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.^[1] Archimedes of Syracuse (Greek: c. ... For other uses, see Universe (disambiguation). ... Syracuse (Italian Siracusa, Sicilian Sarausa, Greek , Latin Syracusae) is an Italian city on the eastern coast of Sicily and the capital of the province of Syracuse. ... Grave monument of Hiëro II in Syracuse Hiero II, tyrant of Syracuse from 270 to 215 BC, was the illegitimate son of a Syracusan noble, Hierocles, who claimed descent from Gelo. ... Archimedes of Syracuse (Greek: c. ...

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., . This article or section is not written in the formal tone expected of an encyclopedia article. ... Look up myriad in Wiktionary, the free dictionary. ...

After having done this, Archimedes called the numbers he had defined the "numbers of the first period", and called the last one, , 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

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 long and short scales are two different numerical systems used throughout the world: Short scale is the English translation of the French term échelle courte. ... A googol is the large number 10100, that is, the digit 1 followed by one hundred zeros (in decimal representation). ...

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 different letters from the alphabet for the numbers 1,2,...,9,10,20,30,...,100,200,300,... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base (or radix) of that numeral system. ...

Archimedes also discovered and proved the law of exponents Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ...

10^a10^b = 10^{a + b}

necessary to manipulate powers of 10.

Estimation of size of 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.^[2]) The reason for the large size of this model is that, in order for stellar parallax not to be measurable, the stars must be placed a great distance from the Earth. 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 astronomy, heliocentrism is the theory that the Sun is at the center of the Universe and/or the Solar System. ... Statue of Aristarchus at Aristotle University in Thessalonica, Greece Aristarchus (Greek: Ἀρίσταρχος; 310 BC - ca. ... Parallax (Greek: παραλλαγή = alteration) is the change of angular position of two stationary points relative to each other as seen by an observer, due to the motion of said observer. ...

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. Ancient Greek weights and measures - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... A light-year or lightyear (symbol: ly) is a unit of measurement of length, specifically the distance light travels in vacuum in one year. ...

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.^[1] Psychophysics is the branch of cognitive psychology dealing with the relationship between physical stimuli and their perception. ... Psychology (from Greek: ψυχή, psukhē, spirit, soul; and λόγος, logos, knowledge) is both an academic and applied discipline involving the scientific study of mental processes and behavior. ... Hermann Ludwig Ferdinand von Helmholtz (August 31, 1821 – September 8, 1894) was a German physician and physicist. ...

References

1. ^ ^{a b} Archimedes, The Sand Reckoner, by Ilan Vardi, accessed 28-II-2007.
2. ^ Aristarchus biography at MacTutor, accessed 26-II-2007.

External links

• Original Greek text
• The Sand Reckoner
• The Sand Reckoner (annotated)
• Archimedes, The Sand Reckoner, by Ilan Vardi; includes a literal English version of the original Greek text