132 THE GENESIS OF SCIENCE. 



in essential conformity with the order which has sponta 

 neously taken place among the branches of natural j^hiloso 

 phy ; " or, in other words — corresponds with the order of 

 historic development. 



Let us compare these assertions with the facts. That 

 there may be perfect fairness, let us make no choice, but 

 take as the field for our comparison, the succeeding section 

 treating of the first science — Mathematics ; and let us use 

 none but M. Comte's own facts, and his own admissions. 

 Confining ourselves to this one science, of course our com- 

 j^arisons must be between its several parts. M. Comte says, 

 that the parts of each science must be arranged in the 

 order of their decreasing generality; and that this oider 

 of decreasing generality agrees with the order of historic 

 development. Our inquiry must be, then, whether the his- 

 tory of mathematics confirms this statement. 



Carrying out his principle, M. Comte divides Mathe- 

 matics into " Abstract Mathematics, or the Calculus (tak- 

 ing the word in its most extended sense) and Concrete 

 Mathematics, which is composed of General Geometry and 

 of Rational Mechanics.'' The subject-matter of the first of 

 these is number ; the subject-matter of the second includes 

 space^ time^ motion^ force. The one possesses the highest 

 possible degree of generality ; for all things whatever 

 admit of enumeration. The others are less general ; see- 

 ing that there are endless phenomena that are not cogniza- 

 ble either by general geometry or rational mechanics. In 

 conformity with the, alleged law, therefore, the evolution 

 of the calculus must throughout have preceded the evolu- 

 tion of the concrete sub-sciences. Now somewhat awk- 

 wardly for him, the first remark M. Comte makes bearing 

 upon this point is, that " from an historical point of view, 

 mathematical analysis cqjpears to have risen out q/'the con- 

 templation of geometrical and mechanical facts." True, 

 he goes on to say that, " it is not the less independent of 



