in the higher Geometry. 391 
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. 8.) and D a given 
line ; take in a b a part ap , having to AP a given ratio, and 
erect a perpendicular p m, such, that the rectangle 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 a b corresponding to B in AB ; con- 
sequently, the curve amc stops short, and is irrational. There- 
fore pm, its ordinate, has not a finite relation 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 p m ; therefore 
it has not a finite relation to AP, that is, APM cannot be 
found in finite terms of AP, or is incommensurate with AP ; 
wherefore, the curve AMB cannot be squared. Now, AMB is 
any oval ; wherefore no oval can be squared. By an argument 
of precisely the same kind, it may proved, that the rectification, 
also, of every oval is impossible. Wherefore, &c. Q. E. D. 
I shall subjoin three problems, that occurred during the con- 
sideration 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 resolving one of 
the most celebrated ever proposed, Kepler’s problem; and 
