Dividing the Circle: Inscribing Regular Polygons
One rule marks off chords giving triangle, hexagon, octagon and up to 48 equal parts
Both diagrams show methods for dividing a circle into equal parts and thereby inscribing regular polygons. In the first, chords stepped around the arc from a common point — a b, a c, a d, a m, a n — enter the circle six, eight, twelve, twenty-four and forty-eight times respectively, while the segment f r enters three times, so that one figure yields divisions into 3, 6, 8, 12, 24 and 48 parts and, by the same rule, into any number. The second gives a construction using a vertical line tangent to the circle at a, marked off in thirds of m a, from which the compass set at n steps off divisions into 3, 4, 5, 6, 7 and 8 parts.
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Chords dividing the circle into 3, 6, 8, 12, 24, 48 parts (f, a, n, m, d, c, b, r)
From a common point the chords a b, a c, a d, a m and a n enter the whole circle six, eight, twelve, twenty-four and forty-eight times, and the line f r enters three times. One figure therefore divides the circle into 3, 6, 8, 12, 24 and 48 parts, and by the same rule into any number of equal parts.
Dividing a circle with a tangent line marked in thirds (m, n, f, a–e)
A vertical line a e is set tangent to the circle at a and extended upward by one third of m a; the whole line f e is filled with these thirds. Setting one compass foot at n then steps off divisions: n f enters three times, n a four, n b five, n c six, n d seven and n e eight.