We have all heard about parallel lines. First, we are taught that they never intersect, and then somewhere in the electives in the high school area they quietly add that there are exceptions to this rule. For example, in the geometry invented by our compatriot Nikolai Lobachevsky. Is this really so, how is this possible at all, and what does Einstein have to do with it - figured it out together with the editors of the popular science portal "Attic".
What's wrong with the fifth postulate
More than 2300 years ago, the ancient Greek mathematician Euclid collected all the knowledge about geometry that he had before him into one big book - "Beginnings". It was in it that the famous five postulates were contained - unprovable statements, on the foundation of which all further reasoning and theorems were built.
The first four postulates were concise and slender. In their truth, probably, no one doubted in the entire history of the world, but the fifth postulate sounded much more confusing and hardly resembled an indisputable truth.
If a line intersecting two lines forms interior one-sided angles less than two lines, then, extended indefinitely, these two lines will meet on the side where the angles are less than two lines
fifth postulate of Euclid's geometry
Dozens of mathematicians tried to prove this statement in different formulations (the most common of them says that in a plane through a point that does not lie on a given line, one and only one line can be drawn parallel to a given line) by dozens of mathematicians, but they were all drawn into the same story. . Their proofs rested on statements that it was absolutely impossible to prove without the fifth postulate itself.
Lobachevsky was embarrassed by the fifth postulate not so much because of its inaccuracy, but rather because of its philosophical load: it settled matter in some kind of frozen absolute space. A hard-core materialist, he couldn't take it solely on faith that parallel lines didn't intersect somewhere in the infinity of space. The scientist turned to proof by contradiction. He tried to replace the fifth postulate with its mirror image ("Through a point not lying on a given line, there pass at least two lines that lie with the given line in the same plane and do not intersect it"). Lobachevsky was waiting for internal contradictions to appear in the entire system of geometric theorems, indirectly indicating that the original version of the fifth postulate was nevertheless inevitably true in our space? But this did not happen - there were no contradictions.
On February 7, 1826 (according to the old style), Lobachevsky presented his work "A concise presentation of the principles of geometry with a rigorous proof of the parallel theorem" before the scientific commission of Kazan University.
New geometry - old problems
Shortly before the speech, the new emperor Nicholas I dismissed Mikhail Magnitsky from the post of trustee of Kazan University, and all the members of the commission thought about how this would affect their lives, and almost did not pay attention to the odd mathematician, who spoke in French about some kind of alien geometry. Further, the manuscript was given for review to some members of the commission, but they apparently simply forgot about it, and the report itself was never approved for publication. Then all the geometry of Lobachevsky could forever remain inside his head, if not for one surprise: he was soon elected the new rector of the university. It is unlikely that Lobachevsky had less work and more strength after this, but gradually he formalized his ideas into a completed work "On the Principles of Geometry", which was first published in the Kazan Vestnik magazine, and then submitted for review to the Academy of Sciences, where the review went to one of the most powerful Russian mathematicians of that time - Mikhail Ostrogradsky.
Mikhail Ostrogradsky
Academician of the St. Petersburg Academy of Sciences
The new geometry remains incomprehensible. The wandering continues.
Later, Lobachevsky published his works in European journals, where they were noticed by the great German Gauss, who himself had been secretly studying non-Euclidean geometry for more than one year. In order to better understand the Kazan scientist, he promptly learned Russian and then, impressed by the courage and clarity of Lobachevsky's thoughts, he nominated him as a corresponding member of the Gottingen Royal Scientific Society. Recognition meets its genius, although in the homeland Ostrogradsky and people around him over and over again reject all work on non-Euclidean geometry until Lobachevsky's death in 1856.
Delayed recognition
12-15 years pass, and mathematicians find several real models at once, in which it is Lobachevsky's geometry that works. In the simplest of them, projective, the interior of the circle is taken as a plane, and its chord is taken as a straight line. As a result, the fact that any number of chords can be drawn through one point lying inside the circle, which do not intersect with one fixed chord, automatically becomes an illustration of the fifth law of Lobachevsky's geometry.
In 1868, the report of Riemann, another pioneer with a different non-Euclidean geometry, in which it is no longer possible to draw a single parallel line through every point in space, is published, and mathematicians gradually become clear that the geometries of Riemann and Lobachevsky are incredibly similar steps to the left and right from the usual Euclidean geometry. The first works on surfaces with positive curvature, like balls, and the second works on surfaces with negative curvature, like hyperboloids or saddles.
A little later, at the beginning of the 20th century, the new geometry will finally meet with physics. Einstein would formulate his general theory of relativity in terms of Riemannian geometry, and the thoughts of people accustomed to walking on the same parallel rails would open up new routes: space and time are not absolute. Movement changes geometry. And millennial axioms are not always true.
Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify a specific person or contact him.
You may be asked to provide your personal information at any time when you contact us.
The following are some examples of the types of personal information we may collect and how we may use such information.
What personal information we collect:
- When you submit an application on the site, we may collect various information, including your name, phone number, email address, etc.
How we use your personal information:
- The personal information we collect allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send you important notices and messages.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you enter a prize draw, contest or similar incentive, we may use the information you provide to administer such programs.
Disclosure to third parties
We do not disclose information received from you to third parties.
Exceptions:
- In the event that it is necessary - in accordance with the law, judicial order, in legal proceedings, and / or based on public requests or requests from state bodies in the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public interest reasons.
- In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the relevant third party successor.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as from unauthorized access, disclosure, alteration and destruction.
Maintaining your privacy at the company level
To ensure that your personal information is secure, we communicate privacy and security practices to our employees and strictly enforce privacy practices.
Signs of parallelism of two lines
Theorem 1. If at the intersection of two lines of a secant:
diagonally lying angles are equal, or
corresponding angles are equal, or
the sum of one-sided angles is 180°, then
lines are parallel(Fig. 1).
Proof. We restrict ourselves to the proof of case 1.
Suppose that at the intersection of lines a and b by a secant AB across the lying angles are equal. For example, ∠ 4 = ∠ 6. Let us prove that a || b.
Assume that lines a and b are not parallel. Then they intersect at some point M and, consequently, one of the angles 4 or 6 will be the external angle of the triangle ABM. Let, for definiteness, ∠ 4 be the outer corner of the triangle ABM, and ∠ 6 be the inner one. It follows from the theorem on the external angle of a triangle that ∠ 4 is greater than ∠ 6, and this contradicts the condition, which means that the lines a and 6 cannot intersect, therefore they are parallel.
Corollary 1. Two distinct lines in a plane perpendicular to the same line are parallel(Fig. 2).
Comment. The way we just proved case 1 of Theorem 1 is called the method of proof by contradiction or reduction to absurdity. This method got its first name because at the beginning of the reasoning, an assumption is made that is opposite (opposite) to what is required to be proved. It is called reduction to absurdity due to the fact that, arguing on the basis of the assumption made, we come to an absurd conclusion (absurdity). Receiving such a conclusion forces us to reject the assumption made at the beginning and accept the one that was required to be proved.
Task 1. Construct a line passing through a given point M and parallel to a given line a, not passing through the point M.
Solution. We draw a line p through the point M perpendicular to the line a (Fig. 3).
Then we draw a line b through the point M perpendicular to the line p. The line b is parallel to the line a according to the corollary of Theorem 1.
An important conclusion follows from the considered problem:
Through a point not on a given line, one can always draw a line parallel to the given line..
The main property of parallel lines is as follows.
Axiom of parallel lines. Through a given point not on a given line, there is only one line parallel to the given line.
Consider some properties of parallel lines that follow from this axiom.
1) If a line intersects one of the two parallel lines, then it intersects the other (Fig. 4).
2) If two different lines are parallel to the third line, then they are parallel (Fig. 5).
The following theorem is also true.
Theorem 2. If two parallel lines are crossed by a secant, then:
the lying angles are equal;
corresponding angles are equal;
the sum of one-sided angles is 180°.
Consequence 2. If a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.(see Fig.2).
Comment. Theorem 2 is called the inverse of Theorem 1. The conclusion of Theorem 1 is the condition of Theorem 2. And the condition of Theorem 1 is the conclusion of Theorem 2. Not every theorem has an inverse, i.e. if a given theorem is true, then the inverse theorem may be false.
Let us explain this with the example of the theorem on vertical angles. This theorem can be formulated as follows: if two angles are vertical, then they are equal. The inverse theorem would be this: if two angles are equal, then they are vertical. And this, of course, is not true. Two equal angles do not have to be vertical at all.
Example 1 Two parallel lines are crossed by a third. It is known that the difference between two internal one-sided angles is 30°. Find those angles.
Solution. Let figure 6 meet the condition.
On February 7, 1832, Nikolai Lobachevsky presented his first work on non-Euclidean geometry to the judgment of his colleagues. That day was the beginning of a revolution in mathematics, and Lobachevsky's work was the first step towards Einstein's theory of relativity. Today "RG" has collected five of the most common misconceptions about Lobachevsky's theory, which exist among people far from mathematical science
Myth one. Lobachevsky's geometry has nothing in common with Euclidean.
In fact, Lobachevsky's geometry is not too different from the Euclidean geometry we are used to. The fact is that of the five postulates of Euclid, Lobachevsky left the first four without change. That is, he agrees with Euclid that a straight line can be drawn between any two points, that it can always be extended to infinity, that a circle with any radius can be drawn from any center, and that all right angles are equal to each other. Lobachevsky did not agree only with the fifth postulate, the most doubtful from his point of view, of Euclid. His formulation sounds extremely tricky, but if we translate it into a language understandable to a common person, it turns out that, according to Euclid, two non-parallel lines will definitely intersect. Lobachevsky managed to prove the falsity of this message.
Myth two. In Lobachevsky's theory, parallel lines intersect
This is not true. In fact, the fifth postulate of Lobachevsky sounds like this: "On the plane, through a point that does not lie on a given line, there passes more than one line that does not intersect the given one." In other words, for one straight line, it is possible to draw at least two straight lines through one point that will not intersect it. That is, in this postulate of Lobachevsky there is no talk of parallel lines at all! We only talk about the existence of several non-intersecting lines on the same plane. Thus, the assumption about the intersection of parallel lines was born because of the banal ignorance of the essence of the theory of the great Russian mathematician.
Myth three. Lobachevsky geometry is the only non-Euclidean geometry
Non-Euclidean geometries are a whole layer of theories in mathematics, where the basis is the fifth postulate different from Euclidean. Lobachevsky, unlike Euclid, for example, describes a hyperbolic space. There is another theory describing spherical space - this is Riemann's geometry. This is where the parallel lines intersect. A classic example of this from the school curriculum is the meridians on the globe. If you look at the pattern of the globe, it turns out that all the meridians are parallel. Meanwhile, it is worth putting a pattern on the sphere, as we see that all previously parallel meridians converge at two points - at the poles. Together the theories of Euclid, Lobachevsky and Riemann are called "three great geometries".
Myth four. Lobachevsky geometry is not applicable in real life
On the contrary, modern science comes to understand that Euclidean geometry is only a special case of Lobachevsky's geometry, and that the real world is more accurately described by the formulas of the Russian scientist. The strongest impetus for the further development of Lobachevsky's geometry was Albert Einstein's theory of relativity, which showed that the very space of our Universe is not linear, but is a hyperbolic sphere. Meanwhile, Lobachevsky himself, despite the fact that he worked all his life on the development of his theory, called it "imaginary geometry."
Myth five. Lobachevsky was the first to create non-Euclidean geometry
This is not entirely true. In parallel with him and independently of him, the Hungarian mathematician Janos Bolyai and the famous German scientist Carl Friedrich Gauss came to similar conclusions. However, the works of Janos were not noticed by the general public, and Karl Gauss preferred not to be published at all. Therefore, it is our scientist who is considered a pioneer in this theory. However, there is a somewhat paradoxical point of view that Euclid himself was the first to invent non-Euclidean geometry. The fact is that he self-critically considered his fifth postulate not obvious, so he proved most of his theorems without resorting to it.
