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another, and so on ad infinitum. Hence there is an endless 
regress in the mere idea of any assigned motion. Zeno’s 
argument has then to be met by proving that the ‘infinite 
regress ’ in this case is ‘ harmless.’. 
As regards the Achilles, Mr. G. H. Hardy remarks that ‘ the 
kernel of it lies in the perfectly valid proof which it affords 
that the tortoise passes through as many points as Achilles, 
a view which embodies an accepted doctrine of modern mathe 
matics ’- 1 
The argument in the Arrow is based on the assumption that 
time is made up of indivisible elements or instants. Aristotle 
meets it by denying the assumption. ‘For time is not made 
up of indivisible instants (nows), any more than any other 
magnitude is made up of indivisible elements.’ ‘ (Zeno’s result) 
follows through assuming that time is made up of (indivisible) 
instants (notvs); if this is not admitted, his conclusion does 
not follow.’ 2 On the other hand, the modern view is that 
Zeno’s contention is true: ‘ If ’ (said Zeno) ‘ everything is at 
rest or in motion when it occupies a space equal to itself, and 
if what moves is always in the instant, it follows that the 
moving arrow is unmoved.’ Mr. Russell 3 holds that this is 
‘ a very plain statement of an elementary fact ’; 
‘ it is a very important and very widely applicable platitude, 
namely “ Every possible value of a variable is a constant ”. 
If x be a variable which can take all values from 0 to 1, 
all the values it can take are definite numbers such as \ or |, 
which are all absolute constants . . . Though a variable is 
always connected with some class, it is not the class, nor 
a particular member of the class, nor yet the whole class, but 
any member of the class.’ The usual x in algebra ‘denotes 
the disjunction formed by the various members’ . . . ‘The 
values of x are then the terms of the disjunction; and each 
of these is a constant. This simple logical fact seems to 
constitute the essence of Zeno’s contention that the arrow 
is always at rest.’ ‘ But Zeno’s argument contains an element 
which is specially applicable to continua. In the case of 
motion it denies that there is such a thing as a state of motion. 
In the general case of a continuous variable, it may be taken 
as denying actual infinitesimals. For infinitesimals are an 
1 Encyclopaedia Britannica, art. Zeno. 
2 Arist. Phys. vi. 9, 239 b 8, 31. 
3 Russell, Principles of Mathematics, i, pp. 350, 351.