( 3*5 ) 



An Analytical Refolution of certain 

 Equations of the Third^ Fifth % 

 Seventh, Ninth Powers^ and Jb 

 on ad Infinitum, in finite Terms^ 

 after the manner of Cardan'^ 

 Rules for Cubicks. By Mr. A. 

 Moivre, TranfaB. N°. 309. 



LET (w) be any Number, (y) an unknown 

 Quantity, or Root of the Equation, (a) a 

 Quantity altogether known, or what they call 

 Hamogeneutn Comparatiom : And let the Relation 

 of thefe Quantities to each other be expreft by 

 the Equation. 



n n— I nn — I nn — 9 



ny -+ n y 3 X n y> ~r» 



ix^ ax 3 4x5 



nn — I w»— 'O nn — 



X X —7 — — n 1 7, C#c. = a. 



ZX2 4x5* 6x7 ' 



Its plain from the Nature of this Series, that 

 if n be any odd Number (that is an Integer, it 

 matters not whether Affirmative or Negative) 

 then the Series will Terminate, and the Equa- 

 tion arifing will be one of the above defin'd, 

 whole Root is 



