But how could this be done? How could the harmonies be fitted into the scheme of a universe full of elliptic orbits and nonuniform motions, from which, in fact, all symmetry and harmony seemed to have departed? As usual, Kepler takes the reader into his confidence, and for his benefit recapitulates the process by which he arrived at his solution. At first, he tried to assign the harmonic ratios to the periods of revolution of the various planets. He drew a blank:
"We conclude that God the Creator did not wish to introduce harmonic proportions into the durations of the planetary years." 13
Next, he wondered whether the sites or volumes of the various planets form a harmonic series. They do not. Thirdly, he tried to fit the greatest and smallest solar distances of every planet into a harmonic scale. Again no good. In the fourth place, he tried the ratios between the extreme velocities of each planet. Again no good.
Next, the variations in the time needed by a planet to cover a unit length of its orbit. Still no good. Lastly, he hit on the idea of transferring the observer's position into the centre of the world, and to examine the variations in angular velocity, regardless of distance, as seen from the sun. And lo! it worked.
The results were even more gratifying than he had expected. Saturn, for instance, when farthest away from the sun, in its aphelion, moves at the rate of 106 seconds arc per day; when closest to the sun, and its speed is at maximum, at 135 seconds arc per day. The ratio between the two extreme velocities is 106 to 135, which only differs by two seconds from 4:5 – the major third. With similar, very small deviations (which were all perfectly explained away at the end), the ratio of Jupiter's slowest to its fastest motion is a minor third, Mars' the quint, and so forth. So much for each planet considered by itself. But when he compared the extreme angular velocities of pairs of different planets, the results were even more marvellous:
"At the first glance the Sun of Harmony broke in all its clarity through the clouds." 14
The extreme values yield in fact the intervals of the complete scale. But not enough: if we start with the outermost planet, Saturn, in the aphelion, the scale will be in the major key; if we start with Saturn in the perihelion, it will be in the minor key. Lastly, if several planets are simultaneously at the extreme points of their respective orbits, the result is a motet where Saturn and Jupiter represent the bass, Mars the tenor, Earth and Venus the contralto, Mercury the soprano. On some occasions, all six can be heard together:
"The heavenly motions are nothing but a continuous song for several voices (perceived by the intellect, not by the ear); a music which, through discordant tensions, through sincopes and cadenzas, as it were (as men employ them in imitation of those natural discords), progresses towards certain pre-designed, quasi six-voiced clausuras, and thereby sets landmarks in the immeasurable flow of time. It is, therefore, no longer surprising that man, in imitation of his creator, has at last discovered the art of figured song, which was unknown to the ancients. Man wanted to reproduce the continuity of cosmic time within a short hour, by an artful symphony for several voices, to obtain a sample test of the delight of the Divine Creator in His works, and to partake of his joy by making music in the imitation of God." 15
The edifice was complete. Kepler finished the book on 27 May, 1618, in one of the most fateful weeks of European history:
"In vain does the God of War growl, snarl, roar and try to interrupt with bombards, trumpets, and his whole tarantantaran... 16 Let us despise the barbaric neighings which echo through these noble lands, and awaken our understanding and longing for the harmonies." 17
Out of the murky abyss he soared to heights of orphic ecstasies:
"The thing which dawned on me twenty-five years ago before I had yet discovered the five regular bodies between the heavenly orbits ...; which sixteen years ago I proclaimed as the ultimate aim of all research; which caused me to devote the best years of my life to astronomical studies, to join Tycho Brahe and to choose Prague as my residence – that I have, with the aid of God, who set my enthusiasm on fire and stirred in me an irrepressible desire, who kept my life and intelligence alert, and also provided me with the remaining necessities through the generosity of two Emperors and the Estates of my land, Upper Austria – that I have now, after discharging my astronomical duties ad satietatum, at long last brought to light... Having perceived the first glimmer of dawn eighteen months ago, the light of day three months ago, but only a few days ago the plain sun of a most wonderful vision – nothing shall now hold me back. Yes, I give myself up to holy raving. I mockingly defy all mortals with this open confession: I have robbed the golden vessels of the Egyptians to make out of them a tabernacle for my God, far from the frontiers of Egypt. If you forgive me, I shall rejoice. If you are angry, I shall bear it. Behold, I have cast the dice, and I am writing a book either for my contemporaries, or for posterity. It is all the same to me. It may wait a hundred years for a reader, since God has also waited six thousand years for a witness..." 18
6. The Third Law
This last quotation is from the Preface to the Fifth Book of the Harmonice Mundi, which contains, almost hidden among the luxuriant growth of fantasy, Kepler's Third Law of planetary motion.
It says, put into modern terms, that the squares of the periods of revolution of any two planets are as the cubes of their mean distances from the sun. 19 Here is an illustration of it. Let the earth's distance from the sun be our unit measure, then Saturn's distance from the sun will be a little over nine units. The square root of 1 is 1; the square root of 9 = 3. The cube of 1 is 1, the cube of 9 is 27. Thus a Saturn year will be a little over twenty-seven earth years; in fact it is thirty years. Apologies for the coarse example – it is Kepler's own. 20
Unlike his First and Second Laws, which he found by that peculiar combination of sleepwalking intuition and wide-awake alertness for clues – a mental process on two levels, which drew mysterious benefits out of his apparent blunderings – the Third Law was the fruit of nothing but patient, dogged trying. When after endless trials, he had at last hit on the square-to-cube ratio, he of course promptly found a reason why it should be just that and none other; I have said before that Kepler's a priori proofs were often invented a posteriori.
The exact circumstances of the discovery of the Third Law were again faithfully recorded by Kepler:
"On March 8 of this present year 1618, if precise dates are wanted, [the solution] turned up in my head. But I had an unlucky hand and when I tested it by computations I rejected it as false. In the end it came back again to me on May 15, and in a new attack conquered the darkness of my mind; it agreed so perfectly with the data which my seventeen years of labour on Tycho's observations had yielded, that I thought at first I was dreaming, or that I had committed a petitio principi..." 21
He celebrated his new discovery, as he had celebrated his First Law, with a quotation from Virgil's Eclogues; in both cases Truth appears in the shape of a teasing hussy who surrenders unexpectedly to her pursuer when he has already given up hope. And in both cases also, the true solution was rejected by Kepler when it first occurred to him, and was only accepted when it crept in a second time, "through a back-door of the mind".
He had been searching for this Third Law, that is to say, for a correlation between a planet's period and its distance, since his youth. Without such a correlation, the universe would make no sense to him; it would be an arbitrary structure. If the sun had the power to govern the planets' motions, then that motion must somehow depend on their distance from the sun; but how? Kepler was the first who saw the problem – quite apart from the fact that he found the answer to it, after twenty-two years of labour. The reason why nobody before him had asked the question is that nobody had thought of cosmological problems in terms of actual physical forces. So long as cosmology remained divorced from physical causation in the mind, the right question could not occur in that mind. Again a parallel to the present situation imposes itself: there is, one suspects, a fragmentation in the twentieth century mind w
hich prevents it from asking the right questions. The offspring of a new synthesis is not a ready solution, but a healthy problem crying lustily for an answer. And vice versa: a one-sided philosophy – whether it be scholasticism or nineteenth-century mechanism, creates sick problems, of the sort "What is the sex of the angels?" or "Is man a machine?"
7. The Ultimate Paradox
The objective importance of the Third Law is that it provided the final clue for Newton; hidden away in it is the essence of the Law of Gravity. But its subjective importance to Kepler was that it furthered his chimerical quest – and nothing else. The Law makes its first appearance as "Proposition No. 8" in a chapter characteristically called "The Main Propositions of Astronomy which are needed for the Investigation of the Celestial Harmonies". In the same chapter (the only one in the book which deals with astronomy proper) the First Law is merely mentioned in passing, almost shamefacedly, and the Second Law not at all. In its place Kepler once more quoted his faulty inverse ratio proposition, whose incorrectness he once knew and then forgot. Not the least achievement of Newton was to spot the Three Laws in Kepler's writings, hidden away as they were like forget-me-nots in a tropical flowerbed.
To change metaphors once more: the three Laws are the pillars on which the edifice of modern cosmology rests; but to Kepler they meant no more than bricks among other bricks for the construction of his baroque temple, designed by a moonstruck architect. He never realized their real importance. In his earliest book he had remarked that "Copernicus did not know how rich he was"; the same remark applies to Kepler himself.
I have stressed this paradox over and again; now it is time to try to resolve it. Firstly, Kepler's obsession with a cosmos built around the Pythagorean solids and the musical harmonies, was not quite as extravagant as it seems to us. It was in keeping with the traditions of Neoplatonism, with the revival of Pythagoreanism, with the teaching of Paracelsians, Rosicrucians, astrologers, alchemists, cabbalists and hermetists, who were still conspicuously in evidence in the early seventeenth century. When we talk of "the age of Kepler and Galileo", we are apt to forget that they were isolated individuals, a generation ahead of the most enlightened men of their time. If the "harmony of the world" was a fantastic dream, its symbols had been shared by a whole dreaming culture. If it was an idée fixe, it was derived from a collective obsession – only more elaborate and precise, enlarged on a grandiose scale, more artful and self-consistent, carried to the ultimate perfection of mathematical detail. The Keplerian cosmos is the crowning achievement of a type of cosmic architecture which began with the Babylonians and ends with Kepler himself.
The paradox, then, is not in the mystic nature of Kepler's edifice but in the modern architectural elements which it employed, in its combination of incompatible building materials. Dream-architects are not worried about imprecisions of a fraction of a decimal; they do not spend twenty years with dreary, heart-breaking computations to build their fantasy towers. Only some forms of insanity show this pedantic method in madness. In reading certain chapters of the Harmonice, one is indeed reminded of the explosive yet painstakingly elaborate paintings by schizophrenics, which would pass as legitimate art if painted by a savage or a child, but are judged by clinical standards if one knows that they are the work of a middle-aged chartered accountant. The Keplerian schizophrenia becomes apparent only when he is judged by the standard of his achievements in optics, as a pioneer of the differential calculus, the discoverer of the three Laws. His split mind is revealed in the manner in which he saw himself in his non-obsessional moments: as a sober "modern" scientist, unaflected by any mystic leanings. Thus he writes about the Scottish Rosicrucian, Robert Fludd:
"It is obvious that he derives his main pleasure from unintelligible charades about the real world, whereas my purpose is, on the contrary, to draw the obscure facts of nature into the bright light of knowledge. His method is the business of alchemists, hermetists and Paracelsians, mine is the task of the mathematician." 22
These words are printed in Harmonice Mundi, which is buzzing with astrological and Paracelsian ideas.
A second point is equally relevant to the Keplerian paradox. The main reason why he was unable to realize how rich he was – that is, to understand the significance of his own Laws – is a technical one: the inadequacy of the mathematical tools of his time. Without differential calculus and/or analytical geometry, the three Laws show no apparent connection with each other – they are disjointed bits of information which do not make much sense. Why should God will the planets to move in ellipses? Why should their speed be governed by the area swept over by the radius vector, and not by some more obvious factor? Why should the ratio between distance and period be mixed up with cubes and squares? Once you know the inverse square law of gravity and Newton's mathematical equations, all this becomes beautifully self-evident. But without the roof which holds them together, Kepler's Laws seem to have no particular raison d'être. Of the first he was almost ashamed: it was a departure from the circle sacred to the ancients, sacred even to Galileo and, for different reasons, to himself. The ellipse had nothing to recommend it in the eyes of God and man; Kepler betrayed his bad conscience when he compared it to a cartload of dung which he had to bring into the system as a price for ridding it of a vaster amount of dung. The Second Law he regarded as a mere calculating device, and constantly repudiated it in favour of a faulty approximation; the Third as a necessary link in the system of harmonies, and nothing more. But then, without the notion of gravity and the method of the calculus, it could be nothing more.
Johannes Kepler set out to discover India and found America. It is an event repeated over and again in the quest for knowledge. But the result is indifferent to the motive. A fact, once discovered, leads an existence of its own, and enters into relations with other facts of which their discoverers have never dreamt. Apollonius of Perga discovered the laws of the useless curves which emerge when a plane intersects a cone at various angles: these curves proved, centuries later, to represent the paths followed by planets, comets, rockets, and satellites.
"One cannot escape the feeling," wrote Heinrich Herz, "that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser thin we are, wiser even than their discoverers, that we get more out of them than was originally put into them."
This confession of the discoverer of radio-waves sounds suspiciously like in echo of Kepler, echoing Plato, echoing Pythagoras: "Methinks that all of nature and the graceful sky are set into symbols in geometriam."
X COMPUTING A BRIDE
ONLY one circumstance, but a basic one, relieved the gloom of Kepler's later years: his second marriage, in 1613, to Susanna Reuttinger. He was forty-one, she twenty-four, the daughter of a cabinet-maker. Susanna's parents had died while she was a child; she had been brought up in the household of the Baroness Starhemberg. We do not know what position she occupied in the household, but to judge by the scandalized reactions of Kepler's correspondents, it must have been a lowly one – something between a maid and a companion.
Kepler's first marriage had been engineered by his well-wishers when he was an inexperienced and penniless young teacher. Before his second marriage, friends and go-betweens again played a prominent part – but this time Kepler had to choose between no less than eleven candidates for his hand. In a letter to an unknown nobleman, which extends to eight printed folio pages, Kepler has described in meticulous detail the process of elimination and selection that he followed. It is a curious document, and among the most revealing in his voluminous writings. It shows that he solved the problem of choosing the right wife among the eleven candidates by much the same method by which he found the orbit of Mars: he committed a series of mistakes which might have proved fatal, but cancelled out; and up to the last moment he failed to realize that he held the correct solution in his hands.
The letter is dated from Linz, 23 October, 1613: 1
"Though all Christians start a wedding invitatio
n by solemnly declaring that their marriage is due to special Divine management, I as a philosopher, would like to discourse with you, O wisest of men, in greater detail about this. Was it Divine Providence or my own moral guilt which, for two years or longer, tore me in so many different directions and made me consider the possibilities of such different unions? If it was Divine Providence, to what purpose did it use these various personalities and events? For there is nothing that I would like to investigate more thoroughly, and that I more intensely long to know, than this: can I find God, whom I can almost touch with my hands when I contemplate the universe, also in my own self? If, on the other hand, the fault was mine, in what did it consist? Cupidity, lack of judgment, or ignorance? And why, on the other hand, was there nobody among my advisers to approve of my final decision? Why am I losing their previous esteem or appear to be losing it?
'What could have seemed more reasonable than that I, as a philosopher, past the peak of virility, at an age when passion is extinct, the body dried and softened by nature, should have married a widow who would look after the household, who was known to me and my first wife, and unmistakably recommended to me by her? But if so, why did nothing come of it? ..."
The reasons why this first project came to nothing were, among others, that the prospective bride had two marriageable daughters, that her fortune was in the hands of a trustee; and, as an afterthought,
"also a consideration of health, because, though her body was strong, it was suspect of ill-health because of her stinking breath; to this came my dubious reputation in matters of religion. In addition to this, when I met the woman after everything had been settled (I had not seen her for the last six years), there was nothing about her that pleased me. It is therefore sufficiently clear that the matter could not succeed. But why did God permit that I should be occupied with this project which was doomed to failure? Perhaps to prevent my getting involved in other perplexities while my thoughts were on this person ? ... I believe that things like this happen to others too, not only once but often; but the difference is that others do not worry as much as I do, that they forget more easily and get over things quicker than I do; or that they have more self-control and are less credulous than I am... And now for the others.