[34 THE GENESIS OF SCIENCE. 



respecting algebraic functions that " most functions were 

 concrete in their origin — even those which are at present 

 the most purely abstract ; and the ancients discovered 

 only through geometrical definitions elementary algebraic 

 properties of functions to which a numerical value was not 

 attached till long afterwards, rendering abstract to us 

 what was concrete to the old geometers." How do these 

 statements tally with his doctrine ? Again, having divided 

 the calculus into algebraic and arithmetical, M. Comte 

 admits, as perforce he must, that the algebraic is more 

 general than the arithmetical ; yet he will not say that 

 algebra preceded arithmetic in point of time. And again, 

 having divided the calculus of functions into the calculus 

 of direct functions (common algebra) and the calculus of 

 indirect functions (transcendental analysis), he is obliged 

 to speak of this last as possessing a higher generality than 

 the first ; yet it is far more modern. Indeed, by implica- 

 tion, M. Comte himself confesses this incongruity ; for he 

 says : — " It might seem that the transcendental analysis 

 ought to be studied before the ordinary, as it provides the 

 equations which the other has to resolve ; but though the 

 transcendental is logically independent of the ordinary^ it 

 is best to follow the usual method of study, taking the 

 ordinary first." In all these cases, then, as well as at the 

 close of the section where he predicts that mathematicians 

 will in time " create procedures of a v:ider generality ^'''^ M. 

 Comte makes admissions that are diametrically opposed to 

 the alleged law. 



In the succeeding chapters treating of the concrete de- 

 partment of mathematics, we find similar contradictions. 

 M. Comte himself names the geometry of the ancients spe- 

 cial geometry, and that of moderns the general geometiy. 

 He admits that while " the ancients studied geometry with 

 reference to the bodies under notice, or specially; the 

 moderns study it with reference to the phenomena to be 



