An astronomer charts the story of our quest to measure the distance to a star in this “delightful history of a crucial advance in knowledge” (Booklist).
Since the ancient Greeks, great scientific minds have attempted to determine the distance of stars using the principle of parallax: the apparent shift in an object's position when viewed from different vantage points. Not until the nineteenth century would three astronomers, armed with the best telescopes of the age, race to conquer this astronomical Everest—their contest ending in a virtual dead heat.
Against a sweeping backdrop filled with kidnappings, dramatic rescue, swordplay, madness, and bitter rivalry, Alan Hirshfeld brings to life the heroes of this remarkable story. Meet the destitute boy plucked from a collapsed building who becomes the greatest telescope maker the world has ever seen; the hot-tempered Dane whose nose is lopped off in a duel over mathematics; the merchant's apprentice forced to choose between the lure of money and his passion for astronomy; and the musician who astounds the world by discovering a new planet from his own backyard.
Generously illustrated with diagrams, period engravings, and paintings, Parallax is an unforgettable tale that illuminates the distinctly human side of science.
An astronomer charts the story of our quest to measure the distance to a star in this “delightful history of a crucial advance in knowledge” (Booklist).
Since the ancient Greeks, great scientific minds have attempted to determine the distance of stars using the principle of parallax: the apparent shift in an object's position when viewed from different vantage points. Not until the nineteenth century would three astronomers, armed with the best telescopes of the age, race to conquer this astronomical Everest—their contest ending in a virtual dead heat.
Against a sweeping backdrop filled with kidnappings, dramatic rescue, swordplay, madness, and bitter rivalry, Alan Hirshfeld brings to life the heroes of this remarkable story. Meet the destitute boy plucked from a collapsed building who becomes the greatest telescope maker the world has ever seen; the hot-tempered Dane whose nose is lopped off in a duel over mathematics; the merchant's apprentice forced to choose between the lure of money and his passion for astronomy; and the musician who astounds the world by discovering a new planet from his own backyard.
Generously illustrated with diagrams, period engravings, and paintings, Parallax is an unforgettable tale that illuminates the distinctly human side of science.
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Overview
An astronomer charts the story of our quest to measure the distance to a star in this “delightful history of a crucial advance in knowledge” (Booklist).
Since the ancient Greeks, great scientific minds have attempted to determine the distance of stars using the principle of parallax: the apparent shift in an object's position when viewed from different vantage points. Not until the nineteenth century would three astronomers, armed with the best telescopes of the age, race to conquer this astronomical Everest—their contest ending in a virtual dead heat.
Against a sweeping backdrop filled with kidnappings, dramatic rescue, swordplay, madness, and bitter rivalry, Alan Hirshfeld brings to life the heroes of this remarkable story. Meet the destitute boy plucked from a collapsed building who becomes the greatest telescope maker the world has ever seen; the hot-tempered Dane whose nose is lopped off in a duel over mathematics; the merchant's apprentice forced to choose between the lure of money and his passion for astronomy; and the musician who astounds the world by discovering a new planet from his own backyard.
Generously illustrated with diagrams, period engravings, and paintings, Parallax is an unforgettable tale that illuminates the distinctly human side of science.
Product Details
| ISBN-13: | 9781429931694 |
|---|---|
| Publisher: | Holt, Henry & Company, Inc. |
| Publication date: | 05/01/2024 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 337 |
| File size: | 3 MB |
About the Author
Alan W. Hirshfeld, astronomer at the University of Massachusetts Dartmouth and an Associate of the Harvard College Observatory, received his undergraduate degree in astrophysics from Princeton and his Ph.D. in astronomy from Yale. He is co-author of Sky Catalogue 2000.0, a two-volume astronomical reference book, and a past winner of a Griffith Observatory/Hughes Aircraft Co. national science writing award. He lives outside Boston.
Read an Excerpt
Parallax
Part 1The Sun-centered model of the cosmos. From Copernicus,De Revolutionibus, 1543.Source: Wolbach Library, Harvard University.1Reinventing the CosmosThe sight of stars always sets me dreaming just as naively as those black dots on a map set me dreaming of towns and villages. Why should these points of light in the firmament, I wonder, be less accessible than the dark ones on the map of France?--Vincent van Gogh, The Letters of Vincent van Gogh
We had the sky, up there, all speckled with stars, and we used to lay on our backs and look up at them, and discuss about whether they was made, or only just happened.--Mark Twain, Huckleberry Finn
Free at last! The Earth, wrenched from its central, immobile station in the universe, sailed majestically around the Sun, joining its sister planets and companion Moon in perpetual motion. This according to the Greek philosopher-mathematician Aristarchus of Samos, who proposed the bold rearrangement of the heavens--replacing central Earth with central Sun--more than two thousand years ago. After all, he claimed, the Sun--the "lantern" that illuminates the heavens--more properly belongs on the throne of the universe, radiating its light symmetrically over the family of planets. And the Earth, only a fraction of the Sun's size according to Aristarchus's own calculations, must logically circle the larger body. Once a year, Aristarchus maintained, our planet completes its solar circuit, then retraces the identical course again and again, ad infinitum. The other planets--Mercury, Venus, Mars, Jupiter, and Saturn--likewise move around the Sun; like heavenly fireflies, they reveal their individual paths against the backdrop of the night sky.Imagine yourself in Aristarchus's sandals. The year is 270 B.C. The place is the city of Alexandria at the mouth of the Nile innorthern Egypt. Founded by Alexander the Great in the wake of his campaign of conquest some six decades earlier, Alexandria had grown to become the intellectual and commercial center of the Hellenistic world: a grand galaxy of buildings, monuments, wide ways, and human strivings. Along the boulevardlike Canopic Way, stretching between the Gate of the Sun and the Gate of the Moon, Alexandria's civic vigor manifested itself in spectacular Dionysian processions, one of which "included a hundred-and-eighty-foot golden phallus, two thousand golden-horned bulls, a gold statue of Alexander carried aloft by elephants, and an eighteen-foot statue of Dionysus, wearing a purple cloak and a golden crown of ivy and grapevines."It was here, after the young Macedonian king's death, that his general, Ptolemy I Soter, established the Temple to the Muses--the "Museum"--and its extraordinary Library with as many as 500,000 documents and scrolls. (By comparison, the largest medieval European library, the Sorbonne, had less than two thousand volumes by the fourteenth century A.D.) A later regent, Ptolemy III, was an even more ardent "book" collector. He decreed that all travelers arriving in Alexandria were to relinquish any documents of literary or scientific value; these were then added to the Library's collection. In return, the travelers got cheap papyrus copies of their "donated" works. Ptolemy once paid a hefty deposit to borrow the "state copies" of Aeschylus, Sophocles, and Euripides from the Athenian library, on the premise of transcribing them; the originals never made it back to Athens.Alexandria. Centuries of wisdom gathered in one place, a magnet to the world's most able intellects. In addition to its rich Library, the Alexandrian Museum had research rooms, an observatory, a zoo displaying exotic species, living quarters, and a dining hall where scholars gathered to dine and debate. Here was an ancient think tank devoted to the arts and sciences, a precursor Institute for Advanced Study, whose collective scholarship became its legacy to future generations--and whose eventual decline under Christian authority in the fourth century A.D. and destruction in 642 A.D. at the hands of Islamic invaders marked one of civilization's greatest losses. Into this percolating cauldron of ideas, Aristarchus presented his radical theory of the cosmos.Devising a complete, logical system that explained the movements of the heavenly bodies, the alternation of day and night, the occurrence of eclipses, the uneven lengths of the seasons, the phases of the Moon, and a host of other celestial phenomena wasa challenge to the ancients. Their powers of observation were severely limited; they had their eyes and they had their minds, the latter clouded by preconceptions about how the universe should be. Physicists Albert Einstein and Leopold Infeld described what these early scientists were up against:In our endeavor to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears its ticking, but he has no way of opening the case. If he is ingenious he may form some picture of a mechanism which could be responsible for all things he observes, but he may never be quite sure his picture is the only one which could explain his observations.For the three centuries preceding Aristarchus, virtually every Greek philosopher from Pythagoras to Aristotle had hewn to the belief that the Earth occupied the hub of the universe and that the Sun, Moon, planets, and even the star-studded celestial sphere that was thought to enclose the universe all circled around it. This "geocentric" mindset had gained inexorable momentum, having barged its way into the bedrock beliefs of generations of deep thinkers by the sheer force of its unchallenged longevity. To these proto-scientists of old, the reality of an Earth-centered cosmos was as self-evident as the contrary "heliocentric" scheme apparently was to Aristarchus.The geocentric seed germinated around 600 B.C. in the speculations of Thales of Miletus in Asia Minor. Thales, a merchant who made his fortune in olive oil, traveled widely, investigating natural phenomena and conducting experiments. As Plato tells it, Thales once became so entranced by the sight of the stars that he fell into a well while strolling at night. One of his more impressive feats was a measurement of the height of Egypt's Great Pyramid from the length of its shadow. Herodotus also credits Thales with predicting the solar eclipse of 585 B.C., which occurred during a pitched battle between the Medes and the Lydians near the River Halys. The combatants were so terror-stricken to see day turn into night that they called an immediate truce. The eclipse did take place, but the story of its prediction is questionable; probably no one in 600 B.C. knew how to foretell a solar eclipse.In the arena of cosmology, Thales seems to have taken a cue from his inadvertent dip in the well. He proposed that the universewas constructed around a disk-shaped Earth, which floated serenely in a cosmic ocean. To modern sensibilities, the idea of a water-borne Earth might seem quaint. Yet it marked a turning point in cosmological thought. In previous cosmic models, nothing happened without the intervention of a divine hand. Thales held that his floating Earth formed by natural means, akin to the aggregation of silt that he had observed on the Nile delta. The gods might have initiated the Earth's formation, but once the process was up and running, they benignly looked on.Anaximander, also from Miletus, modified the model proposed by Thales. Anaximander dispensed with the cosmic ocean entirely and suggested that the Earth-disk was free-floating in space, a revolutionary idea at the time. Around 530 B.C., Pythagoras of Samos, whose name adorns the famous theorem relating the sides of a triangle, began to flesh out the crude geocentric model. He proposed that the Earth was not a disk, but rather a stationary globe, surrounded by a series of eight concentric, transparent spheres, on which were affixed the Sun, Moon, planets, and stars. The steady rotations of these spheres led to the observed motions of the heavenly bodies, including their daily rising and setting. (Aristotle, that ancient voice of authority who helped enshrine the geocentric model, lent his support to the spherical-Earth hypothesis in his treatise On the Heavens, written around 350 B.C. His evidence for a round Earth: the curvature of the Earth's shadow during a lunar eclipse, the changing array of constellations as one journeys north or south, and the disappearance of ships sailing over the horizon. Nonetheless, the Flat-Earth Society persists to this day.)Planetary motions proved to be more complex than the constant, predictable movement of the stars. Planets sped up and slowed down as they coursed through the sky, and on occasion even looped backwards. Their brightness varied during the year, implying that their distance from the Earth changed. Neither motion variations nor brightness variations were easy to explain in an Earth-centered universe, where each planet supposedly circled the Earth at a fixed distance. Discrepancies between the observed and predicted positions of planets threatened to undo the relatively simple cosmic plan of Pythagoras and his followers. "Observation ... is the pitiless critic of theory," science historian Agnes Clerke has noted. "[I]t detects weak points, and provokes reforms which may be the beginnings of discovery. Thus, theory and observation mutually act and react, each alternately takingthe lead in the endless race of improvement." To survive in the face of the planetary data, the geocentric system had to become more complex.Around 370 B.C., Eudoxus of Cnidus in Asia Minor, a former student of Plato's and an acquaintance of Aristotle's, raised the heavenly sphere count from eight to twenty-seven. Now each heavenly body had more than one sphere governing its motion. Eudoxus cleverly tilted the spheres at various angles and interlocked their rotations, in order to bring the predictions of his new geocentric model into agreement with the observations. The universe was being transformed into a kind of cosmic machine, a clocklike device that might have sprung from the mind of some divine mathematician.The ancient Greek astronomers did not necessarily believe in the reality of the cosmic spheres. They viewed them as helpful geometrical constructs, which together formed a virtual computer that emulated the true heavens. As such, it was easy, if need arose, to add yet more spheres and otherwise tinker with the workings of the model, all in the name of reaching accord with the observed motions of the planets. Aristotle himself escalated the number of heavenly spheres to an ungainly fifty-five.When Alexander the Great conquered Persia around 330 B.C., the extensive astronomical records of the Babylonians fell into Greek hands. These detailed catalogues of planetary positions and solar and lunar eclipses set more stringent requirements on cosmic tinkerers. As a result, new features were folded into the geocentric model. In one modification, the Earth was shoved slightly off-center while the other planets continued to circle the vacated spot. Another enhancement had each planet move around a small orbit, called the epicycle, which itself circled the Earth in a wider orbit, the deferent. The relative sizes and speeds of all these moving components were adjusted until the model quite effectively simulated what was seen in the sky. The advent of such features eventually reduced the number of cosmic spheres back within reason.Besides the successful use of the geocentric arrangement of the universe in modeling planetary motions, there were powerful "scientific," emotional, and religious reasons to support such an arrangement. Its aesthetic qualities were pleasing, it conformed to Aristotle's universally accepted idea that dense matter accumulates toward the cosmic center (that is, toward the Earth) while the more evanescent material of the heavenly bodies remains aloft,and it presumably reflected the Creator's obvious intention to construct a humanity-centered cosmos.Also, one must not disregard the role human nature played in this tale of two theories. Suppose you had spent years designing and building your house. If it's anything like my house, some of the windows don't quite match, not all of the doors shut tightly, and it has to be patched on occasion. But fundamentally it does what a house is supposed to do; it keeps the rain off your head and the winter chill at bay. Now along comes a fellow who takes one look at your house and tells you it's all wrong, dismantle it, rearrange the pieces, and build it over again! No bigger than before, seemingly no better than before, just--well, different. How would you react? The geocentrists were hearing essentially the same criticism from Aristarchus about the Earth-centered edifice they had labored to create. It's no surprise that during his lifetime Aristarchus made not a single documented convert within the geocentric camp. Nobody was about to question the old model's fundamental rightness. Nobody, it seemed, except Aristarchus.
Aristarchus's newfangled heliocentric universe, on the other hand, was easy for opponents to criticize. Or simply to ignore.To the ancient Greeks, the Earth was an impermanent island of life and death, birth and decay, quite distinct from the unalterable perfection of the heavenly realm above their heads. It was simply unfathomable that a base object like the Earth could circle among the higher cosmic spheres, as Aristarchus would have it. As proof, the geocentrists pointed out that at any given moment precisely half of the celestial sphere is visible to Earthbound observers, no matter where they live. Were our planet offset from the cosmic center, observers living on one side of the Earth would see a smaller volume of the universe than those living on the other side.There was another facet of Aristarchus's theory that disturbed his contemporaries even more: Aristarchus set the Earth spinning. The geocentrists had been taught that the Earth is at rest and that the observed rising and setting of the stars is caused by the continuous turning of the celestial sphere. In the Aristarchian universe, however, the celestial sphere stands motionless, and the nightly progression of the stars from horizon to horizon is revealed for what it is: an illusion. Rotating in an easterly direction, the Earth successively uncovers stars above its eastern limb while eclipsing others behind its western limb. Like a whirling dancer who circles the firelight, our planet revolves about the Sun, spinningall the while--spinning in a twenty-four-hour cycle that creates the ceaseless succession of night and day and the apparent rising and setting of the stars. (In this suggestion, Aristarchus was in fact preceded by Heraclides of Pontus, a contemporary of Aristotle's, who proposed a rotating, albeit geocentric, Earth in the fourth century B.C.)The very concept of a spinning Earth was preposterous to the geocentrists, who appealed to common experience. Stand outside at night, they suggested, and watch the stars drift slowly across the sky; there is absolutely no feeling of motion underfoot. Why deny the credibility of one's own senses? Obviously, the celestial sphere was in motion, not the Earth. That the sphere of the stars was spinning didn't faze the geocentrists in the least; the heavenly realm wasn't subject to the same kinds of physical restrictions to which the Earth was. Furthermore, by the third century B.C. the geocentrists already had a rough idea of the Earth's size. If our planet's surface truly completed a circuit in a mere day's time, they argued, continents would hurtle around the Earth's axis at hundreds of miles an hour. Gale-force winds would perpetually rake the landscape. Oceans would inundate low-lying cities. Anything--or anyone--not firmly anchored to the ground would tumble away into the sky. How could Aristarchus explain the absence of such cataclysmic effects?Or take a more prosaic example. Toss a stone straight upward and watch it inevitably return to the thrower's hand. Were the Earth spinning, the geocentrists reasoned, the stone would veer backward from its starting point, left behind while the Earth sweeps the thrower forward. Only in retrospect is the counterargument clear: The ancients didn't understand the concept of inertia. It would be many centuries before Galileo explained why a stone dropped from the mast of a moving ship strikes the deck directly below its starting point, not several feet back. The stone maintains the forward inertia it acquired from the ship's motion through the water. The same is true for a stone tossed upward from the ground, only this time the role of the moving ship is played by the Earth itself. The thrower, the stone, even the air that envelops them all move in unison with the Earth's rotating surface. The stone maintains this "sideways" velocity--that is, it keeps up with the moving thrower--even while rising or falling through the air. So a rotating Earth is not counter to observation. Then again, neither was the stationary Earth of the geocentrists.The stars themselves posed a special challenge to Aristarchus's heliocentric theory, for now the geocentrists put forth a stringent observational test. Were the Earth truly in orbit, at times it would swing closer to stars on a given side of the celestial sphere. As a result, stars would vary in brightness during the course of the year, brighter when the Earth was nearer, fainter when it was farther away. For example, during winter (or, specifically, winter in the northern hemisphere), when the Earth supposedly swings toward the constellation Canis Major, the prominent star Sirius should brighten; months later, when the Earth is moving away from that part of the sky, Sirius should dim. But Sirius displays no such annual variation in its light. Nor does any other star in the heavens.While the stars were constant, the planets did vary periodically in brightness. Aristarchus could have easily explained why. In a heliocentric cosmos, planet-to-planet separations change all the time as these bodies make their individual orbital rounds; when a planet swings closer to the Earth's current position, it appears more prominent in the sky. At the time, the geocentric model was unable to account for changing planetary brightness because each of its planets maintains a constant distance from the central, fixed Earth.Were the Earth in orbit, there would be yet another observational consequence: parallax. Parallax (from the Greek parallassein, "to change") is the apparent shift in position of an object when viewed alternately from different vantage points. Parallax manifests itself in our everyday lives because each of our eyes has its own distinct perspective on the surrounding world. Our eyes are separated by a couple of inches, and that's enough to make an object's position appear somewhat different to each eye. As you read the words on this page, for example, your eyes inevitably cross a little, a physical consequence of the parallax effect. Except for very nearby objects, such as a finger held right in front of your nose, this convergence of your eyes is imperceptible. Yet our brain has evolved the ability to interpret the degree of eye-crossing as a gauge of an object's parallax, and hence its distance. The farther the object, the smaller its parallax.The words on this page induce a fair degree of eye-crossing; they are close by and consequently have a relatively large parallax. A plant across the room displays a smaller parallax, and a tree down the street a smaller one yet. What about a star? Does a star far off in space exhibit a parallax? The answer depends on whomyou ask. The geocentrists would tell you that stars show no parallax, that with only a few inches separating our eyes--the twin vantage points from which we observe the heavens--both eyes effectively see the star in the identical position.But what if the vantage points were more widely separated? (Sounds painful, but don't worry, it requires no rearrangement of your face.) Suppose you sighted the star from Greece and had a compatriot do the same from, say, Egypt, many hundreds of miles away. Would the star's position appear different to observers at these two locations? Would the star display a parallax? Yes, the geocentrists would answer, in principle the star might show a parallax, given the substantial distance between the vantage points. But the geocentrists would hasten to point out that there is absolutely no observed parallax of stars when viewed from these two locations or indeed from locations even farther apart. Evidently, the celestial sphere is so remote that a pair of observers standing at opposite ends of the Earth--the widest possible "baseline" in a geocentric universe--would never detect a star's parallax.Now let's pose the parallax question to Aristarchus, the heliocentrist. In his universe, there is a much higher expectation of observing stellar parallax, for the Earth's wide-ranging movement around the Sun provides vantage points whose separation is ever so much larger than the Earth itself. An astronomer viewing stars over such a broad baseline--the width of the Earth's orbit--might be able to detect a perceptible change in the position of a star. In fact, during the course of the year, the star should appear to sway to and fro, perhaps even dance a little oval against the firmament, a skewed reflection of our planet's orbital motion around the Sun.Yet to the pretelescopic vision of the ancients, not a single star danced Aristarchus's parallactic "jig" in the heavens. The stars just marched monotonously up from the eastern horizon and down to the western horizon, with not a twist or a twirl among them. Once again, the evidence weighed against Aristarchus's assertion that the Earth revolved about the Sun. Too bad, for the eventual key to plumbing the stellar depths did lie in just that parallactic wobble that Aristarchus presumably had forecast but would never live to see. Today we know that even the most prominent stellar parallax falls far below the threshold of naked-eye vision. But nobody knew that back then.The moth-eaten fabric of recorded history contains not a jot about Aristarchus's response to his critics. Only one of his written works survives, a short treatise on the sizes and distances of theSun and Moon, and it doesn't even mention the heliocentric universe (although it does have implications for the model). For all we know, Aristarchus might have been dismissed as a crank, a fringe thinker whose ideas were too far removed from the mainstream to be taken seriously. Nevertheless, in the spirit of supporting the underdog, let's take up the debate on Aristarchus's behalf. The primary issue at hand: stellar parallax. We will infer Aristarchus's response to the parallax problem--the inexplicable constancy of the stars in the face of a moving Earth--from a single sentence in an unusual work by one of Aristarchus's contemporaries: Archimedes.
Archimedes was born around 280 B.C. in Syracuse, principal city-state of Sicily. His interests ranged widely across mathematics, physics, astronomy, and engineering. Although Archimedes himself valued his theoretical research most, he managed to develop a host of practical inventions, including an arsenal of devastatingly effective engines of war used to defend Syracuse from Roman invaders. Here is an account of what hostile armies faced when going up against the terrifying weaponry of Archimedes:When the Roman legions attacked, they were met with a rain of missiles and immense stones launched from giant catapults ... . Trying to protect themselves under a cover of shields, the helpless Roman infantry was crushed by boulders and large timbers dropped from cranes that swung out over the battlements. Most horrific of all were the enormous clawlike devices that wrecked the Roman fleet as it tried to enter the harbor, shaking the ships about and even plucking them clean out of the water.As Plutarch tells it, "such terror had seized upon the Romans, 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." The Roman general, Marcellus, could do nothing but settle in for a long siege.In the nonmilitary realm, Archimedes developed the water screw that bears his name: an inclined, hand-turned, hollow spiral that is still used in underdeveloped countries to raise water from irrigation ditches. Archimedes also experimented with levers, pulleys, and fulcrums. "Give me where to stand," he wassaid to have boasted to the Syracusan king, Hieron, "and I will move the Earth." To prove his point, Archimedes rigged up a device with which he single-handedly launched a fully loaded ship from dry dock. Afterward Hieron declared that "Archimedes was to be believed in everything he might say." As for Archimedes, he was content to return to his theoretical pursuits; his machines had been designed "not as matters of any importance, but as mere amusements in geometry."Archimedes displayed the eccentricities often associated with genius. In his Lives, Plutarch describes how Archimedes would "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."The most frequently told tale about Archimedes has to do with a royal crown. King Hieron had contracted with a local goldsmith to fashion for the king a new crown out of solid gold. The crown was duly completed, and it looked magnificent. But the king had heard a rumor that the craftsman had squirreled away some of the gold that had been allocated for the crown, substituting an equal weight of a lesser metal like silver or copper. (Nowadays, jewelers routinely mix gold with other metals for strength; common fourteen-karat gold is only 58 percent pure.) Hieron asked Archimedes to assess the "purity" of the new crown.Gold is pretty heavy stuff: A three-inch gold cube tips the scales at nearly nineteen pounds. (Talk about the burdens of high office; no wonder George Washington refused to wear royal trappings.) Archimedes conceived a plan. First he would weigh Hieron's crown. Then he would measure the crown's volume. Finally he would check whether the crown's weight was proper for that volume of gold. But like all crowns, this one was irregularly shaped. How was Archimedes to determine its volume?According to Plutarch, Archimedes was mulling over this royal conundrum one day while at the public baths. As usual, the water rose as Archimedes settled into the tub. But that day this commonplace phenomenon rang with new significance. His own body had pushed aside the volume of water that used to occupy the space where he now lay. The water had risen in proportion to the volume of his body. Archimedes realized how he might measure the volume of Hieron's crown. He'd fill a tub to the brim withwater, immerse the crown, and capture the overflow in a measuring cup. The crown's volume would simply equal the volume of water in the cup. Upon receiving this flash of insight, Archimedes is alleged to have leaped from his bath and run home naked through the streets of Syracuse, shouting that now-famous expletive of scientific discovery, "Eureka!" ("I have found it!") He determined the crown's volume and found that it was indeed lighter than the equivalent volume of gold. Archimedes reported to King Hieron that the crown had been "cut" with silver. The goldsmith was executed.This alleged streaker of bygone days, Archimedes, played an indirect but critical role in the story of stellar parallax. Having spent part of his life in Alexandria concurrently with Aristarchus, Archimedes was familiar with the revolutionary heliocentric system. Like everyone else at the time, he didn't believe it. But there was one instance late in his life where Aristarchus's outlandish idea proved useful to Archimedes, a fact that he recorded. Had he not, we might never have known any details of Aristarchus's heliocentric hypothesis.Around 216 B.C., shortly before his death at the hands of a Roman soldier, the septuagenarian Archimedes completed a mathematical work for King Hieron's successor, Gelon II. This curious tract, written for the layperson and entitled The Sand-Reckoner, was an attempt by Archimedes to demonstrate his facility with the mathematics of large numbers. To wit, Archimedes set out to compute the number of sand grains required to fill the universe! His pitch has the definite ring of the carnival barker: "I will try to show ... 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." Sounds impressive, even today. But just how big was this universe that Archimedes intended to "fill"?To make a lasting impression on his royal patron, Archimedes wanted a universe vaster than the commonly accepted geocentric system, with its tightly nested heavenly spheres. So without passing judgment on the relative merits of the two competing world systems, he selected Aristarchus's alternative model, which he knew to be sizable indeed. It is through Archimedes that we gain insight into the dimensions of the new heliocentric universe by way of Aristarchus's own words, as set down in The Sand-Reckoner.First let's lay the groundwork for what Aristarchus had to say about the extent of the universe, for he approached the issue in an oblique and somewhat problematic fashion. In general, how can one effectively convey the size of an object? One method is to state its dimensions outright: The house is thirty feet wide. Another method is to compare its size to that of another object or, equivalently, to form the size ratio between the objects: The shopping mall is twice the length of a football field, or the ratio between the length of the shopping mall and the length of a football field is 2. Yet a third method is to express the size ratio of two objects in terms of the size ratio of two other objects: The tree's height is to the man's height as the man's height is to the child's height. In this case, if the man is six feet tall and the child three feet tall, the ratio of their heights is 2; because the tree's height bears this same ratio to the man's height, the tree is twelve feet tall. It is the last of these methods that Aristarchus chose to reveal the size of his universe. Here, in modernized form, is what Archimedes claims Aristarchus wrote:The distance of the stars bears the same relation to the diameter of the Earth's orbit as the surface of a sphere bears to its center.To my ear, Aristarchus's statement is eerily reminiscent of one of those thorny problems on the SAT. Parsing it, we see that the statement contains two ratios: The first ratio is a purely physical one, between the distance of the stars and the diameter of the Earth's orbit; the second ratio is a purely mathematical one, between the surface of a sphere and the center of a sphere. (Aristarchus might have been referring to the celestial sphere, but any sphere will do.) These two ratios, Aristarchus claimed, are equal, which we can represent as follows:(distance of the stars) / (diameter of the Earth's orbit) = (surface of a sphere) / (center of a sphere)Compute the second ratio, the mathematical one, and you simultaneously have learned the first ratio, the physical one. Here Aristarchus has apparently offered Archimedes a way to compute the extent of the heliocentric universe ("distance of the stars") in terms of the diameter of the Earth's orbit. Specifically, how many times bigger is the universe compared to the Earth's orbit? That we can answer by computing the second ratio. Or can we?The geometrically savvy reader will already have recognized that the second ratio, between the surface of a sphere and the centerof a sphere, is patently absurd. Yet similar phrasing was often used by Greek astronomers when expressing the enormity of the heavens. (By "surface of a sphere," we'll take Aristarchus's meaning to be the sphere's diameter.) The center of a sphere is a point, which geometrically speaking has no size at all. Thus, a sphere's diameter is infinitely larger than its center, because any number divided by zero yields infinity. In other words, the second ratio turns out to be infinite. Taken literally, Aristarchus's statement implies that the stars are infinitely far away. An infinite universe did not suit Archimedes, who at this point might have identified with the hapless protagonist in physicist George Gamow's limerick:There was a young fellow from Trinity Who took [the square root of infinity] But the number of digits Gave him the fidgets; he dropped Math and took up Divinity.In The Sand-Reckoner, Archimedes wished to perform a calculation that involved the purported extent of the universe. He was unable to deal with one that was infinitely large, so he "fudged" his assumptions a bit. He chose a less literal interpretation of Aristarchus's enigmatic phrase comparing the surface to the center of a sphere. According to Archimedes, Aristarchus must have meant something like this:The distance of the stars bears the same relation to the diameter of the Earth's orbit as the diameter of the Earth's orbit bears to the diameter of the Earth.Or, in ratio form:(distance of the stars) / (diameter of the Earth's orbit) = (diameter of the Earth's orbit) / (diameter of the Earth)Voilá! The second ratio is no longer infinite, but a straightforward comparison between two measurable quantities: the diameter of the Earth's orbit and the diameter of the Earth. These were numbers for which crude estimates already existed (from Aristotle, among others). Using generous assumptions, Archimedes reasoned that the diameter of the Earth's orbit did not exceed 10,000 Earth-diameters. (In reality, he was still too low by a factor of 2.) The previous relation becomes:(distance of the stars) / (10,000 Earth-diameters) = (10,000 Earth-diameters) / (1 Earth-diameter)Therefore, according to Archimedes, the stars on the celestial sphere are 100 million Earth-diameters away, which dwarfs the estimate of 10,000 Earth-diameters proposed by the geocentrists. (Yet it still falls far short of the true distance to even the nearest star: over 3 billion Earth-diameters.) Next Archimedes converted his cosmic distance to stadia, a terrestrial measure used by the ancients and roughly equivalent to a tenth of a mile. For this calculation, he "fudged" his assumptions yet again. Remember, Archimedes was trying to impress King Gelon by "filling" the biggest possible universe, so he unabashedly goosed up the numbers for maximum effect. He took the Earth's diameter to be nearly 1 million stadia, which he freely admitted was tenfold larger than the commonly accepted value at the time. In the end, Archimedes deduced that the Aristarchian universe was about 100 trillion stadia in radius, roughly 10 trillion miles. To fill it required 1,000 trillion trillion trillion trillion trillion sand grains. I imagine that King Gelon was suitably impressed. (In modern "scientific notation," the number of sand grains would be written 1063, that is, 1 followed by sixty-three zeroes.)Archimedes died shortly after completing The Sand-Reckoner, slain by a soldier when the invading army of the Roman general Marcellus surprised drunken sentries and swept into Syracuse. According to Plutarch, the soldier came upon Archimedes scratching geometric shapes in the dirt and became incensed when the old man scolded him for interrupting. More than a century later, in 75 B.C., Cicero restored the neglected tomb of Archimedes, having discovered it by the legendary figure carved on its face: a cylinder enclosing a sphere. The tomb is now lost.Regardless of how Archimedes chose to interpret Aristarchus's original statement about the size of the universe, Aristarchus's intent is clear. He sought to explode the geocentrists' notion of the compact universe, whose not-so-distant celestial sphere coddled humanity like a cosmic eggshell. The celestial sphere, in his view, was enormous compared to the breadth of the Earth's orbit. The stars are exceedingly far away. And that, we may infer, is how Aristarchus would have responded to his critics on the stellar parallax issue: Stars exhibit no discernible change in their light and no measurable parallax because they are so remote. If a star is greater than a million Earth-diameters away, its parallax will beundetectable by the naked eye. The celestial sphere in the Aristarchian universe lay far beyond that threshold, so there was absolutely no chance for the ancients to witness stellar parallax. In the debate between the two world systems, Aristarchus stood on weak ground, for the absence of stellar parallax confirmed nothing one way or the other. Either the Earth was in motion and the stars were immensely distant or the Earth was fixed and central and the stars were relatively nearby. There was simply no way to distinguish an undetectable parallax from a nonexistent parallax.The geocentrists surely railed against the immense void that Aristarchus had imposed between the celestial sphere and the inner realm of the Sun and planets. From their perspective, there could be no considerable gap among the spheres occupied by the heavenly bodies; each sphere brushed up against its neighbors like the layers of an onion. Why would the Creator have built a universe with so much wasted space? It just didn't make sense.
The Sun-centered cosmos, as conceived by Aristarchus, had no real chance of acceptance within the Hellenistic academy. The geocentric model was firmly rooted by this time and accounted well enough for the observed motions of the Sun, Moon, and planets. In hindsight, we know that the geocentric model was ultimately discredited; yet we shouldn't judge its ancient proponents too harshly. Whatever their religious or philosophical biases, their arguments appealed to a most sensitive human touchstone: common sense. To believe the heliocentric picture of the heavens, even today, means distrusting one's own senses, for in truth the concept of a moving, spinning Earth is counterintuitive. Despite decades as an astronomer, I still find it hard to fathom that the Earth streaks through half a million miles of space while we snooze away the night in our beds. Prior to the 1700s, there simply was no definitive proof of the Earth's orbital motion. The heliocentric system of Aristarchus was also largely qualitative, its mathematical underpinnings apparently not worked out. So while in principle it could account for the observed lunar and planetary movements, no one at the time knew for sure. And most important, the heliocentric system lacked unambiguous observational evidence, such as stellar parallax.In the eyes of his fellow philosophers, Aristarchus was at best a radical thinker, at worst a heretic. At least one tract denouncing him for impiety was published during his lifetime, a bilious indictment by the Stoic philosopher Cleanthes of Assus, suitably titledAgainst Aristarchus. The heliocentric model apparently gained only one major adherent, the mathematician Seleucus of Seliucia on the Tigris River, and that was a full century after the new theory had been proposed. No, the time was not yet right for an unbiased hearing of such revolutionary ideas. Aristarchus stood as much chance of reversing the geocentric juggernaut as a mosquito has of turning back a freight train.Had stellar parallax been observable by the ancient Greeks, the case for the Sun-centered universe might have been won in Aristarchus's time. But as the situation played out, the clockwork universe continued to tick for another 1,800 years before the parallax issue emerged once more in a renewed conflict between the two competing world systems. And yet another 300 years passed until, in a nearly simultaneous burst of competitive energy, three astronomers vied to pin down the long-sought parallax.The Earth-centered model of the cosmos. From Hevelius, Selenographia, 1647.Source: Wolbach Library, Harvard University.Copyright © 2001 by W. H. Freeman and Company
Table of Contents
| Introduction | xi | |
| Part 1 | ||
| 1 | Reinventing the Cosmos | 1 |
| 2 | The Circle Game | 19 |
| 3 | What If the Sun Be Center to the World? | 33 |
| 4 | Crossed Gyes and Wobbling Stars | 49 |
| Part 2 | ||
| 5 | The heavens Grupt | 75 |
| 6 | The Turbulent Lens | 95 |
| 7 | The Wrangler of Pisa | 113 |
| 8 | The Archimedean Gngine | 135 |
| 9 | A Coal Cellar with a View | 151 |
| 10 | Double Vision | 171 |
| Part 3 | ||
| 11 | Dismal Swamp | 193 |
| 12 | The Twice-Built Telescope | 207 |
| 13 | Quest for Precision | 223 |
| 14 | So Many Grasshoppers | 235 |
| 15 | The Star in the Lyre | 245 |
| 16 | The Subtle Weave | 257 |
| Epilogue: A Drink from the Well | 269 | |
| Acknowledgments | 282 | |
| Notes and Further Reading | 284 | |
| Index | 305 |
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