The Solution of Erathostenes
Eratosthenes of Cyrene (276 - 194 BC) was a Greek scholar who made remarkable discoveries in many different fields. His work on prime numbers, such as the sieve of Eratosthenes, is still important in modern number theory. He also made a surprisingly accurate measurement of the Earth's circumference.
Here we are going to present his mechanical solution to double mean proportionals problem:
The two red parallel bars are units apart i.e. , as shown in the diagram below. Between them are three congruent triangles and . The first one is fixed and the rest is movable i.e you can drag the red points and to slide those two triangles. Adjust point in the diagram so that . To find the double mean proportionals between and , move the triangles such that the intersection points and are lying on the line segment between and . Then and are the required mean proportionals.