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

70 HISTORY OF THE THEORY OF DETERMINANTS

close of the 17th quarto page, and the opportunity is taken to
ascertain, besides other things, what had incidentally been learned
about the solution of two equations with two unknowns, three
equations with three unknowns, and so forth.
After this introduction there comes a chapter (pp. 22-42) which
establishes the basic properties of determinants; then one (pp.
42-46) which recurs to the work of the first, rounding it off by
consideration of the (n-l)th derived set of equations; and finally
a chapter (pp. 46-61) which treats of the solution of linear equations
by the method of 'undetermined multipliers.'
The procedure throughout is not at all of the stereotyped kind.
PAIGE, C. LE (1877): JAMET, V. (1877).
[Sur la multiplication des determinants. Nouv. Corresp. Math.,
iii. pp. 141-144, 247, 275-276.]
The row-by-row product of D1 by D2 being A, Le Paige, founding
on the fact that A vanishes if two columns of D1 be identical, concludes that D1 is a factor of A; then similarly that D2 is a factor;
and finally that the cofactor of D1D2 in A is 1.
To this Jamet naturally objects that the vanishing of D1 does
not necessarily entail the identity of two of its columns, and that
therefore the demonstration is imperfect.
Le Paige returns to the subject, and from the vanishing of D1
draws this time a less objectionable conclusion which leads him to
the vanishing of A, after which he proceeds with this proof as before.
FROBENIUS, G., (1877).
[Ueber homogene totale Differentialgleichungen. Crelle's Journ.,
lxxxvi. pp. 1-19.]
At the bottom of the first page Frobenius introduces his use of
the word 'Rang' as applied to a determinant, his definition being
that if in a determinant all the minors of the (m+1)"t order vanish
but not all those of the mt7" order, the determinant is said to be of the
mt" 'Rang.' It is seen to be the same as the order of Rouche's
' critical minor,' the latter, however, being from the first not confined
to a determinant but used in connection with any oblong array.