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to a S, be equal to 6 S, and drawing a K parallel to Sd, 6L parallel to es, and so on. Let the normal at a cut the axis at K, and from K as a centre, with KA or Ka as radius, describe the arc A a. Let the normal at b cut the normal at a in the point L, and from L as a centre, with La or Lb as radius, draw the arc ab. Proceeding in this way we can obtain a series of small arcs of circles which will form a tolerably near approach to an arc of a hyperbola ; and the closer the points, a, b, &c., are taken to each other, the nearer will the resulting curve be to a hyperbola.

138. To apply the hyperbola to form the entasis of a column.—Let A Q (fig. 109) be a rectangular hyperbola (131), having CS for its axis, A for its vertex, and S its focus. Let AS be the difference between the radii at the top and bottom of the shaft of the column, SP the vertical height of the shaft. Draw the vertical A B B', and the horizontal PB'. Let S P cut the rectangular hyperbola at the point l, and divide Sl into any number of equal parts, and also A B' into the same number of equal parts at the points, 1, 2, &c. Draw the horizontals le, 2f, &c., cutting the rectangular hyperbola at e, f, &c., and draw the ordinates through those points, producing them to meet horizontals through the divisions on A B' in the points i, h, &c.; then these will be points on the required curve through which it can be drawn by bending a straight-edge. It is not necessary to draw the contour of the rectangular hyperbola, but only to find as many points upon it as we require to find on the entasis hyperbola.

139, Application of the hyperbola to form the outline of mouldings and other architectural forms.—The hyperbola may be used to form an ovolo in classic architecture (fig. 113); taking C A X as the axis, C E and C T as the asympA pointed arch (fig. 114) can be formed by two equal arcs of a hyperbola meeting at the vertex B or b, A D being

totes, S the focus ; then the curve P AQ can be drawn by any of the previous methods.

Fig. 113.


Fig. 114.


the axis, and A or a the vertex of the curve; when used as a stone arch, the voussoirs must have their joints in the directions of the normals to the curve (135).

Leaf ornaments (figs. 115, 116) can be formed by arcs of hyperbolas. Let CD be the axis of the leaf, dividing it

into two equal parts, A X the axis of a hyperbola, D AC (fig. 115) having its tangent D T at D perpendicular to CD; this will form one side of the leaf, and by reversing the outline we get the other side, D A'C having A' X' for

Fig. 116.

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its axis, A X and A' X' cutting C D at equal angles. Any variety of form can be obtained ; thus the curve may be turned in at D (fig. 116), and not have its tangent at that point perpendicular to DC.

THE PARABOLA. 140. DEFINITIONS.—The parabola is a single curve, which is unenclosed, and consists of two equal branches, AQ, A q (fig. 117), divided by a straight line, B A X, called the axis, which it cuts at right-angles in a point A called the vertex. It has only one focus S situate within the curve, and the double ordinate LS l is called the latusrectum and is equal to four times the length of SA, so that Slor SL equals twice S A.

If we produce the axis to B outside the curve, and take A B equal to AS, then a line B C drawn perpendicular to BA is called the directrix; and if P is any point on the curve, and PK is drawn parallel to AB to meet the directrix at K, then P K is equal to PS. The line SP is called the radius-vector, A M is the abscissa, M P the ordinate of the point P on the curve. The parabola differs from the ellipse and hyperbola in having no centre; it may, in fact, be considered as an ellipse whose axis is of an infininite length. The curvature of the parabola is greatest at the vertex, and decreases as it recedes therefrom.

141. To draw the parabola by continuous motion. Let BAX be the axis, A the vertex, BC the directrix, S the

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focus ; A S being equal to A B. Fix a straight-edge against BC, or let B C be the edge of a drawing-board, at right

angles to B X. Let a T-square CR move from the axis along BC, having a thread of length equal to RC attached at one end to R, the other end being fixed to a pin at S. If the string is kept tight by the point of a pencil pressed against the T-square as it moves upwards or downwards from the axis, the pencil will mark out the parabola.

142. To find points on the contour of the parabola.Suppose the axis AM (fig. 118) and the vertex A to be given, and also a point P

Fig. 118. through which the curve is

В f to pass. Draw A B perpen

g h i k p dicular to AM, PB parallel to A M. Divide A B into any number of equal parts by the points a, b, &c.; and also divide B P into the same number of equal parts at f, g, &c. Draw the lines Af, A 9, &c., cutting horizontals through a, b, &c., in the points l, m, &c.: these will be points on the required curve.

143. To find the lengths of ordinates to a parabola by means of a scale of parts.—Let A M (fig. 119) be the axis, A the vertex, S the focus. If we measure S A on any scale as 1000, then the ordinate SL will be 2000 on the same scale. The following table shows the relative proportions of the abscissæ and their corresponding ordinates :

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