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.