Works

Overview

What survives of Archimedes is about a dozen treatises, four hundred pages in Heath’s edition, written at Syracuse in the third century BCE and addressed as letters to mathematicians in Alexandria. He was killed there in 212, during the Roman capture of the city, by a soldier who found him working, and Plutarch reports that the general Marcellus was distressed about it.

Each treatise is short and self-contained, so this is a collection to sample rather than a book to read through. The usual selections are On the Sphere and Cylinder, Measurement of a Circle, Quadrature of the Parabola, The Sand-Reckoner, On the Equilibrium of Planes, On Floating Bodies and The Method.

The prose is worth noticing. Archimedes writes to named colleagues, tells them what he has found, occasionally teases them, and in one preface admits to having sent out two false propositions deliberately, so that anyone claiming to have proved everything would be exposed. The mathematics is formal and the letters around it are not.

The manuscript history is a story in itself. The Method was unknown until 1906, when the Danish scholar Johan Heiberg examined a thirteenth-century prayer book in Constantinople and found that its parchment had been scraped and reused, and that the older writing underneath was Archimedes. The palimpsest disappeared, resurfaced at auction in 1998 in poor condition, and has since been read with X-ray fluorescence, recovering text Heiberg could not see.

The argument

On the Sphere and Cylinder. The result Archimedes was proudest of. The surface of a sphere is exactly four times the area of its great circle, and the volume of a sphere is two thirds of the cylinder that just contains it, with the surfaces in the same ratio. He asked for a sphere inscribed in a cylinder to be carved on his tomb, and Cicero, as quaestor in Sicily in 75 BCE, found the monument neglected in a thicket and had it cleared.

Measurement of a Circle. Three propositions. The area of a circle equals a right triangle with legs equal to the radius and the circumference. And then the calculation: by inscribing and circumscribing regular polygons and doubling the number of sides from six to ninety-six, Archimedes traps the circumference between three and one seventh and three and ten seventy-firsts. The arithmetic required to do this without decimals, using approximations to square roots that he does not explain, is formidable, and how he obtained those roots is still discussed.

Quadrature of the Parabola. The area cut off by a chord is four thirds of the inscribed triangle on the same base. The proof fills the segment with triangles, each generation totalling a quarter of the one before, and sums the series one plus a quarter plus a sixteenth and so on. It is the first summation of an infinite geometric series in mathematics.

On the Equilibrium of Planes. Physics done as geometry. From a few postulates about balance Archimedes derives the law of the lever, that magnitudes balance at distances inversely proportional to their weights, and then finds the centres of gravity of triangles, parallelograms and parabolic segments.

On Floating Bodies. Book I establishes that a body immersed in fluid is pushed up with a force equal to the weight of the fluid it displaces, which is the principle everyone knows and which Vitruvius attached to the story about the crown, the bath and a naked man running through Syracuse. Book II is the surprise: an analysis of the positions in which a floating paraboloid of revolution will be stable, depending on its proportions and density, which is essentially naval architecture and which nothing in the next two thousand years improved on.

The Sand-Reckoner. A letter to King Gelon arguing that the number of grains of sand needed to fill the universe is not infinite and can be named. To do it Archimedes has to invent a notation for very large numbers, since Greek numerals gave out at a myriad myriads, and he builds a system of orders and periods reaching what we would write as ten to the power of eighty thousand million million. To size the universe he uses Aristarchus’ heliocentric hypothesis, and this letter is the main surviving source for the fact that Aristarchus proposed it.

The Method. A letter to Eratosthenes, and the most remarkable thing here. Archimedes explains that he found several of his theorems, including the volume of the sphere, by a procedure he does not consider a demonstration: he imagines the figure cut into indivisible slices, hangs the slices on one arm of a lever, balances them against a figure whose volume he knows, and reads off the answer from the law of the lever. Then he says plainly that this gives knowledge and not proof, and that the geometrical demonstration must still be supplied, which is what the other treatises do.

Two things make this extraordinary. It treats areas as composed of lines and solids of planes, which is an infinitesimal method twenty centuries before the calculus. And it is a mathematician describing his own heuristics, which is almost unheard of in Greek mathematics, where the finished proofs conceal how anything was found.

How to read it

Start with Measurement of a Circle. It is three propositions, the argument is completely intelligible, and it shows the exhaustion method at work.

Then read The Method’s preface, which is two pages and is the best thing in the collection for a non-specialist, and then the Sand-Reckoner, which is a joke conducted with total rigour.

Understand exhaustion before attempting the harder treatises. The technique is to show that the quantity in question can be neither greater nor smaller than the claimed value, each by deriving a contradiction, and therefore must be equal. It is how Greek mathematics handles limits without having them, and once you see it you can follow the structure of any of these proofs even where the details are heavy.

Use Heath, who converts the arguments into modern notation and adds the commentary that makes them readable. If you want the Greek argument as Archimedes shaped it, Reviel Netz’s newer translation is the scholarly alternative.

Do not expect the engineering. The war machines at the siege of Syracuse, the claw, the burning mirrors, the screw, are all from later writers; Plutarch says Archimedes regarded such work as beneath the dignity of pure geometry and left no treatise on any of it.

Questions it raises

• The Method finds results by a technique Archimedes will not accept as proof. What made the mechanical argument illegitimate, and was he right to distrust it?
• Exhaustion gets the answers a limit would give, by a double contradiction. What is missing that the calculus supplies, and why did it take so long?
• He bounds pi between two fractions rather than computing it. What does that tell you about what Greek mathematics thought a number was?
• The stability analysis in On Floating Bodies had no use anyone could make of it for two millennia. Why work on it?
• Archimedes sent out false propositions to catch plagiarists. What does that say about how results circulated?
• If The Method had not been scraped off and forgotten, would the calculus have come earlier?