IN THE HIGHER GEOMETRY. 17 



PROP. 1C. Porism. Fig. 16. If between two straight lines 

 making a right angle, an infinite number of parabolas of any 

 order whatever be described ; a conic 

 parabola may be drawn, such, that if 

 tangents be drawn to it at its intersec- 

 tions with the given curves, these tan- - ^ /& B 



gents shall always cut, in a given 



ratio, the areas contained by the given curves, the curve 

 found, and the axis of the given curves. Let AMN be one of 

 the given parabolas ; D M o the parabola found, and T M its 

 tangent at M : ATM shall have to T M R a given ratio. 



PROP. 1 7. Porism. A parabola of any order being given : 

 two straight lines may be found, between which if innu- 

 merable hyperbolas of any order be described ; the areas cut 

 off by the hyperbolas and the given parabola at their inter- 

 sections, shall be divided, in a given ratio, by the tangents 

 to the given curve at the intersections ; and conversely, if 

 the hyperbolas be given, a parabola may be found, &c. 



PROP. 18. Porism. A parabola of any order (m + ) being 

 given, another of an order (m -f- In) may be found, such, that 

 the rectangle under its ordinate and a given line, shall have 

 always a given ratio to the area (of the given curve) whose 

 abscissa bears to that of the curve found a given ratio. 



Example. Let m = 1, n = 1, and let the given ratios be 

 those of equality ; the proposition is this : a conic parabola 

 being given, a semi-cubic one may be found, such, that the 

 rectangle under its ordinate and a given line, shall be always 

 equal to the area of the given conic parabola, at equal 

 abscissas. 



Scholium. A similar general proposition may be enunciated 

 and exemplified, with respect to hyperbolas ; and as these are 

 only cases of a proposition applying to all curves whatever, I 

 shall take this opportunity of introducing a very simple, and 

 I think perfectly conclusive demonstration, of the 28th 

 lemma, " Principia," Book i., " that no oval can be squared." 

 It is well known, that the demonstration which Sir Isaac 

 Newton gives of this lemma is not a little intricate; and, 



