320 
FROM PIATO TO EUCLID 
SiopLcrfioç indicated in the famous geometrical passage of the 
Meno 1 ; no doubt, too, the geometrical solution by the Pytha 
goreans of the quadratic equation would incidentally make 
clear to them the limits of possibility corresponding to the 
Siopia-pos in the solution of the most general form of quad 
ratic in Eucl. VI. 27-9, where, in the case of the ‘deficient 5 
parallelogram (Prop. 28), the enunciation states that ‘ the 
given rectilineal figure must not be greater than the parallelo 
gram described on half of the straight line and similar to the 
defect Again, the condition of the possibility of constructing 
a triangle out of three given straight lines (Eucl. I. 22), 
namely that any two of them must be together greater than 
the third, must have been perfectly familiar long before Leon 
or Plato. 
Proclus continues : 2 
‘ Eudoxus of Cnidos, a little younger than Leon, who had 
been associated with the school of Plato, was the first to 
increase the number of the so-called general theorems ; he 
also added three other proportions to the three already known, 
and multiplied the theorems which originated with Plato 
about the section, applying to them the method of analysis. 
Amyclas [more correctly Amyntas] of Heraclea, one of the 
friends of Plato, Menaechmus, a pupil of Eudoxus who had 
also studied with Plato, and Dinostratus, his brother, made 
the whole of geometry still more perfect. Theudius of 
Magnesia had the reputation of.excelling in mathematics as 
well as in the other branches of philosophy; for he put 
together the elements admirably and made many partial (or 
limited) theorems more general. Again, Athenaeus of Cyzicus, 
who lived about the same time, became famous in other 
branches of mathematics and most of all in geometry. These 
men consorted together in the Academy and conducted their 
investigations in common. Hermotimus of Colophon carried 
further the investigations already opened up by Eudoxus and 
Theaetetus, discovered many propositions of the Elements 
and compiled some portion of the theory of Loci, Philippus 
of Medma, who was a pupil of Plato and took up mathematics 
at his instance, not only carried out his investigations in 
accordance with Plato’s instructions but also set himself to 
do whatever in his view contributed to the philosophy of 
Plato. 5 
1 Plato, Meno, 87 a. 2 Proclus on Eucl. L, p. 67. 2-68. 4.