It is generally accepted that the Russian mathematician of the 19th century in its theory forced to cross parallel direct, contrary to classical, Euclidean geometry. We checked how fair this statement is.

In the famous novel by Boris Akunin "Turkish Gambit" One of the heroes says to the other: “Who are you to judge who carries the civilization good and who is death!? He studied the state mechanism, got acquainted with the leaders! And with Count Tolstoy, you met Fedor Mikhailovich Dostoevsky? Have you read Russian literature? What, there was not enough time? Twice two is always four, but three times three nine, right? Two parallel lines never intersect? It’s not at your euclide that they do not intersect, but our Lobachevsky crossed! ”

A similar opinion was expressed at one time by such famous people as writers Alexander Prokhanov, Evgeny Vodolazkin, Max Fry, journalists Alexey Venediktov And Julia Latynina, political scientist Leonid Radzikhovsky (although the same Venediktov later Call This is a specific statement of stupidity), head of the Moscow Health Department Leonid Pechatnikovas well as less famous figures on the pages TASS, "Izvestia", on "Radio Liberty" And in many other media.

To begin with, I would like to clarify what the statement about the parallel lines from the “beginnings” of Euclid is the same with which Lobachevsky argued in his theory. How Writes Doctor of Physical and Mathematical Sciences Vladimir Uspensky in his book “Apology of Mathematics”, “Almost everyone heard about the axiom about parallel lines, because it is held at school.” And the vast majority of those surveyed by him for its content of random people were given the same clear answer: the axiom is that parallel lines do not intersect.

However, let's look at the source. The first systematic presentation of planimetry (the section of the geometry studying the figures on the plane) was given by the ancient Greek mathematician Euclid in his work "Beginning". Euclid put five axioms (or postulates) - allegations that do not require evidence.

1. From any point to any point you can draw a straight line.

2. Limited straight line can be continuously continued in a straight line.

3. From every center, every radius can be described by a circle.

4. All straight angles are equal to each other.

5. If the straight line, crossing the two straight lines, forms internal unilateral corners, smaller two lines, then, continued unlimitedly, these two straight lines will meet from the side where the angles are less than two lines.

Nothing like a popular wording, right? We will draw our attention to the fifth postulate, which in modern sources is often formulated So (such a presentation is attributed to prison, and also sometimes called an axioma playifer): “In the plane through a point that does not lie on this straight line, one can draw one and only one line parallel to this.”

And what are parallel direct? By definition (and not at all according to any axiom) These are direct, which lie in the same plane and do not intersect. Thus, the fact that a huge number of people take as an axiom about parallel lines is just their definition. And the notorious fifth postulate of Euclid looks completely different.

So what did Nikolai Lobachevsky do? Many of his predecessors have tried to prove or refute the fifth postulate of Euclid for centuries, but for this they needed a support point, a base, which only this postulate itself could act. At the same time, at first glance it seems that his truth is obvious. However, the 28-year-old mathematician from Kazan University was not so sure of that. Lobachevsky tried to replace the fifth postulate with his opposite: "Through a point that does not lie on a given line, at least two lines lying from a given line in one plane and do not cross it pass."

Lobachevsky did not touch the remaining four axioms. Mathematics InterestedWhat will happen after this with the entire system of geometric theorems and other statements, will the contradictions that indirectly prove that the assumption of Euclid, albeit unproven, was the only and inevitably true will come out. But it turned out that the world did not collapse. All basic statements of classical geometry have survived perfectly on this foundation.

But it is quite difficult to visually imagine a similar situation. Someone will ask: “Why represent it? It is important that everything is correct in theory ” - and formally will be right. However, apparently, the members of the scientific commission of Kazan University, who listened to the young mathematician report on February 7, 1826, reasoned. Presentation of work under the name "Compressed presentation began geometry with strict proof of the theorem about the parallel" It failed, and the manuscript did not even fall into print.

Three years will pass, and Nikolai Lobachevsky, now the rector of the university, will find the opportunity to publish his work “On the Principle of Geometry” in the journal Kazan Bulletin. Unfortunately, the science of that time was not yet ready to accept this approach - in particular, the work was awarded a negative review Mikhail Ostrogradsky, one of the largest Russian mathematicians. And only a few years later, they pay attention to Lobachevsky in Europe - in particular, the “King of Mathematicians” Karl Friedrich Gauss Put it Its in the corresponding members of the Gettingen Royal Scientific Society, and at the same time it will study the ideas of a colleague in the original, In Russian. Gauss himself adhered to such ideas, according to him, for many years.

And only one and a half decades later, mathematical models will appear in which Lobachevsky’s theory will work without causing endless disputes. In particular, Projective model, where the inside of the circle is taken beyond the plane, and for the line - its chord. As a result, the obvious fact is that through one point lying inside the circle, you can draw as many chords that do not intersect with one fixed chord, in such rules of the game it becomes an illustration of the fifth beginning of Lobachevsky’s geometry. Another example the implementation of the theory of Lobachevsky - the pseudo -sphere, the surface of the rotation of the curve:

And in 1868 a report will be released Bernhard Riman, a person who will offer his approach in non -Euclidean geometry is somewhat different than that of Lobachevsky. However, the success of his theory will become an extra confirmation of the greatness of Kazan, since two mathematicians took similar steps, only in different spaces. Speaking with mathematical terms, then Euclidean Gausov Krivno zero (1), in Lobachevsky - negative (2), in Riman - positive (3):

But this is a completely different story, and here it is necessary to conclude that in the geometry of Lobachevsky, parallel directs also do not intersect - the Russian mathematician did not say anything similar. In the geometry of Riemann, contrary to the statements of some Media, Just There are no parallel lines.