History of Math-Janos Bolyia

There are so many mathematicians that have made huge contributions to mathematics. One of the mathematicians that I find the most interesting is Janos Bolyai. Bolyai was a Hungarian mathematician with an interesting story. His father, Farkas Bolyai, wanted Janos to grow up to be a mathematician. In 1816, Farkas wrote to his friend and great mathematician, Carl Gauss, and asked if Janos could study under him. By 1817, Janos went to Calvinist College in Marosvásárhely to study under Gauss. 

Around 1820, Bolyai started to take after his father and work on Euclid's fifth postulate. The goal for Bolyai was to change to postulate so that it could be derived from the other postulates. In 1823, Janos wrote to his father saying

"...created a new, another world out of nothing..."

When Janos was trying to change Euclid's fifth postulate to be easier to use, he instead found the "opposite" of Euclidean geometry which is now known as non-Euclidean geometry. 

Non-Euclidean geometry is essentially the opposite of Euclidean geometry. One of the main forms of non-Euclidean geometry that is used is hyperbolic geometry. This is the type of non-Euclidean geometry that Janos Bolyai discovered.

What ever axioms or postulates are true in strictly Euclidean geometry, the negation of that axiom or postulate is true in hyperbolic geometry. A lot of hyperbolic geometry is done on a Poincare Disk Model to show the curvature of the geometry. 

Although Bolyai had studied under Gauss and considered him a friend, when Bolyai told Gauss about his discovery Gauss was unimpressed. According to Gauss, he had discovered everything that Bolyai had been working on and said that he never wrote it down. Bolyai was so distraught by this that he gave up all of his work in math and left the country. He then became a very bitter man who was hard to be around. 

Sources

Janos Bolyai Biography-http://www-history.mcs.st-andrews.ac.uk/Biographies/Bolyai.html