Johann Heinrich Lambert's memoir "Theorie der Parallellinien": A review with commentary
Athanase Papadopoulos (IRMA), Guillaume Th\'eret (ICB)

TL;DR
This review explores Lambert's 1766 memoir on the parallel postulate, highlighting its foundational role in non-Euclidean geometry and its implicit discovery of hyperbolic geometry principles.
Contribution
It provides a detailed analysis of Lambert's early work, emphasizing its significance in the development of non-Euclidean geometries and its historical context.
Findings
Lambert's memoir implicitly contains fundamental results of hyperbolic geometry.
The work establishes deep relationships between Euclidean, spherical, and hyperbolic geometries.
Lambert's attempts to prove the parallel postulate contributed to the foundations of non-Euclidean geometry.
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…
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