Roche geometry (a non-Euclidean geometry) corresponds to Euclidean geometry. In Euclidean geometry, there are five basic kilometers, such as the shortest straight line between two points, and so on. Because this geometric system conforms to the plane situation in our daily life, it is also called plane geometry But the fifth axiom, the sum of angles inside a triangle = 180 degrees, or a point outside a straight line can only be a straight line parallel to a known straight line, seems to be proved by the other four axioms. There are indeed many mathematicians in history who have tried to prove this conclusion, but without success. Later, there was a man named Bao Ye who seemed to be Hungarian. He assumed that the sum of the internal angles of the triangle was less than 180 degrees, thus establishing a system. However, mathematicians at that time were rather conservative, and thought that your statement was too different from that of the master, so we didn't accept it. Master Bao is depressed. His works seem to have died young and not been recognized.

At that time, the speed of media and information dissemination was different from today, and few people knew about Bao Ye's work. Later, it should be18th century. Lobachevsky independently established this kind of non-Euclidean geometry on the basis of four axioms of plane geometry+Roche's fifth axiom. If I remember correctly, this non-Euclidean geometry is also called Bao Ye-Robard Chevsky geometry.

The system was applied in navigation and geodesy later.

Moreover, if the sum of the internal angles of the triangle is greater than 180 degrees, a system can also be established. But I don't know what kind of application it is.

What originality does this need? Does justice need to be proved? I just told a fact. Mathematicians say that the fifth axiom can't be proved by the first four axioms. It is independent.