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

DETERMINANTS IN GENERAL (SYLVESTER, 1850) 51
similarly 'Second Minor Determinant' is explained; and then he
adds,
"and so in general we can form a system of 1.th minor determinants by
the exclusion of r lines and r columns, and such system in general will
contain
qn(n-1)... (n-r+l) 2
1.2... r
distinct determinants."
It is thus seen that 'minor determinant is used as 'partial
determinant' had already been used by Lebesgrle (1837), and as
'determinant of a derived system' had been used by Cauchy
(1812), but that, whereas Cauchy added a distinguishing epithet
to specify the order of the determinant, Sylvester did so to
indicate how many lines or columns fewer it had than the
'principal' or 'complete' determinant originally started w" h.
The following proposition or 'law' is next given, viz.: The
whole of a system of rtS minors being zero implies only (r +1)2
eqzbations, that is, by making (r + 1)2 of these nminors zero, all will
become zero: and this is true, no matter what may be the
dimensions or form of the complete determinant. Then, after
some geometrical applications concerned with first minors of a
symmetrical determinant, there follows the explanation-."The law which I have stated for assigning the number of independent or, to speak more accurately, non-coevanescent determinants
belonging to a given system of minors, I call the Homaloidal law,
because it is a corollary to a proposition which represents analytically
the indefinite extension of a property, common to lines and surfaces,
to all loci (whether in ordinary or transcendental space) of the first
order, all of which loci may, by an abstraction derived from the idea
of levelness common to straight lines and planes, be called Homaloids."
A further advance is made just before the close of his paper.
Leaving the square array and taking m lines and n columns, he
says
"This will not represent a determinant, but is, as it were, a Matrix
out of which we may form various systems of determinants by fixing
upon a number p and selecting at will p lines and p columns, the
squares corresponding to which may be termed determinants of the
ith order."
Here there is to be noticed the first use of the word matrix in