Archimedes Psammites, the Sand Reckoner

Michael Lahanas

Αρχιμήδης Ψαµµίτης

Archimedes of Syracuse (287-212 BC) according to a legend was killed while drawing circles on the ground. Let us supposed it was sand (Gr. psammos). Reading about Aristarchus heliocentric world he was thinking how many sand grains are required to fill the entire then known universe.

There are some, King Gelon, who think that the number of the sand is infinite in multitude; and I mean the sand not only 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 this multitude. 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 a mass equal in magnitude to the universe. Archimedes, (c. 220 BC) letter to Gelon II, tyrant of Syracuse.

Archimdes knew that Aristarchus of Samos in his model expanded the size of the universe:

... Ἀρίσταρχος δὲ ὁ Σάμιος ὑποθέσιών τινῶν ἐξέδωκεν γραφάς, ἐν αἷς ἐκ τῶν ὑποκειμένων συμβαίνει τὸν κόσμον πολλαπλάσιον εἶμεν τοῦ νῦν εἰρημένου. Ὑποτίθεται γὰρ τὰ μὲν ἀπλανέα τῶν ἄστρων καὶ τὸν ἅλιον μένειν ἀκίνητον, τὰν δὲ γᾶν περιφέρεσθαι περὶ τὸν ἅλιον κατὰ κύκλου περιφέρειαν, ὅς ἐστιν ἐν μέσῳ τῷ δρόμῳ κείμενος, τὰν δὲ τῶν ἀπλανέων ἄστρων σφαῖραν περὶ τὸ αὐτὸ κέντρον τῷ ἁλίῳ κειμέναν τῷ μεγέθει τηλικαύταν εἶμεν, ὥστε τὸν κύκλον, καθ᾿ ὃν τὰν γᾶν ὑποτίθεται περιφέρεσθαι, τοιαύταν ἔχειν ἀναλογίαν ποτὶ τὰν τῶν ἀπλανέων ἀποστασίαν, οἵαν ἔχει τὸ κέντρον τᾶς σφαίρας ποτὶ τὰν ἐπιφάνειαν ...
...
Aristarchus of Samos has set forth writings of certain hypotheses, in which from the things that are established it follows that the universe is many times greater than that now told. For it is proposed that the fixed [ones] of the stars and the sun remain motionless, and that the earth is borne around the sun along the circumference of a circle, [the sun] remaining in the middle of the course, and the sphere of the fixed stars lying around the same center as the sun being of so great a magnitude, that the circle, along which the earth is presumed to be borne, has such a proportion to the distance of the fixed [ones], as the center of the sphere has to the surface...
Archimedes THE SAND RECKONER

For a finite world there cannot be infinite number of sand grains. But what is the maximum number required to fill the entire Universe?

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 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..
One can also show that the diameter of the universe is less than a line equal to a myriad diameters of the earth and that, moreover, the diameter of the universe is less than a line equal to one hundred myriad myriad stadia ( 1012 m). As soon as one has accepted the fact that the diameter of the sun is not greater than thirty moon diameters and that the diameter of the earth is greater than the diameter of the moon, it is clear that the diameter of the sun is less than thirty diameters of the earth.

... this number is the eighth of the eight numbers, which is one thousand myriads of eight numbers... It is therefore obvious that the number of grains of sand filling a sphere of the size that Aristarchus lends to the sphere of fixed stars is less than one thousand myriad myriad eighth numbers.
Archimedes THE SAND RECKONER


Gillian Bradshaw, Archimedes Sand Reckoner, A Science Fiction Story

Archimedes concludes that the maximum number of sand grains is 8*1063. If we consider that a sand grain contains some 1017 atoms then we get an order of 1080 atoms which is an approximation of the number of atoms of the entire Universe that current modern science estimates. Of course the Universe is almost empty such that as one could say that statistical to find somewhere matter is so small that we could even say that there is no matter in the Universe and the Universe is rather a empty desert (just as some other quantities could sum to 0).

I conceive that these things, King Gelon, will appear incredible to the great majority of people who have not studied mathematics, but that to those who are conversant therewith and have given thought to the question of the distances and sizes of the earth, the sun and moon and the whole universe, the proof will carry conviction. And it was for this reason that I thought the subject would not be inappropriate for your consideration. Archimedes

Question: Number of grains of sand on all the beaches of the earth? : Answer : Around 1019 for a grain size of 1mm3 (See link)

Estimated stars in the Universe: 70 sextillion (7 *1022) estimated to be more than 10 times the sand grains on all beaches and deserts of the Earth

For more details see T. L. Heath, THE SAND-RECKONER (requires free available DJVU viewer)

To see a World in a Grain of Sand
And Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour

Auguries of Innocence

Recursive Functions (including Archimedes Psammites) Stanford Encyclopedia of Philosophy

A Science Fiction Story: Sand Reckoner by Gillian Bradshaw

How Is the Universe Built? Grain by Grain. The New York Times

Notable Properties of Specific Numbers

21st Century Problem Solving: Estimate the number of grains of sand on all th beaches of the earth

Reports in Greek

Αρχιμήδους Ψαμμίτης

Books about Archimedes


Sand-Reckoner Gillian Bradshaw , Doherty, Tom Associates, LLC 2001

The Works of Archimedes: Translation and Commentary, Vol. 1 Reviel Netz, Cambridge University Press 2004

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