Archytas was a mathematician, statesman and philosopher. He worked on the harmonic mean and gave it that name. The reason he worked on this was his interest in the problem of duplicating the cube, finding the side of a cube with volume twice that of a given cube. Hippocrates reduced the problem to finding mean proportionals. Archytas solved the problem with a remarkable geometric solution (not of course a ruler and compass construction).
He is sometimes called the founder of mechanics and Plato, a close friend, made use of his mathematics. Euclid also borrowed his work for Book 8 of the Elements .
