CONIC SECTIONS. 



Fig. 33. If an ordinate be drawn to a 

 second diameter of opposite hyperbolas: 

 the square of this second diameter is to 

 the square of the conjugate diameter, as 

 the sum of the squares of' half the second 

 diameter, and the part of it between the 

 ordinate and the centre is to the square 

 of the ordinate. 



Let A B and MN be conjugate diame- 

 ters of opposite hyperbolas, H K an ordi- 

 nate to the second diameter M N, and 

 draw K D S parallel to MN : then K D S 

 is ordinately applied to AB (18.) ; there- 

 fore 



MC':CB'::KR',orCL':ADxDB, 

 orC D 1 C B' (Cor. 3. Def. 15.) 

 therefore, M C> : C B 1 :: M C -f 

 CL*:CD',orKL. 



PBQP. XXI. 



Fig. 34. If two parallel lines be drawn 

 from two points in the diameter of a 

 parabola to cut or touch the curve : then, 

 as the rectangle under the segments of 

 the secant, or the square of the tangent, 

 drawn from one point, is to the rectangle 

 under the segment of the secant, or the 

 square of the tangent drawn from the 

 other "point, so is the abscissa of the dia- 

 meter between the first point and the 

 curve to the abscissa between the second 

 point and the curve. 



Let the parallel secants M N and P Q 

 meet the diameter of a parabola in D and 

 E : it has been shewn (Prop. 15.) that the 

 diameters of a parabola meet the curve 

 only in one point ; and therefore (Cor. 

 1st. Def. 7.) they are all parallel to a line 

 in the surface of the cone by the section 

 of which the parabola is produced (viz. 

 to the line V B (fig. 13.) in which the 

 touching plane, parallel to the plane of 

 the parabola, meets the conic surface) : 

 therefore, Prop. 8, 



MDXDN:PEXEQ::BD:BE. 



Cor. 1. The squares of the ordinates 

 drawn to a diameter of a parabola are 

 proportional to the abscissas of the 

 diameter between the ordinates and the 

 vertex. 



For the double ordinates RDG and 

 H E K are parallel to one another : there- 

 fore, by this proposition, 



R D X D G, or R D 1 : H E X E K or 

 HE'::BD:BE. 



Cor. 2. If the square of one ordinate, 

 of the diameter ot a parabola, as K D, be 

 made equal to a rectangle contained by 

 the corresponding abscissa B D and the 

 line P: then, it is manifest, from the 



last corollary, that the square of any 

 other ordinate of the same diameter, as 

 H E, will be equal to a rectangle under 

 the corresponding abscissa B E, and the 

 same line P. 



The line P is called the parameter of 

 the diameter to which the ordinates are 

 drawn. 



Fig. 35. Def. 16. If two right lines, as 

 G C S and F C T, be drawn through the 

 centre of two opposite hyperbolas, so as 

 to be parallel to the two lines in the conic 

 surface, which are the intersections of 

 that surface, and a plane drawn through 

 the vertex of the cone, parallel to the 

 plane of the hyperbolas, (viz. to the lines 

 V m and V n, in fig 14 :) these two lines 

 G S and F T are called the asymptotes of 

 the hyperbolas. 



Cor. 1. Every line drawn through 

 the centre, within the angles of the 

 asymptotes that are turned to the hy- 

 perbolas, is a transverse diameter; and 

 every line drawn through the centre 

 within the adjacent angle is a second dia- 

 meter. 



For the former lines are parallel to 

 lines (such as V O in fig. 14.) drawn with- 

 in the cone in the angle contained by the 

 two lines (mVandnV, fig. 14.) in the 

 conic surface, that are parallel to the 

 asymptotes j and the latter lines are paral. 

 lei to lines (such as R V S, fig 14.) with- 

 out the cone : whence the truth of the 

 corollary is manifest by Cor. 2, Def. 7. and 

 Prdp. 14. 



ruop. xxn. 



The asymptotes do not meet either of 

 the opposite hyperbolas. 



For if an asymptote be supposed to 

 meet one of the hyperbolas, being drawn 

 through the centre, it will likewise meet 

 the other hyperbola (Cor. 12) : and thus 

 a line, drawn parallel to a line contained 

 in the surface of a cone, would meet both 

 the opposite conic surfaces, which is im- 

 possible (Pr. 2.) 



PROP. XXIII. 



Fig. 35 and 36. If a point be assumed 

 without a hyperbola, but within the 

 asymptotes, and a right line be drawn 

 from it to touch or cut the hyperbola, or 

 opposite hyperbolas ; then the square of 

 the tangent, or the rectangle under the 

 segments of the secant, is less than the 

 square of the semi-diameter parallel to 

 the tangent or secant ; but if the point be 

 assumed without both the hyperbola and 

 tUe asymptotes, the square of the tangent, 



