Johann Heinrich Lambert's memoir "Theorie der Parallellinien": A review with commentary
Athanase Papadopoulos, Guillaume Théret
TL;DR
The paper reviews Lambert's Theorie der Parallellinien as a pivotal but historical attempt to prove Euclid's parallel postulate, revealing how replacing it with its negation early in Lambert's analysis yields foundational ideas later central to hyperbolic geometry. It situates Lambert within a long historical arc—from Greek and Arab predecessors to Renaissance and modern mathematicians—and emphasizes the deep connections among Euclidean, spherical, and hyperbolic geometries, including the use of trirectangular and Khayyām–Saccheri quadrilaterals. The analysis shows that Lambert's work already contains essential elements of non-Euclidean geometry, such as monotonicity and area- versus angle-based arguments, and that his explorations influenced later figures like Taurinus, Lobachevsky, Bolyai, and Gauss. By tracing the structure of Lambert's three-part memoir and its three hypotheses about parallelism, the paper highlights how historical attempts to resolve the parallel postulate converged on ideas that would redefine geometry in the nineteenth century. The discussion also situates the Lambert memoir within spherical geometry and underscores its role as a bridge between historical inquiry and modern non-Euclidean frameworks.
Abstract
We review the memoir \emph{heorie 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 \emph{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 the 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. This paper will appear as a chapter in the book ``Spherical Geometry in the Eighteenth Century I: Euler, Lagrange and Lambert'', ed. R. Caddeo and A. Papadopoulos, Springer Nature Switzerland, 2026.