The Solution of Menaechmus
Menaechmus (380 - 320 BC) was a Greek mathematician. In fact, he was the mathematics tutor of Alexander the Great. His most famous discovery was on conic sections - He was the first to show that the three special curves - parabola, hyperbola and ellipse - are obtained by cutting a cone in a plane not parallel to the base.
He gave two solutions to double mean proportionals problem. Both involve the intersection of special curves. Given any two positive real numbers and , we want to find and such that
.
First solution: We construct a parabola and a hyperbola (the blue and black curves in the diagram below) that satisfy the following equations respectively:The intersection point of these two curves is . Its coordinates are the required values of and . Second solution:We construct two parabolas (the blue and red ones in the diagram below) that satisfy the following equations respectively:
The intersection of these two curves is . Its coordinates are the required values of and .