324 
FROM PLATO TO EUCLID 
at the beginning of Euclid’s Book X similarly refer to magni 
tudes in general, and the proposition X. 1 or its equivalent 
was actually used by Eudoxus in his method of exhaustion, 
as it is by Euclid in his application of the same method to the 
theorem (among others) of XII. 2 that circles are to one 
another as the squares on their diameters. 
The three 4 proportions ’ or means added to the three pre 
viously known (the arithmetic, geometric and harmonic) have 
already been mentioned (p. 86), and, as they are alterna 
tively attributed to others, they need not detain us here. 
Thirdly, we are told that Eudoxus 4 extended ’ or 4 increased 
the number of the (propositions) about the section (ra ?repl 
rrjv To/x-qv) which originated with Plato, applying to them 
the method of analysis’. What is the section"? The sugges 
tion which has been received with most favour is that of 
Bretschneider, 1 who pointed out that up to Plato’s time there 
was only one 4 section ’ that had any real significance in 
geometry, namely the section of a straight line in extreme 
and mean ratio which is obtained in Eucl. II. 11 and is used 
again in Eucl. IV, 10-14 for the construction of a pentagon. 
These theorems were, as we have seen, pretty certainly Pytha 
gorean, like the whole of the substance of Euclid, Book II. 
Plato may therefore, says Bretschneider, have directed atten 
tion afresh to this subject and investigated the metrical rela 
tions between the segments of a straight line so cut, while 
Eudoxus may have continued the investigation where Plato 
left off. Now the passage of Proclus says that, in extending 
the theorems about 4 the section ’, Eudoxus applied the method 
of analysis; and we actually find in Eucl. XIII. 1-5 five 
propositions about straight lines cut in extreme and mean 
ratio followed, in the MSS., by definitions of analysis and 
synthesis, and alternative proofs of the same propositions 
in the form of analysis followed by synthesis. Here, then, 
Bretschneider thought he had found a fragment of some actual 
work by Eudoxus corresponding to Proclus’s description. 
But it is certain that the definitions and the alternative proofs 
were interpolated by some scholiast, and, judging by the 
figures (which are merely straight lines) and by comparison 
1 Bretschneider, Die Geometrie und ’die Geometer vor Eukleides, pp. 
167-9.