ZENO’S ARGUMENTS ABOUT MOTION 
279 
in the Dichotomy. He observes that time is divisible in 
exactly the same way as a length; if therefore a length is 
infinitely divisible, so is the corresponding time; he adds 
‘ this is why {816} Zeno’s argument falsely assumes that it is 
not possible to traverse or touch each of an infinite number of 
points in a finite time V thereby implying that Zeno did not 
regard time as divisible act infinitum like space. Similarly, 
when Leibniz declares that a space divisible ad infinitum 
is traversed in a time divisible ad infinitum, he, like Aristotle, 
is entirely beside the question. Zeno was perfectly aware that, 
in respect of divisibility, time and space have the same 
property, and that they are alike, always, and concomitantly, 
divisible ad infinitum. The question is how, in the one as 
in the other, this series of divisions, by definition inexhaustible, 
can be exhausted; and it must be exhausted if motion is to 
be possible. It is not an answer to say that the two series 
are exhausted simultaneously. 
The usual mode of refutation given by mathematicians 
from Descartes to Tannery, correct in a sense, has an analogous 
defect. To show that the sum of the infinite series 1 + \ + % + ... 
is equal to 2, or to calculate (in the Achilles) the exact moment 
when Achilles will overtake the tortoise, is to answer the 
question when ? whereas the question actually asked is how ? 
On the hypothesis of divisibility ad infinitum you will, in the 
Dichotomy, never reach the limit, and, in the Achilles, the 
distance separating Achilles from the tortoise, though it con 
tinually decreases, will never vanish. And if you introduce 
the limit, or, with a numerical calculation, the discontinuous, 
Zeno is quite aware that his arguments are no longer valid. 
We are then in presence of another hypothesis as to the com 
position of the continuum; and this hypothesis is dealt with 
in the third and fourth arguments. 2 
It appears then that the first and second arguments, in their 
full significance, were not really met before G. Cantor formu 
lated his new theory of continuity and infinity. On this I 
can only refer to Chapters xlii and xliii of Mr. Bertrand 
Russell’s Principles of Mathematics, vol. i. Zeno’s argument 
in the Dichotomy is that, whatever motion we assume to have 
taken place, this presupposes another motion ; this in turn 
1 Ih. vi. 2, 2B8 a 16-23. 2 Brochard, Joe. cit., p. 9.