66 A SHORT HISTORY OF SCIENCE 



thus led to the invention of this new curve, the first of which 

 we have any definite record. 



THE CRITICISM OF ZENO. The Stoic philosopher Zeno, teach- 

 ing in Athens about this time, though not himself a mathematician, 

 represents an important phase of philosophical criticism of mathe- 

 matics. Every manifold, he says, is a number of units, but a 

 true unit is indivisible. Each of the many must thus be itself 

 an indivisible unit, or consist of such units. That which is in- 

 divisible however can have no magnitude, for everything which 

 has magnitude is divisible to infinity. The separate parts have 

 therefore no magnitude, etc. Again, as to the possibility of mo- 

 tion, he maintains that before the body can reach its destination 

 it must reach the middle point, before it can arrive there it must 

 traverse the quarter, and so on without end. Motion is thus 

 impossible ; so the tortoise, if he have any start, cannot be over- 

 taken by the swift runner Achilles, for while Achilles is covering 

 that distance the tortoise will have attained a second distance, and 

 so on. Such specious criticism was naturally, and in a measure 

 justly, evoked by misguided efforts of certain mathematicians to 

 show that a line consists of a multitude of points, etc. These or 

 similar controversies as to the interpretation of the infinite and 

 the infinitesimal have persisted till our own day, resembling in that 

 respect the classical problems of circle squaring and angle tri- 

 section to which reference has been made above. The more or less 

 mystical statements about the new discoveries of the Pythagoreans 

 also invited sceptical epigrams. 



Zeno was concerned with three problems. . . . These are the 

 problem of the infinitesimal, the infinite, and continuity. . . . From 

 him to our own day, the finest intellects of each generation in turn 

 attacked these problems, but achieved, broadly speaking, nothing. . . . 



B. Russell. 



Aristotle accordingly solves the problem of Zeno the Eleatic, which 

 he propounded to Protagoras the Sophist. Tell me, Protagoras, said 

 he, does one grain of millet make a noise when it falls, or does the 

 ten-thousandth part of a grain? On receiving the answer that it 

 does not, he went on : Does a measure of millet grains make a noise 



