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

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## John Fauvel and Jeremy Gray:

## 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**