Zero - Charles Seife [20]
Archimedes first glimpsed the infinite in the polish of his war mirrors. For centuries the Greeks had been fascinated with conic sections. Take a cone and cut it up; you get circles, ellipses, parabolas, and hyperbolas, depending on how you slice it. The parabola has a special property: it takes the rays of light from the sun, or any distant source, and focuses them to a point, concentrating all the light’s energy on a very small area. Any mirror that could set ships afire must be in the shape of a parabola. Archimedes studied the properties of the parabola, and it is here that he first started toying with the infinite.
To understand the parabola, Archimedes had to learn how to measure it; for instance, nobody knew how to determine the area of a section of a parabola. Triangles and circles were easy to measure, but slightly more irregular curves like the parabola were beyond the ken of the Greek mathematicians of the day. However, Archimedes figured out a way to measure the parabola’s area by resorting to the infinite. The first step was to inscribe a triangle inside the parabola. In the two little gaps left, Archimedes inscribed more triangles. This left four gaps, which were filled with more triangles, and so on (Figure 12). It’s like Achilles and the tortoise—an infinite series of steps, each getting smaller and smaller. The areas of the little triangles quickly approach zero. After a long, involved set of calculations, Archimedes summed the areas of the infinite triangles and divined the area of the parabola. However, any mathematician of the day would have scoffed at this line of reasoning; Archimedes used the tools of the infinite, which were so expressly disallowed by his mathematical colleagues. To satisfy them, Archimedes also included a proof, based upon the accepted mathematics of the time, that relied upon the so-called axiom of Archimedes, although Archimedes himself mentioned that earlier mathematicians deserved the credit. As you may recall, this axiom says that any number added to itself over and over again can exceed any other number. Zero, clearly, was not included.
Figure 12: Archimedes’ parabola
Archimedes’ proof by triangles was as close as you could come to the idea of limits—and calculus—without actually discovering them. In later works Archimedes figured out the volumes of parabolas and circles, rotated around a line, which any math student knows are early homework problems in a calculus course. But the axiom of Archimedes rejected zero, which is the bridge between the realms of the finite and the infinite, a bridge that is absolutely necessary for calculus and higher mathematics.
Even the brilliant Archimedes occasionally scorned the infinite along with his contemporaries. He believed in the Aristotelian universe; the universe was contained within a giant sphere. On a whim he decided to calculate how many grains of sand could fit in the (spherical) universe. In his “Sand Reckoner,” Archimedes first calculated how many grains of sand would fit across a poppy seed, and then how many poppy seeds would span a finger’s breadth. From finger breadths to the length of a stadium (the standard Greek unit of long distance) and on to the size of the universe, Archimedes figured out that 1051 grains of sand would fill the entire universe, packed full even unto the outermost sphere of fixed stars. (1051 is a really, really big number. Take 1051 molecules