XXIV. Directions for making a machine for finding the roots of equations universally, with the manner of using it

Abstract

Sir, Perusing a discourse in the memoirs of the Royal Academy at Petersburgh, Tome vii. page 211, by the leanred John Andrew de Segner, containing an universal method of discovering the roots of equations, which you was so kind as to recommend to my consideration; I found, that the author's method, as you observed to me, consisted in finding several ordinates of a parabolic curve, such, that its abscissas being taken equal to any assumed values of the unknown quantity in the equation, the ordinates corresponding to those abscissas, should be equal to the values of all the terms in the equation (when brought to one side) that is, in other words, in finding several ordinates of a parabolic curve defined by the equation proposed: in which case, as is well known, if a curve be drawn through the extemities of the said ordinates, the points upon the axis, where the curve shall cut it, will necessarily give the several values of the real roots of the equation; and the several points, where the curve shall approach the base, but shall return without reaching it, will shew the impossible ones.