PYTHAGOREAN GEOMETRY 
The special service rendered by Pythagoras to geometry is 
thus described in the Proclus summary : 
‘After these (Thalesand Ameristusor Mamercus) Pythagoras 
transformed the study of geometry into a liberal education, 
examining the principles of the science from tliOj,beginning 
and probing the theorems in an immaterial and intellectual 
manner : he it was who discovered the theory of irrationals ’ 
(or ‘ proportions ’) ‘ and the construction of the cosmic figures A 
These supposed discoveries will claim our attention pre 
sently ; the rest of the description agrees with another 
passage about the Pythagoreans : 
‘ Herein ’, says Proclus, ‘ I emulate the Pythagoreans who 
even had a conventional phrase to express what I mean, 
“a figure and a platform, not a figure and sixpence”, by 
which they implied that the geometry which is deserving of 
study is that which, at each new theorem, sets up a platform to 
ascend by, and lifts the soul on high instead of allowing it 
to go down among sensible objects and so become subser 
vient to the common needs of this mortal life ’. 2 
In like manner we are told that ‘ Pythagoras used defini 
tions on account of the mathematical nature of the subject ’, 3 
which again implies that he took the first steps towards the 
systematization of geometry as a subject in itself. 
A comparatively early authority,Callimachus (about 250 b.c.), 
is quoted by Diodorus as having said that Pythagoras dis 
covered some geometrical problems himself and was the first 
to introduce others from Egypt into Greece. 4 Diodorus gives 
what appear to be five verses of Callimachus minus a few words ; 
1 Proclus on End. I, p. 65. 15-21. 2 lb., p. 84. 15-22. 
3 Pavorinus in Diog. L. viii. 25. 
4 Diodorus x. 6. 4 (Vors. i s , p. 346. 23).