The Pythagorean school of ancient Greece believed that any number in the world could be expressed by integer or fraction, which was their creed. One day, hippasus, a member of this school, suddenly found that the diagonal of a square with a side length of 1 was a strange number, so he studied hard and finally proved that it could not be expressed by integers or fractions. But it broke the Pythagorean creed. So Pythagoras ordered him not to reveal it. But Siberus revealed the secret. Pythagoras was furious and wanted to put him to death Siberus ran away at once, but he was caught and thrown into the sea, giving his precious life for the development of science. The numbers discovered by Siberus are called irrational numbers. The discovery of irrational numbers led to the first mathematical crisis and made great contributions to the development of mathematics.

The Father of Geometry —— Euclid

The geometry we are studying now was founded by the ancient Greek mathematician Euclid (330 BC-275 BC). The Elements of Geometry, written by him in 300 BC, has been regarded as the standard textbook for studying geometry for more than 2,000 years, so Euclid is called the father of geometry.

Born in Athens, Euclid accepted Greek classical mathematics and various scientific cultures and became a famous scholar at the age of 30. At the invitation of the king of Egypt, he stayed in Alexandria to teach and do research.

Mathematics research in ancient Greece has a very long history, and there have been some works on geometry, but all of them discuss a certain aspect and the content is not systematic enough. Euclid collected predecessors' achievements and adopted an unprecedented and unique writing method. First, he put forward definitions, axioms and postulates, then proved a series of theorems from simple to complex, and discussed plane figures and three-dimensional figures, as well as integers, fractions and proportions. , finally completed the masterpiece "Geometry".

After the publication of the original, its manuscript has been circulated for 1800 years. After 1482 was printed and published, it was reprinted about 1000 times, and it was also translated into major languages in the world. /kloc-was introduced to China in the 3rd century and soon lost. The first six volumes were retranslated in 1607, and the last nine volumes were retranslated in 1857.

Euclid was good at solving complex problems with simple methods. He measured the length of the shadow of the pyramid at the moment when the figure of a person was just equal to the height, and solved the big problem of the height of the pyramid that no one could solve at that time. He said: "At this time, the length of the tower shadow is the height of the pyramid."

Euclid was a gentle and honest educator. Euclid was also a rigorous scholar. He opposes opportunism and the pursuit of fame and fortune in his studies, and the style of opportunism and quick success. Although Euclid simplified his geometry, the king (Ptolemy) still didn't understand and wanted to find a shortcut to learn geometry. Euclid said: "In geometry, everyone can only take one road, and there is no paved road for the king." This sentence has become an eternal learning motto. Once, one of his students asked him, what are the benefits of studying geometry? He said humorously to his servant, "Give him three coins because he wants to get real benefits from his study." Vedas (1540- 1603) is a French mathematician. Born in poitou in eastern France, he died in Paris. He studies law as a lawyer and is a member of parliament. Mathematics is his hobby. He is regarded as the greatest mathematician in16th century. He studied the works of many famous mathematicians, paying special attention to the idea of using symbols in mathematics by Diophantine and others. He was the first person who consciously and systematically used letters in mathematics. He has written many works on algebra, such as Introduction to Analytical Methods (159 1), which is the earliest work on symbolic algebra. His representative work on the identification and correction of equations (159 1 writing, 16 15 publishing) is an important symbol of the development of equation theory. He discovered the famous relationship between the roots and coefficients of algebraic equations-"Vieta Theorem". His work laid an important foundation for the development of modern algebra and promoted the development of equation theory. He also accurately predicted that there would be a deductive science about quantity using symbols in the future. He also studied trigonometry, geometry and astronomy, published trigonometry works, designed and improved the calendar, and cracked the other party's password for the government in the war, which won a high reputation.

De Zag (1593- 1662) is a French mathematician. One of the founders of projective geometry. Born in Lyon, died in the same place. Former military engineer and architect. He had contact with mathematicians Mei Sen and Descartes. 1636 published a pamphlet "On Perspective Festival" and began to talk about perspective. 1639, the first draft of trying to deal with the intersection of cone and plane was published. The terminology in this book is strange and difficult to understand. At that time, the emerging analytic geometry was more attractive to people, which led to this important work being quickly forgotten. Until 1845, Schaller stumbled upon the manuscript of this book, which attracted the general attention of mathematicians and listed it as a classic work of pure geometry in the Middle Ages. Infinite elements are introduced into the book, and problems such as poles and epipolar lines, transmission and perspective are discussed, which lays a solid foundation for projective geometry. The "Dezag Theorem" he discovered (two triangles correspond to vertex lines and points correspond to edge intersection lines) is the basic theorem of all projective geometry.

Descartes (1596- 1650) is a French philosopher, mathematician, physicist and physiologist, and one of the founders of analytic geometry. Born in Toulon, France, died in Stockholm, Sweden. Born in a famous family, he lost his mother when he was young and was weak since childhood. When he was studying at school in his early years, the principal chartered him to lie in bed every morning to study and think, and formed the habit of "thinking in the morning" until his later years. 16 12 went to the university of poitiers in Paris to study law, and received his doctorate four years later. 16 18 joined the army and has been to Holland, Denmark and Germany. /kloc-returned to China in 0/621year, traveled to Holland, Switzerland and Italy during the civil strife in France, and returned to Paris in 0/625. From 65438 to 0628, he moved to Holland. Since then, he has gained a relatively quiet and free academic environment, devoted himself to studying philosophy, mathematics, astronomy, physics, chemistry, physiology and many other fields, and devoted himself to writing for more than 20 years. 1649 winter, invited to give a lecture to Christina (1626- 1689) in Queen Christina. She died of pneumonia a few months later because her living habits were destroyed. (16 years later, the body was transported back to Paris). His contributions are various, especially in philosophy and mathematics. For example, emphasizing that the purpose of science is to "benefit the people"; Oppose scholasticism and advocate the method of "systematic doubt"; Putting forward the principle of "I think therefore I am" became the first principle in his philosophy. He emphasized making people "masters and rulers of nature". He advocates rationalism, and his mathematical thoughts are closely related to philosophical thoughts. He applied the reasoning method or deductive method of geometry to philosophy, and clearly declared that the essence of science is mathematics. After he took the concept of material movement as the philosophical basis of natural science, he brought movement into mathematics, mathematics and other natural sciences and got dialectics. There are many important works, among which Methodology (1637) is a classic of literature and philosophy, followed by three famous appendices: Refractive Optics, On Atmospheric Phenomena and Geometry. Geometry is his only mathematical work, but it has established his lofty position in the history of mathematics. He clearly expressed the idea of analytic geometry, which marked the birth of analytic geometry. He also played an important role in promoting the establishment of calculus.

Fermat (160 1- 1665) is a French mathematician. Born in Beaumont de Lomagne in the south of France, he died in castel. My father is in business and has good learning conditions since childhood. He studied law at university and became a lawyer. At the age of 30, he was a member of parliament in Toulouse until his death. He is well read, proficient in several languages, has a lot of natural science knowledge, and especially loves classical literature. Mathematics is just a hobby, but it has made great contributions to number theory, analytic geometry and probability theory, and is known as the "king of amateur mathematicians". He was cowardly, humble, upright, just and honest, and he didn't want to publish his works before his death. After his death, many of his expositions were left in old paper piles, in the margins of pages or in letters to friends. His son compiled these contents into two volumes of mathematical essays and published them in Toulouse (1679). His research on modern number theory was unparalleled before Euler. The famous Fermat's last theorem (no positive integer x, y, z, n satisfies Xn+Yn = Zn, n > 2) has aroused the interest of later mathematicians, but it has not been universally proved so far. He discovered the basic principles of analytic geometry independently of Descartes. He is considered as one of the founders of differential calculus, because he proposed a method to find the tangent of a curve and its maximum and minimum points. He was also one of the pioneers of probability theory in the17th century. He put forward the "Fermat principle" of optics, which has great enlightenment to the later study of variational method.

Wallis (16 13- 1703) is a British mathematician. Born in Kent, he studied in the theology department of Cambridge University in his early years and taught himself mathematics in his spare time. Become one of the famous mathematicians at that time. In his early years, he studied the works of some ancient mathematicians and translated and published some famous works of ancient Greek mathematicians. In arithmetica infinitorum, he calculated some algebraic functions and definite integrals, and obtained the expression of infinite product of π:

4/π=∫ 10√-( 1-x2)dx =( 1 3 3 5 5 7……)/(2 4 4 6 6 8……)

This is the first example of infinite continuous product in the history of mathematics, which is of great significance. He also wrote books such as conic curves, general mathematics or arithmetic, algebra, mechanics or geometric movements. Among them, he made unique creations in arithmetic, algebra, calculus, geometry and so on, and became one of the representatives of English mathematics thought in the17th century.

Pascal (1623- 1662),17th century French mathematician and physicist, once known as "the intermediate link between Archimedes and Newton". Pascal grew up in an academic atmosphere and showed great interest and outstanding talent in mathematics since he was a child. /kloc-at the age of 0/3, he was proficient in Euclid's Elements of Geometry and accidentally got the so-called Pascal Triangle in the same year. He discovered the ingenious structure of figures, studied the properties of number matrices, and wrote Trigonometric Matrix Arithmetic, which laid the foundation of combinatorial theory and mathematical induction. Pascal's triangle corresponds to the coefficient of Newton's binomial theorem expansion, but it is about 600 years later than Jiaxian's triangle in Northern Song Dynasty. Pascal gave a series of conclusions such as Pascal theorem in projective geometry, and also laid the foundation of probability theory and combinatorial theory with Fermat. Pascal made an in-depth study of cycloid, from which he drew a series of conclusions that the rotation of the E-axis of the curve determines the solid center of gravity, which stimulated Leibniz's work on calculus. Pascal once made the first addition and subtraction calculator that can calculate six figures in history, which laid the foundation for the later development of calculators. Pascal also showed outstanding talents in physics, hydrostatics and other fields, and was also a French prose master and theological debater. Unfortunately, he was tired of religious belief all his life, studied mathematics intermittently and returned to religious meditation intermittently. The ascetic self-torture made him live only 39 years old.

Guan Xiaohe (about 1642- 1708) is a Japanese mathematician. Born in Fujioka, Gunma Prefecture, he died in Edo (now Tokyo). Born into a samurai family, he was in charge of wealth for the Edo aristocratic family for a long time until he resigned in 1706. He is the founder of Japanese traditional mathematics-sum calculation, and also the founder of Guan School, and is honored as a saint in Japan. His only published work before his death was Differential Algorithm (1674), and after his death, his disciples compiled and published Ansai Algorithm (17 12). His other major works are secret biographies of the school in the form of manuscripts. His main mathematical achievements are: improving the celestial sphere method introduced by China in algebra, creating and calculating a unique written algebra; Make the summator master the solution of higher-order numerical equation introduced by China; The concept of determinant and its preliminary theory are established. The conditions for the existence of positive and negative roots of the equation and the solution similar to Newton iteration method are found. In addition, he also studied the "circularity" of magic square and overlapping square (infinite series expression of the relationship between radius, arc and vector). He also wrote some books about astronomical calendars. His thoughts were inherited and applied by his disciple Kenbu Guang (1664- 1739) and others, which had an important influence on the development of Japanese mathematics.

Newton (1642- 1727) was an English mathematician, physicist, astronomer and natural philosopher. Born in Woolsop village, about 13 km south of Lincolnshire, England, he died in Kensington, London and was buried in Westminster Abbey, London. Teenagers study in rural primary schools and have been farming for many years. However, he works hard and studies hard. 166 1 in June, 2006, he was admitted to Trinity College of Cambridge University with excellent results.

1665 Bachelor of Arts. Because the plague in London spread to Cambridge, the university was temporarily closed and he returned to China for two years. In the countryside, he thinks about all kinds of problems all day long, and uses his wisdom and accumulated knowledge for several years to create a blueprint for science. His three great inventions [flow number (calculus), gravity and optical analysis] all originated in this period. 1667 returned to Cambridge, was elected as a member of Trinity College, and obtained his master's degree in 1668. Professor Barrow taught him for many years. At this time, he assisted Barrow in writing lectures and papers on calculus and optics, which was highly praised by Barrow. 1669, Barrow frankly claimed that Newton's knowledge had surpassed himself, and he gave Newton the position of "Professor Lucas" that year. It was a good story for a while. Newton kept this position until 170 1. 1696, he worked as the supervisor of the Royal Mint, then moved to London and became the director in 1699. 1703 was elected president of the royal society. 1705 was knighted by Queen Anne (1665- 17 14). He devoted himself to natural philosophy and theology in his later years. In mathematics, his most outstanding contribution is the creation of calculus. As early as 1665, there was a record of "flow counting" in his manuscript. 1669, he wrote his first calculus paper, Analysis with Infinite Equations, and submitted it to the Royal Society for the record (published in 17 1 1). His official book "Flow Numerology and the Method of Infinite Series" was completed at 167 1. 1676, he wrote the third important calculus paper "Quadrature of Curves" (later published as the appendix of 1704 "Optics"). 1687, with Harley's urging and help, he published his masterpiece Mathematical Principles of Natural Philosophy. Starting from the definition and axiom (the law of motion) as the basis of mechanics, this book bases the whole mechanics on strict mathematical deduction, not only deeply applies its own analytical tools, but also formally publishes its own calculus theory for the first time. He also made a series of important discoveries in the field of algebra, such as the m-th square formula of the roots of algebraic equations of degree n, the paired proof of imaginary roots of real coefficient equations, etc. In addition, he also involves number theory, analytic geometry, curve classification, variational method, probability theory and other branches. He has made great contributions in many natural science fields such as mechanics, optics and astronomy, and is regarded as one of the greatest scientists. But his genius is often exaggerated to the extent of deification, ignoring his long-term efforts.

Leibniz (1646- 17 16) is a German mathematician, philosopher and natural scientist. Born in Leipzig, died in Hanover. He lost his father when he was a child, but as the father of an ethics professor at Leipzig University, he left him a rich collection of books. His mother is very knowledgeable and far-sighted, and sent him to the best school in Leipzig to study, so that he received a good family and school education from an early age. He learned to express his thoughts in many languages since he was a child, showing extraordinary philosophical talent. /kloc-at the age of 0/4, he became interested in logic and often put forward his own independent opinions. 166 1 year, he studied law at Leipzig University and geometry at Jena University, and was exposed to the philosophy of science of Galileo, Bacon, Hobbes and Descartes. 1666 obtained the doctor of law degree from Ortoff University in Nuremberg. His thesis "The Art of Combinatorial" already included the early AE of mathematical logic. Thought, and a series of subsequent work made him the founder of mathematical logic. After 1667, he devoted himself to the diplomatic field, and had the opportunity to travel around Europe and get in touch with celebrities in mathematics, especially Huygens, which aroused his interest in mathematics. He once made a calculator that can do multiplication, which is another progress of calculation tools after Pascal's addition mechanism (1642). 1673 During his visit to London, he presented this machine to the Royal Society of England and sent a copy to Emperor Kangxi of China. Unfortunately, no such machine has been found in the Forbidden City. 1676, he went to hanover, where he worked as a consultant and librarian in the duke's office. For the next 40 years, he often lived in Hanover until his death. He founded the Brandenburg Science Association (later renamed the Berlin Academy of Sciences) and served as its chairman. Although he was involved in various political struggles, he never stopped his scientific research. His research fields are very extensive, involving logic, mathematics, mechanics, geology, law, history, linguistics, biology, diplomacy, theology and so on. His most important contribution in the field of mathematics is to independently establish calculus with Newton. Newton established calculus mainly from the viewpoint of kinematics, while Leibniz considered it from the perspective of geometry, which is closely related to Barrow's differential triangle. His first partial differential paper was published in Yi Xue magazine 1684, which is the earliest calculus document in the world. The calculus symbols he created are still widely used in the field of advanced mathematics. In addition, he also made important discoveries in combinatorial analysis, algebraic determinant, curve family envelope and other theories. He systematically expounded the binary notation and connected it with China's gossip. In philosophy, he advocates the monism of objective idealism, which contains dialectical factors. In 17 14, he wrote "monism" and summarized his philosophical views.