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In love with whole numbers, the Pythagoreans believed all things could be derived from them, certainly all other numbers. A crisis in doctrine arose when they discovered that the square root of two (the ratio of the diagonal to the side of a square) was irrational, that √2 cannot be expressed accurately as the ratio of any two whole numbers, no matter how big these numbers are. Ironically this discovery (reproduced in Appendix 1) was made with the Pythagorean theorem as a tool. “Irrational” originally meant only that a number could not be expressed as a ratio. But for the Pythagoreans it came to mean something threatening, a hint that their world view might not make sense, which is today the other meaning of “irrational.” Instead of sharing these important mathematical discoveries, the Pythagoreans suppressed the knowledge of and the dodecahedron. The outside world was not to know.* Even today there are scientists opposed to the popularization of science: the sacred knowledge is to be kept within the cult, unsullied by public understanding.
The Pythagoreans believed the sphere to be “perfect,” all points on its surface being at the same distance from its center. Circles were also perfect. And the Pythagoreans insisted that planets moved in circular paths at constant speeds. They seemed to believe that moving slower or faster at different places in the orbit would be unseemly; noncircular motion was somehow flawed, unsuitable for the planets, which, being free of the Earth, were also deemed “perfect.”
The pros and cons of the Pythagorean tradition can be seen clearly in the life’s work of Johannes Kepler (Chapter 3). The Pythagorean idea of a perfect and mystical world, unseen by the senses, was readily accepted by the early Christians and was an integral component of Kepler’s early training. On the one hand, Kepler was convinced that mathematical harmonies exist in nature (he wrote that “the universe was stamped with the adornment of harmonic proportions”); that simple numerical relationships must determine the motion of the planets. On the other hand, again following the Pythagoreans, he long believed that only uniform circular motion was admissible. He repeatedly found that the observed planetary motions could not be explained in this way, and repeatedly tried again. But unlike many Pythagoreans, he believed in observation and experiment in the real world. Eventually the detailed observations of the apparent motion of the planets forced him to abandon the idea of circular paths and to realize that planets travel in ellipses. Kepler was both inspired in his search for the harmony of planetary motion and delayed for more than a decade by the attractions of Pythagorean doctrine.
A disdain for the practical swept the ancient world. Plato urged astronomers to think about the heavens, but not to waste their time observing them. Aristotle believed that: “The lower sort are by nature slaves, and it is better for them as for all inferiors that they should be under the rule of a master.… The slave shares in his master’s life; the artisan is less closely connected with him, and only attains excellence in proportion as he becomes a slave. The meaner sort of mechanic has a special and separate slavery.” Plutarch wrote: “It does not of necessity follow that, if the work delight you with its grace, the one who wrought it is worthy of esteem.” Xenophon’s opinion was: “What are called the mechanical arts carry a social stigma and are rightly dishonoured in our cities.” As a result of such attitudes, the brilliant and promising Ionian experimental method was largely abandoned for two thousand years. Without experiment, there is no way to choose among contending hypotheses, no way for science to advance. The anti-empirical taint of the Pythagoreans survives to this day. But why? Where did this distaste for experiment come from?
An explanation for the decline of ancient science has been put forward by the historian of science, Benjamin Farrington: The mercantile tradition, which led to Ionian science, also led to a slave economy. The owning of slaves was the road to wealth and power. Polycrates’ fortifications were built by slaves. Athens in the time of Pericles, Plato and Aristotle had a vast slave population. All the brave Athenian talk about democracy applied only to a privileged few. What slaves characteristically perform is manual labor. But scientific experimentation is manual labor, from which the slaveholders are preferentially distanced; while it is only the slaveholders—politely called “gentle-men” in some societies—who have the leisure to do science. Accordingly, almost no one did science. The Ionians were perfectly able to make machines of some elegance. But the availability of slaves undermined the economic motive for the development of technology. Thus the mercantile tradition contributed to the great Ionian awakening around 600 B.C., and, through slavery, may have been the cause of its decline some two centuries later. There are great ironies here.
Approximate lifetimes of Ionian and other Greek scientists between the seventh century B.C. and the fifth century. The decline of Greek science is indicated by the relatively few individuals shown after the first century B.C.
Similar trends are apparent throughout the world. The high point in indigenous Chinese astronomy occurred around 1280, with the work of Kuo Shou-ching, who used an observational baseline of 1,500 years and improved both astronomical instruments and mathematical techniques for computation. It is generally thought that Chinese astronomy thereafter underwent a steep decline. Nathan Sivin believes that the reason lies at least partly “in increasing rigidity of elite attitudes, so that the educated were less inclined to be curious about techniques and less willing to value science as an appropriate pursuit for a gentleman.” The occupation of astronomer became a hereditary office, a practice inconsistent with the advance of the subject. Additionally, “the responsibility for the evolution of astronomy remained centered in the Imperial Court and was largely abandoned to foreign technicians,” chiefly the Jesuits, who had introduced Euclid and Copernicus to the astonished Chinese, but who, after the censorship of the latter’s book, had a vested interest in disguising and suppressing heliocentric cosmology. Perhaps science was stillborn in Indian, Mayan and Aztec civilizations for the same reason it declined in Ionia, the pervasiveness of the slave economy. A major problem in the contemporary (political) Third World is that the educated classes tend to be the children of the wealthy, with a vested interest in the status quo, and are unaccustomed either to working with their hands or to challenging conventional wisdom. Science has been very slow to take root.
Plato and Aristotle were comfortable in a slave society. They offered justifications for oppression. They served tyrants. They taught the alienation of the body from the mind (a natural enough ideal in a slave society); they separated matter from thought; they divorced the Earth from the heavens—divisions that were to dominate Western thinking for more than twenty centuries. Plato, who believed that “all things are full of gods,” actually used the metaphor of slavery to connect his politics with his cosmology. He is said to have urged the burning of all the books of Democritus (he had a similar recommendation for the books of Homer), perhaps because Democritus did not acknowledge immortal souls or immortal gods or Pythagorean mysticism, or because he believed in an infinite number of worlds. Of the seventy-three books Democritus is said to have written, covering all of human knowledge, not a single work survives. All we know is from fragments, chiefly on ethics, and secondhand accounts. The same is true of almost all the other ancient Ionian scientists.
In the recognition by Pythagoras and Plato that the Cosmos is knowable, that there is a mathematical underpinning to nature, they greatly advanced the cause of science. But in the suppression of disquieting facts, the sense that science should be kept for a small elite, the distaste for experiment, the embrace of mysticism and the easy acceptance of slave societies, they set back the human enterprise. After a long mystical sleep in which the tools of scientific inquiry lay moldering, the Ionian approach, in some cases transmitted through scholars at the Alexandrian Library, was finally rediscovered. The Western world reawakened. Experiment and open inquiry became once more respectable. Forgotten books and fragments were again read. Leonardo and Columbus and Copernicus were inspired by or independently retraced parts of this ancient Gree
k tradition. There is in our time much Ionian science, although not in politics and religion, and a fair amount of courageous free inquiry. But there are also appalling superstitions and deadly ethical ambiguities. We are flawed by ancient contradictions.
The Platonists and their Christian successors held the peculiar notion that the Earth was tainted and somehow nasty, while the heavens were perfect and divine. The fundamental idea that the Earth is a planet, that we are citizens of the Universe, was rejected and forgotten. This idea was first argued by Aristarchus, born on Samos three centuries after Pythagoras. Aristarchus was one of the last of the Ionian scientists. By this time, the center of intellectual enlightenment had moved to the great Library of Alexandria. Aristarchus was the first person to hold that the Sun rather than the Earth is at the center of the planetary system, that all the planets go around the Sun rather than the Earth. Typically, his writings on this matter are lost. From the size of the Earth’s shadow on the Moon during a lunar eclipse, he deduced that the Sun had to be much larger than the Earth, as well as very far away. He may then have reasoned that it is absurd for so large a body as the Sun to revolve around so small a body as the Earth. He put the Sun at the center, made the Earth rotate on its axis once a day and orbit the Sun once a year.
It is the same idea we associate with the name of Copernicus, whom Galileo described as the “restorer and confirmer,” not the inventor, of the heliocentric hypothesis.* For most of the 1,800 years between Aristarchus and Copernicus nobody knew the correct disposition of the planets, even though it had been laid out perfectly clearly around 280 B.C. The idea outraged some of Aristarchus’ contemporaries. There were cries, like those voiced about Anaxagoras and Bruno and Galileo, that he be condemned for impiety. The resistance to Aristarchus and Copernicus, a kind of geocentrism in everyday life, remains with us: we still talk about the Sun “rising” and the Sun “setting.” It is 2,200 years since Aristarchus, and our language still pretends that the Earth does not turn.
The separation of the planets from one another—forty million kilometers from Earth to Venus at closest approach, six billion kilometers to Pluto—would have stunned those Greeks who were outraged by the contention that the Sun might be as large as the Peloponnesus. It was natural to think of the solar system as much more compact and local. If I hold my finger before my eyes and examine it first with my left and then with my right eye, it seems to move against the distant background. The closer my finger is, the more it seems to move. I can estimate the distance to my finger from the amount of this apparent motion, or parallax. If my eyes were farther apart, my finger would seem to move substantially more. The longer the baseline from which we make our two observations, the greater the parallax and the better we can measure the distance to remote objects. But we live on a moving platform, the Earth, which every six months has progressed from one end of its orbit to the other, a distance of 300,000,000 kilometers. If we look at the same unmoving celestial object six months apart, we should be able to measure very great distances. Aristarchus suspected the stars to be distant suns. He placed the Sun “among” the fixed stars. The absence of detectable stellar parallax as the Earth moved suggested that the stars were much farther away than the Sun. Before the invention of the telescope, the parallax of even the nearest stars was too small to detect. Not until the nineteenth century was the parallax of a star first measured. It then became clear, from straightforward Greek geometry, that the stars were light-years away.
There is another way to measure the distance to the stars which the Ionians were fully capable of discovering, although, so far as we know, they did not employ it. Everyone knows that the farther away an object is, the smaller it seems. This inverse proportionality between apparent size and distance is the basis of perspective in art and photography. So the farther away we are from the Sun, the smaller and dimmer it appears. How far would we have to be from the Sun for it to appear as small and as dim as a star? Or, equivalently, how small a piece of the Sun would be as bright as a star?
An early experiment to answer this question was performed by Christiaan Huygens, very much in the Ionian tradition. Huygens drilled small holes in a brass plate, held the plate up to the Sun and asked himself which hole seemed as bright as he remembered the bright star Sirius to have been the night before. The hole was effectively* 1/28,000 the apparent size of the Sun. So Sirius, he reasoned, must be 28,000 times farther from us than the Sun, or about half a light-year away. It is hard to remember just how bright a star is many hours after you look at it, but Huygens remembered very well. If he had known that Sirius was intrinsically brighter than the Sun, he would have come up with almost exactly the right answer: Sirius is 8.8 light-years away. The fact that Aristarchus and Huygens used imprecise data and derived imperfect answers hardly matters. They explained their methods so clearly that, when better observations were available, more accurate answers could be derived.
Between the times of Aristarchus and Huygens, humans answered the question that had so excited me as a boy growing up in Brooklyn: What are the stars? The answer is that the stars are mighty suns, light-years away in the vastness of interstellar space.
The great legacy of Aristarchus is this: neither we nor our planet enjoys a privileged position in Nature. This insight has since been applied upward to the stars, and sideways to many subsets of the human family, with great success and invariable opposition. It has been responsible for major advances in astronomy, physics, biology, anthropology, economics and politics. I wonder if its social extrapolation is a major reason for attempts at its suppression.
The legacy of Aristarchus has been extended far beyond the realm of the stars. At the end of the eighteenth century, William Herschel, musician and astronomer to George III of England, completed a project to map the starry skies and found apparently equal numbers of stars in all directions in the plane or band of the Milky Way; from this, reasonably enough, he deduced that we were at the center of the Galaxy.* Just before World War I, Harlow Shapley of Missouri devised a technique for measuring the distances to the globular clusters, those lovely spherical arrays of stars which resemble a swarm of bees. Shapley had found a stellar standard candle, a star noticeable because of its variability, but which had always the same average intrinsic brightness. By comparing the faintness of such stars when found in globular clusters with their real brightness, as determined from nearby representatives, Shapley could calculate how far away they are—just as, in a field, we can estimate the distance of a lantern of known intrinsic brightness from the feeble light that reaches us—essentially, the method of Huygens. Shapley discovered that the globular clusters were not centered around the solar neighborhood but rather about a distant region of the Milky Way, in the direction of the constellation Sagittarius, the Archer. It seemed to him very likely that the globular clusters used in this investigation, nearly a hundred of them, would be orbiting about, paying homage to, the massive center of the Milky Way.
Shapley had in 1915 the courage to propose that the solar system was in the outskirts and not near the core of our galaxy. Herschel had been misled because of the copious amount of obscuring dust in the direction of Sagittarius; he had no way to know of the enormous numbers of stars beyond. It is now very clear that we live some 30,000 light-years from the galactic core, on the fringes of a spiral arm, where the local density of stars is relatively sparse. There may be those who live on a planet that orbits a central star in one of Shapley’s globular clusters, or one located in the core. Such beings may pity us for our handful of naked-eye stars, because their skies will be ablaze with them. Near the center of the Milky Way, millions of brilliant stars would be visible to the naked eye, compared to our paltry few thousand. Our Sun or suns might set, but the night would never come.
Well into the twentieth century, astronomers believed that there was only one galaxy in the Cosmos, the Milky Way—although in the eighteenth century Thomas Wright of Durban and Immanuel Kant of Königsberg each had a premonition that the exquisite lum
inous spiral forms, viewed through the telescope, were other galaxies. Kant suggested explicity that M31 in the constellation Andromeda was another Milky Way, composed of enormous numbers of stars, and proposed calling such objects by the evocative and haunting phrase “island universes.” Some scientists toyed with the idea that the spiral nebulae were not distant island universes but rather nearby condensing clouds of interstellar gas, perhaps on their way to make solar systems. To test the distance of the spiral nebulae, a class of intrinsically much brighter variable stars was needed to furnish a new standard candle. Such stars, identified in M31 by Edwin Hubble in 1924, were discovered to be alarmingly dim, and it became apparent that M31 was a prodigious distance away, a number now estimated at a little more than two million light-years. But if M31 were at such a distance, it could not be a cloud of mere interstellar dimensions; it had to be much larger—an immense galaxy in its own right. And the other, fainter galaxies must be more distant still, a hundred billion of them, sprinkled through the dark to the frontiers of the known Cosmos.
As long as there have been humans, we have searched for our place in the Cosmos. In the childhood of our species (when our ancestors gazed a little idly at the stars), among the Ionian scientists of ancient Greece, and in our own age, we have been transfixed by this question: Where are we? Who are we? We find that we live on an insignificant planet of a humdrum star lost between two spiral arms in the outskirts of a galaxy which is a member of a sparse cluster of galaxies, tucked away in some forgotten corner of a universe in which there are far more galaxies than people. This perspective is a courageous continuation of our penchant for constructing and testing mental models of the skies; the Sun as a red-hot stone, the stars as celestial flame, the Galaxy as the backbone of night.