Chapter five — THE MUSIC OF THE SPHERES

Mathematics is in many ways the most elaborated and sophisticated of the sciences — or so it seems to me, as a mathematician. So I find both a special pleasure and constraint in describing the progress of mathematics, because it has been part of so much human speculation: a ladder for mystical as well as rational thought in the intellectual ascent of man. However, there are some con­cepts that any account of mathematics should include: the logical idea of proof, the empirical idea of exact laws of nature (of space particularly), the emergence of the concept of operations, and the movement in mathematics from a static to a dynamic description of nature. They form the theme of this essay.

Even very primitive peoples have a number system; they may not count much beyond four, but they know that two of any thing plus two of the same thing makes four, not just sometimes but always. From that fundamental step, many cultures have built their own number systems, usually as a written language with similar conventions. The Babylonians, the Mayans, and the people of India, for example, invented essentially the same way of writing large numbers as a sequence of digits that we use, although they lived far apart in space and in time.

So there is no place and no moment in history where I could stand and say ‘Arithmetic begins here, now’. People have been counting, as they have been talking, in every culture. Arithmetic, like language, begins in legend. But mathematics in our sense, reasoning with numbers, is another matter. And it is to look for the origin of that, at the hinge of legend and history, that I went sailing to the island of Samos.

In legendary times Samos was a centre of the Greek worship of Hera, the Queen of Heaven, the lawful (and jealous) wife of Zeus. What remains of her temple, the Heraion, dates from the sixth century before Christ. At that time there was born on Samos, about 580 BC, the first genius and the founder of Greek mathematics, Pythagoras. During his lifetime the island was taken over by the tyrant, Polycrates. There is a tradition that before Pythagoras fled, he taught for a while in hiding in a small white cave in the mountains which is still shown to the credulous.

Samos is a magical island. The air is full of sea and trees and music. Other Greek islands will do as a setting for The Tempest, but for me this is Prospero’s island, the shore where the scholar turned magician. Perhaps Pythagoras was a kind of magician to his followers, because he taught them that nature is commanded by numbers. There is a harmony in nature, he said, 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. The story of his discovery survives only in garbled form, like a folk tale. But what he discovered was precise. A single stretched string vibrating as a whole produces a ground note. The notes that sound harmonious with it are produced by dividing the string into an exact number of parts: into exactly two parts, into exactly three parts, into exactly four parts, and so on. If the still point on the string, the node, does not come at one of these exact points, the sound is discordant.

As we shift the node along the string, we recognize the notes that are harmonious when we reach the prescribed points. Begin with the whole string: this is the ground note. Move the node to the midpoint: this is the octave above it. Move the node to a point one third of the way along: this is the fifth above that. Move it to a point one fourth along: this is the fourth, another octave above. And if you move the node to a point one fifth of the way along, this (which Pythagoras did not reach) is the major third above that.

Pythagoras had found that the chords which sound pleasing to the ear — 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. For example, Pythagoras or his follow­ers believed that we should be able to calculate the orbits of the heavenly bodies (which the Greeks pictured as carried round the earth on crystal spheres) by relating them to the musical intervals. 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. It is likely that many of the later followers of Pythagoras were slaves; they believed in the transmigration of souls, which may have been their way of hoping for a happier life after death.

I have been speaking of the language of numbers, that is arithmetic, but my last example was the heavenly spheres, which are geometrical shapes. The transition is not acci­dental. Nature presents us with shapes: a wave, a crystal, the human body, and it is we who have to sense and find the numerical relations in them. 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.

Pythagoras had proved that the world of sound is governed by exact numbers. He went on to prove that the same thing is true of the world of vision. That is an extraordinary achievement. I look about me; here I am, in this marvellous, 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; so that if I were to turn this right angle of experience (the direction of ‘down’ and the direction of ‘sideways’) four times, back I come to the cross of gravity and the horizon. The right angle is defined by this fourfold operation, and is distinguished by it from any other arbitrary angle.

In the world of vision, then, in the vertical picture plane that our eyes present to us, a right angle is defined by its fourfold rotation back on itself. The same definition holds also in the horizontal world of experience, in which in fact we move. Consider that world, the world of the flat earth and the map and the points of the compass. 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. These cultures already knew in a practical sense that there is a builder’s set square in which the numerical relations dictate and make the right angle. The Babylonians knew many, perhaps hundreds of formulae for this by 2000 BC. The Indians and the Egyptians knew some. The Egyptians, it seems, almost always used a set square with the sides of the triangle made of three, four, and five units. 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. (It is not the proof that stands in the school books.) The four leading points — south, west, north, east — of the triangles that form the cross of the compass are the corners of a square. I slide the four triangles so that the long side of each ends at the leading point of a neighbour. Now I have constructed a square on the longest side of the right-angled triangles — on the hypotenuse. Just so that we should know what is part of the enclosed area and what is not, I will fill in the small inner square area that has now been uncovered with an additional tile. (I use tiles because many tile patterns, in Rome, in the Orient, from now on derive from this kind of wedding of mathematical relation to thought about nature.)

Now we have a square on the hypotenuse, and we can of course relate that by calculation to the squares on the two shorter sides. But that would miss the natural structure and inwardness of the figure. We do not need any calcula­tion. A small game, such as children and mathematicians play, will reveal more than calculation. Transpose two triangles to new positions, thus. Move the triangle that pointed south so that its longest side lies along the longest side of the triangle that pointed north. And move the triangle that pointed east so that its longest side lies along the longest side of the triangle that pointed west.

Now we have constructed an L-shaped figure with the same area (of course, because it is made of the same pieces) whose sides we can see at once in terms of the smaller sides of the right-angled triangle. Let me make the composition of the L-shaped figure visible: put a divider down that separates the end of the L from the upright part. Then it is clear that the end is a square on the shorter side of the triangle; and the upright part of the L is a square on the longer of the two sides enclosing the right angle.

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. He had proved that the square on the longest side or hypotenuse is equal to the square on one of the other two sides plus the square on the other if, and only if, the angle they contain is a right angle. For instance, the sides 3:4:5 compose a right-angled triangle because

5² = 5X5 = 25

= 16+9 = 4x4+3x3

= 4²+3²

And the same is true of the sides of triangles found by the Babylonians, whether simple as 8:15:17, or forbidding as 3367:3456:4825, 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. That seems a bold and extraordinary thing to say, yet it is not extravagant; because what Pythagoras established is a fundamental characterization of the space in which we move, and it is the first time that is translated into numbers. And 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.

The point is that the theorem of Pythagoras in the form in which I have proved it is an elucidation of the symmetry of plane space; the right angle is the element of symmetry that divides the plane four ways. If plane space had a different kind of symmetry, the theorem would not be true; some other relation between the sides of special triangles would be true. And space is just as crucial a part of nature as matter is, even if (like the air) it is invisible; that is what the science of geometry is about. Symmetry is not merely a descriptive nicety; like other thoughts in Pythagoras, it penetrates to the harmony in nature.

When Pythagoras had proved the great theorem, he offered a hundred oxen to the Muses in thanks for the inspiration. It is a gesture of pride and humility together, such as every scientist feels to this day when the numbers dovetail and say, ‘This is a part of, a key to, the structure of nature herself.

Pythagoras was a philosopher, and something of a religious figure to his followers as well. The fact is there was in him something of that Asiatic influence which flows all through Greek culture and which we commonly over­look. 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, from one city to another. 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. As one example among many, 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.

Euclid evidently belonged to the Pythagorean tradition. When a listener asked him what was the practical use of some theorem, Euclid is reported to have said contemp­tuously to his slave, ‘He wants to profit from learning — give him a penny’. The reproof was probably adapted from a motto of the Pythagorean brotherhood, which translates roughly as ‘A diagram and a step, not a diagram and a penny’ — ‘a step’ being a step in knowledge or what I have called the Ascent of Man.

The impact of Euclid as a model of mathematical reasoning was immense and lasting. His book Elements of Geometry was translated and copied more than any other book except the Bible right into modern times. I was first taught mathematics by a man who still quoted the theorems of geometry by the numbers that Euclid had given them; and that was not uncommon even fifty years ago, and was the standard mode of reference in the past. When John Aubrey about 1680 wrote an account of how Thomas Hobbes in middle age had suddenly fallen ‘in love with geometry’ and so with philosophy, he explained that it began when Hobbes happened to see ‘in a gentlman’s library, Euclid’s Elements lay open, and ‘twas the 47 Element libri I’. Proposition 47 in Book I of Euclid’s Elements is the famous theorem of Pythagoras.

The other science practised in Alexandria in the centuries around the birth of Christ was astronomy. Again, we can catch the drift of history in the undertow of legend: when the Bible says that three wise men followed a star to Bethlehem, there sounds in the story the echo of an age when wise men are stargazers. 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. His work came to Europe in Arabic texts, for the original Greek manuscript editions were largely lost, some in the pillage of the great library of Alexandria by Christian zealots in AD 389, others in the wars and invasions that swept the Eastern Mediterranean throughout the Dark Ages.

The model of the heavens that Ptolemy constructed is wonderfully complex, but it begins from a simple analogy. The moon revolves round the earth, obviously; and it seemed just as obvious to Ptolemy that the sun and the planets do the same. (The ancients thought of the moon and the sun as planets.) The Greeks had believed that the perfect form of motion is a circle, and so Ptolemy made the planets run on circles, or on circles running in their turn on other circles. To us, that scheme of cycles and epicycles seems both simple-minded and artificial. Yet in fact the system was a beautiful and a workable invention, and an article of faith for Arabs and Christians right through the Middle Ages. It lasted for fourteen hundred years, which is a great deal longer than any more recent scientific theory can be expected to survive without radical change.

It is pertinent to reflect here why astronomy was devel­oped so early and so elaborately, and in effect became the archetype for the physical sciences. In themselves, the stars must be quite the most improbable natural objects to rouse human curiosity. The human body ought to have been a much better candidate for early systematic interest. Then why did astronomy advance as a first science ahead of medicine? Why did medicine itself turn to the stars for omens, to predict the favourable and the adverse influences competing for the life of the patient — surely the appeal to astrology is an abdication of medicine as a science? In my view, a major reason is that the observed motions of the stars turned out to be calculable, and from an early time (perhaps 3000 BC in Babylon) lent themselves to math­ematics. 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 six hundred years after Christ was the new, powerful impulse. It started as a local event, uncertain in its outcome; but once Mahomet conquered Mecca in ad 630, it took the southern world by storm. In a hundred years, Islam captured Alexandria, established a fabulous city of learning in Baghdad, and thrust its frontier to the east beyond Isfahan in Persia. 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. At the same time, there was a liberation of simple, local skills that had been despised. For instance, the first domed mosques were built with no more sophisticated apparatus than the ancient builder’s set square — that is still used. The Masjid-i-Jomi (the Friday Mosque) in Isfahan is one of the statuesque monuments of early Islam. In centres like these, the knowledge of Greece and of the east was treasured, absorbed and diversified.

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. Mohammedan writers depersonalized and formalized the godhead: the mysticism of Islam is not blood and wine, flesh and bread, but an unearthly ecstasy.

Allah is the light of the heavens and the earth. His light may be compared to a niche that enshrines a lamp, the lamp within a crystal of star-like brilliance, light upon light. In temples which Allah has sanctioned to be built for the remembrance of his name do men praise him morning and evening, men whom neither trade nor profit can divert from remembering him.

One of the Greek inventions that Islam elaborated and spread was the astrolabe. As an observational device, it is primitive; it only measures the elevation of the sun or a star, and that crudely. But by coupling that single observation with one or more star maps, the astrolabe also carried out an elaborate scheme of computations that could determine latitude, sunrise and sunset, the time for prayer and the direction of Mecca for the traveller. And over the star map, the astrolabe was embellished with astrological and religious details, of course, for mystic comfort.

For a long time the astrolabe was the pocket watch and the slide rule of the world. When the poet Geoffrey Chaucer in 1391 wrote a primer to teach his son how to use the astrolabe, he copied it from an Arab astronomer of the eighth century.

Calculation was an endless delight to Moorish scholars. They loved problems, they enjoyed finding ingenious methods to solve them, and sometimes they 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 thirteenth century. The calculations it makes are not deep, an alignment of dials for prognostication, yet it is a testimony to the mechanical skill of those who made it seven hundred years ago, and to their passion for playing with numbers.

The most important single innovation that the eager, inquisitive, and tolerant Arab scholars brought from afar was in writing numbers. The European notation for numbers then was still the clumsy Roman style, in which the number is put together from its parts by simple addition: for example, 1825 is written as MDCCCXXV, because it is the sum of M = 1000, D = 500, C+C+C= 100+100+100, XX = 10+10, and V = 5. Islam replaced that by the modern decimal notation that we still call ‘Arabic’. In the note in an Arab manuscript, the numbers in the top row are 18 and 25. We recognize 1 and 2 at once as our own symbols (though the 2 is stood on end). To write 1825, the four symbols would simply be written as they stand, in order, running straight on as a single number; because it is the place in which each symbol stands that announces whether it stands for thousands, or hundreds, or tens, or units.

However, a system that describes magnitude by place must provide for the possibility of empty places. The Arabic notation requires the invention of a zero. The symbol for zero occurs twice on this page, and several more times 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 five hundred 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.

Seen from the outside, the Alhambra is a square, brutal fortress that does not hint at Arab forms. Inside, it is not a fortress but a palace, and a palace designed deliberately to prefigure on earth the bliss of heaven. The Alhambra is a late construction. It has the lassitude of an empire past its peak, unadventurous and, it thought, safe. The religion of meditation has become sensuous and self-satisfied. It sounds with the music of water, whose sinuous line runs through all Arab melodies, though they are based fair and square on the Pythagorean scale. Each court in turn is the echo and the memory of a dream, through which the Sultan floated (for he did not walk, he was carried). 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. The last Moorish king reigned here until 1492, when Queen Isabella of Spain was already backing the adventure of Columbus. It is a honey­comb of courts and chambers, and the Sala de las Camas is the most secret place in the palace. Here the girls from the harem came after the bath and reclined, naked. Blind musicians played in the gallery, the eunuchs padded about. And the Sultan watched from above, and sent an apple down to signal to the girl of his choice that she would spend the night with him.

In a western civilization, this room would be filled with marvellous drawings of the female form, erotic pictures. Not so here. The representation of the human body was forbidden to Mohammedans. Indeed, even the study of anatomy at all was forbidden, and that was a major handicap to Moslem science. So here we find coloured but extraordinarily simple geometric designs. The artist and the mathematician in Arab civilization have become one. And I mean that quite literally. These patterns represent a high point of the Arab exploration of the subtleties and symmetries of space itself: the flat, two-dimensional space of what we now call the Euclidean plane, which Pythagoras first characterized.

In the wealth of patterns, I begin with a very straightforward one. It repeats a two-leaved motif of dark horizontal leaves, and another of light vertical leaves. The obvious symmetries are translations (that is, parallel shifts of the pattern) and either horizontal or vertical reflections. But note one more delicate point. The Arabs were fond of designs in which the dark and the light units of the pattern are identical. And so, if for a moment you ignore the colours, you can see that you could turn a dark leaf once through a right angle into the position of a neighbouring light leaf. Then, always rotating round the same point of junction, you can turn it into the next position, and (again round the same point) into the next, and finally back on itself. And the rotation spins the whole pattern correctly; every leaf in the pattern arrives at the position of another leaf, however far from the centre of rotation they lie.

Reflection in a horizontal line is a twofold symmetry of the coloured pattern, and so is reflection in a vertical. But if we ignore the colours, we see that there is a fourfold symmetry. It is provided by the operation of rotating through a right angle, repeated four times, by which I earlier proved the theorem of Pythagoras; and thereby the uncoloured pattern becomes in its symmetry like the Pythagorean square.

I turn to a much more subtle pattern. These windswept triangles in four colours display only one very straightfor­ward kind of symmetry, in two directions. You could shift the pattern horizontally or you could shift it vertically into new, identical positions. Being windswept is not irrelevant. It is unusual to find a symmetrical system which does not allow reflection. However, this one does not, because these windswept triangles are all right-handed in movement, and you cannot reflect them without making them left-handed.

Now suppose you neglect the difference between the green, the yellow, the black, and the royal blue, and think of the distinction as simply between dark triangles and light triangles. Then there is also a symmetry of rotation. Fix your attention again on a point of junction: six triangles meet there, and they are alternately dark and light. A dark triangle can be rotated there into the position of the next dark triangle, then into the position of the next, and finally back into the original position — a threefold symmetry which rotates the whole pattern.

And indeed the possible symmetries need not stop there. If you forget about the colours at all, then there is a lesser rotation by which you could move a dark triangle into the space of the light triangle beside it because it is identical in shape. This operation of rotation then goes on into the dark, into the light, into the dark, into the light, and finally back into the original dark triangle — a sixfold symmetry of space which rotates the whole pattern. And the sixfold symmetry in fact is the one we all know best, because it is a symmetry of the snow crystal.

At this point, the non-mathematician is entitled to ask, ‘So what? Is that what mathematics is about? Did Arab professors, do modern mathematicians, spend their time with that kind of elegant game?’ To which the unexpected answer is — Well, it is not a game. It brings us face to face with something which is hard to remember, and that is that we live in a special kind of space — three-dimensional, flat — and the properties of that space are unbreakable. In asking what operations will turn a pattern into itself, we are discovering the invisible laws that govern our space. There are only certain kinds 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.

The structures that enshrine, as it were, the natural patterns of space are the crystals. And when you look at one untouched by human hand — say, Iceland spar — there is a shock of surprise in realizing that it is not self-evident why its faces should be regular. It is not self-evident why they should even be flat planes. This is how crystals come; we are used to their being regular and symmetrical; but why? They were not made that way by man but by nature. That flat face is the way in which the atoms had to come together — and that one, and that one. The flatness, the regularity has been forced by space on matter with the same finality as space gave the Moorish patterns their symmetries that I analysed.

Take a beautiful cube of pyrites. Or to me the most exquisite crystal of all, fluorite, an octahedron. (It is also the natural shape of the diamond crystal.) Their symmetries are imposed on them by the nature of the space we live in — the three dimensions, the flatness within which we live. And no assembly of atoms can break that crucial law of nature. Like the units that compose a pattern, the atoms in a crystal are stacked in all directions. So a crystal, like a pattern, must have a shape that could extend or repeat itself in all directions indefinitely. That is why the faces of a crystal can only have certain shapes; they could not have anything but the symmetries in the patterns. For example, the only rotations that are possible go twice or four times for a full turn, or three times or six times — not more. And not five times. You cannot make an assembly of atoms to make triangles which fit into space regularly five at a time.

Thinking about these forms of pattern, exhausting in practice the possibilities of the symmetries of space (at least in two dimensions), was the great achievement of Arab mathematics. And it has a wonderful finality, a thousand years old. The king, the naked women, the eunuchs and the blind musicians made a marvellous formal pattern in which the exploration of what exists was perfect, but which, alas, was not looking for any change. 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 1000from footholds like the village of Santillana in a coastal strip which the Moors never conquered. It is a religion of the earth there, expressed in the simple images of the village — the ox, the ass, the Lamb of God. The animal images would be unthinkable in Moslem worship. And not only the animal form is allowed; the Son of God is a child, His mother is a woman and is the object of personal worship. When the Virgin is carried in procession, we are in a different universe of vision: not of abstract patterns, but of abounding and irrepressible life.

When Christianity came to win back Spain, the excite­ment of the struggle was on the frontier. Here Moors and Christians, and Jews too, mingled and made an extraordi­nary culture of different faiths. In 1085 the centre of this mixed culture was fixed for a time in the city of Toledo. Toledo was 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 of Italy as the birthplace of the Renaissance. But the conception was in Spain in the twelfth century, and it is symbolized and expressed by the famous school of translators at Toledo, where the ancient texts were turned from Greek (which Europe had forgotten) through Arabic and Hebrew into Latin. In Toledo, amid other intellectual advances, an early set of astronomical tables was drawn up, as an encyclopedia of star positions. It is characteristic of the city and the time that the tables are Christian, but the numerals are Arabic, and are by now recognizably modern.

The most famous of the translators and the most brilliant was Gerard of Cremona, who had come from Italy specifi­cally to find a copy of Ptolemy’s book of astronomy, the Almagest, and who stayed on in Toledo to translate Archimedes, Hippocrates, Galen, Euclid — the classics of Greek science.

And yet, to me personally, the most remarkable and, in the long run, the most influential man who was translated was not a Greek. That is because I am interested in the perception of objects in space. And that was a subject about which the Greeks were totally wrong. It was understood for the first time about the year AD 1000 by an eccentric mathematician whom we call Alhazen, who was the one really original scientific mind that Arab culture produced. 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. The Greek view could not explain how an object, my hand say, seems to change size when it moves. 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. And that, and only that, accounts for the difference in size. It is so simple a notion that it is astonishing that scientists paid almost no attention to it (Roger Bacon is an exception) for six hundred years. But artists attended to it long before that, and in a practical way. 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 fifteenth century. A manuscript of Alhazen’s Optics in translation in the Vatican Library in Rome is annotated by Lorenzo Ghiberti, who made the famous bronze perspectives for the doors of the Baptistry in Florence. He was not the first pioneer of perspective — that may have been Filippo Brunelleschi — and there were enough of them to form an identifiable school of Perspectivi. 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.

The movement is evident as soon as we contrast a work by the Perspectivi with an earlier one. Carpaccio’s painting of St Ursula leaving a vaguely Venetian port was painted in 1495. The obvious effect is to give to visual space a third dimension, just as the ear about this time hears another depth and dimension in the new harmonies in European music. But the ultimate effect is not so much depth as movement. Like the new music, the picture and its inhabitants are mobile. Above all, we feel that the painter’s eye is on the move.

Contrast a fresco of Florence painted a hundred years earlier, about AD 1350. It is a view of the city from outside the walls, and the painter looks naively over the walls and the tops of the houses as if they were arranged in tiers. But this is not a matter of skill; it is a matter of intention. There is no attempt at perspective because the painter thought of himself as recording things, not as they look, but as they are: a God’s eye view, a map of eternal truth.

The perspective painter has a different intention. He deliberately makes us step away from any absolute and abstract view. Not so much a place as a moment is fixed for us, and a fleeting moment: a point of view in time more than in space. All this was achieved by exact and math­ematical means. The apparatus has been recorded with care by the German artist, Albrecht Dürer, who travelled to Italy in 1506 to learn ‘the secret art of perspective’. Dürer of course has himself fixed a moment in time; and if we re-create his scene, we see the artist choosing the dramatic moment. He could have stopped early in his walk round the model. Or he could have moved, and frozen the vision at a later moment. But he chose to open his eye, like a camera shutter, understandably at the strong moment, when he sees the model full face. Perspective is not one point of view; for the painter, it is an active and continuous operation.

In early perspective it was customary to use a sight and a grid to hold the instant of vision. The sighting device comes from astronomy, and the squared paper on which the picture was drawn is now the stand-by of mathematics. All the natural details in which Dürer delights are expres­sions of the dynamic of time: the ox and the ass, the blush of youth on the cheek of the Virgin. The picture is The Adoration of the Magi. The three wise men from the east have found their star, and what it announces is the birth of time.

The chalice at the centre of Dürer’s painting was a test — piece in teaching perspective. For example, we have Uccello’s analysis of the way the chalice looks; we can turn it on the computer as the perspective artist did. His eye worked like a turntable to follow and explore its shifting shape, the elongation of the circles into ellipses, and to catch the moment of time as a trace in space.

Analysing the changing movement of an object, as I can do on the computer, was quite foreign to Greek and to Islamic minds. They looked always for what was unchang­ing and static, a timeless world of perfect order. The most perfect shape to them was the circle. Motion must run smoothly and uniformly in circles; that was the harmony of the spheres.

This is why 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 seventeenth century — Isaac Newton and Gottfried Wilhelm Leibniz. It is now so familiar to us that we think of time as a natural element in a description of nature; but that was not always so. It was they who brought in the idea of a tangent, the idea of acceleration, the idea of slope, the idea of infinitesimal, of differential. There is a word that has been forgotten but that is really the best name for that flux of time that Newton stopped like a shutter: Fluxions was Newton’s name for what is usually called (after Leibniz) the differ­ential calculus. 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 technical concept that makes it work is, oddly enough, the concept of an infinitesimal step; and the intellectual breakthrough came in giving a rigorous meaning to that. But we may leave the technical concept to the professionals, and be content to call it the mathematics of change.

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.

Date: 2016-01-14; view: 701