Page 89 - Foundations Of Differential Calculus

72 4. On the Nature of Differentials of Each Order
differential calculus contains a method for finding all differentials of each
order. From the word differential, which denotes an infinitely small differ-
ence, we derive other names that have come into common usage. Thus we
have the word differentiate, which means to find a differential. A quantity
is said to be differentiated when its differential is found. Differentiation
denotes the operation by which differentials are found. Hence differential
calculus is also called the method of differentiating, since it contains a way
of finding differentials.
139. Just as in differential calculus the differential of any quantity is
investigated, so there is a kind of calculus that consists in finding a quantity
whose differential is one that is already given, and this is called integral
calculus. If any differential is given, that quantity whose differential is the
proposed quantity is called its integral. The reason for this name is as
follows: Since a differential can be thought of as an infinitely small part by
which a quantity increases, that quantity with respect to which this is a
part can be thought of as a whole, that is, integral, and for this reason is
called an integral. Thus, since dy is the differential of y, y, in turn, is the
2 2
integral of dy. Since d y is the differential of dy, dy is the integral of d y.
2 3 3 4
Likewise, d y is the integral of d y, and d y is the integral of d y, and so
forth. It follows that any differentiation, from an inverse point of view, is
also an example of integration.

140. The origin and nature of both integrals and differentials can most
clearly be explained from the theory of finite differences, which has been
discussed in the first chapter. After it was shown how the difference of
any quantity should be found, going in reverse, we also showed how, from
a given difference, a quantity can be found whose difference is the one
proposed. We called that quantity, with respect to its difference, the sum
of the difference. Just as when we proceed to the infinitely small, differences
become differentials, so the sums, which there were called just that, now
receive the name of integral. For this reason integrals are sometimes called
sums. The English call differentials by the name fluxions, and integrals are
called by them fluents. Their mode of speaking about finding the fluent of
a given fluxion is the same as ours when we speak of finding the integral
of a given differential.
141. Just as we use the symbol d for a differential, so we use the symbol

to indicate an integral. Hence if this is placed before a differential, we
are indicating that quantity whose differential is the one given. Thus, if the
differential of y is pdx, that is, dy = pdx, then y is the integral of pdx.

This is expressed as follows: y = pdx, since y = dy. Hence, the integral

of pdx, symbolized by pdx, is that quantity whose differential is pdx.
2
2
In a similar way if d y = qdx, where dp = qdx, then the integral of d y

is dy, which is equal to pdx. Since p = qdx,wehave dy = dx qdx,

and hence y = dx qdx. If in addition, dq = rdx, then q = rdx and