CONCLUSION 221 



only be demonstrated that a curvilinear area is con- 

 tained between two rectilinear areas whose surfaces 

 differ by a quantity as small as desired. A circle, 

 for example, is contained between the increasing 

 surface of an inscribed polygon and the decreasing 

 surface of a circumscribed polygon. 



Because of their distinctive characteristics, it was the 

 mathematical sciences alone which could realize the 

 Greek ideal of axiomatic science, namely a number of 

 logical principles whose rigorous consequences are 

 ensured by reasoned deduction. 



The physical and astronomical sciences, in as far as 

 they have attempted to realize this ideal, have been 

 obliged to limit the field of their investigations. 



Astronomy, for instance, extricated itself from 

 meteorology, with which it was at first mingled, and 

 attempted, with the Pythagoreans, to unite physics and 

 rnathematics. This effort having but imperfectly 

 succeeded, there arose a division between the mechanics 

 of the eternal celestial bodies and that of the terrestrial 

 bodies subject to birth and death. Astronomy then 

 attributed to the celestial bodies a circular motion, 

 and limited its ambition to a geometrical representation 

 of their movement in the heavens. It mattered little 

 whether this representation was physically realizable ; 

 it was sufficient that it accounted for the appearances 

 of the celestial phenomena. This being so, the theory 

 of axioms is satisfied, because the circular movement 

 is the only regular and periodic movement which can be 

 logically conceived for a body in space. In fact, if this 

 body did not move circularly, either it would set off 

 at a tangent and go away into infinity, which is 

 impossible in a finite universe ; or it would fall to the 

 centre of the universe and everything would be motion- 

 less, which is contrary to appearances. 



Similar observations apply to mechanics. Being 

 desirous of constituting this science according to an 



