Diocles's Solution of the Delian Problem

The Demonstration constructs a cissoid and and uses it to show Diocles's solution of the problem of doubling the cube, also known as the Delian problem.
Suppose that a cube of side length is given; it has volume . To double the cube means to construct another cube with twice the volume as the original, , so the side of the new cube would be . Using an unmarked ruler and compass, it is impossible to construct a line segment as long as a given line segment. However, Diocles solved the problem with the aid of a cissoid.
Let be a circle of radius and center . Let and be points on equidistant to the diameter and on opposite sides of . Let be the diameter perpendicular to and let and be the perpendicular projections of and onto the diameter . Then , the point of intersection of the lines and , lies on a cissoid.
Since is a mean proportional between and , . By similarity, . It follows , since and .
Let be the intersection of and . Move so that is the midpoint of and (the cyan point). It follows that . Then and . So and .


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[1] T. Heath, A History of Greek Mathematics, Vol. 1, Oxford: Clarendon Press, 1921 pp. 264–266.
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