ZENO’S ARGUMENTS ABOUT MOTION 281
attempt to extend to the values ot' a variable the variability
which belongs to it alone. When once it is firmly realized
that all the values of a variable are constants, it becomes easy
to see, by taking any two such values, that their difference is
always finite, and hence that there are no infinitesimal differ
ences. If x be a variable which may take all real values
from 0 to 1, then, taking any two of these values, we see that
their difference is finite, although a? is a continuous variable.
It is true the difference might have been less than the one we
chose; but if it had been, it would still have been finite. The
lower limit to possible differences is zero, but all possible
differences are finite; and in this there is no shadow of
contradiction. This static theory of the variable is due to the
mathematicians, and its absence in Zeno’s day led him to
suppose that continuous change was impossible without a state
of change, which involves infinitesimals and the contradiction
of a body’s being where it is not.’
In his later chapter on Motion Mr. Russell concludes as
follows: 1
‘ It is to be observed that, in consequence of the denial
of the infinitesimal and in consequence of the allied purely
technical view of the derivative of a function, we must
entirely reject the notion of a state of motion. Motion consists
merely in the occupation of different places at different times,
subject to continuity as explained in Part V. There is no
transition from place to place, no consecutive moment or
consecutive position, no such thing as velocity except in the
sense of a real number which is the limit of a certain set
of quotients. The rejection of velocity and acceleration as
physical facts (i. e. as properties belonging at each instant to
a moving point, and not merely real numbers expressing limits
of certain ratios) involves, as we shall see, some difficulties
in the statement of the laws of motion; but the reform
introduced by Weierstrass in the infinitesimal calculus has
rendered this rejection imperative.’
We come lastly to the fourth argument (the Stadium).
Aristotle„’s representation of it is obscure through its extreme
brevity of expression, and the matter is further perplexed by
an uncertainty of reading. But the meaning intended to be
conveyed is fairly clear. The eight M’s, B’s and G’s being
1 Op. clt., p. 478.