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LOOKING BACKWARD: from EULER to RIEMANN 11 at the Age of 24

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LOOKING BACKWARD: from EULER to RIEMANN 11 at the Age of 24 LOOKING BACKWARD: FROM EULER TO RIEMANN ATHANASE PAPADOPOULOS Il est des hommes auxquels on ne doit pas adresser d’´eloges, si l’on ne suppose pas qu’ils ont le goˆut assez peu d´elicat pour goˆuter les louanges qui viennent d’en bas. (Jules Tannery, [241] p. 102) Abstract. We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and the notion of space. We shall see that among Riemann’s predecessors in all these fields, one name occupies a prominent place, this is Leonhard Euler. The final version of this paper will appear in the book From Riemann to differential geometry and relativity (L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017. AMS Mathematics Subject Classification: 01-02, 01A55, 01A67, 26A42, 30- 03, 33C05, 00A30. Keywords: Bernhard Riemann, function of a complex variable, space, Rie- mannian geometry, trigonometric series, zeta function, differential geometry, elliptic integral, elliptic function, Abelian integral, Abelian function, hy- pergeometric function, topology, Riemann surface, Leonhard Euler, space, integration. Contents 1. Introduction 2 2. Functions 10 3. Elliptic integrals 21 arXiv:1710.03982v1 [math.HO] 11 Oct 2017 4. Abelian functions 30 5. Hypergeometric series 32 6. The zeta function 34 7. On space 41 8. Topology 47 9. Differential geometry 63 10. Trigonometric series 69 11. Integration 79 12. Conclusion 81 References 85 Date: October 12, 2017. 1 2 ATHANASEPAPADOPOULOS 1. Introduction More than any other branch of knowledge, mathematics is a science in which every generation builds on the accomplishments of the preceding ones, and where reading the old masters has always been a ferment for new dis- coveries. Examining the roots of Riemann’s ideas takes us into the history of complex analysis, topology, integration, differential geometry and other mathematical fields, not to speak of physics and philosophy, two domains in which Riemann was also the heir of a long tradition of scholarship. Riemann himself was aware of the classical mathematical literature, and he often quoted his predecessors. For instance, in the last part of his Habili- tation lecture, Uber¨ die Hypothesen, welche der Geometrie zu Grunde liegen [231] (1854), he writes:1 The progress of recent centuries in the knowledge of mechanics de- pends almost entirely on the exactness of the construction which has become possible through the invention of the infinitesimal cal- culus, and through the simple principles discovered by Archimedes, Galileo and Newton, and used by modern physics. The references are eloquent: Archimedes, who developed the first differ- ential calculus, with his computations of length, area and volume, Galileo, who introduced the modern notions of motion, velocity and acceleration, and Newton, who was the first to give a mathematical expression to the forces of nature, describing in particular the motion of bodies in resisting media, and most of all, to whom is attributed a celebrated notion of space, the “Newtonian space.” As a matter of fact, the subject of Riemann’s habili- tation lecture includes the three domains of Newton’s Principia: mathemat- ics, physics and philosophy. It is interesting to note also that Archimedes, Galileo and Newton are mentioned as the three founders of mechanics in the introduction (Discours pr´eliminaire) of Fourier’s Th´eorie analytique de la chaleur ([117], p. i–ii), a work in which the latter lays down the rigor- ous foundations of the theory of trigonometric series. Fourier’s quote and its English translation are given in 10 of the present paper. In the his- torical part of his Habilitation dissertation,§ Uber¨ die Darstellbarkeit einer Function durch eine trigonometrische Reihe (On the representability of a function by a trigonometric series) [216], a memoir which precisely concerns trigonometric series, Riemann gives a detailed presentation of the history of the subject, reporting on results and conjectures by Euler, d’Alembert, Lagrange, Daniel Bernoulli, Dirichlet, Fourier and others. The care with which Riemann analyses the evolution of this field, and the wealth of his- torical details he gives, is another indication of the fact that he valued to a high degree the history of ideas and was aware of the first developments of the subjects he worked on. In the field of trigonometric series and in others, he was familiar with the important paths and sometimes the wrong tracks that his predecessors took for the solutions of the problems he tackled. Rie- mann’s sense of history is also manifest in the announcement of his memoir Beitr¨age zur Theorie der durch die Gauss’sche Reihe F (α,β,γ,x) darstell- baren Functionen (Contribution to the theory of functions representable by 1In all this paper, for Riemann’s habilitation, we use Clifford’s translation [232]. LOOKING BACKWARD:FROMEULERTORIEMANN 3 Gauss’s series F (α,β,γ,x)), published in the G¨ottinger Nachrichten, No. 1, 1857, in which he explains the origin of the problems considered, mentioning works of Wallis, Euler, Pfaff, Gauss and Kummer. There are many other examples. Among Riemann’s forerunners in all the fields that we discuss in this paper, one man fills almost all the background; this is Leonhard Euler. Riemann was an heir of Euler for what concerns functions of a complex variable, elliptic integrals, the zeta function, the topology of surfaces, the differential geometry of surfaces, the calculus of variations, and several topics in physics. Riemann refers to Euler at several places of his work, and Euler was him- self a diligent reader of the classical literature: Euclid, Pappus, Diophantus, Theodosius, Descartes, Fermat, Newton, etc. All these authors are men- tioned all along his writings, and many of Euler’s works were motivated by questions that grew out of his reading of them.2 Before going into more de- tails, I would like to say a few words about the lives of Euler and Riemann, highlighting analogies, but also differences between them. Both Euler and Riemann received their early education at home, from their fathers, who were protestant ministers, and who both were hoping that their sons will become like them, pastors. At the age of fourteen, Euler attended a Gymnasium in Basel, while his parents lived in Riehen, a village near the city of Basel.3 At about the same age, Riemann was sent to a Gymnasium in Hanover, away from his parents. During their Gymnasium years, both Euler and Riemann lived with their grandmothers.4 They both enrolled a theological curriculum (at the Universities of Basel and G¨ottingen respectively), before they obtain their fathers’ approval to shift to mathematics. There are also major differences between the lives of the two men. Eu- ler’s productive period lasted 57 years (from the age of 19, when he wrote his first paper, until his death at the age of 76). His written production comprises more than 800 memoirs and 50 books. He worked on all domains of mathematics (pure and applied) and physics (theoretical and practical) that existed at his epoch. He also published on geography, navigation, ma- chine theory, ship building, telescopes, the making of optical instruments, philosophy, theology and music theory. Besides his research books, he wrote elementary schoolbooks, including a well-known book on the art of reckon- ing [64]. The publication of his collected works was decided in 1907, the year of his bicentenary, the first volumes appeared in 1911, and the edition is still in progress (two volumes appeared in 2015), filling up to now more than 80 large volumes. Unlike Euler’s, Riemann’s life was short. He pub- lished his first work at the age of 25 and he died at the age of 39. Thus, his productive period lasted only 15 years. His collected works stand in a single slim volume. Yet, from the points of view of the originality and the 2Cf. for instance Euler’s Problematis cuiusdam Pappi Alexandrini constructio (On a problem posed by Pappus of Alexandria), [97], 1780. 3Today, Riehen belongs to the Canton of the city of Basel, and it hosts the famous Beyeler foundation. 4In 1842, at the death of his grandmother, Riemann quitted Hanover and attended the Gymnasium at the Johanneum L¨uneburg. 4 ATHANASEPAPADOPOULOS impact of their ideas, it would be unfair to affirm that either of them stands before the other. They both had an intimate and permanent relation to mathematics and to science in general. Klein writes in his Development of mathematics in the 19th century ([163], p. 231 of the English translation): After a quiet preparation Riemann came forward like a bright me- teor, only to be extinguished soon afterwards. On Euler, I would like to quote Andr´eWeil, from his book on the his- tory of number theory, Number Theory: An approach through history from Hammurapi to Legendre [256]. He writes, in the concluding section: [...] Hardly less striking is the fact that Euler never abandoned a problem after it has once aroused his insatiable curiosity. Other mathematicians, Hilbert for instance, have had their lives neatly divided into periods, each one devoted to a separate topic. Not so Euler. All his life, even after the loss of his eyesight, he seems to have carried in his head the whole of the mathematics of his day, both pure and applied. Once he has taken up a question, not only did he come back to it again and again, little caring if at times he was merely repeating himself, but also he loved to cast his net wider and wider with never failing enthusiasm, always expecting to uncover more and more mysteries, more and more “herrliche pro- prietates” lurching just around the next corner.
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