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General structure. 2-, alpha-, or α-amino acids have the generic formula in most cases, where R is an organic substituent known as a "side chain". Of the many hundreds of described amino acids, 22 are proteinogenic ("protein-building"). It is these 22 compounds that combine to give a vast array of peptides and proteins assembled by ribosomes. Non-proteinogenic or modified amino acids may arise from post-translational modification or during nonribosomal peptide synthesis. Chirality. The carbon atom next to the carboxyl group is called the α–carbon. In proteinogenic amino acids, it bears the amine and the R group or side chain specific to each amino acid, as well as a hydrogen atom. With the exception of glycine, for which the side chain is also a hydrogen atom, the α–carbon is stereogenic. All chiral proteogenic amino acids have the configuration. They are "left-handed" enantiomers, which refers to the stereoisomers of the alpha carbon. A few -amino acids ("right-handed") have been found in nature, e.g., in bacterial envelopes, as a neuromodulator (-serine), and in some antibiotics. Rarely, -amino acid residues are found in proteins, and are converted from the -amino acid as a post-translational modification.
Side chains. Polar charged side chains. Five amino acids possess a charge at neutral pH. Often these side chains appear at the surfaces on proteins to enable their solubility in water, and side chains with opposite charges form important electrostatic contacts called salt bridges that maintain structures within a single protein or between interfacing proteins. Many proteins bind metal into their structures specifically, and these interactions are commonly mediated by charged side chains such as aspartate, glutamate and histidine. Under certain conditions, each ion-forming group can be charged, forming double salts. The two negatively charged amino acids at neutral pH are aspartate (Asp, D) and glutamate (Glu, E). The anionic carboxylate groups behave as Brønsted bases in most circumstances. Enzymes in very low pH environments, like the aspartic protease pepsin in mammalian stomachs, may have catalytic aspartate or glutamate residues that act as Brønsted acids. There are three amino acids with side chains that are cations at neutral pH: arginine (Arg, R), lysine (Lys, K) and histidine (His, H). Arginine has a charged guanidino group and lysine a charged alkyl amino group, and are fully protonated at pH 7. Histidine's imidazole group has a pKa of 6.0, and is only around 10% protonated at neutral pH. Because histidine is easily found in its basic and conjugate acid forms it often participates in catalytic proton transfers in enzyme reactions.
Polar uncharged side chains. The polar, uncharged amino acids serine (Ser, S), threonine (Thr, T), asparagine (Asn, N) and glutamine (Gln, Q) readily form hydrogen bonds with water and other amino acids. They do not ionize in normal conditions, a prominent exception being the catalytic serine in serine proteases. This is an example of severe perturbation, and is not characteristic of serine residues in general. Threonine has two chiral centers, not only the (2"S") chiral center at the α-carbon shared by all amino acids apart from achiral glycine, but also (3"R") at the β-carbon. The full stereochemical specification is (2"S",3"R")--threonine. Hydrophobic side chains. Nonpolar amino acid interactions are the primary driving force behind the processes that fold proteins into their functional three dimensional structures. None of these amino acids' side chains ionize easily, and therefore do not have pKas, with the exception of tyrosine (Tyr, Y). The hydroxyl of tyrosine can deprotonate at high pH forming the negatively charged phenolate. Because of this one could place tyrosine into the polar, uncharged amino acid category, but its very low solubility in water matches the characteristics of hydrophobic amino acids well.
Special case side chains. Several side chains are not described well by the charged, polar and hydrophobic categories. Glycine (Gly, G) could be considered a polar amino acid since its small size means that its solubility is largely determined by the amino and carboxylate groups. However, the lack of any side chain provides glycine with a unique flexibility among amino acids with large ramifications to protein folding. Cysteine (Cys, C) can also form hydrogen bonds readily, which would place it in the polar amino acid category, though it can often be found in protein structures forming covalent bonds, called disulphide bonds, with other cysteines. These bonds influence the folding and stability of proteins, and are essential in the formation of antibodies. Proline (Pro, P) has an alkyl side chain and could be considered hydrophobic, but because the side chain joins back onto the alpha amino group it becomes particularly inflexible when incorporated into proteins. Similar to glycine this influences protein structure in a way unique among amino acids. Selenocysteine (Sec, U) is a rare amino acid not directly encoded by DNA, but is incorporated into proteins via the ribosome. Selenocysteine has a lower redox potential compared to the similar cysteine, and participates in several unique enzymatic reactions. Pyrrolysine (Pyl, O) is another amino acid not encoded in DNA, but synthesized into protein by ribosomes. It is found in archaeal species where it participates in the catalytic activity of several methyltransferases.
β- and γ-amino acids. Amino acids with the structure , such as β-alanine, a component of carnosine and a few other peptides, are β-amino acids. Ones with the structure are γ-amino acids, and so on, where X and Y are two substituents (one of which is normally H). Zwitterions. The common natural forms of amino acids have a zwitterionic structure, with ( in the case of proline) and functional groups attached to the same C atom, and are thus α-amino acids, and are the only ones found in proteins during translation in the ribosome. In aqueous solution at pH close to neutrality, amino acids exist as zwitterions, i.e. as dipolar ions with both and in charged states, so the overall structure is . At physiological pH the so-called "neutral forms" are not present to any measurable degree. Although the two charges in the zwitterion structure add up to zero it is misleading to call a species with a net charge of zero "uncharged". In strongly acidic conditions (pH below 3), the carboxylate group becomes protonated and the structure becomes an ammonio carboxylic acid, . This is relevant for enzymes like pepsin that are active in acidic environments such as the mammalian stomach and lysosomes, but does not significantly apply to intracellular enzymes. In highly basic conditions (pH greater than 10, not normally seen in physiological conditions), the ammonio group is deprotonated to give .
Although various definitions of acids and bases are used in chemistry, the only one that is useful for chemistry in aqueous solution is that of Brønsted: an acid is a species that can donate a proton to another species, and a base is one that can accept a proton. This criterion is used to label the groups in the above illustration. The carboxylate side chains of aspartate and glutamate residues are the principal Brønsted bases in proteins. Likewise, lysine, tyrosine and cysteine will typically act as a Brønsted acid. Histidine under these conditions can act both as a Brønsted acid and a base. Isoelectric point. For amino acids with uncharged side-chains the zwitterion predominates at pH values between the two p"K"a values, but coexists in equilibrium with small amounts of net negative and net positive ions. At the midpoint between the two p"K"a values, the trace amount of net negative and trace of net positive ions balance, so that average net charge of all forms present is zero. This pH is known as the isoelectric point p"I", so p"I" = (p"K"a1 + p"K"a2).
For amino acids with charged side chains, the p"K"a of the side chain is involved. Thus for aspartate or glutamate with negative side chains, the terminal amino group is essentially entirely in the charged form , but this positive charge needs to be balanced by the state with just one C-terminal carboxylate group is negatively charged. This occurs halfway between the two carboxylate p"K"a values: p"I" = (p"K"a1 + p"K"a(R)), where p"K"a(R) is the side chain p"K"a. Similar considerations apply to other amino acids with ionizable side-chains, including not only glutamate (similar to aspartate), but also cysteine, histidine, lysine, tyrosine and arginine with positive side chains. Amino acids have zero mobility in electrophoresis at their isoelectric point, although this behaviour is more usually exploited for peptides and proteins than single amino acids. Zwitterions have minimum solubility at their isoelectric point, and some amino acids (in particular, with nonpolar side chains) can be isolated by precipitation from water by adjusting the pH to the required isoelectric point.
Physicochemical properties. The 20 canonical amino acids can be classified according to their properties. Important factors are charge, hydrophilicity or hydrophobicity, size, and functional groups. These properties influence protein structure and protein–protein interactions. The water-soluble proteins tend to have their hydrophobic residues (Leu, Ile, Val, Phe, and Trp) buried in the middle of the protein, whereas hydrophilic side chains are exposed to the aqueous solvent. (In biochemistry, a residue refers to a specific monomer "within" the polymeric chain of a polysaccharide, protein or nucleic acid.) The integral membrane proteins tend to have outer rings of exposed hydrophobic amino acids that anchor them in the lipid bilayer. Some peripheral membrane proteins have a patch of hydrophobic amino acids on their surface that sticks to the membrane. In a similar fashion, proteins that have to bind to positively charged molecules have surfaces rich in negatively charged amino acids such as glutamate and aspartate, while proteins binding to negatively charged molecules have surfaces rich in positively charged amino acids like lysine and arginine. For example, lysine and arginine are present in large amounts in the low-complexity regions of nucleic-acid binding proteins. There are various hydrophobicity scales of amino acid residues.
Some amino acids have special properties. Cysteine can form covalent disulfide bonds to other cysteine residues. Proline forms a cycle to the polypeptide backbone, and glycine is more flexible than other amino acids. Glycine and proline are strongly present within low complexity regions of both eukaryotic and prokaryotic proteins, whereas the opposite is the case with cysteine, phenylalanine, tryptophan, methionine, valine, leucine, isoleucine, which are highly reactive, or complex, or hydrophobic. Many proteins undergo a range of posttranslational modifications, whereby additional chemical groups are attached to the amino acid residue side chains sometimes producing lipoproteins (that are hydrophobic), or glycoproteins (that are hydrophilic) allowing the protein to attach temporarily to a membrane. For example, a signaling protein can attach and then detach from a cell membrane, because it contains cysteine residues that can have the fatty acid palmitic acid added to them and subsequently removed. Table of standard amino acid abbreviations and properties.
Although one-letter symbols are included in the table, IUPAC–IUBMB recommend that "Use of the one-letter symbols should be restricted to the comparison of long sequences". The one-letter notation was chosen by IUPAC-IUB based on the following rules: Two additional amino acids are in some species coded for by codons that are usually interpreted as stop codons: In addition to the specific amino acid codes, placeholders are used in cases where chemical or crystallographic analysis of a peptide or protein cannot conclusively determine the identity of a residue. They are also used to summarize conserved protein sequence motifs. The use of single letters to indicate sets of similar residues is similar to the use of abbreviation codes for degenerate bases. Unk is sometimes used instead of Xaa, but is less standard. Ter or * (from termination) is used in notation for mutations in proteins when a stop codon occurs. It corresponds to no amino acid at all. In addition, many nonstandard amino acids have a specific code. For example, several peptide drugs, such as Bortezomib and MG132, are artificially synthesized and retain their protecting groups, which have specific codes. Bortezomib is Pyz–Phe–boroLeu, and MG132 is Z–Leu–Leu–Leu–al. To aid in the analysis of protein structure, photo-reactive amino acid analogs are available. These include photoleucine (pLeu) and photomethionine (pMet).
Occurrence and functions in biochemistry. Proteinogenic amino acids. Amino acids are the precursors to proteins. They join by condensation reactions to form short polymer chains called peptides or longer chains called either polypeptides or proteins. These chains are linear and unbranched, with each amino acid residue within the chain attached to two neighboring amino acids. In nature, the process of making proteins encoded by RNA genetic material is called "translation" and involves the step-by-step addition of amino acids to a growing protein chain by a ribozyme that is called a ribosome. The order in which the amino acids are added is read through the genetic code from an mRNA template, which is an RNA derived from one of the organism's genes. Twenty-two amino acids are naturally incorporated into polypeptides and are called proteinogenic or natural amino acids. Of these, 20 are encoded by the universal genetic code. The remaining 2, selenocysteine and pyrrolysine, are incorporated into proteins by unique synthetic mechanisms. Selenocysteine is incorporated when the mRNA being translated includes a SECIS element, which causes the UGA codon to encode selenocysteine instead of a stop codon. Pyrrolysine is used by some methanogenic archaea in enzymes that they use to produce methane. It is coded for with the codon UAG, which is normally a stop codon in other organisms.
Several independent evolutionary studies have suggested that Gly, Ala, Asp, Val, Ser, Pro, Glu, Leu, Thr may belong to a group of amino acids that constituted the early genetic code, whereas Cys, Met, Tyr, Trp, His, Phe may belong to a group of amino acids that constituted later additions of the genetic code. Standard vs nonstandard amino acids. The 20 amino acids that are encoded directly by the codons of the universal genetic code are called "standard" or "canonical" amino acids. A modified form of methionine ("N"-formylmethionine) is often incorporated in place of methionine as the initial amino acid of proteins in bacteria, mitochondria and plastids (including chloroplasts). Other amino acids are called "nonstandard" or "non-canonical". Most of the nonstandard amino acids are also non-proteinogenic (i.e. they cannot be incorporated into proteins during translation), but two of them are proteinogenic, as they can be incorporated translationally into proteins by exploiting information not encoded in the universal genetic code.
The two nonstandard proteinogenic amino acids are selenocysteine (present in many non-eukaryotes as well as most eukaryotes, but not coded directly by DNA) and pyrrolysine (found only in some archaea and at least one bacterium). The incorporation of these nonstandard amino acids is rare. For example, 25 human proteins include selenocysteine in their primary structure, and the structurally characterized enzymes (selenoenzymes) employ selenocysteine as the catalytic moiety in their active sites. Pyrrolysine and selenocysteine are encoded via variant codons. For example, selenocysteine is encoded by stop codon and SECIS element. "N"-formylmethionine (which is often the initial amino acid of proteins in bacteria, mitochondria, and chloroplasts) is generally considered as a form of methionine rather than as a separate proteinogenic amino acid. Codon–tRNA combinations not found in nature can also be used to "expand" the genetic code and form novel proteins known as alloproteins incorporating non-proteinogenic amino acids.
Non-proteinogenic amino acids. Aside from the 22 proteinogenic amino acids, many "non-proteinogenic" amino acids are known. Those either are not found in proteins (for example carnitine, GABA, levothyroxine) or are not produced directly and in isolation by standard cellular machinery. For example, hydroxyproline, is synthesised from proline. Another example is selenomethionine). Non-proteinogenic amino acids that are found in proteins are formed by post-translational modification. Such modifications can also determine the localization of the protein, e.g., the addition of long hydrophobic groups can cause a protein to bind to a phospholipid membrane. Examples: Some non-proteinogenic amino acids are not found in proteins. Examples include 2-aminoisobutyric acid and the neurotransmitter gamma-aminobutyric acid. Non-proteinogenic amino acids often occur as intermediates in the metabolic pathways for standard amino acids – for example, ornithine and citrulline occur in the urea cycle, part of amino acid catabolism (see below). A rare exception to the dominance of α-amino acids in biology is the β-amino acid beta alanine (3-aminopropanoic acid), which is used in plants and microorganisms in the synthesis of pantothenic acid (vitamin B5), a component of coenzyme A.
In mammalian nutrition. Animals ingest amino acids in the form of protein. The protein is broken down into its constituent amino acids in the process of digestion. The amino acids are then used to synthesize new proteins and other nitrogenous biomolecules, or they are further catabolized through oxidation to provide a source of energy. The oxidation pathway starts with the removal of the amino group by a transaminase; the amino group is then fed into the urea cycle. The other product of transamidation is a keto acid that enters the citric acid cycle. Glucogenic amino acids can also be converted into glucose, through gluconeogenesis. Of the 20 standard amino acids, nine (His, Ile, Leu, Lys, Met, Phe, Thr, Trp and Val) are called essential amino acids because the human body cannot synthesize them from other compounds at the level needed for normal growth, so they must be obtained from food. Semi-essential and conditionally essential amino acids, and juvenile requirements. In addition, cysteine, tyrosine, and arginine are considered semiessential amino acids, and taurine a semi-essential aminosulfonic acid in children. Some amino acids are conditionally essential for certain ages or medical conditions. Essential amino acids may also vary from species to species. The metabolic pathways that synthesize these monomers are not fully developed.
Non-protein functions. Many proteinogenic and non-proteinogenic amino acids have biological functions beyond being precursors to proteins and peptides. In humans, amino acids also have important roles in diverse biosynthetic pathways. Defenses against herbivores in plants sometimes employ amino acids. Examples: Roles for nonstandard amino acids. However, not all of the functions of other abundant nonstandard amino acids are known. Uses in industry. Animal feed. Amino acids are sometimes added to animal feed because some of the components of these feeds, such as soybeans, have low levels of some of the essential amino acids, especially of lysine, methionine, threonine, and tryptophan. Likewise amino acids are used to chelate metal cations in order to improve the absorption of minerals from feed supplements. Food. The food industry is a major consumer of amino acids, especially glutamic acid, which is used as a flavor enhancer, and aspartame (aspartylphenylalanine 1-methyl ester), which is used as an artificial sweetener. Amino acids are sometimes added to food by manufacturers to alleviate symptoms of mineral deficiencies, such as anemia, by improving mineral absorption and reducing negative side effects from inorganic mineral supplementation.
Chemical building blocks. Amino acids are low-cost feedstocks used in chiral pool synthesis as enantiomerically pure building blocks. Amino acids are used in the synthesis of some cosmetics. Aspirational uses. Fertilizer. The chelating ability of amino acids is sometimes used in fertilizers to facilitate the delivery of minerals to plants in order to correct mineral deficiencies, such as iron chlorosis. These fertilizers are also used to prevent deficiencies from occurring and to improve the overall health of the plants. Biodegradable plastics. Amino acids have been considered as components of biodegradable polymers, which have applications as environmentally friendly packaging and in medicine in drug delivery and the construction of prosthetic implants. An interesting example of such materials is polyaspartate, a water-soluble biodegradable polymer that may have applications in disposable diapers and agriculture. Due to its solubility and ability to chelate metal ions, polyaspartate is also being used as a biodegradable antiscaling agent and a corrosion inhibitor.
Synthesis. Chemical synthesis. The commercial production of amino acids usually relies on mutant bacteria that overproduce individual amino acids using glucose as a carbon source. Some amino acids are produced by enzymatic conversions of synthetic intermediates. 2-Aminothiazoline-4-carboxylic acid is an intermediate in one industrial synthesis of -cysteine for example. Aspartic acid is produced by the addition of ammonia to fumarate using a lyase. Biosynthesis. In plants, nitrogen is first assimilated into organic compounds in the form of glutamate, formed from alpha-ketoglutarate and ammonia in the mitochondrion. For other amino acids, plants use transaminases to move the amino group from glutamate to another alpha-keto acid. For example, aspartate aminotransferase converts glutamate and oxaloacetate to alpha-ketoglutarate and aspartate. Other organisms use transaminases for amino acid synthesis, too. Nonstandard amino acids are usually formed through modifications to standard amino acids. For example, homocysteine is formed through the transsulfuration pathway or by the demethylation of methionine via the intermediate metabolite "S"-adenosylmethionine, while hydroxyproline is made by a post translational modification of proline.
Microorganisms and plants synthesize many uncommon amino acids. For example, some microbes make 2-aminoisobutyric acid and lanthionine, which is a sulfide-bridged derivative of alanine. Both of these amino acids are found in peptidic lantibiotics such as alamethicin. However, in plants, 1-aminocyclopropane-1-carboxylic acid is a small disubstituted cyclic amino acid that is an intermediate in the production of the plant hormone ethylene. Primordial synthesis. The formation of amino acids and peptides is assumed to have preceded and perhaps induced the emergence of life on earth. Amino acids can form from simple precursors under various conditions. Surface-based chemical metabolism of amino acids and very small compounds may have led to the build-up of amino acids, coenzymes and phosphate-based small carbon molecules. Amino acids and similar building blocks could have been elaborated into proto-peptides, with peptides being considered key players in the origin of life.<ref name="10.1021/acs.chemrev.9b00664"></ref>
In the famous Urey-Miller experiment, the passage of an electric arc through a mixture of methane, hydrogen, and ammonia produces a large number of amino acids. Since then, scientists have discovered a range of ways and components by which the potentially prebiotic formation and chemical evolution of peptides may have occurred, such as condensing agents, the design of self-replicating peptides and a number of non-enzymatic mechanisms by which amino acids could have emerged and elaborated into peptides. Several hypotheses invoke the Strecker synthesis whereby hydrogen cyanide, simple aldehydes, ammonia, and water produce amino acids.<ref name="10.1016/j.gsf.2017.07.007"></ref> According to a review, amino acids, and even peptides, "turn up fairly regularly in the various experimental broths that have been allowed to be cooked from simple chemicals. This is because nucleotides are far more difficult to synthesize chemically than amino acids." For a chronological order, it suggests that there must have been a 'protein world' or at least a 'polypeptide world', possibly later followed by the 'RNA world' and the 'DNA world'. Codon–amino acids mappings may be the biological information system at the primordial origin of life on Earth. While amino acids and consequently simple peptides must have formed under different experimentally probed geochemical scenarios, the transition from an abiotic world to the first life forms is to a large extent still unresolved.
Reactions. Amino acids undergo the reactions expected of the constituent functional groups. Peptide bond formation. As both the amine and carboxylic acid groups of amino acids can react to form amide bonds, one amino acid molecule can react with another and become joined through an amide linkage. This polymerization of amino acids is what creates proteins. This condensation reaction yields the newly formed peptide bond and a molecule of water. In cells, this reaction does not occur directly; instead, the amino acid is first activated by attachment to a transfer RNA molecule through an ester bond. This aminoacyl-tRNA is produced in an ATP-dependent reaction carried out by an aminoacyl tRNA synthetase. This aminoacyl-tRNA is then a substrate for the ribosome, which catalyzes the attack of the amino group of the elongating protein chain on the ester bond. As a result of this mechanism, all proteins made by ribosomes are synthesized starting at their "N"-terminus and moving toward their "C"-terminus. However, not all peptide bonds are formed in this way. In a few cases, peptides are synthesized by specific enzymes. For example, the tripeptide glutathione is an essential part of the defenses of cells against oxidative stress. This peptide is synthesized in two steps from free amino acids. In the first step, gamma-glutamylcysteine synthetase condenses cysteine and glutamate through a peptide bond formed between the side chain carboxyl of the glutamate (the gamma carbon of this side chain) and the amino group of the cysteine. This dipeptide is then condensed with glycine by glutathione synthetase to form glutathione.
In chemistry, peptides are synthesized by a variety of reactions. One of the most-used in solid-phase peptide synthesis uses the aromatic oxime derivatives of amino acids as activated units. These are added in sequence onto the growing peptide chain, which is attached to a solid resin support. Libraries of peptides are used in drug discovery through high-throughput screening. The combination of functional groups allow amino acids to be effective polydentate ligands for metal–amino acid chelates. The multiple side chains of amino acids can also undergo chemical reactions. Catabolism. Degradation of an amino acid often involves deamination by moving its amino group to α-ketoglutarate, forming glutamate. This process involves transaminases, often the same as those used in amination during synthesis. In many vertebrates, the amino group is then removed through the urea cycle and is excreted in the form of urea. However, amino acid degradation can produce uric acid or ammonia instead. For example, serine dehydratase converts serine to pyruvate and ammonia. After removal of one or more amino groups, the remainder of the molecule can sometimes be used to synthesize new amino acids, or it can be used for energy by entering glycolysis or the citric acid cycle, as detailed in image at right.
Complexation. Amino acids are bidentate ligands, forming transition metal amino acid complexes. Chemical analysis. The total nitrogen content of organic matter is mainly formed by the amino groups in proteins. The Total Kjeldahl Nitrogen (TKN) is a measure of nitrogen widely used in the analysis of (waste) water, soil, food, feed and organic matter in general. As the name suggests, the Kjeldahl method is applied. More sensitive methods are available.
Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer science, providing a formalisation of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general-purpose computer. Turing is widely considered to be the father of theoretical computer science. Born in London, Turing was raised in southern England. He graduated from King's College, Cambridge, and in 1938, earned a doctorate degree from Princeton University. During World War II, Turing worked for the Government Code and Cypher School at Bletchley Park, Britain's codebreaking centre that produced Ultra intelligence. He led Hut 8, the section responsible for German naval cryptanalysis. Turing devised techniques for speeding the breaking of German ciphers, including improvements to the pre-war Polish bomba method, an electromechanical machine that could find settings for the Enigma machine. He played a crucial role in cracking intercepted messages that enabled the Allies to defeat the Axis powers in many engagements, including the Battle of the Atlantic.
After the war, Turing worked at the National Physical Laboratory, where he designed the Automatic Computing Engine, one of the first designs for a stored-program computer. In 1948, Turing joined Max Newman's Computing Machine Laboratory at the University of Manchester, where he contributed to the development of early Manchester computers and became interested in mathematical biology. Turing wrote on the chemical basis of morphogenesis and predicted oscillating chemical reactions such as the Belousov–Zhabotinsky reaction, first observed in the 1960s. Despite these accomplishments, he was never fully recognised during his lifetime because much of his work was covered by the Official Secrets Act. In 1952, Turing was prosecuted for homosexual acts. He accepted hormone treatment, a procedure commonly referred to as chemical castration, as an alternative to prison. Turing died on 7 June 1954, aged 41, from cyanide poisoning. An inquest determined his death as suicide, but the evidence is also consistent with accidental poisoning.
Following a campaign in 2009, British prime minister Gordon Brown made an official public apology for "the appalling way [Turing] was treated". Queen Elizabeth II granted a pardon in 2013. The term "Alan Turing law" is used informally to refer to a 2017 law in the UK that retroactively pardoned men cautioned or convicted under historical legislation that outlawed homosexual acts. Turing left an extensive legacy in mathematics and computing which has become widely recognised with statues and many things named after him, including an annual award for computing innovation. His portrait appears on the Bank of England £50 note, first released on 23 June 2021 to coincide with his birthday. The audience vote in a named Turing the greatest person of the 20th century. Early life and education. Family. Turing was born in Maida Vale, London, while his father, Julius Mathison Turing, was on leave from his position with the Indian Civil Service (ICS) of the British Raj government at Chatrapur, then in the Madras Presidency and presently in Odisha state, in India. Turing's father was the son of a clergyman, the Rev. John Robert Turing, from a Scottish family of merchants that had been based in the Netherlands and included a baronet. Turing's mother, Julius's wife, was Ethel Sara Turing (), daughter of Edward Waller Stoney, chief engineer of the Madras Railways. The Stoneys were a Protestant Anglo-Irish gentry family from both County Tipperary and County Longford, while Ethel herself had spent much of her childhood in County Clare. Julius and Ethel married on 1 October 1907 at the Church of Ireland St. Bartholomew's Church on Clyde Road in Ballsbridge, Dublin.
Julius's work with the ICS brought the family to British India, where his grandfather had been a general in the Bengal Army. However, both Julius and Ethel wanted their children to be brought up in Britain, so they moved to Maida Vale, London, where Alan Turing was born on 23 June 1912, as recorded by a blue plaque on the outside of the house of his birth, later the Colonnade Hotel. Turing had an elder brother, John Ferrier Turing, father of Dermot Turing, 12th Baronet of the Turing baronets. Turing's father's civil service commission was still active during Turing's childhood years, and his parents travelled between Hastings in the United Kingdom and India, leaving their two sons to stay with a retired Army couple. At Hastings, Turing stayed at Baston Lodge, Upper Maze Hill, St Leonards-on-Sea, now marked with a blue plaque. The plaque was unveiled on 23 June 2012, the centenary of Turing's birth. Very early in life, Turing's parents purchased a house in Guildford in 1927, and Turing lived there during school holidays. The location is also marked with a blue plaque.
School. Turing's parents enrolled him at St Michael's, a primary school at 20 Charles Road, St Leonards-on-Sea, from the age of six to nine. The headmistress recognised his talent, noting that she "...had clever boys and hardworking boys, but Alan is a genius". Between January 1922 and 1926, Turing was educated at Hazelhurst Preparatory School, an independent school in the village of Frant in Sussex (now East Sussex). In 1926, at the age of 13, he went on to Sherborne School, an independent boarding school in the market town of Sherborne in Dorset, where he boarded at Westcott House. The first day of term coincided with the 1926 General Strike, in Britain, but Turing was so determined to attend that he rode his bicycle unaccompanied from Southampton to Sherborne, stopping overnight at an inn. Turing's natural inclination towards mathematics and science did not earn him respect from some of the teachers at Sherborne, whose definition of education placed more emphasis on the classics. His headmaster wrote to his parents: "I hope he will not fall between two stools. If he is to stay at public school, he must aim at becoming "educated". If he is to be solely a "Scientific Specialist", he is wasting his time at a public school". Despite this, Turing continued to show remarkable ability in the studies he loved, solving advanced problems in 1927 without having studied even elementary calculus. In 1928, aged 16, Turing encountered Albert Einstein's work; not only did he grasp it, but it is possible that he managed to deduce Einstein's questioning of Newton's laws of motion from a text in which this was never made explicit.
Christopher Morcom. At Sherborne, Turing formed a significant friendship with fellow pupil Christopher Collan Morcom (13 July 1911 – 13 February 1930), who has been described as Turing's first love. Their relationship provided inspiration in Turing's future endeavours, but it was cut short by Morcom's death, in February 1930, from complications of bovine tuberculosis, contracted after drinking infected cow's milk some years previously. The event caused Turing great sorrow. He coped with his grief by working that much harder on the topics of science and mathematics that he had shared with Morcom. In a letter to Morcom's mother, Frances Isobel Morcom (née Swan), Turing wrote: Turing's relationship with Morcom's mother continued long after Morcom's death, with her sending gifts to Turing, and him sending letters, typically on Morcom's birthday. A day before the third anniversary of Morcom's death (13 February 1933), he wrote to Mrs. Morcom: Some have speculated that Morcom's death was the cause of Turing's atheism and materialism. Apparently, at this point in his life he still believed in such concepts as a spirit, independent of the body and surviving death. In a later letter, also written to Morcom's mother, Turing wrote:
University and work on computability. After graduating from Sherborne, Turing applied for several Cambridge colleges scholarships, including Trinity and King's, eventually earning an £80 per annum scholarship (equivalent to about £4,300 as of 2023) to study at the latter. There, Turing studied the undergraduate course in Schedule B from February 1931 to November 1934 at King's College, Cambridge, where he was awarded first-class honours in mathematics. His dissertation, "On the Gaussian error function", written during his senior year and delivered in November 1934 (with a deadline date of 6 December) proved a version of the central limit theorem. It was finally accepted on 16 March 1935. By spring of that same year, Turing started his master's course (Part III)—which he completed in 1937—and, at the same time, he published his first paper, a one-page article called "Equivalence of left and right almost periodicity" (sent on 23 April), featured in the tenth volume of the "Journal of the London Mathematical Society". Later that year, Turing was elected a Fellow of King's College on the strength of his dissertation where he served as a lecturer. However, and, unknown to Turing, this version of the theorem he proved in his paper, had already been proven, in 1922, by Jarl Waldemar Lindeberg. Despite this, the committee found Turing's methods original and so regarded the work worthy of consideration for the fellowship. Abram Besicovitch's report for the committee went so far as to say that if Turing's work had been published before Lindeberg's, it would have been "an important event in the mathematical literature of that year".
Between the springs of 1935 and 1936, at the same time as Alonzo Church, Turing worked on the decidability of problems, starting from Gödel's incompleteness theorems. In mid-April 1936, Turing sent Max Newman the first draft typescript of his investigations. That same month, Church published his "An Unsolvable Problem of Elementary Number Theory", with similar conclusions to Turing's then-yet unpublished work. Finally, on 28 May of that year, he finished and delivered his 36-page paper for publication called "On Computable Numbers, with an Application to the Entscheidungsproblem". It was published in the "Proceedings of the London Mathematical Society" journal in two parts, the first on 30 November and the second on 23 December. In this paper, Turing reformulated Kurt Gödel's 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with the formal and simple hypothetical devices that became known as Turing machines. The "Entscheidungsproblem" (decision problem) was originally posed by German mathematician David Hilbert in 1928. Turing proved that his "universal computing machine" would be capable of performing any conceivable mathematical computation if it were representable as an algorithm. He went on to prove that there was no solution to the "decision problem" by first showing that the halting problem for Turing machines is undecidable: it is not possible to decide algorithmically whether a Turing machine will ever halt. This paper has been called "easily the most influential math paper in history".
Although Turing's proof was published shortly after Church's equivalent proof using his lambda calculus, Turing's approach is considerably more accessible and intuitive than Church's. It also included a notion of a 'Universal Machine' (now known as a universal Turing machine), with the idea that such a machine could perform the tasks of any other computation machine (as indeed could Church's lambda calculus). According to the Church–Turing thesis, Turing machines and the lambda calculus are capable of computing anything that is computable. John von Neumann acknowledged that the central concept of the modern computer was due to Turing's paper. To this day, Turing machines are a central object of study in theory of computation. From September 1936 to July 1938, Turing spent most of his time studying under Church at Princeton University, in the second year as a Jane Eliza Procter Visiting Fellow. In addition to his purely mathematical work, he studied cryptology and also built three of four stages of an electro-mechanical binary multiplier. In June 1938, he obtained his PhD from the Department of Mathematics at Princeton; his dissertation, "Systems of Logic Based on Ordinals", introduced the concept of ordinal logic and the notion of relative computing, in which Turing machines are augmented with so-called oracles, allowing the study of problems that cannot be solved by Turing machines. John von Neumann wanted to hire him as his postdoctoral assistant, but he went back to the United Kingdom.
Career and research. When Turing returned to Cambridge, he attended lectures given in 1939 by Ludwig Wittgenstein about the foundations of mathematics. The lectures have been reconstructed verbatim, including interjections from Turing and other students, from students' notes. Turing and Wittgenstein argued and disagreed, with Turing defending formalism and Wittgenstein propounding his view that mathematics does not discover any absolute truths, but rather invents them. Cryptanalysis. During the Second World War, Turing was a leading participant in the breaking of German ciphers at Bletchley Park. The historian and wartime codebreaker Asa Briggs has said, "You needed exceptional talent, you needed genius at Bletchley and Turing's was that genius." From September 1938, Turing worked part-time with the Government Code and Cypher School (GC&CS), the British codebreaking organisation. He concentrated on cryptanalysis of the Enigma cipher machine used by Nazi Germany, together with Dilly Knox, a senior GC&CS codebreaker. Soon after the July 1939 meeting near Warsaw at which the Polish Cipher Bureau gave the British and French details of the wiring of Enigma machine's rotors and their method of decrypting Enigma machine's messages, Turing and Knox developed a broader solution. The Polish method relied on an insecure indicator procedure that the Germans were likely to change, which they in fact did in May 1940. Turing's approach was more general, using crib-based decryption for which he produced the functional specification of the bombe (an improvement on the Polish Bomba).
On 4 September 1939, the day after the UK declared war on Germany, Turing reported to Bletchley Park, the wartime station of GC&CS. Like all others who came to Bletchley, he was required to sign the Official Secrets Act, in which he agreed not to disclose anything about his work at Bletchley, with severe legal penalties for violating the Act. Specifying the bombe was the first of five major cryptanalytical advances that Turing made during the war. The others were: deducing the indicator procedure used by the German navy; developing a statistical procedure dubbed "Banburismus" for making much more efficient use of the bombes; developing a procedure dubbed "Turingery" for working out the cam settings of the wheels of the Lorenz SZ 40/42 ("Tunny") cipher machine and, towards the end of the war, the development of a portable secure voice scrambler at Hanslope Park that was codenamed "Delilah". By using statistical techniques to optimise the trial of different possibilities in the code breaking process, Turing made an innovative contribution to the subject. He wrote two papers discussing mathematical approaches, titled "The Applications of Probability to Cryptography" and "Paper on Statistics of Repetitions", which were of such value to GC&CS and its successor GCHQ that they were not released to the UK National Archives until April 2012, shortly before the centenary of his birth. A GCHQ mathematician, "who identified himself only as Richard," said at the time that the fact that the contents had been restricted under the Official Secrets Act for some 70 years demonstrated their importance, and their relevance to post-war cryptanalysis:
Turing had a reputation for eccentricity at Bletchley Park. He was known to his colleagues as "Prof" and his treatise on Enigma was known as the "Prof's Book". According to historian Ronald Lewin, Jack Good, a cryptanalyst who worked with Turing, said of his colleague: Peter Hilton recounted his experience working with Turing in Hut 8 in his "Reminiscences of Bletchley Park" from "A Century of Mathematics in America:" Hilton echoed similar thoughts in the Nova PBS documentary "Decoding Nazi Secrets". While working at Bletchley, Turing, who was a talented long-distance runner, occasionally ran the to London when he was needed for meetings, and he was capable of world-class marathon standards. Turing tried out for the 1948 British Olympic team, but he was hampered by an injury. His tryout time for the marathon was only 11 minutes slower than British silver medallist Thomas Richards' Olympic race time of 2 hours 35 minutes. He was Walton Athletic Club's best runner, a fact discovered when he passed the group while running alone. When asked why he ran so hard in training he replied:
Due to the problems of counterfactual history, it is hard to estimate the precise effect Ultra intelligence had on the war. However, official war historian Harry Hinsley estimated that this work shortened the war in Europe by more than two years and saved over 14 million lives. At the end of the war, a memo was sent to all those who had worked at Bletchley Park, reminding them that the code of silence dictated by the Official Secrets Act did not end with the war but would continue indefinitely. Thus, even though Turing was appointed an Officer of the Order of the British Empire (OBE) in 1946 by King George VI for his wartime services, his work remained secret for many years. Bombe. Within weeks of arriving at Bletchley Park, Turing had specified an electromechanical machine called the bombe, which could break Enigma more effectively than the Polish "bomba kryptologiczna", from which its name was derived. The bombe, with an enhancement suggested by mathematician Gordon Welchman, became one of the primary tools, and the major automated one, used to attack Enigma-enciphered messages.
The bombe searched for possible correct settings used for an Enigma message (i.e., rotor order, rotor settings and plugboard settings) using a suitable "crib": a fragment of probable plaintext. For each possible setting of the rotors (which had on the order of 1019 states, or 1022 states for the four-rotor U-boat variant), the bombe performed a chain of logical deductions based on the crib, implemented electromechanically. The bombe detected when a contradiction had occurred and ruled out that setting, moving on to the next. Most of the possible settings would cause contradictions and be discarded, leaving only a few to be investigated in detail. A contradiction would occur when an enciphered letter would be turned back into the same plaintext letter, which was impossible with the Enigma. The first bombe was installed on 18 March 1940. Action This Day. By late 1941, Turing and his fellow cryptanalysts Gordon Welchman, Hugh Alexander and Stuart Milner-Barry were frustrated. Building on the work of the Poles, they had set up a good working system for decrypting Enigma signals, but their limited staff and bombes meant they could not translate all the signals. In the summer, they had considerable success, and shipping losses had fallen to under 100,000 tons a month; however, they badly needed more resources to keep abreast of German adjustments. They had tried to get more people and fund more bombes through the proper channels, but had failed.
On 28 October they wrote directly to Winston Churchill explaining their difficulties, with Turing as the first named. They emphasised how small their need was compared with the vast expenditure of men and money by the forces and compared with the level of assistance they could offer to the forces. As Andrew Hodges, biographer of Turing, later wrote, "This letter had an electric effect." Churchill wrote a memo to General Ismay, which read: "ACTION THIS DAY. Make sure they have all they want on extreme priority and report to me that this has been done." On 18 November, the chief of the secret service reported that every possible measure was being taken. The cryptographers at Bletchley Park did not know of the Prime Minister's response, but as Milner-Barry recalled, "All that we did notice was that almost from that day the rough ways began miraculously to be made smooth." More than two hundred bombes were in operation by the end of the war. Hut 8 and the naval Enigma. Turing decided to tackle the particularly difficult problem of cracking the German naval use of Enigma "because no one else was doing anything about it and I could have it to myself". In December 1939, Turing solved the essential part of the naval indicator system, which was more complex than the indicator systems used by the other services.
That same night, he also conceived of the idea of "Banburismus", a sequential statistical technique (what Abraham Wald later called sequential analysis) to assist in breaking the naval Enigma, "though I was not sure that it would work in practice, and was not, in fact, sure until some days had actually broken". For this, he invented a measure of weight of evidence that he called the "ban". "Banburismus" could rule out certain sequences of the Enigma rotors, substantially reducing the time needed to test settings on the bombes. Later this sequential process of accumulating sufficient weight of evidence using decibans (one tenth of a ban) was used in cryptanalysis of the Lorenz cipher. Turing travelled to the United States in November 1942 and worked with US Navy cryptanalysts on the naval Enigma and bombe construction in Washington. He also visited their Computing Machine Laboratory in Dayton, Ohio. Turing's reaction to the American bombe design was far from enthusiastic: During this trip, he also assisted at Bell Labs with the development of secure speech devices. He returned to Bletchley Park in March 1943. During his absence, Hugh Alexander had officially assumed the position of head of Hut 8, although Alexander had been "de facto" head for some time (Turing having little interest in the day-to-day running of the section). Turing became a general consultant for cryptanalysis at Bletchley Park.
Alexander wrote of Turing's contribution: Turingery. In July 1942, Turing devised a technique termed "Turingery" (or jokingly "Turingismus") for use against the Lorenz cipher messages produced by the Germans' new "Geheimschreiber" (secret writer) machine. This was a teleprinter rotor cipher attachment codenamed "Tunny" at Bletchley Park. Turingery was a method of "wheel-breaking", i.e., a procedure for working out the cam settings of Tunny's wheels. He also introduced the Tunny team to Tommy Flowers who, under the guidance of Max Newman, went on to build the Colossus computer, the world's first programmable digital electronic computer, which replaced a simpler prior machine (the Heath Robinson), and whose superior speed allowed the statistical decryption techniques to be applied usefully to the messages. Some have mistakenly said that Turing was a key figure in the design of the Colossus computer. Turingery and the statistical approach of Banburismus undoubtedly fed into the thinking about cryptanalysis of the Lorenz cipher, but he was not directly involved in the Colossus development.
Delilah. Following his work at Bell Labs in the US, Turing pursued the idea of electronic enciphering of speech in the telephone system. In the latter part of the war, he moved to work for the Secret Service's Radio Security Service (later HMGCC) at Hanslope Park. At the park, he further developed his knowledge of electronics with the assistance of REME officer Donald Bayley. Together they undertook the design and construction of a portable secure voice communications machine codenamed "Delilah". The machine was intended for different applications, but it lacked the capability for use with long-distance radio transmissions. In any case, Delilah was completed too late to be used during the war. Though the system worked fully, with Turing demonstrating it to officials by encrypting and decrypting a recording of a Winston Churchill speech, Delilah was not adopted for use. Turing also consulted with Bell Labs on the development of SIGSALY, a secure voice system that was used in the later years of the war. Early computers and the Turing test.
Between 1945 and 1947, Turing lived in Hampton, London, while he worked on the design of the ACE (Automatic Computing Engine) at the National Physical Laboratory (NPL). He presented a paper on 19 February 1946, which was the first detailed design of a stored-program computer. Von Neumann's incomplete "First Draft of a Report on the EDVAC" had predated Turing's paper, but it was much less detailed and, according to John R. Womersley, Superintendent of the NPL Mathematics Division, it "contains a number of ideas which are Dr. Turing's own". Although ACE was a feasible design, the effect of the Official Secrets Act surrounding the wartime work at Bletchley Park made it impossible for Turing to explain the basis of his analysis of how a computer installation involving human operators would work. This led to delays in starting the project and he became disillusioned. In late 1947 he returned to Cambridge for a sabbatical year during which he produced a seminal work on "Intelligent Machinery" that was not published in his lifetime. While he was at Cambridge, the Pilot ACE was being built in his absence. It executed its first program on 10 May 1950, and a number of later computers around the world owe much to it, including the English Electric DEUCE and the American Bendix G-15. The full version of Turing's ACE was not built until after his death.
According to the memoirs of the German computer pioneer Heinz Billing from the Max Planck Institute for Physics, published by Genscher, Düsseldorf, there was a meeting between Turing and Konrad Zuse. It took place in Göttingen in 1947. The interrogation had the form of a colloquium. Participants were Womersley, Turing, Porter from England and a few German researchers like Zuse, Walther, and Billing (for more details see Herbert Bruderer, "Konrad Zuse und die Schweiz"). In 1948, Turing was appointed reader in the Mathematics Department at the University of Manchester. He lived at "Copper Folly", 43 Adlington Road, in Wilmslow. A year later, he became deputy director of the Computing Machine Laboratory, where he worked on software for one of the earliest stored-program computers—the Manchester Mark 1. Turing wrote the first version of the Programmer's Manual for this machine, and was recruited by Ferranti as a consultant in the development of their commercialised machine, the Ferranti Mark 1. He continued to be paid consultancy fees by Ferranti until his death. During this time, he continued to do more abstract work in mathematics,<ref name="doi10.1093/qjmam/1.1.287"></ref> and in "Computing Machinery and Intelligence", Turing addressed the problem of artificial intelligence, and proposed an experiment that became known as the Turing test, an attempt to define a standard for a machine to be called "intelligent". The idea was that a computer could be said to "think" if a human interrogator could not tell it apart, through conversation, from a human being. In the paper, Turing suggested that rather than building a program to simulate the adult mind, it would be better to produce a simpler one to simulate a child's mind and then to subject it to a course of education. A reversed form of the Turing test is widely used on the Internet; the CAPTCHA test is intended to determine whether the user is a human or a computer.
In 1948, Turing, working with his former undergraduate colleague, D.G. Champernowne, began writing a chess program for a computer that did not yet exist. By 1950, the program was completed and dubbed the Turochamp. In 1952, he tried to implement it on a Ferranti Mark 1, but lacking enough power, the computer was unable to execute the program. Instead, Turing "ran" the program by flipping through the pages of the algorithm and carrying out its instructions on a chessboard, taking about half an hour per move. The game was recorded. According to Garry Kasparov, Turing's program "played a recognizable game of chess". The program lost to Turing's colleague Alick Glennie, although it is said that it won a game against Champernowne's wife, Isabel. His Turing test was a significant, characteristically provocative, and lasting contribution to the debate regarding artificial intelligence, which continues after more than half a century. Pattern formation and mathematical biology.
Although published before the structure and role of DNA was understood, Turing's work on morphogenesis remains relevant today and is considered a seminal piece of work in mathematical biology. One of the early applications of Turing's paper was the work by James Murray explaining spots and stripes on the fur of cats, large and small. Further research in the area suggests that Turing's work can partially explain the growth of "feathers, hair follicles, the branching pattern of lungs, and even the left-right asymmetry that puts the heart on the left side of the chest". In 2012, Sheth, et al. found that in mice, removal of Hox genes causes an increase in the number of digits without an increase in the overall size of the limb, suggesting that Hox genes control digit formation by tuning the wavelength of a Turing-type mechanism. Later papers were not available until "Collected Works of A. M. Turing" was published in 1992. A study conducted in 2023 confirmed Turing's mathematical model hypothesis. Presented by the American Physical Society, the experiment involved growing chia seeds in even layers within trays, later adjusting the available moisture. Researchers experimentally tweaked the factors which appear in the Turing equations, and, as a result, patterns resembling those seen in natural environments emerged. This is believed to be the first time that experiments with living vegetation have verified Turing's mathematical insight.
Personal life. Treasure. In the 1940s, Turing became worried about losing his savings in the event of a German invasion. In order to protect it, he bought two silver bars weighing and worth £250 (in 2022, £8,000 adjusted for inflation, £48,000 at spot price) and buried them in a wood near Bletchley Park. Upon returning to dig them up, Turing found that he was unable to break his own code describing where exactly he had hidden them. This, along with the fact that the area had been renovated, meant that he never regained the silver. Engagement. In 1941, Turing proposed marriage to Hut 8 colleague Joan Clarke, a fellow mathematician and cryptanalyst, but their engagement was short-lived. After admitting his homosexuality to his fiancée, who was reportedly "unfazed" by the revelation, Turing decided that he could not go through with the marriage. Homosexuality and indecency conviction. In December 1951, Turing met Arnold Murray, a 19-year-old unemployed man. Turing was walking along Manchester's Oxford Road when he met Murray just outside the Regal Cinema and invited him to lunch. The two agreed to meet again and in January 1952 began an intimate relationship. On 23 January, Turing's house in Wilmslow was burgled. Murray told Turing that he and the burglar were acquainted, and Turing reported the crime to the police. During the investigation, he acknowledged a sexual relationship with Murray. Homosexual acts were criminal offences in the United Kingdom at that time, and both men were charged with "gross indecency" under Section 11 of the Criminal Law Amendment Act 1885. Initial committal proceedings for the trial were held on 27 February during which Turing's solicitor "reserved his defence", i.e., did not argue or provide evidence against the allegations. The proceedings were held at the Sessions House in Knutsford.
Turing was later convinced by the advice of his brother and his own solicitor, and he entered a plea of guilty. The case, "Regina v. Turing and Murray," was brought to trial on 31 March 1952. Turing was convicted and given a choice between imprisonment and probation. His probation would be conditional on his agreement to undergo hormonal physical changes designed to reduce libido, known as "chemical castration". He accepted the option of injections of what was then called stilboestrol (now known as diethylstilbestrol or DES), a synthetic oestrogen; this feminization of his body was continued for the course of one year. The treatment rendered Turing impotent and caused breast tissue to form. In a letter, Turing wrote that "no doubt I shall emerge from it all a different man, but quite who I've not found out". Murray was given a conditional discharge. Turing's conviction led to the removal of his security clearance and barred him from continuing with his cryptographic consultancy for the Government Communications Headquarters (GCHQ), the British signals intelligence agency that had evolved from GC&CS in 1946, though he kept his academic post. His trial took place only months after the defection to the Soviet Union of Guy Burgess and Donald Maclean, in summer 1951, after which the Foreign Office started to consider anyone known to be homosexual as a potential security risk.
Turing was denied entry into the United States after his conviction in 1952, but was free to visit other European countries. In the summer of 1952 he visited Norway which was more tolerant of homosexuals. Among the various men he met there was one named Kjell Carlson. Kjell intended to visit Turing in the UK but the authorities intercepted Kjell's postcard detailing his travel arrangements and were able to intercept and deport him before the two could meet. It was also during this time that Turing started consulting a psychiatrist, Dr Franz Greenbaum, with whom he got on well and who subsequently became a family friend. Death. On 8 June 1954, at his house at 43 Adlington Road, Wilmslow, Turing's housekeeper found him dead. A post mortem was held that evening, which determined that he had died the previous day at age 41 with cyanide poisoning cited as the cause of death. When his body was discovered, an apple lay half-eaten beside his bed, and although the apple was not tested for cyanide, it was speculated that this was the means by which Turing had consumed a fatal dose.
Turing's brother, John, identified the body the following day and took the advice given by Dr. Greenbaum to accept the verdict of the inquest, as there was little prospect of establishing that the death was accidental. The inquest was held the following day, which determined the cause of death to be suicide. Turing's remains were cremated at Woking Crematorium just two days later on 12 June 1954, with just his mother, brother, and Lyn Newman attending, and his ashes were scattered in the gardens of the crematorium, just as his father's had been. Turing's mother was on holiday in Italy at the time of his death and returned home after the inquest. She never accepted the verdict of suicide. Philosopher Jack Copeland has questioned various aspects of the coroner's historical verdict. He suggested an alternative explanation for the cause of Turing's death: the accidental inhalation of cyanide fumes from an apparatus used to electroplate gold onto spoons. The potassium cyanide was used to dissolve the gold. Turing had such an apparatus set up in his tiny spare room. Copeland noted that the autopsy findings were more consistent with inhalation than with ingestion of the poison. Turing also habitually ate an apple before going to bed, and it was not unusual for the apple to be discarded half-eaten. Furthermore, Turing had reportedly borne his legal setbacks and hormone treatment (which had been discontinued a year previously) "with good humour" and had shown no sign of despondency before his death. He even set down a list of tasks that he intended to complete upon returning to his office after the holiday weekend. Turing's mother believed that the ingestion was accidental, resulting from her son's careless storage of laboratory chemicals. Turing biographer Andrew Hodges theorised that Turing deliberately made his death look accidental in order to shield his mother from the knowledge that he had killed himself.
Doubts on the suicide thesis have been also cast by John W. Dawson Jr. who, in his review of Hodges' book, recalls "Turing's vulnerable position in the Cold War political climate" and points out that "Turing was found dead by a maid, who discovered him 'lying neatly in his bed'—hardly what one would expect of "a man fighting for life against the suffocation induced by cyanide poisoning." Turing had given no hint of suicidal inclinations to his friends and had made no effort to put his affairs in order. Hodges and a later biographer, David Leavitt, have both speculated that Turing was re-enacting a scene from the Walt Disney film "Snow White and the Seven Dwarfs" (1937), his favourite fairy tale. Both men noted that (in Leavitt's words) he took "an especially keen pleasure in the scene where the Wicked Queen immerses her apple in the poisonous brew". It has also been suggested that Turing's belief in fortune-telling may have caused his depressed mood. As a youth, Turing had been told by a fortune-teller that he would be a genius. In mid-May 1954, shortly before his death, Turing again decided to consult a fortune-teller during a day-trip to St Annes-on-Sea with the Greenbaum family. According to the Greenbaums' daughter, Barbara:
Government apology and pardon. In August 2009, British programmer John Graham-Cumming started a petition urging the British government to apologise for Turing's prosecution as a homosexual. The petition received more than 30,000 signatures. The prime minister, Gordon Brown, acknowledged the petition, releasing a statement on 10 September 2009 apologising and describing the treatment of Turing as "appalling": In December 2011, William Jones and his member of Parliament, John Leech, created an e-petition requesting that the British government pardon Turing for his conviction of "gross indecency": The petition gathered over 37,000 signatures, and was submitted to Parliament by the Manchester MP John Leech but the request was discouraged by Justice Minister Lord McNally, who said: John Leech, the MP for Manchester Withington (2005–15), submitted several bills to Parliament and led a high-profile campaign to secure the pardon. Leech made the case in the House of Commons that Turing's contribution to the war made him a national hero and that it was "ultimately just embarrassing" that the conviction still stood. Leech continued to take the bill through Parliament and campaigned for several years, gaining the public support of numerous leading scientists, including Stephen Hawking. At the British premiere of a film based on Turing's life, "The Imitation Game", the producers thanked Leech for bringing the topic to public attention and securing Turing's pardon. Leech is now regularly described as the "architect" of Turing's pardon and subsequently the Alan Turing Law which went on to secure pardons for 75,000 other men and women convicted of similar crimes.
On 26 July 2012, a bill was introduced in the House of Lords to grant a statutory pardon to Turing for offences under section 11 of the Criminal Law Amendment Act 1885, of which he was convicted on 31 March 1952. Late in the year in a letter to "The Daily Telegraph", the physicist Stephen Hawking and 10 other signatories including the Astronomer Royal Lord Rees, President of the Royal Society Sir Paul Nurse, Lady Trumpington (who worked for Turing during the war) and Lord Sharkey (the bill's sponsor) called on Prime Minister David Cameron to act on the pardon request. The government indicated it would support the bill, and it passed its third reading in the House of Lords in October. At the bill's second reading in the House of Commons on 29 November 2013, Conservative MP Christopher Chope objected to the bill, delaying its passage. The bill was due to return to the House of Commons on 28 February 2014, but before the bill could be debated in the House of Commons, the government elected to proceed under the royal prerogative of mercy. On 24 December 2013, Queen Elizabeth II signed a pardon for Turing's conviction for "gross indecency", with immediate effect. Announcing the pardon, Lord Chancellor Chris Grayling said Turing deserved to be "remembered and recognised for his fantastic contribution to the war effort" and not for his later criminal conviction. The Queen pronounced Turing pardoned in August 2014. It was only the fourth royal pardon granted since the conclusion of the Second World War. Pardons are normally granted only when the person is technically innocent, and a request has been made by the family or other interested party; neither condition was met in regard to Turing's conviction.
In September 2016, the government announced its intention to expand this retroactive exoneration to other men convicted of similar historical indecency offences, in what was described as an "Alan Turing law". The Alan Turing law is now an informal term for the law in the United Kingdom, contained in the Policing and Crime Act 2017, which serves as an amnesty law to retroactively pardon men who were cautioned or convicted under historical legislation that outlawed homosexual acts. The law applies in England and Wales. On 19 July 2023, following an apology to LGBT veterans from the UK Government, Defence Secretary Ben Wallace suggested Turing should be honoured with a permanent statue on the fourth plinth of Trafalgar Square, describing Turing as "probably the greatest war hero, in my book, of the Second World War, [whose] achievements shortened the war, saved thousands of lives, helped defeat the Nazis. And his story is a sad story of a society and how it treated him."
Area Area is the measure of a region's size on a surface. The area of a plane region or "plane area" refers to the area of a shape or planar lamina, while "surface area" refers to the area of an open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). Two different regions may have the same area (as in squaring the circle); by synecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area". The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.
There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus. For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus. Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable if one supposes the axiom of choice. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.
Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists. Formal definition. An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties: It can be proved that such an area function actually exists. Units. Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m2), square centimetres (cm2), square millimetres (mm2), square kilometres (km2), square feet (ft2), square yards (yd2), square miles (mi2), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units. The SI unit of area is the square metre, which is considered an SI derived unit. Conversions.
Calculation of the area of a square whose length and width are 1 metre would be: 1 metre × 1 metre = 1 m2 and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as: 3 metres × 2 metres = 6 m2. This is equivalent to 6 million square millimetres. Other useful conversions are: Non-metric units. In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units. the relationship between square feet and square inches is where 144 = 122 = 12 × 12. Similarly: In addition, conversion factors include: Other units including historical. There are several other common units for area. The are was the original unit of area in the metric system, with: Though the are has fallen out of use, the hectare is still commonly used to measure land: Other uncommon metric units of area include the tetrad, the hectad, and the myriad. The acre is also commonly used to measure land areas, where
An acre is approximately 40% of a hectare. On the atomic scale, area is measured in units of barns, such that: The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics. In South Asia (mainly Indians), although the countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different results, depending on what reference that has been used. Some traditional South Asian units that have fixed value: History. Circle area. In the 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates, but did not identify the constant of proportionality. Eudoxus of Cnidus, also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared.
Subsequently, Book I of Euclid's "Elements" dealt with equality of areas between two-dimensional figures. The mathematician Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book "Measurement of a Circle". (The circumference is 2"r", and the area of a triangle is half the base times the height, yielding the area "r"2 for the disk.) Archimedes approximated the value of (and hence the area of a unit-radius circle) with his doubling method, in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular hexagon, then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with circumscribed polygons). Quadrilateral area. In the 7th century CE, Brahmagupta developed a formula, now known as Brahmagupta's formula, for the area of a cyclic quadrilateral (a quadrilateral inscribed in a circle) in terms of its sides. In 1842, the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula, known as Bretschneider's formula, for the area of any quadrilateral.
General polygon area. The development of Cartesian coordinates by René Descartes in the 17th century allowed the development of the surveyor's formula for the area of any polygon with known vertex locations by Gauss in the 19th century. Areas determined using calculus. The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas of various curved three-dimensional objects. Area formulas. Polygon formulas. For a non-self-intersecting (simple) polygon, the Cartesian coordinates formula_1 ("i"=0, 1, ..., "n"-1) of whose "n" vertices are known, the area is given by the surveyor's formula: where when "i"="n"-1, then "i"+1 is expressed as modulus "n" and so refers to 0. Rectangles. The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length and width , the formula for the area is: That is, the area of the rectangle is the length multiplied by the width. As a special case, as in the case of a square, the area of a square with side length is given by the formula:
The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom. On the other hand, if geometry is developed before arithmetic, this formula can be used to define multiplication of real numbers. Dissection, parallelograms, and triangles. Most other simple formulas for area follow from the method of dissection. This involves cutting a shape into pieces, whose areas must sum to the area of the original shape. For an example, any parallelogram can be subdivided into a trapezoid and a right triangle, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle: However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of the parallelogram: Similar arguments can be used to find area formulas for the trapezoid as well as more complicated polygons.
Area of curved shapes. Circles. The formula for the area of a circle (more properly called the area enclosed by a circle or the area of a disk) is based on a similar method. Given a circle of radius , it is possible to partition the circle into sectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is , and the width is half the circumference of the circle, or . Thus, the total area of the circle is : Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of the areas of the approximate parallelograms is exactly , which is the area of the circle. This argument is actually a simple application of the ideas of calculus. In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus. Using modern methods, the area of a circle can be computed using a definite integral:
Ellipses. The formula for the area enclosed by an ellipse is related to the formula of a circle; for an ellipse with semi-major and semi-minor axes and the formula is: Non-planar surface area. Most basic formulas for surface area can be obtained by cutting surfaces and flattening them out (see: developable surfaces). For example, if the side surface of a cylinder (or any prism) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone, the side surface can be flattened out into a sector of a circle, and the resulting area computed. The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by Archimedes in his work "On the Sphere and Cylinder". The formula is: where is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus.
General formulas. Bounded area between two quadratic functions. To find the bounded area between two quadratic functions, we first subtract one from the other, writing the difference as formula_21 where "f"("x") is the quadratic upper bound and "g"("x") is the quadratic lower bound. By the area integral formulas above and Vieta's formula, we can obtain that formula_22 The above remains valid if one of the bounding functions is linear instead of quadratic. General formula for surface area. The general formula for the surface area of the graph of a continuously differentiable function formula_36 where formula_37 and formula_38 is a region in the xy-plane with the smooth boundary: An even more general formula for the area of the graph of a parametric surface in the vector form formula_40 where formula_41 is a continuously differentiable vector function of formula_42 is: List of formulas. The above calculations show how to find the areas of many common shapes. The areas of irregular (and thus arbitrary) polygons can be calculated using the "Surveyor's formula" (shoelace formula).
Relation of area to perimeter. The isoperimetric inequality states that, for a closed curve of length "L" (so the region it encloses has perimeter "L") and for area "A" of the region that it encloses, and equality holds if and only if the curve is a circle. Thus a circle has the largest area of any closed figure with a given perimeter. At the other extreme, a figure with given perimeter "L" could have an arbitrarily small area, as illustrated by a rhombus that is "tipped over" arbitrarily far so that two of its angles are arbitrarily close to 0° and the other two are arbitrarily close to 180°. For a circle, the ratio of the area to the circumference (the term for the perimeter of a circle) equals half the radius "r". This can be seen from the area formula "πr"2 and the circumference formula 2"πr". The area of a regular polygon is half its perimeter times the apothem (where the apothem is the distance from the center to the nearest point on any side). Fractals. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). But if the one-dimensional lengths of a fractal drawn in two dimensions are all doubled, the spatial content of the fractal scales by a power of two that is not necessarily an integer. This power is called the fractal dimension of the fractal.
Area bisectors. There are an infinitude of lines that bisect the area of a triangle. Three of them are the medians of the triangle (which connect the sides' midpoints with the opposite vertices), and these are concurrent at the triangle's centroid; indeed, they are the only area bisectors that go through the centroid. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle. Any line through the midpoint of a parallelogram bisects the area. All area bisectors of a circle or other ellipse go through the center, and any chords through the center bisect the area. In the case of a circle they are the diameters of the circle. Optimization. Given a wire contour, the surface of least area spanning ("filling") it is a minimal surface. Familiar examples include soap bubbles. The question of the filling area of the Riemannian circle remains open. The circle has the largest area of any two-dimensional object having the same perimeter.
A cyclic polygon (one inscribed in a circle) has the largest area of any polygon with a given number of sides of the same lengths. A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral. The ratio of the area of the incircle to the area of an equilateral triangle, formula_45, is larger than that of any non-equilateral triangle. The ratio of the area to the square of the perimeter of an equilateral triangle, formula_46 is larger than that for any other triangle.
Astronomical unit The astronomical unit (symbol: au or AU) is a unit of length defined to be exactly equal to . Historically, the astronomical unit was conceived as the average Earth-Sun distance (the average of Earth's aphelion and perihelion), before its modern redefinition in 2012. The astronomical unit is used primarily for measuring distances within the Solar System or around other stars. It is also a fundamental component in the definition of another unit of astronomical length, the parsec. One au is approximately equivalent to 499 light-seconds. History of symbol usage. A variety of unit symbols and abbreviations have been in use for the astronomical unit. In a 1976 resolution, the International Astronomical Union (IAU) had used the symbol "A" to denote a length equal to the astronomical unit. In the astronomical literature, the symbol AU is common. In 2006, the International Bureau of Weights and Measures (BIPM) had recommended ua as the symbol for the unit, from the French "unité astronomique". In the non-normative Annex C to ISO 80000-3:2006 (later withdrawn), the symbol of the astronomical unit was also ua.
In 2012, the IAU, noting "that various symbols are presently in use for the astronomical unit", recommended the use of the symbol "au". The scientific journals published by the American Astronomical Society and the Royal Astronomical Society subsequently adopted this symbol. In the 2014 revision and 2019 edition of the SI Brochure, the BIPM used the unit symbol "au". ISO 80000-3:2019, which replaces ISO 80000-3:2006, does not mention the astronomical unit. Development of unit definition. Earth's orbit around the Sun is an ellipse. The semi-major axis of this elliptic orbit is defined to be half of the straight line segment that joins the perihelion and aphelion. The centre of the Sun lies on this straight line segment, but not at its midpoint. Because ellipses are well-understood shapes, measuring the points of its extremes defined the exact shape mathematically, and made possible calculations for the entire orbit as well as predictions based on observation. In addition, it mapped out exactly the largest straight-line distance that Earth traverses over the course of a year, defining times and places for observing the largest parallax (apparent shifts of position) in nearby stars. Knowing Earth's shift and a star's shift enabled the star's distance to be calculated. But all measurements are subject to some degree of error or uncertainty, and the uncertainties in the length of the astronomical unit only increased uncertainties in the stellar distances. Improvements in precision have always been a key to improving astronomical understanding. Throughout the twentieth century, measurements became increasingly precise and sophisticated, and ever more dependent on accurate observation of the effects described by Einstein's theory of relativity and upon the mathematical tools it used.
Improving measurements were continually checked and cross-checked by means of improved understanding of the laws of celestial mechanics, which govern the motions of objects in space. The expected positions and distances of objects at an established time are calculated (in au) from these laws, and assembled into a collection of data called an ephemeris. NASA Jet Propulsion Laboratory HORIZONS System provides one of several ephemeris computation services. In 1976, to establish a more precise measure for the astronomical unit, the IAU formally adopted a new definition. Although directly based on the then-best available observational measurements, the definition was recast in terms of the then-best mathematical derivations from celestial mechanics and planetary ephemerides. It stated that "the astronomical unit of length is that length ("A") for which the Gaussian gravitational constant ("k") takes the value when the units of measurement are the astronomical units of length, mass and time". Equivalently, by this definition, one au is "the radius of an unperturbed circular Newtonian orbit about the sun of a particle having infinitesimal mass, moving with an angular frequency of "; or alternatively that length for which the heliocentric gravitational constant (the product "G") is equal to ()2 au3/d2, when the length is used to describe the positions of objects in the Solar System.
Subsequent explorations of the Solar System by space probes made it possible to obtain precise measurements of the relative positions of the inner planets and other objects by means of radar and telemetry. As with all radar measurements, these rely on measuring the time taken for photons to be reflected from an object. Because all photons move at the speed of light in vacuum, a fundamental constant of the universe, the distance of an object from the probe is calculated as the product of the speed of light and the measured time. However, for precision the calculations require adjustment for things such as the motions of the probe and object while the photons are transiting. In addition, the measurement of the time itself must be translated to a standard scale that accounts for relativistic time dilation. Comparison of the ephemeris positions with time measurements expressed in Barycentric Dynamical Time (TDB) leads to a value for the speed of light in astronomical units per day (of ). By 2009, the IAU had updated its standard measures to reflect improvements, and calculated the speed of light at (TDB).
In 1983, the CIPM modified the International System of Units (SI) to make the metre defined as the distance travelled in a vacuum by light in 1 / . This replaced the previous definition, valid between 1960 and 1983, which was that the metre equalled a certain number of wavelengths of a certain emission line of krypton-86. (The reason for the change was an improved method of measuring the speed of light.) The speed of light could then be expressed exactly as "c"0 = , a standard also adopted by the IERS numerical standards. From this definition and the 2009 IAU standard, the time for light to traverse an astronomical unit is found to be "τ"A = , which is slightly more than 8 minutes 19 seconds. By multiplication, the best IAU 2009 estimate was "A" = "c"0"τ"A = , based on a comparison of Jet Propulsion Laboratory and IAA–RAS ephemerides. In 2006, the BIPM reported a value of the astronomical unit as . In the 2014 revision of the SI Brochure, the BIPM recognised the IAU's 2012 redefinition of the astronomical unit as .
This estimate was still derived from observation and measurements subject to error, and based on techniques that did not yet standardize all relativistic effects, and thus were not constant for all observers. In 2012, finding that the equalization of relativity alone would make the definition overly complex, the IAU simply used the 2009 estimate to redefine the astronomical unit as a conventional unit of length directly tied to the metre (exactly ). The new definition recognizes as a consequence that the astronomical unit has reduced importance, limited in use to a convenience in some applications. This definition makes the speed of light, defined as exactly , equal to exactly × ÷ or about , some 60 parts per trillion less than the 2009 estimate. Usage and significance. With the definitions used before 2012, the astronomical unit was dependent on the heliocentric gravitational constant, that is the product of the gravitational constant, "G", and the solar mass, . Neither "G" nor can be measured to high accuracy separately, but the value of their product is known very precisely from observing the relative positions of planets (Kepler's third law expressed in terms of Newtonian gravitation). Only the product is required to calculate planetary positions for an ephemeris, so ephemerides are calculated in astronomical units and not in SI units.
The calculation of ephemerides also requires a consideration of the effects of general relativity. In particular, time intervals measured on Earth's surface (Terrestrial Time, TT) are not constant when compared with the motions of the planets: the terrestrial second (TT) appears to be longer near January and shorter near July when compared with the "planetary second" (conventionally measured in TDB). This is because the distance between Earth and the Sun is not fixed (it varies between and ) and, when Earth is closer to the Sun (perihelion), the Sun's gravitational field is stronger and Earth is moving faster along its orbital path. As the metre is defined in terms of the second and the speed of light is constant for all observers, the terrestrial metre appears to change in length compared with the "planetary metre" on a periodic basis. The metre is defined to be a unit of proper length. Indeed, the International Committee for Weights and Measures (CIPM) notes that "its definition applies only within a spatial extent sufficiently small that the effects of the non-uniformity of the gravitational field can be ignored". As such, a distance within the Solar System without specifying the frame of reference for the measurement is problematic. The 1976 definition of the astronomical unit was incomplete because it did not specify the frame of reference in which to apply the measurement, but proved practical for the calculation of ephemerides: a fuller definition that is consistent with general relativity was proposed, and "vigorous debate" ensued until August 2012 when the IAU adopted the current definition of 1 astronomical unit = metres.
The astronomical unit is typically used for stellar system scale distances, such as the size of a protostellar disk or the heliocentric distance of an asteroid, whereas other units are used for other distances in astronomy. The astronomical unit is too small to be convenient for interstellar distances, where the parsec and light-year are widely used. The parsec (parallax arcsecond) is defined in terms of the astronomical unit, being the distance of an object with a parallax of . The light-year is often used in popular works, but is not an approved non-SI unit and is rarely used by professional astronomers. When simulating a numerical model of the Solar System, the astronomical unit provides an appropriate scale that minimizes (overflow, underflow and truncation) errors in floating point calculations. History. The book "On the Sizes and Distances of the Sun and Moon", which is ascribed to Aristarchus, says the distance to the Sun is 18 to 20 times the distance to the Moon, whereas the true ratio is about . The latter estimate was based on the angle between the half-moon and the Sun, which he estimated as (the true value being close to ). Depending on the distance that Albert van Helden assumes Aristarchus used for the distance to the Moon, his calculated distance to the Sun would fall between and Earth radii.
Hipparchus gave an estimate of the distance of Earth from the Sun, quoted by Pappus as equal to 490 Earth radii. According to the conjectural reconstructions of Noel Swerdlow and G. J. Toomer, this was derived from his assumption of a "least perceptible" solar parallax of . A Chinese mathematical treatise, the "Zhoubi Suanjing" (), shows how the distance to the Sun can be computed geometrically, using the different lengths of the noontime shadows observed at three places "li" apart and the assumption that Earth is flat. According to Eusebius in the "Praeparatio evangelica" (Book XV, Chapter 53), Eratosthenes found the distance to the Sun to be "σταδιων μυριαδας τετρακοσιας και οκτωκισμυριας" (literally "myriads ten hundreds and eighty thousands of stadia", where in the Greek text the numerals "myriads", "ten hundreds" and "eighty thousands" are all accusative plural, while "stadia" is the genitive plural of "stadion".) This has been translated either as () stadia (1903 translation by Edwin Hamilton Gifford), or as () stadia (edition of Édouard des Places, dated 1974–1991). Using the Greek stadium of 185 to 190 metres, the former translation comes to to , which is far too low, whereas the second translation comes to 148.7 to 152.8 billion metres (accurate within 2%).
In the 2nd century CE, Ptolemy estimated the mean distance of the Sun as times Earth's radius. To determine this value, Ptolemy started by measuring the Moon's parallax, finding what amounted to a horizontal lunar parallax of 1° 26′, which was much too large. He then derived a maximum lunar distance of Earth radii. Because of cancelling errors in his parallax figure, his theory of the Moon's orbit, and other factors, this figure was approximately correct. He then measured the apparent sizes of the Sun and the Moon and concluded that the apparent diameter of the Sun was equal to the apparent diameter of the Moon at the Moon's greatest distance, and from records of lunar eclipses, he estimated this apparent diameter, as well as the apparent diameter of the shadow cone of Earth traversed by the Moon during a lunar eclipse. Given these data, the distance of the Sun from Earth can be trigonometrically computed to be Earth radii. This gives a ratio of solar to lunar distance of approximately 19, matching Aristarchus's figure. Although Ptolemy's procedure is theoretically workable, it is very sensitive to small changes in the data, so much so that changing a measurement by a few per cent can make the solar distance infinite.
After Greek astronomy was transmitted to the medieval Islamic world, astronomers made some changes to Ptolemy's cosmological model, but did not greatly change his estimate of the Earth–Sun distance. For example, in his introduction to Ptolemaic astronomy, al-Farghānī gave a mean solar distance of Earth radii, whereas in his "zij", al-Battānī used a mean solar distance of Earth radii. Subsequent astronomers, such as al-Bīrūnī, used similar values. Later in Europe, Copernicus and Tycho Brahe also used comparable figures ( and Earth radii), and so Ptolemy's approximate Earth–Sun distance survived through the 16th century. Johannes Kepler was the first to realize that Ptolemy's estimate must be significantly too low (according to Kepler, at least by a factor of three) in his "Rudolphine Tables" (1627). Kepler's laws of planetary motion allowed astronomers to calculate the relative distances of the planets from the Sun, and rekindled interest in measuring the absolute value for Earth (which could then be applied to the other planets). The invention of the telescope allowed far more accurate measurements of angles than is possible with the naked eye. Flemish astronomer Godefroy Wendelin repeated Aristarchus’ measurements in 1635, and found that Ptolemy's value was too low by a factor of at least eleven.
A somewhat more accurate estimate can be obtained by observing the transit of Venus. By measuring the transit in two different locations, one can accurately calculate the parallax of Venus and from the relative distance of Earth and Venus from the Sun, the solar parallax (which cannot be measured directly due to the brightness of the Sun). Jeremiah Horrocks had attempted to produce an estimate based on his observation of the 1639 transit (published in 1662), giving a solar parallax of , similar to Wendelin's figure. The solar parallax is related to the Earth–Sun distance as measured in Earth radii by The smaller the solar parallax, the greater the distance between the Sun and Earth: a solar parallax of is equivalent to an Earth–Sun distance of Earth radii. Christiaan Huygens believed that the distance was even greater: by comparing the apparent sizes of Venus and Mars, he estimated a value of about Earth radii, equivalent to a solar parallax of . Although Huygens' estimate is remarkably close to modern values, it is often discounted by historians of astronomy because of the many unproven (and incorrect) assumptions he had to make for his method to work; the accuracy of his value seems to be based more on luck than good measurement, with his various errors cancelling each other out.
Jean Richer and Giovanni Domenico Cassini measured the parallax of Mars between Paris and Cayenne in French Guiana when Mars was at its closest to Earth in 1672. They arrived at a figure for the solar parallax of , equivalent to an Earth–Sun distance of about Earth radii. They were also the first astronomers to have access to an accurate and reliable value for the radius of Earth, which had been measured by their colleague Jean Picard in 1669 as "toises". This same year saw another estimate for the astronomical unit by John Flamsteed, which accomplished it alone by measuring the martian diurnal parallax. Another colleague, Ole Rømer, discovered the finite speed of light in 1676: the speed was so great that it was usually quoted as the time required for light to travel from the Sun to the Earth, or "light time per unit distance", a convention that is still followed by astronomers today. A better method for observing Venus transits was devised by James Gregory and published in his "Optica Promata" (1663). It was strongly advocated by Edmond Halley and was applied to the transits of Venus observed in 1761 and 1769, and then again in 1874 and 1882. Transits of Venus occur in pairs, but less than one pair every century, and observing the transits in 1761 and 1769 was an unprecedented international scientific operation including observations by James Cook and Charles Green from Tahiti. Despite the Seven Years' War, dozens of astronomers were dispatched to observing points around the world at great expense and personal danger: several of them died in the endeavour. The various results were collated by Jérôme Lalande to give a figure for the solar parallax of . Karl Rudolph Powalky had made an estimate of in 1864.
Another method involved determining the constant of aberration. Simon Newcomb gave great weight to this method when deriving his widely accepted value of for the solar parallax (close to the modern value of ), although Newcomb also used data from the transits of Venus. Newcomb also collaborated with A. A. Michelson to measure the speed of light with Earth-based equipment; combined with the constant of aberration (which is related to the light time per unit distance), this gave the first direct measurement of the Earth–Sun distance in metres. Newcomb's value for the solar parallax (and for the constant of aberration and the Gaussian gravitational constant) were incorporated into the first international system of astronomical constants in 1896, which remained in place for the calculation of ephemerides until 1964. The name "astronomical unit" appears first to have been used in 1903. The discovery of the near-Earth asteroid 433 Eros and its passage near Earth in 1900–1901 allowed a considerable improvement in parallax measurement. Another international project to measure the parallax of 433 Eros was undertaken in 1930–1931.
Direct radar measurements of the distances to Venus and Mars became available in the early 1960s. Along with improved measurements of the speed of light, these showed that Newcomb's values for the solar parallax and the constant of aberration were inconsistent with one another. Developments. The unit distance (the value of the astronomical unit in metres) can be expressed in terms of other astronomical constants: where is the Newtonian constant of gravitation, is the solar mass, is the numerical value of Gaussian gravitational constant and is the time period of one day. The Sun is constantly losing mass by radiating away energy, so the orbits of the planets are steadily expanding outward from the Sun. This has led to calls to abandon the astronomical unit as a unit of measurement. As the speed of light has an exact defined value in SI units and the Gaussian gravitational constant is fixed in the astronomical system of units, measuring the light time per unit distance is exactly equivalent to measuring the product × in SI units. Hence, it is possible to construct ephemerides entirely in SI units, which is increasingly becoming the norm.
A 2004 analysis of radiometric measurements in the inner Solar System suggested that the secular increase in the unit distance was much larger than can be accounted for by solar radiation, + metres per century. The measurements of the secular variations of the astronomical unit are not confirmed by other authors and are quite controversial. Furthermore, since 2010, the astronomical unit has not been estimated by the planetary ephemerides. Examples. The following table contains some distances given in astronomical units. It includes some examples with distances that are normally not given in astronomical units, because they are either too short or far too long. Distances normally change over time. Examples are listed by increasing distance.
Artist An artist is a person engaged in an activity related to creating art, practicing the arts, or demonstrating an art. The most common usage (in both everyday speech and academic discourse) refers to a practitioner in the visual arts only. However, the term is also often used in the entertainment business to refer to musicians and other performers. Artiste (French) is a variant used in English in this context, but this use has become rare. The use of the term "artist" to describe writers is valid, but less common, and mostly restricted to contexts such as critics' reviews; "author" is generally used instead. Dictionary definitions. The "Oxford English Dictionary" defines the older, broader meanings of the word "artist": History of the term. The Greek word , often translated as "art", implies mastery of any sort of craft. The adjectival Latin form of the word, , became the source of the English words technique, technology, and technical. In Greek culture, each of the nine Muses oversaw a different field of human creation:
No muse was identified with the visual arts of painting and sculpture. In ancient Greece, sculptors and painters were held in low regard, the work often performed by slaves and mostly regarded as mere manual labour. The word "art" derives from the Latin "" (stem "art-"), which, although literally defined means "skill method" or "technique", also conveys a connotation of beauty. During the Middle Ages the word "artist" already existed in some countries such as Italy, but the meaning was something resembling "craftsman", while the word "artisan" was still unknown. An artist was someone able to do a work better than others, so the skilled excellency was underlined, rather than the activity field. In this period, some "artisanal" products (such as textiles) were much more precious and expensive than paintings or sculptures. The first division into major and minor arts dates back at least to the works of Leon Battista Alberti (1404–1472): "De re aedificatoria, De statua, De pictura", which focused on the importance of the intellectual skills of the artist rather than the manual skills (even if in other forms of art there was a project behind).
With the academies in Europe (second half of 16th century) the gap between fine and applied arts was definitely set. Many contemporary definitions of "artist" and "art" are highly contingent on culture, resisting aesthetic prescription; in the same way, the features constituting beauty and the beautiful cannot be standardized easily without moving into kitsch. Training and employment. The US Bureau of Labor Statistics classifies many visual artists as either "craft artists" or "fine artists". A craft artist makes handmade functional works of art, such as pottery or clothing. A fine artist makes paintings, illustrations (such as book illustrations or medical illustrations), sculptures, or similar artistic works primarily for their aesthetic value. The main source of skill for both craft artists and fine artists is long-term repetition and practice. Many fine artists have studied their art form at university, and some have a master's degree in fine arts. Artists may also study on their own or receive on-the-job training from an experienced artist.
The number of available jobs as an artist is increasing more slowly than in other fields. About half of US artists are self-employed. Others work in a variety of industries. For example, a pottery manufacturer will employ craft artists, and book publishers will hire illustrators. In the US, fine artists have a median income of approximately US$50,000 per year, and craft artists have a median income of approximately US$33,000 per year. This compares to US$61,000 for all art-related fields, including related jobs such as graphic designers, multimedia artists, animators, and fashion designers. Many artists work part-time as artists and hold a second job.
Actaeon Actaeon (; "Aktaiōn"), in Greek mythology, was the son of the priestly herdsman Aristaeus and Autonoe in Boeotia, and a famous Theban hero. Through his mother he was a member of the ruling House of Cadmus. Like Achilles, in a later generation, he was trained by the centaur Chiron. He fell to the fatal wrath of Artemis (later his myth was attached to her Roman counterpart Diana), but the surviving details of his transgression vary: "the only certainty is in what Aktaion suffered, his pathos, and what Artemis did: the hunter became the hunted; he was transformed into a stag, and his raging hounds, struck with a 'wolf's frenzy' (Lyssa), tore him apart as they would a stag." The many depictions both in ancient art and in the Renaissance and post-Renaissance art normally show either the moment of transgression and transformation, or his death by his own hounds. Story. Among others, John Heath has observed, "The unalterable kernel of the tale was a hunter's transformation into a deer and his death in the jaws of his hunting dogs. But authors were free to suggest different motives for his death." In the version that was offered by the Hellenistic poet Callimachus, which has become the standard setting, Artemis was bathing in the woods when the hunter Actaeon stumbled across her, thus seeing her naked. He stopped and stared, amazed at her ravishing beauty. Once seen, Artemis got revenge on Actaeon: she forbade him speech – if he tried to speak, he would be changed into a stag – for the unlucky profanation of her virginity's mystery.
Upon hearing the call of his hunting party, he cried out to them and immediately transformed. At this, he fled deep into the woods, and doing so he came upon a pond and, seeing his reflection, groaned. His own hounds then turned upon him and pursued him, not recognizing him. In an endeavour to save himself, he raised his eyes (and would have raised his arms, had he had them) toward Mount Olympus. The gods did not heed his desperation, and he was torn to pieces. An element of the earlier myth made Actaeon the familiar hunting companion of Artemis, no stranger. In an embroidered extension of the myth, the hounds were so upset with their master's death, that Chiron made a statue so lifelike that the hounds thought it was Actaeon. There are various other versions of his transgression: The Hesiodic "Catalogue of Women" and pseudo-Apollodoran "Bibliotheke" state that his offense was that he was a rival of Zeus for Semele, his mother's sister, whereas in Euripides' "Bacchae" he has boasted that he is a better hunter than Artemis:
Further materials, including fragments that belong with the Hesiodic "Catalogue of Women" and at least four Attic tragedies, including a "Toxotides" of Aeschylus, have been lost. Diodorus Siculus (4.81.4), in a variant of Actaeon's "hubris" that has been largely ignored, has it that Actaeon wanted to marry Artemis. Other authors say the hounds were Artemis' own; some lost elaborations of the myth seem to have given them all names and narrated their wanderings after his loss. A number of ancient Greek vases depicting the metamorphosis and death of Actaeon include the goddess Lyssa in the scene, infecting his dogs with rabies and setting them against him. According to the Latin version of the story told by the Roman Ovid having accidentally seen Diana (Artemis) on Mount Cithaeron while she was bathing, he was changed by her into a stag, and pursued and killed by his fifty hounds. This version also appears in Callimachus' Fifth Hymn, as a mythical parallel to the blinding of Tiresias after he sees Athena bathing.
The literary testimony of Actaeon's myth is largely lost, but Lamar Ronald Lacy, deconstructing the myth elements in what survives and supplementing it by iconographic evidence in late vase-painting, made a plausible reconstruction of an ancient Actaeon myth that Greek poets may have inherited and subjected to expansion and dismemberment. His reconstruction opposes a too-pat consensus that has an archaic Actaeon aspiring to Semele, a classical Actaeon boasting of his hunting prowess and a Hellenistic Actaeon glimpsing Artemis' bath. Lacy identifies the site of Actaeon's transgression as a spring sacred to Artemis at Plataea where Actaeon was a " hero archegetes" ("hero-founder") The righteous hunter, the companion of Artemis, seeing her bathing naked in the spring, was moved to try to make himself her consort, as Diodorus Siculus noted, and was punished, in part for transgressing the hunter's "ritually enforced deference to Artemis" (Lacy 1990:42). Names of dogs. Notes: The "bed of Actaeon". In the second century AD, the traveller Pausanias was shown a spring on the road in Attica leading to Plataea from Eleutherae, just beyond Megara "and a little farther on a rock. It is called the bed of Actaeon, for it is said that he slept thereon when weary with hunting and that into this spring he looked while Artemis was bathing in it."
"As to Actæon there is a tradition at Orchomenus, that a spectre which sat on a stone injured their land. And when they consulted the oracle at Delphi, the god bade them bury in the ground whatever remains they could find of Actæon: he also bade them to make a brazen copy of the spectre and fasten it with iron to the stone. This I have myself seen, and they annually offer funeral rites to Actæon." Parallels in Akkadian and Ugarit poems. In the standard version of the "Epic of Gilgamesh" (tablet vi) there is a parallel, in the series of examples Gilgamesh gives Ishtar of her mistreatment of her serial lovers: You loved the herdsman, shepherd and chief shepherd Who was always heaping up the glowing ashes for you, And cooked ewe-lambs for you every day. But you hit him and turned him into a wolf, His own herd-boys hunt him down And his dogs tear at his haunches. Actaeon, torn apart by dogs incited by Artemis, finds another Near Eastern parallel in the Ugaritic hero Aqht, torn apart by eagles incited by Anath who wanted his hunting bow.
The virginal Artemis of classical times is not directly comparable to Ishtar of the many lovers, but the mytheme of Artemis shooting Orion, was linked to her punishment of Actaeon by T.C.W. Stinton; the Greek context of the mortal's reproach to the amorous goddess is translated to the episode of Anchises and Aphrodite. Daphnis too was a herdsman loved by a goddess and punished by her: see Theocritus' First Idyll. Symbolism regarding Actaeon. In Greek Mythology, Actaeon is widely thought to symbolize ritual human sacrifice in attempt to please a God or Goddess: the dogs symbolize the sacrificers and Actaeon symbolizes the sacrifice. Actaeon may symbolize human curiosity or irreverence. The myth is seen by Jungian psychologist Wolfgang Giegerich as a symbol of spiritual transformation and/or enlightenment. Actaeon often symbolizes a cuckold, as when he is turned into a stag, he becomes "horned". This is alluded to in Shakespeare's "Merry Wives", Robert Burton's "Anatomy of Melancholy", and others. Cultural depictions.
The two main scenes are Actaeon surprising Artemis/Diana, and his death. In classical art Actaeon is normally shown as fully human, even as his hounds are killing him (sometimes he has small horns), but in Renaissance art he is often given a deer's head with antlers even in the scene with Diana, and by the time he is killed he has at the least this head, and has often completely transformed into the shape of a deer.
Anglicanism Anglicanism, also known as Episcopalianism, is a Western Christian tradition which developed from the practices, liturgy, and identity of the Church of England following the English Reformation, in the context of the Protestant Reformation in Europe. It is one of the largest branches of Christianity, with around 110 million adherents worldwide , most of whom are members of national or regional ecclesiastical provinces of the international Anglican Communion, which forms one of the largest Christian communions in the world, and the world's largest Protestant communion. When united churches with Anglican heritage (such as the Church of South India) were not counted, there were an estimated 97.4 million Anglicans worldwide in 2020. Adherents of Anglicanism are called "Anglicans"; they are also called "Episcopalians" in some countries. The provinces within the Anglican Communion are in full communion with the See of Canterbury and thus with the archbishop of Canterbury, whom the communion refers to as its (Latin, 'first among equals'). The archbishop calls the decennial Lambeth Conference, chairs the meeting of primates, and is the president of the Anglican Consultative Council. Some churches that are not part of the Anglican Communion or recognised by it also call themselves Anglican, including those that are within the Continuing Anglican movement and Anglican realignment.
Anglicans base their Christian faith on the Bible, traditions of the apostolic church, apostolic succession ("historic episcopate"), and the writings of the Church Fathers, as well as historically, the "Thirty-nine Articles of Religion" and "The Books of Homilies". Anglicanism forms a branch of Western Christianity, having definitively declared its independence from the Holy See at the time of the Elizabethan Religious Settlement. Many of the Anglican formularies of the mid-16th century correspond closely to those of historical Protestantism. These reforms were understood by one of those most responsible for them, Thomas Cranmer, the archbishop of Canterbury, and others as navigating a middle way between Catholicism and two of the emerging Protestant traditions, namely Lutheranism and Calvinism. In the first half of the 17th century, the Church of England and the associated Church of Ireland were presented by some Anglican divines as comprising a distinct Christian tradition, with theologies, structures, and forms of worship representing a different kind of middle way, or "via media", originally between Lutheranism and Calvinism, and later between Protestantism and Catholicism – a perspective that came to be highly influential in later theories of Anglican identity and expressed in the description of Anglicanism as "catholic and reformed". The degree of distinction between Protestant and Catholic tendencies within Anglicanism is routinely a matter of debate both within specific Anglican churches and the Anglican Communion. The "Book of Common Prayer" is unique to Anglicanism, the collection of services in one prayer book used for centuries. The book is acknowledged as a principal tie that binds the Anglican Communion as a liturgical tradition.
After the American Revolution, Anglican congregations in the United States and British North America (which would later form the basis for the modern country of Canada) were each reconstituted into autonomous churches with their own bishops and self-governing structures; these were known as the American Episcopal Church and the Church of England in the Dominion of Canada. Through the expansion of the British Empire and the activity of Christian missions, this model was adopted as the model for many newly formed churches, especially in Africa, Australasia, and the Asia-Pacific. In the 19th century, the term "Anglicanism" was coined to describe the common religious tradition of these churches and also that of the Scottish Episcopal Church, which, though originating earlier within the Church of Scotland, had come to be recognised as sharing this common identity. Terminology. The word "Anglican" originates in , a phrase from Magna Carta dated 15 June 1215, meaning 'the English Church shall be free'. Adherents of Anglicanism are called "Anglicans". As an adjective, "Anglican" is used to describe the people, institutions, churches, liturgical traditions, and theological concepts developed by the Church of England.
As a noun, an Anglican is a church member in the Anglican Communion. The word is also used by followers of separated groups that have left the communion or have been founded separately from it. The word originally referred only to the teachings and rites of Christians throughout the world in communion with the see of Canterbury but has come to sometimes be extended to any church following those traditions or rites rather than actual membership in the Anglican Communion. Although the term "Anglican" is found referring to the Church of England as far back as the 16th century, its use did not become general until the latter half of the 19th century. In British parliamentary legislation referring to the English Established Church, there is no need for a description; it is simply the Church of England, though the word "Protestant" is used in many legal acts specifying the succession to the Crown and qualifications for office. When the Union with Ireland Act created the United Church of England and Ireland, it is specified that it shall be one "Protestant Episcopal Church", thereby distinguishing its form of church government from the Presbyterian polity that prevails in the Church of Scotland.
The word "Episcopal" ("of or pertaining to bishops") is preferred in the title of the Episcopal Church (the province of the Anglican Communion covering the United States) and the Scottish Episcopal Church, though the full name of the former is "The Protestant Episcopal Church in the United States of America". Elsewhere, however, the term "Anglican Church" came to be preferred as it distinguished these churches from others that maintain an episcopal polity. Definition. In its structures, theology, and forms of worship, Anglicanism emerged as a distinct Christian tradition representing a middle ground between Lutheran and Reformed varieties of Protestantism; after the Oxford Movement, Anglicanism has often been characterized as representing a "via media" ('middle way') between Protestantism as a whole, and Catholicism. The faith of Anglicans is founded in the Scriptures and the Gospels, the traditions of the Apostolic Church, the historical episcopate, the first four ecumenical councils, and the early Church Fathers, especially those active during the five initial centuries of Christianity, according to the "quinquasaecularist" principle proposed by the English bishop Lancelot Andrewes and the Lutheran dissident Georg Calixtus.
Anglicans understand the Old and New Testaments as "containing all things necessary for salvation" and as being the rule and ultimate standard of faith. Reason and tradition are seen as valuable means to interpret scripture (a position first formulated in detail by Richard Hooker), but there is no full mutual agreement among Anglicans about "exactly how" scripture, reason, and tradition interact (or ought to interact) with each other. Anglicans understand the Apostles' Creed as the baptismal symbol and the Nicene Creed as the sufficient statement of the Christian faith. Anglicans believe the catholic and apostolic faith is revealed in Holy Scripture and the ecumenical creeds (Apostles', Nicene and Athanasian) and interpret these in light of the Christian tradition of the historic church, scholarship, reason, and experience. Anglicans celebrate the traditional sacraments, with special emphasis being given to the Eucharist, also called Holy Communion, the Lord's Supper, or the Mass. The Eucharist is central to worship for most Anglicans as a communal offering of prayer and praise in which the life, death, and resurrection of Jesus Christ are proclaimed through prayer, reading of the Bible, singing, giving God thanks over the bread and wine for the innumerable benefits obtained through the passion of Christ; the breaking of the bread, the blessing of the cup, and the partaking of the body and blood of Christ as instituted at the Last Supper. The consecrated bread and wine, which are considered by Anglican formularies to be the true body and blood of Christ in a spiritual manner and as outward symbols of an inner grace given by Christ which to the repentant convey forgiveness and cleansing from sin. While many Anglicans celebrate the Eucharist in similar ways to the predominant Latin Catholic tradition, a considerable degree of liturgical freedom is permitted, and worship styles range from simple to elaborate.

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