Page 8 - Foundations Of Differential Calculus

Preface vii
receives. Indeed, the investigation of this kind of ratio of increments is not
only very important, but it is in fact the foundation of the whole of analysis
of the infinite. In order that this may become even clearer, let us take up
2 2
again the example of the square x with its increment of 2xω + ω , which
it receives when x itself is increased by ω. We have seen that the ratio here
is 2x + ω to 1. From this it should be perfectly clear that the smaller the
increment is taken to be, the closer this ratio comes to the ratio of 2x to
1. However, it does not arrive at this ratio before the increment itself, ω,
completely vanishes. From this we understand that if the increment of the
2
variable x goes to zero, then the increment of x also vanishes. However,
the ratio holds as 2x to 1. What we have said here about the square is to
be understood of all other functions of x; that is, when their increments
vanish as the increment of x vanishes, they have a certain and determinable
ratio. In this way, we are led to a definition of differential calculus:Itis
a method for determining the ratio of the vanishing increments that any
functions take on when the variable, of which they are functions, is given a
vanishing increment. It is clearly manifest to those who are not strangers
to this subject that the true character of differential calculus is contained
in this definition and can be adequately deduced from it.
Therefore, differential calculus is concerned not so much with vanishing
increments, which indeed are nothing, but with the ratio and mutual pro-
portion. Since these ratios are expressed as finite quantities, we must think
of calculus as being concerned with finite quantities. Although the values
seem to be popularly discussed as defined by these vanishing increments,
still from a higher point of view, it is always from their ratio that conclu-
sions are deduced. In a similar way, the idea of integral calculus can most
conveniently be defined to be a method for finding those functions from the
knowledge of the ratio of their vanishing increments.
In order that these ratios might be more easily gathered together and
represented in calculations, the vanishing increments themselves, although
they are really nothing, are still usually represented by certain symbols.
Along with these symbols, there is no reason not to give them a certain
name. They are called differentials, and since they are without quantity,
they are also said to be infinitely small. Hence, by their nature they are
to be so interpreted as absolutely nothing, or they are considered to be
equal to nothing. Thus, if the quantity x is given an increment ω, so that
2 2 2
it becomes x + ω, its square x becomes x +2xω + ω , and it takes the
2
increment 2xω + ω . Hence, the increment of x itself, which is ω, has the
2
ratio to the increment of the square, which is 2xω+ω ,as1to2x+ω. This
ratio reduces to 1 to 2x, at least when ω vanishes. Let ω = 0, and the ratio
of these vanishing increments, which is the main concern of differential
calculus, is as 1 to 2x. On the other hand, this ratio would not be true
unless that increment ω vanishes and becomes absolutely equal to zero.
Hence, if this nothing that is indicated by ω refers to the increment of