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these sciences logically speaking ; " for that " analytical 

 ideas are, above all others, universal, abstract, and simple • 

 and geometrical conceptions are necessarily founded on 

 them." 



We will not take advantage of this last passage to 

 charge M. Comte with teaching, after the fashion of Hegel, 

 that there can be thought without things thought of. We 

 are content simply to compare the two assertions, that 

 analysis arose out of the contemplation of geometrical and 

 mechanical facts, and that geometrical concei^tions are 

 founded upon analytical ones. Literally interpreted they 

 exactly cancel each other. Interpreted, however, in a 

 liberal sense, they imply, what we believe to be de- 

 monstrable, that the two had a simultaneous origin. The 

 passage is either nonsense, or it is an admission that 

 abstract and concrete mathematics are coeval. Thus, 

 at the very first step, the alleged congruity between the 

 order of generality and the order of evolution, does not 

 hold good. 



But may it not be that though abstract and concrete 

 mathematics took their rise at the same time, the one 

 afterwards developed more rapidly than the other ; and 

 has ever since remained in advance of it ? ISTo : and again 

 we call M. Comte himself as witness. Fortunately for his 

 argument he has said nothing respecting the early stages 

 of the concrete and abstract divisions after their diver- 

 gence from a common root ; otherwise the advent of 

 Algebra long after the Greek geometry had reached a high 

 development, would have been an inconvenient fact for 

 him to deal with. But passing over this, and limiting 

 ourselves to his own statements, we find, at the opening of 

 the next chapter, the admission, that "the historical de- 

 velopment of the abstract portion of mathematical science 

 has, since the time of Descartes, been for the most part 

 determined by that of the concrete." Further on we read 



