Non-Euclidean Geometry

Chapter 14 talks about Non-Euclidean geometry, which comes from the Greek mathematician Euclid.  It starts off stating that Euclid’s Elements led to disagreements because of his assumptions, one of them being the non-euclidean geometry. Even to Bertrand Russell, Euclid’s Elements to him was known to be dull and a “tissue of nonsense.” Euclid wanted people to accept his not so famous fifth postulate, the parallel postulate that no line can intersect a point, without providing proof because it was so simple to understand.

 Non-Euclidean geometry consists of two geometries created on axioms. It is a system of definitions, proofs, and rules that define the points, lines, and its planes.  Two of the non-Euclidean geometries are spherical and hyperbolic geometry.  The difference is in their parallel lines.  In the Euclidean geometry, there is only one line through the point that is in the same line and never intersects.  In spherical geometry, there are no lines, and in hyperbolic geometry there are at least two different lines that pass through the point.

This topic of non-Euclidean geometry has been a controversy for many mathematicians for almost 2,000 years, which is to be able to remove the postulate by making it a theorem. For example, in the book “The Colossal Book of Mathematics”, there were many obsessions over this problem. Farkas Bolyai’s son, Janos became obsessed with this problem and did not give up until he could solve it. 

Resulting later on, he convinced himself that not only the postulate was independent from the axioms, but there was also consistent geometry that through a point, an infinite number of lines are parallel to that. It was Gauss, the Prince of Mathematicians and a really close friend of Janos, who actually created the term “non-Euclidean geometry”, which is now known as hyperbolic geometry. 

In the book “Euclidean and Non-Euclidean Geometries”, Marvin Jay Greenberg states that “according to Euclid, two lines in a plane either meet or are parallel. There is no other possible relation.” Gauss pointed out the error that was wrong in Janos’ discovery.  Even though Janos had discovered most of the non-Euclidean research, Gauss was seen as the mathematician to receive credit.    

According to D.M.Y. Sommerville’s book of “The Elements of Non-Euclidean Geometry,” Non-Euclidean geometry consists of infinite areas, from Bertrand’s proof.  The misconception is when applying the principle of superposition to an infinite of areas.

This chapter really had me thinking, compared to the other blogs I have worked on and read.  I had never heard of Non-Euclidean geometry until working on this section.  I found it interesting on how many mathematicians did not put much attention to this kind of geometry because for 2,000 years, no one had enough research and results. Still at this point, I stay in the middle of fully understanding this chapter, but that is why I continue to read on the topic.