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The Theory of General Determinants in the Historical Order of Development up to 1852
Published online by Cambridge University Press: 15 September 2014
Extract
The nature of the connection of this with the theory of determinants is evident from the title. Some of the elementary portions of the memoir had in fact already appeared in Cauchy's determinant papers of the years 1812, 1840, 1841, and have been noted in our accounts of the latter. In these papers, as was natural, only such isolated properties were given as might be of immediate application to the main subject: here we have a methodically arranged and lucidly written treatise. As, however, in dealing with permutations the question of signature is not taken up, there is no explicit reference to determinants: and all that is therefore necessary is to direct attention to a storehouse of information regarding a subject closely connected with them.
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- Copyright © Royal Society of Edinburgh 1906
References
page 915 note * These are λ rs = -λ sr (r ≠ s), λ rr =0.
page 920 note * The details of the proof not being given, one cannot guess how it was that a second theorem was not obtained, viz., the theorem
page 925 note * It may be worth noting that while both Vandermonde and Schweins used the recurrent law of formation as a definition, they did not write it in exactly the same form. Schweins followed closely the form used by the original discoverer, Bezout, putting for example
the connecting signs being in all cases alternately positive and negative; whereas Vandermonde wrote
where the cyclical change of suffixes causes the connecting signs to be alternately positive and negative when the order of the determinant is even, and to be uniformly positive when the order is odd.
page 927 note * Spottiswoode, like Joachimsthal, it will be observed, deduces nothing from a comparison of the numerators. Thus, by equating the two cofactors of v 1 he might have obtained
page 932 note * See Trans. R.S.E., XXX. p. 4.
page 934 note * Apropos of this happy coinage, Sylvester adds in a footnote the general remark:—“Progress in these researches is impossible without the aid of clear expression; and the first condition of a good nomenclature is that different things shall be called by different names. The innovations in mathematical language here and elsewhere (not without high sanction) introduced by the author, have been never adopted except under actual experience of the embarrassment arising from the want of them, and will require no vindication to those who have reached that point where the necessity of some such additions becomes felt.” The truth of the remark is not appreciably diminished by the occurrence of the word ‘meso-catalecticism’ in another footnote two pages further on. The year of the paper (1851) was for Cayley and Sylvester a year teeming with fresh ideas as well as with fresh words.