Proposition 3, Book XII, Euclid's Elements

This Demonstration is a dynamic illustration and a generalization of Proposition 3, Book XII of Euclids's Elements.



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


Proposition 3, Book XII of Euclid's Elements states: Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another, similar to the whole and having triangular bases, and into two equal prisms; and the two prisms are greater than the half of the whole pyramid [3]. ("Equal" means "have the same volume".)
The generalization includes lemma 14 [1].
[1] V. G. Boltyanskii, Tretja Problema Hilberta, Moscow: Nauka, 1977. Translated by R. A. Silverman as Hilbert's Third Problem (New York: John Wiley & Sons, Inc., 1978).
[2] D. E. Joyce, "Euclid's Elements: Introduction."
[3] Euclid, Elements, T. L. Heath, trans., as The Thirteen Books of Euclid’s Elements, 2nd ed., Book 3, New York: Dover Publications, 1956 pp. 378.
    • 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.