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5. The two given points (E and F) are called, the focuses; the point (C), wherein a right line joining them is bisected, the centre of the hyperbola or opposite hyperbolas; any right line (QP), passing through the centre, and terminated both ways by the opposite hyperbolas, a transverse diameter; the points (Q and P), wherein it meets the hyperbolas, the vertices of the diameter; the diameter (AB), which produced passes through the focuses, the transverse or principal axis, and its vertices (A and B), the principal vertices; the right line, which passes through the centre, at right angles to the transverse axis, and the distance of either extreme of which from either of the principal vertices, is equal to that of either focus from the centre, the second axis. The distance (CE or CF), of the centre from either focus, the eccentricity of the hyperbola or opposite hyperbolas.

6. Four hyperbolas (AQ, BP, MO, NU) are said to be conjugate; when the transverse axis of two of them, is the second axis of the other two, and the contrary.

7. Any right line (OU), passing through the centre, and terminated both ways, by hyperbolas conjugate to those which pass through the principal vertices, is called, a second diameter. 8. A Parabola, is a conick section, formed by a line (BPS), the distances of every point of which, from a given point (F), and a given right line (DK), are equal.

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9. That given point (F), is called the focus, and that given right line (DK), the directrix of the parabola; every right line D (as KPQ), perpendicular to the directrix, is called, a diameter; the point (P), wherein it meets the parabola, the vertex of that diameter; the diameter (DBF), which passses through the focus, the axis of the parabola; and its vertex, (B), the principal vertex.

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10. A right line (RPX, sce the 3 prec. fig.), which meets a conick section in any point (P), and, being produced both ways, falls wholly without the section, is said to be a tangent to the section, or to touch it in that point.

11. But if a right line, meeting a conick section, is on one side of its concourse with the section, within, and on the other, without the section, it is called, a secant.

12. A right line (ST), drawn from any point (S) of a conick section, meeting a diameter (PQ) of the section, and parallel

to a tangent (RPX) to the section, passing through the vertex (P) of the diameter, is said to be ordinately applied to that diameter; and the part (SH or TH), of the right line so applied, between the section and the diameter, is said to be an ordinate to the diameter.

And a right line, terminated by opposite hyperbolas, is said to be ordinately applied to the diameter, whose vertex is in the contact, of a tangent, parallel to the right line so terminated.

cholium. Hence, in the circle, perpendiculars let fall from the circumference to a diameter, are ordinates to that diameter. 15. A segment (PH), of a diameter (PQ), between an ordinate thereto, and a vertex of the diameter, is called an abscissa.

14. Two diameters of an ellipse or hyperbola, each of which is parallel to a tangent, passing through a vertex of the other, are called, conjugate diameters.

From the 12th definition it is manifest, that either of two conjugate diameters is ordinately applied to the other.

15. A right line, which is a third proportional to two conjugate diameters of an ellipse or hyperbola, is said to be, the parameter or latus rectum, of that diameter, which is the first of the three proportionals.

16. A right line, which is four-fold the distance of the vertex of a diameter of a parabola, either, from the directrix or the focus, is said to be, the parameter or latus rectum, of that diameter. 17. The parameter, which belongs to the axis of a parabola, or the transverse axis of an ellipse or hyperbola, is said to be the principal parameter or latus rectum of the section.

18. A right line perpendicularly cutting the transverse axis of an ellipse or hyperbola, produced in the ellipse; and whose distance from the centre, is a third proportional, to the eccentricity and the transverse semiaxis, is called a directrix of the scction.

19. Right lines (CY and CZ, See fig. to Def. 4.), passing thro' the centre (C) of a hyperbola, and the extremes of a right line (YZ) equal to the second axis (MN), and perpendicularly bisected by the transverse axis (AB) in a vertex (A), are called, asymp

totes.

Cor. The angle (YCZ) formed by the asymptotes (CY, CZ) towards either of the opposite hyperbolas (as AQ) is bisected by the axis (AB).

For, in the triangles CAY, CAZ, the sides AY and AZ are equal, being each equal to CM (by this def.), AC common, and

the angles at A equal, being right angles; therefore the angle ACY is equal to ACZ [4. 1. Eu.]

20. Hyperbolas are said to be, right angled, when the asymptotes are at right angles to each other.

Schol. That, in this case, any two conjugate diameters are equal, is demonstrated in prop. 54 of this: whence the hyperbolas in this case are also said to be equilateral, as is in that proposition observed.

21. Two conick sections, which touch the same right line in the same point, are said to touch each other in that point.

22. A circle, which so touches a conick section in any point, that, between it and the section, no other circle, described through that point, can pass, is said, to have the same curvature with the section in the point of contact, or to osculate the section in that point.

23. If the ratio of the principal axis of an ellipse or hyperbola to its second axis, be the same, as that of the principal to the second axis, of another ellipse or hyperbola, these two ellipses or hyperbolas are said to be similar.

A C D B

24. If a right line (AB) be so divided in two points (C and D, that the whole (AB) is to either extreme part AC), as the other extreme part (DB) is to the middle part (CD), that right line (AB) is said to be, harmonically divided.

POSTULATES.

1. That, from any two given points as focuses, and through any other given point not in the right line joining them, an ellipse may be described.

The genesis or formation of the ellipse may be thus conceived. Let E and F (see figure to Def. 2. above), be the points from which, as focuses, and P the point, through which, the ellipse is to be described. Let the extremities E and F of a thread or flexible line EPF whose length is equal to the two right lines EP and PF, be fastened in the points E and F, and by means of a pin P, let the thread be extended, and the pin P be moved round, the thread remaining continually extended, till it return to the

place from which it began to move; then is the sum of the distances of every point of the line described by the pin P, from the points E and F, always equal, as is manifest, and therefore that line will form an ellipse described from the points E and F as focuses, and through the point P (by Def. 2. 1. Sup).

2. That, from any two given points as focuses, and through any other given point neither in the production of a right line joining them, nor in a right line perpendicularly bisecting them, a hyperbola and its opposite may be described.

The genesis or formation of the hyperbola or opposite hyperbolas may be thus conceived. Let E and F (see fig. to Def. 4. above), be the two points, from which, as focuses, and P the point, through which, the hyperbola is to be described, the description of its opposite being also required.

Let one extreme of a ruler EI, be so affixed at the point E, that it may be freely moved round that point as a centre; let the ruler be so placed, as to pass by the point P, through which the figure is to be described, let a pin be so fixed in the ruler, as to be moveable to and from the point I at pleasure, and let one extreme of a thread or flexible line FPI, whose length is equal to the right lines FP and PI together, be affixed at the point F, and go close round the pin in the situation P, so that the pin may be within the angle FPI, let its other extremity be affixed to the extremity of the ruler, and let the ruler be moved round the point E, towards S or T, and, the thread or line still remaining extended, let the pin, affixed to the side of the ruler, describe the line BPS; and since the point P is neither in the production of the right line EF, nor in a right line, as MN perpendicularly bisecting it, it is manifest, that the difference of EP and PF is always, during the above description, equal to a given right line, and because EF and FP together are greater than EP (20. 1 Eu.), taking FP from each, EF is greater than the excess of EP above PF, or than the given difference of EP and PF (Ax. 5. 1 Eu.), and therefore the line described by the pin intersects the right line EF somewhere between E and F as in B, (for if it could intersect the right line EF produced, EF would be equal to the given difference of EP and PF, contrary to what has been proved), whence, the points E and F being on different sides of the described line BPS, and the difference of the distances of every

point of that line from the same points E and F always equal to a given right line, that described line BPS is a hyperbola, described from the points E and F as focuses, and through the point P (by Def. 4. 1 Sup).

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Let now EA be taken on EF equal to BF, and the extreme of the ruler, which was before affixed at the point E, be now affixed at the point F, and a hyperbola QA be described in like manner as above, from the points E and F as focuses, through the point A ; and since EA is equal to BF, the difference of EF and EA is equal to the difference of EF and BF, and therefore the hyperbolas AQ and BP described from the focuses E ond F, are opposite hyperbolas (Def. 4. 1. Sup). And these lines may be extended to a distance from the points E and F, greater than any given distance, namely, if a thread be taken, whose length is greater than that distance.

3. That to a given right line, as directrix, and from any given point not therein, as focus, a parabola may be described.

The genesis or formation of the parabola may be thus conceived. Let LG (See fig. to Def. 8. above), be the right line, to which, as directrix, and F the point, from which, as focus, the parabola is to be described let one side KG, of a square GKQ, be so applied to the directrix LG, that the side KQ, may pass through the point F, and the point K may coincide with the point D; bisect the right line DF in B, and to the extremity Q, of the side KQ, let one extremity of a thread of the same length as KQ be fastened, and let its other extremity, the thread going round a pin, in the side KQ of the square at the point B, be fastened at the point F, which it is manifest, it would reach, because DB and BF are equal [Constr.]; let the side KG of the square be moved along the right line LG, and, the thread remaining extended, let the pin, affixed to the side KQ of the square, describe the line OPS. And since, in every situation of the pin, during the description of the line OPS, the side KQ is equal in length to the thread FPQ, taking from each PQ, which is common, KP is equal to PF [Ax. 3. 1. Eu.]; and so the distances of every point of the line OPS from the point F and right line LG are equal, therefore the line OPS is a parabola (Def. 8. 1. Sup). And this line may be extended to a distance from the point F greater than any given distance, namely, if a square be taken, the length of whose side KQis greater than that distance.

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