Marcu 25, 1897 | 
NATURE 
493 
as a slight consideration renders it apparent. Sylvester, 
in the paper quoted, established the validity of the rule 
for algebraical equations as far as the fifth degree in- 
clusive. The method employed was that of “infinitesimal 
substitution,” which he himself initiated, and had pre- 
viously employed in an essay, ‘On the Theory of Forms,” 
in the Cambridge and Dublin Mathematical Journal. 
It proceeded upon the principle that every finite linear 
substitution may be regarded as the result of an indefinite 
number of simple and separate infinitesimal variations 
impressed upon the variables. He also discussed the 
probability of the specific superior limit to the number 
of real roots in a superlinear equation equalling any 
assigned integer. This valuable memoir contained only 
a small instalment of the desired result. It was not till 
the following year—1865—that he fully established and 
generalised the conjectured theorem of Newton. On 
June 19, he communicated the substance of his dis- 
coveries to the Mathematical Society of London, Prof. 
de Morgan being in the chair as its first president ; and 
on the following June 28 he gave a public lecture in 
King’s College, London, taking as his title, “On an 
elementary proof and generalisation of Sir Isaac Newton’s 
hitherto undemonstrated rule for the discovery of 
imaginary roots.” Sylvester’s fame with posterity will, 
perhaps, be principally associated with this great in- 
tellectual triumph. It may be observed that, subsequent 
to the demonstration, Dr. J. R. Young claimed to have 
proved Newton’s rule twenty years before. Sylvester 
contested this assertion in a characteristic manner, and 
mathematicians are, I think, in agreement that he showed 
it to be without basis. He always wrote well and with 
considerable power of expression ; but, perhaps, he was 
strongest when attempting to demolish any one who 
questioned or denied his claim to priority in a particular 
mathematical discovery. In the case in point he wrote : 
“Tt is such stuff as dreams are made of, and culminating 
as it does ina palpable Jefctio principiz does not need 
a detailed refutation at the hands of the author of this 
lecture. It is not by such vague rhetorical processes, 
but by quite a different kind of mental toil, that the 
truths of science are ‘fon, or a way opened to the inner 
recesses of the reason.” 
When the British Association for the Advancement 
of Science met at Exeter, in 1869, Sylvester was the 
President of the Mathematical and Physical Section. 
Huxley had recently written in JWacmillan’s Magazine ; | 
“Mathematical training is almost purely deductive. The 
mathematician starts with a few simple propositions the 
proof of which is so obvious that they are called self- 
evident, and the rest of his work consists of subtle de- 
ductions from them”; and again, in the Fortnightly 
Review: “Mathematics is that study which knows 
nothing of observation, nothing of experiment, nothing 
of induction, nothing of causation.” It may be safely 
said that any man engaged constantly in mathematical 
research would find no difficulty in refuting these state- 
ments to the satisfaction of any representative body of 
scientific men. Sylvester devoted a considerable portion 
of his address to the Section to contesting Huxley’s state- 
ments, and put in a powerful and eloquent plea for 
mathematics as being a science of observation and 
experiment, and as affording a boundless scope for the 
exercise of the highest efforts of imagination and inven- 
tion. Huxley, I believe, made no reply; and I think 
there can be no doubt that, like many other remarkable 
men in other branches of science, he had no conception 
of the real nature of the life-work of mathematicians of 
the high order to which Sylvester belonged. Amongst 
other matters in his address, he remarks upon the 
extraordinary longevity of the masters of mathematics. 
Amongst these long-lived ones he himself now takes an 
honourable place. 
He left Woolwich (for years he occasionally wrote from 
NO. 1430, VOL. 55 
his house on the Common, over the mom de pluie 
Lani Vicencis”) in 1870, and for some years was with- 
out a professorship. During this time he was much 
interested in the problems of link-motion and conversior 
of motion generally. He wrote several valuable papers 
and invented the skew pantigraph. The title of one» 
his papers of this period is characteristic—‘ Mode 9 
construction and properties of a new sort of lady’s fan 
and on the expression of the curves generated by any 
given system whatever of link work under the form of an 
irreducible determinant.” 
He gave a Friday evening lecture at the Royal 
Institution, entitled “On Recent Discoveries in Me- 
chanical Conversion of Motion.” 
His acceptance, in the year 1875, of an invitation to 
become the first Professor of Mathematics in the new 
Johns Hopkins University at Baltimore, in Maryland, 
may be regarded as concluding the second period of his 
career. He could hardly expect to further increase his 
reputation, which was extraordinarily high, and most of 
the honours that can fall to the lot of a scientific man 
had long been in his possession. 
In Baltimore he soon founded the American Journal 
of Mathematics, and was surrounded by a knot of 
enthusiastic students, whose researches he was able to 
influence, and in some cases to entirely direct. His final 
investigations in the theory of algebraic invariants, 
various questions in diophantine analysis, the construc- 
tive theory of partitions, the theory of universal algebra, 
and the commencement of his researches on differential 
invariants, were principally the outcome of his residence 
in Baltimore. He was assisted, followed up, and 
frequently also inspired by his students in an ideal 
manner. Perhaps the most permanent impress he left 
on the path of American research was in the subject of 
universal algebra, the vigorous offspring of Cayley’s 
memoir, of 1858, on matrices. He established the no- 
menclature of the subject and surveyed the unknown 
country. He showed the connection between linear 
transformation and quaternions, and further arrived 
easily at a generalisation of quaternions. Since then 
Taber, Metzler, and others in the New World, have made 
valuable additions to the theory. 
In 1883 he was elected to succeed Henry J. Stephen 
Smith in the chair of the Savilian Professorship of 
Geometry at Oxford. His inaugural lecture was on the 
subject of differential invariants, termed by him recipro- 
cants. This work was extensive and important, and its 
elaboration, with the able assistance of James Hammond, 
was the last valuable contribution he made to mathe- 
matics. With increasing age infirmities came upon 
him. He suffered from partial loss of sight and 
memory, and in 1892 he obtained permanent leave from 
his duties, and the University appointed a deputy 
professor. 
Henceforth he lived for the most part in London, and 
was a familiar figure in the Athenzeum Club, but he was 
never in good health. At intervals he would go down 
to Tunbridge Wells and live at the Spa Hotel, but he 
did no mathematical work, and his frame of mind was 
not happy. Early in 1896, his condition caused alarm 
to his friends. In August he quite suddenly became 
again interested in mathematical subjects, and this 
appeared to make him calmer and happier. On 
February 26, whilst working at the theory of numbers, 
he had a paralytic stroke and never spoke again. He 
died peacefully at 3.30 a.m. on Monday, March 15, 1897, 
at 5 Hertford Street, Mayfair. 
His work was not so voluminous as that of many of 
his great contemporaries. It may amount to about 1250 
octavo pages and about 1550 quarto pages. Its quality, 
however, is of a very high order, as he always preferred 
to labour at difficult questions; problems which for 
centuries have been a challenge to the human intellect 
