Page 84 - Foundations Of Differential Calculus

4. On the Nature of Differentials of Each Order 67
by one speck from the truth. They want such a ratio between a finite
quantity and one infinitely small to be as is the ratio between the whole
earth and the smallest speck. If there is someone for whom this difference
is not sufficiently large, then let the ratio be magnified by even more than
a thousand, so that the smallness cannot possibly be observed. However,
they are forced to admit that geometrical rigor has been a bit compromised,
and to meet this objection they turn to such examples as they may find
from geometry or analysis of the infinite; from any agreement between these
latter methods they try to draw some good. This argument does not work,
since they frequently try to draw the truth from erroneous arguments.
In order that an argument avoid this difficulty and even be completely
successful, those quantities that we neglect in our calculations must not
be just incomprehensibly small, but they must be actually nothing, as we
have assumed. In this way geometric rigor suffers absolutely no violence.
124. Let us move on to an explanation of differentials of the second order.
These arise from second differences, which were treated in the first chapter,
when we let ω become the infinitely small dx. If we suppose that the variable
I
x increases by equal increments, then the second value x becomes equal to
II III
x + dx, and the following will be x = x +2dx, x = x +3dx,... . Since
the first differences dx are constant, the second differences vanish, and so
2
the second differential of x, that is, d x, is equal to 0. For this reason all of
3 4 5
the other differentials of x are equal to 0, namely, d x =0, d x =0, d x =
0,... . One could object that since differentials are infinitely small, for that
reason alone they are equal to 0, so that there is nothing special about
the variable x, whose increments are considered to be equal. However, this
2
3
vanishing should be interpreted as due not only to the fact that d x, d x,...
are nothing in themselves, but also by reason of the powers of dx, which
vanish when compared to dx itself.
125. In order that this may become clearer, let us recall that the second
2
4
3
difference of any function y of x can be expressed as Pω +Qω +Rω +··· .
4
3
Hence, if ω should be infinitely small, then the terms Qω , Rω ,... vanish
2
when compared with the first term Pω , so that with ω = dx, the second
2 2
differential of y will be equal to Pdx , where dx means the square of the
differential dx. It follows that although the second differential of y, namely
2 2 2 2
d y, by itself is equal to 0, still, since d y = Pdx , d y has a finite ratio
2
to dx , that is, as P to 1. However, since y = x,wehave P =0, Q =0,
R =0,... , so that in this case the second differential of x vanishes, even
2
with respect to dx , and so do the other higher powers of dx. This is the
sense in which we should understand what was stated previously, namely,
2 3
d x =0, d x =0,... .
126. Since the second difference is just the difference of the first difference,
the second differential, or, as it is frequently called, the differentiodifferen-
tial, is the differential of the first differential. Now, since a constant function