Mathematical Thought From Ancient to Modern Times, Volume 1

by Morris Kline

Overview

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:	9780199840410
Publisher:	Oxford University Press
Publication date:	03/01/1990
Sold by:	Barnes & Noble
Format:	eBook
File size:	6 MB

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

1	Mathematics in Mesopotamia	3
1	Where Did Mathematics Begin?	3
2	Political History in Mesopotamia	4
3	The Number Symbols	5
4	Arithmetic Operations	7
5	Babylonian Algebra	8
6	Babylonian Geometry	10
7	The Uses of Mathematics in Babylonia	11
8	Evaluation of Babylonian Mathematics	13
2	Egyptian Mathematics	15
1	Background	15
2	The Arithmetic	16
3	Algebra and Geometry	18
4	Egyptian Uses of Mathematics	21
5	Summary	22
3	The Creation of Classical Greek Mathematics	24
1	Background	24
2	The General Sources	25
3	The Major Schools of the Classical Period	27
4	The Ionian School	28
5	The Pythagoreans	28
6	The Eleatic School	34
7	The Sophist School	37
8	The Platonic School	42
9	The School of Eudoxus	48
10	Aristotle and His School	51
4	Euclid and Apollonius	56
1	Introduction	56
2	The Background of Euclid's Elements	57
3	The Definitions and Axioms of the Elements	58
4	Books I to IV of the Elements	60
5	Book V: The Theory of Proportion	68
6	Book VI: Similar Figures	73
7	Books VII, VIII, and IX: The Theory of Numbers	77
8	Book X: The Classification of Incommensurables	80
9	Books XI, XII, and XIII: Solid Geometry and the Method of Exhaustion	81
10	The Merits and Defects of the Elements	86
11	Other Mathematical Works by Euclid	88
12	The Mathematical Work of Apollonius	89
5	The Alexandrian Greek Period: Geometry and Trigonometry	101
1	The Founding of Alexandria	101
2	The Character of Alexandrian Greek Mathematics	103
3	Areas and Volumes in the Work of Archimedes	105
4	Areas and Volumes in the Work of Heron	116
5	Some Exceptional Curves	117
6	The Creation of Trigonometry	119
7	Late Alexandrian Activity in Geometry	126
6	The Alexandrian Period: The Reemergence of Arithmetic and Algebra	131
1	The Symbols and Operations of Greek Arithmetic	131
2	Arithmetic and Algebra as an Independent Development	135
7	The Greek Rationalization of Nature	145
1	The Inspiration for Greek Mathematics	145
2	The Beginnings of a Rational View of Nature	146
3	The Development of the Belief in Mathematical Design	147
4	Greek Mathematical Astronomy	154
5	Geography	160
6	Mechanics	162
7	Optics	166
8	Astrology	168
8	The Demise of the Greek World	171
1	A Review of the Greek Achievements	171
2	The Limitations of Greek Mathematics	173
3	The Problems Bequeathed by the Greeks	176
4	The Demise of the Greek Civilization	177
9	The Mathematics of the Hindus and Arabs	183
1	Early Hindu Mathematics	183
2	Hindu Arithmetic and Algebra of the Period A.D. 200-1200	184
3	Hindu Geometry and Trigonometry of the Period A.D. 200-1200	188
4	The Arabs	190
5	Arabic Arithmetic and Algebra	191
6	Arabic Geometry and Trigonometry	195
7	Mathematics circa 1300	197
10	The Medieval Period in Europe	200
1	The Beginnings of a European Civilization	200
2	The Materials Available for Learning	201
3	The Role of Mathematics in Early Medieval Europe	202
4	The Stagnation in Mathematics	203
5	The First Revival of the Greek Works	205
6	The Revival of Rationalism and Interest in Nature	206
7	Progress in Mathematics Proper	209
8	Progress in Physical Science	211
9	Summary	213
11	The Renaissance	216
1	Revolutionary Influences in Europe	216
2	The New Intellectual Outlook	218
3	The Spread of Learning	220
4	Humanistic Activity in Mathematics	221
5	The Clamor for the Reform of Science	223
6	The Rise of Empiricism	227
12	Mathematical Contributions in the Renaissance	231
1	Perspective	231
2	Geometry Proper	234
3	Algebra	236
4	Trigonometry	237
5	The Major Scientific Progress in the Renaissance	240
6	Remarks on the Renaissance	247
13	Arithmetic and Algebra in the Sixteenth and Seventeenth Centuries	250
1	Introduction	250
2	The Status of the Number System and Arithmetic	251
3	Symbolism	259
4	The Solution of Third and Fourth Degree Equations	263
5	The Theory of Equations	270
6	The Binomial Theorem and Allied Topics	272
7	The Theory of Numbers	274
8	The Relationship of Algebra to Geometry	278
14	The Beginnings of Projective Geometry	285
1	The Rebirth of Geometry	285
2	The Problems Raised by the Work on Perspective	286
3	The Work of Desargues	288
4	The Work of Pascal and La Hire	295
5	The Emergence of New Principles	299
15	Coordinate Geometry	302
1	The Motivation for Coordinate Geometry	302
2	The Coordinate Geometry of Fermat	303
3	Rene Descartes	304
4	Descartes's Work in Coordinate Geometry	308
5	Seventeenth-Century Extensions of Coordinate Geometry	317
6	The Importance of Coordinate Geometry	321
16	The Mathematization of Science	325
1	Introduction	325
2	Descartes's Concept of Science	325
3	Galileo's Approach to Science	327
4	The Function Concept	335
17	The Creation of the Calculus	342
1	The Motivation for the Calculus	342
2	Early Seventeenth-Century Work on the Calculus	344
3	The Work of Newton	356
4	The Work of Leibniz	370
5	A Comparison of the Work of Newton and Leibniz	378
6	The Controversy over Priority	380
7	Some Immediate Additions to the Calculus	381
8	The Soundness of the Calculus	383
	List of Abbreviations
	Index

Editorial Reviews

"I have always had great regard for this book as the one which relates the development of modern mathematical ideas in a readable fashion."—Michael F. O'Reilly, University of Minnesota in Morris

"Outstanding scholarship and readability. One of only a couple of books available in English for in-depth historical studies at the fourth year/graduate level."—Charles V. Jones, Ball State University

"The consistently high quality of presentation, the accuracy, the readable style, and the stress on the conceptual development of mathematics make [these volumes] a most desirable reference."—Choice

"Without a doubt a book which should be in the library of every institution where mathematics is either taught or played."—The Economist

"What must be the definitive history of mathematical thought....Probably the most comprehensive account of mathematical history we have yet had."—Saturday Review

From the Publisher