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).