THE IRRATIONAL 
155 
proper sense. We are told in the Theaetetus 1 that Theodorus 
of Gyrene (a pupil of Protagoras and the teacher of Plato) 
proved the irrationality of Vs, Vs, &c., up to Vl7, and this 
must have been at a date not much, if anything, earlier than 
400 b. c.; while it was Theaetetus who, inspired by Theodorus’s 
investigation of these particular ‘roots’ (or surds), was the 
first to generalize the theory, seeking terms to cover all such 
incommensurables; this is confirmed by the continuation of 
the passage from Pappus’s commentary, which says that the 
theory was 
‘considerably 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 
It follows from all this that, if Pythagoras discovered any 
thing about irrationals, it was not any ‘ theory ’ of irrationals 
but, at the most, some particular case of incommensurability. 
Now the passage which states that Theodorus proved that 
Vs, Vo, &c. are incommensurable says nothing of V2. The 
reason is, no doubt, that the incommensurability of V2 had# 
been proved earlier, and everything points to the probability 
that this was the first case to be discovered. But, if Pytha 
goras discovered even this, it is difficult to see how the theory 
that number is the essence of all existing things, or that all 
things are made of number, could have held its ground for 
any length of time. The evidence suggests the conclusion 
that geometry developed itself for some time on the basis of 
the numerical theory of proportion which was inapplicable to 
any but commensurable magnitudes, and that it received an 
unexpected blow later by reason of the discovery of the irra 
tional. The inconvenience of this state of things, which 
involved the restriction or abandonment of the use of propor 
tions as a method pending the discovery of the generalized 
theory by Eudoxus, may account for the idea of the existence 
of the irrational having been kept secret, and of punishment 
having overtaken the first person who divulged it. 
1 Plato, Theaetetus, 147 n sq.