The Sand-Reckoner (opening and conclusion) Original title: Ψαμμίτης (Ancient Greek) Archimedes of Syracuse Ἀρχιμήδης (c. 287–212 BCE) Translated by Thomas Little Heath (1861–1940), 1897 3rd century BCE Cold Library · Open Stacks · S-006002 · https://coldlibrary.com/s/006002 Rights: Ancient Greek text. The translator, T. L. Heath, died in 1940; the translation was published in 1897. Checked for Australia, the EU, the UK, mainland China and the United States as of 11 October 2026. Guidance, not legal advice. Source: Internet Archive (scan of Heath, The Works of Archimedes, 1897), https://archive.org/details/worksofarchimede00arch (retrieved 2026-10-11) Generated 2026-10-14 from https://coldlibrary.com/en/open-stacks/archimedes-sand-reckoner ---- The opening 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 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 recognising 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. Now you are aware that ‘universe’ is the name given by most astronomers to the sphere whose centre is the centre of the earth and whose radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account, as you have heard from astronomers. But Aristarchus of Samos brought out a book consisting of some hypotheses, in which the premisses lead to the result that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre 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 centre of the sphere bears to its surface. Now it is easy to see that this is impossible; for, since the centre of the sphere has no magnitude, we cannot conceive it to bear any ratio whatever to the surface of the sphere. We must however take Aristarchus to mean this: since we conceive the earth to be, as it were, the centre of the universe, the ratio which the earth bears to what we describe as the ‘universe’ is the same as the ratio which the sphere containing the circle in which he supposes the earth to revolve bears to the sphere of the fixed stars. For he adapts the proofs of his results to a hypothesis of this kind, and in particular he appears to suppose the magnitude of the sphere in which he represents the earth as moving to be equal to what we call the ‘universe.’ I say then that, even if a sphere were made up of the sand, as great as Aristarchus supposes the sphere of the fixed stars to be, I shall still prove that, of the numbers named in the Principles, some exceed in multitude the number of the sand which is equal in magnitude to the sphere referred to, provided that the following assumptions be made. 1. The perimeter of the earth is about 3,000,000 stadia and not greater. It is true that some have tried, as you are of course aware, to prove that the said perimeter is about 300,000 stadia. But I go further and, putting the magnitude of the earth at ten times the size that my predecessors thought it, I suppose its perimeter to be about 3,000,000 stadia and not greater. 2. The diameter of the earth is greater than the diameter of the moon, and the diameter of the sun is greater than the diameter of the earth. In this assumption I follow most of the earlier astronomers. 3. The diameter of the sun is about 30 times the diameter of the moon and not greater. It is true that, of the earlier astronomers, Eudoxus declared it to be about nine times as great, and Pheidias my father twelve times, while Aristarchus tried to prove that the diameter of the sun is greater than 18 times but less than 20 times the diameter of the moon. But I go even further than Aristarchus, in order that the truth of my proposition may be established beyond dispute, and I suppose the diameter of the sun to be about 30 times that of the moon and not greater. 4. The diameter of the sun is greater than the side of the chiliagon inscribed in the greatest circle in the (sphere of the) universe. I make this assumption because Aristarchus discovered that the sun appeared to be about 1/720th part of the circle of the zodiac, and I myself tried, by a method which I will now describe, to find experimentally the angle subtended by the sun and having its vertex at the eye. The conclusion 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 be not inappropriate for your consideration.