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 (1837-1853)
  The Collected Mathematical Papers, Volume 2 (1854-1873)
  The Collected Mathematical Papers, Volume 3 (1870-1883)
  The Collected Mathematical Papers, Volume 4 (1882-1897)
• 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), 138--142
  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), 446--456, 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), 138--140,
  Note on the new rule of limits ibid. VI (1853), 210--213.
  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 key-note 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), 407--548.
• "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.