THE MATHEMATICAL SCIENCES 127 



in the year 1575 that he became known to the Western 

 world. 1 



3. THE IRRATIONAL a/2. THE ARGUMENTS OF 

 ZENO OF ELEA. PROPORTIONS AND THE METHOD 

 OF EXHAUSTION. INTEGRAL CALCULUS. 2 



The arithmetical realism, naively proclaimed by the 

 Pythagoreans, was checked by the discovery that in 

 a square the diagonal and the side are incommensurable. 

 If space be number or ratio of numbers, this discovery 

 is disconcerting. The Pythagoreans doubtless did not 

 pretend to estimate the number of points which com- 

 pose a segment of a straight line, but they affirmed 

 that this number exists, and that it is necessarily a 

 whole number, since the point is indivisible. Between 

 two straight lines A and B of unequal length, there 

 must be the ratio A/B, in which A and B, representing 

 a sum of points, are necessarily two whole numbers. 

 This ratio leads in fact to a more simple ratio w/N, 

 if a suitable unit of measurement be chosen to estimate 

 the lengths A and B, since this now plays the part of 

 common factor. Let us now suppose that the sides of 

 a square each have 10 times the unknown number of 

 points. According to the so-called theorem of Pytha- 

 goras, the square described on the diagonal will contain 

 200 times this number. The diagonal must therefore 

 be equal to a whole number which, multiplied by itself, 

 gives exactly 200. Now 14 is too small, for 14 x 14 = 

 196, and 15 is too great, since 15 x 15 = 225. Then 

 let us take the side of a square equal to not 10 times 

 but 100 times, to 1,000 times, to n times the number 

 of points, etc. Whatever be the figure chosen, we 

 shall never find for the diagonal a number which 



1 23 Rouse Ball, History of Mathematics, I, p. 118. 



3 See our book, Logique et Mathematiques, Delachaux, 

 Neuchatel, 1900, and our article in the Revue de Metaphy- 

 sique et Morale, July 191 1, " Infini et science grecque." 



