The original concept of geometry gradually formed a relatively shallow knowledge of geometry, which is the need of production practice. Although this knowledge is scattered and mostly empirical, geometry is based on these scattered, empirical and superficial geometric knowledge.
Geometry is one of the oldest branches of mathematics and one of the most basic branches in this field. Ancient China, ancient Babylon, ancient Egypt, ancient India and ancient Greece are all important cradles of geometry.
A large number of unearthed cultural relics prove that in prehistoric times in China, people have mastered a lot of basic knowledge of geometry. A look at many exquisite and symmetrical patterns drawn on the articles used by ancient people, as well as some vessels with simple design but paying attention to volume and volume ratio, is enough to show how rich geometric knowledge people mastered at that time.
The work of Greek scholars played a key role in making geometry a systematic discipline. More than 2,000 years ago, the commerce in ancient Greece was prosperous and the production was relatively developed. A group of scholars are eager to pursue scientific knowledge, and the study of geometry is the most interesting content. What I want to mention here is the contribution of the philosophers Plato and Aristotle to the development of geometry.
Plato introduced the thinking method of logic into geometry, so that the original geometric knowledge gradually developed in a systematic and rigorous direction under the guidance of logic. Plato taught his students geometry in Athens and demonstrated some propositions in geometry with logical reasoning. Aristotle is recognized as the founder of logic, and his syllogism deductive reasoning method has great influence on the development of geometry. Until today, in elementary geometry, syllogism is still used for reasoning.
However, despite the abundant knowledge of geometry at that time, this knowledge was still scattered, isolated and unsystematic. It is the outstanding Greek mathematician Euclid who really sums up geometry as a discipline with strict theory.
Euclid went to Alexandria to teach around 300 BC. He is a respected, gentle and honest educator. He loves mathematics and knows some geometric principles of Plato. He collected all the geometric facts that he could know at that time in great detail, and compiled a set of rigorous theories according to the logical reasoning method put forward by Plato and Aristotle, and wrote the early masterpiece in the history of mathematics-Geometry Elements.
The great historical significance of Geometrical Elements is that it is the first model to establish deductive mathematical system by axiomatic method. In this book, all geometric knowledge is developed and described from the initial division hypothesis and logical reasoning. That is to say, geometry has really become a discipline with a relatively strict theoretical system and scientific methods since the publication of Geometry Elements.
Euclid's Elements of Geometry
Euclid's Elements of Geometry has thirteen volumes, in which the first volume talks about the conditions of congruence of triangles, the relationship between sides and angles of triangles, the theory of parallel lines, and the conditions of equal product (equal area) of triangles and polygons; The second book is about how to turn a triangle into a square with equal products; The third volume talks about circles; The fourth volume discusses inscribed and circumscribed polygons; The sixth volume talks about the theory of similar polygons; The fifth, seventh, eighth, ninth and tenth volumes describe the theory of proportion and arithmetic gain; Finally, the content of solid geometry is described.
From these contents, we can see that the main content of elementary geometry in middle school curriculum has been completely contained in geometric elements. Therefore, for a long time, people think that Geometry Elements is the standard textbook for spreading geometry knowledge for more than two thousand years. Geometry that belongs to geometric elements is called Euclidean geometry, or Euclidean geometry for short.
The most important feature of Elements of Geometry is the establishment of a strict geometric system, which mainly includes definition, axiom, postulate and proposition (including drawing and theorem). The first volume of Geometry Elements has 23 definitions, 5 axioms and 5 postulates. (The last postulate is the famous parallel postulate, or the fifth postulate. It triggered the most famous discussion on the theory of parallel lines in the history of geometry for more than two thousand years, and finally gave birth to non-Euclidean geometry. )
These definitions, axioms and postulates are the basis of the whole book Elements of Geometry. Based on these definitions, axioms and assumptions, the whole book develops its various parts logically. For example, every theorem that appears later explains what is known and what is verification. We should make logical reasoning according to the previous definitions, axioms and theorems, and give careful proof.
As for the methods of geometric argument, Euclid put forward analysis, synthesis and reduction to absurdity. The so-called analysis method is the step of assuming that what is required has been obtained and analyzing the conditions that are established at this time, so as to realize the proof; The comprehensive method is based on the facts that have been proved before, and gradually deduces the matters to be proved; The method of reduction to absurdity is to deny the conclusion under the assumption of retaining the proposition, and from the opposite side of the conclusion, to deduce the results that contradict the proved facts or known conditions, thus confirming the correctness of the conclusion of the original proposition, also known as reduction to absurdity.
The birth of Euclid's Elements of Geometry is of great significance in the history of geometry development. It indicates that geometry has become a discipline with a relatively strict theoretical system and scientific methods.
It has been more than two thousand years since Euclid published Elements of Geometry. Despite the rapid development of science and technology, Euclidean geometry is still a good textbook for middle school students to learn the basic knowledge of mathematics.
Euclidean geometry is a good textbook for teenagers to cultivate and improve their logical thinking ability in long-term practice because of its intuitive and clear features and strict logical deduction methods. I wonder how many scientists in history have benefited from studying geometry and made great contributions.
When he was a teenager, Newton bought a copy of Geometry in a nightclub near Cambridge University. At first, he thought that the content of the book was not beyond the scope of common sense, so he didn't read it carefully, but he was very interested in Descartes' Coordinate Geometry and read it wholeheartedly. Later, Newton lost the scholarship exam in April. 1664. Dr. Barrow, the examiner at that time, said to him, "Because your basic knowledge of geometry is so poor, no matter how hard you try." This conversation gave Newton a great shock. Then, Newton studied the Elements of Geometry from beginning to end, which laid a solid mathematical foundation for future scientific work.
Einstein, a scientific superstar of modern physics, is also a scientist who is proficient in geometry and uses geometric thinking methods to create his own research work. Einstein recalled the road he had traveled, especially when he was twelve years old. "The clarity and reliability of geometry left an indescribable impression on me." Later, the thinking method of geometry really inspired his research work. He has repeatedly proposed that in physics research, we should also proceed from several basic assumptions of the so-called axiom and make logical arguments. In the special theory of relativity, Einstein used this way of thinking to build the whole theory on two axioms: the principle of relativity and the principle of invariance of light speed.
In the history of geometry development, Euclid's Elements of Geometry played an important historical role. This function comes down to one point, that is, the "foundation" of geometry and its logical structure are put forward. In his Elements of Geometry, he used logical chains to expand all geometry, which is unprecedented.
However, in the long river of human understanding, no matter how brilliant the predecessors and famous artists are, it is impossible to solve all the problems. Due to the limitation of historical conditions, the "basic" problem of geometry put forward by Euclid in the Elements of Geometry has not been completely solved, and his theoretical system is not perfect. For example, the definition of a straight line is actually an unknown definition to explain another unknown definition, which plays no role in logical reasoning. For another example, Euclid used the concept of "continuity" in logical reasoning, but never mentioned it in the Elements of Geometry.
Axiomatic system of modern geometry
The discovery of some loopholes and flaws in the logical results in the Elements of Geometry is an opportunity to promote the continuous development of geometry. Finally, the German mathematician Hilbert put forward a relatively perfect geometric axiom system in the book "Geometry Basis" published in 1899, on the basis of summarizing the previous work. This axiom system is called Hilbert axiom.
Hilbert not only put forward a perfect geometric system, but also put forward the principle of establishing an axiomatic system. That is, in a geometric axiom system, which axioms should be adopted and how many axioms should be included should be considered from the following three aspects:
First, the existence of * * * (harmony) means that in an axiomatic system, all axioms should be non-contradictory, they are harmonious and exist in the same system.
Second, independence, every axiom in the axiomatic system should be independent and mutually independent, and no axiom can be deduced from other axioms.
Third, completeness, the axioms contained in the axiom system should be enough to prove any new proposition of this discipline.
This research method of defining basic objects and their relationships in geometry with axiomatic system becomes the so-called "axiomatic method" in mathematics, and the system proposed by Euclid in the Elements of Geometry is called the classical axiomatic method.
Axiomatic method brings a new perspective to geometry research. In axiomatic theory, because the basic object is not defined, it is not necessary to explore what the intuitive image of the object is, but only to study the relationship and nature between abstract objects. From the point of view of axiomatic law, we can use points, lines and surfaces to represent concrete things at will, as long as these concrete things meet the combination relationship, sequence relationship and contract relationship in axioms and make these relationships meet the requirements stipulated in axiomatic system, geometry is formed.
Therefore, all elements that conform to the axiomatic system can form geometry, and there is not only one intuitive image of each geometry, but there may be infinitely many. We call each intuitive image a geometric explanation or geometric model. Familiar geometric figures are not necessary when learning geometry, it is just an intuitive image.
In this respect, the object of geometry research is more extensive, and the meaning of geometry is more abstract than Euclid's time. All these have brought far-reaching influence to the development of modern geometry.