BC the semi-minor-axis of the ellipse, S and H the foci. Find any number of points E, F, G, &c., upon the curve by the foregoing methods (105) and draw the normals (110) EL, FM, GN, &c., at those points. Let the normal at E intersect A C at L, and from L as a centre, with LA or LE as radius, describe an arc AE. Let the normal at F meet that from E at the point M, and from M as a centre, with ME or MF for radius, describe the arc EF. Let the normal at G intersect the normal at F in the point N, and from N as a centre, with NF or N G as radius, describe the arc F G. Proceeding in this way at the other points we can obtain a contour consisting of a number of arcs of circles varying in curvature which will form a near approach to the outline of an ellipse.

The larger the number of points that are taken on the ellipse the nearer will the resulting curve appear to approach to a true ellipse.

114. DEFINITIONS.-Any line A B (fig. 90) drawn across an ellipse and passing through the centre (C) is called a diameter; if a second diameter DE is drawn parallel to the tangents FAG, HBI, at the two ends of the first, the two diameters are said to be conjugate. Any line drawn from one diameter to the curve and parallel to the other diameter is an ordinate. The tangents at the two ends of any diameter are parallel to one another. Each conjugatediameter bisects every chord of the ellipse which is drawn parallel to the other conjugate-diameter.

The tangents at the ends of a pair of conjugate-diameters form a circunscribing parallelogram FGHI to the ellipse, which is always the same in area, and is divided into four equal parallelograms by the conjugate axes. Two lines drawn from a point on an ellipse to the extremities of a diameter are called supplemental chords, as a E, a D.

115. A pair of conjugate-axes being given to find points on the ellipse by means of the supplemental chords.—Let AB, DE (fig. 90) be the given axes, C the centre of

the ellipse. Divide A C, BC, into any number, as four, equal parts by the points 1, 2, 3. Also divide AF, BI, into the same number of equal parts, and join the points 1, 2, 3, on those lines with the point D. From E draw chords passing through the points on A C, BC, and cutting those drawn from D in a, b, c, which will be points on the perimeter of the ellipse.

Another method.

Let A B, DE (fig. 91) be the given axes, C the centre of the ellipse. Draw HAF parallel to DE, and make A F and A H each equal to DE. Draw FG parallel and equal to A B. Divide FG into any number, say four, equal parts by the points 1, 2, 3; also divide AH into the same number of equal parts, and draw lines to B from each

dividing point on A H. Through A draw lines from each of the points on FG intersecting the chords drawn from B

in a, b, c, which will be points on the circumference of the required ellipse.

116. In a given parallelogram to inscribe an ellipse.Let FGHI (fig. 90) be the given parallelogram. Bisect the sides in the points A, B, D, E; and join DE, AB, by lines intersecting at C, which will be the centre of the required ellipse, AB and DE being conjugate diameters. Proceed to find points on the circumference as before described (115), and draw the curve touching the parallelogram at A, B, D, and E.

117. Any pair of conjugate diameters of an ellipse being given to find the axes.-Let A B, DE (fig. 90) be the given diameters, C the centre of the ellipse. Draw the tangent GET parallel to A B, and erect the line E K perpendicular to ET and equal to CB. Now find the centre of the circle which shall pass through C and K and have its centre on

the tangent ET; this is done by joining CK and bisecting CK at right angles by the line OR meeting the tangent at O. Join OK, then from O as a centre, with OK as radius, describe a circle cutting the tangent in the points T and t. Join CT, Ct, and these will be the directions of the axes of the ellipse, since BC is a mean proportional between ET and Et. To find the points N and P where the axes cut the ellipse, join KT, Kt, and on them take Kn, K m, each equal to K E. Draw n N, m P, parallel to KC; then CN, CP are the lengths of the half-major and minor axes of the ellipse.

118. To apply the ellipse to form the contour of architectural mouldings.-Let it be required to draw an ovolo of which the tangent at one point D (fig. 92) is given, and

Draw the vertical line
Draw A C parallel to

also the greatest projection at A. AF, and also DC parallel to AF. FD; then C is the centre of the ellipse, CA and CD semiconjugate diameters, and the ellipse can be drawn by points as described above (115).

Other forms of the ovolo can be drawn by taking A C, BC (figs. 93, 94), as the semi-axes, the flatness or round

ness of the curves being regulated by the ratio of CA to CB.

The cavetto or hollow moulding (fig. 95) can be formed