EUDOXUS 
325 
with the remarks on analysis and synthesis quoted from 
Heron by An-Nairizi at the beginning of his commentary on 
Enel. Book II, it seems most likely that the interpolated defini 
tions and proofs were taken from Heron. Bretschneider’s 
argument based on Eucl. XIII. 1-5 accordingly breaks down, 
and all that can be said further is that, if Eudoxus investi 
gated the relation between the segments of the straight line, 
he would find in it a case of incommensurability which would 
further enforce the necessity for a theory of proportion which 
should be applicable to incommensurable as well as to com 
mensurable magnitudes. Proclus actually observes that 
‘ theorems about sections like those in Euclid’s Second Book 
are common to both [arithmetic and geometry] except that in 
which the straight line is cut in extreme and mean ratio ’ 1 
(cf. Eucl, XIII. 6 for the actual proof of the irrationality 
in this case). Opinion, however, has not even in recent years 
been unanimous in favour of Bretschneider’s interpretation ; 
Tannery 2 in particular preferred the old view, which pre 
vailed before Bretschneider, that ‘ section ’ meant section of 
solids, e. g. by planes, a line of investigation which would 
naturally precede the discovery of conics ; he pointed out that 
the use of the singular, rr/r To/irju, which might no doubt 
be taken as ‘ section ’ in the abstract, is no real objection, that 
there is no other passage which speaks of a certain section 
par excellence, and that Proclus in the words just quoted 
expresses himself quite differently, speaking of ‘sections’ of 
which the particular section in extreme and mean ratio is 
only one. Presumably the question will never be more defi 
nite^ settled unless by the discovery of new documents. 
(a) Theory of proportion. 
The anonymous author of a scholium to Euclid’s Book Y, 
who is perhaps Proclus, tells us that ‘ some say ’ that this 
Book, containing the general theory of proportion which is 
equally applicable to geometry, arithmetic, music and all 
mathematical science, ‘ is the discovery of Eudoxus, the teacher 
of Plato’. 3 There is no reason to doubt the truth of this 
1 Proclus on Eucl. I. p. 60. 16-19. 
2 Tannery, ha géométrie grecque, p. 76. 
s Euclid, ed. Heib., vol. v, p. 280.