The Eratosthenes Machine for Finding the Cube Root of Two

This Demonstration shows Eratosthenes's machine for finding two mean proportionals; that is, given lines and , find and such that . If and , the solution is . Let the lengths of and be 2 and 1, respectively. Move the second and third triangles so that points and lie on the straight line , giving the length of as (approximately 1.25992…).

Eratosthenes's machine consists of a parallel frame and three congruent triangles. Let and . Move the triangles on the right so that and are on the line . By similarity .

The problem of doubling the cube was to find by ruler and compass, which was proved impossible [1]. This Demonstration shows a solution by sliding a line, which is not an allowable operation in a construction by ruler and compass.