DEMOCRITUS 
179 
given rectilineal angle. Euclid proved (in III. 16) that the 
‘ angle of contact ’ is less than any rectilineal angle, thereby 
setting the question at rest. This is the only reference in 
Euclid to this angle and the ‘ angle of a semicircle ’, although 
he defines the ‘angle of a segment’ in III, Def. 7, and has 
statements about the angles of segments in III. 31. But we 
know from a passage of Aristotle that before his time ‘ angles 
of segments ’ came into geometrical text-books as elements in 
figures which could be used in the proofs of propositions 1 ; 
thus e.g. the equality of the two angles of a segment 
(assumed as known) was used to prove the theorem of 
Eucl. I. 5. Euclid abandoned the use of all such angles in 
proofs, and the references to them above mentioned are only 
survivals. The controversies doubtless arose long before his 
time, and such a question as the nature of the contact of 
a circle with its tangent would probably have a fascination 
for Democritus, who, as we shall see, broached other questions 
involving infinitesimals. As, therefore, the questions of the 
nature of the contact of a circle with its tangent and of the 
character of the ‘ hornlike ’ angle are obviously connected, 
I prefer to read yooviys (‘of an angle’) instead of yvdgiys ; this 
would give the perfectly comprehensible title, ‘ On a difference 
in an angle, or on the contact of a circle and a sphere ’. We 
know from Aristotle that Protagoras, who wrote a book on 
mathematics, ire pi rcov ¡xadygaToov, used against the geometers 
the argument that no such straight lines and circles as 
they assume exist in nature, and that (e. g.) a material circle 
does not in actual fact touch a ruler at one point only 2 ; and 
it seems probable that Democritus’s work was directed against 
this sort of attack on geometry. 
We know nothing of the contents of Democritus’s book 
On Geometry or of his Geometrica. One or other of these 
works may possibly have contained the famous dilemma about 
sections of a cone parallel to the base arid very close together, 
which Plutarch gives on the authority of Chrysippus. 3 
‘ If’, said Democritus, ‘a cone were cut by a plane parallel 
to the base [by which is clearly meant a plane indefinitely 
1 Arist. Anal. Pr. i. 24, 41 b 18-22. 
2 Arist. Metaph. B. 2, 998 a 2. 
3 Plutarch, De comm. not. aclv. Stoicos, xxxix. 8. 
N 2