Pythagorus

by Kitty Ferguson

  Most significant for the history of astronomy, Book V began with an ecstatic statement about discovering the relationship between the planets’ orbital radiuses and their orbital periods, even though Kepler had not yet made this discovery when he began writing his book:

  At last I brought it into the light, and beyond what I had ever been able to hope, I laid hold of Truth itself: I found among the motions of the heavens the whole nature of Harmony, as large as that is, with all of its parts. It was not in the same way which I had expected – this is not the smallest part of my rejoicing – but in another way, very different and yet at the same time very excellent and perfect.5

  Kepler felt that this discovery – his third law of planetary motion, or ‘harmonic law’ – was so important that it was essential to go back and insert those sentences to let his readers know what was coming. It was also in Book V that he included a list giving a view of the astonishing mind of Kepler exactly where he was, beginning with a flat statement that sounds completely unremarkable to modern ears but was a blockbuster in his time. The list included his three planetary laws, which are still celebrated among the greatest discoveries in astronomy, but also – surprisingly – his old polyhedral theory.

  It is absolutely settled that the planets and Earth orbit the Sun. The Moon orbits the Earth.

  The planets’ orbits are eccentric. On one side of the Sun their orbits are furthest from the Sun, while on the other side of the Sun the orbits are closest to the Sun. A planet in orbit passes through all other distances between that maximum and minimum.

  There are six planets; that number is dictated by the fact that there are five regular polyhedra.

  Since the planets’ orbits are eccentric (proposition 2), the polyhedra by themselves cannot determine their distances from the Sun without other principles to establish the orbits, diameters, and eccentricities.

  A planet’s velocity is inversely proportional to its distance from the Sun. A planet’s orbit is an ellipse, with the Sun, ‘the source of motion’, being one focus of the ellipse.

  If two objects move the same actual distance, but one of them is farther away from the observer than the other, the movement of the one farther away appears smaller than the one nearer. So, if a planet never changed its speed, then, viewed from the Sun, its motions when it is farthest away would appear smaller than its motions when it is nearest. However, a planet does change its speed. Its motion is not the same at its nearest and farthest points, and the difference is in proportion to the distance from the Sun. In other words, the apparent sizes of the motions are different for two reasons: The actual sizes of the motions are different. Distance makes the sizes of the motions look different. The apparent sizes of the motions are very nearly the inverse square of the proportion of their distances from the Sun.

  When it comes to celestial harmony, it is motions as seen from the Sun that are important. Motions as seen from the Earth are irrelevant.

  The ratio of the squares of the orbital periods of two planets is equal to the ratio of the cubes of their average distances from the Sun. [This is the great ‘harmonic law’, one of Kepler’s most significant discoveries. Kepler made the discovery as he was finishing the book and came back and inserted it in this list.]

  9–13. [These propositions have to do with applying the harmonic law. Kepler tried to spell out more clearly that the ratio of the motions of two planets as they draw closer or move apart, together with the ratio of their periodic times, determine the extreme distances they can have (closest and farthest from the Sun), and this determines how eccentric their orbits are.]6

  Kepler eventually came to the conclusion that celestial harmony could not possibly be audible. There were no sounds in the heavens. How could they be enjoyed? Knowing or calculating the path lengths was too complicated to give pleasure in an instinctive way. The harmony of the cosmos could be best appreciated from the Sun itself, in the visible arcs of the planetary motions as they would be seen from there.7 (Hence number 7 in his list.)

  Think of the Sun, with yourself standing on it, as being at the centre of a huge clock face with the planets moving on large, nearly circular pathways near the rim of the clock face. The entire orbit of a planet is 360 degrees, all the way around the clock. The distance between one and two o’clock, viewed from the centre of the clock, is 30 degrees. You, on the Sun, see Earth circling you – though ‘circling’ is not quite the right word, since Earth’s orbit is not round but slightly elliptical. Earth is at aphelion (the part of its orbit farthest from the Sun and you). You watch for a twenty-four-hour period and find that Earth has moved 57'3" (57 ‘minutes’ and 3 ‘seconds’). Since there are sixty minutes in a degree, Earth has moved almost one degree. Suppose, instead, you are viewing Earth when it is at perihelion (the part of its orbit closest to the Sun). Now you find that Earth’s motion is faster, 61'18" in twenty-four hours, more than 1 degree. Those two measurements – Earth’s apparent diurnal motions at aphelion (57'3") and perihelion (61'18") – are not far different from one another. Earth’s orbit is not very eccentric.

  Kepler pondered how these two numbers might be adjusted so as to produce a harmonious interval in music. By changing 57'3" to 57'28" (a very small adjustment) he could make the interval a concinna, an interval that sounded pleasant in a melody though not when the two notes were played simultaneously. Kepler made similar tiny adjustments for the other planets’ orbits. The most troublesome was Venus, whose motion varied so slightly that its musical interval was a diesis. That was small indeed but still fell into Kepler’s category of concinna.

  Having worked with each planet individually, calculating and adjusting the relationship between its motion at aphelion and at perihelion, Kepler turned to studying the motions of pairs of planets, and was pleased to find fairly good harmony. The small adjustments necessary could, Kepler wrote, easily be ‘swallowed’ without detriment to the astronomy he had constructed using Tycho’s observational data. Again Venus was a problem, and so was Mercury, but their motions were not yet well established anyway.

  Satisfied with the way things were going so far, Kepler proceeded to assign actual notes to each of the planets at aphelion and perihelion and found that when he built a scale with Saturn (the lowest note) at aphelion, the result was a durus scale, a major scale. With Saturn at perihelion the result was a mollis scale, a minor scale. Planetary motion apparently did involve both types of scale. Using other planets as the starting note produced the different modes used in ancient music and church music.[11]

  Thus far, all of these combinations had the planets at the extremes of their motions, at aphelion or perihelion. Particularly for the planets most distant from the Sun, such opportunities would actually occur only rarely. However, if the planets involved in the harmony did not have to be at those extreme positions, the harmonic opportunities were much more numerous. For example: With Saturn moving between the pitches G and B (its pitches at perihelion and aphelion) and Jupiter between B and D, Kepler found, along the way, intervals of an octave, an octave plus a major or minor third, a fourth, and a fifth. Mercury, the true coloratura in the company, offered even more opportunities because the difference between its pitches at perihelion and aphelion was greater than an octave, and it made that change in only forty-four days. The result was that Mercury as it moved along sang every harmonic interval at least once with each of the other planets.9

  As Kepler calculated it, two-note harmonies of this sort occur almost every day, and Mercury, Earth, and Mars even sing three-part harmony fairly often. Venus, with so little eccentricity to its orbit, hardly varies its pitch at all, making it a sort of Johnny One-Note in the choir. If there is to be harmony with Venus, it must be when another planet slides into harmony with her, not the other way around. Four-note harmonies occur either because Mercury, Earth, and Mars are in adjustment with Venus’ monotone, or because they have waited long enough for the slow-changi
ng bass voice of Jupiter or Saturn to ease into the right note. ‘Harmonies of four planets’, wrote Kepler, ‘begin to spread out among the centuries; those of five planets, among myriads of years.’10 As for harmony among all six planets – that grand and greatest ‘universal harmony’ – the chord would be huge, spanning more than seven octaves. (You could not play it on most modern pianos. You would need an organ.) Kepler thought it might be possible for it to occur in the heavens only once in the entire history of the universe. Perhaps one might determine the moment of creation by calculating the past moment when all six planets joined in harmony. Kepler thought about the words of God to Job: ‘Where were you when I laid the Earth’s foundation . . . while the morning stars sang together?’

  Kepler dared to move ahead to what he felt was the true test of his theory: ‘Let us therefore extract, from the harmonies, the intervals of the planets from the Sun, using a method of calculation that is new and never before attempted by anyone.’11 If you did not know the astronomy of the solar system, could you deduce it correctly from the harmonic scheme he believed he had discovered? Starting with the best harmony and figuring out what planetary orbits and motions this harmony implied, what would be the observable consequences of the cosmos’ adhering to this harmony? Kepler used Tycho Brahe’s data for comparison and concluded that ‘all approach very closely to those intervals which I found from the Brahe observations. In Mercury alone there is a small difference’.

  Kepler proceeded to compare the solar system as dictated by his harmonic scheme with the solar system as dictated by his polyhedral theory. His conclusion was that the polyhedra, nested in the way he had earlier suggested, had been God’s rather loose model for the solar system. It dictated how many planets there would be and the approximate dimensions of the spheres within which they moved. It was a sort of sketch, with the final dimensions filled out by the harmonic proportions among the planets’ apparent motions as viewed from the Sun. The concept of ‘harmonies’ was required to reflect an eternally fluid system like that found among real planets in motion, and the real solar system could not be understood apart from its motion.

  Nearing the end of his book, Kepler imagined himself drifting off to sleep to the strains of the planetary harmony, ‘warmed by having drunk a generous draught . . . from the cup of Pythagoras’.12 He is soon dreaming about pure, simple beings who might live on the Sun, in the right position to appreciate the harmony, and of creatures on the other planets: It would be a terrible waste if there were none. They, like Earth dwellers, have no way of appreciating the harmony directly and can only learn of it, as humans had, by a combination of observation and reasoning. Kepler wrote a prayer that God would be praised by the heavens, by the Sun, Moon, and planets, by the celestial harmonies and their beholders – ‘by you above all, happy old Mästlin, for you used to inspire these things I have said, and you nourished them with hope’ – and by his, Kepler’s, own soul. He ended with a return to the old idea that was inherent from the start in the Pythagorean discovery of musical ratios: that one does not have to know about them to be moved by music. There is a mysterious inherent connection between human souls and the underlying pattern of the universe that affects us without our understanding why or how. The same was true, Tycho Brahe had thought, of the design of his palace/observatory. Kepler wrote:

  It does not suffice to say that these harmonies are for the sake of Kepler and those after him who will read his book. Nor indeed are aspects of planets on Earth for the sake of astronomers, but they insinuate themselves generally to all, even peasants, by a hidden instinct.13

  With modern hindsight, it seems Kepler took an odd, eccentric road indeed to arrive at his great ‘harmonic’ 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, ‘My 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.

  [1]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.

  [2]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.

  [3]When astronomers of Kepler’s time and earlier spoke of the ‘spheres’, they did not mean the planets. The Ptolemaic view of the cosmos had the planets travelling in transparent ‘crystalline 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.

  [4]Depending on one’s definition of ‘planet’, Pluto and some other bodies that orbit the Sun may or may not have that status. Hence ‘eight or nine’.

  [5]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 ‘beautiful’ and easy to listen to.

  [6]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.

  [7]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.

  [8]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.

  [9]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.

  [10]In German, dur in music still means ‘major’; moll is ‘minor’.

  [11]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

  Seventeenth–Nineteenth Centuries

  Kepler’s contemporary Galileo wrote that ‘Science’ was to be found ‘in a huge book that stands always open before our eyes – the universe’. But to understand it, one needed to be able to understand the language, and ‘the 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 – 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.2 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.