1 The Birth of Science - MSRI

2.8 Trigonometry and Spherical Geometry 49
limit. 69 Readers of Section 2.7 will have noticed that Archimedes does not
employ “limits” only in the sense that he fails to use a word that matches
ours exactly; a proof in modern analysis needs only to have “limit” replaced
by its definition to become equivalent to his in every way.
Returning to Hellenistic mathematicians, it should be stressed that they
also developed spherical geometry and trigonometry, subjects for which
our main sources of information are the Sphaerica of Theodosius (who
straddled the second and first centuries B.C.), the homonymous work of
Menelaus (first century A.D.), and the Mathematical syntaxis of Ptolemy 70
(second century A.D.). 70a The mathematics developed in these treatises, although
of course instrumental to astronomy and mathematical geography,
has great theoretical interest. It includes not only formulas of spherical
trigonometry (which can be useful to astronomers or geographers), but
also, particularly in the work of Menelaus, a theoretical development of
intrinsic spherical geometry, constructed in analogy with the plane geometry
of Euclid’s Elements. 71 In particular, the theory of spherical triangles
(subsets of the surface of the sphere bounded by three arcs of great circle)
is developed in analogy with the theory of triangles contained in Book I
of the Elements, based on postulates of spherical geometry — some closely
analogous to those of Euclid’s plane geometry and some very different. As
we shall see, those investigations would become important again many
centuries later.
69 For instance, the Encyclopaedia Britannica says: “Although it was a forerunner of the integral calculus,
the method of exhaustion used neither limits nor arguments about infinitesimal quantities”
(15th edition, Micropaedia, sub “exhaustion, method of”).
70 This, the main work of Ptolemy, is better known under the name Almagest, bestowed by the
Arabs. Part of it is devoted to spherical geometry and trigonometry.
70a There are hints, however, that spherical geometry arose even before Theodosius, during the
heyday of the astronomy and mathematical geography, its two most obvious applications. See
note 1a on page 236.
71 Theodosius’ work, by contrast, uses mostly stereometric methods; that is, theorems about
spherical geometry are demonstrated as theorems of solid geometry, using the three-dimensional
space where the surface is immersed. But even Theodosius sometimes uses methods of intrinsic
spherical geometry.
Revision: 1.12 Date: 2003/01/10 06:11:21
page 81