CONIC SECTIONS. 



Prop. XXII. 



143 



If two straight lines be drawn from the foci of an el- 

 lipse perpendicular to a tangent, straight lines drawn 

 from the centre to the points in which they meet the 

 tangent, will each be equal to half the transverse axis. 



Let DPJ be a tangent to the curve at P, and FD, 

 fd perpendiculars to the tangent from the foci ; the 

 straight lines joining the points C, D, and C, d are 

 each equal to AC, half the transverse axis. Join FP, 

 fV, and produce FD,/P till they intersect in E. The 

 triangles FDP, EDP, have the angles at D right angles, 

 and the angles FPD, EPD equal (8.), and the side DP 

 common to both, they are therefore equal, and conse- 

 quently have EDzrDF, and EP=PF, wherefore Ef— 

 FP-f P/=A«. Now, the straight lines FE, F /"being 

 bisected at D and C, the line DC is parallel to Ef 

 (.2 6. E.); and thus the triangles F/E, FCD are simi- 

 lar ; therefore F/:/E or Aa : : FC : CD ; but FC is 

 half of F/; therefore CD is half of Aa. In like manner 

 it may be shewn, that Cd is half of Aa. 



Cor. If the diameter Q7 be drawn parallel to the 

 tangent Do", it will cut off from PF, P/; the segments 

 PG, Pg, each equal to AC, half the' transverse axis. 

 For CdPG, CDPg are parallelograms, therefore PG= 

 dC, and Pg=DC=AC. 



. Prop. XXIII. 

 The rectangle contained by perpendiculars drawn 

 from the foci of an ellipse to a tangent, is equal to the 

 square of half the conjugate axis. 



Let DPd be a tangent, and FD,/V, perpendiculars 

 from the foci, the rectangle contained by FD and fd 

 is equal to the square of CB, half the conjugate axis. 



It is evident from the last proposition, that the points 

 D, d are in the circumference of a circle, whose centre 

 is the centre of the ellipse, and radius CA half the 

 transverse axis ; now FD d being a right angle, if d C 

 be joined, the lines DF, d C, when produced, will meet 

 at H, a point in the circumference (31. 3. E.) ; and 

 since FC=/C, and CH=C d, and the angles FCH,/Ca* 

 are eoual, FH is equal to fd, therefore DF.tf7* = DF FH 

 rrAF.Fa (35. 3. E.) =CB«, (2. Cor. 3.) 



Cor. If PF, Pf be drawn from the point of con- 

 tact to the foci, the square of FD is a fourth propor- 

 tional to / P, FP, and BC J . For the lines /P, FP make 

 equal angles with the tangent (8.), and fd P, FDP are 

 right angles, therefore the triangles fVd, FPD are si- 

 milar, and/P : FP : : {fd : FD : :)fd.FD or CB 1 : FD 1 . 



Prop. XXIV. 



If from C, the centre of an ellipse, a straight line CL 

 be drawn perpendicular to a tangent LD, and from D, 

 the point of contact, a perpendicular be drawn to the 

 tangent, meeting the transverse axis in H, and the con- 

 jugate axis in h, the rectangle contained by CL and 

 DH is equal to the square of CB, the semi-conjugate 

 axis ; and the rectangle contained by CL and D h is 

 equal to the square of CA, the semi-transverse axis. 



Produce the axes to meet the tangent in M and m, 

 and from D draw the semi-ordinates DE, D e, which 

 wiH be perpendicular to the axis. 



The triangles DEH, CL m are evidently equiangular, 

 thereforeDH : DE ::Cm : CL,henceCL.DH=DE.C?w; 

 but DE.C m, or C e.C »ncBC' ; therefore CL.DH=BC\ 

 In the same manner it is shewn, from the triangles Dhe, 

 CLM, thatCL.DA=AC\ 



Cor. 1. If a perpendicular be drawn to a tangent Ellipse. 

 at the point of contact, the segments, intercepted be- "" 

 tween the point of contact and the axes, are to eacli - 

 other reciprocally as the squares of the axes by which 

 they are terminated. 



For AC 1 : BC* : : CL.Dh : : CL.DH : : Dh : DH. 



Cor. 2. If DF be drawn to either focus, and HK 

 be drawn perpendicular to DF, the straight line DK 

 shall be equal to half the parameter of the transverse 

 axis. Draw CG parallel to the tangent at D, meeting 

 DH in N, and DF in G. The triangles GDN, HDK 

 are similar ; therefore GD : DN : : HD : DK, and hence 

 GD.DK=rHD.DN. But GDrzAC, {Cor. 22.) and 

 ND=CL, therefore AC.DK=HD.CL= (by the Prop.) 

 CB* ; wherefore AC : BC : : BC : DK, hence DK is half 

 the parameter of A a, (Def. 15.) 



Prop. XXV. 



Let A a, B b be the transverse aud conjugate axes of p,v $g, 

 an ellipse ; from K, any point in the conjugate axis, let " 

 a straight line KH, which is equal to the sum or differ- 

 ence of the semi-axes CA, CB, be placed so as to meet 

 the transverse axis in H, and in KH, produced beyond 

 H, when KH is the difference of the semi-axis, let HD 

 be taken equal to CB, the point D is in the ellipse. 



Draw DE perpendicular to A a, and through C draw 

 CG parallel to KD, meeting ED in G, then CG=KD 

 =AC by construction; hence G is in the circumference 

 of a circle, of which C is the centre, and CA the radius; 

 and because the triangles CEG, HED are similar 

 GE : DE : : CG : HD : : CA : CB, therefore DE is a se- 

 miordinate, and D a point in the ellipse, (Cor. 5. 16.) 



Scholium. 



The instrument called the trammels, and likewise the 

 elliptical compasses, which workmen use for describing* 

 elliptic curves, are constructed on the property of the 

 curve demonstrated in this proposition. 



Prop. XXVI. Problem. 

 An ellipse being given by position to find its axes. 



Let AB a b be the given ellipse, draw two parallel Fig. 2»j 

 chords H h, K k, and bisect them at L and M ; join 

 LM, and produce it to meet the ellipse in P and p, then 

 Ppis a diameter, (4. Cor. 13.) Bisect Pp in C, the 

 point C is the centre of the ellipse (9.) Take D any 

 point in the ellipse, and on C as a centre, with the dis- 

 tance CD describe a circle. If this circle fall wholly 

 without the curve, then CD must be half the trans- 

 verse axis ; and if it fall wholly within the curve, then 

 CD must be half the conjugate axis ( 1 7. ) If the circle 

 neither fall wholly without the curve nor within it, let 

 the circle meet it again in d; join D d, and bisect^Drf 

 in E, join CE, which produce, to meet the ellipse in A 

 and a, then A a will be one of the axes (5. Cor. 13.), 

 for it is perpendicular to D d (3. 3. E.). which is an or- 

 dinate to A a. The other axes B b will be found, by 

 drawing a straight line through the centre perpendi- 

 cular to A a. 



SECTION III. 



Of the PlYPERBcfcA. 



Definitions. 



1. Let F,/ be two given points, and PQ, pm two F i ff 87 

 Straight lmes between these points, equally distant from ~ 



