Stereographic Projection

This chapter deals with stereographic projection, which is superior to other projections of the sphere because of its angle-preserving and circle-preserving properties; the first property gave instrument makers a huge advantage and the second provides clear astronomical advantages. The earliest text on stereographic projection is Ptolemy's Planisphere, in which he explains how to use stereographic projection to solve problems involving rising times, suggesting that the astrolabe may have existed already. After providing an overview of the astrolabe, an instrument for solving astronomical problems, the chapter considers how stereographic projection is used in solving triangles. It then describes the Cesàro method, named after Giuseppe Cesàro, that uses stereographic projection to project an arbitrary triangle ABC onto a plane. It also examines B. M. Brown's complaint against Cesàro's approach to spherical trigonometry.

Areas, Angles, and Polyhedra

This chapter explains how to find the area of an angle or polyhedron. It begins with a discussion of how to determine the area of a spherical triangle or polygon. The formula for the area of a spherical triangle is named after Albert Girard, a French mathematician who developed a theorem on the areas of spherical triangles, found in his Invention nouvelle. The chapter goes on to consider Euler's polyhedral formula, named after the eighteenth-century mathematician Leonhard Euler, and the geometry of a regular polyhedron. Finally, it describes an approach to finding the proportion of the volume of the unit sphere that the various regular polyhedra occupy.

The Ancient Approach

This chapter discusses the ancient approach to trigonometry, beginning with Hipparchus of Rhodes, the founder of trigonometry. It reconstructs when and where Hipparchus must have lived by taking into account the observations that he made as an astronomer and the references his successors made to him. It then considers the theorems of Menelaus of Alexandria, whose book Sphaerica completely reinvented the mathematical study of the sphere. In particular, it describes Menelaus's Theorem, which became the standard tool of spherical astronomy for the next 900 years. It also examines Abū Sahl al-Kūhī's use of the Menelaus theorems to solve the problem of rising times of arcs of the ecliptic.

Heavenly Mathematics

This chapter reviews topics in plane trigonometry, first by considering ancient ideas that the Earth is round, focusing on tales of Christopher Columbus trying to convince the Spanish court that the Earth is not a disc but a sphere, making it possible to sail westward from Portugal to India. It then discusses the dimensions of the Earth and computes a table of sines. In particular, it looks at the trigonometric table of Claudius Ptolemy, who included a remarkable collection of models for the motions of the heavenly bodies in his astronomical masterpiece, Mathematical Collection. In early Europe, the most prodigious set of trigonometric tables was the Opus palatinum, composed by Georg Rheticus. The chapter concludes with calculations to find the distance to the Moon using only simple measurements.

Navigating by the Stars

This chapter explains how the star is used to find one's position on the Earth while in a ship at sea. Trigonometry was first used for navigation by fourteenth-century Venetian merchant ships. Several navigational techniques can be identified from navigators' personal notebooks, including the table of marteloio. Essentially an application of plane trigonometry, marteloio was part of a group of methods known today as “dead” reckoning. Between 1730 and 1759, a clockmaker by the name of John Harrison constructed a series of four chronometers that could keep remarkably accurate time, even on a ship tossed by waves. The chapter considers the use of the method of Saint Hilaire (also called intercept, cosine-haversine, or Davis's method) to determine three quantities of a star in an astronomical triangle: latitude, declination, and local hour angle. It also discusses the use of the Law of Cosines to solve the star's altitude.

The Modern Approach: Right-Angled Triangles

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.

Exploring the Sphere

This chapter introduces the reader to the celestial sphere, or the Earth's surface. By rotating the sphere, the motions of the heavens can be simulated. There are three features of celestial motion that came to be associated with Aristotle: all objects move in circles; they travel at constant speeds on those circles; the Earth is at the center of the celestial sphere. The chapter shows how the movements of stars and planets on the sphere's surface can be determined by setting up a system of equatorial coordinates. It also explains how the celestial sphere can be set in motion through the day, and how Hipparchus of Rhodes endeavored to determine the eccentricity of the Sun's orbit. Finally, it discusses spherical geometry, with emphasis on finding bounds on the sides and angles of a spherical triangle.

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.

The Medieval Approach

This chapter discusses the medieval approach to trigonometry, beginning with Abū Mahmūd al-Khujandī's theorem and the theorems of Abū Nasr Mansūr ibn ʻAlī ibn ʻIrāq. It then considers the Rule of Four Quantities, the first example of the principle of locality, and its relation to the spherical Law of Sines. It also analyzes Abū 'l-Wafā's proof of the Law of Sines in his Almagest and how the law becomes the planar Law of Sines when reduced to the plane. Finally, it provides an overview of the Indian approach to spherical astronomy, including finding declinations of arcs of the ecliptic, and how to solve the qibla problem to determine the direction of Mecca.