1523 
z 
ARISTOTLE 
337 
and without any assent on the part of the learner, or even 
against his opinion rather than otherwise. As regards defini 
tions, Aristotle is clear that they do not assert existence or 
non-existence; they only require to be understood. The only 
exception he makes is in the case of the unit or monad and 
magnitude, the existence of which has to be assumed, while 
the existence of everything else has to be proved; the things 
actually necessary to be assumed in geometry are points and 
lines only; everything constructed out of them, e.g. triangles, 
squares, tangents, and their properties, e.g. incommensura 
bility, has to be proved to exist. This again agrees sub 
stantially with Euclid’s procedure. Actual construction is 
with him the proof of existence. If triangles other than the 
equilateral triangle constructed in I. 1 are assumed in I. 4-21, 
it is only provisionally, pending the construction of a triangle 
out of three straight lines in I. 22 ; the drawing and producing 
of straight lines and the describing of circles is postulated 
(Postulates 1-3). Another interesting statement on the 
philosophical side of geometry has reference to the geometer’s 
hypotheses. It is untrue, says Aristotle, to assert that a 
geometer’s hypotheses are false because he assumes that a line 
which he has drawn is a foot long when it is not, or straight 
when it is not straight. The geometer bases no conclusion on 
the particular line being that which he has assumed it to be; 
he argues about what it represents, the figure itself being 
a mere illustration. 1 
Coming now to the first definitions of Euclid, Book I, we 
find that Aristotle has the equivalents of Defs. 1-3 and 5, 6. 
But for a straight line he gives Plato’s definition only: 
whence we may fairly conclude that Euclid’s definition 
was his own, as also was his definition of a plane which 
he adapted from that of a straight line. Some terms seem 
to have been defined in Aristotle’s time which Euclid leaves 
undefined, e. g. KtK\da6ou, '■ to be inflected ’, vevew, to ‘ verge ’. 1 2 
Aristotle seems to have known Eudoxus’s new theory of pro 
portion, and he uses to a considerable extent the usual 
1 Arist. Anal. Post. i. 10. 76 b 39-77 a 2 ; cf. Anal. Prior, i. 41. 49 b 34 sq.; 
Metaph. N. 2. 1089 a 20-5. • . 
2 Anal. Post. i. 10. 76 b 9,