Archimedes

Archimedes was born around 287 BCE in Syracuse, a prosperous Greek colony on the southeastern coast of Sicily and one of the great cities of the Mediterranean world ^[1]. The date rests on a report by the Byzantine scholar John Tzetzes that Archimedes lived seventy five years, counted back from his well attested death in 212 BCE ^[2]. His father was an astronomer named Phidias, a detail Archimedes himself mentions in his treatise The Sand Reckoner, and this parentage almost certainly shaped the boy's early exposure to mathematics and the study of the heavens ^[1].

Syracuse in this period was ruled by Hiero II, a capable monarch who kept the city independent and wealthy during decades of conflict between Rome and Carthage. Ancient sources suggest Archimedes was related to, or at least on close terms with, the royal house, which gave him access to patronage and to practical problems worth solving ^[3]. The historian Plutarch, writing some three centuries later, described him as a kinsman and friend of the king ^[3].

Most scholars believe the young Archimedes traveled to Alexandria in Egypt, then the intellectual capital of the Greek world, where the successors of Euclid taught at the Museum and Library ^[2]. He never says so directly, but he addressed several of his mathematical works to Alexandrian scholars, including the astronomer Conon of Samos and the polymath Eratosthenes, and he wrote of Conon with the warmth of a personal friend ^[1]. Whatever the length of his stay, he returned to Syracuse and spent the rest of his life there.

Back in Syracuse, Archimedes built his reputation through a stream of mathematical treatises circulated among the scholars of Alexandria. His habit was to send out theorems without proofs first, letting others struggle with them, then follow with the demonstrations. In the preface to On Spirals he admitted slipping two false propositions into an earlier batch to catch rivals who claimed discoveries they had not actually made ^[1].

His fame among his fellow citizens rested on more tangible work. Hiero II turned to him for engineering problems, and the results entered legend. The best known story, told by the Roman architect Vitruvius, concerns a golden crown the king suspected had been adulterated with silver. Asked to test it without damaging it, Archimedes reportedly noticed while stepping into a bath that the water level rose in proportion to the volume of his body, giving him a way to measure the crown's density. Vitruvius says he ran home naked shouting "Eureka," Greek for "I have found it" ^[4]. The anecdote is likely embellished, and the method described is cruder than the hydrostatic principles Archimedes actually worked out in his treatise On Floating Bodies, but it fixed his name in popular memory ^[2].

Another celebrated feat involved moving a fully loaded ship, possibly the giant vessel Syracusia built for Hiero, using a compound pulley system that let one man do the work of many ^[3]. Plutarch connects the demonstration to Archimedes' boast that given a place to stand, he could move the Earth, a remark that became one of the most quoted lines of ancient science ^[3]. Whether or not the scene happened exactly as told, the mathematics behind it was real: Archimedes had worked out the law of the lever with full geometric rigor in On the Equilibrium of Planes ^[1].

Any honest account of Archimedes achievements must begin with pure mathematics, which he considered his true calling. In Measurement of a Circle he trapped the value of pi between 3 10/71 and 3 1/7 by inscribing and circumscribing regular polygons of 96 sides around a circle, a method of successive approximation that dominated the calculation of pi until early modern times ^[1]. In Quadrature of the Parabola he proved that the area of a parabolic segment equals four thirds the area of a certain inscribed triangle, using a summation of an infinite geometric series ^[2].

His own favorite result came in On the Sphere and Cylinder, where he showed that a sphere has exactly two thirds the volume and surface area of the cylinder that encloses it ^[1]. He asked that a diagram of the sphere and cylinder be carved on his tomb, and the Roman statesman Cicero later wrote that he found the neglected grave near Syracuse in 75 BCE by searching for that very figure ^[5]. Archimedes' method of exhaustion, computing areas and volumes by squeezing them between sums of ever smaller known shapes, anticipated the integral calculus developed by Newton and Leibniz nineteen centuries later ^[2].

In physics he founded two disciplines. On the Equilibrium of Planes derived the law of the lever from simple postulates and introduced the rigorous treatment of centers of gravity ^[1]. On Floating Bodies established what is still called Archimedes' principle: a body immersed in fluid experiences an upward force equal to the weight of the fluid it displaces ^[2]. The second book of that treatise analyzed the stability of floating paraboloids, mathematics with no rival until the age of modern naval architecture ^[1].

A further window into his thinking opened in 1906, when the Danish philologist Johan Ludvig Heiberg identified a palimpsest in Constantinople containing The Method of Mechanical Theorems, a work lost since antiquity. In it Archimedes explained how he used imagined mechanical balancing, weighing geometric figures against each other on a conceptual lever, to discover results he then proved formally ^[6]. The manuscript, now known as the Archimedes Palimpsest, was studied again with modern imaging after its sale at auction in 1998 for two million dollars ^[6].

Almost nothing certain survives about Archimedes' private circumstances. No ancient source records a wife or children, and his own writings reveal only his father's name and his friendships with mathematicians in Alexandria ^[1]. When Conon of Samos died, Archimedes wrote of his grief in the preface to On Spirals, praising his friend's extraordinary gifts ^[1].

The portrait that comes down through Plutarch is of a man consumed by abstraction: forgetting food, neglecting his person, drawing figures in the ashes of the fire and even, after bathing, in the oil anointing his skin ^[3]. Plutarch also claims Archimedes thought so little of his mechanical inventions that he left no written account of them, regarding engineering as ignoble next to pure geometry ^[3]. Modern historians treat that claim cautiously, since Plutarch had his own philosophical reasons for elevating theory over practice, and Archimedes did write at least one lost work on sphere making, describing a mechanical model of the heavens ^[2].

That planetarium was real enough for Cicero to describe it. The Roman general Marcellus took it from Syracuse as his single piece of plunder, and Cicero reports seeing a device attributed to Archimedes that reproduced the motions of the Sun, Moon, and planets ^[5]. The sophistication of the later Antikythera mechanism, recovered from a shipwreck in 1901, has convinced many scholars that such geared astronomical machines genuinely existed in the Greek world ^[6].

The last years of Archimedes' life were dominated by war. After Hiero II died in 215 BCE, Syracuse abandoned its long alliance with Rome and sided with Carthage in the Second Punic War. Rome answered by sending an army and fleet under Marcus Claudius Marcellus, which laid siege to the city in 213 BCE ^[7].

According to the historian Polybius, whose account was written within living memory of the events, the defenses Archimedes had prepared turned the siege into a humiliation for Rome ^[7]. Stone throwing engines of graded ranges battered ships as they approached; beams swung out over the walls to drop weights on vessels below; and clawed devices, remembered as the iron hand or claw of Archimedes, seized the prows of ships, hauled them upward, and capsized them ^[7]. Plutarch says the Romans grew so unnerved that they fled at the sight of any rope or beam projecting above the wall ^[3]. Later legend added a battery of mirrors that focused sunlight to set ships ablaze, but no contemporary source mentions it, and modern reconstructions have found the feat impractical as a weapon ^[2].

Unable to storm the walls, Marcellus settled into a blockade. In 212 BCE, after roughly two years, Roman troops broke into the city during a festival when the watch had relaxed ^[7]. In the sack that followed, Archimedes was killed. The traditional story, given in several versions by Plutarch, holds that a Roman soldier came upon the old mathematician bent over a diagram; Archimedes asked him to wait until the problem was finished, and the soldier ran him through ^[3]. Marcellus, who had given orders to spare him, was said to be distressed by the death and to have honored the family ^[3].

The influence of Archimedes reached its full force only long after his death. His treatises survived antiquity by a narrow thread, copied in Byzantium and translated into Arabic in ninth century Baghdad, where mathematicians such as Thabit ibn Qurra extended his methods ^[2]. Latin translations by William of Moerbeke in 1269 and printed editions in the sixteenth century put his work into the hands of Renaissance scientists ^[2]. Galileo called him superhuman and built directly on his hydrostatics and his analysis of centers of gravity, while the mathematics of Kepler, Cavalieri, and ultimately Newton grew out of problems Archimedes had opened ^[8].

Among plain Archimedes facts, the breadth of his named legacy is striking. Archimedes' principle remains the foundation of buoyancy calculations in ship design and submarine engineering. The Archimedes screw, a helical pump he is credited with introducing (possibly after seeing Egyptian devices), still lifts water and grain and has been adapted in reverse as a turbine for small hydroelectric plants ^[2]. A lunar crater, a mountain range on the Moon, and the Fields Medal, the highest honor in mathematics, all carry his image or name: the medal bears his portrait and a Latin motto urging mathematicians to rise above themselves ^[8].

Historians of science routinely rank him with Newton and Gauss as one of the three greatest mathematicians who ever lived ^[2]. What separates any serious Archimedes biography from the legends of bathtubs and burning mirrors is the surviving mathematics itself: rigorous, audacious, and so far ahead of its era that when the Method resurfaced in 1906, working mathematicians could still learn from a mind that had been silent for more than two thousand years ^[6].