To draw an elliptical figure whose proportion of width to breadth shall remain the same, whatever the length of the major axis may be: Take any square figure and bisect it by the line A in Figure 80. Draw, in each half of the square, the diagonals E F, G H. From P as a centre with the radius P R draw the arc S E R. With the same radius draw from O as a centre the arc T D V. With radius L C draw arc R C V, and from K as a centre draw arc S B T.

Fig. 80.

Fig. 81.

A very near approach to the true form of a true ellipse may be drawn by the construction given in Figure 81, in which A A and B B are centre lines passing through the major and minor axis of the ellipse, of which a is the axis or centre, b c is the major axis, and a e half the minor axis. Draw the rectangle b f g c, and then the diagonal line b e; at a right angle to b e draw line f h, cutting B B at i. With radius a e and from a as a centre draw the dotted arc e j, giving the point j on line B B. From centre k, which is on the line B B and central between b and j, draw the semicircle b m j, cutting A A at l. Draw the radius of the semicircle b m j, cutting it at m, and cutting f g at n. With the radius m n mark on A A at and from a as a centre the point o. With radius h o and from centre h draw the arc p o q. With radius a l and from b and c as centres, draw arcs cutting p o q at the points p q. Draw the lines h p r and h q s and also the lines p i t and q v w. From h as a centre draw that part of the ellipse lying between r and s, with radius p r; from p as a centre draw that part of the ellipse lying between r and t, with radius q s, and from q as a centre draw the ellipse from s to w, with radius i t; and from i as a centre draw the ellipse from t to b and with radius v w, and from v as a centre draw the ellipse from w to c, and one-half of the ellipse will be drawn. It will be seen that the whole construction has been performed to find the centres h, p, q, i and v, and that while v and i may be used to carry the curve around on the other side of the ellipse, new centres must be provided for h p and q, these new centres corresponding in position to h p q. Divesting the drawing of all the lines except those determining its dimensions and the centres from which the ellipse is struck, we have in Figure 82 the same ellipse drawn half as large. The centres v, p, q, h correspond to the same centres in Figure 81, while v', p', q', h' are in corresponding positions to draw in the other half of the ellipse. The length of curve drawn from each centre is denoted by the dotted lines radiating from that centre; thus, from h the part from r to s is drawn; from h' that part from r' to s'. At the ends the respective centres v are used for the parts from w to w' and from t to t' respectively.

Fig. 82.

Fig. 83.

The most correct method of drawing an ellipse is by means of an instrument termed a trammel, which is shown in Figure 83. It consists of a cross frame in which are two grooves, represented by the broad black lines, one of which is at a right angle to the other. In these grooves are closely fitted two sliding blocks, carrying pivots E F, which may be fastened to the sliding blocks, while leaving them free to slide in the grooves at any adjusted distance apart. These blocks carry an arm or rod having a tracing point (as pen or pencil) at G. When this arm is swept around by the operator, the blocks slide in the grooves and the pen-point describes an ellipse whose proportion of width to length is determined by the distance apart of the sliding blocks, and whose dimensions are determined by the distance of the pen-point from the sliding block. To set the instrument, draw lines representing the major and minor axes of the required ellipse, and set off on these lines (equidistant from their intersection), to mark the required length and width of ellipse. Place the trammel so that the centre of its slots is directly over the point or centre from which the axes are marked (which may be done by setting the centres of the slots true to the lines passing through the axis) and set the pivots as follows: Place the pencil-point G so that it coincides with one of the points as C, and place the pivot E so that it comes directly at the point of intersection of the two slots, and fasten it there. Then turn the arm so that the pencil-point G coincides with one of the points of the minor axis as D, the arm lying parallel to B D, and place the pivot F over the centre of the trammel and fasten it there, and the setting is complete.

Fig. 84.

To draw a parabola mechanically: In Figure 84 C D is the width and H J the height of the curve. Bisect H D in K. Draw the diagonal line J K and draw K E, cutting K at a right angle to J K, and produce it in E. With the radius H E, and from J as a centre, mark point F, which will be the focus of the curve. At any convenient distance above J fasten a straight-edge A B, setting it parallel to the base C D of the parabola. Place a square S with its back against the straight-edge, setting the edge O N coincident with the line J H. Place a pin in the focus F, and tie to it one end of a piece of twine. Place a tracing-point at J, pass the twine around the tracing-point, bringing down along the square-blade and fasten it at N, with the tracing-point kept against the edge of the square and the twine kept taut; slide the square along the straight-edge, and the tracing-point will mark the half J C of the parabola. Turn the square over and repeat the operation to trace the other half J D. This method corresponds to the method of drawing an ellipse by the twine and pins, as already described.

Fig. 85.

To draw a parabola by lines: Bisect the width A B in Figure 85, and divide each half into any convenient number of equal divisions; and through these points of division draw vertical lines, as 1, 2, 3, etc. (in each half). Divide the height A D at one end and B E at the other into as many equal divisions as the half of A B is divided into. From the points of divisions 1, 2, 3, etc., on lines A D and B E, draw lines pointing to C, and where these lines intersect the corresponding vertical lines are points through which the curve may be drawn. Thus on the side A D of the curve, the intersection of the two lines marked 1 is a point in the curve; the intersection of the two lines marked 2 is another point in the curve, and so on.