VI.42 James Joseph Sylvester
b. London, 1814; d. London, 1897
Algebra
As a Jew, Sylvester could neither take the degree he earned at St John’s College, Cambridge, in 1837 nor compete for positions at England’s Anglican universities. This effectively forced him down a convoluted path toward his personal goal of a career as a research mathematician. He worked as an actuary in London in the 1840s and 1850s before qualifying as a lawyer by passing the English Bar. He was unemployed for some six years in the 1870s, but held professorships at various times, both of natural philosophy and of mathematics, in England and in the United States. Most notably, Sylvester served as the first Professor of Mathematics at Johns Hopkins University in Baltimore, Maryland, from 1876 to 1883 and, thanks to an 1871 law that finally made it possible for non-Anglicans to hold professorships at Oxbridge, was eligible for and won the appointment as Oxford’s Savilian Professor of Geometry in 1883. He held the Oxford chair until ill health forced his retirement in 1894. The program Sylvester set up at Johns Hopkins established his pivotal place in the history of American research-level mathematics, while his mathematical accomplishments had garnered him an international reputation as early as the 1860s.
Sylvester entered the research arena in the late 1830s with work on the problem of determining when two polynomial equations have a common root. This naturally led not only to questions in the theory of determinants but also to an explicit, pioneering, and self-consciously algebraic analysis of the intermediate expressions that arise in Charles François Sturm’s algorithm for determining the number of real roots of a polynomial equation that lie between two given real numbers (1839, 1840). Sylvester followed this up with what he called the dialytic method of elimination: a new criterion in terms of DETERMINANTS [III.15] for detecting whether two polynomial equations have a common root (1841).
His next major research push came in the 1850s when, together with CAYLEY [VI.46], he formulated a theory of invariants. This involved an associated and slightly more general theory of “covariants.” More concretely, given a binary form of a particular degree, Sylvester and Cayley devised techniques both for explicitly finding invariants and covariants of that form and for determining algebraic relations, or “syzygies,” between them. Sylvester tackled these questions in two important papers: “On the principles of the calculus of forms” (1852) and “On a theory of the syzygetic relations of two rational integral functions” (1853). In the latter, he proved, among other results, Sylvester’s law of inertia: if Q (x1, . . . , xn) is a real QUADRATIC FORM [III.73] of rank r, then there exists a (real) nonsingular linear transformation that takes Q to 
where p is uniquely determined.
Sylvester surprised the mathematical world in 1864 and 1865 with the first proof of NEWTON’S [VI.14] rule (Newton had only stated it) for determining bounds on the number of positive and negative roots of a polynomial equation. However, he then entered a fallow period that ended only with his move to Baltimore. While there, he returned to invariant theory, and specifically to the problem of inductively determining, for binary forms first of degree 2 then of degree 3 then of degree 4, etc., the number of covariants in a minimum generating set associated with the form. In 1868, Paul Gordan had proved that this number is always finite and, in so doing, had proved wrong an earlier result of Cayley, who claimed to have shown that a minimum generating set of covariants for the binary quintic form (that is, the binary form of degree 5) was infinite. By 1879, Sylvester had explicitly calculated minimum generating sets of covariants associated with binary forms of degrees two through ten. He had also succeeded in recognizing and filling (1878) a critical gap in the proof that Cayley had given of a theorem on the maximal number of linearly independent covariants associated with a binary form of any given degree.
Sylvester was the founding editor of the American Journal of Mathematics, and indeed much of this invariant-theoretic work, as well as results on partitions (1882), on rational points on a cubic curve (1879-80), and on matrix algebras (1884), appeared there.
Further Reading
Parshall, K. H. 1998. James Joseph Sylvester: Life and Work in Letters. Oxford: Clarendon.
———. 2006. James Joseph Sylvester: Jewish Mathematician in a Victorian World. Baltimore, MD: Johns Hopkins University Press.
Sylvester, J. J. 1904-12. The Collected Mathematical Papers of James Joseph Sylvester, four volumes. Cambridge: Cambridge University Press. (Reprint edition published in 1973. New York: Chelsea.)