Archimedes: The Works

Archimedes lived in Syracuse, a Greek city on the island of Sicily, about 2,250 years ago. He was a mathematician who loved to measure things nobody else could: the space inside a ball, the number of grains of sand it would take to fill the whole sky, and the reason a ship floats.

He worked out the rule of the lever. A small weight far from the balance point can lift a big weight close to it. That is why a long crowbar can move a heavy rock, and why a small child at the far end of a seesaw can balance a grown-up sitting near the middle.

There is a famous story that Archimedes jumped out of his bath shouting "Eureka!" ("I have found it!") when he saw how to test whether a king's crown was pure gold. The story was written down about 200 years after he died, by a Roman builder named Vitruvius, so nobody knows whether it happened. But the science is real: water pushes up on anything in it, and a crown with cheap silver mixed in takes up more room than pure gold of the same weight.

Who, when and why

Archimedes (c. 287–212 BC) lived in Syracuse when it was a wealthy Greek city allied with Rome. He corresponded with mathematicians at Alexandria, sending them his results with letters that open many of his books. He died when Roman forces captured Syracuse during the Second Punic War. Plutarch, writing about 300 years later, gives several versions of how a Roman soldier killed him, and says the Roman general Marcellus was grieved by it.

His problem was measurement. Euclid had shown how to prove things about shapes, but many questions stayed open. How do you find the area of a curved shape, or the volume of a sphere? Why does a lever work? What makes some things float? Archimedes answered them with proofs as careful as Euclid's, often in short books addressed to friends.

The law of the lever

On the Equilibrium of Planes starts from a few postulates, like "equal weights at equal distances balance", and proves the law of the lever as a theorem.

Pappus of Alexandria, five centuries later, reports Archimedes saying, in effect, give me a place to stand and I will move the Earth. The words are a later report, but the idea is exactly the law of the lever: with a long enough arm, any weight can be balanced.

Why things float

Proposition 5 deals with floating: a floating solid sinks until it has pushed aside its own weight of fluid. Together they give Archimedes' principle: anything in a fluid is pushed up by a force equal to the weight of the fluid it pushes aside. A floating ship sinks until it has pushed aside its own weight of water. A stone is pushed up too, just not enough to float.

Trapping pi

Archimedes drew regular polygons inside and outside a circle, starting with hexagons and doubling the sides four times, to 96. The circle's perimeter is caught between the two polygons. He had no decimals and no algebra; he worked with whole-number fractions and careful estimates of square roots, rounding in the safe direction each time.

The other books, briefly

• On the Sphere and Cylinder: a sphere has two-thirds the volume and two-thirds the surface of the cylinder that just holds it. Archimedes was so proud of this that, according to Cicero, a sphere in a cylinder was carved on his tomb; Cicero found the overgrown tomb in 75 BC.
• Quadrature of the Parabola: a slice of a parabola is $\tfrac43$ of the triangle inside it, found by adding up an endless series of triangles: $1 + \tfrac14 + \tfrac1{16} + \dots = \tfrac43$.
• The Sand-Reckoner: a way to name huge numbers, used to show that even the grains of sand needed to fill the universe can be counted. It also preserves the only surviving report of Aristarchus's idea that the Earth goes around the Sun.
• On Spirals, On Conoids and Spheroids, and The Method, which explains how he found results by imagining shapes balanced on a lever before proving them.

What later science kept, and what it changed

• Kept: the law of the lever and Archimedes' principle, unchanged, in every physics book.
• Extended: his method of adding up thin slices and endless series points straight toward the integral calculus of Newton and Leibniz.
• Changed: we now compute $\pi$ by series, not polygons, to trillions of digits. But 3 10/71 < π < 3 1/7 is still correct.

The lost book found under a prayer book

The Method was lost for centuries. In 1906 the Danish scholar J. L. Heiberg found it in Constantinople, in a prayer book whose parchment had been scraped and reused in the 1200s: beneath the prayers lay a copy of Archimedes from the 900s. This book, the Archimedes Palimpsest, was imaged with ultraviolet and X-ray light from 1999 to 2008, and more of Archimedes' text came back. Heath's 1912 supplement translated The Method soon after Heiberg's find.

The Method shows Archimedes' private way of discovery. He imagined a shape as made of countless thin slices and balanced them, slice by slice, against a known shape on a lever. He says plainly that this only suggests the answer, and that a proper proof must follow. The distinction he draws, between finding a result and proving it, is one later scientists kept returning to.

Legends and engines

Ancient historians (Polybius, Livy, Plutarch) say Archimedes designed catapults and cranes that held off the Roman siege for many months. The later story that he set ships on fire with mirrors first appears centuries afterward, and modern tests have not shown it would work in battle. Plutarch also says Archimedes thought little of his machines compared with pure mathematics; that is Plutarch's view, and Archimedes' own books do take up practical questions such as floating.

Big questions

• Is guessing an answer first, then proving it, a fair way to do mathematics?
• Archimedes counted the grains of sand needed to fill the universe. Why might it matter to show that a number is finite even when it is too big to write out?
• Which matters more, a clever machine or a clever proof?

Galileo, later on this ladder, admired Archimedes above the other ancient mathematicians and built on his statics and floating bodies. Newton's calculus finished the work Archimedes' slices began.