1 The Birth of Science - MSRI

348 11. The Age-Long Recovery
It seems remarkable to me that the second hypothesis [namely, that
the fourth angle is obtuse] holds if instead of plane triangles we take
spherical ones, for in this case not only the angles of a triangle add up
to more than 180 degrees, but also the excess is proportional to the
area of the triangle. 183
This is a truly remarkable observation: it is (though he was unaware
of that) equivalent to the statement that his results consequent upon the
assumption of an obtuse fourth angle coincide with the spherical geometry
of ancient times. Lambert had in fact redemonstrated some classical
theorems.
The Sphaerica of Menelaus, from the first century A.D., is the oldest work
in non-Euclidean geometry that has come down to us. It studies the surface
of the sphere not as something immersed in three-dimensional space,
but through its intrinsic properties (to use the technical term). 184 Each
theorem, including those on spherical triangles, is proved following the page 426
scheme used in the Elements for plane geometry, but interpreting Euclid’s
straight lines (line segments) as arcs of great circles. Naturally, those theorems
of plane geometry that depend on the existence of a parallel to a
line through a given point are not present, being replaced by different
theorems valid in the spherical case. In particular, the theorem cited by
Lambert on the excess of the angle sum of a spherical triangle appears as
proposition 11 in Book I of the Sphaerica.
To a Hellenistic mathematician, there would be no point even in posing
the question whether one can construct a consistent geometry containing
a theory of parallels different from the one in the Elements. This is because
spherical geometry makes it obvious that the answer is yes. 185 It is reasonable
to think, then, that Menelaus’ Sphaerica, containing as it does an
explicit alternative to the geometry of the Elements, may have pointed the
way toward the recognition that such alternatives exist. And sure enough,
183
J. H. Lambert, Die Theorie der Parallellinien, in [Stäckel, Engel], p. 202.
184
However, some theorems of intrinsic sphere geometry were already present in the work of
Theodosius; see note 71 on page 49.
185
For precision’s sake we ought to point out that in spherical geometry (which has no “parallel”
lines) what must be abandoned is not the fifth postulate in its original form, but an assumption that
Euclid makes implicitly in the proof of proposition 16 of Book I (see note 38a on page 159). It may
be objected that one must also modify the first postulate, since by interpreting lines as great circles
in spherical geometry, uniqueness fails for lines passing through antipodal points. This fault can
be easily remedied by treating each pair of antipodal points as a single point (spherical geometry
on the projective plane).
Revision: 1.11 Date: 2003/01/06 07:48:20