内容简介

本书作者是世界上最著名的数学史家和教育家之一，他通过本书向读者展示了从古代到近代再到现代数学发展的历史，其中包括数学在东方和西方世界的发展历程。本书第一版因为其通俗易懂、引人入胜，曾获得美国科学史学会颁发的1995年度Watson Davis奖。本书适合作为高等院校数学专业相关课程的教材，同时也适合对数学史感兴趣的读者阅读。本书的主要特点●灵活的组织：本书主要按年代顺序来介绍各地域各时间段数学的发展，而且一直叙述到20世纪。●天文学：因为天文学的发展与数学有着密切的联系，所以书中包含了丰富的天文学方面的内容。●全球视野：书中不仅介绍了欧洲数学，而且还包括中国、印度和伊斯兰世界的数学发展。●典型的习题及部分习题答案：每章都包含很多习题，而且书中还给出了部分习题的答案，通过这些习题读者可以更充分地理解各章的内容。●附加的教学法：附录中给出了在数学教学中如何使用本书内容的细节。

作品目录

preface

chapter one egypt and mesopotamia

1.1 egypt

1.1.1 introduction

1.1.2 number systems and computations

1.1.3 linear equations and proportional reasoning

1.1.4 geometry

1.2 mesopotamia

1.2.1 introduction

1.2.2 methods of computation

1.2.3 geometry

1.2.4 square roots and the pythagorean theorem

1.2.5 solving equations

1.3 conclusion

exercises

references

chapter two greek mathematics to the time of euclid

2.1 the earliest greek mathematics

2.1.1 thales, pythagoras, and the pythagoreans

2.1.2 geometric problem solving and the need for proof

.2.2 euclid and his elements

2.2.1 the pythagorean theorem and its proof

2.2.2 geometric algebra

2.2.3 the pentagon construction

2.2.4 ratio, proportion, and incommensurability

2.2.5 number theory

2.2.6 incommensurability, solid geometry, and the method

of exhaustion

exercises

references

chapter three greek mathematics from archimedes to ptolemy

3.1 archimedes

3.1.1 the determination ofrr

3.1.2 archimedes' method of discovery

3.1.3 sums of series

3.1.4 analysis

3.2 apollonius and the conic sections

3.2.1 conic sections before apollonius

3.2.2 definitions and basic properties of the conics

3.2.3 asymptotes, tangents, and foci

3.2.4 problem solving using conics

3.3 ptolemy and greek astronomy

3.3.1 astronomy before ptolemy

3.3.2 apollonius and hipparchus

3.3.3 ptolemy and his chord table

3.3.4 solving plane triangles

3.3.5 solving spherical triangles

exercises

references

chapter four greek mathematics from diophantus to hypatia

4.1 diophantus and the arithrnetica

4.1.1 linear and quadratic equations

4.1.2 higher-degree equations

4.1.3 the method of false position

4.2 pappus and analysis

4.3 hypatia

exercises

references

chapter five ancient and medieval china

5.1 calculating with numbers

5.2 geometry

5.2.1 the pythagorean theorem and surveying

5.2.2 areas and volumes

5.3 solving equations

5.3.1 systems of linear equations

5.3.2 polynomial equations

5.4 the chinese remainder theorem

5.5 transmission to and from china

exercises

references

chapter six ancient and medieval india

6.1 indian number systems and calculations

6.2 geometry

6.3 algebra

6.4 combinatorics

6.5 trigonometry

6.6 transmission to and from india

exercises

references

chapter seven mathematics in the islamic world

7.1 arithmetic

7.2 algebra

7.2.1 the algebra of al-khwarizmi

7.2.2 the algebra of aba kamil

7.2.3 the algebra of polynomials

7.2.4 induction, sums of powers, and the pascal triangle

7.2.5 the solution of cubic equations

7.3 combinatorics

7.3.1 counting combinations

7.3.2 deriving the combinatorial formulas

7.4 geometry

7.4.1 the parallel postulate

7.4.2 volumes and the method of exhaustion

7.5 trigonometry

7.5.1 the trigonometric functions

7.5.2 spherical trigonometry

7.5.3 values of trigonometric functions

7.6 transmission of islamic mathematics

exercises

references

chapter eight mathematics in medieval europe

8.1 geometry

8.1.1 abraham bar .hiyya's treatise on mensuration

8.1.2 leonardo of pisa's practica geometriae

8.2 combinatorics

8.2.1 the work of abraham ibn ezra

8.2.2 leviben gerson and induction

8.3 medieval algebra

8.3.1 leonardo of pisa's liber abbaci

8.3.2 the work of jordanus de nemore

8.4 the mathematics of kinematics

exercises

references

chapter nine mathematics in the renaissance

9.1 algebra

9.1.1 the abacists

9.1.2 algebra in northern europe

9.1.3 the solution of the cubic equation

9.1.4 bombelli and complex numbers

9.1.5 viete, algebraic symbolism, and analysis

9.2 geometry and trigonometry

9.2.1 art and perspective

9.2.2 the conic sections

9.2.3 regiomontanus and trigonometry

9.3 numerical calculations

9.3.1 simon stevin and decimal fractions

9.3.2 logarithms

9.4 astronomy and physigs

9.4.1 copernicus and the heliocentric universe

9.4.2 johannes kepler and elliptical orbits

9.4.3 galileo and kinematics

exercises

references

chapter ten pre. calculus in the seventeenth century

10.1 algebraic symbolism and the theory of equations

10.1.1 william oughtred and thomas harriot

10.1.2 albert girard and the fundamental theorem of algebra

10.2 analytic geometry

10.2.1 fermat and the introduction to plane and solid loci

10.2.2 descartes and the geometry

10.2.3 the work of jan de witt

10.3 elementary probability

10.3.1 blaise pascal and the beginnings of the theory of probability

10.3.2 christian huygens and the earliest probability text

10.4 number theory

exercises

references

chapter eleven calculus in the seventeenth century

11.1 tangents and extrema

11.1.1 fermat's method of finding extrema

11.1.2 descartes and the method of normals

11.1.3 hudde's algorithm

11.2 areas and volumes

11.2.1 infinitesimals and indivisibles

11.2.2 torricelli and the infinitely long solid

11.2.3 fermat and the area under parabolas and hyperbolas

11.2.4 wallis and fractional exponents

11.2.5 the area under the sine curve and the rectangular hyperbola

11.3 rectification of curves and the fundamental theorem

11.3.1 van heuraet and the rectification of curves

11.3.2 gregory and the fundamental theorem

11.3.3 barrow and the fundamental theorem

11.4 isaac newton

11.4.1 power series

11.4.2 algorithms for calculating fluxions and fluents

11.4.3 the synthetic method of fluxions and newton's physics

11.5 gottfried wilhelm leibniz

11.5.1 sums and differences

11.5.2 the differential triangle and the transmutation theorem

11.5.3 the calculus of differentials

11.5.4 the fundamental theorem and differential equations

exercises

references

chapter twelve analysis in the eighteenth century

12.1 differential equations

12.1.1 the brachistochrone problem

12.1.2 translating newton's synthetic method of fluxions into

the method of differentials

12.1.3 differential equations and the trigonometric functions

12.2 the calculus of several variables

12.2.1 the differential calculus of functions of two variables

12.2.2 multiple integration

12.2.3 partial differential equations: the wave equation

12.3 the textbook organization of the calculus

12.3.1 textbooks in fluxions

12.3.2 textbooks in the differential calculus

12.3.3 euler' s textbooks

12.4 the foundations of the calculus

12.4.1 george berkeley's criticisms and maclaurin's response

12.4.2 euler and d'alembert

12.4.3 lagrange and power series

exercises

references

chapter

thirteen probability and statistics in the eighteenth century

13.1 probability

13.1.1 jakob bernoulli and the ars conjectandi

13.1.2 de moivre and the doctrine of chances

13.2 applications of probability to statistics

13.2.1 errors in observations

13.2.2 de moivre and annuities

13.2.3 bayes and statistical inference

13.2.4 the calculations of laplace

exercises

references

chapter

fourteen algebra and number theory in the eighteenth century

14.1 systems of linear equations

14.2 polynomial equations

14.3 number theory

14.3.1 fermat's last theorem

14.3.2 residues

exercises

references

chapter fifteen geometry in the eighteenth century

15.1 the parallel postulate

15.1.1 saccheri and the parallel postulate

15.1.2 lambert and the parallel postulate

15.2 differential geometry of curves and surfaces

15.2.1 euler and space curves and surfaces

15.2.2 the work of monge

15.3 euler and the beginnings of topology

exercises

references

chapter sixteen algebra and number theory in the nineteenth century

16.1 number theory

16.1.1 gauss and congruences

16.1.2 fermat's last theorem and unique factorization

16.2 solving algebraic equations

16.2.1 cyclotomic equations

16.2.2 the theory of permutations

16.2.3 the unsolvability of the quintic

16.2.4 the work of galois

16.2.5 jordan and the theory of groups of substitutions

16.3 groups and fields -- the beginning of structure

16.3.1 gauss and quadratic forms

16.3.2 kronecker and the structure of abelian groups

16.3.3 groups of transformations

16.3.4 axiomatizafion of the group concept

16.3.5 the concept of a field

16.4 matrices and systems of linear equations

16.4.1 basic ideas of matrices

16.4.2 eigenvalues and eigenvectors

16.4.3 solutions of systems of equations

16.4.4 systems of linear inequalities

exercises

references

chapter

seventeen analysis in the nineteenth century

17.1 rigor in analysis

17.1.1 limits

17.1.2 continuity

17.1.3 convergence

17.1.4 derivatives

17.1.5 integrals

17.1.6 fourier series and the notion of a function

17.1.7 the riemann integral

17.1.8 uniform convergence

17.2 the arithmetization of analysis

17.2.1 dedekind cuts

17.2.2 cantor and fundamental sequences

17.2.3 the theory of sets

17.2.4 dedekind and axioms for the natural numbers

17.3 complex analysis

17.3.1 geometrical representation of complex numbers

17.3.2 complex functions

17.3.3 the riemann zeta function

17.4 vector analysis

17.4.1 surface integrals and the divergence theorem

17.4.2 stokes's theorem

exercises

references

chapter

eighteen statistics in the nineteenth century

18.1 the method of least squares

18.1.1 the work of legendre

18.1.2 gauss and the derivation of the method of least squares

18.2 statistics and the social sciences

18.3 statistical graphs

exercises

references

chapter

nineteen geometry in the nineteenth century

19.1 non-euclidean geometry

19.1.1 taurinus and log-spherical geometry

19.1.2 the non-euclidean geometry of lobachevsky and bolyai

19.1.3 models of non-euclidean geometry

19.2 geometry in n dimensions

19.2.1 grassmann and the ausdehnungslehre

19.2.2 vector spaces

19.3 graph theory and the four-color problem

exercises

references

chapter twenty aspects of the twentieth century

20.1 the growth of abstraction

20.1.1 the axiomatization of vector spaces

20.1.2 the theory of rings

20.1.3 the axiomatization of set theory

20.2 major questions answered

20.2.1 the proof of fermat's last theorem

20.2.2 the classification of the finite simple groups

20.2.3 the proof of the four-color theorem

20.3 growth of new fields of mathematics

20.3.1 the statistical revolution

20.3.2 linear programming

20.4 computers and mathematics

20.4.1 the prehistory of computers

20.4.2 turing and computability

20.4.3 von neumann's computer

exercises

references

appendix using this textbook in teaching mathematics

courses and topics

sample lesson ideas for incorporating history

time line

answers to selected problems

general references in the history of mathematics

index