Diophantus of Alexandria: a Text and its History

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Norbert Schappacher


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Diophantus’s Arithmetica is one of the most influential works in the history of mathematics. For instance, it was in the margin of his edition of Diophantus that Pierre de Fermat, some day between 1621 and 1665, wrote the statement of his so-called Last Theorem (which was proved only a few years ago). But Fermat was not the first to derive inspiration from Diophantus’s collection of algebraic/arithmetic problems: the Arabs had profited from reading the Arithmetica when developing Algebra as a mathematical discipline. Nor was he the last: at the end of this paper we present the example of a Berkeley Ph.D. thesis of 1998 which is directly inspired by a problem of Diophantus.
But for all the influence that this author had on various mathematicians at various times, we know almost nothing about him,and even the text of the “Arithmetica” betrays very little of whatDiophantus actually knew — or did not know….

Norbert Schappacher 

2010c    [XX]

Leo Corry & N. Sch. :

Zionist Internationalism through Number Theory. Edmund Landau at the Opening of the Hebrew University in 1925.

Science in Context  23 (4) (2010), 427-471.

2010b    [XIX / XX]

Rewriting Points. (Invited talk, History of Mathematics section)

Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010,

pp. 3258-3291.

2010a    [XX]

Eckhard Wirbelauer & N. Sch. :

Zwei Siegeruniversitäten: Die Straßburger Universitätsgründungen von 1872 und 1919.

Jahrbuch für Universitätsgeschichte 13 (2010); pp. 45-72.

2009   [XVIII]

Von Klangfarben, Predigtmaschinen und den Grenzen der Analysis bei Leonhard Euler.

In: Mathesis und Graphé, Leonhard Euler und die Entfalting der Wissenssytseme (H. Bredekamp, W. Velminski, ed.), Akademie Verlag (Berlin) 2009; pp. 209-224.

See also my earlier publication 2007d.

2008b    [XX]

How to describe the transtion towards new mathematical practice: the example of Algebraic Geometry 1937 – 1954.

In  : Report 24/2008 of Oberwolfach Reports.

This is just a summary of my lecture at the Oberwolfach History of Mathematics meeting on 26 May 2008.

2008a    [Φιλ]

Bemerkungen zu Wittgensteins Philosophie der Mathematik in ihrer Beziehung zur Mathematikgeschichte.

In:  « Ein Netz von Normen », Wittgenstein und die Mathematik, M. Kross (ed.). Parerga Verlag (Berlin) 2008; pp. 199-234.

This write-up of my contribution to the 2005 roundtable on Wittgenstein’s philosophy of mathematics at Einstein Forum, Potsdam, Germany, has two parts: in the first, longer section I explore the influence of H. Weyl’s foundational views on Wittgenstein; the second, rather aphoristic section sketches potential applications of Wittgenstein’s approach to mathematical practice to the writing of the history of mathematics.

2007d   [XVIII]

Der nahe und der ferne Euler.

Elemente der Mathematik  62 (2007), 134-154.

On the occasion of the tricentenary of Leonhard Euler’s birth (7 April 1707), I gave talks on Euler in Germany, Italy, Switzerland, and France. Two publications have come from this: 2007d and 2009. They attempt to describe facets of Euler’s thought  within his century, and they owe a lot to my reading of  The Sciences in Enlightened Europe (W. Clark, J. Golinski, S. Schaffer, eds.), Chicago & London 1999. Furthermore, I am preparing the Correspondence between L. Euler and J.H. Lambert for Euler’s Opera Omnia, ser. IV A.

2007c    [XX]

Sigrid Oehler-Klein & N. Sch. :

Siegfried Koller und die neuen Herausforderungen der Statistik im Nationalsozialismus.

In : Die Medizinische Fakultät der Universität Gießen im Nationalsozialismus und in der Nachkriegszeit: Personen und Institutionen, Umbrüche und Kontinuitäten, S. Oehler-Klein (ed.). Franz Steiner Verlag (Stuttgart) 2007; pp. 247-262.

Published on the occasion of Giessen University’s 400th anniversary, the weighty book on the Giessen Medical Faculty under National Socialism contains this short joint paper with Mrs. Oehler-Klein on the first career of the applied statistician Siegfried Koller. This career led him from a thesis under Felix Bernstein in Göttingen, via a second Ph.D. in Medicine at Giessen and thanks to politically well-tuned publications, to a professorship at Berlin University (with his own institute). But this appointment took place only shortly before the end of WW II. (After a number of years in prison, Koller’s second career then made him one of the most infuential statisticians of the Federal Republic of Germany.)

2007b    [XX]

A Historical Sketch of B.L. van der Waerden’s Work on Algebraic Geometry 1926-1946.

In : Episodes in the History of Modern Algebra (1800-1950), J.J. Gray & K.H. Parshall (eds.).

History of mathematics series, vol. 32. AMS / LMS 2007; pp. 245-283.

The proceedings volume of a 2003 conference at MSRI Berkeley, California, on facets of the History of Modern Algebra provided me with the occasion to write up a first installment of my bigger project on the history of Algebraic Geometry between 1919 and 1954. (The pdf-file linked to this entry is word-identical with the text printed in the  volume, but differs in page layout.)

2007a    [XIX]

The Shaping of Arithmetic after C.F. Gauss’s Disquisitiones Arithmeticae.

Catherine Goldstein, N. Sch., Joachim Schwermer (eds.).

Springer Verlag (Berlin, Heidelberg, etc.) 2007.  xii+578 pp.

At long last, this book which we had been working on for several years came out of the press. It grew out of two meetings at Oberwolfach, but several chapters have been worked out only later, in particular the three chapters listed below. In the two opening chapters written wth Catherine Goldstein, we address the subtle notion of what a scientific discipline is, by following very carefully all the multiple and variegated influences which Gauss’s book exerted throughout the 19th century.

2007a1    chap. I.1

Catherine Goldstein & N. Sch. : A book in search of a discipline (1801-1860); pp. 3-65.

2007a2     chap. I.2

Catherine Goldstein & N. Sch. : Several disciplines and a book (1860-1901); pp. 67-103.

2007a3     chap. V.2

Birgit Petri & N. Sch. : On arithmetization; pp. 343-374.

2006b    [XX]

Nazisme, science et médecine.

Christian Bonah, Anne Danion-Grillat, Josiane Olff-Nathan, N. Schappacher (ed.s) :

Paris: Glyphe 2006.  366 pp.

Based on a conference which we organized at Strasbourg in November 2005, the following little book is among the first books in French which contain some of the new perspectives and results elaborated by the research commission for the history of the Kaiser-Wilhlem-Gesellschaft which the President of Max-Planck-Gesellschaft had created in 1997. My chapter within this book:

N. Schappacher: De Felix Bernstein à Siegfried Koller – des implications politiques des statisticiens; pp. 15-40 & 291-295.

2006a    [XX]

Seventy years ago : The Bourbaki Congress at El Escorial and other mathematical (non)events of 1936.

The Mathematical Intelligencer, Special issue International Congress of Mathematicians Madrid August 2006, 8-15.

2008c     Spanish translation by Adolfo Quirós : El Congreso Bourbaki en El Escorial y otros (no) acontecimientos matemáticos de 1936. La Gaceta de la RSME 11-4 (2008), 721-735.

Asked for a contribution by my Spanish friends, I wrote a little piece about the politics and mathematics of 1936, for the special issue of the Mathematical Intelligencer which was distributed to all participants of the Madrid ICM in 2006. As I found out after the event, this special issue never made it to the regular subscribers of the Math Intelligencer; this is one more reason to post my article here. [In retrospect, I regret to have passed over 1936  French politics in silence: Front populaire, congés payés, etc. On the other hand, their consequences for French academia are best appreciated by looking at the following  years….]

2005c    [XX]

Felix Bernstein.

International Statistical Review 73/1 (2005), 3-7.

Just a short biographical note; a bigger project on Bernstein is under way in collaboration with Reinhard Siegmund-Schultze.

2005b    [XIX-XX]

N. Sch. & Klaus Volkert:

Heinrich Weber, un mathématicien à Strasbourg 1895 – 1913.

In La science sous influence. L’université de Strasbourg, enjeu des conflits franco-allemands 1872-1945.

(sous la direction d’Elisabeth Crawford et Josiane Olff-Nathan).

Strasbourg: La Nuée Bleue 2005; pp. 37-47.

This article had actually been written quite a bit earlier, the version posted here is more complete than the one printed in the book. In the meantime, I also wrote the entry on Heinrich Weber for the Nouveau Dictionnaire de Biographie Alsacienne. – The book La science sous influence took a long time to appear, not least because of Elisabeth Crawford’s fatal illness. – This book also contains a very brief report (pp. 253-256), which we wrote with Josiane Olff-Nathan, on then recent research concerning the Nazi Reichsuniversität Strassburg (1941-1944/45). This subject was taken up again in 2006b.

2005a    [XIX]

David Hilbert, report on algebraic number fields (“Zahlbericht”).

In Landmark Writings in Western Mathematics (I. Grattan-Guinness, ed.).

Amsterdam – Boston – etc.: Elsevier 2005; pp. 700-709.

Cf. our introduction 1998c, with Franz Lemmermeyer, to the first english edition of the “Zahlbericht.” See also 2007a3.

2004b    [XIX]

Birgit Petri & N. Sch. :

From Abel to Kronecker. Episodes from 19thCentury Algebra.

In The Legacy of Niels Henrik Abel (O.A. Laudal & R. Piene, ed.s).

Berlin, Heidelberg, etc.: Springer 2004, pp. 227-266.

Studying Kronecker always provides an alternative view of 19th century mathematical developments which one thought one had well understood….

2003     [XX]

Politisches in der Mathematik. Versuch einer Spurensicherung.

Mathematische Semesterberichte 50/1 (2003), 1-27.

A slightly expanded version of my inaugurational lecture at Darmstadt. This text was subsequently translated into Spanish by José Ferreirós :

2004a    Lo político en matemáticas: un intento de rastreo.

La Gaceta de la Real Sociedad Matemática Española 8/1 (2004), 129-157.

2002      [XIX]

Jacqueline Boniface & N. Sch. :

‘Sur le concept de nombre en mathématique’.

Cours inédit de Leopold Kronecker à Berlin (1891).

Revue d’histoire des mathématiques 7 (2001) [printed July 2002], 207-275.

This is an edition, with introduction and commentary, of one of the handwritten lecture notes of Kronecker’s lecture courses from Kurt Hensel’s private mathematical library. This library was sold to Strasbourg during WW II.  Its finest pieces are being digitized and put on the web; downloading them should soon be rendered more practicable. Click here for the original lecture notes from 1891. – Incidentally, an unidentified quote (see p. 240, footnote 41) has meanwhile been supplied; see footnote 80 of 2007a3.

2000      [Μαθ]

Alexander Reznikov, N. Sch. (ed.s) :

Regulators in Analysis, Geometry and Number Theory.

Progress in Mathematics 171 (Birkhäuser) 2000.

In memoriam Саша Резников : И я выхожу из пространства / В запущенный сад величин, /  И минмое рву постоянство / И самосознанье причин. // И твой, бесконечность, учебник / читаю один, без людей – / Беслиственный. дикий лечебник, – / Задачник огромных корней.  (О.М.)

1999c     [Ιστ – μετ]

Gerd Gigerenzer, Zeno Swijtink, Theodore Porter, Lorrraine Daston, Joan Beatty, Lorenz Krüger,

Das Reich des Zufalls – Wissen zwischen Wahrscheinlichkeiten, Häufigkeiten und Unschärfen.

Aus dem Englischen übersetzt von Christa Krüger und N. Schappacher.

Heidelberg – Berlin (Spektrum Verlag), 1999.

1999b     [Μαθ – μετ]

Jürgen Neukirch, Algebraic Number Theory,

translated from the German by N. Schappacher,

Springer Grundlehren 322, 1999.

1999a     [Μαθ]

Jan Nekovar & N. Sch. :

On the asymptotic behaiviour of Heegner points

Turkish Journal of Mathematics 23/4 (1999), 549-556.

This result of mathematical armchairing with Jan was still a long way from proving Mazur’s Conjecture on Heegner points. The latter was, however, established within a few years, using Ratner’s theorems on closures of unipotent flows on p-adic Lie groups, by Nike Vatsal, and by Christophe Cornut in his Strasbourg thesis.

1998f     [XIX-XX]

On the History of Hilbert’s Twelfth Problem, A Comedy of Errors.

In Matériaux pour l’histoire des mathématiques au XXe siècle,

Actes du colloque à la mémoire de Jean Dieudonné (Nice, 1996),

Séminaires et Congrès (Société Mathématique de France) 3 (1998); pp. 243-273.

1998e     [XIX-XX]

H.G.W. Begehr, H. Koch, J. Kramer, N. Sch., E.-J. Thiele

(eds. on behalf of the Berliner Mathematische Gesellschaft) :

Mathematics in Berlin.

Berlin – Basel – Boston: Birkhäuser 1998.  xii+200 pp.

Apart from being coeditor, I wrote two chapters for this booklet produced for historically interested participants at the Berlin ICM of 1998:

1998e1    Gotthold Eisenstein, 16 April 1823 – 11 October 1852; pp. 55-60.

1998e2    The Nazi era: the Berlin way of politicizing mathematics; pp. 127-136.

1998d     [XIX-XX]

Franz Lemmermeyer & N. Sch. :

Introduction to the first English edition of Hilbert’s Zahlbericht.

In D. Hilbert, The Theory of Algebraic Number Fields.

Berlin, Heidelberg, etc.: Springer 1998; pp. XXIII – XXXVI.

See also 2005a. On some delicate points of Adamson’s translation, cf. 2007a3, p. 365.

1998c     [Ιστ]

“Wer war Diophant?”

Mathematische Semesterberichte 45/2 (1998), 141-156.

This is essentially the talk I gave on 29 May 1998 at the Euler-Vorlesung in Sanssouci (Potsdam). It is an exercise in style about some basic methodological problems in the history of mathematics. I subsequently translated and reworked the piece, but never published this improved version in print :

Diophantus of Alexandria. A Text and its History.

1998b     [Μαθ]

Klaus Rolshausen & N. Sch. :

On the second K-group of an elliptic curve.

Journal für die reine und angewandte Mathematik 495 (1998), 61-77.

1998a     [XX]

Beispiele und Gedanken zu den Auswirkungen des Kriegsendes auf die Mathematik in Deutschland.

Sitzungsberichte der Berliner Mathematischen Gesellschaft 1993-1996 (published in 1998), 153-167.

1997     [XIX-XX]

Some Milestones of Lemniscatomy.

In Algebraic Geometry (S. Sertöz, ed.), Proc. Bilkent Summer School, Ankara 1995.

Lecture Notes in Pure and Applied Mathematics Series 193, New York: M. Dekker 1997; pp. 257-290.

1996     [XX]

N. Sch. & René Schoof :

Beppo Levi and the arithmetic of elliptic curves.

Mathematical Intelligencer 18 (1996), 57-69.

1994b     [Μαθ]

C. Kassel, J-L. Loday. N. Sch. (eds.) :

K-theory Strasbourg 1992.

Astérisque 226, 1994.

1994a     [Μαθ]

CM motives and the Taniyama group.

In: Motives (Jannsen, Kleiman, Serre, eds.).

Proceedings of Symposia in Pure Mathematics 55, part I, AMS 1994; pp. 485-508.

1993-94   [Παρ]

Les 350 ans du “Grand Théorème de Fermat”.

L’Ouvert (Journal de l’A.P.M.E.P. d’Alsace et de l’I.R.E.M. de Strasbourg)

73 (déc. 1993), 1-9; 75 (juin 1994), 32-44.

1993     [XX]

Questions politiques dans la vie des mathématiques en Allemagne 1918-1935.

In La Science du Troisième Reich (sous la direction de J. Olff-Nathan).

Paris (Seuil) 1993; pp. 51-89.6.

1992     [XX]

N. Sch. & Erhard Scholz (ed.s) :

Oswald Teichmüller – Leben und Werk,

mit Beiträgen von K. Hauser, F. Herrlich, M. Kneser, H. Opolka, N. Schappacher, E. Scholz.

Jahresbericht der Deutschen Mathematiker-Vereinigung 94 (1992), 1-39.

1991e     [Μαθ]

N. Sch. & Anthony J. Scholl :

The boundary of the Eisenstein symbol.

Mathematische Annalen 290 (1991), 303-321.

1991d     [Μαθ]

.Les Conjectures de Beilinson pour les courbes elliptiques.

Actes des Journées Arithmétiques Luminy 1989.

Astérisque 198–199–200 (1991); pp. 305-317.

1991c     [Μαθ]

Jean-Francois Mestre & N. Sch. :

Séries d’Eisenstein-Kronecker et fonctions L des puissances symétriques de courbes elliptiques sur Q.

In van de Geer, Oort, Steenbrink (eds.), Arithmetic Algebraic Geometry,

Progress in Mathematics 89 (Birkhäuser) 1991; pp. 209-245.

1991b     [XX]

Edmund Landau’s Göttingen – From the life and death of a great mathematical center.

Mathematical Intelligencer 13 (1991), 12-18.

1991a     [XIX – XX]

Développement de la loi de groupe sur une cubique.

Séminaire de Théorie des Nombres Paris 1988/89,

Progress in Mathematics 91. Basel etc (Birkhäuser) 1991, 159-184.

1990     [XIX – XX]

N. Sch., unter Mitwirkung von Martin Kneser : Fachverband – Institut – Staat,

Streiflichter auf das Verhältnis von Mathematik zu Gesellschaft und Politik in Deutschland seit 1890

unter besonderer Berücksichtigung der Zeit des Nationalsozialismus.

In Ein Jahrhundert Mathematik 1890-1990 (G. Fischer, F. Hirzebruch, W. Scharlau, W. Törnig, ed.s).

Braunschweig: Vieweg 1990; pp. 1-82.

1989      [Παρ]

Neuere Forschungsergebnisse in der Arithmetik elliptischer Kurven.

Didaktik der Mathematik 17 (1989), 149-158.

1988b    [Μαθ]

M. Rapoport, N. Sch., P. Schneider (eds.) :

Beilinson’s Conjectures on L-values.

Perspectives in Mathematics 4 (Academic Press) 1988.

Within these proceedings of a memorable Arbeitsgemeinschaft at Oberwolfach, I coauthored the chapter :

1988b1    N.Sch. & Anthony J. Scholl :

Beilinson’s Theorem on Modular Curves; pp. 273-304.

1988a     [Μαθ]

Periods of Hecke Characters.

Lecture Notes in Mathematics 1301 (Springer) 1988.

Except for minor changes, this was my 1986 Göttingen Habilitationsschrift.

1987       [XX]

Das Mathematische Institut der Universität Göttingen 1929 – 1950.

In  Becker, Dahms, Wegeler (ed.s) :

Die Universität Göttingen unter dem Nationalsozialismus.

München: K.G. Saur 1987; pp. 345-373.

1998g    Second, enlarged edition München: K.G. Saur 1998; pp. 523-551.

As far as my chapter is concerned, there is very little difference between the two editions. But the original paper that I wrote in 1983 – my first ever on a historical subject – was much longer and quite different in style from what appeared in the books. I reedited this original version in 2000, and post it here.

1985b      [XX]

Max-Planck-Institut für Mathematik – Historical Notes on the New Research Institute at Bonn.

Mathematical Intelligencer 7 (1985), 41-52.

1985a      [Μαθ]

Günter Harder & N. Sch. :

Special Values of Hecke L-functions and Abelian Integrals.

In Arbeitstagung Bonn 1984, Lecture Notes in Mathematics 1111 (Springer) 1985; pp. 17-49.

1984c       [Παρ]

Gelber Regen. Zur Geschichte einer Kampagne.

Wechselwirkung 22 (1984), 39-42.

A report on my readings – partially prompted by exchanges with Neil Koblitz – on the American campaign about communist biochemical weapons allegedly tried out in South-East Asia, and the alternative explanation by US and British biochemists, that these agents were in fact but fungi-infected bee-faeces. Looking at more recent articles on the subject, like J.B. Tucker’s 2001 Lessons for arms control compliance, makes me wonder whether the right lessons have been learned.

1984b      [Μαθ]

Tate’s Conjecture on the Endomorphisms of Abelian Varieties.

In G. Faltings, G. Wüstholz et al. : Rational Points, Seminar Bonn/Wuppertal 1983/84.

Aspects of Mathematics E6 (Vieweg) 1984; pp. 114-153.

1984a      [Μαθ]

John Tate :

Les conjectures de Stark sur les fonctions L d’Artin en s=0.

Notes d’un cours à Orsay rédigées par Dominique Bernardi et Norbert Schappacher.

Progress in Mathematics 47 (Birkhäuser) 1984.

1983c      [Μαθ]

Catherine Goldstein & N. Schappacher :

Conjecture de Deligne et gamma-hypothèse de Lichtenbaum sur les corps quadratiques imaginaires.

CRAS Paris, t. 296 (25 Avril 1983) Sér. I, 615-618.

1983b      [Μαθ]

L’inégalité de Lojasiewicz ultramétrique.

CRAS Paris, t. 296 (14 Mars 1983) Sér. I, 439-442.

1983a      [Μαθ]

Propriétés de rationalité de valeurs spéciales de fonctions L attachées aux corps CM.

Séminaire de Théorie des Nombres Paris 1981/82, Progress in Mathematics 38 (Birkhäuser) 1983; pp. 267-282.

1982b      [Μαθ]

Rationalité de valeurs spéciales de certaines fonctions L.

Séminaire de Théorie des Nombres 1981/82, exposé 23 ; Université de Bordeaux I, UER Mathématique.

1982a      [Μαθ]

Une classe de courbes elliptiques à multiplication complexe.

Séminaire de Théorie des Nombres Paris 1980/81, Progress in Mathematics 22 (Birkhäuser) 1982; pp. 273-279.

1981       [Μαθ]

Catherine Goldstein & N. Sch. :

Séries d’Eisenstein et fonctions L de courbes elliptiques à multiplication complexe.

Journal für die reine und angewandte Mathematik 327 (1981), 184-218.

As it happened, and as John Coates was the first to point out to us in February of 2010 (!), the very first numbered formula of the paper, i.e., the definition (1.0) of the Kronecker double series on page 186, is in fact wrong : the first argument of the character χ on the right hand side has to be, not z, but ω. In the sequel of the paper, the Kronecker series are almost always employed with the second argument z0 equal to 0, which makes the χ-factor disappear. This may explain why the mistake went undetected for so long.

1980       [Μαθ]

Some Remarks on a Theorem of M.J. Greenberg.

In Proceedings of the Queen’s Number Theory Conference 1979.

Queen’s Papers in Pure and Applied Mathematics 54 (1980); pp. 101-113.

1978       [Μαθ]

Eine diophantische Invariante von Singularitäten über nichtarchimedischen Körpern.

Dissertation Göttingen 1978

1977       [Μαθ]

Zur Existenz einfacher abelscher Varietäten mit komplexer Multiplikation.

Journal für die reine und angewandte Mathematik 292 (1977), 186-190.