CONIC SECTIONS. 



of the two lines V m or V n will not meet 

 either of the two curves again in another 

 point. (Pr.2.) 



Cor. 2 Every right line drawn in the 

 plane of the hyperbolas, which meets one 

 of the curves, but is not a tangent, nor 

 parallel to V m nor V w, will meet the 

 same, or the opposite hyperbola, again in 

 another point. (Pr. 4.) If it be parallel 

 to VO, a line contained in the angle mV n 

 will meet the opposite hyperbola : but 

 if it be parallel to RVS, without the an- 

 gle m V n, it will meet the same hyper- 

 bola again. 



Def. 8 A right line drawn in the plane 

 of a conic section, so as to meet the curve 

 of the section in one point only, and 

 which, being produced both ways, is con- 

 tained on one and the same side of the 

 section, is called a tangent of the section. 



Cor. 1. A tangent of a conic section is 

 a tangent of the conic surface. For it can 

 meet the conic surface only in the point 

 in which it meets the section. 



Cor. 2. There cannot be more than one 

 tangent of a conic section at the same 

 point of the curve. For if there were two 

 tangents, then two planes drawn through 

 them and the vertex of the cone would 

 meet the conic surface in the same right 

 line without cutting the conic surface, 

 which is absurd. (Cor. 2. Pr. 3.) 



If a point be assumed without or with- 

 in a conic section, and two straight lines 

 be drawn through it to cut the section, or 

 opposite sections, and so as to be parallel 

 to two lines given by position : then the 

 rectangle under the segments of the se- 

 cant, or the square of the tangent, paral- 

 lel to one of the lines giv^n by position, 

 will have to the rectangl ander the seg- 

 ments of the secant, or to the square of 

 the tangent, parallel to the other line 

 given by position, a ratio that is always 

 the same, wherever the point (through 

 which the line is drawn) is assumed with- 

 1 out or within the section. For secants 

 and tangents of a conic section are se- 

 cants and tangents of a conic surface : and 

 thus this proposition is included in Pro- 

 position VII. 



LEMMA 1. 



Fig. 15. If there be any number of 

 right lines, as DE, PQ, and FG, all paral- 

 lel to one another, and all terminating in 

 the same two right lines DF and EG ; 

 then a right line, as BC, which bisects two 

 ef the parallels, will bisect all the rest.- 



VOL. IV. 



Draw DHL and EKR parallel to B6 : 

 because DB = BE and FC CG, there- 

 fore FL = RG. It is plain ''<at FL : PH : : 

 RG : KQ ; and therefore PH = KQ, con- 

 sequently PO = OQ. 



LEMMA II. 



Fig. 16. If a right line AB, or a right 

 line produced, be so divided in C and D, 

 that AC X CB = AD X DB : then AC = 

 BD, and AD = CB. 



Bisect AB in O. Then the difference 

 of AO 2 and AC X CB is equal to CO (5 

 and 6. 2. E) : and the difference of AO a 

 and AD X DB is equal to DO 1 : therefore 

 00=00% whence the lemma is mani- 

 fest. 



PROP. XI. 



Fig. 17, 18, 19, 20. If a right line, as 

 BC, bisect two parallel right lines, DE 

 and FG, terminated both ways by a co- 

 nic section, or opposite sections ; the 

 same right line BC will bisect every other 

 right line, as PQ, terminated by the sec- 

 tion, or opposite sections, and parallel to 

 the two former right lines. 



Join DF and EG : then these lines are 

 either parallel to one another, or, being 

 produced, if necessary, they will meet. 



I. When FE and EG are parallel : (fig. 

 17.) let PQ meet these lines in M and N: 

 then DM x MF : PM X MQ : : EN X NG: 

 PN X NG (Pr. 10) : but it is plain that 

 DM X MF = EN x NG ; therefore PMX 

 MQ = PN X NG. Therefore PM = NQ 

 (Lem. 2.) ; and it is obvious that the right 

 line BC, which bisects DE and FG, like- 

 wise bisects PQ (Lem. 1.) 



II. Let FD and EG meet in a point II : 

 (fig. 18, 19,20.) assume any point, O, iw 

 the plane of the conic section, and 

 through it draw TK, RS, and LI, termi- 

 nated by the conic section, and respec- 

 tively parallel to EG, DF, and DE or FG : 

 let PQ meet DF and EG in M and N. It 

 is manifest that DF and EG are similarly 

 divided in M and N, and also in the point 

 of concourse H. Therefore. 



DM X MF : ENxNG : . FH X HD : GN 



XHE. 



Because TK is parallel to EG, and RS to 

 DF: therefore, 



FHX HD: GHXHE::ROX OS : TO 



XOK. 



Consequently DM X MF : EN X NG : 

 ROxOS: TOxOK. Hence, and by Prop. 

 10, we have the following proportions : 



PM XMQ : DM X MF : : LO X 01 : RO 



xos. 



DM X MF : EN X NG : : RO X OS : T 



XOK. 

 E 



