Page 71 - Foundations Of Differential Calculus

54 3. On the Infinite and the Infinitely Small
91. Anyone who denies either of these arguments will find himself in great
difficulties, with the necessity of denying even the most certain principles
of analysis. If someone claims that the fraction a/0 is finite, for example
equal to b, then when both parts of the equation are multiplied by the
denominator, we obtain a =0 · b. Then the finite quantity b multiplied
by zero produces a finite a, which is absurd. Much less can the value b of
the fraction a/0 be equal to 0; in no way can 0 multiplied by 0 produce
the quantity a. Into the same absurdity will fall anyone who denies that
a/∞ = 0, since then he would be saying that a/∞ = b, a finite quantity.
From the equation a/∞ = b it would legitimately follow that ∞ = a/b, but
from this we conclude that the value of the fraction a/b, whose numerator
and denominator are both finite quantities, is infinitely large, which of
course is absurd. Nor is it possible that the values of the fractions a/0 and
a/∞ could be complex, since the value of a fraction whose numerator is
finite and whose denominator is complex cannot be either infinitely large
or infinitely small.

92. An infinitely large quantity, to which we have been led through this
consideration, and which is treated only in the analysis of the infinite, can
best be defined by saying that an infinitely large quantity is the quotient
that arises from the division of a finite quantity by an infinitely small quan-
tity. Conversely, we can say that an infinitely small quantity is a quotient
that arises from division of a finite quantity by an infinitely large quantity.
Since we have a geometric proportion in which an infinitely small quantity
is to a finite quantity as a finite quantity is to an infinitely large quantity,
it follows that an infinite quantity is infinitely greater than a finite quan-
tity, just as a finite quantity is infinitely greater than an infinitely small
quantity. Hence, statements of this sort, which disturb many, should not
be rejected, since they rest on most certain principles. Furthermore, from
the equation a/0= ∞, it can follow that zero multiplied by an infinitely
large quantity produces a finite quantity, which would seem strange were
it not the result of a very clear deduction.

93. Just as when we compare infinitely small quantities by a geometric ra-
tio, we can find very great differences, so when we compare infinitely large
quantities the difference can be even greater, since they differ not only by
geometric ratios, but also by arithmetic. Let A be an infinite quantity that
is obtained from division of a finite quantity a by the infinitely small dx,
so that a/dx = A. Likewise 2a/dx =2A and na/dx = nA. Now, since nA
is an infinite quantity, it follows that the ratio between two infinitely large
quantities can have any value. Hence, if an infinite quantity is either multi-
plied or divided by a finite number, the result will be an infinite quantity.
Nor can it be denied that infinite quantities can be further augmented.
It is easily seen that if the geometric ratio that holds between two infinite