340 PHILOSOPHICAL TRANSACTIONS. PaNNO 1707. 



That is, every thing comes out as above supposed. 



^ III. Hitherto has been concerning the analytical solution of cubic and hU 

 quadratic equations. But because their geometrical efFection by the parabola is 

 commonly taught, and is much valued by some, I shall exhibit it here more 

 generally, and yet more compendiously. 



Any cubic or biquadratic equation being proposed, a comparison must be 

 made between its terms, and the corresponding terms of this following equation, 



~ 1 - 29 + <% 

 by means of which the values p, 9, r, 5, t will be easily found ; any one of them 

 being assumed at pleasure. Then in any given parabola avb, (fig. 1, pi. JO) 

 of which the principal vertex is v, axis vs, and perpendicular to the axis vt ; 

 take vs = /) within the parabola, and in the angle svt inscribe st = q, which 

 being produced cuts the parabola in two points n and o. Bisect no in m, and 

 through M draw ma parallel to the axis, and meeting the curve in a. Parallel 

 to NO draw al, such that al may be the latus-rectum of the parabola to the 

 diameter am, calling it unity. In al, produced both ways if necessary, take 

 AG = r, and from g draw gr parallel to the axis, to cut the curve in b, from 

 whence take br = s. From r draw re parallel and equal to vt, to the left 

 hand when 9 is a positive quantity, or to the right when g is negative. And 

 the same thing is to be understood of ag and br, which must be drawn on the 

 contrary side when r and s are negative. Lastly with centre e, and radius 

 EC = t, describe the circle ck^c, which will cut the parabola in as many points, 

 as there are real roots of the given equation. For, from those points p, k, &c, 

 draw CP, KII, &c. parallel to st, and terminating in the line gr, produced if 

 necessary; then each of these will be a value of .r, or the required root of the 

 given equation, viz. those lying towards the right-hand being affirmative roots, 

 but those on the left negative ones. And when there is a point of contact, in- 

 stead of an intersection, it is considered as two points of intersection that are 

 infinitely near each other. 



The only diflference between cubic and biquadratic equations, constructed 

 after this manner, will be this, that in the former, because of the last term be- 

 ing absent in the forego ing equation, it will always he p^ — g^ ^ s^ -\- t^ =z o, 

 or t = ^5^ + g^ — p'^. Therefore with the centre e and radius ec = 

 V^br'* + (er^) st':* — vs* any circle cKkc being described, one of the roots cp, 

 in the above construction, becomes nothing. 



