Diocles's Solution of the Delian Problem

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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.

[more]

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 .

[less]

Contributed by: Izidor Hafner (September 2012)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Reference

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



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send