Dr T. Muir on the Theory of Determinants. 
501 
whose numbers are II. , III., IV., XII., XIV.; and we likewise see 
that information regarding them all will be got at p. 53 of the History. 
Speaking generally, more importance ought to be attached to the 
existence of numbers at the corner of a gnomon than elsewhere, 
because these indicate fresh departures in the theory. Sometimes, 
however, a fresh departure may have been very trivial, the real 
advance being indicated by a number well removed from the 
corner of a subsequent gnomon. Thus if we examine the history 
of the multiplication-theorem (Xos. XVII., XVIII.), we find the first 
step in the direction of it credited by the table to Lagrange, and 
subsequent steps to Gauss, Binet, and Cauchy ; whereas careful 
investigation at the pages mentioned shows that what Lagrange 
accomplished was of exceedingly little moment, in comparison with 
the magnificent generalisation of Binet and Cauchy. Again, it 
must be borne in mind that all the results numbered in Roman 
figures are not of equal importance, it being well known that one 
theorem in any mathematical subject may have vastly more 
influence on the after development of the subject than half a dozen 
others. Such imperfections, however, being allowed for, the table 
will be found to afford a very ready means of estimating with 
considerable accuracy the proportionate importance to be assigned 
to the various early investigators of the theory. 
If we look for a moment, in conclusion, at the nationality of the 
authors, one outstanding fact immediately arrests attention, viz., 
that almost every important advance is due to the mathematicians 
of France. Were the contributions of Bezout, Vandermonde, 
Laplace, Lagrange, Monge, Binet, and Cauchy left out, there would 
be exceedingly little left to any one else, and even that little would 
be of minor interest. 
GERGONNE (1813). 
[Developpement de la theorie donn6e par M. Laplace pour 
relimination au premier degre. Annates de Mathematiques, 
iv. pp. 148-155.] 
This is such an exposition of the primary elements of the theory 
of determinants and their application to the solution of a set of 
simultaneous linear equations as might be given in the course of an 
