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
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 characterisation 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 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.
Page from an Arabic version of AD 1258, and a Chinese block print of the theorem.
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 gentleman’s library, Euclid’s Elements lay open, and ‘twas the 47 Element libri I’. Proposition 47 in Book 1 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 S
outhern France to the borders of China and India: an empire of spectacular strength and grace, while Europe lapsed in the Dark Ages.
In this proselytising 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 depersonalised and formalised 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 recognise 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 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 civilisation 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 civilisation, 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 civilisation 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 characterised.
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 i
rrelevant. 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 Ascent of Man Page 11