[b][color=#0000ff]Eratosthenes of Cyrene[/color][/b] (276 - 194 BC) was a Greek scholar who made remarkable discoveries in many different fields.  His work on prime numbers, such as [b][color=#0000ff]the sieve of Eratosthenes[/color][/b], is still important in modern number theory.  He also made a surprisingly accurate measurement of the Earth's circumference.  [br][br]Here we are going to present his mechanical solution to double mean proportionals problem:[br][br]The two red parallel bars are [math]a[/math] units apart i.e. [math]AE=a[/math], as shown in the diagram below.  Between them are three congruent triangles [math]\Delta AFM,\Delta M'GN[/math] and [math]\Delta N'HQ[/math].  The first one is fixed and the rest is movable i.e you can drag the red points [math]N[/math] and [math]Q[/math] to slide those two triangles.  Adjust point [math]D[/math] in the diagram so that [math]DH=b[/math].  To find the double mean proportionals between [math]a[/math] and [math]b[/math], move the triangles such that the intersection points [math]B[/math] and [math]C[/math] are lying on the line segment between [math]A[/math] and [math]D[/math].  Then [math]x=BF[/math] and [math]y=CG[/math] are the required mean proportionals.