THEODORUS OF CYRENE 
205 
of proof can be adapted to the cases of U3, V5, &c., if 3, 5 ... 
are substituted for 2 in the proof; e.g. we can prove that, 
if \/3 is commensurable with 1, then the same number will 
be both divisible and not divisible by 3. One suggestion, 
therefore, is that Theodorus may have applied this method 
to all the cases from V3 to V17. We can put the proof 
quite generally thus. Suppose that N is a non-square number 
such as 3,5..., and, if possible, let VN = m/n, where m, n 
are integers prime to one another. 
Therefore m 2 = N .n 2 ; 
therefore m 2 is divisible by N, so that m also is a multiple 
of N. 
Let m — jx.N, (1) 
and consequently n 2 — N. fi 2 . 
Then in the same way we can prove that n is a multiple 
of N. 
Let n — v. N (2) 
It follows from (1) and (2) that m/n — ¡x/v, where ¡i < m 
and v < n; therefore m/n is not in its lowest terms, which 
is contrary to the hypothesis. 
The objection to this conjecture as to the nature of 
Theodorus’s proof is that it is so easy an adaptation of the 
traditional proof regarding' V2 that it would hardly be 
important enough to mention as a new discovery. Also it 
would be quite unnecessary to repeat the proof for every 
case up to V17; for it would be clear, long before V17 was 
reached, that it is generally applicable. The latter objection 
seems to me to have force. The former objection may or may 
not; for I do not feel sure that Plato is necessarily attributing 
any important new discovery to Theodorus. t The object of 
the whole context is to show that a definition by mere 
enumeration is no definition; e.g. it is no definition of em- 
arijfjiT] to enumerate particular kTnarrj[jLaL (as shoemaking, 
carpentering, and the like); this is to put the cart before the 
horse, the general definition of eiriarri^-q being logically prior. 
Hence it was probably Theaetetus’s generalization of the 
procedure of Theodorus which impressed Plato ag bein$ 
original and important rather than Theodorus’s proofs them 
selves.