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The History of Mathematics: A Brief Course

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  • World Part 1. The World of Mathematics and the Mathematics of the

    • Chapter 1. The Origin and Prehistory of Mathematics



        1. Numbers

          • 1.1. Animals'use of numbers

          • 1.2. Young children's use of numbers

          • 1.3. Archaeological evidence of counting







        1. Continuous magnitudes

          • 2.1. Perception of shape by animals

          • 2.2. Children's concepts of space

          • 2.3. Geometry in arts and crafts







          1. Symbols





          1. Mathematical inference



          • 4.1. Visual reasoning

          • 4.2. Chance and probability



        • Questions and problems





    • Chapter 2. Mathematical Cultures I



        1. The motives for creating mathematics

          • 1.1. Pure versus applied mathematics







        1. India



        • 2.1. The Sulva Sutras

          • 2.2. Buddhist and Jaina mathematics

          • 2.3. The Bakshali Manuscript

          • 2.4. The siddhantas

          • 2.5. Aryabhata I

          • 2.6. Brahmagupta

          • 2.7. Bhaskara II



        • 2.8. Muslim India

          • 2.9. Indian mathematics in the colonial period and after







        1. China

          • 3.1. Works and authors

          • 3.2. China's encounter with Western mathematics







        1. Ancient Egypt





        1. Mesopotamia





        1. The Maya



        • 6.1. The Dresden Codex



      • Questions and problems vi CONTENTS





  • Chapter 3. Mathematical Cultures II



      1. Greek and Roman mathematics

        • 1.1. Sources

        • 1.2. General features of Greek mathematics

        • 1.3. Works and authors







      1. Japan

        • 2.1. Chinese influence and calculating devices

        • 2.2. Japanese mathematicians and their works







      1. The Muslims

        • 3.1. Islamic science in general

        • 3.2. Some Muslim mathematicians and their works







      1. Europe

        • 4.1. Monasteries, schools, and universities

        • 4.2. The high Middle Ages

        • 4.3. Authors and works







      1. North America

        • 5.1. The United States and Canada before

          • 5.2. The Canadian Federation and post Civil War United States



        • 5.3. Mexico







      1. Australia and New Zealand

        • 6.1. Colonial mathematics







      1. The modern era

        • 7.1. Educational institutions

        • 7.2. Mathematical societies

        • 7.3. Journals





    • Questions and problems

    • Chapter 4. Women Mathematicians



        1. Individual achievements and obstacles to achievement

          • 1.1. Obstacles to mathematical careers for women







        1. Ancient women mathematicians





        1. Modern European women

          • 3.1. Continental mathematicians

          • 3.2. Nineteenth-century British women

          • 3.3. Four modern pioneers







          1. American women





          1. The situation today



        • Questions and problems





    • Part 2. Numbers

    • Chapter 5. Counting



        1. Number words





        1. Bases for counting

          • 2.1. Decimal systems

          • 2.2. Nondecimal systems







        1. Counting around the world

          • 3.1. Egypt

          • 3.2. Mesopotamia CONTENTS vii

          • 3.3. India

            • 3.4. China

            • 3.5. Greece and Rome

            • 3.6. The Maya









        1. What was counted?

          • 4.1. Calendars

          • 4.2. Weeks





      • Questions and problems





  • Chapter 6. Calculation



      1. Egypt

        • 1.1. Multiplication and division

        • 1.2. "Parts"

        • 1.3. Practical problems







      1. China

        • 2.1. Fractions and roots

        • 2.2. The Jiu Zhang Suanshu







      1. India





      1. Mesopotamia





      1. The ancient Greeks





      1. The Islamic world





      1. Europe





      1. The value of calculation





      1. Mechanical methods of computation

        • 9.1. Software: prosthaphaeresis and logarithms

        • 9.2. Hardware: slide rules and calculating machines

        • 9.3. The effects of computing power





    • Questions and problems



  • Chapter 7. Ancient Number Theory



      1. Plimpton





      1. Ancient Greek number theory

        • 2.1. The Arithmetica of Nicomachus

        • 2.2. Euclid's number theory

        • 2.3. The Arithmetica of Diophantus







        1. China





        1. India

          • 4.1. Varahamihira's mystical square

          • 4.2. Aryabhata I

          • 4.3. Brahmagupta

          • 4.4. Bhaskara II









      1. The Muslims





      1. Japan





        1. Medieval Europe



      • Questions and problems



    • Chapter 8. Numbers and Number Theory in Modern Mathematics



        1. Modern number theory

          • 1.1. Fermat

          • 1.2. Euler viii CONTENTS

          • 1.3. Lagrange

          • 1.4. Legendre

          • 1.5. Gauss

          • 1.6. Dirichlet

          • 1.7. Riemann

          • 1.8. Fermat's last theorem

          • 1.9. The prime number theorem







        1. Number systems

          • 2.1. Negative numbers and zero

          • 2.2. Irrational and imaginary numbers

          • 2.3. Imaginary and complex numbers

          • 2.4. Infinite numbers







          1. Combinatorics

            • 3.1. Summation rules





        • Questions and problems







  • Part 3. Color Plates

  • Part 4. Space

  • Chapter 9. Measurement



      1. Egypt

        • 1.1. Areas

        • 1.2. Volumes







      1. Mesopotamia

        • 2.1. The Pythagorean theorem

        • 2.2. Plane figures

        • 2.3. Volumes







      1. China

        • 3.1. The Zhou Bi Suan Jing

        • 3.2. The Jiu Zhang Suanshu

        • 3.3. The Sun Zi Suan Jing

        • 3.4. Liu Hui

        • 3.5. Zu Chongzhi







        1. Japan

          • 4.1. The challenge problems

          • 4.2. Beginnings of the calculus in Japan







        1. India

          • 5.1. Aryabhata I

          • 5.2. Brahmagupta





      • Questions and problems



    • Chapter 10. Euclidean Geometry



        1. The earliest Greek geometry

          • 1.1. Thales

          • 1.2. Pythagoras and the Pythagoreans

          • 1.3. Pythagorean geometry

          • 1.4. Challenges to Pythagoreanism: unsolved problems





        • 1.6. Challenges to Pythagoreanism: incommensurables CONTENTS ix

        • 1.7. The influence of Plato

        • 1.8. Eudoxan geometry

        • 1.9. Aristotle





        1. Euclid

          • 2.1. The Elements

          • 2.2. The Data







        1. Archimedes

          • 3.1. The area of a sphere

          • 3.2. The Method







        1. Apollonius

          • 4.1. History of the Conies

          • 4.2. Contents of the Conies

            • 4.3. Apollonius' definition of the conic sections

            • 4.4. Foci and the three- and four-line locus







      • Questions and problems





  • Chapter 11. Post-Euclidean Geometry



      1. Hellenistic geometry

        • 1.1. Zenodorus

        • 1.2. The parallel postulate

        • 1.3. Heron

        • 1.4. Pappus







      1. Roman geometry

        • 2.1. Roman civil engineering







      1. Medieval geometry

        • 3.1. Late Medieval and Renaissance geometry







      1. Geometry in the Muslim world

        • 4.1. The parallel postulate

          • 4.2. Thabit ibn-Qurra

          • 4.3. Al-Kuhi

          • 4.4. Al-Haytham

          • 4.5. Omar Khayyam

          • 4.6. Nasir al-Din al-Tusi





          1. Non-Euclidean geometry

            • 5.1. Girolamo Saccheri

            • 5.2. Lambert and Legendre

            • 5.3. Gauss

            • 5.4. Lobachevskii and Janos Bolyai

            • 5.5. The reception of non-Euclidean geometry

            • 5.6. Foundations of geometry











        1. Questions and problems





    • Chapter 12. Modern Geometries



        1. Analytic and algebraic geometry

          • 1.1. Fermat

          • 1.2. Descartes

          • 1.3. Newton's classification of curves

          • 1.5. Challenges to Pythagoreanism: the paradoxes of Zeno of Elea





        • 1.4. Algebraic geometry x CONTENTS





        1. Projective and descriptive geometry

          • 2.1. Projective properties

          • 2.2. The Renaissance artists

          • 2.3. Girard Desargues

          • 2.4. Blaise Pascal

          • 2.5. Newton's degree-preserving mappings

          • 2.6. Charles Brianchon

          • 2.7. Monge and his school

          • 2.8. Jacob Steiner

          • 2.9. August Ferdinand Mobius

          • 2.10. Julius Plucker

          • 2.11. Arthur Cayley







        1. Differential geometry

          • 3.1. Huygens

          • 3.2. Newton

          • 3.3. Leibniz

          • 3.4. The eighteenth century

          • 3.5. Gauss

          • 3.6. The French and British geometers

          • 3.7. Riemann

          • 3.8. The Italian geometers









      1. Topology

        • 4.1. Early combinatorial topology

        • 4.2. Riemann

        • 4.3. Mobius

        • 4.4. Poincare's Analysis situs

        • 4.5. Point-set topology





    • Questions and problems



  • Part 5. Algebra

  • Chapter 13. Problems Leading to Algebra



      1. Egypt





      1. Mesopotamia

        • 2.1. Linear and quadratic problems

        • 2.2. Higher-degree problems







        1. India

          • 3.1. Jaina algebra

          • 3.2. The Bakshali Manuscript









      1. China

        • 4.1. The Jiu Zhang Suanshu

        • 4.2. The Suanshu Shu

        • 4.3. The Sun Zi Suan Jing

        • 4.4. Zhang Qiujian





      • Questions and problems





  • Chapter 14. Equations and Algorithms



      1. The Arithmetica of Diophantus

        • 1.1. Diophantine equations CONTENTS xi

        • 1.2. General characteristics of the Arithmetica

        • 1.3. Determinate problems

        • 1.4. The significance of the Arithmetica

        • 1.5. The view of Jacob Klein







      1. China

        • 2.1. Linear equations

          • 2.2. Quadratic equations

          • 2.3. Cubic equations

          • 2.4. The numerical solution of equations









      1. Japan

        • 3.1. SekiK5wa







      1. Hindu algebra

        • 4.1. Brahmagupta

          • 4.2. Bhaskara II









      1. The Muslims

        • 5.1. Al-Khwarizmi

        • 5.2. Abu Kamil

        • 5.3. Omar Khayyam

        • 5.4. Sharaf al-Din al-Muzaffar al-Tusi







      1. Europe

        • 6.1. Leonardo of Pisa (Fibonacci)

        • 6.2. Jordanus Nemorarius

        • 6.3. The fourteenth and fifteenth centuries

        • 6.4. Chuquet

        • 6.5. Solution of cubic and quartic equations

        • 6.6. Consolidation





    • Questions and problems



  • Chapter 15. Modern Algebra



      1. Theory of equations

        • 1.1. Albert Girard

        • 1.2. Tschirnhaus transformations

          • 1.3. Newton, Leibniz, and the Bernoullis

          • 1.4. Euler, d'Alembert, and Lagrange

          • 1.5. Gauss and the fundamental theorem of algebra

          • 1.6. Ruffini

          • 1.7. Cauchy

          • 1.8. Abel

          • 1.9. Galois









      1. Algebraic structures

        • 2.1. Fields, rings, and algebras

        • 2.2. Abstract groups

        • 2.3. Number systems





      • Questions and problems





  • Part 6. Analysis

    • Chapter 16. The Calculus



        1. Prelude to the calculus xii CONTENTS

          • 1.1. Tangent and maximum problems

          • 1.2. Lengths, areas, and volumes

          • 1.3. The relation between tangents and areas

          • 1.4. Infinite series and products







        1. Newton and Leibniz

          • 2.1. Isaac Newton

          • 2.2. Gottfried Wilhelm von Leibniz

          • 2.3. The disciples of Newton and Leibniz







        1. Branches and roots of the calculus

          • 3.1. Ordinary differential equations

          • 3.2. Partial differential equations

          • 3.3. Calculus of variations

          • 3.4. Foundations of the calculus





      • Questions and problems



    • Chapter 17. Real and Complex Analysis



        1. Complex analysis

          • 1.1. Algebraic integrals

          • 1.2. Cauchy

          • 1.3. Riemann

          • 1.4. Weierstrass







        1. Real analysis

          • 2.1. Fourier series, functions, and integrals

          • 2.2. Completeness of the real numbers

            • 2.3. Uniform convergence and continuity

            • 2.4. General integrals and discontinuous functions



          • 2.5. The abstract and the concrete

          • 2.G. Discontinuity as a positive property





      • Questions and problems





  • Part 7. Mathematical Inferences

  • Chapter 18. Probability and Statistics



      1. Probability

        • 1.1. Cardano

        • 1.2. Fermat and Pascal

        • 1.3. Huygens

        • 1.4. Leibniz

        • 1.5. The Ars Conjectandi of Jakob Bernoulli

        • 1.6. De Moivre

        • 1.7. Laplace

        • 1.8. Legendre

        • 1.9. Gauss

        • 1.10. Philosophical issues

        • 1.11. Large numbers and limit theorems







      1. Statistics

        • 2.1. Quetelet

        • 2.2. Statistics in physics





      • 2.3. The metaphysics of probability and statistics CONTENTS xi"

      • 2.4. Correlations and statistical inference



    • Questions and problems



  • Chapter 19. Logic and Set Theory



      1. Logic

        • 1.1. From algebra to logic

        • 1.2. Symbolic calculus

        • 1.3. Boole's Mathematical Analysis of Logic

        • 1.4. Boole's Laws of Thought

        • 1.5. Venn

        • 1.6. Jevons







      1. Set theory



      • 2.1. Technical background

        • 2.2. Cantor's work on trigonometric series



      • 2.3. The reception of set theory

        • 2.4. Existence and the axiom of choice

        • 2.5. Doubts about set theory







      1. Philosophies of mathematics

        • 3.1. Paradoxes

        • 3.2. Formalism

        • 3.3. Intuitionism

        • 3.4. Mathematical practice





    • Questions and problems

    • Literature



  • Subject Index

    • Name Index



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