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Heavenly Mathematics

Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry

Glen Van Brummelen
Copyright Date: 2013
Pages: 240
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  • Book Info
    Heavenly Mathematics
    Book Description:

    Spherical trigonometry was at the heart of astronomy and ocean-going navigation for two millennia. The discipline was a mainstay of mathematics education for centuries, and it was a standard subject in high schools until the 1950s. Today, however, it is rarely taught.Heavenly Mathematicstraces the rich history of this forgotten art, revealing how the cultures of classical Greece, medieval Islam, and the modern West used spherical trigonometry to chart the heavens and the Earth. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation for its elegant proofs and often surprising conclusions.

    Heavenly Mathematicsis illustrated throughout with stunning historical images and informative drawings and diagrams that have been used to teach the subject in the past. This unique compendium also features easy-to-use appendixes as well as exercises at the end of each chapter that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

    eISBN: 978-1-4008-4480-7
    Subjects: Mathematics, Astronomy, Transportation Studies

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Preface
    (pp. vii-xvi)
  4. 1 Heavenly Mathematics
    (pp. 1-22)

    We’re not ancient anymore. The birth and development of modern science have brought us to a point where we know much more about how the universe works. Not only do we know more; we also have reasons to believe what we know. We no longer take statements on faith. Experiments and logical arguments support us in our inferences and prevent us from straying into falsehood.

    But how true is this, really? Do we really know, for instance, why the trajectory of a projectile is a parabola? In fact, anyone who has seen a soccer goalkeeper kick a ball downfield is...

  5. 2 Exploring the Sphere
    (pp. 23-41)

    At first glance there’s not much to see on a sphere; every point on its surface looks the same as any other. But give it some physical meaning—call it the celestial sphere, or the Earth’s surface—place some identifying marks on it, and set it in motion, and visualizing what’s happening can become rather complicated. This is one reason why armillary spheres (plate 3, figure 2.1), movable models of the celestial sphere with various signposts labeled, were invented by the ancient Greeks. Rotating the sphere simulates the motions of the heavens and provides a tactile experience that cannot easily...

  6. 3 The Ancient Approach
    (pp. 42-58)

    We tend to think of the growth of mathematical knowledge like that of a glacier. The boundaries spread outward gradually as new bits of knowledge are added to the existing structure. But a flag planted at a particular spot will stay there, and the features of its immediate environment stay essentially unchanged. Other than the accretion of new functions and identities, the basic theory remains the same. After all, how could trigonometry look any different from how it looks today?

    Part of the goal of the next couple of chapters is to refute this charge of intellectual lifelessness. Spherical trigonometry,...

  7. 4 The Medieval Approach
    (pp. 59-72)

    Reading al- Kūhī’s statement defending the advantages of Menelaus’s Theorem in the previous chapter is a bit like eavesdropping on someone holding a telephone conversation. We have a rough idea of what was said, but important parts of the debate are a blank to us. We are never told the name of the advocate of the new theorem, nor even what the new theorem was. There is one hint. The traveling theorem salesman claimed that his result “freed” him from having to know Menelaus’s Theorem to solve astronomical problems. This is precisely the word and meaning that became attached to...

  8. 5 The Modern Approach: Right-Angled Triangles
    (pp. 73-93)

    The word “trigonometry” means “triangle measurement,” which is how we’ve thought of the subject for the past several centuries. The term comes from Bartholomew Pitiscus’s 1600 bookTrigonometria(figure 5.1), a variant of the phrase “the science of triangles” that had been used for a number of decades previously. But considering a triangle on its own, as millions of high school students do every day in trigonometry classes, is a relatively recent idea. From what we’ve seen so far of ancient and medieval trigonometry only the spherical Law of Sines works this way, and it wasn’t used particularly often. There...

  9. 6 The Modern Approach: Oblique Triangles
    (pp. 94-109)

    So far spherical trigonometry hasn’t looked much like the plane theory we learned in high school. However, the parallels often lie just below the surface. For instance, cosc= cosacosbdoesn’t resemble the Pythagorean Theoremc² =a² +b², but the latter is just the planar special case of the former. The similarities also apply at the larger scale of the development of the theory. Plane trigonometry begins with a study of right-angled triangles, and when we turn to oblique triangles, we piggyback our analysis on what we have learned already about right-angled triangles (usually...

  10. [Illustrations]
    (pp. None)
  11. 7 Areas, Angles, and Polyhedra
    (pp. 110-128)

    The first goal of trigonometry—to solve any triangle given some information about its sides and angles—has been accomplished, so it is at this point that most textbooks stop. This is a pity, because while the straightforward practical work has been completed, a wealth of mathematical pleasures that might have spurred a lot of curiosity lies just around the corner. Fortunately we are not bound by early 20th-century mathematics curricula, so we shall press onward and taste some of these delights.

    We begin with a seemingly practical problem that we have so far carefully ignored: to find the area...

  12. 8 Stereographic Projection
    (pp. 129-150)

    Astronomers needed to compute and observe long before the computer and the telescope. Before time-saving devices like logarithms and slide rules rescued astronomers from hours of drudgery, calculations were done by hand and were simply part of the job description. In fact, the word “computer” referred originally to a person, not a machine. But even in ancient times there were still tools that could aid the weary scientist by generating at least approximate solutions to astronomical problems. We have already seen the armillary sphere, a model of the celestial sphere that rotates in the same way the heavens do. By...

  13. 9 Navigating by the Stars
    (pp. 151-172)

    B. M. Brown’s complaint in the previous chapter against Cesàro’s remarkable approach to spherical trigonometry might have been made by an astronomer or navigator. For the practitioner already in command of the important theorems and looking ahead to their uses in science, a pit stop to examine elegant alternative approaches is a restless, impatient exercise. While we may value the charm of beautiful mathematics on its own, its charm can only be enhanced by witnessing what it can do in some physical realization. Thus, it seems appropriate to conclude this book with an account of the life-and-death application that gave...

  14. Appendix A. Ptolemy’s Determination of the Sun’s Position
    (pp. 173-178)
  15. Appendix B. Textbooks
    (pp. 179-181)
  16. Appendix C. Further Reading
    (pp. 182-188)
  17. Index
    (pp. 189-192)