IN THE HIGHER GEOMETRY. 



19 



PROP. 19. Problem. Fig. 18. A common logarithmic being 

 given ; to describe 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 logarithmic, G its modula; 

 A B the ordinate at its origin A ; 

 let c be the given point ; A x o B 

 the given area ; M : x the given ratio : draw B Q parallel to A N ; 

 find D a 4th proportional to M, the rectangle BQ . OQ, and 

 M + x. From A D cut off a part A L, equal to A c together with 

 twice G ; at L make L H perpendicular to A D, and between the 

 asymptotes A L, H L, with a parameter, or constant rectangle, 

 twice (D + 2 . A B . 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 A B be the transverse axis, E F 

 the semi-conjugate ; E, of conse- 

 quence, the centre ; c and L the foci. 

 On A B describe a semicircle. Divide 

 the quadrant AK in o in the given 

 ratio of M to x, in which the area is 

 to be cut, and describe the cycloid 

 G M R, such, that the ordinate P M may be always a 4th propor- 

 tional to the arc o Q, the rectangle A B x 2 F E, and the line c L ; 

 this cycloid shall cut the ellipse in M, so that, if M c be joined, 

 the area ACM shall be to c M B : : M : x. 



Demonstr. Let A p = x, P M = y, A c = c, A B = a, and 2 E F 

 = b ; then, by the nature of the cycloid G M R, P M : o Q : 



M 



2 F E x AB : c L, and Q o = A o AQ=by const. x 



M -f- X 



(A K A Q) ; also, CL = AB 2AC, since A c = L B. There- 



c 2 



Fxg-.i9 . 



