CHAP. H.] SUPPLEMENT TO CHAP. VII. 105 



The whole of the calculus is but the deduction of rules for finding from 

 given equations as (1) their " differential equations " as (5), or inversely 

 of finding from the differential equation by "integration," or summation, 

 the equation between the variables themselves. 



Such of these rules as we need for our purpose we can now deduce. 



5. Differentiation and integration of powers of a iugle 



variable. We have already seen that the / d y = y and / 2 x d x = x*, 



hence d (a?) = 2 x d x. 

 If we should take y = a?, we should have, in like manner, as before, 



y + d y = (x + dx)* = x s + SaPdx + Sxdx' + dx*, 

 or dy = 3x*dx + Sxdx* + dx* t 



or % = 3 * + 3 * d x + d x*, 



d x 



and passing to the limits, as before, 



j^ = 3 **, or d y=S x 1 d x. Hence the differential of ' or d (z 3 ) =3 x 1 d x, 



CL (K 



and reversely, the integral of3x t dxorl3x t dx = x 3 . In similar man- 

 ner, we might find 



d (x s ) = 5 x* d x and / 5 x* d x = x". 



Comparing these expressions, we may easily deduce general rules which 

 will enable us at once upon sight to " differentiate," that is, find the rela- 

 tion connecting the increments ; and " integrate " or sum up the successive 

 consecutive values of the variable; for any expression containing the 

 power of a single variable. 



These rules are as follows: 



To differentiate : 



" Diminish the exponent of the power of the variable Try unity, and then 

 multiply by the primitive exponent and ~by the increment of the variable." 



Thus, d(x*) = 2 x d x, <Z(a> 3 ) = 3 tfdx, d(x') = Ix'dx, d(x%) = xidx, 



d (of) = n as"" 1 d x, etc. 



To integrate : 



" Multiply the variable with its primitive exponent increased by unity, by 

 the constant factor, if there is any, and divide the result by the new exponent." 



before the end of that time approach nearer than any given difference, are egual." 

 There can be little doubt that Newton saw clearly that although the quantities 

 might never be able to actually reach their limits, yet that those limits them- 

 selves were equal, and hence the increment could be left out in the equation, 

 but not because by any means it was of insignificant size. His terms ' ' ultimate 

 ratio" and ' l fluent " are alone sufficient to indicate that he understood the 

 true logic of the method he discovered ; while Liebnitz seems to have stood 

 gazing with wonder at the workings of the machine he had found, but whose 

 mechanism he did not understand. [See Philosophy of Mathematics. Bledsoe. 

 Lippincott & Co., 1868.] 



