The half circle laid straight

In 1685 the Jesuit Adam Adamandy Kochański, mathematician at the court of Jan III Sobieski, printed in the Acta Eruditorum of Leipzig a way to lay a half circle out straight, “for use by mechanics”. Draw a diameter and the radius at right angles to it. At both ends of the diameter set the radius off along the tangent. From the end of the upright radius cut the radius off once more on the circle, toward the first tangent, and lay a line from the centre through that mark to the tangent. On the second tangent lay the diameter on past the radius. The line between the two points is the half round, 0.0019 percent short: 3.14153 radii, where the truth is 3.14159. Kochański fixed the error by the year: his ratio falls short of the true one by less than one part in 16850, ten times the year 1685, and by more than one part in 16860, ten times the year to come. Every point here is found from the figure; only the claim that the line is as long as the half round is approximate.