MATH 
 QA 
 485 
 .G85 
 
 
 
 
CORNELL 
 
 UNIVERSITY 
 
 LIBRARY 
 
 MATHEMATICS 
 
CORNELL UNIVERSITY LIBRARY 
 
 3 1924 068 659 097 
 
B Cornell University 
 M Library 
 
 The original of tliis bool< is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924068659097 
 
THE 
 
 PAEABOLA, ELLIPSE, AND HYPEEBOLA, 
 
 TREATED GEOMETRICALLY 
 
DUBLIN UNIVERSITY PRESS SERIES. 
 
 The Provost and Senior Fellows of Trinity College have undertaken the publication 
 of a Series of "Works, chiefly Educational, to be entitled the Dublin University 
 Press Series. 
 
 The following volumes of the Series are now r6ady, viz. : — 
 
 An Introduction to the Systematic ZoologT- and SEorpholoery of 
 
 Vertebrate Animals. By Alexander Macalister, M.D., Dubl., Professor of 
 Comparative Anatomy and Zoology in the University of Dublin. 
 
 The Correspondence of Cicero : a revised Text, with Notes and Prole- 
 gomena. — Vol. I., The Letters to the end of Cicero's Exile. By Robert Y. 
 Tyrrell, M. A., Fellow of Trinity College, and Professor of Latin in the Uni- 
 versity of Dublin. 
 
 Essays in Political and Moral Philosophy. By T. E. Cliffe Leslie, 
 Hon. LL.D., Dubl., of Lincoln^s Inn, Barrister-at-Law, late Exam-iner in Poli- 
 tical Economy in the University of London^ Professor of Jurisprudence and 
 Political Economy in the Queen's University. 
 
 The following will soon appear : — 
 
 Six Lectures on Physical G-eo^aphy. By the Rev. S. Haughton, 
 M.D., Dubl., D.CL., Oxon., F.R.S., Fellow of Trinity College, and Pro- 
 fessor of Geology in the University of Dublin. 
 
 The Mathematical and other Tracts of the late James M'Cullag-h, 
 F.T.C.B., Professor of Natural Philosophy in the University of Dublin. Now 
 first collected, and edited by Rev. J. H. Jellett, B.D., and Rev. Samuel 
 Haughton, M.D., Fellows of Trinity College^ Dublin. 
 
 An Introduction to Logric. By William Henry Stanley Monck, 
 
 M. A., Prof essor of Moral Philosophy in the University of Dublin. 
 The Veil of Isis, By Thomas E. Webb, LL.D., Q.C, Regiics 
 
 Professor of Ldws in the University of Dublin, 
 
 DUBLIN : HODGES, FOSTER, AND FIGGIS. 
 LONDON ■ LONGMANS, GREEN, AND CO. 
 
DUBLIN UNIVERSITY PRESS SERIES. 
 
 THE 
 
 PARABOLA, ELLIPSE, AND HYPERBOLA, 
 
 TREATED GEOMETRICALLY. 
 
 BY 
 
 ROBERT WILLIAM GRIFFIN, A.M., LLC, 
 
 EX-SCHOLAR, TRINITY COLLEGE, DUBLIN. 
 
 DUBLIN : HODGES, FOSTER, & FIGGIS, GRAFTON-ST. 
 LONDON: LONGMANS, GREEN, & CO., PATERNOSTER-ROW. 
 
 1879. 
 
DUBLIN : 
 PRINTED AT THE UNITERSITY PRESS. 
 
PREFACE. 
 
 Long experience has proved to me that it would be a great 
 practical advantage for the general class of students to 
 acquire a knowledge of even the elementary Geometrical 
 properties of the Parahola, Ellipse, and Hyperhola — a 
 knowledge which may be attained, in a very short time, 
 by any one acquainted with the six Books of Euclid, 
 though many have neither the ability nor the perseverance 
 necessary for the Analytical investigation. 
 
 In the following Treatise I have endeavoured to demon- 
 strate, on strictly geometrical principles, the most useful 
 properties of these curves. In the definition of a tan- 
 gent I have avoided the notion of a limit, adopting, in 
 preference, Euclid's definition; and the demonstrations 
 depending thereon will, I think, be found somewhat new 
 and interesting. 
 
 The similar properties of the three curves will be found 
 to be treated in such a manner as to need scarcely any 
 change of either words or notation. 
 
vi Preface. 
 
 In Chap. III. Proposition XIX. et seq. and Corollaries, 
 I have traced the analogy between the properties of con- 
 jugate diameters of the Hyperbola, and those of the El- 
 lipse, further than has been done in any other Treatise. 
 
 To the Board of Trinity College I owe my grateful 
 thanks, for extending to me that liberal support with 
 which they have ever shown themselves ready to assist 
 the humblest efforts. 
 
 EOBEET WM. GEIFFIN. 
 
 19, Teiniit College, 
 September, 1879. 
 
CONTENTS. 
 
 -♦- 
 
 CHAPTEE I. 
 
 FADE 
 
 Thle Paeabola, 1 
 
 Peoblems on the Pababola 45 
 
 CHAPTEE II. 
 
 The Ellipse, 46 
 
 Peoblems ON THE Ellipse, 110 
 
 CHAPTEE III. 
 The Htpeebola, Ill 
 
 APPENDIX, 178 
 
CHAPTER I. 
 
 The Parabola. 
 
 definitions. 
 
 A Parabola is the curve traced out by a point, which 
 moves in such a way that its distance from a fixed point 
 is always equal to its perpendicular distance from a fixed 
 right line. 
 
 The fixed point is called the Focus, and the fixed right 
 line the Directrix. 
 
 Any right line perpendicular to the Directrix is called 
 a Diameter. 
 
 The right line drawn through the Focus perpendicular 
 to the directrix is called the Axis, and the point at which 
 it meets the curve the Vertex, 
 
 A right line which meets the curve, and, being produced, 
 does not cut it, is called a Tangent. 
 
 A right line drawn through any point on the curve 
 perpendicular to the tangent at that point is called a 
 Normal. 
 
 If a right line be drawn parallel to any tangent, the 
 part intercepted on it, between the curve and a diameter 
 passing through the, point of contact of the tangent, is 
 called an Ordinate to that diameter. 
 
 The part intercepted on any diameter between the ordi- 
 nate and the curve is called the Abscissa. 
 
 B 
 
2 The Parabola. [chap. i. 
 
 The right line joining any two points on the curve is 
 called a Chord. 
 
 The chord drawn through the Focus parallel to the 
 ordinates to any diameter is called the Parameter of that 
 diameter. 
 
 The chord drawn through the focus at right angles to 
 the axis is called the Latus rectum. 
 
 Proposition I. 
 
 The focus and directrix of a parabola being given, to 
 determine any number of points on the curve. 
 
 Pig. 1. 
 
 Let F be the focus, and Oy the directrix. 
 
 Draw FO 1. the directrix ; bisect FO at A. 
 
 Then OF is the axis, and A the vertex of the curve. 
 
 On the directrix take any point p ; join F^ ; draw 
 AH X FO ; through H draw HP J_ Fp, to meet pP 
 drawn J. the directrix; joia Pi^. 
 
 Then a" FHP and jsSP are equal ; (4 I. Euclid.) 
 .■. PF^Pp. Hence P is a point on the curve. 
 
CHAP. I.] The Parabola. 3 
 
 In like manner, by joining the focus with other points 
 on the directrix, toy number of points on the curve may 
 be determined. 
 
 Cor. 1. — If Og be taken = Op, another point Q may be 
 found in a similar manner to P, which will be at the same 
 distance both from the axis and the directrix ; hence the 
 curve is symmetrical with regard to the axis. 
 
 Cor. 2. — Since Pp = Qq; 
 
 .-. FP = FQ, also I OFP = L OFQ ; 
 
 .". right lines drawn from the focus to the curve, making 
 equal angles with the axis, are equal. 
 
 Cor. 3. — The Latus rectum is equal to four times the 
 distance of the focus from the vertex. 
 
 Fig- 2. 
 
 Draw PP' through the focus ± the axis ; and Pp ± 
 the directrix. 
 
 Then PP' = 2PF {Cor. 2.) 
 
 = 2Pp 
 
 = 2F0 = 4FA. 
 b2 
 
4 The Parabola. [cbup. i. 
 
 Cor. 4. — If any parallel qR be drawn to the axis be- 
 tween the vertex A and any point P "on the curve, the 
 segment qQ, intercepted on tms line between the curve 
 and the directrix is less than the segment qR intercepted 
 on it between the line AP and the directrix ; and, there- 
 fore, the curve ia convex towards the directrix. 
 
 Fig. 3. 
 
 Since L FEP is a right /.;.•. L AHF+ L CEP = 90°. 
 Hence a AHF is similar to a CPH ; 
 
 .-. FA: AH = EC: CP ; 
 .-. FAx CP = AHx HC. 
 But FA-AO; .: JFlff = Hp ; .-. AS = HC ', 
 
 .-. FAxCP = AH' = iAC\ 
 Similarly, FAxBQ = \AD' ; 
 
 .-. CP : DQ = AC : AD' 
 
 = CP'.DR'; (Simflar A».) 
 
 .-. CP' :CPxDQ=CP': BR' ; 
 .-. CPxBQ = RB' ; 
 
 .: BQiRB =RB:CP^AB: AC. (SimUar a',) 
 But by hyp., AB<AC; .-. BQ<BR; .-. qQ < qR. 
 
CHAP. I.J 
 
 The Parabola. 
 
 Cor. 5. — Also if a perpendicular QN be drawn to the 
 axis between the vertex A and any point P on the ciirve, 
 the segment QN intercepted on this line between the 
 curve and the axis is greater than the segment SiV^ inter- 
 cepted on it between the line AP and the axis; and, 
 therefore, the curve is concave towards the axis. 
 
 For CP : BQ = AC^ : AD' ; [Cor. 4.) 
 
 AM-.AN^PM': QN'; 
 PM : 8N = PM' : QN' ; (Similar A'.) 
 PM' : PM y SN= PM' : QN'; 
 PMySN= QN'; 
 PM: QN= QN: SN; 
 
 but it was proved above that 
 
 PM' : QN = AM : AN, 
 and by hyp., 
 
 AN< AM; .-. QN<PM, and .-. 8N < QN. 
 
6 The Parabola. [chap. i. 
 
 Proposition II. 
 
 The distance of any point from the focus is greater or 
 less than its distance from the directrix according as the 
 point is on the convex or concave side of the curve. 
 
 Fig- 6. 
 
 First let the point Q he on the convex side ; join QF, 
 and let the perpendicular Qp on the directrix be produced 
 to meet the curve in P ; join PF. 
 
 Then QF+QP>PF; (20 I. Euclid.) 
 
 .-. QF+ QP>Pp; 
 
 QF>Qp. 
 
 Next let Q' be on the concave side ; join QF. 
 
 Then QfF< Q'P + PF; (20 I. Euclid.) 
 
 .-. Q'F< Q'P + Pp; 
 
 .-. Q!F< Q'p. 
 
 Cor. — Conversely a point will be on the concave or 
 convex side of a parabola according as its distance from 
 the focus is less or greater than its perpendicular distance 
 from the directrix. 
 
CHAP. I.J The Parabola. 
 
 Proposition III. 
 
 1°. The line wHch bisects the angle between that drawn 
 from any point on a parabola to the focus, and that drawn 
 perpendiciilar to the directrix, falls wholly without the 
 curve. 
 
 2°. Any other line drawn through the point will cut 
 the curve. 
 
 Fig. 6. 
 
 1°. Let PH bisect the angle between PF and Pp, the 
 perpendicular on the directrix. Take any point Q on PH; 
 join QF, and draw Qq _L the directrix. 
 
 Then the a» QPF and QPp are equal ; (4 I. Euclid.) 
 
 .-. QF= Qp, 
 but Qp >Qq; (19 I. Euclid.) 
 
 .-. QF>Qq; 
 .-. Q is a point without the curve. {Cor. Prop, ii.) 
 
8 The Parabola. [chap. i. 
 
 2°. Let PK be any line other than the bisector of the 
 tFPp. 
 
 Draw PL, making the I KPL = z EPF ; cut off 
 PL = PF. Through L draw QLq ± the directrix ; join 
 QF. 
 
 Then the a » i^PQ and LPQ are equal ; (4 1. Euclid.) 
 
 .-. QL=QF; 
 
 .: QF< Qq; 
 
 .'. the point Q is on the concave side of the curve. 
 
 {Cor. Prop. II.) 
 
 Cor. 1. — Hence the line which bisects the angle be- 
 tween that dravm from any point on the curve to the 
 focus, and that drawn perpendicular to the directrix, is a 
 Tangent to the curve. 
 
 Cor. 2. — The tangent at the vertex is perpendicular to 
 the axis, and .'. ordinates to the axis are also perpendi- 
 cular to it. 
 
 Cor. 3. — The normal to a parabola at any point bisects 
 the external angle between the line drawn from that point 
 to the focus,, and the perpendioidar let fall on the directrix. 
 
CHAP. I.J The Parabola. 9 
 
 Pboposttion IV. 
 
 The locus of the foot of the perpendicular let fall from 
 the foouB on any tangent to a parahola is the tangent at 
 the vertex. 
 
 Fig. 8. 
 
 Let PT be any tangent, and FT the perpendicular let 
 fall from the focuB. Draw Pp _L the directrix ; join pT. 
 
 Then the A'FPT and pPT are equal ; (4 I. Euclid.) 
 
 .-. /. pTP = I FTP ^ 90° ; 
 
 .: Tp and Ti^'are in directum. (14 I. Euclid.) 
 
 Also Tp=TF; .: TAisW Op. (^ VI. EucM). 
 
 Hence TA is a tangent at the vertex. 
 
 {Cor. 2, Prop, iii.) 
 
 Cor. 1. — ^If the vertex of a right z FTP move along a 
 fixed right line, while one leg passes through affixed point 
 F, the other leg will always touch a parabola, of which 
 the fixed point is the focus, and the fixed right line the 
 tangent at the vertex. 
 
 Cor. 2. — If a perpendicular Pp be let fall on the directrix 
 from any point P on a parabola, the line I^ joining the 
 foot of this perpendicular with the focus is JL the tangent 
 at the point P. 
 
10 The Parabola, [chap. i. 
 
 Cor. 3. — By the aid of this Prop, we can draw a pair 
 of tangents to a parabola from any point Q without it. 
 
 Fig. 9. 
 
 On the line joining Q with the focus as diameter 
 describe a circle ; the line joining the given point Q with 
 the points of intersection of this circle with the tangent at 
 the vertex will be the required tangents.. 
 
 Note. — The circle described on QF as diameter will 
 always intersect the tangent at the vertex. 
 
 For, bisect QFinC/ and draw ffN _L AM. 
 
 Then Qq = QM+Mq=QM+AO=QM+ AF= 2aN, 
 
 but QF > Qq (Prop, n.) ; .■.QF>2aNox O'F > ON. 
 
 That is, the radius of the circle is always greater than the 
 perpendicular let fall from its centre on the tangent at the 
 vertex. 
 
CHAP. I.J The Parabola. 11 
 
 Proposition V. 
 
 Any line drawn through the focus is out harmonically 
 by the curve, the focus, and the directrix. 
 
 Fig. 10. 
 
 For PD:DQ = Pp: Qq (Similar a».) 
 
 = PF:FQ. 
 Hence DP, DF, and BQ are in harmonic proportion. 
 
 Cor. 1. Pp : FO : Qq = DP : DF : DQ. 
 
 (Similar a».) 
 
 Hence Pp, FO, and Qq, are in harmonic proportion. 
 • "5"' Bin' ^^^ TT' ^^® ^^ Arith. Progression. 
 
 .-. — - + -TT- = -^^', and .•. constant. 
 Pp Qq FO 
 
 Cor. 2. — Hence, also, =^ + ^ is constant. 
 
12 The Parabola. [chap. i. 
 
 Proposition VI. 
 
 The locus of the intersection of tangents to a parabola 
 which cut at right angles is the directrix. 
 
 Fig. 11. 
 
 Let QP, QP, he tangents which cut at right angles. 
 From the focus draw the perpendiculars FT, FT\ on 
 the tangents. 
 
 Then T and T' are points on the "tangents at the vertex. 
 
 (Prop. IV.) 
 
 .-. TT' is ± the axis. [Cor. 2, Prop, in.) 
 
 Also since FTQT' is a rectangle, its diagonal QF is 
 bisected at B. 
 
 Draw QO 11 TT'. 
 
 Then FA^AO; (2 VI. BucUd.) 
 
 and QO being || TT', is .-. ± the axis. 
 Hence .•. QO is the directrix. 
 
CHAP. 1.] The Parabola. 13 
 
 Proposition VII. 
 
 If PN be a normal to a parabola at any point P, and 
 PM the ordinate to the axis, the subnormal MN is of a 
 constant length and equal to half the latus rectum. 
 
 Fig. 12. 
 
 Draw Pp ± the directrix. 
 
 Then iiJj will be ± the tangent at P ; and .'. || PIf. 
 
 {Cor. 2, Prop, iv.) 
 
 Hence A 1^)0= A NPM ; (26 I. Euclid.) 
 
 .-. MN =F0 = 2FA = I Latus rectum. 
 
 {Cor. 3, Prop, i.) 
 
 Def. — The portion of the axis intercepted between the 
 normal at any point on the curve, and the ordinate to 
 the axis drawn through the same point, is called the Sub- 
 normal. 
 
 The portion of the axis intercepted between the tangent 
 at any point on the curve, and the ordinate to the axis 
 drawn through the same point, is called the Subtangent. 
 
14 
 
 The Parabola. 
 
 [chap. I. 
 
 Peoposition VIII. 
 
 If a tangent at any point P on a parabola meet the 
 axis produced in T, and the ordinate PM be drawn, then 
 the subtangent MT will be bisected at the vertex. 
 
 Fig. 13. 
 
 Draw Pp _L the directrix ; join FP. 
 
 Then z FPT = z TPp = z PTF; 
 
 .•.FT=FP^Pp = MO; 
 but AF=AO; 
 
 .•.AT=AM. 
 Cor. 1.— If PN be a normal, then FT = FN. 
 For FT = MO = MF+ FO = MF+ MN = FN. 
 Cor. 2. L PFN = 2 z PTF. 
 
 (Prop. VII.) 
 
CHAP. I.J The Parabola. 15 
 
 Pkoposition IX. 
 
 The rectangle under the Latus rectum and the abscissa 
 is equal to the square of the ordinate. 
 
 1 - Fig. t4. 
 
 Draw PT the tangent and PN the normal at the point P. 
 
 Then PM'' = TM x MN ; (8 VI. EucHd.) 
 
 but TM=2AM; and MN = 2 AF; 
 
 (Props, vn. and viii.) 
 
 .-. PJf' = 2AM - 2AF= iAF x AM. 
 
 Cor. 1. — The squares of the ordinates to the axis are 
 proportional to the abscissas. 
 
 Cor. 2. — The square of the tangent = 4:FP x AM. 
 
 For TP" = TNxTM (8 VI. EucM.) 
 
 = 2TFx2AM 
 
 = 4i?!P X AM. 
 Cor. 3. — The square of the normal = 4:AF x FN. 
 For PN^ =TNx NM= 2FN x 2AF= iAF x FN 
 
16 The Parabola. [chap. i. 
 
 Proposition X. 
 
 The locus of the intersection of any tangent to a para- 
 bola with the line drawn through the foQus perpendicular 
 to the radius vector drawn from the focus to the point of 
 contact of the tangent is the directrix. 
 
 I'ig. 15. 
 
 Let PT he the tangent and FT the ± the radius vector 
 FP. 
 
 Draw TO ± the axis, and Pp ± TO produced. 
 
 Then ATFP = ATjaP ; (26 I. Euclid.) 
 
 .-. Pp = PF; 
 
 hence OT is the directrix. 
 
 Cor. 1. — Conversely, if from any point on the directrix 
 a tangent be drawn to a parabola, the line joining that 
 point to the focus is perpendicular to the radius vector 
 drawn from the focus to the point of contact of the 
 taipgent. 
 
 Cor. 2. — Tangents at the extremities of a focal chord 
 intersect at right angles on the directrix. 
 
 For / FTP = /.pTPaniz. FTQ = iqTQ; 
 
 ,: QTP = i[LFTp + /.FTq) 
 
 = a right /. 
 
CHAP I.J The Parabola. 17 
 
 Proposition XI. 
 
 If any chord PP' of a paratola out the directrix in D, 
 then Fl> is the external bisector of the angle PFP'. 
 
 Fig. 16 
 
 Draw Pp, P'p' ± the directrix. 
 
 Then P'P : PF = Fp' : Pp 
 
 = P'D:DP; (Similar A'.) 
 
 .-. FB is the bisector of the l PFQ. 
 
 (3 VI. Euclid.) 
 
 Cor. 1. — Hence, being given the focus i?" and two points 
 P, P', on a parabola, we can find the directrix and axis. 
 
 For, draw FB bisecting the Z between FP and FP" 
 produced ; the point B where this line meets PP' produced 
 is a point on the directrix. The tangent drawn from B to 
 the circle described with the centre P and the radius PF, 
 will be the directrix, and the perpendicidar let fall on it 
 from the focus, the axis. 
 
 If a tangent to a curve be defined as the line Joining two 
 indefinitely near points on the curve, it will foUow imme-. 
 diately from this Prop, that the right line drawn from the 
 focuB to the point of intersection of any tangent with the 
 
18 The Parabola. [chap. i. 
 
 directrix is perpendicular to the radius vector drawn from 
 the focus to the point of contact of the tangent ; also, that 
 the tangent makes equal angles with the lines drawn from 
 the point of contact to the focus and the perpendicular to 
 the directrix. 
 
 For when P is indefinitely near to P', PD becomes a 
 tangent at the point P, and the z. PFP' is indefinitely 
 smaU, and .-. PFQ = 180°; but L PFD = \ L PFQ, 
 .: L PFD = 90°. 
 
 Also, APZZ) = ^.PpD ; 
 
 .-. L DPF= L DPp. 
 
 Pkoposition XII. 
 
 If two fixed points PP' on a parabola be joined with a 
 third variable point on the parabola, the segment pp' 
 intercepted on the directrix by the produced chords sub- 
 tends a constant angle at the focus. 
 
 ■Fig. 17. 
 
 Since (Prop, xi.) Fp is the bisector of the z between 
 FO and FP produced ; 
 
 .-. LOFp + \ L OFP = 90°; 
 
CHAP. I.] The Parabola. 19 
 
 also, since (Prop, xi.) Fp' is the bisector of L between 
 FO and FF produced ; 
 
 .-. I OF^' -^^L OFF' = 90°; 
 
 .. lOIi) + i OFF = L OFp' + i / OFF ; 
 
 .-. z OFp - I OFp =\L OFF ~\L OFF ; 
 
 .-. LpFp\= \ L PFF and .-. constant. 
 
 Cor. — Hence the anharmonic ratio of the pencil formed 
 by joining four fixed points on a parabola to any fifth 
 variable point is constant. 
 
 For . PFF'F" = 0.pp'p"p"' = F.pp'p"p"'. 
 
 Proposition XIII. 
 
 The line joining the focus to the point of intersection of 
 any two tangents to a parabola bisects the angle between 
 the radii vectores drawn from the focus to the points of 
 contact of the tangents. 
 
 fig. 18. 
 
 Let OF, QF be tangents. Join PF, FF, QF. 
 Draw Fp', Pp ± the directrix. Join Qp, Qp. 
 
 c2 
 
20 The Parabola. [chap. i. 
 
 Then A» QPF and Qp'F are equal ; (4 I. EucM.) 
 .-. QF^ Q/, also t QFP = / Qp'F. 
 
 Again, A» QFP and QpP are equal; (4 I. Euclid.) 
 .-. QF= Qp, and l QFP = l QdP. 
 
 Hence, Qp = Qp', .■.lQpp'=l Qp'p, .•.LQpP=LQpF; 
 
 .■.lQFP = L QFP'. 
 
 Cor, 1. — ^If Qq be drawn || the axis and produced it will 
 bisect the chord PP'. 
 
 For A» Qpq and Qp'q are equal. (26 I. Euclid.) 
 
 ■'■ pq = qp' \ but Pp, Qq, and P'p' are parallel; 
 
 .-. Pilf=i[fP'." 
 
 Cor. 2. — Since QF= Qp = Qp'; /. the circle described 
 win centre Q, and radius QPwiU^ass through^ and jo'. 
 
 Cor. 3. — By the aid of Cor. 2 we can draw a pair of 
 tangents to a parabola from a point Q without it. 
 
 For, with centre Q, and radius QF, describe a circle 
 from the points where this circle cuts the directrix ; draw 
 perpendiculars meeting the curve in P and P' ; the lines 
 joining Q with P and P' will be the required tangents. 
 
 Cor. 4.— A'P'Si? and QPP are similar, and QPis a 
 mean proportional between FP and FP', 
 
 For I FQF = z FQp', L FQP = / PQp, 
 and ^pQq = Lp'Qq; 
 
 .-. L FQF + PQq = 180° ; 
 but LpPQ or L FPQ +'z PQq = 180° ; (29 I. Euclid.) 
 
 .-. L FQF ^L FPQ, 
 also L QFP = L QFF. 
 
 Hence a P'QPis similar to a QPF; 
 
 .-. FF:FQ::FQ:FP. 
 
CHAP I.] The Parabola. 21 
 
 Pkoposition XIV. 
 
 The chord of contact of any two tangents to a parabola 
 is an ordinate to the diameter passing through their point 
 of intersection. 
 
 Fig. 19. 
 
 Let QP, QP', be two tangents, QM a parallel to the 
 axis passing through their intersection, meeting the curve 
 in V. At V draw a tangent meeting QP in B, and QP' 
 in P^ ; join VP, VP" ; through B and Ji' draw parallels 
 to the axis. 
 
 Then VP will be bisected in 8, and VP' in 8'. 
 
 {Cor. 1, Prop, xiii.) 
 
 Hence QP will be bisected at B, and QP' at B^ ; 
 
 (2 VI. Euclid.) 
 
 .-. BI^ is II PF. (2 VI. EucUd.) 
 
 Hence PP' is an ordinate to diameter QM. (Def.) 
 
 Cor. QV=VM. 
 
22 The Parabola. [chap. i. 
 
 Proposition XV. 
 
 The angle at the focus, subtended by the segment in- 
 tercepted on a variable tangent to a parabola by two fixed 
 tangents, is equal to half the angle which the chord of 
 contact of the fixed tangents subtend at the focus, and 
 therefore constant. 
 
 Fig. 20. 
 
 Let QP, QP', be the fixed tangents, RS the variable ; 
 join F with F, S, T, R, and P. 
 
 Th&nl 8FT = ^ I TFF ; &ho iTFR^^/L TFP; 
 
 (Prop, xm.) 
 
 .-. I RFS =^lPFF, which is constant. 
 
 Cor. — The anhaxmonic ratio of the four points in 
 which a variable tangent to a parabola is cut by four fixed 
 tangents is constant. 
 
 For the segments intercepted on the variable tangents 
 subtend constant angles at the focus. 
 
 Proposition XVI. 
 
 The circle circumscribirig the triangle formed by any 
 three tangents to a parabola passes through the focus. 
 
CHAP. I.J The Parabola. ^3 
 
 For L PFM = 2 / PBF, {Cor. 2, Prop, viii.) 
 
 and L FFM= 2 z P'DF; {Cor. 2, Prop, viii.) 
 
 .• lFFM-lPFM=2iPDF-2lPBF, 
 or L FFP = 2l BQD ; 
 
 .-. lSFR =lBQD. (Prop. XV.) 
 
 HenoQ a circle will pass through SQRF. 
 
 Pkopositiox XVII. 
 
 If a tangent at any point P on a parabola meet the axis 
 in T and P, be joined with the vertex, the area of the 
 triangle PAT thus formed wiU be equal to the area of the 
 triangle PAp, formed by drawing Pp parallel to the axis, 
 to meet the tangent at the vertex. 
 
 Fig. 21. 
 
 Draw the ordinate PM. 
 
 Then, since TA = AM; (Prop, viii.) 
 
 .-. A TAP = A MAP = A PAp. 
 
 (38 I. EucM.) 
 
 Cor. A TMP = parallelogram PMAp. 
 
24 The Parabola. [chap. i. 
 
 Proposition XVIII. 
 
 If QG be an ordinate drawn to any diameter Pp, from 
 any point Q on a parabola cutting the axis in L, and QS 
 be an ordinate drawn to the axis from the same point, AR 
 a tangent at the vertex, then the triangle QLS = parallel- 
 ogram ITS AM. 
 
 Fig. 22. 
 
 Draw PT the tangent at P. 
 Then a' QLS and PTM are similar ; 
 
 .-. A QLS : A PTM= QS' : PM' (19 VI. Euclid.) 
 = AS : AM {Cor. 1, Prop, ix.) 
 = D ASKS : □ AMPS; 
 but A PTM== □ AMPS; (Cor. Prop, xvn.) 
 
 .-. A QLS = a J/Sf^S^. 
 Cor. 1. — In like manner it may be proved that 
 
 A Q'S'L = □ AS'K'E. 
 Cor. 2. — The chord QQ'is bisected by the diameter P??. 
 
GHAP. I.J The Parabola. 25 
 
 Since a QLS = □ KSAE; .-. a QGK= trap. lLii<? ; 
 
 also A Q'Z-8' = □ K'&AH ; . • . a <^GK' = trap. HAL G ; 
 
 .". A QGK^ A Q'GK' ; they are also similar; 
 
 .-. qg = q;g. 
 
 Cor. 3. A QG'Z'= a ©iTP. 
 
 For A rPif = □ AMPH; {Cor. Prop, xvii.) 
 
 .-. □ TPGL = trap. SALG = a QGZ". (Con 2.) 
 
 Cor. 4. — Chords drawn parallel to the tangent at any 
 point will be bisected by the diameter passing through that 
 point. 
 
 This follows immediately from Cor. 2. 
 
 Proposition XIX. 
 
 If a line be drawn parallel to the chord of contact of two 
 tangents, the parts intercepted on it between the curve and 
 the tangents are equal. 
 
 Fig. 23. 
 
 Draw the diameter through Q. 
 
 Then PP' is an ordinate to QM; (Prop, xiv.) 
 
26 The Parabola. [chap. i. 
 
 ,'. 88', which is || PP', is also an ordinate ; 
 
 . . PP and 88' are bisected by QMN; 
 
 {Cor. 4, Prop, xvm.) 
 .-. NT=NT'. 
 
 Hence T8 = T'S'. 
 
 Proposition XX. 
 
 The squares of the ordinates to any diameter are pro- 
 portional to the abscissas. 
 
 Fig. 24. 
 
 Let T8 be the axis, PN any diameter, QG, RH ordi- 
 nates II the tangent at P. 
 
 Then a QGK = a PGLT, 
 
 {Cor. 3, Prop, xviii.) 
 
 and A BSJSr = □ PECT; 
 
 .-. A QGK: A iiiffiV^ D PGLT: a P^CT. 
 
 Hence QG*' : EH' = PG : PH. 
 
 (19 & 1 VI. Euclid.) 
 
CHAP. I.J 
 
 The Parabola. 
 
 27 
 
 Proposition XXI. 
 
 The parameter of any diameter is four times the line 
 joining thfe focus with the point where that diameter 
 meets the curve. 
 
 Fig. 26. 
 
 Let QH be the focal ordinate to the diameter PH. 
 Draw AG \\ QS. 
 
 Then QH' : AG' = PH : PG = TF: TA; 
 
 = 4:TF^ TFATFxTA; 
 
 but AG' = PT' = iTF X TA ; 
 
 {Cor. 2, Prop, ix.) 
 .-. QW = 4TFx TF; 
 
 .-. QH = 2TF 
 
 = 2FP. (Prop, viii.) 
 
 Hence, parameter = 4:FP {Cor. 4, Prop, xviii.) 
 
 Co^._The abscissa intercepted on any diameter by a 
 double focal ordinate = ;^ of the parameter of that dia- 
 meter. 
 
 For PE =TF=FP = \ parameter. 
 
28 
 
 The Parabola 
 
 [chap. I. 
 
 Proposition XXII. 
 
 The square of the ordinate to any diameter is equal to 
 the rectangle under the parameter of that diameter and 
 the abscissa. 
 
 Fig. 26t 
 
 Let R8 be any ordinate to the diameter PS, and QH 
 the focal ordinate. 
 
 Then 
 
 R8'': QH' = PS:PH; 
 
 (Prop. XX.) 
 
 but 
 
 QH=^FP 
 
 (Prop. XXI.) 
 
 
 '2FT 
 
 (Prop, vili.) 
 
 
 = 2PH; 
 
 
 
 .-. RS' : 4PH' = P8 : PS 
 
 
 
 = 4:PJB:xPS: 
 
 iPS X PH; 
 
 
 .-. RS' = iPH X PS 
 
 
 
 = 4FT X PS 
 
 
 
 = iFP X PS 
 
 
 
 = Parameter x 
 
 Abscissa. 
 
 (Prop. XXI.) 
 
CHAP. I.J The Parabola. 29 
 
 Pkoposition XXIII. 
 
 If from any point on a parabola a perpendicular be let 
 fall on any diameter, the square of this perpendicular will 
 be equal to the rectangle under the Latus rectum and the 
 
 Fig. 27. 
 
 Draw AC the tangent at the vertex ; join CF. 
 
 Then a' QPO and CAT are similar ; 
 
 .-. QE^ : QG' - CA^ : Cr ; 
 
 .-. QH'' : 4FP X PG = FAx AT: FT x TA 
 
 (Props. XXII. and iv.) 
 
 = FA: FT or FP 
 
 = 4FA X PG : iFP X PG ; 
 
 -•. QH' = 4FA X PG = Latus rectum x Abscissa. 
 
 UoTE. The parameter of any diameter is proportional to the square of 
 
 the cosecant of lie angle which its ordinates make with the axis, 
 
 QG* iFP X FG iFF 
 oosec" L QGS = -^,= ^pj,^ pq = 4^! 
 
 .-. Latus rectum x oosec' L QOff= 4^2* = parameter of diameter FH. 
 
30 The Parabola. [chap. i. 
 
 Proposition XXIV. 
 
 If a tangent be drawn at any point P on a parabola, and 
 also an ordinate PJf from same point to any diameter ^M, 
 the portion intercepted on that diameter between the tan- 
 gent and the ordinated will be bisected by the curve. 
 
 Fig. 28. 
 
 Let QQ' be the chord || the tangent at P. 
 Draw QH J. the diameter passing through P, 
 and P8 JL Q'ilf produced. 
 
 Then GQ = GQ' ; {Cor. 4, Prop, xviii.) 
 
 .-. QH=Q'H' = PS; 
 but QH' = Latus rectum x PG, (Prop, xxiii.) 
 
 and PS" = Latus rectum x Q'M; 
 
 .-. QM=PG^ TQ'. 
 
 Cor. — The tangents at the extremities of a double ordi- 
 nate intersect on the diameter. 
 
CHAP. I.J 
 
 The Parabola. 
 
 31 
 
 Proposition XXV. 
 
 If any diameter intersect two parallel chords, the rect- 
 angles under the segments of these chords are proportional 
 to the segments of the diameter intercepted between the 
 chords and the curve. 
 
 Fig. 29. Fig. 80. 
 
 Draw the diameter 8M, bisecting the parallel chords 
 PF, QQf. 
 
 Then PM^ = 4SFx 8M, (Prop, xxii.) 
 
 and VD' = 48F x SB ; 
 
 .-. PIS/P - W = ^8F X DM, 
 
 or PBxBF=4:8Fx VB. 
 
 Similarly QB x FQ' = A8F - VF ; 
 
 .: PB X BF : QF X FQ: = VB : VF. 
 
 Cor. — ^If a chord be drawn through any point, the rect- 
 angle under the segments is equal to the rectangle under 
 the parameter of the diameter bisecting the chord, and the 
 part intercepted on a diameter passing through the point 
 between the curre and the point. 
 
32 The Parabola.. [chap. 
 
 Proposition XXVI. 
 
 If two chords of a parabola, intersect, the rectangles 
 under the segments are proportional to the parameters of 
 the diameters bisecting the chords. 
 
 Fig. 31. Fig. 32. 
 
 Let CM, DN, be the diameters bisecting the chords 
 QQ', PP'. Draw the diameter B V through the inter- 
 section of the chords. 
 
 Then PB x BP = AFJD x FB, {Cor., Prop, xxv.) 
 
 and QB x BCt = 4FC x VB ; {Cor., Prop, xxv.) 
 
 .-. PB X BP' :QBx BQ : : 'iFD : 4FC. 
 
 Cor. 1. — The rectangles under the segments of any two 
 intersecting chords are proportional to the lengths of the 
 parallel focal chords. 
 
 Cor. 2. — If two intersecting chords be parallel to two 
 others, the rectangles under the segments of the one pair 
 are proportional to the rectangles under the segments of 
 the other pair. 
 
CHAP. I.J Tlie Parabola. 33 
 
 Co)'. 3. — If the points P and P' coincide, the line BP 
 becomes a tangent. Hence, if from any point two tan- 
 gents be drawn to a parabola, and from any other point 
 two parallel chords, the rectangles under the segments of 
 the chords are proportional to the squares of the tangents. 
 
 Proposition XXVII. 
 
 If TP and TQ be two tangents to a parabola, OC aline 
 drawn parallel to either, cutting the other in 0, the curve 
 in A and C, and the chord of contact of the tangents in B, 
 then OA, OB, and OC are in geometric proportion. 
 
 Fig. 33. 
 
 For OAx OC: OQ' = TP' : TQ' 
 
 = OB' : OQ' ; 
 
 (Cor. 3, Prop, xxvi.) 
 
 .-. OAx 0C== OB'; 
 .: OA:OB=OB: OC. 
 
 D 
 
34 The Parabola. [chap. i. 
 
 Proposition XXYIII. 
 
 If a circle intersect a parabola in four points, the common 
 chords will be equally inclined to the axis. 
 
 Fig. 34. 
 
 Let PF, QQ be the common chords intersecting in 0. 
 
 Then OP x OP' : OQ x OQ = the ratio of the parallel 
 focal chords, {Cor. 1, Prop, xxvi.) 
 
 but OPx OF=OQxOQ;; 
 
 (35 III. EucHd.) 
 
 .•. the focal chords drawn parallel to Q(jf and PP are 
 equal. 
 
 Hence the focal chords, and .'. the chords QQ' and PP' 
 are equally inclined to the axis. {Cor. 2, Prop, i.) 
 
 Cor. I. — In like manner it can be shown that PQ and 
 P'Q, also PQ' and P'Q are equally inclined to the axis. 
 
CHAP. I.J 
 
 The Parabola. 
 
 35 
 
 Proposition XXIX. 
 
 Any line OB drawn through the intersection of two 
 tangents to a parahpla is out harmonically by the curve, 
 the point, and the chord of contact of the tangents. 
 
 Fig. 35. 
 
 Through £ and 5' draw TS and T'& \\ PQ, the chord 
 of contact of the tangents. 
 
 Then BO : OB' = BT : FT' = B8 : B'S' ; 
 
 (Similar a'.) 
 .-. BO' : OB" = BTxB8: BIT : S&, 
 
 = BTxTA: FT' : T'A', 
 
 (Prop. XIX.). 
 = TP" : PT'\ 
 
 = Ba' : O'F" ; 
 
 .-. BO : OF =Ba : OF. 
 
 d2 
 
36 The Parabola. [chap. i. 
 
 Cor. — If through any point without a parabola a line 
 be drawn cutting the curve in B and ff ; and Off be taken 
 equal to the harmonic mean between OB and OB, the 
 locus of C is a straight Une, namely, the chord of contact 
 of the tangents drawn from 0. 
 
 Proposition XXX. 
 
 If PM, QN be any diameters, BE a Hne drawn parallel 
 to the ordinates to either, meeting the curve in D and QP 
 in C, then BC, BD, and BE are in geometric proportion. 
 
 Pig. 36. 
 
 Draw the ordinate QG. 
 
 Then BE^ : BIP = GQ' : BB% 
 
 = PG : PB, (Prop, xx.) 
 
 = PQ : PC, (2 VI. Euclid.) 
 
 = BE : BC, 
 
 = BE' : BE X BC ; 
 .-. BD' = BEx BC. 
 
CHAP. I.J The Parabola. 37 
 
 Proposition XXXI. 
 
 If through any point P on a parabola lines POO, QPQ, 
 be drawn parallel to any two adjacent sides DA, DC, of an 
 inscribed quadrilateral, meeting opposite sides in 0, €f, and 
 Q, Q', then PO x PC : PQ x P(^ in a constant ratio. 
 
 Fig. 37. 
 
 Through 5 and (7 draw 5^ and CG\\\joAD; \ova.AG, 
 and produce it to meet EB produced in H. 
 OL : BH ^ LA : AH 
 
 = OD : DK. (Parallel lines.) 
 
 Alternation OL : OD = BM : DK 
 
 Also SC or Pff: SQ = BK: KC. 
 
 .-. OL X Pff: aDx 8Q = BRx BK 
 
 .-. OLxPa-.PQ X SQ = EKxBK 
 
 = avxap 
 
 = PL X ffP 
 
 (Similar A".) 
 DKxKC; 
 DK X KC* 
 aCx OD 
 
 PS X pq ;* 
 
 * The diameter which bisects the parallel chords OCsoA AD will bisect 
 the chords 5^ and PV (Cor. 4, Prop, xviii.), also the parallel lines HK and 
 ZO: Hence £H = SK, and (rV^JPZ. 
 
38 The Parabola. [chap. i. 
 
 .-. OLxPa + PLx ffP: PQ' ^ SQ + PSx PQ' 
 
 = EKxBK:BKxKC; 
 .-. POxPa .PQx PQ' = EKxBK:DKx EC. 
 
 Now it is evident that the points A, B, C, B being fixed, 
 E is also fixed, and .•. EKx BK: BK x KG in a constant 
 ratio, and .•. PO x PC/ -. PQ x PQ' in a constant ratio. 
 
 Proposition XXXII. 
 
 If from any point on a parabola lines PR, PK, PS, 
 PS' be drawn to the sides of an inscribed quadrilateral, 
 making with them any constant angles, then the rect- 
 angles under the lines drawn to the opposite sides will be 
 in a constant ratio. 
 
 Fig. 38. 
 
 Take any other point p on the curve. 
 
CHAP. I.J The Parabola. 39 
 
 Through the points P, ^, draw QPQ' a,ndi pqq' \\ DC and 
 POa, <ypo' II AD, also pr, pr\ ps, ps', \\ PR, PB!, PS, 
 PS', respectively. 
 
 Then PR-.pr^PQ: pq, (Similar a'). 
 
 and PB;:pr' = P^:pq'; 
 
 .-. PMx PR : pr x pr' = PQ x PQ : pq x pq'. 
 
 Similarly it may be proved that 
 
 PS X PS' : ps X ps' = POx pa : po x po' ; 
 
 but PQ X PQ' -.POxPa =pqxpq' :po x po' ; 
 
 .: PR X PK : PS x PS' = pr x pr' :ps xps'. 
 
 Cor. 1. — The rectangle under the perpendiculars let fall 
 from any point of a parabola, on two opposite sides of an 
 inscribed quadrilateral, is a constant ratio to the rectangle 
 under the perpendiculars let fall on the other two sides. 
 
 Cor. 2. — If the points A and P coincide, also the points 
 C and P, then the sides A£ and CP become tangents, and 
 the sides PC and AD coincide and become the chord of 
 contact. Then the rectangle under the perpendiculars let 
 fall from any point on a parabola on two fixed tangents 
 is in a constant ratio to the square of the perpendicular 
 let fall on their chord of contact. 
 
 Def. — Two curves are said to be similar and 
 situated, when a radius vector drawn from a fixed point in 
 any direction to the first curve bears a constant ratio to the 
 radius vector drawn from a fixed point in a parallel direc- 
 tion to the second curve. 
 
 Def. — Two curves are similar, when a radius vector 
 drawn from a fixed point to the first curve bears a 
 constant ratio to the radius vector drawn from a fixed 
 point to the second curve, in a direction inclined at a con- 
 stant angle to the former. 
 
 The two fixed points are called centres of similariti/. 
 
40 The Parabola. [chap. i. 
 
 Pkoposition XXXIII. 
 A.11 parabolas are similar figures. 
 
 Fig. 39. 
 
 Let P,f, be the foci, A, a, the vertices. Draw any two 
 lines FP,fp, making equal L' with the axes. Draw tan- 
 gents to the curves at P and p ; let fall perpendiculars 
 FT, ft, on the tangents from the foci; join TA, ta. 
 
 Then TA, ta, will be tangents at the vertices. 
 
 (Prop. IV.) 
 
 Hence also z PFT = ^ /. PFA, am^ipft = ^ Lpfa ; 
 
 (Prop. XIII.) 
 .-. lPFT= Lpft. 
 
 Hence A PFT is similar to pft. 
 
 Also A TFA is similar to Atfa; 
 
 .-. PF: FT = pf:ft, and TF : FA = tf:fa; 
 
 .: PF:FA::pf:fa. 
 Alt. PF:pf::FA:fa; 
 
 .'. The curves are similar. 
 
CHAP. 1. 1 The Parabola. 41 
 
 Proposition XXXIV. 
 
 The area included between any ordinate PM, the ab- 
 scissa AM, and the curve, is two-thirds of the parallelogram 
 which has the ordinate and abscissa for its adjacent sides. 
 
 Fig. 40. 
 
 Draw tangents at A and P; join AP 
 
 Since TA = AM, and .-. TH = HP, (Prop, xxiv.) 
 
 .-. A TAH= i A TAP = I A APM. 
 
 Similarly, if through Hvfe draw a diameter, and at the 
 point Q where it meets the curve draw a tangent, we can 
 prove that A HQJi = i a QGP, and A HQh' = i A QGA. 
 Continuing in this way to form new A' by drawing 
 diameters through h and h', we can prove that the areas 
 of the exterior A' formed by the tangents are the halves 
 of the areas of the interior A', formed by joining the 
 points of contact with the extremities of the chords. The 
 same will hold good though the number of A' be increased 
 indefinitely ; 
 
42 The Parabola. [chap. i. 
 
 .•. the sum of all the exterior A' = half the sum of the 
 interior A*. 
 
 Hence the area included between the curve and the lines 
 AM and AP = f area of A TPM = f parallelogram 
 AMPB. 
 
 Otherwise thus — 
 
 Fig. 41. 
 
 Draw the adjacent ordinate QN; join PQ and produce 
 it; draw the tangent at A; draw TC\\ AB, and QD 
 II TM. 
 
 Then, since PJf is || AB, the en QM= QC. 
 
 Now when Q is indefinitely near P, PT becomes the 
 tangent at P, and (Prop, xiv., Cor.) AM = AT. 
 
 .-. CD BQ= aBD; 
 
 .: a QM=2 a BQ. 
 
 If the parabolic arc AP be divided into an indefinite 
 number of small parts, it can be similarly shown that each 
 internal parallelogram is double the external, and hence 
 the total internal area APM is double of the external area 
 APB, and .-. =4 of a AMPB. 
 
CHAP. I.J The Farahola. 43 
 
 Def. — If a right-angled triangle be made to revolve 
 round one of the sides containing the right angle, the sur- 
 face generated by the hypothenuse is called a right cone, 
 and the hypothenuse the generating line. 
 
 Proposition XXXV. 
 
 The section of a right cone by a plane parallel to the 
 generating line is a parabola. 
 
 Fig. 42. 
 
 Let the plane -B F.S, drawn through the axis of the cone, 
 perpendicular to the plane of the section, coincide with the 
 plane of the paper, then both the section PAP and the 
 base jBP^will be perpendicular to the plane of the paper ; 
 therefore the line MP in which the section cuts the base is 
 perpendicular to the plane of the paper, and, therefore, 
 perpendicular to BE, the diameter of the base. 
 
 Hence BM x ME = MP\ (35 III. EucHd.) 
 
44 The Parabola. [chap. i. 
 
 If now any other plaiie bpe be drawn parallel to the 
 base, meeting the section in pm, it can similarly be shown 
 that mp is perpendicular to he, and .•,hm x me = mp'; 
 
 .-. MI" : mp^ = BM x ME : bm x me 
 
 = ME : me 
 
 = MA : mA. (Similar A'.) 
 
 Hence the section P'AP is a parabola. 
 
 Cor. 1. — The latus rectum is a third proportional to 
 YD and DA. 
 
 For VD:DA= AM : ME 
 
 = AM X MB : ME x MB 
 = AM X MB : MP' 
 
 = AM X DA : Latus rectum x AM ; 
 
 (Prop. IX.) 
 .•.' VD : DA ^ DA : Latus rectum. 
 
 Cor. 2. — If from a perpendicular be let fall on the 
 section, the foot of this perpendicular wiU be the focus. 
 
 For A" VDO and O^i^ are equiangular ; 
 
 .-. VD : 2D0 = 20A : iAF; 
 
 .: VD: DA =DA: 4AF; 
 
 .-. 4^2?"= 4 Latus rectum. {Cor. 1.) 
 
 Hence Fib the focus. (Cor. 2, Prop, i.) 
 
CHAP. I.J The Parabola. 45 
 
 Problems on the Parabola. 
 
 1. If from any point on a tangent to a parabola, a per- 
 pendicular be let fall on the radius vector drawn from the 
 focus to the point of contact of the tangent, the segment 
 intercepted on the radius vector, between the foot of the 
 perpendicular and the focus, is equal to the perpendicular 
 let fall from the point on the directrix. 
 
 2. The focus and a tangent being given, the locus of 
 the vertex is a circle. 
 
 3. Find the locus of the point of intersection of any- 
 tangent to a parabola, with the line drawn from the foous, 
 making a constant angle with the tangent. 
 
 If the vertex of a triangle of given species lie fixed, while one hase angle 
 moves along a fixed right line, the locus of the other base angle will he a 
 right line. (See also Oor. 1, Prop, iv.) 
 
 4. Given four tangents to a parabola, determine the 
 focus and directrix. (See Prop, xvi.) 
 
 5. Normals at the extremities of a focal chord intersect 
 on the diameter which bisects the chord. (See Prop, x.) 
 
 6. The circle described on any focal chord as diameter 
 touches the directrix. 
 
 7. If a triangle ABC circumscribe a parabola whose focus 
 is F, the lines drawn through A, B, C, perpendicular to 
 FA, FB, and i?'C respectively, will pass through a point. 
 
 8. If a circle intersect a parabola in four points, the 
 sums of the ordinates of the points of intersection on oppo- 
 site sides of the axis are equal to each other. 
 
 9. Show how to cut from a cone a parabola having a 
 given Latus rectum. 
 
 10. Prove that the lines joining the middle points of the 
 sides of a triangle self-conjugate to a parabola are tangents 
 to the curve. (See Props, xxix. and xxiv.) 
 
CHAPTEE II. 
 
 The Ellipse. 
 
 definitions. 
 
 An Ellipse is the curve traced out by a point, whioh moves 
 in such a way that its distance from a fixed point is to its 
 perpendicular distance from a fixed right line in a con- 
 stant ratio £ : 1 (t being less than unity). 
 
 The fixed point is called the Focus, and the fixed right 
 line the Directrix. 
 
 The right line drawn through the Focus perpendicular 
 to the directrix is called the Axis. 
 
 The points at which the axis meets the curve are called 
 the Vertices. 
 
 The middle point of the portion of the axis intercepted 
 between the vertices is called the Centre. 
 
 Any line drawn through the centre is called a Diameter. 
 
 A right line which meets the curve, and being produced, 
 does not cut it, is called a Tangent. 
 
 A right line drawn through any point on the curve 
 perpendicular to the tangent at that point is called a 
 Normal. 
 
 The right line joining any two points on the curve is 
 called a Chord. 
 
 For Definitions of Ordinate and Abscissa see Definitions, 
 Chapter I. 
 
CHAP. II. J 
 
 The Ellipse. 
 
 47 
 
 Lemma. 
 
 If a right line AB be divided internally at in any ratio, and externally 
 at 0' in the same ratio, and a circle be described on 00' as diameter, the 
 right lines joining any point P on this circle with the extremities of the 
 line AB will have the same ratio. 
 
 Fig. 1. 
 
 Bisect 00' in 0; join CP, PO. 
 
 Then Aff : ffB ^ AO : OB; 
 
 ■ Aa + AO : AO' ^A0= O'B + OB : O'B - OB ; 
 .: 2A0: 200=200: 2BC; 
 .-. AC: CP=CP: OB; 
 
 . . A ACP is similar to A PCB ; (6 VI. Euclid.) 
 
 .-. LGPB= I CAP; 
 but /: CPO = L COP (5 I. EucUd.) 
 
 = L OAF + L OPA (32 I. EucUd.) 
 
 = L CPB + L OPA ; 
 .: lBP0 = l0PA; 
 .: AP:PB = AO: OB. (3 VI. EucUd.) 
 
48 The Ellipse. [chap. ii. 
 
 Proposition I. 
 
 The focus, directrix, and eccentricity of an ellipse being 
 given, to determine any number of points on the curve. 
 
 Fig. 2. 
 
 Let J'' be the focus, and Oy the directrix. Draw FO ± 
 the directrix ; divide FO internally at A, and externally 
 at A', so that FA : AO : : FA' : A'Oin the given ratio, 
 i : 1 ; then A and A' are the vertices of the curve, and 
 AA' the axis. (-De/.) 
 
 On the directrix take any point p ; join Fp ; draw AS 
 and A'S' ± AA', meeting ^i?" in B'and H' ; on SS as 
 diameter describe a circle ; through p draw pPP' | ( the 
 axis cutting the circle in P, P' ; join PF, P'F. 
 
 Then since ^J?is || Op ; .-. HF : Hp : : AF : AO; 
 
 also since A'H' is || Op; .-. H'F : H'p : : A'F : A'O ; 
 
 .: H'F : H'p : : HF : Hp. 
 
 Hence PF : Pp = FH : Hp (Lemma.) 
 
 = FA:AO = i:\. 
 
 (2 VI. Euclid.) 
 Also FF : P'p = FH : Hp (Lemma.) 
 
 = FA:AO = i.l; 
 
 (2 VI. Euclid.) 
 .-. P and P' are points on the curve. 
 
CHAP. it.J The Ellipse. 49 
 
 In like manner, by taking other points on the directrix, 
 any number of points on the curve may be determined. 
 
 Cor. 1. — ^If Op' be taken equal to Op, and another point 
 Q found in a similar manner to P, it is obvious that Qp' 
 will be equal to Pp ; therefore corresponding to any point 
 P, at a perpendicular distance from the axis = Op, there 
 is another point Q on the other side of the axis at a dis- 
 tance from it = Op, and also at the same distance as P from 
 the directrix ; hence the curve is symmetrical with regard 
 to the axis. 
 
 Cor. 2.— Bisect AA' in C ; through C draw LC8 ± 
 AA' and PF, meeting HE' in 8, and PF in L. 
 
 Then since AH, C8, and A'H are parallel ; 
 .-. 8S= 8E'. 
 
 Hence 8 is the centre of the circle H'P'PH ; 
 and .-. LP = LF. (3 III. Euclid.) 
 
 Therefore corresponding to any point P on the curve 
 there is another point F on the other side CL, situated in 
 precisely the same manner with regard to it as P ; hence 
 the curve is symmetrical also with regard to CL. 
 
 If, therefore, CO be measured off = CO, and CF = CF, 
 and Cy be drawn \. CfC, the curve could be equally well 
 described with F as focus, and Qfy' directrix. 
 
 Cor. 3. PF\Pp = AF:AO = AF-.A'O; 
 .-. PF:Pp = AF+ AF : AO + A'O 
 
 = 2CA:2CO; 
 .-. PF:Pp = CA: CO, 
 
 that is, the distance of any point on the curve from the 
 focus is to its distance from the directrix as CA : CO. 
 
 Cor. 4. AF:A'0 = AF:AO; 
 
 .-, A'F+ AF : AF- AF= A'O + AO : A'O - AO; 
 
 2CA : 2CF = 2C0 : 2CA ; 
 
 .: CFx CO=CA\ 
 
 £ 
 
50 The Ellipse. [chap. ii. 
 
 (Jor, 5. — From the symmetry of the curve it is evident 
 that any chord drawn through C is bisected at that point. 
 From this property the point C is called the centre of the 
 curve. 
 
 Def. — The chord drawn through the centre at right- 
 angles to the Axis, or Aids major, as that portion of the 
 axis intercepted between the vertices is frequently termed, 
 is called the Axis minor. 
 
 Proposition II. 
 
 The sum of the distances of any point P on an ellipse 
 from the foci is constant and equal to the axis-major. 
 
 Fig. 3. 
 
 PF:Pp= CA : CO. 
 
 {Cor. 3, Prop, i.) 
 
 Also PF' : Pp' =CA:CO; 
 
 ... PF+ PF' : Pp + Pp' = CA : CO. 
 But Pp + Pp' = 2C0 ; .-. PF+ PF' = 2CA = AA'. 
 
 Cor. 1. — By the help of the preceding theorem we can 
 describe an ellipse mechanically. 
 
 If the extremities of a thread, equal in length to AA', 
 be fastened to the fixed points F, I", it is obvious that a 
 pencil, moved about so as to keep the thread always 
 stretched, will describe an ellipse whose foci are F and J". 
 
CHAP. II. J The Ellipse. 51 
 
 Cor. 2. — The distance of the extremity of the axis 
 minor from either focus is equal to the semiaxis major. 
 Let CB he the semiaxis minor ; join BF, BF. 
 
 Then the A' BCF and BCF' are equal ; 
 
 (4 I. EucM.) 
 .-. BF== Br = i {BF+ BF') = iAA' = CA. 
 
 Cor. 3—BC' = BF' - FC 
 
 = CA'- CF' = A'Fy FA. 
 
 Cor. 4. — The sum of the distances of any point from the 
 foci of an ellipse is greater or less than the axis major, 
 according as thp point is Avithout or within the ellipse. 
 
 Fig. 4. 
 
 First let the point Q be without the ellipse ; join QF, 
 QF', and let (W meet the ellipse iu P ; join PF. 
 
 Then QF+QP>PF; 
 
 .-. QF+ QF'>PF+PF' 
 
 >AA'. 
 
 Next, let Q' be within the ellipse ; join Q'J?, Q'J?", and 
 let P'Q produced meet the elUpse in P ; join PF. 
 
 Then qF<Q!P + PF; 
 
 .-. QF+ QF' < PF' + PF 
 
 <AA'. 
 
 e2 
 
52 The ElKpse. [chap. ii. 
 
 Cor. 5. — Conversely, a point will be within or without 
 an ellipse according as the sum of its distances from the 
 foci is less than or greater than the axis major. 
 
 Proposition III. 
 
 1°. The line which bisects the external angle formed by 
 drawing lines from any point on an eUipse to the foci 
 faUs wholly without the curve. 
 
 2°. Any other line drawn through the point will cut 
 the curve. 
 
 Fig. 5. 
 
 1°. Let PS^ bisect the angle between PF&iiA PV pro- 
 duced. 
 
 Let Q be any point on PS. 
 
 Cut o£E PE = PF. 
 
 Join OF, QF', QB. 
 
 Then the a" QFP and QBP are equal ; (4 I. Euclid.) 
 
 .-. QR = QF; 
 
 but F'Q +QB> F'B ; (20 I. Euclid.) 
 
 .-. F'Q+QF>FT + PF 
 
 > axis major ; (Prop, ii.) 
 
 .'. Q is a point without the curve. 
 
 {Cor. 5, Prop, ii.) 
 
CHAP. ii.J The Ellipse. 53 
 
 2°. Let PK be any line other than the bisector of the 
 angle FPR. 
 
 Draw PL, making the z KPL = t KPF; cut off 
 PL = PF; join F'L, and where it cuts P^ join with F. 
 
 Fig. 6. 
 
 Then the A« FPQ and iPQ are equal ; (4 I. Euclid.) 
 .-. QF=QL; 
 but F'P + PL> F'L ; (20 I. Euclid ) 
 
 . jr'p + PF> F'Q + QF; 
 AA' > F'Q + OF; 
 
 .-. the point Q is within the ellipse. {Cor. 5, Prop, ii.) 
 Therefore the line PK will cut the curve. 
 
 Cor. 1. — Hence the line which bisects the external angle 
 between lines drawn from any point on an ellipse to the 
 foci is a tangent to the curve. 
 
 Cor. 2. — The tangent at the vertex is perpendicular to 
 the axis, and hence ordinates to the axis are also perpen- 
 dicular to the axis. 
 
 Cor. 3. — The normal to an ellipse at any point bisects 
 the angle between the focal radii vectores drawn to the 
 point. 
 
54 The Ellipse. [chap. ii. 
 
 Proposition IV. 
 
 The locus of the foot of the perpendicular let faU from 
 either focus on any tangent to an ellipse is a circle de- 
 scribed on the axis major as diameter. 
 
 Fig. 7. 
 
 Let PS be any tangent, and FT a perpendicular let 
 fall on it from the focus. 
 
 Produce F'P to meet FT produced in R ; join TC. 
 
 Then the A" PFT and PR T are equal ; 
 
 (26 I. Euclid.) 
 
 Now in A F'FR, F'F is bisected in C, and FR in T. 
 .-. CT is \\F'R, (2 VI. Euclid.) 
 
 and CT=iF'R (Similar a».) 
 
 = i {F'P + FP) 
 
 = GA. 
 
 Hence the locus of T is a circle described with the 
 centre C, and radius CA. 
 
 This circle is called the Auxiliary circle of the ellipse. 
 
CHAP. ii.J The Ellipse. 55 
 
 Cor. 1. — Conversely the right line drawn from either 
 foous to the adjacent point of intersection of any tangent 
 with the auxiliary circle is perpendicular to the tangent. 
 
 Cor. 2. — If the vertex of a right angle FTP move 
 along a fixed circle, while one leg passes through a fixed 
 poiat jP within that circle, the other leg wiU always touch 
 an ellipse. 
 
 Cor. 3. — By the aid of this Prop, we can draw a pair 
 of tangents to, an ellipse from any point Q without it. 
 
 On the line joining the given point Q, with either focus 
 as diameter, describe a circle ; the lines joining Q with the 
 points of intersection of this circle with the auxiliary circle 
 will be the required tangents. 
 
 Note. — The circle described on QF as diameter wiU 
 always cut the auxiliary circle. 
 
 Fig. 8. 
 
 For, bisect QFinO; join OC, QF'. 
 
 Then, since QF+QF'>AA'; (Con 4, Prop, m.) 
 
 .-. OF+OC > CA; 
 
 .-. 0C> CA- OF. 
 
 That is, the distance between the centres of the circles 
 is greater than the difference of their radii ; .-. they will 
 intersect. 
 
 Cor. 4. — The right line drawn from the centre parallel 
 to either focal radius vector, to meet the tangent, is equal 
 to the semiaxis major. 
 
56 The Ellipse. [chap. ii. 
 
 Cor. 5. — The circle described on any focal radius vector 
 as diameter will touch the auxiliary circle. 
 
 Proposition V. 
 
 The rectangle under the focal perpendiculars on any 
 tangent to an ellipse is equal to the square of the semi- 
 axis minor. 
 
 Fig. 9. 
 
 Let FT, FT, he perpendiculars let fall from the foci 
 on the tangent at any point P. 
 Join TC, and produce it to meet T'F' produced in 8. 
 
 Then the A» F'CS and FCT are equal ; (26 I. EuoUd.) 
 
 .-. F'8 = FT, and C8=CT=CA; (Prop, iv.) 
 
 .'. 8ia a point on the auxiliary circle ; 
 
 .-. A'F' X F'A = 8F' X FT' (35 III Euclid.) 
 
 = FTxFT'; 
 
 .: A'C - CF" = FT x F'T, (5 II. EucUd.) 
 
 or 5C^ = FT X FT. [Cor. 3, Prop, ii.) 
 
CHAP. II.] The Ellipse. 57 
 
 Proposition VI. 
 
 The locus of the intersection of the tangents to an 
 ellipse which cut at right angles is a circle. 
 
 Kg. 10. 
 
 Let QP, QP', be tangents, which cut at right angles. 
 Describe the auxiliary circle cutting the tangent QP in 
 T and T', and the tangent QF in H and S' ; join FS, 
 FT, F'H', F'T' ; draw Q^ touching the auxiliary circle ; 
 join CK, CQ. 
 
 Then FT and F'T' are both ± QP, and .-. || QP'. 
 
 {Cor. 1, Prop. IV.) 
 Similarly, F'H' and FH are both J. QP', and .-. || QP ; 
 
 [Cor. 1, Prop. IV.) 
 hence QT = FH, and QT' = F'H. 
 
 Now CQ = CK^ + QK^ 
 
 = CA' + QT X QT' (36 III. Euclid.) 
 = CA^ + FH X F'H' 
 = CA^ + CB' (Prop. V.) 
 
 = AB'; 
 .-. CQ = AB. 
 
 Hence the locus of Q is a circle described with the centre 
 C, and radius = AB. 
 This circle is called the Director Circle of the ellipse. 
 
58 
 
 The Ellipse. 
 
 [chap. it. 
 
 Pkoposition VII. 
 
 If PN be a normal to an ellipse at any point P, and 
 PM an ordinate to the axis, then 
 
 GM:CN= CA': OF'. 
 
 Fig. 11. 
 
 Join PF, PF' ; draw p'Pp \\ the axis major, meeting 
 the directrices in p', p. 
 
 Since {Cor. 3, Prop, iii.) PiV bisects the lF'PF; 
 
 .-. WF':JVF= PF' : PF (3 YI. Euclid.) 
 
 = Pp' : Pp. 
 Comp. and div. 
 
 JSfF' + NF : NF' -NF^ Pp' + Pp : Pp' - Pp, 
 or 2CF : 2CJSf = 2C0 : 2CM ; 
 
 .-. CM: CN=CO: CF 
 
 = COx CF: CF\ 
 
 Hence CM : CN = CA' : CFK 
 
 {Cor. 4, Prop, i.) 
 
 Cor. I.— CM: MN = CA' : C£'- 
 
 For CM:CM-CN = CA' : CA' - CF\ 
 
 or CM : MN = CA' : CB\ 
 
CHAP. ii.J The Ellipse. 59 
 
 Cor. 2. F'P : PF= F'N : NF 
 
 .-. F'P + PF:PF= F'N + NF : NF, 
 or CA : CF = PF : NF. 
 
 Similarly, CA:CF= PF' : NF' ; 
 
 .-. CA' : CF^-.-.PFx PF' : NF x NF' 
 .-. CA' : CA'- CF' : : PFxPF'-.PFx PF'-NFxNF' 
 CA- ■.CB'::PFx PF' : PN\ 
 
 Proposition YIII. 
 
 If a tangent at any point P of an ellipse he produced to 
 meet the axis in T, and the ordinate PM be drawn, then 
 CA win be a mean proportional between CM and CT. 
 
 Fig. 12. 
 
 Through P draw/Pjo || the axis ; join PF, PF'. 
 Then since PT is the bisector of the z. FPM ; 
 
 . p,j, . j,p^ prp . pp ^Qg^^ 3^ Yj_ Euclid.) 
 
 = /P : Pp. 
 Oomp. and div. 
 
 F'T+ TF: F'T- TF = Pp' + Pp : Pp' - Pp, 
 or 2CT : 2CF = 2C0 : 2CM ; 
 
 .-. CTx CM= CFx CO 
 
 = CA\ {Cor. 4, Prop, i.) 
 
60 The Ellipse. [chap. ii. 
 
 Cor. 1. — If the auxiliaxy circle be described, and PM 
 produced to meet it in Q, then QT will be a tangent to 
 the circle. 
 
 For CT: CA = CA: CM; 
 
 .-. CT: CQ=CQ: CM; 
 
 .-. the a'CQT and CMQ are similar ; 
 
 (6 VI. EuoUd.) 
 
 .-. lCQT=l CMQ ^90°; 
 
 .". QT is a tangent to the circle. 
 
 Cor. 2. — Tangents to the ellipse and auxiliary circle, re- 
 spectively, at the points where they are out by any ordinate 
 to the axis, will meet the axis produced in the same 
 point. 
 
 Cor. 3. — Corresponding ordinates PM, QM, of the 
 ellipse and auxiliary circle are in a constant ratio. 
 
 Since the A JSTPT is right-angled, and PM ± NT ; 
 
 .-. NMx MT= PM^- 
 
 Also, since ACQTia right-angled, and QM ± CT; 
 
 .-. CMx MT= QM'. 
 
 Hence PM' : QM' = iVJf : CM 
 
 = CB' : CA'. 
 
 (Cor. 1, Prop. VII.) 
 
 .-. PM:QM=CB: CA. 
 
 Note. — The angle QCA is called ths excentric angle of 
 the point P 
 
CHAP. ii.J The Ellipse. 61 
 
 Cor. 4. — The ellipse is concave towards the axis major. 
 
 Fig. 13. 
 
 QM : PM = AC : BC = qm : pm. {Cor. 3.) 
 
 But QM : FM = sm : rm ; (Similar a'.) 
 
 .■. qm : pm = sm : rm. 
 
 But qm is > sm by property of the circle ; 
 
 .•. pm is > rm. 
 
 That is, if a perpendicular be drawn to the axis major, 
 between the vertex A and any point P on the curve, the 
 segment intercepted on it between the curve and the axis 
 major is greater than the segment intercepted on it be- 
 tween the line AP and the axis major ; and, therefore, the 
 curve is concave towards the axis major. 
 
 Proposition IX. 
 
 The rectangle under the segments of the axis made by 
 any ordinate is to the square of the ordinate in a constant 
 ratio. (See Fig. 12.) 
 
 PM' : QM' = CB' : CA\ (Prop, viii.) 
 
 but QM' = A'M X MA ; (35 III. EucUd.) 
 
 .-. PM' : A'M xMA = CB': CA'. 
 
62 The Ellipse. [chap. ii. 
 
 Cor. 1. — The Latus rectum is a third proportional to the 
 axis major and axis minor. (See Fig. 12.) 
 
 For (I'Y : AF x FA' = CB' : CA^ 
 
 (^y : CE' = CB" : CA ; {Cor. 3, Prop, ii.) 
 
 hence ' CA : CB ^ CB : '^. 
 
 Cor. 2. — If from the foot of any ordinate a line be 
 drawn parallel to the line joining the extremities of the 
 axes, the square of the ordinate will be equal to the rect- 
 angle under the segments of the axis minor. 
 
 Fig. 14. 
 
 CA' : CM} = CB' : CQ' ; (ParaUel lines.) 
 •. CA' : CA' - CM' = CB' : CB' -CQ'; 
 ■. CA' : AM X MA' = CB' : BQ x QB ; 
 •. CB' : PM'= CB' -.BQxQF; 
 PM' = BQ X QF. 
 
CHAP. II.] Ttie Ellipse. 63 
 
 Cor. 3. — If PN be a normal at point P, meeting the 
 axis major in N, and NH a perpendicular let fall on the 
 focal radius vector, then PH is equal to half the Latus 
 rectum. 
 
 Fig. 15. 
 
 For FT : FP = PH : PN. (Similar A».) 
 
 Also F'T' : F'P = PH : PN; (Similar A».) 
 
 .-. FT X F'T' : FP x F'P = PH'' : PN'' 
 
 CB' : FP X F'P = PH' : PN" ; (Prop, v.) 
 but CA' : FP X F'P = GR : PN' ; {Cor. 2, Prop, vii.) 
 .-. CA : CB= CB-.PH; 
 .-. PH = ^ Latus rectum. {Cor. 1, Prop, ix.) 
 
 Cor. 4. — The normal and the focal perpendiculars on 
 the tangent at any point are in Harmonic proportion. 
 Let the tangent T'T, when produced, meet the axis 'in Q. 
 
 Then T'Q : QT = F'T' : FT. (Siinilar A'.) 
 
 Also F'T' : FT= T'P : PT; (Similar A'.) 
 
 .-. T'Q,: Q,T=T'P : PT; 
 
 .-. T'Q, PQ, and fQ; and .-. F'T', NP, and FT, are in 
 Harmonic proportion. 
 
64 The Ellipse. [chap, il 
 
 Proposition X. 
 
 The locus of the intersection of any tangent to an eUipse, 
 with the line drawn through the focus perpendicular to the 
 radius vector drawn from the focus to the point of contact 
 of the tangent, is the directrix. 
 
 Fig. 16. 
 
 Let PT be any tangent, FT A. FP. 
 
 Draw TR J. F'P produced, and TO ± i^'P produced ; 
 join TF'. 
 
 The A» TFP and TEP are equal ; (26 I.B uclid.) 
 
 .-. FT= rS'and PF= PS; 
 
 .-. F'ff=F'P + PF = AA'. 
 
 Now F'S' = FT - TH^ = FT - TF' 
 
 = F'O' - OF'' 
 
 = 2C0 X 2CF; 
 
 .-. AA'^ = 400 X CF, or CA' = CO x OF. 
 
 Hence OTia the directrix. {Cor. 4, Prop, i.) 
 
 Cor. 1. — Conversely, if from any point on the directrix 
 a tangent be drawn to an ellipse, the line joining that 
 point to the focus is perpendicular to the radius vector 
 drawn from the focus to the point of. contact of the tan- 
 gent. 
 
 Cor. 2. — Tangents at the extremities of a focal chord 
 intersect on the directrix. 
 
CHAP. ii.J _ The Ellipse. 65 
 
 Proposition XI. 
 
 If any chord PP' of an ellipse cut the directrix in B, 
 and if F be the focus corresponding to the directrix on 
 which B is situated, then FB is the external bisector of the 
 angle PFF. 
 
 Fig.. 17. 
 
 Produce FFio Q ; draw Pp, P'p JL the directrix. 
 Then P'F: PF ^ P'p' : Pp 
 
 = P'B : PB; (SimHar a'.) 
 .-. FB bisects the z PFG. (Prop. B., VI.. Euclid.) 
 
 Cor. 1. — Hence, being given one focus F, and three 
 points P, P', P", on an ellipse, we can find the directrix 
 and the axes. 
 
 For, draw FB bisecting the l between PF and P'P pro- 
 duced : the point where this line meets P'P produced wiU 
 be one point on the directrix. Similarly, by bisecting 
 the external z. between PF and P"F, another point on 
 the directrix may be found, and hence the directrix itself. 
 
 Also, since the eccentricity is the ratio of PF to the 
 perpendicular distance of P from the directrix, the axes 
 may be found by Prop, i., and Cor. 
 
66 
 
 The Ellipse. 
 
 [chap. II. 
 
 If a tangent to a curve be defined as The line joining two 
 indefinitely near points on the curve, it will follow imme- 
 diately from this Prop, that the right lAne drawn from the 
 focus to the point of intersection of any tangent with the di- 
 rectrix is perpendicular to the radms vector drawn form the 
 focus to the point of contact of the tangent; also that the 
 tangent makes equal /.' with the radii vectores drawn from the. 
 foci to the point of contact. 
 
 See Fig. 17. _ 
 
 For, when P is indefinitely near to P', FB becomes a 
 tangent at the point P; and the L PFP' is indefinitely 
 small ; 
 
 .-. PPQ = 180°; 
 
 but 
 
 I PFD = \t PFQ ; 
 L PFD = 90°. 
 
 Fig. 18. 
 
 Again, PF : PF' = Pp : Pp' = PD : PD', 
 
 (2, VI. Euclid.) 
 
 and L PFD = z PF'D', each being 90°. 
 
 Hence a PPZ) is similar to a PP'D' ; (7, VI. Euclid.) 
 .-. lFPD^ lF'PD'. 
 
CHAP, ii.J The Ellipse. 67 
 
 Proposition XII. 
 
 If two fixed points P, P', on an ellipse be joined with 
 a third variable point 0, the segment pp', intercepted on 
 either directrix hy the produced chords, subtends a con- 
 stant angle at the focus corresponding to that directrix. 
 
 Fig. 19. 
 
 Since (by Prop, xi.) I^ is the bisector of the external 
 z between FO and FP ; 
 
 .-. LOIi) + il OFP = 90°. 
 
 Also Fp' is the bisector of the external z. between FO 
 and FP' ; 
 
 .-. lOFp' + ^l OFP' = 90°; 
 
 .-. L OFp + I / OFP = L OFp' + \L OFP'; 
 
 .-. LOFp - L OFp' = \{L OFP' - L OFP), 
 
 or LpFp' = ^l PFF, and .-. constant. 
 
 Cor. 1. — Hence the anharmonio ratio of the pencil 
 formed by joining four fixed points on an ellipse to any 
 fifth variable point is constant. 
 
 For • PFP"P"' = ■ pp'p"p"' = F- pp'p"p"'. 
 f2 
 
68 The Ellipse. [chap. ii. 
 
 Pboposition XIII. 
 
 The line joining the focus to the intersection of two 
 tangents to an ellipse bisect the angle ■which the points of 
 contact subtend at the focus. 
 
 Fig. 20. 
 
 Let QP, QP', be tangents ; join PF, PF', P'F', P'F. 
 
 Produce FP tiU PR = PF', and FP' till P'R' = P'F' ; 
 join QR, QR'. 
 
 The A' QF'P' and QR'P' are equal; .-. QRf = QF'. 
 
 (4, 1. Euclid.) 
 The A» QPF' and QPR are equal ; .-. QR = QF' ; 
 
 (4, I. EucM.) 
 hence QR = QR. 
 
 But FR' = FP' + P'F' = FP + PF' = FR ; 
 
 hence the ^'FQR and FQR' are equal ; (8, I. Euclid.) 
 
 .-. t QFR = L QFR'. 
 
 Cor. 1. — It is obvious from the above demonstration that 
 
 L QF'P' = L QR'P' = L QRP = L QF'P. 
 
CHAP. ii.J The Ellipse. 69 
 
 Note. — If the ordinate at any point P on an ellipse be produced to meet 
 the Auxiliary circle in Q, then 
 
 tan i PrC = J\—^ • tan i QC[F; and taniPFA =J]-^ tan * QCF. 
 y 1 + e ^ 1 ~ e 
 
 Pig. 21. 
 
 Draw the tangent at the vertex meeting the tangents at P and Q in S 
 and K respectively ; join KC, SF", SF. Let GA = a. 
 Then i HF'C =ilPF'Cajid. l\SFA =i l PFA. (Prop, xm.) 
 
 Also t_ KOA = \L QGF.\ 
 
 TTA 
 Now im.\PFC=\«D.HF'0 = — , 
 
 tan \PFA = tan BFA = — , 
 AF 
 
 KA 
 
 tan \ QCF = tan KCA = ; 
 
 ^C 
 
 tan iPF'C SA x AC PM AC CB AC 
 
 ■ ■• tan^QCJ = KA^AF = QM' aF = -CA' AF' ^^'"'- ^' ^"^"P" ^"'"^ 
 
 C£ 
 
 " AF' 
 
 -/CA^-CF^ _ ICA-OF _ l a- ae 
 CA + CF'~SCA + CF~ '^a + ae ' 
 
 .-. tan iPFC= ^\~ ■ tan \ QCF. 
 
 , . ta.niPFA JSA x AC PM AC CB AC 
 
 ^s^'"' t^dm = Ta7af= qm-af^- CA -AF (C-'S-P^P- V".) 
 
 Cor. 1 
 
 CB -ya^ - gV 
 ~ AF a - ae 
 
 tan J PFA = a/^^ ■ tan i QCi^. 
 
 tanJ ^fJ'C _ 1 -e 
 tan i Pi^Lii ~ 1 + «■ 
 
 ^[5 
 
70 The Ellipse. [chap. ii. 
 
 Cor. 2. — ^If a quadrilateral be circumscribed to an ellipse, 
 its opposite sides subtend supplementary angles at either 
 focus. 
 
 Cor. 3. — By the aid of this Prop., also, we can draw a 
 pair of tangents to an ellipse from any point without it. 
 
 With Q as centre and QP' as radius, describe a circle ; 
 with F as centre and a radius = the axis major, describe a 
 circle ; the lines joining the points of intersection of these 
 circles with F will determine the points of contact of the 
 tangents. 
 
 Proposition XIY. 
 
 The angle at either focus, subtended by the segment 
 intercepted on a variable tangent to an ellipse by two 
 fixed tangents, is constant. 
 
 Fig. 22. 
 
 Let QP, QP', be the fixed tangents, BS the variable 
 tangent ; join FP', Fit, FT, F8, FP 
 
 Then iTFS = \lTFP, &\so LTFR = \lTFP' ; 
 
 (Prop. XIII.) 
 .-. ^RFS = \L FFP, and .-. constant. 
 
 Cor. 1. — The anharmonic ratio of the four points in 
 which a variable tangent to an ellipse is cut by four fixed 
 tangents is constant. 
 
 For the segments intercepted on the variable tangent 
 subtend constant angles at the focus. 
 
CHAP. II. J The Ellipse. 71 
 
 Proposition XV. 
 
 If a tangent at the extremity of any diameter CP meet 
 the axis produced in T, then the area of the a CPT thus 
 formed will he equal to the area of the a CAH, formed 
 by producing the diameter CP to meet the tangent at the 
 vertex A. 
 
 Fig: 23. 
 Draw the ordinate PM; join AP, HT. 
 Then TC : CA = CA : CM (Prop, viii.) 
 
 = CR:CP; (ParaUel lines.) 
 .-. PAiB\\ET; (2, VI. Euclid.) 
 
 .-. A PEA = A PTA ; (37, I. Euclid.) 
 ... A CHA = A CPT. 
 
 Cor. 1.— A Pilfr= trap. PMAE. 
 
 Bef. — The chords which join the extremities of any 
 diameter to any point on the curve are called supplemental 
 chords. 
 
 The axis major is sometimes called the Transverse axis, 
 and the axis minor the Conjugate axis. 
 
 Note. — Throughout the following Props. CA denotes 
 the semi-axis major, and CB the semi-axis minor. 
 
72 The Ellipse. [chap. ii. 
 
 Proposition XVI. 
 
 If an ordinate, drawn to the axis from any point P on 
 an ellipse, be produced to meet the auxiliary circle in Q, 
 and CQ' be drawn perpendicular to CQ, the perpendicular 
 drawn from Q' to the axis will meet the ellipse in a point 
 P', through which, if a diameter be drawn, it will be 
 parallel to the tangent at P. 
 
 Fig. 24. 
 
 The tangent to the circle at Q will intersect the tangent 
 to the ellipse at P on the axis. {Cor. 2, Prop, vm.) 
 
 Then, since QTand Q'(7 are both j. CQ,; .-. they are 
 parallel. 
 
 Hence a' QMT and Q'TM' are similar ; 
 
 .-. MT:M'C= QM: Q'M' (Similar a».) 
 
 = PM : P'M'. {Cor. 3, Prop, viii.) 
 
 Hence a ' PMT and P'M'C are similar ; 
 
 .-. P'C is II PT. 
 
 Bef. — The diameter which is parallel to the tangent at 
 any point is said to be conjugate to the diameter which 
 passes through that point. 
 
 Cor. 1. — If we draw tangents at P' and Q', it can be 
 similarly shown that CP is parallel to the tangent at P' : 
 hence, if the diameter which passes through P be conju- 
 gate to that through P ; conversely, the diameter which 
 passes through P will be conjugate to that through P'. 
 
CHAP. ii.J The Ellipse. 73 
 
 Cor. 2. CM= Q'M', and CM' = QM. 
 
 Since z M'Q^C + L M'Cq = 90°, 
 
 and also L M'C(^ + /. MCQ = 90° ; 
 
 .-. lM'QC=lMCQ. 
 
 Hence the A » M'Q'C and Jf CQ are similar ; 
 but CQ' = CQ ; .-. Cilf = QM', and Cilf ' = QJf. 
 
 Cor. 3. Cilf' + CJf '^ = CA\ 
 
 For C^' = CQ' = Ci!f' + QJf ' = CJf + OJf'^ (Cor. 2.) 
 
 Cor. 4. PJf '^ + P'Jf' ^ = CB'. 
 
 Since Qilf : Q'Jf ' = PM : F'M' ; (Cor. 3, Prop, viii.) 
 
 .-. QM^ + Q'Jf'^ : QJr= = PM' + F'M'' : PM\ 
 
 Alt. .-. C^' : PJf + Filf'' = QJf' : PM' [Cars. 2 & 3.) 
 
 = CA' : CB' ; 
 
 {Cor. 3, Prop, viii.) 
 .-. PM' +P'M" = CB\ 
 
 Cor. 5. — The sum of the squares of any pair of semi- 
 conjugate diameters is equal to the sum of the squares of 
 the semi-axes. 
 
 For CM' + CM" = CA', {Cor. 2.) 
 
 and PM' + F'M" = CB ; {Cor. 3.) 
 
 .-. CP" + CP" = CA' + CB. 
 Cor. 6. PM : CM' = CB : CA : : FM' : CM. 
 For PM : QM ^ CB : CA; but QM =^ CM' ; 
 
 .-. PM : CM' = CB : CA. 
 Again F'M' : Q'if' = CB : CA; but Q'Jf' = CM; 
 
 .-. P'M' ■.CM= CB: CA. 
 Cor. 7. — The A^ CPilf and CPM' are equal in area. 
 For PM : P'M = QM : Q'M' = CM' : CM; {Coi: 2.) 
 .■. PMx CM = P'M' X CM'. 
 
74 The Ellipse. [chap. ii. 
 
 Proposition XVII. 
 
 If QGhe an ordinate drawn to any diameter Pp from 
 any point Q on an ellipse, cutting the transverse axis in L, 
 QS an ordinate drawn to the axis from the same point, 
 and produced, if necessary, to meet the diameter Pp in K; 
 AS a tangent at the vertex : then a QL8 = trap. K8AH. 
 
 Pig. 25. 
 
 Draw the ordinate PM to the axis. 
 
 Then CA : CS : CM = AH : SK : MP ; 
 
 (Similar a'.) 
 .-. CA + 08 : OA + CM= AH + SE : AH + MP, 
 
 or A'8 : A'M= AH + 8K : AH + MP ; 
 
 .-. A'8 X A8 : A'M x AM = {AH + 8K) A8 : 
 
 {AH + MP) AM; 
 
 . . 80': PM' = trap. K8AH : trap. PMAH; 
 
 (Prop. IX.) 
 
 .-.A Q8L : A PMT = trap. KSAH: trap. PMAH; 
 
 (19, VI. Euclid.) 
 
 but A PMT = trap. PMAH ; {Cor. 1, Prop, xv.) 
 
 .-. A QSL = trap. ESAH. 
 
CHAP. II. ] The MUpse. 75 
 
 Cor. 1.— A QG'Jr=trap. TPGL. 
 
 For A QSL = trap. K8AH ; 
 
 .-. ' A QGK = tra.^. GLAH = iisip. TPGL. 
 
 Cor. 2. — In like manner it can be shown that 
 
 A GQ'K' = trap. TPGL ; 
 
 .-. A GQX= ^GQ'K'; 
 
 but they are also similar ; 
 
 .-. GQ=GQf. 
 
 Hence any diameter bisects all chords parallel to the 
 tangents at its extremities. 
 
 Proposition XVIII. 
 
 If CP' be the semi-diameter conjugate to CP, QG any 
 ordinate to CP ; then QG' : PG x Gp = CP" : CP\ 
 
 For A CPT : a CGL = CP" : CG\ 
 
 Division, a CPT : trap. TPGL = CP' : CP" - CG\ 
 
 or A CFV : a QGK= CP" -.PGxGp; 
 
 (Cor. 1, Prop, xviii.) 
 
 .-. CF' : GQ' = CP' -.PGxGp, 
 
 (19, VI. Euclid.) 
 
 or QG' ■.PGxGp= CP" : CP\ 
 
 Cor. 1. — Ordinates to any diameter at equal distances 
 from the centre are equal. 
 
 For the rectangles are the same for equal distances from 
 the centre. 
 
 Cor. 2. — The squares of the ordinates to any diameter 
 are proportional to the rectangles under the segments of 
 the diameter. 
 
76 The Ellipse. [chap. ii. 
 
 Proposition XIX. 
 
 If a normal at any point P of an ellipse meet the axis 
 major in N, and the axis minor in N' ; then 
 
 PN : CF : : CB : CA : : CF : PN', 
 
 CF being the semi-diameter conjugate to OP. 
 
 Fig. 26. 
 
 Since CF is || TP, the A FCM' is similar to the 
 A PMT, and .-. simHaj to AJSTPM. (8, VI. Euclid.) 
 
 .-. FC:NP= CM' : PM 
 
 = QM:PM {Cor. 2, Prop, xvi.) 
 
 = CA : CB. {Cor. 2, Prop, viii.) 
 
 Also PN' : PN^ CM: MN. (Similar a».) 
 
 = CA'' : CR ; (Prop, vii.) 
 
 .-. PN' X PN : PN' = CJ' : C& 
 
 = FC : PN' ; 
 
 .: PN' xPN=FC'; 
 
 .-. PN' iFC^FC-.PN 
 
 = CA : CB. 
 
 Cor. PN X PN = FCy 
 
CHAP, ir.] The Ellipse. 77 
 
 Proposition XX. 
 
 The rectangle under the distances of the foci, from any 
 point P on an ellipse, is equal to the square of the semi- 
 diameter CP', conjugate to that passing through the 
 point. 
 
 Fig. 27. 
 
 Circumscribe the A F'PF hy a circle cutting the axis 
 minor in N' and L ; join PN' and N'F. 
 
 Since CL Msects FF', and is also ± it ; .•. LN' is the 
 diameter of the circle, and arc F'N' = arc FN' ; 
 
 .-. L F'PN' = I FPN'. (26, III. Euclid.) 
 Also L PF'JSf = z PN'F; 
 
 hence the 2^' F'PN and N'PF are similar ; 
 .■. F'P:PN=N'P:PF; 
 
 .-. F'PxPF=PNxN'P 
 
 = 0P'\ {Oor. 1, Prop. XIX.) 
 
 Cor. L—PN" = PN' X PN - N'N x NP 
 
 (3, II. Euclid.) 
 = CF' - F'N X NF. 
 
78 The Ellipse. [chap. it. 
 
 Otherwise thus : — 
 
 2CA = PF+PF'; 
 .-. 4:CA' = PF' + PF^ + 2PF X PF' (4, II. EucHd.) 
 = 2CP' + 201" + 2PFx PF' ; 
 .-. PFxPF' = 2CA' - CF' - CP' 
 
 = CA' + CB' - CP" {Cor. 3, Prop, ii.) 
 
 = CP' + OF' - GF [Cor. 3, Prop, xvi.) 
 
 = CF^ 
 
 Cor. 2. — Since PN' bisects the lFPF' it is a normal, 
 
 and .". Pi, which is J. PiV', is a tangent. Hence the circle 
 
 which passes through the foci, and any point P on the 
 
 ellipse, passes also through the points in which the tangent 
 
 and normal at P intersect the axis minor. 
 
 Cor. 3. — If K he the foot of the perpendicular let fall 
 from N' on either PF' or PP produced, 
 
 then PK = \ {PF + PF) = semi-axis major. 
 
 This follows immediately from the well-known properties 
 of the triangle FPF', and its circumscribing circle. (See 
 Geometrical Gymnasium, Exercise 179, Galbraith and 
 Haughton's Manual of Euclid, books i., ii., iii.) 
 
 Cor. 4. — If K' be the foot of the perpendicular let fall 
 from L on PF', 
 
 then F'E' = ^ [PF + PF') = semi-axis major. 
 
 Also if K" be the foot of the perpendicular let fall from 
 L on FP produced, 
 
 then FK" = | {PF + PF) = semi-axis major. 
 
 (See as in Cor. 3.) 
 
 Cor. 5. CA' : CB' ^PF xPF: PN' 
 
 {Cor. 2, Prop, vii.) 
 = CF' : PN' ; 
 
 .-. CA: CB= CF : PN. 
 
CHAP. ii.J The Ellipse. 79 
 
 Proposition XXI. 
 
 If CK be a perpendicular let fall from the centre on a 
 tangent to an ellipse at any poiat P, and CP' the semi- 
 diameter conjugate to CP, then CK : CA :: CB : CP'. 
 
 Fig. 28. 
 
 Draw FT, F'T' i. the tangent at P. 
 
 Since the A" FPT and F'PT' are similar ; 
 .-. FP.F'P^FT.FT'; 
 
 =:FP + F'P:FT+F'T' 
 = CA : CK; 
 . pp X p'p : FT X F'T' = CA' : CK ; 
 but FP X F'P = CP", (Prop. XX.) 
 
 and FT x F'T' = BC ; (Prop, v.) 
 
 .-. CP" :BC'= CA': CK' ; 
 .- CP' :BC= CA: CK. 
 
 Cor. — The area of the triangle formed by joining the 
 extremities of any pair of conjugate semi-diameters is 
 constant. 
 
 For CP' X CK^CBx CA; 
 
 .-. area of a P'CP = area of a BCA, and .-. constant. 
 
80 The Ellipse. ■ • [chap. ii. 
 
 Proposition XXII. 
 
 If two ohords of an ellipse intersect one another, tte 
 rectangles under the segments are proportional to the 
 squares of the parallel semi-diameters. 
 
 Fig. 29. 
 
 Let FP' be any chord drawn through 0, and CE the 
 parallel semi-diameter. Draw the ordinates PM, P'M', 
 PN to the axis, and produce them to meet the auxiliary 
 circle in Q, Q', S. 
 
 Then PM : FM' = QM : Q'M' ; {Cor. 2, Prop, viii.) 
 
 .•. PP' and QQ', if produced, will meet the axis produced 
 in the same point T. 
 
 Through draw ffOL J. the axis. 
 
 Again JSfC: MT^RN: FM' (Sinular a'.) 
 
 = 8N : Q'M' ; 
 
 {Cor. 3, Prop, viii.) 
 
 .-. SC is II QT; hence a SCR is similar to a Q'TF. 
 
 Now OF:OrQr = FT: Q'T, (ParaUel Unes.) 
 
 and OP '.aO, -'FT: QT; 
 
 .: OPx OF : 0Qx GQ^ = FT^ : Q'T' 
 
 = RC' : C8\ 
 
 Alt. OPx OF : CR' = CQ X CQ' : CS\ 
 
CHAP. II.J 
 
 The Ellipse. 
 
 81 
 
 Similarly, if any other chord pp be drawn through 0, 
 and the ordinates pm, p'm' produced to meet the auxiliary 
 circle in g', g^ : then, since 
 
 OL:aL = CB: CA 
 
 = pm : qm =p'm' : q'm' ; 
 
 {Cor. 3, Prop, viii.) 
 .•. the chord qcf will pass through Cf ; 
 and if Cr he the semi-diameter parallel to pp', 
 
 then Op >i Op' : Cr" = (/q X ffq' : CAK 
 
 But aO, X O'Q' = Cg X aq' ; (35, III. EucHd.) 
 
 hence OP ^ OF : Op x Op' = CR^ : Cr\ 
 
 {See Note, page 83.) 
 Otherwise thus : — 
 
 Fig. 30. 
 
 Fig. 31. 
 
 Let PP' he any chord drawn through the point 0, and 
 CR the parallel semi-diameter, OK the semi-diameter 
 conjugate to CR; draw the diameter through 0, meeting 
 the curve in j8 ; draw 8M an ordinate to vR', and Sw, 
 PN' ordinates to CR. 
 
 Then jSC" - CN'^ : RC - CM" = PN" : 8M'^ ; 
 
 [Cor. 2, Prop, xviii.) 
 
 .-. RC - PN' : RC - SW = CN' : CM^ 
 
 ^ON': 8M'. 
 
 Alt. RC - PN' : ON' = RC - SM' : 8M'. 
 
 Comp. RC - PN + ON' : ON' = RC : 8M'. 
 
82 The Ellipse. iciiw. m 
 
 Alt. RC^ -POx or : BC = ON' : SM' 
 
 = OC : CS\ 
 
 Div. POx OF : HO' = C8' - OC : CS' 
 
 = SOx OS' : C8\ 
 
 If pp' be any other chord drawn through 0, and Cr the 
 parallel semi-diameter ; then 
 
 pO X Op' : Cr' = 80 x 08' : C8\ 
 Hence POxOF -.pOx Op' = C^* : Cr". 
 
 Cor. 1. — The rectangle under the segments of any 
 diameter made by an ordinate is to the square of the 
 ordinate as the square of the semi-diameter is to the square 
 of itg^ semi-conjugates 
 
 This follows by supposing one of the chords to pass 
 through the centre, and the other parallel to the tangent 
 at its extremity. 
 
 Ctir. 2.— If be without the ellipse, and P, F coincide, 
 also jo, pf ; then the tangents to an ellipse from any point 
 without it are proportional to the parallel semi-diameters. 
 
 Cor. Z.—rli two parallel tangents OP, OF, be met by 
 any third tangent, OO ; then OP : ffF = OQ : OQ. 
 
 Fig. 32. 
 For OP-.OQ^CE: Cr = (/P' : O'Q. (Co/'. 2.) 
 
CHAP. II. J The Ellipse. 83 
 
 Cor. 4. — If from a point without an ellipse a secant and 
 also a tangent be drawn, the rectangle under the whole 
 secant and its external segment is to the square of the 
 tangent as the squares of the parallel semi-diameters. 
 
 Note. — If for 0, ff, L, we read o, o', I, the demonstration, 
 page 80, will apply to the case when the point in which the 
 chords intersect lies without the ellipse. 
 
 Proposition XXIII. 
 
 If from a point without an ellipse two tangents OP, 
 OR be d;rawn, any Une AQ' drawn parallel to either will 
 be out by the curve and chord of contact of the tangents 
 in Geometric proportion. 
 
 Fig. 33. 
 
 Draw the semi-diameters CE and CD \\ OP and OR 
 respectively. 
 
 Then AQ x AQ : AP^ = CD"- : CE' 
 
 {Cor. 4, Prop, xxii.) 
 = OE" : OP' 
 
 {Cor. 2, Prop, xxii.) 
 
 = AB' : AP' ; 
 
 (Parallel Hues.) 
 .-. AQ X AQ' = AB\ 
 
 g2 
 
84 The Ellipse. [chap. ii. 
 
 Cor. — If ^Q' be a tangent : then^Q = AB. 
 ■ That is, if two parallel tangents to an ellipse be cut by 
 any third tangent, the segment intercepted on either of 
 the parallel tangents, between the chord of contact of the 
 other two and the curve, is bisected by the non-parallel 
 tangent. 
 
 Peoposition XXIV. 
 
 If a circle intersect an ellipse in four points, the common 
 chords will be equally inclined to the axis. 
 
 Fig. 34. 
 
 Let PP', QQ Ipe the common chords intersecting in 0. 
 Draw the Semi-diameters CB, C8 || PP', QQf respec- 
 tively. 
 
 Then OP ^ OP' : OQ ^ OQl = CR^ : 08' ; 
 
 (Prop. XXII.) 
 
 but OP X or = OQ X OQf ; (35, III Euclid.) 
 
 .■.CB=C8. 
 
 Hence CB and 08, and .•. OP and OQ', are equally 
 inclined to the axis. {Cor. 1, Prop, i.) 
 
 Cor. 1. — In like manner it can be shown that PQ and 
 P'Q, also PQ[ and P'Q are equally inclined to the axis. 
 
CHAP. ii.J The Ellipse. 85 
 
 Proposition XXY. 
 
 If a tangent to an ellipse at any point P meet any 
 diameter AA' produced in T, and the ordinate PM be 
 drawn ; then CA is a mean proportional between CM and 
 CT. 
 
 Fig. 35. 
 
 Draw tangents at A and A', meeting the tangent at P 
 in G and H. 
 Then A'T : TA = A'H -.AG; (Parallel lines.) 
 
 and A'M : MA = HP : PG; 
 
 but A'E:AG = HP:PG; 
 
 {Cor. 3, Prop, xxii.) 
 .-. A'T: TA = A'M:MA; 
 
 hence CMxCT=CA\ 
 
 Cor. 1. — Conversely, if PM be an ordinate to any 
 diameter AA', and CT be taken a third proportional to 
 CM and CA ; then PT will be a tangent to the ellipse. 
 
 Cor. 2. — The tangents at the extremities of any double 
 ordinate iatersect on the diameter corresponding to that 
 ordinate. 
 
 Cor. 3. — If a diameter be drawn through the inter- 
 section of two tangents to an ellipae, it will bisect the 
 chord of contact. 
 
86 
 
 The Ellipse. 
 
 [chap. II. 
 
 Proposition XXVI. 
 
 If a variable tangent to an ellipse meet two fixed 
 parallel tangents, it mil intercept segments on them whose 
 rectangle is constant, and equal to the square of the 
 parallel semi-diameter. 
 
 (See Fig. 35.) 
 
 Let GPH be the variable tangent meeting the fixed 
 tangents AG, A'Hin the points G, H. 
 
 Draw PE an ordinate to BC. 
 
 (Prop. XXV.) 
 
 Then CT : CA = CA : CM. 
 
 Division CT:TA^CA: AM. 
 Alt. CT:CA=TA: AM. 
 
 Comp. CT : TA' = TA : TM; 
 
 .-. CD : A'H=AG : PM; (Similar a».) 
 
 .-. A'SxAG= CDxPM 
 
 = CBxCE=C£'. (Prop. XXV.) 
 
 Cor. 1. — ^The rectangle under the segments of the 
 variable tangent is equal to the square of the semi- 
 diameter CQ drawn parallel to it. 
 
 For A'S :HP=CB:CQ = AG:PG; 
 
 {Cor. 3, Prop, xxii.) 
 
 .-. A'H xAG:HPxPG= GB^ : CQ\ 
 
 But 
 
 A'HxAG^ GB'; 
 .-. HP X PG = CQ\ 
 
CHAP. ii.J The Ellipse. 87 
 
 Cor. 2. — Any tangent to an ellipse ■will be cut harmoni- 
 cally by two parallel tangents and the diameter passing 
 through their points of contact. 
 
 For MT: TG = A'H : AG (Similar a'.)' 
 
 = HP : PG. 
 
 {Cor. 3, ftop. XXII.) 
 
 Cor. 3. — If two parallel tangents AC, BF, be cut by 
 any two other tangents EF, CB ; then 
 
 AE : BD = EG : OF = CO : CD. 
 
 Fig. 36. 
 
 For CA X BD = AE X BF; (Prop, xxvi.) 
 
 .-. AE:AC=BD:BF; 
 
 . . AE:AC-AE = BD:BF- BD. 
 
 Alt. AE: BD = EC: DF 
 
 = EO : OF, 01= CO: OD. 
 
 Cor. 4.— The lines joining F with C, and D with E, 
 would, if produced, intersect on the diameter BA pro- 
 duced. 
 
88 
 
 The Ellipse. 
 
 [chap. II. 
 
 Proposition XXVII. 
 
 The triangles CPT and GAK, formed by drawing tan- 
 gents at the extremities of any two semi-diameters of an 
 ellipse, are equal in area. 
 
 Pig. 37. 
 
 Draw the ordinate TM. 
 Then TC : CA = CA : CM. 
 
 = CK: CP; 
 
 .-. KT is UFA; 
 
 .: A PTA = A PKA ; 
 
 .-. A CPT = A CAK. 
 
 (Prop. XXV.) 
 (Parallel lines.) 
 
 (37, I. EucM.) 
 
 Cor. 1. — If the ordinate AN be drawn from A to the 
 semi-diameter CP, then the area of the a CMP = area of 
 A CAN. 
 
 For ^iVisllPT; 
 
 .-. TC:CA = PC : CN; 
 hence CA : CM = PC : CN; 
 
 .-. MN'vs, II PA; 
 
 .: A NPM= A NAM; (37, I. Euclid.) 
 
 • ■. A CMP = A CAN. 
 
CHAP. H. 
 
 The EUime. 
 
 89 
 
 Proposition XXVIII. 
 
 If from the extremities of any two conjugate diameters 
 CP, CQ, the ordinates PM, QJ!^ he drawn to any other 
 diameter CA ; then 
 
 CN' = AM X MA', and CM' = AN x NA'. 
 
 Fig. 38. 
 
 Mr : OZV' = MP' : QN' (Similar a».) 
 
 = MA X MA' : NA x NA'. 
 
 [Cor. 2, Prop, xviii.) 
 But MA X ilf^' = ^C" - CM' (6, II. Euclid.) 
 
 = TCxCM- CM' (Prop, xxv.) 
 = Cif X if T ; 
 .-. MT' : CN' = CMx MT: AN x NA!. 
 
 Alt. Jf 2" :CMxMT== CN' : CA' - CN' ; 
 
 (5, II. EucM.) 
 
 .-. MT: CM=- CN' : CA' - CN'- 
 
 Comp. MT;CT = CN' : CA' ; 
 
 .-. CM X MT : CM X CT = CN' : CA'. 
 
90 The Ellipse. [chap. ii. 
 
 But CMxCT^CA'; (Prop, xxv.) 
 
 •. CN" = CMx MT= AM x MA'. 
 
 In like manner it can Tae proved that CM^ = ^JVx NA'. 
 
 Cor. 1.— CM' + CN^ = CA'^ 
 
 For CM' + CN' = AN x JSTA' + AM x MA' 
 
 = CA' - CN' + CA^ - CM' ; 
 (5, II. Euclid.) 
 .■.2CM' + 2CJV'='2CA'; 
 
 or, CM' + CN' = CA'. 
 
 Cor. 2.—PM' + QN' = BC, BC being the semi- 
 diameter conjugate to CA. 
 
 For CW : CA' = PM' : AM x MA' or CN' ; 
 
 (Prop. XVIII.) 
 
 also, CB' : CA' = QN' : AN x NA' or CM' ; 
 
 .-.CB': CA' = PM' + QN' : CN' + CM'. 
 
 But CA' =CN'.+ CM', .: CB' = PM' + QN\ 
 
 Cor. 3.—CA ■.CB=CM.QN=CN : PM. 
 
 For CA' : CB' = AM x MA', or CN' : PM' 
 
 = ANx NA', or CM' : QN' ; 
 
 .: CA: CB=CN:PM=CM: QN. 
 
 Cor. 4.— The a CMP is = a CNQ in area. 
 
 Produce QiVto Q'. Then QiV= Q'iV^; 
 
 .-. CN: CM= PM : Q'iV; (Cor. 3.) 
 
 .-. aCMP= aCNQ'= aCNQ. 
 
 (15, VI. Euclid.) 
 
CHAP. ii.J The Ellipse. 91 
 
 Proposition XXIX. 
 
 If any tangent to an ellipse meet any two conjugate 
 diameters CP, CQ, the rectangle under its segments is 
 equal to the square of the parallel semi-diameter. 
 
 Fig, 39. 
 
 Draw the ordinates, PM, QK, to the diameter passing 
 through the point of contact of the tangent. 
 
 Then CM : PM = CA : AT' ; (Similar A».) 
 
 .-. CMxAT' = PMx CA 
 
 = CB X CN; 
 
 {Cor. 3, Prop, xxviii.) 
 
 .-. CM : CN = CB: AT'. 
 
 Again, CN : NQ^CA.AT; (Similar A'.) 
 
 .-. CN X AT=CA xNQ 
 
 = CB X CM; 
 
 {Cor. 3, Prop, xxviii.) 
 
 .•. CM : CN =AT: CB; 
 
92 The Ellipse. 
 
 
 hence CB : AT' = AT : 
 
 CB; 
 
 .: AT X AT' = CB". 
 
 
 Otherwise thus : — 
 
 
 [chap. n. 
 
 Fig. 40. 
 
 Draw PM, QM' ordinates to CA, and PN an ordinate 
 to CB. 
 Then C8 x CM' = CA' ^CMx CT; (Prop, xxv.) 
 
 .-. CS : CT = CM : CM'. 
 
 But C8 : CT = CQ: PT; (Similar A'.) 
 
 .-. GQ: PT=CM: CM' 
 
 = PN : CM' 
 
 = PT' : CQ ; 
 PTxPT'= CQ\ 
 
 (Similar A'.) 
 
 Cor. — Conversely, if on any tangent segments AT, 
 AT' be measured from the point of contact whose rect- 
 angle is equal to the square of the parallel semi-diameter, 
 then the diameters drawn through T and T' will be con- 
 jugate. 
 
CHAP. 11. J The Ellipse. 93 
 
 Peoposition XXX. 
 
 t 
 
 ' Given in magnitude and position any two conjugate 
 semi-diameters CA', CiB, of an ellipse, to find the axes. 
 
 Fig. 41. 
 
 On Ci5' produced take BD a third proportional to CS 
 and CA ; bisect GB in H; draw KO -L GB to meet 
 BO drawn {| CA! \ with the centre and radius OC 
 describe a circle cutting B!0 produced in T and T'\ join 
 GT, Cr. Draw FM and BfN \\ GT and GT respec- 
 tively. Take CA a mean proportional between GT and 
 GM\ also GB a mean proportional between GT' and 
 GN. Then (Li and GB are the semi-axes. 
 
 For GA'' ^CBT ^B'D (Const.) 
 
 = r^ X ^r ; (35, III. EucM.) 
 
 .•. CA and C5 are conjugate diameters ; 
 
 [Cor., Prop. XXIX.) 
 
 but L TGT' = 90° ; hence CA and CB are the semi-axes. 
 
94 
 
 The Ellipse. 
 
 [chap. II. 
 
 Proposition XXXI. 
 
 If a straight line cut two tangents TP, TQ to an 
 ellipse in the points A, A', the curve in B, B', and the 
 chord of contact in ; then 
 
 A(P : A'(y ^AB yAB': A'B x A'F. 
 
 Fig,42, 
 
 Fig. 43. 
 
 Draw AM \\ TQ ; also the semi-diameters CH, CK, and 
 CL II TP, TQ and AA', respectiTely. 
 
 Then AO'-.A'O'^ AJSP : A'Q' (ParaUellines.) 
 
 I AM' : AP" 
 ( AP' : A'Q' 
 
 ( TQ" : TP' 
 ( AP" : A'Q' 
 
 (Similar A'.) 
 
 ( CK' : CH' \ 
 ( AP' : A'Q' ) 
 
 {Co7'. 2, Prop. XXII.) 
 
 f CK' : CL' 
 CL' : CH' 
 AP' : A'Q' 
 
CHAP. II.J 
 
 The Ellipse. 
 
 95 
 
 A'Q' : A'B X A'B' 
 
 A£ xAB' : AP' 
 
 AF'iA'Q'; 
 
 {Cor. 4, Prop, xxii.) 
 
 AO' : A'O' = ABxAB': A'B x A'B'. 
 
 Proposition XXXII. 
 
 If any line T8 be drawn parallel to the chord of contact 
 of the tangents to an ellipse, the segments AT, B8 inter- 
 cepted between the curve and the tangents will be equal. 
 
 Pig. 44. 
 
 Draw the diameter bisecting the parallel chords PQ, 
 AB. This diameter will pass through 0. 
 
 {Cor. 2, Prop, xxv.) 
 Then, since PM=MQ, 
 
 but 
 
 , TJ}f=NS 
 
 AN^NB 
 
 .AT=BS. 
 
96 The Ellipse. [chap. ii. 
 
 Proposition XXXIII. 
 
 Any line OB, drawn through the intersection of two 
 tangents to an ellipse, is cut harmonically by the curve 
 and the chord of contact of the tangents. 
 
 Fig. 46. 
 
 Through! B and ^ draw T8 and T'& \\ PQ, the chord 
 of contact of the tangents. 
 
 Then TF^ : T'F^ = TB x TA : T'B' x T'A'; 
 
 {Cor. 4, Prop, xxii.) 
 but TB = A8, and T'F = A' 8' ; (Prop, xxxii.) 
 
 . yp2 . rp'p2 = TBx B8: TB' x B'8' 
 
 = OB' : OB" ; (Similar a».) 
 
 but TP : T'P = O'B : O'B' ; (Parallel Hnes.) 
 
 hence OB : OB' = OB : OB. 
 
 Cor. — If through any point without an ellipse a 
 line be drawn cutting the curve in B, S, and 0O\>q taken 
 equal to the harmonic mean between OB and OB!, the 
 locus of 0' is a straight line ; namely, the chord of contact 
 of the tangents drawn from 0. 
 
CHAP. n.J The Ellipse. 97 
 
 Proposition XXXIV. 
 
 Any line drawn through the middle point of the chord 
 of contact of two tangents to an ellipse, will be cut har- 
 monically by the curv.e, and the line drawn through the 
 intersection of the tangents parallel to their chord of 
 contact. 
 
 Fig. 46. 
 
 Let TP, TQ be tangents, BB' any chord drawn through 
 0', the middle point of PQ, and meeting the line drawn 
 through T \\ PQ in 0. Through B" draw B'B" || PQ ; 
 join rC. 
 
 Then TC will pass through 0'. {Cor. 3, Prop, xxv.) 
 
 i B"M : (XT' ) 
 T}"M ■ B"B' ^)^ -^ ■ ^ ^ [ 
 B m.Ji Ji (OT : B"B' ) 
 
 = ^2" • BJff' 1 (Similar a'.) 
 
 = B"T X BT' : TT x BB". 
 
 But TT X BB" = 2B"T x BT' ; 
 
 .-. B'ff' = 2B"M ; .'. B'M is an ordinate, and B' a point 
 
 on the curve. , „ ^ 
 
 Now PT : TB" = BT' : T'B" ; (Prop, xxxiii.) 
 
 .-. BO : OB' = B(y : &B'. (Parallel lines.) 
 
 H 
 
98 The Ellipse. [chap. ii. 
 
 Otherwise thus :— 
 
 Let TP, TQ be tangents, ££' any chord drawn through 
 O', the middle point of PQ, and produced to meet the 
 line drawn through T\\ PQ in 0; join OC, TO ; then 
 (Cor. 3, Prop, xxv.) TC will pass through C. Draw i2iV 
 a tangent a,t B, BM || QP, and .■. an ordinate to CT' ; 
 through a draw BE || ^iV. 
 
 Fig. 47. 
 
 Then, CT x Off ^ CA' =CNxCM; (Prop, xxv.) 
 . . CT: OM =CN: CO' ; 
 .-. 00: OR =CR: OH; (ParaUel lines.) 
 
 .-. the tangents atX) and E will intersect at ; 
 
 (Prop, xxv.) 
 and>enoe BO : OB' = Bff : OB. 
 
 (Prop. XXXIII.) 
 
 Cor. 1. — ^If through any point Of within an ellipse any 
 line be drawn cutting the curve in B, B', and be taken 
 the harmonic conjugate to C with regard to B and B, 
 the locus of is a straight line. 
 
 Cor. 2. — If from tangents be drawn to the ellipse, 
 the chord of contact will pass through ff. 
 
 For (Prop, xxxiii.) the chord of contact must divide :the 
 line OB harmonically. 
 
CHAP. II. 
 
 The Ellipse. 
 
 99 
 
 Proposition XXXV. 
 
 If through any point P on an ellipse lines FOG, QfPQ 
 be drawn parallel to any two adjacent sides J) A, DC oi 
 an insoribed quadrilateral, meeting the opposite sides in 
 0, 0', and Q, Q'; then PO x P(/ : PQ x PQ! in a con- 
 stant ratio. 
 
 Fig. 48. 
 
 Through B and C draw BE and CG \\ AD ; join AG, 
 and produce it to meet EB produced in H; draw the 
 diameter iV!MB bisecting the parallel chords AD, GO, BE. 
 
 Then 
 
 
 MR = RlK', 
 
 (Parallel lines.) 
 
 hence 
 
 
 BH=EE 
 
 
 Now 
 
 
 OL:EB = LA: AH 
 = O'D : DK. 
 
 
 Alt. 
 
 
 OLiffD^HB: DK. 
 
 
 Also 
 
 P'O 01 
 
 : SO: Q8 = BK:KC. 
 
 (Similar a'.) 
 
 Compounding, 
 
 
 OL 
 
 xPG 
 
 : 0'i)x Q8 = SBxBK: 
 
 DKxKC; 
 
 h2 
 
100 The ElUpse. [chap. ii. 
 
 .-. OLxPa -.Pi^ X Q8 = I!KxBK:l)Kx KC 
 
 = avx (yp-.acxffi) 
 = PLx ap.psxpq-, 
 .-. oLxpa + PLxpa.pq xQ8+P8xPQ 
 
 = EKxBK:DKxKO, 
 
 or PO X PO' : PQ X PQf = UK X BK : BK x KC. 
 
 Now it is evident that the points A, B, G, D, teing 
 fixed, E is also fixed, and 
 
 .•. EK X BK : DK x KG in a constant ratio ; 
 
 .•. PO X P<y : PQ X PQ' in a constant ratio. 
 
 Proposition XXXVI. 
 
 If from any point P on an ellipse, lines PJR, PR', P8, PS', 
 be drawn to the sides of an inscribed quadrilateral, making 
 with them any constant angles, then the rectangles under 
 the lines drawn to the opposite sides wUl be in a constant 
 ratio. 
 
 Fig. 49. 
 
 Take any other point p on the curve. 
 
CHAP. ii.J The Ellipse. 101 
 
 Through the points P and 79 draw Q-PQ'and qp^ II -DC, 
 and PO(y,poo' \\ AD, sho pr, pr' , ps, ps' || PR, PR', P8, 
 P8', respectively. 
 
 Then PR : pr = PQ : pq, (Similar a*.) 
 
 and PR'-.pr'-PQ! -.p^; ( Do. ) 
 
 .-. PR X PR! : pr x pr' = PQ x PQ' : pq x pqf. 
 
 Similarly it may be proved that 
 
 P8 X PiS' : ps X ps' = POx pa : po x po' ; 
 
 but PQ X pq : PO X pa =pq xpq' : po x po' ; 
 
 (Prop. XXXV.) 
 PR X PR : PS X PS' = pi- X pr' : ps x ps'. 
 
 Cor. 1. — The rectangle under the perpendiculars let 
 fall from any point of an ellipse on two opposite sides of 
 an inscribed quadrilateral is in a constant ratio to the 
 rectangle under the perpendiculars let fall on the other 
 two sides. 
 
 Cor. 2. — If the points A and B coincide, also the points 
 C and D, then the sides AB and CD become tangents, 
 and the sides BC and AD coincide, and become the chord 
 of contact. Then the rectangle under the perpendiculars 
 let fall from any point of an ellipse on two fixed tangents 
 is in a constant ratio to the square of the perpendicular 
 let fall on their chord of contact. 
 
 Cor. 3. — If we suppose AB, CD to intersect in X and 
 AC, BD in F; also that PR, PR', PS, PS', pr, pr, 
 ps,ps' lie in the line XY; then, regarding ACBD as the 
 quadrilateral, the above proportion becomes 
 
 PX' : PY"- = pX' : pY' ; 
 
 .-. XY is cut harmonically by the curve. 
 
102 The Ellipse. [chap. it. 
 
 Proposition XXXVII. 
 
 If two fixed tangents AD, DB, to an ellipse be cut by 
 a diameter AB, parallel to their chord of contact, and by 
 a third variable tangent EF, the rectangle under the seg- 
 ments of the two fixed tangents, intercepted between the 
 diameter and the variable tangent, is constant. 
 
 Fig. 50. 
 
 Join PC; produce it to meet the curve in R ; join BR. 
 Since CA = CB; (Prop, xxxii.) 
 
 .-. the i\'AOP and BCR are equal ; (4, 1. Euclid.) 
 
 .-. J5i2is = and|| AP ; 
 
 .'. BR is a tangent. 
 Then RB : PE = BF : FJD ; {Cor. 2, Prop, xxvi.) 
 
 .. AP : PE = BF : FD ; 
 
 . . AP : AP + PE = BF : BF + Fl) ; 
 
 .-. AP -.AE^BF: BD; 
 
 . . AEx BF= BD x.AP,mi ■■. .constant. 
 
CHAP. II. J 
 
 The Ellipse. 
 
 103 
 
 Lemma. 
 
 If on any right line we have segments AJB, BC, &c., 
 and on any other right line inclined to the former at any 
 angle segments A'£^, B'C, &o., such that AB : A'& 
 = AC : SC, &c. = m:n; then the locus of the middle 
 points of AA!, BS, CC, &c., will be a right line. 
 
 Fig. 51. 
 
 Measure off CC/ and AO', such that 
 
 CO : CO' = Aff' : A'O = m:n. 
 
 Bisect 00" in M and Off in M' ; join MM'. 
 This line will pass through the middle points of AA', 
 BB', and CC. 
 
 For AB: A'B' = BC : B'C = CO : CC = AO : A'O; 
 
 .■.i{AB + BC+CO + A&'):iiA'B' + B'C+Cff + A'0) 
 
 = m : n; 
 
 or OM: OM' = m : n. 
 
 Again, AB + AC : A'B' + A'0 = m:n = MC : M'O ; 
 
 .-. AB + AC - MCiA'B' + A'O - M'0 = m : »; 
 
 or MB : M'B' = m : n. 
 
104 The Ellipse. [chap. ii. 
 
 Now consider the a OBB^, the sides of which are cut 
 by the transversal MM' ; 
 
 BF B'M' OM 
 
 FB' ■ M'O ' MB 
 
 h 
 
 BF B'M' OM 
 *"■ FB' ' MB ' M'O ' ' 
 
 BF n m 
 \ ''FB''m'n~'' 
 .■.BF=FB'. 
 
 In like manner it can be shown that AA' and CC are 
 bisected by MM'. 
 
 Proposition XXXVIII. 
 
 The right line joining the middle points of the dia- 
 gonals of a quadnlateral circumscribing an ellipse will 
 pass through the centre. 
 
 Fig. 52. 
 
 Let ABA'B^ be the quadrilateral. Through the centre 
 C draw Dlf || PQ, the chord of contact of the opposite 
 sides AB, A'W. 
 
CHAP. II. J The Ellipse. lOo 
 
 Then AD x ffD' = BD x A'D' ; 
 
 (Prop. XXXVII.) 
 .■.AB-.A'iy-.-.BD-.B'iy. 
 
 Hence M, iV, and C, the middle points oiAA', BB', and 
 IJiy, lie on a right line. 
 
 (Prop, xxxii;, and Lemma, page 103.) 
 
 Pkopqsition XXXIX. 
 
 If a quadrilateral be ciroumscrihed to an ellipse, the dia- 
 gonals will intersect on the chords of contact of the sides. 
 
 Fig. 53. 
 
 For AC will be cut by PQ, the chord of contact of the 
 tangents DA, BC, so that 
 
 AO' : OC = AH X AG : CG x CH; 
 
 (Prop. XXXI.) 
 
 but it will also be cut by the chord of contact of the 
 tangents AB, CD in the same ratio ; 
 
 .■. PQ and B8 will intersect J^C in the same point. 
 
 In like manner it may be shown that the diagonal DB 
 will pass through the intersection of PQ and RS. 
 
106 The Ellipse. [chap. ii. 
 
 Proposition XL. 
 
 The areas of the ellipse and auxiliary circle are in the 
 ratio CJB-.VA. 
 
 Let the ellipse and aijxiliary circle he divided by any 
 number of lines drawn parallel to the axis minor. 
 
 Fig. 54. 
 Then, since 
 
 FM:P'M= QN:QN=CB:CA; 
 
 {Cor. 3, Prop, viii.) 
 
 .-. quadrl. PMNQ : quadrl. FMNC^ = CB : CA. 
 
 The same will be true of aU the figures similarly 
 described in the ellipse and auxiliary circle. 
 
 Hence the sum of all the quadrilateral figures inscribed 
 in the ellipse : the sum of all the corresponding quadri? 
 lateral figures inscribed in the circle = BO : CA. 
 
 Now this will be true whatever be the number of the 
 figures. 
 
 Let the number of the figures be increased, and the 
 breadth of each indefinitely diminished. Then the sum of 
 the areas of the figures inscribed in the ellipse will be 
 
CHAP. IT. J The Ellipse. 107 
 
 equal to the area of the ellipse, and the sum of the areas 
 of those inscribed in the circle equal to the area of the 
 circle ; 
 
 .'. area of ellipse : area of circle = CB : CA. 
 
 Pkoposition XLI. 
 
 The section of a right cone by a plane, which intersects 
 all the generating lines on the same side of the vertex, 
 will be an elKpsfe. 
 
 Fig. 55. 
 
 Let the plane B VE, drawn through the axis of the cone 
 perpendicular to the plane of the section, coincide with the 
 plane of the paper ; then both the section APA' and the 
 base BPE will be perpendicular to the plane of the paper ; 
 therefore the line MP, in which the section cuts the base, 
 is perpendicular to the plane of the paper, and therefore 
 perpendicular to BE, the diameter of the base ; hence 
 
 BMxME = MF". (35, III. Euclid.) 
 
 If, now, any other plane bpe be drawn parallel to the 
 base, meeting the section in pm, it can similarly be shown 
 that mp is perpendicular to be; r. hm ^ me = mp^. 
 
108 The Ellipse. [chap. ii. 
 
 But BM : bm = A'M : A'm ; (Similar a'.) 
 
 also ME : me = MA : mA ; 
 
 .-. BM X ME :bm y. me = A'M ^ MA : A'm x mA, 
 or MP' : mp' = A'M x MA : A'm x mA. 
 
 Hence the section APA' is an ellipse. (Prop, ix.) 
 
 Fig. 66. 
 
 Gor. 1. — The axis minor is a mean proportional between 
 the diameters of the frustrum of the pone which just 
 includes the section. ' 
 
 Draw the section EBG parallel to the base, bisecting 
 AA' in C. 
 
 Then CE = I AD and CG ^ i A'D' ; (Similar A^) 
 
 .■.CEx CG or CB" = \ AD x A'D' ; 
 
 hence 4GB'^ or the square of the axis minor = AD x A'D'. 
 
 Cor. 2. — The Latus-rectum is a fourth proportional to 
 AA', AD and A'D'. 
 
 Since ABA' is an ellipse, 
 
 L X AA' = iBG' {Gor. 1, Prop, ix.) 
 
 = AD X A'D' ; 
 
 .-. AA' : AD = A'D' : L. 
 
CHAP. ii.J The ElU/pse. 109 
 
 Cor. 3.— If CF and OF' be measured off = AG ov 
 i ALT, then F and J?" are the foci. 
 
 Draw AH ± A'D'. 
 
 Then AA'^ - AD"' = AW - ED'^ 
 
 = A'JDT X AD 
 
 (Con, 5, 11. Euclid.) 
 
 = iBC' ; {Cor. 1.) 
 
 .-. CA' -AG' = £C' ; 
 
 hence C^' - BC = ^(?'- = Ci?"' = CF" ; 
 
 .-. i^and F' are the foci. (Con 3, Prop, ii.) 
 
 Cor. 4. — The spheres inscribed in a cone, to touch the 
 plane of the section, will determine the foci. 
 
 For AF=AC- CF 
 
 = AC-AG {Cor. 3.) 
 
 = A {AA' - A'D) 
 
 = i{AA' + AV.- VD-A'D) 
 
 = i{AA' + AV- VA'); 
 
 .-. l?* is the point of contact of the circle inscribed mthe 
 triangle AA'V. (See Galbraith and Haughton's Manual 
 of Euclid, Appendix, Book iv.) 
 
 In like manner it may be shown that IP' is the point of 
 contact of the cii'cle exscribed to the triangle AA' V. 
 
( 110 ) 
 
 Problems on the Ellipse. 
 
 1. The axis major is the longest straight line that can 
 be drawn in an EUipse. 
 
 Join the two foci with the extremities of the chord ; then 
 twice the chord wUl be less than the sum of the four focal 
 radii vectores, and therefore less than twice the axis major. 
 
 2. The minor axis is the least diameter of an Ellipse. 
 Connect Problem 1 with Cor. 5, Prop. xvi. 
 
 3. Given the foci of an Ellipse, describe it so as to touch 
 a given line. 
 
 See Prop. iv. 
 
 4. Find the locus of the point of intersection of any 
 tangent to an EUipse, with the line drawn from the focus 
 making a constant angle with the tangent. 
 
 If the vertex of a triangle of a given species be fixed, 
 while one base angle moves along a fixed circle, the locus 
 of the other base angle will be a circle. See also Prop. vi. 
 
 5. The angle between the tangents at the extremities of 
 a chord which passes through either focus is half the sup- 
 plement of the angle which the chor^ subtends at the 
 other focus. 
 
 Draw the normals at the extremities of the focal chord, 
 and see Cor. 3, Prop. iii. 
 
 6. Diameters parallel to supplemental chords are 
 conjugate. 
 
 See Def., page 71, and Cor. 2, Prop. xvii. 
 
 7. If a rectangle be circumscribed to an Ellipse, its 
 diagonals determine the equi-conjugate diameters. 
 
 8. Show that Prop, xxxiii. follows from Prop, xxxvi. 
 
 9. The line drawn parallel to the axis major through 
 the intersection of nonnals at the extremities of a focal 
 chord will bisect the chord. 
 
 See Prop. xiii. ; also note properties of circles inscribed, 
 exscribed, and circumscribed to a triangle. , , 
 
 10. 'WTien is the sum of two conjugate diameters a mioi- 
 mum ; and when is it a maximum ? 
 
 See Cor. 5, Prop. xvi. ; Cor., Prop, xxi., and Prop. xxix. 
 
CHAPTEE III. 
 
 The Hyperbola. 
 
 uefinitions. 
 
 A Hyperbola is the curve traced out by a point which 
 moves in such a way that its distance from a fixed point 
 is to its distance from a fixed right line in a constant ratio 
 E : 1 (£ being greater than unity) . 
 
 The fixed point is called the Focus, and the fixed right 
 line the Directrix. 
 
 For Definitions of Axis, Vertex, Centre, Diameter, Tan- 
 gent, Normal, Chord, Ordinate, Abscissa, Latus rectum, &c., 
 see Definitions, Chapters I. and II. 
 
 Conjugate Diameters may be defined, as on page 72. 
 
 Sxippkmental Chords, Transverse Axis, Conjugate Axis, as 
 
 on page 71. , 
 
 Two hyperbolas are said to be conjugate when the 
 
 transverse axis of each is the conjugate axis of the other- 
 
 If the axes be equal the hyperbola is said to be Meet- 
 angular or Equilateral. 
 
 Tangents to the curve whose points of contact are at 
 infinity are called Asymptotes. 
 
112 The Hyperbola. [chap. in. 
 
 Peoposition I. 
 
 The focus, directrix, and eccentricity of a hyperbola 
 being given, to determine any number of points on the 
 curve. 
 
 Let i^be the focus, and Oy the directrix. 
 
 Draw FO _L the directrix. Divide FO internally at A, 
 and externally at A', so that FA : AO = FA' : A'Oin the 
 given ratio e : 1; then A and A' are the vertices of the 
 curve, and AA' the axis. 
 
 On the directrix take any point p ; join FP, and pro- 
 duce it ; draw AH, A'H' perpendiculars to AA', meeting 
 i^ in JJ, H'. On HH' as diameter describe a circle; 
 through p draw PpP' \\ to the axis, cutting the circle in 
 P, F ; join PF, FF. 
 
 Then, since AH is || Op; .-. HF : Hp = AF : AO. 
 Also, since A'H is || Op; .-. H'F i H'p = A'F : A'6 ; 
 
 .■.H'F:H'p = HF:Hp. 
 
CHAP. iii.J The Hyperbola. 113 
 
 Hence PF : Pp = FH : Hp (Lemma, page 47.) 
 
 = FA:AO=t:l. (2, VI. Euclid.) 
 
 Also FF : Fp = FH : Hp (Lemma, page 47.) 
 
 ^FA:A0 = ,:1. (2, VI. Euclid.) 
 
 .•. P and P' are points on the curve. 
 
 In like manner, by taking other points' on the directrix, 
 any number of points on the curve may be determined: 
 
 CoTi 1. — If Op' betaken equal to Op, and another point 
 Q found in a similar manner to P, it is obvious that Qp' 
 ■wiU be equal to Pp. Therefore, corresponding to any 
 point P on the curve, at a perpendicular distance from the 
 axis = Op, there is another point Q on the other side of 
 the axis, at a distance from it = Op, . and also at the same 
 distance as P from the directrix ; hence the curve is sym- 
 metrical with regard to the axis. 
 
 Co^. 2.— Bisect ^^' in C; through Odraw CL8 J. AA', 
 and .-. J. PF, meeting HH' in 8, and PP' in L. 
 
 Then, since AH, 08 and AH' are parallel, .-.SH^ SH; 
 
 .-. 8 ia the centre of the circle H'FHP ; 
 
 .. LP = LF. (3, III. Euclid.) 
 
 Therefore, corresponding to any point P on the curve, 
 there is another point F on the other side of CL, situated 
 in precisely the same manner with regard to it as P ; hence 
 the curve is symmetrical also with regard to CL. 
 
 If therefore CCf be measured off = CO, and CF' = CF, 
 and O^Y' be drawn 1. ffC, the curve could be equally 
 well described with F' as focus, and ffY' as directrix. 
 
 Cor. 3.— Since PF : Pp ^ AF : AO = A'F : A'O ; 
 
 .-. PF:Pp = A'F- AF : A'O - AO 
 
 ^ 2CA : 2C0; 
 
 . . PF/. Pp^CA: CO. 
 
 1 
 
114 
 
 The Hypm-hola. 
 
 [chap. Ill, 
 
 That is, the distance of any point on the curve from the 
 focus is to its distance from the directrix as CA : CO. 
 
 Cor. 4.— Since A'F : A'O = AF: AO; 
 
 .-. A'F+ AF : A'F- AF = A'0 + AO : A'O - AO ; 
 
 or 20F : 2CA = 20A : 200 ; 
 
 .: CFx 00= 0A\ 
 
 Oor. 5. — From the symmetry of the curve it is evident 
 that any line drawn through to meet the curve will be 
 bisected at that point ; hence is the centre of the curve. 
 
 Proposition II. 
 
 The difference of the distances of the foci from any 
 point on a hyperbola is equal to the transverse axis. ' 
 
 Fig: 2. , 
 
 PF' : P/ = OA : 00; {Oor. 3, Prop. i. 
 also PF : Pp = OA : CO ; 
 
 .-. PF' - PF : Pp' -Pp=OA:00; 
 but Pp -Pp=Oa = 200 ; 
 
 ... PF' - PF=20A =AA'. 
 
CHAP. III. J The Hyperbola. 115 
 
 Cor. 1. — By the help of the preceding theorem we can 
 construct a hyperbola mechanically. 
 
 If two of the extremities of two threads, the difference 
 of whose lengths is equal to AA', be fastened to the two 
 fixed points A, A', and the other extremities, having been 
 passed through the eye of a needle, be joined, it is 
 obvious that the eye of the needle, moved about so as to 
 keep the threads always stretched, will describe a hyber- 
 bola whose foci are F and F'. 
 
 Cor. 2. — The difference of the distances of any point 
 from the focii of a hyperbola is greater or less than the 
 transverse axis, according as the poiut is on the concave 
 or convex side of the curve. 
 
 Fig. 3. 
 
 First let! Q be any point on the concave side of the 
 curve. 
 
 Join QF, QF', and let QF' meet the curve in P ; 
 join PF. 
 
 Then 4F" = QP + PF', 
 
 and QF <QP + PF; (20, 1. EucM.) 
 
 .-. QF'- QF>PF' -PF; 
 
 but PF' -PF^ AA' ; .-. QF' -QF> AA'. 
 
 i2 
 
116 The Hyperbola. [chap. hi. 
 
 Next, let Q' be on the convex side. 
 
 Join QF, <^F' ; produce QF' to meet the curve in P ; 
 join PF. 
 
 Then F'Qf = F'F - PQf, 
 
 and ^F> PF~PQ; {Cor., 20, I. Euclid.); 
 
 .-. ^F'-&F<PF'-PF; 
 
 but PF' -PF= AA' ; .-. QF' -QF< AA'. 
 
 Cor. 3, — Conversely, a point will be on the concave or 
 convex side of a hyperbola, according as the difference, of 
 its distances from the foci is greater or less than the 
 transverse axis. 
 
 Pkoposition III. 
 
 1°. The line which bisects the angle formed by drawing 
 lines from any point P on a hyperbola to the foci lies 
 altogether without the curve. 
 
 2°. Any other line drawn through the. point P wiU cut 
 the curve. 
 
 Fig. 4. 
 
 1°. Let PS" bisect the / between Pi?'and PF'. 
 Cut off PR = PF; join F, F' and R with any point Q 
 on the line PH. • 
 
CHAP. iii.J The Hyperbola. 117 
 
 Then the a' QlFR and QJ2P are equal ; (4, 1. EuclidO 
 .-. QE^QF; 
 but F'Q -QR< F'R ; {Cor., 20, 1. Euclid.) 
 
 .-. F'Q - QF< F'R 
 
 < F'P - PR 
 
 < F'P - PF (Const.) 
 
 < AA' ; (Prop, ii.) 
 
 .'. Q is a point without the curve. {Cor. 3, Prop, ii.) 
 
 2°. Let PK he any line other than the bisector of 
 ihQLFPF'. 
 
 Pig. 5. 
 Draw PL, making the L KPL = L KPF; out off 
 
 PL = Pi?: 
 
 Join F'L ; produce F'L to meet P^in Q ; join QP. 
 Then the a» FPQ, and iPQ are equal ; (4, 1. Euclid.) 
 
 .-. QF=QL; 
 
 but F'L > F'P - PL 
 
 (Cor., 20, 1. Euclid.) 
 
 > F'P - PF 
 
 > AA' ; (Prop, n.) 
 
 .-. F'Q - QL or F'Q -QF> AA' ; 
 
 .■. Q is a point on the concave side of the curve. 
 
 {Cor. 3, Prop, ii.) 
 
118 The Hyperbola. [chap. hi. 
 
 Cor. 1. — Hence the Una which bisects the angle between 
 lines drawn from any point on a hyperbola to the foci is 
 a tangent. 
 
 Cor. 2. — The tangent at the vertex is perpendicular to 
 the axis. 
 
 Cor. 3. — The normal to a hyperbola at any point bisects 
 the external angle between the focal radii vectores drawn 
 to the point. 
 
 Proposition IV. 
 
 The locus of the foot of the perpendicular let fall from 
 either focus on any tangent to a liyperbola is a circle 
 described on the transverse axis as diameter. 
 
 Fig. 6. 
 
 Let PT be any tangent, and FT a perpendicular let fall 
 from the focus. 
 
 Produce FT to meet F'P in R ; join TC. 
 
 Then the a» FTF and BTF are equal ; (26, 1. Euclid.) 
 
 .-. FT= TR. 
 
 Now, in A F'FR, F'F is bisected in C, and FR in T; 
 
CHAP. iii.J The Hyperbola. 1 19 
 
 .-. CT=\FR = l{F'P-PR) 
 = ^{F'P-PF) 
 = i AA' (Prop. 11.) 
 
 = CA. 
 
 Hence the locus of T is a circle described with the 
 centre C and radius CA. 
 
 Cor. 1. — If the vertex of a right angle FTL move along 
 a fixed circle, while one leg passes through a fixed point F 
 without that circle, the other leg wiU always touch a 
 hyperbola. 
 
 Draw the diameter of the circle passing through F; cut 
 off CF' = CF; produce J'T till TR^ TF; join F'R, and 
 produce it to meet TL in P ; then P is the point at which 
 TL wiU touch the hyperbola. 
 
 Join FP, CT. 
 
 Then the a' FTP and RTP are equal ; (4, 1. Euclid.) 
 
 .-. FP = RP. 
 
 Also, since FT = TR and FC = CF' ; 
 
 .-. CT \\ F'R and = ^ F'R; 
 
 ... PF' - PF = PF' - PR 
 
 = F'R 
 
 = 2CT 
 
 = 2CA; 
 
 hence P is a point on the hyperbola of which F and F' are 
 the foci, and AA' the transverse axis. 
 
120 The Hyperbola.. [chap. ni. 
 
 Cor. 2. — If FT become a tangent to the circle, then CT 
 will be i. FT, iand .•. in directum with TL ; and .•. F'R, 
 which is II CT, will meet TL at infinity ; heno e CL is an 
 asymptote to the curve. 
 
 Fig. 7. 
 
 If FT' be the other tangent to the circle drawn from F, 
 then CT' will also be an asymptote. 
 
 Cor. 3.— If TO be drawn J. the axis, then TO is the 
 directrix. 
 
 For CFxCO= CA' ; (8, VI. EucM.) 
 
 .■. is a point on the directrix. • {Cor. 4, Prop, i.) 
 
 Cor. 4. — If AR be drawn ± CF, meeting CL in H, and 
 SB be drawn || CF, meeting CB drawn J. CF in B, then 
 CB is the semi-conjugate axis. 
 
 For the a" CFT and CHA are equal ; (26, I. Euclid.) 
 
 .-. TF =AH=CB; 
 
 ... CB' = FT' = FA X FA'. 
 
 (36, III. Euclid.) 
 
CHAP. III.] The Hyperbola. 121 
 
 Hence the point B, together with the corresponding 
 point B^, obviously correspond to the extremities of the 
 axis minor in the Ellipse. (See Cor. 3, Prop, ii., page 51.) 
 
 Cor. 5. CF'' =. CA' + CR. 
 
 For CJf' = CH^ = CA^ + AH^ = CA^ + CW. 
 
 Cor. 6.— If we cut off ^and Cf each = CF or CE, 
 and consider/, /' the foci, and BW the transverse axis of a 
 hyperbola, then AA' will be its conjugate axis. 
 
 For the square of its semi-conjugate axis 
 
 = Cr- CW {Cor. 5.) 
 
 = CH' - AH' 
 
 = CA\ 
 
 The hyperbola described with BB" as the transverse 
 axis, and /, /' as foci, is called the hyperbola conjugate to 
 the original hyperbola. 
 
 Cor. 7. — The Unes joining the extremities of the axes 
 are bisected by one asymptote and parallel to the other. 
 
 For CASB is a rectangle ; .'. AB is bisected by CM. 
 
 Also, CB^AH= AH' and CB \\ AS' ; 
 
 .-. ^J5 II CS'. 
 
 (33, 1. Euclid.) 
 
 Cor. 8. — The perpendicular from the focus on either 
 asymptote is equal to the semi-conjugate axis. 
 
 For A CAH= A CTF; (26, I. EucM.) 
 
 .■.ft=ab:=cb. 
 
 Note. — When the axis only is mentioned it is always 
 understood to be the transverse axis. 
 
122 The Hyperbola,. [chap. m. 
 
 Proposition V. 
 
 The distance of any point P on the curve from the focus 
 J^is equal to the length of the line Pq drawn from the 
 point parallel to an asymptote to meet the directrix. 
 
 Pig. 8. 
 
 On the transverse axis as diameter describe a circle. 
 Draw the tangent i^T; join CT, and draw TO J. CF. 
 Then OT is the directrix, and CT an asymptote. 
 
 {Cors. 2 & 3, Prop, iv.) 
 Also Pq.Pp = CT: 00 (Similar a».) 
 
 = CA:CO 
 
 = PF : Pp. {Cor. 3, Prop, i.) 
 Hence, PF^Pq. 
 
 Cor. — Hence, being given the eccentricity of a hy- 
 perbola, we can find the angle between the asymptotes. 
 
 For CT:CO = Pq: Pp (Similar a».) 
 
 = PF: Pp 
 = £ : 1. 
 
 Now in the A CTO we know L&t = 90° ; also the ratio 
 of the sides {CT : CO) ; hence the / TOO, or half the 
 L between the asymptotes, is known. 
 
CHAP. III. J The Hyperbola. 123 
 
 Proposition VI. 
 
 The rectangle under the focal perpendiculars on any- 
 tangent to a hyperbola is equal to the square of the semi- 
 conjugate axis. 
 
 rig- 9- 
 
 Let FT, F'T', be perpendiculars from the foci on the 
 tangent at any point P- 
 
 Join TC, and produce it to meet F'T' in S. 
 
 Then the a » FCT and F'CS are equal ; 
 
 (26, 1. EucM.) 
 
 .-. F'S = FT, and CS=CT=CA; (Prop, iv.) 
 
 .•. S is a point on the auxiliary circle ; 
 
 .-. A'F' X F'A = F'T' X F'8 (35, III. Euclid.) 
 
 = F'T' X FT; 
 .-. F'C - CA" = F'T' X FT, (5, II. Euclid.) 
 
 or ' BO' = F'T' X FT. {Cor. 5, Prop, iv.) 
 
124 The Hyperbola. [chap. m. 
 
 Proposition VII. 
 
 The locus of the intersection of tangents to a hyperbola 
 which out at right angles is a circle. 
 
 Fig. 10. 
 
 Describe the auxiliary circle cutting the tangent QP in 
 T and T', and the tangent QF in H and H'. Join FH, 
 FT, FH', FT', QP; produce ^JC to meet the auxiHary 
 circle in K and L. ' 
 
 Then FT,. F'T, are ± Qi*, and .-. || QP' ; 
 
 (Prop. IV.) 
 
 and FH, F'H' 1. QF ; (Prop, iv.) 
 
 hence F'T' = QH, and FT= QH. (34, I. EucM.) 
 
 Now CQ' = CL^ -KQx QL (5, II. Euclid.) 
 
 = CL' - QH' X QH (35, III. EucM.) 
 
 = CA' - FT X F'T' 
 
 = CA' - BG\ (Prop. V.) 
 
 Hence the locus of Q is a circle described with the centre 
 C, and is called the director circle of the hyperbola. 
 
CHAP. III. 
 
 The Hyperbola. 
 
 125 
 
 Pboposition VIII. 
 
 If PiVbe a normal at any point P on a hyperbola, and 
 PM an ordinate to the transverse axis ; then 
 
 CA": CF':: CM: ON. 
 
 Fig. 11. 
 
 Join PF, PF' ; draw Ppp' || the axis, meeting the 
 directrices in p, p'. 
 
 Since PN bisects the external z between PF and PF' ; 
 
 {Cor. 3, Prop, iii.) 
 
 .-. F'N:NF=PF':PF 
 
 (Prop. B., VI. Euclid.) 
 
 = Pp' : Pp. 
 
 Comp. F'N+ NF : F'N- NF = Pp' + Pp : Pp' - Pp ; 
 
 .-. 2CN:2CF=2CM:2CO. 
 
 Alt. CN:CM=CF:CO 
 
 = CF' : CF X CO 
 
 = CF' : CA\ 
 
 {Cor. 4, Prop, i.) 
 Cor. CM : MN = CA' : CE". 
 
 For CM:CN-CM= CA' : CF' - CA' ; 
 
 or CM:MN=CA':CB'. 
 
 {Cor. 4, Prop, iv.) 
 
126 
 
 The Hyperbola. 
 
 [chap. III. 
 
 Proposition IX. 
 
 If a tangent at any point P of a hyperbola be produced 
 to meet the axis in T, and the ordinate PM be drawn ; then 
 CA will be a mean proportional between CM and OT. 
 
 or 
 
 Fig. 12. 
 Through P draw Ppp' \\ to the axis ; join PF, PF'. 
 
 Then, since PT is the bisector of the z F'PF; 
 
 {Cor. 1, Prop, ni.) 
 .-. F'T.TF= F'P : PF (3, YI. Euclid.) 
 
 = Pp' : Pp. 
 Comp. and div. 
 
 F'T+ TF: F'T -TF= Pp' + Pp : Pp' - Pp ; 
 2CF:2CT=2CM:2CO; 
 .-. CTx CM= CFx CO 
 
 = CA\ {Cor. 4, Prop, i.) 
 
 Cor. 1. — If the auxiliary circle be described, and TQ 
 drawn ± the axis ; then QM is a tangent to the circle. 
 Join CQ. 
 
 Since CT : CA = CA : CM; 
 
 . CT:OQ=CQ: CM; 
 
 .-. the a' CQT and CMQ are similar ; 
 
 (6, VI. Euclid.) 
 .-. L CQM^L CTQ ; .-. QM is a tangent. 
 
CHAP, in.] 
 
 The Hyperbola. 
 
 127 
 
 Cor. 2. — If from the foot of any ordinate of a hyper- 
 bola a tangent be drawn to the auxiliary circle, this tangent 
 is to the ordinate in a constant ratio. 
 Draw the normal FN. 
 
 Then, since the a TFN\s. right-angled, and FM j. TN\ 
 
 .-. TMy.MN=FM\ (8, VI. Euclid. ) 
 
 Also, since a CQM\s, right-angled, and QT _L CM; 
 
 :. CMxTM= QW 
 
 Hence QM' : FM^ = CM : MN 
 
 = CA' : CB' ; 
 
 {Cor., Prop, vni.) 
 
 .-. QM:FM=CA: CB. 
 
 Cor. 3. — If QA be drawn and produced to meet FM 
 produced in K, then 
 
 ME'' MQ, and FM: MK = CB : CA. 
 
 Fig. 13. 
 
128 The Hyperbola. [chap. hi. 
 
 For L MQA = z QA'A (32, III. EudM.) 
 
 = I AQT (13, VI. Euclid.): 
 - L ARM; 
 .-. MK=MQ; 
 Also PM:QM=CB:CA; 
 
 .-. PM-.MK^ CB: CA. 
 
 Cor. 4. — The curve is concave towards the transverse 
 axis. 
 
 (See Fig. 13.) 
 
 Draw any, line, perpendicular to the axis between the 
 vertex A and the point P ; let it meet the curve in jo, A.P in 
 S, the axis in m, and QA produced in r. From m draw the 
 tangent mq to the aiixiliary circle ; draw qn i to the 
 axis ; join np ; produce qA to meet pr produced in A. 
 
 Then PM : MK= CB : CA =pm : mk ; {Cor. 2.) 
 
 but PM : MK = sm : rm ; (Similar A'.) 
 
 .•. pm : mk = sm : rm. 
 
 Alt. pm : sm^ mk : rm 
 
 = qn : vn; 
 
 but qn is > vn by property of the circle ; 
 
 .". pm is > sm. 
 
 That is, if a perpendicular be drawn to the transverse 
 axis produced, between the vertex A and any point P on 
 the branch of the curve of which A is the vertex, the 
 segment intercepted on it between the curve and the 
 transverse axis is greater than the segment intercepted on 
 it between the line AP and the transverse axis; and, 
 therefore, the curve is concave towards the transverse axis. 
 
CHAP, ni.] 
 
 The Hyperbola. 
 
 129 
 
 Proposition X. 
 
 The rectangle under the segments of the axis made by 
 any ordinate is to the square of the ordinate in a constant 
 ratio. 
 
 See Fig. 12. 
 
 For PJf » : QM^ = CE" : CA' ; {Cor. 2, Prop, ix.) 
 but QM^ = MAx MA' ; (36, III. Euclid.) 
 
 .-. PM^ : MA X MA' = CE" : CA\ 
 
 Cor. 1. — The Latus rectum is a third proportional to 
 the transverse and conjugate axes. 
 
 ^\' :AFxFA'= CE: CA'; 
 
 
 ■■•©■ 
 
 : CE = CB' : CA\ 
 
 Hence 
 
 CA 
 
 ■.CB=CB:^. {Cor. 4, Prop.' iv.) 
 
 Cor. 2. 
 
 
 CR = CM' X cr. 
 
 For 
 
 CM' 
 CM' - CA' 
 PM' 
 
 Fig. Hv 
 
 CA'=CM:CT; (Prop, ix.) 
 
 CA'= CM- CT: CT. 
 
 CE = TM: CT (Prop, x.) 
 
 '^PM: CT (Similar a'.) 
 
 = PM' : PM X CT' ; 
 CB' = CM' X CT'. 
 
 K 
 
130 The Hyperbola. [chap. hi. 
 
 Peoposition XI. 
 
 The locus of the intersection of any tangent to a hyper- 
 bola, with the line drawn through the focus perpendicular 
 to the radius vector drawn from the focus to the point of 
 contact of the tangent, is the directrix. 
 
 Fig. 15. 
 
 Let PT be any tangent, and FT J. FP. 
 Draw TH ± PF', and TO 1. the axis. 
 The a' PFT and PET are equal. (26, I. Euclid.) 
 
 .-. FT = TH, and PH = PF; 
 
 .■ . F'H = F'P -PH= F'P -PF= AA'. 
 
 (Prop. II.) 
 Now F'm = F'T"^ - TH^ (47, I. Euclid.) 
 
 = F'T^ - TF^ 
 = F'O" - OF^ ; 
 .-. AA'^ = ACF X CO ; {Cor. 6, II. BuoUd.) 
 .-. CA^^CFx CO. 
 
 Hence OT is the directrix. (Cor. 4, Prop, i.) 
 
 Cor. — Conversely, if from any point on the directrix 
 
 a tangent be drawn to a hyperbola, the line joining that 
 
 point to the focus is perpendicular to the radius vector 
 
 drawn from the focus to the point of contact of the tangent. 
 
xIAP, 
 
 III.J 
 
 The Hyperbola. 
 
 131 
 
 Proposition XII. 
 
 If any chord PP' of a hyperbola cut the directrix in D, 
 and if JF* be the focus corresponding to the directrix, on 
 which D is situated ; then FD is the external bisector of 
 the LPPr. 
 
 Fig. 16. 
 
 Draw Pp, P'p' ± the directrix ; produce P'F. 
 Then PF.Pp =FF: Pp'. 
 
 Alt. PF:PF=Pp:Fp' 
 
 ^PB-.TJ); (Similar A'.) 
 .-. FD bisects the /.PFQ. (Prop, b., TI. Euclid.) 
 
 Cor. 1. — Hence, being given one focus F, and three 
 points P, P', P", on a hyperbola, we can find the direc- 
 trix and axes. 
 
 For, draw FI) bisecting the angle between PF and P'F 
 produced ; the point where this fine meets P'P produced 
 wiU be one point on the directrix. Similarly, by bisecting 
 the angle between PF&ni P"i^ produced, another point 
 on the directrix may be found ; and hence the directrix 
 and axes. 
 
 k2 
 
132 The Hyperbola. [chap. hi. 
 
 If a tangent be defined as the line joining two indefi- 
 nitely near points on the ciirve, it will follow immediately 
 from the proposition that the right line drawn from the 
 focus to the point of intersection of any tangent with the 
 directrix is perpendicular to the radius vector drawn from 
 the focus to the point of contact of the tangent ; also, that 
 the tangent bisects the angle between the radii vectores 
 drawn from the foci to the point of contact. 
 
 For when P is indefinitely near to F, PD becomes a 
 tangent at the point P, and the I PPP is indefinitely 
 small (see Fig. 16) ; 
 
 .-. I PFQ = 180°, 
 
 but lPFD^^lPFQ; 
 
 .-. I PFD =90°. 
 
 Also (see Fig. 17), 
 
 Fig. 17. , 
 PF-.P'F' = Pp:Pp' 
 
 = PB : Pjy, (Similar a~.) 
 
 and / PFL = PF'K, each being 90°; 
 
 .-.A PFB is similar to A PFIf; 
 hence l FPD = z F'PI/. 
 
CHAP. III.] The Hyperhota. 133 
 
 Proposition XIII. 
 
 If two fixed points, P, P', on a hyperbola be joined 
 with a third variable point 0, the segment pp' intercepted 
 on either directrix by the produced chords subtend a con- 
 stant angle' at the focus corresponding to that directrix. 
 
 Fig. 18. 
 
 Since Fp is the external bisector of the L PFO ; 
 
 (Prop. XII.) 
 
 .-. i.OFp-,\LOFP = m°. 
 
 Also Fp' is the external bisector of the L OFP' ; 
 
 (Prop-. XII.) 
 .-. lOF^' + \lOFF = Q0°; 
 
 .-. L OFp + \tOFP = L OFp' + lL OFF ; 
 
 .-. LOFp-L OFp' = i (z OFF - L OFP) 
 
 or LpFp' ^\L PFF, and .•. constant. 
 
 Cor. — The anharmonic ratio of the pencil formed 
 by joining four fixed points on a hyperbola to any fifth 
 variable point is constant. 
 
 For . PFF'F" = . pp'fp'" = F . pp'p'p'". 
 
134 The Hyperbola. [chap. hi. 
 
 Proposition XIV. 
 
 The line joining the focus to the intersection of two 
 tangents to a hyperbola bisects the angle which the points 
 of contact subtend at the focus. 
 
 Fig. 19. 
 
 Let QP, OF be tangents; join PF, PF', FF, FF. 
 Cut off PR = PF, and FR = FF; join Q with R, F, 
 F,F'. 
 
 Then the a» QFP and QRP are equal ; 
 
 (4, 1. Euclid.) 
 •. QR = QF. 
 
 Again, the a' QFF and QR'F are equal ; 
 
 (4, 1. Euclid.) 
 .-. QR'= QF; 
 
 hence QR = QR' ; 
 
 but F'R=^F'P-PR 
 
 = F'P-PF 
 
 = FF'-FF (Prop. XI. j 
 
 = FF'-FR' 
 
 = F'R'. 
 
 Hence the a'RF'Q and R'F'Q are equal ; 
 
 (8, 1. Euclid.) 
 .-. iRF'Q = iRF'Q. 
 
CHAP, iii.j The Hyperbola. 135 
 
 Cor. — It is obvious from the above demonstration, also, 
 that 
 
 lPFQ=lPRQ=lQRF 
 = L QFF. 
 
 Pkoposition XV 
 
 The angle subtended at either focus by the segment in- 
 tercepted on a variable tangent to a hyperbola by two 
 fixed tangents is constant. 
 
 Fig. 20. 
 
 Let QP, QP' be the two fixed tangents, and PS the 
 variable tangent. 
 
 Join FP, F8, FT, FB, FP'. 
 
 Then z TF8 = ^ L TFP; also t TFR = i z TFF; 
 
 (Prop. XIII.) 
 
 .-. z TF8 + L TFR = i z TFP + i z TFP\ 
 or Z RFS = a constant. 
 
 Cor. — The anharmonic ratio of the four points in 
 which a variable tangent to a hyperbola is cut by four fixed 
 tangents is constant. 
 
 For the segments intercepted on the variable tangent 
 subtend constant angles at the focus. 
 
136 The Hyperhoh. [chap. in. 
 
 Proposition XVI. 
 
 If a tangent at the extremity of any diameter meet the 
 axis in T, the area of the triangle OPT thus formed will 
 be equal to the area of the triangle CAH, formed by draw- 
 ing a tangent at the vertex. 
 
 Fig. 21. 
 
 Draw the ordinate PM; join PA, TH. 
 
 Then 
 
 TO: CA^CA: CM (Prop, ix.) 
 
 
 = CH:CP; (ParaUel lines.) 
 
 
 .-. PAi^WET; (2, VI. Euclid.) 
 
 
 .-.A PHT= A HTA ; (37, 1. Euclid.) 
 
 hence 
 
 A CPT = A CAH. 
 
 Cor. 
 
 ^ PMT = tm^. PMAS. 
 
 If we take the equal A' CPT and CAS, in succession 
 from the A CPM; 
 
 then A PMT = trap. PMAH. 
 
CHAP. 111.] The Hyperhota. 1-37 
 
 Proposition XVII. 
 
 If an ordinate to the axis drawn from any point P on 
 a hyperbola be produced to meet the asymptotes in D and 
 B', then PD x PD' = square of semi-conjugate axis. 
 
 Fig. 22. 
 
 For MB" : MC = AH' or BC':AC'; 
 
 (Similar a'.) 
 
 but PM' : MC - CA' = PC' : CA' ; (Prop, x.) 
 
 .-. MD' - PM' : CA"" = PC : CA' ; 
 
 hence MB' - PM' = PC, 
 
 or PD X PB' = PC (5, II. Euclid.) 
 
 Note. — In the figures connected with some of the fol- 
 lowing Propositions, in order to avoid confusion, occasion- 
 ally lines are not actually drawn : when a letter is omitted, 
 reference to any preceding figure, where the same letter 
 has been used, will at once indicate the point. 
 
138 
 
 The Hyperbola. 
 
 [chap. III. 
 
 Proposition XVIII. 
 
 The segments PQ, P'Qf intercepted on any line between 
 the curve and its asymptotes are equal. 
 
 Fig. 23. 
 
 Through P and P' draw _L" to the axis. 
 
 Then QP : PR = QF : P 8, (Similar A*.) 
 
 and PQ' : Pr = QF : F8' ; 
 
 .-. QP xPQ :PRxPr=QP' X QfF : FS x P'S'; 
 but PR- Pr = BC^ = F8xFS' (Prop, xvii.) 
 
 .-. QP V pq = QF X Q'P', 
 
 or QPxPF + QPx qF = QF - PF + QF x QP ; 
 
 (1, II. Euclid.) 
 .-. QPxPF = QF' X PP', 
 
 or 
 
 QP = QF. 
 
CHAP. iTi.] The Hyperbola. 139 
 
 Cor. 1. — If a line drawn from C through the middle 
 point of QQ meet the curve in i), then the line drawn 
 through D parallel to QQ' will be a tangent. 
 
 For, suppose it meet the curve again in If, and let it 
 meet the asymptotes in E and E'. 
 
 Then, by this Proposition, DE = D'E'. 
 
 But in the A QCQ', since QQ' is bisected in G, and 
 EE' II QQ; 
 
 .-. DE = DE'. 
 
 Hence BE' = IfE', which is impossible ; .'. &c. 
 
 Cor. 2. — Any line PP' is an ordinate to the diameter 
 passing through its middle point, and the portion of any 
 tangent intercepted between the asymptotes is bisected at 
 its point of contact. 
 
 Cor. 3. — If any line be drawn parallel to PP' or EE', 
 it is obvious that the portion intercepted between the 
 asymptotes will be bisected by the diameter CD ; but the 
 segments intercepted between the curve and the asymp- 
 totes are also equal, and hence any diameter bisects all its 
 ordinates. 
 
 Cor. 4. PQ X Pq = DE\ 
 
 For, if through D a perpendicular be drawn to the axis, 
 m eeting the asymptotes in K, K'. 
 
 Then BK : DE = PP : PQ ; (Similar a'.) 
 
 also BK':BE' = Pr:Pq; ( Do. ) 
 
 .-. DK^ BK' : BE x BE'= PExPr.PQx PQ' ; 
 
 but DK X BK' = BC = PRxPr; (Prop, xvii.) 
 
 .-. PQ X PC^ = BE X BE' 
 
 = BE\ (See Cor. 1.) 
 
 Cor. 5. — If through any point on a hyperbola a line be 
 drawn in a constant direction, the rectangle under the 
 segments intercepted between the point and the asymptotes 
 is constant. 
 
140 
 
 ITie H'lfperbola. 
 
 [chap. rn. 
 
 Proposition XIX. 
 
 If CP be the semi-diameter conjugate to CP, and tan- 
 gents be drawn at P, P', meeting the transverse and- con- 
 jugate axes in T, T' respectively ; TQ, T'Q perpendicu- 
 lars to the axes meeting the circles described on the axe_s as 
 diameters in Q, Q ; then L QCA = L Q'CB. 
 
 Fig. 24. 
 
 Join QM, Q'm'. 
 
 Then QM, Q'«j' are tangents to the circles at Q, Q', 
 respectively. (Cor. 1, Prop, ix.) 
 
 PM:MQ=CB: CA, {Cor. 2, Prop, ix.) 
 
 and P'm':m'Q'=CA : CB; 
 
 .-. Fm'xPM= Q'm X QM. 
 
CHAP. III.] The Hyperbola. 141 
 
 But since CF is || PT; 
 
 .-. PM:MT=Cm' -.m'P; 
 .-. PM xm'P'= Cm' xMT; 
 .-. Q^m' X QM = Cm' x MT; 
 .-. Cm' : m'Q=QM:MT 
 = CM:MQ; 
 
 (8, VI. Euclid.) 
 hence the a '»«'CQ' and ilfCQ are similar ; 
 
 .^ (7, VI. EucKd.) 
 and .-. iQCA = lQ'CB. 
 
 Cor. 1. — :CP is parallel to the tangent at P'. 
 
 For /-w.^-r.j^^-;*! 
 
 [m Q : m T ) 
 
 _ iCA:CB I {Cor. 2, Prop, ix.) 
 " ICm'-.m'Q'] (8, VI. Euclid.) 
 
 ^{QM: MPi (Cor. 2, Prop, ix.) 
 
 \CM:Qm] (Prop. XIX.) 
 
 = CM;MP= Pm: Cm; 
 
 .-. A' P'm'T' and CPmave similar, and .-. CP \\ PT. 
 Hence, if one diameter CF' of an hyperbola be con- 
 jugate to another CP, then, conversely, CP will be con- 
 jugate to CP'. 
 
 Cor. 2. PM = Q'm' = Cm ; and P'm'- =-QM= CM'. 
 Since the a' Cm'Qf and CMQ are similar, 
 m'Q' :MQ^CQ': CQ 
 = CB : CA. 
 But PM:MQ=CB:CA;{ Cor. 2, Prop. ix. ) 
 
 .-. m'Q:MQ = PM:MQ; 
 .-. m'Q = PM. 
 Also Fm' : m'q= CA : CB 
 
 = QM:Q'm'; (SimHar A'.) 
 .-. Fm' ^QM. 
 
142 ine Hyperbola. [chap. iir. 
 
 Cor. 3. OM^ - CM" = CA\ 
 
 For CA' = CQ' = CM^ - QM^ = CM' - CM". {Cor. 2.) 
 
 Cor. 4. P'M" - PM' = CB". 
 
 For CB' = CQ" = Cm" - m'Q" = PM" - PM". {Cor. 2.) 
 
 Cor. 6. — ^The difference of the squares of any pair of 
 conjugate semi-diameters is equal to the difference of the 
 squares of the semiaxes. 
 
 For (X4' - CB" = CM' - CM" - {P'M" - PM") 
 
 {Cors. 3 and 4.) 
 = CM' + PM' - {CM" - P'M") 
 = CP' - CF'. 
 Cor. 6. PM : CM' ^ CB : CA = P'M' : CM. 
 For PM:QM = CB:CA; {Cor. 2, Prop, ix.) 
 
 but QM = m'P' = CM' ; 
 
 .-. PM: CM'=CB: CA. 
 Again, Cm' : CM = CQ' : CQ; (Similar A'.) 
 
 .-. FM':CM=CB: CA. 
 Cor. 7.— The A* CPilf and CFM' are equal in area. 
 For PM>^ CM = P'M' x CM'. {Cor. 6.) 
 
 Cor. 8. Mm' is parallel to an asymptote. 
 For Cm' iCm^CQiCQ (Similar a'.) 
 
 = CB:CA; 
 
 .*. Mm' is II AB, and .*. || an asymptote. 
 
 {Cor. 7, Prop. iv») 
 Cor. 9. — If MP, m'P' be produced to meet in 0, then 
 CO will be an asymptote. 
 
 Since CMOm' is a parallelogram, CO will bisect Mm', 
 and .*. it will bisect AB, which is || Mm'. 
 
 Hence CO is an asymptote. {Cor. 7, Prop, iv.) 
 
CHAP. III.] The Hyperbola. 143 
 
 Proposition XX. 
 
 If CP, CF be conjugate semi-diameters, PM, PM' 
 ordinates to tlie axis, PT a tangent at P ; then 
 
 A CVM' = A PTM, and a CP'F= a CPT. 
 
 Fig. 25. 
 
 For A CPM: a CHA : a CVM'=CM^: CA' : CM" ; 
 
 (19, VI. Euclid.) 
 
 .-. A CPM- A CHA : A CVM' = CM' - CA' : CM"; 
 
 but CM' - CA' = CM" ; {Cor. 3, Prop, xix.) 
 
 .. ACPM-^CHA = ACrM', 
 
 or trap. PiLi^ = A CFilf' ; 
 
 .-. APTM^^ACVM'. 
 
 {Cor., Prop. XVI.) 
 
 Again, A CPJf = A C'P'Jf' ; 
 
 {Cor, 7, Prop, xix.) 
 
 .-. A CPilf - A PTJf = A CFM' - A OFJf' ; 
 
 . . aCPV= a CPT. 
 
144 
 
 The Hyperbola. 
 
 [chap. III. 
 
 Proposition XXI. 
 
 If QC be an ordinate drawn to any diameter Pp, from 
 any point Q on a hyperbola, cutting the transverse axis in 
 L ; QS an ordinate drawn to the axis from the same point 
 and produced, if necessary, to meet the diameter Pjo in ^; 
 AS a tangent at the vertex ; then A QL8 = trap. K8AH. 
 
 Fig. 26. 
 
 Let CP" be the diameter conjugate to CP ; draw the 
 ordinates PM, P'M' to the axis. 
 
 Then CA : CS : CM = AH : SK : MP; 
 
 (Similar A'.) 
 
 .-. CA+CS: CA + CM=AH + SK: AH+ MP, 
 
 or A'S : A'M= AH + SK : AH + MP ; 
 
 .-.A'SxSA: A'Mx MA = {AH+ SK) 8 A : [AH + MP) MA ; 
 
 .-. QS' : PM' = trap. KSAH : trap. PMAH ; 
 
 (Prop. X.) 
 
CHAP. iii.J The Hyperbola. 145 
 
 .-.A QSL : A PMT = trap. K8AS : trap. PMAH, 
 
 (19, VI. Euclid.) 
 but A PMT = trap. PMAH; {Cor. Prop, xvi.) 
 
 .-. A QSL = trap. ZS^S^. 
 
 Cor. 1. AQe-r=trap. TPffi. 
 
 For A QSL = trap. ZS^JT. 
 
 .-. AQ(?-Zr=trap. (Ji^JT = trap. TPGi. 
 
 (Prop. XVI.) 
 Cor. 2. — ^In like manner it can be shown that 
 
 A GQ'JT' = trap. TPGL ; 
 
 .-. A GQK = A GQfK' ; 
 
 but they are also similar ; .•. GQ = GQ^. 
 
 Hence, any diameter bisects all chords parallel to the 
 tangents at its extremities. (See also Cor. 3^ Prop, xviii.) 
 
 Proposition XXII. 
 
 If CP' be the semi-diameter conjugate to CP, QG any 
 ordinate to CP ; then QG^ : PG ^ Gp = CP" : CP\ 
 (See Eig. 26.) 
 
 For A CPT : A CGL = CP^ : CG\ 
 
 (19, VI. EucUd.) 
 
 Division, A CPT : trap. TPGL = CP' : CG^ - CP\ 
 or A CP'F: A QGK= CP'-.PGxGp; 
 
 (Pj-Op. XX.) 
 
 .-. CF' : QG" = CP' -.PGxGp; 
 
 (19, VI. Euclid.) 
 
 hence QG' : PG ^ Gp = CP" : CI". 
 
 L 
 
146 The Hyperbola. [chap. hi. 
 
 Cor. 1. — Ordinates to any diameter at equal distances 
 from the centre are equal. 
 
 For the rectangles are the same for equal distances 
 from the centre. 
 
 Cor. 2.— QG^ : CQ"" - CP" = CP" : CP'- 
 
 For PGx Gp= CCP - CP\ 
 
 Cor. 3. — If the ordinate GQ he produced to meet the 
 conjugate hyperhola in Q' ; then 
 
 Q'G» : C(?^ + CP = CP" : CP\ 
 
 Fig. 27. 
 
 Draw QG' an ordinate to CP. 
 
 Then QfG" : CG" - GP^ = CP' : CP\ {Cor. 2.) 
 
 Alt. CG" : GP = CG'^ - CP' : CP' ; 
 
 .-. CG' + CP : GP = CG" : CP' ; 
 
 .-. QG' : GG' + GP' = CP' : CP 
 
 Cor. 4. — The squares of the ordinates to any diameter 
 are proportional to the rectangles under the segments of 
 that diameter. 
 
CHAP. III.] 
 
 The Hyperbola. 
 Pkoposition XXIII. 
 
 147 
 
 If a normal at any point P of a hypertola meet the 
 transverse axis in N, and the conjugate axis in N' ; then 
 
 PN : CP' =CB:CA = CP' : PN', 
 CP' being the semi-diameter conjugate to CP. 
 
 Fig. 28. 
 
 Since CF is || PT, the A P'Cm ' is similar to A PMT, 
 and .-. similar to A NPM; 
 
 .-. FC-.NP^ Pm' : PM 
 
 = MQ : PM {Cor. 2, Prop, xix.) 
 = CA : C£. [Cor. 2, Prop, ix.) 
 Also PJSf' :PN=CM: MN (Parallel lines.) 
 
 = CA"" : CB" ; {Cor., Prop, viii.) 
 .-. PN' X PR : PJSf' = CA' : CB' = FC' : PN' ; 
 hence PN x PN' = FC\ 
 
 or PN' ■.FC = FC:PN=CA: CB. 
 
 Cor.— PN^PN'^FC\ 
 
 l2 
 
148 The Hyperbola. [chap. hi. 
 
 Pkoposition XXIV. 
 
 The rectangle under the distances of the foci, from any 
 point P on a hyperbola, is equal to the square of the semi- 
 diameter CP" conjugate to that passing through the point. 
 
 Fig. 29. 
 
 Circumscribe the A FPP by a circle, cutting the con- 
 jugate axis in L and N" ; join PN', FN' \ produce PN' 
 to meet' the transverse axis in N. 
 
 Then, since LN' bisects FF', and is also perpendicular 
 to it ; .•. LN' is the diameter of the circle, and arc FN' 
 = arc FN' ; 
 
 hence /. F'PL = FPL ; .. PZ is a tangent. (Prop, iii.) 
 
 Also L LPN' = 90° ; .-. PiV is a normal. 
 
 Now I FPN = I F'PN', and / PFN= l PN'F ; 
 
 (26, III. Euclid.) 
 .-. the A' FPN and F'PN' are simHar ; 
 
 .-. F'P:PN'=PN:PF; 
 
 .-. PFx PF = PNx PN' = CP". 
 
 {Cor., Prop. XXIII.) 
 
CHAP. iiT.J The Hyperbola. 149 
 
 Otherwise thus — 
 
 2CA = PF' - PF; 
 .-. WA^ = PV^ + PF"" - 2PF' X PF 
 
 (4, II. EucHd.) 
 = 2CP' + 2CF' - 2PF' X PF; 
 
 {Cor., 13, II. Euclid.) 
 .-. PF' X PF=^ OF'' + CP' - 2CA^ 
 
 = CF^ - CA^ + CP' - CA' 
 = OB" + CP" - CA' {Cor. 5, Prop, iv.) 
 = CP' - CP' + CP" {Cor. 5, Prop, xix.) 
 = CP'^ 
 
 Cor. 1. — Since PL bisects the lFPF', it is a tangent, 
 and .•. PN, which is ± PL, is a normal. Hence the 
 circle which passes through the foci and any point P on a 
 hyperhola, passes also through the points in which the 
 tangent and normal at P intersect the conjugate axis. 
 
 Cor. '2. — If jE'be the foot of the perpendicular let fall 
 from N' on either PF' or PF produced ; then 
 
 PK= J {PF' - PF) = semi- transverse axis. 
 
 Cor. 3. — If K' he the foot of the perpendicular let fall 
 from L on PF' ; then 
 
 F'K' = i {PF' - PF) = semi-transverse axis. 
 Cor. 4.— CA-.CB^CF: PN. 
 (See Fig. 11.) 
 
 For F'P:PF^F'N:NF. (Prop, b., VI. Eaclid.) 
 Div. F'P -PF:PF= F'N - NF : NF. 
 Alt. 2CA : 2CF = PF : N'F. 
 
 Similarly, 2CA : 2CF = PF' : NF' ; 
 
 .: CA^ : CF"" = PF x PF' : NF x NF'- 
 .-. CA': CF' - CA' = PFx PF' : NF x NF'- PF v PF'; 
 .-. CA' : CB' = CP" : PN\ 
 
150 
 
 TheHyperboh. 
 
 [chap. ni. 
 
 Proposition XXV. 
 
 If CK be a perpendicular let fall from the centre on the 
 tangent to a hyperbola at any point P, and OP' the semi- 
 diameter conjugate to OP; then CE:CA = GB: CP'. 
 
 but 
 ^nd 
 
 Fig. 30. 
 
 Draw FT, FT ± the tangent at P. 
 Since the i^'FPT and F'PT are similar ; 
 
 . j^p . 2?'p = FT : F'T' = F'P - FP : F'T' - FT 
 
 = CA : CK; 
 .-. FPx F'P : FT - F'T' = CA' : CIP, 
 
 FP X F'P = CP", (Prop. XXIV.) 
 
 FTxF'T' =CR; (Prop, vi.) 
 
 .-. CI"' : CB' = CA' : CK' ; 
 .-. CP' : BC = CA: CK. 
 
 Cor. — The area of the triangle formed by joining the 
 extremities of any pair of conjugate gemi-diameters is 
 constant. 
 
 For CP'x CK= CBxCA; 
 
 .'. area of A P'CP = area ot ABCA, and therefore constant. 
 
CHAP. III.] The Hyperbola. 151 
 
 Proposition XXVI. 
 
 If a chord PP' of a hyperbola pass through a fixed 
 point 0, the rectangle under its segments is to the square 
 of the parallel semi-diameter CB in a constant ratio. 
 
 Fig. 31. 
 
 Draw the diameter passing through 0, meeting the curve 
 in (8; draw the or4inates Sif, S^f'. 
 
 Then PN"" : CE' = CiV= - CBf' : 0A'\ (Prop, xxii.) 
 
 Comp. PN' + CE" : CE' = ON' : GE'' 
 
 Alt. PJT + CE : ON^ = CE : CE\ 
 
 SimHarly, SM^ + CE' : CJP = CE' : CE^ ; 
 
 .-. PN' + CE' : SM' + CE = CN' : CM' 
 
 = ON' : 8M^ ; 
 
 (Similar a^) 
 
 .-. PN' + CE' - ON' : ON' = 8M' + CE' - 8M' : 8M' ; 
 .-. CE' + PO X OP : ON' = CE' : 8M'- 
 Alt. CE' + PO X OP' : CE' = ON' : SM' 
 
 = CO' : C8'. 
 Division, POxOE : CE' = CC - CS' : CS'. 
 
152 
 
 The Hyperbola. 
 
 [chap. III. 
 
 Case 2. — (The importance of this proposition renders it 
 advisable to give a separate demonstration of each case). 
 
 As before, draw the diameter passing through ; also 
 PN' and 8M ordinates to the diameters CR and CR 
 respectively. 
 
 Fig. 32. 
 
 8M' : CM' - CR' = CR" : CR' ; (Prop, xxii.) 
 
 .-. 8M' + CR : CM' = CR' : CR". 
 
 But CJY" - CR' : PN" = CR' : CR" ; (Prop, xxii.) 
 
 .-. 8M' + CR' : CN" - CR' = CM' : PUT", 
 
 or 8M' + CR' : PN' - CR' = CM' : CN' 
 
 = 8M': ON'; 
 
 (Similar a'.) 
 
 .-. 8M' + CR' - 8M' : 8M' = PN' - ON' - CR' : ON', 
 or CR' iPOx OR - CR = 8M' : ON' 
 
 = C8' : CO' ; 
 .-. POx OR : CR' = CS' + CO' : C8'. 
 
CHAP. iii.J The Hyperbola. 
 
 Case 3. — Same construction. 
 
 153 
 
 Kg. 33. 
 
 PN'^ : CN'^ - CR = CR^ : CR\ (Prop, xxii.) 
 Also 8M''' : CM" - CE' = C^' : CE'; (Prop, xxii.) 
 .-. CM" - Clf : CN" - CE" = 8M" : FN", 
 8M' - CR' : FN' - CE' = CM' : CN' 
 
 = 8M' : ON' ; 
 8M' + CR' - 8M' : ON' - FN' + CR' = 8M' : ON', 
 
 CR' -.OFx 0F'+ CR' = C8' : CO'; 
 
 (Similar a».) 
 
 .-. OFx OF' : CR" = CO' - C8' : C8'. 
 
 Cor. 1. — If two chords FF', pp' of a hyperbola inter- 
 sect, the rectangles under the segments are proportional to 
 the squares of the parallel semi-diameters. 
 
 Let Cr be the semi-diameter || pp'. 
 Then OFx OF': CR' = CO' - C8' : CS' = Op x Op' : Cr' ; 
 .-. OF X OF: Op X Op = CE : Cr'. 
 
 or 
 
 or 
 
154 The Hyperbola. [chap. hi. 
 
 Cor. 2. — If from any point two tangents be drawn to 
 a hyperbola, the tangents will be proportional to the 
 parallel semi-diameters. 
 
 Pkoposition XXVII. 
 
 If from a point without a hyperbola two tangents OP, 
 OR be drawn, any line A.(^ drawn parallel to either will 
 be cut by the curve and chord of contact of the tangents in 
 geometric proportion. 
 
 Draw the semi-diameters CE and CB \\ OP and OR 
 respectively. 
 
 Then AQ x AQ : AP' = CD^ : CE' 
 
 {Cor. 1, Prop. XXVI.) 
 
 = OR' : OP' 
 
 [Cor. 2, Prop, xxvi.) 
 
 = AS' : AP' ; (Similar A».) 
 
 .-. AQ X AQi = AB\ 
 
CHAP, m.] The Hyperbola. 155 
 
 Pkoposition XXVIII. 
 
 If a circle intersect a hyperbola in four points, the 
 common chords will he equally inclined to the axis. 
 
 Fig. 35. 
 
 Let PP", QQ be the common chords intersecting in 0. 
 
 Draw the semi-diameters CR, CS || PP', QQ' respec- 
 tively. 
 
 Then OP X OP' lOQx OQ' = CB' : C8' ; 
 
 {Cor. 1, Prop. XXVI.) 
 
 but OP X OP' =OQx OQ' ; (35, III. Euclid.) 
 
 .-. CR = CS. 
 
 Hence CR and CS, and .'. PP' and QQ', are equally 
 inclined to the axis. {Cor. 1, Prop, i.) 
 
 Cor. — In like manner it can be shown that the common 
 chords PQ and FQ', also PQ' and P'Q, are equally in- 
 clined to the axis. 
 
156 The Hyperbola. [chap. hi. 
 
 Pkoposition XXIX. 
 
 If a tangent to a hyperbola at any point P meet any 
 diameter AA' in T, and the ordinate PMhe drawn ; then 
 CA is a mean proportional between CM and CF. 
 
 Fig. 36. 
 
 Draw tangents at A and A', meeting the tangent at P 
 in G and H. 
 Then A'T : TA = A'H : AG, (Similar a".) 
 
 and A'M -.MA^ HP : PG; (Parallel Hnes.) 
 
 but A'H : AG = HP : PG ; 
 
 {Cor. 2, Prop, xxvi.) 
 
 .-. A'T : TA = A'M : MA ; 
 
 .-. A'T+ TA : A'T- TA = A'M+ MA : A'M- MA, 
 
 or 2CA : 2CT = 2CM : 2CA. 
 
 hence CMx CT= CA\ 
 
 Cor. 1. — Conversely, if PM be an ordinate to any 
 diameter AA', and CT be taken a third proportional to 
 CM and CA; then PT will be a tangent to the hyperbola. 
 
CHAP. iii.J The Hyperhola. 157 
 
 Cor. 2. — The tangents at the extremities of any douhle 
 ordinate intersect on the diameter corresponding to that 
 ordinate. 
 
 Cor. 3. — If a diameter be drawn through the intersec- 
 tion of two tangents to a hyperhola, it will bisect the chord 
 of contact. 
 
 Cor. 4. — If the tangent at P meet a diameter of the 
 conjugate hyperbola in 2), and the ordinate PJE be drawn ; 
 then CB is a mean proportional between CD and CH. 
 
 For CT:CM=TD: DP (Parallel lines.) 
 
 = DC:DE; (Similar A^ 
 
 - .-. CT^ CM: CM'' = DC : DE 
 
 or CA' : CM^ = DC : DE ; 
 
 .: CA" : CM' - CA'' = DC : DE -DC ; 
 
 .-. CB' : PM' or CE'==DC : CE 
 
 (Cor. 2, Prop, xxii.) 
 
 = DCx CE : CE' ; 
 hence CB' ^ DC x CE. 
 
 Cor. 5. — Any diameter is cut harmonically by a tangent 
 and the ordinate to the diameter drawn from the point of 
 contact of the tangent. 
 
 Since CM : CA = CA : CT. 
 
 Comp. and Div. 
 
 CM+CA: CM- CA = CA + CT : CA - CT, 
 
 or A'M : MA = AT : TA. 
 
 (Joj.^ 6. — An y tangent to a hyperbola will be cut har- 
 monically by two parallel tangents and the diameter pass- 
 ing through their points of contact. 
 
 For MA' : MT : MA ^ PH : PT : PG. 
 
 (2, VI. Euclid.) 
 
158 
 
 The Hyperbola, 
 
 [chap. III. 
 
 Proposition XXX. 
 
 If a variable tangent to a hyperbola meet two fixed 
 parallel tangents, it will intercept segments on them whose 
 rectangle is constant, and equal to the square of the 
 parallel semi-diameter. 
 
 See Fig. 36. 
 
 Let POn be the variable tangent meeting the fixed 
 tangents AG, A'S in the points G, H. 
 
 Draw the semi-diameter CB || ^G* or A'S; also draw 
 PE an ordinate to BC. 
 
 Then CT : CA = CA : CM. (Prop, xxix.) 
 
 Division, CT : TA = CA : AM. 
 
 Alt. CT: CA =TA: AM. 
 
 Comp. CT : TA' = TA:TM; 
 
 .-. CD-.A'E^AG.PM; (Similar A'.) 
 
 ■.A'SxAG= CBxPM 
 
 = CBx CJE 
 
 = CB". [Cor. 4, Prop, xxix.) 
 
 Cor. — The rectangle imder the segments of the variable 
 tangent is equal to the square of the semi-diameter CQ 
 drawn parallel to it. 
 
 For A'ff :HP=CB:CQ = AG:PG; 
 
 {Cor. 2, Prop, xxvi.) 
 
 .-. A'H X AG : HPx PG = CB' : CQ\ 
 
 But A'H X AG = CB" ; .-. HP x PG = CQ\ 
 
CHAP. III.] The Hyperbola. 159 
 
 Proposition XXXI. 
 
 The triangles CPT and CAK, formed by drawing tan- 
 gents at the extremities of any two semi- diameters of a 
 hyperbola, are equal in area. 
 
 Fig. 37. 
 
 Draw the ordinate PM. 
 
 Then TO : CA = CA : CM; (Prop. xXix.) 
 
 = CE:CP; (Similar a ».) 
 
 .-. ETisWPA; (2, VI. Euclid.) 
 
 .-. APT.4 = A PKA ; (37, I. Euclid.) 
 
 hence A CPT = A CAK. 
 
 Cor. — If the ordinate AN be drawn from A to the 
 semi-diameter CP produced; then 
 
 Area of A CMP = area of A CAN. 
 
 For ^iVis II PT; .: TC : CA = PC : CN; 
 
 hence CA : CM = PC : CN; 
 
 .-. MN is II PA ; (2, VI. Euclid.) 
 
 .-. A PAM= A PAN; (31, 1. EucM.) 
 
 hence A CMP = A CAN. 
 
160 The Hyperbola. [chap. iit. 
 
 Proposition XXXII. 
 
 If from the extremities of any two conjugate diameters 
 CP, CQ, the ordinates FM, QN he drawn to any other 
 diameter CA ; then 
 
 CN' = AM X MA' ; and CM' = AM x MA' + CAK 
 
 Fig. 38. 
 
 For PM" : CJf' - CA" = CB" : C4^ 
 
 {Cor. 2, Prop, xxii.) 
 
 and ON' : CN + CA^ = CB' : CA' ; 
 
 (Cor. 3, Prop, xxii.) 
 
 .-. PW : QiV" = CJf - CA' : CN' + CA' ; 
 
 but PM : ON = TM : CN ; (Similar a'.) 
 
 .-. TM' : CN' = CM' - CA' : CN'+ CA' 
 
 = CM' - CTx CM : CN' + CA' 
 
 (Prop. XXIX.) 
 = CMx TM: CN' + CA' ; 
 
 (2, II. EucM.) 
 .-. TM' :CMxTM= CN' : CN' + CA'; 
 
 .: TM : CM= CN' : CN' + CA'; 
 Div. TM : CT = CN' : CA' ; 
 
CHAP. iii.J The Hyperbola. 161 
 
 .-. rjf X CM: CTx CM= CiV" : GA^; 
 but CTxCM= CA^ ; (Prop, xxix.) 
 
 .-. TMx CM=CN'; 
 hence GiV" = CM"- CMx CT (2, II. Euclid.) 
 
 = CM^-CA^ 
 = AMxMA'. 
 Also CM' = CJV' + C^' 
 
 Cor. 1.— CJf » - CiV^' = CA\ 
 
 Cor. 2.— ON' - PM" = C£\ 
 
 For QN' : {ON" + CA') or CM' = CB" : CA\ 
 
 and PM' : Ci!f ' - CA' = CB' :CA'; 
 
 {Cor. 2, Prop, xxii.) 
 
 .-. QN' - PM' : CA' = CB' : CA' ; 
 
 hence QN' - PM' = CB'. 
 
 Cor. 3.— CN:PM=CA:CB=CM: QN. 
 
 For CA' : CB' = CM' - CA' : PM' = CN' + CA" : QN\ 
 
 or CA' : C^= CN' : PM'= CM' : QN'; {Cor. 1.) 
 
 .-. CA •.CB= CN : PM = CM : QN. 
 
 Cor. 4.— The a CMP = a CNQ in area. 
 
 For CN:PM=CM: QN, {Cor. 3.) 
 
 and /.CMP = lCNQ; 
 
 .-. A OilfP = A CNQ. (15, VI. Euclid.) 
 
 M 
 
162 The Hyperbola. [chap. hi. 
 
 Pkoposition XXXIII. 
 
 If any tangent to a hyperbola meet any two conjugate 
 diameters CP, CQ, the rectangle under its segments is 
 equal to the square of the parallel semi-diameter CB. .. 
 
 Fig. 39. 
 
 Draw the ordinates PM, QN, to the diameter passing 
 through the point of contact. 
 
 Then CM : PM = CA : AT' ; (Similar A'.) 
 
 .-. CJf X AT' == PMx CA 
 
 = C£x ON; 
 
 {Cor. 3, Prop, xxxii.) 
 .-. CM : CN = CB: AT'. 
 Again, CN-.NQ^CA: AT. (SimHax a'.) 
 
 .-. CN xAT = CAy^NQ 
 = CBx CM; 
 
 (Cor. 3, Prop, xxxii.) 
 .-. CM : CJSr -^ATiCB; 
 hence CB : AT' = AT : CB ; 
 
 .-. AT X AT' = CB'. 
 
CHAP, m.] The Hyperbola. 163 
 
 Proposition XXXIV. 
 
 Given in magnitude and position any two conjugate 
 semi-diameters GA', CS, of a hyperbola, to find the axes. 
 
 Kg. 40. 
 
 Take AD a third proportional to CA and CK. 
 
 Bisect CD in B. Draw SO J. CD, meeting AO 
 drawn || CB' in 0. With the centre 0, and radius OC, 
 describe a circle cutting OA! in T, T' ; join CT, CT ; draw 
 AM, A'N ± CT, CT', produced respectively. Take CA 
 a mean proportional between CTand CM; also CBaJmean 
 proportional between CT and CN, Then CA, CB are the 
 axes. 
 
 For CF^ = CA' x A'D (Const.) 
 
 = TA' X A'T' ; (36, III. Euclid.) 
 .-. CA and CB are conjugate diameters; (Prop.'xxxiii.) 
 but /. TCT' = 90° ; .-. CA and CB are the axes in position. 
 Also CA^= CT X CM, and CB' = CT' x CJV; 
 
 .-. CA and CB represent the semiaxes in magnitude. 
 
 (Prop. IX.) 
 M 2 
 
164 The Eyperhoh. [chap. m. 
 
 Proposition XXXV. 
 
 If any line T8 be drawn parallel to the chord of contact 
 of two tangents to a hyperbola, the segments AT, B8, in- 
 tercepted between the curve and the tangents will be 
 equal. 
 
 Fig. 41. 
 
 Draw the diameter of the parallel chords BA, PQ. 
 This diameter will pass through 0. 
 
 {Cor. 2, Prop, xxix.) 
 
 Then, since PM= MQ ; {Cor. 2, Prop, xxi.) 
 
 .-. TN=NS; 
 but AN = NB ; {Cor. 2, Prop, xxi.j 
 
 hence AT = B8. 
 
 Cor. 1. — The tangent drawn parallel to the chord of 
 contact of two other tangents is bisected at its point of 
 contact. 
 
 Cor. 2. — ^If a line be drawn parallel to the chord of 
 contact of two tangents, the segment intercepted between 
 the tangents is bisected by the diameter passing through 
 their intersection. 
 
CHAP. iii.J The Hyperbola. 165 
 
 Proposition XXXVI. 
 
 Any line OA drawn through the intersection of two 
 tangents to a hyperbola is cut hannomcally by the curve 
 and the chord of contact of the tangents. 
 
 Fig. 42. 
 
 Through A and A' draw T8, T'8' || PQ, the chord of 
 contact of the tangents. Then 
 
 TP" : T'P' ^ TAx TB: T'A' x T'^ ; 
 
 {Cor., Prop. XXVI.) 
 
 but TB = AS, and T'ff = A' 8' ; (Prop, xxxv.) 
 
 . ypj . jT/pj ^TAxA8: T'A' x A'8' ; 
 
 but TA : T'A' = OA : OA' = A8:A8'; 
 
 (Similar a*.) 
 
 . rj^p, . rptpr^ ^ QJ2. Q^n. 
 
 .: TP : T'P = OA : OA' ; 
 hence Aff : A'ff = OA : OA'. (Parallel lines.) 
 
166 The Hyperbola. [chap. hi. 
 
 Pkoposition XXXVII. 
 
 Any line drawn through the middle point of the chord 
 of contact of two tangents to a hyperbola will he cut har- 
 monically by the curve and the line drawn through the 
 intersection of the tangents parallel to their chord of con- 
 tact. 
 
 Jig. 43. 
 
 Let TP, TQ, be tangents, BB' any chord drawn through 
 (y, the middle point of PQ, and meeting the line drawn 
 through r II PQ ia 0. Through JB" draw 3"^ \\ to PQ; 
 join TC. 
 
 Then TC produced will pass through C. 
 
 {Cor. 3, Prop, xxix.) 
 
 \B"T : TT'\ .^. .. ,. 
 
 = 1^2" . Bm (Similar a'.) 
 
 = H'T X BT' : BB" x TT'. 
 But TT' X BB" = 2B"T x BT ; 
 
 .-. B'B" = 2B"M; 
 .*. B'Mia an ordinate, and B' a point on the curve. 
 Now TB : BT' = B"T : B"T'; (Prop, xxxvii.) 
 
 .-. OB:Ba=OF: EG. (Parallel Hnes.) 
 
CHAP. iii.J The Hyperbola. 167 
 
 Proposition XXXVIII. 
 
 If any line cut the asymptotes of a hyperbola in Q, Q', 
 and the curve in P, then the rectangle under PQ and PQ', 
 is equal to the square of the semi-diameter CB^ drawn 
 parallel to QQ'. 
 
 Fig. 44. 
 
 Draw the semi-conjugate axis CB. Through Q draw 
 QBR II CB. 
 
 Then QP x QF : QB x QBf = CB" : CB" ; 
 
 {Cor. 1, Prop. XXVI.) 
 
 but QP' = PQ', and QB> = BQf' ; (Prop, xvin.) 
 
 .-. QPxPC^-.QBx BQf' = CF' : CB' ; 
 
 but QB X BQ" = CF ; (Prop, xvii.) 
 
 .-. QPxPq = CF\ 
 
 Cor. — The portion of any tangent intercepted between 
 the asymptotes is = the parallel diameter. 
 
 For CF^ =QPx PQ = DE\ (See Cor. 4, Prop, xviii., 
 and Fig. 23.) 
 
168 The Hyperbola. [chap. hi. 
 
 Proposition XXXIX. 
 
 The parallelogramB inscribed between the curve and its 
 asymptotes are equal. 
 
 Fig. 45. 
 
 Let CMPN, CM' FN' be the paraUelograms. 
 
 Join PP, and produce it to meet the asymptotes in 
 
 Then □ CMPN : □ CM'FN' = { J^ ; J^^j 
 
 (23, VI. EucHd.) 
 
 ^ \PQ : QP'l 
 \PQ' : P'Q'j ■ 
 
 (Similar A'.) 
 
 But PQ = P'Qf,a.niPQr=QP'; (Prop, xvin.) 
 
 henoe * a CMPN = □ CM'FN'. 
 
CHAP, HI. J The Hyperbola. 169 
 
 Cor. 1. — The area of the sector of a hyperbola, made by 
 joining any two points on the curve with the centre, is 
 equal to the area included between the curve, one asymp- 
 tote, and the lines drawn through the points parallel to the 
 other asymptote. 
 
 For the a' CPN' and CPN axe equal in area, being 
 halves of equal parallelograms ; take them successively 
 from the figure CPFN', and sector CPF = figure NPPN'. 
 
 Cor. 2. — The hyperbola continually approaches its 
 asymptotes, but can never meet either at any finite dis- 
 tance. ' 
 
 For, the area of the □ CMPN being constant ; 
 
 .•. PJf varies inversely as PN ox CM; 
 
 hence, as CM increases, PM will diminish. 
 
 And .*. when CM becomes infinitely great, PJf will be 
 infinitely small. 
 
 Cor. 3. — The area of the a formed by any tangent to 
 a hyperbola with the asymptotes is constant. 
 
 For the portion of a tangent at any point P intercepted 
 between the asymptotes mil be bisected at its point of 
 contact (Prop, xviii.) ; 
 
 .•. area of A RC8 = 2 area of a CMPN, and .•. constant. 
 
 Cor. 4. — If two hyperbolas have the same asymptotes, 
 the segments of any line, drawn parallel to either asymp- 
 tote, intercepted between the curves and the other asymp- 
 tote, are in a constant ratio. 
 
 If the second hyperbola be supposed to cut PiV in p, and 
 P'N'mp'; 
 
 then zdCP: a Cp= a CP' : aCp'; 
 
 .-. PN:pN=FN' :pN'. 
 
170 The Hyperbola. [chap. hi. 
 
 Proposition XL. 
 
 If through any point P on a hyperhola lines OPff, 
 QPQ' be drawn parallel to any two adjacent sides AD, DC, 
 of an inscribed quadrilateral, meeting the opposite sides in 
 0, a, and Q, Q': then 
 
 PO X PCy : PQ X PQ' in a constant ratio. 
 
 Fig. 46. 
 
 Through B and C draw BS and CG || AD ; join AG, 
 and produce it to meet JEB in H. 
 
 The diameter which bisects GO and BE will also bisect 
 AD, and will .-. bisect SK and Lff. Hence BS = EK, 
 and (/V = PL. 
 
 Now OL : BE = LA : AH 
 
 = (Xi) : D^. (ParaUel lines.) 
 
 Alt. OL : O'D = BH : DK. 
 
 Also POT or SC:8Q = BK : EC ; (Similar A'.) 
 
 .-. OL X pa : aD X 8Q = BH X BK : DK x KC ; 
 .-. OLxPff : PQ' X SQ = EKxBK:DKxKC 
 
 = arx aP: acx aD 
 = PLx ap -.psxpq; 
 
CHAP. iii.J The Hyperbola. 171 
 
 .-. 0LxPa+PLx0'P:P(^x8Q + F8xPQf 
 
 = EKxBK:BKx CK ; 
 
 or POxPa .PQx PQ = EK-BE.DExEC; 
 
 Now it is evident that the points A, B, C, B being fixed, 
 U is also fixed, and .•. UK x BK : BK x KC in a constant 
 ratio, and .■- PO x PO' : PQ x PQ; in a constant ratio. 
 
 Pkoposition XLI. 
 
 If from any point P on a hyperbola lines PB, PR', 
 PS, PS' be drawn to the sides of an inscribed quadri- 
 lateral, making with them any constant angles ; then the 
 rectangles under the lines drawn to the opposite sides will 
 be in a constant ratio. 
 
 rig. 47. 
 Take any other point p on the curve. 
 
172 The Hyperbola. [chap. hi. 
 
 Through the points P and^ draw QPQ and qp(][ \\ DC, 
 and OPff, poo' \\ AD ; also pr, pr, ps, ps' || PR, PR, PS, 
 PS' respectively. 
 
 Then PB : pr = PQ : pq, (Similar a'.) 
 
 and PR':pr' = Pq:p(i; 
 
 .;. PR X pr; : pr x p/ = PQ x Pq -. pq v p^. 
 
 Similarly it may be proved that 
 
 PS X PS' : ps X ps' = POx PCX : po x po' ; 
 but PQ X PC^ : PO X PC =pqxpgf :pox po' ; 
 
 .-. PR X PR' -.PSx PS' =pr X pr' : ps x ps'. 
 
 Cor. 1. — The rectangle under the perpendiculars let 
 fall from any point of a hyperbola on two opposite sides of 
 an inscribed quadrilateral is in a constant ratio to the rect- 
 angle under the perpendiculars let fall on the other two 
 
 Cor. 2. — If the points A and Z) coincide, also the points 
 B and C, then the sides AB and BC become tangents, and 
 the sides AB and CD coincide and become the chord of 
 contact. Then the rectangle under the perpendiculars let 
 fall from any point of an hyperbola on two fixed tangents 
 is in a constant ratio to the square of the perpendicular let 
 fall on their chord of contact. 
 
 Cor. 3. — If we suppose AB, CD to intersect in X, and 
 AC,BDhi.Y; also that Pi2, PK, PS, PS',pr,p/,ps, 
 psf lie in the lines XY; then, regarding ACBD as the 
 quadrilateral, the above proportion becomes 
 
 PX^:PY'^pX^:pY'; 
 
 .". XF is cut harmonically by the curve. 
 
CHAP, ui.j The Hyperbola. 173 
 
 Proposition XLII. 
 
 If two fixed tangents DP, BQ, to a hyperbola be cut 
 by a diameter AB parallel to their chord of contact, and 
 by a third variable tangent EF, the rectangle under the 
 segments of the two fixed tangents intercepted between the 
 diameter and the variable tangent is constant. 
 
 Fig. 48. 
 
 Join PC; produce it to meet the curve in B; join BR. 
 
 Since CA=CB; (Cor. 2, Prop, xxxv.) 
 
 .-. A' ACP and BCR are eqaal ; (4, I. Euclid.) 
 
 .-. BR is = and || AP, and .-. a tangent. 
 
 If BR be supposed to be produced to meet FB produced 
 in^; 
 then RB-.PE^BZ: PD (Prop, xxx.) 
 
 = RZ-RB:PD-PE = BZ:EB; 
 
 .-. AP : PE = BF : FB ; 
 
 .: AP : AP- PE = BF: BF-FB; 
 
 .■.AP:AE=BF:BB; 
 
 .-. AE X BF'- BD X AP, and .■. constant. 
 
174 The Hyperbola. [chap. hi. 
 
 Pkoposition XLIII. 
 
 To describe a conic section to pass through five given 
 points. 
 
 Fig. 49. Fig. 50. 
 
 Join BD and CE. Through A draw ^Q || BD, and AS 
 II CE; in .4Q find a point P, and in AS a point T, such 
 that 
 
 BE>( BD: OBx EE = AQy QP : CQ X QE, 
 
 and BR X ED : CB X BE = £S X 8D : AS X ST. 
 
 The poiats P and T will (Prop. xxii. Chap. li., and 
 Cor. 1, Prop. XXVI. Chap, iii.) be points on the curve. 
 
 If .-. we bisect AT in 1^, and CE in M, and join MJSF, 
 this line wiU evidently pass through the centre. Likewise 
 bisect AP in K, and BD in H, and joia KS; this line 
 will also pass through the centre. Should MN and KS 
 happen to be parallel the curve will be a parabola ; if not, 
 their point of intersection will determine the centre of the 
 curve. 
 
 Now determine OV, such that 
 
 PK' : DS" =07^" OK" : OV - Om ; 
 
 then {Cor. 2, Prop, xxii.) OV will be a semi-diameter to 
 which PK and DS are ordinates ; hence any number of 
 ordinates and points on the curve may be found. Also the 
 axes (by Prop. xxx. Chap, ii., and Prop, xxxiv. Chap. iii.). 
 
CHAP. III.J 
 
 The Hyperbola. 
 
 175 
 
 Proposition XLIV. 
 
 To describe a conic section to toiich five given right 
 lines. 
 
 Fig. 61. 
 
 First, consider the quadrilateral BFQS formed hy any 
 four of the five tangents ; draw the diagonals ; let them in- 
 tersect in K. Next consider the quadrilateral AGHL 
 formed by leaving out another tangent ; draw the 
 diagonals ; let them intersect in M. 
 
 d this way go round the figure, leaving out in succession 
 each of the tangents, and three other points N, 0, P, may 
 be similarly found. The line joining K, M, produced will 
 intersect the tangents in o, /3 ; and the line joining KP 
 produced will intersect the tangents in 7, S, points on the 
 curve. 
 
176 The Hyperbola. [chap. hi. 
 
 For the intersection K of the diagonals of the quadri- 
 lateral BFGR is (by Prop, xx.) a point on the chord of 
 contact of the reqiiired conic with the lines FG, LH; also 
 the intersection M of diagonals of the quadrilateral 
 AGHL is, by same Prop., a point on the chord of contact 
 with the same lines ; .•. a, /3, are the points of contact of 
 the required conic with these Hnes. Similarly it may be 
 shown that y, S, and e, are the points of contact with the 
 other right lines. 
 
 Proposition XLV. 
 
 If a right cone be out by a plane which, when produced, 
 cuts the opposite cone, the section will be a hyperbola. 
 
 Kg. 62. 
 
 Let the plane B VE drawn through the axis of the cone 
 perpendicular to the plane of the section coincide with the 
 plane of the paper, then both the section ^PJf and the 
 base ^P-E" will be ± the plane of the paper; .-. the line 
 
CHAP. III.J 
 
 The Hyperbola. 
 
 177 
 
 MP in which the section cuts the hase is J. the plane of 
 the paper, and ..J. BE. 
 
 Hence 
 
 BMxME = MF\ (35, III. Euclid.) 
 
 If now any other plane bpe be drawn || the base, meet- 
 ing the section in pm, it can similarly be shown that mp 
 is J_ be, and .•. bm x me = mp''. 
 
 But BM : bm = A'M : A'm ; (Similar a'.) 
 
 also ME : me = MA : mA ; 
 
 .-. BM X ME : bm X me = A'M x MA : A'm x mA, 
 or MP' : mp" = A'M x MA : A'M x mA. 
 
 Hence the section is a hyperbola. (Prop, x.) 
 
 Cor. 1. — The conjugate axis is a mean proportional 
 between the diameters of the sections drawn through A and 
 A' parallel to the base. 
 
 This follows, as iu Chap, ii., page 108. 
 
 Cor. 2. — The spheres inscribed in the cone, to touch the 
 plane of the circle, will determine the foci. 
 This follows, as in Chap, ii., page 109. 
 
( 178 ) 
 
 APPENDIX. 
 
 Many solutions having beeii given of Problem 9, page 110, 
 I consider it advisable to add the following : — 
 
 Let PF be the focal chord. 
 
 Draw the tangents, and also the normals at P and P'. 
 Suppose the former to intersect in T, and the latter in N. 
 Let the line drawn through N \\ to the axis intersect PP' 
 in 0, andPTin G. 
 
 Draw NR 1. PF. 
 
 Then Tis the centre exscribed to the a PFP", and N 
 is the centre of the inscribed circle ; 
 
 .•. F', N, F, are in directum ; 
 also TFis ± PF ; (Prop, xm.) 
 
 .-. FF=PE. 
 
 (G. & H.'s Euclid, Appendix,. Book IV.) 
 
 Now F . F'NPT is a harmonic pencil ; 
 .'. F . F'NPT is a harmonic pencil. 
 .-. N6, which is || F'Fis bisected at ; 
 hence FO=OH; (26, 1. Euclid.) 
 
 .-. FO = OP. 
 
 In Cors. 2 and 3, Prop. ,^I., Chap. III.,^ in order to 
 describe" fctfet'-p06fl;it)jl J of rtlie "fSbinlj.it i& ^SauBied that the 
 curve is'.GOiicsuBfi HowarSs ih^^raiiftierspVjip^t This pro- 
 perty is hil\j denaonstrated in Cor. 4, Prop. IX. ^ It is to 
 be n&tet iiA tHetessen^al fropaHifj oi\ipj«f[t4|i^'enl;, proved 
 in Prop'. IILi'does hot^depfei£d'<to'13ie'8testiBiJ)ti6a. 
 
 THE END. 
 
GENERAL LISTS OF NEW WORKS 
 
 PUBLISHED BY 
 
 Messrs. LONGMANS, GEEEN & CO. 
 
 PATERI^OSTER ROW, LONDON. 
 
 HISTORY, POLITICS, HISTORICAL MEMOIRS &c. 
 
 Aimitage'3 Childhood of the English Nation. Fcp. 8to. 2*. 6d. 
 Arnold's Lecture on Modem History. 8to. 7s. Gd. 
 Bagehot's literary Studies. 2 vols. 8vo. 28^. 
 BucMe's History of Civilisation. 3 vols, crown 8vo. 24«. 
 Chesney's Indian Polity. 8vo. 21j. 
 
 — Waterloo Lectures. 8vo. 10«. Gd. 
 Digby's Famine Campaign in India. 2 vols. 8vo. 32.;. 
 Durand's First Afghan War. Crown 8vo, 
 Epochs of Ancient History : — 
 
 Beesly's G-racohi, Marius, and Sulla, 2s. Gd. 
 
 Capes's Age of the Antonines, 2s. Gd. 
 
 — Early Boman Empire, 2j. Gd. 
 Cox's Athenian Empire, 2s. Gd. 
 
 — (Greeks and Persians, 2s. Gd. 
 
 Curteis's Rise of the Macedonian Empire, 2s. Gd. 
 
 Ihne's Home to ifa Capture by the Gauls, 2s. Gd. 
 
 Merivale's Homan Triumvirates, 2s. Gd. 
 
 Sankey's Spartan and Theban Supremacies, 2^. Gd. 
 Bpocha of English History : — 
 
 Creighton*s Shilling History of England (Introductory Volume). 
 Pop. 8vo. Is, 
 
 Browning's Modem England, 1820-1875, 9d. 
 
 Cordery's Sianiggle against Absolute Monarchy, 1603-1688, 9d. 
 
 Creighton's (Mrs.) England a Continental Power, 1066-1216, 9d. 
 
 Crdghton's (Rev. M.) Tudors and tae Reformation, 1485-1603, 9d. 
 
 Rowley's Rise of the People, 1215-1485, 9d. 
 
 Rowley's Settlement of the Constitution, 1688-1778, 9^. 
 
 Tancock's England during the American & European Wars, 
 1778-1820, 9d. 
 
 York-Powell's Early England to the Conquest, Is. 
 Epochs of Modem History : — 
 
 Church's Spinning of the Middle Ages, 2s. Gd. 
 
 Cox's Crusades, 2s. Gd. 
 
 Creighton's Age of Elizabeth, 2s. Gd. 
 
 Chtirdner's Houses of Lancaster and York, 2s. Gd. 
 
 Qardiner's Puritan Revolution, 2s. Gd. 
 
 — Thirty Years' War, 2s. Gd. 
 Hale's Fall of the Stuarts, 2s. Gd. 
 Johnson's Normans in Europe, 2s. Gd. 
 
 London, XiONaMANS & CO. 
 
General Lists of Ifew Works, 
 
 Epochs of Modem Hlstoiy — continued, 
 
 Ludlow's War of American Independence, 2s. 6d. 
 
 Morris's Age of Queen Anne, 2*. Gd. 
 
 Seebohm's Protestant Eevolution, 2s. 6(f. 
 
 Stubbs's Early Flantagenets, 2s. Gd. 
 
 Warbnrton's Edward ni., 2«. Bd. 
 ftoude's English in Ireland in the 18th Century. 3 vols. 8vo. 48s. 
 
 — History of England. 12 vols. 8to. £8. 18s. 12 vols, crown 8vo. 72s. 
 
 — Julius Cffisar, a Sketch. 8vo. 16s. 
 
 Q-airdner's Eiohard III. and PerMn Warbeok. Crown 8yo. 10s. 6d. 
 
 Gardiner's England undsr Buckingham and Charles I., 1624-1628. 2 vols. 8vo. 24s. 
 
 — Personal Government of Charles I., 1628-1637. 2 vols. Svo. 24s. 
 Greville's Journal of the Reigns of G«orge IV. & William IV. 3 vols. 8vo. 36s. 
 Hayward's Selected Essays. 2 vols, crown Svo. 12s. 
 
 Hearn's Aryan Household. 8vo. 16s. 
 
 Howorth's History of the Mongols. Vol. I. Royal 8vo. 28s. 
 
 Ihne's History of Rome. 3 vols. 8vo. 45s. 
 
 Leck/s History of England. Vols. I. & 11., 1700-1760. Svo. 36s. 
 
 — — — European Morals. 2 vols, crown Svo. 16s. 
 
 — Spirit of Rationalism in Europe. 2 vols, crown Svo. 16s. 
 liBwes's History of Philosophy. 2 vols. Svo. 32s. 
 Longman's Lectures on the History of England. Svo. ISs. 
 
 — Life and Times of Edward m. 2 vols. Svo. 28s. 
 Macanlay's Complete Works. S vols. Svo. £5. 5s. 
 
 — History of England : — 
 
 Student's Edition. 2 vols. or. Svo. 12s. I Cabinet Edition. 8 vols, post Svo. 4Ss 
 
 People's Edition. 4 vols. or. Svo. 16s. | Library Edition. 8 vols. Svo. £4. 
 Macaulay's Critical and Historical Essays. Cheap Edition. Crown Svo. 3s. Gd. 
 
 Cabinet Edition. 4 vols, post Svo. 24s. I Library Edition. 3 vols. Svo. 36j. 
 
 People's Edition. 2 vols. cr. Svo. 8s. | Student's Edition. 1 vol. or. Svo. 6s. 
 May's Constitutional History of England. 3 vols, crown Svo. ISs. 
 
 — Democracy in Europe. 2 vols. Svo. 32s. 
 Merivale's Fall of the Roman Republic. 12mo. 7s. Gd. 
 
 — General History of Rome, B.C. 763— A.D. 476. Crown Svo. 7s. 6d. 
 
 — History of the Romans under the Empire. S vols, post Svo. 48s. 
 Phillips's Civil War in Wales and the Marches, 1642-1649. Svo. 16s. 
 Prothero's Life of Simon de Montfort. Crown Svo. 9s. 
 
 Rawlinson's Seventh Great Oriental Monarchy — ^The Sassanians. Svo, 28 s, 
 
 — Sixth Oriental Monarchy — Parthia. Svo. 16s. 
 Seebohm's Oxford Reformers — Oolet, Erasmus, & More. Svo. 14s. 
 Sewell's Popular History of France. Crown Svo. 7s. 6d. 
 Short's History of the Church of England. Crown Svo. 7s. Bd. 
 Smith's Carthage and the Carthaginians. Crown Svo. 10s. 6d. 
 Taylor's Manual of the History of India. Crown Svo. 7s. Sd. 
 Todd's Parliamentary Government in England. 2 vols. Svo. 37s. 
 Trench's Realities of Irish Life. Crown Svo. 2s. Bd. 
 Walpole's History of England. Vols. I. & II. Svo. 36s. 
 
 BIOGRAPHICAL WORKS. 
 
 Burke's Vicissitudes of Families. 2 vols, crown Svo. 21s. 
 Oates's Dictionary of General Biography. Medium Svo. 25s, 
 
 London, LONGMAITS & CO. 
 
General lists of New Works. 
 
 Gleig'a Life of the Duke of WeUington. Crown 8vo. 6s. 
 Jerrold'B Lite of Napoleon III. Vols. I. to ni. 8vo. price 18s. each. 
 Jones's Life of Admiral Frobisher. Crown 8to. 6j. 
 Leck^B Leaders of Public Opinion in Ireland. Crown 8vo. 7s. Bd. 
 Life (The) of Sir William Fairbaim, Crown 8vo. 18s. 
 Life (The) of Bishop Frampton. Crown 8vo. 10s. M. 
 
 Life (The) and Letters of Lord Maoaulay. By his Nephew, G. Otto Trevelyan, 
 M.P. Cabinet Edition, 2 vols, post 8vo. 12s. Library Edition, 2 to1s.8to. 36s. 
 Uarshman's Memoirs of Havelock. Crown 8vo. 3s. Gd. 
 Memoirs of Anna Jamesen, by Oerardine Maopherson. 8to. 12s. 6d. 
 Memorials of Charlotte Williams-Wynn. Crown 8vo. 10s. 6d!. 
 Mendelssohn's Letters. Translated by Lady Wallace. 2 vols. or. 8vo. 6s. each. 
 Mill's (John Stuart) Autobiography. 8vo. 7s. M. 
 Newman's Apologia pro Vita Sua. Crown 8to. 6s. 
 
 Nohl's Life of Mozart. Translated by Lady Wallace. 2 vols, crown 8vo. 21s. 
 Fattison's Life of Casanbon. 8vo. 18s. 
 
 Spedding'a Letters and Life of Francis Bacon. 7 vols. 8vo. £4. 4s. 
 Stephen's Essays in Ecclesiastical Biography. Crown 8to. 7s. 6d, 
 Stigaud's Lite, Works &c. of Heinrich Heine. 2 vols. 8vo. 28s. 
 Zinunem's Life and Works of Lessing. Crown 8to. IDs. 6d. 
 
 CRITICISM, PHILOSOPHY POLITY &c. 
 
 Amoe's View of the Science of Jurisprudence. 8to. 18s. 
 
 — Primer of the English Constitution. Crown 8vo. 6s. 
 Arnold's Manual of English Literature. Crown 870. 7s. Gd. 
 Bacon's Ifesays, with Annotations by Whately. 8vo. 10s. Gd. 
 
 — Works, edited by Speddlng. 7 vols. 8vo. 73j. Gd. 
 Bain's Logic, Deductive and Inductive. Crown 8vo. 10s. Gd. 
 
 Pabt I. Deduction, 4s. | Paet II. Induction, 6s. Gd. 
 Bladdey's German and English Dictionary. Post 8vo. 7s. Gd. 
 Bolland & Lang's Aristotle's Politics. Crown 8vo. 7s. Gd. 
 Bollinger's Lexicon and Concordance to the New Testament. Medium 8to. 30s. 
 Comte's System of Positive Polity, or Treatise upon Sociology, translated : — 
 
 Vol. I. GeneralViewof Positivism and its Introductoiy Principles. 8vo. 21s. 
 
 Vol. n. Social Statics, or the Abstract Laws of Human Order. 14s. 
 
 Vol*. lit. Social Dynamics, or General Laws of Human Progress. 21s. 
 
 Vol. rV. Theory o£ the Future of Man ; with Early ifesays. 24j. 
 CongTOve's Politics of Aristotie ; Greek Text, English Notes. 8vo. 18s. 
 Contanseau's Practical French & English Dictionary. Post 8vo. 7s. 6if. 
 
 — Pocket French and English Dictionary, Square 18mo. 3s. Gd. 
 
 Dowell's Sketch of Taxes in England. Vol. I. to 1643. 8vo. lOs. Gd. 
 Farrar's Language and Languages. Chrown 8vo. Gs. 
 Grant's Ethics of Aristotle, Greek Text, English Notes. 2 vols. 8vo. 32s. 
 Hodgson's Philosophy of Keflecbion. 2 vols. 8vo. 21s. 
 Kalisoh's Historical and Critical Commentary on the Old Testament ; with a 
 
 New Translation. Vol. I. Genesis, 8vo. 18s. or adapted for the G«nersd 
 
 Header, 12s. Vol. II. Exodus, 16s. or adapted for the General Header, 12s. 
 
 Vol. in. Leviticus, Part I. 16s. or adapted for the General Header, 8s. 
 
 Vol. IV. Leviticus, Fart n. 16s. or adapted for the General Header, 8s. 
 
 London, LONGMANS & CO. 
 
General Lists of New Works. 
 
 Latham's Handbook of the Englfeh Lan^age. Crown 8vo. 6*. 
 
 — English Dictionary. 1 toI. medium 8vo. 24j. 4 vols. 4to. £7. 
 Lewis on Authority in Matters of Opinion. 8vo. 14*. 
 
 Liddell & Scott's Greek-English Lexicon. Crown 4to. 36s. 
 
 — — — Abridged Greek-English Lexicon. Square 12mo. 7s. Bd. 
 Longman's Pocket (Jerman and English Dictionary. 18mo. 5j. 
 Macaulay*s Speeches corrected by Himself. Crown 8vo. 3s. Gd. 
 Macleod's Economical Philosophy. Vol. I. 8vo. 16*, Vol. II. Part 1. 12*. 
 Mill on Eepresentative Government. Crown 8vo. 2s. 
 
 Liberty. Post 8vo. 7*. 6d. Crown 8vo. 1*. 4d. 
 
 Mill's Dissertations and Discussions. 4 vols. 8vo. 46j. Bd. 
 
 — Essays on Unsettled Questions of Political Economy. 8vo. Bs. Bd. 
 
 — Examination of Hamilton's Philosophy. 8vo. 16*. 
 
 — Logic, Ratiocinative and Inductive. 2 vols. 8vo. 25*. 
 
 — Phenomena of the Human Mind. 2 vols. Svo. 28*. 
 
 — Principles of Political Economy. 2 vols. Svo. 30*. 1 vol. or. Svo. 5*. 
 
 — Subjection of Women. Crown Svo. 6*. 
 
 — TJtzHtarianisra. Svo. 5*. 
 
 Morell's Philosophical Fragments. Crown Svo. 5*. 
 
 Mailer's (Max) Lectures on the Science of Language. 2 vols, crown Svo. 16*. 
 
 — Hibbert Lectures on the Origin and Growth of Religion. Svo. 10*. Bd. 
 K"oir6 on Max MUUer's Philosophy of Language. Svo. 6*. 
 
 Rich's Dictionary of Roman and Greek Antiquities. Crown Svo. 7*. Bd. 
 Roget's Thesaurus of English Words and Phrases. Crown Svo. 10*. Bd. 
 Sandars's Institute of Justinian, wiUi English Notes. Svo. 18*. 
 Swinboume's Picture Logic. Post Svo. 5*. 
 
 Thomson's Outline of Necessary Laws of Thought. Crown Svo. 6*. 
 Tocqneville's Democraqy in America, translated by Reeve. 2 vols, crown Svo. 1 Bs, 
 Twiss's Law of Nations, Svo. in Time of Peace, 12*. in Time of War, 21*. 
 Whately's Elements of Logic. Svo. 10*. Bd. Crown Svo. 4*. Bd. 
 
 — — — Rhetoric. Svo. 10*. Bd. Crown Svo. 4*. Bd. 
 
 — English Synonymes. Fcp. Svo. 3*. 
 
 White & Riddle's Large Latin-English Dictionary. 4to. 28*. 
 White's Coll^^ Latin-English Dictionary. Medium Svo. 16*. 
 
 — Junior Student's Complete Latin-English and English-Latin Dictionary. 
 Square 12mo. 12*. 
 
 a^___-i.-i_ f The English-Latin Dictionary, 5s. Bd. 
 beparateiy j ^j^^ Latin-English Dictionary, 7*. Bd. 
 
 White's Middle-Class Latin-English Dictionary. Fcp. Svo. 3*. 
 
 Williams's Nicomachean Ethics of Aristotle translated. Crown Svo. 7*. Bd 
 
 Yonge's Abridged English-Greek Lexicon. Square 12mo. 8*. Bd. 
 
 — Large English-Greek Lexicon. 4to. 21*. 
 
 Zeller's Socrates and the Socratic Schools. Crown Svo. 10*. Bd, 
 
 — Stoics, Epicureans, and Sceptics. Crown Svo. 14*. 
 
 — Plato £md the Older Acadrany. Crown Svo. 18*. 
 
 MISCELLANEOUS WORKS & POPULAR METAPHYSICS. 
 
 Arnold's (Dr. Thomas) Miscellaneous Works. Svo. 7*. Bd. 
 Bain's Emotions and the WiU. Svo. 15*. 
 
 London, LONGMANS & CO. 
 
General Lists of New Works. 
 
 Bain's Mental and Moral Science. Crown 8vo. 10s, 6d. Or separately : Part^I, 
 Mental Science, 6s. Gd. Part II. Moral Science, 4«. 6d. 
 
 — Senses and the Intellect. 8to. 15s. 
 
 Buckle's Miscellaneous and Postlinmous "Works. 3 vols. 8vo. 52s. Gd. 
 
 Conington's Miscellaneous Writings. 2 vols. 8to. 28*, 
 
 Edwards's Specimens of English Prose. 16mo. 2*. Gd. 
 
 Froude's Short Studio on Great Subjecte. 3 vols, crown 8yo. IBs. 
 
 Qerman Home Life, reprinted fi-om Eraser's Magazine. CSrown 8vo. 6*. 
 
 Hume's Essays, edited by Green & Grose. 2 vols. 8vo. 28*. 
 
 ■ — Treatise of Human Nature, edited by Green & Grose. 2 vols. 8vo. 28«. 
 Macaulay's Miscellaneous Writings. 2 vols. 8vo. 2l5. 1 vol. crown 8vo. 4s. Gd, 
 
 — Writings and Speeches. Crown 8vo. Gs. 
 
 Mill's Analysis of the Phenomena of the Human Mind. 2 vols. 8vo. 28j. 
 MiUler's (Max) Chips from a German Workshop. 4 vols. 8vo. 58j. 
 Mullinger's Schools of Ohai-les the Great. 8vo. ts. Gd. 
 Rogers's Defence of the Eclipse of Faith Fcp. 8vo. Zs, Gd, 
 
 — Eclipse of P^th. Pep. 8vo, 5*. 
 
 Selections from the Writings of Lord Macaulay. Crown 8vo. Gs. 
 The Essays and Contributions of A, K. H. B. Crown 8vo. 
 
 Autumn Holidays of a Country Parson. Zs. Gd. 
 
 Changed Aspects of Unchanged Truths. 3*. Gd. 
 
 Common-place Philosopher in Town and Country. 35. Gd. 
 
 Counsel and Comfort spoken from a City Pulpit. 3.t. Gd. 
 
 Critical Essays of a Country Pai^on. 3*. Gd, 
 
 Graver Thoughts of a Country Parson. Thi-ee Seri^, Zs. Gd, each. 
 
 Landscapes, Churches, and Moralities. Zs. Gd. 
 
 Leisure Hours in Town. Zs. Gd. 
 
 Lessons of Middle Age. 3 *. Gd. 
 
 Present-day Thoughta. Zs. Gd. 
 
 Becreations of a Country Parson. Three Series, Zs. Gd. each. 
 
 Seaside Musings on Sundays and WeeTc-Days. ds. Gd. 
 
 Sunday Afternoons in the Parish Chui ch of a University City, Zs. Gd. 
 
 Wit and Wisdom of the Rev. Sydney Smith 16mo. 3*. Gd. 
 
 ASTRONOMY, METEOROLOGY, POPULAR GEOGRAPHY &c. 
 
 Dove's Law of Storms, translated by Scott. 8vo. IQs. Gd, 
 Herschd's Outlines of Astronomy. Square crown 8vo. 125. 
 Keith Johnston's Dictionary of Geography, or Gazetteer. 8vo. 42j. 
 Nelson's Work on the Moon. Medium 8vo. Zls. Gd. 
 Proctor's Essays on Astronomy. 8vo. 12*. 
 
 — Larger Staf Atlas. Folio, 15s. or Maps only,-12j. Gd. 
 
 — Moon. Crown 8vo. 10*. Gd. 
 
 — New Star Atlas. Crown 8vo. 5s. 
 
 — Orbs Around Us. Crown 8v0. 7s. Gd. 
 
 — Other Worlds than Ours. Crown 8vo. IOj, Gd. 
 
 — Saturn and its System. 8vo. 14j. 
 
 — Sun. Crown 8vo. 14*. 
 
 — Transits of Venus, Past and Coming. Crown 8vo. 8s. Gd, 
 
 — Treatise on the Cycloid and Cydoidal Curves. Crown 8vo. 10*. Gd. 
 
 London, LONGMANS & CO. 
 
General Lists of New Works. 
 
 Proctor's Universe of Stars. 8vo. 10s. Bd. 
 
 Schellen's Spectrum Analysis. 8to. 2Ss. 
 
 Smitli's Air and Hain. Sto. 24j. 
 
 The,Public Schools Atlas of Ancient Geography* Imperii^ Svo. 75. 6d. 
 
 — — — Atlas of Modern Geography. Imperial 8to. 5*. 
 
 Webb's Celestial Objects for Common Telescopes. New Edition in preparation. 
 
 NATURAL HISTORY 8c POPULAR SCIENCE. 
 
 Amott's Elements of Physics or Natural Philosophy. Crown 8to. 12*. 6d. 
 Brando's Dictionary of Science, Literature, and Art. 3 vols, medium Svo. 63*. 
 Decaisne and Le Maout's G«ieral System of Botany. Imperial Svo. Bis. 6d. 
 Evans's Aacient Stone Implements of Great Britain. Svo. 2Bs. 
 Ganot's Elementary Treatise on Physics, by Atkinson. Large crown Svo. 15*. 
 
 — Natural Philosophy, by Atkinson. Crown Svo. 7s. 6d. 
 Gore's Art of Scientific Discovery. Crown Svo. 15s. 
 Grove's'Correlation of Physical Forces. Svo. IBs. 
 Hartwig's Aerial World. Svo. 10*. Gd. 
 
 — Polar World. Svo. IOj. 6d. 
 
 — Sea and its Living Wonders. Syo.lOs.Gd. 
 
 — Subterranean World. Svo. 10*. Gd. 
 
 — Tropical World. Svo. 10*. Bd. 
 Haughton's Principle! of Animal Mechanics. Svo. 21*. 
 Heer's PrimieTal World of Switzerland. 2 vols. Svo. 16*. 
 Helmholtz's Lectures on Scientific Subjects. Svo. 12*. Bd. 
 Helmholtz on the Sensations of Tone, by Ellis. Svo. 36*. 
 
 Heiiasley's Handbook of Trees, Shrubs, & Herbaceous Plants. Medium Sto. 12*. 
 Hullah's Lectures on the History of Modem Music. Svo. S*. Gd, 
 
 ■ — Transition Period of Musical History. Svo. 10*. Gd. 
 Keller's Lake Dwellings of Switzerland, by Lea. 2 vols, royal Svo. 42*. 
 Elirby and Spence's Introduction to Entomology. Crown Svo. 5*. 
 Lloyd's Treatise on Magnetism. Svo. 10*. Gd. 
 
 — — on the Wave-Theory of Light. S70. 10*. Gd. 
 Loudon's Encyclopsedia of Plants. Svo. 42*. 
 
 Lubbock on the Origin of Civilisation & Primitive Condition of Man. Svo. 18*. 
 Macalister'B Zoology and Morphology of Vertebrate Animals. Svo, 10*. Gd. 
 Nicola' Puzzle of Life. Crown Svo. 3*. 6d. 
 Owen's Comparative Anatomy and Physiology of the "Vertebrate Animals. 3 vols. 
 
 8to. 73*. Gd. 
 Proctor's Light Science for Leisure Hours. 2 vols, crown Svo. 7*. 6d. each. 
 Rivers's Rose Amateur's Guide. Pep. Svo. 4*. Gd. 
 Stanley's Familiar History of Birds. Fcp. Svo. 8*. Gd. 
 Text-Books of Science, Mechanical and Phyiical, 
 
 Abuey's Photography, small Svo. 8*. Gd. 
 
 Anderson's (Sir John) Strength of Mattrials, 3*. Gd. 
 
 Armstrong's Organic Chemistry, 8*. Gd. 
 
 Barry's Railway Appliances, 8*. Gd. 
 
 Bloxam's Metals, 3*. Gd. 
 
 Gooderc's Elements of Mechanism, S*. 9d. 
 — Principles of Mechanics, 8*. Gd. 
 
 Gore's Electro-Metallurgy, 6*. 
 
 Griflan's Algebra and Trigonometry, 8*. Gd. 
 
 London, LONGMANS & CO. 
 
General Lists of New Works. 
 
 Text-Books of Saeace— continued, 
 
 Jenkin'a Electaicity and Magnetism, ds. Sd. 
 
 Maxwell's Theory of Heat, 3s. 6d. 
 
 Merrifield's Technical Arithmetic and Mensuration, Bs. Gd. 
 
 Miller's Inorganic Chemistry, Ss. Gd. 
 
 Preece & SiTewright's Tel^praphy, 3*. ^d. 
 
 Eutley's Study of Bocke, 4^. Gd, 
 
 Shelley's Workshop Appliances, 3^. Gd. 
 
 Thom6's Structiiral and Physiological Botany, 6a. 
 
 Thorpe's Quantitative Chaaical Jjialysis, is. Bd* 
 
 Thorpe & Muir's QuaJltatiTe Analysis, ds. 6d. 
 
 Tilden's Chemical Philosophy, Ss. Gd. 
 
 TJnwin'e Machine Deeign, 3s. Gd. 
 
 Watson's Plane and Solid Gl^ometa^y, 3^. Gd. 
 Tyndall on Sound. Crown 8vo. 10*. Gd. 
 
 — Contributions to Molecular Physira. 8to. 16*. ■ 
 
 — Fragments of Science. New Edit. 2 vols, crown Svo. [/n the press. 
 
 — Heat a Mode of Motion. Crown Svo. 
 
 — Lectures on Electrical Phenomena. Crown Svo. Is. sewed, 1*. Gd. doth. 
 
 — Lectures on Light. Crown Svo. 1*. sewed, Is. Gd. doth, 
 
 — Lectures on Light delivered in Anaerica. Crown Svo. 7s. Gd. 
 
 — Lessons in Electricity. Crown Svo. 2s. Gd. 
 Von Gotta on Eocks, by Lawrence. Post Svo. lis. 
 Woodward's Q«ology of England and Wales. Crown Svo. 14*. 
 Wood's BiUeAmmals. With 112 Vignettes. Svo. 14s. 
 
 — Homes Without Hands. Svo. lis. 
 
 — Insects Abroad. Svo. lis. 
 
 — Insects at Home. With 700 Ulustratious. Svo. 145. 
 
 — Out of Doors, or Articles on Natural History. Crown Svo. Is, Gd. 
 
 — Strange Dwellings. With 60 Woodcuts. Crown Svo. 7s. Gd. 
 
 CHEMISTRY & PHYSIOLOGY. 
 
 Auerbach'a Anthriuxai, translated by W. Crookes, F.K.S. Svo. 12*. 
 
 Buckton's Health in the House ; Lectures on Elementary Physiology. Fcp. Svo. 2a. 
 
 Orookes's Handbook of Dyeing and Calico Printing. Svo. 42«. 
 
 — Select Methods in Chemical Analysis. Crown Svo. 12 j. Gd. 
 Kingzett's Animal Chemistry. Svo. IS*. 
 
 — History, Products and Process^ of the Alkali Trade. Svo. 12a. 
 Miller's Elements of Chemistry, Theoretical and Practical. 3 vols. Svo. Part I. 
 
 Chemical Physics, 16*. Part II. Inorganic Chemistry, 24*. Part III. Organic 
 Chemistry, New Edition in the press. 
 Watts's Dictionary of Chemistry. 7 vols, medium Svo. £10. 16a. Gd. 
 — Third Supplementary Volume, in Two Parts. Part I. 36a. 
 
 THE FINE ARTS &. ILLUSTRATED EDITIONS, 
 
 Bewick's Select Fables of ^sop and others. Crown Svo. 7*. Gd. demy Svo. ISa. 
 Doyle's Fairyland ; Pictures from the Elf- World, Folio, 15a. 
 Jameson's Sacred and Legendary Art. 6 vols, square crown Svo. 
 Legends of the Madonna. 1 vol. 21a. 
 
 — — — Monastic Orders. 1 vol. 21s. 
 
 — — — Saints and Martyrs. 2 vols. 81a. Gd. 
 
 — — — Saviour. Completed by Ladj Eastlake. 2 vols, 42a. 
 
 LondoP, LONGMANS & CO. 
 
General Lists of New Works. 
 
 Longman's Three Cathedrals Dedicated to St. Paul. Square crown 8to. 21*. 
 Uacaulay'S' Lays of Ancient Borne. With 90 Illustrations. Pop. 4to. 21s. 
 Macfarren's Lectures on Harmony. Sto. 12s, 
 
 Miniature Edition of Macaulay's Lays of Ancient Bome. Imp, 16 mo. 10s. 6(2. 
 Moore's Irish Melodies. With 161 Plates by D. Maclise, B.A. Super-royal 8vo. 21s. 
 
 — Lalla Bookh. Tennid's Edition. With 68 Illustrations. Pep. 4to. 21s. 
 Northcote and Brownlow's Boma Sotterranea. Part I. 8vo. 24s. 
 
 Perry on Grieek and Boman Sculpture. 8vo. [In preparation. 
 
 Eedgrave's Dictionary of Artists of the English School. 8vo. 16s. 
 
 THE USEFUL ARTS, MANUFACTURES &c. 
 
 Bourne's Catechism of the Steam Engine. Fcp. 8vo. 6s. 
 
 — Examples of Steam, Air, and Gas Engines. 4to. 70s. 
 
 — Handbook of the Steam Engine. Fcp. 8to. 9s. 
 
 — Eecent Improvements in the Steam Engine. Fcp. 8to. 6s. 
 
 — Treatise on the Steam Engine. 4to. 42s. 
 Cresy's Encydopsedia of Civil Engineering. 8vo. 42s. 
 Oulley's Handbook of Practical Tdegraphy. 8vo. 16s. 
 
 Eastlake's Household Taste in Furniture, &c. Square crown 8vo. 14s. 
 Fairbaim's TTseful Information for Engineers. 3 vols, crown 8vo. 31s. M. 
 
 — Applications of Cast and Wrought Iron. 8vo. 16s. 
 
 — Mills and Millwork. 1 vol. 8vo. 25s. 
 Gwilt's Encyclopaedia of Architecture. 8vo. 52s. 6ii. 
 
 Hobson's Amateur Mechanics Practical Handbook. Crown 8vo, 2s. ed. 
 Hoskold's Engineer's Valuing Assistant. 8vo. 31s. Bd. 
 Kerl's Metallurgy, adapted by Crookes and EShrig. 3 vols. 8vo. £4. 19s. 
 Loudon's Encydopsedia of Agriculture. 8vo. 21s. 
 
 — — — Gardening. 8vo. 21s. 
 Mitchell's Manual of Practical Assaying. 8vo. 31s. M. 
 Northcott's Lathes and Taming. 8vo. 18s. 
 
 Payen's Industrial Chemistry, translated from Stohmann and Bugler's Gterman 
 
 Edition, by Dr. J. D. Barry. Edited by B. H. Paul, Ph.D. 8vo. 42s. 
 Stoney's Theory of Strains in Girders. Boy. 8vo. 36s. 
 Thomas on Coal, Mine-Gases and Ventilation. Crown 8vo. 10s. M. 
 ■Ore's Dictionary of Arte, Manufactures, & Mines. 4 vols, medium 8vo. £7. 7s. 
 
 RELIGIOUS &, MORAL WORKS. 
 
 Abbey & Overton's English Church in the Eighteenth Century. 2 vols. 8to.36s. 
 Arnold's (Eev. Dr. Thomas) Sermons. 6 vols, crown 8vo. 6s. each. 
 Bishop Jeremy Taylor's Entire Works. With Lite by Bishop Heber. Edited bv 
 the Eev. 0. P. Eden. 10 vols. 8vo. £5. 5s. " 
 
 Bonltbee's Commentary on the 39 Articles. Crown 8vo. es. 
 Browne's (Bishop) Exposition of the 39 Artides. 8vo. les. 
 Conybeare & Howson'sLife and Letters of St. Paul :— 
 
 Librray Edition, with aU the Original Ulustcations, Maps, Landscapes on 
 
 Sted, Woodcuts, &c. 2 vols. 4to. 42s. -=voi.» u^i 
 
 Intermediate Edition, with a Sdeotion of Maps, Plates, and Woodcuts 
 
 2 vols, square crown 8vo. 21s. •J^^,'^.«a. 
 
 1 vd"raown f'^*'g''' '^^'^ »"* condensed, with 46 Illustrations and Maps. 
 Colenso's Lectures on the Pentateuch and the Moabite Stone. 8vo. 12s. 
 
 London, LONGMAN'S & CO. 
 
General Lists of New Works. 
 
 Colenso on the Pentateuch, and Book of Joshua. Crown 8to. Gs. 
 
 — — Part VII. completion of the larger Work. 8vo. 24s. 
 D'Aublgnfi's Reformation in Europe in the Time of Calvin. 8 vols. 8vo. £6. 12*. 
 Drommond's Jewish Messiah. Svo. 15s. 
 
 BUicott's (Bishop) Commentary on St. Paul's Epistles, Svo. Q-alatiana, Ss. 6d. 
 
 Ephesians, 8*. Gd. Pastoral Epistles, 10s. Gd. Philippians, Colossians, and 
 
 Philemon, 10*. 6d. Thessalonians, 7s. Gd. 
 BUicott's Lectures on the Life of our Lord. Svo. 12*. 
 Bwald's History of Israel, translated by Carpenter. 5 vols. Svo. 63*. 
 
 — Antiquitira of Israel, translated by Solly. Svo. 12*. Gd. 
 Ck)ldziher*s Mythology among the Hebrews. Svo. 16*. 
 Jukes's Types of Genesis. Crown Svo, 7*. Gd. 
 
 — Second Death and the Restitution of aU Things. Crown Svo. 3*. Gd, 
 Kalisch's Bible Studies. Part I. the Prophecies of Balaam. Svo. 10*. Gd. 
 
 — — — Part n. the Book of Jonah. Svo. 10*. Gd. 
 
 Keith's Evidence of the Truth of the Christian Religion derived from the Fulfil- 
 ment of Prophecy. Square Svo. 12*. Gd. Post Svo. 6*. 
 Kuenen on the Prophets and Prophecy in Israel. Svo. 21*. 
 Lyra Q^rmanica. Hymns translated by Miss Winkworth. Fcp. Svo. 5s. 
 Manning's Temporal Mission of the Holy Ghost. Svo. 8*. Gd. 
 Martineau's Endeavours after the Christian Life. Crown Svo. 7*. 6d. 
 
 — Hymns of Praise and Prayer. Crown Svo. 4*. Gd, 32mo. 1*. Gd. 
 
 — Sermons; Hours of Thought on Sacred Things. Crown Svo. 7*. 6d. 
 Merivale's (Dean) Lectures on Early Church History. Crown Svo. 
 
 Mill's Three Essays on Religion. Svo. 10*. Gd. 
 
 Monseli's Spirituid Songs for Sundays and Holidays. Pep. Svo. 6*. 18mo. 2*. 
 
 MUller*s (Max) Lecture on the Science of Religion. Crown Svo. 10*. Gd. 
 
 Newman's Apologia pro Vita Sua. Crown Svo. 6*. 
 
 O'Conor's New Testament CoEomentarira. Crown Svo. Epistle to the Romans, 
 
 3*. Gd. Epistle to the Hebrews, 1*. Gd. St. John's Gospel, 10*. 6^. 
 One Hundred Holy Songs, &c. Square fcp. Svo. 2*. Gd. 
 Passing Thoughts on Religion. By Miss SeweU. Fcp. Svo. 3*. Gd, 
 SeweU's (Miss) Preparation for the Holy Communion. 32mo. 3*. 
 Shipley's Ritual of the Altar. Imperial Svo. 42*. 
 Supernatural Religion. 3 vols. Svo. 38*. 
 Thoughts for the Age. By Miss Sewell. Fcp. Svo. 3*. Gd. 
 Vaughan's Trident, Crracent, and Cross ; the Religious History of India, 8vo.9*.6rf. 
 Whately's Lessons on the Christian Evidences. ISmo. Gd. 
 "White's Four Gospels in Greek, with Greek-English Lexicon. 32mo. 5*. 
 
 TRAVELS, VOYAGES 5tc. 
 
 Ball's Alpine Guide. 3 vols, post Svo. with Maps and Ulustrationa :— I. "Wratem 
 
 Alps, 6*. Gd. U. Central Alps, 7*. Gd. Ill, Eastern Alps, 10*. Gd. 
 BaU on Alpine Travelling, and on the Geology of the Alps, 1*, 
 Baker's Rifle and the Hound in Ceylon. Crown Svo, 7*. Gd. 
 
 — Bight Tears in Ceylon, Crown Svo. 7*, Gd, 
 
 Bent's Freak of Freedom, or the Republic of San Marino. Crown Svo. 
 Brassey's Voyage in the. Yacht * Sunbeam.* Crown Svo. 7*. Gd. Svo. 21*. 
 Edwards's (A, B.) Thousand Miles up the Nile. Imperial Svo, 42*. 
 
 London, LONGMANS fo CO, 
 
10 
 
 General lists of New Works. 
 
 Etsus's niyiiaii Letters. Post 8to. 7s. Sd. 
 
 Grohman'a Tyrol and the Tyrolese. Crown 8vo. 6*. 
 
 Indian Alps (The). By a Lady Pioneer. Imperial 8to. iSs. 
 
 Lefroy*s Discovery and Early Settlement of the Bermuda Islands. 2 vols. 
 
 royal Svo. 60*. 
 Miller and Skertchley's Fenland Past and Present. Hoyal 8to. BU, 6d, 
 
 Paper, 50*. 
 Noble's Cape and South Africa. Fcp. 8to. Ss. Gd. 
 Packe's Guide to the Pyrenees, for Mountaineers. Crown 8to. 7s. 6d. 
 The Alpine Club Map of Switzerland. In four sheets. 42«. 
 Wood's Discoveries at Bphesus. Imperial Svo. GBs. 
 
 Large 
 
 WORKS OF FICTION. 
 
 Becker's Charides ; Private Life among the Ancient Greeks. Post Svo. 7s, Gd. 
 — Gallus ; Homan Scenes of the Time of Augustus. Post Svo. 7s. 6d. 
 
 Cabinet Edition of Stories and TaJra by Miss Sewell : — 
 
 Amy Herbert, 2s. Gd. 
 Cleve Hall, 2s. Gd. 
 The Earl's Daughter, 2s. Gd. 
 Experience of Life, 28. Gd. 
 Gertrude, 2s. Gd. 
 
 Ivors, 2s. Gd. 
 
 Katharine Ashton, 2«. Gd. 
 Laneton Parsonage, 3*. Gd, 
 Margaret Percive^, 3s. Gd, 
 Ursula, 3s, Gd. 
 
 Novels and Tales by the Bight Hon. the Earl of Beaconsfleld, £.G. 
 Edition, complete in Ten Volumes, crown Svo. price £3. 
 
 Lothair, 6s. Henrietta Temple, Gs. 
 
 Coningsby, 6s. 
 Sybil, Gs. 
 Taacxed, Gs. 
 Teuetia, 6s. 
 
 Cabinet 
 
 Contarini Fleming, 6s. 
 Alroy, Izion, &c. 6«, 
 The Young Duke, &c.6j. 
 Vivian Grey, 6*. 
 
 The Modem Novelist's Library. Each 
 complete in itself, price 2s. boards, 
 By the Earl of Beaconsfleld, K.G. 
 
 Lothalr. 
 
 Coningsby. 
 
 Sybil. 
 
 TEmcred. 
 
 Venetia. 
 
 Henrietta Temple. 
 
 Contarini Fleming. 
 
 Alroy, Ddon, Sic. 
 
 The Young Duke, &c. 
 
 Vivian Grey. 
 By Anthony Trollope. 
 
 Barchester Towers. 
 
 The Warden. 
 By the Author of ' the Rose Garden. 
 
 ITnawares. 
 
 Lord ^eaconsfleld's Novels and Tales. 
 
 Work in crown Svo. A Single Volume, 
 or2». Gd, doth:— 
 
 By Major Whyte-Mdville. 
 
 Digby Grand. 
 
 Genial Boimce. 
 
 Eate Coventry. 
 
 The Gladiators. 
 
 Good for Nothing. 
 
 Holmby House. 
 
 The Interpreter. 
 
 The Queen's !Kfories. 
 By the Author of ' the Atelier du Lys.' 
 
 Mademoiselle Mori. 
 
 The Atelier du Lys. 
 By Various Writras. 
 
 Atherstone Priory, 
 
 The Burgom^ter's Family. 
 
 Elsa and her Vulture. 
 
 The Six Sisters of the Valley. 
 10 vols, cloth extra, gilt edges, 30j. 
 
 Whispers from Fairy Land. By the Right Hon. B. H. EnatchboU-Hugessen 
 
 M.P. With Nine Illustrations. Crown Svo. 3s. 6d. 
 Higgledy-P^gledy ; or. Stories for Everybody and Everybody's Children. By 
 
 the Bight Hon. E. M. EnatchbuU-Hugessen, M.P. Witii Nine Illustrations 
 
 from Designs by B. Doyle. Crown Svo. 3*. 6d. 
 
 London, LONGMAIfS & CO. 
 
General Lists of New "Works. 11 
 
 POETRY & THE DRAMA. 
 
 Bailey's Festusj a Poem. Crown 8to. 12*. Gd. 
 
 Bowdler's Family Shakspeare. Medium 8to. 14j. 6 vols. fop. 8vo. 2ls. 
 
 Brian Boru, a Tragedy, by J. T. B. Crown 8to. Bs. 
 
 Cayley's Iliad of Horn er, Homometrically iaranslated. 8to. 12s. 6d. 
 
 Oonington's ^neid of Yirgil, translated into English Yerse. Crown 8to, 9j, 
 
 Cooper's Tales from Euripides. Small 8vo. 
 
 Edwards's Poetry-Book of Elder Poets. 16mo, 28. Qd. 
 
 — Poetry-Book of Modem Poets. 16mo. 2s. Gd. 
 
 Ingelow's Poems. First Series. Illustrated Edition. Fcp. 4to. 21s. 
 Macaulay's Lays of Ancient Rome, with Ivry and the Armada. 16mo. Bs. Gd. 
 Petrarch's Sonnets and Stanzas,translated by C. B. Cayley, B.A, Crown 8vo.lO*.6rf'. 
 Poems. By Jeiin Ingelow. 2 vols. fcp. 8vo. 10*. 
 
 First Series. ' Divided,* ' The Star's Monument/ &;c. 5s. 
 
 Second Series. ' A Story of Doom,' ' Q-ladys and her Island,' &c. 5s. 
 Southey's Poetical "Works. Medium 8vo. 145. 
 Tonge's Horatii Opera, Library Edition, 8vo. 21s. 
 
 RURAL SPORTS, HORSE & CATTLE MANAGEMENT &c. 
 
 Blaine's EncyclopBedia of Bural Sports. 8Vo. 21j. 
 
 Dobson on the Ox, Ms Diseases and thar Treatment. Crown 8vo. 7s. Gd. 
 
 Fitzwygram's Horses and Stables. 8vo. 10s. Gd. 
 
 Francis's Book on Angling, or Treatise on Fishing. Post 8vo. 15*. 
 
 Malet's Annals of the Road, and Nimrod's Essays on the Road. Medium 8vo. 21*. 
 
 MUes's Horse's Foot, and How to Keep it Sound. Imperial 8vo. 12*. Gd, 
 
 — Plain Treatise on Horsfr-Shoeing, Post 8vo. 2s. Gd. 
 
 — Stables and Stable-Fitting. Imperial 8vo. 15*. 
 
 — Remarks on Horses' Teeth. Post 8vo. 1*. Gd. 
 Nevile's Horses and Riding. Crown 8vo. 6*. 
 Beynardson's Down the Road. Medium 8vo. 21*. 
 Ronalds's Fly-Fisher's Entomology. 8vo. 14*. 
 
 Stoneheuge's Dog in Health an4 Disease. Square crown 8vo. 7*. Gd. 
 
 — Greyhound. Square crown 8vo. 15*. 
 
 Touatt's "Work on the Dog. 8vo. 12*. Gd, 
 
 — — — — Horse. 8vo. 6*. 
 "Wilcocks's Sea-Fisherinan. Post 8vo. 12*. Gd. 
 
 WORKS OF UTILITY & GENERAL INFORMATION. 
 
 Acton's Modem Cookery for Private Families. Fcp. 8vo, 63. 
 
 Black's Practical Treatise on Brewing. 8vo. 10*. Gd. 
 
 Buckton's Food and Home Cookery. Crown 8vo. 2*. 
 
 Bull on the Maternal Management of Children. Fcp. 8vo. 2*. Gd. 
 
 Bull's Hints to Mothers on the Management of their Health during the Period of 
 
 Pregnancy and in the Lying-in Room. Fcp. 8vo. 2*. Gd. 
 Campbell-'Walker's Correct Card, or How to Play at Whist. 92mo. 2*. Gd. 
 Cramp's English Manual of BanMng. 8vo. 16*. 
 Cunningham's Conditions of Social Well-Being. 8vo. 10*. G4. 
 Handbook of Gold and Silver, by an Indian Official. 8to. 12*. Gd. 
 Johnson's (W. & J. H.) Patentee's Manual. Fourth Edition. 8vo. 10*. Gd. 
 Longman's Chess Openings. Fcp. 8vo. 2*. Gd. 
 
 London, LONGMANS & CO. 
 
12 General Lists of New "Works. 
 
 Macleod's Economics for Begirmera. Small crown 8vo. 2j. Sd. 
 
 — Theory and Practice of BanMng-. 2 Tola. 8vo. 26*. 
 
 — Elements of Banking. Fourth Edition. Crown Sto. Ss. 
 M'ChiUoch's Dictionary of Commerce and Commercial Navigation. 8to. 63j, 
 Maunder's Biographical Treasury. Fcp. 8vo. 6*. 
 
 — Historical Treasury. Pep. Svo. 6*. 
 
 — Scientific and Literary Treasury. Fcp. Svo. Bs. 
 
 — Treasury of Bible Knowledge. Edited by the Rev, J. Ayre, M.A. Fcp. 
 
 8to. 6*. 
 
 — Treasury of Botany. Edited by J. Lindley, F.B.S. and T. Moore, F.L.S. 
 
 Two Parts, fcp. Svo. 12*. 
 
 — Treasury of Geography. Fcp. Svo. Gs. 
 
 — Treasury of Knowledge and Library of Bfiferenoe. Fcp. Svo. 6*. 
 
 — Treasury of Natural History. Fcp. Svo. Gs. 
 Pereira's Materia Medica, by Bentley and Redwood. Svo. 25*. 
 
 Pcwtner's Comprehensive Spedfla: ; Building-Artificers' Work. Conditions and 
 
 Agreements. Crown Svo. 6* . 
 Pierce's Three Hundred Chess Problems and Studies. Fcp. Svo. 73. 6d. 
 Pole's Theory of the Modem Scientific Q-ame of Whist. F<^. Svo, 2s, Gd. 
 Scott's Farm Valuer. Crown S70. 5s. 
 Smith's Handbook for Midwives. Crown Svo. 5s. 
 
 The Cabinet Lawyer, a Popular Digest of the Laws of England. Fcp. Svo. 9j. 
 West on the Diseases of Infancy and Childhood. Svo. 18;. 
 Willich's Popular Tables for ascertaining the Value of Property. Post Svo. 10*. 
 Wilson on Banking Reform, Svo. 7s. Gd, 
 ' — on the Resources of Modem jOountriea 2 vols. Svo. 24*. 
 
 MUSICAL WORKS BY JOHN HULLAH, LL.D. 
 
 Chromatic Scale, with the Inflected Syllables, on Large Sheet. 1*. Gd. 
 
 Card of Chromatic Scale. Id. 
 
 Exercises for the Cultivation of the Voice. For Soprano or Tenor, 2s. Gd. 
 
 Grammar of Musical Harmony. Royal Svo. 2 Parts, each 1*. Gd. 
 
 Exercises to Grammar of Musical Harmony. 1*. 
 
 Grammar of Counterpoint. Part I. super-royal Svo. 2s. Gd. 
 
 Hullah'8 Manual of Singing. Parte I. & II. 2*. Gd. ; or together, 6*. 
 
 Exercises and Figures contained in Parts I. and n. of the Manual. Books 
 
 I. & n. each 8d. 
 Large Sheets,' containing the Figures in Part I. of the Manual. Nos. 1 to 8 in 
 
 a Parcel. 6*. 
 Large Sheets, containing the Exercises in Part I. of the Manual. Nos. 9 to 40, 
 
 in Four Parcels of Bight Nos. each, per Parcel. 6*. 
 Large Sheets, the Figures in Part II. Nos. 41 to 62 in a Parcel, '9*. 
 Hymns for the Yotmg, set to Music, Royal Svo. Bd. 
 Infant School Songs. Gd. 
 Notation, the Musical Alphabet. Crown Svo. Gd. 
 Old English Songs for Schools, Harmonised, Gd. 
 Rudiments of Musical Grammar. Royal Svo. 3*. 
 School Songs for 2 and 3 Voices. 2 Books, Svo. each Gd. 
 Time and Tune in the Elementary School. Crown Svo. 2*. Qd. 
 Exercises and Figures in the same. Crown Svo. 1*. or 2 Farts, Gd each. 
 
 London, LONGMANS & CO. 
 
 Spottiswoode cfc Co., .PHntersj New-street Square^ London.