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

28 HISTORY OF THE THEORY OF DETERMINANTS

Hamilton's widely accepted term. This it was unnecessary to
linger over, as his predecessors had already dealt satisfactorily
with it. Then came the question of multiplication of vectors.
Seeing that when a and b represent two lines in magnitude only,
in other words, are scalars and not vectors, the product ab
represents the rectangle of which a and b are adjacent sides,
Grassmann ventured to denote by the product ab, when a and b
are vectors, a parallelogram having the vectors for adjacent sides.
This definition of multiplication manifestly entailed the result.
a2 = 0;
and along with the definition of addition required further that
a(b+c) = ab+ac.
These two again involved a third, viz.,
ab = -ba;
for from the two we have
0 = (a+b)2,
= (a+b)a+(a+b)b,
= a 2+ba+ab+b2,
= ba+ab.
The remaining steps of the building up of the theory need not be
told, as these laws of outer multiplication (" zussere Multiplication ") suffice for the purpose we have in view.
The exposition of the theory itself is broken up into an
introduction and nine chapters, all of them marked by ability
and much originality. It is the second chapter which deals
specially with outer multiplication, and at the end of it (pp.
70-73) occurs the application which concerns determinants. The
matter is introduced by a sentence or two pointing out that it is
scarcely to be expected that outer multiplication can be so
directly applied to ordinary algebra as to geometry and dynamics,
because in ordinary algebra the quantities are essentially alike
(gleichartige, in the sense of the Ausdehnungslehre), and outer
multiplication presupposes the idea of unlikeness. In certain
circumstances, however, we are told that we may impose
distinctions upon the quantities, and then outer multiplication
may be applied with notable results.