|
Arabic
Bengali
Chinese
English
French
German
Hebrew
Hindi
Italian
Japanese
Korean
Malay
Polish
Portuguese
Spanish
Turkish
Ukrainian
Vietnamese|
MOST IMPORTANT SCIENTIFIC DISCOVERIES
Non-Euclidean geometry. History and essence of scientific discovery
Directory / The most important scientific discoveries On definition of Euclid parallel lines are straight lines that lie in the same plane and never meet, no matter how far we extend them. But already the most ancient commentators of Euclid, Posidonius (II century BC), Geminus (I century BC), Ptolemy (II century AD) - did not consider the fifth postulate of Euclid to have the same evidence as other postulates and axioms of Euclid, and tried either deduce it as a consequence of other provisions, or replace the definition of parallel given by Euclid with another definition. In the second half of the XNUMXth century Leibniz also critical of the main provisions of Euclid. As is well known, he also wanted to construct a purely geometric analysis that would directly express the properties of position, just as algebra expresses magnitude. But only in the first half of the XNUMXth century did the idea come to apply to the question of parallel lines and systematically carry out in the theory of parallel lines that method of proof by contradiction, which was so often used by Greek mathematicians. This brilliant idea belonged to Saccheri. In the work, which appeared in the year of his death, "Euclid, Delivered from Every Spot," Saccheri takes as a starting point a quadrilateral whose two opposite sides, perpendicular to the base, are equal to each other. In such a quadrilateral, the angles formed by equal sides with the side opposite to the base are equal, and the proof of this property of the quadrilateral does not depend on Euclid's postulate. If they are straight lines, then Euclid's postulate is proven, since in this case the sum of the angles of a triangle is equal to two right angles. But Saccheri (and this is his original brilliant idea) also makes two other hypotheses - the hypothesis of an acute angle and the hypothesis of an obtuse angle, deduces the ensuing consequences from these hypotheses and tries to prove the impossibility of these consequences, i.e., the admissibility of only one hypothesis of a right angle. He easily manages to prove that the obtuse angle hypothesis is invalid, since it leads to contradictions. In order to find the same contradiction in the acute angle hypothesis, he deduces a number of remarkable theorems, which were later proved again by Legendre. Such, for example, are the theorems according to which if one or another or a third hypothesis holds for one quadrilateral, then it also holds for any other. Three years after its appearance, in 1766, Lambert poses the same problem as Saccheri. Instead of a quadrilateral with two right angles and two equal sides, Lambert considers a quadrilateral with three right angles and makes three hypotheses about the fourth angle. His exposition has some peculiarities in comparison with Saccheri's: he avoids resorting to arguments based on continuity. From the fact that in the hypotheses of an obtuse and acute angle there is no similarity of figures, Lambert deduces the conclusion about the existence of an absolute measure. In 1799, the brilliant mathematician Carl Gauss went along the path that Saccheri and Lambert had gone before him - along the path of a systematic derivation of all the consequences of the acute angle hypothesis. But his reflections led to doubts about the possibility of proving Euclid's axiom, and by 1816 the mathematician was convinced that such a proof was impossible. Gauss's public opinion about the unprovability of Euclid's axiom had no influence and was even subjected to rude attacks. This was one of the reasons why he decided not to publish his research and thoughts on the question of foundations, "for fear of the cry of the Boeotians" (letter to Bessel dated January 27, 1829). But he did not interrupt his research and with the greatest interest and sympathy welcomed those works and thoughts that coincided with his research and views. How far he went along this path is shown by his letter to Wolfgang Bolyai dated March 6, 1832, in which Gauss says that between 1797 and 1802 he found the results that Johann Bolyai arrived at. For example, a purely geometric proof of the theorem that in non-Euclidean geometry the difference of the sum of the angles of a triangle from 180 degrees is proportional to the area of the triangle. Wolfgang Bolyai, a school friend of Gauss, showed great interest in the theory of parallel lines. This extraordinary interest, according to his letter to his son in 1820, poisoned him with all the joys of life, made him a martyr to the desire to free geometry from stain, "remove the cloud that obscures the beauty of the virgin-truth." But while the efforts of almost the entire life of his father were directed to the proof of the 5th postulate, and he failed to achieve the goal, his talented son was one of the creators of non-Euclidean geometry. Johann Bolyai was born in 1802 in Klausenburg. Already in 1807, his father wrote with delight and pride to Gauss about the extraordinary mathematical abilities of the boy, who by the age of thirteen had already studied planimetry, stereometry, trigonometry, conic sections, and at the age of 14 he was already solving problems of differential and integral calculus with ease. Wolfgang failed to send his son to study in Göttingen with the "mathematical colossus", and in 1818 Johann entered the Vienna Engineering Academy, where much attention was paid to higher mathematics. In 1823, he completed his course at the academy and, as a military engineer, was sent to the Temetvar fortress. It is quite natural that Johann, who possessed extraordinary mathematical abilities, almost as a boy, decided to try his hand at solving the problem over which his father was tormented, but about which his father told him that whoever solved it was worthy of a diamond the size of the globe. In 1820, Johann informs his father that he has already found a way to prove the axiom, and then his father writes him a heated letter warning him against engaging in the theory of parallel lines. On a winter night in 1823, he found that basic relationship between the length of a perpendicular dropped from a point to a straight line and the angle that the asymptote (parallel line) makes with this perpendicular Lobachevsky), which is the key to non-Euclidean trigonometry. Enthusiastic about his discovery, which seemed to him to open the way to the proof of Axiom XI, he writes on November 3 from Temetvar to his father: “I created a new, different world out of nothing. Everything that I have sent so far is only a house of cards in comparison with the tower now being erected." In 1829, Wolfgang completed a large mathematical essay, on which he worked for about twenty years. As an appendix to this book, the immortal work of Johann Boliai was also published. Of course, Boliai did not suspect that at the same time in distant Kazan Lobachevsky was publishing his first work "On the Principles of Geometry" (1829). Nikolai Ivanovich Lobachevsky (1792–1856) was born in the Makaryevsky district of the Nizhny Novgorod province. His father occupied the place of a district architect and belonged to the number of petty officials who received a meager content. The poverty that surrounded him in the first days of his life turned into poverty when in 1797 his father died and his twenty-five-year-old mother was left alone with the children without any means. In 1802, she brought three sons to Kazan and assigned them to the Kazan Gymnasium, where the phenomenal abilities of her middle son were quickly noticed. When in 1804 the senior class of the Kazan gymnasium was transformed into a university, Lobachevsky was included in the number of students in the natural science department. The young man studied brilliantly. Lobachevsky received an excellent education. Lectures on astronomy were read by Professor Litroff. He listened to lectures on mathematics by Professor Bartels, a pupil of such a prominent scientist as Carl Friedrich Gauss. Already in 1811, Lobachevsky received a master's degree, and he was left at the university to prepare for a professorship. In 1814, Lobachevsky received the title of associate of pure mathematics, and in 1816 he was made a professor. From 1819 Lobachevsky taught astronomy. The administrative activity of the scientist began in 1820, when he was elected dean. Despite the exhausting practical activity that did not leave a single moment of rest, Lobachevsky never stopped his scientific studies and during his rectorship published his best works in the Scientific Notes of Kazan University. If Johann Bolyai began to study the theory of parallel lines under the influence of his father, then Lobachevsky could begin to study it only because interest in this theory was especially revived at the end of the XNUMXth and beginning of the XNUMXth century. In the twenty-fifth anniversary preceding the appearance of Lobachevsky's first work, not a year passed without the appearance of one or more works on the theory of parallel lines. Up to 30 works are known, printed only in German and French from 1813 to 1827. Legendre's work aroused interest in the theory of parallel lines among Russian mathematicians as well. The first Russian academician who earned an honorable place in the history of Russian mathematical teaching with his published works, CE. Gur'ev, in his most important work, An Essay on the Improvement of the Elements of Geometry, published in 1798, paid special attention to the theory of parallel lines and to the proofs given by Legendre. Criticizing these proofs, Guriev offers his own. Based on the assertion that, under certain conditions, lines that seem parallel to us can intersect, Lobachevsky came to the conclusion that it is possible to create a new, consistent geometry. Since its existence was impossible to imagine in the real world, the scientist called it "imaginary geometry." But he, like I. Boliai, did not come to this idea right away. The lectures of 1815–1817, the geometry textbook of 1823, and the "Exposition succincte des principes de la geometrie", which has not come down to us, read at a meeting of the Physics and Mathematics Department on February 12, 1826 - these are the three stages of Lobachevsky's thought in the field of the theory of parallel lines. In lectures, he gives three different ways to justify it; in a textbook of 1823, he declares that all the proofs so far given do not deserve to be honored in the full sense of mathematics, and, finally, three years later he already gives that system for constructing geometry on a position different from Euclid's postulate, which immortalized his name. "Exposition" has not reached us. The first printed work of Lobachevsky, which he calls an extract from the Exposition, was published in the Kazan Vestnik in 1829-1830. This date establishes the priority of the publication of Lobachevsky's discovery in comparison with I. Boliai, since the latter's "Appendix" was published in 1831, and went out of print only in 1832. As the title "Exposition" shows, it had as its subject not only the exact theory of parallel lines, but was also devoted to the question of the principles of geometry. Although both I. Boliai and Lobachevsky were elected members of the Hannover Academy of Sciences for this discovery, it was Lobachevsky's geometry that received citizenship rights in Western Europe. In 1837 Lobachevsky's works were published in French. In 1840 he published in German his theory of parallels, which deserved the recognition of the great Gauss. In Russia, Lobachevsky did not see the evaluation of his scientific works. Obviously, Lobachevsky's research was beyond the understanding of his contemporaries. Some ignored him, others greeted his work with rude ridicule and even scolding. While our other highly talented mathematician Ostrogradsky enjoyed well-deserved fame, no one knew Lobachevsky; Ostrogradsky himself treated him either mockingly or hostilely. Quite correctly, or rather, thoroughly, one geometer called Lobachevsky's geometry stellar geometry. One can form an idea of infinite distances if one remembers that there are stars from which light reaches the Earth for thousands of years. So, the geometry of Lobachevsky includes the geometry of Euclid not as a particular, but as a special case. In this sense, the first can be called a generalization of the geometry known to us. Now the question arises, does Lobachevsky own the invention of the fourth dimension? Not at all. The geometry of four and many dimensions was created by the German mathematician, a student of Gauss, Riemann. The study of the properties of spaces in a general form now constitutes non-Euclidean geometry, or the geometry of Lobachevsky. The Lobachevsky space is a space of three dimensions, which differs from ours in that the postulate of Euclid does not take place in it. The properties of this space are now being understood by assuming a fourth dimension. But this step already belongs to the followers of Lobachevsky. Naturally, the question arises, where is such a space. The answer to it was given by the largest physicist of the XX century Albert Einstein. Based on the works of Lobachevsky and Riemann's postulates, he created the theory of relativity, which confirmed the curvature of our space. According to this theory, any material mass curves the surrounding space. Einstein's theory was repeatedly confirmed by astronomical observations, as a result of which it became clear that Lobachevsky's geometry is one of the fundamental ideas about the Universe around us. Author: Samin D.K.
▪ Saussure's linguistic concept
Artificial leather for touch emulation
15.04.2024 Petgugu Global cat litter
15.04.2024 The attractiveness of caring men
14.04.2024
▪ Construction of the road from solar panels is being completed ▪ Augmented Reality Contact Lenses
▪ section of the site Metal detectors. Article selection ▪ article Turgenev Ivan Sergeevich. Famous aphorisms ▪ article Stony raspberry. Legends, cultivation, methods of application ▪ article The disappearance of the drink. Focus Secret
Home page | Library | Articles | Website map | Site Reviews
www.diagram.com.ua |
We recommend interesting articles Section 
See other articles Section 
Latest news of science and technology, new electronics:
All languages of this page