Page 70 - Foundations Of Differential Calculus

3. On the Infinite and the Infinitely Small 53
n
n
equals 1 + dx , and since dx = 0, the ratio is that of equals. Hence, if we
follow the usage of exponents, we call dx infinitely small of the first order,
2 3
dx of the second order, dx of the third order, and so forth. It is clear that
in comparison with an infinitely small quantity of the first order, those of
higher order will vanish.
89. In a similar way it is shown that an infinitely small quantity of the
third and higher orders will vanish when compared with one of the second
order. In general, an infinitely small quantity of any higher order vanishes
when compared with one of lower order. Hence, if m is less than n, then

m
n
m
adx + bdx = adx ,
n
m
since dx vanishes compared with dx , as we have shown. This is true also
√ 1
with fractional exponents; dx vanishes compared with dx or dx 2 , so that
√ √
a dx + bdx = a dx.
0
Even if the exponent of dx is equal to 0, we have dx = 1, although dx =0.
n
Hence the power dx is equal to 1 if n = 0, and from being a finite quantity
becomes infinitely small if n is greater than 0.
Therefore, there exist an infinite number of orders of infinitely small
quantities. Although all of them are equal to 0, still they must be carefully
distinguished one from the other if we are to pay attention to their mutual
relationships, which has been explained through a geometric ratio.
90. Once we have established the concept of the infinitely small, it is
easier to discuss the properties of infinity, or the infinitely large. It should
be noted that the fraction 1/z becomes greater the smaller the denominator
z becomes. Hence, if z becomes a quantity less than any assignable quantity,
that is, infinitely small, then it is necessary that the value of the fraction1/z
becomes greater than any assignable quantity and hence infinite. For this
reason, if 1 or any other finite quantity is divided by something infinitely
small or 0, the quotient will be infinitely large, and thus an infinite quantity.
Since the symbol ∞ stands for an infinitely large quantity, we have the
equation
a
= ∞.
dx
The truth of this is clear also when we invert:
a
= dx =0.
∞
Indeed, the larger the denominator z of the fraction a/z becomes, the
smaller the value of the fraction becomes, and if z becomes an infinitely
large quantity, that is z = ∞, then necessarily the value of the fraction
a/∞ becomes infinitely small.