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.