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Mr. Brougham's general Theorems 
PROP. XVII. PORISM. 
A parabola of any order being given, two 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 intersections ; 
and, conversely, if the hyperbolas be given, a parabola may be 
found, &c. &c. 
PROP. XVIII. PORISM. 
A parabola of any order [m -j- n) being given, another of 
an order (m 2, n) 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=i,n = i, and let the given ratios be those 
of equality ; the proposition is this ; a conic parabola being 
given, a semicubic 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 whatsoever, I 
shall take this opportunity of introducing a very simple, and, I 
think, perfectly conclusive demonstration, of the 28th lemma, 
Prmcipia,Bookl. “ that no oval can be squared T It is well known, 
that the demonstration which Sir Isaac Newton gives of this 
lemma, is not a little intricate ; and, whether from this diffi- 
culty, or from some real imperfection, or from a very natural 
