Archimedes

• 1. Introduction

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  Archimedes of Syracuse left almost no trace of his ordinary life. We do not know for certain whether he married, whether he had children, or whether he ever set foot in Alexandria. Yet a Roman writer named Cicero, centuries later, went looking for his grave near the Agrigentine gate in Syracuse and found it overgrown with bushes, neglected and forgotten. On the stone sat a sphere and a cylinder. Archimedes himself had asked for those two shapes, because he counted that single relationship between them as his finest discovery. How does a man whose biography is mostly lost become, by near-universal agreement, the greatest mathematician of antiquity? Why does a parchment of prayers, written over in the 13th century, still matter to scientists today? And what does it mean that he could prove the area of a circle, weigh the sands of the universe, and still die over a diagram he refused to abandon?

• 2. The Method Behind The Proofs

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  In Quadrature of the Parabola, Archimedes leaned on a lemma now called the Archimedean property: the excess by which a greater region exceeds a lesser, added to itself, can exceed any bounded region. Eudoxus of Cnidus and earlier mathematicians had used this idea, the method of exhaustion, to find the volumes of the tetrahedron, cylinder, cone, and sphere. Archimedes took the technique further than anyone before him. In Measurement of a Circle, he showed that a circle's area equals that of a right triangle whose base and height match the circle's circumference and radius. To pin down the value of pi, he drew a regular hexagon outside a circle and another inside it. He then doubled the sides again and again, measuring each polygon as he went. After four doublings, when each polygon had 96 sides, he placed pi between roughly 3.1408 and 3.1429. The true value sits near 3.1416. He also developed a second, stranger approach laid out fully in The Method of Mechanical Theorems. There he used the law of the lever to weigh shapes against one another and find their areas and volumes by physical reasoning. As he put it, it is more feasible to provide a proof when one already has some knowledge of the matter than to investigate it knowing nothing. He confessed that he found his results this way first, then went back and proved them rigorously by exhaustion.

• 3. Weighing The Universe

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  In The Sand Reckoner, also called Psammites, Archimedes set out to count something absurd: the grains of sand needed to fill the entire universe. Greek numbers stalled at the myriad, their word for 10,000. So he built a new system using powers of a myriad of myriads, that is 100 million, and pushed it until it could name truly enormous quantities. His conclusion ran to 8 vigintillion grains. The point was not the sand. He had shown that mathematics could represent arbitrarily large numbers at all. The same treatise is the only surviving work where Archimedes discusses astronomy. It records the heliocentric model of Aristarchus of Samos, which placed the Sun at the center, and it preserves contemporary ideas about the size of the Earth and the distances between celestial bodies. Without trigonometry or any table of chords, Archimedes measured the Sun's apparent diameter using a straight rod fitted with pegs. He applied correction factors and gave his answer as upper and lower bounds to account for observational error. Ptolemy, quoting Hipparchus, later cited Archimedes' solstice observations in the Almagest. That citation makes Archimedes the first known Greek to have recorded multiple solstice dates and times across successive years.

• 4. The Engineer Of Syracuse

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  King Hiero II of Syracuse, the ruler whom Plutarch claimed was a relative of Archimedes, kept the mathematician busy with practical demands. Athenaeus of Naucratis, quoting a writer named Moschion, describes how Hiero commissioned the Syracusia, said to be the largest ship built in classical antiquity, and how Archimedes launched it. Plutarch tells a different version, in which Archimedes boasted he could move any great weight, and Hiero dared him to move a ship. The accounts disagree on how it was done. Plutarch credits a block-and-tackle pulley system, Hero of Alexandria credits the baroulkos windlass, and Pappus of Alexandria credits the principle of leverage, attaching to Archimedes the famous boast: Give me a place to stand on, and I will move the Earth. The screw used to clear water leaking through the Syracusia's hull is often tied to his name. Yet the device likely predates him, and Philo of Byzantium, Strabo, and Vitruvius, his closest sources on it, never credit him with inventing it. The wreath problem is more vivid. Vitruvius, writing about two centuries after Archimedes died, says Hiero suspected a goldsmith of swapping silver for some of the pure gold in a temple wreath. Stepping into a bath, Archimedes noticed the water rose higher the deeper he sank. He saw he could use this to find the wreath's volume, and ran into the streets without dressing, crying Eureka, meaning I have found it. Galileo Galilei, who built a hydrostatic balance in 1586, thought a different account closer to the truth, judging it probable that Archimedes followed the balance method described in the Carmen de Ponderibus.

• 5. Defense And Death

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  In 214 BC, during the Second Punic War, Syracuse switched its loyalty from Rome to Carthage, and the Roman army under Marcus Claudius Marcellus moved to take the city. The greatest fame Archimedes earned in antiquity came from what happened next. According to Plutarch, he had built war machines for Hiero II but never used them in the old king's lifetime. Now he allegedly oversaw them himself. Plutarch, Livy, and Polybius, three separate historians, describe improved catapults and cranes that dropped heavy lead onto Roman ships, or seized them with an iron claw, lifting them from the water before letting them sink. The famous burning mirrors appear in none of those three earliest accounts. The satirist Lucian of Samosata, in the 2nd century, mentions ships set ablaze by artificial means but says nothing of mirrors. Galen, writing later that century, was the first to name them. Nearly four hundred years on, Anthemius tried despite his doubts to reconstruct the supposed reflector. René Descartes later rejected the heat ray as false. The Romans captured Syracuse only after a long siege, and during the sack Archimedes was killed. The oldest account, from Livy, says a soldier who did not recognize him struck him down as he drew figures in the dust. Plutarch reports the soldier ordered Archimedes to come, and Archimedes refused, needing to finish his problem. Valerius Maximus recorded last words close to a plea: I beg of you, do not disturb this. The line now attributed to him, Do not disturb my circles, appears in no ancient source. Marcellus was reportedly furious, for he had called Archimedes a geometrical Briareus and ordered that he not be harmed.

• 6. Machines Of The Heavens

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  Cicero recorded that Marcellus carried two planetariums built by Archimedes back to Rome. These devices showed the motion of the Sun, the Moon, and five planets. Marcellus gave one to the Temple of Virtue in Rome and kept the other as his only personal loot from Syracuse. Pappus of Alexandria mentions a now-lost treatise, On Sphere-Making, which may have described how such mechanisms were constructed. Building them would have demanded a sophisticated grasp of differential gearing. For a long time scholars assumed such gearing lay beyond ancient technology. Then in 1902 came the Antikythera mechanism, another device built around 100 BC for a similar purpose. Its discovery confirmed that the ancient Greeks knew how to build geared instruments of this kind. Some scholars regard Archimedes' planetarium as a precursor to it. There may even be a faint visual echo. The numismatists Filippo Paruta and Leonardo Agostini reported a bronze coin from Sicily bearing Archimedes' portrait on one side and a sphere resting on a base on the other. Ivo Schneider read that image as a rough picture of one of the planetaria, perhaps minted in Rome for the man who had carried two of them home.

• 7. A Palimpsest Reborn

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  In 1906, the Danish professor Johan Ludvig Heiberg traveled to Constantinople to study a 174-page goatskin parchment of prayers written in the 13th century. He had read a short transcription published seven years earlier by Papadopoulos-Kerameus. Heiberg confirmed it was a palimpsest, a document whose older text had been scraped away and written over, a thrifty practice in the Middle Ages when vellum was costly. Beneath the prayers lay 10th-century copies of treatises by Archimedes, several long thought lost. The parchment holds seven of his works. It carries the only surviving Greek copy of On Floating Bodies, the work containing his principle of buoyancy: any body immersed in fluid feels an upthrust equal to the weight of the fluid it displaces. It is the sole source for The Method of Mechanical Theorems, once believed gone forever. It also preserved a fuller version of the Stomachion, the dissection puzzle also called Archimedes' Box. In 2003, Reviel Netz of Stanford argued the puzzle was an early problem in combinatorics, calculating that its 14 pieces form a square in 17,152 ways, or 536 once rotations and reflections are set aside. The parchment sat in a Constantinople monastery library for centuries before passing to a private collector in the 1920s. On the 29th of October 1998, it sold at auction to an anonymous buyer for 2.2 million dollars. At the Walters Art Museum in Baltimore, ultraviolet and X-ray light coaxed out the hidden text before the manuscript returned to its owner. This same recovery feeds a long line of admiration. Galileo called Archimedes superhuman and my master, Christiaan Huygens said he was comparable to no one, and Carl Friedrich Gauss named him among only three epoch-making mathematicians, beside Newton and Eisenstein.