26 NEWTON S PEINCIPIA. 



manner we can only obtain the exact integral where the 

 expression submitted to us is a complete differential Thus, 



though such an expression as == is integrable, such 



an expression as is not integrable, for want 



v 1 -f-# 2 



of the x in the numerator; and various approximations 

 and other contrivances are resorted to in order to ac 

 complish or, at least, approach this object, of which the 

 methods of series^ of logarithms, and circular arcs are the 

 most frequently used. The simplest case of integration 

 by series may be understood in examples like the last; 

 for if the square root be extracted by a series, we may be 

 able to integrate each term, and so by the sum of the 

 integrals to approach the real value of the whole. 



From the doctrine as now explained, and the original 

 foundations of the method as traced above, it follows that a 

 variety of the most important problems may be solved with 

 ease and certainty, which by the ancient geometry could 

 only in certain cases, or by a happy accident, be investigated. 

 Thus the tangents of curves may be found. For as the 



subtangent S P : P M :: M N : T N, S p = 



= ~j~ And so the perpendicular may always be drawn ; 



for the subnormal EP = -- = ~^ = ^ There ~ 



fore we have only to insert the one of these quantities in 

 terms of the other from the equation between x and y (the 

 equation to the curve), and we get the expressions for the 



