Page 72 - Foundations Of Differential Calculus

3. On the Infinite and the Infinitely Small 55
quantities shows them to be unequal, then even less will an arithmetic ratio
show them to be equal, since their difference will always be infinitely large.
94. Although there are some for whom the idea of the infinite, which we
use in mathematics, seems to be suspect, and for this reason think that
analysis of the infinite is to be rejected, still even in the trivial parts of
mathematics we cannot do without it. In arithmetic, where the theory of
logarithms is developed, the logarithm of zero is said to be both negative
and infinite. There is no one in his right mind who would dare to say
that this logarithm is either finite or even equal to zero. In geometry and
trigonometry this is even clearer. Who is there who would ever deny that
the tangent or the secant of a right angle is infinitely large? Since the
rectangle formed by the tangent and the cotangent has an area equal to
the square of the radius, and the cotangent of a right angle is equal to 0,
even in geometry it has to be admitted that the product of zero and infinity
can be finite.

95. Since a/dx is an infinite quantity A, it is clear that the quantity A/dx
will be a quantity infinitely greater than the quantity A. This can be seen
from the proportion a/dx : A/dx = a : A, that is, as a finite number to
one infinitely large. There are relations of this kind between infinitely large
quantities, so that some can be infinitely greater than others. Thus, a/dx 2
2
is a quantity infinitely greater than a/dx;ifwelet a/dx = A, then a/dx =
3
A/dx. In a similar way a/dx is an infinite quantity infinitely greater than
2
a/dx , and so is infinitely greater than a/dx. We have, therefore an infinity
of grades of infinity, of which each is infinitely greater than its predecessor.
If the number m is just a little bit greater than n, then a/dx m is an infinite
n
quantity infinitely greater than the infinite quantity a/dx .
96. Just as with infinitely small quantities there are geometric ratios indi-
cating inequalities, but arithmetic ratios always indicate equality, so with
infinitely large quantities we have geometric ratios indicating equality, but
whose arithmetic ratios still indicate inequality. If a and b are two finite
quantities, then the geometric ratio of two infinite quantities a/dx + b and
a/dx indicates that the two are equal; the quotient of the first by the sec-
ond is equal to 1+b dx/a = 1, since dx = 0. However, if they are compared
arithmetically, due to the difference b, the ratio indicates inequality. In a
2
2
similar way, the geometric ratio of a/dx + a/dx to a/dx indicates equal-
ity; expressing the ratio, we have 1 + dx = 1, since dx = 0. On the other
hand, the difference is a/dx, and so this is infinite. It follows that when we
consider geometric ratios, an infinitely large quantity of a lower grade will
vanish when compared to an infinitely large quantity of a higher grade.

97. Now that we have been warned about the grades of infinities, we will
soon see that it is possible not only for the product of an infinitely large
quantity and an infinitely small quantity to produce a finite quantity, as