\ii Mifcellanea Curio fa . 



Curve Line, by the Evolution of which the 

 Parabola is defcrib'd (as Hugenim has demon- 

 ■ ftrated) and which, the Line DF (Fig. 2.) 

 which is perpendicular to the Parabola in 

 the point p, is always a Tangent to. But 

 the point P (that is, that in which the right 

 Line DF touches the Paraboloid) is the Cen- 

 ter of a Circle, which (being defcrib'd with 

 the Radius DP) coincides with the Parabola 

 in the point D, or has the fame Curvature 

 with it, as is manifeft. 



Having therefore defciib'd fiich Parabo- 

 loids UXP, VNa (Fig. 2.) on either fide 

 the Axis, 'tis clear ? that unlefs the Center 

 of the Circle be placed within thefe Limits, 

 it cannot interfecl the Parabola in more than 

 two points. From whence we may deter- 

 mine, under what conditions, the Coeffici- 

 ents of the intermediate Terms are retrain- 

 ed, in Biquadratick Equations, that fb there 

 may be four Roots. And at firft fight 'tis 

 plain that p cannot be greater than | bb y 

 (viz. in thofe Forms where 'tis dr.p) nor 

 than % b\ But in General, \ 6 b 3 -\~ipb^ 

 |$ that is EG the diftance of the Center 

 from the.A&is, ought to be lefs than EH^ 



4 Vi 7 VE% that is (becaufe VE^ I 6 ^r 



I p) than ; bb'\-%p \f\ 6 b2 ~1- or - f p, the 

 Sines and — being left doubtful, that fo 

 they may be varied according to the nature 

 of any Equation-, as was fhewn above in 

 Cubicks. Neither would I be ofFenfively te- 

 dious to the Learned on the one hand, nor 

 deprive Learners on the other, of the Ex- 



ercife 



