easy to trace a process of abridgment from
" For the repealing of my banish'd brother,"—J. C. iii. 1. 51.
to (2) " Punish my life for (89) tainting of my love,"
T. N. v. 1. 141.
down to ourmodern (3) "for tainting my love." And hence the E. E. (William of Palerne, edit. Skeat), "for drede of descuverynge of that was do," 1. 1024, "of kastyng of lokes," 1. 942, are abbreviated in modern English into "disclosing that which was done" and "casting looks." This abbreviation is also remarkably illustrated by Bacon in his third Essay. He first uses the abbreviated form, and then, with a verbal noun that could not so easily have a verbal force, he adopts the full form : "Concerning the Means of procuring Unity. Men must beware that in the Procuring or Muniting of Religious Unity, they do not dissolve and deface the Laws of Charity." It is perhaps this feeling that the verbal was an ordinary noun, which allows Shakespeare to make an adjective qualify it even though of is omitted after it.
"He shall have old turning the key."—Macbeth, ii. 3. 2.
The substantival use of the verbal with "the" before it and "of" after it seems to have been regarded as colloquial. Shakespeare puts into the mouth of Touchstone :
"I remember the kissing of her batlet and ... the wooing of a peascod instead of her."—A. Y. L. ii. 4. 49-51.
"Did these bones cost no more (in) the breeding?"
Hamlet, v. 1. 190.
94. The (in Early Eng. thi, thy) is used as the ablative of the demonstrative and relative, with comparatives to signify the measure of excess or defect.
This use is still retained. "The sooner the better," i.e. "By how munch the sooner by so much the better." (Lat. "quo citius, eo melius.")
It is sometimes stated that "the better" is used by Shakespeare for "better," &c. : but it will often, perhaps always, be found that the has a certain force.
" The good conceit I hold of thee
Makes me the better to confer with thee."—T. G. of V. iii. 2. 19.
" The rather
For that I saw."—Macb. iv. 3. 184.
In both passages "the" means "on that account." In
" Go not my horse the better
I must become a borrower of the night,"—Macb. iii. 1. 25.
Banquo is perhaps regarding his horse as racing against night, and " the better " means " the better of the two." The following passage has been quoted by commentators on the passage just quoted, to show that "the" is redundant. "And hee that hit it (the quintain) full, if he rid not the faster, had a sound blow in his neck, with a bag full of sand hanged on the other end."—STOWE'S Survey of London, 1603. But the rider is perhaps here described as endeavouring to anticipate the blow of the quintain by being "the faster" of the two. Or more probably, "the faster" may mean the faster because he had struck the quintain, which, if struck, used to swing round and strike the striker on the back, unless he rode the (" on that account") faster. In either case it is unscholar-like to say that the is redundant.
# CONJUNCTIONS.
95. And (in old Swedish œn [Wedgewood] is used for "and," "if," and "even") emphatically used for "also," "even," "and that too." We still use "and that" to give emphasis and call attention to an additional circumstance, e.g. "He was condemned, and that unheard." This construction is most common in participial phrases. The "that" is logically unnecessary, and is omitted by Shakespeare.
" Suffer us to famish and their storehouses crammed with grain."
—Coriol. i. 1. 82.
"And shall the figure of God's majesty
Be judged by subject and inferior breath,
And he himself not present?"—Rich. II. iv. 1. 129.
"When I have most need to employ a friend,
And most assured that he is a friend,
Deep, hollow, treacherous, and full of guile
Be he unto me."—Rich. III. ii. 1. 37.
In the last two passages an ellipsis of "be" or "to be" might be understood, but scarcely in the following :
'So may he ever do and ever flourish
When I shall dwell with worms, and my poor name
Banish'd the kingdom."—Hen. VIII. iv. 2. 126.
"The friends thou hast, and their adoption tried,
Grapple them to the soul with hoops of steel."
Hamlet, i. 3. 62.
Compare 3 Hen. VI. i. 2. 47 ; Tr. and Cr. i. 3. 51.
So perhaps Hamlet, iii. 3. 62; T. N. i. 1. 38; and in the following irregular sentence :
" But a man that were to sleep your sleep, and a hangman to help him to bed, I think he (redundant pronoun : see 243) would change places with his officer."—Cymb. v. 4. 179.
i.e. "and that too a hangman being ready to help him to bed."
96. And. This use, though most frequent with participles, is also found without them :
" Here comes a spirit of his, and to torment me."
Temp. ii. 2. 15.
"He that has and a little tiny wit."—Lear, iii. 2. 74.
i.e. " a little and that a very little." So
" When that I was and a little tiny boy."—T. N. v. 1. 398.
97. And is frequently found in answers in the sense of "you are right and" or "yes and," the "yes" being implied. Hence the "and," introducing a statement in exact conformity with a previous statement, comes almost to mean "exactly." It is frequently found before "so."
" Hamlet. Will the king hear this piece of work ?
Pol. (Yes) And the queen too."—Hamlet, iii. 2. 53.
" Cass. This rudeness is a sauce to his good wit.
Brut. And so it is."—J. C. i. 2. 307.
i.e. "you are right, and so it is;" or "just so," "even so."
"Pompey. I'll try you on the shore.
Antony. And shall, sir. "—A. and C. ii. 7. 134.
i.e. "You say well, and you shall," or "So you shall," "that you shall," emphatically.
"Sir M. And there's... a head of noble gentlemen.
Archbishop. And so there is. "—I Hen. IV. iv. 4. 27.
"Parolles. After them, and take a more dilated farewell.
Bertram. And I will do so."—A. W. ii. 1. 60.
i.e. "that is just what I will do."
"Mayor. But I'll acquaint our duteous citizens
With all your just proceedings in this cause.
Glouc. And to that end we wish'd your lordship here."
Rich. III. iii. 4. 67.
i.e. "To that very end," "even to that end."
98. And is often found in this emphatic sense after statements implied by ejaculations, such as "faith," "sooth," "alas," &c. Thus
"Catesby. Your friends at Pomfret, they do need the priest.
Hastings. Good faith (it is so), and when I met this holy man
Those men you talk of came into my mind."
Rich. III. iii. 2. 117.
"Faith, and so we should."—I Hen. IV. iv. 1. 52.
This use is found in A.-S.
99. "And" emphatic in questions. When a question is being asked, "and," thus used, does not express emphatic assent, but emphatic interrogation:
"Alas! and would you take the letter of her?"—A. W. iii. 4. 1.
i.e. "is it so indeed, and further would you actually &c.?" So
"And wilt thou learn of me?"—Rich. III. iv. 4. 269.
i.e. "do you indeed wish to learn of me?"
Hence Ben Jonson, who quotes Chaucer:
"What, quoth she, and be ye wood?"
adds that
"And, in the beginning of a sentence, serveth for admiration."—B. J. 789.
It is common in ballads, and very nearly redundant :
"The Perse owt of Northumberlande,
And a vow to God made he."—Percy (MÄTZNER).
(Mr. Furnivall suggests "an avow," the original form of the word "vow.")
100. "And" for "also" in Early English. We find "and" often used for "also," "both," &c., and standing at the beginning of a sentence in earlier English. Wickliffe has, 2 Cor. xi. 21, 22 :
"In what thing ony man dare, and I dare. Thei ben ebreus, and I."
"And" is used for "even" or "also" in Acts xiv. 15 :
"And we ben deedli men like you."
In "I almost die for food, and let me have it,"
A. Y. L. ii. 7. 104.
"I pray you" may perhaps be understood after and, implied in the imperative "let."
101. And or an (= if). (The modern and is often spelt an in E. E.) This particle has been derived from an, the imperative of unnan, to grant. This plausible but false derivation was originated by Horne Tooke, and has been adopted by the editors of the Cambridge Shakespeare. But the word is often written and in Early English (Stratmann), as well as in Elizabethan authors.
" For and I shulde rekene every vice
Which that she hath ywiss, I were to nice."—CHAUC. Squire's Prol.
" Alcibiades bade the carter drive over, and he durst. "—N.P. 166.
" They will set an house on fire and it were but to roast their eggs."—B. E. 89.
" What knowledge should we have of ancient things past and history were not ? "—Lord BERNERS, quoted by B. J. 789.
102. "And" with the subjunctive. The true explanation appears to be that the hypothesis, the if, is expressed not by the and, but by the subjunctive, and that and merely means with the addition of, plus, just as but means leaving out, or minus.
The hypothesis is expressed by the simple subjunctive thus :
"Go not my horse the better
I must become a borrower of the night."—Macb. iii. 1. 25.
This sentence with and would become, " I must become a borrower of the night and my horse go not the better," i.e. "with, or on, the supposition that my horse go not the better." Similarly in the contrary sense, " but my horse go the better," would mean "without or excepting the supposition that my horse, &c." Thus Chaucer, Pardonere's Tale, 275 :
" It is no curtesye
To speke unto an old man vilonye
But he trespas."
So also Mandeville (Prologue) :
" Such fruyt, thorgh the which every man is saved, but it be his owne defaute."
103. And if. Latterly the subjunctive, falling into disuse, was felt to be too weak unaided to express the hypothesis ; and the same tendency which introduced "more better," "most unkindest," " &c., superseded and by and if, an if, and if. There is nothing remarkable in the change of and into an. And, even in its ordinary sense, is often written an in Early English. (See Halliwell.)
And or an is generally found before a personal pronoun, or "if," or "though ;" rarely thus :
"And should the empress know."—T. A. ii. 1. 69.
In the Elizabethan times the indicative is often used for the subjunctive.
The following is a curious passage :—
" O. Will it please you to enter the house, gentlemen?
D. And your favour, lady."—B. J. Sil. Wom. iii. 2. med.
Apparently, "And your favour (be with us)," i.e. "if you please."
104. An't were was wrongly said by Horne Tooke to be put for " as if it were."
"Cress. O ! he smiles valiantly.
Pand.
Does he not?
Cress. O yes ; and 'twere a cloud in autumn."
Tr. and Cr. i. 2. 139.
" He will weep you an't were a man born in April."
Ib. i. 2. 189.
" I will roar you and 'twere any nightingale. "—M. N. D. i. 2. 86.
"'A made a fairer end and went away, and it had been a Christom
child."—Hen. V. ii. 3. 10.
Some ellipsis is probably to be understood. " I will roar you, and if it were a nightingale (I would still roar better)."
The same construction is found in E. E.
"Ye answer and ye were twenty yere olde."
Cov. Myst. p. 80 (MÄTZNER).
It is illustrated by the use of " ac," " atque," after " similis," " pariter," &c. thus :
" (Homo) qui prosperis rebus æque ac tu ipse (gauderes) gaud-eret ."—CIC. De Amicitia, vi. 1.
i.e. "a man who would rejoice at your prosperity, and you yourself (would rejoice as much and no more)." " You answer in such and such a way, and were you twenty years old you would answer similarly."
105. And if represents both " even if " and " if indeed (i.c. both κα ε and ε κα ).
And if is used emphatically for "even if" in
" It dies and if it had a thousand lives."—1 Hen. VI v. 4. 75. So 1 Hen. IV. i. 3. 125.
" What and if
His sorrows have so overwhelm'd his wits."—Tit. And. iv. 4. 10.
" He seems to be of great authority, give him gold. And though authority be a stubborn bear, yet he is oft led by the nose with gold."—W. T. iv. 4. 831.
On the other hand, and if seems to mean "if indeed" in the following passages:—
"Percy. Seize it if thou darest.
Aum. And if I do not, may my hands rot off!"
Rich. II. iv. 1. 49.
" Oh father!
And if you be my father, think upon
Don John my husband."—MIDDLETON and ROWLEY (Waiker).
"Prince. I fear no uncles dead (419).
Glou.
Nor none that live, I hope.
Prince. And if they live, I hope I need not fear,"
Rich. III. iii. 2. 148,
where the Prince is referring to his maternal uncles who have been imprisoned by Richard, and he says, "if indeed they live I need not fear."
Thus probably we must explain :
" O full of danger is the duke of Gloucester !
And the queen's sons and brothers haught and proud ;
And were they to be ruled, and not to rule,
This sickly land might solace as before."—Rich. III. ii. 3. 29.
Here, at first sight, "but" seems required instead of "and." But "and were they " means " if indeed they were."
It is not easy to determine whether and though is used for "even though" or for " though indeed " in the following—
"I have now
(And though perhaps it may appear a trifle)
Serious employment for thee."—MASSINGER (Walker).
In all these passages an or and may be resolved into its proper meaning by supplying an ellipsis. Thus in the passage from Rich. II. iv. 1. 49, "And if I do not," &c. means, " I will seize it, and, if I do not seize it, may my hands rot off."
106. As (A.-S. "eall-swa," with the sense "just as") is a contraction of al(l)-so. In Early English we find "so soon so he came." The al(l) emphasized the so, " al(l)-so soon al(l)-so he came." Hence through different contractions, alse, als, ase, we get our modern as. (Comp. the German als.) The dropping of the / is very natural if alse was pronounced like "half." The broad pronunciation of as may throw light upon the pun in
" Sir And. And your horse now would make him an ass.
Mar. Ass I doubt not."—T. N. ii. 3. 185.
It follows that as originally meant both our modern so, "in that way," and our modern as, "in which way." The meaning of so is still retained in the phrases " as soon as" and "I thought as much," &c., but generally as has its second meaning, viz. " in which way."
107. As, like "an" (102), appears to be (though it is not) used by Shakespeare for as if. As above (102), the " if" is implied in the subjunctive.
"To throw away the dearest thing he owed
As 'twere a careless trifle."—Macb. i. 4. 11. So v. 5. 13.
i.e. "in the way in which (he would throw it away) were it a careless trifle." Often the subjunctive is not represented by any inflection :
" One cried, 'God bless us,' and ' Amen' the other,
As they had seen me with these hangman's hands."
Macbeth, ii. 1. 28 ; Rich. III. iii. 5. 63.
Sometimes the as is not followed by a finite verb :
" As gentle and as jocund as (if I were going) to jest,
Go I to fight."—Rich. II. i. 3. 95.
108. As, like "who," "whom," "which" (see below, Relative), is occasionally followed by the supplementary " that."
" Who fair him 'quited as that courteous was."
SPENS. F. Q. i. I. 30.
109. As for " that" after " so." (" In which way ;" "As the result of which.") This is a consequence of the original connection of as with " so."
"You shall be so received
As you shall deem yourself lodged in my heart."
L. L. L. ii. 1. 174.
"Catesby . . . finds the testy gentleman so hot
As he will lose his head ere give consent."
Rich. III. iii. 4. 41.
After "such:"
"Yet such deceit as thou that dost beguile
Art juster far."—Sonn.
This occurs less commonly without the antecedent so :
" My lord, I warrant you we'll play our part
As he shall think by our true diligence
He is no less than what we say he is."—T. of Sh. Ind. i. 68.
This points out an important difference between the Elizabethan and modern uses of as. We almost always apply it, like "because" (117), to the past and the present ; Shakespeare often uses it of the future, in the sense of " according as."
"And, sister, as the winds give benefit
And convoy is assistant, do not sleep,
But let me hear from you."—Hamlet, i. 3. 2.
Here a modern reader would at first naturally suppose as to mean " since" or " because ;" but . the context shows that it means "according as."
110. As, in its demonstrative meaning of so, is occasionally found parenthetically = "for so."
" This Jacob from our holy Abraham was
(As his wise mother wrought in his behalf)
The third possessor. "—M of V. i. 3. 73.
" Who dares receive it other—
As we shall make our griefs and clamours roar
Upon his death?"—Macb. i. 7. 78.
i.e. "so did his mother work;" "so will we make our griefs roar."
"The fixure of her eye has motion in 't,
As we are mock'd with art. "—W. T. v. 3. 68.
There seems some confusion in the difficult passage
"Speak truly, on thy knighthood and thy oath,
As so defend thee heaven and thy valour."
Rich. II. i. 3. 15.
In the similar line 34 as is omitted. This would lead us to conjecture "and." But perhaps the marshal was beginning to say "speak truly as may heaven defend thee," but diverged into the more ordinary "so," which was the customary mode of invocation. In that case the meaning will be "as thou wouldst desire the fulfilment of thy prayer, 'so help me heaven.'"
So in
"Duke. If this be so (as, yet, the glass seems true)
I shall have share in this most happy wreck."
T. N. v. 1. 272.
The Duke has called the appearance of the twins "a. natural perspective that is and is not" (ib. 224), i.e. a glass that produces an optical delusion of two persons instead of one. He now says : "it they are two, brother and sister (and indeed, spite of my incredulity, the perspective or glass seems to be no delusion), then I shall," &c. The curious introduction of the "wreck" suggests that the glass called up the thought of the "pilot's glass." (M. for M. ii. 1. 168.)
An ellipsis must be supplied in
" Had I but time (which I have not)—as this fell sergeant,
Death,
Is strict in his arrest."—Hamlet, v. 2. 347.
111. As = "as regards which," "though," "for," was sometimes used parenthetically in a sense oscillating between the relative "which," "as regards which," and the conjunction "for," "though," "since." It is used as a relative in
" But say or he or we, (as neither have [pl. see 12, Neither],)
Received that sum."—L. L. L. ii. 1. 133.
As is used in a transitional manner for " as regards which" or " for indeed," in
"Though I die for it, as no less is threatened me."
Lear, iii. 3. 19.
"When I was young, as, yet, I am not old."
I Hen. VI. iv. 4. 17.
" If you will patch a quarrel
As matter whole you've not to make it with."
A. and C. ii. 1. 53.
Here in the second example, " When I was young as I yet, or still, am," would have retained the relatival signification of as, but the addition of " not old" obliges us to give to as the meaning not of "which," but "as regards which" or "for." So in
"She dying, as it must be so maintained."
M. Ado, iv. 1. 216.
112. As, owing to its relatival signification, is sometimes loosely used for "which." This is still usual with us, but rarely except when preceded by " such " or " the same."
" That gentleness as I was wont to have."—J. C. i. 2. 33.
" Under these hard conditions as this time
Is like to lay upon us."—J. C. i. 2. 174.
This is still common in provincial language. See 280.
As is used for "where" in
"Here as I point my sword the sun arises."—J. C. ii. 1. 106.
113. As is frequently used (without such) to signify "namely:"
" And that which should accompany old age,
As honour, love, obedience, troops of friends."
Macb. v. 3. 25.
"Tired with all these for restful death I cry,
As to behold desert a beggar born
And needy nothing trimm'd in jollity
And, &c."—Sonn. 66.
So C. of E. i. 2. 98; Hen. VIII. iv. 1. 88; M. of V. iii. 2. 109.
"Two Cliffords, as the father and the son."
3 Hen. VI. v. 7. 7.
So A. Y. L. ii. 1. 6; Rich. II. ii. 1. 18; and Hamlet, i. 1. 117, where however a line has probably dropped out between 11,6 and 117.
114. As is apparently used redundantly with definitions of time (as ς is used in Greek with respect to motion). It is said by Halliwell to be an Eastern Counties' phrase :
"This is my birth-day, as this very day
Was Cassius born."—J. C. v. 1. 72.
"One Lucio as then the messenger."—M. for M. v. 1. 74.
The as in the first example may be intended to qualify the statement that Cassius was born on "this very day," which is not literally true, as meaning "as I may say." Here, and in our Collect for Christmas Day, "as at this time to be born," as seems appropriate to an anniversary. In the second example the meaning of "as then" is not so clear; perhaps it means "as far as regards that occasion." Compare
"Yet God at last
To Satan, first in sin, his doom applied,
Though in mysterious terms, judg'd as then best."
MILTON, P. L. x. 173.
where "as then" seems to mean "for the present." So "as yet " means "as far as regards time up to the present time." So in German "als dann" means "then," and "als" is applied to other temporal adverbs.
As in E. E. was often prefixed to dates:
"As in the year of grace," &c.
"As now" is often used in Chaucer and earlier writers for "as regards now," "for the present :"
"But al that thing I must as now forbere."
CHAUC. Knighte's Tale, 27.
In "Meantime I writ to Romeo
That he should hither come as this dire night,"
R. and J. v. 3. 247.
as perhaps means "as (he did come)."
115. As was used almost but not quite redundantly after "seem" (as it is still, after "regard," "represent") :
"To prey on nothing that doth seem as dead."
A. Y. L. iv. 3. 119.
and even after "am:"
"I am but as a guiltless messenger."—
A. Y. L. iv. 3. 12.
" I am here in the character of," &c.
As is also used nearly redundantly before participles to denote a cause, "inasmuch as:"
"If he be now return'd
As checking at his voyage."—Hamlet, iv. 7. 63.
116. As, like "that" (see 287), is used as a conjunctional suffix: sometimes being superfluously added to words that are already conjunctions. In the case of "when as," "where as," it may be explained from a desire to give a relative meaning to words interrogative by nature:
" (I am) one that was a woeful looker-on
When as the noble duke of York was slain."
3 Hen. VI. ii. 1. 46; i. 2. 75.
So " Whereas."—2 Hen. VI. i. 2. 58, for "where."
117. Because ("for this reason that") refers to the future instead of, as with us, to the past, in
"The splitting rocks cower'd in the sinking sands
And would not dash me with their rugged sides,
Because thy flinty heart, more hard than they,
Might in thy palace perish (act. 291), Margaret."
2 Hen. VI. iii. 2. 100.
i.e. " in order that thy flinty heart might have the privilege of destroying me."
118. But (E. E. and modern northern English "bout ") is in Old Saxon "bi-utan," where "bi" is our modern "by," and "utan" means "without." Thus but is a contraction for "by-out," and is formed exactly like "with-out." Hence but means excepted or excepting. This use of out in compounds may be illustrated by "outstep (except) the king be miserable."
" It was full of scorpyones and cocadrilles out-takene in the foresaid monethes."
" Alle that y have y grant the, out-take my wyfe."
The two latter passages illustrate the difficulty of determining whether but is used as a passive participle with nominative absolute, or as an active participle with the objective case. In the same way we find " excepted " and " except " placed (a) after a noun or pronoun, apparently as passive participles, and (b) before, as prepositions. Thus—
* (a) " Only you excepted."—M. Ado, i. 1. 126.
" Richard except."—Rich. III. v. 3. 242.
Then, on the other hand,—
* (b) "Always excepted my dear Claudio."—M. Ado, iii. 1. 93.
"Except immortal Cæsar."—J. C. i. 2. 60.
(For the confusion between "except" and "excepted" compare " defect" for "rejected," &c. See below, 342.)
The absence of inflections, however, in the above instances leaves us uncertain whether "except" is a preposition or participle. But "save" seems to be used for "saved" and "he" to be the nominative absolute in
"All the conspirators save only he." —J. C. v. 5. 69.
So "Save thou."—Sonn. 109.
"Nor never none
Shall mistress be of it save I alone."—T. N. iii. 1. 172.
" What stays had I but they."—Rich. III. ii. 2. 76, iv. 4. 34; Cymb. ii. 3. 153; Macbeth, iii. 1. 54; R. and J. i. 2. 14.
On the other hand, Shakespeare does not agree with modern usage in the inflections of the pronouns (see 206—216).
119. But is almost always used in Layamon for "unless" or "without" (prep.), or "without" (adv.) in the sense of "outside." Thus (i. 159): "that a queen should be king in this land and their sons be buten," (1. t. boute), i.e. "without (the land)." So (i. 215) "buten laeve," i.e. "without leave." It occurs adversatively in (i. 353) a passage which illustrates the transition, "If thou wilt receive his reconciliation, it will be well; but, he will never deliver Evelin to thee." Here but is the preposition "without," used adverbially as "otherwise."
120. But, in all its uses, may be explained from the meaning of "out-take" or except. It is sometimes used (like and, see above) to except or " out-take" a whole clause, the verb being occasionally in the subjunctive.
"And, but thou love me, let them find me here."
R. and J. ii. 2. 76.
i.e. "except or without thou love me." R. and J. ii. 2. 76.
"And, but I be deceived, Signior Baptista may remember me."—T. of Sh. iv. 2. 2.
Compare 1 Hen. VI. iii. 1. 34 : "Except I be provoked."
So
"Not without the prince be willing."—M. Ado, iii. 3. 86.
We now use "unless" in this sense, and by a comparison of Wickliffe with Tyndale and Cranmer it will be seen that but was already often superseded by "except."
But with the subjunctive is, however, more common in Early than in Elizabethan English. Sometimes without the subjunctive—
" And, but she spoke it dying, I would not
Believe her lips."—Cymb. v. 5. 41.
" And, but he's something stain'd
With grief that's beauty's canker, thou might'st call him
A goodly person."—Tempest, i. 2. 414.
" The common executioner
Falls not the axe upon the humbled neck
But first begs pardon."—A. Y. L. iii. 5. 5.
" And, but infirmity hath something seized
His wish'd ability, he had himself
The lands and waters 'twixt your throne and his
Measured, to look upon you."—W. T. v. 1. 141.
121. But. Transition of meaning. These last passages illustrate the transition of but from except to "on the contrary," "by way of prevention." The transition is natural, inasmuch as an exception may well be called contrary to the rule. The first passage is a blending of two constructions: "if she had not spoken it dying I would not believe," and " I would not believe, but she spoke it dying." Similarly: "Except infirmity had seized—he had (would have) measured," and " He had (would have) measured, but (by way of prevention) infirmity hath seized."
The different usages of but arise, (1) from its variations between the meaning of "except," " unless," and the adversative meaning "on the other hand;" (2) from the fact that the negative before but, in the sense of " except," is sometimes omitted and at other times inserted. Thus " but ten came" may mean " ten however came," or " (none) but ten, i.e. only ten, came." But is now much more confined than it was, to its adversative meaning. We still say " it never rains but it pours" (where the subject is the same before and after but); and, even where a new subject is introduced, we might say, " I did not know but you had come," " You shall not persuade me but you knew," &c. ; but this use is colloquial, and limited to a few common verbs. We should scarcely write
" I never saw but Humphrey duke of Gloucester
Did bear him like a noble gentleman."—2 Hen. VI. i. 1. 83.
122. "But" signifying prevention. The following passages illustrate the "preventive" meaning of but:
" Have you no countermand for Claudio yet
But he must die to-morrow?"—M. for M. iv. 2. 97.
i.e. "to prevent that he must die." If " but " were the ordinary adversative, it would be "but must he die?"
" That song to-night
Will not go from my mind: I have much to do
But (to prevent myself) to go hang my head all at one side
And sing it, like poor Barbara."—Othello, iv. 3. 32.
" Have you no wit, manners, nor honesty but to gabble like tinkers at this time of night?"—T. N. ii. 3. 95.
i.e. "to prevent you from gabbling," or, as Shakespeare could write, " to gabble." See 349.
After verbs of "denying" and "doubting" which convey a notion of hindrance, but is often thus used :
" I doubt not but to ride as fast as York."—Rich. II. ii. 5. 2.
" I have no doubt (i.e. fear) about being prevented from riding.'
So I Hen. IV. ii. 2. 14:
" It must not be denied but I am a plain dealing villain."
M. Ado, i. 3. 32.
" There must be no denial to prevent my being supposed a plain-dealing villain." In the last passage, however, but is used transitionally, almost as an adversative. Compare
"It cannot be but I am pigeon-livered,"—Hamlet, ii. 2. 605.
which approximates to " It cannot be (that I am otherwise than a coward)," i.e. "it cannot be that I am courageous ; on the contrary (but adversative), I am pigeon-liver'd."
The variable nature of but is illustrated by the fact that "believe not but," and "doubt not but" are used in the same signification :
" We doubt not but every rub is smoothed."—Hen. V. ii. 2. 187. i.e. "we have no doubt of a nature to prevent our believing that," &c. So Rich. II. v. 2. 115. But, on the other hand,
" I'll not believe but they ascend the sky."—Rich. III. i. 3. 287. i.e. " I'll not believe anything except (or 'otherwise than') that they ascend. "
In the first of these passages but is semi-adversative.
" She is not so divine
But with as humble lowliness of mind
She is content to be at your command."—I Hen. VI. v. 5. 18.
i.e. "not so divine as to prevent that she should be content."
" But " and " but that " are still thus used.
123. But (in phrases like "there is no man but hates me," where a subject immediately precedes but) often expels the subject from the following relative clause. This perhaps arose in part from a reluctance to repeat a subject which was already emphatically expressed. See 244. For the same reason the relative is omitted in such expressions as
" There is no creature loves me."—Rich. III. v. 3. 200.
In such cases we still sometimes omit the subject, but perhaps not often where but is separated from the preceding subject, as in
" There is no vice so simple but assumes
Some mark of virtue in its outward parts."
M. of V. iii. 2. 81.
On the other hand, this omission is not found in the earliest stages of the language (Mätzner, iii. p. 469), and thus we find the subject frequently retained in Shakespeare :
" I found no man but he was true to me."—J. C. v. 5. 35.
" There's ne'er a villain dwelling in all Denmark
But he's an arrant knave."—Hamlet, i. 5. 124.
Less frequently but expels the object in the relative clause :
"No jocund health that Denmark drinks to-day
But the great cannon to the clouds shall tell."
Hamlet, i. 2. 126.
124. But meaning except may apply to an expressed contingency, as (1)
"God defend but I should still be so."—I Hen. IV. iv. 3. 38.
i.e. "God forbid everything except (I should, &c.)"
"But being charged we will be still by land."
A. and C. iv. II. 1.
i.e. "Excepting the supposition of our being charged."
(2) Sometimes the contingency is merely implied.
" I should sin
To think but (except I should think) nobly of my grandmother."
Temp. i. 2. 119.
"Her head's declined and death will seize her, but
Your comfort makes her rescue."—A. and C. iii. 11. 48.
i.e. "only your comfort."
The last passage illustrates the connection between but meaning only, and but used adversatively.
125. But thus varying between an adversative and an exceptional force causes many ambiguities. Thus :
' Whenever Buckingham doth turn his hate
On you and yours, but with all duteous love
Doth cherish you and yours, God punish me."
Rich. III. ii. 1. 33.
Here but means "without," or "instead of, cherishing you."
"You salute not at the court but you kiss your hands. "
A. Y. L. iii. I. 50.
i.e. "without kissing your hands."
126. But is not adversative, but means "if not," after "beshrew me," &c.:
" Beshrew my soul but I do love," &c.—K. J. v. 4. 51.
So 3 Hen. VI. i. 4. 150.
" The Gods rebuke me but it is tidings
To wash the eyes of kings."—A. and C. v. 1. 27; ib. 103.
Thus we explain:
" I'll plead for you myself but you shall have him."
T. of Sh. ii. 1. 15.
i.e. "I'll plead for you myself if you shall not have him otherwise ;" but it must be admitted that the above construction may be confused with " I may have to plead for you myself, but (adversative) in any case you shall have him." So
" I should woo hard but be your groom,"—Cymb. iii. 6. 70.
is, perhaps, a confusion between " if I could not be your groom otherwise " and "but in any case I would be your groom." In the last example, however, it is possible that there is an additional confusion arising from the phrase : " It would go hard with me but. "
127. But in the sense of except frequently follows negative comparatives, where we should use than.
"No more but instruments."—M. for M. v. 1. 237.
Here two constructions are blended, " Nothing except instruments" and "only instruments ; no more." So—
"No more dreadfully but as a drunken sleep."
M. for M. iv. 2. 150.
"The which no sooner had his prowess confirm'd,
But like a man he died. "—Macbeth, v. 8. 42.
"I think it be no other but even so."—Hamlet, i. 1. 108.
" No more but that."—A. W. iii. 7. 30.
"With no worse nor better guard but with a knave."
Othello, i. 1. 126.
"Thou knowest no less but all."—T. N. i. 4. 13.
Sometimes but follows an adjective qualified by the negative with "so."
"Not so dull but she can learn."—M. of V. iii. 2. 164.
So Chaucer :
"I nam but dede,"—Knighte's Tale.
where, omitting the negative n, we should say " I am but dread."
128. But passes naturally from "except" to "only," when the negative is omitted. (" No-but" or "nobbut" is still used provincially for "only.") Thus :
" No more but that,"—A. W. iii. 7. 30.
becomes " but that."
"Glouc. What, and wouldst climb a tree ?
Simple. But that in all my life."—2 Hen. VI. ii. 1. 99.
i.e. "no more but that one tree," or " only that one tree."
" Cleo. Antony will be himself.
Ant. But stirr'd by Cleopatra."—A. and C. i. 1. 142.
i.e. "not except stirr'd," "only if stirr'd."
"But sea-room, and (if Fol.) the brine and billow kiss the moon, I care not."—P. of T. iii. 1. 45.
" Where Brutus may but find it."—J. C. i. 3. 144.
i.e. " Where Brutus can (do nothing) but find it," i.e., as we say, "cannot but find it." Possibly, however, but (see 129) may be transposed, and the meaning may be "Brutus only," i.e. "Brutus alone may find it."
" He that shall speak for her is afar off guilty
But that he speaks."—W. T. ii. 1. 105.
i.e. "simply in that he speaks," "merely for speaking."
The effect of the negative on but is illustrated by
" But on this day let seamen fear no wreck."—K. J. iii. 1. 92.
Here, at first, but might seem to mean "only," but the subsequent negative gives it the force of " except."
But perhaps means "only" in
" He boasts himself
To have a worthy feeding : but I have it
Upon his own report, and I believe it."—W. T. iv. 4. 169.
i.e. " I have it merely on his own report, and I believe it too."
There is, perhaps, a studied ambiguity in the reply of Hamlet:
" Guild. What should we say, my lord ?
Hamlet. Anything but to the purpose. "—Hamlet, ii. 2. 287.
The ellipsis of the negative explains "neither" in the following difficult passage:
"To divide him inventorily would dizzy the authentic of memory and yet but yaw neither (i.e. do nothing but lag clumsily behind neither) in respect of his quick sail."—Hamlet, v. 2. 120.
"Neither" for our "either" is in Shakespeare's manner, after a negative expressed or implied.
But means "setting aside" in
"What would my lord, but that (which) he may not have,
Wherein Olivia may seem serviceable."—T. N. v. 1. 104.
Such instances as this, where but follows not a negative but a superlative, are rare :
"Pistol. Sweet knight, thou art now one of the greatest men in this realm.
Silent. By're lady, I think 'a be, but goodman Puff of Barson." 2 Hen. IV v. 3. 93.
But seems used for "but now" in
"No wink, sir, all this night,
Nor yesterday : but (but now) slumbers."—B. J. Fox, i. 1.
129. But (like excepted and except) varies in its position. Similarly "only" varies with us : we can say either "one only" or "only one."
" This very morning but."—B. J. Sad Sh. ii. 2.
i.e. "only this morning."
" Where one but goes abreast."—Tr. and Cr. iii. 3. 155.
for " but one" or " one only."
"But in these fields of late."—Tr. and Cr. iii. 3. 188.
for " but of late."
" A summer's day will seem an hour but short."—V. and A.
"Betwixt them both but was a little stride."
SPENS. F. Q. ii. 7. 24.
"And when you saw his chariot but appear."—J. C. i. 1. 48.
i.e. " his chariot merely" or "but his chariot."
"Your oaths are words and poor conditions but unseal'd."
A. W. iv. 2. 30.
i.e. "merely unsealed agreements."
130. The same forgetfulness of the original meaning of words which led to "more better," &c., led also to the redundant use of but in " but only," " merely but," " but even," &c.
"Merely but art."—L. C. 25.
"He only lived but till he was a man."—Macbeth, v. 8. 40.
"My lord, your son had only but the corpse."
2 Hen. IV. i. 1. 192.
" Even but now " for "but now."
M. of V. v. 1. 272; A. Y. L. ii. 7. 3.
"But a very prey to woe."—Rich. III. iv. 4. 106.
"Augustus,
In the bestowing of his daughter, thought
But even of gentlemen of Rome."—B. J. Sejan. iii. 2.
Probably like " merely but."
So
"Even just."—Hen. V. ii. 3. 12.
"But now," like "even now" (38), is capable of different meanings : "a moment ago" and "at the present moment."
"But now I was the lord
Of this fair mansion, and even now, but now
This house, these servants, and this same myself
Are yours."—M. of V. iii. 2. 171.
For. See 151.
131. Or (before). Or in this sense is a corruption of A.-S. œr (Eng. ere), which is found in Early English in the forms er, air, ar, ear, or, eror.
" Or (before) he have construed."—ASCH. 95.
As this meaning of or died out, it seems to have been combined with ere for the sake of emphasis. Thus :
"Dying or ere they sicken."—Macbeth, iv. 3. 173 ;
K. J. v. 6. 44 ; Temp. v. 1. 103.
We find in E. E. "erst er," "bifore er," "before or" (Mätzner, iii. 451).
Another explanation might be given. Ere has been conjectured to be a corruption of e'er, ever, and "or ever" an emphatic form like "whenever," "wherever." "Ever" is written "ere" in Sonn. 93, 133. And compare "Or ever your pots be made hot with thorns. "—Ps. lviii.
Against the latter explanation is the fact that "ever" is much more common than "ere." It is much more likely that "ever" should be substituted for "ere" than "ere" for "ever." For Or . . . or, see 136.
132. Since seems used for when in—
" Beseech you, sir,
Remember since you owed no more to time
Than I do now."—W. T. v. 1. 219.
"Remember the time past when you," &c.
"We know the time since he was mild and affable."
2 Hen. VI. iii. 1. 9.
"Thou rememberest
Since once I sat upon a promontory. "—M. N. D. ii. 1. 149.
"This fellow I remember
Since once he play'd a farmer's eldest son."
T. of Sh. Ind. i. 84.
So 2 Hen. IV. iii. 2. 206.
This meaning of since arises from the omission of "it is" in such phrases as "it is long since I saw you," when condensed into "long since, I saw you." Thus since acquires the meaning of "ago," "in past time," adverbially, and hence is used conjunctively for "when, long ago."
Since (like the adverb) is found connected with a simple present where we use the complete present (so in Latin) :
"Since the youth of the count was to-day with my lady, she is much out of quiet."—T. N. ii. 3. 144.
More remarkable is the use of the simple past for the complete present:
" I was not angry since I came to France
Until this instant."—Hen. 'V. iv. 7. 58.
Note
" Whip him . . .
So saucy with the hand of she here,—what's her name?
Since she was Cleopatra."—A. and C. iii. 13. 99.
Perhaps the meaning is "Whip him for being saucy with this woman, since (though she is not now worthy of the name) she once was (emphatical) Cleopatra." Else "What is her new name since she ceased to be Cleopatra ?" If since, in the sense of " ago," could be used absolutely for "once," a third interpretation would be possible: "What's her name ? Once she was Cleopatra."
133. So is used with the future and the subjunctive to denote "provided that."
"I am content so thou wilt have it so. "—R. and J. iii. 5. 18.
"So it be new, there's no respect how vile. "—Rich. II. ii. 1. 25.
So seems to mean "in this way," "on these terms," and the full construction is "be it (if it be) so that." "Be it" is inserted in
"Be it so (that) she will not."—M. N. D. i. 1. 39.
"That" is inserted in Chaucer, Piers Ploughman, &c.
"(Be it) So that ye be not wrath."—CHAUCER, C. T. 7830.
means "provided you will not be angry." So
" Poor queen ! So that thy state might be no worse
I would my skill were subject to thy curse."
Rich. II. iii. 4. 102.
So, thus meaning "on condition that," is sometimes used where the context implies the addition of "even."
" Messenger. Should I lie, madam?
Cleopatra.
O, I would thou didst
So (even if) half my Egypt were submerged."—A. and C. ii. 5. 94.
Sometimes the subjunctive inflection is neglected and "so as" is used for "so that."
"So as thou livest in peace, die free from strife."
Rich. II. v. 5. 27.
We must distinguish the conditional "so heaven help me" from the optative "so defend thee heaven" (Rich. II. i. 3. 34), where the order of the words indicates that "be it . . . that" cannot be understood. Here so means "on the condition of my speaking the truth," and is not connected with defend. Compare Rich. III. ii. 1. 11, 16. See also 275—283.
That. See Relative.
That omitted before the subjunctive. See 311.
134. Where is frequently used metaphorically as we now use whereas.
"It (the belly) did remain
I' the midst o' the body idle and unactive
. . . . .where the other instruments
Did see and hear, devise," &c.—Coriol. i. 1. 105.
for "whereas the other instruments did," &c. Comp. Coriol. i. 10. 13. So Lear, i. 2. 89 ; Rich. II. iii. 2. 185.
135. Whereas, on the other hand, is used for where in
" Unto St. Alban's
Whereas the king and queen do mean to hawk."
2 Hen. VI. i. 2. 58.
"They back returned to the princely place ;
Whereas . . . a knight . . . they new arrived find."
SPENS. F. Q. i. 4. 38.
So "where-that."—Hen. V. v. Prologue, 17. Probably both "as" and "that" were added to give a relative meaning to the (originally) interrogative adverb where. See 287.
136. Whether is sometimes used after "or" where we should omit one of the two :
" Or whether doth my mind, being crown'd with you,
Drink up the monarch's plague, this flattery?
Or whether shall I say mine eye saith true," &c.—Sonn. 114.
"Move those eyes?
Or whether riding on the balls of mine
Seem they in motion?"—M. of V. iii. 2. 18.
" Or whether his fall enraged him, or how it was."
Coriol. i. 3. 69.
The first example is perhaps analogous to the use of " or . . . or," as in
"Why the law Salique which they have in France
Or should or should not bar us in our claim."
Hen. V. i. 2. 12 ; T. N. iv. 1. 65.
There is, perhaps, a disposition to revert to the old idiom in which the two particles were similar: "other . . . other." (The contraction of "other" into "or" is illustrated by "whe'r" for " whether" in O.E. and the Elizabethan dramatists.) Perhaps, also, additional emphasis is sought by combining two particles. We find "whether . . . or whether ?" to express direct questions in Anglo-Saxon. In the second example a previous "whether" is implied in the words "move those eyes?"
137. While (originally a noun meaning "time"). Hence "a-while," "(for) a time;" "the while," "(in) the (mean) time;" " whil-om" ("om" being a dative plural inflexion used adverbially), "at a (former) time;" "while-ere" (Temp. iii. 2. 127), "a time before," i.e. "formerly."
So whiles (genitive of while) means "of, or during, the time." The earliest use of while is still retained in the modern phrase "all the while that he was speaking." " The while that," from a very early period, is used in the condensed form "the while" or " while that" or while; and whiles was similarly used as a conjunction.
While now means only "during the time when," but in Elizabethan English both while and whiles meant also "up to the time when." (Compare a similar use of "dum" in Latin and ως in Greek.)
" We will keep ourself
Till supper-time alone. While (till) then, God be with you."
Macbeth, iii. 1. 43.
"I'll trust you while your father's dead."
MASSINGER (Nares).
"He shall conceal it
Whiles you are willing it shall come to note."—T. N. iv. 3. 28.
"Let the trumpets sound
While we return these dukes what we decree.
[A long flourish.
Draw near, &c."—Rich. II. i. 3. 122.
# PREPOSITIONS.
138. Prepositions primarily represent local relations; secondarily and metaphorically, agency, cause, &c. A preposition (as after, see below) may be used metaphorically in one age and literally in the next, or vice versâ. This gives rise to many changes in the meaning ot prepositions.
The shades of different meaning which suggest the use of different prepositions are sometimes almost indistinguishable.
We say, "a canal is full of water." There is no reason why we should not also say " full with water," as a garden is " fair with flowers." Again, " a canal is filled with water," the verb in modern English preferring witch to signify instrumentality, but "filled of water " is conceivable ; and, as a matter of fact, Shakespeare does write "furnished of, provided of, supplied of," for with. Lastly the water may be regarded as an agent, and then we say, "the canal is filled by the water." But an action may be regarded as "of" the agent, as well as "by" the agent, and "of" is frequently thus used in the A. V. of the Bible and in Elizabethan authors, as well as in E. E. For these reasons the use of prepositions, depending upon the fashion of metaphor in different ages, is very variable. It would be hard to explain why we still say, "I live on bread," but not " Or have we eaten on the insane root ? " (Macb. i. 3. 84) ; as hard as to explain why we talk of a "high" price or rate, while Beaumont and Fletcher speak of a " deeper rate."
139. Prepositions: modern tendency to restrict their meaning.
One.general rule may be laid down, that the meanings of the prepositions are more restricted now than in the Elizabethan authors : partly because some of the prepositions have been pressed into the ranks of the conjunctions, e.g. "for," "but," "after;" partly because, as the language has developed, new prepositional ideas having sprung up and requiring new prepositional words to express them, the number of prepositions has increased, while the scope of each has decreased. Thus many of the meanings of "by" have been divided among " near," "in accordance with," " by reason of," "owing to;" "but" has divided some of its provinces among "unless," " except ;" " for" has been in many cases supplanted by " because of," "as regards ;" "in" by " during."
140. A. Ben Jonson in his Grammar, p. 785, writes thus :—" A hath also the force of governing before a noun—' And the Protector had layd to her for manner's sake that she was a council with the Lord Hastings to destroy him.'—Sir T. MORE."
" Forty and six years was this temple a building."
St. John ii. 20.
The present text is in, but Cranmer and Tyndale bad "a."
This a, which still exists in alive, afoot, asleep, &c. is a contraction of A.-S. on or the less common form an. We find in Early English " on live," " on foot," " on hunting," " on sleep;" "a morrow and eke an eve," for " by morning and also by evening ; " "a land and a water," Piers Pl. (where some MSS. have on), " a (for in) God's name," " an end " for " on the (at the) end."
In the Folio we sometimes find a where we write o':
"What is 't a clocke ?"—Rich. III. v. 3. 47.
See Adverbs, 24
141. After ("following," Latin "secundum," hence "according to ").
"Say, you chose him,
More after our commandment than as guided
By your own true affections. "—Coriol. ii. 3. 238.
'After my seeming."—2 Hen. IV. v. 2. 128.
Compare " Neither reward us after our iniquities," in our Prayer-book.
After is now used only of space or time, except in "after the pattern, example, &c.," where the sense requires the metaphorical meaning.
142. Against used metaphorically to express time. This is now restricted to colloquial language :
"I'll charm his eyes against he do appear."—M. N. D. iii. 2. 99. i.e. "against the time that he do appear." Any preposition, as "for," "in," can thus be converted into a conjunction by affixing " that," and the "that" is frequently omitted.
" Against (the time that) my love shall be as I am now."—Sonn. 63.
"'Gainst that season comes."—Hamlet, i. 1. 158.
"As against the doom."—Ib. iii. 4. 50.
i.e. "as though expecting doom's-day."
143. At. The use of a mentioned in 140 was becoming unintelligible and vulgar in Shakespeare's time, and he generally uses at instead. The article is generally omitted in the following and similar adverbial forms.
"All greeting that a king at friend can send his brother."
W. T. v. 1. 140.
"The wind at help."—Hamlet, iv. 3. 46.
"At shore."—MONTAIGNE. "At door."—W. T. iv. 4. 352.
"(A ship) that lay at rode."—N. P. 177.
"As true a dog as ever fought at head "—T. A. v. 1. 102.
"Bring me but out at gate."—Coriol. iv. 1. 47.
"At point."—Coriol. v. 4. 64; Cymb. iii. 6. 17.
But "When they were fallen at a point for rendering up the hold."
HOLINSHED, Duncane.
The at of price generally requires an adjective or article, as well as a noun, after it, except in "at all." We have, however,
" If my love thou hold'st at aught,"—Hamlet, iv. 3. 60.
i.e. "at a whit."
In Early English at does not seem to have been thus extensively used. It then was mostly used (Stratmann) in the sense of "at the hands of (πρóς with gen.) : "I ask at, take leave at, learn at a person," &c.
At is used like "near" with a verb of motion where we should use "up to :"
"I will delve one yard below their mines,
And blow them at the moon."—Hamlet, iii. 4. 209.
In "Follow him at foot,"—Ib. iv. 3. 56. at is not "on" but "near," as in "at his heels."
144. At, when thus used in adverbial expressions, now rejects adjectives and genitives as interfering with adverbial brevity. Thus we can say "at freedom," but not
"At honest freedom."—Cymb. iii. 4. 71.
"At ample view."—T. N. i. 1. 27.
"At a mournful war."—Sonn. 46.
"At heart's ease."—J. C. i. 2. 207.
We say "at loose," but not "Time . . . often at his very loose decides
That which long process could not arbitrate,"—L. L. L. v. 2. 752. where "loose" means "loosing" or "parting."
So we say " aside," but not
"To hang my head all at one side."—Othello, iv. 3. 22.
We say "at the word," but, with the indefinite article, "in a word," not
"No, at a word, madam."—Coriol. i. 3. 122.
It is, perhaps, on account of this frequent use of at in terse adverbial phrases that it prefers monosyllables to dissyllables. Thus we have "at night" and "at noon," and sometimes "at eve" and "at morn," but rarely "at evening" or "at morning," except where "at morning" is conjoined with "at night," as in
"At morning and at night."—M. of V. iii. 2. 279.
London was not so large as it now is when Shakespeare wrote
"Inquire at London."—Rich. II. v. 3. 51.
145. By (original meaning "near"). Hence our "to come by a thing," i.e. "to come near" or "attain."
"(How) cam'st thou by this ill tidings ?"—Rich. II. iii. 4. 80.
"I'll come by (i.e. acquire) Naples."—Temp. ii. 1. 292.
By is used in a manner approaching its original meaning in
" Fed his flocks
By (on) the fat plains of fruitful Thessaly."
B. and F. Fair Sh. i. I.
" At a fair vestal throned by the west. "—M. N. D. ii. I. 58.
So Wickliffe: "By (on) everi Saboth," Acts xiii. 27. Somewhat similar is our present colloquial "by this" of time; an expression which is found in
"Of the poor suppliant who by this I know
Is here attending."—A. W. v. 3. 134; Lear, iv. 6. 45.
This is illustrated by the play on "by your favour," where favour means also "complexion," "face," in
" Duke.
Thine eye
Hath stay'd upon some favour that it loves,
Hath it not, boy?
Viola.
A little, by your favour"—T. N. ii. 4. 26.
Compare also the puns in T. N. iii. I. 2—10.
Hence "about," "concerning."
"How say you by the French lord?"—M. of V. i. 2. 60.
"Tell me, sirrah, but tell me true, I charge you,
By him and by this woman here what know you ?"
A. W. v. 3. 237.
" I would not have him know so much by me."
L. L. L. iv. 3. 150.
" I know nothing by myself," I Cor. iv. 4 (no harm about myself).
" Many may be meant by (to refer to) the fool multitude."
M. of V. ii. 9. 25.
Compare B. J. Poetast. v. I :
" Lupus. Is not that eagle meant by Cæsar, ha ? ....
Cæsar. Who was it, Lupus, that inform'd you first
This should be meant by us ?"
Hence from near came the meaning like, according to.
" It lies you on to speak
Not by your own instruction, nor by the matter
Which your own heart prompts you. "—Coriol. iii. 2.52.
"And him by oath they duly honoured. "—R. of L. 410.
i.e. " according to their oath."
"Not friended by his wish, to your high person
His will is most malignant."—Hen. VIII. i. 2. 40.
i.e. " in accordance with his wish," "to his heart's content."
" If my brother wrought by my pity it should not be so."
M. for M. iii. 2. 224.
" I will believe you by the syllable
Of what you shall deliver."—P. of T. v. 1. 170.
So, where we say "to the sound of : "
"Sound all the lofty instruments of war,
And by that music let us all embrace."
By seems to mean "near," hence "with," in
"(My daughter) hath his solicitings,
As they fell out by time, by means and place,
All given to mine ear. "—Hamlet, ii. 2. 127.
Perhaps we may thus explain :
" I'll trust by leisure him that mocks me once."—T. A. i. 1. 301. i.e. " in accordance with, to suit, my leisure."
The use of by in
" The people . . . by numbers swarm to us,"
3 Hen. VI. iv. 2. 2.
is the same as in
"By ones, by twos, by threes."—Coriol. ii. 3. 47.
By, in the sense of "near," like our "about" (Acts xiii. 21, Wick. " by fourti yeeris," the rest "about "), Greek κατ , was used from the first in rough distributive measurements in E. E. : " He smote to the ground by three, by four," " by nine and ten," " by one and one." So
" I play the torturer by small and small
To lengthen out the worst that must be said."
Rich. II. iii. 2. 189.
i.e. "in lengthening out by little and little." Hence, perhaps, from " by one by one " sprang our shorter form, " one by one," " little by little ;" though it is possible that "one by one" means "one next to or after one."
By is used as a noun in the expression "on the by" (as one passes by).—B. J. 746.
We still use by as an adverb after "close," "hard," &c., but we should scarcely say,
"I stole into a neighbour thicket by."—L. L. L. v. 2. 94.
146. By ("near," "following close after," hence "as a consequence of").
"The bishop of York,
Fell Warwick's brother, and, by that, our foe."
3 Hen. VI. iv. 4. 12.
"Lest, by a multitude
The new-heal'd wound of malice should break out."
Rich. III. ii. 2. 124.
" So the remembrance of my former love
Is by a newer object quite forgotten. "—R. and J. ii. 4 194.
"Fear'd by their breed and famous by their birth."
Rich. II. ii. 1. 52.
Hence sometimes it seems to be (but is not) used instrumentally with adjectives which appear to be (but are not) used as passive verbs.
By does not mean " by means of," but " as a consequence of," in
" An eagle sharp by fast. "—V. and A. 55.
" Oh how much more does beauty beauteous seem
By that sweet ornament which truth doth give. "—Sonn.
" Laer. Where is my father?
King.
Dead!
Queen.
But not by him."
Hamlet, iv. 5. 128.
147. For (original meaning "before," "in front of"). A man who stands in front of another in battle may either stand as his friend for him or as his foe against him. Hence two meanings of for, the former the more common.
148. (I.) For, meaning "in front of," is connected with "instead of," "in the place of," " as being."
" Or for the lawrell he may gain a scorne."
B. J. on Shakespeare
i.e. " instead of the laurel."
" See what now thou art,
For happy wife, a most distressed widow,
For joyful mother, one that wails the name,
For queen, a very caitiff crown'd with care."
Rich. III. iv. 4. 98.
"Thyself a queen, for me that was a queen. "—Ib. i. 3. 202.
Between this and the following meanings we may place
" Learn now, for all. "—Cymb. ii. 3. 111.
" This is for all "—Hamlet, i. 3. 131.
i.e. "once instead of, or in the place of, all."
"I abjure
The taints and blames I laid upon myself
For (as being) strangers to my nature."—Macbeth, iv. 3. 125.
"Conscience ... is turned out of all towns and cities for a dangerous thing. "—Rich. III. i. 4. 146.
" How often have I sat crown'd with fresh flowers
For summer's queen !"—B. and F. Fair Sh. i. I.
Hence for is nearly redundant in
"Let the forfeit
Be nominated for an equal pound."—M. of V. i. 3. 150.
There is a play on the word in
" On went he for a search, and away went I for (packed up in a basket and treated like) old clothes. "—M. W. of W. iii. 5. 100.
" Three dukes of Somerset three-fold renown'd
For hardy and undoubted champions. "—3 Hen. VI. v. 7. 6.
(Where probably hardy means Fr. hardi, " bold ;" and " undoubted" means " not frightened," " doubt" like "fear" being used for "frighten.")
Perhaps for comes under this head in
" What is he for a fool that betroths himself to unquietness."
M. Ado, i. 3. 49.
i.e. " What is he, as being a fool." It is more intelligible when the order is changed: " For a fool, what is he," i.e. " considered as a fool—it being granted that he is a fool—what kind of fool is he?"
So
" What is he for a vicar?"—B. J. Sil. Wom. iii. 1. med.
So in German "was für ein?"
149. For is hence loosely used in the sense " as regards."
" It was young counsel for the persons and violent counsel for the matter."—B. E. 75.
Very commonly this for stands first, before an emphatic subject or object, which is intended to stand in a prominent and emphatic position :
" For your desire to know what is between us,
O'er-master it as you may. "—Hamlet, i. 5. 39 ; 2. 112.
" Now, for the taking of Sicily, the Athenians did marvellously covet it. "—N. P. 171.
" For your intent,
It is most retrograde to our desires."
Hamlet, i. 2. 112 ; Rich. II. v. 3. 137.
"For a certain term," "for seven days, a day" (or even "for the day" where one day is meant), is still customary, but not
"Doom'd for a certain term to walk the night,
And for the day confined to fast in fires."—Hamlet, i. 4. 11.
150. For, from meaning "in front of," came naturally to mean "in behalf of," "for the sake of," "because of."
"Yet I must not (kill Banquo openly),
For certain friends that are both his and mine."
Macbeth, iii. 1. 120.
i.e. "because of certain friends."
This use was much more common than with us. When we refer to the past we generally use "because of," reserving for for the future. Compare, on the other and:
" O be not proud, nor brag not of thy might,
For mastering her that foil'd the God of fight."
V. and A. 114.
" He gave it out that he must depart for certain news."
N. P. 179.
" No way to that, for weakness, which she enter'd."
I Hell. VI iii. 2. 25.
i.e. " no way can be compared for weakness with that," &c.
" Of divers humours one must be chiefly predominant, but it is not with so full an advantage but, for the volubilitie and supplenes of the mind, the weaker may by occasion reobtaine the place again." —MONTAIGNE, 116.
For is similarly used with an ellipse of " I lay a wager " in
"Now, for my life, she's wandering to the Tower."
Rich. III. iv. 1. 3.
151. For, in the sense of "because of," is found not only governing a noun, but also governing a clause :
" You may not so extenuate his offence
For I have had such faults."—M. for M. ii. 1. 28.
i.e. "because I have had such faults."
"('Tis ungrateful) to be thus opposite with heaven,
For (because) it requires the royal debt it lent you."
Rich. III. ii. 2. 95.
So Uthello, i. 3. 269; Cymb. iv. 2. 129. And parenthetically very frequently:
"The canker-blossoms have as deep a dye
As the perfumed tincture of the roses,
But for their virtue only is their shew,
They live unwoo'd, and unrespected fade."—Sonn. 54.
" Oh, it is as lawful,
For we would give much, to use violent thefts."
Tr. and Cr. v. 1. 21.
i.e. to rob, "because we wish to be generous."
With the future, for meant "in order that."
" And, for the time shall not seem tedious,
I'll tell thee what befel me. "—3 Hen. VI. iii. I. 11
The desire of clearness and emphasis led to the addition of because.
" But for because it liketh well our eyes."—N. P. Pref.
"And for because the world is populous."—Rich. II. v. 5. 3.
Comp. " but only" "more better," &c.
For, when thus followed by a verb, like after, before, &c. (" after he came," "before he went"), is called a conjunction. It is often, like other prepositions (287) thus used, followed by "that." Coriol. iii. 3. 93, &c. The two uses occur together in the following passage, which well illustrates the transition of for :
" I hate him for he is a Christian,
But more for that . . . he lends," &c.—M. of V. i. 3. 43.
152. For to, which is now never joined with the infinitive except by a vulgarism, was very common in E. E. and A.-S., and is not uncommon in the Elizabethan writers. It probably owes its origin to the fact that the prepositional meaning of " to" was gradually weakened as it came to be considered nothing but the sign of the infinitive. Hence for was added to give the notion of motion or purpose. Similarly in Danish and Swedish (Mätzner, ii. p. 54) "for at " is used. Both in E. E. and in Elizabethan writers the for is sometimes added to the latter of two infinitives as being, by a longer interval, disconnected from the finite verb, and therefore requiring an additional connecting particle :
" First, honour'd Virgin, to behold thy face
Where all good dwells that is ; next for to try," &c.
B. and F. Fair Sh. v. I.
For the same reason:
" Let your highness
Lay a more noble thought upon mine honour
Than for to think that I would sink it here."—A. W. v. 3. 181.
From the earliest period " for to," like " to," is found used without any notion of purpose, simply as the sign of the infinitive. So in Shakespeare:
"Forbid the sea for to obey the moon.—W. T. i. 2. 427.
153. For, variable. The following passage illustrates the variableness of for :
" Princes have but their titles for (to represent) their glories,
An outward honour for (as the reward of) an inward toil,
And for (for the sake of gaining) unfelt (unsubstantial) imagi-
nation
They often feel a world of restless cares."—Rich.III. i. 4. 78-80
154. (IL) For (in opposition to) : hence "to prevent."
" And over that an habergeon for percing of his herte."
CHAUCER, Sire Thopas, 13790.
" Love. Is there an officer there?
Off. Yes, two or three for failing."—B. J. Alch. v. 3.
"The which he will not every hour survey
For blunting the fine point of seldom pleasure."—Sonn. 52.
" We'll have a bib for spoiling of thy doublet."
B. and F. (Nares).
So it is said of Procrustes, that if his victim was too long for the bed, "he cut off his legs for catching cold."—Euphues (Malone).
It can be proved that Sir T. North regarded for as meaning "in spite of," since he translates " Mais, nonobstant toutes ces raisons," by "But, for all these reasons," (N. P. 172); where the context also shows beyond dispute that for has this meaning. On the other hand, in
"All out of work and cold for action,"—Hen. V. i. 2.
for seems to mean "for want of," unless "out of work and cold " can be treated as equivalent to " eager," which would naturally be followed by for.
For is found in E. E. in this sense, but perhaps always with the emphatic "all."
For in this sense is sometimes used as a conjunction:
"For all he be a Roman. "—Cymb. v. 4. 109.
i.e. " despite that he be a Roman."
For may either mean " against " or (149) " for what concerns" in
"I warrant him for drowning."—Temp. i. I. 47.
We still retain the use of for in the sense of in spite of, as in "for all your plots I will succeed." Such phrases, however, frequently contain a negative, in which case it is difficult to ascertain whether for means Because of or in spite of.
" My father is not dead for all your saying."
Macbeth,, iv. 2. 36.
" (The stars) will not take their flight
For all the morning light."—MILTON, Hymn on the Nativity.
It is a question how to punctuate
"To fall off
From their Creator and transgress his will
For one restraint lords of the world besides."
MILTON, P. L. i. 32.
If a comma be placed after "will," and not after "restraint," then "besides" should be treated as though it were "except" or "but:" " Lords of the world but for one restraint."
155. For is sometimes ready for, fit for. (See 405.)
"He is for no gallants' company without them."
B. J. E. in &c. i. I.
"Your store is not for idle markets."—T. N. iii. 3. 46.
Compare our " I am for (going to) Paris."
Some ellipsis, as " I pray," must be understood in
"(I pray) God for his mercy."—Rich. II. ii. 2. 98 ; v. 2. 75.
156. Forth is used as a preposition (from) :
"Steal forth thy father's house."—M. N. D. i. 1. 164.
" Loosed them forth their brazen caves." 2 Hen. VI. iii. 2. 89, and 1 Hen VI. i. 2. 54.
Sometimes with "of" or " from :"
"That wash'd his father's fortunes forth of France." 3 Hen. VI. ii. 2. 157.
So Rich. II, iii. 2. 204—5 ; Temp. v. 1. 160. The "of" in itself implies motion from. (See 165.)
"From forth the streets of Pomfret. "—K. iv. 2. 148.
So Rich. II. ii. 1. 106.
Forth, being thus joined with prepositions less emphatic than itself, gradually assumed a prepositional meaning, displacing the prepositions. Forth is not found as a preposition in E. E. See also Prepositions omitted.
157. From is sometimes joined with out, to signify outward motion, where we use out of.
"In purchasing the semblance of my soul
From out the state of hellish cruelty."—M. of V. iii. 4. 20.
"From out the fiery portal of the East."—Rich. II. iii. 3. 64.
158. From is frequently used in the sense of "apart from," "away from," without a verb of motion.
"From thence (i.e. away from home) the sauce to meat is ceremony."—Macbeth, iii. 4. 36.
"I am best pleased to be from such a deed."—K. iv. 1. 86.
"Which is from (out of) my remembrance."—Temp. i. 1. 65.
"They run themselves from breath."—B. J. Cy.'s Rev. i. 1.
"Clean from the purpose."— C. i. 3. 35.
"This discourse is from the subject."—B. and F. Eld. B. v. 1.
"This is from my commission."—T. N. i. 5. 208.
"Anything so overdone is from the purpose of playing." Hamlet, iii. 2. 22.
"This is from the present."—A. and C. ii. 6. 30.
Hence " differently from: "
"Words him a great deal from the matter."—Cymb. i. 4. 16.
i.e. "describes him in a manner departing from the truth."
"This label on my bosom whose containing
Is so from sense in hardness."—Cymb. v. 5. 431.
" Write from it, if you can, in hand and phrase."
T. N. v. 1. 340.
"For he is superstitious grown of late
Quite from the main opinion he held once."— C. ii. 1. 196.
" So from himself impiety hath wrought. "—R. of L.
"To be so odd and from all fashions."—M. 4do, iii. 1. 72.
" Particular addition from the bill
That writes them all alike."—Macbcth, iii. 1. 100.
This explains the play on the word in
"Queen. That thou dost love thy daughter from thy soul." Rich. III. iv. 4. 258.
" I wish you all the joy that you can wish,
For I am sure you can wish none from me."
M. of V. iii. 2. 192.
i.e. "none differently from me," "none which I do not wish you." This is probably the correct interpretation of the last passage. So Othello, i. 1. 132.
"If aught possess thee from me."—C. of E. ii. 2. 180.
Also "apart from :"
"Nay, that's my own from any nymph in the court."
B. J. Cy.'s Rev. ii. 1.
"From thee to die were torture more than death."
2 Hen. VI. iii. 2. 401.
159. In, like the kindred preposition on (Chaucer uses "in a hill" for "on a hill"), was used with verbs of motion as well as rest. We still say "he fell in love," "his conduct came in question."
" He fell in a kind of familiar friendship with Socrates."
N. P. 192.
"Duncane fell in fained communion with Sueno."
HOLINSHED.
"In so profound abysm I throw all care."—Sonn. 112.
"Cast yourself in wonder. "— C. i. 3. 60.
"Sounds of music creep in our ears."—M. of V. v. 1. 56.
"They who brought me in my master's hate."
Rich. III. iii. 2. 56.
"But first I'll turn yon fellow in his grave."
Ib. i. 2. 262; 3. 88.
" And throw them in the entrails of a wolf."—Ib. iv. 3. 23.
"If ever ye came in hell."—UDALL.
In (for "into") with "enter," Rich. II. ii. 3. 160 ; Rich. III. v. 3. 227.
Into is conversely sometimes found with verbs of rest implying motion. "Is all my armour laid into my tent?"—Rich. III. v. 5. 51.
"Confin'd into this rock."—Tempest, i. 2. 361.
" To appear into the world."—-MONTAIGNE, 224.
And earlier "Hid into three measures of meal."—WICKLIFFE, Luke xiii. 21.
160. In for on :
" What in your own part (side) can you say to this ?"
Othello, i. 3. 74.
So in the phrase "in the neck," where we should say "on the neck " or "on the heels."
" Soon after that depriv'd him of his life
And, in the neck of that, task'd the whole state."
1 Hen. IV. iv. 3. 92.
The same phrase occurs Sonn. 131 ; MONTAIGNE, 17; N. P. 172.
" In pain of your dislike."—2 Hen. VI. iii. 2. 257.
161. In for "during" or "at." In has now almost lost its metaphorical use applied to time. As early as the sixteenth century "In the day of Sabbath " (WICKLlFFE, Actsxiii. 14) was replaced by "on." It is still retained where the proper meaning of "in," " in the limits of," is implied, as with plurals, "Once in ten days" or "for once in my life," or "he does more in one day than others in two." Thus A. V. Gen. viii. 4, "In the seventh month, on the eighteenth day." We also find frequently in the A. V. "In the day of the Lord, in the day when," &c. "in the day of judgment." This may in part be due to a desire to retain the more archaic idiom, as being more solemn and appropriate; but perhaps the local meaning of in may be here recognized. We still say "in this calamity, crisis," &c. where we mean "entangled in, surrounded by the perils of this calamity;" and some such meaning may attach to "in" when we say "In the day of tribulation, vengeance," &c. Occasionally, however, we find "at the day of judgment" (Matt. xi. 22), as also in Shakespeare in the only passage where this phrase occurs. Shakespeare frequently uses in for "at" or "during."
"How ! the duke in council
In this time of the night."—Othello, i. 2. 93.
" In night."—V. any A. 720.
"In all which time."—Rich. III. i. 3. 127.
" In such a night as this."—M. of V.v. 1. 1, 6, 9.
"This is, sir, a doubt
In such a time as this, nothing becoming you."
Cymb. iv. 4. 15.
" Nay, we will slink away in supper-time."—M. of V. ii. 4. 1.
162. In metaphorically used for "in the case of," "about," &c.
" Triumph in so false a foe. "—R. of L.
"In second voice we'll not be satisfied." .
Tr. and Cr. ii. 3. 149
"Almost all
Repent in their election. "—Coriol. ii. 3. 263.
"Our fears in Banquo stick deep."—Macb. iii. 1. 49.
"(We) wear our health but sickly in his life
Which in his death were perfect."—Ib. iii. 1. 107.
We say "in my own person" or "by myself," not
"Which in myself I boldly will defend."—Rich. II. i. 1.145.
So "But I bethink me what a weary way
In Ross and Willoughby . . . will be found."—Ib. ii. 2. 10.
i.e "in the case of Ross," equivalent to "by Ross."
In is used metaphorically where we should say "in the thought of" in
"Strengthen your patience in our last night's speech."
Hamlet, v. 1. 317.
163. In. We still say "it lies in your power." But we find also—
"And the offender's life lies in the mercy
Of the duke only,"—M. of V. iv. 1. 355.
where we now should use at. This example illustrates the apparently capricious change in the use of prepositions.
We should now use at instead of in and of, in
"In night and on the court and guard of safety."
Othello, ii. 3. 216.
and "What! in a town of war." —Ib. 213.
"In-round" (O. Fr. "en rond") is used for the more modern "a-round" in
"They compassed him in round among themselves. "—N. P. 192. But probably "round" is for "around." Compare "compassed him in."—A. V. 2 Chron. xxi. 9.
164. In is used with a verbal to signify "in the act of" or "while."
"He raves in saying nothing."—Tr. and Cr. iii. 3. 247.
" When you cast
Your stinking greasy caps in hooting at
Coriolanus' exile."—Coriol. iv. 6. 131.
"Mine eyes, the outward watch
Whereto my finger like a dial's point
Is pointing still, in cleansing them from tears."—Rich. II v. 5. 54. "The fire that mounts the liquor till't run o'er,
In seeming to augment it, wastes it."—Hen. VIII. i. 1. 145. " And may ye both be suddenly surprised
By bloody hands in sleeping on your beds."—1 Hen. VI. v. 3. 41.
"As patches set upon a little breach
Discredit more in hiding of the fault."—K. J. iv. 2. 30.
It is probable, as the last example suggests, that these verbals are nouns after which "of" is sometimes expressed. Hence "in sleeping" may simply be another form of "a-sleeping." But the in brings out, more strongly than the a-, the time in which, or while, the action is being performed. It is also probable that the influence of the French idiom, "en distant ces mots," tended to mislead English authors into the belief that in was superfluous, and that the verbals thus used were present participles. (See also 93.) In is used thus with a noun :
"Wept like two children in (during) their deaths' sad stories."
Rich. III. iv. 3. 8.
" (These blazes) giving more light than heat, extinct in both,
Even in their promise, while it is a-making."
Hamlet, i. 3. 119.
165. Of (original meaning "off" or "from"). Comp. π ; "ab," Mceso-Gothic "af."
In Early English of is used for "from," "out of," "off," as in "He lighted of his steed, arose of the dead," "The leaves fall of the tree." This strong meaning of motion was afterwards assigned to "off" (which is merely an emphatic form of of), and hence of retained only a slight meaning of motion, which frequently merged into causality, neighbourhood, possession, &c.
Off is, perhaps, simply of in
" Over-done or come tardy off."—Hamlet, iii. 2. 28.
i.e. "fallen short of." Compare στρε ν. Otherwise "come off" is a passive participle, 295.
Of retains its original meaning in
" Overhear this speech
Of vantage."—Hamlet, iii. 3. 33.
i.e. "from the vantage-ground of concealment."
"Therefore of all hands must we be forsworn."
L. L. L. iv. 3. 219.
i.e. "from all sides," "to which ever side one looks," hence "in any case."
"Being regarded of all hands by the Grecians."—N. P. 176.
So our modern "off hand," applied to a deed coming from the hand, and not from the head. Hence "of hand" is used where we use "on" (175) in
"Turn of no hand."—M. of V. ii. 2. 45.
Of also retains this meaning with some local adjectives and adverbs, such as "north of," "south of," "within fifteen hundred paces of" (Hen. V. iii. 7. 136). We could say "the advantage of," but not "You should not have the eminence of him."
Tr. and Cr. ii. 2. 266.
'There is a testril of (from) me too."—T. N. ii. 3. 34.
166. Of used for "out of," "from," with verbs that signify, either literally or metaphorically, depriving, delivering, &c.
" We'll deliver you of your great danger. —Coriol. v. 6. 14.
" I may be delivered of these woes."—K. J. iii. 4. 56.
This use of of is still retained in the phrase " to be delivered of a child."
"Heaven make thee free of it."—Hamlet, v. 2. 342.
"To help him of his blindness."—T. G. of V. iv. 2. 45.
"Unfurnish me of reason."—W. T. v. 1. 123.
"Take of me my daughter."—M. Ado, ii. 1. 311.
"Rid the house of her."—T. Sh. i. 1. 150.
" Scour me this famous realm of enemies."—B. and F.
"That Lepidus of the triumvirate
Should be deposed."—A. and C. iii. 6. 28.
"His cocks do win the battle still of mine."—A. and C. ii. 3.36.
" Get goal for goal of youth. "—A. and C. iv. 8. 22.
"I discharge thee of thy prisoner."—M. Ado, v. 1. 327.
In virtue of this meaning, of is frequently placed after forth and out, to signify motion.
Hence, metaphorically,
"He could not justify himself of the unjust accusations."—N. P. 173.
Of is also used with verbs and adjectives implying motion from, such as "fail," "want," &c. Hence—
"But since you come too late of our intents."—Rich. III. iii- 5. 69.
167. Of thus applied to time means "from." So still "of late."
" I took him of a child up."—B. J. E. in &c. ii. 1.
i.e. "from a child, when a mere child." So in E. E. "of youth."
" Of long time he had bewitched them with sorceries."
Acts viii. 11.
" Being of so young days brought up with him."
Hamlet, ii. 2. 11.
168. Of, meaning "from," passes naturally into the meaning " resulting from," "as a consequence of."
"Of force."—M. of V. iv. 1. 421 ; 1 Hen. IV. iii. 2. 120.
"Of no right."—I Hen. IV. iii. 2. 100.
"Bold of your worthiness. "—L. L. L. ii. 1. 28.
"We were dead of sleep."—Temp. v. 1. 221.
" And of that natural luck
He beats thee 'gainst the odds."—A. and C. ii. 3. 26.
Hence "What shall become of this?" M. Ado, iv. 1. 211; T. N ii. 1. 37, means "what will be the consequence of this?"
So "by means of: "
" And thus do we of wisdom and of reach
By indirection find direction out."—Hamlet, ii. 1. 64.
While by is used of external agencies, of is used of internal motives, thus:
" Comest thou hither by chance, or of devotion ?"
2 Hen. VI. ii. 1. 88.
" The king of his own royal disposition."—Rich. III. i. 3. 63.
" Of purpose to obscure my noble birth. "—1 Hen. VI. v. 4. 22.
" Art thou a messenger, or come of pleasure ?"
2 Hen. VI. v. 1. 16.
Sometimes "out of" is thus used:
" But thou hast forced me,
Out of thy honest truth, to play the woman."
Hen. VIII. iii. 2. 431.
Of, "as a result of," is used as a result for "with the aid of," "with," or "at."
"That . . . she be sent over of the King of England's cost."
2 Hen. VI. i. 1. 61.
"Of the city's cost, the conduit shall run nothing but claret wine."
Ib. iv. 6. 3.
Hence the modern phrase "To die of hunger."
169. Of hence is used in appeals and adjurations to signify "out of."
" Of charity, what kin are you to me?"—T. N. v. 1. 237.
Hence, the sense of "out of" being lost, = "for the sake of," "boy."
" Speak of all loves."—M. N. D. ii. 2. 154.
This explains
" Let it not enter in your mind, of love."—M. of V. ii. 9. 42.
Similar is the use of of in protestations:
"Leon. We'll have dancing afterwards.
Ben. First, of my word."—T. N. v. 4. 123.
"A proper man, of mine honour."—2 Hen. VI. iv. 2. 103.
170. Of meaning "from" is placed before an agent (from whom the action is regarded as proceeding) where we use "by."
"Received of (welcomed by) the most pious Edward."
Macb. iii. 6. 27.
"Like stars ashamed of day."—V. and A.
i.e. "shamed by day."
Of is frequently thus used with "long," "'long," or "along." —LAYAMON. "Along of" = "from alongside of" (παρ with gen.).
"The good old man would fain that all were well
So 'twere not 'long of him."—3 Hen. VI. iv. 7. 32.
"'Long all of Somerset."—I Hen. VI. iv. 3. 46, 33.
"I am so wrapt and throwly lapt of jolly good ale and
old."—STILL.
171. Of is hence used not merely of the agent but also of the instrument. This is most common with verbs of construction, and of filling ; because in construction and filling the result is not merely effected with the instrument, but proceeds out of it. We still retain of with verbs of construction and adjectives of fulness; but the Elizabethans retained of with verbs of fulness also.
" Supplied of kernes and gallow-glasses."—Macb. i. 2. 13.
"I am provided of a torch-bearer."—M. of V. ii. 2. 24.
"You are not satisfied of these events."—Ib. v. 1. 297.
" Mettle—whereof thy proud child arrogant man is puffed."
T. of A. iv. 3. 180.
"Mixt partly of Mischief and partly of Remedy."—B. E. 114.
Hence
" Flies Whose woven wings the summer dyes
Of many colours."—B. and F. Fair Sh. v. 1.
Of with verbs of construction from "out of" sometimes assumes the meaning of "instead of."
"Made peace of enmity, fair love of hate."—Rich. III. ii. 1. 50. And with "become:"
"(Henry) is of a king become a banish'd man."—3 Hen. VI. iii. 3. 25.
172. Of is hence used metaphorically with verbs of construction, as in the modern
"They make an ass of me."—T. N. v. 1. 19.
But of is also thus found without verbs of construction, as .
'Apem. Or thou shalt find—
Timon. A fool of thee. Depart."
T. of A. iv. 3. 232.
"E'en such a husband
Hast thou of me as she is for a wife."—M. of V. iii. 5. 89.
"We should have found a bloody day of this."—1 Hen. VI. iv. 7. 34.
" We shall find of him
A shrewd contriver."—J. C. ii. 1. 157.
"We lost a jewel of her."—A. W. v. 3. 1.
"You have a nurse of me."—P. of T. iv. 1. 25.
"You shall find of the king, sir, a father."—A. W. i. 1. 7.
i.e. "in the king."
173. Of is hence applied not merely to the agent and the instrument, but to any influencing circumstance, in the sense of "as regards," "what comes from."
" Fantasy,
Which is as thin of substance as the air."—R. and J. i 4. 99.
"Roses are fast flowers of their smells."—B. E. 188.
" A valiant man of his hands."—N. P. 614.
"But of his cheere did seem too solemn-sad."—SPEN. F. Q. i. 1.
Under this head perhaps come :
"Niggard of question ; but of our demands
Most free in his reply."—Hamlet, iii. 1. 13.
"Of his own body he was ill, and gave
The clergy ill example."—Hen. VIII. iv. 2. 43.
"That did but show thee, of a fool, inconstant
And damnable ungrateful."—W. T. iii. 2. 187.
i.e. " as regards a fool," " in the matter of folly."
This may almost be called a locative case, and may illustrate the Latin idiom "versus animi." It is common in E. E. We still say, in accordance with this idiom, "swift of foot," " ready of wit," &c.
174. Of passes easily from meaning "as regards" to " concerning," "about."
"Mine own escape unfoldeth to my hope
The like of him."—T. N. i. 2. 21.
"You make me study of that."—Temp. ii. 1. 81.
"'Tis pity of him."—M. for M. ii. 3. 42 ; A. and C. i. 4. 71.
"'Twere pity of my life."—M. N. D. iii. 1. 44.
"I wonder of there being together."—Ib. iv. 1. 128.
"Wise of (informed of) the payment day."—B. E.
" He shall never more
Be fear'd of doing harm."—Lear, ii. 2. 113.
"The same will, I hope, happen to me, of death."
MONTAIGNE, 36.
i.e. "with respect to death."
" I humbly do desire your grace of pardon."
M. of V. iv. 1. 402.
"I shall desire you of more acquaintance."
M. N. D. iii. 1. 183 ; A. Y. L. v. 4. 56.
For this use of "desire" compare A. V. St. John xii. 21, "they desired him saying," where Wickliffe has "preieden," "prayed."
"I humbly do beseech you of your pardon."—O. iii. 3. 212.
"The dauphin whom of succours we entreated."
Hen. V. iii. 3. 45.
"Yet of your royal presence I'll adventure
To borrow of a week."—W. T. i. 2. 38.
"We'll mannerly demand thee of thy story."—Cymb. iii. 6. 92.
"Enquire of him."—Rich. II. 3. 186.
i.e. "about him."
"Discern of the coming on of years."—B. E. 105.
" Having determined of the Volsces and," &c.—Coriol. ii.2.41.
"I'll venture so much of my hawk or hound."
T. of Sh. v. 2. 72.
"Since of your lives you set
So slight a valuation."—Cymb. iv. 4. 48.
In "No more can you distinguish of a man
Than of his outward show,"—Rich. III. iii. 1. 9, 10.
the meaning seems to be, "you can make no distinctions about men more than," i.e. "except, about their appearances." So
" Since my soul could of men distinguish."—Hamlet, iii. 2. 69.
In the following passages we should now use "for:"—
"France whereof England hath been an overmatch."—B. E. 113.
"I have no mind of feasting."—M. of V. ii. 5. 37.
"In change of him."—Tr. and Cr. iii. 3. 27.
" Of this my privacy I have strong reasons."
Tr. and Cr. iii. 3. 190.
"In haste whereof, most heartily I pray
Your highness to assign our trial day."—Rich. II. i. 1. 150.
As we say "what will become of (about) me !" so
"What will betide of me."—Rich. III. i. 3. 6.
We say "power over us," not
"The sovereign power you have of us."—Hamlet, ii. 2. 27.
"I have an eye on him," not
"Nay, then, I have an eye of you."—Ib. 301.
175. Of signifying proximity of any kind is sometimes used locally in the sense of " on." The connection between of and on is illustrated by M. of V. ii. 2, where old Gobbo says : "Thou hast got more haire on thy chin than Dobbin my philhorse has on his taile;" and young Gobbo retorts, "I am sure he had more haire of his taile than I have of my face."
"Gra. My master riding behind my mistress—
Cart. Both of one horse."—T. of Sh. iv. 1. 71.
Of is sometimes used metaphorically. for "on."
Compare
"A plague of all cowards !"—1 Hen. IV. ii. 4. 127.
with
"A plague upon this howling."—Temp. i. 1. 39.
"Who but to-day hammer'd of this design."—W. T. ii. 2. 49.
"I go of message."—2 Hen. VI. iv. 1. 113.
A message may be regarded as a motive from which, or as an object towards which, an action proceeds, and hence either of or "on" may be used.
Compare "He came of an errand."—M. W. of W. i. 4. 80.
with
"I will go on the slightest errand."—M. Ado, ii. 1. 272.
"Sweet mistress, what your name is else I know not,
Nor by what wonder you do hit of mine."—C.of E. iii. 2. 30.
Add also—
"And now again Of him that did not ask, but mock, bestow
Your sued-for tongues."—Coriol. ii. 3. 215.
" I shall bestow some precepts of this virgin." A. W. iii. 5. 103 ; T. N. iii. 4. 2.
"Trustyng of (comp. "depending on") the continuance."
ASCH. Ded.
176. Of, signifying "coming from," "belonging to," when used with time, signifies "during."
"These fifteen years : by my fay a goodly nap!
But did I never speak of all that time?"—T. of Sh. Ind. 2. 84.
"There sleeps Titania sometime of the night."—M. N. D. ii. 1. 253. i.e. "sometimes during the night."
"My custom always of the afternoon."—Hamlet, i. 5. 60.
" And not be seen to wink of ali the day."—L. L. L. i. 1. 43.
"Of the present."—Tempest, i. 1. 24.
So often "Of a sudden."
177. Of is sometimes used to separate an object from the direct action of a verb : (a) when the verb is used partitively, as " eat of," "taste of," &c ; (b) when the verb is of French origin, used with "de," as "doubt," "despair," "accuse," "repent," "arrest," "appeal," "accept," "allow;" (c) when the verb is not always or often used as a transitive verb, as "hope" or "like," especially in the case of verbs once used impersonally.
* (a)"King. How fares our cousin Hamlet ?
Hamlet. Excellent, i' faith : of the chameleon's dish."
Hamlet, iii. 2. 98.
* (b) "To appeal each other of high treason."—Rich. II. i. 1. 27.
" Of capital treason we arrest you here."—Ib. iv. 1. 151.
* (c)"So then you hope of pardon from Lord Angelo?" M. for M. iii. 1. 1.
" I will hope of better deeds to-morrow."—A. and C. i 1. 62.
The of after "to like" is perhaps a result of the old impersonal use of the verb, "me liketh," "him liketh," which might seem to disqualify the verb from taking a direct object. Similarly "it repents me of" becomes "I repent of;" "I complain myself of" becomes " I complain of." So in E. E. "it marvels me of" becomes "I marvel of." Hence—
"It was a lordling's daughter that liked of her master."
P. P. 16.
"Thou dislikest of virtue for the name."—A. W. ii. 3. 131.
"I am a husband if you like of me."—M. Ado, v. 4. 59.
So L. L. L. i. 1. 107 ; iv. 3. 158 ; Rich. III. iv. 4. 354.
"To like of nought that would be understood."
BEAUMONT on B. J.
178. Of naturally followed a verbal noun. In many cases we should call the verbal noun a participle, and the of has become unintelligible to us. Thus we cannot now easily see why Shakespeare should write—
"Dick the shepherd blows his nail."—L. L. L. v. 2. 923.
and on the other hand—
" The shepherd blowing of his nails."—3 Hen. VI. ii. 5. 3.
But in the latter sentence blowing was regarded as a noun, the prepositional "a," "in," or "on" being omitted.
" The shepherd was a-blowing of his nails."
In the following instances we should now be inclined to treat the verbal as a present participle because there is no preposition before it :
" Here stood he (a-)mumbling of wicked charms. "—Lear, ii. 1. 41.
"We took him (a-)setting of boys' copies."—2 Hen. VI. iv. 2. 96.
"And then I swore thee, (a-)saving of thy life."—J. C. v. 3. 38.
"Here was he merry (a-)hearing of a song."—A. Y. L. ii. 7. 4.
where "hear of" does not mean, as with us, "hear about." So Lear, v. 3. 204. In all the above cases the verbal means "in the act of."
In most cases, however, a preposition is inserted, and thus the substantival use of the verbal is made evident. Thus :
"So find we profit by losing of our prayers."—A. and C. ii. 1. 8.
" Your voice for crowning of the king."
Rich. III. iii. 4. 29 ; Hamlet, i. 5. 175 ; Lear, i. 3. 1.
" With halloing and singing of anthen-is."—2 Hen. IV. i. 2. 213.
"What, threat you me with telling of the king?"
Rich. III. i. 3. 113.
" About relieving of the sentinels."—1 Hen. VI. ii. 1. 70 ; iii. 4. 29.
If it be asked why "the" is not inserted before the verbal,—e.g. "about the relieving of the sentinels,"—the answer is that relieving is already defined, and in such cases the article is generally omitted by Shakespeare. (See 89.)
When the object comes before the verbal, of must be omitted :
" Ophelia. Hamlet . . . shaking of mine arm
And thrice his head thus waving."—Hamlet, ii. 1. 92.
The reason is obvious. We can say "in shaking of mine arm," but not "in his head thus waving."
Compare C. of E. v. 1. 153 ; A. Y. L. ii. 4. 44, iv. 3. 10 ; W. T. iii. 3. 69 ; 1 Hen. IV. ii. 4. 166 ; R. and J. v. 1. 40.
"Yet the mother, if the house hold of our lady."—ASCH. 40.
"Hold," by itself, would mean "actually hold" (capiat). "Hold of" means "be of such a nature as to hold" (capax sit), "holding of."
179. Of is sometimes redundant before relatives and relatival words in dependent sentences, mostly after verbs intransitive.
"Make choice of which your highness will see first."
M. N. D. v. 1. 43.
"What it should be ... I cannot dream of."
Hamlet, ii. 2. 10.
"Making just report
Of how unnatural and bemadding sorrow
The king hath cause to plain."—Lear, iii. 2. 38.
"He desires to know of you of whence you are,"
P. of T. ii. 3. 80.
where, however, "whence" is, perhaps, loosely used for "what place," and of strictly used for "from."
The redundant and appositional of, which we still use after "town," "city," "valley," &c., is used after "river" (as sometimes by Chaucer and Mandeville) in
"The river of Cydnus."—A. and C. ii. 2. 192.
180. On, upon (interchanged in E. E. with "an"), represents juxtaposition of any kind, metaphorical or otherwise. It was in Early English a form of the preposition "an" which is used as an adverbial prefix (see 141) ; and as late as Ascham we find—
" I fall on weeping."—ASCH. iii. 4.
"For sorrow, like a heavy-hanging bell
Once set on ringing, with his own weight goes."—R.of L. 1494.
Compare also our a-head with
"Hereupon the people ran on-head in tumult together."—N. P. 191.
"Why runnest thou thus on head?"—Homily on Matrimony.
The metaphorical uses of this preposition have now been mostly divided among of, in, and at, &c. We still, however, retain the phrase, "on this," "on hearing this," &c. where on is "at the time of," or " immediately after." But we could not say—
"Here comes (333) the townsmen on (in) procession."
2 Hen. VI. ii. 1. 68.
"Read on (in) this book."—Hamlet, iii. 1. 44. So MONTAIGNE, 227: "To read on some book."
"Blushing on (at) her."—R. of L. st. 453.
" On (at) a moderate pace."—T. N. ii. 2. 3.
"The common people being set on a broile."—N. P. 190.
(Comp. our " set on fire.")
"Horses on ('in' or 'of') a white foam."—N. P. 186.
" On (of) the sudden."—Hen. VIII. iv. 2. 96.
"And live to be revenged on ('for' or 'about') her death."
R. of L. 1778.
" Be not jealous on (of) me."
" Fond on her. "—M. N. D. ii. 1. 266.
"Nod on (at) him."— . C. i. 2. 118.
"Command upon me."-Macbeth, iii. 1. 17.
On, like "upon," is used metaphorically for "in consequence of" in
" Lest more mischance
On plots and errors happen."—Hamlet, v. 2. 406 ;
for "in dependence on " in
" I stay here on my bond."—M. of V. iv. 1. 242.
In
" She's wandering to the tower On pure heart's love to greet the tender princes,"
Rich. III. iv. 1. 4.
there is a confusion between "on an errand of love" and "out of heart's love."
181. On is frequently used where we use "of" in the sense of "about," &c. Thus above, "jealous on," and in Sonn. 84, "Fond on praise." In Early English (Stratmann) we have " On witchcraft I know nothing." " What shall become on me?" "Denmark won nothing on him." Compare—
" Enamour'd on his follies."—I Hen. IV v. 2. 71.
" His lands which he stood seized on."—Hamlet, i. 1. 88.
"Or have we eaten on the insane root ? "—Macbeth, i. 3. 84.
" He is so much made on here. "—Coriol. iv. 5. 203.
" What think you on't. "—Hamlet, i. 1. 55.
Note the indifferent use of on and "of" in
" God have mercy on his soul
And of all Christian souls. "—Hamlet, iv. 5. 200.
The use of on in
" Intended or committed was this fault ?
If on the first,—Ipardon thee,"—Rich. II. v. 3. 34.
is illustrated by
"My gracious uncle, let me know my fault,
On what condition stands it. "—Ib. ii. 3. 107.
182. On, being thus closely connected with " of," was frequently used even for the possessive "of," particularly in rapid speech before a contracted pronoun.
" One on's ears. "—Coriol. ii. 2. 85. So Coriol. i. 3. 72 ; it. 1. 202.
" The middle on's face. "—Lear, iv. 5. 20.
"Two on's daughters. "—Ib, i. 4. 114.
" Two on's."—Cymb. v. 5. 311.
" My profit on't. "—Temp. i. 2. 365, 456.
" You lie out on't, sir."—Hamlet, v. 1. 132 ; Lear, iv. 1. 52.
" He shall hear on't."—B. J. E. in &c.
"I am glad on't."— . C. i. 3. 137.
In the two last examples on may perhaps be explained as meaning "concerning," without reference to "of."
The explanation of this change of "of" to " on " appears to be as follows. "Of " when rapidly pronounced before a consonant became "o'."
"Body o' me."—Hen. VIII. v. 2. 22.
" O' nights."—T. N. i. 3. 5.
Hence the o' became the habitual representative of "of" in colloquial language, just as " a- " became the representative of " on " or "an." But when o' came before a vowel, what was to be done? Just as the "a-" was obliged to recur to its old form "an" before a vowel or mute h (compare Hamlet, i. 4. 19, "to stand an-end," and see 24), so before a vowel o' was forced to assume a euphonic n. (Compare the Greek custom.)
And even when the pronoun is not contracted, we find in Corial. iv. 5. 174, the modern vulgarism—
"Worth six on him."
"To break the pate on thee."—1 Hen. IV. ii. 1. 34.
183. Out (out from) is used as a preposition like forth.
" You have push'd out your gates the very defender of them."
Coriol. v. 2. 41.
(Early Eng. " Come out Ireland," " Out this land.")
" Out three years old. "—Temp. i. 2. 41, " beyond three years."
Explained by Nares, " completely."
From out. See 157.
184. Till is used for to :
"From the first corse till he that died to-day,"
Hamlet, i. 2. 105.
where probably till is a preposition, and "he" for "him." See He.
"Lean'd her breast up till a thorn."—P. P. st. 21.
Early Eng. "He said thus til (to) him," and, on the other hand, "To (till) we be gone." So "unto" in Chaucer for " until"
"I need not sing this them until (for ' unto them')."
HEYWOOD.
" We know whereuntil (whereto) it doth amount."
L. L. L. v. 2. 494.
" And hath shipped me intil (into) the land."—Hamlet,v. 1. 81.
185. To (see also Verbs, Infin.). Radical meaning motion towards. Hence addition. This meaning is now only retained with verbs implying motion, and only the strong form " too (comp. of and off) retains independently the meaning of addition. But in Elizabethan authors too is written to, and the prepositional meaning " in addition to " is found, without a verb of motion, and sometimes without any verb.
" But he could read and had your languages
And to't as sound a noddle," &c.—B. J. Fox, ii. I.
"If he ... to his shape, were heir of all this land."
K. i. 1. 144.
" And to that dauntless temper of his mind
He hath a wisdom that doth guide his valour."
Macbeth, iii. 1. 52.
i.e. "in addition to that dauntless temper." To, in this sense, has been supplanted by "beside." Compare also
" Nineteen more, to myself."—B. J. E. in &c. iv. 5.
To is used still adverbially in "to and fro," and nautical expressions such as "heave to," "come to." This use explains "Go to," M. of V. ii. 2. 169. "Go did not in Elizabethan or E. E. necessarily imply motion from, but motion generally. Hence "go to" meant little more than our stimulative " come, come."
186. To hence means motion, "with a view to," "for an end," &c. This is of course still common before verbs, but the Elizabethans used to in this sense before nouns.
" He which hath no stomach to this fight."—Hen. V. iv. 3. 35.
" For to that (to that end)
The multiplying villanies of Nature
Do swarm upon him."—Macbeth, i. 2. 10.
" Prepare yourself to death. "—W. T. iii. 1. 167.
" Arm you to the sudden time."—K. J. v. 6. 26.
" The impression of keen whips I 'ld wear as rubies
And strip myself to (for) death as to a bed."
M. for M. ii. 4. 102.
"Giving to you no further personal power
To (for the purpose of) business with the king."
Hamlet, i. 2. 37.
" Pawn me to this your honour. "—T A. i. 1. 147.
" Few words, but, to effect, more than all yet."
Lear, iii. 1. 52.
"He is frank'd up to fatting for his pains."
Rich. III. i. 3. 314.
Hence it seems used for for in
"Ere I had made a prologue to my brains
They had begun the play."—Hamlet, v. 2. 30
And perhaps in
"This is a dear manakin to you, Sir Toby."—T. N. iii. 2. 57.
But see 419 a, for this last example.
187. To hence, even without a verb of motion, means " motion to the side of." Hence "motion to and consequent rest near," as in
" Like yourself
Who ever yet have stood to charity."—Hen. VIII. ii. 4. 86.
" To this point I stand."—Hamlet, iv. 5. 187.
"I beseech you, stand to me."—2 Hen. IV. ii. 1. 70.
i.e. "Come and stand by me, help me."
Motion against in:
"The lady Beatrice hath a quarrel to yow."—M. Ado, ii. 1. 44.
So T. N. iii. 4. 248 ; Coriol. iv. 5. 113.
Motion to meet :
" To her doom she dares not stand."—B. and F. Fair Sh. v. 1.
Motion toward:
"What wouldst thou have to Athens ?"—T. of A. iv. 3. 287.
"To Milan let me hear from thee by letters."
T. G. of V. i. 1. 57.
Hence "by the side of," "in comparison with."
"Impostors to true fear."—Macb. iii. 4. 64.
i.e. "Impostors when brought to the side of, and compared with, true fear."
" There is no woe to his correction,
Nor to his service no such joy on earth."
T. G. of V. ii. 4. 138, 139.
"The harlot's cheek, beautied with plastering art,
Is not more ugly to the thing that helps it
Than is my deed to my most painted word."
Hamlet, iii. 1. 51—53.
In " Treason can but peep to what it would,
Acts little of his will,"—Ib. iv. 5. 125.
either to means "towards," an unusual construction with " peep," or the meaning is "treason can do nothing more than peep in comparison with what it wishes to do."
" Undervalued to tried gold.—M. of V. ii. 7. 53.
Hence "up to," "in proportion to," "according to."
"The Greeks are strong and skilful to their strength."
Tr. and Cr. i. 1. 7.
"That which we have we prize not to the worth."
M. Ado, iv. 1. 220.
" To's power he would
Have made them mules. "—Coriol. ii. 1. 262.
"Perform'd to point the tempest that I bade thee."
Temp. i. 2. 194.
" He needs not our mistrust, since he delivers
Our offices and what we have to do
To the direction just."—Macb. iii. 3. 4.
Hence "like."
" My lady, to the manner of the days,
In courtesy gives undeserving praise. "—L. L. L. v. 2. 365.
" Looked it of the hue
To such as live in great men's bosoms?"—B. J. Sejan. v. I.
" This is right to (exactly like) that (saying) of Horace."
B. J. E. out &c. ii. 1.
To seems to mean " even up to " in
" And make my senses credit thy relation
To points that seem impossible."—P. of T. v. 2. 125.
188. To is sometimes used without any sense of motion for " near."
"It would unclog my heart
Of what lies heavy to't."—Coriol. iv. 2. 48.
" Sits smiling to my heart."—Hamlet, i. 2. 124.
for " by " in
"Where . . . the best of all her sex
Doth only to her worthy self abide."—B. and F. F. Sh. ii. 1.
In the difficult passage (W. T. iv. 4. 550) :
"But, as the unthought on accident is guilty
To what we wildly do."
" Guilty" seems used for " responsible," and chance is said to be "responsible to" rashness (personified). (Or is to "as to," i.e. as regards ?)
In N. P. 175 there is "to the contrary," (but this is a translation of "au contraire,") for "on the contrary."
To is inserted after "trust" (whereas we have rejected it in parenthetical phrases, probaby for euphony's sake).
" And, trust to me, Ulysses,
Our imputation will be oddly poised. "—Tr. and Cr. i 3. 339.
To seems "up to," "as much as," in
"I'll part sooner with my soul of reason than yield to one foot of land."—B. and F. Elder Brother, iii. 5.
188a. "To," with Adjectives signifying obedience, &c. To is still used in the sense of "towards" after some adjectives, such as (1) "gentle," (2) "disobedient," (3) "open." But we could not say
1. " If thou dost find him tractable to us."—Rich. III. iii. 1. 74.
2. " A will most incorrect (unsubmissive) to heaven."
Hamlet, i. 2. 95.
"The queen is stubborn to justice."—Hen. VIII. ii. 4. 122.
3. " Penetrable to your kind entreats."—Rich. III. iii. 7. 225.
" Vulgar to sense." —Hamlet, i. 2. 99.
i.e. "open to ordinary observation."
Similarly to is used after nouns where we should use "against," "in the sight of:"
" Fie ! 'tis a fault to heaven,
A fault against the dead, a fault to nature,
To reason most absurd."—Hamlet, i. 2. 103.
189. To, from meaning "like," came into the meaning of "representation," "equivalence," "apposition." (Comp. Latin " Habemus Deum amico.")
"I have a king here to my flatterer."—Rich. II. iv. I. 306.
"To crave the French king's sister
To wife for Edward."—3 Hen. VI. iii. 1. 31.
"Now therefore would I have thee to my tutor."
T. G. of V. iii, 1. 84.
"Destiny . . . that hath to instrument this lower world."
Temp. iii. 2. 54.
"And with her to dowry some petty dukedoms."
Hen. V. iii. Prol. 31.
"Lay their swords to pawn."—M. W. of W. iii. 1. 113.
"Had I admittance and opportunity to friend."—Cymb. i. 4. 118.
"Tunis was never graced before with
Such a paragon to their queen."—Temp. ii. 1. 75.
Compare also Macb. iv. 3. 10 ; . C. iii. 1. 143.
"The king had no port to friend."—CLARENDON, Hist. 7.
"A fond woman to my mother (i.e. who was my mother) taught me so."—WAGER.
Thus "to boot" means "by way of, or for, addition." So in E. E. "to sooth" is used for "forsooth."
190. To, in the phrase " I would to God," may mean "near," " in the sight of;" or there may be a meaning of motion: "I should desire (even carrying my desire) to God. " In the phrase " He that is cruel to halves" (B. J. Disc. 759), to means, perhaps, "up to the limit of." Possibly, however, this phrase may be nothing but a corruption of the more correct idiom " Would God that," which is more common in our version of the Bible than " I would." The to may be a remnant and corruption of the inflection of "would," " wolde; " and the I may have been added for the supposed necessity of a nominative. Thus
"Now wolde God that I might sleepen ever."
CHAUCER, Monke's Tale, 14746.
So "thou wert best" is a corruption of "it were best for thee."
This theory is rendered the more probable because, as a rule, in Wickliffe's version of the Old Testament, " Wolde God" is found in the older MSS., and is altered into "we wolden" in the later. Thus Genesis xvi. 3; Numbers xx. 3; Joshua vii. 7 ; Judges ix. 29; 2 Kings v. 3 (Forshall and Madden, 1850). However, Chaucer has "I hoped to God" repeatedly.
To was used, however, without any notion of "motion toward the future " in to-night (last night).
" I did dream to-night."—M of V. ii. 5.18 ; 2 Hen. VI. iii. 2. 31.
So in E. E. "to year" for "this year," "to summer," &c. Perhaps the provincial "I will come the night, the morn," &c. is a corruption of this "to." It is, indeed, suggested by Mr. Morris that to is a corruption of the demonstrative. On the other hand, to in E. E. was " often used with a noun to form adverbs."—LAYAMON (Glossary).
"He aras to þan mid-nihte,"—LAYAMON, i. 324.
is used for " he arose in the midnight."
Unto, like To, 185, is used for "in addition to :"
"Unto my mother's prayers I bend my knee."
Rich. II. v. 3. 97.
191. Upon (" for the purpose of") is still used in "upon an errand," but not, as in
" Upon malicious bravery dost thou come ?"—Othello, i. 1. 100.
We should use "over" in
" I have no power upon you,"—A. and C. i. 3. 23
. and we should not use upon in
" And would usurp upon my watery eyes."—T. A. iii. 1. 269.
" Let your highness
Command upon me."—Macbeth, iii. 1. 1'7
. though after "claim" and "demand" upon is still used. So "an attack upon" is still English, but not
" I have o'erheard a plot of death upon him."—Lear, iii. 6. 9 6.
nor " I am yours . . . upon your will to suffer."—A. W. iv. 4. 30. i.e. "in dependence on." It would seem that the metaphorical use of upon is now felt to be too bold unless suggested by some strong word implying an actual, and not a possible influence. Thus "claim" and " demand" are actual, while " power" may, perhaps, not be put in action. So "attack" and "assault" are the actual results of " plot." Yet the variable use of prepositions, and their close connection with particular words, is illustrated by the fact that we can say, "I will wait upon him," but not
" I thank you and will stay upon your leisure. "—A. W. iii. 5. 48. Even here, however, our "wait upon" means, like "call upon," an actual interview, and does not, like " stay upon," signify the " staying in hope of, or on the chance of, audience."
Upon also means " in consequence of."
" When he shall hear she died upon (i.e. not 'after,' but 'in consequence of") his words."—M. Ado, iv. 1. 225.
" And fled is he upon this villany."—Ib. v. 1. 258.
" Break faith upon commodity."—K. J. ii. 1. 597.
" Thy son is banish'd upon good advice. "—Rich. II. i. 3. 233.
In " You have too much respect upon the world,"
M. of V. i. 1. 74.
there is an allusion to the literal meaning of "respect." "You look too much upon the world." The upon is connected with "respect," and is not used like our "for" in "I have no respect for him. "
The use of "upon" to denote "at" or "immediately after" is retained in "upon this;" but we could not say
"You come most carefully upon your hour."—Hamlet, i. 1.6.
192. Upon is often used like on adverbially after the verb "look."
" Nay, all of you that stand and look upon."—Rich. II. iv. 1. 237.
" Why stand we like soft-hearted women here
And look upon, as if," &c.—3 Hen. VI. ii. 3. 27.
" Strike all that look upon with marvel, come."—W. T. v. 3. 100.
" Near upon" is adverbial in
" And very near upon
The duke is entering."—M. for M. iv. 6. 14.
" Indeed, my lord, it followed- hard upon."—Hamlet, i. 2. 179.
Upon, from meaning superposition, comes to mean "in accordance with " (like " after ") :
" Upon my power I may dismiss this court."
M. of V. iv. 1. 104.
193. With (which, like "by," signifies juxtaposition) is often used to express the juxtaposition of cause and effect.
"I live with (on) bread like you."—Rich. II. iii. 2. 175.
We could say "he trembles with fear," "fear" being regarded as connected with the trembler, but not
My inward soul
With nothing trembles : at something it grieves
More than with parting from my lord the king."
Rich. IL ii. 2. 12, 13.
" As an unperfect actor on the stage
Who with his fear is put besides his part. "—Sonn. 23.
We should say "in his fear (or "by his fear," personifying Fear) ; or append the clause to the verb, "put beside his part with fear."
" It were a better death than die with mocks,
Which is as bad as die with tickling."—M. Ado, iii. 1. 79, 80.
"Another choaked with the kernell of a grape, and an emperour die by the scratch of a combe, and Aufidius with stumbling against the doore, and Lepidus with hitting his foot."—MONTAIGNE, 32.
Here the use of "by" seems intended to distinguish an external from an internal cause.
We say "so far gone in fear," but not
" Thus both are gone with conscience and remorse."
Rich. III. iv. 3. 20.
" This comes with seeking you."—T. N. iii. 4. 366.
"I feel remorse in myself with his words."—2 Hen. VI. iv. 7. 111. More rarely, with is used with an agent :
" Rounded in the ear
With that same purpose-changer, that sly devil."—K. J. ii. 1. 567.
" We had like to have had our two noses snapped off with two old men without teeth."—M. Ado, v. 1. 116.
" Boarded with a pirate. "—2 Hen. VI. iv. 9. 33.
" He was torn to pieces with a bear."—W. T. v. 2. 66.
" Assisted with your honoured friends."—Ib. v. 1. 13.
This explains
" Since I am crept in favour with myself
I will maintain it with some little cost."—Rich. III i. 2. 260.
The obvious interpretation is, " since I have crept into the good graces of myself ;" but the second line shows the "I" to be superior to "myself," which is to be maintained by the "I." The true explanation is, "since I have crept into (Lady Anne's) favour with the aid of my personal appearance, I will pay some attention to my person." Add, probably, Hamlet, iii. 2. 207.
This meaning is common in E. E. :
"He was slayn wyþ (by) Ercules."
R. OF BRUNNE, Chron. i. 12. 340.
With = "by means of."
" He went about to make amends with committing a worse fault." —N. P. 176, where the French is "par une autre." So N. P. 176.
With = "in addition to," even when there are not two nouns to be connected together :
"Very wise and with his wisdome very valiant."—N. P. 664.
With is, perhaps, used for " as regards," " in relation to," as in our modern " this has not much weight with me," in
"Is Cæsar with Antonius priz'd so slight ? "—A. and C. i. 1. 56. though here, perhaps, as above, with may mean "by." At all events the passage illustrates the connection between " with" and " by." Compare
" His taints and honours
Wag'd equal with (i.e, in) him."—A. and C. v. 1. 31.
"So fond with gain."—R. of L. 134.
194. With is hence loosely used to signify any connection with an action, as in "to change with" (MONTAIGNE, 233), where we should say "to exchange for." So, though we still say "I parted with a house," or "with a servant (considered as a chattel)," we could not say
" When you parted with the king."—Rich. II. ii. 2. 2.
" As a long-parted mother witch her child." Ib. iii. 2. 8 ; Rich. III. i. 4. 251.
where with is connected with parting. See 419a. So
" I rather will suspect the sun with cold
Than thee with wantonness. "—M. W. of W. iv. 4. 5.
as we say "I charge him with."
" Next them, with some small distance, follows a gentleman bearing the purpose."—Hen. VIII. ii. 4, stage direction.
" Equal with," 3 Hen. VI. iii. 2. 137, is like our " level with." In
" The violence of either grief or joy
Their own enactures with themselves destroy,"
Hamlet, iii. 2. 207.
"with themselves" seems to mean "by or of themselves."
Note "They have all persuaded with him."—M. of V. iii. 2. 283. i.e. "argued with." So "flatter" is used for "deal flatteringly" in T. N. i. 5. 322, and in the first of the following lines :
"K. Rich. Should dying men flatter with those that live?
Gaunt. No, no, men living flatter those that die."
Rich. II. ii. 1. 88, 89.
"(She) married with my uncle."—Hamlet, i. 2. 151.
" will break with her."—M. Ado, i. 1. 311.
i.e. " open the matter in conversation with."
195. With is used by Ben Jonson for like.
" Not above a two shilling.
B. 'Tis somewhat with the least."—B. J. E. in &c. i. 4.
" Something like, very near the least.
"He is not with himself."-T. A. i. 1. 368. i.e. "in his senses."
Ben Jonson also uses without in the sense of "unlike," "beyond."
" An act without your sex, it is so rare."—B. J. Sejan. ii. 1.
196. Withal, the emphatic form of " with " (see " all "), is used for with after the object at the end of a sentence. Mostly, the object is a relative.
" These banish'd men that I have kept withal."
T. G. of V. v. 4. 152.
i.e. " With whom I have lived."—K. J. iii. 1. 327.
"And this is false you burden me withal."—C. of E. v. 1. 268. i.e. "this with which you burden me."
" Such a fellow is not to be talk'd withal."—M. for M. v. 1. 347.
Sometimes "this" is understood after withal, so that it means "with all this," and is used adverbially :
" So glad of this as they I cannot be
Who are surprised withal."—Temp. iv. 1. 217.
i.e. "surprised with, or at, this." Here however, perhaps, and elsewhere certainly, with means "in addition to," and "with-all (this)" means "besides."
" I must have liberty withal."—A. Y. L. ii. 7. 48.
" Adding withal."—Rich. II. iv. 1. 18, &c.
But in " I came hither to acquaint you withal,"—A. Y. L. i. 1. 139. there is no meaning of "besides," and withal means "therewith," "with it."
Withal follows its object, but is (on account of the "all" at the end of the previous verse) not placed at the end of the sentence, in
"Even all I have, yea, and myself and all
Will I withal endow a child of thine."—Rich. III. iv. 4. 249.
197. Without (used locally for " outside ").
" What seal is that that hangs Without thy bosom?"
Rich. II. v. 1. 56.
" Without the peril of the Athenian law."—M. N..D. iv. 1. 150.
"A mile without the town."—Ib. i. 1. 104.
This explains the pun :
"Val. Are all these things perceived in me ?
Speed. They are all perceived without ye."—T. G. of V. ii. 1. 35.
Reversely, "out of" is used metaphorically for "without."
"neither can anything please God that we do if it be done out of charity."—HALLIWELL.
198. Prepositions are frequently omitted after verbs of motion. Motion in :
"To reel the streets at noon."—A. and C. i. 4. 20.
" She warzder'd many a wood."—SPENS. F. Q. i. 7. 28.
" To creep the ground." "Tower the sky."-MILTON, P. L, vii. 441.
Motion to or from :
"That gallant spirit hath aspired the clouds."
R. and J. iii. 1. 122.
" Ere we could arrive the point proposed. "—J. C. i. 2. 110.
"Arrived our coast."—3 Hen. VI. v. 3. 8.
" Some sailors that escaped the wreck."—M. of V. iii. 1. 110.
"When we with tears parted Pentapolis."—P. of T. v. 3. 38.
"Depart the chamber and leave us."—2 Hen. IV. iv. 4. 91.
" To depart the city. "—N. P. 190.
" Since presently your souls must part your bodies."
Rich. II. iii. 1. 3.
We can still say "to descend the hill," but not "to descend the summit," nor
" Some (of her hair) descended her sheav'd hat."—L. C. 31.
These omissions may perhaps illustrate the idiom in Latin, and in Greek poetry.
Verbs of ablation, such as "bar," "banish," "forbid," often omit the preposition before the place or inanimate object. Thus
"We'll bar thee from succession."—W. T. iv. 4. 440.
Or " Of. succession."—Cymb. iii. 3. 102. becomes "Bars me the right."
M. of V. ii. 1. 16 ; Rich. III. iv. 4. 400 ; A. Y. L. i. 1. 20.
Where a verb can take either the person or thing as an object, it naturally takes an indirect object without a preposition. Compare
" Therefore we banish you our territories."—Rich. II. i. 3. 139.
198 a. The preposition is omitted after some verbs and adjectives that imply "value," "worth," &c.
" The queen is valued thirty thousand strong."
3 Hen. VI. v. 3. 14.
" Some precepts worthy the note."—A. W. iii. 5. 104.
An imitation of this construction is, perhaps, to be traced in
" Guilty so great a crime."—B. and F. F. Sh. iv. 1.
The omission of a preposition before "good cheap" (A.-S. ceáp, "price," "bargain"), 1 Hen. IV. iii. 3. 50, may perhaps be thus explained without reference to the French "bon marché." And thus, without any verb or adjective of worth,
"He has disgraced me and hindered me half a million."
M. of V. iii. 1. 57.
"Semblative" (unless adverbial [1]) is used with the same construction as "like" in
" And all is semblative a woman's part."—T. N. i. 4. 34.
199. The preposition is also sometimes omitted before the thing heard after verbs of hearing:
"To listen our purpose."—M. Ado, iii. 1. 12.
"List a brief tale."—Lear, v. 3. 181.
So J. C. v. 5. 15; Hamlet, i. 3. 30; J. C. iv. 1. 41.
"Listening their fear." "—Macbeth, ii. 2. 28.
Hence in the passive,
"He that no more must say is listen'd more." Rich. II. ii. 1. 9.
"Hearken the end."—2 Hen. IV. ii. 4. 305 ; Temp. i. 2. 122.
200. The preposition is omitted after some verbs which can easily be regarded as transitive. Thus if we can say "plot my death," there is little difficulty in the licence.
" That do conspire (for) my death. "—Rich. III. iii. 4. 62.
" (In) Which from the womb I did participate."—T.N.v. 1.245.
" She complain'd (about) her wrongs."—R. of L. 1839.
" And his physicians fear (for) him mightily."
Rich. III. i. 1. 137.
So 1 Hen. IV. iv. 1. 24; T. of A. ii. 2. 12; T. A. ii. 3. 305; M. of V. iii. 2. 29.
This explains
" O, fear me not."—Hamlet, i. 3. 52; iii. 4. 7.
" That he would labour (for) my delivery."—Rich. III. i. 1. 253.
"To look (for) your dead."—Hen. V. iv. 7. 76.
" I must go look (for) my twigs."—A. W. iii. 6. 115.
" He hath been all this day to look (for) you."—A. Y. L. ii. 5. 34.
And in the difficult passage—
"O, whither hast thou led me, Egypt? See
How I convey my shame out of thine eyes
By looking back what I have left behind
'Stroy'd in dishonour."—A. and C. iii. 10. 53.
While turning away from Cleopatra, Antony appears to say, that he is looking back (for) the fleet that he has left dishonoured and destroyed.
So
"Scoffing (at) his state."—Rich. III. iii. 2. 163.
" Smile you (at) my speeches as I were a fool!"—Lear, ii. 2. 88.
" Thou swear'st (by) thy gods in vain."—Ib. i. 1. 163.
"Yet thus far, Griffith, give me leave to speak (of) him."
Hen. VIII. iv. 2. 32.
Both here and in L. L. L. v. 2. 349 ; Macbeth, iv. 3. 159; T. N. i. 4. 20, "speak" is used for describe. In Macbeth, iv. 3. 154, "'tis spoken" is used for "'tis said." Again, "said" is used for " called" in
"To be said an honest man and a good housekeeper."
T. N. iv. 2. 10 ; so Macbeth, iv. 3. 210.
"Talking that" is used like "saying that" in Tempest, ii. 1. 96. "Speak," however, in R. and J. iii. 1. 158, "Spake him fair" means" speak to: " but in the same expression M. of V. iv. 1. 271 it means "speak of." Similarly, "whisper" is often used without a preposition before a personal object.
" He came to whisper Wolsey."—Hen. VIII. i. 1. 179.
" They whisper one another in the ear."—K. J. iv. 2. 189.
" Your followers I will whisper to the business."
W. T. i. 2. 437.
Rarely, "whisper her ear."—M. Ado, iii. 1. 4.
In some cases, as in
"She will attend it better,"
T. N. i. 3. 27, 2. 453; M. of V. v. 4. 103.
the derivation may explain the transitive use.
"Despair thy charm,"—Macbeth, v. 8. 13.
is, perhaps, a Latinism. So "sympathise," meaning "suffer with," is used thus:
" The senseless brands will sympathise
The heavy accent of thy moving tongue."
Rich. II. v. 1. 47.
"Deprive," meaning " take away a thing from a person," like "rid," can dispense with "of" before the impersonal object.
"'Tis honour to deprive dishonour'd life."—R. of L. 1186.
This explains how we should understand—
"Which might deprive your sovereignty of 'reason."
Hamlet, i. 4. 73.
i.e. "which might take away your controlling principle of reason." So, perhaps, "Frees all faults. "—Tempest, Epilogue, 18.
This seems to have arisen from the desire of brevity. Compare the tendency to convert nouns, adjectives, and neuter verbs into active verbs (290).
201. The preposition was also omitted before the indirect object of some verbs, such as "say," "question," just as we still omit it after the corresponding verbs, "tell" and "ask."
" Sayest (to) me so, friend?"—T. of Sh. i. 2. 190.
" You will say (to) a beggar, nay."—Rich. III. iii. 1. 119.
" Still question'd (of) me the story of my life."—Othello, i. 3. 129.
In "Hear me a word,"—Rich. III. iv. 4. 180.
it must be a question whether me or word is the direct object. In
"I cry thee mercy,"—Rich. III. iv. 4. 515.
"mercy" is the direct object. This is evident from the shorter form
"(I) Cry mercy."—Rich. III. v. 3. 224.
After "give," we generally omit "to," when the object of "to" is a personal noun or pronoun. But we could not write—
" A bed-swerver, even as bad as these
That (to whom) vulgars (the vulgar) give bold'st titles."
W. T. ii. 1. 94.
" Unto his lordship, (to) whose unwished yoke
My soul consents not to give sovereignty."—M. N. D. i. 1. 81. Somewhat similar is
"This 'longs the text."—P. of T. ii. Gower, 40.
for "belongs (to) the text."
202. Preposition omitted in adverbial expressions of time, manner, &c.
"Forbear to sleep the nights, and fast the days."
Rich. III. iv. 4. 118.
This is illustrated by our modern
"(Of) What kind of man is he ?"—T. N. i. 5. 159.
"But wherefore do not you a mightier way Make war upon this bloody tyrant, time?"—Sonn. 16.
"My poor country
(Shall) More suffer, and more sundry ways, than ever."
Macbeth, iv. 3. 48 ; so Ib. i. 3. 154.
"Revel the night, rob, murder, and commit
The newest sins the newest kind of ways."—2 Hen. IV. iv. 5. 126.
"And ye sad hours that move a sullen pace."
B. and F. F. Sh. iv. 1.
"I will a round unvarnish'd tale deliver
Of my whole course of life; what drugs, what charms,
What conjuration, and what nightly magic
(For such proceeding I am charg'd withal)
I won his daughter."—Othello, i. 3. 91.
" How many would the peaceful city quit
To welcome him! Much more, and much more cause,
Did they this Harry."—Hen. V. v. Prol. 34.
"To keep Prince Harry in continual laughter the wearing out of six fashions, which is four terms."—2 Hen. IV. v. 1. 84.
"Why hast thou not served thyself into my table so many meals?" —Tr. and Cr. ii. 3. 45: i.e. "during so many meals."
"To meet his grace just distance 'tween our armies."
2 Hen. IV. iv. 1. 225.
"That I did suit me all points like a man."—A. Y. L. i. 3. 118.
" But were I not the better part made mercy."—Ib. iii. 1. 2.
" And when such time they have begun to cry."—Coriol. iii. 3. 19.
"Where and what time your majesty shall please."
Rich. III. iv. 4. 450.
"What time we will our celebration keep."—T. N. iv. 3. 30.
" A while they bore her up,
Which time she chanted snatches of old tunes."—Ham. iv. 7. 178.
In the following cases it would seem that a prepositional phrase is condensed into a preposition, just as "by the side of" (Chaucer, "byside Bathe") becomes "be-side," and governs an object.
"On this side Tiber. "— . C. iii. 2. 254.
"Fasten'd ourselves at either end the mast."—C. of E. i. 1. 86.
" A sheet of paper writ o' both sides the leaf."—L. L. L. v. 2. 8.
" On each side her the Bishops of London and Winchester." Hen. VIII. iv. 1 (order of coronation).
" She is as forward of our breeding as
She is in the rear our birth."—W. T. iv. 4. 522.
"Our purpose " seems to mean "for our purpose," in
"Not to know what we speak to one another, so we seem to know, is to know straight, our purpose: chough's language, gabble enough and good enough."—A. W. iv. 1. 21.
This seems the best punctuation. " Provided we seem to know what we say to one another, ignorance is exactly as good as knowledge, for our purpose."
Hence the use of this for "in this way" or "thus" is not so bold as it seems:
"What am I that thou shouldst contemn me this?
What were thy lips the worse for one poor kiss?"
V. and A. 203.
Perhaps, however, "contemn" is confused with "refuse." But this is used for "thus" in E. E.
All constantly repeated adverbial expressions have a tendency to abbreviate or lose their prepositions. Compare "alive" for "on live," "around" for "in round," "chance" for "perchance," "like " for "belike," &c. In some adverbial expressions the preposition can be omitted when the noun is qualified by an adjective, but not otherwise. Thus we can use "yester-day," "last night," "this week," adverbially, but not "day," "night," "week," because in the latter words there is nothing to indicate how time is regarded. In O. E. the inflections were sufficient to justify an adverbial use, "dayes," "nightes." (Compare νυκτóς.) But the inflections being lost, the adverbial use was lost with them.
203. Prepositions: transposed. (See also Upon.) In A.-S. and E. E. prepositions are often placed after their objects. In some cases the preposition may be considered as a separable part of a compound transitive verb. Thus in
"Ne how the Grekes with a huge route
Three times riden all the fire aboute,"—CHAUC. C. T. 2954.
"ride about" may be considered a transitive verb, having as its object "fire." Naturally, emphatic forms of prepositions were best suited for this emphatic place at the end of the sentence; and therefore, though "to," "tyll," "fro," "with," "by," "fore," were thus transposed, yet the longer forms, "untylle," "before," "ben-hind," "upon," "again," were preferred. Hence in the Elizabethan period, when the transposition of the weaker prepositions was not allowed, except in the compound words "whereto," "herewith," &c. (compare "se-cum, quo-cum") the longer forms are still, though rarely, transposed.
For this reason, "with," when transposed, is emphasized into "withal." The prepositions "after," "before," and "upon," are thus transposed by Shakespeare:
"God before."—Hen. V. i. 2. 307; iii. 6. 55, for "'fore God."
"Hasten your generals after. "—A. and C. ii. 4. 2.
So
" I need not sing this them until (unto)."—HEYWOOD.
" For fear lest day should look their shames upon."
M. N. D. iii. 2. 385.
"That bare-foot plod I the cold ground upon."—A. W. iii. 4. 6.
"For my good will is to't,
And yours it is against."—Tempest, iii. 1. 31.
The use of prepositions after the relative, which is now somewhat avoided, but is very common in E. E., is also common in Shakespeare, and is evidently better adapted to the metre than the modern idiom, as far as regards the longer forms. " Upon which " is not so easily metricized as
"Ten thousand men that fishes gnawed upon."—Rich. III. i. 4. 25.
"The pleasure that some fathers feed upon."—Rich. II. ii. 1. 79.
204. Prepositions transposed. "It stands me upon." This phrase cannot be explained, though it is influenced, by the custom of transposition. Almost inextricable confusion seems to have been made by the Elizabethan authors between two distinct idioms: (1) "it stands on" (adv.), or "at hand," or "upon " (comp. "instat," πρoσ κει), i.e. "it is of importance," "it concerns," "it is a matter of duty ;" and (2) "I stand upon" (adj.), i.e. "I in-sist upon."
In (1) the full phrase would be, "it stands on, upon, to me," but, owing to the fact that "to me" or " me " (the dative inflection) is unemphatic, and "upon" is emphatic and often used at the end of the sentence, the words were transposed into "it stands me upon." "Me" was thus naturally mistaken for the object of upon.
Hence we have not only the correct form—
"It stands me (dative) much upon (adverb)
To stop all hopes."—Rich. III. iv. 2. 59.
(So Hamlet, v. 2. 63, where it means "it is imperative on me."
But also the incorrect—
" It stands your grace upon to do him right."
Rich. II. ii. 3. 138.
"It only stands
Our lives upon to use our strongest hands." —A. and C. ii. 1. 51.
where "grace" and "lives" are evidently intended to be the objects of "upon," whereas the Shakespearian use of "me" (220) renders it possible, though by no means probable, that "me," in the first of the above examples, was used as a kind of dative.
Hence by analogy—
" It lies you on to speak."—Coriol. iii. 2. 52.
The fact that this use of upon in "stand upon is not a mere poetical transposition, but a remnant of an old idiom imperfectly understood, may be inferred from the transposition occurring in Elizabethan prose:
"Sigismund sought now by all means (as it stood him upon) to make himself as strong as he could."—NARES.
Perhaps this confusion has somewhat confused the meaning of the personal verb "I stand on." It means "I trust in" (M. W. of W. ii. 1. 242), "insist on" (Hen. V. v. 2. 93), and "I depend on" (R. and J. ii. 2. 93), and in
"The moist star
Upon whose influence Neptune's empire stands."
Hamlet, i. 1. 119.
# PRONOUNS.
205. Personal, Irregularities of (omission of, insertion of, see Relative and Ellipses). The inflections of Personal Pronouns are frequently neglected or misused. It is perhaps impossible to trace a law in these irregularities. Sometimes, however, euphony and emphasis may have successfully contended against grammar. This may explain I in "and I," "but I," frequently used for me. "'Tween you and I" seems to have been a regular Elizabethan idiom. The sound of d and t before me was avoided. For reasons of euphony also the ponderous thou is often ungrammatically replaced by thee, or inconsistently by you. This is particularly the case in questions and requests, where, the pronoun being especially unemphatic, thou is especially objectionable. To this day many of the Friends use thee invariably for thou, and in the Midland and North of England we have "wilta?" for "wilt thou?" Compare E. E. "wiltow?" for "wilt thou?" "þinkestow?" for "thinkest thou?" and similarly, in Shakespeare, thou is often omitted after a questioning verb. Again, since he and she could be used (see below) for "man" and "woman," there was the less harshness in using he for him and she for her. Where an objective pronoun is immediately followed by a finite verb, it is sometimes treated as the subject, as below, "no man like he doth grieve."
206. He for him :
"Which of he or Adrian, for a good wager, begins to crow ?"
Tempest, ii. 1. 28.
Some commentators insert "them" after "which of." (See 408.)
" I would wish me only he."—Coriol. i. 1. 236.
"And yet no man like he doth grieve my heart." R. and J. iii. 5. 84.
"From the first corse till he that died to-day."—Ham. i. 2. 104. where "till" is a preposition. See Prepositions, Till, 184.
207. He for him precedes its governing verb in the following examples :
"Thus he that over-ruled I over-sway'd."—V. and A. 109.
"and he my husband best of all affects."—M. W. of W. iv. 4. 87.
So probably he depends upon "within" in
"'Tis better thee without than he within."—Macbeth, iii. 3. 14.
208. Him for he.
Him is often put for "he," by attraction to "whom" understood, for "he whom."
"Him (he whom) I accuse
By this the city ports hath enter'd."—Coriol. v. 6. 6.
"Ay, better than him (he whom) I am before knows me."
A. Y. L. i. 1. 46.
"When him (whom) we serve's away"—A. and C. iii. 1. 15.
"Your party in converse, him (whom) you would sound,
He closes with you," &c.—Hamlet, ii. 1. 42.
Sometimes the relative is expressed :
" His brother and yours abide distracted—but chiefly him that you term'd Gonzalo."—Temp. v. i. 14.
Sometimes he is omitted:
" Whom I serve above is my master."—A. W. ii. 3. 26.
" To (him to) whom it must be done."—J. C. ii. 2. 331.
In "Damn'd be him,"—Macbeth, v. 8. 34.
perhaps let, or some such word, was implied.
209. I for me (for euphony : see 205):
"Here's none but thee and I."—2 Hen. VI. i. 2. 69.
"All debts are cleared between you and I."—M. of V. iii. 2. 321.
"You know my father hath no child but I."—A. Y. L. i. 1. 46.
"Unless you would devise some virtuous lie
And hang some praise upon deceased I."—Sonn. 72.
The rhyme is an obvious explanation of the last example. But, in all four, I is preceded by a dental.
So " Which may make this island
Thine own for ever, and I, thy Caliban,
For aye thy foot-licker."—Temp. iv. 1. 217.
210. Me for I :
"No mightier than thyself or me."—J. C. i. 3. 76.
"Is she as tall as me?"—A. and C. iii. 3. 14.
Probably than and as were used with a quasi-prepositional force.
211. She for her:
"Yes, you have seen Cassio and she together."—O. iv. 2. 3
"So saucy with the hand of she here—what's her name ?"
A. and C. iii. 13. 98.
She was more often used for "woman" than "he" for "man." Hence, perhaps, she seemed more like an uninflected noun than "he" and we may thus extenuate the remarkable anomaly
"Praise him that got thee, she that gave thee suck."
Tr. and Cr. ii. 3. 252.
212. Thee for thou. Verbs followed by thee instead of thou have been called reflexive. But though "haste thee," and some other phrases with verbs of motion, may be thus explained, and verbs were often thus used in E. E., it is probable that "look thee," "hark thee," are to be explained by euphonic reasons. Thee, thus used, follows imperatives which, being themselves emphatic, require an unemphatic pronoun. The Elizabethans reduced thou to thee. We have gone further, and rejected it altogether. (See 205.)
"Blossom, speed thee well."—W. T. iii. 3. 46.
"Look thee here, boy."—Ib. 116.
"Run thee to the parlour."—M. Ado, iii. 1. 1.
"Haste thee."—Lear, v. 3. 251.
"Stand thee by, friar."—M. Ado, iv. 1. 24.
"Hark thee a word."—Cymb. i. 5. 32.
"Look thee, 'tis so."—T. of A. iv. 3. 530.
"Come thee on."—A. and C. iv. 7. 16.
"Now, fellow, fare thee well."—Lear, iv. 6. 41.
"Hold thee, there's my purse."—A. W. iv. 5. 46; . C. v. 3. 85.
" Take thee that too."—Macbeth, ii. 1. 5.
In the two latter instances thee is the dative.
Thee is probably the dative in
"Thinkst thee?"—Hamlet, v. 2. 63.
or, at all events, there is, perhaps, confusion between "Thinks it thee?" i.e. "does it (E.E.) seem to them?" and "thinkst thou ?" Very likely "thinkst" is an abbreviation of "thinks it." (See 297.) Compare the confusion in
"Where it thinkst best unto your royal selfe."
Rich. III. iii. 1. 63 (Folio).
213. Thee for thou is also found after the verb to be, not merely in the Fool's mouth :
"I would not be thee, nuncle."—Lear, i. 4. 204.
but also Timon :
"I am not thee."—T. of A. iv. 3. 277.
and Suffolk :
" It is thee I fear."—2 Hen. VI. iv. 1. 117.
where thee is, perhaps, influenced by the verb, "I fear," so that there is a confusion between "It is thou whom I fear" and "Thee I fear." In these cases thee represents a person not regarded as acting, but about whom something is predicated. Hence thou was, perhaps, changed to thee according to the analogy of the sound of he and she, which are used for "man" and "woman."
214. Them for they:
"Your safety, for the which myself and them
Bend their best studies."—K. . iv. 2. 50.
Perhaps them is attracted by "myself," which naturally suggests the objective "myself and (they) them(selves)."
215. Us for we in "shall's." "Shall" (315), originally meaning necessity or obligation, and therefore not denoting an action on the part of the subject, was used in the South of England as an impersonal verb. (Compare Latin and Greek.) So Chaucer, "us oughte," and we also find "as us wol," i.e. "as it is pleasing to us." Hence in Shakespeare
"Say, where shall's lay him?"—Cymb. iv. 3. 233.
"Shall's have a play of this?"—Ib. v. 5. 28.
" Shall's attend you there?"—W. T. i. 2. 178.
"Shall's to the Capitol?"—Coriol. iv. 6. 148.
216. After a conjunction and before an infinitive we often find I, thou, &c., where in Latin we should have "me," "te," &c. The conjunction seems to be regarded as introducing a new sentence, instead of connecting one clause with another. Hence the pronoun is put in the nominative, and a verb is, perhaps, to be supplied from the context.
" What he is indeed
More suits you to conceive than I (find it suitable) to speak of."
A. Y. L. i. 2. 279.
i.e. "than that I should speak of it."
"A heavier grief could not have been imposed
Than I to speak my griefs unspeakable."—C. of E. i. 1. 33.
"The soft way which thou dost confess
Were fit for thee to use as they to claim."—Coriol. iii. 2. 83.
" Making night hideous, and we fools of nature
So horridly to shake our disposition."—Hamlet, i. 4. 54.
"Heaven would that she these gifts should have,
And I to live and die her slave."—A. Y. L. iii. 2. 162.
Sometimes the infinitive is implied, but not expressed:
"To beg of thee it is my more dishonour
Than thou of them."—Coriol. iii. 2. 125.
I, thou, and he, are also used for me, thee, and him, when they stand quasi-independently at some distance from the governing verb or preposition.
" But what o' that ? Your majesty and we that have free souls, it touches us not."—Hamlet, iii. 2. 252.
"I shall think the better of myself and thee during my life; 1 for a valiant champion, and thou for a true prince."—1 Hen. IV. ii. 4. 300.
"(God) make me that nothing have with nothing griev'd,
And thou with all pleas'd that hast all achieved."
Rich. II. iv. 1. 217.
"With that same purpose-changer, that sly devil,
That daily break-vow, he that wins of all."—K. J. ii. 1. 568.
" Now let me see the proudest,
He that dares most, but wag his finger at thee."
Hen. VIII. v. 3. 131.
(To punctuate, as in the Globe, "the proudest he," is intolerably harsh.)
"Justice, sweet prince, against that woman there,
She whom thou gavest to me to be my wife,
That hath abused and dishonour'd me."—C. of E. v. 1. 198.
"Why, Harry, do I tell thee of my foes
Which art my near'st and dearest enemy,
Thou that art like enough," &c. ?—1 Hen. IV. iii. 2. 123.
217. His was sometimes used, by mistake, for's, the sign of the possessive case, particularly after a proper name, and with especial frequency when the name ends in s. This mistake arose in very early times. The possessive inflection's (like the dative plural inflection um) was separated by scribes from its noun. Hence after the feminine name "Guinivere," we have in the later text of LAYAMON, ii. 511, "for Gwenayfer his love." The h is no more a necessary part of this separate inflection than it is of "his," the third pers. sing. indic. pres. of "beon" ("be"). "His" is constantly found for "is" in Layamon. No doubt the coincidence in sound between the inflection's and the possessive "his" made the separation seem more natural, and eventually confused's with his.
"Mars his sword . . . nor Neptune's trident nor Apollo's bow."
B. J. Cy.'s Rev. i. 1.
Also, by analogy,
"Pallas her glass."—BACON, Adv. of L. 278.
This is more common with monosyllables than with dissyllables, as the's in a dissyllable is necessarily almost mute. Thus
"The count his gallies."—T. N. iii. 3. 26.
"Mars his true moving."—1 Hen. VI. i. 2. 1.
So Tr. and Cr. iv. 5. 176, 255, &c.
"Charles his gleeks."—1 Hen. VI. iii. 2. 123.
but never, or very rarely, "Phœbus his."
The possessive inflection in dissyllables ending in a sibilant sound is often expressed neither in writing nor in pronunciation.
"Marry, my uncle Clarence (Folio) angry ghost."
Rich. III. iii. 1. 144 ; ii. 1. 137.
"For justice sake."— . C. iv. 3. 19.
"At every sentence end."—A. Y. L. iii. 2. 144.
"Lewis" is a monosyllable in
"King Lewis his satisfaction all appear."—Hen. V. i. 2. 88.
His is used like "hie" (in the antithesis between "hic . . . ille ").
"Desire his (this one's) jewels and this other's house."
Macb. iv. 3. 80 ; M. of V. iii. 2. 54-5 ; Sonn. xxix. 5, 6.
This explains
"And, at our stamp, here o'er and o'er one falls :
He murder cries, and help from Athens calls."
M. N. D. iii. 1. 25.
His, being the old genitive of it, is almost always used for its.
218. His, her, &c. being the genitives of he, she (she in E. E. had, as one form of the nom., "heo," gen. "hire"), &c. may stand as the antecedent of a relative. Thus :
"In his way that comes in triumph over Pompey's blood."
. C. i. 1. 55.
i.e. "in the way of him that comes."
" Love make his heart of flint that you shall love."—T. N. i. 5. 305.
"Unless her prayers whom heaven delights to hear."—A. W.iii.4.27.
"If you had known . . . her worthiness that gave the ring."
M. of V. v. 1. 200.
"Armies of pestilence, and they shall strike
Your children yet unborn and unbegot
That lift your vassal hands against my head."
Rich. II. iii. 2. 89.
i.e. "the children of you who lift your hands."
"Upon their woes whom fortune captivates."
3 Hen. VI. i. 4. 115. So Lear, v. 3. 2.
"And turn our impress'd lances in our eyes
Which do command them."—Lear, v. 3. 50.
In " Alas, their love may be call'd appetite,
No motion of the liver, but the palate,
That suffer surfeit, cloyment and revolt,"—T.N. ii. 4. 100-2.
it seems better to take that as the relative to "them," implied in "their (of them)," rather than to suppose "suffer" to be the subjunctive singular (367), or that to be the relative to "liver" and "palate" by confusion. It is true that is not often so far from its antecedent, but the second line may be treated as parenthetical.
This is perhaps not common in modern poetry, but it sometimes occurs :
"Poor is our sacrifice whose eyes
Are lighted from above."—NEWMAN.
219. Your, our, their, &c., are often used in their old signification, as genitives, where we should use "of you," &c.
"We render you (Coriolanus) the tenth to be ta'en forth
At . . . your only choice."—Coriol. i. 9. 36.
i.e. "at the choice of you alone."
"To all our lamentation."—Coriol. iv. 6. 34.
i.e. "to the lamentation of us all."
"Have I not all their letters to meet me in arms ?"
1 Hen. IV. ii. 3. 28.
i.e. "letters from them all."
220. Me, thee, him, &c. are often used, in virtue of their representing the old dative, where we should use for me, by me, &c. Thus:
"I am appointed (by) him to murder you."—W. T. i. 2. 412.
"John lays you plots."—K. . iii. 4. 145.
This is especially common with me.
Me is indirect object in
"But hear me this."—T. N. v. 1. 123.
"What thou hast promis'd—which is not yet perform'd me."
Tempest, i. 2. 244.
We say "do me a favour," but not "to do me business."—Tempest, i. 2. 255.
" Give me your present to one Master Bassanio."
M. of V. ii. 2. 115.
"Who does me this?"—Hamlet, ii. 2. 601.
"Sayest thou me so?"—2 Hen. VI. ii. 1. 109.
Me seems to mean "from me" in
"You'll bear me a bang for that."— . C. iii. 2. 20.
"with me" in
"And hold me pace in deep experiment."—1 Hen. IV. iii. 1. 48.
Me means "to my injury" in
"See how this river comes me cranking in,
And cuts me, from the best of all my land,
A huge half-moon."—1 Hen. IV. iii. 1. 100.
"at my cost" and "for my benefit" in
" The sack that thou hast drunk me could have bought me lights as good cheap at the dearest chandler's in Europe."—1 Hen. IV. iii. 3. 50.
Me in narrative stands on a somewhat different footing :
"He pluck'd me ope his doublet."— . C. i. 2. 270.
"He steps me to her trencher."—T. G. of V. iv. 4. 9.
"The skilful shepherd peel'd me certain wands."
M. of V. i. 3. 85.
" He presently, as greatness knows itself,
Steps me a little higher than his vow."—1 Hen. IV. iv. 3. 75.
Falstaff, when particularly desirous of securing the attention of the Prince ("Dost thou hear me, Hal?"), indulges twice in this use of me.
"I made me no more ado, ... I followed me close."
1 Hen. IV. ii. 4. 233, 241.
Here, however, the verbs are perhaps used reflexively, though this would seem to be caused by the speaker's intense desire to call attention to himself. So in
"Observe me judicially, sweet sir ; they had planted me three
demi-culverins,"—B. J. E. in &c. iii. 2.
the me seems to appropriate the narrative of the action to the speaker, and to be equivalent to "mark me," "I tell you." In such phrases as
"Knock me here,"—T. of Sh. i. 2. 8.
the action, and not merely the narrative of the action, is appropriated.
You is similarly used for "look you :"
"And 'a would manage you his piece thus, and come you in and come you out."—2 Hen. IV. iii. 2. 304.
In "Study me how to please the eye indeed
By fixing it upon a fairer eye,"—L. L. L. i. 1. 80.
me probably means "for me," "by my advice," i.e. "I would have you study thus." Less probably, "study" may be an active verb, of which the passive is found in Macb. i. 4. 9.
There is a redundant him in
"The king, by this, is set him down to rest."—3 Hen. VI. iv. 3. 2. where there is, perhaps, a confusion between "has set him(self) down" and "is set down."
Her seems used for "of her," "at her hands," in
"I took her leave at court."—A. W v. 3. 79.
i.e. "I bade her farewell."
Us probably is used for " to us" in
" She looks us like
A thing made more of malice than of duty."—Cymb. iii. 5. 32.
But possibly as "look" in Hen. V. iv. 7. 76, A. and C. iii. 10. 53, is used for " look for," so it may mean " look at." So
"Twa brooks in which I look myself."—B. J. Sad Sh. ii. 1.
i.e. "I view myself."
Us seems equivalent to "for us" in
" We have not spoke us yet of torch-bearers."
M. of V. ii. 4. 5.
i.e. "spoken for ourselves about torch-bearers."
221. Your, like "me" above (Latin, iste), is used to appropriate an object to a person addressed. Lepidus says to Antony :
"Your serpent of Egypt is lord now of your mud by the operation of your sun : so is your crocodile."—A. and C. ii. 7. 29.
Though in this instance the your may seem literally justified, the repetition of it indicates a colloquial vulgarity which suits the character of Lepidus. So Hamlet, affecting madness :
" Your worm is your only emperor for diet ; your fat king and your lean beggar is but variable service."—Hamlet, iv. 3. 24.
Compare
" But he could read and had your languages."—B. J. Fox, ii. I.
i.e. " the languages which you know are considered important."
So : "I would teach these nineteen the special rules, as your punto, your reverso, your stoccata, your imbroccato, your passada, your montanto."—Bobadil, in B. J. E. in &c. iv. 5.
Hence the apparent rudeness of Hamlet is explained when he says to the player :
"But if you mouth it as many of your players do."—Ham. iii. 2. 3. i.e. "the players whom you and everybody know."
222. Our is used, like "my," vocatively:
" Our very loving sister, well be -met,"—Lear, v. I. 20.
"Tongue-tied our queen, speak thou."—W. T. i. 1. 27.
" Our old and faithful friend, we are glad to see you."
M. for M. v. 1. 2.
In all these cases our is used in the royal style, for "my," by a single speaker referring merely to himself.
223. Him, her, me, them, &c. are often used in Elizabethan, and still more often in Early English, for himself, herself, &c.
" How she opposes her (sets herself ) against my will."
T. G. of V. iii. 2. 26.
"My heart hath one poor string to stay it by."—K. . v. 6. 55. "And so I say I'll cut the causes off
Flattering me with impossibilities."—3 Hen. VI. iii. 2. 143.
224. He and she are used for "man" and "woman."
"And that he
Who casts to write a living line must sweat."
B. J. on Shakespeare.
" I'll bring mine action on the proudest he
That stops my way in Padua."—T. of Sh. iii. 2. 236.
" Lady, you are the cruellest she alive. "-T. N. i. 5. 259.
" I think my love as rare
As any she belied with false compare."—Sonn. 130.
"That she belov'd knows nought that knows not this."
Tr. and Cr. i. 2. 314.
" With his princess, she
The fairest I have yet beheld."—W. T. v. I. 86.
" Betwixt two such shes." "—Cymb. i. 6. 40 ; ib. i. 3. 29.
This makes more natural the use of "he that," with the third person of the verb, in
" Are not you he
That frights the maidens? "—M. N. D. ii. I. 34.
So A. Y. L. iii. 2. 411.
225. Pronoun for pronominal adjective. The pronominal adjectives his, their, being originally possessive inflections of he, they, &c., were generally used in E. E. possessively or subjectively, i.e. " his wrongs " would naturally mean then " the wrongs done by him," not "to him." Hence, for objective genitives, "of" was frequently introduced, a usage which sometimes extended to subjective genitives. Hence
"The kindred of him hath been flesh'd upon us."—Hen. V. ii. 4.50.
"Tell thou the lamentable tale of me."—Rich. II. v. I. 44.
"The native mightiness and fate of him."—Hen. V. ii. 4. 64.
"Against the face of them."—Psalm xxi. 12.
It is used, perhaps, for antithesis in
" Let her be made
As miserable by the death of him
As I am made by my poor lord and thee."
Rich. III. i. 2. 21.
" O world, thou wast the forest to this heart,
And this indeed, 0 world, the heart of thee."
. C. iii. I. 208.
226. It is sometimes used indefinitely, as the object of a verb, without referring to anything previously mentioned, and seems to indicate a pre-existing object in the mind of the person spoken of.
" Courage, father, fight it out. "-3 Hen. VI. i. 4. 10.
i.e. " the battle."
" Ber. She never saw it.
King. Thou speak'st it falsely."—A. W. v. 3. 113.
i.e. " what thou sayest. "
" Dangerous peer,
That smooth'st it so with king and commonweal."
2 Hen. VI. ii. I. 22.
where it = " matters."
"To revel it with him and his new bride." (So C. of E. iv. 4 66.) —3 Hen. VI. iii. 3. 225.
i.e. "to take part in the intended bridal revels."
"I cannot daub it further." "—Lear, iv. I. 54.
i.e. " continue my former dissembling."
But it is often added to nouns or words that are not generally used as verbs, in order to give them the force of verbs.
" Foot it."—Tempest, i. 2. 380.
"To queen it."—Hen. VIII. ii. 3. 37.
"To prince it."—Cymb. iii. 4. 85.
" Lord Angelo dukes it well."—M. for M. iii. 2. 100.
And, later,
" Whether the charmer sinner it or saint it,
If folly grow romantic, I must paint it."
POPE, Moral Essays, ii. 15.
The use of it with verbs is now only found in slang phrases.
227. It is sometimes more emphatically used than with us. We have come to use it so often superfluously before verbs that the emphatic use of it for "that" before "which" is lost.
" There was it
For which my sinews shall be stretched upon him."
Coriol. v. 6. 44.
"That's it that always makes a good voyage of nothing."
T. N. ii. 4. 80.
" An if it please me which thou speak'st."—T. A. v. I. 59.
" It holds current that I told you of."—I Hen. IV. ii. I. 59.
So Isaiah (A. V. ) Ii. 9 : "Art thou not it that hath cut Rahab?"
Perhaps we must explain it as the antecedent of "what" (and not as in 226) in
"Deign it, Goddess, from my hand
To receive whate'er this land
From her fertile womb doth send."—B. and F. Fair Sh. i. 1.
228. Its was not used originally in the Authorized Version of the Bible, and is said to have been rarely used in Shakespeare's time. It is, however, very common in Florio's Montaigne. His still represented the genitive of It as well as of He. Its is found, however, in M. for M. i. 2. 4, where it is emphatic ; in W. T. i. 2 (three times, 151, 152, 266) ; Hen. VIII. i. 1.18 ; Lear, iv. 2. 32, and elsewhere. Occasionally it, an early provincial form of the old genitive, is found for its, especially when a child is mentioned, or when any one is contemptuously spoken of as a child. Ben Jonson (Sil. Wom. ii. 3) uses both forms—
" Your knighthood shall come on its knees."
And then, a few lines lower down—
" It knighthood shall fight all it friends."
Comp. W. T. iii. 2. 109 :
" The innocent milk in it most innocent mouth."
" The hedge-sparrow fed the cuckoo so long,
That it's had it head bit off by it young. "—Lear, i. 4. 235.
But also of an unknown person :
" The corse they follow did with desperate hand
Fordo it own life."—(Folio.) Hamlet, v. 1. 245.
"Woman it pretty self. "—(Folio.) Cymb. iii. 4. 160.
And of the ghost:
"It lifted up it head."—(Folio.) Hamlet, i. 2. 216.
Perhaps the dislike of its, even in the eighteenth century, aided the adoption of the French idiom " lever la tête."
"Where London's column, pointing at the skies,
Like a tall bully lifts the head and lies."
POPE, Moral Essays, iii. 340.
"It-selfe" is found referring to "who." (See 264.)
" The world who of it-selfe is peised well."—K. . ii. I. 575.
229. Her is very often applied by Shakespeare to the mind and soul.
" Whose soul is that which takes her heavy leave ?"
3 Hen. VI. ii. 6. 42.
"Since my dear soul was mistress of her choice."
Hamlet, iii. 2. 68.
So Rich. III. iii. 5. 28 ; Hamlet, ii. 2. 580.
" Our mind partakes
Her private actions to your secrecy. "—P. of T. i. 1. 153.
So Montaigne, 117.
The former passage from Hamlet shows the reason of this. The soul, when personified, is regarded as feminine, like Psyche. The body of a woman is also thus personified in
" And made thy body bare
Of her two branches, those sweet ornaments."—T. A. ii. 4. 18.
Milton occasionally uses its; often her for its; seldom, if ever, his for its.
" His form had not yet lost
All her original brightness. "—Milton, P. L. i. 592.
In this, and some other passages, but not in all, Milton may have been influenced by the Latin use of the feminine gender. " Form " represents "forma," a feminine Latin noun.
Personification will explain
" That Tiber trembled underneath her banks."
. C. i. I. 50.
230. Ungrammatical remnants of ancient usage . In Chaucer and earlier writers, preference is expressed, both by our modern "I had, or would, rather (i.e. sooner)," and by " (To) me (it) were lever (German lieber)," i.e. "more pleasant." These two idioms are confused in the following example :
" Me rather had my heart might feel your love."
Rich. II. iii. 3. 192.
In the earliest writers "woe !" is found joined with the dative inflection of the pronoun, "woe is (to) us," "woe is (to) me."
" Wa worthe (betide) than monne (the man, dat.)."
LAYAMON, i. 142.
As early as Chaucer, and probably earlier, the sense of the inflection was weakened, and "woe" was used as a predicate : "I am woe," " we are woe," &c. Hence Shakespeare uses " sorrow " thus. Similarly our "I am well" is, perhaps, an ungrammatical modification of " well is me," Ps. cxxviii. 2 (Prayer-book). In Early English both constructions are found. In Anglo-Saxon, Mätzner "has only met with the dative construction."
"I am sorrow for thee."—Cymb. v. 5. 297.
"I am woe for't, sir."—Temp. v. I. 139.
" Woe is my heart."—Cymb. v. 5. 2.
" Woe, woe are we, sir."—A. and C. iv. 14. 133.
On the other hand,
" Woe is me."—Hamlet, iii. I. 168.
" Woe me."—M. for M. iv. I. 26.
Similarly, the old "(to) me (it) were better," being misunderstood, was sometimes replaced by " I were better."
" I were better to be eaten to death. "—2 Hen. IV. i. 2. 245.
"I were best to leave him."—1 Hen.VI . v. 3. 82.
"Poor lady, she were better love a dream."—T. N. i. 2. 27.
" Thou'rt best."—Tempest, i. 2. 366.
And when the old idiom is retained, it is generally in instances like the following:
"Answer truly, you were best."— . C. iii. 3. 15.
" Madam, you're best consider. "—Cymb. iii. 2. 79.
where you may represent either nominative or dative, but was almost certainly used by Shakespeare as nominative.
231. Thou and You. Thou in Shakespeare's time was, very much like "du" now among the Germans, the pronoun of (1) affection towards friends (2) good-humoured superiority to servants, and (3) contempt or anger to strangers. It had, however, already fallen somewhat into disuse, and, being regarded as archaic, was naturally adopted (4) in the higher poetic style and in the language of solemn prayer.
(1) This is so common as to need no examples. It should be remarked, however, that this use is modified sometimes by euphony (the ponderous thou, art, and terminations in est being avoided) and sometimes by fluctuations of feeling. Thus in the T. G. of V. Valentine and Proteus in the first twenty lines of earnest dialogue use nothing but thou. But as soon as they begin to jest, "thou art" is found too seriously ponderous, and we have (i. 1. 25) "you are over boots in love," while the lighter thee is not discarded in (i. 1. 28) "it boots thee not." So in the word-fencing of lines 36-40, you and your are preferred, but an affectionate farewell brings them back again to thou. The last line presents an apparent difficulty :
" Proteus. All happiness bechance to thee in Milan !
Valentine. As much to you at home, and so farewell."
T. G. of V. i. 1. 61-2.
But while thee applies to the single traveller, you is better suited to Proteus and his friends at home. It may be added, that when the friends meet after their long parting, there is a certain coldness in the frequent you. (T. G. of V. ii. 5. 120.)
Fathers almost always address their sons with thou; sons their fathers with you. Thus in the dialogue between Henry IV. and the Prince (l Hen. IV. iii. 2), line 118, " What say you ?" is perhaps the only exception to the rule. So in the dialogue between Talbot and his son (1 Hen. VI. iv. 5) before the battle. In the excitement of the battle (1 Hen. VI. iv. 6. 6-9) the son addresses his father as thou: but such instances are very rare. (A. Y. L. ii. 3. 69 is a rhyming passage, and impassioned also.) A wife may vary between thou and you when addressing her husband. Lady Percy addresses Hotspur almost always in dialogue with you : but in the higher style of earnest appeal in I Hen. IV. ii. 3. 43-67, and in the familiar "I'll break thy little finger, Harry," ib. 90, she uses thou, throughout.
In the high Roman style, Brutus and Portia use you.
Hotspur generally uses thou to his wife, but, when he becomes serious, rises to you, dropping again to thou.
" Hotspur. Come, wilt thou see me ride?
And when I am o' horse-back, I will swear
I love thee innnitely——But hark you, Kate ;
I must not have you henceforth question me :
This evening must I leave you, gentle Kate.
I know you wise ; but yet no further wise
Than Harry Percy's wife: constant you are,
But yet a woman: and for secrecy
No lady closer——For I well believe
Thou wilt not utter what thou dost not know;
And so far will I trust thee, gentle Kate."
I Hen. IV. ii. 3. 103-115.
Mark the change of pronoun as Bassanio assumes the part of a friendly lecturer:
"Gra. I have a suit to you.
Bass. You have obtain'd it.
Gra. You must not deny me ; I must go with you to Belmont.
Bass. Why, then you must.—But hear thee, Gratiano ;
Thou art too wild, too rude and bold of voice," &c.
M. of V. ii. 2. 187-90.
232. Thou is generally used by a master to a servant, but not always. Being the appropriate address to a servant, it is used in confidential and good-humoured utterances, but a master finding fault often resorts to the unfamiliar you (much as Cæsar cut his soldiers to the heart by giving them the respectful title of Quirites). Thus Valentine uses you to Speed in T. G. of V. ii. I. 1-17, and thou, Ib. 47-69. Compare
" Val Go to , sir: tell me, do you know madam Silvia?"—Ib. 14. with
" Val. But tell me : dost thou know my lady Silvia ? "—Ib. 44.
Similarly to the newly-engaged servant Julia, who says "I'll do what I can," Proteus blandly replies :
"I hope thou wilt. [To Launce.] How now, you whore-
son peasant,
Where have you been these two days loitering?"
T. G. of V. iv. 4. 48.
When the appellative "sir" is used, even in anger, thou generally gives place to you.
" And what wilt thou do ? Beg, when that is spent?
Well, sir, get you in."—A Y L. i. 1. 79, 80.
"Ay, ay, thou wouldst begone to join with Richmond :
I will not trust you, sir."—Rich. III. iv. 4. 492.
Compare " Speak, what trade art thou ? "— . C. i. 1. 5.
with " You, sir, what trade are you ?"—Ib. 9.
This explains the change from thou to you in Tempest, i. 2. 443. Throughout the scene Prospero, addressing Ferdinand as an impostor, "speaks ungently" with thou. In Tempest, v. I. 75-79, Prospero, who has addressed the worthy Gonzalo in the friendly thou, and the repentant Alonso in the impassioned thou, turning to his unnatural brother says,
" Flesh and blood
You brother mine,"
but, on pronouncing his forgiveness immediately afterwards, he says,
"I do forgive thee,
Unnatural though thou art."
So
"For you, most wicked sir, whom to call brother Would even infect my mouth, I do forgive
Thy rankest fault."-Tempest, v. I. 230-2.
" Worthy sir, thou bleed'st."—Coriol. i. 5. 15.
is easily explained by the admiring epithet "worthy." Compare Ib. 24 : "Bold gentleman, prosperity be thy page."
The difference between thou and you is well illustrated by the farewell addressed by Brutus to his schoolfellow Volumnius, and his servant Strato :
" Farewell to you ; and you ; and you, Volumnius ;
Farewell to thee, too, Strato."— C. v. 5. 33.
Compare also the farewell between the noble Gloucester and Edgar "dressed like a peasant :"
" Edg. Now fare you well, good sir."—Lear, iv. 6. 32.
" Glouc. Now, fellow, fare thee well. "—Ib. 41.
It may seem an exception that in sc. iv. 1, Edgar uses thou to Gloucester, but this is only because he is in the height of his assumed madness, and cannot be supposed to distinguish persons. Afterwards, in sc. vi., he invariably uses you—a change which, together with other changes in his language, makes Gloucester say:
" Thou speak'st
In better phrase and manner than thou didst."—Lear, iv. 6. 8.
It may be partly this increased respect for Edgar, and partly euphony, which makes Gloucester use you in ll. 10 and 24.
Thus Clarence to the Second Murderer:
" Clar. Where art thou, keeper? Give me a cup of wine.
Sec. Murd. You shall have wine enough, my lord, anon.
Clar. In God's name, what art thou?
Sec. Murd. A man, as you are.
Clar. How darkly and how deadly dost thou speak!
Your eyes do menace me: why look you pale?
Who sent you hither? Wherefore do you come?"
Rich. III. i. 4. 167–176.
The last two lines seem discrepant: but they are not. Clarence is addressing both murderers, and both reply:
"Both. To, to, to——
Clar. To murder me?
Both. Ay, ay."
Afterwards, when the murderers reproach Clarence with his faults, they address him as thou.
233. Thou towards strangers who were not inferiors was an insult. "If thou thouest him some thrice, it shall not be amiss," (T. N. iii. 2. 48,) is the advice given to Sir Andrew Aguecheek when on the point of writing a challenge.
In addressing Angelo, whose seat he occupies, the Duke in the following passage begins with ironical politeness, but passes into open contempt:
" Duke (to Escalus). What you have spoke I pardon; sit you down;
We'll borrow place of him. (To Angelo.) Sir, by your leave,
Hast thou or word or wit or impudence,
That now can do thee office ?"—M. for M. v. 1. 358.
Thou is also used in a contemptuous " aside."
" Hastings. 'Tis like enough for I stay dinner there.
Buckingham (aside). And supper too, although thou know'st
it not.
Come, will you go?"—Rich. III. iii. 2. 122.
And, where there is no contempt, Cassius passes into thou when he addresses Brutus absent, whereas in his presence he restricts himself to you ( . C. i. 2. 311). The former is the rhetorical, the latter the conversational pronoun. So
" Be thou my witness,
You know that I held Epicurus strong."— C. v. 1. 74-7.
This explains the apparent liberty in
"O wise young judge, how I do honour thee!"
M. of V. iv. 1. 224.
234. Thou is often used in statements and requests, while you is used in conditional and other sentences where there is no direct appeal to the person addressed. Similarly the somewhat archaic ye is distinguished by Shakespeare from you by being used in rhetorical appeals. (See Ye, 236.)
Come thou on my side, and entreat for me
As you would beg, were you in my distress."
Rich. III. i. 4. 273.
" But tell me now
My drown'd queen's name, as in the rest you said
Thou hast been god-like perfect."—P. of T. v. 1. 208.
" I go, and if you plead as well to them
As I can say nay to thee for myself."—Rich. III. iii. 7. 52.
" Give me thy hand, Messala;
Be thou my witness that against my will, &c.
You know that I held Epicurus strong."— . C. v. 1. 74–7.
235. Thou. Apparent exceptions.
" If he be leaden, icy-cold, unwilling,
Be thou so too, and so break off your talk."
Rich. III. iii. 1. 177.
Here "your talk" means the talk between "thee and him."
In Homlet, i. 2. 41-49, the King, as he rises in his profession of affection to Laertes, passes from you to thou, subsequently returning to you.
In the following instance a kiss induces the speaker to pass from your to thou :
" Goneril. Decline your head. (Kisses Edmund.) This kiss,
if it durst speak,
Would raise thy spirits up into the air. "—Lear, iv. 2. 23.
The most difficult passage is :
"If thou beest not immortal, look about you"— . C. ii. 3. 8, 9.
In this short scene Cæsar is six times addressed by the soothsayer in the solemn and prophetic thou and thee, but once, as above, you. I can only suggest that "look about you" may mean "look about you and your friends."
In almost all cases where thou and you appear at first sight indiscriminately used, further considerations show some change of thought, or some influence of euphony sufficient to account for the change of pronoun.
The French Herald addresses Henry V. as thou, not for discourtesy (Hen. V. iv. 7. 74), but in the "high style" appropriate between heralds and monarchs. Few subjects would address their lords as thou. Only a Caliban addressing his Stephano would in the ordinary language say:
"Good my lord, give me thy favour still."—Temp. iv. 1. 204. Caliban almost always thou's unless he is cursing (Temp. i. 2. 363), or when he is addressing more than one person.
236. Ye. In the original form of the language ye is nominative, you accusative. This distinction, however, though observed in our version of the Bible, was disregarded by Elizabethan authors, and ye seems to be generally used in questions, entreaties, and rhetorical appeals. Ben Jonson says: "The second person plural is for reverence sake to some singular thing." He quotes—
"O good father dear,
Why make ye this heavy cheer?"—GOWER.
Compare:
" I do beseech ye, if you bear me hard."— . C. iii. 1. 157.
"You taught me how to know the face of right,
And come ye now to tell me John hath made
His peace with Rome ?"—K. . v. 2. 91.
"The more shame for ye; holy men I thought ye."
Hen. VIII. iii. 1. 102.
"Therein, ye gods, you make the weak most strong." " I' the name of truth,
. C. i. 3. 91.
"I' the name of truth,
Are ye fantastical? ... My noble partner
You greet with present grace."—Macbeth, i. 3. 53-55.
Ye and your seem used indiscriminately in Temp. v. 1. 33-8, "Ye elves ... and ye that ... you demi-puppets ... and you whose pastime is, &c."
The confusion between you and ye is illustrated by the irregularity of the following:
"What mean you ... do ye not know? ... If, therefore, at the first sight ye doe give them to understand that you are come hither ... do you not think? Therefore, if you looke . . ."—N. P. 170.
Sometimes ye seems put for you when an unaccented syllable is wanted:
"I never loved you much; but I ha' prais'd ye."
A. and C. ii. 6. 78.
and perhaps in
" Ye shall, my lord,"—Rich. III. iv. 2. 86.
the "shall" being emphatic, and ye unemphatic, but the Folio varies here, as frequently in this play.
237. Mine, my. Thine, thy. The two forms, which are interchangeable in E. E. both before vowels and consonants, are both used by Shakespeare with little distinction before vowels.
Though there are probably many exceptions, yet the rule appears to be that mine and thine are used where the possessive adjective is to be unemphatic, my and thy in other cases.
Mine is thus used before words to which it is so frequently prefixed as to become almost a part of them, as "mine host" (M. W. of W. i. 3. 1), but my in the less common
"Unto my hostess of the tavern."—1 Hen. IV. i. 2. 53.
So we have almost always " mine honour," the emphatic
"By my honour
He shall depart untouched,"—J. C. iii. 1. 141.
being an exception. Mine is almost always found before "eye," "ear," &c. where no emphasis is intended. But where there is antithesis we have my, thy:
"My ear should catch your voice, my eye your eye."
M. N. D. i. 1. 188.
and also in the emphatic
"To follow me and praise my eyes and face."—M. N. D. iii. 2. 223.
Euphony would dictate this distinction. The pause which we are obliged to make between my, thy, and a following vowel, serves for a kind of emphasis. On the other hand, mine, pronounced "min," glides easily and unemphatically on to the following vowel.
238. Mine, hers, theirs, are used as pronominal adjectives before their nouns. That mine should be thus used is not remarkable, as in E. E. it was interchangeable with my, and is often used by Shakespeare where we should use my.
"Mine and my father's death come not upon thee."
Hamlet, v. 2. 341.
"The body is dead upon mine and my master's false accusation." —M. Ado, v. 1. 249. So P. of T. i. 2. 92; Cymb. v. 5. 230.
In the following, mine is only separated by an adjective from its noun "And his and mine lov'd darling."—Tempest, iii. 3. 93.
More remarkable are
" What to come is yours and my discharge."—Temp. ii. 1. 253.
"By hers and mine adultery."—Cymb. v. 5. 186.
'Even in theirs and in the commons' ears."—Coriol. v. 6. 4.
It is felt that the ear cannot wait till the end of the sentence while so slight a word as her or their remains with nothing to depend on. The same explanation applies to mine, which, though unemphatic immediately before its noun, is emphatic when separated from its noun.
239. This of yours is now, as in E. E., generally applied to one out of a class, whether the class exist or be imaginary. We could say "this coat of yours," but not (except colloquially) "this head of yours." It is, however, commonly used by Shakespeare where even the conception of a class is impossible.
"Nor scar that whiter skin of hers than snow."—Othello, v. 2. 4.
" Will not a calf-skin stop that mouth of thine?"—K. J. iii. 1. 299.
"This of hers, thine," &c. seem used as an adjective, like the Latin "iste." "This mouth of you" was felt to be harsh, the "you" being too weak to stand in such a position. "This your mouth" requiring a forced and unnatural pause after "this," was somewhat more objectionable to Shakespeare, than to the Latin style of Milton and Addison. Hence "this of you" was used but modified. It is rare that we find such a transposition as
'O then advance of yours that phraseless hand."—L. C. 225.
240. Pronouns transposed. A feeling of the unemphatic nature of the nominatives we and they prevents us from saying "all we."
"Into the madness wherein now he raves
And all we mourn for."—Hamlet, ii. 2. 151.
So "all we "in the A. V. of the Bible, and "all they," Mark xii. 44.
"Find out" is treated as a single word in
"Cass. Cinna, where haste you so?
Cinna. To find-out you."— . C. i. 3. 134.
So
"To belch-up you."—Tempest, iii. 3. 56.
"And leave-out thee."—Rich. III. i. 3. 216.
"Both they (i.e. both of them)
Match not the high perfection of my loss."—Ib. iv. 4. 65.
No modern poet would be allowed to write, for the sake of rhyme,
"All days are nights to see till I see thee,
And nights bright days when dreams do show thee me."
Sonn. 43.
We could only say "give him me," when we meant "give him, not to so-and-so, but to me," emphatically, which is not the meaning here.
241. Omission of Thou. (See also 399, 402.) After a verb ending with the second person singular inflection, the thou is sometimes omitted in questions, as:
"Didst not mark that?"—Othello, ii. 1. 260.
"How dost that pleasant plague infest ? "—DANIEL.
"Wilt dine with me, Apemantus ?"—T. of A. i. 1. 206.
Thou is often omitted after "wouldst," or perhaps merged, in the form "woo't," as "wilt thou" becomes "wilta."
"Noblest of men, woo't die?"—A. and C. iv. 15. 59.
"Woo't weep? Woo't fight?... I'll do it"—Hamlet, v. 1. 299.
Sometimes thou is inserted:
" Woo't thou fight well?"—A. and C. iv. 2. 7.
242. Insertion of Pronoun. When a proper name is separated by an intervening clause from its verb, then for clearness (see 248) the redundant pronoun is often inserted.
"Sueno, albeit he was of nature verie cruell, yet qualified he his displeasure."—HOLINSHED, Duncane.
" Demeratus—when on the bench he was long silent . . . one asking him ... he answered."—B. J. Disc. 744.
"For the nobility, though they continued loyal unto him, yet did they not co-operate with him."—B. E.
243. Insertion of Pronoun. Even where there is no intervening conjunctional clause, the pronoun is frequently inserted after a proper name as the subject. More rarely, the subject is a common noun. Still more rarely, the pronoun is inserted after the object.
The subject or object stands first, like the title of a book, to call the attention of the reader to what may be said about it. In some passages the transition may be perceived from the exclamatory use
"O thy vile lady!
She has robbed me of my sword,"—A. and C. iv. 14. 22.
to the semi-exclamation:
"For God he knows."—Rich. III. iii. 7. 236; 1. 10; 1. 26.
" Where Heaven he knows how we shall answer him." K. J. v. 7. 59.
(So T. G. of V. iv. 4. 112, and
"God, I pray him."—Rich. III. i. 3. 212.
The object (as in the last example) precedes in
" My sons, God knows what has bechanced them." 3 Hen. VI. i. 4. 6.
" Senseless trees they cannot hear thee,
Ruthless beasts they will not cheer thee."—P. P. 393.)
and hence to passages of simple statement:
"The skipping king he ambled up and down."
1 Hen. IV. iii. 2. 60.
" Of six preceding ancestors that gem
Conferr'd by testament to the sequent issue
Hath it been owed and worn."—A. W. v. 3. 198.
"But this same Cassio, though he speak of comfort
Touching the Turkish loss, yet he looks sadly."
Othello, ii. 1. 31.
But many such passages of simple statement may be regarded as abridgments of the construction with "for," "of," or some other preposition:
"For your intent ... it is most retrograde to our desires."
Hamlet, i. 2. 112.
"For my voice, I have lost it with halloing and singing of anthems."—2 Hen. IV. i. 2. 213.
So "For (as regards) your brother, he shall go with me," might become
"Your brother he shall go along with me."
A. W. iii. 6. 117 ; Rich. II. ii. 2. 80; 1 Hen. IV. ii. 4. 442.
So "Of Salisbury, who can report of him?"—2 Hen. VI. v. 3. 1.
# RELATIVE PRONOUNS.
244. Omission of the Relative. The relative is frequently omitted, especially where the antecedent clause is emphatic and evidently incomplete. This omission of the relative may in part have been suggested by the identity of the demonstrative that and the relative that:—
"We speak that (dem.) that (rel.) we do know,' may naturally be contracted into—
"We speak that we do know."
Thus—
"And that (that) most deeply to consider is
The beauty of his daughter."—Temp. iii. 2. 106.
"Thy honourable metal may be wrought
From that (to which) it is disposed."— . C. i. 2. 314.
"Now follows that (that) you know, young Fortinbras," &c.
Hamlet, i. 2. 17.
"And that (that) is worse—the Lords of Ross are fled."
Rich. II. ii. 2. 52.
i.e. "which is worse." So often in the A. V. of the Bible, " that is, being interpreted," meins "which is" (as the Greek shows), though a modern reader would suppose that to be the demonstrative.
In many cases the antecedent immediately precedes the verb to which the relative would be the subject.
" I have a brother (who) is condemned to die."
M. for M. ii. 2. 33; C. of E. v. 1. 283.
"I have a mind (which) presages."—M. of V. i. 1. 175.
"The hate of those (who) love not the king."
Rich. II. ii. 2. 128.
"In war was never lion (that) raged more fierce."
Ib. ii. 1. 173.
"And sue a friend (who) 'came debtor for my sake."
Sonn. 139.
"What wreck discern you in me (that)
Deserves your pity?"—Cymb. i. 6. 84; W. T. iv. 4. 378, 512.
"You are one of those (who)
Would have him wed again."—W. T. v. 1. 23.
"I'll show you those (who) in troubles reign,
Losing a mite, a mountain gain. "—P. of T. ii. Gower, 8.
"Of all (who have) 'say'd (tried) yet, may'st thou prove prosperous." —P. of T. i. 1. 59.
"And they are envious (that) term thee parasite."—B. J. Fox, i. 1.
"For once (when) we stood up about the corn, he himself stuck not to call us the many-headed multitude."
Coriol. ii. 3. 16.
i.e. "On one occasion (on which) we stood up," &c. Compare—
"Was it not yesterday (on which) we spoke together?"
Macbeth, iii. 1. 74.
"Off with his head,
And rear it in the place (in which) your father's stands."
3 Hen. VI. ii. 6. 86.
"Declare the cause
(for which) My father, Earl of Cambridge, lost his head."
1 Hen. VI. ii. 5. 55.
"O that forc'd thunder (that) from his breath did fly !—
O that sad breath (that) his spongy lungs bestow'd! "
L. C. 46.
"And being frank she lends to these (who) are free."
Sonn. 4.
So explain:
"To me (whom) you cannot reach you play the spaniel."
Hen. VIII. v. 2. 126.
"That's to you sworn (that) to none was ever said."
L. C. 25. So M. for M. iii. 2. 165.
Most of these examples (except those in which when and why are omitted) omit the nominative. Modern usage confines the omission mostly to the objective. "A man (whom) I saw yesterday told me," &c. We must either explain thus :
"Myself and Toby
Set this device against Malvolio here (which device),
Upon some stubborn and discourteous parts,
We had conceiv'd against him,"—T. N. v. 1. 370.
or suppose (more probably), that there is some confusion between. "conceiving enmity" and "disliking parts."
In "To her own worth
She shall be prized: but that you say 'Be' t so,'
I'll speak it in my spirit and honour 'No,'"
Tr. and Cr. iv. 4. 136.
that probably means " as to that which."
Other instances are:
"My sister ... a lady, sir (who), though it was said she much resembled me, was yet of many accounted beautiful."—T. N. ii. 1. 27.
"What should I do (that) I do not?"—A. and C. i. 3. 8.
"Of every virtue (that) gives renown to men."—P. of T. i. 1. 13.
Either a relative or a nominative (see 399) is omitted in
"These are my mates that make their wills their law
(Who) have some unhappy passenger in chace."
T. G. of V. v. 4. 15.
In "And curse that justice did it,"—Coriol. i. 1. 179.
either the relative is omitted after "justice," or "that" is used for "because" (284).
So, after disobeying King Cymbeline by allowing Posthumus to speak to the King's daughter, the Queen, while purposing to betray Posthumus, says aside :
"Yet I'll move him (the king)
To walk this way : I never do him (the king) wrong
But he (who, like Posthumus) does buy my injuries to be friends,
Pays dear for my offences."—Cymb. i. 1. 105.
The relative adverb where is omitted in
"From that place (where) the morn is broke
To that place (where) day doth unyoke. "—B. and F. F. Sh. i. 1.
That, meaning "when," is omitted after "now." (See 284.)
245. The Relative is omitted (as well as the verb "is," "are," &c.) between a pronominal antecedent and a prepositional phrase, especially when locality is predicated.
"And they in France of the best rank and station."
Hamlet, i. 3. 129.
"He made them of Greece (i.e. the Grecians) to begin warre." —N. P. 175.
So "What is he at the gate ?"—T. N. i. 5. 125.
So in Early English and Anglo-Saxon. We make the same omission, but only after nouns : "The babes in the wood."
246. The Relative is omitted in the following example, and the antecedent is attracted into the case which the relative, if present, would have :
"Him (he whom) I accuse,
By this, the city ports hath enter'd."—Coriol. v. 6. 6.
Apparently there is an ellipsis of "that (relative) is" before participles in the following:
"Not that devour'd, but that which doth devour,
Is worthy blame,"—R. of L. 451.
where "that devour'd" seems used for "that that is devour'd."
"Why have you not proclaim'd Northumberland,
And all the rest (that are) revolted, faction-traitors?"
Rich. II. ii. 2. 57.
And in
"I hate the murderer, love him murdered,"
Rich. II. v. 5. 40.
the meaning seems to be, not" I love the fact that he is murdered," but "I love him (who is) murdered." Compare the harsh construction in
"But you must know your father lost a father,
That father (who was) lost, lost his."—Hamlet, i. 2. 90.
"A little riper and more lusty red
Than that (which is) mixed in his cheek."
A. Y. L. iii. 5. 222.
The relative is attracted to a subsequent implied object in the following :
" Thou shalt not lack
The leaf of eglantine, whom not to slander,
Outsweetened not thy breath."—Cymb. iv. 2. 223.
i.e. "the leaf which, not to slander it, would not outsweeten," &c.
247. The Relative (perhaps because it does not signify by inflection any agreement in number or person with its antecedent) frequently (1) takes a singular verb, though the antecedent be plural, and (2) the verb is often in the third person, though the antecedent be in the second or first.
(1) "All things that belongs" (so Folio; Globe, belong).—T. of Sh. ii. 1. 357.
" Whose wraths to guard you from,
Which here in this most desolate isle else falls
Upon your head."—Temp. iii. 2. 80.
"Contagious fogs which falling on our land
Hath every pelting river made so proud."—M: N. D. ii. 1. 91.
This, however, might be explained by 337.
"'Tis not the many oaths that makes the truth."
A. W. iv. 2. 21; K. J. ii. 1. 216.
"With sighs of love that costs the fresh blood dear."
M. N. D. iii. 2. 97.
" My observations
Which with experimental seal doth warrant
The tenour of my book."—M. Ado, iv. 1. 168.
"'Tis your graces that charms."—Cymb. i. 6. 117.
"So, so, so : they laugh that wins" (Globe, win).
Othello, iv. 1. 125.
" So are those crisped snaky golden locks
Which makes."—M. of V. iii. 2. 92.
" Those springs
In chalic'd flowers that lies."—Cymb. ii. 3. 24.
" Each substance of a grief hath twenty shadows
Which shows like grief itself."—Rich. II. ii. 2. 15.
"It is not words, that shakes me thus."—Othello, iv. 1. 43.
" But most miserable
Is the desires that's glorious." (Globe, " desire.")
Cymb. i. 6. 6.
"'Tis such fools as you
That makes the world full of ill-favour'd children."
A. Y. L. iii. 5. 53.
"(The swords) That makes such waste in brief mortality."
Hen. V. i. 2. 28.
"There are some shrewd contents in yon same paper
That steals the colour from your cheeks.—M. of V. iii. 2. 246.
" Is kindling coals that fires all my heart."—3 Hen. VI. ii. 1. 83.
" With such things else of quality and respect
As doth import you. "—thello, i. 3. 283.
"Such commendations as becomes a maid."—I Hen. VI. v. 3. 177.
" Such thanks as fits a king's remembrance. "—Hamlet, ii. 2. 26.
" Like monarch's hands that lets not bounty fall."
L. C. 41 (Globe, let).
"If it be you (you gods) that stirs these daughters' hearts."
Lear, ii. 4. 275 (Globe, stir).
"To be forbod the sweets that seems so good."
L. C. 164 (Globe, seem).
The distance of the relative from the antecedent sometimes makes a difference, as in
"I that please some, try all, both joy and terror
Of good and bad, that makes and unfolds error.
W. T. iv. 1. 2.
This construction is found as late as 1671 :
"If it be true that monstrous births presage
The following mischiefs that afflicts the age."
The Rehearsal, Epilogue.
(2)
"Antiochus, I thank thee who hath taught."—P. of T. i. 1. 41.
" Casca, you are the first that rears your hand. "—J. C. iii. 1. 30
"Rears his" or "rear your" would be right.
"To make me proud that jests."—L. L. L. v. 2. 66.
"For it is you that puts us to our shifts."—T. A. iv. 2. 176.
So Temp. v. 1. 79.
" O Lord, that lends me life !"—2 Hen. VI. i. 1. 9.
" They do but greatly chide thee who confounds."—Sonn. 8.
The last two examples may also be explained (see 340) by the northern inflection of s for st : and the examples in (1) might come under the cases of plural nominative with apparently singular inflection considered in 333. But taking all the examples of (1) and (2) we are, I think, justified in saying that the relative was often regarded like a noun by nature third person singular, and, therefore, uninfluenced by the antecedent.
On the other hand, the verb is irregularly attracted into the second person in
" That would I learn of you
As one that are best acquainted with her person."
Rich. III. iv. 4. 268.
248. Relative with Supplementary Pronoun. With the Germans it is still customary, when the antecedent is a pronoun of the first or second person, to repeat the pronoun for the sake of defining the person, because the relative is regarded as being in the third person. Thus "Thou who thou hearest," &c. The same repetition was common in Anglo-Saxon (and in Hebrew) for all persons. "That (rel.) through him " = "through whom, " "a tribe that they can produce " = " a tribe who can produce," &c.
Hence in Chaucer, Prol. 43-45 :
" A knight ther was, and that a worthy man,
That, from the tymë that he first began
To ryden out, he lovede chyvalrye ; "
and in the same author "that his" = "whose," "that him" = "whom," &c.
In the same way in Elizabethan authors, when the interrogative who (251) had partially supplanted that as a relative, we find who his for whose, whom him for whom, which it for which, &c.
The following is probably not a case of the supplementary pronoun :
"Bardolph and Nym had ten times more valour than this roaring devil i' the old play, that every one may pare his nails with a wooden dagger."—Hemy V. iv. 4. 76.
That... his is not elsewhere used in Shakespeare, that I know of. The above probably means "than this (fellow, who is) a mere devil-in-the-play, so that every one may beat him."
249. The Supplementary Pronoun is generally confined to cases (as above, 242) where the relative is separated from its verb by an intervening clause, and where on this account clearness requires the supplementary pronoun.
" Who, when he lived, his breath and beauty set
Gloss on the rose, smell on the violet."—V. and A.
" Which, though it alter not love's sole effect,
Yet doth it steal sweet hours from love's delight."
Sonn. 36.
" And who, though all were wanting to reward,
Yet to himself he would not wanting be."—B. J. Cy.'s Rev.
"Whom,
Though bearing misery, I desire my life
Once more to look on him."—W. T. v. 1. 138.
" (The queen) whom Heavens in justice both on her and hers
Have laid most heavy hand."—Cymb. v. 5. 464.
Here the construction is further changed by the addition of "both ... and hers."
"You are three men of sin whom Destiny
(That hath to instrument this lower world,
And what is in't) the never-surfeited sea
Hath caused to belch up you."—Temp. iii. 2. 53.
In the following passage the which may almost with better right be regarded as supplementary than the noun which follows :
" Our natural goodness
Imparts this; which if you or stupified
Or seeming so in skill, cannot or will not
Relish a truth like us, inform yourselves
We need no more of your advice."—W. T. ii. 1. 165.
Here which means "as regards which," and in this and in other places it approximates to that vulgar idiom which is well known to readers of "Martin Chuzzlewit." (See 272.)
The following seems at first as though it could be explained thus; but "who" is put for "whom" (see 274), and "exact the penalty" is regarded as a transitive verb :
"Who, if he break, thou may'st with better face
Exact the penalty."—M. of V. i. 3. 137.
Or this may be an imitation of the Latin idiom which puts the relative before the conjunction, thus :
"Who, when they were in health, I tell thee, herald,
I thought upon one pair of English legs
Did walk three Frenchmen."—Hen. V. iii. 6. 157.
250. Which that.
"Spite of his spite which that in vain
Doth seek to force my fantasy."—INGELEND (A.D. 1560).
This use of which that consecutively is common in Chaucer, but not in Elizabethan authors. When it is remembered that which was originally an interrogative, it is easier to understand how that may have been added to give a relative force to which.
251. Who and what. In Early English who was the masc. or fem. and what the neut. interrogative (or used as the indefinite relative who-so, what-so), that being both the demonstrative and relative, except in the oblique cases.
The transition of the interrogative to the relative can easily be explained. Thus, the sentence
" O now who will behold
The royal captain of this ruin'd band?
Let him cry 'Praise and glory on his head,"'
Hen. V. iv. Prologue.
may easily become "now let him who will behold," &c.
We can now only use who-ever in this sense, but the Germans still use their interrogative (wer) thus. In such cases the who mostly retains a trace of its interrogative meaning by preceding the antecedent clause :
" Who steals my purse (he) steals trash,"—Othello, iii. 3. 157.
and hence referring to a definite past :
" Who was the thane (he) lives yet."—Macbeth, i. 3. 109.
In this and other examples (as in Greek) the antecedent pronoun is often omitted owing to the emphatic position of the relative.
" Whom we raise we will make fast. "—2 Hen. VI. i. 4. 25.
"Is proclamation made that who finds Edward
Shall have a high reward ?"—3 Hen. VI. v. 3. 9.
"Fixing our eyes on whom our care was fixed."
C. of E. i. 1. 85.
"We are going to whom it must be done."—J. C. ii. 1. 331.
252. What, being simply the neuter of the interrogative who, ought consistently to be similarly used. As, therefore, who is used relatively, we may expect what to be used so likewise. And so it is; but, inasmuch as the adjective which very early took the force of the relative pronoun, what was supplanted by which, and is rarely used relatively. Even when it is thus used, it generally stands before its antecedent (like the transitional use of who above), thereby indicating its interrogative force, though the position of the verb is altered to suit a statement instead of a question.
"What our contempt doth often hurl from us
We wish it ours again."—A. and C. i. 2. 127. So Rich. II. i. 1. 87.
"What you have spoke it may be so perchance."
Macbeth, iv. 3. 11.
" Look, what I speak, my life shall prove it true."
Rich. II. i. 1. 87.
"It is true that what is settled by custom, though it be not good, yet at least it is fit. "—B. E. 99
An unemphatic antecedent precedes what in
"And I do fearfully believe 'tis done
What we so feared he had a charge to do."—K. J. iv. 1. 75.
I cannot remember any instance where what has for its antecedent a noun, as in the modern vulgarism, "The man what said." In
"And let us once again assail your ears,
That are so fortified against our story,
What we have two nights seen."—Hamlet, i. 1. 33.
What depends on a verb of speech, implied either in "assail your ears " or in "story," i.e. "let us tell you what we have seen," or "our story describing what we have seen."
The antecedent was mostly omitted:
" What is done (that) cannot be undone."—Macb. v. 1. 74.
This use is common now, but we could not say
" To have his pomp and all what (that which) state compounds." T. of A. iv. 2. 35.
The following is a curious use of what:
" That Julius Cæsar was a famous man :
With what his valour did enrich his wit
He did set down to make his valour live."
Rich. III. iii. 1. 85 : i.e. "(that) with which."
253. What is used for "for what," " why " (quid), as in
" What (why) shall I don this robe and trouble you?"
Cymb. iii. 4. 34.
" What need we any spur but our own cause?"
J. C. ii. 1. 123.
" What shall I need to draw my sword?"—T. A. i. 1. 189.
" What should I stay?"—A. and C. v. 2. 317.
and in some other passages where the context shows this to be the meaning:
" Falstaff. This apoplexy is, as I take it, a kind of lethargy.
Justice. What tell you me of it: be it as it is."
2 Hen. IV. i. 2. 130.
The following use of what for " in what state," i.e. "how far advanced," should be noticed :
"M. What is the night ?
Lady M. Almost at odds with morning, which is which."
Macbeth, iii. 4. 126.
These adverbial uses of what are illustrated by
" His equal mind I copy what I can
And, as I love, would imitate the man."
POPE, Imit. Hor. ii. 131.
254. What = "whatever."
" What will hap more to-night, safe scape the king,"
Lear, iii. 6. 121.
where the construction may be "Happen what will," a comma being placed after "will," or "Whatever is about to happen." Probably the former is correct and "will" is emphatic, "hap" being optative.
What = "whoever."
" There's my exchange. What in the world he is
That names me traitor, villain-like he lies. "—Lear, v. 3. 97.
What is often used apparently with no sense of "of what kind or quality" where we should use who, especially in the phrase "what is he?"
" Chief Justice. What's he that goes there?
Servant. Falstaff, an't please your lordship."
2 Hen. IV. i. 2. 66.
" What's he that wishes so? My cousin Westmoreland?" Hen. V. iv. 3. 18.
"Ros. What is he that shall buy his flock and pasture?
Cor. That young swain."—A. Y. L. ii. 4. 88-9.
" Captain. He did see the love of fair Olivia !
Vio. What's she?
Captain. A virtuous maid, the daughter of a count."
T. N. i. 2. 35 ; ib. i. 5. 124.
So Lear, v. 3. 125 ; Macbeth,, v. 7. 2 ; Rich. II. v. 5. 69.
But in the Elizabethan and earlier periods, when the distinction between ranks was much more marked than now, it may have seemed natural to ask, as the first question about anyone, " of what condition or rank is he ?" In that case the difference is one of thought, not of grammar.
255. What hence in elliptical expressions assumes the meaning "any."
" I love thee not a jar of the clock behind
What lady-she (224) her lord."—W. T. i. 2. 44.
i.e. "less than any lady whatsoever loves her lord." So
"With promise of his sister and what else."
3 Hen. VI. iii. 1. 51 ; Tempest, iii. 1. 72.
i.e. "whatever else may be conceived," or "everything else."
" What not " is still used in this sense, as
" He that dares approach
On him, on you, who not? I will maintain
Mine honour firmly."—Lear, v. 3. 100: i.e. "on everybody."
Like the Latin "qua—qua," so " what—what " is used for "partly—partly," mostly joined to "with." In this collocation perhaps the alliteration of the two w's has had some influence : for what is not thus used except before "with."
" And such a flood of greatness fell on you
What with our help, what with the absent king,
What with the injuries of a wanton time."
1 Hen. IV. v. 1. 50.
So Tr. and Cr. v. 1. 103.
Originally this may have been "considering what accrued from our help, what from the king's absence," &c. but "what" is used by Spenser in the sense of "part," "her little what." (See p. 5.)
256. What is sometimes used before a noun without the appended indefinite article in exclamations. (See Article, 86.) It is also used without a noun in this sense :
'O father Abram, what these Christians are ! "
M. of V. i. 3. 162.
" What mortality is !"—Cymb. iv. 1. 16.
i.e. "what a thing mortality is !"
257. Who for any one :
"The cloudy messenger turns me his back
And hums as who should say, 'You'll rue the time
That clogs me with this answer.'"—Macbeth, iii. 6. 42.
"He doth nothing but frown, as who should say, 'If you will
not have me, choose.'"—M. of V. i. 2. 45.
Comp. M. of V. i. 1. 93, Rich. II. v. 4. 8. In these passages it is possible to understand an antecedent to 'who,' "as, or like (one) who should say." But in the passages
"Timon surnamed Misantropos (as who should say Loup-garou, or the man-hater)."—N. P. 171.
" She hath been in such wise daunted
That they were, as who saith, enchanted."
GOWER, C. A. 1. (quoted by Clarke and Wright).
it is impossible to give this explanation. And in Early Eng. (Morris, Specimens, p. xxxii.) "als wha say" was used for "as any one may say." Comp. the Latin quis after si, num, &c. Possibly an if is implied after the as by the use of the subjunctive. (See 107.)
Littré explains "comme qui dirait " by supplying "celui." "Il portait sur sa teste comme qui dirait un turban; c'est-à-dire, il portait, comme dirait celui qui dirait un turban." But this explanation seems unsatisfactory, in making a likeness to exist between "carrying" and "saying." But whatever may be the true explanation of the original idiom, Shakespeare seems to have understood who as the relative, for the antecedent can be supplied in all passages where he uses it, as J. C. i. 2. 120, "As who goes farthest."
258. That, which, who, difference between. Whatever rule may be laid down for the Elizabethan use of the three relative forms will be found to have many exceptions. Originally that was the only relative; and if Wickliffe's version of the New Testament be compared with the versions of the sixteenth century and with that of 1611, that will be found in the former replaced by which and who in the latter, who being especially common in the latest, our Authorized Version. Even in Shakespeare's time, however, there is great diversity of usage. Fletcher, in the Faithful Shepherdess (with the exception of a few lines containing the plot, and probably written by Beaumont), scarcely uses any relative but the smooth that throughout the play (in the first act which is only used once) ; and during the latter half of the seventeenth century, when the language threw off much of its old roughness and vigour, the fashion of Wickliffe was revived. That came into favour not because, as in Wickliffe's time, it was the old-established relative, but because it was the smoothest form : the convenience of three relative forms, and the distinctions between their different shades of meaning, were ignored, and that was re-established in its ancient supremacy. Addison, in his "Humble Petition of Who and Which," allows the petitioners to say: "We are descended of ancient families, and kept up our dignity and honour many years, till the jack-sprat That supplanted us." But the supplanting was a restoration of an incapable but legitimate monarch, rather than a usurpation. Since the time of Addison a reaction has taken place ; the convenience of the three distinct forms has been recognized, and we have returned somewhat to the Elizabethan usage.
259. As regards the Shakespearian use, the following rules will generally hold good :—
(1) That is used as a relative (a) after a noun preceded by the article, (b) after nouns used vocatively, in order to complete the description of the antecedent by adding some essential characteristic of it.
(2) Who is used (a) as the relative to introduce a fact about the antecedent. It may often be replaced by "and he," "for he," "though he," &c. (b) It is especially used after antecedents that are lifeless or irrational, when personification is employed, but not necessarily after personal pronouns.
(3) Which is used (a) in cases where the relative clause varies between an essential characteristic and an accidental fact, especially where the antecedent is preceded by that; (b) where the antecedent is repeated in the relative clause; (c) in the form "the which," where the antecedent is repeated, or where attention is expressly called to the antecedent, mostly in cases where there is more than one possible antecedent and care is required to distinguish the real one; (d) where "which" means "a circumstance which," the circumstance being gathered from the previous sentence.
260. That. (a) Since that introduces an essential characteristic without which the description is not complete, it follows that, even where this distinction is not marked, that comes generally nearer to the antecedent than who or which.
"To think of the teen that I have turn'd you to
Which is from my remembrance ! "—Temp. i. 2. 65.
I to the world am like a drop of water
That in the ocean seeks another drop,
Who falling there to seek his fellow forth,
Unseen, inquisitive, confounds himself."—C.of E. i. 2. 37.
"You have oft enquired
After the shepherd that complain d of love,
Who you saw sitting by me on the turf."—A. Y. L. iii. 4. 52.
"And here's a prophet that I brought with me
From forth the streets of Pomfret, whom I found
With many hundreds treading on his heels. "—K.F. iv. 2.148.
The same order is preserved in A. Y. L. iii. 5. 13 ; 2 Hen. IV. i. 3. 59 ; Lear, iii. 4. 134-139 ; 2 Hen. VI. iv. 1. 3 ; Lear, iv. 2. 51-53 (where we find that, who, that, consecutively) ; Lear, iii. 7. 89, 90 ; 1 Hen. IV. ii. 1. 80 (that, the which, that); Tempest, iv. 1. 76.
The distinction between that and which is preserved in
" It is an heretic that (by nature, of necessity) makes the fire,
Not she which (as an accidental fact) burns in it."
W. T. ii. 3. 115.
" And he doth sin that doth belie the dead,
Not he which (as you do) says the dead is not alive."
2 Hen. IV. i. 1. 99.
In the latter passage "he that" = "who-so," and refers to a class, "he which" to the single person addressed. Thus Wickliffe (Matt. xxiii. 21) has "he that sweareth," whereas the other versions have "whoso" or "whosoever sweareth."
That is generally used after he, all, aught, &c. where a class is denoted. This is so common as not to require examples, and it is found even where that is objective.
"He that a fool doth very wisely hit."—A. Y. L. ii. 7. 53.
In "The great globe itself,
Yea, all which it inherit,"—Temp. iv. 1. 154.
euphony perhaps will not allow "that it." (See Which, 265.)
The following is not an exception :
"It was the swift celerity of his death,
Which I did think with slower foot came on,
That brain'd my purpose."—M. for M. v. 1. 400.
for here which is used parenthetically (see 271). So Rich. II. iii. 4. 50.
In "He that no more must say is listen'd more
Than they whom youth and ease have taught to glose."
Rich. II. ii. 1. 9, 10.
a distinction appears to be drawn between the singular nominative represented by the uninflected that, and the objective plural represented by the inflected whom.
261, That. (b) After nouns used vocatively.
"Hail, many-coloured messenger ! that ne'er
Dost disobey the wife of Jupiter :
Who with thy saffron wings upon my flowers
Diffusest honey-drops, refreshing showers."
Temp. iv. 1. 76-79.
"Hast thou conspired with thy brother, too,
That for thine own gain shouldst defend mine honour?"
K. F. i. 1. 242.
"Yon brother mine, that entertain'd ambition,
Expell'd remorse and nature ; who with Sebastian
Would here have kill'd your king."
Tempest, v. 1. 79 ; 33-9.
This close dependence of that on the antecedent, wherein it differs from who and which, is a natural result of its being less emphatic, and therefore less independent, than the two other forms. When the relative is necessarily emphatic, as at the end of a verse, we may sometimes expect that to be replaced by which, for that and no other reason.
"Sometimes like apes that mow and chatter at me,
And after bite me ; then like hedgehogs which
Lie tumbling in my bare-foot way."—Temp. ii. 2. 10.
262. That is sometimes, but seldom, separated from the antecedent, like who. (See 263.)
"As if it were Cain's jawbone that did the first murder." Hamlet, v. 1. 85.
It is perhaps not uncommon after the possessive case of nouns and pronouns. (See 218.) The antecedent pronoun is probably to be repeated immediately before the relative.
" Cain's jawbone, (him) that did," &c.
Less commonly as in
" They know the corn
Was not our recompense, resting well assured
That ne'er did service for it."—Coriol. iii. 1. 122.
The use of that for who = "and they" is archaic. Acts xiii. 43: "They sueden Paul and Barnabas that spakun and counceileden hym." Tyndale, Cranmer, and Geneva have which; Rheims and A. V. who.
263. Who (a) for "and he," "for he," &c.
"Now presently I'll give her father notice
Of their disguising and pretended flight ;
Who (and he), all enraged, will banish Valentine."
T. G. of V. ii. 6. 38.
"My name is Thomas Mowbray, duke of Norfolk,
Who (and I) hither come engaged by my oath
Against the duke of Norfolk that (because he) appeals me."
Rich. II. i. 3. 17.
" Caius Ligarius doth bear Caesar hard
Who (since he) rated him for speaking well of Pompey."
. C. ii. 1. 216.
Hence who is often at some distance from the antecedent.
"Archbishop. It was young Hotspur's case at Shrewsbury.
Lord Bardolph. It was, my lord : who (for he) lined himself with hope."—2 Hen. IV. i. 3. 27.
"To send the old and miserable king
To some retention and appointed guard,
Whose (for his) age has charms in it."—Lear, v. 3. 48.
"I leave him to your gracious acceptance; whose (for his) trial shall better publish his commendation."—M. of V. iv. 1. 165.
"In Ephesus I am but two hours old,
As strange unto your town as to your talk,
Who (and I), every word by all my wit being scann'd,
Want wit, in all, one word to understand."
C. of E. iii. 2. 153.
So Temp. iii. 1. 93 ; A. and C. i. 3. 29 ; Hen. V. i. Prologue, 33.
264. Who personifies irrational antecedents. (b) Who is often used of animals, particularly in similes where they are compared to men.
"I am the cygnet to this pale faint swan,
Who chants a doleful hymn to his own death."—K. . v. 7.22.
"Or as a bear encompass'd round with dogs,
Who having pinch'd a few and made them cry."
3 Hen. VI. ii. 1. 16.
So 1 Hen. IV. v. 2. 10 ; 2 Hen. VI. iii. 1. 253, v. 1. 153 ; but also in other cases where action is attributed to them, e.g.
"A lion who glared."— . C. i. 3. 21.
"A lioness who quickly fell before him."—A. Y. L. iv. 2. 13.
Who is also used of inanimate objects regarded as persons.
"The winds
Who take the ruffian billows by the tops."—2 Hen. IV. iii. 1. 22.
So R. and F. i. 1. 119 ; i. 4. 100 : "The winds . . . who."
"Rotten opinion, who hath writ me down
After my seeming."—2 Hen. IV. v. 2. 128.
"Night . . . who."—Hen. V. iv. Prol. 21.
" Your anchors, who
Do their best office if they can but stay you."—W. T. iv. 4. 581.
"A queen
Over her passion, who most rebel-like
Sought to be queen o'er her."—Lear, iv. 2. 16.
So probably in
"Your eye
Who hath cause to wet the grief on't."—Tempest, ii. 1. 127.
i.e. "your eye which has cause to give tearful expression to the sorrow for your folly."
"My arm'd knee
Who bow'd but in my stirrups."—Coriol. iii. 2. 119.
But is who the antecedent here to "me" implied in "my?" (See 218.)
"The heart
Who great and puff'd up with this retinue."
2 Hen. IV. iv. 3. 120.
So V. and A. 191 and 1043, "her heart . . . who ;" T. A. iii. 2. 9, "my breast . . . who."
The slightest active force, or personal feeling, attributed to the antecedent, suffices to justify who. Thus :
" The dispers'd air who answer'd."—R. of L. 1805.
"Applause
Who like an arch reverberates."—Tr. and Cr. iii. 3. 120.
" Therefore I tell my sorrows to the stones
Who though they cannot answer," &c.—T. A. iii. 1. 38.
" Bushes,
As fearful of him, part, through whom he rushes."
V. and A. 630.
So "her body . . . who," R. of L. 1740 ; "the hairs who wave," V. and A. 306 ; "lips who . . . still blush," R. and . iii. 3. 38 ; "sighs who," R. and . iii. 5. 136 ; "mouths who," P. of T. i. 4. 33 ; "palates who," P. of T. i. 4. 39 ; "her eyelids who like sluices stopped," V. and A. Sometimes who is used where there is no notion of personality:
" The world, who of itself is peised well,"—K. . ii. 1. 575.
where perhaps who is used because of the pause after "world," in the sense "though it." (See 263.) If there had been no comma between "world" and the relative, we should have had that or which.
Perhaps in this way we may distinguish in
" The first, of gold, who this inscription bears ;
The second, silver, which this promise carries."
M. of V. ii. 7. 4.
i.e. "the first of gold, and it bears this inscription ; the second, (silver,) which carries," &c. In the first the material, in the second the promise, is regarded as the essential quality. [Or does euphony prefer which in the accented, who in the unaccented syllables ?]
In almost all cases where who is thus used, an action is implied, so that who is the subject.
Whom is rare.
" The elements
Of whom your swords are temper'd."—Temp. iii. 2. 62.
265. Which (E. E. adj. hw-ilc, "wh(a)-like") is used interchangeably with Who and That. It is interchanged with who in
"Then Warwick disannuls great John of Gaunt,
Which did subdue the greatest part of Spain;
And, after that wise prince, Henry the Fifth,
Who by his power conquered all France."
3 Hen. VI. iii. 3. 87.
Like who (263), which implies a cause in
"Deposing thee before thou wert possess'd,
Which (for thou) art possess'd now to depose thyself."
Rich. II ii. 1. 108.
It is often used for that (see 261), where the personal antecedent is vocatively used or preceded by the article :
" The mistress which I serve."—Temp. iii. 1. 6.
So M. for M. v. 1. 305 ; W. T. i. 2. 455, v. 2. 60.
" Abhorred slave,
Which any point of goodness will not take."—Temp. i. 2. 352.
" And thou, great goddess Nature, which hast made it."
W. T. ii. 3. 104.
So in our version of the Lord's Prayer.
266. Which, like that, is less definite than who. Who indicates an individual, which a "kind of person;" who is "qui," which "qualis."
" I have known those which (qualis) have walked in their sleep who (and yet they, 263) have died holily in their beds."—Macb. v. 1. 66.
"For then I pity those I do not know
Which (unknown persons) a dismiss'd offence would after gall."
M. for M. ii. 2. 102.
"They have—as who have not, that their great stars
Throned and set high?—servants, who seem no less,
Which are to France the spies and speculations
Intelligent of our state."—Lear, iii. 1. 24.
Here "who seem no less" is parenthetical, and for who might be written "they." Which means "of such a kind that." Where "so dear," "such," &c. is implied in the antecedent, we may expect the corresponding which (278) in the relative :
"Antonio, I am married to a wife
Which is as dear to me as life itself."—M. of V. iv. 1. 283.
When the antecedent is personal and plural, which is generally preferred to who. Which, like that (260), often precedes who.
" I am Prospero, and that very duke
Which was thrust from Milan, who," &c.—Tempest, v. 1. 160.
267. The . . . that; that . . . which. In A.-S. " e" (the) was the relative and "se" the article. When the form " e" (the) became the article, "that" became the relative. In the same way it perhaps arises that when that was applied to the antecedent, the relative form preferred by Shakespeare was which. " The man that says" = "whoever says," and the indefinite that is sufficient; but " that man," being more definite, requires a more definite relative. After a proper name, who would answer the purpose ; but after " that man," that being an adjective, " which man" was the natural expression, which being originally also an adjective. Hence the marked change in
" If he sees aught in you that makes him like
That anything he sees which moves his liking."—K. . ii. 1. 52.
" When living blood doth in these temples beat
Which owe the crown that thou o'er-masterest."—Ib. ii. 1. 109.
Possibly "that" is a demonstrative, and "he" is used for "man" in the following, which will account for the use of which ; but more probably which is here used for that, and there is a confusion of constructions.
" Rather proclaim it, Westmoreland, through our host,
That he which hath no stomach to this fight,
Let him depart."—Hen. V. iv. 3. 34.
268. Which more definite than That. Generally it will be found that which is more definite than that. Which follows a name, that a pronoun:
" Here's the Lord Say which sold the towns in France ; he that made us pay one-and-twenty fifteens."—2 Hen. VI. iv. 5. 23.
Sometimes which is used in this sense to denote an individual or a defined class, while that denotes a hypothetical person or an indefinite class. Hence
" And such other gambol faculties a' has, that show a weak mind and an able body, for the which the Prince admits him."—2 Hen. IV. ii. 4. 74.
And compare
" She that was ever fair and never proud, &c.
She was a wight, if ever such wight were."—Othello, ii. 1. 149.
with " I find that she which late
Was in my nobler thoughts most base, is now
The praised of the king : who (263), so ennobled,
Is as'twere born so."—A. W. ii. 3. 179.
" It is a chance which does redeem all sorrows
That I have ever felt."—Lear, v. 3. 266.
Which states a fact, that a probability, in
"Why, Harry, do I tell thee of my foes,
Which art my near'st and dearest enemy ?
Thou that art like enough."—1 Hen. IV. iii. 2. 124.
In "Cut off the heads of too fast growing sprays
That look too lofty in our commonwealth :
You thus employ'd, I will go root away
The noisome weeds which, without profit, suck
The soil's fertility from wholesome flowers."—Rich. II. iii 4.37.
We must explain "all the heads that may happen to look too lofty, and the weeds which, as a fact, suck the fertility," &c.
So that introduces an essential, and which an accidental, or at all events a less essential quality, in the two following passages :—
"(Thou) commit'st thy anointed body to the cure
Of those physicians that first wounded thee."
Rich. II. ii. 1. 99.
" Now for our Irish wars.
We must supplant those rough, rug-headed kerns,
Which live like venom where no venom else,
But only they, have privilege to live."—Ib. 157.
That may state a fact with a notion of purpose :
"Now, sir, the sound that tells (i.e. to tell) what hour it is
Are clamorous groans which strike upon my heart,
Which is the bell."—Rich. II. v. 5. 57.
269. Which with repeated antecedent. Which being an adjective frequently accompanies the repeated antecedent, where definiteness is desired, or where care must be taken to select the right antecedent.
" Salisbury. What other harm have I, good lady, done
But spoke the harm that is by others done ?
Canstance. Which harm within itself so heinous is—"
K. F. iii. 1. 39.
"And, if she did play false, the fault was hers, K. F. iii. 1. 39.
Which fault lies," &c.—K. F. i. 1. 119 ; Rich. II. i. 1. 104.
This may sometimes explain why which is used instead of that, and why that is preferred after pronouns :
"Let my revenge on her that injured thee
Make less a fault which I intended not. "—F. Sh. v. 1.
An antecedent noun ("fault") can be repeated, and therefore can be represented by the relative which ; an antecedent pronoun "her" cannot.
Sometimes a noun of similar meaning supplants the antecedent:
" Might'st bespice a cup
To give mine enemy a lasting wink,
Which draught to me were cordial."—W. T. i. 2. 318
270. The Which. The above repetition is, perhaps, more common with the definite "the which" :
"The better part of valour is discretion ; in the which better part I have saved my life."—1 Hen. IV. v. 4. 125.
Sometimes the noun qualified by which is not repeated, and only slightly implied in the previous sentence :
"Under an oak . . . to the which place."—A. Y. L. ii. 1. 33.
"Let gentleness my strong enforcement be,
In the which hope I blush."—Ib. ii. 7. 119.
The question may arise why "the" is attached to which and not to who. (The instance
" Your mistress from the whom I see
There's no disjunction,"—W. T. iv. 4. 539.
is, perhaps, unique in Shakespeare.) The answer is, that who is considered definite already, and stands for a noun, while which is considered as an indefinite adjective ; just as in French we have "lequel," but not "lequi." "The which" is generally used either as above, where the antecedent, or some word like the antecedent, is repeated, or else where such a repetition could be made if desired. In almost all cases there are two or more possible antecedents from which selection must be made. (The use of "lequel" is similar.)
"To make a monster of the multitude, of the which (multitude) we being members should bring ourselves to be monstrous members." —Coriol. ii. 3. 10.
" Lest your justice
Prove violence, in the which (violence) three great ones suffer."
W. T. ii. 1. 128.
"Eight hundred nobles
In name of lendings for your highness' soldiers,
The which (nobles) he hath detain'd for lewd employments."
Rich. II. i. 1. 90.
"The which" is also naturally used after a previous "which."
"The present business
Which now's upon us : without the which this story
Were most impertinent."—Temp. i. 1. 138.
"The chain
Which God he knows I saw not, for the which
He did arrest me."—C. of E. v. 1. 230.
271. Which for "which thing," often parenthetically.
" Camillo,
As you are certainly a gentleman, thereto
Clerk-like experienced, which no less adorns
Our gentry, than our parents' noble names."—W. T. i. 2. 392.
Very often the " thing" must be gathered not from what precedes but from what follows, as in
" And, which became him like a prince indeed,
He made a blushing 'cital of himself."—I Hen. IV. v. 2. 62.
" And, which was strange, the one so like the other
As could not be distinguished."—C. of E. i. 1. 53.
That is rarely thus used by Shakespeare:
" And, that is worse,
The Lord Northumberland, his son young Henry Percy,
With all their powerful friends, are fled to him."
Rich. II. ii. 2. 55.
Often, however, in our A. V. that in "that is, being interpreted," is the relative, though a modern reader would not perceive it.
"I was never so berhymed since Pythagoras' time that (when) I was an Irish cat, which I can hardly remember."—A. Y. L. iii.2.188.
" I'll resolve you,
Which to you shall seem probable, of every
These happen'd accidents."—Temp. v. 1. 249.
i.e. "I will explain to you (and the explanation shall seem probable) every one of these accidents."
"My honour's at the stake, which (danger) to defeat
I must produce my power."—A. W. ii. 3. 156.
"Even as I have tried in many other occurrences, which Cæsar affirmed (ce que dit César), that often," &c.—MONTAIGNE, 36.
272. Which for "as to which." Hence which and " the which" are loosely used adverbially for "as to which." So in Latin, "quod" in "quod si."
" Showers of blood,
The which how far off from the mind of Bolingbroke
It is such crimson tempest should bedew," &c.
Rich. II. iii. 3. 45.
" With unrestrained loose companions—
Even such, they say, as stand in narrow lanes,
And beat our watch, and rob our passengers;
Which he, young, wanton, and effeminate boy,
Takes on the point of honour, to support
So dissolute a crew. "—Rich. II. v. 3. 10.
"But God be thanked for prevention;
Which I in sufferance heartily will rejoice."
Hen. V. ii. 2. 159.
273. Which. It is hard to explain the following:
" A mote will turn the balance which Pyramus which Thisbe is the better."—M. N. D. v. 1. 325. unless which is used for the kindred "whether."
In " My virtue or my plague, be it either which,"
Hamlet, iv. 7. 13.
there is perhaps a confusion between "be it either" and "be it whichever of the two." Perhaps, however, " either" may be taken in its original sense of "one of the two," so that " either which is "which-one-so-ever of the two."
274. Who for whom. The inflection of who is frequently neglected.
" Who I myself struck down."—Macheth. iii. 1. 123.
" Who does the wolf love ? The lamb."—Coriol. ii. 1. 8.
Compare W. T. iv. 4. 66, v. I. 109.
Apparently it is not so common to omit the m when the whom is governed by a preposition whose contiguity demands the inflection:
"There is a mystery with whom relation
Durst never meddle."—Tr. and Cr. iii. 3. 201.
Compare especially,
"Consider who the king your father sends,
To whom he sends."—L. L. L. ii. 1. 2.
The interrogative is found without the inflection even after a preposition:
"C. Yield thee, thief.
Gui. To who ? "—Cymb. iv. 2. 75 ; Othello, i. 2. 52.
"With who?"—Othello, iv. 2. 99.
And in a dependent question:
"The dead man's knell
Is there scarce asked for who."—Macbeth, iv. 3. 171.
In the following, who is not the object of the preposition:
"This is a creature . . . might make proselytes
Of who she but bid follow. "—W. T. v. 1. 109.
# RELATIVAL CONSTRUCTIONS.
275.—So as. Bearing in mind that as is simply a contraction for "all-so" ("alse," "als," "as"), we shall not be surprised at some interchanging of so and as.
We still retain "as . . . so ": "As I had expected so it happened," but seldom use "so . . . as," preferring "as . . . as;" except where so (as in the above phrase) requires special emphasis. The Elizabethans frequently used so before as.
" So well thy words become thee as thy wounds." Macbeth, i. 2. 43.
"Look I so pale, Lord Dorset, as the rest?" Rich. III. ii. 1. 83.
"And with a look so piteous in purport
As if he had been loosed out of hell."—Hamlet, ii. 1. 82.
" Thou art so full of fear
As one with treasure laden."—V. and A.
"Fair and fair and twice so fair
As any shepherd may be."—PEELE.
"All so soon as."—R. and i. 1. 140.
This is not very common in Shakespeare. Nor is it common to find so for as where the clause containing the second as is implied but not expressed.
"Make us partakers of a little gain,
That now our loss might be ten times so much."
1 Hen. VI. ii. 1. 53.
If the relatival as precedes, so, not as, must follow as the demonstrative. The exception below is explicable as being a repetition of a previous as used demonstratively:
"As little joy, my lord, as you suppose
You should enjoy, were you this country's king,
As little joy may you suppose in me
That I enjoy."—Rich. III. i. 3. 153.
"That " is the relative.
Ben Jonson (p. 789) writes as follows on so and as: " When the comparison is in quantity, then so goeth before and as followeth.
'Men wist in thilk time none
So fair a wight as she was one.'—GOWER, lib. 1.
But if the comparison be in quality, then it is contrary.
'For, as the fish, if it be dry,
Mote, in default of water dye:
Right so without air or live,
No man ne beast might thrive.'—GOWER."
So as is frequently used for so that. (See 109.)
This construction is generally found with the past and future indicative, but we sometimes find "so as he may see," for "so that he may see." "So as" is followed by the subjunctive in
"And lead these testy rivals so astray
As one come not within another's way."—M. N. D. iii. 2. 359.
Compare the use of s with the subjunctive in Greek. There is no more reason for saying, "I come so that (i.e. in which way) I may see," than for saying, "I come so as (i.e. in which way) I may see." We sometimes find so as that for so as in this sense.
The so is omitted after as in the adjurations
"As ever thou wilt deserve well at my hands, (so) help me to a candle,"—T. N. iv. 2. 86.
where as means "in which degree," and so "in that degree." Hence as approximates to " if."
It would seem that "as . . . so " are both to be implied from the previous verse in
" Had you been as wise as bold,
(As) young in limbs, (so) in judgment old."
M. of V. ii. 7. 71.
276. AS . . . as. The first As is sometimes omitted:
"A mighty and a fearful head they are
As ever offered foul play in a state."—1 Hen. IV. iii. 2. 168.
"He pants and looks (as) pale as if a bear were at his heels."
T. N. iii. 4. 323; Tempest, v. 1. 289.
In the expression "old as I am," &c. we almost always omit the first as. Shakespeare often inserts it:
"As near the dawning, provost, as it is."—M. for M. iv. 2. 97.
"But I believe, as cold a night as 'tis, he could wish himself in Thames up to the neck."—Hen. V. iv. 1. 118.
The expression is elliptical: "(be it) as cold as it is."
277. That ... that, that. . . (as) to. That is still used provincially for such and so: e.g. "He is that foolish that he understands nothing." So
"From me whose love was of that dignity
That it went hand in hand even with the vow
I made to her in marriage."—Hamlet, i. 5. 48.
That is more precise than "of that kind" or "such."
That, meaning "such," is used before the infinitive where we use the less emphatic "the."
"Had you that craft to reave her
Of what should stead her most?"—A. W. v. 3. 86.
So T. N. i. 1. 33; Rich. III. i. 4. 257; and Macheth, iv. 3. 374:
"There cannot be
That vulture in you to devour so many."
This omission of "as " after that meaning " so," is illustrated by the omission of "as" after "so " (281).
278. Such which. Such (in Early English, "swulc," "suilc," "suilch," "sich") was by derivation the natural antecedent to whick; such meaning "so-like," "so-in-kind ;" which meaning "what-like," "what-in-kind?" Hence—
"Such sin
For which the pardoner himself is in. "—M. for M. iv. 2. 111.
"There rooted between them such an affection which cannot
choose but branch now. "—W. T. i. 1. 26.
So W. T. iv. 4. 783; Coriol. iii. 2. 105.
Compare "Duty so great which wit so poor as mine
May make seem bare. "—Sonn. 26.
Similarly which is irregularly used after "too :"
"And salt too little which may season give
To her foul-tainted flesh."—M. Ado, iv. 1. 144.
Whom follows such in
"Such I will have whom I am sure he knows not."
A. W. iii. 6. 24.
279. Such that; so . . . that (rel.); such . . . where. Hence such is used with other relatival words:
"Such allowed infirmities that honesty
Is never free of."—W. T. i. 2. 263.
" To such a man
That is no flaming tell-tale. "— . C. i. 3. 116.
"For who so firm that cannot be seduced."— . C. i. 2. 316.
"His mother was a witch, and one so strong
That could control the moon. "—Temp. v. 1. 270; ib. 315
" But no perfection is so absolute
That some impunity doth not pollute."—R. of L.
" Who's so gross
That seeth not this palpable device ?"—Rich. III. iii. 6. 11.
" Such things were
That were most precious to me. "—Macbeth, iv. 3. 222.
"For no man well of such a salve can speak
That heals the wound and cures not the disgrace."
Sonn. 34.
Coriol, iii. 2. 55; T. G. of V. iv. 4. 70; A. W. i. 3. 221; Lear, ii. 2. 127; Othello, iii. 3. 417.
Hence it seems probable that that is the relative, having for its antecedent the previous sentence, in the following passages from Spenser:—
" Whose loftie trees yclad with summer's pride
Did spred so broad that heaven's light did hide."—F. Q. i. 1. 'T.
"(He) Shook him so hard that forced him to speak."—Ib. 42.
Similarly
"And the search so slow Which could not trace them. "—Cymb. i. 1. 65.
The licence in the use of these words is illustrated by—
"In me thou seest the twilight of such day
As, after sunset, fadeth in the west,
Which by and by black night doth take away.
In me thou seest the glowing of such fire
That on the ashes of his youth doth lie
As on the death-bed."—Sonn. 73.
In the first case such as is used, because which follows; in the second, such that, because as follows. So Hamlet, iii. 4. 41—46:
"Such an act that . . . . such a deed as."
Such, so, where:
"Soch a schoole where the Latin tonge were properly and perfitlie spoken."—ASCH. 45.
" In no place so unsanctified
Where such as thou mayest find him."—Macbeth, iv. 2. 81.
"So narrow where one but goes abreast."
Tr. and Cr. iii. 3. 155.
280. That as. We now use only such with as, and only that with which. Since, however, such was frequently used with which, naturally that was also used with as (in which way) used for which. Thus as approaches the meaning of a relative pronoun.
" I have not from your eyes that gentleness
As I was wont to have."— . C. i. 2. 33.
" Under these hard conditions as this time
Is like to lay upon us."—Ib. 174.
"Those arts they have as I could put into them." Cymb. v. 5. 338.
"Methinks the realms of England, France, and Ireland
Bear that proportion to my flesh and blood
As did the fatal brand Althea burned
Unto the prince's heart at Calydon."—2 Hen. VI. i. 1. 233.
"With that ceremonious affection as you were wont." Lear, i. 4. 63.
So after this :
" I beseech you do me this courteous office as to know what my offence is."—T. N. iii. 4. 278.
Similarly
" With hate in those where I expect most love."
Rich. III. ii. 1. 33.
Either (1) the nominative is omitted (see 399), or (2) as is put for who, the relative to an implied antecedent, in:
"Two goodly sons,
And, which was strange, the one so like the other
As could not be distinguish'd but by names."
C. of E. i. 1. 52.
i.e. (1) "so like that (they) could not be," as being used for that (see 109) ; or (2) "the one so like the other," &c. is loosely used for "the two so like each other as could not be distinguished."
Similarly as is used as a relative after an antecedent implied, but not expressed, by so with an adjective:
"I cannot but be sad, so heavy-sad
As ... makes me faint."—Rich. II. ii. 2. 31.
i.e. " I feel such sadness as."
281 So (as). Under the Relative we have seen that sometimes the antecedent, sometimes the relative, is omitted, without injury to the sense. Similarly in relatival constructions, e.g. so . . . as, so . . . that, &c. one of the two can be omitted.
The as is sometimes omitted :
" I wonder he is so fond
(as) To trust the mockery of unjust slumbers."
Rich. III. ii. 3. 26.
"So fond [i.e. foolish] (as) to come abroad."
M. of V. iii. 3. 10.
"No woman's heart
So big (as) to hold so much."—T. N. ii. 4. 99.
"Shall I so much dishonour my fair stars
(as) On equal terms to give him chastisement?"
Rich. II. iv. 1. 21.
R. and ii. 3. 91 ; Macbeth, ii. 3. 55; Rich. II. iii. 3. 12.
As or who is omitted in:
"And while it is so, none so dry or thirsty
Will deign to sip or touch one drop of it."—T. of Sh. v. 2. 144.
i.e. "None is so thirsty (who) will deign" where we should say "as to deign." Less probably, "none (be he how) so (ever) dry."
So and as are both omitted in :
"Be not (so) fond
(As) To think that Cæsar bears such rebel blood."— . C. iii. 1. 40.
282. So (that). The that is sometimes omitted.
"I am so much a fool (that) it would be my disgrace."
Macb. iv. 2. 27.
283. (So) that. So before that is very frequently omitted:
"Ross. The victory fell onus. Dunc. Great happiness!
Ross. (So) that now Sueno, the Norway's king, craves composi-
tion."—Macbeth, i. 2. 59.
Compare Macb. i. 7. 8, ii. 2. 7, ii. 2. 24; .C. i. 1. 50.
In all these omissions the missing word can be so easily supplied from its correspondent that the desire of brevity is a sufficient explanation of the omission.
" A sheet of paper
Writ o' both sides the leaf, margent and all,
That he was fain to seal on Cupid's name."—L. L. L. v. 2. 9.
284. That, for because, when. Since that represents different cases of the relative, it may mean " in that," " for that," "because" ("quod"), "or at which time" ("quum").
In, or for that:
"Unsafe the while that we must lave our honours," &c.
Macheth, iii. 2. 32.
"O, spirit of love! How quick and fresh art thou
That (in that), ... nought enters there but," &c.
T. N. i. 1. 10.
"Like silly beggars
Who sitting in the stocks refuge their shame,
That (because) many have and others must sit there,
And in this thought they find a kind of ease."
Rich. II. v. 5. 27.
At which time; when :
"In the day that thou eatest thereof. "—Gen. ii. 17.
"Now it is the time of night
That the graves all gaping wide,
Every one lets forth his sprite."—M. N. D. v. 1. 387.
"So wept Duessa until eventyde,
That shynyng lamps in Jove's high course were lit."
SPENS. F. Q. i. 5. 19.
"Is not this the day
That Hermia should give answer of her choice?"
M. N. D. iv. 1. 133.
"So, till the judgment that yourself arise,
You live in this and dwell in lovers' eyes."—Sonn. 55.
Compare "Then that," apparently "then when." (2 Hen. IV. iv. 1. 117.)
These uses of that are now superseded by the old interrogatives why and when, just as, even in Shakespeare's time, many of the uses of that had been transferred to the interrogatives who and which.
"Albeit I will confess thy father's wealth
Was the first motive that I wooed thee, Anne."
M. W. of W. iii. 4. 14.
i.e. "for which, or why, I wooed thee."
The use of that for when is still not uncommon, especially in the phrase "now that I know," &c. It is omitted after "now" in
"But now (that) I am return'd, and that war thoughts
Have left their places vacant, in their rooms
Come thronging soft and delicate desires."—M. Ado, i. 1. 303. So Rich. III. i. 2. 170; M. N. D. iv. 1. 67, 109.
That = "in which in
"Sweet Hero, now thy image doth appear
In the sweet semblance that I loved it first."—M. Ado, v. 1. 260.
285. That omitted and then inserted. The purely conjunctional use of that is illustrated by the Elizabethan habit of omitting it at the beginning of a sentence, where the construction is obvious, and then inserting it to connect a more distant clause with the conjunction on which the clause depends. In most cases the subjects of the clauses are different.
"Though my soul be guilty and that I think," &c.
B. J. Cy.'s Rev. iii. 2.
"Were it not thy sour leisure gave sweet leave,
And that thou teachest. "—Sonn. 39.
"If this law
Of nature be corrupted through affection,
And that great minds, of partial indulgence
To their benumbed wills, resist the same."
Tr. and Cr. ii. 2. 179.
This may explain (without reference to "but that," 122):
"If frosts and fasts, hard lodging and thin weeds
Nip not the gaudy blossoms of your love,
But that it bear this trial."—L. L. L. v. z. 813.
For "if that," see 287.
"Think I am dead, and that even here thou takest,
As from my death-bed, my last living leave."
Rich. II. v. 1. 38.
So T. N. v. 1. 126; W. T. i. 2. 84; A. and C. iii. 4. 31; P. of T. i. Gower, 11.
"I love and hate her, for she's fair and royal,
And that she hath all worthy parts more exquisite."
cymb. iii. 5. 71.
i.e. "for that" or "because."
"She says I am not fair, that I lack manners;
She calls me proud, and that she could not love me."
A. Y. L. iv. 2. 16.
In the above example the that depends upon a verb of speech implied in "calls." This construction is still more remarkable in—
"But here's a villain that would face me down
He met me on the mart, and that I beat him."—C. of E. iii. 1. 7.
Compare the French use of "que" instead of repeating "si," "quand," &c.
286. Whatsoever that. In the following there is probably an ellipsis :
"This and what needful else (there be)
That calls upon us. "—Macbeth, v. 8. 72.
"Till whatsoever star (it be) that guides my moving
Points on me graciously with fair aspect. "—Sonn. 26.
"As if that whatsoever god (it be) who leads him
Were slily crept into his human powers."—Cariol. ii. 1. 235.
In the latter, that is probably the demonstrative. It might, however, be the conjunctional that. See "if that," 287.
287. That as a conjunctional affix. Just as so and as are affixed to who (whoso), when (whenso), where (whereas, whereso), in order to give a relative meaning to words that were originally interrogative, in the same way that was frequently affixed.
"When that the poor have cried."
. C. iii. 2. 96; T. N. v. 1. 398.
"Why that:"—Hen. V. v. 2. 34.
"You may imagine him upon Blackheath,
Where that his lords desire him to have borne
His bruised helmet and his bended sword
Before him through the city."—Hen. V. v. Prologue, 17.
So A. Y. L. ii. 7. 75; iv. 3. 117. This, with the above, explains
"Edmund. When by no means he could.
Gloucester. Pursue him, ho ! go after. By no means what?
Edmund. Persuade me to the murder of your lordship,
But that I told him," &c.—Lear, ii. 1. 47.
Gradually, as the interrogatives were recognized as relatives, the force of that, so, as, in "when that," "when so," "when as," seems to have tended to make the relative more general and indefinite; "who so" being now nearly (and once quite) as indefinite as "whosoever." The "ever" was added when the "so" had begun to lose its force. In this sense, by analogy, that was attached to other words, such as "if," "though," "why," &c.
"If that the youth of my new interest here
Have power to bid you welcome. "—M. of V. iii. 2. 224.
Compare "If that rebellion
" If that rebellion
Came like itself, in base and abject routs."
2 Hen. IV. iv. 1. 32; T. N. i. 5. 324, v. 1. 375.
So Lear, v. 3. 262; Rich. III. ii. 2. 7.
The fuller form is found, CHAUC. Pard. Tale, 375: "If so were that I might;" and Lodge writes, "If so I mourn." Similarly, "If so be thou darest."—Coriol. v. 14. 98.
Compare:
"While that."—Hen. V. v. 2. 46.
"Though that." Coriol. i. 1. 144; Lear, iv. 6. 219; T. N. i. 3. 48.
"Lest that."—Hen. V. ii. 4. 142; T. N. iii. 4. 384.
" Whether that."—1 Hen. VI. iv. 1. 28.
"So as that," frequently found.
"Since that."—Macb. iv. 3. 106; Rich. III. v. 3. 202.
"How that" is also frequent. We also find that frequently affixed to prepositions for the purpose of giving them a conjunctival meaning: "For that" (Macb. iv. 3. 185); "in that; " "after that," &c.
The Folio has
"Your vertue is my priuiledge : for that
It is not night when I doe see your face.
Therefore I thinke I am not in the night."
M. N. D. ii. 1. 220.
The Globe omits the full stop after "face," making "for that " (because) answer to "therefore." Others remove the stop after "privilege " and place it after " for that."
Hence we find "but that" where we should certainly omit that
" The breath no sooner left his father's body
But that his wildness, mortified in him,
Seem'd to die too."—Hen. V. i. 1. 26.
288. That, origin of. Is that, when used as above, demonstrative or relative? The passage quoted above from Chaucer, "If so were that," renders it probable that a similar ellipsis must be supplied with the other conjunctions: "Though (it be) that," "Since (it is) that," &c. With prepositions the case is different, e.g. "for that," "in that," "after that." For this use of that can be traced to A.-S., where we find "for þam þe," i.e. "for this purpose that," "after þam þe," &c. Here "þam" is more emphatic than "be," and evidently gave rise to the English that. But " Pam " was the A. -S. demonstrative. It follows that the that is (by derivative use, at all events) demonstrative in "for that," or, perhaps we should say, stands as an abridgment for " that (demonst.) that (rel.)." In fact, we can trace the A.-S. "after þam þe" to the E. E. "after that that," and so to the later "after that. " Hence we must explain
"The rather
For that I saw the tyrant's power afoot. "—Macb. iv. 3. 185.
as "for that (that), i.e. for that, because, I saw." It would be wrong, however, to say that that in "since that" is, by derivative use, demonstrative. On the contrary, "since " in itself (siþ-þan) contains the demonstrative, and "since that" corresponds to "sib-ban þat" where that (þat) is relative. And similarly " though that" corresponds to the A.-S. "beah be," where that(þe) is the relative. The that in "after that," "before that," invites comparison with the "quam" in "postquam" and "antequam," though in the Latin it is the antecedent, not the relative, that is suppressed. The tendency of the relative to assume a conjunctional meaning is illustrated by the post-classical phrase, "dico quod (or quia) verum est," in the place of the classical "dico id verum esse." Many of the above Elizabethan phrases, which are now disused, may be illustrated from French: "since that," "puisque;" "though that," "quoi que;" "before that," "avant que," &c. Instead of "for that," we find in French the full form, "par ce que," i.e. "by that (dem.) that (rel.)." It is probable that Chaucer and Mandeville, if not earlier writers, were influenced in their use of the conjunctional that by French usage. Even in the phrase " I say that it is true," that may be explained as having a relatival force (like τι, " quod," and the French "que"), meaning, "I say in what way, how that, it is true." In the phrase, "I come that (in the way in which; 'ut,' ς, 'afin que') I may see," the relatival force of that is still more evident.
289. As is used in the same way as a conjunctional affix. Thus "while as:"
"Pirates . . . still revelling like lords till all be gone
While as the silly owner of the goods
Weeps over them."—2 Hen VI. i. 1. 225.
"When as:"
" When as the enemy hath been ten to one. "—3Hen VI. i. 2. 75.
" When as the noble Duke of York was slain."—Ib. ii. 1. 46.
So Ib. v. 7. 34.
"Where as" is used by us metaphorically. But Shakespeare has
" Unto St. Alban's,
Where as the king and queen do mean to hawk."
2 Hen VI. i. 2. 57.
"They back retourned to the princely Place,
Whereas an errant knight . . . they new arrived find."
SPENS. F. Q. i. 4. 38.
So " there as" is used in earlier English. "There that" is also found in Chaucer in a local sense.
Of course the" so " in "whenso," "whereso" &c., is nearly the same in meaning, just as it is the same in derivation, with the as in " whenas," &c.
# VERBS, FORMS OF.
290. Verbs, Transitive (formation of). The termination en (the infinitive inflection) is sufficient to change an English monosyllabic noun or adjective into a verb. Thus "heart" becomes "hearten;" "light," "lighten;" "glad," "gladden," &c. The licence with which adjectives could be converted into verbs is illustrated by
"Eche that enhauncith hym schal be lowid, and he that mekith hymself shall be highid."—WICKLIFFE, St. Luke xiv. 11.
In the general destruction of inflections which prevailed during the Elizabethan period, en was particularly discarded. It was therefore dropped in the conversion of nouns and adjectives into verbs, except in some cases where it was peculiarly necessary to distinguish a noun or adjective from a verb. (So strong was the discarding tendency that even the e in "owen," "to possess," was dropped, and Shakespeare continually uses "owe" for "owen" or "own"(T. N. i. 5. 329; Rich. II. iv. 1. 185). The n has now been restored.) But though the infinitive inflection was generally dropped, the converting power was retained, undiminished by the absence of the condition. Hence it may be said that any noun or adjective could be converted into a verb by the Elizabethan authors, generally in an active signification, as—
" Which happies (makes happy) those that pay the willing lover."
Sonn. 11.
"Time will unfair (deface) that (which) fairly doth excel." —Ib. 5.
So:
Balm'd (healed).—Lear, iii. 6. 105.
Barn.—"Barns a harvest. "—R. of L.
Bench (sit).—Lear, iii. 6. 40.
Bold (embolden).—"Not bolds the king."—Lear, v. 1. 26.
Brain.
"Such stuff as madmen
Tongue and brains not."—Cymb. v. 4. 147.
i.e. "such stuff as madmen use their tongues in, but not their brains. "
Child.—" Childing autumn."—M. N. D. ii. 1. 112: i.e. "autumn producing fruits as it were children."
Climate.—"Climates (neut.) [lives] here."—W. T. v. 1. 170.
Cowarded.—"That hath so cowarded and chased your blood. "—Hen. V. ii. 2. 75.
Coy (to be coy).—"Nay, if he coy'd."—Coriol. v. 1.6.
Disaster (make disastrous-looking).—"The holes where eyes should be which pitifully disaster the cheeks."—A. and C. ii. 7. 18.
False.—"Has falsed his faith."—SPENS. F. Q. i. 19. 46.
Fame.—"Fames his wit."—Sonn. 84.
Fault.—"Cannot fault (neut.) twice."—N. P. Pref.; B. J. Alch.iii. 1.
Feeble.—"And feebling such as stand not in their liking."
Coriol. i. 1. 199.
Fever (give a fever to).—"The white hand of a lady fever thee, Shake thou to look on't."—A. and C. iii. 13. 138.
Fond.
" My master loves her truly,
And I, poor monster, fond as much on him."—T. N. ii. 2. 35.
Fool (stultify).
"Why, that's the way
To fool their preparations. "—A. and C. v. 2. 225.
This explains
"Why old men fool and children calculate."— . C. i. 3. 65.
Foot.—"Foots" (kicks).—Cymb. iii. 5. 148. On the other hand, in "A power already footed " (Lear, iii. 2. 14), it means " set on foot;" and in "the traitors late footed in the kingdom" (Ib. iii. 7. 45), it means "that have obtained a footing."
Force (to urge forcibly).—"Why force you this?"—Coriol. iii. 2. 51. Also (to attach force to, regard) :
"But ah ! who ever shunn'd by precedent
The destin'd ills she must herself assay,
Or forced examples 'gainst her own content,
To put the by-past perils in her way?"—L. C. 157.
i.e. "whoever regarded examples." So L. L. L. v. 2. 441.
Furnace.—"Furnaces sighs."—Cymb. i. 6. 66.
Gentle.—"This day shall gentle his condition. "—Hen. V. iv. 3. 63.
God.—"He godded me."—Coriol. v. 3. 11.
Honest.—"Honests (honours) a lodging."—B. J. Sil. Wom. i. 1.
Inherit (make an inheritor).
"That can inherit us So much as of a thought of ill in him."—Rich. II. ii. 1. 85.
Knee (kneel).—" Knee the way."—Coriol. v. 1. 5.
Lesson (teach).—"Lesson me."—T.G. of V.ii 7. 5; Rich. III. i. 4. 246.
Linger (make to linger).
"Life
Which false hope lingers in extremity."
Rich. II. ii. 2. 72; M. N. D. i. 1. 4.
Mad.—"Mads" (makes angry).-Rich. II. v. 5. 61.
Mallow (ripen, trans.).—T. N. i. 3. 43.
Mist (cover with mist).—" If that her breath will mist or stain the stone."—Lear, v. 3. 262.
Malice.—"Malices" (bears malice to).—N. P.
Pale (make pale).—"And 'gins to pale his uneffectual fire."
Hamlet, i. 5. 90.
Panging (paining).
"'Tis a sufferance panging
As soul and body's severing."—Hen. VIII. ii. 3. 15.
Path (walk).—" For if thou path (neuter), thy native semblance on."— . C. ii. 1. 83.
Plain (make plain).—"What's dumb in show I'll plain in speech."
P. of T. iii. Gower, 14.
Property (treat as a tool).—"They have here propertied me."
T. N. iv. 2. 100; K. . v. 2. 79.
Rag'd (enraged).—There is no corruption (though the passage is marked as corrupt in the Globe) in
"For young colts being rag'd do rage the more."
Rich. II. ii. 1. 70.
Safe.—"And that which most with you should safe my going, Fulvia is dead."—A. and C. i. 3. 55.
i.e. "make my departure unsuspected by you of dangerous consequences. "
Scale (weigh, put in the scale).—"Scaling his present bearing with his past."—Coriol. ii. 3. 257.
Stage (exhibit).—" I do not like to stage me to their eyes." M. for M. i. 1. 69.
Stock (put in the stocks).—"Stocking his messenger." Lear, ii. 2. 139.
Stream (unfurl).—"Streaming the ensign."—Rich. II. iv. 1. 94.
Toil (give labour to).—Probably in
" Why this same toil and most observant watch
So nightly toils the subject of the land."—Hamlet, i. 1. 72.
So "toil'd," passive.—Rich. II. iv. 1. 96.
Tongue.—"How might she tongue me?"—M. for M. iv. 4. 28.
i.e. "speak of, or accuse, me." " Tongue" means " speak " in
" Such stuff as madmen
Tongue, and brain not."—Cymb. v. 4. 147.
Trifle.—" Trifles (renders trifling) former knowing."—Macb. ii. 4. 4.
Undeaf.—" My death's sad tale may yet undeaf his ear." Rich. II. ii. 1. 6.
Verse (expressing in verse).—" Versing love. "—M. N. D. ii. 1. 67.
Violent (act violently).—"And violenteth in a sense as strong." Tr. and Cr. iv. 4. 4.
Wage (pay : so E. E.).—" He waged me."—Coriol. v. 6. 40.
Womb (enclose).—" The close earth wombs or the profound sea hides." W. T. iv. 4. 501.
Worthied (ennobled).—"That worthied him."—Lear, ii. 2. 128.
The dropping of the prefix be was also a common licence. We have recurred to " bewitch " and "belate," but Shakespeare wrote—
" And witch the world with noble horsemanship." 1 Hen. IV. iv. 1. 110.
" Now spurs the lated traveller apace. "—Macbeth, iii. 3. 6.
"Disorder, that hath spoil'd us, friend us now." Hen. V. iv. 5. 17.
291. Sometimes an intransitive verb is converted into a transitive verb.
Cease.—" Heaven cease this idle humour in your honour ! " T. of Sh. Ind. 2. 13. So Cymb. v. 5. 255.
Expire.—Time "expires a term."—R and . i. 4. 109-
Fall—An executioner " falls an axe."—A. Y. L. iii 5.5 and pro. bably (though fall may be the subjunctive) in
"Think on me, and fall thy edgeless axe."—Rich. III. v. 3. 135.
Peer.—"Peers (causes to peer) his chin."—R. of L.
Perish.—"Thy flinty heart . . . might perish (destroy) Margaret." 2 Hen. VI. iii. 2. 100.
Quail (make to quail).—"But when he meant to quail and shake the orb."—A. and C. v. 1. 85.
Relish.—"Relishes (makes acceptable) his nimble notes to pleasing ears."—R. of L.
Remember (remind : so Fr.).—"Every stride I take Will but remember me what," &c.—Rich. II. i. 3. 269.
Retire (so Fr.).—" That he might have retired his power " Rich. II. ii. 2. 46.
Shine.—"God doth not shine honourupon all men equally."-B.E.45,
Squint.—" Squints the eye and makes the harelip."—Lear,iii.4.122. i.e. "makes the eye squint."
Fear. This word is not in point. It had the signification of " frighten " in A. -S. and E. E. Hence,
" Thou seest what's past : go fear thy king withal." 3 Hen. VI. iii. 3. 226.
"This aspect of mine hath fear'd the valiant." M. of V. ii. 1. 9.
So in Spenser, " Words fearen babes."
The same remark applies to " learn," which meant " teach."
"The red plague rid you
For learning me your language."—Tempest, i. 2. 365.
292. The licence in the formation of verbs arose partly from the unfixed nature of the language, partly from the desire of brevity and force. Had it continued, it would have added many useful and expressive words to the language. In vigorous colloquy we still occasionally use such expressions as—
"Grace me no grace, nor uncle me no uncles."—Rich.II. ii. 3. 87.
" Thank me no thankings, nor proud me no prouds." R. and J. iii. 5. 153.
As it is, we can occasionally use the termination -fy, as in "stultify," and sometimes the suffix -en or the prefix be-. But for the most part we are driven to a periphrasis.
293. Transitive verbs are rarely used intransitively. Eye (appear).
"But, sir, forgive me Since my becomings kill me, when they do not
Eye well to you."—A. and C. i. 3. 97.
Lack (to be needed).—" And what so poor a man as Hamlet is
May do to express his love and friending to you,
God willing, shall not lack."—Hamlet, i. 5. 186. So E. E.
Need (to be needed).—"These ceremonies need not." B. J. E. in & c. iii. 2.
This is perhaps a remnant of the ancient love for impersonal verbs. Such verbs would be appropriate to express "need." Hence in Matt. xix. 20, Mark x. 21, Wickliffe has "faileth to me" and "to thee," where the A. V. has "what do I lack " and " thou lackest." Similarly, Milton (Areopagitica) uses "what wants there?" for "what is needed?" and this use still exists in conversation. So often Shakespeare, e.g.
"There wanteth now our brother Gloucester here." Rich. III. ii. 1. 43.
Show (like our " look : " compare German " schauen ").
" Each substance of a grief hath twenty shadows
Which shows like grief itself. "—Rich. II. ii. 2. 15.
294. Verbs Passive (formation of). Hence arose a curious use of passive verbs, mostly found in the participle. Thus "famous'd for fights" (Sonn. 25) means "made famous ;" but in
"Who, young and simple, would not be so lover'd?"—L. C.
laver'd means "gifted with a lover." And this is the general rule. A participle formed from an adjective means "made (the adjective)," and derived from a noun means "endowed with (the noun)." On the other hand, stranger'd below means, not "gifted with a stranger," but "made a stranger." This use will be best illustrated by the following examples :—
Childed (provided with children).—" He childed as I father'd." Lear, iii. 6. 117.
Faith'd (believed).—" Make thy words faith'd."—Ib. ii. 1. 72.
Father'd (provided with a father). See above, Lear, iii. 6. 117.
Feebled (enfeebled).—K. \- v. 2. 146.
Fielded (encamped in the field).—" Our fielded friends." Coriol. i. 4. 12.
Grav'd (entomb'd).—" Grav'd in the hollow ground." Rich. II. iii. 2. 140.
Guiled (deceitful).—"A guiled shore."—M. of V. iii. 2. 97.
Compare:
"Beguiled (i.e. made plausible) With outward honesty, but yet defiled
With inward vice."—R. of L.
Inhabited (made to inhabit).—" O, knowledge ill-inhabited, worse than Jove in a thatch'd house."—A. Y. L. iii. 3. 10.
King'd (ruled).—"King'd of our fears, until our fears, resolv'd, Be by some certain king purged and deposed."—K. .ii. 1. 371. i.e. "ruled by our fears."
Look'd (looking).—"Lean-look'd prophets."—Rich. II. ii. 4. 11.
Lorded (made a lord).—"He being thus lorded."—Tempest, i. 2. 97.
Contrast this with "king'd" above, which means not " made a king," but " ruled as by a king."
Meered. " When half to half the world opposed, He being the meered question."—A. and C. iii. 13. 10.
The word "meered" is marked as corrupt by the Globe : but perhaps it is the verb from the adj. "meere" or "mere," which in Elizabethan English means "entire." Hence, "he being the entire question," i.e. "Antony, being the sole cause of the battle, ought not to have fled."
Million'd.—"The million'd accidents of time."—Sonn. 115.
Mouthed.—"Mouthed graves."—Ib. 77.
Necessited.—
" I bade her, if her fortunes ever stood Necessited to help, that by this token
I would relieve her."—A. W. v. 3. 85.
i.e. "made necessitous."
Nighted (benighted).—" His nighted life."—Lear, iv. 5. 13 ; "Thy nighted colour."—Hamlet, i. 2. 68: i.e. "thy night-like colour. "
Paled.—"Paled cheeks."—L. C. 28.
Pensived.—Ib. 31.
Pined.—" His pined cheek."—Ib. 5.
Practised (plotted against).—"The death-practised duke." Lear, iv. 6. 284.
Servanted (made subservient).—Coriol. v. 2. 89.
Slow'd (retarded).—" I would I knew not why it should be slow'd." R. and . iv. 1. 16.
Stranger'd (made a stranger).—" Dower'd with our curse, and stranger'd with our oath."—Lear, i. 1. 207.
Toil'd.—I have been so toil'd."—B. J. E. out &c. iii. 1.
Traded.—" Traded pilots."—Tr. and Cr. ii. 2. 64.
Unlook'd (unloolced for).—Rich. III. i. 3. 214 : compare look (seek). Hen. V. iv. 7. 76.
Unsured (unassured).—"Thy now unsured assurance to the crown." K. . ii. 1. 471.
Vouchsafed (?).—" To your most pregnant and vouchsafed ear." T. N. iii. 1. 190.
i.e. capable of conceiving and graciously bestowed.
Window'd (placed in a window).
" Wouldest thou be window'd in great Rome." A. and C. iv. 14. 72.
Woman' d (accompanied by a woman).
" To have him see me woman'd."—Othello, iii. 4. 195.
Year'd.—" Year'd but to thirty."—B. J. Sejan. i. 1.
In many cases a participle seems preferred where an adjective would be admissible, as "million'd." So in Tempest, v. 1. 43, " the azured vault."
295. Verbs Passive. With some few intransitive verbs, mostly of motion, both be and have are still used. "He is gone," "he has gone." The is expresses the present state, the has the activity necessary to cause the present state. The is is evidently quite as justifiable as has (perhaps more so), but it has been found more convenient to make a division of labour, and assign distinct tasks to is and has. Consequently is has been almost superseded by has in all but the passive forms of transitive verbs. In Shakespearian English, however, there is a much more common use of is with intransitive verbs.
" My life is run his compass."— . C. v. 3. 25.
" Whether he be scaped."—3 Hen. VI. ii. 1. 2.
" Being sat."—L. C. st. x.
" Being deep stept in age."—ASCH. 189.
" An enter'd tide."—Tr. and Cr. iii. 3. 159.
" I am arrived for fruitful Lombardy."—T. of Sh. i. 1. 3.
" Pucelle is entered into Orleans." 1 Hen. VI. i. 5. 36 ; Cymb. v. 4. 120.
" Five hundred horse . . . are marched up." 2 Hen. IV. ii. 1. 186.
" The king himself is rode to view their battle." Hen. V. iv. 3. 1.
" His lordship is walk'd forth."—2 Hen. IV. i. 1. 3.
" The noble Brutus is ascended."— . C. iii. 2. 11.
" You now are mounted
Where powers are your retainers."—Hen. VIII. ii. 4. 112.
" I am descended of a gentler blood."—I Hen. VI. v. 4. 8.
" Through his lips do throng
Weak words, so thick come (particip.) in his poor heart's aid."
R. of L. 1784.
Compare our "welcome."
" How now, Sir Proteus, are you crept before us?" T. G. of V. iv. 1. 18.
So Rich. III. i. 2. 259.
" Prince John is this morning secretly stolen away." M. Ado, iv. 2. 63.
This idiom is common with words of "happening :"
" And bring us word . . . how everything is chanced." . C. v. 4. 32; 2 Hen. IV. i. I. 87.
" Things since then befallen. "—3 Hen. VI. ii. 1. 106.
" Of every one these happen'd accidents."—Temp. v. 1. 249.
"Sad stories chanced in the days of old."—T. A. iii. 2. 83.
Hence a participial use like " departed " in
" The treachery of the two fled hence."—W. T. ii. 1. 195.
In some verbs that are both transitive and intransitive this idiom is natural:
" You were used to say." —Coriol. iv. 1. 3.
Perhaps this is sometimes a French idiom. Thus, " I am not purposed" (MONTAIGNE, 38), is a translation of "je ne suis pas délibéré. "
This constant use of "be" with participles of verbs of motion may perhaps explain, by analogy, the curious use of " being " with the present participle in
" To whom being going."—Cymb. iii. 6. 63.
As above mentioned, the tendency to invent new active verbs increased the number of passive to the diminution of neuter verbs :
" Poor knave, thou art overwatch'd. "— . C. iv. 3. 241.
"Be wreak'd (i.e. avenged) on him."—V. and A. So, N. P. 194.
" Possess" was sometimes used for to "put in possession," as in
"Possess us, possess us " (T. N. ii. 3. 149) : i.e. "inform us." So M. of V. iv. 1. 35. Hence the play on the word.
" Deposing thee before thou wert possess'd (of the throne),
Which art possessed (with a spirit of infatuation) to destroy
thyself."—Rich. II. ii. 1. 107–8 ; M. of V. i. 3. 65.
We still say a man "is well read." But in Macb. i. 4. 9, there is—
" As one that had been studied in his death."
"For Clarence is well-spoken."—Rich. III. i. 3. 348.
" I am declined into the vale of years."—Othello, iii. 3. 265.
" How comes it, Michael, you are thus forgot?" Ib. ii. 3. 188.
i.e. "you have forgotten yourself."
" If I had been remembered."—Rich. III. ii. 4. 22.
We still say "well-behaved," but not
" How have I been behaved. "—Othello, iv. 2. 108.
It was perhaps already considered a vulgarity, for Dogberry says (M. Ado, iv. 2. 1) :
" Is our whole dissembly appear'd? "
and in a prose scene (Coriol. iv. 3. 9)–
" Your favour is well appear'd (fol.) by your tongue."
Perhaps, however, appear was sometimes used as an active verb. See Cymb. iv. 2. 47, iii. 4. 148, quoted in 296.
296. Verbs Reflexive. The predilection for transitive verbs was perhaps one among other causes why many verbs which are now used intransitively, were used by Shakespeare reflexively. Many of these were derived from the French.
"Advise you."—T. N. iv . 2. 102.
" Where then, alas! may I complain myself?"—Rich. II. i. 2. 42.
"Endeavour thyself to sleep."—T. N. iv. 2. 104.
" I do repent me."—Ib. v. 3. 52.
" Repose you."—Ib. ii. 3. 161.
" He . . . retired himself."—Rich. II. iv. 2. 96 ; Coriol. i. 3. 30, which is in accordance with the original meaning of the word.
It has been shown above that "fear" is used transitively for "frighten." Hence, perhaps, as in Greek φoβo μαι,
" I fear me."—2 Hen. VI. i. 1. 150.
Appear is perhaps used reflexively in
" No, no; we will hold it as a dream till it appear itself." M. Ado, i. 2. 22.
" If you could wear a mind
Dark as your fortune is, and but disguise
That which to appear itself must not yet be."—Cymb. iii. 4. 148.
i.e. "that which, as regards showing itself, must not yet have any existence." Though these passages might be perhaps explained without the reflexive use of appear, yet this interpretation is made more probable by
" Your favour is well appear'd."—Coriol. iv. 3. 9.
297. Verbs Impersonal. An abundance of Impersonal verbs is a mark of an early stage in a language, denoting that a speaker has not yet arrived so far in development as to trace his own actions and feelings to his own agency. There are many more impersonal verbs in Early English than in Elizabethan, and many more in Elizabethan than in modern English. Thus—
" It yearns me not."—Hen. V. iv. 3. 26.
"It would pity any living eye."—SPENS. F. Q. i. 6. 43.
Comp. 2 Maccabees iii. 21 : "It would have pitied a man."
"It dislikes me."—Othello, ii. 3. 49.
So "it likes me," "meseems," " methinks," &c.
"Which likes me."—Hen. V. iv. 3. 77.
And therefore like is probably (not merely by derivation, but consciously used as) impersonal in
"So like you, sir."—Cymb. ii. 3. 59.
Want is probably not impersonal but intransitive, "is wanting," in
"There wants no diligence in seeking him?"*—Cymb.iv.2.20.
The singular verb is quite Shakespearian in
"Though bride and bridegroom wants (are wanting)
For to supply the places at the table. "—T. of Sh. iii. 2. 248.
So in
"Sufficeth my reasons are both good and weighty. "—Ib.i.1.252. "Sufficeth I am come to keep my word."—Ib. iii. 2. 108.
the comma after "sufficeth" is superfluous ; " that I am come to keep my word sufficeth."
In
"And so betide to me As well I tender you and all of yours,"—Rich. III. ii. 4. 71.
betide may be used impersonally. But perhaps so is loosely used as a demonstrative for "such fortune," in the same way in which as (280) assumes the force of a relative. If betide be treated as impersonal, befal in "fair befal you" may be similarly treated, and in that case "fair" is an adverb. But see (5). The supposition that " betide " is impersonal and " fair " an adverb is confirmed by " Well be (it) with you, gentlemen."—Hamlet, ii. 2. 398.
The impersonal needs (which must be distinguished from the adverbial genitive needs) often drops the s; partly, perhaps, because of the constant use of the noun need. It is often found with "what," where it is sometimes hard to say whether "what" is an adverb and need a verb, or "what" an adjective and need a noun.
" What need the bridge much broader than the flood ?" M. Ado, i. 1. 318.
either "why need the bridge (be) broader?" or "what need is there (that) the bridge (be) broader ?"
See 293.
Comp. the old use of " thinketh "(seemeth) :
" Where it thinks best unto your royal self."—Rich. III. iii. 1. 63.
The Folio has thinkst : and perhaps this is the true reading, there being a confusion between "it thinks" and "thinkest thou." Compare "thinkst thee " in
"Doth it not, thinkst thee, stand me now upon?"—Hamlet, v. 2. 63.
The impersonal and personal uses of think were often confused. Chapman (Walker) has "methink." S seems to have been added to assimilate the termination to that of " methinks " in "methoughts" (W. T. i. 2. 154; Rich. III. i. 4. 9).
It is not easy, perhaps not possible, to determine whether, in the phrase "so please your highness," please is used impersonally or not; for on the one hand we find, " So please him come,"
( . C. iii. 1. 140) ;
and on the other,
" If they please."—W. T. ii. 3. 142.
"I do repent: but Heaven hath pleased it so. "—Ham. iii. 4. 173.
# VERBS, AUXILIARY.
298. Be, Beest, &c., was used in A.-S. (beon) generally in a future sense. Hence, since the future and subjunctive are closely connected in meaning, be assumed an exclusively subjunctive use ; and this was so common, that we not merely find "if it be" (which might represent the proper inflected subjunctive of be), but also "if thou beest," where the indicative is used subjunctively.
" If, after three days' space, thou here beest found." 2 Hen. VI. iii. 2. 295.
" Beest thou sad or merry,
The violence of either thee becomes."—A. and C. i. 5. 59.
And (Mätzner, vol. i. p. 367), bee, beest, bee, pl. bee, is stated by Wallis to be the regular form of the subjunctive. Hence, from the mere force of association, be is often used (after though, if, and other words that often take the subjunctive) without having the full force of the subjunctive. Indeed any other verb placed in the same context would be used in the indicative. Thus:
" Though Page be a secure (careless) fool, and stands so firmly on his wife's frailty."—M. W. of W. ii. 1. 242.
" If Hamlet from himself be ta'en away
And, when he's not himself, does wrong Laertes."—Ham.v.2.245.
" If he be a whoremonger and comes before him,
He were as good go a mile on his errand."—M. for M. iii. 2. 38.
299. Be in questions and dependent sentences.
So, as a rule, it will be found that be is used with some notion of doubt, question, thought, &c. ; for instance, (a) in questions, and (b) after verbs of thinking.
* (a)
"Be my horses ready?"—Lear, i. 5. 36.
" Be the players ready?"—Hamlet, iii. 2. 111.
This is especially frequent in questions of appeal :
" Where be his quiddities?"—Hamlet, v. 1. 107.
" Where be thy brothers?"—Rich. III. iv. 4. 92.
" Where be the bending knees that flatter'd thee?
Where be the thronging troops that follow'd thee?"
Ib. iv. 4. 95-6.
And in questions implying doubt, e.g. " where can they be ?"
" Where be these bloody thieves?"—Othello, v. 1. 64.
Partly, perhaps, by attraction to the previous be, partly owing to the preceding where, though not used interrogatively, we have
" Truths would be tales,
Where now half-tales be truths."—A. and C. ii. 2. 137.
* (b)
" I think it be, sir ; I deny it not."—C. of E. v. 1. 379.
" I think this Talbot be a fiend of hell. "—1 Hen. VI. ii. 1. 46.
" I think he be transformed into a beast. "—A. Y. L. ii. 7. 1.
" I think it be no other but even so."—Hamlet, i. 1. 108.
So 1 Hen. IV. ii. 1. 12 ; T. G. of V. ii. 3. 6.
Be expresses more doubt than is after a verb of thinking. In the following, the Prince thinks it certain that it is past midnight, the Sheriff thinks it may possibly be two o'clock :
"Prince. I think it is good morrow, is it not ?
Sheriff. Indeed, my lord, I think it be two o'clock."
1 Hen. IV. ii. 4. 573.
Very significant is this difference in the speech of the doubtful Othello—
" I think my wife be honest, and think she is not," Othello, iii. 3. 384.
where the is is emphatic and the line contains the extra dramatic syllable. Be is similarly used by a jealous husband after "hope:"
" Ford. Well, I hope it be not so."—M. W. of W. ii. 1. 113.
where the hope is mixed with a great deal of doubt.
" I kissed it (the bracelet):
I hope it be not gone to tell my lord
That I kiss aught but he,"—Cymb, ii. 3. 153.
where, though the latter part is of course fanciful, there is a real fear that the bracelet may be lost.
Also, in a dependent sentence like the following :
" Prove true
That I, dear brother, be now ta'en for you."—T. N. iii. 4. 410.
Be follows " when," as "where" above, especially where when alludes to a future possibility.
" Haply a woman's voice may do some good
When articles too nicely urged be stood on."—Hen. V. v.2.93.
In " Alas, our frailty is the cause, not we,
For such as we are made, of such we be,"—T. N. ii. 2. 33.
it can scarcely be asserted that "for" is "for that" or "because." It is more probable that the scene originally ended there, and that Shakespeare used be in order to get the rhyme, which so often terminates a scene.
300. Be is much more common with the plural than the singular. Probably only this fact, and euphony, can account for
" When blood is nipp'd and ways be foul."—L. L. L. v. 2. 926.
In "When he sees reason of fears, as we do, his fears out of doubt be of the same relish as ours,"—Hen. V. iv. 1. 113. the be may partly be explained as not stating an independent fact, but a future event, dependent on the clause " when," &c. Partly, perhaps, " out of doubt" is treated like " there is no doubt that," and be follows in a kind of dependent clause.
Be is also used to refer to a number of persons, considered not individually, but as a kind or class.
"O, there be players that I have seen play, and heard others praise, and that highly, that," &c.—Hamlet, iii. 2. 32 ; ib. 44.
" There be some sports are painful."—Tempest, iii. 1. 1.
But it cannot be denied that the desire of euphony or variety seems sometimes the only reason for the use of be or are.
"Where is thy husband now ? Where be thy brothers ?
Where are thy children?"—Rich. III. iv. 4. 92.
301. Were. What has been said above of be applies to were, that it is often used as the subjunctive where any other verb would not be so used, and indeed where the subjunctive is unnecessary or wrong, after "if," "though," &c., and in dependent sentences.
In early authors there seems to have been a tendency to use should for shall, and were for be after "that" in subordinate sentences : " Go we fast that we were there." " Let us pray that he would. " My will is that it were so." In these sentences a wish is implied, and were, perhaps, indicates the desire that the wish should be fulfilled, not hereafter, but at once, as a thing of the past.
" I am a rogue, if I were not at half-sword with a dozen of them two hours together."—1 Hen. IV. ii. 4. 182.
"If there were anything in thy pocket but tavern reckonings, I am a villain."—1 Hen. IV. iii. 3. 180.
" What if we do omit
This reprobate till he were well inclined ? "—M. for M. iv. 3. 78.
In some of these passages there may be traced, perhaps, a change of thought : "I am a rogue (that is, I should be), if it were true that I was not," &c. " What if we omit (what if we were to omit) this reprobate till he were well inclined?"
"Duchess. I pray thee, pretty York, who told thee this ?
York. Grandam, his nurse.
Duchess. His nurse ! Why, she was dead ere thou wert born.
York. If 'twere not she, I cannot tell who told me."
Rich. III. ii. 4. 34.
"If ever Bassianus, Cæsar's son,
Were gracious in the eyes of royal Rome,
Keep then this passage to the Capitol."—T. A. i. 1. 11.
Comp. 2 Hen. IV. v. 2. 85 ; A. and C. i. 3. 41.
"No marvel, then, though he were ill-affected."—Lear, ii. 1. 100.
where the meaning is: "It is no wonder, then, that he was a traitor," and no doubt or future meaning is implied.
Somewhat similar is an idiom common in good authors even now: " It is not strange that he should have succeeded," for the shorter and simpler, " It is not strange that he succeeded."
"Lamachus, ... whom they sent hither, though he were waxen now somewhat old."—N. P. 172.
So, but with a notion of concession,
"And though (granting that) he were unsatisfied in getting,
Which was a sin, yet in bestowing, madam,
He was most princely."—Hen. VIII. iv. 2. 55.
" If it were so it was a grievous fault. "— . C. iii. 2. 84.
So, beginning with certainty :
" She that was ever fair and never proud. "—Othello, ii. 1. 149.
and ending with doubt :
" She was a wight, if ever such wight were."—Ib. ii. 1. 159.
In dependent sentences even after "know," as well as "think :"
" I would I had thy inches : thou shouldst know
There were a heart in Egypt."—A. and C. i. 3. 41.
" Which of your friends have I not strove to love,
Although I knew he were mine enemy."—Hen. VIII. ii. 4. 31.
"Imagine 'twere the right Vincentio."—T. of Sh. iv. 4. 12.
" As who should say in Rome no justice were."—T. A. iv. 3. 20.
" But that it eats our victuals, I should think
Here were a fairy."—Cymb. iii. 6. 42.
" He will lie, sir, with such volubility that you would think truth were a fool."—A. W. iv. 3. 285.
302. Were is used after " while" in
"If they would yield us but the superfluity while it were wholesome." —Coriol. i. 1. 18.
and, still more remarkably, after " until," referring to the past, in
"It hath been taught us from the primal state
That he which is, was wish'd until he were."
A. and C. i. 4. 42.
The following is contrary to our usage, though a natural attraction :
" And they it were that ravished our sister."—T. A. v. 3. 99.
for "it was they." See 425 at end.
Can. See May, 307.
303. Do, Did: original use. In Early as in modern English, the present and past indefinite of the indicative were generally represented by inflected forms, as "He comes," "He came, without the aid of do or did. Do was then used only in the sense of "to cause," " to make," &c. ; and in this sense was followed by an infinitive.
"They have done her understonde."—GOWER.
i.e. "they have caused her to understand."
Similarly it is used like the French " faire " or " laisser" with the ellipsis of the person who is " caused " to do the action, thus—
"Do stripen me and put me in a sakke,
And in the nexte river do me drenche."
CHAUCER, Marchante's Tale, 10,074.
i.e. "cause (some one) to strip me—to drench me."
In the same way "let " is repeatedly used in Early English :
" He let make Sir Kay seneschal of England."—Morte d'Arthur. where a later author might have written "he did make."
Gradually the force of the infinitive inflection en was weakened and forgotten; thus "do stripen" became "do strip," and do was used without any notion of causation.
Sometimes do is reduplicated, as :
" And thus he did do slen hem alle three."—CHAUCER, C. T. 7624. or used with " let," as in
" He let the feste of his nativitee
Don crien."—CHAUCER, C. T. 10, 360.
The verb was sometimes used transitively with an objective noun, as :
" He did thankingys."—WICKLIFFE, St. Matt. xv. 36.
and so in Shakespeare in
" Do me some charity."—Lear, iii. 4. 61.
" This fellow did the third (daughter) a blessing." Lear, i. 4. 115.
" Do my good-morrow to them."—Hen. V. iv. 1. 26.
" To do you salutation from his master." . C. iv. 2. 5 ; Rich. III. v. 3. 210.
"After the last enchantment you did here. "—T. N. iii. 1. 123.
and in the words " to don," i.e. " put on," and " dout," i.e. " put out."
But as a rule do had become a mere auxiliary, so that we even find it an auxiliary to itself, as in
" Who does do you wrong?"—T. N. v. 1. 143.
304. Do, did. How used by Shakespeare? In St. Matt. xv. 37, Wickliffe has "and alle eten ;" Tyndal, &c., "all dia eat." It is probable that one reason for inserting the did here was the similarity between the present and past of "eat," and the desire to avoid ambiguity. In the following verse, however, Wickliffe has "etun," Tyndal "ate," and the rest "did eat." This shows how variable was the use of did in the sixteenth century, and what slight causes determined its use or non-use. The following passage in connection with the above would seem to show that did was joined to eat to avoid ambiguity, and when it was not joined to other verbs :
" And the Peloponnesians did eat it up while the Byzantines died."—N. P. 180.
It can hardly be denied that in such lines as
" It lifted up it (so Folio) head, and did address
Itself to motion,"—Hamlet, i. 2. 216.
the did is omitted in the first verb and inserted in the second simply for the sake of the metre. Did is commonly used in excited narrative :
" Horses did neigh, and dying men did groan,
And ghosts did shriek and squeal about the streets."
. C. ii. 2. 23.
" The sheeted dead
Did squeak and gibber in the Roman streets."
Hamlet, i. 1. 116.
But in both the above passages the inflection in -ed is also used.
305. Verbs: "Do" omitted before "Not." In Early English the tenses were represented by their inflections, and there was no need of the auxiliary " do." As the inflections were disused, "do" came into use, and was frequently employed by Elizabethan authors. They, however, did not always observe the modern rule of using the auxiliary whenever not precedes the verb. Thus—
"I not doubt."—Temp. ii. 1. 121.
"Whereof the ewe not bites."—Ib. v. 1. 38.
"It not belongs to you."—2 Hen. IV. iv. 1. 98.
" It not appears to me."—Ib. 107.
" Hear you bad writers and though you not see." BEAUMONT on B. .
" On me whose all not equals Edward's moiety." Rich. III. i. 2. 259.
" Neat Terence, witty Plautus, now not please." B. J. on Shakespeare.
Less commonly in a subordinate sentence
" I beseech you . . . that you not delay."—Coriol. i. 6. 60.
Later, a rule was adopted that either the verb, or the auxiliary part of it, must precede the negative: "I doubt not," or "I do not doubt." Perhaps this may be explained as follows. The old English negative was "ne." It came before the verb, and was often supplemented by a negative adverb "nawicht," "nawt," "noht" (which are all different forms of "no whit" or "naught"), coming after the verb.
" His hors was good, but he ne was not gaie." CHAUCER, C. T. 74.
(Compare in French "ne . . . pas," in Latin, "non (nenu)," i.e. "ne . . . unum.") In the fifteenth century (Mätzner) this reduplication began to pass out of fashion. In Shakespeare's time it had been forgotten; but, perhaps, we may trace its influence in the double negative "nor will not," &c., which is common in his works.
" Vex not yourself, nor strive not with your breath." Rich. II. ii. 1. 3.
Possibly the idiom now under consideration is also a result of the Early English idiom. The not, which had ousted the old dual negative "ne" . . . "not," may have been thought entitled to a place either before or after the verb. Latin, moreover, would tend in the same direction. It must further be remembered that not is now less emphatic than it was, when it retained the meaning of "naught" or "no-whit." We can say, "I in-no-way trust you," or, perhaps, even " I no-whit trust you," but not is too unemphatic to allow us to say "I not trust you." Hence the "do" is now necessary to receive a part of the emphasis.
Not is sometimes found in E. E. and A.-S. between the subject and the verb, especially in subordinate sentences where the not, " no-whit," is emphatic.
306. Do, Did, omitted and inserted. In modern English prose there is now an established rule for the insertion and omission of do and did. They are inserted in negative and interrogative sentences, for the purpose of including the "not" or the subject of the interrogation between the two parts of the verb, so as to avoid ambiguity. Thus: "Do our subjects revolt?" "Do not forbid him." They are not inserted except for the purpose of unusual emphasis in indicative sentences such as "I remember." In Elizabethan English no such rule had yet been established, and we find—
"Revolt our subjects?"—Rich. II. iii. 2. 100.
"Forbid him not."—Mark ix. 39. E. V.
On the other hand—
"I do remember."—T. N. iii. 3. 48.
This licence of omission sometimes adds much to the beauty and vigour of expression.
"Gives not the hawthorn-bush a sweeter shade?" 3 Hen. VI. ii. 5. 42.
is far more natural and vigorous than
"Does not the hawthorn-bush give sweeter shade?"
307. Can, May, Might. May originally meant "to be able " (E. E. "mag;" A.-S. "magan;" German "mögen"). A trace of this meaning exists in the noun "might," which still means "ability." Thus we find
"I am so hungry that I may (can) not slepe." CHAUCER, Monke's Tale, 14, 744.
"Now help me, lady, sith ye may and can." Knighte's Tale, 2,314.
In the last passage may means "can," and "ye can " means "ye have knowledge or skill." This, the original meaning of "can," is found, though very rarely, in Shakespeare:
"I've seen myself and served against the French,
And they can well on horseback."—Hamlet, iv. 7. 85.
i.e.
" they are well skilled."
"And the priest in surplice white
That defunctive music can."—Phœnix and Turtle, 14.
And perhaps in
"The sum of all I can, I have disclosed;
Why or for what these nobles were committed
Is all unknown to me, my gracious lady."
Rich. III. ii. 4. 46.
"The strong'st suggestion
Our worser genius can "—Tempest, iv. 1. 27.
A trace of this emphatic use of can is found in
"What can man's wisdom
In the restoring his bereaved sense ?"—Lear, iv. 4. 8.
But, as "can" (which even in A.-S. meant "I know how to" and therefore "I am able") gradually began to encroach on may, and to assume the meaning "to be able," may was compelled to migrate from "ability" to " possibility " and "lawfulness. Thus "mögen" signifies moral, "können" physical, possibility. In the following passage:
"From hence it comes that this babe's bloody hand
May not be cleansed with water of this well,"—F. Q. ii. 10. it is not easy at once to determine whether may means "can" or "is destined," "must," "ought." Hence we are prepared for the transition which is illustrated thus by Bacon :
"For what he may do is of two kinds, what he may do as just and what he may do as possible."
308. May in "I may come" is therefore ambiguous, since it may signify either "lawfulness," as in "I may come if I like," or " possibility," as in "I may come, but don't wait for me." In the latter sentence the "possibility" is transposed so as to include the whole sentence "it is possible that I may come," just as—
"He needs not our mistrust,"—Macb. iii. 3. 2.
means "it is not necessary that we should mistrust him."
309. May is used with various shades of the meaning of "permission," "possibility," &c.:
"He shall know you better, sir, if I may live to report you." M. for M. iii. 2. 172.
i.e. "if I am permitted by heaven to live long enough."
It is a modest way of stating what ought to be well known, in
" If you may please to think I love the king."—W. T. iv. 4. 532.
"A score of ewes may be worth ten pounds."—2 Hen. IV. iii. 2.57. i.e. "is possibly worth ten pounds." "May be" is often thus used almost adverbially for possibly.
In
"Season your admiration for awhile Till I may deliver,"—Hamlet, i. 2. 193.
may means "can," "have time to."
" May (can) it be possible?"—Hen. V. ii. 2. 100.
310. May with a Negative. Thus far Elizabethan and modern English agree; but when a negative is introduced, a divergence appears.
In "I may not-come" may would with us mean "possibility," and the "not" would be connected with "come" instead of may; "my not-coming is a possibility." On the other hand, the Elizabethans frequently connect the "not" with may, and thus with them "I may-not come" might mean "I can-not or must-not come." Thus may is parallel to "must" in the following passage :—
"Yet I must not,
For certain friends that are both his and mine,
Whose loves I may not drop."—Macb. iii. I. 122.
Probably this disuse of may in "may not" (in the sense of "must not") may be explained by the fact that "may not" implies compulsion, and may has therefore been supplanted in this sense by the more compulsory "must."
311. May used for the old subjunctive in the sense of purpose.
If we compare Wickliffe's with the sixteenth-century Versions of the New Testament, it appears that, in the interval, the subjunctive had lost much of its force, and consequently the use of auxiliary verbs to supply the place of the subjunctive had largely increased.
In I Cor. iv. 8, Wickliffe has, "And I wold that ye regne, that also we regnen with you," where the later Versions, "And I would to God that ye did reign, that we also might reign." So also Col. i. 28: "Techynge eche man in al wisdom; that we offre eche man perfight," where the rest have "that we may offer" or " to offer." So ib. 25, "that I fille the word of God" for "that I may fulfil." But may is found very early used with its modal force
The subjunctive of purpose is found in—
"Go bid thy mistress. . . she strike upon the bell."—Macb. ii. I. 31.
"Sir, give me this water that I thirst not."—St. ohn iv. 15.
" He wills you, in the name of God Almighty,
That you divest yourself."—Hen. V. ii. 4. 78.
But it was not easy to distinguish the subjunctive representing an object, from the indicative representing a fact, since both were used after "that," and there was nothing but their inflections (which are similar in the plural) to distinguish the two. The following is an instance of the indicative following "that:"—
"But freshly looks and over-bears attaint
With cheerful semblance and sweet majesty,
That every wretch pining and pale before,
Beholding him, plucks comfort from his looks."
Hen. V. iv. Prologue, 39.
Hence arose the necessity, as the subjunctive inflections lost their force, of inserting some word denoting "possibility" or "futurity" to mark the subjunctive of purpose. "Will" is apparently used in this sense as follows:—
"Therefore in fierce tempest is he coming,
In thunder and in earthquake like a Jove,
That, if requiring fail, he will compel."—Hen. V. ii. 4. 99.
But, as a rule, may was used for the present subjunctive and might for the past, according to present usage. "That" is omitted in " Direct mine arms I may embrace his neck."—1 Hen. VI. ii. 5. 37. i.e. "that I may embrace."
In "Lord marshal, command our officers at arms
Be ready to direct these home alarms,"—Rich. II. i. 1. 204–5.
it is doubtful whether "be" is the subjunctive or the infinitive with "to" omitted (349). I prefer the former hypothesis, supplying "that" after "command." Compare
"Some one take order Buckingham be brought
To Salisbury."—Rich. III. iv. 4. 539.
So "that" is omitted before "shall :"
"The queen hath heartily consented he shall espouse Elizabeth." Rich. III. iv. 5. 18.
312. Might, the past tense of may, was originally used in the sense of "was able" or "could."
"He was of grete elde and might not travaile."—R. BRUNNE.
So
"That mought not be distinguish'd."—3 Hen. VI. v. 2. 45.
" So loving to my mother,
That he might not beteem the winds of heaven
Visit her face too roughly."—Hamlet, i. 2. 141.
i.e. "could not bring himself to allow the winds," &c.
It answers to "can" in the following :—
"Ang. Look, what I will not that I cannot do.
Isab. But might you do't, and do the world no wrong?"
M. for M. ii. 2. 52.
"Might you not know she would do as she has done?" A. W. iii. 4. 2.
i.e. "Could you not know."
" I might not this believe
Without the sensible and true avouch
Of mine own eyes."—Hamlet, i. 1. 56.
"But I might see young Cupid's fiery shaft quench'd in the chaste beams of the wat'ry moon."—M. N. D. ii. 1. 161.
"In that day's feats,
When he might act the woman in the scene,
He proved best man i' the field."—Coriol. ii. 2. 100.
i.e. "when he was young enough to be able to play the part of a woman on the stage."
Might naturally followed may through the above-mentioned changes. Care must be taken to distinguish between the indicative and the conditional use of might. "How might that be?" (indicative) would mean "How was it possible for that to take place?" On the other hand, "How might that be?" (subjunctive) would mean " How would it be possible hereafter that this should take place?" The same ambiguity still attends "could." Thus "How could I thus forget myself yesterday!" but "How could I atone to-morrow for my forgetfulness yesterday ? "
313. May, Might, like other verbs in Elizabethan English, are frequently used optatively. We still use may thus, as in "May he prosper!" but seldom or never might. But it is clear that—
"Would I might
But ever see that man,"—Temp. i. 2. 168.
naturally passes into "Might I but see that man," Thus we have—
"Lord worshipped might he be."—M. of V. ii. 2. 98.
314. Must (E. E. moste) is the past tense of the E. E. present tense mot, which means "he is able," "he is obliged." From meaning "he had power to do it," or "might have done it," the word came to mean "ought," and it is by us generally used with a notion of compulsion. But it is sometimes used by Shakespeare to mean no more than definite futurity, like our "is to" in "He is to be here to-morrow."
"He must fight singly to-morrow with Hector, and is so prophetically proud of an heroical cudgelling that he raves in saying nothing."—Tr. and Cr. iii. 3. 247.
So, or nearly so, probably in
"Descend, for you must be my sword-bearer." M. of V. ii. 6. 40.
And somewhat similar, without the notion of compulsion, is the use in M. of V. iv. 1. 182; M. N. D. ii. 1. 72.
It seems to mean "is, or was, destined" in
"And I must be from thence."—Macbeth, iv. 3. 212.
So
"A life which must not yield To one of woman born."—Ib. v. 8. 12.
315. Shall. Shall for will. Shall meaning "to owe" is connected with "ought," "must," "it is destined."
Thus,
"If then we shall shake off our slavish yoke,
Imp out our drooping country's broken wing,
Away with me."—Rich. II. ii. 2. 291.
i.e. "if we are to, ought to."
"Fair Jessica shall be my torch-bearer."—M. of V. ii. 4. 40.
i.e. "is to be."
Hence shall was used by the Elizabethan authors with all three persons to denote inevitable futurity without reference to "will" (desire).
"If much you note him,
You shall offend him and extend his passion."—Macb. iii. 4. 57.
i.e. "you are sure to offend him."
So probably,
"Nay, it will please him well, Kate, it shall (is sure to) please him." Hen. V. v. 2. 369.
"My country
Shall have more vices than it had before."—Macb. iv. 3. 47.
" And, if I die, no man shall pity me. "—Rich. III. v. 3. 201.
i.e. "it is certain that no man will pity me."
There is no notion of compulsion on the part of the person speaking in
"They shall (are sure to) be apprehended by and by." Hen. V. ii. 2. 2.
"If they do this (conquer),
As, if please God, they shall (are destined to do)."
Hen. V. iv. 3. 120.
The notion of necessity, must, seems to be conveyed in
"He that parts us shall bring a brand from heaven,
And fire us hence like foxes."—Lear, v. 3. 22.
In " He shall wear his crown,"— C. i. 3. 87. shall means "is to." So in
"Your grace shall understand."—M. of V. iv. 1. 149.
"What is he that shall (is to) buy? "—A. Y. L. ii. 4. 88.
"Men shall deal unadvisedly sometimes." Rich. III. iv. 4. 292.
i.e. "men cannot help making mistakes."
" He that escapes me without some broken limb shall (must, will have to), acquit him well."—A. Y. L. i. 1. 134.
"K. Desire them all to my pavilion.
Glost. We shall, my lord."—Hen. V. iv. 1. 27.
In the last passage, " I shall" has a trace of its old meaning, " I ought:" or perhaps there is a mixture of " I am bound to" and "I am sure to." Hence it is often used in the replies of inferiors to superiors.
" King Henry. Collect them all together at my tent:
I'll be before thee.
Erpingham. I shall do't, my lord."—Hen. V. iv. 1. 305.
"Fear not, my lord, your servant shall do so." M. N. D. ii. 1. 268.
So A. W. v. 3. 27; A. and C. iii. 12. 36, iv. 6. 3, v. 1. 3; Hen. V. iv. 3. 126 ; M. for M. iv. 4. 21; A. and C. v. 1. 68.
" You shall see, find," &c., was especially common in the meaning "you may," "you will," applied to that which is of common occurrence, or so evident that it cannot but be seen.
"You shall mark
Many a duteous and knee-crooking slave,
That, doting on his own obsequious bondage,
Wears out his time. Whip me such honest knaves."
Othello, i. 1. 440.
Shall is sometimes colloquially or provincially abbreviated into se, s:
"Thou's hear our counsel."—R. and . i. 3. 9.
"I'se try."—Lear, iv. 6. 246. (See 461.)
316. Will. You will. He will. Later, a reluctance to apply a word meaning necessity and implying compulsion to a person addressed (second person), or spoken of (third person), caused post-Elizabethan writers to substitute will for shall with respect to the second and third persons, even where no will at all, i.e. no purpose, is expressed, but only futurity. Thus will has to do duty both as will proper, implying purpose, and also as will improper, implying merely futurity. Owing to this unfortunate imposition of double work upon will, it is sometimes impossible to determine, except from emphasis or from the context, whether will signifies purpose or mere futurity. Thus (1) "He will come, I cannot prevent him," means " He wills (or is determined) to come ;" but (2) "He will come, though unwillingly," means "His coming is certain."
Will is seldom used without another verb:
" I will no reconcilement."—Hamlet, v. 2. 258.
So in "I will none of it." (See 321.)
317. Shall. You shall. He shall. On the other hand shall, being deprived by will of its meaning of futurity, gradually took up the meaning of compulsory necessity imposed by the first person on the second or third. Thus: "You shall not go," or even "You shall find I am truly grateful." (Not "you will find," but "I will so act that you shall perforce find," &c.)
The prophetic shall ("it shall come to pass ") which is so common in the Authorized Version of the Bible, probably conveyed to the original translators little or nothing more than the meaning of futurity. But now with us the prophetic shall implies that the prophet identifies himself with the necessity which he enunciates. Thus the Druid prophesying the fall of Rome to Boadicea says—
"Rome shall perish. "—COWPER.
318. Shall. I shall. When a person speaks of his own future actions as inevitable, he often regards them as inevitable only because fixed by himself. Hence " I shall not forgive you " means simply, "I have fixed not to forgive you;" but "I shall be drowned," "My drowning is fixed." (See 315.)
319. Will. "I will." Some passages which are quoted to prove that Shakespeare used will with the first person without implying wish, desire, &c., do not warrant such an inference.
In Hamlet, v. 2. 183, "I will win for him, if I can; if not, I will gain nothing but my shame and the odd hits," the will is probably used by attraction with a jesting reference to the previous "will:" "My purpose is to win if I can, or, if not, to gain shame and the odd hits."
" There is no hope that ever I will stay
If the first hour I shrink and run away."—I Hen. VI. iv. 5. 30.
i.e. " Trere is no hope of my ever being willing to stay."
" I'll do well yet."—Coriol. iv. 1. 21.
i.e. "I intend to do well yet."
" I will not reason what is meant hereby,
Because I will (desire to) be guiltless of the meaning."
Rich. III. i. 4. 95.
In "I will sooner have a beard grow in the palm of my hand than he shall get one on his cheek,"—2 Hen. IV. i. 2. 23. there is a slight meaning of purpose, as though it were, "I will sooner make a beard grow," derived from the similarity in sound of the common phrase " I will sooner die, starve, than, &c."
In "Good argument, I hope, we will not fly,"—Hen. V. iv. 3. 113. the meaning appears to be "good argument, I hope, that we have no intention of flying."
There is a difficulty in the expression "perchance I will;" but, from its constant recurrence, it would seem to be a regular idiom. Compare the following passages :—
"Perchance, Iago, I will ne'er go home."—Othello, v. 2. 197.
"Perchance I will be there as soon as you."—C. of E. iv. 1. 39.
"Perhaps I will return immediately."—M. of V. ii. 5. 52.
In all these passages "perchance" precedes, and the meaning seems to be in the last example, for instance: "My purpose may, perhaps, be fulfilled," and "my purpose is to return immediately," or, in other words, "If possible, I intend to return immediately." In all these cases, the "perhaps" stands by itself. It does not qualify "will," but the whole of the following sentence.
In "I will live to be thankful to thee for't,"—T. N. iv. 2. 88. the will refers, not to live, but to "live-to-be-thankful," and the sentence means "I purpose in my future life to prove my thankfulness."
320. Will is sometimes used with the second person (like the Greek optative with ν) to signify an imperative. It is somewhat ironical, like our "You will be kind enough to be quiet." Perhaps originally an ellipsis, as in Greek, was consciously understood, "You will be quiet (if you are wise)," &c.
"You'll leave your noise anon, ye rascals."—Hen. VIII. v. 4. 1.
In "Gloucester, thou wilt answer this before the pope," 1 Hen. VI. i. 3. 52. there is no imperative, but there is irony.
On the other hand, "you will," perhaps, means "you are willing and prepared" in :
"Portia. You know I say nothing to him : he hath neither Latin, French, nor Italian, and you will come into court and swear that I have a poor pennyworth in the English."—M. of V. i. 2. 75.
321. Will, with the third person. Difficult passages.
The following is a perplexing passage :—
" If it will not be (i.e. if you will not leave me) I'll leave you."—M. Ado, ii. 1. 208.
Here the meaning seems to be "if it is not to be otherwise," and in Elizabethan English we might expect shall. But probably "it" represents fate, and, as in the phrase, "come what will," the future is personified: "If fate will not be as I would have it." And this explains
"What shall become of (as the result of) this? What will this do?"—M. Ado, iv. 1. 211.
The indefinite unknown consequence is not personified, the definite project is personified. "What is destined to result from this project? What does this project intend to do for us?"
"My eye will scarcely see it,"—Hen. V. ii. 2. 104.
means " can scarcely be induced to see it."
"He will" means "he will have it that," "he pretends," in
" This is a riddling merchant for the nonce;
He will be here, and yet he is not here."—1 Hen. VI. ii. 3. 58.
In " She'll none of me,"—T. N. i. 3. 113.
"will" means "desires," "none" "nothing," and "of" "as regards" (173), "to do with."
322. Should. Should is the past tense of shall, and underwent the same modifications of meaning as shall. Hence should is not now used with the second person to denote mere futurity, since it suggests a notion, if not of compulsion, at least of bounden duty. But in a conditional phrase, "If you should refuse," there can be no suspicion of compulsion. We therefore retain this use of should in the conditional clause, but use would in the consequent clause :
" If you should refuse, you would do wrong."
On the other hand, Shakespeare used should in both clauses:
"You should refuse to perform your father's will if you should
refuse to accept him."—M. of V. i. 2. 100.
And should is frequently thus used to denote contingent futurity.
"They told me here, at dead time of the night,
Ten thousand swelling toads, as many urchins,
Would make such fearful and confused cries,
As any mortal body hearing it
Should straight fall mad."—T.A. ii. 3. 102, 104,
"Would" = "were in the habit." Comp. φíλoυν.
"(In that case) Strength should be lord of imbecility,
And the rude son should strike the father dead ;
Force should be right."—Tr. and Cr. i. 3. 114.
323. Should for ought. Should, the past tense, not being so imperious as shall, the present, is still retained in the sense of ought, applying to all three persons. In the Elizabethan authors, however, it was more commonly thus used, often where we should use ought:
" You should be women ;
And yet your beards forbid me to interpret
That you are so."—Macbeth, i. 3. 45,
"So should he look that seems to speak things strange." Ib. i. 2. 46.
"I should report that which I say I saw,
But know not how to do it. "—Ib. v. 5. 31.
"Why 'tis an office of discovery, love,
And I should be obscured."—M. of V. ii. 6. 44.
i.e. "A torch-bearer's office reveals (439) the face, and mine ought to be hidden."
324. Should is sometimes used as though it were the past tense of a verb "shall," meaning "is to," not quite "ought." Compare the German "sollen."
"About his son that should (was to) have married a shepherd's daughter."—W. T. iv. 4. 795.
" The Senate heard them and received them curteously, and the people the next day should (were to) assemble in counsell to give them audience."—N. P. Alcibiades, 170.
In the following, should is half-way between the meaning of "ought" and "was to." The present, shall, or "am to," might be expected ; but there is perhaps an implied past tense, "I (you said) was to knock you."
" Petruchio. And rap me well, or I'll knock your knave's pate.
Grumio. My master is grown quarrelsome: I should knock you, And then I know after who comes by the worse."
T. of Sh. i. 1. 131.
325. Should was hence used in direct questions about the past, where shall was used about the future. Thus, "How shall the enemy break in?" i.e. "How is the enemy to break in?" became, when referred to the past, "How was the enemy to break in?"
" I was employ'd in passing to and fro
About relieving of the sentinels.
Then how or which way should they first break in?"
1 Hen. VI. ii. 1. 71.
" What should this mean?"—Hen. VIII. iii. 2. 160.
i.e. "what was this (destined, likely) to mean?" It seems to increase the emphasis of the interrogation, since a doubt about the past (time having been given for investigation) implies more perplexity than a doubt about the future. So we still say, "Who could it be?" "How old might you be?"
"What should be in that Cæsar?"— . C. i. 2. 142.
i.e. "what could there be," "what might there be." "Shall," "may," and the modern "can," are closely connected in meaning.
" Where should he have this gold?"—T. of A. iv. 3. 398.
In the following instance, should depends upon a verb in the present ; but the verb follows the dependent clause, which may, therefore, be regarded as practically an independent question.
" What it should be . . . I cannot dream of."—Hamlet, ii. 2. 7.
But also
" Put not yourself into amazement how should these things be." M. for M. iv. 2. 220.
326. Should was used in a subordinate sentence after a simple past tense, where shall was used in the subordinate sentence after a simple present, a complete present, or a future. Hence we may expect to find should more common in Elizabethan writers than with us, in proportion as shall was also more common. We say "I will wait till he comes," and very often, also, "I intended to wait till he came." The Elizabethans more correctly, " I will wait till he shall come ;" and therefore, also, "I intended to wait till he should come." Thus, since it was possible to say "I ask that I shall slay him," Wickliffe could write "They axeden of Pilate that thei schulden sle hym " (Acts xiii. 28); "They aspiden hym that thei schulden fynde cause" (Luke vi. 7). In both cases we should now say "might."
So
" She replied, It should be better he became her guest."—A. and C. ii. 2. 226.
" Thou knew'st too well
My heart was to thy rudder tied by the strings,
And thou shouldst tow me after."—Ib. iii. 11. 58.
The verb need not be expressed, as in
"A lioness lay crouching . . . with cat-like watch,
When that the sleeping man should stir."—A. Y. L. iv. 2. 117.
"She has a poison which shall kill you," becomes
" She did confess she had
For you a mortal mineral, which being took
Should by the minute feed on life."—Cymb. v. 5. 51.
This perhaps explains
"Why, 'tis well known that whiles I was protector,
Pity was all the fault that was in me,
For I should melt at an offender's tears,
And lowly words were ransom for their fault."
2 Hen. VI. iii. 1. 126.
" All my fault is that I shall melt (am sure to melt)," would become "all my fault was that I should melt ;" "for" meaning " for that" or "because."
"And (Fol.) if an angel should have come to me,
And told me Hubert should put out mine eyes,
I would not have believed him."—K. . iv. 1. 68-70.
Here, since the Elizabethans could say "Hubert shall," they can also say "he told me Hubert should."
So since the Elizabethans could say "To think that deceit shall steal such gentle shapes," they could also say, regarding the subor. dinate clause as referring to the past,
"Oh, that deceit should steal such gentle shapes!" Rich. III. ii. 2. 27.
" Good God, (to think that) these nobles should such stomachs bear !"—1 Hen. VI. i. 3. 90.
327. "Should have with the second and third persons. The use of "should have " with the second and third persons is to be noted. It there refers to the past, and the should simply gives a conditional force to "have." It is incongruous to use should in connection with the past, and hence we now say "If an angel had come" in this sense. When we use "should have," it refers to a question about the past which is to be answered in the future. "If he should have forgotten the key, how should we get out," i.e "if, when he comes, it should turn out that he had forgotten." Compare, on the other hand, the Shakespearian usage.
" Gods, if you
Should have ta'en vengeance on my faults, I never
Had lived to put on this."—Cymb. v. 1. 8.
In M. Ado, ii. 3. 81, the "should have" is inserted, not in the conditional clause, but in a dependent relative clause. "If it had been a dog that should have howled thus, they would have killed him."
328. " Should," denoting a statement not made by the speaker. (Compare "sollen" in German.) There is no other reason for the use of should in
" But didst thou hear without wonder how thy name should be so hanged and carved about these trees."—A. Y. L. iii. 2. 182.
Should seems to indicate a false story in George Fox's Journal :
"From this man's words was a slander raised upon us that the Quakers should deny Christ," p. 43 (Edition 1765). "The priest of that church raised many wicked slanders upon me : 'That I rode upon a great black horse, and that I should give a fellow money to follow me when I was on my black horse.' "
"Why should you think that I should woo in scorn? " M. N. D. iii. 2. 122.
329. Would for will, wish, require. Would, like should, could, ought, (Latin "potui," "debui,") is frequently used conditionally. Hence "I would be great" comes to mean, not "I wished to be great," but "I wished (subjunctive)," i.e. "I should wish." There is, however, very little difference between "thou wouldest wish" and "thou wishest," as is seen in the following passage :—
"Thou wouldst (wishest to) be great,
Art not without ambition, but without
The illness should (that ought to) attend it: what thou
wouldst highly
That thou wouldst holily, wouldst not play false,
And yet wouldst wrongly win."—Macbeth, i. 5. 20.
As will is used for "will have it," "pretends," so would means "pretended," "wished to prove."
" She that would be your wife. "—C. of E. iv. 4. 152.
i.e. " She that wished to make out that she was your wife."
So "One that would circumvent God."—Hamlet, v. 1. 87.
Applied to inanimate objects, a "wish" becomes a " requirement:"
"I have brought
Golden opinions from all sorts of people,
Which would (require to) be worn now in their newest gloss."
Macbeth, i. 7. 32.
"Words
Which would (require to) be howled out in the desert air."
Ib. iv. 3. 194.
"And so he goes to heaven,
And so am I revenged. That would (requires to) be scann'd."
Hamlet, iii. 3. 75.
"This would (requires to) be done with a demure abasing of your eye sometimes."—B. E. 92.
It is a natural and common mistake to say, "Would is used for should, by Elizabethan writers."
Would is not often used for "desire" with a noun as its object :
"If, duke of Burgundy, you would the peace."
Hen. V. v. 2. 68.
330. Would often means "liked," "was accustomed." Compare φíλει.
"A little quiver fellow, and a' would manage his piece thus : and a' would about and about, and come you in and come you out ; rah-tah-tah would a' say, bounce would a' say: and away again would a' go, and again would a' come."—2 Hen. IV. iii. 2. 200.
" It (conscience) was wont to hold me only while one would tell twenty."—Rich. III. i. 4. 122.
"But still the house affairs would draw her hence." Othello, i. 3. 147.
So, though more rarely, will is used for "is accustomed."
"Sometimes a thousand twangling instruments
Will hum about mine ears.—Tempest, iii. 2. 147.
331. " Would " not used for "should." Would seems on a superficial view to be used for should, in
"You amaze me; I would have thought her spirit had been invincible against all assaults of affection."—M. Ado, ii. 3. 119. But it is explained by the following reply : "I would have sworn it had," i.e. "I was ready and willing to swear." So, "I was willing and prepared to think her spirit invincible."
So in
" What power is in Agrippa, If I would say, 'Agrippa, be it so,'
To make this good?"—A. and C. ii. 2. 144.
'If I would say" means "If I wished, were disposed, to say."
" Alas, and would you take the letter of her?"—A. W. iii. 4. 1.
i.e. "Were you willing," "Could you bring yourself to."
To take would for should would take from the sense of the following passage:
"For I mine own gain'd knowledge should profane
If I would time expend with such a snipe,
But for my sport and profit."—Othello, i. 3. 390.
i.e. "If I were willing to expend."
Would probably means "wish to" or "should like to," in
" You could, for a need, study a speech which I would set down and insert in't, could you not ?"—Hamlet, ii. 2. 567.
In
"Prince. What wouldest thou think of me, if I should weep? Poins. I would think thee a most princely hypocrite."
2 Hen. IV. ii. 2. 59.
the second would is attracted to the first, and there is also a notion of determination, and voluntary "making up one's mind" in the reply of Poins.
So "be triumphant" is equivalent to "triumph," in which willingness is expressed, in
"Think you, but that I know our state secure,
I would be so triumphant as I am ?"—Rich. III. iii. 2. 84.
i.e. "think you I would triumph as I do? "
In
" I would be sorry, sir, but the fool should be as oft with your master as with my mistress,"—T. N. iii. 1. 44.
it must be confessed there seems little reason for would. Inasmuch, however, as the fool is speaking of something that depends upon himself, i.e. his presence at the Count's court, it may perhaps be explained as "I would not willingly do anything to prevent," &c., just as we can say "I would be loth to offend him," in confusion between "I should be loth to offend him" and " I would not willingly, or I would rather not, offend him."
In
" And how unwillingly I left the ring, When nought would be accepted but the ring,"
M. of V. v. 1. 197.
there seems, as in our modern "nothing would content him but," some confusion between "he would accept nothing" and "nothing could make itself acceptable."
# VERBS, INFLECTIONS OF.
332. Verbs: Indicative Present, old forms of the Third Person Plural. There were three forms of the plural in Early English—the Northern in es, the Midland in en, the Southern in eth : "they hop-es," "they hop-en," "they hop-eth." The two former forms (the last in the verbs "doth," " hath," and possibly in others) are found in Shakespeare. Sometimes they are used for the sake of the rhyme; sometimes that explanation is insufficient :
En.—
" Where, when men be-en, there's seldom ease." Pericles, ii. Gower, 28.
"O friar, these are faults that are not seen,
Ours open and of worst example be-en."—B. J. S Sh. i. 2.
" All perishen of men of pelf,
Ne aught escapen but himself."—Pericles, ii. Gower, 36.
" As fresh as bin the flowers in May."—PEELE.
" Words fearen (terrify) babes."—SPENS. F. Q.
" And then the whole quire hold their hips and laugh,
And waxen in their mirth."—M. N. D. ii. 1. 56.
This form is rarely used by Shakespeare, and only archaically. As an archaic form it is selected for constant use by Spenser.
333. Third person plural in -s. This form is extremely common in the Folio. It is generally altered by modern editors, so that its commonness has not been duly recognized. Fortunately, there are some passages where the rhyme or metre has made alteration impossible. In some cases the subject-noun may be considered as singular in thought, e.g. "manners," &c. In other cases the quasi-singular verb precedes the plural object; and again, in others the verb has for its nominative two singular nouns or an antecedent to a plural noun (see 247). But though such instances are not of equal value with an instance like "his tears runs down," yet they indicate a general predilection for the inflection in -s which may well have arisen from the northern E. E. third person plural in -s.
"The venom clamours of a jealous woman
Poisons more deadly than a mad dog's tooth."
C. of E. v. 1. 69.
"The great man down, you mark his favourites flies,
The poor advanced makes friends of enemies."
Hamlet, iii. 2. 214-5.
Here the Globe reads "favourite;" completely missing, as it seems to me, the intention to describe the crowd of favourites scattering in flight from the fallen patron.
"The extreme parts of time extremely forms
All causes to the purpose of his will."—L. L. L. v. 2. 750.
"Manners" is, perhaps, used as a singular in
"What manners is in this ?"—R. and . v. 3. 214.
"Which very manners urges."—Lear, v. 3. 234.
So
"Whose church-like humours fits not for a crown." 2 Hen. VI. i. 1. 247.
" Riches" may, perhaps, be considered a singular noun (as it is by derivation, " richesse ") in
"The riches of the ship is come ashore."—Othello, ii. 1. 83.
But not
" My old bones aches" (Globe, ache).—Tempest, iii. 2. 2.
" His tears runs down his beard like winter-drops" (Globe, run). Ib. v. 1. 16.
" We poor unfledg'd
Have never wing'd from view o ' the nest, nor knows not
What air's from home" (Globe, know).—Cymb. iii. 3. 27.
" And worthier than himself
Here tends (Globe and Quarto, tend) the savage strangeness he
puts on,
Disguise the holy strength of their command," &c.
Tr. and Cr. ii. 3. 135.
" These naughty times
Puts (Globe, put) bars between the owners and their rights."
M. of V. iii. 2. 19.
" These high wild hills and rough uneven ways
Draws out our miles, and makes them weansome."
Rich. II. ii. 3. 5.
" Not for all the sun sees, or
The close earth wombs, or the profound seas hides."
(Globe, sea.)—W. T. iv. 4. 501.
" The imperious seas breeds monsters" (Globe, breed). Cymb. iv. 2. 35.
" Untimely storms makes men expect a dearth (Globe, make). Rich. III. ii. 3. 33.
Numbers, perhaps, sometimes stand on a different footing :
" Eight yards of uneven ground is three score and ten miles afoot with me."—1 Hen. IV. ii. 2. 28. i.e. "A distance of eight yards;" and compare
" Three parts of him is ours already."— .C. i. 3. 154.
" Two of both kinds makes up four."—M. N. D. iii. 2. 438.
But no such explanation avails in
" She lifts the coffer-lids that close his eyes,
Where, lo ! two lamps burnt out in darkness lies."
V. and A. 1128.
" Whose own hard dealings teaches them suspect
The deeds of others. "—M. of V. i. 3. 163.
" Those pretty wrongs that liberty commits
Thy beauty and thy years full well befits."—Sonn. 41.
There is some confusion in
"Fortune's blows
When most struck home, being gentle wounded craves
A noble cunning."—Coriol. iv. 4. 8.
On the whole, it is probable that though Shakespeare intended to make "blows" the subject of " craves," he afterwards introduced a new subject, "being gentle," and therefore "blows" must be considered nominative absolute and "when" redundant: "Fortune's blows (being) struck home, to be gentle then requires a noble wisdom."
"Words to the heat of deeds too cold breath gives,"
in a rhyming passage.
Macbeth, ii. 1. 61.
It is perhaps intended to be a sign of low breeding and harsh writing in the play of Pyramus and Thisbe.
"Thisbe, the flowers of odours savours sweet." M. N. D. iii. 1. 84.
334. Third person plural in -th.
"Those that through renowne hath ennobled their life."
See, however, Relative, 247.
MONTAIGNE, 32.
"Their encounters, though not personal, hath been royally encountered" (Globe, have).—W. T. i. 1. 29.
"Where men enforced doth speak anything."—M. of V. iii. 2. 33.
"Hath all his ventures fail'd?" (Globe, have.)—Ib. iii. 2. 270.
This, however, is a case when the verb precedes the subject. (See below, 335.)
335. Inflection in -s preceding a plural subject. Passages in which the quasi-singular verb precedes the plural subject stand on a somewhat different footing. When the subject is as yet future and, as it were, unsettled, the third person singular might be regarded as the normal inflection. Such passages are very common, particularly in the case of "There is," as—
"There is no more such masters."—Cymb. iv. 2. 371.
"There was at the beginning certaine light suspitions and accusations put up against him."—N. P. 173.
"Of enjoin'd penitents there's four or five."—A. W. iii. 5. 98.
"The spirit upon whose weal depends and rests
The lives of many." —Hamlet, iii. 3. 14.
"Then what intends these forces thou dost bring?" 2 Hen. VI. v. 1. 60.
"There is no woman's sides can," &c.—T. N. ii. 4. 96.
"Is there not charms?"—Othello, i. 1. 172.
"Is all things well?"—2 Hen. VI. iii. 2. 11.
"Is there not wars? Is there not employment ?" 2 Hen. IV. i. 2. 85.
So 1 Hen. VI. iii. 2. 123; R. and . i. 1. 48; 2 Hen. IV. iii. 2. 199; 1 Hen. VI. iii. 2. 9.
"Here comes the townsmen."—2 Hen. VI. ii. 1. 68.
"Here comes the gardeners" (Globe, come).—Rich. II. iii. 4. 24.
"There comes no swaggerers here."—2 Hen. IV. ii. 4. 83.
This, it is true, comes from Mrs. Quickly, but the following are from Posthumus and Valentine:
"How comes these staggers on me?"—Cymb. v. 5. 233.
" Far behind his worth
Comes all the praises that I now bestow."—T. G. of V. ii. 4. 72.
And in the Lover's Complaint, where the rhyme makes alteration impossible:
" And to their audit comes
Their distract parcels in combined sums."—L. C. 230.
"What cares these roarers for the name of king?"—Temp. i. 1. 17.
"There grows all herbs fit to cool looser flames." B. and F. F. Sh. i. 1.
"There was the first gentlemanlike tears that ever we shed." W. T. v. 2. 155.
"Has his daughters brought him to this pass?" (Globe, have.) Lear, iii. 4. 65.
"What means your graces?" (Globe, mean.)—Ib. iii. 7. 30.
" But most miserable
Is the desires that's (247) glorious" (Globe, desire).—Cymb. i. 6. 6.
("Few" and "more" might, perhaps, be considered nouns in
"Here's a few flowers."—Cymb. iv. 2. 283.
"There is no more such masters."—Ib. iv. 2. 371.
A sum of money also can be considered as a singular noun:
"Forthy three thousand ducats here is six."—M. of V. iv. 1. 84.)
" There lies
Two kinsmen (who) digged their graves with weeping eyes."
Rich. II. iii. 3. 168.
"Sir, there lies such secrets in this fardell and box." W. T. iv. 4. 783.
"At this hour
Lies at my mercy all mine enemies" (Globe, lie).
Tempest, iv. 1. 264.
336. Inflection in "s" with two singular nouns as subject.
The inflection in s is of frequent occurrence also when two or more singular nouns precede the verb:
" The heaviness and guilt within my bosom
Takes off my manhood."—Cymb. iv. 2. 2.
"Faith and troth bids them."—Tr. and Cr. iv. 5. 170.
"Plenty and peace breeds cowards."—Cymb. iii. 5. 21.
"For women's fear and love holds quantity."—Hamlet, iii. 2. 177.
"Where death and danger dogs the heels of worth." A. W. iii. 4 15.
"Scorn and derision never comes (Globe and Quarto, come) in tears."—M. N. D. iii. 2. 123.
"Thy weal and woe are both of them extremes,
Despair and hope makes thee ridiculous."—V. and A. 988.
" My hand and ring is yours."—Cymb. ii. 4. 57.
"O, Cymbeline, heaven and my conscience knows." Ib. iii. 3. 99.
" Hanging and wiving goes by destiny."—M. of V. ii. 9. 83.
"The which my love and some necessity
Now lays upon you."—M. of V. iii. 4. 34.
337. Apparent cases of the inflection in "s."
Often, however, a verb preceded by a plural noun (the apparent nominative) has for its real nominative, not the noun, but the noun clause.
"The combatants being kin
Half stints their strife before they do begin. "—Tr. and Cr. iv. 5. 93.
i.e. "The fact that the combatants are kin."
"Wherein his brains still beating puts him thus
From fashion of himself."—Hamlet, iii. 1: 182.
i. e. "The beating of his brains on this."
"And our ills told us
Is as our earing."—A. and C. i. 2. 115.
i.e. "The telling us of our faults is like ploughing us."
"And great affections wrestling in thy bosom
Doth make an earthquake of nobility."—K. . v. 2. 42.
" To know our enemies' minds we 'ld rip their hearts:
(To rip) Their papers is more lawful."—Lear, iv. 6. 266.
So in
"Blest be those,
How mean soe'er, that have their honest wills,
Which seasons comfort,"—Cymb. i. 6. 8.
"which" has for its antecedent "having one's honest will."
Conversely, a plural is implied, and hence the verb is in the plural, in
"Men's flesh preserv'd so whole do seldom win." 2 Hen. VI. iii. 1. 301.
i.e. "when men are too careful about their safety they seldom win."
"Smile heaven (the gods, or the stars) upon this fair conjunction,
That long have frowned upon their enmity."—Rich. III. v. 5. 21.
It may be conjectured that this licence, as well as the licence of using the -s inflection where the verb precedes, or where the noun clause may be considered the nominative, would in all probability not have been tolerated but for the fact that -s was still recognized as a provincial plural inflection.
The following is simply a case of transposition:
"Now, sir, the sound that tells what hour it is
Are clamorous groans."—Rich. II. v. 5. 56.
338. S final misprinted. Though the rhyme and metre establish the fact that Shakespeare used the plural verbal inflection in s, yet it ought to be stated that -s final in the Folio is often a misprint. Being indicated by a mere line at the end of a word in MS., it was often confused with the comma, full stop, dash or hyphen.
" Comes (,) shall we in ?"—T. of A. i. 1. 284.
"At that that I have kil'd my lord, a Flys."—T. A. iii. 2. 53.
"Good man, these joyful tears show thy true hearts." Hen. VIII. v. 3. 175.
Conversely, in one or two places the dash or hyphen has usurped the place of the s.
"Unkle, what newe—?"—1 Hen. IV. v. 2. 30.
" With gobbets of thy Mother-bleedíng heart." 2 Hen. VI. iv. 1. 85.
Sometimes (even without the possibility of mistake for a comma) the -s is inserted:
"Sir Protheus, your Fathers call's for you."—T. G. of V. i. 3. 88. "Sawcie Lictors
Will catch at us like Strumpets, and scald Rimers
Ballads us out of tune."—A. and C. v. 2. 216.
Yet in many passages the -s is probably correct, though we should now omit it, especially at the end of nouns. As we still use "riches," "gains," almost as singular nouns, so Shakespeare seems to have used "lands," "wars," "stones," "sorrows," "flatteries," "purposes," "virtues," "glories," "fortunes," "things," "attempts," "graces," "treasons," "succours," "behaviours," "duties," "funerals," " proceedings," &c. as collective nouns.
In other cases there seems at least a method in the error. The -s is added to plural adjectives and to adjectives or nouns dependent upon nouns inflected in "s," as
"The letters patents."—Rich. II. ii. 1. 202 (Folio).
It is common in E. E. for plural adjectives of Romance origin to take the plural inflection. But see 430. The Globe reads "patents" in Rich. II. ii. 3. 130.
The following are selected, without verification, from Walker:
"Kings Richards throne."—Rich. II. i. 3.
"Smooth and welcomes newes."—1 Hen. IV. i. 1.
"Lords Staffords death."—Ib. v. 3.
"The Thicks-lips."—Othello, i. 1.
A word already plural sometimes receives an additional plural inflection:
"Your teethes."— . C. v. 1.
"Others faults."—1Hen. IV. v. 2.
"Men look'd . . . each at others."—Coriol. v. 5.
"Boths."—T. A. ii. 4. "On others grounds."—Othello, i. 1.
339. Past indicative forms in u are very common in Shakespeare. Thus, "sang" does not occur, while "sung" is common as a past indicative. "Sprang" is less common as a past tense than "sprung" (2 Hen. IV. i. 1. 111). "Begun" (Hamlet, iii. 2. 220) is not uncommon for "began," which is also used. We also find
" I drunk him to his bed."—A. and C. i. 5. 21.
Past indicative tenses in u were common in the seventeenth century, but the irregularity dates from the regular Early English idiom.
In A.-S. the second person singular, and the three plural persons of some verbs, e.g. "singan," had the same vowel u, while the first and third persons singular had a. Hence, though the distinction was observed pretty regularly in E. E., yet gradually the u and a were used indiscriminately in the past tense without distinction of person.
340. Second Person Singular in -ts. In verbs ending with -t, -test final in the second person sing. often becomes -ts for euphony. Thus: "Thou torments," Rich. II. iv. 1. 270 (Folio); "Thou requests," Rich. III. ii. 1. 98 (Folio); "revisits" Hamlet, i. 4. 53; "splits," M. for M. ii. 2. 115; "exists," Ib. iii. 1. 20 (Folio); "solicites," Cymb. i. 6. 147 (Folio); "refts," Cymb. iii. 3. 103 (Folio). "Thou fleets," Sonn. 19; this is marked in
"What art thou call'st . . . and affrights?" B. and F. F. Sh. iv. 1.
This termination in -s contains perhaps a trace of the influence of the northern inflection in -s for the second pers. sing.
341. Past Indicative: -t for -ted. In verbs in which the infinitive ends in -t, -ed is often omitted in the past indicative for euphony.
" I fast and prayed for their Intelligence."—Cymb. iv. 2. 347
"There they hoist us."—Tempest, i. 2. 147.
"Plunged in the foaming brine and quit the vessel."—Ib. 211.
"When service sweat for duty, not for meed."—A. Y. L. ii. 3. 58.
" Stood Dido ... and waft her love
To come again to Carthage."—M. of V. v. 1. 10.
Compare Hen. VIII. ii. 1. 33; M. of V. iii. 2: 205.
We find "bid" for "bided," i.e. "endured," in
"Endured of (by) her for whom you bid like sorrow." Rich. III. iv. 4. 304.
This is, of course, as natural as "chid," "rid," &c., which are recognized forms. On the other hand, the termination in -ed is sometimes used for a stronger form:
"I shaked."—Tempest, ii. 1. 319.
342. Participle: -ed omitted after d and t. Some verbs ending in -te, -t, and -d, on account of their already resembling participles in their terminations, do not add -ed in the participle. The same rule, naturally dictated by euphony, is found in E. E. "If the root of a verb end in -d or -t doubled or preceded by another consonant, the -de or -te of the past tense, and -d or -t of the past participle, are omitted." Thus—
Acquit.—"Well hast thou acquit thee."—Rich. III. v. 5. 3.
Addict.—Mirror for Magistrates (NARES).
Articulate.—"These things indeed you have articulate." 1 Hen. IV. v. 1. 72.
Betid.—Tempest, i. 2. 31.
Bloat(ed).—"Let the bloat king tempt you."—Hamlet, iii. 4. 182.
Contract.—" He was contract to lady Lucy."—Rich. III. iii. 7. 179.
Degenerate.—" They have degenerate."—B. E. 38.
Deject.—"And I of ladies most deject and wretched." Hamlet, iii. 1. 163.
Devote.—T. of Sh. i. 1. 32.
Disjoint for disjointed.—Hamlet, i. 2. 20.
Enshieid.—" An enshield beauty."—M. for M. ii. 4. 80.
Exhaust.—" Their means are less exhaust."—B. E. 16.
Graft.—"Her noble stock graft with ignoble plants." Rich. III. iii. 7. 127.
Compare "An ingraft infirmity."—Othello, ii. 3. 144.
Heat.—" The iron of itself, though heat red-hot."—K. . iv. 1. 61.
Hoist.—" For 'tis the sport to have the enginer
Hoist with his own petard."—Hamlet, iii. 4. 207.
Infect.—" Many are infect."—Tr. and Cr. i. 3. 188.
Quit.—"The very rats instinctively have quit it."—Temp. i. 2. 147.
Suffocate.—"Degree is suffocate."—Tr. and Cr. i. 3. 125.
Taint.—"Unspotted heart never yet taint with love." Wed.—Hen. VIII. ii. 1. 141. 1 Hen. VI. v. 3. 183.
Wed.—Hen. VIII. ii. 1. 141.
Waft.
"A braver choice of dauntless spirits Than now the English bottoms have waft o'er."—K. . ii. 1. 73.
Wet.—Rich. III. i. 2. 216.
Whist (for" whisted," which is used by Surrey in the indicative).
"The wild waves whist."—Tempest, i. 2. 379.
i.e. "being whisted or made silent." So, in imitation,
"The winds, with wonder whist,
Smoothly the waters kist."—MILTON, Hymn on the Nativity.
Words like "miscreate," Hen. V. i. 2. 16; "create," M. N. D. v. 1. 412, "consecrate," Ib. 422, being directly derived from Latin participles, stand on a different footing, and may themselves be regarded as participial adjectives, without the addition of d.
343. Participles, Formation of. Owing to the tendency to drop the inflection en, the Elizabethan authors frequently used the curtailed forms of past participles which are common in Early English: "I have spoke, forgot, writ, chid," &c.
"Have you chose this man? "—Coriol. ii. 3. 163.
Where, however, the form thus curtailed was in danger of being confused with the infinitive, as in "taken," they used the past tense for the participle:
Arose.—"And thereupon these errors are arose."—C. of E. v. 1. 388.
Drove for driven.—2 Hen. VI. iii. 2. 84.
Eat.—"Thou . . . hast eat thy bearer up."—2 Hen. IV. iv. 5. 165;
M. Ado, iv. 1. 196.
Froze for frozen.—C. of E. v. 1. 313; 2 Hen. IV. i. 1. 199.
Holp.—"We were. . . holp hither."—Temp. i. 2. 63.
(In this case, however, the en is merely dropped.)
Took.—"Where I have took them up."— . C. ii. 1. 50.
Mistook.—"Then, Brutus, I have much mistook your passion." Ib. i. 2. 48.
Rode for ridden.—2 Hen. IV. v. 3. 98; Hen. V. iv. 3. 2.
Smit for smitten.—T. of A. ii. 1. 123.
Smote for smitten.—Coriol. iii. 1. 319.
Strove for striven.—Hen. VIII. ii. 4. 30.
Writ.—Rich. II. ii. 1. 14.
Wrote for written.—Lear, i. 2. 93; Cymb. iii. 5. 21.
Or sometimes the form in ed:
"O, when degree is shaked."—Tr. and Cr. i. 3. 101.
So Hen. V. ii. 1. 124 ; Temp. ii. 1. 39; 1 Hen. IV. iii. 1. 17. But shook for shaken is also common.
"The wind-shaked surge."—Othello, ii. 1. 13.
"Ope" in "The gates are ope," Coriol. i. 4. 43, seems to be the adjective "open" without the -n, and not a verb.
344. Irregular participial formations. The following are irregular :—
"You have swam."—A. Y. L. iv. 1. 38.
"I have spake."—Hen. VIII. ii. 4. 153.
"Misbecomed."—L. L. L. v. 2. 778.
"Becomed."—Cymb. v. 5. 406.
"Which thou hast perpendicularly fell"—Lear, iv. 6. 54.
"We had droven them home."—A. and C. iv. 7. 5.
"Sawn" for "seen" is found as a rhyme to "drawn," L. C. 91.
"Sirucken."—C. of E. i. 1. 46; L. L. L. iv. 3. 224; . C. iii. 1. 209.
"When they are fretten with the gusts of heaven." M. of V. iv, 1. 77.
"Sweaten."—Macbeth, iv. 1. 65. (So Quartos.)
Caught seems to be distinguished as an adjective from the participle catch'd in
"None are so surely caught when they are catch'd
As wit turned fool. "—L. L. L. v. 2. 69.
The following are unusual:—
"Splitted."—C. of E. i. 1. 105, v. 1. 308; A. and C. v. 1. 24.
"Seated."—Sonn. 62.
The following are archaic :—
"Marcus, unknit that sorrow-wreathen knot. "—T.A. iii. 2. 4.
"Foughten."—Hen. V. iv. 6. 18.
345. The participial prefix y- is only two or three times used in Shakespeare's plays: "y-clept," "y-clad," "y-slaked." In E. E. y- is prefixed to other forms of speech beside participles, like the German ge-. But in Elizabethan English the y- was wholly disused except as a participial prefix, and even the latter was archaic. Hence we must explain as follows:
" The sum of this
Brought hither to Pentapolis
Yravished the regions round. "—P. of T. iii. Gower, 35.
Shakespeare was probably going to write (as in the same speech, line 1, "yslaked hath") "yravished the regions hath," but the necessity of the rhyme, and the diminished sense of the grammatical force of the participial prefix, made him alter the construction. The y- is used by Sackville before a present participle, "y-causing." In M. of V. ii. 9. 68, and elsewhere, we find "I wiss" apparently for the old "y-wiss."
# VERBS, MOODS AND TENSES.
346. Indicative simple present for complete present with adverbs signifying "as yet," &c.
This is in accordance with the Latin idiom, "jampridem opto," &c., and it is explicable on the ground that, when an action continued up to the present time is still continuing, the speaker may prefer the verb to dwell simply on the fact that the action is present, allowing the adverb to express the past continuousness:
"That's the worst tidings that I hear of yet." 1 Hen. IV. iv. 1. 127.
"How does your honour for this many a day?"—Hamlet, iii. 1. 91.
347. Simple past for complete present with "since," &c.
This is in accordance with the Greek use of the aorist, and it is as logical as our more modern use. The difference depends upon a difference of thought, the action being regarded simply as past without reference to the present or to completion.
"I saw him not these many years, and yet
I know 'tis he."—Cymb. iv. 2. 66.
"I saw not better sport these seven years' day."—2 Hen. VI. ii. 1. 3.
"Since death of my dear'st mother
It did not speak before."—Cymb. iv. 2. 190.
"I did not see him since."—A. and C. i. 3. 1.
"I was not angry since I came in France
Until this instant."—Hen. V. iv. 7. 58.
"I can tell you strange news that you yet dreamed not of."—M. Ado, i. 2. 4.
It will be noticed that the above examples all contain a negative. The indefinite tense seems to have peculiar propriety when we are denying that an action was performed at any time whatever. Hence the contrast:
"Judges and senates have been bought with gold,
Esteem and love were never to be sold."
POPE, Essay on Man, iv. 187.
But we have also, without a negative,
" And since I saw thee,
The affliction of my mind amends."—Tempest, v. 1. 114.
The simple present is in the following example incorrectly combined with the complete present. But the two verbs are so far apart that they may almost be regarded as belonging to different sentences, especially as "but" may be regarded as semi-adversative.
"And never since the middle summer's spring
Met we. . . but. . . thou hast disturbed our sport."
M. N. D. ii. 1. 83–7.
On the other hand, the complete present is used remarkably in—
"D. Pedro. Runs not this speech like iron through your blood?
Claud. I have drunk poison whiles he utter'd it."
M. Ado, v. 1. 253.
This can only be explained by a slight change of thought: "I have drunk poison (and drunk [339] poison all the) while he spoke."
348. Future for Subjunctive and Infinitive. The future is often used where we should use the infinitive or subjunctive.
A comparison of Wickliffe with the versions of the sixteenth century would show that in many cases the Early English subjunctive had been replaced by the Elizabethan "shall."
" And I will sing that they shall hear I am not afraid." M. N. D. iii. 1. 126.
"That you shall surely find him
Lead to the Sagittary the raised search."—Othello, i. 1. 158.
"That thou shalt see the difference of our spirits,
I pardon thee thy life before thou ask it."—M. of V. iv. 1. 368.
"Therefore in fierce tempest is he coming
That, if requiring fail, he will compel."—Hen. V. ii. 4. 101.
Here, however (283), "so" may be omitted before "that," i.e. "so that he purposes compulsion if fair means fail."
" Reason with the fellow,
Lest you shall chance to whip your information."
Coriol. iv. 6. 53.
" If thou refuse and wilt encounter with my wrath." W. T. ii. 3. 138.
" The constable desires thee thou wilt mind
Thy followers of repentance."—Hen. V. iv. 3. 84.
" Will you permit that I shall stand condemn'd ?" Rich. II. ii. 3. 119.
So with "for" used for "because" (117) in the sense of "in order that."
"And, for the time shall not seem tedious,
I'll tell thee what befel me."—3 Hen. VI. iii. 1. 10.
As in Latin, the future is sometimes correctly and logically used with reference to future occurrences; but we find it side by side with the incorrect and modern idiom.
" Farewell till we shall meet again."—M. of V. iii. 4. 40.
" He that outlives this day and comes safe home,
He that shall live this day and see old age."
Hen. V. iv. 3. 40.
"All France will be replete with mirth and joy,
When they shall hear how we have play'd the men."
1 Hen. VI. i. 6. 16.
"When they shall know."—Rich. II. i. 4. 49.
"If you shall see Cordelia."—Lear, iii. 1. 46.
" Till your strong hand shall help to give him strength." K. J. ii. 1. 133.
The future seems used (perhaps with reference to the original meaning of "shall") to signify necessary and habitual recurrence in
" Good Lord, what madness rules in brain-sick men
When for so slight and frivolous a cause
Such factious emulations shall arise."—IHen. VI. iv. 1. 113.
So "Men shall deal unadvisedly sometimes." Rich. III. iv. 4. 293.
349. Infinitive. " To " omitted and inserted. In Early English the present infinitive was represented by -en (A.-S. -an), so that "to speak" was "speken," and "he is able to speak" was "he can speken," which, though very rare, is found in Pericles, ii. Prologue, 12. The -en in time became -e, and the -e in time became mute; thus reducing "sing-en" to "sing." When the en dropped into disuse, and to was substituted for it, several verbs which we call auxiliary, and which are closely and commonly connected with other verbs, retained the old licence of omitting to, though the infinitival inflection was lost. But naturally, in the Elizabethan period, while this distinction between auxiliary and non-auxiliary verbs was gradually gaining force, there was some difference of opinion as to which verbs did, and which did not, require the "to," and in Early English there is much inconsistency in this respect. Thus in consecutive lines "ought" is used without, and "let" with, "to."
"And though we owe the fall of Troy requite,
Yet let revenge thereof from gods to light."
Mirror for Magistrates (quoted by Dr. GUEST).
"You ought not walk."—J. C. i. 1. 3.
"Suffer him speak no more."—B. J. Sejan. iii. 1.
"If the Senate still command me serve."—Ib. iii. 1,
"The rest I wish thee gather."—I Hen. VI. ii. 5. 96.
"You were wont be civil."—Othello, ii. 3. 190.
"I list not prophesy."—W. T. iv. 1. 26.
"He thought have slaine her"—SPENS. F. Q. i. 1. 50.
"It forst him slacke."—Ib. 19.
"Stay" is probably a verb in
"How long within this wood intend you (to) stay?" M. N. D. ii. 1. 138.
"Desire her (to) call her wisdom to her."—Lear, iv. 5. 35.
"As one near death to those that wish him (to) live." A. W. ii. 1. 134.
"What might'st thou do that honour would (wished) thee (to) do?"—Hen. V. Prologue, 18.
"That wish'd him in the barren mountains (to) starve." 1 Hen. IV. i. 3. 159.
So M. for M. iv. 3. 138; M. Ado, iii. 1. 42. Hence "overlook " is probably not the subjunctive (see however 369) but the infinitive in
"Willing you (to) overlook this pedigree."—Hen. V. ii. 4. 90.
So after "have need :"
"Thou hadst need send for more money,"—T. N. ii. 3. 99.
"Vouchsafe me speak a word. "—C. of E. v. 1. 282.
"To come view fair Portia."—M. of V. ii. 7. 43.
"We'll come dress you straight."—M. W. of W. iv. 2. 80.
" I will go seek the king."—Hamlet, ii. 1. 101.
We still retain a dislike to use the formal to after "go" and "come," which may almost be called auxiliaries, and we therefore say, "I will come and see you."
We cannot reject now the to after "know" (though after this word we seldom use the infinitive at all, and prefer to use the conjunction "that"), but Shakespeare has
"Knowing thy heart (to) torment me with disdain."—Sonn. 132.
A similar omission is found in
"That they would suffer these abominations
By our strong arms from forth her fair streets (to be) chased."
R. of L. 1634.
So
"Because, my lord, we would have had you (to have) heard The traitor speak."—Rich. III. iii. 5. 56.
To is inserted after "let" both in the sense of "suffer" and in that of "hinder."
"And let (suffer) no quarrel nor no brawl to come." T. N. v. 1. 364.
"If nothing lets (prevents) to make us happy both."—Ib. 256.
On the other hand, to is omitted after "beteem " in the sense of "suffer:"
"He might not beteem the winds of heaven
Visit her face too roughly."—Hamlet, i. 2. 141.
After "durst:"
"I durst, my lord, to wager she is honest."—Othello, iv. 2. 11.
The to is often inserted after verbs of perceiving,—"feel," "see," "hear," &c.
"Who heard me to deny it?"—C. of E. v. 1. 25.
"Myself have heard a voice to call him so." 2 Hen. VI. ii. 1. 94.
" Whom when on ground she grovelling saw to roll." SPENS. F. Q. v. 7. 32.
"Methinks I feel this youth's perfections
To creep in at mine eyes."—T. N. i. 5. 317.
" I had rather hear you to solicit that."—Ib. iii. 1. 120.
"To see great Hercules whipping a gig,
And profound Solomon to tune a jig,
And Nestor play at push-pin with the boys."
L. L. L. iv. 3. 167-9.
This quotation shows that, after "see," the infinitive, whether with or without "to," is equivalent to the participle. "Whipping," "to tune," and "play," are all co-ordinate. The participial form is the most correct: as in Latin, "Audivi illam canentem;" modern English, "I heard her sing;" Elizabethan English, "I heard her to sing." The infinitive with to after verbs of perception occurs rarely, if ever, in Early English (Mätzner quotes Wickliffe, St. John xii. 18, but ?). It seems to have been on the increase towards the end of the sixteenth century, for whereas Wickliffe (St. Matt. xv. 31) has "The puple wondride seynge dumb men spekynge and crokid men goynge, blynde men seyinge," Tyndale (1534) has "The people wondred to se the domme speak, the maymed whole, the halt to go, and the blynde to se ;" and the A. V. (1611) has to throughout. This idiom is also very common in North, and Florio's "Montaigne." We have recurred to the idiom of Early English.
Compare William of Palerne, 1. 871: "and whan he bat semly sitte him bi-fore," i.e. "and when he saw her in her beauty sit before him." In this quotation we might render "sitte" by the participle " sitting," as the girl is regarded as "in the state of sitting." This opens the question of the origin of the phrase "to see great Hercules whipping" Is "whipping," by derivation, a verbal abbreviated for "a-whipping," as in 93, or a present participle? The common construction after "see" and "hear" in Layamon and William of Palerne seems to be neither the participle nor the verbal, but the infinitive in -e or -en. Probably, when the infinitive inflection died out, it was felt that the short uninflected form was not weighty enough to express the emphatic infinitive, and recourse was had to the present participle, a substitution which was aided by the similarity of the terminations -en and -ing. This is one of the many cases in which the terminations of the infinitive and present participle have been confused together (93), and the -ing in this construction represents the old infinitive inflection -en. This may explain:
" I my brother know
Yet living (to live) in my glass."—T. N. iii. 4. 415.
i.e. "that my brother lives."
Hence, perhaps, also -ing was added as a reminiscence of the old gerundive termination -ene, in such expressions as
"Put the liveries to making."—M. of V. ii. 2. 124.
Similarly we find, side by side, in Selden's " Table Talk," "He fell to eating" and he " fell to eat."
350. "To" omitted and inserted in the same sentence. The to is often omitted in the former of two clauses and inserted in the latter, particularly when the finite principal verb is an auxiliary, or like an auxiliary.
" Whether hadst thou rather be a Faulconbridge
And, like thy brother, to enjoy thy land."—K. J. i. 1. 134.
" I would no more
Endure this wooden slavery than to suffer
The flesh-fly blow my mouth."—Tempest, iii. 1. 62.
" Who would be so mock'd with glory, or to live
But in a dream of friendship?"—T. of A. iv. 2. 33.
So K. J. v. 2. 138-9 ; J. C. iv. 3. 73 ; T. N. v. I. 346.
" Sir, I desire you (to) do me right and justice,
And to bestow your pity on me."—Hen. VIII. ii. 4. 14.
" Bids you
Deliver up the crown and to take pity."—Hen. V. ii. 4. 104.
" Makes both my body pine and soul to languish." P. of T. i. 1. 31.
" Make thy two eyes like stars start from their spheres,
Thy knotted and combined locks to part. —Hamlet, i. 4. 18.
" Brutus had rather be a villager
Than to repute himself a son of Rome."—J. C. i. 2. 173.
" She tells me she'll wed the stranger knight,
Or never more to view nor day nor night. "—P. of T. ii. 5. 17.
"Some pagan shore,
Where these two Christian armies might combine
The blood of malice in a vein of league,
And not to spend it so unneighbourly."—K. J. v. 2. 39.
Thus probably we must explain:
"And let them all encircle him about,
And fairy-like to pinch the unclean knight."
M. W. of W. iv. 4. 57.
The common explanation "to-pinch," attributes to Shakespeare an archaism which is probably nowhere found in his works (not even in P. of T. iii. 2. 17). See All to, 28.
It is a question how to explain
" She is abus'd, stol'n from me and corrupted
By spells and medicines bought of mountebanks:
For nature so preposterously to err,
Being not deficient, blind or lame of sense,
Sans witchcraft could not."—Othello, i. 3. 62.
Here, either as above, (1) " to err" depends on "could," i. e. "Nature was not able to err;" or (2) " could not might perhaps stand for "could not be," "was impossible," having for its subject "Nature to err." (See 354.) In (2) "for" may be either (a) a conjunction, or (b) a preposition: "It was not possible for Nature thus to err." I prefer (1).
In
"For little office The hateful commons will perform for us
Except, like curs, to tear us all to pieces," Rich. II. ii. 2. 139.
"to tear" may be considered as a noun, the object of "except."
351. It were best (to). To is often omitted after "best" in such phrases as "it were best," "thou wert best," &c. Perhaps there is in some of these cases an unconscious blending of two constructions, the infinitive and imperative, exactly corresponding to the Greek o σθ' o ν δρ σoν.
" 'Tis best put finger in the eye."—T. of Sh. i. 1. 78.
" I were best not call."—Cymb. iii. 6. 19.
" 'Twere best not know myself."—Macbeth, i. 2. 73.
" Best draw my sword."—Cymb. iii. 6. 25.
In most of these cases the speaker is speaking of himself: but often it is impossible, without the context, to tell whether the verb is in the infinitive or imperative. Thus in
" Better be with the dead,"—Macbeth, iii. 2. 20.
it is only the following line,
" Whom we, to gain our peace, have sent to peace,"
that shows that be is infinitive. When we now use this idiom, we generally intend the verb to be used imperatively.
352. I were best (to). The construction
" Thou wert better gall the devil."—K. J. iv. 3. 94.
" I were best leave him. "—1 Hen. VI. v. 3. 82.
" Madam, you're best consider."—Cymb. iii. 2. 79.
like the modern construction "if you please," (in which we should now say, and be correct in saying, that "you," is the subject, though it was originally the object, of "please,") represents an old impersonal idiom: "Me were liefer," i.e. "it would be more pleasant to me;" " Me were loth;" " Him were better." Very early, however, the personal construction is found side by side with the impersonal. The change seems to have arisen from an erroneous feeling that " Me were better" was ungrammatical. Sometimes the to is inserted :
" You were best to go to bed."—2 Hen. VI. v. 1. 196.
"You were best to tell Antonio what he said."—M. of V. ii. 8. 33.
353. " To" omitted after Conjunctions.
Where two infinitives are coupled together by a conjunction, the to is still omitted in the former, where the latter happens to be nearer to the principal verb, e.g. after "rather than." " Rather than see himself disgraced, he preferred to die." But we could not say
" Will you be so good, scauld knave, as eat it ?"—Hen. V. v. 1. 31.
This is probably to be explained, like the above, as a blending of two constructions—the infinitive, " Will you be so good as to eat it and the imperative, "Eat it, will you be so good ?"
In
" Under the which he shall not choose but fall." Hamlet, iv. 7. 66.
"Nay then, indeed she cannot choose but hate thee." Rich. III. iv. 4. 289.
" Thou shalt not choose but go"—T. N. iv. 1. 61.
the obvious and grammatical construction is "he shall not choose anything except (to) fall;" "she cannot choose anything except (to) hate thee;" but probably (contrary to Mätzner's view, iii. 18) the explanation of the omission is, that Shakespeare mentally supplies "shall," "can," &c. " He shall not choose anything else, but (shall) fall." This is supported by
" Who ... cannot choose but they must blab."—Othello, iv. 1. 28.
354. Noun and infinitive used as subject or object.
It might be thought that this was a Latinism. But a somewhat similar use of the infinitive with a noun in impersonal sentences is often found in E. E. and, though rarely, in A.-S.
" No wondur is a lewid man to ruste."—CHAUCER, C. T. 504.
"It is ful fair a man to bear him even."—Ib. 1525.
"It spedith one man for to die for be puple."—WICKLIFFE, St John xviiii. 14.
(So Matzner, but Bagster has " that o man,") i.e. "that one man should die."
" It is the lesser fault, modesty finds,
Women to change their shapes than men their minds."
T. G. of V. v. 4. 109.
" As in an early spring
We see the appearing buds which to prove fruit
Hope gives not so much warrant as despair
That frosts will bite them. "—2 Hen. IV. i. 3. 39.
" This to prove true
I do engage my life."—A. Y. L. v. 4. 171.
" Be then desir'd
A little to disquantity your train,
And the remainder that shall still depend
To be such men that shall besort your age." —Lear, i. 4. 272.
In the following instance " brags of" is used like "boasts :"
" Verona brags of him
To be a virtuous and well-govern'd youth."—R. and J. i. 5. 70.
"I have deserv'd
All tongues to talk their bitterest."—W. T. iii. 2. 217.
"(This) is all as monstrous to our human reason
As my Antigonus to break his grave."—Ib. v. 1. 42.
" O that self-chain about his neck
Which he foreswore most monstrously to have."
C. of E. v. 1. 11; Rich. III. iv. 4. 337.
Add perhaps
"The duke Will never grant this forfeiture to hold"—M. of V. iii. 3. 25.
though" forfeiture" may be personified, and " grant "used like "allow." We retain this use, but transpose "for" in "for to "(see the example from Wickliffe above) and place it before the noun or pronoun:
"For me to put him to his purgation would perhaps plunge him into far more choler."—Hamlet, iii. 2. 317.
355. The Infinitive used as a Noun. This use is still retained when the Infinitive is the subject of a verb, as "To walk is pleasant ; " but we should not now say—
" What's sweet to do to do will aptly find."—L. C. 13.
" My operant powers their functions leave to do." Hamlet, iii. 2. 184 ; ib. iii. 4. 66.
" Have not to do with him."—Rick. III. i. 3. 292.
So 3 Hen. VI. iv. 5. 2.
" Metaphors far-fet hinder to be understood."—B. J. Disc. 757.
Apparently to is omitted in the following curious passage :—
" For to (to) have this absolute power of Dictator they added never to be afraid to be deposed."—N. P. 611.
It is doubtful whether the infinitive is a noun in the objective in
"Nor has he with him to supply his life."—T. of A. iv. 1. 46.
i.e. "the power of supplying;" or whether "anything" is understood: "He has not anything to supply his livelihood."
We can say " I was denied my rights," but not
" I am denied to sue my livery here."—Rich. III. ii. 3. 129.
356. Infinitive, indefinitely used. To was originally used not with the infinitive but with the gerund in -e, and, like the Latin " ad" with the gerund, denoted a purpose. Thus "to love" was originally "to lovene," i.e. "to (or toward) loving" (ad amandum). Gradually, as to superseded the proper infinitival inflection, to was used in other and more indefinite senses, "for," "about," "in," " as regards," and, in a word, for any form of the gerund as well as for the infinitive.
"To fright you thus methinks I am too savage." Macb. iv. 2. 70.
Not " too savage to fright you," but " in or for frighting you."
"I was too strict to make mine own away."—Rich. II. i. 3. 243.
i.e. "I was too severe to myself in sacrificing my son."
"Too proud to be (of being) so valiant."—Coriol. i. 1. 263.
"I will not shame myself to give you (by giving you) this." M. of V. iv. 1. 431.
" Make moan to be abridged." —Ib. i. r. 126.
Not, "in order to be," but, "about being abridged."
" Who then shall blame
His pester'd senses to recoil and start."—Macb. v. 2. 22.
i.e. "for recoiling." Comp. T. of Sh. iii. 2. 27; A. Y. L. v. 2. 110.
" O, who shall hinder me to wail and weep?" Rich. III. ii. 2. 27.
i.e. "as regards, or from, wailing."
"But I shall grieve you to report (by reporting) the rest." Rich. II. ii. 2. 95.
"You might have saved me my pains to have taken away the ring." T. N. ii. 2. 6.
i.e. "by having taken away."
" I the truer, so to be (for being) false with you." Cymb. i. 5. 44.
"Lest the State shut itself out to take any penalty for the same."—B. E. 158.
i.e. "as regards taking any penalty." We still say, "I fear to do it," where "to" has no meaning of purpose; but Bacon wrote—
" Young men care not to innovate."—B. E. 161.
"are not cautious about innovating." So Tr. and Cr. v. 1. 71.
This gerundive use of the infinitive is common after the verb " to mean :"
" What mean these masterless and gory swords
To lie discolour'd by this place of peace ?"—R. and J. v. 3.143.
" What mean you, sir,
To give them this discomfort ? "—A. and C. iv. 1. 34.
So Tr. and Cr. v. 1. 30.
"To weep to have that which it fears to lose."—Sonn. 64.
i.e. "to weep because of having, because-it has."
We say, " I took eleven hours to write it," or " I spent eleven hours in writing," not
"Eleven hours I spent to write it over." Rich. III. iii. 6. 5 ; M. of V. i. 1. 154.
" But thou strik'st me
Sorely, to say (in saying) I did."—W. T. v. 1. 18.
"You scarce can right me throughly then to say You did mistake."—Ib. ii. 1. 99.
i.e. "by saying."
" I know not what I shall incur to pass it."—Ib. ii. 2. 57.
i.e. "I know not what penalty I shall incur as the consequence of, or for, letting it pass."
" You're well to live. "—W. T. iii. 3. 121.
i.e. "You are well off as regards living," resembles our modern, "You are well to do." The infinitive thus used is seldom preceded by an object :
"So that, conclusions to be as kisses, if your (221) four negatives
Make your two affirmatives, why then," &c.—T. N. v. 1. 22.
" What ! I, that kill'd her husband and his father,
To take her in her heart's extremest hate!" "
Rich. III. i. 2. 231-2.
From 216 it will be seen that the English pronoun, when it represents the Latin accusative before the infinitive, is often found in the nominative. The following is a curious instance of the ambiguity attending this idiom :—
" I do beseech your grace
To have some conference with your grace alone."
Rich. II. v. 3. 27.
i.e. "about having some conference," and here, as the context shows, "that I may have some conference."
Equally ambiguous, with a precisely opposite interpretation, is
"Sir, the queen
Desires your visitation, and to be
Acquainted with this stranger."—Hen. VIII. v. 1. 169.
i.e. "and that you will become acquainted."
"Of him I gather'd honour
Which he to seek (seeking) of me again perforce
Behoves me keep at utterance. "—Cymb. iii. 2. 73.
Probably we must thus explain:
"Thou'lt torture me to leave unspoken that
Which, to be spoke, would torture thee."—Ib. v. 5. 139.
i.e. "You wish to torture me for leaving unspoken that which, by being spoken, would torture you."
"Foul is most foul being foul to be a scoffer," A. Y. L. iii. 5. 62.
seems to mean " foulness is most foul when its foulness consists in being scornful."
357. "To" frequently stands at the beginning of a sentence in the above indefinite signification. Thus Macb. iv. 2. 70, quoted above, and—
" To do this deed,
Promotion follows."—W. T. i. 2. 356.
" To know my deed, 'twere best not know myself." Macbeth, ii. 2. 73.
" To say to go with you, I cannot."—B. J. E. out &c. iv. 6.
" To belie him I will not. "—A. W. iv. 3. 299.
" Other of them may have crooked noses, but to owe (as regards owning) such straight arms, none. "—Cymb. iii. 1. 38.
"For of one grief grafted alone,
To graft another thereupon,
A surer crab we can have none. "—HEYWOOD.
" To lack or lose that we would win
So that our fault is not therein,
What woe or want end or begin? "—Ib.
" To sue to live, I find I seek to die,
And seeking death find life,"—M. for M. iii. 1. 43.
where "to sue to live" means "as regards suing to live," and corresponds to "seeking death."
This indefinite use of the infinitive in a gerundive sense seems to be a continuation of the old idiom which combined to with the gerund.
Less frequently the clause depends on "that :"
" But that I'll give my voice on Richard's side,
God knows I will not do it."—Rich. III. iii. 1. 53.
358. For to. When the notion of purpose is to be brought out, for to is often used instead of to, and in other cases also. Similarly the Danish and Swedish languages (Mätzner) have "for at," and the old French has "por (pour) à," with the infinitive. For to is still more common in Early English than in Elizabethan.
359. Infinitive active is often found where we use the passive, as in
" Yet, if men moved him, was he such a storm
As oft 'twixt May and April is to see."—L. C. 102.
This is especially common in "what's to do" (T. N. iii. 3. 18; &c.) for "what's to be done. " See Ellipses, 405, and compare
"Savage, extreme, rude, cruel, not to trust"—Sonn. 129.
i.e. "not to be trusted."
360. Infinitive, complete Present. It is now commonly asserted that such expressions as "I hoped to have seen him yesterday" are ungrammatical. But in the Elizabethan as in Early English authors, after verbs of hoping, intending, or verbs signifying that something ought to have been done but was not, the Complete Present Infinitive is used. We still retain this idiom in the expression, " I would (i.e. wished to) have done it." " I ought (i.e. was bound) to have done it." But we find in Shakespeare—
" I hoped thou shouldst have been my Hamlet's wife;
I thought thy bride-bed to have deck'd, sweet maid."
Hamlet, v. 1. 268.
" Thought to have begg'd."—Cymb. iii. 6. 48.
In
"Levied an army weening to redeem, And have install'd me in the diadem,"—Hen. VI. ii. 5. 89,
it is difficult to explain the juxtaposition of the simple present with an apparently complete present infinitive. Probably have is here used in the sense of "cause," i.e. "thinking to redeem me and to have me install'd," "to cause me to be install'd." So in
"Ambitious love hath so in me offended
That barefoot plod I the cold ground upon
With sainted vow my faults to have amended,"
A. W. iii. 4. 7.
"to have amended" seems to mean "to cause to be amended." But possibly there is no need for this supposition of transposition. The thought of unfulfilment and disappointment growing on the speaker might induce her to put the latter verb in the complete present infinitive.
" Pharnabazus came thither thinking to have raised the siege."—N. P. 179.
Sometimes the infinitive is used without a verb of "thinking," to imply an unfulfilled action.
"I told him of myself, which was as much
As to have ask'd him pardon."—A. and C. ii. 2. 79.
But often it seems used by attraction to "have," expressed or implied in a previous verb.
" She would have made Hercules to have turned spit." M. Ado, ii. 1. 261.
"I had not (i.e. should not have) been persuaded to have hurled
These few ill-spoken lines into the world."
BEAUMONT on Faithful Shepherdess.
So Milton : " He trusted to have equall'd the Most High."
The same idiom is found in Latin poetry (Madvig, 407. Obs. 2) after verbs of wishing and intending. The reason of the idiom seems to be a desire to express that the object wished or intended is a completed fact, that has happened contrary to the wish and cannot now be altered.
361. Subjunctive, simple form. See also Be, Were, An, But, If, &c. The subjunctive (a consequence of the old inflectional form) was frequently used, not as now with would, should, &c., but in a form identical with the indicative, where nothing but the context (in the case of past tenses) shows that it is the subjunctive, as :
" But, if my father had not scanted me,
Yourself, renowned prince, then stood as fair."
M. of V. ii. 1. 17.
" Preferment goes by letter and affection,
And not by old gradation where each second
Stood heir to the first."—Othello, i. 1. 38.
If it be asked what is the difference between "stood" here and " would have stood," I should say that the simple form of the subjunctive, coinciding in sound with the indicative, implied to an Elizabethan more of inevitability (subject, of course, to a condition which is not fulfilled). "Stood" means "would certainly have stood." The possibility is regarded as an unfulfilled fact, to speak paradoxically. Compare the Greek idiom of να with the indicative.
" If he did not care whether he had their love or no, he waived indifferently'twixt doing them neither good nor harm ; but he seeks their hate with greater devotion than they can render it him."—Coriol. ii. 2. 17.
" If they
Should say, 'Be good to Rome,' they charged him even
As those should do," &c.—Coriol. iv. 6. 112.
" (If I rebuked you) then I check'd my friends." Rich. III. iii. 7. 150.
"Till" is used varyingly with the indicative present, future, and the subjunctive.
The subjunctive is found after "so" in the sense of "so (that)," i.e. " (if it be) so (that)."
" I will . . . endow a child of thine,
So in the Lethe of thy angry soul
Thou drown the sad remembrance of these wrongs."
Rich. III iv. 4. 251.
Sometimes the presence of the subjunctive, used conditionally (where, as in the case of did, the subjunctive and indicative are identical in inflections), is indicated by placing the verb before the subject :
"Did I tell this . . . who would believe me?" M. for M. ii. 4. 171.
"Live Roderigo, M. for M. ii. 4. 171.
He calls me to a restitution. "—Othello, v. 1. 14.
" Live a thousand years,
I shall not find myself so fit to die. "—J. C. iii. 1. 159.
"Live thou, I live."—M. of V. iii. 2. 61.
Where we should say, "Should I tell, live," &c.
The indicative is sometimes found where the subjunctive might be expected:
"Pleaseth you walk with me down to his house,
I will discharge my bond,"—C. of E. iv. 1. 12.
where the first clause might be taken interrogatively, " Is it your pleasure to walk with me? In that case I will," &c. So 2 Hen. IV. iv. 1. 225. Perhaps we may thus explain the so-called imperative in the first person plural:
" Well, sit we down,
And let us hear Bernardo speak of this."—Hamlet, i. 1. 33.
i.e. "suppose we sit down?" "what if we sit down?" Compare Ib. 168.
So
"Alcib. I'll take the gold thou giv'st me, not all thy counsel. Timon. Dost thou, or dost thou not, Heaven's curse upon
thee!"—T. of A. iv. 3. 131.
So "willy-nilly" and
"He left this ring behind him, would I or not. "-T. N. i. 5. 321,
" Please" is, however, often found in the subjunctive, even interrogatively.
"Please it you that I call?"—T. of Sh. iv. 4. 1.
It then represents our modern "may it please?" and expresses a modest doubt.
The subjunctive is also found, more frequently than now, with if, though, &c. The subjunctive " he dare" is more common than "he dares" in the historical plays, but far less common in the others. The only difference between the two is a difference of thought, the same as between "he can jump six feet" and "he could jump six feet," i.e. if he liked.
Compare
"For I know thou darest, But this thing dare not." —Tempest, iii. 2. 62–3.
i.e. "would not dare on any consideration :" stronger than "dares."
The indiscriminate use of "dare" and "dares" (regulated, perhaps, by some regard to euphony) is illustrated by
" Here boldly spread thy hands, no venom'd weed
Dares blister them, no slimy snail dare creep."
B. and F. F. Sh. iii. 1.
362. Subjunctive auxiliary forms. The simple form of the subjunctive is sometimes interchanged and co-ordinate with the auxiliary form.
"If thou wert the ass, thy dulness would torment thee, and still thou livedst but as a breakfast to the wolf ; if thou wert the wolf, thy greediness would afflict thee, and oft thou shouldst hazard thy life for a dinner ; wert thou a horse, thou wouldest be seized by the leopard ; wert thou a leopard, thou wert german to the lion."—T. of A. iv. 3. 385–94.
Note here that " livedst " and "shouldst " imply inevitability and compulsion. " Wouldest" is used in the passive because the passive in itself implies compulsion. " Would " is used after " dulness " and "greediness" because they are quasi-personified as voluntary persecutors. Why not "hazardedst" as well as "livedst?" Perhaps to avoid the double d.
"Do," "did," are often used with verbs in the subjunctive :
" Better far, I guess,
That we do make our entrance several ways."—I Hen. VI. ii. 1. 30.
"Lest your retirement do amaze your friends."—I Hen. IV. v. 4. 5.
363. The Subjunctive is replaced by the Indicative after "if," where there is no reference to futurity, and no doubt is expressed, as in "if thou lovest me."
"O Nell, sweet Nell, if thou dost love thy lord,
Banish the cankers of ambitious thoughts."
1 Hen. VI. i. 2. 17.
"An thou canst not smile as the wind sits, thou'lt catch cold shortly."—Lear, i. 4. 112.
" Ah, no more of that, Hal, an thou lovest me."—I Hen. IV. ii. 4. 312.
In the last example Falstaff is assuming the Prince's love as a present fact in order to procure the immediate cessation of ridicule. But in the following he asks the Prince to do him a favour regarded as future, and as somewhat more doubtful :—
"If thou love me, practise an answer."—1 Hen. IV. ii. 4. 411.
Incredulity is expressed in
"If thou have power to raise him, bring him hither." Ib. iii. 1. 60.
In " If thou dost nod thou break'st thy instrument," J. C. iv. 3. 271.
the meaning is "you are sure to break," and the present indicative being used in the consequent, is also used in the antecedent. So in
" I am quickly ill and well
So (almost 'since') Antony loves."—A. and C. i. 3. 73.
In
" It (my purpose) is no more But that your daughter, ere she seems as won,
Desires this ring,"—A. W. iii. 7. 32.
the purpose is regarded graphically as a fact in the act of being completed. However, the indiscriminate use of the indicative and subjunctive at the beginning of the seventeenth century is illustrated by the A. V. St. Matt. v. 23:
" Therefore, if thou bring thy gift to the altar, and there rememberest."
364. Subjunctive used optatively or imperatively. This was more common then than in modern poetry.
" Who's first in worth, the same be first in place." B. J. Cy.'s Rev. v. 1.
(May) " Your own good thoughts excuse me, and farewell." L. L. L. ii. 1. 177.
" O heavens, that they were living both in Naples,
The king and queen there! (provided) that they were, I wish
Myself were mudded in the oozy bed."—Tempest, v. 1. 150.
" No man inveigh against the wither'd flower,
But chide rough winter that the flower hath kill'd."
R. of L.
"In thy fats our cares be drowned,
With thy grapes our hairs be crowned."—A. and C. ii. 7. 122.
The juxtaposition of an imperative sometimes indicates the imperative use.
"Touch you the sourest points with sweetest terms,
Nor (let) curstness grow to the matter."—A. and C. ii. 2. 25.
" Good now, sit down, and tell me he that knows," &c. Hamlet, i. 1. 70.
" Take Antony Octavia to his wife."—A. and C. ii. 2. 129.
"Run one before, and let the queen know."—Ib. iv. 8. 1.
"Thus time we waste, and longest leagues make short; Sail seas in cockles, have an wish but for 't." F. of T. iv. 4. Gower, 2.
i.e. "Let any one but wish it, and we will sail seas in cockles."
Sometimes only the context shows the imperative use :
"For his passage,
(See that) The soldiers' music and the rites of war
Speak loudly for him."—Hamlet, v. 2. 411.
The "and" is superfluous, or else "question" is imperative, in
" Question, your grace, the late ambassadors,
And you shall find. "—Hen. V. ii. 4. 31.
So in " Hold out my horse and I will first be there." Rich. II. ii. 1. 300.
" Then (see that) every soldier kill his prisoners." Hen. V. iv. 6. 37.
On the other hand, "prove" is conditional (or "and" is omitted) in
" O my father !
Prove you that any man with me conversed,
Refuse me, hate me, torture me to death."
M. Ado, iv. 1. 182–6.
Often it is impossible to tell whether we have an imperative with a vocative, or a subjunctive used optatively or conditionally.
"Melt Egypt into Nile, and kindly creatures
Turn all to serpents."—A. and C. ii. 5. 78.
" That I shall clear myself,
Lay all the weight ye can upon my patience,
I make as little doubt as," &c.—Hen. VIII. v. 1. 66.
"Now to that name my courage prove my title." A. and C. v. 2. 291.
"Sport and repose turn from me day and night." Hamlet, iii. 2. 218.
365. This optative use of the subjunctive dispensing with "let," "may," &c. gives great vigour to the Shakespearian line :
" Judge me the world."—Othello, i. 2. 72.
i.e. "let the world judge for me."
"Disorder, that hath spoil'd us, friend us now." Hen. V. iv. 5. 17.
"Long die thy happy days before thy death." Rich. III. i. 3. 207.
"The worm of conscience still begnaw thy soul."—Ib. 222.
The reader of Shakespeare should always be ready to recognize the subjunctive, even where the identity of the subjunctive with the indicative inflection renders distinction between two moods impos. sible, except from the context. Thus:
" Therefore take with thee my most heavy curse,
Which in the day of battle tire thee more
Than all the complete armour that thou wear'st !
My prayers on the adverse party fight,
And there the little souls of Edward's children
Whisper the spirits of thine enemies,
And promise them success and victory. "—Rich. III. iv. 4. 190.
Here, in the second line, "tire," necessarily subjunctive, impresses upon the reader that the co-ordinate verbs, "fight," &c., are also subjunctive. But else, it would be possible for a careless reader to take " fight," &c. as indicative, and ruin the passage.
This optative or imperative use of the subjunctive, though common in Elizabethan writers, had already begun to be supplanted by auxiliaries. Thus Wickliffe has (Coloss. ii. 16) " No man juge you," while all the other versions have " Let no man judge you."
366. Subjunctive, complete present. (See Should for "if he should have.") The subjunctive with "have" is not very frequent. It is used where a past event is not indeed denied, but qualified conditionally, in an argumentative manner :
"If, sir, perchance
She have restrain'd the riots of your followers,
'Tis on such ground . . . as clears her from all blame."
Lear, ii. 4. 145.
i.e. "If it should hereafter be proved that she have," "if so be that she have."
So " If this young gentleman have done offence." T. N. iii. 4. 344.
" Though it have" is somewhat similarly used to express a concession for the sake of argument, not a fact.
" For though it have holp madmen to their wits." Rich. II. v. 5. 62.
367. Subjunctive used indefinitely after the Relative.
"In her youth
There is a prone and speechless dialect
Such as move men."—M. for M. i. 2. 189
" And the stars whose feeble light
Give a pale shadow."—B. and F.
" But they whose guilt within their bosom lie
Imagine every eye beholds their blame."—R. of L. ii. 1344.
" Thou canst not die, whilst any zeal abound." DANIEL (quoted by WALKER).
" I charge you to like as much of this play as please you." A. Y. L. Epilogue.
" And may direct his course as please himself." Rich. III. ii. 2. 129.
Perhaps (but see 218)
" Alas, their love may be called appetite,
No motion of the liver, but the palate
That suffer surfeit."—T. N. ii. 4. 102.
In the subordinate clauses of a conditional sentence, the relative is often followed by the subjunctive :
"A man that were to sleep your sleep."—Cymb. v. 4. 179.
i.e. "If there were a man who was destined to sleep your sleep."
" If they would yield us but the superfluity while it were wholesome." —Coriol. i. 1. 18.
368. Subjunctive in a subordinate sentence. The subjunctive is often used with or without "that," to denote a purpose (see above, That). But it is also used after "that," "who," &c. in dependent sentences where no purpose is implied, but only futurity.
" Be it of less expect
That matter needless of importless burden
Divide thy lips."—Tr. and Cr. i. 3. 71.
No "purpose" can be said to be implied in "please," in the following :—
" May it please you, madam,
That he bid Helen come to you."—A. W. i. 3. 71.
" Yet were it true
To say this boy were like me."—W. T. i. 2. 135.
" Thou for whom Jove would swear
Juno but an Æthiop were."—L. L. L. iv. 3. 118.
" Would you not swear that she were a maid?" M. Ado, iv. 1. 40.
" One would think his mother's milk were scarce out of him." T. N. i. 5. 171.
In the last four passages the second verb is perhaps attracted to the mood of the first.
"Proteus. But she is dead.
Silv. Say that she be: yet," &c.
T. G. of V. iv. 2. 109.
" With no show of fear,
No, with no more than if we heard that England
Were busied with a Whitsun Morris-dance."
Hen. V. ii. 4. 25.
"I pray (hope) his absence proceed by swallowing that." Cymb. iii. 5. 58.
" If it be proved against an alien
That by direct or indirect attempt
He seek the life of any citizen."—M. of V. iv. 1. 351.
" One thing more rests that thyself execute."—T. of Sh. i. 1. 251.
where, however, "that" may be the relative, and " execute" an imperative.
I know of no other instance in Shakespeare but the following, where the subjunctive is used after "that" used for "so that," of a fact :
" Through the velvet leaves the wind
All unseen can passage find,
That the lover sick to death
Wish himself the heaven's breath."—L. L. L. iv. 3. 108.
The metre evidently may have suggested this licence : or -es or -d may have easily dropped out of "wishes" or "wish'd."
The subjunctive is used where we should use the future in
"I doubt not you (will) sustain what you're worthy of by your attempt."—Cymb, i. 4. 125.
"Think" seems used subjunctively, and "that" as a conjunction in
" And heaven defend (prevent) your good souls that you
(should) think
I will your serious and great business scant
For (because) she is with me."—Othello, i. 3. 267.
The "that" is sometimes omitted :
" It is impossible they bear it out."—Ib. ii. 1. 19.
Here "bear" is probably the subjunctive. The subjunctive is by no means always used in such sentences. We may contrast
" No matter then who see it."—Rich. II. v. 2. 59.
"I care not who know it."—Hen. V. iv. 7. 118.
with
" I care not who knows so much."—T. N. iii. 4. 300.
369. The Subjunctive after verbs of command and entreaty is especially common ; naturally, since command implies a purpose.
"We enjoin thee that thou carry."—W. T. ii. 3. 174.
"I conjure thee that thou declare."—Ib. i. 2. 402.
So M. for M. v. 1. 50.
"Tell him from me
He bear himself with honourable action."
T. of Sh. Ind. i. 1. 110.
"Thy dukedom I resign, and do entreat
Thou pardon me my wrongs."—Temp. v. 1. 119.
So after "forbid."
"Fortune forbid my outside have not charmed her." T. N. ii. 2. 19.
Sometimes an auxiliary is used :
"I do beseech your majesty may salve."—1 Hen. IV. iii. 2. 155.
Hence in such passages as
"Go charge my goblins that they grind their joints," Temp. iv. 1. 259.
the verb is to be considered as in the subjunctive.
After a past tense " should" is used :
" She bade me . . . I should teach him."—Othello, i. 3. 165.
370. Irregular sequence of tenses. Sometimes the sequence of tenses is not observed in these dependent sentences :
"Therefore they thought it good you hear a play." T. of Sh. Ind. 2. 136.
"'Twere good you do so much for charity."—M. of V. iv. 1. 261.
In both cases a present is implied in the preceding verb : "They thought and think," "It were and is good."
Reversely in
"But do not stain
The even virtue of our enterprise
To think that or our cause or our performance
Did need an oath."—J. C. ii. 1. 136.
"Did need" means "ever could need," and is stronger than "need" or "can need." In
"Is it not meet that I did amplify my judgment?"—Cymb. i. 5. 17. as in "It is time he came," the action is regarded as one "meet" in time past, as well as in the future.
"It hath been taught us from the primal state
That he which is is wished until he were."—A. and C. i. 3. 42.
Here "were" is used partly for euphony and alliteration, partly. because the speaker is speaking of the past, "is and was always wished until he were."
371. Conditional sentences, The consequent does not always answer to the antecedent in mood or tense. Frequently the irregularity can be readily explained by a change of thought.
"And that I'll prove on better men than Somerset,
(Or rather, I would) Were growing time once ripen'd to
my will."—1 Hen. VI. ii. 4. 98.
So 3 Hen. VI. v. 7. 21.
"If we shall stand still
(Or rather, if we should, for we shall not) We should take root."
Hen. VIII. i. 2. 86.
" I will find
Where truth is hid, (and I would find it) though it were hid
indeed
Within the centre."—Hamlet, ii. 2. 157–8.
Compare Ezek. xiv. 14, A. V. :
"Though these three men, Noah, Daniel, and Job, were in it, they should deliver but their own souls."
with ib. 20, "they shall deliver."
" But if the gods themselves did see her then
(If they had seen her) The instant burst of clamour that she
made
Would have made milch the burning eyes of heaven."
Hamlet, ii. 2. 535-40.
"Till I know 'tis done,
Howe'er my hopes (might be), my joys were ne'er begun."
Ib. iv. 3. 70.
Sometimes the consequent is put graphically in the present merely for vividness:
" If he should do so,
He leaves his back unarm'd ; ... never fear that."
2 Hen. IV. i. 3. 80.
Or else the speaker rises in the tone of confidence :
" I am assured, if I be measured rightly,
Your majesty hath no just cause to hate me."—Ib. v. 2. 66.
# PARTICIPLES.
372. Participles, Active. Our termination -ing does duty for (1) the old infinitive in -an; (2) the old imperfect participle in end, ende, ande; and (3) a verbal noun in -ung. Hence arises great confusion. It would sometimes appear that Shakespeare fancied that -ing was equivalent to -en, the old affix of the Passive Participle. Thus—
" From his all-obeying breath
I hear the doom of Egypt. "—A. and C. iii. 13. 77.
i.e. "obeyed by all."
" Many a dry drop seemed a weeping tear."—R. of L. i. 1375. So " His unrecalling crime " (R. of L.) for "unrecalled."
(In " Many excesses which are owing a man till his age,"—B. E. 122. i.e. " own, or, belonging to a man," owing is not a participle at all, but an adjective, "agen," "âwen," "ôwen," "owenne," "owing;" which was mistaken for a participle.
" There is more owing her than is paid."—A. W. i. 3. 107.
" Wanting, "as in Coriol. ii. 1. 217, "One thing is wanting," can be explained from the use of the verb wanteth in the following passage :
" There wanteth now our brother Gloucester here
To make the period of this perfect peace."—R. III. ii. 1. 43.)
The same explanation may apply to " I am much beholding to you," which is sometimes found for "beholden," Rich. III. ii. 1. 129, J. C. iii. 2. 70–3, and even to
" Relish your nimble notes to pleasing ears."—R. of L.
In the following, -ing might be supplanted, without altering the sense, by the infinitive or the verbal preceded by a\- :
" Women are angels, wooing:
Things won are done."—Tr. and Cr. i. 2. 312.
i.e. "women are considered angels to woo, or a-wooing," where wooing, if treated as an ordinary present participle, would give the opposite to the intended meaning. Probably in the above, as in the following, a\- is omitted.
" Be brief, lest that the process of thy kindness
Last longer (a-, or in) telling than thy kindness date."
Rich. III. iv. 4. 254.
The "in" is inserted in
" Pause a day or two
Before you hazard ; for in choosing wrong I lose your com-
pany."—M. of V. iii. 2. 2.
i.e. "in the event of your choosing wrong, I lose your company." The two constructions occur together in
" Come, come, in wooing sorrow let's be brief,
Since, (a-)wedding it, there is such length in grief."
Rich. II. v. 3. 72.
It is perhaps a result of this confusion between the verbal and the infinitive that, just as the infinitive with "to" is used independently at the beginning of a sentence (357) in a gerundive signification, so is the infinitive represented by -ing :
" Why, were thy education ne'er so mean,
Having thy limbs, a thousand fairer courses
Offer themselves to thy election."—B. J. E. in &c. ii. I.
i.e. "since thou hast thy limbs." This explains the many instances in which present participles appear to be found agreeing with no noun or pronoun.
Part of this confusion may arise from the use of the verbal in -ing as a noun in compounds. We understand at once that a " knedyng trowh" (CHAUCER, C. T. 3548) means "a trough for kneading;" but "spending silver" (Ib. 12946) is not quite so obviously "money for spending." Still less could we say
" Sixth part of each! A trembling contribution."
Hen. VIII. i. 2. 95.
Somewhat different is
" Known and feeling sorrows,"—Lear, iv. 6. 226.
where "feeling" seems to be used like "known," passively, "known and realized sorrows."
So "loading" is used for "laden," BACON, Essays, p. 49 (Wright).
" Your discontenting father,"—W. T. iv. 4. 543. may perhaps be explained by the use of the verb "content you;" "I discontent (me) " meaning "I am discontented."
373. The Verbal differs in Elizabethan usage from its modern use. (a) We do not employ the verbal as a noun followed by "of," unless the verbal be preceded by "the," or some other defining adjective. But such phrases as the following are of constant occurrence in Elizabethan English :
" To disswade the people from making of league. "—N. P. 170.
" He was the onely cause of murdering of the poor Melians."
Ib. 171.
" By winning only of Sicilia."—N. P. 171.
" Enter Clorin the Shepherdess, sorting of herbs."
B. and F. F. Sh. ii. 1.
i.e. "a-sorting, or in sorting of herbs."
For instances from Shakespeare, see 178 and 93.
(b) On the other hand, when the verbal is constituted a noun by the dependence of "the," or any other adjective (except a possessive adjective) upon it, we cannot omit the of. The Elizabethans can.
" To plague thee for thy foul misleading me."
3 Hen. VI. v. 1. 97.
We should prefer now to omit the "thy" as well as "foul," though we have not rejected such phrases as
" Upon his leaving our house."—Goldsmith.
For instances of "of" omitted when "the" precedes the verbal, see Article, 93. In this matter modern usage has recurred to E. E.
374. Participles, Passive. It has been shown (294) that, from the licence of converting nouns, adjectives, and neuter verbs into active verbs, there arose an indefinite and apparently not passive use of Passive Participles. Such instances as
" Of all he dies possess'd of, "—M. of V. v. 1. 293.
(possess being frequently used as an active verb,) may thus be explained.
Perhaps,
" And, gladly quaked (made to quake), hear more,"
Coriol. i. 9. 6.
may be similarly explained. Compare also :
" All the whole army stood agazed on him."
1 Hen. VI. i. 1. 126.
But, in the following, we can only say that, in the excessive use of this licence, -ed is loosely employed for -ful, -ing, or some other affix expressing connection.
" Revenge the jeering and disdain'd contempt."
1 Hen. IV. i. 3. 183.
" Brooded-watchful day."—K. J. iii. 3. 52.
As we talk of "watching (during) the night," this may explain
" The weary and all-watched night."—Hen. V. iv. Prologue, 38.
But more probably "all-watched" (like "o'er-watched," J. C. iv. 3. 241) resembles "weary," and means "tired with watching." For this use of adjectives see 4.
" Grim-look'd night."—M. N. D. v. 1. 171.
" The ebbed man."—A. and C. i. 4. 43.
It is perhaps still not unusual to say "the tide is ebbed."
" A moulten raven."—1 Hen. IV. iii. 1. 152.
" With sainted vow."—A. W. iii. 4. 7. (= saintly).
" And at our more considered time we'll read."—Hamlet, ii. 2. 81.
" Unconstrained gyves."—L. C. 242.
Sometimes passive participles are used as epithets to describe the state which would be the result of the active verb. Thus:
" Why are you drawm?"—Temp. ii. 1. 308; M.N.D. iii. 2. 402.
i.e. " Why do I find you with your swords drawn?"
" Under the blow of thralled discontent."—Sonn. 124.
"The valued file" (Macb. iii. 1. 95) perhaps means "the file or catalogue to which values are attached."
375. The Passive Participle is often used to signify, not that which was and is, but that which was, and therefore can be hereafter. In other words, -ed is used for -able.
" Inestimable stones, unvalued jewels."—Rich. III. i. 4. 27.
i.e. "invaluable."
" All unavoided is the doom of destiny."—Ib. iv. 4. 217.
i.e. "inevitable." So
" We see the very wreck that we must suffer,
And unavoided is the danger now."—Rich. II. ii. 2. 268.
" With all imagined (imaginable) speed."—M. of V. iii. 4. 52.
" The murmuring surge
That on the unnumber'd idle pebbles chafes."—Lear, iv.6.21.
So, probably, Theobald is right in reading
" The twinn'd stone upon th' unnumber'd beach,"
Cymb. i. 6. 36.
though the Globe retains "number'd."
" Unprized " in
"This unprized precious maid,"—Lear, i. 1. 262.
may mean "unprized by others, but precious to me."
" There's no hoped for mercy with the brothers."
3 Hen. VI. v. 4. 35.
i.e. "to be hoped for."
It has been conjectured that "delighted" means "capable of being delighted" in
" This sensible warm motion to become
A kneaded clod, and the delighted spirit
To bathe in fiery floods."—M. for M. iii. 1. 121.
More probably, "delighted" here means the spirit "that once took its delight in this world;" but "kneaded" seems used for "kneadable."
376. Participle used with a Nominative Absolute. In Anglo-Saxon a dative absolute was a common idiom. Hence, even when inflections were discarded, the idiom was retained; and indeed, in the case of pronouns, the nominative, as being the normal state of the pronoun, was preferred to its other inflections. The nominative absolute is much less common with us than in Elizabethan authors. It is often used to call attention to the object which is superfluously repeated. Thus in
" The master and the boatswain,
Being awake, enforce them to this place,"—Temp. v. 1. 100.
there is no need of "them." So "he" is superfluous in
" Why should he then protect our souvereign,
He being of age to govern of himself?"—2 Hen. VI. i. 1. 166.
It is common with the relative and relative adverbs.
" Then Deputy of Ireland ; who remov'd,
Earl Surrey was sent thither."—Hen. VIII. ii. 1. 42.
" My heart,
Where the impression of mine eye infixing,
Contempt his scornful perspective did lend me."
A. W. v. 3. 47.
" Thy currish spirit
Govern'd a wolf, who hang'd for human slaughter,
Even from the gallows did his fell soul fleet."
M. of V. iv. 1. 134.
" Emblems
Laid nobly on her ; which perform'd, the choir
Together sung 'Te Deum."—Hen. VIII. iv. 1. 91.
The participle with a nominative originally intended to be absolute seems diverted into a subject in
" The king . . . aiming at your interior hatred
Makes him send."—Rich. III. i. 3. 65-8.
i.e. "the fact that the king guesses at your hatred makes him send."
377. The Participle is often used to express a condition where, for perspicuity, we should now mostly insert "if."
" Requires to live in Egypt, which not granted,
He lessens his requests."—A. and C. iii. 12. 12.
" That whoso ask'd her for his wife,
His riddle told not, lost his life."—P. of T. i. Gower, 38.
" For I do know Fluellen valiant,
And, touch'd with choler, hot as gunpowder."
Hen. V. iv. 7. 188.
" Your honour not o'erthrown by your desires,
I am friend to them and you."—W. T. v. 1. 230.
" Admitted" is probably a participle in
" This is the brief of money, plate and jewels
I am possess'd of: 'tis exactly valued,
Not petty things admitted."—A. and C. v. 1. 146.
i.e. "exactly, if petty things be excepted."
The participle is sometimes so separated from the verb that it seems to be used absolutely.
" Resolve me with all modest haste which way
Thou might'st deserve, or they impose this usage,
Coming from us."—Lear, ii. 4. 27.
i.e. "since thou comest."
" But being moody give him line and scope."
2 Hen. IV. iv. 4. 39.
" And" is sometimes joined to a participle or adjective thus used. See And, 95. " What remains
" What remains
But that I seek occasion how to rise,
And yet the king not privy to my drift."—3 Hen. VI. i. 2. 47.
" But when the splitting wind
Makes flexible the knees of knotted oaks,
And flies (being) fled under shade."—Tr. and Cr. i. 3. 51.
i.e. "the flies also being (295) fled."
378. Participle without Noun. This construction is rare in earlier English.
"My name is gret and merveylous, treuly you telland."—Cov. Myst. (Mätzner).
Here again, as in 93, we must bear in mind the constant confusion between the infinitive, the present participle, and the verbal. In the above example we should expect the infinitive, "to tell you the truth," and perhaps "telland" is not exactly used for, but confused with, "tellen."
It is still a usual idiom with a few participles which are employed almost as prepositions, e.g. "touching," "concerning," "respecting," "seeing." "Judging" is also often thus incorrectly used, and sometimes "considering ;" but we could scarcely say—
" Or in the night imagining (if one imagines) some fear,
How easy is the bush suppos'd a bear."—M. N. D. v. 1. 21.
" Here, as I point my sword, the sun arises,
Which is a great way growing on the south,
Weighing the youthful season of the year."—J. C. ii. 1. 108.
Note especially—
" I may not be too forward,
Lest (I) being seen thy brother, tender George,
Be executed."—Rich. III. v. 3. 95.
" (It must be done) something from the palace, always thought
That I require a clearness."—Macbeth, iii. 1. 132.
i.e. "it being always borne in mind."
" (Death sits) infusing him (man) with self and vain conceit,
And, (man having been) humour'd thus,
(Death) comes at the last."—Rich. II. iii. 2. 168.
This use is common in prose.
" He was presently suspected, judging (since men judged) the ill success not in that he could not, but . . . for that he would not."—N. P. 182.
So "being," i.e. "it being the fact," is often used where we use " seeing."
" You loiter here too long, being you are to take soldiers up in
counties as you go."—2 Hen. IV. ii. 1. 200 ; M. Ado, iv. 1. 51.
" Through I with death and with
Reward did threaten and encourage him,
Not doing't and (it) being done."—W. T. iii. 2. 166.
i.e. "I threatened him, not doing it, with death, and encouraged him with reward, (it) being done;" a specimen of irregular terseness only to be found in Elizabethan authors and in Mr. Browning's poems.
The context often suggests a noun or pronoun :
"If not that, I being queen, you bow like subjects,
Yet that, (I being) by you deposed, you quake like rebels."
Rich. III. i. 3. 162.
"But her eyes—
How could he see to do them ? Having made one,
Methinks it should have power to steal both his."
M. of V. iii. 2. 125.
i.e. "when he had made one."
"Had, having, and in quest to have, extreme."—Sonn. 129.
i.e. "when an object is had, possessed," unless it is still more irregularly used for "having had."
This irregularity is perhaps in some cases explained by 372.
379. Participle with Pronoun implied. Sometimes a pronoun on which a participle depends can be easily understood from a pronominal adjective. Compare
"Nostros vidisti flentis ocellos."
So
"Not helping, death's my fee."—A. W. ii. 1. 192.
i.e. "death is the fee of me not helping."
"Men
Can counsel speak and comfort to that grief
Which they themselves not feel ; but, tasting it,
Their counsel turns to passion."—M. Ado, v. 1. 22.
"She dares not look, yet, winking, there appears
Quick-shifting antics ugly in her eye."—R. of L. 458.
"Coming (as we came) from Sardis, on our former ensign
Two mighty eagles fell."—J. C. v. 1. 80.
380. Instead of the Participle an Adjective is sometimes found.
"I would not seek an absent argument
Of my revenge, thou present."—A. Y. L. iii. 1. 4.
"And (she), her attendants absent, swallowed fire."—J. C. iv. 3. 156.
" Joy absent, grief is present for that time."—Rich. II. i. 3. 259.
Sometimes the adjective depends on an implied pronoun:
"Thy word is current with him for my death,
But dead, thy kingdom cannot buy my breath."
Rich. II. i. 3. 232.
i.e. "the breath of me when dead."
" It is an obvious conjecture from this use of "absent," "present," "dead," that their quasi-participial terminations favoured this participial use. But add
"Thence,
A prosperous south-wind friendly, we have cross'd."
W. T. v. 1. 161.
381. The Participle is sometimes implied in the case of a simple word, such as "being."
"I have heard him oft maintain it to be fit that sons (being) at perfect age and fathers declining, the father should be as ward to the son."—Lear, i. 2. 77.
"And be well contented
To make your house our tower. You (being) a brother of us,
It fits we thus proceed, or else no witness
Would come against you."—Hen. VIII. v. 1. 106.
i.e "Since you are our brother." (Or (?) "though you were our brother, it [would be and] is fit to proceed thus.")
"(Those locks are) often known
To be the dowry of a second head,
The skull that bred them (being) in the sepulchre."
M. of V. iii. 2. 96.
We retain this use in antithetical phrases, such as "face to face," "sword against sword," but we should rarely introduce an adjective into such an antithetical compound. Shakespeare, however, has
"And answer me declined sword 'gainst sword."
A. and C. iii. 13. 27.
# ELLIPSES.
382. Several peculiarities of Elizabethan language have already been explained by the desire of brevity which characterised the authors of the age. Hence arose so many elliptical expressions that they deserve a separate treatment. The Elizabethan authors objected to scarcely any ellipsis, provided the deficiency could be easily supplied from the context.
"Vouchsafe (to receive) good-morrow from a feeble tongue."
J. C. ii. 1. 313.
"When shall we see (one another) again?"
Cymb. i. 1. 124 ; Tr. and Cr. iv. 4. 59.
Just so we still use "meet."
"You and I have known (one another), sir."
A. and C. ii. 6. 86 ; Cymb. i. 4. 36.
"On their sustaining garments (there is) not a blemish,
But (the garments are) fresher than before."
Tempest, i. 2. 219.
Thus also, as in Latin, a verb of speaking can be omitted where it is implied either by some other word, as in
"She calls me proud, and (says) that
She could not love me."—A. Y. L. iv. 3. 16.
"But here's a villain that would face me down
He met me on the mart."—C. of E. iii. 1. 7.
i.e. "maintain to my face that he met me;" or by a question as in
"What are you?
(I ask) Your name and quality; and why you answer
This present summons."—Lear, v. 3. 120.
(The Globe inserts a note of interrogation after quality.)
"Enforce him with his envy to the people,
And (say) that the spoil got on the Antiates
Was ne'er distributed."—Coriol. iii. 3. 4.
Thus, by implying from "forbid" a word of speaking, "bid," and not by a double negative, we should perhaps explain
"You may as well forbid the mountain pines
To wag their high tops and (bid them) to make no noise."
M. cf V. iv. 1. 76.
Thus "I know not whether to depart in silence
Or bitterly to speak in your reproof
Best fitteth my degree or your condition.
If (I thought it fittest) not to answer, you might haply
think," &c.—Rich. III. iii. 7. 144.
After "O!" "alas !" and other exclamations, a verb of surprise or regret is sometimes omitted.
"O (it is pitiful) that deceit should steal such gentle shapes."
Rich. III. ii. 2. 27.
"Good God! (I marvel that) these nobles should such
stomachs bear:
I myself fight not once in forty years."—1 Hen. VI. i. 3. 90.
Sometimes no exclamation is inserted :
"Ask what thou wilt. (I would) That I had said and done."
2 Hen. VI. i. 3. 31.
Ellipses in Conjunctional Sentences. The Elizabethans seem to have especially disliked the repetition which is now considered necessary, in the latter of two clauses connected by a relative or a conjunction.
383. And:
" Have you
Ere now denied the asker, and now again
Of him that did not ask but mock (do you) bestow
Your sued-for tongues?"—Coriol. ii. 3. 213.
Here in strictness we ought to have "bestowed," or "do you bestow."
An ellipse must be supplied proleptically in
"(Beggars) Sitting in the stocks refuge their shame,
That (i.e. because) many have (sat), and many must sit
there."—Rich. II. v. 5. 27.
"Of (such) dainty and such picking grievances."
2 Hen. IV. iv. 1. 198.
"It (i.e. love) shall be (too) sparing and too full of riot."
V. and A. 1147.
"It shall be (too) merciful and too severe."—Ib. 1155.
384. As:
"His ascent is not so easy as (the ascent of) those who," &c
Coriol. ii. 2. 30.
"Returning were as tedious as (to) go o'er."—Macb. iii. 4. 138.
"They boldly press so far as (modern Eng. that) further none
(press)."—B. J. Cy.'s Rev. v. 3.
"O, 'tis sweating labour
To bear such idleness so near the heart
As Cleopatra (bears) this."—A. and C. i. 3. 95.
"And I, that haply take them from him now,
May yet ere night yield both my life and them
To some man else, as this dead man doth (to) me."
3 Hen. VI. ii. 5. 60.
"Return those duties back as (they) are most fit (to be returned)."
Lear, i. 1. 99.
As can scarcely, in the above, be taken for "which."
"This is a strange thing (as strange) as e'er I look'd on."
Temp. v. 1. 289.
385. But (after but the finite verb is to be supplied without the negative) :
"The tender nibbler would not take the bait
But (would) smile and jest."—P. P. 4.
"To be thus is nothing,
But to be safely thus (is something)."—Macbeth, iii. 1. 47.
"And though I could
With barefaced power sweep him from my sight
And bid my will avouch it, yet I must not,
(For certain friends that are both his and mine,
Whose loves I may not drop,) but (I must) wail his fall
Who I myself struck down."—Macbeth, iii. 1. 119.
Sometimes but itself is omitted :
"'Tis not my profit that doth lead mine honour,
(But it is) Mine honour (that doth lead) it (i.e. profit)."
A. and C. ii. 7. 83.
Sometimes the repeated varies slightly from the original proposition :
"'Tis not enough to help the feeble up,
But (it is necessary) to support him after."— . of A. i. 1.107.
In the following, the negative is implied in the first verb through the question, "Why need we?" i.e. "We need not." The second verb must not be taken interrogatively, and thus it omits the negative.
"Why, what need we
Commune with you of this, but rather follow
Our forceful indignation?"—W. T. ii. 1. 162.
i.e. "Why need we commune with you? we need rather follow our own impulse." Else, if both verbs be taken interrogatively, "but" must be taken as "and not :" "Why need we commune with you, and not follow our own impulse?"
Where the negative is part of the subject, as in "none," a new subject must be supplied :
"God, I pray him
That none of you may live your natural age
But (each of you) by some unlook'd accident cut off."
Rich. III. i. 3. 214.
386. Ere:
"The rabble should have first unroof'd the city
Ere (they should have) so prevail'd with me."—Coriol. i. 1. 223.
"I'll lean upon one crutch and fight with the other
Ere (I will) stay behind this business."—Coriol. i. 1. 246.
387. If :
" I am more serious than my custom ; you
Must be so too, if (you must or intend to) heed me."
Temp. ii. 1. 220.
See "must," 314.
" I yet beseech your majesty
If (it is) for (i.e. because) I want that glib and oily art
. . . That you make known," &c.—Lear, i. 1. 227.
" O, if (you be) a virgin
And your affection (be) not gone forth, I'll make you
The queen of Naples."—Tempest, i. 2. 447–8.
" Haply you shall not see me more, or if (you see me),
(You will see me) A mangled shadow."—A. and C. iv. 1. 27.
This is a good Greek idiom. So
"Not like a corse : or if, not to be buried,
But quick, and in mine arms."—W. T. iv. 4. 131.
In the following hypothetical sentence there is a curious ellipsis :
" Love, loving not itself, none other can."—Rich. II. v. 2. 88.
i.e. "if a man does not love his own flesh and blood he cannot (love) a stranger."
388. Like (i.e. resembling) :
"But you like none, none (like) you, for constant heart."—Sonn.
388a, Or :
"For women's fear and love holds quantity ;
In neither (is) aught, or (it is) in extremity."
Hamlet, iii. 2. 178.
i.e. "women's fear and love vary together, are proportionable : they either contain nothing, or what they contain is in extremes."
389. Since:
"Be guilty of my death since (thou art guilty) of my crime."
R. of L.
390. Than:
"To see sad sights moves more than (to) hear them told."
R. of L. 451.
"It cost more to get than (was fit) to lose in a day."
B. J. Poetaster.
" Since I suppose we are made to be no stronger
Than (that) faults may shake our frames."
M. for M. ii. 4. 133.
"But I am wiser than (I should be were I) to serve their
precepts. "—B. J. E. out &c. i. 1.
"My form
Is yet the cover of a fairer mind
Than (that which is fit) to be butcher of an innocent child."
K. J. iv. 2. 258.
"This must be known ; which being kept close might move
More grief to hide, than hate to utter (would move) love."
Hamlet, i. 1. 108–9.
i.e. "this ought to be revealed, for it (273), by being suppressed, might excite more grief in the king and queen by the hiding (356) of the news, than our unwillingness to tell bad news would excite love."
"What need we any spur but our own cause
To prick us to redress? What other bond
Than (that of) secret Romans?"—J. C. ii. 1. 125.
As in the case of "but" (385), so in the following, the verb must be repeated without its negative force :
" I heard you say that you had rather refuse
The offer of an hundred thousand crowns
Than (have) Bolingbroke's return to England."
Rich. II. iv. I. 17.
Here, perhaps, the old use of the subjunctive "had" for "would have" exerts some influence.
The word "rather" must be supplied from the termination er in
" The rarer action is
In virtue (rather) than in vengeance."—Temp. v. 1. 28.
" You are well understood to be a perfecter giber for the table than a necessary bencher in the Capitol."—Coriol. ii. 1. 91.
391. Though:
"Saints do not more, though (saints) grant for prayers' sake."
R. and J. i. 5. 107.
"I keep but two men and a boy (as) yet, till my mother be dead.
But what though? Yet I live like a poor gentleman Lorn."
M. W. of W. i. 1. 287.
392. Till:
" He will not hear till (he) feel."—T. of A. ii. 2. 7.
393. Too.... to :
"His worth is too well known (for him) to be forth-coming."
B. J. E. out &c. v. 1.
394. Relative. (In relative sentences the preposition is often not repeated.)
"Most ignorant of what he's most assured (ot)."
M. for M. ii. 2. 119.
"A gift of all (of which) he dies possess'd."—M. of V. iv. 1. 389.
"Err'd in this point (in) which now you censure him."
M. for M. ii. 1. 15.
"For that (for) which, if myself might be his judge,
He should receive his punishment in thanks."—Ib. i. 4, 28.
" I do pronounce him in that very shape (In which) He shall appear in proof."—Hen. VIII.i. 1. 196.
" As well appeareth by the cause (for which) you come."
Rich. II. i. 1. 26.
"In this (in or of) which you accuse her."—W. T. ii. 1. 133.
" In that behalf (in) which we have challenged it."
K. J. ii. 1. 264.
" To die upon the bed (upon which) my father died."
W. T. iv. 4. 466.
" In such a cause as fills mine eyes with tears,
And stops my tongue while (my) heart is drown'd in cares."
3 Hen. VI. iii. 3. 14.
There is a proleptic omission in
"Or (upon) whom frown'st thou that I do fawn upon." Sonn. 149.
395. Antithetical sentences frequently do not repeat pronouns, verbs, &c.
"What most he should dislike seems pleasant to him,
What (he should) like, (seems) offensive."—Lear, iv. 2. 10.
Sometimes the verb has to be repeated in a different tense.
" To know our enemies' minds we'ld rip their hearts:
(To rip) Their papers is more lawful."—Lear, iv. 6. 266.
" To be acknowledg'd, madam, is (to be) overpaid."
Ib. iv. 7. 4.
The antithesis often consists in the opposition between past and present time.
" I meant to rectify my conscience, which
I then did feel full sick, and yet (do feel) not well."
Hen. VIII. ii. 4. 204.
" And may that soldier a mere recreant prove
That means not (to be), hath nut (been), or is not in love."
Tr. and Cr. i. 3. 288.
"She was beloved, she loved ; she is (beloved) and doth (love)."
Ib. iv. 5. 292.
396. Ellipsis of Neither before Nor, One before Other.
"(Neither) He nor that affable familiar ghost."—Sonn. 86.
"But (neither) my five wits nor my five senses can
Dissuade one foolish heart from seeing thee."—Ib. 141.
"A thousand groans . . .
Came (one) on another's neck."—Ib. 131.
"Pomp. You will not bail me then, sir.
Lucio. (Neither) Then, Pompey, nor now."
M. for M. iii. 2. 86.
397. Ellipsis of Adverbial and other Inflections.
"The duke of Norfolk sprightfully and bold(ly)."
Rich. II. i. 3. 3.
"Good gentlemen, look fresh(ly) and merrily."—J. C. ii. 1. 224.
"Apt(ly) and willingly."—T. N. v. 1. 135.
"With sleided silk, feat(ly) and affectedly."—L. C. 48.
"His grace looks cheerfully and smooth(ly) this morning."
Rich. III. iii. 4. 50.
"And she will speak most bitterly and strange(ly)."
M. for M. v. 1. 36.
"How honourable(y) and how kindly we
Determine."—A. and C. v. 1. 58.
"And that so lamely and unfashionable(y)."—Rich. III. i. 1. 22.
It will not escape notice (1) that in all but two of these instances the -ly is omitted after monosyllabic adjectives, which can be more readily used as adverbs without change; (2) that "honourable," " unfashionable," &c., in their old pronunciation would approximate to "honourably," "unfashionably," and the former is itself used as an adverb. (See 1.) Nevertheless it seems probable that this, like the following idiom, and like many others, arises partly from the readiness with which a compound phrase connected by a conjunction is regarded as one and inseparable. Compare
" Until her husband('s) and my lord's return."—M. of V. iii. 4. 30.
" As soul('s) and body's severing."—Hen. VIII. ii. 3. 16.
where "soul-and-body" is a quasi-noun.
" Shall be your love('s) and labour's recompense."
Rich. II. ii. 3. 62.
398. Ellipsis of Superlative Inflection.
"The generous and gravest citizens."—M. for M. iv. 6. 13.
"Only the grave and wisest of the land."—HEYWOOD (Walker).
"The soft and sweetest music."—B. J. (Ib.).
"The vain and haughtiest minds the sun e'er saw."
GOFFE (Ib.).
"To mark the full-fraught man and best endued."
Hen. V. ii. 2. 139.
"The humble as the proudest sail doth bear."—Sonn. 80.
The est of the second adjective modifies the first.
Reversely we have—
"The best condition'd and unwearied spirit,"—M. of V. iii. 2. 295. where "best" modifies the second adjective.
"Call me the horrid'st and unhallow'd thing
That life and nature tremble at."—MIDDLETON (Walker).
In
"I took him for the plainest harmless creature,"
Rich. III. iii. 4. 25.
though the meaning may be "the plainest, (the most) harmless creature," it is more likely a compound word, "plainest-harmless" (see 2).
399. Ellipsis of Nominative. Where there can be no doubt what is the nominative, it is sometimes omitted.
"It was upon this fashion bequeathed me by will, but poor a thousand crowns, and as thou sayest charged my brother, on his blessing, to breed me well."—A. Y. L. i. I. 3.
" They call him Doricles : and boasts himself
To have a worthy feeding."—W. T. iv. 4. 168.
"Who loved her so, that speaking of her foulness
(He) Washed it with tears."—M. Ado, iv. 1. 156.
" (It) shall not be long but I'll be here again."
Macbeth, iv. 2. 23.
" Nor do we find him forward to be sounded,
But with a crafty madness keeps aloof."—Hamlet, iii. 1. 8.
This explains K. J. ii. 1. 571, and
"When I am very sure, if they should speak,
(They) Would almost damn those ears which," &c.
M. of V. i. 1. 97.
Compare
"Come, fortune's a jade, I care not who tell her,
(Who, i.e. since she) Would offer to strangle a page of the
cellar."—B. and F.
" The king must take it ill
That he's so slightly valued in his messenger,
(That he or ? you) Should have him thus restrained."
Lear, ii. 2. 154.
So Hen. VIII. i. 2. 197.
The following might be explained by transposition, "may all" for "all may :" but more probably "they" is implied :
"That he awaking when the other do,
May all to Athens back again repair."
M. N. D. iv. 1. 72. See also Ib. v. i. 98.
400. The omission of the Nominative is most common with "has," "is," "was," &c.
"He has" is frequently pronounced and sometimes written "has," and "he" easily coalesces with "was," "will," &c. Hence these cases should be distinguished from those in the preceding paragraph.
" And to the skirts of this wild wood he came,
Where, meeting with an old religious man,
After some question with him was converted."
A. Y. L. v. 4. 167.
"This young gentlewoman had a father whose skill was almost as great as his honesty : had it stretch'd so far, would have made nature immortal."—A. W. i. 1. 20.
"Hero. I'll wear this.
Marg. By my troth, 's not so good."—M. Ado, iii. 4. 9 and 18.
" For Cloten
There wants no diligence in seeking him,
And (he) will no doubt be found."—Cymb. iv. 3. 21.
" For I do know Fluellen valiant.
And, touch'd with choler, hot as gunpowder;
And quickly will return an injury."—Hen. V. iv. 7. 188.
"This is that banish'd haughty Montague,
And here is come."—R. and J. v. 3. 52.
" As for Cromwell,
Beside that of the jewel-house, (he) is made master
O' the rolls."—Hen. VIII. v. i. 34; 50.
" I know the gentleman ; and, as you say,
There (he) was a' gaming."—Hamlet, ii. 1. 58.
" Bring him forth; has sat in the stocks all night," &c.
A. W. iv. 3. 116.
So Ib. 114, 298 ; T. N. i. 5. 156.
"'Tis his own blame: hath put himself from rest."
Lear, ii. 4. 293.
Ib. iii. 1. 5; Othello, iii. 1. 67; T. of A. iii. 2. 39, iii. 3. 23, iv. 3. 463. This omission is frequent after appellatives or oaths.
" Poor jade, is wrung in the withers out of all 'cess."
I Hen. IV. ii. 1. 6.
" Poor fellow, never joyed since the price of oats rose."—Ib. 11.
" Richard. Send for some of them.
Ely. Marry, and will, my lord, with all my heart."
Rich. III. iii. 4. 36.
In " And the fair soul herself,
Weigh'd between loathness and obedience, at
Which end o' the beam should bow,"—Tempest, ii. 1. 131.
either "she" is omitted, or "should" is for "she would," or "o' " has been inserted by mistake.
401. A Nominative in the second person plural or first person is less commonly omitted.
"They all rush by
And leave you hindermost ;
Or like a gallant horse, fall'n in first rank,
(You) Lie there for pavement to the abject rear."
Tr. and Cr. iii. 3. 162.
"They . . . gave me cold looks,
And, meeting here the other messenger,
Having more man than wit about me, (I) drew."
Lear, ii. 4. 42.
The I before "pray thee," "beseech thee," is constantly omitted. (Tempest, ii. 1. 1.)
"Good-morrow, fair ones ;
(I) pray you if you know."—A. Y. L. iv. 3. 76.
i.e. "I ask you whether you know."
The inflection of the second person singular allows the nominative to be readily understood, and therefore justifies its omission.
"Art any more than a steward ?"—T. N. ii. 3. 122.
"It was she
First told me thou wast mad; then (thou) cam'st in smiling."
Ib. v. 1. 357.
402. Ellipsis of Nominative explained. This ellipsis of the nominative may perhaps be explained partly (1) by the lingering sense of inflections, which of themselves are sometimes sufficient to indicate the person of the pronoun understood, as in Milton—
"Thou art my son beloved : in him am pleased ;"
partly (2) by the influence of Latin ; partly (3) by the rapidity of the Elizabethan pronunciation, which frequently changed "he" into "'a" (a change also common in E. E.),
" 'a must needs,"—2 Hen. VI. iv. 2. 59.
and prepared the way for dropping "he" altogether. Thus perhaps in
"Who if alive and ever dare to challenge this glove, I have sworn to take him a box o' th' ear,"—Hen. V. iv. 7. 132.
we should read "'a live and ever dare." In the French of Rabelais the pronouns are continually dropped : but the fuller inflections in French render the omission less inconvenient than in English. In the following instance there is an ambiguity which is only removed by the context :—
"We two saw you four set on four; and (you) bound them and were masters of their wealth."—1 Hen. IV. ii. 4. 278.
403. Ellipsis of It is, There is, Is.
"So beauty blemish'd once (is) for ever lost. "—P. P. 13.
"I cannot give guess how near (it is) to day. "—J. C. ii. 1. 2.
" Seldom (is it) when
The steeled gaoler is the friend of men."
M. for M. iv. 2. 90.
"And (it is) wisdom
To offer up a weak poor innocent lamb."—Macb. iv. 3. 16.
"Since [there is neither (163)] brass nor stone nor earth nor
boundless sea,
But sad mortality o'ersways their power."—Sonn. 64.
"'Tis certain, every man that dies ill, the ill (is) upon his
own head."—Hen. V. iv. 1. 197.
" Many years,
Though Cloten (was) then but young, you see, not wore him
From my remembrance."—Cymb. iv. 4. 23.
So Hen. V. iv. 7. 132 (quoted in 402), if the text be retained.
It is a question whether "are" is omitted, or whether (less probably) (And, 95) "and" is used for "also" with a nom. absolute, in
" But 'tis not so above ;
There is no shuffling, there the action lies
In his true nature : and we ourselves (? are) compelled
To give in evidence."—Hamlet, iii. 3. 62 ; T. N, i. 1. 38;
Hen. V. i. 1. 57.
" Which I did store to be my foster-nurse,
When service should in my old limbs lie lame,
And unregarded age (? should be) in corners thrown."
A. Y. L. ii. 2. 42.
As the verb is omitted by us constantly after "whatever," e.g. "anything whatever," so Shakespeare could write,
" Beyond all limit of what else (is) in the world."
Temp. iii. 1. 172.
Thus also "however" is for " however it may be," i.e. "in any case :"
" If haply won perhaps a hapless gain ;
If lost, why then a grievous labour won ;
However (it be), but a folly bought with wit."
T. G. of V. i. 1. 34.
We have passed in the use of "however" from the meaning "in spite of what may happen in the future," to "in spite of what happened in the past," i.e. "nevertheless."
"There is" is often omitted with "no one but," as
"(There is) no one in this presence
But his red colour hath forsook his cheeks."
Rich. III. ii. 1. 84.
" Who is" (244) is omitted in
"Here's a young maid (who is) with travel much oppressed,
And faints for succour."—A. Y. L. ii. 4. 75.
Otherwise the nominative (399) is omitted before "faints."
404. Ellipsis of It and There.
" Whose wraths to guard you from,
Which here in this most desolate isle else falls
Upon your head, (there) is nothing but heart-sorrow,
And a clear life ensuing."—Temp. iii. 2. 82.
" Satisfaction (there) can be none but by pangs of death."
T. N. iii. 4. 261
" D. Pedro. What! sigh for the toothache?
Leon. Where (there) is but a humour or a worm."
M. Ado, iii. 2. 27; Ib. ii. 2. 20.
" At the Elephant (it) is best to lodge."—T. N. iii. 3. 40.
" Be (it) what it is."—Cymb. v. 4. 149.
" The less you meddle with them the more (it) is for your honesty."—M. Ado, iii. 3. 56.
The omission is common before "please."
"So Please (it) him (to) come unto this place."—J. C. iii. 1. 140.
" Is (it) then unjust to each his due to give ?"
SPENS. F. Q. i. 9. 38.
"(It) remains
That in the official marks invested you
Anon do meet the Senate."—Coriol. ii. 3. 147.
This construction is quite as correct as our modern form with "it." The sentence "That in . . . . Senate," is the subject to " remains." So—
" And that in Tarsus (it) was not best
Longer for him to make his rest."—Pericl. ii. Gower, 25.
" Happiest of all is (it or this), that her gentle spirit
Commits itself to you to be directed."—M. of V. iii. 2. 166.
We see how unnecessary and redundant our modern "it" is from the following passage :—
"Unless self-charity be sometimes a vice,
And to defend ourselves it be a sin." —Othello, ii. 3. 203.
This is (if the order of the words be disregarded) as good English as our modern " Unless it be a sin to defend ourselves." The fact is, this use of the modern "it" is an irregularity only justified by the clearness which it promotes. "It" at the beginning of a sentence calls attention to the real subject which is to follow. "It is a sin, viz. to defend oneself."
The sentence is sometimes placed as the object, "it" being omitted.
"But long she thinks (it) till he return again."—R. of L. 454.
"Being" is often used for "it being," or "being so," very much like ν and its compounds in Greek.
"That Lepidus of the triumvirate
Should be deposed; and, (it) being (so), that we detain
All his revenue."—A. and C. iii. 6. 30.
" I learn you take things ill which are not so
Or, being (so), concern you not."—A. and C. ii. 2. 30.
405. Ellipses after will and is.
"I will," i.e. "I purpose," when followed by a preposition of motion, might naturally be supposed to mean "I purpose motion." Hence, as we have
"He purposeth to Athens,"—A. and C. iii. 1. 35.
so
"I'll to him."—R. and J. iii. 2. 141.
"Will you along?"—Coriol. ii. 3. 157.
"Now we'll together."—Macbeth, iv. 3. 136.
" I will to-morrow,
And betimes I will, to the weird sisters."—Ib. iii. 4. 133.
"Strange things I have in head that will to hand."
Ib. iii. 4. 139.
Compare
"Give these fellows some means (of access) to the king."
Hamlet, iv. 6. 13.
Similarly, as we have
"I must (go) to Coventry."—Rich. II. i. 2. 56.
" I must (go) a dozen mile to-night."—2 Hen. IV. iii. 2. 310.
so "And he to England shall along with you."—Hamlet, iii. 3. 4.
We still say, "He is (journeying) for Paris," but not
"He is (ready) for no gallants' company without them."
B. J. E. out & c. i. 1.
"Any ordinary groom is (fit) for such payment."
Hen. VIII. v. 1. 174.
So T. N. iii. 3. 46; A. W. iii. 6. 109.
"I am (bound) to thank you for it."—T. of A. i. 2. 111.
Such an ellipsis explains
"Run from her guardage to the sooty bosom
Of such a thing as thou, (a thing fit) to fear (act.), not to
delight."—Othello, i. 2. 71.
Again, we might perhaps say, "This is not a sky (fit) to walk under," but not
"This sky is not (fit) to walk in."—J. C. i. 3. 39.
The modern distinction in such phrases appears to be this: when the noun follows is, there is an ellipse of "fit," "worthy:" when the noun precedes is, there is an ellipse of "intended," "made." Thus : "this is a book to read" means "this is a book worthy to read;" but, "this book is to read and not to tear," means "this book is intended or made for the purpose of reading." This distinction was not recognized by the Elizabethans. When we wish to express "worthy" elliptically, we insert a: "He is a man to respect," or we use the passive, and say, "He is to be respected." Shakespeare could have written "He is to respect" in this sense. The Elizabethans used the active in many cases where we should use the passive. Thus—
"Little is to do."—Macbeth, v. 7. 28.
"What's more to do."—Ib. v. 8. 64; A. and C. ii. 6. 60; J. C. iii. 1. 26; 2 Hen. VI. iii. 2. 3.
Hence "This food is not to eat" might in Shakespeare's time have meant "This food is not fit to eat;" now, it could only mean "intended to eat." Similarly "videndus" in Cicero meant "one who ought to be seen," "worthy to be seen;" but in poetry and in later prose it meant "one who may be seen," "visible."
The following passages illustrate the variable nature of this ellipsis :—
" I have been a debtor to you For curtesies which I will be ever to pay you,
And yet pay still."—Cymb. i. 4. 39.
i.e. "kindnesses which I intend to be always ready to pay you, and yet to go on paying."
We still retain an ellipsis of "under necessity" in the phrase
"I am (yet) to learn."—M. of V. i. 1. 5.
But we should not say:
"That ancient Painter who being (under necessity) to represent the griefe of the bystanders," &c.—MONTAIGNE, 3.
We should rather translate literally from Montaigne: "Ayant à représenter."
In
" I am to break with thee of some affairs,"
T. G. of V. iii. 1. 59.
the meaning is partly of desire and partly of necessity: "I want."
So Bottom says to his fellows :
"O, masters, I am (ready) to discourse wonders."
M. N. D. iv. 2. 29.
The ellipsis is "sufficient" in
"Mark Antony is every hour in Rome
Expected ; since he went from Egypt 'tis
A space (sufficient) for further travel."—A. and C. ii. 1. 31.
# IRREGULARITIES.
406. Double Negative.—Many irregularities may be explained by the desire of emphasis which suggests repetition, even where repetition, as in the case of a negative, neutralizes the original phrase :
" First he denied you had in him no right."
C. of E. iv. 2. 7.
"You may deny that you were not the cause."
Rich. III. i. 3. 90.
" Forbade the boy he should not pass these bounds. "—P. P. 9.
" No sonne, were he never so old of yeares, might not marry."—ASCH. 37.
This idiom is a very natural one, and quite common in E. E.
Double Comparative and Superlative. See Adjectives, 11.
407. Double Preposition. Where the verb is at some distance from the preposition with which it is connected, the preposition is frequently repeated for the sake of clearness.
" And generally in all shapes that man goes up and down in, from fourscore to thirteen, this spirit walks in."
T. of A. ii. 2. 119.
" For in what case shall wretched I be in. "—DANIEL.
"But on us both did haggish age steal on."—A. W. i. 2. 29.
"The scene wherein we play in."—A. Y. L. ii. 7. 139.
"In what enormity is Marcius poor in?"—Coriol. ii. 1. 18.
" To what form but that he is, should wit larded with malice, and malice forced with wit, turn him to?"—Tr. and Cr. v. 1. 63.
408. "Neither ... nor," used like "Both . . . and," followed by "not."
" Not the king's crown nor the deputed sword,
The marshal's truncheon nor the judge's robe,
Become them," &c.—M. for M. ii. 2. 60.
This very natural irregularity (natural, since the unbecomingness may be regarded as predicated both of the " king's crown," the " deputed sword," and the " marshal's truncheon") is very common.
"He nor that affable familiar ghost
That nightly gulls him with intelligence
As victors of my silence cannot (406) boast."—Sonn. 86.
The following passage may perhaps be similarly explained :
"He waived indifferently 'twixt doing them neither good nor harm."—Coriol. ii. 2. 17.
But it is perhaps more correct to say that there is here a confusion of two constructions, " He waived 'twixt good and harm, doing them neither good nor harm." The same confusion of two constructions is exemplified below in the use of the superlative.
409. Confusion of two Constructions in Superlatives.
" This is the greatest error of all the rest."—M. N. D. v. 1. 252.
" Of all other affections it is the most importune. "—B. E. Envy.
"York is most unmeet of any man."—2 Hen. VI. i. 3. 167.
" Of all men else I have avoided thee."—Macbeth, v. 8. 4.
" He hath simply the best wit of any handicraft-man in Athens."
M. N. D. iv. 2. 9.
" To try whose right,
Of thine or mine, is most in Helena. "—Ib. iii. 2. 337.
" I do not like the tower of any place. "—Rich. III. iii. 1. 68.
This (which is a thoroughly Greek idiom, though independent in English) is illustrated by Milton's famous line—
" The fairest of her daughters Eve."
The line is a confusion of two constructions, " Eve fairer than all her daughters," and " Eve fairest of all women." So " I dislike the tower more than any place," and " most of all places," becomes "of any place."
Our modern " He is the best man that I have ever seen," seems itself to be incorrect, if "that" be the relative to "man." It may, perhaps, be an abbreviation of "He is the best man of the men that I have ever seen."
410. Confusion of two constructions with "whom."
"Young Ferdinand whom they suppose is drown'd."
Temp. iii. 2. 92.
" Of Arthur whom they say is killed to-night."—K. J. iv. 2. 165.
" The nobility ... whom we see have sided."—Coriol. iv. 2. 2.
So in St. Matt. xvi. 13, all the versions except Wickliffe's have " Whom do men say that I, the son of man, am ?" Wickliffe has " Whom seien men to be mannes sone?"
The last passage explains the idiom. It is a confusion of two constructions, e.g. " Ferdinand who, they suppose, is drowned" and "whom they suppose to be crowned."
411. Other confusions of two constructions.
"Why I do trifle thus with his despair
Is done to cure it,"—Lear, iv. 6. 33.
combines " Why I trifle is to cure" and "My trifling is done to cure." In itself it is illogical.
"The battle done, and they within our power
Shall never see his pardon,"—Lear, v. 1. 67.
is a confusion of "let the battle be done, and they" and "the battle (being) done, they."
" I saw not better sport these seven years day. "—2 Hen. VI. ii. 1. 3. A combination of "since this day seven years" and "during these seven years."
" Out of all 'cess (excess),"—1 Hen. IV. ii. 1. 6.
is a confusion of "to excess," or "in excess," and "out of all bounds." " So late ago," T. N. v. 1. 22, seems a combination of "so lately" and "so short a time ago,"
"Marry that, I think, be young Petruchio,"—R. and J. i. 5. 133. is a confusion of "That, I think, is and "I think that that be." For the subjunctive after "think," see Subjunctive, 368 and 299.
So, perhaps,
"This youth, howe'er distressed, appears he hath had
Good ancestors,"—Cymb. iv. 2. 47.
is a confusion of " He hath had, (it) appears, good ancestors," and "He appears to have had." This is, perhaps, better than to take "appears" as an active verb. See 295. Precisely similar is :
" Let what is meet be said, it must be meet."—Coriol. iii. 1. 170.
combining "Let what is meet be said to be" and " Let it be said (that) what is meet must be meet."
Compare 353, and. add, as a confusion of the infinitive and imperative,
"There is no more but (to) say so. "—Rich. III. iv. 2. 81.
In "We would have had you heart," Ib. III. iii. 5. 56, there may be some confusion between " you should have heard and " we would have had you hear;" but more probably the full construction is "We would have had you (to have) heard (360)," and " to have " is omitted through dislike of repetition. So Coriol. iv. 6. 35 (415) : " We should . . . found it so."
Compare also
"He would have had me (to have) gone into the steeple-house."
Fox's Journal (ed. 1765), p. 57.
" He would have had me (to have) had a meeting."—Ib. p. 60.
412. Confusion of proximity. The following (though a not uncommon Shakespearian idiom) would be called an unpardonable mistake in modern authors :—
" The posture of your blows are yet unknown. "—J. C. v. 1. 33.
" Whose loss of his most precious queen and children
Are even now to be afresh lamented."—W. T. iv. 1. 26.
"Which now the loving haste of these dear friends
Somewhat against our meaning have prevented."
Rich. III. iii. 5. 56.
" The venom of such looks, we fairly hope,
Have lost their quality."—Hen. V. v. 2. 19.
" But yet the state of things require."—DANIEL, Ulysses and Siren.
"The approbation of those . . . are," &c.—Cymb. i. 4. 17.
" How the sight
Of those smooth rising cheeks renew the story
Of young Adonis."—B. F. F. Sh. i. 1.
" Equality of two domestic powers
Breed scrupulous faction."—A. and C. i. 3. 48.
" The voice of all the gods
Make heaven drowsy."—L. L. L. iv. 3. 345.
Here, however, "voice" may be (471) for "voices."
" Then know
The peril of our curses light on thee."—K. J. iii. 1. 295.
" The very thought of my revenges that way
Recoil upon myself."—W. T. ii. 3. 20.
" More than the scope
Of these delated articles allow."—Hamlet, i. 2. 38.
The subjunctive is not required, and therefore " have "is probably plural, in
" If the scorn of your bright eyne
Have power to raise such love in mine."—A. Y. L. iv. 3. 51.
In these cases the proximity of a plural noun seems to have caused the plural verb, contrary to the rules of grammar. The two nouns together connected by "of" seem regarded as a compound noun with plural termination. So
" These kind-of-knaves."—Lear, ii. 2. 107.
" Those blest-pair-of-fixed-stars."—B. and F. F. Sh. ii. 1.
" These happy-pair of lovers meet straightway. "—Ib.
Similarly—
" Where such as thou mayest find him."—Macbeth, iv. 2. 81.
In the following instance the plural nominative is implied from the previous singular noun—
" As every alien pen hath got my use,
And under thee their poesy disperse."—Sonn. 78.
In " And the stars whose feeble light
Give a pale shadow to the night,"—B. and F. F. Sh. iii. 1.
perhaps "give" may be subjunctive after the relative. (See 367.)
413. Implied nominative from participial phrases. Sometimes a nominative has to be extracted ungrammatically from the meaning of a sentence. This is often the case in participial phrases :
" Beaten for loyalty
Excited me to treason."—Cymb. v. 5. 343.
i.e. may having been beaten."
" The king of his own virtuous disposition,
Aiming belike at your interior hatred,
Which in your outward actions shews itself,
Makes him to send."—Rich. III. i. 2. 63.
i.e. " the fact that the king aims makes him to send."
414. The redundant Object. Instead of saying "I know what you are," in which the object of the verb "I know" is the clause "what you are," Shakespeare frequently introduces before the dependent clause another object, so as to make the dependent clause a mere explanation of the object.
" I know you what you are."—Lear, i. 1. 272.
" I see you what you are."—T. N. i. 4. 269.
" Conceal me what I am."—Ib. i. 2. 53.
" You hear the learn'd Bellario what he writes."
M. of V. iv. 1. 167.
" We'll hear him what he says."—A. and C. v. 1. 51.
" To give me hearing what I shall reply."
IHen. VI. iii. 1. 28.
" But wilt thou hear me how I did proceed?"
Hamlet, v. 2. 27.
" March on and mark King Richard how he looks."
Rich. II. iii. 3. 61 ; Ib. v. 4. 1.
" Sorry I am my noble cousin should
Suspect me that I mean no good to him."
Rich. III. iii. 7. 89.
" See the dew-drops, how they kiss
Every little flower that is."—B. and F. F. Sh. ii. r.
Hence in the passive :
" The queen's in labour,
(They say in great extremity) and fear'd
She'll with the labour end,"—Hen. VIII. v. 1. 19.
where the active would have been " they fear the queen that she will die." For "fear" thus used, see Prepositions, 200.
So "no one asks about the dead man's knell for whom it is " becomes in the passive
" The dead man's knell
Is there scarce asked, for who,"—Macbeth, iv. 3. 171.
and "about which it is a wonder how his grace should glean it" becomes
" Which is a wonder how his grace should glean it."
Hen. V. i. 1. 53.
This idiom is of constant occurrence in Greek ; but it is very natural after a verb of observation to put, first the primary object of observation, e.g. " King Richard," and then the secondary object, viz. " King Richard's looks." There is, therefore, no reason whatever for supposing that this idiom is borrowed from the Greek. After a verb of commanding the object cannot always be called redundant, as in
" (She) bade me, if I had a friend that loved her,
I should but teach him how to tell my story."
Othello, i. 3. 165.
i.e. " she commanded me (that) I should," &c. But it is redundant in
" The constable desires thee thou wilt mind
Thy followers of repentance."—Hen. V. iv. 3. 85.
" He wills you . . . that you divest yourself."—Ib. ii. 4. 77–8.
Compare
" Belike they had some notice of (about) the people
How I had moved them."—J. C. iii. 2. 275.
A somewhat different case of the redundant object is found in
" Know you not, master, to some kind of men
Their graces serve them but as enemies?
No more do yours,"—A. Y. L. ii. 3. 10.
where the last line means, "your graces are not more serviceable to you."
415. Construction changed by change of thought.
" One of the prettiest touches was when, at the relation of the queen's death, . . . how attentiveness wounded his daughter."—W. T. v. 2. 94.
The narrator first intends to narrate the point of time, then diverges into the manner, of the action.
"Purpose is but the slave to memory,
Which now, like fruit unripe, sticks on the tree,
But fall unshaken when they mellow be. "—Hamlet, iii. 2. 201.
The subject, which is singular, is here confused with, and lost in, that to which it is compared, which is plural. Perhaps this explanation also suits :
" And then our arms, like to a muzzled bear,
Save in aspect hath all offence sealed up,"—K. J. ii. 1. 250.
though this may be a case of plural nominative with singular verb. (See 334.)
In the following, Henry V. begins by dictating a proclamation, but under the influence of indignation passes into the imperative of the proclamation itself :
" Rather proclaim it, Westmoreland, through our host
That he which hath no stomach to this fight
Let him depart."—Hen. V. iv. 3. 35-6.
This is more probable than that "he" (224) is used for "man."
" Should " is treated as though it were " should have " (owing to the introduction of the conditional sentence with " had ") in the following anomalous passage :
" We should by this to all our lamentation,
If he had gone forth consul, found it so."—Coriol. iv. 6. 35.
So Rich. III. iii. 5. 56 (411).
The way in which a divergence can be made from the subject to the thing compared with the subject is illustrated by
" So the proportions of defence are filled :
Which, of a weak and niggardly projection,
Doth, like a miser, spoil his coat with scanting
A little cloth."—Hen. V. ii. 4. 46.
" Whose veins, like a dull river far from spring
Is still the same, slow, heavy, and unfit
For stream and motion, though the strong winds hit
With their continual power upon his sides."
B. and F. F. Sh. i. 1.
" But, good my brother,
Do not, as some ungracious pastors do,
Show me the steep and thorny way to heaven,
Whiles, like a puffed and reckless libertine,
Himself the primrose path of dalliance treads."
Hamlet, i. 3. 50.
instead of "whiles you tread." But in
"Those sleeping stones
That, as a waist, doth girdle you about,
Had been dishabited,"—K. J. ii. 1. 216.
"doth," probably, has "that" for its subject. See Relative, 247. In
" Are not you he That frights the maidens of the villagery,
Skim milk, and sometimes labour in the quern
And bootless make the breathless housewife churn?"
M. N. D. ii. 1. 35–9.
the transition is natural from " Are not you the person who ?" to " Do not you ?"
416. Construction changed for clearness. (See also 285.) Just as (285) that is sometimes omitted and then inserted to connect a distant clause with a first part of a sentence, so sometimes "to" is inserted apparently for the same reason—
"That God forbid, that made me first your slave,
I should in thought control your times of pleasure,
Or at your hand the account of hours to crave. "—Sonn. 58.
Here "to" might be omitted, or "should" might be inserted instead, but the omission would create ambiguity, and the insertion would be a tedious repetition.
" Heaven would that she these gifts should have,
And I to live and die her slave."—A. Y. L. iii. 2. 162.
" Keep your word, Phœbe, that you'll marry me,
Or else, refusing me, to wed this shepherd."
Ib. v. 4. 21-2.
" But on this condition, that she should follow him, and he not to follow her. "—BACON, Adv. of L. 284.
" The punishment was, that they should be put out of commons and not to be admitted to the table of the gods. "—Ib. 260.
" That we make a stand upon the ancient way, and look about us and discover what is the straight and right way, and so to walk in it."—B. E. 100.
In the following, the infinitive is used in both clauses, but the "to" only in the latter :—
" In a word, a man were better relate himself to a Statue or Picture, than to suffer his thoughts to pass in smother."
B. E. 103.
417. Noun Absolute. See also Redundant Pronoun, 243. Sometimes a noun occurs in a prominent position at the beginning of a sentence, to express the subject of the thought, without the usual grammatical connection with a verb or preposition. In some cases it might almost be called a vocative, only that the third person instead of the second is used, and then the pronoun is not redundant. Sometimes the noun seems the real subject or object of the verb, and the pronoun seems redundant. When the noun is the object, it is probably governed by some preposition understood, "as for," " as to."
" My life's foul deed, my life's fair end shall free it."—R. of L.
" The prince that feeds great natures, they will slay him."
B. J. Sejanus, iii. 3.
" But virtue, as it never will be moved,
So lust," &c.—Hamlet, i. 5. 53.
" Look when I vow, I weep; and vows so born,
In their nativity all truth appears."—M. N. D. iii. 2. 124.
But this may be explained by 376.
"'Tis certain, every man that dies ill, the ill upon his own head." —Hen. V. iv. 1. 197.
"But if I thrive, the gain of my attempt
The least of you shall share his part thereof."
Rich. III. v. 3. 267.
" That thing you speak of I took it for a man."—Lear, iv. 6. 77.
The following may be thus explained :—
"Rather proclaim it, Westmoreland, through our host,
That he which hath no stomach to this fight,
Let him depart."—Hen. V. iv. 3. 34.
" That can we not . . . but he that proves the king
To him will we prove loyal."—K. J. ii. 1. 271.
" He" being regarded as the normal form of the pronoun, is appropriate for this independent position. So
"But I shall laugh at this a twelve-month hence,
That they who brought me in my master's hate
I live to look upon their tragedy."—Rich. III. iii. 2. 57.
These three examples might, however, come under the head of Construction changed, 415, as the following (which closely resembles the first) certainly does :
" My lord the emperor,
Sends thee this word that, if thou love thy son,
Let Marcius, Lucius, or thyself, old Titus,
Or any one of you, chop off your hand."—T. A. iii. 1.151
In this, and perhaps in the first example, the "that," like τι in Greek, is equivalent to inverted commas.
"May it please your grace, Antipholus, my husband,
Whom I made lord of me, . . . this ill day
A most outrageous fit of madness took him."
C. of E. v. 1. 138.
" The trumpery in my house, go bring it hither."—Temp. iv. 1. 186.
It is, of course, possible to have an infinitive instead of a noun :
" To strike him dead, I hold it not a sin."—R. and J. i. 4. 61.
For the noun absolute with the participle, see Participle, 376.
418. Foreign Idioms. Several constructions in Bacon, Ascham, and Ben Jonson, such as "ill," for "ill men" (Latin 'mali'), " without all question" ('singe omni dubitatione'), seem to have been borrowed from Latin. It is questionable, however, whether there are many Latinisms in construction (Latinisms in the formation of words are of constant occurrence) in Shakespeare. We may perhaps quote—
" Those dispositions that of late transform you
From what you rightly are."—Lear, i. 4. 242.
Compare
"He is ready to cry all this day,"—B. J. Sil. Wom. 4.
as an imitation of the Latin use of "jampridem" with the present in the sense of the perfect. But it is quite possible that the same thought of continuance may have prompted the use of the present, both in English and Latin. " He is and has been ready to cry,"&c. The use of "more better," &c., the double negative, and the infinitive after 'than," are certainly of English origin. The following—
" Whispering fame
Knowledge and proof doth to the jealous give,
Who than to fail would their own thought believe,"—
B. J. Sejan. 2.
in the omission of "rather" after "would," reminds us of the omission of "potius" after "malo." Perhaps also
"Let that be mine,"—M. for M. ii. 2. 12.
is an imitation of "meum est," "It is my business."
The following resembles the Latin idiom, "post urbem conditam," except that there is also an ellipsis of a pronoun :
"'Tis our hope, sir,
After (our being) well enter'd (as) soldiers, to return
And find your grace in health."—A. W. ii. 1. 6.
I cannot recall another such an instance, and it is doubtful whether "after" does not here mean "hereafter:" "It is our hope to return hereafter well-apprenticed soldiers." But such participial phrases preceded by prepositions seem to be of classical origin, as in Milton :
" Nor delay'd
The winged saint after his charge received."
MILTON, P. L. v. 248.
"He, after Eve seduced, unminded slunk
Into the wood fast by."—Ib. 332.
and even, contrary to the particular Latin idiom :
"They set him free-without his ransom paid."—1 Hen. VI. iii. 3.72.
The following resembles the Latin use of "qui si," for the English "and if he."
" Which parti-coated presence of loose love
Put on by us, if in your heavenly eyes
Have misbecomed our oaths and gravities."—L. L. L. v. 2. 778.
419. Transposition of Adjectives.
The adjective is placed after the noun :
(1) In legal expressions in which French influence can be traced :
" Heir apparent."—1 Hen. IV. i. 2. 65.
" Heir general."—Hen. V. i. 2. 66.
"Thou cam'st not of the blood-royal."—Ib. 157.
"In the seat royal."—Rich. III. iii. 1. 164.
" Sport royal."—T. N. ii. 3. 187.
" Or whether that the body public be a horse."
M. for M. i. 2. 163.
" My letters patents (Fol.) give me leave."—Rich. II. ii. 3. 130.
(2) Where a relative clause, or some conjunctional clause, is understood between the noun and adjective :
"Duncan's horses,
(Though) Beauteous and swift, the minions of their race,
Turned wild in nature."—Macbeth, ii. 4. 15.
"Filling the whole realm . . . with new opinions
(That are) Divers and dangerous."—Hen. VIII. v. 3. 18.
Hence, where the noun is unemphatic, as "thing," "creature," this transposition may be expected :
"In killing creatures (that were) vile."—Cymb. v. 5. 252.
"He look'd upon things (that are) precious as they were
The common muck of the world."—Coriol. ii. 2. 129.
Hence, after the name of a class, the adjective is more likely to be transposed than in the case of a proper name. Thus
"Celestial Dian, goddess argentine."—P. of T. v. 2. 251.
i.e. "goddess (that bearest) the silver bow." The difference between a mere epithet before the noun, and an additional statement conveyed by an adjective after the noun, is illustrated by
" If yet your gentle souls fly in the air
And be not fix'd in (a) doom (that is) perpetual."
Rich. III. iv. 4. 11, 12.
Similarly in
"With eyes severe, and beard of formal cut."—A. Y. L. ii. 7. 155.
"My presence like a robe pontifical."—1 Hen. IV. iii. 2. 56.
"eyes" and "a robe" are unemphatic, their existence being taken for granted, and the essence of the expression is in the transposed adjective.
The "three" is emphatic, and the divorcing of some "souls and bodies" is taken as a matter of course, in
" Souls and bodies hath he divorced three."—T. N. iii. 4. 260 Somewhat similar—
"Satisfaction there can be none."—Ib. 262.
This relative force is well illustrated by
" Prince. I fear no uncles dead.
Glou.
Nor none that live. I hope."
Rich. III. iii. 1. 146.
(3) Hence participles (since they imply a relative), and any adjectives that from their terminations resemble participles, are peculiarly liable to be thus transposed.
Similarly adjectives that end in -ble, -ite, and -t, -ive, -al, are often found after their nouns, e.g. "unspeakable," "unscaleable," "impregnable;" "absolute," "devout," "remote," "infinite" (often), "past," "inveterate;" "compulsative," "invasive," "defective;" "capital," "tyrannical," "virginal," "angelical," "unnatural."
(4) Though it may be generally said that when the noun is unemphatic, and the adjective is not a mere epithet but essential to the sense, the transposition may be expected, yet it is probable that the influence of the French idiom made this transposition especially common in the case of some words derived from French. Hence, perhaps, the transposition in
"Of antres vast and deserts idle."—Othello, i. 1. 140.
And, besides "apparent" in the legal sense above, we have
"As well the fear of harm as harm apparent."
Rich. III. ii. 2. 130.
Hence, perhaps, the frequent transposition of " divine," as
"By Providence divine."—Tempest, i. 2. 158.
So
" Ful wel sche sang the service devyne."
CHAUCER, C. T. 122.
"Men devout."—Hen. V. i. 1. 9.
"Unto the appetite and affection common."—Coriol. i. 1. 108
Latin usage may account for some expressions, as
"A sectary astronomical."—Lear, i. 2. 164.
419a. Transposition of adjectival phrases.
It has been shown above (419), that when an adjective is not a mere epithet, but expresses something essential, and implies a relative, it is often placed after the noun. When, however, connected with the adjective, e.g. "whiter," there is some adverbial phrase, e.g. "than snow," it was felt that to place the adjective after the noun might sometimes destroy the connection between the noun and adjective, since the adjective was, as it were, drawn forward to the modifying adverb. Hence the Elizabethans sometimes preferred to place the adjectival part of the adjective before, and the adverbial part after, the noun. The noun generally being unemphatic caused but slight separation between the two parts of the adjectival phrase. Thus "whiter than snow," being an adjectival phrase, "whiter" is inserted before, and "than snow" after, the noun.
"Nor scar that [whiter] skin-of-hers [than snow]."
Othello, v. 2. 4.
" So much I hate a [breaking] cause to be
[Of heavenly oaths]."—L. L. L. v. 2. 355.
So
" A [promising] face [of manly princely virtues]."
B. and F. (Walker).
" As common
As any [the most vulgar] thing [to sense]."—Ham. i. 2. 99.
i.e. "anything the most commonly perceived."
" I shall unfold [equal] discourtesy
[To your best kindness]."—Cymb. ii. 3. 101.
" The [farthest] earth [removed from thee]."—Sonn. 44.
"Bid these [unknown] friends [to us], welcome."
W. T. iv. 3. 65.
" Thou [bloodier] villain [than terms can give thee out]." Macbeth, v. 8. 7.
"A [happy] gentleman [in blood and lineaments]."
Rich. II. iii. 1. 9.
" As a [long-parted] mother [with her child]."
Ib. iii. 2. 8. (See 194.)
'Thou [little better] thing [than earth]."—Ib. iii. 4. 77.
' You have won a [happy] victory [to Rome]."
Coriol. v. 3. 186.
Hence, even where the adjective cannot immediately precede the noun, yet the adjective comes first, and the adverb afterwards.
"That were to enlard his fat-already-pride."
Tr. and Cr. ii. 2. 205.
"May soon return to this our [suffering] country
[Under a hand accurst]."—Macbeth, iii. 6. 48.
"The [appertaining] rage
[To such a greeting]."—R. and J. iii. 1. 66.
"With [declining] head [into his bosom]."—T. of Sh. Ind. 1. 119. So probably
"Bear our [hack'd] targets [like the men that owe them]."
A. and C. iv. 8. 31.
This is very common in other Elizabethan authors :
"The [stricken] hind [with Shaft]."—LORD SURREY (Walker).
"And [worthie] work [of infinite reward]."
SPENSER, F. Q. iii. 2. 21.
"Of that [too wicked] woman [yet to die]."
B. and F. (Walker).
"Some sad [malignant] angel [to mine honour]."—Ib.
which perhaps explains
"Bring forth that [fatal] screech-owl [to our house]."
3 Hen. VI. ii. 6. 56.
So
"Thou [barren] thing [of honesty] and honour !"—B. and F.
perhaps explains
" Thou perjur'd and thou [simular] man [of virtue]."
Lear, iii. 2. 54.
"Bring me a [constant] woman [to her husband]."
Hen. VIII. iii. 1. 134.
"O, for my sake do you with fortune chide,
The [guilty] goddess [of my harmful deeds]."—Sonn. 111.
" To this [unworthy] husband [of his wife]."—A. W. iii. 4. 30.
"A [dedicated] beggar [to the air]."—T. of A. iv. 2. 13.
This transposition extends to an adverb in
" And thou shalt live [as freely] as thy lord
[To call his fortunes thine]."—T. N. i. 4. 39, 40.
i.e. "as free to use my fortune as I am."
Unless "to" is used loosely like "for," the following is a case of transposition :
"This is a [dear] manakin [to you], Sir Toby."
T. N. iii. 2. 57.
420. Transposition of Adverbs. The Elizabethan authors allowed themselves great licence in this respect.
We place adverbial expressions that measure excess or defect before the adjective which they modify, "twenty times better," &c. This is not always the case in Shakespeare :
"Being twenty times of better fortune."—A. and C. iv. 1. 3.
"Our spoils (that) we have brought home
Do more than counterpoise, a full third part,
The charges of the action."—Coriol. v. 6. 77.
" I am solicited not by a few,
And those of true condition."—Hen. VIII. i. 2. 18.
For not transposed, see also 305.
"Like to a harvest man that's task'd to mow
Or all, or lose his hire."—Coriol. i. 3. 40.
In "All good things vanish less than in a day" (Nash), there is, perhaps, a confusion between "less long-lived than a day" and "more quickly than in a day." At all events the emphatic use of " less" accounts for the transposition.
Such transpositions are most natural and frequent in the case of adverbs of limitation, as but (see But, 54), only, even, &c.
"Only I say,"—Macbeth, iii, 6. 2.
for "I only say."
"Only I yield to die."—J. C. v. 4. 12.
for "I yield only in order to die,"
"And I assure you
Even that your pity is enough to cure me,"—B. J.
for "that even your pity."
" He did it to please his mother and to be partly proud,"
Coriol. i. 1. 40.
for "and partly to be proud."
Somewhat similar is
"Your single bond, "—M. of V. i. 3. 146.
for "the bond of you alone."
421. Transposition of Adverbs. When an adverb is transposed to the beginning for emphasis, it generally transposes the subject after the verb, but adverbs are sometimes put at the beginning of a sentence without influencing the order of the other words.
"Seldom he smiles."—J. C. i. 2. 205.
"For always I am Cæsar."—Ib. i. 2. 212.
"No more that thane of Cawdor shall deceive."
Macbeth, i. 2. 63.
" Of something nearly that concerns yourselves."
M. N. D. i. 1. 126.
422. Transposition of Article. In Early English we sometimes find "a so new robe." The Elizabethan authors, like ourselves, transposed the a and placed it after the adjective: "so new a robe." But when a participle is added as an epithet of the noun, e.g. "fashioned," and the participle itself is qualified by an adjective used as an adverb, e.g. "new," we treat the whole as one adjective, thus, "so new-fashioned a robe." Shakespeare on the contrary writes—
"So new a fashion'd robe."—K. J. iv. 2. 27.
"So fair an offer'd chain."—C. of E. iii. 2. 186.
" Or having sworn too hard a keeping oath."
L. L. L. i. 1. 65.
"So rare a wonder'd father and a wife."
Temp. iv. 1. 123.
"I would have been much more a fresher man."
Tr. and Cr. v. 6. 20.
We still say, "too great a wit," but not with Chaucer, C. T. :
"For when a man hath overgret a wit,"
possibly because we regard "overgreat" as an adjective, and "too great" as a quasi-adverb. Somewhat similar is :
"On once-a-flock-bed, but repair'd with straw,
With tape-ty'd curtains never meant to draw."
POPE, Moral E. iii. 301.
So we can say "how poor an instrument," regarding "how" as an adverb, and "how poor" as an adverbialized expression, but not
"What poor an instrument,"—A. and C. v. 2. 236.
because "what" has almost lost with us its adverbial force.
" So brave(ly) a mingled temper saw I never."
B. and F. (Walker).
" Chaucer, who was so great(ly) a learned scholar."
KINASTON (Walker)
The a is used even after the comparative adjective in
" If you should need a pin,
You could not with more tame a tongue desire it."
M. for M. ii. 2. 46.
423. Transpositions in Noun-clauses containing two nouns connected by "of." It has been observed in 412 that two nouns connected by "of" are often regarded as one. Hence sometimes pronominal and other adjectives are placed before the whole compound noun instead of, as they strictly should be, before the second of the two nouns.
" Yet that thy brazen gates of heaven may ope."
3 Hen. VI. ii. 3. 40.
"My pith of business."—M. for M. i. 4. 70.
"The tribunes have pronounced
My everlasting doom of banishment."—T. A. iii. 1. 51.
" Let it stamp wrinkles in her brow of youth."
Lear, i. 4. 306.
" My latter part of life."—A. and C. iv. 6. 39.
"My whole course of life."—Othello, i. 3. 91.
" I will presently go learn their day of marriage."
M. Ado, ii. 2. 57.
" Thy bruising irons of wrath."—Rich. III. v. 3. 110.
"Thy ministers of chastisement."—Ib. 113.
"In my prime of youth."—Ib. 119.
" Thy heat of lust."—R. of L. 1473.
"My home of love."—Sonn. 109.
" And punish them to your height of pleasure."
M. for M. v. 1. 240.
" His means of death, his obscure funeral."
Hamlet, iv. 5. 213.
i.e. "the means of his death."
"What is your cause of distemper ?"—Hamlet, iii. 2. 350.
" Your sovereignty of reason."—Ib. i. 4. 73. (See 200.)
" My better part of man."—Macbeth, v. 7. 18.
"His chains of bondage."—Rich. II. i. 3. 89.
" Your state of fortune and your due of birth."
Rich. III. iii. 7. 127.
This is perhaps illustrated by
"What country-man ?"—T. N. v. 1. 238; T. of Sh. i. 2. 190.
for "a man of what country?"
The possessive adjective is twice repeated in
"Her attendants of her chamber."—A. Y. L. ii. 2. 5.
So
"This cause of Rome,"—T. A. i. 1. 32.
does not mean "this cause as distinguished from other causes of Rome," but " this, the Roman cause." Somewhat similar is
" Your reproof
Were well deserv'd of rashness,"—A. and C. ii. 2. 124.
where we should say "the reproof of your rashness" (unless "of" here means "about," "for").
"The idea of her life shall sweetly creep
Into his study of imagination."—M. Ado, iv. 2. 27.
i.e. "the study of his imagination."
"Our raiment and state of bodies."—Coriol. v. 3. 95.
"More than ten criers, and six noise of trumpets."
B. J. Sejan. v. 7.
The compound nature of these phrases explains, perhaps, the omission of the article in
" Hath now himself met with the fall-of-leaf."
Rich. II. iii. 4. 49.
424. Transposition of Prepositions in Relative and other clauses. We now dislike using such transpositions as
"The late demand that you did sound me in."-—Rich. III. iv. 2. 87.
"Betwixt that smile we would aspire to."—Hen. VIII. iii. 2. 368.
" A thousand men that fishes gnawed upon."—Rich. III. i. 4. 25.
"Found thee a way out of his wreck to rise in."
Hen. VIII. iii. 2. 438.
But it may be traced to E. E. (203), and is very common in Shakespeare, particularly in Hen. VIII., where we even find
"Where no mention
Of me must more be heard of."—Hen. VIII. iii. 2. 435.
It has been said above (203) that the dissyllabic forms of prepositions are peculiarly liable to these transpositions. Add to the above examples :
" Like a falcon towering in the skies,
Coucheth the fowl below."—R. of L. 506.
425. Transposition after Emphatic Words. The influence of an emphatic word at the beginning of a sentence is shown in the transposition of the verb and subject. In such cases the last as well as the first word is often emphatic.
" In dreadful secrecy impart they did."—Hamlet, i. 2. 207.
"And so have I a noble father lost, A sister driven into desperate terms. "—Ib. iv. 7. 25.
Here note, that though the first line could be re-transposed and Laertes could naturally say "I have lost a father," on the other hand he could not say "I have driven a sister" without completely changing the sense. "Have" is here used in its original sense, and is equivalent to "I find." When "have" is thus used without any notion of action, it is separated from the participle passive.
" But answer made it none."—Hamlet, i. 2. 216.
" Pray can I not."—Ib. iii. 3. 38.
" Supportable
To make the dear loss have I means much weaker."
Temp. v. 1. 146.
The influence of an emphatic adverbial expression preceding is shown in the difference between the order in the second and the first of the two following lines :—
" As every alien pen hath got my use,
And under thee their poetry disperse."—Sonn. 78.
" I did, my lord,
But loath am to produce so bad an instrument."
A. W. v. 3. 201.
"Before the time I did Lysander see,
Seem'd Athens as a paradise to me. "—M. N. D. i. 1. 205.
When the adverbs "never," "ever," are emphatic and placed near the beginning of a sentence, the subject often follows the verb, almost always when the verb is "was," &c. We generally write now "never was," but Shakespeare often wrote "(there) was never."
" Was never widow had so dear a loss."—Rich. III. ii. 2. 77.
Sometimes a word is made emphatic by repetition :
" Sec. O. Peace! We'll hear him.
Third O. Ay, by my beard will we."——T. G. of V. iv. 1. 10.
" Hamlet. Look you, these are the stops.
Guild. But these cannot I command."—Hamlet, iii. 3. 377.
Or partly by antithesis, as well as by its natural importance :
" I your commission will forthwith despatch,
And he to England shall along with you."
Hamlet, iii. 3. 3, 4.
" My soul shall thine keep company to heaven."
Hen. V. iv. 6. 16.
The following is explained by the omission of "there :"
" I am question'd by my fears . . . that (there) may blow
No sneaping winds at home."—W. T. i. 2. 13.
There seems a disposition to place participles, as though used absolutely, before the words which they qualify.
" And these news,
Having been well, that would have made me sick,
Being sick, have in some measure made me well."
2 Hen. IV. i. 1. 138.
It is rare to find such transpositions as
" Then the rich jewell'd coffer of Darius,
Transported shall be at high festivals."—1 Hen. VI. i. 6. 26.
Transpositions are common in prose, especially when an adverb precedes the sentence.
" Yet hath Leonora, my onely daughter, escaped."
MONTAIGNE (Florio), 225.
" And, therefore, should not we marry so young."—Ib.
"Now, sir, the sound that tells what hour it is
Are clamorous groans,"—Rich. II. v. 5. 56.
is rather a case of "confusion of proximity" ("are" being changed to "is") than transposition. (See 302.)
426. Transposition after Relative. The relative subject, possibly as being somewhat unemphatic itself, brings forward the object into a prominent and emphatic position, and consequently throws a part of the verb to the end, not however (as in German) the auxiliary.
" By Richard that dead is."—1 Hen. IV. i. 3. 146.
" But chide rough winter that the flower hath killed."—R. of L.
" That heaven's light did hide."—SPENS. F. Q. i. 1. 7.
427. Other Transpositions. In the second of two passive clauses when the verb "is" is omitted, the subject is sometimes transposed, perhaps for variety.
" When liver, heart, and brain,
These sovereign thrones, are all supplied, and filled
(Are) Her sweet perfections with one self king."
T. N. i. 1. 39.
" Since his addiction was to courses vain,
And never (was) noted in him any study."—Hen. V. i. 1.57.
It is not probable that "perfections" and "study" are here absolutely used with the participle. See, however, And, 95.
In "By such two that would by all likelihood have confounded each other" (Cymb. i. 4. 53), "two" is emphatic, like "a pair." So "we" is emphatic in, "all we like sheep have gone astray," and in Hamlet, ii. 2. 151, in both cases, because of antithesis.
" Into the madness wherein now he raves
And all we mourn for."—Hamlet, ii. 2. 151. (See 240.)
# COMPOUND WORDS.
428. Hybrids. The Elizabethans did not bind themselves by the stricter rules of modern times in this respect. They did not mind adding a Latin termination to a Teutonic root, and vice versâ. Thus Shakespeare has "increaseful," "bodement," &c. Holland uses the suffix -fy after the word "fool" (which at all events does not come to us direct from the Latin), "foolify," where we use "stultify." The following words illustrate the Elizabethan licence :—
" Bi-fold."—Tr. and Cr. v. 2. 144.
" Out-cept."—B. J. (Nares).
" Exteriorly."—K. J. iv. 2. 257.
" Sham'st thou not, knowing whence thou art extraught?"
3 Hen. VI. ii. 2. 142.
where there is a confusion between the Latin "extracted" and the English "raught," past part. of "reach." Compare Pistol's "exhale," Hen. V. ii. 1. 66, i.e. "ex-haul," "draw out," applied to a sword.
There was also great licence in using the foreign words which were pouring into the language.
" And quench the stelled fires."—Lear, iii. 7. 61.
" Be aidant and remediate."—Ib. iv. 4. 17.
" Antres vast and deserts idle."—Othello, i. 3. 140.
429. Adverbial Compounds.
"Till Harry's back-return."—Hen. V. v. Prologue, 41.
"Thy here-approach," Macb. iv. 3. 133, 148 ; "Our hence-going," Cymb. iii. 2. 65 ; "Here-hence," B. J. Poetast. v. 1 ; "So that men are punish'd for before-breach of the king's laws in now-the-king's-quarrel ," Hen. V. iv. 1. 179, i.e. "the king's now (present) quarrel." This last extraordinary compound is a mere construction for the occasion, to correspond antithetically to "before-breach," but it well illustrates the Elizabethan licence.
" The steep-up heavenly hill."—Sonn. 7.
" I must up-fill this osier cage of ours."—R. and J. ii. 3. 7.
" Up-hoarded."—Hamlet, i. 1. 136.
"With hair up-staring."—Tempest, i. 2. 213.
430. Noun-Compounds. Sometimes the first noun may be treated as a genitive used adjectively. (See 22.) Thus, "thy heart-blood " (Rich. II. iv. 1. 38) is the same as "thy heart's blood ;" " brother-love" (Hen. VIII. v. 3. 73), i.e. brother's love.
So
"Any-moment-leisure."—Hamlet, i. 3. 133.
" This childhood-proof."—M. of V. i. 1. 144.
" Childhood-innocence."—M. N. D. iii. 2. 202.
"All the region-kites."—Hamlet, ii. 2. 607.
"A lion-fell."—M. N. D. v. 1. 227, i.e. "a lion's skin."
So probably
" Faction-traitors."—Rich. II. ii. 2. 57.
"Self" is used as a compound noun in "self-conceit," and this explains
"Infusing him with self-and-vain-conceit. "—Rich. II. iii. 2. 166.
"Every minute-while,"—1 Hen. VI. i. 4. 54.
where "while" has its original force as a noun = "time."
But often when a noun is compounded with a participle, some preposition or other ellipse must be supplied, as "like" in our "stone-still," &c., and the exact meaning of the compound can only be ascertained by the context.
" Wind-changing Warwick."—3 Hen. VI. v. 1. 57.
" My furnace-burning heart."—Ib. ii. 1. 80.
i.e. "burning like a furnace."
" Giant-rude," A. Y. L. iv. 3. 34 ; "marble-constant," A. and C. v. 2. 240; "honey-heavy-dew," J. C. ii. 1. 230; so "flower-soft hands," A. and C. ii. 2. 215; "maid-pale peace," Rich. II. iii. 3. 98 ; "an orphan's water-standing eye," 3 Hen. VI. v. 6. 40, i.e. "standing with water;" "weeping-ripe," L. L. L. v. 2. 274, "ripe for weeping;" "thought-sick," Hamlet, iii. 4. 51, i.e. "as i.e. the result of thought ;" so "lion-sick," Tr. and Cr. ii. 3. 13, is explained lower down, "sick of proud heart;" "pity-pleading eyes," R. of L. 561, i.e. "pleading for pity ;" "peace-parted souls," Hamlet, v. i. 261, i.e. "souls that have departed in peace;" "fancy-free," M. N. D. ii. 1. 164, i.e. "free from fancy (love) ;" "child-changed father," Lear. iv. 7. 17, i.e., "changed to a child."
Or the noun is put for a passive participle or an adjective.
" Upon your sword sit laurel(led) victory."—A. and C.i. 3.100.
"The honey of his music(al) vows."—Hamlet, iii. 1. 164.
" The venom(ous) clamours of a jealous woman."
C. of E. v. 1. 69; so R. of L. 850.
"The Carthage queen."—M. N. D. i. 1. 173.
"Your Corioli walls."—Coriol. i. 1. 8 ; ii. 1. 180.
"Our Rome gates."—Ib. iii. 3. 104.
For similar examples, see 22.
Sometimes the genitive is used :
"I'll knock your knave's pate."
T. of Sh. i. 2. 12 ; C. of E. iii. 1. 74.
431. Preposition-Compounds.
"An after-dinner's (comp. 'afternoon's') breath."
Tr. and Cr. ii. 3. 120.
"At after-supper."—Rich. III. iv. 3. 31 ; M. N. D. v. i. 34.
" At over-night."—A. W. iii. 4. 23.
"The falling-from of his friends."—T. of A. iv. 3. 400.
The preposition usually attached to a certain verb is sometimes appended to the participle of the verb in order to make an adjective.
"There is no hoped-for mercy."—3 Hen. VI. v. 4. 35.
" Some never-heard-of torturing pain,"—T. A. ii. 3. 285.
for "unheard-of."
" Your sued-for tongues."—Coriol. ii. 3. 216.
" Bemock'd-at stabs."—Temp. iii. 3. 63.
" The unthought-on accident."—W. T. iv. 4. 549.
"Your unthought-of Harry."—1 Hen. IV. iii. 2. 141.
432. Verb-Compounds. Verbs were compounded with their objects more commonly than with us.
" Some carry-tale, some please-man, some slight zany,
Some mumble-news."—L. L. L. v. 2. 463-4.
" All find-faults."—Hen. V. v. 2. 398.
We still use "mar-plot" and "spoil-sport." Such compounds seem generally depreciatory. "Weather-fend" in
" In the lime grove which weather-fends your cell,"
Temp. v. 1. 10.
means "defend from the weather," and stands on a somewhat different footing.
One is disposed to treat "wilful-blame" as an anomalous compound in
" In faith, my lord, you are too wilful-blame."
I Hen. IV. iii. 1. 177.
like "A false-heart traitor."—2 Hen. VI. v. 1. 143.
But " heart" is very probably a euphonious abbreviation of "hearted." The explanation of "too wilful-blame" is to be sought in the common expression "I am too blame," Othello, iii. 3. 211, 282 ; M. of V. v. 1. 166. "I am too too blame," is also found in Elizabethan authors. It would seem that, the "to" in " I am to blame" being misunderstood, "blame" came to be regarded as an adjective, and "to" (which is often interchanged in spelling with "too") as an adverb. Hence "blame," being regarded as an adjective, was considered compoundable with another adjective.
433. Participial Nouns. A participle or adjective, when used as a noun, often receives the inflection of the possessive case or the plural.
" His chosen's merit."—B. and F. F. Sh. iii. 1.
" All cruels else subscribed."—Lear, iii. 7. 65.
i.e. "All cruel acts to the contrary being yielded up, forgiven." Compare for the meaning Lear, iv. 7. 36, and for "subscribe," Tr. and Cr. iv. 5. 105. Another explanation is, "all other cruel animals being allowed entrance."
So "Vulgars," W. T. ii. 1. 94 ; "Severals," Hen. V. i. 1. 86, i.e. "details."
"Yon equal potents."—K. J. ii. 1. 357.
"To the ports
The discontents repair."—A. and C. i. 4. 39.
" Lead me to the revolts (revolters) of England here."
K. J. v. 4. 7 : so Cymb. iv. 4. 6.
Add, if the text be correct :
" The Norways' king. "—Macbeth, i. 2. 59.
i.e. "the king of the Norwegians."
It would appear as though an adjective in agreement with a plural noun received a plural inflection in
" Letters-patents."—Hen. VIII. iii. 2. 249 ; Rich. II. ii. 1. 202 (Folio), 3. 130.
More probably the word was treated by Shakespeare as though it were a compound noun. But in E. E. adjectives of Romance origin often take the plural inflection.
" Lawless resolutes."—Hamlet, i. 1. 98.
"Mighty opposites."—Ib. v. ii. 62.
434. Phrase-Compounds. Short phrases, mostly containing participles, are often compounded into epithets.
" The always-wind-obeying deep."—C. of E. i. 1. 64.
" My too-much-changed son."—Hamlet, ii. 2. 36.
" The ne'er-yet-beaten horse of Parthia."—A. and C. iii. 1. 33.
" Our past-cure malady."—A. W. ii. 1. 124.
" A past-saving slave."—Ib. iv. 3. 158.
" The none-sparing war."—Ib. iii. 2. 108.
" A jewel in a ten-times-barred-up chest."—Rich. II i. 1.180.
" A too-long-wither'd flower."—Ib. ii. 1. 134.
" Tempt him not so too-far."—A. and C. i. 3. 11.
" The to-and-fro-conflicting wind."—Lear, iii. 1. 11.
" You that have turn'd off a first-so noble wife."
A. W. v. 3. 220.
" Of this yet-scarce-cold battle."—Cymb. v. 5. 469.
" A cunning thief, or a-that-way-accomplished courtier."
Ib. i. 4. 101.
" In this so-never-needed help."—Coriol. v. 1. 34.
" A world-without-end bargain."—L. L. L. v. 2. 799.
See .Sonn. 5.
" Our not fearing Britain."—Cymb. ii. 4. 191.
" The ne'er-lust-wearied Antony."—A. and C. ii. 1. 38.
" A twenty-years-removed thing ."—T. N. v. i. 92.
435. Anomalous Compounds, We still, though rarely, abbreviate "the other" into "t'other," but we could not say
" The t'other."—B. J. Cy's. Rev. iv. 1; v. 1 (a corruption of E. E. ).
" Yea, and furr'd moss when winter flowers are none,
To winter-ground thy corpse."—Cymb, iv. 2. 229.
i.e. perhaps "to inter during winter." So "to winter-rig" is said (Halliwell) to mean " to fallow land during winter."
"And" is omitted in
"At this odd-even and dull watch of the night."
Othello, i. 1. 124.
Cicero says, that the extreme test of a man's honesty is that you can play at odd and even with him in the dark. And perhaps "odd-(and-)even" here means, a time when there is no distinguishing between odd and even.
As there is a noun " false-play," there is nothing very remarkable in its being converted thus into a verb :
"Pack'd cards with Cæsar and false-played my glory."
A. and C. iv. 14. 19.
A terse compound is often invented for special use, made intelligible by the context. Thus, the profit of excess is called
"Poor-rich gain."—R. of L. 140.
" Where shall I live now Lucrece is unlived."—Ib. 1754.
# PREFIXES.
A-. See 24.
436. All-to (see 28) is used in the sense of " completely asunder" as a prefix in
" And all-to-brake his skull. "—Judges ix. 53.
"Asunder" was an ordinary meaning of the prefix "to" in E. E. It must be borne in mind that all had no necessary connection with to, till by constant association the two syllables were corrupted into a prefix, all-to, which was mistaken for altogether and so used. Hence, by corruption, in many passages, where all-to or all-too is said to have the meaning of "asunder," it had come to mean "altogether," as in
" Mercutio's ycy hand had al-to frozen mine.—HALLIWELL.
It has been shown (73) that too and to are constantly interchanged in Elizabethan authors. Hence the constant use of all too for " quite," "decidedly too," as in Rich. II. iv. 1. 28, "all too base," may have been encouraged by the similar sound of all-to. Shakespeare does not use the archaic all-to in the sense of "asunder," nor does Milton probably in
"She plumes her feathers and lets grow her wings,
That in the various bustle of resort
Were all too ruffled."—MILTON, Comus, 376.
437. At-in "attask'd," Lear, i. 4. 366 ("task'd," "blamed"), perhaps represents the O.E. intensive prefix "of," which is sometimes changed into "an-," "on-," or "a-." But the word is more probably a sort of imitation of the similar words " attach" and "attack."
438. Be. The prefix be is used, not merely with verbs of colouring, "smear," "splash," &c., to localize and sometimes to intensify action, but also with nouns and adjectives to convert the nouns into verbs :
" Bemonster. "—Lear, iv. 2. 63.
" Be-sort."—Ib. i. 4. 272.
"All good be-fortune you."—T. G. of V. iv. 3. 41.
"Bemadding."—Lear, iii. 1. 38.
It is also used seemingly to give a transitive signification to verbs that, without this prefix, mostly require prepositions :
"Begnaw."—Rich. III. i. 3. 221.
"Behowls the moon."—M. N. D. v. 1. 379.
"Bespeak" and "address" in Hamlet, ii. 2. 140.
"Beweep."—Rich. III. ii. 2. 49 ; Lear, i. 4. 324.
In participles, like other prefixes, it is often redundant, and seems to indicate an unconscious want of some substitute for the old participial prefix.
" Well be-met."—Lear, v. 1. 20.
But the theory that be- in "become," "believe," "belove," &c., represents the old ge-, does not seem to be sound.
439. Dis- was sometimes used in the sense of un-, to mean " without," as
" Discompanied," Cy.'s Rev. iii. 3, for "unaccompanied," i.e. "without company."
" A little to disquantity your train."—Lear, i. 4. 270.
"Dishabited," K. J. ii. 1. 220, = "Caused to migrate."
"Dislived," CHAPMAN, = "Deprived of life."
"Disnatured," Lear, i. 4. 305, for "Unnatural."
"Disnoble," HOLLAND ; "Distemperate," RALEIGH ; for " ignoble " and " intemperate."
"Being full of supper and distempering draughts."
Othello, i. 1. 99.
"Discovery" is often used for "uncovering," i.e. "unfold," whether literally or metaphorically. "So shall my anticipation prevent your discovery," Hamlet, ii. 2. 305, i.e. "render your disclosure needless by anticipation." So Rich. III. iv. 4. 240.
440. En- was frequently used, sometimes in its proper sense of enclosing, as "enclosed," "enguard," Lear, i. 4. 349; "encave," Othello, iv. 1. 82 ; "How dread an army hath enrounded him," Hen. V. iv. Prol. 36 ; "enwheel thee round," Othello, ii. 1. 87 ; "enfetter'd," ib. ii. 3. 351 ; "enmesh," ib. 368 ; "enrank," 1 Hen. VI. i. 1. 115 ; "enshelter'd and embay'd," Othello, ii. 1. 18 ; "en-steep' d," ib. 70 ; "engaol'd," Rich. II. i. 3. 166 ; "enscheduled," Hen. V. v. 2. 73 ; "enshelled," Coriol. iv. 6. 45. So "em-bound," "envassell'd," DANIEL on Florio ; "embattle" (to put in battle array); "enfree" (to place in a state of freedom); " en-tame," A. Y. L. iii. 5. 48 (to bring into a state of tameness). But the last instances show that the locative sense can be metaphorical instead of literal, and scarcely perceptible. There is little or no difference between " free" and "enfree." So " the enridged sea," Lear, iv. 6. 71 ; " the enchafed flood," Othello, ii. 1. 17, are, perhaps, preferred by Shakespeare merely because in participles he likes some kind of prefix as a substitute for the old participial prefix. In some cases the en- or in- seems to take a person as its object, "endart," R. and J. i. 3. 98 ("to set darts in," not "in darts"). So "enpierced," R, and J. i. 4. 19 ; and so, perhaps, "empoison." The word "impale" is used by Shake-peare preferably in the sense of "surrounding:"
"Impale him with your weapons round about,"
Tr. and Cr. v. 7. 5.
means "hedge him round with your weapons." So
"Did I impale him with the regal crown," "—3 Hen. VI. iii, 3. 189,
441. For- is used in two words now disused:
"Forslow no longer."—3 Hen. VI. ii. 3. 56.
" She fordid herself. "—Lear, v. 3. 255 ; M. N. D. v. 1. 381. In both words the prefix has its proper sense of "injury."
442. Un- for modern in-; in- for un-. (Non- only occurs twice in all the plays of Shakespeare, and in V. and A. 521.)
Incharitable, infortunate, incertain, ingrateful, incivil, insubstantial.
Unpossible, unperfect, unprovident, unactive, unexpressive, unproper, unrespective, unviolable, unpartial, unfallible, undividable, unconstant, uncurable, uneffectual, unmeasurable, undisposed, unvincible (N. P. 181), unreconciliable (A. and C. v. 1. 47).
We appear to have no definite rule of distinction even now, since we use ungrateful, ingratitude ; unequal, inequality. Un\- seems to have been preferred by Shakespeare before p and r, which do not allow in- to precede except in the form im-. In- also seems to have been in many cases retained from the Latin, as in the case of "ingratus," "infortunium," &c. As a general rule, we now use in\- where we desire to make the negative a part of the word, and un\- where the separation is maintained—"untrue," "infirm." Hence un- is always used with participles—" untamed," &c. Perhaps also un- is stronger than in-. "Unholy" means more than " not holy," almost "the reverse of holy." But in "inattentive," "intemperate," in\- has nearly the same meaning, "the reverse of."
" You wrong the reputation of your name
In so unseeming to confess receipt."—L. L. L. ii. 1. 156.
Here "unseeming" means "the reverse of seeming" more than "not seeming" (like o φημι): "in thus making us as though you would not confess."
# SUFFIXES.
443. -Er is sometimes appended to a noun for the purpose of signifying an agent. Thus—
" A Roman sworder. "—2 Hen. VI. iv. 1. 135.
" O most gentle pulpiter. "—A. Y. L. iii. 2. 163.
" A moraler. "—Othello, ii. 3. 301.
" Homager."—A. and C. i. 1. 31. (O. Fr. "homagier.")
"Justicers."—Lear, iv. 2. 79. (Late Lat. "justitiarius.")
In the last two instances the -er is of French origin, and in many cases, as in " enchanter," it may seem to be English, while really it represents the French -eur.
"Joinder," T. N. v. 1. 160, perhaps comes from the French "joindre."
The -er is often added to show a masculine agent where a noun and verb are identical :
"Truster."—Hamlet, i. 2. 172.
"The pauser reason."—Macbeth, ii. 3. 117.
"Causer."—Rich. III. iv. 4. 122.
"To you, my origin and ender."—L. C. ii. 22.
Note the irregular, "Precurrer" (for " precursor").—P. P.
We have "windring" from "winder," Tempest, iv. 1. 128, formed after the analogy of "wander," " clamber," "waver," the er having apparently a frequentative force.
444. -En, made of (still used in golden, &c.), is found in—
"Her threaden fillet."—L. C. 5.
"A twiggen bottle."—Othello, iii. 3. 152.
445. -Ive, -ble. (See 3.) -Ive is sometimes used in a passive instead of, as now, in an active signification. Thus : "Incomprehens ive depths;" "plausive," "worthy to be applauded;" "directive," " capable of being directed;" "insuppressive metal;" "the fair, the inexpressive she" (similarly used by Milton in the Hymn on the Nativity). On the other hand, -ble is sometimes used actively, as in "medicinable" (which is also used passively), and in "unmerita ble."
"This is a slight unmeritable man. "—J. C. iv. 1. 12.
So "defensible," "deceivable," "disputable," and "tenable."
In "Intenible sieve," A. W. i. 2. 208, not only does -ble convey an active meaning, but Shakespeare uses the Latin instead of the English form of the termination, just as we still write "terrible," not "terrable." I imagine we have been influenced in our -able by the accidental coincidence of meaning between the word "able" and the termination -ble. But French influence must have had some weight.
446. -Less. Sometimes found with adjectives, as "busyless," " siekless," " modestless."
-Less used for "not able to be."
" That phraseless hand."—L. C. 225 ; i.e. " in-describable."
"That termless skin."—Ib. 94.
"Sumless treasuries."—Hen. V. i. 2. 165.
"My careless crime."—R. of L. 771.
"Your great opposeless wills."—Lear, iv. 6. 38.
It is commonly used with words of Latin or Greek origin, as above. Add "reasonless," Hen. V. v. 4. 137 ; "crimeless," 2 Hen. VI. ii. 4. 63.
447. -Ly found with a noun, and yet not appearing to convey an adjectival meaning. "Anger-ly," Macb. iii. 5. 1 ; T. G. of V. i. 2. 62. Compare "wonder-ly'" in the Morte d' Arthur, and "cheer-ly," Tempest, i. 1. 6. This is common in E. E.
The -ly represents " like," of which it is a corruption. Compare :
" Villain-like he lies."—Lear, v. 3. 97.
So "masterly," adv., W. T. v. 3. 65 ; Othello, i. 1. 26 ; "hungerly," adv., ib. iii. 4. 105 ; "exteriorly," adv., K. J. iv. 2. 257 ; "silverly," adv., ib. v.2. 46. "Fellowly," Temp. v. 1. 64, and "traitorly," W. T. iv. 4. 822, are used as adjectives. Perhaps a vowel is to be supplied in sound, though omitted, in "unwield(i)ly," Rich. II. iv. 1. 205; "need(i)ly," R. and J. iii. 2. 117 ; and they may be derived from "unwieldy" and "needy." Add "orderly," Rich. II. i. 3. 9; "manly," Macbeth, iv. 3. 235.
448. -Ment. We seldom use this suffix except where we find it already existing in Latin and French words adopted by us. Shakespeare, however, has "intendment," "supplyment," "designment," "denotement," and "bodement."
449. -Ness is added to a word not of Teutonic origin :
"Equalness."—A. and C. v. 1. 47.
450. -Y is found appended to a noun to form an adjective.
"Slumbery agitation. "—Macbeth, v. 1. 12.
" Unheedy haste."—M. N. D. i. 1. 237.
In "Batty wings," M. N. D. iii. 2. 365, "batty" seems to mean "like those of bats." "Wormy beds," ib. iii. 2. 384, is "worm-filled." "Vasty," in "the vasty fields of France," Hen. V. Prologue, 12 ; 1 Hen. IV. iii. 1. 52, is perhaps derived from the noun "vast," Tempest, i. 2. 327 ; Hamlet, i. 2. 198. " Womby vaultages," Henry V. ii. 4. 124: i.e. "womb-like."
Y appended to adjectives of colour has a modifying force like -ish :
"Their paly flames."—Hen. V. iv. Prol. 8.
"His browny locks."—L. C. 85.
451. Suffixes were sometimes influenced by the Elizabethan licence of converting one part of speech into another. We should append -ation or -ition, -ure or -ing, to the following words used by Shakespeare as nouns: "solicit," "consult," "expect," &c. ; "my depart," 2 Hen. V1. i. 1. 2 ; 3 Hen. VI. iv. 1. 92, ii. 1. 110 ; " uncurable discomfort," 2 Hen. VI. v. 2. 86 ; "make prepare for war," 3 Hen. VI. iv. 1. 131 ; "a smooth dispose," Othello, i. 3. 403 ; "his repair," 3 Hen. VI. v. 1. 20 ; "deep exclaims," Rich. III. i. 2. 52, iv. 4. 135 ; "his brow's repine," V. and A. 490 ; " a sweet retire," Hen. V. iv. 3. 86 ; "false accuse," 2 Hen. VI. iii. 1. 160 ; "your ladyship's impose," T. G. of V. iv. 3. 8 ; " the sun's appear," B. and F. F. Sh. v. 1 ; "from suspect," 2 Hen. VI. iii. 2. 139 ; " manage," M. of V. iii. 4. 25 ; " commends," ib. ii. 1. 90 ; "the boar's annoy," Rich. III. v. 3. 156 ; "the disclose," Hamlet, iii. 1. 174 "commends," Rich. II. iii. 3. 126.
Almost all of these words come to us through the French.
Note " O heavenly mingle."—A. and C. i. 5. 59.
"Immoment toys."—Ib. v. 1. 106.
# PROSODY.
452. The ordinary line in blank verse consists of five feet of two syllables each, the second syllable in each foot being accented.
"We both have féd as wéll, and wé can both
Endúre the wínt er's cóld as well as hé."
J. C. i. 2. 98–9.
This line is too monotonous and formal for frequent use. The metre is therefore varied, sometimes (1) by changing the position of the accent, sometimes (2) by introducing trisyllabic and monosyllabic feet. These licences are, however, subject to certain laws. It would be a mistake to suppose that Shakespeare in his tragic metre introduces the trisyllabic or monosyllabic foot at random. Some sounds and collections of sounds are peculiarly adapted for monosyllabic and trisyllabic feet. It is part of the purpose of the following paragraphs to indicate the laws which regulate these licences. In many cases it is impossible to tell whether in a trisyllabic foot an unemphatic syllable is merely slurred or wholly suppressed, as for instance the first e in "different." Such a foot may be called either dissyllabic or quasi-trisyllabic.
453. The accent after a pause is frequently on the first syllable. The pause is generally at the end of the line, and hence it is on the first foot of the following line that this, which may be called the "pause-accent," is mostly found. The first syllable of initial lines also can, of course, be thus accented. It will be seen that in the middle of the line these pause-accents generally follow emphasized monosyllables. (See 480–6.)
"Cómfort, my liége! why loóks your gráce so pále?"
Rich. II. iii. 2. 75.
Examples of the "pause-accent" not at the beginning.
* (1) "Feéd and regárd him nót. Aré you a man ?"
Macbeth, iii. 4. 58.
Sometimes the pause is slight, little more than the time necessary for recovery after an emphatic monosyllable.
* (2) "Be ín their flów ing cups freshly remémber'd."
Hen. V. iv. 3. 55.
So arrange
" And thése flátter ing stréams, and make our fáces. "
Macbeth, iii. 2. 33.
" These " may be emphasized. (See 484.)
* (3) "Whó would I beliéve me. O′ ! péril ous móuths."
M. for M. ii. 4. 172.
* (4) " Afféc | tion, poóh! | You spéak —líke a green girl."
Hamlet, i. 3. 101.
" Wé shall be cáll'd —púrgers, not múr derérs."
J. C. ii. 1. 180.
* (5) "The lífe of cóm fort. Bút for thée, féllow."
Cymb. iv. 3. 9.
The old pronunciation " fellów " is probably not Shakespearian.
In (3) (4) and (5) "O," "speak," "call'd," and "thee" may, perhaps, be regarded as dissyllables (see 482–4), and the following foot a quasi-trisyllabic one. There is little practical difference between the two methods of scansion.
* (6) " Sénseless | línen ! Háppier | thereín than I."
Cymb. i. 3. 7.
Here either there is a pause between the epithet and noun, or else "senseless" may possibly be pronounced as a trisyllable, " Sénse (486) less linen." The line is difficult.
" Therefore, mérchant, I'll lím | it thee this dáy,"
C. of E. i. 1. 151.
seems to begin with two trochees, like Milton's famous line :
" U'ni vérsal reproách far wórse to béar."—P. L. vi. 34.
But " therefore" may have its accent, as marked, on the last syllable.
The old pronunciation " merchant is not probable. Or " there " may be one foot (see 480) : "Thére fore mérchant ."
* (7) "Ant. Obéy it ón all cáuse. Cleop.
Párdon, párdon.
A. and C. iii. 11. 68.
is, perhaps, an instance of two consecutive trochees. (There seems no ground for supposing that "pardon" is to be pronounced as in French.) But if the diphthong "cause" be pronounced as a dissyllable (see 484), the difficulty will be avoided.
We find, however, a double trochee (unless "my" has dropped out) in
"Sec. Cit. Cæ'sar has hád great wróng.
Third Cit.
Hás he, másters?"
J. C. iii. 2. 115.
Even here, however, " wrong " may be a quasi-dissyllable (486).
(8) Between noun and participle a pause seems natural. Often the pause represents "in" or " a-" (178).
"Thy knée bússing | the stónes."—Coriol. iii. 2. 75.
"The smíle mócking the sígh. "—Cymb. iv. 2. 54.
"My wínd cóoling my bróth."—M. of V. i. 1. 22.
In these lines-the foot following the emphasized monosyllable may (as an alternative to the " pause-accent ") be regarded as quasi-trisyllabic.
453 a. Emphatic Accents. The syllable that receives an accent is by no means necessarily emphatic. It must be emphatic relatively to the unaccented syllable or syllables in the same foot, but it may be much less emphatic than other accented syllables in the same verse. Thus the last syllable of " injuries," though accented, is unemphatic in
"The ín juríes that théy themsélves procúre." Lear, ii. 4. 303.
Mr. Ellis (Early English Pronunciation, part i. p. 334) says that " it is a mistake to suppose that there are commonly or regularly five stresses, one to each measure." From an analysis of several tragic lines of Shakespeare, taken from different plays, I should say that rather less than one of three has the full number of five emphatic accents. About two out of three have four, and one out of fifteen has three. But as different readers will emphasize differently, not much importance can be attached to such results. It is of more importance to remember, (1) that the first foot almost always has an emphatic accent ; (2) that two unemphatic accents rarely, if ever, come together ("for" may perhaps be emphatic in
" Heár it not, Dún can ; fór it is a knéll."
Macbeth, ii. 1. 63) ;
and (3) that there is generally an emphatic accent on the third or fourth foot.
The five emphatic accents are common in verses that have a pause-accent at the beginning or in the middle of the line.
" Náture seems déad, and wíck ed dréams abúse."
Macbeth, ii. 1. 50.
" The hánd le tóward my hánd. Cóme, let me clútch thee."—Ib. ii. 1. 34.
And in antithetical lines :
" I háve thee nót, and yét I sée thee stíll."
Macbeth, ii. 1. 35.
" Bríng with thee aírs from héaven or blásts from héll."
Hamlet, i. 4. 41.
454. An extra syllable is frequently added before a pause, especially at the end of a line :
* (a) "'Tis nót alóne my ínk y clóak, good móther."
Hamlet, i. 2. 77.
but also at the end of the second foot :
* (b) "For míne own sáfeties ; you máy be ríght ly júst."
Macbeth, iv. 3. 30.
and, less frequently, at the end of the third foot :
* (c) " For góod ness dáres not chéck thee; wear thoú thy wróngs."—Macbeth, iv. 3. 33.
and, rarely, at the end of the fourth foot :
* (d) "With áll my hón ours ón my bróther: whereón."
But see 466.
Temp. i. 2. 127.
" So déar the lóve my peó ple bóre me: nor sét."
Ib. i.2.141.
455. The extra syllable is very rarely a monosyllable, still more rarely an emphatic monosyllable. The reason is obvious. Since in English we have no enclitics, the least emphatic monosyllables will generally be prepositions and conjunctions. These carry the attention forward instead of backward, and are therefore inconsistent with a pause, and besides to some extent emphatic.
The fact that in Henry VIII., and in no other play of Shakespeare's, constant exceptions are found to this rule, seems to me a sufficient proof that Shakespeare did not write that play.
" Go gíve 'em wél come ; yóu can spéak the Frénch tongue."—Hen. VIII. i. 4. 57.
"Féll by our sérv ants, by those mén we lóv'd most."
Ib. ii. 1. 122.
" Be súre you bé not lóose; for thóse you máke friends."—Hen. VIII. ii. 1. 127.
"To sí lence én vious tóngues. Be júst and fear not.
Ib. iii. 2. 447.
So Hen. VIII. ii. 1. 67, 78, 97 ; and seven times in iii. 2. 442–451 ; eight times in iv. 2. 51–80.
Even where the extra syllable is not a monosyllable it occurs so regularly, and in verses of such a measured cadence, as almost to give the effect of a trochaic line with an extra syllable at the beginning, thus :
" In áll my míser íes ; but thóu hat |fórced me
Out óf (457 a) thy hónest trúth to pláy the wóman.
Let's drý our éyes : and thús far héar me, Crómwell :
And whén 1 | ám for- gótten, ás I sháll be,
And sleep in dúll could | márble whére no méntion
Of mé must móre be héard of, sáy I táught thee.
Say, Wólsey, that once tród the wáys of glóry
And sóunded áll the dépths and shóals of hónour,
Found thée a way, out óf (457 a) his wréck, to | ríse in
A sure and sáfe one, thóugh thy máster míssed it."
Hen. VIII. iii. 2. 430–9.
It may be safely said that this is not Shakespearian.
" Boy" is unaccented and almost redundant in
" I párt ly knów the mán : go cáll | him híther, boy."
(Folio) Rich. III. iv. 2. 41.
(Hither, a monosyllable, see 189.) And even here the Globe is, perhaps, right in taking " Boy exit" to be a stage direction.
In " Bíd him make háste and meét me át the North gate,"—T. G. of V. iii. 1. 258.
"gate" is an unemphatic syllable in "Nórthgate," like our "Néw-gate." So
" My mén should cáll me lord : I ám yoúr good-man."
T. of Sh. Ind. 2. 107.
"A hált er grát is : nó thing élse, for Gód's-sake."
M. of V. iv. 1. 379.
"parts," like " sides," is unemphatic, and "both is strongly emphasized, in
" Ráther to shów a nób | le gráce to bóth parts."
Coriol. v. 3. 121.
So " out " is emphatic in
" We'll háve a swásh ing ánd a márt | ial oútside."
A. Y. L. i. 3. 122.
The's for "is" is found at the end of a line in
" Perceive I speak sincerely, and high note 's
Ta'en of your many virtues."—Hen. VIII. ii. 3. 59.
456. Unaccented Monosyllables. Provided there be only one accented syllable, there may be more than two syllables in any foot. "It is he" is as much a foot as "'tis he;" "we will serve " as " we'll serve ;" " it is over " as "'tis o'er."
Naturally it is among pronouns and the auxiliary verbs that we must look for unemphatic syllables in the Shakespearian verse. Sometimes the unemphatic nature of the syllable is indicated by a contraction in the spelling. (See 460.) Often, however, syllables must be dropped or slurred in sound, although they are expressed to the sight. Thus in
"Provide thee | two prop er pal freys, black as jet,"
T. A. v. 2. 50.
" thee" is nearly redundant, and therefore unemphatic.
" If" and " the " are scarcely pronounced in
"And in it are the lórds of Yórk, Bérkeley, and Séymour." —Rich. II. ii. 3. 65.
" Mir. I év er sáw so nóble. Prosp. It goes ón, | I sée."—Temp, i. 2. 419.
"Bút that the séa, moúnting to the wél kin's chéek."
Ib. i. 2. 4.
(" The" need not be part of a quadrisyllabic foot, nor be suppressed in pronouncing
" The cur iósi ty of ná tions tó depríve me."
Lear, i. 2. 4.
Compare, possibly,
" But I have ever had that cúriós(i)ty."—B. and F. (Nares). )
So " to," the sign of the infinitive, is almost always unemphatic, and is therefore slurred, especially where it precedes a vowel. Thus :
" In séeming to augmént it wástes it. Bé advís'd."
Hen. VIII. i. 1. 145.
where " in" before the participle is redundant and unemphatic.
"For trúth to (t') over(o'er)péer. Rather than fóol | it só."
Coriol. ii. 3. 128.
So the "I" before "beseech" (which is often omitted, as Temp. ii. 1. 1), even when inserted, is often redundant as far as sound goes.
"(I) beseéch your mares ty, give me léave to gó."
2 Hen. VI. ii. 3. 20.
" (I) beséech your grác es bóth to pár don mé."
Rich. III. i. 1. 84. So Ib. 103.
Perhaps
" (I) pray thee (prithee) stáy with ús, go nót to Wítt enbérg,"
Hamlet, i. 2. 119.
though this verse may be better scanned
" I práy these stáy with us, go nót to Wíttenberg." See 469.
" Let me sée, let me sée; ís not the léaf turn'd dówn ? "
J. C. iv. 3. 273.
So (if not 501)
"And I' will kíss thy fóot : (I) prithee bé my gód."
Temp. ii. 2. 152.
"With you" is "wi' you" (as in "good-bye" for "God be with you") ; "the" is thy', and "of" is slurred in
"Two nó ble párt ners wíth you ; the old dúch ess of Nórfolk."—Hen. VIII. v. 3. 168.
To write these lines in prose, as in the Folio and Globe, makes an extraordinary and inexplicable break in a scene which is wholly verse.
For the quasi-suppression of of see
"The bás tard of O'r leáns with hím is join'd,
The dúke of Alén çon flí eth tó his síde."
1 Hen. VI. i. 1. 92, 93.
In the Tempest this use of unaccented monosyllables in trisyllabic feet is very common.
"Go make thysélf like a nýmph o' the séa; be súbject
To no síght , but thíne and míne."—Temp. i. 2. 301.
Even in the more regular lines of the Sonnets these superfluous syllables are allowed in the foot. Thus :
" Excúse not sí lence só; for't lies in thée. "—Sonn. 101.
And even in rhyming lines of the plays :
"Cáll them agaín, sweet prínce, accépt their suít ;
I'f you dený them, áll the lánd will rúe 't."
Rich. III. iii. 7. 221.
This sometimes modifies the scansion. " Hour " is a dissyllable, and 't is absorbed, in
"You knów I gáve 't you hálf an hoú r sínce."
C. of E. iv. 1. 65.
Almost any syllables, however lengthy in pronunciation, can be used as the unaccented syllables in a trisyllabic foot, provided they are unemphatic. It is not usual, however, to find two such unaccented syllables as
" Which most gíb inglý, ungráve ly hé did fáshion."
Coriol. ii. 3. 233.
457. Accented monosyllables. On the other hand, sometimes an unemphatic monosyllable is allowed to stand in an emphatic place, and to receive an accent. This is particularly the case with conjunctions and prepositions at the end of the line. We still in conversation emphasize the conjunctions "but," "and," "for," &c. before a pause, and the end of the line (which rarely allows a final monosyllable to be light, unless it be an extra-syllable) necessitates some kind of pause. Hence
" This my mean task
Would be as heavy to me as odious, but
The mistress which I serve quickens what's dead."
Temp. iii. 1. 5.
" Or ere
It should the good ship so have swallow'd and
The fraughting souls within her."—Ib. i. 2. 12.
" Freed and enfranchised, not a party to
The anger of the king, nor guilty of
(If any be) the trespass of the queen."—W. T. ii. 2. 62, 63.
So Temp. iii. 2. 33, iv. 1. 149 ; W. T. i. 2. 372, 420, 425, 432, 449, 461, &c.
The seems to have been regarded as capable of more emphasis than with us :
" Whose shadow the dismissed bachelor loves."—Temp. iv. 1. 67.
"With silken streamers the young Phœbus fanning."
Hen. V. iii. Prol. 6.
" And your great uncle's, Edward the Black Prince."
Ib. i. 1. 105, 112.
"And Prosp'ro (469) the prime duke, being (470) so reputed. "—Temp. i. 2. 72.
" Your breath first kindled the dead coal of war."—K. J. v. 2. 83,
" Omitting the sweet benefit of time."—T. G. of V. ii. 4- 65.
" So doth the woodbine, the sweet honeysuckle."
M. N. D. iv. 1. 47.
" Then, my queen, in silence sad,
Trip we after the night's shade."—Ib. iv. 1. 101.
" His brother's death at Bristol the Lord Scroop."
1 Hen. IV. i. 3. 271.
" So please you something touching the Lord Hamlet."
Hamlet, i. 3. 89.
" Thou hast affected the fine strains of honour."
Coriol. v. 3. 149, 151.
In most of these cases the precedes a monosyllable which may be lengthened, thus :
"Your breath first kíndled the déa d (484) cóal | of wár."
So Temp. i. 2. 196, 204 ; ii. 2. 164 ; iv. 1. 153.
Compare
" Oh, weep for Adonais. The quick dreams."
SHELLEY, Adonais, 82.
But this explanation does not avail for the first example, nor for
" That heals the wound and cures not the disgrace.—Sonn. 34.
" More needs she the divine than the physician."—Macb. v. 1. 82. (Unless, as in Rich. II. i. 1. 154, "physician" has two accents :
"More néeds she the divíne thán the physí cián.")
On the whole there seems no doubt that "the" is sometimes allowed to have an accent, though not (457 a) an emphatic accent.
Scan thus :
"A dévil (466), a bór n (485) dév | il (475), ón whose náture."—Tempest, iv. 1. 188.
avoiding the accent on a.
The in
"Then méet and joín. Jove's líght nings, thé precúrsors,"
Tempest, i. 2. 201.
seems to require the accent. But "light(e)nings" is a trisyllable before a pause in Lear, iv. 7. 35 (see 477), and perhaps even the slight pause here may justify us in scanning—
" Jove's líght (e)níngs, the precúrsors."
457 a. Accented Monosyllabic Prepositions. Walker (Crit. on Shakespeare, ii. 173–5) proves conclusively that "of" in " out of" frequently has the accent. Thus:
" The fount out of which with their holy hands. "—B. and F.
" Into a relapse ; or but suppose out of."—MASSINGER.
" Still walking like a ragged colt,
And oft out of a bush doth bolt."—DRAYTON.
Many other passages quoted by Walker are doubtful, but he brings forward a statement of Daniel, who, remarking that a trochee is inadmissible at the beginning of an iambic verse of four feet, instances :
" Yearly out of his wat'ry cell,"
which shows that he regarded "out óf" as an iambus. Walker conjectures " that the pronunciation (of monosyllabic prepositions) was in James the First's time beginning to fluctuate, and that Massinger was a partisan of the old mode." Hence, probably, the prepositions received the accent in
" Such mén | as hé | be né | ver át | heart's éase."
J. C. i. 1. 208.
"Therefóre (490), | out óf | thy lóng | exper | ienc'd time."
R. and J. iv. 1. 60 ; Coriol. i. 10. 19.
" Vaunt cóur | iers tó | oak-cléav | ing thún | der-bólts."
Lear, iii. 2. 5.
So Hen. VIII. iii. 2. 431, 438.
" To bríng | but fíve | and twén | ty ; tó | no móre."
Lear, ii. 4. 251.
" Lor. Who únd | ertákes | you tó | your end. | Vaux.
Prepare there."—Hen. VIII. ii. 2. 97.
For this reason I think it probable that " to " in " in-to," "un-to," sometimes receives the accent, thus :
" That év | er lóve | did máke | thee rún | intó."
A. Y. L. ii. 4. 35.
" Came thén | intó | my mínd, | and yét | my mind."
Lear, iv. 1. 36.
"Fán you intó despair. Have the pów er stíll."
Coriol. iii. 3. 127.
"I had thóught, by mák ing thís well knówn untó you."
Lear, i. 4. 224 ; M. of V. v. 1. 169.
" By this | vile cón | quest sháll | attaín | untó."
J. C. v. 5. 38 ; Rich. III. iii. 5. 109.
"Discúss | untó | me. A'rt | thou óff | icér?"
Hen. V. iv. 1. 38. (But this is Pistol.)
With in "without" seems accented in
" That wón | you wíth | out blóws."—Coriol. iii. 3. 133.
458. Two extra syllables are sometimes allowed, if unemphatic, before a pause, especially at the end of the line. For the details connected with this licence see 467—9, and 494, where it will be seen that verses with six accents are very rare in Shakespeare, and that therefore the following lines are to be scanned with five accents.
" Perúse | this létter. | Nóthing | almóst | sees míracles."
Lear, ii. 2. 172.
" Must be a fáith | that réa | son wíth | out míracle."
Ib. i. 1. 225.
" Like óne | that méans | his pró | per hárm | in mánacles."
Coriol. i. 9. 57.
"Was dúke dom large enóugh : of témp(o) ral róyalties."—Tempest, i. 2. 110.
" I dáre | avóuch | it, sir. | What, fíf | ty fóllowers!"
Lear, ii. 4. 240.
" You fóol | ish shép | herd, whére | fore dó | you fóllow her?"—A. Y. L. iii. 5. 49.
" Of whóm | he's chief, | with áll |the síze | that vérity."
Coriol. v. 2. 18.
"Ely. Incline | to it, | or nó. |
Cant. He séems | indífferent."—Hen. V. i. 1. 72.
" As if | I lóv'd | my lítt | le shóuld | be díeted."
Coriol. i. 9. 52.
"Why, só | didst thóu. | Come théy | of nó | ble fámily ?"
Hen. V. ii. 2. 129.
"That né | ver máy | ill óff | ice ór | fell jéalousy"
Ib. v. 2. 491.
" That hé | suspécts | none ; ón | whose fóol | ish hónesty."
Lear, i. 2. 197.
"Withín | my tént | his bónes | to-níght | shall líe
Most líke | a sóld | ier, órd | er'd hón | (ou)rablý."
J. C. v. 5. 79.
Compare
"Young mán, | thoucóuld'st | notdíe | morehén | (ou)rable."
Ib. v. 1. 60.
If "ily" were fully pronounced in both cases, the repetition would be intolerable in the following :—
"Cor. But whát | is líke | me fór | merlý. |
Men.
That's wórthily."—Coriol. iv. 1. 53.
"The rég | ion óf | my héart : | be Ként | unmánnerly."
Lear, i. 1. 147.
"Lóok, where | he cómes ! | Not póp | py nór | mandrágora." —Othello, iii. 3. 330.
" A's you | are óld | and réverend, | you shóuld | be wíse."
Lear, i. 4. 261.
"To cáll | for récompense : | appeár | it tó | your mind."
Tr. and Cr. iii. 3. 8.
"Is nót | so ést | imable, próf | itáb | le neither."
M. of V. i. 3. 167.
"Agé is | un-néc | essary : ón | my knées | I bég."
Lear, ii. 4. 157.
"Our múst | y sú | perflúity. | Sée our | best élders."
Coriol. i. 1. 230.
459. The spelling (which in Elizabethan writers was more influenced by the pronunciation, and less by the original form and derivation of the word, than is now the case) frequently indicates that many syllables which we now pronounce were then omitted in pronunciation.
460. Prefixes are dropped in the following words :—
'bolden'd for "embolden'd."—Hen. VIII. i. 2. 55.
'bove for "above."—Macbeth, iii. 5. 31.
'bout for "about."—Temp. i. 2. 220.
'braid for "upbraid."—P. of T. i. 1. 93.
'call for "recall."—B. and F.
'came for "became."—Sonn. 139.
'cause for "because."—Macbeth, iii. 6. 21.
'cerns for "concerns."
"What 'cerns it you."—T. of Sh. v. 1. 77.
'cide for "decide."—Sonn. 4.6.
'cital for "recital."
"He made a blushing'cital of himself."—1 Hen.IV. v.2.62.
'collect for "recollect."—B. J. Alch. i. 1.
'come for "become."
" Will you not dance ?
How 'come you thus estranged ?"—L. L. L. v. 2. 213.
'coraging for "encouraging."—ASCH. 17.
'count for "account."
"Why to a public 'count I might not go."
Hamlet, iv. 7. 17.
'dear'd for "endear'd."—A. and C. i. 4. 44.
'fall for "befall."—Ib. iii. 7. 40. So in O. E.
'friend for "befriend."—Hen. V. iv. 5. 17.
'gain-giving for "against-giving," like our "misgiving."—Hamlet, v. 2. 226.
'gave for "misgave."—Coriol. iv. 5. 157 (perhaps).
So "My minde 'gives me that all is not well" (Nares). But the dropping of this essential prefix seems doubtful. "Gave" would make sense, though not such good sense. In
"Then sáy | if théy | be trúe. | This (mis-)shá | pen knáve,"
Temp. v. 1. 268.
Walker with great probability conjectures "mís-shap'd." In
"Told thee no lies, made thee no mistakings, serv'd,"
Temp. i. 2. 248.
it is more probable that the second "thee," not mis-, is slurred.
'get for "beget."—Othello, i. 3. 191.
'gree for "agree."—M. of V. ii. 2. 108; T. G. of V. ii. 4. 183 ; A. and C. ii. 6. 38.
'haviour for "behaviour."—Hamlet, i. 2. 81.
'joy for "enjoy."—2 Hen. VI. iii. 2. 365.
'larum for "alarm."
" Then shall we hear their'larum and they ours."
Coriol. i.4. 9.
Folio, "their Larum."
'las for "alas."—Othello, v. 1. 111.
'lated for "belated."—A. and C. iii. II. 3.
'less for "unless."—B. J. Sad Sh. iii. I.
'longs for "belongs."—Per. ii. Gow. 40.
'longing for "belonging."—Hen. VIII. i. 2. 32 ; W. T. iii. 2. 104 ; Hen. V. ii. 4. 80.
'miss for "amiss."—V. and A.
'mong (pronounced) for "among."
"Be bríght and jóv ial amóng your gúests to-níght."
Macbeth, iii. 2. 28.
"Cel. That líved | amongst mén. |
Oliv.
And wéll | he míght | do só."
A. Y. L. iv. 3. 124.
'nighted for "benighted."—Lear, iv. 5. 13.
'nointed for "anointed."—W. T. iv. 4. 813.
'noyance for "annoyance."—Hamlet, iii. 3. 13.
'pairs for "impairs."—B. E. 91. So in O. E.
'pale* for "impale," "surround."
" And will you 'pale your head in Henry's glory,
And rob his temples of the diadem."—3 Hen. VI. i. 4. 103.
'parel for "apparel."—Lear, iv. 1. 51.
'plain for " complain." (Fr. plaindre.)
"The king hath cause to plain."
Lear, iii. 1. 39 ; Rich. II. i. 3. 175.
'rag'd for "enraged."—Rich. II. ii. 1. 70.
'ray for "array."—B. J. Sad Sh. ii. " Battel ray."
N. P. 180. O. E.
'rested for "arrested."—C. of E. iv. 2. 42. Dromio uses whichever form suits the metre best.
" I knów | not át | whose súit | he ís | arrés | ted wéll ;
But hé's | in a suit | of búff | which résted | him, that can |
I téll."—C. of E. iv. 2. 43.
So should be read
"King. Or yield up Aquitaine.
Princess.
We (ar)rest your word."
L. L. L. ii. 1. 160.
It has been objected that 'rested is a vulgarism only fit for a Dromio. But this is not the case. It is used by the master Antipholus E. (C. of E. iv. 4. 3).
'say'd for "assay'd."—Per. i. 1. 59. Comp. B. J. Cy.'s Rev. iv. 1.
'scape for " escape " freq.
'scuse for "excuse."—Othello, iv. 1. 80 ; M. of V. iv. 1. 444.
'stall'd apparently for "forestalled."—B. J. Sejan. iii. 1; for "install'd."—Rich. III. i. 3. 206.
'stonish'd for " astonish'd."
"Or'stonish'd as night-wanderers often are."—V.and A.825.
'stroy'd for "destroy'd."
"'Stroy'd in dishonour."—A. and C. iii. 11. 54.
'tend for "attend."—Hamlet, iv. 3. 47.
'turn for "return;" 'lotted for "allotted."
unsisting for "unresisting" (explained in the Globe Glossary as "unresting").
* "Did I impale him with the regal crown?"—3 Hen. VI. iii. 3. 189.
" That wounds the unsisting postern with these blows."
M. for M. iv. 2. 92.
This explains how we must scan M. for M. iv. 2. 92.
"Prevént | it, resíst ('sist) | it, lét | it nót | be só."
Rich. III. iv. 1. 148.
"A sóoth | sayer bíds | you bewáre ('ware) | the ídes |of Márch."—J. C. i. 2. 19.
"Envíron'd ('viron'd) | me abóut | and hów | led ín | mine éars."—Rich. III. i. 4. 59.
"At án | y tíme | have recóurse ('course) | untó | the prínces."—Ib. iii. 5. 109.
"Lest I' | revenge ('venge)—whát? | Mysélf | upón |mysélf?" —Ib. v. 3. 185.
The apostrophe, which has been inserted above in all cases, is only occasionally, and perhaps somewhat at random, inserted in the Folio. It is therefore not always possible to tell when a verb is shortened, as "comes" for "becomes," or when a verb may, perhaps, be invented. For instance, "dear'd" may be a verbal form of the adjective "dear," or a contraction of the verb "endear'd."
"Comes (becomes) dear'd (endear'd) by being lack'd."
A. and C. i. 4. 44.
Sometimes, perhaps, the prefix, though written, ought scarcely to be pronounced:
" How fáres | the kíng | and's fóllow | ers ? (Con) | fíned | togéther."—Temp. v. 1. 7.
"O (de)spiteful love ! unconstant womankind,"
T. of Sh. iv. 2. 14.
unless the "O" stands by itself. (See 512.)
"(Be)lónging | to a mán. | O bé | some óth | er mán."
R. and J. ii. 2. 42.
461. Other Contractions are :
Barthol'mew (T. of Sh. Ind. i. 105); Ha'rford for "Haverford" (Rich. III. iv. 5. 7) ; dis'ple for " disciple" (B. J. Fox, iv. 1 ; so SPENSER, F. Q. i. 10. 27); ignomy for "ignominy" (M. for M. ii. 4. 111, 1 Hen. IV. v. 4. 100 [Fol.] ; genman (UDALL) ; gentl'man (Ham. [1603] i. 5) ; gent (SPENSER) freq. for "gentle" (so in O. E.); easly (CHAPMAN, Odyss.) for "easily;" par'lous for "perilous" (Rich. III. ii. 4. 35) ; inter'gatories for "interrogatories" (M. of V. v. 1. 298) ; canstick for "candlestick,"—
" I had rather hear a brazen canstick turned." 1 Hen. IV. iii. 1. 131.
Marle (B. J. E. out &c. v. 4) for " marvel ;" whe'er for " whether " (0. E.) ; and the familiar contraction good-bye, "God be with you," which enables us to scan Macbeth, iii. 1. 44. We also find in's for " in his ;" th'wert for "thou wert ;" you're for "you were ; " h'were for "he were." So "she were" is contracted in pronunciation :
"'Twere goód | she were spó | ken wíth : | for shé | may stréw."—Hamlet, iv. 5. 14.
Y'are for "you are ;" this' for " this is :"
" O this' the poison of deep grief; it springs
All from her father's death."—Hamlet, iv. 5. 76.
" This' a | good blóck."—Lear, iv. 6. 187.
So we ought to scan
" Lear. This is a | dull síght. | Aré you | not Ként ? |
Kent.
The same."—Lear, v. 3. 282.
" Sir, this is | the gent | lemán I I tóld | you óf."
T. of Sh. iv. 4. 20.
" Sir, this is | the hoúse. | Please it | you thát I cáll?"
Ib. 1.
This, for " this is," is also found in M. for M. v. 1. 131 (Fol. this 'a) ; Temp. iv. 1. 143 ; T. of Sh. i. 2. 45. Many other passages, such as T. G. of V. v. 4. 93, M. for M. iv. 2. 103, T. of Sh. iii. 2. 1, require is to be dropped in reading. This contraction in reading is common in other Elizabethan authors ; it is at all events as early as Chaucer, Knighte's Tale, 233.
Shall is abbreviated into 'se and 's in Lear, iv. 6. 246 ; R. and J. i. 3. 9. In the first of these cases it is a provincialism, in the second a colloquialism. A similar abbreviation "I'st," for "I will," "thou'st" for "thou wilt," "thou shalt," &c., seems to have been common in the early Lincolnshire dialect (Gill, quoted by Mr. Ellis). Even where not abbreviated visibly, it seems to have been sometimes audibly, as,
" If thát | be trúe I I shall sée | my bóy | agáin."
K. J. iii. 4. 78.
" I shall give | worse páy | ment."—T. N. iv. 1. 21.
" He is, | Sir Jóhn : | I féar | we shall stáy | too lóng."
1 Hen. IV. iv. 2. 83.
With seems often to have been pronounced wi', and hence combined with other words. We have "w'us," (B. and F. Elder Brother, v. 1) for "with us," and "take me w'ye" (ib.) for " with ye."
Beside the well-known "doff" "do-off," and "don" "do-on," we also find "dout" for "do-out" (hamlet, iv. 7. 192) ; " probal " for "probable" (Othello, ii. 3. 344).
# WORDS CONTRACTED IN PRONUNCIATION.
462. Sometimes the spelling does not indicate the contracted pronunciation. For instance, we spell nation as though it had three syllables, but pronounce it as though it had two. In such cases it is impossible to determine whether two syllables coalesce or are rapidly pronounced together. But the metre indicates that one of these two processes takes place.
Syllables ending in vowels are also frequently elided before vowels in reading, though not in writing. Thus :
" Prosp. Agaínst | what shoúld | ensúe. |
Mir.
How cáme | we ashóre ? "
Temp. i. 2. 158.
" You gíve | your wife | too unkínd | a cáuse | of grief."
M. of V. v. 1. 175.
"No (i)mpéd | imént | betwéen, |bút that | you múst."
Coriol, ii. 3. 236.
" There wás | a yíeld | ing ; thís | admíts | no (e)xcúse."
Ib. v. 6. 69.
Here even the Folio reads " excuse."
" It ís | too hárd | a knót | for mé | to untie."
T. N. ii. 2. 42.
The is often elided before a vowel, and therefore we may either pronounce this is, this' (461), or write th' for the, in
"O worthy Goth, this is the incarnate devil."—T. A. v. 1. 40.
Remembering that "one" was pronounced without its present initial sound of w, we shall easily scan (though "the " is not elided in many modern texts)—
" Th' one swéet | ly flátt | ers, th'óth | er féar | eth hárm."
R. of L. 172.
" One hálf | of mé | is yóurs, | th' óther | half yóurs."
M. of V. iii. 2. 16.
"Ránsom | ing him (217) | or píty | ing, thréate | ning th' other."—Coriol. i. 6. 36.
And this explains
" And óf | his óld | expér(i) (467) | ence th(e) ón | ly dárling."
A. W. ii. 1. 110.
" Has shóok | and trém | bled át | the ill néigh | bourhóod."
Hen. V. i. 2. 154.
" Whére should | this mú | sic bé? | I'the áir, | or the earth?"
Temp. i. 2. 387, 389.
(Folio " i' th' air, or th' earth.")
463. R frequently softens or destroys a following vowel (the vowel being nearly lost in the burr which follows the effort to pronounce the r).
" Whén the | alárum | were strúck than í dly sít."
Cor. ii. 2. 80.
" Ham. Perchánce t'will wálk agáin.
Hor. I wárrant | it will."—Hamlet, i. 2. 3.
"I' have | cast óff | for éver; | thou shált, I wárrant thee."
Lear, i. 4. 332.
" I bét | ter broók than floúrish | ing péo | pled tówns."
T. G. of V. v. 4. 3.
" Whiles I | in Ire land nóurish | a míght | y bánd."
2 Hen. VI. iii. 1. 348.
" Place bárrels of pítch upón the fát al stáke."
1 Hen. VI. v. 4. 57.
"'Tis márle | he stább' | d you nót."
B. J. E. out &c. v. 4; Rich. III. i. 4. 64.
" A bárren detést ed vále | you sée it is."
T. A. ii. 3. 92 ; 2 Hen. VI. ii. 4. 3.
So "quarrel," Rich. III. i. 4. 209.
This is very common with "spirit," which softens the following i, or sometimes the preceding i, in either case becoming a monosyllable.
" And thén, | they sáy, | no spírit | dares stír abróad."
Hamlet, i. 1. 161.
So scan
" How now, | spírit, whither I wánder | you?"—M.N.D. ii. 1. 1.
(" Whither" is a monosyllable. See 466.)
This curtailment is expressed in the modern " sprite." So in Lancashire, " brid " for " bird." Hence we can scan
"In aíd | Whereóf, | wé of | the spírits | ualty."
Hen. V. i. 2. 132.
Instances might be multiplied.
464. R often softens a preceding unaccented vowel.
This explains the apparent Alexandrine
"He thinks | me nów | incáp | ablé ; | conféd(e)rates."
Temp. i. 2. 111, iv. 1. 140.
465. Er, el, and le final dropped or softened, especially before vowels and silent h. The syllable er, as in letter, is easily interchangeable with re, as lettre. In O. E. "bettre" is found for "better." Thus words frequently drop or soften -er; and in like manner -el and -le, especially before a vowel or h in the next word :
* (1) "Repórt | should rénd | er him hoúr | ly tó | your eár."
Cymb. iii. 4. 153.
"Intó | a góod | ly búlk. | Good tímes |encoúnter her."
W. T. ii. 1. 20.
"This léft | er he eár | ly báde | me gíven | his fáther."
R. and f. v. 3. 275.
" You'll bé | good cómpany, | my síst | er and yoú."
MIDDLETON, Witch, ii. 2.
" Than e'ér |the mást | er of árts | or gív | er of wít."
B. J. Poetast.
* (2) " Trável you | far ón, | or áre | you át | the fárthest?"
T. of Sh. iv. 2. 73.
* (3) " That máde | great Jóve | to húmb | le him tó | her hand." Ib. i. 1. 174.
" Géntlemen | and friénds, | I thánk | you fór | your páins."
Ib. iii. 2. 186.
" I' am | a géntle | man óf | a cóm | paný."
Hen. V. iv. 1. 39, 42.
" Needle," which in Gammer Gurton rhymes with " feele," is often pronounced as a monosyllable.
"Deep clerks she dumbs, and with her needle (Folio) composes."
P. of T. v. Gower, 5; Cymb. i. 1. 168.
" Or when she would with sharp needle (Folio) wound
The cambric which she made more sound
By hurting it. "—P. of T. iv. Gower, 23.
In the latter passage " needle wóund" " is certainly harsh, though Gower does bespeak allowance for his verse. Mr. A. J. Ellis suggests "'ld " for " "would," which removes the harshness.
" And gríp | ing ít | the néedle | his fing | er pricks."
R. of L. 319.
" Their néedles | to láne | ces, ánd | their gént | le héarts."
K. J. v. 2. 157.
" To thréad | the póst | ern óf | a smáll | needle's éye."
Rich. II. v. 5. 17.
" Needle's " seems harsh, and it would be more pleasing to modern ears to scan "the póst | ern óf a | small née | dle's éye." But this verse in conjunction with P. of T. iv. Gower, 23, may indicate that " needle" was pronounced as it was sometimes written, very much like "neeld," and the d in "neeld" as in "vild" (vile) may have been scarcely perceptible.
" A sámple | to the yoúng | est, tó | the móre | matúre."
Cymb. i. 1. 48.
" The cómm | on peóple | by númb | ers swarm | to ús."
3 Hen. VI. iv. 2. 2 ; T. A. i. 1. 20.
And, even in the Sonnets:
" And troúble | deaf heáv | en wíth | my bóot | less cries."
Sonn. 29.
" Uncle Már | cus, sínce | it ís | my fá | ther's mínd."
T. A. v. 3. 1.
" Duke F. And gét | you fróm | our cóurt. |
Ros. Me, uncle? |
Duke F. You, cousin ? "
A. Y. L. i. 3. 44.
466. Whether and ever are frequently written or pronounced whe'r or where and e'er. The th is also softened in either, hither, other, father, &c., and the v in having, evil, &c.
It is impossible to tell in many of these cases what degree of " softening " takes place. In " other," for instance, the th is so completely dropped that it has become our ordinary " or," which we use without thought of contraction. So "whether" is often written "wh'er" in Shakespeare. Some, but it is impossible to say what, degree of " softening," though not expressed in writing, seems to have affected th in the following words:—
Brother.
"But fór | our trust | y bróther | -in-láw, | the ábbot."
Rich. II. v. 3. 137.
Either.
"Either léd | or drív | en ás | we point | the wáy."
J. C. iv. 1. 23 ; Rich. III. i. 2. 64, iv. 4. 82.
" Are híred | to béar | their stáves; | either thóu, | Macbéth."
Macbeth, v. 7. 18; M. N. D. ii. 1. 32.
Further.
"As íf | thou never (né'er) | walk'dst fúrther | than Fins bury."
1 Hen. IV. iii. 1. 257.
Hither.
"'Tis hé | That sént us ('s) | hither nów | to slaúght | er thée."
Rich. III. i. 4. 250.
So the Quartos. The Folio, which I have usually followed in other plays, differs greatly from the Quartos in Rich. III. Its alterations generally tend to the removal of seeming difficulties. Neither.
" Neither háve | I món | ey nor | commód | itý."
M. of V. i. 1. 178.
Rather.
"Ráther than | have máde | that sáv | age dúke | thine héir."
3 Hen. VI. i. 1. 224. So Othello, iii. 4. 25; Rich. II. iv. 1.16.
Thither.
" Thither gó | these news | as fast | as horse | can carry 'em."
2 Hen. VI. i. 4. 78.
Whether.
"Good sir, | say whéther | you'll áns | wer mé | or nó."
C. of E. iv. 1. 60.
Perhaps
"Which hé | desérves | to lóse. | Whether he wás (h' was: 461) | combíned."—Macbeth, i. 3. 111.
"But see, | whether Brút | us bé | alíve | or déad."
J. C. v. 4. 30 ; Rich. III. iv. 2. 120.
" A héart | y wélcome. | Whether thóu | beest hé | or nó."
Tempest, v. 1. 111.
Whither.
" What meáns | he now? | Go ásk | him whíther | he góes."
1 Hen. VI. ii. 3. 28.
" Glouc. The king | is in | high ráge. |
" Corn.
Whíther is | he góing?"—Lear, ii. 4. 299.
So scan
"Hów now, | spírit ! whíther | wánder | yoú?"
M. N.D. ii. 1.
This perhaps explains :
"To fínd | the (462) other fórth, | and boý | advént | uring
bóth."—M. of V. i. 1. 143.
But see 501.
Having.
"Hów could | he sée | to dó | them? Háving | made óne."
M. of V. iii. 2. 124.
" Having lóst | the faír | discóv | ery óf | her wáy."
V. and A. 828.
" Our grán | dam éarth |having th s | | distémp | eratúre."
1 Hen. IV. iii. 1. 34.
So Rich. III. i. 2. 235; T. of A. v. 1. 61 ; A. W. v. 3. 123; Cymb. v. 3. 45.
In all of these verses it may seem difficult for modern readers to understand how the v could be dropped. But it presents no more difficulty than the v in "ever," "over."
Evil.
It is also dropped in " evil" " and "devil" (Scotch "de'il").
" The évils |she hátch'd |were nót | efféct | ed, só."
Cymb. v. 5. 60.
" Of hórr | id héll | can cóme | a dévil | more dámn'd."
Macbeth, iv. 3. 56.
"Evil-éyed | untó |you ; y' áre (461) | my pr son | er, bút."
Cymb. i. 1. 72.
So Rich. III. i. 2. 76. Of course, therefore, the following is not an Alexandrine:
"Repróach | and d ss | olú | tion háng | eth óver him."
Rich. II. ii. 1. 258.
Similarly the d is dropped in "madam," which is often pronounced "ma'am," a monosyllable.
The v is of course still dropped in hast for havest, has for haveth or haves. In the Folio, has is often written ha's, and an omission in other verbs is similarly expressed, as "sit's" for "sitteth" (K. J. ii. 1. 289).
467. I in the middle of a trisyllable, if unaccented, is frequently dropped, or so nearly dropped as to make it a favourite syllable in trisyllabic feet.
* (1) "Jud | cious púnish | ment! 'Twás | this flésh | begót."
Lear, iii. 4. 76; M. for M. i. 3. 39.
" Our rév | (e)rend cárdi | nal cárried. | Like it, | your gráce."—Hen. VIII. i. 1. 100, 102, 105, &c.
"With whóm | the Ként | ishmén | will wíll | ingly rise."
3 Hen. VI. i. 2. 41.
" Which áre | the móv | ers óf | a lánguish | ing déath."
Cymb. i. 5. 9.
" My thóught | whose múr | der yét | is bút | fantástical."
Macbeth, i. 3. 139.
" That lóv'd | your fáther: | the rési | due óf | your fórtune."
A. Y. L. ii. 7. 196.
" Prómising | to bríng | it tó | the Pór | pentíne."
C. of E. v. 1. 222.
So 1 Hen. VI. iv. 1. 166.
* (2) Very frequently before ly :
" The méa | sure thén | of óne | is éasi | ly tóld."
L. L. L. v. 2. 190.
" His shórt | thick néck | cannót | be eás | ily hármed."
V. and A. 627.
" Préttily | methóught | did pláy | the ór | atór."
1 Hen. VI. iv. 1. 175.
* (3) And before ty :
" Such bóld | hostíli | ty, teach | ing his ('s) dú | teous lánd."
1 Hen. IV. iv. 3. 44.
" Of gód- | like ámi | ty, which | appéars | most stróngly."
M. of V. iii. 4. 3.
" Ariel | and áll | his quáli | ty.
" Prosp.
Hást | thou, spírit? "—Tempest, i. 2. 193.
"Of smóoth | civíli | ty yét | am I ín | land bréd."
A. Y. L. ii. 7. 96.
Compare BUTLER, Hudibras, part ii. cant. 3. 945 :
"Which ín | their dark | fatál | 'ties lúrk | ing
At dés | tin'd pér | iods fáll | a-wórk | ing."
This explains the apparent Alexandrines:
" Thóu wilt | prove hís. | Táke him | to prí | son, ófficer."
M. for M. iii. 2. 32.
" Some trícks | of dés | perát | ion, áll | but máriners."
Temp. i. 1. 211.
"One dówle | that's ín | my plúme, | my féll | ow mínisters."
Temp. iii. 2. 65, v. 1. 28 ; M. for M. iv. 5. 6 ; Macb. i. 5. 49.
" Thís is | the gént | lemán | I tóld | your ládyship."
T. G. of V. ii. 4. 87.
" A vírt | uous gént | lewóm | an, míld | and beaútiful."
T. G. of V. iv. 4. 184.
" And té | diousnéss | the limbs | and oút | ward flóurishes."
Hamlet, ii. 2. 91.
Sometimes these contractions are expressed in writing, as "par'lous," Rich. III. ii. 4. 35. This is always a colloquial form.
468. Any unaccented syllable of a polysyllable (whether containing i or any other vowel) may sometimes be softened and almost ignored. Thus—
a
"Hóld thee, | from this, | for éver. | The bárb | arous Scýthian."—Lear, i. 1. 118. " Sáy by | this tó | ken I' | desíre | his cómpany."
M. for M. iv. 3. 144.
ed
" With thém | they think | on. Things | withoút | all rémedy."—Macbeth, iii. 2. 11. " Men. Yoú must | retúrn | and ménd | it. Sen.
Thére's | no rémedy."
Coriol. iii. 2. 26 ; T. N. iii. 4. 367.
em
" All bró | ken ímple | ments óf | a rú | ined hóuse."
T. of A. iv. 2. 16.
" Joín'd with | an enemy | procláim'd; | and fróm | his cóffers." Hen. V. ii. 2. 168 ; M. for M. ii. 2. 180; Macb. iii. 1. 105.
en
" The méss | engers fróm | our sís | ter ánd | the kíng."
Lear, ii. 2. 54.
"'Tis dóne | alréa | dy, ánd | the méss | enger góne."
A. and C. iii. 6. 31 ; A. W. iii. 2. 111.
Passenger is similarly used.
er
" In oúr | last cónference, | páss'd in | probá | tion with you."—Macbeth, iii. 1. 80.
es
"This ís | his máj | esty, sáy | your mind | to him."
A. W. ii. 1. 98.
"I thát | am rude | ly stámped, | and wánt | love's májesty."
Rich. III. i. 1. 16.
Majesty is a quasi-dissyllable in Rich. III. i. 3. 1, 19, ii. 1. 75 ; Rich. II. ii. 1. 141,147, iii. 2. 113, v. 2. 97, 3. 35 ; Macbeth, iii. 4. 2, 121.
ess
" Our púr | pose néc | essary ánd | not én | vious."
J. C. ii. 1. 178.
i
"Lét us | be sácrific | ers ánd | not bút | chers, Caíus."
Ib. ii. 1. 166.
"The ínn | ocent mílk | in ít | most ínn | ocent moúth."
W. T. iii. 2. 101.
"There táke | an ín | ventorý | of áll | I háve."
Hen. VIII. iii. 2. 452.
ua
" Go thóu | to sánctua | ry [sanctu'ry or sanct'ry], ánd | good thóughts | posséss thee."—Rich. III. iv. 1. 94.
"Shall flý | out óf (457a) | itsélf; | nor sléep | nor sánctuary."
Coriol. i. 10. 19.
" Some réad | Alvár | ez' Hélps | to Gráce,
Some Sánctua | ry óf | a tróub | led sóul."
COLVIL'S Whig Supplication, i. 1186 (Walker).
u
" When lí v | ing líght | should kíss | it ; 'tís | unnátural."
Macbeth, ii. 4. 10 ; Hen. V. iv. 2. 13.
"Thoughts spécu | latíve | their ún | sure hópes | reláte."
Macbeth, v. 4. 19.
"And né | ver líve | to shów | the incrédu | lous wórld."
2 Hen. IV. iv. 5. 153.
"Hów you | were bórne | in hánd, | how cróss'd, | the ín-st ruments."—Macbeth, iii. 1. 81, iv. 3. 239.
469. Hence polysyllabic names often receive but one accent at the end of the line in pronunciation.
Proper names, not conveying, as other nouns do, the origin and reason of their formation, are of course peculiarly liable to be modified ; and this modification will generally shorten rather than lengthen the name.
" To yoúr | own cón | science, sír, | befóre | Políxenes."
W. T. iii. 2. 47.
" That ére | the sún | shone bríght | on. O'f | Hermíone."
Ib. v. 1. 95.
" The rár | est óf | all wó | men. Gó, | Cleómenes."
Ib. 112.
" To oúr | most fáir | and prínce | ly cóus | in Kátharine."
Hen. V. v. 2. 4.
" My bróth | er ánd | thy ún | cle, cálled | António."
Temp. i. 2. 66.
"My lórd | Bassán | io, sínce | you have fóund | António."
M. of V. i. 1. 59 : so often in this play.
" Then all | a-fíre | with mé | ; the kíng's | son Férdinand."
Temp. i. 2. 212.
"I rát | ifý | thís my |rich gíft. | O Férdinand."—Ib.iv.1.8.
"Then pár | don mé | my wróngs. | But hów | should Próspero?"—Ib. v. 1. 119.
" I'll áf | ter, móre | to bé | revenged | on E'glamour."
T. G. of V. v. 2. 51.
"Whát it | contáins. | I'f you | shall sée | | Cordélia."
Lear, iii. 1. 46.
" Upón | such sácr | ifíc | es, mý | Cordélia."
Ib. v. 3. 20, 245.
So throughout the play.
" When thóu | liest hów | ling. Whát! | the fair | Ophélia."
Hamlet, v. 1. 265.
" At Gré | cian swórd | contémn | ing. Téll | Valéria,"
Coriol. i. 3. 46.
" Here, íf | it like | you hón | our. Sée | that Cláudio."
M. for M. ii. 1. 33, iii. 1. 48.
" So thén | you hópe | | of pár | don fróm | lord A'ngelo ?"
Ib. iii. 1. 1, iv. 3. 147, i. 4. 79.
" I sée | my són | Antíph | olús | and Drómio."
C. of E. v. 1. 196.
" The fórm | of déath. | Meantíme | I wrít | to Rómeo."
R. and J. v. 3. 246.
" Lóoks it | not líke | the king? | Márk it, | Horátio."
Hamlet, i. 1. 43.
" They lóve | and dóte | on ; cáll | him boúnt | (e)ous Búckingham." —Hen. VIII. ii. 1. 52; Rich. III. iv. 4. 508, ii. 2. 123.
" Vaux. The greát | ness óf | his pér | son.
Buck.
Náy, | Sir Nícolas."
Hen. VIII. ii. 1. 100.
" But I' | beséech | you, whát's I becóme | of Kátharine?"
Ib. iv. 1. 22.
"Sáw'st thou | the mél | anchól | y Lórd | Northúmberland?" —Rich. III. v. 3. 68.
" Thérefore | presént | to hér, | as sóme | time Márgaret."
Ib. iv. 4. 274.
" And yóu our nó | less lóv | ing són | of A'lbany."
Lear, i. 1. 43.
" Exásp | erátes, | makes mád | her sís | ter Góneril."
Ib. v. 1. 60.
" As fít | the bríd | al. Beshréw | me múch, | Emília."
Othello, iii. 4. 150.
" Is cóme | from Cæ's | ar ; thére fore héar | it, A'ntony."
A. and C. i. 1. 27, i. 5. 21, &c.
" Than Clé | opátr | a, nór | the quéen | of Ptólemy."
Ib. i. 4. 6.
" With thém, | the twó | brave beárs, | Wárwick | and Móntague."—3 Hen. VI. v. 7. 10.
Less frequently in the middle of the line:
" My lórd | of Búckingham, | if mý | weak ór | atóry."
Rich. III. iii. 1. 37.
" Cóusin | of Búck | ingham ánd | you ságe, | grave mén."
Ib. iii. 7. 217.
" Lóoking | for A'ntony. | But áll | the chárms | of lóve."
A. and C. ii. 1. 20.
" Did sláy | this Fórtinbras; | who, bý | a seál'd | compáct (490)."—Hamlet, i. 1. 86.
" Thrift, thríft, | Horátio, | the fú | nerál | bak'd méats." Ib. i. 2. 180.
" He gáve | to Alexánder ; to Ptólem | y hé | assígned."
Ib. iii. 6. 15.
" Thou árt | Hermíone; | or ráth | er, thoú | art shé."
W. T. v. 3. 25.
" To sóft | en A'ngelo, | and thát's | my píth | of business."
M. for M. i. 4. 70.
Enobárbus in A. and C. has but one accent, wherever it stands in the verse:
" Bear háte | ful mémo | ry, póor | Enobár | bus did."
A. and C. iv. 9. 9, &c.
" Of yóur | great pré | decéssor, | King E'dward | the Third."
Hen. V. i. 2. 248.
It may here be remarked that great licence is taken with the metre wherever a list of names occurs:
" That Harry duke of Hereford, Rainold lord Cobham,
Sir Thomas Erpingham, Sir John Ramston,
Sir John Norbery, Sir Robert Waterton, and Francis Quoint."
Rich. II. ii. 1. 279, 283, 284.
" The spirits
Of valiant Shirley, Stafford, Blunt, are in my arms."
1 Hen. IV. v. 4. 4.
" Whither away, Sir John Falstaffe, in such haste?"
1 Hen. IV. iii. 2. 104.
" John duke of Norfolk, Walter Lord Ferrers."
Rich. III. v. 5. 13.
" Lord Cromwell of Wingfield, Lord Furnival of Sheffield."
Ib, iv. 7. 166.
" Sir Gilbert Talbot, Sir William Stanley."—Ib. iv. 5. 10.
In the last examples, and in some others, the pause between two names seems to license either the insertion or omission of a syllable.
470. Words in which a light vowel is preceded by a heavy vowel or diphthong are frequently contracted, as power, jewel, lower, doing, going, dying, playing, prowess, &c.
"The which | no sóon | er hád | his prówess | confírm'd."
Macbeth, v. 8. 41.
Comp.
" And he that routs most pigs and cows, The fórm | idáb | lest mán | of prówess."
Perhaps
Hudib. iii. 3. 357.
" Which bóth | thy dú | ty ówes | and óur | power cláims."
A. W. ii. 3. 168.
(This supposes "our" emphasized by antithesis, but "and our pów | er cláims" (ELLIS) may be the correct scanning.)
Being.—"That with | his pér | emptór | y "sháll" | being pút."
Coriol. iii. 1. 94, 2. 81.
" The sóv | ereigntý | of eí | ther béing | so great."
This explains the apparent Alexandrines:
R. of L. 69.
" And béing | but a tóy | that ís | no grief | to give."
Rich. III. ii. 1. 114.
"Withóut I a párall | el, thése | being áll | my stúdy."
Tempest, i. 2. 74.
Doing.—"Can láy | to béd | for éver: | whiles yóu, | doing thús."
Ib. ii. 1. 284.
Seeing.—" Or séeing | it óf | such child | ish friend | linéss."
Coriol. ii. 3. 183.
" I'll in | mysélf | | to sée, | and in thée | seeing íll."
Rich. II. ii. 1. 94.
" That yóu | at súch | times séeing | me né | ver sháll."
Hamlet, i. 5. 173.
-ying.—" And próph | esýing | with ác | cents tér | rible."
This may explain
Macbeth, ii. 3 62.
" Lóck'd in | her món(u) [468] | ment. Shé'd | a próph(e)- | sying féar."—A. and C. iv. 14. 120.
So with other participles, as
"They, knówing | dame E'l | eanór's | aspír | ing húmour."
2 Hen. VI. i. 2. 97.
The rhythm seems to demand that " coward " should be a quasi-monosyllable in
" Wrong ríght, | base nóble, | old yoúng, | coward vál | iánt."
T. A. iv. 1. 29.
"Noble" a monosyllable. (See 465.)
"Yét are | they páss | ing cówardly. | But I' | beséech you."
Coriol. i. 1. 207.
471. The plural and possessive cases of nouns in which the singular ends in s, se, ss, ce, and ge, are frequently written, and still more frequently pronounced, without the additional syllable :
"A's the | dead cár | casses óf | unbúr | ied mén."
Coriol. iii. 3. 122.
" Thínking | upón | his sér | vices tóok | from yóu."
Ib. ii. 2. 231.
" Their sénse | are [Fol. sic] shút. "—Macbeth, v. 1. 29.
" My sénse | are stópped."—Sonn. 112.
"These vérse."—DANIEL.
"I'll tó | him ; hé | is híd | at Láwr | ence' céll."
R. and J. iii. 2. 141.
" Great kings of France and England ! That I have laboured,
Your might | inéss | on bóth | parts bést | can witness."
Hen. V. v. 2. 28.
" Place " is probably used for " places " in
" The frésh | springs, bríne- | pits, bár | ren pláce | and fértile."—Tempest, i. 2. 338.
" These twó | Antíph | olús [Folio], | these twó | so líke."
C. of E. v. 1. 357.
"Are there balance?"—M. of V. iv. 1. 255.
" (Here) have I, thy schoolmaster, made thee more profit
Than óth | er prín | cess [Folio] cán | that háve | more time."—Temp. i. 2. 173.
" Sits on his horse back at mine hostess door."
K. J. ii. 1. 289 (Folio).
" Looked pále | when théy | did héar | of Clár | ence (Folio) déath."—Rich. III. ii. 1. 137, iii. 1. 144.
Probably the s is not sounded (horse is the old plural) in
" And Duncan's horses (a thing most strange and certain)."
Macbeth, ii. 4. 14.
" Lies in their purses, and whoso empties them."
Rich. II. ii. 2. 130.
Even after ge the s was often suppressed, even where printed. Thus:
" How many ways shall Carthage's glory grow !"
SURREY'S Æneid IV. (Walker).
But often the s was not written. So
" In violating marriage sacred law."
Edward III. (1597 A.D.) (LAMB.)
The s is perhaps not pronounced in
" Conjéct | (u)ral márr | iage(s); mák | ing párt | ies stróng."
Coriol. i. 1. 198.
" Are brá | zen im | ages óf | canón (491) | iz'd sáints."
2 Hen. VI. i. 3. 63.
"The im | ages óf | revólt | and flý | ing óff!"
Lear, ii. 4. 91.
" O'ff with | his són | George's héad."—Rich. III. v. 3. 344.
" Létters | should nót | be knówn, | riches póv | ertý. "
Tempest, ii. 1. 150.
This may perhaps explain the apparent Alexandrines :
" I próm | is'd yóu | redréss | of thése | same griévances."
2 Hen. IV. iv. 2. 113.
" This déi | ty in | my bós | om twén | ty cónsciences."
Temp. ii. 1. 278.
" And stráight | discláim | their tóngues? | Whát are | your óffices?"—Coriol. iii. 1. 35.
" Popíl | ius Lé | na spéaks | not óf | our púr | poses."
J. C. iii. 1. 23.
" She lév | ell'd át | our púr | poses, ánd | being (470) róyal,"
A. and C. v. 2. 339.
(or " | our púrpose(s), and bé | ing róyal.")
" A thíng | most brú | tish, I' | endówed | thy púrposes."
Tempest, i. 2. 357.
" Nor whén | she púrposes | retúrn. | Beséech | your highness."
Cymb. iv. 3. 15.
" As blánks, | benévo | lences ánd | I wót | not what."
Rich. II. ii. 1. 250.
" My sérv | ices whích | I have ('ve) dóne | the Sígn | iorý."
Othello, i. 2. 18.
" These pípes | and thése | convéy | ances óf | our blóod."
Coriol. v. 1. 54.
" Profésses | to persuáde | the kíng | his són's | alive."
Temp. ii. 1. 236.
Either "whom I" is a detached foot (499) or s is mute in
" Whom I', | with thís | obéd | ient stéel, | three ínches of it (inch of 't)."—Tempest, ii. I. 285.
472. Ed following d or t is often not written (this elision is very old: see 341, 342), and, when written, often not pronounced.
" I hád | not quóted him. I I féar'd | he díd | but trífle."
Hamlet, ii. 1. 112.
" Reg. That ténded (Globe, 'tend') | upón | my father.
Glou. I knów | not, mádam."—Lear, ii. 1. 97.
" Since nót | to bé | avóided | it fálls | on mé."
1 Hen. IV. v. 4. 13.
" But júst | ly ás | you háve | excéeded all prómise."
A. Y. L. i. 2. 156.
" For tréas | on éxe | cuted ín | our láte | king's dáys."
1 Hen. VI. ii. 4. 91.
" And só, | ríveted | with faíth | untó (457) | your flésh."
M. of V. v. 1. 169.
" Be sóon | colléct | ed and áll | things thóught | upón."
Hen. V. i. 2. 305.
"I's to | be freíghted | out of féar : | and ín | that móod."
A. and C. iii. 13. 196.
" Was ápt | ly fítted | and nát | (u)rally | perfórm'd."
T. of Sh. Ind. 1. 87.
" Is nów | convérted : | but nów | I wás the lórd."
M. of V. iii. 2. 169.
" Which I' | mistrústed | not : fáre | well thére | fore, Héro." M. Ado, ii. 1. 189.
" All ún | avóided | is the dóom | of dést | iný."
Rich. III. iv. 4. 217.
but here "destiny" (467) may be a dissyllable, and -ed sonant.
This explains the apparent Alexandrine :
" I thús | negléct | ing wórld | ly énds | all dédicated."
Temp. i. 2. 89.
" Shóuting | their ém | ulá | tion. Whát | is gránted them?"
Coriol. i. 1. 218.
So strong was the dislike to pronouncing two dental syllables together, that "it" seems nearly or quite lost after "set" and " let " in the following :
" I húmb | ly sét it | at your wíll ; but fór | my mistress."
Cymb. iv. 3. 9.
" To his | expér | ienced tóngue ; | yet lét it | please bóth."
Tr. and Cr. i. 3. 68.
" Yóu are a | young húnt | sman, Már | cus : lét it alone."
T. A. iv. 2. 101.
" You sée | is kíll'd | in hím : | and yét it | is dánger."
Lear, iv. 7. 79.
So perhaps " Of éx | cellént | dissémb | ling ; ánd | let it lóok."
A. and C. i. 3. 79.
But more probably, "dissémbling ; | and lét | it lóok."
473. Est in superlatives is often pronounced st after dentals and liquids. A similar euphonic contraction with respect to est in verbs is found in E. E. Thus " bindest " becomes "binst," " eatest " becomes " est." Our " best " is a contraction for " bet-est."
"Twó of | the swéet'st | compán | ions ín | the wórld."
Cymb. v. 5. 349.
" At yóur | kind'st léisure."—Macbeth, ii. 1. 24.
" The stérn'st | good níght."—Ib. ii. 2. 4.
" Secret'st."—Ib. iii. 4. 126.
" Thís is | thy éld'st | son's són."—K. J. ii. 1. 177.
So Temp. v. 1. 186.
" Since déath | of mý | dear'st móth | er."—Cymb. iv. 2. 190.
" The lóy | al'st hús | band thát | did é'er | plight tróth."
Ib. i. 1. 96.
A. W. ii. 1. 163, "great'st." " The sweet'st, dear'st."—W. T. iii. 2. 202. "Near'st."—Macb. iii. 1. 118. " Unpleasant'st."—M. of V. iii. 2. 254. "Strong'st."—Rich. II. iii. 3. 201. "Short'st." —Ib. v. 1. 80. "Common'st."—Ib. v. 3. 17. "Faithfull'st."—T. N. v. 1. 117.
This lasted past the Elizabethan period.
" Know there are rhymes which fresh and fresh apply'd
Will cure the arrant'st puppy of his pride."
POPE, Imit. Hor. Epist. i. 60.
The Folio reads "stroakst," and "made" in
" Thou stróakedst | me ánd | madest múch | of mé, would'st give me."—Tempest, i. 2. 333.
But the accent on "and" is harsh. Perhaps "and má | dest."
# VARIABLE SYLLABLES.
474. Ed final is often mute and sonant in the same line. Just as one superlative inflection -est does duty for two closely connected adjectives (398):
"The generous and gravest citizens."—M. for M. iv. 6. 13.
and the adverbial inflection ly does duty for two adverbs (397) :
"And she will speak most bitterly and strange."
M. for M. v. 1. 36.
so, when two participles ending in -ed are closely connected by " and," the ed in one is often omitted in pronunciation.
"Despis'd, | distréss | ed, hát | ed, márt | yr'd, killed."
R. and J. iv. 5. 59.
"We have with | a léav | en'd ánd | prepár | ed chóice."
M. for M. i. 1. 52.
" To this | unlóok'd | for, ún | prepár | ed pómp."
K. J. ii. 1. 560.
In the following the -ed sonant precedes :
" That wére | embátt | ailéd | and ránk'd | in Ként."
K. J. iv. 2. 200.
" We are | impréss | ed ánd | engág'd | to fight."
1 Hen. IV. i. 1. 21.
"For this | they háve | engróss | ed ánd | pil'd úp."
2 Hen. IV. iv. 5. 71.
"Thou cháng | ed ánd | self-cóv | er'd thíng, | for sháme."
Lear, iv. 2. 62.
At the end of a line ed is often sounded after er:
" Which his | hell-góv | ern'd árm | hath bútc | heréd." Rich. III. i. 2. 74.
See J. C. ii. 1. 208 ; iii. 1. 17 ; iii. 2. 7, 10 ; iv. 1. 47; v. 1. 1. So Rich. III. iii. 7. 136 ; iv. 3. 17 ; v. 3. 92 ; M. N. D. iii. 2. 18, &c. This perhaps arises in part from the fact that " er " final in itself (478) has a lengthened sound approaching to a dissyllable.
Ed is very frequently pronounced in the participles of words ending in fy, "glorify," &c.
" Most pút | rifí | ed córe, | so fair | withóut."
Tr. and Cr. v. 9. 1.
" My mórt | ifí | ed spirit. | Now bíd | me rún."
J. C. ii. 1. 324.
"Váughan | and áll | that háve | miscárr | iéd."
Rich. III. v. 1. 5.
" The French | and E'ng | lish thére | miscár | riéd."
M. of V. ii. 8. 29.
"That cáme | too lág | to sée | him bú | riéd."—Ib. ii. 1. 90. So frequently in other Elizabethan authors. Also when preceded by rn, rm, "turned," "confirmed," &c., and in " followed : "
" As théy | us tó | our trénch | es fóll | owéd."
Coriol. i. 4. 42.
On the other hand, -ed is mute in
" By whát | by-páths | and ín | diréct | crook'd wáys."
2 Hen. IV. iv. 5. 185.
In
" Warder. We dó | no óth | erwíse | than wé | are will'd.
Glou. Who wíll | ed yóu? | Or whóse | will stánds |
but míne,"—I Hen. VI. i. 3. 11.
it would seem that the latter "willed" is the more emphatic of the two, and it will probably be found that in many cases where two participles are connected, the more emphatic has ed sonant. Thus the former "banished" is the more emphatic of the two in
"Hence bán | ishéd | is bánish'd fróm | the wórld." R. and J. iii. 3. 19.
475. A word repeated twice in a verse often receives two accents the first time, and one accent the second, when it is less emphatic the second time than the first. Or the word may occupy the whole of a foot the first time, and only part of a foot the second. Thus in
" Fáre (480) | well, gen | tle mís | tress : fáre | well, Nán."
M. W. of W. iii. 4. 97.
"Fáre (480) | well, gén | tle cóus | in. Cóz, | farewéll."
K. F. iii. 2. 17.
" Of gréat | est júst | ice. Wrí | te (484), wríte, | Rináldo."
A. W. iii. 4. 29.
"These ví | olént | desíres | have vio | lent énds."
R. and F. ii. 6. 9.
"With hér | that hát | eth thée | and hátes | us áll."
2 Hen. VI. ii. 4. 52.
Here the emphasis is on "ends" and "us all."
"Duke. stíll (486) so crú | el ?
Oliv. Still so cón | stant, lórd."—T. N. v. 1. 113.
"Com. Knów (484), | I práy you.
Coriol. I' | 'll knów | no fúrther."—Coriol. iii. 3. 87.
"Déso | late, dés | olate, wíll | I hénce | and díe."
Rich. II. i. 2. 73.
The former " Antony " is the more emphatic in
"But wére | I Brútus
And Brú | tus A'n | toný, | thére were | an A'ntony."
J. C. iii. 2. 231.
So, perhaps, the more emphatic verb has the longer form in
"He róus | eth úp | himsélf | and mákes | a páuse."
R. of L. 541.
This is often the case with diphthongic monosyllables. See 484. Compare
"Nów | it schéy | neth, nów | it réyn | eth fáste."
CHAUCER, C. T. 1537.
476. On the other hand, when the word increases in emphasis, the converse takes place.
"And lét | thy blóws, | dóubly | redóub | (e)
léd." Rich. II. i. 3. 80.
" Virg. O, héavens, | O, héav | ens.
Coriol. Náy, | I prí | thee, wóman."
Coriol. iv. 1. 12.
"Wás it | his spírit | by spír | its táught | to wríte?"
Sonn. 86.
" And wíth | her pérson | age, hér | tall pér | sonáge.
" M. N. D. iii. 2. 292.
"Március | would háve | all fróm | you—Már | ciús,
Whom láte | you have námed | for cónsul."
Coriol. iii. 1. 95.
Even at the end of the verse Marcius has but one accent, as a rule. But here it is unusually emphasized.
"And whé'r | he rún | or flý | they knów | not Whéther."
V. and A. 304.
"King. Be pát | ient, gént | le quéen, | and I' | will stay.
Queen. Whó can | be pát | iént | in thése | extrémes."
3 Hen. VI. i. 1. 215—6.
" Yield, my lórd | protéct | or, yí | eld, Wínch | estér."
1 Hen. VI. iii. 1. 112.
" Citizens. Yield, Már | cius, yí | eld.
Men. Hé | ar (480) mé, | one wórd."
Coriol. iii. 1. 215.
" A dévil (466), | a bór | n (485) dé | vil, ín | whose náture."
Tempest, iv. 1. 188.
So arrange
"You heavens (512), | Give me | that pát | ience, pát | iénce | I néed."
Lear, ii. 4. 274.
(" Patient" was treated as a trisyllable by the orthoepists of the time.)
"Being hád, | to trí | umph bé | ing (on the other hand) lack'd, to hópe."—Sonn. 52.
Similarly
" Which árt | my néar'st | and déar est én | emý."
I Hen. IV. iii. 2. 123.
On the other hand, perhaps, "sire," and not "cowards," is a dissyllable in
"Cowards fá | ther cówards, | and báse | things sí | re base."
Cymb. iv. 2. 26.
So, perhaps,
"Pánting | he lies | and bréath | eth ín | her fáce."
V. and A. 62.
Here "lies" is unemphatic, "breatheth" emphatic.
For diphthongic monosyllables see 484.
The same variation is found in modern poetry. In the following line there is, as it were, an antithetical proportion in which the two middle terms are emphatic, while the extremes are unemphatic :
"Tówer be | yond tów | er, spí | re bé | yond spíre."—TENNYSON.
# LENGTHENING OF WORDS.
477. R, and liquids in dissyllables, are frequently pronounced as though an extra vowel were introduced between them and the preceding consonant:
"The párts | and grá | ces óf | the wrés | t(e)lér."
A. Y. L. ii. 2. 13.
" In séc | ond ácc | ent óf | his órd | (i)nánce."
Hen. V. ii. 4. 126.
The Folio inserts i here, and e, Ib. iii. Prologue, 26. In the latter passage the word is a dissyllable.
"If yóu | will tár | ry, hó | ly pílg | (e)rím."—A. W.iii. 5.43.
" While shé | did cáll | me rás | cal fíd | d(e)lér."
T. of Sh. ii. 1. 158.
"The lífe | of him. | Knów'st thou | this cóun | t(e)rý ?" T. N. i. 2. 21. So Coriol i. 9. 17; 2 Hen. VI. i. 1. 206.
" And thése | two Dróm | ios, óne | in sémb | (e)lánce."
C. of E. v. 1. 358 ; T. G. of V. i. 3. 84.
" Yóu, the | great tóe | of thís | assémb | 1(e)ý."
Coriol. i. 1. 158.
" Cor. Be thús | to thém. |
Patr. You dó | the nó | b(e)lér."—Ib. iii. 2. 6.
" Edm. Sir, you | speak nó | b(e)lý. |
Reg. Why is | this réason'd? "—Lear, v. 1. 28.
(?) " Go search | like nó | b(e)lés, | like nó | ble súbjects."
P. of T. ii. 4. 50.
The e is actually inserted in the Folio of Titus Andronicus in "brethren :"
"Give Mú | cius búr | ial with |his bréth | erén."
T. A. i. 1. 347.
And this is by derivation the correct form, as also is " childeren."
" These áre | the pár | ents óf | these chíl | d(e)rén."
C. of E. v. 1. 360.
"I gó. | Wríte to | me vér | y shórt | (e)lý."
Rich. III. iv. 4. 428.
" A rót | ten case | abides | no hánd | (e)líng."
2 Hen. IV. iv. 1. 161.
"The friends | of Fránce | our shrouds | and táck | (e)língs."
3 Hen. VI. v. 3. 18.
"Than Ból | ingbróke's | retúrn | to E'ng | (e)lánd."
Rich. II. iv. 1. 17.
"And mean | to máke | her quéen | of E'ng | (e)lánd."
Rich. III. iv. 4. 263.
So in E. E. "Engeland."
"To bé | in án | ger ís | impí | etý;
But whó | is mán | that ís | not án | g(e)rý?"
T. of A. iii. 5. 56.
in which last passage the rhyme indicates that angry must be pronounced as a trisyllable.
" And stréngth | by límp | ing swáy | disá | b(e)léd."—Sonn. 66.
So also in the middle of lines—
"Is Cáde | the són | of Hén | (e)rý | the Fifth?"
2 Hen. VI. iv. 8. 36.
This is common in Hen. VI., but not I think in the other plays—not for instance in Rich. II.
"That cróaks | the fá | tal én | t(e)ránce | of Dúncan."
Macbeth, i. 5. 40.
"Cárries | no fá | vour ín't | but Bért | (e)rám's."
A. W. i. 1. 94.
" O mé! | you júgg | (e)lér! | you cán | ker blóssom."
M. N. D. iii. 2. 282.
"'Tis mónst | (e)róus. | Iá | go, who | begán it?"
Othello, ii. 3. 217.
"And thát | hath dázz | (e)léd | my réa | son's light."
T. G. of V. ii. 4. 210.
"Béing | so frús | t(e)ráte. | Téll him | he mócks."
A. and C. v. 1. 2.
"Lord Dóug | (e)lás, | go yóu | and téll | him só."
1 Hen. IV. v. 2. 33.
"Gráce and | remém | b(e)ránce | be tó | you bóth."
W. T. iv. 4. 76.
" Of quíck | cross líght | (e)ning? | To wátch, | poor pérdu."
Lear, iv. 7. 35.
" Thou kíll'st | thy míst | (e)réss : | but wéll | and frée."
A. and C. ii. 5. 27.
"To táunt | at slack | (e)néss. | Caníd | ius wé."
Ib. iii. 7. 28.
So also probably "sec(e)ret," "monst(e)rous" (Macbeth, iii. 6. 8), "nob(e)ly," "wit(e)ness," T. G. of V. iv. 2. 110, and even "cap(i)tains" (French " capitaine :" Macbeth, i. 2. 34, 3 Hen. VI. iv. 7. 32, and perhaps Othello, i. 2. 53).
Spenser inserts the e in some of these words, as "handeling," F. Q. i. 8. 28 ; "enterance," ib. 34.
478. Er final seems to have been sometimes pronounced with a kind of " burr," which produced the effect of an additional syllable ; just as "Sirrah" is another and more vehement form of "Sir." Perhaps this may explain the following lines, some of which may be explained by 505–10, but not all :
"Corn. We'll téach | you—
Kent. Sír, | ′I'm | too óld | to léarn."
Lear, ii. 2. 135.
(But? "I′ am.")
" Lénds the tongue vóws ; | these blá | zes dáugh | tér."
Hamlet, i. 3. 117.
"And thére | upón, | gíve me your dáugh | tér."
Hen. V. v. 2. 475.
"Bru. Spread fúr | thér. |
Menen. One wó | rd (485) móre, | one wórd."
Coriol. iii. I. 311.
"Líke a | ripe sís | tér : | the wóm | an lów."
A. Y. L. iv. 3. 88.
"Of óur | dear sóuls. | Meantíme, | sweet sís | tér."
T. N. v. 1. 393.
"I práy | you, úncle (465), | give me | this dág | gér."
Rich. III. iii. 1. 110.
"A bróth | er's múr | dér. | Práy can | I nót."
Hamlet, iii. 3. 38.
"Fríghted | each óth | ér. | Whý should | he fóllow?"
A. and C. iii. 13. 6.
" And só | to árms, | victór I ious fá | thér."
2 Hen. VI. v. 1. 211.
"To céase. | Wast thóu | ordáin'd, | dear fá | thér?"
Ib. v. 2. 45.
"Corn. Whére hast | thou sént | the king? |
Glouc.
To Dó | vér."—Lear, iii. 7. 51.
"Will I' | first wórk. | Hé's for | his más | tér."—Cymb. i. 5. 28.
"Lear. Than the | sea-móns | tér. |
Alb.
Pray, sir, | be pátient."—Lear, i. 4. 283.
But perhaps "patient" may have two accents. In that case "ter " is a pause-extra syllable.
In the two following lines s follows the r:
"To speak | of hór | rórs, | he cómes | befóre me."
Hamlet, ii. 1. 84.
"Públius, | how nów ? | How nów, | my más | térs?" T. A. iv. 3. 35; and perhaps Macbeth, iii. 4. 133.
"And give him hálf: | and fór | thy víg | óur."
Tr. and Cr. ii. 2. 272.
"Téll me, how fáres l our lóv | ing móth | ér?"
Rich. III. v. 3. 82.
"Cass. Good níght, | my lórd. |
Brut. Good night, | good bróth | ér."
J. C. iv. 3. 237.
"He whóm | my fáth | er námed? | Your E'd | gár."
Lear, ii. 1. 94.
(? "ná(484) | med? Yoú | r (480) E'dgar.")
" I'll fól | low yóu | and téll | what án | swér."
3 Hen. VI. iv. 3. 55.
"I have six | ty sáil : | Cæ′sar | none bét | tér."
A. and C. iii. 7. 50.
" This woód | en slá | very, thán | to súff | ér."
Temp. iii. 1. 62.
Sometimes this natural burr on r influences the spelling. In Genesis and Exodus (Early English Text Society, Ed. Morris) we have "coren" for "corn," "boren" for "born." Thus the E. E. "thurh" is spelt "thorugh" by early writers, and hence even by Shakespeare in
"The fálse | revólt | ing Nór | mans thó | rough thée."
2 Hen. VI. iv. 1. 87.
So M. N. D. ii. 1. 3, 5; Coriol. v. 3. 115.
In the following difficult lines it may be that r introduces an extra syllable :
"I′gnomy | in rán | som ánd | free pá | rdón
A're of two hóu | ses, láw | ful mé | rcý."
M. for M. ii. 4. 111, 112.
It would of course save trouble to read "ignominy," against the Folio. But compare
" Thy íg | nomý (Fol.) | sleep wíth | thee ín | thy grave."
1 Hen. IV. v. 4. 100.
" Hence, brók | er láck | ey ! I'g | nomý | and sháme."
Tr. and Cr. v. 11. 33.
and in T. A. iv. 2. 115 (where the Folio reads "ignominy") the i is slurred.
"No mán | knows whíther. | I crý | thee mé | rcý."
Rich. III. iv. 4. 515.
" It is | my són, | young Hár | ry Pé | rcý."
Rich. II. ii. 3. 21.
"Thou, Rích | ard, shált | to the dúke | of Nór | fólk."
3 Hen. VI. i. 2. 38.
So we sometimes find the old comparative "near" for the modern "nearer."
"Bétter | far óff | than néar | be né'er | the néar."
Rich. II. v. 1. 88.
"The néar | in blóod |
The néar | er blóody."—Macbeth, ii. 3. 146.
"Nor near nor farther off . . . than this weak arm."
Rich. II. iii. 2. 64.
And "far" for "farther," the old "ferror."
"Fár than | Deucá | lion óff."—W. T. iv. 4. 442.
479. The termination "ion" is frequently pronounced as two syllables at the end of a line. The i is also sometimes pronounced as a distinct syllable in soldier, courtier, marriage, conscience, partial, &c. ; less frequently the e in surgeon, vengeance, pageant, creature, pleasure, and treasure.
The cases in which ion is pronounced in the middle of a line are rare. I have only been able to collect the following:
"With ób | servá | tión | the whích | he vénts."
A. Y. L. ii. 7. 41.
"Of Hám | let's tráns | formá | tión : | so cáll it."
Hamlet, ii. 2. 5.
"Be chósen | with pró | clamá | tións | to-dáy."
T. A. i. 1. 190.
Gill, 1621, always writes "ti-on" as two syllables. But there is some danger in taking the books of orthoepists as criteria of popular pronunciation. They are too apt to set down, not what is, but what ought to be. The Shakespearian usage will perhaps be found a better guide.
Tión, when preceded by c, is more frequently prolonged, perhaps because the c more readily attracts the t to itself, and leaves ion uninfluenced by the t.
"It wére | an hón | est áct | ión | to say so."
Othello, ii. 3. 145; Tr. and Cr. i. 3. 340.
"Her swéet | perféct | ións | with óne | self king."
T. N. i. 1. 39.
"Yet háve | I fiérce | afféct | ións | and think."
A. and C. i. 5. 17.
" With sóre | distráct | ión what I' | have dóne."
Hamlet, v. 2. 241.
" To ús | in oúr | eléct | ión | this dáy."—T A. i. 1. 235.
In
" That sháll | make áns | wer tó | such quést | ións. It is enóugh. | I'll thínk | upón | the quést | ións,"
2 Hen. VI. i. 2. 80, 82.
it seems unlikely that "questions" is to be differently scanned in two lines so close together. And possibly, "it is (it's) enóugh," is one foot. Still, if " questions" in the second verse be regarded as an unemphatic (475) repetition, it might be scanned :
"It ís | enóugh. | I'll thínk | upón | the quéstions."
The Globe has
"Jóin'd in | commíss | ion with him ; | but either (466) |
Had bórne || the action of yourself, or else
To him || had left it solely."—Coriol. iv. 6. 14.
But better arrange as marked above, avoiding the necessity of laying two accents on "commission." So Folio—which, however, is not of much weight as regards arrangement.
I is pronounced in "business" in
"To sée | this bús | inéss. | To-mór | row néxt."
Rich. II. ii. 1. 217; Rich. III. ii. 2. 144; M. of V. iv.
1. 127 ; Coriol. v. 3. 4.
"Divín | est cré | atúre, | Astræ′ | a's dáughter."
I Hen. VI. i. 6. 4.
So probably
" Than thése | two cré | . | Whích is Sebástian?"
T. N. v. 1. 231.
"But hé's | a tríed | and vál | iant sóld | iér."—J. C. iv. 1. 28.
"Your sís | ter ís | the bét | ter sól | diér."—Lear, iv. 5. 3.
"Máking | them wóm | en óf | good cárr | iáge."
R. and J. i. 4. 94.
"Márri | age ís | a mát | ter óf | more wór | th."
1 Hen. VI. v. 5. 55, v. 1. 21.
" To wóo a máid | in wáy | of márr | iáge."
M. of V. ii. 9. 13.
" While I′ | thy ám | iá | ble chéeks | I do cóy."
M. N. D. iv. 1. 2.
"Young, vál | iánt, | wise, and, | no dóubt, | right róyal."
Rich. III. i. 1. 245; Tempest, iii. 2. 27.
"With th' án | ciént | of wár | on óur | procéedings."
Lear, v. 1. 32.
"You have dóne | our plé | asúres | much gráce, | fair ládies."—T. of A. i. 2. 151.
So
" Táke her | and úse | her át | your plé | asúre."
B. and F. (Walker).
" We'll léave | and thínk | it ís | her plé | asúre."—Ib.
" But 'tís | my lórd | th' Assist ant's plé | asúre."—Ib.
"He dáre | not sée | you. A′t | his plé | asúre."—Ib.
"Yóu shall | have ránsom. | Lét me | have súr | geóns."
Lear, iv. 6. 196.
" If ón | ly to gó | (484) wárm | were górg | eóus."
Ib. ii. 4. 271.
"Your mínd | is tóss | ing ón | the ó | ceán."
M. of V. i. 1. 8; Hen. V. iii. 1. 14.
"The néw | est státe. | Thís is | the sér | geánt."
Macbeth, i. 2. 3.
Similarly
"But théy | did sáy | their práy | ers ánd | addréss'd them."—Ib. ii. 2. 25; Coriol. v. 3. 105.
"Hath túrn'd | my féign | ed práy | er ón | my héad."
Rich. III. v. 1. 21, ii. 2. 14.
Even where "prayer" presents the appearance of a monosyllable, the second syllable was probably slightly sounded.
For i and e sonant in " -ied," see 474.
479 a. Monosyllabic feet in Chaucer. Mr. Skeat (Essay on Metres of Chaucer, Aldine Edition, 1866) has shown that Chaucer often uses a monosyllabic foot, but the instances that have been pointed out are restricted to the first foot.
"May, | with all thyn floures and thy greene."—C. T. 1512.
" Til | that deeth departe schal us twayne."—Ib. 1137.
" Ther | by aventure this Palamon."—Ib. 1518.
"Now | it schyneth, now it reyneth fast."—Ib. 1537.
"Al | by-smoterud with his haburgeon. "—Ib. 77.
It will be shown in paragraphs 480—6 that Shakespeare uses this licence more freely, but not without the restrictions of certain natural laws.
480. Fear, dear, fire, hour, your, four, and other monosyllables ending in r or re, preceded by a long vowel or diphthong, are frequently pronounced as dissyllables. Thus "fire" was often spelt and is still vulgarly pronounced "fier." So "fare" seems to have been pronounced "fa-er;" "ere," "e-er;" "there," "the-er," &c.
It is often emphasis, and the absence of emphasis, that cause this licence of prolongation to be adopted and rejected in the same line :
Fair.—
"Ferd. Or night | kept cháin'd | belów. | Prosp. Fáir | ly spóke."
Tempest, iv. 1. 31.
(or perhaps (484) "belów. | Fáir | ly spóke.")
Fare.—
"Póison'd, | ill fá | re, déad, | forsóok, | cast óff."
K. J. v. 7. 35.
"Lóath to | bid fá | rewéll, | we táke | our léaves."
P. of T. ii. 5. 13.
"Lúcius, | may gówn. | Fáre | well, góod | Méssala."
J. C. iv. 3. 231.
"Died év | ery dáy | she lív'd (Fol.). | Fáre | thee wéll."
Macbeth, iv. 3. 111.
"Fáre | well, kins | man ! I′ | will tálk | with yóu."
1 Hen. IV. i. 3. 234.
"For wórms, | brave Per | cy. Fá | rewéll (so Folio), | great héart."—Ib. v. 4. 87.
"Why thén | I wí | ll (483). Fá | rewéll, | old Gáunt."
Rich. II. i. 2. 44.
So J. C. iv. 3. 231; 1 Hen. IV. iv. 3. 111 (Folio); M. W. of W. iii. 4. 97; K. J. iii. 2. 17. (See 475.)
Ere.—
"For I′ | inténd | to háve | it ér | e (é-er) lóng."
1 Hen. VI. i. 3. 80.
I should prefer to prolong the emphatic here, rather than "our," in
"What shóuld | be spók | en hé | re (hé-er) whére | our fáte."
Macbeth, ii. 3. 128.
Mere.—The pause after "night" enables us to scan thus :
"They have tráv | ell'd áll | the night (484)· | 'Mé | re fétche ."—Lear, ii. 4. 90.
There.—
"Hath déath | lain wíth | thy wífe. | Thére | she lies."
R. and J. iv. 5. 36.
"Towards Cálais ; | now grant | him thé | re, thé | re seen."
Hen. V. v. Prol. 7.
(I have not found a Shakespearian instance of "Caláis." Otherwise at first sight it is natural to scan "Towárds | Caláis.")
"Exe. Like mú | sic.
Cant.
Thé | refóre | doth héav'n | divíde."
Hen. V. i. 2. 183.
Where.—
"I knów | a bánk, | whére | the wíld | thyme blóws."
M. N. D. ii. 1. 249.
"Hor. Whére, | my lórd? |
Ham. I'n may | mind's eýe, | Horátio."
Hamlet, i. 2. 185.
(But Folio inserts "Oh" before "where.")
Rarely.—
"I′s not | this búck | led wéll? Ráre | ly, rárely."
A. and C. iv. 4. 11.
(The first "rarely" is the more emphatic: or? (483), "well.")
Dear.—
"As dóne: | persév | eránce, | déar | my lórd."
Tr. and Cr. iii. 3. 150.
"Déar | my lórd, | íf you, in yóur | own próof."
M. Ado, iv. 1. 46.
"The kíng | would spéak | with Córnwall : | the dé | ar fáther."—Lear, ii. 4. 102.
"Oliv. Than mú | sic fróm | the sphé | res.
Viol.
Dé | ar lády."
T. N. iii. 1. 1-1.
Fear.—
"Féar | me nót, | withdráw, | I héar | him cóming."
Hamlet, iii. 4. 7.
Hear.—
"Hear, Ná | ture, hé | ar, dé | ar Gód | dess, hear."
Lear, i. 4. 297.
(The emphasis increases as the verse proceeds.)
Near.—
"Néar, | why thén | anóth | er tíme | I'll héar it."
T. of A. i. 2. 184.
Tears.—
"Auf. Name not | the Gód, | thou bóy | of té | ars. Coriol.
Há !"
Coriol. v. 6. 101.
"Téar | for téar, | and lóv | ing kiss | for kíss."
T. A. v. 3. 156.
Year.—
"Twelve yé | ar since, | Mirán | da, twélve | year sínce."
Tempest, i. 2. 53.
(The repeated "year" is less emphatic than the former.)
And, perhaps, if the line be pronounced deliberately,
"Mány | yéars | of háp | py days | befál."—Rich.II. i.1.21.
It might be possible to scan as follows :
"Well struck | in yé | ars, fá | ir ánd | not jéalous."
Rich. III. i. 1. 92.
But the Folio has "jealious," and the word is often thus written (Walker) and pronounced by Elizabethan authors.
Their (?).—If the text be correct, in.
"The commons hath he pill'd with grievous taxes,
And quite lóst | their héarts. | The nó | bles háth | he fin'd
For án | cient quarrels (463), | and quíte | lost thé | ir
hearts,"—Rich. II. ii. 1. 247-8.
it is almost necessary to suppose that the second their is more emphatic than the first. Else the repetition is intolerable. See 475, 476. But even with this scansion the harshness is so great as to render it probable that the text is corrupt.
Hire.—
" A shíp | you sént | me fór | to hí | re waftage."
C. of E. iv. 1. 95.
Sire.—
"And ís | not like | the sí | re: hón | ours thrive."
A. W. ii. 3. 142.
Door.—
" And wíth | my swórd | I'll kéep | this dó | or sáfe."
T. A. i. 1. 288.
More.—
"If móre, | the mó | re hást | thou wróng'd | (èd) mé."
Lear, v. 3. 168.
(The second "more" is the more emphatic.)
"As máy | compáct | it mó | re. Gét | you góne."
Ib. i. 4. 362.
" Who hádst | desérv | ed mó | re thán | a príson."
Temp. i. 2. 362.
Our (perhaps).—
"To líst | en óu | r púr | pose. This is (461) | thy office."—M. Ado, iii. 1. 12.
("This is" is a quasi-monosyllable. See 461.)
"And bý | me, hád | not óu | r háp | been bád."
C. of E. i. 1. 39.
"First Sen. Which wé | devíse | him,
Corn. O'u | r spóils | he kíck'd at."
Coriol. ii. 2. 128.
"First" requires emphasis in
"Sic. In óu | r fírst | way.
Men.
I' | 'll bríng | him tó you.
Ib. iii. 1. 334.
Hour (often).—
" A't the | sixth hóu | r, át | which tíme | my lórd."
Tempest, v. 1. 4.
Your.—
" And só, | though yóu | rs, nót | yours—próve | it só."
M. of V. iii. 2. 20.
"Lart. My hórse | to yóu | rs, nó! |
Mart. 'Tis dóne! |
Lart.
Agréed."
Coriol. i. 4. 2.
"And pún | ish thém | to yoú | r héight | of pléasure."
M. for M. v. i. 240.
Unless "pleasure" is a trisyllable. (See 479.)
"Is he párd | on'd ánd | for yóu | r lóve | ly sáke."—Ib. 496.
There is an emphatic antithesis in
"Whó is | lost tóo. | Take yóu | r pá | tience to you,
And I'll say nothing."—W. T. iii. 2. 232.
"And sháll | have yóu | r wíll, | becáuse | our king."
3 Hen. VI. iv. 1. 17.
481. Monosyllables which are emphatic either (1) from their meaning, as in the case of exclamations, or (2) from their use in antithetical sentences, or (3) which contain diphthongs, or (4) vowels preceding r, often take the place of a whole foot. This is less frequent in dissyllabic words. In (1) and (2) as well as (3) the monosyllables often contain diphthongs, or else long vowels.
In many cases it is difficult, perhaps impossible, to determine whether a monosyllable should be prolonged or not. Thus, in
"On thís | unwórth y scáff | old tó | bring fórth,"
Hen. V. Prologue, 10.
many may prefer to scan "| -old to brí | ng fórth," and to prolong the following monosyllable rather than to accent "to ;" and in
"Came póur | ing líke | the tíde | intó a a bréach,"
Hen. V. i. 2. 149.
it is possible to prolong the preceding monosyllable, "the tí | de in | to a bréach." Such cases may often be left to the taste of the reader (but for the accent of " into " see 457a). All that can safely be said is, that when a very unemphatic monosyllable, as "at," " and," "a," "the," &c. has the accent, it is generally preceded or followed by a very strongly accented monosyllable, as
"Assume the port of Mars; and at his heels."
Hen. V. Prologue, 6.
It is equally a matter of taste whether part of the prolonged monosyllable should be considered to run on into the following foot, or whether a pause be supposed after the monosyllable, as
" Gírding | with gríev | ous síege | cástles | and tówns."
Hen. V. i. 2. 152.
" As knóts | bý the | conflúx | of méet | ing sáp."
Tr. and Cr. i. 3. 7.
482. Monosyllabic exclamations.
Ay.—
" Polon. Whérefore | should yóu | do thís? | Reg.
A'y, | my lórd?"
Hamlet, ii. 1. 36.
"King. Will you | be rúled | by mé? |
Laert.
A'y, | my lórd."
Ib. iv. 7. 60.
"A'y, | what élse? | And bút | I bé | decéiv'd."
T. of Sh. iv. 4. 2.
"Vol. That bróught | thee tó | this wórld. |
Vir.
A'y, | and mine."
Coriol. v. 3. 125.
" Corn. I's he | pursú | ed (474)?
Glou.
A' | y, mý | good lórd."
Lear, ii. 1. 111.
Nay.—
" What sáys | he? Ná | y, nó | thing; áll | is said."
Rich. II. ii. 1. 148.
"Cor. How, trái | tor!
Com.
Ná | y, tém | p(e)ratelý ; | your prómise."
Coriol, iii. 3. 67.
Stay.—
" Stáy, | the kíng | hath thrówn | his wárd | er dówn."
Ib. i. 3. 118.
Yea.—
" Yéa, | my lórd. | How bróoks | your gráce | the aír?"
Ib. iii. 2. 2.
Hail.—
" 'Gaínst mý | captív | itý. | Háil, | brave friend."
Macbeth, i. 2. 5.
O.—
"Cass. , | 'tis trúe. |
Hect
. Ho! bid | my trúm | pet sóund."
Tr. and Cr. v. 3. 13.
" Cleo. , | 'tis tréa | son.
Charm
. Mádam, | I trúst | not só."
A. and C. i. 5. 7.
"To híde the sláin. | ˙, from this | time fórth."
Hamlet, iv. 4. 65.
"Mir. ˙, | good sír, | I dó. |
Prosp. I práy | thee, márk me."
Tempest, i. 2. 80.
Perhaps
"Pol. The dévil | himsélf. |
King.
˙, | 'tis (it is) | too trúe."
Ib. iii. 1. 49.
" Self a | gainst sélf. | ˙, | prepós | teróus."
Rich. III. ii. 4. 63.
"Their cléa | rer réa | son. ˙, | góod | Gonzalo."
Temp. v. 1. 68.
I have not found "reason " a trisyllable in Shakespeare.
" ˙, | my fóllies! | Then E'd | gar wás | abúsed."
Lear, iii. 7. 91.
" ˙, | the diff | erénce | of mán | and mán."
Ib. iv. 2. 26.
? " The héart | of wó | man is.| , | (453) Brútus."
J. C. ii. 4. 40.
"Struck Cæ′ | sar ón | the néck. | ˙, | you flátterers."
Ib. v. 1. 44.
Soft.—
" But só | ft! com | paný | is cóm | ing hére."
T. of Sh. iv. 5. 26.
Come.—
" Cóme, | good féll | ow, pút | mine ir | on ón."
A. and C. iv. 4. 3.
What.—
" Whére be | these knáves? | Whát, | no mán | at dóor!"
T. of Sh. iv. 1. 125.
"Whát, | unjúst! | Bé not | so hót; | the dúke."
M. for M. v. 1. 315.
Well.—
" Wéll, | gíve her | that ring, | and thére | withál."
T. G. of V. iv. 4. 89.
"Gon. Rémem | her whát | I téll | you.
Osw.
Wé | ll, mádam."
Lear, i. 3. 21.
483. Monosyllables emphasized by position or antithesis. A conjunction like "yet" or "but," implying hesitation, may naturally require a pause immediately after it ; and this pause may excuse the absence of an unaccented syllable, additional stress being laid on the monosyllable.
But.—
"Of góod | ly thóus ands. Bú | t, fór | all this."
Macbeth, iv. 3. 44.
" The Góds | rebúke | me bú | t ít | is tídings."
A. and C. v. 1. 27.
Yet.—
"Thóugh I | condémn | not, yé | t, I, ún | der párdon."
Lear, i. 4. 365.
"Yét (as yet), | I thínk, | we áre | not bróught | so lów."
T. A. iii. 2. 76.
" Brut. When Cæ's | ar's héad | is óff. |
Cass. Yét | I féar him."
J. C. ii. 1. 183.
Pronouns emphasized by antithesis or otherwise, sometimes dispense with the unaccented syllable.
" Shów | men dú | tifúl ?
Why, só I didst thó | u. Séem | they gráve | and learned?
Why, só | didst thóu."—Hen. V. ii. 2. 128.
(Possibly, however, "seem" may be prolonged instead of "thou.")
" When yóu | shall pléase | to pláy | the thieves for wives.
I'll wátch as lóng | for yó | u thén. | Approach."
M. of V. ii. 6. 24.
" Were yó | u in | my stéad, would yóu | have héard?"
Coriol. v. 3. 192.
You is emphatic from Desdemona to Othello in Coriol. v. 3. 192.
" Othello. Tis a | good hand,
A fránk | one.
Desd. Yó | u máy | indéed | say só."
Othello, iii. 4. 44.
So in
" How in | my stréngth | you pléase. | For yó | u, E'dmund."
Lear, ii. 1. 114.
and in the retort of Brutus on Cassius,
" Lét me | tell yó | u, Cáss | ius, yóu | yoursélf
Are múch | condémn'd | to háve | an ítch | ing pálm."
J. C. iv. 3. 9.
Perhaps aware of Ferdinand's comment on his emotion, "your father's in some passion," Prospero turns to Ferdinand and says, " it is you who are moved" in
"Yo'u do lóok, | my són, | ín a | mov'd sórt."
Temp. iv. 1. 146.
Otherwise the reading of the line so as to avoid accenting "my" seems difficult.
There is no prolongation, though there is antithetical emphasis, in
"Lóok up | on hím, | love hím, | he wór | ships yóu."
A. Y. L. v. 2. 88.
The repeated "thence" seems to require a pause in
" Thénce to | a wátch, | thénce | intó (457a) |a wéakness."
Hamlet, ii. 1. 148.
But possibly, like "ord(i)nance," "light(e)ning" (see 477), so " weakness " may be pronounced a trisyllable.
484. Monosyllables containing diphthongs and long vowels, since they naturally allow the voice to rest upon them, are often so emphasized as to dispense with an unaccented syllable. When the monosyllables are imperatives of verbs, as " speak," or nouns used imperatively, like "peace," the pause which they require after them renders them peculiarly liable to be thus emphasized. Whether the word is dissyllabized, or merely requires a pause after it, cannot in all cases be determined. In the following examples the scansion is marked throughout on the former supposition, but it is not intended to be represented as necessary.
A (long).
"Júst as | you léft | them, á | ll prís | 'ners, sír."
Temp. v. 1. 8.
"Try mán | y, á | ll góod, | serve trú | ly néver."
Cymb. iv. 2. 373.
"Yea, lóok'st | thou pá | le ? Lét | me sée | the writing."
Rich. II. v. 2. 57.
" Duke. Like the | old á | ge.
Clown. A're | you réad | y, sír ?"
T. N. ii. 4. 50.
"Yéa, his | dread trí | dent sháke. | My brá | ve spirit."
Temp. i. 2. 206.
Ai.
" 'Gainst mý | captív | itý. | Háil, | brave friend."
Macbeth, i. 2. 5.
" I'll bé | with (wi') you strái | ght. Gó | a líttle | befóre."
Hamlet, iv. 4. 31.
I should prefer to avoid laying an accent on " the " in
"To fá | il in the | dispós | ing óf | these chances."
Coriol. iv. 7. 40.
"Which ís | most fá | int. Nów | 'tis true
I múst | be hére | confín'd | by yóu."—Temp. Epilogue, 3.
Ay.
"Sáy | agáin, | whére didst | thou léave | these varlets?"
Temp. iv. 1. 170.
So in the dissyllable " payment."
" He húmb | ly práys | you speed | y páy | mént."
Perhaps
T. of A. ii. 2. 28.
Perhaps
" What sá | y yóu, | my lórd ? | Aré you | contént."
1 Hen. VI. iv. 1. 70.
Perhaps
E.
"Senators. Wé | 'll súre | ty him.
Com.
A′g | ed sír, | hands óff."
Coriol. iii. 1. 178.
" Men. The cón | sul Córi | olán | us——
Bru.
Hé | cónsul ! "—Ib. iii. 1. 280.
Ea.
"Péace, | I sáy. | Good é | ven tó | you, friend."
A. Y. L. ii. 3. 70.
" Antón ius dé | ad! I'f | thou sáy | so, villain."
A. and C. ii. 5. 26.
"Doct. But, thóugh | slow, dé | adlý. |
Queen.
I wón | der, dóctor."
Cymb. i. 5. 10.
"Whý dost | not spéak ? | What, dé | af: nót | a wórd?"
T. A. v. 1. 46.
"Spéak, | Lavín | ia, what | accúrs | ed hánd?"
Ib, iii. 1. 66.
" Which wás | to plé | ase. Nów | I wánt
Spirits to | enfórce, | nót to | enchánt."
Temp. Epilogue, 13.
" Eárth's in | créase, | fóison | plénty,
Barns and | gárners | néver | émpty."—Ib. iv. 1. 110.
Perhaps
" Glou. Aláck, | the night | comes ón, | and the (457) blé | ak wínds."—Lear, ii. 4. 303.
Perhaps
" Truly | to spé | ak, ánd | with nó | addítion,"
Hamlet, iv. 4. 17.
or
"Trúly | to spéak, | and with nó | addít | ión."
" Be frée | and hé | althfúl. | So tárt | a fávour."
A. and C. ii. 5. 38.
" The safety and health of this whole state,"
Hamlet, i. 3. 21.
could not be scanned without prolonging both "health" and "whole." Such a double prolongation is extremely improbable, considering the moderate emphasis required. More probably "sanity" should be read, as has been suggested, for "sanctity," the reading of the Folio.
Ee.
"Fórward, | not per | manént, | swéet, | not lásting."
Hamlet, i. 3. 8.
" Séek | me óut, | and thát | way I′ | am wife in."
Hen. VIII iii. 1. 39.
" The cúrt | ain'd slé | ep wítch | craft cél | ebrátes."
Macbeth, ii. 1. 51.
" Doth cóm | fort thee in | thy slé | ep ; líve, | and flóurish."
Rich. III. v. 3. 130.
" This íg | norant prés | ent ánd | I fé | el nów."
Ib. i. 5. 58.
" Enóugh | to fétch | him in. | Sée | it dóne."
A. and C. iv. 1. 14.
"Yét but | thrée. | Cóme one | móre,
Two of | bóth kinds | máke up fóur."
M. N. D. iii. 2. 437.
"When sté | el gróws | sóft as | the pára | site's silk."
Coriol. i. 9. 45.
"Soft" is emphasized as an exclamation (see 481), but perhaps on the whole it is better to emphasize " steel " here.
" Ferd. Makes thís | place Pár | adíse.
" Prosp.
Swéet | now, sílence."
Temp. iv. 1. 124.
Eo. The eo in the foreign-derived word "leopard" stands on a different footing:
" Or hórse | or óx | en fróm | the lé | opárd."
1 Hen. VI. i. 5. 31.
So, often, in Elizabethan authors.
I.
"Mén for | their wí | ves : wí | ves fór | their húsbands."
3 Hen. VI. v. 6. 41.
"Of gréat | est just | ice. Wrí | te, wríte, | Rináldo."
A. W. iii.. 4. 29.
"Hórri | ble sí | ght ! Nów | I sée | 'tis trúe."
Macbeth, iv. 1. 122.
" Full fíf | teen húndred, | besí | des cóm | mon mén."
Hen. V. iv. 8. 84.
I know of no instance where "hundred," like (477) "Henry," receives two accents. Else the "be-" in "besides" might (460) be dropped, and the verse might be differently scanned.
"Each mán's | like mí | ne: yóu | have shéwn | all Héctors."
A. and C. iv. 8. 7.
"At a póor | man's hóuse: | he ús'd | me kí | ndlý."
Coriol. i. 9. 83. But see 477.
Ie.
Possibly "friends" may require to be emphasized, as its position is certainly emphatic, in
"Till déath | unlóads | thee. Frí | ends hást | thou nóne."
M. for M. iii. 1. 28.
"No, sáy'st | me so, | fríend ? | what cóun | trymán ?"
T. of Sh. i. 2. 190.
"Yield, my lord, | protéct | or yí | eld, Wín | chestér."
1 Hen. VI. iii. 1. 112.
("My" is dropped, 497.)
"Mórt de | ma ví | e ! I′f | they ríde | alóng."
Hen. V. iii. 5. 11.
0.
"Dríve him | to Ró | me: 'tis (it | is) tíme | we twaín."
A. and C. i. 4. 73.
" Card. Róme | shall réme | dy thís. |
Glou.
Roam thí | ther, thén."
1 Hen. VI. iii. 1. 51.
"While hé | himsélf | kéeps in | the có | ld fíeld."
3 Hen. VI. iv. 3. 14.
"Tóad that | únder | cóld | stóne
Days and | nights has | thírty | óne."—Macbeth, iv. 1. 6.
So scan
" Go to the | creáting | a whó | le tribe | of fóps."
Lear, i. 2. 14.
Oa.
"Is gó | ads, thó | rns (485), nét | tles, táils | of wasps."
W. T. i. 2. 329.
Oi.
" Jóint | by jóint, | but wé | will knów | his púrpose."
M. for M. v. 1. 314.
"What whéels, | racks, fíres? | What flay | ing, bó | ilíng?"
W. T. iii. 2. 177.
"God save | you, sir. | Where have yóu | been bró | ilíng ?"
Hen. VIII. iv. 1. 56.
"Of their | own chó | ice : óne | is Jún | ius Brútus."
Coriol. i. 1. 220.
" What sáy | you, bó | ys ? Will | you bide | with hím ?"
T. A. v. 2. 13.
Oo.
"Than in | my thóught | it lies. | Góod | my lórd."
A. W. v. 3. 184.
It might be thought that in the above the prolongation rests on lies (lieth), but that we have also
" Góod | my lórd, | gíve me | thy fáv | our stíll."
Temp. iv. 1. 204.
"The gó | od góds | will móck | me prés | entlý."
A. and C. iii. 4. 15.
" He stráight | declín | ed, dró | op'd, tóok | it déeply."
W. T. ii. 3. 14.
"Tó it, | boy ! Már | cus, ló | ose whén | I bíd."
T. A. iv. 3. 58.
"Hours, mín | utes, nó | on, mid | night, ánd | all eýes."
W. T. i. 2. 290.
" But ró | om, fái | ry, here | comes O'b | erón."
M. N. D. ii. 1. 58.
"Bóot | less hóme | and wéath | er-béat | en báck."
1 Hen. IV. iii. 1. 67.
" Pull óff | my bó | ot: hárd | er, hárd | er, só."
Lear, iv. 6. 177.
"But mó | ody | and dú | ll mél | anchóly."
C. of E. v. 1. 79.
Some may prefer to read "dull" as a monosyllable; but I can find no instance of "meláncholý" to justify such a scansion.
In
" Lear. To thís | detést | ed gró | om. Gon.
A't | your choice, sir,'
Lear, ii. 4. 220.
either "groom " or " your " should be dissyllabized.
" I' do | wánder | évery | whére
Swifter | thán the | móon's | sphére."—M. N. D. ii. 1. 7.
Ou.
"Which élse | would frée | have wró | ught. A'll | is wéll."
Macbeth, ii. 1. 19.
In
" Should drink | his blóod—| móunts | up tó | the áir."
MARLOW, Edw. II.
Collier (Hist. of British Stage, vol. iii.) thinks " mounts" the emphatic word to be dwelt on for the length of a dissyllable.
Ow.
"Own" is perhaps emphasized by repetition (or "Are" is a dissyllable, as " fare," " ere," " where," 480) in
" Hel. Mine ówn | and not | mine ó | wn.
Dem.
A're | you súre?"
M. N. D. iv. 1. 189.
Oy.
The last syllable of " destroy " seems prolonged in "To fright | them ére | destró | y. But | come ín."
Coriol. iv. 5. 149.
U.
It may be that "fume" is emphasized in "She's tíck | led nów. | Her fú | me néeds | no spúrs."
2 Hen. VI. i. 3. 153.
(Unless "needs" is prolonged either by reason of the double vowel or because "needs" is to be pronounced "needeth.")
" Trúe | nobíl | ity is | exémpt | from féar."
2 Hen. VI. iv. 1. 129.
Titania speaks in verse throughout, and therefore either "and" must be accented and "hoard" prolonged, or we must scan as follows :
" The squír | rel's hóard, | and fétch | thee néw | ' núts."
M. N. D. iv. 1. 40.
" Cord. That wánts | the méans | to lead it. |
Mess. Néws, | mádam."
Lear, iv. 4. 20.
485. Monosyllables containing a vowel followed by " r " are often prolonged.
A.
" Thyr. Hear it | apár | t. Cleo. Nóne | but fríends : | say bóldly."
A. and C. iii. 13. 47.
"Hó | ly séems | the quárrel
Upón | his grá | ce's pá | rt ; bláck | and fearful
O'n the | oppó | ser."—A. W. iii. 1. 5.
" Well fítt(ed) | in á |rts, gló | | rióus | in árms."
L. L. L. ii. 1. 45.
"Strikes his | breast há | rd, ánd | anón | he cásts."
Hen. VIII. iii. 2. 117.
" But cóuld | be wílling | to má |rch ón | to Cálais."
Hen. V. iii. 6. 150.
" Hárk | ye, lórds, | ye sée | I have gíven | her phýsic."
T. A. iv. 2. 162.
"Lóok how | he mákes | to Cæ's | ar, már | k him."
J. C. iii. 2. 18.
Ei.
" I dréamt | last night | óf the | three wé | ird sisters."
Macbeth, ii. 1. 20 (Folio, "weyard").
" A'nd be | times I' | will tó | the wé | ird sisters."
Ib. iii. 4. 133, iv. 1. 136.
Or " will " is perhaps emphasized and the prefix in " betimes " ignored. In either case " weird" is a dissyllable.
" The wé | ird sís | ters hánd |in hánd."—Macbeth, i. 3. 32.
I.
" A thí | rd thinks | withóut | expénse | at áll."
1 Hen. VI. i. 1. 76.
"Of Líon | el dúke | of Clárence, | the thí | rd són."
Ib. ii. 5. 75.
"To kíng | Edwárd | the thí | rd, whére | as hé."—Ib. 76.
0.
"Bru. Spread fúr | thér (478). Men.
One wó | rd móre, | one wórd."
Coriol. iii. 1. 311.
" Máke the | prize líght. | One wór | d móre, | I chárge thee."—Temp. i. 2. 452.
"Ham. One wór | d móre, | good lády. |
Queen.
Whát shall | I dó ?"
Hamlet, iii. 4. 180.
" Do móre | than thís | in spó | rt; fá | ther, fáther ! "
Lear, ii. 1. 37.
" Wórse | and wórse ! | She wíll | not cóme ! | O, víle !"
T. of Sh. v. 2. 93.
"Nót in | the wó | rst ránk | of mán | hood, sáy't."
Macbeth, iii. 1. 103.
"Why só, | brave ló | rds, whén | we joín | in léague."
T. A. iv. 2. 136.
"My ló | rd, wíll | it pléase | you páss | alóng."
Rich. III. iii. 1. 110.
" Of góod old A′ | brahám. | Lórds | appéllants."
Rich. II. iv. 1. 104.
(" A′ppellants " is not Shakespearian.)
"But téll | me, ís | young Geór | ge Stán | ley líving?"
Ib. v. 5. 9.
or, possibly,
" But téll me, |
Is yóung | George Stán | ley líving?"
Ou.
"Henry doth claim the crown from John of Gaunt, The fóu | rth són : | York cláims | it fróm | the thírd."
2 Hen. VI. ii. 2. 55.
So, perhaps,
"And lóng | live Hén | ry fóu | rth óf | that náme."
Rich. II. iv. 1. 112.
("Four" was often spelt "fower." "Henry" is not pronounced " Hén(e)rý" in Richard II.)
"Heart," not "you," ought to be emphatic in
" Nót by | the mát | ter whích | your héar | t prómpts you."
Coriol. iii. 2. 54.
Probably we ought to arrange the difficult line, Macbeth, iv. 1. 105, thus :
"A′nd an | etérn | al cú | rse fáll | on yóu.
Let me know.
Why sínks," &c. ?
486. Monosyllables are rarely prolonged except as in the above instances. In some cases, however, as in "bath," "dance," a vowel varies very much in its pronunciation, and is often pronounced (though the incorrectness of the pronunciation would now be generally recognized) in such a way as to give a quasi-dissyllabic sound.
" Yóu and | your crá | fts, yóu | have cráft | ed fáir."
Coriol. iv. 6. 118.
" I′f that | yóu will | Fránce | wín,
Then witch | Scótland | fírst be | gín."—Hen. V. i. 2. 167.
In a few other cases monosyllables are, perhaps, prolonged :
" You sháll | read ús | the wí | ll. Cæ's | ar's wíll ! "
J. C. iii. 2. 153.
" Cas. Cícer | o ón | e ?
Mes. Cíc | eró | is déad."—Ib. iv. 3. 179.
"I′ will | éver | bé your | héad,
So be | góne; | yóu are | spéd."—M. of V. ii. 9. 72.
"Then sháll | the réalm | of A′lb | ión
Cóme | to gréat | confús | ión."—Lear, iii. 2. 92.
"For óur | best áct. | I′f we | shall stá | nd still."
Hen. VIII. i. 2. 85.
(Can "all" have dropped out after " shall?")
"The thánk | ings óf | a kí | ng. I′ | am, sír."
Cymb. v. 5. 407.
" Hére she | cómes, | cúrst and | sád : Cymb. v. 5. 407.
Cúpid | ís a | knávish | lád."—M. N. D. iii. 2. 439.
"Well" (481) is prolonged as an exclamation, and perhaps there is a prolongation of the same sound in
" Mélt | ed ás | the snów | séems to | me nów."
M. N. D. iv. 1. 163.
So, in
" The gó | ds, nót | the patríc | ians, máke | it, ánd,"
Coriol. i. 1. 75.
" gods " is probably prolonged by emphasis, and the second " the " is not accented. So " most " in
" With Tí | tus Lárcius, | a mó | st vál | iant Róman."
Coriol. i. 2. 14.
" Larcius" has probably but one accent. However, "a" appears sometimes to have the accent.
So, perhaps,
" Ang. Where práy | ers cró | ss.
Isab.
A't | what hóur | to-mórrow?"
M. for M. ii. 2. 159.
" Drachm " (Folio " Drachme ") is a dissyllable in
" A′t a | crack'd drách | m ! Cúsh | ions, léad | en spóons."
Coriol. i. 5. 6.
487. E mute pronounced. This is a trace of the Early English pronunciation.
Es, s.
" Your gráce | misták | es: ón | ly tó | be bríef."
Rich. II. iii. 3. 9.
"Who's thére, | that knóck | (e)s só | impér | iouslý ?"
1 Hen. VI. i. 3. 5.
"Well, lét | them rést : | come híth | er, Cát | esbý."
Rich. III. iii. 1. 157.
" Here cómes | his sérv | ant. Hów | now, Cát | esbý ?"
Ib. 7. 58.
"Till áll | thy bónes | with ách | es máke | thee róar."
Temp. i. 2. 370.
" A′ches | contráct, | and stárve | your súp | ple jóints."
T. of A. i. 1. 257, v. 1. 202.
But this word seems to have been pronounced, when a noun, " aatch." At least it is made by Spenser, Sh. Cal. Aug. 4, to rhyme with "matche."
"Send Có | levíle | with hís | conféd | erátes."
2 Hen. IV. iv. 3. 79.
So
" Wórces | ter, gét | thee góne ! | For I' | do sée." 1 Hen. IV. i. 3. 15, iii. 1. 5, v. 5. 14 (Fol. omits "thee").
" We háve ; | whereupón (497) | the éarl | of Wórc | estér."
Rich. II. ii. 2. 58.
So
" Glóucestér," 1 Hen. VI. i. 3. 4, 6, 62, and " O lóv | ing uncle (465), | kind dúke | of Glóu | cestér."
1 Hen. VI. iii. 1. 142.
" This is the flower that smiles on every one
To shów | his téeth | as whíte | as whá | le's bóne."
L. L. L. v. 2. 332.
So, in a rhyming passage,
" Whose shád | ow thé | dismíss | ed báche | lor loves
Being | lass-lórn ; | thy póle | -clipt vín | e-yárd
And thý | sea-márge, | stérile | and róck | y-hárd."
Temp. iv. 1. 69.
"She név | er hád | so swéet | a cháng | elíng."
M. N. D. ii. 1. 23.
Perhaps
"Fran. They ván | ish'd stráng | ely. Seb.
No mát | ter, since."
Temp. iii. 3. 40. But see 506.
Possibly "cradles" may approximate to a trisyllable, "crad(e)les " (so "jugg(e)ler," &c. 477), in
"Does thóughts | unvéil | in théir | dumb crá | dlés."
Tr. and Cr. iii. 3. 200.
The e is probably not of French but of Latin origin in " statue:"
"She dréamt | to-níght | she sáw | my stát | ué."
J. C. ii. 2. 76.
" E′ven at | the báse | of Póm | pey's stát | ué."
(Folio) Ib. iii. 2. 192.
Globe " statua."
So in the plural :
"But líke | dumb stát | ués | of bréath | ing stónes."
Rich. III. iii. 7. 25.
Globe, "statuas."
" No marble statua nor high
Aspiring pyramid be raised."—HABINGTON (Walker).
488. The "e " in commandment, entertainment, &c., which originally preceded the final syllable, is sometimes retained, and, even where not retained, sometimes pronounced.
" Be vál | ued 'gainst | your wífe's | commánd | (e)mént."
M. of V. iv. 1. 451.
"From hím | I háve | expréss | commánd | (e)mént."
The e is inserted in
1 Hen. VI. i. 3. 20.
" If to women he be bent
They have at commandement."—P. P. 418.
" Good sír, | you'll gíve | them én | tertáin | (e)mént."
B. J. Fox, iii. 2.
Perhaps an e is to be sounded between d and v in
" A′nton | y Wóod | (e)vílle, | her bróth | er thére."
Rich. III. i. 1. 67.
489. E final in French names is often retained in sound as well as spelling :
"The mél | anchól | y Jáq | ues gríeves | at thát."
A. Y. L. ii. 1. 26.
"O m | Paróll | es, théy | have márr | ied mé."
A. W. ii. 3. 289.
"His gráce | is át | Marséill | es, tó | which pláce."
Ib. iv. 3. 9 ; T. of Sh. ii. 1. 377.
"Dáughter | to Chár | lemáin, | who wás | the són."
Hen. V. i. 2. 75.
" Guiénne, | Champág | ne, Rhé | ims, O′r | leáns."
1 Hen. VI. i. 1. 60.
"This prince | Montáig | ne, íf |he bé | no móre."
"He cán | not sáy | but thát | Montáig | ne yét."
DANIEL (on Florio).
"Now E'sp | eránc | e, Pér | cy, ánd | set ón."
1 Hen. IV. v. 2. 97.
"Cáll'd then | brave lórd | Pónton | de Sáu | traillés."
1 Hen. VI. i. 4. 28.
"Díeu de | battái | lles! Whére | have théy | this méttle?"
Hen. V. iii. 5. 15.
So in " Vive :"
" ' Vive | le roí,' | as I′ | have bank'd | their tówns."
K. J. v. 2. 104.
Thus, perhaps, we may explain the apparent trisyllabic "marshal" by a reference to "mareschal :"
"Great már (e)shál | to Hén | (e)rý (477) | the Síxth."
1 Hen. VI. iv. 7. 70.
"With wíng | ed háste | tó then | lord már | (e)shál."
1 Hen. IV. iv. 4. 2.
On the other hand, the influence of the r (see 463) seems to make "marshall" a quasi-monosyllable in
"Lord márshal, | commánd | our óff | icérs | at árms."
Rich. II. i. 1. 204.
The i in the French " capitaine" is invisibly active in
"A wíse | stout cáp | (i)táin, | and sóon | persuáded."
3 Hen. VI. iv. 7. 30 ; Macbeth, i. 2. 34.
# ACCENT.
490. Words in which the accent is nearer the end than with us.
Many words, such as "edict," "outrage," "contract," &c., are accented in a varying manner. The key to this inconsistency is, perhaps, to be found in Ben Jonson's remark that all dissyllabic nouns, if they be simple, are accented on the first. Hence " edict" and "outrage" would generally be accented on the first, but, when they were regarded as derived from verbs, they would be accented on the second. And so, perhaps, when "exile" is regarded as a person, and therefore a "simple" noun, the accent is on the first; but when as "the state of being exiled," it is on the last. But naturally, where the difference is so slight, much variety may be expected. Ben Jonson adds that "all verbs coming from the Latin, either of the supine or otherwise, hold the accent as it is found in the first person present of those Latin verbs ; as from célebro, célebrate," Without entering into the details of this rule, it seems probable that "edict," "precépt," betray Latin influence. The same fluctuation between the English and French accent is found in CHAUCER (Prof. Child, quoted by Ellis, E. E. Pronunc. i. 369), who uses "batáille," C. T. 990, and "bátail," ib. 2099 : "For-túne," ib. 917, and "fórtune," ib. 927 ; " daungér, " and " dáunger. "
Abjéct (Latin).—
" Wé are | the quéen's | abjécts, | and múst | obéy."
Rich. III. i. 1. 106.
But if the monosyllable "queen" be emphasized, we may scan
" Wé are | the qué | en's ábjects, | and múst | obéy."
Accéss (Latin).—W. T. v. 1. 87.
Aspéct (Latin).—A. and C. i. 5. 33 ; T. N. i. 4. 28.
Charácters.—
"I sáy | withóut | charác | ters fáme | lives lóng."
Rich. III. iii. 1. 81 ; Hamlet, i. 3. 59.
Comméndable.
"Thanks fáith, | for sílence | is ónly | comménd | ablé
In a néat's | tongue dried | and a máid | not vend | iblé."
M. of V. i. 1. 111.
This shows how we must scan
"'Tis sweet and (497) | comménd | able ín | your ná | ture, Hámlet."—Hamlet, i. 2. 87.
But, on the other hand,
"And pówer, | untó | itsélf | most cóm | mendáble."
Coriol. iv. 7. 51.
Commérce (Latin).—So arrange
"Péaceful | commérce | from dí | vidá- | ble shóres."
Tr. and Cr. i. 3. 105.
Confiscate (Latin).—C. of E. i. 1. 21; but "cónfiscáte," ib. i. 2. 2.
Consórt (Latin).—
"What sáy'st | thou? Wílt | thou bé | of óur | consórt ? "—T. G. of V. iv. 1. 64.
" Edmund. of thát | Yes, madam,
He wás | of thát | consórt.
Reg.
No már | vel, thén."
Lear, ii. 1. 99.
Contráry (Latin).—
" Our wílls | and fátes | do só | contrá | ry rún."
Hamlet, iii. 2. 221.
Contráct (Latin).
"Márk our | contráct. | Márk your | divórce, | young sír."
W. T. iv. 4. 428 ; A. W. ii. 3. 185 ; 1 Hen. VI. iii.
1. 143, v. 4. 156 ; Rich. III. iii. 7. 5, 6 ; Temp.
ii. 1. 151.
Compáct (Latin, noun).—Rich. III. ii. 2. 133 ; J. C. iii. 1. 215.
Différent (Latin).—
" And múch | différ | ent fróm | the mán | he wás."—C. of E. v. 1. 46.
Here, however, by emphasizing the monosyllable "much," the word " different" may be pronounced in the usual way.
Edict (Latin).—2 Hen. VI. iii. 2. 258, and
"It stánds | as án | edíct | in dés | tiný."
M. N. D. i. 1. 151.
Effígies (Latin unaltered).
" And ás | mine éye | doth hís | effí | gies witness."
A. Y. L. ii. 7. 193.
Envý (verb ; noun, énvy).
"I's it | for hím | you dó | envy | me só?"—T. of Sh. ii. 1. 18.
Execútors.—Hen. V. i. 2. 203 is not an instance, for it means
"executioners." In its legal sense, Ib. iv. 2. 51, it is accented as with us.
Exíle (Latin).—R. and J. v. 3. 211 (frequent).
Instinct (noun, Latin).
"Háth, by | instínct, | knówledge | from óth | ers' éyes."
2 Hen. IV. i. 1. 86.
" Bý a | divíne | instínct | men's minds | mistrúst."
Rich. III. ii. 3. 42 ; Coriol. v. 3. 35.
Intó.—See 457a.
Miséry.—Some commentators lay the accent on the penultimate in
"Of súch | misér | y dóth | she cút | me óff,"
M. of V. iv. 1. 272.
but much more probably "a" has dropped out after "such."
The passage
"And búss | thee ás | thy wife. | Míser | y's lóve,"
K. J. iii. 4. 35.
proves nothing. The pause-accent is sufficient to justify "mísery." Nothíng.—See Something, below.
Obdúrate (Latin).—3 Hen. VI. i. 4. 142 ; M. of V. iv. 1. 8 ; T. A. ii. 3. 160 ; R. of L. 429.
"A′rt thou | obdú | rate, flín | ty, hárd | as stéel ?"
V. and A. 198.
Oppórtune (Latin)—
" And móst | oppórt | une tó | our néed | I háve."—W. T. iv. 4. 511.
"The móst | oppórt | une pláce, | the stróng'st | suggéstion."
Temp. iv. 1. 26.
Outráge.—1 Hen. VI. iv. 1. 126.
Perémptory (perhaps).
"Yea, mís | tress, áre | you só | perémp | tóry ?"
P. of T. ii. 5. 73.
This accentuation is not found elsewhere in Shakespeare : but the author of Pericles of Tyre may have used it. It is possible, however, to scan
"Yea, mís | t(e)réss (477), | are you | so pé | rempt(o)rý ?"
Porténts.—
" Thése are | porténts : | but yét | I hópe, | I hópe."
Othello, v. 2. 45.
So 1 Hen. IV. ii. 3. 65 ; Tr. and Cr. i. 3. 96.
Hence "fear" is not a dissyllable in
"A pród | igý | of féar, | ánd a | portént."
1 Hen. IV. v. 1. 20.
If " and "is correct, we must probably scan as follows :
" And thése | doth she applý | for wárn | ings ánd | porténts."
J. C. ii. 2. 80.
Precépts (Latin).—Hen. V. iii. 3. 26; but "précepts," Hamlet, ii. 2. 142.
Prescience retains the accent of science, indicating that the word was not familiar enough as yet to be regarded as other than a compound :
" Forestáll | prescí | ence ánd | estéem | no áct."
Tr. and Cr. i. 3. 199.
Recórd (noun, Latin).—
Rich. III. iii. 1. 72, iv. 4. 28 ; T. N. v. 1. 253.
Sepúlchre (Latin).—
" Bánish'd | this fráil | sepúl | chre óf | our flésh."—Rich. II. i. 3. 194.
" Or, át | the léast, | in hérs | sepúl | chre thine."
T. G. of V. iv. 2. 118.
" May líke | wise bé | sepúl | chred ín | thy sháde." R. of L. 805 ; and, perhaps, Lear, ii. 4. 134.
Sinister (Latin).—
"'Tis nó | sinís | ter nór | no áwk | ward cláim."
Hen. V. ii. 4. 85.
So, but comically, in
" And thís | the crán | ny ís, | ríght and | siníster,
Through whích | the féar | ful lóv | ers áre | to whísper."
M. N. D. v. 1. 164.
Sojóurn'd (perhaps) in
" My héart | to hér | but ás | guest- wíse | sojóurn'd."
Ib. iii. 2. 171,
But (?) emphasize "her," and scan
" My héart | to hér | ' bút | as gúest - | wise sójourn'd."
Somethíng (sometimes perhaps).
"My ínward | sóul At nó | thing trémb | les : át | somethíng | it gríeves."
Rich. II. ii. 2. 12.
Compare perhaps
" And I′ | nothíng | to báck | my súit | at áll."
Rich. III. i. 1. 236.
But, if "I" be emphasized, "nothing" may be pronounced as usual.
" I féar | nothing | what máy | be sáid | agáinst me."
Hen, VIII. i. 2. 212.
But " fear " may be a dissyllable, 480.
Sweethéart.—
Hen. VIII. i. 4. 94 : heart being regarded as a noun instead of the suffix -ard.
Triúmphing (Latin) sometimes.
" As 't wére | triúmph | ing át | mine én | emíes."
Rich. III. iii. 4. 91.
Untó.—See 457 a.
Welcóme.—
"Nor friends, | nor fóes, | to mé | welcóme | you áre."
Rich. II. ii. 3. 170.
This particular passage may be explained by a pause, but "welcóme" is common in other authors.
Wherefóre (in some cases), though it can often be taken as "thérefore," and explained by a preceding pause.
" O′ft have | you (óft | en háve | you thánks | therefóre)."
Tr. and Cr. iii. 3. 20.
" And wé | must yéarn | therefóre."—Hen. V. ii. 3. 6.
" Hate mé ! | Wherefóre? | O mé ! | what néws, | my lóve."
M. N. D. iii. 2. 272.
Perhaps
" Fór the | sound mán. | Déath on | my státe, | wherefóre? "
Lear, ii. 4. 113.
But better
"Death on my state : (512)
Whérefore | should hé | sit hére? | This áct | persuades
me."
491. -Ised, when ending polysyllables, generally has now a certain emphasis. This is necessary, owing to the present broad pronunciation of i. Such polysyllables generally have now two accents, the principal accent coming first. But in Shakespeare's time it would seem that the i approximated in some of these words to the French i, and, the -ed being pronounced, the i in -ised was unemphatic. Hence the Elizabethan accent of some of these words differs from the modern accent.
Advértised.—
" As I′ | by friends | am wéll | advért | iséd."
Rich. III. iv. 4. 501.
" Whereín | he might | the kíng | his lórd | advértise."
Hen. VIII. ii. 4. 178.
" I was | advért | ised théir | great gén | eral slépt."
Tr. and Cr. ii. 2. 111.
So M. for M. i. 1. 42.
Chástised.—
" And whén | this árm | of míne | hath chás | tiséd."
Rich. III. iv. 4. 331.
"This cáuse | of Róme, | and chás | tiséd | with arms."
T. A. i. 1. 32.
This explains:
Canónized.—
" Canón | izéd, | and wór | shipp'd ás | a sáint."
K. J. iii. 1. 177.
" Whý thy | canón | iz'd bónes, | héarsed | in déath."
Hamlet, i. 4. 47.
"Are brá | zen ím | age(s) [471] óf | canón | iz'd sáints."
2 Hen. VI. i. 3. 63.
Authórized.—
"Authór | iz'd bý | her grán | dam. Sháme | itsélf."
Macbeth, iii. 4. 66.
" Authór | izíng | thy trés | pass wíth | compáre."—Sonn. 35.
" His rúde | ness só | with hís | authór | iz'd yóuth."
L. C. 104.
So once :
Solémnised.—
"Of Já | ques Fál | conbrídge | solém | niséd."
L. L. L. ii. 1. 42.
But in M. of V. "sólemnised."
492. Words in which the accent was nearer the beginning than with us. Ben Jonson (p. 777) says all nouns, both dissyllabic (if they be "simple") and trisyllabic, are accented on the first syllable. Perhaps this accounts for the accent on cónfessor, &c. The accent on the first syllable was the proper noun accent ; the accent on the second (which in the particular instance of conféssor ultimately prevailed) was derived from the verb.
Archbishop.—
" The már | shal ánd | the árch | bishóp | are stróng."
2 Hen. IV. ii. 3. 42, 65.
Cément (noun).
"Your tém | ples búrn | ed ín | their cé | ment ánd."
Coriol. iv. 6. 85.
So the verb, A. and C. ii. 1. 48 ; iii. 2. 29.
Cómpell'd (when used as an adjective).
" This cóm | pell'd fór | tune, háve | your móuth | fill'd úp."
Hen. VIII. ii. 3. 87.
" I tálk | not of | your sóul : | our cóm | pell'd sins."
M. for M. ii. 4. 57.
Cómplete.—
"A máid | of gráce | and cóm | plete máj | estý."
L. L. L. i. 1. 137.
So Hamlet, i. 4. 52 ; Hen. VIII. i. 2. 118 ; Rich. III. iii. r. 189.
Cónceal'd.—
" My cón | ceal'd lá | dy tó | her cán | cell'd lóve."
R. and J. iii. 3. 98.
Cónduct.—
The verb follows the noun "safe-cónduct" in " Safe-cón | ductíng | the réb | els fróm | their ships."
Rich. III. iv. 4. 483.
But the noun is condúct in T. A. iv. 3. 65.
Cónfessor.—
Hen. VIII. i. 2. 149 ; R. and J. ii. 6. 21, iii. 3. 49.
" O′ne of | our có (sic) | vent ánd | his cón | fessór."
M. for M. iv. 3. 133.
Cóngeal'd—
"O′pen | their cón | geal'd móuths | and bléed | afrésh."—Rich. III. i. 2. 56.
Cónjure (in the sense of "entreat ").—T. G. of V. ii. 7. 2 ; frequent.
Cónsign'd.—
" With dís | tinct bréath, | and cón | sign'd kíss | es to them."—Tr. and Cr. iv. 4. 47.
See "dístinct" below.
Córrosive.—
"Cáre is | no cúre, | but rá | ther cór | rosíve."
I Hen. VI. iii. 3. 3 ; 2 Hen. VI. iii. 2. 403.
Délectable.—
"Máking the hárd | way sóft | and dé | lectáble. "
Rich. II. ii. 3. 7.
Détestable.—
" And I′ | will kíss | thy dé | testá | ble bónes."
K. J. iii. 4. 29 ; T. of A. iv. 1. 33.
Dístinct.—
"To offénd | and júdge | are dís | tinct óff | icés."
M. of V. ii. 9. 61.
See "cónsign'd" above.
Fórlorn.—
" Now fór | the hón | our óf | the fór | lorn Frénch."
1 Hen. VI. i. 2. 19.
Húmane.—" It ís | the húm | ane wáy, | the óth | er cóurse."
Coriol. iii. 1. 327.
Máintain.—
"That hére | you máin | tain sév | eral fác | tións."
1 Hen. VI. i. 1. 71.
Máture.—So apparently in
" Of múrder | ous léchers : | ánd in | the má | ture time."
Lear, iv. 6. 228.
This is like" náture," but I know no other instance of "máture."
Méthinks (sometimes).
" So yóur | sweet húe | which mé | thinks stíll | doth stánd."
Sonn. 104.
I cannot find a conclusive instance in Shakespeare, but this word is often (Walker) thus accented in Elizabethan writers.
Mútiners.—Coriol. i. 1. 492. See Píoners below.
Mýself (perhaps, but by no means certainly, in)
" I mý | self fíght | not ónce | in fór | ty yéar."
1 Hen. VI. i. 3. 91.
But certainly hímself, mýself, &c. are often found in Elizabethan authors, especially in Spenser :
" Mourns inwardly and makes to hímselfe mone."
SPENS. F. Q. ii. 1. 42.
The reason for this is that self, being an adjective and not a noun, is not entitled to, and had not yet invariably received, the emphasis which it has acquired in modern times.
And so, perhaps :
"And bánd | ing thém | selves ín | contrá (490) | ry párts."
1 Hen. VI. iii. 1. 81.
Nórthampton.—
"Last night | I héar | they láy | at Nórth- |
amptón."—Rich. III. ii. 4. 1.
O′bscure (adj.; as a verb, obscúre).
"To rib | her cére | cloth in | the ób | scure gráve."
M. of V. ii. 7. 51.
"His méans | of déath, | his ób | scure fú | nerál."
Hamlet, iv. 5. 213.
O′bservant.—
"Than twén | ty sill | y dúck | ing ób | servánts."
Lear, ii. 2. 109.
Perséver—
"Ay, dó, | persév | er, count | erféit | sad lóoks." M. N. D. iii. 2. 236 ; A. W. iii. 7. 31 ; K. J. ii. 1. 421 ;
Hamlet, i. 2. 92.
This is the Latin accent in accordance with Ben Jonson's rule.
"Bóunty, | persév | (e)rance, mér | cy, lów | linéss."
Macbeth, iv. 3. 93.
Pérspective.—A. W. v. 3. 48 ; Rich. II. ii. 2. 18.
The double accent seems to have been disliked by the Elizabethans. They wrote and pronounced "muleters" for "muleteers," "enginer" (Hamlet, iii. 4. 206) for "engineer," "pioners" for "pioneers." This explains :
Píoners.—
" A wórth | y píoner. | Once more | remóve, | good fríends."—Hamlet, i. 5. 162.
Plébeians (almost always).
" The pléb | eiáns | have gót | your fél | low-tribune."
Coriol. v. 4. 39 ; i. 9. 7, &c.
This explains
" Lét them | have cúsh | ions bý you. | You're pléb | eiáns."
Ib. iii. 1. 101.
Exceptions: Hen. V. v. Chorus, 27 ; T. A. i. 1. 231.
So " Epicurean " in Elizabethan authors and A. and C. ii. 1. 24. The Elizabethans generally did not accent the e in such words.
Púrsuit.—
" In púr | suit óf | the thing | she wóuld | have stáy."
Sonn. 143.
" We trí | fle tíme. | I prí | thee púr | sue séntence."
M. of V. iv. 1. 298.
Púrveyor.—
" To bé | his púr | veyór : | but hé | rides wéll."
Macbeth, i. 6. 22.
Quíntessence.—
" Téaching | áll that | réad to | knów The quínt | essénce | of év | ery spríte."—A. Y. L. iii. 2. 147.
Récordér (?).—
"To bé | spoke tó | but by | the ré | cordér."
Rich. III. iii. 7. 30.
So also Walker, who quotes from DONNE'S Satires, v. 248, Ed. 1633:
" Recorder to Destiny on earth, and she."
But this line might be scanned otherwise.
Rélapse.—
"Kílling | in ré | lapse óf | mortál | itý."
Hen. V. iv. 3. 107.
Rhéumatic.—
"O'erwórn, | despís | ed, rhéu | matíc, | and óld."
V. and A. 135 ; M. N. D. ii. 1. 105.
So
"These prág | matíc | young mén | at théir | own wéapons."
B. J.
Sécure.—" Upón | my sé | cure hóur | thy ún | cle stóle."
Hamlet, i. 5. 61 ; Othello, iv. 1. 72.
Séquester'd.—
"Whý are | you sé | questér'd | from áll | your tráin ?"
T. A. ii. 3. 75.
Súccessor (rare).
" For béing | not própp'd | by án | cestrý | whose grace
Chalks súcc | essórs | their wáy, | nor cáll'd | upón," &c.
Hen. VIII. i. 1. 60.
Súccessive (rare).—
" Are nów | to háve | no súcc | essíve | degrées."
M. for M. ii. 2. 98.
Tówards (sometimes).
"And sháll | contín | ue our grác | es tó | wards him."
Macbeth, i. 6. 30.
"I gó, | and tó | wards thrée | or fóur | o'clóck."
Rich. III. iii. 5. 101.
Compare
"Should, líke | a swáll | ow préy | ing tó | wards stórms."
B. J. Poetast. iv. 7.
"O' the plágue, | he's sáfe | from thínk | ing tó | ward Lóndon."
B. J. Alchemist, i. 1.
So, perhaps,
"I ám | infórmed | that hé | comes tó | wards Lóndon."
3 Hen. VI. iv. 4. 26.
"And tó | ward Lón | don théy | do bénd | their cóurse."
Rich. III. iv. 5. 14.
U'tensils (perhaps).
" He has brave útensils ; for so he calls them."
Temp. iii. 2. 104.
Wíthout.—See 457a,
The English tendency, as opposed to the Latin, is illustrated by the accentuation of the first syllable of "ígnominy," and its consequent contraction into "ígnomy" (1 Hen. IV. v. 4. 100, &c.).
# VERSES.
493. A proper Alexandrine with six accents, such as—
"And nów | by wínds | and wáves | my lífe | less límbs | are tóssed,"—DRYDEN.
is seldom found in Shakespeare.
494. Apparent Alexandrines. The following are Alexandrines only in appearance. The last foot contains, instead of one extra syllable, two extra syllables, one of which is slurred (see 467—9) :—
"The núm | bers óf | our hóst | and máke | discóvery (dis-cov' ry)."—Macbeth, v. 4. 6.
"He thínks | me nów | incáp | ablé; | conféderates."
Tempest, i. 2. 111.
" In vír | tue thán | in vén | geance : théy | being pénitent."
Ib. v. 1. 28.
" And móre | divérs | itý | of sóunds | all hórrible."—Ib. 235.
"In bítt | ernéss. | The cómm | on éx | ecútioner."
A. Y. L. iii. 5. 3.
"I sée | no móre | in yóu | than ín | the órdinary."—Ib. 42.
" Were rích | and hón | ouráble ; | besídes | the géntlemen."
T. G. of V. iii. 1. 64.
"Which sínce | have steád | ed múch ; | so, óf | his géntle-ness ."—Temp. i. 2. 165 ; Rich. III. v. 3. 245 ; Hen. V. ii. 2. 71.
For the contraction of "gentleman" to "gentl'man," or even "genman," see 461.
" Are yóu | not gríeved | that A'r | thur ís | his prísoner (468) ?"—K. J. iii. 4. 123.
" And I' | must frée | ly háve | the hálf | of ánything."
M. of V. iii. 2. 251.
"To másk | thy mónst | rous vísage. | Seek nóne | con-spíracy." —J. C. ii. 1. 81.
"Had hé | been vánq | u(i)sher, ás, | bý the | same cóvenant." —Hamlet, i. 1. 93.
"My lórd, | I cáme | to sée | your fá | ther's fúneral."
Ib. i. 2. 176.
"Untáint | ed, ún | exám | in'd, frée, | at líberty."
Rich. III. iii. 6. 9.
"And só | doth míne. | I múse | why shé's | at líberty."
Ib. i. 3. 305.
So, perhaps,
" From tóo | much lí | bertý, | my Lú | cio, líberty."
M. for M. i. 2. 129.
"A′bso | lute Mí | lan. Mé, | poor mán, | my líbrary."
Tempest, i. 2. 109.
" Shall sée | advánt | ageá | ble fór | our dígnity."
Hen. V. v. 2. 88.
unless "advántage | able fór | ."
495. Sometimes the two syllables are inserted at the end of the third or fourth foot—
"The flúx | of cómpany. | Anón | a cáre | less hérd."
A. Y. L. ii. 1. 52.
" To cáll | for récompense; | appéar | it tó | your mínd."
Tr. and Cr. iii. 3. 3.
"Is nót | so éstima | ble, pró | fitá | ble néither."
M. of V. i. 3. 167.
"O'erbéars | your ófficers ; | the ráb | ble cáll | him lórd."
Hamlet, iv. 5. 102.
"To mé | invéterate, | héarkens | my bróth | er's súit."
Temp. i. 2. 122.
" With áll | prerógative. | Hénce his ambít | ion grówing."
Ib. i. 2. 105.
"In báse | applíance(s) (471). | This óut | ward sáint | ed députy (468)."—M. for M. iii. 1. 89.
"Than wé | bring mén | to cómfort them ('em). | The fáult's | your ówn."—Tempest, ii. 1. 134-5.
496. In other cases the appearance of an Alexandrine arises from the non-observance of contractions—
" I dáre | abíde | no longer (454). | Whíther (466) should |
I flý?"—Macbeth, iv. 2. 73.
"She lé | vell'd át | our púr | pose(s) (471), ánd, | béing(470) roýal."—A. and C. v. 2. 339.
" All mórt | al cónse | quence(s) (471) háve | pronóunced | me thús."—Macbeth, v. 3. 5.
"As mís | ers dó | by beggars (454) ; | neither (466) gáve | to mé."—Tr. and Cr. iii. 3. 142.
497. Apparent Alexandrines. The following can be explained by the omission of unemphatic syllables :—
"Hor. Hail to | your lórdship. |
Ham.
I am (I'm) glád | to sée | you wéll."
Hamlet, i. 2. 160.
"Whereóf | he is the (he's th') héad ; | then íf | he sáys | he loves you."—Ib. i. 3. 24.
"Thou art swórn | as déeply | to (t') efféct | what wé | inténd."—Rich. III. iii. 1. 158.
"I had thóught, | my lórd, | to have léarn'd | his héalth | of yóu."—Rich. II. ii. 3. 24.
" That trace him | in his (in's) líne. | No bóast | ing like I a fóol."—Macbeth, iv. 1. 153.
"In séeming | to augmént | it wástes | it. Bé | advís'd."
Hen. VIII. i. 1. 145.
"When mír(a) | cles háve | by the gréat | est béen | deníed."
A. W. ii. 1. 144.
" Persuádes | me it is (t's) óth | erwíse ; | howe'ér | it bé."
Rich. III. ii. 2. 29.
"A wórth | y óff(i)cer | i' the wár, | but ín | solént."
Coriol. iv. 6. 30.
"I prómise | you I' am ('m) | afráid | to héar | you téll it."
Ib. i. 4. 65.
"Come, sís | ter, cóusin | I would ('ld) sáy, | pray pár | don mé."—Rich. II. ii. 2. 105.
"That máde | them do it ('t). | They are ('re) wíse | and hón | (ou)ráble."—J. C. iii. 2. 218.
"With áll | preróg(a)tive ; | hénce his | ambít | ion grówing." —Tempest, i. 2. 105.
"Mine éyes | even sóc | iablé | to the shów | of thine."
Ib. v. 1. 63.
"As gréat | to mé | as láte ; | and suppórt | ablé."
Temp. v. 1. 146.
unless "supportable" can be accented on the first.
"Ostentation" is perhaps for "ostention" (Walker), and "the" is "th'," in
"The ostentation of our love which, left unshown."
A. and C. iii. 6. 52.
"Is" ought probably to be omitted in
"With gól | den chéru | bims (is) frétted ; | her án | diróns."
Cymb. ii. 4. 88.
"So sáucy | with the hánd | of shé | here—whát's | her náme?"—A. and C. iii. 13. 98.
"Come Lám | mas éve | at níght | shall she bé | fourtéen."
R. and J. i. 3. 17.
"Of óffic(467) | er, (465) and óff | ice sét | all héarts | in the (i' th') state."—Tempest, i. 2. 84.
"Uncóup | le (465) in the (i' th') wést | ern váll | ey, lét | them
gó."—M. N. D. iv. 1. 112.
"Cóme to | one márk ; | as many | ways meet in | one tówn."—Hen. V. i. 2. 208.
" Verbátim | to rehéarse | the méth | od óf | my pén."
1 Hen. VI. iii. 1. 13.
The following is intended to be somewhat irregular :
"Now bý | mine hón | our, bý | my lífe, | by my tróth."
Rich. II. v. 2. 78.
We must probably scan as an ordinary line,
"That séeming | to be móst | which wé | indéed | least áre,"
T. of Sh. v. 2. 175.
since it rhymes with an ordinary line,
" Our stréngth | as weak, | our wéak | ness pást | compáre."
The following can be explained by the quasi-omission of unemphatic syllables :
" Awáy ! | though párt | ing bé | a dréad | ful córr(o)sive."
2 Hen. VI. iii. 2. 403.
"Córrosive," as in 1 Hen. VI. iii. 3. 3, is accented on the first, and here pronounced "corsive."
"Bút with | a knáve | of cómm | on hire, | a gónd(o)lier."
Othello, i. 1. 126.
"Our" is not a dissyllable, but "ag'd" is a monosyllable in
"But lóve, | dear lóve, | and óur | ag'd fá | ther's right."
Lear, iv. 4. 28.
So perhaps
"An ág'd | intér | pretér | though yóung | in yéars."
T. of A. v. 3. 6.
498. Alexandrines doubtful. There are several apparent Alexandrines, in which a shortening of a preposition would reduce the line to an ordinary line. "Upon," for instance, might lose its prefix, like " 'gainst" for "against."
" To lóok | upon my sóme | time más | ter's róy | al face."
Rich. II. ii. 5. 75.
" Forbíds | to dwell up | on ; yét | remém | ber this."
Rich. III. v. 3. 239.
" Upon óur | house('s) (471) thátch, | whíles a | more fróst | y péople."—Hen. V. iii. 5. 24.
"Upon the sís | terhóod, | the vó | tarists óf | St. Cláre."
M. for M. i. 4. 5.
" Brut. "Is líke | to láy upon us (on's). |
Cass. I'm glád | that mý | weak wórds."
J. C. i. 2. 176.
" Is góne | to práy | the hó | ly kíng | upon his (on's) áid."
Macbeth, iii. 6. 30.
So "to" (or "in," 457a) in "into" may be dropped in
" Fall ínto | the cóm | pass óf | a præ′ | muníre."
Hen. VIII. iii. 2. 340.
"The wátches | on únto | mine éyes | the óut | ward wátch."
Rich. II. v. 4. 52.
(?) " Ráther | a dítch | in E′gypt
Be géntle | grave únto | me. Ráther | on Ní | lus' múd."
A. and C. v. 2. 58.
"Gentle" is a quasi-monosyllable, see 465; "rather," see 466.
So Walker reads "to" for "unto" in
"Unto a póor, | but wórth | y gént | lemán. | She's wédded,"—Cymb. i. 1. 7.
and observes, "Unto and into have elsewhere, I think, taken the place of to."
Perhaps the second line of the rhyming couplet is purposely lengthened in
" I′ am | for the áir ; | this níght | I'll spend
Un'to | a dís | mal ánd | a fát | al énd."—Macb. iii. v. 21.
In " Better to leave undone, than by our deed
Acquire too high a fame when him we serve's away,"
A. and C. iii. 1. 15.
we might arrange
" Better léave | undóne, | than bý | our déed | acqúire."
Or the latter line might be (but there is not pause enough to make it probable) a trimeter couplet. (See 501.)
" At Má | rián | a's hóuse | to-níght. | Her cáuse | and yóurs,"
M. for M. iv. 3. 145.
must be an Alexandrine, unless in the middle of the line "Mariana" can be shortened like " Marian," as " Helena" becomes " Helen " (M. N. D. i. 1. 208). Compare
" For Már | iana's sáke : | but ás | he adjúdg'd | your brother."
M. for M. v. 1. 408.
The following seem pure Alexandrines, or nearly so, if the text be correct:—
"How dares (499) | thy hársh | rude tóngue | sound this | unpléas | ing néws."—Rich. II. iii. 4. 74.
" Suspíc | ion, áll | our líves, | shall bé | stuck fúll | of eyes."
1 Hen. IV. v. 2. 8.
" A chér | ry líp, | a bón | ny éye, | a páss | ing pléas | ing tongue. "—Rich. III. i. 1. 94.
" Tó the | young Ró | man bóy | she hath sóld | me ánd | I fáll."—A. and C. iv. 12. 48.
" And thése | does shé | applý | for wárn | ings ánd | porténts ."—J. C. iii. 1. 23.
This is the Shakespearian accent of "portent" (490), but perhaps " and" should be omitted.
" Oút of | a gréat | deal óf | old ír | on I′ | chose fórth."
1 Hen. VI. i. 2. 101.
It is needless to say that Shakespeare did not write this line, whether it be read thus or
" Oút of | a great déal | of óld | iron I′ | chose fórth."
In
" 'Tis hé | that sént | us híth | er nów | to slaugh | ter thée,"
Rich III. i. 4. 250.
"hither" (466) may be a monosyllable, and then we can read
" 'Tis hé | that sent us |."
The latter line in the following couplet seems to be an Alexandrine :
" Of whát | it ís | not : thén, | thrice-grác | ious quéen,
More than | your lórd's | depárt | ure wéep | not : móre's
I not séen."—Rich. II. ii. 2. 25, v. 4. 110.
Sometimes apparent Alexandrines will be reduced to ordinary lines, if exclamations such as "O," "Well," &c. be considered (512) as detached syllables.
" Vol. That théy | combíne | not thére. |
Cor. ( Tush, tush !)
Men. A good demánd."
Coriol. iii. 2. 45.
" Coriol. The óne | by the óther. |
Com. (Well,) | O'n to | the márk | et pláce."
Ib. iii. 1. 112.
" Sic. 'Tis hé, | 'tis hé : | (O,) he's grówn | most kind | of láte. "—Ib. iv. 6. 11.
" Upón | the Brít | ish pàrty. | (O,) untíme | ly death."
Lear, iv. 6. 25.
In the last two examples " O " might coalesce with the following vowel. But see also 503 and 512.
499. Apparent Alexandrines are sometimes regular verses of five accents preceded or followed by a foot, more or less isolated, containing one accent.
"(Shall I) With bated breath and whispering humbleness
Say this. Fair sír, | you spit | on mé | on Wéd | nesday
last."—M. of V. i. 3. 126.
"Háve I No friend will ríd | me óf | this lív | ing féar?"
Rich. II. v. 4. 2.
The "No" is emphatic, and there is a slight pause after "I."
" Whip him, Were't twén | ty óf | the gréat | est tríb | u-táries." —A. and C. iii. 13. 96.
" Come, cóme, No móre | of this | unpróf | itá | ble chát."
1 Hen. IV. iii. 1. 63.
" There cannot be those numberless offences
'Gáinst me, that I' | cannót | take péace | with : nó |
black énvy."—Hen. VIII. ii. 1. 85.
" A′s you | are cért | ainlý | a gén | tlemán, theretó,
Clerk-líke | expéri | énced."—W. T. i. 2. 391.
" Besides, I like | you nót. | I′f you | will knów | my hóuse."
A. Y. L. iii. 5. 74.
" Which to | dený concérns | móre than aváils,
For ás thy brát hath béen | cast óut | like to | itsélf."
W. T. iii. 2. 87.
"Só it | should now,
Were there | necéss | itý | in yóur | requést, || althóugh
'Twere néed | ful I' | denied it."—Ib. i. 2. 22.
"Máking | práctis'd | smiles
A's in | a a lóok | ing gláss, | and thén | to sigh, || as 'twére
The mórt | o' the déer."—W. T. i. 2. 117.
The context might perhaps justify a pause after "well" in
" Flor. To háve | them ré | compénsed | as thóught | on.
Cam. Wéll, my lórd."
W. T. iv. 4. 532.
But better "To have them (t' have 'em) ré | compénsed."
" His traín | ing such
That hé | may fúrn | ish ánd | instrúct | great teachers,
And név | er séek | for áid | óut of | himsélf.
Yet see," &c.—Hen. VIII. i. 2. 114.
"Whát, girl ! | though gréy
Do sóme | thing ming | le with | our yóung | er brówn,
A bráin," &c.—A. and C. iv. 8. 21.
yet há' we
" A cértain number,
Though thánks | to áll, | múst I | I seléct | from áll. The
rést
Shall béar," | &c.—Coriol. i. 6. 81 ; i. 7. 2.
" And the buildings of my fancy.
Only—
There's one thing wanting which I doubt not but."
Ib. ii. 1. 216.
Collier transposes "only" and "but" to the respectively following lines. The line
" So to esteem of us and on our knees we beg," ought probably to be arranged thus :
" So to | estéem | of ús, and ón | our knees
We bég | as ré | compénse | of óur | dear services (471)."
W. T. ii. 3. 150.
So
" Whom I′ | with this | obé | dient stéel, three inches (471) of it."—Temp. ii. 1. 283 ; i.e. " three inch of't."
So transpose " 'tis," i. e. "it is," to the preceding line in
" York. I féar, | I féar,—|
Duch. What should | you féar? | It is ('Tis) Nothing bút | some bónd | that hé | is ént | er'd
into. "—Rich. II. v. 2. 65.
"I do " must be omitted (456) before "beseech you" in
"(I do) beséech | you, pár | don mé, | I máy | not show it."
Ib. 70.
So Cymb. i. 6. 48.
500. Trimeter Couplet. Apparent Alexandrines are often couplets of two verses of three accents each. They are often thus printed as two separate short verses in the Folio. But the degree of separateness between the two verses varies greatly. Thus perhaps—
" Where it may sée | itsélf ; thís is | not stránge | at áll."
Tr. and Cr. iii. 3. 111.
"That hás | he knóws | not what. Náture, | what thíngs | there áre."—Ib. iii. 3. 127.
And certainly in the following :—
"Anne. I wóuld | I knéw | thy héart. Glou. 'Tis fíg | ured ín | my tóngue.
Anne. I féar | me bóth | are false. Glou. Then név | er mán | was true.
Anne. Well, wéll, | put úp | your swórd. Glou. Say thén | my péace | is máde."—Rich. III. i. 2. 193.
"Jul. I wóuld I knéw | his mind. Luc. Perúse | this pá | per, madam.
Jul. 'To Jú | lia.' Sáy, | from whóm? Luc. Thát the | conténts | will shéw.
Jul. Say, sáy, | who gáve | it thée ?"—T. G. of V. i. 2. 33-7.
"Luc. Go tó ; |'tis wéll ; | awáy ! Isab. Heaven kéep | your hón | our safe. "—M. for M. ii. 2. 156.
"Isab. Sháll I | atténd | your lórdship? A. At án | y tíme | 'fore nóon."—Ib.ii. 2. 60; ii. 4. 104, 141.
"Ros. The hóur | that fóols | should ásk. B. Now fáir | befáll | your mask.
Ros. Fair fáll | the fáce | it cóvers. B. And sénd | you má | ny lóvers."—L. L. L. ii. 1. 123.
" Ang. Why dóst | thou ásk | again? Prov. Lést I | might bé | too rash.
Prov. Repént | ed ó'er | his doom. Ang. Go tó, | let thát | be míne!
Ang. And yóu | shall wéll | be spared. Prov. I cráve | your hón | our's párdon."—M. for M. ii. 2. 9—12; Othello, iii. 3.
28-31; Temp. iii. 1. 31, 59.
Shakespeare seems to have used this metre mostly for rapid dialogue and retort. But in the ghost scene in Hamlet :
"Ghost. To whát | I sháll | unfóld.
Ham. Speak; I′ | am bóund | to héar."
Hamlet, i. 5. 6.
501. The trimeter couplet, beside being frequent in dialogue, is often used by one and the same speaker, but most frequently in comic, and the lighter kind of serious, poetry. It is appropriate for Thisbe :
" Most rád | iant Pý | ramús, most líl | y-whíte | of húe."
M. N. D. iii. 1. 94, 97.
And for Pistol, when he rants :
"An óath | of míck | le might; and fú | ry sháll | abáte."
Hen. V. ii. 1. 70, 44 ; ii. 3. 4, 64; v. 1. 93.
" He ís | not vé | ry táll : yet fór | his yéars | he's táll."
A. Y. L. iii. 5. 118.
" And ′I'll | be swórn | 'tis trúe: trávell | ers né'er | did líe."—Temp. iii. 2. 26.
" Coy lóoks | with héart- | sore síghs; one fád | ing mó- | ment's mírth."—T. G. of V. i. 1. 30.
" He wóuld | have gív'n | it yóu, but I′ | being ín the way
Did ín | your náme | recéive it: párdon | the fáult, | I
práy."—Ib. 39, 40.
" A frée- | stone cól | our'd hánd ; I vér | ilý | did thínk."
A. Y. L. iv. 3. 25.
" Then lét's | make háste | awáy, and lóok | untó | the máin."—2 Hen. VI. i. 1. 208.
" Am I′ | not wítch'd | like her? Or thóu | not fálse | like hím ?"—Ib. iii. 2. 119.
" Why ring | not óut | the bélls alóud | throughóut | the tówn ?"—1 Hen. VI. i. 6. 12.
" As Æ′th | | ióp | ian's tooth, ór the | fann'd snów | that's bólted."—W. T. iv. 4. 375.
" This páus | inglý | ensúed. Néither | the kíng | nor's héirs."—Hen. VIII. i. 2. 168.
"The mónk | might bé | decéiv'd; and thát | 'twas dáng(e) | rous fór him."—Ib. 179.
" Anón | expéct | him hére ; but if | she bé | obdúrate (490)."—Rich. III. iii. 1. 39.
This metre is often used by the Elizabethan writers in the translation of quotations, inscriptions, &c. It is used for the inscriptions the caskets :
" Who chóos | eth mé shall gain what mán | y mén | desire.
Who chóos | eth mé | must give and ház | ard áll | he háth."—M. of V. ii. 7. 5, 9.
In the pause between a comparison and the fact such a couplet may be expected.
"A's | Æné | as díd
The óld | Anchí | ses béar, so fróm | the wáves | of Tiber
Did I′ | the tír | ed Cæsar."—J. C. i. 2. 114.
"To háve | what wé | would have, we spéak | not whát | we méan."—M. for M. ii. 4. 118.
Sometimes the first trimeter has an extra syllable, which takes the place of the first syllable of the second trimeter.
" Shall thére | by bé | the swéeter. Reá | son thús | with lífe."—M. for M. iii. 1. 5.
" Envél | ope yóu, | good Próvost ! Whó | call'd hére | of late?"—Ib. iv. 2. 78.
" Mátters | of néed | ful válue. Wé | shall wríte | to yóu."
Ib. i. 1. 56.
Sometimes the first trimeter, like the ordinary five-accent verse, has an extra syllable. In the following examples the two verses are clearly distinct. They might almost be regarded as separate lines of three accents rather than as a couplet :
" Hypér | ion tó | a sátyr. | So lóv | ing tó | my móther." Hamlet, i. 2. 140.
" For énd | ing thée | no sóoner. Thou hást | nor yóuth | nor áge."—M. for M: iii. 1. 32.
" That I′ | am tóuch'd | with mádness. Make nót | im-póss | iblé."—Ib. v. 1. 51. (But? 494.)
" Ariel. And dó | my spírit | ing gently.
Prosp. Do só, | and áfter | two days."
Tempest, i. 2. 298.
" Belów | their cób | bled shóes.
They say | there's gráin | enough."
Coriol. i. 1. 200.
502. The comic trimeter. In the rhyming parts of the Comedy of Errors and Love's Labour Lost, there is often great irregularity in the trimeter couplet. Many of the feet are trisyllabic, and one-half of the verse differs from the other. Often the first half is trochaic and the second iambic.
" Ant. E. Whérefore? | fór my | dínner : I háve | not dín'd I to-dáy."—C. of E. iii. 1. 40.
" Ant. E. Dó you | héar, you | mínion ? You'll lét | us ín, | I hópe."—Ib. 54.
In the following, the former half is iambic and the latter anapæstic :
" Thou wóuldst | have cháng'd | thy face || for a náme, | or thy náme | for an áss."—C.of E. iii. 1. 47.
And conversely :
" It would máke | a man mád | as a búck || to bé | so bóught | and sóld."—Ib. 72.
There are often only five accents.
" Bal. G od méat, s r, | s cómm n : | that é | very chúrl | affórds.
Ant. E. And wélc me | m re cómm n ; | four thát | is nó-th ng | but wórds."—Ib. iii. 1. 24, 25.
Sometimes it is hard to tell whether the verse is trisyllabic with four accents, or dissyllabic with five.
" Have át | you wíth | a próverb—| Shall I′ | set ín | my stáff?"
Ib. 51.
may be scanned with six accents, but the line to which it rhymes seems to have four :
" And só | tell your máster. | O Lórd, I I must láugh,"
Ib. 50.
and the following line also :
" Have at yóu | with anóther; | that's whén | can you téll,"
Ib. 52.
and it is therefore possible that we ought to accent thus :
" Have at yoú | with a próverb—| Shall I sét | in my stáff?"
503. Apparent trimeter couplets. Some apparent trimeter couplets are really ordinary dramatic lines.
For example, in the last line but two of 501 (M. for M. v. i. 51), "impóssible" may easily be one foot with two superfluous syllables. It is often a matter of taste which way to scan a line, but it must be borne in mind, that the trimeter couplet is rarely used to express intense emotion. Hence in an impassioned address like that of Henry V. at Harfleur, we should probably read
" Defý us | to our wórst : | four ás I ám | a sóldier,"
Hen. V. iii. 3. 5.
or, better (479), "for as ′I'm | a sól | diér."
So
" And wél | come, Sómerset ; | I hóld | it ców | ardíce."
2 Hen. VI. iv. 2. 7.
Or, less probably, " Sómersét " may have two accents and " cówardice" (470) one.
" As chíl | dren fróm | a béar, | the Vóls | ces shúnning him."
Coriol. i. 3. 34.
" So tédiously | awáy. | The póor | condém ned E′nglish." Hen. V. iv. Prol. 221 ; but ib. 28 is a trimeter couplet.
" And húgg'd me | in his árm | and kínd | ly kíss'd | my chéek."—Rich. III. ii. 2. 24.
" Than thát | míx'd in | his chéek. | 'Twas júst | the díff(e)rence."—A. Y. L. iii. 5. 122.
" He is ('s) my bróth | er tóo. | But fítt | er tíme | for thát."
M. for M. v. 1. 498.
" And nót | the pún(i)sh | ment ; thérefore, | indéed | my fáther."—M. for M. i. 3. 39.
The following are doubtful, but probably ordinary lines :
" I knów him | as mysélf, | fór from | our ín | fancý."
T. G. of V. ii. 3. 62.
Or "ínfancy" may have only one accent (467).
" Máy a | free fáce, | put ón, | deríve | a líberty."
W. T. i. 2. 112.
"Either" may be a monosyllable (see 466) in
" Your sénse | pursúes | not míne : | either yóu | are ígnorant."
M. for M. ii. 4. 74.
" For ín | equál(i)ty : | but lét | your réa | son sérve."
Ib. v. 1. 65.
In
" Alexas did revolt ; and went to Jewry on Affairs of Antony,"—A. and C. iv. 6. 12.
"on" may be transposed to the second line; or, considering the licence attending the use of names and the constant dropping of prefixes, we might perhaps read " Aléxas | did (re)vólt | ."
In " Cálls her | a nón | paréil ; I né | ver sáw | a wóman,"
Temp. iii. 2. 108.
though it is against Shakespearian usage to pronounce "non-pareil" a dissyllable, as in Dorsetshire, "a núnprel apple," yet Caliban here may be allowed to use this form. I believe " nonp'rel type " is still a common expression.
Sometimes an exclamation, as "O," gives the appearance of a trimeter couplet :
" Fór the | best hópe I I háve. | (O,) do not wísh | one móre."—Hen. V. iv. 3. 33.
See also 498 ad fin.
504. The verse with four accents is rarely used by Shakespeare, except when witches or other extraordinary beings are introduced as speaking. Then he often uses a verse of four accents with rhyme.
"Dóuble, | dóuble, | tóil and | trouble,
Fíre | búrn and | cáuldron | búbble."—Macbeth, iv. 1. 20.
The iambic metre in such lines is often interchanged with the trochaic :
M. for M. iii. 2. 274–8.
(The last line means "he ought to have grace for the purpose of standing upright, and virtue [for the purpose of] walking in the straight path." "Go" is often used for "walk." "To" is omitted before "go.")
Sometimes in the same couplet we find one line iambic and the other trochaic :
" And hére | the mái | den sléep | ing sóund
O′n the | dánk and | dírty | gróund."—M. N. D. ii. 2. 74–5.
It would be, perhaps, more correct to say that both lines are trochaic, but in one there is an extra syllable at the beginning, as well as at the end. So apparently
" Thís is | hé my | máster | sáid,
(De)spísed | thé A | thénian | máid."—M. N. D. 72-3 :
but the prefix "de-" might (460) be dropped.
So
" (De)spísed | in na | tív | i | tý Shall úp | ón their chíldren | bé."—Ib. v. i. 420.
There is difficulty in scanning
" Prétty | sóul, she | dúrst not | líe
Near this lack-love, this kill-courtesy."—Ib. 76—7.
It is of course possible that " kill-curt'sy " may have the accent on the first : but thus we shall have to accent the first " this " and "love" with undue emphasis. It is also more in Shakespeare's manner to give "courtesy" its three syllables at the end of a line. I therefore scan
" (Near this) láck-love, | thís kill | cóurte | sý."
Perhaps, however, as in Macbeth, iii. 5. 34, 35, and ? 21, a verse of five accents is purposely introduced.
505. Lines with four accents are, unless there is a pause in the middle of the line, very rare. The following, however, seem to have no more than four accents :
"Let's éach | one sénd | únto | his wífe."—T. of Sh. v. 2. 66.
" No wórse | than I′ | upon sóme | agreément."—Ib. iv. 4. 33.
"He sháll | you fínd | réady | and wílling."—Ib. 34.
"The mátch | is máde, | and áll | is dóne."—Ib. 46.
"Go fóol, | and whóm | thou kéep'st |commánd."
Ib. ii. 1. 259.
The frequent recurrence of these lines in the Taming of the Shrew will not escape notice.
" And pút | yoursélf | únder | his shrówd." (? corrupt.)
A. and C. iii. 13. 71.
" A lád | of lífe, | an ímp | of fáme."
Hen. V. iv. 1. 45 (Pistol).
" We knew not
The dóc | trine óf | ill-dóing, | nor dréam'd
That any did."—W. T. i. 2. 70.
"Go téll | your cóusin | and bring | me wórd."
1 Hen. IV. v. 1. 109.
"For áught I I knów, | my lórd, | they dó."
Rich. II. v. 1. 53.
But perhaps the lines may be arranged :
" Aum. For áught | I knów,
My lórd, | they dó. |
York. You wíll | be thére, | I knów.
Aum. If Gód | prevént | (it) nót, | I púrpose | só."
" With " may be, perhaps (457), transposed to the former of the following verses, thus :
" With ád | orá | tions, fér | tile té | ars, (480) wíth
Groans (484) | that thún | der lóve, | with síghs | of fire."
T. N. i. 5. 274.
But the enumerative character of the verse (509) may justify it as it stands.
It is difficult to scan
"Lock'd in her monument. She had a prophesying fear,"
A. and C. iv. 14. 120.
without making the latter portion a verse of four accents.
(Perhaps
"Lóck'd in | her món(u) | ment. Shé'd | a próphe | sying féar," making "sying" a monosyllable like "being," "doing." See 470.)
"Should fróm | yond cloúd | speak di | vine things."
Coriol. iv. 5. 110.
But I should prefer
" If Jupiter
Shóuld, from | yond clóud, | spéak di | vine thíngs | and sáy
"Tis trúe,'—| (507) I'd nót | belíeve them móre
Than thée, | all-nó | ble Március."
Shakespeare would have written "things divine," not "divine things " at the end of a verse. (See 419, at end.)
"Is nót | much míss'd | bút with | his fríends."—Coriol. iv. 6. 13.
" Befóre | the kíngs | and quéens | of Fránce."
1 Hen. VI. i. 6. 27.
"And éven | these thrée | days háve | I wátch'd."
Ib. i. 4. 16.
"Here throúgh | this gáte | I cóunt | each óne. "—Ib. 60.
" Think nót | the kíng | did bán | ish thée,"
Rich. II. i. 3. 279.
is not found in the Folio, which also varies, ib. i. 3. 323; iii. 7. 70. Perhaps
"They thús | diréct | ed, wé | will fóllow
I'n the | main báttle | whose púissance | on éi | ther
side."—Rich. III. v. 3. 298.
(But the second line is harsh, and perhaps part of it ought to be combined with the first in some way. "Puissance" is a dissyllable generally in Shakespeare, except at the end of the line. I know no instance in Shakespeare where, as in Chaucer, "battle" is accented on the last. Remembering that ed is often not pronounced after t and d, we might scan the first line thus, with three accents :
"They thús | diréct(ed), | we'll fóllow.")
If "ed" is not pronounced (472) in "divided," that may explain
"The archdéa | con háth | divíded it."—1 Hen. IV. iii. 1. 72.
The following may seem a verse of four accents :
"Whereas the contrary bringeth bliss."—1 Hen. VI. v. 5. 64.
But "contráry" is found in Hamlet, iii. 2. 221. And as "country" (see 477) is three syllables, so, perhaps, " contrary" is four :
"Whereás | the cónt | (e)rár | y bring | eth bliss."
A verse of four accents is exceedingly discordant in the formal and artificial speech of Suffolk, in which this line occurs.
Somewhat similarly, Shakespeare has" cursoráry" for" cursory:"
" I have but with a cursorary eye. "—Hen. V. v. 2. 77.
In " Antony Woodville, her brother there,"—Rich. III. i. 1. 67. "Woodville" is probably to be pronounced a trisyllable, a semi-vowel inserting itself between the d and v—"Wood-e-ville." The e final (see 488) would not be sounded before "her."
" Valiant" is a trisyllable in
" Young, vál | iánt, | wíse, and | no dóubt | right róyal."
Rich. III. i. 2. 245.
506. Lines with four accents, where there is an interruption in the line, are not uncommon. It is obvious that a syllable or foot may be supplied by a gesture, as beckoning, a movement of the head to listen, or of the hand to demand attention, as in
"He's tá'en. | [Shóut.] | And hárk, | they shóut | for jóy."
J. C. v. 3. 32.
"Knéel thou | down, Phílip. | (Dubs hím knight.) | But ríse | more gréat."—K. J. i. 1. 161.
"Márry | to——(Enter O'thello.) | Come, cáp | tain, wíll | you gó?"—Othello, i. 2. 53.
Here, however, as in
"A wíse | stout cáp | (i)táin, | and sóon | persuáded."
3 Hen. VI. iv. 7. 32.
"Our cáp | (i)táins, | Macbéth and Bán | quo? Yés."
Macbeth, i. 2. 34.
we may scan
"Márry | to——Cóme, | cáp(i) | tain, wíll | you gó,"
but very harshly and improbably.
"Cass. Flátter | ers !" (Turns tó Brutus.) | Now, Brú | tus, thánk | yoursélf."—J. C. v. 1. 45.
An interruption may supply the place of the accent:
"And fálls | on th' óth | er——(Enter Lády Macbeth.) | How nów, | what néws?"—Macbeth, i. 7. 28.
The interval between two speakers sometimes justifies the omission of an accent, even in a rhyming passage of regular lines :
"Fairy. Aré not | you hé? | ' Puck. | Thou spéak'st | aríght, I ám | that mér | ry wán | derer óf | the níght."
M. N. D. ii. 1. 42.
"Mal. As thóu | didst léave | it. ′ Serg. | Dóubtful | it stóod."
Macbeth, i. 2. 7.
" Cass. Messá | la ! ′ Mess. | What sáys | my gén | erál?"
J. C. v. 1. 70.
"Dun. Who cómes | | here? ′ Mal. | The wórth | y tháne | of Róss."—Macbeth i. 2. 45.
" Sic. Withóut | assístance. | | Men. I think | not só."
Coriol. iv. 6. 33.
The break caused by the arrival of a new-comer often gives rise to a verse with four accents.
"Than yóur | good wórds. | ′ | But whó | comes hére?"
Rich. II. ii. 3. 20.
"Stánds for | my bóunty. | ′ | But whó | comes hére?"
Ib. 67.
"Agáinst | their wíll. | ′ | But whó | comes | hére?"
Ib. iii. 3. 19.
So, perhaps, arrange
"High be our thoughts !
I know my uncle York hath power enough
To sérve | our túrn. | ′ | But whó | comes hére?"
Ib. iii. 2. 90.
It is possible that in some of these lines "comes" should be pronounced "cometh." "Words," "turn," and "will" might be prolonged by 485, 486.
507. Lines with four accents where there is a change of thought are not uncommon. In some cases the line is divided into two of two accents each, or into one line of three accents, and another of one.
* (1) Change of thought from the present to the future :
"Háply | you sháll | not sée | me móre ; | or if,
A máng | led shádow. | ′ | Perchánce | to-mórrow
You'll serve | anóther | máster."—A. and C. iv. 1. 28.
"I'll sénd | her stráight | awáy. | ′ | To-mórrow
I'll' to | the wárs : | shé to | her sing | le sórrow."
A. W. ii. 3. 313.
"Fresh kíngs | are cóme | to Tróy. | ′ | To-mórrow
We múst | with áll | our máin | of pówer | stand fást."
Tr. and Cr. ii. 2. 272.
* (2) From a statement to an appeal, or vice versâ :
"You have | not sought it. | ′ | How cómes | it thén?"
1 Hen. IV. v. 1. 27.
Unless "comes" is "cometh." See 506 at end.
"Lórd of | his réason. | ′ | Whát through | you fléd?"
A. and C. iii. 13. 4.
(I do not remember an instance of "ré | asón." See, however, 479.)
Perhaps "Come híth | er, cóunt. | ′ | Do you (d' you) knów | these wómen?"—A. W. v. 3. 165.
But possibly:
"Come híth | er, cóu | nt (486). Dó | you knów | these women?"
"But stáy. | Here comes (Fol.) | the gár | denérs."
Rich. II. iii. 4. 24.
("gárdeners" may have but one accent.)
"Néver | belíeve | me. ′ | Bóth are | my kínsmen."
Ib. ii. 2. 111.
The pause may account for
"As hé | would dráw it. | ′ | Long stáy'd | he só."
Hamlet, ii. 1. 91.
(As ed is pronounced after i and u, so it might be after y in "stáyed," but the effect would be painful.)
"Which hás | no néed | of yóu.
Begóne,"
is the best way of arranging A. and C. iii. 11. 10.
"And léave | eightéen. | ′ | Alás, poor | Príncess."
A. and C. ii. 1. 61.
"A prínc | e's cóurage. | ′ | Awáy, | I príthee."
Cymb. iii. 4. 187.
"Lét us | withdráw. | ′ | 'Twill bé a stórm."
Lear, ii. 4. 290.
* (3) Hence after vocatives :
"Títus, | ′ | I (am)'m cóme | to tálk | with thée."
T. A. v. 2. 16.
"Géntle | men, ′ | impórt | une mé | no fúrther."
T. of Sh. i. 1. 48.
"Géntle | men, ′ | that I′ | may sóon | make góod."—Ib. 74.
"Géntle | men, ′ | contént | ye, ′I'm | resólved."—Ib. 90.
"Géntle | men, ′ | wíll you | go mús | ter mén?"
Rich. II. ii. 2. 108.
" Géntle | men, ′ | go mús | ter úp | your mén."
Rich. II. ii. 2. 118
"Good Már | garét. | Rún | thee tó | the párlour."
M. Ado, iii. 1. 1.
Either a pause may explain
"But téll | me, ′ | is yóung | Géorge Stán | ley líving?"
Rich. III. v. 5. 9.
or "George" (485) may be a quasi-dissyllable.
508. A foot or syllable can be omitted where there is any marked pause, whether arising from (1) emotion, (2) antithesis, or (3) parenthesis, or (4) merely from the introduction of a relative clause, or even a new statement.
* (1)
"Wére't | my fitness To lét | these hánds | obéy | my blóod, |—′ |
They're ápt | enóugh | to dís | locáte | and téar
Thy flésh | and bónes."—Lear, iv. 2. 64.
"O′ | dislóy | al thing
That shóuld'st | repáir | my yóuth, |—′ | thou héap'st
A yéar's | age ón | me."—Cymb. i. 1. 132.
There is an intended solemnity in the utterances of the ghosts in
"Let fáll | thy lánce. | ′ | Despáir | and die."
Rich. III. v. 3. 143.
and "Thínk on | lord Hástings. | ′ | Despáir | and díe."—Ib. 148.
* (2)
"Scarce án | y jóy Did év | er só | long live. | | No sorrow
But kíll'd | itsélf | much sóon | er."—W. T. v. 3. 53.
* (3)
"He quit his fórt | unes here (Which yóu | knew gréat) | ′ | ánd to | the hazard."
Ib. iii. 2. 169.
* (4) "Mark whát | I sáy, | ′ | which yóu | shall fínd."
* M. for M. iv. 3. 130.
Perhaps
"Is my kíns | man, ′ | whóm | the kíng | hath wróng'd,"
Rich. II. ii. 2. 114.
in a very irregular passage, part of which is nearly prose.
"I′nto | his títle | which | the | we fínd."
1 Hen. IV. iv. 3. 104.
"That shé | did gíve me, | ′ | whose pó | sy wás."
M. of V. v. 1. 148.
"Cáll our cares féars, | ′ | which wíll | in time."
Coriol. iii. 1. 137.
" 'Tis súre | enóugh |—án you | knew hów."
T. A. iv. 1. 95.
A pause may, perhaps, be expected before an oath, as in
" As yoú | shall gíve th' advíce. | Bý | the fíre
That quíck | ens E | gypt's slíme."—A. and C. i. 3. 68.
(But "vice" or "by" may be prolonged.)
"That mý | most jéal | ous ánd | too dóubt | ful heart
May live | at péace. | ′ | He sháll | concéal it."
T. N. iv. 3. 28; Macbeth, i. 5. 6.
"To wátch, | poor pérdu !
With thís | thin hélm. | ′ | Mine éne | my's dóg,
Though he | had bít | me, shóuld | have stood | that níght
Agáinst | my fíre."—Lear, iv. 7. 36.
" Last níght | 'twas ón | mine árm. | ′ | I kíss'd it."
Cymb. ii. 3. 151.
(Certainly not "I kíss | ed it.")
" Would thén | be nóthing. | ′ | Trúths would | be táles."
A. and C. ii. 2. 137.
"Póint to | rich énds. | ′ | Thís my | mean tásk."
Temp. iii. 1. 4.
" Must gíve | us páuse (484). | ′ | Thére's the | respéct."
Hamlet, iii. 1. 68.
509. Lines with four accents are found where a number of short clauses or epithets are connected together in one line, and must be pronounced slowly :
" Earth gapes, hell burns, fiends roar, saints pray."
Rich. III, iv. 4. 75.
" Witty, courteous, liberal, full of spirit."
3 Hen. VI. i. 2. 43.
The last line is very difficult. " And," or a pause equal to "and," after "witty," would remove the difficulty.
It is remarkable that Shakespeare ventures to introduce such a line even in a rhyming passage :
" Youth, beauty, wisdom, courage, all
That happiness and prime can happy call."
M. for M. ii. 1. 184.
" Ho ! héarts, | tongues, fígures, | scribes, bárds, | poéts |
cannot
Think, spéak, | cast, wríte, | sing núm | ber, ho!
His love to Antony."—A. and C. iii. 2. 17.
" Is goads, thorns, nettles, tails of wasps. "—W. T. i. 2. 329.
(Here, however, "goads" and "thorns" may be prolonged. See 484, 485.)
" With thát | harsh, nó | ble, sím | ple—| nóthing."
Cymb. iii. 4. 135.
The following occurs amid regular verse :
" These drums! these trumpets! flutes! what."
A. and C. ii. 7. 138.
" When you do dance, I wish you
A wave of the sea, that you might ever do
Nóthing | but thát; | move stíll, | still só."
W. T. iv. 4. 142.
Here still, which means "always," is remarkably emphatic, and may, perhaps, be pronounced as a quasi-dissyllable. So "til" is a monosyllabic foot in CHAUCER, C. T. 1137.
510. Apparent lines of four accents can sometimes be explained by giving the full pronunciation to contractions, such as s for eth, 'd for ed, 'll for will, 've for have, 't for it, &c. ; or they are lines of three accents with a detached foot.
"Silv. Whát's (is) | your wíll? |
Prot. That I′ | may cóm | pass yóurs."
T. G. of V. iv. 2. 92.
"And wére | the king | on't (of ít), what wóuld | I dó?"
Temp, ii. 1. 145.
"In whát | you pléase. | ′I'll (will) | do whát | I cán."
Ib. iv. 4. 47.
" You've ádd | ed wó | rth (485) ún | to ít | and lústre."
T. of A. i. 2. 154.
" Dríve him | to Rö | me ; 't (it) is tíme | we twáin."
A. and C. i. 4. 73.
" Whence cóm | est thóu ? | What wóuld | est thóu? | Thy náme?"—Coriol. iv. 5. 58.
But the pauses between the abrupt questions may be a sufficient explanation.
"And ne'er (név | er) á | true óne. | In súch | a níght."
M. of V. v. 1. 148.
The first " a " may be emphatic, meaning "one." Else 508.
"Our thighs | páck'd (ed) | with wáx, | our móuths | with hóney."—2 Hen. IV. iv. 5. 77.
" So múch | as lán | k'd (ed) nót. | 'Tis pít | y óf him."
A. and C. i. 4. 71.
" 's" = "his" in
" Vincént | ió | 's (his) són | brought úp | in Flórence."
T. of Sh. i. 1. 14.
In
" Sal. My lord, I long to hear it at full,"
2 Hen. VI. ii. 2. 6.
"hear" is a dissyllable (485), or "the" omitted after "at." Compare "atte" in E. E. for "at the."
I feel confident that "but would" must be supplied in
"And what poor duty cannot do, noble respect
Takes it in might, not merit,"—M. N. D. v. 1. 91.
and we must read :
" And what poor duty cannot do, but would,
Noble respect takes not in might but merit."
" And, ere our coming, see thou shake the bags
Of hoarding abbots ; imprisoned angels
Set at liberty. The fat ribs of peace
Must by the hungry now be fed upon,"—K. J. iii. 3. 8.
ought probably to be arranged :
" Of hoarding abbots ;
Imprisoned angels set at liberty.
The fat ribs of peace
Must," &c.
Or (Walker) invert "imprisoned angels" and "set at liberty."
Arrange thus :
" Your Coriolanus
Is nót | much míss'd,
Bút with | his fríends. | The cóm | monwéalth | doth stánd,
And só | would dó, | were hé | more áng | ry át it."
Coriol, iv. 6. 13.
Similarly
" Most cért | ain. Síst | er, wélcome.
Práy you (see 512)
Be év | er knówn | to pát | ience, mý | dear'.st síster."
A. and C. iii. 6. 97.
So arrange
" That won you without blows.
Despising (499),
For you, the city, thus I turn my back."
Coriol. iii. 3. 133.
" Cel. Look, whó | comes hére? |
Silv. My érr | and ís | to yóu :
Fair yóuth (512), |
My gént | le Phœ′ be bíd | me gíve | you thís."
A. Y. L. iv. 3. 6.
"Got'twéen | asléep | and wáke.
Wéll, then (512),
Legít(i) | mate E'd | gar, I' | must háve | your lánd."
Lear, i. 2. 15.
" As péarls | from día | monds drópp'd.
In brief (511)."—Lear, iv. 3. 24.
Hen. V. ii. Prologue, 32, is corrupt.
"I live with bread like you :
Feel want, taste grief, need friends : subjected thus,
How can you say to me I am a king?"—Rich. II. iii. 2. 175.
511. Single lines with two or three accents are frequently interspersed amid the ordinary verses of five accents. They are, naturally, most frequent at the beginning and end of a speech.
These lines are often found in passages of soliloquy where passion is at its height. Thus in the madness of Lear, iv. 6. 112-29, there are eight lines of three accents, and one of two ; and the passage terminates in prose. And so perhaps we should arrange
" Would use his heav'n for thunder ; nothing but thunder !
Merciful heaven (512),
Thou rather with thy sharp and sulphurous bolt
Split'st the unwedgeable and gnarled oak
Than the soft myrtle.
But man, proud man,
Drest in a little brief authority," &c.
M. for M. ii. 2. 110—19.
So in the impassioned speech of Silvius :
" If thou remember'st not the slightest folly
That ever love did make thee run into,
Thou hast not loved,"—A. Y. L. ii. 4. 36.
which is repeated in 1. 39 and 42.
The highest passion of all expresses itself in prose, as in the earful frenzy of Othello, iv. 1. 34—44, and Lear, iv. 6. 130.
Rarely we have a short line to introduce the subject.
" York. Then thus:
Edward the third, my lords, had seven sons."
2 Hen. VI. ii. 2. 9, 10.
" Into his ruin'd ears, and thus deliver :
'Henry Bolingbroke,
On both his knees,' " &c.—Rich. II. iii. 3. 32.
" Ross. (So) That now
Sweno, the Norways' king, craves composition."
Macbeth, i. 2. 59.
"For Cloten :
There wants no diligence in seeking him."—Cymb. iv. 3. 19.
Sometimes the verse (which is often written as prose in the Folio) closely resembles prose. It is probable that the letter J. C. ii. 3. 1—10 is verse, the last two words, "thy lover, Artemidorus," being irregular. So A. Y. L. iii. 2. 268—74.
The irregular lines uttered by Cassius, when he is cautiously revealing the conspiracy to Casca, looking about to see that he is not overheard, and also pausing to watch the effect of his words on Casca, are very natural.
"Unto some monstrous state.
Now could I, Casca, name to thee a man
Most like this dreadful night,
That thunders, lightens, opens graves, and roars."
J. C. i. 3. 71—74.
It will also not escape notice that "now could I, Casca," and "that thunders, lightens," are amphibious sections. See 513.
The following pause may be explained by the indignation of Macduff, which Malcolm observes and digresses to appease :
" Why in that rawness left you wife and child
Without leave-taking?
I pray you (512)
Let not my jealousies be your dishonours."
Macbeth, iv. 3. 28.
A pause is extremely natural before Lear's semi-confession of infirmity of mind :
" A'nd, to | deal pláinly,
I féar | I ám | not ín | my pérf | ect mínd."
Lear, iv. 7. 62.
A stage direction will sometimes explain the introduction of a short line. The action takes up the space of words, and necessitates a broken line, thus :
"Macb. This is a sorry sight. [Looking on his hands.]
Lady M. A foolish thought, to say a sorry sight."
Macbeth, ii. 2. 21.
Macbeth may be supposed to draw his dagger after the short line:
"As thís | which nów | I dráw."—Macbeth, ii. 1. 41.
So after Lady Macbeth has openly proposed the murder of Duncan in the words—
" Oh, never
Shall sun that morraw see,"—Macbeth, i. 5. 62.
she pauses to watch the effect of her words till she continues :
" Your face, my thane, is as a book where men," &c.
The irregular lines in the excited narrative of the battle—
" Like valour's minion, carv'd out his passage
Till he faced the slave,"—Macbeth, i. 2. 20 (so ib. 51).
are perhaps explained by the haste and excitement of the speaker.
This is illustrated by
" Except they meant to bathe in reeking wounds,
Or memorize another Golgotha,
I cannot tell.
But I am faint, my wounds cry out for help."
Macbeth, i. 2. 41.
In
"As cannons overcharged with double cracks ; || so they Doubly redoubled strokes upon the foe,"—Ib. i. 2. 37.
there may be an instance of a short line. But more probably we must scan " As cánnons | o'erchárged | ."
Such a short line as
" Only to herald thee into his sight,
Not pay thee,"—Macbeth, i. 3. 103.
is very doubtful. Read (though somewhat harshly) :
" On'ly | to hér(a)ld (463) | thee ín | to's síght, | not pay thee."
So
"Lét's (us) | awáy ; | our téars | are nót | yet bréw'd,"
Macbeth, ii. 3. 129, 130.
and the following lines must be arranged so as to make 1. 132 an interjectional line.
There is a pause after " but let " in
" But let—
The fráme | of thíngs | disjóint, | bóth the | worlds súffer."
Macbeth, iii. 2. 16 ; iv. 3. 97.
and in the solemn narrative preparatory to the entrance of the Ghost :
"Last night of all,
When yond same star that's westward from the pole."
Hamlet, i. 1. 35.
So
"And are upon the Mediterranean flote
Bound sadly home for Naples,
Supposing that they saw the king's ship wreck'd."
Temp. i. 2. 235.
So M. N. D. iii. 2. 49.
"Lastly,
If I do fail in fortune of my choice
Immediately to leave you and be gone."—M. of V. ii. 9. 14.
" Yet I,
A dull and muddy-mettled rascal, peak."
Hamlet, ii. 2. 593.
" I, his sole son, do this same villain send
To heaven."—Ib. iii. 3. 78.
In
"Dost thou hear?"—Temp. i. 2. 106.
"thou" is unemphatic, and scarcely pronounced. Or else these words must be combined with the previous, thus :
" Hénce his | ambít | ion grów |—ing—Dóst | thou héar?"
512. Interjectional lines. Some irregularities may be explained by the custom of placing ejaculations, appellations, &c. out of the regular verse (as in Greek φε , &c.).
" Yes. |
Has he | affections in him ?"—M. for M. iii. 1. 107.
" Alack
I love myself. Wherefore? for any good?"
Rich. III. v. 3. 187.
" What,
Are there no posts despatch'd for (480) Ireland?"
Rich. II. ii. 2. 103.
So arrange
" North. Why ?
I's he | not wíth | the quéen? |
Percy. Nó, my | good lórd."
Ib. ii. 3. 512.
" Fie,
There's no such man; it is impossible."
Othello, iv. 2. 134.
" And such a one do I profess myself,
For, sir,
It is as sure as you are Roderigo."
Othello, i. 1. 55; Lear, i. 1. 56.
Perhaps we ought thus to arrange
" O, sir,
Your presence is too bold and péremptory."
I Hen. IV. i. 3. 17.
This is Shakespeare's accentuation of " peremptory."
" Farewell. [Exit Banquo.]
Let every man be master of his time."—Macbeth, iii. 1. 40.
" Sir,
I have upon a high and pleasant hill."—T. of A. i. I. 63.
" Sirrah,
Get thee to Plashy, to my sister Gloucester."
Rich. II. ii. 2. 90.
So Rich. III. i. 2. 226 ; i. 4. 218.
" Great king,
Few love to hear the sin they love to act."—P. of T. i. 1. 91.
" My dismal scene I needs must act alone.
Come, vial."—R. and J. iv. 3. 20.
" Come, Hastings, help me to my lodging. O !
Poor Clarence."—Rich. III. ii. 1. 133.
" For Hecuba !
What's Héc | | ubá | to hím, | or he | to Hecuba (469)?"
Hamlet, ii. 2. 584.
" If thou hast any sound or use of voice,
Speak to me."—Ib. i. 1. 129.
So ib. 132, 135 : and " O vengeance," ib. 610 ; "A scullion!" ib. 616.
So we should read
" I'll wait upon you instantly. (Exeunt.) [To FLAV.] Come hither.
Pray you,
How goes," &c.—T. of A. ii. 1. 36.
Similarly "Nay, more," C. of E. i. 1. 16; "Stay," T. N. iii. 1. 149; " Who's there?" Hamlet, i. 1. 1; "Begone," J. C. i. 1. 57 ; " O, Cæsar," J. C. iii. 1. 281; "Let me work," J. C. ii. 1. 209 ; "Here, cousin," Rich. II. iv. 1. 182; "What's she?" T. N. i. 2. 35; "Draw," Lear, ii. 1. 32 ; "Think," Coriol. iii. 3. 49.
So arrange
"Viol. Hold, there's hálf | my cóffer. |
Anton. Will your | dený | me nów?"
T. N. iii. 4. 38.
"So, I am sát | isfíed, | gíve me | a bówl | of wíne."
Rich. III. v. 3. 72.
" Ratcliffe, abóut | the míd | of níght | cóme to | my tént."
Rich. III. 77, 209.
The excitement of Richard gives rise to several interjectional lines of this kind in this scene.
A short line sometimes introduces a quotation :
" If Cæsar hide himself, shall they not whisper,
Lo, Cæsar is afraid ?"—J. C. ii. 2. 101.
" Did scowl on gentle Richard. No man cried
'God save him.'"—Rich. II. v. 2. 28.
Perhaps we should arrange as follows :
" He'll spend that kiss
Which is my heaven to have.
Come [applying the asp to her bosom]
Thou mortal wretch,
With thy sharp teeth this knot intrinsicate
Of life at once untie. "—A. and C. v. 2. 306.
This seems better than scanning the words from "which" to " wretch" as one line, either (1) as an ordinary line, with "come, thou mór | tal wretch," or (2) as a trimeter couplet, making "come " a dissyllable.
So it is better to arrange :
" Buckingham,
I prithee pardon me
That I have giv'n no answer all this while."
2 Hen. VI. v. 1. 32.
Merely with a special view to mark a solemn pause Shakespeare writes :
"So, as a painted tyrant Pyrrhus stood,
And, like a neutral to his will and matter,
Did nothing.
But, as we often see," &c.—Hamlet, ii. 2. 504.
Such irregularities are very rare.
" Sirrah,
A word with you. Attend those men our pleasure ?"
is the right way to arrange Macb. iii. 1. 45, 46. Shakespeare could not possibly (as Globe) make "our pleasure " a detached foot.
The ejaculation seems not a part of the verse in
"Hath séiz'd | the wáste | ful kíng. | [O,] what pít | y is it."
Rich. II. iii. 4. 55.
"And hé | himsélf | not présent. | [O,] forefénd | it, Gód !"
Rich. II. iv. 1. 129.
See also 498, at end ; 503.
513. The Amphibious Section. When a verse consists of two parts uttered by two speakers, the latter part is frequently the former part of the following verse, being, as it were, amphibious—thus :
"S. The E'ng | lish fórce, | so pléase you.
M. Táke thy | face hénce. Séyton, | I'm síck | at héart."
Macbeth, v. 3. 19.
"M. News, my | good lórd, | from Róme.
Ant. Grátes me : | the súm.
Cleo. Nay, héar | them, A′n | toný."—A. and C. i. 1. 19.
"B. Who's thére? |
M. A fríend.
B. Whát, sir, | not yét | at rést ? The kíng's | abéd."
Macbeth, ii. 1. 10.
" Kent. This óff | ice tó you.
Gent. I' will | talk fúr | ther wíth you.
Kent. Nó, | do not."—Lear, iii. 1. 42.
" Gent. Which twáin | have bróught | her tó.
Edg.
Hail, gént | le sír. |
Gent. Sir, spéed | you, whát's | your wíll ?"
Lear, iv. 6. 212.
"Prosp. Agáinst | what shóuld | ensue.
Mir. How cáme | we ashóre ?
Prosp. By Pró | vidénce | divíne."
Temp. i. 2. 158.
"Claud. And húg | it ín | my árms.
hére | my fá | ther's gráve."
M. for M. iii. 1. 86.
" E. How fáres | the prínce?
Mess. Well, mád | am, ánd | in héalth. Duch. What is |
thy news, then ?"—Rich. III. ii. 4. 40.
"Brut. That óth | er mén | begín.
Cas. Then léave | him óut. Casca. Indéed | he ís | not fít."
J. C. ii. 1. 153.
Probably—
"Macb. And bréak it | to our hópe. I will | not fíght | with thée.
Macd. Then yíeld | thee, cóward."—Macbeth, v. 8. 22.
Compare also Macbeth, i. 4. 43, 44; ii. 3. 75, 101—2; iii. 1. 18 19, 2. 12-13, 4. 12, 15, 20, 151 ; J. C. ii. 4. 16, 17 ; Coriol. iii. 2. 6 ; Othello, iii. 3. 282, &c.
In the following instance the first " still " is emphatic :
"Oliv. As hówl | ing áft | er músic.
Duke. Stíll | so crú el !
Oliv. Stíll | so cón | stant, lórd."
T. N. v. 1. 113.
Sometimes a section will, on the one side, form part of a regular line, and, on the other, part of a trimeter couplet.
" Hor. Of míne | own éyes. Mar. I′s it | not like | the kíng ?
Hor. As thóu | art tó | thysélf."—Hamlet, i. 1. 58, 59.
"Ophel. In hón | ourá | ble fáshion. | Pol. Ay, fásh | ion yóu | may cáll it. Go to, go to."—Ib. i. 3. 112.
Ham. Nó, it | is strúck. Hor. Indéed, | I héard | it nót ; then ít | draws néar | the séason.—Ib. i. 4. 4.
In the last example, "indeed," when combined with what follows, is a detached interjection (512).
514. Interruptions are sometimes not allowed to interfere with the completeness of the speaker's verse.
This is natural in dialogue, when the interruption comes from a third person :
"Polon. Práy you | be róund | with hím. |
(Ham. [Within] Mother, mother, mother!)
Queen. I'll wár | rant yoú."
' Hamlet, iii. 4. 5, 6.
Or, when a man is bent on continuing what he has to say :
" Ham. Rashly—and that should teach us
There's a divinity that shapes our ends,
Rough-hew them how we will—
(Hor. That's certain.)
Ham. Up from my cabin," &c.
Hamlet, v. 2. 11, 12.
"Shy. This is (461) kínd | I óffer—
(Bass. This were kindness.)
Shy. This kínd | ness wíll | I shów."
M. of V. i. 3. 143.
" King R. Rátcliffe—|
(Rat. My lord.)
King R. The sún | will nót | be séen | to-day."
Rich. III. v. 3. 281.
"Brutus. Awáy, | slight mán. |
(Cassius. Is't possible ?)
Brutus. Héar me, | for I′ | will speak."
J. C. iv. 3. 37, 38.
Or, when a speaker is pouring forth his words, endeavouring to break through the obstacle of unintelligence, as Kent trying to make himself intelligible to the mad Lear :
"Kent. Nó, my | good lórd; |I ám | the vér | y mán—
(Lear. I'll see that straight.)
Kent. Thát from | your first | of dif | ference ánd | decáy
Have fóll | ow'd your | sad stéps, |—
(Lear. You're welcome hither.)
Kent. Nor nó | man élse."
i.e. " I and no one else." Then, in despair of making himself understood, Kent continues :
"All's cheerless, dark, and deadly."
Sometimes the interlocutor's words, or the speaker's continuation, will complete the line :
"Cæsar. So múch | as lánk | ed nót. | (Folio has lank'd.)
Lep. 'Tis pít | y óf him.
Cæsar. Let his | shames quíckly."—A. and C. i. 4. 71.
If there are two interlocutors, sometimes either interlocution will complete the line :
"Gent. Than ís | his úse. |
Widow. Lord, hów | we lóse | our páins !
Helena. All's wéll | that énds | well yét."
A. W. v. 1. 24, 25.
"Bru. Good Márc | ius | hóme | again. |
Sic. The vé | ry tríck on't.
Men. Thís is | unlíkely."
Coriol. iv. 6. 71.
515. Rhyme. Rhyme was often used as an effective termination at the end of the scene. When the scenery was not changed, or the arrangements were so defective that the change was not easily perceptible, it was, perhaps, additionally desirable to mark that a scene was finished. The rhyme in T. N. ii. 2. 32 is perhaps a token that the scene once concluded with these lines, and that the nine lines that follow are a later addition.
Rhyme was also sometimes used in the same conventional way, to mark an aside, which otherwise the audience might have great difficulty in knowing to be an aside. Thus, in a scene where there are no other rhyming lines, Queen Margaret is evidently intended to utter Rich. III. iv. 4. 16, 17; 20, 21, as asides, though there is no notice of it. One of the lines even rhymes with the line of another speaker:
" Q. Eliz. When didst thou sleep, when such a deed was
done ?
Q. Marg. When holy Harry died, and my sweet son."
Rich. III. iv. 4. 24, 25.
Queen Margaret does not show herself till line 35, as also in Rich. III. i. 3. till line 157, though in the latter scene the asides do not rhyme.
515 a. Prose. Prose is not only used in comic scenes ; it is adopted for letters (M. of V. iv. 1. 149-66), and on other occasions where it is desirable to lower the dramatic pitch : for instance, in the more colloquial parts of the household scene between Volumnia and Virgilia, Coriol. i. 3, where the scene begins with prose, then passes into verse, and returns finally to prose. It is also used to express frenzy, Othello, iv. 1. 34—44 ; and madness, Lear, iv. 6. 130 ; and the higher flights of the imagination, Hamlet, ii. 2. 310-20
# SIMILE AND METAPHOR.
516. Similarity.—In order to describe an object that has not been seen we use the description of some object or objects that have been seen. Thus, to describe a lion to a person who had never seen one, we should say that it had something like a horse's mane, the claws of a cat, &c. We might say, "A lion is like a monstrous cat with a horse's mane." This sentence expresses a likeness of things, or a similarity:
517. Simile.—In order to describe some relation that cannot be seen, e.g. the relation between a ship and the water, as regards the action of the former upon the latter, to a landsman who had never seen the sea or a ship, we might say, "The ship acts upon the water as a plough turns up the land." In other words, "The relation between the ship and the sea is similar to the relation between the plough and the land." This sentence expresses a similarity of relations, and is called a simile. It is frequently expressed thus :
"As the plough turns up the land, so the ship acts on the sea."
Def. A Simile is a sentence expressing a similarity of relations.
Consequently a simile is a kind of rhetorical proportion, and must, when fully expressed, contain four terms :
A : B C : D.
518. Compression of Simile into Metaphor.—A simile is cumbrous, and better suited for poetry than for prose. Moreover, when a simile has been long in use, there is a tendency to consider the assimilated relations not merely as similar but as identical. The simile modestly asserts that the relation between the ship and the sea is like ploughing. The compressed simile goes further, and asserts that the relation between the ship and the sea is ploughing. It is expressed thus : "The ship ploughs the sea."
Thus the relation between the plough and the land is transferred to the ship and the sea. A simile thus compressed is called a Metaphor, i.e. transference.
Def. A Metaphor is a transference of the relation between one set of objects to another, for the purpose of brief explanation.
519. Metaphor fully stated or implied.—A metaphor may be either fully stated, as " The ship ploughs (or is the plough of) the sea," or implied, as "The winds are the horses that draw the plough of the sea." In the former case it is distinctly stated, in the latter implied, that the "plough of the sea " represents a ship.
520. Implied Metaphor the basis of language.—A great part of our ordinary language, all that relates to the relations of invisible things, necessarily consists of implied metaphors; for we can only describe invisible relations by means of visible ones. We are in the habit of assuming the existence of a certain proportion or analogy between the relations of the mind and those of the body. This analogy is the foundation of all words that express mental and moral qualities. For example, we do not know how a thought suggests itself suddenly to the mind, but we do know how an external object makes itself felt by the body. Experience teaches us that anything which strikes the body makes itself suddenly felt. Analogy suggests that whatever is suddenly perceived comes in the same way into contact with the mind. Hence the simile—"As a stone strikes the body, so a thought makes itself perceptible to the mind." This simile may be compressed into the full metaphor thus, "The thought struck my mind," or into the implied metaphor thus, " This is a striking thought." In many words that express immaterial objects the implied metaphor can easily be traced through the derivation, as in " excellence," " tribulation," " integrity," " spotlessness," &c.
N.B. The use of metaphor is well illustrated in words that describe the effects of sound. Since the sense of hearing (probably in all nations and certainly among the English) is less powerful and less suggestive of words than the senses of sight, taste, and touch, the poorer sense is compelled to borrow a part of its vocabulary from the richer senses. Thus we talk of "a sweet voice," "a soft whisper," "a sharp scream," " a piercing shriek," and the Romans used the expression "a dark-coloured voice," where we should say "a rough voice."
521. Metaphor expanded.—As every simile can be compressed into a metaphor, so, conversely, every metaphor can be expended into its simile. The following is the rule for expansion. It has been seen above that the simile consists of four terms. In the third term of the simile stands the subject (" ship," for instance) whose unknown predicated relation (" action of ship on water ") is to be explained. In the first term stands the corresponding subject (" plough ") whose predicated relation (" action on land ") is known. In the second term is the known relation. The fourth term is the unknown predicated relation which requires explanation. Thus—
Sometimes the fourth term or unknown predicate may represent something that has received no name in the language. Thus, if we take the words of Hamlet, " In my mind's eye," the metaphor when expanded would become—
For several centuries there was no word in the Latin language to describe this "perceptive faculty of the mind." At last they coined the word " imaginatio," which appears in English as " imagination." This word is found as early as Chaucer ; but it is quite conceivable that the English lan guage should, like the Latin, have passed through its best period without any single word to describe the "mind's eye."
522. The details of the expansion will vary according to the point and purpose of the metaphor. Thus, when Macbeth (act iii. sc. 1) says that he has " given his eternal jewel to the common enemy of man," the point of the metaphor is apparently the pricelessness of a pure soul or good conscience, and the metaphor might be expanded thus—
" As a jewel is precious to the man who wears it, so is a good conscience precious to the man who possesses it."
But in Rich. II. i. 1. 180, the same metaphor is expanded with reference to the necessity for its safe preservation :—
" A jewel in a ten-times barr'd-up chest
Is a bold spirit in a loyal breast."
523. Personal Metaphor.—There is a universal desire among men that visible nature, e.g. mountains, winds, trees, rivers and the like, should have a power of sympathising with men. This desire begets a kind of poetical belief that such a sympathy actually exists. Further, the vocabulary expressing the variable moods of man is so much richer than that which expresses the changes of nature that the latter bor rows from the former. Hence the morn is said to laugh, mountains to frown, winds to whisper, rivulets to prattle, oaks to sigh. Hence arises what may be called Personal Metaphor.
Def. A Personal Metaphor is a transference of personal relations to an impersonal object for the purpose of brief explanation.
524. Personal Metaphors expanded.—The first term will always be " a person ;" the second, the predicated relation properly belonging to the person and improperly transferred to the impersonal object; the third, the impersonal object. Thus—
" As a person frowns, so an overhanging mountain (looks gloomy).
" As a child prattles, so a brook (makes a ceaseless cheerful clatter)."
525. Personifications.—Men are liable to certain feelings, such as shame, fear, repentance and the like, which seem not to be originated by the person, but to come upon him from without. For this reason such impersonal feelings are in some languages represented by impersonal verbs. In Latin these verbs are numerous, "pudet," "piget," "tædet," "pœnitet," "libet," &c. In Early English they were still more numerous, and even now we retain not only "it snows," " it rains," but also (though more rarely) " methinks," "meseems," "it shames me," " it repents me." Men are, however, not contented with separating their feelings from their own person ; they also feel a desire to account for them. For this purpose they have often imagined as the causes of their feelings, Personal Beings, such as Hope, Fear, Faith, &c. Hence arose what may be called Personification.
In later times men have ceased to believe in the personal existence of Hope and Fear, Graces and nymphs, Flora and Boreas ; but poets still use Personification, for the purpose of setting before us with greater vividness the invisible operations of the human mind and the slow and imperceptible processes of inanimate nature.
Def. Personification is the creation of a fictitious Person in order to account for unaccountable results, or for the purpose of vivid illustration.
526. Personifications cannot be expanded.—The process of expansion into simile can be performed in the case of a Personal Metaphor, because there is implied a comparison between a Person and an impersonal object. But the process cannot be performed where (as in Personifications) the impersonal object has no material existence, but is the mere creation of the fancy, and presents no point of comparison. " A frowning mountain " can be expanded, because there is implied a comparison between a mountain and a person, a gloom and a frown. But " frowning Wrath " cannot be expanded, because there is no comparison.
It is the essence of a metaphor that it should be literally false, as in " a frowning mountain." It is the essence of a personification that, though founded on imagination, it is conceived to be literally true, as in " pale fear," " dark dishonour." A painter would represent " death" as " pale," and " dishonour" as "dark," though he would not represent a " mountain" with a " frown," or a " ship" like a " plough."
527. Apparent Exception.—The only case where a simile is involved and an expansion is possible is where a person, as for instance Mars, the God of War, is represented as doing something which he is not imagined to do literally. Thus the phrase " Mars mows down his foes " is not literally true. No painter would represent Mars (though he would Time) with a scythe. It is therefore a metaphor and, as such, capable of expansion thus :—
" As easily as a haymaker mows down the grass, so easily does Mars cut down his foes with his sword."
But the phrase " Mars slays his foes" is, from a poet's or painter's point of view, literally true. It is therefore no metaphor, and cannot be expanded.
528. Personification analysed.—Though we cannot expand a Personification into a simile, we can explain the details of it. The same analogy which leads men to find a correspondence between visible and invisible objects leads them also to find a similarity between cause and effect. This belief, which is embodied in the line—
" Who drives fat oxen should himself be fat,"
is the basis of all Personification. Since fear makes men look pale, and dishonour gives a dark and scowling expression to the face, it is inferred that Fear is " pale," and Dishonour " dark." And in the same way Famine is " gaunt ;" Jealousy " green-eyed ;" Faith " pure-eyed ;" Hope " white-handed."
529. Good and bad Metaphors—There are certain laws regulating the formation and employment of metaphors which should be borne in mind.
(1.) A metaphor must not be used unless it is needed for explanation or vividness, or to throw light upon the thought of the speaker. Thus the speech of the Gardener, Rich. II. iii. 4. 33,—
" Go then, and like an executioner
Cut off the heads of our fast-growing sprays," &c.
is inappropriate to the character of the speaker, and conveys an allusion instead of an explanation. It illustrates what is familiar by what is unfamiliar, and can only be justified by the fact that the gardener is thinking of the disordered condition of the kingdom of England and the necessity of a powerful king to repress unruly subjects.
(2.) A metaphor must not enter too much into detail: for every additional detail increases the improbability that the correspondence of the whole comparison can be sustained. Thus, if King Richard (Rich. II. v. 5. 50) had been content, while musing on the manner in which he could count time by his sighs, to say—
"For now hath Time made me his numbering clock,"
there would have been little or no offence against taste. But when he continues—
"My thoughts are minutes, and with sighs they jar
Their watches on unto mine eyes, the outward watch,
Whereto my finger, like a dial's point,
Is pointing still, in cleansing them from tears.
Now, sir, the sound that tells what hour it is
Are clamorous groans which strike upon my heart,
Which is the bell,"—
we have an excess of detail which is only justified because it illustrates the character of one who is always " studying to compare," and "hammering out" unnatural comparisons.
(3.) A metaphor must not be far-fetched nor dwell upon the details of a disgusting picture :
" Here lay Duncan,
His silver skin laced with his golden blood ;
. . . . .there the murderers
Steep'd in the colours of their trade, their daggers
Unmannerly breech'd with gore."—Macbeth, ii. 3. 117.
There is but little, and that far-fetched, similarity between gold lace and blood, or between bloody daggers and breach'd legs. The slightness of the similarity, recalling the greatness of the dissimilarity, disgusts us with the attempted comparison. Language so forced is only appropriate in the mouth of a conscious murderer dissembling guilt.
(4.) Two metaphors must not be confused together, particularly if the action of the one is inconsistent with the action of the other.
It may be pardonable to surround, as it were, one metaphor with another. Thus, fear may be compared to an ague-fit, and an ague-fit passing away may be compared to the overblowing of a storm. Hence, " This ague-fit of fear is overblown" (Rich. II. iii. 2. 190) is justifiable. But
" Was the hope drunk
Wherein you dressed yourself? Hath it slept since ?"
Macbeth, i. 7. 36.
is, apart from the context, objectionable ; for it makes Hope a person and a dress in the same breath. It may, however, probably be justified on the supposition that Lady Macbeth is playing on her husband's previous expression—
" I have bought
Golden opinions from all sorts of people,
Which would be worn now in their newest gloss,
Not cast aside so soon."
(5.) A metaphor must be wholly false, and must not combine truth with falsehood.
"A king is the pilot of the state," is a good metaphor. "A careful captain is the pilot of his ship," is a bad one. So
" Ere my tongue
Shall wound mine honour with such feeble wrong,
Or sound so base a parle,"—Rich. II. i. 1. 190.
is objectionable. The tongue, though it cannot "wound," can touch. It would have been better that "honour's" enemy should be intangible, that thereby the proportion and the perfection of the falsehood might be sustained. Honour can be wounded intangibly by "slander's venom'd spear" (Rich. II. i. 1. 171) ; but, in a metaphor, not so well by the tangible tongue. The same objection applies to
"Ten thousand bloody crowns of mothers' sons
Shall ill-become the flower of England's face,
Change the complexion of her maid-pale peace
To scarlet indignation, and bedew
Her pastures' grass with faithful English blood."
Rich. II. iii. 3. 96.
If England is to be personified, it is England's blood, not the blood of ten thousand mothers, which will stain her face. There is also a confusion between the blood which mantles in a blush and which is shed ; and, in the last line, instead of "England's face," we come down to the literal "pastures' grass."
(6.) Personifications must be regulated by the laws of personality. No other rule can be laid down. But exaggerations like the following must be avoided :—
" Comets, importing change of times and states,
Brandish your crystal tresses in the sky,
And with them scourge the bad revolting stars."
1 Hen. VI. i. 1. 2.
The Furies may be supposed to scourge their prostrate victims with their snaky hair, and comets have been before now regarded as scourges in the hand of God. But the liveliest fancy would be tasked to imagine the stars in revolt, and scourged back into obedience by the crystal hair of comets.
# NOTES AND QUESTIONS.
# MACBETH, ACT III.
# SCENE 1.
LINE
3. "Thou play'dst most foully for't." Expand the metaphor into its simile. (Grammar, 521.)
14. "And all-thing unbecoming." See "All" (Grammar). What is there remarkable in this use of all? Comp. iii. 2. 11—
"Things without all remedy."
15. "A solemn supper." Modernize. Trace the present meaning from the derivation. Compare
" A solemn hunting is in hand."—T. A. ii. 1. 112.
17. "'To the which." What is the antecedent to the which? Why do we say the which, but never the who ? (Grammar,
"Which," 270.)
25. "The better." When do we add the to a comparative ? (Grammar, 94.) Can the be explained here?
44. "While then." (See 137.) Compare
" He shall conceal it
Whiles you are willing it shall come to note."
T. N. iv. 3. 29.
Illustrate from Greek and Latin.
49. "To be thus thus is nothing but to be safely thus." Explain the grammatical construction of the last clause. (See 385.)
51. " Which would be feared. Modernize would. Explain (Grammar, 329) the Elizabethan usage.
"'Tis much he dares." Is there any object to " he dares "? (244.)
52. "And to that dauntless temper of his mind." Meaning of? (See Grammar, "To.")
54. "None but he." Illustrate this construction by Shakespeare's use of except. (See Grammar, "But.")
56.
" . . . And, under him, My genius is rebuked ; as, it is said,
Mark Antony's was by Cæsar."
See A. and C. ii. 3. 20—30. Trace the meaning of genius from its derivation.
65. " For Banquo's issue have I filed my spirit." Meaning of? Give similar instances of the dropping of the prefix. (See Prosody, 460.)
72. " Champion me to the utterance." Meaning of? Trace the meaning of champion and utterance from the derivation. What historical inference may be drawn from the fact that both these words are derived from the French? Mention a similar inference contained in the dialogue between Gurth and Wamba in " Ivanhoe."
75. "So please your highness." Parse please. (See 297.)
81. "How you were borne in hand, how cross'd, the instruments." Is this an Alexandrine ? (See Prosody, 468 ; and compare
" My books and instruments shall be my company."
T. of Sh. i. 1. 82.)
"Like labour with the rest, where the other instruments."
coriol. i. 1. 104.
"I. But now thou seem'st a coward.
P. Hence, vile instrument."—Cymb. iii. 4. 75.
"Borne in hand." Meaning?
" The Duke
Bore many gentlemen, myself being one,
In hand and hope of action. "—M. for M. i. 4. 52.
We do not now say "to bear in hope," but "to keep a person in hope, suspense," &c. So a rich hypocrite, pretending illness to squeeze presents out of his expectant legatees, is said to—
" Look upon their kindness, and take more
And look on that, still bearing them in hand,
Letting the cherry knock against their lips."
B. J. Fox, i. 1. init.
We still say, to "bear in mind," but we generally use "a hand " in this sense.
83. " To half a soul and to a notion crazed." Meaning of notion here? Compare
" His notion weakens, his discernings
Are lethargied."—Lear, i. 4. 248.
Trace the double meaning of the word from the derivation.
84. "M. Say 'Thus did Banquo.' Murd. You made it known to us." Scan. (See 454.)
87. "Your patience so predominant in your nature." Scan.
88. " Are you so gospell'd to pray for this good man." Modernize. (See 282.)
91. "M. And beggar'd yours for ever. Murd. We are men, my liege. " Scan.
95. "The valued file." Trace this and other meanings of file from the derivation. Explain the meaning and use of valued (374). Could we say "a valued catalogue?"
99. "The gift which bounteous nature hath in him closed." Parse closed. (See 460.) Compare
" Dance, sing, and in a well-mixed border
Close this new brother of our order."—ROWLEY.
What is now the difference between "I have him caught," and "I have caught him" ? Compare
" And when they had this done."—St. Luke v. 6.
100. "Particular addition from the bill that writes them all alike." Meaning of from ? (See Prepositions.)
103. " Not in the worst rank of manhood, say't." Scan. (See 485.)
108. " Who wear our health but sickly in his life
Which in his death were perfect. Murd. I am one, my liege."
What is the antecedent to which ? Scan the second line.
112. "So weary with disasters, tuag'd with fortune." Parse and explain tugg'd. How does the meaning differ from the modern meaning? Compare
" Both tugging to be victors, breast to breast."
3 Hen. VI. ii. 5. 12.
and, for the construction :
" And, toil'd with works of war, retired himself
To Italy."—Rich. II. iv. I. 96.
113. "That I would set my life on any chance." Expand the metaphor. Compare
" Who sets me else ? By heaven I'll throw at all."
Rich. II. iv. I. 57.
116.
"And in such bloody distance, That every minute of his being thrusts
Against my near'st of life."
Expand the metaphor. What is meant by "my near'st of life ? " Illustrate by " home-thrust," and υ κε oς.
120. "And bid my will avouch it." Trace the meaning from the derivation.
121. " For certain friends." Meaning of for here ? How did for become a conjunction?
122. " Whose loves I may not drop." What is the meaning of may? Derive the modern from the original meaning.
123.
" But wail his fall Who I myself struck down."
What is the antecedent to who ? What is there remarkable in the sentence? (Gram. 274.)
127. " Perform what you command us. First Murd. Though our lives—"
What do you suppose the First Murderer intended to say?
Why did Macbeth interrupt him ?
128. "Your spirits shine through you. Within this hour at most.' Scan.
130. "The perfect spy of the time." Apparently in this difficult passage spy is put for "that which is spied," "knowledge."
132. "Always thought." Parse thought. Illustrate the construction from Greek.
"From the palace." From, how used?
138. "I'll come to you anon. We are resolved, my lord."
Perhaps "t' you anón" is to be considered as one foot. If not, how can this verse be scanned? (See 500.) What is the emphatic word in the Murderer's reply?
# SCENE 2.
3. "Say to the king, I would attend his leisure" Modernize the latter words. Trace the different meanings of attend from the derivation. What is the exact meaning of would?
9.
" Lady M. 'Tis safer to be that which we destroy Than by destruction dwell in doubtful joy.
Enter MACBETH.
How now, my lord ! Why do you keep alone?"
Illustrate the character of Lady Macbeth from her words before and after the entrance of her husband. Why and when, for the most part, does Shakespeare use rhyme?
11. "With them they think on. Things without all remedy." Scan. What is the object of on ? (See 242.) How is all used ?
16. "But let the frame of things disjoint, both the worlds suffer." Perhaps a pause is intended after "let :" "But let—yes, even the frame," &c. In that case "But let" is an unfinished verse, and the rest is a complete verse. In the Fol. 1623 the first line ends with "disjoint," containing four accents. When does Shakespeare use verses with four accents (505–9) ?
19. "That shake us nightly; better be with the dead." Scan. How can you justify an accent on the first syllable in the foot "bétter ?"
21.
" Than on the torture of the mind to lie In restless ecstasy. Duncan is in his grave."
What suggested the expression " to lie on the torture of the mind "? Trace this, as well as the modern, meaning of ecstasy from the derivation. Compare
"Where violent sorrow seems
A modern ecstasy."—Macbeth, iv. 3. 170.
Give instances of classical words restricted in meaning by modern, compared with Elizabethan, usage. (See Introduction.) Scan the latter line.
27. "Gentle my lord." Explain and illustrate the position of my. (See 13.)
29. "Be bright and jovial among your guests to-night." Trace the meaning from the derivation. Give words similarly derived. Scan.
30. "Let your remembrance apply to Banquo." Scan. (See Prosody, 477.)
38. "Nature's copy." Meaning of? Comp. T. N. i. 5. 257 :
"'Tis beauty truly blent whose red and white
Nature's own sweet and cunning hand laid on."
40.
"Ere the bat hath flown His cloister'd flight."
What is alluded to?
42. "The shard-borne beetle." Shard is scale. Ben Jonson talks of "scaly beetles with their habergeons." And in Cymb. iii. 2. 20, " The sharded beetle" is opposed to " the full-winged eagle."
46. " Seeling night." To seel was " to close the eyelids of hawks partially or entirely by passing a fine thread through them ; siller, Fr. This was done to hawks till they became tractable."—NARES.
48. " Cancel and tear to pieces that great bond. " Comp. Rich. III. iv. 4. 77 : " Cancel his bond of life." Macbeth iv. I. 99 : " Shall live the lease of nature. " And—
" Through her wounds doth fly
Life's lasting date from cancell'd destiny."—R. of L.
Explain the meaning of the expression here, and trace the meaning of cancel from the derivation.
54. "Hold thee still." Modernize. (See 20.)
# SCENE 3.
3, 4. " To the direction just." Meaning of to? (See 187.)
5. " Now spurs the lated traveller apace." Modernize. Illustrate by similar instances the shortening of the word.
INE
10. "Within the note of expectation." This may perhaps mean, "the memorandum or list of expected guests," Compare
"I come by note."—M. of V. iii. 2. 140.
"That's out of my note."—W. T. iv. 3. 49.
Otherwise it may mean "the boundary," "limit." Compare
"Within the prospect of belief."—Macbeth, i. 3. 74.
# SCENE 4.
1.
"Sit down: at first And last the hearty welcome."
Compare 1 Hen. VI. v. 5. 102 :
"Ay grief I fear me both at first and last."
Meaning of? What distinction is now made between first and at first, last and at last ?
5.
"Our hostess keeps her state, but in best time We will require her welcome."
Show, from the antithesis implied in but, what is meant by "keeping her state." Compare
"The king caused the queene to keepe the estate, and then sate the ambassadors and ladies, as they were marshalled by the king, who would not sit, but walked from place to place making cheare."—HOLINSHED, quoted by CLARK and WRIGHT.
The "state" was used technically to mean "a canopy."
11. "Be large in mirth." Modernize. Illustrate from largess.
12. "The table round. There's blood upon thy face. M. 'Tis Banquo's then." What name has been given, and why, to this arrangement of the parts of verses? Compare lines 15, 20, 51, 69, which are similarly arranged. (See Prosody, 513.)
13. "'Tis better thee without than he within." Meaning? Comment on the syntax. (See 206, 212.)
23. "As broad and general as the casing air." Compare 2 Hen. VI. v. 2. 43 :
"Now let the general trumpet blow his blast.'" Meaning of general ? Modernize. What is the difference between "general," "universal," and "common"?
34.
" The feast is sold That is not often vouch'd, while 'tis a-making,
'Tis given with welcome: to feed were best at home."
Analyse the sentence, and show the confusion of two constructions. Whence arose the use of a, as in a-making? (See 140.) Scan the last line.
36. "From thence." Meaning of? (See 158.)
42. "Who may I rather challenge for unkindness." Is who always used for whom? Whence arises the difference between may, in "may I challenge," as here, and "I may challenge" ?
57. "You shall offend him." Modernize. What is the present rule for the use of shall with respect to the second and third persons? How did the rule arise? (See 317.)
61. "This is the very painting of your fear." Modernize. Trace from the derivation the Elizabethan meaning, and hence the modern meaning, as in "His very dog deserted him."
64. "Impostors to true fear." Meaning of to ? (See 187.)
66. "Authorized by her grandam." Compare for the accent—"His madness so with his authorized youth."—L. C. 15. "Authorizing thy trespass with compare."—Sonn. 35.
75. "Ere human statutes purged the gentle weal." How is gentle used? If the weal was already gentle, how did it require to be purged ?
79.
"The times have been
That, when the brains were out, the man would die."
Modernize that. Illustrate this use. (See 284.)
81. "With twenty mortal murders on their crowns." Why twenty ? (See above, line 27.)
87. "To those that know me. Come, love and health to all." Scan this and the previous line.
91. "We thirst." Thirst is not used elsewhere by Shakespeare in the sense of "drinking a health." [? "first."]
95. "Thou hast no speculation in those eyes." Illustrate from this use of speculation the general difference between the Elizabethan and the modern use of classical words. (See Introduction.)
98. "Only." Probably transposed. (See Grammar, 420.)
99. "What man dare." Why not dares? Compare
" Let him that is no coward
But dare maintain. "—I Hen. VI. ii. 4. 32.
(Dare occurs thus three times in the unhistorical plays, dares thirty times. In the historical plays dare eight, dares seven times.)
105. "If trembling I inhabit, then protest me." No other instance has been given where inhabit means "linger at home." Shakespeare may, however, have derived this use of the word from o κoυρε ν (" to be a stay-at-home" as opposed to "going out to war") through NORTH'S Plutarch, 100 :—
"The home-tarriers and house-doves," &c.
Trace this and the modern meaning of protest from the derivation. Comp. M. Ado, v. 1. 149 :
" I will protest your cowardice."
106. "The baby of a girl." Baby was sometimes used for "doll : "
" And now you cry for't
As children do for babies back again."
B. and F. (HALLIWELL).
109. "You have displaced the mirth, broke the good meeting." What is here contrary to common usage? (See 343.)
112.
" You make me strange Even to the disposition that I owe."
Comp. C. of E. ii. 2. 151 :
" As strange unto your town as to your talk."
Owe is frequently used for ow(e)n, as ope for open. Comp. debeo from de and habeo.
122. Why does not Lady Macbeth continue her expostulations when she is alone with her husband ?
124. "Augurs and understood relations." Comp. below, iv. 3. 173 :
"O, relation
Too nice, and yet too true."
The utterances of birds are apparently called relations.
126. "What is the night?" Illustrate this use of what. (See 252.)
129. "Did you send to him, sir ?" Why does Shakespeare here make Lady Macbeth thus address her husband?
133. "And betimes I will to the weird sisters." This line must probably be scanned by pronouncing weird as two syllables. (See Prosody.) In the Folio weird is spelt weyard. Comp. ii. 1. 20 :
" I dreamt last night of the three weird sisters."
138. "Returning were as tedious as go o'er." Parse returning and go.
141. "You lack the season of all natures, sleep." Illustrate from this and other passages the practical and unimaginative character of Lady Macbeth, as contrasted with her husband. Compare with this v. 1. Compare also ii. 2. 67 : "A little water clears us of this deed;" and v. 1. 35 : "Yet here's a spot," and, in the same scene, "What, will these hands ne'er be clean?" In what sense may such lines as ii. 2. 67, iii. 4. 141, be called specimens of "irony"?
Compare also Duncan speaking of the first (not of the second) Thane of Cawdor :
"There's no art
To find the mind's construction in the face.
He was a gentleman on whom I built
An absolute trust."—i. 4. 11.
In the same scene, 1. 58, Duncan says of Macbeth, "It is a peerless kinsman."
Other instances of Shakespearian "irony" may be found in Rich. III. iii. 2. 67; Coriol. iii. 1. 19; 1 Hen. IV. ii. 4. 528, compared with 2 Hen. IV. v. 5. 51; A. and C. i. 2. 32, compared with Ib. v. 2. 330, T. of A. i. 2. 92, Rich. III. i. 2. 112, and Ib. iv. 1. 82; Macbeth, ii. 3. 97-100, and Ib. v. 2. 22; Rich. III. iii. 1. 110.
# SCENE 5.
1. Why does Shakespeare make the witches speak in a different metre from the rest of the play? Illustrate from the Midsummer Night's Dream and the Tempest.
7. "Close contriver of all harms." Meaning of close? Comp. Cymb. iii. 5. 85: "Close villain, I'll have thy secret."
11.
"All you have done Hath been but for a wayward son."
Illustrate this from Lady Macbeth's description of her husband, i. 5. Contrast the character of Macbeth with that of Richard III.
24. "There hangs a vaporous drop profound." Perhaps mysterious.
32.
"And you all know security Is mortals' chiefest enemy."
Trace the modern meaning of security from the derivation. What does it mean here? Illustrate from Milton's Allegro.
# SCENE 6.
2. "Only I say." Probably transposed as above.
4. "Was pitied of Macbeth." Modernize. Account for this use of of.
8. "Who cannot want the thought how monstrous." Scan. (See Prosody, 477.) Compare, for the meaning of want, W. T. iii. 2. 55.
19. "I think . . . they should find." Modernize. Explain the difference between the Elizabethan and the modern should. (See 326.)
"An't please heaven." Explain an't (See 101.)
21. "He fail'd his presence." Comp. Lear, ii. 4. 143:
"I cannot think my sister in the least
Would fail her obligation."
How is fail now used when it takes an object after it ?
27. "Received of the most pious Edward." (See line 4.)
30. "Is gone to pray the holy king upon his aid." Unless it can be shown that upon is sometimes used for on, this line, as it stands, is an Alexandrine.
35. "Free from our feasts and banquets bloody knives." Comp. Timon of A. v. 1. :
"Rid me these villains from your companies."
Also perhaps Tempest, Epilogue: "Prayer which frees all faults."
36. "Do faithful homage." Trace the modern and ancient meaning from the derivation.
38. "Hath so exasperate the king." Why is the d omitted? (See 343.)
40. "And with an absolute 'Sir, not I.'" Compare "an absolute 'shall.'"—Coriol. iii. 1. Also, "an absolute and excellent horse."—Hen. V. iii. 7; "I am absolute 'twas very Cloten."—Cymb. iv. 2. Trace the different meanings from the derivation.
42. "As who should say." (See 257.)
# INDEX TO THE QUOTATIONS
# FROM SHAKESPEARE'S PLAYS.
The references are to the numbered paragraphs, and to the scenes and lines of the "Globe" edition.
References marked thus (†) will not be found quoted in the paragraph referred to, but similar references will be found explaining the difficulty of the reference in question.
References in parentheses thus (6) refer to the explanatory notes at the end of the play.
# VERBAL INDEX.
# A
A- abbreviated preposition adverbial prefix
A, an (article)
omitted after "like," "as" " " "what" " " "so"
in archaic poetry
"A many men," "an eight days"
used for "one"
"Many a man"
transposed
'A for " he "
Accent, pause accent
on monosyllabic prepositions
on other monosyllables, especially "the"
emphatic accent, or "stress"
Elizabethan, on some words thrown forward
thrown back
variable, why?
Accents, five
six apparently
four apparently
emphatic
Acc'ess
Accuse (noun)
Active participles, confusion in
Addict (participle)
Adjectival phrases transposed
Adjectives
both active and passive
combined together
anomalously formed
transposed
as adverbs
transposition of
used for nouns
Adverbs, formation of
transposed
adverbial compounds
After (adv.)
= "according to "
Again = "on the other hand"
Against, used of time
Alar(u)m
Alexandrines, very rare
apparent
Alive
All for "any"
for "every"
used adverbially
"Without all question"
All-obeying = all-obeyed
All-to
Almost="mostly," "generally"
Alone = "above all"
Along
Amphibious section, the
An, one, pronunciation of
"And if" = if indeed
"And though"
And = "and that too"
in answers
used for "also" by Wickliffe
with the subjunctive
= "even"
= "if"
Ang(e)ry
Anon. "Ever and anon"
Another
Antecedent, plural with singular verb
An't were
Anything, (add.)
Archbishop
Arose for "arisen"
Arrived. "Arrived our coast"
Arrived. "I am arrived"
Article. See "a," "the." indefinite, transposition of
Artificial, adj. active
As
a contraction of "al(l)so"
= "as if"
= "lamely"
"So as"
"As that"
That as
"As then"
= "which," "where"
= "for so"
= "though"
"when as"
"as-as"
"so...as," omitted in
"that...(as) to," omitted in
Asp'ect
At. "At friend"
"At the first"
"At first" = "at the first"
-Ation, -ition, suffix omitted
Auth'orize
Auxiliary verbs
Away. "I cannot away with"
A-weary
Awful = "awe-struck"
# B.
Back. "To and back"
Backward (noun)
Bad (noun)
Banish. " I banish you the land"
Bar. " I bar you your rights"
Barn (verb)
Barr(e)ls
Barr(e)en
Be (verb), how used
Be-, prefix
dropped
Beaten
Because = "in order that"
"for because"
Be-en, plural of "be"
Befal. "Fair befal"
Behaved. "Have I been behaved "
Beholding
Being used like "seeing"
Beshrew. "Beshrew my soul but"
Besides = " For the rest"
Best. "I were best"
Bestow. "I bestow this of you"
Better. "I were better"
Bin, plural of "be"
Blame. "Too blame"
-Ble, suffix active
Bloat = "bloated"
Bodement
Both
= only
"But only"
transposed
with subjunctive = "sunless"
for "each"
Brain (verb)
Briefly = "recently"
But
meaning and derivation of
transition of
signifying prevention
"I doubt not but"
"No more but"
But-en, E. E. = "without"
By, adv
prep
"to come by"
prep. = "about"
# C.
Call for "recal"
'Came for "became"
Can. "And they can well on horseback"
Can'onized
Canstick = "candlestick"
Care. "I care not who hnow it"
Careless (passive) = " ncared four"
Catch'd and "caught"
-Ce' final for -ce's
Cease = "cause to cease"
Chance. "How chance?"
Chanced (partic.-pass.)
Chaucer, varies in accentuation
uses French transpositions
Cheap. "Good cheap"
Chid (participle)
Child(e)ren
Childing. "Chikding autumn"
Chose for "chosen"
'Cide for decide
Climate = "live"
(verb)
Come. "To come view fair Portia"
Command(e)ment
Comm'erce
Comp'act (noun)
Comparative in er after dentals and liquids
doubled
Com'pell'd
Complain. "Complain myself"
Com'plete
Compound words
phrase compounds
anomalous
Condition, expressed by participle
Conditional sentences, irregularities of
Confusion of constructions
in superlatives
with "whom"
Conjunctions
"that" a conjunctional affix
conjunctional sentences, ellipses in
Construction, irregularities of
Consult (noun)
Contemptible="contemptuous"
Contract, for "contracted"
Contraction or slurring of syllables in pronunciation
Couplets, trimeter
# D.
Dare. "He dare," "he dares"
Dazz(e)led
Dear (dissyllable)
Declined. " I am declined"
Degenerate (participle)
Deject (participle)
for "dejected"
Denied. " First he denied you had in him no right"
Desire. "I desire you of pardon"
Devote for " devoted "
Dialogue in verses of three accents
Dipthongs dissyllabled
Dis-, prefix
Disdamed = disdainful
Dishabited
Disjoint (participle)
Dislikes. "It dislikes me"
Disnatured
Disnoble
Dispose (noun)
Divine, adj. transposed by
Chaucer and Shakespeare
Do
"Little is to do"
"What's more to do"
"I (do) not know"
"To do salutations"
omitted and inserted
"Don," "dout" from "do"
Door (dissyllable)
Dreadful = "awe-struck"
Drove for "driven"
Droven for "driven"
# E.
E final pronounced
of French origin pronounced
Each for "both"
for "each other"
Eas(i)ly
Eat for " eaten"
-Ed final for -ful, -ing
in participles dropped after "t," "te," &c
Either (monosyllable)
Ejaculation, not reckoned in the verse
Elision of "the," "to," &c before vowels
Ellipses
of a verb of speech
after "will" and "is"
in conjunctional sentences
of "it"
of "it is"
of "there is"
of "is"
of "neither" before "nor"
of nominative
of "one" before "other"
of superlative inflection
of a verb of motion
in Antithetical sentences
in Relative sentences
Emphasis, different in different accented syllables
prolongs words
prolongs monosyllables
En (third person plural inflection)
prefix
suffix
termination
Endeavour. "Endeavours thyself"
Eng(e)land
Enshield for " enshielded"
(participle)
Entertain(e)ment
Env'y (verb)
-Er, -el, and -le final dropped
final, a dissyllable
suffix
-Es. Final es dropped after ss, ce, ge
third person plural inflection if pres. indic
Escaped. "Was escaped"
-Est, dropped in superlatives after dentals and liquids
-Eth (third person plural inflection)
Even, transposed
"But even now"
Ever
Ever
Every, one, other, neither (plural nouns)
Evil (monosyllable)
Eye = "appear"
Except, excepted
Ex'ile
Expect (noun)
Expire (verb transitive)
Exteriorly
Extra syllable before a pause
two pause-extra syllables
# F.
Fair. "Fair befal
(noun)
Fairies speak in verses of four accents
Fall (verb transitive)
False (verb)
Fame (verb)
Famous'd (participle)
Far = "very"
for "farther"
Fastly
Fault (verb)
Fear (dissyllable)
"fear me not" = "fear
not for me"
(Verb act.)
Fell. "Thou hast fell"
Felt (adjective)
Fidd(e)ler
Flour(i)sh
Folio reads "and" for "an"
(see Index to Plays)
has the 3rd pers. pl. indic.
pres. in -s
Folio writes "it" for "its"
misprints in
Fool. "Why old men fool (verb)
Foot (verb)
For. "For all this"
= "as regards"
= "because"
= "because of"
"for because"
"I am for"
= "instead of"
"For that"
= "to prevent"
prep
For-, prefix
"For to"
Force (verb)
Foreign idioms
Forgot (participle)
"You are forgot" = "have forgotten yourself "
Forth, without verb of motion
= " from "
French, transposition of adjectives.
Fretten
Frighten
From. "From out"
without verb of motion
Froze for "frozen".
-Ful, suffix active and passive
Furnace (verb)
Future for subjunctive.
# G.
Gave for "misgave"
General (noun)
Glad (noun)
Go. "Go along" = "come along"
= "walk" in Wickliffe
"go to"
Good. "Good my lord".
"good now".
"good chea"
Graft (participle).
Grav'd = "entomb'd"
Guiled. "Guiled shore"
# H.
Hand(e)ling
Happen'd (partic. pass.)
Happily = "haply"
Happy (verb).
Hardy "bold"
Have. "Should have"
"to have" omitted after "would have"
"Thought to have begged"
Hear. "Who heard me to deny it?"
Heat (participle)
He for "him"...
"man"
Hence, without verb of motion.
Hen(e)ry
Henry VIII. not written by Shakespeare
Her, antecedent of relative
for "herself"
"its"
"'s"
Here. "Thy here-approach"
Hers, used for "her" adj.
Him, dative
for "he"
"himself".
= "he whom".
Hinder. "Who shall hinder me
to weep?"
His, antecedent of a relative
for "its"
"'s".
Hither, without verb of motion
monosyllable
Hitherto, used of space
Hoist (participle)
Holp for "holpen".
Homager
Home. "Speak him home"
Honest (verb)
Hour (dissyllable)
How. "How chance?"
for "however," for "as"
However (it be)
Hybrid compounds
# I.
I for "me"
unaccented dropped
" (I) beseech you"
slurred in "minister," &c
If. "If that"
Ignomy.
Impersonal verbs
Importless
-In for "-un," prefix
In. "He fell in love"
= "in the case of"
"In round"
= during: "in night"
with the verbal, "in sleeping"..
Indicative
Simple Present for Complete
Simple Past for Complete Present
Present,Third Pers. Pl. in -en
in -es, th
Past in u
Second Per. Sing. in -ts
Future for Subjunctive
"And thou lovest me"
Infect(participle)
Infinitive
active for passive
indefinitely used
perfect, "He thought to have done it"
used as a noun.
"To" omitted ; inserted
omitted and inserted
after the same verb
with noun, used as subject or object
Inflections.
-Ing, termination
confused with the old inflection "en".
Inhabited = " housed"
In's for "in his"
Interjectional lines
Interrogative Pronouns, transition from to Relative
Into, with verbs of rest
accent of
Inward (noun)
Irregularities of construction
Is, ellipses after
ellipse of
-Iséd final in polysyllables
It.
ellipse of
for "its".
" it is," ellipse of
"To voice it with claims"
emphatic as antecedent
-Ition, -ation, suffix omitted
Its, post-Shakespearian
substitutes for
-Ive, suffix passive
# J.
Jugg(e)ler
Just, adj. = "exact"
Justicers
# K.
Know. " I know you what you are"
# L.
Lack = "to be wanting"
Laid (adjective)
Lated (verb)
Latinisms
Learn (verb act.)
Lengthening of words in pronunciation
Less, suffix
Let = "did".
Like. " If you like of me"
Likes. "It likes me"
Lines, see Verses.
Liquids introduce a semi-vowel
List. "List a brief tale"
'Longs for "belongs"
Look. "To look your dead"
Lover'd.
-Ly, suffix
# M.
Mad (verb)
Maj(es)ty, (dissyllable)
Malice (verb).
Many. "Many a man"
"A many men"
a noun
an adjective adverbially used
Mark. "Mark King Richard how he looks"
Marle for "marvel"
May
"May not" = "must not"
used for the subjunctive in
the sense of purpose.
Me for "I"
= "for me," "by me"
= "myself"
"Of me" for "my"
"Me rather had"
Mean. "What mean ye to
weep?".
Meered (particip.)
Meiny = "train"
derivation of
-Ment, suffix
Mere, adj. = "complete"
Mered (particip.)
Might
= "could"
Million'd (participle passive)
Mine, how differs from "my"
used for "my".
Misbecomed for "misbecame"
Mistook (participle)
Monosyllables accented
unaccented
prolonged so as to make up a foot
Monosyllabic prepositions, accent of
Moods
Moraler
More, most = "greater" "greatest"
"More better"
"More fearful"
"No more but"
Most = "greatest"
"Most best"
Mouthed (participle passive)
Much = "great"
Must, original use of
= "is to"
My, how differs from "mine"
"Good my lord
Myself (derivation of)
# N.
Names, used as adjectives
polysyllabic, receive but one accent
Near for "nearer"
Necessited
Neck. "In the neck of that"
Need (verb intr.)
" What need?"
Needs (adverb)
Negative, double
Neither, ellipse of, before "nor"
a monosyllable
used for "both"
-Ness, suffix
Never. "Never so "
No. "No more but"
Nominative absolute
Nominative, ellipsis of
implied from participial phrases.
None. "I will none of it "
Nor, used for "and"
Not = "not only"
"I not doubt"
Nothing (adv.)
(perhaps).
Noun absolute
noun-compounds
of French origin formed from verbs without change
Nour(i)sh
# O.
Object, redundant
Objective following intransitive verbs
Ob'scure
Of.
accented in " out of"
= about
= as a consequence of
= as regards
"Blowing of his nails"
= by
= from
= on
original meaning "from"
"To admit of".
with verbs of filling
Off
connected with "of"
On
= about
"I fall on weeping"
"One on's ears"
"On sleep"
Once = "above all;" "once for all"
"At once" = "once for all"
One, ellipse of, before "other"
= "above all "
how pronounced
(adjective)
Only transposed
= "mere"
Ope for "open" (adj.)
Or, "or or"
= "before,' "or ere," "or ever"
Other for "others"
monosyllable
(singular pronoun)
Ought. "You ought not walk"
Our, antecedent of relative
"Come, our queen"
= "of us"
Ourselves, derivation of
Out (preposition)
Over = " over again "
Overwatched = fatigued
Owe for "own"
Owing (adjective)
# P.
Pale (noun)
Paled (passive)
Parted. "Parted with" for "parted from"
Participles.
-ed omitted after d and t
-en dropped
irregular formations of
prefix y\-
imply a condition
used absolutely without Noun or Pronoun
Passive with some verbs of motion
Past for Present tense
Path (verb)
Pause, effect of an accent
the pause-extra-syllable
two pause-extra-syllables
frequently prolongs a monosyllable
in verses of four accents
Peer = "cause to peer"
Peers (verb transitive)
Pensived (passive)
Perchance. "Perchance I will"
Perfect infinitive
Perish = "destroy"
Perishen = they perish
Pers'ever
Pined (passive)
Pitied. "It would have pitied
a man"
Plain = "make plain"
Pleaseth. "Pleaseth it"
Pleasure, has two accents
Possess = " inform "
Practised = "plotted against"
Prefixes
dropped
"en-"
Prepositional compounds
Prepositions
doubled
Prepositions omitted before indirect object
omitted after verbs of motion, worth, hearing, and other verbs
omitted in adverbial phrases
transposed
accent of
local and metaphorical meaning
restricted in meaning
transition of into conjunctions
Present, Simple for Complete
Presently = "at once"
Private (noun)
Probable (adj.), active
Pronoun, personal
redundant
relative
omitted
anomalies of.
between conjunction and infinitive
transposed
Proper = "own"
Prose, when used
Prosody
Prowess, (quasi-monosyllable)
# Q.
Quail = "make to quail"
Quit (participle)
# R.
R softens or destroys a following
or preceding vowel
prolongs -er.
when following a vowel prolongs a monosyllable
-r and -re final dissyllabize monosyllables
after dentals introduces a quasi-vowel
Recall. "Unrecailing" for "unrecalled"
Relatival constructions
Relative
with plural antecedent and singular verb
omitted
with supplementary pronoun
See "who," "which," "that."
Relish (verb transitive)
Remains for "it it remains"
Remember = "remind"
Rememb(e)rance
Retire (verb act.)
Rhyme, when used
Right used for "true"
Rode for " ridden "
Round = "straightforwardly "
Royal, why transposed
Run. "Is run"
# S.
'S, adverbial suffix
-S final dropped after se, ce.
S misprinted in Folio
Sanctuary pronounced "sanct'ry"
Sat. "Being sat"
Save. "Save he"
Sawn for "seen"
Say used for "call"
'Say'd for "assayed"
Scaling = "weighing"
'Se for "shall"
Se'cure
Seldom (adjective)
Self (adjective)
omitted
Semb(e)lance
Sense for "senses"
Several (noun)
Severally = " separately"
Shaked for "shaken"
Shall
"I shall, my lords"
= "is sure to"
"It shall come to pass"
"Mark you his absolute shall"
She for "her"
"woman"
Shine (verb act.)
(verb transitive)
Should
denotes contingent futurity
= "ought," " was to"
" should have "
like German "sollen"
after past, corresponds to "shall" after present
Show = "appear"
Sightless (passive)
Since, difference of tenses with.
"A year since"
"Since that"
= "when".
Sir, a mark of anger
Sith
Smit for " smitten "
Smote for "smitten"
So inserted
omitted
for "also"
for "then"
"So long time"
= "provided that"
"So that," "so as"="provided that"
"So defend thee heaven"
"So (as)" omitted
"So as".
"So that:" that omitted
so omitted
"So where"
Solicit (noun)
Some
Something, adv.
Sometimes = "formerly"
Sorrow. "I am sorrow"
Spake for "spoken"
Speak. "Speak him fair"
Splitted
Spoke (participle)
Squint (verb act.)
Stand. "It stands me upon"
"Stand on tip-toe"
Stept. "Being deep stept"
Still for "constantly"
Streaming = "unfurling"
Strove for "striven"
Strucken
Studied. "As one that had been
studied"
Subjunctive
in a dependent sentence
of purpose
used indefinitely after the relative
used optatively or imperatively
with "an" or "and"
Such. "Such as" = "Such that"
"Such that"
"Such where"
"Such which"
Suffixes.
"-ation," "-ition," omitted
"-ble" (active)
"-en"
"-ive" (passive)
"-less"
"-ly"
"-ment ".
"-ness"
Suffix, "-y"
Suffocate (participle)
for "suffocated"
Superlative in -est for "very"
after dentals and liquids
confusion in
double
inflection, ellipse of
Swam for " swum "
Sweaten
Sworder
Syllables dropped in writing
dropped or slurred in pronunciation
# T.
Taint (participle)
Tear (dissyllable)
Tenses, irregularities of
Terrible = "frightened"
Than, with comparative,
explained.
for "then"
Thankful = "thank-worthy "
That, demonstrative, "tkat as"
"thaw which"
difference between "that," "who," and "which"
relative
less definite than "which".
"whatsoever tkat"
a conjunctional affix
= "because," "when"
in "after that," &c. like "quam" in "postquam"
omitted and then inserted
omitted after "so"
"So that:" "so" omitted.
"Such that"
= that which
before a verbal
"The better"
omitted
"The which"
The omitted in archaic poetry.
"the that"
apparently accented
" Lifts the head"
"The Talbot;" "the death"
Thee, dative
for "thou"
Their the genitives of "they"
Them for them"
Then for "than"
for (?) "when".
There, for " thereupon," " then"
"There is," ellipse of
They. "They in France"
Thinks (verb impersonal).
"Where it thinkst best"
"Methoughts," "methink"
"I think it be "
This for "this is"
Thorough for "through"
Thou
omitted
between equals
to servants
as an insult
rhetorical
apparent exceptions
Γhough. "Though that"
Thought. "Thought to have begged"
Thyself, derivation of
Till = to
To
used after." see," "feel"
= according to.
"To be abridged"= "about being abridged ".
"To be" for "being"
" To give you" = "by giving you".
"I would to God"
= "in addition to"
= "in comparison with"
inserted for connection
inserted, omitted
="like"
prefix
= "with a view to"
"With God to friend"
"To-fore"
Toil (verb act.)
Toil'd (passive)
Tongue (verb)
" To-night"
Too = "very"
" Too blame"
Took (participle)
Towards, sometimes
Traded (passive)
Transpositions
of adjectives and participles
of indefinite article
of adverbs
of possessive adjectives.
of prepositions
Trifle (verb transitive).
Trimeter Couplets
# U.
Un- for "in-," prefix
Unaccented syllable of a trisyllable softened
Under (adjective)
Undoubled = "undaunted"
Unfair (verb)
Unsisting for "unresisting".
Until with subjunctive
for "unto"
Upon
" It stands me upon"
Us for "we"
Utensils
# V.
Verbal preceded by "the" and not followed by "of"
preceded by "in"
followed by "of" and not preceded by " the "
Verb-compounds
Verbs, auxiliary
intransitive used transitively
impersonal
inflection of third for second person
intransitive followed by the objective
of filling with "of"
passive, formation of.
singular inflection with plural subject
reflexive
formed from nouns and and-jectives.
transitive used intransitively
passive to express motion
Indicative mood
Infinitive
Subjunctive
Participles
Tenses
Verses of five accents
of six accents apparently
of four accents apparently
of three and two accents
the Amphibious verse
in four accents spoken by fairies, witches, &c.
Versing (writing in verse)
Very = true
Vouchsafed
Vowels, when unaccented in a polysyllable, slurred
affected by γ.
# W.
Waft (participle)
Waged = "paid"
Wanteth (impersonal verb)
Warr(a)nt
Were, subjunctive use off.
What, exclamation of impatience
semi-transition to relative, how checked
= "any"
= "whatever," "who"
= "why"
followed by antecedent
= " of what a nature ? "
Whatsoever, "whatsoever that"
Whe'er for "whether"
When. "When that"
exclamation of impatience
Where. "Sowhere"
"Suchwhere"
= "whereas"
Whereas = " where"
Whether. "Or whether"
Which, anomalies of
" Such which"
"which that"
difference between "which," "who," and "that"
interchanged with "who" and "that"
less definite than "who"
more definite than "that"
with repeated antecedent
"The which"
= "which thing," parenthetical.
= "as to which"
While, whiles
"While that"
"a-while," "whilom"
= till
with subjunctive
Whilst. "The whilst"
Whist for "whisted"
Who, transition from relative to interrogative
"As who should say"
difference between "who," "which," and "that"
= "and he," "for he"
personifies irrational antecedents
Who for "whom"
more definite than "which"
Why. "Why that"
"For why ?"
"Wky and for what"
"Why and wherefore'
Wilful blame
Will, ellipses after
substituted for "shall"
"That he will"
"I will not" = "I shall" in Shakespeare
difficult passages
Wish. "The rest I wish thee gather"
Witch (verb)
Witches speak in verses of four accents
With
= "like"
= "by"
"I live with (on) bread"
Withal
Without = "unlike" "outside of"
Woe. "I am woe"
Wont, derivation of
Would = "was wont to"
not used for "should"
for "wish, require"
" I would to God"
in the consequent clause
Wreathen (participle)
Wrest(e)ler
Writ (participle)
Wrote for "written"
# Y.
Y-, (participial prefix)
V-ravished
-Y, suffix
Ye, differs from "you"
Year'd (passive participle)
Yearns. "It yearns me not"
Yet = "as yet" before a negative
You, differs from " thou "
a mark of anger to servants, "you, sir"
differs from "ye"
Youngly (adverb)
Your, antecedent of relative
= "of you"
colloquial use of
dissyllable
Yours. "This of yours"
Yourselves, derivation of
A CATALOG OF SELECTED
# DOVER BOOKS
IN ALL FIELDS OF INTEREST
CONCERNING THE SPIRITUAL IN ART, Wassily Kandinsky. Pioneering work by father of abstract art. Thoughts on color theory, nature of art. Analysis of earlier masters. 12 illustrations. 80pp. of text. 5 × 8½.
23411-8
ANIMALS: 1,419 Copyright-Free Illustrations of Mammals, Birds, Fish, Insects, etc., Jim Harter (ed.). Clear wood engravings present, in extremely lifelike poses, over 1,000 species of animals. One of the most extensive pictorial sourcebooks of its kind. Captions. Index. 284pp. 9 × 12.
23766-4
CELTIC ART: The Methods of Construction, George Bain. Simple geometric techniques for making Celtic interlacements, spirals, Kells-type initials, animals, humans, etc. Over 500 illustrations. 160pp. 9 × 12. (Available in U.S. only.)
22923-8
AN ATLAS OF ANATOMY FOR ARTISTS, Fritz Schider. Most thorough reference work on art anatomy in the world. Hundreds of illustrations, including selections from works by Vesalius, Leonardo, Goya, Ingres, Michelangelo, others. 593 illustrations. 192pp. 7 × 10¼.
20241-0
CELTIC HAND STROKE-BY-STROKE (Irish Half-Uncial from "The Book of Kells"): An Arthur Baker Calligraphy Manual, Arthur Baker. Complete guide to creating each letter of the alphabet in distinctive Celtic manner. Covers hand position, strokes, pens, inks, paper, more. Illustrated. 48pp. 8¼ × 11.
24336-2
EASY ORIGAMI, John Montroll. Charming collection of 32 projects (hat, cup, pelican, piano, swan, many more) specially designed for the novice origami hobbyist. Clearly illustrated easy-to-follow instructions insure that even beginning paper-crafters will achieve successful results. 48pp. 8¼ × 11.
27298-2
THE COMPLETE BOOK OF BIRDHOUSE CONSTRUCTION FOR WOODWORKERS, Scott D. Campbell. Detailed instructions, illustrations, tables. Also data on bird habitat and instinct patterns. Bibliography. 3 tables. 63 illustrations in 15 figures. 48pp. 5¼ × 8½.
24407-5
BLOOMINGDALE'S ILLUSTRATED 1886 CATALOG: Fashions, Dry Goods and Housewares, Bloomingdale Brothers. Famed merchants' extremely rare catalog depicting about 1,700 products: clothing, housewares, firearms, dry goods, jewelry, more. Invaluable for dating, identifying vintage items. Also, copyright-free graphics for artists, designers. Co-published with Henry Ford Museum & Greenfield Village. 160pp. 8¼ × 11.
25780-0
HISTORIC COSTUME IN PICTURES, Braun & Schneider. Over 1,450 costumed figures in clearly detailed engravings-from dawn of civilization to end of 19th century. Captions. Many folk costumes. 256pp. 8 × 11¾.
23150-X
ANATOMY: A Complete Guide for Artists, Joseph Sheppard. A master of figure drawing shows artists how to render human anatomy convincingly. Over 460 illustrations. 224pp. 8 × 11¼.
27279-6
MEDIEVAL CALLIGRAPHY: Its History and Technique, Marc Drogin. Spirited history, comprehensive instruction manual covers 13 styles (ca. 4th century through 15th). Excellent photographs; directions for duplicating medieval techniques with modern tools. 224pp. 8 × 11¼.
26142-5
DRIED FLOWERS: How to Prepare Them, Sarah Whitlock and Martha Rankin. Complete instructions on how to use silica gel, meal and borax, perlite aggregate, sand and borax, glycerine and water to create attractive permanent flower arrangements. 12 illustrations. 32pp. 5 × 8½.
21802-3
EASYTO-MAKE BIRD FEEDERS FOR WOODWORKERS, Scott D. Campbell. Detailed, simple-to-use guide for designing, constructing, caring for and using feeders. Text, illustrations for 12 classic and contemporary designs. 96pp. 5 × 8½.
25847-5
SCOTTISH WONDER TALES FROM MYTH AND LEGEND, Donald A. Mackenzie. 16 lively tales tell of giants rumbling down mountainsides, of a magic wand that turns stone pillars into warriors, of gods and goddesses, evil hags, powerful forces and more. 240pp. 5 × 8½.
29677-6
THE HISTORY OF UNDERCLOTHES, C. Willett Cunnington and Phyllis Cunnington. Fascinating, well-documented survey covering six centuries of English undergarments, enhanced with over 100 illustrations: 12th-century laced-up bodice, footed long drawers (1795), 19th-century bustles, 19th-century corsets for men, Victorian "bust improvers," much more. 272pp. 5 × 8¼.
27124-2
ARTS AND CRAFTS FURNITURE: The Complete Brooks Catalog of 1912, Brooks Manufacturing Co. Photos and detailed descriptions of more than 150 now very collectible furniture designs from the Arts and Crafts movement depict davenports, settees, buffets, desks, tables, chairs, bedsteads, dressers and more, all built of solid, quarter-sawed oak. Invaluable for students and enthusiasts of antiques, Americana and the decorative arts. 80pp. 6½ × 9¼.
27471-3
WILBUR AND ORVILLE: A Biography of the Wright Brothers, Fred Howard. Definitive, crisply written study tells the full story of the brothers' lives and work. A vividly written biography, unparalleled in scope and color, that also captures the spirit of an extraordinary era. 560pp. 6 × 9¼.
40297-5
THE ARTS OF THE SAILOR: Knotting, Splicing and Ropework, Hervey Garrett Smith. Indispensable shipboard reference covers tools, basic knots and useful hitches; handsewing and canvas work, more. Over 100 illustrations. Delightful reading for sea lovers. 256pp. 5 × 8½.
26440-8
FRANK LLOYD WRIGHT'S FALLINGWATER: The House and Its History, Second, Revised Edition, Donald Hoffmann. A total revision-both in text and illustrations—of the standard document on Fallingwater, the boldest, most personal architectural statement of Wright's mature years, updated with valuable new material from the recently opened Frank Lloyd Wright Archives. "Fascinating"—The New York Times. 116 illustrations. 128pp. 9¼ × 10¾.
27430-6
PHOTOGRAPHIC SKETCHBOOK OF THE CIVIL WAR, Alexander Gardner. 100 photos taken on field during the Civil War. Famous shots of Manassas Harper's Ferry, Lincoln, Richmond, slave pens, etc. 244pp. 10 × 8¼.
22731-6
FIVE ACRES AND INDEPENDENCE, Maurice G. Kains. Great back-to-the-land classic explains basics of self-sufficient farming. The one book to get. 95 illustrations. 397pp. 5 × 8½.
20974-1
SONGS OF EASTERN BIRDS, Dr. Donald J. Borror. Songs and calls of 60 species most common to eastern U.S.: warblers, woodpeckers, flycatchers, thrushes, larks, many more in high-quality recording.
Cassette and manual 99912-2
A MODERN HERBAL, Margaret Grieve. Much the fullest, most exact, most useful compilation of herbal material. Gigantic alphabetical encyclopedia, from aconite to zedoary, gives botanical information, medical properties, folklore, economic uses, much else. Indispensable to serious reader. 161 illustrations. 888pp. 6½ × 9¼. 2-vol. set. (Available in U.S. only.)
Vol. I: 22798-7
Vol. II: 22799-5
HIDDEN TREASURE MAZE BOOK, Dave Phillips. Solve 34 challenging mazes accompanied by heroic tales of adventure. Evil dragons, people-eating plants, bloodthirsty giants, many more dangerous adversaries lurk at every twist and turn. 34 mazes, stories, solutions. 48pp. 8¼ × 11.
24566-7
LETTERS OF W. A. MOZART, Wolfgang A. Mozart. Remarkable letters show bawdy wit, humor, imagination, musical insights, contemporary musical world; includes some letters from Leopold Mozart. 276pp. 5 × 8½.
22859-2
BASIC PRINCIPLES OF CLASSICAL BALLET, Agrippina Vaganova. Great Russian theoretician, teacher explains methods for teaching classical ballet. 118 illustrations. 175pp. 5 × 8½.
22036-2
THE JUMPING FROG, Mark Twain. Revenge edition. The original story of The Celebrated Jumping Frog of Calaveras County, a hapless French translation, and Twain's hilarious "retranslation" from the French. 12 illustrations. 66pp. 5 × 8½.
22686-7
BEST REMEMBERED POEMS, Martin Gardner (ed.). The 126 poems in this superb collection of 19th- and 20th-century British and American verse range from Shelley's "To a Skylark" to the impassioned "Renascence" of Edna St. Vincent Millay and to Edward Lear's whimsical "The Owl and the Pussycat." 224pp. 5 × 8½.
27165-X
COMPLETE SONNETS, William Shakespeare. Over 150 exquisite poems deal with love, friendship, the tyranny of time, beauty's evanescence, death and other themes in language of remarkable power, precision and beauty. Glossary of archaic terms. 80pp. × 8¼.
26686-9
THE BATTLES THAT CHANGED HISTORY, Fletcher Pratt. Eminent historian profiles 16 crucial conflicts, ancient to modern, that changed the course of civilization. 352pp. 5 × 8½.
41129-X
THE WIT AND HUMOR OF OSCAR WILDE, Alvin Redman (ed.). More than 1,000 ripostes, paradoxes, wisecracks: Work is the curse of the drinking classes; I can resist everything except temptation; etc. 258pp. 5 × 8½.
20602-5
SHAKESPEARE LEXICON AND QUOTATION DICTIONARY, Alexander Schmidt. Full definitions, locations, shades of meaning in every word in plays and poems. More than 50,000 exact quotations. 1,485pp. 6½ × 9¼. 2-vol. set.
Vol. 1: 22726-X
Vol. 2: 22727-8
SELECTED POEMS, Emily Dickinson. Over 100 best-known, best-loved poems by one of America's foremost poets, reprinted from authoritative early editions. No comparable edition at this price. Index of first lines. 64pp. × 8¼.
26466-1
THE INSIDIOUS DR. FU-MANCHU, Sax Rohmer. The first of the popular mystery series introduces a pair of English detectives to their archnemesis, the diabolical Dr. Fu-Manchu. Flavorful atmosphere, fast-paced action, and colorful characters enliven this classic of the genre. 208pp. × 8¼.
29898-1
THE MALLEUS MALEFICARUM OF KRAMER AND SPRENGER, translated by Montague Summers. Full text of most important witchhunter's "bible," used by both Catholics and Protestants. 278pp. 6 × 10.
22802-9
SPANISH STORIES/CUENTOS ESPAÑOLES: A Dual-Language Book, Angel Flores (ed.). Unique format offers 13 great stories in Spanish by Cervantes, Borges, others. Faithful English translations on facing pages. 352pp. 5 × 8½.
25399-6
GARDEN CITY, LONG ISLAND, IN EARLY PHOTOGRAPHS, 1869-1919, Mildred H. Smith. Handsome treasury of 118 vintage pictures, accompanied by carefully researched captions, document the Garden City Hotel fire (1899), the Vanderbilt Cup Race (1908), the first airmail flight departing from the Nassau Boulevard Aerodrome (1911), and much more. 96pp. 8 × 11¾.
40669-5
OLD QUEENS, N.Y., IN EARLY PHOTOGRAPHS, Vincent F. Seyfried and William Asadorian. Over 160 rare photographs of Maspeth, Jamaica, Jackson Heights, and other areas. Vintage views of DeWitt Clinton mansion, 1939 World's Fair and more. Captions. 192pp. 8 × 11.
26358-4
CAPTURED BY THE INDIANS: 15 Firsthand Accounts, 1750-1870, Frederick Drimmer. Astounding true historical accounts of grisly torture, bloody conflicts, relentless pursuits, miraculous escapes and more, by people who lived to tell the tale. 384pp. 5 × 8½.
24901-8
THE WORLD'S GREAT SPEECHES (Fourth Enlarged Edition), Lewis Copeland, Lawrence W. Lamm, and Stephen J. McKenna. Nearly 300 speeches provide public speakers with a wealth of updated quotes and inspiration-from Pericles' funeral oration and William Jennings Bryan's "Cross of Gold Speech" to Malcolm X's powerful words on the Black Revolution and Earl of Spenser's tribute to his sister, Diana, Princess of Wales. 944pp. 5 × 8 .
40903-1
THE BOOK OF THE SWORD, Sir Richard F. Burton. Great Victorian scholar/adventurer's eloquent, erudite history of the "queen of weapons"—from prehistory to early Roman Empire. Evolution and development of early swords, variations (sabre, broadsword, cutlass, scimitar, etc.), much more. 336pp. 6 × 9¼.
25434-8
AUTOBIOGRAPHY: The Story of My Experiments with Truth, Mohandas K. Gandhi. Boyhood, legal studies, purification, the growth of the Satyagraha (nonviolent protest) movement. Critical, inspiring work of the man responsible for the freedom of India. 480pp. 5 × 8½. (Available in U.S. only.)
24593-4
CELTIC MYTHS AND LEGENDS, T. W. Rolleston. Masterful retelling of Irish and Welsh stories and tales. Cuchulain, King Arthur, Deirdre, the Grail, many more. First paperback edition. 58 full-page illustrations. 512pp. 5 × 8½.
26507-2
THE PRINCIPLES OF PSYCHOLOGY, William James. Famous long course complete, unabridged. Stream of thought, time perception, memory, experimental methods; great work decades ahead of its time. 94 figures. 1,391pp. 5 × 8½. 2-vol. set.
Vol. I: 20381-6 Vol. II: 20382-4
THE WORLD AS WILL AND REPRESENTATION, Arthur Schopenhauer. Definitive English translation of Schopenhauer's life work, correcting more than 1,000 errors, omissions in earlier translations. Translated by E. F. J. Payne. Total of 1,269pp. 5 × 8½. 2-vol. set.
Vol. 1: 21761-2 Vol. 2: 21762-0
MAGIC AND MYSTERY IN TIBET, Madame Alexandra David-Neel. Experiences among lamas, magicians, sages, sorcerers, Bonpa wizards. A true psychic discovery. 32 illustrations. 321pp. 5 × 8½. (Available in U.S. only.)
22682-4
THE EGYPTIAN BOOK OF THE DEAD, E. A. Wallis Budge. Complete reproduction of Ani's papyrus, finest ever found. Full hieroglyphic text, interlinear transliteration, word-for-word translation, smooth translation. 533pp. 6½ × 9¼.
21866-X
MATHEMATICS FOR THE NONMATHEMATICIAN, Morris Kline. Detailed, college-level treatment of mathematics in cultural and historical context, with numerous exercises. Recommended Reading Lists. Tables. Numerous figures. 641pp. 5 × 8½.
24823-2
PROBABILISTIC METHODS IN THE THEORY OF STRUCTURES, Isaac Elishakoff. Well-written introduction covers the elements of the theory of probability from two or more random variables, the reliability of such multivariable structures, the theory of random function, Monte Carlo methods of treating problems incapable of exact solution, and more. Examples. 502pp. 5 × 8½.
40691-1
THE RIME OF THE ANCIENT MARINER, Gustave Dore, S. T. Coleridge. Doré's finest work; 34 plates capture moods, subtleties of poem. Flawless full-size reproductions printed on facing pages with authoritative text of poem. "Beautiful. Simply beautiful."—Publisher's Weekly. 77pp. 9¼ × 12.
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NORTH AMERICAN INDIAN DESIGNS FOR ARTISTS AND CRAFTSPEOPLE, Eva Wilson. Over 360 authentic copyright-free designs adapted from Navajo blankets, Hopi pottery, Sioux buffalo hides, more. Geometrics, symbolic figures, plant and animal motifs, etc. 128pp. 8 × 11. (Not for sale in the United Kingdom.)
25341-4
SCULPTURE: Principles and Practice, Louis Slobodkin. Step-by-step approach to clay, plaster, metals, stone; classical and modern. 253 drawings, photos. 255pp. 8 × 11.
22960-2
THE INFLUENCE OF SEA POWER UPON HISTORY, 1660-1783, A. T. Mahan. Influential classic of naval history and tactics still used as text in war colleges. First paperback edition. 4 maps. 24 battle plans. 640pp. 5 × 8½.
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THE STORY OF THE TITANIC AS TOLD BY ITS SURVIVORS, Jack Winocour (ed.). What it was really like. Panic, despair, shocking inefficiency, and a little heroism. More thrilling than any fictional account. 26 illustrations. 320pp. 5 × 8½.
20610-6
FAIRY AND FOLK TALES OF THE IRISH PEASANTRY, William Butler Yeats (ed.). Treasury of 64 tales from the twilight world of Celtic myth and legend: "The Soul Cages," "The Kildare Pooka," "King O'Toole and his Goose," many more. Introduction and Notes by W B. Yeats. 352pp. 5 × 8½.
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BUDDHIST MAHAYANA TEXTS, E. B. Cowell and others (eds.). Superb, accurate translations of basic documents in Mahayana Buddhism, highly important in history of religions. The Buddha-karita of Asvaghosha, Larger Sukhavativyuha, more. 448pp. 5 × 8½.
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ONE TWO THREE . . . INFINITY: Facts and Speculations of Science, George Gamow. Great physicist's fascinating, readable overview of contemporary science: number theory, relativity, fourth dimension, entropy, genes, atomic structure, much more. 128 illustrations. Index. 352pp. 5 × 8½.
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EXPERIMENTATION AND MEASUREMENT, W.J. Youden. Introductory manual explains laws of measurement in simple terms and offers tips for achieving accuracy and minimizing errors. Mathematics of measurement, use of instruments, experimenting with machines. 1994 edition. Foreword. Preface. Introduction. Epilogue. Selected Readings. Glossary. Index. Tables and figures. 128pp. 5 × 8½.
40451-X
DALÍ ON MODERN ART: The Cuckolds of Antiquated Modern Art, Salvador Dali. Influential painter skewers modern art and its practitioners. Outrageous evaluations of Picasso, Cézanne, Turner, more. 15 renderings of paintings discussed. 44 calligraphic decorations by Dali. 96pp. 5 × 8½. (Available in U.S. only.)
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ANTIQUE PLAYING CARDS: A Pictorial History, Henry Rene D'Allemagne. Over 900 elaborate, decorative images from rare playing cards (14th-20th centuries): Bacchus, death, dancing dogs, hunting scenes, royal coats of arms, players cheating, much more. 96pp. 9¼ × 12¼.
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MAKING FURNITURE MASTERPIECES: 30 Projects with Measured Drawings, Franklin H. Gottshall. Step-by-step instructions, illustrations for constructing handsome, useful pieces, among them a Sheraton desk, Chippendale chair, Spanish desk, Queen Anne table and a William and Mary dressing mirror. 224pp. 8 × 11¼.
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THE FOSSIL BOOK: A Record of Prehistoric Life, Patricia V. Rich et al. Profusely illustrated definitive guide covers everything from single-celled organisms and dinosaurs to birds and mammals and the interplay between climate and man. Over 1,500 illustrations. 760pp. 7½ × 10 .
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Paperbound unless otherwise indicated. Available at your book dealer, online at www.doverpublications.com, or by writing to Dept. GI, Dover Publications, Inc., 31 East 2nd Street, Mineola, NY 11501. For current price information or for free catalogues (please indicate field of interest), write to Dover Publications or log on to www.doverpublications.com and see every Dover book in print. Dover publishes more than 500 books each year on science, elementary and advanced mathematics, biology, music, art, literary history, social sciences, and other areas.
The somewhat grotesque name of "amphibious verse " (Par. 513) sprang in this way from class-teaching. I have retained it, as answering its purpose, by communicating its meaning readily and impressively.
Of course it is possible to study Shakespeare with great advantage, and yet without any reference to textual criticism. Only, it should be distinctly understood in such cases that textual criticism is not attempted.
In correcting the proof-sheets I have gained much from consulting Mr. Walker's " Criticisms on Shakespeare."
Compare "More by all mores."—T. N. v. 1. 139.
It should, however, be stated that the n is often dropped in Early English.
Morris, "Specimens of Early English," p. xxxiii. Inf. "loven." Gerund, "to lovene."
Collapse is accented on the last syllable in most dictionaries.
" How brave lives he that keeps a fool, although the rate be deeper,
But he that is his own fool, sir, does live a great deal cheaper."
Exceptions are "eternal" used for "infernal" (O. iv. 2, 130; J. C. i. 2. 160; Hamlet, i. 4. 21); "triple" for "third" (A. W. ii. 1. 111); "temporary" for "temporal" (M. for M. v. 1. 145); "important " for "importunate" (Lear, iv.4.26); "expiate" for "expired" (Rich. III. iii. 3. 23); "colleagued" (Hamlet, i. 2. 21) for " co-leagued ;" "importing" (ib. 23) for "importuning." The Folio has "Pluto's" for "Plutus" (J. C. iv. 3. 102).
It is used as a singular adjective, without the article, in Cymb. iii. 4. 144: "You think of other place."
Myself seems used for our "by myself" in
" I had as lief been myself alone."- A. Y. L. iii. 2. 269.
Compare "Shall stand a tip-toe."—Hen. V. iv. 3. 42.
Or, adopting this construction, we may take all to mean "the whole house." "The principals did seem to rend, and the whole house to topple."
Compare "A noble peer of mickle trust and power."—MILTON, Comus.
"Stands off" is used for "stands out, i.e. in relief."—Hen. V. ii. 2. 103.
Sith for sither, like "mo" for " mo-er." (See 17.)
The verb "hear" may be supplied from the context.
So γ ρ in Greek.
So almost always in the Folio. See Index to Plays.
So Folio.
So Folio.
Comp. ς, στε, for the various meanings.
Comp. ο ον ξαρτ ετα. γ μον γαμε ν .— SCH. Prom. Vinct. 90S.
Halliwell's Dictionary.
Similarly "sauf" was used in French in agreement with a noun placed in the nominative absolute.
The old form sith occurs several times in Shakespeare, and mostly in the metaphorical meaning "because." Sith in Hamlet, ii. 2. 12, is an exception. Sith in A.-S. meant "late," "later;" "sith-than," "after that." Sithence (Chaucer, "sethens," "sins") is found once in Shakespeare.
COMP. ντ , which in composition denotes against, and at other times instead of, for.
But " towns of war," Hen. V. ii. 4. 7, means " garrisoned towns," and so probably here, like our "man of war."
Compare "Too late of our intents."—Rich. III. iii. 5. 69
Globe, "of."
Comp. πpóς throughout.
So "retentive to" . C. i. 3. 96.
" To see great Pompey pass the streets of Rome."—J. C. i. 1. 47.
The Globe inserts "at."
But "and (there was) much more cause " may be a parenthesis.
"Condemning some to death, and some to exile ;
Ransoming him, or pitying, threatening the other."—Coriol 1. 6. 36.
Hence a "lady-she," W. T. i. 2. 44, means " a well-born woman."
The Elizabethan distinction between thou and you is remarkably illustrated by the usage in E. E., as detailed by Mr. Skeat in (William of Palerne, Preface, p. xli.
See, however—"How many ages hence
Shall this our lofty scene be acted over!"— . C. ii. 1. 112.
See 415 and compare T. A. iii. 1. 151 ; Lear, ii. 1. 63.
Hence "such-like" (Temp. iii. 3. 59) is a pleonasm.
St. Mark iii. 35. Where our Version has "Whosoever shall do the will of my Father," Wickliffe has "Who that doth."
Compare "if so be that."
Compare "The gates are ope," Coriol. i. 4. 43.
In this and many other instances the verb in the second clause may be attracted into the subjunctive by the subjunctive in the first clause.
Quoted from Richardson's Dictionary.
The question may arise why do was preferred to let as an auxiliary verb. Probably the ambiguity of let, which meant both "suffer" and "hinder," was an obstacle to its general use.
Quoted from Todd's "Johnson."
So in ante-Elizabethan English, and in Spenser, we find "nill," "not," for "will not," "wot not," "nam" for "am not," &c. "Cannot" is also a trace of the close connection between the verb and the accompanying negative.
"Thou shalt not," &c.
Coriol. iii. 1. 90, "Mark you his absolute 'shall.'" A similar feeling suggested the different methods of expressing an imperative in Latin and Greek, and the substitution of the optative with ν for the future in Greek.
Madvig, 348. 1.
Morris, Specimens of Early English, xxxv.
"This thing " means "this creature Trinculo," and is antithetical to "thou."
I have found no instance in Shakespeare like the following, quoted by Walker from Sidney's Arcadia:
"And I think there she do dwell."
Comp. " Returning were as tedious as (to) go o'er,"—Macb. iii. 4. 138. in which the ing perhaps qualifies " go" as well as " return," and might be supplanted by " to."
It would be interesting to trace the corresponding process in French by which the gerund "dicendo" and the participle "dicens" were blended in "disant." It was not till the beginning of the eighteenth century that the Academy definitely pronounced " La règle est faite. On ne fera plus accorder les participes présents." But from the earliest times the d of the gerund became t.
Compare the Greek idiom.—Jelf, ii. 863. 2. 2.
"That" might (but for, 260) be treated as a relative pronoun.
See 461.
The use of "never so" is to be explained (as in Greek, θαυμαστ ν σον) by an ellipsis. Thus—
"Though ne'er so richly parted (endowed)."—E. out &c. iii. I. means—" Though he were endowed richly—though never a man were endowed so richly."
Comp. if the reading be retained—
"Which, of he or Adrian, begins to crow?"—Temp. i. 1. 29.
This however is perhaps explained below, In- is a part of the noun "ingratitude;" un- in the adjective "ungrateful" means "not."
The words " trochaic" and "iambic" are of course used, when applied to English poetry, to denote accent, not quantity.
Globe, "this is."
Compare nourrice, nurse.
The same tendency is still more noticeable in E. E. See Essay on the Metres of Chaucer, by the Rev. W. W. Skeat (Aldine Series).
It is a matter of taste which yours should receive the emphasis.
The words "iambic " and "trochaic" here and elsewhere refer to accent, not quantity.
I think I have met with this conjecture in some commentator.
"Vox fusca."
" I have been studying how I may compare
This prison where I live unto the world ;
I cannot do it; yet I'll hammer it out."—Rich. II. v. 5. 1.
The numbers refer to the paragraphs of the Grammar.
Liddell and Scott : δoκ , ii. 4.
Neither of these passages is conclusive, as authorize coming at the beginning of the verse may have the accent on the first syllable. Add therefore:
"His rudeness so with his authorized youth."—L. C. 15.
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Buzz Kill: The Knapper twins' favorite street artists
Jenny An October 7, 2011 12:06PM
5. Shepard Fairey He created the Obama "Hope" poster now hanging in the National Portrait Gallery in Washington, D.C. The street artist whose primary form is wheat-pasting (a Knappers favorite) has also been arrested on multiple occasions. 4. Banksy The man behind the Academy Award-nominated documentary Exit Through the Gift Shop is an elusive street-art figure from Bristol, England. He mostly uses stencils to create his visual messages, which are usually political. 3. Posterboy Most of his work plays off visuals and words found on posters. Sometimes it's funny, sometimes it's cutting and sometimes it borders on obscene.
Jenny An
Contact: Jenny An
Denver Tattoo Industry Had a Colorful History Before the Dark Events of December 27
Tattoo Artists Made Their Mark on Denver
By Emily Ferguson | {
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A decline in the reverse repo rate is seen as positive or bullish for the Rupee while an increase is seen as negative or bearish. However, AutoTrade was also designed to benefit the experienced traders as each time their trades are followed they are rewarded with up to half a pip per standard lot traded by their followers. A high reading is is seen as positive or bullish for the Zloty, while a low reading is seen as negative or Bearish.
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Click on that link and it will take you to our account opening form where you will be able to select the Myfxbook account. Once your account has been funded, you will receive an email notification from us. Click on the link in that notification to finalize the procedure. Hey, thank you for your feedback. Hope you keep enjoying it: Cool indicator dude, I am already using MA so nice to have another addition into my list while I am very pleased with this alert system, it helps me spot any good opportunities rather easily without having to research for it.
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Q: Mysql update takes data only from first row, is it bug? I have query like this:
UPDATE `portal_dyslektyczny`.`questions` AS `Question` SET `Question`.`order` = `Question`.`order` - (`Question`.`order` - 3) + 1 WHERE `Question`.`order` > 3 AND `Question`.`question_group_id` = 1;
and even though all rows have ascending order like
+----+-------+-------------------+
| id | order | question_group_id |
+----+-------+-------------------+
| 74 | 6 | 1 |
| 75 | 7 | 1 |
| 76 | 8 | 1 |
+----+-------+-------------------+
but when i run query table ends like this
+----+-------+-------------------+
| id | order | question_group_id |
+----+-------+-------------------+
| 74 | 4 | 1 |
| 75 | 4 | 1 |
| 76 | 4 | 1 |
+----+-------+-------------------+
as i get it for each row value change should look like this
ID: 74
`Question`.`order` = 6 - ( 6 - 3) + 1;
ID: 75
`Question`.`order` = 7 - ( 7 - 3) + 1;
so on.
Bu it isn't. It looks like it get values only from first row. Does any one know why?
A: Check your formula - it resolves to 4 in all cases.
x - (x - 3) + 1 = 4
| {
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{"url":"https:\/\/www.studyadda.com\/question-bank\/hydrogen_q56\/1456\/110870","text":"\u2022 # question_answer In Bosch?s process which gas is utilised for the production of hydrogen gas A) Producer gas B) Water gas C) Coal gas D) None of these\n\nCorrect Answer: B\n\nSolution :\n\n$\\underbrace{CO+{{H}_{2}}}_{\\text{water gas}}+{{H}_{2}}O\\xrightarrow{\\text{catalyst}}C{{O}_{2}}+2{{H}_{2}}$\n\nYou need to login to perform this action.\nYou will be redirected in 3 sec","date":"2021-01-28 15:50:06","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\": 0.7781419157981873, \"perplexity\": 9042.114578658342}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"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-2021-04\/segments\/1610704847953.98\/warc\/CC-MAIN-20210128134124-20210128164124-00150.warc.gz\"}"} | null | null |
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{"url":"https:\/\/www.semanticscholar.org\/paper\/Entropy%2C-Invertibility-and-Variational-Calculus-of-%C3%9Cst%C3%BCnel\/6885ffbc6f849b8865b48f1ff72a5d71322a679e","text":"# Entropy, Invertibility and Variational Calculus of the Adapted Shifts on Wiener space\n\n@article{stnel2009EntropyIA,\ntitle={Entropy, Invertibility and Variational Calculus of the Adapted Shifts on Wiener space},\nauthor={A. S. {\\\"U}st{\\\"u}nel},\njournal={arXiv: Probability},\nyear={2009}\n}\n\u2022 A. \u00dcst\u00fcnel\n\u2022 Published 23 March 2009\n\u2022 Mathematics\n\u2022 arXiv: Probability\nLocal Invertibility of Adapted Shifts on Wiener Space and Related Topics\n\u2022 Mathematics\n\u2022 2013\nIn this article we show that the invertibility of an adapted shift on the Brownian sheet is a local property in the usual sense of stochastic calculus. Thanks to this result we give a short proof of\nParametric Regularity of the Conditional Expectations via the Malliavin Calculus and Applications\nLet (W, H, \u03bc) be the classical Wiener space and assume that $$U_{\\lambda } = I_{W} + u_{\\lambda }$$ is an adapted perturbation of identity where the perturbation u \u03bb is an H-valued map, defined up to\nVariational calculus on Wiener space with respect to conditional expectations\nWe give a variational formulation for $-\\log\\mathbb{E}_\\nu\\left[e^{-f}|\\mathcal{F}_t\\right]$ for a large class of measures $\\nu$. We give a refined entropic characterization of the invertibility of\nLocal invertibility of adapted shifts on Wiener space, under finite energy condition\nIn this paper we study the connection between local invertibility and global invertibility of adapted shifts on Wiener space. First we go from the global to the local and we obtain an explicit\nA general framework for variational calculus on Wiener space\nWe provide a framework to derive a variational formulation for $-\\log\\mathbb{E}_\\nu\\left[e^{-f}\\right]$ for a large class of measures $\\nu$. We use a family of perturbations of the identity $(W^u)$\nA comparison theorem for stochastic differential equations under a Novikov-type condition\nWe consider a system of stochastic differential equations driven by a standard n-dimensional Brownian motion where the drift coefficient satisfies a Novikov-type condition while the diffusion\nMartingale representation for degenerate diffusions\nOn the form of the relative entropy between measures on the space of continuous functions\n\u2022 Mathematics\n\u2022 2013\nIn this paper we derive an integral (with respect to time) representation of the relative entropy (or Kullback-Leibler Divergence) between measures mu and P on the space of continuous functions from\n\n## References\n\nSHOWING 1-10 OF 17 REFERENCES\nSufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space\n\u2022 Mathematics\n\u2022 2006\nLet (W, H, \u03bc) be the classical Wiener space. Assume that U\u00a0=\u00a0IW\u00a0+\u00a0u is an adapted perturbation of identity, i.e., u : W \u2192 H is adapted to the canonical filtration of W. We give some sufficient\nThe realization of positive random variables via absolutely continuous transformations of measure on Wiener space\n\u2022 Mathematics\n\u2022 2006\nLet $\\mu$ be a Gaussian measure on some measurable space $\\{W=\\{w\\},{\\mathcal{B}}(W)\\}$ and let $\\nu$ be a measure on the same space which is absolutely continuous with respect to $\\nu$. The paper\nMonge-Kantorovitch Measure Transportation and Monge-Amp\u00e8re Equation on Wiener Space\n\u2022 Mathematics\n\u2022 2004\nAbstractLet (W,\u03bc,H) be an abstract Wiener space assume two \u03bdi,i=1,2 probabilities on (W,\u212c(W)). We give some conditions for the Wasserstein distance between \u03bd1 and \u03bd2 with respect to the\nTransformation of Measure on Wiener Space\n\u2022 Mathematics\n\u2022 2000\nThis book gives a systematic presentation of the main results on the transformation of measure induced by shift transformations on Wiener space. This topic has its origins in the work of Cameron and\nA variational representation for certain functionals of Brownian motion\n\u2022 Mathematics\n\u2022 1998\nIn this paper we show that the variational representation - log Ee -f(W) = inf E{1\/2\u222b 0 1 \u2225v s \u2225 2 ds + f(w + \u222b 0 v s ds)} holds, where W is a standard d-dimensional Brownian motion, f is any bounded\nTransportation cost for Gaussian and other product measures\nAbstractConsider the canonical Gaussian measure \u03b3N on \u211d, a probability measure \u03bc on \u211dN, absolutely continuous with respect to \u03b3N. We prove that the transportation cost of \u03bc to \u03b3N, when the cost of\nAn Introduction to Analysis on Wiener Space\nPreliminaries.- Gross-Sobolev derivative, divergence and Ornstein-Uhlenbeck operator.- Meyer inequalities.- Hypercontractivity.- L p -multipliers theorem, meyer inequalities and distributions.- Some\nThe Notion of Convexity and Concavity on Wiener Space\n\u2022 Mathematics\n\u2022 2000\nAbstract We define, in the frame of an abstract Wiener space, the notions of convexity and of concavity for the equivalence classes of random variables. As application we show that some important\nStochastic differential equations for the non linear filtering problem\n\u2022 Mathematics\n\u2022 1972\nThe general nonlinear filtering or estimation problem may be described as follows. xty (0<t<T)y called the signal or system process is a stochastic process direct observation is not possible. The\nAnalysis on Wiener Space and Applications\nThe aim of this book is to give a rigorous introduction for the graduate students to Analysis on Wiener space, a subject which has grown up very quickly these recent years under the new impulse of","date":"2022-07-07 17:10:39","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\": 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\": 0.7921419739723206, \"perplexity\": 916.5222366642384}, \"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-2022-27\/segments\/1656104495692.77\/warc\/CC-MAIN-20220707154329-20220707184329-00353.warc.gz\"}"} | null | null |
Porsche has confirmed how quickly its upcoming 918 Spyder can go around the famed Nürburgring in Germany. The 770-horsepower hybrid supercar has lapped the 'Ring in seven minutes and 14 seconds.
The official lap time for the 918 Spyder is among the fastest for a road legal car at the Nurburgring's northern circuit, undercutting the fastest time of its predecessor, the Carrera GT, achieved back in 2004 by some 18 seconds making it the fastest Porsche yet around the Nürburgring. | {
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Q: fermat's little theorem and residue classes
I am trying to understand fermat's little theorem in residue classes but the below slides make absolutely no sense to me. In computer classes a' means if you have 3 then 3' would be 6 because 3+6=9 so I am really confused here about what they are doing..
I know that residue classes mod m basically means the remainder when mod by m but I dont really understand what they mean or do by [a][b]=[a'][b']
Also, really not sure how they are finding the inverse...the general equation is a congruent to b (mod m)
A: The primes in the statement of prop 4.2.2 are not operators; $a'$ and $b'$ are simply names of variables which are different from $a$ and $b$, although suggestively named.
It is, however, not very clearly written. It appears that the author is using $[a]$ and $[a']$ denote the residue classes of the integers $a$ and $a'$, and so forth. But it is strictly speaking nonsense to write "if $[a]=[a']$ and $[b]=[b']$ modulo $m$ ...". For then $[a]=[a']$ simply asserts that the residue classes are the same, and this identity is just an identity between sets of numbers; there is nothing modular about the way these sets are equal.
But if the author does mean the premises to be $[a]=[a']$ and $[b]=[b']$, then the conclusions $[a]+[b]=[a']+[b']$ and $[a][b]=[a'][b']$ are completely vacuous, because of course we're allowed to substitute equals for equals.
What the proposition ought to have been, in order to be meaningful, is
If $a\equiv a'\pmod m$ and $b\equiv b'\pmod m$, then $(a+b)\equiv(a'+b')\pmod m$ and $ab\equiv a'b' \pmod m$.
or, equivalently,
If $[a]=[a']$ and $[b]=[b']$, then $[a+b]=[a'+b']$ and $[ab]=[a'b']$.
... and because of this fact it is possible and meaningful to define the sum and product of residue classes by $[a]+[b]=[a+b]$ and $[a]\cdot[b]=[ab]$.
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"redpajama_set_name": "RedPajamaStackExchange"
} | 7,912 |
"""Create volume
"""
from profitbricks.client import ProfitBricksService, Volume
datacenter_id = '700e1cab-99b2-4c30-ba8c-1d273ddba022'
client = ProfitBricksService(
username='username', password='password')
i = Volume(
name='Explicitly created volume',
size=56,
image='<IMAGE/SNAPSHOT-ID>',
bus='VIRTIO')
response = client.create_volume(
datacenter_id=datacenter_id, volume=i)
"""Create snapshot
"""
from profitbricks.client import ProfitBricksService # noqa
datacenter_id = '700e1cab-99b2-4c30-ba8c-1d273ddba022'
volume_id = '700e1cab-99b2-4c30-ba8c-1d273ddba025'
client = ProfitBricksService(
username='username', password='password')
volume = client.create_snapshot(
datacenter_id=datacenter_id,
volume_id=volume_id,
name='<URLENCODED_SNAPSHOT_NAME>',
description='<URLENCODED_SNAPSHOT_DESCRIPTION>')
"""Restore Snapshot
"""
from profitbricks.client import ProfitBricksService # noqa
datacenter_id = '700e1cab-99b2-4c30-ba8c-1d273ddba022'
volume_id = '700e1cab-99b2-4c30-ba8c-1d273ddba025'
snapshot_id = '7df81087-5835-41c6-a10b-3e098593bba4'
client = ProfitBricksService(
username='username', password='password')
response = client.restore_snapshot(
datacenter_id=datacenter_id,
volume_id=volume_id,
snapshot_id=snapshot_id)
"""Update Volume
"""
from profitbricks.client import ProfitBricksService # noqa
datacenter_id = '700e1cab-99b2-4c30-ba8c-1d273ddba022'
volume_id = '700e1cab-99b2-4c30-ba8c-1d273ddba025'
client = ProfitBricksService(
username='username', password='password')
volume = client.update_volume(
datacenter_id=datacenter_id,
volume_id=volume_id,
size=100,
name='Resized storage to 100 GB')
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,384 |
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