# What is Euclid's life story

## Short lecture 400 years of JLU

Bolyai János, son of Bolyai Farkas, was born in Cluj-Napoca (Transylvania) in 1802 and was a mathematical genius even as a child. At 14 he was already studying higher mathematics. After completing the college, the father is faced with the problem of finding a suitable place to study for his gifted son. At that time there was no mathematics professor at the universities of Vienna and Budapest with whom the boy could have learned something. Bolyai asked Gauss whether his son could study with him in Göttingen, his request went unanswered. For financial reasons, Bolyai János can only study at the Military Engineering Academy in Vienna.

Bolyai János was confronted with the problem of parallels early on, as his father tried to prove the parallel axiom. He often said to his son: "Anyone who can prove the axiom of parallels deserves a diamond the size of the earth". From 1818 Bolyai János studied the essence of the axiom of parallels. His father had tried every possible way to dissuade him so as not to waste time and energy on sterile investigations. Bolyai János finished his studies in 1823; in the meantime he had worked out a new theory of parallels. He wrote to his father (from Timisoara): "I have created a new world out of nothing." In 1826 he handed over the German-language work on non-Euclidean geometry to his former teacher at the military engineering academy. In 1831 his work appeared in Latin, the “Scientiam spatii” (Appendix), the solution to the 2200 old problem of parallels: There is no problem at all, there is a Euclidean and a non-Euclidean geometry.

The non-Euclidean geometry differs from the Euclidean geometry in that the axiom of parallels does not apply here, since different geometries are based on different axioms. In non-Euclidean geometry, there is no exactly parallel through a point outside of a straight line.

How do two straight lines become parallel? According to Bolyai we can imagine it as follows: Let AB and CD be two straight lines that intersect. If the straight line CD is rotated around the point C, this straight line “jumps” at some point into the position CF, in which the straight lines AB and CF no longer intersect. In this position the lines AB and CF are parallel.