Mathematical Work
Bolyai's main interests were the foundations of geometry and the parallel axiom.
His main work, the Tentamen (Tentamen iuventutem studiosam in elementa matheosos introducendi), was an attempt at a rigorous and systematic foundation of geometry, arithmetic, algebra and analysis. In this work, he gave iterative procedures to solve equations which he then proved convergent by showing them to be monotonically increasing and bounded above. His study of the convergence of series includes a test equivalent to Raabe's test, which he discovered independently and at about the same time as Raabe. Other important ideas in the work include a general definition of a function and a definition of an equality between two plane figures if they can both be divided into a finite equal number of pairwise congruent pieces. See also Bolyai-Gerwien theorem.
He first dissuaded his son from the study of non-Euclidean geometry, but by 1830 he became enthusiastic enough to persuade his son to publish his way-breaking thoughts.
Read more about this topic: Farkas Bolyai
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