János Bolyai (1802-1860), was born in Kolgsvár on December 15, 1802. His father took special care in his physical and intellectual education in this order so that János's intellect could have a healthy body at his disposal. From an early age, Alnos endowed with an extremely observant spirit revealed superior intellectual abilities. At age 9, when his father decided to send him to school he had already acquired deep knowledge of various subjects with primacy for the exact sciences, for example at age 4 could distinguish certain geometric figures, knew the sine function, identified the known constellations , at age 5 he had learned to read, practically on his own and was well above average in language and music learning, at age 7 he began to play the violin and made such good progress that he was soon playing difficult concert pieces.
Farkas Bolyai had the most talented of his disciples to teach his son various subjects but reserved for himself the teaching of mathematics, in a letter written to Gauss he expressed the wish that his son should be a mathematician. At the age of 12, János became a normal student at Marosvásárhely Calvinist College skipping the first three years, started in 4th grade and often happened to follow lessons directed at older students, rated himself "excellent" when he submitted to the rigorousum examination on 6/30/1817, which gave him the right to study the Latin classics. However, this was not the way to even study mathematics with Gauss because he refused his father's request to do so.
He decided to pursue a career in military engineering at the Vienna Academy of Engineering. Farkas Bolyai took special care in preparing his son for the entrance exam because his results were crucial for any of the seven years of study the candidate was admitted to. But the lack of money forced János to stay another year in the Department of Philosophy in Marosvásárhely and so only in August 1818 after receiving financial help from some people entered that Academy in the 4th year, the most advanced possible by regulation the following year. He was already the 2nd best student in his class with the highest grades in everything except drawing and handwriting. That year Archduke Johann von Hausburg, Commander-in-Chief of the Academy and Superintendent of Engineers, during a visit learned of János Bolyai's mathematical talent, and endeavored to send a message to Farkas Bolyai expressing his recognition and conviction of We could already expect a rapid advance in the military career if we continued to work diligently. A year before finishing his academic studies, which happened on 9/6/1822, his mother died. He was not sent to the detachment service, but along with six other distinguished cadets, he was allowed to attend an additional course to receive special training in architecture and military fortifications.
While in Vienna, János Bolyai showed special interest in certain fields of mathematics, in particular Euclid's 5th postulate. Indeed, his interest had been piqued by his father, who selflessly passed on his splendid knowledge and laid the foundations for the wonderful deeds described in TENTAMEN. This was one of the undeniable merits of Farkas Bolyai even though he did not achieve what his son would create. , Non-Euclean geometry. During his years at the János Academy he further deepened his knowledge of the subject, his ambition was heightened by the inspiring interest of his mathematics teacher Johan Walter von Eckwehr and the enthusiasm of Károly Szasz, a Hungarian tutor in Vienna.
János Bolyai's own goal was to prove the 5th postulate by an indirect path, his discussions with Szasz resulted in the recognition that assuming that the circumference of infinite radius is a straight line is equivalent to Euclid's axiom of parallelism when they left, promised each other that if one of them got serious axiom test results, they would declare a joint success. Later, we have already clarified in his writings that the agreement only involved that proof but was not valid for the creation of a new system of geometry. In September 1823, János-Bolyai was commissioned as lieutenant and sent to Temesvár Fortification. shortly thereafter on 3 November he wrote in a letter to his father that he "discovered the basic idea of a new geometric system", that "created a new, another world out of nothing", his hypothesis was based on a definition of parallelism rather than Euclid's geometry, his investigations were recalled in an extremely structured "Appendix" work consisting of 43 sections.
János's troubled military career, with constant displacement, greatly interfered with his life and in 1833 retired with his captain's pension. And a new time seemed to happen, but it was by no means happier than before. The unfavorable reception of the "Appendix", both published and sent to various places and particularly Gauss' laconic and ambiguous appreciation, made János irritable and turned it into a misatropo. In the last period of his life, János Bolyai barely dealt with mathematics, feeling so unhappy with it, died on January 27, 1860, suffering from pneumonia.