I.2 The Sand Reckoner

In the courtyard, in the shade, the world was made of sand and was about to end inside a number.

Archimedes sat on the low stool before the raked surface, and he did not count the grains — no one is so foolish as to count the grains — but he tamed them. For days now. The city believed him at rest after the triumph of the ship; they imagined him weary, an old man taking his ease in the shade. They did not know that this was the only labour that gave him joy, and that beside it the dragging of a ship had been like the lifting of a feather.

On the little table, the wax tablet waited. At the top he had already incised the name of the addressee, and each time he read it over again something in him softened: βασιλεῦ Γέλων1. King Gelon. The boy of the grandstand, the son of Hiero, who one day would rule Syracuse and who in the meantime — a thing rarer than a kingdom — had the desire to understand.

To him he was writing. Not a treatise: a letter. But the most reckless letter he had ever dared.

"Some believe, O Gelon" — so it began — "that the number of the sand is infinite." Ἄπειρον.2 The word with which men surrender. They say it and close their eyes, like children who, not knowing how to count beyond their fingers, decide that beyond their fingers there is nothing. "Others" — he went on — "do not believe it infinite, but think that no number has been named great enough to surpass it."

And here, in the wax, Archimedes laid his finger on the wound of the world: he would name that number.

Not for the sand of a single beach. Too easy, unworthy. Instead he imagined — and wrote it in tight Doric, the tongue of the island, the one the learned men of Athens found coarse and he found his own — a mass of sand as great as the whole earth, with all the seas and all the hollows of the ground filled to the height of the tallest mountains3. Then he multiplied it again. Then he made it as great as the cosmos.

To make the cosmos great, however, he needed the vastest cosmos a human mind had ever dared to draw. And here his hand paused an instant above the wax.

Because the vastest cosmos was offered to him by a man whom nearly everyone judged mad, if not impious. Aristarchus of Samos. Aristarchus maintained that the fire stood still at the centre — the sun, not the earth — and that the earth turned about the sun, and that the sphere of the fixed stars was so preposterously distant as to make the earth and its orbit a point, a grain, a nothing. At the Portico, to whoever said it aloud, they answered with the charge of disturbing the gods.

Archimedes did not disturb the gods. He disturbed only numbers, and with delicacy. It did not matter to him whether Aristarchus was right — that was his affair, and the sky's. What mattered was that Aristarchus's hypothesis handed him the largest possible box: even were the world immense to that very degree, into it he would pour his sand and write out its number. He took the heretical idea as one takes a tool from the wall: without reverence and without fear. The sphere of the fixed stars, the one that Aristarchus supposes.4 He wrote it so, that dangerous name, in the midst of the calculation, as one names a supplier.

The number was there. It was not infinite. It was enormous, a number to make the voice tremble in pronouncing it — but it had a name, it had an order, it had an end. To utter it he had had to invent new words, orders beyond the myriads, myriads of myriads stacked like the steps of a stair that no one before had had the courage to climb. And at the top of the stair, there where the many saw only the white darkness of the word infinite, there was instead a landing, and a door, and behind the door still more number.

He set down the stylus. He closed his eyes.

This, O Gelon, to the many who have no part in the mathematics will not seem credible; but to those who do have part in them, and have meditated upon the distances and the magnitudes of the earth and the sun and the moon and the whole cosmos, it will be believed by way of the demonstration.5

By way of the demonstration. Not by the authority of the king. Not by the wonder of the crowd upon the quay. Not because Archimedes said it. Only διὰ τὰν ἀπόδειξιν6 — by way of the demonstration: the one thing in the world that had no need of armies to impose itself, nor of temples, nor of fathers. A true thing was true even if all fell silent. Even if all died. Truth held no citizenship, and for this no city could fall far enough to carry it away.

He rose, and to loosen his limbs he climbed the few steps up to the terrace.

Below him lay Syracuse at the height of her beauty. The greatest of the Greek cities, the most beautiful of them all7 — so they would call her, one day, even the Romans, with that melancholy one grants only to what one has destroyed. He saw the whole of her: Ortygia clasped in her embrace of water, the two harbours like two joined hands, Achradina teeming, Tyche with its roofs, Neapolis with its theatres, and high above Epipolae climbing all the way to Euryalus8, where the walls closed off the sky. The noonday sun held her still, gilded, eternal as a theorem.

Eternal. The word left in his mouth a taste he did not like.

Because that morning a ship had come in from the west, and with the ship a piece of news, and the news came from a plain of Italy whose name no one in Syracuse yet knew how to pronounce: Cannae. A Carthaginian consul had closed a Roman army in a pocket and snuffed it out as one snuffs a lamp. Thousands of dead in a single afternoon. Rome was bleeding. And when Rome bleeds, old Hiero had thought at once, gripping the arms of his throne, every city of the sea must choose on which side of the blood to stand.

Archimedes looked at his perfect city and, for the first time in years, became aware of a calculation he had never made.

He knew how to name the number of the grains that would fill the universe. He could write it, that number, in Doric characters upon a wax tablet, and hand it to a child king so that he might learn that nothing, not even the sky, is truly without measure.

But he did not know — he had no instrument, no hypothesis, no Aristarchus to lend him one — he did not know how to count how many mornings of sun remained to Syracuse.

That, perhaps, was the only infinite number ever to fall into his hands. And infinite, he suspected now with a sudden chill in the shade of his own city, was only the name we give to what will end too soon for us to have time to count it.

Eighteen hundred years later, a compiler would write, under the entry Arénaire: "opuscule où Archimède se propose de compter les grains de sable9" — a little work in which Archimedes sets himself to count the grains of sand. A curiosity. A feat of ingenuity. He would not hear, beneath the wax, the tremor: that this game was the last serene thing the man would write before the world asked its hands back of him.

He went back into the shade. He took up the stylus again. There was still a demonstration to close, and as long as a demonstration is open the rest of the world can wait.

Rome too. Carthage too.

For a little while yet.