376 PHILOSOPHICAL TRANSACTIONS. [aNNO iQS? . 



On the Construction of Solid Problems, or of Equations of the third and fourth 

 Degree, by means of only one given Parabola and a Circle. By Mr. Edmund 

 Halley. N° 188, p. 335. Translated from the Latin. 

 How all equations, that involve the third or fourth power of the unknown 

 quantity, may be constructed by means of any given parabola and a circle, is 

 shown and clearly demonstrated by Descartes in the third book of his geometry : 

 but first he directs to take away the second term of the equation, if such there 

 be, and then by a rule there given, to find the roots of the equation so reduced. 

 But since that operation appeared too laborious, some have devised a like con- 

 struction, without any such previous reduction ; among whom Fr. Schooten 

 might be thought to have discovered a very easy and simple method for con- 

 structing cubifc equations howsoever afiected, if by unfolding the principle from 

 whence he derived his rule, he had paid a greater regard for the memory of his 

 readers, which he overcharges with many perplexed cautions. But our country- 

 man Mr. Thomas Baker, in an entire treatise written on these constructions, 

 has comprised in one general rule, not only cubic, but also biquadratic equations 

 of any kind ; and this rule he has sufficiently illustrated by demonstrations and 

 examples in all cases ; and towards the end he subjoined a method of investi- 

 gating that general rule ; but he has not shown the very method, by means of 

 which he obtained his Universal Geometrical Clavis, or at least might have ob- 

 tained it with much more ease ; and since this rule of Baker's is no less per- 

 plexed with cautions about the signs -)- and — , than that of Schooten, so that 

 no person can pretend to perform these constructions aright without having the 

 book by him ; I thought it would neither be unpleasant nor unprofitable to 

 young students, to explain the foundations of both, and by some amendment 

 of the method, to clear up as much as possible so difficult a subject, Descartes's 

 construction, which very easily discovers the roots of all cubic and biquadratic 

 equations, where the second term is wanting, may be supposed as known ; but 

 as it is the key to what is to follow, it may not be improper to add here his rule, 

 with some alteration for the better. When the second term is wanting, all 

 cubic equations are reduced to this form z^ >|<: a/)z. aaq = o ; and biquadratic 

 ones to this form z* ■>^ apzz. aaqz. d'^ r = o ; where a denotes the parameter 

 of any given parabola, used in the construction ; or else taking a for unity, 

 the equations are reduced to these forms; viz. 3^ >Kpz-9 = O, or z* ^ pzz. 

 qz. r = 0. Now the parabola FAG, fig. 7, pi. 10, being given, whose axis 

 is ACDKL, and latus rectum a or 1, let AC be taken = J^a, and let it be 

 always set off from the vertex A towards the inner parts of the figure ; then 

 take CD = \p, in that line AC produced towards C, if it be — j!> in the equa- 



