The Sand Reckoner

"The Sand Reckoner" (Greek: Ψαμμίτης, "Psammites") is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the then-current model, and invent a way to talk about extremely large numbers. The work, about 8 pages long in translation, is addressed to the Syracusan king Gelo II (son of Hiero II) and is probably the most accessible work of Archimedes; in some sense, it is the first research-expository paper. [http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/sand_reckoner.ps Archimedes, The Sand Reckoner, by Ilan Vardi] , accessed 28-II-2007.]

Naming large numbers

First, Archimedes had to invent a system of naming large numbers. The number system in use at that time could express numbers up to a myriad (10,000), and by utilizing the word "myriad" itself, one can immediately extend this to naming all numbers up to a myriad myriads (10⁸). Archimedes called the numbers up to 10⁸ "first numbers" and called 10⁸ itself the "unit of the second numbers". Multiples of this unit then became the second numbers, up to this unit taken a myriad myriad times, 10⁸·10⁸=10¹⁶. This became the "unit of the third numbers", whose multiples were the third numbers, and so on. Archimedes continued naming numbers in this way up to a myriad myriad times the unit of the 10⁸-th numbers, i.e., $(10^8)^\{(10^8)\}=10^\{8cdot\; 10^8\}$.

After having done this, Archimedes called the numbers he had defined the "numbers of the first period", and called the last one, $(10^8)^\{(10^8)\}$, the "unit of the second period". He then constructed the numbers of the second period by taking multiples of this unit in a way analogous to the way in which the numbers of the first period were constructed. Continuing in this manner, he eventually arrived at the numbers of the myriad myriadth period. The largest number named by Archimedes was the last number in this period, which is::$left((10^8)^\{(10^8)\}\; ight)^\{(10^8)\}=10^\{8cdot\; 10^\{16.$ Another way of describing this number is a one followed by (short scale) eighty quadrillion (80·10¹⁵) zeroes; compared to this number the otherwise enormous googol, or one followed by one hundred zeroes, seems paltry.

The system is reminiscent of a positional numeral system with base 10⁸, which is remarkable because the Greeks at the time used a very primitive system for writing numbers, simply employing 27 different letters from the alphabet for the units 1, 2, ... 9, the tens 10, 20, ... 90 and the hundreds 100, 200, ... 900.

Archimedes also discovered and proved the law of exponents ::$10^a\; 10^b\; =\; 10^\{a+b\}$necessary to manipulate powers of 10.

Estimation of the size of the Universe

Archimedes then estimated an upper bound for the number of grains of sand required to fill the Universe. To do this, he used the heliocentric model of Aristarchus of Samos. (Since this work by Aristarchus has been lost, Archimedes' work is one of the few surviving references to his theory. [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Aristarchus.html Aristarchus biography at MacTutor] , accessed 26-II-2007.] ) The reason for the large size of this model is that the Greeks were unable to observe stellar parallax with available techniques which implies that any parallax is extremely subtle and so the stars must be placed at great distances from the Earth (assuming heliocentrism to be true).

According to Archimedes, Aristarchus did not state how far the stars were from the Earth. Archimedes therefore had to make an assumption; he assumed that the Universe was spherical and that the ratio of the diameter of the Universe to the diameter of the orbit of the Earth around the Sun equaled the ratio of the diameter of the orbit of the Earth around the Sun to the diameter of the Earth. (This assumption can also be expressed by saying that the stellar parallax caused by the motion of the Earth around its orbit equals the solar parallax caused by motion around the Earth.)

In order to obtain an upper bound, Archimedes used overestimates of his data:
* He assumed that the perimeter of the Earth was no bigger than 300 myriad stadia (~5·10⁵ km.)
* He assumed that the Moon was no larger than the Earth, and that the Sun was no more than thirty times larger than the Moon.
* He assumed that the angular diameter of the Sun, as seen from the Earth, was greater than 1/200th of a right angle.Archimedes then computed that the diameter of the Universe was no more than 10¹⁴ stadia (in modern units, ~2 light years), and that it would require no more than 10⁶³ grains of sand to fill it.

Archimedes made some interesting experiments and computations along the way. One experiment was to estimate the angular size of the Sun, as seen from the Earth. Archimedes' method is especially interesting as it takes into account the finite size of the eye's pupil and therefore may be the first known example of experimentation in psychophysics, the branch of psychology dealing with the mechanics of human perception, whose development is generally attributed to Hermann von Helmholtz. Another interesting computation accounts for solar parallax and the different distances between the viewer and the Sun, whether viewed from the center of the Earth or from the surface of the Earth at sunrise. This may be the first known computation dealing with solar parallax.

References

External links

* [http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/psammites.ps Original Greek text]
* [http://web.fccj.org/~ethall/archmede/sandreck.htm The Sand Reckoner]
* [http://www.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/SandReckoner/SandReckoner.html The Sand Reckoner (annotated)]
* [http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/sand_reckoner.ps Archimedes, The Sand Reckoner, by Ilan Vardi; includes a literal English version of the original Greek text]