1. Zu Chongzhi and Pi

Zu Chongzhi was not only proficient in astronomical calendars, but also made great contributions to mathematics, especially his outstanding achievements in the study of pi, which surpassed the previous generation and made him shine in the history of mathematics in the world.

We all know that pi is the ratio of the circumference of a circle to the diameter of the same circle, and this ratio is a constant, which is now commonly expressed by the Greek letter "π". Pi is an infinite decimal that can never be divided, and it cannot be completely and accurately expressed by fractions, finite decimals or cyclic decimals. Due to the progress of modern mathematics, pi of more than two thousand decimal places has been calculated.

Pi is widely used. Especially in astronomy and calendars, all problems involving circles should be calculated by pi. The earliest value of pi obtained by working people in ancient China in production practice is "3", which is of course inaccurate, but it has been used until the Western Han Dynasty. Later, with the development of astronomy, mathematics and other sciences, more and more people studied pi. At the end of the Western Han Dynasty, Liu Xin first abandoned the inaccurate pi value of "3", and the pi he once adopted was 3.547. Zhang Heng of the Eastern Han Dynasty also calculated pi as **=3. 1622. Of course, these values have made great progress compared with π=3, but they are far from accurate. At the end of the Three Kingdoms, mathematician Liu Hui created the method of secant to find pi, and the research on pi has made great progress.

The method of secant to find pi is roughly as follows: first make a circle, and then make a regular hexagon inscribed in the circle. Suppose the diameter of this circle is 2, then the radius is equal to 1. The inscribed side of a regular hexagon must be equal to the radius, so it is also equal to1; Its circumference is equal to 6. If the circumference 6 inscribed with a regular hexagon is taken as the circumference of a circle and divided by the diameter 2, the ratio of circumference to diameter π=6/2=3, which is the ancient value π=3. However, this value is incorrect. We can clearly see that the perimeter of an inscribed regular hexagon is much smaller than that of a circle.

If we double the number of inscribed sides of a regular hexagon to become an inscribed regular dodecagon, and then find its circumference by an appropriate method, then we can see that this circumference is closer to the circumference of a circle than that of an inscribed regular hexagon, and the area of this inscribed regular dodecagon is closer to the area of a circle. From this, we can draw a conclusion that the more sides in a circle are connected with a regular polygon, the smaller the difference between the total length (circumference) of its sides and the circumference of the circle. Theoretically, if the number of inscribed sides of a regular polygon increases to infinity, the perimeter of the regular polygon will closely coincide with the circumference, and the inscribed area of the infinite regular polygon calculated from this will be equal to the area of the circle. But in fact, it is impossible for us to increase the number of inscribed sides of a regular polygon to infinity, so that the circumference of this infinite regular polygon coincides with the circumference. We can only increase the number of sides inscribed with a regular polygon, so that its perimeter and circumference almost coincide. Therefore, by increasing the number of inscribed regular polygon sides of a circle, the number of pi is always slightly less than the true value of π. According to this principle, Liu Hui begins with a circle inscribed with a regular hexagon, and the number of sides gradually doubles until the inscribed regular hexagon is calculated, and the pi is 3. 14 1O24. Turn this number into a fraction, which is 157/50.

The pi obtained by Liu Hui was later called "Hui rate". His calculation method actually has the concept of limit in modern mathematics. This is a brilliant achievement in the study of ancient pi in China.

Zu Chongzhi has made great achievements in deducing pi. According to "Sui Shu Law and Discipline", Zu Chongzhi changed ten feet into one hundred million feet, so as to find pi. He calculated the result of * * * and got two numbers: one is the abundant remainder (approximate value of surplus), which is 3.1415927; One is the number (that is, the approximate value of the loss), which is 3. 14 15926. The true value of pi is just between these two numbers. "Sui Shu" has only such a simple record, without specifying how he calculated it. However, judging from the mathematical level at that time, there was no better method except Liu Hui's cyclotomy. Zu Chongzhi probably adopted this method. Because of Liu Hui's method, when the number of inscribed sides of a regular polygon of a circle increases to 24,576, Zu Chongzhi's result can be obtained accurately.

The remainder can be listed as inequality, such as: 3. 14 15926 (*) < π (true pi) < 3. 14 15927 (remainder), indicating that pi should be between the remainder. According to the habit of using fractions in calculation at that time, Zu Chongzhi also adopted two fractional values of pi. One is 355/ 1 19 (about equal to 3. 14 15927), which is relatively accurate, so Zu Chongzhi calls it "the secret rate". The other is (about 3. 14), which is rough, so Zu Chongzhi calls it "approximate rate". In Europe, it was not until 1573 that the German mathematician Walter worked out the value of 355/ 1 19. Therefore, Japanese mathematician Mishima suggested that the value of pi of 355/ 1 19 be called "ancestral rate" to commemorate China, a great mathematician.

2. Newton and calculus

Most modern historians believe that Newton and Leibniz independently developed calculus and created their own unique symbols for it. According to people around Newton, Newton came up with his method several years earlier than Leibniz, but he hardly published anything before 1693, and didn't give his complete explanation until 1704. At the same time, Leibniz published a complete description of his method in 1684. In addition, Leibniz's symbol and "differential method" were completely adopted in continental Europe, and this method was also adopted in Britain about 1820 years later. Leibniz's notebook records the development process of his thought from early stage to mature stage, but only Newton's final result is found in the known records. Newton claimed that he had been reluctant to publish his calculus because he was afraid of being laughed at. Newton was closely related to the Swiss mathematician Nicolas Fadio Diu Lei, who was attracted by Newton's law of universal gravitation from the beginning. 169 1 year, Diu Lei intended to compile a new edition of Newton's mathematical principles of natural philosophy, but he never finished it. Some biographers who study Newton think there may be love in their relationship. However, the relationship between them cooled down on 1694. At that time, Diu Lei also exchanged several letters with Leibniz.

At the beginning of 1699, other members of the Royal Society (Newton is one of them) accused Leibniz of plagiarizing Newton's achievements, and the debate broke out in 17 1 1. The Royal Newton Society announced that a survey showed that Newton was the real discoverer and Leibniz was denounced as a liar. However, it was later found that Newton wrote the conclusion of the investigation and comment on Leibniz himself, so the investigation was questioned. This led to a heated debate between Newton and Leibniz about calculus and ruined their lives until the latter died in 17 16. This debate has drawn a gap between British and continental European mathematicians, and may have hindered the development of British mathematics for at least a century.

3. Euclid and Elements of Geometry

Little is known about his life now. I probably studied in Athens in my early years and I know Plato's theory very well. Around 300 BC, he came to Alexandria at the invitation of Ptolemy (364-283 BC) and worked there for a long time. He is a gentle and honest educator. He always persuades people who are interested in mathematics. However, we are opposed to the style of refusing to study hard and being opportunistic, and we are also opposed to narrow and practical views. According to Proclus (about 4 10 ~ 485), King Ptolemy once asked Euclid if there were any other shortcuts to learn geometry besides his Elements. Euclid replied, "Geometry has no king's road." It means that in geometry, there is no road paved for kings. This sentence later became an eternal learning motto. Stobeus (about 500) told another story, saying that a student had just started to learn the first proposition and asked Euclid what he would get after learning geometry. Euclid said: Give him three coins because he wants to get real benefits from his study.

Born in Athens, Euclid was a student of Plato. His scientific activities were mainly carried out in Alexandria, where he established a school of mathematics headed by him.

Euclid is famous for his main work "Elements of Geometry". He systematically sorted out and summarized the mathematical achievements of predecessors, and formed a strict system with strict deductive logic based on some axioms, which is of great significance.

The geometric system established by Euclid was so rigorous and complete that even Einstein, the most outstanding great scientist in the 20th century, could not help but look at him with new eyes.

Einstein said: "When he first came into contact with Euclidean geometry, if he was not moved by its clarity and reliability, then he would not have become a scientist."

Perhaps he didn't create much mathematical content in the Elements of Geometry, but he undoubtedly made contributions to the selection of axioms, the arrangement of theorems and some rigorous proofs. In this respect, his work is excellent.

Euclid's Elements of Geometry has 13 articles, the first of which is definitions and axioms. For example, he first defined the concepts of point, line and surface.

He compiled five axioms, including:

1. It is possible to make a straight line from one point to another arbitrary point;

2. All right angles are equal;

3. if a = b and b=c, then a = c；;

4. if a=b, A+C = B+C and so on.

Another axiom put forward by Euclid himself is that the whole is greater than the parts.

Although this axiom is not as easy to be recognized and accepted as other axioms, it is necessary and indispensable in Euclidean geometry. It just shows his genius that he can bring it up.

Chapter 1 ~ 4 of Elements of Geometry mainly talks about the basic properties of polygons and circles, such as congruent polygon theorem, parallel line theorem, pythagorean chord theorem and so on.

The second article talks about geometric algebra, which replaces numbers with geometric line segments, and solves the contradiction that the Greeks do not recognize irrational numbers, because some irrational numbers can be expressed as graphic methods.

The third chapter discusses the properties of a circle, such as chord, tangent, secant and central angle.

Chapter four discusses the inscribed circle and circumscribed circle of a circle.

The fifth part is the theory of proportion. This article is of great significance to the future history of mathematics development.

The sixth article is about similarity. One of the propositions is that the area of a rectangle on the hypotenuse of a right triangle is equal to the sum of the areas of two similar rectangles on two right angles. Readers may wish to have a try.

Chapters 7, 8 and 9 are number theory, which describes the nature of the ratio of integers to integers.

Article 10 is to classify irrational numbers.

The article 1 1 ~ 13 is about solid geometry.

All 13 articles * * * contain 467 propositions. The appearance of geometric elements shows that human beings have reached a scientific state in geometry and established a scientific logic theory on the basis of experience and intuition.

Euclid, a professor of mathematics at the University of Alexandria, transformed the earth and heaven into a huge pattern composed of intricate graphics.

He also used his amazing clever fingers to disassemble the pattern into simple components: points, lines, angles, surfaces and solids-translating an endless picture into the limited language of elementary mathematics.

Although Euclid simplified his geometry, he insisted on studying the principle of geometry thoroughly so that his students could fully understand it.

It is said that Dorothy, king of Alexandria, learned geometry from Euclid and was impatient with Euclid's explanation of his principles over and over again.

The king asked, "Is there a simpler way to learn geometry than yours?"

Euclid replied, "Your Majesty, there are two roads in the country, one is the hard road for ordinary people, and the other is the royal road. But geometrically, everyone can only go the same way. Learning to be excellent is an official, please understand. "

Euclid's sentence was later popularized as "there is no shortcut to knowledge" and became an eternal proverb.

Due to lack of information, we know little about the details of Euclid's life. There is a story about Euclid's quarrel with his wife, who was very angry.

The wife said, "put away your messy photos." Does it bring you bread and beef? "

Euclid was born with a foolish temper. He just smiled and said, "Do you know what women think? What I write now will be of great value to future generations! "

The wife sneered: "Can we reunite in the afterlife? You bookworm. "

Euclid was about to argue when his wife picked up a part of his Elements and threw it into the stove. Euclid rushed to catch it, but it was too late.

It is said that his wife burned the last and most wonderful chapter of Geometry. But this regret is irreversible. She burned not only some useful books, but also the crystallization of Euclid's sweat and wisdom.

If the above story is true, then Euclid may not have caused his wife's anger. Because ancient writers told us that he was a "gentle and kind old man."

Because of Euclid's profound knowledge, his students almost worshipped him. When Euclid was teaching students, he guided them and cared for them like a real father.

However, sometimes he whips arrogant students with bitter satire to tame them. After learning the first theorem, a student asked, "What are the benefits of learning geometry?"

So Euclid turned to the servant and said, "Grumma, give this gentleman three coins because he wants to get real benefits from his study."

Euclid advocated that learning must be gradual and diligent, not in favor of opportunism and against narrow practical concepts. Papos, a latecomer, especially appreciated his modesty.

Like most scholars in ancient Greece, Euclid did not care much about the "practical" value of his scientific research. He likes studying for the sake of research.

He is shy and humble, aloof from the world and lives quietly in his own home. In that world full of intrigue, people are allowed to perform noisy and vulgar performances.

He said: "these fleeting things will eventually pass, but the patterns of stars are eternal."

In addition to writing an important geometric masterpiece "The Elements of Geometry", Euclid also wrote works such as data, graphic segmentation, wrong conclusions about mathematics, optics and the book "Reflective Optics".