1. Greek mathematician Euclid. Born about 330 BC, he died in 260 BC.
Euclid was one of the most famous and influential mathematicians in ancient Greece. He is a member of the Alexandria school. Euclid wrote a book called "The Original" * * *, 13. This work has a great influence on the future development of geometry, mathematics and science, and on the whole thinking method of westerners. The main object of Geometry Elements is geometry, but it also involves other topics such as number theory and irrational number theory. Euclid used the axiomatic method. Axioms are some basic propositions that do not need to be proved, and all theorems are derived from them. In this deductive reasoning, every proof must be based on axioms or theorems that have been proved. This method later became a model of establishing any knowledge system, and for almost 2000 years, it was regarded as a model of rigorous thinking that must be followed. The Elements of Geometry is the pinnacle of the development of ancient Greek mathematics.
Euclid (active in about 300-? )
Ancient Greek mathematician. He is famous for his 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, "In geometry, there is no paved road 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.
Euclid arranged the rich achievements accumulated in Greek geometry since the 7th century BC in a strict logical system, making geometry an independent and deductive science. Besides the Elements of Geometry, he has many other works, but most of them have been lost. Known Numbers is the only Greek work except the original among his pure geometry works. Its style is similar to the first six volumes of the original work, and it contains 94 propositions. It has been pointed out that if some elements in a graph are known, others can be determined. Graphics are divided into existing Latin texts and Arabic texts. This paper discusses dividing known figures into equal parts or equal parts by straight lines. Optics is one of the early works of geometric optics. It studies perspective, stating that the incident angle of light is equal to the reflection angle, and that vision is the result of light reaching the object from the eyes. There are still some works that are not sure whether they belong to Euclid or not, and have been lost.
Euclid's Elements of Geometry contains 23 definitions, 5 axioms and 5 postulates, from which 48 propositions are derived (Volume I).
2. Liu Hui's life
(Born around 250 AD), wei ren was an outstanding mathematician in ancient China and one of the founders of China's classical mathematical theory in the late Three Kingdoms period. History books rarely record his birth, death and life story. According to limited historical data, he was born in Linzi or Zichuan, Shandong Province in the Wei and Jin Dynasties. Never been an official.
work
Liu Hui's mathematical works are rarely handed down to later generations, and all of them have been copied over and over again. His main works are:
Nine arithmetic notes (10);
The weight difference (1) was renamed as island calculation in the Tang Dynasty.
"Nine Chapters Heavy Difference Map" L volume, but unfortunately the last two were lost in the Song Dynasty.
Mathematical achievement
Liu Hui's mathematical achievements are roughly in two aspects:
First, clarify the ancient mathematical system of China and lay its theoretical foundation. This aspect is embodied in Nine Chapters of Arithmetic Notes. It has actually formed a relatively complete theoretical system:
(1) In number system theory
This paper expounds the general division, simplification, four operations and simplification rules of complex fractions with the same sign and different sign. In the annotation of prescription, he discussed the existence of irrational roots from the infinite meaning of prescription, introduced new numbers, and created a method of infinitely approaching irrational roots with decimals.
(2) In convex calculus theory.
First of all, he gave a clear definition of rate, and based on three basic operations, such as multiplication and division, he established a unified theoretical basis for the operation of numbers and formulas. He also defined the "equation" in China's ancient mathematics by rate, that is, the augmented matrix of linear equations in modern mathematics.
③ In Pythagoras' theory.
The pythagorean theorem and the calculation principle of solving pythagorean form are demonstrated one by one, the theory of similar pythagorean form is established, and pythagorean measure is developed. Through the analysis of typical figures such as "crossing in the hook" and "straight in the stock", a similar theory with China characteristics was formed.
④ In the area and volume theory.
Liu Hui's principle is put forward by using the principle of complement, the deficiency of complement and the limit method of "cyclotomy", which solves the problem of calculating the area and volume of various geometric shapes and geometries. The theoretical value of these aspects is still shining.
Second, on the basis of inheritance, put forward your own ideas. This aspect is mainly reflected in the following representative innovations:
① Circumcision and Pi
He is in "Nine Chapters Arithmetic? In the annotation of roundness field, the exact formula of circle area is proved by secant technique, and the scientific method of calculating pi is given. 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.
② Liu Hui principle
Chapter 9 Arithmetic? Yang Equestrian Notes, when he solved the volume of cone by infinite division, he put forward Liu Hui's principle of calculating the volume of polyhedron.
③ the theory of "concerted housing reform"
Chapter 9 Arithmetic? He pointed out the inaccuracy of the formula V=9D3/ 16(D is the diameter of the ball) and introduced the famous geometric model "Mouhe Square Cover". "Mouhe Square Cover" refers to the intersection of inscribed cylinders with two perpendicular axes.
④ New technology of equation
Chapter 9 Arithmetic? Equation ",he put forward a new method to understand linear equations, using the idea of ratio algorithm.
⑤ gravity difference operation.
In his white paper "Island Calculations", he put forward the complex difference technique, which used complex tables, continuous cables, cumulative moments and other methods to measure the height and distance. He also developed gravity difference technology from two observations to three observations and four observations by analogy. In the 7th century, India and Europe only began to study the problem of two observations in15 ~16th century.
Contribution and status
Liu Hui's work not only had a far-reaching impact on the development of ancient mathematics in China, but also established a lofty historical position in the world. In view of Liu Hui's great contribution, many books call him "Newton in the history of Chinese mathematics".
Fermat
Fermat (160 1 ~ 1665)
pierre de fermat
Fermat, a French mathematician, 160 17 was born in Beaumont de Lomagne near Toulouse in southern France in August. His father Dominic Fermat opened a large leather goods store in the local area, and the industry was very rich, which made Fermat live in a rich and comfortable environment since he was a child.
Fermat's father was respected by people because he was rich and well-run, so he was awarded the title of local affairs consultant. However, Fermat did not feel much superiority because of his rich family when he was young. Fermat's mother's name is Clara de Rogge, and she is a nobleman in a robe. Dominic's great wealth and Rogge's big noble constitute Fermat's extremely rich social status.
Fermat was taught by his uncle Pierre when he was a child, and received a good enlightenment education, which cultivated his extensive interests and hobbies and also had an important influence on his character. It was not until 14 years old that Fermat entered Beaumont de Lomagne College. After graduation, he studied law at the University of Orleans and the University of Toulouse.
In France in the17th century, men's most exquisite occupation was to be lawyers, so it became a fashion and admirable for men to study law. Interestingly, France has created good conditions for those "quasi-lawyers" who are productive and lack qualifications to become lawyers as soon as possible. 1523, Francois I organized and established a special organ to sell officials and titles, and publicly sold officials and titles. Once this social phenomenon of selling officials and titles came into being, in order to meet the needs of the times, it was out of control and continued to this day.
On the one hand, selling official titles caters to the rich, allowing them to obtain official positions and improve their social status, on the other hand, it also improves the government's financial situation. Therefore, in the17th century, except court officials and military attache, any official position can be bought and sold. Until today, the duties of court clerks, notaries, messengers, etc. I haven't completely got rid of the nature of buying and selling. France's expertise in buying officials has benefited many middle classes, and Fermat is no exception. Before graduating from college, Fermat bought the positions of "lawyer" and "senator" in Beaumont de Lomagne. Fermat returned to his hometown after graduation and easily became a member of Toulouse Parliament for a term of 163 1 year.
Although Fermat didn't lose his official position from entering the society to his death, and he was promoted year by year, according to records, Fermat didn't have any achievements, and his coping ability in officialdom was very general, let alone his leadership ability. However, Fermat did not interrupt his promotion. After serving as a member of the local Council for seven years, Fermat was promoted to an investigative senator and had the right to investigate and question the administrative authorities.
1642 There was an authoritative person named Boris who was an adviser to the Supreme Court. Boris recommended Fermat to the Supreme Criminal Court and the main court of Dali Palace in France, which gave Fermat a better chance of promotion in the future. 1646, Fermat was promoted as the chief speaker of the parliament and later served as the chairman of the Catholic Union. Fermat's official career has nothing outstanding to commend, but Fermat has never used his power to extort money from people, never accepted bribes, and has won people's trust and praise.
Fermat's marriage made Fermat rank among the noblesse de robe, and Fermat married his cousin Louise de Rogge. Fermat is proud of his mother's noble blood, and now he just adds the symbol "Germany" before his name.
Fermat has three daughters and two men. Except for Clara, the eldest daughter, all four children made Fermat feel respectable. Two daughters became priests, and the second became an assistant bishop of Fermares. Especially the eldest son, Clement samore, who not only inherited Fermat's public office and became a lawyer in 1665, but also compiled Fermat's mathematical works. If Fermat's eldest son hadn't actively published Fermat's mathematical works, it would be hard to say that Fermat could have had such a great influence on mathematics, because most of the papers were published after Fermat's eldest son died. In this sense, Samuel can also be called the heir of Fermat's career.
For Fermat, the real career is academic, especially mathematics. Fermat is familiar with French, Italian, Spanish, Latin and Greek, and he also has a lot of research. The erudition of language provided Fermat with language tools and convenience for his mathematical research, enabling him to learn and understand Arabic and Italian algebra and ancient Greek mathematics. Perhaps it is these that laid a good foundation for Fermat's attainments in mathematics. In mathematics, Fermat can not only roam freely in the kingdom of mathematics, but also stand outside the world of mathematics and have a bird's eye view of mathematics. This can not be absolutely attributed to his mathematical talent, but also to his erudition.
Fermat is introverted, modest and quiet, and is not good at selling and showing himself. Therefore, he rarely published his own works or even published a complete book. Some of his articles are always anonymous. After Fermat's death, his eldest son compiled his notes, notes and letters into a book and published a Mathematical Paper. We have long recognized the importance of timeliness to science, and even in the17th century, this problem is very prominent. Fermat's mathematical research results were not published in time and could not be disseminated and developed. It was not entirely a loss of personal reputation, but affected the pace of mathematical progress in that era.
Fermat was healthy all his life, but almost died of the plague of 1652. 1665 after the new year's day, Fermat began to feel his body changed, so he stopped playing on 65438 10/0. On the third day, Fermat died. Fermat was buried in Casterly Cemetery and later in the family cemetery in Toulouse.
Fermat has never received a special mathematics education in his life, and mathematics research is just a hobby. However, in France in the17th century, no mathematician can match it: he was one of the inventors of analytic geometry; The contribution to the birth of calculus is second only to Newton, Leibniz, the main founder of probability theory, and the person who inherited the world of number theory in17th century. In addition, Fermat also made important contributions to physics. Fermat was the greatest mathematician in France in the17th century.
At the beginning of17th century, it predicted a rather spectacular mathematical prospect. In fact, this century is also a glorious era in the history of mathematics. Geometry first became the most attractive pearl of this era, and the application of algebraic method, a new geometric method, directly led to the birth of analytic geometry. As a brand-new method, projective geometry has opened up a new field. Infinitesimal division caused by the ancient quadrature problem was introduced into geometry, which led to a new research direction of geometry and finally promoted the invention of calculus. The revival of geometry is inseparable from a generation of mathematicians who are diligent in thinking and brave in creation, and Fermat is one of them.
Contribution to analytic geometry
Fermat discovered the basic principle of analytic geometry independently of Descartes.
Before 1629, Fermat began to rewrite the book Plane Trajectory, which was lost by the ancient Greek geometer Apollonius in the third century BC. He supplemented some lost proofs of Apollonius trajectory with algebraic methods, summarized and sorted out ancient Greek geometry, especially Apollonius' conic curve theory, and made a general study of curves. 1630, he wrote an 8-page paper "Introduction to Plane and Solid Trajectory" in Latin.
Fermat began to correspond with the great mathematicians Mei Sen and Robewal at that time in 1636, and talked about his own mathematical work. However, Introduction to Plane and Solid Trajectory was published after Fermat's death 14 years ago, so few people knew Fermat's work before 1679, but now it seems that Fermat's work is groundbreaking.
Fermat's discovery was revealed in Introduction to Plane and Solid Trajectory. He pointed out: "An equation determined by two unknowns corresponds to a trajectory and can describe a straight line or curve." Fermat's discovery was seven years earlier than Descartes' discovery of the basic principles of analytic geometry. Fermat also discussed the equations of general straight lines and circles, hyperbolas, ellipses and parabolas.
Descartes looks for its equation from the trajectory, while Fermat studies the trajectory from the equation, which are two opposite aspects of the basic principle of analytic geometry.
In a letter from 1643, Fermat also talked about his analytic geometry thought. He talked about cylinder, elliptic paraboloid, hyperboloid and ellipsoid, and pointed out that an equation containing three unknowns represents a surface, and further studied it.
Contribution to calculus
16 and 17 centuries, calculus is the brightest pearl after analytic geometry. As we all know, Newton and Leibniz were the founders of calculus. Before them, at least dozens of scientists did basic work for the invention of calculus. But among many pioneers, Fermat is worth mentioning, mainly because he provided the inspiration closest to the modern form for the derivation of the concept of calculus, so that in the field of calculus, Fermat, as the founder after Newton and Leibniz, will also be recognized by the mathematical community.
The tangent of a curve and the minimum of a function are one of the origins of calculus. This work is relatively old, dating back to ancient Greece. Archimedes used the exhaustive method to find the area of any figure surrounded by curves. Because method of exhaustion was troublesome and clumsy, he was gradually forgotten, and was not taken seriously until16th century. When Kepler explored the laws of planetary motion, he encountered the problem of how to determine the ellipse area and ellipse arc length. The concepts of infinity and infinitesimal are introduced to replace the tedious exhaustive method. Although this method is not perfect, it has opened a very broad thinking space for mathematicians since cavalieri came to Fermat.
Fermat founded tangent method, maximum method, minimum method and definite integral method, which made great contributions to calculus.
Contribution to probability theory
As early as in ancient Greece, the relationship between contingency and inevitability aroused the interest and debate of many philosophers, but it was after15th century to describe and deal with it mathematically. In the early16th century, Italian mathematicians such as cardano studied the game opportunities in dice and explored the division of gambling funds in game points. /kloc-In the 7th century, French Pascal and Fermat studied the abstraction of Italian Pachauri, and established corresponding relations, thus laying the foundation of probability theory.
Fermat considers that there are 2× 2× 2× 2 = 16 possible outcomes for four gambling, except one outcome, that is, the opponent wins all four gambling, and the first gambler wins all other situations. Fermat has not used the word probability yet, but he has come to the conclusion that the probability of the first gambler winning is 15/ 16, that is, the ratio of the number of favorable situations to the number of all possible situations. This condition can generally be met in combination problems, such as card games, throwing silver and modeling balls from jars. This study actually laid a game foundation for the abstraction of probability space, a mathematical model of probability, although this summary was made by Kolmogorov in 1933.
Fermat and Pascal established the basic principle of probability theory-the concept of mathematical expectation in their mutual communication and work. This should start with the mathematical problem of integral: in an interrupted game, how to determine the division of gambling funds between players with the same assumed skills, and how to know the scores of two players when they are interrupted and the scores needed to win the game. Fermat discussed the situation that player A needs 4 points to win and player B needs 3 points to win, which is Fermat's solution to this special situation. Because it can be decided four times at most.
The concept of generalized probability space is a thorough axiomatization of people's intuitive ideas about concepts. From the point of view of pure mathematics, the finite probability space seems unremarkable. But once random variables and mathematical expectations are introduced, it becomes a magical world. This is Fermat's contribution.
Contribution to number theory
/kloc-At the beginning of the 7th century, the book Arithmetic written by Diophantu, an ancient Greek mathematician in the 3rd century A.D., spread in Europe. Ma Fei bought this book in Paris. He studied the indefinite equations in the book in his spare time. Fermat limited the study of indefinite equations to the range of integers, thus creating a mathematical branch of number theory.
Fermat's achievements in the field of number theory are enormous, including:
(1) All prime numbers can be divided into 4n+ 1 and 4n+3.
(2) The prime number in the form of 4n+1can and can only be expressed as the sum of two squares in one direction.
(3) No prime number in the form of 4n+3 can be expressed as the sum of two squares.
(4) The prime number in the form of 4n+1can and can only be used as the hypotenuse of a right triangle with integer right angles; The square of 4n+ 1 is and can only be the hypotenuse of two such right triangles; Similarly, the m power of 4n+ 1 is and can only be the hypotenuse of m such right triangles.
(5) The area of a right triangle with a rational number side length cannot be a square number.
(6) The prime number of 4n+1and its square can only be expressed as the sum of two squares in one direction; Its cubic and fourth power can only be expressed as the sum of two squares in two ways; The 5th power and 6th power can only be expressed as the sum of two squares in three ways, and so on until infinity.
Contribution to optics
Fermat's outstanding contribution to optics is that he put forward principle of least action, also called the principle of shortest time action. This principle has a long history. As early as ancient Greece, Euclid put forward the laws of linear propagation and phase reflection of light. Later, Helen revealed the theoretical essence of these two laws-light takes the shortest path. After several years, this law was gradually extended to natural law, and then became a philosophical concept. In the end, a more general conclusion was drawn that "nature works in the shortest possible way" and influenced Fermat. Fermat's genius lies in turning this philosophical concept into a scientific theory.
Fermat also discussed the situation that the path of light takes the minimum curve when it propagates in a point-by-point changing medium. Some problems are explained by principle of least action. This has greatly inspired many mathematicians. Euler, in particular, used this principle to find the extreme value of a function by variational method. This leads directly to Lagrange's achievement and gives the concrete form of principle of least action: for a particle, the integral of the product of its mass, velocity and the distance between two fixed points is a maximum and a minimum; That is to say, for the actual path taken by particles, it must be the maximum or minimum.