Archimedes

The life of Archimedes

Timeline c. 287 BCBorn in Syracuse, the son of the astronomer Phidias. c. 270 BCHiero II becomes master of Syracuse; later he proclaims himself king. c. 250 BCAround this time, he writes to Conon of Samos and to Dositheus, in Alexandria, sending them his results. c. 240 BCWrites to Eratosthenes, in Alexandria, the letter we know as The Method. 215 BCHiero dies. Syracuse allies itself with Carthage against Rome. 214 BCThe Roman general Marcellus begins the siege of Syracuse. Archimedes’s machines defend the city. 212 BCSyracuse falls. Archimedes is killed by a Roman soldier. 75 BCCicero, a Roman magistrate in Sicily, finds his tomb again. A Greek city in Sicily Archimedes was born around 287 BC in Syracuse, on the east coast of Sicily. The date comes from a twelfth-century Byzantine author, John Tzetzes, who says he died at 75. Syracuse was then one of the largest cities in the Greek world, with two harbours and a fortified island, Ortygia, at the heart of the ancient city.

October 29, 2026 Read

Eureka: the king's crown and Archimedes's principle

The most famous story about Archimedes does not appear in any of his books. It was told two centuries later by the Roman architect Vitruvius, in book IX of his treatise On Architecture. Hiero’s crown King Hiero had commissioned a votive crown to offer to the gods and had given the goldsmith an exact quantity of gold. The crown was finished and weighed the same as the gold supplied. But a rumour spread that the goldsmith had kept part of the gold and put the same mass of silver in its place. Hiero asked Archimedes to find out the truth without damaging the crown.

October 29, 2026 Read

The law of the lever

Levers had been used long before Archimedes, to lift stones, draw water from wells and weigh goods. What Archimedes did that was new was to turn the lever into a theorem. In the treatise On the Equilibrium of Planes, he starts from a few simple axioms, such as “equal weights at equal distances balance”, and deduces the general law in the manner of the Elements of Euclid. The law Magnitudes, whether commensurable or incommensurable, balance at distances inversely proportional to their weights. — Archimedes, On the Equilibrium of Planes, book I, propositions 6 and 7

October 29, 2026 Read

The measurement of the circle and the number π

That every circle has the same ratio between circumference and diameter was known before Archimedes. What was uncertain was the value of this ratio, which we now call $\pi$: the Babylonians sometimes used $3\frac{1}{8}$, the Egyptians something close to $3.16$. In the short treatise Measurement of a Circle, Archimedes did two things: he related the area of the circle to its circumference, and he showed how to compute $\pi$ to any desired precision.

October 29, 2026 Read

The quadrature of the parabola

To “square” a figure, for the Greeks, was to find a square, or another rectilinear figure, with the same area. Squaring the circle was the impossible problem par excellence. Archimedes managed to square a curved figure: the segment of a parabola, the region between a parabola and a chord. Every segment bounded by a straight line and a section of a right-angled cone is four thirds of the triangle that has the same base and the same height as the segment. — Archimedes, Quadrature of the Parabola, letter to Dositheus

October 29, 2026 Read

The sphere and the cylinder

Greek geometers knew how to compute the volume of prisms, pyramids, cylinders and cones. The sphere resisted. In the two-book treatise On the Sphere and Cylinder, Archimedes solved the problem and found relations of surprising simplicity. The theorems Call a great circle the circle that passes through the centre of the sphere, like the equator. Archimedes proved that: the surface of the sphere is four times the area of the great circle: $S = 4\pi r^2$; the volume of the sphere is four times that of a cone whose base is the great circle and whose height is the radius: $V = 4 \cdot \frac{1}{3}\pi r^2 \cdot r = \frac{4}{3}\pi r^3$. And from this he drew the consequence that delighted him most. Consider the cylinder that exactly encloses the sphere, with radius $r$ and height $2r$. Its volume is $\pi r^2 \cdot 2r = 2\pi r^3$, and its total surface, counting both lids, is $2\pi r \cdot 2r + 2\pi r^2 = 6\pi r^2$. So

October 29, 2026 Read

The Method and the palimpsest

Archimedes’s books present perfect proofs, but they do not say how he reached the results. For centuries, mathematicians suspected he had a secret method. They were right, and the method was found in the most unlikely place. The letter to Eratosthenes Certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards, because their investigation by this method did not furnish an actual proof. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge. — Archimedes, The Method, letter to Eratosthenes

October 29, 2026 Read

The spiral of Archimedes

In the treatise On Spirals, also dedicated to Dositheus, Archimedes studies a new curve, defined by a motion. If a straight line in a plane revolves at a uniform rate about one of its extremities, which remains fixed, and returns to the position from which it started, and if, at the same time, a point moves at a uniform rate along the line, starting from the fixed extremity, the point will describe a spiral in the plane. — Archimedes, On Spirals, definition

October 29, 2026 Read

The Sand Reckoner

There are some, King Gelon, who think that the number of the sand is infinite. I will try to show you that some of the numbers I have named exceed not only the number of the sand that would fill the Earth, but also that of the sand that would fill the universe. — Archimedes, The Sand Reckoner, letter to King Gelon The Sand Reckoner is a letter to Gelon, son of King Hiero, and the only book by Archimedes written for a reader who was not a mathematician. The problem is to prove that there is a number larger than the quantity of grains of sand needed to fill the whole universe.

October 29, 2026 Read

The screw, the claw and the mirrors

Plutarch writes that Archimedes considered engineering a lesser activity, and that he left no book about his machines. Even so, it was they that made him famous among his contemporaries. The ancient sources describe several, with very different degrees of reliability. The screw of Archimedes An inclined cylinder, with a helix inside or a tube wound spirally around the outside, has its lower end dipped in water. As the cylinder turns, each portion of water is trapped in a fold of the helix and rises, turn by turn, until it comes out at the top. The historian Diodorus of Sicily, in the first century BC, attributes the invention to Archimedes, during a stay in Egypt, where the machine was used to irrigate the fields along the Nile and to drain water from mines.

October 29, 2026 Read

The legacy of Archimedes

Whoever understands Archimedes and Apollonius will admire less the discoveries of the greatest men of today. — Gottfried Wilhelm Leibniz The manuscripts What remains of Archimedes reached us by a thread. Almost all of his work depended on three Byzantine Greek manuscripts. Two of them, which scholars call codices A and B, were lost after being copied or translated in the Renaissance; the third is the palimpsest, which stayed hidden under a prayer book. In 1269 the Dominican William of Moerbeke translated the Greek texts into Latin, and through this translation Archimedes came to be read in European universities. The first printed edition of the Greek text came out in Basel in 1544.

October 29, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the files or the thumbnails provided by Commons, reduced to at most 800 pixels wide, with no other editing. Archimedes thoughtful, painting by Domenico Fetti (c. 1620): source page · public domain. Ortygia, Syracuse, photo by Andrea Schaffer: source page · CC BY 2.0. Archimedes and the Roman soldier, eighteenth-century mosaic in the ancient style: source page · public domain. Archimedes moving the Earth with a lever, engraving from the Mechanics Magazine (London, 1824): source page · public domain. Latin manuscript of On the Sphere and Cylinder (c. 1270): source page · public domain. Cicero discovering the tomb of Archimedes, painting by Pierre-Henri de Valenciennes (1787): source page · public domain. A leaf of the Archimedes palimpsest, The Walters Art Museum: source page · CC BY 3.0. The screw of Archimedes, illustration from Chambers’s Twentieth Century Dictionary (1908): source page · public domain. The Fields Medal, photo by Stefan Zachow for the International Mathematical Union: source page · public domain. Figures made for the blog The cover and the four interactive figures (the lever, the polygons that enclose π, the triangles of the quadrature of the parabola and the spiral) are original illustrations, with text in each of the site’s languages.

October 30, 2026 Read