Johann Heinrich Lambert’s Memoir Theorie der Parallellinien: A Review with Commentary

Athanase Papadopoulos () and Guillaume Théret ()
Additional contact information
Athanase Papadopoulos: Université de Strasbourg et CNRS, Institut de Recherche Mathématique Avancée
Guillaume Théret: Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB)

Chapter Chapter 7 in Spherical Geometry in the Eighteenth Century I, 2026, pp 161-193 from Springer

Abstract: Abstract We review the memoir Theorie der Parallellinien by Johann Heinrich Lambert, written in 1766. Lambert, a victim of the prejudices of his time, conceived this memoir as an attempt to prove the so-called parallel postulate of Euclid’s Elements, and consequently, the non-existence of the geometry that we now call hyperbolic geometry. In fact, by developing the foundations of a geometry obtained by replacing theParallel postulate parallel postulate with its negation while keeping Euclid’s other postulates unchanged, Lambert was hoping to arrive at a contradiction. Of course, he failed in his endeavor, but these attempts at proving the parallel postulate implicitly contain, without Lambert having foreseen it, fundamental results of hyperbolic geometry, the discovery of which, by Lobachevsky, Bolyai and Gauss, was not to take place until the following century. Thus, Lambert’s memoir (which he did not intend to publish but which was eventually published in 1895) constitutes one of the founding texts of non-Euclidean geometry. Spherical geometry is one of the three geometries of constant curvature, the other two being Euclidean geometry and hyperbolic geometry. In this sense, along with hyperbolic geometry, spherical geometry constitutes one of the two non-Euclidean geometries. In fact, Lambert, like Lobachevsky and others after him, understood the deep relationships between the three geometries: Euclidean, spherical, and hyperbolic, in particular the formal and the more profound analogies between the trigonometric formulae, the properties of birectangular isosceles quadrilaterals and of trirectangular quadrilaterals, the monotonicity properties (which can be formulated in terms of convexity properties) which hold in opposite senses in spherical and hyperbolic geometry which at some points he calls a sphere of imaginary radius. It is for these reasons that we decided to include in this volume, dedicated to spherical geometry, a chapter on this important memoir by Lambert, trying to highlight its most important ideas.

Keywords: Johann Heinrich Lambert; Neutral geometry; Parallel axiom; Spherical geometry; History of geometry; Hyperbolic geometry; Theory of parallel lines; Lambert quadrilateral; Ibn al-Haytham–Lambert quadrilateral; Greek mathematics; Arabic mathematics; Eighteenth century mathematics; Khayyām–Saccheri quadrilateral; Spherical trigonometry; Parallel postulate; 01A20; 01A30; 01A50; 51-03 (search for similar items in EconPapers)
Date: 2026
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-12466-1_7

Ordering information: This item can be ordered from
http://www.springer.com/9783032124661

DOI: 10.1007/978-3-032-12466-1_7

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().