Why is it called Sylvester's law of inertia?

Question

In Sylvester's law of inertia, how does the name "law of inertia" fit with the statement of the theorem? I guess it's from physics, but I just don't see the connection.

Answer 1

The quote in Mariano's answer is from the introduction to Sylvester's paper. Typical of Sylvester's mathematical papers, he used so many nonstandard terms in that paper that he appended a five-page "Glossary of new or unusual Terms, or of Terms used in a new or unusual sense in the preceding Memoir". There he lists:

Inertia. -- The unchangeable number of integers in the excess of positive over negative signs which adheres to a quadratic form expressed as the sum of positive and negative squares, notwithstanding any real linear transformations impressed upon such form.

Sylvester did similarly for many mathematical terms, i.e. coined them or used them in a "new or unusual ways" mathematically. You can find many such examples in Jeff Miller's Earliest Known Uses of Some of the Words of mathematics, including: allotrious factor, anallagmatic, Bezoutiant, catalecticant, combinant covariant cumulant cyclotomy, cyclotomic, dialytic, discriminant, Hessian, invariant, isomorphic, Jacobian, latent, law of intertia of quadratic forms, matrix, minor, nullity, plagiograph, quintic, Schur complement, sequence, syzygy, totient, tree, umbral calculus, umbral notation, universal algebra, x/y/z-coordinate, zero matrix, zetaic multiplication. Please see each entry for Sylvester's role - some are major, others are minor.

Apparently Sylvester's penchant for colorfully naming mathematical objects arose from his love of language and poetry. Indeed, Karen Parshall wrote:

Sylvester's love of poetry and language manifested itself in notable ways even in his mathematical writings. His mastery of French, German, Italian, and Greek was often reflected in the mathematical neologisms - like "meicatecticizant" and "tamisage" - for which he gained a certain notoriety. Moreover, literary illusions, poetic quotations, and unfettered hyperbole spiced his published papers and lectures.

Sylvester wrote about such:

Perhaps I may without immodesty lay claim to the appellation of Mathematical Adam, as I believe that I have given more names (passed into general circulation) of the creatures of mathematical reason than all the other mathematicians of the age combined.-- James Joseph Sylvester, Nature 37 (1888), p. 152.

You can find a short Sylvester biography here.

Answer 2

From Sylvester's On the Theory of the Syzygetic Relations:

This constant number of positive signs which attaches to a quadratic function under all its transformations [...] may be termed conveniently its inertia, until a better word is found.