A square by compass alone

Not one line is laid with the straightedge: every step is a swing of the compass, and the square shows itself only when its four corners are found. Georg Mohr showed in his Euclides Danicus of 1672, and Lorenzo Mascheroni again in La geometria del compasso of 1797, that every point compass and straightedge can find, the compass alone can find too. To find the corners of a square in a given circle by compass alone is a problem that bears Napoleon’s name. Walk the radius round the circle three times, from one corner to the corner opposite. Open the compass to a long chord of that walk, two steps wide, and swing it about both ends: the two swings cross at a point whose distance from the centre is the side of the square. Swung about the corner opposite, that opening cuts the circle at the last two corners. The sides are drawn in only to show the square.