the intersecting lines, all of which are tangents, and each one produces a point in the curve.

To Describe the Ellipse by Trammelling Fig. 4.— Draw the major and minor axes perpendicular to each other. Take a strip of paper or wood ami mark off from one end the lengths of the semi-major and semi minor axes respectively, as shown at M}, M*. If now the strip is placed across the axes so that these points rest on them as shown in two positions in Fig. 4, the end of the rod will be in the curve at that particular point, and if it is marked with a pencil a sufficient number of such points maybe found, through which the curve may be drawn either freehand or by aid of wood curves. This method is especially useful in producing small ellipses in which the trammel itself is ralher cumbersome.

Another method of drawing an ellipse by means of intersecting lines is shown in Fig. 8, and it is, perhaps, the clearest and best for draughtsmen. Draw the major axis A-B, and the semi minor axis x-c perpendicular to it. Describe two circles, one on the major axis, the other on the minor (one-half only is shown). Divide each semicircle into the same number of equal parts, and this is best done by drawing radials from the centre to the divisions on the large circle ; these will divide the smaller one .similarly, as shown by the dotted lines in the left quadrant. From the points on the outer circle draw ordinates parallel with the minor axis, and from the corresponding points on the inner circle draw lines parallel to the major axis. The intersections of these lines will be points in the elliptic curve, which may then be drawn through them.

To find the Centre and Axes of a given Ellipse.—Let

M-M, m -m, Fig. 5, be the given ellipse; draw any two lines parallel and cutting opposite sides of the ellipse, as A-A and B II. Bisect each of these and draw the line D -D through their centres. Bisect D D in point C, which is the centre of the ellipse. To determine the direction of the axes. With C as centre, and any radius, describe a circle cutting the ellipse in points x-2-3. Join 1-2 and 2-3, and lines drawn parallel to these through the centre will be the major and minor axis respectively.

To find the foci, normal and tangent, of any ellipse; (see Fig. 6). With the seini major axis as radius and either end (jf the minor axis as centre, describe an arc cutting the major axis in F.F. These are the two focal points.

Normals.—Let n, Fig. 6, be a ¡joint in the curve to which we desire to find a normal or perpendicular. From this point draw lines to the foci F-F., and bisect the angle contained between them. This is done by describing an arc from point n, and from the ends of the arc, with same radius, describe intersecting arcs. Draw a line through this point and the given point n. This line is normal to the curve at that particular point, and any other may be found in like manner. This method is used to obtain the joint lines of the voussoirs in elliptic arches, and for the ribs of centering.

A tangent a.t the same point is readily found by drawing a line at right angles to the normal and touching the curve.

To describe an Ellipse by Means of a Looped String, Fig. 7.—Draw the conjugate axes to dimensions required, and find the focal points as described in Fig. 6. Drive two pins in the foci, as at 1-2, and a third pin at A, one end of the major axis. Form a tight loop with thread or string around the two pins at x and A ; having fastened the loop, remove the pin at A and substitute a pencil. The loop will now lie around the two pins 1-2. and a regular and continuous curve will be produced by keeping the string tant and moving the pencil around from point A, as shown (the original of this drawing was actually produced by the method described). In practice, with large curves, a difficulty is experienced in preventing the string stretching, and so interrupting the continuity of the curve, hence trammelling is more often resorted to in the workshop. For draughtsmen's work a silk thread is often employed.

The False or Three-Centred " Ellipse," Fig. 9.--This figure is drawn with compasses, and is therefore not a true elliptic figure, but is an approximation thereto, much used

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