Page 69 - Foundations Of Differential Calculus

52 3. On the Infinite and the Infinitely Small
adx = 0, where a is any finite quantity. Despite this, the geometric ratio
adx : dx is finite, namely a : 1. For this reason these two infinitely small
quantities dx and adx, both being equal to 0, cannot be confused when
we consider their ratio. In a similar way, we will deal with infinitely small
quantities dx and dy. Although these are both equal to 0, still their ratio
is not that of equals. Indeed, the whole force of differential calculus is
concerned with the investigation of the ratios of any two infinitely small
quantities of this kind. The application of these ratios at first sight might
seem to be minimal. Nevertheless, it turns out to be very great, which
becomes clearer with each passing day.
87. Since the infinitely small is actually nothing, it is clear that a finite
quantity can neither be increased nor decreased by adding or subtracting an
infinitely small quantity. Let a be a finite quantity and let dx be infinitely
small. Then a + dx and a − dx, or, more generally, a ± ndx, are equal to a.
Whether we consider the relation between a ± ndx and a as arithmetic or
as geometric, in both cases the ratio turns out to be that between equals.
The arithmetic ratio of equals is clear: Since ndx = 0,wehave

a ± ndx − a =0.

On the other hand, the geometric ratio is clearly of equals, since

a ± ndx
=1.
a
From this we obtain the well-known rule that the infinitely small vanishes
in comparison with the finite and hence can be neglected. For this reason
the objection brought up against the analysis of the infinite, that it lacks
geometric rigor, falls to the ground under its own weight, since nothing is
neglected except that which is actually nothing. Hence with perfect jus-
tice we can affirm that in this sublime science we keep the same perfect
geometric rigor that is found in the books of the ancients.

88. Since the infinitely small quantity dx is actually equal to 0, its square
2
3
n
dx , cube dx , and any other dx , where n is a positive exponent, will
be equal to 0, and hence in comparison to a finite quantity will vanish.
2
However, even the infinitely small quantity dx will vanish when compared
2
to dx. The ratio of dx±dx to dx is that of equals, whether the comparison
is arithmetic or geometric. There is no doubt about the arithmetic; in the
geometric comparison,
2
dx ± dx
2
dx ± dx : dx = =1 ± dx =1.
dx
3 n+1
In like manner we have dx ± dx = dx and generally dx ± dx = dx,
n+1
provided that n is positive. Indeed, the geometric ratio dx ± dx : dx