# Non-Euclidean Geometries

June 28, 2021Euclid, his work called Elements, BC. He wrote about 300 years. This book, which is one of the most famous and perhaps the most read books in human history, contained the proof of many geometric theorems based on five axioms.

There is a difference between an axiom and a postulate. Euclides spoke of postulates as acknowledgments that some operations can be done. We will use the term “axiom” here, following a general convention.

1. A line can be drawn from one point to another (or a line passes through two points).

2. A finite and continuous line segment can be drawn on a straight line (or a continuous line between two points is finite).

3. Taking a given point as the center and any length as the radius, a circle can be drawn (or the geometric locus of points equidistant from a point is a circle).

4. All right angles are equal to each other.

5. When two lines are cut by a straight line, if the sum of the two interior angles formed on one side of the sac is less than 180 degrees, these two lines intersect on the side with angles less than 180 degrees.

In addition, Euclid and the geometers following him thought that straight lines were infinite or could be extended infinitely in either direction. They didn’t think it might have anything to do with the fifth axiom. As can be seen, despite the simplicity and clarity of the first four axioms, the fifth axiom seemed more complex. Perhaps Euclid himself had certain concerns about the fifth axiom. In the first 28 proofs he gave, he did not use this fifth axiom.

Proclus (410-485) wrote a commentary on the Elements and included efforts to derive the fifth axiom from the first four. He also stated that Ptolemy (Ptolemy) gave a false proof. He himself tried to give a proof, but this proof was also wrong. On the other hand, he proposed another axiom equivalent to the fifth axiom:

Given a line and a point not on that line, one and only one line can be drawn through that point and parallel to the line.

This postulate of Proclus became known as Playfair’s axiom after a commentary by John Playfair in 1795. In his commentary, Playfair suggested replacing the fifth axiom with his own.

An attempt to prove historically important was undertaken by Girolomo Saccheri in 1697. What made Saccheri important was the difference in his approach. Saccheri assumed that the fifth axiom was false and tried to derive a contradiction from it. He proved many theorems that did not belong to Euclidean geometry, under the assumption that the line intersecting two parallel lines makes an angle of less than 90 degrees. After all, he also claimed to have found a contradiction. Under the assumption that there is a point infinitely distant on a plane; the narrow angle assumption led to a contradiction.

In 1766 Lambert followed a similar method to Saccheri. But it was not his purpose to introduce a contradiction. Lambert showed that under the acute angle assumption, the sum of the interior angles increases as the area of a triangle decreases.

Legendre devoted almost 40 years of his life to research on the parallel axiom. In the appendix to his book Eléments de Géométrie, Euclid’s fifth axiom “The sum of the interior angles of a triangle is equal to the sum of two right angles.” He gave a proof showing that it is equivalent to the axiom. Legendre’s proof, like Saccheri’s proof, was based on the assumption that straight lines are infinite, but Legendre did not realize this error.

In the face of all these fruitless efforts, D’Alembert called the problems of the parallel axiom “the scandal of elementary geometry.” Perhaps Gauss was the first mathematician to understand the problem behind the parallel axiom. Gauss began working on the parallel axiom from 1792. In 1813, “We are still no better than Euclid in the theory of parallels. This is an embarrassing part of the math…” he had to say.

By 1817 Gauss had been convinced that the fifth axiom was independent of the other four. He began working on the consequences of a geometry idea that allowed a line to be drawn more than one parallel from a point other than itself. However, Gauss did not publish these works and kept them secret. In those years, Kant’s synthetic a priori understanding was dominant, and the idea that Euclidean geometry was a necessity of pure intuition was widespread. According to historians, Gauss was not an academic who liked to engage in conflict on such issues.

Gauss would consult with his colleague Farkas Bolyai on the parallel axiom. Farkas Bolyai had also given proofs in the past that later turned out to be false. He knew how difficult the subject was and the harms of obsessing over it. Bolyai advised his son, János Bolyai, whom he had raised as a mathematician, not to spend an hour on the fifth axiom problem. However, János Bolyai did not listen to his father. Father in 1823 to inform his father about the results of his studies.