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3. Give Me a Lever Long Enough and I Will Move the World
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23 “GIVE ME A LEVER LONG ENOUGH AND I WILL MOVE THE WORLD” III Humankind has long known about the fundamental properties of the fulcrum point through the use of the balance and of levers of various types and sizes. In book I of his treatise On the Equilibrium of Planes, Archimedes treated the topic, in terms of mathematics and physics, with clarity: Commensurate sizes are in equilibrium if they are suspended at distances inversely proportional to their weights. [book I, proposition 6] The Syracusan scientist offered a demonstration of this concept that was criticized by the late nineteenth-century scholar Ernst Mach, whose arguments are not entirely convincing.1 Archimedes is here referring to a balance (i.e., a rod suspended or supported at its midpoint in a horizontal position when it is in balance, on the ends of which weights are stationed). In response to the success he achieved by a complex system of levers, pulleys, and winches used to launch 24 THE GREAT ARCHIMEDES III the great ship Syracusia (cf. chap. 7), Archimedes, amazed at the achievement, is said to have cried out enthusiastically: Give me a lever long enough and I will move the world.2 Archimedes had realized that one could easily reduce the force required to move a given weight simply by increasing the distance in proportion to the point of application, that is, the fulcrum. It would thus be possible to lift a very heavy weight with limited exertion, even, theoretically, one as heavy as that of the earth, were it possible to find a fixed fulcrum point external to the earth [plate 3]. With regard to the basic propositions of static objects (i.e., the mechanics of the balance of solid bodies) this demonstration is rigorously deductive and scientifically innovative: to explain this Archimedes had to expound on the concepts of balance and center of gravity in geometric terms and in terms of a theory of proportions. We are therefore confronted with an assertion of a quantitative measurement that is itself subject to physics, namely the “law of the lever,” according to which two variables are in equilibrium when the ratio of their distances from the fulcrum is the reciprocal of the ratio between their weights. For example, if the weights standing in relation to one another are at 2 to 3, their position on the lever ought to be in a ratio of 3 to 2 [fig. 6]. In the remainder of books I and II of On the Equilibrium of Planes, Archimedes treats the establishment of centers of gravity of various shapes of planes, such as the parallelogram (“the center of gravity of any parallelogram is the point where the diagonals intersect,” book I, proposition 10), the triangle (“the center of gravity of any triangle is the point at which lines drawn from the vertices to the mid-points of the sides intersect,” proposition 14), the trapezoid (proposition 15), and the parabolic segment or polygon inscribed on it (to their centers of gravity Archimedes dedicated the entirety of book II). [34.236.152.203] Project MUSE (2024-03-19 09:07 GMT) “GIVE ME A LEVER LONG ENOUGH AND I WILL MOVE THE WORLD” 25 III Polyhedrons are prominent among the geometric shapes that Archimedes explicitly engages. Less than a century earlier, Euclid had described the five regular polyhedrons in the thirteenth book of his Elements. He thus described these regular three-dimensional shapes, with all their sides of equal length, as formed from a single regular polygon. They are: • the tetrahedron, with 4 equal faces formed by equilateral triangles (book XIII, proposition 13); • the hexahedron (or cube), with 6 equal square faces (book XIII, proposition 15); • the octahedron, with 8 equal faces formed by equilateral triangles, four of which meet at each vertex (book XIII, proposition 14); Figure 6 The lever is a simple device that consists of a stiff “pole” that turns upon a fixed point called a fulcrum (F). At each extremity of the pole one force is applied. One end bears “resistance” (R) and the other countervailing “power” (P). The distance from the fulcrum to the resistance is called the “resistance arm” (br), and that between the power and the fulcrum is called the “power arm” (bp). 26 THE GREAT ARCHIMEDES III • the dodecahedron, with 12 faces made up of equal regular pentagons, and 20 vertices and 30 edges altogether (book XIII, proposition 17); • the icosahedron, with 20 faces formed by regular equilateral triangles, and 12...