Take the case of a famous problem which plays a great part in the history of Greek geometry, the doubling of the cube, or its equivalent, the finding… - Thomas Little Heath

Hippocrates... is said to have proved the theorem that circles are to one another as the squares on their diameters, and it is difficult to see how he could have done this except by some form, or anticipation, of the method [of exhaustion].

Eudoxus was perhaps the greatest of all Archimedes's predecessors, and it is his achievements, especially the discovery of the method of exhaustion, which interest us in connexion with Archimedes.

Eudoxes... not only based the method [of exhaustion] on rigorous demonstration... 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, 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 methods of Eudoxes. Archimedes adds that we must give no small share of the credit for these theorems to Democritus... 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". ...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 equal or unequal... Democritus was already close on the track of infinitesimals.