DATE AND TRADITIONS 
357 
could advance mathematical science to however small an 
extent. 1 Although, as I have indicated, Pappus’s motive was 
rather to represent Apollonius in a relatively unfavourable 
light than to state a historical fact about Euclid, the state 
ment accords well with what we should gather from Euclid’s 
own works. These show no sign of any claim to be original; 
in the Elements, for instance, although it is clear that he 
made great changes, altering the arrangement of whole Books, 
redistributing propositions between them, and inventing new 
proofs where the new order made the earlier proofs inappli 
cable, it is safe to say that he made no more alterations than 
his own acumen and the latest special investigations (such as 
Eudoxus’s theory of proportion) showed to be imperative in 
order to make the exposition of the whole subject more 
scientific than the earlier efforts of writers of elements. His 
respect for tradition is seen in his retention of some things 
which were out of date and useless, e. g. certain definitions 
never afterwards used, the solitary references to the angle 
of a semicircle or the angle of a segment, and the like; he 
wrote no sort of preface to his work (would that he had!) 
such as those in which Archimedes and Apollonius introduced 
their treatises and distinguished what they claimed as new in 
them from what was already known: he plunges at once into 
his subject, ‘ A point is that which has no part ’! 
And what a teacher he must have been! One story enables 
us to picture him in that capacity. According to Stobaeus, 
‘ some one who had begun to read geometry with Euclid, 
when he had learnt the first theorem, asked Euclid, “what 
shall I get by learning these things ? ” Euclid called his slave 
and said, “ Give him threepence, since he must, make gain out 
of what he learns ”.’ 2 
Ancient commentaries, criticisms, and references. 
Euclid has, of course, always been known almost exclusively 
as the author of the Elements. -From Archimedes onwards 
the Greeks commonly spoke of him as o aroLxeicoTTjs, the 
writer of the Elements, instead of using his name. This 
wonderful book, with all its imperfections, which indeed are 
slight enough when account is taken of the date at which 
1 Pappus, vii, pp. 676. 25-678. 6. - Stobaeus, Floril. iv. p. 205.