Archimedes' Law of the Lever

We start with the simplest machine known to mankind, the lever. Supposedly, ever since man developed beyond the level of the ape, he has used sticks to lever stones. The Egyptians in building their pyramids used elaborate machines consisting of a combination of levers; yet their knowledge of levers appears to have remained largely inarticulated. We all know that in pushing a door shut, the nearer the point at which we push it is to the line of the hinges, the harder we need push. Yet how many of us realize that this common experience exemplifies the law of the lever? The hinge is the fulcrum about which the turning moment of our push counterbalances the opposing turning moment of friction at the hinge. We have the experience, but not the articulation.

It seems that Archimedes (287–212 B.C.) was the first in history to ask for a precise mathematical formulation of the conditions of equilibrium of the lever. To ask this question was itself a tremendous step—to ask for mathematical laws for the behavior of a combination of sticks and stones; for here is a crucial novelty—that number plays a role in understanding and predicting nature.

We now retrace the essential steps by which Archimedes derived his formulation. He started with the simplest case: a weightless lever with equal arms suspending equal weights. See Figure 1.1.