Chapter Two: Archimedes Mechanicus

Archimedes of Syracuse (ca. 287–212 B.C.) was, by reputation at least, the most accomplished mechanic of antiquity. He was also perhaps the first to formulate what I call the mechanical challenge as the general goal of mechanics—to move any weight, however large, with any power, however small—and the first apparently to have found its general solution. From this insight he allegedly boasted that, given a place to stand, he could move the earth (the motto to this chapter), and he was also said to have devised numerous ingenious mechanical instruments and weapons of war. His contribution to theoretical mechanics, on the other hand, is usually taken to be his rigorous, formal proof of the law of the equilibrium of the balance—that two weights are in equilibrium on a balance when their magnitudes are inversely as their distances from the pivot. It has generally been inferred that the law of equilibrium was in fact his theoretical solution to the mechanical challenge, and that his main purpose in proving this law was to establish on sound foundations the science of mechanics—the science of designing machines to accomplish great works with small powers. This may be true, although there is no direct evidence that he actually applied the law of equilibrium to the design of machines; Hero of Alexandria was perhaps the first to do this explicitly. Rather, what one finds in Archimedes’ mechanical works is an entirely different purpose. Instead of proving the law of equilibrium in order to apply it to practical machines, he seems to have proved it in order to apply it to purely geometrical problems. By identifying the pivot of the balance with the common centre of gravity of two weights, he extended the law of equilibrium into the more general law of centres of gravity—that all the parts of a magnitude, taken together in balanced pairs in accordance with the law of equilibrium, act as though at a single point called the centre of gravity of that magnitude. And with the law of centres of gravity, he devised a mechanical method in geometry to determine the areas, volumes, and centres of gravity of curvilinear surfaces and bodies. How he did this is the subject of this chapter.