The Sand Reckoner (Greek: Ψαμμίτης, Psammites) is a work by Archimedes, an Ancient Greek mathematician of the 3rd century BC, 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, Archimedes had to estimate the size of the universe according to the contemporary model, and invent a way to talk about extremely large numbers. The work, also known in Latin as Arenarius, is about eight pages long in translation and is addressed to the Syracusan king Gelo II (son of Hiero II). It is considered the most accessible work of Archimedes.
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 (μυριάς — 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 order" and called 108 itself the "unit of the second order". Multiples of this unit then became the second order, up to this unit taken a myriad-myriad times, 108·108=1016. This became the "unit of the third order", whose multiples were the third order, and so on. Archimedes continued naming numbers in this way up to a myriad-myriad times the unit of the 108-th order, i.e., (108)^(108) After having done this, Archimedes called the orders he had defined the "orders of the first period", and called the last one, ( 10 8 ) ( 10 8 ) {\displaystyle (10^{8})^{(10^{8})}} , the "unit of the second period". He then constructed the orders of the second period by taking multiples of this unit in a way analogous to the way in which the orders of the first period were constructed. Continuing in this manner, he eventually arrived at the orders of the myriad-myriadth period. The largest number named by Archimedes was the last number in this period, which is
( ( 10 8 ) ( 10 8 ) ) ( 10 8 ) = 10 8 ⋅ 10 16 . {\displaystyle \left(\left(10^{8}\right)^{(10^{8})}\right)^{(10^{8})}=10^{8\cdot 10^{16}}.}
Another way of describing this number is a one followed by (short scale) eighty quadrillion (80·1015) zeroes. Archimedes' system is reminiscent of a positional numeral system with base 108, which is remarkable because the ancient Greeks used a very simple system for writing numbers, which employs 27 different letters of the alphabet for the units 1 through 9, the tens 10 through 90 and the hundreds 100 through 900.
Law of exponents Archimedes also discovered and proved the law of exponents, b m b n = b m + n {\displaystyle b^{m}b^{n}=b^{m+n}} , necessary to manipulate powers of some base b {\displaystyle b} . (Specifically, for this case, b = 10 {\displaystyle b=10} for 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. The original work by Aristarchus has been lost. This work by Archimedes however is one of the few surviving references to his theory, whereby the Sun remains unmoved while the Earth orbits the Sun. In Archimedes's own words:
His [Aristarchus'] hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the orbit, and that the sphere of fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface. 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 small 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 the following assumptions:
The Universe was spherical The ratio of the diameter of the Universe to the diameter of the orbit of the Earth around the Sun equalled 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. Put in a ratio:
Diameter of Universe Diameter of Earth orbit around the Sun = Diameter of Earth orbit around the Sun Diameter of Earth {\displaystyle {\frac {\text{Diameter of Universe}}{\text{Diameter of Earth orbit around the Sun}}}={\frac {\text{Diameter of Earth orbit around the Sun}}{\text{ Diameter of Earth}}}}
In order to obtain an upper bound, Archimedes made the following assumptions of their dimensions:
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