THEODORUS OF CYRENE 
209 
as with the triangle ABC, and so on. This could not have 
escaped Theodorus if his proof in the cases of A3, V5 ... 
took the form suggested by Zeuthen; but he was presumably 
content to accept the traditional proof with regard to V2. 
The conjecture of Zeuthen is very ingenious, but, as he 
admits, it necessarily remains a hypothesis. 
Theaetetus 1 (about 415-369 b. c.) made important contribu 
tions to the body of the Elements'. These related to two 
subjects in particular, (a) the theory of irrationals, and (b) the 
five regular solids. 
That Theaetetus actually succeeded in generalizing the 
theory of irrationals on the lines indicated in the second part 
of the passage from Plato’s dialogue is confirmed by other 
evidence. The commentary on Eucl. X, which has survived 
in Arabic and is attributed to Pappus, says (in the passage 
partly quoted above, p. 155) that the theory of irrationals 
‘had its origin in the school of Pythagoras. It was con 
siderably developed by Theaetetus the Athenian, who gave 
proof in this part of mathematics, as in others, of ability 
which has been justly admired. ... As for the exact dis 
tinctions of the above-named magnitudes and the rigorous 
demonstrations of the propositions to which this theory gives 
rise, I believe that they were chiefly established by this 
mathematician. For Theaetetus had distinguished square 
roots 2 commensurable in length from those which are incom 
mensurable, and had divided the well-known species of 
irrational lines after the different means, assigning the medial 
to geometry, the binomial to arithmetic, and the apotome to 
harmony, as is stated by Eudemus the Peripatetic.’ 3 
1 On Theaetetus the reader may consult a recent dissertation, De Theae 
teto Atheniensi mathematico, by Eva Sachs (Berlin, 1914). 
2 ‘ Square roots ’. The word in Woepcke’s translation is ‘ puissances 
which indicates that the original word was Swa/ieis. This word is always 
ambiguous; it might mean ‘ squares ’, but I have translated it ‘ square 
roots ’ because the bvvafus of Theaetetus’s definition is undoubtedly the 
square root of a non-square number, a surd. The distinction in that case 
would appear to be between ‘square roots,’ commensurable in length and 
square roots commensurable in square only; thus \S3 and \/l2 are 
commensurable in length, while and */1 are commensurable in 
square only. I do not see how Swiifxeis could here mean squares; for 
‘squares commensurable in length’ is not an intelligible phrase, and it 
does not seem legitimate to expand fit into ‘ squares (on straight lines) 
commensurable in length 
3 For an explanation of this see The Thirteen Books of Euclid's Elements 
vol. iii, p. 4. 
1523 
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