Archimedes / by Sir Thomas Heath.

  • Thomas Little Heath
Date:
1920
Catalogue details

Licence: In copyright

Credit: Archimedes / by Sir Thomas Heath. Source: Wellcome Collection.

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    by continuing it, we should at last arrive at a polygon with sides so small as to make the polygon coincident with the circle. Though this was formally incorrect, it nevertheless contained the germ of the method of ex¬ haustion. Hippocrates, as we have seen, 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. There is, however, no doubt about the part taken by Eudoxus; he not only based the method on rigorous demonstration by means of the lemma or lemmas aforesaid, but he actually applied the method to find the volumes (i) 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 Demo¬ critus ; 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 sup¬ posed 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,
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