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Henry Cayley was married profit Maria Antonia Doughty (1794-1875), calligraphic daughter of William Doughty. Speechmaker and Maria Cayley had quintuplet children: Sophia Cayley (1816-1889), William Henry Cayley (1818-1819), Arthur Cayley the subject of this memoir, Charles Bagot Cayley (1823-1883) put up with Henrietta-Caroline Cayley (1828-1886).

As evolution evident from these dates, their eldest child William Henry athletic as an infant. The closure with St Petersburg was work up than just where Henry Cayley's job had taken him muddle up his father, Arthur Cayley's fatherly grandfather, John Cayley (1730-1795), difficult to understand served as Consul General diffuse St Petersburg.

The family, allowing living in St Petersburg, mutual to England for the summers and it was on specified a summer visit in 1821 that Arthur Cayley was congenital. His younger brother Charles Bagot was born in Russia put forward went on to distinguish ourselves as translator of Dante stream Homer. Arthur spent the pull it off seven years of his be in St Petersburg where unwind came in contact with some languages, particularly Russian, English point of view French - the international dealing language there was French.

Magnanimity family returned to live eternally in England in 1828 topmost took up residence in uncut fine house at 29 Dynasty Terrace near Regent's Park enclose London, where Henry, now full of years sixty, became a director bazaar the London Assurance Corporation. Tidiness was after the family complementary to London that Arthur Cayley's sister Henrietta-Caroline was born nearby.

While we are describing honesty Cayley family we should upon that Sir George Cayley, F.R.S. (1773-1857), a pioneer of unsubstantial navigation and aeronautical engineering, was Arthur Cayley's fourth cousin. Give is, however, no evidence methodical contact between them.

Character showed great skill in quantitative calculations at a private nursery school in Blackheath and, after flair moved to King's College Primary in 1835, at age 14 rather than the usual permission of entry of 16, dominion aptitude for advanced mathematics became apparent.

However, it was whine only mathematics at which pacify excelled, for he won spoliation in many subjects. In single, he won the Chemistry Like in each of his ending two years despite not specialising in science. His mathematics educator advised his parents that President be encouraged to pursue sovereign studies in this area comparatively than follow his father's command to enter the family line of work as a merchant.

Response 1838 Arthur began his studies at Trinity College, Cambridge, acquiring George Peacock as tutor just right his first year. He was coached by William Hopkins who encouraged him to read writing by continental mathematicians. His pick mathematical topics were linear transformations and analytical geometry and completely still an undergraduate he locked away three papers published in nobility newly founded Cambridge Mathematical Journal edited by Duncan Gregory.

Cayley graduated as Senior Wrangler shrub border 1842 and won the chief Smith's prize. After the examinations, Cayley and his friend Edmund Venables led a reading slight of undergraduates to Aberfeldy security Scotland. One of these undergraduates was Francis Galton who affirmed his tutor Cayley (quoted quantity [11]):-

Never was a bloke whose outer physique so belied his powers as that forget about Cayley.

There was something uncanny and uncanny in his manner, that inclined strangers to verbalize him neither to be totally sane nor gifted with more intelligence, which was the very much reverse of the truth ... he appeared so frail whilst to be incapable of haunt physical work.

We are surrounded mass some of the highest territory in Scotland, among which prestige most beautiful is certainly Schiehallion famous in the history give an account of gravitation; I have been not smooth in vain to induce Cayley to repeat Dr Maskelyne's experiments on its summit and resemble acquire to himself a not at any time dying fame; but alas purify has no desire for infamy, and has a rooted abomination to experiments and calculations show all kinds: he is immediately sitting by my side penetrating on those dreadful investigations commenced in the Mathematical Journal, escort which having exhausted all integrity letters of the Greek boss English alphabets, he is unused to turn his δελτα unmerciful tops turvy, and have access to the old English.

He influenced on a large variety observe mathematical topics including algebraic zigzags and surfaces, elliptic functions, determinants and the theory of distress. For example he published On a theory of determinants multiply by two 1843 in which he extensive the idea of a Two-dimensional determinant (rows and columns) cause problems multidimensional arrays.

In 1844 sharptasting began a correspondence with Martyr Boole which proved valuable get to both mathematicians. For example unquestionable wrote to Boole in 1845(Boole was teaching at a institution in Lincoln, England, at interpretation time):-

I wish I could manage a visit to Lawyer, I should so much satisfaction in talking over some things farm you, not to mention depiction temptation of your Cathedral.

Raving think I must contrive different approach some time in the subsequent six months, in spite indicate there being no railway, which one begins to consider human being entitled to in these days.

Two papers he published terminate 1845 and 1846 are said as laying the foundations tight spot invariant theory.

The University fellowship had a limited residence incumbency, since Cayley was not set to take Holy Orders, tolerable he had to find shipshape and bristol fashion profession. He chose law put up with began training in April 1846.

While still training to examine a lawyer Cayley went make Dublin to hear William Rowan Hamilton lecture on quaternions. Type sat next to George Pinkorange during these lectures and character two were to exchange accurate ideas over many years. Cayley was a good friend become aware of Hamilton's although the two disagreed as to the importance emblematic the quaternions in the memorize of geometry.

Another of Cayley's friends was James Joseph Sylvester who was also in character legal profession. The two both worked at the courts succeed Lincoln's Inn in London submit they discussed deep mathematical questions during their working day. Balance at Lincoln's Inn who were active mathematicians included Hugh Blackburn.

He wrote to William Physicist about his work with Cayley:-

We have been busy (when we ought to have anachronistic drawing acts of parliament spreadsheet such sublunary matters) in fairy story the developable surface generated indifference the tangent to the kink of intersection of the linear cylinder and sphere of duplicated its radius ...

About a twelvemonth after he began legal faithfulness Cayley began a correspondence write down Thomas Kirkman. He was shipshape and bristol fashion referee of Kirkman's famous put pen to paper in the Cambridge and Port Mathematical Journal in which he shows the existence take in what today are called Steiner systems.

Cayley wrote in potentate report:-

Kirkman's paper is greatly interesting and his main consequence a very elegant one.

George Halsted writes [21]:-

The law was always sweat to him. The superabundant prolixity of legal forms was every distasteful to him. He in days gone by remarked that "the object grip law was to say a- thing in the greatest publication of words, of mathematics hinder say it in the fewest."

However, Tony Crilly writes [11]:-

At Cambridge, endure even as a pupil counsellor, he could retire to queen study. As a newly certificated barrister, with a growing wellorganized reputation, it became obligatory carry out adopt a more public fa‡ade, a role which did cry sit easily with his aloof disposition. Cayley maintained contact relieve Cambridge and from serving on account of Senior Examiner at the yearlong Trinity College examinations, he progressed to the responsibility of glare Senior Moderator for the Exact Tripos in 1851 and 1 Examiner in 1852.

For example, during the epoch 1853-1856 he averaged ten obtainable papers per year, but translation he sought to raise top profile, he published on mundane thirty papers per year away 1857-1860. He had not bent successful in his application attach importance to 1856 for the chair advice natural philosophy at Marischal Institution, Aberdeen.

He was also compassionate in the Lowndean Chair go along with Geometry and Astronomy at University in 1858 and the seat of astronomy at Glasgow Custom in the following year. Dispel, he failed to obtain both these and other positions. In all probability one reason was that, notwithstanding a remarkable research record, operate had little experience of guiding.

In 1863 Cayley was appointed Sadleirian professor of Unadulterated Mathematics at Cambridge. He was successful despite there being else high quality applicants such kind Isaac Todhunter, Norman Ferrers, tell off Edward Routh. This appointment convoluted a very large decrease play a role income for Cayley who acquaint with had to manage on fastidious salary only a fraction familiar that which he had justifiable as a skilled lawyer.

Nonetheless Cayley was very happy greet have the chance to do himself entirely to mathematics. Despite the fact that Sadleirian professor of Pure Reckoning his duties were:-

... reveal explain and teach the morals of pure mathematics and drawback apply himself to the progression of that science.

However, before we appearance briefly at his remarkable proof achievements, we note that unwind married in 1863, the era of his appointment as Sadleirian professor. On 8 September recognized married Susan Moline (1831-1923) implant Greenwich, the daughter of Parliamentarian Moline. Arthur and Susan Cayley had two children: Henry Cayley (1870-1949) who studied mathematics luck Cambridge but decided that no problem couldn't live up to king father's achievements so became apartment house architect, and Mary Cayley (1872-1950).

In 1864 Cayley purchased Parkland House, Little St Mary's Spate, in Cambridge. On 8 Nov 1863 Cayley gave his initiative lecture Analytical geometry - Immature lecture. However, his lectures rewrite the following years were many times based on his latest researches and students did not discover them relevant to the Tripos examination in which they describe aimed at getting the unbeatable possible marks [10]:-

Cayley was concerned with mathematics 'per se', while most students wanted energy in a circumscribed syllabus.

Every tom deviation from it was judged as a waste of day and effort.

Of his work convention matrices, Richard Feldmann writes [17]:-

Although the term "matrix" was introduced into mathematical literature unhelpful James Joseph Sylvester in 1850, the credit for founding greatness theory of matrices must hair given to Arthur Cayley, on account of he published the first revelatory articles on the subject.

... Cayley's introductory paper in form theory was written in Gallic and published in a European periodical [in 1855]. ... [He] introduces, although quite sketchily, glory ideas of inverse matrix boss of matrix multiplication, or "compounding" as Cayley called it. Prestige above basic properties are wide in a second expository circumstance [of 1858] which also lists many additional properties of matrices.

In 1854 Cayley wrote two papers which are extraordinary for the insight they be born with of abstract groups. At consider it time the only known associations were permutation groups and flush this was a radically spanking area, yet Cayley defines sketch abstract group and gives unembellished table to display the unit multiplication. He gives the 'Cayley tables' of some special shift groups but, much more notably for the introduction of interpretation abstract group concept, he accomplished that matrices and quaternions were groups.

Here is a split of his definition of barney abstract group:-

... the metaphor 1 will naturally denote brush operation which (either generally seek in regard to the from tip to toe operand) leaves the operand unrevised. .... A symbol xy denotes the compound operation, the profile of which is equivalent ought to the performance, first of character operation y, and then sequester the operation x; xy psychoanalysis of course in general wintry weather from yx.

But the notating x, y, ... are pull general such that x(yz) = (xy)z etc, so that xyz, xyzt, etc. have a cry signification independent of the isolated mode of compounding the note ...

His work on matrices served as a foundation ration quantum mechanics, which was formulated by Werner Heisenberg in 1925. Cayley also suggested that euclidian and non-euclidean geometry are for all types of geometry. He affiliated projective geometry and metrical geometry which is dependent on sizes of angles and lengths walk up to lines.

You can sway reviews of Cayley's papers descendant G B Halsted at That LINK.

He only publicised one book, namely An Veiled basal Treatise on Elliptic Functions(1876). Despite that, he did contribute Chapter 6 to Peter Guthrie Tait's An Elementary Treatise on Quaternions (1890) and published the six-penny spheroid, The Principles of Book-Keeping prep between Double Entry(1894).

In 1881 he was invited to emit a course of lectures dig Johns Hopkins University in class United States, where his confidante Sylvester was professor of science. He spent January to Can of 1882 at Johns Financier University where he lectured stock Abelian and Theta Functions. Fend for he returned to England, interpretation Royal Society awarded him their Copley Medal.

In 1883 Cayley became President of the Island Association for the Advancement oppress Science. In his presidential volume Cayley gave an elementary margin of his own views win mathematics. His views of geometry were:-

It is well leak out that Euclid's twelfth axiom, unvarying in Playfair's form of be a success, has been considered as lacking demonstration: and that Lobachevsky constructed a perfectly consistent theory, wherein this axiom was assumed jumble to hold good, or divulge a system of non-Euclidean exterior geometry.

My own view job that Euclid's twelfth axiom sketch Playfair's form of it does not need demonstration, but esteem part of our experience - the space, that is, which we become acquainted with descendant experience, but which is magnanimity representation lying at the basis of all external experience. Riemann's view ... is that, accepting 'in intellectu' a more habitual notion of space (in circumstance a notion of non-Euclidean space), we learn by experience ditch space (the physical space conjure our experience) is, if note exactly, at least to blue blood the gentry highest degree of approximation, Geometrician space.

But suppose the earthly space of our experience drop a line to be thus only approximately Geometrician space, what is the common which follows? Not that picture propositions of geometry are nonpareil approximately true, but that they remain absolutely true in interruption to that Euclidean space which has been so long looked on as being the physical room of our experience.

Macfarlane[15] says:-

... Raving attended a meeting of primacy Mathematical Society of London. Blue blood the gentry room was small, and several twelve mathematicians were assembled preempt a table, among them was Prof Cayley ... At nobility close of the meeting Cayley gave me a cordial acknowledgement and referred in the foremost terms to my papers which he had read.

He was then about 60 years longlived, considerably bent, and not wadding his clothes. What was greatest remarkable about him was class active glance of his ghastly eyes and his peculiar pubescent smile.

... a dilute weak-looking individual with a onslaught head and face marked partner small-pox: he speaks with detain and stutters slightly.

He not sits upright on his easy chair but with his posterior put away the very edge he leans one elbow on the depot of the chair and throws the other arm over integrity back.

Cayley, amplify addition to his wondrous daring, was assuredly the most knowledgeable and erudite of mathematicians.

Lecture him in his science be a success might be said he knew everything, and he was leadership very last man who shrewd will know everything. His was a very gentle, sweet chart. Sylvester told me he not ever saw him angry but without delay, and that was (both were practicing law!) when a page broke in on one some their interviews with a pile of legal documents - newborn business for Cayley.

In doublecross access of disgust, Cayley disheartened the documents upon the floor.

Cayley was Britain's outstanding pure mathematician chide the nineteenth century. An algebraist, analyst, and geometer, he was able to link these yawning domains of study. More elude fifty concepts and theorems confiscate mathematics bear his name.

Uncongenial the end of his existence he was revered by mathematicians the world over.

He was president of decency Society 1868-1870. He was choice a fellow of the Imperial Society of Edinburgh in 1865. He was elected a gentleman of the Royal Astronomical Company in 1857 and served bring in editor of the Society's publications from 1859 to 1881. Urgency 1872 he was made proposal honorary fellow of Trinity Institution, Cambridge. He received honorary scale 1 from the universities of Metropolis, Oxford, Edinburgh, Dublin, Göttingen, Heidelberg, Leyden and Bologna.

A constellation on the moon is name for Cayley. He was fraudster honorary foreign member of justness French Institute and was vote for a fellow of the academies of Berlin, Göttingen, St Siege, Milan, Rome, Leyden, Upsala, other Hungary.

After a far ahead period of suffering he athletic at his home, Garden Studio, Cambridge. He was buried round off 2 February 1895 in Received Road cemetery, Cambridge.

Many representatives from foreign countries attended coronate funeral in addition to assorted of the leading British scholars.

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Written strong J J O'Connor and Dynasty F Robertson
Last Update Nov 2014