UC-NRLF 3k^> i ^ few 4c \ & 4S? J^L^cr*^ TREATISE CONSTRUCTION, PROPERTIES, AND ANALOGIES THREE CONIC SECTIONS BY THE Rev. B. BRIDGE, B. D. F. R. S. FELLOW OF ST. PETER'S COLLEGE, CAMBRIDGE. FROM THE SECOND LONDON EDITION, WITH ADDITIONS AND ALTERATIONS BY THE AMERICAN EDITOR. NEW HAVEN: DURRIE AND PECK. NEW-YORK : COLLINS, KEESE, AND CO, 1839. t \ Entered according to the Act of Congress, in the year 1831, by Hezekiaii Howe, in the Clerk's Office of the District Court of Connecticut. STEREOTYPED BY FRANCIS F. RIPLEY, New Yokk. ADVERTISEMENT, The present edition of Bridge's Conic Sections, is reprinted, with a few alterations, from the second London edition. Such changes only have been made, as seemed necessary to adapt the work to the purpose for which it is. intended, namely, that of furnishing a text- book for recitation in Colleges. To this end, the propositions have been enunciated without the use of letters and without reference to particular diagrams. As it is true, however, that a proposition is more readily comprehended, when it is asserted of the lines and an- gles of some particular figure, immediately before the eye, than when expressed in general terms ; it has been thought proper to introduce the general enunciations at the close of the demonstrations to which they belong, and to leave the author's statements at the commence- ment of those demonstrations unaltered. In this respect the conven- ience of students has been consulted, rather than the usual practice of writers. The demonstrations, as they stand in the original work of Bridge, are in general so much distinguished for conciseness 'and simplicity, and leave so little to be supplied by the student, (a circumstance of great importance in a book designed for large classes,) that it has been thought best to vary from the author only in a very few instances. Alterations have, however, been made, when they seemed likely to be attended by any material advantage. The number of the propositions has been somewhat increased, not for the purpose of completing the enumeration of particular properties of the Conic Sections, which, in a work like this, considering the fer- tility of the subject, would be equally impracticable and useless ; but in order to exhibit to the student how far many truths may be gener- alized, which he is apt to consider as limited by particular circum- stances. A few propositions, not of this kind, have been added, as 4 ADVERTISEMENT. being among the more curious of those which Bridge has omitted to notice. For the convenience of students, some references, particularly to- wards the close of the book, have been made to the mathematical treatises of President Day. The original numbering of the Properties and of the Articles has been suffered to stand ; and whenever any thing has been inserted in the body of the work, the number of the preceding article has been repeated with a letter annexed. The additional Properties are dis- tinguished by the capitals A, B, C, &c. A few notes contain what- ever else is peculiar to this Edition. F. A. P. BARNARD. Yale College, June 20, 1831. CONTENTS. CHAPTER I. Introduction. Page. Sect. I. On the nature of the Curves arising from the cutting of a Cone obliquely to its base, ..... 9 II. On the mode of describing the Conic Sections upon a plane, 12 CHAPTER II. On the Parabola. III. Definitions, . 18 IV. On the Properties of the Parabola, .... 19 CHAPTER III. On the Ellipse. V. Definitions, 34 VI. On the Properties, of the Ellipse, .... 35 CHAPTER IV. On the Hyperbola. VII. Definitions, . 57 VHI. Properties of the Hyperbola analogous to those of the Ellipse, 59 CONTENTS. Page. IX. On the Properties of the Hyperbola derived from its re- lation to the Asymptote, 73 X. On the Properties of the Equilateral Hyperbola, . 81 CHAPTER V. On the Curvature of the Conic Sections. XI. On Curvature, and the Variation of Curvature, • .85 XII. On the Curvature of the Parabola, • 92 XIII. On the Curvature of the Ellipse, 94 m XIV. On the Curvature of the Hyperbola, ... 97 CHAPTER VI. On the analogous Properties of the Three Conic Sections. XV. On the changes which take place in the nature of the * Curve described upon the surface of a Cone, during the revolution of the plane of intersection, . . .99 XVI. On the mode of constructing the Three Conic Sections by means of a Directrix, and the Properties derived therefrom, . . 102 XVII. On the analogous properties of the Normal, Latus-rec- tum, Radius of Curvature, &c. &c. in all the Conic Sections, Ill CHAPTER VII. On the method of finding the dimensions of Conic Sections whose Later a-recta are given, and of describing such as shall pass, through certain given points. XVIIL On the method of finding the dimensions of Conic Sec- tions, whose latera-recta are given, 117 CONTENTS. 7 Page. XIX. On the method of describing Conic Sections which shall pass through three given points, 121 CHAPTER VIII. On the Quadrature of the Conic Sections. XX. On the relation which obtains between the areas of Conic Sections of the same kind, having the same vertex and axis ; and on the Quadrature of the Parabola, Ellipse, and Hyperbola, 126 XXI. On the Quadrature of the Parabola, according to the method of the Ancients, 133 CONIC SECTIONS. CHAPTER I. INTRODUCTION. A cone is a solid figure formed by the revolution of a right an- gled triangle about one of its sides. (Euc. Def. 11. 3. Sup.) From the manner in which this solid is generated, it is evident that if it be cut by a plane parallel to its base, the intersection of the plane with the solid, will be a circle, since this section will coincide with the revolution of a perpendicular to the fixed side of the triangle ; and if it be cut by a plane passing through its vertex, the intersection will be a triangle, the sides of which will correspond to the hypothenuse of the generating triangle, in different positions, or at different periods of the revolution. If the plane by which the cone is cut be not par- allel to the base, or do not pass through the vertex, then the line tra- ced out upon its surface will be one of those curves more particular- ly distinguished by the name of Conic Sections, the properties of which are to be made the subject of the following Treatise. (1.) Let BEFGp be a cone, and let it be cut by a plane EEnG perpendicular to its base and passing through its vertex ; then the section BEG will be a triangle. Next, let it be cut by a plane pAon at right angles to the plane BEwG, and parallel to a plane touching the side BE of the cone ; then the curve line pPAOo, which is form- ed by the intersection of this latter plane with the surface of the cone, is called a Parabola. C.S 2 10 INTRODUCTION. For the purpose of investigating the nature of this curve, let CPDON be a plane parallel to the base of the cone ; the intersection CPDO of this plane with the cone will be a circle. Since the plane BEnG divides the cone in- to two equal parts, CD (the com- mon intersection of the planes BEnG, CPDON) will be * di- ameter of that circle ; and for the same reason EG will be a diameter of the circle EpGoF. Let ANn be the common inter- section of the planes BEnG, p Aon, and PNO, pno, those of the plane pAon, with the planes CPDON, E^GoF respective- ly. Because the planes pAon, p CPDON are perpendicular to the plane BEnG, PNO must be perpendicular to the plane BEnG, (Euc. 18. 2. Sup.) and conse- quently perpendicular to the two lines AN, ND drawn in that plane ; (Euc. Def. 1. 2. Sup.) for the same reason pno is perpendicular to the two lines An, nG. Hence by the property of the circle CNx PN 2 pit- ND=PN 2 , or ND= TW f and EnxnG=pn 2 , or nG=%— CN En Now since An is parallel to BE, and CD parallel to EG, the fig- ure CNnE is a parallelogram; .\CN=E?&. By similar triangles AND, AnG ; AN : An : : ND : nG : : ^J ;?£l : : (since CN=En) CN En v ' PN 2 :pn 2 . (2.) Hence the nature of the curve APp is such, that if it begins to be generated from the given point A, and PN is drawn always at right angles to AN, AN will vary as PN 2 . And the same may be said with respect to the relation of AN and NO on the other side of ANn. INTRODUCTION. 11 (3.) Next, let the plane MPAoM be drawn, as before, perpen- dicular to the plane BEG, but passing through the sides of the cone BE, BG ; then the curve MPAoM, formed by the intersection of this plane with the surface of the cone, is called an Ellipse. In this case, draw two planes, CPDON, UpKon, parallel to the base of the cone ; then, for the same reason as before, PN will be perpendicular both to AN and ND, and pn will be perpendicular both to An and nK ; .-. NCxND=PN 2 , and rcHxrcK=^w 2 . By sim. triangles AND, AriK ; we have AN NM ANxNM An rcM AnxnM. MNC, JVbiH; ND : nK, NC : nR ; NCxND : nUxnK PN 2 : pri*. (4.) The nature of the curve APM therefore is such, that if A and M are given points, and PN be always drawn at right angles to AM between the points A, M, ANxNM will vary as PN 2 ; and the same with respect to the relation between ANxNM and NO 2 . 12 INTRODUCTION. (5.) Lastly, let the plane pAon be drawn, as before, perpendicu- lar to the plane BEG, but cutting the side BG in A, and, when produced, meet- ing a plane drawn touching the other side EB produ- ced, in M ; then the curve pVAOo formed by the in- tersection of the plane pAon with the surface of the cone, is called an Hyperbola. Let the plane CPDON be drawn parallel to the base ; then, by similar triangles, AND, AnG ; we have MNC, MrcE ; F:( c/^ /\ N>\ D -""rX / JLL— ps \v n \ ^X AN NM An wM ND NC wG, wE; ANxNM AwxnM : NDxNC : nGxnE : : PN 2 : pn\ (6.) Hence the nature of the curve AVp is such, that if A and M are given points, and PN be always drawn at right angles to AN, the point A lying between M and N, then ANxNM will vary as PN 2 ; and the same with respect to the relation of ANxNM and NO 2 . II. Having thus explained the nature of the curves arising from the intersection of a plane with the surface of a cone, we now proceed to show how these curves may be constructed geometrically. INTRODUCTION. 13 (7.) Let ELF be a line given in position, and LZ another line drawn at right angles to it in the point L. In LZ take any point S, and bisect SL in A. Let a point P move from A, in such a manner that it may always be at equal distances from S and the line ELF (or, in other words, let the line SP revolve round S as a cen- ter, and intersect another line PM moving parallel to LZ, in such a manner that SP may be always equal to PM ;) then the point P will trace out a curve OAP, having two similar branches, AP, AO, one on each side of the line AZ ; which curve will be a Parabola. To show that this curve will be a parabola^ draw PNO at right angles to AZ ; then LNPM will be a parallelogram, and LN=PM =SP ; but LN=AN+AL=AN+AS (since AL=AS by construc- tion,) .-.SP=AN+AS. Let AN=:r, PN=y, SA=a; thenSP=AN+AS =x-\-a, and SN=AN— -AS, =x — a. Now PN 2 =SP 2 — SN 2 , (Euc. 47. 1.) or y 2 =(#-fa) 2 — (x— a) 2 * x*+2ax+a 2 — x*+2ax— a 8 14 INTRODUCTION. Since Aa is a constant quantity, x varies as y 2 , or AN ccPN 2 ; the relation between AN and PN is therefore the same as in Art. 2. ; hence the curve AP is a Parabola.* (8.) Next, take any line SH, and produce it both ways towards A and M. Let a point P begin to move from A, in such a manner that the sum of its distances from S and H may be always the same, (or, in other words, let two lines, SP, PH, intersecting each other in P, revolve round the fixed points S and H, in such a manner that SP+PH may be a constant quantity ;) then the curve APMO tra- ced out by the point P will be an Ellipse. To prove this, it may be observed that when P is at A, then HA-f AS or HS-f-2AS, is equal to that constant quantity ; and when P is at M, SM-fMH or HS+2HM, is equal to the same quantity. Hence HS+2AS=HS+2HM, from which it appears that 2AS = 2HM, or AS = HM. Now SP+PH = HA+AS= HA+HM=AM ; bisect therefore SH in C, and make CM equal to * Geometrically demonstrated thus : Since SP=AN+AS, SP 2 =(AN+AS) 2 = (Euc. 8. 2.) SN 2 +4AS.AN. But (Euc. 47. 1.) SP 2 =SN 2 +PN 2 ; .-. SN 2 +PN 2 =SN 2 -f4AS.AN, or PN 2 =4AS.AN. INTRODUCTION. 15 CA, then M will be the point where the curve cuts the line SH produced ; and AM will be the constant quantity to which SP-f PH is equal. Let Then AC or CM=a, SC or CH=6, and SP+PH=AM= 2AC=2a; if .-.SP=a— z, HP will be equal to a+z. AN=AC-CN=a— *, NM=CM+CN=a+^ CN=-r, [-SN -SC— CN=6— x, and PN=y; j NH =CH+ CN=6 -{-#, Draw PNO at right angles to AM, then (Euclid, 47. 1.) we have, HP 2 =PN 2 +NH*, or (a+z)* =y 2 +(b+x) 2 , (A) and SP 2 =PN 2 -j-NS 2 , or (a— zy=y*+(b— x)\ (B) bx Subtract (B) from (A), then 4.az=4.bx, or #=— ; substitute this value for z in equation (A), and it becomes (bx\ 2 which reduced is a 4 +2a 2 ^+6 2 ^ 2 =a 2 y 2 +a?b 2 +2a' i bx+a' i x*, or a 4 — a 2 6 2 — a 2 ^ 2 +6 3 ^=a 2 y 2 , i.e. (a 2 — 6 2 )x(« 2 — # 2 )=a 2 y 2 . But since a and 6 are constant quantities, a 2 — # 2 varies as y 2 ; now a*—a;*=(a—x) X (a+#) ; .-. (a—x)x(a+x) ocy 2 , or ANxNM ocPN 2 ; hence, as N lies between A and M, the relation between ANxNM and PN 2 is such, that the curve APM is an Ellipse. (9.) Lastly, Take any line SH, and let the two lines SP, HP, inter- secting each other in P revolve round the fixed points, S, H, in such a manner that the difference of the lines HP and SP (viz. HP — SP) may be a constant quantity ; then the curve traced out by the point P will be an Hyperbola. 16 INTRODUCTION. In this case, let A be the point where the curve cuts SH ; bisect SH in C, and take CM=CA. Since CH=CS, and CM=CA, HM will be equal to AS. Now when P comes to A, HA — AS= a con- stant quantity ; but HA — AS=HA — HM=AM ; v AM is that con- stant quantity. Hence AM=HP — SP. Let Then AC or CM=a, i AN =-CN-CA=^— a, SC or CH=6, I NM-ON+CM=:r+a, CN-ir, j NS =ON— CS-^— 6, i and PN=y; NH =CN+CH=^+6, and HP— SP=AM= 2AO=2a; if ..BF=z-\-a J SP will be equal to z — a. Draw PNO at right angles to AN, then we have HP 2 =PN 2 +NH 2 , or (*+a) 2 =y 2 -f(:r-H>) 2 3 (A) SP 2 =PN 2 +NS 2 , or {z-af^y^x-bf. (B) bx Subtract (B) from (A), then Aaz=ibx, and *— — ; substitute this (bx \ 2 \-a 1 ^yi+^x+bf, which reduced is b*x*+2a?bx+a*=a 2 i/*+a 9 x*+2a*bx+a*b*, or b*x*—a 2 x*— a*b*+a*=a*y\ i. e. (6*-— a 2 ) x (*•— a 2 )=a 2 y 2 . INTRODUCTION. 17 Hence x 2 — a? ooy 2 , or (ar— a) x (x+a) ccy 2 ; i. e. ANxNM ocPN 2 ; and since A lies between N and M, the relation between ANxNM, and PN 2 , is the same with that in the Hyperbola.* Having thus established the identity of the curves generated by these two different methods, we now proceed to demonstrate their properties, beginning with the parabola. * The same may be proved geometrically, as follows. The dem- onstration is applicable either to the Ellipse or Hyperbola. Take AI=SP. Then IM=HP. .-. HP=CI+CA, and SP= CI co C A. Now (Euc.47. 1.) (CI+CA) 2 (=HP 2 )=PN 2 -f(CN-fCS) 2 (=HN 2 ); and, (CI^CA) 2 (=SP 2 )=PN 2 +(CN^CS) 2 (==SN 2 ). That is, CI 2 + 2CA.CI+CA 2 =PN 2 +CN 2 +2CN.CS+CS 2 , an d CI 2 — 2CA.CI+CA 2 =PN 2 +CN 2 — 2CN.CS+CS 2 . Subtract, and 4CA.CI=4CN.CS or CA.CI-CN.CS, .-. CA : CN : : CS : CI, and CA 2 ; CN 2 : : CS 2 :CI 2 . From A, draw AG at right angles to AC ; make AG a mean proportional between AS and SM, and join CG, meeting PN in D. Then AG 2 =AS.SM=CS 2 ^ CA 2 , (Euc. 5. 2. cor.) and CS 2 =CA 2 ± AG 2 .* By sim. tri. CA 2 : CN 2 :: CA 2 ±AG 2 (CS 2 ) : CN 2 ±ND 2 ,* but (as above) CA 2 : CN 2 : : CS 2 : CI 2 , .-. CI 2 =CN 2 ±ND 2 . In the first equation as expanded above, therefore, let CS 2 ± AG 2 1 be substituted for CA 2 , CS.CN for CA.CI, and CN 2 ±ND 2 for CI 2 , and we have CS 2 ±AG 2 +2CS.CN+CN 2 ±ND 2 =PN 2 +CS 2 +2CS. CN+CN 2 , or ±AG 2 ±ND 2 =PN 2 , that is, AG 2 ^ND 2 =PN 2 . But (sim. tri.) AC 2 : AG 2 :: CN 2 ^ CA 2 ( AN.NM) ; AG 2 * ND 2 (PN 2 ). But the ratio AC 2 : AG 2 is constant. Hence AN.NMx PN 2 , which (N being between A and M) is the property of the Ellipse, and (A being between N and M) is the property of the Hyperbola. * The sign — for the Ellipse, and + for the Hyperbola. t The sign -f for the Ellipse, and — for the Hyperbola. C. S. 3 18 ON THE PARABOLA. CHAPTER II. ON THE PARABOLA. III. DEFINITIONS. (10.) Let pAP be a parabola generated by the lines SP, PM, moving according to the law prescribed in Art. 7. ; then the line ELF, which regulates the motion of the line PM, is called the Di- rectrix ; the point S, about which the line SP revolves, the Focus ; the line AZ, which passes through the middle of the curve, the Axis ; and the highest point A, the Vertex of the parabola. ON THE PARABOLA. 19 (11.) Let fall the perpendicular PN upon the axis AZ, and through the focus S draw BC parallel to it, and meeting the curve in the points B and C. PN is then called the Ordinate to the axis, AN the Abscissa ; and the line BC is called the Principal Latus- rectum, or the Parameter to the Axis. (12.) Produce MP in the direction PW, or, in other words, draw PW parallel to the axis AZ ; from any point Q, of the parabola draw QVq parallel to a tangent at P ; and through S draw be parallel to Q,V. PW is called the diameter to the point P ; Q,V the ordinate, PV the abscissa, and be the parameter, to the diameter PW. (13.) Let PT touch the curve in P, and meet the axis produced in T, draw PO at right angles to PT, and let it cut the axis in O. PT is called the tangent, TN the subtangent, PO the normal, and NO the subnormal, to the point P. IV. On the Properties of the Parabola. Property 1. (14.) The Latus-rectum BC is equal to 4AS. Draw BD (Fig. in page 13.) parallel to LZ, then SB=BD=SL. But since SA=AL, SL is equal to 2AS ; hence SB=2AS, and 2SBorBC=4AS. This proposition may be thus enunciated. The latus-rectum is equal to four times the distance from the fo- cus to the vertex. Property 2. (15.) The tangent PT bisects the angle MPS. Take Vp so small a part of the curve, that it may be considered as coinciding with the tangent, and consequently as a right line. Join Sjo, and draw pm parallel to AZ ; let fall po, pn, perpendiculars upon SP, PM. 20 ON THE PARABOLA. The figure Mnpm is a parallelogram, .:nM==pm ; and since po is at right angles to SP, it may be considered as a small circular arc described with radius Sp, .-. So=Sp. Also SP=--PM, and Sp=pm. Now Po-SP— So=SP— Sp, i and P«=PM— wM=SP- pm, V ..Vo=Vn. =SP— Sp, S In the small right-angled triangles Vpo, Vpn, we have therefore "Pp common, and Po=P/i, .-. (47. 1.) po=pn ; having .:. their three sidss equal, the angle pVo must be equal to the m M F A ^ C,Z s Eln N ^4p (16.) Cor. Since the angle MPS continually increases as P moves towards A, and at A becomes equal to two right angles, the tangent at A must be perpendicular to the axis. For, if not, let it take some other direction as P#, cutting RQ, in M.* By Hyp. AN : AH : : AH : AR (Euc. 12. 5.) AN : AH : : AN-f AH(NH) : AR+AH(RH) and AN 2 : AH 2 : : NH 2 : RH 2 : : (sim. tri.) PN 2 : MR 2 (Euc. Def. 11. 5.) AN 2 : AH 2 : : AN : AR : : (7.) PN 2 : QR 2 , .-. PN 2 : MR 2 : : PN 2 J QR 2 , or MR 2 =QR 2 , which is impossi- ble, unless HP produced pass through Q. Therefore, &c. Cor. Hence if HQ, cuts the curve, AN : AH : : AH : AR. The demonstration is not essentially dhferent from any other ar- rangement of the points A, P and Q,. To prove that the tangent bisects the angle SPM, let the ordinate PN be drawn. * M is taken between Q, and R, because, if taken on the other side of &, P# must cut the curve. 22 ON THE PARABOLA. I Property 3. If PT meets the axis produced in T, then SP=ST, and TN= 2AN. (17.) Since PM is parallel to TZ, the angle MPT=alternate an- gle STP ; but (by Prop. 2.) T, it will di- vide the triangle PST into two equal triangles ; consequently TY= YP ; but (18.) TA is also equal to AN ; .-. TY : YP : : TA : : AN ; hence, (Euc. 6. 2.) AY is parallel to PN, and consequently perpen- dicular to the line AZ. That is, the vertical tangent intersects any other tangent, in the point where a perpendicular from the focus upon that tangent intersects it. (30.) Cor. Since the normal PO is perpendicular to PT, it is parallel to SY, .-.TS : SO :: TY : YP; but TY=YP, .-.TS=SO. Hence, (since SP=TS=SO,) if a circle be described with center S at the distance SP, it will pass through the points P, T, and O ; and the < OSP at the center will be double of the angle OTP at the circumference. Property 12. (31.) PO is a mean proportional between BS and bg. Since PY-YT, OS=ST=SP, and TO=2SP=6^ (24.) Also (21.) ON=BS. But (Euc. 8. 0. Cor.) ON : OP : : OP : OT .•.BS:OP::OP:&g-; that is, the normal is a mean proportional between the semiparame- ters of the axis and the diameter at the point of contact. ON THE PARABOLA. 31 (32.) Cor. I. SA, SY and ST=SP, are severally halves of ON, OP and OT. .-. SA : SY : : SY : SP ; and SY 2 =SA.SP ; or SY= V(SA.SP), and as SA is constant, SY ocV(SP.) (32.a.) Cor. 2. Since OP 2 =BS.^, and BS is constant ; OP 2 oc bg cc2bg. And OPccV(2&g\) That is, the normal varies as the square root of the parameter to the diameter at the point of contact. (32.6.) Cor. 3. Since SO=SP, M comes to B, SB+BH=2AC ; hence 2SB or 2BH=2AC ; .-. SB or BH=AC. That is, the distance from either focus to the extremity of the axis minor is equal to the semi-axis major. Property 2. (37.) MSxSA=BC 2 For BC 2 =SB 2 — SC 2 (Euc. 47. 1.) =AC 2 — SO 2 (by Prop. 1.) =(AC+SC)X(AC— SO) =(CM+SC)x(AC— SC) (forCM=AC) =MSxSA. That is, the rectangle of the focal distances from the vertices is equal to the square of the semi-axis minor. 36 ON THE ELLIPSE. (38.) Cor. In the same manner, it might be shown, that AH x HM=BC 2 . Property 3. (39.) The latus-rectum LST is a third proportional to the major and minor axes. For SL+LH=2AC(by const: .-.LH=2AC — SL, and LH 2 =4AC 2 — 4AC xSL+SL 2 , Again, LH 2 =SL 2 +SH 2 (Euc. 47. 1.) =SL 2 +4SC 2 (for SH=2SC) =SL 2 +4(SB 2 — BC 2 ) =SL 2 +4(CA 2 — BC 2 ). Hence 4AC 2 — 4ACxSL+SL 2 =SL 2 +4AC 2 — 4BC 2 ; .-.4ACxSL=4BC 2 . And putting this equation into a ; 2AC : 2BC : : 2BC : 2SL, proportion, we have \ or AM : BO : : BO : LT. Therefore the latus-rectum is a third proportional to the major and minor axes. (39a.) AS.SL=iLT.AM. For (39) B0 2 =LT.AM. .-. iB0 2 (=BC 2 )=(37) AS.SM=^LT. AM. Property 4. (40.) Produce SP to p ; then if YZ bisects the angle HP/?, it will be a tangent to the Ellipse in P. (Fig. in page 29.) For if YZ does not touch the ellipse, let it cut it in Q, ; take Pp= PH, and join pB, QS, QH, and Qp. Since P^=PH, PZ com- mon, and /(|^) «\/(|f)> BC being constant ' Property D. (66.a.) Let the vertical tangents AE, MF be drawn ; then EA.FM= BC 2 and EF is the diameter of a circle, passing through S and H. If P coincide with B, then E A and FM each =BC, and E A.FM= BC 2 . But if not, let the tangent PE intersect the axis in T. Then (sim. tri.) EA : SY : : TA : TY, and HZ : FM : : TZ ; TM. But (Euc. 36, 3. Cor.) TA.TM=TY.TZ, or TA:TY::TZ:TM; .-. EA : SY : : HZ : FM, and EA.FM=SY.HZ=(65.)BC*. Again, (38.) AH.HM=BC 2 =EA.FM, .-. AH : EA : : FM : HM. Hence (Euc. 6. 6.) the triangles EAH and HFM are similar, and < QHS, ?S> QS. (Euc. 24. 1.) In the right angled triangles SyP, syV } Vy is common, and H?— qs=AM ; .-.Hd— Gls> E.$. Or HQ,>Hs+Q,s ; which (Euc. 20. 1.) is impossible. Hence TP does not cut the curve, that is, it touches it. Therefore, if from the foci two straight lines be drawn to any point in the curve, the straight line bisecting the angle contained by these lines, is a tangent. (77.) Cor. When P comes to A, the N 2 — PN 2 = 2BC 2 ; in the same manner, if nip be produced to q, it may be proved that qm 2 - pm*=2AC\ That is, the square of any ordinate to either axis is less than the square of the same ordinate produced to the conjugate Hyperbola, by twice the square of the semi-axis, to which it is parallel. Property 7. (Prop. 8. of Ellipse.) (84.) If the ordinate Vn be drawn to the minor axis, then BC 2 -}- Cn 2 : Prc 2 : : BC 2 : AC 2 . In this case, Prc=CN, and C/i = PN; therefore ANxNM :CN, .CTa (CN 2 — AC 2 =)Prc 2 — ACT and PN 2 = Cti 2 ; hence, by substitution in Art. 81, Cor. 1, we have, P^—AC 2 : Cw 2 : .-. Prc 2 — AC 2 : AC 2 : Compone?ido, Pra 2 : AC 2 : or BC 2 +Cw 2 : Pn 2 : AC 2 :BC 2 ; Crc 2 : BC 2 . BC 2 +Cw 2 :BC 2 , BC 2 : AC 2 . ON THE HYPERBOLA. 65 That is, as the square of the minor axis is to the square of the major, so is the sum of the squares of the semi-minor, and of the distance from the center to any ordinate upon the minor, to the square of that ordinate. Property 8. (Prop. 11. of Ellipse;) % (85.) If the tangent PT cuts the major axis in T, and the minor axis in t, then CNxCT=AC 2 , and C*xCrc=BC 2 . (See last Fig.) Since PT bisects the angle UPS, by Euc. 3, 6. we have HT: TS :: HP :PS; .-. HT— TS(2CT)* : HT+TS(SH) : : HP— PS(2AC) : HP+PS. (A) But by Euc. 12. 2, HP 2 =HS 2 + PS 2 +2HSxSN ; .-. HP 2 — PS 2 =HS 2 +2HSxSN, =(HS+SN) 2 — SN 2 , =HN 2 — SN 2 ; .-. HN— SN(a^HP— PS(2AC) : : HP+PS ; HN+SN(2CN).| (B) Hence we have, / § / I : : 2AC : HP+PS, (A) tH ^kY>^ m j • SH : 2AC . 2CT : 2AC d # CT : AC HP+PS : 2CN ; (B) 2AC : 2CN, AC ; CN, or CNxCT=AC 2 . C*t \ ( C**CTk A AC : CT (and first : third CN 2 : CA 2 ; CN 2 — AC 2 : AC 2 :: PN 2 : BC 2 . <- N\ tj- But bv sim. As, PTfr, TC*, NT : CT : : PN : Ct. Hence, PN : Ct : : PN 2 : BC 2 , or PNxC*=BC 2 , but PN=Crc, .-. C?*xC*=BC 2 . (86.) Since CN : AC first 2 : second 2 ) CN : CT dividendo, NT : CT * For HT— TS=HC+CT— T%=SC+CT— (SC-CT)=2CT. t For HN+SN=HS+2SN=2CS+2SN=2(CS+SN)=2CN. c.s. 66 ON THE IIYPERBOLA. Therefore, if a tangent and ordinate be drawn from any point of the curve to either of the axes, half that axis will be a mean pro- portional between the distances of the two intersections from the center. (87.) Coi^CN 2 — AC 2 =CN 2 — CN xCT=CNx(CN— CT)=CN x NT. Property A. (Prop. C. of Ellipse.) (87. a.) Let TLG be the focal tangent, or the tangent drawn at the extremity of SL, the ordinate from the focus. Let NPG be any ordinate, produced to meet the tangent TLG. Then SP=NG. *» If AI be taken equal to SP, then IM=HP. z ''' 1 *SZ r n *h ".• SP=Cf-CA, and HP=CI+CA. X P X * P H V * J^ - (Euc. 47. h) (SP 2 )(CI— CA) 2 =PN 2 +(CN— CS) 2 (NS 2 ). H? * ^ And (HP 2 )(CI+CA) 2 =PN 2 +(C^+CS) 2 (NH 2 ). fexjmnd and subtract 4CI.CA=4CN.CS, and ^ ? y C\« \ % CI.CA=CN.CS. .-. CN : CI(SP+AC) : : CA : CS : : CT : CA (85.) ... CN— CT(TN) : SP : : CT : CA. (Euc. 19. 5.) Again, CS.CT=AC 2 , and (70.) CS 2 — AC 2 (=CS.ST)=BC 2 . .-. CS.CT : CS.ST, or CT : ST : : AC 3 : BC 2 : : AC ; SL (75.) .-. ST : SL : : CT : AC : : TN ; SP. ptJ i s P+*£~ A C r *4 /■ tC^c m* /AS ' ON THE HYPERBOLA. 67 But (sim. tri.) ST : SL : : TN : NG. .-. SP=NG ; That is, the distance from the focus to any point of the curve is equal to the ordinate to that point, produced until it meets the focal tangent. Cor. 1. AS=AE and SM=MF. Also C£=AC. For (sim. tri.) CT : C* : : ST : SL : : CT : AC. Cor. 2. Hence* also, since TN : NG is a constant ratio, TN : SP is a constant ratio. •Therefore, if a line be drawn through T per- pendicular to AC, the distance of the point P from that line (=TN) is in a constant ratio to SP, the distance of the same point 'from the focus. This ratio, being (by demonstration above) =CT : CA, is a ratio of less inequality. This perpendicular is the directrix of the Hyperbola. (See Art. 138, et seq.) Property 9. (Prop. 12. of Ellipse.) (88.) If PCG, DCK, be conjugate diameters of the Hyperbola, and OPF be drawn perpendicular to CD produced *if necessary, then POxPF=BC 2 . (See next Fig.) all^| < POX, i Draw Cy parallel to PF. Then because PO is parallel^ Cy, and Ct parallel to PN, the right-angled triangles tCy, PON7 are similar ; .-. Ct : Cy : : PO : PN. But Cy=PF, and PN=Crc, being opposite sides oQk parallelogram ; .-. C* : PF : : PO ;Crc, or POxPF=C*xC?z=BC 2 (86.) Therefore, if from the extremity of any diameter, a perpendicular i , w drawn to its conjugate ; then the rectangle of that perpendicular and the part of it intercepted by the axis major, will be equal to the square of the semi-axis minor. Property 10. (Prop. 13. of Ellipse.) (89.) Draw the ordinates DL, PN, to the major axis, then CN 2 — CL 2 =AC 2 , and DL 2 — PN 2 =BC 2 . (See next Fig.) 68 ON THE HYPERBOLA. • Draw the ordinate mD, and produce it to meet the hyperbola in q, and draw qk perpendicular to the major axis. By Cor. 1. Art. 81. Or 2 -— AC 2 : kq % : and CN 2 — AC 2 : PN 2 : .-.C& 2 — AC 2 : kf\ AC 2 : BC 2 , AC 2 : BC 2 , CN 2 — CA 2 : PN 2 . \/c=mqj mD=CL, and qk=T>Ij ; and by Art. 83. mq 2 - m* JI^AC 2 ; .-.^ 2 — AC 2 =(CA; 2 ~AC 2 =)AC 2 -fmD 2 =AC 2 +CL s hence, by substitution, we have, AC 2 +CL 2 : DL 2 : : CN 2 — AC 2 : PN 2 (A.) By sim. As, DCL, PTN, / 1/4***] DL 2 :CL 2 aequo, AC 2 -f CL 2 ♦ CL 2 Hence AC 2 -f CL 2 : AC 2 PN 2 : NT 2 , CN 2 — AC 2 :NT 2 , CNxNT :NT 2 , (87.) ffl CN : NT. : CN— NT(CT) :CTxCN, : AC 2 (85.) ON THE HYPERBOLA. 69 .-. AC 2 +CL 2 =CN 2 , j. or CN 2 — CL 2 =AC. (90.) Since AC 2 +CL 2 =CN 2 , by substitution in proportion (A), we have, CN 2 : DL 2 : or CN 2 : CN 2 — AC 2 : and div d0 , AC 2 : CN 2 — AC 2 : .-. AC 2 : DL 2 — PN 2 : CN 2 — AC 2 : PN 2 , DL 2 : PN 2 , DL 2 — PN 2 : PN 2 , CN 2 — AC 2 : PN 2 , AC 2 : BC 2 ; -,&* .-. DL 2 — PN 2 =BC 2 . Hence, if ordinates to either axis be drawn from the extremities of any two conjugate diameters, the difference of their squares will be equal to the square of half the other axis. Property 11. (Prop. 14. of Ellipse.) (91.) PC 2 ^CD 2 =ACVBC 2 , PG and DK being conjugate diameters. For by Arts. 89, 90. CN 2 — CL 2 =AC 2 , and DL 2 — PN 2 =BC 2 , .-. CN*-f PN 2 — (CL 2 +DL 2 )=AC 2 — BC 2 , or CP 2 — CD 2 =AC 2 — BC 2 . Hence the din%§ence of the squares of any two semi-conjugate di- ameters is equal to the difference of the squares of the semi-axes. Property 12. (Prop. 15. of Ellipse.) CDxPF-ACxBC. Tn Art. 90. it was proved that CN 2 : DL 2 : .-.CN :DL : or CN :AC : AC 2 : BC 2 ; AC :BC, DL : BC. 70 ON THE HYPERBOLA. But by sim. *s, TCy, DCL, CT : Cy(PF) : : CD : DL. P Hence we have, CN : AC and CT: PF .-. CNxCT(AC 2 ) : ACxPF or AC: ' PF fr-* 1 **** i .-. CDxPF=ACxBC. DL : BC, CD : DL ; CD : BC, CD:BC; That is, if from the extremity of any diameter, a perpendicular be drawn to its conjugate, the rectangle of that perpendicular and the semi-conjugate, is equal to the rectangle of the semi-axes. (92.) Cor. Hence it appears, that all the parallelograms inscri- bed in the Hyperbolas, and having their sides drawn through the ex- tremities of any diameter and its conjugate, are equal to each other and to the parallelogram described about the major and minor axes ; for the parallelogram eghf (see Fig. in page 61.) described about the conjugate diameters PCG, DCK, is equal to four times eDCP= 4CD X PF=4AC x BC= right-angled parallelogram whose sides are 2AC and 2BC= parallelogram described about the major and minor axes. Property 13. (Prop. 16. of Ellipse.) (93.) If SY, HZ, be perpendiculars drawn from the foci to the tangent PYZ, then the points Y and Z are in the circumference of circle described upon the major axis AM. Join YC, and produce SY to meet HP in W. Since the tangent PYZ bisects the < HPS ; in the right-angled triangles WPY, SPY, we shall have PW=PS, and WY=YS. Now since WY=YS, and HC=CS, CY must be parallel to HW, and .-. SC : SH : : CY : HW ; but SC=|SH ; .-. CY=£HW=4(HP — PW)=^(HP— PS)=4AM=AC ; .-. Y is a point in the circle whose center is C, and radius C A. In the same manner it might be proved that Z is a point in the same circle. ON THE HYPERBOLA. 71 Hence, if perpendiculars be dropped from the foci upon any tan- gent to the hyperbola, the intersections of those perpendiculars with the tangent will be in the circumference of a circle described upon the axis major. Property 14. (Prop. 17. of Ellipse.) (94.) SYxHZ=BC 2 . Since the Property B. (Prop. D. of Ellipse.) (95.a.) Let the vertical tangents AE', MF' be drawn ; then E'A.F'M=BC 2 , and E'F' is the diameter of a circle passing through S and H. By sim. tri. E'A : SY : : TA : TY, and HZ:FM::TZ:TM. But (Euc. 35. 3.) TA.TM=TY.TZ. Or, TA : TY : : TZ : TM. .-.E'A: SY::HZ : F'M, and E'A.F'M=SY.HZ=(94.)BC 2 . : Again, (74.) AH.HM=BC 2 =E'A.F'M, .-. AH : E'A : : PI : HM. Hence (Euc. 6. 6.) the triangles E'AH and HF'M are similar, and Z.pL=BC 2 ; also PL.L/)= BC 2 and pZ.ZP=BC 2 . By Cor. 1. Art. 81, CN 2 — CA 2 : PN 2 : : AC 2 : BC 2 . By sim. As, LNC, aAC, CN 2 : LN 2 : : AC 2 : Aa 2 (BC 2 ) ... ON 2 : CN 2 — CA 2 : and dividendo, AC 2 : CN 2 — CA 2 : or AC 2 : LN 2 — PN 2 : LN 2 : PN 2 , LN 2 — PN 2 : PN 2 ; ON 2 — CA 2 : PN 2 , AC 2 : BC 2 i ... LN 2 — PN 2 =BC 2 . But LN 2 — PN 2 =(LN— PN) x (LN+PN)=PLxPZ .-. PLxPZ=BC 2 or PL.Lp=BC 2 . For the same reason, p/xpL=BC 2 or ^/./P=BC 2 . ON THE HYPERBOLA. 77 Therefore, if an ordinate to the axis-major be produced to meet the asymptotes, then the rectangle of the segments intercepted be- tween the curve and either asymptote will be equal to the square of the semi-axis minor. (101.) Cor. 1. Hence VLxP l=plxpL=YL.Lp=pl.lP. Cor. 2. Draw any other ordinate Qq, and produce it to meet the asymptotes in X and Y, then will Q,XxQY=Aa 2 ; hence we have QXxGlY=PLxPZ. Property 17. (102.) Draw any diameter PCG, and produce it tog) draw the ordinate Q,T to that diameter, and produce it to meet the asymptotes R, r ; then QRxQr=TrxTR. (See last Fig.) Through the points P, Q,, draw hi, XY perpendicular to the axis of the hyperbola, and draw the tangent ef at P. By sim. triangles Q,XR, PLe ; QrY, Vfl ; we have aX : aR :: PL : Pe, andqY :__<^ : : PI t Vf) .-. Q,XxQ,Y : QRxQr : : PLxPZ : PexP/. But by Art. 101, aXxdY=PLxPZ ; hence QRxQr=PexP/. In the same manner, by drawing an ordinate through T perpen- dicular to the axis, it might be shown that TrxTR=PexP/; hence QRxQr=TrxTR. Therefore, if an ordinate to any diameter be produced to meet the asymptotes, the rectangle of the segments intercepted between the curve and one asymptote, will be equal to the rectangle of the segments intercepted between the curve and the other. (103.) Cor. 1. Since Q,r=Q/T + Tr, and TR=Q,T-f-QR, we have aRx(dT-fTr) =Trx(QT+QR), or QRxaT+QRxTr=TrxQ,T+TrxQR. Subtract QRxTr from each side of this latter equation, and there results QRxQ,T=TrxQ,T, from which it appears that QR=Tr ; 78 ON THE HYPERBOLA. in the same manner it may be proved that Q,X= Sm / \\ KX / L.^^/ dicular to the diameter AL J then will Ae be equal to DE, and A/ will be equal to DF. Now (by Euc. 8. 6.) in the right- angled triangles AEL, AFK, we have Ae : AE : : AE : AL ; AE 8 .-. Ae or DE= AL - ; also A/ : AF : : AF : AK ; .-. A/ or DF= ON CURVATURE IN GENERAL. 91 AF 2 AE 2 AF 2 -r^=r : hence DE : DF : : -r^- : -r-== . But the curvature of the AK AL AK circles AEL, AFK, (see note page 88,) is measured by the relation which obtains between DE and DF just at the point of contact ; and at that point, AE and AF both become equal to AD (by Art. 120.) and consequently equal to each other. At the point of contact, there- fore, (since AE 2 =AF 2 ) we have DE : DF : : J* : -?- : : AK : AL AK AL ; i. e. curvature of circle AEL : curvature of circle AFK : : di- ameter of AFK : diameter of AEL : : radius of AFK : radius of AEL ; i. e. the curvature in different circles varies inversely as their radii. Theorem 4. (123.) Let now APQ, be any curve, PVO the circle op cur- vature to the point P ; take any arc PCI and through Q, draw RQ,q parallel to the chord P V passing through some given point S ; then (assuming the relation of the quantities PQ, and Q,R at the point of PQ, 2 contact) PV will be equal to :=-=-, P<7 2 By Theorem 2, PV is equal to -*- ; but since the curve and qix circle of curvature coincide at the point of contact, at that point 92 ON THE CURVATURE OF THE PARABOLA. P^ will become equal to PQ,, and qR equal to Q,R, and consequently pa 8 PV= qr (124.) Draw now VO at right angles to PV, and join PO ; then (PVO being a right angle, consequently in a semi-circle) PO will be the diameter of curvature to the point P. Bisect PO in r, then Vr will be the radius, and r the center of curvature to the point P. XII. On the Curvature of the Parabola. Let AQ,P be a Parabola, whose axis is AZ, and focus S ; and let PVO be the circle of a curvature to any point P. Join SP, and produce it to meet the circle of curvature in V, then PV is the chord of curvature passing through the focus. (125.) The Chord PV=4SP. Take any arc QP, so small that it may be considered as coinciding with the circle of curvature, and ON THE CURVATURE OF THE PARABOLA. 93 draw Q,R parallel to SP ; draw also Qiv parallel to the tangent PT, cutting SP in x, and the diameter PW in v ; then Q,RP# will be a parallelogram, and P# will be equal to Q,R. Now since xv is par- allel to PT, and Vv parallel to TS, the &Vxv is similar to the APST ; but by Art. 17, SP is equal to ST ; .-. P#=Pv ; hence Vv is equal to Q,R. Let Qiv move up towards P parallel to itself, then, at the point of contact, Qiv will become equal to Q,P ;* since therefore Vv= (Q P 2 v Q v 2 %s- by Art. 123. = )^~. But U,K / Pv by Art. 22, 4SPxPv=Q,i; 2 ; .-. ^=4SP ; hence PY=4SP= pa- rameter to the point P. That is, the chord of curvature, passing through the focus, is equal to the parameter of the diameter at the point of contact. (126.) SA.P0 2 =16SP 5 . Draw YO at right angles to PV and join PO. Then (124.) PO is the diameter of curvature, and there- fore parallel to SY, which is perpendicular to the tangent PT. Hence the triangles PYO, SYP are similar .-. PO : PY(=4SP) : : SP : SY, PO 2 : 16SP 2 : : SP 2 : SY 2 (=SA.SP, by Art. 32.) PO 2 : 16SP 2 ::SP : SA, .-.SA.P0 2 =16SP 3 . Therefore, a parallelopiped, whose base is the square of the diam- eter of curvature, and whose height is the distance from the focus to the vertex, is equal to 16 times the cube of the dist;mce from the focus to the point of contact. 4gpf Cor. 1. The diameter of curvature ■» ;>^ >v . V(SA) # By Art. 120, Qx becomes equal to QP ; but at the point of con- tact P, the points x and v coincide ; therefore at that point the three lines QP, Q#, Qiv become equal to each other. 94 ON THE CURVATURE OF THE ELLIPSE. PV 3 pvi Cor. 2. SA.P0 2 =-.~ and PO= £r7 ^ r -r V 4 2V(SA) , Cor. 3. The diameter PO (and of course the radius Pr) ocSP* 3. or PV 2 , because SA is constant. (127.) At the vertex A, where SP becomes perpendicular to the tangent, the chord and diameter of curvature will of course coincide ; and in this case each of them becomes equal to 4SA, i. e. to the la- tus-rectum.* The diameter (and consequently the radius) of cur- vature is therefore the least at A ; hence, by Art. 122, the curvature itself will be greatest at A ; and since it varies as =-, i. e. , it will keep continually decreasing as the point P recedes from A. XIII. Chi the Curvature of the Ellipse. (Fig. in next page.) Let APM be an ellipse, PVLO the circle of curvature to the point P ; join PS, PC, and produce them to meet the circle of curvature in the points V, L ; draw YO, LO at right angles to PV, PL, and join PO ; then PV is the chord of curvature passing through the fo- cus ; PL the chord passing through the center ; and PO the diame- ter of curvature. Draw the conjugate diameter DCK ; then, (128.) The chord of curvature (PL) passing through the center 2CD 2 is equal to -«*• . Take any small arc Q,P as before ; draw Q,R parallel to PC, and Q,v parallel to RP ; then will Pv be equal to RQ,. Suppose Qv to move up towards P, then, at the point of con- * For at A, SP becomes equal to SA ; .-. PV=»4SP=~4SA ; and „> 4SP* 4SA* ... ON THE CURVATURE OP THE ELLIPSE. 95 tact, Qv becomes equal to Q,P (120.), and vG becomes equal to PG, i. e. to 2PC. Now by Art. 53, FvxvG : Qv 2 : : PC 2 : CD 2 ; substituting therefore for Pv, vG and Q,v, their values at the point of contact, we have QRx2PC : GIP 2 : : PC 2 : CD 2 , or 2PC :^~ :: 2CD 5 123.), .•.PL=-p^-. (129.) The diameter of curvature (PO)= 2CD S PF The triangles PCF, PLO have a common < at P, and right * .„ ; the radius of curvature (Pr) will con- •.PO= PCxPF PF sequently be equal to „-. 96 ON THE CURVATURE OP THE ELLIPSE. (129.a.) The chord of curvature (PY) passing through the focus U ™—. The triangles PEF, PVO, have a common < at P and AC right v "per) :: 2CD 2 CD 2 PC : PF, .-. PO=— «^- ; and Pr the radius of curvature =^=. -pp- ) :: PF SCD "AC 2CD 2 : PE(AC), .-. PV= --^-= chord of curvature passing through the focus. r 2PT) 2 2RC 2 (134.) At the vertex A, the diameter of curvature — - «■ — r-^ = latus-rectum. Here the analogy between the Ellipse and the Hyperbola ends ; for with respect to the variation of curvature, since the normal PI keeps continually increasing from the point A,* the curvature will continually decrease as the point P recedes from A. (135.) In the equilateral hyperbola (see Fig. in page 82) the latus- rectum is equal to the major axis ; the curvature therefore at the vertex A is the same with the curvature of the circle described upon the major axis. In this case PI=PC (Art. 115); .-.PI 3 ocPC 3 , and in the recess of P from the point A. the curvature varies in the same ratio, viz. ( - or ^pr 3 ) with respect to the two sides of the isosceles triangle CPI, one of which (PC) revolves round the fixed point C, and the other (PI) round the moveable point I, at right an- gles to the curve. Here then is an instance of great symmetry in the curvature of the equilateral hyperbola. * That the radius of curvature varies as the cube of the normal, is proved in the same manner as in Note f, page 96. ANALOGOUS PROPERTIES, &C. 99 CHAPTER VI. ON THE ANALOGOUS PROPERTIES OF THE THREE CONIC SECTIONS. Hitherto we have noticed no other analogies than those which take place between the Ellipse and Hyperbola ; but as the three Conic Sections are derived from the same solid merely by changing the position of the plane which intersects its surface, it may naturally be expected that they will possess many properties common to them all. Previous to the investigation of these analogous properties, it may be worth while to consider the changes which take place in the nature of the section, during the revolution of the plane of intersec- tion from a position parallel to the base of the cone, till it becomes a tangent to one of its sides. XV. On the changes which take place in the nature of the curve describ- ed upon the surface of a cone, during the revolution of the plane of intersection. (136.) Let the triangle BEZG represent the section of a cone perpendicular to its base, and passing through the vertex ; then if the cone be cut by a plane perpendicular to BEZG, and parallel to the base, the section AFD will be a circle. Draw the diameter AD of the circle AFD, and draw AZ parallel to the side BE of the cone. Conceive a plane (at right angles to the plane BEZG) to pass through AD, and afterwards to revolve through the angle DAG till it becomes a tangent to the side BG of the cone. From what was shown in Chapter I. it is evident that whilst this plane revolves 100 ANALOGOUS PROPERTIES OF through the angle DAZ, its intersection APM with the surface of the cone will be an Ellipse, whose major axis is AM ; when it comes into the position AZ, it will be a Parabola, whose axis is AZ ; and that whilst it revolves through the angle ZAG, it will be an Hyper- bola, whose major axis is AM', M' being the intersection of z A and EB produced. It may further be observed, that in the revolution of the plane through the angle DAZ, so long as it cuts the side BE between D, and E, a whole ellipse will be formed upon the surface of the cone. When it comes into such a position as to cut the base, a part only of an ellipse will be formed ; and when it arrives at the position AZ, the point M moves off to an infinite distance, so that the Parabola thus formed may be considered as a part of an Ellipse, whose axis major is infinite. And as at the instant the plane leaves the position AZ in direction Zz, the curve of intersection becomes an Hyperbola, THE THREE CONIC SECTIONS. 101 the Parabola may also be regarded as an Hyperbola, whose major axis is infinite. These three curves therefore approach to identity at the same time that the plane approaches to parallelism with the side BE of the cone. (137.) The same conclusion may be drawn from the algebraic construction of these curves. Let the angle MAZ be equal to the angle ZAz, then the major axis (AM') of the Hyperbola will be equal to the major axis (AM) of the Ellipse.* In each case, find the center C or C, and let the abscissas AN or AN' =#, the ordin- ate PN or P'N'=y, semi-axis major (AC or AC')=a, semi-axis minor =&, AS or AS' (S or S' being focus) =c. Then in the Ellipse NM=AM— AN=2a— x, and MS=AM— AS-2a— c ; in the Hyperbola, N'M'=AM'+AN'==2a+^, and M'S'=AM'+AS'= 2a-fc. Now by Arts. 46, 81. (see Figs, in pp. 40, 64.) we have ANxNM or AN'xN'M' : PN 2 or P'N' 2 : : AC 2 : BC 2 , or xx{2a±x) : y 2 : : a % ; 6 2 . V- Hence y 2 =-j x(2ax±x*) is the general equation between the ab- a scissa and ordinate of the ellipse and hyperbola. But in the Ellipse MSxSA = BC 2 , and in the Hyperbola M'S'xS'A = BC 2 , or (2a±c) Xc=l» 2 , hence by substitution y 2 = 2ac4-c z "L " . 2c*x 2cx 2 , cV _ . * ^— x(2ax + x 2 )4.cx4- = 4- h — w • Conceive now the a 2 v ' a a a* angles MAZ, ZA# to be continually diminished, then the axis major both in the Ellipse and Hyperbola is continually increased, and just at the instant of their approach to coincidence with the line AZ, each of them becomes indefinitely great ; in which case, (supposing x and c to be finite quantities) the three fractional terms of the last equa- * Since AZ is parallel to M'BE, the angles MAZ, ZAz are re- spectively equal to the angles of the triangle MAM' ; which triangle is therefore isosceles. 102 ANALOGOUS PROPERTIES OF tion become equal to nothing; .-. y 2 =4cr, or PN 2 =4ASxAN, which is the property of the Parabola. Hence it appears that a fi- nite part of an Ellipse or Hyperbola whose latus-rectum is finite, but whose axis major is infinite, may be considered as a Parabola ; and vice versa, that a finite part of a parabola may be considered as a part of an ellipse or hyperbola, whose axis major is infinite, and latus-rectum finite. XVI. On the mode of constructing the Three Conic Sections by means of a Directrix, and the Properties derived therefrom. In Chap. I. we have already shown the method of constructing the Parabola by means of a directrix ; we now proceed to show that the Ellipse and Hyperbola may also be constructed by lines revolving in a similar manner. (138.) Let MED be a line given in position ; and from the point E, draw CEC at right < s to MED ; in CEC take any point A, and set off AS : AE : : m : 1. Let the line SP begin to revolve from A round S, and PM move parallel to EC, in such manner that SP may be always to PM as AS to AE (i. e. in the given ratio of m : 1.) ; then the curve generated by the point of intersection P will be one of the Conic Sections. Let fall PN at right < s to AC, and let AN=#, PN=y, AS=c ; then, since AS (c) : AE : : m : 1, we have AE=— ; now PM = m ' NE=AE + AN=— +ar, and SP : Pm(— +A : : m : 1 ; ... SP= m \m / c+mx ; also SN=AN— AS=:r— c. Hence we have, SP 2 = (c+ra:r) 2 =c 2 -f2cra.r+raV 2 , SN 2 = (x— cf = c*—2 cx+x* ; . 7 . SP 2 — SN 2 (=PN 2 )-y 2 =(w-f-l)2c2:-l-(m 2 — l).r«r ON THE THREE CONIC SECTIONS. 103 (139.) Let ra=l, or SP = PM, then m + l=2, and m 2 -— 1 = 0, ... y 2 =4c#, or PN 2 =4ASxAN; hence ALP is a parabola, whose vertex is A, focus S, and axis AC. B M •'B C / P E D a! s T N c (140.) Let m be less than 1, or SP less than PM. On the same side of A with PN, take AC : SC : : 1 : m, or AC : AC— SC(= SA=c) : : 1 : 1 — m, then c=(l — m).AC ; hence (m + \)2cx = (m -f 1) (l—m)x2AC.x=(l—m 2 ).2AC.a;. From C draw BC at right angles to AC, and take BC 2 : AC 2 : : 1— m 2 : 1, then 1— m % = BC 2 B ° 2 A . 1 — 2 , and m 2 — 1= AC 2 ' Substitute these values for 1 — m 2 and BC 5 m 2 — 1, and we have (m + l)2c#= T7 ^x2AC.#, and (m 2 — \)x 2 = AC 5 BC ""AC 2 *^ 5 nowlet AC== «> BC=6, then y 2 =((ra-hl)2c#-f(ra 2 -^ l)^ 2 =)^x(2aa:— x*). Hence by (Art. 137) ALP is an ellipse, whose semi-axis major =*AC, semi-axis minor =BC, and focus S.* * To prove that S is the focus, we have AC : SC : : 1 : m, .-. AC 2 : SC 2 : : 1 : m\ and AC 2 : AC 2 — SC 2 : : 1 : 1— m* ; but AC 2 : BC 2 :: 1 : 1— m\ .-. BC 2 = AC 2 — SC 2 , and SC 2 = AC 2 — BC 2 . 104 ANALOGOUS PROPERTIES OF (141.) Let m be greater than 1, or SP greater than PM. Take C on the other side of A in such a manner that AC : SC : : 1 : m, or AC : SC— AC(=AS=c) : : 1 : m— 1, then c=(wfr— 1).AC, and (m + l)2c=(m-f l)(m— l).2AC = (ra 2 — 1).2AC. From C draw BC at right < s to AC, and take BC 2 : AC 2 : : m 2 — 1 : 1, then RC 2 m 2 — I — xt^' ^ et BC=6, AC=a, and substituting as before, we b 2 have y*——(2ax J r a; 2 )] hence ALP is an hyperbola, whose semi- axis major is AC, semi-axis minor BC, and focus S. Prom this mode of describing the three Conic Sections we deduce the following properties. Property 1. If a tangent be drawn to the extremity of the latus-rectum of any conic section, it will cut the axis, or the axis produced, in the same point with the directrix. Draw the latus-rectum LST ; let LE be a tangent to the curve at L, and cut the axis in E ; then (142.) In the Parabola, SP = PM, .-. AS = AE; hence SE = 2 AS. But by Art. 18, the sub-tangent SE= twice the abscissa AS, .-. E is the extremity of the sub-tangent, and also a point in the di- rectrix. (143.) In the Ellipse, AS : AE : : m : 1, m being less than 1. By construction (140) SC : AC : : m : 1. .-. (Euc. 12. 5.) SC : AC : : SC-fAS(AC) ; AC+AE(EC). Therefore EC is a third proportional to SC and AC ; which is also true (55.) if E be the point where the tangent cuts the axis pro- duced. Hence E is a point both in the directrix and tangent. (144.) In the Hyperbola, AS : AE : : m : 1, m being greater than 1. THE THREE CONIC SECTIONS. 105 By construction (141.) SC : AC : : m : 1 ; .-. (Euc. 19. 5.) SC : AC : : SC— AS(AC) : AC— AE(EC). * Therefore, EC is a third proportional to SC and AC ; which is also true (85.) if E be the point where the tangent cuts the axis. Hence, E is a point both in the directrix and tangent. (145.) This line LE, which is drawn touching the curve at the extremity of the latus-reclum, is called the focal tangent ; from what has just now been proved, it follows therefore that if a line be drawn at right angles to the axis from the point where it is intersected by the focal tangent, that line will be the directrix.* Property A. (145.a.) In the cone YYZ, let APQ be any conic section, and BHG an inscribed sphere, touching the cone in the circle BDG, and the plane of the conic section APQ, in S. Then S is the focus of the conic section APQ,. Also, if the plane of the circle BDG be * The substance of Arts. 138 to 145, inclusive, may very readily be inferred, without the aid of Algebra, from Arts. 18.a., 57.a., and C. S 14 106 ANALOGOUS PROPERTIES OF produced to intersect the plane of the conic section APQ in EF, then EP is the directrix of the conic section APQ,. Let YYZ be a plane passing through the axis VC of the cone, cut- ting the plane of the conic section APQ, perpendicularly in AW, the axis of the conic section (1, 3 and 5,) and cutting the circle BDG in the line BG. Since YB and VG are tangents to the sphere from the same point Y, they are equal* and the axis VC of the cone, which bisects the angle BVG, cuts BG at right angles. For the same rea- son, the axis YC cuts all other lines passing through K in the plane of the circle BDG at right angles, and this plane is, therefore, per- pendicular to the axis YC, and consequently, to the plane YYZ, which passes through it. Since, therefore, the planes of the circle BDG, and the conic section APQ are both perpendicular to the plane YYZ, their common intersection EF is perpendicular to VYZ, and therefore to the lines BX, WX, which it meets in that plane. Draw YL, in the plane YYZ, parallel to AW, intersecting GB, produced if necessary, in L. From any point P in the curve APQ, draw PM at right angles to EF. PM is parallel to WX, and con- sequently to VL. Join VP, intersecting the circumference of the circle BDG in D. Join LD, DM. Since D is in the plane of the parallels PM, VL, the lines LD, DM are in that plane. But they are also in the plane of their circle BDG. Therefore they are in the common intersection of the two planes, and are in the same straight line. Now VD=VB, because both are tangents to the sphere from the same point Y. For the same reason PS=PD. And (sim. tri.) PD : PM : : YD : YL, PS : PM : : VB : YL, a constant ratio. * For the plane of the lines YB VG cuts the sphere in a circle, to which YB and YG are tangents. Hence it follows from Euc. 36. 3, that VB=YG. THE THREE CONIC SECTIONS. 107 Hence, the distance SP, of any point of the curve P, from S, is in a constant ratio to the perpendicular PM, to the line EF ; which is the property of the focus and directrix of the conic section APQ,. Therefore S is the focus, and EF the directrix. Property 2. In any Conic Section, the distance SP= half latus-rectum on • 1— ~,cos. . 110 ANALOGOUS PROPERTIES OP From the extremity of the latus-rectum LST draw LK at right < s to the directrix ; and from p draw pm parallel, and pn perp'en- M Ts L p K 1 n ft aI s in P T\ 1 \^v dicular, to the axis. From the nature of the construction we have, SP : PM (or NE) : : SL : LK (or SE), or SP : SL : : NE : SE ; .-. SP— SL : SL : : NE— SE (SN) : SE, i. e. SP— SL : SN : : SL : SE ; for the same reason, ST— S/> (SL— S^) : Sn : : SL : SE ; .-. SP— SL : SL— $p : : SN : Sn : : (by sim. As) SP : Sp. Hence SPxS/?— SLxSp=SL xSP— SPxSp, or 2SPxSp=SL x(SP+Sp)=SLxPp ; .-. 4SPxSp=2SLxPp=latus-rectumxPp. Cor. 1. Hence p+^| (151.) Cor. 1. Hence fp+o-=dT-* For since SLx ( sp + s P)= ocm a u SP+S^/ 1 , 1 \ 2 2SPxS P ; we have g ^ +_)-_. THE THREE CONIC SECTIONS. . Ill Cor. 2. Since SP— SL : SL— Sp : : SP : Sp, SP, SL and Sp are in harmonical proportion. Or, half the latus-rectum is an harmo- nical mean between the segments, into which the focus of a conic section divides any line which passes through it. XVII. On the analogous Properties of the Normal, Latus-rectum, Radius of Curvature, $*c. $*. in all the Conic Sections. If S be the focus, A the vertex, and P any point in the Parabola, then (Arts. 125, 126.) 4SP= chord of curvature passing through 3 3 4SP^ 2SP 2 the focus : —^-rr = diameter, and -t-&~t «= radius of curvature V(SA) V(bA) to the same point. In the Ellipse and Hyperbola, if C be center, S the focus, AC the semi-axis major, CD the semi-conjugate to the semi-diameter PC, and PF a perpendicular let fall from the point P 2CD 2 to the conjugate diameter, then (Arts. 128, 129.) -p~- = chord of 2CD 2 curvature passing through the center from the point P ; — r^r = chord through focus ; -p^r = diameter, and p^~ = radius of curvature to the same point. Property 1. In every conic section, the cube of the normal divided by the ra- dius of curvature is equal to the square of half the latus-rectum. (152. In the Parabola, (see Fig. in page 30.) since normal PO : SY::TP:TY, and TP = 2TY, .-.PO = 2SY; hence cube of normal = 8SY 3 = (by Art. 32.) 8SP* x SA^ ; radius of curvature 2SP^ cube of normal 8SP*xSA*xSA* f,A rad. of curvature 2SP£ square of 2S A = square of half the latus-rectum. = 4SA 2 112 * ANALOGOUS PROPERTIES OP (153.) In the Ellipse and Hyperbola (see Figures and Properties RC 2 in pp. 49, 67, 68.) POxPF=BC 3 ; / N I»©-5=r, and cube of , BC 8 *S i CD 2 . cube of normal normal = 5==- ; radius of curvature =.™-; hence — = — ? — PF 3 PF rad. of curvature = CD^PF^ = CD 2 ^PF 2== ( by Pr ° pertieS m PP - 5L 69 ') BC« BC* ,BC 2 _, .. . , , square of -r— r =square of half the latus-rectum. AC 3 xBC 2 AC 2 u AC (154.) Cor. Since half the latus-rectum is a constant quantity, the radius of curvature varies as the cube of the normal ; the curvature therefore varies inversely as the cube of the normal in all the Conic Sections ; which accords with what has already been demonstrated in Sections XIII and XIV. Property 2. In any conic section, if a perpendicular (OX) be let fall upon the line SP from the point O, where the normal intersects the axis, then the part PX cut off by this perpendicular is equal to half the latus- rectum. (155.) In the Parabola. Draw the ordinate PN ; then, since (by Art. 30.) SP = SO, the angle SOP = SPO, and PO is common to the two right-angled triangles PXO, PON ; these two triangles are therefore equal and similar ; T A S N o hence PX = NO = (Art. 21.) half the latus-rectum. (156.) In the Ellipse and Hyperbola. (Fig. in p. 113.) Draw the conjugate diameter DCK, then the right-angled trian- gles PEF, PXO are similar; .-.PE(AC) : PF :: PO : PX; hence "PC 2 AC x PX - PO x PF - BC 2 ; .-. PX = ~ = half the latus-rectum. THE THREE CONIC SECTIONS. 113 (157.) Cor. By means of- this Property, if SP be given in length and position, and the latus-rectnm and position of the tangent be also given, we can determine geometrically the position of the axis ; for we have only to make PX equal to half the latus-rectum, and draw XO at right angles to SP, and PO at right angles to the tangent at P, then O (the intersection of XO, PO) is a point in the axis, which being joined to S, gives SO the position of the axis. Property 3. In any conic section, take the arc PQ,, and from the point Q, draw GIT perpendicular and Q,R parallel to SP ; then (assuming the rela- tion of Q,T and Q,R just at the point of contact) the latus-rectum is Q,T a equal to -^ (158.) In the Parabola.' Draw the perpendicular SY upon the tangent PY; then, since the arc Q,P coincides with the tangent at P, the triangle Q,PT continually approaches towards similarity with the triangle SPY as Q, moves up towards P ; and at the point of contact QP : QT : : SP : SY ; .-. QP 2 : QT 2 : : SP 2 : SY 2 , and (dividing the first two terms by C. S. 15 114 ANALOGOUS PROPERTIES OF QR)^ :~:: SP 2 : SY 2 :: (by Art. 32.) SP 2 : SPxSA : : Q,P 2 SP : SA. Now -^-=-(=chord of curvature passing through the fo- cus) = 4SP; hence we have 4SP : ^=-::SP.-.SA, : ^— = 4SPxSA ACj . . ; — — - — = 4S A = latus-rectum. (159.) In the Ellipse and Hyperbola. Draw the conjugate diam- eter DCK, and the perpendiculars PF and SY upon it and the tan- gent ; then the triangles QPT, PEF are similar, .-. QP 2 : Q/T 2 : : PE 2 (AC 2 ) : PF 2 , and ^~ (-jfr^**^ l~i; AC 2 : PF 5 Q,K V AG / U,ri QT 2 2CD 2 xPF 2 2AC 2 xBC 2 2BC 2 QR AC 3 AC ^ AC the latus-rectum. The demonstration of this property of the Conic Sections forms the substance of the first three Propositions of the third Section (B. 1 .) of Sir Isaac Newton's Principia. Property 4. (160.) In every conic section,, the chord of curvature passing through the focus is to the latus-rectum in the duplicate ratio of i THE THREE CONIC SECTIONS. 115 SP : SY ; and the diameter of curvature is to the same in the trip- licate ratio of SP : S Y. QP2 For the chord of curvature passing through the focus = ; =^ 5 -; and Q,T 2 by Property 3, the latus-rectum = ? -=- ; hence the , , * " &P 2 T& 2 chord of curvature : latus-rectum : : — — - : — — — : : GlP 2 : Q.T 2 : : SP 2 : SY 2 : (dK. CIR but diameter : chord of curvature (see Fig. in page 95.) : : SP : SY ; .'. diameter of curvature : latus-rectum : : SP* : SY 3 . Property 5. Let L = latus-rectum of any conic section ; then, in the Parabola, LxSP-=4SY 2 ; in the Ellipse, LxSP is less than 4SY 2 ; and in the Hyperbola, L xSP is greater than 4SY 2 . (161.) In the Parabola. (32) SAxSP = SY 2 , .-.4SAxSP=4SY 2 , or LxSP=4SY 2 , for L=4SA. RC 2 v SP (162.) In the Ellipse. By Art. 66. SY 2 = * , ... 4SY 2 HP 4BC 2 x SP / , 2BC 2 HP = ( for -xc ~ h ' and ' ' ' 4BC2 = 2AC X L ) ^^— P 5 hence L x SP : 4SY 2 : : HP : 2AC : : 2AC— SP* : 2 AC, and as 2 AC — SP is less than 2 AC, L x SP must be less than 4SY 2 . (163.) In the Hyperbola, by a similar process we have LxSP : 4SY 2 : : HP : 2AC : : 2AC+SPt : 2AC, and as 2AC+SP is greater than 2AC, LxSP must be greater than 4SY 2 . * For SP+HP=2AC, .-. HP=2AC— SP. t For HP— SP=2AC, .. HP-2AC+ SP. 116 ANALOGOUS PROPERTIES, &C. (164.) Before we conclude this Section, it will be proper to show the method of expressing the relation between SP and SY, in the form of an algebraic equation. In the Parabola, therefore, let SA— a,' SP=x, SY=y; then since SY 2 =SAxSP, we have y 2 =ax, or y=V(ax,) for the equation to the curve, in terms of the distance from the focus, and the perpendicular from the focus upon the tan- gent. In the Ellipse and Hyperbola, let AC=a, BC=^ SP=:r, SY = y; then HP = 2AC±SP=2a±;r, .-.since SY 2 = — -- — , b 2 x b 2 x 2 bx we have y - ^— = g-^ and y- v(2g ^ where the negative or positive sign must be used according as the section is an Ellipse or an Hyperbola.* (165.) To investigate the relation between CP and Cy (see Fig- ures in pages 49, 68,) let CP=x, Cy or PF=y ; then in the Ellipse, since AC 2 + BC 2 = CD 2 + PC 2 , we have a 2 -{- b 2 = CD 2 + x 2 , ... CD 2 - a 2 + b 2 —x 2 or CD = V (a 2 + b 2 — x 2 .) Again, since ACxBC=CDxPF, we have ab = CT)xy, .-. CD= — ; hence — = V(a 2 -f6 2 — x*,) or y = — — — — — is the equation to the y K ' V(a 2 + Z> 2 — x 2 ) * curve in terms of the distance from the center, and 'perpendicular from the center upon the tangent. In the Hyperbola, PC 2 ^CD 2 = AC 2 ^CB 2 , or # 2 ^CD 2 = a 2 ^b 2 ; ,-. CD 2 = * 2 -a 2 +& 2 , and y- "V^ -t V (# 2 — a 2 -fo 2 ) * These expressions are the equations of the several conic sec- tions, considered as spirals, described by the revolution of the radius vector SP, about the focus. t In these equations, the curves are considered as described by a radius vector CP, revolving about the center. This mode of con- sideration is, of course, inapplicable to the Parabola. METHOD OF FINDING, &C. 117 CHAPTER VII. ON THE METHOD OP FINDING THE DIMENSIONS OF CONIC SECTIONS WHOSE LATERA-RECTA ARE GIVEN, AND OF DESCRIBING SUCH AS SHALL PASS THROUGH CERTAIN GIVEN POINTS. XVIII. On the method of finding the dimensions of Conic Sections, whose later a-recta are given. (166.) Let S be the focus of any conic section, P some point in the curve at a given distance from S ; join SP, and let it meet the tangent PT in the given angle SPT ; let the latus-rectum = L, and take PX=£L ; from X draw XO at right angles to SP, and from P draw PO at right angles to PT, then by Art. 157, O will be a point in the axis join SO, and it will give the po sition of the axis. (167.) We are thus furnished with the means of determining geo- metrically the position of the axis of any conic section whose latus- rectum is given, and whose tangent at a given point meets a line drawn from the focus to that point, in a given angle. The position of the axis being found, its dimensions may be ascertained from the properties of each particular curve. In the Parabola, the latus-rec- tum is equal to four times the distance of the focus from the vertex ; if therefore in OS produced, we take SA equal to £L, A will be the vertex of the Parabola. In the Ellipse and Hyperbola, it will be necessary to find the center, as also the major and minor axis ; which is done in the following manner. 118 METHOD OF FINDING THE (168.) In the Ellipse, the lines drawn from the foci to any point in the curve make equal angles with the tangent at that point ; if therefore the angle HPZ be made equal to the angle SPY, and SO be produced to meet PH in the point H, that point will be the other focus ; and this determines the length (SP+PH) of the major axis. Now by Art. 45, the conjugate diameter DCK cuts off from SP a part equal to the semi-axis major ; hence if PE be taken equal to J(SP-l-PH), and through E we draw DC parallel to the tangent at P, C will be the center of the ellipse. It only remains therefore to produce SH both ways, and make CA, CM each equal to PE, and we have AM the major axis of the curve. But (39) the latus-rec- tum is a third proportional to the major and minor-axis ; the minor axis is therefore a mean pro- portional between the major axis and the latus-rectum ; from C then draw BC at right angles to AM, make BC a A mean proportional between AC and ^L, and B will be the extremity of the minor axis ; thus the dimensions of the el- lipse are determined. BY (169.) In the Hyperbola, the tangent bisects the angle SPH; in this case, therefore, the angle HPY must be made equal to the an- gle SPY on the opposite side of the* tangent ; then if OS is produced till it meets PH in Ihe point H, that point will be the other focus. Produce SP to E, and take PE equal to J(HP— SP) ; through E DIMENSIONS OF CONIC SECTIONS. 119 draw EC parallel to the tangent at P, and C will be the center. Take OA, CM, each equal to PE, then AM will be the major axis. The minor axis is determined precisely in the same manner as in the Ellipse. (170.) We have thus shown the method of solving this Problem, when the nature of the curve is given. Suppose now that the latus- rectum, the distance SP, and the position of the tangent be given as before, and it is required to find not only the dimensions, but the na- ture of the conic section. In this case we have recourse to Arts. 161, 162, 163 ; from which, when the latus-rectum and the relation between SP and SY are given, we can determine the particular na- ture of the curve. For it is there proved, that if LxSP be equal to 4SY 2 , the curve is a Parabola ; if LxSP be less than 4SY 2 , it is H M AS N H M an Ellipse ; and if LxSP be greater than 4SY 2 , it is an Hyperbola. In order to affect this general solution of the Problem, let the sine of the given < SPY=s, radius=l, then (by Trigonometry) SP : SY : : 1:5; .-. SY=s . SP, and SY 2 =s 2 . SP 2 ; consequently 4SY 2 =4s 2 . SP 2 . Having therefore found the position of the axis, as in the former case ; then, to know whether the conic section, whose di- mensions are required, be a Parabola, Ellipse, or Hyperbola, we must compare LxSP with 4s 2 . SP 2 . If LxSP be equal to 4s 2 . SP 2 , i. e. if L be equal to 4s 2 . SP, then the curve is a Parabola; take therefore SA=|L, and A is the vertex. If L be less than 4s f . SP, the curve is an Ellipse ; in which case, make the < HPZ (on the same side of the tangent with SP) equal to SPY, and proceed as in Art. 168. If L be greater than 4s 2 . SP, the curve is an Hy- perbola ; make therefore HPY (on the other side of the tangent) equal to SPY, and proceed as in Art. 169. 120 METHOD OF FINDING THE (171.) By Art. 160, the chord of curvature passing through the focus : the latus-rectum : : SP 2 : SY 2 : : 1 : s 2 ; .-. the latus-rec- tum =s 2 x chord of curvature ; if therefore the chord of curvature and the relation of SP to SY be given, the latus-rectum will also be given. We are thus enabled to give the trigonometrical solution of the following PROBLEM. (172.) Given the chord of curvature passing from any point through the focus of a conic section, the distance of that point from the focus, and the position of the tangent ; it is required to find the nature and dimensions of the conic section. e Let the chord of curvature to the point P=40, SP=12, the angle SPY=30° ; then since the sine of 30°=half radius, s=£ ; .-. L«= (s 2 X chord of curvature =)£ x 40=10 ; also 4s 2 xSP = 4 x^SP = SP=12 ; hence L is less than 4s 2 xSP, and consequently the conic section is an Ellipse. Z Since the < SPY -= 30°, the b , V °\ •b M N If GIN, PN, which are to each other in a given ratio ; then the areas AGIN, APN, generated by those ordinates, will be to each other in the same given ratio. For take the ordinate qpn indefinitely near to Q,PN, and draw dm, Vo parallel to the axis, then the flnxional or incremental areas QNn$ PNrcjo will approach to equality with the parallelograms QJSnm, PNno, as the ordinate qpn approaches to Q,PN ; but (by Euc. 1. 6.) these parallelograms are to each other in the ratio of GIN : PN ; the nascent increments therefore of the areas AGIN, APN are to each other in the ratio of GIN : PN ; and as these areas begin together from A, the areas themselves must also QUADRATURE OF THE CONIC SECTION. 127 be to each other in that ratio, i. e. area AGIN : area APN : : GIN : PN. (178.) Suppose now the curves AQq, APp to be two Conic Sec- tions of the same kind whose latera-recta are respectively L and I ; for instance, let them be two Parabolas ; then by the property of the parabola LxAN = QN 2 , and ZxAN = PN 2 , hence AQN : APN (::QN ; PN) : : V (LxAN) : V(ZxAN)::VL : W. If they be Ellipses or Hyperbolas which have the same major axis AM, and whose minor axes are respectively BC and bC, then ' ANxNM : PN 2 : : AC 2 : 6C 2 , and GIN 2 : ANxNM : : BC 2 : AC 2 5 or>r<2 Q.hC* 2 .-.QN 2 : PN 2 ::BC 2 :&C 2 " AC * AC : :L : I Hence in this case also AGIN : APN( : : aN : PN) : : VL : Vl ; i. e. in Conic Sections of the same kind, having the same vertex and axis, the areas AGIN, APN are to each other in the given subdupli- cate ratio of their latera-recta. (179.) Take any point S in the axis, and join SGI, SP ; then we have area AQN : area APN : : QN : PN, andASaN : A SPN ::QN:PN; .•.AQN— SQN : APN— SPN ::GtN : PN, or area AGIS : area APS : : GIN : PN ::VL; Vl in all the Conic Sections. (180.) But Ellipses and Hyperbolas having the same vertex and axis, will also have the same center.* Let C be that center, and in each case join GIC, PC ; then * Although they have the same center, it should be recollected that (since the minor axes are not equal) they cannot have the same focus. 128 QUADRATURE OF THE PARABOLA. In the Ellipse, AQN : APN : : QN and A QCN : A PON : : QN .-. AQN+QCN : APN+PCN : : QN or sector ACQ, : sector ACP : : QN ::VL PN, PN; PN, PN, Vl. In the Hyperbola, AQN : APN : : QN : PN, and A QCN : A PCN : : QN : PN ; .-. QCN— AQN : PCN— APN : : QN : PN, or sector ACQ : sector ACP : : Q^N : PN, :: VL : Vl. On the Quadrature of the Parabola. (181.) Let AP be a Parabola whose latus-rectum (BC)=a, ab- scissa (AN) =#, ordinate (PN) = y ; then, by the Property of the Parabola, y z =ax, .:y=a 2 x 2 j and y X =a 2 x 2 ' x now the fluent of %xYa 2 x 2 =\xy\ hence the area APN=fANxPN=f yx = %a 2 x 2 = the circumscribing parallelogram. QUADRATURE OF THE PARABOLA. 129 (182.) Draw a tangent to the point P, and produce NA to meet it in T ; then since AN = |NT, the A PNT = (£TNxPN= )AN x PN; hence the area ANP=§ANxPN=f A PNT. Now sup- pose, in the Figure, at page 28, that a tangent be drawn to the point G, and that the line MS drawn parallel to the axis meets it in S, then the area AZG = |A TZG ; but A TZG : A SMG : : ZG 2 : MG 2 : : 1 : 4, .-.A SMG = 4 times A TZG= 6 times area AZG = 3 times area MAG ; hence area MAG =i A SMG. (183.) But the area of a Parabola may be ascertained in terms of the square of its latus-rectum. For let AN : AS : : n : 1, then AN= n . AS ; but PN 2 = 4AS x AN = An . AS 2 , .-. PN = 2AS. Vn ; 3 A?i 2 hence area APN (| ANxPN)= f X n. AS X 2AS xVrc=-^-xAS 2 = 3, n 2 (for AS=£BC, and .-. AS 2 = T V BC 2 V~ >cBC 2 ; or if the whole 3 n 2 Parabola is taken (as in Fig. p. 28,) then the area MAG =— x square of latus-rectum. (184.) Not only the area ANP contained between the abscissa and ordinate, but also the area ASP described by the revolution of the line SP round the focus S, may be ascertained in the same man- ner. For since AN=n.AS, SN = (AN— AS=)(rc— 1).AS ; hence A SPN = (iSN X PN = ) n ^- . AS x PN. Now area APS = area APN— ASPN=fra.ASxPN ^- xASxPN= -y-.ASx PN=(for PN = 2AS Vn) £+J^ xAS»- ( -^±J^ xBC*. (185.) Hence it appears, that if the latus-rectum be given, the par- abolic areas ANP, ASP may be found without any other irrationality C.S. 17 130 QUADRATURE OP THE ELLIPSE. than that which arises from extracting the square root of numbers 5 for iff*-* 1, 4, 9, 16, &c. then Area ANP = fe f, f, y, &c. of the square of the latus-rectum j and Area ASP = fa fa f, if, &c. of the same ; but if n be not a square number, then the expression for these areas will involve an irrational quantity. On the Quadrature of ^Ellipse. (186.) Let ABMO be an Ellipse, and upon the major axis AM describe the circle ARML ; draw any ordinate Q,PN, then by Prop- erty 9, of the Ellipse, Q,N : PN in the given ratio of RC or AC : BC. But from what was proved in Sect. 20, area AQN : area APN : : QN : PN : : AC : BC ; and for the same reason, the semicircle ARM will be to the semi-ellipse ABM in the same ratio ; hence the whole Ellipse ABM : circle ARML described upon its major axis : : BC : AC : : minor axis : major axis. (187.) As the area of the Ellipse bears this given ratio to the area of its circumscribing circle, the quadrature of the Ellipse must therefore depend upon the quadrature of the circle. Let QUADRATURE OF THE HYPERBOLA. 131 p =3.1416 (=*areaof a circle whose radius is 1), then the area of the circle whose radius is AC -■ p X AC 2 ; hence the area of the Ellipse : /;xAC 2 :: BC : AC, .-.area of the Ellipse = p x AC X BC, i. e. the area of an Ellipse is found by multiplying the rectangle under its semi-axes by the same decimal number (p) as the square of the radius is multiplied by, to find the area of a circle. From this it also appears, that the area of an Ellipse is equal to the area of a circle whose radius is a mean proportional! between its semi- axes ; for the area of that circle is equal to (p x(rad.) 2 =p xthe square of V ( AC x BC) - ) p x AC X BC. (188.) The area of the parallelogram circumscribing the Ellipse is equal to 4 AC x BC, .-. area of Ellipse : area of that parallelo- gram : : p x AC xBC ; 4AC xBC : : 2 h or 3.1416 : 4 : : .7854 : 1 ; i. e. the area of an Ellipse has the same ratio to the area of its cir- cumscribing parallelogram as the area of a circle has to its circum- scribing square. On the Quadrature of the Hyperbola. (189.) Let AP/> be an Hyperbola whose semi-axis major AC = a, semi-axis minor bC=b ; and let CN = .r, PN = ?/; then by Cor. 1. Prop. 6. of Hyperbola, CN 2 — CA 2 : PN 2 : : AC 2 : BC 2 , or x % — a 1 iifiia*: 6 2 , .-.y = - V(^ 2 ~« 2 ); hence yx ' = --. x V(^ 2 — a 2 ), a a whose fluent found by a series and properly corrected would give the value of the area APN ; but this area may be ascertained by means of logarithms, when we have found the value of the hyperbo- lic sector ACP. (See Fig. in p. 126.) (190.) Now the area of this sector is thus found. The area of A CPN = -£CN xPN = % .-. the fluxion of the A CPN = &p* . * See Day's Mensuration, &c, Art. 30. t Let x — mean proportional between AC and BC, then AC : x : : x : BC, .-. x> = AC x BC, or x= V (AC x BC.) 132 QUADRATURE OP THE HYPERBOLA. but sector ACP = A PCN — area APN, .-. fluxion of sector ACP = fluxion of A PCN— fluxion of area APN= — ^— — V x = xy ~ vx ; we must therefore find the values of ~ and y ~. Since y = b , „ \ • bxx xy bx°- x u » -a. ion -V(^-a 2 ) 5 , = ^- 2 -_ z ^,.^== 5 ^ 7 ^ 2 -_ z ^; by Art. 189. a^x*^a?f ' ~2 ... 2a V (x^d ■ ; hence x v — y x or ^ bx*x bx V (# 2 — a 2 ) aJa;' ^=^ V( f— } ; hence 2=£ or fluxion of sector ACP = 2 2a 2 the fluent or sec- 2aV(^ 2 — a 2 ) 2a 2v(x 2 — a 2 )' tor ACP = u- X hyp. log. (x -f V (# 2 — a 2 ) ) + Cor. ; when x = a, ACP = 0, ... sector ACP = $ x hyp. log. * + v (*'— g ') . 2 ° a (191.) The triangle CPN=^=^^- 2 ^^ ) J .-.area APN /& /*a (= A CPN- sector ACP)= tov (^- g)_g» x hyp . log . j-L-v (# 2 a 2 ) * — ' i. i. Suppose AQ,a to be an equilateral hyperbola, in which a = 6 = l, then the area AQ,N=£rV(:r 2 — 1) — £ hyp. log. (x-{-V(z 2 — 1) ). A portion of this hyperbola, whose abscissa is equal to its semi-axis major (in which case x = 2) will be numerical- ly expressed by the quantity V3— \ hyp. log. (2-f V3) = 1.7320 — .6584=1.0736 ; thus in Figure page 82, if the abscissa AN be taken equal to AC, then the area (APN) corresponding to this abscissa : square ACBa : : 1.0736 : 1, and area APN : quadrant ACB : : 1.0736 : .7854 : : 1.3669 : 1. QUADRATURE OF THE PARABOLA, &C. 133 XXI. On the Quadrature of the Parabola, according to the method of the Ancients. (192.) Let BQAPC be any portion of a Parabola cut off by the straight line BC ; bisect BC in the point D, and draw DA parallel to the axis ; then AD will be the diameter to the point A, and (by converse of Art. 23.) BC will be an ordinate to that diameter. Moreover, since a tangent to the point A is parallel to BC, A will be the highest point or vertex of the figure B&APC ; if therefore BA, AC, be joined, then this figure and the triangle ABC will have the same base and vertex. (193.) Bisect BD in E, and draw EQ, parallel to DA ; through Q, draw GtNP parallel to BC, and from P draw PF parallel to AD ; then QNP will be an ordinate to the diameter AD in the point N, and Q,E, PF will be diameters to the points Q,, P respectively ; and since Q,EDN is a parallelogram, Q,N will be equal to ED, i. e. to £BD; hence* QN 2 : BD 2 : : 1 : 4 ; but by the property of the Parabola, AN : AD : : QN 2 : BD 2 , .-. AN : AD : : 1 : 4, or AN = £AD ; hence ND or Q,E = f AD. Again, since EG is parallel to DA, and BE = ^BD, EG must be equal to £AD, .-. Q.G = { f AD, and EG: GQ,:: 2:1. (194.) Join AE, Ad, GIB ; then since BD is bisected in E, the triangle ABE is equal to half the triangle ABD (by Euc. 1. 6. ;) and since GQ, is equal to £GE, the triangles A&G, BQ,G are res- pectively half of the triangles AGE, BGE ; hence the triangle AQ,B is half of the triangle ABE, and consequently ^th of the triangle ABD. In the same manner (if AP, PC, AF, be joined,) it may 134 QUADRATURE OF THE PARABOLA. be proved that the triangle APC is {th the triangle ADC ; hence the two triangles AQB, APC, taken together, are equal to one fourth of the triangle ABC. (195.) Now suppose BE, ED were bisected, and from the points of bisection lines were drawn parallel to DA (which will evidently bisect BG, G/.,») then the sum of the triangles formed within the parabolic spaces*' BQ, QA (by drawing lines from the points where those parallel lines cut the curve to the extremities of the chords BQ, QA) will be equal to ]th of the triangle AQBt ; and the sum of the triangles formed in a similar manner within the parabolic spaces AP, PC, will be equal to {th of the triangle APC ; .-. the sum of the tri- angles formed within the four parabolic spaces BQ,, QA, AP, PC is equal to ith of A AQB+Z^VPC, i. e. to T yth of the triangle ABC. By bisecting the halves of BE, ED, &c. and drawing lines as be- fore, parallel to DA, and joining the points of their intersection with the curve to the extremities of the chords, a series of eight triangles would be formed in the remaining parabolic spaces, the sum of Which would be equal to {th of the sum of the triangles formed within the parabolic spaces BQ, QA, AP, PC, i. e. to T * T th of the triangle ABC. We might thus go on bisecting the successive parts of the base BC, and forming triangles in a similar manner, till the whole parabolic figure BAC was exhausted, in which case it is evident that the area of that figure would be equal to the sum of the areas of all the triangles thus formed within it. (196.) Let the triangle ABC = a, then to find the sum of the areas of all these triangles, we have merely to sum the series * By parabolic spaces, we mean such portions of the Parabola as are contained between the arcs BQ, QA. AP, PC, and the straight lines BQ, QA, AP, PC respectively. t For the same reason that the sum of the triangles AQB, APC is equal to ith the triangle ABC, this conclusion being evidently true for the triangles thus inscribed in any portion of a Parabola. BY THE METHOD OF THE ANCIENTS. 135 a _|_ a _j-iL_|-iL -f&c. continued ad infinitum, which is a geometric series, whose first term is a, and common ratio J, Now the sum of this series* = 1 7— =li =-tt 5 •'• tne area °f tne parabola VI— r /l—\ a BAC is equal to f x area of the A BAC. If a tangent was drawn to the point A, and from B, C, lines were drawn parallel to DA, then the triangle ABC would be the half of the parallelogram thus formed ; the parabolic area BAC is therefore fds of the circumscribing paral- lelogram ; which accords with what has already been proved respecting the quadrature of the Parabola in Section XX ; for it is evident the foregoing demonstration is true for the axis, since AD is any diameter. (197.) From the given ratio which subsists between the parabolic area and its inscribed triangle, we may prove, that such portions of a Parabola as are cut oif by ordinates to equal diameters, are equal to one another. Let oAQ (Fig. in p. 136) be any Parabola, and draw the diameters P W, pw to the points P, p ; take PW=p«?, and through W, w, draw the ordinates OWQ, owq ; draw the axis AD ; take AD equal to PW or pw, and through D draw the ordinate BC ; and in the parabolic spaces BAC, OPQ, inscribe the triangles BAC, OPQ. Draw the tangent to the point P, and produce the axis to meet it in the point T ; let S be the focus, and join SP ; from.S let fall SY perpendicular upon the tangent, and draw QF perpendicular upon PW produced. Now 4SAxAD = CD 2 , and 4SPxPW=WQ, 2 ; therefore WQ 2 : CD 2 :: 4SPxPW : 4SAxAD : : (since PW= AD) SP : SA. Again, since the ordinate WQ. is parallel to the tan- gent TP, and the diameter PW is parallel to the axis AD, the tri- angles WQF, STY are similar, .-. WQ 2 : QF 2 j : ST 2 (or SP 2 ) : SY 2 : : (Euc. Def. 11. 5.) SP : SA ; hence WQ 2 : CD 2 : : WQ 2 : QF 2 , .-. CD=QF. But the A PWQ=iPWxQF and the A ADC=JADxCD; since therefore PW, QF, are respectively equal to AD, CD, the A PWQ must be equal to the A ADC. * See Day's Algebra, Art. 442. 136 QUADRATURE OF THE PARABOLA, &C. T Now these A s are the halves of the triangles OPQ and ABC ; hence the A OPQ, is equal to the triangle ABC, and consequently the Parabolic area OPQ, to the parabolic area BAC* In the same manner it might be proved that the parabolic area opq is equal to the area BAC ; .-. the area opq is equal to the area OPQ. These observations upon the quadrature of the Parabola accord- ing to the method of the Ancients, contain the substance of the last seven propositions (viz. from 18 to 24 inclusive) of Archimedes De Quadratures Parabola, and of the fourth Proposition of his book De Conoidibus et Sphceroidibus. * For by Art. 195, the parabolic areas OPQ, BAC are f ds of the triangles OPQ, BAC respectively. THE END. Or THfc "hut THIS BOOK IS DUE ON THE LAST DATE STAM PED Bf LOW ]unl 714777/ AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. 12^ !$$$&&$ 6Jun'55AM 5KkW & — =B18'S4-I2M 24MAR59* 8 L0 TU B . Cm JUL lft'75 MAR 24 1959 gp f g^BsSP **t LD 21-100m-12,'43 (8796s) ii \H-5ZHO THE UNIVERSITY OF CALIFORNIA UBRARY