Fig. 1. Draw a straight line, equal in length to the semicircle A B C.
With A and C as centres, and for radius A C, strike the two arcs to
intersect each other in S. Join S A and S C extended, to cut the line
through B in D and E. Then, D E 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 D E, 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
D E, to cross each other at C. Draw from B square to D E, and from C
parallel to D E to meet in O; then O B will be the required radius.
Figs. 3, 4 and 5 show how to bisect any given angle. Let A B C be
the given angle. With B as centre, strike the arc D D to any radius.
With D D as centres, and for radius more than half the distance D D,
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 A B and B C be the two lines forming the
given angle, and it is required to connect those lines from A to C.
Divide A B and B C into any number of equal parts, connect those parts,
and the curve will be formed if A B and B C has been divided into a
sufficient number of parts.
Fig. 9 shows a semi-ellipse, A B being the semi-major axis, and B D the
semi-minor axis. Let A B and D B, Fig. 10, equal A B and D B, 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 F F. With D as centre, and for radius A B strike
arcs of circles at F F. At any point on the curve, say at S, draw lines
to F F and bisect the angle. Now draw through S, square to this line
that bisects the angle for the required normal tangent.
[Illustration: PLATE 2.]