Trigonometric Delights

Trigonometry has always been an underappreciated branch of mathematics. It has a reputation as a dry and difficult subject, a glorified form of geometry complicated by tedious computation. In this book, Eli Maor draws on his remarkable talents as a guide to the world of numbers to dispel that view. Rejecting the usual arid descriptions of sine, cosine, and their trigonometric relatives, he brings the subject to life in a compelling blend of history, biography, and mathematics. He presents both a survey of the main elements of trigonometry and a unique account of its vital contribution to science and social development. Woven together in a tapestry of entertaining stories, scientific curiosities, and educational insights, the book more than lives up to the title __Trigonometric Delights.__ ? Maor, whose previous books have demystified the concept of infinity and the unusual number "e," begins by examining the "proto-trigonometry" of the Egyptian pyramid builders. He shows how Greek astronomers developed the first true trigonometry. He traces the slow emergence of modern, analytical trigonometry, recounting its colorful origins in Renaissance Europe's quest for more accurate artillery, more precise clocks, and more pleasing musical instruments. Along the way, we see trigonometry at work in, for example, the struggle of the famous mapmaker Gerardus Mercator to represent the curved earth on a flat sheet of paper; we see how M. C. Escher used geometric progressions in his art; and we learn how the toy Spirograph uses epicycles and hypocycles. Maor also sketches the lives of some of the intriguing figures who have shaped four thousand years of trigonometric history. We meet, for instance, the Renaissance scholar Regiomontanus, who is rumored to have been poisoned for insulting a colleague, and Maria Agnesi, an eighteenth-century Italian genius who gave up mathematics to work with the poor--but not before she investigated a special curve that, due to mistranslation, bears the unfortunate name "the witch of Agnesi." The book is richly illustrated, including rare prints from the author's own collection. __Trigonometric Delights__ will change forever our view of a once dreaded subject. Trigonometry has always been the black sheep of mathematics. Too advanced to be part of "elementary math," yet too elementary for the higher branches of the profession, it has been looked upon as a glorified form of geometry, complicated by tedious computation. Nothing could be further from the truth. Uniquely positioned as a meeting point between pure and applied mathematics, its rich history shows how different branches of science - among them geography, astronomy, physics, and even music - have influenced one another. In this book, Eli Maor rejects the usual arid descriptions of the sine and cosine functions and their trigonometric relatives. He brings the subject to life in a compelling blend of mathematics, history, and biography. From the "proto-trigonometry" of the Egyptian pyramid builders to Renaissance Europe's quest for more accurate artillery; from the earliest known trigonometric table, carved on a clay tablet by an unknown Babylonian scholar, to Fourier's famous theorem, which finally explained the source of musical harmony, here is a rich tapestry of almost four thousand years of trigonometric history. Contents Preface Prologue. Ahmes the Scribe, 1650 B.C. Recreational Mathematics in Ancient Egypt 1. Angles 2. Chords Plimpton 322. The Earliest Trigonometric Table? 3. Six Functions Come of Age Johann Miille, alias Regiomontanus 4. Trigonometry Becomes Analytic François Vieté 5. Measuring Heaven and Earth Abraham De Moivre 6. Two Theorems from Geometry 7. Epicycloids and Hypocycloids Maria Agnesi and Her "Witch" 8. Variations on a Theme by Gauss 9. Had Zeno Only Known This! 10. (sin x)/x 11. A Remarkable Formula Jules Lissajous and His Figures 12. tan x 13. A Mapmaker's Paradise 14. sin x = 2: Imaginary Trigonometry Edmund Landau: The Master Rigorist 15. Fourier's Theorem Appendixes 1. Let's Revive an Old Idea 2. Barrow's Integration of sec φ 3. Some Trigonometric Gems 4. Some Special Values of sin α Bibliography Credits for Illustrations Index In 1858 a Scottish lawyer and antiquarian, A. Henry Rhind (1833-1863), on one of his trips to the Nile valley, purchased a document that had been found a few years earlier in the ruins of a small building in Thebes (near present-day Luxor) in Upper Egypt.