132 SCIENCE IN GRECO-ROMAN ANTIQUITY 



unlimited, it would not be able to accomplish its daily 

 revolution in 24 hours. Further, what is infinite is 

 imperfect, unfinished, and unthinkable ; yet the world 

 is a finite whole which can be conceived in the mind. 



But if magnitude be not infinitely great, it is per 

 contra infinitely divisible, and, in this sense, there is 

 an infinity of magnitudes, but only potentially and not 

 actually, since the division is never completed. Con- 

 tinuity must be defined thus : that which is divisible 

 into parts which are always divisible (de coelo, 268 to 6). 

 If this be so, the arguments of the Eleatic school 

 against the reality of motion lose all their force, for 

 it is not necessary that the possible divisions of time 

 and space should be performed in order that motion 

 may really take place. 



With regard to number, Aristotle adopts a quite 

 opposite attitude. He admits the virtual existence of 

 an infinite number, in this sense that after each whole 

 number there is always another. But a numerical 

 infinitely small is inconceivable, since unity is an 

 element below which it is impossible to go. 



To sum up, Aristotle considers all magnitude as 

 finite, but he admits its infinite divisibility, thus 

 rejecting spatial atomism. On the other hand, he 

 affirms the extensible infinity of number, but not its 

 infinite divisibility. 



We see that though the views of Aristotle have 

 undeniable metaphysical interest, they do not present 

 any method of symbolizing and using, mathematically, 

 continuity and infinity. From this point of view the 

 problem discussed by Zeno remained untouched. 



In order to avoid running counter to this problem, 

 Greek science, with Eudoxus, had recourse to strata- 

 gem. This geometer begins by enunciating a theory 

 of proportions which, taking into consideration geo- 

 metrical continuity, is applicable to all ratios of mag- 

 nitude, whether commensurable or not. If (A, B) 



