My younger brother Zhuge Liang

Chapter 127: The sky is high and the earth is thick. To others, it is an adjective, but to Zhuge Liang, it is a math problem.
When Cui Yan heard Zhuge Liang's name, he was stunned for a moment. Finally, he roughly understood his identity through the prefix "Langya Zhuge".

It was only because he had not yet officially taken up an official position, and in the past one or two years he had been protecting Zheng Xuan in running schools, often living in the secluded depths of Mount Buqi, and was not very familiar with the recent up-and-coming talents in the outside world.

Gu Yanwu of the Ming Dynasty wrote a poem called "Buqishan" to describe the difficulty of Zheng Xuan's running a school: "There are people cultivating the academy in the barren mountain, and I don't remember the name of the mountain or the county. I wonder if the Yellow Turbans are everywhere in the world, and whether there is room for Zheng Kangcheng."

But no matter what Zhuge Liang's identity was, since he easily solved Zheng Xuan's door-blocking problem, Cui Yan would not make things difficult for him and simply led him into the inner courtyard.

After turning a few steps and passing through several bamboo fences, Zhuge Liang finally saw an old man with a long beard sitting on a small stone in a courtyard with a yellow earth floor. The old man placed the scroll on another large stone in front of him and was pondering over it.

There was no table or chair in the yard, and the seats and things were all made of stone. The door of the room at the back of the yard was also wide open. The room was very small, and you could see the whole room through the door and window at a glance. There was only a bed but no table, so it was probably just used for sleeping.

Zhuge Liang was a very observant person and he noticed something unusual at first glance: there was no table in the room and no oil lamp on the bedside chest, so there should be no oil lamp in the entire room.

The big stone in front of the old man was very wide, and there were two pottery bowls in one corner that had not been cleared away, which meant that he had just finished his meal.

It seems that this old man's lifestyle is very simple and healthy. He works at sunrise and rests at sunset. He reads outdoors with the help of natural light and goes to bed as soon as it gets dark.

The house was built on a mountain, so the yard didn't even need an earthen wall, just a simple wooden stake and bamboo fence.

Zhuge Liang couldn't help but feel a little respect in his heart. This is a true hermit. At this time last year, he was still farming in Longzhong. The environment was also very simple at that time, and he completely let nature take its course.

Cui Yan went up to the old man and said a few words to him. The old man seemed to have poor hearing and only then noticed the visitor. Then Zhuge Liang came forward to salute, and Sun Qian also hurried to greet his mentor.

After Cui Yan finished helping with the introduction, he actually stepped aside temporarily and stopped disturbing everyone. Instead, he took Zheng Xuan's bowl and went outside the yard to wash the dishes himself.

Zheng Xuan looked up and saw Sun Qian. He was still somewhat pleased. He asked him a few questions about his experiences in the past few years, and then turned to Zhuge Liang:
"My dear Zhuge, are you here to ask me how to calculate academic degrees? I haven't seen you for several years, and I am so nostalgic that I have neglected you.

Well, looking at your appearance, you seem to be able to understand the way of heaven. If you study hard, your future will be limitless. However, if jade is not polished, it will not become a useful tool. Don't rely on your talent and be lazy. "

Zhuge Liang bowed to show that he was willing to learn.

Zheng Xuan picked up the scroll and the answer sheet that Cui Yan had just handed back, glanced at them briefly, and showed a look of appreciation:

"This solution is more concise than mine. It does not go beyond the scope of the Nine Chapters, but it also gives a concise and refreshing answer."

When Zhuge Liang saw that Zheng Xuan mentioned the math problem itself, he spoke freely and no longer stuck to it: "Mr. Zheng's question seems a bit redundant. The common multiples of the rotation periods of the five planets, gold, wood, water, fire and earth, are fixed. Since we are only looking for the common multiple, why do we need to tell you the initial positions of the stars?

Is it just to confuse the problem solver, deliberately giving more useless conditions? If you want to use the initial position condition, it is better to change it. Don't ask how many years it will take to return to the initial position, but ask how many years it will take from a random initial position for the five planets to appear in a conjunction. "

Zheng Xuan was stunned. What he had learned had not gone beyond the scope of "Nine Chapters". He knew that the questions after being modified according to Zhuge Liang's description had solutions, but there were only a few special solutions in "Nine Chapters".

He himself did not conclude whether "from any initial position, after an infinite amount of time, there will always be a five-star chain", that is, he did not know whether there was a general solution, so he did not dare to make a random solution, but just played it safe and asked people to find the lowest common multiple.

Besides, this was just a challenge for those who were seeking fame and reputation to meet him. It didn't have to be too difficult. With questions like this, only Zhuge Liang had met him directly in the past two years by solving problems in the name of discussing mathematics. It would be meaningless to make it more difficult.

The rest who wanted to see him were either Yuan Tan and Kong Rong who had official positions and could not be stopped, or they had to work hard in the outer sect to show their sincerity, bring their own food and study with their senior brothers for a year or even longer before they could see Zheng Xuan.

Seeing Zhuge Liang pointing out this problem, Zheng Xuan did not have any competitive spirit and just answered casually:

"This question is indeed a bit redundant. Thirty years ago, when I was in Chang'an, I studied under my master, Ji Changgong. I was not able to see my master in person for three years, and my senior brother taught me everything. In the end, my master couldn't solve this celestial trajectory diagram, so he asked me to come in and help solve it.

For decades, I have been thinking about the past. When I retired to my own school, I wanted to leave a quick way for young people with mathematical talent to seek an interview, so I made some changes and came up with these questions. I didn’t want to use the same questions year after year, for fear that people would copy the answers, so I often changed some figures to confuse the eyes, and some conditions were indeed redundant. ”

Zhuge Liang smiled and talked to Zheng Xuan: If you change the useless initial conditions every time someone solves the problem, it will still be easy for people to see through it, and you can gain fame and reputation next time. Why not change it like this...

Then, Zhuge Liang casually set up a set of equations (but changed X/Y/Z to A, B, and C), and asked Zheng Xuan to draw an initial position of the five stars. Zhuge Liang calculated it for him on the spot, and he could work out how many years it would take for the five stars to align.

Anyway, Zhuge Liang still remembered the basic conditions of the revolution period of gold, wood, water, fire and earth very clearly. Some of them were written by the ancients, and some were taught to him by his elder brother. As long as he could find common multiples and solve multivariate equations, he would definitely be able to solve the whole thing.

Moreover, Zhuge Liang also proved that "I don't care how messy your initial position is, you can always return to the state of five stars in a row in the end."

This goes one step further than the Nine Chapters, which does not provide a rigorous argument for the general solution.

Zheng Xuan initially thought that this boy was really ignorant of the immensity of heaven and earth, but as he watched, his expression became serious.

While Zhuge Liang was solving the system of equations, Cui Yan, who was outside the door, came back after washing the dishes. Seeing Zhuge Liang speaking so eloquently in front of his mentor, his face turned a little ugly.

How can this young man be so frivolous and have no respect for the elderly?

Finally, when Zhuge Liang finished writing and began to talk about teaching Zheng Xuan principles, Cui Yan could no longer hold back:

"How can you not know the immensity of the world? You are so arrogant as to say that you can calculate the positions of the stars in the future. You must know that the days of the year are changing. Although the movement of the stars has its own rules, there are also many exceptions. How can you generalize? Duke Zheng has been the leader of the world in calendar studies for decades..."

"Ji Gui! Don't be rude. You just don't understand." Zheng Xuan suddenly spoke up and stopped Cui Yan from helping him. He didn't want his favorite student to make a fool of himself. Counting was not Cui Yan's forte.

Although, just a few minutes ago, Zheng Xuan also felt that Zhuge Liang was ignorant, he held back and watched for a while longer, and found that the other party was not arrogant, but really had the ability. Cui Yan, however, could not understand, so he said what was in his heart.

Fortunately, Zhuge Liang was not angry. He first explained the principle to Zheng Xuan at his leisure, and then turned to Cui Yan: "Brother Cui, your love of learning is really admirable. Although you are not good at mathematics at present, you have this desire to explore. As long as you are willing to spend time, you will definitely achieve something in the future." Cui Yan was stunned. He did not expect this to be the case. What does "have this desire to explore" mean? When did he show his desire to explore?

Even Zheng Xuan was stunned. He thought that Zhuge Liang would fight back or just be indifferent to show his magnanimity, but he never expected that Zhuge Liang would encourage Cui Yan to "maintain his curiosity about mathematics."

Zhuge Liang saw that they had been stunned for a long time, so he said leisurely:
"Brother Cui, you are asking such questions, how can you not be eager to learn? How many people in the world can keep exploring these questions that are useless to their careers? It just so happens that these two questions are simple, and Liang has seen my brother do experiments, so I can answer them for you. After reading this, if you don't believe it, you can go to the beach and do the experiment yourself."

Then Zhuge Liang took a piece of paper and began to calculate: "To calculate the thickness of the earth, we must first use Zhang Heng's theory of the spherical sky as a basis. The sky is like a spherical qi, and the earth is like a chicken egg floating in the sky. If it is the theory of the spherical sky and square earth, there will be no sky high and earth thick. Zheng Gong learned the spherical sky algorithm from the fifth spherical sky. I don't need to explain this point, right?"

Zheng Xuan and Cui Yan nodded quickly. They had no problem understanding the Hun Tian Theory. Although they did not have a clear concept of gravity, they had already vaguely acknowledged that the earth was suspended in the celestial sphere.

Since we have acknowledged that the earth is a sphere, the rest is easy. To calculate the radius of the earth, we only need the Pythagorean theorem, which is enough for the fourth grade of elementary school. Ptolemy of ancient Greece calculated it hundreds of years ago. If Ptolemy's book had been passed down to the Han Dynasty, the people of the Han Dynasty could have directly copied the answer.

However, Zhuge Liang would certainly not copy the answer. He still chose the experimental method to prove it, but he did not do the experiment now. Instead, he had done it before when he followed his elder brother in studying.

"Suppose the ground is as thick as armor, and a pole is erected on the seashore, three feet high. Then walk to a distance of about four thousand feet away, lie on the ground, and you can no longer see the top of the pole on the horizon. But if you move closer, when you are three thousand seven hundred or eight hundred feet away from the pole, you can vaguely see the top of the pole again.

In this way, we can roughly estimate that at a distance of four thousand feet, the curvature of the earth is enough to block objects five feet high.

So, make a right triangle, the hypotenuse is the thickness of the ground; the leg is four thousand zhang; the chord is the thickness of the ground plus three zhang, which is the height of the pole. So the square of the thickness of the ground plus three minus the square of the thickness of the ground equals four thousand zhang square - the thickness of the ground is about three million zhang. "

Zheng Xuan and Cui Yan were stunned: "The ground is three million feet thick?"

Zhuge Liang: "I'm talking about the radius. The diameter is six million feet. If you don't believe me, you can go to the beach and set up a log to do an experiment. As long as the observation points are at the beach and the altitude is zero, it will be absolutely accurate."

Then, Zhuge Liang casually calculated the height of the sky.

This time he was very satisfied with the calculation, but Zheng Xuan and others had some twists and turns in their understanding, because even those who believed in the Hun Tian theory still had problems with the "heliocentric theory" and the "geocentric theory" -

Zhang Heng's original theory of the spherical sky was obviously close to the geocentric theory, and astronomers at that time also observed that the five planets were sometimes far away from the earth and sometimes close to it. Although they did not calculate the precise orbits of the superposition of epicycles and deferents like Ptolemy, they at least knew the revolution periods of the planets and the magnification relationship between the farthest and closest distances of the planets to the ground.

Zhuge Liang relied on the only thing Zheng Xuan could understand, "the extreme values of the distance changes of the five stars", and made a brief deduction, and then gave Zheng Xuan a brief demonstration of the "heliocentric theory" taught to him by his elder brother.

“So, the Hun Tian theory is not precise enough and is not as simple as the heliocentric theory. In my opinion, if the sun is the center of the sky, then the orbits of gold and water are between the earth and the sun, and the orbits of fire, earth and wood are outside the earth and the sun.

Because the "mean wheels" of Venus and Mercury, that is, the average distances of these two stars from the earth as determined on the armillary sphere, are equal, it can be seen that they must be inside the earth. Therefore, the mean distances of Venus and Mercury from the earth is exactly the distance between the earth and the sun. The farthest point is the earth and the sun plus the sun and Venus, or the earth and the sun plus the sun and Mercury. The closest point is the earth and the sun minus the sun and Venus, or the earth and the sun minus the sun and Mercury.

In this way, the distances between the Sun and Mercury and between the Sun and Venus cancel each other out, so that the average distance between Venus and Mercury and the Earth is almost equal to the distance between the Earth and the Sun.

Fire, earth, and wood are outside the earth, so the average distance between the earth and fire is the distance between fire and the sun. The difference between the maximum and minimum distances is twice the distance between the earth and the sun.

My brother once taught me the geocentric theory of the great Western scholar Ptolemy. After verifying it, I concluded that the stars inside the earth are based on the center and the earth is the mean; the stars outside the earth are based on the earth and the center is the mean.

Zheng Xuan could no longer understand what was said afterwards, and Cui Yan had long since lost interest in understanding either.

Zhuge Liang made another messy calculation. Although he could not figure out how far the sun was from the earth, he did figure out that "the distance between the Mercury Sun and the Earth is 40% of that between the Earth and the Earth, the distance between the Metal Sun and the Earth is about 70% of that between the Metal Sun and the Earth, and the distance between the Mars Sun and the Earth is about one and a half times that between the Earth and the Earth."

"So, although I haven't figured out the distance between the earth and the sun yet, I can still figure out the ratio of the height of the sky. If Lord Zheng has another trick that can calculate the height of the sun and the sky, I will humbly ask for your advice. The height of the sky and the thickness of the earth are roughly like this."

After hearing this, Cui Yan was already very regretful. Why did he have to say that “young people don’t know their place”?

To others, "the sky is high and the earth is vast" may be an adjective, but to Zhuge Liang, "the sky is high and the earth is vast" is just a math problem.

After all, Zheng Xuan was a man who was truly devoted to studying. After the initial shock, he quickly abandoned his prejudices and his desire to save face. Regardless of his advanced age and high prestige, he turned to Zhuge Liang to ask for advice on his spherical celestial view and various imperfect calculations.

Zhuge Liang also answered every question, and occasionally he would ask Zheng Xuan for advice on how to calculate some major evil astrological signs, such as "Mars in the heart" and "solar eclipse".

Zheng Xuan was shocked at first, thinking that such a thing must be impossible to calculate, but after discussing with Zhuge Liang, he was impressed by the complicated yet mathematically beautiful calculation process.

Before they knew it, it was already late at night, and Zheng Xuan's courtyard had lit an oil lamp for the first time in several years, which was borrowed from the courtyard where Cui Yan lived downstairs.

As Zheng Xuan was calculating, he suddenly realized a possibility: it seemed that Zhuge Liang was asking and answering questions to himself the whole time. Was he here to study or to challenge others?

Although this question was not very important, Zheng Xuan couldn't help but ask it, so he asked it indirectly.

Zhuge Liang also answered very openly: "I am here to learn sincerely. After I calculated privately, I found that some of the most ominous disasters, such as Mars guarding the heart, were not recorded in the "Book of Han" and the results of my calculations, nor were they consistent with some fragments of the "Dongguan Hanji".

So I really want to know how those years when celestial disasters shouldn't have occurred, where did those celestial disasters come from, or did Ban Gu fabricate things, because something terribly catastrophic happened in that year and he just made it up and wrote that a disaster happened in that year?

For example, according to my calculation, the year when Emperor Cheng of Han died, Mars should not have been in the heart. Could it be that because the emperor died and the situation in the world deteriorated after his death, leading to Wang Mang's entry into politics, Ban Gu deliberately said that Mars was in the heart that year?"

Zheng Xuan was even more shocked. He had never expected that Zhuge Liang could use mathematics to overturn the fabrications of previous historians. Ban Gu had a very high status in the history of the Eastern Han Dynasty. How could Zhuge Liang even figure out Ban Gu's fabrications?
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PS: Because there is some pretentious mathematical content, there will be three updates today to avoid readers who don’t like reading mathematics saying it’s nonsense.

(End of this chapter)