- 科学史研究 (ISSN:21887535)
- vol.26, no.164, pp.193-206, 1987 (Released:2021-09-21)
Archimedes gave geometrical demonstrations to find the volume of a sphere and the ellipsoids of revolution in totally different ways, although both lead to the same integral,∫[2a,0]x (2a — x)dx. Nicolas Bourbaki -a group of French mathematicians-state their views on why Archimedes had no concept of integral calculus as follows: "Might it not be that Archimedes regarded such a standpoint as extreme 'abstraction,'and dared to concentrate on studying characteristic properties of each figure he was working on ?" Certainly there is something in what Nicolas Bourbaki say. However, their views do not answer fully the question To the solution of this difficult problem, in my opinion, an important clue can be found by considering Archimedes' scholastic career in chronological order. It was not until Archimedes wrote On Spirals in his late forties or early fifties that he could work out the summing of the series 1²+2²+…n². In his later work On Conoids and Spheroids Archimedes could obtain for the first time the sum of a series ∑Xk(2a—Xk), necessary to give geometrical proofs about the volume of the ellipsoids of revolution However,it was difficult for Archimedes,in writing On the Sphere and Cylinder I, to obtain the sum. Therefore, he proved the theorem about the volume of a sphere in a way not making use of such summation When the same integral appeared, Archimedes could not notice the internal connection unifying them. This may be because, for one thing, he excluded from geometry, due to their mechanical nature, the discussions using indivisibles found in The Method which could have been a clue toward noticing the internal connection. Secondly, obtaining the sum of a series was not a simple matter to Archimedes who lacked the necessary algebraic symbols.