9772

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 .

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

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

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+