**APPENDIX
III**

**The
Construction of the Perfect Heptagon**

Modern
mathematics has produced proof of the impossibility of geometrically
constructing a perfect heptagon.
Nevertheless, there are two constructions that produce a heptagon so
close to perfection that in practice it is often difficult to discover any
disparity, for the accuracy depends upon the thickness of a line.

1. Within the circle to be divided into
seven equal arcs, an octogram is inscribed consisting
of a right square UVQT and a turned square MONP.

2. Within the turned square a right square
YZCA is inscribed.

3. The vertical diameter MN is drawn
cutting AC at B.

4. AC is extended to meet the
circumference at E. (The accuracy of
this operation can be checked by drawing the arc BD down to E.)

5. From centre E and radius EB an arc is
drawn to cut the circumference at F and K.

6. EF and EK are then rotated around the
circumference to create the polygon EFGHLJK.

**The Ad Triangulum Construction**

1. Within the circle to be divided into
seven equal arcs a hexagram AECFBD is inscribed consisting of two triangles ABC
and DEF.

2. The vertical diameter AF is drawn,
bisecting BC at G.

3. From centre C and radius CG an arc is
drawn to cut the circumference at H and J.

4. CH and CJ are rotated round the
circumference to create the polygon CHKLMNJ.

This
construction is much the simplest, but both produce an identical result, for GC
and BE are identical. The AD Triangulum, in simplified form, was the construction used
by medieval masons if it was ever required.