There is a world-famous Nile in northeast Africa. It crosses the Sahara desert in northern Africa and flows into the Mediterranean Sea, and the narrow strips on both sides of the strait become fertile oases. Where the river flows downstream, Egypt, one of the oldest civilizations, was born.
The Nile Delta is rich in an aquatic plant called papyrus. The ancient Egyptians tore the stems of papyrus into thin slices layer by layer, and then glued them together one by one, which became writing paper. Many ancient Egyptian papyrus have been preserved to this day, which has become precious materials for us to inspect Egyptian history and culture.
The Egyptians wrote about 3500 BC. The earliest preserved papyrus for recording mathematical knowledge is now in the British Museum. This papyrus was written by Amos who lived between 1600 BC and 1800 BC. According to him, the content of papyrus paper was transcribed from the old paper before 2200 BC. On this papyrus, there are some descriptions of four operations of fractions and arithmetic, as well as the rules of measurement.
The emperor of ancient Egypt was called "Pharaoh", and the famous pyramid was the tomb of Pharaoh. Today, there are more than 70 pyramids scattered in the south of the Nile Delta. Among them, the largest is the Pyramid of the King of Qi: the tower height146.5m; Each side of the tower foundation is about 240 meters long, which surrounds the tower for about one kilometer. There are tunnels, stone steps and graves in the tower. This pyramid was built in 2800 BC, and it was the tallest building in the world for more than 4,600 years before the completion of the Eiffel Tower in Paris in 1889. This is really an amazing miracle! In the process of building these huge buildings, the ancient Egyptians accumulated rich geometric knowledge.
We assume that we must draw a plan before building the pyramid. It is estimated that this painting was painted on a clay tablet, which is probably the first floor plan in the world. From the analysis, the cartographer must know that the pattern and the completed building are different in size, but the shape is the same. It can be judged that the Egyptians at that time had mastered the knowledge of proportion and similarity.
After drawing the floor plan, a large clearing should be leveled out, and the actual size should be set out on the ground to prepare for construction. Building materials are large stones weighing several tons, and a pyramid needs many such stones. At that time, there were no means of transportation and no decent roads. Stones can only be transported as close as possible along the Nile by boat, and then transported to the construction site by rolling logs. Every stone must be chiseled and ground according to a certain shape in advance. Every corner of the stone should be repeatedly corrected to a right angle with a T-square or a triangular plate. Then, lay a huge stone as the foundation. The second floor should be smaller according to a certain proportion, and each floor should be placed in the middle of the next floor. In this way, add one layer at a time, reduce equally on all sides, and finally meet accurately at the top of the tower.
A pyramid, with hundreds of thousands of people and millions of boulders, can be built in decades without making mistakes. Look how clever the ancient Egyptians were in designing, calculating, measuring and building!
How to draw a right angle accurately is probably the biggest problem that the ancient Egyptians have to solve. Because the foundation of the pyramid must be strictly square, and the four corners must be strictly right angles; No matter which angle is slightly deviated, the whole building will be deformed. At that time, measuring instruments had not been invented, and it was not easy to make a square as big as one kilometer in circumference!
They probably solved this problem by driving two stakes into the ground first, and then tightening the rope between the stakes to draw a straight line and become an edge of the pyramid. Then, tie a rope to each of the two stakes, and the length of the rope should exceed half of the distance between the two stakes. Tighten the end of the rope, rotate with the stake as the origin, and draw two intersecting arcs. Cross the intersection of these two arcs and draw a straight line. When it intersects the first straight line, the included angle is the exact right angle. The latter straight line is another sideline of the foundation.
Then, to check whether one side of the wall or boulder is upright, how to make a right angle in the air? The ancient Egyptians skillfully used a hammer to aim. This method has been used to this day. The hammer line swings freely, draws an arc in the air, and stops at right angles to the ground. If the wall can be parallel to the hammer line, it is perpendicular to the ground.
Now, we all know that the simple way to draw a right angle is to use a right triangle. However, this must first make a right triangle.
The ancient Egyptians used ropes to measure the land. The job of a professional knotter is to tie knots at equal intervals on the measuring rope. Perhaps they first discovered some triangles composed of three ropes with a certain length, and the angle corresponding to the longest side is a right angle. One of them consists of three, four and five equally spaced knots; The other is to take 5 knots at equal intervals, 12 knots and 13 knots. Saw the narrow wooden strips to this length, and then connect them end to end to make a right triangle. With this triangle, it is convenient to measure and draw pictures in the future.
Farmers build their own huts, so they can say, "My house is six steps long and four steps wide, and the roof is taller than my head by an oak tree." Designing a large building pyramid can't be like this. Because there are thousands of workers, everyone's steps are different from those of tussah. Therefore, they stipulate the length of someone-a certain part of the king's body is said to be a standard unit; Then, according to this standard unit, a certain length of wooden strips or metal strips are made as general measuring tools. This is the earliest ruler.
In Egypt, the main unit of length is the wrist ruler, that is, the length from the elbow to the tip of the middle finger. Smaller units are: palm ruler, equal to one-seventh of wrist ruler; A ruler is equal to a quarter of a ruler. These small units were very useful because it was difficult for Egyptians at that time to understand the meaning of fractions. Today, people are already familiar with fractions, but in habit, everyone likes to use small units. For example, the British and Americans always say seven inches, not seven twelfth feet. In our country, some people say half a foot, but no one says five tenths of a foot.
Every harvest season, Egyptian monks levy taxes on farmers. Farmers mainly hand in their own agricultural products and need standard weight units to weigh millet, oil, wine and so on. The amount of tax is determined by the amount of land, which requires measuring the land area.
The method of finding the area is probably learned by craftsmen when they first started laying square brick floors. They found that if a piece of land is three bricks long and three bricks wide, it needs to be paved with nine bricks (3× 3); Another piece of land, three bricks long and five bricks wide, needs fifteen bricks (3×5). In this way, to calculate the area of square and rectangle, just multiply the length by the width.
But the problem is that not all land is square or rectangular. Some land, as if angular and irregular in shape, can be easily divided into several triangles. How can I find the area of a triangle? In fact, once you master the solution of rectangular and square areas, the triangular area is not difficult to find.
A square piece of linen can be folded into two triangles of equal size, and the area of each triangle is exactly half of the square. It is estimated that from this simple clue, the ancient Egyptians learned how to find the area of a triangle: length times width, and then divided by two.
I think the work of surveying land is very heavy. Because the land in Egypt is mainly distributed along the Nile, the river begins to flood in mid-July every year, flooding a lot of land, and it does not begin to recede until 1 1 month. After the flood receded, a layer of fertile silt was left in the field to help farmers get a bumper harvest; But the flood washed away the land boundary, and the land was re-measured every year. Therefore, people often attribute the origin of geometry in Egypt to the flooding of the Nile.
In a large number of surveying work, Egyptians will certainly encounter the incomprehensible figure of "circle". They find it difficult to divide the circle into many triangles, and each triangle is a standard triangle composed of three straight lines. Therefore, the ancient Egyptians believed that the circle was a sacred figure given to people by God. Today, we are all familiar with the circle and deal with it every day, but it is not easy to understand and master the essence of the circle.
Experience is the mother of wisdom. The early Egyptians must have drawn a circle by winding a rope around a stake. They drew a big circle with a long rope and a small circle with a short rope, knowing that the area of the circle is determined by the distance from the circumference to the center of the circle. This is what we often call radius.
About 3,500 years ago, when the pyramids had become monuments, an Egyptian document named Ahmet wrote such a rule: the area of a circle is very close to 3 7 1 times the area of a square with a radius of 1. This was a great discovery at that time!
How Ahmet got this method of calculating the area of a circle, I'm afraid we'll never know. We can only guess that he probably used the method of drawing triangles. Now, his papyrus manuscript is put in an exquisite frame and hung in the British Museum in London.
Although papyrus manuscripts scattered in museums around the world can help us understand the mathematics of ancient Egypt, most of the existing materials are obtained from the investigation of ancient buildings along the Nile River.
Some pyramids face east, west, north and south accurately on all sides, which shows that the ancient Egyptians were very clever in determining the direction. They may determine the east, west, north and south according to the shadow of a tall stone pillar.
There is the ruins of a big temple, and a row of columns stand today. In the 365 days of the year, only the morning sun from summer solstice can shine on this row of columns. Count the days when the sun shines into the temple twice along this pillar. This is the length of a year.
In measuring time, the Egyptians also determined time according to the positions and shadows of the sun, moon and stars. However, they are much more advanced than the original hunters and gatherers. In the morning, when primitive people saw long shadows, they could only say, "It's still early!" "Egyptians have a daily rule. Looking at the shadow on the scale bar, you can say, "The second hour in the morning is coming! " "
Since then, people have real science. However, many pictures left by ancient Egypt show the busy scene of God who is in charge of day and night. It seems that they are carrying a very heavy burden of superstition and are groping hard on the road of science.