# Non-Euclidean Geometries

Euklides, Elements (Eng. Elements), M.Ö. He wrote in 300. This book, one of the most famous and perhaps the most widely read in the history of humanity, included the proof of many geometry theorems based on five axioms.

**There is a difference between axiom and postulate. Euclides mentioned postulates as acceptances that some operations could be done. Here, we will use the term “axiom erek following a general custom.**

1. A line can be drawn from one point to another (or crosses from one point to another).

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

3. A circle can be drawn by taking a given point as a center and any length as a radius (or the geometric location of points equal to a point is a circle).

4. All right angles are equal to each other.

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

In addition, Euclides and the following geometricians felt that the right lines were infinite, or that they could be infinitely extended in either direction. They did not think this had 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 his first 28 proofs, he did not use this fifth axiom.

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

When a line and a point that is not on this line are given, one and only a line passing through this point and parallel to the line can be drawn.

This postulate of Proclus became known as the axiom of Playfair after a commentary by John Playfair in 1795. In its commentary, Playfair was proposing to replace the fifth axiom with its own axiom.

An attempt at proof of historical significance was undertaken in 1697 by Girolomo Saccheri. What made Saccheri important was the difference in his approach. Saccheri assumed that the fifth axiom was wrong and tried to derive a contradiction from this. Under the assumption that the line that cuts two parallel lines makes an angle of less than 90 degrees, he proved many theorems not belonging to Euclidean geometry. In the end, he also claimed to have found a contradiction. Under the assumption that there is a point at an infinite distance on a plane; the conjecture of the narrow angle led to a contradiction.

In 1766, Lambert followed a method similar to Saccheri. However, his aim was not to reveal a contradiction. Lambert showed that as the area of a triangle decreases, the sum of the inner angles increases, assuming a narrow angle.

Legendre devoted almost 40 years of his life to research on parallel axioms. In the appendix of his book Eléments de Géométrie, Euklides’s proof that the fifth axiom is equivalent to the axiom of inci the sum of the inner angles of a triangle equals the sum of two right angles. Ver Legendre’s proof, like Saccheri’s proof, was based on the assumption that the right lines were infinite, but Legendre did not notice this mistake.

In the face of all these inconclusive efforts, D’Alembert called the problems of the parallel axiom a sk scandal of basic geometry.. Perhaps Gauss was the first mathematician to understand the problem behind the parallel axiom. Gauss had begun to work on the parallel axiom since 1792. In 1813, he said, konusunda We are not even ahead of Euclides about the parallel theory. This is a shameful part of mathematics.

By 1817 Gauss had been convinced that the fifth axiom was independent of the other four. He began to work on the results of a geometry idea that allowed multiple lines to be drawn from one point outside of one line. However, Gauss did not publish his work and kept it a secret. Kant’s synthetic a priori conception was dominant in those years, and the idea that Euclidean geometry was a necessity of pure appearance was widespread. According to historians, Gauss was not an academician who liked to engage in conflict on such issues.

Gauss exchanged views on his parallel axiom with his colleague Farkas Bolyai. Farkas Bolyai, in the past, had proved to be false later. He knew how difficult the subject was and the harms of obsessing it. Bolyai advised his son, János Bolyai, whom he had trained as a mathematician, not to spend an hour on the fifth axiom problem. However, János Bolyai did not listen to his father. The letter he wrote to his father in 1823 to inform his father of the results of his studies, he wrote, ler The things I discovered were so wonderful that I was astonished. It took two more years for Son Bolyai to write and complete his study of this wonderful new world, and he published it in the appendix of his father’s book of geometry.

Bolyai’s leap was the possibility of a new geometry. Gauss read this work, and in a letter to a friend, remembered Bolyai as a genius. On the other hand, he said that Bolyai had discovered it before, but did not publish it, so as to discourage it.

In parallel with these developments, a Russian mathematician Lobachevsky published a study on non-Euclidean geometries in 1829, unaware of the work of Gauss and Bolyai. The study could not reach a large audience because it was written in Russian and was published in a local university journal. Lobachevsky later published his work in French in the Crelle’s Journal in 1837. He then wrote a booklet describing his work in detail in 1840, where he changed the parallel axiom of Euclides as follows: From a point not on a line, two parallel lines can be drawn. From this axiom, Lobachevsky studied trigonometric identities and showed that as the triangles shrink, the trigonometric identities converge to the form in Euclides geometry.

Riemann, one of Gauss’ doctoral students, spoke of a new understanding of geometry in his opening lecture on 10 June 1854. In this speech, Riemann described the “spherical” geometry in which no parallel can be drawn from a point other than himself.

No proof was given that Bolyai and Lobachevsky’s geometries were consistent. It has not been shown yet that the theorem of a theorem cannot be given both for itself and for its not. On the other hand, there was no such evidence about Euclidean geometry. There was no inconsistency in the evidence made for centuries. Riemann then showed that non-Euclidean geometries are relatively consistent: Non-Euclidean geometries are only consistent if and only if Euclidean geometry is consistent.

Eugenio Beltrami (1835 – 1900) showed for the first time that non-Euclidean geometries could have a model. In 1868, in his essay Essay on the interpretation of non-euclidean geometry, Beltrami introduced a model for non-Euclidean geometry in 3-dimensional Euclidean geometry. This model was a surface, also called a false sphere, obtained by rotating a tractrix on the asymptote.

Later, in 1871, Klein introduced models for all non-Euclidean geometries, including Riemann’s non-Euclidean geometry. Klein showed that there were basically three different geometries: Bolyai has two infinitely distant points in Lobachevsky geometry. In Riemannian geometry, there is no such infinite distance. In Euklides geometry, there are two infinitely distant points overlapping for each line.

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Prepared by:** Sociologist Ömer YILDIRIM

**Source:**Ömer YILDIRIM’s Personal Lecture Notes. Atatürk University Department of Sociology 1st Grade Giriş Introduction to Philosophy ”and 2nd, 3rd, 4th Grade Tarihi History of Philosophy” Lecture Notes (Ömer YILDIRIM); Open Education Philosophy Textbook