128 SCIENCE IN GRECO-ROMAN ANTIQUITY 



when squared will exactly equal 2 x io\ Of this 

 fact the Pythagoreans were able to give the following 

 demonstration. Let a be the diagonal and b the side 

 of the square. These two numbers may be supposed 

 to be prime to one another, for if they were not, they 

 could always become so by the suppression of their 

 common factors. From the equation a 2 = 2b 2 we 

 must conclude that a 2 and consequently a is an even 

 number. Since a and b are prime to one another, 

 b can only be odd. But if a be even, we can postulate 

 a = 2,a x and the original relation becomes 4a x 2 = 2b 2 

 or 2a x 2 = b 2 . In this case b is even, but then a and b 

 are no longer prime to one another, which is contrary 

 to the hypothesis. The side and the diagonal of a 

 square are thus incommensurable. 



Although disconcerted by this discovery, the Pytha- 

 goreans regarded it as an isolated instance ; it did not 

 cause them to modify their arithmetical-spatial con- 

 ceptions, and they were not able to glimpse the 

 relationship between the continuum and infinity. 

 Zeno of Elea was the first to propound this problem 

 with precision. According to a generally accepted 

 opinion, he desired, in discussing this question, to prove 

 first of all the impossibility of motion, and, indirectly, 

 to deny the plurality of Being. But, as we have seen, 

 a passage of Plato (Parmenides, 128 C) shows that Zeno 

 simply sought to oppose the idea of plurality as 

 affirmed by the Pythagoreans. The testimony of 

 Plato is the more convincing since the argument of 

 Zeno has no significance if it denies the fact of motion, 

 but is, on the contrary, decisive in showing that motion 

 is incompatible with the hypothesis of plurality. Of 

 this argument briefly summed up by Aristotle (Phys. 

 239 b 9) we only possess the parts which deal with 

 continuity in its relations with infinity. 



According to Zeno it must be admitted that either 

 the division of space, time and motion can be continued 



