the tangents at the extremities of the chord p, p' will meet one another in some point of the line LG (III. 53. Cor.); but the straight line Ghp is the tangent at p: therefore Gp' is the tangent at p'. And, because from the point G, in which the tangents Gp, Gp meet one another, the line Gq is drawn to cut the circumference in the points q, r and the chord pp in n, Gq is harmonically divided in these points (Lem.). Therefore (II. def. 20.) the four straight lines V G, V rR, V nN, V q Q are harmonicals; and, consequently, because QR is parallel to V G, it is bisected by V N in N (II. 49. Cor. 1.). Lp (Lem.), and QR is bisected in the point N, as before. But, further, in the ellipse and hyperbola,the diameter P P' is bisected by the centre C. For, since Kp passes through the point c, and is cut by the straight line K L, which is drawn perpendicular to the diameter Oc from a point L so taken that O c produced is harmonically divided, Kp is harmonically divided by K L and the circumference (III. 52. Cor.). Therefore, the four straight lines VK, VpP', V CC, V p P, are harmonicals (II. def. 20.); and, because P P is parallel to V K, it is bisected by VC in C (II. 49. Cor. 1.). Therefore, &c. Cor. 1. In the ellipse and hyperbola, the tangents at the extremities P, P' of any diameter are parallel to one another. For they are the originals of Gp and Gp' in the plane of the circular section; and all straight lines, the projections of which pass through G, are parallel to VG and to one another (IV. 10.). Cor. 2. If any diameter of a conic section bisects a straight line which is not a diameter, the bisected straight line shall be an ordinate of the diameter by which it is bisected. PROP. 18. In every conic section the tangents at the extremities of any ordinate, QR, meet the diameter, PN, in the same point, T; and that in such a manner, that, in the ellipse and hyperbola, C N, CP, and CT are proportionals, and, in the parabola, N P is equal to PT. Let the construction remain as in the last proposition. Then, in the ellipse and hyperbola, because qr, the projection of QR, passes through G, and that Gp, G p' are tangents drawn from G to the circle, the tangents at q and meet one another in some point of pp' produced (Lem.). But these tangents are the projections of the tangents at the points Q and R of the ellipse or hyperbola (6.), and pp' produced is the projection of P P' produced. Therefore, the tangents at Q and R meet one another in some point, T, of P P' produced. Again, because tq and trare tangents drawn from t to the circle, the line tp which passes through t is harmonically divided by qr and the circumference (Lem.). Therefore, the four straight lines V p' P', Vn N, V pP, Vt T, are harmonicals (II. def. 20.), and divide P P' produced harmonically (II. 49.); and, because the mean P P is bisected in C, CN, CP, and C T are proportionals (II. 46.). In the parabola, because the projection qr of the ordinate QR passes through G, and that G L, Gp are tangents drawn from G to the circle, the tangents at q and r meet one another in some point of Lp produced (Lem.), and consequently, as before, the tangents at Q and R meet one another in some point, T, of N P produced. Again, be cause tq and tr are tangents drawn from t to the circle, the line tp is harmonically divided by qr and the circumference (Lem.). Therefore, the four straight lines V L, V nN, V pP,VtT, are harmonicals (II. def. 20.); and beis bisected by V P in P (II. 49. Cor. 1)., cause N T is parallel to V L (IV. 6.), it that is, NP is equal to PT. axis of the conic section, these demonWhen the diameter in question is the strations will be modified, and appear under a more simple form, to which they are easily reduced by substituting A M for PN, A F for PH, &c. Therefore, &c. *See note, page 221. PROP. 19. In the ellipse and hyperbola, the squares of any two semiordinates of the same diameter are to one another as the rectangles under the corresponding abscissa: in the parabola, the squares of any two semiordinates of the same diameter, are to one another as the abscissa. Let P Q R be an ellipse or hyperbola, PU any diameter, and QR, Q'R' any two ordinates, cutting the diameter PU in the points N, N respectively; the square of QN shall be to the square of Q'N' as the rectangle PNXNU to the rectangle P N'x N'U. Through N draw a plane parallel to the base of the cone, and, therefore, cutting the cone in a circular section Pqur (11.), and let pu, qr be the projections of PU, Q R respectively, upon this plane, by straight lines drawn from the vertex Vof the cone; then, because pqur is a circle, the rectangle under q N, Nr is equal to the rectangle under pN, Nu (III.20.). ThroughV drawVK parallel to and VG parallel to QR to meet the same PU to meet the plane of the circle in K, plane in G; then, because pu is the projection of PU, it will, if produced, pass through the point K, and, for the like reason, qr produced will pass through the point G (2. Cor. 4.). the triangles V Kp and V Ku are simiThen, because V K is parallel to PU, lar to the triangles PNp and UNu respectively (I. 15.); therefore (II. 31.) PNpN::VK: Kp and NU: Nu: : VK: Ku, and, consequently (II. 37. Cor. 3.), PNxNU: pNxNu:: VK Kpx Ku. And, in the same manner, because the triangles QNq, RN r are similar to the triangles V Gq, V Gr respectively, it may be shown that QNX NR or (17.) Q N2: qNxNr or pNx Nu:: VG2 GqxGr. But, from the former proportion, invertendo (II. 15.) pNx Nu: PN× NU:: Kp × Ku : V K2. Therefore (II. def. 12.), the ratio of QN to PN × NU is compounded of the ratios of V G2 to Gq × Grand Kpx Kuto V K2. Now, if through N' there be likewise drawn a plane parallel to the base of the cone, and which, therefore, likewise cuts the cone in a circle (11.), p' q'u' r', and if VK and VG are produced to meet this new plane in the points K' and G' respectively, the projections p' u' and q' of the diameter PU and the ordinate Q'R' upon this plane will pass through the points K' and G' respectively (2. Cor. 4.), because PU is parallel to VK' as before, and Q'R' to Q R, that is (IV. 6.), to VG'. Therefore, as before, it may be shown, that the ratio of Q'N' to PN' × N'U' is compounded of the ratios of V G' 2 to G'q'xG'r' and K' p' x K'u' to V K' 2. But, if x y is the projection of QR upon the plane p' q'u'r', xy produced will pass through the point G', because QR is parallel to VG' (2. Cor. 4.): and, because p'q'u' r' is a circle, G' q' × G' r' is equal to Gxx G'y (III. 20.). Also, because the triangles VKp, VK u are similar to the triangles V K'p', V K'u' respec tively, V K: Kp :: V K': K'p' (II.31.), * The straight lines VqQx, VRry, VqQ', and VR' are omitted in each of the figures of this proposition, and VK: Ku:: V K': K'u', and, consequently (II. 37. Cor. 3.), V K: Kpx Ku:: V K': K' p' x K'u', or, invertendo (II. 15.), KpжKu: VK:: K'p' × K'u' V K'2; and, in like manner, because the triangles V Gq, V Gr are similar to the triangles V G' x, VG'y respectively, VG: Gq × Gr:: VG2: Gx × G'y or G'q' × G'r'. Therefore the ratio of V G to Gqx Gris the same with the ratio of V G': to G'q'x G'r', and the ratio of KpxKu to V K is the same with the ratio of K' p' x K'u' to V K. Therefore, because ratios which are compounded of the same ratios are the same with one another (II. 27.), the ratio of QN to PN NU is the same with the ratio of Q'N' to PN'N'U; and, alternando (II. 19.), Q N2 : Q'N' 2 :: PN xNU: PN'x N' U. 2 any two ordinates cutting the diameter PN in the points N, N' respectively: the square of QN shall be to the square of Q'N' as PN to P'N'. Through N draw a plane parallel to the base of the cone, and therefore cutting it in a circular section qpr (11.), and let p N, qr be the projections of PN, QR respectively upon this plane, by straight lines drawn from V; also, let V L be the slant side of the cone, which is parallel to the axis of the parabola (13. Cor. 2.), and therefore (IV. 6.) likewise parallel to the diameter PN, and let it meet the plane qpr in the point L of the circumference qpr: then, because V L is parallel to PN, p N produced passes through the point L (2. Cor.4.), and, because qpr is a circle, the rectangle under q N, Nr, is equal to the rectangle under p N, N L. (III. 20.) Through V draw VG parallel to QR, to meet the plane of the circle p q r in G; then, because qr is the projection of QR, it will, if produced, pass through the point G (2. Cor. 4.). Now, as in the former part of the proposition, it may be shown that Q N° qNxNr or pNx NL:: VG2: Gqx Gr; also, because the triangles VLp, PNp are similar, p N: PN:: Lp: VL (II. 31.), and consequently, since pNxN L is to PN × N Las pÑ to PN (II. 35.), pN × NL: PÑx NL:: Lp: VL(II. 12.); therefore the ratio of QN 2 to PNNL is compounded of the ratios of V G2 to Gq x Gr and Lp to VL. And in the same manner, if through N' there be drawn a plane parallel to the base of the cone, and if V Land V G be produced to meet it in the points L' and G' respectively, it may be shown that the projections p' N' and q'r' of the diameter PN or PN' and the ordinate QR upon this plane pass through the points L' and G' respectively, and accordingly that the ratio of Q'N' to PN'N' L' is compounded of the ratios of V G2 to G'q' × G' r', and L'p' to V L. But if xy is the projection of QR upon the plane of the circle p' q'u'r', it may be shown, as in the former part of the proposition, that xy produced will pass through G', and that G'q'x G'r is equal to G x x Gy. Also, because the triangle V Lp is similar to the triangle V L'p', the ratio of Lpto V L is the same with the ratio of Lp' to V.L' (II. 31.); and, because the triangles VGq, V Gr are similar to the triangles V G'x, V G'y respectively, the ratio of V G2 to Gq x Gr is the same with the ratio of VG to G'x ×G'y or G'q'x G' r'. Therefore (II. 27.) Q N PNNL:: Q'N': PN× N' L', and alternando (II. 19.), QN2: Q'N/2 :: PNXNL: PN'×Ñ' L'. But N L is equal to N'L' (I. 22.), and consequently (II. 35.), PN-NL: PN' × N' L:: PN: PN'. Therefore Q N 2 QN/2 :: PN: PN'. The foregoing demonstrations are not applicable, in the above form, to the case in which the diameter P N is also the axis of the conic section. They become, however, much more simple when they are adapted to this particular case, and the manner in which this is to be done is obvious. Therefore, &c. Cor. 1. In the ellipse and hyperbola, the square of the semiordinate varies as the rectangle under the abscissæ; in the parabola, the square of the semiordinate varies as the abscissa (II. 35. Schol.). Cor. 2. If, in the ellipse, a diameter DZ is drawn parallel to the ordinates of the diameter PU, (see the first figure of the Scholium which follows this proposition,) the square of the semiordinate QN is to the rectangle PN x NU under the abscissæ, as the square of the semidiameter CP to the square of the semidiameter C D. being understood instead of the rectangle under its segments, when it touches the conic section instead of cutting it. For, let P U and QR cut one another in N, and P' U' and Q'R' in N'; also, because Q'R' is parallel to QR, let it cut P U in M (I. 14. Cor. 3.). Then, by the last corollary, QNXNR: Q'M×MR'::PN-NU : PMXMU, and Q'M MR': Q'N'N'R' :: PMx MU: P'N'x N'U'; And the same may be directly inferred from the demonstration of the proposition: for the projection of P' U' will pass through the point K' in the same manner as the projection of Q'R' passes through G. Cor. 3. It is not necessary, in the demonstration of the first part of the therefore, ex æquali (II. 24.), QN × NR proposition, that the conic section should: Q'N'x NR:: PNXNU; P'N'x be an ellipse or an hyperbola, or N'U'. PU a diameter having the ordinates QR, Q'R'; but only that PU should be a straight line cutting the conic section P Q R in two points, and QR, QR two parallel straight lines likewise cutting the conic section, each in two points, (in which case the part shewing that Q NxNR and Q'N'x N'R' are equal to Q N2 and Q'N', will have to be omitted,) or even one or both touching the conic section in a single point, the only difference being that in this case the points Q and Q' coincide with the points R and R' respectively. Therefore, generally, if a straight line PU cuts a conic section in two points, and is cut by any two parallel straight lines which likewise cut the conic section each in two points, or one or both of them touch the conic section, the rectangle under the segments of one parallel, or its square, if it be a tangent, shall be to the rectangle under the segments of the other, or to its square if it be a tangent, in the same ratio as the rectangles under the corresponding segments of the straight line which is cut by them. N Cor. 4. And hence, in any conic section, if two straight lines, PU, QR cut one another, and likewise other two P'U', Q'R', which are parallel to the two first respectively, and if each of them cuts the conic section in two points, or one or more touch it in a single point, the rectangle under the segments of either of the first shall be to the rectangle under the segments of its parallel as the rectangle under the segments of the remaining one of the first to the rectangle under the segments of its parallel; the square of any of the straight lines R R M Cor. 5. It is indifferent, also, in the second part of the demonstration, whether PQR be a parabola and PN a diameter, or PQR be an hyperbola, and PN a straight line parallel to one of the asymptotes, for in this case also PN will be parallel to a slant side of the cone (14.. Cor. 1. and IV. 6.); and in either case, QR and Q'R' may be any two parallel straight lines cutting the conic section each in two points, or one or both of them touching it in a single point. Therefore, generally, if a straight line PN, which is a diameter of a parabola or parallel to one of the asymptotes of an hyperbola, be cut by any two parallel straight lines which likewise cut the parabola or hyperbola each in two points, or one (or both of them, as is possible in the case of the hyperbola) touch it in a single point; the rectangle under the segments of one parallel, or its square if it be a tangent, shall be to the rectangle under the segments of the other, or to its square if it be a tangent, in the same ratio as the parts PN, PN of the line PN, which are cut off by the parallels respectively. Scholium. The diameter D Z, which is supposed in Cor. 2. to be drawn in the ellipse parallel to the ordinates of the diameter PU, is said to be conjugate to PU. It is easy to perceive that the diameter PU is in this case likewise parallel to the ordinates of DZ: for, let QR he any ordinate of the diameter PU, and let QC be joined and produced to meet the ર rallel to PU (II. 29.), because QR and QX are bisected in N and C respectively (17.); and RX is an ordinate to the diameter D Z, (17. Cor. 2.) because RY is to Y X as QC to CX, that is, in a ratio of equality. From this reciprocal relation such diameters PU and DZ are called conjugate diameters; and the diameter which is conjugate to the transverse axis (and therefore (13. Cor. 1.) perpendicular to that axis) is called the conjugate axis of the ellipse, for, being perpendicular to its ordinates, it divides the figure symmetrically, and therefore is a second axis of the figure. In the hyperbola, which, as we have seen, although so different in form, is very like the ellipse in its properties, there are no diameters, properly speak ing, except such as lie in the angle made by the asymptotes. Let us, however, define the conjugate diameter of any diameter PU to be a straight line DZ, which is drawn through the centre C parallel to the ordinates of P U, is bisected in the centre, and is such that CD is to CP as the square of the ordinate QN to the rectangle under the abscissæ PN, NU. Such a straight line D Z will, it is evident, as in the ellipse, bisect all straight lines which are drawn parallel to the diameter P U, and terminated by the hyperbola. But there The conjugate axis of the ellipse being always less than the principal or transverse axis, the former is frequently called the minor axis, and the latter the major axis of the ellipse. + There is, however, no other straight line which divides the figure symmetrically, that is, no third axis. For, if C Q be joined in the first figure of prop. 17, and if P C P' be supposed to represent the transverse axis, then if it were possible that C Q could divide the figure symmetrically, or (which is the same thing) bisect its ordinates at right angles, C Q T would be a right angle, and, consequently, because QN is perpendicular to CT, CQ would be a mean proportional between CN and C T, (II. 34. Cor.) and therefore equal to CP (18.), so that Q N2 would be to C P2-C N2 or P N X N P' in a ratio of equality, and consequently (19.) the square of every other semi-ordinate of the axis would be equal to the rectangle under its abscissæ, and the figure would be a circle, not an ellipse. is yet a more striking analogy, to which we are led by observing that, in the ellipse, the curve in which the extremity of a conjugate diameter so defined lies, is a part of the ellipse itself, or, as we may consider it, an ellipse having the same centre and axes with the given ellipse. It might be expected that the same, or something similar, would obtain in the hyperbola; and such, we are about to demonstrate, is the case; viz. that "the locus of the extremities of all the conjugate diameters of a given hyperbola is an hyperbola which has the same centre and the same axes as the first." To demonstrate this: Let C B be drawn parallel to the tangent at the principal vertex A, and therefore perpendicular to the transverse axis A'CA (13. Cor. 1.): let P be any point in the hyperbola, and draw PM likewise parallel to the tangent at A, to meet CA produced in M, so that PM is a semiordinate to the transverse axis A'CA (17. and def. 17.); take C B such that CB2 shall be to CA2 as PM2 to AM × MA', and make CB' equal to CB, so that, according to the above definition, B B is the conjugate axis of the hyperbola. Let PT be drawn touching the hyperbola in P to meet CA in T; through A draw A Q parallel to PT, and therefore (def. 17.) an ordinate to the diameter PU, by which it is consequently bisected (17.) in the point of intersection N; through C draw CD parallel to P T, and take CD such that CD2 shall be to CP2 as Q N2 to PN × NU, and make C Z equal to CD, so that D Z is the diameter which is conjugate to the diameter C P. The points Ď, Z shall lie in an hyperbola which has BB' for its transverse axis and A A for its conjugate axis. From D draw DE perpendicular to CB produced. Then, because the sides of the triangles CDE, PTM are parallel, each to each, those triangles are similar (I. 18.): therefore, CE : PM :: CD: PT (II. 31.), and, consequently |