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

DETERMINANTS IN GENERAL (SPOTTISWOODE, 1851) 55
l(,1) (2,2) (,3) =n 1,)(212) (2 3) (2 3) (2 1) (2,1) (2,2)
((2l) (2,2) (,3) (I (,1) + (1,2) (+ (1,3)
(3,2) (3,3) (3,3) (3,1) (3,1) (3,2)
(3,1) (3,2) (3,3)
The quantities (1,1), (1, 2),... which Cauchy called 'terms'
and Jacobi 'elements,' are named 'constituents'; and the determinant of the nth order having these constituents is denoted
shortly by
~(1,1)(2, 2)... (, n)
The first result, deduced in somewhat loose fashion from the
definition, is "Cramer's rule"; but the first that is actually
formulated and numbered is one of much later date than Cramer,
viz."Theorem I. —If the whole of a vertical or horizontal row be multiplied
by the same quantity, the determinant is multiplied by that quantity."
In this form, as is well known, it afterwards became almost
stereotyped. The second result, which is of about the same age,
is that regarding a determinant whose vertical row consists of
p-termed expressions, second vertical row of q-termed expressions
third vertical row of r-termed expressions, and so on, being to the
effect that such a determinant is expressible as a sum of pqr
determinants with monomial constituents. The next seven results
are, like the first, new only in form, the wording being, as in
Theorem I., more topographical in character than formerly, on
account of the determinant being now more consciously viewed
as connected with a square holding n-, quantities situate in n
vertical rows and at the same time in n horizontal rows. The
tenth result, which is a converse of the ninth, is new but
unimportant, viz."Theorem X.-If a determinant of the nth order vanishes, a system of n
homogeneous linear equations, the coefficients of which are the constituents of
the given determinant, may always be established."
The ninth he establishes by the method of so-called 'mathematical
induction,' deducing from it the solution of
(rl) x + (r2)X2 +... + (rn)x' = ur }i