18 GENERAL THEOREMS, CHIEFLY PORISMS, 



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 remain 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 a & a part a p, having to A p a given ratio, and 

 erect a perpendicular pm, such, that 

 the rectangle pm . D shall have to 

 the area A p M a given ratio ; it is 

 evident that m will describe a curve 

 a me, which can never cut the axis, 

 unless in a. Now because p m is pro- 



APM 

 portional 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, p m is a 

 maximum at 6, the point of a b corresponding to B in A B ; 

 consequently the curve a me stops short, and is irrational. 

 Therefore pm, its ordinate, has not a finite relation to ap, its 

 abscissa ; but a p has a given ratio to A p ; therefore p m has 

 not a finite relation to A p, and A p M has a given ratio to p m ; 

 therefore it has not a finite relation to A P, that is, A p M cannot 

 be found in finite terms of AP, or is incommensurate with 

 A P ; therefore the curve A M B cannot be squared. Xow A M B 

 is any oval ; therefore no oval can be squared. By an argu- 

 ment of precisely the same kind, it may be proved, that the 

 rectification, also, of every oval is impossible. Therefore, 

 &c. Q. E. D. 



I shall subjoin three problems, that occurred during the 

 consideration of the foregoing propositions. The first is an 

 example of the application of the porisms to the solution of 

 problems. The second gives, besides, a new method of re- 

 solving 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. 



