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

6 HISTORY OF THE THEORY OF DETERMINANTS

a, /3', y' = a 'Y + a' / Y + a" /1 /3 Y.,.
a1, a3 +a. a I, &c.
the signs + being used when the number of terms in the side of the
square is odd, and + and - alternately when it is even. Then the
theorem in question is
pa + o-3 +r..., pa' + (-' + y..., pa" o- +T +...
pa + o-/I + ry, pa' +/3' 7 -'y..., pla ' a, pa " + r'" r'
p"a + cr/ + 3r"y..., p"a' + o-' + Try'..., p"a" + ~ " + T" 7 '
p r, T I... a, /3, 7..., 7... a,, y...
= p, cr~ T... a", y...
"This theorem admits of a generalisation which we shall not have
occasion to make use of, and which therefore we may notice at another
opportunity."
Here, then, we have for the first time in the notation of
determinants the pair of upright lines so familiar in all the later
work. The introduction of them marks an epoch in the history,
so important to the mathematician is this apparently trivial
matter of notation. By means of them every determinant be-.came representable, no matter how heterogeneous or complicated
its elements might be; and the most disguised member of the
family could be exhibited in its true lineaments. While the
common characteristic of previous notations is their ability to
represent the determinant of such a system as
a1 a2 a3 1, 1 1. 2 a1,3
b1 b2 b, or a2,1 a2,2 a2,3
C1 2 c3 C3, 1 C3,2 C3,3
and failure to represent in the case of systems like
a b c a b c 4 56
c ab 1 ab 32 7
b c a, 0 I a, 8 1 0: