212 THE ELEMENTS DOWN TO PLATO’S TIME 
lines in all, we may perhaps assume that the subdivision of 
the three species of irrationals distinguished by Theaetetus 
into thirteen was due to Euclid himself, while the last words 
of the quotation seem to refer to Eucl. X. 115, where it is 
proved that from the medial straight line an unlimited number 
of other irrationals can be derived which are all different from 
it and from one another. 
It will be remembered that, at the end of the passage of the 
Theaetetus containing the definition- of ‘ square roots ’ or surds, 
Theaetetus says that ‘ there is a similar distinction in the case 
of solids’. We know nothing of any further development 
of a theory of irrationals arising from solids; but Theaetetus 
doubtless had in mind a distinction related to VIII. 12 (the 
theorem that between two cube numbers there are two mean 
proportional numbers) in the same way as the definition of 
a ‘ square root ’ or surd is .related to VIII. 11; that is to say, 
he referred to the incommensurable cube root of a non-cube 
number which is the product of three factors. 
Besides laying the foundation of the theory of irrationals 
as we find it in Eucl., Book X, Theaetetus contributed no less 
substantially to another portion of the Elements, namely 
Book XIII, which is devoted (after twelve introductory 
propositions) to constructing the five regular solids, circum 
scribing spheres about them, and finding the relation between 
the dimensions of the respective solids and the circumscribing 
spheres. We have already mentioned (pp. 159, 162) the tradi 
tions that Theaetetus was the first to ‘ construct’ or ‘write upon’ 
the five regular solids, 1 and that his name was specially 
associated with the octahedron and the icosahedron. 2 There 
can be little doubt that Theaetetus’s ‘ construction ’ of, or 
treatise upon, the regular solids gave the theoretical con 
structions much as we find them in Euclid. 
Of the mathematicians of Plato’s time, two others are 
mentioned with Theaetetus as having increased the number 
of theorems in geometry and made a further advance towards 
a scientific grouping of them, Leodamas of Thasos and 
Archytas of Taras. With regard to the former we are 
1 Suidas, S.V. ©ecu-njrof. 
2 Schol. 1 to Fuel. XIII (Euclid, ed. Heiberg, vol. v, p. 654).