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In 1551, Georg Rheticus published a compact set of tables that effectively completed the set of all six trigonometric functions. However, his work was not widespread, and may not have been known to Francesco Maurolico when he published a secant table in 1558. Before the end of the century, several authors argued whether Maurolico had borrowed the notion of the secant, and his table, from Rheticus. We present Maurolico’s text on his table (named the tabula benefica in a nod to Regiomontanus’s tangent table, the tabula foecunda), as well as a translation and analysis. Finally, we demonstrate that Maurolico’s table is too accurate to have derived from Rheticus’s work, absolving him of the historical accusation.

We do not study the history of science to determine winners and losers in the game of truth. Rather, we wish to understand what it meant for humans to explore their world and what science was to them.

International audience

‘Spheres and more’ considers the ten formulas for right-angled spherical triangles (and how they can be generated), the spherical Pythagorean theorem, and Napier’s rules. Spherical trigonometry was intended originally for astronomers, but medieval Islamic scholars used it to predict the beginning of the sacred month of Ramadan and the times of the five daily prayers. The impact of spherical geometry on Euclid’s axioms resulting in two types of non-Euclidean geometries—elliptical geometry and hyperbolic geometry—is also considered.

Technological advances, so pervasive in almost every aspect of our modern lives, become mundane to us almost overnight. How does a calculator find out, apparently effortlessly, that sin 33° = 0.5446? There are no right triangles drawn inside of the calculator, so where did that number come from? ‘Building a sine table with your bare hands’ looks at how different sine values have been calculated through history without any mechanical aids. It aintroduces the sine sum and difference laws, the half- and double-angle formulas, the golden triangle, and one of the most remarkable numbers in all of mathematics, the golden ratio, which appears in a dazzling array of areas of mathematics, natural phenomena, works of art, and architecture.

‘ … and beyond, to complex things’ first considers the Taylor series for the exponential function. One of the most famous, yet enigmatic, numbers in mathematics, e is an irrational number equal to 2.718281828. … Exponential functions deal with the phenomena of growth and decay. As calculus was starting to become established, curious parallels between the apparently disparate worlds of trigonometry and exponential functions were starting to appear. Imaginary numbers, Euler’s formula, and Euler’s identity are discussed along with the Argand diagram, De Moivre’s formula, hyperbolic trigonometric functions, and the catenary curve. Imaginary numbers are now at the heart of science and technology, and are used in the study of electromagnetic waves, cellular and wireless technologies, and fluid dynamics.

‘To infinity … ’ looks at how infinite trigonometric series are used to compute π. It shows how Machin’s formula used the inverse tangent series to compute π to a hundred places. Lord Kelvin’s use of Fourier analysis in studying tide behaviour is also explained along with the Gibbs phenomenon. The invention of Cartesian coordinates and calculus in the 17th and 18th centuries reflects a major shift in mathematics. Geometry gradually changed from being synthetic (in the style of Euclid) to being analytic. The basic mathematical objects―originally points, lines, and shapes as well as numbers―became functions that accept input quantities and produce output quantities.

‘Sines, cosines, and their relatives’ begins by defining the basic trigonometric functions—sine, cosine, and tangent—and explaining their use. These functions are geometric quantities defined using the ratios of the Opposite, Adjacent, and Hypotenuse sides of the right triangle. Less common functions are the cosecant, secant, and cotangent functions. The history of the naming of the trigonometric functions is discussed along with an explanation of even more obscure functions: the versed sine, versed cosine, exsecant, and excosecant. The versed sine was used frequently in practical applications like astronomy, navigation, and surveying. Finally, inverse trigonometric functions and graphs of trigonometric functions are considered.

Trigonometry: A Very Short Introduction draws together the full history of trigonometry, stretching across two millennia and several cultures such as ancient Greece, medieval India, and the Islamic world. It introduces the key concepts of trigonometry, drawing readers beyond the basic relationships first encountered in school to reveal the richness of the entire subject of trigonometry and ideas such as curved space. It also explores connections with genuine modern applications, including navigation, the analysis of music, computer graphics, and powerful modelling tools in science, and shows how trigonometry has participated in big questions about the world, including the shape of the universe and the nature of infinity.

‘Why?’ considers some of the mathematical problems faced by scientists in the past: Hipparchus of Rhodes trying to predict the times of eclipses; Maurice Bressieu, the 16th-century French mathematician and humanist, calculating the height of a tower; and Lord Kelvin trying to predict ocean tide behaviour. Each of these scientists was faced with the same difficulty: the mathematical tools they had at their disposal were not up to the task of quantifying the phenomenon they were studying. This is the problem at the heart of trigonometry: how can we bring together geometry and computation to solve real physical problems? Today, examples of bridges between geometry and measurement are everywhere.

The world of trigonometry is full of identities: some of them extremely useful, others beautiful, and a few that are simply bizarre. ‘Identities, and more identities’ takes a tour of the menagerie of identities, viewing a little from each of these categories. The first two examples are known as triangle identities, because they refer to angles and lengths in a given triangle. The Law of Sines and the Law of Cosines are discussed, along with Mollweide’s formulas, the Law of Tangents, Morrie’s Law, and the introduction of logarithms, which became the preferred computing tool in mathematical astronomy, and then in practical disciplines like surveying and architecture in the early 17th century.

Preface vii 1 Heavenly Mathematics 1 2 Exploring the Sphere 23 3 The Ancient Approach 42 4 The Medieval Approach 59 5 The Modern Approach: Right- Angled Triangles 73 6 The Modern Approach: Oblique Triangles 94 7 Areas, Angles, and Polyhedra 110 8 Stereographic Projection 129 9 Navigating by the Stars 151 Appendix A. Ptolemy's Determination of the Sun's Position 173 Appendix B. Textbooks 179 Appendix C. Further Reading 182 Index 189

International audience; Giovanni Bianchini's 15 th-century Tabulae primi mobilis is a collection of 50 pages of canons and 100 pages of tables of spherical astronomy and mathematical astrology, beginning with a treatment of the conversion of stellar coordinates from ecliptic to equatorial. His new method corrects a long-standing error made by a number of his antecedents, and with his tables the computations are much more efficient than in Ptolemy's Almagest. The completely novel structure of Bianchini's tables, here and in his Tabulae magistrales, was taken over by Regiomontanus in the latter's Tabulae directionum. One of the tables Regiomontanus imported from Bianchini contains the first appearance of the tangent function in Latin Europe, which both used as an auxiliary quantity for the calculation of stellar coordinates.

Claudius Ptolemy was not a philosopher. His reputation relies especially on his Almagest, the astronomical masterwork that dominated medieval Islam and Europe until Copernicus; and his Geography, a...

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.

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.

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.

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.

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.

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.

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.

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.

This book traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the Earth. Once at the heart of astronomy and ocean-going navigation for two millennia, the discipline was also a mainstay of mathematics education for centuries and taught widely until the 1950s. The book explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. The book conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. It is illustrated throughout with stunning historical images and informative drawings and diagrams. It also features easy-to-use appendices as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

The stars and planets move not on a flat surface but on the celestial sphere. So ordinary plane trigonometry isn’t adequate to describe their motion.

The Almagest model of the motions of the planets is one of the monumental achievements in the history of science. Its ability to predict the locations of some planets to within a couple of degrees of their true positions centuries into the future would give even modern scientists pause; it reigned as the planetary paradigm for well over a millennium. Nevertheless a number of Muslim astronomers revised or even replaced it with various alternatives, designed not to improve its predictive success, but rather to correspond better to the requirement of uniform circular motion. Although Jamshid al-Kāshi came after many of these reformers, he did not follow them. Rather than overturning Ptolemy, he revised the planetary model and applied his prodigious mathematical skills to help it conform more closely to his conception of Ptolemy’s wishes. The most characteristic feature of a planet’s motion is its regular retrograde arcs, illustrated in Fig. 1 (the motion of Mars from May to December 2003). Ptolemy’s epicyclic model of Fig. 2 does a remarkable job of predicting the planet’s longitude, its position projected onto the ecliptic.1 However, as Fig. 1 makes clear, planets exhibit considerable motions above and below the ecliptic. A complete planetary theory must also incorporate these movements; thus, it must handle the planet’s latitude. Ptolemy attempted this in Book XIII, the last book of the Almagest, by tilting the deferent and epicycle out of the plane of the ecliptic, and setting them in back and forth motions in ways that he considered best suited to reproduce the phenomena (see Fig. 3). For the inferior planets Venus and Mercury he defined three effects. (i) The deferent circle is set in a bobbing motion, starting flat on the ecliptic when the epicycle is at points A and C, and reaching its maximum inclination when the epicycle is at B and D. The epicycle is itself tilted with respect to the plane of the deferent as follows: (ii) the first diameter,2 in the line of sight as seen from Earth, has zero tilt when the center of the epicycle is

The 10th-century mathematician Abū Sahl al-Kūhī, one of the best geometers of medieval Islam, wrote several treatises on the first three books of Euclid's Elements . We present an edition and translation of al-Kūhī's revision of Book I of the Elements , in which he altered the book's focus to the theorems and rearranged the propositions. The most dramatic of the changes is the complete absence of all geometric constructions, suggesting that al-Kūhī preferred instead to use Euclid's Data , and the related procedures of analysis and synthesis, for constructions. Other novel aspects include a proof of Euclid's fourth postulate, the use of the parallel postulate early in the work, and a different proof of the Pythagorean theorem.

(2013). Seeking the Divine on Earth: The Direction of Prayer in Islam. Math Horizons: Vol. 21, No. 1, pp. 15-17.

Al-Kūhī est connu pour ses travaux concernant la determination de l'angle par rapport a l'horizon. Toutefois, Al-Samaw'al a remarque un certain nombre d'erreurs critiques par rapport au traite d'Al-Kūhī et donne des solutions geometriques a partir de ses propres calculs