A History of Mathematics 3rd Edition by Victor J. Katz, ISBN-13: 978-0321387004
[PDF eBook eTextbook]
- Publisher: Pearson; 3rd edition (July 12, 2008)
- Language: English
- 992 pages
- ISBN-10: 0321387007
- ISBN-13: 978-0321387004
A History of Mathematics, Third Edition, provides students with a solid background in the history of mathematics and focuses on the most important topics for today’s elementary, high school, and college curricula. Students will gain a deeper understanding of mathematical concepts in their historical context, and future teachers will find this book a valuable resource in developing lesson plans based on the history of each topic.
This book is ideal for a junior or senior level course in the history of mathematics for mathematics majors intending to become teachers.
Table of Contents:
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . xi
PART ONE Ancient Mathematics
Chapter 1 Egypt and Mesopotamia 1
1.1 Egypt . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Mesopotamia . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 27
Exercises . . . . . . . . . . . . . . . . . . . . . . . 28
References and Notes . . . . . . . . . . . . . . . . . . 30
Chapter 2 The Beginnings of Mathematics in Greece 32
2.1 The Earliest Greek Mathematics . . . . . . . . . . . . . . 33
2.2 The Time of Plato . . . . . . . . . . . . . . . . . . . . 41
2.3 Aristotle . . . . . . . . . . . . . . . . . . . . . . . . 43
Exercises . . . . . . . . . . . . . . . . . . . . . . . 47
References and Notes . . . . . . . . . . . . . . . . . . 48
Chapter 3 Euclid 50
3.1 Introduction to the Elements . . . . . . . . . . . . . . . . 51
3.2 Book I and the Pythagorean Theorem . . . . . . . . . . . . 53
3.3 Book II and Geometric Algebra . . . . . . . . . . . . . . 60
3.4 Circles and the Pentagon Construction . . . . . . . . . . . . 66
3.5 Ratio and Proportion . . . . . . . . . . . . . . . . . . . 71
3.6 Number Theory . . . . . . . . . . . . . . . . . . . . . 77
3.7 Irrational Magnitudes . . . . . . . . . . . . . . . . . . 81
3.8 Solid Geometry and the Method of Exhaustion . . . . . . . . 83
3.9 Euclid’s Data . . . . . . . . . . . . . . . . . . . . . . 88
Exercises . . . . . . . . . . . . . . . . . . . . . . . 90
References and Notes . . . . . . . . . . . . . . . . . . 92
Chapter 4 Archimedes and Apollonius 94
4.1 Archimedes and Physics . . . . . . . . . . . . . . . . . 96
4.2 Archimedes and Numerical Calculations . . . . . . . . . . . 101
4.3 Archimedes and Geometry . . . . . . . . . . . . . . . . 103
4.4 Conic Sections before Apollonius . . . . . . . . . . . . . . 112
4.5 The Conics of Apollonius . . . . . . . . . . . . . . . . . 115
Exercises . . . . . . . . . . . . . . . . . . . . . . . 127
References and Notes . . . . . . . . . . . . . . . . . . 131
Chapter 5 Mathematical Methods in Hellenistic Times 133
5.1 Astronomy before Ptolemy . . . . . . . . . . . . . . . . 134
5.2 Ptolemy and the Almagest . . . . . . . . . . . . . . . . . 145
5.3 Practical Mathematics . . . . . . . . . . . . . . . . . . 157
Exercises . . . . . . . . . . . . . . . . . . . . . . . 168
References and Notes . . . . . . . . . . . . . . . . . . 170
Chapter 6 The Final Chapters of Greek Mathematics 172
6.1 Nicomachus and Elementary Number Theory . . . . . . . . . 173
6.2 Diophantus and Greek Algebra . . . . . . . . . . . . . . . 176
6.3 Pappus and Analysis . . . . . . . . . . . . . . . . . . . 185
6.4 Hypatia and the End of Greek Mathematics . . . . . . . . . . 189
Exercises . . . . . . . . . . . . . . . . . . . . . . . 191
References and Notes . . . . . . . . . . . . . . . . . . 192
PART TWO Medieval Mathematics
Chapter 7 Ancient and Medieval China 195
7.1 Introduction to Mathematics in China . . . . . . . . . . . . 196
7.2 Calculations . . . . . . . . . . . . . . . . . . . . . . 197
7.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . 201
7.4 Solving Equations . . . . . . . . . . . . . . . . . . . . 209
7.5 Indeterminate Analysis . . . . . . . . . . . . . . . . . . 222
7.6 Transmission To and From China . . . . . . . . . . . . . . 225
Exercises . . . . . . . . . . . . . . . . . . . . . . . 226
References and Notes . . . . . . . . . . . . . . . . . . 228
Chapter 8 Ancient and Medieval India 230
8.1 Introduction to Mathematics in India . . . . . . . . . . . . 231
8.2 Calculations . . . . . . . . . . . . . . . . . . . . . . 233
8.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . 237
8.4 Equation Solving . . . . . . . . . . . . . . . . . . . . 242
8.5 Indeterminate Analysis . . . . . . . . . . . . . . . . . . 244
8.6 Combinatorics . . . . . . . . . . . . . . . . . . . . . 250
8.7 Trigonometry . . . . . . . . . . . . . . . . . . . . . . 252
8.8 Transmission To and From India . . . . . . . . . . . . . . 259
Exercises . . . . . . . . . . . . . . . . . . . . . . . 260
References and Notes . . . . . . . . . . . . . . . . . . 263
Chapter 9 The Mathematics of Islam 265
9.1 Introduction to Mathematics in Islam . . . . . . . . . . . . 266
9.2 Decimal Arithmetic . . . . . . . . . . . . . . . . . . . 267
9.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . 271
9.4 Combinatorics . . . . . . . . . . . . . . . . . . . . . 292
9.5 Geometry . . . . . . . . . . . . . . . . . . . . . . . 296
9.6 Trigonometry . . . . . . . . . . . . . . . . . . . . . . 306
9.7 Transmission of Islamic Mathematics . . . . . . . . . . . . 317
Exercises . . . . . . . . . . . . . . . . . . . . . . . 318
References and Notes . . . . . . . . . . . . . . . . . . 321
Chapter 10 Mathematics in Medieval Europe 324
10.1 Introduction to the Mathematics of Medieval Europe . . . . . . 325
10.2 Geometry and Trigonometry . . . . . . . . . . . . . . . . 328
10.3 Combinatorics . . . . . . . . . . . . . . . . . . . . . 337
10.4 Medieval Algebra . . . . . . . . . . . . . . . . . . . . 342
10.5 The Mathematics of Kinematics . . . . . . . . . . . . . . 351
Exercises . . . . . . . . . . . . . . . . . . . . . . . 359
References and Notes . . . . . . . . . . . . . . . . . . 362
Chapter 11 Mathematics around the World 364
11.1 Mathematics at the Turn of the Fourteenth Century . . . . . . . 365
11.2 Mathematics in America, Africa, and the Pacific . . . . . . . . 370
Exercises . . . . . . . . . . . . . . . . . . . . . . . 379
References and Notes . . . . . . . . . . . . . . . . . . 380
PART THREE Early Modern Mathematics
Chapter 12 Algebra in the Renaissance 383
12.1 The Italian Abacists . . . . . . . . . . . . . . . . . . . 385
12.2 Algebra in France, Germany, England, and Portugal . . . . . . 389
12.3 The Solution of the Cubic Equation . . . . . . . . . . . . . 399
12.4 Vi`ete, Algebraic Symbolism, and Analysis . . . . . . . . . . 407
12.5 Simon Stevin and Decimal Fractions . . . . . . . . . . . . 414
Exercises . . . . . . . . . . . . . . . . . . . . . . . 418
References . . . . . . . . . . . . . . . . . . . . . . 420
Chapter 13 Mathematical Methods in the Renaissance 423
13.1 Perspective . . . . . . . . . . . . . . . . . . . . . . . 427
13.2 Navigation and Geography . . . . . . . . . . . . . . . . 432
13.3 Astronomy and Trigonometry . . . . . . . . . . . . . . . 435
13.4 Logarithms . . . . . . . . . . . . . . . . . . . . . . 453
13.5 Kinematics . . . . . . . . . . . . . . . . . . . . . . . 457
Exercises . . . . . . . . . . . . . . . . . . . . . . . 462
References and Notes . . . . . . . . . . . . . . . . . . 464
Chapter 14 Algebra, Geometry, and Probability in the Seventeenth Century 467
14.1 The Theory of Equations . . . . . . . . . . . . . . . . . 468
14.2 Analytic Geometry . . . . . . . . . . . . . . . . . . . 473
14.3 Elementary Probability . . . . . . . . . . . . . . . . . . 487
14.4 Number Theory . . . . . . . . . . . . . . . . . . . . . 497
14.5 Projective Geometry . . . . . . . . . . . . . . . . . . . 499
Exercises . . . . . . . . . . . . . . . . . . . . . . . 501
References and Notes . . . . . . . . . . . . . . . . . . 504
Chapter 15 The Beginnings of Calculus 507
15.1 Tangents and Extrema . . . . . . . . . . . . . . . . . . 509
15.2 Areas and Volumes . . . . . . . . . . . . . . . . . . . 514
15.3 Rectification of Curves and the Fundamental Theorem . . . . . 532
Exercises . . . . . . . . . . . . . . . . . . . . . . . 539
References and Notes . . . . . . . . . . . . . . . . . . 541
Chapter 16 Newton and Leibniz 543
16.1 Isaac Newton . . . . . . . . . . . . . . . . . . . . . . 544
16.2 Gottfried Wilhelm Leibniz . . . . . . . . . . . . . . . . 565
16.3 First Calculus Texts . . . . . . . . . . . . . . . . . . . 575
Exercises . . . . . . . . . . . . . . . . . . . . . . . 579
References and Notes . . . . . . . . . . . . . . . . . . 580
PART FOUR Modern Mathematics
Chapter 17 Analysis in the Eighteenth Century 583
17.1 Differential Equations . . . . . . . . . . . . . . . . . . 584
17.2 The Calculus of Several Variables . . . . . . . . . . . . . . 601
17.3 Calculus Texts . . . . . . . . . . . . . . . . . . . . . 611
17.4 The Foundations of Calculus . . . . . . . . . . . . . . . . 628
Exercises . . . . . . . . . . . . . . . . . . . . . . . 636
References and Notes . . . . . . . . . . . . . . . . . . 639
Chapter 18 Probability and Statistics in the Eighteenth Century 642
18.1 Theoretical Probability . . . . . . . . . . . . . . . . . . 643
18.2 Statistical Inference . . . . . . . . . . . . . . . . . . . 651
18.3 Applications of Probability . . . . . . . . . . . . . . . . 655
Exercises . . . . . . . . . . . . . . . . . . . . . . . 661
References and Notes . . . . . . . . . . . . . . . . . . 663
Chapter 19 Algebra and Number Theory in the Eighteenth Century 665
19.1 Algebra Texts . . . . . . . . . . . . . . . . . . . . . 666
19.2 Advances in the Theory of Equations . . . . . . . . . . . . 671
19.3 Number Theory . . . . . . . . . . . . . . . . . . . . . 677
19.4 Mathematics in the Americas . . . . . . . . . . . . . . . 680
Exercises . . . . . . . . . . . . . . . . . . . . . . . 683
References and Notes . . . . . . . . . . . . . . . . . . 684
Chapter 20 Geometry in the Eighteenth Century 686
20.1 Clairaut and the Elements of Geometry . . . . . . . . . . . . 687
20.2 The Parallel Postulate . . . . . . . . . . . . . . . . . . 689
20.3 Analytic and Differential Geometry . . . . . . . . . . . . . 695
20.4 The Beginnings of Topology . . . . . . . . . . . . . . . . 701
20.5 The French Revolution and Mathematics Education . . . . . . 702
Exercises . . . . . . . . . . . . . . . . . . . . . . . 706
References and Notes . . . . . . . . . . . . . . . . . . 707
Chapter 21 Algebra and Number Theory in the Nineteenth Century 709
21.1 Number Theory . . . . . . . . . . . . . . . . . . . . . 711
21.2 Solving Algebraic Equations . . . . . . . . . . . . . . . . 721
21.3 Symbolic Algebra . . . . . . . . . . . . . . . . . . . . 730
21.4 Matrices and Systems of Linear Equations . . . . . . . . . . 740
21.5 Groups and Fields—The Beginning of Structure . . . . . . . . 750
Exercises . . . . . . . . . . . . . . . . . . . . . . . 759
References and Notes . . . . . . . . . . . . . . . . . . 761
Chapter 22 Analysis in the Nineteenth Century 764
22.1 Rigor in Analysis . . . . . . . . . . . . . . . . . . . . 766
22.2 The Arithmetization of Analysis . . . . . . . . . . . . . . 788
22.3 Complex Analysis . . . . . . . . . . . . . . . . . . . . 795
22.4 Vector Analysis . . . . . . . . . . . . . . . . . . . . . 807
Exercises . . . . . . . . . . . . . . . . . . . . . . . 813
References and Notes . . . . . . . . . . . . . . . . . . 815
Chapter 23 Probability and Statistics in the Nineteenth Century 818
23.1 The Method of Least Squares and Probability Distributions . . . 819
23.2 Statistics and the Social Sciences . . . . . . . . . . . . . . 824
23.3 Statistical Graphs . . . . . . . . . . . . . . . . . . . . 828
Exercises . . . . . . . . . . . . . . . . . . . . . . . 831
References and Notes . . . . . . . . . . . . . . . . . . 831
Chapter 24 Geometry in the Nineteenth Century 833
24.1 Differential Geometry . . . . . . . . . . . . . . . . . . 835
24.2 Non-Euclidean Geometry . . . . . . . . . . . . . . . . . 839
24.3 Projective Geometry . . . . . . . . . . . . . . . . . . . 852
24.4 Graph Theory and the Four-Color Problem . . . . . . . . . . 858
24.5 Geometry in N Dimensions . . . . . . . . . . . . . . . . 862
24.6 The Foundations of Geometry . . . . . . . . . . . . . . . 867
Exercises . . . . . . . . . . . . . . . . . . . . . . . 870
References and Notes . . . . . . . . . . . . . . . . . . 872
Chapter 25 Aspects of the Twentieth Century and Beyond 874
25.1 Set Theory: Problems and Paradoxes . . . . . . . . . . . . 876
25.2 Topology . . . . . . . . . . . . . . . . . . . . . . . 882
25.3 New Ideas in Algebra . . . . . . . . . . . . . . . . . . 890
25.4 The Statistical Revolution . . . . . . . . . . . . . . . . . 903
25.5 Computers and Applications . . . . . . . . . . . . . . . . 907
25.6 Old Questions Answered . . . . . . . . . . . . . . . . . 919
Exercises . . . . . . . . . . . . . . . . . . . . . . . 926
References and Notes . . . . . . . . . . . . . . . . . . 928
Appendix A Using This Textbook in Teaching Mathematics 931
A.1 Courses and Topics . . . . . . . . . . . . . . . . . . . 931
A.2 Sample Lesson Ideas to Incorporate History . . . . . . . . . . 935
A.3 Time Line . . . . . . . . . . . . . . . . . . . . . . . 939
General References in the History of Mathematics . . . . . . . . . 945
Answers to Selected Exercises . . . . . . . . . . . . . . . . . 949
Index and Pronunciation Guide . . . . . . . . . . . . . . . . . 961
Victor J. Katz received his PhD in mathematics from Brandeis University in 1968 and has been Professor of Mathematics at the University of the District of Columbia for many years. He has long been interested in the history of mathematics and, in particular, in its use in teaching. He is the editor of The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Sourcebook (2007). He has edited or co-edited two recent books dealing with this subject, Learn from the Masters (1994) and Using History to Teach Mathematics (2000). Dr. Katz also co-edited a collection of historical articles taken from MAA journals of the past 90 years, Sherlock Holmes in Babylon and other Tales of Mathematical History. He has directed two NSF-sponsored projects to help college teachers learn the history of mathematics and learn to use it in teaching. Dr. Katz has also involved secondary school teachers in writing materials using history in the teaching of various topics in the high school curriculum. These materials, Historical Modules for the Teaching and Learning of Mathematics, have now been drafted by the MAA. Currently, Dr. Katz is the PI on an NSF grant to the MAA that supports Convergence, an online magazine devoted to the history of mathematics and its use in teaching.
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