Why is Riemannian geometry a more general geometry of Euclidean geometry and Robachevsky's non-Euclidean geometry?

1. The history of the development of non-Euclidean geometry

1. 1 the problem raised

The development of non-Euclidean geometry originated more than 2,000 years ago in the ancient Greek mathematician's Euclid's "Origin of Geometry". One of the axiom five is Euclid himself, it reads "If a straight line intersects two straight lines, and if the sum of the two interior angles intersected by the same side is less than the two right angles, then the two straight lines after infinite extension must intersect at a point on the side", this axiom caused extensive discussion, because it is not as simple as the other axioms, axioms, Euclid Euclid himself was not satisfied with this axiom, he used it after proving all the theorems that do not require the parallel axiom, suspecting that it may not be an independent axiom, and perhaps it can be replaced by other axioms or axioms. Mathematicians have always been obsessed with this axiom for more than 2,000 years, starting from the Ancient Greek era and continuing through the 19th century, and have worked tirelessly to try to solve the problem. Mathematicians have mainly advanced along 2 avenues of research: One way is to find a more self-evident proposition instead of the parallel axiom; the other way is to try to derive the parallel axiom from the other 9 axioms, axioms, along the first way to find the fifth axiom of the simplest expression is the Scottish mathematician in 1795, Playfair (J, Playfair1748-1819) gave: "Through a point outside a straight line, there is and is only one straight line parallel to the original straight line" also known as the axiom of parallelism which we use in our secondary school textbooks today, but which was actually stated by the ancient Greek mathematician Procrustes in the 5th century AD. The problem is, however, that all these alternative axioms are no more acceptable or "natural" than the original fifth axiom. The first major attempt to prove the fifth postulate in history was made by the ancient Greek astronomer Ptolemy (c. 150 A.D.), and Proclus later pointed out that Ptolemy's "proof" inadvertently assumed that only one line can be made parallel to a known line from a point outside the line, which is the above mentioned Preferential postulate. Axiom

1.2 Solution of the Problem

1.2.1 The Birth of Non-Euclidean Geometry

A breakthrough was made in the 18th century along the second avenue of proof of the fifth axiom. First, the Italian Sacchairn (Sacchairn1667-1733) proposed to prove the fifth postulate by reductio ad absurdum, Sacchairn began with the quadrilateral ABCD, if the angle A and the angle is a right angle, and AC = BD, it is easy to prove that the angle C is equal to the angle D. Thus the fifth postulate is equivalent to the assertion that the angle C and angle D is a right angle. Sakkari proposed another two assumptions: (1) obtuse angle assumption: angle C and angle D are obtuse; (2) acute angle assumption: angle C and angle D are acute. Finally, under the acute angle hypothesis, Saceri derived a series of results which, because they were contrary to empirical knowledge, caused him to abandon his final conclusions. But objectively for the creation of non-Euclidean geometry provides valuable ideas and methods, opened up a new way different from the previous. Later the Swiss mathematician Lambert (Lambetr1728-1777) did similar work with Sakkari. He also examined a class of quadrilaterals in which three of the angles are right angles and the fifth angle has three possibilities: right, obtuse and acute. He also obtained, under the assumption of acute angles, that "the area of a triangle depends on the sum of its interior angles; and that the area of a triangle is proportional to the difference between a square angle and the sum of its interior angles. He believed that a set of hypotheses provides a geometrical possibility as long as they do not contradict each other. The famous French mathematician Legendre (A. M. Legendre1752-1833) was also very concerned about the problem of parallel axioms, and he obtained an important theorem: "the sum of the interior angles of a triangle cannot be greater than two right angles", which predicted that there might exist a new geometry, and at the beginning of the 19th century, the German Sawekat (A. M. Legendre 1752-1833) was a mathematician, and the German Sawekat (A. M. Legendre 1752-1833) was a mathematician. At the beginning of the 19th century, the German Schweikart (schweikart1780-1859) made this idea clearer by studying "star geometry" and pointing out that: "There are two types of geometry: narrow geometry (Euclidean geometry) and star geometry. In the latter, the triangle has the characteristic that the sum of the interior angles of the triangle is not equal to the two right angles",

1.2,2 The birth of non-Euclidean geometry

Some of the mathematicians mentioned before, especially Lambert, were pioneers of non-Euclidean geometry, but none of them formally mentioned a new geometry and established its systematic theory, and the famous mathematician Gauss (Gauss), who was the first to mention a new geometry and established its systematic theory. Gauss (Gauss1777-1855), Boyo (Bolyai1802-1860), and Lobatchevsky (Lobatchevsky1793-1856) did so, and became the founders of non-Euclidean geometry, and Gauss was the first to point out that the Euclid's fifth axiom is independent of the others, and as early as 1792 he already had an idea to go for a logical geometry in which Euclid's fifth axiom does not hold.In 1794 Gauss found that in his kind of geometry the area of a quadrilateral is directly proportional to the difference between the sum of 2 flat angles and the sum of the quadrilateral's interior angles, and from this derived that the area of a triangle is not more than a constant, no matter how far apart its vertices are. Later he further developed his new geometry, called non-Euclidean geometry. He was convinced that this geometry was logically non-contradictory and that it was true and could be applied, and for this reason he measured the interior angles of triangles formed by three

peaks, believing that the deficit of the sum of the interior angles could be revealed only in very large triangles. But his measurements failed because of an instrumental error. Unfortunately, Gauss did not write any treatise on non-Euclidean geometry during his lifetime. It was only after his death that people learned of his results and views on non-Euclidean geometry from his correspondence with friends.

2. Insights from the History of Non-Euclidean Geometry

The birth of non-Euclidean geometry was a major step of innovation in mathematics since the Greek era． Here we will follow the historical development of things to describe the importance of this history. M. Klein (M. Klein) in the evaluation of this period of history, said: "The history of non-Euclidean geometry in a striking form to show that mathematicians are influenced by the spirit of their times is so powerful. At that time Saceri had rejected the singular theorem of Euclidean geometry and had concluded that Euclidean geometry was the only correct one. But a hundred years later Gauss, Lobachevsky, and Boyle embraced the new geometry with confidence."

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2.1 On the Mathematical Discipline Itself

2.1.1 The Relative Independence of the Development of Mathematics

The non-Euclidean geometrical system, established by logical deduction, provided a model for the development of mathematics, and it became clear that mathematics could exist in its own logical system and thus develop independently. The relative independence of the development of mathematics is highlighted as: the development of mathematical theory is often ahead of its time, it can be carried out independently of the physical world, can be ahead of the social practice, and the reaction of social practice, to promote the development of mathematics and even the entire science. 19th century, mathematics has always been closely integrated with applied mathematics, that is, mathematics can not leave the practical subjects and independent development, the ultimate goal of the study of mathematics The ultimate purpose of studying mathematics is to solve practical problems, but non-Euclidean geometry for the first time to make the development of mathematics ahead of the practical sciences, beyond people's experience, non-Euclidean geometry for mathematics to create a whole new world: human beings can use their own thinking, in accordance with the logical requirements of the mathematics of free thinking. Mathematics was then considered to be an arbitrary structure that did not arise directly or indirectly from the study of nature. The gradual understanding of this view led to the present-day split between pure and applied mathematics.

2.1.2 The essence of mathematics lies in its full freedom

The creation of non-Euclidean geometry brought to light a distinction that had always been recognized, but not clearly understood: the difference between mathematical space and physical space. Mathematicians create m geometric theory, and then determine their view of space, this based on mathematical theory of space, the view of nature, in general, does not deny the existence of the objective world and other content, it only emphasizes the fact that people on the space of the judgment of a series of conclusions obtained purely by their own creation. The reality of the material world and the theory of this reality are always two different things. Because of this, the cognitive activity of human beings to explore knowledge and establish theories never ends. The creation of non-Euclidean geometry makes people realize that mathematics is a creation of the human spirit, not a direct copy of the objective reality, which makes mathematics gain a great white south, but also makes mathematics lose the certainty of reality. Mathematics was freed from nature and science and continued its own journey. In this regard, M. Klein says: "This stage in the history of mathematics has freed mathematics from its close connection with reality and has separated mathematics itself from science, just as science has separated itself from philosophy, philosophy from religion, and religion from animism and superstition. It is now possible to make use of the words of Georges Cantor: 'Mathematics is a science. Cantor's words: 'The essence of mathematics lies in its full freedom"'.

2.1.3 Renewal of the concept of geometry

The emergence of non-Euclidean geometry broke the dominance of Euclidean geometry and renewed the concept of geometry. The traditional Euclidean geometry that space is unique, and the emergence of non-Euclidean geometry to break this concept, prompting people to Euclidean geometry and even the entire geometry of the foundations of the problem of in-depth discussion.

2.2 Cultural and Educational Aspects

2.2.1 Non-Euclidean geometry is a product of the human spirit that dares to challenge the tradition and dedicate itself to science. Gauss, Boyle, and Lobachevsky discovered the non-Euclidean geometry almost at the same time, but the three of them treated the new geometry with different attitudes. Gauss realized the existence of the new geometry at an early stage, but he did not announce his new ideas to the world, he was influenced by Kant's (Kant) idealistic ideas, and did not dare to challenge Euclidean geometry, which had been in the traditional geometric circles for 2000a long time, and thus delayed the birth of non-Euclidean geometry. Boyle devoted himself to the study of the parallel axiom, and finally discovered the new geometry. There is also a story that when Gauss decided to keep his discovery secret, Boyle was eager to publicize his research through Gauss's comments, but Gauss wrote back to his father, F. Boyle, saying, "To praise him is to praise myself. The content of the whole article, the line of thought adopted by your son and the results obtained coincide with my thinking of 30 to 35 years ago"], Boyle was y disappointed with Gauss's reply. Thinking that Gauss was trying to plagiarize his own results, especially after the publication of Lobachevsky's work on non-Euclidean geometry, he decided never to publish a paper again．

Robachevsky in 1826 after the publicity of the new geometrical ideas, did not get the understanding and praise of his contemporaries, but was satirized and attacked, "but no force can shake Robachevsky's confidence, he is like a lighthouse standing in the sea, the shock of the shock waves, full of show his tough will, his whole life has always been for the new idea of the struggle f4Jj''

Robachevsky's new geometrical ideas. f4Jj', in his blindness, but also orally completed the Pan-Geometry.

The discovery of the new geometry of the 3 people reveals that: only by breaking through the superstition of tradition and authority can we give full play to the creativity of science; only by braving hardship and suffering, and by dedicating oneself to science can we pursue and defend truths that transcend the times. It is generally believed that Gauss, Boyle, Robachevsky three people at the same time discovered the new geometry, which is the people of the history of justice, but people prefer to call the new geometry for the geometry of Roche, which is the people of Robachevsky for the dedication of the spirit of science

high praise.

2,2,2 the spirit of non-Euclidean geometry prompted people to establish tolerance, tolerance of all products

The creation of non-Euclidean geometry, the liberation of human thought, new insights, new ideas continue to emerge, "mathematics is revealed as a free creation of human thought" 5]. The development of mathematics makes Kantor said: "the essence of mathematics lies in its freedom". This artistic atmosphere of intellectual vigor and democracy enabled mathematics to move forward at an unprecedented rate. The creation of non-Euclidean geometry and the development of mathematics brought about by the twists and turns of the course, so that people realize the importance of free creativity, a hundred schools of thought on the development of science, prompting people to establish tolerance, tolerance of all the spirit and virtues [6.

2.3 Philosophical Thought

2.3.1 Epistemological Changes

French philosopher and mathematician Henri Poincare (1912-1992), who was a member of the French Academy of Sciences, said: "The essence of mathematics lies in its freedom. HenriPoincare) said7: the discovery of non-Euclidean geometry was the source of a revolution in epistemology. In short, one could say that this discovery has triumphantly broken the dilemma that traditional logic requires to bind any theory: that is, the principles of science are either necessary truths (logical conclusions synthesized a priori) or asserted truths (facts observed by the senses). He points out that principles may be simple arbitrary conventions, but these conventions are by no means independent of our mind and nature; they can only exist by the tacit agreement of all men, and they are closely dependent on the actual external conditions of the environment in which we live. In fact, it is precisely for this reason that, for exploring the unknown or the presently imperceptible, we can rely on our knowledge of nature in the field of philosophy to make a kind of "tacit agreement", which is the beginning and the foundation of the knowledge of all things. In addition, we in the theoretical judgment, give up the either/or judgment, Einstein said 8]: this either/or judgment is not correct. These critics, mathematicians, the judgment is undoubtedly non-Euclidean geometry after the creation of its ideas, theories, especially on the epistemology of the most direct impact; further modern theory and technological advances can not be separated from its intrinsic influence, such as the "theory of relativity" of the production, especially on the further understanding of space and time, the theory of sets, modern The creation of "relativity", especially the further understanding of space-time, set theory, the foundations of modern analysis, mathematical logic, quantum mechanics and other disciplines to establish and develop can be regarded as a direct result of non-Euclidean geometry. The shock of the creation of non-Euclidean geometry has not yet subsided.

2.3.2 Breaking the traditional way of human thinking

The primary basis for analyzing and evaluating a theory should be to see whether it has "compatibility", that is, whether it has or will come to self-contradictory conclusions, if a theory can not yet be "self-explanatory If a theory is not yet "self-explanatory". This theory is either a simple expression and enumeration of human experience, has not evolved to the height of the "theory"; or at least need to be further refined and improved. Originally non-Euclidean geometry and Euclidean geometry theory is established on the premise of contradiction, and Euclidean geometry has been generally accepted. The acceptance of non-Euclidean geometry inevitably raises the question of whether contradictory premises can necessarily lead to contradictory results. The traditional way of thinking is that this is certain, i.e., contradictory premises necessarily lead to contradictory results. Accepting non-Euclidean geometry means breaking through this traditional way of thinking. With the passage of time, especially the wide application of the results of non-Euclidean geometry, people realize that: we can not guarantee that the contradictory premise will lead to the contradictory result in the process of establishing the theory. Therefore, in the process of theory building, compatibility is a must have ...], especially in the process of deriving a certain conclusion, we must be aware of the establishment of the theoretical system of non-contradictory, whether the exclusion of neutrality.

2-4 to the mathematical researcher

2.4.1 Courage to face the storm on the road of scientific exploration

In the journey of scientific exploration, a person can withstand the momentary setbacks and blows is not difficult, but is the courage to long-term and even lifelong struggle in the face of adversity. Lobachevsky's new doctrine, contrary to more than 2,000 a traditional thinking, shaking the Euclidean geometry "sacrosanct" basis of authority, but also contrary to people's "common sense". As soon as his theory was published, social ridicule, attacks, and even insults and abuse rained down on him: the Academy of Sciences refused to accept his thesis; the archbishop declared his theory "heresy"; most of the authorities called Lobachevsky's theory "pseudoscience", a "pseudo-science", and a "pseudo-science", and "pseudo-science". Most authorities called Lobachevsky's doctrine "pseudo-science" and a "joke"; even those who were more kind-hearted could at best only hold "a tolerant and regretful attitude towards an errant eccentric"; even many famous literary figures rose up to oppose this new geometry, such as the German poet Goethe, who, in a statement to the press, rejected his thesis. Such as the German poet Goethe, in his famous book (Fotoderm) wrote the following lines: "There is geometry, the name of the day: 'non-European', their own ridicule, inexplicable". In the face of all kinds of attacks, ridicule, Lobachevsky undaunted, inch by inch, he is like a lighthouse standing in the sea, showing a scientist "the pursuit of science requires a special bravery". Robachevsky firmly believed in the correctness of his own doctrine, for which he fought all his life. After the publication of his non-Euclidean geometrical system in 1826, he published eight books, including "On the Origin of Geometry". Before his death, he was almost blind, but he still dictated and wrote his famous book "Pan-Geometry" in Russian and French. Lobachevsky is in the adversity of the lifelong struggle of the warrior. Similarly, a mathematical workers, especially the high reputation of academic experts, correctly identify those who have matured or have obvious practical significance of scientific and technological achievements is not difficult, the difficulty is to identify ff; those who have not matured or practical significance has not yet dew Chuan to the results of science. The development of mathematics is never smooth sailing, in more cases is full of hesitation, wandering, to go through difficult twists and turns or even face more crises. Each of our scientific T-authors, not only should be a brave in the face of adversity tenacious point

head of the scientific explorers, and should be a scientific field of new things in the firm supporters.

2_4_2Correct treatment of achievements in mathematics

Mathematics is a highly historical or cumulative subject. Major mathematical theories are always built on the basis of the inheritance and development of the original theories, and they not only do not overturn the original theories, but also always contain the original theories. For example, non-Euclidean geometry can be regarded as a broadening of Euclidean geometry. Therefore, some historians of mathematics believe that "in most disciplines, the buildings of one generation are demolished by the next, and the creations of one are destroyed by the next. In mathematics alone, each generation adds a floor to the ancient edifice" [1]. Krein, examining the history of the study of the fifth axiom, especially the transition from "latent" to "manifest" non-Euclidean geometry in the 18th-19th centuries for more than 100 a, said: "Any larger branch of mathematics or larger special result will not be the work of a single individual. Any larger branch of mathematics or larger special result will not be the work of a single individual, but at best some decisive steps or proofs can be attributed to the individual. This mathematical accumulation applies especially to non-Euclidean geometry". In fact, since the Principia Geometria and up to the nineteenth century, the problem of the fifth axiom has acted as a magnet, attracting and inspiring a wide range of talented mathematicians of all ages. This resulted in the formation of a mathematical ****some with the longest time span and the largest number of members in the history of science, and with the dissemination and study of the fifth axiom as its paradigm. In this ****some, mathematicians exchanged ideas, exchanged research results, and commented on them, creating a system of constant competition and incentives. Lobachevsky, too, was inspired by his predecessors and his own failures, and he ventured to think of the opposite formulation of the problem: that there might not be a proof of the fifth axiom at all. So he turned his mind to the search for an answer to the question of the unprovability of the fifth axiom. It was along this path that Lobachevsky discovered a new world of geometry in the process of trying to prove the unfalsifiability of the fifth axiom. It can also be said that Roche's geometry owes its m-presentation to the study of the fifth axiom by Saceri, Lambert, and others. In today's mathematical field, which is becoming more and more subdivided, there are fewer and fewer mathematicians who are proficient in more than one field. In this regard, mathematical researchers should be united and communicate with each other, and treat their achievements with a calm mind, without arrogance or impatience.

2.5 to math teachers and math learners

2.5.1 in the questioning to cultivate creative thinking

Robachevsky believes that as a good math teacher, teaching mathematics must be precise, rigorous narrative, all concepts should be completely clear. Because, in his view, the math curriculum is based on concepts, and this is especially true of geometry. Therefore, in preparing his lessons, he found the flaws in the logical system of Euclidean geometry through a thorough consideration of its logical structure, which puzzled him greatly. He was determined to eliminate those defects in his teaching practice. Later he did prepare a textbook on geometry, the Institutes of Geometry (1883). Not only did he develop and implement his ideas on non-Euclidean geometry in his textbook, but his research on non-Euclidean geometry

was always combined with teaching activities. Many of his theorems on non-Euclidean geometry were derived in the course of lectures, and exchanged, modified and perfected among students. We can say with certainty that his great achievement of founding non-Euclidean geometry was cut from the perspective of reforming geometry education, and is a successful example of a great breakthrough by a mathematics educator. As the historian of mathematics Borgas points out, "Lobachevsky's desire to establish a geometry that was irreproachable in a pedagogical sense" was "an important reason for his reform of Neogeometry". "His pedagogical investigations led to scientific conclusions that were colorful, that opened a new stage in the development of geometry, and that served as a new method for mankind to study and conquer the world around him". So as a 2l century math teacher, in the usual teaching process to continue to learn the new knowledge of this era, be brave enough to question the knowledge you have mastered; teaching to guide students to open up their minds, the importance of divergent thinking; teachers should select some typical problems, encourage students to be new and different, bold guessing, exploring, and cultivating the students' sense of innovation.

2.5_2 in the teaching and training of students' innovative thinking

Robachevsky just began to follow the thinking of the predecessor, trying to give Ⅲ the proof of the fifth axiom. In the only surviving notes of his students' lectures, there are several proofs given by him in his geometry teaching in the school year 1816-1817. But he soon realized that the proofs were wrong. The failures of his predecessors and of himself inspired him to think of the opposite formulation of the problem: that there might not be a proof of the fifth postulate at all. So he turned his mind to the search for an answer to the question of the unprovability of the fifth axiom. It was along this path that Lobachevsky discovered a new geometric world in the process of trying to prove the unfalsifiability of the fifth axiom. "Learning from thinking, thinking from doubt", we are exploring the knowledge of the thinking process is always from the beginning of the problem, and in the problem solving in the development. Teachers should not only be good at asking questions, but also to stimulate students to question. Teaching, to encourage students to encounter problems in the learning process and discuss with classmates, so that there is an opportunity for students to fully perform. First of all, to provide the same idea to solve different problems, and then put forward changes in individual conditions, requiring new ideas to solve, in order to break the original thinking stereotypes, so that thinking is flexible and creative.

2.5.3 The significance of the history of non-Euclidean geometry to the study of mathematics in colleges and universities

College and university students can learn through the study of mathematical culture, to understand the interaction between the development of human society and the development of mathematics, and to recognize the inevitable laws of the occurrence and development of mathematics; to understand the process of human understanding of the objective world from the mathematical point of view; the development of the emotions and attitudes of the quest for knowledge, realism, and courageous exploration; and the systematic, rigorous, and rigorous development of mathematics, the development and development of mathematics. Mathematics systematic, rigorous, application of a wide range of understanding of the relativity of mathematical truth; improve the interest in learning mathematics. The birth and development of non-Euclidean geometry is a tortuous and difficult process, and mathematicians have made great efforts for it. It has far-reaching and positive significance and influence on the present and future learners of mathematics.

There is no end to knowledge learning

and research, and it is only through continuous innovation and exploration that new knowledge can be created and new areas of knowledge discovered.

"Reading history makes one wise", learning the history of the development of non-Euclidean geometry for the revelation of the reality of mathematical knowledge and application, for guiding students to experience the real mathematical thinking process, creating a mathematical learning atmosphere of exploration and research, for

stimulating students' interest in mathematics, cultivating the spirit of exploration, are of great significance. have important significance.

The emergence of non-Euclidean geometry

By the 1820s, Professor Lobachevsky of the University of Kazan, Russia, in the process of proving the fifth axiom, he took another route. He came up with a proposition that contradicted the axiom of Euclidean parallelism, used it in place of the fifth axiom, and then combined it with the first four axioms of Euclidean geometry to form a system of axioms that launched a series of deductions. He thought that if there was a contradiction in the reasoning based on this system, it would be equivalent to proving the fifth axiom. As we know, this is really the counterfactual in mathematics.

But in the course of his extremely careful and in-depth reasoning, he arrived at proposition after proposition that was intuitively mind-boggling but logically without contradiction. In the end, Lobachevsky came to two important conclusions:

First, the fifth axiom cannot be proved.

Second, the chain of reasoning that unfolds in the new system of axioms yields a new series of theorems that are logically free of contradictions and form a new theory. This theory is as perfect and rigorous a geometry as Euclidean geometry.

This geometry is known as Lobachevsky's geometry, or Roche's geometry for short. It was the first non-Euclidean geometry to be proposed.

From the non-Euclidean geometry created by Lobachevsky, an extremely important and universal conclusion can be drawn: a set of logically non-contradictory assumptions has the potential to provide a geometry.

Almost simultaneously with the creation of non-Euclidean geometry by Lobachevsky, the Hungarian mathematician Bowyer Janosz discovered the unprovability of the fifth postulate and the existence of non-Euclidean geometry. Bowyer was also met with indifference from his family and society in his study of non-Euclidean geometry. His father, the mathematician Boyer Farkasch, considered the study of the Fifth Axiom to be a laborious and foolish endeavor and advised him to abandon it. But Bowyer-Janos insisted on working hard to develop a new geometry. Finally, in 1832, the results were published in an appendix to one of his father's books.

Gauss, who was known as the "Prince of Mathematics" in his time, also found that the fifth postulate could not be proved, and studied non-Euclidean geometry. However, Gauss was afraid that this theory would be attacked and persecuted by the church forces at that time, and did not dare to publicly publish his research results, but only expressed his views in letters to his friends, and did not dare to come forward to publicly support the new theories of Lobachevsky, Bowyer and theirs.