Book:John Fauvel/The History of Mathematics: A Reader

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