Geometry is one of the oldest subjects of mathematics. It was developed by Euclid, a Greek mathematician, who was born in 300 BCE. His most famous treatise of geometry was compiled into the book titled, Elements, which inspired many of the mathematics found in the common core of general education. Included in his book, Euclid describes Euclidean geometry as the study of plane and solid figures on the basis of a small set of his axioms and theorems. Euclid's axioms and common notions are also a part of his famous work Elements. These axioms and theorems of geometric figures make up the foundation of modern geometry. Euclid's standard of deductive reasoning and findings of geometry lasted for more than 2,000 years unchanged.

Inspired by Euclid's findings on geometry, Felix Klein, a German mathematician, connected geometry, group theory and function theory to make non-Euclidean geometry. While attending the University of Bonn, Klein was an assistant to Plucker who excelled in mathematics and experimental physics and geometry. Within the year Klein received his doctorate from the University of Bonn and his mentor Plucker passed away and left his work, the Foundations of Line Geometry, incomplete. Klein along with Alfred Clebsch, another German mathematician, worked together to add and finish Plucker's work on the Foundations of Line Geometry. Years later, Klein became a professor at Erlangen in southern Germany and continued his studies of geometry and geometric transformations.

The process of mathematicians studying Euclid's work, The Elements, and specifically the truthfulness of Euclid's fifth postulate throughout history led other mathematical subjects came to light like the euclidean plane, calculus and many more which impacted other areas of study like physics.

I would highly recomend viewing this series "The History of Non-Euclidean Geometry" provided by the youtube channel Extra Credits. The five videos regarding non-euclidean geometry give a detailed and entertaining summarization of the history of geometry.



Klein's findings of non-Euclidean geometry consist of transformations, which only hold if the Euclidean geometry or Euclid's axioms is also satisfied. Klein published his geometric transformations in his treatise, Erlanger Programm, which is now the standard view for geometry. "Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. In this way the Erlanger Programm defined geometry so that it included both Euclidean geometry and non-Euclidean geometry" (Maths History 2021).