  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).
