Page 9 - Foundations Of Differential Calculus

viii Preface
2
the quantity x, since this has the ratio to the increment of the square x
2
as1to2x, the increment of the square x is equal to 2xω and for this
reason is also equal to zero. Although both of these increments vanish
simultaneously, this is no obstacle to their ratios being determined as 1
to 2x. With respect to this nothing that so far has been represented by
the letter ω, in differential calculus we use the symbol dx and call it the
differential of x, since it is the increment of the quantity x. When we put
2
dx for ω, the differential of x becomes 2xdx. In a similar way, it is shown
3
2
that the differential of the cube x will be equal to 3x dx. In general, the
n
differential of any quantity x will be equal to nx n−1 dx. No matter what
other functions of x might be proposed, differential calculus gives rules for
finding its differential. Nevertheless, we must constantly keep in mind that
since these differentials are absolutely nothing, we can conclude nothing
from them except that their mutual ratios reduce to finite quantities. Thus,
it is in this way that the principles of differential calculus, which are in
agreement with proper reasoning, are established, and all of the objections
that are wont to be brought against it crumble spontaneously; but these
arguments retain their full rigor if the differentials, that is, the infinitely
small, are not completely annihilated.
To many who have discussed the rules of differential calculus, it has
seemed that there is a distinction between absolutely nothing and a special
order of quantities infinitely small, which do not quite vanish completely
but retain a certain quantity that is indeed less than any assignable quan-
tity. Concerning these, it is correctly objected that geometric rigor has been
neglected. Because these infinitely small quantities have been neglected, the
conclusions that have been drawn are rightly suspected. Although these in-
finitely small quantities are conceived to be few in number, when even a
few, or many, or even an innumerable number of these are neglected, an
enormous error may result. There is an attempt wrongfully to refute this
objection with examples of this kind, whereby conclusions are drawn from
differential calculus in the same way as from elementary geometry. Indeed,
if these infinitely small quantities, which are neglected in calculus, are not
quite nothing, then necessarily an error must result that will be the greater
the more these quantities are heaped up. If it should happen that the er-
ror is less, this must be attributed to a fault in the calculation whereby
certain errors are compensated by other errors, rather than freeing the cal-
culation from suspicion of error. In order that there be no compensating
one error by a new one, let me fix firmly the point I want to make with
clear examples. Those quantities that shall be neglected must surely be
held to be absolutely nothing. Nor can the infinitely small that is discussed
in differential calculus differ in any way from nothing. Even less should this
business be ended when the infinitely small is described by some with the
example wherein the tiniest mote of dust is compared to a huge mountain
or even to the whole terrestrial globe. If someone undertakes to calculate