492 
NATURE 
[Marcu 25, 1897 
the charm of his personal influence. Throughout his 
long tenure of the chair of Natural Philosophy, he has 
carried lightly like a flower the weight of honour which 
the scientific world has united to render to him. He 
has remained ever the same kind friend of his students, 
and his interest in them, old and young, and in every 
scientific worker, has found many quietly sympathetic 
modes of expression. The enthusiastic testimony to his 
pre-eminence as a scientific man, and to his admirable 
personal qualities, which was borne by the whole world 
at the magnificent celebration last June, will not soon be 
forgotten by those who had the privilege of taking part 
in that great ceremonial: it was an emphatic tribute to 
the greatness of the part which the Physical Laboratory 
at Glasgow has played in science during the last fifty 
years. A. GRAY. 
JAMES JOSEPH “SS VEVE SIDR. 
E is dead, and it becomes a sad duty to give a brief 
account of his long life and great work. 
Born in London September 3, 1814, he was the 
youngest but one of seven children of Abraham Joseph 
Sylvester. He was the last survivor. Three sisters 
lived for many years at Norwood, and of his three 
\brothers two, Frederick and Joseph, lived for the most 
part in America, whilst George resided at Worcester. 
He obtained his early education at private schools in 
London; thence he went to the Liverpool Institution, 
and in 1837 graduated at St. John’s College, Cambridge, 
as Second Wrangler. The first five names in the Mathe- 
matical Tripos of the year are Griffin, Sylvester, Brumell, 
Green, Gregory. It is astonishing to think that Green, of 
immortal memory, has been dead for nearly fifty years! 
Sylvester was keenly disappointed at his failure to be 
senior of the year. He was always of an excitable dis- 
position, and it is currently reported that, on hearing the 
result of the examination, he was much agitated. Being 
of the Jewish persuasion, he was unable to take his 
degree at Cambridge, but later he obtained a degree at 
the University of Dublin. On leaving Cambridge he at 
once commenced the long series of mathematical papers, 
which he was to contribute to scientific periodicals all 
over the world, by the publication, in vol. x1. of the PAz/o- 
sophical Magagine, of an analytical development of 
Fresnel’s optical theory of crystals. 
This was followed by some articles upon subjects of 
applied mathematics, and it was not until 1839 that he 
brought his intellect to bear upon the analysis of con- 
tinuous and of discontinuous quantity, departments of 
pure mathematics which well-nigh monopolised his 
attention for the remainder of his life. He was appointed 
Professor of Natural Philosophy at University College, 
London, and later on held the post of Professor of 
Mathematics in the University of Virginia. He returned 
to England in the year 1845, and the first period of his 
scientific career may be said to have closed. He had 
published some thirty papers, and was already well 
known in both hemispheres as an original and imaginative 
man of science. The subjects dealt with comprise 
““Dialytic Method of Algebraical Elimination,” “ Sturm’s 
Functions,” “Criteria for Determining the Roots of 
Numerical Equations,” “ The Calculus of Forms” (after- 
wards known as the “Theory of Invariants”), ‘“ The 
Equation in Integers 4x? + Sy3 + Cz? = Dayz.” The 
latter problem was a favourite subject of thought through- 
out his life, and the first problem in the theory of numbers 
that he attacked. The theory of invariants sprang into 
existence under the strong hand of Cayley, but that it 
emerged finally a complete work of art, for the admiration 
of future generations of mathematicians, was largely 
owing to the flashes of inspiration with which Sylvester’s 
intellect illuminated it. The nomenclature of the theory 
NO. 1430, VOL. 55 | 
is almost entirely due to him. The words “invariant,” 
“covariant,” ‘“ Hessian,” “discriminant,” ‘ contra- 
variant,” ‘combinants,” “commutant,” ‘ concomitant,” 
are a few of those introduced by him at this time, which 
have been part of the stock-in-trade of mathematicians 
ever since. 
A beautiful theory of the rotation of a rigid body 
about a fixed point, after Poinsot, should be mentioned. 
It is one of the few papers that he wrote on dynamics. 
For ten years after his return from Virginia he was 
occupied with a firm of actuaries. He founded the Law 
Reversionary Interest Society, and also accomplished a 
considerable amount of mathematical research. In 1853 
appeared his first important memoir in the Phzlesophical 
Transactions of the Royal Society, bearing the title, “On 
a theory of the syzygetic relations of the rational in- 
tegral functions, comprising an application to the theory 
of Sturm’s functions and that of the greatest algebraical 
common measure.” This isa masterly exposition, covering 
170 quarto pages. 
In 1855 he was appointed Professor of Mathematics 
at the Royal Military Academy, Woolwich. This was a 
great relief, as the work of an actuary was manifestly un- 
suitable, and had indeed been most distasteful to him. 
He held this professorship for fifteen years. It was a 
time of great activity. Year by year his fame increased, 
and recognition by foreign academies was liberally 
bestowed. In addition to continual work at the theory 
of invariants, he laboured at some of the most difficult 
questions in the theory of numbers. 
Cayley had reduced the problem of invariant enumer- 
ation to that of the partition of numbers. Sylvester 
may be said to have revolutionised this part of mathematics 
by giving a complete analytical solution of the problem, 
which was in effect to enumerate the solutions in positive 
integers of the indeterminate equation— 
axt+by+ez...+ld=m 
Thereafter he attacked the similar problem connected 
with two such simultaneous equations (known to Euler 
as the Problem of the Virgins), and was partially and 
considerably successful. In June 1859, he delivered a 
series of seven lectures on compound partition in general 
at King’s College, London. The outlines of these lectures, 
printed at the time for distribution amongst his audience, 
are now being published for the first time by the London 
Mathematical Society. He was assisted in the preparation 
of these lectures by Captain (now Sir Andrew) Noble, 
with whom from that time forth he was in sympathetic 
friendship. 
The year 1864 may be regarded as the time of his 
greatest intellectual achievement, which caused him to 
be considered as one of the foremost of living mathe- 
maticians. On April 7, 1864, he read a paper before the 
Royal Society of London, bearing the title “ Algebraical 
Researches, containing a disquisition on Newton’s rule 
for the discovery of imaginary roots, and an allied rule 
applicable to a particular class of equations, together 
with a complete invariantive determination of the char- 
acter of the roots of the general equation of the fifth 
degree, &c.” In the “ Arithmetica Universalis,” Newton 
gave a rule for discovering an inferior limit to the 
number of imaginary roots in an equation of any degree, 
but without demonstration. Neither did he give any 
indication of the mental process by which he was led to 
conjecture the truth of the rule, nor did he set forth the 
evidence upon which it rests. For years the question 
of proving or disproving the rule had been a crux of the 
science. Euler, Waring, Maclaurin and Campbell were 
amongst those who sought in vain to unravel the 
mystery. The only step that had been gained was to 
show that if ay negative terms occur in the quadratic 
elements involved in the statement, there must be some 
imaginary roots. This, however, was not a great step, 
