352 PHILOSOPHICAL TRANSACTIONS. [ANNO 1798. 



which Sir Isaac Newton gives of this lemma, is not a little intricate ; and, whether 

 from this difficulty, or from some real imperfection, or from a very natural wish 

 not to believe that the most celebrated desideratum in geometry must for ever re- 

 main a desideratum, certain it is, that many have been inclined to call in question 

 the conclusiveness of that proof. 



Let amc be any curve whatever, (fig. 17,) and d a given line ; take in ab a part 

 ap, having to ap a given ratio, and erect a perpendicular pm, such, that the rect- 

 angle pm . d shall have to the area apm a given ratio ; it is evident that m will 

 describe a curve amc, which can never cut the axis, unless in a. Now because pm 



is proportional to — -, or to apm, pm will always increase ad infinitum, if amc is 

 infinite ; but if amc stops or returns into itself, that is, if it is an oval, pm is a 

 maximum at b, the point of ab corresponding to b in ab ; consequently the curve 

 amc stops short, and is irrational. Therefore pm, its ordinate, has not a finite re- 

 lation to ap, its abscissa ; but ap has a given ratio to ap ; therefore pm has not a 

 finite relation to ap, and apm has a given ratio to pm ; therefore it has not a finite 

 relation to ap, that is, apm cannot be found in finite terms of ap, or is incom- 

 mensurate with ap ; therefore the curve amb cannot be squared. Now amb is 

 any oval ; therefore no oval can be squared. By an argment of precisely the 

 same kind, it may be proved, that the rectification, also, of every oval is impossible. 

 Therefore, &c. <a. e. d. 



I shall subjoin 3 problems, that occurred during the consideration of the fore- 

 going propositions. The first is an example of the application of the porisms to 

 the solution of problems. The 2d gives, besides, a new method of resolving one 

 of the most celebrated ever proposed, Kepler's problem ; and the last exhibits a 

 curve before unknown, at least to me, as possessing the singular property of a 

 constant tangent. 



Prop. 19. Problem. Fig. 18. — A common logarithmic being given ; to de- 

 scribe a conic hyperbola, such, that if from its intersection with the given curve a 

 straight line be drawn to a given point, it shall cut a given area of the logarithmic 

 in a given ratio. The analysis leads to this construction. Let bme be the loga- 

 rithmic, g its modula ; ab the ordinate at its origin a ; let c be the given point ; 

 anob the given area ; if : N the given ratio : draw bg parallel to an ; find d a 4th 

 proportional to m, the rectangle bq . oq, and m -f- n. From ad cut off a part 

 al, equal to ac together with twice g ; at l make lh perpendicular to ad, and be- 

 tween the asymptotes al, hl, with a parameter, or constant rectangle, twice 

 (d -f- 2 . ab . g) describe a conic hyperbola : it is the curve required. 



Prop. 20. Problem. Fig. 19. — To draw, through the focus of a given ellipse, a 

 straight line that shall cut the area of the ellipse in a given ratio. — Const. Let ab 

 be the transverse axis, ef the semi-conjugate ; e, of consequence, the centre ; c 

 and l the foci. On ab describe a semicircle. Divide the quadrant ak in the given 

 ratio in which the area is to be cut, and describe the cycloid gmr, such, that the 



