Free Is God a Mathematician? by Mario Livio Page A

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 theseas 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