Is God a Mathematician_ - Mario Livio [25]
Archimedes’ opus covers an astonishing range of mathematics and physics. Among his many achievements: He presented general methods for finding the areas of a variety of plane figures and the volumes of spaces bounded by all kinds of curved surfaces. These included the areas of the circle, segments of a parabola and of a spiral, and volumes of segments of cylinders, cones, and other figures generated by the revolution of parabolas, ellipses, and hyperbolas. He showed that the value of the number, the ratio of the circumference of a circle to its diameter, has to be larger than 3 10/71 and smaller than 3 1/7. At a time when no method existed to describe very large numbers, he invented a system that allowed him not only to write down, but also to manipulate numbers of any magnitude. In physics, Archimedes discovered the laws governing floating bodies, thus establishing the science of hydrostatics. In addition, he calculated the centers of gravity of many solids and formulated the mechanical laws of levers. In astronomy, he performed observations to determine the length of the year and the distances to the planets.
The works of many of the Greek mathematicians were characterized by originality and attention to detail. Still, Archimedes’ methods of reasoning and solution truly set him apart from all of the scientists of his day. Let me describe here only three representative examples that give the flavor of Archimedes’ inventiveness. One appears at first blush to be nothing more than an amusing curiosity, but a closer examination reveals the depth of his inquisitive mind. The other two illustrations of the Archimedean methods demonstrate such ahead-of-his-time thinking that they immediately elevate Archimedes to what I dub the “magician” status.
Archimedes was apparently fascinated by big numbers. But very large numbers are clumsy to express when written in ordinary notation (try writing a personal check for $8.4 trillion, the U.S. national debt in July 2006, in the space allocated for the figure amount). So Archimedes developed a system that allowed him to represent numbers with 80,000 trillion digits. He then used this system in an original treatise entitled The Sand Reckoner, to show that the total number of sand grains in the world was not infinite.
Even the introduction to this treatise is so illuminating that I will reproduce a part of it here (the entire piece was addressed to Gelon, the son of King Hieron II):
There are some, king Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the earth, including in it all the seas and the hollows of the earth filled up to a height equal to that of the highest of the mountains, would be many times further still from recognizing that any number could be expressed which exceeds the multitude of the sand so taken. But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus [a work that has unfortunately been lost], some exceed not only the number of the mass of sand equal in magnitude to the earth filled up in the way described, but also that of a mass equal in magnitude to the universe. Now you are aware that “universe” is the name given by most astronomers to the sphere whose center is the center of the earth and whose radius is