At the end of Wei dynasty and the beginning of Jin dynasty, after a long period of exclusion and respect for Confucianism, speculative thoughts in academic circles revived. The "Seven Sages of Bamboo Forest", led by Ruan Ji and Ji Kang, became the typical representatives of the quiet school with informal manners and laws. They respect nature, stand aloof from the world, and like to talk openly or mysteriously. Under the influence of this unique "Wei-Jin demeanor", there is also a wave of debate in China's mathematics circle. Liu Hui, who experienced the transition from chaos to unity, demonstrated and annotated some problems and solutions in Nine Chapters of Arithmetic.
Nine Chapters of Arithmetic is the most important of the ten arithmetic classics. It is a classic work compiled by many scholars from the pre-Qin Dynasty to the Western Han Dynasty, and its composition is similar to that of the Bible, a classic work of Western Christianity. It covers a wide range, and records 246 application problems related to production and life practice in 9 categories, such as square field, millet, decline, shortage, business work, average loss, surplus and deficiency, equation and Pythagorean.
You may not understand this very well. Explain it briefly. For example, square field, small area and business work are geometric problems such as current area and volume, millet, score reduction and flat loss are what we are talking about now, and profit and loss are current profit and loss problem. This primary school has learned that equations and Pythagoras are easier to understand, and middle school students should be able to understand them.
"Nine Chapters Arithmetic" has made wonderful examples and solutions in solving simultaneous equations, calculating four fractions, calculating positive and negative numbers, and calculating the volume area of geometric figures. , are all advanced in the world. However, because the solution is primitive, it lacks necessary proof. Liu Hui made supplementary proofs for all of them, and wrote Nine Chapters Arithmetic Notes with a volume of 10. In these proofs, he showed his creative contributions in many aspects.
In algebra, he correctly put forward and defined many mathematical concepts, such as power (area), equation (linear equations), positive and negative numbers and so on. He was the first person in the world to put forward the concept of decimal, and used decimal to represent the cube root of irrational numbers. In solving linear equations, he created a simpler method of mutual multiplication and elimination than direct division, which is basically consistent with the current solution, and he also put forward the "indefinite equation problem" for the first time in the history of Chinese mathematics. He also established the summation formula of arithmetic progression's first n terms.
Geometrically, Liu Hui's main contributions are the proposal of secant and the calculation of emblem rate. Since the pre-Qin period, China has always taken the value of "three circumference and one diameter" (that is, the ratio of circumference to diameter is 3: 1) to calculate the circle. However, the results calculated with this value often have great errors. As Liu Hui said, the circumference of a circle calculated by "Three Circumferences and One Diameter" is actually not the circumference of a circle, but the circumference of a circle inscribed by a regular hexagon, and its value is far less than the actual circumference. Zhang Heng of the Eastern Han Dynasty was not satisfied with this result. He began to get pi by studying the relationship between a circle and its circumscribed circle. This value is better than "three-circumference diameter one", but Liu Hui thinks that the calculated circumference must be longer than the actual circumference, which is inaccurate.
By chance, Liu Hui saw a stonemason cutting stones. It looked interesting, so she stood by and observed it carefully. Liu Hui saw that a square stone was first cut off by a stonemason, and the stone with four corners had eight corners in an instant, and then the eight corners were cut off, and so on. The stonemason kept cutting off these corners one by one until there were no corners to cut. In the end, Liu Hui found that the square stone had become a smooth pillar unconsciously. Masons are polishing stones every day, but it is such a trivial matter that Liu Hui suddenly sees something that others have not seen-the idea of "infinite approximation". Like a stonemason, Liu Hui kept dividing the circle and finally invented the "tangent circle".
In Liu Hui's view, because the circumference calculated by "three-circumference diameter one" is actually the circumference of a circle inscribed by a regular hexagon, it is quite different from the circumference; Then we can divide the circumference into six arcs on the basis of inscribed regular hexagon, and then continue to divide each arc into two, thus making a regular dodecagon inscribed in the circle. Isn't the circumference of this regular dodecagon closer to the circumference than that of a regular hexagon? If the circumference is further divided into circles inscribed with a regular quadrangle, then the circumference of a regular quadrangle must be closer to the circumference than that of a regular dodecagon. This shows that the finer the circumference is divided, the smaller the error is, and the closer the circumference of inscribed regular polygon is to the circumference. This continuous division continues until the circumference can no longer be divided, that is, when the number of sides inscribed with a regular polygon in a circle is infinite, its circumference is "combined" with the circumference, which is exactly the same.
He first cuts a circle from the hexagon inscribed in the circle, and every time the number of sides is doubled, he calculates the area of 192 polygon, π= 157/50=3. 14, and then calculates the area of 3072 polygon, π = 3927/1. This result was the most accurate data for calculating pi in the world at that time. Liu Hui has great confidence in this new method of secant circle, and has extended it to all aspects of circle calculation, thus greatly promoting the development of mathematics since the Han Dynasty. Later, in the Northern and Southern Dynasties, Zu Chongzhi continued to work hard on the basis of Liu Hui, and finally made Pi accurate to the seventh place after the decimal point, which was more than 1 100 years earlier than the west. History will never forget the great contribution of Liu Hui's new method of secant circle to the development of ancient mathematics in China.
Liu Hui also added the tenth chapter to the Notes on Nine Chapters of Arithmetic, which was published separately in the Tang Dynasty, and later changed its name to Archipelago Arithmetic. It was pointed out that it was this masterpiece that brought China's surveying to its peak, which was earlier than Europe's 1400 years.
This book has nine topics, which mainly solve the problems of height, depth and breadth. Liu Hui developed the ancient "gravity difference technique", that is, repeatedly observing from different positions with a ruler, taking the difference and calculating the height or depth of the mountain. For example, the first title of the book "Island Calculation" is to find the height of the island: today there is an island with two tables, three feet high and a thousand steps in front and back, so that the back table and the front table are in a line. Walking 123 steps from the front table, the man looked at the ground and took Wang Daofeng to meet the bottom table. Walking 127 steps from the back table, the man looked at the ground and walked Wang Daofeng, also touching the end of the table. Ask about the height of the island and the geometry of the table.
It means that if we want to measure an island, we should set up two 3-foot rulers to measure it. The distance between front and back is 1000 step, and the front and back rulers are all in a straight line. We walk back from the former ruler 123 steps, and people's eyes just look at the top of the island through the end of the ruler, and observers' eyes just walk back from the back ruler 127 steps. How far is this island from the previous scale? In fact, this problem is a triangle-like application problem that we have learned in junior high school mathematics now. The solution is relatively simple, so I won't do it here.
Liu Hui has made such great achievements in mathematics for the following reasons:
First of all, Liu Hui is a picky person. Liu Hui will learn from his predecessors when he studies mathematics, but he will not be superstitious about their conclusions. He criticized the idea of sticking to the rules and pointed out: "Scholars learn from the past and from mistakes." It is this critical spirit that supports Liu Hui's in-depth study of Nine Chapters Arithmetic, and on this basis, he wrote Nine Chapters Arithmetic Immortal Notes.
Secondly, Liu Hui is a person who is good at discovering the essence of problems. Liu Hui classified 264 problems in Nine Chapters of Nine Chapters of Arithmetic according to his own ideas, and gave his own solutions, such as: he solved geometric problems with the method of in-and-out complement, solved various measurement problems with the method of multiple difference, and solved proportional problems with modern techniques ... so that "birds of a feather flock together, and each has its own home."
Finally, Liu Hui is a man who is good at using tools. Faced with boring and empty math problems, Liu Hui is good at solving practical problems with graphics. Whether it is the previous secant technique, the chess measuring method recorded in Nine Chapters Arithmetic Notes (that is, the three-dimensional geometric model method), or the coloring of various geometric figures, these are Liu Hui's clever use of tools, transforming abstraction into intuitive performance.
Liu Hui's life is a life of hard exploration of mathematics. Although his position is low, his personality is noble. He is not a mediocre man who seeks fame and fame, but a great man who never tires of learning. He left us a valuable fortune. Because of his outstanding contribution in the history of mathematics, some people call him "Newton in the history of Chinese mathematics".