imt.i.j DISSERTATION SECOND. 29 



course, removes all those causes of uncertainty which pre- 

 vailed when the results of continued action could alone be 

 estimated. It is not even by the effects produced in a 

 short time, but by effects taken in their nascent or eva- 

 nescent state, that the true proportion of causes must be 

 ascertained. 



Thus, though the astronomers had proved that the planets 

 describe ellipses round the sun as the common focus, and 

 that the line from the sun to each planet sweeps over areas 

 proportional to the time ; had not the geometer resolved the 

 elliptic motion into its primary elements, and compared them 

 in their state of evanescence, it would never have been dis- 

 covered that these bodies gravitate to the sun with forces 

 which are inversely as the square of their distances from the 

 centre of that luminary. Thus, fortunately, the first dis- 

 covery of Newton was the instrument which was to conduct 

 him safely through all the intricacies of his future investiga- 

 tions. 



The calculus, as already remarked, necessarily divides 

 itself into two branches ; one which, from the variable quan- 

 tities, finds the relation of their fluxions or differentials ; 

 another which, from the relation of these last, investigates 

 the relation of the variable quantities themselves. The first 

 of these problems is always possible, and, in general, easy to 

 be resolved ; the second is not always possible, and when 

 possible, is often very difficult, but in various degrees, ac- 

 cording to the manner in which the differentials and the va- 

 riable quantities are combined with one another. 



If the function, into which the differential stands multi- 

 plied, consist of a single term, or an aggregate of terms, in 

 each of which the variable quantity is raised to a power ex- 

 pounded by a number positive, negative, or fractional, the 

 integration can be effected with ease, either in algebraic or 

 logarithmic terms ; and the calculus had not been long known 

 before this problem was completely resolved. 



