The Cave and the Light

by Arthur Herman

  The other Platonic figure was Eratosthenes the geographer, a sometime Academy student who came to Alexandria and became director of the Great Library around 267. His nickname in the ancient world was Pentathlos, the all-around scholar-athlete. Indeed, his writings show an impressively wide range of interests, from geometry, astronomy, and mathematics (including a work on the philosophy of mathematics that he called, strikingly, the Platonicus) to history, metaphysics, and poetry. Using his observations of the different shadows cast by sundials along the same meridian and a little number crunching, Eratosthenes made a calculation of the earth’s diameter that was amazingly accurate: 7,850 miles, only about 60 off the actual mark.

  It was in the study of geography, however, that Eratosthenes made his most amazing discovery. While working at the Great Library, he grew fascinated by a manuscript left by an intrepid mariner named Pytheas from the Greek colony of Massalia (today’s Marseilles). Beginning around 320 BCE, Pytheas had made several voyages at the far end of the western Mediterranean, including beyond the famed Pillars of Hercules, or Strait of Gibraltar. Pytheas also sailed around the coast of Spain and made at least one trip across the English Channel, including circumnavigating the British Isles. In addition, he gathered what information he could about lands lying still farther west.

  Pytheas had published his extraordinary voyages as the History of the Ocean (now lost). His account fit none of the accepted conceptions about the shape of the world, and Aristotelian scholars in particular branded him a liar. Eratosthenes, however, instantly saw its value. He already understood that Alexander’s conquests had changed the Greeks’ traditional map of India and Asia (Eratosthenes’s own map was the first to divide the globe into meridians of longitude and parallels of latitude).17 Assumptions about what new inhabited lands might exist, and where, had to change as well. Perhaps Plato’s account in the Timaeus of a great continent lying to the west, Atlantis, encouraged the Academy-trained geographer to keep an open mind.

  In any event, Eratosthenes took Pytheas’s book and did some quick calculations based on his own estimate of the earth’s diameter. He concluded that if the Indian Ocean was not a landlocked sea, as most Greeks supposed, but opened up onto a still larger ocean extending to the shores of the Pillars of Hercules, as Pytheas’s voyage indicated, then it might be possible for a sailor to sail west from Spain to India, although Eratosthenes calculated it would take at least thirteen thousand miles.§ Furthermore, he speculated, perhaps there was even another “inhabited world” (oikoumenē) to be found between Spain and India, one that covered at least part of the western hemisphere of the globe—a hemisphere that mathematics proved had to exist since the earth was round, but which was still entirely unknown.18

  His Aristotelian colleagues scoffed. How could one predicate the existence of something no one had ever seen, especially an inhabited landmass; and when everyone knew the Indian Ocean ended at the western shores of India? So Eratosthenes’s stunning thesis of a possible New World located between Europe and Asia never caught on, even after the Romans discovered there was indeed an ocean on the far side of India. His idea of a western continent faded from the science books. It would take Columbus’s accidental discovery in 1492 to finally prove that the Aristotelians at Alexandria had been wrong and Eratosthenes right all along.

  In some ways, this was not surprising. For all their formidable brainpower, the Alexandrians were content to work inside the box, as we would say, instead of trying to think outside it. Like good Aristotelians, they were content to be specialists in the true modern sense: more concerned with uncovering the how, whether it was in geography or astronomy or medicine, than pondering the why. They fit perfectly the character of the modern scientist as described by philosopher Thomas Kuhn: expert puzzle solvers, for whom the challenge of the puzzle, not a thirst for breakthough discoveries, is the name of the game.19

  A passion for solving puzzles certainly describes Alexandria’s most famous scientist, who probably came to the city around 265 BCE. He almost certainly studied mathematics with a leading figure there, Conon of Samos, and he may have created his first important invention in Alexandria as well.

  He is the founder of Western technology as an intellectual discipline—one might even say as a passion.

  He was Archimedes of Syracuse.

  As with most ancient scientists, details on his life are scanty. It’s generally agreed that Archimedes was born in Syracuse in 287, the same city where Plato had launched his failed utopia a century before. He was the son of an astronomer named Pheidias and related by blood (so it is alleged) to the city’s ruler Hiero II.20 At some point, it is clear that Pheidias’s son went to Alexandria to study with the distinguished array of scientists and scholars at the Museum.‖

  It must have been a dazzling, exciting experience. When the young Archimedes arrived, he would have found eminent mathematicians like Aristarchus and Eratosthenes and his future teacher Conon of Samos meeting under Alexandria’s covered walks to escape the hot Egyptian sun or taking their ease in the Museum’s arcade with its stone benches and recessed seating along the walls. Nearby was the Great Library, with its daunting collection of scientific and literary works, where its director, Callimachus of Cyrene, had recently finished his largest and most Aristotelian project, the Collection of the Wonders of the World, a veritable encyclopedia covering topics from geography and history to the miraculous properties of plants and minerals.21

  The spirit of Aristotle was alive in another way, which seized Archimedes’s attention from the start. Aristotle’s praise for technology, or technē, knowledge for a practical purpose rather than just theoretic understanding, had found a congenial audience in Ptolemaic Alexandria. Researchers there were working on a host of practical technologies. The most important was the science of ballistics, indispensable for siege warfare in the age of the warring states of Hellenistic Greece.

  The key figure was yet another former Strato student named Ctesibius, who was devising a series of improved catapults, including one that worked on the principle of compressed air. We don’t know how practical Ctesibius’s machines were; in a historical sense it doesn’t matter. The point is that in Archimedes’s day, Alexandria was becoming the center of Greek military technology research, and technicians like Ctesibius were as renowned as the city’s great theoretical mathematicians.22

  The result was a conceptual revolution—not the last time a military industrial complex has served as the spur to intellectual and technological growth. Plato had seen it coming in his own lifetime. According to the historian Plutarch, he had violently opposed it, objecting that the best geometers were giving up pure philosophy in order to work on material projects and “instruments which require much base and manual labor.”23 Aristotle’s own reservation about purely technological knowledge was that it didn’t encourage the discovery of any new principles of nature but only applied what was already known.

  His heirs in Alexandria were now proving him wrong. By constantly devising new formulae for calculating the angle of falling catapult shot, including the shot’s speed and direction, the city’s geometers were reinventing the theory of mechanics. Thanks to men like Ctesibius, the use of mechanical demonstrations to establish new scientific principles had come to stay. And one of those caught up in the new science of applied mathematics and geometry was Archimedes, who watched and learned and picked up principles and tricks he would use to create his own set of high-tech weapons for the most famous siege in the history of technology: the siege of Syracuse in 214 BCE.

  The other area of technical study in Alexandria that drew Archimedes’s attention was hydraulic engineering. This was essential for keeping the wheat and cotton fields of the Nile valley irrigated and keeping the city of Alexandria clothed and fed. The reigning figure, again, was Ctesibius, who wrote an important treatise on hydraulics and created a working water clock.24 It was probably in Alexandria that Archimedes came up with the invention that bore his name throughout the ancient world: th
e Archimedes water screw. Called “the snail” because of its shape, the screw enabled farmers to move water over long distances and even uphill by means of its continuous twisting motion. The ancient historian Diodorus asserts that Archimedes’s invention enabled the Ptolemies to irrigate most of the Nile Delta.25 It has remained in use in the poorer parts of rural Egypt until today.

  When Archimedes returned to Syracuse, his interest in things aquatic did not stop. He continued his research in the principles of hydraulics and another field he largely invented, hydrostatics, or the study of fluids at rest. This is the origin of the most famous story about Archimedes. It tells how the king of Syracuse, Hiero II, once asked him to figure out the gold content of a certain sacrificial wreath or crown. Archimedes was pondering the problem as he eased himself into his morning bath, noticed the water spilling over the sides of the bath, and realized that by weighing the volume of water displaced by an object, he could determine how much gold or silver it contained. According to the story, he then leaped out and ran down the street stark naked, shouting: “I have found it! I have found it!” (Eureka! Eureka!)26

  A charming story, which dates back at least to the Roman architect Vitruvius. Most modern scholars doubt it happened, which doesn’t mean it isn’t true. What is far more revealing is that Archimedes, in his own treatise on hydrostatics, never mentions it or tells us anything about how he discovered or researched any of his amazing works.

  This is a striking omission, but typical of the Aristotelian spirit of the ancient Greek scientist. What ultimately fascinates us most about the history of science is its Platonic side—the dramatic process of discovery and intuition, when what is true is at once clear and vice versa; those moments of insight that Socrates and Plato would recognize as seeing with the inner eye. What fascinates us most interested the scientist of the Alexandrian age least.

  Instead, all the emphasis in Greek science is on laying out new mathematical or scientific proofs, including using visual diagrams, so logically and so beyond contradiction that there could be no possibility of dissent. This is precisely what all of Archimedes’s writings do. That is why they are so important, and so boring. What Aristotle had formalized and Euclid codified—the power of clear, logical reasoning and presentation—Archimedes made his intellectual slave and the servant of science.

  And as his research continued in Syracuse, Archimedes made sure word of what he was doing got back to friends in Alexandria. Among his correspondents was a former Croton pupil named Dositheusa to whom Archimedes would send one major treatise after another that would revolutionize mathematics. There was Quadrature of the Parabola, then two books on Sphere and Cylinder, one on Spiral Lines, and finally a treatise on Conoids and Spheroids.27

  By dividing volumes and areas into infinite numbers of smaller pieces, Archimedes worked out the principles of integral calculus nineteen hundred years before it was invented.

  Taken together, they laid the future cornerstone of what comes to be called calculus, or the mathematics of infinity (Archimedes was the first mathematician to use the concept of infinity in his work). Without it, modern math and science as we know it would not exist.

  In one sense, the impetus to Archimedes’s math research was proving that Plato had been wrong: you can’t form a sphere out of a series of triangles or pentagons, as Plato claimed in the Timaeus, any more than you can form a circle out of a square. Archimedes showed this by demonstrating that you can’t measure the area of a circle or sphere or cylinder using any kind of straight line. However, you can measure it by slicing it into pieces that can be bounded by straight lines, then slicing that part left over into measurable pieces again and again, until finally there is nothing left unmeasured.28 This is what calculus, and the concept of infinity, enabled Archimedes to do.

  He discovered, for example, that the area of a sphere is four times the area of its greatest circle; and that the area of a parabola (like the one described by a stone flung from a catapult) is four-thirds the area of the triangle enclosed within it. Archimedes’s work was also establishing in passing another scientific landmark for the future. If you can’t measure something, then it probably doesn’t exist. It’s a notion that Aristotle would have rejected at once. It was, however, another logical outgrowth of his own scientific method.b

  Archimedes also liked to lay traps for the unaware and the uninitiated. He enjoys leading the reader to a conceptual dead end in order to force him to go back and reassess his own original assertions. Archimedes even seems to have sent out fraudulent proofs to mathematical rivals for their approval in order to expose them as fools. In spite of the bath story, he can’t have been a very lovable man. All the same, he fully grasped a basic Aristotelian principle from his years at Alexandria—that knowledge is power—and he was determined to live by it.

  This is why, despite Plutarch’s later disclaimer that Archimedes “regarded the business of engineering as a base and sordid activity,” he remained so fascinated with technology. He invented a working hydraulic musical organ. He also created a working mechanical planetarium, an enclosed globe in which replicas of the sun, moon, and planets performed the same motions relative to the moving sphere of the stars that they do in the sky, so that astonished observers could even watch the successive phases of the moon.29

  He experimented with pulleys, by which he demonstrated his famous leverage principle, “that with any given force it was possible to lift any given weight.” This enabled him to stage one of his most famous coups de théâtre in front of King Hiero and the entire city of Syracuse. Archimedes tied a series of pulleys to a dry-docked three-masted ship loaded with cargo and passengers, and then, to the crowd’s stunned amazement, he lifted it into the harbor by himself. This led Hiero to declare, “From this day forward, Archimedes is believed no matter what he says,” which led Archimedes to reply in the full flush of triumph, “Give me a lever and a place to stand, and I shall move the earth.”30

  As the applause stopped and the crowd went home, no one in Syracuse could have doubted Archimedes’s superhuman genius and skill. Still, even his most devoted admirers could not have imagined that his miraculous machines were about to spell the difference between life and death for their city.

  As a city-state, Syracuse had survived many crises. It had survived tyrants like Dionysius II and Dion; it had survived sieges by the Athenians (led by Socrates’s friend Alcibiades) and the Carthaginians. It had even survived Plato’s disastrous hopes of creating a real-life Republic out of its laws and citizens.

  Shortly after Archimedes’s masterful show in Syracuse harbor, however, the city faced the deadliest threat of all. There was a new regional power in the central Mediterranean basin, the imperial republic to the north: Rome. Breaking Syracuse to its will was the key to Rome’s hegemony over Sicily and the rest of Italy, and in 214 BCE an enormous Roman fleet and army gathered at the entrance to the harbor. Syracuse went to battle stations, and its most famous citizen, now in his seventies, sprang into action.

  The best description of what happened next is by the Greek historian Polybius. He wrote it not long after the siege of Syracuse, when some of the participants may still have been alive. He tells us how the Syracusans put Archimedes in charge of constructing their defenses and how he laid them out so that “they would have everything ready to hand, and could respond to any attack by the enemy with a countermove.”31

  The Romans were hardly amateurs when it came to besieging cities. Their plan was a simultaneous land and sea assault, with boats equipped with massive mobile siege towers rowing up to storm Syracuse’s walls. However, as Polybius relates, “Archimedes had constructed artillery which could cover a whole variety of ranges, so that while the attacking ships were still at a distance he scored so many hits with his catapults and stone-throwers” that the Roman general Marcus Claudius Marcellus had to call off the attack before his men reached the walls.

  On the landward side, the Romans fared no better. Their main objective was the eastern Hexa
ply gate. But every attempt to get scaling ladders and grappling hooks up to the gate’s walls met a hailstorm of fire from the catapults and other war engines Archimedes had devised. Stunned and exhausted, the Roman legionnaires finally had to retreat.32

  The desperate Romans then tried a night attack. Archimedes was waiting for them. As the Roman triremes glided up in the darkness, great wooden beams suddenly swung out from the walls of the harbor and hovered over the water. A rain of catapult stones drove the Roman marines back from the bows of their ships, while grappling irons attached to chain pulleys dropped from the hovering wooden beams. At Archimedes’s command, each engineer in charge of a beam then seized the prow of a ship, like an eagle or hawk seizing its prey.

  When Archimedes’s engineers were sure they had a good grip, they pulled down on the lever controlling the beam’s pulley. With a powerful jerk, each Roman ship was lifted by its prow and made to stand on its stern. As the Romans on board jumped for their lives, the pulleys would suddenly be released.

  “The result was that some of the vessels heeled over and fell on their sides,” Polybius relates, “and others capsized,” while still others hit the water so hard that they began to flood and became useless to steer.33 All the same, a few intrepid Roman ships managed to get close enough to the walls to avoid the catapult barrage, and Roman troops made ready to jump ashore and begin their attack.

  It was a tragic mistake. Archimedes had carved out a series of loopholes along the city walls, and as the Romans scrambled for shore, Syracusans opened up with “scorpions,” or iron dart Gatling guns, another Archimedes invention, which fired one devastating volley after another into the Roman ranks.

  Men screamed as they died under the deadly curtain of fire. The handful who reached the foot of the walls found themselves bombarded by rocks and beams thrown from above. Marcellus, realizing his attack had once again failed, sounded the retreat.