As a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, Cambridge, where he excelled in Greek, French, German, and Italian, as well as mathematics. He worked as a lawyer for 14 years.
He was consequently able to prove the Cayley-Hamilton theorem—that every square matrix is a root of its own characteristic polynomial. He was the first to define the concept of a group in the modern way—as a set with a binary operation satisfying certain laws. Formerly, when mathematicians spoke of "groups", they had meant permutation groups.
See also Cayley's theorem.
Arthur Cayley was born in Richmond, London, England, on August 16 1821. His father, Henry Cayley, was a distant cousin of Sir George Cayley the aeronautics engineer innovator, and descended from an ancient Yorkshire family. He settled in Saint Petersburg, Russia, as a merchant. His mother was Maria Antonia Doughty, daughter of William Doughty. According to some writers she was Russian, but her father's name indicates an English origin. His brother was the linguist Charles Bagot Cayley. Arthur spent his first eight years in Saint Petersburg. In 1829 his parents settled permanently at Blackheath, near London. Arthur was sent to a private school. He early showed great liking for, and aptitude in, numerical calculation. At age 14 he was sent to King's College School. The school's master observed indications of mathematical genius and advised the father to educate his son not for his own business, as he had intended, but to enter the University of Cambridge.
At the unusually early age of 17 Cayley began residence at Trinity College, Cambridge. The cause of the Analytical Society had now triumphed, and the Cambridge Mathematical Journal had been instituted by Gregory and Leslie Ellis. To this journal, at the age of twenty, Cayley contributed three papers, on subjects which had been suggested by reading the Mécanique analytique of Lagrange and some of the works of Laplace.
Cayley finished his undergraduate course by winning the place of Senior Wrangler, and the first Smith's prize. His next step was to take the M.A. degree, and win a Fellowship by competitive examination. His tutor at Cambridge was George Peacock and his private coach was William Hopkins. He continued to reside at Cambridge for four years; during which time he took some pupils, but his main work was the preparation of 28 memoirs to the Mathematical Journal.
Because of the limited tenure of his fellowship it was necessary to choose a profession; like De Morgan, Cayley chose law, and at age 25 entered at Lincoln's Inn, London. He made a specialty of conveyancing. It was while he was a pupil at the bar examination that he went to Dublin to hear Hamilton's lectures on quaternions.
His friend Sylvester, his senior by five years at Cambridge, was then an actuary, resident in London; they used to walk together round the courts of Lincoln's Inn, discussing the theory of invariants and covariants. During this period of his life, extending over fourteen years, Cayley produced between two and three hundred papers.
At Cambridge University the ancient professorship of pure mathematics is denominated the Lucasian, and is the chair which had been occupied by Isaac Newton. Around 1860, certain funds bequeathed by Lady Sadleir to the University, having become useless for their original purpose, were employed to establish another professorship of pure mathematics, called the Sadlerian. The duties of the new professor were defined to be "to explain and teach the principles of pure mathematics and to apply himself to the advancement of that science." To this chair Cayley was elected when 42 years old. He gave up a lucrative practice for a modest salary; but he never regretted the exchange, for the chair at Cambridge enabled him to end the divided allegiance between law and mathematics, and to devote his energies to the pursuit which he liked best. He at once married and settled down in Cambridge. More fortunate than Hamilton in his choice, his home life was one of great happiness. His friend and fellow investigator, Sylvester, once remarked that Cayley had been much more fortunate than himself; that they both lived as bachelors in London, but that Cayley had married and settled down to a quiet and peaceful life at Cambridge; whereas he had never married, and had been fighting the world all his days.
At first the teaching duty of the Sadlerian professorship was limited to a course of lectures extending over one of the terms of the academic year; but when the University was reformed about 1886, and part of the college funds applied to the better endowment of the University professors, the lectures were extended over two terms. For many years the attendance was small, and came almost entirely from those who had finished their career of preparation for competitive examinations; after the reform the attendance numbered about fifteen. The subject lectured on was generally that of the memoir on which the professor was for the time engaged.
The other duty of the chair - the advancement of mathematical science - was discharged in a handsome manner by the long series of memoirs which he published, ranging over every department of pure mathematics. But it was also discharged in a much less obtrusive way; he became the standing referee on the merits of mathematical papers to many societies both at home and abroad.
In 1876 he published a Treatise on Elliptic Functions, which was his only book. He took great interest in the movement for the University education of women. At Cambridge the women's colleges are Girton and Newnham. In the early days of Girton College he gave direct help in teaching, and for some years he was chairman of the council of Newnham College, in the progress of which he took the keenest interest to the last.
In 1872 he was made an honorary fellow of Trinity College, and three years later an ordinary fellow, which meant stipend as well as honour. About this time his friends subscribed for a presentation portrait. Maxwell wrote an address to the committee of subscribers who had charge of the Cayley portrait fund. The verses refer to the subjects investigated in several of Cayley's most elaborate memoirs; such as, Chapters on the Analytical Geometry of dimensions; On the theory of Determinants; Memoir on the theory of Matrices; Memoirs on skew surfaces, otherwise Scrolls; On the delineation of a Cubic Scroll, etc.
In 1881 he received from the Johns Hopkins University, Baltimore, where Sylvester was then professor of mathematics, an invitation to deliver a course of lectures. He accepted the invitation, and lectured at Baltimore during the first five months of 1882 on the subject of the Abelian and Theta Functions.
I wish to speak to you to-night upon Mathematics. I am quite aware of the difficulty arising from the abstract nature of my subject; and if, as I fear, many or some of you, recalling the providential addresses at former meetings, should wish that you were now about to have from a different President a discourse on a different subject, I can very well sympathize with you in the feeling. But be that as it may, I think it is more respectful to you that I should speak to you upon and do my best to interest you in the subject which has occupied me, and in which I am myself most interested. And in another point of view, I think it is right that the address of a president should be on his own subject, and that different subjects should be thus brought in turn before the meetings. So much the worse, it may be, for a particular meeting: but the meeting is the individual, which on evolution principles, must be sacrificed for the development of the race.
Cayley doubtless felt that he was addressing not only the popular audience then and there before him, but the mathematicians of distant places and future times; for the address is a valuable historical review of various mathematical theories, and is characterized by freshness, independence of view, suggestiveness, and learning.
In 1889 the Cambridge University Press requested him to prepare his mathematical papers for publication in a collected form—a request which he appreciated very much. They are printed in magnificent quarto volumes, of which seven appeared under his own editorship. While editing these volumes, he was suffering from a painful internal malady, to which he succumbed on January 26, 1895, in the 74th year of his age. When the funeral took place, a great assemblage met in Trinity Chapel, comprising members of the University, official representatives of Russia and America, and many of the most illustrious philosophers of Britain.
The remainder of his papers were edited by Prof. Forsyth, his successor in the Sadlerian chair. The Collected Mathematical papers number thirteen quarto volumes, and contain 967 papers. His writings are his best monument, and certainly no mathematician has ever had his monument in grander style. De Morgan's works would be more extensive, and much more useful, but he did not have behind him a University Press. As regards fads, Cayley retained to the last his fondness for novel-reading and for travelling. He also took special pleasure in paintings and architecture, and he practiced water-color painting, which he found useful sometimes in making mathematical diagrams.
To the third edition of P. G. Tait's Elementary Treatise on quaternions, Cayley contributed a chapter entitled "Sketch of the analytical theory of quaternions." In it the √−1 reappears in all its glory, and in entire, so it is said, independence of i, j, k.
In 1894 there arose a brisk discussion between Tait and Cayley on "Coordinates versus quaternions," the record of which is printed in the Proceedings of the Royal Society of Edinburgh. Cayley maintained the position that while coordinates are applicable to the whole science of geometry and are the natural and appropriate basis and method in the science, quaternions seemed a particular and very artificial method for treating such parts of the science of three-dimensional geometry as are most naturally discussed by means of the rectangular coordinates x, y, z. In the course of his paper Cayley says:
I have the highest admiration for the notion of a quaternion; but, as I consider the full moon far more beautiful than any moonlit view, so I regard the notion of a quaternion as far more beautiful than any of its applications. As another illustration, I compare a quaternion formula to a pocket-map—a capital thing to put in one's pocket, but which for use must be unfolded: the formula, to be understood, must be translated into coordinates.
He goes on to say,
I remark that the imaginary of ordinary algebra—for distinction call this θ—has no relation whatever to the quaternion symbols i, j, k; in fact, in the general point of view, all the quantities which present themselves, are, or may be, complex values , or in other words, say that a scalar quantity is in general of the form a + θb. Thus quaternions do not properly present themselves in plane or two-dimensional geometry at all; but they belong essentially to solid or three-dimensional geometry, and they are most naturally applicable to the class of problems which in coordinates are dealt with by means of the three rectangular coordinates x, y, z.
To the pocketbook illustration it may be replied that a set of coordinates is an immense wall map, which you cannot carry about, even though you should roll it up, and therefore is useless for many important purposes. In reply to the arguments, it may be said, first, √−1 has a relation to the symbols i, j, k for each of these can be analyzed into a unit axis multiplied by √−1; second, as regards plane geometry, the ordinary form of complex quantity is a degraded form of the quaternion in which the constant axis of the plane is left unspecified. Cayley took his illustrations from his experience as a traveller. Tait brought forward an illustration from which you might imagine he had visited the Bethlehem Iron Works, and hunted tigers in India. He says,
A much more natural and adequate comparison would, it seems to me, liken Coordinate Geometry to a steam-hammer, which an expert may employ on any destructive or constructive work of one general kind, say the cracking of an eggshell, or the welding of an anchor. But you must have your expert to manage it, for without him it is useless. He has to toil amid the heat, smoke, grime, grease, and perpetual din of the suffocating engine-room. The work has to be brought to the hammer, for it cannot usually be taken to its work. And it is not in general, transferable; for each expert, as a rule, knows, fully and confidently, the working details of his own weapon only. quaternions, on the other hand, are like the elephant's trunk, ready at any moment for anything, be it to pick up a crumb or a field-gun, to strangle a tiger, or uproot a tree; portable in the extreme, applicable anywhere—like in the trackless jungle and in the barrack square—directed by a little native who requires no special skill or training, and who can be transferred from one elephant to another without much hesitation. Surely this, which adapts itself to its work, is the grander instrument. But then, it is the natural, the other, the artificial one.
The reply which Tait makes, so far as it is an argument, is: There are two systems of quaternions, the i, j, k one, and another one which Hamilton developed from it; Cayley knows the first only, he himself knows the second; the former is an intensely artificial system of imaginaries, the latter is the natural organ of expression for quantities in space. Should a fourth edition of his Elementary Treatise be called for i, j, k will disappear from it, excepting in Cayley's chapter, should it be retained. Tait thus describes the first system:
From "the most intensely artificial of systems, arose, as if by magic, an absolutely natural one" which Tait thus further describes. "To me quaternions are primarily a Mode of Representation:—immensely superior to, but of essentially the same kind of usefulness as, a diagram or a model. They are, virtually, the thing represented; and are thus antecedent to, and independent of, coordinates; giving, in general, all the main relations, in the problem to which they are applied, without the necessity of appealing to coordinates at all. Coordinates may, however, easily be read into them:—when anything (such as metrical or numerical detail) is to be gained thereby. quaternions, in a word, exist in space, and we have only to recognize them:—but we have to invent or imagine coordinates of all kinds."
To meet the objection why Hamilton did not throw i, j, k overboard, and expound the developed system, Tait says:
To Cayley's presidential address we are indebted for information about the view which he took of the foundations of exact science, and the philosophy which commended itself to his mind. He quoted Plato and Kant with approval, J. S. Mill with faint praise. Although he threw a sop to the empirical philosophers at the beginning of his address, he gave them something to think of before he finished.
He first of all remarks that the connection of arithmetic and algebra with the notion of time is far less obvious than that of geometry with the notion of space; in which he, of course, made a hit at Hamilton's theory of Algebra as the science of pure time. Further on he discusses the theory directly, and concludes as follows:
So you will observe that doctors differ—Tait and Cayley—about the soundness of Hamilton's theory of couples. But it can be shown that a couple may not only be represented on a straight line, but actually means a portion of a straight line; and as a line is unidimensional, this favors the truth of Hamilton's theory.
As to the nature of mathematical science Cayley quoted with approval from an address of Hamilton's:
It is the aim of the evolution philosopher to reduce all knowledge to the empirical status; the only intuition he grants is a kind of instinct formed by the experience of ancestors and transmitted cumulatively by heredity. Cayley first takes him up on the subject of arithmetic:
Then he takes him up on the subject of geometry, where the empiricist rather boasts of his success.
In his address he remarks that the fundamental notion which underlies and pervades the whole of modern analysis and geometry is that of imaginary magnitude in analysis and of imaginary space (or space as a locus in quo of imaginary points and figures) in geometry. In the case of two given curves there are two equations satisfied by the coordinates (x, y) of the several points of intersection, and these give rise to an equation of a certain order for the coordinate x or y of a point of intersection. In the case of a straight line and a circle this is a quadratic equation; it has two roots real or imaginary. There are thus two values, say of x, and to each of these corresponds a single value of y. There are therefore two points of intersection, viz., a straight line and a circle intersect always in two points, real or imaginary. It is in this way we are led analytically to the notion of imaginary points in geometry. He asks, What is an imaginary point? Is there in a plane a point the coordinates of which have given imaginary values? He seems to say No, and to fall back on the notion of an imaginary space as the locus in quo of the imaginary point.