Mathematical Thought From Ancient to Modern Times, Volume 2

Mathematical Thought From Ancient to Modern Times, Volume 2

by Morris Kline

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This comprehensive history traces the development of mathematical ideas and the careers of the men responsible for them. Volume 1 looks at the disciplines origins in Babylon and Egypt, the creation of geometry and trigonometry by the Greeks, and the role of mathematics in the medieval and early modern periods. Volume 2 focuses on calculus, the rise of analysis in the 19th century, and the number theories of Dedekind and Dirichlet. The concluding volume covers the revival of projective geometry, the emergence of abstract algebra, the beginnings of topology, and the influence of Godel on recent mathematical study.

Product Details

ISBN-13: 9780199840427
Publisher: Oxford University Press
Publication date: 03/01/1990
Sold by: Barnes & Noble
Format: NOOK Book
Sales rank: 968,657
File size: 11 MB
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About the Author

Morris Kline is Professor of Mathematics, Emeritus, at the Courant Institute of Mathematical Sciences, New York University, where he directed the Division of Electromagnetic Research for twenty years.

Table of Contents

1Mathematics in Mesopotamia3
1Where Did Mathematics Begin?3
2Political History in Mesopotamia4
3The Number Symbols5
4Arithmetic Operations7
5Babylonian Algebra8
6Babylonian Geometry10
7The Uses of Mathematics in Babylonia11
8Evaluation of Babylonian Mathematics13
2Egyptian Mathematics15
2The Arithmetic16
3Algebra and Geometry18
4Egyptian Uses of Mathematics21
3The Creation of Classical Greek Mathematics24
2The General Sources25
3The Major Schools of the Classical Period27
4The Ionian School28
5The Pythagoreans28
6The Eleatic School34
7The Sophist School37
8The Platonic School42
9The School of Eudoxus48
10Aristotle and His School51
4Euclid and Apollonius56
2The Background of Euclid's Elements57
3The Definitions and Axioms of the Elements58
4Books I to IV of the Elements60
5Book V: The Theory of Proportion68
6Book VI: Similar Figures73
7Books VII, VIII, and IX: The Theory of Numbers77
8Book X: The Classification of Incommensurables80
9Books XI, XII, and XIII: Solid Geometry and the Method of Exhaustion81
10The Merits and Defects of the Elements86
11Other Mathematical Works by Euclid88
12The Mathematical Work of Apollonius89
5The Alexandrian Greek Period: Geometry and Trigonometry101
1The Founding of Alexandria101
2The Character of Alexandrian Greek Mathematics103
3Areas and Volumes in the Work of Archimedes105
4Areas and Volumes in the Work of Heron116
5Some Exceptional Curves117
6The Creation of Trigonometry119
7Late Alexandrian Activity in Geometry126
6The Alexandrian Period: The Reemergence of Arithmetic and Algebra131
1The Symbols and Operations of Greek Arithmetic131
2Arithmetic and Algebra as an Independent Development135
7The Greek Rationalization of Nature145
1The Inspiration for Greek Mathematics145
2The Beginnings of a Rational View of Nature146
3The Development of the Belief in Mathematical Design147
4Greek Mathematical Astronomy154
8The Demise of the Greek World171
1A Review of the Greek Achievements171
2The Limitations of Greek Mathematics173
3The Problems Bequeathed by the Greeks176
4The Demise of the Greek Civilization177
9The Mathematics of the Hindus and Arabs183
1Early Hindu Mathematics183
2Hindu Arithmetic and Algebra of the Period A.D. 200-1200184
3Hindu Geometry and Trigonometry of the Period A.D. 200-1200188
4The Arabs190
5Arabic Arithmetic and Algebra191
6Arabic Geometry and Trigonometry195
7Mathematics circa 1300197
10The Medieval Period in Europe200
1The Beginnings of a European Civilization200
2The Materials Available for Learning201
3The Role of Mathematics in Early Medieval Europe202
4The Stagnation in Mathematics203
5The First Revival of the Greek Works205
6The Revival of Rationalism and Interest in Nature206
7Progress in Mathematics Proper209
8Progress in Physical Science211
11The Renaissance216
1Revolutionary Influences in Europe216
2The New Intellectual Outlook218
3The Spread of Learning220
4Humanistic Activity in Mathematics221
5The Clamor for the Reform of Science223
6The Rise of Empiricism227
12Mathematical Contributions in the Renaissance231
2Geometry Proper234
5The Major Scientific Progress in the Renaissance240
6Remarks on the Renaissance247
13Arithmetic and Algebra in the Sixteenth and Seventeenth Centuries250
2The Status of the Number System and Arithmetic251
4The Solution of Third and Fourth Degree Equations263
5The Theory of Equations270
6The Binomial Theorem and Allied Topics272
7The Theory of Numbers274
8The Relationship of Algebra to Geometry278
14The Beginnings of Projective Geometry285
1The Rebirth of Geometry285
2The Problems Raised by the Work on Perspective286
3The Work of Desargues288
4The Work of Pascal and La Hire295
5The Emergence of New Principles299
15Coordinate Geometry302
1The Motivation for Coordinate Geometry302
2The Coordinate Geometry of Fermat303
3Rene Descartes304
4Descartes's Work in Coordinate Geometry308
5Seventeenth-Century Extensions of Coordinate Geometry317
6The Importance of Coordinate Geometry321
16The Mathematization of Science325
2Descartes's Concept of Science325
3Galileo's Approach to Science327
4The Function Concept335
17The Creation of the Calculus342
1The Motivation for the Calculus342
2Early Seventeenth-Century Work on the Calculus344
3The Work of Newton356
4The Work of Leibniz370
5A Comparison of the Work of Newton and Leibniz378
6The Controversy over Priority380
7Some Immediate Additions to the Calculus381
8The Soundness of the Calculus383
List of Abbreviations

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