The Modern Approach: Oblique Triangles

This chapter discusses the modern approach to solving oblique triangles. Two important theorems about planar oblique triangles are the spherical and planar Law of Sines and the Law of Cosines, which is an extension of the Pythagorean Theorem applied to oblique triangles. Book I of Euclid's Elements deals primarily with the Pythagorean Theorem (Proposition 47) and its converse (Proposition 48), while Book II contains theorems that may be translated directly into various algebraic statements. The chapter considers two of the last three theorems of Book II: Proposition 12, which deals with obtuse-angled triangles, and Proposition 13, which is concerned with acute-angled triangles. It also extends the Law of Cosines to the sphere and uses it to solve astronomical and geographical problems, such as finding the distance from Vancouver to Edmonton. Finally, it describes Delambre's analogies and Napier's analogies.