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Nikolai Ivanovich Lobachevsky, the second son of a minor official, was born on November 2, 1793, in the Makariev region of Novgorod Oblast, Russia. In 1807 he entered Kazan University. He then spent 40 years of his life at the school as a student, associate professor, professor, and finally president.

Kazan university leaders hope to rival european universities. They brought in several distinguished professors from Germany, among them the astronomer Littrow, who later became director of the Vienna Observatory. German professors quickly recognized Lobachevsky's talent and gave him plenty of encouragement. In 1811, at the age of 18, Lobachevsky received a master's degree. Two years later, at the age of 21, Lobachevsky served as a trainee "supernumerary professor," that is, an assistant professor.

In 1816, at the age of 23, Lobachevsky was promoted to general professor. In addition to his mathematical work, he taught astronomy and physics. Lobachevsky soon became director of the university library and director of the university museum. Lobachevsky was appointed rector in 1827.

When the government decided to modernize the university buildings and add new ones, Lobachevsky took it upon himself to do his job well without wasting a penny. To qualify himself for this task, he studied architecture. His mastery of the subject and his great attention to practicality made these buildings not only beautiful and functional, but also less expensive than the money allocated, which is unique in the history of official architecture. A few years later, in 1842, a disastrous fire destroyed half of Kazan and also Lobachevsky's finest buildings, including his newly built observatory, his proud masterpiece. But thanks to his cool head, the instruments and library survived. Immediately after the fire, he set about rebuilding. Two years later, the disaster has left no trace.

In 1842, the year of the fire, and also under Gauss's mediation, Lobachevsky was elected a Fellow of Foreign Communications of the Royal Society of Göttingen for his invention of non-Euclidean geometry. It is incredible that a man like Lobachevsky, burdened with such a heavy teaching and administrative workload, could find time to create one of the greatest masterpieces of all mathematics and a milestone in the human mind. He did the work off and on for twenty years. His first public communication in non-Euclidean geometry was in 1826. Gauss did not hear of this achievement until around 1840.

Lobachevsky's Non-Euclidean Geometry To understand Lobachevsky's work, we must first look at Euclid's outstanding achievements. His Elements contained, in addition to a systematic account of elementary geometry, all the knowledge of number theory known at his time. Geometry education was dominated by Euclid for more than 2200 years.

Euclid admits that his fifth postulate (parallel postulate) is a pure hypothesis. The fifth postulate can be stated in many equivalent forms, each of which can be inferred from the others using the remaining postulates of Euclidean geometry. Perhaps the simplest of these equivalent statements is this:

Given any line I and a point P not on I, then on the plane determined by l and P, one can draw exactly one line I'passing through P, such that I' and l do not intersect no matter how far they extend (in either direction).

We say that two lines in a plane that never intersect are parallel. Thus Euclid's fifth postulate asserts that there is exactly one line parallel to l through point P. Euclid's penetrating insight into geometric properties convinced him that, in his day, this postulate had not been deduced from other postulates, although there had been many attempts to prove it. Since Euclid himself could not deduce this postulate from his other postulates, and wished to use it in the proofs of many of his theorems, he dutifully placed it with his other postulates.

We have already mentioned the "equivalent" statements of the parallel postulate. One of these, called the "right-angle hypothesis," implies two possibilities, neither of which is identical with Euclid's hypothesis: the introduction of Lobachevsky geometry and the introduction of Riemann geometry.

Consider a shape AXYB that "looks like" a rectangle, containing four lines AX, XY, YB, BA, where BA is the base, AX and YB are drawn equal in length and perpendicular to AB, and on the same side of AB. The important thing to remember about this figure is that the angles XAB and YBA are right angles, and the sides AX and BY are of equal length. Without the postulate of parallelism, it is possible to prove that the angles AXY and BYX are equal, but without this postulate it is impossible to prove that the angles AXY and BYX are right angles, although they look like right angles. If we assume that the angles AXY and BYX are right angles, we can prove that the angles AXY and BYX are right angles, and vice versa. Thus, the assumption that angles AXY and BYX are right angles is equivalent to the postulate of parallelism. This assumption is known today as the "right-angle hypothesis."

We know that the right-angle hypothesis leads to a geometry that is free of contradictions and practical, in fact, to a geometry that is reformed to meet modern standards of logical rigor. But this graph offers two other possibilities: equal angles AXY, BYX are smaller than the right-angle-acute hypothesis; equal angles AXY, BYX are larger than the right-angle-obtuse hypothesis. Since any angle satisfies one of the three requirements of being equal to, less than, or greater than a right angle, and only one of them is satisfied, these three assumptions are the right angle hypothesis, the acute angle hypothesis, and the obtuse angle hypothesis.

Common experience leads us first to the first hypothesis. To see that other assumptions may not be as implausible as they seem at first sight, we need to consider something closer to actual human experience than Euclid's highly idealized "flat" drawing of figures. But we note first that neither the acute angle hypothesis nor the obtuse angle hypothesis enables us to prove Euclid's postulate of parallelism, because, as we have said, Euclid's postulate is equivalent to the right angle hypothesis (which is both necessary and sufficient for inference of parallelism in the sense of mutual inference). Thus, even if we succeed in constructing geometries on one of the two new hypotheses, we will not find parallelism in the Euclidean sense in these geometries.

To make the other assumptions less unreasonable than they seem at first sight, assume that the earth is a perfect sphere. Draw a plane through the center of this ideal earth, which makes a great circle with the surface of the earth. Suppose we want to travel from point A to point B on the surface of the earth always on a sphere, and suppose further that we want to take the shortest possible path.

The above example introduces an important definition, namely the definition of geodesics on surfaces. We have just seen that the shortest distance connecting two points on the earth is itself a distance measured on the sphere, an arc of a great circle connecting them. We also see that the longest distance connecting two points is another arc on the same great circle, unless the two points are opposite ends of a diameter, in which case the shortest distance is equal to the longest distance. We recall now the definition of a line segment joining two points in the plane-"the shortest distance between two points." Applying this definition to the sphere, we say that a great circle on the sphere corresponds to a line in the plane. Since the Greek word for earth is the first syllable ge of a geodesic, we call all limits connecting any two points on an arbitrary surface geodesic of a surface. Thus, on the plane, geodesics are Euclidean lines; on the sphere, geodesics are great circles. A geodesic can be thought of as the position of a line between two points on a surface when it is as taut as possible.

Now, at least in navigation, and even when considering intermediate distances, the ocean cannot be considered as a plane; rather, it is to be regarded as something very similar to it, i.e., a part of a sphere. The geometry of the law of navigation of great circles is not Euclidean geometry, and therefore Euclidean geometry is not the only geometry available to man. On the plane, two geodesics intersect at exactly one point unless they are parallel; but on the sphere, any two geodesics always intersect at exactly two points. Moreover, on the plane, no two geodesics can enclose a space; on the sphere, any two geodesics always enclose a space.

Now imagine the equator on the sphere and two geodesics perpendicular to the equator through the north pole. In the Northern Hemisphere, this produces a curved triangle WNE with two sides that are equal. Each side of this triangle is a geodesic arc. Make arbitrarily another geodesic line intersecting two edges of equal length, such that the two sections cut between the equator and this line are equal. Now we have AXYB on the sphere corresponding to the quadrilateral we just had on the plane. The two corners at the bottom of this figure are right angles and the corresponding two sides are equal. But the two equal angles at X and Y are now larger than right angles. Thus, in the geometry of great circle navigation (which is closer to real human experience than the idealized figure obtained by elementary geometry), what is real is not Euclid's postulate, but the geometry obtained by the obtuse angle hypothesis.

In the same way, observing a less familiar surface, we can make the acute angle hypothesis reasonable. This curved surface looks like two infinitely long horns welded together at one end.

To describe it more precisely, we must introduce planar curves called drag lines, which are generated by drawing two straight lines XOX', YOY', on a plane, intersecting at right angles at zero. Imagine an inextensible fiber along YOY with a heavy ball tied to one end; the other end of the fiber is at point 0. Pull this end outward along the line OX. Since the ball follows, it draws half of the drag line; the other half is drawn by pulling the end of the fiber along OX', which of course is only a reflection or mirror image of the first half on OY. Suppose that in each case the process of pulling apart goes on indefinitely-"to infinity." Now imagine the drag line rotating around the line XOX'. A two-horn surface is created; it has constant negative curvature and is called a pseudosphere. If we use geodesics to draw a quadrilateral with equal sides and two right angles on this surface, we find that the acute angle hypothesis holds.

Thus, the assumptions of right angles, obtuse angles, and acute angles hold for the Euclidean plane, sphere, and pseudosphere, respectively, and in all cases the "line" is geodesic or extreme. Euclidean geometry is the extreme case of spherical geometry, and when the radius of the sphere is infinite, Euclidean geometry is obtained.

Euclid did not construct a geometry suitable for the earth, he based it on the assumption that the earth was flat. It took more than 2000 years until Lobachevsky appeared.

In Einstein's words, Lobachevsky was challenging an axiom. Anyone who challenges a truth that for more than 2000 years has been regarded by most as undeniable risks his scientific reputation, if not his life.

Einstein himself challenged the axiom that two events can occur in different places at the same time, and analysis of this ancient assumption led to the discovery of special relativity.

Lobachevsky challenges the axiom that Euclid's parallelism postulate, or its equivalent right angle hypothesis, is necessary for a consistent geometry. He supported his challenge by creating a geometric system based on the acute angle hypothesis, in which not one but two lines parallel to a given line pass through a fixed point. Neither of Lobachevsky's parallel lines intersects the line parallel to them, nor does any line passing through the fixed point and falling within the angle formed by the two parallel lines intersect it. This apparently strange situation is "realized" by geodesics on the pseudosphere.

For any ordinary purpose (measuring distances, etc.), the difference between Euclidean geometry and Lobachevsky geometry is insignificant, but that is not the point: both geometries are self-consistent, both conform to human experience. Lobachevsky rejects the undeniable "truth" of Euclidean geometry. His geometry was only the first of several constructed by his successors. Some of these alternative geometries-for example, Riemann geometry of general relativity-are today at least as important in the still active and developing parts of physical science as Euclidean geometry was and is in the relatively static and classical parts. For some purposes Euclidean geometry is best, or at least sufficient, but for others it is not, and non-Euclidean geometry is needed.

For 2200 years it has been believed, in a sense, that Euclid discovered an absolute truth or necessary pattern of human perception in his geometric system. Lobachevsky's creation actually proves this view wrong. His bold challenge, and its consequences, inspired mathematicians and scientists to challenge other axioms, such as causality.

Perhaps we have not yet experienced the full impact of Lobachevsky's method of challenging axioms. It is not an exaggeration to call Lobachevsky the Copernicus of geometry.

This article comes from Weixin Official Accounts: Laohu Science (ID: LaohuSci), Author: I am Lao Hu

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