Page 68 - Foundations Of Differential Calculus

3. On the Infinite and the Infinitely Small 51
such a way that it is said to be an infinitely long ellipse, we can correctly
say that the axis of the parabola is an infinitely long straight line.
83. This theory of the infinite will be further illustrated if we discuss
that which mathematicians call the infinitely small. There is no doubt that
any quantity can be diminished until it all but vanishes and then goes
to nothing. But an infinitely small quantity is nothing but a vanishing
quantity, and so it is really equal to 0. There is also a definition of the
infinitely small quantity as that which is less than any assignable quantity.
If a quantity is so small that it is less than any assignable quantity, then
it cannot not be 0, since unless it is equal to 0 a quantity can be assigned
equal to it, and this contradicts our hypothesis. To anyone who asks what
an infinitely small quantity in mathematics is, we can respond that it really
is equal to 0. There is really not such a great mystery lurking in this idea as
some commonly think and thus have rendered the calculus of the infinitely
small suspect to so many. In the meantime any doubts that may remain
will be removed in what follows, where we are going to treat this calculus.
84. Since we are going to show that an infinitely small quantity is really
zero, we must first meet the objection of why we do not always use the
same symbol 0 for infinitely small quantities, rather than some special
ones. Since all nothings are equal, it seems superfluous to have different
signs to designate such a quantity. Although two zeros are equal to each
other, so that there is no difference between them, nevertheless, since we
have two ways to compare them, either arithmetic or geometric, let us look
at quotients of quantities to be compared in order to see the difference.
The arithmetic ratio between any two zeros is an equality. This is not the
case with a geometric ratio. We can easily see this from this geometric
proportion 2:1=0:0, in which the fourth term is equal to 0, as is the
third. From the nature of the proportion, since the first term is twice the
second, it is necessary that the third is twice the fourth.

85. These things are very clear, even in ordinary arithmetic. Everyone
knows that when zero is multiplied by any number, the product is zero and
that n·0 = 0, so that n :1=0:0. Hence, it is clear that any two zeros can
be in a geometric ratio, although from the perspective of arithmetic, the
ratio is always of equals. Since between zeros any ratio is possible, in order
to indicate this diversity we use different notations on purpose, especially
when a geometric ratio between two zeros is being investigated. In the
calculus of the infinitely small, we deal precisely with geometric ratios of
infinitely small quantities. For this reason, in these calculations, unless we
use different symbols to represent these quantities, we will fall into the
greatest confusion with no way to extricate ourselves.
86. If we accept the notation used in the analysis of the infinite, then
dx indicates a quantity that is infinitely small, so that both dx = 0 and