Heavenly Mathematics by Van Brummelen Glen

Author:Van Brummelen, Glen
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2012-06-14T16:00:00+00:00

6

The Modern Approach: Oblique Triangles

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 = cosacosb doesn’t resemble the Pythagorean Theorem c2 = 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 by breaking the oblique triangle into two right triangles). We shall do the same on the sphere. Our goal in this chapter mirrors the goal of plane trigonometry on oblique triangles: to solve triangles, that is, given values for certain sides and angles, to find values for the other sides and angles. We begin with a brief exploration of the fundamental theorem of planar oblique triangles, and its extension to the sphere.

Most students encounter two important theorems about planar oblique triangles: the Law of Sines, which we saw in chapter 4 in both its planar and spherical incarnations; and its more powerful sibling the Law of Cosines, which we shall find profitable to revisit for a few moments:

c2 = a2 + b2 – 2abcosC.

Written in this way, we see that this statement is an extension of the Pythagorean Theorem applied to oblique triangles. Before extending the Law of Cosines to the sphere we should understand why the planar version is true; and since the beginning, this connection to Pythagoras has been the proof’s starting point.

But when was the beginning for the Law of Cosines? I’ve been asked this question before, and it sounds like the answer should be a simple fill-in-the-blank. The answer, however, turns out to be anything but straightforward. As a historian of mathematics, my first instinct in answering many questions is to turn to Euclid. As a compendium of much of the mathematics up to its composition in the 3rd century BC, the Elements is an amazingly rich source of answers to historical questions, even (paradoxically) for subjects that came along later like trigonometry. This time, Euclid again comes through.

The Pythagorean Theorem (Proposition 47) and its converse (Proposition 48) are the climax of the Elements’ opening book. The much shorter Book II is also the most controversial. Its theorems, which appear to be statements about squares and rectangles, may be translated directly into various algebraic statements, such as (a + b)2 = a2 + 2ab + b2. For this reason, some of Euclid’s readers have referred to Book II as “geometric algebra.” Historians of mathematics bridle at this interpretation; it imposes a modern layer of understanding on the book that the ancient Greeks could not possibly have intended. If you want to treat the Elements as a textbook of modern mathematics, then “geometric algebra” is fine. But if you want to treat it as a historical record, thinking of Book II as modern algebra is a serious distortion.

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