Lobachevsky's axiom of parallelism, main consequences. Nikolai Lobachevsky: parallel lines do intersect! How Lobachevsky proved that parallel lines intersect

An even deeper study of the issue will lead us to such a concept as space curvature. Without going into details, we only pay attention to the fact that the surface can be curved at each point in two qualitatively different ways. In one case, the surface resembles part of an ellipsoid, and the curvature is assumed to be positive. In another case, the surface looks like a saddle and its curvature is negative. The pseudosphere, as can be seen in its image (and hence the Lobachevsky plane), has a negative curvature, and it turns out that this curvature is constant (does not depend on a point on the surface). This, by the way, clarifies the origin of the name "pseudosphere": an ordinary sphere is a surface with constant positive curvature.

Lobachevsky's geometry, created in the 19th century, was the most important step towards the creation of the field of mathematics, which is now called differential geometry. It is engaged in the study of arbitrary curved spaces, and its mathematical apparatus is the foundation of such an important area of ​​modern physics as the general theory of relativity (GR). The fact is that, according to general relativity, the space-time in which we live has curvature, and the curvature of space corresponds to the presence of a gravitational field at this point in space.

General relativity has undergone numerous experimental checks (see: Centenary of General Relativity, or the Anniversary of the First November Revolution, Elements, 11/25/2015), and the corrections associated with it have to be taken into account for accurate satellite navigation. In addition, it describes the physics of massive objects such as ordinary and neutron stars, supernovae and black holes (the list goes on). Finally, general relativity underlies the modern science of the universe, cosmology.

According to common sense, as well as to all available observational data, the Universe on large scales is homogeneous and isotropic. In any case, this means that it is a space of constant spatial curvature. In this regard, three possibilities have been considered since the earliest years of cosmology: a flat universe, a universe of positive curvature ("spherical universe"), and a universe of negative curvature ("Lobachevsky's universe"). At the moment, however, it is believed that the curvature of the Universe is zero (within the limits of modern measurement accuracy). This finds an explanation in the modern theory of inflation. According to the latter, the Universe in the initial stage of its evolution experienced a very rapid expansion and as a result increased many times over (this is called inflation). It is quite possible that before inflation the Universe was spherical, the "Lobachevsky Universe" or had some other complex geometry. However, the expansion has led to the fact that now only a very small part of the entire Universe is accessible to observations, and its geometry should be indistinguishable from a flat one.

The fifth postulate of Euclid “If a line falling on two lines forms interior one-sided angles that are less than two lines in sum, then, continued indefinitely, these two lines will meet on the side where the angles in sum are less than two lines,” it seemed to many mathematicians back in antiquity somehow not very clear, partly due to the complexity of its formulation.

It seemed that only elementary sentences, simple in form, should be postulates. In this regard, the 5th postulate has become the subject of special attention of mathematicians, and research on this topic can be divided into two areas, in fact, closely related to each other. The first sought to replace this postulate with a simpler and more intuitive one, such as, for example, the statement formulated by Proclus “Through a point that does not lie on a given line, only one line can be drawn that does not intersect with a given one”: it is in this form that the 5th postulate , or rather, the equivalent axiom about parallel appears in modern textbooks.

Representatives of the second direction tried to prove the fifth postulate on the basis of others, that is, turn it into a theorem. Attempts of this kind were initiated by a number of Arab mathematicians of the Middle Ages: al-Abbas al-Jawhari (beginning of the 9th century), Sabit ibn Korra, Ibn al-Khaytham, Omar Khayyam, Nasireddin at-Tusi. Later, Europeans joined these studies: Levi Ben Gershon (XIV century) and Alfonso (XV century), who wrote in Hebrew, and then the German Jesuit H. Clavius ​​(1596), the Englishman J. Wallis (1663), and others. interest in this problem arose in the 18th century: from 1759 to 1800, 55 works were published analyzing this problem, including the very important works of the Italian Jesuit J. Saccheri and the German J. G. Lambert.

Proofs were usually carried out by the method of "contradiction": from the assumption that the 5th postulate is not fulfilled, they tried to derive consequences that would contradict other postulates and axioms. In reality, however, one ended up with a contradiction not with other postulates, but with some explicit or implicit "obvious" proposition, which, however, was impossible to establish on the basis of other postulates and axioms of Euclidean geometry: thus, the proofs did not achieve their goal , - it turned out that in place of the 5th postulate, some other statement equivalent to it was again put. For example, the following provisions were taken as such a statement:

Rice. 2. There are straight lines equally spaced from each other

Rice. 4. Two converging lines intersect

The geometry in which these statements do not hold is, of course, not the one we are used to, but it still does not follow from this that it is impossible or that these statements follow from other postulates and axioms of Euclid, so that there were gaps in all the proofs. or stretches. Clavius ​​justified the assumption that there are lines equidistant from each other by the Euclidean "definition" of a line as a line equidistant with respect to points on it. Wallis was the first to base his proof of the 5th postulate on the “natural” position, according to which for any figure there is a similar one of an arbitrarily large size, and substantiated this statement by the 3rd postulate of Euclid, affirming from any center and any solution can describe a circle ( in fact, the statement about the existence of, for example, unequal similar triangles or even circles is equivalent to the 5th postulate). A. M. Legendre in successive editions of the textbook "Principles of Geometry" (1794, 1800, 1823) cited new proofs of the 5th postulate, but a careful analysis showed gaps in these proofs. Having subjected Legendre to fair criticism, our compatriot S. E. Guryev in the book "An experiment on the improvement of the elements of geometry" (1798), however, himself made a mistake in proving the 5th postulate.

Quite quickly, the connection between the sum of the angles of a triangle and a quadrilateral and the 5th postulate was realized: the 5th postulate follows from the statement that the sum of the angles of a triangle is equal to two right angles, which can be deduced from the existence of rectangles. In this regard, an approach has become widespread (it was followed by Khayyam, at-Tusi, Wallis, Sakkeri), in which a quadrilateral is considered, resulting from laying off equal segments on two perpendiculars to one straight line. Three hypotheses are being investigated: the two upper angles are acute, obtuse or straight; in this case, an attempt is made to show that the hypotheses of obtuse and acute angles lead to a contradiction.

Another approach (used by Ibn al-Haytham, Lambert) analyzed three similar hypotheses for a quadrilateral with three right angles.

Saccheri and Lambert showed that the obtuse angle hypotheses do indeed lead to a contradiction, but they failed to find a contradiction when considering the acute angle hypotheses: Saccheri concluded such a contradiction only as a result of an error, and Lambert concluded that the apparent absence of a contradiction in the acute angle hypothesis is due to for some fundamental reason. Lambert found that, when accepting the hypothesis of an acute angle, the sum of the angles of each triangle is less than 180 ° by an amount proportional to its area, and compared with this the opening at the beginning. 17th century the position according to which the area of ​​a spherical triangle, on the contrary, is greater than 180 ° by an amount proportional to its area.

In 1763, G. S. Klugel published "Overview of the most important attempts to prove the theory of parallel lines", where he reviewed about 30 proofs of the 5th postulate and identified errors in them. Klugel concluded that Euclid was right in placing his statement among the postulates.

Nevertheless, attempts to prove the 5th postulate played a very important role: trying to bring the statements opposite to it to a contradiction, these researchers actually discovered many important theorems of non-Euclidean geometry - in particular, such geometry, where the 5th postulate is replaced by the statement about the possibility draw through a given point at least two lines that do not intersect the given one. This statement, which is equivalent to the acute angle hypothesis, was taken as the basis by the discoverers of non-Euclidean geometry.

The idea that the assumption of an alternative to the 5th postulate leads to the construction of a geometry different from Euclidean, but equally consistent, was independently proposed by several scientists: K. F. Gauss, N. I. Lobachevsky and J. Boyai (as well as F K. Schweikart and F. A. Taurinus, whose contribution to the new geometry, however, was more modest and who did not publish their research). Gauss, judging by the notes preserved in his archive (and published only in the 1860s), realized the possibility of new geometry as early as the 1810s, but also never published his discoveries on this topic: “I fear the cry of the Boeotians (that is, fools: the inhabitants of the region of Boeotia were considered the most stupid in ancient Greece), if I express my views in full, ”he wrote in 1829 to his friend mathematician F. W. Bessel. Misunderstanding fell entirely on Lobachevsky, who made the first report on the new geometry in 1826 and published the results in 1829. In 1842, Gauss succeeded in electing Lobachevsky a corresponding member of the Göttingen Scientific Society: this was the only recognition of Lobachevsky's merits during his lifetime . J. Boyai's father, the mathematician Farkas Boyai, who also tried to prove the 5th postulate, warned his son against research in this direction: “... it can deprive you of your leisure, health, peace, all the joys of life. This black abyss is able, perhaps, to swallow a thousand such titans as Newton, on Earth this will never clear up ... ". Nevertheless, J. Boyai published his results in 1832 in an appendix to a geometry textbook written by his father. Boyai also did not achieve recognition, moreover, he was upset that Lobachevsky was ahead of him: he did not study non-Euclidean geometry anymore. So only Lobachevsky during the rest of his life, firstly, continued research in the new field, and secondly, propagated his ideas, published a number of books and articles on new geometry.

So, in the Lobachevsky plane, through the point C outside the given line AB, there are at least two lines that do not intersect AB. All lines passing through C are divided into two classes - those that intersect and those that do not intersect AB . These latter lie in some angle formed by the two extreme lines that do not intersect AB. It is these lines that Lobachevsky calls parallel to the line AB, and the angle between them and the perpendicular is the angle of parallelism. This angle depends on the distance from point C to line AB: the greater this distance, the smaller the angle of parallelism. Lines lying inside the angle are called divergent with respect to AB.

Any two divergent lines p and q have a single common perpendicular t, which is the shortest line segment from one to the other. If the point M moves along p in the direction from t, then the distance from M to q will increase to infinity, and the bases of the perpendiculars dropped from M to q will fill only the final segment.

If the lines p and q intersect each other, then the projections of the points of one of them onto the other also fill the bounded segment.

If the lines p and q are parallel, then the distances between their points decrease indefinitely in one direction, and increase indefinitely in the other; one straight line is projected onto the ray of the other.

The figures show various mutual positions of the lines p and q, which are possible in Lobachevsky's geometry; r and s are perpendiculars parallel to q. (We are forced to draw a curved line q, although we are talking about a straight line. Even if our world as a whole obeyed the laws of Lobachevsky geometry, we would still not be able to depict on a small scale without distortion how everything looks on a large scale: in Lobachevsky geometry there are no similar figures that are not equal).

Inside the angle there is a line parallel to both sides of the angle. It divides all points inside the angle into two types: through the points of the first type one can draw straight lines intersecting both sides of the angle; no such straight line can be drawn through points of the second type. The same is true for the space between parallel lines. Between two divergent lines there are two lines parallel to both of them; they divide the space between the divergent lines into three regions: through points in one region, lines can be drawn that intersect both sides of the angle; no such straight lines can be drawn through points in the other two regions.

The diameter of a circle is always based on an acute, not a right angle. The side of a regular hexagon inscribed in a circle is always greater than its radius. For any n > 6, one can construct a circle such that the side of a regular n-gon inscribed in it is equal to its radius.

Lobachevsky was interested in the question of the geometry of physical space, in particular, using the data of astronomical observations, he calculated the sum of the angles of large, interstellar triangles: however, the difference between this sum of angles and 180 ° lay entirely within the error of observations. The misunderstanding that befell Lobachevsky, who himself called his geometry "imaginary", is largely due to the fact that in his time such ideas seemed to be pure abstractions and a game of the imagination. Is the new geometry really consistent? (After all, even if Lobachevsky failed to meet contradictions, this does not guarantee that it will not be discovered later). How does it relate to the real world, as well as to other areas of mathematics? This became clear not immediately, and the success that eventually fell to the lot of new ideas was associated with the discovery of models of new geometry.

Lobachevsky geometry is a geometric theory based on the same basic assumptions as ordinary Euclidean geometry, with the exception of the parallel axiom, which is replaced by the Lobachevsky parallel axiom. The Euclidean axiom of parallel states: through a point not lying on a given line, there passes only one line that lies with the given line in the same plane and does not intersect it. In Lobachevsky geometry, instead of it, the following axiom is accepted: through a point not lying on a given line, there pass at least two lines that lie with the given line in the same plane and do not intersect it. It would seem that this axiom contradicts extremely common ideas. Nevertheless, both this axiom and the entire Lobachevsky geometry have a very real meaning. Lobachevsky geometry was created and developed by N. I. Lobachevsky, who first reported it in 1826. Lobachevsky geometry is called non-Euclidean geometry, although the term “non-Euclidean geometry” is usually given a broader meaning, including here other theories that arose after Lobachevsky geometry and also based on a change in the basic premises of Euclidean geometry. Lobachevsky geometry is called specifically hyperbolic non-Euclidean geometry (as opposed to Riemann's elliptic geometry).

Lobachevsky geometry is a theory rich in content and having applications both in mathematics and in physics. Its historical significance lies in the fact that by constructing it, Lobachevsky showed the possibility of geometry other than Euclidean, which marked a new era in the development of geometry and mathematics in general (see Geometry). From a modern point of view, one can give, for example, the following definition of Lobachevsky's geometry on a plane: it is nothing but geometry inside a circle on an ordinary (Euclidean) plane, only expressed in a special way. Namely, we will consider a circle on an ordinary plane (Fig. 1) and its interior, i.e., the circle, with the exception of the circle that bounds it, we will call the “plane”. The point of the "plane" will be the point inside the circle. "Direct" we will call any chord (for example, a, b, b`, MN) (with excluded ends, since the circumference of a circle is excluded from the "plane"). "Movement" is any transformation of a circle into itself, which transforms chords into chords.

Accordingly, the figures inside the circle are called equal, which are translated one into another by such transformations. Then it turns out that any geometric fact described in such a language represents a theorem or axiom of Lobachevsky geometry. In other words, any statement of Lobachevsky geometry on the plane is nothing but a statement of Euclidean geometry, referring to figures inside a circle, only retelling in the indicated terms. The Euclidean axiom about parallels is clearly not satisfied here, because through the point O, which does not lie on a given chord a (i.e., “straight line”), there pass any number of chords (“straight lines”) that do not intersect it (for example, b, b`). Similarly, the Lobachevsky geometry in space can be defined as the geometry inside the ball, expressed in appropriate terms (“straight lines” - chords, “planes” - flat sections of the inside of the ball, “equal” figures - those that are translated one into another by transformations that translate the ball into itself and chords to chords). Thus, Lobachevsky's geometry has a completely real meaning and is as consistent as the geometry of Euclid. The description of the same facts in different terms, or, conversely, the description of different facts in the same terms, is a characteristic feature of mathematics. It appears clearly, for example, when the same line is given in different coordinates by different equations or, on the contrary, the same equation in different coordinates represents different lines.

The emergence of Lobachevsky geometry

The source of Lobachevsky's geometry was the question of the axiom of parallels, which is also known as the V postulate of Euclid (under this number, the statement equivalent to the axiom of parallels given above appears in the list of postulates in Euclid's Elements). This postulate, in view of its complexity in comparison with others, caused attempts to give its proof on the basis of other postulates.

Here is an incomplete list of scientists who were engaged in the proof of the 5th postulate until the 19th century: the ancient Greek mathematicians Ptolemy (2nd century), Proclus (5th century) (Proclus's proof is based on the assumption that the distance between two parallel ones is finite), Ibn al-Khaytham from Iraq ( late 10th - early 11th centuries) (Ibn al-Khaytham tried to prove the fifth postulate, based on the assumption that the end of the moving perpendicular to the straight line describes a straight line), Tajik mathematician Omar Khayyam (2nd half of the 11th - early 12th centuries), the Azerbaijani mathematician Nasiraddin Tuei (13th century) (Khayyam and Nasiraddin, when proving the fifth postulate, proceeded from the assumption that two converging lines cannot become divergent as they continue), the German mathematician C. Clavius ​​(Schlussel, 1574), the Italian mathematicians P. Cataldi (who first published in 1603 a work entirely devoted to the question of parallels), J. Borelli (1658), J. Vitale (1680), the English mathematician J. Wallis (1663, published in 1693) (Wallis founded the dock The proof of the V postulate is based on the assumption that for any figure there is a figure similar to it, but not equal to it). The proofs of the geometers enumerated above amounted to the replacement of Postulate V by another assumption, which seemed more obvious.

The Italian mathematician J. Saccheri (1733) made an attempt to prove the fifth postulate by contradiction. Having accepted a proposal that contradicted the postulate of Euclid, Saccheri developed rather extensive consequences from it. Mistakenly recognizing some of these consequences as leading to contradictions, Saccheri concluded that Euclid's postulate had been proved. The German mathematician I. Lambert (circa 1766, published in 1786) undertook similar studies, but he did not repeat Saccheri's mistakes, but admitted his impotence to discover a logical contradiction in the system he constructed. Attempts to prove the postulate were also made in the 19th century. Here we should note the work of the French mathematician A. Legendre; one of his proofs (1800) is based on the assumption that through each point inside an acute angle one can draw a line intersecting both sides of the angle, i.e., like all his predecessors, he replaced the postulate with another assumption. The German mathematicians F. Schweikart (1818) and F. Taurinus (1825) came quite close to the construction of Lobachevsky's geometry, but they did not have a clearly expressed idea that the theory they outlined would be logically as perfect as Euclid's geometry.

The question of the fifth postulate of Euclid, which occupied geometers for more than two millennia, was resolved by Lobachevsky. This solution boils down to the fact that the postulate cannot be proved on the basis of other premises of Euclidean geometry and that the assumption of a postulate opposite to the postulate of Euclid allows one to construct a geometry that is as meaningful as Euclidean and free from contradictions. Lobachevsky made a report about this in 1826, and in 1829-30 he published the work On the Principles of Geometry, outlining his theory. In 1832 a work of the same content was published by the Hungarian mathematician J. Bolyai. As it turned out later, the German mathematician K. F. Gauss also came to the idea of ​​the possibility of the existence of a consistent non-Euclidean geometry, but hid it, fearing to be misunderstood. Although Lobachevsky's geometry developed as a speculative theory and Lobachevsky himself called it "imaginary geometry", nevertheless, it was Lobachevsky who considered it not as a game of the mind, but as a possible theory of spatial relations. However, the proof of its consistency was given later, when its interpretations were indicated and thus the question of its real meaning, logical consistency, was completely resolved.

Lobachevsky geometry studies the properties of the "Lobachevsky plane"(in planimetry) and "Lobachevsky space" (in stereometry). The Lobachevsky plane is a plane (a set of points) in which straight lines are defined, as well as the movements of figures (at the same time, distances, angles, etc.), obeying all the axioms of Euclidean geometry, with the exception of the parallel axiom, which is replaced by the above axiom Lobachevsky. The Lobachevsky space is defined in a similar way. The task of clarifying the real meaning of Lobachevsky geometry consisted in finding models of the plane and Lobachevsky space, i.e., in finding such objects in which the appropriately interpreted positions of planimetry and stereometry of Lobachevsky geometry would be realized.

Here are some facts of Lobachevsky's geometry that distinguish it from Euclid's geometry and established by Lobachevsky himself.

1) In Lobachevsky geometry there are no similar but unequal triangles; triangles are congruent if their angles are equal. Therefore, there is an absolute unit of length, i.e., a segment distinguished by its properties, just as a right angle is distinguished by its properties. Such a segment can be, for example, the side of a regular triangle with a given sum of angles.

2) The sum of the angles of any triangle is less than p and can be arbitrarily close to zero. This is directly visible in the Poincaré model. The difference p - (a + b + g), where a, b, g are the angles of the triangle, is proportional to its area.

3) Through a point O, not lying on a given line a, there are infinitely many lines that do not intersect a and are in the same plane with it; among them there are two extreme b, b`, which are called parallel to the straight line a in the sense of Lobachevsky. In the Klein (Poincaré) models, they are represented by chords (arcs of circles) having a common end with the chord (arc) (which, by definition of the model, is excluded, so that these lines do not have common points) (Fig. 1.3). Its angle between the straight line b (or b`) and the perpendicular from O to a - the so-called. angle of parallelism - as the point O moves away from the line, it decreases from 90° to 0° (in the Poincaré model, the angles in the usual sense coincide with the angles in the sense of Lobachevsky, and therefore this fact can be seen directly on it). Parallel b on the one hand (and b` on the opposite side) asymptotically approaches a, and on the other hand it infinitely moves away from it (distances are difficult to determine in models, and therefore this fact is not directly visible).

4) If the lines have a common perpendicular, then they diverge infinitely on both sides of it. To any of them it is possible to restore perpendiculars that do not reach the other line.

5) A line of equal distances from a straight line is not a straight line, but a special curve called an equidistant line, or a hypercycle.

6) The limit of circles of infinitely increasing radius is not a straight line, but a special curve called the limit circle, or horocycle.

7) The limit of spheres of infinitely increasing radius is not a plane, but a special surface - a limiting sphere, or horosphere; it is remarkable that Euclidean geometry holds on it. This served Lobachevsky as the basis for the derivation of trigonometry formulas.

8) The circumference is not proportional to the radius, but grows faster.

9) The smaller the region in space or on the Lobachevsky plane, the less the geometric relations in this region differ from the relations of Euclidean geometry. We can say that in an infinitesimal region, the Euclidean geometry takes place. For example, the smaller the triangle, the less the sum of its angles differs from p; the smaller the circle, the less the ratio of its length to radius differs from 2p, etc. A decrease in the area is formally equivalent to an increase in the unit of length, therefore, with an infinite increase in the unit of length, the Lobachevsky geometry formulas turn into the formulas of Euclidean geometry. Euclidean geometry is in this sense a "limiting" case of Lobachevsky geometry.

Lobachevsky geometry continues to be developed by many geometers; it studies: solving construction problems, polyhedra, regular systems of figures, the general theory of curves and surfaces, etc. A number of geometers also developed mechanics in Lobachevsky space. These studies did not find direct applications in mechanics, but gave rise to fruitful geometric ideas. In general, Lobachevsky geometry is a vast field of study, like the geometry of Euclid.

On February 7, 1832, Nikolai Lobachevsky presented his first work on non-Euclidean geometry to the judgment of his colleagues. That day was the beginning of a revolution in mathematics, and Lobachevsky's work was the first step towards Einstein's theory of relativity. Today "RG" has collected five of the most common misconceptions about Lobachevsky's theory, which exist among people far from mathematical science

Myth one. Lobachevsky's geometry has nothing in common with Euclidean.

In fact, Lobachevsky's geometry is not too different from the Euclidean geometry we are used to. The fact is that of the five postulates of Euclid, Lobachevsky left the first four without change. That is, he agrees with Euclid that a straight line can be drawn between any two points, that it can always be extended to infinity, that a circle with any radius can be drawn from any center, and that all right angles are equal to each other. Lobachevsky did not agree only with the fifth postulate, the most doubtful from his point of view, of Euclid. His formulation sounds extremely tricky, but if we translate it into a language understandable to a common person, it turns out that, according to Euclid, two non-parallel lines will definitely intersect. Lobachevsky managed to prove the falsity of this message.

Myth two. In Lobachevsky's theory, parallel lines intersect

This is not true. In fact, the fifth postulate of Lobachevsky sounds like this: "On the plane, through a point that does not lie on a given line, there passes more than one line that does not intersect the given one." In other words, for one straight line, it is possible to draw at least two straight lines through one point that will not intersect it. That is, in this postulate of Lobachevsky there is no talk of parallel lines at all! We only talk about the existence of several non-intersecting lines on the same plane. Thus, the assumption about the intersection of parallel lines was born because of the banal ignorance of the essence of the theory of the great Russian mathematician.

Myth three. Lobachevsky geometry is the only non-Euclidean geometry

Non-Euclidean geometries are a whole layer of theories in mathematics, where the basis is the fifth postulate different from Euclidean. Lobachevsky, unlike Euclid, for example, describes a hyperbolic space. There is another theory describing spherical space - this is Riemann's geometry. This is where the parallel lines intersect. A classic example of this from the school curriculum is the meridians on the globe. If you look at the pattern of the globe, it turns out that all the meridians are parallel. Meanwhile, it is worth putting a pattern on the sphere, as we see that all previously parallel meridians converge at two points - at the poles. Together the theories of Euclid, Lobachevsky and Riemann are called "three great geometries".

Myth four. Lobachevsky geometry is not applicable in real life

On the contrary, modern science comes to understand that Euclidean geometry is only a special case of Lobachevsky's geometry, and that the real world is more accurately described by the formulas of the Russian scientist. The strongest impetus for the further development of Lobachevsky's geometry was Albert Einstein's theory of relativity, which showed that the very space of our Universe is not linear, but is a hyperbolic sphere. Meanwhile, Lobachevsky himself, despite the fact that he worked all his life on the development of his theory, called it "imaginary geometry."

Myth five. Lobachevsky was the first to create non-Euclidean geometry

This is not entirely true. In parallel with him and independently of him, the Hungarian mathematician Janos Bolyai and the famous German scientist Carl Friedrich Gauss came to similar conclusions. However, the works of Janos were not noticed by the general public, and Karl Gauss preferred not to be published at all. Therefore, it is our scientist who is considered a pioneer in this theory. However, there is a somewhat paradoxical point of view that Euclid himself was the first to invent non-Euclidean geometry. The fact is that he self-critically considered his fifth postulate not obvious, so he proved most of his theorems without resorting to it.