THEAETETUS 
211 
of the theorem of Eucl. X. 9, was the achievement which Plato 
intended to hold up to admiration. 
This conjecture is of great interest, but it is, so far as 
I know, without any positive confirmation. On the other 
hand, there are circumstances which suggest doubts. For 
example, Zeuthen himself admits that Hippocrates, who re 
duced the duplication of the cube to the finding of two mean 
proportionals, must have had a proposition corresponding to 
the very proposition VIII. 11 on which X. 9 formally depends. 
Secondly, in the extract from Simplicius about the squaring 
of lunes by Hippocrates, we have seen that the proportionality 
of similar segments of circles to the circles of which they form 
part is explained by the statement that ‘ similar segments are 
those which are the same part of the circles ’ ; and if we may 
take this to be a quotation by Eudemus from Hippocrates’s 
own argument, the inference is that Hippocrates had a defini 
tion of numerical proportion which was at all events near 
to that of Eucl. VII, Def. 20. Thirdly, there is the proof 
(presently to be given) by Archytas of the proposition that 
there can be no number which is a (geometric) mean between 
two consecutive integral numbers, in which proof it will 
be seen that several propositions of Eucl., Book VII, are 
pre-supposed ; but Archytas lived (say) 430-S65 B.C., and 
Theaetetus was some years younger. I am not, therefore, 
prepared to give up the view, which has hitherto found 
general acceptance, that the Pythagoreans already had a 
theory of proportion of a numerical kind on the lines, though 
not necessarily or even probably with anything like the 
fullness and elaboration, of Eucl., Book VII. 
While Pappus, in the commentary quoted, says that Theae 
tetus distinguished the well-known species of irrationals, and 
in particular the medial, the binomial, and the apotome, he 
proceeds thus : 
‘As for Euclid, he set himself to'give rigorous rules, which 
he established, relative to commensurability and incommen 
surability in general ; he made precise the definitions and 
distinctions between rational and irrational magnitudes, he 
set out a great number of orders of irrational magnitudes, 
and finally he made clear their whole extent.’ 
As Euclid proves that there are thirteen irrational straight