Page 11 - Foundations Of Differential Calculus

x Preface
duced, it is not immediately known what method is to be used to compare
the vanishing increments of absolutely all kinds of functions. Gradually
this discovery has progressed to more and more complicated functions. For
example, for the rational functions, the ultimate ratio that the vanishing
increments attain could be assigned long before the time of Newton and
Leibniz, so that the differential calculus applied to only these rational func-
tions must be held to have been invented long before that time. However,
there is no doubt that Newton must be given credit for that part of differ-
ential calculus concerned with irrational functions. This was nicely deduced
from his wonderful theorem concerning the general evolution of powers of
a binomial. By this outstanding discovery, the limits of differential calculus
have been marvelously extended. We are no less indebted to Leibniz insofar
as this calculus at that time was viewed as individual tricks, while he put
it into the form of a discipline, collected its rules into a system, and gave a
crystal-clear explanation. From this there followed great aids in the further
development of this calculus, and some of the open questions whose an-
swers were sought were pursued through certain definite principles. Soon,
through the studies of both Leibniz and the Bernoullis, the bounds of dif-
ferential calculus were extended even to transcendental functions, which
had in part already been discussed. Then, too, the foundations of integral
calculus were firmly established. Those who followed in the elaboration of
this field continued to make progress. It was Newton who gave very com-
plete papers in integral calculus, but as to its first discovery, which can
hardly be separated from the beginnings of differential calculus, it cannot
with absolute certainty be attributed to him. Since the greater part has yet
to be developed, it is not possible to say at this time that this calculus has
absolutely been discovered. Rather, let us with a grateful mind acknowl-
edge each one according to his efforts toward its completion. This is my
judgment as to the attribution of glory for the discovery of this calculus,
about which there has been such heated controversy.
No matter what name the mathematicians of different nations are wont
to give to this calculus, it all comes to this, that they all agree on this
outstanding definition. Whether they call the vanishing increments whose
ratios are under consideration by the name differentials or fluxions, these
are always understood to be equal to zero, and this must be the true notion
of the infinitely small. From this it follows that everything that has been
debated about differentials of the second and higher orders, and this has
been more out of curiosity then of usefulness, comes back to something
very clear, namely, that when everything vanishes together we must con-
sider the mutual ratio rather than the individual quantities. Since the ratio
between the vanishing increments of the functions is itself expressed by
some function, and if the vanishing increment of this function is compared
with others, the result must be considered as the second differential. In this
way, we must understand the development of differentials of higher orders,