Travelling Mathematics - The Fate of Diophantos' Arithmetic

214 Pages·2010·7.32 MB·English

by  Ad Meskens (auth.)

Science Networks . Historical Studies
Founded by Erwin Hiebert and Hans Wußing
Volume 41
Edited by Eberhard Knobloch, Helge Kragh and Erhard Scholz
Editorial Board:
K. Andersen, Aarhus R. Halleux, Liège
D. Buchwald, Pasadena S. Hildebrandt, Bonn
H.J.M. Bos, Utrecht Ch. Meinel, Regensburg
U. Bottazzini, Roma J. Peiffer, Paris
J.Z. Buchwald, Cambridge, Mass. W. Purkert, Bonn
K. Chemla, Paris D. Rowe, Mainz
S.S. Demidov, Moskva A.I. Sabra, Cambridge, Mass.
E.A. Fellmann, Basel Ch. Sasaki, Tokyo
M. Folkerts, München R.H. Stuewer, Minneapolis
P. Galison, Cambridge, Mass. H. Wußing, Leipzig
I. Grattan-Guinness, London V.P. Vizgin, Moskva
J. Gray, Milton Keynes
Ad Meskens
Travelling
Mathematics -
The Fate of Diophantos'
Arithmetic
Ad Meskens
Artesis Hogeschool Antwerpen
Department Bedrijfskunde,
Lerarenopleiding en Sociaal Werk
Verschansingstraat 29
2000 Antwerpen
Belgium
e-mail: [email protected]
Fig. 1.1 © Stephen Chrisomalis, published with kind permission. All drawings by Paul Tytgat, with
kind permission.
ISBN 978-3-0346-0642-4 e-ISBN 978-3-0346-0643-1
DOI 10.1007/978-3-0346-0643-1
Library of Congress Control Number: 2010935851
© Springer Basel AG 2010
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Cover illustration: From Waller Ms de-00215, August Beer: Über die Correction des Cosinusgesetzes bei der
Anwendung des Nicol’schen Prismas in der Photometrie, after 1850. With friendly permission by The Waller
Manuscript Collection (part of the Uppsala University Library Collections).
Cover design: deblik
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Contents
Preface vii
1 Arithmetic and the beginnings of algebra 1
1.1 In the beginning... . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Classical Greece. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 The Greek written heritage . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Numbers in Classical Greece. . . . . . . . . . . . . . . . . . . . . . 16
1.5 All is number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 Alexandria ad Aegyptum 27
2.1 The capital of memory . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Diophantos’ Alexandria . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Education and the culture of paideia . . . . . . . . . . . . . . . . . 35
2.4 Heron of Alexandria: a Diophantine precursor? . . . . . . . . . . . 37
3 Diophantos and the Arithmetika 43
3.1 The manuscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Diophantos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 The book On Polygonal Numbers and the lost books . . . . . . . . 49
3.4 Symbolism in the Arithmetika . . . . . . . . . . . . . . . . . . . . . 52
3.5 The structure of a Diophantine problem . . . . . . . . . . . . . . . 57
3.6 The Arithmetika . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.7 The algebra of the Arithmetika . . . . . . . . . . . . . . . . . . . . 78
3.8 Interpretations of algebra in the Arithmetika . . . . . . . . . . . . 88
3.9 Negative numbers? . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4 Sleeping beauty in the Dark Ages 103
4.1 The sins of the Fathers... . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 From Alexandria to Baghdad . . . . . . . . . . . . . . . . . . . . . 107
4.3 The Byzantine connection . . . . . . . . . . . . . . . . . . . . . . . 113
4.4 Diophantos reinvented: Fibonacci . . . . . . . . . . . . . . . . . . . 117
vi Contents
5 New vistas 123
5.1 Printed by ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 Wherefore art thou number? . . . . . . . . . . . . . . . . . . . . . 125
5.3 From the rule of coss to algebra . . . . . . . . . . . . . . . . . . . 128
6 Humanism 133
6.1 Trait d’union: Bessarion and the humanists . . . . . . . . . . . . . 133
6.2 Diophantos goes north: Regiomontanus. . . . . . . . . . . . . . . . 135
7 Renaissance or the rebirth of Diophantos 139
7.1 Xylander: A sphinx to solve a riddle . . . . . . . . . . . . . . . . . 139
7.2 Coincidence of traditions: Rafael Bombelli . . . . . . . . . . . . . . 142
7.3 The great art: Guillaume Gosselin . . . . . . . . . . . . . . . . . . 146
7.4 The marvel is no marvel: Simon Stevin . . . . . . . . . . . . . . . . 150
8 Fair stood the wind for France 155
8.1 Diophantos’ triangles: François Viète and the New Algebra . . . . . 155
8.2 Emulating the Ancients: Claude-Gaspar Bachet de Méziriac . . . . 161
8.3 This margin is too small... . . . . . . . . . . . . . . . . . . . . . . 166
9 Coda: Hilbert’s tenth problem 171
10 Stemma 173
Bibliography 177
Index 201
Preface
AnyonestudyingtheworkofDiophantosofAlexandriaisimmediatelyconfronted
withthreequestions:‘WhowasDiophantos?’,‘Howlargewashisalgebraicknowl-
edge?’ and ‘To what extent are Diophantos’ writings unique within the classical
mathematical corpus?’
Unfortunately, none of these questions can be answered satisfactorily. Not only
is there scant biographical evidence to go on, but the Diophantine corpus also
remains rather elusive. Depending on one’s position, it lends itself to either mini-
mization or hineininterpretierung. Nonetheless, Diophantos’ writings have, along-
sidetheworkofHeron,cometooccupyaspecialpositionwithintheGreekmath-
ematical corpus. Or rather within the known ancient mathematical corpus. This
qualification is not unimportant, particularly as nearly all of our knowledge of
ancient mathematics has come to us via copies, implying that it is to an extent
filtered by the choices and judgments of mediæval scribes.
Be that as it may, Diophantos’ Arithmetika, despite its relative isolation within
the classical Greek corpus, represents a phase in the evolution from a syntactic to
a symbolic algebra, which we may refer to as syncoptic algebra. In the opening
chapters,weshallthereforefocusfirstonthedevelopmentofalgebrainEgyptand
Babylonbeforeattemptingtoformulate answers totheaforementionedquestions.
Subsequently,weshallconsiderhowtheArithmetika influencedtheworkofPierre
de Fermat.
The classic English study in this field is still Sir Thomas Heath’s translation of
Diophantos’sixGreekbooks.However,sincethepublicationofthisseminalwork,
many new facts have come to light, not in the least through the discovery of some
Arabic manuscripts. It would therefore seem worthwhile to provide an updated
account of our understanding of Diophantos and his writings.
The idea for this study came to me during my work on sixteenth-century
Netherlandishalgebra,moreinparticularthewritingsofSimonStevin,whotrans-
lated four of the six Greek Diophantos books into French. This first and foremost
providedtheinspirationforanewDutchtranslation(thefirst,infact,inover250
years) of the Greek books of the Arithmetika. This Dutch version was produced
by my wife, a classical scholar. In addition to the translation, however, we felt
it necessary also to explain the mathematics. Our edition, published by Antwerp
viii Preface
University College Press (now Artesis University College Antwerp Press) in 2006,
therefore has a special layout: the right-hand pages contain the translation, while
theleft-handpagesprovideamathematicalelucidationinmodernnotation1.This
is preceded by an introduction, which –due to space restrictions– had to be short-
ened,leavinguswiththefeelingthatDiophantosandthefateofhisbooksmerited
a more extensive account. One thing led to another, and hence this monograph.
There are essentially three ways a contemporary author can render ancient
mathematics.Hecanprovideafull-lengthtranslationoftheoriginaltextandleave
theinterpretationtothereader,orhecanrenderthetextinmodernmathematical
notation, or he can use a semi-symbolic notation that some argue can capture the
flavour of the ancient text while also offering the conciseness of modern notation.
The latter option, however, would leave the reader with such cumbersome formu-
√
lae as: @A ((cid:3)a,(cid:3)b) =@A (a,b), which a contemporary mathematician would
√
immediately translate as a2b2 =ab.
We feel that such a semi-symbolic notation fails on both counts: it is too far re-
moved from the original flavour of the text as well as its mathematical content,
and consequently there is a risk it will merely alienate the modern reader from
the essence of Diophantos. We have therefore chosen instead to cite the origi-
nal text where appropriate and to indicate which pitfalls may present themselves
when reading it. However, as excessive citing would become tedious and distract
the modern reader, we also rely on anachronistic mathematical formulae. We feel
this approach offers closer insight into the thinking underlying the writings dis-
cussedhere,whilealsodoingfulljusticetotheirancientauthors’accomplishments.
Although the contemporary reader must always bear in mind the Italian adage
traduttore traditore – in respect of both the translation of the ancient text and
its mathematical reformulation – we believe that the combination of text and for-
mulae provides a good basis for a better understanding of the Arithmetika and
related texts, as well as the issues that ancient mathematicians faced in noting
down their work.
In the ten-or-so years it has taken to conclude this study, we have been sup-
ported by many people and institutions. Over this period, we have contracted
debtsofgratitudewecanhardlycontemplateeverbeingabletorepayadequately.
We were also fortunate enough to see our Dutch translation of Diophantos being
awardedtheJanGillisprizeoftheRoyalFlemishAcademyofBelgiumforScience
and the Arts, which provided the necessary funds for further research. We are
also indebted to Artesis University College for their support through a scientific
projectgrantandfortheirpermissiontousepartsofourintroductiontotheDutch
Diophantos translation in this monograph. The present study would never have
materialized either without the cooperation of various libraries, particularly the
libraryoftheTeacherTrainingCollegeoftheArtesisUniversityCollegeAntwerp,
theErfgoedbibliotheekHendrikConscience(Antwerp)andMuseumPlantinMore-
1Heath’s translation is in fact a compromise between a translation of the Greek text and a
retranslation into mathematical symbolism, which has left us with neither a good edition, nor
goodmathematics.Nonetheless,itsucceedsbrilliantlyinbringingacrosstheDiophantineflavour.
Preface ix
tus (Antwerp). We are also grateful to Stephen Chrisomalis (Wayne State Uni-
versity) for his kind permission to reproduce the illustration of demotic figures.
Special thanks are due to two colleagues for reading drafts of the manuscript and
for making useful suggestions: Guido Vanden Berghe and, as always, Jean Paul
van Bendegem. The long-standing cooperation with Paul Tytgat (Department of
Industrial Sciences) once again proved invaluable: most of the drawings in this
book are by his hand. Stephen Windross read and corrected the English draft.
Finally,ImustthankmywifeNicoleforhercontinuoussupportoverthepast
twenty years. Her contribution was a genuine labour of love.
Ad Meskens, 2010