278 
ZENO OF ELEA 
lengths which are indeterminate. In the first and third there 
is only one moving object, and it is shown that it cannot even 
begin to move. The second and fourth, comparing the motions 
of two objects, make the absurdity of the hypothesis even 
more palpable, so to speak, for they prove that the movement, 
even if it has once begun, cannot continue, and that relative 
motion is no less impossible than absolute motion. The first 
two establish the impossibility of movement by the nature of 
space, supposed continuous, without any implication that time 
is otherwise than continuous in the same way as space; in the 
last two it is the nature of time (considered as made up of 
indivisible elements or instants) which serves to prove the 
impossibility of movement, and without any implication that 
space is not likewise made up of indivisible elements or points. 
The second argument is only another form of the first, and 
the fourth rests on the same principle as the third. Lastly, the 
first pair proceed on the hypothesis that continuous magni 
tudes are divisible ad infinitum; the second pair give the 
other horn of the dilemma, being directed against the assump 
tion that continuous magnitudes are made up of indivisible 
elements, an assumption which would scarcely suggest itself 
to the imagination until the difficulties connected with the 
other were fully realized. Thus the logical order of the argu 
ments corresponds exactly to the historical order in which 
Aristotle has handed them down and which was certainly the 
order adopted by Zeno. 
Whether or not the paradoxes had for Zeno the profound 
meaning now claimed for them, it is clear that they have 
been very generally misunderstood, with the result that the 
criticisms directed against them have been wide of the mark. 
Aristotle, it is true, saw that the first two arguments, the 
Dichotomy and the Achilles, come to the same thing, the latter 
differing from the former only in the fact that the ratio of 
each space traversed by Achilles to the preceding space is not 
that of 1 ; 2 but a ratio of 1 : n, where n may be any number, 
however large; but, he says, both proofs rest on the fact that 
a certain moving object ‘ cannot reach the end of the course if 
the magnitude is divided in a certain way’. 1 But another 
passage shows that he mistook the character of the argument 
1 Arist. Phys. vi. 9, 289 b 18-24.