Why indeed an oval? There is something in the perfect symmetry of spheres and circles which has a deep, reassuring appeal to the unconscious – otherwise it could not have survived two millennia. The oval lacks all such archetypal appeal. It has an arbitrary form. It distorts that eternal dream of the harmony of the spheres, which lay at the origin of the whole quest. Who art thou, Johann Kepler, to destroy divine symmetry? All he has to say in his own defence is, that having cleared the stable of astronomy of cycles and spirals, he left behind him "only a single cartful of dung": his oval. 24
At this point, the sleepwalker's intuition failed him, he seems to be overcome by dizziness, and clutches at the first prop that he can find. He must find a physical cause, a cosmic raison d'être for his oval in the sky – and he falls back on the old quack remedy which he has just abjured, the conjuring up of an epicycle! To be sure, it is an epicycle with a difference: it has a physical cause. We have heard earlier on that while the sun's force sweeps the planet round in a circle, a second, antagonistic force, "seated in the planet itself " makes it turn in a small epicycle in the opposite direction. It all seems to him "wonderfully plausible", 25 for the result of the combined movement is indeed an oval. But a very special oval: it has the shape of an egg, with the pointed end at the perihelion, the broad end at the aphelion.
No philosopher had laid such a monstrous egg before. Or, in Kepler's own words of wistful hindsight:
"What happened to me confirms the old proverb: a bitch in a hurry produces blind pups... But I simply could not think of any other means of imposing an oval path on the planets. When these ideas fill upon me, I had already celebrated my new triumph over Mars without being disturbed by the question ... whether the figures tally or not... Thus I got myself into a new labyrinth... The reader must show tolerance to my gullibility." 26
The battle with the egg goes on for six chapters, and took a full year of Kepler's life. It was a difficult year; he had no money, and was down with "a fever from the gall". A threatening new star, the nova of 1604 had appeared in the sky; Frau Barbara was also ill, and gave birth to a son – which provided Kepler with an opportunity for one of his dreadful jokes:
"Just when I was busy squaring my oval, an unwelcome guest entered my house through a secret doorway to disturb me." 27
To find the area of his egg, he again computed series of one hundred and eighty Sun-Mars distances and added them together; and this whole operation he repeated no less than forty times. To make the worthless hypothesis work, he temporarily repudiated his own, immortal Second Law – to no avail. Finally, a kind of snowblindness seemed to descend on him: he held the solution in his hand without seeing it. On 4 July, 1603, he wrote to a friend that he was unable to solve the geometrical problems of his egg; but "if only the shape were a perfect ellipse all the answers could be found in Archimedes' and Appollonius' work". 28 A full eighteen months later, he again wrote to the same correspondent that the truth must he somewhere half-way between egg-shape and circular-shape "just as if the Martian orbit were a perfect ellipse. But regarding that I have so far explored nothing." 29 What is even more astonishing, he constantly used ellipses in his calculations – but merely as an auxiliary device to determine, by approximation, the area of his egg-curve – which by now had become a veritable fixation. Was there some unconscious biological bias behind it? Apart from the association between the squaring of the egg and the birth of his child, there is nothing to substantiate that attractive hypothesis. *
____________________
*
It will be remembered that Copernicus, too, stumbled on the ellipse and kicked it aside; but Copernicus, who firmly believed in circles, had much less reason to pay attention to it than Kepler, who had progressed to the oval.
And yet, these years of wandering through the wilderness were not entirely wasted. The otherwise sterile chapters in the New Astronomy devoted to the egg hypothesis, represent an important further step towards the invention of the infinitesimal calculus. Besides, Kepler's mind had by now become so saturated with the numerical data of the Martian orbit, that when the crucial hazard presented itself, it responded at once like a charged cloud to a spark.
This hazard is perhaps the most unlikely incident in this unlikely story. It presented itself in the shape of a number which had stuck in Kepler's brain. The number was 0.00429.
When he at last realized that his egg had "gone up in smoke" 30 and that Mars, whom he had believed a conquered prisoner "securely chained by my equations, immured in my tables", had again broken loose, Kepler decided to start once again from scratch.
He computed very thoroughly a score of Mars-Sun distances at various points of the orbit. They showed again that the orbit was some kind of oval, looking like a circle flattened inward at two opposite sides, so that there were two narrow sickles or "lunulae" left between the circle and the Martian orbit. The width of the sickle, where it is thickest, amounted to 0.00429 of the radius:
At this point Kepler, for no particular reason, became interested in the angle at M – the angle formed between the sun and the centre of the orbit, as seen from Mars. This angle was called the "optical equation". It varies, of course, as Mars moves along its orbit; its maximum value is 5° 18′. This is what happened next, in Kepler's own words: 31
"... I was wondering why and how a sickle of just this thickness (0.00429) came into being. While this thought was driving me around, while I was considering again and again that ... my apparent triumph over Mars had been in vain, I stumbled entirely by chance on the secant * of the angle 5° 18′, which is the measure of the greatest optical equation. When I realized that this secant equals 1.00429, I felt as if I had been awakened from a sleep..."
It had been a true sleepwalker's performance. At the first moment, the reappearance of the number 0.00429 in this unexpected context must have appeared as a miracle to Kepler. But he realized in a flash that the apparent miracle must be due to a fixed relation between the angle at M and the distance to S, a relation which must hold true for any point of the orbit; only the manner in which he had stumbled on that relation was due to chance. "The roads that lead man to knowledge are as wondrous as that knowledge itself."
Now at last, at long last, after six years of incredible labour, he held the secret of the Martian orbit. He was able to express the manner in which the planet's distance from the sun varied with its position, in a simple formula, a mathematical Law of Nature. But be still did not realize that this formula specifically defined the orbit as an ellipse. † Nowadays, a student with a little knowledge of analytical geometry would realize this at a glance; but analytical geometry came after Kepler. He had discovered his magic equation empirically, but he could no more identify it as the shorthand sign for an ellipse than the average reader of this book can; it was nearly as meaningless to him. He had reached his goal, but he did not realize that he had reached it.
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*
The "secant" of the angle at M is the ratio MC: MS.
†
In modern denotation, the formula is: R = 1 e cos β where R is the distance from the sun, β the longitude referred to the centre of the orbit, and e the eccentricity.
The result was that he went off on one more, last, wild goose chase. He tried to construct the orbit which would correspond to his newly discovered equation; but he did not know how, made a mistake in geometry, and arrived at a curve which was too bulgy; the orbit was a via buccosa, chubby-faced, as he noted with disgust.
What next? We have reached the climax of the comedy. In his despair, Kepler threw out his formula (which denoted an elliptic orbit) because he wanted to try out an entirely new hypothesis: to wit, an elliptic orbit. It was as if the tourist had told the waiter, after studying the
menu: "I don't want côtelette d'agneau, whatever that is; I want a lamb chop."
By now he was convinced that the orbit must be an ellipse, because countless observed positions of Mars, which he knew almost by heart, irresistibly pointed to that curve; but he still did not realize that his equation, which he had found by chance-plus-intuition, was an ellipse. So he discarded that equation, and constructed an ellipse by a different geometrical method. And then, at long last, he realized that the two methods produced the same result.
With his usual disarming frankness, he confessed what had happened:
"Why should I mince my words? The truth of Nature, which I had rejected and chased away, returned by stealth through the backdoor, disguising itself to be accepted. That is to say, I laid [the original equation] aside, and fell back on ellipses, believing that this was a quite different hypothesis, whereas the two, as I shall prove in the next chapter, are one and the same ... I thought and searched, until I went nearly mad, for a reason why the planet preferred an elliptical orbit [to mine]... Ah, what a foolish bird I have been!" 32
But in the List of Contents, in which he gives a brief outline of the whole work, Kepler sums up the matter in a single sentence:
"I show [in this chapter] how I unconsciously repair my error."
The remainder of the book is in the nature of a mopping-up operation after the final victory.
8. Some Conclusions
It was indeed a tremendous victory. The great Ferris wheel of human delusion, with its celestial catwalks for the wandering planets, this phantasmagoria which had blocked man's approach to nature for two thousand years, was destroyed, "banished to the lumber-room". Some of the greatest discoveries, as we saw, consist mainly in the clearing away of psychological road-blocks which obstruct the approach to reality; which is why, post factum, they appear so obvious. In a letter to Longomontanus, 33 Kepler qualified his own achievement as the "cleansing of the Augean stables".
But Kepler not only destroyed the antique edifice; he erected a new one in its place. His Laws are not of the type which appear self-evident, even in retrospect (as, say, the Law of Inertia appears to us); the elliptic orbits and the equations governing planetary velocities strike us as "constructions" rather than "discoveries". In fact, they make sense only in the light of Newtonian Mechanics. From Kepler's point of view, they did not make much sense; he saw no logical reason why the orbit should be an ellipse instead of an egg. Accordingly, he was more proud of his five perfect solids than of his Laws; and his contemporaries, including Galileo, were equally incapable of recognizing their significance. The Keplerian discoveries were not of the kind which are "in the air" of a period, and which are usually made by several people independently; they were quite exceptional one-man achievements. That is why the way he arrived at them is particularly interesting.
I have tried to re-trace the tortuous progress of his thought. Perhaps the most astonishing thing about it is the mixture of cleanness and uncleanness in his method. On the one hand, he throws away a cherished theory, the result of years of labour, because of those wretched eight minutes of arc. On the other hand he makes impermissible generalizations, knows that they are impermissible, yet does not care. And he has a philosophical justification for both attitudes. We heard him sermonising about the duty to stick rigorously to observed fact. But on the other hand he says that Copernicus "sets an example for others by his contempt for the small blemishes in expounding his wonderful discoveries. If this had not been always the usage, then Ptolemy would never have been able to publish his Almagest, Copernicus his Revolutions, and Reinhold his Prutenian Tables... It is not surprising that, when he dissects the universe with a lancet, various matters emerge only in a rough manner." 34
Both precepts have, of course, their uses. The problem is to know when to follow one, when the other. Copernicus had a one-track mind; he never flew off at a tangent; even his cheatings were heavy-handed. Tycho was a giant as an observer, but nothing else. His leanings toward alchemy and astrology never fused, as in Kepler, with his science. The measure of Kepler's genius is the intensity of his contradictions, and the use he made of them. We saw him plod, with infinite patience, along dreary stretches of trial-and-error procedure, then suddenly become airborne when a lucky guess or hazard presented him with an opportunity. What enabled him to recognize instantly his chance when the number 0.00429 turned up in an unexpected context was the fact that not only his waking mind, but his sleepwalking unconscious self was saturated with every conceivable aspect of his problem, not only with the numerical data and ratios, but also with an intuitive "feel" of the physical forces, and of the Gestalt configurations which it involved. A locksmith who opens a complicated lock with a crude piece of bent wire is not guided by logic, but by the unconscious residue of countless past experiences with locks, which lend his touch a wisdom that his reason does not possess. It is perhaps this intermittent flicker of an overall vision which accounts for the mutually compensatory nature of Kepler's mistakes, as if some balancing reflex or "backfeed" mechanism had been at work in his unconscious mind.
Thus, for instance, he knew that his inverse ratio "law" (between a planet's speed and solar distance) was incorrect. His thirty-second chapter ends with a short, almost off-hand admission of this. But, he argues, the deviation is so small, that it can be neglected. Now this is true for Earth with its small eccentricity, yet not at all true for Mars, with its large eccentricity. Yet even toward the end of the book (in chapter 60), long after he had found the correct law, Kepler speaks of the inverse-ratio postulate as if it were true not only for earth, but also for Mars. He could not deny, even to himself, that the hypothesis was incorrect; he could only forget it. Which he promptly did. Why? Because, though he knew that the postulate was bad geometry, it made good physics to him, and therefore ought to be true. The problem of the planetary orbits had been hopelessly bogged down in its purely geometrical frame of reference, and when Kepler realized that he could not get it unstuck, he tore it out of that frame and removed it into the field of physics. This operation of removing a problem from its traditional context and placing it into a new one, looking at it through glasses of a different colour as it were, has always seemed to me of the very essence of the creative process. 35 It leads not only to a revaluation of the problem itself, but often to a synthesis of much wider consequences, brought about by a fusion of the two previously unrelated frames of reference. In our case, the orbit of Mars became the unifying link between the two formerly separate realms of physics and cosmology.
It may be objected that Kepler's ideas of physics were so primitive that they ought to be regarded merely as a subjective stimulus to his work (like the five perfect solids), without objective value. In fact, however, his was the first serious attempt at explaining the mechanism of the solar system in terms of physical forces, and once the example was set, physics and cosmology could never again be divorced. And secondly, whereas the five solids were indeed merely a psychological spur, his sky-physics played, as we saw, a direct part in the discovery of his laws.
For, although the functions of gravity and inertia are reversed in the Keplerian cosmos, his intuition that there are two antagonistic forces acting on the planets, guided him in the right direction. A single force, as previously assumed (that of the Prime Mover or kindred spirits) could never produce oval orbits and periodic changes of speed. These could only be the result of some dynamic tug-of-war going on in the sky – as indeed they are; though his ideas about the nature of the "sun's force" and the planet's "laziness" or "magnetism" were pre-Newtonian.
9. The Pitfalls of Gravity
I have tried to show that without his invasion into the territory of physics Kepler could not have succeeded. I must now discuss briefly Kepler's particular brand of physics. It was, as to be expected, physics-on-the-watershed, half-way between Aristotle and Newton. The essential concept of impetus or momentum, which makes a moving body persist in its motion without the help of an external force, is absent fro
m it; the planets must still be dragged through the ether like a Greek oxcart through the mud. In this respect Kepler had not advanced further than Copernicus, and both were unaware of the progress made by the Ockhamists in Paris.
On the other hand, he came very near to discovering universal gravity, and the reasons for his failure to do so are not only of historical, but also of topical interest. Over and again he seems to balance on the brink of the idea and yet, as if pulled back by some unconscious resistance, to shrink from the final step. One of the most striking passages is to be found in the introduction to the Astronomia Nova. There Kepler starts by demolishing the Aristotelian doctrine that bodies which are by nature "heavy" strive toward the centre of the world, whereas those which are "light" strive toward its periphery. His conclusions are as follows:
"It is therefore clear that the traditional doctrine about gravity is erroneous... Gravity is the mutual bodily tendency between cognate [i.e. material] bodies towards unity or contact (of which kind the magnetic force also is), so that the earth draws a stone much more than the stone draws the earth...
Supposing that the earth were in the centre of the world, heavy bodies would be attracted to it, not because it is in the centre, but because it is a cognate [material] body. It follows that regardless where we place the earth ... heavy bodies will always seek it...
If two stones were placed anywhere in space near to each other, and outside the reach of force of a third cognate body, then they would come together, after the manner of magnetic bodies, at an intermediate point, each approaching the other in proportion to the other's mass (my italics).