The Thomas L. Heath translation can be read online for free, or downloaded (or read online) at

(if you don’t want to tackle the whole)

Archimedes (287-212 BC)
“The Sand-Reckoner”
translated by Thomas L. Heath

Much of The Sand-Reckoner is taken up with geometric proofs. In the two excerpts below, Archimedes begins by offering an alternative definition of the universe: it is not centered on the earth, but on the sun. He then explains the new numbering system needed to calculate the volume of this heliocentric universe.

About this translation

Thomas L. Heath, a Cambridge-educated British civil servant, translated both Archimedes and Euclid into English. His translation of Euclid’s Elements is still the most commonly used of this classic text.
In the excerpts below, Heath has transformed Archimedes’s Greek numeral system into exponential numbers readable by English speakers. In places, he has retained the Greek in brackets.

Because this website does not support exponential notation, the exponential numbers have been indicated using the strike-through (so, for example, 10 to the second power would be written 102).

The Heliocentric 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…..[H]e 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 [a work of Archimedes that has not survived], some exceed in multitude the number of the sand….

New Orders and Periods of Numbers

I. We have traditional names for numbers up to a myriad (10,000); we can therefore express numbers up to a myriad myriads (100,000,000). Let these numbers be called numbers of the first order.
Suppose the 100,000,000 to be the unit of the second order, and let the second order consist of the numbers from that unit up to (100,000,000)2.

Let this again be the unit of the third order of numbers ending with (100,000,000)3; and so on, until we reach the 100,000,000th order of numbers ending with (100,000,000)100,000,000, which we will call P.

II. Suppose the numbers from 1 to P just described to form the first period.

Let P be the unit of the first order of the second period, and let this consist of the numbers from P up to 100,000,000 P.

Let the last number be the unit of the second order of the second period, and let this end with (100,000,000)2 P.

We can go on in this way till we reach the 100,000,000th order of the second period ending with (100,000,000)100,000,000 P, or P2.

III. Taking P2 as the unit of the first order of the third period, we proceed in the same way till we reach the 100,000,000th order of the third period ending with P3.

IV. Taking P3 as the unit of the first order of the fourth period, we continue the same process until we arrive at the 100,000,000th order of the 100,000,000th period ending with P100,000,000. This is a myriad-myriad units of the myriad-myriad-th order of the myriad-myriad-th period [?? ????????????????? ???????? ????????????????? ??????? ?????? ????????] [100,000,000 times the product of (100,000,000)99,999,999 and P99,999,999 i.e. P100,000,000].

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