THE MATHEMATICAL SCIENCES 131 



and atomism could take shape, thanks to the work of 

 Leucippus and Democritus. 



From a mathematical point of view the problem to 

 be solved is the following : no longer to identify dis- 

 continuous number with continuous magnitude, and 

 yet to find a means of adapting number to the study 

 of geometrical figures. This problem is difficult, for 

 the reasoning of Zeno seems to be faultless, and the 

 impossibility of reconciling it with the data of spatial 

 intuition seems to condemn for ever the rational and 

 direct use of mathematical infinity. On the other 

 hand, in practical applications, certain sophists such 

 as Antiphon affirm, on the basis of these reasonings, 

 an identity between curvilinear and rectilinear elements 

 which is inaccep table. 



Thus, in spite of the efforts of Aristotle to render 

 legitimate the notion of continuity, the confidence of 

 Greek mathematicians in directly infinitesimal specu- 

 lations was for ever shaken. Besides, the formulae 

 enunciated by Aristotle were not of any practical use 

 in mathematics ; they belonged to a treatise on physics 

 which had in the highest degree a metaphysical 

 character. To Aristotle, indeed, the question which 

 presented itself is the following : "If infinity be a 

 given reality, the enumeration of all the whole numbers 

 must have a limit, which is logically impossible (Phys., 

 204 b 4-10). But to reject infinity is to declare that 

 time has a beginning, that magnitude is discontinuous 

 and that the power to reckon has a limit (Phys., 206 

 a 9-12). To remove these difficulties it is necessary, 

 according to Aristotle, to distinguish between magni- 

 tude and number in the problem of infinity. 1 An 

 infinite magnitude could no more exist than an in- 

 finite space. As a matter of fact, space could not 

 extend beyond the material world of which it forms 

 the boundary (Phys., 212-31). If the universe were 

 1 Cf. 21 Milhaud, Etudes, p. 120. 



