...there is no study in the world which brings into more harmonious action all the faculties of the mind than [mathematics], ... or, like this, seems to raise them, by successive steps of initiation, to higher and higher states of conscious intellectual being....

So long as a man remains a gregarious and sociable being, he cannot cut himself off from the gratification of the instinct of imparting what he is learning, of propagating through others the ideas and impressions seething in his own brain, without stunting and atrophying his moral nature and drying up the surest sources of his future intellectual replenishment.

[on graph theory...]

The theory of ramification is one of pure colligation, for it takes no account of magnitude or position; geometrical lines are used, but these have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation.

Time was when all the parts of the subject were dissevered, when algebra, geometry, and arithmetic either lived apart or kept up cold relations of acquaintance confined to occasional calls upon one another; but that is now at an end; they are drawn together and are constantly becoming more and more intimately related and connected by a thousand fresh ties, and we may confidently look forward to a time when they shall form but one body with one soul.

The world of ideas which it [mathematics] discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connexion of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the surest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.

I know, indeed, and can conceive of no pursuit so antagonistic to the cultivation of the oratorical faculty ... as the study of Mathematics. An eloquent mathematician must, from the nature of things, ever remain as rare a phenomenon as a talking fish, and it is certain that the more anyone gives himself up to the study of oratorical effect the less will he find himself in a fit state to mathematicize.

Mathematics is the music of reason.

May not music be described as the mathematics of the sense, mathematics as music of the reason? The musician feels mathematics, the mathematician thinks music: music the dream, mathematics the working life.

The object of pure physics is the unfolding of the laws of the intelligible world; the object of pure mathematics that of unfolding the laws of human intelligence.

The early study of Euclid made me a hater of geometry.

Aspiring to these wide generalizations, the analysis of quadratic functions soars to a pitch from whence it may look proudly down on the feeble and vain attempts of geometry proper to rise to its level or to emulate it in its flights.

[A quotation by one of Sylvester's students at Johns Hopkins University]

... the substance of his lectures had to consist largely of his own work, and, as a rule, of work hot from the forge. The consequence was that a continuous and systematic presentation of any extensive body of doctrine already completed was not to be expected from him. Any unsolved difficulty, any suggested extension, such would have been passed by with a mention by other lecturers, became inevitably with him the occasion of a digression which was sure to consume many weeks, if indeed it did not take him away from the original object permanently. Nearly all of the important memoirs which he published, while in Baltimore, arose in this way. We who attended his lectures may be said to have seen these memoirs in the making.

Surely with as good reason as had Archimedes to have the cylinder, cone and sphere engraved on his tombstone might our distinguished countrymen [Arthur Cayley and George Salmon] leave testamentary directions for the cubic eikosiheptagram to be engraved on theirs. Spirit of the Universe! wither are we drifting, and when, where, and how is all this to end?

Writing to his old friend Arthur Cayley, Sylvester confessed, "I expect Poincarre [sic] tomorrow and he will have rooms in College. I rather dread the encounter as there is so little in the way of Mathematics upon which I can hope to talk to him!"

All roads are said to lead to Rome, so I find, in my own case at least, that all algebraical inquiries sooner or later end at that Capital of Modern Algebra over whose shining portal is inscribed "Theory of Invariants".

James Joseph Sylvester's father was Abraham Joseph who was a merchant. The first strange fact (which the careful reader may already have noticed!) is that Sylvester's father had the surname Joseph, and not Sylvester. Indeed the subject of this biography grew up with the name James Joseph and it was only shortly before he began his university studies that he decided to add the surname Sylvester. One might ask at this point why he added an extra name. The reason was that his eldest brother decided at this time to emigrate to the United States and he was required to have at least three names before he was allowed to gain residence. We should also mention at this point the fact that Sylvester was born into a Jewish family, and brought up in the Jewish faith, which would lead to difficulties later in his life which we describe below.

Sylvester attended two schools in London, the first one being a boarding school in Highgate which he attended up to 1827, after which he undertook a further eighteen months study at a school in Islington. In 1828, at the age of fourteen, he entered University College London, and began his studies in the first year that the College received students. This was a sensible choice since, unlike some other British universities, it was non-sectarian. It also had the very talented De Morgan as its professor of mathematics. However things did not go well for Sylvester for, when only five months into his studies, he was accused of threatening a fellow student with a knife in the College refectory. Sylvester's family withdrew him from University College immediately after the allegations were made. Realising that, although Sylvester was able to profit from university education, he was not yet mature enough for university life, his family sent him to continue his schooling at the Royal Institution in Liverpool.

On 7 July 1831 Sylvester matriculated as a student at St John's College, Cambridge, although his studies were interrupted when he was forced to take most of the two years 1833-34 and 1834-35 out due to a lengthy illness. After regaining his health he took the mathematical tripos examination in 1837. Two other famous mathematicians took the tripos examination in the same year as Sylvester, namely Duncan Gregory and George Green. Sylvester came second, Green who was 20 years older than the other two came fourth, with Duncan Gregory fifth. (The mathematician who came first, William Griffin, did little work of importance after graduating: this was not at all uncommon a result of the 'speed test' which the tripos was at that time.)

We were careful to say that Sylvester took the tripos examination in 1837 rather than saying he graduated in that year, for he was not able to graduate. At this time it was necessary for a student to sign up to the Thirty-Nine Articles of the Church of England before graduating and Sylvester, being Jewish, naturally refused to take the necessary oath so could not graduate. For the same reason he was not eligible for a Smith's prize nor was he eligible to compete for a Fellowship.

For the three years from 1838 Sylvester held the chair of natural philosophy at the University of London, one of the few places which did not bar him because of his religion. His former teacher De Morgan was one of his colleagues. However Sylvester was a mathematician and it was mathematics, not physics, that he wished to teach. He was a very active researcher and by the time he resigned the chair of natural philosophy in 1841 he had published fifteen papers on fluid dynamics and algebraic equations. Sylvester might have been a professor for three years but he still had no degree for the reason we indicated above. He remedied this in 1841 when he was awarded a B.A. and an M.A. by Trinity College, Dublin. In fact it was legislation which allowed Roman Catholics to take degrees at Trinity College which also meant that Jews could graduate there.

At the age of 27 he applied for, and was appointed to, the chair of mathematics in the University of Virginia in Charlottesville in the United States. His application was strongly supported by De Morgan, John Herschel and Charles Babbage. For example De Morgan wrote:-No person of his years in this country has more reputation than Mr Sylvester as an original mathematician, or bids fairer to extend the exact sciences by his labours. From my own knowledge of what he has done, I can safely say that he is a mathematician of great power, well acquainted with the most modern forms of the science, and very zealous in the persecution of his inquiries.However he resigned after only a few months in post. The following version of events is recorded by Feuer in [11]. A student who had been reading a newspaper in one of Sylvester's lectures insulted him and Sylvester struck him with a sword stick. The student collapsed in shock and Sylvester believed (wrongly) that he had killed him. He fled to New York where one of his elder brothers was living. Although this story is probably accurate in one sense, in another it gives a quite false impression of events. There is no doubt that the students at the University of Virginia were very badly behaved and their behaviour was made worse by drink. It is also likely that they reacted more against Sylvester as a foreigner although there are many records of abusive behaviour before Sylvester arrived. Certainly Sylvester complained to the Faculty on 1 February 1842 about the behaviour of a particular first year student. The abuse suffered by Sylvester from this student got worse after this. Although the student was reprimanded by the university authorities, no sanctions were made against him. Sylvester objected to the lenient attitude of the Faculty on 19 March and resigned three days later.

In New York he began to look for university positions. After two unsuccessful bids for posts at Columbia College and Harvard, he boarded a ship back to England on 20 November 1843. He suffered another upset while in New York. He met a local girl, Miss Marston, and became very fond of her. He proposed marriage but she turned him down on the grounds that he was of the Jewish religion.

On his return to England Sylvester worked as an actuary and was secretary at the Equity Law and Life Assurance Company. However, he also gave mathematics tuition with his pupils including Florence Nightingale. He decided to study law and, by good fortune, Cayley was also studying to become a lawyer. Both met at the courts of Lincoln's Inn in London and they discussed mathematics as they walked around the courts and, although very different in temperament, they became life long friends.

Sylvester tried hard to return to being a professional mathematician and he applied for a lectureship in geometry at Gresham College, London in 1854 but he was not appointed. Another failed application was for the chair in mathematics at the Royal Military Academy at Woolwich, but, after the successful applicant died within a few months of being appointed, Sylvester became professor of mathematics at Woolwich.

Sylvester did important work on matrix theory, a topic in which he became interested during walks with Cayley while they were at the courts of Lincoln's Inn. In 1851 he discovered the discriminant of a cubic equation and first used the name 'discriminant' for such expressions of quadratic equations and those of higher order.

You can see more about the discriminant at THIS LINK.

He published important papers in 1852 and 1853, namelyOn the principle of the calculus of formsandOn the theory of syzygetic relations and two rational integer functions. In particular he used matrix theory to study higher dimensional geometry. He also contributed to the creation of the theory of elementary divisors of lambda matrices.

De Morgan was the first president of the London Mathematical Society. Sylvester became the second president of that Society in 1866. He was the first recipient of the gold medal which the Society awarded to honour De Morgan. He had also been elected to the Paris Academy of Sciences in 1863 and had been a fellow of the Royal Society of London since 1839. Being at a military academy Sylvester was forced to retire at age 55. Parshall writes [21]:-During his last five years there, in fact, Sylvester frequently found himself at odds with the military authorities over his teaching duties and increasingly sensed the diminishment of his mathematical talents.After retiring in 1870 he lived in London, spending most days at the Athenaeum Club (he had been elected a member of the Club in 1856). At first it looked as though he might give up mathematics since he published his only book at this time and it was on poetry. Clearly Sylvester was proud of this work, entitledThe Laws of Verse, since after this he sometimes signed himself "J J Sylvester, author of The Laws of Verse". Parshall writes [21]:-Sylvester's love of poetry and language manifested itself in notable ways even in his mathematical writings. His mastery of French, German, Italian, and Greek was often reflected in the mathematical neologisms - like "meicatecticizant" and "tamisage" - for which he gained a certain notoriety. Moreover, literary illusions, poetic quotations, and unfettered hyperbole spiced his published papers and lectures.For three years Sylvester appears to have done no mathematical research but then Chebyshev visited London and the two discussed mechanical linkages which can be used to draw straight lines. After working on this topic Sylvester lectured on it at an evening lecture entitledOn recent discoveries in mechanical conversion of motionwhich he gave at the Royal Institution. One mathematician in the audience at this lecture was Kempe and he became absorbed by this topic. Kempe and Sylvester worked jointly on linkages and made important discoveries.

In 1877 Sylvester accepted a chair at Johns Hopkins University and he founded in 1878 theAmerican Journal of Mathematics, the first mathematical journal in the United States. In fact the seven years that Sylvester spent at Johns Hopkins saw a resurgence in his mathematical interests. For the first time in his career he was teaching and undertaking research in a proper university environment and, moreover, he was able to give leadership in a way that had not been possible throughout the rest of his career. For the first time he had research students around him whose studies he could supervise in directions which his interests took him. In fact he supervised the doctorates of nine students during his seven years at Johns Hopkins. His research interests reinvigorated, he involved his students in important ideas in the theory of partitions which he undertook during these years. He published this work in theAmerican Journal of Mathematicswhich he had founded. Given that he was so successful at Johns Hopkins, one has to ask why he left when he did. We know of his reasons through a personal letter he wrote to Klein who considered succeeding him but found the salary and conditions insufficient. Sylvester certainly wanted to return to his native land and had some personal reasons to return. However, his main reason was (see [5]):-... because I did not consider that my mathematical rerudition was sufficiently extensive nor the vigour of my mental constitution adequate to keep me abreast of the continually advancing tide of mathematical progress to that extent which ought to be expected from one on whom practically rests the responsibility of directing and moulding the mathematical education of 55 million of one of the most intellectual races of men upon the face of the earth.When Smith died in 1883 Sylvester, although 68 years old at this time, was appointed to the Savilian chair of Geometry at Oxford. However Sylvester only liked to lecture on his own research and this was not well liked at Oxford where students wanted only to do well in examinations. In 1892, at the age of 78, Oxford appointed a deputy professor in his place and Sylvester, by this time partially blind and suffering from loss of memory, returned to London where he spent his last years at the Athenaeum Club.

Macfarlane [17] describes Sylvester in the following way:-Sylvester was fiery and passionate ... Sylvester never wrote a paper without foot-notes, appendices, supplements, and the alterations and corrections in his proofs were such that the printers found their task well-nigh impossible. ... Sylvester satisfied the popular idea of a mathematician as one lost in reflection, and high above mundane affairs. ... Sylvester was an orator, and if not a poet, he at least prided himself on his poetry.One of Sylvester's students at Johns Hopkins University describes his teaching there:-... the substance of his lectures had to consist largely of his own work, and, as a rule, of work hot from the forge. The consequence was that a continuous and systematic presentation of any extensive body of doctrine already completed was not to be expected from him. Any unsolved difficulty, any suggested extension, such would have been passed by with a mention by other lecturers, became inevitably with him the occasion of a digression which was sure to consume many weeks, if indeed it did not take him away from the original object permanently. Nearly all of the important memoirs which he published, while in Baltimore, arose in this way. We who attended his lectures may be said to have seen these memoirs in the making.The following quote, from Thomas Hirst, tells us something about Sylvester's personality:-On Monday having received a letter from Sylvester I went to see him at the Athenaeum Club. ... He was, moreover, excessively friendly, wished we lived together, asked me to go live with him at Woolwich and so forth. In short he was eccentrically affectionate.Sylvester sent the following puzzle to theEducational Times. It tells us of one of his hobbies as well as his interest in puzzles:-I have a large number of stamps to the value of 5d and 17d only. What is the largest denomination which I cannot make up with a combination of these two different values.[The answer is 63d. Can you prove this!]

In the "Fortnightly Review" [1869] we are told that "Mathematics is that study which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation" (Huxley, "Lay sermons, addresses and reviews" (London 1870), p. 185). I think no statement could have been made more opposite to the undoubted facts of the case, that mathematical analysis is constantly invoking the aid of new principles, new ideas, and new methods, not capable of being defined by any form of words, but springing direct from the inherent powers and activity of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer physical world (to which the inner one in each individual man may, I think, be conceived to stand in somewhat the same general relation of correspondence as a shadow to the object from which it is projected, or as the hollow palm of one hand to the closed fist which it grasps of the other), that it is unceasingly calling forth the faculties of observation and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verification, and that it affords a boundless scope for the exercise of the highest efforts of imagination and invention.

During a conversation with the writer in the last weeks of his life, Sylvester remarked as curious that notwithstanding he had always considered the bent of his mind to be rather analytical than geometrical, he found in nearly every case that the solution of an analytical problem turned upon some quite simple geometrical notion, and that he was never satisfied until he could present the argument in geometrical language.

P A MacMahon (1898)