 Archytas’ solution has been rightly hailed as “the most remarkable of all [the solutions]” and as a “bold construction in three dimensions” (Heath 1921, 246); Mueller calls it “a tour de force of the spatial imagination” (1997, 312 n. 23). We owe the preservation of Archytas’ solution to Eutocius, who in the sixth century AD collected some eleven solutions to the problem as part of his commentary on the second book of Archimedes’ On the Sphere and Cylinder. Eutocius’ source for Archytas’ solution was ultimately Aristotle’s pupil Eudemus, who in the late fourth century BC wrote a history of geometry. The solution is complex and it is not possible to go through it step by step here (see Huffman 2005, 342–401 for a detailed treatment of the solution). Archytas proceeds by constructing a series of four similar triangles (see Figure 1 below) and then showing that the sides are proportional so that AM : AI :: AI : AK :: AK : AD, where AM was equal to the side of the original cube (G) and AD was twice AM. Thus the cube double the volume of the cube on AM should be built on AI. The real difficulty was in constructing the four similar triangles, where the given length of the side of the original cube and a length double that magnitude were two of the sides in the similar triangles. The key point for the construction of these triangles, point K, was determined as the intersection of two rotating plane figures. The first figure is a semicircle, which is perpendicular to the plane of the circle ABDZ and which starts on the diameter AED and, with point A remaining fixed, rotates to position AKD. The second is the triangle APD, which rotates up out of the plane of the circle ABDZ to position ALD. As each of these figures rotates, it traces a line on the surface of a semicylinder, which is perpendicular to the plane of ABDZ and has ABD as its base. The boldness and the imagination of the construction lies in envisioning the intersection at point K of the line drawn by the rotating semicircle on the surface of the semicylinder with the line drawn by the rotating triangle on the same surface. We simply don’t know what led Archytas to produce this amazing feat of spatial imagination, in order to construct the triangles with the sides in appropriate proportion. For a recent attempt to situate Archytas’ solution in the mathematics of his time and make it less “miraculous,” see Menn 2015. 