# The Eratosthenes Machine for Finding the Cube Root of Two

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…).

a

b

x

y

a:x=x:y=y:b

a=1

b=2

x=

3

2

EA

HD

B

C

AD

GC

3

2

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 .

|EA|=b

|HD|=a

B

C

AD

EA:FB=FB:GC=GC:HD