Book:John Fauvel/The History of Mathematics: A Reader
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John Fauvel and Jeremy Gray: The History of Mathematics: A Reader
Published $\text {1987}$, Macmillan
- ISBN 0-333-42790-4
Subject Matter
An anthology of extracts of mathematical writings from antiquity to recent times
Contents
- Acknowledgements
- Introduction
- Chapter 1 Origins
- 1.A On the Origins of Number and Counting
- 1.A1 Aristotle
- 1.A2 Sir John Leslie
- 1.A3 K. Lovell
- 1.A4 Karl Menninger
- 1.A5 Abraham Seidenberg
- 1.B Evidence of Bone Artefacts
- 1.B1 Jean de Heinzelin on the Ishango bone as evidence of early interest in number
- 1.B2 Alexander Marshack on the Ishango bone as early lunar phase count
- 1.C Megalithic Evidence and Comment
- 1.C1 Alexander Thorn on the megalithic unit of length
- 1.C2 Stuart Piggott on seeing ourselves in the past
- 1.C3 Euan MacKie on the social implications of the megalithic yard
- 1.C4 B. L. van der Waerden on neolithic mathematical science
- 1.C5 Wilbur Knorr's critique of the interpretation of neolithic evidence
- 1.D Egyptian Mathematics
- 1.D1 Two problems from the Rhind papyrus
- 1.D2 More problems from the Rhind papyrus
- 1.D3 A scribe's letter
- 1.D4 Greek views on the Egyptian origin of mathematics
- 1.D5 Sir Alan Gardiner on the Egyptian concept of part
- 1.D6 Arnold Buffum Chace on Egyptian mathematics as pure science
- 1.D7 G. J. Toomer on Egyptian mathematics as strictly practical
- 1.E Babylonian Mathematics
- 1.E1 Some Babylonian problem texts
- 1.E2 Sherlock Holmes in Babylon: an investigation by R. Creighton Buck
- 1.E3 Jöran Friberg on the purpose of Plimpton $\mathit { 322 }$
- 1.E4 The scribal art
- 1.E5 Marvin Powell on two Sumerian texts
- 1.E6 Jens Høyrup on the Sumerian origin of mathematics
- Chapter 2 Mathematics in Classical Greece
- 2.A Historical Summary
- 2.A1 Proclus's summary
- 2.A2 W. Burkert on whether Eudemus mentioned Pythagoras
- 2.B Hippocrates' Quadrature of Lunes
- 2.C Two Fifth-century Writers
- 2.C1 Parmenides' The Way of Truth
- 2.C2 Aristophanes: Meton squares the circle
- 2.C3 Aristophanes: Strepsiades encounters the New Learning
- 2.D The Quadrivium
- 2.D1 Archytas
- 2.D2 Plato
- 2.D3 Proclus
- 2.D4 Nicomachus
- 2.D5 Boethius
- 2.D6 Hrosvitha
- 2.D7 Roger Bacon
- 2.D8 W. Burkert on the Pythagorean tradition in education
- 2.E Plato
- 2.E1 Socrates and the slave boy
- 2.E2 Mathematical studies for the philosopher ruler
- 2.E3 Theaetetus investigates incommensurability
- 2.E4 Lower and higher mathematics
- 2.E5 Plato's cosmology
- 2.E6 Freeborn studies and Athenian ignorance
- 2.E7 A letter from Plato to Dionysius
- 2.E8 Aristoxenus on Plato's lecture on the Good
- 2.F Doubling the Cube
- 2.F1 Theon on how the problem (may have) originated
- 2.F2 Proclus on its reduction by Hippocrates
- 2.F3 Eutocius's account of its early history, and an instrumental solution
- 2.F4 Eutocius on Menaechmus's use of conic sections
- 2.G Squaring the Circle
- 2.G1 Proclus on the origin of the problem
- 2.G2 Antiphon's quadrature
- 2.G3 Bryson's quadrature
- 2.G4 Pappus on the quadratrix
- 2.H Aristotle
- 2.H1 Principles of demonstrative reasoning
- 2.H2 Geometrical analysis
- 2.H3 The distinction between mathematics and other sciences
- 2.H4 The Pythagoreans
- 2.H5 Potential and actual infinities
- 2.H6 Incommensurability
- Chapter 3 Euclid's Elements
- 3.A Introductory Comments by Proclus
- 3.B Book I
- 3.B1 Axiomatic foundations
- 3.B2 The base angles of an isosceles triangle are equal (Proposition 5)
- 3.B3 Propositions 6, 9, 11 and 20
- 3.B4 The angles of a triangle are two right angles (Proposition 32)
- 3.B5 Propositions 44, 45, 46 and 47
- 3.C Books II -- VI
- 3.C1 Book II: Definitions and Propositions 1, 4, 11 and 14
- 3.C2 Book III: Definitions and Proposition 16
- 3.C3 Book V: Definitions
- 3.C4 Book VI: Definitions and Propositions 13, 30 and 31
- 3.D The Number Theory Books
- 3.D1 Book VII: Definitions
- 3.D2 Book IX: Propositions 20, 21 and 22
- 3.D3 Book IX: Proposition 36
- 3.E Books X -- XIII
- 3.E1 Book X: Definitions, Propositions 1 and 9, and Lemma 1
- 3.E2 Book XI: Definitions
- 3.E3 Book XII: Proposition 2
- 3.E4 Book XIII: Statements of Propositions 13--18 and a final result
- 3.F Scholarly and Personal Discovery of Euclid's Text
- 3.F1 A. Aaboe on the textual basis
- 3.F2 Later personal impacts
- 3.G Historians Debate Geometrical Algebra
- 3.G1 B. L. van der Waerden
- 3.G2 Sabetai Unguru
- 3.G3 B. L. van der Waerden
- 3.G4 Sabetai Unguru
- 3.G5 Ian Mueller
- 3.G6 John L. Berggren
- Chapter 4 Archimedes and Apollonius
- 4.A Archimedes
- 4.A1 Measurement of a circle
- 4.A2 The sand-reckoner
- 4.A3 Quadrature of the parabola
- 4.A4 On the equilibrium of planes: Book I
- 4.A5 On the sphere and cylinder: Book I
- 4.A6 On the sphere and cylinder: Book II
- 4.A7 On spirals
- 4.A8 On conoids and spheroids
- 4.A9 The method treating of mechanical problems
- 4.B Later Accounts of the Life and Works of Archimedes
- 4.B1 Plutarch
- 4.B2 Vitruvius
- 4.B3 John Wallis (1685)
- 4.B4 Sir Thomas Heath
- 4.C Diocles
- 4.C1 Introduction to On Burning Mirrors
- 4.C2 Diocles proves the focal property of the parabola
- 4.C3 Diocles introduces the cissoid
- 4.D Apollonius
- 4.D1 General preface to Conics: to Eudemus
- 4.D2 Prefaces to Books II, IV and V
- 4.D3 Book I: First definitions
- 4.D4 Apollonius introduces the parabola, hyperbola and ellipse
- 4.D5 Some results on tangents and diameters'
- 4.D6 How to find diameters, centres and tangents
- 4.D7 Focal properties of hyperbolas and ellipses
- Chapter 5 Mathematical Traditions in the Hellenistic Age
- 5.A The Mathematical Sciences
- 5.A1 Proclus on the divisions of mathematical science
- 5.A2 Pappus on mechanics
- 5.A3 Optics
- 5.A4 Music
- 5.A5 Heron on geometric mensuration
- 5.A6 Geodesy: Vitruvius on two useful theorems of the ancients
- 5.B The Commentating Tradition
- 5.B1 Theon on the purpose of his treatise
- 5.B2 Proclus on critics of geometry
- 5.B3 Pappus on analysis and synthesis
- 5.B4 Pappus on three types of geometrical problem
- 5.B5 Pappus on the sagacity of bees
- 5.B6 Proclus on other commentators
- 5.C Problems Whose Answers Are Numbers
- 5.C1 Proclus on Pythagorean triples
- 5.C2 Problems in Hero's Geometrica
- 5.C3 The cattle problem
- 5.C4 Problems from The Greek Anthology
- 5.C5 An earlier and a later problem
- 5.D Diophantus
- 5.D1 Book I.7
- 5.D2 Book I.27
- 5.D3 Book II.8
- 5.D4 Book III.10
- 5.D5 Book IV(A).3
- 5.D6 Book IV(A).9
- 5.D7 Book VI(A).11
- 5.D8 Book V(G).9
- 5.D9 Book VI(G).19
- 5.D10 Book VI(G).21
- Chapter 6 Islamic Mathematics
- 6.A Commentators and Translators
- 6.A1 The Banu Musa
- 6.A2 Al-Sijzi
- 6.A3 Omar Khayyam
- 6.B Algebra
- 6.B1 Al-Khwarizmi on the algebraic method
- 6.B2 Abu-Kamil on the algebraic method
- 6.B3 Omar Khayyam on the solution of cubic equations
- 6.C The Foundations of Geometry
- 6.C1 Al-Haytham on the parallel postulate
- 6.C2 Omar Khayyam's critique of al-Haytham
- 6.C3 Youshkevitch on the history of the parallel postulate
- Chapter 7 Mathematics in Mediaeval Europe
- 7.A The Thirteenth and Fourteenth Centuries
- 7.A1 Leonardo Fibonacci
- 7.A2 Jordanus de Nemore on problems involving numbers
- 7.A3 M. Biagio: A quadratic equation masquerading as a quartic
- 7.B The Fifteenth Century
- 7.B1 Johannes Regiomontanus on triangles
- 7.B2 Nicolas Chuquet on exponents
- 7.B3 Luca Pacioli
- Chapter 8 Sixteenth-century European Mathematics
- 8.A The Development of Algebra in Italy
- 8.A1 Antonio Maria Fior's challenge to Niccolò Tartaglia (1535)
- 8.A2 Tartaglia's account of his meeting with Gerolamo Cardano in 1539
- 8.A3 Tartaglia versus Ludovico Ferrari (1547)
- 8.A4 Gerolamo Cardano
- 8.A5 Rafael Bombelli
- 8.B Renaissance Editors
- 8.B1 John Dee to Federigo Commandino
- 8.B2 Bernardino Baldi on Commandino
- 8.B3 Paul Rose on Francesco Maurolico
- 8.C Algebra at the Turn of the Century
- 8.C1 Simon Stevin
- 8.C2 François Viète
- Chapter 9 Mathematical Sciences in Tudor and Stuart England
- 9.A Robert Record
- 9.A1 The Ground of Artes
- 9.A2 The Pathway to Knowledge
- 9.A3 The Castle of Knowledge
- 9.A4 The Whetstone of Witte
- 9.B John Dee
- 9.B1 Mathematicall Praeface to Henry Billingsley's Euclid
- 9.B2 Bisecting an angle, from Henry Billingsley's Euclid
- 9.B3 Views of John Dee
- 9.C The Value of Mathematical Sciences
- 9.C1 Roger Ascham (1570)
- 9.C2 William Kempe (1592)
- 9.C3 Gabriel Harvey (1593)
- 9.C4 Thomas Hylles (1600)
- 9.C5 Francis Bacon (1603)
- 9.D Thomas Harriot
- 9.D1 Dedicatory poem by George Chapman
- 9.D2 A sonnet by Harriot
- 9.D3 Examples of Harriot's algebra
- 9.D4 Letter to Harriot from William Lower
- 9.D5 John Aubrey's brief life of Harriot
- 9.D6 John Wallis on Harriot and Descartes
- 9.D7 Recent historical accounts
- 9.E Logarithms
- 9.E1 John Napier's Preface to A Description of the Admirable Table of Logarithms
- 9.E2 Henry Briggs on the early development of logarithms
- 9.E3 William Lilly on the meeting of Napier and Briggs
- 9.E4 Charles Hutton on Johannes Kepler's construction of logarithms
- 9.E5 John Keil on the use of logarithms
- 9.E6 Edmund Stone on definitions of logarithms
- 9.F William Oughtred
- 9.F1 Oughtred's Clavis Mathematicae
- 9.F2 John Wallis on Oughtred's Clavis
- 9.F3 Letters on the value of Oughtred's Clavis
- 9.F4 John Aubrey on Oughtred
- 9.G Brief Lives
- 9.G1 Thomas Allen (1542-1632)
- 9.G2 Sir Henry Savile (1549-1622)
- 9.G3 Walter Warner (1550-1640)
- 9.G4 Edmund Gunter (1581-1626)
- 9.G5 Thomas Hobbes (1588-1679)
- 9.G6 Sir Charles Cavendish (1591-1654)
- 9.G7 René Descartes (1596-1650)
- 9.G8 Edward Davenant
- 9.G9 Seth Ward (1617-1689)
- 9.G10 Sir Jonas Moore (1617-1679)
- 9.H Advancement of Mathematics
- 9.H1 John Pell's Idea of Mathematics
- 9.H2 Letters between Pell and Cavendish
- 9.H3 Hobbes and Wallis
- 9.H4 The mathematical education of John Wallis
- 9.H5 Samuel Pepys learns arithmetic
- Chapter 10 Mathematics and the Scientific Revolution
- 10.A Johannes Kepler
- 10.A1 Planetary motion
- 10.A2 Celestial harmony
- 10.A3 The regular solids
- 10.A4 The importance of geometry
- 10.B Galileo Galilei
- 10.B1 On mathematics and the world
- 10.B2 The regular motion of the pendulum
- 10.B3 Naturally accelerated motion
- 10.B4 The time and distance laws for a falling body
- 10.B5 The parabolic path of a projectile
- Chapter 11 Descartes, Fermat and their Contemporaries
- 11.A René Descartes
- 11.A1 Descartes's method
- 11.A2 The elementary arithmetical operations
- 11.A3 The general method for solving any problem
- 11.A4 Pappus on the locus to three, four or several lines
- 11.A5 Descartes to Marin Mersenne
- 11.A6 Descartes's solution to the Pappus problem
- 11.A7 'Geometric' curves
- 11.A8 Permissible and impermissible methods in geometry
- 11.A9 The method of normals
- 11.A10 H. J. M. Bos on Descartes's Geometry
- 11.B Responses to Descartes's Geometry
- 11.B1 Florimond Debeaune's inverse tangent problem
- 11.B2 Philippe de la Hire on conic sections
- 11.B3 Philippe de la Hire on the algebraic approach
- 11.B4 Hendrik van Heuraet on the rectification of curves
- 11.B5 Jan Hudde's rules
- 11.C Pierre de Fermat
- 11.C1 On maxima and minima and on tangents
- 11.C2 A second method for finding maxima and minima
- 11.C3 Fermat to Bernard de Frenicle on 'Fermat primes'
- 11.C4 Fermat to Marin Mersenne on his 'little theorem'
- 11.C5 Fermat's evaluation of an 'infinite' area
- 11.C6 Fermat's challenge concerning $x^2 = A y^2 + 1$
- 11.C7 On problems in the theory of numbers: a letter to Christaan Huygens
- 11.C8 Fermat's last theorem
- 11.D Girard Desargues
- 11.D1 Preface to Rough Draft on Conics
- 11.D2 The invariance of six points in involution
- 11.D3 Desargues's involution theorem
- 11.D4 Descartes to Desargues
- 11.D5 Pascal's hexagon
- 11.D6 Desargues's theorem on triangles in perspective
- 11.D7 Philippe de la Hire's Sectiones Conicae
- 11.E Infinitesimals, Indivisibles, Areas and Tangents
- 11.E1 Gilles Personne de Roberval on the cycloid
- 11.E2 Blaise Pascal to Pierre de Carcavy
- 11.E3 Isaac Barrow on areas and tangents
- Chapter 12 Isaac Newton
- 12.A Newton's Invention of the Calculus
- 12.A1 Tangents by motion and by the $o$-method
- 12.A2 Rules for finding areas
- 12.A3 The sine series and the cycloid
- 12.A4 Quadrature as the inverse of fluxions
- 12.A5 Finding fluxions of fluent quantities
- 12.A6 Finding fluents from a fluxional relationship
- 12.B Newton's Principia
- 12.B1 Prefaces
- 12.B2 Axioms, or laws of motion
- 12.B3 The method of first and last ratios
- 12.B4 The nature of first and last ratios
- 12.B5 The determination of centripetal forces
- 12.B6 The law of force for an elliptical orbit
- 12.B7 Gravity obeys an inverse square law
- 12.B8 Motion of the apsides
- 12.B9 Against vortices
- 12.B10 Rules of reasoning in philosophy
- 12.B11 The shape of the planets
- 12.B12 General scholium
- 12.B13 Gravity
- 12.C Newton's Letters to Leibniz
- 12.C1 From the Epistola Prior
- 12.C2 From the Epistola Posterior
- 12.D Newton on Geometry
- 12.D1 On the locus to three or four lines
- 12.D2 The enumeration of cubics
- 12.D3 On geometry and algebra
- 12.D4 Newton's projective transformation
- 12.E Newton's Image in English Poetry
- 12.E1 Alexander Pope, Epitaph
- 12.E2 Alexander Pope, An Essay on Man, Epistle II
- 12.E3 William Wordsworth, The Prelude, Book III
- 12.E4 William Blake, Jerusalem, Chapter 1
- 12.F Biographical and Historical Comments
- 12.F1 Bernard de Fontenelle's Eulogy of Newton
- 12.F2 Voltaire on Descartes and Newton
- 12.F3 Voltaire on gravity as a physical truth
- 12.F4 John Maynard Keynes on Newton the man
- 12.F5 D. T. Whiteside on Newton, the mathematician
- Chapter 13 Leibniz and his Followers
- 13.A Leibniz's Invention of the Calculus
- 13.A1 A notation for the calculus
- 13.A2 Debeaune's inverse tangent problem
- 13.A3 The first publication of the calculus
- 13.B Johann Bernoulli and the Marquis de l'Hôpital
- 13.B1 Bernoulli's lecture to l'Hôpital on the solution to Debeaune's problem
- 13.B2 Bernoulli on the integration of rational functions
- 13.B3 Bernoulli on the inverse problem of central forces
- 13.B4 O. Spiess on Bernoulli's first meeting with l'Hôpital
- 13.B5 Preface to l'Hôpital's Analyse des Infiniment Petits
- 13.B6 L'Hôpital on the foundations of the calculus
- 13.B7 Stone's preface to the English edition of l'Hôpital's Analyse des Infiniment Petits
- Chapter 14 Euler and his Contemporaries
- 14.A Euler on Analysis
- 14.A1 A general method for solving linear ordinary differential equations
- 14.A2 Euler's unification of the theory of elementary functions
- 14.A3 Logarithms
- 14.A4 The algebraic theory of conics
- 14.A5 The theory of elimination
- 14.B Euler and Others on the Motion of the Moon
- 14.B1 Pierre de Maupertuis on the figure of the Earth
- 14.B2 Correspondence between Euler and Alexis-Claude Clairaut
- 14.B3 Clairaut on the system of the world according to the principles of universal gravitation
- 14.B4 Euler to Clairaut, 2 June 1750
- 14.C Euler's Later Work
- 14.C1 A general principle of mechanics
- 14.C2 Fermat's last theorem and the theory of numbers
- 14.C3 The motion of a vibrating string
- 14.C4 Nicolas Condorcet's Elogium of Euler
- 14.D Some of Euler's Contemporaries
- 14.D1 Jean-Paul de Gua on the use of algebra in geometry
- 14.D2 Gabriel Cramer on the theory of algebraic curves
- 14.D3 Jean d'Alembert on algebra, geometry and mechanics
- 14.D4 Joseph Louis Lagrange on solvability by radicals
- 14.D5 Joseph Louis Lagrange's additions to Euler's Algebra
- 14.D6 Johann Heinrich Lambert on the making of maps
- Chapter 15 Gauss, and the Origins of Structural Algebra
- 15.A Gauss's Mathematical Writings
- 15.A1 Gauss's mathematical diary for 1796
- 15.A2 Critiques of attempts on the fundamental theorem of algebra
- 15.A3 The constructibility of the regular $17$-gon
- 15.A4 The charms of number theory
- 15.A5 Curvature and the differential geometry of surfaces
- 15.B Gauss's Correspondence
- 15.B1 Three letters between Gauss and Sophie Germain
- 15.B2 Three letters between Gauss and Friedrich Wilhelm Bessel
- 15.C Two Number Theorists
- 15.C1 Adrien Marie Legendre on quadratic reciprocity
- 15.C2 E. E. Kummer: Ideal numbers and Fermat's last theorem
- 15.D Galois Theory
- 15.D1 Evariste Galois's letter to Auguste Chevalier
- 15.D2 An unpublished preface by Galois
- 15.D3 Augustin Louis Cauchy on the theory of permutations
- 15.D4 Camille Jordan on the background to his work on the theory of groups
- Chapter 16 Non-Euclidean Geometry
- 16.A Seventeenth- and Eighteenth-century Developments
- 16.A1 John Wallis's lecture on the parallel postulate
- 16.A2 From Gerolamo Saccheri's Euclides Vindicatus
- 16.A3 Johann Heinrich Lambert to Immanuel Kant
- 16.A4 Kant on our intuition of space
- 16.A5 Lambert on the consequences of a non-Euclidean postulate
- 16.A6 Two attempts by Legendre on the parallel postulate
- 16.B Early Nineteenth-century Developments
- 16.B1 Ferdinand Karl Schweikart's memorandum to Gauss
- 16.B2 Gauss on Janos Bolyai's Appendix
- 16.B3 Nicolai Lobachevskii's theory of parallels
- 16.B4 Correspondence between Wolfgang and Janos Bolyai
- 16.B5 Janos Bolyai's The Science Absolute of Space
- 16.C Later Nineteenth-century Developments
- 16.C1 Roberto Bonola on the spread of non-Euclidean geometry
- 16.C2 Bernhardt Riemann on the hypotheses which lie at the basis of geometry
- 16.C3 Eugenio Beltrami on the interpretation of non-Euclidean geometry
- 16.C4 Felix Klein on non-Euclidean and projective geometry
- 16.C5 J. J. Gray on four questions about the history of non-Euclidean geometry
- 16.D Influences on Literature
- 16.D1 Fyodor Dostoevsky, from The Brothers Karamazov
- 16.D2 Gabriel Garcia Marquez, from One Hundred Years of Solitude
- Chapter 17 Projective Geometry in the Nineteenth Century
- 17.A Developments in France
- 17.A1 Jean Victor Poncelet on a general synthetic method in geometry
- 17.A2 Michel Chasles
- 17.A3 Joseph Diaz Gergonne on the principle of duality
- 17.A4 M. Paul on students' studies at the Ecole Polytechnique
- 17.B Developments in Germany
- 17.B1 August Ferdinand Mobius
- 17.B2 Julius Plücker on twenty-eight bitangents
- 17.B3 From Alfred Clebsch's obituary of Julius Plücker
- 17.B4 Christian Wiener's stereoscopic pictures of the twenty-seven lines on a cubic surface
- Chapter 18 The Rigorization of the Calculus
- 18.A Eighteenth-century Developments
- 18.A1 George Berkeley's criticisms of the calculus
- 18.A2 Colin MacLaurin on rigorizing the fluxional calculus
- 18.A3 D'Alembert on differentials
- 18.A4 Lagrange on derived functions
- 18.A5 Lagrange on algebra and the theory of functions
- 18.B Augustin Louis Cauchy and Bernard Bolzano
- 18.B1 Bolzano on the intermediate value theorem
- 18.B2 Cauchy's definitions
- 18.B3 Cauchy on two important theorems of the calculus
- 18.B4 J. V. Grabiner on the significance of Cauchy
- 18.C Richard Dedekind and Georg Cantor
- 18.C1 Dedekind on irrational numbers and the theorems of the calculus
- 18.C2 Cantor's definition of the real numbers
- 18.C3 The correspondence between Cantor and Dedekind
- 18.C4 Cantor on the uncountability of the real numbers
- 18.C5 Cantor's statement of the continuum hypothesis
- Chapter 19 The Mechanization of Calculation
- 19.A Leibniz on Calculating Machines in the Seventeenth Century
- 19.B Charles Babbage
- 19.B1 Babbage on Gaspard de Prony
- 19.B2 Dionysius Lardner on the need for tables
- 19.B3 Anthony Hyman's commentary on the analytical engine
- 19.B4 Ada Lovelace on the analytical engine
- 19.C Samuel Lilley on Machinery in Mathematics
- 19.D Computer Proofs
- 19.D1 Letter from Augustus De Morgan to William Rowan Hamilton
- 19.D2 Donald J. Albers
- 19.D3 F. F. Bonsall
- 19.D4 Thomas Tymoczo
- Sources
- Name Index
- Subject Index