Mathematical Thought from Ancient to Modern Times 1st edition by Morris Kline ISBN 0195061357 9780195061352

Table of Contents:

1. Mathematics in Mesopotamia

   1. Where Did Mathematics Begin?
   2. Political History in Mesopotamia
   3. The Number Symbols
   4. Arithmetic Operations
   5. Babylonian Algebra
   6. Babylonian Geometry
   7. Uses of Mathematics in Babylonia
   8. Evaluation of Babylonian Mathematics

2. Egyptian Mathematics

   1. Background
   2. The Arithmetic
   3. Algebra and Geometry
   4. Egyptian Uses of Mathematics
   5. Summary

3. Classical Greek Mathematics

   1. Background
   2. General Sources
   3. Major Schools of the Classical Period
   4. Ionian School
   5. Pythagoreans
   6. Eleatic School
   7. Sophist School
   8. Platonic School
   9. School of Eudoxus
   10. Aristotle and His School

4. Euclid and Apollonius

   1. Introduction
   2. Background of Euclid’s Elements
   3. Definitions and Axioms
   4. Books I to IV of the Elements
   5. Book V: Theory of Proportion
   6. Book VI: Similar Figures
   7. Books VII, VIII, IX: Theory of Numbers
   8. Book X: Classification of Incommensurables
   9. Books XI, XII, XIII: Solid Geometry
   10. Merits and Defects of the Elements
   11. Other Mathematical Works by Euclid
   12. Mathematical Work of Apollonius

5. Alexandrian Greek Period

   1. Founding of Alexandria
   2. Character of Alexandrian Greek Mathematics
   3. Areas and Volumes in Archimedes’ Work
   4. Areas and Volumes in Heron’s Work
   5. Exceptional Curves
   6. Creation of Trigonometry
   7. Late Alexandrian Activity in Geometry

6. Alexandrian Period: Reemergence of Arithmetic and Algebra

   1. Symbols and Operations of Greek Arithmetic
   2. Independent Development of Arithmetic and Algebra

7. Greek Rationalization of Nature

   1. Inspiration for Greek Mathematics
   2. Rational View of Nature
   3. Mathematical Design
   4. Greek Mathematical Astronomy
   5. Geography
   6. Mechanics
   7. Optics
   8. Astrology

8. Demise of the Greek World

   1. Review of Greek Achievements
   2. Limitations of Greek Mathematics
   3. Problems Bequeathed by the Greeks
   4. Demise of Greek Civilization

9. Mathematics of the Hindus and Arabs

   1. Early Hindu Mathematics
   2. Hindu Arithmetic and Algebra (A.D. 200-1200)
   3. Hindu Geometry and Trigonometry (A.D. 200-1200)
   4. The Arabs
   5. Arabic Arithmetic and Algebra
   6. Arabic Geometry and Trigonometry
   7. Mathematics circa 1300

10. Medieval Period in Europe

    1. Beginnings of European Civilization
    2. Materials Available for Learning
    3. Role of Mathematics in Early Medieval Europe
    4. Stagnation in Mathematics
    5. First Revival of Greek Works
    6. Revival of Rationalism and Interest in Nature
    7. Progress in Mathematics Proper
    8. Progress in Physical Science
    9. Summary

11. The Renaissance

    1. Revolutionary Influences in Europe
    2. New Intellectual Outlook
    3. Spread of Learning
    4. Humanistic Activity in Mathematics
    5. Clamor for Reform of Science
    6. Rise of Empiricism

12. Mathematical Contributions in the Renaissance

    1. Perspective
    2. Geometry Proper
    3. Algebra
    4. Trigonometry
    5. Major Scientific Progress
    6. Remarks on the Renaissance

13. Arithmetic and Algebra in the 16th and 17th Centuries

    1. Introduction
    2. Status of Number System and Arithmetic
    3. Symbolism
    4. Solutions of Third and Fourth Degree Equations
    5. Theory of Equations
    6. Binomial Theorem and Topics
    7. Theory of Numbers
    8. Relationship of Algebra to Geometry

14. Beginnings of Projective Geometry

    1. Rebirth of Geometry
    2. Problems from Perspective Work
    3. Work of Desargues
    4. Work of Pascal and La Hire
    5. Emergence of New Principles

15. Coordinate Geometry

    1. Motivation for Coordinate Geometry
    2. Coordinate Geometry of Fermat
    3. Rene Descartes
    4. Descartes’s Work in Coordinate Geometry
    5. Seventeenth-Century Extensions
    6. Importance of Coordinate Geometry

16. Mathematization of Science

    1. Introduction
    2. Descartes’s Concept of Science
    3. Galileo’s Approach
    4. Function Concept

17. Creation of the Calculus

    1. Motivation for the Calculus
    2. Early Seventeenth-Century Work
    3. Work of Newton
    4. Work of Leibniz
    5. Comparison of Newton and Leibniz
    6. Controversy over Priority
    7. Immediate Additions to the Calculus
    8. Soundness of the Calculus