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THE INFINITESIMAL CALCULUS 69 

of tangents at the beginning of this chapter. Again, if y = 
a . x, where a is constant ; dy = a.dx. If y = x . z, then dy = 

X 

(x + dx}(z + dz)x . z = x . dz + z . dx. If y = , x = y . z, so 

Z 

dx = y.dz + z.dy; hence dy y ' Z . Since the integral 

m 

calculus is the inverse of the differential calculus, we have at once 
f2x . dx = x z , \a . dx = afdx, 

fx . dz + / z . dx = xz, 

and so on. More fully, from d(x 3 ) = 3cc 2 . dx, we conclude, not 
that fx 2 . dx = $x 3 , but that fx 2 . dx = $x 3 + c, where <: c " de- 
notes some constant depending on the fixed value for x from 
which the integration starts. 

Consideraparabola?/ 2 = a#; then 2y.dy a. dx, or dx = ' . 

a 

tf) O j 

Thus the area from the origin to the point x is j^L-l ? + c ; 

8 

2ty 3 ^i/ 2 cfo/ 2V 3 

but d- = y ' y ; thus the area is - + c, or, since y 2 ax, 
3ct a on 

%x .y + c. To determine c when we measure the area from to x, 
we have the area zero when x = ; hence the above equation 
gives c = 0. This whole result, now quite simple to us, is one 
of the greatest discoveries of Archimedes. 

Let us now make a few short reflections on the infinitesimal 
calculus. First, the extraordinary power of it in dealing with 
complicated questions lies in that the question is split up into 
an infinity of simpler ones which can all be dealt with at once, 
thanks to the wonderfully economical fashion in which the 
calculus, like analytical geometry, deals with variables. Thus, 
a curvilinear area is split up into rectangular elements, all the 
rectangles are added together at once when it is observed that 
integral is the inverse of the easily acquired practice of differ- 
entiation. We must never lose sight of the fact that, when we 
differentiate y or integrate y . dx, we are considering, not a par- 
ticular x or y, but any one of an infinity of them. Secondly, 
we have seen that what in the first place had been regarded 
but as a simple method of approximation, leads at any rate in 
certain cases to results perfectly exact. The fact is that the