3()8 PHILOSOPHICAL TRANSACTIOKS. [aNNO I687. 



HR, or the distance of the centre R of the circle from the axis, is always the 

 difference between ^p and -^h^ + 4-9 ; which if they are equal, then the centre 

 falls in the axis ; \f ibp be greater than -^b^ + -^q, then it falls on the left side 

 of the axis, if less, on the right. If therefore the square root, or \^ddd, of 

 the cube of ■§■ VH, that is, o( -^bb — ^p, calling it d, be greater than HR, or 

 the difference between -^'-j-b^ + t9 and ■^bp ; the centre R is found within the 

 space NPV, circumscribed by the paraboloids VPX, VNL, and the indefinite 

 right line DNP; and therefore the circle will cut the parabola in three points 

 y, Y, Y, on the right side of the line BK, and so the equation have three affir- 

 mative roots. But the centre being without this space NVP, it is explicable 

 by only one affirmative root. Here we are to observe, by the bye, that the 

 right line DP touches the paraboloid VPX in the point P, EP being -^b^ ; 

 but cuts the other paraboloid VNL in the point N, such that letting fall NF 

 perpendicular to the axis, VF is 4- of EV, or -^bb, and NF ttt^'- But VW, 

 which, from the point V being raised perpendicular to the axis, meets the line 

 DP in W, is = ^\b^ or 4- EP. 



Hence we may safely conclude, that if in the equation, either /> be greater 

 than 4-^% or q be greater than -rV^^> that there will be found only one root, 

 and that an affirmative one. Cartes's rule then, p. 70, Edit. Amsterd. 1659, 

 fails, in which he determines that there are always as many true roots, as there 

 are changes of the signs + and — in the equation ; though Schooten in his 

 commentaries, would excuse this mistake; for innumerable more equations of the 

 preceding form, with three changes of signs may be devised, which have rather 

 one than three roots. Also Prop. 5, Sect. 5, of our countryman Harriot's Ars 

 Analytica, and Prob. 1 8 of Vieta's Numer. Potest. Resol. is hardly founded ; 

 since, from the limitations they have there set down, that must agree to the 

 whole parallelogram PIV W, which we have just now proved to agree only to 

 the space NVP alone, that is, to afford a centre to the circle intersecting the 

 parabola in three other points, besides the vertex. 



But the quantity q, or the last term, {b and p given, so that p be less than 

 4-6^) is accurately limited from the preceding equation V ddd = Vt^* + -s-y 

 ^■^bp; viz. when the circle touches the parabola; therefore-]^ should be less 

 than -^bp — ^'^i^ -\- t/d^; but if p be greater than -^bb, and that 4^ should 

 be greater than -^bp — ^b^ — -/d^, then the centre does not fall in the space 

 NV\^ : and under these conditions the equation will always be explicable by 

 three roots ; otherwise by one only. But whether there be three or one, they 

 are always affimative, because of the position of the centre R to the right hand 

 of the line DP. 



And this is the most difficult case; so that those who understand what has 

 been said above, will easily comprehend what follows. . Now let the equation 



