Non-Euclidean geometry: denial of the fifth postulate
A geometry in which Euclid's fifth postulate (of a unique parallel) is dropped and replaced by the assumption that through a point off a line there passes more than one non-intersecting line. From such a change one obtains a coherent body of theorems in which no contradiction arises.
The conclusion was reached independently and almost simultaneously by the Russian Nikolai Lobachevsky (1829) and the Hungarian Janos Bolyai (1832). The two-thousand-year programme of proving the fifth postulate was closed: it turned out to be a choice, not a theorem.
From here came the understanding of axioms as freely chosen premises rather than self-evident truths about the world. The question of the geometry of physical space became empirical, not a priori.
Why this date
The date follows the first publication: 'On the principles of geometry', Kazan Messenger, 1829-1830. Lobachevsky's Kazan talk of 11 (23) February 1826 was never printed. Bolyai issued his Appendix in 1832 (an offprint was sent to Gauss on 20 June 1831), Gauss published nothing. Saccheri (1733) and Lambert (written 1766, printed 1786) derived many theorems of this geometry but did not accept it as a system of its own, so the count starts from 1829 and not from the 18th century.
What it grew from
-
The Elements: definition, axiom, proof
influenced · Confidence: high
The fifth postulate of the Elements and the centuries of attempts to derive it from the other axioms set the very problem that Lobachevsky and Bolyai solved in the 1820s by the opposite move, adopting the denial of the postulate and meeting no contradiction.
Dispute: Priority is contested, and priority of publication diverges from priority of discovery. Lobachevsky printed first (1829-1830), Bolyai in 1832, while Gauss kept his conclusions to himself from the 1790s, which the Stanford Encyclopedia of Philosophy explains thus: 'he refrained from publishing for fear of scandal'. The Dictionary of Scientific Biography awards precedence to Lobachevsky on the fact of printing: 'Since Gauss did not publish his work on the subject, and since Bolyai published only at a later date, Lobachevsky clearly holds priority'. Wolfe separates the two things: Lobachevsky published earlier, but on Wolfe's account made the discovery later than Bolyai did, 'But there is ample evidence that he made his discoveries later than Bolyai made his', where the pronoun refers to Lobachevsky and not to Gauss. Awareness of one another needs a date attached to it. At the time of the Kazan publication of 1829-1830 it was unknown outside Kazan: MacTutor writes 'Neither Bolyai nor Gauss knew of Lobachevsky's work, mainly because it was only published in Russian in the Kazan Messenger a local university publication'. Later this changed: the German exposition of 1840 reached Gauss, 'This last publication greatly impressed Gauss', while Bolyai read Lobachevsky in 1848 and suspected Gauss stood behind it. Lobachevsky, on Wolfe's account, never learned of Bolyai at all: 'there seems to be no evidence that Lobachewsky ever heard of Bolyai'. The role of the precursors is disputed as well: Saccheri (1733) and Lambert (1766) derived a large part of the theorems of this geometry, and Papadopoulos and Théret call Lambert's text the closest to hyperbolic geometry among everything written before Lobachevsky, yet neither of them accepted it as a system of its own.
Link sources
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
In the 1820's, Nikolai I. Lobachevsky (b. 1793, d. 1856) and Janos Bolyai (b. 1802, d. 1860) independently tackled this question in a radically new way.
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
Girolamo Saccheri (b. 1667, d. 1733) tried reductio. He inferred a long series of propositions from the negation of Euclid's Postulate, until he reached one which he pronounced 'repugnant to the nature of the straight line'.
- Harold E. Wolfe, Introduction to Non-Euclidean Geometry, ch. 3 (Dryden Press) (1945) Verified 20 Sep 2026
Slowly but surely they had directed the speculations of geometers to the point where the discovery of Non-Euclidean Geometry could not long be delayed.
- Mathematicians Reconsider Euclid's Parallel Postulate, Science and Its Times (via Encyclopedia.com) (2001) Verified 20 Sep 2026
Work on the parallel postulate in the eighteenth century set the stage for the development of non-Euclidean geometries in the nineteenth century.
- Nikolai Ivanovich Lobachevsky, MacTutor History of Mathematics Archive (2000) Verified 20 Sep 2026
This last publication greatly impressed Gauss but much has been written about Gauss's role in the discovery of non euclidean geometry which is just simply false.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
Other links
-
Geometry of manifolds and curvature (Riemann)
parallel to · Confidence: medium
Riemann independently arrived at spaces of constant negative curvature and absorbed Lobachevskian geometry as a special case of a general scheme of manifolds, although historians do not confirm any direct influence of Lobachevsky and Bolyai on the 1854 lecture.
Dispute: Historians disagree on whether Riemann knew the works of Lobachevsky and Bolyai. Scholz denies it: 'he apparently had no knowledge of Bolyai's or Lobachevsky's studies in the foundations of geometry', and traces the roots of the lecture to Gauss's theory of surfaces and Herbart's philosophy. Bottazzini and Tazzioli cautiously allow the opposite: 'However one can argue that Riemann probably knew Lobachevsky's geometry, at least in the version that appeared in that Journal', while noting that the lecture itself carries no references to Bolyai and Lobachevsky. The widespread notion of a kinship between non-Euclidean geometry and the special theory of relativity of 1905 is likewise contested: Walter writes that 'Up until the time of Minkowski's space-time theory, however, non-Euclidean geometry appeared to be irrelevant to physics', and that in 1905 'physicists generally presented their results in mechanics and electrodynamics using either Cartesian coordinate or vector methods'. Non-Euclidean language entered relativistic physics through Minkowski (1907-1908) and Varicak (1910), that is after 1905, and Riemannian geometry became physics only in the general theory of relativity: in the words of the Stanford Encyclopedia of Philosophy, 'Klein's Erlangen program failed to cover the differential geometry of Riemann (§5), which Einstein (1915, 1916) placed at the core of his general theory of relativity'.
Link sources
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
Regarded as Riemannian 3-manifolds, Euclidean space has constant zero curvature, Lobachevskian space has constant negative curvature, and elliptic space has constant positive curvature.
- Erhard Scholz, The Concept of Manifold, 1850-1950 (in History of Topology, ed. I. M. James) (1999) Verified 20 Sep 2026
Riemann even sketched such a reduction in the last section of his talk, although he apparently had no knowledge of Bolyai's or Lobachevsky's studies in the foundations of geometry.
- Umberto Bottazzini, Rossana Tazzioli, Naturphilosophie and its role in Riemann's mathematics, Revue d'histoire des mathématiques 1 (1995) Verified 20 Sep 2026
Nor did he refer to the work of Bolyai and Lobachevsky, even though a paper of the latter had appeared in 1837 in Crelle's Journal.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
Node sources
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
In the 1820's, Nikolai I. Lobachevsky (b. 1793, d. 1856) and Janos Bolyai (b. 1802, d. 1860) independently tackled this question in a radically new way.
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
Since Lobachevsky was the first to publish, the system of geometry based on the said 'absolute geometry' plus the negation of Euclid's Postulate is properly called Lobachevskian geometry.
- Rosenfeld B. A. «Lobachevsky, Nikolai Ivanovich», Dictionary of Scientific Biography (via Encyclopedia.com) (1973) Verified 20 Sep 2026
Since Gauss did not publish his work on the subject, and since Bolyai published only at a later date, Lobachevsky clearly holds priority.
- Nikolai Ivanovich Lobachevsky, MacTutor History of Mathematics Archive (2000) Verified 20 Sep 2026
On 11 February 1826, in the session of the Department of Physico-Mathematical Sciences at Kazan University, Lobachevsky requested that his work about a new geometry was heard and his paper A concise outline of the foundations of geometry was sent to referees.
- Athanase Papadopoulos, Guillaume Théret, Hyperbolic geometry in the work of J. H. Lambert (2015) Verified 20 Sep 2026
This was sixty years before the first writings of Lobachevsky and Bolyai appeared in print.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.