THE ASCENT OF MAN

 JACOB BROWNOWSKI

MACDONALD FUTURA PUBLISHERS                        1973

PART V

 

Chapter 5: The Music of the Spheres

• Even very primitive peoples have a number system.
• Born on Samos, about 580 BC, the first genius and founder of Greek mathematics, was Pythagoras. He said there is a harmony in nature, a unity in her variety, and it has a language: numbers are the language of nature.
• Pythagoras found a basic relation between musical harmony and mathematics, finding that the chords which sound pleasing to the western ear correspond to exact divisions of the string by whole numbers.
• To the Pythagoreans that discovery had a mystic force. The agreement between nature and number was so cogent that it persuaded them that not only the sounds of nature, but all her characteristic dimensions, must be simple numbers that express harmonies.
• They felt that all the regularities in nature are musical; the movements of the heavens were, for them, the music of the spheres.
• These ideas gave Pythagoras the status of a seer in philosophy, almost a religious leader, whose followers formed a secret and perhaps revolutionary sect. Pythagoras was a pioneer in linking geometry with numbers, and since it is also my choice among the branches of mathematics, it is fitting to watch what he did.
• Having proved that the world of sound is governed by numbers, Pythagoras  went on to prove that the same thing is true of the world of vision.
• Here I am, in this marvelous, coloured landscape of Greece, among the wild natural forms, the Orphic dells, the sea. Where under this beautiful chaos can there lie a simple, numerical structure?
• The question forces us back to the most primitive constants in our perception of natural laws. To answer well, it is clear that we must begin from universals of experience.
• There are two experiences on which our visual world is based: that gravity is vertical, and that the horizon stands at right angles to it.
• And it is that conjunction, those cross wires in the visual field, which fixes the nature of the right angle.
• Here I am looking across the straits from Samos to Asia Minor, due south. I take a triangular tile as a pointer and I set it pointing there, south. (I have made the pointer in the shape of a right-angled triangle, because I shall want to put its four rotations side by side.)
• If I turn that triangular tile through a right angle, it points due west. If I now turn it through a second right angle, it points due north.
• And if I now turn it through a third right angle, it points due east. Finally, the fourth and last turn will take it due south again, pointing to Asia Minor, in the direction in which it began.
• Not only the natural world as we experience it, but the world as we construct it is built on that relation. It has been so since the time that the Babylonians built the Hanging Gardens, and earlier, since the time that the Egyptians built the pyramids.
• The Babylonians knew many, perhaps hundreds of formulae for this by 2000 BC. The Indians and the Egyptians knew some.
• It was not until 550 BC or thereabouts that Pythagoras raised this knowledge out of the world of empirical fact into the world of what we should now call proof.
• That is, he asked the question, ‘How do such numbers that make up these builder’s triangles flow from the fact that a right angle is what you turn four times to point the same way?
• His proof, we think, ran something like this ….
• Pythagoras had thus proved a general theorem: not just for the 3:4:5 triangle of Egypt, or any Babylonian triangle, but for every triangle that contains a right angle.
• And the same is true of the sides of triangles found by the Babylonians, whether simple 8:15:17, or forbidding as 3367: 3456: 4827, which leave no doubt that they were good at arithmetic.
• To this day, the theorem of Pythagoras remains the most important single theorem in the whole of mathematics. The exact fit of the numbers describes the exact laws that bind the universe.
• In fact, the numbers that compose right-angled triangles have been proposed as messages which we might send out to planets in other star systems as a test for the existence of rational life there.
• Pythagoras was a philosopher, and something of a religious figure to his followers as well. We tend to think of Greece as part of the west; but Samos, the edge of classical Greece, stands one mile from the coast of Asia Minor. From there much of the thought that inspired Greece first flowed; and, unexpectedly, it flowed back to Asia in the centuries after, before ever it reached Western Europe.
• Knowledge makes prodigious journeys, and what seems to us a leap in time often turns out to be a long progression from place to place. The caravans carry with their merchandise the methods of trade of their countries – the weights and measures, the methods of reckoning – and techniques and ideas went where they went, through Asia and North Africa.
• The mathematics of Pythagoras has not come to us directly. It fired the imagination of the Greeks, but the place where it was formed into an orderly system was the Nile city, Alexandria.
• The man who made the system, and made it famous, was Euclid, who probably took it to Alexandria around 300 BC.
• The impact of Euclid as a model of mathematical reasoning was immense and lasting. His Elements of Geometry was translated and copied more than any other book except the Bible right into modern times.
• The other science practiced in Alexandria in the centuries around the birth of Christ was astronomy. The secret of the heavens that wise men looked for in antiquity was read by a Greek called Claudius Ptolemy, working in Alexandria about AD 150.
• The model of the heavens that Ptolemy constructed is wonderfully complex, but begins with a simple analogy: the moon revolves round the earth ….
• The pre-eminence of astronomy rests on the peculiarity that it can be treated mathematically; and the progress of physics, and most recently of biology, has hinged equally on finding formulations of their laws that can be displayed as mathematical models.
• Every so often, the spread of ideas demands a new impulse. The coming of Islam 600 years after Christ was the new, powerful impulse. By AD 730 the Moslem empire reached from Spain and Southern France to the borders of China and India: an empire of spectacular strength and grace, while Europe lapsed in the Dark Ages.
• In this proselytizing religion, the science of the conquered nations was gathered with a kleptomaniac zest and there was a liberation of simple, local skills that had been despised.
• Mahomet had been firm that Islam was not to be a religion of miracles; it became in intellectual content a pattern of contemplation and analysis.
• One of the Greek inventions that Islam elaborated and spread was the astrolabe, which for a long time was the pocket watch and the slide rule of the world.
• Moorish scholars loved problems, enjoyed finding ingenious methods to solve them, and sometimes turned their methods into mechanical devices.
• A more elaborate ready-reckoner than the astrolabe is the astrological or astronomical computer, something like an automatic calendar, made in the Caliphate of Baghdad in the 13^th century.
• The most important single innovation that the eager, inquisitive, and tolerant Arab scholars brought from afar was in writing numbers.
• The European notation for writing numbers was still the clumsy Roman style. Islam replaced that by the modern decimal notation that we still call ‘Arabic’. The Arabic notation requires the invention of a zero.
• The symbol for zero occurs twice on this page, and several times more on the next, looking just like our own.
• The words zero and cipher are Arab words; so are algebra, almanac, zenith, and a dozen others in mathematics and astronomy.
• The Arabs brought the decimal system from India about AD 750, but it did not take hold in Europe for another 500 years after that.
• It may be the size of the Moorish Empire that made it a kind of bazaar of knowledge, whose scholars included heretic Nestorian Christians in the east and infidel Jews in the west.
• It may be a quality in Islam as a religion, which, though it strove to convert people, did not despise their knowledge. In the east the Persian city of Isfahan is its monument. In the west there survives an equally remarkable outpost, the Alhambra in southern Spain.
• The Alhambra is most nearly the description of Paradise from the Koran.

 

Blessed is the reward of those who labour patiently and put their trust in Allah. Those that embrace the true faith and do good works shall be forever lodged in the mansions of paradise, where rivers will roll at their feet …. And honoured shall they be in the gardens of delight, upon couches face to face. A cup shall be borne round among them from a fountain, limpid, delicious to those who drink … Their spouses on soft green cushions and on beautiful carpets shall recline.

• The Alhambra is the last and most exquisite monument of Arab civilization in Europe.
• In asking what operations will turn a pattern into itself, we are discovering the invisible laws that govern our space. There are only cerain kind of symmetries which our space can support, not only in man-made patterns, but in the regularities which nature herself imposes on her fundamental, atomic structures.
• Thinking about these forms of pattern was the great achievement of Aab mathematics.
• There is nothing new in mathematics, because there is nothing new in human thought, until the ascent of man moved forward to a different dynamic.
• Christianity began to surge back in northern Spain about AD 1000. Here Moors, Christians and Jews mingled and made extraordinary culture of different faiths.
• In 1085 the center of this mixed culture was fixed for a time in the city of Toledo, the intellectual port of entry into Christian Europe of all the classics that the Arabs had brought together from Greece, from the Middle East, from Asia.
• We think f Italy as the birthplace of the Renaissance. But the conception was in Spain in the 12^th century, and it is symbolized and expressed by the famous school of translators at Toledo, where ancient texts were turned from Greek (which Europe had forgotten) through Arabic and Hebrew and Latin.
• In Toledo, amid other intellectual advances, an early set of astronomical tables was drawn up, as an encyclopedia of start positions.
• The Greeks had thought that light goes from the eyes to the object. Alhazen first recognized that we see an object because each point of it directs and reflects a ray into the eye.
• In Alhazen’s account it is clear that the cone of rays that comes from the outline and shape of my hand grows narrower as I move my hand away from you. As I move it towards you, the cone of rays that enters your eye becomes larger and subtends a larger angle.
• That, and only that, accounts for the difference in size. The concept of the cone of rays from object to the eye becomes the foundation of perspective. And perspective is the new idea which now revivifies mathematics.
• The excitement of perspective passed into art in north Italy, in Florence and Venice, in the 15^th century. It was a school of thought, for its aim was not simply to make the figures lifelike, but to create the sense of their movement in space.
• Analyzing the changing movement of an object, as I can do on the computer, was quite foreign to Greek and Islamic minds. The Ptolemaic system was built up of circles, along which time ran uniformly and imperturbably.
• But movements in the real world are not uniform. They change direction and speed at every instant, and they cannot be analysed until a mathematics is invented in which time is a variable.
• That is a theoretical problem in the heavens, but it is practical and immediate on earth – in the flight of a projectile, in the spurting growth of a plant, in the single splash of a drop of liquid that goes through abrupt changes of shape and direction.
• The Renaissance did not have the technical equipment to stop the picture frame instant by instant. But the Renaissance had the intellectual equipment: the inner eye of the painter, and the logic of the mathematician.
• In this way Johannes Kepler after the year 1600 became convinced that the motion of a planet is not circular and not uniform. It is an ellipse along which the planet runs at varying speeds.
• That means that the old mathematics of static patterns will no longer suffice, nor the mathematics of uniform motion. You need a new mathematics to define and operate with instantaneous motion.
• The mathematics of instantaneous motion was invented by two superb minds of the late 17^th century – Isaac Newton and Gottfried Wilhelm Leibniz. It was they who brought in the idea of a tangent, the idea of acceleration, the idea of slope, the idea of infinitesimal and differential.
• To think of it merely as a more advanced technique is to miss its real content. In it, mathematics becomes a dynamic mode of thought, and that is a major mental step in the ascent of man.
• The laws of nature had always been made of numbers since Pythagoras said that was the language of nature. But now the language of nature had to include numbers which described time. The laws of nature become laws of motion and nature herself becomes not a series of static frames but a moving process.

 

Chapter 6: The Starry messenger