Part 4 (1/2)
It was Pythagoras who pioneered the quest for a link between numbers and the cosmos. Pondering the hidden meanings of the world around him as he played on his lyre he began to wonder whether the laws of harmony depended on numbers. He found that to play tones an octave apart, the length of the strings needed to be in specific ratios: the keynote of the octave sounded when the ratio was 1:2; a fifth required a ratio of 2:3; and a fourth, 3:4.
Perhaps numbers might belong to a world beyond perception, which could only be fully apprehended by thought. His striking conclusion was that the numerals 1, 2, 3, and 4 represented all known objects: 1 represents a point; 2 points can be connected by a line; 3 points make a triangle, in particular a perfect equilateral triangle; and 4 points make a tetrahedron, a pyramid of three perfect triangles. From these could be constructed the five ”Pythagorean” solids (later ”Platonic” solids after Plato): the tetrahedron, cube, octahedron (eight equilateral triangles), dodecahedron (twelve pentagons), and icosahedron (twenty equilateral triangles). Each could be circ.u.mscribed by a sphere, with each point of the solid touching its surface, and each could also contain a sphere whose surface touched each of its sides.
Represented as dots, 1, 2, 3, and 4 form an equilateral triangle set out in four rows, known as the tetraktys (tetras is Greek for ”four”): Pythagoras's tetraktys.
To Pythagoras this a.n.a.lysis made sense of our world, in which he recognized four elements (earth, water, air, and fire), four seasons, four points of the compa.s.s, and four rivers of paradise (the Pishon, the Gihon, the Tigris, and the Euphrates). His followers swore an oath ”by him who has committed to our soul the tetraktys, the original source and root of eternal Nature.” The sum of the numbers that made up the tetraktys is ten, which Pythagoras considered the perfect number. Once we have counted to ten, we return to one, the number of creation.
Pythagoras's claim was that numbers were the fabric of our universe and existed independently of us. Numbers were the keys through which could be heard the harmony of the cosmos.
The Kabbalah.
The Egyptian G.o.d Thoth, known to the Greeks as Hermes Trismegistus (”Thrice-great Hermes”), was credited with a huge number of writings on philosophy, astrology, and magic. Over the centuries Hermetic literature incorporated elements of whatever science existed as well as the teachings of Pythagoras.
In Kepler's time Hermetic literature was enthusiastically embraced as an antidote to the rational approach of Greek philosophy and science. It was full of mystery and magic and spoke in terms of a vital or living force at the heart of the cosmos. Hermetic literature also included kabbalistic texts.
Versions of the Kabbalah had begun to appear in the thirteenth century. A princ.i.p.al theme was how one might see the invisible in the visible and the spiritual in matter. The Kabbalah discussed the clash between opposites like light and darkness to produce the world in which we live. Someone like Kepler, who was interested in the teachings of Proclus, was naturally drawn to the Kabbalah with its similar theme.
A central notion of kabbalistic philosophy is the Sephirot. The Sephirot is usually represented as the tree of life with ten branches rooted in the earth and extending to Heaven, signifying the earth as a microcosm reflecting the universe, the macrocosm. It is made up of five pairs of opposites-beginning and end, good and evil, above and below, east and west, and north and south-and thus has ten emanations, ten being a holy number in Judaism as well as in Pythagoreanism.
By the end of the fifteenth century the Kabbalah had been integrated into Christian theology, though the Christian Kabbalah emphasized the Trinity rather than the Sephirot. Christian thinkers were especially fascinated by the Gematria, which a.s.signed numbers to letters of the Hebrew alphabet. This concept of numbers for language opened up the possibility of a.s.signing numbers to the various names of G.o.d, thereby further revealing His celestial powers and His mystery. Thus the Kabbalah became identified with magic and numerology. (Until the nineteenth century the Hebrew alphabet had no numbers; letters were used for numbers. Thus in Roman times 666 happened to be the letters for Nero's name.) By Kepler's day the Christian Kabbalah was considered one of the ”handmaidens” of true wisdom, along with alchemy and astrology. But all this clashed with the onset of a new, materialistic science that claimed to be able to predict the course of cannonb.a.l.l.s and planets using mathematics, but only if a division were made in nature between dead and live matter. For mathematics could be applied only to the former, not to the latter.
Kepler's model of the universe.
When Kepler was growing up, there was a flood of astrological, kabbalistic, and alchemical texts being published. Anything attributed to Hermes Trismegistus was hailed as a revelation. They held readers spellbound, the vaguer the better. Kepler was hooked; his enormous imagination was sparked.
Why was the world as it was? Why were there six planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn)? Why were they at certain distances from the sun? What was the relations.h.i.+p between their distances from the sun and their speeds? Might the answers to these questions lie in certain arrangements of geometrical figures?
By this time Kepler was a district mathematician, teaching mathematics and astronomy at the Protestant Seminary in Graz. During one of his cla.s.sroom lectures on geometry he happened to draw an equilateral triangle. Inside it he drew a circle touching all three sides and around it another circle touching its points, just as Pythagoras had described. Suddenly it all fell into place. It was a model of the universe.
Clearly the reason why there were six planets was because there were five perfect solids symbolizing the five intervals between the planets. The planets moved on spheres that circ.u.mscribed the five solids. Calculating the distances of the planets from the sun, Kepler drew up a new image of Copernicus's universe with the sun at the center and nested planetary spheres on which the planets moved. The sphere of Mercury inscribed an octahedron, that of Venus an icosahedron, that of Earth a dodecadron, that of Mars a cube, and that of Saturn a tetrahedron, which the orbit of Jupiter circ.u.mscribed.
Kepler's 1596 model of the universe. (Kepler, Mysterium Cosmographic.u.m [1596].).
Kepler attributed this revelation to ”divine ordinance.” He had ”always prayed to G.o.d [that] Copernicus had told the truth.” In his diary, he noted the fateful day when G.o.d spoke to him: July 19, 1595.
He was convinced he had discovered G.o.d's geometrical plan of the cosmos, which G.o.d had made in his own image. He published his work in 1596 in a book ent.i.tled Mysterium Cosmographic.u.m (The Mystery of the Universe).
However, his model was not in total agreement with Copernicus's data, especially the data for the orbit of Mercury. Despite his mystical leanings, Kepler was a new breed of scientist. He required theories to be supported by data. He decided that what he needed was more precise data. Copernicus's were not good enough.
From circles to ellipses.
Among the people to whom Kepler sent his new book was the greatest observational astronomer of the day, Tycho Brahe. Kepler by now was a handsome twenty-five-year-old with a high forehead, immaculate goatee, aquiline nose, and a look of piercing intelligence. Tycho, as he was always known, was twice Kepler's age. He sported a mustache so immense that it looked like a walrus's tusks and was famous for his prosthetic nose, having had his real one cut off in a duel. He had a copper nose for everyday and a gold and silver one for special occasions.
Tycho achieved his world-renowned accuracy by making all his observations from his monstrous-looking observatory, Uraniburg, on the island of Hveen, off the coast of Denmark. But to Kepler's annoyance, Tycho refused to reveal his data to anyone until he had refined his own model of the universe-in which every planet except the earth orbited the sun and the entire a.s.semblage, in turn, orbited the earth.
Impressed with Kepler, Tycho offered him a position as his a.s.sistant so that he could help him with the mathematics of his model, little realizing that Kepler simply wanted to lay his hands on his data. Kepler accepted the offer and joined Tycho in Prague, where he was imperial mathematician to the court of Emperor Rudolf II.
Johannes Kepler.
Tycho Brahe.
Tycho set Kepler to work to improve his observations of Mars, the most difficult of the planets due to its p.r.o.nounced retrograde motion. Astronomers described the orbit of Mars as having a large ”eccentricity”-the distance that the sun had to be moved from the center of Mars's...o...b..t to improve agreement with Tycho's data of the complicated system of circles rolling on circles. This displacement was a mathematical device used in every model of the universe-in Ptolemy's it was the earth whose position was displaced from the center of the universe. In reality, of course, in Copernicus's system, the sun was at the center of the universe. The models of Ptolemy, Copernicus, and Tycho could not deal adequately with Mars's eccentricity. Kepler bet his colleagues that he could straighten it out in eight days. In fact it took him eight years.
In October 1601 Tycho suddenly died at the age of fifty-five and Kepler was appointed imperial mathematician. He inherited all of Tycho's data and more important, no longer had to waste time fiddling with Tycho's model of the universe.
To start with, Kepler a.n.a.lyzed Tycho's data on the orbit of Mars, trying to preserve the old model of the universe by explaining the orbits of the planets in terms of circles. Taking Tycho's best data for Mars, he used the mathematical device of displacing the sun from the center of Mars's...o...b..t by a certain distance to allow for eccentricity. Then, by adroit mathematics, he moved himself from the earth to Mars and found that the earth also moved in an orbit similar to Mars's, with varying speeds.
Supposing the orbit of Mars was not a circle but an oval? Kepler spent 1604 struggling with the mathematics of an oval. That year was full of problems. Both he and his wife fell ill; when he became short of money his wealthy wife refused to dip into her funds; and she also gave birth to yet another child whom Kepler saw as yet another problem. And an ominous new star appeared in the sky-the nova of 1604.
Then he tried replacing the oval with an ellipse. An ellipse is a circle that has been squashed at its north and south poles. It has two centers, or focii, neither of which is in the middle. When the two centers are moved together the ellipse becomes a circle. This worked perfectly. The curve went through all of Tycho's data points for the orbit of Mars and also fitted Mars's measured eccentricity. Kepler had discovered his first law of planetary motion: that every planet moves in an ellipse with the sun at one of its centers. The sun is no longer at the center of the universe but at one of the ellipse's foci.
Soon after, he discovered his second law: that a line drawn from the sun to a planet sweeps out equal areas in equal times. This meant that a planet's speed varied as it traveled in an ellipse around the sun: the planet sped up as it neared the sun and slowed down as it moved away.
Kepler had overthrown the two-thousand-year-old a.s.sumptions that the complicated orbits of planets could only be explained by adding circles moving on circles in uniform circular motion and that the planets move with a uniform speed. He published his new laws of astronomy in his 1609 Astronomia nova (The New Astronomy).
But what kept the planets from escaping altogether and flying off into the void? Perhaps there were tentacles emanating from the sun, grasping a planet and whipping it around in its...o...b..t. Kepler imagined the attraction to be magnetism. Newton would later discover that it was gravity. Kepler, however, could only conceive of it as some sort of vital or living force.
As Pauli points out, Kepler was caught between two worlds. His laws of planetary motion were an accurate description of the paths of the planets around the sun, but they emerged from mathematical calculations that wrenched the sun out of its true place at the center of the universe. Using mathematics meant he had to treat the earth as dead matter. However, according to his Renaissance beliefs the earth was not dead at all, it had a soul, an anima terrae, akin to the human soul. It was a living organism. Sulphur and volcanic products were its excrement, springs coming from mountains its urine, metals and rainwater its blood and sweat, and sea water its nourishment. Kepler's attempts to link such animistic beliefs to scientific data made him a new breed-a scientific alchemist. He had no choice but to compartmentalize his work: ellipses were confined to the scientific side of his life, circles and spheres to the religious and alchemical side.
Kepler's third law.
In 1611 Emperor Rudolf abdicated. To escape the dangerous political intrigues that followed, Kepler moved to Linz, the capital of Upper Austria, a charming city on the Danube. Before leaving Prague, however, his wife fell ill with typhus and died.
Kepler's marriage had not been happy. Nevertheless, after her death he was lonely. He also had three young children to look after, two girls and a boy. He looked around for another wife in the same way he had discovered his two laws-by trial and error. He ended up with eleven choices, some of whom he had advertised for, others whom he had tried out, sometimes boarding his children with them to see if they all got along. One was attractive but too young, another fat, another was of poor health. Kepler finally settled on number five and she gave him the peace of mind to resume his scientific research.
His first two laws had been essentially geometrical-number was missing. Now he turned his attention to numbers. If the sun controlled the planets, he thought there had to be a relations.h.i.+p between the planets' distances from the sun and their speeds.
Meanwhile Europe was heading for the Thirty Years' War. Troops were on the move causing famine, havoc, and plague. Then one of his daughters died. In his grief he turned inward to ”contemplation of the Harmony,” which he believed to exist in nature. Thinking of the musical harmonies explored by Pythagoras, he pondered the eternal reality of numbers, which revealed the very essence of the soul.
How did this numerical harmony relate to the planets in a sun-centered system? Kepler tried to find a way to work out whether harmonious ratios could be formed out of the planets' periods of revolution, their volumes, their sizes, or their velocities when they were furthest from and closest to the sun. But he failed. Then he thought of examining the ratios of a planet's angular velocities at its extremes from the sun, that is, its change in angle at any period as it moved across the sky. And finally the astral music of the Divine Composer began to emerge.
Little by little Kepler worked out the ratios that produced the melodies played by the planets as they moved in their elliptical orbits. It was a heavenly symphony ”perceived by the intellect, not by the ear,” he wrote. But for Kepler it was much more. To him the planets sang in ”imitation of G.o.d” in different voices-soprano, contralto, tenor, and ba.s.s. But on the earth there was only discord: ”The Earth sings Mi-Fa-Mi, so we can gather even from this that Misery and Famine reign on our planet,” he wrote despondently.
Kepler's third law a.s.serts that the following two quant.i.ties are proportional: the time needed for the earth to go once around the sun, multiplied by itself (that is, squared); and the earth's average distance from the sun, multiplied by itself three times (that is, cubed). It completed for him what had been the goal of Pythagoras: to explain the universe in terms of geometry and number. He scoured tables of numbers until he found the pattern but he never revealed precisely how he had discovered this capstone of his life's work. He recorded the date: March 8, 1618. ”At first I thought I was dreaming,” he wrote in the book he published the following year, Harmonices Mundi.
Sure that G.o.d had spoken through him, he wrote that he did not mind if his book had to ”wait a hundred years for a reader. Did not G.o.d wait six thousand years for one to contemplate His works?”
All this took place at a time of great personal difficulty. Kepler's mother, Katharina, had been put on trial for witchcraft. Her sister had been burned at the stake as a witch, and this, together with her husband's disappearance, rendered her very suspicious to the gullible populace. In old age she was far from lovely and had a nasty temperament that made her an easy target in the witch-hunting mania in Germany of the early seventeenth century, so much so that she came close to sharing her sister's fate. In 1615, she was in the middle of a feud with another old woman. This neighbor persuaded an influential relative to accuse Katharina of making her extremely ill by feeding her a witch's potion. Others soon began to remember becoming seriously ill after having accepted drinks from Katharina.
Not only was his mother in danger, but so was the family name. Kepler had to take time off from pondering the universe to defend his mother for whom he felt affection and pity, despite his horrendous childhood. The proceedings took over six years. At one point jailers flourished instruments of torture and execution in front of Katharina's face, as was customary. Unusually for the time, the story has a happy ending. Kepler finally succeeded in obtaining her release.
Robert Fludd.
Robert Fludd-a universe made up of fours.
Two years before he finished Harmonices, in 1617, Kepler happened to see a highly ill.u.s.trated book at the Frankfurt book fair: A Metaphysical, Physical and Technical History of the Macro-and the Micro-Cosm, by the English physician and Rosicrucian, Robert Fludd.
While Kepler's family was low cla.s.s, Fludd's was n.o.ble. His father, Thomas Fludd, had been knighted by Queen Elizabeth I for his services as war treasurer in the Netherlands and paymaster to English troops in Provence. In portraits Fludd looks rather plump and well fed, with a pointed goatee. He holds his two middle fingers pressed together, perhaps in some sort of secret sign. In one portrait he has fingernails as long as a mandarin's.
Fludd studied at Oxford and became intrigued by Greek philosophy. As was the custom for wealthy young gentlemen, he toured France and Germany, meeting and sometimes tutoring n.o.bility. In Germany he became acquainted with a secret society who called themselves Brothers of the Rosy Cross-Rosicrucians. They called for a reform of knowledge in preparation for Armageddon and claimed access to deep secrets and truths in medicine, philosophy, and science. Governments deemed their mysticism and apocalyptic message dangerous and they were often charged with heresy and religious innovation, serious offenses in those days.