A portion of an ellipse D A E (figs. 96, 97) will serve to

form the section of a Gothic torus or base moulding, C

cb the semi-axes of the other ellipse.

In fig. 99 is shown how the ellipse can be applied to

form the sections of flutings for columns; EBF being the outline of the fluting rather less than half an ellipse, whose centre is at C, and whose semi-axes are CA, CB.

119. To apply the ellipse to form the contour of leaf ornaments. Let BD (fig. 100) be the axis of the leaf; draw A C, A'C' in

tersecting the axis at

E, and making

equal angles with it. Draw BC perpendicular to AC, BC' to A'C'; then AC, BC are the semi-axes of an ellipse, and A'C', BC' those of an equal and similar ellipse. The ellipses

BAD, BA'D can be drawn by either of the previous methods (104, 105), and the contour of a leaf will be formed.

Fig. 100.

Let PQ (fig. 101) be the axis of the leaf, QT a line at

120. To apply the ellipse to form the entasis of a column. -Let BC (fig. 102) be the axis of the column, A C its half-diameter at base, BD the half-diameter at summit.

G

Draw DE parallel to BC, then AE is the difference be

tween the top and bottom radii. From C as a centre, with CA as radius, describe a circle cutting DE at K. Divide BC into any number of equal parts at R, S, T, &c., and draw RQ, SP, &c., at right angles to BC. Divide EK into the same number of equal parts as BC is divided into, at the points a, b, c, &c., and draw al, bm, &c., at right angles to EK, and cutting the circle in the points l, m, &c. Through l, m, n, &c., draw lines parallel to BC intersecting those through R, S, T, &c., at the points Q, P, &c.; then Q, P, &c. will be points upon the circumference of an ellipse, and the curvé can be drawn by bending a straight-edge through them.

121. To apply the ellipse to form the outline of a Tudor arch. -Let A C, BC (fig. 103) be the semi-axes of an ellipse of considerable eccentricity, BC being greater than the vertical height of the proposed arch from the springing, and AC greater than its half-span. Draw AL at right angles to AC, and let AK or AL be the height of the arch,

AN or A M its half-span. Draw NQ or MP at right angles to A C, and KQ, LP parallel to A C; then the arc

AQ or AP forms one side of the arch required. The opposite side, Q E, or P C, will be drawn by repeating A Q, or A P.

By this method, a continuous curve is obtained from the springing to the vertex, the change of curvature taking place gradually throughout, and not suddenly, as is the case when the arch is struck by two different arcs of circles, forming an abrupt change in the rate of curvature at their point of junction.

122. A diameter of an ellipse and an ordinate being given, to find the conjugate diameter.-Let A B (fig. 104) be

G 2

the given diameter, F G the given ordinate. Bisect A B in C. At F, draw F H at right angles to A B; and from C as a centre, with CA as radius, describe a circle cutting FH in H. Take A I, equal to F H, and draw IK parallel and equal to F G. Draw DC E parallel to FG, join A K, and produce it to meet DE at D. Make CE equal to CD, and DE is the conjugate diameter required.

123. To inscribe an ellipse in a given trapezium.—Let ABCD (fig. 105) be the given trapezium. Produce the

sides to meet at the points P and Q. Draw the diagonals intersecting at I. Through I draw PEIG, QFIH; then the ellipse will touch the sides of the trapezium at the points E, F, G, H. Bisect E G at T, and F H at V. Draw PVO, QTO intersecting in O; then O is the centre of the required ellipse.

Join EO, and produce E O to K, making O K equal to OE. Then E K is a diameter of the ellipse, and H L, MF, drawn parallel to A D, are ordinates. Therefore the conjugate diameter can be found by help of the last proposition (122), and the ellipse drawn by the method before described (115).