30 NEWTON S PRINCIPIA. 



Thus in the parabola z/ 2 = 2 a x, the radius of curvature is 



a 



In all these operations, however, it must be observed, 

 that as constant or invariable quantities have no dif 

 ferentials, so when we reverse the operation and find in 

 tegrals from given differential expressions, we never can 

 tell whether a constant must not be added in order to com 

 plete that quantity, by taking whose differential the given 

 expression was originally obtained. The determining of 

 this constant quantity, and the finding whether there be 

 any or not, depends upon the particular conditions of each 

 problem. It is always added as a matter of course. 

 Thus when we integrate d x + dy,we cannot tell whether 

 this quantity arose from taking the differentials of x and y 

 only, or from taking the differential of x + y + c ; and it 

 must depend upon the nature of the question whether c is 

 to be added to the integral or no ; and if to be added, how 

 it shall be ascertained. 



Having explained this important method of investigation, 

 by the help of which Newton was enabled to make his 

 greatest mathematical discoveries, and by the principles 

 of which he demonstrates them in the Principia, it only 

 remains, before proceeding to the analysis of those dis 

 coveries, that we should remark the preference which he 

 gives to the geometrical methods, improved and adapted 

 to his purpose by the doctrine of Prime and Ultimate ratios. 

 He uses this doctrine similar in principle to, and the foun 

 dation of, the noble and refined calculus which we have 

 been considering ; but he does not at all employ that 

 calculus. 



The First book treats of the motion of bodies with- 



