CLASSIFICATION OF MATHEMATICS 
13 
accounting for 
list of subjects 
bus, Theon of 
3i', arithmetic, 
his order was 
l with number 
3 with number 
. sphaeric were 
dry with mag- 
on. In Plato’s 
same subjects, 
metry, appear, 
Dlid geometry, 
of stereometry 
was, however, 
for of course 
as a part of 
3 and others, 
ogical. Astro- 
ire is therefore 
for, after con- 
e third dimen- 
before passing 
motion. But 
n, namely that 
studied. ‘ The 
en discovered.’ 
cause no State 
ffi are difficult, 
ho do investi- 
vithout whose 
;ries. But, to 
intendent, and 
>w stand, those 
prevented by 
> i 
now stand ’) in 
tances’, i.e. so 
long as the director has not the authority of the State behind 
him: this seems to be the best interpretation in view of the 
whole context; but it is possible, as a matter of construction, 
to connect the phrase with the preceding words, in which case 
the meaning would-be ‘and, even when such a superintendent 
has been found, as is the case at present’, and Plato would 
be pointing to some distinguished geometer among his con 
temporaries as being actually available for the post. If Plato 
intended this, it would presumably be either Archytas or 
Eudoxus whom he had in mind. 
It is again on a logical ground that Plato made harmonics 
or music follow astronomy in his classification. As astronomy 
is the motion of bodies (фора (Забои?) and appeals to the eye, 
so there is a harmonious motion (kvappovto? фора), a motion 
according to the laws of harmony, which appeals to the ear. 
In maintaining the sisterhood of music and astronomy Plato 
followed the Pythagorean view (cf. the passage of Archytas 
above quoted and the doctrine of the ‘ harmony of the 
spheres ’), 
(a) Arithmetic and logistic. 
By arithmetic Plato meant, not arithmetic in our sense, but 
the science which considers numbers in themselves, in other 
words, what we mean by the Theory of Numbers. He does 
not, however, ignore the art of calculation (arithmetic in oui* 
sense); he speaks of number and calculation {арсврог ка1 
Хоу игрой) and observes that ‘ the art of calculation [Хоуиткр) 
and arithmetic {арсврдтскд) are both concerned with number 
those who have a natural gift for calculation (oi фугас Xoyc- 
(ttlkol) have, generally speaking, a talent for learning of all 
kinds, and even those who are slow are, by practice in it, 
made smarter. 1 But the art of calculation (Хоусапкд) is only 
preparatory to the true science; those who are to govern the 
city are to get a grasp of Хоуихтскр, not in the popular 
sense with a view to use in trade, but only for the pui’pose of 
knowledge, until they are able to contemplate the nature^of 
number in itself by thought alone. 2 This distinction between 
арсврдтскр (the theory of numbers) and Xоусатскд (the art of 
«■ 1 Republic, vii. 522 c, 525 A, 526 в. 
2 lb. vii. 525 в, c.