The Young Arthur Cayley

The Young Arthur CayleyAuthor(s): Tony CrillySource: Notes and Records of the Royal Society of London, Vol. 52, No. 2 (Jul., 1998), pp. 267-282Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/531861 .

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Notes Rec. R. Soc. Lond. 52 (2), 267-282 (1998)

THE YOUNG ARTHUR CAYLEY

by

TONY CRILLY

Middlesex University Business School, The Burroughs, London NW4 4BT, UK ([email protected])

Arthur Cayley, F.R.S. (1821-1895) is widely regarded as Britain's leading pure mathematician of the 19th century. From his start as a Cambridge prodigy, he built up a formidable record of publication, from incidental notes to extensive memoirs, on a wide range of mathematics.' His emergence as a mathematician, before he became the sober-minded eminence of the 1860s and 1870s, is largely unknown. The primary guide to his whole life remains, more than a century later, the obituary written by his student and successor at Cambridge, A.R. Forsyth.2 The folklore surrounding Cayley's life is dominated by his collaboration with James Joseph Sylvester, F.R.S. (1814-1897). This partnership, tempered by Sylvester's obsession with the apportionment of credit, began in earnest in the 1850s. In this paper, I attempt to signpost Cayley's formative years, when he was at the beginning of his long mathematical journey, the period which ends on 3 June 1852, the date of his election to the Royal Society, when he formally came of age as a Victorian man of science.

INTRODUCTION

Cayley gives the impression of a taciturn character, a view encouraged by the severe style of his recorded mathematical research. Transported to modem times he would be a clear example of a 'no-nonsense' mathematician. He dealt with the mathematics at hand, without much commentary or clues to its development-much to the consternation of his contemporaries. The style of his published work contrasts sharply with that of Sylvester, whose papers were enhanced with an eloquence and passion rarely found in mathematical writing.3 Without Sylvester's ebullience, Cayley was just as passionate about his subject, but his feelings were often camouflaged by a blanket of Victorian reserve. His austerity and what he considered 'proper' in published work carried over to private letters. Though his correspondence with Sylvester lasted almost 50 years, there was just one occasion when he abandoned restraint-and then barely. In the mid-1850s a single epithet 'eureka' preceded his discovery of a theorem in invariant theory (a branch of algebra which focuses on the study of algebraic functions left unaltered by transformations of their variables).4 It was not a casual aside but an exclamation justly deserved, and the praiseworthy theorem duly became the centrepiece of his later work in invariant theory.

With his friends of youth, Cayley was more expansive. In his letters to William

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Thomson and George Boole, as well as the glimpses offered offstage through the letters of others, we perceive the young tyro at work. Cayley's single-mindedness at the outset of his solitary mathematical voyage is notable, as is his disregard for those who treated mathematics as an intellectual pastime. The story of the 'Irish boatman' will reveal a mischievous sense of humour.

BACKGROUND

Arthur Cayley was born on 16 August 1821 in Richmond, Surrey, the second son of five children of Henry Cayley (1768-1850), Russia merchant, and his wife Maria Antonia Doughty (1794-1875). His parents lived in the British community in St Petersburg and Arthur was born on one of their home visits. His paternal grandfather, John Cayley (1730-1795), served as Consul General in St Petersburg. Arthur Cayley was the fourth cousin of Sir George Cayley, F.R.S. (1773-1857), inventor and aeronautical pioneer, though I have found no evidence of social contact between the 'St Petersburg Cayleys' and the mainstream Cayleys from Yorkshire.5

Henry Cayley and his young family returned permanently to England in 1828. Sophia was the eldest child, but the next, William Henry, died in infancy. Arthur's younger brother Charles was born in Russia and the family circle was completed by the birth of Henrietta-Caroline on their return.6 Henry, then aged 60, established himself in the City of London, became a director of the London Assurance Corporation, and was active in the reorganization of the Baltic Exchange.

After attending a private school at Blackheath, Arthur entered the Senior Department of King's College London two years earlier than was normal, at the age of 14. He consistently gained school prizes and in his third and final year won the Chemistry Prize in competition with specialist science students. His father, having expected his eldest son to follow him into a commercial career, was eventually persuaded that Arthur should be allowed to enter university.

CAMBRIDGE

Cayley went up to Trinity College, Cambridge in October 1838. His first-year tutor was George Peacock, who left in the following year to become the Dean of Ely. Peacock was a pioneering Cambridge mathematician, but in his year of being in loco parentis to the young Cayley, his contact may have been limited to pastoral care. A greater mathematical influence was the celebrated mathematical coach William Hopkins. He encouraged his pupils to read leading continental writers, and, significantly, the advanced reading he suggested resulted in research topics for Cayley's early mathematical papers.

In the severe competition of the Mathematical Tripos, a contest which exacted a severe mental and physical strain on the leading contenders, Cayley appeared unaffected. According to Albert Pell, an undergraduate at the time, Cayley relished the 'Derby Day' atmosphere of the January examination:

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The young Arthur Cayley

Figure 1. Presentation of the senior wrangler [Arthur Cayley] to the vice chancellor. From Victor A. Huber, The English Universities, 3 vols (London, 1843).

While the examination was going on in the Senate House a small crowd was frequently in attendance outside by the door, discussing the merits of the examined and waiting to get the latest intelligence of their work. I, among them, went for this purpose, and on a Johnian in our group saying, 'I wonder how our man is getting on,' [T.I.] Barstow [another examinee] in a loud, contemptuous voice said he did not know, nor did he care, but he could tell him that 'Cayley had finished his papers a quarter of an hour ago, and was now licking his lips for more'.7

Cayley became the senior wrangler of 1842 and won the First Smith's Prize, the result of a further competitive examination. It was no more than had been predicted by the tutors and undergraduates in Cambridge, but the actual event made him an instant

celebrity. He was widely talked about and judged either surprisingly normal, or else the inhabitant of an intellectual plane unattainable by mere mortals.

In the summer of 1842, with his lifelong friend, the future churchman Edmund

Venables, Cayley escorted an undergraduate reading party to Aberfeldy in Scotland.8 Francis Galton was a member of the party and took note of his tutor: 'Never was a man whose outer physique so belied his powers as that of Cayley', he wrote, '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.'9

There was certainly little about Cayley which conformed to the popular image of the dusty and remote Cambridge Don. Galton wrote home to his father: 'Cayley is

unanimously voted a brick and most gentlemanly-minded man.' 0 Venables was more

prosaic in his remembrance of his companion: 'Cayley gained a hold on the reverence and affections of his pupils which he never lost.'1 He failed to mention his own

observations, made at the time, that none of the young 'cantabs' in Scotland was 'in

any danger of killing themselves by work' or that Cayley's teaching consisted of

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elementary subjects on a par with the theory of logarithms. During their free time in Scotland, Cayley and Venables went walking in the countryside and engaged in the collecting of new botanical specimens. Venables wrote back to George Stokes in Cambridge:

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 Maskelynes experiments on its summit,12 and to acquire to himself a never dying fame; but alas he has no desire for notoriety, and has a rooted aversion to experiments & 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 & English alphabets, he is fain to turn his AeXcra s topsy turvy, & have recourse to the old English.'3

Sitting beneath Schiehallion, Cayley interrupted his calculations only to send Stokes

greetings and to refer him to Cauchy's work on exact differentials which might be

applied to hydrodynamics. George Stokes, the senior wrangler of the previous year, had also been taught by Hopkins, and at his suggestion took hydrodynamics as his research subject after completing his degree.'4 Hydrodynamics was a Cambridge subject taught to undergraduates in the final year of the Tripos. Though Venables reinforces the view that Cayley was at heart a pure mathematician, Cayley's wide mathematical interests drew on the extensive 'mixed mathematics' taught at Cambridge during the early 1840s, before the reforms of the Tripos of 1848 reduced this content. His contributions to mathematics generally include many associated with such 'applied' subjects as optics, dynamics and astronomy.

In October 1842, the 22-year-old Cayley became a fellow of Trinity College, one of the youngest to be elected during the 19th century. A fellowship included a modest

income, but the real advantage for a young academic was the complete freedom to pursue any interests. Cayley had before him the world of mathematics and ambitiously chose it all. The age of intense specialization had yet to arrive; hard choices were unnecessary and Cayley exercised free rein. For the first few years, his work included elliptic functions, algebraic curves and surfaces, the theory of integration and determinants.

On his return from an extensive tour of the Swiss Alps and Italy in early 1844, again with Venables, Cayley settled back to Cambridge and research. He discovered the work of George Boole on linear transformations, and took the first steps in what was to become his best-known contribution to mathematics: invariant theory. In June, the earnest young mathematician opened his correspondence with Boole:

Will you allow me to make an excuse of the pleasure afforded me by a paper of yours published some time ago in the Mathematical Journal 'On the theory of linear transformations' and of the interest I take in the subject, for sending you a few formulae relative to it, which were suggested to me by your very interesting paper. I should be delighted if they were to prevail upon you to resume the subject which really appears inexhaustible.'5

The founding of the Cambridge Mathematical Journal in 1837 played a critical part in stimulating mathematical activity in Cambridge. It gave a voice to many unpublished mathematicians and publicly placed an importance on mathematical

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Figure 2. 2 Stone Buildings, Lincoln's Inn (photograph taken c. 1980).

investigation. Through his refereeing duties, it brought Cayley into contact with a wide circle of British mathematicians. But his ambition went beyond a Fenland, or even a British reputation. He was an entrepreneurial mathematician and, from these early days, the only one to publish consistently in the prestigious Journal de Mathematiques Pures et Appliquees founded by Joseph Liouville and the Journal fir die reine und angewandte Mathematik founded by A.L. Crelle. His first two papers published by Crelle, for instance, were on invariant theory, which he then called the theory of 'Hyperdeterminants'. This work encapsulated Cayley's two great methods of invariant theory, methods which formed the core of his own work and which anticipated the powerful symbolic method of the German mathematicians of the 1860s.

The alertness of the young British mathematicians of the 1840s to Continental advances owed much to the Cambridge Analytical Society, active in the early part of the century. The pioneering attitudes of George Peacock, John Herschel and Charles Babbage inspired succeeding generations. The immediate beneficiaries included Augustus de Morgan, George Biddell Airy and William Whewell, all of whom became eminent Victorians who combined wide scientific interests with mathematical expertise. But it was Cayley in the generation following these who became the specialized mathematician and perhaps gained most. He was cast in a different mould. Whereas Peacock, Herschel and Babbage received Continental mathematics, Cayley stemmed the one-way flow and took his mathematics to the Continent.

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In 1845, a year which could be regarded as Cayley's youthful annus mirabilis, he published 13 papers on a wide range of subjects. Cayley recognized promising ideas quickly and it was typical that he should be the first to write a paper on quaterions after their discovery by Sir William Rowan Hamilton in October 1843. Invariant theory and quaterions became the two most intensively studied algebraic subjects in Britain during the second half of the century. Cayley made important contributions to each and, perhaps uniquely, because of his breadth of mathematical knowledge, was especially alive to the connection between these two vigorous branches of mathematics. It is Cayley's ability to perceive links that is especially noticeable.

LINCOLN'S INN

In April 1846 Cayley, the young man of 24 years, was admitted to Lincoln's Inn to read for the Bar. His father, Henry Cayley, had prospered in business and, in his late seventies, was about to take a less active role in the London commercial world. There was precedence in the family for a Law career as Henry's own grandfather, Cornelius Cayley (1692-1779), had been a prominent member of the Bar and Recorder of Kingston-upon-Hull in Yorkshire.

Cayley became a pupil of the celebrated conveyancing counsel J.H. Christie, a widely respected Real Property lawyer and a man of literary accomplishment. Christie had put a notorious youth behind him. In February 1821 he had acquired the distinction of being one of the last men to fight a duel by pistols on English soil.16 As a man of the world, Christie could hardly have been overly impressed by the shy youth arriving for interview at Lincoln's Inn:

Mr. Cayley arrived at Stone Buildings, sent in his card, was admitted and asked to be taken as a pupil. Christie inquired whether he had any introduction; the reply was, No. Had he been at a University? Yes. Christie, who seldom had a vacant chair in his pupil room, used to describe himself as not having been favourably inclined towards this monosyllabic applicant. However, he had been at the University and might be worth while to inquire further. Christie did so, and by successive and separate questions elicited the information that Cayley's University was Cambridge; his college Trinity; that he had taken a degree; in honours; in mathematical honours; that he had been a Wrangler; that he had been Senior Wrangler."7

For Christie this was sufficient. The halo of the 'Senior Wrangler' was a prized asset in the barristers' chambers which ringed Lincoln's Inn Fields. In entering Christie's

pupils' room, Cayley became part of the practical legal culture and was among men who could sharpen the qualities which the Tripos had inculcated. They could rapidly concentrate their attention on the matter at hand, and, with a clear eye to detail, ensure that a piece of work was gone through once and once only. The legal draft was then 'settled' and the next document considered. These lessons on efficiency were well learnt by the callow youth who sat in the pupils' room of their mentor. In Cayley's case, they carried over to his mathematical life.

Fortunately for Cayley, barristers who engaged with the drafting of legal documents had no need to be flamboyant or effective public speakers. Remarking on Cayley's

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retiring nature, a fellow pupil recollected: 'He had -

known; jokes, and the badinage of the pupil-room, seemed to be delightful novelties to him, and his face beamed with amusement as he listened to them without taking much part in the conversation, being content to devote his time 4i assiduously to work which I suspect was not r

altogether congenial to his taste."' The immature face he presented to the legal world might well be contrasted with his assuredness in his dealings with the mathematical world. This unimposing legal novice sitting in pupil-rooms off Chancery Lane drew inspiration directly from such 18th century masters as Laplace, Cramer and :---- Vandermonde, and confidently compared his own published work with such established senior mathematicians as A.L. Cauchy, Otto Hesse and * i. Sir W.R. Hamilton.

Around Lincoln's Inn, the young lawyer- Figure 3. Arthur Cayley, aged mathematicians educated in London and in approximately 30 (held at St John's Oxford and Cambridge were in frequent contact College, Cambridge). with each other. Sylvester was one, but others included Charles Hargreave, who also worked for Christie, and who won a Royal Society Gold Medal in 1844 for his work on differential equations. Hugh Blackburn, who became professor of mathematics at Glasgow, wrote to Thomson about a meeting with Cayley around the Inns of Court:

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 st[raight] cylinder and sphere of double its radius, but though we have got a tolerable notion of it, it is rather difficult to make it of paper as it runs through itself in 6 curves.19

Also in the group were men like Archibald Smith, founder of the Cambridge Mathematical Journal, Benjamin Gray, James Cockle and Robert Moon. They had all been to Cambridge and their associations and allegiances followed them to the Inns of Court.20

Cayley maintained close contact with William Thomson. Though Thomson was three years behind Cayley at Cambridge they had been on friendly terms, and it was Cayley who introduced him to the French mathematicians when Thomson visited Paris after graduation in 1845. In June 1847, Thomson, then editor of the revamped Cambridge and Dublin Mathematical Journal, and recently appointed professor of natural philosophy at Glasgow, was planning a Swiss mountaineering expedition. After his own sojourn in Switzerland, Cayley was able to offer useful advice:

If you want to immortalise yourself, go over the Weissenthor-or you may try my more

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Tony Crilly

modest & unpretending scheme (which turned out a failure however [on his visit of 1843] thanks to a soft morning) of getting up to the top of it on the Zermatt side and returning. But I hope I shall be able to talk all these routes over with you, and give you my best advice upon them.-you know a story of an Irish boatman wishing to show his respect to somebody, for the love he bore the family- 'I'll take you nearer the falls-than any one ever went before' -and so I can promise to recommend you all the sois disant (or hurleur disant) dangerous and difficult passes. Heigh ho. I wish I was going too; I should like to finish off with the Col de Erin, and the Col de Geant for instance.2'

Instead, Cayley planned a tour in the highlands of Scotland, playfully suggesting to Thomson: 'it is very unpatriotic of you not to come there too'. On his return to

Chancery Lane after a five-week trek in the heart of north-west Scotland, Cayley recounted his extensive figure-of-eight route with its crossover at Inverness and its northernmost point at Ullapool. The tour began and ended with Loch Lomond:

Considering LochGoilhead as my starting point I went by Inverary, Glen Etive &c. to Fort William-then a long cut across the moor of Rannoch & by Loch Ericht & across country to Foyers & then to Beauly, Ullapool, Loch Maree-the isle of Skye (charming weather it was there, a perfect hurricane of rain & wind. [...] From Skye thro' Glen Affaric to Inverness, thence to Aviemore, & across to Braemar.22

After meeting 'horizontal rain' he progressed to Aberfeldy, which he had visited with the reading party in 1842. Cayley's restraint of later years is absent and he uses his letters to convey more than mathematics. The solitary mathematical quest is apparent:

There is a good deal of pleasure in meeting with some human beings one knows, I must confess travelling by one's self is occasionally a little dreary-& so it made an agreeable episode, coming to a place one knew & where one was known. I walked thro' Glen Lyon to Tyndrum-and then, for I hate spinning out a tour when one gets out of the country one likes, in short not finishing when one has done, I came as fast as I could home.2

In marching through open country mathematics was always present in his thoughts but underplayed. Cayley could not resist the urge to tease: 'I have been guiltless of

any Mathematics all the time except a note for Crelle which is subscribed "Loch Rannoch" & which will puzzle his geography a little I expect, & that was something I had in my head before I started, so I really was not visited with an idea the whole time.24

Sylvester had begun his law pupillage in the Autumn of 1846, several months after

Cayley. Mathematically, it was a desultory period for him, but on 24 November 1847, he sat down to write his first letter to Cayley: 'It occurs to me that this investigation [on the theory of numbers] is likely to prove very congenial to your present line of thought and I make you a Cadeau of the subject not doubting that it will turn to good account in your able hands.'25 The 'cadeau' proved to be an isolated gift and for much of his pupillage, Sylvester was mathematically silent.

Cayley remained close to Boole. Boole regarded his young follower highly and recognized in him a greater enthusiasm for the techniques of mathematics than he himself possessed. To Boole, Cayley was something of a phenomenon 'that most indefatigable and I may add most favoured wooer of the Nymph Mathesis'.26 In 1847, Boole published The Mathematical Analysis of Logic and Cayley was an ardent

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Figure 4. Arthur Cayley's certificate of election to Royal Society. Certificate No. The Royal Society).

0 (held at

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Tony Crilly

reader. Boole wanted his friend's opinion on account of its critical thoroughness: 'I

hope now you have set to work to examine my principles you will not stop short but prove them to the bottom. I do not fear the result', Boole wrote, 'I had rather have one such reader as you than a thousand who take everything for granted.'27

Cayley had already visited Paris and Berlin where he made the acquaintance of several continental mathematicians. In his efforts to make himself known to a wide circle of mathematicians he was unconstrained by any snobbery which might derive from social class or university pedigree. To Cayley, they were brother workers in a great mathematical endeavour. He opened a correspondence with the solitary mathematician Thomas Penyngton Kirkman (1806-1895). Kirkman was then 42- years-old and from a completely separate social milieu-the son of a cotton dealer from Bolton, Lancashire. Kirkman had been successful at Bolton Grammar school and after nine years working for his father became a student at Trinity College, Dublin, where he obtained a degree in 1833. After six years as curate at Croft in Lancashire he was made the rector in 1845.

Cayley became aware of Kirkman's existence in 1847, when he acted as a referee for a paper Kirkman submitted to the Cambridge and Dublin Mathematical Journal on a problem in combinatorial mathematics, to calculate the greatest number of triads that can be formed with x symbols, so that no duad shall be twice employed. Cayley judged that 'Kirkman's paper is decidedly interesting and his main result a very elegant one'.28 He later wrote to Kirkman about the new algebras which were then being enthusiastically explored by British mathematicians. Kirkman introduced the

topic of 'pluquaterions' to the readers of the Philosophical Magazine, which 'is the fruit of my meditations on Professor Sir W.R. Hamilton's elegant theory of quaternions, and on a pregnant hint kindly communicated to me, without proof, by Arthur Cayley, Esq., Fellow of Trinity College, Cambridge'.29

In June 1848, Cayley travelled to Dublin where he attended Hamilton's lectures on quaternions, and heard the material which would form the basis of Hamilton's Lectures on Quaternions (1853).30 Hamilton publicly praised Cayley's work and acknowledged him as the first mathematician to publish on the subject of quaternions - after himself. Whereas Hamilton regarded quaternions as 'universal' and thereby a natural tool for studying geometry, Cayley admired quaternions as a beautiful mathematical theory, but he never wavered in his belief that geometry should be studied using coordinates. Hamilton, as discoverer, was champion of the quaternion camp but even then, Cayley, as a leading algebraic geometer, naturally favoured coordinates, as he used to say, the 'wonderful creation of Descartes'. While maintaining this opposing position, Cayley always remained on the best terms with Hamilton.31 Their mathematical relationship was actually asymmetrical for though Cayley knew Hamilton's work, Hamilton's preoccupation with quaterions left him relatively ignorant of Cayley's researches on algebra and algebraic geometry.

While in Dublin, Cayley met the twin brothers Michael and William Roberts, John Jellett and Richard Townsend. They were all devotees of geometry, while Michael Roberts also acquired a research interest in invariant theory. Cayley generalized one of Jellett's results on integration and added to William Roberts's work

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on the transformation of curves, results which he published shortly afterwards.32 Most importantly, Cayley met George Salmon (1819-1904). He was two years older

than Cayley, and in 1848 was appointed as Donnegal lecturer in mathematics at Trinity College, Dublin. Cayley was struck by Salmon's ability when he refereed the latter's 'On the degree of a surface reciprocal to a given one' for the Cambridge and Dublin Mathematical Journal. In dealing with one of Cayley's favourite geometrical topics, the paper provoked Cayley's admiration: 'I have been very much interested with it & have learnt a good deal from it' .33 Salmon's close personal friendship with Cayley endured. They worked together and Cayley contributed whole chapters to Salmon's textbooks. With Sylvester there existed a competitive element, but with Salmon cooperation was the guiding spirit.

BARRISTER-AT-LAW

Cayley received the call to the Bar on 3 May 1849, with his father acting as his sponsor. He left behind the years of apprenticeship, and migrated from his rooms in Chancery Lane to the inner quadrangle of Lincoln's Inn, which accommodated the chambers of his sole conduit for legal work, Mr Christie. Number 2, Stone Buildings would become his mark in the years to follow.34 From this address he began to think in terms of writing not only single papers but series of papers united under a single theme. They would constitute measured responses in distinction to previous 'bulletins from the front'. They would consolidate his youthful gains and break new ground and were in keeping with a man beginning his mature years.

The Inns of Court in 19th-century London provided a convenient shelter for scholars quietly pursuing erudite scientific, literary and bibliographic research. A man like Trollope's Mr Wharton Q.C., an habitue of Stone Buildings, could reflect on hours spent as he wanted 'in the centre of the metropolis, but in the perfect quiet as far as the outside world was concerned'.35 Increasingly this would be more difficult for Cayley to maintain. 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 1852.36

While the legal profession would provide some stability, Cayley had a career decision to make. Would it be law or mathematics? Would he want a regulated well- paid existence coupled with progression from barrister to judge and beyond? It was a choice between the path taken by James Cockle, who progressed to become Chief Justice for Queensland, and the academic route chosen by Hugh Blackburn. Perhaps Cayley realized that the ambitious lawyer's path inevitably meant little time for mathematics. If he accepted the prospect of 'legal diversions' he might embark on a career like Cockle's and the life of a part-time mathematician. This, Cayley could not

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endure. He coasted in the legal profession and while he continually sought the right niche in Academe, he had to wait until 1863 when he was elected to the Sadleirian Chair of Pure Mathematics at Cambridge.37

At the beginning of the 1850s, his link with Sylvester, which had begun hesitantly in 1847, began to flower. In December 1851, Cayley made a new discovery which appeared to hold promise for invariant theory, a subject which had languished after its bright beginning in 1844. In a letter to Sylvester he announced his discovery and accompanied it with an unadorned prediction: 'This will constitute the foundation of a new theory of Invariants'. It was certainly important, though not enough to merit a 'eureka' from Cayley.38

THE ROYAL SOCIETY

In April 1852, Cayley submitted his first paper to the Royal Society, 'Analytical researches connected with Steiner's extension of Malfatti's problem', a geometrical problem of describing three circles, each of them touching the two others and the two sides of a given triangle. It was a problem which had interested him previously.39 As a referee, Peacock harboured misgivings about the appearance of pure mathematics in the Philosophical Transactions, but he was prepared to make an exception for Cayley: 'In the case of Cayley's papers his character as an analyst would exempt it from the application of this principle [of excluding pure mathematics], more especially as it is the first which he has presented to the Society' .0 At the Royal Society Council meeting two weeks later, Cayley was included in the 15 candidates proposed for election, along with the physicist John Tyndall and the Oxford mathematician Bartholomew Price.4' Cayley would be a welcome acquisition, so necessary was it for the Royal Society to elect active scientists in the first years following reform of its constitution, which took place in 1847.

Cayley's candidate's certificate is completed in the unmistakable hand of Sylvester, his principal proposer. For the answer to the question 'Discoverer of ?', Sylvester simply entered, 'Hyperdeterminants', and to the question 'Eminent as a ?', he added 'Geometer and Analyst'. Of the 21 other supporters, many were people who had played a part in his intellectual life: close friends from Cambridge, William Thomson, and the senior wranglers who flanked each side of his own year, George Stokes and John Couch Adams; his contacts from University College, London, Charles Hargreave and John Graves; and from King's College, London, Charles Wheatstone, who had been one of his school teachers. Signatories from Cambridge included his mathematical coach William Hopkins and the formidable Master of Trinity, William Whewell, who regarded Cayley as one of the finest mathematicians Trinity had ever produced. At their Council meeting on the 3 June 1852, Cayley was elected at the first ballot.

On election, the first paper read to the Royal Society by Cayley was 'Introductory memoir on quantics [algebraic forms]'. He signalled his intention of formulating invariant theory afresh and from 1854 to 1878 he published his path-breaking

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'memoirs on quantics' in which he remodelled and extended his earlier work. A few days after his election, Cayley was in Sylvester's rooms in Lincoln's Inn Fields. In the outer office Thomson was writing a letter to Stokes. He did not share Cayley's enthusiasm for pure algebra and actually thought it wasteful of Cayley's mathematical talent which he so admired. For Stokes's amusement, he added in a postscript that his letter 'was written in the dark, with Cayley and Sylvester talking about invariants all the time'.42 The incessant talk in the inner office was concerned with the establishment of mathematical priority of Cayley's discovery of the new foundation for invariant theory. After his election, Cayley embarked on the most productive part of his mathematical career. He maintained the level of production in Continental journals, but his production in home-based journals increased and he began to publish extensive memoirs in the Philosophical Transactions.

After one year's membership of the Royal Society, Cayley was in the running for the award of the Society's prestigious Royal Medal. On 2 June 1853, James Booth, who worked on analytical geometry and elliptic functions, proposed him for the award and he was seconded by William Hopkins. Following his paper on Malfatti's problem, published in the Philosophical Transactions, Cayley had submitted nothing further to the journal, but the proposal for a Royal Medal was in recognition of previous work. At the same meeting, Charles Darwin and the chemist August Wilhelm von Hofmann were also proposed. Ster opposition and more candidates were to follow. Edward Frankland, John Tyndall, the botanist John Lindley and Robert Bunsen were added to the list and, surprisingly, Booth also proposed Sylvester. Sylvester had been a Fellow for 14 years as against Cayley's one year, and Booth's move was perhaps calculated to smooth ruffled feathers. In the event, Charles Darwin and John Tyndall were recommended to the Queen, but to be proposed so soon after membership was significant recognition.43

For Cayley's position as a London mathematician at the centre of the Cambridge academic network in the 1850s, membership of the Royal Society was pivotal. He joined the Royal Astronomical Society, but there were no specialist mathematical societies in the London of the 1850s. Activities were informally coordinated around the Inns of Court and the Royal Society. In the early 1850s, the Society's meeting rooms in Somerset House were only a five-minute walk from Cayley's chambers in Lincoln's Inn-along the Strand, then one of the busiest thoroughfares of Victorian London. He was on familiar territory. Between the twin centres of his scientific and legal activity lay King's College, which he had attended as a school pupil. He adroitly combined his legal profession with mathematical research and as the decade progressed, he was recruited to serve on various Royal Society committees. As a new member Cayley was formally a 'man of science', and, on his way to becoming the representative pure mathematician of the Victorian Age.

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ACKNOWLEDGEMENTS

I thank the following for their generous help and for allowing me to quote from the manuscript collections under their care: The Royal Society of London, the Cambridge University Library, the Master and Fellows of Trinity College, Cambridge, and the Master and Fellows of St John's College, Cambridge. I am grateful to the British Academy for a study grant which has allowed me to carry out research in the history of mathematics.

NOTES

1 With a few exceptions, Cayley's mathematical works are contained in The collected mathematical papers of Arthur Cayley (eds A. Cayley and A.R. Forsyth), 13 vols +

supplement (Cambridge University Press, 1889-98). This work is abbreviated here to Coll. Math. Papers.

2 A.R. Forsyth, Arthur Cayley, 'Obituary Notices' in Proc. R. Soc. Lond. 58, 1-43, or Coll. Math. Papers 8, ix-xliv. J.D. North, Arthur Cayley, Dict. Sci. Biog. 3, 162-170 (1971).

3 I.M. James, 'James Joseph Sylvester, F.R.S. (1814-1897)', Notes Rec. R. Soc. Lond. 51, 247-261 (1997). Karen Hunger Parshall, Selected correspondence ofJ.J. Sylvester (Oxford University Press, 1998, in the press).

4 This remark is based on the cache of letters written between Cayley and Sylvester and held at St John's College, Cambridge.

5 Charles H. Gibbs-Smith, 'Sir George Cayley "Father of Aerial Navigation" (1773-1857)', Notes Rec. R. Soc. Lond. 17, 36-56 (1962).

6 Arthur Cayley's brothers and sisters were Sophia (1816-1889), William Henry (1818-1819), Charles Bagot (1823-1883) and Henrietta-Caroline (1828-1886). C.B. Cayley became an associate of the Pre-Raphaelite Brotherhood and a respected translator of Dante and Homer. He was a frequent visitor to the Rossetti family home and was romantically linked with Christina in the 1860s.

7 Thomas Mackay (ed.), The reminiscences ofAlbert Pell (London: John Murray, 1908), pp. 71-72.

8 Both were members of the Cambridge Camden Society which became engaged in the controversial restoration of the Round Church in Cambridge in the 1840s. The church had been left in a perilous position after vaulting collapsed in August, 1841.

9 Francis Galton, Memories of my life (London, Methuen, 1908), p. 72. 10 Francis Galton to Samuel Tertius Galton, 19 June 1842 in Karl Pearson, The life of Francis

Galton, 4 vols (Cambridge, 1914), vol. 1, p. 168. 11 The Guardian, 6 February, 201 (1895). 12 Nevil Maskelyne, F.R.S (1732-1811), who became Astronomer Royal in 1764, was a fellow

of Trinity College. In 1774 his plumb line experiments on Schiehallion, the 'hill of the Caledonians', gained him attention in the scientific world. From his determination of the gravitational constant G, he calculated the density of the earth.

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13 Edmund Venables to G.G. Stokes, 7 July 1842, CUL Manuscripts, Add 7656.V14. Cayley's adoption of the symbol A and the 'topsy-turvy' V (was no doubt influenced by Laplace and by the French school involved with the calculus of operations (Maria Panteki, Relationships between algebra, differential equations and logic in England 1800-1860, unpublished Ph.D. thesis, Council for National Academic Awards, 1991, pp. 24-46). In his early papers, Cayley used V in several different ways and he also experimented with versions of 'topsy-turvy' Js and Fs. Though he used subscripts in his early papers, he grew to dislike them and the purely alphabetic notation he used in algebra and geometry placed an unnecessary limitation on the precision of his mathematical expressions. His inability to deal effectively with the n- dimensional case is noticeable.

14 David B. Wilson, Kelvin and Stokes, a comparative study in Victorian physics (Bristol, Adam Hilger, 1987), pp. 30-35.

15 Cayley to Boole, 13 June 1844, Trin.Coll.Camb. R.2.88[1]. 16 Jonathan Henry Christie (c. 1793-1876). The moonlight duel involving the London literati

took place at Chalk Farm where his opponent was John Scott, editor of the London Magazine. Scott was mortally wounded and Christie, who fought for the honour of his close friend John Gibson Lockhart, was indicted for murder but acquitted. John Savill Vaizey, The Institute- a club of conveyancing counsel. Memoirs offormer members, (Lincoln's Inn, 1895-1907), pp. 74-83. John G. Millingen, The history of duelling, 2 vols. (London, 1841), vol. 2, pp. 244-252.

17 Vaizey, op. cit. (note 16), p. 78. 18 Forsyth, op. cit. (note 1), p. xiv. 19 Hugh Blackburn to William Thomson, 11 March 1849, CUL Manuscripts, Add. 7342.B 102. 20 Charles Hargreave (1820-1866), Archibald Smith (1813-1872), Benjamin Gray (1820-1886),

James Cockle (1819-1895), Robert Moon (1817-1889) and Hugh Blackburn (1823-1909). 21 Cayley to William Thomson, 5 June 1847, CUL Manuscripts, Add 7342.C50. 22 Cayley to William Thomson, 17 August 1847, CUL Manuscripts, Add 7342.C55. 23 Cayley to William Thomson, 17 August 1847, Kelvin Collection, CUL, Add 7342.C55. 24 Possibly, 'Note sur les fonctions elliptiques', Journ. f die r und a. Math. 37, 58-60 (1848),

or Coll. Math. Papers 1,402-404. 25 Sylvester to Cayley, 24 November 1847, St John's College, Cambridge. 26 Boole to William Thomson, [15?] September 1846, CUL Manuscripts, Add MS7432.B 155.

Boole letters in Kelvin Papers in CUL, Add MS7432. Letter B155 Arthur Cayley, 'On a multiple integral connected with the theory of attractions', Camb. Dub. Math. Journ. 2, 219-223 (1847), or Coll. Math. Papers 1,285-289.

27 Boole to Cayley, 8 December 1847, Royal Society of London, Mss 782, E-13. 28 Cayley was a discriminating referee, and in the same report rejected a paper on geometry

which treated the subject in 'that way without any reference to general geometrical theories or without any attempt to make a "Zusammengesetzung" of the whole mass of theorems one obtains, it is very uninteresting work'. Cayley to William Thomson, 8 February 1847, CUL Manuscripts, Add 7342.C45.

29 Cayley to Thomas P. Kirkman [1848] in Thomas P. Kirkman, 'On pluquaternions, and homoid products of sums of n squares', Phil. Mag. 33, 447-459,494-509 (1848).

30 The lectures were held on the 21, 23, 26 and 28 June 1848. England's great Chartist demonstration took place on 10 April 1848.

31 'On Professor MacCullagh's theorem of the polar plane', Proc. R. Irish Acad. 6, 481-491 (1858), or Coll. Math. Papers 4, 12-20. Cayley to Sir W.R. Hamilton, 3 April 1865, in Robert Percival Graves, Life of Sir William Rowan Hamilton, 3 vols (London, Longmans, Green; Dublin, Hodges, Figgis, 1882-89), vol. 3, p.199.

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32 Michael Roberts (1817-1882), William Roberts (1817-1883), John Hewitt Jellett (1817-1888), Richard Townsend (1821-1884). 'Sur la generalisation d'une theoreme de M. Jellett qui se rapporte aux attractions', Journal de Mathematiques Pures et Applique'es 13, 264-268 (1848), or Coll. Math. Papers 1,388-391. 'Sur quelques transmutations des lignes courbes', Journal de Mathematiques Pures et Appliquees 14, 40-46 (1849), or Coll. Math. Papers 1,471 f.n.

33 Cayley to William Thomson, [4] December 1846, CUL Manuscripts, Add 7342.C43. 34 Stone Buildings in Lincoln's Inn was built during the years 1775-85. It was severely hit by

enemy action during the Second World War but has been restored to its original style. 35 Anthony Trollope, The Prime Minister (London, Penguin, 1994. First edn 1875), p. 28. 36 The Cambridge mathematician N.M. Ferrers (1829-1903) and Scottish mathematician/

physicist P.G. Tait (1831-1901) were the senior wranglers in 1851 and 1852. Both were in Cayley's circle of academic friends.

37 This chair replaced the university lectureships which had been funded by the Sadleirian endowment, as set out in Lady Sadleir's will of 25 September 1707.

38 Cayley to Sylvester, 5 December 1851, St John's College, Cambridge. 39 'On a system of equations connected with Malfatti's problem, and on another algebraical

system', Camb. Dub. Math. Journ. 4,270-275 (1849), or Coll. Math. Papers 1,465-470. 40 Ref. Rep., 21 June 1852, Royal Society of London, RR.2.41. 41 John Tyndall (1820-1893) and the Oxford mathematician Bartholomew Price (1818-1898). 42 W. Thomson to G.G. Stokes, 8 June 1852, in The correspondence between Sir George

Gabriel Stokes and Sir W Thomson, Baron Kelvin of Largs, 2 vols. (ed. D.B. Wilson), vol. 1, p.131 (Cambridge University Press, 1990).

43 He was proposed for a Royal Medal again in 1854, 1858 and awarded the medal in 1859. Cayley was awarded the Copley Medal in 1882.

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The Young Arthur Cayley