Coming of Age in the Milky Way - Timothy Ferris (2003)
Part I. SPACE
Chapter 2. RAISING (AND LOWERING) THE ROOF
Aristarchus of Samos supposed that the heavens remained immobile and that the earth moved through an oblique circle, at the same time turning about its own axis.
Now see that mind that searched and made
All Nature’s hidden secrets clear
Lie prostrate prisoner of night.
The earth-centered universes of Eudoxus, Aristotle, Callippus, and Ptolemy were small by today’s standards. Ptolemy’s appears to have been the most generous. Certainly he thought it grand, and he liked to remark, with an astronomer’s fondness for wielding big numbers, that in his universe the earth was but “a point” relative to the heavens. And, indeed, it was enormous by the standards of a day when celestial objects were assumed to be small and to lie close at hand; Heraclitus and Lucretius thought the sun was about the size of a shield, and Anaxagoras the atomist was banished for impiety when he suggested that the sun might be larger than the Peloponnesus. Nevertheless, the Ptolemaic universe is estimated to have measured only some fifty million miles in radius, meaning that it could easily fit inside what we now know to be the dimensions the earth’s orbit around the sun.
The diminutive scale of these early models of the cosmos resulted from the assumption that the earth sits, immobile, in the center of the universe. If the earth does not move, then the stars do: The starry sphere must rotate on its axis once a day in order to bring the stars trooping overhead on schedule, and the larger the sphere, the faster it must rotate. Were such a cosmos very large, the speed mandated for the celestial sphere would become unreasonably high. The stars of Ptolemy’s universe already were obliged to hustle along at better than ten million miles per hour, and were the celestial sphere imagined to be a hundred times larger it would have to be turning faster than the velocity of light. One did not have to be an Einstein, or even to know the velocity of light, to intuit that that was too fast—a point that had begun to worry cosmologists by the sixteenth century. All geocentric, immobile-earth cosmologies tended to inhibit appreciation of the true dimensions of space.
To set the earth in motion would be to expand the universe, a step that seemed both radical and counterintuitive. The earth does not feel as if it is spinning, nor does the observational evidence suggest any such thing: Were the earth turning on its axis, Athens and all its citizens would be hurtling eastward at a thousand miles per hour. If so, the Greeks reasoned, gale-force easterlies ought constantly to sweep the world, and broad jumpers in the Olympics would land in the stands well to the west of their jumping-off points. As no such effects are observed, most of the Greeks concluded that the earth does not move.
The problem was that the Greeks had only half the concept of inertia. They understood that objects at rest tend to remain at rest—a context we retain today when we speak of an “inert object” and mean that it is immobile—but they did not realize that objects in motion, including broad jumpers and the earth’s atmosphere, tend to remain in motion. This more complete conception of inertia would not be achieved until the days of Galileo and Newton. (Even with amendments by Einstein and intimations of others by the developing superunified theories, plenty of mystery remains in the idea of inertia today.) Its absence was a liability for the ancient Greeks, but it was not the same thing as the religious prejudice to which many schoolbooks still ascribe the motives of rational and irrational geocentrists alike.
If one goes further and imagines that the earth not only spins on its axis but orbits the sun, then one’s estimation of the dimensions of the cosmos must be enlarged even more. The reason for this is that if the earth orbits the sun, then it must alternately approach and withdraw from one side of the sphere of stars—just as, say, a child riding a merry-go-round first approaches and then recedes from the gold ring. If the stellar sphere were small, the differing distance would show up as an annual change in the apparent brightness of stars along the zodiac; in summer, for instance, when the earth is on the side of its orbit closer to the star Spica, its proximity would make Spica look brighter than it does in winter, when the earth is on the far side of its orbit. As no such phenomenon is observed, the stars must be very far away, if indeed the earth orbits the sun.
The astonishing thing, then, given their limited understanding of physics and astronomy, is not that the Greeks thought of the universe in geocentric terms, but that they did not all think of it that way. The great exception was Aristarchus, whose heliocentric cosmology predated that of Copernicus by some seventeen hundred years.
Aristarchus came from Samos, a wooded island near the coast of Asia Minor where Pythagoras, three centuries earlier, had first proclaimed that all is number. A student of Strato of Lampsacus, the head of the Peripatetic school founded by Aristotle, Aristarchus was a skilled geometer who had a taste for the third dimension, and he drew, in his mind’s eye, vast geometrical figures that stretched not only across the sky but out into the depths of space as well. While still a young man he published a book suggesting that the sun was nineteen times the size and distance of the moon; his conclusions were quantitatively erroneous (the sun actually is four hundred times larger and farther away than the moon) but his methods were sound.
It may have been this work that first led Aristarchus to contemplate a sun-centered cosmos: Having concluded that the sun was larger than the earth, he would have found that for a giant sun to orbit a smaller earth was intuitively as absurd as to imagine that a hammer thrower could swing a hammer a hundred times his own weight. The evolution of Aristarchus’ theory cannot be verified, however, for his book proposing the heliocentric theory has been lost. We know of it from a paper written in about 212 B.C. by Archimedes the geometer.
In a small, heliocentric universe, the earth would be much closer to a summer star like Spica in summer than in winter, making Spica’s brightness vary annually. As there is no observable annual variation in the brightness of such stars, Aristarchus concluded that the stars are extremely distant from the earth.
Archimedes’ paper was titled “The Sand Reckoner,” and its purpose was to demonstrate that a system of mathematical notation he had developed was effective in dealing with large numbers. To make the demonstration vivid, Archimedes wanted to show that he could calculate even such a huge figure as the number of grains of sand it would take to fill the universe. The paper, addressed to his friend and kinsman King Gelon II of Syracuse, was intended as but a royal entertainment or a piece of popular science writing. What makes it vitally important today is that Archimedes, wanting to make the numbers as large as possible, based his calculations on the dimensions of the most colossal universe he had ever heard of—the universe according to the novel theory of Aristarchus of Samos.
Archimedes, a man of strong opinions, had a distaste for loose talk of “infinity,” and he begins “The Sand Reckoner” by assuring King Gelon that the number of grains of sand on the beaches of the world, though very large, is not infinite, but can, instead, be both estimated and expressed:
I will try to show you, by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me … some exceed not only the number of the mass of sand equal in magnitude to the earth filled up in the way described, but also that of a mass equal in magnitude to the universe.1
Continuing in this vein, Archimedes adds that he will calculate how many grains of sand would be required to fill, not the relatively cramped universe envisioned in the traditional cosmologies, but the much larger universe depicted in the new theory of Aristarchus:
Aristarchus of Samos brought out a book consisting of certain hypotheses, in which it appears, as a consequence of the assumptions made, that the universe is many times greater [in size] than that now so called. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same center as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface.2
Here Archimedes has a problem, for Aristarchus is being hyperbolic when he says that the size of the universe is as much larger than the orbit of the sun as is the circumference of a sphere to its center. “It is easy to see,” Archimedes notes, “that this is impossible; for, since the center of the sphere has no magnitude, we cannot conceive it to bear any ratio whatever to the surface of the sphere.”3 To plug hard numbers into Aristarchus’ model, Archimedes therefore takes Aristarchus to mean that the ratio of the size of the earth to the size of the universe is comparable to that of the orbit of the earth compared to the sphere of stars. Now he can calculate. Incorporating contemporary estimates of astronomical distances, Archimedes derives a distance to the sphere of stars of, in modern terminology, about six trillion miles, or one light-year.*
This was a stupendous result for its day—a heliocentric universe with a radius more than a hundred thousand times larger than that of the Ptolemaic model, proposed four centuries before Ptolemy was born! Although we know today that one light-year is but a quarter of the distance to the nearest star, and less than one ten-billionth of the radius of the observable universe, Aristarchus’ model nonetheless represented a tremendous increase in the scale that the human mind had yet assigned to the cosmos. Had the world listened, we today would speak of an Aristarchian rather than a Copernican revolution in science, and cosmology might have been spared a millennium of delusion. Instead, the work of Aristarchus was all but forgotten; Seleucus the Babylonian championed the Aristarchian system a century later, but appears to have been lonely in his enthusiasm for it. Then came the paper triumph of Ptolemy’s shrunken, geocentric universe, and the world stood still.
Writing “The Sand Reckoner” was one of the last acts of Archimedes’ life. He was living in his native Syracuse on the southeast coast of Sicily, a center of Greek civilization, and the city was besieged by the Roman general Marcus Claudius Marcellus. Though his last name means “martial” and he was nicknamed the Sword of Rome, Marcellus for all his mettle was getting nowhere in Syracuse. Credit for holding his army at bay went to the frightening machines of war that Archimedes had designed. Roman ships approaching the city walls were seized in the jaws of giant Archimedean cranes, raised high into the air while the terrified marines aboard clung to the rails, then dashed on the rocks below. Troops attacking on foot were crushed by boulders rained down on them by Archimedean catapults. As Plutarch recounts the siege, the Romans soon were so chagrined that “if they did but see a little rope or a piece of wood from the wall, instantly crying out, that there it was again, Archimedes was about to let fly some engine at them, they turned their backs and fled.”4
“Who,” Marcellus asked in his frustration as the siege wore on, “is this Archimedes?”
A good question. The world remembers him as the man who ran naked through the city streets shouting “Eureka” after having realized, while lowering himself into a bath, that he could measure the specific gravity of a gold crown (a gift to King Hieron, one that he suspected of being adulterated) by submerging it and weighing the amount of water it displaced. Remembered, too, is his invention of the Archimedes’ screw, still widely used to pump water today, and his fascination with levers and pulleys. “Give me a place to stand,” he is said to have boasted to King Hieron, “and I shall move the earth.”5 The king requested a demonstration on a smaller scale. Archimedes commandeered a ship loaded with freight and passengers—one that normally would have required a gang of strong men to warp from the dock—and pulled the ship unassisted, employing a multiple pulley of his own design. The king, impressed, commissioned Archimedes to build the engines of war that were to hold off the Romans.
Plutarch writes that although he was famous for his technological skills, Archimedes disdained “as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit,” preferring to concentrate upon pure mathematics. His passion for geometry, Plutarch adds,
made him forget his food and neglect his person, to that degree that when he was occasionally carried by absolute violence to bathe, or have his body anointed, he used to trace geometrical figures in the ashes of the fire, and diagrams in the oil on his body, being in a state of entire preoccupation, and, in the truest sense, divine possession with his love and delight in science.6
Archimedes determined the value of pi to three decimal places, proved that the area of the surface of a sphere equals four times that of a circle of the same size (the rule of 4πr2), and discovered that if a sphere is circumscribed within a cylinder, the ratio of their volumes and surfaces is 3:2. (He was so proud of this last feat that he asked friends to have a sphere within a cylinder inscribed on his tombstone. Cicero, quaestor of Sicily in 75 B.C., located and restored the tomb; it has since vanished.)
Marcellus’ invasion came while the Syracusans were celebrating the feast of Diana, traditionally an excuse for heavy drinking. Marcellus had ordered that no free citizens be injured, but his men had seen many of their compatriots killed by Archimedes’ war machines, and they were not in a conciliatory mood. As the story is told, Archimedes was absorbed in calculations when a Roman soldier approached and addressed him in an imperative tone. Archimedes was seventy-five years old and no fighter, but he was also one of the freest men who ever lived, and unaccustomed to taking orders. Drawing geometrical diagrams in the sand, Archimedes waved the soldier aside, or told him to go away, or otherwise dismissed him, and the angry man cut him down. Marcellus damned the soldier as a murderer, writes Plutarch, adding that “nothing afflicted Marcellus so much as the death of Archimedes.”7
Greek science was mortal, too. By the time of Archimedes’ death the world center of intellectual life already had shifted from Athens to Alexandria, the city Alexander the Great had established a century earlier with the charter—inspired, I suppose, by his boyhood tutor Aristotle—that it be a capital of learning modeled on the Greek ideal. Here Ptolemy I, the Macedonian general and biographer of Alexander, established with the wealth of empire a vast library and a museum where scientists and scholars could carry on their studies, their salaries paid by the state. It was in Alexandria that Euclid composed his Elements of geometry, that Ptolemy constructed his eccentric universe, and that Eratosthenes measured the circumference of the earth and the distance of the sun to within a few percent of the correct values. Archimedes himself had studied at Alexandria, and had often ordered books from the library there to be sent to Syracuse. But the tree of science grew poorly in Alexandrian soil, and within a century or two had hardened into the dead wood of pedantry. Scholars continued to study and annotate the great books of the past, and roomfuls of copiers laboriously duplicated them, and historians owe a great debt to the anonymous clerks of the library of Alexandria, but they were the pallbearers of science and not its torchbearers.
The Romans completed their conquest of the known world on the day in 30 B.C. that Cleopatra, last of the Ptolemies, bared her breast to the asp. Theirs was a nonscientific culture. Rome revered authority; science heeds no authority but that of nature. Rome excelled in the practice of law; science values novelty over precedent. Rome was practical, and respected technology, but science at the cutting edge is as impractical as painting and poetry, and is exemplified more by Archimedes’ theorems than by his catapults. Roman surveyors did not need to know the size of the sun in order to tell time by consulting a sundial; nor did the pilots of Roman galleys concern themselves overmuch with the distance of the moon, so long as it lit their way across the benighted Mediterranean. Ceramic stars ornamented the ceilings of the elegant dining rooms of Rome; to ask what the real stars were made of would have been as indelicate as asking one’s host how the roast pig on the table had been slaughtered. When a student Euclid was tutoring wondered aloud what might be the use of geometry, Euclid told his slave, “Give him a coin, since he must gain from what he learns.”8 This story was not popular in Rome.
Roman rule engendered among those it oppressed a growing scorn for material wealth, a heightened regard for ethical values, and a willingness to imagine that their earthly sufferings were but a preparation for a better life to come. The conflict between this essentially spiritual, otherworldly outlook and the stolid practicality of Rome crystallized in the interrogation that Pontius Pilate, a prefect known for his ruthlessness and legal acumen, conducted of the obscure Jewish prophet Jesus of Nazareth.
The world knows the story. Pilate asked Jesus, “Are you the king of the Jews?”
“My kingdom is not of this world,” Jesus replied.
“Are you a king, then?”
“You say I am a king,” Jesus replied. “To this end was I born, and for this cause I came into the world, that I should bear witness to the truth. Everyone that is of the truth hears my voice.”
“What is truth?” asked Pilate.9
Jesus said nothing, and was led off to execution, and his few followers dropped from sight. Yet within two centuries his eloquent silence had swallowed up the words of the law, and Christianity had become the state religion of Rome.
Science, however, fared no better in Christian than in pagan Rome. Christianity, in its emphasis upon asceticism, spirituality, and contemplation of the afterlife, was inherently uninterested in the study of material things. What difference did it make whether the world was round or flat, if the world was corrupt and doomed? As Saint Ambrose put it in the fourth century, “To discuss the nature and position of the earth does not help us in our hope of the life to come.” Wrote Tertullian the Christian convert, “For us, curiosity is no longer necessary.”
To the Christians, the fall of Rome illustrated the futility of putting one’s trust in the here and now. “Time was when the world held us fast to it by its delight,” declaimed Pope Gregory the Great, seated on a marble chair amid the flickering candles of the chapel of the Catacomb of St. Domitilla in Rome at the close of the sixth century (by which time the city had been sacked five times). “Now ’tis full of such monstrous blows for us, that of itself it sends us home to God at last. The fall of the show points out to us that it was but a passing show,” he said, advising the somber celebrants to “let your heart’s affections wing their way to eternity, that so despising the attainments of this earth’s high places, you may come unto the goal of glory which ye shall hold by faith through Jesus Christ, our Lord.”10
Christian zealots are alleged to have burned the pagan books in the library of Alexandria, and Muslims to have burned the Christian books, but the historical record of this great crime is subject to dispute on both counts; in any event, the books went up in smoke. The old institutions of learning and philosophy, most of them already in decline, collapsed under the rising winds of change. Plato’s Academy was closed by Justinian in A.D. 529; the Sarapeum of Alexandria, a center of learning, was razed to the ground by Christian activists in A.D. 391; and in 415 the geometer Hypatia, daughter of the last known associate of the museum of Alexandria, was murdered by a Christian mob. (“They stripped her stark naked,” an eyewitness reported. “They raze[d] the skin and ren[t] the flesh of her body with sharp shell, until the breath departed out of her body; they quartered] her body; they [brought] her quarters unto a place called Cinaron and burn[ed] them to ashes.”11)
Scholars fled from Alexandria and Rome and headed for Byzantium—followed closely by the Roman emperor himself, after whom the city was renamed Constantinople—and the pursuit of science devolved to the province of Islam. Encouraged by the Koran to practice taffakur, the study of nature, and taskheer, the mastery of nature through technology, Islamic scholars studied and elaborated upon classics of Greek science and philosophy forgotten in the West. Evidence of their astronomical research is written in the names of stars—names like Aldebaran, from Al Dabaran, “the follower;” Rigel, from Rijl Jauzah al Yusra, “the left leg of the Jauzah;” and Deneb, from Al Dhanab al Dajajah, “the hen’s tail.”
But the Arabs were enchanted by Ptolemy, and envisioned no grander cosmos. Aristarchus’ treatise on astronomical distances was translated in the early tenth century by a Syrian-Greek scholar named Questa ibn Luqa, and an Arabic secret society known as the Brethren of Purity published an Aristarchian table of wildly inaccurate but robustly expansive planetary distances, but otherwise little attention was paid to the concept of a vast universe. The generally accepted authority on the scale of what we today call the solar system was al-Farghani, a ninth-century astronomer who, by assuming that the Ptolemaic epicycles fit as tightly as ball bearings between the planetary spheres—“there is no void between the heavens,” he asserted—estimated that Saturn, the outermost known planet, was eighty million miles away.12 Its true distance is more than ten times that.
The Islamic devotees of Ptolemy, however, inadvertently undermined the very cosmology they cherished, by transmuting Ptolemaic abstractions into real, concrete celestial spheres and epicycles. So complex and unnatural a system, palatable if regarded as purely symbolic, became hard to swallow when represented as a genuine mechanism that was actually out there moving the planets around. The thirteenth-century monarch King Alfonso (“the Learned”) of Castile is said to have remarked, upon being briefed on the Ptolemaic model, that if this was really how God had built the universe, he might have given Him some better advice.
But that was many long, dark centuries later. The last classical scholar in the West was Ancius Boethius, who enjoyed power and prestige in the court of the Gothic emperor Theodoric at Ravenna until he backed the losing side in a power struggle and was jailed. In prison he wrote The Consolation of Philosophy, a portrait of the life of the mind illuminated by the fading rays of a setting sun. There, Boethius contrasts the constancy of the stars with the unpredictability of human fortune:
Creator of the starry heavens,
Lord on thy everlasting throne,
Thy power turns the moving sky
And makes the stars obey fixed laws
All things thou holdest in strict bounds,—
To human acts alone denied
Thy fit control as Lord of all.
Why else does slippery Fortune change
So much, and punishment more fit
For crime oppress the innocent?13
In words the Greek Stoics would have appreciated, the muse of philosophy upbraids Boethius for his self-pity. “You are wrong if you think Fortune has changed towards you,” she tells him. “Change is her normal behavior, her true nature. In the very act of changing she has preserved her own particular kind of constancy towards you.”14
In Boethius, the universe of Ptolemy is reduced to a symbol of resignation to the vicissitudes of fate:
Consider how thin such fame is and how unimportant. It is well known, and you have seen it demonstrated by astronomers, that beside the extent of the heavens, the circumference of the earth has the size of a point; that is to say, compared with the magnitude of the celestial sphere, it may be thought of as having no extent at all. The surface of the world, then, is small enough, and of it, as you have learnt from the geographer Ptolemy, approximately one quarter is inhabited by living beings known to us. If from this quarter you subtract in your mind all that is covered by sea and marshes and the vast area of desert by lack of moisture, then scarcely the smallest of regions is left for men to live in. This is the tiny point within a point, shut in and hedged about, in which you think of spreading your fame and extending your renown.15
Boethius was executed in 524, and with the extinguishing of that last guttering lamp the darkness closed in. The climate during the Dark Ages grew literally colder, as if the sun itself had lost interest in the mundane. The few Western scholars who retained any interest in mathematics wrote haltingly to one another, trying to recall such elementary facts of geometry as the definition of an interior angle of a triangle. The stars came down: Conservative churchmen modeled the universe after the tabernacle of Moses; as the tabernacle was a tent, the sky was demoted from a glorious sphere to its prior status as a low tent roof. The planets, they said, were pushed around by angels; this obviated any need to predict celestial motions by means of geometrical or mechanical models. The proud round earth was hammered flat; likewise the shimmering sun. Behind the sky reposed eternal Heaven, accessible only through death.
*Archimedes concluded that it would take 1063 grains of sand to fill the Aristarchian universe. The American cosmologist Edward Harrison points out that 1063 grains of sand equals 1080 atomic nuclei, which is “Eddington’s number”—the mass of the universe as calculated in the 1930s by the English astrophysicist Arthur Stanley Eddington. So Archimedes, in underestimating the size of the universe but imagining it to have a matter density much higher than it does, arrived at a total amount of cosmic matter that wasn’t far from Eddington’s twentieth-century estimate.