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 contemporary model, and invent a way to talk about extremely large numbers. The work, also known in Latin as Archimedis Syracusani Arenarius & Dimensio Circuli, which is 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.
Naming large numbers
|Periods and orders with their intervals in modern notation|
|Period||Order||Interval||log10 of interval|
|1||1||(1, Ơ ], where the
unit of the second order,
Ơ = 108
|2||(Ơ, Ơ 2]||(8, 16]|
|k||(Ơ k − 1, Ơ k ]||(8k − 8, 8k ]|
|Ơ||(Ơ Ơ − 1, Ƥ ], where the
unit of the second period,
Ƥ = Ơ Ơ = 108×108
|(8×108 − 8, 8×108]
= (799 999 992, 800 000 000]
|2||1||(Ƥ, ƤƠ ]||(8×108, 8 × (108 + 1)]
= (800 000 000, 800 000 008]
|2||(ƤƠ, ƤƠ 2]||(8 × (108 + 1), 8 × (108 + 2)]|
|k||(ƤƠ k − 1, ƤƠ k ]||(8 × (108 + k − 1), 8 × (108 + k )]|
|Ơ||(ƤƠ Ơ − 1, ƤƠ Ơ ]
= (2ƤƠ −1, 2Ƥ ]
|(8 × (2×108 − 1), 8 × (2×108)]
= (1.6×109 − 8, 1.6×109]
= (1 599 999 992, 1 600 000 000]
|Ơ||1||(Ƥ Ơ − 1, Ƥ Ơ − 1Ơ ]||(8×108 × (108 − 1),
8 × (108 × (108 − 1) + 1)]
= (79 999 999 200 000 000,
79 999 999 200 000 008]
|2||(Ƥ Ơ − 1Ơ, Ƥ Ơ − 1Ơ 2]||(8 × (108 × (108 − 1) + 1),
8 × (108 × (108 − 1) + 2)]
|k||(Ƥ Ơ − 1Ơ k − 1, Ƥ Ơ − 1Ơ k ]||(8 × (108 × (108 − 1) + k − 1),
8 × (108 × (108 − 1) + k )]
|Ơ||(Ƥ Ơ − 1Ơ Ơ − 1, Ƥ Ơ − 1Ơ Ơ ]
= (Ƥ Ơ Ơ −1, Ƥ Ơ ]
|(8 × (2×108 − 1), 8 × (2×108)]
= (8×1016 − 8, 8×1016]
= (79 999 999 999 999 992,
80 000 000 000 000 000]
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., .
After having done this, Archimedes called the orders he had defined the "orders of the first period", and called the last one, , 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
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.
Archimedes also discovered and proved the law of exponents, , 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. 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 revolves about the Sun. In Archimedes' 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 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 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.
In order to obtain an upper bound, Archimedes made the following assumptions:
- that the perimeter of the Earth was no bigger than 300 myriad stadia (5.55·105 km).
- that the Moon was no larger than the Earth, and that the Sun was no more than thirty times larger than the Moon.
- that the angular diameter of the Sun, as seen from the Earth, was greater than 1/200 of a right angle (π/400 radians = 0.45° degrees).
Archimedes then concluded that the diameter of the Universe was no more than 1014 stadia (in modern units, about 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.
Coincidental equality between Archimedes' number and Eddington's number
The total number of nucleons in the observable universe of roughly the Hubble radius is the Eddington number, currently estimated at 1080. Archimedes' 1063 grains of sand contain roughly 1080 nucleons, making the two numbers relatively equal.
"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 magnitude. 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 the seas 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 exceeded 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, 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 the mass equal in magnitude to the universe."— Archimedis Syracusani Arenarius & Dimensio Circuli
- Archimedes, The Sand Reckone, by Ilan Vardi, accessed 28-II-2007.
- Alan Hirshfeld. "Eureka Man: The Life and Legacy of Archimedes". Retrieved 17 February 2016.
- Aristarchus biography at MacTutor, accessed 26-II-2007.
- Arenarius, I., 4–7
- Smith, William — A Dictionary of Greek and Roman Biography and Mythology (1880), p. 272
- Harrison, Edward Robert ♦ Cosmology: The Science of the Universe Cambridge University Press, 2000, pp. 481, 482
- Newman, James R. — The World of Mathematics (2000), p. 420
- The Sand-Reckoner, by Gillian Bradshaw. Forge (2000), 348pp, ISBN 0-312-87581-9. This is a historical novel about the life and work of Archimedes.
- Original Greek text
- The Sand Reckoner (annotated)
- The Sand Reckoner (Arenario) italian annotated translation, with notes about Archimede and the greek mathematical notation and the ellenic unit of measure. Source file of the Arenarius Greek text (for LaTeX).
- Archimedes, The Sand Reckoner, by Ilan Vardi; includes a literal English version of the original Greek text