János Bolyai’s father, Farkas (Wolfgang) was a friend of Gauss and an author of numerous defences of Euclidean geometry. His son, János, however, fully accepted the idea of a new geometry and published it as a 24-page appendix to his father’s two-volume work on geometry in 1832. The content of this remarkable 24-page essay is described, noting the importance of the study of three-dimensional geometry (without which there can be no claim that the new geometry applies to physical space). János Bolyai also gave trigonometric formulae for his geometry, noting reassuringly that they reduced to the familiar Euclidean ones for small triangles.
Extract: Section 32 of Bolyai’s essay, which gives an approach that is close to that of intrinsic differential geometry.
KeywordsEuclidean Geometry Small Triangle Trigonometric Formula Unique Parallel Euclidean Straight Line
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