Squared Circle

A circle and a square of equal area, or of equal perimeter, drawn together. In geometry the first is impossible with compass and straightedge, and that impossibility is part of what the figure means: the circle is the infinite and the square the manifest, and the operation that would make them equal cannot be completed. The alchemists took it as the emblem of the work, the making of the fixed out of the volatile.

No exact construction exists. A square equal in area to a given circle would require a length equal to the radius times the square root of pi, and pi is transcendental, so no finite compass-and-straightedge construction can produce it; this was settled by Lindemann in 1882, after Wantzel had shown in 1837 that the older problems were insoluble. Approximate constructions exist and are old, and the alchemical emblem is drawn by eye rather than proved.