In October, he arrived back in Prague with his wife – but without his furniture and chattels, which he had to leave behind in Linz as he had no money to pay for the transport. He was again ill with intermittent fever, and again thought that he was suffering from consumption. The imperial nod of consent to his employment was not followed by concrete action, so Kepler and his wife had to live entirely on Tycho's bounty. At the Emperor's request, who wanted his mathematicus close at hand, Tycho had given up the splendours of Benatek and moved to a house in Prague, where the Keplers, having no money for rent, were forced to take up quarters. During the next six months, Kepler had little time for astronomy, as he was fully occupied with writing the accursed polemics against Ursus and Craig, and nursing his real and imaginary ailments. Frau Barbara, who even in better days had not been a cheerful soul, hated the alien ways and narrow, winding streets of Prague, whose stench was strong enough "to drive back the Turks", as a contemporary English traveller wrote. 11 The Keplers were drinking the bitter cup of refugee existence to the dregs.
In the spring of 1601, Frau Barbara's rich father died back in Styria – he had paid the price of conversion to die in his country. This gave Kepler a welcome pretext to leave his family in Tycho's charge, and to go back to Gratz to save the inheritance. In this he did not succeed; but he stayed in Gratz for another four months, and seems to have had a wonderful time, dining in the houses of the Styrian nobles as a kind of distinguished exile on home leave, climbing mountains to measure the curvature of the earth, writing infuriating letters to Tycho whom he reproached for not giving enough money to Frau Barbara, and thoughtfully asking her whether Elisabeth Brahe, who was at last allowed to marry the Junker Tengnagel, was "showing signs of the baby" – which arrived three months after the ceremony. He returned to Prague in August, his mission unaccomplished, but his health fully restored, and in radiant spirits. He now only had to mark time for another two months till the decisive turn in his life.
On the 13th of October, 1601, Tycho was a guest at supper at Baron Rosenberg's table in Prague. Among the other guests was an Imperial Councillor, so it must have been an illustrious company; but since Tycho had been in the habit of entertaining royalty, and was accustomed to vast amounts of drink, it is difficult to understand why he was unable to cope with the predicament in which he found himself. Kepler has carefully recorded what happened in the Diary of Observations – a kind of logbook where all important events of the Brahe household were entered:
"On October 13, Tycho Brahe, in the company of Master Minkowitz, had dinner at the illustrious Rosenberg's table, and held back his water beyond the demands of courtesy. When he drank more, he felt the tension in his bladder increase, but he put politeness before his health. When he got home, he was scarcely able to urinate.
At the beginning of his illness, the moon was in opposition to Saturn ... [follows the horoscope of the day].
After five sleepless nights, he could still only pass his water with the greatest pain, and even so the passage was impeded. The insomnia continued, with internal fever gradually leading to delirium; and the food he ate, and from which he could not be kept, exacerbating the evil. On October 24, his delirium ceased for several hours; nature conquered and he expired peacefully among the consolations, prayers and tears of his people.
So from this date the series of celestial observations was interrupted, and his own observations of thirty-eight years have come to an end.
On his last night in his gentle delirium, he repeated over and again these words, like someone composing a poem:
Let me not seem to have lived in vain.
No doubt he wished that these words should be added to the title-page of his works, thus dedicating them to the memory and uses of posterity." 12
During his last days, whenever the pain subsided, the great Dane had refused to keep to a diet, ordered and ate ravenously whatever dish came to his mind. When delirium set in again, he kept repeating softly that he hoped his life had not been wasted (ne frusta vixisse videar). The meaning of these words becomes clear through his last wish addressed to Kepler. 13 It was the same wish which he had expressed in his first letter to him: that Kepler should build the new universe not on the Copernican, but on Tycho's system. Yet he must have known, as his delirious complaint revealed, that Kepler would do just the opposite, and put the Tychonic legacy to his own use.
Tycho was buried with great pomp in Prague, his coffin carried by twelve imperial Gentlemen-at-Arms, preceded by his coat of arms, his golden spurs and favourite horse.
Two days later, on 6 November, 1601, the Emperor's privy councillor, Barwitz, called on Kepler at his lodgings, to appoint him, as Tycho's successor, to the post of Imperial Mathematicus.
VI THE GIVING OF THE LAWS
1. Astronomia Nova
KEPLER stayed in Prague as Imperial Mathematicus from 1601 to 1612, to the death of Rudolph II. It was the most fruitful period of his life, and brought him the unique distinction of founding two new sciences: instrumental optics, which does not concern us, and physical astronomy. His 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 1
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. The third law, published later, does not concern us at this point.
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 lo
st. Hence I shall not be blamed if, prompted by the same affection for the reader, I follow the same method." 1a
Before embarking on the story, it will be prudent to add my own apology to Kepler's. Prompted by the same "affection for the reader" I have tried to simplify as far as possible a difficult subject: even so, the present chapter must of necessity be slightly more technical than the rest of this book. If some passages tax his patience, even if occasionally he fails to grasp a point or loses the thread, he will, I hope, nevertheless get a general idea of Kepler's odyssey of thought, which opened up the modern universe.
2. Opening Gambits
It will be remembered that at the partitioning of the cosmos which followed 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." 2
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 that most famous of Latins, the priest of nature Pliny, specially indicted him: MARS IS A STAR WHO DEFIES OBSERVATION." 3
Thus Kepler, in his dedication of the New Astronomy to the Emperor Rudolph II. The dedication is written in the form of an allegory of Kepler's war against Mars, begun under "Tycho's supreme command", patiently pursued in spite of the warning example of Rheticus who went off his head over Mars, in spite of other dangers and terrible handicaps, such as a lack of supplies owing to Rudolph's failure to pay Kepler's salary – and so on to the triumphant end when the Imperial Mathematicus, riding a chariot, leads the captive enemy to the Emperor's throne.
Thus 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; and that 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, the whole system should be centred on the body of the sun itself. 4
But in fact it was not. The sun occupies not the exact centre of the orbit at C; it occupies one of the two foci of the ellipse at S.
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 in the sun. Accordingly, the question arose in his mind: if the force which moves the planets comes from S, why do they insist on turning round C? 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 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, 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. Here he met, for once, with immediate success. 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 1° 50′. He was delighted, and remarked smugly that "the observations took the side of my preconceived ideas, as they often did before." 5
The third innovation was the most radical. To gain more elbow room, he had to get out of the straight-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 th
is 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 was really 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." 6