Is God a Mathematician_ - Mario Livio [26]
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 1063 (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