A history of mathematics
(Book)

Origins

Egypt

Mesopotamia

Ionia and the Pythagoreans

Heroic age

Age of Plato and Aristotle

Euclid of Alexandria

Archimedes of Syracuse

Apollonius of Perga

Greek trigonometry and mensuration

Revival and decline of Greek mathematics

China and India

Arabic hegemony

Europe in the Middle Ages

Renaissance

Prelude to modern mathematics

Time of Fermat and Descartes

Transitional period

Newton and Leibniz

Bernoulli era

Age of Euler

Mathematicians of the French Revolution

Time of Gauss and Cauchy

Geometry

Analysis

Algebra

Poincare and Hilbert

Aspects of the twentieth century

References

General bibliography

Appendix: Chronological table

Index.

1. Origins -- The concept of number -- Early number bases -- Number language and the origin of counting -- Origin of geometry -- 2. Egypt -- Early records -- Hieroglyphic notation -- Ahmes papyrus -- Unit fractions -- Arithmetic operations -- Algebraic problems -- Geometric problems -- A trigonometric ratio -- Moscow papyrus -- Mathematical weaknesses -- 3. Mesopotamia -- Cuneiform records -- Positional numeration -- Sexagesimal fractions -- Fundamental operations -- Algebraic problems -- Quadratic equations -- Cubic equations -- Pythagorean triads -- Polygonal areas -- Geometry as applied arithmetic -- Mathematical weaknesses -- 4. Ionia and the Pythagoreans -- Greek origins -- Thales of Miletus -- Pythagoras of Samos -- The Pythagorean pentagram -- Number mysticism -- Arithmetic and cosmology -- Figurate numbers -- Proportions -- Attic numeration -- Ionian numeration -- Arithmetic and logistic --

5. The Heroic Age -- Centers of activity -- Anaxagoras as Clazomenae -- Three famous problems -- Quadrature of lunes -- Continued proportions -- Hippias of Elis -- Philolaus and Archytas of Tarentum -- Duplication of the cube -- Incommensurability -- The golden section -- Paradoxes of Zeno -- Deductive reasoning -- Geometric algebra -- Democritus of Abdera -- 6. The age of Plato and Aristotle -- The seven liberal arts -- Socrates -- Platonic solids -- Theodorus of Cyrene -- Platonic arithmetic and geometry -- Origin of analysis -- Eudoxus of Cnidus -- Method of exhaustion -- Mathematical astronomy -- Menaechmus -- Duplication of the cube -- Dinostratus and the squaring of the circle -- Autolycus of Pitane -- Aristotle -- End of the Hellenic period -- 7. Euclid of Alexandria -- Author of the Elements -- Other works -- Purpose of the Elements -- Definitions and postulates -- Scope of Book I -- Geometric algebra -- Books III and IV -- Theory of proportion -- Theory of numbers -- Prime and perfect numbers -- Incommensurability -- Solid geometry -- Apocrypha -- Influence of the Elements --

8. Archimedes of Syracuse -- The siege of Syracuse -- Law of the lever -- The hydrostatic principle -- The Sand-Reckoner -- Measurement of the circle -- Angle trisection -- Area of a parabolic segment -- Volume of a paraboloidal segment -- Segment of a sphere -- On the sphere and cylinder -- Books of Lemmas -- Semiregular solids and trigonometry -- The Method -- Volume of a sphere -- Recovery of The Method -- 9. Apollonius of Perga -- Lost works -- Restoration of lost works -- The problem of Apollonius -- Cycles and epicycles -- The Conics -- Names of the conic sections -- The double-napped cone -- Fundamental properties -- Conjugate diameters -- Tangents and harmonic division -- The three- and four-line locus -- Intersecting conics -- Maxima and minima, tangents and normals -- Similar conics -- Foci of conics -- Use of coordinates -- 10. Greek trigonometry and mensuration -- Early trigonometry -- Aristarchus of Samos -- Eratosthenes of Cyrene -- Hipparchus of Necaea -- Menelaus of Alexandria -- Ptolemy's Almagest -- The 360-degree circle -- Construction of tables -- Ptolemaic astronomy -- Other works by Ptolemy -- Optics and astronomy -- Heron of Alexandria -- Principle of least distance -- Decline of Greek mathematics --

11. Revival and decline of Greek mathematics -- Applied mathematics -- Diophantus of Alexandria -- Nicomachus of Gerasa -- The Arithmetica of Diophantus -- Diophantine problems -- The place of Diophantus in algebra -- Pappus of Alexandria -- The Collection -- Theorems of Pappus -- The Pappus problem -- The Treasury of analysis -- The Pappus-Guldin theorems -- Proclus of Alexandria -- Boethius -- End of the Alexandrian period -- The Greek anthology -- Byzantine mathematicians of the sixth century -- 12. China and India -- The oldest documents -- The Nine chapters -- Magic squares -- Rod numerals -- The abacus and decimal fractions -- Values of pi -- Algebra and Horner's method -- Thirteenth-century mathematicians -- The arithmetic triangle -- Early mathematics in India -- The Sulvas�utras -- The Siddh�antas -- Aryabhata -- Hindu numerals -- The symbol for zero -- Hindu trigonometry -- Hindu multiplication -- Long division -- Brahmagupta -- Brahmagupta's formula -- Indeterminate equations -- Bhaskara -- The Lilavati -- Ramanujan --

13. The Arabic hegemony -- Arabic conquests -- The House of Wisdom -- Al-jabr -- Quadratic equations -- The father of algebra -- Geometric foundation -- Algebraic problems -- A problem from Heron -- 'Abd al-Hamid ibn-Turk -- Thabit ibn-Qurra -- Arabic numerals -- Arabic trigonometry -- Abu'l-Wefa and al-Karkhi -- Al-Biruni and Alhazen -- Omar Khayyam -- The parallel postulate -- Nasir Eddin -- Al-Kashi -- 14. Europe in the Middle Ages -- From Asia to Europe -- Byzantine mathematics -- The Dark Ages -- Alcuin and Gerbert -- The century of translation -- The spread of Hindu-Arabic numerals -- The Liber abaci -- The Fibonacci sequence -- A solution of a cubic equation -- Theory of numbers and geometry -- Jordanus Nemorarius -- Campanus of Novara -- Learning in the thirteenth century -- Medieval kinematics -- Thomas Bradwardine -- Nicole Oresme -- The latitute of forms -- Infinite series -- Decline of medieval learning --

15. The Renaissance -- Humanism -- Nicholas of Cusa -- Regiomontanus -- Application of algebra to geometry -- A transitional figure -- Nicolas Chuquet's Triparty -- Luca Pacioli's Summa -- Leonardo da Vinci -- Germanic algebras -- Cardan's Ars magna -- Solution of the cubic equation -- Ferrari's solution of the quartic equation -- Irreducible cubics and complex numbers -- Robert Recorde -- Nicholas Copernicus -- Georg Joachim Rheticus -- Pierre de la Ram�ee -- Bombelli's Algebra -- Johannes Werner -- Theory of perspective -- Cartography -- 16. Prelude to modern mathematics -- Fran�cois Vi�ete -- Concept of a parameter -- The analytic art -- Relations between roots and coefficients -- Thomas Harriot and William Oughtred -- Horner's method again -- Trigonometry and prosthaphaeresis -- Trigonometric solution of equations -- John Napier -- Invention of logarithms -- Henry Briggs -- Jobst B�urgi -- Applied mathematics and decimal fractions -- Algebraic notations -- Galileo Galilei -- Values of pi -- Reconstruction of Apollonius' On Tangencies -- Infinitesimal analysis -- Johannes Kepler -- Galileo's Two new sciences -- Galileo and the infinite -- Bonaventure Cavalieri -- The spiral the and parabola --

17. The time of Fermat and Descartes -- Leading mathematicians of the time -- The Discours de la m�ethode -- Invention of analytic geometry -- Arithmetization of geometry -- Geometric algebra -- Classification of curves -- Rectification of curves -- Identification of conics -- Normals and tangents -- Descartes' geometric concepts -- Fermat's loci -- Higher-dimensional analytic geometry -- Fermat's differentiations -- Fermat's integrations -- Gregory of St. Vincent -- Theory of numbers -- Theorems of Fermat -- Gilles Persone de Roberval -- Evangelista Torricelli -- New curves -- Girard Desargues -- Projective geometry -- Blaise Pascal -- Probability -- The cycloid -- 18. A transitional period -- Philippe de Lahire -- Georg Mohr -- Pietro Mengoli -- Frans van Schooten -- Jan De Witt -- Johann Hudde -- Ren�e Fran�cois de Sluse -- The pendulum clock -- Involutes and evolutes -- John Wallis -- On conic sections -- Arithmetica infinitorum -- Christopher Wren -- Wallis' formulas -- James Gregory -- Gregory's series -- Nicolaus Mercator and William Brouncker -- Barrow's method of tangents --

19. Newton and Leibniz -- Newton's early work -- The binomial theorem -- Infinite series -- The Method of fluxions -- The Principia -- Leibniz and the harmonic triangle -- The differential triangle and infinite series -- The differential calculus -- Determinants, notations, and imaginary numbers -- The algebra of logic -- The inverse square law -- Theorems on conics -- Optics and curves -- Polar and other coordinates -- Newton's method and Newton's parallelogram -- The Arithmetica universalis -- Later years -- 20. The Bernoulli era -- The Bernoulli family -- The logarithmic spiral -- Probability and infinite series -- L'Hospital's rule -- Exponential calculus -- Logarithms of negative numbers -- Petersburg paradox -- Abraham De Moivre -- De Moivre's theorem -- Roger Cotes -- James Stirling -- Colin Maclaurin -- Taylor's series -- The Analyst controversy -- Cramer's rule -- Tschirnhaus transformations -- Solid analytic geometry -- Michel Rolle and Pierre Varignon -- Mathematics in Italy -- The parallel postulate -- Divergent series --

21. The age of Euler -- Life of Euler -- Notation -- Foundation of analysis -- Infinite series -- Convergent and divergent series -- Life of d'Alembert -- The Euler identities -- D'Alembert and limits -- Differential equations -- The Clairauts -- The Riccatis -- Probability -- Theory of numbers -- Textbooks -- Synthetic geometry -- Solid analytic geometry -- Lambert and the parallel postulate -- B�ezout and elimination -- 22. Mathematicians of the French Revolution -- The age of revolutions -- Leading mathematicians -- Publications before 1789 -- Lagrange and determinants -- Committee on Weights and Measures -- Condorcet on education -- Monge as administrator and teacher -- Descriptive geometry and analytic geometry -- Textbooks -- Lacroix on analytic geometry -- The organizer of victory -- Metaphysics of the calculus and geometry -- G�eom�etrie de position -- Transversals -- Legendre's Geometry -- Elliptic integrals -- Theory of numbers -- Theory of functions -- Calculus of variations -- Lagrange multipliers -- Laplace and probability -- Celestial mechanics and operators -- Political changes --

23. The time of Gauss and Cauchy -- Nineteenth-century overview -- Gauss : early work -- Number theory -- Reception of the Disquisitiones arithmeticae -- Gauss's contributions to astronomy -- Gauss's middle years -- The beginnings of differential geometry -- Gauss's later work -- Paris in the 1820s -- Cauchy -- Gauss and Cauchy compared -- Non-Euclidean geometry -- Abel and Jacobi -- Galois -- Diffusion -- Reforms in England and Prussia -- 24. Geometry -- The school of Monge -- Projective geometry : Poncelet and Chasles -- Synthetic metric geometry : Steiner -- Synthetic nonmetric geometry : von Staudt -- Analytic geometry -- Riemannian geometry -- Spaces of higher dimensions -- Felix Klein -- Post-Riemannian algebraic geometry -- 25. Analysis -- Berlin and G�ottingen at mid-century -- Riemann in G�ottingen -- Mathematical physics in Germany -- Mathematical physics in the English-speaking countries -- Weierstrass and students -- The arithmetization of analysis -- Cantor and Dedekind -- Analysis in France --

26. Algebra

Introduction

British algebra and the operational calculus of functions

Boole and the algebra of logic

De Morgan

Hamilton

Grassmann and Ausdehnungslehre

Cayley and Sylvester

Linear associative algebras

Algebraic geometry

Algebraic and arithmetic integers

Axioms of arithmetic

27. Poincar�e and Hilbert

Turn-of-the-century overview

Poincar�e

Mathematical physics and other applications

Topology

Other fields and legacy

Hilbert

Invariant theory

Hilbert's Zahlbericht

The foundations of geometry

The Hilbert problems

Hilbert and analysis

Waring's problem and Hilbert's work after 1909

28. Aspects of the twentieth century

General overview

Integration and measure

Functional analysis and general topology

Algebra

Differential geometry and tensor analysis

The 1930s and World War II

Probability

Homological algebra and category theory

Bourbaki

Logic and computing

Future outlook

References

General bibliography

Appendix : Chronological table

Index.