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 .



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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.
[1] J. J. O'Connor and E. F. Robertson. "Doubling the Cube." MacTutor History of Mathematics archive. (Jun 26, 2012) www-history.mcs.st-and.ac.uk/HistTopics/Doubling_the_cube.html.
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