IV.2 Sphere and Cylinder

The news reached Marcellus toward midday, when the city was his at last and his men were dividing its spoils among themselves. It concerned neither a fortress nor a harbour, but an old man; and to carry it, through all the din of the sack, there came a single voice.

Of everything that day had laid upon his shoulders — the fire, the plunder he had not wholly managed to prevent, the booty his men claimed as a wage owed to them — it was that death that wounded him most deeply: Μάλιστα δὲ τὸ Ἀρχιμήδους πάθος ἠνίασε Μάρκελλος.1

Of the death itself Marcellus wished to know no more than he already knew, and that already was enough to rob him of sleep: an old man bent over a drawing, a soldier who had seen in that stooped back nothing worth a moment's patience, a sword swifter than an order given too late. He did not ask which of the tales already running through the camp was the truest. He asked, instead, who was left to mourn him, and who kept safe what that man had loved more than his own life — for of an entire life there remained to Marcellus, by now, only the last unspoken wish of the one who had lived it.

He had the soldier who had struck the blow sought out, and turned from him as from a man defiled; then he found the old man's kin, and did them honour2 — as one honours those who remain of a man who deserved more than the death he had been given. To the killer, for all the rest of the campaign, he would never again speak a word.

They found no widow. They found no children: the old man, said the neighbours with that grimace one reserves for things one does not wholly understand, had married only his circles. What they found instead, in the workshop near Ortygia, was a boy seated among the dismantled pulleys and the mirrors still warped from the last assault — and he was not weeping for the city, though it burned but a few streets away: he was weeping for him, the old man, with the face of one who has lost a father without ever having had the right to call him so. His name was Damon. For years he had drawn taut the cords of those machines without ever fully possessing the reason of them, lending his hands to a thought that stayed always a step ahead of him; and now, before Marcellus's soldiers, he was the one voice left that knew what that thought had truly wanted for itself.

He said it, and repeated it when they led him before Marcellus, in the strained voice of one who has learned a sentence by heart because he fears he will forget it just now when it counts: the master, for years, had asked friends and kin for one thing only, for when he should be dead. Not a eulogy. Not his name cut in great letters, as is done for kings.

It is said that he had asked his friends and his kinsmen to place upon his tomb, after his death, the cylinder that encloses the sphere within it, and to inscribe there the ratio by which the containing solid exceeds the one contained.3

Not a ship. Not a crown of gold wrested from a goldsmith's fraud. Not a machine to hurl stones or lift prows. A cylinder, and within the cylinder a sphere, and above the two the one thing for which he had ever asked, explicitly, to be remembered: a ratio.

Marcellus did not understand geometry — he had never studied it, it had never served him in a life of sieges, and now it seemed to him too late to begin — but he understood orders, and this, though given to no soldier, was an order. He summoned the stonecutters who had followed the camp for the headstones of the centurions, and told them to carve exactly what the boy described, without altering a single letter of a language that neither of them, neither the Roman nor the craftsman, could read all the way through.

What they would carve, a compiler would rediscover centuries later in another book, in yet another language, among the propositions of a treatise that few, outside Syracuse, had ever taken the trouble to read to the end — the corollary to the proposition with which Archimedes had just proved that the sphere is four times the cone of equal base and of radius as height. It said, in Doric first of all:

"πᾶς κύλινδρος βάσιν μὲν ἔχων τὸν μέγιστον κύκλον τῶν ἐν τῇ σφαίρᾳ, ὕψος δὲ ἴσον τῇ διαμέτρῳ τῆς σφαίρας, ἡμιόλιός ἐστι τῆς σφαίρας καὶ ἡ ἐπιφάνεια αὐτοῦ μετὰ τῶν βάσεων ἡμιολία τῆς ἐπιφανείας τῆς σφαίρας.4".

And then, for the learned who would not stoop to read Doric, the same property in the cold tongue of the commentators: sesquialterum esse sphaerae, et superficiem eius sesquialteram superficiei sphaerae5. Or, put more simply still, in the tongue of a boy who had never had the wish to finish Euclid: the sphere is two thirds of the cylinder that contains it. No more. No less. Forever — even when the stone that tells of it shall one day lie buried in brambles.

It was, of all that he had proved in a lifetime, the thing he was proudest of: not the ships moved with a finger, not the mirrors that, it was said, set fire to the sails, not the iron arms that lifted the Roman prows like fish on a hook. Those he had built against his will, for a king who wished to see and for a city that needed to live one year more, one month more, one day more; and he had never wished to leave the method of them in writing, having judged ignoble and vulgar the mechanical art and every craft that touches upon need6. The circle within the cylinder, on the other hand, he had sought for himself alone, in a courtyard, in the shade, with no king to ask it of him — and for that alone he had asked for a signature in stone.

A compiler, should he ever write of him under the letter that was his due, would list the machines: the screw, the claw, the burning mirror, the ship moved by a single old man seated on a stool. Every gear documented, drawn with the precision of one who has never held in his hands a true cylinder of stone nor felt the weight of a sphere that fills it exactly. Of the tomb, in those crowded pages, there would remain but a single line, and not even that certain: it is said to have been found, centuries later, covered in brambles, by a Roman magistrate who could read numbers better than he could read men.

The city was still burning, in some quarter far from the harbour. The tomb, for now, was only an order shouted at weary stonecutters and a promise made by a victor in tears. But the ratio between the sphere and the cylinder had no need of being carved to be true, nor even of being remembered by the Romans to remain so. And this, perhaps, the old man had always known — better than anyone else, better even than the one who now wept for him.