* 
CLASSIFICATION OF MATHEMATICS 11 
it God cannot 
ant of human 
ies them, men 
t. And he is 
s, and that we 
r t , number and 
mid show the 
ik of would in 
a man capable 
e unreservedly 
’ i 
eek conviction 
;ion and scien- 
impiety in the 
el to the State- 
Г. 
ematics. 
not appear to 
ial meaning of 
bematical until 
general, mean- 
speaks of KaXa 
<aXd e?nrrjSev- 
posed to men’s, 
hat, he asks in 
be answers that 
. 2 But in the 
jects, as fit for 
the science of 
1 no doubt the 
bs in his scheme 
ng the habit of 
adij/xara. The 
dal use of the 
eas such things 
lar [ioixtlkti can 
; them, the sub- 
,nnot be known 
laws, vii. 817 E. 
by any one who has not first gone through a course of instruc 
tion in them ; they concluded that it was for this reason that 
these studies were called /xadiухапку. 1 The special use of the 
word [хабт][idTLKTj seems actually to have originated in the 
school of Pythagoras. It is said that the esoteric members 
of the school, those who had learnt the theory of know 
ledge in its most complete form and with all its elaboration 
of detail, were known as ¡хабу¡хатlkol, mathematicians (as 
opposed to the dKovcr/xariKOL, the exoteric learners who were 
entrusted, not with the inner theory, but only with the prac 
tical rules of conduct) ; and, seeing that the Pythagorean 
philosophy was mostly mathematics, the term might easily 
come to be identified with the mathematical subjects as 
distinct from others. According to Anatolius, the followers 
of Pythagoras are said to have applied the term ¡хабщхалкг] 
more particularly to the two subjects of geometry and 
arithmetic, which had previously been known by their own 
separate names only and not by any common designation 
covering both. 2 There is also an apparently genuine frag 
ment of Archytas, a Pythagorean and a contemporary and 
friend of Plato, in which the word /хавгцхата appears as 
definitely appropriated to mathematical subjects : 
‘ The mathematicians (roî nepl та ¡хавгцхата) seem to me to 
have arrived at correct conclusions, and it is not therefore 
surprising that they have a true conception of the nature of 
each individual thing ; for, having reached such correct con 
clusions regarding the nature of the universe, they were 
bound to see in its true light the nature of particular things 
as well. Thus they have handed down to us clear knowledge 
about the speed of the stars, their risings and settings, and 
about geometry, arithmetic, and sphaeric, and last, not least, 
about music; for these /хавгцхата seem to be sisters.’ 3 
This brings us to the Greek classification of the different 
branches of mathematics. Archytas, in the passage quoted, 
specifies the four subjects of the Pythagorean quadrivium, 
geometry, arithmetic, astronomy, and music (for ‘sphaeric’ 
means astronomy, being the geometry of the sphere con- 
1 Anatolius in Hultscli’s Heron, pp. 276-7 (Heron, vol. iv, Heiberg, 
p. 160. 18-24). 
2 Heron, ed. Hultsch, p. 277 ; vol. iv, p. 160. 24-162. 2, Heiberg. 
3 Diels, Vorsokratiker, i 3 , pp. 330-1.