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

DETERMINANTS IN GENERAL (CAYLEY, 1841)

7

Cayley's notation is equally suitable for all. To illustrate by
analogy,-the infinitesimal calculus supplied with Lagrange's
notation for the differential coefficient of p(x), but unable to
symbolise the differential coefficients of such a special function
as ax3 +bx2, or log(1 - x)/(1 + x) would be in the exact predicament of the theory of determinants prior to Cayley.
Of less importance is the fact, which the quotation indicates,
that Cayley had discovered for himself the multiplicationtheorem, but characteristically hesitated to proclaim it new:
also, that, probably following Yandermonde, he took the recurrent law of formation for his definition, making the signs
all + in one case and + and - alternately in the next, exactly
as Yandermonde did.
He then proceeds to the seemingly geometrical problem:
"To find the relation that exists between the distances of five points
in space."
"We have, in general, whatever xI, y1, I1, w1, &c., denote,
x12 + Y12 + z12~wu,12, - 2x1, -2y, -2z1, - 2w1, 1,2 + Y~2 + 22 + W~- 2X21 - 2Y2' - 2z2, - 2w2, I
%2 +y52 + z52 +52, - 2x5 - 2y5, -2z, -2w5, 1
1, 0, 0, 0, 0,I0
multiplied into
1 i, X1, y1l, z1, w1, x12~+ Y12 +wz,2 + w12
1, x2, y2, z2, w2, X,2+y2"~X2+z+W2
l, X5, X/5 Z5 Ww X52+j?52+Z ~2 + W152
0, 0, 0, 0, 0, 1
2 2 -2 2 "
xi -xi +Yi - yi' -tzi- z01 + - 2~'i, xIt -x 2+2... ~~, X1 -X3~...,~ X1 4+.. Xi-X6+..., 1
2 2 2 2
X21 1..... x2x2~..., 2)2+2, X2+-4..., Y 2 +..., 1..Cj -;C1 aII -5) Y51 Z51 W51 X5 2 + Y52 + 7 15 2 + qV 52
1 *1., 1 1, 1,O.