An Analytical Resolution of certain Equations of the Third, Fifth, Seventh, Ninth Powers, and so on ad Infinitum , in finite Terms, after the manner of Cardan 's Rules for Cubicks. By Mr. A. Moivre , Transact. N o 309 .
LET (n) be any Number, (y) an unknown Quantity, or Root of the Equation, (a) a Quantity altogether known, or what they call Homogeneum Comparationis: And let the Relation of these Quantities to each other be exprest by the Equation.
ny +
nn - 1 / 2 × 3 ny3 +
nn - 1 / 2 × 3 ×
nn - 9 / 4 × 5 ny5 +
nn - 1 / 2 × 3 ×
nn - 9 / 4 × 5 ×
nn - 25 / 6 × 7 ny7,
&c. = a.
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 Equation arising will be one of the above defin'd, whose Root is