396 PHILOSOPHICAL TKANSACTIONS. [aNNO I687. 



a plane one. Therefore, all cubic equations, however affected, are explicable 

 either by one or by three roots, and which are always possible, if we admit ne- 

 gative roots for true ones ; so biquadratics, whose last term r is affected with 

 the sign — , are explicable by two or four; but if it be + '" in the equation, 

 and it be so great, that \/GD-—ar, fig. 8, pi. ]0, be less than that the circle, 

 described with that radius, and on the centre G, can touch the parabola in some 

 point, the given equation is altogether impossible, nor is it explicable by any 

 affirmative or negative root; but of this more hereafter. 



Since then there is so great a difference between the cases of cubic and biqua- 

 dratic equations, that they cannot be comprised together ; we shall first treat of 

 cubics, and then of the others. Now cubic equations are constructed by an infi- 

 nite number of circles in a given parabola ; but biquadratics by one only, at 

 least, by these methods, and because, putting z — e, or any other indetermi- 

 nate, equal to nothing, a cubic equation is reduced to a biquadratic, having the 

 same roots with the cubic ; and besides another equal to e ; whence it is, that 

 a cubic equation may be constructed by as many different circles as we can ima- 

 gine quantities, e, that is an infinite number; and of all these constructions, 

 the easiest is that which I gave towards the end of my preceding discourse on 

 the construction of cubic and biquadratic equations ; yet the following is little 

 inferior to it, which seems better accommodated to the design of determining 

 the number of the roots and their limits, and which arises from the taking away 

 of the second term, by putting, after the common manner, x= z -\- or — -^of 

 the co-efficient of the second term, and it is this: having given the parabola 

 A BY, fig. 10, whose vertex is A, axis AE, and latus rectum o, reduce the 

 equation to the usual form, viz. z". iz*. apz. aaq = O; then at the distance of' 

 4-i, let BK be drawn parallel to the axis, to the right hand if it be -|- b, other- 

 wise to the left, meeting the parabola in B; and let the indefinite line DP be 

 raised perpendicular to the supposed line AB, meeting the axis in G. From the 

 point B, let fall upon the axis the perpendicular BC, and let GE be always = 

 to AC, and be set off towards the lower side. From E, set off EH = 4/) ; 

 upwards if it be -}- J5 in the equation, but downwards if — jb ; and from the 

 point H (or E, if the quantity p be wanting) let the perpendicular H Q be pro- 

 duced meeting the indefinite line DP in the point O. Lastly, in the indefinite 

 lineHQ, take OR = ^q, to be set off from O to the right hand if it be 

 — q, but to the left if -J- 9 ; and a circle, described from the centre R, and 

 with the radius R A, will cut the parabola in so many points as the proposed 

 equation has true roots ; and they will be the perpendiculars ZY let fall from the 

 intersections Y to the line BK parallel to the axis; of which those on the right 

 hand of the line BK are affirmative, and those on the left, negative. 



The advantage of this construction consists in this, that it is performed by 



