"A... fallacy is contained in all proofs [of the Parallel Postulate] based upon the idea of direction. ... Another class of demonstrations is based upon considerations of infinite areas. [In] Bertrands Proof... The fallacy... consists in applying the principle of superposition to infinite areas as if they were finite magnitudes."
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or consideration of quadratic forms other than the definite quadratic forms ass
"The chief mathematical work of Wolfgang Bolyai appeared in two volumes, 1832-1833 entitled Tentamen juventutem studiosam in elementa matheseos puræ... introducendi. It is followed by an appendix composed by his son Johann. Its twenty-six pages make the name of Johann Bolyai immortal. He published nothing else but he left behind one thousand pages of manuscript."
"Propositio XXXIII. Hypothesis anguli acuti est absolute falsa; quia repugnans naturae lineae rectae. [Proposition 33. The hypothesis of acute angle is absolutely false; because repugnant to the nature of the straight line.]"
", "Geometry as a Branch of Physics" (1949) from Albert Einstein: Philosopher-Scientist, ed. ."
"Helmholtzs investigations were carefully examined by S. Lie who reduced the Riemann-Helmholtz problem to the following form: To determine all the continuous groups in space which, in a bounded region, have the property of displacements. There arose three types of groups which characterize the three geometries of Euclid, of N. I. Lobachevski and J. Bolyai and of F. G. B. Riemann."
"While Lobachevski enjoys priority of publication, it may be that Bolyai developed his system somewhat earlier. Bolyai satisfied himself of the non-contradictory character of his new geometry on or before 1825; there is some doubt whether Lobachevski had reached this point in 1826. Johann Bolyais father seems to have been the only person in Hungary who really appreciated the merits of his sons work. For thirty-five years this appendix, as also Lobachevskis researches, remained in almost entire oblivion. Finally Richard Baltzer of the University of Giessen, in 1867, called attention to the wonderful researches."
"Although K. F. Gauss, one if the spiritual fathers of non-Euclidean geometry... proposed a possible test of the flatness of space by measuring the interior angles of a terrestrial triangle, it remained for... K. Schwarzschild to formulate the procedure and to attempt to evaluate [curvature] K on the basis of astronomical data... Schwarzschilds pioneer attempt is so inspiring in its conception and so beautiful in its expression...[!]"