THE IRRATIONAL 
157 
suggests that dXoyoy does not here mean irrational or incom 
mensurable at all. but that the book was an attempt to con 
nect the atomic theory with continuous magnitudes (lines) 
through ‘ indivisible lines ’ (cf. the Aristotelian treatise On 
indivisible lines), and that Democritus meant to say that, 
since any two lines are alike made up ot‘ an infinite number 
of the (indivisible) elements, they cannot be said to have any 
expressible ratio to one another, that is, he would regard them 
as ‘ having no ratio ’! It is, however, impossible to suppose 
that a mathematician of the calibre of Democritus could have 
denied that any two lines can have a ratio to one another; 
moreover, on this view, since no two straight lines would have 
a ratio to one another, dXoyot y pappat would not be a class of 
lines, but all lines, and the title would lose all point. But 
indeed, as we shall see, it is also on other grounds inconceiv 
able that Democritus should have been an upholder of * indi 
visible lines ’ at all. I do not attach any importance to the 
further argument used in support of the interpretation in 
question, namely that dXoyos in the sense of ‘irrational’ is 
not found in any other writer before Aristotle, and that 
Plato uses the words dpppros and ao-vpperpos only. The 
latter statement is not even strictly true, for Plato does in 
fact use the word dXoyoL specifically of y pappat in the passage 
of the Republic where he speaks of youths not being dXoyoL 
ooa-rrep ypappat, ‘ irrational like lines Poor as the joke is, 
it proves that dXoyot ypappat was a recognized technical 
term, and the remark looks like a sly reference to the very 
treatise of Democritus of which we are speaking. I think 
there is no reason to doubt that the book was on ‘ irrationals ’ 
in the technical sense. We know from other sources that 
Democritus was already on the track of infinitesimals in 
geometry; and nothing is more likely than that he would 
write on the kindred subject of irrationals. 
I see therefore no reason to doubt that the. irrationality 
of V2 was discovered by some Pythagorean at a date appre 
ciably earlier than that of Democritus; and indeed the simple 
proof of it indicated by Aristotle and set out in the propo 
sition interpolated at the end of Euclid’s Book X seems 
appropriate to an early stage in the development of geometry. 
1 Plato, Republic, 534 n.