The main content of geometry research, in order to discuss the properties of different patterns, can be said to be the most inseparable from human life. As far away as Babylon and Egypt, people have known to use the nature of some graphics to measure land and divide pastoral areas. However, it is not regarded as an independent knowledge, but only as some basic common sense in human life. Only since the ancient Greek era did we really study it seriously. So, let's briefly introduce it.

Ancient Greek geometry

Analytic geometry

Perspective geometry

noneuclidean geometry

differential geometry

Axiomatization of geometry

The development of ancient Greek geometry

1 development stage

2. The reasons for the development of ancient Greek geometry

3. Okilade's contribution-introducing "elements"

4. Archimedes' contribution

5. apollonius's contribution

6. Famous problems in ancient Greek geometry

(1) Fiona Fang problem

(2) the dual product problem

(3) Angle trisection.

(4) Parallel postulate

7. People who influence the development of mathematics

8. Reasons for the decline of ancient Greek mathematics

9. Applied science related to geometry

10. Criticism of ancient Greek mathematics

1. Development stage:

Geometry developed in ancient Greece is the driving force of all modern mathematics. To understand the whole mathematical structure, we must first understand the development of geometry in ancient Greece. We can divide it into three stages:

(1) Enlightenment:

The main characters are Thales, Pythagoras and Edoux.

Thales:

As the father of ancient Greek astronomy and geometry, he correctly predicted the time of solar eclipse. He began to do systematic research on some geometric figures.

Pythagoras (Shi Bi School):

Pioneering collective creation, known as Bibi School, is also a musician who invented Bibi scale. Bibi theorem is an important theorem in geometry. This school believes that "number" is the basis of everything in the universe.

Especially the dollar:

An exhaustive method was established. The so-called exhaustive method is the concept of "infinite approximation". The main idea is to get the approximate value of pi. Theoretically, Dollas is the founder of calculus.

Another contribution of You Dollas is the systematic study of the proportion problem.

(2) Peak period:

Important figures are: Euclid.

Archimedes

Apollonius (Apollonos)

Okil de:

He sorted out some predecessors' achievements in mathematics and wrote a book "Elements" (translated into "Elements of Geometry" in Chinese). This book is the first mathematics textbook in history and the best seller. Every branch of mathematics in the future began with this book. At present, the content of plane geometry in junior and middle schools is mainly the book Element. The details of this book will be introduced separately later. Another advantage of this book is that it is easy to read. Okil himself has not made any major breakthrough in mathematics, but he is a master of mathematics. This book was not introduced to China until after the middle of Ming Dynasty.

Archimedes:

He was born in Sicily and studied in Alexandria, Egypt. He is one of the three great mathematicians in history, and has invented countless things. We will introduce him and his contribution respectively in the future.

Apollonius:

Contemporary with Archimedes, his greatest contribution was the study of conic curve, which had a direct impact on the later invention of analytic geometry and even calculus. The application of conic curve was not developed until16th century.

(3) recession:

Since Archimedes and apollonius, Greek mathematics has gradually entered a period of decline.

Ptolemy:

Carry forward trigonometric functions, thus making astronomy hot.

Pabbs:

It can be said that it is a representative figure of the last period.

2. The reasons for the development of ancient Greek geometry:

We can't help asking: Why did ancient Greece develop such great mathematical achievements? What motivated them to regard nature as disorderly, mysterious, pluralistic and terrible in all civilizations before Greece? Natural phenomena are controlled by God. People's life and luck are determined by God's will. However, in the period of Greek civilization, intellectuals adopted a new attitude towards nature, which was rational, evaluative and realistic. They admire nature.

Bi School first put forward the following concept: "Erase mystery and uncertainty from natural activities, rearrange seemingly chaotic natural phenomena into understandable order and pattern, and the decisive key lies in the application of mathematics." Plato inherited the idea of Bi School:

Plato thought: "Only by following mathematics can we understand the real face of the real world. Science becomes science because it contains mathematics." It is because some scholars in the Greek era held this view of nature and established the practice of studying nature according to mathematics, which provided great incentives for the mathematical innovation of the wax-eating era itself and later generations. In mathematics, geometry is the closest description. For the Greeks, the principle of geometry is a concrete expression of the structure of the universe.

3. O 'Keerid's contribution:

The book Elements has a volume of 13, and its contents are as follows:

(1) 1-6 Volume: Plane geometry, based on the following five postulates:

A any two points can be a straight line.

B, straight lines can be extended at will.

C, you can draw a circle with any point as the center and any length as the radius.

D, right angles are equal.

parallel postulate

Study the following attributes:

The essence of triangle-consistency, similarity and so on.

Properties of parallel lines-internal dislocation angle, congruence angle.

Bi theorem.

The nature of a circle-inscribed circle, circumscribed circle.

The question of proportion.

Properties of parallelogram.

(2) Books 7, 8 and 9: Integer Theory.

This paper discusses odd numbers, even numbers and prime numbers, and discusses the application of exhaustive method.

(3) Volume 1 1, 12, 13: solid geometry.

The properties of pyramid, cone and cylinder are discussed, and the application of exhaustive method is introduced.

(4) Volume 10: an unmeasurable problem

Properties similar to irrational numbers.

The biggest feature of this book is:

It just quoted a few simple assumptions, and then derived a series of theorems based on these assumptions, and finally became a complete theory, establishing strict logical reasoning between cause and effect, thus establishing the status of mathematics as a deductive science. This book also has some shortcomings, and in fact, these shortcomings are the driving force for the future development of mathematics. For example, Article 5 (Parallel postulate), countless mathematicians revolve around this hypothesis. Finally, in the19th century, non-Euclidean geometry was created, which directly produced Einstein's theory of relativity. Element is the first mathematical work. Through the study, I have an understanding of the basic concepts of mathematics, the way of proof and the logic of theorem layout.

Other works of O 'Keerid:

Cone, whose content is the "conic curve" skeleton of aroni AS.

Phenomena discuss astronomical problems.

4. The contribution of Archimedes:

Archimedes was born in Narkus, Sicily in 287 BC. He is studying in Alexandria. His attitude towards learning starts from some simple axioms, and then deduces other theorems with impeccable logic to describe physics and mathematics together. He is the first man, so we can also call him the father of physics. He is the first engineer with scientific spirit. He looked for universal principles. And then applied to special engineering problems. His most important contribution is to carry forward the "exhaustive method", which has been equal to the concept of "arbitrary approximation" and entered the field of modern calculus. He used the exhaustive method to calculate the approximate value of π, and got:

3. 1408 & lt； π& lt； 3. 142858

Archimedes founded hydrostatics (buoyancy principle is the most important achievement) and discovered lever principle, so he can be regarded as a technical expert (an expert in art). The death of Archimedes can represent the beginning of the decline of Greek mathematics, and we will discuss the reasons for the decline later. One of the shortcomings of Archimedes' works is that the content is very difficult to understand and unreadable. Therefore, it has not been as widely circulated as the book elements. By the way, 1906, Archimedes' book Method was discovered in Turkey, which caused a sensation at that time.

5. apollonius's contribution:

He and Archimedes live in Alexandria at the same time. His main research object is conic curve, and there were some sporadic achievements before him, but he began to define and discuss conic curve strictly. From the point of view of geometry, his book "Conic Curve" can be said to be the pinnacle of ancient Greek geometry. This book consists of eight volumes and 487 items. Its real practicality was not developed until16th century. In fact, after that, mathematicians in any period probably started with O 'Keeled's "Element" and apollonius's "Conic Curve".

6. Famous problems in Greek mathematics:

The so-called problem is the following question, whether it can be solved only by compasses and rulers without scales:

Fiona Fang asked:

Can you turn a known circle into a square so that the two areas are equal?

This problem was studied by many people in Dollas's time, and it was not proved impossible until19th century. However, during the research period, many other branches of mathematics appeared.

Dual product problem:

For a known regular cube, its length, width and height should be expanded to make the volume of the new cube twice that of the original cube.

Equiangular problem:

How to divide any angle into three equal parts?

Problems 2 and 3 were not solved until19th century and proved impossible.

Parallel hypothesis:

Some people think that parallel postulate is not postulate, so some people remove it and create a new set of geometry without violating the original Euclidean geometry, which is non-Euclidean geometry and the basis of Einstein's theory of relativity.

Some people may think that the Greeks are unrealistic. These three questions were completely unrealistic at that time, and they can only be said to be used by some leisure class people to hone their brains. However, it is precisely because so many people put their energy into the research that the trend of geometry research is indirectly led, which leads to the vigorous development of mathematics in the future.

7. An influential figure in the development of mathematics

(1) Alexander the Great

(2) the Ptolemaic dynasty:

Alexandria was established, and the Alexandria Library was established, which was the largest library in the world at that time. In this library, many influential scholars have been produced. (Archimedes et al.)

King Hiro:

King of Sicily, direct patron of Archimedes.

Socrates, Plato, Aristotle.

Cleopatra

The last figure of Ptolemy dynasty, the first fire in the library of Alexandria, started because of it.

Christian and Muslim leaders:

The main role in the second and third destruction of Greek mathematics.

8. The decline of Greek mathematics

After Archimedes, apollonius and others, Greek mathematics began to decline, and later we will discuss the disaster it suffered:

The first disaster:

With the arrival of the Romans, Greek mathematics was destroyed. The Romans were very practical. They designed many projects, but refused to think deeply about the principle of use. The Roman emperor was not keen on supporting mathematicians. In14th century BC, Greece was completely conquered by Rome. At that time, Cleopatra, the last monarch of Ptolemy dynasty, had a good relationship with Caesar. Caesar set fire to the battleship in Alexandria to help her argue with her brother. As a result, the fire got out of control and the Alexandria Library was burned down. Probably millions of books and manuscripts were set on fire, which caused great losses. This time, the destruction consumed a lot of Greek mathematics.

The second disaster:

The rise of Christianity made Greek mathematics face the second catastrophe. Because they oppose research outside the church and laugh at mathematics, astronomy and physics, Christians are forced to ban them from participating in Greek research to prevent pollution. So thousands of Greek books were destroyed.

The third disaster:

After the Muslims conquered Alexandria, even the last books were burned. At that time, there was a saying in the Muslim Conquest: If the contents of these books are already in the Koran, we don't have to read them. If they are not in the Koran, we should not read them, so all the books were burned.

Remaining part:

At this time, some scholars moved to Constantinople and put their trust in the Eastern Roman Empire. Although I still feel the unfriendly atmosphere of Christians, it is always safer, which makes the knowledge stock increase slowly until 14 century Renaissance.

9. Science related to geometry

Astronomy:

For the Greeks, the principle of geometry is the concrete expression of space, so almost every mathematician has worked hard on astronomy. In fact, the invention of trigonometry is a technology developed for studying astronomy. Many mathematicians have designed models of motion between celestial bodies and planets. The popular Heliocentrism who knew the center of the earth was put forward by Aristak (he was the first great astronomer in Alexandria). However, there were many people who opposed it at that time. Geocentric theory was put forward by Ptolemy. This theory was not overthrown until16th century. In the Ptolemaic era, that is, the peak of the development of astronomy. Another great astronomer was apollonius, who described the motion of planets from a quantitative perspective, which was close to the research field of astronomy in18th century. Ptolemy's Great Astronomy is a classic.

In addition, China mathematicians have also made great contributions in geometry, which are listed as follows:

Geometric history of China

From the late Ming Dynasty (/kloc-6th century) to the publication of some Chinese versions of Euclid's Elements of Geometry, China's geometry has developed independently for a long time. We should pay attention to many ancient handicrafts and achievements in architectural engineering and water conservancy engineering, which contain rich geometric knowledge.

China's geometry has a long history, and reliable records can be traced back to BC15th century. In Oracle Bone Inscriptions, there are two words: rules and moments. Rules are used to draw circles and moments are used to draw squares.

The shape of moments in stone carvings in the Han Dynasty is similar to that of right-angled triangles. Around the 2nd century BC, China recorded the famous Pythagorean Theorem (Pythagoras originated relatively late).

The study of circle and square plays an important role in the development of ancient geometry in China. Mozi's definition of a circle is: "A circle is equal in length." A circle whose center is equal to the circumference is called a circle, which was explained more than 100 years before Euclid.

And Liu Xin (? 23), Zhang Heng (78- 139), Liu Hui (263), Wang Fan (2 19-257), Zu Chongzhi (429-500), Zhao Youqin (A.D.13rd century) and others, among whom Liu Fan.

Zu Chongzhi got the result π=355/ 133 more than one thousand years earlier than Europe.

In Liu Hui's notes on Nine Chapters of Arithmetic, his genius for the concept of limit has been revealed many times.

In plane geometry, right-angled triangles or squares are used, and in solid geometry, cones and rectangular cylinders are used for displacement, which constitute the characteristics of ancient geometry in China.

Chinese mathematicians are good at applying algebraic results to geometry, and using geometric figures to prove the organic combination of algebra, numerical algebra and intuitive geometry, which has achieved good results in practice.