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

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Carl B. Boyer and Uta C. Merzbach: A History of Mathematics (2nd Edition)

Published $\text {1991}$, John Wiley & Sons, Inc.

ISBN 0-471-54397-7

Revised edition of A History of Mathematics by Carl B. Boyer from 1968.


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
Attic numeration
Ionian numeration
Arithmetic and logistic
Chapter 5. The Heroic Age
Centers of activity
Anaxagoras of Clazomenae
Three famous problems
Quadrature of lunes
Continued proportions
Hippias of Elis
Philolaus and Archytas of Tarentum
Duplication of the cube
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
Platonic solids
Theodorus of Cyrene
Platonic arithmetic and geometry
Origin of analysis
Eudoxus of Cnidus
Method of exhaustion
Mathematical astronomy
Duplication of the cube
Dinostratus and the squaring of the circle
Autolycus of Pitane
End of the Hellenic period
Chapter 7. Euclid of Alexandria
Author of The Elements
Other works
Purpose of The Elements
Definitions and postulates
Scope of Book l
Geometric algebra
Books III and IV
Theory of proportion
Theory of numbers
Prime and perfect numbers
Solid geometry
Influence of The Elements
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
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
Hindu numerals
The symbol for zero
Hindu trigonometry
Hindu multiplication
Long division
Brahmagupta's formula
Indeterminate equations
The Lilavati
Chapter 13. The Arabic Hegemony
Arabic conquests
The House of Wisdom
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
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
Nicholas of Cusa
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
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
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
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
Theory of numbers
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
Lacroix on analytic geometry
The Organizer of Victory
Metaphysics of the calculus and geometry
Géométrie de position
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
Gauss and Cauchy compared
Non-Euclidean geometry
Abel and Jacobi
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
British algebra and the operational calculus of functions
Boole and the algebra of logic
De Morgan
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
Mathematical physics and other applications
Other fields and legacy
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
Differential geometry and tensor analysis
The 1930s and World War II
Homological algebra and category theory
Logic and computing
Future outlook
General Bibliography
Appendix: Chronological Table