1. Euclid (Euclide of Alexander), Greek mathematician. Born around 330 BC and died around 260 BC. Euclid was one of the most famous and influential mathematicians in ancient Greece. He was a member of the Alexandrian school. Euclid wrote the Elements, a 13-volume book. This work had a huge impact on the future development of geometry, mathematics, and science, as well as on the entire way of thinking in the West. The main object of "The Elements" is geometry, but other topics are also covered, such as number theory and the theory of irrational numbers. Euclid used an axiomatic approach. Axioms (postulates) are certain basic propositions that do not require proof. All theorems are derived from axioms. In this kind of deductive reasoning, every proof must be predicated on an axiom or a proven theorem. This approach later became the paradigm for building any body of knowledge, and for nearly 2,000 years was held up as a model of rigorous thinking that must be followed. "Elements of Geometry" is the pinnacle of the development of ancient Greek mathematics. Euclid (about 300-?) is famous for his "Elements" (referred to as "Principia"). Little is known about his life. He may have studied in Athens in his early years and was very familiar with Plato's theory. Around 300 BC, at the invitation of King Ptolemy (364-283 BC), he came to Alexandria and worked there for a long time. He was a gentle and generous educator with a passion for mathematics. Interested people are always kind. However, he opposed the industrious and opportunistic style and the narrow pragmatism. According to Proclus (c. 410-485), King Ptolemy once asked. Euclid, besides his Principia, is there any other shortcut to learning geometry? Euclid replied, "In geometry there are no roads paved for kings." This sentence became popular. Axioms of learning through the ages. Stobaeus (c. 500) also tells the story of a student who was beginning to learn his first proposition and asked Euclid what he could gain by studying geometry. He was told to give him three coins because he wanted to make real profits from learning. Euclid organized the rich results accumulated in Greek geometry since the 7th century BC into a rigorous logical system, making geometry an independent discipline. Deductive science. In addition to "Elements", he also wrote many other works. Unfortunately, most of them have been lost. "Elements" is his only surviving pure geometric work in Greek. The book is similar in style to the first six volumes of the Principia and consists of 94 propositions stating that if certain elements of a figure are known, other elements can also be determined. The Partition of Figures, now available in Latin and Arabic, discusses. Dividing a Known Figure into Equal or Proportional Parts by Straight Lines "Optics" is one of the early works on geometric optics, which explores the problem of perspective, describes the angle of incidence of light as equal to the angle of reflection, and demonstrates that vision is the arrival of light rays from the eye. The result of objects. Several other works have not yet been attributed to Euclid and have been lost. Euclid's Elements contains 23 definitions, 5 axioms and 5 postulates, from which 48 propositions are derived (Volume 1). ). Liu Hui (born around 250 AD), a native of Wei in the late Three Kingdoms, was an outstanding mathematician in ancient China and one of the founders of classical Chinese mathematical theory. His birth and death dates and life are rarely found in historical records. He was a native of Linzi or Zichuan, Shandong during the Wei and Jin Dynasties. He never became an official. Very few of Liu Hui's mathematical works were handed down to the public. His main works include "Nine Chapters on Arithmetic". "Notes" has 10 volumes; "Chongcha" has 1 volume, which was renamed "Haidao Suanjing" in the Tang Dynasty; "Nine Chapters of Chongcha" has 1 volume, but unfortunately the latter two volumes were lost in the Song Dynasty. Mathematics Achievements Liu Hui's mathematical achievements are roughly the same. There are two aspects: First, it cleans up the ancient Chinese mathematics system and lays its theoretical foundation. This aspect is concentrated in "Nine Chapters on Arithmetic".
A relatively complete theoretical system has been formed: ① In the number system theory, the similarities and differences of numbers are used to discuss general divisions, reductions, four arithmetic operations, and the simplified algorithm of traditional fractions; in the notes on the square root, the meaning of the square root is different. Finally, he discussed the irrational existence of roots, introduced new numbers, and created an irrational method for decimal fractions to infinitely approximate the roots. He first gave a clear definition of "rate" in the theory of formula arithmetic, and based on the three basic operations of multiplication, communion, and odd and even, he established a theoretical basis for the unification of numbers and formula operations, and used "rate" Defines the "equation" in ancient Chinese mathematics, which is the linear equation system in modern mathematics. He also used "rate" to define the "equation" in ancient Chinese mathematics, which is the extended matrix of the linear equation system in modern mathematics. (3) Regarding the Pythagorean Theorem, he demonstrated the calculation principle and solution of the Pythagorean Theorem one by one through the analysis of typical graphics such as "Hook in the Horizontal" and "Hook in the Straight", and established the Pythagorean Theorem. The stock similarity theory developed the Pythagorean metric and formed a similarity theory with Chinese characteristics. Through the analysis of typical graphics such as "Hook in the Horizontal" and "Stock in the Straight", a similarity theory with Chinese characteristics was formed. (4) In terms of area and volume theory, Liu Hui solved the calculation problems of area and volume of various geometric figures and geometric bodies by using the principle of complementation, substituting excess for emptiness, and the limit method of "cutting a circle". The theoretical value of these aspects is still shining brightly today. Secondly, on the basis of inheritance, he put forward his own creative insights. It is mainly reflected in the following aspects, which are representative: ① Cut circles and pi. In "Nine Chapters of Mathematics", he used "circle field technique" to describe the calculation of the area and volume of various geometric figures. He used the method of cutting circles to prove the precise formula of the area of ??a circle, and gave a scientific method for calculating the circumference of a circle in the notes of "Nine Chapters of Arithmetic-Circle Art". He first inscribed the circle from the hexagon, doubling the number of sides each time, calculating the area of ??192 sides, and obtained π=157/50=3.14, and then calculated the area of ??3072 sides, and obtained π=3927/1250=3.1416. , called "return rate". Liu Hui's principle is annotated in "Nine Chapters of Arithmetic? Yang Ma Ji". He used the infinite division method to solve the volume of a cone. Liu Hui's principle is used to calculate the volume of a polyhedron. (3) "Nine Chapters of Arithmetic" says "Mou He Fang Gai". In the notes of "Nine Chapters on Arithmetic", he pointed out the inaccuracy of the sphere volume formula V=9D3/16 (D is the diameter of the sphere), and proposed the famous geometric model "Muhe Square Cover". Mouhekangai "refers to the mutually perpendicular intersection of the two axes of a square tangent cylinder. ④ "Nine Chapters of Arithmetic" describes the new art of equations. He proposed a new method of solving linear equations using the idea of ??ratio algorithm. Bai Juyi in "Nine Chapters of Arithmetic" The "Double Difference Technique" was proposed in "The Book of Sea Island Algorithms" and he used height measurement and distance measurement methods such as heavy tables, connected cables, and product moments. He also used the method of "analogy" to develop the "Double Difference Technique" from two measurements. to "three measurements" and "four measurements." The contribution and status of Liu Hui's works not only had a profound impact on the development of ancient Chinese mathematics, but also in India in the 7th century and in Europe in the 15th and 16th centuries. , and also established a lofty historical position in the history of world mathematics. In view of Liu Hui's great contribution, many books call him "Fermat Fermat (1601-1665)", Pierre de - Fermat, French mathematician, was born on August 17, 1601 in Beaumont-de-Lomagne, near Toulouse, southern France. His father, Dominique Fermat, owned a large leather shop there and had a considerable fortune. The inheritance allowed Fermat to grow up in a wealthy and comfortable environment. Fermat's father was highly respected for his wealth and business skills and received the title of local affairs advisor, but Fermat did not grow up because of his wealthy family. How much sense of superiority. Fermat's mother was named Clair de Rogge, who was born into a noble family. Dominic's huge wealth and Rogge's aristocratic status made Fermat extremely wealthy. He received a good education and developed a wide range of interests and hobbies, which had an important impact on his character. Before the age of 14, Fermat attended the public school of Beaumont-de-Lomagne and graduated in Orleans. He studied law at the University and the University of Toulouse. In 17th-century France, the most demanding profession for men was that of a lawyer, so it became fashionable for men to study law and was envied by others.
Interestingly, France has created good conditions for those "quasi-lawyers" who are promising but lack qualifications to become lawyers as soon as possible. In 1523, Francois I organized the establishment of an agency dedicated to selling official positions and sold official positions publicly. Once this social phenomenon of selling one's official position and getting one's title arose, it got out of hand in response to the needs of the times and has been passed down to this day. On the one hand, selling official positions and conferring titles caters to the needs of those wealthy people, allowing them to obtain official positions and thus improve their social status; on the other hand, selling official posts and conferring titles also improves the government's financial situation. Thus, by the 17th century, any official position other than that of a court or military official could be bought and sold. To this day, positions such as court clerks, notaries and correspondents have not completely escaped the nature of buying and selling. The professionalization of French office-buying benefited many middle-class people, and Fermat was no exception. Before Fermat graduated from university, he bought the positions of "lawyer" and "senator" in Beaumont-de-Lomagne. After graduating, Fermat returned to his hometown and easily became a member of the Toulouse Parliament in 1631. Although Fermat did not lose his position until his death and was promoted every year, according to records, Fermat did not make any great achievements in politics, and his ability to deal with officialdom was also average, not to mention his leadership skills. However, Fermat's promotion was not interrupted. Fermat was recommended to the Supreme Court in 1642 by an authority named Brisas, of whom he was an adviser. In 1646 he was elevated to the rank of Chief Speaker of Parliament and later served as president of the Catholic League, among other posts. Fermat's official career did not have any outstanding political achievements, but Fermat never used his position to extort money, never accepted bribes, was generous and aboveboard, and won people's trust and praise. Fermat's marriage brought Fermat into the ranks of the noble family. Fermat married his maternal uncle's cousin, Lois de Ruge. Fermat was already proud of his mother's aristocratic lineage, and now he only needed to add the aristocratic surname "de" to his name. Fermat had three daughters and two sons. Except for his eldest daughter, Clarisse, who was married, the other four children made Fermat even more proud. Two of his daughters became priests, and his second son became Archbishop of Fimares. Especially the eldest son Clement Samour, who not only inherited Fermat's public office and became a lawyer in 1665, but also compiled Fermat's mathematical papers. If Fermat's eldest son had not actively published Fermat's mathematical papers, it is difficult to say that Fermat would have had such a significant impact on mathematics. In this sense, Samuel is also the heir to Fermat's career. For Fermat, the real career was academics, especially mathematics. Fermat was fluent in French, Italian, Spanish, Latin and Greek and was extremely knowledgeable. Linguistic erudition provided Fermat with linguistic tools and convenience for his mathematical research, enabling him to learn and understand Arabic, Italian algebra, and ancient Greek mathematics. It is these that may have laid a good foundation for Fermat's mathematical attainments. In terms of mathematics, Fermat could not only roam freely in the kingdom of mathematics, but also stand outside the mathematical world and have a bird's eye view of the mathematical world. This cannot be attributed absolutely to his mathematical talent, but has something to do with his erudition. Fermat was introverted by nature, modest and reticent, and was not good at promoting himself or showing himself off. Therefore, he rarely published his papers, not even a complete work. He published a few articles, but always anonymously. After Fermat's death, his eldest son compiled his notes, annotations, and letters into a book and published "Mathematical Essays." We have now long recognized the importance of temporality to science, and the issue was prominent even in the seventeenth century. Fermat's mathematical discoveries were not published in time to be disseminated and developed. This was not entirely a loss of personal reputation, but affected the pace of mathematical progress of that era. Fermat remained in good health throughout his life, but nearly died from the plague of 1652. Just after New Year's Day, 1665, Fermat began to feel changes in his health and was suspended from work on January 10. On the third day, Fermat died. Fermat was buried in Castres Cemetery and later reburied in the family cemetery in Toulouse. Fermat never received any special education in mathematics, and his mathematical research was just a hobby. However, in 17th-century France, no mathematician could compare with him: he was one of the inventors of analytic geometry, second only to Newton and Leibniz in the birth of calculus, and the main originator of probability theory Man and the sole author of number theory in the 17th century.
Fermat also made important contributions to physics. Fermat was the greatest French mathematician of the 17th century, and the beginning of the 17th century heralded a brilliant future for mathematics. In fact, this century was one of the most glorious eras in the history of mathematics. Geometry first became the crown jewel of this era, because the application of algebraic methods in geometry directly led to the birth of analytic geometry; projective geometry opened up new fields as a new method; the infinitesimal division method emerged from the ancient product problem The introduction of geometry led to new directions in geometric research and ultimately led to the invention of calculus. The renaissance of geometry was the result of a generation of thoughtful and creative mathematicians, one of whom was Fermat. Contribution to analytic geometry Fermat discovered the basic principles of analytic geometry independently of Descartes. Before 1629, he had begun to rewrite the lost book on plane trajectories by the ancient Greek geometer Apollonius from the third century BC. . He used algebraic methods to supplement some of Apollonius's lost proofs about trajectories, summarized and organized ancient Greek geometry, especially Apollonius' theory of conic sections, and conducted general research on curves. In 1630, he wrote an 8-page treatise in Latin called "Introduction to Loci in Planes and Solids." In 1636, Fermat began to correspond with Mersenne and Robert Wahl, the great mathematicians of the time, in which he talked about some of his mathematical work. However, "Introduction to Plane and Solid Trajectories" was not published until 14 years after Fermat's death, so few people knew about Fermat's now-seemingly seminal work before 1679. Fermat's discoveries are presented in Introduction to Trajectories in Planes and Solids. "Two unknown quantities determine an equation, which corresponds to a trajectory, which can be drawn as a straight line or a curve," he said. Fermat's discovery preceded Descartes' discovery of the basic principles of analytic geometry by seven years. Fermat also discussed the general equations of straight lines and circles, as well as hyperbolas, ellipses, and parabolas. Descartes found his equations by observing trajectories, while Fermat studied trajectories from equations. These are two opposite aspects of the basic principles of analytic geometry. In a letter from 1643, Fermat also discussed his ideas in analytic geometry. He talked about cylindrical surfaces, elliptical paraboloids, hyperboloids with two lobes and elliptical surfaces. He pointed out that an equation containing three unknown quantities represents a curved surface, and carried out further research on this. Contributions to Calculus In the 16th and 17th centuries, calculus was the most dazzling jewel after analytic geometry. As we all know, Newton and Leibniz were the creators of calculus. Before them, at least dozens of scientists performed pioneering work and ultimately invented calculus. But among the many pioneers, Fermat is still worth mentioning, mainly because he provided the inspiration closest to the modern form for the introduction of the concept of calculus, so that in the field of calculus, following Newton and Leibniz, Fermat The addition of Ma as the originator will also be recognized by the mathematics community. The problem of tangents to curves and the problems of large and small values ??of functions are one of the origins of calculus. This work has a long history, dating back to ancient Greece. Archimedes once used the method of exhaustion to find the area of ??any figure enclosed by a curve. Due to its cumbersome nature, the method of exhaustion fell into oblivion and was not taken seriously again until the 16th century. When Kepler was exploring the laws of planetary motion, he encountered the problem of how to determine the area and arc length of an ellipse, so he introduced the concepts of infinity and infinitesimal to replace the cumbersome exhaustive method. Although this method is not perfect, it has opened up a very broad thinking space for mathematicians from Cavalieri to Fermat. Fermat made significant contributions to calculus, establishing tangent lines, methods of finding large and small values, and methods of definite integrals. Contribution to probability theory As early as the ancient Greek period, the issue of contingency, inevitability and their relationship has aroused the interest and debate of many philosophers, but its mathematical description and treatment only happened after the 15th century. At the beginning of the 16th century, Italian mathematicians such as Cardano appeared to explore the division of bets in point games while studying the contingency in dice games. In the 17th century, the French Pascal and Fermat studied the Italian Pacioli's "Abstract" and established a correspondence, thus laying the foundation for probability theory. Fermat considered four gambles with 2 x 2 x 2 x 2 = 16 possible outcomes. The first gambler won in all but one of the outcomes, i.e. the opponent won in all four gambles. .
Fermat did not use the word probability at this time, but he did arrive at the probability that the first gambler would win, which is 15/16, which is the ratio of the number of favorable situations to the number of all possible situations. This condition is generally satisfied in combinatorial problems such as card games, throwing silver coins, and loading balls from a bottle. In fact, this research laid the foundation for the abstraction of mathematical models of probability - probability spaces - in games, although Kolmogorov did not make this summary until 1933. Fermat and Pascal established the basic principle of probability theory - the concept of mathematical expectation - in their correspondence and writings. It starts with the mathematical problem of points: how to determine the distribution of bets on an interrupted game between players who are assumed to be equally skilled, knowing the scores of both players at the time of the interruption and the number of points required to win the game. Fermat discussed it like this: Player A needs 4 points to win, and player B needs 3 points to win. This is Fermat's solution to this special situation. Obviously, up to four points can decide the outcome. The concept of a general probability space is a thorough axiomaticization of people's intuitive ideas about this concept. From a purely mathematical perspective, finite probability spaces seem banal. But once random variables and mathematical expectations are introduced, they become a magical world. This is where Fermat's contribution lies. Contribution to Number Theory In the early 17th century, the book "Arithmetic" written by Dioptus, an ancient Greek mathematician in the 3rd century AD, was circulated in Europe. In 1621, Fermat bought this book in Paris, and he used his spare time to study the indefinite equations in the book in depth. Fermat limited the study of indefinite equations to the range of integers, thus creating the branch of mathematics known as number theory. Fermat's achievements in the field of number theory are huge, the main ones are (1) All prime numbers can be divided into two forms, namely 4n+1 and 4n+3. (2) A prime number of the form 4n+1 can be, and can only be, expressed in one way as the sum of two square numbers. (3) No prime number of the form 4n+3 can be expressed as the sum of two square numbers. (4) A prime number of the form 4n+1 can and can only be expressed as the hypotenuse of a right triangle, and the right side of the right triangle is an integer; the square of 4n+1 can and can only be expressed as 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 rational sides cannot be a square number. (6) A prime number with shape 4n+1 and its square number can only be expressed as the sum of two square numbers in one way; its third power and fourth power can only be expressed as two square numbers in two ways. its 5th and 6th powers can only be expressed in three ways as the sum of two square numbers, and so on, until infinity. Contribution to optics Fermat's outstanding contribution to optics was the proposal of the principle of minimum action, also called the principle of shortest time action. This principle has been proposed for a long time. As early as the ancient Greek period, Euclid proposed the law of linear propagation of light and the law of reflection. Later, Helen revealed the theoretical essence of these two laws - light takes the shortest path. Over time, this law expanded into a natural law and then into a philosophical concept. -Eventually, people came to the more general conclusion that "nature moves by the shortest path", which had an impact on Fermat. Fermat's brilliance lay in transforming this philosophical concept into a scientific theory. Fermat also discussed the propagation of light in a medium that changes point by point. The path of light is a very small curve. And explained some problems using the principle of least action. This gave many mathematicians great encouragement. Euler, in particular, raced against time to use the variational technique of calculus and use this principle to find the extreme value of a function. This led directly to Lagrange's achievements, which gave a specific form of the principle of least action: for a mass, the integral of the product of its mass, velocity and the distance between two fixed points is an extremum and an extremum; That is, for the path the mass actually takes, it must be an extremum or extremum.