Shulou(Shulou.com)11/24 Report--

It is found that the honor of non-Euclidean geometry belongs to two people, ∶, a Hungarian, Polyo, and a Russian, Robachevsky. They did very similar studies on the subject independently of each other. In particular, both men describe a difference from Euclidean geometry in both two-dimensional and three-dimensional cases. Robachevsky's results were first published in a little-known Russian publication in 1829, in French in 1837, in German in 1840, and finally in French in 1855. Polyo published his paper as an appendix in a two-volume geometry book written by his father in 1831.

It's easiest to put their achievements together. Both men define parallel lines in a novel way: given a point P and a line m in ∶, some of the lines passing through P intersect m and some do not. The two sets are separated by two straight lines passing through P, which do not intersect m very much, but the two lines are arbitrarily close to m from P to the right and one to the left. They are the straight lines n 'and n "in the following figure.

Lines n 'and n "separate lines that intersect or do not intersect the line m through P and Robachevsky calls these two boundaries the parallel lines of the pair of lines m that pass through point P. this definition is found in a pamphlet he wrote in 1840. in fact, all the lines between the two boundaries pass through point P and do not intersect the line m.

In such a discussion, the perpendicular from point P to line m can still be defined. The two parallel lines to the left and right (that is, n 'and n ") are equilateral to this vertical line, which is called a parallel angle. If this angle is a right angle, then Euclidean geometry is obtained. However, if it is less than the right angle, there may be new geometry. The result is that the size of the angle depends on the length of the vertical line from point P to the line m. Neither Polyo nor Robachevsky took the trouble to prove that a parallel angle less than a right angle would not cause contradiction. On the contrary, they all assume that there will be no contradiction, and then use a lot of force to calculate the size of the parallel angle from the length of the vertical line.

They all proved that if ∶ gives a family of parallel lines (pointing in the same direction) and specifies a point on one of the lines, it must pass through this point to make a curve perpendicular to all these lines.

A curve perpendicular to a family of parallel lines in Euclidean geometry, this curve is a straight line, it is perpendicular to all parallel lines in the family and passes through this point. If it is still in Euclidean geometry, but take a family of straight lines passing through the common point Q and another point P, then the curve perpendicular to all these lines is the circle with Q as the center and passing through P.

Curves perpendicular to Euclidean parallel lines these curves defined by Polyo and Robachevsky have similar points to the above two Euclidean drawings ∶ it is orthogonal to all these parallel lines, but it is curved rather than straight. Polyo called this curve the L curve, while Robachevsky called it the horocycle, a term that is more helpful and has been used to this day.

A curve perpendicular to the Euclid line passing through a point their complex argument led them to 3D geometry. Here, Robachevsky's argument is a little clearer than Polyo, and both men are significantly better than Gauss. If the figure that defines the limit circle is rotated around one of the parallel lines, these lines will become a family of parallel lines in three-dimensional space, and the limit circle will sweep out a cup-shaped surface, which Poll approximately calls the F surface, while Robachevsky calls it the limit ball (horosphere). They all made it clear that something noteworthy would happen. A circle or an limit circle will be cut through the plane of the limit ball, and if a triangle is made on the limit sphere with the limit circle as the side, the sum of the three interior angles is equal to two right angles. To put it another way, although the space containing the limit ball is a three-dimensional version of occasion L, so it must be non-Euclidean, but if you limit the limit sphere, you will get two-dimensional Euclidean geometry.

Polyo and Robachevsky also know that balls can be made in their three-dimensional space, and prove that the formula of spherical geometry still holds, and has nothing to do with the axiom of parallel lines (although they are not so original in this respect). Robachevsky chose to use a very clever approach related to his parallel line to prove that a triangle on a sphere must determine a triangle on a plane and is also determined by this plane triangle; conversely, a triangle on a plane also determines a triangle on a sphere and is also determined by this spherical triangle. This means that the formula of spherical geometry must determine the formula of triangles that can be used to limit the sphere. When examining the details, Robachevsky proved that triangles on the limit sphere can be described by the formula of hyperbolic trigonometry, and Polyo did it more or less.

The formulas of the spherical triangles of Gauss, Polyo and Robachevsky depend on the radius of the ball. Similarly, the formula of a hyperbolic triangle must depend on a real parameter. However, there is no clear geometric explanation for this parameter. Despite this drawback, these formulas have some properties that can help us reconfirm something. For example, when the sides of triangles are very small, they are very close to the well-known formulas of plane geometry, which helps to explain why these formulas have not been discovered for such a long time-they are very different from Euclidean geometry in small areas of space.

Under this new background, the formulas of length and area can be given, which show that the area of a triangle is proportional to the interior angle of the triangle and the difference between the two right angles. In particular, Robachevsky felt that the good reason for accepting this new geometry was that ∶ had such credible formulas. In his view, all geometry is about measurement, and each geometric theorem is to express the reliable relationship between these measurements in formulas. Since his method has given this formula, in his view, it is enough as a sufficient reason for the existence of this new geometry.

Since Polyo and Robachevsky proposed a novel 3D geometry, they also raised a question: ∶, which kind of geometry is true? Is it Euclidean geometry, or is it new geometry that contains a value of a parameter that can be determined experimentally? At this point, Polyo left the problem behind, but Robachevsky made it clear that the problem could be solved by measuring the parallax of the constellation. Here, he did not succeed either, because the experiment was notoriously meticulous.

On the whole, the reaction to the thoughts of Polyo and Robachevsky was contempt and hostility while they were alive, and they themselves did not foresee the ultimate success of their discovery. Polyo and his father sent their work to Gauss. But Gauss replied in 1832 that he could not praise the work. Because "praising it is praising myself" (Gauss means that he himself got such a result a long time ago), this is not enough. Gauss also added his simpler proof of the result of little Polyo at the beginning of the article. However, he said he was very happy because the son of his old friend surpassed him. Little Poljo was furious and refused to publish his work, thus depriving himself of the opportunity to guarantee his priority by publishing it in mathematical journals. Oddly enough, there is no evidence that Gauss knew the details of the young Hungarian's work in advance. It is likely that Gauss saw the beginning of Polyo's work and knew what it would do next.

A more tolerant interpretation of the existing evidence is that ∶ believed in the 1830s that physical space could be described by non-Euclidean geometry, and he certainly knew how to use hyperbolic triangles to master two-dimensional non-Euclidean geometry (although he did not leave any detailed discussion). But the three-dimensional situation was first known by Polyo and Robachevsky, while Gauss did not know until after reading their research.

Ostrograzki Robachevsky fared slightly better than Polyo. His earliest article was saved by Ostrograzki, a higher-ranking figure and a mathematician in St. Petersburg, while Robachevsky was in Kazan in another province. His article published in the German Journal of Pure and Applied Mathematics sadly quoted articles published in Russian, which were adapted from these Russian articles. He got only one book review for a small book in 1840, which was more stupid than average. Robachevsky gave this little book to Gauss, who thought it was excellent and arranged for Robachevsky to be elected to the Gottingen Academy of Sciences. However, Gauss's enthusiasm came to an end, and Robachevsky never received Gauss's support again.

Such a terrible response to a major discovery has naturally led to analysis at all levels. It should be said that the definition of the parallel line that these two people rely on is inadequate in terms of their actual situation, but the criticism of their work is not here, but dismisses them contemptuously, as if they are self-evident that they are wrong, so wrong that it is not worth the effort to find out the mistakes that must be made, and the right response is to pile ridicule on the author's head or ignore it. No comment. It can be used to measure the degree of control of Euclidean geometry over the minds of people at that time, and even the discoveries of Copernicus and Galileo received better treatment from the experts of the time.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

Welcome to subscribe "Shulou Technology Information " to get latest news, interesting things and hot topics in the IT industry, and controls the hottest and latest Internet news, technology news and IT industry trends.