Dr T. Muir on the Theory of Determinants. 499 
of the multii^lication-theorem. In the first place, he gave the 
generalisation from which the mnltiplication-theorein is got by 
putting m = n, or, as we nowadays say, by substituting two square 
matrices for two rectangular matrices, and then he gave the theorem 
which we have been comparing with Cauchy’s and which degener- 
ates into his own first theorem when s is put equal to 1. Now the 
first of these generalisations Cauchy could justly have laid claim to. 
His identity (xviii. 5) is not indeed stated or viewed as a general- 
isation of the multiplication-theorem, but it is unquestionably so in 
reality. Ostensibly the identity concerns any minor of a product- 
determinant, but every such minor is obtained by multiplying 
together two rectangular matrices, and, conversely, every determinant 
which is the product of two rectangular matrices may be viewed as 
a minor of the product of two determinants. 
On looking back, however, at Cauchy’s memoir as a whole, one 
cannot but be struck with admiration both at the quality and the 
quantity of its contents. Supposing that none of its theorems 
had been new, and that it had not even presented a single old 
theorem in a fresh light, the memoir would have been most valuable, 
furnishing as it did, to the mathematicians of the time, an almost 
exhaustive treatise on the theory of general determinants. It is not 
too much to say, although it may come to many as a surprise, that 
the ordinary text-books of determinants supplied to university 
students of the present day do not contain more of the general 
theory than is to be found in Cauchy’s memoir of about eighty years 
ago. One apparently trivial instrument, which Cauchy had not 
received from his predecessors, and which he did not make for 
himself, viz., a notation for determinants whose elements had special 
values, is at the foundation of the whole difference between his 
treatise and those at present employed. When this want came to 
be supplied later on, the functions crept steadily into everyday use, 
and a fresh impetus was consequently given to the study of them. 
But if from the work of the said eighty years all researches regard- 
ing special forms of determinants be left out, and all investigations 
which ended in mere rediscoveries or in rehabilitations of old ideas, 
there is a surprisingly small proportion left. If one bears this in 
mind, and recalls the fact, temporarily set aside, that Cauchy, 
instead of being a compiler, presented the entire subject from a 
