VOL. LXXXVIII J PHILOSOPHICAL TRANSACTIONS. 351 



a given ratio a given area of the logarithmic ; the point where this curve meets the 

 logarithmic is always situated in a conic hyperbola, which may be found. 



Scholium. This proposition is, properly speaking, neither a porism, a theorem, 

 nor a problem. It is not a theorem, because something is left to be found, or, as 

 Pappus expresses it, there is a deficiency in the hypothesis : neither is it a porism; 

 for the theorem, from which the deficiency distinguishes it, is not local. 



Prop. 15. Porism. Fig. 15. — A conic hyperbola being given ; 2 points may be 

 found, from which if straight lines be inflected, to the innumerable intersections 

 of the given curve with parabolas or hyperbolas, of any given order whatever, 

 described between given straight lines ; and if co-ordinates be drawn to the inter- 

 sections of these curves with another conic hyperbola, which may be found ; the 

 lines inflected shall always cut off areas that have to one another a given ratio, from 

 the areas contained by the co-ordinates. — Let x and y be the points found ; hd 

 the given hyperbola, fe the one to be found ; adc one of the curves lying between 

 ab and ag, intersecting hd and fe ; join xd, yd ; then the area a yd : xdcb in a 

 given ratio. 



Prop. 16. Porism. Fig. 16. — If between 2 straight lines making a right angle, 

 an infinite number of parabolas of any order whatever be described ; a conic para- 

 bola may be drawn, such, that if tangents be drawn to it at its intersections with 

 the given curves, these tangents shall always cut, in a given ratio, the areas con- 

 tained by the given curves, the curve found, and the axis of the given curves. — Let 

 amn be one of the given parabolas ; dmo the parabola found, and tm its tangent at 

 m : atm shall have to tmr a given ratio. 



Prop. 17. Porism. — A parabola of any order being given ; 2 straight lines 

 may be found, between which if innumerable hyperbolas of any order be described; 

 the areas cut off" by the hyperbolas and the given parabola at their intersections, 

 shall be divided, in a given ratio, by the tangents to the given curve at the inter- 

 sections ; and conversely, if the hyperbolas be given, a parabola may be found, 

 &c. &c. 



Prop. 18. Porism. — A parabola of any order (m -{■ n) being given, another 

 of an order (m + 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 abscissae. 



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 



