Glen Van Brummelen

Quest University Canada

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.

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 n...

In my 2018 article in this journal, I described 15th-century Italian astronomer Giovanni Bianchini's treatment of the problem of stellar coordinate conversion in his Tabulae primi mobilis, the first correct European solution. In this treatise Bianchini refers to a book he had written previously, containing the same error that had plagued his predec...

‘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 function...

‘ … 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 betwe...

‘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...

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...

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...

‘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...

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...

‘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...

Giovanni Bianchini’s fifteenth-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 anteceden...

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 t...

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 cent...

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 fundamen...

Len Berggren’s two surveys of the mathematical sciences in medieval Islam, the two preceding articles in this volume, synthesized a vast body of literature. Their scope extended well beyond mathematics itself to include overlapping disciplines such as optics, geography and astronomy. In addition, the surveys recognized questions in the literature r...

The practice of the history of mathematics is in flux. This statement may seem ironic or even paradoxical, for a discipline that relies seemingly on logic and precision. However, trends in the scholarly practice of history are gradually causing substantial changes in the questions raised by practitioners of the discipline, and the methods used to t...

The widely practiced Muslim ritual of facing Mecca to pray turns out to involve the less widely practiced tools of spherical trigonometry.

Spherical trigonometry was at the heart of astronomy and ocean-going navigation for two millennia. The discipline was a mainstay of mathematics education for centuries, and it was a standard subject in high schools until the 1950s. Today, however, it is rarely taught.Heavenly Mathematicstraces the rich history of this forgotten art, revealing how t...

Hippocrates of Chios, one of the earliest authors in the written tradition of Greek mathematics, was active in the second half of the fifth century bce.

Introduction Does anyone care about trigonometry? Certainly many of our students don't, aside from the exigency of getting through their exams. As mathematics teachers, we have passion for our subject for its own sake — but we often justify ourselves to our students in terms of what the mathematics can accomplish elsewhere. For trigonometry as for...

As any high school math teacher will tell you, the word "trigonometry" means "triangle measurement." The more perspicacious teacher might even know that the word was first coined by Bartholomew Pitiscus with his Trigonometriae [16], a study of the so-called "science of triangles" (Figure 24.1). This sounds familiar, even comfortable to modern teach...

Writing a history of early trigonometry brought me to confront in a practical way some difficult historiographic questions. What does it mean to know a theorem? How does one determine what belongs to trigonometry, and doesn't? To what extent can one legitimately talk about knowledge crossing cultural boundaries intact? Although these questions do n...

Without Abstract

In terms of complexity, planetary latitudes are the culmination of Ptolemy's mathematical astronomy. Al-Kashi's remarkable system removes its mathematical flaws, and demonstrates that Muslim astronomers not only mastered this apex of Ptolemaic astronomy, but also perfected its mathematics. The remainder of this paper is devoted first to a brief des...

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 dra...

Kenneth Ownsworth May (1915-1977) was a brilliant and influential mathematician, historian, and educator who founded the journal Historia Mathematica as well as the Canadian Society for the History and Philosophy of Mathematics. He viewed the practice of the history of mathematics as a unique melding of the crafts of mathematician and historian. Th...

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

The integration of history is not confined to traditional teaching delivery methods, but can often be better achieved through a variety of media which add to the resources available for learner and teacher.

Kūshyār ibn Labbān, an Iranian scientist who flourishedca.A.D. 1000, composed an astronomical handbook entitled theJāmicZı̄j. It has been considered to be derivative of al-Battānı̄'sZı̄j al-S·ābı̄(ca.A.D. 900), and through it, back to Ptolemy'sAlmagest(ca.A.D. 150). We analyze the tables of planetary motion and consider the possible dependences. We...

The authorship of portions of some ancient and medieval astronomical manuscripts is a current historical issue. This article develops a statistical method for determining whether or not a link exists between a pair of mathematically defined historical astronomical tables. The existence or nonexistence of such links aids the analysis of the origins...