DEMOCRITUS 
181 
of Democritus’s works of/he treatises// n Geometry, Geometrica, 
Numbers, and On irrational lines and solids corresponds to 
the order of the separate sections of Euclid’s Elements, Books 
I-VT (plane geometry), Books YII-IX (on numbers), and 
Book X (on irrationals). With regard to the work On irra 
tional Lines and solids it is to be observed that, inasmuch as 
his investigation of the cone had brought Democritus con 
sciously face to face with infinitesimals, there is nothing 
surprising in his having written on irrationals; on the con 
trary, the subject is one in which he would be likely to take 
special interest. It is useless to speculate on what the treatise 
actually contained ; but of one thing we may be sure, namely 
that the âXoyoL y pa y y at, 'irrational lines’, were not aroyot 
ypayyai, ‘indivisible lines’. 1 Democritus was too good a 
mathematician to have anything to do with such a theory. 
We do not know what answer he gave to his puzzle about the 
cone ; but his statement of the dilemma shows that he was 
fully alive to the difficulties connected with the conception of 
the continuous as illustrated by the particular case, and he 
cannot have solved it, in a sense analogous to his physical 
theory of atoms, by assuming indivisible lines, for this would 
have involved the inference that the consecutive parallel 
sections of the cone are unequal, in which case the surface 
would (as he said) # be discontinuous, forming steps, as it were. 
Besides, we are told by Simplicius that, according to Demo 
critus himself, his atoms were, in a mathematical sense 
divisible further and in fact ad infinitum, 2 while the scholia 
to Aristotle’s De caelo implicitly deny to Democritus any 
theory of indivisible lines : ‘ of those who have maintained 
the existence of indivisibles, some, as for example Leucippus 
and Democritus, believe in indivisible bodies, others, like 
Xenocrates, in indivisible lines ’. 3 
With reference to the ’EKTreTaaryara it is to be noted that 
this word is explained in Ptolemy’s Geograq)hy as the projec 
tion of the armillary sphere upon a plane. 4 This work and 
that On irrational lines would hardly belong to elementary 
geometry, 
1 On this cf. 0. Apelt, Beitrcige zur Geschichte der griechischen Philo 
sophie, 1891, p. 265 sq. 
2 Sim pi. in Phys., p. 83. 5. 3 Scholia in Arist., p. 469 b 14, Brandis. 
4 Ptolemy, Geogr. vii. 7.