The Eratosthenes Machine for Finding the Cube Root of Two

move second triangle

move third triangle

d(G,C) = 1.256

This Demonstration shows Eratosthenes's machine for finding two mean proportionals; that is, given lines

a

and

b

, find

x

and

y

such that

a:x=x:y=y:b

. If

a=1

and

b=2

, the solution is

x=

3

2

. Let the lengths of

EA

and

HD

be 2 and 1, respectively. Move the second and third triangles so that points

B

and

C

lie on the straight line

AD

, giving the length of

GC

as

3

2

(approximately 1.25992…).

Eratosthenes's machine consists of a parallel frame and three congruent triangles. Let

|EA|=b

and

|HD|=a

. Move the triangles on the right so that

B

and

C

are on the line

AD

. By similarity

EA:FB=FB:GC=GC:HD

.