Page 81 - Foundations Of Differential Calculus

64 4. On the Nature of Differentials of Each Order
will vanish when compared with its predecessor (paragraph 88 and follow-
ing), so that only Pω will remain. For this reason, in the present case where
ω is infinitely small, the difference of y,∆y, is equal to Pω.
114. The analysis of the infinite, which we begin to treat now, is noth-
ing but a special case of the method of differences, explained in the first
chapter, wherein the differences are infinitely small, while previously the
differences were assumed to be finite. Hence, this case, in which the whole
of analysis of the infinite is contained, should be distinguished from the
method of differences. We use special names and notation for the infinitely
small differences. With Leibniz we call infinitely small differences by the
name differentials. From the discussion in the first chapter on the different
orders of differences, we can easily understand the meaning of first, second,
third, and so forth, differentials of any function. Instead of the symbol ∆,
by which we previously indicated a difference, now we will use the symbol
2
d, so that dy signifies the first differential of y, d y the second differential,
3
d y the third differential, and so forth.
115. Since the infinitely small differences that we are now discussing we
call differentials, the whole calculus by means of which differentials are
investigated and applied has usually been called differential calculus. The
English mathematicians (among whom Newton first began to develop this
new branch of analysis, as did Leibniz among the Germans) use different
names and symbols. They call infinitely small differences, which we call
differentials, fluxions and sometimes increments. These words seem to fit
better in Latin, and they signify reasonably well the things themselves.
A variable quantity by continuously increasing takes on various different
values, and for this reason can be thought of as being in flux, from which
comes the word fluxion. This was first used by Newton for the rate of
change, to designate an infinitely small increment that a quantity receives,
as if, by analogy, it were flowing.

116. It might be uncivil to argue with the English about the use of words
and a definition, and we might easily be defeated in a judgment about the
purity of Latin and the adequacy of expression, but there is no doubt that
we have won the prize from the English when it is a question of notation. For
differentials, which they call fluxions, they use dots above the letters. Thus,
˙ y signifies the first fluxion of y, ¨ is the second fluxion, the third fluxion
y
has three dots, and so forth. This notation, since it is arbitrary, cannot
be criticized if the number of dots is small, so that the number can be
recognized at a glance. On the other hand, if many dots are required, much
confusion and even more inconvenience may be the result. For example, the
tenth differential, or fluxion, is very inconveniently represented with ten
10
dots, while our notation, d y, is very easily understood. There are cases
where differentials of even much higher order, or even those of indefinite