Dürer’s spiral

In the first book of his Underweysung der Messung, printed in 1525, the first line Albrecht Dürer draws, once he has said what a line, a surface and a body are, is a spiral “mit dem zirckel”, with the compass. Under the woodcut he says it again: “Dise Schnecken lini ist mit dem zirckel zogen”, this spiral is drawn with the compass. It is useful for many things, he writes, and among them “zuo einem horneiffen”, a volute, on a capital; to make that clearer he draws two level lines back from it. His next spiral he plots point by point along a marked ruler. This one is his own construction, step for step, and it is exact. An upright line is divided into four equal parts, and the part second from the bottom is halved; here every division is a halving by a vesica. About the middle of the upright, opened to its top, a half circle swings round to its foot; about the halving point, from the foot back up to the first quarter; about the middle again, down to the third quarter; about the halving point again, up to the middle; and last about the point between the middle and the halving point, down to the halving point. Every centre stands on the upright, so each half circle sets off the way the last one arrived, and the line winds inward without a break. Here the line swells with the radius it turns on, and the two level lines are as wide as the line where they leave it.