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

420 HISTORY OF THE THEORY OF DETERMINANTS
then
((1,..., ( am+,,..., n+) = (al,,.. a)(am+1 *,.* + C. mn)
+ (a..., an_-1)(am+2, * * n)
-a possibly new result which he considers "the fundamental
theorem in the theory of continued fractions." This, he says, is
an immediate consequence of the fact that (a,,.., a,,,j) can
be expressed as a determinant, all that is further necessary being
the application of the "well-known simple rule for the decomposition of determinants. Thus, e.g., the determinant
a I
-1 b 1
-1 c 1
-1 d 1
-1 e 1

is obviously decomposable into
a 1 x d 1
-1 b 1 -I e 1
-1 c -1 f
or into
al x c 1
-1 b -1 d 1
-1 e 1
-1 f
or into
a x b 1
-1 c 1
-1 dc 1
-1 e 1
-1 f

-1 f
+ a 1 x e 1
-1 b -1 f,
+ a x d 1
-1 e 1
-1 f,

+ c 1
-1 d 1
-1 e 1
-1 f.

Following this is what is called " Corollary I.," namely,
a1, 2,..., am) (a, a*,.,,, - (, * *...., a ) * (al, a,..., +)
= (-))m(a,+iam+-... to i- 1 factors),
its connection with the expression for the difference of two convergents being illustrated by the instances i= 1, 2, 3, 4,...