Dividing the Circle and Building an Octagon from a Square
Proof that chord a d is one-eighth of the circle; laying a regular octagon on a square
The upper diagram and its note prove that the chord a d is one eighth of the circle: since a e is one sixth, and that sixth is divided into four parts (four times six being twenty-four), a d spans three of those parts, and because three goes eight times into twenty-four, a d fits eight times into the circle. The lower figure gives a compass construction for inscribing a regular octagon in the square r f m S: with one compass foot at corner f and the other at the square's centre, an arc struck to the side at b and joined to f meets the side at n, marking one face of the octagon between n and b.
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Proof that chord a d is one-eighth of the circle (a, b, c, d, e)
a e is one sixth of the circle, and that sixth is divided into four parts, so four times six is twenty-four. The chord a d is three of those four parts, and since three goes eight times into twenty-four, a d enters eight times into the circle.
Inscribing a regular octagon in the square r f m S (f, n, b, r, a, m, S)
To divide the square r f m S into eight faces, set one compass foot at corner f and the other over the centre, carrying that foot from the centre out to the side at b. Draw b to f; where it crosses at n lies one face of the octagon, between n and b.