At the beginning of the Elements of Geometry, Euclid gave 23 definitions, 5 postulates and 5 axioms. In fact, the commune he said is what we later called axiom, and his axiom is some methods used in calculation and proof (such as axiom 1: the same quantity is equal, axiom 5: the whole is greater than the parts, and so on). The five postulates he gave are very close to geometry and are axioms in our later textbooks. They are:

Postulate 1: You can draw a straight line from any point to any other point.

Postulate 2: A finite line segment can extend continuously.

Postulate 3: A circle can be drawn at any distance with any point as the center.

Postulate 4: All right angles are equal to each other.

Postulate 5: One straight line and the other two straight lines intersect in the same plane. If the sum of the two internal angles of a certain side is less than the sum of the two right angles, then the two straight lines extend indefinitely and intersect at that side.

In these five postulates, Euclid did not naively assume the existence and compatibility of definitions. Aristotle pointed out that the first three postulates say that straight lines and circles can be constructed, so he is a statement of the existence of two things. In fact, Euclid proved many propositions by this construction method. The fifth postulate is verbose, not as concise as the first four. What is declared is not what exists, but Euclid's idea. This is enough to show his genius. From Euclid's putting forward this axiom to 2 100 in 1800, people have been worried about this fifth postulate, although they have never doubted the correctness of the whole system. Many mathematicians want to remove this postulate from this system, but after several efforts, it is fruitless to generalize the fifth postulate from other postulates.

At the same time, mathematicians have also noticed that this postulate is not only a discussion of the concept of parallelism (so it is called the axiom of parallelism), but also a discussion of the sum of interior angles of triangles (that is, the axiom of interior angles). Gauss knows this very well. He thinks that Euclidean geometry, the geometry of material space, which he said in a letter to a friend in 1799, shows that he thinks that parallel kilometers cannot be derived from other postulates, and he begins to seriously develop a new geometry that can be applied. 18 13 developed other geometries, first called anti-Euclidean geometry, then called starry geometry, and finally called non-Euclidean geometry. In his geometry, the inner angle of a triangle can be greater than 180 degrees. Of course, Gauss is not the only one who got this geometry. There are three people in history. One was his partner, and the other was discovered independently by the son of Gauss's friend. One of the interesting problems is that parallel lines passing through a point outside a straight line in non-Euclidean geometry can be infinite.