*Glen Van Brummelen*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175997
- eISBN:
- 9781400844807
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175997.003.0006
- Subject:
- Mathematics, History of Mathematics

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 ...
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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.Less

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.

*Glen Van Brummelen*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175997
- eISBN:
- 9781400844807
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175997.003.0005
- Subject:
- Mathematics, History of Mathematics

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 ...
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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.Less

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.

*Chris Bleakley*

- Published in print:
- 2020
- Published Online:
- October 2020
- ISBN:
- 9780198853732
- eISBN:
- 9780191888168
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198853732.003.0001
- Subject:
- Mathematics, History of Mathematics, Logic / Computer Science / Mathematical Philosophy

Chapter 1 traces the origins of algorithms from ancient Mesopotamia to Greece in the 2th century BC. The oldest known algorithms were inscribed on clay tablets by the Babylonians more than 4,000 ...
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Chapter 1 traces the origins of algorithms from ancient Mesopotamia to Greece in the 2th century BC. The oldest known algorithms were inscribed on clay tablets by the Babylonians more than 4,000 years ago. The clay tablets document algorithms ranging from geometry to accountancy. One tablet in particular - YBC 7289 - indicates knowledge of the Pythagorean Theorem thousands of years before its supposed invention by the ancient Greeks. The Greeks made other advances in algorithms. Euclid’s algorithm determines the greatest common divisor of two numbers. The Sieve of Eratosthenes finds prime numbers. Both algorithms proved to be important stepping stones to modern cryptography - the mathematics of secret messages.Less

Chapter 1 traces the origins of algorithms from ancient Mesopotamia to Greece in the 2^{th} century BC. The oldest known algorithms were inscribed on clay tablets by the Babylonians more than 4,000 years ago. The clay tablets document algorithms ranging from geometry to accountancy. One tablet in particular - YBC 7289 - indicates knowledge of the Pythagorean Theorem thousands of years before its supposed invention by the ancient Greeks. The Greeks made other advances in algorithms. Euclid’s algorithm determines the greatest common divisor of two numbers. The Sieve of Eratosthenes finds prime numbers. Both algorithms proved to be important stepping stones to modern cryptography - the mathematics of secret messages.