10 



GENERAL THEOREMS, CHIEFLY PORISMS, 



PROP. 5. Local Theorem. Fig. 11. If from a given point A, 

 a straight line DE move parallel to itself, and another cs, 

 from a given point c, move along with 

 it round c ; and a point i move along A B, 

 from H, the middle point of A B, with a 

 velocity equal to half the velocity of D E ; 

 then, if A P be always taken a third pro- 

 portional to A s and B c, and through p, 

 with asymptotes D' E' and A B, a conic 

 hyperbola be described ; also with focus I 

 and axis A B, a conic parabola be de- 

 scribed ; then the radius vector from c to M, the intersection of 

 the two curves, moving round c, shall describe a given ellipse. 

 PROP. 6. Theorem. A common logarithmic being given, 

 and a point without it, a parabola, hyperbola, and ellipse may 

 be described, which shall intersect the logarithmic and each 

 other in the same points ; the parabola shall cut the logarith- 

 mic orthogonally ; and if straight lines be drawn from the 

 given point to the common intersections of the four curves, 

 these lines shall be normals to the logarithmic. 



PROP. 7. Porism. Two points in a circle being given, but 

 not in one diameter, another circle may be described, such, 

 that if from any point of it to the given points straight lines 

 be drawn, and a line touching the given circle, the tangent 

 shall be a mean proportional between the lines so inflected. 

 Or, more generally, the square of the tangent shall have a 

 given ratio to the rectangle under the inflected lines. 



PROP. 8. Porism. Fig. 12. Two straight lines AB, AP, 

 not parallel, being given in position, a conic 

 parabola MN may be found, such, that if, 

 from any point of it M, a perpendicular M p 

 be drawn to the one of the given lines 

 nearest the curve, and this perpendicular be 

 produced till it meets the other line in B ; 

 and if from B a line be drawn to a given 

 point c ; then M p shall have to P B together 

 with c B, a given ratio. 



Kg.12. 



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