TOL. XXV.] PHILOSOPHICAL TRANSACTIONS. 341 



The demonstration of the foregoing is as follows. Every thing remaining as 

 in the construction, and cp iDcing produced, if necessary, to cut am in h, then 

 CH will be an ordinate of the parabola to the diameter ah, and therefore ch^ = 

 AL X AH = AH because AL = 1. But CH = CP + AG, and ah = gb + bp, 

 therefore cp^ + 2ag X cp + ag"^ = gb -f bp* But, by the nature of the 

 parabola, ag^ = bg : hence cp^ + 1kg X cp = bp. Now from the point c 

 let CD be perpendicular to bp, which may also meet ei parallel to bp in the 

 point I. Then, because of the similar triangles cdp and tvs, it will be 



DP = , and CD = ; therefore cp + ^ag X cp = bp = dp + 



ST ST ' 



VS X CP , 11^... VS -rt . 



db = h BR — IE ; or cp + 2ag X cp cp — BR = — IE. But 



ST ' ST 

 liL^ zrz CE?^ — CI^ := CE'^ — CD^ — Vt'^ — 2CD X VT = CE^ — r — 



ST* 

 2VT* X CP ,, , , l\ 2 II SV» 



VT = (because vt* = st — sv*) ce — cp A ; cp* — st* 



ST ^ ST* 



_j_ sv^ — 2sT X CP H CP ; this therefore will be equal to the square 



whose side is cp* + 2ag X cp cp — br. And when this equation is re- 



' ST * 



duced to the terms j&, q, r, s, /, it becomes the very equation proposed. 



Hence it appears that any one biquadratic equation will admit of innumerable 

 different constructions by the parabola, according to the different values of that 

 quantity which we said might be assumed at pleasure. But the most simple case 

 is, by making vs = p = O, and then the construction coincides with the com- 

 mon one, in which the right lines cp, &c. representing the roots, are perpendi- 

 cular to the axis ; in which case the equation becomes 



a?* = — Ara^ — Ar^x^ -\- 4rsx — q* 



+ 25 — 29 — .5^ ' ;^ 



- 1 + <S 



which is easily constructed as above. 



§ IV. But lest the organical description of the parabola should seem too 

 difficult, we may have recourse to a certain mechanical artifice, to be performed 

 by means of a plummet, or thread with a weight hanging at the end of it; by 

 help of which the last equation may be constructed very easily and exactly, and 

 thence the roots of any cubic or biquadratic equations may be found; and that 

 without drawing any other than right lines and a circle. Now this construction 

 which may be called a mechanical one, is in the following manner. 



Against a smooth and upright wall, or any other plane perpendicular to the 

 horizon, at any point p, (fig. 2, pi. lO) let there be hung a very fine flexible 

 thread fp, with any weight p at its extremity. In this thread mark any point 



