String, Straight-edge and Shadow

by Julia E. Diggins

  What had started as astrology became a science when people collected the first long series of astronomical observations. It lasted more than 300 years, the longest uninterrupted table ever kept to this day!

  Through these records a constant pattern could be seen in the periodic movements of the sun, moon and planets, a pattern that enabled people to foretell the time of a future eclipse and the future position of these celestial bodies.

  The wheel and the arch

  Apart from their astronomy, the Mesopotamians left us other great monuments to their division of the circle: the arch and the wheel.

  They were probably the earliest people to use the wheel. And when they changed from their older solid wheels to wheels with spokes – the divided circle – they created light chariots for their wars.

  They were also probably the first people to invent the arch. Nowhere on the broad plains could they find mountains to supply stone, or forests to supply wood. For building material they had to use sun-baked brick. They had to discover a way to use this brick as a support for a door or gateway.

  Again their knowledge of the circle came to the rescue. They found that if they placed bricks in the form of a half-circle with a wedge-shaped brick in the centre, the wedge-shaped brick (keystone) worked as a force against the supporting bricks. Such an arch had the strength to hold the weight of a wall. By interlocking arches, they created a dome. The arch and the dome are still characteristic of the architecture of the Middle East.

  These circular forms – the arch and the dome – followed the trade routes from Babylon all around the Mediterranean. In time they became the basis of the domes, bridges and aqueducts of the Roman Empire centuries later. Of course we use them in our constructions today.

  Navigation today

  The mementos of the division of the circle, thousands of years ago, still guide our daily life. For many centuries, ships on the high seas – ancient sailing ships, modern steamers – continued to use these dependable signs, the stars. Old, but not old-fashioned, the stars today are the same ever-present guideposts that directed nomads across the trackless deserts.

  Today, pilots in the air or on the sea are giving up star navigation for the latest electronic devices. Yet the very instruments by which they plot their courses still reflect the division of the circle by the Mesopotamian stargazers: the compass, divided into 360° – north at 0°, east at 90°, south at 180° and west at 270° – enables a navigator to mark off in degrees an accurate flight plan.

  Likewise, the instruments with which we measure distances on the earth recall the ancient achievement. Global lines of longitude and latitude, spaced through 360°, enable us to place a position on the earth by means of their graph-pattern. And our modern surveyors use a theodolite which can measure angles both horizontally (for directions) and vertically (for heights).

  Even our clock is based on the division of the circle. Look at its circular face divided into 12 hours, each hour into 60 minutes and each minute into 60 seconds. In these ways and many others, we still use the great accomplishments of Babylonian astronomers after thousands of years.

  They help take us back in our imagination to those unknown stargazers of long ago – who first divided the circle and made possible the charting of land and sky, the map-making and astronomy, that were the crowning achievements of ancient Middle Eastern geometry.

  The Ionian Greeks

  Geometry and Thought

  7. The Sixth Century BC

  At the start of the sixth century BC, the Mediterranean world was changing. The centre of civilisation was about to shift to Greece, or Hellas as it was then called. Geometry, too, was about to undergo a radical change – from a purely useful art to a new kind of abstract thinking.

  The ancient peoples of the river valleys, the Nile and the Tigris and Euphrates, had done marvellous things with practical geometry. They had used it to mark off their fields and irrigation canals, to construct beautiful buildings and gigantic pyramids, to measure the travels of the stars, to find directions on land and sea.

  But by now, both Egypt and Mesopotamia had passed their zenith. Their creative time was over, though they were enjoying a final blaze of splendour.

  The Egyptians had developed a sumptuous civilisation in their fertile valley, hot and low-lying, with the slow river flowing through it and the limitless desert beyond. Now, under Psamtik II, the land was more prosperous than it had been for almost ten centuries. But Psamtik was a collector of art and antiquities, and his Egypt was hardly more than a museum of past glories.

  Mesopotamian culture, too, had flourished between the ‘two rivers’ and across the vast, warm plains. During the reign of Ashurbanipal, most famous of the Assyrian kings, Nineveh had become the largest and most magnificent city in the world. Yet its huge library had also been a collection of learning from the past. And now Nineveh was utterly destroyed and Nebuchadnezzar was beautifying his great capital city of Babylon. Here he built the famous Hanging Gardens, like a verdant mountain on the flat Babylonian plain, to please his wife, who was homesick for the hills of Media. He also encouraged the compilation of star records in the temples. But Babylon’s days of power were numbered.

  While these ancient centres were basking in revived glory, something fresh and new was stirring further west.

  The Hellenes, a people who streamed down from the chilly forests and mountains to the north, had established their cities on the rocky Greek peninsula. From its cliffs they faced the open sea, and looked out at the snow-capped isles of the dark blue Aegean. It was a craggy land of stubborn soil, where strong people had to work hard to get food. Soon they were overflowing from Greece proper, taking to their ships and colonising the islands and the shores of the Mediterranean and the Black Sea.

  So the sixth century BC was an age of expansion, trade, travel, exploration and the mixing of the old cultures with the newly awakened one.

  During this century of changes the spotlight of ancient history was starting to swing west. For the next 300 years, the mainspring of civilisation would be Greece in her creative age. The Greeks were to introduce a new element into culture: reason. Their love of reason would transform art and architecture, philosophy, literature, science and, in the first place, mathematics.

  We have seen what the old Middle Eastern civilisations accomplished in practical geometry with the help of the circle and the right angle, and how they read the sun’s messages in the shadows. Now we shall see how theoretical geometry was established by Greeks on these same elements. Its foundations were set firmly, by means of the observation of shadows, on the circle, the right angle, the right-angled triangle, and the relationships within and between these forms. The elements were the same, but the approach was entirely different.

  This new approach began in the Greek colony of Ionia in Asia Minor, where the brilliant city of Miletus was a crossroad between the East and the West. These Ionian Greeks were keen and imaginative. They asked questions about everything and began to collect old answers and frame new ones. Their lively Ionian temperament, their crossroads location, the times they lived in and the new Greek spirit all combined to produce a stimulating intellectual environment. Here, in the sixth century BC, there flourished a remarkable group of individuals. Among them were great poets, and Aesop of fable fame. But most fascinating were those whom we would call the world’s first scientists.

  The Greeks had a different name for them. They were the Ionian ‘philosophers’. Philosophy meant ‘the love of knowledge’, and the term fitted them well.

  These early philosophers made discoveries in astronomy, physics, mathematics and geography. Perhaps the earliest was Thales, who studied magnets and measurements. Later came Anaximander, who wrote the first treatise on natural history and made the first map of the world. On the isle of Samos lived Pythagoras, who was credited with inventing multiplication tables (though he probably didn’t). But it wasn’t only discoveries they cared about. They asked searching questions about the universe. What wa
s the prime substance in everything: was it water or air, was it mind, was it the boundless unknown?

  Most important of all, they started a new kind of thinking – rational thought: thought based on careful reasoning. The Egyptians and Babylonians had discovered new ways of doing things. The Greeks found new ways of thinking about them. They observed nature, put their observations in order and tried to find abstract rules.

  The first to do this was Thales, whom we have just named as the first Ionian philosopher. He is important in our story for another reason. According to tradition, Thales of Miletus was the founder of geometry.

  Miletus: a rich crossroads

  The great harbour of Miletus faced west and welcomed the sailing fleets of Greek and Phoenician merchants. Her rich market was a trading place for the overland caravans of the East, of Persia and Babylon and Egypt. Milesian sailors and merchants travelled to all parts of the known world and came home with strange tales and strange knowledge. And in their teeming city, different peoples and traditions mixed every day. It was natural that the inhabitants of Miletus and the nearby islands should trade not only merchandise but ideas.

  8. Thales at Home and Abroad

  Thales, the ‘father of geometry’, was a sort of Greek Benjamin Franklin. The known facts of his life are few. He was a merchant. He travelled extensively to the older centres of civilisation and learned much on his travels. He said ‘the magnet has a soul because it moves the iron’, showing he had studied lodestones, natural magnets. And he is believed to have been the first to experiment with electricity, the static kind in a piece of rubbed amber.

  But Thales was also an interesting character and inspired some of the choicest of Aesop’s Fables. Many stories of his accomplishments were told by later writers, some serious and some quite fanciful. True or not, these tales teach us much about Thales’ way of thought. He was forever asking ‘why?’ working out his answer from what he saw, and standing ready to prove it. Even the anecdotes about his business ventures show how his mind worked. He was a great observer. He would study the pattern of repeated occurrences and then prophesy the natural path.

  Another famous story, told by Aesop, illustrates the same mental traits. It shows that Thales was not above trying to follow the reasoning of a little donkey.

  Thales had inherited a salt mine. The salt was transported from the mines by donkeys. They were weighted down with bags of salt at the deposits, and then had to carry them to the market. This donkey train had a long journey in the hot sun.

  As they crossed a stream en route, one little donkey was so warm and fatigued that he lay down in the cool water and rolled over. Afterwards he not only felt refreshed for the rest of the trip, but realised that a great weight had been removed from his back. On every trip thereafter, he repeated this same stunt.

  His master Thales was surprised at the beast’s fresh appearance, disappointed in his scant cargo, and very puzzled as to how it had been dissolved. For a while the donkey outsmarted Thales, but in the long run Thales outsmarted him by using some simple deductive reasoning. Thales asked himself, ‘What sort of thing refreshes the donkey and dissolves the salt? … A cool stream … Is there a stream along the route? … Yes! … What will absorb the water and fatigue the donkey? … Sponges!’ So on the next trip Thales filled the saddle bags with sponges instead of salt, and the little donkey’s happy habit was broken.

  So as a business man in Ionia, Thales was already using a new type of thinking. But two other interests led him to establish the science of geometry: his travels in the Middle East and his study of shadows.

  Thales and the oil presses

  One afternoon as Thales and his friends were discussing money – coins had just recently been invented – Thales made the remark, ‘Anybody can make money if he puts his mind to it.’

  His friends immediately said, ‘Prove it.’

  Thales was in an awkward spot and he had to think and think. He said to himself, ‘What item is useful to everybody?’ His answer was, ‘Oil.’

  In 600 BC oil didn’t mean petroleum, but olive oil. Olive oil was used for soap. It provided fuel for lamps. It was used for cooking. And it was prized as a skin-softener.

  Thales decided to study oil from the tree to the oil press. During this investigation, the first stumbling block he found was the fact that for several seasons the trees had not been producing olives. Why? Thales thought next about weather conditions. In fact he had to do research on the weather of past seasons – the kind favourable to the ripening of the olive and the kind unfavourable.

  After that, he also had to try to discover a pattern in the weather conditions, so that he could see what to expect in the future. From his diligent work in laying out the pattern from the past, he calculated that favourable weather conditions were due the next season.

  Now he made a tour among the discouraged olive growers and bought up all their olive presses. Of course they were delighted to sell them because the presses had been useless for several seasons. Besides, in the past, a grower without a press could always borrow one from a neighbour if the need arose.

  But when the big crop came the following year, there were no presses to borrow and none to buy either – Thales had bought them all. Thales cornered the oil market and made a fortune. Some say he gave the presses back afterwards because he didn’t have time to go into the oil business.

  Thales’ travels

  As Thales was walking in his garden one night, enraptured by the sparkling splendour of the stars, suddenly the silent stillness of the night was broken by the sound of a great splash and a gurgle. Thales had stepped majestically into a well!

  Fishing him out, his servant remarked with a chuckle, ‘Master, while you are trying to pry into the mysteries of the sky, you overlook the common objects under your feet.’

  Nobody likes to be damp and laughted at. In the days that followed, Thales decided to look at the hot dry earth beneath his feet. He would study the shadow patterns that lay there, speaking so eloquently of the sun’s messages upon the earth! And he would see more of the earth itself, by travelling to the ancient countries of Mesopotamia and Egypt (from what we know of Thales, we can guess that he probably decided to engage in some shipping and foreign trade on the side).

  The first stop on his journey was Babylon, a glamorous city with a long history and a large library of cuneiform tablets. There he was fascinated by the impressive records of the stargazers. He stayed for quite a while – poring over the charts, studying the methods of sky measurement, learning the use of the circle and its divisions for measuring angles and directions.

  Then he crossed over into Egypt. There he mastered the construction of engineering works. He studied the irrigation canals, the well-laid-out fields, the wall decorations showing the history of Egypt in pictures, the designs in Egyptian decorations.

  He was absorbing all the old practical geometry of Egypt and Mesopotamia. This was typically Greek, busily learning from older civilisations.

  And he brought to his travels another trait that was typical of the Greeks and the future civilisation they were building. He had a new kind of inquiring mind.

  Everywhere he went, Thales studied the shadows traced on these flat ancient lands by ziggurats, obelisks, buildings and people. He saw these shadows as people had never seen them before. We might say he had an x-ray eye, because he developed the habit of ‘seeing through’ the obvious to find new meanings – of looking into and beyond visible externals to discover an abstract form and relation.

  Here was a remarkable traveller. If we can believe the tales of Thales at home and abroad, he took with him his fresh Ionian insight, even as he absorbed the old practical lore of the Babylonians and Egyptians. Out of this combination was to come the new theoretical geometry.

  9. How High is the Pyramid?

  In the Land of the Nile – so the legend goes – Thales amazed and frightened his guides by telling them, as if by magic, the exact height of the Great Pyramid.

  T
he story is worth telling in some detail. It shows us Thales’ new geometry in action and enables us to compare it with the old Egyptian kind.

  Naturally, Thales’ visit to Egypt was not complete without a sightseeing trip to the desert at Giza, to see the three pyramids and the Sphinx half-buried in the sand nearby. In 600 BC the pyramids were about 2000 years old.

  Thales engaged guides and took a Greek friend along. When they reached the mighty monuments, the guides seemed proud to boast that the Egyptian pyramids had been standing when the ancestors of the Greeks were ‘long-haired barbarians’.

  Thales stood for a time admiring the most gigantic of the tombs: the Great Pyramid of Cheops, which covers more than twelve acres! He looked up the great slope, rising to a peak against the cloudless Egyptian sky, and noticed how the brilliant sunlight hit directly against one face and drew a pointed shadow over the desert sands. Then he asked his celebrated question.

  ‘How high is this pyramid?’

  The guides were astonished and got into a lengthy discussion. No sightseer had ever asked them that before. Visitors were always content to know the dimensions of the pyramid’s square base – 252 paces along each side. Sometimes the Greek tourists didn’t believe that, and had to pace it off for themselves. But this one wanted to know something more: the height. Nobody knew the height of the Great Pyramid. Perhaps, long ago, the builders had known, but by the present dynasty, everyone had forgotten. And, of course, you couldn’t measure it. A rope dragged all the way up to the top (and who was going to risk that?) would just give the length of the sloping side. They couldn’t think of any way to find out the height, short of boring a hole from the top of the pyramid down to its base, but that was impossible.