CONIC SECTIONS. 



not be both transverse, nor both second 

 diameters. 



PROP. XYIII. 



Fig-. 30 and 31 If a diameter of an 

 ellipse, or of opposite hyperbolas, be pa- 

 rallelto the ordinates of another diameter, 

 these two are conjugate diameters. 



Let the diameter E D be parallel to 

 P Q S, an ordinate of the diameter F H ; 

 draw the diameter P R and join S R cut- 

 ting E D in T. Because P Q = Q S, and 

 P G = G R ; therefore S R is parallel to 

 F H. And because E D is parallel to P 

 Q S and P G = G R ; therefore R T = 

 T S. Therefore R S is an ordinal e of the 

 diameter E D, and it is parallel to F H ; 

 therefore^ D and F H are conjugate di- 

 ameters. Def. 14. 



Cor. If a diameter of an ellipse, as E D, 

 be parallel to F O, a tangent at a vertex 

 of another diameter F H ; then F H is 

 parallel to D I, a tangent at a vertex of 

 ED. 



For a tangent at a vertex of a diameter 

 is parallel to the ordinates of that diame- 

 ter. 



PROP. XIX. 



If a point be assumed without or within 

 an ellipse, and two right lines, parallel to 

 two diameters, be drawn from it to cut 

 or touch the ellipse ; then, as the rec- 

 tangle under the segments of the secant, 

 or the square of the tangent, parallel to 

 one of the diameters, is to the rectangle 

 under the segments of the secant, or the 

 square of the tangent, parallel to the other 

 diameter, so is the square of the first dia- 

 mefertothe square of the second diame- 

 ter. And the same thing is true of two 

 transverse diameters of opposite hyperbo- 

 las, and any two lines, parallel to these, 

 drawn through a point to cut the two 

 curves. 



For diameters of an ellipse, and of op- 

 posite hyperbolas, are secants that inter- 

 sect in the centre : and because they are 

 bisected there, this proposition is mani- 

 fest from Pr. 10. 



Def 15. Fig. 32. Let a point, as O, be 

 assumed in the plane of two opposite hy- 

 perbolas, and let the secant, O H K, be 

 drawn through it parallel to a transverse 

 diameter B A ; and the secants R O S, 

 G O L, &c. parallel to any second diame- 

 ters, M N, P Q, &c. : in these diameters 

 take the segments M N, P Q, &c. all bi- 

 sected in the centre, such that the 

 squares of M N, P Q, &c. may severally 

 be to the square of the transverse diame- 

 tei-s A B, as the rectangles R O x OS, 



G O x O H, &c. contained by the seg- 

 ments of the secants parallel to the se- 

 cond diameters are to K O x OH, the 

 rectangle under the segments of the 

 secant parallel to the transverse diame- 

 ter : then the magnitudes of the second 

 diameters are the segmenls MN, P Q, &c. 



Because the ratios of the rectangles 

 KO x OH, SO X OR, G O x OH, 

 &c. are invariably the same wherever 

 the point O is assumed, (10,) it is plain 

 that the magnitudes of the second dia- 

 meters M N, P Q, &c. are also invaria- 

 bly the same wherever the point O is 

 assumed. 



And because the ratio of the rectan- 

 gles K O X O H to the square o the 

 transverse diameter A 13 is the same as 

 the ratio of the rectangle, contained by 

 the segments of any secant drawn 

 through O, parallel to a transverse dia- 

 meter, to the square of the transverse 

 diameter to which it is parallel, (19,) it is 

 also manifest that the magnitudes of the 

 second diameters are the same, from what- 

 ever transverse diameter they are de- 

 duced. 



Cor. 1. And hence, taking the magni- 

 tudes of the transverse diameters as here 

 defined, Prop. 19, may be enunciated 

 for the hyperbola as generally as it is 

 for the ellipse : that is, the rectangle 

 under the segments of a secant, or the 

 square of a tangent parallel to one dia- 

 meter (whether a transverse or a second 

 diameter) of opposite hyperbolas, is to 

 the rectangle under the segments of a 

 secant, or the square of a tangent, paral- 

 lel to another diameter, as the square of 

 the first diameter is to the square of the 

 second diameter. 



Cor. 2. If two tangents be drawn to an 

 ellipse, or a hyperbola, oropposite hyper- 

 bolas, from the same point, then these 

 tangents are proportional to the diame- 

 ters, or semi-diameters, drawn parallel to 

 the tangents. 



For the squares of the tangents are 

 proportional to the squares of the dia- 

 meters. 



Cor. 3. If a right line be ordinately ap- - 

 plied to a diameter of an ellipse, or to a 

 transverse diameter of a hyperbola ; 

 then as the square of the diameter is 

 to the square of the conjugate diame- 

 ter, so is the rectangle contained by the 

 abscisses of the diameter, between the 

 vertices and ordinate, to the square of the 

 ordinate. 



For the double-ordinate is bisected by 

 the diameter, and it is parallel to the con- 

 jugate diameter. 



