Democritus wrote a large number of mathematical treatises, the titles only of which are preserved. We gather from one of these titles, 'On irrational lines and solids', that he wrote on irrationals. Democritus realized as fully as Zeno, and expressed with no less piquancy, the difficulty connected with the continuous and the infinitesimal. This appears from his dilemma about the circular base of a cone and a parallel section; the section which he means is a section 'indefinitely near' (as the phrase is) to the base, i. e. the _very next_ section, as we might say (if there were one). Is it, said Democritus, equal or not equal to the base? If it is equal, so will the very next section to it be, and so on, so that the cone will really be, not a cone, but a cylinder. If it is unequal to the base and in fact less, the surface of the cone will be jagged, like steps, which is very absurd. We may be sure that Democritus's work on 'The contact of a circle or a sphere' discussed a like difficulty. Lastly, Archimedes tells us that Democritus was the first to state, though he could not give a rigorous proof, that the volume of a cone or a pyramid is one-third of that of the cylinder or prism respectively on the same base and having equal height, theorems first proved by Eudoxus. We come now to the time of Plato, and here the great names are Archytas, Theodoras of Cyrene, Theaetetus, and Eudoxus. Archytas (about 430-360 B. C.) wrote on music and the numerical ratios corresponding to the intervals of the tetrachord. He is said to have been the first to write a treatise on mechanics based on mathematical principles; on the practical side he invented a mechanical dove which would fly. In geometry he gave the first solution of the problem of the two mean proportionals, using a wonderful construction in three dimensions which determined a certain point as the intersection of three surfaces, (1) a certain cone, (2) a half-cylinder, (3) an anchor-ring or _tore_ with inner diameter _nil_. Theodorus, Plato's teacher in mathematics, extended the theory of the irrational by proving incommensurability in certain particular cases other than that of the diagonal of a square in relation to its side, which was already known. He proved that the side of a square containing 3 square feet, or 5 square feet, or any non-square number of square feet up to 17 is incommensurable with one foot, in other words that √3, √5 ... √17 are all incommensurable with 1. Theodorus's proof was evidently not general; and it was reserved for Theaetetus to comprehend all these irrationals in one definition, and to prove the property generally as it is proved in Eucl. X. 9. Much of the content of the rest of Euclid's Book X (dealing with compound irrationals), as also of Book XIII on the five regular solids, was due to Theaetetus, who is even said to have discovered two of those solids (the octahedron and icosahedron).