Hyperbolic geometry in the work of Johann Heinrich Lambert

TL;DR

Johann Heinrich Lambert's 1766 work laid foundational principles of hyperbolic geometry, predating Lobachevsky and Bolyai, by exploring the negation of Euclid's parallel postulate.

Contribution

The paper highlights Lambert's early development of hyperbolic geometry, including key results, and provides historical context and commentary on his pioneering work.

Findings

Lambert's results predate Lobachevsky and Bolyai.

Fundamental hyperbolic geometry principles were established by Lambert.

The work challenged Euclidean assumptions on parallels.

Abstract

The memoir Theorie der Parallellinien (1766) by Johann Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though its author's aim was, like many of his pre-decessors', to prove that such a geometry does not exist. In fact, Lambert developed his theory with the hope of finding a contradiction in a geometry where all the Euclidean axioms are kept except the parallel axiom and that the latter is replaced by its negation. In doing so, he obtained several fundamental results of hyperbolic geometry. This was sixty years before the first writings of Lobachevsky and Bolyai appeared in print. In the present paper, we present Lambert's main results and we comment on them. A French translation of the Theorie der Parallellinien, together with an extensive commentary, has just appeared in print (A. Papadopoulos and G. Th{\'e}ret, La th{\'e}orie des lignes parall{\`e}les de Johann Heinrich Lambert. Collection Sciences dans l'Histoire, Librairie Scientifique et Technique Albert Blanchard, Paris, 2014).