Page 73 - Foundations Of Differential Calculus

56 3. On the Infinite and the Infinitely Small
we have already seen, but also that a product of this kind can also be
either infinitely large or infinitely small. Thus, if the infinite quantity a/dx
is multiplied by the infinitely small dx, the product will be equal to the
2 3
finite a. However, if a/dx is multiplied by the infinitely small dx or dx or
2 3
another of higher order, the product will be adx, adx , adx , and so forth,
and so it will be infinitely small. In the same way, we understand that if
2
the infinite quantity a/dx is multiplied by the infinitely small dx, then the
n
m
product will be infinitely large. In general, if a/dx is multiplied by bdx ,
the product ab dx m−n will be infinitely small if m is greater than n; it will
be finite if m equals n; it will be infinitely large if m is less than n.
98. Both infinitely small and infinitely large quantities often occur in
series of numbers. Since there are finite numbers mixed in these series, it is
clearer than daylight, how, according to the laws of continuity, one passes
from finite quantities to infinitely small and to infinitely large quantities.
First let us consider the series of natural numbers, continued both forward
and backward:

..., −4, −3, −2, −1, +0, +1, +2, +3, +4, ....

By continuously decreasing, the numbers approach 0, that is, the infinitely
small. Then they continue further and become negative. From this we un-
derstand that the positive numbers decrease, passing through 0 to increas-
ing negative numbers. However, if we consider the squares of the numbers,
since they are all positive,

..., +16, +9, +4, +1, +0, +1, +4, +9, +16, ...,

we have 0 as the transition number from the decreasing positive numbers
to the increasing positive numbers. If all of the signs are changed, then
0 is again the transition from decreasing negative numbers to increasing
negative numbers.
√
99. If we consider the series with general term x, which is continued
both forwards and backwards, we have
√ √ √ √ √ √ √
..., + −3, + −2, + −1, +0, + 1, + 2, + 3, + 4, ...,

and from this it is clear that 0 is a kind of limit through which real quantities
pass to the complex.
If these terms are considered as points on a curve, it is seen that if they
are positive and decrease so that they eventually vanish, then continuing
further, they become either negative, or positive again, or even complex.
The same happens if the points were first negative, then also vanish, and if
they continue further, become either positive, negative, or complex. Many