[ Pobierz całość w formacie PDF ]

It is easily seen that if the geometric ratio that holds between two infinite
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/dx2
is a quantity infinitely greater than a/dx; if we let a/dx = A, then a/dx2 =
A/dx. In a similar way a/dx3 is an infinite quantity infinitely greater than
a/dx2, 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/dxm is an infinite
quantity infinitely greater than the infinite quantity a/dxn.
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
similar way, the geometric ratio of a/dx2 + a/dx to a/dx2 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
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
finite a. However, if a/dx is multiplied by the infinitely small dx2 or dx3 or
another of higher order, the product will be adx, adx2, adx3, and so forth,
and so it will be infinitely small. In the same way, we understand that if
the infinite quantity a/dx2 is multiplied by the infinitely small dx, then the
product will be infinitely large. In general, if a/dxn is multiplied by bdxm,
the product ab dxm-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
3. On the Infinite and the Infinitely Small 57
examples of phenomena of this kind are found in the theory of plane curves,
treated in a preceding book.1
100. In the same way infinite terms often occur in series. Thus, in the
harmonic series, whose general term is 1/x, the term corresponding to the
index x = 0 is the infinite term 1/0. The whole series is as follows:
1 1 1 1 1 1 1 1
. . . , - , - , - , - , + , + , + , + , . . . .
4 3 2 1 0 1 2 3
Going from right to left the terms increase, so that 1/0 is infinitely large.
Once it has passed through, the terms become decreasing and negative.
Hence, an infinitely large quantity can be thought of as some kind of limit,
passing through which positive numbers become negative and vice versa.
For this reason it has seemed to many that the negative numbers can be
thought of as greater than infinity, since in this series the terms continu-
ously increase, and once they have reached infinity, they become negative.
However, if we consider the series whose general term is 1/x2, then after
passing through infinity, the terms become positive again,
1 1 1 1 1 1 1
. . . , + , + , + , + , + , + , + , . . . ,
9 4 1 0 1 4 9
and no one would say that these are greater than infinity.
101. Frequently, in a series an infinite term will constitute a limit sep- [ Pobierz całość w formacie PDF ]