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

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.

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 of 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 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

Incommensurability

Solid geometry

Apocrypha

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

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