How can it be done to divide the entire plot of land into three-fifths? This was the thought in the minds of everyone present. Pen "Fun" Pavilion www.biquge.info

The giant Tian Jiu also thought this in his heart, and he immediately spoke: "How can you make people do the impossible?" The size of a piece of land, which is a square field, is a square field, then I can know; It is a guitian (isosceles triangle), which I can also know; It is an inclined field (isosceles trapezoidal), I can know. However, the terrain is like a dog's tooth, and I really don't know its size. Your Excellency, is it difficult for a strong man? ”

The child of the Le family immediately replied: "I am not difficult for the strong, but what I ask for is nothing more than the word fairness, I don't want more than other people's land, but others can't forcibly take my land." That's what I'm asking for, isn't that excessive? ”

Of course excessively! The giant thought so in his heart, the Mo family is the most proficient in surveying, carpentry and other foreign miscellaneous arts among the sons, if he can't even do it as a giant of the Mo family, who in the world can do it?

"Is all you want is fairness? I'll give you justice! At this moment, a calm voice sounded, like thunder, in the ears of the giant.

The people who were arguing also looked over in unison, wanting to see who dared to boast such a huge mouth here and could do such an impossible thing.

The person who came was Dai Yan. He has actually been here for a while, but he has been silently observing everyone's disputes, wanting to see where things will go under the mediation of the Mo family, but when he found that even if the Mo family also took action, it failed to completely solve the problem after all, he finally made a move.

When the children of the Lejia saw it, wasn't this person the Xuanzi who had been spreading in Fengyi recently? There was some fear of it in his heart, but after all, the desire for his own land prevailed, and he still asked: "Childe, the villain wants three-fifths of the exact size of this land, are you sure you can do it?" ”

"What's so difficult about this, this son can definitely do it, and I can guarantee you accuracy!" Dai Yan said indifferently.

"Forgive me for taking the liberty of asking how Childe can accurately measure the exact size of this place?" The giant Tian Jiu asked, he actually didn't believe that anyone could do this.

"Before we do this, we will first determine the size of the land. Said: Three hundred steps and one mile, the name is Jingtian. The well field, 900 acres, the public land is one. Let's first determine the first point, one step is long, one step is one step wide, and one step is one square step, which I call a flat step. From the ancient method, it can be seen that the size of an acre of land is 90,000 steps, and the size of the land is 900 acres, so an acre of land is 100 square steps, that is, it is ten steps long and ten steps wide is one acre. Do you have any questions about this? Dai Yan asked.

"We have no doubt about this basic common sense." Everyone applauded in unison.

"Secondly, this place seems to be uneven and intertwined, but we can be sure that the land before the two families is straight at the junction with the neighborhood, and the place with its back to Surabaya is also straight, so that we can draw a line at the shortest part of the field and hang down from the boundary between the two original lands, and this land is a square field. In this way, we first measure the length and width of this square field, then we can know the size of this square field, you should have no doubt? ”

Everyone present felt that this matter was too simple, and there was no doubt about it. So Dai Yan asked Tang Ying to take people to measure the size of the square field that was demarcated, and measured that its width was 450 steps, and its length was 40 steps, so it can be seen that the size of this square field is 180 acres.

Then Dai Yan said: "Then the rest of the matter is to measure the rest of the land." Perhaps this is where you find the most difficult, the rest of the land is a mess, how should we measure it? Boy, let's draw a few more lines, and each line will perpendicular to the line we just drawn, so I will draw the rest of the land into five parts, then two of them are Guitian, and three are Xietian."

"Wait." This was the giant Tian Jiu who interrupted Dai Yan's words. "Childe, please forgive me for taking the liberty of taking the liberty of the land, which is not a Guitian or a Xietian. Guitian, semi-wide, to make up for the void for the straight field; And the same is true for the sloping field, which can make up for the void for the sloping field. ”

Now it's Dai Yan's turn to be speechless, can you make up for the void as a straight field to be a Guitian, and a sloping field? He always thought that what Guitian was talking about was a triangular field, and the inclined field was a trapezoidal field, but it was necessary to make up for the deficiency with surplus? Doesn't this mean that Keita is an isosceles triangle, and a slanted field is an isosceles trapezoid? So what are the shapes of triangles and trapezoids in general?

In fact, this is an inherent defect of Chinese civilization, Chinese civilization only began to have the germ of categorical ideas until this time, and began to appear the proposition of hard white debate, white horse is not horse, but after the Warring States Period, these debates also began to disappear from history, but Dai Yan did not know. He felt bitter in his heart, how could he feel that he was teaching elementary school students?

That's it, work harder, Dai Yan comforted himself in his heart. "Giant, do you think that the fields enclosed by the three borders are only the fields that can make up for the void with surpluses?" Dai Yan asked Tian Jiu carefully.

"Yes, yes." Tian Gu replied.

"Then the kid here asks the giant to draw three lines on the ground at will, and form a block, this shape has three corners, since the giant thinks that this shape is not a Guitian, then I will call it a triangle." Dai Yan said.

Tian Jiu did as Dai Yan instructed, and randomly drew a triangle on the ground.

"The giant thinks that this form can't make up for the void with a profit, but on the surface it is true, so how about the kid take one more step?" After Dai Yan finished speaking, he drew a congruent triangle that was inverted from the original triangle with one side of the triangle as the common side and one side as the bottom edge, so a parallelogram was formed.

"We see the shape of the two triangles as a whole, which is different from the square field and the oblique field, but this field can also be used to make up for the void." After Dai Yan finished speaking, he made a vertical line at the bottom corner of the parallelogram, which was directly perpendicular to the bottom edge, so that a right triangle was cut.

"Then, with the surplus of this shape to make up for the void of the other place, then the whole shape becomes a square field, so that the size (area) of the whole square field is wide (bottom) multiplied by the positive slave (the height of the triangle), then the size of the triangle is half of this square, that is, the half width is multiplied by the positive slave." Dai Yan said lightly.

Half wide to multiply from! Juzi Tian Jiu's heart was shocked, of course he knew what this meant, in fact, this is the method of measuring the size (area) of Keita that has been handed down for thousands of years, but he randomly drew a so-called triangle, and the young man in front of him easily added a stroke and easily concluded that the size of this shape is the same as half a wide multiplication of positive follow, and Keita's algorithm is exactly the same, what do they have in common? Tian Jiu's brows were tightly locked, and his heart fell into deep thought.

"What do you mean by this? Childe means that the size of the triangle can be calculated in this way? Just now, Childe only measured the size of this triangle, but Childe decided that the size of all triangles can be calculated according to this method? Is there such a truth in the world that one law can lead to ten thousand laws? This was a question from a Mo disciple behind Tian Jiu.

"Entanglement, shut up, of course this son has his reasons for doing this." Tian Jiu spoke. Then he asked Dai Yan: "However, I look at Gongzi's method one ring after another, and it is very precise. I can't say why, but in my heart, I feel that Childe's method seems to be taken for granted, and I hope Childe will teach me. After speaking, he bowed to Dai Yan to show respect.

"Sir, never." Dai Yan hurriedly avoided Tian Jiu's salute, joking, knowing that the person in front of him was a famous Mo family giant in the whole world, a famous scholar in the world, how could Dai Yan dare to accept his salute. Then he said: "What I just said is to show that as long as a triangle is drawn at random, then its size is half wide and multiplied by positive and subordinate, and the calculation method of the irregular slope field can also be calculated in the same way, do you think so?" ”

Everyone present shook their heads again and again, expressing that they couldn't understand, and Tian Jiu also shook his head and said nothing. Dai Yan also began to scold his mother in his heart, why could so many people present not understand the geometric knowledge that the third grade of primary school knew in later generations?

In fact, this is why Dai Yan doesn't know the origin of geometry. The origin of geometry is later recognized as originating in ancient Egypt, where the ancient Egyptians had a wealth of experience in surveying techniques and some geometric knowledge because the Nile flooded once a year, and each time it flooded, flooding the land on both banks; At the same time, because the pharaohs of ancient Egypt had to build pyramids as soon as they ascended the throne, it is impossible to imagine how such regular geometric figures as the pyramids could be built without rich knowledge of mathematics and geometry.

But is the geometry of ancient Egypt the geometry of later generations? Not. Although the ancient Egyptians had a lot of experience in mathematics and geometry, they did not elevate it to a systematic theory. It was Thales, a merchant from Greece, who really established the foundations of geometry. Thales spent his early years studying in ancient Egypt, from which he learned the rudiments of geometry. Then he traveled to ancient Babylon, where the priestly class was extremely developed, and the ancient Babylonians worshipped the moon, the god of the moon, so the ancient Babylonian priests needed to explain the lunar eclipse, so they accumulated a wealth of algebraic knowledge, they could count the days of the lunar eclipse to the decimal place, and Thales learned algebra from here. He then returned to his hometown of Mirito, a port city where the ancient Greeks were confronted with a terrible problem: ships had to enter and leave the harbor every day, but the harbor was of different depths, and there could be reefs on the seabed, and the inability to determine the distance between the ships could lead to serious disasters. So how do you measure distances at sea?

If you measure the distance on land, it can't be easier, you can measure it directly with a ruler, so can you do it at sea? Thales solved this problem by applying the law of similar triangles, based on his deep mathematical knowledge accumulated over thousands of years from the two original civilizations of ancient Egypt and ancient Babylon. At the same time as solving this problem, Thales and the ancient Greek philosophers produced a breakthrough: it had to repeatedly use speculation, argumentation, and certainty for a long time, and this is logical proof.

The emergence of a logical proof is no less than a mutation in the genes of human civilization, because it means that as long as you can give known conditions and settings, then you can deduce certain unknowns. Historically, only ancient Greece evolved this way of thinking, and none of ancient Egypt, ancient Babylon, ancient India, or ancient China were able to evolve this way of thinking. With this way of thinking, ancient Greek mathematics and geometry seemed to have a framework, and subsequent mathematicians continued to add to it, and finally formed a systematic geometry - Euclidean geometry in the hands of Euclid.

And even geometry as simple as Euclidean geometry had a famous donkey bridge theorem in medieval Europe: the first five theorems of the first part of the Geometry Primitives. The fifth principle is: the two base angles of an isosceles triangle are equal, and it is such a simple theorem that has become the most famous "fool's difficulty" in history, that is, the "donkey bridge", and those who can understand this theorem can be regarded as crossing the donkey bridge.

And Dai Yan wanted to prove the area formula of triangles before he had built the framework of the whole geometry, and it must also be applied to all triangles, how can people in this era understand? This is definitely not a gap in intelligence, etc., this is a gap in more than 2,000 years of civilization, and it is also a gap in cognition.