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Part 1. Let PQ, see fig. 1, 2 and 3, terminated by a conick section, be ordinately applied to the principal axis BH of the section, meeting that axis in H, PQ is bisected in H.

Let F be a focus, and ST a directrix, and, in fig. 1 and 2, those which are adjacent to PQ; let PS and QT be perpendiculars, let fall from P and Q on ST, and join FP and FQ; and, because both PQ and ST are perpendicular to the axis BH (Cor. 3. 11. 1 and Def. 8 and 18. 1 Sup.), they are parallel to each other (28. 1 Eu.), as are also PS and QT, being each perpendicular to ST (by the same), therefore PT is a parallelogram, and PS and QT are equal, and therefore FP and Q, having the same ratio to them (6. 1 Sup.), are also equal; whence the triangles FHP and FHQ, having FH common, and the angles at H right, PH is equal to HQ (Cor. 7. 6. Eu.), and so PQ is bisected in H.

Part 2. Let now PO, see fig. 1 and 2, terminated by an ellipse or opposite hyperbolas, be ordinately applied to the second axis MN, meeting that axis in K, OP is bisected in K.

The right line OP is perpendicular to MKN (Cor. 3. 11. 1 Sup.), and therefore parallel to AB (Def. 3. and 5. 1 Sup. and 28. 1 Eu.); draw OL and PH at right angles to AB, and produce them to meet the section again in R and Q, they are ordinately applied to the axis AB (Cor. 2. 11. 1 Sup.), and the rectangle OLR is to the rectangle ALB, as the rectangle PHQ is to the rectangle AHB (14. 1 Sup.); but PQ and OR being bisected in H and L (by part 1), and OL and PH being, because of the parallelogram OH, equal, the rectangles OLR and PHQ are equal, therefore the rectangles ALB and AHB are equal, (14. 5 Eu.), and therefore AL is equal to HB (Cor. 1 and 2. 7. 2 Eu.); but AC is equal to CB (1.1 Sup.), therefore LC and CH are equal (Ax. 2 and 3. 1 Eu.); but, because of the paral

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lelograms OC and CP, the right lines OK and KP arc equal to LC and CH (34. 1 Eu.), therefore OK and KP are equal, and so OP is bisected in K.

Scholium. A perpendicular, let fall from any point of a conick section, on an axis, which is not the second axis of a hyperbola, meets the axis within the section, for it is parallel to a tangent, drawn through the nearer vertex of the axis (Cor. 1. 11. 1 Sup. and 28. 1 Eu), which tangent falling wholly without the section (Def. 10. 1 Sup.), if the perpendicular did not meet the axis within the section, it would meet the tangent, contrary to the definition of parallel right lines.

And this perpendicular is an ordinate to the axis (Cor. 2. 11. 1 and Def. 12. 1 Sup.), and if it be produced beyond the axis, 90 that the part produced may be equal to the ordinate, its other extreme is in the section, for otherwise, a right line ordinately applied to the axis, and terminated by the section, would not be bisected by the axis, contrary to this proposition.

Cor. A tangent to a conick section, which is perpendicular to an axis, which is not the second axis of a hyperbola, touches the section in a vertex of that axis; for a perpendicular to the axis, drawn from any other point of the section, meets the axis within the section, by the prec. schol. and would not therefore be a tangent (Def. 10. 1 Sup).

PROP. XXIII. THEOR.

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

Part 1. When the figure is an ellipse, (see fig. 1 preceding prop.), PH being an ordinate to the axis AB, and PK to the axis MN; these ordinates are perpendicular to their respective axes (Cor. 5. 11. 1 Sup.), and it is manifest from the 1st, 14th and 22. 1 Sup. that the square of PH is to the rectangle AHB, as the square of CM is to the square of CB, and the square of PK to the rectangle NKM, as the square of CB to the square of CM:

Part 2. When the ordinate, as PH, (see fig. 2. preceding prop.), meets the principal axis AB of a hyperbola. The square of PH is to the rectangle AHB, as the square of CM is to the square of CB.

From the focus F, draw FD at right angles to AH, meeting the hyperbola in D, draw DG at right angles to the adjacent directrix TS, which let AB meet in Z.

FD is to DG or FZ, as FB is to BZ (6. 1 Sup.), and by alternating, FD is to FB, as FZ is to BZ (16. 5 Eu.); but since CF is to CB, as FB is to BZ (Schol. 6. 1 Sup.), by compounding, CF and CB together, or AF is to CB, as FB and BZ together, or FZ is to BZ (18. 5 Eu.); but it is above shewn, that FD is to FB, as FZ to BZ; therefore FD is to FB, as AF is to CB (11. 5 Eu.), and therefore the rectangle under FD and CB is equal to the rectangle AFB (16. 6 Eu.), or, which is equal (2. 1 Sup.), to the square of CM; therefore CM is a mean proportional between CB and FD, and so the square of FD is to the square of CM, or (2. 1 Sup.), the rectangle AFB, as the square of CM is to the square of CB (20. 6 and Cor. 3. 22. 5′ Eu.), but the square of PH is to the rectangle AHB, as the square of FD is to the rectangle AFB (14 and 22. 1 Sup.), therefore the ratios of the square of PH to the rectangle AHB, and of the square of CM to the square of CB, being each equal to that of the square of FD to the rectangle AFB, are equal to each other (11.5 Eu).

Part S. When the ordinate, as PK, meets the second axis MN of a hyperbola. The square of PK, is to the sum of the squares of CM and CK, as the square of CB is to the square of CM.

For, (by the prec. part and inverting), the rectangle AHB is to the square of PH or CK, as the square of CB is to the square of CM; therefore the rectangle AHB with the square of CB, or, which is equal (6. 2 Eu.), the square of CH or PK, is to the square of CM and CK together, as the square of CB is to the square of CM (12. 5 Eu).

Part 4. When the figure is a parabola, see fig. 3 of prec. prop. The square of PH, is equal to the rectangle under BH, and the principal parameter.

From the focus F draw FD at right angles to BH, meeting the parabola in D, let fall the perpendicular DG on the directrix ST, which let the axis BH meet in Z.

The right line DF is equal to DG (Def. 8. 1 Sup.), therefore the square of DF is equal to the square of GD or of ZF, or, ZB and BF being equal (Def. 8. 1 Sup.), to four times the square of

BF (Cor. 4. 2 Eu.), or to the rectangle under BF and four times BF, or, the principal parameter being equal to four times BF (Def. 16. and 17. 1 Sup.), to the rectangle under BF and the principal parameter; but the square of DF is to the square of PH, as BF is to BH (22 and 20. 1 Sup.), or, as the rectangle under BF and the principal parameter, is to the rectangle under BH and the same parameter (1. 6 and 11. 5 Eu.); whence, the square of DF having been just proved equal to the rectangle under BF and the principal parameter, the square of PH is equal to the rectangle under BH and the same parameter (14.5 Eu).

Cor. 1. Hence, in ellipses, the rectangle under the abscissas of the greater axis is greater, of the second axis, less, than the square of the ordinate.

Cor 2. And, in ellipses and hyperbolas, a right line, drawn from the centre, at right angles to the principal axis, whose square, is to the square of the principal semiaxis, as the square of an ordinate to the principal axis, is to the rectangle under the abscissas, is the second semiaxis.

PROP. XXIV. THEOR.

A right line, terminated by a conick section, passing through a focus, and ordinately applied to the principal axis, is equal to · the principal parameter.

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Let BD be a conick section, see fig. 1, 2 and 3, whose principal axis is BF, and DG a right line passing through a focus F,ordinately applied to the same axis, and terminated both ways by the section; DG is equal to the principal parameter of the section.

First, let the section be an ellipse or hyperbola, see fig. 1 and 2, let A and B be the principal vertices, C the centre, and MN the second axis. The square of CB is to the square of CM, as the rectangle AFB, or, which is equal (2. 1 Sup.), the square of CM, is to the square of F (23. 1 Sup); therefore the right lines CB, CM and FD are continually proportional (22. 6 Eu.), and therefore also their doubles (1 & 22. 1 Sup.), AB, MN and DG (15. 5 Eu.; whence, AB and MN being conjugate diameters (Def. 14 3 and 5, and Cor. 1. 11. 1 Sup.), DG is equal to the parameter of the principal axis AB, or the principal parameter of the section (Def. 15. and 17. 1 up).

Let now DG, see fig. 3, be a right line, terminated by a parabola, passing through its focus F, and ordinately applied to its axis BF; let KL be the directrix, which let FB produced, meetin O, and draw DK and GL at right angles to KL; BF is equal to OB (Def. 8. 1. Sup), and therefore OF, or either of its equals (34. 1 Eu.), DK or GL, double to OB; whence, FD being equal to DK, and FG to GL(Def. 8. 1 Sup), DG is fourfold of OB or BF, and therefore equal to the principal parameter (Def. 16. and 17. 1 Sup).

PROP. XXV. THEOR.

If a right line, touching a hyperbola, or cutting a hyperbola or opposite hyperbolas, and parallel to either axis, meet both asymptotes; the square of the segment of the tangent, between the hyperbola, and either asymptote, or rectangle under the segments of the secant, between an intersection with the hyperbola, or either of the opposite hyperbolas, and the asymptotes, or between either asymptote, and the hyperbola or opposite hyperbolas, is equal to the square of the semiaxis, which is parallel to the tangent or

secant.

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