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.