Basic concepts of Lobachevsky geometry. Some. In what geometry do parallel lines intersect? Lobachevsky lines intersect

Lobachevsky plane

Lobachevsky geometry (hyperbolic geometry) is one of the non-Euclidean geometries, a geometric theory based on the same basic premises as ordinary Euclidean geometry, with the exception of the parallel axiom, which is replaced by Lobachevsky's parallel axiom.

The Euclidean Parallel Axiom says:

through a point not lying on a given straight line, there is only one straight line that lies with a given straight line in one plane and does not intersect it.

In Lobachevsky's geometry, the following axiom is accepted instead:

through a point that does not lie on the given straight line, there are at least two straight lines that lie with the given straight line in the same plane and do not intersect it.

Lobachevsky's geometry has extensive applications in both mathematics and physics. Its historical significance lies in the fact that by constructing it, Lobachevsky showed the possibility of a geometry different from Euclidean, which marked a new era in the development of geometry and mathematics in general.

History

Attempts to prove the fifth postulate

The starting point of Lobachevsky's geometry was Euclid's V postulate - an axiom equivalent to the parallel axiom. He was included in the list of postulates in Euclid's Elements). The relative complexity and unintuitiveness of its formulation caused a feeling of its secondary nature and gave rise to attempts to deduce it from the rest of Euclid's postulates.

Among those trying to prove were the following scientists:

• ancient Greek mathematicians Ptolemy (II century), Proclus (V century) (based on the assumption that the distance between two parallel ones is finite),
• Ibn al-Haytham from Iraq (late - early centuries) (based on the assumption that the end of a moving perpendicular to a straight line describes a straight line),
• Iranian mathematicians Omar Khayyam (2nd half - early 12th centuries) and Nasir ad-Din at-Tusi (13th century) (based on the assumption that two converging straight lines cannot become divergent without intersection when they continue),
• German mathematician Clavius ​​(),
• Italian mathematicians
  • Cataldi (for the first time in 1603 he published a work entirely devoted to the question of parallel),
• English mathematician Wallis (, published in) (based on the assumption that for every figure there is a similar, but not equal figure),
• French mathematician Legendre () (based on the assumption that through each point inside an acute angle a straight line can be drawn that intersects both sides of the angle; he also had other attempts to prove it).

In these attempts to prove the fifth postulate, mathematicians introduced a new statement that seemed more obvious to them.

Attempts have been made to use proof by contradiction:

• Italian mathematician Saccheri () (having formulated a statement contradicting the postulate, he derived a number of consequences and, mistakenly recognizing some of them as contradictory, he considered the postulate proven),
• German mathematician Lambert (about, published in) (after conducting research, he admitted that he could not find contradictions in the system he built).

Finally, an understanding began to emerge that it is possible to build a theory based on the opposite postulate:

• German mathematicians F. Schweickart () and Taurinus () (however, they did not realize that such a theory would be logically equally coherent).

Creating non-euclidean geometry

Lobachevsky in his work "On the Principles of Geometry" (), his first published work on non-Euclidean geometry, clearly stated that the V postulate cannot be proved on the basis of other premises of Euclidean geometry, and that the assumption of a postulate opposite to that of Euclidean allows one to construct geometry in the same way meaningful, like Euclidean, and free from contradictions.

At the same time and independently, Janos Bolyai came to similar conclusions, and Karl Friedrich Gauss came to such conclusions even earlier. However, Boyai's writings did not attract attention, and he soon abandoned the topic, and Gauss generally refrained from publishing, and his views can only be judged by a few letters and diary entries. For example, in a letter from 1846 to astronomer G. H. Schumacher, Gauss speaks of Lobachevsky's work as follows:

This work contains the foundations of the geometry that should have taken place and, moreover, would have constituted a strictly consistent whole, if the Euclidean geometry were not true ... Lobachevsky calls it "imaginary geometry"; You know that for 54 years (since 1792) I have shared the same views with a certain development of them, which I do not want to mention here; Thus, I did not find for myself anything practically new in Lobachevsky's work. But in the development of the subject, the author did not follow the path that I myself followed; it is executed by Lobachevsky masterfully in a truly geometric spirit. I consider myself obliged to draw your attention to this composition, which, probably, will give you absolutely exceptional pleasure.

As a result, Lobachevsky acted as the first brightest and most consistent propagandist of this theory.

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.

Assertion of Lobachevsky's geometry

the angle is even more difficult.

Poincaré model

Content of Lobachevsky geometry

A sheaf of parallel lines in the geometry of Lobachevsky

Lobachevsky built his geometry, starting from the basic geometric concepts and his axiom, and proved theorems with a geometric method, similar to how it is done in the geometry of Euclid. The theory of parallel lines served as the basis, since it is here that the difference between the geometry of Lobachevsky and the geometry of Euclid begins. All theorems that do not depend on the parallel axiom are common to both geometries and form the so-called absolute geometry, which includes, for example, the theorems on the equality of triangles. Following the theory of parallels, other sections were built, including trigonometry and the beginnings of analytic and differential geometry.

Let us cite (in modern notation) several facts of Lobachevsky's geometry that distinguish it from Euclid's geometry and were established by Lobachevsky himself.

Through point P not lying on the given line R(see figure), there are infinitely many straight lines that do not intersect R and are in the same plane with it; among them there are two extreme x, y, which are called parallel line R in the sense of Lobachevsky. In the Klein (Poincaré) models, they are depicted by chords (circular arcs) having a chord (arc) R a common end (which, by the definition of the model, is excluded, so that these lines do not have common points).

Angle between perpendicular PB from P on R and each of the parallel (called parallelism angle) as the point is removed P from a straight line decreases from 90 ° to 0 ° (in the Poincaré model, the angles in the usual sense coincide with the angles in the Lobachevsky sense, and therefore this fact can be seen directly on it). Parallel x on the one hand (a y with the opposite) asymptotically approaches a, and on the other hand, it infinitely moves away from it (in models, distances are difficult to determine, and therefore this fact is not directly visible).

For a point located from a given straight line at a distance PB = a(see figure), Lobachevsky gave a formula for the angle of parallelism P (a) :

Here q- some constant associated with the curvature of the Lobachevsky space. It can serve as an absolute unit of length in the same way as in spherical geometry a special position is occupied by the radius of a sphere.

If the lines have a common perpendicular, then they diverge infinitely in both directions from it. To any of them, you can restore perpendiculars that do not reach another straight line.

In Lobachevsky's geometry there are no similar, but unequal triangles; triangles are equal if their angles are equal.

The sum of the angles of any triangle is less than π and can be arbitrarily close to zero. This can be seen directly in the Poincaré model. The difference δ = π - (α + β + γ), where α, β, γ are the angles of the triangle, is proportional to its area:

The formula shows that there is a maximum area of ​​a triangle, and this is a finite number: π q 2 .

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

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

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

The circumference is not proportional to the radius, but grows faster. In particular, in the geometry of Lobachevsky, the number π cannot be defined as the ratio of the circumference of a circle to its diameter.

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

Applications

• Lobachevsky himself applied his geometry to the calculation of definite integrals.
• In the theory of functions of a complex variable, Lobachevsky's geometry helped to construct the theory of automorphic functions. The connection with the geometry of Lobachevsky was here the starting point of the research of Poincaré, who wrote that "non-Euclidean geometry is the key to solving the whole problem."
• Lobachevsky's geometry is also used in number theory, in its geometric methods, united under the name "geometry of numbers".
• A close connection was established between Lobachevsky's geometry and the kinematics of the special (particular) theory of relativity. This connection is based on the fact that equality, which expresses the law of propagation of light

when divided by t 2, that is, for the speed of light, gives - equation of a sphere in space with coordinates v x , v y , v z- speed components along the axes NS, at, z(in the "space of velocities"). The Lorentz transformations preserve this sphere and, since they are linear, transform the straight lines of the velocity space into straight lines. Therefore, according to the Klein model, in the space of velocities inside the sphere of radius with, that is, for speeds less than the speed of light, the Lobachevsky geometry takes place.

• Lobachevsky's geometry found a remarkable application in the general theory of relativity. If we consider the distribution of matter masses in the Universe to be uniform (this approximation is permissible on a cosmic scale), then it turns out that under certain conditions space has the Lobachevsky geometry. Thus, Lobachevsky's assumption about his geometry as a possible theory of real space was justified.
• Using the Klein model, a very simple and short proof is given

LV 1. (Axiom of parallelism of Lobachevsky). In any plane there is a straight line a 0 and a point A 0 that does not belong to this line, such that at least two straight lines that do not intersect a 0 pass through this point.

The set of points, lines and planes satisfying the axioms of membership, order, congruence, continuity and the Lobachevsky axiom of parallelism will be called the three-dimensional Lobachevsky space and denoted by A 3. Most of the geometric properties of figures will be considered by us on the plane of the space Л 3, i.e. on the Lobachevsky plane. Let us pay attention to the fact that the formal logical negation of the axiom V 1, the axiom of parallelism of Euclidean geometry, has exactly the formulation that we gave as axiom LV 1. On the plane, there is at least one point and one straight line for which the statement of the axiom of parallelism of Euclidean geometry does not hold. Let us prove a theorem from which it follows that the statement of the Lobachevsky parallelism axiom is valid for any point and any straight line of the Lobachevsky plane.

Theorem 13.1.Let a be an arbitrary straight line and A a point not lying on this straight line. Then in the plane defined by point A and line a, there are at least two lines passing through A and not intersecting line a.

Proof. We carry out the proof by contradiction, using Theorem 11.1 (see § 11). Suppose that in the Lobachevsky space there is a point A and a straight line a such that in the plane defined by this point and a straight line, through point A there is a single straight line that does not intersect a. Let us drop the point A perpendicular AB to the straight line a and at the point A we raise the perpendicular h to the straight line AB (Fig. 50). As follows from Theorem 4.2 (see § 4), the lines h and a do not intersect. The straight line h, by virtue of the assumption, is the only straight line passing through A and not intersecting a. Let us choose an arbitrary point C on the straight line a. Let us set aside from the ray AC in the half-plane with the boundary AB, which does not contain point B, the angle CAM equal to ACB. Then, as follows from the same Theorem 4.2, the line AM does not intersect a. It follows from our assumption that it coincides with h. Therefore, the point M belongs to the line h. Triangle ABC - rectangular,. Let's calculate the sum of the angles of the triangle ABC:. It follows from Theorem 11.1 that the condition of the axiom of parallelism of Euclidean geometry is satisfied. Therefore, in the plane under consideration there can be no points A 0 and a straight line a 0 such that at least two straight lines that do not intersect a 0 pass through this point. We have come to a contradiction with the condition of the Lobachevsky parallel axiom. The theorem is proved.

It should be noted that in what follows we will use the assertion of Theorem 13.1, in fact, replacing the assertion of Lobachevsky's axiom of parallelism. By the way, in many textbooks it is this statement that is accepted as the axiom of parallelism of Lobachevsky's geometry.

It is easy to obtain the following corollary from Theorem 13.1.

Corollary 13.2. In the Lobachevsky plane, through a point that does not lie on a given straight line, there are infinitely many straight lines that do not intersect the given one.

Indeed, let a is a given straight line, and A is a point that does not belong to it, h 1 and h 2 are straight lines passing through A and not intersecting a (Fig. 51). Obviously, all lines that pass through point A and lie in one of the corners formed by h 1 and h 2 (see Fig. 51) do not intersect line a.

In Chapter 2, we proved a number of statements that are equivalent to the parallel axiom of Euclidean geometry. Their logical negations characterize the properties of figures on the Lobachevsky plane.

First, on the Lobachevsky plane, the logical negation of the fifth postulate of Euclid is valid. In Section 9, we formulated the postulate itself and proved a theorem on its equivalence to the axiom of parallelism of Euclidean geometry (see Theorem 9.1). Its logical negation is:

Statement 13.3.On the Lobachevsky plane, there are two non-intersecting straight lines, which, when they intersect with the third straight line, form internal one-sided angles, the sum of which is less than two right angles.

In § 12 we formulated Posidonius' proposal: on the plane there are at least three collinear points located in one half-plane from the given line and equidistant from it. We also proved Theorem 12.6: Posidonius' proposal is equivalent to the assertion of the parallelism axiom of Euclidean geometry. Thus, the negation of this statement acts on the Lobachevsky plane.

Statement 13.4. The set of points equidistant from a straight line on the Lobachevsky plane and located in one half-plane relative to it, in turn, do not lie on one straight line.

On the Lobachevsky plane, a set of points equidistant from a straight line and belonging to one half-plane relative to this straight line form a curved line, the so-called equidistant line. We will consider its properties later.

Consider now Legendre's proposal: n Theorem 11.6 we proved (see § 11) asserts that From this it follows that on the Lobachevsky plane the logical negation of this proposition is valid.

Statement 13.5. On the side of any acute angle, there is such a point that the perpendicular to it, raised at this point, does not intersect the second side of the angle.

Let us note the properties of triangles and quadrangles of the Lobachevsky plane, which follow directly from the results of Sections 9 and 11. First of all, Theorem 11.1. States that the assumption of the existence of a triangle, the sum of the angles of which coincides with the sum of two right angles, is equivalent to the axiom of parallelism of the Euclidean plane. From this and Legendre's first theorem (see Theorem 10.1, § 10) the following statement follows

Statement 13.6. On the Lobachevsky plane, the sum of the angles of any triangle is less than 2d.

This immediately implies that the sum of the angles of any convex quadrilateral is less than 4d, and the sum of the angles of any convex n - gon is less than 2 (n-1) d.

Since on the Euclidean plane the angles adjacent to the upper base of the Saccheri quadrangle are equal to right angles, which, in accordance with Theorem 12.3 (see § 12), is equivalent to the axiom of parallelism of Euclidean geometry, we can draw the following conclusion.

Statement 13.7. The corners adjacent to the upper base of the Saccheri quadrangle are acute.

It remains for us to consider two more properties of triangles on the Lobachevsky plane. The first is related to Wallis's proposal: on the plane there is at least one pair of triangles with correspondingly equal angles, but not equal sides. In Section 11 we proved that this proposition is equivalent to the parallel axiom of Euclidean geometry (see Theorem 11.5). The logical denial of this statement leads us to the following conclusion: on the Lobachevsky plane there are no triangles with equal angles, but not equal sides. Thus, the following proposition is true.

Statement 13.8. (the fourth criterion for the equality of triangles on the Lobachevsky plane).Any two triangles on the Lobachevsky plane, having correspondingly equal angles, are equal to each other.

Consider now the next question. Can a circle be described around any triangle on the Lobachevsky plane? The answer is given by Theorem 9.4 (see § 9). In accordance with this theorem, if a circle can be described around any triangle on the plane, then the condition of the axiom of parallelism of Euclidean geometry is satisfied on the plane. Therefore, the logical negation of the assertion of this theorem leads us to the following proposition.

Statement 13.9. On the Lobachevsky plane, there is a triangle around which a circle cannot be described.

It is easy to construct an example of such a triangle. Let us choose some straight line a and point A, which does not belong to it. Let us drop the perpendicular h from point A to line a. By virtue of Lobachevsky's axiom of parallelism, there is a straight line b passing through A and not perpendicular to h, which does not intersect a (Fig. 52). As you know, if a circle is circumscribed around a triangle, then its center lies at the point of intersection of the median perpendiculars of the sides of the triangle. Therefore, it is enough for us to give an example of such a triangle, the median perpendiculars of which do not intersect. Let's choose a point M on the line h, as shown in Figure 52. We symmetrically display it relative to the lines a and b, we get points N and P. Since the line b is not perpendicular to h, the point P does not belong to h. Therefore, the points M, N and P are the vertices of the triangle. Lines a and b serve as perpendiculars by construction. They, as mentioned above, do not intersect. The triangle MNP is the required one.

It is easy to construct an example of a triangle in the Lobachevsky plane around which a circle can be described. To do this, it is enough to take two intersecting lines, select a point that does not belong to them, and reflect it relative to these lines. Conduct the detailed construction yourself.

Definition 14.1. Let there be given two directed straight lines and. They are called parallel if the following conditions are met:

1. straight lines a and b do not intersect;

2. for arbitrary points A and B of straight lines a and b, any internal ray h of angle ABB 2 intersects line a (Fig. 52).

We will denote parallel lines in the same way as is customary in the school geometry course: a || b. Note that parallel lines on the Euclidean plane satisfy this definition.

Theorem 14.3. Let a directed straight line and a point B, which does not belong to it, be given on the Lobachevsky plane. Then a single directed straight line passes through this point such that straight line a is parallel to straight line b.

Proof. Let us drop from point B the perpendicular BA to the straight line a and from point B we will restore the perpendicular p to the straight BA (Fig. 56 a). The straight line p, as has already been noted many times, does not intersect the given straight line a. Let us choose an arbitrary point C on it, divide the points of the segment AC into two classes and. The first class will include such points S of this segment for which the ray BS intersects the ray AA 2, and the second class includes such points T for which the ray BT does not intersect the ray AA 2. Let us show that such a division into classes produces a Dedekind section of the segment AC. In accordance with Theorem 4.3 (see § 4), we should check that:

2. and classes and contain points other than A and C;

3. any point of the class other than A lies between point A and any point of the class.

The first condition is obvious, all points of the segment belong to one or another class, while the classes themselves, based on their definition, do not have common points.

The second condition is also easy to check. Obviously, and. The class contains points other than A; to verify this statement, it is enough to choose some point of the ray AA 2 and connect it to point B. This ray will intersect the segment BC at the point of the first class. The class also contains points other than C, otherwise we will come to a contradiction with Lobachevsky's axiom of parallelism.

Let us prove the third condition. Let there exist a point S of the first class, different from A, and such a point T of the second class such that the point T lies between A and S (see Fig. 56 a). Since, then ray BS intersects ray AA 2 at some point R. Consider ray BT. It intersects the AS side of the ASR triangle at point T. According to Pasha's axiom, this ray must intersect either the AR side or the SR side of this triangle. Suppose that the ray BT intersects the side SR at some point O. Then two different straight lines BT and BR pass through the points B and O, which contradicts the axiom of Hilbert's axiom. Thus, the ray BT intersects the side AR, which implies that the point T does not belong to the class K 2. The resulting contradiction leads to the statement that the point S lies between A and T. The condition of Theorem 4.3 is fully verified.

In accordance with the conclusion of Theorem 4.3 on the Dedekind section on the segment AC, there exists a point for which any point lying between A and belongs to the class, and any point lying between and C belongs to the class. Let us show that the directed line is parallel to the line ... In fact, it remains for us to prove that it does not intersect the straight line a, since, due to the choice of points of class K 1, any inner ray of the angle intersects. Suppose that the straight line intersects the straight line a at some point H (Figure 56 b). Let us choose an arbitrary point P on ray HA 2 and consider ray BP. Then it intersects the segment М 0 С at some point Q (prove this statement yourself). But the interior points of the segment М 0 С belong to the second class, the ray BP cannot have common points with the line a. Thus, our assumption about the intersection of lines BM 0 and a is incorrect.

It is easy to check that the line is the only directed line passing through point B and parallel. Indeed, let another directed straight line pass through point B, which, as well, is parallel. In this case, we will assume that M 1 is the point of the segment AC. Then, based on the definition of the class K 2,. Therefore, the ray BM 0 is the inner ray of the angle, therefore, by virtue of Definition 14.1, it intersects the straight line. We have come to a contradiction with the statement proved above. Theorem 14.3 is completely proved.

Consider point B and a directed line that does not contain it. In accordance with the proved Theorem 14.3, a directed straight line parallel to a passes through point B. Let us drop the perpendicular BH from point B to line a (Fig. 57). It is easy to see that angle HBB 2 - acute... Indeed, if we assume that this angle is a straight line, then it follows from Definition 14.1 that any straight line passing through point B intersects the straight line a, which contradicts Theorem 13.1, i.e. Lobachevsky's parallelism axiom LV 1 (see § 13). It is easy to see that the assumption that this angle is obtuse also leads to a contradiction now with Definition 14.1 and Theorem 4.2 (see §4), since the inner ray of angle HBB 2, perpendicular to BH, does not intersect ray AA 2. Thus, the following statement is true.

Theorem 14.4. Let the directed line be parallel to the directed line. If from point B of the straight line we lower the perpendicular VN to the straight line, then the angle HBB 2 is acute.

The following corollary obviously follows from this theorem.

Consequence.If there is a common perpendicular to the directed lines and, then the line is not parallel to the line.

Let us introduce the concept of parallelism for undirected lines. We will assume that two undirected straight lines are parallel if directions can be chosen on them so that they satisfy Definition 14.1. As you know, the straight line has two directions. Therefore, from Theorem 14.3 it follows that through a point B, which does not belong to the line a, there are two undirected straight lines parallel to this line. Obviously, they are symmetric about the perpendicular dropped from point B to line a. These two straight lines are the very boundary lines separating the bundle of straight lines passing through point B and intersecting a, from the bundle of straight lines passing through B and not intersecting line a (Fig. 57).

Theorem 15.2. (The property of symmetry of parallel lines on the Lobachevsky plane).Let the directed line be parallel to the directed line. Then the directed line is parallel to the line.

The symmetry property of the concept of parallelism of lines on the Lobachevsky plane allows us not to indicate the order of directed parallel lines, i.e. do not specify which line is the first and which is the second. Obviously, the symmetry property of the concept of parallelism of straight lines also holds on the Euclidean plane. It follows directly from the definition of parallel lines in Euclidean geometry. In Euclidean geometry, the transitivity property is also fulfilled for parallel lines. If line a is parallel to line b and line b is parallel to line c. then straight lines a and c are also parallel to each other. A similar property is also true for directed straight lines on the Lobachevsky plane.

Theorem 15.3. (Property of transitivity of parallel lines on the Lobachevsky plane).Let three different directed straight lines be given,. If and , then .

Consider a directed line parallel to a directed line. Let's cross them with a straight line. Points A and B, respectively, are the intersection points of the straight lines, and, (Fig. 60). The following theorem is true.

Theorem 15.4. The angle is greater than the angle.

Theorem 15.5. The outer corner of a degenerate triangle is greater than an inner corner that is not adjacent to it.

The proof follows immediately from Theorem 15.4. Do it yourself.

Consider an arbitrary segment AB. Through point A we draw a straight line a, perpendicular to AB, and through point B, a straight line b parallel to a (Fig. 63). As follows from Theorem 14.4 (see § 14), the line b is not perpendicular to the line AB.

Definition 16.1. An acute angle formed by straight lines AB and b is called the angle of parallelism of the segment AB.

It is clear that a certain parallelism angle corresponds to each line segment. The following theorem is true.

Theorem 16.2. Equal segments correspond to equal angles of parallelism.

Proof. Let two equal segments AB and A ¢ B ¢ be given. Let us draw through points A and A ¢ directed straight lines and, perpendicular to AB and A ¢ B ¢, respectively, and through points B and B ¢ directed straight lines and, parallel, respectively, and (Fig. 64). Then and respectively, the angles of parallelism of the segments AB and A ¢ B ¢. Let's pretend that

Let us set aside the angle a 2 from the VA beam in the BAA 2 half-plane (see Fig. 64). By virtue of inequality (1), ray l is the inner ray of angle ABB 2. Since ½1, then l intersects the ray AA 2 at some point P. Let us put on the ray A ¢ A 2 ¢ from the point A the segment A ¢ P equal to AP. Consider triangles ABP and A ¢ B ¢ P ¢. They are rectangular, according to the hypothesis of the theorem, they have equal legs AB and A ¢ B ¢, by construction, the second pair of legs AP and A ¢ P are equal to each other. Thus, the right-angled triangle ABP is equal to the triangle A ¢ B ¢ P ¢. That's why . On the other hand, ray B ¢ P ¢, intersects ray A ¢ A 2 ¢, and the directed straight line B 1 ¢ B 2 ¢ is parallel to the straight line A 1 ¢ A 2 ¢. Therefore, the ray B ¢ P ¢ is the inner ray of the angle A ¢ B ¢ B 2 ¢, ... The resulting contradiction refutes our assumption, inequality (1) is false. Similarly, it is proved that the angle cannot be less than the angle. The theorem is proved.

Let us now consider how the angles of parallelism of unequal segments are related to each other.

Theorem 16.3. Let the segment AB be greater than the segment A ¢ B ¢, and the angles and, accordingly, their angles of parallelism. Then .

Proof. The proof of this theorem follows directly from Theorem 15.5 (see § 15) on the external angle of a degenerate triangle. Consider segment AB. Let's draw a directed straight line through point A, perpendicular to AB, and through point B, a directed straight line, parallel (Fig. 65). Let us put on the ray AB the segment AP equal to A ¢ B ¢. Since, then P is the inner point of the segment AB. Let's draw a directed line C 1 C 2 through P, also parallel. The angle serves as the angle of parallelism of the segment A ¢ B ¢, and the angle is the angle of parallelism of the segment AB. On the other hand, it follows from Theorem 15.2 on the symmetry of the concept of parallelism of lines (see § 15) that the line С 1 С 2 is parallel to the line. Therefore, the triangle RBC 2 A 2 is degenerate, is external, and its internal corners. Theorem 15.5 implies the truth of the assertion being proved.

The converse is easy to prove.

Theorem 16.4.Let and the angles of parallelism of the segments AB and A ¢ B ¢. Then, if, then AB> A ¢ B ¢.

Proof. Suppose the opposite,. Then it follows from Theorems 16.2 and 16.3 that , which contradicts the hypothesis of the theorem.

And so we proved that each segment corresponds to its own angle of parallelism, and the larger segment corresponds to a smaller angle of parallelism. Consider a statement proving that for any acute angle there is a segment for which this angle is the angle of parallelism. This will establish a one-to-one correspondence between the segments and acute angles on the Lobachevsky plane.

Theorem 16.5. For any acute angle, there is a line segment for which this angle is the parallel angle.

Proof. Let an acute angle ABC be given (Fig. 66). We will assume that all points considered in what follows on rays BA and BC lie between points B and A and B and C. Let's call a ray admissible if its origin belongs to the side of the angle BA, it is perpendicular to the straight line BA and is located in the same half-plane relative to the straight line BA as the side BC of the given angle. Let us turn to Legendre's proposal: n A perpendicular drawn to a side of an acute angle at any point on that side intersects the second side of the angle. We have proved Theorem 11.6 (see § 11), which states that Legendre's proposal is equivalent to the parallel axiom of Euclidean geometry. From this we concluded that on the Lobachevsky plane the logical negation of this statement is valid, namely, on the side of any acute angle there is such a point that the perpendicular to it, raised at this point, does not intersect the second side of the angle(see § 13). Thus, there is such an admissible ray m with the origin at point M, which does not intersect the BC side of the given angle (see Fig. 66).

Let's split the points of the segment VM into two classes. Class will belong to those points of this segment for which admissible rays with origins at these points intersect the BC side of this angle, and the class belong those points of the BC segment for which the admissible rays with origins at these points do not cross the BC side. Let us show that such a partition of the segment BM forms a Dedekind section (see Theorem 4.3, § 4). To do this, check that

5. and classes and contain points other than B and M;

6. any point of the class other than B lies between point B and any point of the class.

The first condition is obviously fulfilled. Any point of the segment BM belongs to either the class K 1 or the class K 2. Moreover, a point, by virtue of the definition of these classes, cannot belong to two classes at the same time. Obviously, we can assume that the point M belongs to K 2, since the admissible ray with the origin at the point M does not intersect BC. Class K 1 contains at least one point different from B. To construct it, it is sufficient to select an arbitrary point P on the BC side and drop from it the perpendicular PQ onto the beam BA. If we assume that the point Q lies between the points M and A, then the points P and Q lie in different half-planes with respect to the line containing the ray m (see Fig. 66). Therefore, the segment PQ intersects the ray m at some point R. We obtain that two perpendiculars are dropped from the point R onto the line BA, which contradicts Theorem 4.2 (see § 4). Thus, the point Q belongs to the segment BM, the class K 1 contains points other than B. It is easy to explain why on the ray BA there is a segment containing at least one point belonging to the class K 2 and different from its end. Indeed, if the class K 2 of the segment BM under consideration contains a single point M, then we choose an arbitrary point M ¢ between M and A. Consider an admissible ray m ¢ with the origin at the point M ¢. It does not intersect ray m, otherwise two perpendiculars are dropped from the point to line AB, therefore m ¢ does not intersect ray BC. The segment VM ¢ is the desired one, and all further reasoning should be carried out for the segment VM ¢.

Let us check the validity of the third condition of Theorem 4.3. Suppose that there are such points and that point P lies between point U and M (Fig. 67). Let us draw the admissible rays u and p with origins at the points U and P. Since, the ray p intersects the side BC of a given angle at some point Q. The straight line containing the ray u intersects the side BP of the triangle BPQ, therefore, according to Hilbert's axiom (Pasha's axiom , see § 3) it intersects either side BQ or side PQ of this triangle. But, therefore, the ray u does not intersect the side BQ, therefore, the rays p and u intersect at some point R. We again come to a contradiction, since we have built a point from which two perpendiculars are dropped on the line AB. The condition of Theorem 4.3 is fully satisfied.

M. It follows that. We have obtained a contradiction, since we have constructed a point of class K 1, located between the points and M. It remains for us to show that any inner ray of the angle intersects the ray BC. Consider an arbitrary inner ray h of this angle. Let us choose an arbitrary point K on it, belonging to the corner, and drop the perpendicular from it onto the line BA (Fig. 69). The base S of this perpendicular obviously belongs to the segment BM 0, i.e. class K 1 (prove this fact yourself). It follows that the perpendicular KS intersects the BC side of the given angle at some point T (see Fig. 69). Ray h crossed the side ST of triangle BST at point K, according to the axiom (Pasha's axiom), it must intersect either side BS or side BT of this triangle. It is clear that h does not intersect the segment BS, otherwise two lines, h and BA, pass through two points, and this intersection point. Thus, h crosses the BT side, i.e. beam VA. The theorem is completely proved.

And so, we have established that each segment in Lobachevsky's geometry can be associated with an acute angle - its angle of parallelism. We will assume that we have introduced the measure of angles and segments; note that the measure of segments will be introduced by us later, in §. We introduce the following definition.

Definition 16.6. If x is the length of the segment, and j is the value of the angle, then the dependence j = P (x), which associates the length of the segment with the value of its parallelism angle, is called the Lobachevsky function.

It's clear that . Using the properties of the parallelism angle of a segment proved above (see Theorems 16.3 and 16.4), we can draw the following conclusion: the Lobachevsky function is monotonically decreasing. Nikolai Ivanovich Lobachevsky obtained the following remarkable formula:

,

where k is some positive number. It is important in the geometry of the Lobachevsky space, and is called its radius of curvature. Two Lobachevsky spaces with the same radius of curvature are isometric. From the above formula, as is easy to see, it also follows that j = P (x) is a monotonically decreasing continuous function whose values ​​belong to the interval.

On the Euclidean plane, we fix a circle w centered at some point O and with a radius equal to one, which we will call absolute... The set of all points of the circle bounded by the circle w will be denoted by W ¢, and the set of all interior points of this circle by W. Thus,. The points of the set W will be called L-dots The set W of all L-points is L-plane, on which we will construct the Cayley-Klein model of the Lobachevsky plane. We will call L ‑ straight arbitrary chords of the circle w. We will assume that the L-point X belongs to the L-line x if and only if the point X as a point of the Euclidean plane belongs to the chord x of the absolute.

L-plane, the Lobachevsky axiom of parallelism holds: through an L ‑ point B that does not lie on the L ‑ line a pass at least two L ‑ lines b and c that have no common points with the L ‑ line a. Figure 94 illustrates this statement. It is also easy to understand what the parallel directed lines of the L-plane are. Consider Figure 95. The L-line b passes through the intersection of the L-line a with the absolute. Therefore, the directional L-line A 1 A 2 is parallel to the directional L-line B 1 A 2. Indeed, these lines do not intersect, and if we choose arbitrary L-points A and B belonging to these lines, respectively, then any inner ray h of angle A 2 BA intersects line a. Thus, two L-lines are parallel if they have a common point of intersection with an absolute. It is clear that the symmetry and transitivity property of the concept of parallelism of L-lines is satisfied. In paragraph 15, we proved the property of symmetry, while the property of transitivity is illustrated in Figure 95. Line A 1 A 2 is parallel to line B 1 A 2, they intersect the absolute at point A 2. Lines B 1 A 2 and C 1 A 2 are also parallel, they also intersect the absolute at the same point A 2. Therefore, straight lines A 1 A 2 and C 1 A 2 are parallel to each other.

Thus, the basic concepts defined above satisfy the requirements of axioms I 1 -I 3, II, III, IV of groups of Hilbert's axioms and the axiom of parallelism of Lobachevsky, therefore they are a model of the Lobachevsky plane. We have proved the substantial consistency of the Lobachevsky planimetry. Let us formulate this statement as the following theorem.

Theorem 1. Lobachevsky's geometry is consistent in terms of content.

We have built a model of the Lobachevsky plane, but with the construction of a spatial model similar to that considered on a plane, you can get acquainted in the manual.

The most important conclusion follows from Theorem 1. The parallelism axiom is not a consequence of axioms I - IV of Hilbert's axioms. Since the fifth postulate of Euclid is equivalent to the axiom of parallelism of Euclidean geometry, this postulate also does not depend on the rest of Hilbert's axioms.

”Devoted to the relationship between Russian and British science, mathematician Valentina Kirichenko tells PostNauka about the revolutionary nature of Lobachevsky's ideas for the geometry of the 19th century.

Parallel lines do not intersect even in the geometry of Lobachevsky. Somewhere in the films you can often find the phrase: "And our Lobachevsky's parallel lines intersect." Sounds beautiful, but not true. Nikolai Ivanovich Lobachevsky really came up with an extraordinary geometry, in which parallel lines behave quite differently than we are used to. But still they do not overlap.

We are used to thinking that two parallel lines do not converge and do not move away. That is, no matter what point we take on the first line, the distance from it to the second line is the same, it does not depend on the point. But is it really so? And why is this so? And how can this be verified at all?

If we are talking about physical straight lines, then only a small section of each straight line is available to us for observation. And given the measurement errors, we will not be able to draw any definite conclusions about how the straight lines behave very, very far from us. The ancient Greeks had similar questions. In the III century BC, the ancient Greek geometer Euclid very accurately outlined the main property of parallel lines, which he could neither prove nor disprove. Therefore, he called it a postulate - a statement that should be taken on faith. This is the famous fifth postulate of Euclid: if two straight lines on the plane intersect with the secant, so that the sum of the inner one-sided angles is less than two straight lines, that is, less than 180 degrees, then with sufficient continuation these two straight lines will intersect, and it is on the side of the secant along which the sum is less than two right angles.

The key words in this postulate are "with enough continuation." It is because of these words that the postulate cannot be verified empirically. Maybe the lines will intersect in the line of sight. Maybe after 10 kilometers or beyond the orbit of Pluto, or maybe even in another galaxy.

Euclid outlined his postulates and results, which logically follow from them, in the famous book "Beginnings". From the ancient Greek name of this book comes the Russian word "elements", and from the Latin name - the word "elements". Euclid's Beginnings is the most popular textbook of all time. In terms of the number of editions, it is second only to the Bible.

I would especially like to note the wonderful British edition of 1847 with very clear and beautiful infographics. Instead of dull designations in the drawings, they use colored drawings - not like in modern school geometry textbooks.

Until the last century, Euclid's "Beginnings" were required for study in all educational programs, which implied intellectual creativity, that is, not just learning a craft, but something more intellectual. The non-obviousness of the fifth postulate of Euclid raised a natural question: is it possible to prove it, that is, deduce it logically from the rest of Euclid's assumptions? Many mathematicians, from the contemporaries of Euclid to those of Lobachevsky, tried to do this. As a rule, they reduced the fifth postulate to some more visual statement, which is easier to believe.

For example, in the 17th century, the English mathematician John Wallis reduced the fifth postulate to this statement: there are two similar, but unequal triangles, that is, two triangles whose angles are equal, but the sizes are different. It would seem, what could be simpler? Let's just change the scale. But it turns out that the ability to change the scale while maintaining all angles and proportions is an exclusive property of Euclidean geometry, that is, geometry in which all Euclidean postulates are fulfilled, including the fifth.

In the 18th century, the Scottish scholar John Playfair reformulated the fifth postulate in the form in which it usually appears in modern school textbooks: two straight lines intersecting each other cannot be simultaneously parallel to the third line. It is in this form that the fifth postulate appears in modern school textbooks.

By the early 19th century, many were under the impression that proving the fifth postulate was like inventing a perpetual motion machine — a completely useless exercise. But even to assume that Euclid's geometry is not the only possible one, no one had the courage: Euclid's authority was too great. In such a situation, Lobachevsky's discoveries were, on the one hand, natural, and on the other, absolutely revolutionary.

Lobachevsky replaced the fifth postulate with the exact opposite statement. Lobachevsky's axiom sounded like this: if from a point that does not lie on a straight line, release all the rays intersecting this straight line, then on the left and right these rays will be limited by two limiting rays, which will no longer intersect the straight line, but will become closer and closer to it. Moreover, the angle between these limiting rays will be strictly less than 180 degrees.

It immediately follows from Lobachevsky's axiom that through a point that does not lie on a given straight line, it is possible to draw not one straight line parallel to a given one, as in Euclid, but as many as you like. But these straight lines will behave differently from Euclid's. For example, if we have two parallel straight lines, then they can first approach, and then move away. That is, the distance from a point on the first line to the second line will depend on the point. It will be different for different points.

Lobachevsky's geometry contradicts our intuition in part because at the small distances we usually deal with, it differs very little from Euclidean. Similarly, we perceive the curvature of the Earth's surface. When we walk from house to store, it seems to us that we are walking in a straight line, and the Earth is flat. But if we fly, say, from Moscow to Montreal, then we already notice that the plane is flying in an arc of a circle, because this is the shortest path between two points on the Earth's surface. That is, we notice that the Earth looks more like a soccer ball than a pancake.

Lobachevsky's geometry can also be illustrated with the help of a soccer ball, not ordinary, but hyperbolic. A hyperbolic soccer ball is glued together like a regular one. Only in an ordinary ball, white hexagons are glued to black pentagons, and in a hyperbolic ball, instead of pentagons, you need to make heptagons and also glue them with hexagons. In this case, of course, it will not turn out to be a ball, but rather a saddle. And on this saddle the geometry of Lobachevsky is realized.

Lobachevsky tried to talk about his discoveries in 1826 at Kazan University. But the text of the report has not survived. In 1829 he published an article on his geometry in a university journal. Lobachevsky's results seemed meaningless to many - not only because they destroyed the usual picture of the world, but because they were not presented in the most understandable way.

However, Lobachevsky also had publications in high-rating journals, as we call them today. For example, in 1836 he published an article entitled "Imaginary Geometry" in French in the famous journal Crell, in the same issue with articles by the most famous mathematicians of the time - Dirichlet, Steiner and Jacobi. And in 1840 Lobachevsky published a small and very understandably written book entitled "Geometric Investigations in the Theory of Parallel Lines." The book was in German and published in Germany. A devastating review immediately appeared. The reviewer especially scoffed at Lobachevsky's phrase: "The further we continue straight lines in the direction of their parallelism, the more they approach each other." "This statement alone," the reviewer wrote, "already characterizes the work of Mr. Lobachevsky enough and frees the reviewer from the need for further evaluation."

But the book also has one unbiased reader. This was Karl Friedrich Gauss, also known by the nickname King of Mathematicians, one of the greatest mathematicians in history. He praised Lobachevsky's book in one of his letters. But his review was published only after his death, along with the rest of the correspondence. And then the real boom of Lobachevsky's geometry began.

In 1866, his book was translated into French, then into English. Moreover, the English edition was reprinted three more times due to its extraordinary popularity. Unfortunately, Lobachevsky did not live up to this time. He died in 1856. And in 1868, a Russian edition of Lobachevsky's book appeared. It was published not as a book, but as an article in the oldest Russian journal "Mathematical Collection". But then this magazine was very young, it was not yet two years old. But the more famous is the Russian translation of 1945, made by the remarkable Russian and Soviet geometer Veniamin Fedorovich Kagan.

By the end of the 19th century, mathematicians were divided into two camps. Some immediately accepted Lobachevsky's results and began to further develop his ideas. Others could not give up the belief that Lobachevsky's geometry describes something that does not exist, that is, Euclid's geometry is the only true one and nothing else can be. Unfortunately, the latter included the mathematician, better known as the author of "Alice in Wonderland" - Lewis Carroll. His real name is Charles Dodgson. In 1890, he published an article entitled "A New Theory of Parallels," where he defended a highly visual version of the fifth postulate. Lewis Carroll's axiom sounds like this: if you inscribe a regular quadrangle in a circle, then the area of ​​this quadrangle will be strictly larger than the area of ​​any of the segments of the circle lying outside the quadrilateral. In the geometry of Lobachevsky, this axiom is not true. If we take a sufficiently large circle, then no matter what quadrangle we inscribe into it, no matter how long the sides of this quadrangle are, the area of ​​the quadrilateral will be limited by a universal physical constant. In general, the presence of physical constants and universal measures of length is an advantageous difference between Lobachevsky's geometry and Euclid's geometry.

But Arthur Cayley, another famous English mathematician, in 1859, that is, just three years after the death of Lobachevsky, published an article that later helped legalize Lobachevsky's postulate. Interestingly, Cayley was moonlighting as a lawyer in London at that time and only then received a professorship at Cambridge. In fact, Cayley constructed the first model of Lobachevsky's geometry, although at first glance he was solving a completely different problem.

And another wonderful English mathematician, whose name was William Kingdon Clifford, was deeply imbued with the ideas of Lobachevsky. And in particular, he was the first to put forward the idea long before the creation of general relativity that gravity is caused by the curvature of space. Clifford assessed Lobachevsky's contribution to science in one of his lectures on the philosophy of science: "Lobachevsky became for Euclid what Copernicus became for Ptolemy." If before Copernicus, mankind believed that we know everything about the Universe, now it is clear to us that we observe only a small part of the Universe. Likewise, before Lobachevsky, mankind believed that there is only one geometry - Euclidean, everything about it has long been known. Now we know that there are many geometries, but we do not know everything about them.

Euclid's fifth postulate "If a straight line falling on two straight lines forms internal one-sided angles, in total less than two straight lines, then, continued indefinitely, these two straight lines will meet on the side where the angles in the sum are less than two straight lines" to many mathematicians in antiquity it seemed 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 directions, in fact, are closely related to each other. The first sought to replace this postulate with a simpler and more intuitively clear one, as, for example, the statement formulated by Proclus “Through a point that does not lie on a given straight line, only one straight line can be drawn that does not intersect with the given one”: it is in this form that the 5th postulate or rather, the parallel axiom, equivalent to it, 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-Jauhari (early 9th century), Sabit ibn Korra, Ibn al-Khaisam, Omar Khayyam, Nasireddin at-Tusi. Later, Europeans joined these studies: Levi Ben Gershon (14th century) and Alfonso (15th 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 very important works by the Italian Jesuit G. Saccheri and the German I. G. Lambert.

Proofs were usually carried out by the method "by contradiction": from the assumption that the 5th postulate is not fulfilled, they tried to deduce consequences that would contradict other postulates and axioms. In reality, however, in the end they did not get a contradiction with other postulates, but with some explicit or implicit "obvious" proposition, which, however, could not be established 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, again, some other equivalent statement was put. For example, the following provisions were taken as such a statement:

Rice. 2. There are straight lines equidistant from each other

Rice. 4. Two converging lines intersect

The geometry in which these statements do not hold is, of course, not the same as we are used to, but it does not yet follow that it is impossible or that these statements follow from other postulates and axioms of Euclid, so that all the proofs had some gaps. or stretching. Clavius ​​substantiated the assumption that there are straight lines, equidistant from each other, by the Euclidean “definition” of a straight line as a line, equally spaced in relation 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 arbitrarily large size, and substantiated this statement with the 3rd postulate of Euclid, which asserts 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). AM Legendre in successive editions of the textbook "Principles of Geometry" (1794, 1800, 1823) gave new proofs of the 5th postulate, but a careful analysis showed gaps in these proofs. Having subjected Legendre to just criticism, our compatriot S. Ye. Guriev in his book "Experience on the improvement of 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 quadrangle 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 straight lines, 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 quadrangle is considered, which is obtained as a result of laying off equal segments on two perpendiculars to one straight line. Three hypotheses are investigated: the two upper corners are sharp, obtuse or straight; an attempt is made to show that the hypotheses of obtuse and acute angles lead to contradiction.

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

Saccheri and Lambert showed that the hypotheses of obtuse angles do lead to a contradiction, but they failed to find contradictions when considering the hypotheses of acute angles: Saccheri made the conclusion about such a contradiction only as a result of an error, and Lambert concluded that the apparent absence of contradiction in the acute angle hypothesis was 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 what was discovered in the beginning. XVII century the position according to which the area of ​​a spherical triangle, on the contrary, is more than 180 ° by an amount proportional to its area.

In 1763, GS Klugel published "A Review of the Most Important Attempts to Prove the Theory of Parallel Lines", where he examined about 30 proofs of the 5th postulate and revealed errors in them. Klugel concluded that Euclid quite reasonably placed his statement among the postulates.

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

Several scientists independently came to the idea that the assumption of an alternative to the 5th postulate leads to the construction of a geometry different from the Euclidean, but equally consistent: K.F. Gauss, N.I. Lobachevsky and J. Boyai (as well as F K. Schweickart 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 records preserved in his archive (and published only in the 1860s), realized the possibility of a new geometry in the 1810s, but also never published his discoveries on this topic: “I am afraid of the cry of the Boeotians (ie, fools: the inhabitants of the region of Boeotia were considered the most stupid in Ancient Greece), if I express my views entirely, ”he wrote in 1829 to his friend mathematician FV Bessel. Misunderstanding fell to the lot of Lobachevsky, who made the first report on the new geometry in 1826 and published the results obtained in 1829. In 1842, Gauss achieved the election of Lobachevsky as a corresponding member of the Göttingen Scientific Society: this was the only recognition of Lobachevsky's merits during his lifetime ... Father J. Boyai - mathematician Farkash 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 absorb a thousand such titans as Newton, on Earth this will never be cleared 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 was no longer engaged in non-Euclidean geometry. So only Lobachevsky, for the rest of his life, firstly, continued research in a new field, and secondly, he promoted his ideas, published a number of books and articles on new geometry.

So, in the Lobachevsky plane, at least two straight lines that do not intersect AB pass through the point C outside the given line AB. All lines passing through C are divided into two classes - intersecting and non-intersecting AB. These latter lie in a certain angle formed by two extreme straight lines that do not intersect AB. It is these lines that Lobachevsky calls parallel to the straight 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 that lie inside the corner are called divergent with respect to AB.

Any two diverging lines p and q have a single common perpendicular t, which is the shortest line segments 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 a finite 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 straight lines p and q are parallel, then in one direction the distances between their points decrease indefinitely, while in the other they increase indefinitely; one straight line is projected onto another ray.

The figures show various mutual positions of the straight lines p and q, which are possible in the geometry of Lobachevsky; 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's geometry, we still would not be able to depict on a small scale without distortions how everything looks in large: in Lobachevsky's geometry there are no similar figures that are not equal).

Inside the corner, there is a straight line parallel to both sides of the corner. It divides all points inside the corner into two types: through the points of the first type, you can draw straight lines that intersect both sides of the corner; 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 diverging lines there are two lines parallel to both of them; they divide the space between diverging lines into three areas: through points in one area, you can draw lines that intersect both sides of the corner; such lines cannot be drawn through the points in the other two regions.

An acute angle, not a right angle, always rests on the diameter of a circle. The side of a regular hexagon inscribed in a circle is always larger than its radius. For any n> 6 it is possible to 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 from 180 ° lay entirely within the observation error. The misunderstanding that fell to the lot of 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 play of the imagination. Is the new geometry really consistent? (After all, even if Lobachevsky did not manage to meet a contradiction, this does not guarantee that it will not be discovered later). How does it relate to the real world, as well as other areas of mathematics? This became clear far from immediately, and the success that ultimately fell to the lot of new ideas was associated with the discovery of models of new geometry.