There are some, King Gelon, who think that the number of the sand is infinite. I will try to show you that some of the numbers I have named exceed not only the number of the sand that would fill the Earth, but also that of the sand that would fill the universe.

— Archimedes, The Sand Reckoner, letter to King Gelon

The Sand Reckoner is a letter to Gelon, son of King Hiero, and the only book by Archimedes written for a reader who was not a mathematician. The problem is to prove that there is a number larger than the quantity of grains of sand needed to fill the whole universe.

Numbers without names

The difficulty was not calculating, it was naming. The Greek numeral system used letters of the alphabet and went, in practice, up to the myriad, 10 000. A “myriad of myriads”, $10^8$, was already an enormous number. Archimedes proposed to go on:

  • the numbers up to $10^8$ form the first order;
  • taking $10^8$ as a new unit, the numbers up to $10^8 \cdot 10^8 = 10^{16}$ form the second order;
  • and so on, up to order number $10^8$, which ends at $\left(10^8\right)^{10^8} = 10^{800\,000\,000}$.

All these orders form the first period. Archimedes went on with periods, up to $10^{8 \cdot 10^{16}}$. It is, in essence, the idea of using powers of the base as steps, the same idea behind scientific notation.

The size of the universe

To fill the universe, one had to know its size. Archimedes took the largest universe the astronomers of his time proposed: that of Aristarchus of Samos, who put the Sun at the centre with the Earth going around it. In this model, the stars had to be so far away that no apparent shift in them could be noticed over the year.

Archimedes then estimated, always erring on the safe side: how many grains of sand fit in a poppy seed, how many seeds fit across the width of a finger, how many fingers fit in a stadium (about 180 m), how many stadia across the diameter of the universe. Multiplying it all, he concluded that Aristarchus’s universe would hold fewer than

$$10^{63} \text{ grains of sand}.$$

It is a number of the eighth order of the first period, far from the limits of the system he had invented.

Why would a king want to know this?

The book is a lesson about infinity itself. Many people confused “very large” with “infinite”. Archimedes shows that even a quantity that seems unimaginable has a number, and that mathematics can name it and go beyond it. At the same time, the book gathers what was known in astronomy: the sizes of the Sun, the Moon and the Earth, and even a measurement of the apparent diameter of the Sun made by Archimedes himself.

Exercise: myriads

Write as powers of 10 a myriad, a myriad of myriads and the last number of the second order.

Myriad: $10^4$. Myriad of myriads: $10^4 \cdot 10^4 = 10^8$, the end of the first order. The second order uses $10^8$ as its unit and goes up to $10^8 \cdot 10^8 = 10^{16}$.

Exercise: a modern number

The observable universe is estimated to contain about $10^{80}$ atoms. In which order of Archimedes’s system would this number fall?

Order $k$ runs from $10^{8(k-1)}$ to $10^{8k}$. For $10^{80}$, we need $8(k-1) < 80 \le 8k$, that is, $k = 10$: $10^{80}$ is exactly the last number of the tenth order of the first period. Archimedes’s system, created to count sand, is still more than enough to count the atoms of the universe.

Sources: Archimedes, The Sand Reckoner · Thomas L. Heath, The Works of Archimedes (1897) and Aristarchus of Samos (1913) · Reviel Netz, A New History of Greek Mathematics (2022) · Owen Gingerich, “Did Copernicus Owe a Debt to Aristarchus?”, Journal for the History of Astronomy 16 (1985).

All topics on Archimedes · ← The spiral of Archimedes · The screw, the claw and the mirrors →

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