J.J. SYLVESTER

J.J.Sylvester
Link to entry in the St. Andrews'
MacTutor
History of Mathematics archive.
 J.J.Sylvester
Wikipedia entry.
 The Collected Mathematical Papers, Volume 1 (18371853)
The Collected Mathematical Papers, Volume 2 (18541873)
The Collected Mathematical Papers, Volume 3 (18701883)
The Collected Mathematical Papers, Volume 4 (18821897)

Sylvester's Law of Inertia: A demonstration of the theorem that every homogeneous
quadratic polynomial is reducible by real orthogonal substitutions to the form of a
sum of positive and negative squares
Philosophical Magazine IV (1852), 138142
Theorem of the Day.
Google Scholar search result
for Sylvester's Law of Inertia.
Google search result
for Sylvester's Law of Inertia.

On a remarkable modification of Sturm's theorem
Philosophical Magazine V (1853), 446456,
and the subsequent papers on the subject:
On the new rule for finding superior and inferior limits to the real
roots of any algebraical equation ibid. VI (1853), 138140,
Note on the new rule of limits ibid. VI (1853), 210213.
As an artist delights in recalling the particular time and atmospheric
effects under which he has composed a favourite sketch, so I hope to be
excused putting upon record that it was in listening to one of the magnificent
choruses in the' Israel in Egypt' that, unsought and unsolicited, like a ray
of light, silently stole into my mind the idea (simple, but previously unperceived)
of the equivalence of the Sturmian residues to the denominator
series formed by the reverse convergents. The idea was just what was
wanting,the keynote to the due and perfect evolution of the theory.
 On a theory of the syzygetic relations
of two rational integral functions, comprising an application to the theory
of Sturm's functions, and that of the greatest algebraical common measure.
Philosophical Transactions of the Royal Society of London CXLIII
(1853), 407548.

"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." (Sylvester, 1850)

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."
 Laws of Verse (book by J.J.Sylvester, 1870)
 Chapter on Sylvester from
Ten British Mathematicians of the Nineteenth Century
by A. Macfarlane.

Cayley, Sylvester, and Early Matrix Theory by Nicholas J. Higham.