''The Sand Reckoner'' ( el, Ψαμμίτης, ''Psammites'') is a work by
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
, an
Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
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
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
. 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 eight pages long in translation, is addressed to the
Syracusan king
Gelo II
Gelon also known as Gelo (Greek: Γέλων ''Gelon'', ''gen.'': Γέλωνος; died 478 BC), son of Deinomenes, was a Greek tyrant of the Sicilian cities Gela and Syracuse, and first of the Deinomenid rulers.
Early life
Gelon was the son o ...
(son of
Hiero II
Hiero II ( el, Ἱέρων Β΄; c. 308 BC – 215 BC) was the Greek tyrant of Syracuse from 275 to 215 BC, and the illegitimate son of a Syracusan noble, Hierocles, who claimed descent from Gelon. He was a former general of Pyrrhus of Epirus a ...
), and is probably the most accessible work of Archimedes; in some sense, it is the first
research-expository paper.
[Archimedes, The Sand Reckoner 511 R U, by Ilan Vardi](_blank)
accessed 28-II-2007.
Naming large numbers
First, Archimedes had to invent a system of naming
large numbers
Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical m ...
. 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 (10
8).
Archimedes called the numbers up to 10
8 "first order" and called 10
8 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, 10
8·10
8=10
16. 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 10
8-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
The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes.
For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one million, the t ...
) eighty quadrillion (80·10
15) zeroes.
Archimedes' system is reminiscent of a
positional numeral system
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral syste ...
with base 10
8, 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
Heliocentrism (also known as the Heliocentric model) is the astronomical model in which the Earth and planets revolve around the Sun at the center of the universe. Historically, heliocentrism was opposed to geocentrism, which placed the Earth at ...
of
Aristarchus of Samos
Aristarchus of Samos (; grc-gre, Ἀρίσταρχος ὁ Σάμιος, ''Aristarkhos ho Samios''; ) was an ancient Greek astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or ...
. The original work by Aristarchus has been lost. This work by Archimedes however is one of the few surviving references to his theory,
[Aristarchus biography at MacTutor](_blank)
accessed 26-II-2007. whereby the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
remains unmoved while the
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
orbits
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
the Sun. In Archimedes's own words:
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:
In order to obtain an upper bound, Archimedes made the following assumptions of their dimensions:
* that the perimeter of the Earth was no bigger than 300 myriad
stadia
Stadia may refer to:
* One of the plurals of stadium, along with "stadiums"
* The plural of stadion, an ancient Greek unit of distance, which equals to 600 Greek feet (''podes'').
* Stadia (Caria), a town of ancient Caria, now in Turkey
* Stadi ...
(5.55·10
5 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
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s = 0.45
° degrees).
Archimedes then concluded that the diameter of the Universe was no more than 10
14 stadia (in modern units, about 2
light years), and that it would require no more than 10
63 grains of sand to fill it. With these measurements, each grain of sand in Archimedes's thought-experiment would have been approximately 19 μm (0.019 mm) in diameter.
Calculation of the number of grains of sand in the Aristarchian Universe
Archimedes claims that forty poppy-seeds laid side by side would equal one Greek dactyl (finger-width) which was approximately 19 mm (3/4 inch) in length. Since volume proceeds as the cube of a linear dimension ("For it has been proved that spheres have the triplicate ratio to one another of their diameters") then a sphere one dactyl in diameter would contain (using our current number system) 40
3, or 64,000 poppy seeds.
He then claimed (without evidence) that each poppy seed could contain a myriad (10,000) grains of sand. Multiplying the two figures together he proposed 640,000,000 as the number of hypothetical grains of sand in a sphere one dactyl in diameter.
To make further calculations easier, he rounded up 640 million to one billion, noting only that the first number is smaller than the second, and that therefore the number of grains of sand calculated subsequently will exceed the actual number of grains. Recall that Archimedes's meta-goal with this essay was to show how to calculate with what were previously considered impossibly large numbers, not simply to accurately calculate the number of grains of sand in the universe.
A Greek stadium had a length of 600 Greek feet, and each foot was 16 dactyls long, so there were 9,600 dactyls in a stadium. Archimedes rounded this number up to 10,000 (a myriad) to make calculations easier, noting again that the resulting number will exceed the actual number of grains of sand.
The cube of 10,000 is a trillion (10
12); and multiplying a billion (the number of grains of sand in a dactyl-sphere) by a trillion (number of dactyl-spheres in a stadium-sphere) yields 10
21, the number of grains of sand in a stadium-sphere.
Archimedes had estimated that the Aristarchian Universe was 10
14 stadia in diameter, so there would accordingly be (10
14)
3 stadium-spheres in the universe, or 10
42. Multiplying 10
21 by 10
42 yields 10
63, the number of grains of sand in the Aristarchian Universe.
Following Archimedes's estimate of a myriad (10,000) grains of sand in a poppy seed; 64,000 poppy seeds in a dactyl-sphere; the length of a stadium as 10,000 dactyls; and accepting 19mm as the width of a dactyl, the diameter of Archimedes's typical sand grain would be 18.3 μm, which today we would call a grain of
silt
Silt is granular material of a size between sand and clay and composed mostly of broken grains of quartz. Silt may occur as a soil (often mixed with sand or clay) or as sediment mixed in suspension with water. Silt usually has a floury feel when ...
. Currently, the smallest grain of sand would be defined as 50 μm in diameter.
Additional calculations
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's method is especially interesting as it takes into account the finite size of the eye's pupil,
[Smith, William — A Dictionary of Greek and Roman Biography and Mythology (1880), p. 272] and therefore may be the first known example of experimentation in
psychophysics
Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, m ...
, the branch of
psychology
Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries betwe ...
dealing with the mechanics of human perception, whose development is generally attributed to
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
. 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.
Quote
References
Further reading
* ''The Sand-Reckoner'', by
Gillian Bradshaw. Forge (2000), 348pp, . This is a historical novel about the life and work of Archimedes.
External links
Original Greek text Source file of the Arenarius Greek text (for LaTeX).
*
ttp://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
{{DEFAULTSORT:Sand Reckoner, The
Ancient Greek astronomy
Mathematics manuscripts
Works by Archimedes
Astronomy books
Large numbers