"The subject of s was forced upon the Greek mathematicians so soon as they came to close grips with the problem of the quadrature of the circle. Antiphon the Sophist was the first to [inscribe] a series of successive regular polygons in a circle, each of which had double as many sides as the preceding, and he asserted that, by continuing this process, we should at length exhaust the circle: [according to Simplicius, on Aristotle, Physics] 'he thought that in this way the area of the circle would sometime be used up and a polygon would be inscribed in the circle the sides of which on account of their smallness would coincide with the circumference.' Aristotle roundly said that this was a fallacy... Antiphon's argument.. as early as the time of Antiphon himself (a contemporary of Socrates) had been subjected to a destructive criticism expressed with unsurpassable piquancy and force. No wonder that the subsequent course of Greek geometry was profoundly affected by the arguments of Zeno on motion. Aristotle... called them 'fallacies', without being able to refute them. The mathematicians, however, knew better, and, realizing that Zeno's arguments were fatal to infinitesimals, they saw that they could only avoid the difficulties connected with them by once for all banishing the idea of the infinite, even the potentially infinite, altogether from their science; thenceforth, therefore, they made no use of magnitudes increasing or diminishing ad infinitum, but contented themselves with finite magnitudes that can be made as great or as small as we please. If they used infinitesimals at all, it was only as a tentative means of discovering propositions; they proved them afterwards by rigorous geometrical methods. An illustration of this is furnished by the Method of Archimedes. ...Archimedes finds (a) the areas of curves, and (b) the volumes of solids, by treating them respectively as the sums of an infinite number (a) of parallel lines, i.e. infinitely narrow strips, and (b) of parallel planes, i.e. infinitely thin laminae; but he plainly declares that this method is only useful for discovering results and does not furnish a proof of them, but that to establish them scientifically a geometrical proof by the , with its double ' is still necessary."