The Ascent of Man
It seems very strange that an architecture that moved large building stones on rollers could miss the use of the wheel; we forget that what is radical about the wheel is the fixed axle. It seems strange to make suspension bridges and miss the arch. And it seems strangest of all to have a civilisation that kept careful records of numerical information, yet did not put them in writing – the Inca was as illiterate as his poorest citizen, or as the Spanish gangster who overthrew him.
The messages in the form of numerical data came to the Inca on pieces of string called quipus. The quipu only records numbers (as knots arranged like our decimal system) and I would dearly like to say, as a mathematician, that numbers are as informative and human a symbolism as words; but they are not. The numbers that described the life of a man in Peru were collected on a kind of punched card in reverse, a braille computer card laid out as a knotted piece of string. When he married, the piece of string was moved to another place in the kinship bundle. Everything that was stored in the Inca’s armies, granaries and warehouses was noted on these quipus. The fact is that Peru was already the dreaded metropolis of the future, the memory store in which an empire lists the acts of every citizen, sustains him, assigns him his labours, and puts it all down impersonally as numbers.
It was a remarkably tight social structure. Everyone had a place; everyone was provided for; and everyone – peasant, craftsman or soldier – worked for one man, the supreme Inca. He was the civil head of state and he was also the religious incarnation of godhead. The artisans who lovingly carved a stone to represent the symbol of the link between the sun and its god and king, the Inca, worked for the Inca.
So, necessarily, it was an extraordinarily brittle empire. In less than a hundred years, from 1438 onwards, the Incas had conquered three thousand miles of coastline, almost everything between the Andes and the Pacific. And yet, in 1532 an almost illiterate Spanish adventurer, Francisco Pizarro, rode into Peru with no more than sixty-two terrible horses and a hundred and six foot soldiers; and overnight he conquered the great empire. How? By cutting the top off the pyramid – by capturing the Inca. And from that moment, the empire sagged, and the cities, the beautiful cities, lay bare for the gold plunderer and the vultures.
But, of course, a city is more than a central authority. What is a city? A city is people. A city is alive. It is a community which lives on a base of agriculture, so much richer than in a village, that it can afford to sustain every kind of craftsman and make him a specialist for a lifetime.
The specialists are gone, their work has been destroyed. The men who made Machu Picchu – the goldsmith, the coppersmith, the weaver, the potter – their work has been robbed. The woven fabric has decayed, the bronze has perished, the gold has been stolen. All that remains is the work of the masons, the beautiful craftsmanship of the men who made the city – for the men who make a city are not the Incas but the craftsmen. But naturally, if you work for an Inca (if you work for any one man) his tastes rule you and you make no invention. These men still worked to the end of the empire with the beam; they never invented the arch. Here is a measure of the time lag between the New World and the Old, because this is exactly the point which the Greeks reached two thousand years earlier, and at which they also stopped.
Paestum in Southern Italy was a Greek colony whose temples are older than the Parthenon: they date from about 500 BC. Its river has silted up and it is now separated from the sea by dull salt-flats. But its glory is still spectacular. Although it was ransacked by Saracen pirates in the ninth century, and by Crusaders in the eleventh, Paestum in ruins is one of the marvels of Greek architecture.
Paestum is contemporary with the beginning of Greek mathematics; Pythagoras taught in exile in another Greek colony at Crotone not far from here. Like the mathematics of Peru two thousand years later, the Greek temples were bounded by the straight edge and the set square. The Greeks did not invent the arch either, and therefore their temples are crowded avenues of pillars. They seem open when we see them as ruins, but in fact they are monuments without spaces. That is because they had to be spanned by single beams, and the span that can be sustained by a flat beam is limited by the strength of the beam.
If we picture a beam lying across two columns, then a computer analysis will show the stresses in the beam increase as we move the columns farther apart. The longer the beam, the greater the compression that its weight produces in the top, and the greater the tension it produces in the bottom. And stone is weak in tension; the columns will not fail, because they are compressed, but the beam will fail when the tension becomes too great. It will fail at the bottom unless the columns are kept close together.
The Greeks could be ingenious in making the structure light, for example by using two tiers of columns. But such devices were only makeshifts; in any fundamental sense, the physical limitations of stone could not be overcome without a new invention. Since the Greeks were fascinated by geometry, it is puzzling that they did not conceive the arch. But the fact is that the arch is an engineering invention, and very properly is the discovery of a more practical and plebeian culture than either Greece or Peru.
The circle remained the basis of the arch when it went into mass-production in Arab countries.
The Great Mosque at Cordoba.
The aqueduct at Segovia in Spain was built by the Romans about AD 100, in the reign of the emperor Trajan. It carries the waters of the Rio Frio that flows from the high Sierra ten miles away. The aqueduct spans the valley for almost half a mile in more than a hundred double-tiered round arches made of rough-hewn granite blocks, laid without lime or cement. Its colossal proportions so awed the Spanish and Moorish citizens in later and more superstitious ages that they named it El Puente del Diablo, the devil’s bridge.
The structure seems to us also prodigious and splendid out of proportion to its function of carrying water. But that is because we get water by turning a tap, and we lightly forget the universal problems of city civilisations. Every advanced culture that concentrates its skilled men in cities depends on the kind of invention and organisation that the Roman aqueduct at Segovia expresses.
The Romans did not invent the arch in the first place in stone, but as a moulded construction made of a kind of concrete. Structurally the arch is simply a method of spanning space which does not load the centre more than the rest; the stress flows outward fairly equally throughout. But for this reason the arch can be made of parts: of separate blocks of stone which the load compresses. In this sense, the arch is the triumph of the intellectual method which takes nature apart and puts the pieces together in new and more powerful combinations.
The Romans always made the arch as a semicircle; they had a mathematical form that worked well, and they were not inclined to experiment. The circle remained the basis of the arch still when it went into mass-production in Arab countries. This is plain in the cloistered, religious architecture that the Moors used; for instance, in the great mosque at Cordoba, also in Spain, built in AD 785 after the Arab conquest. It is a more spacious structure than the Greek temple at Paestum, and yet it has visibly run into similar difficulties; that is, once again it is filled with masonry, which cannot be got rid of without a new invention.
Theoretical discoveries that have radical consequences can usually be seen at once to be striking and original. But practical discoveries, even when they turn out to be far-reaching, often have a look that is more modest and less memorable. A structural innovation to break the limitation of the Roman arch did come, probably from outside Europe, and arrived almost by stealth at first. The invention is a new form of the arch based not on the circle, but on the oval. This does not seem a great change, and yet its effect on the articulation of buildings is spectacular. Of course, a pointed arch is higher, and therefore opens more space and light. But, much more radically, the thrust of the Gothic arch makes it possible to hold the space in a new way, as at Rheims. The load is taken off the walls, which can therefore be pierced with glass, and the total effect is to ha
ng the building like a cage from the arched roof. The inside of the building is open, because the skeleton is outside.
John Ruskin describes the effect of the Gothic arch admirably.
Egyptian and Greek buildings stand, for the most part, by their own weight and mass, one stone passively incumbent on another; but in the Gothic vaults and traceries there is a stiffness analogous to that of the bones of a limb, or fibres of a tree; an elastic tension and communication of force from part to part, and also a studious expression of this throughout every visible line of the building.
Of all the monuments to human effrontery, there is none to match these towers of tracery and glass that burst into the light of Northern Europe before the year 1200. The construction of these huge, defiant monsters is a stunning achievement of human foresight – or rather, I ought to say, since they were built before any mathematician knew how to compute the forces in them, of human insight. Of course it did not happen without mistakes and some sizeable failures. But what must strike the mathematician most about the Gothic cathedrals is how sound the insight in them was, how smoothly and rationally it progressed from the experience of one structure to the next.
The cathedrals were built by the common consent of townspeople, and for them by common masons. They bear almost no relation to the everyday, useful architecture of the time, and yet in them improvisation becomes invention at every moment. As a matter of mechanics, the design had turned the semicircular Roman arch into the high, pointed Gothic arch in such a way that the stress flows through the arch to the outside of the building. And then in the twelfth century also came the sudden revolutionary turning of that into the half arch: the flying buttress. The stress runs in the buttress as it runs in my arm when I raise my hand and push against the building as if to support it – there is no masonry where there is no stress. No basic principle of architecture was added to that realism until the invention of steel and reinforced concrete buildings.
One has the sense that the men who conceived these high buildings were intoxicated by their new-found command of the force in the stone. How else could they have proposed to build Vaults of 125 feet and 150 feet at a time when they could not calculate any of the stresses? Well, the vault of 150 feet – at Beauvais, less than a hundred miles from Rheims – collapsed. Sooner or later the builders were bound to run into some disaster: there is a physical limit to size, even in cathedrals. And when the roof of Beauvais collapsed in 1284, some years after it was finished, it sobered the high Gothic adventure: no structure as tall as this was attempted again. (Yet the empirical design may have been sound; probably the ground at Beauvais was simply not solid enough, and shifted under the building.) But the vault of 125 feet at Rheims held. And from 1250 onwards Rheims became a centre for the arts of Europe.
The arch, the buttress, the dome (which is a sort of arch in rotation) are not the last steps in bending the grain in nature to our own use. But what lies beyond must have a finer grain: we now have to look for the limits in the material itself. It is as if architecture shifts its focus at the same time as physics does, to the microscopic level of matter. In effect, the modern problem is no longer to design a structure from the materials, but to design the materials for a structure.
The masons carried in their heads a stock, not so much of patterns as of ideas, that grew by experience as they went from one site to the next. They also carried with them a kit of light tools. They marked out with compasses the oval shapes for the vaults and the circles for the rose windows. They defined their intersections with callipers, to line them up and fit them into repeatable patterns. Vertical and horizontal were related by the T-square, as they had been in Greek mathematics, using the right angle. That is, the vertical was fixed with the plumb-line, and the horizontal was fixed, not with a spirit-level, but with a plumb-line joined to a right angle.
The wandering builders were an intellectual aristocracy (like the watchmakers five hundred years later) and could move all over Europe, sure of a job and a welcome; they called themselves freemasons as early as the fourteenth century. The skill that they carried in their hands and their heads seemed to others to be as much a mystery as a tradition, a secret fund of knowledge that stood outside the dreary formalism of pulpit learning that the universities taught. When the work of the freemasons petered out, by the seventeenth century, they began to admit honorary members, who liked to believe that their craft went back to the pyramids. That was not really a flattering legend, because the pyramids were built with a much more primitive geometry than the cathedrals.
The masons carried with them a kit of light tools. The vertical was fixed with the plumb-line; and the horizontal was fixed, not with a spirit level, but with a plumb-line joined to a right angle.
Masons at work, 13th century.
Yet there is something in the geometrical vision which is universal. Let me explain my preoccupation with beautiful architectural sites – such as the cathedral at Rheims. What does architecture have to do with science? Particularly, what does it have to do with science the way we used to understand it at the beginning of this century, when science was all numbers – the coefficient of expansion of this metal, the frequency of that oscillator?
The fact of the matter is that our conception of science now, towards the end of the twentieth century, has changed radically. Now we see science as a description and explanation of the underlying structures of nature; and words like structure, pattern, plan, arrangement, architecture constantly occur in every description that we try to make. I have by chance lived with this all my life, and it gives me a special pleasure: the kind of mathematics I have done since childhood is geometrical. However, it is no longer a matter of personal or professional taste, for now that is the everyday language of scientific explanation. We talk about the way crystals are put together, the way atoms are made of their parts – above all we talk about the way that living molecules are made of their parts. The spiral structure of DNA has become the most vivid image of science in the last years. And that imagery lives in these arches.
What did the people do who made this building and others like it? They took a dead heap of stones, which is not a cathedral, and they turned it into a cathedral by exploiting the natural forces of gravity, the way the stone is laid naturally in its bedding planes, the brilliant invention of the flying buttress and arch and so on. And they created a structure that grew out of the analysis of nature into this superb synthesis. The kind of man who is interested in the architecture of nature today is the kind of man who made this architecture nearly eight hundred years ago. There is one gift above all others that makes man unique among the animals, and it is the gift displayed everywhere here: his immense pleasure in exercising and pushing forward his own skill.
A popular cliché in philosophy says that science is pure analysis or reductionism, like taking the rainbow to pieces; and art is pure synthesis, putting the rainbow together. This is not so. All imagination begins by analysing nature. Michelangelo said that vividly, by implication, in his sculpture (it is particularly clear in the sculptures that he did not finish), and he also said it explicitly in his sonnets on the act of creation.
When that which is divine in us doth try
To shape a face, both brain and hand unite
To give, from a mere model frail and slight,
Life to the stone by Art’s free energy.
‘Brain and hand unite’: the material asserts itself through the hand, and thereby prefigures the shape of the work for the brain. The sculptor, as much as the mason, feels for the form within nature, and for him it is already laid down there. That principle is constant.
The best of artists hath no thought to show
Which the rough stone in its superfluous shell
Doth not include: to break the marble spell
Is all the hand that serves the brain can do.
By the time Michelangelo carved the head of Brutus, other men quarried the marble for him. But Michelangelo had begun as one of the quarrymen in
Carrara, and he still felt that the hammer in their hands and in his was groping in the stone for a shape that was already there.
The quarrymen work in Carrara now for the modern sculptors who come here – Marino Marini, Jacques Lipchitz and Henry Moore. Their descriptions of their work are not as poetic as Michelangelo’s, but they carry the same feeling. The reflections of Henry Moore are particularly apposite as they run back to the first genius of Carrara.
To begin with, as a young sculptor, I could not afford expensive stone, and I got my stone by going round the stone-yards and finding what they would call a ‘random block’. Then I had to think in the same way that Michelangelo might have done, so that one had to wait until an idea came that fitted the shape of the stone and that was seen, the idea, in that block.
Of course, it cannot be literally true that what the sculptor imagines and carves out is already there, hidden in the block. And yet the metaphor tells the truth about the relation of discovery that exists between man and nature; and it is characteristic that philosophers of science (Leibniz in particular) have turned to the same metaphor of the mind prompted by a vein in the marble. In one sense, everything that we discover is already there: a sculptured figure and the law of nature are both concealed in the raw material. And in another sense, what a man discovers is discovered by him; it would not take exactly the same form in the hands of someone else – neither the sculptured figure nor the law of nature would come out in identical copies when produced by two different minds in two different ages. Discovery is a double relation of analysis and synthesis together. As an analysis, it probes for what is there; but then, as a synthesis, it puts the parts together in a form by which the creative mind transcends the bare limits, the bare skeleton, that nature provides.