1888-89.] Dr T. Muir on the Theory of Determinants. 
401 
GAKNIER (1814). 
[Analyse Algebrique, faisant suite a la premiere section de l’algebre. 
2 e edition, revue et consid6rablement augmentee. xvi. +668 
pp. Paris.] 
The title of Garnier’s chapter xxvii. (pp. 541-555) is “ Developpe- 
ment de la theorie donnee par M. Laplace pour V elimination au 
premier degre.” It consists, however, of nothing but a simple 
exposition, confessedly borrowed from Gergonne’s paper of 1813, 
and six pages of extracts from Laplace’s original memoir of 1772. 
As forming part of a popular text-book, it probably did more service 
in bringing the theory to the notice of mathematicians than a 
memoir in a recondite serial publication could have done ; and we 
certainly know that Sylvester, who afterwards did so much to 
advance the theory, expresses himself indebted to it. 
SYLVESTER (1839). 
[On Derivation of Coexistence : Part I.* Being the Theory of 
simultaneous simple homogeneous Equations. Philosophical 
Magazine , xvi. pp. 37-43.] 
Sylvester was apparently first brought into contact with deter- 
minants while investigating the subject of the elimination of x 
between two equations of the m th and n th degrees. At the close of 
a paper on this subject (Phil. Mag. f xv. p. 435) he says— “ I trust to 
be able to present the readers of this magazine with a direct and 
symmetrical method of eliminating any number of unknown 
quantities between any number of equations of any degree, by a 
newly invented process of symbolical multiplication, and the use of 
compound symbols of notation.” These last words, indicative of the 
method, exactly describe the matter dealt with in the paper we 
have now come to, and as will soon be seen, the functions which 
are the outcome of the said “ compound symbol ” of operations are 
determinants. 
It would also appear that Sylvester was unacquainted with any 
* Misprint for II., as an expression in the paper itself shows. 
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