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

18 HISTORY OF THE THEORY OF DETERMINANTS

the necessary justification for its use but a fuller and better understanding of the whole process. Thus, the given determinant being
a. a. a,3 a4 a5
b, b2 b3 b4 b5
C1 C2 C3 C4 C5
d 2 d2 d3 d4 d5
el e2 e3 e4 e5

the successive derived arrays are
I ab2I I 2bI a 3b41 a4b5
b1c2l 1 bc bc,4l 1lb4c5
d e2 d2e3 d3e4 d4e,
I alb2c3d4 II a2bc4d5 I
bI bcsdae4 I I b2cd4e5 i,

I Ib2c3 I
I blC213, I
I cld()2e3 I

Cbdc4 I
I b2c3d
I c(3e4

I 3b4C I
I b3c4d5 I
I c3d4e5 I

I alb2cd4e5 I.

In this way we see that the determinants of the intermediate
arrays are not equivalents of the given determinant: indeed, if
the given determinant be of the nt' degree in the elements, the
second is of the degree (n-1)2, the third of the degree (n-2)3,
the fourth of the degree (n-3)4, and so on by a rise and a subsequent fall until the degree 1.n is reached.
TAIT, P. G. (1866): MARTIN, H. (1867).
[Note on determinants of the third order. Proceed. R. Soc. Edinburgh, vi. pp. 59-61.]
[Note on Professor Tait's 'quaternion path' to determinants of
the third order. Proceed. R. Soc. Edinburgh, vi. pp. 121-125.]
As the scalar portion of the product of three vectors
ix+jy+ kz, ix' jy' kz', ix" +jy'"+ kz"
is the negative of the determinant
x y z
x' y' z'
x" y" z",
it is natural that workers with quaternions should have properties
of three-line determinants suggested to them. Tait draws attention