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The memoir Theorie der Parallellinien (1766) by Johann Heinrich Lambert isone of the founding texts of hyperbolic geometry, even though its author's aimwas, like many of his pre-decessors', to prove that such a geometry does notexist. In fact, Lambert developed his theory with the hope of finding acontradiction in a geometry where all the Euclidean axioms are kept except theparallel axiom and that the latter is replaced by its negation. In doing so, heobtained several fundamental results of hyperbolic geometry. This was sixtyyears before the first writings of Lobachevsky and Bolyai appeared in print. Inthe present paper, we present Lambert's main results and we comment on them. AFrench translation of the Theorie der Parallellinien, together with anextensive commentary, has just appeared in print (A. Papadopoulos and G.Th{\'e}ret, La th{\'e}orie des lignes parall{\`e}les de Johann HeinrichLambert. Collection Sciences dans l'Histoire, Librairie Scientifique etTechnique Albert Blanchard, Paris, 2014).

Hyperbolic geometry in the work of Johann Heinrich Lambert