Page 83 - Foundations Of Differential Calculus

66 4. On the Nature of Differentials of Each Order
120. If y is any function of x, as we have seen, its first differential will have
the form Pω. Since ω = dx,wehave dy = Pdx. Whatsoever function of x
y might be, its differential is expressed by the product of a certain function
of x that we call P and the differential of x, that is, dx. Although the
differentials of x and y are both infinitely small, and hence equal to zero,
still there is a finite ratio between them. That is, dy : dx = P : 1. Once we
have found the function P, then we know the ratio between the differential
dx and the differential dy. Since differential calculus consists in finding
differentials, the work involved is not in finding the differentials themselves,
which are both equal to zero, but rather in their mutual geometric ratio.
121. Differentials are much easier to find than finite differences. For the
finite difference ∆y by which a function increases when the variable quan-
tity x increases by ω, it is not sufficient to know P, but we must investigate
also the functions Q,R,S,... that enter into the finite difference that we
have expressed as
2 3
Pω + Qω + Rω + ··· .
For the differential of y we need only to know the function P. For this
reason, from our knowledge of the finite difference of any function of x
we can easily define its differential. On the other hand, from a function’s
differential it is not possible to figure out its finite difference. Nevertheless,
we shall see (in paragraph 49 of the second part) that from a knowledge
of the differentials of all orders it is possible to find the finite difference of
any given function. Now, from what we have seen, it is clear that the first
differential dy = Pdx gives the first term of the finite difference, that is,
Pω.
122. If the increment ω by which the variable x is considered to be in-
2
creased happens to be very small, so that in the expression Pω + Qω +
3 2 3
Rω +··· the terms Qω and Rω , and even more so the remaining terms,
become so small in comparison to Pω that they can be neglected in compu-
tations where rigor is not so important, in this case when we know the dif-
ferential Pdx we also know approximately the finite difference Pω. Hence,
in many cases we can use calculus in applications with no little profit. There
are some who judge that differentials are very small increments, but they
deny that they are actually equal to zero, and so they say that they are only
indefinitely small. This idea presents to others an occasion to blame anal-
ysis of the infinite for not obtaining exact, but only approximate, results.
This objection has some justification unless we insist that the infinitely
small is absolutely equal to zero.

123. Those who are unwilling to admit that the infinitely small becomes
nothing, in order that they might seem to meet the objection, compare
differentials to the very smallest speck of dust in relation to the whole earth.
One is thought not to have given the true bulk of the earth who departs