whence FD being equal to DK, and FG to GL (Def. 8. 1 Sup.), DG is double of OH, and fourfold of OP or PF, and therefore equal to the parameter of the diameter PH (Def. 16. 1 Sup).

Cor. The segment (OH), of a diameter of a parabola, intercepted between the directrix, and the ordinate (GD) which passes through the focus, is bisected in its vertex.

PROP. XL. THEOR.

If from any point of a conick section, an ordinate be drawn to a diameter; the square of the ordinate is, in the case of an ellipse or transverse diameter of a hyperbola, to the rectangle under the abscissas, and in the case of the second diameter of a hyperbola, to the sum of the squares of the second semidiameter, and the segment thereof between the centre and ordinate, as the square of the semidiameter which is parallel to the ordinate, to the square of that which it meets ; and, in the case of a parabola, the square of the ordinate, is equal to the rectangle, under the abscissa, and the parameter of the diameter, to which the ordinate is drawn.

Part 1. Let DH be an ordinate, drawn from any point D of an ellipse, to a diameter QP, and let ST be the diameter parallel to DH, C being the centre. The square of DH is to the rectangle QHP, as the square of SC is to the square of CP.

S D

P

H

G

Let DH produced meet the ellipse again in G; the rectangle DHG is to the rectangle QHP, as the rectangle SCT is to the rectangle QCP (14. 1 Sup.); but DG is bisected in H (31. 1 Sup.), and QP and ST are bisected in C (5.1 Sup.); therefore the square of DH is the rectangle QHP, as the square of SC is to the square of CP. Part 2. Let now DH be an ordinate, drawn from any point D of a hyperbola, to a transverse diameter QPH, C being the centre, and SC the second semidiameter to which DH is parallel. The square of DH is to the rectangle QHP, as the square of SC is to the square of CP.

Produce GD to meet the

K

H

P

T

asymptote CL in L, and through L, draw XLZ parallel

to CP, meeting the opposite hyperbolas in X and Z; the rectangle DIIG is to the rectangle QHP, as the rectangle DLG is to the rectangle XLZ (14. 1 Sup.); but the rectangle DLG is equal to the square of CS (37. 1 Sup.), and the rectangle XLZ is equal to the square of CP (by the same); therefore the rectangle DHG, or, DG being bisected in H (31. 1 Sup.), the square of DH, is to the rectangle QHP, as the square of CS is to the square of CP.

Part 3.

Let an ordinate DK meet a second diameter TS. The square of DK is to the squares of CS and CK together, as the square of CP is to the square of CS.

For (by the preceding part and inverting), the rectangle QHP is to the square of DH or CK, as the square of CP is to the square of CS; therefore, the rectangle QHP with the square of CP, or, which is equal (6. 2 Eu.), the square of CH or DK, is to the squares of CS and CK together, as the square of CP is to the square of CS (12. 5 Eu).

Part 4. Lastly, let DH be an ordinate to a diameter PII of a parabola. The square of DH, is equal to a rectangle, under the abscissa PH, and the parameter of the diameter PH.

F

K

D

Find the focus F (35. 1 Sup.), through which, P draw KL parallel to DG, meeting the parabola in K and L, and PH in O; KL is ordinately applied to the diameter PH (Constr. and Def. 12. 1 Sup.), and therefore equal to the parameter of that diameter (39. 1 Sup), and therefore fourfold of PO (Def. 16. 1 Sup. and Cor. 39. 1 Sup), whence, OK, being the half of KL (31. 1 Sup.), is double to PO, and so PO, OK and KL are continually proportional (Theor. 1. 15. 5 Eu.), therefore the square of KO is equal to the rectangle under PO and KL (17. 6 Eu.); but the square of KO is to the square of DH, as PO is to PH (20 and 31. 1 Sup.), or, (1.6 Eu.), as the rectangle under PO and KL is to the rectangle under PH and KL; whence, the square of KO having been just proved equal to the rectangle under PO and KL, the square of DH is equal to the rectangle under PH and KL (14. 5 Eu.), or, KL being equal to the parameter of the diameter PH (39. 1 Sup.), to the rectangle under PH and that parameter.

Cor. 1. If from two points of an ellipse, hyperbola or opposite hyperbolas, ordinates be drawn to the same diameter; the ratios of the squares of the ordinates, in the case of an ellipse or transverse diameter of a hyperbola, to the rectangles under their respective abscissas, and, in the case of a second diameter of a hyperbola, to the sums of the squares of the second semidiameter and the segments thereof between the centre and ordinates, are equal, being each, by this proposition, equal to the ratio of the square of the semidiameter which is parallel to the ordinates, to the square of that which they meet. (

Cor. 2. And if from two points of a conick section or opposite sections, ordinates be drawn to the same semidiameter; the ratio of the squares of the ordinates to each other is, in the case of an ellipse or transverse diameter of a hyperbola, equal to that of the rectangles under their respective abscissas, in the case of a second diameter of a hyperbola, to that of the sums of the squares of the semidiameter and the segments thereof between the centre and ordinates, and, in the case of a parabola, to that of the abscissas; the two former cases being manifest from the prec. cor. and 16. 5 Eu, and the last from this prop. and 1.6 Eu.

Cor. 3. If from any point of a conick section, an ordinate be drawn to a diameter which is not the second diameter of a hyperbola, and from any point of the diameter within the section or opposite sections, a right line be drawn parallel to the ordinate, of which the square is to the square of the ordinate, in the case of an ellipse or hyperbola, as the rectangles under the segments of the diameter between the parallels and its vertices, and, in the case of a parabola, as the segments of the diaineter, between the parallels and its vertex, the extreme of the right line drawn parallel to the ordinate, which is remote from the diameter, is in the section, within which the point in the diameter is taken.

For if the right line drawn parallel to the ordinate met the section in any other point, its square would not be to the square of the ordinate, in the ratio demonstrated in the preceding corollary.

Cor. 4. If to any diameter of an ellipse or hyperbola, an ordinate be drawn; the rectangle under the abscissas, in the case of an ellipse or transverse diameter of hyperbola, and the sum of the squares of the semidiameter and the segment thereof between the centre and ordinate, in the case of a second diameter of a hyperbola, is to the square of the ordinate, as the diameter is to its parameter, as is manifest from this proposition, Def. 15. 1. Sup. and Cor. 2. 20. 6 Eu.

Scholium. Let GH, sec fig. 1 and 2, be an ordinate to any diameter of an ellipse or a transverse one of a hyperbola, let that diameter be QP, and from its vertex P draw PD at right angles to QP equal to its parameter; complete the parallelogram PR, join QD, and draw HKO parallel to PD, meeting QD and RD in K and O, and through K, SKL parallel to QP, meeting QR and PD, produced if necessary, in S and L.

In the case of the ellipse, fig. 1, the rectangle QHP is to the square of HG, as the diameter QP to its parameter PD Cor. 4. to this prop.), or, because of the parallels, as QH is to HK, or (1.6 Eu.), as the same rectangle QHP to the rectangle KHP; therefore the square of HG is equal to the rectangle KHP (9. 5 Eu.), and, of course, being applied to the parameter PD with the attitude of the abscissa PH adjacent to PD, is deficient by a figure OL similar to the rectangle RP under the diameter QP and its parameter PD (Def. 5. 6 Eu). A similar consequence would follow, if the square of GH were applied to the parameter QR, from the point Q, with the attitude QH. On account of this deficiency, Appolonius named this line, an Ellipse or Ellipsis, which, in the Greek language, signifies a deficiency.

In like manner, in the case of the hyperbola, fig. 2, the rectangle QHP is to the square of GH, as the diameter QP to its parameter PD (Cor. 4 to this prop.), or, because of the parallels, as QH to HK, or (1. 6 Eu.), as the same rectangle QHP to the rectangle KHP; therefore the square of GH is equal to the rectangle KHP (9.5 Eu.), and of course being applied to the parameter PD with the attitude of the less abscissa HP, exceeds by figure LO similar to the rectangle RP under the diameter QP

and its parameter PD (Def. 6. 6 Eu). On account of this excess, Apollonius named this line, a Hyperbola or Hyperbole, which, in the Greek language, signifies a redundancy.

And the square of an ordinate to any diameter of a parabola is, by this prop. equal to the rectangle under the abscissa and the parameter of the diameter. For this reason, Apollonius named this line, a Parabola or Parabole, which, in the Greek language, signifies similitude or equality.

PROP. XLI. THEOR.

If a right line, touching a conick section, or cutting in two points a conick section or opposite sections, meet any diameter; the square of the segment of the tangent, or rectangle under the segments of the secant, between the diameter, and the point or points in which it meets the section or sections, is, in the case of an ellipse or transverse diameter of a hyperbola, to the rectangle under the segments of the diameter, between the tangent or secant and its vertices, and, in the case of a second diameter of a hyperbola, to the sum of the squares of the semidiameter, and the segment thereof between the centre, and the tangent or secant, as the square of the semidiameter to which the tangent or secant is parallel, to the square of the semidiameter which it meets; and, in the case of a parabola, is equal to the rectangle under the segment of the diameter, between its vertex, and the tangent or secant, and the parameter of the diameter, whose ordinates are parallel to the tangent or secant.

First, let the figure be an ellipse; let KL be a tangent, touching it in K, and meeting a diameter QP in L, let OCR be a diameter parallel to L, C being the centre, and let DG be a secant, parallel to the diameter OR, meeting the ellipse in D and G and the diameter QP in H; the square of KL is to the rectangle PLQ, and the rectangle DHG to the rectangle

QIP, as the rectangle OCR is to the rectangle OCP (14. 1 Sup.), or, OR and QP being bisected in C (5.1 Sup.), as the square of CO is to the square of CP.