Mifcellanea Curiofa. 195 



to denote the whole preceding Term, which 

 will be the fame Series as Mr. Newton has 

 hrft found. 



By the fame Method, this general Pro- 

 blem may be folv ? d^ the AbfcilTe correfpond- 

 ing to a certain Area in any Curve being gi- 

 ven, to find the Abfcilfe, whofe correfpond- 

 ing Area fhall be to the firft in a given Ra- 

 tio. 



The Logarithmick Series might alfo be 

 found without borrowing any other Idea, 

 than that Logarithms are the Indices of 

 Powers : Let the Number, whofe Logarithm 

 we inquire, be i+S fuppofe its Log. to be 

 az. ~\- bzz. -\- czj i &c. Let there be another 

 Number i -\-y *, thereof its Logarithm will 

 be ay -\- byy -\- cy* 7 &c* Now if 1 ~|~ t — 



1 -\-y* , it follows, that az, ~|- bzjL -|- cz}^ &c. 

 ay~\- byy*\- cy 3 y &c. :: /?, 1. that is, az~\~ 

 bzjL-\-cz} 9 &c. = nay -|- nbyy «ry 3 , ■ &g. 

 Therefore we may find a Value of & exprelfc 

 by the Powers of y \ again, fince 1 -j- & == 



1 -\-y [U \ therefore z, = 1 -\-y^> — > 1, that is, 



n n—\ . n n> — i _ n — 2 



Z = #V + — X yy J- — X X 



I 2 1 Z 3 



&c Therefore is doubly expreft by 

 the Powers of y. Compare thefe two Values 

 together, and the Coefficients a, b 7 c, &c* 

 will be determined, except the firft a which 

 may be taken at pleafure, and gives accord- 

 ingly, all the different Species of Logarithms* 



Q % A* 



