272 
ZENO OF ELEA 
himself (a contemporary of Socrates), 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 indeed 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 diffi 
culties 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 con 
tented themselves with finite magnitudes that can be made as 
great or as small as we please. 1 If they used infinitesimals 
at all, it was only as a tentative means of discovering proposi 
tions ; they proved them afterwards by rigorous geometrical 
methods. An illustration of this is furnished by the Method of 
Archimedes. In that treatise Archimedes finds {a) the areas 
of curves, and (6) 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 (6) 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 scientific 
ally a geometrical proof by the method of exhaustion, with 
its double reductio ad absurdum, is still necessary. 
Notwithstanding that the criticisms of Zeno had so impor 
tant an influence upon the lines of development of Greek 
geometry, it does not appear that Zeno himself was really 
a mathematician or even a physicist. Plato mentions a work 
of his (та rov Zrjvcouos ypaypara, or to avyypaypa) in terms 
which imply that it was his only known work. 2 Simplicius 
too knows only one work of his, and this the same as that 
mentioned by Plato 3 ; when Suidas mentions four, a Commen 
tary on or Exposition of Empedocles, Controversies, Against 
the philosophers and On Nature, it may be that the last three 
titles are only different designations for the one work, while 
the book on Empedocles may have been wrongly attributed 
1 Cf. Arist. Fhys. iii. 7, 207 b 81. 2 Plato, Parmenides, 1^7 C sq. 
3 Simpl. in Fhys., pp. 139. 5, 140. 27 Diels.