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68 THE NATURE OF MATHEMATICS 

we see that it must be y . dx Hence 



and hence the sign " d " destroys, so to speak, the effect of the 
sign " / ". We also have fdx = x, and find that this summation 
is the inverse process to differentiation. Thus the problems of 
tangents and quadratures are inverses of one another. The 
quantity which by its differentiation produces a proposed dif- 
ferential, is called the " integral " of this differential ; since we 
consider it as having been formed by infinitely small continual 
additions : each of these additions is what we have named the 
differential of the increasing quantity, it is a fraction of it : 
and the sum of all these fractions is the entire quantity which 
we are in search of. For the same reason we call "integrat- 
ing" or "taking the sum of" a differential the finding the 
integral of the sum of all the infinitely small successive 
additions which form the series, the differential of which, 
properly speaking, is the general term. 

It is evident that two variables which constantly remain 
equal increase the one as much as the other during the same 
time, and that consequently their differences are equal : and 
the same holds good even if these two quantities had differed 
by any quantity whatever when they began to vary ; provided 
that this primitive difference be always the same, their differ- 
entials will always be equal. 

Eeciprocally, it is clear that two variables which receive at 
each instant infinitely small equal additions must also either 
remain constantly equal to one another, or always differ by the 
same quantity : that is, the integrals of two differentials which 
are equal can only differ from each other by a constant quantity. 
For the same reason, if any two quantities whatever differ in 
an infinitely small degree from each other, their differentials 
will also differ from one another infinitely little : and recipro- 
cally, if two differential quantities differ infinitely little from 
one another, their integrals, putting aside the constant, can 
also differ but infinitely little one from the other. 

Now, some of the rules for differentiation are as follows. 
If y=f(x), dy=f(x + dx)f(x\ in which higher powers of 
differentials added to lower ones may be neglected. Thus, 
if y = x z , then dy = (x + dx) 2 -x 2 = '2x.dx+(dx)'* = l 2x .dx. 
Here it is well to refer back to the treatment of the problem