Read Is God a Mathematician? Online
Authors: Mario Livio
Is God a Mathematician? (7 page)
Wars have always been popular with historians. Consequently, the events of the Roman siege on Syracuse during the years 214–212 BC have been lavishly chronicled by many historians. The Roman general Marcus Claudius Marcellus (ca. 268–208 BC), by then of considerable military fame, anticipated a rapid victory. He apparently failed to consider a stubborn King Hieron, aided by a mathematical and engineering genius. Plutarch gives a vivid description of the havoc that Archimedes’ machines inflicted upon the Roman forces:
He [Archimedes] at once shot against the land forces all sorts of missile weapons, and immense masses of stone that came down with incredible noise and violence; against which no man could stand; for they knocked down those upon whom they fell in heaps, breaking all their ranks and files. At the same time huge poles thrust out from the walls over the ships sunk some by great weights which they let down from on high upon them; others they lifted up into the air by an iron hand or beak like a crane’s beak and, when they had drawn them up by the prow, and set them on end upon the poop, they plunged them to the bottom of the sea…A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.
The fear of the Archimedean devices became so extreme that “if they [the Roman soldiers] did but see a piece of rope or wood projecting above the wall, they would cry ‘there it is again,’ declaring that Archimedes was setting some engine in motion against them, and would turn their backs and run away.” Even Marcellus was deeply impressed, complaining to his own crew of military engineers: “Shall we not make an end of fighting against this geometrical Briareus [the hundred-armed giant, son of Uranus and Gaia] who, sitting at ease by the sea, plays pitch and toss with our ships to our confusion, and by the multitude of missiles that he hurls at us outdoes the hundred-handed giants of mythology?”
According to another popular legend that appeared first in the writings of the great Greek physician Galen (ca. AD 129–200), Archimedes used an assembly of mirrors that focused the Sun’s rays to burn the Roman ships. The sixth century Byzantine architect Anthemius of Tralles and a number of twelfth century historians repeated this fantastic story, even though the actual feasibility of such a feat remains uncertain. Still, the collection of almost mythological tales does provide us with rich testimony as to the veneration that “the wise one” inspired in later generations.
As I noted earlier, Archimedes himself—that highly esteemed “geometrical Briareus”—attached no particular significance to all of his military 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 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 equal to the straight line between the center of the Sun and the center of the Earth. This is the common account, as you have heard from astronomers. But Aristarchus of Samos brought out a book consisting of some hypotheses, in which the premises lead to the result that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun in the circumference of a circle, the Sun lying in the middle of the orbit.
This introduction immediately highlights two important points: (1) Archimedes was prepared to question even very popular beliefs (such as that there is an infinity of grains of sand), and (2) he treated with respect the heliocentric theory of the astronomer Aristarchus (later in the treatise he actually corrected one of Aristarchus’s hypotheses). In Aristarchus’s universe the Earth and the planets revolved around a stationary Sun that was located at the center (remember that this model was proposed 1,800 years before Copernicus!). After these preliminary remarks, Archimedes starts to address the problem of the grains of sand, progressing by a series of logical steps. First he estimates how many grains placed side by side it would take to cover the diameter of a poppy seed. Then, how many poppy seeds would fit in the breadth of a finger; how many fingers in a stadium (about 600 feet);
and continuing up to ten billion stadia. Along the way, Archimedes invents a system of indices and a notation that, when combined, allow him to classify his gargantuan numbers. Since Archimedes assumed that the sphere of the fixed stars is less than ten million times larger than the sphere containing the orbit of the Sun (as seen from Earth), he found the number of grains in a sand-packed universe to be less than 10
63
(one followed by sixty-three zeros). He then concluded the treatise with a respectful note to Gelon:
I conceive that these things, king Gelon, will appear incredible to the great majority of people who have not studied mathematics, but that to those who are conversant therewith and have given thought to the question of the distances and sizes of the Earth and Sun and the Moon and the whole universe the proof will carry conviction. And it was for this reason that I thought the subject would not be inappropriate for your consideration.
The beauty of
The Sand Reckoner
lies in the ease with which Archimedes hops from everyday objects (poppy seeds, sand, fingers) to abstract numbers and mathematical notation, and then back from those to the sizes of the solar system and the universe as a whole. Clearly, Archimedes possessed such intellectual flexibility that he could comfortably use his mathematics to discover unknown properties of the universe, and use the cosmic characteristics to advance arithmetical concepts.
Archimedes’ second claim to the title of “magician” comes from the method that he used to arrive at many of his outstanding geometrical theorems. Very little was known about this method and about Archimedes’ thought process in general until the twentieth century. His concise style gave away very few clues. Then, in 1906, a dramatic discovery opened a window into the mind of this genius. The story of this discovery reads so much like one of the historical mystery novels by the Italian author and philosopher Umberto Eco that I feel compelled to take a brief detour to tell it.
The Archimedes Palimpsest
Sometime in the tenth century, an anonymous scribe in Constantinople (today’s Istanbul) copied three important works of Archimedes:
The Method, Stomachion,
and
On Floating Bodies.
This was probably part of a general interest in Greek mathematics that was largely sparked by the ninth century mathematician Leo the Geometer. In 1204, however, soldiers of the Fourth Crusade were lured by promises of financial support to sack Constantinople. In the years that followed, the passion for mathematics faded, while the schism between the Catholic Church of the west and the Orthodox Church of the east became a
fait accompli.
Sometime before 1229, the manuscript containing Archimedes’ works underwent a catastrophic act of recycling—it was unbound and washed so the parchment leaves could be reused for a Christian prayer book. The scribe Ioannes Myronas finished copying the prayer book on April 14, 1229. Fortunately, the washing of the original text did not obliterate the writing completely. Figure 12 shows a page from the manuscript, with the horizontal lines representing the prayer texts and the faint vertical lines the mathematical contents. By the sixteenth century, the palimpsest—the recycled document—somehow made its way to the Holy Land, to the monastery in St. Sabas, east of Bethlehem. In the early nineteenth century, this monastery’s library contained no fewer than a thousand manuscripts. Still, for reasons that are not entirely clear, the Archimedes palimpsest was moved yet again to Constantinople. Then, in the 1840s, the famous German biblical scholar Constantine Tischendorf (1815–74), the discoverer of one of the earliest Bible manuscripts, visited the Metochion of the Holy Sepulcher in Constantinople (a daughter house of the Greek Patriarchate in Jerusalem) and saw the palimpsest there. Tischendorf must have found the partially visible underlying mathematical text quite intriguing, since he apparently tore off and stole one page from the manuscript. Tischendorf’s estate sold that page in 1879 to the Cambridge University Library.