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This method is of use in drawing a perspective representation of a circle, the trapezium being the representation of a square circumscribing the circle. The point K is not necessarily upon the line B C.

THE HYPERBOLA. 124. DEFINITIONS.—The name hyperbola is given to a class of curves which are not enclosed, or that do not enclose a space. These curves have two branches, as A L, Al (fig. 107), each of which start from the same point, A, and go off to infinity; the two branches are exactly alike, and are divided from each other by a right-line, C S, passing through the vertex, A, of the curve, which is the point where the two branches meet; this line, CS, is called the axis of the curve, and lines SL, SI, drawn at right angles to the axis, cut the two branches of the curve at equal distances therefrom. The hyperbola has two fuci, s, (fig. 106), and the point C, halfway between the foci, is called the centre. The ratio of the distance of the centre from the focus S, to its distance from the vertex A or CS:C A, is called the eccentricity of the curve, which is increased as A is moved towards C, and decreased as A is moved away from C; thus (fig. 107) there are three hyperbolas of different degrees of eccentricity, but having a common centre and focus. Any line drawn from the focus S to the curve, as S P (fig. 106), is called the radius-vector. Two lines intersecting at the centre C (fig. 107), and making equal angles with the axis Cs, as CB, Cb, to which the curve is continually approaching without ever reaching, are called asymptotes. When the asymptotes make a right-angle with each other, the curve to which they belong is called the rectangular-hyperbola. The ordi

nate drawn through the focus S, as SL (fig. 107), at right angles to the axis, is called the latus-rectum. Any distance measured along the axis, either from the centre C or the vertex A, is called an abscissa.

125. To draw an hyperbola by continuous motion.Draw a straight line, HCS (fig. 106), as an axis, and let

Fig. 106.

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HS be the given distance apart of the foci, C the centre, A the given vertex. Between A and C take A X, equal to A S. Let a ruler, as H Q, be made to revolve about a pin at one end, H, in the axis ; and from H measure on HQ the length H Y, equal to H X. At Q, fasten a thread of length equal to Q Y, and attach the other end to a pin at the focus S. Let the ruler lie upon the axis, and the thread be tightened by a pencil pressed against the edge of the ruler. Then, as the ruler revolves about the point H, the pencil keeping the thread tight will mark out the curve beginning at the vertex A.

This method is derived from the property of the hyperbola that the difference between the focal distances of any point P on the curve are always the same; that is, HP -SP is always equal to H X or H Y.

126. To find any number of points upon the contour of a hyperbola, whose vertex and foci are given.—Let HS (fig. 106) be the axis, H and S the given foci, A the given vertex. Make A X equal to A S, take any point E on the axis to the right of the vertex A, and from Sas a centre, with X E as radius, describe a circle ; also from H as a centre, with H E as radius, draw an arc cutting the circle at the point P, which will then be a point upon the required curve. In the same manner we can find any number of points on the curve, by taking different points along the axis from A, and

Fig. 107. proceeding as before. Having found a sufficient number of points on the curve, we can draw the curve itself by hand, or by bending a ruler through those points.

127. To drau the asymptotes of a hyperbola, whose vertex and foci are given. — Let HCS be the given axis (fig. 107), H and s the foci, C the centre, A the given vertex. Draw 6 A B at right angles to A C, and from C as a centre, with CS as radius, describe

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a circle cutting b A B in the points B, 6. Draw the rightlines CB, Cb, which will be the asymptotes of the hyperbola l A L.

In the rectangular-hyperbola, A B is equal to A C, and the asymptotes make a right angle with each other at the centre C.

128. To find points on a hyperbola by means of the asymptotes.---Let C A (fig. 108) be the axis, CR, Cr the

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asymptotes found as described above (127). Draw any line q A Q making an acute angle with the axis, and cutting the asymptotes in Q and q. Take Q P, equal to A q, then P is a point on the hyperbola. So also by drawing any other lines through the vertex and intersecting the asymptotes, we can find as many points as we please upon the re

quired curve, which can then be drawn in by hand, or by bending a ruler through those points.

Having determined one point P on the part of the curve above the axis, we can find points on the part below by drawing any line, R P r, cutting the asymptotes in R and r. Take rp, equal to RP, then p is a point on the lower curve; and in the same way, by drawing other lines through P or A intersecting the asymptotes, any number of points on the lower curve can be found.

The two portions of the hyperbola, above and below the axis, are exactly equal and similar, so that when one is drawn the other can be drawn from it.

129. To draw any hyperbola by means of the rectangular hyperbola.Let C A (fig. 109) be the axis, C the centre, and A the vertex of the required curve. Draw A B at right angles to A C, and equal thereto, then the line C B is an asymptote of the rectangular hyperbola, and taking CS equal to CB, S will be one of its foci. Let C B' be a given asymptote of the required hyperbola, draw A B' at right angles to A C, and make C S' equal to C B', then S' is the focus of the required curve. Find points on the rectangular hyperbola, as f, l, Q, by any of the previous methods, and draw the ordinates, M fh, SIP, SQL. Then, by taking Mh: Mf, Sl:SP, SQ:SL in the ratio of AB': A B, we find corresponding points h, P, L, on the required curve. This can be done geometrically (16), or by using proportional compasses or a scale of parts.

Suppose it be required to find points on the hyperbola from A to P, draw the ordinate SIP, which will be equal to A B'; divide S l into any number of equal parts at 1, 2, 3, &c., and also divide SP or A B' into the same number of

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