The word "Geometry" in English evolved from Greek, and its original meaning was land survey, which was later adopted by the Ming DynastyXu GuangqiTranslated into geometry.According to a large number of empirical studies, it was the Egyptians who created geometry, which was generated by land survey.Geometry is a science of studying shape, which takes people's visual thinking as the leading factor, and cultivates people's observation ability, spatial imagination and insight.The first development of geometry isEuclidOfeuclid geometry , followed by the first half of the 19th century,Non Euclidean geometryThe birth ofProjective geometryThe prosperity of, and finally the unity of geometry.
The word "geometry" originated from Arabic and refers to the measurement of land, that is, geodesy.Later, the Latin pronunciation was translated as "geometria".The word "geometry" in Chinese dates back to the Ming DynastyMatteo RicciWhen Xu Guangqi co translated the Geometric Elements, it was created by Xu Guangqi.At that time, no basis was given. Later generations believed that on the one hand, geometry might be the transliteration of Latin Greek GEO, and on the other hand, because Geometric Elements also used geometric methods to explain the content of number theory, or it might be the free translation of magnitude, so it is generally believed that geometry is a combination of sound and meaning.
The translation of geometry in the "Elements of Geometry" published in 1607 was not popular at that time, and there was another translation at the same time - metaphysics, such asDicowenAt that time, Xing Xue Bei Zhi, compiled by Zou Liwen and Liu Yongxi, also had a certain impact.In 1857Li ShanlanAfter the publication of the last nine volumes of Geometry Original, which was further translated by Wei Lie Yali, the name of geometry received certain attention, but it was not until the beginning of the 20th century that there was an obvious trend to replace the word "metaphysics". For example, Xu Shuxun changed the name of Geometry Original to "Continuation of Geometry" after the 11th printing of the Chengdu reprinted edition of "Metaphysics Purpose" in 1910.Until the middle of the 20th century, there were few“Metaphysics”The use of the word appears.
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Because of the need of human production and life, geometry came into being.
In primitive society, human beings accumulated a lot of knowledge about the shape, size and position relationship of objects in production and life.For example, ancient people knew the shape and size of their prey, remembered the distance between their place of residence and their place of hunting, and the position of the hunting place in their place of residence.
With the continuous development of human society, people have more and more knowledge about the shape, size and position relationship of objects, and gradually accumulated more and more geometric knowledge.
the Nile
It is said that four thousand years agothe NileEvery year, the flood always submerges the land on both sides of the river, and the water recedes, making the boundaries of the land unclear.At that time, the working people of Egypt had to carry out land survey every year in order to re measure the boundaries of the flooded land, so they accumulated a lot of knowledge about land survey.This gave rise to a rudimentary knowledge of geometry.
Later, the Greeks learned the elementary knowledge of geometry such as surveying and painting from Egypt due to their trade with the Egyptians.On the basis of these preliminary knowledge of geometry, the Greeks gradually enriched and improved into a complete geometry.The word "geometry" comes from Greek, and its original meaning is "surveying land technology".The word "geometry" is still used today.
In 338 BC, Euclid, a Greek, systematically summarized and sorted out the geometry knowledge of Egypt and the Greeks before him, and wrote a book called The Elements of Geometry.In 1607, Xu Guangqi, a mathematician in China, cooperated with Matteo Ricci, a Westerner, to introduce Euclid's "Elements of Geometry" to China for the first time.Euclid's The Elements of Geometry is a book with profound influence in the history of geometry.Nowadays, most of the geometry textbooks we study are based on the "Original Geometry".
Notes on Nine Chapters of Arithmetic
The study of geometry has a long history in China.Before 1000 BCBlack Pottery CultureDuring this period, the patterns on pottery werediamond, square and circle inscribed square, etcGeometry。In 500 BCMozhaiAuthored《Ink classics》It has some knowledge of geometry.On《Chapter Nine Arithmetic》It records the calculation method of land area and object volume.On《Zhou Bi Suanjing》It recordsright triangleThe relationship between the three sides of.This is the famous "Gou San Gu Si Xian Wu"Pythagorean theorem, also known as“Quotient height theorem”。ShanggaoPythagorean theorem of right triangle was discovered.Zu Chongzhi'sPiIt is also famous in the world.And ancient Chinese mathematiciansLiu Hui、Wang XiaotongHe has made great contributions to geometry.
With the continuous development of industrial and agricultural production and science and technology, the knowledge of geometry is becoming more and more abundant, and the research area is becoming more and more extensive.
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Xu Guangqi
The word "geometry" originates from the translation of "The Origin of Geometry".Geometric Elements is the most influential and largest mathematics textbook in the history of world mathematics.The introduction of Geometric Elements into China should first be attributed to Xu Guangqi, a scientist at the end of the Ming Dynasty.Xu GuangqiheMatteo RicciOne of the great contributions of the Chinese version of Geometry Original is to determine the Chinese name of the subject of studying figures as "geometry" and the translation of some basic terms in geometry.The original text of "geometry" is "geometry" (English geometry). When Xu Guangqi and Matteo Ricci translated "geo", they took the pronunciation of "geo" as "geometry" (Ming Dynasty pronunciation: gi ho), and the Chinese original meaning of "geometry" is "measuring size".The translation of "geometry" (English geometry) from "geometry" to "geometry" is really a stroke of genius.Some of the most basic terms in geometry, such as point, line, straight line, parallel line, angle, triangle, quadrangle and other Chinese translation terms, are determined by this translation.These translated names have been handed down to the present day, and have been eastward toChinese character cultural circleJapan, Korea, etc“Metaphysics(h ì nh h ọ c) "), which has a far-reaching impact.
Rhind Papyrus
The earliest record of the beginning of geometry can be traced back to ancient Egypt andMesopotamia。Early geometry is the collection of empirical laws about length, angle, area and volume, which are developed for practical needs (such as exploration, construction, astronomy and some handicrafts).The earliest known text on geometry isEgyptianRhind Papyrus (2000-1800 BC) andMoscow papyrus (English:Moscow Mathematical Papyrus）(circa 1890 BC), andMudstone slabs in Babylon, Cuba(e.g“Plimpton 322(English: Plimpton 322) "(1900 BC).For example, Moscow papyrus gives how to calculatePrismatic platformFormula for volume.Southern EgyptianAncient NubiansA geometry system was established, including an early version of the solar clock.
Geometry has a long history.The oldesteuclid geometry Based on a set of assumptions and definitions, people use basiclogical reasoningConstruct a series ofproposition。It can be said that "The Elements of Geometry" is the first example of axiomatic system, which has a profound impact on the development of western mathematical thought.
Descartes
A thousand years later,DescartesOn《Methodology》The introduction of coordinates into geometry has brought revolutionary progress.From this geometric problemAlgebraThe form of expression.
Euclidean geometry The fifth postulate of, because it is not self-evident, has attracted the attention of mathematicians of all ages.Finally, two non Euclidean geometries were established by Lobachevsky and Riemann.
The modernization of geometry is attributed toKlein, Hilbert et al.Klein, under the influence of Pruck, appliedgroup theoryThe view ofTransformation group。Hilbert laid a true scientific axiomatic foundation for geometry.It should be pointed out that the axiomatization of geometry has a profound influence, and it plays an extremely important leading role in the strictness of the whole mathematics.It's rightmathematical logicThe enlightenment of the scholars is also quite profound.
Ancient geometry
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Pythagorean theorem
The earliest recorded beginnings of geometry can be traced back to ancient Egypt (see ancient Egyptian mathematics), ancient India (see ancient Indian mathematics), andBabylon, Cuba(See the mathematics of ancient Babylon). Its age began about 3000 BC.The early geometry is the empirical principle of length, angle, area and volume, which is used to meet the practical needs in surveying and mapping, architecture, astronomy, and various crafts.Among them, there are surprisingly complex principles, which are difficult for modern mathematicians to useCalculusTo derive them.For example, Egypt and Babylon are both inPythagorasI knew it 1500 years agoPythagoras theorem(Pythagorean theorem);Egyptians have squaresPyramidThe correct formula for the volume of the frustum (truncated pyramid);And Babylon has oneTrigonometric function table。
The degree of development of Chinese civilization is similar to that of its corresponding period, so it may also have developed mathematics, but there is no relic of that period that can make us confirm this.Perhaps this is partly due to the use of primitive paper in early China, rather than recording their achievements with clay or stone carvings.
1. The rainy season every year,the NileThe water flooded the farmland on both banks.After the rainy season, the river receded, leaving fertile land, and ancient Egypt began to cultivate.In order to restore the boundaries of each person's farmland, it was necessary to re measure it, which made the geometry of the ancient Egyptians gradually developed.
2. Tombs built by ancient Egyptians for dead pharaohsPyramidIt is made of huge stones.During the construction activities over the yearsSolid geometryLearning has also developed. They can cut many huge stones accurately and transport them to the construction site, and then build a magnificent pyramid.The stones are closely connected with each other, and the whole pyramid is integrated, showing a superb technical level.
In the middle of the third century BC, the king of EgyptPtolemy I Soter Ask a mathematician: Is there a shortcut to mastering geometry without learning the Elements of Geometry.Mathematicians assert that there is no easy way to geometry in the world.This mathematical expert is the author of "The Elements of Geometry" and the famous Euclid of ancient Greece.
School life
Euclid in Raphael's famous painting Athens School
Euclid was a famous mathematician in ancient Greece and the pioneer of Euclidean geometry.Euclid was born in Athens, which was the center of ancient Greek civilization at that time.The rich cultural atmosphere deeply infected Euclid. When he was a teenager, he couldn't wait to enter“Plato Academy ”Learn.
One day, a group of young people came to the“PlatoXueyuan ".The gate of the school park was closed, and there was a wooden board hanging at the door, which said: "Those who do not know mathematics are not allowed to enter!" This was a rule set by Plato himself in those years, in order to let students know that he attached importance to mathematics, but he confused young people who came to ask for advice.Some people are thinking that it is because I don't know mathematics that I want to come here to seek advice. If I do, what else can I do here?As people looked at each other, Euclid stepped out of the crowd. He adjusted his clothes, looked at the sign, and resolutely opened the gate of the school. He walked in without looking back.
Plato School Park is a school founded by Plato when he was 40 years old, which mainly taught mathematics.In the school park, teaching between teachers and students is conducted through dialogue, so students are required to have a high degree of abstract thinking ability.Mathematics, especially geometry, involves the object of truth, universality and abstraction.They are related to things in life, but do not come from these specific things, so learning geometry is considered to be the most effective way to seek truth.Plato even claimed that "God is a geometer." This view has not only become the dominant idea of the school, but also has been accepted by more and more Greek people.People gradually fell in love with mathematics, and Euclid was no exception.After climbing into the school park, he immersed himself in the kingdom of mathematics.He devotes himself to seeking and takes Plato's learning as his goal. In addition, he never goes anywhere and does nothing.After reading and studying all Plato's works and manuscripts, it can be said that no one can be as familiar with Plato's academic thoughts and mathematical theories as Plato.After a thorough exploration of Plato's thought, he came to the conclusion that figures are drawn by God, and all abstract logical laws are embodied in figures.Therefore, the training of wisdom should begin with the geometry of graphics as the main research object.He really understood the essence of Plato's thought, and began to follow the path that Plato had gone through, taking the study of geometry as the main task of self-study, and finally achieved the achievements that the world admired.
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Parthenon Temple
The earliest geometry originated in ancient Egypt in the 7th century BC, and was later developed by ancient Greek mathematiciansThalesAnd others spread to ancient GreeceMili Capital, borrowed againPythagorean schoolclassic.Before Euclid, people had accumulated a lot of knowledge of geometry, but there was a big shortcoming and deficiency in these knowledge, that is, the lack of systematicness.Most of them are fragmentary and fragmentary knowledge. There is no strong connection between axioms and axioms, and between proofs and proofs, let alone strict logical argumentation and explanation of formulas and theorems.Therefore, with the prosperity and development of social economy, especially with the development of agriculture, forestry and animal husbandry, and the increase of land development and utilization, it is urgent to systematize and systematize these geometric knowledge into a complete set of knowledge system that can be self justified and coherent. It has become the general trend of scientific progress.Euclid has been keenly aware of the development trend of geometric theory through his early systematic and thorough study of Plato's mathematical thought, especially geometric theory.He is determined to finish the work in his lifetime.In order to accomplish this task, Euclid spared no pains to travel from the Aegean seasideAncient City of Athens, came to the new port of Egypt in the Nile Basin——Alexandria, in order to realize their original intention in this new but culturally rich foreign city.For countless days and nights here, while collecting previous mathematical monographs and manuscripts and consulting relevant scholars, he tried to write books and explain his understanding of geometry, even if it was superficial.After Euclid's selfless labor, it finally yielded fruitful results in 300 BC. This is the book Geometric Elements, which was finalized after several drafts were changed.This is a masterpiece handed down from generation to generation. Geometry has not only realized systematization and organization for the first time, but also bred a new research field - Euclidean geometry, or "Euclidean geometry" for short[1]。
Plane geometry
The Original Preface of Engraving Geometry written by Xu Guangqi
The Elements of Geometry is an immortal work integrating the ideas of predecessors and Euclid's personal creativity.There are not many works written by Euclid today, but we can see his true thoughts from the detailed writing style of this book.
The book is divided into 13 volumes.The book contains five "axioms", five "postulates", 23 definitions and 467 propositions.In each volume, Euclid adopted a completely different narrative approach from his predecessors, that is, he first proposed axioms, postulates and definitions, and then proved them from simple to complex.This makes the whole book more compact and lively.In terms of the content arrangement of the whole book, he also carried out his ingenious arrangement.From shallow to deep, from simple to complex, it successively discusses straight edge, circle, proportion theory, similar shape, number, solid geometry andExhaustion methodAnd so on.Among them, the discussion on exhaustion method became the source of modern calculus thought.From the arrangement of these voluminous contents, we can easily find that this book has basically covered the 400 years of mathematical development history of geometry from ancient Egypt in the 7th century BC to the 4th century BC - the period of Euclid's life.Among them, the representative is Euclid's exposition of straight edges and circles in Volumes 1 to 4.It is in these volumes that he summarized and developed the thinking achievements of predecessors, and skillfully demonstratedPythagoras theorem, also known as Pythagorean Theorem.In a right trianglesquareThe area of is equal to the sum of the areas of two squares on two right angles.His proof confirmed the correctness of Pythagorean theorem and lasted for more than 2000 years.The Elements of Geometry is a masterpiece that has been popular for thousands of years in the history of science.It not only preserved many early Greek geometric theories, but also carried forward these ancient mathematical ideas through Euclid's pioneering systematic arrangement and complete exposition.It initiated the study of classical number theory, and on the basis of a series of axioms, definitions and postulates, established Euclidean geometry systemaxiomatic method The earliest model of mathematical deduction system established.According to the system of Euclidean geometry, all theorems are deduced from some basic propositions, namely axioms, which are certain, unproved and true.In this caseDeductive reasoningIn, every proof of a theorem must be based on either axioms or axioms that have been previously proved, and finally make a conclusion.This method later became a strict way to establish any knowledge system. People not only applied it to mathematics, but also to science. It was also applied to theology, even philosophy and ethics, which had a profound impact on later generations.Although Euclid's geometry has been regarded as an almost impeccable example of strict thinking for almost 2000 years, in fact it is not always correct.It is found that some Euclideans, as axioms that are self-evident, are difficult to be self-evident and are increasingly suspected.Ratio "FifthAxiom of parallelism”In his book "Elements of Geometry", Euclid asserted that "by knowing an external known point, one and only one straight line can be made parallel to the known straight line." This result can still be empirically verified in the ordinary plane, so that in the ubiquitous closuresphereThe parallel axiom of the middle (the earth is a big curved surface) is not true.It was founded by Guizhou Russian Robchevsky and German RiemannSpherical geometryThat is to say, take a few miles to get geometry.
The way to seek knowledge
Euclid was not only a learned mathematician, but also an educator known as a "gentle and kind kind elder".In the process of writing books and educating people, he never forgot the warning sign hanging at the gate of the "Plato Academy"PlatonismThe rigorous and realistic traditional style of study inherited from ancient times.He treats students kindly and strictly, but he never publicizes his contributions.He always inspires and educates those students who are interested in exhausted mathematical mysteries, while he criticizes those who are eager for quick success and quick benefits and unwilling to study hard.In the Outline of the Development of Geometry by Proclos, the late tutor of Plato School, there is a story that mathematics, driven by Euclid, has gradually become a fashionable topic in people's lives (which is diametrically opposite to today's society), so that at that timePtolemy The king also wanted to follow the fashion and learn geometry.Although the king was well informed, Euclidean geometry was outside his intellectual range.So he asked Euclid if there were any shortcuts to learning geometry.In this respect, the king and ordinary people are the same. "
Euclid is a beacon in the history of human scientific thought.For the first time, he systematized mathematical theory and gradually made geometry a formal discipline system with independent development.He was rightHistory of mathematicsMany difficult propositions and theorems have made groundbreaking arguments and explanations, laying a solid theoretical foundation for the development of mathematics. His shock in theory has also become an indispensable step for future generations to climb the peak of wisdom.Both positive and negative aspects have promoted the progress of human mathematical thinking, thus providing a more effective tool for later human beings to better and deeper understand the nature.Therefore, later generations honored him as the "father of geometry" to remember his outstanding contribution to several stomach phases.
We can no longer examine Euclid's life, but only know that he left a book and two sentences for the world, one of which was to face a young man's question about geometry, who asked, "What's the use of your geometry?".His answer was: "Please give the young man three coins, because he wants to get practical benefits from geometry." From this, Euclid is also a great philosopher!
European Modern
Yang Zhenning once made a speech, thinking that modern science did not take place in China but in the West, because of the influence of the Geometric Elements and the Book of Changes.This influence directly leads to two ways of thinking and two cultures.Yang Zhenning's speech once caused strong dissatisfaction among scholars who supported the Book of Changes.However, Nie Wentao, also a supporter of Chinese culture, believes that the intuitive thinking caused by Euclid has led Western scholars to be keen on anatomy and the study of object motion trajectory, so there will be two books that will affect the world, namely, "On the Movement of Heart and Blood" and "On the Movement of Celestial Bodies".However, the oriental thinking will be more conducive to respect and understanding of life, so once integrated with modern technology, it will inevitably lead tolife sciencesHuge development in the field.
In short, Euclid's influence transcends time and space and will continue to have an impact in the foreseeable future.
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geometrySimple and beautiful learning, but exactlygeometryThe core existence of learning is perfect——Newton
geometryIt seems that sometimes it is necessary to be ahead of analysis, but in fact, geometry is ahead of analysis, just like a servant walking in front of the master, it is to open the way for the master——Sylvester
fractalgeometryIt not only shows the beauty of mathematics, but also reveals the essence of the world, and also changes the way people understand the mysteries of nature;It can be said that fractalgeometryIt really describes naturegeometryThe study of science has also greatly expanded the cognitive domain of human beings——Zhou Haizhong
Descartes' analysisgeometryYu Newton's calculus has been extended to the singular mathematical methods of Lobachevsky, Riemann, Gauss and Selvestor.In fact, mathematics is not only an indispensable tool for all disciplines, but also flies freely without considering the constraints of intuitive feelings——Nicholas Murray Butler
