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A History of Mathematics From Mesopotamia to Modernity

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  • Introduction Picture Credits xiv

    • Why this book?

    • On texts, and on history

    • Examples

    • Historicism and ‘presentism’

    • Revolutions, paradigms, and all that

    • External versus internal

    • Eurocentrism





    1. Babylonian mathematics





      1. On beginnings





      1. Sources and selections





      1. Discussion of the example





      1. The importance of number-writing





      1. Abstraction and uselessness





      1. What went before





      1. Some conclusions



      • Appendix A. Solution of the quadratic problem

      • Solutions to exercises







    1. Greeks and ‘origins’





      1. Plato and theMeno





      1. Literature





      1. An example





      1. The problem of material





      1. The Greek miracle





      1. Two revolutions?





      1. Drowning in the sea of Non-identity





      1. On modernization and reconstruction





      1. On ratios



      • Appendix A. From theMeno

      • Appendix B. On pentagons, golden sections, and irrationals

      • Solutions to exercises







    1. Greeks, practical and theoretical viii Contents





      1. Introduction, and an example





      1. Archimedes





      1. Heron or Hero





      1. Astronomy, and Ptolemy in particular





      1. On the uncultured Romans





      1. Hypatia



      • Appendix A. From Heron’sMetrics

      • Appendix B. From Ptolemy’sAlmagest

      • Solutions to exercises







    1. Chinese mathematics





      1. Introduction





      1. Sources





      1. An instant history of early China



    • 4.The Nine Chapters



      1. Counting rods—who needs them?





      1. Matrices





      1. The Song dynasty and Qin Jiushao





      1. On ‘transfers’—when, and how?





      1. The later period



      • Solutions to exercises







    1. Islam, neglect and discovery





      1. Introduction





      1. On access to the literature





      1. Two texts





      1. The golden age





      1. Algebra—the origins





      1. Algebra—the next steps





      1. Al-Samaw’al and al-K ̄ash ̄i





      1. The uses of religion



      • Appendix A. From al-Khw ̄arizm ̄i’s algebra

      • Appendix B. Th ̄abit ibn Qurra

      • Appendix C. From al-K ̄ash ̄i,The Calculator’s Key, book 4, chapter

      • Solutions to exercises







    1. Understanding the ‘scientific revolution’





      1. Introduction





      1. Literature





      1. Scholastics and scholasticism





      1. Oresme and series





      1. The calculating tradition





      1. Tartaglia and his friends





      1. On authority





        1. Descartes Contents ix





        1. Infinities







      1. Galileo

        • Appendix A

        • Appendix B

        • Appendix C

        • Appendix D

        • Solutions to exercises









    1. The calculus



        1. Introduction





        1. Literature





        1. The priority dispute





        1. The Kerala connection





        1. Newton, an unknown work





        1. Leibniz, a confusing publication





        1. ThePrincipiaand its problems





        1. The arrival of the calculus





        1. The calculus in practice









      1. Afterword

        • Appendix A. Newton

        • Appendix B. Leibniz

        • Appendix C. From thePrincipia

        • Solutions to exercises









    1. Geometries and space



        1. Introduction





        1. First problem: the postulate





        1. Space and infinity





        1. Spherical geometry





        1. The new geometries





        1. The ‘time-lag’ question





        1. What revolution?



        • Appendix A. Euclid’s proposition I.16

        • Appendix B. The formulae of spherical and hyperbolic trigonometry

        • Appendix C. From Helmholtz’s 1876 paper

        • Solutions to exercises









    1. Modernity and its anxieties



        1. Introduction





        1. Literature





        1. New objects in mathematics





        1. Crisis—what crisis?





        1. Hilbert





        1. Topology









      1. Outsiders x Contents

        • Appendix A. The cut definition

        • Appendix B. Intuitionism

        • Appendix C. Hilbert’s programme

        • Solutions to exercises









    1. A chaotic end?



        1. Introduction





        1. Literature





        1. The Second World War





        1. Abstraction and ‘Bourbaki’





        1. The computer





        1. Chaos: the less you know, the more you get





        1. From topology to categories





        1. Physics





        1. Fermat’s Last Theorem



        • Appendix A. From Bourbaki, ‘Algebra’, Introduction

        • Appendix B. Turing on computable numbers

        • Solutions to exercises







  • Conclusion

  • Bibliography

  • Index

  • Introduction List of figures



      1. Euclid’s proposition II.1





  • Chapter 1. Babylonian mathematics



      1. A mathematical tablet





      1. Tally of pigs





      1. The ‘stone-weighing’ tablet YBC4652





      1. Cuneiform numbers from 1 to





      1. How larger cuneiform numbers are formed





      1. The ‘square root of 2’ tablet





      1. Ur III tablet (harvests from Lagash)





  • Chapter 2. Greeks and ‘origins’



      1. TheMenoargument





      1. Diagram for Euclid I.35





      1. The five regular solids





      1. Construction of a regular pentagon





      1. The ‘extreme and mean section’ construction





      1. How to prove ‘Thales’ theorem’





  • Chapter 3. Greeks, practical and theoretical



      1. Menaechmus’ duplication construction





      1. Eratosthenes’ ‘mesolabe’





      1. Circumscribed hexagon





      1. Angle bisection for polygons





      1. Heron’s slot machine





      1. The geocentric model





      1. The chord of an angle





      1. The epicycle model





      1. Figure for ‘Heron’s theorem’





      1. Diagram for Ptolemy’s calculation





      1. The diagram for Exercise





  • Chapter 4. Chinese mathematics



      1. Simple rod numbers





      1. 60390 as a rod-number





      1. Calculating a product by rod-numbers





      1. Li Zhi’s ‘round town’ diagram





      1. Diagram for Li Zhi’s problem





      1. Watchtower from theShushu jiuzhang xii List ofFigures





      1. Equation as set out by Qin





      1. The ‘pointed field’ from Qin’s problem





      1. Chinese version of ‘Pascal’s triangle’





  • Chapter 5. Islam, neglect and discovery



      1. MS of al-K ̄ash ̄i





      1. Ab ̄u-l-Waf ̄a’s construction of the pentagon





      1. Al-Khw ̄arizm ̄i’s first picture for the quadratic equation





      1. Diagram for Euclid’s proposition II.6





      1. Table from al-Samaw’al (powers)





      1. Table from al-Samaw’al (division of polynomials)





      1. Al-Khw ̄arizm ̄i’s second picture





      1. The figure for Th ̄abit ibn Qurra’s proof





      1. Al-K ̄ash ̄i’s seven regular solids





      1. Al-K ̄ash ̄i’s table of solids





      1. The method of finding the qibla





  • Chapter 6. Understanding the ‘scientific revolution’



      1. Arithmetic book from Holbein’sThe Ambassadors





      1. Graph of a cubic curve





      1. Kepler’s diagram fromAstronomia Nova





      1. Descartes’ curve-drawing machine





      1. Kepler’s infinitesimal diagram for the circle





      1. Archimedes’ proof for the area of a circle





  • Chapter 7. The calculus



      1. Indian calculation of the arc





      1. Tangent at a point on a curve





      1. Infinitely close points, infinite polygons, and tangents





      1. The exponential/logarithmic curve of Leibniz





      1. Newton’s diagram forPrincipiaI, proposition





      1. The catenary, and the problem it solves





      1. Cardioid and an element of area





      1. Newton’s picture of the tangent





      1. Newton’s ‘cissoid’





      1. Leibniz’s illustration for his 1684 paper





  • Chapter 8. Geometries and space



      1. The figure for Euclid’s postulate





      1. Saccheri’s three ‘hypotheses’





      1. ‘Circle Limit III’ by Escher





      1. Geometry on a sphere





      1. Ibn al-Haytham’s idea of proof for postulate





      1. Descriptive geometry





      1. Perspective and projective geometry





      1. Lambert’s quadrilateral





      1. Lobachevsky’s diagram List ofFigures xiii





      1. The parallax of a star





      1. The diagram for Euclid I.16





      1. A ‘large’ triangle on a sphere, showing how proposition I.16 fails





      1. The elements for solving a spherical triangle





      1. Proof of the ‘angles of a triangle’ theorem





      1. Figure for Exercise 1(b)





      1. Figure for Exercise





      1. Figure for Exercise





  • Chapter 9. Modernity and its anxieties



      1. Dedekind cut





      1. The Brouwer fixed point theorem





      1. Circle, torus and sphere





      1. Torus and knotted torus





      1. The ‘dodecahedral space’





      1. A true lover’s knot





      1. Elementary equivalence of projections





      1. The three Reidemeister moves





      1. Two equivalent knots—why?





      1. Graph of a hyperbola





  • Chapter10. A chaotic end?



      1. A ‘half-line angle’





      1. Trigonometric functions from Bourbaki





      1. The ‘butterfly effect’ (Lorenz)





      1. ‘Douady’s rabbit’





      1. The Smale horseshoe map





      1. A string worldsheet, or morphism





      1. The classical helium atom





      1. Elliptic curve (real version)





      1. Torus, or complex points on a projective elliptic curve





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