In his definition of distance Cayley has frequently been accused of circularity recently, for example, by Max Jammer, in Concepts of Space [Cambridge, Mass. A degenerate conic gives rise to the familiar Euclidean geometry.

Whereas during the first half of the century geometry had seemed to be becoming increasingly fragmented, Cayley and Klein, through the medium of these ideas, apparently succeeded for a lime in providing geometers with a unified view of their subject. Thus, although the so-called Cayley-Klein metric is now seldom taught, to their contemporaries it was of great importance.

Cayley is responsible for another branch of algebra over and above invariant theory, the algebra of matrices. I, no. They later suggested to him the analytical geometry of n dimensions. In his first systematic memoir on the subject C. He later derived many important theorems of matrix theory. Thus, for example, he derived many theorems of varying generality in the theory of those linear transformations that leave invariant a quadratic or bilinear form.

His formulas, however, do not include all orthogonal transformations except as limiting cases see E.

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The theory of matrices was developed in two quite different ways: the one of abstract algebraic structure, favored by Cayley and Sylvester; the other, in the geometrical tradition of Hamilton and Grassmann. Benjamin Peirce whose study of linear associative algebras, published in but evolved by him much earlier, was a strong influence on Cayley and Cayley himself were notable for their ability to produce original work in both traditions.

It is on the strength of his work on linear associative algebras that Peirce is often regarded as cofounder of the theory of matrices. In his many informal comments on the relation between matrices and quaternions see, for example, his long report to the British Association, reprinted in C.

Tail, printed in C.

## The Collected Mathematical Papers of Arthur Cayley, ScD, FRS

Tait [Cambridge, ], pp. He had no significant part in the controversy between Tait and J. Gibbs, author of the much simpler vector analysis. That Cayley found geometrical analogy of great assistance in his algebraic and analytical work—and conversely—is evident throughout his writings; and this, together with his studied avoidance of the highly physical interpretation of geometry more typical of his day, resulted in his developing the idea of a geometry of n dimensions.

But only Moebius made his fourth dimension spatial, as opposed to temporal. II [], 55— 62 might have been considered at the time to have a misleading title, for it contained little that would then have been construed as geometry. It concerns the nonzero solutions of homogeneous linear equations in any number of variables.

By Cayley had made use of four dimensions in the enunciation of specifically synthetic geometrical theorems, suggesting methods later developed by Veronese C. Cayley and Sylvester subsequently developed these ideas. Cayley wrote copiously on analytical geometry, touching on almost every topic then under discussion.

There Cayley gave an almost complete proof to be supplemented by Bacharach, in Mathematische Annalen , 26 [], — that when a plane curve of degree r is drawn through the mn points common to two curves of degrees m and n both less than r , these do not count for mn conditions in the determination of the curve but for mn reduced by.

The Cayley-Bacharach theorem was subsequently generalized by Noether. Cayley systematically showed the relations between the two schemes C. It is possible only to hint at that set of interrelated theorems in algebraic geometry which Cayley did so much to clarify, including those on the twenty-eight bitangents of a nonsingular quartic plane curve and the theorem first announced in on the twenty-seven lines that He on a cubic surface in three dimensions C. Strictly speaking, Cayley established the existence of the lines and Salmon, in a correspondence prior to the paper, established their number.

See the last page of the memoir and G. Salmon, The Geometry of Three Dimensions , 2nd ed.

## Arthur Cayley

Although no longer in vogue this branch of geometry, in association with Galois theory, invariant algebra, group theory, and hyperelliptic functions, reached a degree of intrinsic difficulty and beauty rarely equaled in the history of mathematics. As might have been expected from his contributions to the theory of invariants, Cayley made an important contribution to the theory of rational transformation and general rational correspondence.

For a history of the subject see C. Cayley devoted a great deal of his time to the projective characteristics of curves and surfaces. Apart from his intricate treatment of the theory of scrolls where many of his methods and his vocabulary still survive , the Cayley-Zeuthen equations are still a conspicuous reminder of the permanent value of his work. Given an irreducible surface in three-dimensional space, with normal singularities and known elementary projective characters, many other important characteristics may be deduced from these equations, which were first found empirically by Salmonan and later proved by Cayley and Zeuthen.

In the abstract theory of groups, where nothing is said of the nature of the elements, the group is completely specified if all possible products are known or determinable. This formulation differed from those of earlier writers to the extent that he spoke only of symbols and multiplication without further defining either. He went on to give what has since been taken as the first set of axioms for a group, somewhat tacitly postulating associativity, a unit element, and closure with respect to multiplication.

The axioms are sufficient for finite, but not infinite, groups. There is some doubt as to whether Cayley ever intended his statements in the paper to constitute a definition, for he not only failed to use them subsequently as axioms but later used a different and unsatisfactory definition. See, for instance, an article for the English Cyclopaedia , in C.

## DML: Digital Mathematics Library: Retrodigitized Mathematics Journals and Monographs

In a number of historical articles G. In addition to his part in founding the theory of abstract groups, Cayley has a number of important theorems to his credit: perhaps the best known is that every finite group whatsoever is isomorphic with a suitable group of permutations see the first paper of This is often reckoned to be one of the three most important theorems of the subject, the others being the theorems of Lagrange and Sylow.

But perhaps still more significant was his early appreciation of the way in which the theory of groups was capable of drawing together many different domains of mathematics: his own illustrations, for instance, were drawn from the theories of elliptic functions, matrices, quantics, quaternions, homographic transformations, and the theory of equations.

If Cayley failed to pursue his abstract approach, this fact is perhaps best explained in terms of the enormous progress he was making in these subjects taken individually. The property was appreciated by Poncelet and was discussed analytically by Jacobi using elliptic functions when the conics were circles. Using his first paper of and gradually generalizing his own findings, by Cayley was discussing the problem of the number of polygons which are such that their vertices lie on a given curve or curves of any order and that their sides touch another given curve or curves of any class.

That he was able to give a complete solution even where the polygons were only triangles is an indication of his great analytical skill.

Cayley wrote little on topology, although he wrote on the combinatorial aspect, renewed the discussion of the four-color-map problem, and corresponded with Tait on the topological problems associated with knots. He wrote briefly on a number of topics for which alone a lesser mathematician might have been remembered. He wrote to great effect on the theory of the numbers of partitions, originated by Euler. His interest in this arose from his need to apply it to invariant theory and is first evident in his second memoir on quantics, C.

This technique is linked to that of characterizing invariants and covariants of binary quantics as the polynomial solutions of linear partial differential equations. The differential operators were in this context known as annihilators, following Sylvester. He wrote occasionally on dynamics, but his writings suggest that he looked upon it as a source of problems in pure mathematics rather than as a practical subject. Thus in five articles he considered that favorite problem of the time, the attraction of ellipsoids; and in a paper of he extended a certain problem in potential theory to hyperspace C.

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That he kept himself informed of the work of others in dynamics is evident from two long reports on recent progress in the subject which he wrote for the British Association C. P , III, no. Cayley looked into the matter independently, found a new and simpler method for introducing the variation of the eccentricity, and confirmed the value Adams had previously found C. Original Works. The printing of the first seven vols, and part of the eighth was supervised by Cayley himself.

The editorial task was assumed by A.

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Forsyth when Cayley died. His excellent biography of Cayley is in vol. The list of writings in vol. XIV includes the titles of several articles which Cayley contributed to the Encyclopaedia Britannica. See, e. Upward of twenty sections and the whole of ch. Cayley frequently gave advice and assistance to other authors. Thus he contributed ch. Secondary Literature.

General histories of mathematics are not listed here, nor are mathematical works in which historical asides are made. The best biographical notice is by A. Cayley, Arthur, The collected mathematical papers of Arthur Cayley : supplementary volume containing titles of papers and index. Cambridge [Eng. Cambridge: at the University Press, Cain, Wm. William , Cajori, Florian, Selected pages Page S, Vol.

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