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

DETERMINANTS IN GENERAL (SYLVESTER, 1851) 61
I aCb2c3f4 I alb2c,5 ab2c3f6 I cb2c 7
aI b2c34 I ab2C3.5 a 1b2c3g a b2CIg7
= Ca1 3 '3. a1b2c3d4e5f67
we see that if we delete a b2c everywhere on both sides we are
left with
d4 d5 d6 d7
'4 C5 e6 e7 = de5f6g71;
'f4 f5 f,6 f7
g4 g5 96 g7
so that the theorem is seen to be the Extensional of a manifest
identity.*
SYLVESTER, J. J. (1851, Aug.).
[On a certain fundamental theorem of determinants. Philos.
Magazine (4), ii. pp. 142-145; Collected Math. Papers, i.
pp. 252-255.]
After a characteristic introductory paragraph about the importance of the new theorem and his reasons for publishing it,
Sylvester proceeds"The theorem is as follows:-Suppose that there are two determinants
of the ordinary kind, each expressed by a square array of terms made
up of n lines and n columns, so that in each square there are n2 terms.
Now let n be broken up in any given manner into two parts p and q,
so that p + q= n. Let, firstly, one of the two given squares be divided
in a given definite manner into two parts, one containing p of the n
given lines, and the other part q of the same; and secondly, let the
other of the two given squares be divided in every possible way into two
parts, consisting of q and p lines respectively, so that on tacking on
the part containing q lines of the second square to the part containing
p lines of the first square, and the part containing p lines of the second
square to the part containing q of the first, we get back a new couple
of squares, each denoting a determinant different from the two given
determinants: the number of such new couples will evidently be
n(n- 1)... (n -p+ 1).
1.2... p
* See Trans. R. Soc. Edinburgh, xxx. p. 4.