Free Is God a Mathematician? by Mario Livio

toys; he basically regarded them as diversions of geometry at play. Unfortunately, this aloof attitude may have eventually cost Archimedes his life. When the Romans finally captured Syracuse, Archimedes was so busy drawing his geometrical diagrams on a dust-filled tray that he failed to notice the tumult of war. According to some accounts, when a Roman soldier ordered Archimedes to follow him to Marcellus, the old geometer retorted indignantly: “Fellow, stand away from my diagram.” This reply infuriated the soldier to such a degree that, disobeying his commander’s specific orders, he unsheathed his sword and slew the greatest mathematician of antiquity. Figure 11 shows what is believed to be a reproduction (from the eighteenth century) of a mosaic found in Herculaneum depicting the final moments in the life of “the master.”
    Archimedes’ death marked, in some sense, the end of an extraordinarily vibrant era in the history of mathematics. As the British mathematician and philosopher Alfred North Whitehead remarked:
    The death of Archimedes at the hands of a Roman soldier is symbolical of a world change of the first magnitude. The Romans were a great race, but they were cursed by the sterility which waits upon practicality. They were not dreamers enough to arrive at new points of view, which could give more fundamental control over the forces of nature. No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.

    Figure 11
    Fortunately, while details of Archimedes’ life are scarce, many (but not all) of his incredible writings have survived. Archimedes had a habit of sending notes on his mathematical discoveries to a few mathematician friends or to people he respected. The exclusive list of correspondents included (among others) the astronomer Conon of Samos, the mathematician Eratosthenes of Cyrene, and the king’s son, Gelon. After Conon’s death, Archimedes sent a few notes to Conon’s student, Dositheus of Pelusium.
    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 discoveredthe 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