222 SCIENCE IN GRECO-ROMAN ANTIQUITY 



axiomatic type similar to that which characterizes the 

 Elements of Euclid, Archimedes confined his studies 

 to statics. In doing this, he thought to find in a purely 

 logical principle — the principle of symmetry — a suffi- 

 cient foundation for the law of the lever and that of 

 the equilibrium of bodies. If he did not attempt to 

 found dynamics, it was probably for fear of being 

 obliged to have recourse to an obscure sensible intuition. 

 The study of a body in motion implies notions of con- 

 tinuity and indefinite divisibility in time and space, 

 notions which are always in some degree irreconcilable 

 with logic. 



Aristotle was more venturesome ; but his dynamic 

 theses are rendered obscure by a notion of force which is 

 borrowed from biological conceptions. 



Greek science directed along these lines was bound to 

 come to a standstill. 



In the first place, the field assigned to mathematics 

 is too restrained and too arbitrary, since the curves 

 called mechanical are excluded from it. Then, within 

 these limits, the demonstrations become more and 

 more complicated from fear of making a direct appeal 

 to infinity. Doubtless the use of infinity offers advan- 

 tages which are inappreciable from the standpoint of 

 demonstrative rigour, but it is difficult and incon- 

 venient to manipulate, and it lacks generality and 

 necessitates, in its progressive application, more and 

 more complicated geometrical constructions. 



This mistrust of infinity, already so great as concerns 

 integration, appears again and in a more marked degree 

 in questions relating to geometrical space. The Greeks 

 refused to think of this as infinite. Consequently 

 they never imagined as possible the geometrical exis- 

 tence of points and straight lines removed to infinity. 

 We know how much these ideas have vivified modern 

 geometry ; they have rendered possible generalizations 



