  We review and comment on some works of Euler and his followers on spherical
geometry. We start by presenting some memoirs of Euler on spherical
trigonometry. We comment on Euler's use of the methods of the calculus of
variations in spherical trigonometry. We then survey a series of geometrical
resuls, where the stress is on the analogy between the results in spherical
geometry and the corresponding results in Euclidean geometry. We elaborate on
two such results. The first one, known as Lexell's Theorem (Lexell was a
student of Euler), concerns the locus of the vertices of a spherical triangle
with a fixed area and a given base. This is the spherical counterpart of a
result in Euclid's Elements, but it is much more difficult to prove than its
Euclidean analogue. The second result, due to Euler, is the spherical analogue
of a generalization of a theorem of Pappus (Proposition 117 of Book VII of the
Collection) on the construction of a triangle inscribed in a circle whose sides
are contained in three lines that pass through three given points. Both results
have many ramifications, involving several mathematicians, and we mention some
of these developments. We also comment on three papers of Euler on projections
of the sphere on the Euclidean plane that are related with the art of drawing
geographical maps.
