demonstration by means of the lemma or lemmas aforesaid, but he actually applied the method to find the volumes (1) of any pyramid, (2) of the cone, proving (1) that any pyramid is one third part of the prism which has the same base and equal height, and (2) that any cone is one third part of the cylinder which has the same base and equal height. Archimedes, however, tells us the remarkable fact that these two theorems were first discovered by Democritus (who flourished towards the end of the fifth century B.C.), though he was not able to prove them (which no doubt means, not that he gave no sort of proof, but that he was not able to establish the propositions by the rigorous method of Eudoxus). Archimedes adds that we must give no small share of the credit for these theorems to Democritus; and this is another testimony to the marvellous powers, in mathematics as well as in other subjects, of the great man who, in the words of Aristotle, “seems to have thought of everything”. We know from other sources that Democritus wrote on irrationals; he is also said to have discussed the question of two parallel sections of a cone (which were evidently supposed to be indefinitely close together), asking whether we are to regard them as unequal or equal: “for if they are unequal they will make the cone irregular as having many indentations, like steps, and unevennesses, but, if they are equal, the cone will appear to have the property of the cylinder and to be made up of equal, not unequal, circles, which is very absurd”. This explanation shows that Democritus was already close on the track of infinitesimals.

Archimedes says further that the theorem that spheres are in the triplicate ratio of their diameters was proved by means of the same lemma. The proofs of the propositions about the volumes of pyramids, cones and spheres are, of course, contained in Euclid, Book XII. (Props. 3-7 Cor., 10, 16-18 respectively).

It is no doubt desirable to illustrate Eudoxus’s method by one example. We will take one of the simplest, the proposition (Eucl. XII., 10) about