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E-raamat: Mathematics of the Heavens and the Earth: The Early History of Trigonometry

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  • Ilmumisaeg: 10-Aug-2021
  • Kirjastus: Princeton University Press
  • Keel: eng
  • ISBN-13: 9781400833313
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  • Formaat: PDF+DRM
  • Ilmumisaeg: 10-Aug-2021
  • Kirjastus: Princeton University Press
  • Keel: eng
  • ISBN-13: 9781400833313
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The Mathematics of the Heavens and the Earth is the first major history in English of the origins and early development of trigonometry. Glen Van Brummelen identifies the earliest known trigonometric precursors in ancient Egypt, Babylon, and Greece, and he examines the revolutionary discoveries of Hipparchus, the Greek astronomer believed to have been the first to make systematic use of trigonometry in the second century BC while studying the motions of the stars. The book traces trigonometry's development into a full-fledged mathematical discipline in India and Islam; explores its applications to such areas as geography and seafaring navigation in the European Middle Ages and Renaissance; and shows how trigonometry retained its ancient roots at the same time that it became an important part of the foundation of modern mathematics.

The Mathematics of the Heavens and the Earth looks at the controversies as well, including disputes over whether Hipparchus was indeed the father of trigonometry, whether Indian trigonometry is original or derived from the Greeks, and the extent to which Western science is indebted to Islamic trigonometry and astronomy. The book also features extended excerpts of translations of original texts, and detailed yet accessible explanations of the mathematics in them.

No other book on trigonometry offers the historical breadth, analytical depth, and coverage of non-Western mathematics that readers will find in The Mathematics of the Heavens and the Earth.

Arvustused

"Fans of the history of mathematics will be richly rewarded by this exhaustively researched book, which focuses on the early development of trigonometry... Finally, the generous and lucid explanations provided throughout the text make Van Brummelen's history a rewarding one for the mathematical tourist."--Mathematics Teacher "[ T]his new and comprehensive history of trigonometry is more than welcome--even more so because it is the first in English... [ T]his book will be appreciated by many with an interest--general or more specific--in the history of mathematics."--Steven Wepster, Centaurus "[ T]his book will have wide appeal, for students, researchers, and teachers of history and/or trigonometry. The excerpts selected are balanced and their significances well articulated... It is a book written by an expert after many years of exposure to individual sources and in this way Van Brummelen uniquely advances the field. The book will no doubt become a necessary addition to the libraries of mathematicians and historians alike."--Clemency Montelle and Kathleen M. Clark, Aestimatio "Van Brummelen's history does far more than simply fill a vacant spot in the historical literature of mathematics. He recounts the history of trigonometry in a way that is both captivating and yet more than satisfying to the crankiest and most demanding of scholars... The Mathematics of the Heavens and the Earth should be a part of every university library's mathematics collection. It's also a book that most mathematicians with an interest in the history of the subject will want to own."--Rob Bradley, MAA Reviews "I highly recommend the book to all those interested in the way in which the ancient people solve their practical problems and hope that the next volume of this interesting history of spherical and plane trigonometry will appear soon."--Cristina Blaga, Studia Mathematica

Muu info

There does not seem to have been a book-length history of trigonometry in English before this fine book. Van Brummelen takes us from the unnamed Egyptians and Babylonians who created trigonometry to the subject's first few centuries in Europe. In between, he deftly traces how it was studied by the astronomers Hipparchus and Ptolemy in classical Greece, and later by a host of scholars in India and the Islamic world. -- John H. Conway, coauthor of "The Book of Numbers" This book is the first detailed history of trigonometry in more than half a century, and it far surpasses any earlier attempts. The Mathematics of the Heavens and the Earth is an extremely important contribution to scholarship. It will be the definitive history of trigonometry for years to come. There is nothing like this out there. -- Victor J. Katz, professor emeritus, University of the District of Columbia A pleasure to read. The Mathematics of the Heavens and the Earth is destined to become the standard reference on the history of trigonometry for the foreseeable future. Although other authors have attempted to tell the story, I know of no other book that has either the breadth or the depth of this one. Van Brummelen is one of the leading experts in the world on this subject. -- Dennis Duke, Florida State University Van Brummelen presents a history of trigonometry from the earliest times to the end of the sixteenth century. He has produced a work that rises to the highest standards of scholarship but never strays into pedantry. His extensive bibliography cites every work of consequence for the history of trigonometry, copious footnotes and diagrams illuminate the text, and reproductions from old printed works add interest and texture to the narrative. -- J. Lennart Berggren, professor emeritus, Simon Fraser University This book presents, for the first time in more than a century, a concise history of plane and spherical trigonometry, an important field within applied mathematics. It will appeal to a wide audience thanks to the pleasant style in which it is written, but at the same time it adheres to a very high scholarly standard. -- Benno van Dalen, Ludwig Maximilians University, Munich
Preface xi
The Ancient Heavens 1(8)
Precursors
9(24)
What Is Trigonometry?
9(1)
The Seqed in Ancient Egypt
10(2)
Text 1.1 Finding the Slope of a Pyramid
11(1)
Babylonian Astronomy, Arc Measurement, and the 360° Circle
12(6)
The Geometric Heavens: Spherics in Ancient Greece
18(2)
A Trigonometry of Small Angles? Aristarchus and Archimedes on Astronomical Dimensions
20(13)
Text 1.2 Aristarchus, the Ratio of the Distances of the Sun and Moon
24(9)
Alexandrian Greece
33(61)
Convergence
33(1)
Hipparchus
34(3)
A Model for the Motion of the Sun
37(4)
Text 2.1 Deriving the Eccentricity of the Sun's Orbit
39(2)
Hipparchus's Chord Table
41(5)
The Emergence of Spherical Trigonometry
46(3)
Theodosius of Bithynia
49(4)
Menelaus of Alexandria
53(3)
The Foundations of Spherical Trigonometry: Book III of Menelaus's Spherics
56(7)
Text 2.2 Menelaus, Demonstrating Menelaus's Theorem
57(6)
Spherical Trigonometry before Menelaus?
63(5)
Claudius Ptolemy
68(2)
Ptolemy's Chord Table
70(4)
Ptolemy's Theorem and the Chord Subtraction/Addition Formulas
74(2)
The Chord of 1°
76(1)
The Interpolation Table
77(1)
Chords in Geography: Gnomon Shadow Length Tables
77(3)
Text 2.3 Ptolemy, Finding Gnomon Shadow Lengths
78(2)
Spherical Astronomy in the Almagest
80(2)
Ptolemy on the Motion of the Sun
82(4)
Text 2.4 Ptolemy, Determining the Solar Equation
84(2)
The Motions of the Planets
86(2)
Tabulating Astronomical Functions and the Science of Logistics
88(2)
Trigonometry in Ptolemy's Other Works
90(3)
Text 2.5 Ptolemy, Constructing Latitude Arcs on a Map
91(2)
After Ptolemy
93(1)
India
94(41)
Transmission from Babylon and Greece
94(1)
The First Sine Tables
95(4)
Aryabhata's Difference Method of Calculating Sines
99(3)
Text 3.1 Aryabhata, Computing Sines
100(2)
Bhaskara I's Rational Approximation to the Sine
102(3)
Improving Sine Tables
105(2)
Other Trigonometric Identities
107(4)
Text 3.2 Varahamihira, a Half-angle Formula
108(1)
Text 3.3 Brahmagupta, the Law of Sines in Planetary Theory?
109(2)
Brahmagupta's Second-order Interpolation Scheme for Approximating Sines
111(2)
Text 3.4 Brahmagupta, Interpolating Sines
111(2)
Taylor Series for Trigonometric Functions in Madhava's Kerala School
113(8)
Applying Sines and Cosines to Planetary Equations
121(3)
Spherical Astronomy
124(5)
Text 3.5 Varahamihira, Finding the Right Ascension of a Point on the Ecliptic
125(4)
Using Iterative Schemes to Solve Astronomical Problems
129(4)
Text 3.6 Paramesvara, Using Fixed-point Iteration to Compute Sines
131(2)
Conclusion
133(2)
Islam
135(88)
Foreign Junkets: The Arrival of Astronomy from India
135(2)
Basic Plane Trigonometry
137(3)
Building a Better Sine Table
140(9)
Text 4.1 Al-Samaw'al ibn Yahya al-Maghribi, Why the Circle Should Have 480 Degrees
146(3)
Introducing the Tangent and Other Trigonometric Functions
149(7)
Text 4.2 Abu'l-Rayhan al-Biruni, Finding the Cardinal Points of the Compass
152(4)
Streamlining Astronomical Calculation
156(2)
Text 4.3 Kushyar ibn Labban, Finding the Solar Equation
156(2)
Numerical Techniques: Approximation, Iteration, Interpolation
158(8)
Text 4.4 Ibn Yunus, Interpolating Sine Values
164(2)
Early Spherical Astronomy: Graphical Methods and Analemmas
166(7)
Text 4.5 Al-Khwarizmi, Determining the Ortive Amplitude Geometrically
168(5)
Menelaus in Islam
173(6)
Text 4.6 Al-Kuhi, Finding Rising Times Using the Transversal Theorem
175(4)
Menelaus's Replacements
179(7)
Systematizing Spherical Trigonometry: Ibn Mucadh's Determination of the Magnitudes and Nasir al-Din al-Tusi's Transversal Figure
186(6)
Applications to Religious Practice: The Qibla and Other Ritual Needs
192(9)
Text 4.7 Al-Battani, a Simple Approximation to the Qibla
195(6)
Astronomical Timekeeping: Approximating the Time of Day Using the Height of the Sun
201(4)
New Functions from Old: Auxiliary Tables
205(4)
Text 4.8 Al-Khalili, Using Auxiliary Tables to Find the Hour-angle
207(2)
Trigonometric and Astronomical Instruments
209(6)
Text 4.9 Al-Sijzi (?), On an Application of the Sine Quadrant
213(2)
Trigonometry in Geography
215(2)
Trigonometry in al-Andalus
217(6)
The West to 1550
223(61)
Transmission from the Arab World
223(1)
An Example of Transmission: Practical Geometry
224(6)
Text 5.1 Hugh of St. Victor, Using an Astrolabe to Find the Height of an Object
225(2)
Text 5.2 Finding the Time of Day from the Altitude of the Sun
227(3)
Consolidation and the Beginnings of Innovation: The Trigonometry of Levi ben Gerson, Richard of Wallingford, and John of Murs
230(12)
Text 5.3 Levi ben Gerson, The Best Step Size for a Sine Table
233(4)
Text 5.4 Richard of Wallingford, Finding Sin(1°) with Arbitrary Accuracy
237(5)
Interlude: The Marteloio in Navigation
242(5)
Text 5.5 Michael of Rhodes, a Navigational Problem from His Manual
244(3)
From Ptolemy to Triangles: John of Gmunden, Peurbach, Regiomontanus
247(17)
Text 5.6 Regiomontanus, Finding the Side of a Rectangle from Its Area and Another Side
254(1)
Text 5.7 Regiomontanus, the Angle-angle-angle Case of Solving Right Triangles
255(9)
Successors to Regiomontanus: Werner and Copernicus
264(9)
Text 5.8 Copernicus, the Angle-angle-angle Case of Solving Triangles
267(3)
Text 5.9 Copernicus, Determining the Solar Eccentricity
270(3)
Breaking the Circle: Rheticus, Otho, Pitiscus and the Opus Palatinum
273(11)
Concluding Remarks 284(3)
Bibliography 287(36)
Index 323
Glen Van Brummelen is professor of mathematics at Quest University Canada. He is the coeditor of "Mathematics and the Historian's Craft" and the coauthor of "Calculus Explorations Powered by Technology: Tales of History and Imagination".