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Mathematical Treasure: Bolyai’s Essay on Euclid’s 5th Postulate

Frank J. Swetz (The Pennsylvania State University)

János Bolyai (1802-1860) was a gifted Hungarian mathematician who is considered one of the founders of non-Euclidean geometry. János published his theories concerning Euclid’s troublesome “Fifth postulate” as an Appendix, “The Science of Absolute Space: Independent of the Truth or Falsity of Euclid’s Axiom XI (which can never be decided a priori)” in his father’s book, Tenamen Juventutem Studiosum in Elementa Matheseos Purae Introducendi [An Attempt to Introduce Studious Youths to the Elements of Pure Mathematics] (1832). His father, Farkas Bolyai (1775-1856), was also a noted mathematician and friend of Carl Friedrich Gauss (1777-1855). János Bolyai’s classical essay on absolute space was translated from Latin into English by George Halsted of the University of Texas and first published in 1891. Here, we examine the fourth edition of 1896, which can be read in its entirety in the Biodiversity Library Digital Collection.

The wordy, 30-page “Translator’s Introduction” supplies a detailed discussion of the “Parallel Postulate Controversy” and supplies excellent background. A few pages are given here; an interested reader would do well to read the whole Introduction.



The title page for the Appendix:

János defined parallelism of lines on the first page of the essay:

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Bolyai’s Essay on Euclid’s 5th Postulate," Convergence (May 2017)