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

8 HISTORY OF THE THEORY OF DETERMINANTS
Putting the:'w's equal to 0, each factor of the first side of the
equation vanishes, and therefore in this case the second side of the
equation becomes equal to zero. Hence xi,yl,z1, x2,y2, 2, &c., being
the coordinates of the points 1, 2, &c., situated arbitrarily in space,
and 122, 132, &c., denoting the squares of the distances between these
points, we have immediately the required relation
0, -12 B22 2 2, 1
2 2 -2 2 1
212, 0, 23, 242 252 ' 1
312, 322, 0 342, 352 1 2O
412, 422, 43, 0, 452 1
512, 522, 532, 542 0, 1
1, 1, 1, " 1, 1, 0
which is easily expanded, though from the mere number of terms the
process is somewhat long."
Than this no better example could have been chosen to
illustrate what has just been said above regarding the great
advantages of Cayley's notation. As is well known, the result
arrived at had been given in forms, lengthy and forbidding,
many years before by Lagrange and Carnot. What Cayley did
was to rob it of all disguise, by expressing it as the vanishing
of an elegantly formed determinant; and secondly, to show that
the said determinant vanished because it was eight times the
square* of another determinant whose zero character could not
be overlooked. As has been implied, the result is purely algebraical, its geometrical character only appearing when x, y, z
are taken to denote the coordinates of a point.
The corresponding identities for the cases of four points in a
plane and three points in a straight line are given; and the latter
of the two is most interestingly shown to be deducible also from
the general theory of elimination. This is done as follows:"Let x, -x,,,,, -X, =, x, -,,=
then -12 23, -_ 2 312 =, and,a + + 7=0;
from which a, /3, 7 'are to be eliminated. Multiplying the last equation
byy, y, a7, af, and' reducing by the three first,
" The first factor being 16 times the second, and the w's unnecessary.