Page 92 - Foundations Of Differential Calculus

4. On the Nature of Differentials of Each Order 75
tity. In the latter a way of investigating integrals of given differentials is
shown. In both parts there are indications of the best applications to both
analysis itself and higher geometry. For this reason even this first part of
analysis has already grown so that to cover it requires no small book. In
the integral calculus both new methods of integration are being discovered
every day, as well as the revelation of new aids for the solution of different
kinds of problems. Due to the new discoveries that are continuously being
made, we could never exhaust, much less describe and explain perfectly, all
of this. Nevertheless I will make every effort in these books to make sure
that either everything that has so far been discovered shall be presented,
or at least the methods by which they can be deduced are explained.

149. It is common to give other parts of analysis of the infinite. Besides
differential and integral calculus, one sometimes finds differentio-differential
calculus and exponential calculus. In differentio-differential calculus the
methods of finding second and higher differentials are usually discussed.
Since the method of finding differentials of any order will be discussed in
this differential calculus, this subdivision, which seems to be based more
on the importance of its discovery rather than the thing itself, we will
omit. The illustrious Johann Bernoulli, to whom we are eternally grateful
for innumerable and great discoveries in analysis of the infinite, extended
the methods of differentiating and integrating to exponential quantities by
means of exponential calculus. Since I plan to treat in both parts of calculus
not only algebraic but also transcendental quantities, this special part has
become superfluous and outside our plan.
150. I have decided to treat differential calculus first. I will explain the
method by which not only first differentials but also second and higher
differentials of variable quantities can be expeditiously found. I will begin
by considering algebraic quantities, whether they be explicitly given or im-
plicitly by equations. Then I will extend the discovery of differentials to
nonalgebraic quantities, at least to those which can be known without the
aid of integral calculus. Quantities of this kind are logarithms and expo-
nential quantities, as well as arcs of circles and in turn sines and tangents of
circular arcs. Finally, we will teach how to differentiate compositions and
mixtures of all of these quantities. In short, this first part of differential
calculus will be concerned with differentiating.

151. The second part will be dedicated to the explanation of the applica-
tions of the method of differentiating to both analysis and higher geometry.
Many nice things spill over into ordinary algebra: finding roots of equations,
discussing and summing series, discovering maxima and minima, defining
and discovering values of expressions that in some cases seem to defy de-
termination. Higher geometry has received its greatest development from
differential calculus. By its means tangents to curves and their curvature
can be defined with marvelous facility. Many other problems concerned