II THE "COSMIC MYSTERY"
1. The Perfect Solids
FROM the frustrations of his first year in Gratz, Kepler escaped into the cosmological speculations which he had playfully pursued in his Tuebingen days. But now these speculations were becoming both more intense, and more mathematical in character. A year after his arrival – more precisely on 9 July, 1595, for he has carefully recorded the date – he was drawing a figure on the blackboard for his class, when an idea suddenly struck him with such force that he felt he was holding the key to the secret of creation in his hand. "The delight that I took in my discovery," he wrote later, "I shall never be able to describe in words." 1 It determined the course of his life, and remained his main inspiration throughout it.
The idea was, that the universe is built around certain symmetrical figures – triangle, square, pentagon, etc. – which form its invisible skeleton, as it were. Before going into detail, it will be better to explain at once that the idea itself was completely false; yet it nevertheless led eventually to Kepler's Laws, the demolition of the antique universe on wheels, and the birth of modern cosmology. The pseudo-discovery which started it all is expounded in Kepler first book, the Mysterium Cosmographicum, * which he published at the age of twenty-five.
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The full title reads: A Forerunner (Prodromus) to Cosmographical Treatises, containing the Cosmic Mystery of the admirable proportions between the Heavenly Orbits and the true and proper reasons for their Numbers, Magnitudes and Periodic Motions, by Johannes Kepler, Mathematicus of the Illustrious Estates of Styria, Tuebingen, anno 1596.
In the Preface to the work, Kepler explained how he came to make his "discovery". While still a student in Tuebingen, he had heard from his teacher in astronomy, Maestlin, about Copernicus, and agreed that the sun must be in the centre of the universe "for physical, or if you prefer, for metaphysical reasons". He then began to wonder why there existed just six planets "instead of twenty or a hundred", and why the distances and velocities of the planets were what they were. Thus started his quest for the laws of planetary motion.
At first he tried whether one orbit might perchance be twice, three of four times as large as another. "I lost much time on this task, on this play with numbers; but I could find no order either in the numerical proportions or in the deviations from such proportions." He warns the reader that the tale of his various futile efforts "will anxiously rock thee hither and thither like the waves of the sea". Since he got nowhere, he tried "a startlingly bold solution": he inserted an auxiliary planet between Mercury and Venus, and another between Jupiter and Mars, both supposedly too small to be seen, hoping that now he would get some sensible sequence of ratios. But this did not work either; nor did various other devices which he tried.
"I lost almost the whole of the summer with this heavy work. Finally I came close to the true facts on a quite unimportant occasion. I believe Divine Providence arranged matters in such a way that what I could not obtain with all my efforts was given to me through chance; I believe all the more that this is so as I have always prayed to God that he should make my plan succeed, if what Copernicus had said was the truth." 2
The occasion of this decisive event was the aforementioned lecture to his class, in which he had drawn, for quite different purposes, a geometrical figure on the blackboard. The figure showed (I must describe it in a simplified manner) a triangle fitted between two circles; in other words, the outer circle was circumscribed around the triangle, the inner circle inscribed into it.
As he looked at the two circles, it suddenly struck him that their ratios were the same as those of the orbits of Saturn and Jupiter. The rest of the inspiration came in a flash. Saturn and Jupiter are the "first" (i.e. the two outermost) planets, and "the triangle is the first figure in geometry. Immediately I tried to inscribe into the next interval between Jupiter and Mars a square, between Mars and Earth a pentagon, between Earth and Venus a hexagon..."
It did not work – not yet, but he felt that he was quite close to the secret. "And now I pressed forward again. Why look for two-dimensional forms to fit orbits in space? One has to look for three-dimensional forms – and, behold dear reader, now you have my discovery in your hands! ..."
The point is this. One can construct any number of regular polygons in a two-dimensional plane; but one can only construct a limited number of regular solids in three-dimensional space. These "perfect solids", of which all faces are identical, are: (1) the tetrahedron (pyramid) bounded by four equilateral triangles; (2) the cube; (3) the octahedron (eight equilateral triangles); (4) the dodecahedron (twelve pentagons) and (5) the icosahedron (twenty equilateral triangles).
They were also called the "Pythagorean" or "Platonic" solids. Being perfectly symmetrical, each can be inscribed into a sphere, so that all of its vertices (corners) lie on the surface of the sphere. Similarly, each can be circumscribed around a sphere, so that the sphere touches every face in its centre. It is a curious fact, inherent in the nature of three-dimensional space, that (as Euclid proved) the number of regular solids is limited to these five forms. Whatever shape you choose as a face, no other perfectly symmetrical solid can be constructed except these five. Other combinations just cannot be fitted together.
So there existed only five perfect solids – and five intervals between the planets! It was impossible to believe that this should be by chance, and not by divine arrangement. It provided the complete answer to the question why there were just six planets "and not twenty or a hundred". And it also answered the question why the distances between the orbits were as they were. They had to be spaced in such a manner that the five solids could be exactly fitted into the intervals, as an invisible skeleton or frame. And lo, they fitted! Or at least, they seemed to fit, more or less. Into the orbit, or sphere, of Saturn he inscribed a cube; and into the cube another sphere, which was that of Jupiter. Inscribed in that was the tetrahedron, and inscribed in it the sphere of Mars. Between the spheres of Mars and Earth came the dodecahedron; between Earth and Venus the icosahedron; between Venus and Mercury the octahedron. Eureka! The mystery of the universe was solved by young Kepler, teacher at the Protestant school in Gratz.
Model of the universe; the outermost sphere is Saturn's. Illustration in Kepler Mysterium cosmographicum.
Detail, showing the spheres of Mars, Earth, Venus and Mercury with the Sun in the centre.
"It is amazing!" Kepler informs his readers, "although I had as yet no clear idea of the order in which the perfect solids had to be arranged, I nevertheless succeeded ... in arranging them so happily, that later on, when I checked the matter over, I had nothing to alter. Now I no longer regretted the lost time; I no longer tired of my work; I shied from no computation, however difficult. Day and night I spent with calculations to see whether the proposition that I had formulated tallied with the Copernican orbits or whether my joy would be carried away by the winds... Within a few days everything fell into its place. I saw one symmetrical solid after the other fit in so precisely between the appropriate orbits, that if a peasant were to ask you on what kind of hook the heavens are fastened so that they don't fall down, it will be easy for thee to answer him. Farewell!" 3
We had the privilege of witnessing one of the rare recorded instances of a false inspiration, a supreme hoax of the Socratic daimon, the inner voice that speaks with such infallible, intuitive certainty to the deluded mind. That unforgettable moment before the figure on the blackboard carried the same inner conviction as Archimedes' Eureka or Newton's flash of insight about the falling apple. But there are few instances where a delusion led to momentous and true scientific discoveries and yielded new Laws of Nature. This is the ultimate fascination of Kepler – both as an individual and as a case history. For Kepler's misguided belief in the five perfect bodies was not a passing fancy, but rem
ained with him, in a modified version, to the end of his life, showing all the symptoms of a paranoid delusion; and yet it functioned as the vigor motrix, the spur of his immortal achievements. He wrote the Mysterium Cosmographicum when he was twenty-five, but he published a second edition of it a quarter century later, towards the end, when he had done his life work, discovered his three Laws, destroyed the Ptolemaic universe, and laid the foundations of modern cosmology. The dedication to this second edition, written at the age of fifty, betrays the persistence of the idée fixe:
"Nearly twenty-five years have passed since I published the present little book... Although I was then still quite young and this publication my first work on astronomy, nevertheless its success in the following years proclaims with a loud voice that never before has anybody published a more significant, happier, and in view of its subject, worthier first-book. It would be mistaken to regard it as a pure invention of my mind (far be any presumption from my intent, and any exaggerated admiration from the reader's, when we touch the seven-stringed harp of the Creator's wisdom). For as if a heavenly oracle had dictated it to me, the published booklet was in all its parts immediately recognised as excellent and true throughout (as it is the rule with obvious acts of God).'
Now, Kepler's style is often exuberant and sometimes bombastic, but rarely to this extent. The apparent presumption is in fact the radiance of the idée fixe, an emanation of the immense emotive charge which such ideas carry. When the patient in a mental home declares that he is the mouthpiece of the Holy Ghost, he means it not as a boast but as a flat statement of fact.
Here we have, then, a young man of twenty-four, an aspirant of theology, with only a sketchy knowledge of astronomy, who hits upon a crank idea, convinced that he has solved the "cosmic mystery". "There is no great ingenuity," to quote Seneca, "without an admixture of dementedness," but as a rule the dementedness devours the ingenuity. Kepler's history will show how exceptions to this rule may occur.
2. Contents of the Mysterium
Leaving its crankish leitmotif aside, Kepler's first book contains the seeds of his principal future discoveries. I must therefore briefly describe its content.
The Mysterium has an overture, a first and a second movement. The overture consists of the Introduction to the Reader, which I have already discussed, and the first chapter, which is an enthusiastic and lucid profession of faith in Copernicus. 4 It was the first unequivocal, public commitment by a professional astronomer which appeared in print fifty years after Canon Koppernigk's death, and the beginning of his posthumous triumph. 5 Galileo, by six years Kepler's senior, and astronomers like Maestlin, were still either silent on Copernicus, or agreed with him only in cautious privacy. Kepler had intended to add to his chapter a proof that there was no contradiction between the teaching of Copernicus and Holy Scripture; but the head of the theological faculty in Tuebingen, whose official consent to the publication of the book had to be obtained, directed him to leave out any theological reflections and – in the tradition of the famous Osiander preface – to treat the Copernican hypothesis as a purely formal, mathematical one. * Kepler accordingly postponed his theological apologia to a later work, but otherwise did the exact opposite of what he was advised to do, by proclaiming the Copernican system to be literally, physically and incontrovertibly true, "an inexhaustible treasure of truly divine insight into the wonderful order of the world and all bodies therein". It sounded like a fanfare in praise of the brave new heliocentric world. The arguments in its favour which Kepler adduced could mostly be found in Rheticus' Narratio Prima, which Kepler reprinted as an appendix to the Mysterium, to save his readers the labour of toiling through Copernicus' unreadable book.
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It was, as we know, Kepler himself who, a few years later, discovered that the preface to the Revolutions was written by Osiander and not by Copernicus.
After this overture, Kepler gets down to his "principal proof that the planetary spheres are separated from each other, or fenced in, as it were, by the five perfect solids. (He does not mean, of course, that the solids are really present in space, nor does he believe in the existence of the spheres themselves, as we shall see.) The "proof" consists, roughly, in the deduction that God could only create a perfect world, and since only five symmetrical solids exist, they are obviously meant to be placed between the six planetary orbits "where they fit in perfectly". In fact, however, they do not fit at all, as he was soon to discover to his woe. Also, there are not six planets but nine (not to mention the small fry of asteroids between Jupiter and Mars), but at least Kepler was spared in his lifetime the discovery of the three others, Uranus, Neptune and Pluto.
In the next six chapters (III to VIII), it is explained to us why there are three planets outside and two inside the earth's orbit; why that orbit is placed just where it is; why the cube lies between the two outermost planets and the octahedron between the two innermost; what affinities and sympathies exist between the various planets and the various solids, and so on – all this by a priori deductions derived straight from the Creator's secret thoughts, and supported by reasons so fantastic that one can hardly believe one is listening to one of the founders of modern science. Thus, for instance, "the regular solids of the first order [i.e. those which lie outside the earth's orbit] have it in their nature to stand upright, those of the second order to float. For, if the latter are made to stand on one of their sides, the former on one of their corners, then in both cases the eye shies from the ugliness of such a sight." By this kind of argument young Kepler succeeds in proving everything that he believes and in believing everything that he proves. The ninth chapter deals with astrology, the tenth with numerology, the eleventh with the geometrical symbolism of the Zodiac; in the twelfth, he alludes to the Pythagorean harmony of the spheres, searching for correlations between his perfect solids and the harmonic intervals in music – but it is merely one more arabesque to the dream. On this note ends the first half of the book.
The second is different. I have talked of a work in two movements, because they are written in different moods and keys, and are held together only by their common leitmotif. The first is medieval, aprioristic and mystical; the second modern and empirical. The Mysterium is the perfect symbol of the great watershed.
The opening paragraph of the second half must have come as a shock to his readers:
"What we have so far said served merely to support our thesis by arguments of probability. Now we shall proceed to the astronomical determination of the orbits and to geometrical considerations. If these do not confirm the thesis, then all our previous efforts have doubtless been in vain." 6
So all the divine inspiration and a priori certitude were merely "probabilities"; and their truth or falsehood was to be decided by the observed facts. Without transition, in a single startling jump, we have traversed the frontier between metaphysical speculation and empirical science.
Now Kepler got down to brass tacks: the checking of the proportions of his model of the universe against the observed data. Since the planets do not revolve around the sun in circles but in oval-shaped orbits (which Kepler's First Law, years later, identified as ellipses), each planet's distance from the sun varies within certain limits. This variation (or eccentricity) he accounted for by allotting to each planet a spherical shell of sufficient thickness to accommodate the oval orbit between its walls (see the model on p. 250). The inner wall represents the planet's minimum distance from the sun, the outer wall its maximum distance. The spheres, as already mentioned, are not considered as physically real, but merely as the limits of space allotted to each orbit. The thickness of each shell and the intervals between them, were laid down in Copernicus' figures. Were they spaced in such a way that the five solids could be exactly fitted between them? In the Preface, Kepler had confidently announced that they could. Now he found that they could not. There was fairly good agreement for the orbits of Mars, Earth and Venus, but not for Jupiter an
d Mercury. The trouble with Jupiter Kepler dismissed with the disarming remark that "nobody will wonder at it, considering the great distance". As for Mercury, he frankly resorted to cheating. 7 It was a kind of Wonderland croquet through mobile celestial hoops.
In the following chapters Kepler tried various methods to explain away the remaining discrepancies. The fault must lie either in his model or in the Copernican data; and Kepler naturally preferred to blame the latter. First, he discovered that Copernicus had placed into the centre of the world not really the sun, but the centre of the earth's orbit, "in order to save himself trouble and so as not to confuse his diligent readers by dissenting too strongly from Ptolemy." 8 Kepler undertook to remedy this, hoping thereby to obtain more favourable Lebensraum for his five solids. His mathematical knowledge was as yet insufficient for this task, so he turned for help to his old teacher, Maestlin, who willingly complied. The new figures did not help Kepler at all; yet he had at one stroke, and almost inadvertently, shifted the centre of the solar system where it belonged. It was the first momentous by-product of the phantom chase.
His next attempt to remedy the disagreement between his dream and the observed facts concerned the moon. Should her orbit be included into the thickness of the earth's sphere, or not? He explained frankly to his dear readers that he would choose the hypothesis which best fits his plan; he will tuck the moon into the earth's shell, or banish her into the outer darkness, or let her orbit stick half-way out, for there are no a priori reasons in favour of either solution. ( Kepler's a priori proofs were mostly found a posteriori.) But fiddling with the moon did not help either, so young Kepler proceeded to a frontal attack against the Copernican data. He declared them with admirable impertinence to be so unreliable that Kepler's own figures would be strongly suspect if they agreed with Copernicus'. Not only were the tables unreliable; not only was Copernicus inexact in his observations, as reported by Rheticus (from whom Kepler quotes long, damning passages); but the old Canon also cheated: