The Sand Reckoner

Note to readers: The text of Archimedes’ work continues to the point where a paraphrase becomes easier to understand than a direct translation. Note that this translation uses modern number notation, whereas the Greeks were hampered by a notation closely akin to Roman numerals.

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 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 the 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 premises 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 a 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 put the magnitude of the Earth at ten times the size that my predecessors thought it.

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 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 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 the diameter of the moon and not greater.

4. the diameter of the sun is greater than the side of the chilleagon inscribed in the greatest circle in the (sphere of) the universe.

"I state this proposition (later to be proved) because Aristarchus discovered that the sun appeared to be one 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.

[Archimedes next establishes this proposition and continues to his final assumption.]

5. Suppose a quantity of sand [be] taken not greater than a poppy seed, and suppose that it contains not more than 10,000 grains. Next, suppose the diameter of the poppy seed to be not less than one 40th of a finger-breadth.

Paraphrase of Orders and Periods of Numbers

Given the traditional names for numbers up to a myriad, 10,000, one can express numbers up to a myriad of myriads, or 100,000,000. Such numbers belong to the first order.

Suppose the myriad of myriads be the fundamental unit of the second order and let the second order consist of numbers from that unit up to as many (a myriad of myriads) units, or 100,000,000².

Let this last number be the unit for the third order and let that order end with 100,000,000³, continuing this process until we reach the 100,000,000th order of numbers, thus ending with the number

         100,000,000^100,000,000,

which one may call P.

Next, let the first period consist of the numbers from 1 to P. Now let P be the unit of the first order of the second period and let this period consist of the numbers from P up to 100,000,000P. This last unit will be taken as the unit of the second order of the second period, ending with 100,000,000²P. We continue in this manner until we come to the 100,000,000th order of the second period, ending with 100,000,000^100,000,000P, or P².

Next, taking P² as the unit of the first order of the third period, we continue until we reach the 100,000,000th order of the third period, ending with P³.

Next, taking P³ as the unit of the first order of the fourth period, we continue the process until we reach the 100,000,000th order of the 100,000,000th period, ending with P^100,000,000. [This last number was expressed by Archimedes as "a myriad-myriad units of the myriad-myriad-th order of the myriad-myriad-th period."]

Taking advantage of modern notation (which we have been doing all along, of course), we can display Archimedes’ scheme more simply as follows:

First Period

Second Period

.
.

Qth Period

It follows that the last number in this scheme is the last number in the Qth order of the Qth period, or (in words) P to the 10th to the 8th. Since P may be written as a 1 followed by 800 million zeros, the final number in Archimedes’ scheme would be written as 1 followed by 800 million times Q zeros, or 1 followed by 8 billion billions of zeros, a large number by anyone’s reckoning!

Filling the Universe with Sand

Turning now to Archimedes’ reckoning, he proceeds to fill up the (then) known universe with sand by considering a succession of spheres, each 100 times the diameter of its predecessor in the succession. He uses a fact well known to Greek geometers: the ratio of the volumes of two spheres is the third power of the ratio of their diameters.

He begins with a poppy seed which, you will recall, was not less than one 40th of a finger-breadth. A sphere of diameter 40 poppy seeds would therefore have a volume no greater than 64,000 poppy seeds. Since each poppy seed contains no more than 10,000 grains of sand, a sphere of one finger-breadth contains at most 640,000,000 grains of sand. The latter number consists of 6 units of the second order plus 40,000,000 units of the first order, a quantity that is not more than 10 units of the second order in Archimedes’ numbering scheme.

A sphere of one finger-breadth contains no more than 10 units of the second order of sand grains.

The Greek measure of larger distances, the stadium, is less than 10,000 finger-breadths, according to Archimedes. Thus,

Now, as we have already seen, the diameter of the (then) known universe is less than 10,000,000,000 stadia. It follows that the number of grains of sand sufficient to fill up that universe is less than 1,000 units of the seventh order of numbers, or 10⁵¹.

A sphere of the size attributed by Aristarchus to the sphere of fixed stars would contain a quantity of sand no greater than 10,000,000 units of the eighth order of numbers. (i.e., 10⁶³) This follows from the hypothesis that the ratio

(diameter of Earth):(diameter of universe)

equals the ratio

(diameter of universe):(diameter of sphere of fixed stars).

As we saw earlier, the diameter of the universe is not greater than 10,000 Earth diameters, so that the diameter of the sphere of fixed stars is not greater than 10,000 x the diameter of the ‘universe.’ It follows that the volume of the sphere of fixed stars is not greater than (10,000)³ x the volume of the ‘universe.’

Finally, the number of grains of sand that would fill up the sphere of fixed stars would be

< (10,000)³ x 1,000 units of the seventh order
< 10,000,000 units of the eight order (i.e., 10⁶³)

Archimedes concludes:

"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, the 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."

Notes

1. Archimedes’ estimate for the number of grains of sand sufficient to fill the then known universe, namely 10⁶³ grains of sand, is eerily similar to a well-known estimate for the total number of fundamental particles in the visible universe, namely, 2⁶⁴. These do not by any means fill our ‘universe,’ however.

2. The Greek unit of the stadium equals 185 metres. The Greek universe therefore had a diameter of 10⁹ stadia or 185,000,000 kilometres. A sphere of this diameter, centered on the Sun, would barely contain the orbit of Mars. Given Note 1, the totality of fundamental particles in the universe could be packed into a sphere a little over twice this size!

3. Archimedes met an untimely death while deep in thought, pondering a figure he had drawn in the sand. He did not see the Roman soldier approach, sword in hand. The mosaic below portrays this historical event.

The foregoing text is freely adapted from The World of Mathematics (James R. Newman, ed.) Vol. 1, Simon & Schuster, New York. 1956.