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*
Hence the short-cuts, or short-circuits, between different sets of symbols in Pythagorean mystic number-lore, such as the correlation of odd and even numbers with male and female, right and left; or the magic quality attributed to the pentagram.
It did not last long. Within a few centuries, the unitary awareness faded, religious and rational philosophizing split apart – were partially reunited, then divorced again; with results that will become apparent as the story unfolds.
The Pythagorean synthesis would have been incomplete had it not also included precepts for a way of life.
The Brotherhood was a religious order, but at the same time an academy of science, and a power in Italian politics. The ascetic rules of life seem to have anticipated the Essenes', which in turn served as a model to primitive Christian communities. They shared all property, led a communal existence, and gave equal status to women. They observed rites and abstinences, gave much time to contemplation and examinations of conscience. According to the degree of purification which a Brother achieved, he was gradually initiated into the higher mysteries of musical, mathematical and astronomical theoria. The secrecy surrounding these was partly due to the tradition of the older mystery cults, whose adepts had known that the Bacchic, and even the Orphic, ecstasies would cause havoc if offered to all and sundry. But the Pythagoreans also realized that similar dangers inhered in the orgies of reasoning. They apparently had an intuition of the hybris of science, and recognized it as a potential means both of man's liberation and destruction; hence their insistence that only those purified in body and spirit should be trusted with its secrets. In a word, they believed that scientists ought to be vegetarians, as Catholics believe that priests ought to live in celibacy.
It may be thought that this interpretation of the Pythagorean insistence on secretiveness is far-fetched, or that it implies prophetic foresight on their part. The answer to this is that Pythagoras was, by personal experience, well aware of the immense technological potentialities of geometry. I have mentioned already that Polycrates, and the islanders he ruled, were devoted to engineering. Herodotus, who knew the island well, reports: 10
"I have written thus at length of the Samians, because they are the makers of the three greatest works to be seen in any Greek land. First of these is the double-mouthed tunnel they pierced for an hundred and fifty fathoms through the base of a high hill ... through which the water, coming from an abundant spring, is carried by its pipes to the city of Samos".
Herodotus is fond of telling tall stories, and his report was not taken very seriously, until, at the beginning of our century, the tunnel was actually found and excavated. It is no less than nine hundred yards long, complete with water-course and inspection-pathway, and its shape shows that it was begun from both ends. It further shows that the two digging parties, one working from the north, the other from the south, had met in the centre only a couple of feet apart. Having watched this fantastic feat being performed (by Eupalinos, who also built the second marvel mentioned by Herodotus, a huge mole to protect the Samian war-fleet), even a lesser genius than Pythagoras might have realized that Science may become a hymn to the creator or a Pandora's box, and that it should be trusted only to saints. It is said, incidentally, that Pythagoras, like St. Francis, preached to animals, which would seem rather odd behaviour in a modern mathematician; but in the Pythagorean view nothing could be more natural.
5. Tragedy and greatness of the Pythagoreans
Towards the end of the Master's life, or shortly after his death, two misfortunes befell the Pythagoreans, which would have meant the end of any sect or school with a less universal outlook. They triumphantly survived both.
One blow was the discovery of a type of numbers such as
– the square root of 2 – which could not be fitted into any dot-diagram. And such numbers were common: they are, for instance, represented by the diagonal of any square. Let the side of the square be called a, and the diagonal d. It can be proved that if I assign to a any precise numerical value, then it becomes impossible to assign a precise numerical value to d. The side and the square are "incommensurable"; their ratio a/d cannot be represented by any real numbers or fractions thereof; it is an "irrational" number; it is both odd and even at the same time. * I can easily draw the diagonal of a square, but I cannot express its length in numbers – I cannot count the number of dots it contains. The point-to-point correspondence between arithmetic and geometry has broken down – and with it the universe of numbershapes.
____________________
*
The simplest manner of proving this is as follows. Let d be represented by a fraction
,
where m and n are unknown. Let a = 1, then d2 = 12 12 and
Then
If m and n have a common factor, divided it out, then either m or n must be odd. Now m2 = 2n2, therefore m2 is even, therefore m is even, therefore n is odd. Suppose m = 2p. Then 4p2 = 2n2, therefore n2 = 2p2 and therefore n is even, contra hyp. Therefore no fraction
will measure the diagonal.
It is said that the Pythagoreans kept the discovery of irrational numbers – they called them arrhētos, unspeakable – a secret, and that Hippasos, the disciple who let the scandal leak out, was put to death. There is also another version, in Proclos: 11
"It is told that those who first brought out the irrationals from concealment into the open perished in shipwreck, to a man. For the unutterable and the formless must needs be concealed. And those who uncovered and touched this image of life were instantly destroyed and shall remain forever exposed to the play of the eternal waves."
Yet, Pythagoreanism survived. It had the elastic adaptability of all truly great ideological systems which, when some part is knocked out of them, display the self-regenerating powers of a growing crystal or a living organism. The mathematization of the world by means of atom-like dots proved a premature shortcut; but on a higher turn of the spiral, mathematical equations proved once again the most serviceable symbols for representing the physical aspect of reality. We shall meet with further examples of prophetic intuition supported by the wrong reasons; and we shall find that they are rather the rule than the exception.
Nobody before the Pythagoreans had thought that mathematical relations held the secret of the universe. Twenty-five centuries later, Europe is still blessed and cursed with their heritage. To non-European civilizations, the idea that numbers are the key to both wisdom and power, seems never to have occurred.
The second blow was the dissolution of the Brotherhood. We know little of its causes; it probably had something to do with the equalitarian principles and communist practices of the order, the emancipation of women, and its quasi-monotheistic doctrine – the eternal messianic heresy. But persecution remained confined to the Pythagoreans as an organized body – and probably prevented them from degenerating into sectarian orthodoxy. The Master's principal pupils – among them Philolaus and Lysis – who had gone into exile, were soon allowed to return to Southern Italy and to resume teaching. A century later, that teaching became one of the sources of Platonism, and thus entered the mainstream of European thought.
In the words of a modern scholar: " Pythagoras is the founder of European culture in the Western Mediterranean sphere." 12 Plato and Aristotle, Euclid and Archimedes, are landmarks on the road; but Pythagoras stands at the point of departure, where it is decided which direction the road will take. Before that decision, the future orientation of Greco-European civilization was still undecided: it may have taken the direction of the Chinese, or Indian, or pre-Columbian cultures, all of which were still equally unshaped and undecided at the time of the great sixth-ce
ntury dawn. I do not mean to say that if Confucius and Pythagoras had exchanged their places of birth, China would have beaten us to the Scientific Revolution, and Europe become a land of tea-sipping mandarins. The interactions of climate, race and spirit, the directional influence of outstanding individuals on the course of History, are so obscure that no predictions are possible even in reverse; all "if " statements about the past are as dubious as prophecies of the future are. It seems fairly plausible that if Alexander or Ghengis Khan had never been born, some other individual would have filled his place and executed the design of the Hellenic or Mongolic expansion; but the Alexanders of philosophy and religion, of science and art, seem less expendable; their impact seems less determined by economic challenges and social pressures; and they seem to have a much wider range of possibilities to influence the direction, shape and texture of civilizations. If conquerors be regarded as the engine-drivers of History, then the conquerors of thought are perhaps the pointsmen who, less conspicuous to the traveller's eye, determine the direction of the journey.
III THE EARTH ADRIFT
I HAVE tried to give a brief general description of Pythagorean philosophy, including aspects of it that are only indirectly related to the subject of this book. In the following sections, some important schools of Greek philosophy and science – Eleatics and Stoics, Atomists and Hippocratics – will hardly be mentioned at all, until we arrive at the next turning point in cosmology, Plato and Aristotle. The development of man's views about the cosmos cannot be treated in isolation from the philosophical background which coloured these views; on the other hand, if the narrative is not to be swallowed up by the background, the latter can only be sketched in at certain turning points of the tale, where the general philosophical climate had a direct impact on cosmology and altered its course. Thus, for instance, the political views of Plato, or the religious convictions of Cardinal Bellarmine, profoundly influenced astronomical developments for centuries, and must accordingly be discussed; whereas men like Empedokles and Democritus, Socrates and Zeno, who had a lot to say about the stars, but nothing that is really relevant to our subject, must be passed in silence.
1. Philolaus and the Central Fire
From the end of the sixth century B.C. onward, the idea that the earth was a sphere, freely floating in air, made steady head-way. Herodotus 1 mentions a rumour that there exist people far up in the north who sleep six months of the year – which shows that some of the implications of the earth's roundness (such as the polar night) had already been grasped. The next, revolutionary step was taken by a pupil of Pythagoras, Philolaus, the first philosopher to attribute motion to our globe. The earth became air-borne.
The motives which led to this tremendous innovation we can only guess. Perhaps it was the realization that there is something illogical in the apparent movements of the planets. It seemed crazy that the sun and planets should turn round the earth once a day, but at the same time slowly crawl along the Zodiac on their annual revolutions. Everything would be much simpler if one assumed that the daily revolution of the entire sky was an illusion caused by the earth's own motion. If the earth existed free and unattached in space, could she not also move? Yet the apparently obvious idea of letting the earth rotate on her own axis did not occur to Philolaus. Instead, he made her revolve, in twenty-four hours, round an extraneous point in space. By describing one complete circle a day, the observer on earth would have the illusion, like a traveller on a roundabout, that the whole cosmic fair was turning in the opposite direction.
In the centre of his roundabout, Philolaus placed the "watchtower of Zeus", also called "the hearth of the universe" or the "central fire". But this "central fire" is not to be confused with the sun. It could never be seen; for the inhabited part of the earth – Greece and its neighbours – was always turned away from it, as the dark side of the moon is always turned away from the earth. Moreover, between the earth and the central fire Philolaus inserted an invisible planet: the antichton or counter-earth. Its function was, apparently, to protect the antipodes from being scorched by the central fire. The ancient belief that the farwestern regions of the earth, beyond the straits of Gibraltar, are shrouded in eternal twilight 2 was now explained by the shadow which the counter-earth threw on those parts. But it is also possible – as Aristotle contemptuously remarks – that the counterearth was invented merely to bring the number of moving things in the universe up to ten, the sacred number of the Pythagoreans. 3
Around the central fire, then, revolved in concentric orbits these nine bodies: the antichton innermost, then the earth, the moon, the sun and the five planets; then came the sphere carrying all the fixed stars. Beyond this outer shell there was a wall of fiery ether, enclosing the world from all sides. This "outer fire" was the second and main source from which the universe drew its light and breath. The sun served merely as a kind of transparent window or lens, through which the outer light was filtered and distributed. The picture reminds one of Anaximander's holes in the flame-filled tyre. But these fantastic imaginations were perhaps less fantastic than the notion of a ball of fire hurtling across the sky through eternity, without burning out; a preposterous idea at which the mind boggles. Looking at the sky with eyes washed clean of theories, is it not more convincing to regard the sun and stars as holes in the curtain enclosing the world?
The only heavenly object considered to be similar to the earth was the moon. It was supposed to be inhabited by plants and animals fifteen times as strong as ours, because the moon enjoyed daylight for fifteen days in succession. Other Pythagoreans thought that the lights and shadows on the moon were reflections of our oceans. As for eclipses, some were caused by the earth, some by the counter-earth, which also accounted for the faint ashen light on the lunar disc at new moon. Still others seem to have assumed the existence of several counter-earths. It must have been a lively debate.
2. Herakleides and the Sun-Centred Universe
In spite of its poetic oddities, the system of Philolaus opened up a new cosmic perspective. It broke away from the geocentric tradition – the sturdy conviction that this earth occupies the centre of the Universe, from which, massive and immobile, it never budges an inch.
But it was also a landmark in another direction. It separated neatly two phenomena which had previously been mixed up: the succession of day and night, that is, the diurnal rotation of the sky as a whole; and the annual motions of the seven wandering planets.
The next improvement of the model concerned the daily motions. The central fire dropped out; the earth, instead of going round it, was now made to spin on her own axis, like a top. The reason was, presumably, 4 that the Greek seafarers' growing contacts with distant regions – from the Ganges to the Tagus, from the island of Thule to Taprobrana – had failed to produce any sign, or even rumour, of the central fire or the antichton, both of which should have been visible from the other side of the earth. I have said before that the Pythagoreans' world-view was elastic and adaptable. They did not drop the idea of the central fire as a source of heat and energy; but they transferred it from outer space into the core of the earth; and the counter-earth they simply identified with the moon. 5
The next great pioneer in the Pythagorean tradition is Herakleides of Pontus. He lived in the fourth century B.C., studied under Plato, and presumably also under Aristotle; hence, by chronological order, he ought to be discussed after these. But I shall first follow the development of the Pythagorean cosmology, the boldest and most hopeful in antiquity, to its end – which came in the generation after Herakleides.
Herakleides took the earth's rotation round its own axis for granted. This explained the daily round of the skies, but left the problem of the annual motion of the planets untouched. By now, these annual motions had become the central problem of astronomy and cosmology. The multitude of fixed stars presented no problem. They never altered their positions relative to each other or to the earth. 6 They were a permanent guarantee of law and order and regularity in the universe, an
d could be imagined, without much difficulty, as a pattern of pin-heads (or pin-holes) in the celestial pin-cushion which either turned, as a unit, around the earth, or appeared to do so owing to the earth's rotation. But the planets, the tramp stars, moved with a shocking irregularity. Their only reassuring feature was that they all moved along the same narrow belt or lane looped around the sky (the Zodiac): which meant that their orbits all lay very nearly in the same plane.
To get an idea of how the Greeks perceived the universe, imagine all transatlantic traffic – submarines, ships, aircraft – to be confined to the same trade-route. The "orbits" of all craft will then be along concentric circles round the earth's centre, all in the same plane. Let an observer lie on his back in a cavity in the centre of the transparent earth, and watch the traffic. It will appear to him as points moving at different speeds along a single line: his zodiacal lane. If the transparent sphere is set rotating round the observer (who, himself, remains at rest) the traffic-lane will rotate with the sphere, but the traffic will still remain confined to this lane. The traffic consists of: two submarines ploughing the waters at different depths under the lane: they are the "lower" planets, Mercury and Venus; then a single ship with blazing lights: the sun; then three aeroplanes at different heights: the "upper" planets, Mars, Jupiter and Saturn, in that order. Saturn would be very high up in the stratosphere; above it there is only the sphere of the fixed stars. As for the moon, she is so close to the observer in the centre, that she must be considered a ball rolling on the concave wall inside his cavity; but still in the same plane with all the other craft. This, then, in broad outlines, is the antique model of the world ( Fig. A ).