A small soldier of the frontier army in the late Ming Dynasty

Chapter 781 Caused by a Math Problem
It is very easy to use the stacking method to calculate this question, and the next few questions are questions that need to use the "Da Yanqiu Yishu" and "Stacking Technique" algorithms.

These questions involve practical problems in engineering, taxation, military, etc., and cannot be answered without these algorithms.

At this time, the arithmetic problems presented are mostly the content of "Nine Chapters of the Book of Numbers" and "Sun Tzu's Mathematical Classics", which is deeper than "Nine Chapters of Arithmetic". Huang Bowen even skipped two questions, waiting for the answer in time .

Just like this question, it says: "Weiwei's military camp is in the west of the town. I don't know how many soldiers there are in the camp. Three thousand and sixty-four thousand bowls. It's not too late to see if it's exhausted. Three people share one bowl of rice, and four people share one bowl of soup. . May I ask you, sir, who can calculate the number of soldiers in the camp."

Although it is not difficult, but the time is limited, he can't spend more time on each question, and solve the following ones first. After all, these questions with insufficient skills are complicated and time-consuming.

Huang Bowen has passed his thirties, but he still hasn't married a wife. Although his parents hoped that he would study well and get a good reputation since he was a child, he has been a child for many years.

It is because he has been obsessed with arithmetic since he was a child, which is regarded as a small way by others. If he does not do well in stereotyped essays, he will naturally fail in the examination of a scholar.

Moreover, his father was originally a down-and-out scholar, and his family life was poor, with barely enough food and clothing. After his parents passed away, his life became even more difficult. He made a living by selling goods from all over the world. Why don't you have dreams in your heart?
Xuanfu town recruits officials widely, and the emphasis on practical arithmetic gave him hope, the opportunity is in front of him, how can he not work hard?

He hecked his hands that were numb from the cold, roasted them on the stove by the table, rubbed his face vigorously, and continued to concentrate on answering the following questions.

Then he was taken aback, looking at the question below: "A question about shape?"

Liu Dongyang, who was sitting a few rows in front of him, also shrank his eyes: "Geometry problem?"

Liu Dongyang's family experience is happier than Huang Bowen's. After all, his parents are still alive, he is married to a wife, and has a son and a daughter. He is living a happy life.The ancestors were once rich, so they were able to support him to become a scholar, and lived a life of rich and young with clothes to stretch out their hands and food to open their mouths.

Liu Dongyang has received a good family education since he was a child, and he is also very interested in arithmetic. As early as when he was young, he found the ancient mathematics classic "Nine Chapters on Arithmetic" on his father's bookshelf, and he has been fascinated by mathematics since then.

Afterwards, he collected many arithmetic books, including Xu Guangqi's translation of Euclid's "Elements of Geometry".

The characteristics and strengths of Chinese and Western mathematics opened Liu Dongyang's eyes.

Western learning emphasizes strict logic, clear reasoning, advancing layer by layer, and finally getting the result.Middle school places more emphasis on solving methods and pays attention to calculation skills. No matter what your process is, you will get the result. It is a bit like quality education for later generations, while Western learning is a bit like exam-oriented education.

However, their respective charms and the different mathematical ideas of the two parties still made Liu Dongyang lost in the ocean of mathematics.While studying "Nine Chapters of Arithmetic", he also absorbed new ideas from "Elements of Geometry", which made his mathematical attainments increasingly advanced.

It's just that the good times didn't last long, Liu Dongyang's family was in trouble, so he had to switch to business, and then earn some money to barely support his family. After all, being good at math doesn't mean being strong in business.

When Xuanfu Town was recruiting officials, Liu Dongyang's family was engaged in business in the capital, and the family only had a casual discussion, and the whole family resolutely came to apply for the job.

In "The Elements of Geometry", Xu Guangqi named the "morphology" as geometry, and there are also translations, such as "parallel lines", "triangles", "diagonal", "right angles", "acute angles", "obtuse angles", etc. ", "similar" and other Chinese terms are familiar to later generations.

So when he saw this question, Liu Dongyang thought: "Geometry question..."

Like Huang Bowen, he also skipped questions. In fact, not only Liu Dongyang and Huang Bowen, but also Li Tanran, Zhao Zhongju and others kept skipping questions.

They want to see if there is anything they can do for the next questions. After all, unlike yesterday's exam, today's arithmetic questions are still traceable.

Seeing the geometry problem, everyone took out their own squares and compasses one after another. When entering the examination room, the examiner also distributed pencils.The moment, also known as the curved ruler, is often used by carpenters, but it is naturally necessary to use the moment to do arithmetic problems. There are also compasses, which appeared as early as the Xia Dynasty.

Huang Bowen looked at this question carefully: "If there is a round city, the circumference is unknown, and the gates are opened on all sides. There are cross roads outside the gate. The northwestern cross road is designated as dry land, and the northeast cross road is designated as Burgundy. , the southeast cross road head is defined as the land of Xun, and the southwest cross road head is designated as the land of Kun... Or ask: A and B are both in the dry land, and B walks 320 steps east and stands there. A walks [-] steps south and sees B, How far is it?"

"To answer the question, you need to write the solution and act out the script."

Huang Bowen took a deep breath: "I respond with the circle cutting technique!"

He used a rectangle to draw a triangle on the draft, and the three vertices were designated as heaven, earth, and dry, and then he used a compasses to draw an inscribed circle to represent the round city. Out of circle and square, circle out of square, square out of moment, moment out of ninety-nine 81. Therefore, to break the moment, it is thought that Juguang is three, stock is repaired four, and diameter and corner are five..."

He made a three-part and four-point diagram, set the center of the inscribed circle as the center, and intersected the triangle and the inscribed circle with the triangle and the inscribed circle from top to bottom with the vertical line passing through the center at three points: sun, south and north.From left to right, the horizontal line passing through the center intersects with the triangle and the inscribed circle at Sichuan, East, and West points, etc., respectively calculates the hook and the strand, and then finds its chord.

Liu Dongyang also took a deep breath, and began to draw a sketch: "Let's set up a right-angled triangle, and set up three points A, B, and C..."

Huang Bowen was clamoring on the abacus: "Pick up the stock to find the string, use the hook to multiply the stock, and double the amount to realize the result..."

Looking at the result on the abacus, he took up a pen and wrote it down with satisfaction. At this time, Liu Dongyang also used the Euclidean geometry formula to calculate the hook and the stock, and then used the Pythagorean theorem to get the result. He wrote: "Answer: The city diameter is 240 steps. "

Although the content of the "Zhoubi Suanjing" has been tested for the geometry questions at this time, the first question is relatively simple. It was Zhao Zhongju who took a long time to draw the picture and finally made it.

The next few geometry problems are relatively simple, just like this: "Today there is a bamboo that is one foot high, and the end is folded to reach the ground. Go to the original three feet, and ask the person who folded it how tall?"

Huang Bowen and Liu Dongyang used Chinese and Western methods respectively, and calculated them at the same time.

But the next question is more difficult, but it is connected to the round city above, but said: "Maybe if there is a tree 480 steps south of the west gate, and [-] steps east of the north gate to see it, how far is it?"

Huang Bowen rubbed his face hard for a while, and then used the Tianyuan technique: "I set Tianyuan one as the radius, and put it on the ground when walking south..."

He calculated: "Take the multiplication of the two steps as the reality, and the two steps are combined to follow. One step is the usual method, and the radius is obtained."

Liu Dongyang is also drawing a picture: "The remaining radius is unknown..."

Zhao Zhongju went back and did it again, Li Tanran finally figured out the chicken's problem, and rushed to catch up.

"Now there are accumulations multiplied by the sum, subtracted from the accumulation, and the rest by the sum of the equal multiplication, we get 17 steps. It is only said that harmony is beneficial to reality. The fourth is beneficial, the third is from the top to the bottom, and the second is to benefit the bottom from the bottom. One is the right corner, and three squares are divided, such as a quarter of a flat. Ask, what are the lengths and squares?"

Huang Bowen replied: "Twelve steps are flat, and thirty steps are long."

"Today, there are 24 steps of Huang Fang's multiplication and direct accumulation, only cloud strands and nine steps, what is the geometry of the hook?"

Huang Bowen replied: "Three steps."

"Today, there is a stock power reduction string comparison and stock multiplication hook and so on. It is only said that the hook power plus string comparison is the same as the hook multiplication string. Ask the stock geometry?"

Huang Bowen set Tianyuanyi as the stock, and Diyuanyi as the hook string, and finally answered: "Four steps."

He felt dizzy in his head, and his temple was throbbing and tingling.Liu Dongyang also put down his brush and closed his eyes for a while to rest his mind, but Huang Bowen shook his head and looked at the next question: "Today there are strands compared with division of strings and direct product, etc. It is only said that Pythagorean is compared with division of strings with sum and congruence. Ask string geometry?"

Huang Bowen tried his best to sit still, and exhaled slowly. He was deeply proud of being able to answer so many questions so far in the exam. However, there are still the final questions of the Marquis of Yongning, which he must do.

He picked up a square object and a pencil, and drew a picture on the draft: "I set the Tian Yuan as a hook, the Earth Yuan as a stock, the Ren Yuan as a string, and the Wu Yuan as a number!"

Finally, after he finished answering this question, he finally saw the final question set by Yongning Hou Wangdou.

"In the Jingbian army, there is a general Xiancai who is good at using the blunderbuss, and there is a general Xuan who is good at using the cannon. Xiancai shoots the geese and piles them up. The bigger the pile, the higher the pile. Xuan laughed and said: "I shot it with a cannon, and the pile of geese will be exhausted." .”

"You should know that the heap of 鴽 is a sharp cone. You should know that all the sharp cones have the principle of stacking. The number of elements starts from a hair and increases, and when they are stacked, they form a flat cone. A certain number of elements is stacked to form a square. When multiple elements are stacked, they form a flat and pointed cone, and when square numbers start from a hair and gradually increase, they form a pointed cone."

"A certain number of squares are stacked to form a cube, and squares that are stacked up and down will form a pointed cone. The number of cubes starts from a hair and gradually increases to become a surface, and when stacked, it becomes a three-fold pointed cone. Know the three times. The bottoms of the above sharp cones are all square, but the upper four sides are not flat and concave, and the more multiplied, the more concave."

"You should know that the three-fold square starts from a hair and gradually increases to become a surface, and when it is stacked, it becomes a four-fold sharp cone. From then on, it is recursive to infinity. Lines, surfaces, and bodies all have the principle of circulation."

"Excuse me, sir, who knows the calculation, how is the algorithm of this cone? How much is the product of the cone? How much is the inner product of this circle?"

Huang Bowen stared dumbfounded, his mind was buzzing, and he subconsciously thought: "I set up the circle cutting technique, pile up..."

Then he was stunned again, although the circle cutting technique can be used to infinitely approach the area of ​​a circle, it is said that the finer the cutting, the less the loss, and the cutting is cut again, so that it cannot be cut, then it fits with the circle without losing anything, but Not enough to solve the immediate problem.

This question has infinitely small divisions and infinitely large summation, and then set the setting, "you should know that all the squares have sharp cones", "you should know that all the sharp cones have the principle of accumulation", and then in the limit thinking, the sharp cones It seems to have curves and movement again.

Zhao Xuan was also asked to bombard the movement area, which seems to be mutually active.

Because the cone keeps getting bigger and smaller, the area of ​​the shell's trajectory is also different.

This, how to calculate this?
Various images swirled in his mind, and Huang Bowen sat there blankly, making the buzzing in his mind even louder.

"Son, the test of merit is the right way!"

Wet tears in mother's eyes.

"Wen'er, if you go on like this, what should you do?"

The father, who had always been strict, didn't say anything to him, just shook his head, with a look of disappointment in his eyes, then turned around and walked away.

"Huang Bowen, have you been a child for decades?"

The contemptuous and mocking eyes of the classmates were so piercing, humiliating, shocking, and the past flashed quickly in front of them.

"No, I want to solve this problem and become famous in one fell swoop!"

Huang Bowen struggled, he stretched out his hand to calculate, but there were bursts of golden stars in front of his eyes, making him unable to sit still.He tried his best to support the tabletop, but his body was still shaking. Suddenly, he spurted out a mouthful of blood, his eyes were spinning, and he couldn't hold it any longer, so he fell to the ground like that.

There were screams in the examination room, and Zhao Zhongju yelled: "It's terrible, someone vomited blood in the exam..."

(End of this chapter)