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