1888-89.] Dr T Muir on the Theory of Determinants. 
759 
fact that in the second period the pre-eminence belongs to Germany, 
France, however, taking still a fairly good second place. 
CAYLEY (1841). 
[On a theorem in the geometry of position. Cambridge Math. 
Journ ., ii. pp. 267-271 ; or Collected Math. Papers , i. 
pp. 1-4.] 
Of the two English mathematicians whose names are inseparably 
associated with the development of what has been called Modern 
Higher Algebra , Sylvester, as we have seen, was the first to direct 
public attention to the functions then partially known as deter- 
minants, but called by him in the heat of supposed discovery 
“zetaic products of differences.” Cayley it was, however, who gave 
the great impetus to the study of them — an impetus due to two 
different causes, the choice of an exceedingly apt notation and the 
masterly manner in which he put the functions to use. How he 
obtained his knowledge we know not. It may be that Sylvester’s 
two early papers had directed his attention to the matter, and that 
he had then read some of the authors who preceded Cauchy ; but, 
whether this be true or not, it is certain that by his own inde- 
pendent research he had attained in 1841 a powerful and compre- 
hensive grasp of the subject. The little paper to which we have now 
come is ample evidence of this. A peculiar interest attaches to it 
also, as being the first fruits of Cayley’s genius, the earliest of that 
long and varied series of papers which has done so much to extend 
the bounds of pure mathematics.* 
With characteristic directness and concision he opens as follows : — 
“We propose to apply the following (new 1) theorem to the 
solution of two problems in Analytical Geometry. 
* In a strictly chronological arrangement Cayley’s paper would not follow, 
but precede the papers of Craufurd, Cauchy, and Jacobi of the same year. It 
was published in February : Cauchy’s note was presented to the Academy on 
8th March, and Jacobi’s memoir bears the date 17th March, though not pub- 
lished for more than two months afterwards. As Cayley’s first appearance, 
however, marks the beginning of a new T epoch, and as the other papers referred 
to belong by their character to the preceding epoch, a slight deviation from the 
chronological order seems warranted. 
