The history of the creation of Lobachevsky's geometry is at the same time the history of attempts to prove the fifth postulate of Euclid. This postulate is one of the axioms put by Euclid as the basis for the presentation of geometry (see Euclid and his Elements). The fifth postulate is the last and most complex of the sentences included by Euclid in his axiomatics of geometry. Recall the formulation of the fifth postulate: if two lines intersect with a third so that on either side of it the sum of the interior angles is less than two right angles, then on the same side the original lines intersect. For example, if in Fig. 1 angle is a straight line, and the angle is slightly less than a straight line, then the straight lines will certainly intersect, and to the right of the straight line. Many theorems of Euclid (for example, "in an isosceles triangle, the angles at the base are equal") express much simpler facts than the fifth postulate. In addition, it is rather difficult to test the fifth postulate in an experiment. Suffice it to say that if in Fig. 1 the distance is considered equal to 1 m, and the angle differs from the straight line by one arc second, then we can calculate that the lines and intersect at a distance of more than 200 km from the line.

Many mathematicians who lived after Euclid tried to prove that this axiom (the fifth postulate) is superfluous, i.e. it can be proved as a theorem on the basis of the remaining axioms. So, in the 5th c. mathematician Proclus (the first commentator on the works of Euclid) made such an attempt. However, in his proof, Proclus imperceptibly used the following statement: two perpendiculars to one straight line are at a limited distance from each other along their entire length (i.e., two straight lines perpendicular to a third cannot move away from each other indefinitely, like lines on Fig. 2). But with all the seeming visual "obviousness", this statement, with a strict axiomatic presentation of geometry, requires substantiation. In fact, the statement used by Proclus is the equivalent of the fifth postulate; in other words, if it is added to the rest of Euclid's axioms as another new axiom, then the fifth postulate can be proved (which Proclus did), and if the fifth postulate is accepted, then the statement formulated by Proclus can be proved.

Critical analysis of further attempts to prove the fifth postulate revealed a large number of similar "obvious" statements that can replace the fifth postulate in Euclid's axiomatics. Here are some examples of such equivalents of the fifth postulate.

1) Through a point inside an angle smaller than the expanded one, it is always possible to draw a straight line intersecting its sides, i.e. straight lines on a plane cannot be located as shown in Fig. 3. 2) There are two similar triangles that are not equal to each other. 3) Three points located on one side of a straight line at an equal distance from it (Fig. 4) lie on one straight line. 4) For every triangle there is a circumscribed circle.

Gradually, the "proofs" become more and more sophisticated, in them subtle equivalents of the fifth postulate are hidden deeper and deeper. Assuming that the fifth postulate is wrong, mathematicians have tried to arrive at a logical contradiction. They came up with statements that monstrously contradict our geometric intuition, but the logical contradiction did not work out. Or maybe we will never arrive on such a path to a contradiction? Could it be that, replacing the fifth postulate of Euclid by its negation (while retaining the rest of Euclid's axioms), we will come to a new, non-Euclidean geometry, which in many respects does not agree with our usual visual representations, but nevertheless does not contain any logical contradictions ? Mathematicians could not suffer this simple but very daring idea for two millennia after the appearance of Euclid's Elements.

The first to admit the possibility of the existence of non-Euclidean geometry, in which the fifth postulate is replaced by its negation, was K. F. Gauss. The fact that Gauss owned the ideas of non-Euclidean geometry was discovered only after the death of the scientist, when they began to study his archives. The ingenious Gauss, to whose opinions everyone listened, did not dare to publish his results on non-Euclidean geometry, fearing to be misunderstood and drawn into controversy.

19th century brought the solution to the riddle of the fifth postulate. Independently of Gauss, our compatriot, Professor of Kazan University N. I. Lobachevsky, also came to this discovery. Like his predecessors, Lobachevsky initially tried to deduce various consequences from the denial of the fifth postulate, hoping that sooner or later he would come to a contradiction. However, he proved many dozens of theorems without revealing logical contradictions. And then Lobachevsky came up with a conjecture about the consistency of geometry, in which the fifth postulate is replaced by its negation. Lobachevsky called this geometry imaginary. Lobachevsky set out his research in a number of works, beginning in 1829. But the mathematical world did not accept Lobachevsky's ideas. Scientists were not prepared for the idea that there could be a geometry other than Euclidean. And only Gauss expressed his attitude to the scientific feat of the Russian scientist: he achieved the election in 1842 of N. I. Lobachevsky as a corresponding member of the Gottingen Royal Scientific Society. This is the only scientific honor that fell to the lot of Lobachevsky during his lifetime. He died without having achieved recognition of his ideas.

Talking about the geometry of Lobachevsky, one cannot fail to note another scientist who, together with Gauss and Lobachevsky, shares the merit of discovering non-Euclidean geometry. It was the Hungarian mathematician J. Bolyai (1802-1860). His father, the famous mathematician F. Bolyai, who worked all his life on the theory of parallels, believed that the solution of this problem was beyond human strength, and wanted to protect his son from failures and disappointments. In one of his letters, he wrote to him: “I went through all the hopeless darkness of this night and buried every light, every joy of life in it ... it can deprive you of all your time, health, peace, all the happiness of your life ...” But Janos did not heed his father's warnings. Soon the young scientist independently of Gauss and Lobachevsky came to the same ideas. In an appendix to his father's book published in 1832, J. Bolyai gave an independent exposition of non-Euclidean geometry.

Lobachevsky geometry (or Bolyai Lobachevsky geometry, as it is sometimes called) preserves all the theorems that can be proved in Euclidean geometry without using the fifth postulate (or the axiom of parallelism of one of the equivalents of the fifth postulate - included in our days in school textbooks). For instance: vertical angles are equal; the angles at the base of an isosceles triangle are equal; from a given point, only one perpendicular can be lowered to a given line; the signs of equality of triangles are also preserved, etc. However, the theorems, in the proof of which the axiom of parallelism is used, are modified. The theorem on the sum of the angles of a triangle is the first theorem of a school course, the proof of which uses the axiom of parallelism. Here we are in for the first "surprise": in the geometry of Lobachevsky, the sum of the angles of any triangle is less than 180°.

If two angles of one triangle are respectively equal to two angles of another triangle, then in Euclidean geometry the third angles are also equal (such triangles are similar). There are no such triangles in Lobachevsky's geometry. Moreover, in the geometry of Lobachevsky, the fourth criterion for the equality of triangles takes place: if the angles of one triangle are respectively equal to the angles of another triangle, then these triangles are equal.

The difference between 180° and the sum of the angles of a triangle in Lobachevsky's geometry is positive; it is called the defect of this triangle. It turns out that in this geometry the area of ​​a triangle is related in a remarkable way to its defect: , where and mean the area and defect of the triangle, and the number depends on the choice of units for measuring areas and angles.

Let now be some acute angle (Fig. 5). In Lobachevsky geometry, one can choose a point on the side such that the perpendicular to the side does not intersect with the other side of the angle. This fact just confirms that the fifth postulate is not fulfilled: the sum of the angles and is less than the expanded angle, but the lines and do not intersect. If we begin to approximate the point to , then there is such a “critical” point that the perpendicular to the side still does not intersect with the side, but for any point lying between and , the corresponding perpendicular intersects with the side. Straight lines and more and more approach each other, but do not have common points. On fig. 6 these lines are shown separately; It is precisely such straight lines approaching each other indefinitely that Lobachevsky calls parallel in his geometry. And Lobachevsky calls two perpendiculars to one straight line (which move away from each other indefinitely, as in Fig. 2) divergent straight lines. It turns out that this limits all the possibilities of arranging two lines on the Lobachevsky plane: two non-coinciding lines either intersect at one point, or are parallel (Fig. 6), or are divergent (in this case they have a single common perpendicular, Fig. 2).

On fig. 7, the perpendicular to the side of the angle does not intersect with the side, and the lines are symmetrical to the lines with respect to . Further, , so that is the perpendicular to the segment in its middle and, similarly, the perpendicular to the segment in its middle. These perpendiculars do not intersect, and therefore there is no point equidistant from the points , i.e. the triangle has no circumscribed circle.

On fig. Figure 8 shows an interesting arrangement of three lines on the Lobachevsky plane: each two of them are parallel (only in different directions). And in fig. 9 all lines are parallel to each other in one direction (a bundle of parallel lines). The red line in fig. 9 is “perpendicular” to all lines drawn (i.e., the tangent to this line at any point is perpendicular to the line passing through). This line is called the limiting circle, or horocycle. The straight lines of the considered beam are, as it were, its "radii", and the "center" of the limiting circle lies at infinity, since the "radii" are parallel. At the same time, the limiting circle is not a straight line, it is "curved". And other properties that a line has in Euclidean geometry, in Lobachevsky geometry turn out to be inherent in other lines. For example, the set of points located on one side of a given straight line at a given distance from it, in Lobachevsky's geometry, is a curved line (it is called an equidistant line).

We briefly touched only on some facts of Lobachevsky's geometry, without mentioning many other very interesting and meaningful theorems (for example, the circumference and area of ​​a circle of radius grow here depending on according to an exponential law). There is a conviction that this theory, rich in very interesting and meaningful facts, is in fact consistent. But this conviction (which all three creators of non-Euclidean geometry had) does not replace the proof of consistency.

To obtain such a proof, it was necessary to build a model. And Lobachevsky understood this well and tried to find her.

But Lobachevsky himself could no longer do this. The construction of such a model (that is, the proof of the consistency of Lobachevsky's geometry) fell to the lot of mathematicians of the next generation.

In 1868, the Italian mathematician E. Beltrami investigated a concave surface called a pseudosphere (Fig. 10), and proved that Lobachevsky's geometry acts on this surface! If we draw the shortest lines (“geodesics”) on this surface and measure distances along these lines, make triangles from the arcs of these lines, etc., then it turns out that all Lobachevsky geometry formulas are realized exactly (in particular, the sum of the angles of any triangle less than 180°). True, not the entire Lobachevsky plane is realized on the pseudosphere, but only its limited piece, but nevertheless this was the first breach in the blank wall of non-recognition of Lobachevsky. And two years later, the German mathematician F. Klein (1849-1925) proposes another model of the Lobachevsky plane.

Klein takes a certain circle and considers such projective transformations of the plane (see projective geometry) that map the circle onto itself. "Plane" Klein calls the interior of the circle, and he considers these projective transformations to be "movements" of this "plane". Further, each chord of the circle (without ends, since only the internal points of the circle are taken) is considered by Klein to be a "straight line". Since the "movements" are projective transformations, the "direct lines" become "direct lines" under these "movements". Now in this "plane" you can consider segments, triangles, etc. Two figures are called "equal" if one of them can be translated into the other by some "movement". Thus, all the concepts mentioned in the axioms of geometry are introduced, and it is possible to check the fulfillment of the axioms in this model. For example, it is obvious that a single “straight line” passes through any two points (Fig. 11). It can also be seen that through a point that does not belong to the "line" , there are infinitely many "lines" that do not intersect . Further verification shows that all other axioms of Lobachevsky geometry are also satisfied in the Klein model. In particular, for any "line" (i.e. the chord of a circle) and any point of this "line" there is a "movement" that takes it to another given line with a point marked on it. This allows us to verify the validity of all the axioms of Lobachevsky's geometry.

Another model of Lobachevsky's geometry was proposed by the French mathematician A. Poincaré (1854-1912). He also considers the interior of a certain circle; He considers “straight lines” arcs of circles that touch the radii at the points of intersection with the boundary of the circle (Fig. 12). Without speaking in detail about the "movements" in the Poincaré model (they will be circular transformations, in particular, inversions with respect to "straight lines", transforming the circle into itself), we restrict ourselves to indicating Fig. 13, which shows that the Euclidean axiom of parallelism has no place in this model. It is interesting that in this model the circle (Euclidean) located inside the circle turns out to be a "circle" in the sense of Lobachevsky's geometry as well; circle touching the boundary. Then the light will (in accordance with Fermat's principle on the minimum time of movement along the light path) propagate just along the "straight lines" of the considered model. Light cannot reach the boundary in a finite time (since its speed decreases to zero there), and therefore this world will be perceived by its “inhabitants” as infinite, and in its metrics and properties coinciding with the Lobachevsky plane.

Subsequently, other models of Lobachevsky's geometry were also proposed. These models finally established the consistency of Lobachevsky's geometry. Thus, it was shown that Euclid's geometry is not the only possible one. This had a great progressive impact on the entire further development of geometry and mathematics in general.

And in the XX century. it was found that the Lobachevsky geometry is not only important for abstract mathematics, as one of the possible geometries, but is also directly related to the applications of mathematics to physics. It turned out that the relationship between space and time, discovered in the works of X. Lorentz, A. Poincaré, A. Einstein, G. Minkowski and described in the framework of the special theory of relativity, is directly related to Lobachevsky's geometry. For example, Lobachevsky geometry formulas are used in the calculations of modern synchrophasotrons.

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None. By definition, parallel lines have no intersection points.

Now let's get on to geometry and fallacies. Everywhere "planes" will be considered, whatever that means.

Geometry of Euclid. What was taught in school, what is more familiar and almost exactly carried out in everyday life. I will single out those two facts that will be significant later. First: in this geometry there is a distance, between any two points there is a shortest line, and moreover, only one (a line segment). Second: through a point that does not lie on a given line, one can draw a line parallel to the given line and, moreover, only one.

This corresponds to some pair of axioms from Pogorelov's textbook, so it will be more convenient for me to rely on this.

Geometry of Lobachevsky. Everything is fine with the distance in it, but it is difficult for us to imagine it because of the constant negative curvature (we didn’t understand - it’s not scary). Parallelism is more difficult. Through a point outside a line, it is always possible to draw not just one, but infinitely many parallel lines.

spherical geometry. First, what do we consider "straight". Straight lines on the sphere - great circles = circles carved on the sphere by a plane passing through the center = circles of radius equal to the radius of the sphere. They are straight in the sense that they are shortest way between not very distant (later it will become clear which) points. Some may have noticed that if the cities are on the same parallel, then the plane does not fly along this parallel, but along a trajectory convex to the north in the northern hemisphere. If you draw, you will notice that the large circle connecting the two points runs north of the parallel.

Why is distance on a sphere bad? Let us take diametrically opposite points on the sphere, for which there are infinitely many shortest curves. More clearly: I will look at the north and south poles. All merilians pass through them, they all have the same length, any other path will be longer.

There are no parallel lines at all, any two lines intersect at diametrically opposite points.

projective plane. The most important and first difference: there is no distance and cannot be. In principle, it cannot be introduced so that it satisfies some natural conditions (it is preserved during the "movements" of the plane). Thus, geometry itself does not know about any "infinitely distant lines", all this is invented by people in order to somehow understand the projective plane. The most "simple" way: imagine a familiar plane (the so-called "affine map") and add to it a line that is "infinitely removed", and all the lines that were parallel to the given one in the plane that was presented will intersect in one point on this line at infinity. Such a description is quite simple: I wrote something in two sentences, and someone has already presented something. But it is misleading, there is no distinguished line in projective geometry. But this description already shows that parallel lines

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 the five 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 wording sounds extremely tricky, but if you translate it into understandable common man language, it turns out that, according to Euclid, two non-parallel lines will necessarily 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

Against, modern science comes to understand that Euclidean geometry is only a special case of Lobachevsky's geometry, and that in real world is more precisely 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.

Recently, in a post on pseudo-scientific topics, one of the commentators started talking about Lobachevsky's geometry (that he does not understand it) and even asked for an explanation. I then limited myself to the statement that I understand. It seemed to me impossible to explain this theory in the limited framework of the commentary and in one text (without drawings).

However, after thinking, I still decided to try to give a small popular digression into this theory.

A little background. Geometry since the time of Euclid has become axiomatic theory, in which most of the statements were proved on the basis of several postulates (axioms). It was believed that these axioms are "obvious", i.e. reflect the properties of real (physical) space.

One of these axioms aroused suspicion among scientists: could it be derived from the rest of the postulates? The modern formulation of this axiom is as follows:

"Through a point that does not lie on a given line, at most one line can be drawn parallel to it." That one straight line can be drawn is not an axiom, but a theorem.

In this case, a “parallel” is a straight line that does not intersect the given one. So, the essence of the axiom is that such a straight line is one!

(The common statement “Lobachevsky proved that parallel lines can also intersect” is, of course, blatantly wrong! After all, this would contradict their definition!)

Lobachevsky, like many before him, decided to prove that this statement can be deduced from other axioms. To do this, as is often done in mathematics, he chose the method "by contradiction", i.e. assumed that there are more than one line that does not intersect the given one and tried to deduce from this a contradiction with other facts. But the further he developed the theory, the more he became convinced that no contradiction was foreseen! Those. it turned out that the theory with the "wrong" postulate also has the right to exist!

Of course, at first his calculations were not recognized, they laughed at him. That is why the great Gauss (who came to the same conclusions) did not dare to publish his results. But over time, I had to admit that PURELY LOGICALLY Lobachevsky's theory is no worse than Euclidean.

One of the ingenious ways to see this is to come up with such "straight" ones who behave like Lobachevsky's "straight ones". And mathematicians have found such an example, and not just one.

Perhaps the simplest is the Poincaré model. You can build it yourself with simple tools.

Draw a straight line on a piece of paper. Take a compass and, placing its needle on this straight line, draw semicircles that are on one side of the straight line. Now erase the line (and with it the endpoints of the semicircles). So, these semicircles "without ends" will behave like straight lines in Lobachevsky's geometry!

Indeed, we select one semicircle and a point outside it. There are quite a few semicircles that do not intersect with the original and all pass through a given point. Among them, two stand out: they touch our original “line” at the end points (which, as you remember, we erased). there is no real intersection. These two circles set the "boundaries", between which there are all lines that do not intersect the given one. They are an infinite number.

You can see that the triangles in this model are not the same as in the (Euclidean) plane: the sum of their angles is less than 180 degrees! However, the smaller the triangle, the greater the sum of its angles. In the "small", at small distances, the geometry of Lobachevsky practically coincides with the geometry of Euclid. Therefore, generally speaking, we will not be able to “experimentally” distinguish one from the other if it turns out that the (cosmic) distances available to us are small for this purpose.

However, in our time, neither physicists, nor, especially, mathematicians, are trying to perceive Lobachevsky's geometry as a model of "real", physical space. Mathematicians have realized that all they can say is: if such-and-such axioms are true, then such-and-such theorems are also true. Well, what are “sets”, “points”, “straight lines”, “angles”, “distances”, etc.? – we do not know! Just like Stanislav Lem: “Sepulks are objects for sepulking”

“They say that Bertrand Russell defined mathematics as a science in which we never know what we are talking about, and how correct what we say is. It is known that mathematics is widely used in many other areas of science. […] Thus, one of the main functions of mathematical proof is to create a reliable basis for insight into the essence of things.”

(from the book "Physicists are joking")

Interesting information about the relationship between mathematics and empiricism can be found in the work