Every diameter of a conic section bisects its ordinates; and, if the conic section is an ellipse or an hyperbola, the diameter is itself bisected by the centre.

Let A PQ be any conic section, A M* its axis, PN any diameter, PH a tangent at P, QR a parallel to PH, and therefore an ordinate to the diameter PN (def. 17.), and let QR cut PN in N: QR shall be bisected in N.

Let the vertex V of the cone, the circle A p q, its centre O, and diameter Ao a', as also the line AF*, which is the common section of the planes A PQ, Apq, remain as in the foregoing propositions; let the lines grand pn in the circle be the projections of QR and P N in the conic section, and the tangent ph* (6.) the projection of the tangent PH; and let the vertical plane of the conic section cut the plane Ap q of the circle in the line KG, which cuts A a' or Aa' produced in L, and join V L. Then, because V L and A M are sections of parallel planes by the same plane, VL is parallel to A M (IV. 12.); and, for the like reason, K G is parallel to AF, and therefore (III. 2.) perpendicular to A a'.

And first, in the ellipse or hyperbola, let A' be the other principal vertex and C the centre, join V C*, and let V C or V C produced cut A a' or A a' produced in c. Then, because A A is parallel to V L, and bisected by V C in C, the four straight lines V L, VA', V C, VA are harmonicals (II. 49. Cor. 2.). Therefore, A L or A L produced is divided by these straight lines harmonically (II. 49.); and, because the mean A a' is bisected in O, O L, O a' and O c are proportionals (II. 46.). Let the planes VPP', VPH cut the vertical plane in the lines VK, VG, which meet the line KG in the points K, G respectively. Then, because VK and P P are sections of parallel planes by the plane VPP, VK is parallel to P P' (IV. 12.), and, for the like reason, VG is parallel to PH: therefore, because the straight lines V K, V G are parallel to the original straight lines PP', PH respectively, and meet the plane of the circle p Aq

The line VC is wanting in the upper figure, and VN, VP, VP, VQ, VR, and VT, are wanting in each of the figures of this proposition. It has been attempted, however, to supply the assistance which might have been derived to the conception from the visible presence of these lines by placing the corre.

in the points K, G respectively, the projections pp', ph will, if produced, pass (2. Cor. 4.), and, for the like reason, the through the points K, G respectively projection qr of the ordinate QR, which through the point G. is parallel to PH or V G, will also pass

chord of the circle p Aq passing through Join G p* Then, because pp' is a the point c, and that LG is drawn perpendicular to the diameter O c produced from a point L so taken that the diameter produced is divided harmonically,

sponding letters n, p, p, q, r, and t, in the straight lines VN, VP, VP, VQ, VR, and VT respectively. The line AF and the letter F; in the two first figures, the letters M and h; in the second the line Gp', and in the third the line Ghp, have likewise been omitted.

:

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.), Kpx Ku: VK:: K'p' xK'u VK2; and, in like manner, because the triangles V Gq, V Gr are similar to the triangles V G'x, V G'y respectively, VG: Gq x Gr:: VG: Gx × G'y or G'q' × G'r'. Therefore the ratio of V G to Gqx Gr is the same with the ratio of V G'2 to G'q'G' r', and the ratio of K px Ku to V K is the same with the ratio of K'p' x K'u' to V K' 2. Therefore, because ratios which are compounded of the same ratios are the same with one another (II. 27.), the ratio of QN to PNX NU is the same with the ratio of Q'N' to PN'x N'U; and, alternando (II. 19.), Q N2 : Q'N' 2 :: PN xNU PN'×N' U.

In the next place, let Q PR be a parabola, PN any diameter, and QR, Q'R'

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 VL 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 9pr 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 pqr 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 No :qNx Nr or p N x NL:: VG: Gqx Gr; also, because the triangles VLp, PNp are similar, p N: PN:: Lp: VL (II. 31.), and consequently, since pNxN L is to PNNL as pÑ to PN (II. 35.), pN × NL: PÑx NL:: Lp: VL(II. 12.); therefore the ratio of QN to PN x N L is compounded of the ratios of VG 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'2 to PN/XN' L' is compounded of the ratios of V G to G'q' × G'r', and L'p' to V L. But if xy is the projection of Q R 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 Gʻ, and that G'q'x G' is equal to G× Gy. Also, because the triangle V Lp is similar to the_triangle V L'p', the ratio of Lp to V L is the same with the ratio of L'p' to V.L' (II. 31.); and, because the triangles V Gq, V Gr are similar to the triangles V G'x, V G'y respectively, the ratio of V G to Gq x Gr is the same with the ratio of VG2 to G'x ×G'y or G'q'G' r'. Therefore (II. 27.) Q N

PNNL:: Q'N': PN',x N'L', and alternando (II. 19.), QN2': QʻN/* : PNNL: PN'-N' L'. But NL is equal to N'L' (I. 22.), and consequently (II. 35.), PNxNL: PN' x N' L:: PN: PN. Therefore Q N = Q'N :: 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 × NU under the abscissæ, as the square of the semidiameter C P to the square of the semidiameter CD.

Cor. 3. It is not necessary, in the demonstration of the first part of the proposition, that the conic section should be an ellipse or an hyperbola, or PU a diameter having the ordinates QR, Q'R'; but only that PU should be a straight line cutting the conic section PQR in two points, and QR, Q'R' two parallel straight lines likewise cutting the conic section, each in two points, (in which case the part shewing that QNxNR and Q'N' × N'R' are equal to Q N2 and Q'N'2, 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 PU', 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

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 PU in M (I. 14. Cor. 3.). Then, by the last corollary,

QNX NR: Q'M× MR'::PNX NU : PMXMU, and

Q'M× MR': Q'N'× N'R' :: PMx MU: P'N' × N'U' ; therefore, ex æquali (II. 24.), QN × NR : Q'N'x N'R':: PNXNU: P'N'× 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. 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.