The Sand Reckoner (Greek: Ψαμμίτης, Psammites) is a work by
Archimedes
Contents 1 Naming large numbers 2 Estimation of the size of the universe 2.1 Coincidental equality between Archimedes' number and Eddington's number 3 Quote 4 References 5 Further reading 6 External links Naming large numbers[edit] Periods and orders with their intervals in modern notation[2] Period Order Interval log10 of interval 1 1 (1, Ơ ], where the unit of the second order, Ơ = 108 (0, 8] 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
( 10 8 ) ( 10 8 ) = 10 8 ⋅ 10 8 displaystyle (10^ 8 )^ (10^ 8 ) =10^ 8cdot 10^ 8 .[2]
After having done this,
Archimedes
( 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
myriadmyriadth period. The largest number named by
Archimedes
( ( 10 8 ) ( 10 8 ) ) ( 10 8 ) = 10 8 ⋅ 10 16 . displaystyle left((10^ 8 )^ (10^ 8 ) right)^ (10^ 8 ) =10^ 8cdot 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.
Archimedes
10 a 10 b = 10 a + b displaystyle 10^ a 10^ b =10^ a+b , necessary to manipulate powers of 10.
Estimation of the size of the universe[edit]
Archimedes
His [Aristarchus'] hypotheses are that the fixed stars and the Sun
remain unmoved, that the
Earth
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
The
Universe
This assumption can also be expressed by saying that the stellar
parallax caused by the motion of the
Earth
Diameter of Universe
Diameter of
Earth
= Diameter of
Earth
displaystyle frac text Diameter of
Universe
In order to obtain an upper bound,
Archimedes
that the perimeter of the
Earth
Archimedes
"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
References[edit] ^ a b Archimedes, The Sand Reckone, by Ilan Vardi, accessed 28II2007. ^ a b Alan Hirshfeld. "Eureka Man: The Life and Legacy of Archimedes". Retrieved 17 February 2016. ^ Aristarchus biography at MacTutor, accessed 26II2007. ^ 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 Further reading[edit] The SandReckoner, by Gillian Bradshaw. Forge (2000), 348pp, ISBN 0312875819. This is a historical novel about the life and work of Archimedes. External links[edit] Original Greek text
The Sand Reckoner (annotated)
The Sand Reckoner (Arenario) italian annotated translation, with notes
about
Archimedes
v t e Ancient Greek mathematics Mathematicians Anaxagoras Anthemius Archytas Aristaeus the Elder Aristarchus Apollonius Archimedes Autolycus Bion Bryson Callippus Carpus Chrysippus Cleomedes Conon Ctesibius Democritus Dicaearchus Diocles Diophantus Dinostratus Dionysodorus Domninus Eratosthenes Eudemus Euclid Eudoxus Eutocius Geminus Heron Hipparchus Hippasus Hippias Hippocrates Hypatia Hypsicles Isidore of Miletus Leon Marinus Menaechmus Menelaus Metrodorus Nicomachus Nicomedes Nicoteles Oenopides Pappus Perseus Philolaus Philon Porphyry Posidonius Proclus Ptolemy Pythagoras Serenus Simplicius Sosigenes Sporus Thales Theaetetus Theano Theodorus Theodosius Theon of Alexandria Theon of Smyrna Thymaridas Xenocrates Zeno of Elea Zeno of Sidon Zenodorus Treatises Almagest
Archimedes
Problems Problem of Apollonius Squaring the circle Doubling the cube Angle trisection Centers Cyrene Library of Alexandria Platonic Academy Timeline of Ancient Greek mathematicians v t e Archimedes Written works On the Equilibrium of Planes
Measurement of a Circle
On Spirals
On the Sphere and Cylinder
On Floating Bodies
The Quadrature of the Parabola
Ostomachion
The Sand Reckoner
The Method of Mechanical Theorems
Book of Lemmas
Finds and inventions Archimedean solid Archimedes's cattle problem Archimedes's principle Archimedes's screw Claw of Archimedes Miscellaneous
Archimedes
