The octagon from the square

“Der do wil ein gerecht acht ecke machen”: so the workshop booklet Geometria deutsch, attributed to Mathes Roriczer, opens its true octagon. Draw the square, open the compass from its centre to a corner, and set that opening off from every corner along both sides. Join the marks across each corner and cut it off: what is left is the octagon. The opening is half the diagonal, the radius of the square’s own circle, so every swing from a corner runs through the centre, and the octagon is exact.