Doubling the Cube was one of three famous problems which were discussed by the early Greek geometers, the other two being the trisection of an angle and the squaring of the circle. There are several theories as to how the duplication problem originated; the statements of the ancients on this point being quite unsatisfactory. The legendary origin, told by Eratosthenes in a letter to Ptolemy Euergetes, was that King Minos, when he learned that the dimensions of a tomb for his son Glaucus were to be 100 feet each way, complained of them as too small, and commanded the tomb to be doubled and the cubical form to be retained. Another legend, also mentioned by Eratosthenes, was that certain Delians, in obedience to an oracle, attempted to double one of the altars, and finding a difficulty in doing so, consulted the geometers who were with Plato at the academy. The duplication of the cube hence came to be called the Delian problem.
In whatever manner the problem originated, it was much older than Plato's time, and the first contribution to the solution of it was made by Hippocrates of Chios. He showed that the solution could be obtained if between two straight lines, the greater of which was double the less, there could be inserted two mean proportionals; and in this modified form the problem was ever afterwards attacked. Solutions were discovered by various geometers, Archytas, Menæchmus, Eratosthenes, Nicomedes, and others, and an account of them will be found in the commentary of Eutocius on Archimedes's treatise Of the Sphere and Cylinder. This account is translated into English in the Proceedings of the Edinburgh Mathematical Society, vol. iv. pp. 2-17. It is often and inaccurately stated, even in mathematical books, that the duplication of the cube cannot be effected by geometry. The truth is that it cannot be effected by elementary plane geometry, where straight lines and circles are the only lines that are employed. By the use of the conic sections or several other geometrical curves, as well as by mechanical contrivances, the solution can be obtained without much difficulty. Nowadays the problem possesses only an historical interest, except for those persons whom De Morgan calls paradoxers.