12 
INTRODUCTORY 
sidered solely with reference to the problem of accounting for 
the motions of the heavenly bodies) ; the same list of subjects 
is attributed to the Pythagoreans by Nicomachus, Theon of 
Smyrna, and Proclus, only in a different order, arithmetic, 
music, geometry, and sphaeric ; the idea in this order was 
that arithmetic and music were both concerned with number 
(7to(tov), arithmetic with number in itself, music with number 
in relation to something else, while geometry and sphaeric were 
both concerned with magnitude (), geometry with mag 
nitude at rest, sphaeric with magnitude in motion. In Plato’s 
curriculum for the education of statesmen the same subjects, 
with the addition of stereometry or solid geometry, appear, 
arithmetic first, then geometry, followed by solid geometry, 
astronomy, and lastly harmonics. The mention of stereometry 
as an independent subject is Plato’s own idea ; it was, however, 
merely a formal addition to the curriculum, for of course 
solid problems had been investigated earlier, as a part of 
geometry, by the Pythagoreans, Democritus and others. 
Plato’s reason for the interpolation was partly logical. Astro 
nomy treats of the motion of solid bodies. There is therefore 
a gap between plane geometry and astronomy, for, after con 
sidering plane figures, we ought next to add the third dimen 
sion and consider solid figures in themselves, before passing 
to the science which deals with such figures in motion. But 
Plato emphasized stereometry for another reason, namely that 
in his opinion it had not been sufficiently studied. ‘ The 
properties of solids do not yet seem to have been discovered.’ 
He adds : 
‘ The reasons for this are two. First, it is because no State 
holds them in honour that these problems, which are difficult, 
are feebly investigated ; and, secondly, those who do investi 
gate them are in need of a superintendent, without whose 
guidance they are not likely to make discoveries. But, to 
begin with, it is difficult to find such a superintendent, and 
then, even supposing him found, as matters now stand, those 
who are inclined to these researches would be prevented by 
their self-conceit from paying any heed to him.’ 1 
I have translated ¿9 vvv eyei (‘ as matters now stand ’) in 
this passage as meaning ‘in present circumstances’, i.e. so 
1 Plato, Republic, vii. 528 A-c.