Some Think, O King Hiero, That the Grains of Sand Cannot be Counted

Among the many cultural roots of the world in which we live, rational Greek thought and the texts of the Bible are two of the most influential. Since late antiquity, strenuous attempts to reconcile these two strands have been made by Christian intellectuals, with mixed results, and still today their relationship with rational thinking is an issue for Christianity and Islam. Is it possible that elements of this dialogue were already in place two centuries before the birth of Christ? The idea is a small gem that can be found in Giuseppe Boscarino’s recent critical edition and translation of Psammites (The Sand Reckoner) by Archimedes.

Archimedes, who lived in Sicily immediately before its occupation by the Romans, is one of the great scientific figures of antiquity. From Archimedes we have inherited pages of stunning mathematical richness that played an important part in the renaissance of scientific thought in the modern era and continued to inspire the development of mathematics right up until the end of the nineteenth century.

From the writings of Archimedes this strange little book, The Sand Reckoner, in which he literally counts grains of sand, has also survived. Counting grains of sand seems like an undignified occupation for a scientist. What is he up to?

The Sand Reckoner is not really a work of science. It is really more of a ‘popular’ text, as we deduce from a reference to more technical, now lost, work by the same Archimedes.

The problem The Sand Reckoner deals with is the construction of an arithmetical system. The numbers used in the third century BC do not allow us to count very numerous things. The numerical system was the Greek one, similar to the Roman in which X was 10, V was 5, and 15 was written XV. The largest number to have a specific name was 10,000, which was called a ‘myriad’ and indicated in Greek with the letter M. There was no way of writing directly, and hence no way of using, numbers that were much bigger than this. Archimedes tackled the problem and constructed a system that could deal with arbitrarily large numbers. The solution is to call a myriad of myriads ‘a second order number’. In this way two second-order numbers are two myriads of myriads, which is to say 200 million. A myriad of myriads of numbers of the second order results in a number of the third order, which is to say 10 million billion, and so on. It is a solution similar to the one in modern science: we use the powers of ten.

In Psammites, Archimedes gives a demonstration of the usefulness of this system by estimating the total number of grains of sand in the world. In fact, he goes one better than this: he estimates how many grains there would be in the entire universe, if the universe happened to be filled with sand.

First he estimates how many grains of sand would fill a mustard seed, then how many mustard seeds would fill a box the size of his index finger, how many boxes would fill the Earth, how many Earths the solar system, and finally how many solar systems would fit into the universe, according to the astronomy of the time.

During the course of this calculation he reveals the technical accuracy at his disposal to measure the diameter and the distance from us of the sun and the moon, and much else besides about the astronomy that he is familiar with.

Particularly fascinating is his reference to the heliocentric theory of Aristarchus, which anticipates Copernicus by some fifteen centuries. The final outcome of the reckoning is that the number of grains of sand needed to fill the universe is a thousand myriads of numbers of the eighth order, which is to say, in modern terms, ten to the power of sixty-three (10⁶³). A big number, but one that is definite and conceivable.

The game is a refined one, and the playful execution of it by Archimedes is carried out impeccably. Beneath the playfulness you also get the sense that there is something essential at stake. At the beginning of the text, couched in the form of a letter, its polemical objective is made quite explicit: ‘There are those who think, O King Hiero, that the grains of sand cannot be counted.’ And this reminds us of a passage in the Bible:

Who can number the sand of the sea, and the drops of rain, and the days of eternity? Who can find out the height of heaven, the breadth of the Earth, the depth of the abysses? […] There is only one who has this wisdom: the Lord sitting upon his throne.

These are the powerful opening words of the book of Ecclesiasticus, or Sirach. They speak of counting grains of sand, but emphasize the impossibility of such reckoning and such knowledge. Can there be a connection between the two texts?

Ecclesiasticus was probably written in Palestine in the Hellenistic era, which is to say in a period of Greek political and cultural domination, and then translated soon afterwards into Greek in Egypt, as is mentioned in the text itself, probably in Alexandria, where the Greek dynasty of the Ptolemies was involved in an effort to compile, translate, study and conserve the entire knowledge of antiquity. It is largely thanks to this effort that the Bible is available to be appropriated by the Jewish, Christian and Muslim traditions. The Bible that we are familiar with, in other words, was compiled and edited in Alexandria under the initiative of enlightened Greek sovereigns: it was thanks to Greek universalism and multiculturalism that the Bible was transmitted to us, more than to the particular culture of the ancient Jewish world. In that same Alexandria, in the public institutes of research, the celebrated Library and the Museum (prototypes of the modern university), a brilliant young Sicilian – Archimedes – probably studied, and kept in epistolary touch with the Alexandrine intelligentsia for the rest of his life.

On close inspection, the translator of Ecclesiaticus into Greek mentions that he came across the book in the thirty-eighth year of the reign of Euergetes, which is to say 140 years before the birth of Christ – at a time, that is, when Archimedes had already been killed by the Romans. But perhaps we should take these dates with a pinch of salt, since Archimedes could have come into direct contact with the Hebrew text in Alexandria, or with similar Hebrew texts. In Alexandria it had been customary for over a century to systematically translate learned Hebrew works into Greek. The conceit of the grains of sand that cannot be counted, rhetorically standing for irrevocable human limitations, existed prior to Ecclesiasticus. Pindar tells us, for instance, centuries earlier, that ‘Sand escapes from counting.’

Taking all this into account, perhaps the polemical objective of Archimedes starts to become clear. With an enlightenment flourish before the letter, Archimedes rebels against the type of knowledge that insists on mysteries that are intrinsically beyond human understanding. Archimedes does not pretend to know the exact dimensions of the universe, or the precise number of grains of sand. It isn’t the comprehensive nature of his own knowledge that he is defending. On the contrary, he is quite explicit about the approximate and provisional nature of the estimations that he makes. He talks, for instance, of various alternatives regarding the dimensions of the universe, a subject on which he does not have a well-defined position. And he is aware in fact of how yesterday’s ignorance may be enlightened today, and how today’s knowledge may be revised tomorrow.

But he rebels against abandoning the search for knowledge. His is a declaration of faith in the knowability of the world, and a proud rebuke to those who are content with their own ignorance and with delegating knowledge elsewhere.

Many centuries have passed, and the text of Ecclesiasticus, along with the rest of the Bible, can be found in countless homes all over the planet, while Archimedes’ text is read only by a few. Archimedes was slaughtered by the Romans during the sacking of Syracuse, the last proud remnant of Magna Grecia to fall under the Roman yoke, during the expansion of that future empire that would soon adopt Ecclesiasticus as one of the foundational texts of its official religion, a position which it was to occupy there for more than a thousand years. During that millennium, the calculations made by Archimedes languished in a state of incomprehensibility. Near to Syracuse there is one of the most beautiful sites in all Italy, the theatre at Taormina, overlooking from above the Mediterranean Sea and Mount Etna. In the time of Archimedes, the theatre was used to stage plays by Sophocles and Euripides. The Romans adapted it for gladiatorial combat. In other words, the cultural battle between the world of Ecclesiasticus and that of Archimedes saw the complete triumph of the former.

But let’s look again, point by point, at the text from Ecclesiasticus. The number of grains on the shores of the seas was estimated by Archimedes; the number of drops of rain enters into the calculations of climatologists; the number of days since the Big Bang have been determined by cosmology; Aristarchus had already begun to measure the height of the sky; the extension of the Earth had been assessed by Eratosthenes some decades before Archimedes, and today it is known, like the depth of marine abysses, with millimetrical precision. For the supposedly unanswerable questions listed in Ecclesiasticus, we have found answers.

In the meantime, new open questions have arisen.

The issue posed by Archimedes is still current: do we want to go in search of what we still don’t know, or should we simply accept the idea that our knowledge has fixed boundaries?

The subtle, intelligent game played in The Sand Reckoner is not just a demonstration of an audacious mathematical construction, or the virtuosity of one of the most extraordinary intellects of antiquity. It is also a defiant cry of reason, which recognizes its own ignorance but refuses to delegate knowledge to others. It is a small, low key and extremely intelligent manifesto against obscurantism. It has never been more current.