210 THE ELEMENTS DOWN TO PLATO’S TIME 
The irrationals called by the names here italicized are 
described in Enel. X. 21, 36 arid 73 respectively. 
Again, a scholiast 1 on Enel. X. 9 (containing the general 
theorem that squares which have not to one another the ratio 
of a square number to a square number have their sides 
incommensurable in length) definitely attributes the discovery 
of this theorem to Theaetetus. But, in accordance with the 
traditional practice in Greek geometry, it was necessary to 
prove the existence of such incommensurable ratios, and this 
is done in the porism to Eucl. X. 6 by a geometrical con 
struction ; the porism first states that, given a straight line a 
and any two numbers m, n, we can find a straight line x such 
that a:x = m\n\ next it is shown that, if y be taken a mean 
proportional between a and x, then 
a 2 :y 2 = a: x = m: n; 
if, therefore, the ratio m: n is not a ratio of a square to 
a square, we have constructed an. irrational straight line 
a V(n/m) and therefore shown that such a straight line 
exists. 
The proof of Eucl. X. 9 formally depends on YIII. 11 alone 
(to the effect that between two square numbers there is one 
mean proportional number, and the square has to the square 
the duplicate ratio of that which the side has to the side); 
and YIII. 11 again depends on VII. 17 and 18 (to the effect 
that ah: ac = h: c, and a:b = ac:bc, propositions which are 
not identical). But Zeuthen points out that these propositions 
are an inseparable part of a whole theory established in 
Book VII and the early part of Book VIII, and that the 
real demonstration of X. 9 is rather contained in propositions 
of these Books which give a rigorous proof of the necessary 
and sufficient conditions for the rationality of the square 
roots of numerical fractions and integral numbers, notably 
VII. 27 and the propositions leading up to it, as well as 
VIII. 2. He therefore suggests that the theory established 
in the early part of Book VII was not due to the Pytha 
goreans, but was an innovation made by Theaetetus with the 
direct object of laying down a scientific basis for his theory 
of irrationals, and that this, rather than the mere formulation 
1 X, No. 62 (Heiberg’s Euclid, vol. v, p. 450).