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This chapter discusses the modern approach to solving right-angled triangles. After a brief background on John Napier's trigonometric work, in which he referred mostly to right-angled spherical triangles, the chapter describes the theorems for right triangles. It then considers an oblique triangle split into two right triangles and the ten fundamental identities of a right-angled spherical triangle, how the locality principle can be applied to derive the Pythagorean Theorem, and how to find a ship's direction of travel using the theorem. It also looks at Napier's work on logarithms which was devoted to trigonometry, along with Napier's Rules. The chapter concludes with an overview of “pentagramma mirificum,” a pentagram in spherical trigonometry that was discovered by Napier.

*Keywords: *
right-angled triangle;
John Napier;
spherical triangle;
oblique triangle;
locality principle;
Pythagorean Theorem;
logarithms;
Napier's Rules;
pentagramma mirificum;
spherical trigonometry

*Chapter.*
*5941 words.*
*Illustrated.*

*Subjects: *
History of Mathematics

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