156 
PYTHAGOREAN GEOMETRY 
If then it was not Pythagoras but some Pythagorean who 
discovered the irrationality of V 2, at what date are we to 
suppose the discovery to have been made ? A recent writer 1 
on the subject holds that it was the later Pythagoreans who 
made the discovery, not much before 410 B.C. It is impos 
sible, he argues, that fifty or a hundred years would elapse 
between the discovery of the irrationality of V2 and the like 
discovery by Theodorus (about 410 or 400 B.c.) about the other 
surds Vs, Vs, &c. It is difficult to meet this argument 
except by the supposition that, in the interval, the thoughts 
of geometers had been taken up by other famous problems, 
such as the quadrature of the circle and the duplication of the 
cube (itself equivalent to finding £/2), Another argument is 
based on the passage in the Laivs where the Athenian stranger 
speaks of the shameful ignorance of the generality of Greeks, 
who are not aware that it is not all geometrical magnitudes 
that are commensurable with one another; the speaker adds 
that it was only ‘ late ’ (ó\¡se 7rore) that he himself learnt the 
truth. 2 Even if we knew for certain whether ‘ late ’ means 
‘ late in the day ’ or ‘ late in life ’, the expression would not 
help much towards determining the date of the first discovery 
of the irrationality of V2; for the language of the passage is 
that of rhetorical exaggeration (Plato speaks of men who are 
unacquainted with the existence of the irrational as more 
comparable to swine than to humaif beings). Moreover, the 
irrational appears in the Republic as something well known, 
and precisely with reference to V2; for the expressions ‘the 
rational diameter of (the square the side of which is) 5 ’ 
[— the approximation \/(49) or 7] and the ‘irrational 
(dpprjTos) diameter of 5 ’ [= V(50)] are used without any word 
of explanation. 3 
Further, we have a well-authenticated title of a work by 
Democritus (born 470 or 460 B.C.), 7repl dXóycov ypappwv kcc'í 
vaarcov a(3, 1 two books on irrational lines and solids ’ (vaarov 
is nXfjpes, ‘ full as opposed to Kevov. 1 void and Democritus 
called his ‘ first bodies ’ vetará). Of the contents of this work 
we are not informed; the recent writer already mentioned 
1 H. Yogt in Bibliotheca mathematica, x 3 , 1910, pp. 97-155 (cf. ix 3 , 
p. 190 sq.), 
2 Plato, Latvs, 819 n-820 c. 3 Plato, Republic, vii. 546 d.