# Proposition 3, Book XII, Euclid's *Elements*

Requires a Wolfram Notebook System

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

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

Contributed by: Izidor Hafner (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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.

## Permanent Citation