History of Mathematics, A: An Introduction

by Katz, Victor J.

• ISBN13: 9780321016188
• ISBN10: 0321016181
• Edition: 2nd
• Format: Paperback
• Copyright: 1998-01-01
• Publisher: Addison Wesley

Summary

One of the leading historians in the mathematics field, Victor Katz provides a world view of mathematics, balancing ancient, early modern, and modern history.

Table of Contents

Preface	ix
PART ONE Mathematics Before the Sixth Century	1	(191)
CHAPTER 1 Ancient Mathematics	1	(45)
1.1 Ancient Civilizations	2	(2)
1.2 Counting	4	(4)
1.3 Arithmetic Computations	8	(6)
1.4 Linear Equations	14	(5)
1.5 Elementary Geometry	19	(6)
1.6 Astronomical Calculations	25	(2)
1.7 Square Roots	27	(3)
1.8 The Pythagorean Theorem	30	(5)
1.9 Quadratic Equations	35	(11)
CHAPTER 2 The Beginnings of Mathematics in Greece	46	(56)
2.1 The Earliest Greek Mathematics	47	(5)
2.2 The Time of Plato	52	(2)
2.3 Aristotle	54	(4)
2.4 Euclid and the Elements	58	(37)
2.5 Euclid's Other Works	95	(7)
CHAPTER 3 Archimedes and Apollonius	102	(33)
3.1 Archimedes and Physics	103	(5)
3.2 Archimedes and Numerical Calculations	108	(3)
3.3 Archimedes and Geometry	111	(5)
3.4 Conics Before Apollonius	116	(1)
3.5 The Conics of Apollonius	117	(18)
CHAPTER 4 Mathematical Methods in Hellenistic Times	135	(33)
4.1 Astronomy Before Ptolemy	136	(9)
4.2 Ptolemy and the Almagest	145	(11)
4.3 Practical Mathematics	156	(12)
CHAPTER 5 The Final Chapters of Greek Mathematics	168	(24)
5.1 Nicomachus and Elementary Number Theory	171	(2)
5.2 Diophantus and Greek Algebra	173	(10)
5.3 Pappus and Analysis	183	(9)
PART TWO Medieval Mathematics: 500-1400	192	(150)
CHAPTER 6 Medieval China and India	192	(46)
6.1 Introduction to Medieval Chinese Mathematics	192	(1)
6.2 The Mathematics of Surveying and Astronomy	193	(4)
6.3 Indeterminate Analysis	197	(5)
6.4 Solving Equations	202	(8)
6.5 Introduction to the Mathematics of Medieval India	210	(2)
6.6 Indian Trigonometry	212	(6)
6.7 Indian Indeterminate Analysis	218	(7)
6.8 Algebra and Combinatorics	225	(5)
6.9 The Hindu-Arabic Decimal Place-Value System	230	(8)
CHAPTER 7 The Mathematics of Islam	238	(50)
7.1 Decimal Arithmetic	240	(3)
7.2 Algebra	243	(20)
7.3 Combinatorics	263	(5)
7.4 Geometry	268	(6)
7.5 Trigonometry	274	(14)
CHAPTER 8 Mathematics in Medieval Europe	288	(54)
8.1 Geometry and Trigonometry	292	(8)
8.2 Combinatorics	300	(7)
8.3 Medieval Algebra	307	(7)
8.4 The Mathematics of Kinematics	314	(13)
INTERCHAPTER Mathematics Around the World	327	(15)
1.1 Mathematics at the Turn of the Fourteenth Century	327	(5)
1.2 Mathematics in America, Africa, and the Pacific	332	(10)
PART THREE Early Modern Mathematics: 1400-1700	342	(202)
CHAPTER 9 Algebra in the Renaissance	342	(43)
9.1 The Italian Abacists	343	(5)
9.2 Algebra in France, Germany, England, and Portugal	348	(10)
9.3 The Solution of the Cubic Equation	358	(9)
9.4 The Work of Viete and Stevin	367	(18)
CHAPTER 10 Mathematical Methods in the Renaissance	385	(46)
10.1 Perspective	389	(4)
10.2 Geography and Navigation	393	(5)
10.3 Astronomy and Trigonometry	398	(18)
10.4 Logarithms	416	(4)
10.5 Kinematics	420	(11)
CHAPTER 11 Geometry, Algebra, and Probability in the Seventeenth Century	431	(37)
11.1 Analytic Geometry	432	(13)
11.2 The Theory of Equations	445	(3)
11.3 Elementary Probability	448	(10)
11.4 Number Theory	458	(2)
11.5 Projective Geometry	460	(8)
CHAPTER 12 The Beginnings of Calculus	468	(76)
12.1 Tangents and Extrema	469	(6)
12.2 Areas and Volumes	475	(17)
12.3 Power Series	492	(4)
12.4 Rectification of Curves and the Fundamental Theorem	496	(7)
12.5 Isaac Newton	503	(19)
12.6 Gottfried Wilhelm Leibniz	522	(10)
12.7 First Calculus Texts	532	(12)
PART FOUR Modern Mathematics: 1700-2000	544	(313)
CHAPTER 13 Analysis in the Eighteenth Century	544	(52)
13.1 Differential Equations	545	(15)
13.2 Calculus Texts	560	(14)
13.3 Multiple Integration	574	(4)
13.4 Partial Differential Equations: The Wave Equation	578	(4)
13.5 The Foundations of Calculus	582	(14)
CHAPTER 14 Probability, Algebra, and Geometry in the Eighteenth Century	596	(54)
14.1 Probability	597	(13)
14.2 Algebra and Number Theory	610	(11)
14.3 Geometry	621	(16)
14.4 The French Revolution and Mathematics Education	637	(3)
14.5 Mathematics in the Americas	640	(10)
CHAPTER 15 Algebra in the Nineteenth Century	650	(54)
15.1 Number Theory	652	(10)
15.2 Solving Algebraic Equations	662	(8)
15.3 Groups and Fields-The Beginning of Structure	670	(7)
15.4 Symbolic Algebra	677	(10)
15.5 Matrices and Systems of Linear Equations	687	(17)
CHAPTER 16 Analysis in the Nineteenth Century	704	(62)
16.1 Rigor in Analysis	706	(23)
16.2 The Arithmetization of Analysis	729	(8)
16.3 Complex Analysis	737	(9)
16.4 Vector Analysis	746	(7)
16.5 Probability and Statistics	753	(13)
CHAPTER 17 Geometry in the Nineteenth Century	766	(39)
17.1 Differential Geometry	768	(4)
17.2 Non-Euclidean Geometry	772	(13)
17.3 Projective Geometry	785	(7)
17.4 Geometry in N Dimensions	792	(5)
17.5 The Foundations of Geometry	797	(8)
CHAPTER 18 Aspects of the Twentieth Century	805	(52)
18.1 Set Theory: Problems and Paradoxes	807	(7)
18.2 Topology	814	(8)
18.3 New Ideas in Algebra	822	(12)
18.4 Computers and Applications	834	(23)
ANSWERS TO SELECTED PROBLEMS	857	(6)
GENERAL REFERENCES IN THE HISTORY OF MATHEMATICS	863
INDEX AND PRONUNCIATION GUIDE	I-1