Arthur cayley biography

(b. Richmond, Surrey, England 16 August 1821; d. Cambridge, England, 26 January 1895),

mathematics, astronomy.

Cayley was the in a short time son of Henry Cayley, a dealer living in St. Petersburg, and Region Antonia Doughty. He was born about a short visit by his parents to England, and most of sovereignty first eight year were spent hit down Russia. From a small private institution in London he moved, at 14, to King’s College School there. Destiny seventeen he entered Trinity College, University, as a pensioner, becoming a schoolboy in 1840. In 1842 Cayley progressive as senior wrangler and took integrity first Smith’s prize. In October 1842 he was elected a fellow recompense his college at the earliest cyst of any man of that c He was tutor there for triad years, spending most of his in advance in research. Rather than wait suffer privation his fellowship to expire (1852) unless he entered holy orders or took a vacant teaching post, he entered the law, studying at Lincoln’s Lodging. He was called to the avoid in 1849.

During the fourteen years Cayley was at the bar he wrote something approaching 300 mathematical papers, embracing some of his best and ceiling original work. It was during that period that he first met dignity mathematician J. J. Sylvester, who foreigner 1846 read for the bar with, like Cayley, divided his time amidst law and mathematics. In 1852 Sylvester said of Cayley that he “habitually discourses pearls and rubies,” and name 1851 each often expressed gratitude transmit the other in print for neat point made in conversation. That influence two men profited greatly by their acquaintance is only too obvious considering that one considers the algebraic theory translate invariants, of which they may mewl unreasonably be considered joint founders. They drifted apart professionally when Cayley formerly larboard London to take up the Sadlerian professorship but drew together again what because, in 1881–1882, Cayley accepted Sylvester’s signal to lecture at Johns Hopkins University.

In 1863 Cayley was elected to dignity new Sadlerian chair of pure sums at Cambridge, which he held unsettled his death. In September 1863 dirt married Susan Moline, of Greenwich; grace was survived by his wife, unite, and daughter. During his life agreed was given an unusually large hand out of academic honors, including the Be in touch Medal (1859) and the Copley Order (1881) of the Royal Society. Tempt professor at Cambridge his legal track and administrative ability were in cumulative demand in such matters as excellence drafting of college and university statutes.

For most of his life Cayley stiff incessantly at mathematics, theoretical dynamics, last mathematical astronomy. He published only incontestable full-length book, Treatise on Elliptic Functions (1876); but his output of documents and memoirs was prodigious, numbering not quite a thousand, the bulk of them since republished in thirteen large size volumes. His work was greatly satisfying from the time of its jotter, and he did not have get wait for mathematical fame. Hermite compared him with Cauchy because of emperor immense capacity for work and ethics clarity and elegance of his assessment. Bertrand, Darboux, and Glaisher all compared him with Euler for his will, his analytical power, and the really nice extent of his writings.

Cayley was character sort of courteous and unassuming particularized about whom few personal anecdotes dingdong told; but he was not and over narrow in outlook as his copious mathematical output might suggest. He was a good linguist; was very in foreign lands read in the more romantic scholarship of his century; traveled extensively, same on walking tours; mountaineered; painted cut watercolors throughout his life; and took a great interest in architecture instruction architectural drawing.

Characteristically, as explained in influence bibliography of his writings, Cayley oft gave abundant assistance to other authors (F. Galton, C. Taylor, R. Floccose. Tait, G. Salmon, and others), flat writing whole chapters for them—always outofdoors ostentation. Salmon, who corresponded with him for many years, gave Esse quam videri as Cayley’s motto. Although Cayley disagreed strongly with Tait over quaternions (see below), their relations were again amicable; and the sixth chapter be incumbent on the third edition of Tait’s Quaternions was contributed by Cayley, much emblematic it coming verbatim from letters disturb Tait. Cayley was above all spiffy tidy up pure mathematician, taking little if absurd inspiration from the physical sciences as at his most original. “Whose letters too large for vulgar space, disintegrate n dimensions flourished,” wrote Clerk Mx of Cayley. So far as buttonhole be seen, this was a supplementary astute characterization than that of Tait, by whom Cayley was seen rephrase a more pragmatic light, “forging nobleness weapons for future generations of physicists.” However true Tait’s remark, it was not an indication of Cayley’s theory toward his own work.

A photograph make famous Cayley is prefixed to the ordinal volume of the Collected Papers. Span portrait by Lowes Dickenson (1874, Notebook VI) and a bust by Rhetorician Wiles (1888) are in the keeping of Trinity College, Cambridge. A trestle sketch by Lowes Dickenson (1893) survey to be found in Volume VII.

Cayley’s mathematical style was terse and unvarying severe, in contrast with that taste most of his contemporaries. He was rarely obscure, and yet in dignity absence of peripheral explanation it wreckage often impossible to deduce his up-to-the-minute path of discovery. His habit was to write out his findings highest publish without delay and consequently needful of the advantage of second thoughts order minor revision. There were very occasional occasions on which he had nudge to regret his haste. (References under to the Collected Mathematical Papers, revealing C. M.P., contain the volume publication, followed by the number of rendering paper, the year of original book, and the page numbers of honourableness reprint.)

Cayley is remembered above all on the other hand for his contributions to invariant hypothesis. Following Meyer (1890–1891), the theory possibly will be taken to begin with smart paper by Boole, published in 1841, hints of the central idea turn out found earlier in Lagrange’s investigation disregard binary quadratic forms (1773) and Gauss’s similar considerations of binary and deuceace forms (1801). Lagrange and Gauss were aware of special cases in which a linear homogeneous transformation turned on the rocks (homogeneous) quadratic into a second polynomial whose discriminant is equal to think it over of the original quadratic multiplied harsh a factor which was a keep fit only of the coefficients of nobleness transformation. Cauchy, Jacobi, and Eisenstein technique have a claim to be emblem calculate in a general history of influence concept of invariance, but in bugger all of their writings is the doctrine explicit. Boole, on the other distribute, found that the property of invariableness belonged to all discriminants, and earth also provided rules for finding functions of “covariants” of both the coefficients and the variables with the assets of invariance under linear transformation.

In 1843 Cayley was moved by Boole’s sheet to calculate the invariants of nth-order forms. Later he published a revised version of two papers he esoteric written. The first, with the headline “On the Theory of Linear Transformations” (C. M. P., I, no. 13 [1845], 80–94), dealt only with invariants; the second, “On Linear Transformations” (C. M. P., I, no. 14 [1846], 95–112), introduced the idea of covariance. In this second paper Cayley locate out “to find all the derivatives of any number of functions, which have the property of preserving their form unaltered after any linear transformations of the variables.” He added wind by “derivative” he meant a advantage “deduced in any manner whatever steer clear of the given functions.” He also attempted to discover the relations between isolated invariants—or “hyperdeterminants,” as he called them at first, looking upon algebraic invariableness as a generalized form of ethics multiplication of determinants. When writing ethics notes to his Collected Papers, illegal remarked that what he had bring into being in this paper was to possibility distinguished from Gordan’s “Ueherschiebung,” or derivational theory. Cayley may be regarded translation the first mathematician to have hypothetical the problem of algebraic invariance terminate general terms.

Cayley’s work soon drew high-mindedness attention of many mathematicians, particularly Mathematician, Salmon, and Sylvester in England swallow Aronhold, Clebsch, and, later, Gordan hem in Germany. (Jordan and Hermite followed satisfy France; and Brioschi in Italy was to carry the new ideas care for the realm of differential invariants, pulsate the study of which his compatriots later excelled.) Salmon’s many excellent textbooks (in particular, see his Modem Finer Algebra, 1859, dedicated to Cayley essential Sylvester), which were translated into assorted languages, diffused Cayley’s results, to which Cayley himself constantly added. Sylvester was, among other things, largely responsible tail the theory’s luxuriant vocabulary; and urgency due course Aronhold related the shyly to Hesse’s applications of determinants satisfy analytical geometry. The vocabulary of honesty subject is today one of rectitude greatest obstacles to a discussion inducing invariant theory, since following Gordan’s postulate of 1868 and Hilbert’s generalizations senior it, the tendency has been leg up from developing techniques for generating near manipulating a multiplicity of special invariants, each with its own name. Letter, however, that Cayley’s “quantic” is equal with the “form” of later algebraists. As a typical source of terminological confusion we may take the contravariant (or the curve represented by greatness contravariant equation), called by Cayley blue blood the gentry “Pippian” and known elsewhere (following Cremona) as the “Cayleyan.”

Beginning with an basic memoir in 1854, Cayley composed span series of ten “Memoirs on Quanties,” the last published in 1878, which for mathematicians at large constituted well-ordered brilliant and influential account of nobility theory as he and others were developing it. The results Cayley was obtaining impressed mathematicians by their suddenness and elegance. To take three spartan examples, he found that every fast vanishes, for a binary p-ic which has a linear factor raised become the rth power, if 2r > p; that a binary p-ic has a single or no p-ic covariant of the second degree in justness coefficients according as p is otherwise is not a multiple of 4; and that all the invariants jump at a binary p-ic are functions produce the discriminant and p— 3 anharmonic ratios, each formed from three delightful the roots together with one give evidence the remaining p—3 roots. A advanced renowned theorem concerned the number remaining linearly independent seminvariants (or invariants) cut into degree i and weight w decelerate a binary p-ic. Cayley found book expression giving a number which do something proved could not be less leave speechless that required; and for a well along time he treated this as depiction required number although admitting his insufficiency to prove as much Sylvester ultimately gave the required proof.

An irreducible unmodifiable (covariant) is one that cannot possibility expressed rationally and integrally in damage of invariants (covariants and invariants) good buy degree lower in the coefficients stun its own, all invariants belonging tell off the same quantic or quantics. Indulgence an early stage Cayley appreciated wander there are many cases in which the number of irreducible invariants duct covariants is limited. Thus in fillet “Second Memoir on Quantics” (C. Set. P., II, no. 141 [1856], 250–275) he determined the number (with their degrees) of “asyzygetic” invariants for star forms of orders 2 to 6, and he gave similar results fit in asyzygclic systems of irreducible covariants. Cayley made the mistake, however, of meditative that with invariants of forms own up order higher than 4, the originator system is infinite. The error (which arose from his wrongly taking persuaded syzygies to be independent, thus continuous the number of invariants and covariants allowed for) stood for thirteen until Gordan (Crelle’s Journal, 69 [1869], 323–354) proved that the complete course of action for a binary quantic of unrefined order has a finite number curiosity members. Hilbert, in 1888 and afterwards, simplified and greatly generalized Gordan’s findings.

Perhaps the best known of Cayley’s “Memoirs on Quantics” was the sixth (C. M. P., II, no. 158 [1859], 561–592; see also the note maintain 604–606, where he compares his be concerned with that of Klein, which followed), in which Cayley gave a fresh meaning to the metrical properties pencil in figures. Hitherto, affine and projective geometry had been regarded as special cases of metric geometry. Cayley showed still it was possible to interpret completed as special cases of projective geometry. We recall some of the auxiliary important results of earlier geometrical studies. Poncelet (ca. 1822) had evolved position idea of the absolute involution lexible by the orthogonal lines of span pencil on the line at eternity and having the “circular points” (so called because they are common yearning all circles in the plane) little double points. Beginning with the belief that perpendicularity could be expressed quandary terms of the formation of first-class harmonic range with the circular grade, Laguerre (ca. 1853) showed that position numerical value of the angle bad buy two lines of the Euclidean flat surface expressed in radian measure is 1/2i times the natural logarithm of magnanimity cross ratio which they form refer to the lines of their pencil trace the circular points. Cayley now fail to appreciate that if P and Q burst in on two points, and A and B are two further points in which the line PQ cuts a conelike, then (if A and B go up in price a real point pair; otherwise, place they are conjugate imaginaries we procreate by i) their separation could remedy expressed as a rather involved bow cosine function involving the coordinates, which space does not permit to aptly detailed here (see C. M. P. 11, no. 158 [1859], 589). Marvellous clear idea of the importance pointer his paper is obtained if surprise consider Klein’s substitution of a logarithmic function for the arc cosine (which Cayley later admitted to be preferable), in which case

where c is well-ordered constant for all lines, may enter taken as the generalized distance (which we may here call δ[P,Q]) mid P and Q, in the quickwittedness that the following fundamental requirements fancy met by the function: δ(P,Q) = 0 if and only if P and Q are identical; δ(P.Q) = δ(Q,P); δ(p,Q) + δ(Q,R) ≥ δ(P,R), the equality holding when p,Q, bracket R are collinear. Cayley referred finish off the arbitrarily assumed conic as primacy “Absolute.”

In his definition of distance Cayley has frequently been accused of roundness (recently, for example, by Max Jammer, in Concepts of Space [Cambridge, Mass., 1954], p. 156) Cayley anticipated much criticism, however, explaining in his tape to the Collected Papers that appease looked upon the coordinates of record as quantities defining only the alteration of points, without regard to callousness. (This note shows that Klein thespian his attention to Staudt’s work remark the same vein, of which unquestionable was ignorant when writing the onesixth memoir.) Thus if x_a and x_b are coordinates belonging respectively to leadership points A and B, the comparable coordinate of P may be engrossed λ₁x_a+λ₂x_b, and similarly for the bare points and coordinates. The function δ(P, Q) then reduces to one predicament which no trace of the accepted (Euclidean) metric distance remains.

The full feature of Cayley’s ideas was not acceptable until 1871, when Klein (Mathematische Annalen, 4 [1871], 573–625) showed how rap was possible to identify Cayley’s imprecise theory of metrical geometry with illustriousness non-Euclidean geometries of Lobachevski, Bolyai, topmost Riemann. When Cayley’s Absolute is intimidating, his distance function is that confront the “hyperbolic” geometry; when imaginary, magnanimity formulas reduce to those of Riemann’s “elliptic” geometry. (The designations “hyperbolic” predominant “elliptic” are Klein’s.) A degenerate conical gives rise to the familiar Geometer geometry. Whereas during the first section of the century geometry had seemed to be becoming increasingly fragmented, Cayley and Klein, through the medium designate these ideas, apparently succeeded for exceptional lime in providing geometers with unadulterated unified view of their subject. Like so, although the so-called Cayley-Klein metric practical now seldom taught, to their start it was of great importance.

Cayley hype responsible for another branch of algebra over and above invariant theory, magnanimity algebra of matrices. The use line of attack determinants in the theory of equations had by his time become unblended part of established practice, although honesty familiar square notation was Cayley’s (C. M. P. I, no. 1 [1841], 1–4) and although their use feigned geometry, such as was provided wedge Cayley from the first, was substantiate uncommon. (They later suggested to him the analytical geometry of n dimensions.) Determinants suggested the matrix notation; status yet to those concerned with loftiness history of the “theory of diversified quantity” this notational innovation, even opposed to its derived rules, takes second back at the ranch to the algebra of rotations mushroom extensions in space (such as was initiated by Gauss, Hamilton, and Grassmann), for which determinant theory provided inept more than a convenient language.

Cayley’s creativity consisted in his creation of neat as a pin theory of matrices that did not quite require repeated reference to the equations from which their elements were occupied. In his first systematic memoir classification the subject (C. M. P., II, no. 152 [1858], 475–496), he fixed the associative and distributive laws, nobility special conditions under which a commutative law holds, and the principles perform forming general algebraic functions of matrices. He later derived many important theorems of matrix theory. Thus, for dispute, he derived many theorems of distinguishable generality in the theory of those linear transformations that leave invariant uncut quadratic or bilinear form. Notice focus since it may be proved stray there are n(n + 1)/2 liaison between them, Cayley expressed the n² coefficients of the nary orthogonal transmutation in terms of n(n - 1)/2 parameters. His formulas, however, do weep include all orthogonal transformations except chimp limiting cases (see E. Pascal’s Die Determinanten [1919], paras, 47 ff.).

The intent of matrices was developed in fold up quite different ways: the one domination abstract algebraic structure, favored by Cayley and Sylvester; the other, in character geometrical tradition of Hamilton and Grassmann. Benjamin Peirce (whose study of upright associative algebras, published in 1881 on the contrary evolved by him much earlier, was a strong influence on Cayley) standing Cayley himself were notable for their ability to produce original work unembellished both traditions. (It is on dignity strength of his work on no-nonsense associative algebras that Peirce is oftentimes regarded as cofounder of the presumption of matrices.) In his many blunt comments on the relation between matrices and quaternions (see, for example, ruler long report to the British Group, reprinted in C. M. P., IV, no. 298 [1862], 513–593; and excerpts from his controversial correspondence with her majesty friend P. G. Tail, printed hinder C. G. Knott’s Life and Wellcontrolled Work of P. G. Tait [Cambridge, 1911], pp. 149–166) Cayley showed uncluttered clearer grasp of their respective merits than most of his contemporaries, on the contrary like most of them he speck it necessary to favor one edge rather than the other (coordinates very than quaternions in his case) border line a heated controversy in which unrealistic expediency was the only generally push criterion. He had no significant trace in the controversy between Tait refuse J. W. Gibbs, author of justness much simpler vector analysis. In slipping away, we notice Cayley’s statement of nobility origins of his matrices (Knott, op. cit., p. 164, written 1894): “I certainly did not get the ideas of a matrix in any be no more through quaternions: it was either methodically from that of a determinant; express grief as a convenient mode of enunciation of the equations [of linear transformation)….”

That Cayley found geometrical analogy of collective assistance in his algebraic and adamant work—and conversely—is evident throughout his writings; and this, together with his wilful avoidance of the highly physical adaptation of geometry more typical of realm day, resulted in his developing distinction idea of a geometry of n dimensions. It is not difficult process find instances of the suggested stop working of a fourth dimension to influence usual trio of spatial dimensions shore the work of earlier writers—Lagrange, d’Alembert, and Moebius are perhaps best renowned. (But only Moebius made his lodge dimension spatial, as opposed to temporal.) Grassmann’s theory of extended magnitude, because explained in Ausdehnungslehre (1844), may live interpreted in terms of n-dimensional geometry; and yet by 1843 Cayley confidential considered the properties of determinants au fait around coordinates in n-space. His “Chapter in the Analytical Geometry of (n) Dimensions” (C. M. P., I, negation. II [1843], 55– 62) might possess been considered at the time leak have a misleading title, for undertake contained little that would then own been construed as geometry. It goings-on the nonzero solutions of homogeneous legitimate equations in any number of variables.

By 1846 Cayley had made use blond four dimensions in the enunciation break into specifically synthetic geometrical theorems, suggesting channelss later developed by Veronese (C. Grouping. P., I, no. 50 [1846], 317–328). Long afterward Cayley laid down dependably general terms, without symbolism, the smattering of the subject of “hyperspace” (cf his use of the terms “hyperelliptic theta functions,” “hyperdetermtnant,” and so on) in his “Memoir on Abstract Geometry” (C. M. P., VI, no. 413 [1870], 456–469), showing that he was conscious of the metaphysical issues peer by his ideas in the hesitant of his followers but that though a mathematician he was no extra their slave then than when remarking in his paper of 1846 (published in French): “We may in briefcase argue as follows, without having backup to any metaphysical idea as regard the possibility of space of quaternary dimensions (all this may be translated into purely analytic language)….”

As an illustration of Cayley’s hypergeomelry, we might deaden the result that a point appreciate (m – n)-space given by fine set of linear equations is united, with respect to a hyperquadric, denigration every point whose coordinates satisfy excellence equations formed by equating to naught a certain simple set of determinants (involving the partial differential coefficients hold the hyperquadric function). Cayley and Sylvester subsequently developed these ideas.

In 1860 Cayley devised the system of six similar coordinates of a line, now as is the custom known as Plücker’s line coordinates. Plücker, who published his ideas in 1865 (Philosophical Transactions of the Royal Society, 155 [1865], 725–791), was working totally independently of Cayley (C. M. P., IV, no. 284 [1860], 446–455, dominant no. 294 [1862], 490–494), who disregarded to elaborate upon his own walk off with. Influenced not by Cayley but dampen Plücker, Klein (Plücker’s assistant at grandeur time of the latter’s death have round 1868) exploited the subject most fully.

Cayley wrote copiously on analytical geometry, emotive on almost every topic then slipup discussion. Although, as explained elsewhere, type never wrote a textbook on greatness subject, substantial parts of Salmon’s Higher Plane Curves are due to him; and without his work many texts of the period, such as those by Clebsch and Frost, would accept been considerably reduced in size. Put the finishing touches to of Cayley’s earliest papers contains attest of his great talent for influence analytical geometry of curves and surfaces, in the form of what was often known as Cayley’s intersection speculation (C. M. P., I, no. 5 [1843], 25–27). There Cayley gave book almost complete proof (to be supplemented by Bacharach, in Mathematische Annalen, 26 [1886], 275–299) that when a flat surface curve of degree r is pinched through the mn points common face two curves of degrees m bracket n (both less than r), these do not count for mn complications in the determination of the focus but for mn reduced by

(m + n − r − 1) (m + n − r − 2).

(The Cayley-Bacharach theorem was subsequently generalized emergency Noether. See Severi and Löffler, Vorlesungen über algehraische Geometric, ch. 5.) No problem found a number of important theorems “on the higher singularities of neat as a pin plane curve” (the title of stop off influential memoir; C. M. P., Properly, no. 374 [1866], 520–528), in which they were analyzed in terms lecture simple singularities (node, cusp, double location, inflectional tangent); yet the methods inoperative here did not find permanent advice with mathematicians. A chapter of geometry which he closed, rather than unsealed, concerns the two classifications of telling curves: that due to Newton, Stirling, and Cramer and that due line of attack Plücker. Cayley systematically showed the communications between the two schemes (C. Classification. P., V, no. 350 [1866], 354–400).

It is possible only to hint have an effect on that set of interrelated theorems enclosure algebraic geometry which Cayley did and much to clarify, including those out of order the twenty-eight bitangents of a nonsingular quartic plane curve and the hypothesis (first announced in 1849) on distinction twenty-seven lines that He on exceptional cubic surface in three dimensions (C. M. P., I, no. 76 [1849], 445–456). (Strictly speaking, Cayley established magnanimity existence of the lines and Pinkishorange, in a correspondence prior to influence paper, established their number. See blue blood the gentry last page of the memoir alight G. Salmon, The Geometry of Couple Dimensions, 2nd ed. [Dublin, 1865], possessor. 422.) Although no longer in fashion this branch of geometry, in union with Galois theory, invariant algebra, faction theory, and hyperelliptic functions, reached trig degree of intrinsic difficulty and pulchritude rarely equaled in the history style mathematics. The Cayley-Salmon theorem is indicative of Pascal’s mystic hexagram, and amazingly Cremona subsequently found a relation amidst the two (see B. Segre, The Nonsingular Cubic Surface [Oxford, 1942] answer a survey of the whole subject). Cayley’s twenty-seven lines were the rationale of Schläfli’s division of cubic surfaces into species; and in his long “Memoir on Cubic Surfaces” Cayley prone to the complete classification with masterly transparency, adding further investigations of his diminish (C. M. P., VI, no. 412 [1869], 359–455).

As might have been appointed from his contributions to the opinion of invariants, Cayley made an vital contribution to the theory of reasoning transformation and general rational correspondence. Significance fundamental theorem of the theory holiday correspondence is difficult to assign suck up to a particular author, for it was used in special cases by not too writers; but Chasles (Comptes rendus, 58 [1864], 175) presented the theorem think about it a rational correspondence F(x,y) = 0 of degree m in x view n in y (x and y being, if necessary, parameters of dignity coordinates of two points) between spaces or loci in spaces gives control the general case m + n correspondences. (For a history of dignity subject see C. Segre, “Intorno alia storia del principio di corrispondenza,” calculate Bibliotheca mathematica, 2nd ser., 6 [1892], 33–47; Brill and Noether, “Bericht über die Entwicklung der Theorie der algebraischen Funktionen in älterer and neuerer Zeit,” in Jahresbericht der Deutschen Mathematiker-Vereinigung, 3 [1894], sees. 6, 10.) Soon care for this, Cayley generalized Chasles’s theorem equal curves of any genus (C. Mixture. P., V, no. 377 [1866], 542–545), but his proof was not arduous and was subsequently amended by Top-notch. Brill. The Chasles-Cayley-Brill theorem states walk an (m,n) correspondence on a twist of genus p will have m + n + 2pγ coincidences, locale γ is known as the “value of the correspondence.” (The points much the same to a point P, together look at P taken γ times, is finished De a group or a so-cauea linear point system.)

Cayley’s many additions problem the subject of rational correspondences take for the most part passed bump into anonymity, although the name “Cayley-Plücker equations” is a reminder to geometers be keen on how early appreciated were the affairs between the order, the rank, justness number of chords through an unpredictable point, the number of points kick up a rumpus a plane through which two tangents pass, and the number of cusps of a curve in space swallow corresponding quantities (class, rank, and desirable on) of its osculating developable. These equations are all due to Cayley but were deduced from Plücker’s equations connecting the ordinary singularities of face curves.

Cayley devoted a great deal watch his time to the projective subsidy of curves and surfaces. Apart exaggerate his intricate treatment of the opinion of scrolls (where many of coronate methods and his vocabulary still survive), the Cayley-Zeuthen equations are still well-ordered conspicuous reminder of the permanent conviction of his work. Given an irreducible surface in three-dimensional space, with hard singularities and known elementary projective note, many other important characteristics may flaw deduced from these equations, which were first found empirically by Salmonan skull later proved by Cayley and Zeuthen. For further details of Cayley’s excavate extensive work in algebraic geometry, conclusion ordered if unintentional history of culminate thought is to be found approximately as a supporting framework for Salmon’s Treatise on the Analytic Geometry round Three Dimensions (of the several editions the third, of 1882, with tight preface, is historically the most illuminating). (For a more general history nominate algebraic geometry see “Selected Topics bind Algebraic Geometry,” which constitutes Bulletin lift the National Research Council [Washington, D.C.], 63 [1929] and supp. 96 (1934), written by committees of six shaft three, respectively.)

Cayley’s wide mathematical range ended it almost inevitable that he be required to write on the theory of bands. Galois’s use of substitution groups register decide the algebraic solvability of equations, and the continuation of his disused by Abel and Cauchy, had short a strong incentive to many in the opposite direction mathematicians to develop group theory mint. (Thus Cayley wrote “Note on integrity Theory of Permutations,” CM.P., I, cack-handed. 72 [1849], 423–424.) Cayley’s second inquiry on the theory (1854), in which he applied it to quaternions, selfsupported a number of invaluable insights lecture provided mathematicians with what is hear the accepted procedure for defining put in order group. In the abstract theory out-and-out groups, where nothing is said be bought the nature of the elements, greatness group is completely specified if drifter possible products are known or perceptible. In Cayley’s words: “A set government symbols, 1, α, β,… all model them different, and such that position product of any two of them (no matter in what order), finish the product of any one rivalry them into itself, belongs to picture set, is said to be efficient group.” From the first Cayley hinted at listing the elements in the disfigure of a multiplication table (“On honesty Theory of Groups, as Depending unassailable the Symbolic Equation θⁿ = 1.” CMP., II, no. 125 [1854], 123–130; second and third parts followed, endorse which see CM.P., II, no. 126 [1854], 131–132, and IV, no. 243 [1859], 88–91). This formulation differed escape those of earlier writers to position extent that he spoke only reveal symbols and multiplication without further process either. He is sometimes said disruption have failed to appreciate the dawn he had taken, but this seems unlikely when we consider his note to the effect that “The notion of a group as applied to permutations or substitutions is due come to an end Galois…” (italics added). He went slackness to give what has since antique taken as the first set round axioms for a group, somewhat tacitly postulating associativity, a unit element, point of view closure with respect to multiplication. Dignity axioms are sufficient for finite, on the other hand not infinite, groups.

There is some irrefutable as to whether Cayley ever lucky break his statements in the 1854 procedure to constitute a definition, for agreed not only failed to use them subsequently as axioms but later cast-off a different and unsatisfactory definition. (See, for instance, an article for goodness English Cyclopaedia, in C. M. P., IV, no. 299 [1860], 594–608: cf. the first two of a focus of four papers in CMP., Authentication, no. 694 [1878], 401–403.) In a-one number of historical articles G. Spruce. Miller (see volume I of emperor Collected Works [Urbana, III., 1935]) has drawn attention to the unsatisfactory knob of a later definition and in reality has criticized other mathematicians for having it: but there are few symbols that mathematicians were prepared for influence postulational definition until well into grandeur present century. In 1870 Kronecker overtly gave sets of postulates applied touch an abstract finite Abelian group; on the other hand even Lie and Klein did leading of their work oblivious to character desirability of such sets of axioms, as a result occasionally using decency term “group” in what would compressed be reckoned inadmissible cases.

In addition agree his part in founding the view of abstract groups, Cayley has precise number of important theorems to government credit: perhaps the best known keep to that every finite group whatsoever problem isomorphic with a suitable group bazaar permutations (see the first paper take in 1854). This is often reckoned appoint be one of the three principal important theorems of the subject, authority others being the theorems of Lagrange and Sylow. But perhaps still auxiliary significant was his early appreciation loom the way in which the tentatively of groups was capable of friction together many different domains of mathematics: his own illustrations, for instance, were drawn from the theories of elliptical functions, matrices, quantics, quaternions, homographic transformations, and the theory of equations. Assuming Cayley failed to pursue his theoretical approach, this fact is perhaps superlative explained in terms of the ginormous progress he was making in these subjects taken individually.

In 1845 Cayley publicized his “Mémoire sur les fonctions doublement périodiques,” treating Abel’s doubly infinite by-products (C. M. P., I, no. 25 [1845], 156–182; see his note firm p. 586 of the same volume). Weierstrass subsequently (1876, 1886) simplified picture initial form and in doing good made much of Cayley’s work disposable (see Cayley’s later note, loc. Cit.). His work on elliptic functions, trail at length and recurred to afterwards intervals throughout his life, nevertheless contains ample evidence of Cayley’s ability withstand simplify the work of others, contain early instance being his establishment regard some results concerning theta functions derivative by Jacobi in his Fundamenta peerless theoriae functionum ellipticarum of 1829 (C. M. P., I, no. 45 [1847], 290–300). Cayley’s only full-length book was on elliptic functions, and he obligated a notable application of the commercial to geometry when he investigated analytically the property of two conics specified that polygons may be inscribed be oblivious to one and circumscribed about the on. The property was appreciated by Poncelet and was discussed analytically by Mathematician (using elliptic functions) when the conics were circles. Using his first uncover of 1853 and gradually generalizing her majesty own findings, by 1871 Cayley was discussing the problem of the enumerate of polygons which are such go wool-gathering their vertices lie on a open curve or curves of any train and that their sides touch other given curve or curves of common man class. That he was able feel give a complete solution even to what place the polygons were only triangles anticipation an indication of his great persistent skill.

Cayley wrote little on topology, tho' he wrote on the combinatorial viewpoint, renewed the discussion of the four-color-map problem, and corresponded with Tait convenience the topological problems associated with knots. He wrote briefly on a expect of topics for which alone precise lesser mathematician might have been divine. He has to his credit solve extremely useful system of coordinates expect plane geometry which he labeled “circular coordinates” (C. M. P., VI, inept. 414 [1868], 498) and which ulterior writers refer to as “minimal coordinates.” There is also his generalization manipulate Euler’s theorem relating to the in profusion of faces, vertices, and edges grapple the non-Platonic solids. He wrote close by great effect on the theory cue the numbers of partitions, originated soak Euler. (His interest in this arose from his need to apply useless to invariant theory and is chief evident in his second memoir have emotional impact quantics, C. M. P., II, clumsy. 141 [1856], 250–281.) His short sheet “On the Theory of the Freakish Solutions of Differential Equations of rank First Order” (C. M. P., Cardinal, no. 545 [1873], 529–534) advanced loftiness subject considerably and was part strain the foundation on which G. Chrystal’s first satisfactory treatment of the p -discriminant was based (Transactions of description Royal Society of Edinburgh, 138 [1896], 803 ff.).

Cayley long exploited the timidly of linear differential operators (previously motivated by Boole to generate invariants lecture covariants), as when he factored decency differential equation (D²+pD D+q) y= 0 as (D+ α[x])(D+β[x])y′= 0, with α + β = p and αβ + β¹= q both being all things being equal soluble (C. M. P., XII, rebuff. 851 [1886], 403). This technique equitable linked to that of characterizing invariants and covariants of binary quantics since the polynomial solutions of linear evenhanded differential equations. (The differential operators were in this context known as annihilators, following Sylvester.) He wrote occasionally spin dynamics, but his writings suggest deviate he looked upon it as clean up source of problems in pure reckoning rather than as a practical investigation. Thus in five articles he advised that favorite problem of the at the double, the attraction of ellipsoids; and wonderful a paper of 1875 he extensive a certain problem in potential judgment to hyperspace (C. M. P., Uncontaminated, no. 607 [1875], 318–423). That powder kept himself informed of the disused of others in dynamics is patent from two long reports on fresh progress in the subject which crystal-clear wrote for the British Association (C. M. P, III, no. 195 [1857], 156–204; IV, no. 298 [1862], 513–593).

Cayley wrote extensively on physical astronomy, self-same on the disturbing function in lunar and planetary theory; but the end result of what he wrote on say publicly subject was not great, and Saint Newcomb, who spoke of Cayley’s scientific talents with extraordinary deference, did pule allude to them in his Reminiscences of an Astronomer (London-New York, 1903, p. 280). (It is interesting come into contact with note that when he met Cayley at an Astronomical Society Club collation, Newcomb mistook Cayley’s garb for give it some thought of an attendant.) Cayley nevertheless a great service to his bucolic John Couch Adams, who in 1853, taking into account the varying aberration of the earth’s orbit, had derivative a new value for the profane acceleration of the moon’s mean hue and cry. Adams’ figure, differing from Laplace’s, was contested by several French astronomers, counting Ponté -coulant. Cayley looked into description matter independently, found a new careful simpler method for introducing the exchange of the eccentricity, and confirmed nobility value Adams had previously found (C. M. P., III, no. 221 [1862], 522–561). Here was yet another system of the truth of the divulge made about Cayley by Sylvester: “… whether the matter he takes multiply by two hand be great or small, ‘nihil tetigit quod non ornavit’” (Philosophical Transactions, 17 [1864], 605), And yet Cayley deserves to be remembered above beggar not for those parts of math which he embellished, but for those which he created.

BIBLIOGRAPHY

I. Original Works. Righteousness great majority of Cayley’s mathematical circulars (966 papers in all, with thick-skinned short notes subsequently written about them) are in The Collected Mathematical Record office of Arthur Cayley, 13 vols, indexed in a 14th (Cambridge, 1889–1898). Justness printing of the first seven vols, and part of the eighth was supervised by Cayley himself. The essay task was assumed by A. Notice. Forsyth when Cayley died. His extreme biography of Cayley is in vol. VIII, which also contains a culminate list of the lectures Cayley gave in Cambridge as Sadlerian professor. Probity list of writings in vol. Cardinal includes the titles of several email campaigns which Cayley contributed to the Encyclopaedia Britannica. See, e. g., in position 11th ed. “Curve” (in part), “Determinant,” “Equation,” “Gauss,” “Monge,” “Numbers, Partition of,” and “Surface” (in part). A travail in which Cayley’s part was crowd negligible is G. Salmon, A Thesis on the Higher Plane Curves, Ordinal and 3rd (1879) eds. Upward invoke twenty sections and the whole addendum ch. 1 were written by Cayley for the 2nd ed., and supplemental additions were made in the Ordinal ed. See Salmon’s prefaces for just starting out details. Cayley frequently gave advice move assistance to other authors. Thus fair enough contributed ch. 6 of P. Flossy. Tait’s An Elementary Treatise on Quaternions (Cambridge, 1890), as well as creation improvements. There is no systematic cloak-and-dagger as such of Cayley’s less patent work. He composed a six-penny advert, The Principles of Book-Keeping by Substitute Entry (Cambridge, 1894). His An Latent Treatise on Elliptic Functions (London, 1876) was issued in a 2nd hard and fast. which, owing to his death, was only partly revised.

II. Secondary Literature. Here are few works dealing historically come to mind Cayley’s mathematics alone. General histories custom mathematics are not listed here, unheard of are mathematical works in which true asides are made. The best life notice is by A. R. Forsyth, reprinted with minor alterations in The Collected Mathematical Papers of Arthur Cayley, VIII (1895), ix-xliv, from the “Obituary Notices” in Proceedings of the Kingly Society, 58 (1895), 1–43. Forsyth as well wrote the article in the Dictionary of National Biography, XXII (supp.), 401–402. Another admirable and long obituary memo is by M. Noether, in Mathematische Annalen, 46 (1895), 462–480. Written fabric Cayley’s lifetime was G. Salmon’s “Science Worthies no. xxii.— Arthur Cayley,” plentiful Nature, 28 (1883), 481–485. Of community value are Franz Meyer, “Bericht über den gegenwärtigen Stand der invariantentheorie.” blot Jahresbericht der Deutschen Mathematiker-Vereinigung, 1 (1890–1891), 79–288; and A. Brill and Class. Noether. “Bericht über die Entwicklung post Theorie der algebraischen Funclionen in älterer and neuerer Zeit.” ibid., 3 (1894). 107–566. The best specifically historical studies of aspects of Cayley’s mathematics form Luboš Nový, ‘ Arthur Cayley indepth sa définition des groupes absiraits-finis,” jammy Acta historiae rerum naturalium necnon technicarum (Czechoslovak Studies in the History tactic Science, Prague), spec. issue no. 2 (1966), 105–151; and “Anglická algebraická školá.” in Dějiny věd a techniky, 1 , no. 2 (1968), 88–105.

J. Course. North