Arthur showed great skill in numerical calculations at a private school in Blackheath and, after he moved to King's College School in 1835, at age 14 rather than the usual age of entry of 16, his aptitude for advanced mathematics became apparent. However, it was not only mathematics at which he excelled, for he won prizes in many subjects. In particular, he won the Chemistry Prize in each of his final two years despite not specialising in science. His mathematics teacher advised his parents that Arthur be encouraged to pursue his studies in this area rather than follow his father's wishes to enter the family business as a merchant.
In 1838 Arthur began his studies at Trinity College, Cambridge, having George Peacock as tutor in his first year. He was coached by William Hopkins who encouraged him to read papers by continental mathematicians. His favourite mathematical topics were linear transformations and analytical geometry and while still an undergraduate he had three papers published in the newly founded Cambridge Mathematical Journal edited by Duncan Gregory. Cayley graduated as Senior Wrangler in 1842 and won the first Smith's prize. After the examinations, Cayley and his friend Edmund Venables led a reading party of undergraduates to Aberfeldy in Scotland. One of these undergraduates was Francis Galton who described his tutor Cayley (quoted in ):-
Never was a man whose outer physique so belied his powers as that of Cayley. There was something eerie and uncanny in his ways, that inclined strangers to pronounce him neither to be wholly sane nor gifted with much intelligence, which was the very reverse of the truth ... he appeared so frail as to be incapable of ordinary physical work.While on the trip, Venables wrote to George Stokes in Cambridge (quoted in ):-
We are surrounded by some of the highest mountains in Scotland, among which the most beautiful is certainly Schiehallion famous in the history of gravitation; I have been trying in vain to induce Cayley to repeat Dr Maskelyne's experiments on its summit and to acquire to himself a never dying fame; but alas he has no desire for notoriety, and has a rooted aversion to experiments and calculations of all kinds: he is now sitting by my side carrying on those dreadful investigations commenced in the Mathematical Journal, in which having exhausted all the letters of the Greek and English alphabets, he is fain to turn his δελτα s tops turvy, and have recourse to the old English.For four years Cayley taught at Cambridge having won a Fellowship and, during this period, he published 28 papers in the Cambridge Mathematical Journal. He worked on a large variety of mathematical topics including algebraic curves and surfaces, elliptic functions, determinants and the theory of integration. For example he published On a theory of determinants in 1843 in which he extended the idea of a 2-dimensional determinant (rows and columns) to multidimensional arrays. In 1844 he began a correspondence with George Boole which proved valuable to both mathematicians. For example he wrote to Boole in 1845 (Boole was teaching at a school in Lincoln, England, at the time):-
I wish I could manage a visit to Lincoln, I should so much enjoy talking over some things with you, not to mention the temptation of your Cathedral. I think I must contrive it some time in the next six months, in spite of there being no railway, which one begins to consider oneself entitled to in these days.In 1844 Cayley, along with his friend Edmund Venables, made a trip to the Swiss Alps, also visiting Italy. He took a much more international approach than many of his fellow English mathematicians, publishing papers in the French Journal de Mathématiques Pures et Appliquées and in Crelle's journal Journal für die reine und angewandte Mathematik. Two papers he published in 1845 and 1846 are regarded as laying the foundations for invariant theory.
The Cambridge fellowship had a limited tenure, since Cayley was not prepared to take Holy Orders, so he had to find a profession. He chose law and began training in April 1846. While still training to be a lawyer Cayley went to Dublin to hear William Rowan Hamilton lecture on quaternions. He sat next to George Salmon during these lectures and the two were to exchange mathematical ideas over many years. Cayley was a good friend of Hamilton's although the two disagreed as to the importance of the quaternions in the study of geometry. Another of Cayley's friends was James Joseph Sylvester who was also in the legal profession. The two both worked at the courts of Lincoln's Inn in London and they discussed deep mathematical questions during their working day. Others at Lincoln's Inn who were active mathematicians included Hugh Blackburn. He wrote to William Thomson about his work with Cayley:-
We have been busy (when we ought to have been drawing acts of parliament and such sublunary matters) in constructing the developable surface generated by the tangent to the curve of intersection of the straight cylinder and sphere of double its radius ...In fact Cayley and Thomson were also good friends. About a year after he began legal training Cayley began a correspondence with Thomas Kirkman. He was a referee of Kirkman's famous paper in the Cambridge and Dublin Mathematical Journal in which he shows the existence of what today are called Steiner systems. Cayley wrote in his report:-
Kirkman's paper is decidedly interesting and his main result a very elegant one.He was admitted to the bar on 3 May 1849. He spent 14 years as a lawyer but Cayley, although very skilled in conveyancing (his legal speciality), always considered it as a means to make money so that he could pursue mathematics. George Halsted writes :-
The law was always drudgery to him. The superabundant verbiage of legal forms was always distasteful to him. He once remarked that "the object of law was to say a thing in the greatest number of words, of mathematics to say it in the fewest."During these 14 years as a lawyer Cayley published about 250 mathematical papers - how many full time mathematicians could compare with the productivity of this 'amateur'? However, Tony Crilly writes :-
At Cambridge, and even as a pupil barrister, he could retire to his study. As a newly qualified barrister, with a growing scientific reputation, it became obligatory to adopt a more public persona, a role which did not sit easily with his retiring disposition. Cayley maintained contact with Cambridge and from serving as Senior Examiner at the annual Trinity College examinations, he progressed to the responsibility of being Senior Moderator for the Mathematical Tripos in 1851 and Senior Examiner in 1852.As a result he began to look for academic appointments and, to improve his chances of a professorship he began to increase his already remarkably high rate of publication of mathematics papers. For example, during the years 1853-1856 he averaged ten published papers per year, but as he sought to raise his profile, he published on average thirty papers per year during 1857-1860. He had not been successful in his application in 1856 for the chair of natural philosophy at Marischal College, Aberdeen. He was also interested in the Lowndean Chair of Geometry and Astronomy at Cambridge in 1858 and the chair of astronomy at Glasgow University in the following year. However, he failed to obtain both these and other positions. Perhaps one reason was that, despite a remarkable research record, he had little experience of teaching.
In 1863 Cayley was appointed Sadleirian professor of Pure Mathematics at Cambridge. He was successful despite there being other high quality applicants such as Isaac Todhunter, Norman Ferrers, and Edward Routh. This appointment involved a very large decrease in income for Cayley who now had to manage on a salary only a fraction of that which he had earned as a skilled lawyer. However Cayley was very happy to have the chance to devote himself entirely to mathematics. As Sadleirian professor of Pure Mathematics his duties were:-
... to explain and teach the principles of pure mathematics and to apply himself to the advancement of that science.Cayley was to more than fulfil these conditions. However, before we look briefly at his remarkable research achievements, we note that he married in 1863, the year of his appointment as Sadleirian professor. On 8 September he married Susan Moline (1831-1923) from Greenwich, the daughter of Robert Moline. Arthur and Susan Cayley had two children: Henry Cayley (1870-1949) who studied mathematics at Cambridge but decided that he couldn't live up to his father's achievements so became an architect, and Mary Cayley (1872-1950). In 1864 Cayley purchased Garden House, Little St Mary's Lane, in Cambridge. On 8 November 1863 Cayley gave his inaugural lecture Analytical geometry - Introductory lecture. However, his lectures over the following years were often based on his latest researches and students did not find them relevant to the Tripos examination in which they all aimed at getting the greatest possible marks :-
Cayley was concerned with mathematics 'per se', while most students wanted competence in a circumscribed syllabus. Any deviation from it was regarded as a waste of time and effort.He published over 900 papers and notes covering nearly every aspect of modern mathematics. The most important of his work is in developing the algebra of matrices, work on non-euclidean geometry and n-dimensional geometry. Of his work on matrices, Richard Feldmann writes :-
Although the term "matrix" was introduced into mathematical literature by James Joseph Sylvester in 1850, the credit for founding the theory of matrices must be given to Arthur Cayley, since he published the first expository articles on the subject. ... Cayley's introductory paper in matrix theory was written in French and published in a German periodical [in 1855]. ... [He] introduces, although quite sketchily, the ideas of inverse matrix and of matrix multiplication, or "compounding" as Cayley called it. The above basic properties are expanded in a second expository article [of 1858] which also lists many additional properties of matrices.As early as 1849 Cayley had written a paper linking his ideas on permutations with Cauchy's. In 1854 Cayley wrote two papers which are remarkable for the insight they have of abstract groups. At that time the only known groups were permutation groups and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication. He gives the 'Cayley tables' of some special permutation groups but, much more significantly for the introduction of the abstract group concept, he realised that matrices and quaternions were groups. Here is a part of his definition of an abstract group:-
... the symbol 1 will naturally denote an operation which (either generally or in regard to the particular operand) leaves the operand unaltered. .... A symbol xy denotes the compound operation, the performance of which is equivalent to the performance, first of the operation y, and then of the operation x; xy is of course in general different from yx. But the symbols x, y, ... are in general such that x(yz) = (xy)z etc, so that xyz, xyzt, etc. have a definite signification independent of the particular mode of compounding the symbols ...Cayley developed the theory of algebraic invariance, and his development of n-dimensional geometry has been applied in physics to the study of the space-time continuum. His work on matrices served as a foundation for quantum mechanics, which was developed by Werner Heisenberg in 1925. Cayley also suggested that euclidean and non-euclidean geometry are special types of geometry. He united projective geometry and metrical geometry which is dependent on sizes of angles and lengths of lines.
You can see reviews of Cayley's papers by G B Halsted at THIS LINK.
He only published one book, namely An Elementary Treatise on Elliptic Functions (1876). However, he did contribute Chapter 6 to Peter Guthrie Tait's An Elementary Treatise on Quaternions (1890) and published the six-penny booklet, The Principles of Book-Keeping by Double Entry (1894).
In 1881 he was invited to give a course of lectures at Johns Hopkins University in the United States, where his friend Sylvester was professor of mathematics. He spent January to May of 1882 at Johns Hopkins University where he lectured on Abelian and Theta Functions. After he returned to England, the Royal Society awarded him their Copley Medal. In 1883 Cayley became President of the British Association for the Advancement of Science. In his presidential address Cayley gave an elementary account of his own views of mathematics. His views of geometry were:-
It is well known that Euclid's twelfth axiom, even in Playfair's form of it, has been considered as needing demonstration: and that Lobachevsky constructed a perfectly consistent theory, wherein this axiom was assumed not to hold good, or say a system of non-Euclidean plane geometry. My own view is that Euclid's twelfth axiom in Playfair's form of it does not need demonstration, but is part of our experience - the space, that is, which we become acquainted with by experience, but which is the representation lying at the foundation of all external experience. Riemann's view ... is that, having 'in intellectu' a more general notion of space (in fact a notion of non-Euclidean space), we learn by experience that space (the physical space of our experience) is, if not exactly, at least to the highest degree of approximation, Euclidean space. But suppose the physical space of our experience to be thus only approximately Euclidean space, what is the consequence which follows? Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience.Two descriptions of Cayley, both of him as an old man, are interesting. Macfarlane  says:-
... I attended a meeting of the Mathematical Society of London. The room was small, and some twelve mathematicians were assembled round a table, among them was Prof Cayley ... At the close of the meeting Cayley gave me a cordial handshake and referred in the kindest terms to my papers which he had read. He was then about 60 years old, considerably bent, and not filling his clothes. What was most remarkable about him was the active glance of his grey eyes and his peculiar boyish smile.Thomas Hirst, one of his friends, wrote:-
... a thin weak-looking individual with a large head and face marked with small-pox: he speaks with difficulty and stutters slightly. He never sits upright on his chair but with his posterior on the very edge he leans one elbow on the seat of the chair and throws the other arm over the back.Here is G B Halsted's tribute :-
Cayley, in addition to his wondrous originality, was assuredly the most learned and erudite of mathematicians. Of him in his science it might be said he knew everything, and he was the very last man who ever will know everything. His was a very gentle, sweet character. Sylvester told me he never saw him angry but once, and that was (both were practicing law!) when a messenger broke in on one of their interviews with a mass of legal documents - new business for Cayley. In an access of disgust, Cayley dashed the documents upon the floor.We end by giving this tribute from :-
Cayley was Britain's outstanding pure mathematician of the nineteenth century. An algebraist, analyst, and geometer, he was able to link these vast domains of study. More than fifty concepts and theorems of mathematics bear his name. By the end of his life he was revered by mathematicians the world over.In addition to the Copley Medal we mentioned above, we note that Cayley was elected a fellow of the Royal Society in 1852 and received their Royal Medal in 1859. The London Mathematical Society awarded him their De Morgan Medal in 1884. He was president of the Society 1868-1870. He was elected a fellow of the Royal Society of Edinburgh in 1865. He was elected a fellow of the Royal Astronomical Society in 1857 and served as editor of the Society's publications from 1859 to 1881. In 1872 he was made an honorary fellow of Trinity College, Cambridge. He received honorary degrees from the universities of Cambridge, Oxford, Edinburgh, Dublin, Göttingen, Heidelberg, Leyden and Bologna. A crater on the moon is named for Cayley. He was an honorary foreign member of the French Institute and was elected a fellow of the academies of Berlin, Göttingen, St Petersburg, Milan, Rome, Leyden, Upsala, and Hungary.
After a long period of suffering he died at his home, Garden House, Cambridge. He was buried on 2 February 1895 in Mill Road cemetery, Cambridge. Many representatives from foreign countries attended his funeral in addition to many of the leading British scholars.
Article by: J J O'Connor and E F Robertson