178 THE ELEMENTS DOWN TO PLATO’S TIME 
the continuation of the fragment of doubtful authenticity 
already quoted that 
‘in the putting together of lines, with the necessary proof, no 
one has yet surpassed me, not even the so-called harpedon- 
aptae (rope-stretchers) of Egypt ’. 
This does not tell us much, except that it indicates that 
the ‘ rope-stretchers ’, whose original function was land 
measuring or practical geometry, had by Democritus’s time 
advanced some way in theoretical geometry (a fact which the 
surviving documents, such as the book of Ahmes, with their 
merely practical rules, would not have enabled us to infer). 
However, there is no reasonable doubt that in geometry 
Democritus was fully abreast of the knowledge of his day; 
this is fully confirmed by the titles of treatises by him and 
from other sources. The titles of the works classed as mathe 
matical are (besides the astronomical works above mentioned): 
1. On a difference of opinion {yvoofnjs: v. I. yvcoyovos, gno 
mon), or on the contact of a circle and a sphere; 
2. On Geometry; 
3. Geometricorum (? I, II)/ 
4. Numbers; 
5. On irrational lines and solids {volcttoov, atoms 1); 
6. ’EKTreTcctryaTa. 
As regards the first of these works I think that the 
attempts to extract a sense out of Cobet’s reading yva>[iovos 
(on a difference of a gnomon) have failed, and that yvcoyys 
(Diels) is better. But ‘ On a difference of opinion ’ seems 
scarcely determinative enough, if this was really an alternative 
title to the book. We know that there were controversies in 
ancient times about the nature of the ‘ angle of contact ’ (the 
‘angle’ formed, at the point of contact, between an arc of 
a circle and the tangent to it, which angle was called by the 
special name hornlike, KeparoeLSyy), and the ‘angle’ comple 
mentary to it (the ‘angle of a semicircle’). x The question was 
whether the ‘hornlike angle’ was a magnitude comparable 
with the rectilineal angle, i.e. whether by being multiplied 
a sufficient number of times it could be made to exceed a 
* Proclus on End. I, pp. 121, 24-122. 6.