Ancient people did have a regular pattern to turn to, however. Even in their primitive days, they had known how to weave reeds into mats to cover their earthen floors. The characteristic of the warp and woof in weaving is rectangular. And just as a woven mat covered the earthen floor, so could a rectangular form mark off the boundaries of their fields. Yes, the rectangular shape worked in weaving – but how could that shape be marked on the earth?
Marking a right angle
We know part of the early solution. Pictures on tomb and temple walls show how ropes were used as measuring devices. To make a strong rope, people could use a sturdy vine or twist reeds together. And a stretched rope would mark the boundary between two neighbours.
But there were neighbours on four sides, not just on one or two sides. At first, the corners may have been marked off freehand. But people wanted something more dependable than a freehand boundary. The big question was how to mark an accurate right-angle corner.
Exactly how this was first done – in Mesopotamia or Egypt – is a question that historians may never be able to answer. But we have some tantalising clues.
In the wall paintings of thousands of years ago, there are pictures of Egyptian surveyors dragging a knotted rope. From that custom, the ancient surveyors seem to have been called ‘rope-stretchers’. And apparently those ancient rope-stretchers knew that certain dimensions – so many knots or spaces on each side – would make an accurate right-angled triangle.
How did they find out? Perhaps the discovery was made and lost many times. We can only imagine the way it might have happened.
Perhaps the day was very hot and some Egyptian rope-stretchers got more tired than usual, laying out a right angle by an old guesswork method. The idea was to make a straight line and then another crossing it in the middle. This was hard work in the blazing sun – it took three, sometimes four, of their measuring ropes with evenly spaced knots.
First they had to lay the straight line. That meant hammering two end stakes firmly into the ground with a knotted rope stretched tightly between. Then they found the middle of the line and drove in a centre stake.
Next they took a much longer rope, with plenty of play, and fastened it to the two end stakes. Seizing this rope by the middle, they pulled it as far as possible to one side, opposite the centre stake, and drove in a side stake to hold it there. Finally, they stretched a cross rope from this side stake back over the centre stake.
That ought to be a right angle. But the overseer was fussy, so they would have to do the same thing over again on the other side, and then keep at it the rest of the day, until they lined up the ropes perfectly.
Perhaps at noon when the whole crew stopped to rest in the scanty shade of a clump of palm trees, one man stayed behind, wearily studying the knotted ropes stretched on the ground. Long ago, he might remember, they used to find the middle of a rope by laboriously folding it in two, but nowadays they just used ropes with an odd number of knots and counted spaces to the centre knot. Wasn’t there some simple way to count for a right angle?
He began counting. There it was, the 3–4–5 right-angled triangle: three spaces between the knots on one arm, four spaces on the other arm, and five spaces on the long side opposite the right angle. Naturally, he shouted to the others to come and see. These easy dimensions on a single rope would give them the form they needed so badly, and with hardly any work at all. The rope-knotters could even make a special rope, with large knots already spaced in the dimensions of 3, 4 and 5 – and they could peg it anywhere to make a right-angled triangle in a few minutes.
By some such accident, or series of accidents, the string disclosed yet another secret – the perfect right angle.
Measure and repeat!
Now that surveyors could mark off accurately, with rope, all the sides and corners of a field, they had ample practice. For every year, when the floods came and receded, the carpet of rich black soil buried the boundary markers, and the fields had to be remeasured.
Digging level ditches
The rope-stretchers had another big problem, too. The flood season was followed by a long dry season, for which some preparation had to be made. A network of canals was needed to irrigate the land. Digging them created new difficulties. If you have ever tried to dig a ditch, you know how hard it is to keep the bottom level, and the sides at right angles to the bottom. But this is important if the ditch is to hold water. Water does not run uphill and if it runs downhill it flows into one spot. A way had to be found to level the irrigation ditches and straighten their walls.
To do this, the rope-stretchers took two straight sticks of equal length and spliced them together to form an angle. Then they reinforced this angle with a crossbar, which made a shape something like our letter A. Finally they hung a weighted string from the vertex of the angle.
When the two sticks which formed the angle were on level ground, the string hung straight down through the centre of the crossbar. If the ground was not level, the string hung off-centre. To make sure that the side of a ditch was vertical, they hung a weighted string against the side. Thus – with level and plumb line – they performed the all-important work of surveying ditches and canals.
Measuring area
The rope-stretchers also helped with one other major problem of community life – taxes.
Because taxes were paid according to the size of a field, a way of measuring area had to be found. For measuring length, there were two available ways. A rope could be doubled by using it twice, or halved by folding it (a rope with evenly spaced knots was a good measure). Also, parts of the body could be used as units of length – for instance, the breadth of the finger, the width of the palm, or the span of the outstretched hand from tip of thumb to tip of little finger. And do you know that the dimensions of Noah’s Ark are given in cubits? A cubit is the distance from the elbow to the tip of the longest finger.
But the human body does not provide convenient units for measuring area. People had to describe the size of a field by a morning’s work, or a day’s work for a yoke of oxen. But some farmers worked faster than others, and so did some oxen. You can see how disputes would arise. Even early on, people realised that exact measurement was a good way to keep peace.
Again the idea of weaving came to the rescue. In their woven mats, ancient people could see a design of little squares. And they could construct a square by making the sides of a rectangle equal. The square became the unit for measuring area.
Naturally, solving all these problems made the rope-stretchers very important people in the early communities. Conversely, important people were proud to be rope-stretchers.
Rope-stretchers and Pharaohs
In ancient Egypt, when the Nile overflow clogged the irrigation ditches with mud and buried the boundary markers, whole villages would go out together to clear them. The leading rope-stretcher of such groups was the local chieftain who supervised the work. To these local chieftains the people paid their taxes with shares of grain or flax according to the size of their property. They also gave them a respect bordering on veneration.
Since the whole civilisation depended upon water, which nourished the fields and gave food, these early rope-stretching chiefs were looked upon as divine givers of life. From their number came area chiefs and finally a national leader who was the start of the long line of Egyptian Pharaohs.
Building the pyramids
For the burial of their Pharaohs, the ancient Egyptians built the great pyramid tombs that are still admired by travellers today. These pyramids are masterpieces of ancient practical geometry. Even now they are enduring monuments to the accuracy of the string-made right-angled triangle and the square, and the early appreciation of sturdiness in the pyramid form.
With the right angle, the pyramid builders laid out accurate direction lines: they accepted the direction in which a shadow always pointed at noon as the north-south line. By drawing a line at right angles to it, they got an east-west line. These two were always their base li
nes. The sides of the pyramids’ square bases face exactly to north, south, east and west. And this precision was achieved thousands of years ago.
The first pyramid
The world’s oldest man-made stone structure is the step-pyramid tomb at Saqqara. It was built about 2700 BC. Its square base was pegged off with a knotted rope. In receding steps, it tapered to a peak on top; it was the forerunner of the true pyramid form.
The Great Pyramid was built around 2550 BC. A wall drawing has survived that shows us the actual start of the work. It depicts all the pomp and grandeur of the ceremony connected with laying out pyramids and temples in that period. Just as laying the cornerstone of an important government building brings out officials today, so marking off the ground plan of a pyramid was a great occasion almost 5000 years ago.
The place was selected. The rope-stretchers were on hand. Amidst great crowds, the Pharaoh and his retinue marched to the scene to perform the ceremony. This impressive occasion was called the Put-ser, which means ‘to stretch a cord’. The ancient picture shows the Pharaoh holding a golden mallet, and the inscription tells us that the Pharaoh spoke words befitting a royal rope-stretcher:
‘I have grasped the wooden peg; I hold the handle of the mallet; I grasp the cord with Seshata [the goddess of the stars]; I cast my face toward the course of the rising constellation; I let my glance enter the constellation of the Great Bear; I establish the four corners of the temple.’
6. The Stargazers
Meanwhile on the flat plain of the Middle East, civilisation was taking another course. And so was practical geometry, with the accomplishments of the stargazers, who divided the circle and became the world’s first systematic astronomers.
We can imagine how the night sky must have looked to the earliest stargazers in the valley of the Tigris and Euphrates Rivers. Those people of old watched the familiar star patterns move across the heavens. They watched the great drama of the procession of the constellations as if it were a gigantic circus parade.
In that starry sky, the ancients picked out figures of human beings and animals and identified them with heroes and gods. They passed on their ideas to others, and we still call the constellations by names that trace back to those times.
And we still use direction-finding tools that derive from instruments they made while stargazing thousands of years ago.
Early peoples of the plain – the Sumerians, the Chaldeans, the Babylonians – needed a guide for their travels and wanderings and wars across that broad flat region. They found it in the stars.
This was a necessity the ancient Egyptians never experienced. Civilisations of the River Nile and the Tigris and Euphrates had progressed in the same direction in measuring off land and building irrigation canals. But varying geographical conditions impelled them to develop later along different paths.
The great Nile River was itself a well-defined highway between the settlements of Upper and Lower Egypt. Its narrow valley was protected by mountains and deserts on both sides. For centuries this isolated valley was free from foreign invasion. The people were able to pursue the arts of peace. Wall paintings still exist, after more than 5000 years, that show the Egyptians at work and play – living industrious or leisurely lives in efficient, luxurious and artistic surroundings.
But the valley of the Tigris and Euphrates, called Mesopotamia, the Land Between the Two Rivers, extended over broad, flat plains dotted with little cities. Nomadic tribes roamed over these plains and made war on one another and on the city dwellers. And the many city-kingdoms made war on each other, too. Wall pictures from this ancient land show warriors, chariots, weapons and war machines.
In addition to this endless warfare, the peoples of the plain were famous traders. They had no wood or metals, and they needed both for their cities. So caravans of donkeys and camels and flotillas of sailing boats set out constantly to exchange goods with neighbouring and faraway lands. All this created a problem.
Mesopotamia was so extensive and so bare of natural landmarks that people had to find a way to lay out directions for their wars and their travels across and beyond the valley. The solution was finally found by their stargazers – ancient astrologers held in great esteem, as were the rope-stretchers along the Nile.
The power of the stars
The Egyptians, we know, gave credit for their abundant crops to the rope-stretching surveyors who constructed and maintained the life-giving irrigation ditches. For their part, the people of Mesopotamia believed that if messages from the sky regulated the seasons, they must also regulate people’s activities. They thought that movements of heavenly bodies controlled and forecast important human events, so they gave credit to star-gazing priests who studied these movements.
In the Nile Valley people built tombs for their Pharaohs. In the valley of the Tigris and Euphrates, people built temples to their sky gods atop lofty ziggurats (broad, tiered towers) for their stargazers. From these high platforms, the priest-astrologers were better able to watch the whole sky, and study and interpret the star movements that they believed directed human affairs.
These temples supervised the life of the community. In return, the people presented a portion of their livestock and crops to their stargazing priests, part to be used in sacrifice to their gods and part as taxes to be kept in the temple treasury for the support of the government.
In time, the temples became observatories. And the stargazers became astronomers, and solved the problem of measuring the travels of distant stars.
Already these stargazers had noted the messages of time and direction in the shadows; they had observed the changes in the position of the rising and setting sun through the year. They had used the phases of the moon, its periodic growth and waning, to regulate their early calendar. They had watched the steady movement of star-groups across the sky and the travels of the ‘wandering’ planets in relation to a fixed star.
But they needed an instrument to measure more exactly this brilliant parade across the star-studded heavens. The people needed a measure of direction for travel on the earth. The priests needed a measure for the travels of the stars in the sky. These needs led them to find a very important secret from the circle. We do not know just when or where an unknown stargazer (or stargazers) made the discovery. The idea of pointing the sides of an angle at two stars must have been very ancient – holding up a string to measure the distance between them was useless, since the length changed as you brought it closer to your eye. But how could you measure an angle?
The answer lay in the division of the circle into six parts. This was the earliest and easiest circular partition, and Egyptians, as well as Mesopotamians, used it for sky measurement. Perhaps we can imagine how it was discovered.
Dividing the circle
Possibly an old stargazer, looking back on the games of his childhood, recalled tracing a perfect circle on the ground with a string – and used string and stylus to draw a circle on his clay tablet. Now, how would he divide it?
Perhaps he was just toying with his string, or he thought of a field of six-petaled flowers, or he remembered something children did in a game.
After their feet had scuffed out a circle on the ground, sometimes one child stood on the scuff-marks and held the string, while another child went round in a second circle that overlapped the first. The stargazer marked off the curve of his circle with a series of six arcs, using the same length of string with which he had drawn the circle. When he swung these arcs clear through the circle, he got the flower pattern with its six petals. His discovery was simply that these arcs would cut the circumference into exactly six equal parts.
Anyhow, by chance or by intuition, he hit upon the secret of dividing the circle into six equal parts. After that, it was an easy matter for other stargazers to go on dividing each arc in half, and then in half again. But how many times should they redivide it? The ancient Mesopotamians counted by sixes and tens. Their earliest stargazers thought the year had 360 days. So what could be more reasonab
le than to continue dividing this circle, with its six convenient arcs, into smaller and smaller parts – until there were 360 tiny divisions in all! With these tiny divisions, the stargazers had a new convenient unit of measure: the arcs on the circumference of the circle would measure the corresponding angles at its centre.
As soon as this discovery was made, measuring the path of a star or planet was easy. Ancient astronomers attached a movable pointer to the centre of a semicircle. With this device, they could follow the planets and measure off their distant travels in units of angular measure on the semicircle.
Using the same device, land directions were easy to indicate, too. They could mark off directions on the earth from the east – that is, from the position of the rising sun on the day when it rose midway between the farthest points where it had risen during the year.
And these unknown stargazers left us a monument to the division of the circle. From that time on, tables of measurements of the star movements were kept in the temples.
Recording eclipses
As far back as four thousand years ago, these ancient astronomers – with their pointers and semicircles and quarter-circles (quadrants) and sixth-circles (sextants) – observed and recorded eclipses of the moon. But at that remote time such observations were only occasional and unsystematic. Gradually it became the custom to make more frequent observations until in 747 BC the series became continuous and a record was carefully kept.
String, Straight-edge and Shadow Page 3
