Book:Carl B. Boyer/A History of Mathematics/Second Edition

Revised edition of by Carl B. Boyer from 1968.

Contents

 * Foreword by Isaac Asimov
 * Preface to the Second Edition (Uta C. Merzbach, Georgetown Texas, March 1991)
 * Preface to the First Edition (Carl B. Boyer, Brooklyn New York, January 1968)


 * Chapter 1. Origins
 * The concept of number
 * Early number bases
 * Number language and the origin of counting
 * Origin of geometry


 * Chapter 2. Egypt
 * Early records
 * Hieroglyphic notation
 * Ahmes Papyrus
 * Unit fractions
 * Arithmetic operations
 * Algebraic problems
 * Geometric problems
 * A trigonometric ratio
 * Moscow Papyrus
 * Mathematical weaknesses


 * Chapter 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


 * Chapter 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


 * Chapter 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


 * Chapter 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


 * Chapter 7. Euclid of Alexandria
 * Author of
 * Other works
 * Purpose of
 * Definitions and postulates
 * Scope of Book l
 * Geometric algebra
 * Books III and IV
 * Theory of proportion
 * Theory of numbers
 * Prime and perfect numbers
 * Incommensurability
 * Solid geometry
 * Apocrypha
 * Influence of


 * Chapter 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


 * Chapter 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


 * Chapter 10. Greek Trigonometry and Mensuration
 * Early trigonometry
 * Aristarchus of Samos
 * Eratosthenes of Cyrene
 * Hipparchus of Necaea
 * Menelaus of Alexandria
 * Ptolemy's Almagest
 * The 36o-degree circle
 * Construction of tables
 * Ptolemaic astronomy
 * Other works by Ptolemy
 * Optics and astrology
 * Heron of Alexandria
 * Principle of least distance
 * Decline of Greek mathematics 175


 * Chapter 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


 * Chapter 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 Sulvasūtras
 * The Siddhāntas
 * Aryabhata
 * Hindu numerals
 * The symbol for zero
 * Hindu trigonometry
 * Hindu multiplication
 * Long division
 * Brahmagupta
 * Brahmagupta's formula
 * Indeterminate equations
 * Bhaskara
 * The Lilavati
 * Ramanujan


 * Chapter 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


 * Chapter 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 latitude of forms
 * Infinite series
 * Decline of medieval learning


 * Chapter 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 Ramée
 * Bombelli's Algebra
 * Johannes Werner
 * Theory of perspective
 * Cartography


 * Chapter 16. Prelude to Modern Mathematics
 * François Viète
 * 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 Bürgi
 * 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
 * Bonaventura Cavalieri
 * The spiral the and parabola


 * Chapter 17. The Time of Fermat and Descartes
 * Leading mathematicians of the time
 * The Discours de la méthode
 * 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 Tonicelli
 * New curves
 * Girard Desargues
 * Projective geometry
 * Blaise Pascal
 * Probability
 * The cycloid


 * Chapter 18. A Transitional Period
 * Philippe de Lahire
 * Georg Mohr
 * Pietro Mengoli
 * Frans van Schooten
 * Jan De Witt
 * Johann Hudde
 * René François de Sluse
 * The pendulum clock
 * Involutes and evolutes
 * John Wallis
 * On Conic Sections
 * Arithmetics infinitorum
 * Christopher Wren
 * Wallis' formulas
 * James Gregory
 * Gregory's series
 * Nicolaus Mercator and William Brouncker
 * Barrows' method of tangents


 * Chapter 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


 * Chapter 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


 * Chapter 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
 * Bézout and elimination


 * Chapter 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
 * Géométrie 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


 * Chapter 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


 * Chapter 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


 * Chapter 25. Analysis
 * Berlin and Göttingen at mid-century
 * Riemann in Göttingen
 * Mathematical physics in Germany
 * Mathematical physics in the English-speaking countries
 * Weierstrass and students
 * The Arithmetization of analysis
 * Cantor and Dedekind
 * Analysis in France


 * Chapter 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


 * Chapter 27. Poincare and Hilbert
 * Turn-of-the-century overview
 * Poincare
 * 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


 * Chapter 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