Lambert was a colleague of Euler and Lagrange at the Berlin Academy of
In 1766 Lambert wrote Theorie der Parallellinien which was a study of
the parallel postulate. By assumed that the parallel postulate was
false and he managed to deduce a large number of non-euclidean
results. He noticed that in this new geometry the sum of the angles of
a triangle increases as its area decreases.
Lambert is best known, however, for his work on Pi . Euler had already
established in 1737 that e and e ^2 are irrational. Lambert was the
first to provide a rigorous proof that Pi is irrational.
In a paper presented to the Berlin Academy in 1768 Lambert showed
that, if x is a nonzero rational number, then neither e ^x nor tan x
can be rational. Since tan Pi /4 = 1 then Pi /4 must be irrational.
Lambert conjectured that e and Pi are transcendental. This was not
proved for another century when Hermite proved that e is
transcendental and Lindemann proved that Pi is transcendental.
Lambert also made the first systematic development of hyperbolic
functions and is responsible for many innovations in the study of heat
Biographies of mathematicians are from the
Mathematics archive at the University of St. Andrews, and are
used with permission.