Cissoid of Diocles - Delian Problem

Delian Problem

The Greek geometer Diocles used the cissoid to obtain two mean proportionals to a given ratio. This means that given lengths a and b, the curve can be used to find u and v so that u/a=v/u=b/v. As a special case, this can be used to solve the Delian problem: how much must the length of a cube be increased in order to double its volume? Specifically, if a is the side of a cube, and b=2a, then the volume of a cube of side u is

so u is the side of a cube with double the volume of the original cube. Note however that this solution does not fall within the rules of compass and straightedge construction since it relies on the existence of the cissoid.

Let a and b be given. It is required to find u so that u3=a2b, giving u and v=u2/a as the mean proportionals. Let the cissoid

be constructed as above, with O the origin, A the point (2a, 0), and J the line x=a, also as given above. Let C be the point of intersection of J with OA. From the given length b, mark B on J so that CB=b. Draw BA and let P = (x, y) be the point where it intersects the cissoid. Draw OP and let it intersect J at U. Then u=CU is the required length.

To see this, rewrite the equation of the curve as

and let N = (x, 0), so PN is the perpendicular to OA through P. From the equation of the curve,

From this,

By similar triangles PN/ON=UC/OC and PN/NA=BC/CA. So the equation becomes

so

as required.

Diocles did not really solve the Delian problem. The reason is that the cissoid of Diocles cannot be constructed perfectly, at least not with compass and straightedge. To construct the cissoid of Diocles, one would construct a finite number of its individual points, then connect all these points to form a curve. The problem is that there is no well-defined way to connect the points. If they are connected by line segments, then the construction will be well-defined, but it will not be an exact cissoid of Diocles, but only an approximation. Likewise, if the dots are connected with circular arcs, the construction will be well-defined, but incorrect. Or one could simply draw a curve directly, trying to eyeball the shape of the curve, but the result would only be imprecise guesswork.

Once the finite set of points on the cissoid have been drawn, then line PC will probably not intersect one of these points exactly, but will pass between them, intersecting the cissoid of Diocles at some point whose exact location has not been constructed, but has only been approximated. An alternative is to keep adding constructed points to the cissoid which get closer and closer to the intersection with line PC, but the number of steps may very well be infinite, and the Greeks did not recognize approximations as limits of infinite steps (so they were very puzzled by Zeno's paradoxes).

One could also construct a cissoid of Diocles by means of a mechanical tool specially designed for that purpose, but this violates the rule of only using compass and straightedge. This rule was established for reasons of logical — axiomatic — consistency. Allowing construction by new tools would be like adding new axioms, but axioms are supposed to be simple and self-evident, but such tools are not. So by the rules of classical, synthetic geometry, Diocles did not solve the Delian problem, which actually can not be solved by such means.

On the other hand, if one accepts that cissoids of Diocles do exist, then there must exist at least one example of such a cissoid. This cissoid could then be translated, rotated, and expanded or contracted in size (without changing its proportional shape) at will to fit into any position. Then one would readily admit that such a cissoid can be used to correctly solve the Delian problem.