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

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## Carl B. Boyer and Uta C. Merzbach:

## 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*

- Author 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