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

DETERMINANTS IN GENERAL (CAYLEY, 1841)

5;

Modern Higher Algebra, Sylvester, as we have seen, was the
first to direct public attention to the functions then partially
known as determinants, but called by him in the heat of
supposed discovery "zetaic products of differences." Cayley it
was, however, who gave the great impetus to the study of
them-an impetus due to two different causes, the choice of an
exceedingly apt notation and the masterly manner in which he
put the functions to use. How he obtained his knowledge we
know not. It may be that Sylvester's two, early papers had
directed his attention to the matter, and that he had then read
some of the authors who preceded Cauchy; but, whether this
be true or not, it is certain that by his own independent research
he had attained in 1841 a powerful and comprehensive grasp of
the subject. The little paper to which we have now come is
ample evidence of this. A peculiar interest attaches to it also,
as being the first fruits of Cayley's genius, the earliest of that
long and varied series of papers which has done so much to
extend the bounds of pure mathematics.*
With characteristic directness and concision he opens as
follows:" We propose to apply the following (new?) theorem to the solution
of two problems in Analytical Geometry.
"Let the symbols
I aI /1 a, a 3', C
a" f", Y" &c.
denote the quantities
a,- a/3'-a'/, a/'y" - a/3'y' a'/3y - ac/y" + a"'/3y - a"/3'y, &c.
the law of whose formation is tolerably well known, but may be thus
expressed,
Ia =_a,:, / _ a - a,
a, /3'
*In a strictly chronological arrangement Cayley's paper would not follow,
but precede the papers of Craufurd, Cauchy, and Jacobi of the same year. It
was published in February: Cauchy's note was presented to the Academy on
8th March, and Jacobi's memoir bears the date 17th March, though not published
for more than two months afterwards. As Cayley's first appearance, however,
marks the beginning of a new epoch, and as the other papers referred to belong
by their character to the preceding epoch, a slight deviation from the chronological order seems warranted.