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one, of two right lines meeting each other, touch or cut in two points, a hyperbola or opposite hyperbolas, and the other so touch or cut a hyperbola or opposite hyperbolas conjugate to the former ; the square of the segment of the former tangent, or rectangle under the segments of the former secant, between the concourse and the section or sections, is to the square of, or rectangle under, the like segment or segments of the other tangent or secant, when the concourse is in an asymptote, as the squares of the semidiameters, which are parallel to the right lines so meeting each other ; and when the concourse is not in an asymptote, the square or rectangle pertaining to the hyperbola or hyperbolas, to which the diameter passing the concourse of the meeting right lines is a transverse one, is to the square or rectangle pertaining to the conjugate hyperbola or hyperbolas, in a ratio compounded of the ratios of the semidiameters to which the meeting right lines are parallel, and of the ratio of the rectangle under the distances of the concourse from the vertices of the diameter passing through it, to the sum of the squares of the semidiameter which passes through the concourse, and the distance of the concourse from the centre.

Case 1. If the concourse of the right lines so meeting each other be in an asymptote, these squares or rectangles are equal to the squares of the semidiameters to which the same right lines are parallel (37. 1 Sup.), and are therefore to each other, as the squares of the same semidiameters (Schol 7. 5 Eu).

Case 2. Let GH and TD be two right lines meeting each

G other in D, and let one of them cut the hyperbola PG in GT and H, and the other TD the hyperbola . OT conjugate to. PG in S and T; let CO and CR be semidiameters parallel to GH and TD, and QPD the diameter passing through the concourse D; the rectangle

RY GDH is to the rectangle TDS, in a ratio, compounded of the ratios, of the square of Co to the square of CR, and and of the rectangle QDP to the sum of the squares of CP and CD.

P

For the rectangle GDH is to the rectangle QDP, as the square of CO is to the square of CP (41. 1 Sup.), and the rectangle SDT is to the sum of the squares of CP and CD, as the square of CR is to the square of CP (by the same); therefore the rectangle GDH is to the rectangle SDT, in a ratio, compounded of the ratios, of the square of CO to the square of CR, and of the rectangle QDP to the sum of the squares of CP and CD (Cor. 6. 23. 6 Eu).

The demonstration is similar, if one or both of the right lines meeting each other, were tangents, or cut opposite byperbolas.

PROP. XLIV. THEOR.

If from any point of a conick section, an ordinate be drawn to a

diameter, and a tangent meeting the same diameter ; the semidiameter is, in the case of an ellipse or hyperbola, a mean proportional between the segments of the diameter, between the centre and ordinate, and between the centre and tangent ; and, in the case of a parabola, the segment of the diameter, between the ordinate and tangent, is bisected in its vertex.

Let PH, see fig. 1, 2 and 3, of this prop. be an ordinate, drawn from any point P of a conick section, to a diameter HR, which is not a second diameter of a hyperbola, or in the case of a hyperbola, fig. 2, let PT be an ordinate drawn to a second diameter CT, and let PK be a tangent drawn from P, meeting the diameter HK in K, and in fig. 2, the diameter CT in X, CG being in fig. 1 and 2, the semidiameter passing through H and K, and Co in fig. 1, that which passes through T and X; CG is, in fig. 1 and 2, a mean proportional between CH and CK; CO, in fig. 2, between CT and CX; and in fig. 3, KH is bisected in G.

Case 1. When the ordinate PH (see fig. 1 and 2), and tangent PK, meet any diameter of an ellipse, or a transverse one of a hyperbola.

Let D and G be the vertices of that diameter, through which, draw DR and GL parallel to PH, meeting PK in Ř and L, these touch the section in D and G (Def. 12. 1 Sup.) ; on CD, produced in fig. 2, take CS equal to CH, DS is equal to GH (Ax. 2 and 3. 1 Eu). · And because the tangent RPK meets the tangents DR and GL, the square of RP is to the square of PL, as the square of DR is

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R to the square of GL (Cor. 2. 14. 1 Sup.), therefore RP is to PL, as DR is to GL (22. 6 Eu.), or, because of the equiangular triangles RDK and LGK, as DK is to GK (4. 6 and 16.5 Eu.); but, because of the parallels RD, PH and LG, DH is to HG, as RP is to PL (Cor. 2. 10.6 Er.), therefore DH is to HG, as DK is to GK (11. 5 Eu.) ; therefore, by dividing in fig. 1, and compounding in fig. 2, SH is to HG, as DG is to GK (17 and 18 5 E«.); and taking the halves of the antecedents, CH is to HG, as CG is to GK (Theor. 1. 15. 5 and 22. 5 En.), and, by converting, CH is to CG, as CG is to CK.

Case 2. When the ordinate PT (see fig. 2), and tangent PX, meet a second diameter CT of a hyperbola.

Let O be a vertex of the diameter CT, and DG the transverse diameter conjugate to the diameter CT, meeting the tangent PX in K; draw PH an ordinate to the diameter DGH (36. 1 Sup).

By the preceding case, CH, CG and CK are continually proportional, therefore the square of CH is to the square of CG, as CH is to CK (Cor. 2. 20. 6 Eu.), and, by dividing, the excess of the square of CH above that of CG, or, which is equal (6. 2 En.), the rectangle DHG, is to the square of CG, as KH is to CK (17.5 Eu.; whence, the square of PH being to the square of Co, as the rectangle DHG is to the square of ČG (40. 1 Sup. and 16. 5 Eu.), the ratios of the square of PH to the square of CO, and of KH to CK, being each equal to the ratio of the rectangle DHG to the square of CG, are equal to each other (11. 5 Eu.) ; but, because of the equiangular triangles HKP and CKX, PH is to CX, as KH is to CK (4. 6 and 16. 5 Eu), therefore the square of PH or CT is to the square of Co, as PH or UT is to ČX (11. 5 Eu.); therefore CT', CO and CX are continually proportional (20. 6 Eu).

H

Case 3. When the ordinate PH
and tangent PK meet a diameter
GH of a parabola.
Let the ordinate PH be produ-

P

RS ced to meet the parabola again in Q, and let GR, HS and QT, drawn parallel to PK, meet the diameter PT drawn through P; KG GR and QT are ordinates to the diameter PT (Def. 12. 1 Sup). And since PQ is double to PH,

Q (31, 1 Sup.), PT is double to PS (2. 6 Eu.), and QT to HS (4. 6 and 16. 5 Eu.', or GR, therefore the square of QT' is fourfold the square of GR (Cor. 4. 2 Eu.), and therefore the abscissa PT, fourfold the abscissa PR (Cór. 2. 40. 1 Sup. ), and PS or KH double to PR or KG, and so KH is bisected in G.

Cor. If from the vertices (P and G, see fig 3), of two diameters (PT and GH) of a parabola, ordinates (PH and GR) be drawn to the same diameters ; the abscissas (GI and PR) are equal.

For, the tangent PK being drawn from P, meeting Gil in K, because of the parallelogram PAGR, PR is equal to KG (34. 1 Eu.), or, which is equal (by this prop.), to GH.

PROP. XLV. THEOR.

If from any point of an ellipse or hyperbola, an ordinate be drawn

to any diameter, and a tangent from the same point, meet the same diameter ; the rectangle under the segments of the diameter, between the ordinate and centre, and between the ordinate and tangent, is, in the case of an ellipse, or transverse diameter of a hyperbola, equal to the rectangle under the segments of the same diameter, between the ordinate and its vertices; and, in the case of a second diameter of a hyperbola, to the sum of the squares of the second semidiameter, and segment of the same diameter, between the centre and ordinate.

Let PH, see fig. 1 and 2 of the prec. prop. be an ordinate drawn from any point P of a conick section, to any diameter of an ellipse, or a transverse one of a hyperbola, let this diameter be DG, let PT, see fig. 2, be an ordinate drawn from P to a second diameter CT of a hyperbola, and let a tangent P's drawn from P, in fig. 1, meet DG produced in K, and, in fig. 2, meet

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