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