Kepler — Eight Minutes of Arc
Extracted from Arthut Koestler, The Sleepwalkers (1959)
[Kepler’s] magnum opus, published in 1609, bears the significant title:
A NEW ASTRONOMY Based on Causation
or A PHYSICS OF THE SKY
derived from Investigations of the
MOTIONS OF THE STAR MARS
Founded on Observations of
THE NOBLE TYCHO BRAHE
Kepler worked on it, with interruptions, from his arrival at Benatek in 1600, to 1606. It contains the first two of Kepler’s three planetary laws: (1) that the planets travel round the sun not in circles but in elliptical orbits, one focus of the ellipse being occupied by the sun; (2) that a planet moves in its orbit not at uniform speed but in such a manner that a line drawn from the planet to the sun always sweeps over equal areas in equal times. . . .
On the surface, Kepler’s laws look as innocent as Einstein’s E=mc2, which does not reveal, either, its atom-exploding potentialities. But the modern vision of the universe was shaped, more than by any other single discovery, by Newton’s law of universal gravitation, which in turn was derived from Kepler’s three laws. Although (owing to the peculiarities of our educational system), a person may never have heard of Kepler’s laws, his thinking has been moulded by them without his knowledge; they are the invisible foundation of a whole edifice of thought.
Thus the promulgation of Kepler’s laws is a landmark in history. They were the first “natural laws” in the modern sense: precise, verifiable statements about universal relations governing particular phenomena, expressed in mathematical terms. They divorced astronomy from theology, and married astronomy to physics. Lastly, they put an end to the nightmare that had haunted cosmology for the last two millennia: the obsession with spheres turning on spheres, and substituted a vision of material bodies not unlike the earth, freely floating in space, moved by physical forces acting on them.
The manner in which Kepler arrived at his new cosmology is fascinating; I shall attempt to re-trace the zig-zag course of his reasoning. Fortunately, he did not cover up his tracks, as Copernicus, Galileo and Newton did, who confront us with the result of their labours, and keep us guessing how they arrived at it. Kepler was incapable of exposing his ideas methodically, text-book fashion; he had to describe them in the order they came to him, including all the errors, detours, and the traps into which he had fallen. The New Astronomy is written in an unacademic, bubbling baroque style, personal, intimate, and often exasperating. But it is a unique revelation of the ways in which the creative mind works.
“What matters to me,” Kepler explained in his Preface, “is not merely to impart to the reader what I have to say, but above all to convey to him the reasons, subterfuges, and lucky hazards which led me to my discoveries. When Christopher Colombus, Magelhaen, and the Portuguese relate how they went astray on their journeys, we not only forgive them, but would regret to miss their narration because without it the whole, grand entertainment would be lost. Hence I shall not be blamed if, prompted by the same affection for the reader, I follow the same method.”
. . . .
[After] young Kepler’s arrival at Benatek Castle, he was allotted the study of the motions of Mars which had defeated Tycho’s senior assistant, Longomontanus, and Tycho himself.
“I believe it was an act of Divine Providence,” he commented later on, “that I arrived just at the time when Longomontanus was occupied with Mars. For Mars alone enables us to penetrate the secrets of astronomy which otherwise would remain forever hidden from us.”
The reason for this key position of Mars is that, among the outer planets, his orbit deviates more than the others’ from the circle; it is the most pronouncedly elliptical. It was precisely for that reason that Mars had defied Tycho and his assistant: since they expected the planets to move in circles, it was impossible to reconcile theory with observation:
“He [Mars] is the mighty victor over human inquisitiveness, who made a mockery of all the stratagems of astronomers, wrecked their tools, defeated their hosts; thus did he keep the secret of his rule safe throughout all past centuries and pursued his course in unrestrained freedom; wherefore the most famous of Latins, the priest of nature Pliny, specially indicted him: MARS IS A STAR WHO DEFIES OBSERVATION.”
Thus Kepler, in his dedication of the New Astronomy to the Emperor Rudolph II. . . .
. . . Mars held the secret of all planetary motion, and young Kepler was assigned the task of solving it. He first attacked the problem on traditional lines; when he failed, he began to throw out ballast and continued doing so until, by and by, he got rid of the whole load of ancient beliefs on the nature of the universe, and replaced it by a new science.
As a preliminary, he made three revolutionary innovations to gain elbow room, as it were, for tackling his problem. It will be remembered that the centre of Copernicus’ system was not the sun, but the centre of the earth’s orbit; . . . already in the Mysterium Cosmographicum Kepler had objected to this assumption as physically absurd. Since the force which moved the planets emanated from the sun, whe whole system should be centred on the body of the sun itself.
But in fact it was not. The sun occupies not the exact centre of the orbit . . . it occupies one of the two foci of the ellipse . . .
Kepler did not know as yet that the orbit was an ellipse; he still regarded it as a circle. But even so, to get approximately correct results, the centre of the circle had to be placed at C, and not at the sun. Accordingly the question arose in his mind: if the force which moves the planets comes from S [the sun], why do they insist on turning round C [the centre]? Kepler answered the question by the assumption that each planet was subject to two conflicting influences: the force of the sun, and a second force located in the planet itself. This tug-of-war caused it now to approach the sun, now to recede from him.
The two forces are, as we know, gravity and inertia. Kepler, as we shall see, never arrived at formulating these concepts. But he prepared the way for Newton by postulating the two dynamic forces to explain the eccentricity of the orbits. Before him, the need for a physical explanation was not felt; the phenomenon of eccentricity was merely “saved” by the introduction of an epicycle or eccenter [eccentric], which made C turn round S. Kepler replaced the fictitious wheels by real forces.
For the same reason, he insisted on treating the sun as the centre of his system not only in the physical but in the geometrical sense, by making the distances and positions of the planets relative to the sun (and not relative to the earth or the centre C) the basis of his computations. This shift of emphasis, which was more instinctive than logical, became a major factor in his success.
His second innovation is simpler to explain. The orbits of all planets lie very nearly, but not entirely, in the same plane; they form very small angles with each other — rather like adjacent pages of a book which is nearly, but not entirely closed. The planes of all planets pass, of course, through the sun — a fact which is self-evident to us, but not to pre-Keplerian astronomy. Copernicus, once again, misled by his slavish devotion to Ptolemy, had postulated that the plane of the Martian orbit oscillates in space; and this oscillation he made to depend on the position of the earth — which, as Kepler remarks, “is no business of Mars.” He called this Copernican idea “monstrous” (though it was merely due to Copernicus’ complete indifference to physical reality) and set about to prove that the plane in which Mars moves passes through the sun, and does not oscillate, but forms a fixed angle with the plane of the earth’s orbit. He proved, by several independent methods, all based on the Tychonic observations, that the angle between the planes of Mars and Earth remained always the same, and that it amounted to 1o 50'. He was delighted, and remarked smugly that “the observations took the side of my preconceived ideas, as they often did before.”
The third innovation was the most radical. To gain more elbow room, he had to get out of the strait-jacket of “uniform motion in perfect circles” — the basic axiom of cosmology from Plato up to Copernicus and Tycho. For the time being, he still let circular motion stand, but he threw out uniform speed. Again he was guided mainly by physical considerations: if the sun ruled the motions, then his force must act more powerfully on the planet when it is close to the source, less powerfully when away from it; hence the planet will move faster or slower, in a manner somehow related to its distance from the sun.
This idea was not only a challenge to antique tradition; it also reversed the original purpose of Copernicus. It will be remembered that Copernicus’ original motive for embarking on a reform of the Ptolemaic system was his discontent with the fact that, according to Ptolemy, a planet did not move at uniform speed around the centre of its orbit, but only around a point at some distance from the centre. This point was called the punctum equans — the point in space, from which the planet gave the illusion of “equal motion.” Canon Koppernigk regarded this arrangement as an evasion of the command of uniform motion, abolished Ptolemy’s equants, and added, instead, more epicycles to his system. This did not make the planet’s real motion either circular, or uniform, but each wheel in the imaginary clockwork which was supposed to account for it, did turn uniformly — if only in the astronomer’s mind.
When Kepler renounced the dogma of uniform motion, he was able to throw out the epicycles which Copernicus had introduced to save it. Instead, he reverted to the equant as an important calculating device. . . .
Let the circle be the track of a toy train chugging round a room. When near the window it runs a little faster, near the door a little slower. Provided that these periodic changes of speed follow some simple, definite rule, then it is possible to find a punctum equans, “E,” from which the train seems to move at uniform speed. The closer we are to a moving train, the faster it seems to move; hence the punctum equans will be somewhere between the centre C of the track and the door, so that the speed-surplus of the train when passing the window will be eliminated by distance, its speed deficiency at the door compensated by closeness. The advantage gained by the introduction of this imaginary punctum equans is that, seen from E, the train seems to move uniformly, that is, it will cover equal angles at equal times — which makes it possible to compute its various positions 1, 2, 3, etc., at any given moment.
By these three preliminary moves: (a) the shifting of the system’s centre into the sun; (b) the proof that the orbital planes do not “oscillate” in space; and (c) the abolition of uniform motion, Kepler had cleared away a considerable amount of the rubbish that had obstructed progress since Ptolemy, and made the Copernican system so clumsy and unconvincing. In that system Mars ran on five circles; after the clean-up, a single eccentric circle must be sufficient — if the orbit really was a circle. He felt confident that victory was just around the corner, and before the final attack wrote a kind of obituary notice for classical cosmology:
“Oh, for a supply of tears that I may weep over the pathetic diligence of Apianus [author of a very popular textbook] who, relying on Ptolemy, wasted his valuable time and ingenuity on the construction of spirals, loops, helixes, vortices and a whole labyrinth of convolutions, in order to represent that which exists only in the mind, and which Nature entirely refuses to accept as her likeness. And yet that man has shown us that, with his penetrating intelligence, he would have been capable of mastering Nature.”
The First Assault
Kepler’s first attack on the problem is described in great detail in the sixteenth chapter of the New Astronomy.
The task before him was to define the orbit of Mars by determining the radius of the circle, the direction (relative to the fixed stars) of the axis connecting the two positions where Mars is nearest and farthest from the sun (perihelion and aphelion) ..., and the positions of the Sun (S), orbital centre (C), and punctum equans (E), which all lie on that axis. Ptolemy had assumed that the distance between E and C was the same as between C and S, but Kepler made no such assumption, which complicated his task even more.
He chose out of Tycho’s treasure four observed positions of Mars at the convenient dates when the planet was in opposition to the sun. The geometrical problem which he had to solve was, as we saw, to determine, out of these four positions, the radius of the orbit, the direction of the axis, and the position of the three central points on it. It was a problem which could not be solved by rigorous mathematics, only by approximation, that is, by a kind of trial-and-error procedure which has to be continued until all the pieces of the jig-saw puzzle fit together tolerably well. The incredible labour that this involved may be gathered from the fact that Kepler’s draft calculations (preserved in manuscript) cover nine hundred folio pages in small handwriting.
At times he was despairing; he felt, like Rheticus, that a demon was knocking his head against the ceiling, with the shout: “These are the motions of Mars.” At other times, he appealed for help to Maestlin (who turned a deaf ear), to the Italian astronomer, Magini (who did the same), and thought of sending an S.O.S. to Francois, Vieta, the father of modern algebra: “Come, oh Gallic Appolonius, bring your cylinders and spheres and what other geometer’s houseware you have. . . .” But in the end he had to slog it out alone, and to invent his mathematical tools as he went along.
Half-way through that dramatic sixteenth chapter, he burst out:
“If thou [dear reader] art bored with this wearisome method of calculation, take pity on me who had to go through with at least seventy repetitions of it, at a very great loss of time; nor wilst thou be surprised that by now the fifth year is nearly past since I took on Mars. . . .”
Now, at the very beginning of the hair-raising computations in chapter sixteen, Kepler absentmindedly put three erroneous figures for three vital longitudes of Mars, and happily went on from there, never noticing his error. The French historian of astronomy, Delambre, later repeated the whole computation, but, surprisingly, his correct results differ very little from Kepler’s faulty ones. The reason is, that toward the end of the chapter Kepler committed several mistakes in simple arithmetic — errors in division which would bring bad marks to a schoolboy — and these errors happen very nearly to cancel out his earlier mistakes. We shall see, in a moment, that, at the most crucial point of the process of discovering his Second Law, Kepler again committed mathematical sins which mutually cancelled out, and “as if by miracle” (in his own words), led to the correct result.
At the end of that breathtaking chapter, Kepler seems to have triumphantly achieved his aim. As a result of his seventy-odd trials, he arrived at values for the radius of the orbit and for the three central points which gave, with a permissible error of less than 2', the correct positions of Mars for all ten oppositions recorded by Tycho. The unconquerable Mars seemed at last to have been conquered. He proclaimed his victory with the sober words:
“Thou seest now, diligent reader, that the hypothesis based on this method not only satisfies the four positions on which it was based, but also correctly represents, within two minutes, all the other observations. . . . ”
There follow three pages of tables to prove the correctness of his claim; and then, without further transition, the next chapter starts with the following words:
“Who would have thought it possible? This hypothesis, which so closely agrees with the observed oppositions, is nevertheless false. . . .”
Eight Minutes of Arc
In the two following chapters Kepler explains, with great thoroughness and an almost masochistic delight, how he discovered that the hypothesis is false and why it must be rejected. In order to prove it by a further test, he had selected two specially rare pieces from Tycho’s treasury of observations, and lo! they did not fit; and when he tried to adjust his model to them, this made things even worse, for now the observed positions of Mars differed from those which his theory demanded by magnitudes up to eight minutes of arc.
This was a catastrophe. Ptolemy, and even Copernicus, could afford to neglect a difference of eight minutes, because their observations were only accurate within a margin of ten minutes anyway. “But,” the nineteenth chapter concludes, “but for us, who, by divine kindness were given an accurate observer such as Tycho Brahe, for us it is fitting that we should acknowledge this divine gift and put it to use. . . . Henceforth I shall lead the way toward that goal according to my own ideas. For, if I had believed that we could ignore these eight minutes, I would have patched up my hypothesis accordingly. But since it was not permissible to ignore them, those eight minutes point the road to a complete reformation of astronomy: they have become the building material for a large part of this work. . . .”
It was the final capitulation of an adventurous mind before the “irreducible, obstinate facts.” Earlier on, if a minor detail did not fit into a major hypothesis, it was chated away or shrugged away. Now this time-hallowed indulgence had ceased to be permissible. A new era had begun in the history of thought: an era of austerity and rigour. As Whitehead has put it:
“All the world over and at all times there have been practical men, absorbed in ‘irreducible and stubborn facts’” all the world over and at all times there have been men of philosophic temperament who have been absorbed in the weaving of general principles. It is this union of passionate interest in the detailed facts with equal devotion to abstract generalization which forms the novelty in our present society.”
This new departure determined the climate of European thought in the last three centuries, it set modern Europe apart from all other civilizations in the past and present, and enabled it to transform its natural and social environment as completely as if a new species had arisen on this planet.
The turning point is dramatically expressed in Kepler’s work. In the Mysterium Cosmographicum the facts are coerced to fit the theory. In the Astronomia Nova, a theory, built on years of labour and torment, was instantly thrown away because of a discord of eight miserable minutes of arc. Instead of cursing those eight minutes as a stumbling block, he transformed them into the cornerstone of a new science.
What caused this change of heart in him? . . . some of the general causes which contributed to the emergence of the new attitude [are] the need of navigators, and engineers, for greater precision in tools and theories; for stimulating effects on science of expanding commerce and industry. But what turned Kepler into the first law-maker of nature was something different and more specific. It was his introduction of physical causality into the formal geometry of the skies which made it impossible for him to ignore the eight minutes of arc. So long as cosmology was guided by purely geometrical rules of the game, regardless of physical causes, discrepancies between theory and fact could be overcome by inserting another wheel into the system. In a universe moved by real, physical forces, this was no longer possible. The revolution which freed thought from the stranglehold of ancient dogma, immediately created its own, rigorous discipline.
The Second Book of the New Astronomy closes with these words:
“And thus the edifice which we erected on the foundation of Tycho’s observations, we have now again destroyed. . . . This was our punishment for having followed some plausible, but in reality false, axioms of the great men of the past.”
The Wrong Law
The next act of the drama opens with Book Three. As the curtain rises, we see Kepler preparing himself to throw out mor ballast. The axiom of uniform motion has already gone overboard; Kepler feels, and hints, that the even more sacred one of circular motion must follow. The impossibility of constructing a circular orbit which would satisfy all existing observations, suggests to him that the circle must be replaced by some other geometrical curve.
But before he can do that, he must make an immense detour. For if the orbit of Mars is not a circle, its true shape can only be discovered by defining a sufficient number of points on the unknown curve. A circle is defined by three points on its circumference; every other curve needs more. The task before Kepler was to construct Mars’ orbit without any preconceived ideas regarding its shape; to start from scratch, as it were.
To do that, it was first of all necessary to re-examine the motion of the earth itself. For, after all, the earth is our observatory; and if there is some misconception regarding its motion, all conclusions about the motions of other bodies will be distorted. Copernicus had assumed that the earth moves at uniform speed — not, as the other planets, only “quasi-uniformly” relative to some equant or epicycle, but really so. And since observation contradicted the dogma, the inequality of the earth’s motion was explained away by the suggestion that the orbit periodically expanded and contracted, like a kind of pulsating jellyfish. It was typical of those improvisations which astronomers could afford so long as they felt free to manipulate the universe as they pleased on their drawing boards. It was equally typical that Kepler rejected it as “fantastic,” again on the grounds that no physical cause existed for such a pulsation.
Hence his next task was to determine, more precisely than Copernicus had done, the earth’s motion round the sun. For that purpose he designed a highly original method of his own. It was relatively simple, but it so happened that nobody had thought of it before. It consisted, essentially, in the trick of transferring the observer’s position from earth to Mars, and to compute the motions of the earth exactly as an astronomer on Mars would do it.
The result was just as he had expected: the earth, like the other planets, did not revolve with uniform speed, but faster or slower according to its distance from the sun. Moreover, at the two extreme points of the orbit, the aphelion and perihelion, ... the earth’s velocity proved to be, simply and beautifully, inversely proportional to distance.
At this decisive point, Kepler flies off the tangent and becomes airborne, as it were. Up to here he was preparing, with painstaking patience, his second assault on the orbit of Mars. Now he turns to a quite different subject. “Ye physicists, prick your ears,” he warns, “for now we are going to invade your territory.” The next six chapters are a report on that invasion into celestial physics, which had been out of bounds for astronomy since Plato.
A phrase seems to have been humming in his ear like a tune one cannot get rid of; it crops up in his writings over and over again: there is a force in the sun which moves the planet, there is a force in the sun, there is a force in the sun. And since there is a force in the sun, there must exist some beautifully simple relation between the planet’s distance from the sun and its speed. A light shines the brighter the nearer we are to its source, and the same must apply to the force of the sun: the closer the planet to it, the quicker it will move. This is his instinctive conviction, already expressed in the Mysterium Cosmographicum; but now, at last, he has succeeded in proving it.
In fact he has not. He has proved the inverse ratio of speed to distance only for the two extreme points of the orbit; and his extension of his “Law” to the entire orbit was a patently incorrect generalization. Moreover, Kepler knew this, and admitted it at the end of the thirty-second chapter, before he became airborne; but immediately afterwards, he conveniently forgot it. This is the first of the critical mistakes which “as if by a miracle” cancelled out, and led Kepler to the discovery of his Second Law. It looks as if his conscious, critical faculties were anaesthetized by the creative impulse, by his impatience to get to grips with the physical forces in the solar system.
Since he had no notion of the momentum which makes the planet persist in its motion, and only a vague intuition of gravity which bends that motion into a closed orbit, he had to find, or invent, a force which, like a broom, sweeps the planet around its path. And since the sun causes all motion, he let the sun handle the broom. This required that the sun rotate round its own axis — a guess which was only confirmed much later; the force which it emitted rotated with it, like the spokes of a wheel, and swept the planets along. But if that were the only force acting on them, the planets would all have the same angular velocity, they would all complete their revolutions in the same period — which they do not. The reason, Kepler thought, was the laziness or “inertia” of the planets, who desire to remain in the same place, and resist the sweeping force. The “spokes” of that force are not rigid: they allow the planet to lag behind; it works rather like a vortex or whirlpool. The power of the whirlpool diminishes with distance, so that the farther away the planet, the less power the sun has to overcome its laziness, and the slower its motion will be.
It still remained to be explained, however, why the planets moved in eccentric orbits instead of always keeping the same distance from the centre of the vortex. Kepler first assumed that apart from being lazy, they performed an epicyclic motion in the opposite direction under their own steam, as it were, apparently out of sheer cussedness. But he was dissatisfied with this, and at a later stage assumed that the planets were “huge round magnets” whose magnetic axis pointed always in the same direction, like the axis of a top; hence the planet will periodically be drawn closer to, and be repelled by the sun, according to which of its magnetic poles faces the sun.
Thus, in Kepler’s physics of the universe, the roles played by gravity and inertia are reversed. Moreover, he assumed that the sun’s power diminishes in direct ratio to distance. He sensed that there was something wrong here, since he knew that the intensity of light diminishes with the square of the distance; but he had to stick to it, to satisfy his theorem of the ratio of speed to distance, which was equally false.
The Second Law
Refreshed by this excursion into Himmelsphysik, our hero returned to the more immediate task in hand. Since the earth no longer moved at uniform speed, how could one predict its position at a given time? (The method based on the punctum equans had proved, after all, a disappointment.) Since he believed to have proved that its speed depended directly on its distance from the sun, the time it needed to cover a small fraction of the orbit was always proportionate to that distance. Hence he divided the orbit (which, forgetting his previous resolve, he still regarded as a circle) into 360 parts, and computed the distance of each bit of arc from the sun. The sum of all distances between, say 0o and 85o was a measure of the time the planet needed to get there.
But this procedure was, as he remarked with unusual modesty, “mechanical and tiresome.” So he searched for something simpler:
“Since I was aware that there exists an infinite number of points on the orbit and accordingly an infinite number of distances [from the sun] the idea occurred to me that the sum of these distances is contained in the area of the orbit. For I remembered that in the same manner Archimedes too divided the area of a circle into an infinite number of triangles.”
Accordingly, he concluded, the area swept over by the line connecting the planet and sun . . . is a measure of the time required by the planet to get from A to B; hence the line will sweep out equal areas in equal times. This is Kepler’s immortal Second Law (which he discovered before the first) — a law of amazing simplicity at the end of a dreadfully confusing labyrinth.
Yet the last step which had got him out of the labyrinth had once again been a faulty step. For it is not permissible to equate an area with the sum of an infinite number of neighbouring lines, as Kepler did. Moreover, he knew this well, and explained at length why it was not permissible. He added that he had also committed a second error, by assuming the orbit to be circular. And he concluded: “But these two errors — it is like a miracle — cancel out in the most precise manner, as I shall prove further down.”
The correct result is even more miraculous than Kepler realized, for his explanation of the reasons why his errors cancel out was once again mistaken, and he got, in fact, so hopelessly confused that the argument is practically impossible to follow — as he himself admitted. And yet, by three incorrect steps and their even more incorrect defense, Kepler stumbled on the correct law. It is perhaps the most amazing sleepwalking performance in the history of science — except for the manner in which he found his First Law, to which we now turn.