CONIC SECTIONS. 



ENXNG = PNXNQ : : TOxOK : LO 



XOI. 

 Therefore, ex xquo, PMxMQ : PNx 



NQ: :LOXOI:LOXOI. 

 Consequently PMxMQ=PNxNQ ;and 

 PM=QN (Lem.2). Therefore theriglit 

 line BC, which bisects DE and FG, will 

 likewise bisect PQ. (Lem.l). 



Def. 9. A right line which bisects two 

 parallel right lines, both terminated by a 

 conic section or opposite sections, is call- 

 ed a diameter of the section, or opposite 

 sections. This definition relates merely 

 to the position of the diameters, and not 

 to their magnitude. 



Def. 10. The centre of an ellipse, or 

 opposite hyperbolas, is a point in wm'ch 

 is bisected every right line drawn 

 through it, and terminated both ways by 

 the ellipse, or opposite hyperbolas. 



Fig. 21, 22. To find the centre of an 

 ellipse, or opposite hyperbolas, giving by 

 position. 



Draw two parallel right lines, as DE 

 and FG, terminated both ways in the 

 ellipse, or one hyperbola, or one of them 

 in one hyperbola, and the other in the 

 opposite hyperbola : draw the right line 

 BC to bisect both the parallels DE and 

 FG : then it is plain that BC will in all ca- 

 ses meet both the opposite hyperbolas ; 

 for it will bisect all the right lines that 

 can be drawn in both, parallel to DE and 

 FG (11.) : let it meet the ellipse and op- 

 posite hyperbolas in B andC, and bisect 

 BC in A, then is A the centre required. 



Let H be a point in the ellipse, or one 

 of the hyperbolas, and draw HLM paral- 

 lel to DE or FG : take AN=AL, and 

 draw PK through N parallel to DE or 

 FG. Then HM and KP terminated by 

 the ellipse, or opposite hyperbolas, are 

 bisected in L and N : and because BLx 

 LC=BNxNC, therefore HLxLM, or 

 HL'=KNxNP, or KN 1 (Pr. 10) : there- 

 fore HL=KN; and it is plain that HA 

 passes through K, and that HK is bisect- 

 cd in the centre A. 



Cor. It follows from this proposition, 

 that a right line drawn through the 

 centre of two opposite hyperbolas from a 

 point H in one of them will meet the 

 other. 



PROP. XIII. 



Fig. 23. An ellipse, or opposite hyper- 

 bolas, have only one centre. 

 If there were two centres of an ellipse, 



then the right line drawn through them, 

 and terminated by the periphery, would 

 be bisected in two different points (12), 

 which is absurd. 



If it be possible, let A and D be both 

 centres of two opposite hyperbolas, and 

 from C, a point in one of the hyperbolas, 

 darw CAB and CDF through A and D 

 to meet the opposite hyperbola : also from 

 B and F draw B D E and F A G to meet 

 the first hyperbola, and join D A, G C, 

 and C E. Because A and D are both cen- 

 tres, therefore BA= AC, and B D=D 

 E, and C E is parallel to D A. In like 

 manner, because F D.=C D, and F A= 

 A G, therefore C G is parallel to D A. 

 Therefore G C and C E, drawn through 

 the same point and parallel to the same 

 line, make only one right line that meets 

 a conic section in three points, which is 

 absurd. 



Cor. All the diameters of an ellipse, 

 or opposite hyperbolas, intersect in the 

 centre, and mutually bisect one another. 



For if not, then there would be more 

 than one centre. 



PROP. XIV. 



Fig. 24, 25, 26. Every right line drawn 

 through the centre of an ellipse is a dia- 

 meter; and every right line drawn 

 through the centre of two opposite hy- 

 perbolas, so as to be terminated by the 

 opposite hyperbolas, or so as to be paral- 

 lel to a right line terminated by one of the 

 hyperbolas, is a diameter. 



When a line drawn through the centre 

 A of two opposite hyperbolas is parallel to 

 H K (fig. 23.), a line terminated in one hy- 

 perbola, draw the diameters H A G, F A K, 

 and join F H and G K ; and when a line 

 drawn through the centre is terminated 

 by an ellipse (fig. 24, 25), or opposite hy- 

 perbolas, draw H. K parallel to it, and 

 make the same construction as before. 

 Because H A = A G, and K = A F 

 (Def. 10.) the two triangles F A H and 

 G A K are equal in all respects, and it is 

 manifest that F H and G K are parallel, 

 and are bisected by the line through the 

 centre parallel to H K : therefore that 

 line is a diameter. (Def. 9.) 



Cor. A right line drawn through the 

 centre of an ellipse, or opposite hyperbo- 

 las, which bisect one right line not pass- 

 ing through the centre, and terminated 

 by the ellipse, or one of the hyperbolas, 

 or both, will bisect all right lines termi- 

 nated in the like manner, and parallel to 

 the former line. 



