Fig.1. Draw a straight line, equal in length to the semicircle ABC. With A and C as centres, and for radius AC, strike the two arcs to intersect each other in S. Join SA and SC extended, to cut the line through B in D and E. Then, DE is the length of the required line, and if this was bent around the semicircle it would reach from A to C. This line throughout this work is termed the stretch-out of the semicircle.

Fig.2. Given the length DE, find the radius to strike a semi-*circle equal in length to it. Draw a line from E at 60°, and from B at 45° to DE, to cross each other at C. Draw from B square to DE, and from C parallel to DE to meet in O; then OB will be the required radius.

Figs. 3, 4 and 5 show how to bisect any given angle. Let ABC be the given angle. With B as centre, strike the arc DD to any radius. With DD as centres, and for radius more than half the distance DD, describe arcs intersecting in E. Then, a line from B to E will bisect the angle.

Figs. 6, 7 and 8 show how to ease any given angle, that is to form a curve that will connect the two straight lines, from any two given points, on those lines. Let AB and BC be the two lines forming the given angle, and it is required to connect those lines from A to C. Divide AB and BC into any number of equal parts, connect those parts, and the curve will be formed if AB and BC has been divided into a sufficient number of parts. Fig.9 shows a semi-ellipse, AB being the semi-major axis, and BD the semi-minor axis. Let AB and DB, Fig.10, equal AB and DB, Fig.9. To strike the curve, move this rod around, keeping D on the major axis, and A on the minor axis, and mark off points at the end of the rod all round.

Fig.11. Given a semi-ellipse, draw a normal tangent. Determine the foci of the ellipse FF. With D as centre, and for radius AB strike arcs of circles at FF. At any point on the curve, say at S, draw lines to FF and bisect the angle. Now draw through S, square to this line that bisects the angle for the required normal tangent.