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By "Sylvester's Law of Inertia," I mean:

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.

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Well I imagine the term inertia comes from the invariance under coordinate changes since inertia is the tendency of an object to resist a change in motion in physics. – WWright Oct 15 '10 at 22:06
From Sylvester's 1852 article, linked at the end of the Wikipedia article: " view of the physical meaning of quantity of matter inclines me, upon the ground of analogy, to give [this law] the name of the Law of Inertia for Quadratic forms, as expressing the fact of the existence of an invariable number inseparably attached to such forms." So, I think @WWright is right. – Rahul Oct 15 '10 at 23:30
@Rahul: Indeed, Sylvester was one of the founders of invariant theory of forms (the term "invariant" is due to him -- see my answer). – Bill Dubuque Oct 16 '10 at 0:02
up vote 16 down vote accepted

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.

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The deleted version of this post contained many comments. Some were off-topic, but it was pointed out that for example the idea that Sylvester had a role in the naming of Schur complement is highly dubious, and contradicted by the website linked in that paragraph. – Jonas Meyer Oct 17 '10 at 17:05
NOTE My original post was, alas, worded ambiguously - so that could it could possibly be misinterpreted to imply that the all the entries cited from Jeff Miller's list were all strict "coinings". I've updated the post to remove that ambiguity. – Bill Dubuque Oct 17 '10 at 18:25
@Jonas Meyer: Regarding the "Schur Complement", Jeff wrote: "However the historical notes in chapter 0 of Fuzhen Zhang (ed.) The Schur Complement and Its Applications (2005) identify 'implicit manifestations' in the work of Sylvester in 1851". This is not the appropriate place to address any updates to Jeff's nice list - esp. those that have no relation to the topic of this thread. T.. and I are in the process of discussing this and other issues by email. Stay tuned for any updates... – Bill Dubuque Oct 17 '10 at 18:25
This is still misleading. While Sylvester is mentioned in Miller's entry on the Schur complement, Sylvester neither coined the term nor used it "in a new and unusual way". – Robin Chapman Oct 17 '10 at 19:11
According to Miller's compilation, while Sylvester (and Laplace before him) may have considered related concepts, the term "Schur complement" is recent, dating back only to 1968, based on a 1917 paper of Schur. Sylvester died in 1897, four years before Schur attained his doctorate and probably before any of Schur's work appeared in print. Sylvester in this case contributed ideas but not the modern vocabulary. – T.. Oct 17 '10 at 20:08

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.

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But we did find the better word, right? Signature? – Qiaochu Yuan Oct 15 '10 at 22:21
@Qiaochu: indeed, we nowadays say signature but, honestly, I don't know why it is the better word! – Mariano Suárez-Alvarez Oct 15 '10 at 22:54
But imagine how confusing mathematical physics might be if, similarly, many mathematical objects had names borrowed from physics. – Bill Dubuque Oct 16 '10 at 0:51
@Bill, why do you say if?! :) – Mariano Suárez-Alvarez Oct 16 '10 at 15:37
FWIW, I keep seeing "inertia" in the literature of numerical linear algebra, but have never seen "signature". I don't know why. – J. M. Oct 17 '10 at 13:14

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