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Mathematical subjects come and go. If you glance at a textbook from a century ago you may recognize some of the contents, but some will be unfamiliar or even baffling. A high school text in analytic geometry, for instance, once contained topics like involutes of circles, hypocycloids, and auxiliary circles of ellipses: topics that most college students today will never see. But spherical trigonometry may be the most spectacular example of changing fashions in the 20th-century mathematics classroom. Born of the need to locate stars and planets in the heavens, for more than 1500 years it was the big brother to the plane trigonometry that high school students slog through today. Navigators on the open seas relied on spherical trigonometry to find their way; lives were lost when their skills failed them. Its dominance continued through the early 20th century: editions of Euclid's Elements that were designed for classrooms often included appendices devoted to this now-forgotten subject.
During World War II the popularity of spherical trigonometry remained high. Applications in naval and military settings were touted as motivations, and were given a prominent place in the exercises. Through the early 1950s textbooks continued to be published, although gradually spherical trigonometry found itself relegated to the last major section in a textbook mostly devoted to plane trigonometry. Suddenly, mid-decade, it disappeared, dropped in a pedagogical tide that was heading in other directions. Today almost no trigonometry texts even mention the existence of a spherical counterpart. The only book on the subject continuously in print (Clough-Smith 1966) is difficult to obtain and available only from nautical booksellers. This paucity comes strangely at a time when new applications of spherical trigonometry are being found. GPS devices have some of its formulas built in. It's amusing to see bibliographies of research papers in computer graphics and animation (for use in movies like those made by Pixar) referring to nothing older than last week, except for that stodgy old spherical trig text.
So if mathematics teachers have long since given up on spherical trigonometry, why bring it back? The author is not advocating that everyone should dust off the covers of their grandparents' textbooks, but a treasure would be lost if no one did.
Heavenly Mathematics
How Large Is the Earth?
Building a Sine Table with Our Bare Hands
The Distance to the Moon
Exercises
Exploring the Sphere
Introducing the Celestial Sphere
Spherical Geometry
Lunes and Triangles on the Sphere
What Is the Smallest and Largest Possible Perimeter ofa Spherical Triangle?
Exercises
The Ancient Approach
Menelaus and His Theorems
Abu Sahl al-Kuhi and the Winds of Change
Exercises
The Medieval Approach
Delving Beneath the Surface: Indian Spherical Astronomy
Finding the Direction of Mecca
Exercises
The Modern Approach: Right-Angled Triangles
Deriving the Basic Identities
Applying the Locality Principle
Applying our Knowledge to the Sky and Sea
Napier and the Birth of Logarithms
Symmetries Codified: The "Pentagramma mirificum"
Exercises
The Modern Approach: Oblique Triangles
Using the Law of Cosines
Delambre's and Napier's Analogies
Exercises
Areas, Angles, and Polyhedra
Euler's Polyhedral Formula
The Regular Polyhedra
Exercises
Stereographic Projection
Using Stereographic Projection to Solve Triangles
A Crystallographic Breakthrough: The Cesaro Method
Exercises
Navigating by the Stars
Preparing to Navigate: The Observations
A Digression: The Haversine
The Method of Saint Hilaire
Exercises
Where to Go from Here
Appendix
Ptolemy's Determination of the Sun's Position
Textbooks
Further Reading
Index