Pythagorus (39 page)
With modern hindsight, it seems Kepler took an odd, eccentric road indeed to arrive at his great âharmonic' law. He found it twice, at first rejecting it because of a computational error on March 8, 1618, and then discovering that it was correct a few weeks later, on May 15. The comment has sometimes been made that the harmonic law was an accidental discovery in the midst of a labyrinth of worthless musical/mathematical speculation, and that Kepler hardly realised he had made an important discovery. But Kepler definitely knew it was significant. It was in response to this discovery that he fell to his knees and exclaimed, âMy God, I am thinking Thy thoughts after Thee.' Without the underpinning of modern mathematics and the modern scientific method, the convoluted musical path Kepler took may have been the only way he could have got there. After all, he was the one who did get there. Kepler had one of the truest ears in history for the harmony of mathematics and geometry.
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A regular polygon is a flat shape in which all edges are the same length. For example: the triangle, square, pentagon, hexagon, etc. ad infinitum.
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A regular polyhedron is a solid shape in which all the edges have the same length and all the faces the same shape. The Pythagorean or Platonic solids are the regular polyhedra.
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When astronomers of Kepler's time and earlier spoke of the âspheres', they did not mean the planets. The Ptolemaic view of the cosmos had the planets travelling in transparent âcrystalline spheres', nested within one another like the layers of an onion and centred on the Earth. Though Kepler and Mästlin discussed spheres in their correspondence about Kepler's new idea, Kepler (like his predecessor Tycho Brahe) did not believe there were actual glasslike spheres that one could crash through in a space vehicle. Thinking about them in a geometrical sense, not as physical reality, was nevertheless helpful in visualising the movements of the planets.
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Depending on one's definition of âplanet', Pluto and some other bodies that orbit the Sun may or may not have that status. Hence âeight or nine'.
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An example of a third on the piano is the interval from C to E (major third) or C to E-flat (minor third). An example of a sixth is the interval from C to A (major sixth) or C to A-flat (minor sixth). These are intervals that modern ears are most likely to hear as âbeautiful' and easy to listen to.
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The Tychonic system had the Sun and the Moon orbiting the Earth, and all the other planets orbiting the Sun. It was the geometric equivalent of the Copernican system, but retained the unmoving Earth.
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Kepler's first law of planetary motion: A planet moves in an elliptical orbit and the Sun is one focus of the ellipse. Kepler's second law of planetary moton: A straight line drawn from a planet to the Sun sweeps out equal areas in equal times as the planet travels in its elliptical orbit.
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A half-step is the interval between any note on the piano and the one immediately to either side of it, regardless of whether that is a white or black key.
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Imagine you are standing across the corridor from a moving walkway in an airport. A man is walking along the walkway, from your left to your right, but he is going the wrong way and so is actually losing ground. Say he is walking at 5 miles per hour and the walkway is moving, in the opposite direction, at 10 miles per hour. From your vantage point, you see the combined movement, and the man appears to be moving 5 miles an hour towards the left. A woman is walking faster, 8 miles per hour, but also in the wrong direction. Eight miles per hour is not sufficient to avoid losing ground against the 10-mile-an-hour walkway that is moving in the opposite direction, so, again, from your vantage point, you see the combined motion, and this woman appears to be moving 2 miles per hour towards your left. You cannot be faulted for thinking that the man (who appears to be moving 5 miles per hour towards your left) is moving faster than the woman. If the walkway stopped you would find out what the true velocity of each one was, and your finding would contradict your initial impression. Likewise, Kepler concluded that if the daily rotation of the heavens had stopped, Pythagoras would have seen that Saturn is the slowest of the planets, and should be sounding the lowest tone.
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In German,
dur
in music still means âmajor';
moll
is âminor'.
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You get the same result by playing scales using only the white keys on a piano but starting on different notes. The Ionian mode (start on C) is the same as the major scale, the Dorian mode (start on D), the Phrygian mode (E), the Lydian mode (F), and the Mixolydian mode (G). The Aolian mode (start on A) is the same as the minor scale.
CHAPTER 17
Enlightened and Illuminated
SeventeenthâNineteenth Centuries
Kepler's contemporary Galileo
wrote that âScience' was to be found âin a huge book that stands always open before our eyes â the universe'. But to understand it, one needed to be able to understand the language, and âthe language is mathematics'.
1
Galileo was not the first in his family to win a place in history. His father, Vincenzo, appears in textbooks of music history as a prominent musician of the sixteenth century â a composer, one of the best music theorists of his time, and a fine lutenist. One of his areas of research was ancient Greek music, and there is a story that when he read Boethius'
De musica
, the account of Pythagoras hanging weights on lengths of string, plucking the strings, and discovering the ratios of musical harmony piqued his curiosity.
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Amazingly, no records survive, from all the prior centuries during which scholars had been reading Boethius, of anyone trying this to see whether it would work. Vincenzo discovered, of course, that it did not, but he went on experimenting with the physics of vibrating strings. When his son watched a lamp swinging in the Pisa cathedral and first decided to experiment with pendulums, perhaps he had in mind his father's tests with weights and strings.
Two decades later, the younger Galileo, though largely oblivious to the work Kepler was doing, had become personally convinced that the Copernican system was correct, and he was looking for physical evidence to support that opinion and convince other scholars. Copernicus had mentioned in
De revolutionibus
that the planet Venus might supply important evidence in the case against an Earth-centred cosmos. Venus, reflecting the Sun's light, waxes and wanes as the Moon does, but if the Ptolemaic arrangement of the cosmos were correct, Earth dwellers would never be positioned in such a way as to see the face of Venus anywhere near fully lit (the equivalent of a full Moon). As the first decade of the seventeenth century drew to a close, the newly invented telescope (Galileo did not invent it but was putting it to better use than anyone else) made it possible to observe the phases of Venus as never before, and in 1610 Galileo followed up on Copernicus' suggestion. He found that Venus had a full range of phases. How could any scholar fail to see that this was irrefutable evidence in favour of Copernicus? But Galileo's Catholic colleagues included a group of recalcitrant scholars who remind one of an unusually virulent strain of
acusmatici
.
Except in the case of Giordano Bruno, whose offences by church standards were so flagrant and numerous that he would almost surely have been burned at the stake no matter where he thought the centre of the universe was, the Catholic church hierarchy had for centuries been rather sluggishly tolerant of new astronomical theories. Not a murmur was heard when Nicholas of Cusa, in the early fifteenth century, put the Earth in motion and removed it from the centre of the universe, nor when Copernicus published
De revolutionibus
in 1543. Two of Copernicus' strongest supporters were prominent Catholic clergy. But in 1616, when both Galileo and his opponents were pushing the church for a ruling on the Copernican question, a decree was issued condemning the ânew' astronomy, though not actually calling it heresy â a technicality perhaps, but a victory for Galileo and the cardinals who supported him. In this decree, the Pythagoreans took an unfair hit:
And whereas it has also come to the knowledge of the said Congregation that the Pythagorean doctrine â which is false and altogether opposed to the Holy Scripture â of the motion of the Earth and the immobility of the Sun, which is also taught by Nicolaus Copernicus . . . is now being spread abroad and accepted by many, as may be seen from a certain letter of a Carmelite Father.
The Carmelite father who had put the Pythagoreans in the range of fire was the Reverend Father Paolo Antonio Foscarini. His letter, dated the year before the decree, was titled âOn the Opinion of the Pythagoreans and of Copernicus Concerning the Motion of the Earth, and the Stability of the Sun, and the New Pythagorean System of the World'. Foscarini insisted this doctrine was âconsonant with truth and not opposed to Holy Scripture'. The church's âGeneral Congregation of the Index', which made official judgements on such matters, felt differently. Copernicus' book
De revolutionibus
â seventy-three years after its publication â was âsuspended until corrected', and Foscarini's work was âaltogether prohibited and condemned'. It took seventeen more years of on-and-off sparring, and Galileo's book
Dialogo
, for matters to come to a truly dangerous head in his famous trial. The Catholic church, for centuries the guardian and bastion of learning, had turned foolish to the point of malign senility and condemned herself and Italy â the ancient home of Pythagoras â to what was virtually a new scientific dark age. The centre of scientific endeavour and achievement moved, irretrievably, to northern Europe and England.
As the scientific revolution continued north of the Alps in the mid-seventeenth century, Kepler's three laws of planetary motion and his Rudolfine Tables, based on Tycho Brahe's observations, rightly gave him his earthly immortality, but his polyhedral theory and most of
Harmonice mundi
were consigned to the cabinet of curiosities. No one took nested polyhedrons or cosmic chords and scales seriously or followed up on them as science. They had been the odd and unlikely midwives to Kepler's ânew astronomy', helping birth the future, but in doing so had relegated themselves to the past. However, the conviction that numbers and harmony and symmetry were guides to truth because the universe was created according to a rational, orderly plan began to be treated as a given, trustworthy enough to underpin what would later be called the scientific method.
No one was using the words âscience' or âscientific' yet in their modern sense, but the process for determining what was and was not true about nature and the universe was continuing to evolve, and people were discussing and beginning to agree about how this process should work. The French scientist and philosopher René Descartes, one of the first to try to establish a solid foundation for human understanding of the world, chose mathematics as the only trustworthy road to sure knowledge.
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He tried to show that a single, united system of logical mathematical theory could account for everything that happens in the physical universe. Christiaan Huygens, Edmond Halley, and Isaac Newton all shared the conviction that when observations were inadequate, one could even with some confidence go out on a limb on the assumption that the universe is orderly, and discovering new examples of âorder' was beginning to be regarded as a sign that one was on the right track. Robert Hooke, in the field of biology, suggested that crystals like those that may have alerted the Pythagoreans to the existence of the five regular solids occurred because their atoms had an orderly arrangement.
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Robert Boyle wrote his book
The Sceptical Chymist
, which many identify as marking the beginning of modern chemistry, and cited Pythagoras, asserting that the final decisions of science must be made on the basis of both the evidence of the senses and the operation of reason. This balance, on which Kepler had performed such prodigious acrobatics as he struggled to write his
Astronomia nova
â without thinking of it as a âscientific method' â was becoming the balance of science.
Newton, born mid-century, capped off the Copernican revolution with his discovery of the laws of gravity and his 1687 book
Philosophiae Naturalis Principia Mathematica
(âMathematical Principles of Natural Philosophy'), known as his
Principia
. A fervent believer in the harmony and order of the universe, he was convinced that the observable patterns in the cosmos were the visible manifestation of a profound, mysterious, underlying order. His theories of gravitation admirably supported the Pythagorean ideal of unity and simplicity. The same force, gravity, that kept the planets in orbit also dictated the trajectory of a ball thrown on Earth and kept human beings' feet on the ground, and its laws could be stated in a simple formula. Though he was notoriously miserly about giving credit where credit was clearly due among his contemporaries, Newton, in an extraordinary gesture, wrote that his own famous law of universal gravitation could be found in Pythagoras. Nor was this the extent of Newton's unusual attributions. He sought examples among the Greeks, the Hebrews, and other ancient thinkers, of ideas and discoveries that seemed â sometimes it was quite a stretch â to foreshadow his own. This was not modesty. Newton was by no means a modest man. It was more a way of elevating himself to the company of the greatest sages. Better than discovering something new was rediscovering knowledge that God had previously revealed only to extraordinary men of legendary wisdom. Newton thought of another link with Pythagoras when he used a prism and split the light of the Sun into seven colours. There were seven notes in the Pythagorean scale.
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Gottfried Leibniz, Newton's arch-rival and one of those contemporaries to whom Newton should have given considerably more credit, wrote in Pythagorean tones that âmusic is the pleasure the human soul experiences from counting without being aware that it is counting'.
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Leibniz tried to construct a universal language which had no words, that could express all human statements and resolve arguments in a completely unambiguous way, even, he hoped, bring into agreement all versions of Christian faith. His attempts to make good on this scheme included a use of numbers that would have pleased the Pythagoreans and annoyed Aristotle: âFor example, if the term for an “animate being” should be imagined as expressed by the number 2, and the term for ârational' by the number 3, the term for “man” will be expressed by the number 2Ã3, that is 6'.
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Newton's discoveries about
gravity showed the cosmos seeming to operate like a stupendous, dependable mechanism, and, in the eighteenth century, scholars and amateur science aficionados picked up on that idea and became obsessed with mechanisms and machines. The demonstration of a new apparatus to explain or test a scientific principle was likely to cause more excitement than a lecture or a new theory at meetings of the Royal Society of London for Improving Natural Knowledge, or of the Birmingham âLunar Men' of Charles Darwin's grandfather. It was the age of the âclockmaker's universe' and of England's industrial revolution. Careful observation and experiment became the hallmark of science, but cautious generalisation was also encouraged, especially if it led to practical applications.
In other ways, in the eighteenth century, the universe was failing to live up to its promise of simplicity. The Swedish botanist Carl Linnaeus was applying two-word Latin names to more and more species that travellers and voyagers to all corners of the world were discovering. There were a greater number than anyone had ever imagined. Linnaeus saw new plants in his garden, too, and began to suspect, a century before Darwin's
Origin of Species
, that new species were emerging all the time. He decided that these had always existed in the mind of God but were just now coming into material existence, a very Platonic way of assuaging his religious scruples.
Carl Linnaeus
No one's faith in the completeness of universal harmony and the power of numbers surpassed that of the French mathematician Pierre Simon de Laplace, whose lifetime spanned the turn of the eighteenth to the nineteenth century. For him, numbers and mathematics were an unshakably trustworthy bridge to the past and future â if one could know the exact state of everything in the universe at a given moment. His contention was that an omniscient being with that knowledge, with unlimited powers of memory and mental calculation, and with knowledge of the laws of nature, could extrapolate from that the exact state of everything in the universe at any other given moment.
Meanwhile, Pythagorean themes appeared in other than scientific settings. The Whig party praised the governmental structure which brought together king and Parliament by means of ânatural' laws, with these words:
What made the planets in such Order move,
He said, was harmony and mutual Love.
The Musick of his Spheres did represent