I o8 Mifcellanea Curio fa. 



Hand of the Line BK, and fo the Equation will 

 have three Roots. But the Center being with- 

 out this Space NUP, it is explicable but by 

 one Affirmative Root, Here it may be noted 

 by the by, that the Right Line DP may touch 

 • the Paraboloid UPX in the point P, EP be- 

 ing ^ , but wilf cut the other Paraboloid 

 U NL in the point AT, fo that letting fall NF 

 perpendicular to the Axis, UF is 5 EU, or 

 & b\ and NF ; os b\ But UW (which be- 

 ing perpendicularly applied to the Axis at 

 the point meets DP in is jss s \ or 



Hence we may fafely conclude, that if in 

 the Equation either p be greater than f 

 or q greater than 2 - 7 b' : ! , that there will be 

 found but one Root, and that an Affirmative 

 one. Cartas Rule therefore (Page 70. Edit. 

 Amfterd* 1659.) is net true, in which he de- 

 termines that' there are always as many true 

 Roots, as there are changes of the Sines \ 

 and — in the Equation: Schooten in his Corn- 

 men taries vainly endeavouring the defence 

 of this Miftake. Alfo Prof. 5. Sed. 5. of 

 our Country-man Harriot's Ars Andytke (as 

 alfo Prob. 1 8. of Vieta% Numer. Poiefi. Retold) 

 is hardly found ^ fince from the Limitations 

 which they have there fet down, that muft 

 agree to the whole Parallelogram PIUW, 

 which we have prov'd does agree only to 

 the Space NUP ; that is to afford a Center 

 to the Circle interfe&ing the Parabola in 

 three other points befides the Vertex. 



But the quantity q or the lafb Term (b 

 and p being given, lb that p be lefs than 



