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The Transcendence of Pi Has Been Known for About a Century

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Results in Mathematics Contents 117 J. Bair/F. Jongmans Volume 7/No. 2/1984 Some remarks about recent result Pages 117-250 on the asymptotic cone 119 W. Beekmann/S. C Chang On the structure of summability fields 130 P. Bundschuh/1. Shiokawa A measure for the linear inde- pendence of certain numbers 145 P. L. Butzer/R. J. Nessel/E. L. Stark Eduard Helly (1884-1943) in memoriam 154 A. S. Cavaretta jr./H. P. Dikshit/ A. Sharma An extension of a theorem of Walsh 164 R. Fritsch The transcendence of ir has been known for about a century-but who was the man who discovered it? 184 Y. Hirano/H. Tominaga On simple ring extensions gener- ated by two idempotents 190 J. Joussen Eine Bemerkung zu einem Satz von Sylvester 192 H. Karzel/C. J. Maxson Fibered groups with non-trivial centers 209 H. Meiert G. Rosenberger Hecke-Integrale mit rationalen periodischen Funktionen und Dirichlet-Reihen mit Funk• tionalgleichung 234 G. Schiffels/M. Stemel Einbettung von topologischen Ringen in Quotientenringe Short Communications on Mathematical Dissertations 249 G. BaszenskijW. Schempp TAXT Konvergenzbeschleunigung von Orthogonal-Doppelreihen The Journal Copyright RESULTS IN MATHEMATICS It is a fundamental condition of publication that submitted RESULTATE DER MATHEMATIK manuscripts have not been published, nor will be simulta- publishes mainly research papers in all Heids of pure and applied mathematics. In addition, it publishes summaries of any mathematical neously submitted or published elsewhere. By submitting a field and surveys of any mathematical subject provided they are de- manuscript, the authors agree that the Copyright for their signed to advance some recent mathematical development. Finally, it article is transferred to the publisher if and when the arti- publishes short Communications on mathematical dissertations. Such cle is accepted for publication. The Copyright Covers the short Communications should be written by the author of the disserta• exclusive rights to reproduce and distribute the article, in- tions and by the Supervisor. They should not exceed two printed cluding reprints, Photographie reproduetions, microform pages. or any other reproduetions of similar nature, and transla- RESULTS IN MATHEMATICS tions. RESULTATE DER MATHEMATIK Photographie reproduetion, microform, or any other re- veröffentlicht Forschungsbeiträge zu allen Gebieten der reinen und produetion of text, figures or tables from this Journal is angewandten Mathematik. Darüber hinaus können auch in begrenz• prohibited without permission obtained from the pub• tem Ausmass Beiträge, welche eine Zusammenfassung eines mathema• lisher. tischen Gebietes oder eine Übersicht über einen mathematischen Problemkreis geben, veröffentlicht werden, sofern sie geeignet sind, © 1984 Birkhäuser Verlag, P.O. 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Zeller, Tübingen Printed in Switzerland 0378-6218/84/020164-20 $01.50 + 0.20/0 Results in Mathematics, Vol. 7 (1984) © 1984 Birkhäuser Verlag, Basel The transcendence of TT has been known for about a Century - but who was the man who discovered it? RUDOLF FRITSCH Freiburg, April 12, 1882 A very well known German mathematical Institution is the "Mathematische Forschungsinstitut Oberwolf ach". Its director Professor Martin Barner of the University of Freiburg im Breisgau, built his private house on the Loretto hill, a part of the Black Forest belonging to the city of Freiburg. It Stands on a place of extreme mathematical interest, because a young man had an important idea here. It was his 30th birthday, and he was alone on a stroll to Günterstal, a small village with a medieval monastery, today also part of Freiburg. Five years beforehand, in October 1877 as associate professor in Freiburg he had been invited to take similar walks in the Company of the (füll) professors Thomae1 from Freiburg and du Bois-Reymond2 from Tübingen, Thomae's friend and predecessor, who often came back for short visits. On the first of these walks he was unsuitably dressed, hiking through the creeks and brushwood of the Black Forest with top-hat and tails, whereas his colleagues looked like todays equivalent of "green-peacers". Besides enjoying the wonderful landscape, the group engaged in mathematical discussions. Thomae and du Bois-Reymond were specialists in (complex) analysis; our young man was brilliant in geometry, having learned a lot from Clebsch3 and Felix Klein4, especially the tools which he used so effectively a short time later. One topic touched on in their talks was the problem of the transcendence of TT. Euler5 and Lambert6 had conjectured that TT was transcendental. If this could be proved, then the very old question of squaring the circle would be settled7. Thomae and du Bois-Reymond proposed to attack this problem by means of continued fractions, a device which the French mathematician Liouville8 had very successfully used in order to clarify the notion of transcendental numbers and to exhibit some of such numbers. But our young friend was not attracted to this approach. Some years previously he had spent a winter term (1876/77) in Paris, where Hermite9 had shown him how to prove the transcendence of e using integration of real functions. He feit that this must be the right path leading to the goal, although for a long time he had had no idea how to begin. Meanwhile his personal circumstances had changed. On October 1, 1879 he had taken over the 165 166 RUDOLF FRITSCH chair of Thomae who had moved to Jena, and was too far away to come back just for walks. Reminiscing about the past, he frequently made such strolls alone now, just like the one on this pleasant day. Some weeks beforehand he had looked through his collection of reprints for Hermite's famous paper on the Solution of the general equation of degree 5 by means of elliptic functions10. By chance, the proof of the transcendence of e caught his eye and he started to study it again. Hermite himself regarded this paper as his own most significant accomplishment. At the place where Barner's house Stands today, the long hoped for idea flashed through his mind: e™ = — 1! He rushed home and wrote down the paper: "Über die Zahl 7r" (On the number -rr)11.
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