The emergence and development of non-Euclidean geometry

1。 . The development history of non-Euclidean geometry

Put forward the questions of 1 and 1

The development of non-Euclidean geometry originated from Euclid's Elements of Geometry more than 2000 years ago. Among them, postulate five was put forward by Euclid himself. Its content is "If a straight line intersects with two straight lines, and the sum of two internal angles on the same side is less than two right angles, then the two straight lines intersect at a point on that side after infinite extension". This postulate is widely discussed because it is not as concise as other axioms and postulates. Euclid himself was not satisfied with this assumption. He used the parallel postulate after proving all the theorems that don't need it. He suspects that it may not be an independent postulate, and it may be replaced by other postulates or axioms. Mathematicians have been worried about this postulate for more than 2,000 years from the ancient Greek era to the19th century, and they have been trying tirelessly to solve this problem. Mathematicians mainly follow two research approaches: one is to find a more self-evident approach. Another method is to try to deduce the parallel postulate from the other nine axioms and postulates. The simplest expression of the fifth postulate found along the first road is given by the Scottish mathematician J, Playfair1748-1819 in 1795: "Crossing a straight line, only one straight line is parallel to the original straight line", which is a parallel axiom used in our middle school textbooks today, but However, the problem is that all these alternative postulates are not more acceptable and "natural" than the original fifth postulate. The first major attempt to prove the fifth postulate in history was the ancient Greek astronomer Ptolemy (about AD 150). Later, Proclus pointed out that Ptolemy's "proof" inadvertently assumed that only one straight line could be parallel to the known straight line other than the straight line, which is the Prefil postulate mentioned above.

Solution to the problem of 1.2

1.2. 1 the germination of non-euclidean geometry

The work of demonstrating the fifth postulate along the second road made a breakthrough in the18th century. First, Italian Saccheri (Saccharn1667-1733) proposed to prove the fifth postulate by reducing to absurdity. Saccheri starts with the quadrilateral ABCD. If Angle A and Angle D are right angles and AC=BD, it is easy to prove that Angle C is equal to the acute angle of Angle D(2): Angle C and Angle D are acute angles. Finally, under the acute angle hypothesis, Saccheri deduced a series of results, which made him give up the final conclusion, because it was contrary to his experience. But objectively, it provides a very valuable idea for the establishment of non-Euclidean geometry. It has opened up a new road different from its predecessors. Later, the Swiss mathematician Lambert (Llambi TR 1728- 1777) did similar work to Saccheri. He also investigated a quadrilateral in which three angles are right angles and the fifth angle has three possibilities: right angle, obtuse angle and acute angle. He also concluded that "the area of a triangle depends on the sum of its internal angles;" The area of a triangle is proportional to the difference of angles and the sum of internal angles. He believes that as long as a set of assumptions do not contradict each other, it provides a geometric possibility. The famous French mathematician Legendar (A.M., Legendar1752-1833) also paid great attention to the parallel postulate, and he got an important theorem: "The sum of the internal angles of a triangle cannot be greater than two right angles". 1At the beginning of the 9th century, German Schweickart (1780- 1859) made this idea more clear. He pointed out through the study of "star geometry": "There are two kinds of geometry: narrow geometry (Euclidean geometry) star geometry, and in the latter, triangles have one characteristic.

1.2,2 The Birth of Non-Euclidean Geometry

Some mathematicians mentioned above, especially Lambert, are pioneers of non-Euclidean geometry, but they have not formally proposed a new geometry and established their systematic theory. Famous mathematicians Gauss (KOOC-0/777-/KOOC-0/855), Boyue (Bolyai/KOOC-0/802-/KOOC-0/860) and Lobachevsky (Lobachevsky/KOOC-0/793-/KOOC-0/856). Gauss was the first person to point out that Euclid's fifth postulate was independent of other postulates. As early as 1792, he had the idea of establishing logical geometry, in which Euclid's fifth postulate failed. 1794 gauss found that in his geometry, the area of a quadrilateral is directly proportional to the sum of the two right angles of the quadrilateral and the difference between the internal angles, from which it is deduced that the area of a triangle does not exceed a constant. No matter how far apart the vertices are, he further developed his new geometry, which is called non-Euclidean geometry. He firmly believes that this geometry is logically non-contradictory, true and applicable, so he also measured 3.

He thinks that the sum of the internal angles of the triangle formed by mountain peaks can only be displayed in a big triangle. However, due to the instrument error, his measurement failed. Unfortunately, Gauss didn't write anything about non-Euclidean geometry before his death. People learned his research results and views on non-Euclidean geometry from his correspondence with friends after his death.

2。 . Enlightenment from the development history of non-Euclidean geometry

The birth of non-Euclidean geometry is an important innovation step in mathematics since the Greek era. Here we will describe the historical significance along the historical development process of things. When evaluating this history, M. Klein said: "The history of non-Euclidean geometry shows that mathematicians have been greatly influenced by their spirit of the times in an amazing form. At that time, Saccheri rejected the singularity theorem of Euclid's geometry and concluded that Euclid's geometry was the only correct one. however

Mathematics itself 2. 1

2. 1. 1 Relative independence of mathematical development

The non-Euclidean geometry system established by logical deduction provides a model for the development of mathematics, which makes people clearly see that mathematics can have its own logical system and develop independently. The relative independence of mathematical development is highlighted as follows: the development of mathematical theory is often ahead of time, which can be carried out independently of the physical world, ahead of social practice and react to social practice. Promote the development of mathematics and even the whole science. /kloc-before the 0/9th century, mathematics was always closely combined with applied mathematics, that is, mathematics could not develop independently without practical disciplines. The ultimate goal of studying mathematics is to solve practical problems, but for the first time, non-Euclidean geometry makes the development of mathematics ahead of practical science and beyond people's experience. Non-Euclidean geometry has created a whole new world for mathematics: human beings can use their own thinking. Free thinking is required according to the logic of mathematics, so mathematics should be considered as any structure that is not directly or indirectly produced by the need to study nature. This view is gradually understood by people, which leads to the split between pure mathematics and applied mathematics today, LlJ. ..

2. 1.2 The essence of mathematics lies in its full freedom.

The creation of non-Euclidean geometry shows the difference between mathematical space and physical space that people have been aware of but have not clearly understood. Mathematicians created the theory of M-geometry, and then decided their view of space. This view of space and nature based on mathematical theory generally cannot deny the existence of the objective world. It only emphasizes the fact that a series of conclusions drawn from people's judgment on space are purely 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 activities of human beings to explore knowledge and establish theories will be endless. The establishment of non-Euclidean geometry makes people realize that mathematics is the creation of human spirit, not the direct copy of objective reality, which makes mathematics a great success. At the same time, it also makes mathematics lose the certainty of reality. Mathematics is liberated from nature and science and continues its journey. In this regard, M. Klein said: "This stage in the history of mathematics has made mathematics get rid of the close connection with reality and separated mathematics itself from science, just like science and philosophy, philosophy and religion, religion and animism and superstition. Now George Kang can use it.

2. 1.3 update of geometric concepts

The appearance of non-Euclidean geometry broke the situation that Euclidean geometry dominated the world, and the concept of geometry was updated. The appearance of non-Euclidean geometry broke this concept, prompting people to deeply discuss the basic problems of Euclidean geometry and even the whole geometry.

2.2 Cultural and educational aspects

2.2. 1 Non-Euclidean geometry is the spiritual product of human beings who dare to challenge tradition and devote themselves to science. Gauss, Boyle and Lobachevsky discovered non-Euclidean geometry almost at the same time, but their attitudes towards new geometry were different. Gauss realized the existence of new geometry a long time ago, but he didn't announce his new ideas to the world. He was influenced by Kant's idealism. Dare not challenge Euclidean geometry, which lasted for 2000a years, delaying the birth of non-Euclidean geometry. Boyo devoted himself to the study of parallel postulate and finally discovered new geometry. There is another story. When Gauss decided to keep his findings secret, Boyo was eager to make his research public through Gauss's evaluation. However, Gauss wrote back to his father F Boyo, saying, "Praising him means praising myself. The content of the whole article. People think that Gauss wants to plagiarize his own achievements, especially after Lobachevsky's works on non-Euclidean geometry were published, he decided not to publish any papers.

Lobachevsky's new geometry thought was not understood and praised by his contemporaries in 1826, but was satirized and attacked. "But nothing can shake Lobachevsky's confidence. He is like a lighthouse standing in the sea, and the impact of stormy waves shows his resolute will. He has been fighting for new ideas all his life. When he was blind, he also dictated Pan Geometry.

3. The process of discovering new geometry enlightens us that only by breaking through the superstition of tradition and authority can we give full play to scientific creativity; Only by being brave in suffering and devoting oneself to science can we pursue and defend the truth that transcends the times. It is generally believed that Gauss, Boyo and Lobachevsky discovered new geometry at the same time, which is people's justice to history, but people prefer to call it Roche geometry, which is the science that people dedicated to Lobachevsky.

High praise from the spirit.

The spirit of 2,2,2 non-Euclidean geometry urges people to establish a product of tolerance and tolerance.

The establishment of non-Euclidean geometry has liberated human thoughts, and new ideas are constantly emerging. "Mathematics emerged as the free creation of human thought" 5]. The development of mathematics made Cantor sincerely say that "the essence of mathematics lies in its freedom". This active and democratic artistic atmosphere makes mathematics develop at an unprecedented speed. The tortuous establishment of non-Euclidean geometry and the resulting mathematical development make people realize that freedom is created and a hundred schools of thought contend.

2.3 Philosophical thinking

2.3. 1 epistemological changes

French philosopher and mathematician Henri Poincare said 7: The discovery of non-Euclidean geometry is the root of an epistemological revolution. In short, people can say that this discovery has successfully broken the dilemma required by traditional logic that binds any theory: that is, the principle of science is either the inevitable truth (the logical conclusion of transcendental synthesis); Either the truth of assertion (the fact of sensory observation). He pointed out that principles may be simple and arbitrary agreements, but these agreements are by no means irrelevant to our hearts and nature. They can only exist by the tacit understanding of all people, and they are closely dependent on the actual external conditions in the environment where we live. In fact, it is precisely because of this that in the field of philosophy, we can reach a kind of "tacit understanding" based on our understanding of nature, which is the beginning and foundation of understanding everything. In addition, in theoretical evaluation, we give up the either-or judgment. Einstein said [8]: This either-or judgment is incorrect. The judgments of these judges and mathematicians undoubtedly have the most direct influence on the establishment of ideas and theories, especially on the establishment of epistemology. The further theoretical and technological progress in modern times can not be separated from its internal influence, such as the emergence of "Relativity", especially the further understanding of time and space, the establishment and development of set theory, the basis of modern analysis, mathematical logic, quantum mechanics and other disciplines can all be regarded as the direct results of non-Euclidean geometry. The vibration caused by the establishment of non-Euclidean geometry has not disappeared so far.

2.3.2 Breaking the traditional way of thinking of human beings

The primary basis for analyzing and evaluating a theory should be to see whether it is "compatible", that is, whether it has reached or will reach contradictory conclusions, if a theory cannot "justify itself". It shows that this theory is only a simple expression and enumeration of human experience, and has not yet evolved to the height of "theory"; Or at least need to be further improved. Originally, the premise of non-Euclidean geometry and Euclidean geometry theory is contradictory, and Euclidean geometry has been generally accepted. Does accepting non-Euclidean geometry inevitably lead to such problems, and does contradictory premise necessarily lead to contradictory results? The traditional way of thinking thinks that this is certain, that is, contradictory premises will inevitably lead to contradictory results. Accepting non-Euclidean geometry means breaking through the shackles of this traditional way of thinking. With the passage of time, especially the extensive application of non-Euclidean geometry results, people realize that in the process of establishing theories, we cannot guarantee that contradictory premises will lead to contradictory results. Therefore, compatibility is necessary in the process of establishing a theory, especially in the process of deducing a conclusion.

2-4 pairs of mathematics researchers

2.4. 1 bravely face the storm on the road of scientific exploration.

On the journey of scientific exploration, it is not difficult for a person to withstand temporary setbacks and blows. It is difficult to have the courage to struggle for a long time or even for life in adversity. Lobachevsky's new theory violates the traditional thinking of more than two thousand years, shakes the authoritative foundation of Euclid's geometry, and also violates people's "common sense". As soon as his theory was published, it was ridiculed, attacked, even insulted and abused in society. The archbishop declared his theory "heresy"; Most authorities call Lobachevsky's theory "pseudoscience" and "joke"; Even a kind-hearted person can only hold a "tolerant and sorry attitude towards a wrong weirdo" at most; Even many famous writers rose up against this new geometry. For example, the German poet Goethe wrote such a poem in his masterpiece (Faust): "There is geometry, and the Japanese name is' non-European', and I myself laugh inexplicably." In the face of all kinds of attacks and ridicule, Lobachevsky is fearless and indomitable. He is like a lighthouse standing in the sea. It shows a scientist's "special courage to pursue scientific needs". Lobachevsky firmly believed in the correctness of his theory and struggled for it all his life. Since/kloc-0 published "Non-Euclidean Geometry System" in 826, he has published eight books including "Geometry Elements". Before his death, he was almost blind in Los Angeles and dictated his masterpiece Pan Geometry in Russian and French. Lobachevsky struggled in adversity all his life. For a mathematician, especially a prestigious academic expert, it is not difficult to correctly identify those scientific and technological achievements that are mature or have obvious practical significance. It is difficult to identify FF in time. Those scientific achievements that are not yet mature or whose practical significance has not yet been revealed. The development of mathematics is by no means smooth sailing. More often, it is full of hesitation, wandering, experiencing difficulties and twists and turns, and even facing more crises. Every author of science T should have the courage to persevere in adversity.

Scientific explorers should be staunch supporters of new things in the field of science.

2_4_2 Correctly treat the achievements in the field of mathematics

Mathematics is a historical or highly accumulated subject. Important mathematical theories are always based on inheriting and developing the original theories. They will not only overthrow the original theory, but also always contain the original theory. For example, non-Euclidean geometry can be regarded as an extension of Euclidean geometry. Therefore, some mathematical historians believe that "in most disciplines, the structure of one generation is demolished by the next generation, and one's creation is destroyed by the next generation." Only mathematics, each generation adds a floor to the old building "1". When Klein examines the history of the fifth postulate research, especially the historical process of the transformation of non-Euclidean geometry from "potential" to "obvious" in the 18 ~ 19 century, he refers to: "any big branch of mathematics or a big special achievement. Some decisive steps or proofs can be attributed to individuals. This mathematical accumulation is especially suitable for non-Euclidean geometry. "In fact, from" The Elements of Geometry "to19th century, the fifth postulate attracted and inspired talented mathematicians of all ages to fight for it like a magnet. This has formed the longest time span and the largest number of members in the history of science. In this isomorphism, mathematicians exchange ideas, exchange research results and evaluate research results, forming a system of constant competition and encouragement. Lobachevsky was also inspired by his predecessors and his own failures, which made him think boldly: there may be no proof of the fifth postulate at all. So he changed his mind. Set out to find the unprovable solution of the fifth postulate. It is along this road that Lobachevsky discovered a new geometric world while trying to prove the unprovable fifth postulate. It can also be said that M of Roche Geometry is now attributed to Saccheri and Lambert's research on the fifth postulate. Today, the field of mathematics is becoming more and more detailed, and there are fewer and fewer mathematicians who are proficient in many fields. So mathematical researchers should unite and communicate with each other. Treat achievements with a peaceful attitude, without arrogance or rashness.

2.5 math teachers and math learners

2.5. 1 Cultivate innovative thinking in asking questions and posing difficult questions.

Lobachevsky believes that as an excellent math teacher, teaching math must be precise and rigorous, and all concepts should be completely clear. Because in his view, mathematics curriculum is based on concepts, especially geometry, he found the defects of its logical system through comprehensive thinking on the logical structure of Euclid geometry in preparing lessons. He is very confused. He is determined to eliminate these defects in teaching practice. Later, he did write a geometry textbook, Geometry Course (1883). He not only formed and implemented his non-Euclidean geometry thought in textbooks, but also talked about non-Euclidean geometry.

His research is always combined with teaching activities. Many of his theorems about non-Euclidean geometry are derived from M in the teaching process, and exchanged, modified and perfected among students. We can definitely say that his great achievements in creating non-Euclidean geometry are from the perspective of geometry education reform. This is a successful example of a major breakthrough made by a mathematics educator. As Borgas, a historian of mathematics, pointed out, "Lobachevsky hopes to establish a new geometry in the sense of teaching methods" and "this is an important reason for his reform of new geometry". His exploration of teaching methods has reached a scientific conclusion, which is a new way for human beings to study and conquer the Zhou world. Therefore, as a math teacher in the 2 1 century, we should keep learning this in the usual teaching process. Guide students to broaden their thinking and attach importance to divergent thinking in teaching; Teachers should select some typical problems, encourage students to innovate, guess and explore boldly, and cultivate students' innovative consciousness.

2.5 _ 2 Training students' innovative thinking in teaching

At first, Lobachevsky tried to prove the fifth postulate according to the ideas of his predecessors. In the only remaining student's lecture notes, several proofs given by him in geometry teaching in1816-1817 school year were recorded. But he soon realized that the proof was wrong. His predecessor and his own failures inspired him from the opposite side. The opposite formulation that made him think boldly: there may be no proof of the fifth postulate at all. So he changed his mind and began to look for the unprovable fifth postulate. It is along this road that Lobachevsky discovered a new geometric world while trying to prove the unprovable fifth postulate. "Learning begins with thinking, and thinking comes from questioning". Our thinking process of exploring knowledge always starts from problems and develops in solving them. It is also necessary for teachers to inspire students to question and ask difficult questions. In teaching, students should be encouraged to put forward the problems they encounter in the process of learning and discuss them with their classmates, so that students can have the opportunity to fully express themselves. First of all, they should provide the same ideas to solve different problems, and then they should propose changes in personal conditions and ask for new ideas to solve these problems, thus breaking the original thinking mode and making their thinking flexible and creative.

2.5.3 The significance of non-Euclidean geometry history for college students to learn mathematics.

Through the study of mathematics culture, college students can understand the interactive relationship between human social development and mathematics, and understand the inevitable law of mathematics development; Understand the process of human understanding the objective world from the perspective of mathematics; Cultivate the emotion and attitude of seeking knowledge, seeking truth and being brave in exploration; Understand the systematicness, rigor and universality of mathematics and the relativity of mathematical truth; Improve the interest in learning mathematics. The birth and development of non-Euclidean geometry is tortuous and arduous, and mathematicians have made great efforts for it. It has far-reaching and positive significance and influence on today's and future mathematics learners. Knowledge learning

Only through constant innovation and exploration can we create new knowledge and discover new knowledge fields.

"Reading history makes people wise" and studying the development history of non-Euclidean geometry are very important for revealing the realistic source and application of mathematical knowledge, guiding students to experience the real mathematical thinking process and creating a mathematical learning atmosphere for exploration and research.

It is of great significance to stimulate students' interest in mathematics and cultivate the spirit of exploration.

Generation of Non-Euclidean Geometry

1In the 1920s, Lobachevsky, a professor at Kazan University in Russia, took another road in the process of proving the fifth postulate. He put forward a proposition that contradicted the European parallel axiom, replaced the fifth postulate with it, and then combined with the first four postulates of European geometry to form an axiom system and launched a series of reasoning. He believes that if there are contradictions in reasoning based on this system, it is equivalent to proving the fifth postulate. We know that this is actually the reduction to absurdity in mathematics.

However, in his meticulous and in-depth reasoning process, he put forward one proposition after another that is intuitively absurd but logically contradictory. Finally, Lobachevsky drew two important conclusions:

First of all, the fifth postulate cannot be proved.

Secondly, a series of reasoning in the new axiom system has produced a series of logically non-contradictory new theorems and formed new theories. This theory is as perfect and rigorous as Euclidean geometry.

This geometry is called Luo Barczewski geometry, or Roche geometry for short. This is the first non-Euclidean geometry.

From the non-Euclidean geometry founded by Lobachevsky, we can draw an extremely important and universal conclusion: a set of logically contradictory assumptions may provide a geometry.

Almost at the same time that Lobachevsky founded non-Euclidean geometry, Hungarian mathematician Bao Ye Janos also discovered the existence of unprovable fifth postulate and non-Euclidean geometry. In the process of learning non-Euclidean geometry, Baoye was also given a cold shoulder by his family and society. His father, Bao Ye Facas, a mathematician, thought it was a foolish thing to study the Fifth Postulate, and advised him to give up this kind of research. But Bao Ye Janos insisted on developing new geometry. Finally, in 1832, in a book by his father, the research results were published in the form of an appendix.

Gauss, then known as the "prince of mathematics", 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 he dared not publish his research results publicly. He just expressed his views to his friends in his letters, but he didn't dare to stand up and publicly support the new theories of Lobachevsky and Bao Ye.

Roche geometry

The axiom system of Roche geometry differs from Euclid geometry only in that the parallel axiom of Euclid geometry is replaced by "from a point outside a straight line, at least two straight lines can be parallel to this straight line", and other axioms are basically the same. Due to the difference of parallel axioms, a series of new geometric propositions different from Euclidean geometry are derived through deductive reasoning.

As we know, Roche geometry adopts all axioms of European geometry except one parallel axiom. Therefore, any geometric proposition that does not involve parallel axioms, if correct in Euclidean geometry, is also correct in Roche geometry. In European geometry, all propositions involving parallel axioms are not valid in Roche geometry, and they all have new meanings accordingly. Here are a few examples to illustrate:

Riemann geometry

Euclidean geometry and Roche geometry have the same axioms about combination, sequence, continuity and contraction, but the parallel axioms are different. European geometry says that "at a point outside a straight line, only one straight line is parallel to the known straight line". Roche geometry says that "at least two straight lines are parallel to the known straight lines at a point outside the straight line". So is there a geometry that "crosses a point outside a straight line and cannot be parallel to a known straight line"? Riemann geometry answers this question.

German mathematician Riemann founded Riemann geometry. He clearly put forward the existence of another geometry in a paper "On the Hypothesis of Geometry as the Foundation" made by 185 1, which opened a new and broad field of geometry.

A basic law in Riemannian geometry is that any two straight lines on the same plane have common points (intersections). In Riemannian geometry, the existence of parallel lines is not recognized, and its other postulate says that straight lines can extend indefinitely, but the total length is limited. The model of Riemannian geometry is a sphere that has been properly "improved".

Modern Riemannian geometry has been widely used in general relativity. The space geometry in physicist Einstein's general relativity is Riemann geometry. In the general theory of relativity, Einstein gave up the idea of the unification of time and space. He thinks that time and space are only approximately consistent in a small enough space, but the whole time and space are uneven. This explanation in physics is completely similar to the concept of Riemannian geometry.

In addition, Riemannian geometry is also an important tool in mathematics. It is not only the basis of differential geometry, but also applied to differential equations, variational methods and complex variable function theory.

Different assumptions

The perpendicular and diagonal of the same line intersect.

Two lines perpendicular to the same line are parallel to each other.

There are similar polygons.

Crossing three points that are not in a straight line can be done, and only a circle can be made.

Roche geometry

The perpendicular and diagonal of the same line do not necessarily intersect.

Two straight lines perpendicular to the same straight line spread to infinity when both ends are extended.

There are no similar polygons.

Passing three points that are not on the same straight line may not necessarily make a circle.

From some propositions of Roche geometry listed above, we can see that these propositions are contradictory to the intuitive image we are used to. Therefore, some geometric facts in Roche geometry are not as easily accepted as European geometry. However, mathematicians put forward that we can use the facts in European geometry we are used to as an intuitive "model" to explain Roche geometry, which is correct.

1868, Italian mathematician Bertrami published a famous paper "An Attempt to Explain Non-Euclidean Geometry", which proved that non-Euclidean geometry can be realized on the surface of Euclidean space (such as quasi-sphere). In other words, non-Euclidean geometry propositions can be translated into corresponding Euclidean geometry propositions. If there is no contradiction in Euclidean geometry, there is no contradiction in non-Euclidean geometry.

Since people admit that Euclid has no contradiction, it is natural to admit that non-Euclid geometry has no contradiction. Until then, non-Euclidean geometry, which has been neglected for a long time, began to get extensive attention and in-depth research in academic circles, while Lobachevsky's original research was highly praised and praised by academic circles, and he himself was also known as "Copernicus in geometry".

The relationship of three kinds of geometry

Euclidean geometry, Roche geometry and Riemann geometry are three different geometries. All the propositions of these three kinds of geometry constitute a strict axiom system, which meets the requirements of harmony, completeness and independence. So these three geometries are all right.

In our moderate space, that is, in our daily life, European geometry is applicable; In the universe or nuclear world, Roche geometry is more in line with objective reality; Riemann geometry is more accurate in studying practical problems such as navigation and aviation on the earth's surface.

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