REACTIONS OF ABSTRACT AND CONCRETE SCIENCE. 215 



Leaving here these details of astronomical progress, and 

 the philosophy of it, let us observe how the relatively con 

 crete science of geometrical astronomy, having been thus 

 far helped forward by the development of geometry in gen 

 eral, reacted upon geometiy, caused it also to advance, and 

 was again assisted by it. Hipparchus, before making his 

 solar and lunar tables, had to discover rules for calculating 

 the relations between the sides and angles of triangles 

 trigonometry a subdivision of pure mathematics. Further, 

 the reduction of the doctrine of the sphere to the quanti 

 tative form needed for astronomical purposes, required the 

 formation of a spherical trigonometry, which was also 

 achieved by Hipparchus. Thus both plane and spherical 

 trigonometry, which are parts of the highly abstract and 

 simple science of extension, remained undeveloped until 

 the less abstract and more complex science of the celestial 

 motions had need of them. The fact admitted by M. 

 Comte, that since Descartes the progress of the abstract 

 division of mathematics has been determined by that of 

 the concrete division, is paralleled by the still more signifi 

 cant fact that even thus early the progress of mathematics 

 was determined by that of astronomy. 



And here, indeed, we may see exemplified the truth, 

 which the subsequent history of science frequently illus 

 trates, that before any more abstract division makes a fur 

 ther advance, some more concrete division must suggest 

 the necessity for that advance must present the new order 

 of questions to be solved. Before astronomy presented 

 Hipparchus with the problem of solar tables, there was 

 nothing to raise the question of the relations between lines 

 and angles ; the subject-matter of trigonometry had not 

 been conceived. And as there must be subject-matter be 

 fore there can be investigation, it follows that the progress 

 of the concrete divisions is as necessary to that of the ab 

 stract, as the progress of the abstract to that of the concrete. 



