ZENO OF ELEA 
We have already seen how the consideration of the subject 
of infinitesimals 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 indicate the correct road upon which the solution was to 
be found, though he expressed his idea in a crude form which 
was bound to provoke immediate and strong criticism from 
logical minds. Antiphon had inscribed 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 con 
tinuing this process, we should at length exhaust the circle: 
‘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/ 1 Aristotle roundly said that 
this was a fallacy which it was not even necessary for a 
geometer to trouble to refute, since an expert in any science 
is not called upon to refute all fallacies, but only those which 
are false deductions from the admitted principles of the 
science; if the fallacy is based on anything which is in con 
tradiction to any of those principles, it may at once be ignored. 2 
Evidently therefore, in Aristotle’s view, Antiphon’s argument 
violated some ‘geometrical principle’, whether this was the 
truth that a straight line, however short, can never coincide 
with an arc of a circle, or the principle assumed by geometers 
that geometrical magnitudes can be divided ad infinitvjm. 
But Aristotle is only a representative of the criticisms 
directed against the ideas implied in Antiphon’s argument; 
those ideas had already, as early as the time of Antiphon 
1 Simpl. in Arist. Phys., p. 55. 6 Diels. 
2 Arist. Phys. i. 2, 185 a 14-17.