Page 12 - Foundations Of Differential Calculus

Preface xi
in such a way that they always are seen to be truly finite quantities and
that this is the only proper way for them to be represented. At first sight,
this description of analysis of the infinite may seem, for the most part, both
shallow and extremely sterile, although that obscure notion of the infinitely
small hardly offers more. In truth, if the ratios that connect the vanishing
increments of any functions are clearly known, then this knowledge very of-
ten is of the utmost importance and frequently is so important in extremely
arduous investigations that without it almost nothing can be clearly un-
derstood. For instance, if the question concerns the motion of a shot fired
from a cannon, the air resistance must be known in order to know what
the motion will be through a finite distance, as well as both the direction
of the path at the beginning and also the velocity, on which the resistance
depends. But this changes with time. However, the less distance the shot
travels, the less the variation, so that it is possible more easily to come to
knowledge of the true relationship. In fact, if we let the distance vanish,
since in that case both the difference in direction and change in velocity
also are removed, the effect of resistance produced at a single point in time,
as well as the change in the path, can be defined exactly. When we know
these instantaneous changes or, rather, since these are actually nothing,
their mutual relationship, we have gained a great deal. Furthermore, the
work of integral calculus is to study changing motion in a finite space. It is
my opinion that it is hardly necessary to show further the uses of differen-
tial calculus and analysis of the infinite, since it is now sufficiently noted,
if even a cursory investigation is made. If we want to study more carefully
the motion of either solids or fluids, it cannot be accomplished without
analysis of the infinite. Indeed, this science has frequently not been suf-
ficiently cultivated in order that the matter can be accurately explained.
Throughout all the branches of mathematics, this higher analysis has pen-
etrated to such an extent that anything that can be explained without its
intervention must be esteemed as next to nothing.
I have established in this book the whole of differential calculus, deriving
it from true principles and developing it copiously in such a way that noth-
ing pertaining to it that has been discovered so far has been omitted. The
work is divided into two parts. In the first part, after laying the founda-
tions of differential calculus, I have presented the method for differentiating
every kind of function, for finding not only differentials of the first order,
but also those of higher order, and those for functions of a single variable
as well as those involving two or more variables. In the second part, I have
developed very fully applications of this calculus both in finite analysis and
the study of series. In that part, I have also given a very clear explanation
of the theorem concerning maxima and minima. As to the application of
this calculus to the geometry of plane curves, I have nothing new to offer,
and this is all the less to be required, since in other works I have treated
this subject so fully. Even with the greatest care, the first principles of