Hero of Alexandria's Mechanics

    The Mechanics of Hero of Alexandria tells us that the ancient Romans had a fairly good understanding of the use of simple machines. Hero discusses "the wheel and axle, the lever, the pulley, the wedge, and the screw." In his Catoptrica he also shows that light reflecting from a mirrored surface takes a minimum path.

  Hero's Mechanics survived only in an Arabic translation. There doesn't appear to be an English translation available but there are French translations and German translations. In French the simple machines are given as "le treuil, le levier, la  moufle (poulie), le coin et la vis sans fin."

Lagrange in his Analytical Mechanics makes a rather cryptic remark about the Center of Force and for heavy bodies subject to gravity it being the center of the Earth. What is peculiar about it is that he equates the ratio of the potential energy to the sum of the forces acting on a mechanism as the distance to the center of the Earth. This would be true if the distances to the Earth's center are the same or the forces acting on the parts of a body are equivalent to a single force acting on a center of mass. The change in direction of the vertical at sea level would also allow us to compute the distance to the Earth's center. Perhaps this is an allusion to the work in Geodesy that was taking place in the early 19th Century.

Supplemental (Jan 6): The key to Lagrange's system of mechanics are the virtual velocities or equivalently small changes in position which produce no change in their scalar product with the corresponding forces. Perhaps Lagrange was suggesting the problems of the chain and the arch for the "student" of analytical mechanics. In 1829 Gauss proposed "A new general Principle of Mechanics," the principle of least constraint based on least squares.