String, Straightedge, and Shadow the Story of Geometry

by Julia E. Diggins

  While the argument went on, Thales and his friend had been walking around quietly, staying close to the pyramid’s shadow, where it was cool. ‘Never mind my question!’ called Thales suddenly, as the guides approached. ‘I know the answer. The Great Pyramid at Giza rises to a height of 160 paces!’

  Terrified, the guides flung themselves on their faces before Thales, convinced that he was a magician.

  Proportion

  Thales did not, of course, get the answer by magic. He simply measured two shadows on the sand, and then used an abstract rule from his new kind of geometry.

  To show you just what his method was – and the way he probably worked it out, and how different it was from the old geometry of the pyramid builders – we shall go back and imagine an earlier scene.

  When Thales reached Egypt to spend the winter as tourist and student and merchant traveller, he must have had many things to do. Perhaps there was business to transact on the crowded streets and wharves. Of course he wanted to see the famous monuments, the colossal statues and pyramid tombs. But first of all he went to the Temple of Thoth, where the priests were said to have great learning. They proved hospitable, and he spent many days in the cool temple interior, studying the old Egyptian methods.

  Now his studies were over and he was going to start sightseeing.

  It was a warm sunny afternoon and he was sitting outside the temple, waiting to bid farewell to the high priest Thothmes, a most important man. The attendant would take quite a while to fetch him.

  As he waited, Thales studied the scene and thought about the accomplishments of Egyptian geometry. In the great empty space before the temple stood a high gilded obelisk, making a fine ‘shadow pole’, or sundial, in the afternoon sunlight. A few white-clad priests and worshippers were standing about, their shadows very distinct, too. Off to one side was the vast temple, perfectly laid out so that its sides would face the four points of the compass. And through its colonnade he could just glimpse a great wall painting, a master piece of Egyptian proportion.

  Proportion. It ran, he knew, like a golden thread all through the work of the ancients. Proportion was used by the Chaldean astronomers in the angles of their sextants and the corresponding arcs of the movements of distant stars. Proportion was used by Egyptian architects, in designing their buildings and erecting the actual structures. But it was best shown in the vast wall paintings with which the Egyptians decorated their temples and tombs. Those vivid scenes were all painted by artists using an innate feeling for proportion. But as Thales sat there, watching the shadows lengthen outside the temple, he saw something entirely different: abstract proportion.

  Copying tomb paintings

  The Egyptian artist had a very simple means of transferring his small sketch to the huge wall. First he covered his sketch all over with small squares, something like modern graph paper. Then he made squares all over the wall, only large ones this time. Finally he studied where the lines of his sketch crossed the small squares, and then copied these lines in the same relative position over the large squares.

  That was intuitive proportion at its most practical, the Egyptians at their best.

  Where others saw only the men and the structures and their shadows in the hot sunlight, Thales saw abstract right-angled triangles as well! All these triangles were made the same way: an upright object, a pointed obelisk or white-clad Egyptian; a slanting sun ray that hit the top of the object; and the flat shadow that it cast on the ground.

  But Thales saw far more than that. He saw the motion of the lengthening shadows. For as Thales watched, he noticed that all the shadows changed together, in length and direction. At first, they were all half as long as the objects that cast them. Later, they were all the same length as the objects. Later still, the shadows were all twice as long as the height of the objects.

  Probably many people had observed something like that, over the centuries. But the Ionian traveller tried to find a constant pattern. He had to prove it was always so, and to find out why.

  And he did.

  Thales noticed that all the abstract right-angled triangles changed together, too – not the whole triangles. The right angle and the height of the object that made its upright or vertical side – these did not change. But the rest of the triangle changed as the sun changed its position in the sky. The sun was so far away that its rays hit the tops of all the objects and the tips of all the shadows, at the same slant. So, as the sun was higher or lower, the other two angles had to change in all the triangles. And as the angles changed, the other two sides had to change too – the length of the shadow (the flat or horizontal base of the triangle), and the length of the sun’s ray from top to tip (the slanting third side).

  Sun’s rays casting shadows in mid-afternoon

  Sun’s rays casting shadows in late afternoon

  So at each moment, all the sun-made right-angled triangles were exactly the same shape – not the same size, but the same shape: the right angles and heights of the objects remained unchanged, but the other two sides and other two angles changed as the sun seemed to move across the sky.

  Now Thales knew his eyes hadn’t deceived him. The shadow lengths would always change together in the same way, while the heights of the objects must of course stay unchanged. He had his secret for measuring the height of the pyramid.

  Working out heights

  Before you hear exactly how Thales accomplished that feat, you may want to try out his secret for yourself.

  You can watch these same shadow changes on your own playground or sports field by comparing the right-angled triangles formed by a flagpole, a goal post and your own height.

  Start, say, in mid-afternoon when your shadow is as long as you are tall. At that same time, the flagpole’s shadow will be as long as the flagpole is high, and the shadow of the goal post will also be equal to its height. You can pace off the shadows of the flagpole and the goal post to find their heights without bothering to climb up with a measuring tape.

  Of course, you could start earlier, at a time when your shadow is about half of your height. Then you could pace off the other two shadows, double their length, and so find the height of the other objects.

  Or you might wait till later in the day, when your shadow is twice as long as your height. The other shadows would, of course, also be twice the height of their objects. So you could pace off the other two shadows, and take half of the distance – and that would give you the height of the flagpole or the goal post.

  The rule for measuring height from shadows

  The secret to measuring height from shadows is simply a proportion. As Thales did, you will find that at any moment there is a constant relation between one object’s height and its shadow, and the next object’s height and its shadow. In this case, you are using the equal ratios between the height of an object and its shadow, and your height and your shadow.

  You can put it like this:

  How Thales measured the height of the pyramid

  Now that you know the secret, you can imagine exactly how Thales measured the height of the pyramid.

  When he asked his famous question, the guides, you remember, began to talk and argue. Meantime Thales, who already knew the distance along each side of the pyramid’s base, 252 paces, was busy pacing off the length of the pyramid’s shadow. It measured 114 paces. Thales knew his own height, 2 paces (6 feet). So, just as he finished his pacing, his friend measured his shadow for him: it was 3 paces. Now Thales had all the necessary dimensions; three items of the proportion would give him the missing fourth one, the height of the pyramid.

  He made his calculation as shown in the illustration.

  Do you see what Thales did? He used an abstract right-angled triangle! He pictured the height of the Great Pyramid as an imaginary post from its top straight down to its base. Such an imaginary post would cast an imaginary shadow, all the way from where it stood at the centre of the pyramid clear out to the tip of the pyramid’s real shadow: so the length of this imagina
ry shadow would be one-half the length of the base plus the actual projecting shadow. Therefore:

  Of course, the guides promptly spread the news of Thales’ magical solution to this seemingly impossible problem. When the priests of Thoth verified that 160 paces was indeed the height of the Great Pyramid, according to the old records, popular astonishment knew no bounds.

  The tale travelled far and wide, so far and wide that it has come down to us after 2500 years. And the story has even more meaning today – for the Great Pyramid was a sturdy monument to ancient practical geometry, but Thales’ shadow-reckoning of its height was an even more significant monument in the development of reasoning.

  10. Abstract Thinking

  With this ‘new’ thinking, Thales was the first to abstract and formulate the rules of geometry. By his method, he was also liberating people’s minds.

  How well his fellow Greeks realised this can be seen in the most famous of all the stories about Thales. It concerns a sensational event that occurred in the midst of a battle.

  In 585 BC, the Medes and Lydians were in the sixth year of a stubborn war. Suddenly broad daylight turned to darkness: the sun gave no light. Terrified out of their wits by the fearful gloom, the warring hosts stopped killing each other and immediately concluded a peace.

  History says that this was an eclipse of the sun. Legend says that Thales had forecast it accurately, through the pattern in the records he had studied in Babylon.

  Scholars have doubted that Thales predicted the eclipse, but recent studies suggest he used an early imperfect method known to Assyrian court astronomers. Either way, the point of the story is the same. The Greeks believed Thales had made the prediction, in other words, that such a prediction was possible.

  Thales’ style of reasoning had taught them to look for an orderly pattern in nature, instead of imagining that an eclipse came about when Apollo, the sun god, hid his face in displeasure. So great, indeed, was Thales’ fame that he was ranked as the first of the fabled Seven Wise Men of Greece. He was supposed to have orginated the motto ‘Know yourself!’ but ‘Think for yourself!’ sounds more like Thales.

  He taught people to do just that, with his new abstract rules. Before him, geometry had consisted of isolated observations, ways grasped from trial and error, for handling and calculating material things. Thales showed the need of a careful demonstration, based on a logical sequence of geometric concepts abstracted from material things.

  To explain how he did this, we shall show Thales actually teaching. There is no evidence that he did, but tradition gives him at least two famous pupils, and he was not a solitary thinker but a busy man of affairs. So he might have expounded his new concepts in some such scenes as those we are about to describe.

  Thales teaching Babylonian angles and Egyptian measures

  When Thales returned from abroad, he probably brought interesting and even expensive curios for his friends, perhaps Mesopotamian seals and amulets, Egyptian jewellery and glass. But the most valuable contribution he brought back to his homeland was the wealth of new ideas in his head, the string in his pocket and the memory of shadows on the sand.

  Picture the amusement when his friends gathered around him to hear about his travels, and Thales took out a piece of string and began to draw circles on the ground. But not everyone laughed. Some friends were fascinated as he made more circles and showed how the Babylonians divided them. Or he would knot the string, to show how the Egyptians made a right-angled triangle and squared off their rectangular fields. Or he would reveal his new thoughts about the sun-made right-angled triangles – the abstract ones made by the bright ray, the object and the shadow.

  Soon – we may imagine – a small group assembled, Thales teaching and expounding, and his friends gradually joining in the discussion.

  They met in an open-air place, with white sand on the ground and poles of different heights to cast different length shadows. Thales would draw circles and triangles and straight lines on the sand, with a string and straight-edge, tell in which country he had studied these forms, and explain his new ideas about them. The others would sit and stand and pace in the sun. They would study the diagrams, watch the shadows, swing the string, ask questions and argue.

  It was like an exciting new game: the game of string, straight-edge and shadows.

  Every game has its rules. So the eager players would go into huddles to discuss the rules and reasons, and to reach agreements. These rules, Thales showed them, had to be reached through careful step-by-step reasoning, based on agreements and definitions.

  It’s hard to imagine how thrilling this new intellectual game must have been to its participants. For the first time in history, people were doing sustained abstract thinking about the principles of line and form. The Babylonians and Egyptians had used these right angles, levels, divided circles and geometric designs. Thales and his friends were now thinking about them – thinking in the abstract, as he had when he measured the pyramid.

  Angles in a circle and different kinds of triangles

  Thales explained the Babylonian division of the circle into 360 degrees and showed how the same measurements applied to angles. Mostly they could see these things for themselves, with just a few definitions and rules:

  A circle is a closed curve on which every point is equally distant from a point within, called the centre. Such a closed curve is called the circumference, and the radius of the circle is the distance from the centre to the circumference.

  A diameter of the circle is a straight line passing through the centre and dividing the circumference into two equal parts, each of 180 degrees.

  By using a piece of string longer than the radius and swinging equal, intersecting arcs from each end of the diameter, another line called a perpendicular can be drawn through the centre to divide the circle into four equal parts – each of 90 degrees.

  Arcs marked off at successive points on the circumference, with string the same length as the radius, divide the circle into six equal parts – each of 60 degrees.

  From these definitions and rules it was obvious that the space surrounding any point may be divided into 360 degrees.

  A straight line, such as a diameter, drawn through a point is called a straight angle and has 180 degrees.

  A perpendicular to this straight angle forms two angles – each of 90 degrees, called a right angle.

  Angles adding up to a straight line total 180 degrees, whether they are two right angles, two unequal angles or more than two angles.

  Speaking of Egyptian rope triangles, Thales reminded his friends that a long rope, knotted at intervals of ten paces, would form a right-angled triangle when stretched on the earth with sides of 3, 4 and 5 units. And so would a shorter rope knotted at one-pace intervals, as would a small cord knotted at intervals the width of a person’s hand. The triangles would differ greatly in size, but they would be similar: each would be a right-angled triangle with sides in the proportion 3:4:5.

  More carefully now, Thales went over the rules for measuring the height of objects from sun-made imaginary right-angled triangles, as he had done at the Great Pyramid.

  When their corresponding angles are equal, right-angled triangles are similar and their corresponding sides are in proportion.

  The study of the Egyptian level revealed that the string hanging from the vertex of the triangle is perpendicular to the base and divides the triangle into two right-angled triangles of exactly the same size. Then it became clear that any triangle which has two equal sides must have equal base angles. Such a triangle is called isosceles, which in Greek means ‘equal legs’.

  In an isosceles triangle, the two base angles are equal.

  From a familiar pattern, found by dividing the circle into six equal parts, an unexpected new relation emerged. All three sides in each of the triangles are equal to the length of the radius, and all three angles are equal to 60 degrees.

  In an equilateral (equal-sided) triangle, each interior angle is 60 degrees and the sum o
f the three interior angles is 180 degrees.

  Other rules of angles

  Since two lines perpendicular to each other form four equal angles of 90 degrees, it was easy to see that, when any two straight lines cross, the two pairs of opposite angles are equal.

  A diagonal line cuts two parallel lines at the same angle, thus forming two pairs of equal angles smaller than 90 degrees, and two pairs of equal angles larger than 90 degrees.

  The alternate interior angles formed by a diagonal cutting two parallel lines are equal.

  ‘By constructing a line parallel to the base of any triangle,’ said Thales, ‘we can see that the sum of the angles in any triangle is equal to 180 degrees and that in any right-angled triangle, since the right angle is 90 degrees, the sum of the other two angles is 90 degrees.’

  Finally, Thales showed his friends how to combine these rules to make an important discovery about semicircles. He pointed out that two straight lines drawn from any point on a semicircle to the ends of the diameter form an enclosed angle of 90 degrees. Can you see how he might have done it?