The Ascent of Man

by Jacob Bronowski

  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 realising 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 0f 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 1000 from 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 symbolised 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 recognisably 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 recognised 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 the 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 th
e 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.

  The moment of time as a trace in space.

  Paolo Uccello’s perspective analysis of a chalice.

  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 break-through 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.

  CHAPTER SIX

  THE STARRY MESSENGER

  The first science in the modern sense that grew in the Mediterranean civilisation was astronomy. It is natural to come to astronomy straight from mathematics; after all, astronomy was developed first, and became a model for all the other sciences, just because it could be turned into exact numbers. That is not an idiosyncrasy on my part. What is an idiosyncrasy is that I should choose to begin the drama of the first Mediterranean science in the New World.

  The rudiments of astronomy exist in all cultures, and were evidently important in the concerns of early peoples all over the world. One reason for this is clear. Astronomy is the knowledge that guides us through the cycle of the seasons – for example, by the apparent movement of the sun. In this way there can be fixed a time when men should plant, should harvest, move their herds and so on. Therefore all settled cultures have a calendar to guide their plans, and this was true in the New World as it was in the river-basins of Babylon and Egypt.

  An example is the civilisation of the Mayans that flourished before AD 1000 in the isthmus of America between the Atlantic and the Pacific Oceans. It has a claim to be the highest of the American cultures: it had a written language, skill in engineering, and original arts. The Mayan temple complexes, with their steep pyramids, housed some astronomers, and we have portraits of a group of them on a large altar stone that has survived. The altar commemorates an ancient astronomical congress that met in the year AD 776. Sixteen mathematicians came here to the famous centre of Mayan science, the sacred city of Copan in Central America.

  The Mayans had a system of arithmetic which was far ahead of Europe; for example, they had a symbol for zero. They were good mathematicians; nevertheless, they did not map the motions of the stars, except the simplest. Instead, their ritual was obsessed with the passage of time, and this formal concern dominated their astronomy as it did their poems and legends.

  When the great conference met at Copan, the Mayan priest astronomers had run into difficulty. We might suppose that such a major difficulty, calling for learned delegates to come from many centres, would turn on some real problem of observation. But we would be wrong. The congress was called to resolve an arithmetical problem of computation that perpetually troubled the Mayan guardians of the calendar. They kept two calendars, one sacred and one profane, which were never in step for long; and they spent their ingenuity trying to stop the drift between them. The Mayan astronomers had only simple rules for the planetary motions in the heavens, and they had no concept of their machinery. Their idea of astronomy was purely formal, a matter of keeping their calendars right. That was all that was done in AD 776 when the delegates proudly posed for their portraits.

  The point is that astronomy does not stop at the calendar. There is another use among early peoples which, however, was not universal. The movements of the stars in the night sky can also serve to guide the traveller, and particularly the traveller at sea who has no other landmarks. That is what astronomy meant to the navigators of the Mediterranean in the Old World. But so far as we can judge, the peoples of the New World did not use astronomy as a scientific guide for land and ocean voyages. And without astronomy it is really not possible to find your way over great distances, or even to have a theory about the shape of the earth and the land and sea on it. Columbus was working with an old and, to our minds, crude astronomy when he set sail for the other side of the world: for instance, he thought that the earth was much smaller than it really is. Yet Columbus found the New World. It cannot be an accident that the New World never thought that the earth is round, and never went out to look for the Old World. It was the Old World which set sail round the earth to discover the New.

  Astronomy is not the apex of science or of invention. But it is a test of the cast of temperament and mind that underlies a culture. The seafarers of the Mediterranean since Greek times had a peculiar inquisitiveness that combined adventure with logic – the empirical with the rational – into a single mode of inquiry. The New World did not.

  Then did the New World invent nothing? Of course not. Even so primitive a culture as Easter Island made one tremendous invention, the carving of huge and uniform statues. There is nothing like them in the world, and people ask, as usual, all kinds of marginal and faintly irrelevant questions about them. Why were they made like this? How were they transported? How did they get to the places that they are at? But that is not the significant problem. Stonehenge, of a much earlier stone civilisation, was much more difficult to put up; so was Avebury, and many other monuments. No, primitive cultures do inch their way through these enormous communal enterprises.

  The critical question about these statues is, Why were they all made alike? You see them sitting there, like Diogenes in their barrels, looking at the sky with empty eye-sockets, and watching the sun and the stars go overhea
d without ever trying to understand them. When the Dutch discovered this island on Easter Sunday in 1722, they said that it had the makings of an earthly paradise. But it did not. An earthly paradise is not made by this empty repetition, like a caged animal going round and round, and making always the same thing. These frozen faces, these frozen frames in a film that is running down, mark a civilisation which failed to take the first step on the ascent of rational knowledge. That is the failure of the New World cultures, dying in their own symbolic Ice Age.

  Easter Island is over a thousand miles from the nearest inhabited island, which is Pitcairn Island, to the west. It is over fifteen hundred miles from the next, the Juan Fernandez Islands to the east, where Alexander Selkirk, the original for Robinson Crusoe, was stranded in 1704. Distances like that cannot be navigated unless you have a model of the heavens and of star positions by which you can tell your way. People often ask about Easter Island, How did men come here? They came here by accident: that is not in question. The question is, Why could they not get off? And they could not get off because they did not have a sense of the movement of the stars by which to find their way.

  Why not? One obvious reason is that there is no Pole Star in the southern sky. We know that is important, because it plays a part in the migration of birds, which find their way by the Pole Star. That is perhaps why most bird migration is in the northern hemisphere and not in the southern.

  The absence of a Pole Star could be meaningful down here in the southern hemisphere, but it cannot be meaningful for the whole of the New World. Because there is Central America, there is Mexico, there are all sorts of places which also did not have an astronomy and yet which lie north of the equator.