The theory of determinants in the historical order of development, by Sir Thomas Muir.

DETERMINANTS IN GENERAL (CAUCHY, 1847)

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quand le produit K renfermera une seule fois chacune des lettres
a, A, 7Y..., **,
la formule (15) etant relative au cas ou l'on sera oblig6 d'op6rer entre
ces lettres prises deux a deux un nombre pair d'echanges, pour passer
du produit I a3y... I au produit I K J."
The obtaining of the results
a/3y = 3ya = | yaf | a
= - I ay3 I - ay | = -| yj3a I
from transformations of the form
I ao I = - I 3a I
is then shortly considered; and this is followed by a concluding
paragraph, in which the statement occurs that "cette de'composition (des sommes alterndes en facteurs symboliques) une fois
operde on peut s'en servir avec avantage pour d6couvrir ou pour
demontrer les principales propridtes des sommes alterndes."
Cauchy's position is thus seen to be very different from Grassmann's. Grassmann was not concerned with determinants: his
problem was to solve the set of equations
aix +- a2y + a3z = aO
bx + b2y + b3z = bo
C1X + c2y + c3 = co
and he satisfied himself by a curious process of reasoning that
(ao + bo + co)(a2 + b2 + c2)(a + b3 + c3)
-(a+b +b+ cl)(a2 + c2)(a3+b3 +c3)
provided that the multiplications indicated be performed according
to the laws of "outer multiplication." Cauchy, on the other
hand, starts with the determinant formed from
a1 a2 a3
b1 b2 b3
01 (2 03
C1 C2 03,
his problem being to find under what conditions, as regards the
arbitrarily introduced symbols a, /3, y, the product
(ala + a2f + a37)( bla + b23 + 07) (a + c + y)
will be identical with the determinant; and he is led to the need