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The Theory of Determinants in the Historical Order of its Development

Published online by Cambridge University Press:  15 September 2014

Thomas Muir
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Extract

With Grunert it is necessary to take a long step backward. Although the memoirs of Bezout, Vandermonde, and Laplace were known to him, in addition to those of Hindenburg, Rothe, and Scherk, he advances only a short distance into the subject; his aim, indeed, is little more than the establishment of Cramer's rule for the solution of a set of simultaneous linear equations.His mode of presenting the subject, however, is fresh and interesting, the method of “undetermined multipliers” being taken to start with.

Type
Proceedings 1888-89
Information
Proceedings of the Royal Society of Edinburgh , Volume 16 , 1890 , pp. 389 - 448
Copyright
Copyright © Royal Society of Edinburgh 1889

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References

note * page 399 Note, however, the error in sign of λ2 and λ4.

note * page 400 In the proof he is fortunate (or unfortunate) enough to use anotherspecial case in which the statement is true. He says:—“Les deux termes a 7b 6c 1d 3e 5f 2 et e 7f 6a 1b 3c 5d 2 entrent dans D4, et qui se déduisent 1'un de l' autre par une permutation tournante entre les lettres ont meme signe.”

note * page 401 Misprint for II., as an expression in the paper itself shows.

note * page 402 He would not even hesitate to extend the use of the symbol, denoting, for example,.

note * page 411 The third rale is incorrectly stated.

note * note page 413 I.e., aeqnationem finalem

note † page 413 The equations are taken in the form

note * page 414 Euler's, although not called so.

note * page 419 Only the initials A. Q. G. C. are appended to the article. There can be little doubt, however, that they belong to Craafurd, whose name in full appears elsewhere in the Journal.

note * page 423 It would be convenient to say, a term of the cyclical species 2(1)+ 1(2)+ 1(3).

note * page 430 He says, for example (Jour, de I'jtc. Polyt., x. p. 10), “Sien appliquant successivement à la permutation A, les deux substitutions et , on obtient pour résultat la permutation A6; la substitution! sera équiralente au produit des deux autres et j' indiquerai oette équivalence comme il suit .”

note † page 430 This also is a paraphrase of Jaeobi's proof.

note * page 431 In the compounding of reciprocal permutations the order is immaterial. This is the exception hinted at in (d).

note * page 432 This is a paraphrase of Jacobi's demonstration, which is not so simple as it might have been. The notation of substitutions, which Jacobi did not follow Cauchy in using, is here a great help toward clearness.

note * page 433 The best way perhaps of applying Cauchy's rule is to write the primitive permutation, 123456789 say, above the given permutation, 683192457 say, draw the pen through 1 and the figure below it, seek 6 in the upper line and draw the pen through it and the figure below it, and so on, marking dorfnl on the completion of every cycle.

note * page 440 The demonstration in the original is considerably disfigured by misprints.

note * page 446 I.e., permutations of a,b,c,d, …