Greek mathematician, physicist, and engineer of Syracuse, Sicily (c. 287β212 BCE). Widely regarded as the greatest mathematician of antiquity. Bounded the value of Ο, invented methods that anticipated integral calculus by 1,900 years, and founded the fields of statics and hydrostatics. Killed during the Roman siege of Syracuse, reportedly while working on a mathematics problem.
Archimedes lived in Syracuse, a Greek city in Sicily, around 287β212 BCE. He's the source of one of the most famous stories in the history of science: the King of Syracuse asked him to determine whether a crown was made of pure gold or had been mixed with cheaper silver, without damaging it. Stepping into a bath and noticing the water level rise, Archimedes supposedly realised he could measure the crown's volume by water displacement β and ran through the streets naked shouting "Eureka!" ("I have found it!").
Archimedes was also a brilliant engineer. He designed war machines that helped defend Syracuse against Roman siege for two years, including devices for hurling heavy stones and (according to legend) mirrors that could set enemy ships on fire using focused sunlight. He invented the Archimedes screw, a device for lifting water that is still used today in some irrigation systems.
When Syracuse finally fell to the Romans in 212 BCE, a Roman soldier found Archimedes drawing mathematical diagrams in the sand. According to the story (Plutarch, c. 75 CE), Archimedes said "Do not disturb my circles" β and the soldier killed him anyway, despite orders that he be taken alive. He was reportedly given an honourable burial, with a sphere inscribed in a cylinder carved onto his tombstone β representing what he considered his greatest mathematical discovery.
| Born | c. 287 BCE, Syracuse, Sicily |
| Died | 212 BCE, Syracuse (during Roman siege) |
| Studied at | Possibly Alexandria, Egypt (corresponded with Alexandrian mathematicians) |
| Major works | On the Sphere and Cylinder, Measurement of a Circle, On Floating Bodies, The Method, On the Equilibrium of Planes |
Archimedes inscribed and circumscribed regular polygons around a circle, starting with a hexagon (6 sides) and repeatedly doubling the number of sides up to 96. By calculating the perimeters of the inscribed and circumscribed 96-gons, he showed:
This gives 3.1408 < Ο < 3.1429 β the first mathematically rigorous bounds on Ο, correct to two decimal places.
Archimedes proved that a sphere inscribed in a cylinder (touching the cylinder at the equator and both flat ends) has exactly β the volume and β the surface area of the cylinder. He considered this his greatest achievement and requested the diagram be placed on his tombstone.
To find areas and volumes of curved shapes, Archimedes approximated them with a sequence of simpler shapes (like polygons) whose areas he could calculate exactly, then showed the approximation error could be made arbitrarily small. This is conceptually identical to the modern definition of a definite integral as a limit of Riemann sums β but developed roughly 1,900 years before calculus was formalised.
Archimedes' Principle (On Floating Bodies): an object submerged in a fluid experiences an upward force equal to the weight of the fluid displaced. This is the foundation of the science of hydrostatics.
The law of the lever: "Give me a place to stand and I will move the Earth" β Archimedes showed that a lever's mechanical advantage depends on the ratio of the distances from the fulcrum, formalising a principle known informally for millennia.
Archimedes' treatise The Method of Mechanical Theorems was believed lost for centuries. In 1906, Danish scholar Johan Ludvig Heiberg discovered that a 13th-century Byzantine prayer book (a palimpsest β a manuscript reused by scraping off earlier text) contained faint traces of several Archimedes works underneath, including a copy of The Method. The manuscript resurfaced at a 1998 Christie's auction (sold for $2 million) and was subjected to advanced imaging techniques (X-ray fluorescence, multispectral imaging) at Johns Hopkins and Stanford from 1999β2008, recovering text that had been invisible for 700 years.
The Method shows that Archimedes used a mechanical, physically-motivated technique β imagining figures balanced on a lever β to discover results, which he then proved rigorously using the method of exhaustion in his other works. This is historically significant: it shows Archimedes distinguished between the process of discovery (heuristic, physical reasoning) and the process of proof (rigorous, geometric deduction) β a distinction central to how mathematics is still practised today.
In The Sand Reckoner, Archimedes devised a system for naming and working with extremely large numbers β attempting to estimate the number of grains of sand that would fill the universe (as then conceived). He arrived at a number around 10βΆΒ³ in modern notation, requiring him to invent a new numerical naming system beyond the largest number Greek notation could otherwise express (a myriad myriad, 10βΈ).
A problem attributed to Archimedes (surviving in a poem, disputed authorship) asks for the number of cattle in a herd satisfying eight simultaneous conditions. The smallest solution, found only with 20th-century computers (1965), has 206,545 digits β an extraordinary result if genuinely due to Archimedes, though many historians doubt he could have solved it with ancient tools and it may be a later addition to the corpus attributed to him.