D. Ivanenko. Geometry of Lobachevsky and new problems in physics
Shmyrova Irina
“The ideas of our brilliant compatriot, which seemed an unacceptable paradox, are now widely developed and generalized, and are one of the cornerstones of modern science,” wrote a prominent Soviet geometer, Professor P.K. Rashevsky Objective: establish what led to the creation of non-Euclidean geometry.
Download:
Preview:
MKOU VASHUTINSKAYA BASIC EDUCATIONAL SCHOOL
The history of the emergence and significance of non-Euclidean geometry in modern science
Geometry done by:
9th grade student
Shmyrova Irina
Work coordinator:
Mathematic teacher
Sedykh Elena Valerievna
year 2013
1. Introduction………………………………………………………………… 3
2. The history of the creation of a new geometry………………………………. four
3. Non-Euclidean geometry…………………………………………… 8
4. Reviews and evidence …………………………………………. eleven
4. Significance of non-Euclidean geometry…………………………………………………………………………………………………………………………………………………………………………………………………………………………… 15
5. Conclusion…………………………………………………………. 16
6. Literature used…………………………………………. eighteen
7. Glossary of terms…………………………………………………... 19
Introduction
The path that Lobachevsky took for the first time, to a large extent determined the face of modern science, made a real revolution in mathematics.
“The ideas of our brilliant compatriot, which seemed an unacceptable paradox, are now widely developed and generalized, and are one of the cornerstones of modern science,” wrote a prominent Soviet geometer, Professor P.K. Rashevsky [1].
The discovery of non-Euclidean geometry revolutionized not only geometry and even not only mathematics, but, one might say, the development of human thinking in general. And thenthat Euclidean geometry is not the only possible one, made at the beginning of the last century by Gauss, Lobachevsky and Bolyai, had an impact on the worldview of mankind. However, few people know that since the end of the last century, non-Euclidean geometry, along with Euclidean, has been one of the working tools of mathematics, despite the fact that the "space in which we live", within the limits accessible to our understanding, is more Euclidean than non-Euclidean.[ 2].
The nature of mathematical theories is such that in various ways representingthe basic concepts of these theories, in geometry, for example, are points, lines, motions, etc., we can apply them to objects of various kinds. Therefore, geometry can be applied not only to the space in which we live, but also to other spaces that arise in mathematical and physical theories. The geometries of these spaces turn out to be different; in particular, they may not be Euclidean.
Objective : establish what led to the creation of non-Euclidean geometry. Hypothesis : the development of science was at such a stage that it was impossible not to come to the creation of non-Euclidean geometry.
I. The history of the creation of new geometry
Euclid himself can probably be considered the first non-Euclidean geometer (Fig. 1). His reluctance to use the “not self-evident” fifth postulate follows at least from the fact that Euclid proves his first twenty-eight sentences without resorting to this postulate. From the first century BC before 1820, mathematicians tried to deduce the fifth postulate from the rest, but succeeded only in replacing it with various equivalent assumptions, such as "two parallel lines are everywhere equidistant from each other" or "any three points not located on the same straight line belong to a circle" .
Figure 1. Euclid
Lobachevsky, in On the Principles of Geometry (1829), his first printed work on non-Euclidean geometry, clearly stated that postulate V cannot be proved on the basis of other premises of Euclidean geometry, and that the assumption of a postulate opposite to Euclid's postulate allows one to construct a geometry as meaningful as Euclidean and free from contradictions [1].
Simultaneously and independently, Janos Bolyai came to similar conclusions (Fig. 2), and Carl Friedrich Gauss (Fig. 3) came to such conclusions even earlier.
Figure 2. Janos Bolyai
However, Bolyai's writings did not attract attention, and he soon abandoned the subject, while Gauss refrained from publishing at all, and his views can only be judged from a few letters and diary entries.
Figure 3. Carl Friedrich Gauss
Student notes of Lobachevsky's lectures (from 1817) have been preserved, where he made an attempt to prove the fifth postulate of Euclid, but in the manuscript of the textbook "Geometry" (1823), he already abandoned this attempt. In Reviews of the Teaching of Pure Mathematics for 1822 and 1824, Lobachevsky pointed out the "still invincible" difficulty of the problem of parallelism and the need to take in geometry as initial concepts directly acquired from nature.
On February 23, 1826, a brilliant mathematician reads his report on non-Euclidean geometry to an uncomprehending, bored, indifferent audience. The commission, which has not understood anything, does not give any feedback. The work has not been published. And only in 1829 the memoirs "On the Principles of Geometry" were published - the first work on non-Euclidean geometry. The work was not understood.
A devastating review came from the Academy of Sciences, articles appear where Lobachevsky is called a provincial charlatan, an ignorant self-satisfied nonentity. The authors of these reviews relied on the fact that everything stated by Mr. Lobachevsky (Fig. 4) in his works has no place in nature and, therefore, is completely incomprehensible and absurd for the mind. Nobody supported Lobachevsky, but he had the courage to defend his ideas to the end.
Figure 4. Lobachevsky Nikolai Ivanovich
Not finding understanding at home, Lobachevsky tried to find like-minded people abroad. In 1837, Lobachevsky's article "Imaginary Geometry" on French(Géométrieimaginaire) appeared in the authoritative Berlin magazine Crelle, and in 1840 Lobachevsky published on German a small book "Geometric Investigations in the Theory of Parallels", which contains a clear and systematic presentation of his main ideas. Two copies were given to Carl Friedrich Gauss, "the king of mathematicians" of that time. As it turned out much later, Gauss himself secretly developed non-Euclidean geometry, but did not dare to publish anything on this topic [1].
Euclid's fifth postulate became a kind of impetus for the creation of another geometry, or a continuation of Euclid's geometry. Simultaneously, scientists from many countries came to the same conclusions. However, some scientists did not understand, like Lobachevsky, others were afraid to publish their works.
The creators of non-Euclidean geometry were such bright scientists as Euclid himself, Gauss, Boyai, Lobachevsky. For some scientists, discoveries in non-Euclidean geometry occurred simultaneously, independently of each other.
II.Non-Euclidean geometry
Lobachevsky considered Euclid's axiom of parallelism to be an arbitrary constraint. From his point of view, this requirement is too strict, limiting the possibilities of the theory describing the properties of space, and therefore, in creating non-Euclidean geometry, he used Euclid's planar postulates as a special, limiting case and abandoned the V postulate, accepting the independence of the axiom of Euclid's parallel lines from the rest of the axioms .
Instead of the V postulate, he accepts the opposite proposition: in the plane through a point not lying on a given line, there passes more than one line that does not intersect the given one. Along with this proposal, Lobachevsky accepts the remaining axioms of Euclidean geometry and builds a new geometry on this basis. The resulting geometry is logically coherent, there are no contradictions anywhere. Lobachevsky calls it "imaginary".
Through a point C lying outside the line AB, it is possible, Lobachevsky suggested, to draw at least two lines a and b that do not intersect with the line AB (Fig. 5). In the same way, the line AB and the lines m, n, p passing through the point C do not intersect.
Figure 5. Sentence opposite to Euclid's 5th postulate.
The sum of the angles of a triangle in "imaginary geometry" is always less than 180 o (Fig. 6).
Figure 6. Triangle in Lobachevsky geometry.
There is no similarity in the Lobachevsky plane. After all, all similarity theorems are derived only with the help of Euclid's axiom of parallelism. N.I. Lobachevsky established that on the limiting surface, called the orisphere, the internal geometry is Euclidean.
The new geometry developed by Lobachevsky does not include the Euclidean geometry, but the Euclidean geometry can be obtained from it by passing to the limit (as the space curvature tends to zero). In the Lobachevsky geometry itself, the curvature is negative. Already in the first publication, Lobachevsky developed in detail the trigonometry of non-Euclidean space, differential geometry (including the calculation of lengths, areas and volumes) and related analytical issues.
In the geometry of N.I. Lobachevsky, the basic concepts of Euclid are used: perpendiculars, axial symmetries and turns. It stores the properties isosceles triangle, well-known signs of equality of triangles and other elements of "absolute geometry" [2].
In the Lobachevsky space, curvilinear geometric images were identified, subordinate to the geometry of Euclid. Lobachevsky used this remarkable result to derive trigonometric relations between the elements of rectilinear triangles in his space. But the resulting relations are much more complicated than the Euclidean ones. These relations have not only trigonometric functions of angles, not just lengths of sides, but some functions of them [4] .
Having made his famous discovery, N. I. Lobachevsky did not refute Euclidean geometry, but only pushed the boundaries of science that existed in ancient world. Any facts of Lobachevsky's planimetry do not contradict the geometry of Euclid. However, the generated geometry differs significantly from the previous one. Lobachevsky, obviously, wanted to emphasize the contradiction of postulate V: on the plane, through a point lying outside a given line, there passes more than one line that does not intersect the given one. And thus replaced the Euclidean postulate with a more general axiom of parallelism and preserved all the reasoning of Euclid's geometry.
III. Testimonials and evidence
In the last years of his life, Lobachevsky unsuccessfully tried to prove the consistency of his geometry.
To obtain such a proof, it was necessary to construct a model of the geometry. In 1868 (12 years after Lobachevsky's death), the Italian scientist E. Beltrami investigated a concave surface called a pseudosphere and proved that Lobachevsky's geometry acts on this surface (Fig. 7). [ 5].
In 1868 The Italian mathematician E. Beltrami investigated a concave surface called a pseudosphere and proved that Lobachevsky's geometry acts on this surface.
Figure 7. Pseudosphere
And after 2 years, the German mathematician Klein offers another model of the Lobachevsky plane (Fig. 8).
Klein takes some circle. "Plane" Klein calls the interior of the circle. Further, each chord of the circle (without ends, since only the internal points of the circle are taken) is considered by Klein to be a "straight line". Now in this "plane" one can consider segments, triangles, etc. Two figures are called "equal" if one of them can be transferred to the other by some movement. Thus, all the concepts mentioned in the axioms of geometry are introduced, and it is possible to check the fulfillment of the axioms in this model. For example, it is obvious that through any two points A, B there is only one "straight line". It can also be seen that through the point A, which does not belong to the "line" a, there are infinitely many "lines" that do not intersect a. Further verification shows that in the Klein model all other axioms of Lobachevsky geometry are also satisfied[4]
Figure 8. Klein model.
Another model of Lobachevsky's geometry was proposed by the French mathematician A. Poincaré (1854-1912). He also considers the interior of some circle. He considers “straight lines” the arcs of circles, which at the points of intersection with the border of the circle touch the radii (Fig. 9) [1].
Figure 9. The Poincaré model.
At the end of the last century, in the works of Poincaré and Klein, a direct connection was established between the geometry of Lobachevsky and the theory of functions of a complex variable and with the theory of numbers (more precisely, the arithmetic of indefinite quadratic forms). Since then, the apparatus of Lobachevsky's geometry has become an integral component of these branches of mathematics. In the last 15 years, the importance of Lobachevsky geometry has increased even more thanks to the work of the American mathematician Thurston (the winner of the Fields Medal in 1983), who established its connection with the topology of three-dimensional manifolds (Fig. 10). Dozens of papers are published annually in this area. In this regard, we can talk about the end of the romantic period in the history of Lobachevsky's geometry, when the main attention of researchers was drawn to its understanding from the point of view of the foundations of geometry in general. Modern research more and more require business knowledge of Lobachevsky geometry[ 2].
Figure 10. William Paul Thurston
An important note regarding the drawings depicting the behavior of straight lines on the Lobachevsky plane. As experiments show, our physical space is either Euclidean in properties, or differs very little from it. When operating with a drawing, we are forced to limit ourselves to its small size, and the deviation from Euclideanism, if it exists, will be observed only at very large extensions. Therefore, for clarity, it is usually customary to depict straight lines, slightly bending them, in order to more clearly express the nature of their convergence or divergence on the Lobachevsky plane. However, Lobachevsky did not allow himself such liberties [4].
How long did it take scientists to check on various models: Klein's pseudosphere, Poincare's model, mathematician Thurston's three-dimensional manifolds, that Lobachevsky's geometry works? What doubts did Lobachevsky himself have about the correctness of his ideas?! But it was precisely the elements of Lobachevsky's geometry that became the basis of such branches of mathematics as number theory and the theory of functions of a complex variable, and many others.
IV. Significance of Non-Euclidean Geometry
The new geometry was a pure product of the mind, separated from the surrounding reality. Therefore, Lobachevsky called it "imaginary." The advent of non-Euclidean geometry was important step in the transformation of mathematics into the science of logical conceivable forms and relationships. This process went on all fronts, not only in geometry, but also in algebra. Set theory and mathematical logic appeared. In geometry, soon after Lobachevsky's geometry, multidimensional Euclidean geometry appeared [2].
V. Conclusion
The creators of non-Euclidean geometry were such bright scientists as Euclid himself, Gauss, Boyai, Lobachevsky. Euclid made attempts to prove the fifth postulate, but he failed. For some scientists, discoveries in non-Euclidean geometry occurred simultaneously, independently of each other.
N. I. Lobachevsky pushed the boundaries of science that existed at that time. Any facts of Lobachevsky's planimetry do not contradict the geometry of Euclid. However, the generated geometry differs significantly from the previous one. Lobachevsky, obviously, wanted to emphasize the contradiction of postulate V: on the plane, through a point lying outside a given line, there passes more than one line that does not intersect the given one. And thus replaced the Euclidean postulate with a more general axiom of parallelism and preserved all the reasoning of Euclid's geometry.
It took a lot of time for scientists to check on various models: the Klein pseudosphere, the Poincaré model, the three-dimensional manifolds of the mathematician Thurston, that Lobachevsky's geometry works? What doubts did Lobachevsky himself have about the correctness of his ideas?! But it was precisely the elements of Lobachevsky's geometry that became the basis of such branches of mathematics as number theory and the theory of functions of a complex variable, and many others.
Lobachevsky was called the "Copernicus of geometry", but he can also be called the Columbus of science, who discovered a new field of science, followed by a continent of new geometry and, in general, new mathematics. The path that Lobachevsky took for the first time determined to a large extent the face of modern science.
The discovery of new geometry was the beginning of numerous studies of outstanding mathematicians of the 19th century. Geometry served as an impetus for the development of science, and hence the understanding of the world that surrounds us.
And at the beginning of the 20th century, it was discovered that Lobachevsky's geometry is absolutely necessary in modern physics. For example, in Einstein's theory of relativity, in the calculations of modern synchrophasotrons, in astronautics.
Used Books
1. Laptev B.L. N.I. Lobachevsky and his geometry. Student aid. M., "Enlightenment", 1976.
2. Sherbakov R.N., Pichurin L.F. from projective geometry - to non-Euclidean (around the absolute): Book. For extracurricular reading. IX, X class. - M.: Enlightenment, 1979. - 158s., Ill. - (World of Knowledge)
3. Pogorelov A.V. Geometry: Proc. For 7-9 cells. general education institutions / A.V. Pogorelov.-5th ed. - M.: Enlightenment, 2010.-224 p.
4. Alekseevsky D.V., Vinberg E.B., Solodovnikov A.S. Geometry of spaces of constant curvature. In: Itogi nauki i tekhniki. Modern problems of mathematics. Fundamental Directions. M.: VINITI, 1988. T. 29. S. 1 - 146. rostransto - a fundamental (along with time) concept of human thinking, reflecting the multiple nature of the existence of the world, its heterogeneity. A lot of objects, objects, given in human perception at the same time, forms a complex ... ...Philosophical Encyclopedia
- geometry- a geometry based on the same basic assumptions as Euclidean geometry, with the exception of the parallel axiom (see Fifth postulate). In Euclidean geometry, according to this axiom, on the plane through the point P, which lies outside the line A, A passes.
Mathematical Encyclopedia
- Lobachevsky geometry- a geometric theory based on the same basic assumptions as ordinary Euclidean geometry, with the exception of the axiom of parallels, which is replaced by Lobachevsky's axiom of parallels. The Euclidean axiom of parallels says: ... ...
Great Soviet Encyclopedia
- Geometry - a branch of mathematics that studies the properties of various shapes (points, lines, angles, two-dimensional and three-dimensional objects), their size and relative position. For the convenience of teaching, geometry is divided into planimetry and solid geometry. Encyclopedia
~ ~
Nikolai Ivanovich Lobachevsky (1793-1856)
The great Russian geometer, the creator of non-Euclidean geometry, Nikolai Ivanovich Lobachevsky was born on November 2, 1793 in the Nizhny Novgorod province, into a poor family of a petty official. After a childhood filled with need and deprivation, after graduating from the gymnasium, which he managed to enter only thanks to the exceptional energy of his mother Praskovya Alexandrovna, we see him as a fourteen-year-old boy already a student of the newly opened Kazan University, within the walls of which all his further life and work pass. . N. I. Lobachevsky was lucky to study mathematics at the gymnasium with an outstanding person and, apparently, a brilliant teacher - Grigory Ivanovich Kartashevsky. Under his influence, the mathematical abilities of the future great geometer developed. As a student, he studied with the famous Bartels, a professor at first Kazan, then Yuryev University, having seriously mastered the mathematics of his time from primary sources, mainly from the works of Gauss and Laplace. However, despite the early manifestation of mathematical talents, the decision to devote himself to mathematics did not occur to N. I. Lobachevsky immediately; there is evidence that he first prepared himself for medical studies. In any case, by the age of 18 he had already chosen mathematics.
The student years of N. I. Lobachevsky were filled not only with an ardent passion for science and persistent scientific pursuits; they are also full of youthful pranks and pranks, in which his cheerful character manifested itself very early. It is known that he was in a punishment cell for launching a rocket in Kazan at 11 pm, that many other pranks were blamed on him. But besides this, more serious offenses are also noted: "free-thinking and dreamy self-conceit, perseverance" and even "outrageous deeds ... in which, to a large extent, he showed signs of godlessness."
For all this, N. I. Lobachevsky almost paid with exclusion from the university, and only the strengthened petitions of Kazan mathematics professors gave him the opportunity to graduate from it. His further career develops rapidly: 21 years old N. I. Lobachevsky is an adjunct, and 23 years old is an extraordinary professor; in the same years, in connection with lectures on geometry, read by him in 1816-1817, he first approached the question, the solution of which was the glory of his whole life - the question of the axiom of parallels.
The youth of N. I. Lobachevsky was coming to an end. A period of full disclosure of his rich and diverse personality began. Scientific creativity began, exceptional in its mathematical power. His amazingly multifaceted work, full of inexorable energy and passion, began and quickly developed as a professor, soon in all respects the first professor at Kazan University. His ardent participation began in all areas of activity, organization and construction of Kazan University, which then turned into almost twenty years of full and sole leadership of the entire university life. The mere enumeration of the various university posts, successively, and often in parallel, held by him, gives an idea of the scope of his university work. At the end of 1819 he was elected dean; at the same time, he is responsible for putting the university library in order, which was in an incredibly chaotic state. During the same years, his professorial activity received a new content: after Professor Simonov's departure on a round-the-world trip, for two whole academic years he had to read physics, meteorology and astronomy. By the way, N. I. Lobachevsky never lost interest in physics in the future and did not refuse not only to teach it at the university, but also to read popular lectures on physics, accompanied by carefully and interestingly prepared experiments. In 1822, N. I. Lobachevsky became an ordinary professor; at the same time he becomes a member of the building committee for putting the old and building new university buildings in order. In 1825 he was already chairman of this committee. In fact, he is the main builder of the entire set of new buildings of Kazan University and, carried away by these new duties, he carefully studies architecture both from the engineering and technical side, and from the artistic side. Many of the most architecturally successful buildings of Kazan University are the implementation of the construction plans of N. I. Lobachevsky; these are: anatomical theater, library, observatory.
Finally, in 1827, N. I. Lobachevsky became the rector of the university and held this post for 19 years. He understands his duties as a rector very broadly: from the ideological leadership of teaching and the entire life of the university to personal involvement in all daily university needs. Having become rector, he continued to carry out the duties of the university librarian for several more years and laid them down only after he had put the library on the proper height. As an example of the energy and activity shown by N. I. Lobachevsky for the benefit of the university, it should be said about his role during two tragic events that hit Kazan life during his rectorship. The first of these events was the cholera epidemic of 1830, which raged in the Volga region and claimed many thousands of lives. When cholera reached Kazan, N. I. Lobachevsky immediately took heroic measures against the university: the university was actually isolated from the rest of the city and turned into a kind of fortress. Accommodation and meals for students were organized on the university territory itself - all this with the most active participation of the rector. The success was brilliant - the epidemic passed by the university. The energetic selfless work of N. I. Lobachevsky in the fight against cholera made such a great impression on the entire society of that time that even official authorities considered it necessary to note it, N. I. Lobachevsky was expressed "the highest goodwill" for his diligence in protecting the university and other educational institutions from cholera.
Another disaster that broke out over Kazan was a fire in 1842, terrible in its devastating consequences. During this terrible fire, which destroyed a huge part of the city, N. I. Lobachevsky again showed miracles of energy and diligence in saving university property from fire. In particular, he managed to save the library and astronomical instruments.
However, the central point of application of the energy and talents of N. I. Lobachevsky as the rector of the university was his direct concern for the education of youth in the broadest sense of the word. All other aspects of his activities as a rector constituted only a framework for the implementation of this main task. The problems of upbringing attracted him in all their scope and, like everything that interested him, they interested him most ardently. Since 1818, N. I. Lobachevsky was a member of the school committee in charge of secondary and lower educational institutions, and since then he has not lost sight of, along with questions of university teaching, and requests school life. Constantly supervising entrance exams to the university, N. I. Lobachevsky knew perfectly well what knowledge a schoolboy of that time came to a higher educational institution with. Being interested in the whole line of human development - from childhood to late adolescence - he demanded a lot from education, and the ideal of the human personality that was drawn before him was very high. The speech of N. I. Lobachevsky "On the most important subjects of education" is a wonderful monument not only of pedagogical thought, but, if I may say so, of that "educational emotion", that pedagogical pathos, without which she herself pedagogical activity turns into a deadly craft. N. I. Lobachevsky himself possessed in full measure the diversity and breadth of vital interests that were part of his ideal of a harmoniously developed human personality. Naturally, he demanded a lot from a young man who came to the university to study. First of all, he demands from him that he be a citizen "who, with high knowledge, is the honor and glory of his fatherland," that is, sets before him a high and responsible patriotic ideal, based, in particular, on high qualifications within the chosen profession. But he further emphasizes that "mental education alone does not complete education," and makes great demands on intelligent person as a full-fledged representative of intellectual, ethical and aesthetic culture. N. I. Lobachevsky was not only a theorist of education, but in fact an educator, a teacher of youth. He was not only a professor who read his lectures brilliantly and carefully, but also a man who knew the direct road to a youthful heart and knew how, in all cases when it was required, to find those very necessary words that could act on a student who had gone astray, to return him to work, discipline him. The authority of N. I. Lobachevsky among the students was extremely high. The students loved Nikolai Ivanovich, despite his strictness as a professor and, in particular, as an examiner, despite his vehemence, and sometimes harshness.
N. I. Lobachevsky is probably the largest person nominated by almost two hundred years of glorious history of Russian universities. If he had not written a single line of independent scientific research, we, nevertheless, would have to remember him with gratitude as our most remarkable university figure, as a person who gave the high titles of professor and rector of the university such a completeness of content, which they were not given by any other of the persons who bore these titles before him, in his time or after his death. But N. I. Lobachevsky, in addition, was also a brilliant scientist, and if he were not such, if he, along with all his other talents, also had a first-class creative gift and creative experience, he would be in the field of university teaching, and university leadership, and his very educational activities could not be what he really was.
The main scientific merit of N. I. Lobachevsky lies in the fact that he was the first to fully see the logical unprovability of the Euclidean axiom of parallels and made all the main mathematical conclusions from this unprovability. The axiom of parallels, as you know, says: in a given plane to a given line, it is possible to draw only one parallel line through a given point not lying on this line. Unlike the rest of the axioms of elementary geometry, the axiom of parallels does not have the property of immediate evidence, at least for one thing, which is a statement about the entire infinite line as a whole, while in our experience we are faced only with larger or smaller "pieces" (segments ) straight lines. Therefore, throughout the history of geometry - from antiquity to the first quarter of the last century - there have been attempts to prove the axiom of parallels, that is, to derive it from the rest of the axioms of geometry. N. I. Lobachevsky also began with such attempts, having accepted the assumption opposite to this axiom that at least two parallel lines can be drawn to a given line through a given point. N. I. Lobachevsky tried to reduce this assumption to a contradiction. However, as he unfolded from the assumption he made and the totality of the rest of Euclid's axioms an ever longer chain of consequences, it became more and more clear to him that no contradiction not only could not be obtained, but could not be obtained. Instead of a contradiction, N. I. Lobachevsky received, though peculiar, but logically completely harmonious and impeccable system of sentences, a system that has the same logical perfection as ordinary Euclidean geometry. This system of sentences constitutes the so-called non-Euclidean geometry or Lobachevsky geometry.
Having received the conviction of the consistency of the geometric system he constructed, N. I. Lobachevsky did not give a rigorous proof of this consistency, and could not give it, since such a proof went beyond the methods of mathematics at the beginning of the 19th century. The proof of the consistency of Lobachevsky's geometry was given only at the end of the last century by Cayley, Poincare and Klein.
Without giving a formal proof of the logical equality of his geometric system with the usual system of Euclid, N. I. Lobachevsky in essence fully understood the undoubtfulness of the very fact of this equality, expressing with complete certainty that, given the logical impeccability of both geometric systems, the question of which of them is realized in the physical world, can only be decided by experience. N. I. Lobachevsky was the first to look at mathematics as an experimental science, and not as an abstract logical scheme. He was the first to set up experiments to measure the sum of the angles of a triangle; the first who managed to abandon the millennial prejudice of a priori geometric truths. It is known that he often liked to repeat the words: "Leave toil in vain, trying to extract all wisdom from one mind, ask nature, it keeps all secrets and your questions will be answered without fail and satisfactorily." In the point of view of N. I. Lobachevsky, modern science introduces only one amendment. The question of what kind of geometry is realized in the physical world does not have that immediate naive meaning that was attached to it in the time of Lobachevsky. After all, the most basic concepts of geometry - the concepts of a point and a straight line, having been born, like all our knowledge, from experience, are, nevertheless, not directly given to us in experience, but arose only by abstraction from experience, as our idealizations of experimental data, idealizations that only enable applications mathematical method to the study of reality. To clarify this, we will only point out that the geometric line, by virtue of its infinity alone, is not - in the form in which it is studied in geometry - the subject of our experience, but only an idealization of very long and thin rods or light rays directly perceived by us. . Therefore, the final experimental verification of the axiom of parallel Euclid or Lobachevsky is impossible, just as it is impossible to establish the sum of the angles of a triangle absolutely exactly: all measurements of any physical angles given to us are always only approximate. We can only assert that Euclid's geometry is an idealization of real spatial relationships, which fully satisfies us as long as we are dealing with "pieces of space not very large and not very small", i.e., until we go into either one or the other side too far beyond our usual, practical scales, as long as we, on the one hand, say, remain within the solar system, and on the other, do not plunge too deep into the atomic nucleus.
The situation changes when we move on to cosmic scales. The modern general theory of relativity considers the geometric structure of space as something dependent on the masses acting in this space and comes to the need to involve geometric systems that are "non-Euclidean" in a much more complex sense of the word than that associated with Lobachevsky's geometry.
The significance of the very fact of the creation of non-Euclidean geometry for all modern mathematics and natural sciences is colossal, and the English mathematician Clifford, who called N. I. Lobachevsky "Copernicus of geometry", did not fall into exaggeration. NI Lobachevsky destroyed the dogma of "immovable, the only true Euclidean geometry" in the same way as Copernicus destroyed the dogma about the Earth, which is immovable and constituting the unshakable center of the Universe. N. I. Lobachevsky convincingly showed that our geometry is one of several logically equal geometries, equally flawless, equally complete logically, equally true as mathematical theories. The question of which of these theories is true in the physical sense of the word, that is, which is most adapted to the study of one or another circle physical phenomena, is precisely a question of physics, not mathematics, and, moreover, a question whose solution is not given once and for all by Euclidean geometry, but depends on what kind of circle of physical phenomena we have chosen. The only, indeed significant, privilege of Euclidean geometry remains that it continues to be a mathematical idealization of our everyday spatial experience and therefore, of course, retains its main position both in a significant part of mechanics and physics, and even more so in all technology. But the philosophical and mathematical significance of the discovery of N. I. Lobachevsky, this circumstance, of course, cannot belittle.
These are, in brief, the main lines of the versatile cultural activity of Nikolai Ivanovich Lobachevsky. It remains to say a few more words about recent years his life. If the 20s and 30s of the XIX century. were the period of the highest flowering of both the creative and scientific-pedagogical and organizational activities of N. I. Lobachevsky, then from the middle of the forties, and moreover, quite suddenly for N. I. Lobachevsky, a period of inactivity and senile burnout begins. The main event that brought with it this tragic turning point in the life of N. I. Lobachevsky was his dismissal on August 14, 1846 from the post of rector. This dismissal happened without the desire of N. I. Lobachevsky and contrary to the petition of the university council. Almost simultaneously, he was dismissed from the post of professor of mathematics, so that in the spring of 1847, N. I. Lobachevsky found himself removed from virtually all his duties at the university. This suspension had all the features of a rough official disqualification, bordering on a direct insult.
It is quite understandable that N. I. Lobachevsky, for whom his work in the university field was a large and irreplaceable part of his life, took his resignation as a heavy, irreparable blow. This blow was especially hard, of course, because it broke out at that time in the life of N. I. Lobachevsky, when his creative scientific work was basically already completed and, consequently, university activities became the main content of his life. If we add to this the exceptionally active character of N. I. Lobachevsky and the habit, created over decades, of being a leader in organizational affairs, and not an ordinary participant, a habit to which he truly had the right, then the dimensions of the catastrophe that befell him become quite clear. Personal sorrows added to the cup: the beloved son of N. I. Lobachevsky died, an adult young man, according to his contemporaries, very similar to his father in appearance and character. N. I. Lobachevsky was never able to cope with this blow. Old age began - premature, but all the more oppressive, with increasing signs of paradoxically early decrepitude. His health was rapidly declining. He began to lose his sight, and by the end of his life he was completely blind. The last work "Pangeometry" was already dictated to him. Broken by life, a sick, blind old man, he died on February 24, 1856.
As a scientist N. I. Lobachevsky is in the full sense of the word a revolutionary in science. For the first time having made a breach in the idea of Euclidean geometry as the only conceivable system of geometric knowledge, the only conceivable set of proposals on spatial forms, N. I. Lobachevsky did not find not only recognition, but even a simple understanding of his ideas. It took half a century for these ideas to enter into mathematical science, become its integral part and become the turning point that determined to a large extent the entire style of mathematical thinking of the subsequent era and from which, in fact, Russian mathematics begins. Therefore, during his lifetime, N. I. Lobachevsky fell into the difficult position of an "unrecognized scientist." But this non-recognition did not break his spirit. He found a way out in that varied, ebullient activity, which has been briefly outlined above. The strength of Lobachevsky's personality triumphed not only over all the difficulties of the gloomy time in which he lived, it also triumphed over what, perhaps, is the most difficult thing for a scientist to endure: over ideological isolation, over a complete lack of understanding of what was dearest and most necessary to him. - his scientific discoveries and ideas. However, one should not blame his contemporaries, among whom were prominent scientists, for not understanding Lobachevsky. His ideas were far ahead of his time. Among foreign mathematicians, only the famous Gauss understood these ideas. But, in possession of them, Gauss never had the courage to publicly state this. However, he understood and appreciated Lobachevsky. He took the initiative in the only scientific honor that fell to the lot of Lobachevsky: on the proposal of Gauss, Lobachevsky was elected in 1842 a corresponding member of the Göttingen Royal Society of Sciences.
If N. I. Lobachevsky undoubtedly won the right to immortality in the history of science with his geometric works, then we should not forget that in other areas of mathematics he published a number of brilliant works on mathematical analysis, algebra and probability theory, as well as on mechanics, physics and astronomy.
The name of N. I. Lobachevsky entered the treasury of world science. But the brilliant scientist always felt like a fighter for the Russian national culture, her daily builder, living her interests, suffering from her needs.
The main works of N. I. Lobachevsky: Complete Works on Geometry, Kazan, 1833, vol. I (contains: On the Principles of Geometry, 1829; Imaginary Geometry, 1835; Application of Imaginary Geometry to Some Integrals, 1836; New Principles of Geometry with a Complete Theory of Parallels, 1835-1838); 1886, vol. II (contains works in foreign languages, including: Geometrische Untersuchungen zur Theorie der Parallellinien, 1840, in which N. I. Lobachevsky outlined his ideas about non-Euclidean geometry); Geometric research on the theory of parallel lines (Russian translation by A. V. Letnikov of the famous memoir of N. I. Lobachevsky Geometrische Untersuchungen...), "Mathematical Collection", M., 1868, III; Pangeometry, "Scientific Notes of the Kazan University", 1855; Complete works, M. - L., Gostekhizdat, 1946.
About N. I. Lobachevsky:Yanishevsky E., Historical note on the life and work of N. I. Lobachevsky, Kazan, 1868; Vasiliev A. V., Nikolai Ivanovich Lobachevsky, St. Petersburg, 1914; Sintsov D. M., Nikolai Ivanovich Lobachevsky, Kharkov, 1941; Nikolai Ivanovich Lobachevsky (on the 150th anniversary of his birth; articles by P. S. Aleksandrov and A. N. Kolmogorov), M. - L., 1943; Nikolai Ivanovich Lobachevsky (articles by B. L. Laptev, P. A. Shirokov, N. G. Chebotarev), ed. Academy of Sciences of the USSR, M. - L., 1943; Kagan V. F., The great scientist N. I. Lobachevsky and his place in world science, M. - L., 1943; his own, N. I. Lobachevsky, ed. Academy of Sciences of the USSR, M.-L., 1944.
Nikolai Ivanovich Lobachevsky - an outstanding Russian mathematician, for four decades - rector, activist of public education, founder of non-Euclidean geometry.
This is a man who was several decades ahead of his time and remained misunderstood by his contemporaries.
Biography of Lobachevsky Nikolai Ivanovich
Nicholas was born on December 11, 1792 in poor family petty official Ivan Maksimovich and Praskovya Alexandrovna. The birthplace of the mathematician Nikolai Ivanovich Lobachevsky is Nizhny Novgorod. At the age of 9, after the death of his father, he was transported by his mother to Kazan and in 1802 was admitted to the local gymnasium. After graduating in 1807, Nikolai became a student at the newly founded Kazan Imperial University.
Under the tutelage of M. F. Bartels
A special love for the physical and mathematical sciences was able to instill in the future genius Grigory Ivanovich Kartashevsky, a talented teacher who deeply knew and appreciated his work. Unfortunately, at the end of 1806, due to disagreements with the leadership of the university, "for displaying a spirit of disobedience and disagreement," he was dismissed from the university service. Bartels, a teacher and friend of the famous Carl Friedrich Gauss, began to teach mathematics courses. Arriving in Kazan in 1808, he took patronage over a capable but poor student.
The new teacher approved of the success of Lobachevsky, who, under his supervision, studied such classics as "The Theory of Numbers" by Carl Gauss and "Celestial Mechanics" by the French scientist Pierre-Simon Laplace. For disobedience, stubbornness and signs of godlessness in his senior year, the likelihood of expulsion hung over Nikolai. It was the patronage of Bartels that contributed to the removal of the danger hanging over the gifted student.
in the life of Lobachevsky
In 1811, after graduation, Nikolai Ivanovich, whose brief biography is of sincere interest to the younger generation, was approved as a master in mathematics and physics and left at the educational institution. Two scientific studies - in algebra and mechanics, presented in 1814 (earlier than the deadline), led to his elevation to adjunct professor (associate professor). Further, Nikolai Ivanovich Lobachevsky, whose achievements would later be correctly assessed by descendants, began teaching himself, gradually increasing the range of courses he taught (mathematics, astronomy, physics) and seriously thinking about the restructuring of mathematical principles.
The students loved and highly appreciated the lectures of Lobachevsky, who a year later was awarded the title of extraordinary professor.
New orders of Magnitsky
In order to suppress freethinking and revolutionary mood in society, the government of Alexander I began to rely on the ideology of religion with its mystical-Christian teachings. Universities were the first to undergo drastic checks. In March 1819, M. L. Magnitsky, a representative of the main board of schools, arrived in Kazan with an audit, taking care exclusively of his own career. According to the results of his check, the state of affairs at the university turned out to be extremely deplorable: the lack of scholarship of the pupils of this institution entailed harm to society. Therefore, the university needed to be destroyed (publicly destroyed) - with the aim of an instructive example for the rest.
However, Alexander I decided to correct the situation with the hands of the same inspector, and Magnitsky, with particular zeal, began to “put things in order” within the walls of the institution: he removed 9 professors from work, introduced the strictest censorship of lectures and a harsh barracks regime.
The wide activity of Lobachevsky
The biography of Nikolai Ivanovich Lobachevsky describes the difficult period of the church-police system established at the university, which lasted for 7 years. The strength of the rebellious spirit and the absolute employment of the scientist, which did not leave a single minute of free time, helped to withstand difficult tests.
Nikolai Ivanovich Lobachevsky replaced Bartels, who left the walls of the university, and taught mathematics in all courses, also headed the physics room and read given subject, taught students astronomy and geodesy, while I. M. Simonov was in world tour. Enormous work was invested by him in putting the library in order, and especially in filling its physical and mathematical part. Along the way, mathematician Nikolai Ivanovich Lobachevsky, being the chairman of the construction committee, supervised the construction of the main building of the university and for some time served as dean of the Faculty of Physics and Mathematics.
Non-Euclidean geometry of Lobachevsky
The colossal number of current affairs, extensive pedagogical, administrative and research work did not become an obstacle to creative activity mathematics: 2 textbooks for gymnasiums came out from under his pen - “Algebra” (convicted for use and “Geometry” (not published at all). Strict supervision was established for Nikolai Ivanovich by Magnitsky, due to his insolence and violation of established instructions However, even under these conditions, which are degrading to human dignity, Lobachevsky Nikolai Ivanovich worked hard on the strict construction of geometric foundations.The result was the discovery of a new geometry by scientists, accomplished on the path of a radical revision of the concepts of the era of Euclid (3rd century BC).
In the winter of 1826, a Russian mathematician carried out a report on geometric principles, which was submitted for review to several eminent professors. However, the expected review (neither positive nor even negative) was not received, and the manuscript of the valuable report has not survived to our times. The scientist included this material in his first work "On the Principles of Geometry", published in 1829-1830. in the Kazan Bulletin. In addition to presenting important geometric discoveries, Nikolai Ivanovich Lobachevsky described a refined definition of a function (clearly distinguishing between its continuity and differentiability), undeservedly attributed to the German mathematician Dirichlet. Also, the scientists made careful studies of trigonometric series, evaluated several decades later. A talented mathematician is the author of a method for the numerical solution of equations, which over time was unfairly called the “Greffe method”.
Lobachevsky Nikolai Ivanovich: interesting facts
The auditor Magnitsky, who for several years inspired fear with his actions, was expected by an unenviable fate: for many abuses revealed by a special audit commission, he was removed from his post and sent into exile. Mikhail Nikolayevich Musin-Pushkin was appointed the next trustee of the educational institution, who was able to appreciate vigorous activity Nikolai Lobachevsky and recommended him for the post of rector of Kazan University.
For 19 years, starting in 1827, Lobachevsky Nikolai Ivanovich (see photo of the monument in Kazan above) worked hard in this post, achieving the dawn of his beloved offspring. On account of Lobachevsky - a clear improvement in the level of scientific and educational activities in general, the construction of a huge number of office buildings (physics office, library, chemical laboratory, astronomical and magnetic observatory, mechanical workshops). The rector is also the founder of the strict scientific journal "Scientific Notes of the Kazan University", which replaced the "Kazan Vestnik" and was first published in 1834. In parallel with the rector's office for 8 years, Nikolai Ivanovich was in charge of the library, was engaged in teaching activities, and wrote instructions to mathematics teachers.
Lobachevsky's merits include his sincere cordial concern for the university and its students. So, in 1830, he managed to isolate the educational territory and conduct a thorough disinfection in order to save the staff of the educational institution from the cholera epidemic. During a terrible fire in Kazan (1842), he managed to save almost all educational buildings, astronomical instruments and library material. Nikolai Ivanovich also opened free access to the university library and museums to the general public and organized popular science classes for the population.
Thanks to the incredible efforts of Lobachevsky, the authoritative, first-class, well-equipped Kazan University has become one of the best educational institutions in Russia.
Misunderstanding and rejection of the ideas of the Russian mathematician
All this time, the mathematician did not stop in ongoing research aimed at developing new geometry. Unfortunately, his ideas - deep and fresh, went so against the generally accepted axioms that contemporaries failed, and perhaps did not want to appreciate the works of Lobachevsky. Misunderstanding and, one might say, bullying to some extent did not stop Nikolai Ivanovich: in 1835 he published "Imaginary Geometry", and a year later - "The Application of Imaginary Geometry to Some Integrals". Three years later, the world saw the most extensive work, New Principles of Geometry with a Complete Theory of Parallels, which contained a concise, extremely clear explanation of his key ideas.
A difficult period in the life of a mathematician
Not getting an understanding native land, Lobachevsky decided to acquire like-minded people outside of it.
In 1840, Lobachevsky Nikolai Ivanovich (see photo in the review) published his work with clearly stated main ideas in German. One copy of this edition was handed to Gauss, who himself was secretly engaged in non-Euclidean geometry, but did not dare to speak publicly with his thoughts. Having familiarized himself with the works of the Russian colleague, the German recommended that the Russian colleague be elected to the Gottingen Royal Society as a corresponding member. Gauss spoke laudatory about Lobachevsky only in his own diaries and among the most trusted people. The election of Lobachevsky nevertheless took place; this happened in 1842, but it did not improve the position of the Russian scientist in any way: he had to work at the university for another 4 years.
The government of Nicholas I did not want to evaluate the many years of work of Nikolai Ivanovich Lobachevsky and in 1846 suspended him from work at the university, officially naming the reason: a sharp deterioration in health. Formally, the former rector was offered the position of assistant trustee, but without a salary. Shortly before the dismissal and deprivation of the professorial department, Lobachevsky Nikolai Ivanovich, whose brief biography is still being studied in educational institutions, recommended instead of himself the teacher of the Kazan gymnasium A.F. Popov, who perfectly defended his doctoral dissertation. Nikolai Ivanovich considered it necessary to give the right path in life to a young capable scientist and found it inappropriate to occupy the chair under such circumstances. But, having lost everything at once and finding himself in a position that was completely unnecessary for himself, Lobachevsky lost the opportunity not only to lead the university, but also to somehow participate in the activities of the educational institution.
In family life, Lobachevsky Nikolai Ivanovich since 1832 was married to Varvara Alekseevna Moiseeva. In this marriage, 18 children were born, but only seven survived.
last years of life
Forced removal from the business of his whole life, rejection of the new geometry, the rude ingratitude of his contemporaries, a sharp deterioration in the financial situation (due to ruin, the wife's estate was sold for debts) and family grief (the loss of the eldest son in 1852) had a devastating effect on physical and spiritual health Russian mathematician: he noticeably haggard and began to lose his sight. But even the blind Nikolai Ivanovich Lobachevsky did not stop attending exams, came to solemn events, participated in scientific disputes and continued to work for the benefit of science. The main work of the Russian mathematician "Pangeometry" was written by students under the dictation of the blind Lobachevsky a year before his death.
Lobachevsky Nikolai Ivanovich, whose discoveries in geometry were appreciated only decades later, was not the only researcher in the new field of mathematics. The Hungarian scientist Janos Bolyai, independently of his Russian colleague, brought to the court of his colleagues in 1832 his vision of non-Euclidean geometry. However, his works were not appreciated by contemporaries.
The life of an outstanding scientist, wholly devoted to Russian science and Kazan University, ended on February 24, 1856. They buried Lobachevsky, who was never recognized during his lifetime, in Kazan, at the Arsky cemetery. Only after a few decades did the situation in the scientific world change dramatically. A huge role in the recognition and acceptance of the works of Nikolai Lobachevsky was played by the studies of Henri Poincare, Eugenio Beltrami, Felix Klein. The realization that Euclidean geometry had a full-fledged alternative had a significant impact on the scientific world and gave impetus to other bold ideas in the exact sciences.
The place and date of birth of Nikolai Ivanovich Lobachevsky are known to many contemporaries related to the exact sciences. In honor of Nikolai Ivanovich Lobachevsky, a crater on the Moon was named. The name of the great Russian scientist is scientific Library University in Kazan, to which he dedicated a huge chunk of his life. There are also Lobachevsky streets in many cities of Russia, including Moscow, Kazan, Lipetsk.
Send your good work in the knowledge base is simple. Use the form below
Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you.
Posted on http://www.allbest.ru/
Ukhta State Technical University, Ukhta
Life of N.I. Lobachevsky and his scientific activity
“Sometimes a person is given credit even if he did not borrow.”
Nikolai Ivanovich Lobachevsky was born in 1792 in Nizhny Novgorod. Nikolai Ivanovich had older and younger brothers. Nikolai's father, Ivan Maksimovich Lobachevsky, worked as an official in Nizhny Novgorod. His wife, Praskovya Alexandrovna, was the daughter of poor townspeople, nothing more is known about her. Nikolai's parents got married at a young age, both were not yet eighteen at the time of the wedding. Soon after the move, the father of the future great scientist dies at the age of 40, leaving his family in a difficult financial situation. However, the Lobachevsky brothers were brought up in the house of the land surveyor Sergei Stepanovich Shebarshin, and did not live in poverty. In 1802, Praskovya Alexandrovna sent her sons to the Kazan gymnasium, for state support. At first, the University program was not much different from the gymnasium, but the situation changed for the better in 1808 with the arrival of prominent foreign scientists Caspar Renner, professor of mathematics, Martin Bartels, also a professor of mathematics, who was a teacher and friend of Karl Gauss. The latter instilled in Lobachevsky an interest in geometry. Already at the age of 19, Nikolai Ivanovich received a master's degree, and was left at the university to prepare for a professorship. In the same year, together with M. Bartels, they study in depth the classical works of Gauss and Laplace: “The Theory of Numbers” and the first volumes of “Celestial Mechanics”. The study of these works prompted Lobachevsky to start his own research. In 1811 he published "Theory of the elliptical motion of bodies" and in 1813 - "On the resolution algebraic equation x m? 1 = 0". In 1814 he began teaching.
Non-Euclidean Geometry - the main work of Lobachevsky's life, a scientific feat, had a huge impact on the further development of mathematics and mathematical thinking. The first work related to this topic was published by Lobachevsky, already being the rector of Kazan University, in 1826 "A concise presentation of the foundations of geometry with a rigorous proof of parallel theorems." Lobachevsky was the first scientist who presented to the public works on this topic. Other scientists also dealt with this problem, but Lobachevsky made the greatest contribution to its solution, therefore, the geometry he created bears his name. Also, among the published works of the scientist: “On the principles of geometry” (1829-1830), “Imaginary geometry” (1835), “The application of imaginary geometry to certain integrals” (1836), “New principles of geometry with a complete theory of parallel” (1835- 1838), “Geometric studies on the theory of parallel lines” (1840). At the heart of the mathematical discipline is a system of postulates and axioms. Lobachevsky's geometry is no exception. Lobachevsky accepts all the axioms and postulates proposed by the geometry of Euclid and do not depend on the V postulate, and replaces the V postulate with his own: “On the plane, through a point that does not lie on a line, more than one line can be drawn that does not intersect this one.”
Two boundary lines xx" and yy" (Fig. 1) do not intersect the line R and are called parallel to it at the point P.
All lines inside the angle xPy intersect the line R. PB is the perpendicular to the line R.
The angle is called the angle of parallelism.
The lines inside the angles xPy" and yPx" do not intersect the line R- are called diverging from the line R.
This is the main difference between Lobachevsky geometry and Euclidean geometry. It is also important to note that in Lobachevsky geometry:
1) The sum of the angles of a triangle is always less than 2d (two lines)
2) There are no similar figures.
3) The unit of length is given by some geometric construction, that is, the space itself with its geometric properties defines a particular unit of length.
4) The direction of parallelism is set.
The space in which the Lobachevsky axiom is supposed to be fulfilled is called the Lobachevsky space. Mutual arrangement lines and planes in space is characterized by the cone of parallelism, which is an analogue of the concept of the angle of parallelism. Let the Alpha plane and a point P not lying on it (Fig. 2) be given, PP "is perpendicular to Alpha. Pb is a straight line parallel to the Alpha plane and P"B" is its projection onto this plane. Then the angle bPP" is the angle of parallelism at point P with respect to P"B". We will rotate the line Pb around the perpendicular PP", and then Pb will describe a conical surface with a vertex at the point P. This surface is called the cone of parallelism. Thus, all generators of this cone are parallel to the plane alpha. Any line passing through the point P inside the cone intersects the plane alpha passing outside the cone - diverges from alpha.
· Any plane that intersects a cone along two generators intersects Alpha.
· Any plane passing along one generatrix of the cone is parallel to Alpha.
· Any plane that intersects only the top of the cone is called diverging from the Alpha plane.
For the first time, the realization of Lobachevsky's geometry on surfaces was established by the Italian mathematician Beltrami in 1868 (Fig. 3). He noticed that the geometry on a piece of the Lobachevsky plane coincides with the geometry on surfaces of constant negative curvature, the simplest example which the pseudosphere represents. However, only a local interpretation of the geometry is given here, that is, on a limited area, and not on the entire Lobachevsky plane.
Three years later, in 1871, the German mathematician Klein came up with another, full-fledged model (Fig. 4). The plane in it is the inside of the circle, the straight line is the chord, excluding the ends, the point is the point inside the circle. The belonging between them is understood in the usual Euclidean sense, however, the V postulate of Euclid is no longer fulfilled here, but precisely Lobachevsky's axiom is fulfilled: there are infinitely many lines passing through the point P that do not intersect the line a. Also, all the consequences of the axiom are fulfilled.
In 1882, another model of Lobachevsky's geometry was presented by the French mathematician Poincaré (Fig. 5). The role of the Lobachevsky plane is played by the open half-plane P, the role of the straight lines is played by the semicircles contained in it, with centers on the bounding line p, and the rays perpendicular to this line. The “straight” point serves as the beginning of two rays, two arcs of semicircles (with excluded ends). The bounding line is also excluded. An angle is a figure of two rays with a common origin, not contained in one straight line. Half-lines perpendicular to the boundary line are the limits of the considered semicircles (see Fig. b). When the center of the semicircle moves away along the bounding straight line, and the semicircle passes through the point, then in the limit it “straightens out” and also becomes a half-line. Therefore, semicircles of infinite radius are considered as straight lines in this model. All the axioms of Euclidean geometry are satisfied here, except for the parallel axiom. Thus, the Lobachevsky geometry is satisfied in this model. You can build an analytical model of geometry by representing points as coordinates and expressing the distance as a formula in coordinates. Such a model of Lobachevsky's geometry was given by the German mathematician Riemann as a special case of the general geometry defined by him, now called Riemannian.
The scientific ideas of Lobachevsky were not understood by most of his contemporaries, and after the publication of the first work on “imaginary geometry”, Nikolai Ivanovich was subjected to the most severe persecution in his homeland. The only lifetime recognition of his scientific merit was the election to the Göttingen Royal Scientific Society, thanks to the recommendations of Gauss. But, nevertheless, Lobachevsky did not give up, and until the end of his life he believed that the triumph of his ideas was inevitable. In 1855, having lost his sight due to difficult experiences and constant mental stress, he dictates his last work"Pangeometry". He died the following year. However, after the death of Lobachevsky, his ideas attracted the attention of the scientific community, and served as a powerful incentive to revise the views on the foundations of geometry. Its geometry has found application in general and special relativity, in number theory (in its geometric methods). Lobachevsky's geometry also has a philosophical meaning, as it expands our understanding of the structure of the world and space. At the moment, there are many scientific works devoted to the geometry of Lobachevsky, both in domestic literature and in foreign ones. The study of Lobachevsky geometry is a mandatory part of the program of mathematical departments of most of our universities and all pedagogical institutes - familiarization with the basics of this geometric system is considered a necessary part of the training of a future teacher high school. Lobachevsky's geometry classes are also widely cultivated in school mathematical circles.
geometry elliptic lobachevsky
List of used literature
1) Geometry of Lobachevsky [Electronic resource]:
http://en.wikipedia.org/wiki/Lobachevsky_geometry
2) Geometry of Lobachevsky [Electronic resource]:
http://geom.kgsu.ru/index.php
3) Lobachevsky, Nikolai Ivanovich [Electronic resource]:
http://en.wikipedia.org/wiki/Nikolai_Lobachevsky
4) Poincare model [Electronic resource]:
http://geometrie.ru/site/lobachevskiy/m1.htm
5) Shirokov P. A. A brief outline of the foundations of Lobachevsky's geometry [text]: /P. A. Shirokov - 2nd edition - M.: Nauka, 1983 - 80 p.
Hosted on Allbest.ru
...Similar Documents
Origin of non-Euclidean geometry. The emergence of "Lobachevsky geometry". Axiomatics of Lobachevsky planimetry. Three models of Lobachevsky geometry. The Poincaré and Klein model. Mapping of Lobachevsky geometry on a pseudosphere (Beltrami's interpretation).
abstract, added 03/06/2009
Biography of N.I. Lobachevsky. Lobachevsky's activities in organizing a printed university organ and his attempts to found at the university Scientific society. The history of the recognition of geometry by N.I. Lobachevsky in Russia. The emergence of non-Euclidean geometry.
thesis, added 09/14/2011
The history of the emergence of non-Euclidean geometry. Comparison of Euclid's and Lobachevsky's parallel postulates. Basic concepts and models of Lobachevsky geometry. Triangle and polygon defect, absolute unit of length. Definition of a parallel line.
term paper, added 03/15/2011
short biography N.I. Lobachevsky. The history of the discovery of non-Euclidean geometry. Basic facts and consistency of Lobachevsky geometry, its significance and application in mathematics and physics. The way of recognition of the ideas of N.I. Lobachevsky in Russia and abroad.
thesis, added 08/21/2011
Student years N.I. Lobachevsky. The first years of teaching. Organization of a printed university organ. The history of the discovery of non-Euclidean geometry. Recognition of the geometry of N.I. Lobachevsky and its application in mathematics and physics.
thesis, added 03/05/2011
Geometric shapes on the surface of a sphere. Basic facts of spherical geometry. Lobachevsky's concepts of geometry. Surface of constant negative curvature. Lobachevsky's geometry real world. Basic concepts of Riemann's non-Euclidean geometry.
presentation, added 04/12/2015
The Poincaré model of Lobachevsky's geometry: the question of its consistency. Inversion, its analytical task. Transformation of a circle and a straight line, preservation of angles during inversion. Invariant lines and circles. Lobachevsky's system of axioms of geometry.
thesis, added 09/10/2009
An overview of the five groups of axioms on which Lobachevsky's planimetry is based. The essence of the Cayley-Klein model in higher geometry. Features of the proof of the cosine theorem, theorems on the sum of the angles of a triangle, on the fourth criterion for the congruence of triangles.
term paper, added 06/29/2013
Biography of the Russian scientist N.I. Lobachevsky. Hilbert's system of axioms. Parallel lines, triangles and quadrilaterals on the plane and space according to Lobachevsky. The concept of spherical geometry. Proof of theorems on various models.
abstract, added 11/12/2010
The study of the stages of development of geometry - a science that studies spatial relationships and forms, as well as other relations and forms similar to spatial ones in their structure. Geometry ancient egypt, Greece, Middle Ages. Postulates of N.I. Lobachevsky.
N. I. Lobachevsky. His life and scientific activity Litvinova Elizaveta Fedorovna
Chapter VII
Scientific activity of Lobachevsky. – From the history of non-Euclidean or imaginary geometry. – Participation of Lobachevsky in the creation of this science. - Different, modern views on the future of non-Euclidean geometry and its relation to Euclidean. – A parallel between Copernicus and Lobachevsky. – Consequences from the works of Lobachevsky for the theory of knowledge. – Works of Lobachevsky on pure mathematics, physics and astronomy .
The origin of imaginary, or non-Euclidean, geometry originates from the postulate of Euclid, which we all meet in the course of elementary geometry. When studying geometry in childhood, we are usually surprised not by the postulate itself, accepted without proof, but by the statement of the teacher that all attempts to prove it have so far been unsuccessful.
Firstly, it seems obvious to us that the perpendicular and the oblique will intersect with sufficient continuation, and secondly, it seems so easy to prove. And it is difficult to find a person who has studied geometry and has never tried to prove Euclid's postulate. It can be said that talented and mediocre people are equally subject to this temptation, with the only difference that the former soon become convinced of the inconsistency of their evidence, while the latter persist in their opinion. Hence the countless number of attempts to prove the mentioned postulate.
On this postulate, as is known, the theory of parallel lines is built, on the basis of which the Thales theorem is proved on the equality of the sum of the angles of a triangle to two right angles. If it were possible, without resorting to the theory of parallels, to prove that the sum of the angles of a triangle is equal to two right angles, then from this theorem one could derive proofs of Euclid's postulate, and in this case all elementary geometry would be a strictly deductive science.
We know from the history of geometry that a Persian mathematician, who lived in the middle of the thirteenth century, was the first to pay attention to the Thales theorem and tried to prove it without using the theory of parallels. AT basis In this proof, as in all subsequent ones, it was easy to see the silent assumption of the same postulate of Euclid. Of the innumerable subsequent attempts of this kind, only the works of Legendre, who dealt with this issue for almost half a century, deserve attention.
Legendre sought to prove that the sum of the angles of a triangle cannot be more or less than two lines; from this, of course, it would follow that it should be equal to two straight lines. Currently, Legendre's proof is recognized as untenable. Be that as it may, without reaching his main goal, Legendre did a lot to present the geometry of Euclid in the sense of adapting it to the requirements of the new time, and elementary geometry in the form in which it is now passed, with all its advantages and disadvantages, belongs to Legendre .
The Italian Jesuit Saccheri in 1733 in his research approached the ideas of Lobachevsky, that is, he was ready to reject the postulate of Euclid, but did not dare to express this, but strove at all costs prove him, and of course, just as unsuccessfully.
At the end of the last century in Germany, the brilliant Gauss in 1792 for the first time asked himself a bold question: what will happen to geometry if the postulate of Euclid is rejected? This question was born, one might say, together with Lobachevsky, who answered it by creating his own imaginary geometry. Here it seems to us to decide whether this question arose independently in the mind of our Lobachevsky, or whether it was raised by Bartels, having communicated to a gifted student the idea of his friend Gauss, with whom, until his departure for Russia, he maintained an active personal relationship. Some modern Russian mathematicians, prompted probably by the best of feelings, are striving to prove that Gauss' thought arose in Lobachevsky's mind quite independently. Prove it's impossible; everyone knows the letter of Gauss, referring to 1799, in which he says: "It is possible to construct a geometry for which the axiom of parallel lines does not hold."
Let us refer to the words of the Kazan professor Vasiliev, who proved his deep respect for the merits and memory of Lobachevsky; speaking of Bartels' close relationship with Gauss, he remarks:
Therefore, it cannot be considered too risky to suggest that Gauss shared his thoughts on the theory of parallels with his teacher and friend Bartels. Could Bartels, on the other hand, have failed to report Gauss' bold views on one of the fundamental questions of geometry to his inquisitive and talented Kazan student? Of course he couldn't.
But does all this detract from the merits of Lobachevsky? Of course not.
Legendre's works, which we mentioned, appeared in 1794. They did not satisfy, but revived interest in the theory of parallels, and we know that in the first twenty-five years of our century, writings relating to the theory of parallels appeared incessantly. According to Professor Vasiliev, many of them are still preserved in the library of Kazan University and, as it is reliably known, were acquired by Lobachevsky himself.
In 1816, Gauss assessed all these attempts as follows: “There are few questions in the field of mathematics about which so much would be written as about a gap in the principles of geometry, and yet we must admit honestly and frankly that, in essence, we have not gone beyond two thousand years further than Euclid. Such a frank and direct consciousness is more in line with the dignity of science than vain desires to hide the gap ... "
From all this we see that at the time when Lobachevsky entered the mathematical field, everything was prepared for the solution of the problem of the theory of parallels in the sense in which it was done by Lobachevsky. In 1825, the theory of parallels by the German mathematician Taurinus came out, which mentions the possibility of such a geometry in which Euclid's postulate does not hold. Lobachevsky's first work on this subject was presented to the Faculty of Physics and Mathematics in Kazan in 1826; it was published in 1829, and in 1832 a collection of works by Hungarian scientists, father and son Boliay, appeared on non-Euclidean geometry. We know that Father Boliai was a friend of Gauss; from this we can conclude that he was more familiar than Lobachevsky with the thoughts of Gauss; meanwhile, Lobachevsky's geometry received the right of citizenship in Western Europe. Lobachevsky's first work, which appeared in German, deserved, as we said, the approval of Gauss. Regarding him, Gauss wrote to Schumacher: “You know that for fifty-four years I have shared the same views. Actually, I did not find a single fact in Lobachevsky's work that was new to me; but presentation very different from that what am I intended to give this subject. The author talks about the subject like a connoisseur, in a true geometrical spirit. I felt obliged to draw your attention to this book "Geometrische Untersuchungen zur Theorie der Parallellinien", the reading of which will certainly bring you great pleasure. This letter was written in Göttingen and refers to 1846. However, it cannot be concluded that Gauss did not know about Lobachevsky's work from Bartels earlier. We will say more: it is impossible to admit that Bartels kept silent about the successes of his talented student.
From what we have said, it is obvious that the cornerstone of Lobachevsky's geometry is the negation of Euclid's postulate, without which geometry seemed unthinkable for about two thousand years. We know how firmly people have always held on to the heritage of centuries and how much courage is required from a person who destroys age-old delusions. From the sketch of Lobachevsky's life, we saw how little he was appreciated and understood by his contemporaries as a scientist. And now, a hundred years after his birth, ordinary educated people hold a deep prejudice against Lobachevsky's geometry, if only they know of its existence. It is impossible to express this geometry in a popular form, just as it is impossible to explain to a deaf person the delights of nightingale trills. In order to understand the significance of this abstract science, it is necessary to be able to think abstractly, which can be obtained only by long studies in philosophy and mathematics. With this in mind, we will only say about the geometry created by Lobachevsky what it consists of, what significance modern scientists attribute to it, how and by whom it was developed after Lobachevsky, and what these later works were related to the works of Lobachevsky himself. In all this, to the reader who is not initiated into secrets higher mathematics, you have to take the word of the authorities.
In the anniversary speeches and pamphlets dedicated to the memory of Lobachevsky, Russian mathematicians made every effort to explain to the public the nature and significance of Lobachevsky's scientific merits, and since they concerned mainly imaginary geometry, we have to use these efforts in this case. But, having carefully followed the oral and printed reviews of the educated public, we noticed a general dissatisfaction and the following requirements quite clearly stated: for a person who knows only the geometry of Euclid, the most significant question is what relation does Lobachevsky's geometry have to this geometry. And this subject is also discussed in the speeches mentioned, but still here, apparently, the public demands direct answers to the following questions: does Lobachevsky's geometry refute Euclid's geometry, does it replace it, making it redundant, or is it only a generalization of the latter? What does it have to do with the fourth dimension, which has done such a service to spiritists? Should Lobachevsky be considered, despite all his virtues, a dreamer in science, and why is Lobachevsky called the Copernicus of geometry?
We have already said that at first Lobachevsky had in mind only to improve the exposition of Euclidean geometry, to impart greater rigor to its principles, and did not in the least think of undermining these principles. The attempts of such a strong mind as Legendre possessed finally convinced true mathematicians of the impossibility of proving Euclid's postulate logically, that is, deriving it from the properties of a plane and a straight line. Then Lobachevsky, who in general had a penchant for philosophy, came up with the idea of checking whether Euclid's postulate is confirmed by experience within the limits of the greatest distances accessible to us.
Note that in the experiment he was looking for checks, and not proof of postulate.
The greatest distances available to man are those that give him astronomical observations. Lobachevsky made sure that for these distances the results of observations are compatible with Euclid's postulate. It follows from this that the absence of a logical proof of this postulate does not in the least undermine the truth of geometry for available us distances, and at the same time, the laws of mechanics and physics based on it retain their truth.
But it is natural for a person to ask himself with the thought: “What is there, beyond the distances accessible to us? For those that we call infinite, do the properties of our space have absolute significance? Here is the question that Lobachevsky proposed to himself.
Lobachevsky constructed his geometry logically, assuming the axioms known to us relating to the line and the plane, and assuming as a hypothesis that the sum of the angles of a triangle is less than two lines. But even with such an assumption, which can only take place for spaces that are much larger than our solar system, Lobachevsky's geometry for the measurements available to us gives the same results as Euclid's geometry. Quite correctly, or rather, thoroughly, one geometer called Lobachevsky's geometry stellar geometry. One can form an idea of infinite distances if one remembers that there are stars from which light reaches the Earth for thousands of years. So, the geometry of Lobachevsky includes the geometry of Euclid not as private, but as special happening. In this sense, the first can be called a generalization of the geometry known to us. Now the question arises, does Lobachevsky own the invention of the fourth dimension? Not at all. The geometry of four and many dimensions was created by the German mathematician, a student of Gauss, Riemann. The study of the properties of spaces in general view now constitutes a non-Euclidean geometry, or Lobachevsky geometry. The Lobachevsky space is space of three dimensions, which differs from ours in that the postulate of Euclid does not take place in it. The properties of this space are now being understood by assuming a fourth dimension. But this step already belongs to the followers of Lobachevsky. Therefore, non-Euclidean geometry adjoins and constitutes, as it were, a continuation of its geometry of many dimensions, which, while giving great generality and abstractness to many problems of geometry, at the same time is an indispensable tool in solving many problems of analysis.
Riemann, in his treatise On the Hypotheses Underlying Geometry, expressed the idea that Euclid's geometry is not a necessary consequence of our concepts of space in general, but is the result of experience, hypotheses that find their confirmation within the limits of our observations. Riemann gave general formulas, using which and applying which to the study of the so-called pseudospherical surface (glass view), the Italian mathematician Beltrami found that all the properties of lines and figures of geometry Lobachevsky belong to lines and figures on this surface. This is how the geometry of many dimensions was related to the geometry of Lobachevsky.
The works of Beltrami led to the following important conclusions: 1) geometry two dimensions Lobachevsky is not an imaginary geometry, but has an objective existence and a completely real character; 2) what in Lobachevsky's geometry corresponds to our plane is a pseudospherical (glass) surface, and what he calls a straight line is a geodesic line (the shortest distance between two points) of this surface.
The existence of a geometry of two dimensions, different from our planimetry, is easy to imagine. Let us imagine a spherical surface, elliptical or some kind of concave, and imagine lines and figures on it. Convex and concave surfaces are called curves surfaces.
Our plane, a straight surface, has no curvature, and in mathematics it is customary to say: the curvature of the plane is zero. Similarly, our space has no curvature. Curved surfaces have either positive or negative curvature. The glass surface has a negative curvature, while the elliptical surface has a positive one. Similarly, negative curvature is attributed to this Lobachevsky space.
The Lobachevsky space, as differing significantly from ours, cannot be imagined introduce, it is only conceivable. The same applies to spaces of four and many dimensions.
Closely related to Riemann's research are the works of Helmholtz, who rightly says: "While Riemann entered this new field of knowledge, starting from the most general and basic questions, I myself came to similar conclusions."
Riemann proceeded in his research from an algebraic general expression for the distance between two infinitely close points, and from this he deduced various properties of spaces; Helmholtz, proceeding from the fact of the possibility of movement of figures and bodies in our space, finally deduced the Riemann formula. Possessing an extremely clear mind, Helmholtz, as it were, illuminated for us the whole depth of Riemann's thoughts.
In this case, it is especially important for us that, by explaining to us the origin of geometric axioms, he indirectly determined the relationship between Lobachevsky's geometry and ours.
According to Helmholtz, the main difficulty in purely geometric studies is the ease with which we here mix daily an experience With logical thought processes. Helmholtz proves that much of Euclid's geometry relies on experience and cannot be deduced by logical means. It is remarkable that construction problems play such an essential role in geometry. At first glance, they seem to be nothing more than practical actions, but in fact they have the force of provisions. To make the equality clear geometric shapes, usually they are mentally superimposed one on top of the other. In the possibility of such a situation, we early age actually convinced. Helmholtz also proves that the special characteristic features of our space are of experiential origin.
On the basis of physiological data relating to the structure of our sense organs, Helmholtz comes to the conviction, which is very important for us, that all our abilities for sensory perceptions extend to the Euclidean space of three dimensions, any space, although three dimensions, but having a curvature, or space with more than three dimensions, we, by virtue of our very organization, are not able to imagine.
Thus, the teaching of Helmholtz, who is justly considered the genius of our century, confirms, for its part, the results obtained by the mathematicians Riemann and Lobachevsky. But if we are unable by any natural or artificial means to obtain this performance, it's still geometry two dimensions other than ours is available to our representation. Helmholtz gives us the means to penetrate into the essence of pseudo-spherical and spherical geometry, resorting to extremely ingenious methods, which, of course, we will not dwell on. In this case, the most important thing for us is a clear parallel between the origin of experimental and logical truths.
Using the conclusions of Helmholtz, it is easy to understand how to understand the space of more than three dimensions. Helmholtz wondered what would be the geometry of beings who would know by experience only two dimensions, that is, would live in plane, quite compatible with it. Being flat, such beings would know all planimetry in the exact form in which we - beings of three dimensions - know it now; but these same hypothetical beings would not have the slightest idea of the third dimension, and all our solid geometry could have nothing concrete for them. Nevertheless, these flat creatures, deprived of the possibility of actually constructing stereometry, could, using analysis, study it analytically. We, beings of three dimensions, are in exactly the same position in relation to a space of four dimensions and generally different from ours: we cannot create a synthetic geometry of this space, but nothing prevents us from studying its properties analytically. Lobachevsky was the first to give the experience of studying such a space, which lies outside our experience. For people who do not know mathematical analysis, neither the Lobachevsky space nor the geometry of many dimensions exist, just as there are no visible only through a telescope heavenly bodies for people looking at the sky with the naked eye.
After what we have said here, it is not difficult to decide whether Lobachevsky was a dreamer in science? Further scientific research proved the reality of his geometry of two dimensions and showed in general the possibility of an analytical study of spaces that differ from our Euclidean one. And, it can be said, the most powerful minds of our time are working in the spirit of Lobachevsky, and what Lobachevsky's contemporaries considered a dream is now recognized as a deep, truly scientific research.
This work, as Professor Vasiliev says, is now being carried out both in Lobachevsky's homeland and in all the cultural countries of Europe: in England, France, Germany, Italy, in Spain, barely awakening from mental sleep, among the virgin forests of Texas.
It is not our task to expound the doctrine of the spiritualists about the space of four dimensions; we will only notice that it seeks to convince of the real existence of a space of four dimensions, and therefore is diametrically opposed to the views of true mathematicians and philosophers, who, on the contrary, prove the complete impossibility of this for us mortals.
It is gratifying to see that the development of Lobachevsky's ideas is expanding, and not only in the field of mathematics alone; both the physiology of the sense organs and that branch of philosophy that is now customarily called the theory of knowledge must take part in the solution of the questions contained in them. As proof of how far the influence of Lobachevsky's ideas extends, let us cite the words of Mr. Mikhailov, who says in his congratulatory telegram to Kazan University: “I am happy that back in 1888-1889 I could combine the philosophical principles of the great Russian geometer Lobachevsky and the doctrine of symmetry great Frenchman Louis Pasteur in my lectures on physiology given at St. Petersburg University.
From the main scientific merits of Lobachevsky, let's move on to secondary ones. He was not exclusively a geometer, like, for example, the German mathematician Steiner. Modern Russian mathematicians find great interest in his works on algebra and analysis. One of these works complements one of Gauss' thoughts.
Lobachevsky, like Riemann, was not only a mathematician, but also a philosopher, and the significance of his work for the theory of knowledge is almost as great as for mathematics. It is remarkable that not only in mathematics, but also in the philosophy of that time, the question of the essence and origin of geometric axioms was raised.
In general, the era in which Lobachevsky lived was significant in mental activity. Helmholtz speaks of it with delight: "This era was rich in spiritual blessings, inspiration, energy, ideal hopes, creative thoughts." The appearance of Kant's Critique of Pure Reason belongs to this era, which also included a new doctrine of space. Kant, as you know, argued that the idea of space precedes all experience and therefore is a completely subjective form of our view, independent of experience. Such a teaching was opposed to the teachings of Locke and the French sensualists, who denied innate ideas and subjective a priori forms of view. Mathematicians, generally speaking, did not deny the existence of the latter; however, we know the following opinion of Gauss: “Our knowledge of the truths of geometry is devoid of that complete conviction in their necessity (and, therefore, absolute truth), which belongs to the doctrine of quantities; we must modestly admit that if number is only a product of our spirit, then space has a reality besides our spirit, to which we cannot prescribe laws a priori.
From the opinion of Gauss cited here, it is clear that he recognized an essential difference between the concepts about the quantities and representation of space. The former are the results of the laws of our mind, the latter are the consequences of our experience or results physiological properties our sense organs, which determine the nature of all our perceptions of the external world. We meet the same views in Lobachevsky. They are considered diametrically opposed to the views of Kant. In essence, in our opinion, all Kant's views are reduced to the same opinion, if we deeply delve into what he means by synthetic views a priori and translate to modern language. The whole difference is in the language, in the ways of expression. We equally cannot prescribe the laws of both reality and our sensory perception of this reality. This explains the fact that many adherents of Kant are followers of Lobachevsky. his logical construction geometry without the postulate of Euclid, Lobachevsky undoubtedly indirectly proved that it cannot be deduced logically, and that, consequently, Euclidean geometry is not a deductive science and can never, under any effort of the mind, become deductive, therefore all these efforts should be considered fruitless. And Clifford rightly says that after Lobachevsky, the modern geometer, for whom both the form of space studied by Euclid, and the form of space studied by Lobachevsky, and the one with which the name Riemann is associated, are equally logically possible, will not claim that he knows the properties in general spaces at distances inaccessible to us; and will not think that he can judge what properties whatever space and what it will have.
So, the works of Lobachevsky and other scientists who dealt with non-Euclidean geometry, as if they said to a person: “The geometry that really exists for you, in logical relation is only a particular case of absolute geometry; your geometry is terrestrial and human.” After this kind of discovery, the horizon of a person should have expanded in the same way as it increased after the same person stopped thinking that the earth was the center of the world, surrounded by concentric crystal spheres, and suddenly realized himself living on an insignificant grain of sand in the vast ocean of worlds. Such were the results of the revolution in science made by Copernicus. Hence the parallel between Copernicus and Lobachevsky, first introduced by Clifford in his Philosophy of the pure sciences and now illuminated by many of the most eminent scientists. “Lobachevsky’s research,” says Professor Vasiliev, “has posed a question of no less importance to the philosophy of nature, the question of the properties of space: are these properties the same here and in those distant worlds from where light reaches us hundreds of thousands, millions of years? Are these properties now what they were when solar system was formed from a foggy spot, and what will they be like when the world approaches that state of uniformly scattered energy everywhere, in which physicists see the future of the world?
Such is the wide horizon that those scientific investigations open to us, the first foundation of which was laid by the firm hand of our famous compatriot. Lobachevsky, as we have seen, was a true son of a young people, thanks to the good will of an enlightened monarch, he saw the light of science in the remote semi-wild eastern outskirts of Russia.
We have already said that Lobachevsky's geometry in no way undermines Euclid's geometry; consequently, it does not threaten all our knowledge, the basis of which is our geometry, called by Lobachevsky common.
In support of this, let us cite evidence of the high respect for experience that the creator of imaginary geometry himself had. He says in his "New Principles of Geometry": "The first data, no doubt, will always be those concepts that we acquire in nature through our senses. The mind can and must reduce them to the smallest number, so that they later serve as a solid foundation for science. In his speech on "The Most Important Subjects of Education" Lobachevsky draws attention to the words of Bacon:
“Leave to labor in vain, trying to extract all wisdom from the mind; ask nature, she keeps all truths and will answer your questions satisfactorily".
In the form of expressing his philosophical views, Lobachevsky obviously belonged to the followers of Locke - he did not believe in the existence of innate ideas and was a great enemy of any scholasticism.
Despite all this, we, as we have already said, cannot agree that Lobachevsky's discoveries dealt an indirect but fatal blow to Kant's views on space. And from the point of view of a person who, together with Kant, asserts that ideas about space are the result of our organization, that it does not result from experience, but conditions experience - Lobachevsky's geometry retains all its strength. Non-Euclidean geometry serves only as a refutation of the false view that our geometry, that is, geometry in use, can be created by logic alone. The opponents of Locke and the sensualists recognize the usefulness of non-Euclidean geometry for more than just one analysis. Among them is Professor Zinger; he says: “Investigations (of Lobachevsky) can also be very useful for geometry, because, representing a generalization of geometric relations, they can indicate such dependencies and connections between geometry proposals that it would be impossible to notice without their help, and, thus, may open up new avenues for research on real space."
Lobachevsky's works on pure mathematics have not been translated into foreign languages, but it is very likely that if this had been done earlier, they would have been known abroad. In them, Lobachevsky showed the same qualities of mind that he discovered in geometry, delving into the very essence of the subject and defining with great subtlety the difference between concepts. Kazan professor Vasiliev, a student of the famous modern mathematician Weierstrass, finds that Lobachevsky, as early as the thirties, expressed the need to distinguish between the continuity of a function and its differentiability; in the seventies this task was brilliantly accomplished by Weierstrass and revolutionized modern mathematics. Lobachevsky also worked in the field of probability theory and mechanics; he was also very interested in astronomy. In 1842, he observed a total solar eclipse in Penza, and he was very interested in the phenomenon solar corona.
In his report on this astronomical expedition, he sets out and criticizes various views on the explanation of the solar corona. Regarding this, he sets out his view of the theory of light, in which he says, among other things: "The true theory must consist in one simple, single beginning, from which the phenomenon is taken as a necessary consequence with all its diversity." The theory of excitement did not satisfy him, and he tried to combine it with the theory of expiration. So, although Lobachevsky is not in all mathematical sciences with equal success he developed his own views, but the general nature of his activity was the same everywhere: everywhere he strove to establish common principles and separate concepts that were not completely identical with each other. With such a power of mind and with such a desire, he could have made a revolution in other mathematical sciences, if he had the opportunity to devote as much time to them as he gave to geometry.
In one of his writings on geometry, Lobachevsky expresses the idea that, perhaps, the laws of molecular forces unknown to us will be expressed using non-Euclidean geometry. If this thought of the great geometer comes true, then his work will acquire even greater significance. But in any case, all this still belongs to the realm of dreams. Contemporary followers of Lobachevsky are also divided into sober mathematicians and mathematicians-dreamers who are fond of fantasy. The most prominent of the former are Beltrami, Sophus Lie and Poincaré; among the latter, a prominent place is occupied by the astronomer Wallner, who died a few years ago, and who asserted that our space has a curvature. One of his ardent followers in America went even further, trying to explain many natural phenomena by the curvature of space.
“I think,” says Professor Vasiliev, “that Lobachevsky would not approve of (such) speculations about the property of our space.”
And we will conclude our sketch of Lobachevsky's scientific merits by recognizing the validity of these words, which should prevent us from mixing dreams on the basis of non-Euclidean geometry with scientific research this subject, the beginning of which was laid by our compatriot Lobachevsky.
From Biron's book author Kurukin Igor VladimirovichChapter Four "BIRONOVSHINA": A CHAPTER WITHOUT A HERO Although the whole court trembled, although there was not a single nobleman who would not expect misfortune from Biron's anger, but the people were decently controlled. It was not burdened with taxes, the laws were issued clearly, but executed exactly. MM.
From The Real Book of Frank Zappa author Zappa FrankCHAPTER 9 A Chapter for My Father At Edwards Air Force Base (1956-1959), my father had security clearance to the strictest military secrets. At that time, I was being kicked out of school every now and then, and my father was afraid that because of this they would lower the degree of secrecy? or even kicked out of work. He said,
From the book Daniil Andreev - Knight of the Rose author Bezhin Leonid EvgenievichCHAPTER FORTY-ONE ANDROMEDA NEBULAR: CHAPTER RESTORED Adrian, the eldest of the Gorbov brothers, appears at the very beginning of the novel, in the first chapter, and is told about in the final chapters. We will quote the first chapter in its entirety, since this is the only
From the book My Memories. Book one author Benois Alexander NikolaevichCHAPTER 15 Our silent engagement. My chapter in Muter's book About a month after our reunion, Atya decisively announced to her sisters, who still dreamed of seeing her married to such an enviable groom as Mr.
From the book Petersburg Tale author Basina Marianna Yakovlevna"THE HEAD OF LITERATURE, THE HEAD OF POETS" There were various rumors about Belinsky's personality among St. Petersburg writers. A half-educated student, expelled from the university for incompetence, a bitter drunkard who writes his articles without leaving the binge ... The only truth was that
From the book Notes of the ugly duckling author Pomerants Grigory SolomonovichChapter Ten An Unexpected Chapter All my main thoughts came suddenly, unintentionally. So is this one. I read stories by Ingeborg Bachmann. And suddenly I felt that I mortally want to make this woman happy. She has already died. I have never seen her portrait. The only sensual
From the book of Baron Ungern. Dahurian crusader or Buddhist with a sword author Zhukov Andrey ValentinovichChapter 14 The Last Chapter, or the Bolshevik Theater questionnaires”) “prisoner of war Ungern”, reports and reports compiled based on the materials of these
From the book Pages of my life author Krol Moses AaronovichChapter 24 April 1899 came, and I began to feel very bad again. It was still the results of my overwork when I was writing my book. The doctor found that I needed a long rest and advised me
From the book Pyotr Ilyich Tchaikovsky author Kunin Joseph FilippovichChapter VI. THE HEAD OF RUSSIAN MUSIC Now it seems to me that the history of the whole world is divided into two periods, - Pyotr Ilyich teased himself in a letter to his nephew Volodya Davydov: - the first period is everything that happened from the creation of the world to the creation of the "Queen of Spades". Second
From the book Being Joseph Brodsky. Apotheosis of loneliness author Solovyov Vladimir Isaakovich From the book I, Maya Plisetskaya author Plisetskaya Maya MikhailovnaChapter 29 mysterious world connection! What an aching longing, What a misfortune befell! Mandelstam All evil chances have armed themselves with me!.. Sumarokov Sometimes you need to have embittered people against yourself. Gogol It is more profitable to have another among the enemies,
From the author's bookChapter 30. CONFUSION IN TEARS The last chapter, farewell, forgiving and compassionate I imagine that I will soon die: sometimes it seems to me that everything around me is saying goodbye to me. Turgenev Let's take a good look at all this, and instead of indignation, our heart will be filled with sincerity.
From the author's bookChapter 10. Apostasy - 1969 (First chapter about Brodsky) The question of why IB poetry is not published in our country is not a question about IB, but about Russian culture, about its level. The fact that it is not printed is a tragedy not for him, not only for him, but also for the reader - not in the sense that he will not read it yet.
From the author's bookCHAPTER 47 CHAPTER WITHOUT A TITLE What title should I give this chapter?.. I think out loud (I always speak loudly to myself out loud - people who don't know me shy away). "Not my Bolshoi Theater"? Or: “How did the Bolshoi Ballet die”? Or maybe such a long one: “Lord rulers, do not
- The displacement is called the vector connecting the start and end points of the trajectory The vector connecting the beginning and end of the path is called
- Trajectory, path length, displacement vector Vector connecting the initial position
- Calculating the area of a polygon from the coordinates of its vertices The area of a triangle from the coordinates of the vertices formula
- Acceptable Value Range (ODZ), theory, examples, solutions