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The transcendence of π has been known for about a century — but who was the man who discovered it?

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Notes

  1. Johannes Thomae (1840–1921), full professor of mathematics in Freiburg from 1874 to 1879, then in Jena, is nearly forgotten today. In the “Mathematiker-Lexikon” by Herbert Meschkowski [Mannheim 1964], he is not even mentioned. Nevertheless, he left some traces. For instance, the German expression “Mächtigkeit” for “cardinality” was proposed by him to his friend Georg Cantor (1845–1918); more about him may be found in the obituary published in the Jahresbericht der Deutschen Mathematiker-Vereinigung [Jber. DMV 30/1921, 133–144]. Hint: For biographical information on the mathematicians who are mentioned in this article without any further details see the above named encyclopedia.

  2. Paul du Bois-Reymond (1831–1889), full professor of mathematics in Freiburg from 1870 to 1874, then in Tübingen and from 1884 in Berlin.

  3. Rudolf Friedrich Alfred Clebsch (1833–1872).

  4. Felix Klein (1849–1925). 5Leonhard Euler (1707–1783): “Introductio in analysis infinitorum” [Lausanne 1748, Chapter IV; Opera omnia VIII, IX].

  5. Johann Heinrich Lambert (1728–1777): “Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques.” [Hist. Acad. roy. sci. belles lettr. Berlin, 1761, 265–322; Opera Math. II, 112–159].

  6. The problem of squaring the circle had been posed by the Greek philosopher Anaxagoras (500?–428 B.C.) while he was imprisoned by the Athenians under a charge of impiety (±430 B.C.).

  7. Joseph Liouville (1809–1882): “Sur les classes très étundus de quantités dont la valeur n'est ni algébrique, ni même réductible a des irrationelles algébriques” [C.R. Acad. Sci. Paris 18/1844, 883–885, 900–911; J. Math. pures appl. (1) 16/1851, 133–142]. Today the transcendental numbers constructed by Liouville are called Liouville Numbers.

  8. Charles Hermite (1822–1901): “Sur la fonction exponentielle” [C.R. Acad. Sci. Paris 77/1873, 18–24, 74–79, 226–233, 285–293; see: (Œuvres II, 150–181].

  9. Charles Hermite: “Sur la résolution de l'équation du cinquiéme degré” [C.R. Acad. Sci. Paris 46/1858, 508–515; Œuvres II, 5–12].

  10. [Math. Ann. 20/1882, 213–225].

  11. You look as though you have just solved the squaring of the circle.

  12. Ferdinand Johannes Heinrich Lindemann, born in Hannoversch-Münden June 12, 1806, died in Schwerin April 14, 1880. He was teacher at a high school for girls in Hannover from 1843 to 1854; then he moved to Schwerin.

  13. Emilie Crusius, born in Hannover December 18, 1823, married to F. J. H. Lindemann in Hannover on October 3, 1847, died in Hannover May 31, 1907.

  14. Gottlob Christian Crusius (1785–1848). His dictionary of Homer: “Vollst. Griechisch-Deutsches Wörterbuch über die Gedichte des Homer und der Homeriden …” [1. ed. Hannover 1836] was used almost up to the present day.

  15. Ernst Curtius (1814–1896): “Griechische Geschichte bis zur Schlacht bei Chäronea” [3 vol., Berlin 1857–67].

  16. Alexander von Humboldt (1769–1859): “Kosmos, Entwurf einer physischen Weltbeschreibung” [5 vol., Stuttgart-Tübingen 1845–1862].

  17. This is Lindemann's own judgement. At the time this report was written, no further information on Dr. Bastian is available.

  18. In the 19th century the custom of editing the so-called “Schulprogramm” had a certain scientific signficance. In a gymnasium one teacher was asked every year to write a scientific exposition. In general this had nothing to do with a “program” but was often original research. It was combined with the invitation to attend some ceremonies of the school, for instance at the completion of a school year or on the occasion of a visit by monarch, and was intended to demonstrate the quality of the teachers and the teaching.

  19. Moritz Abraham Stern (1807–1894) spent all his academic life in Göttingen with the exception of one student year in Heidelberg: PhD 1829, associate professor 1849, full professor 1859, retired 1884. His oral exam for getting the PhD was the first one in which Gauß examined. Gauß later said that he was more afraid than Stern. His text book: “Lehrbuch der Algebraischen Analysis” [Leipzig-Heidelberg 1860] was the basis for most of his lectures. Bernhard Riemann (1826–1866) and Richard Dedekind (1831–1916) named Stern among their academic teachers. See his curriculum vitae in “Allgemeine Deutsche Biographie, vierundfünfzigster Band” [Leipzig 1908].

  20. Jakob Steiner (1796–1863): “Vorlesungen über synthetische Geometrie, 1. Teil”, ed. by C. F. Geiser [Leipzig 1867].

  21. Oskar Schlömilch (1823–1901 [Biog. Jbuch. Deut. Nekr 6/1904, 119–122]): “Analytische Geometrie des Raumes” [Leipzig 1855]. Almost every textbook on calculus mentions his formula for the remainder in the Taylor expansion.

  22. Alfred Enneper (1830–1895).

  23. Wilhelm Eduard Weber (1804–1891) together with Gauß constructed the electro-magnetic telegraph.

  24. Friedrich Wöhler (1800–1882) produced urea from ammonia, thereby destroying the barrier between organic and inorganic chemistry.

  25. I cannot learn anything in Clebsch's lecture; one is so engrossed by the beauty of his language that one cannot pay heed to the content.

  26. Oskar Perron (1880–1975), we cite from his address delivered at the official celebration of Lindemann's 70th birthday [Jber. DMV 31/1922, 26–28]. Regarding Perron, see the obituary in the Jahrbuch der Bayerischen Akademie der Wissenschaften [Jbuch. Bay. Akad. Wiss. 1976, 217–227].

  27. Oh, this Lindemann thinks in one minute more geometry than I can learn in a complete semester.

  28. Wolfgang Sartorius von Waltershausen (1809–1876): Gauß zum Gedächtnis [Leipzig 1856]. His father Georg Sartorius (1765–1828) had made political economy a subject of academic research and teaching. On recognition of this, King Ludwig I of Bavaria raised him to a hereditary peerage (1827).

  29. Ernst Friedrich Wilhelm Klinkerfues (1827–1884) was the successor of Gauß as head of the Göttingen observatory. He was known as the inventor of an automatic gas lighter and a bifilarhygrometer. He ended his life by shooting himself in the observatory. For his personality see “Briefwechsel zwischen Carl Friedrich Gauss und Christian Ludwig Gerling” ed. by Clemens Schaefer [Berlin 1927].

  30. at a festivity in honour of the students who came back from the war.

  31. Friedrich Neesen (1849–1923) received his PhD from the University of Bonn in 1871 and became professor of physics in Berlin in 1877.

  32. Felix Klein — only three years older than Lindemann — received his habilitation in Göttingen just a year before.

  33. Felix Klein: “Über die sogenannte Nicht-Euklidische Geometrie” [Nachr. kön. Ges. Wiss. 1871, 419–433; Math. Ann. 4/1871, 573–625].

  34. The first one was Franz Joseph Konrad Diekmann (1848–1905), later professor and director of the “Realgymnasium” in Viersen (Rheinland). His thesis had the title: “Über die Modifikationen, welche die ebene Abbildung einer Fläche 3. Ordnung durch Auftreten von Singularitäten erhält” [Math. Ann. 4/1871, 442–475].

  35. [Math. Ann. 7/1874, 56–144].

  36. Klein had been promoted to full professor at the University of Erlangen for the Wintersemester 1872/73. He succeeded Hans Pfaff (1824–1872], who was given the chair in 1869, after Hermann Hankel (1839–1873). Hankel had moved to Tübingen; in Erlangen he was (1867) the successor of Georg Karl Christian v. Staudt (1798–1867), the famous geometer (see: R. Fritsch: Ein Lehrer und zwei Schüler: Buzengeiger, v. Staudt und Feuerbach, in: Auf den Weg gebracht, ed. by H. Sund und M. Timmermann, [Konstanz 1979]). v. Staudt's predecessor was Hans Pfaff's father Wilhelm Pfaff (1774–1835) whose better known older brother, Johann Friedrich Pfaff, (1765–1825) was the doctoral supervisor of Gauß. Klein's inaugural lecture in Erlangen was the famous “Erlanger Programm” which has greatly influenced the subsequent development of geometry, even up to the present time. The precise title was “Vergleichende Betrachtungen über neuere geometrische Forschungen” [Erlangen 1872; Math. Ann. 43/1893, 63–100; Ges. Math. Abh. I, 460–497].

  37. This did not bother him later. One of his PhD students in Königsberg was Arnold Sommerfeld (1868–1951), who became full professor of theoretical physics at the University of München in 1906. Sommerfeld's most famous PhD student was the Nobel-prize winner Werner Heisenberg (1901–1976).

  38. Julius Plücker (1801–1868) let the very young student of botany and physics Felix Klein help him to prepare his lectures on experimental physics. Under obligation to his late teacher, Klein turned to mathematics after Plücker's death.

  39. In particular, Klein took over the editorship of the famous “Mathematische Annalen”, founded by Clebsch and his friend Carl Neumann (1832–1925).

  40. Paul Gordan (1837–1912); Klein succeeded in obtaining a second chair of mathematics at the university of Erlangen, which Gordan got in 1873.

  41. Max Noether (1844–1921), the father of Emmy Noether (1882–1935).

  42. Alfred Clebsch: “Vorlesungen über Geometrie”, bearbeitet und herausgegeben von Dr. Ferdinand Lindemann [Leipzig 1876].

  43. Aurel Edmund Voß (1845–1931) held the second chair of mathematics at the University of München from 1903 to 1925, as successor of Gustav Bauer (1820–1906, see H. Gericke-H. Uebele: “Philipp Ludwig von Seidel und Gustav Bauer, zwei Erneuerer der Mathematik in München” in' Die Ludwig-Maximilians-Universität in ihren Fakultäten I [Berlin 1972]) and as predecessor of Heinrich Tietze (1880–1964, see [Jber. DMV 83/1981, 182–185]). The present chair holder is Bodo Pareigis (*1937).

  44. With youthful daring you have taken over this great and difficult task before finishing your university studies, and you have thus produced a work, which also has become your own, in every way. Whenever I look at it, I still admire the depth and the width of the viewpoint whereby you were able to solder together in a harmonic whole all that which Clebsch in the last years of his life had partially carried out in lectures or even had left unfinished, as e.g. the theory of invariants of connexes. For about 50 years your work has maintained the same importance for every geometer; no nation can offer anything of similar comprehensive importance [Jber. DMV 31/1922, 25–26]. (The notion of “Connex” as introduced by Clebsch [Abhandlungen der Königlichen Gesellschaft der Wissenschften zu Göttingen, mathematische Klasse 17/1872, 11–12] means a polynomial equation containing the coordinates of a variable point and a variable line, each in a homogenous manner.)

  45. Karl Brater (1819–1869) strove for the freedom of the press in Germany; see M. Spindler:” Bayerische Geschichte im 19. und 20. Jahrhundert” [München 1978] and A. Sapper47: “Frau Pauline Brater” [München 1908].

  46. Agnes Brater, married Sapper (1852–1929) and became very famous later as author of books for children and young people.

  47. Anna Hegel (1859–1927), a granddaughter of the philosopher Georg Wilhelm Friedrich Hegel (1770–1831), married Felix Klein in Erlangen 1875.

  48. Every student had to pay for the lectures he was attending. Klein was Professor at the “Polytechnische Schule”, which was named in 1877: “Technische Hochschule”. Today this is the “Technische Universität”.

  49. Philipp Ludwig von Seidel (1821–1896) became full professor in 1855; before this he was recognized for applying statistics to the health sciences and thereby liberating München from regular epidemics of typhoid fever. His most important contribution to mathematics was the discovery of uniform convergence. He retired in 1893 and Lindemann became his successor.

  50. Bernhard Riemann (1826–1866).

  51. like “Hofbäcker” (court baker), “Hofbuchhändler” (court book seller) etc.

  52. Johann Freiherr von Lutz (1826–1890), a friend of the German chancellor Otto v. Bismarck (1815–1898), was Minister of Education in the Bavarian Kingdom from 1869 to 1890, Minister of Justice from 1867 to 1871, from 1880 to 1890 also Prime Minister. He played an essential role in the so-called “Kulturkampf” of the late 19th century.

  53. Friedrich Prym (1841–1915).

  54. Carl Gustav Jacobi (1804–1851).

  55. This, as well as the International Congress of Mathematicians, was held in conjunction with the famous world's fair in Paris 1900. During a reception by Prince Roland Bonaparte (1858–1924) Lindemann, Hubert (1862–1943) and Minkowski (1864–1909) amused themselves by looking at the inner organs of Mrs. Minkowski shown by the newly developed X-ray apparatus; this is how light-headed people were about X-rays then. This was the international congress at which Hilbert posed his famous problems for the 20th century.

  56. Joseph Bertrand (1822–1900).

  57. Camille Jordan (1838–1922).

  58. Gaston Darboux (1842–1917).

  59. Georges Henry Halphen (1844–1889): “Œuvres” [4 vol., Paris 1916–1924].

  60. Justin Bourget (1822–1887, [J. Math. Elem. 11/1887] became rector of the academy of Aix-en-Provence in 1878.

  61. Ole Jacob Broch (1818–1889, [Acta Math. 12/1889 last page]) was professor of mathematics in Christiania (today Oslo, capital of Norway), he lived many years in Paris, he was member of the International Committee of Weights and Measures, in 1879 he became chief of the International Office of Weights and Measure in Sèvres.

  62. Napoléon Haton de la Goupillière (1833–1927 [C.R. Acad. Sci. Paris 184/1827, 50–52]).

  63. Michel Chasles (1793–1880).

  64. Georges Fouret (1845–1923) was also president of the “Société Mathématique de France” in 1887.

  65. Carl Stumpf (1848–1936) later founded the Institute of Psychology at the University of München and was the founder of the ethnology of music.

  66. Ludwig II (1845–1886) built the castles Herrenchiemsee, Linderhof and Neuschwanstein. He liked to be alone and often disappeared for weeks at a time in the mountains or the woods.

  67. Matthias (von) Lexer (1830–1892) was professor of German language and literature.

  68. Hans Carl Friedrich von Mangoldt (1854–1925) got his PhD in 1878 under the supervision of Weierstraß (1815–1897) in Berlin. In 1904, he became the first rector of the new Technical School of Danzig. Thousands of students learned mathematics from his “Einführung in die Mathematik” [3 vol. Leipzig 1911–1914], later revised by Konrad Knopp (1882–1957), familiar as “Mangoldt-Knopp”.

  69. Emil Warburg (1846–1931) became full professor in Freiburg in 1876. He moved to Berlin in 1895 where he was president of the “Physikalisch-Technische Reichsanstalt” from 1905 to 1922. He was the founder of medicinal photochemistry.

  70. “Die Schwingungsformen gezupfter und gestrichener Seiten” [Freiburger Berichte 7/1879, 500–532; see also Jbuch. Fortschr. Math. 11/1879, 716–718].

  71. “Entwicklung der Funktionen einer komplexen Variablen nach Laméschen Funktionen und nach Zugeordneteneter Kugelfunktionen” [Math. Ann. 19/1881, 323–386]. Together with his work on π, this paper was the reason that Lindemann was offered a chair in Königsberg.

  72. “Über das Verhalten der Fourier'schen Reihe an Sprungstellen” [Math. Ann. 19/1881, 517–523]. Here Lindemann improved a result of Seidel.50

  73. “Vorlesungen über Geometrie unter besonderer Benutzung der Vorträge von Alfred Clebsch. Die Flächen erster und zweiter Ordnung oder Klasse und der Lineare Complex” [Leipzig 1891].

  74. Klein moved to Leipzig in 1880 and returned to Göttingen in 1886, where he stayed until his death in 1925.

  75. He did not note the receipt of the paper in his private diary. Later on — as he was convinced — he made a remark misdating the event as Christmas 1881; see: “Felix Klein, Handschriftlicher Nachlaß” edited by Konrad Jacobs [Erlangen 1977].

  76. “Sur le rapport de la circonférence au diamètre, et sur les logarithmes népériens des nombres commensurables ou des irrationelles algébriques”. [C.R. Acad. Sci. Paris 115/1882, 72–74].

  77. “Über die Ludolph'sche Zahl” [Sber. Akad. Wiss. Berlin 1882, 679–686].

  78. Richard Dedekind (1831–1916).

  79. Leopold Kronecker (1821–1891).

  80. Hieronymus Georg Zeuthen (1839–1920); see [Math. Ann. 83/1921, 1–23].

  81. Cyparissos Stephanos (1857–1917) was professor of mathematics in Athens/Greece.

  82. Luigi Cremona (1830–1903).

  83. James Joseph Sylvester (1814–1897): “On the divisors of the sum of a geometrical series whose first term is unity and common ratio any positive or negative integer”. [Nature 37/1888, 417–418]. The battle of Solferino between Napoleon III and the Austrian emperor Franz Joseph was an essential step toward the unification of Italy; the bloody battlefield inspired Henri Dunant to found the International Red Cross. The battle of Sadowa decided the Austrian-Prussian war of 1866; in Germany it is more familiar as the battle of “Königgrätz”.

  84. “Zu Lindemann's Abhandlung: Über die Ludolphsche Zahl” [Sber. Akad. Wiss. Berlin 1885, 1067–1085; Math. Werke II, 341–362]. To make it crystal clear: Lindemann completely proved that ez is transcendental for every non-zero algebraic number z; because of eπi = −1 this implies the transcendence of π. Lindemann stated and Weierstraß published the proof that ez 0, …, e z n are linearly independent over the field of algebraic numbers, whenever z0, …, zn are distinct algebraic numbers n ≧ 1. Weierstraß acknowledged Hermann Amadeus Schwarz (1843–1921) and Dedekind79 for helpful comments. The next important simplification was obtained by Hilbert: “Über die Transzendenz der Zahlen e und π” [Math. Ann 43/1893, 216–219; Ges. Abh. I, 1–4].

  85. Friedrich Wilhelm Bessel (1784–1846).

  86. Franz Neumann (1798–1895) had the chair of physics and mineralogy; he is the father of Carl Neumann40.

  87. Adolf Hurwitz (1859–1919) completed his PhD under the supervision of Felix Klein in Leipzig 1881 and his habilitation in Göttingen 1882. (He could not habilitate in Leipzig because he had been at a “Realgymnasium” instead of a “Classical Gymnasium”).

  88. There is a picture of her as Marina in Shakespeare's “Pericles” from a performance in the royal München theatre which was given solely for King Ludwig II68 in October 20, 1882 (nobody else was in the auditorium!).

  89. Her grandchildren enjoyed listening to her when she vividly narrated fairly tales.

  90. Henri Poincaré (1854–1912): “La Science et l'hypothèse” [Paris 1902] translated: “Wissenschaft und Hypothese” Autorisierte deutsche Ausgabe mit erläuternden Anmerkungen von F. und L. Lindemann [Leipzig 1904], and: “Science et méthode” [Paris 1908] translated: “Wissenschaft und Methode” [Leipzig und Berlin 1914]. The statesman Raymond Poincaré was a cousin of Henri Poincaré.

  91. His successors in the chair were: Constantin Caratheodory (1873–1950, appointed 1924), Eberhard Hopf (1902–1983, appointed 1944, moved 1949 to Indiana University, Bloomington [Notices AMS 28/1981, 508; 30/1983, 683–684]), Robert König (1885–1979, appointed 1950 [Jbuch. Bay. Akad. Wiss. 1981], Karl Stein (*1913, appointed 1954, he developed Stein spaces and Stein manifolds) and today Otto Forster (*1937, appointed 1982).

  92. “oh heilige Quadratur des Zirkels” (oh the holy squaring of the circle) Minkowski wrote in a letter to his friend Hilbert on July 20, 1898; see “Hermann Minkowski, Briefe an David Hilbert” ed. by L. Rüdenberg and H. Zassenhaus [Berlin-Heidelberg-New York 1973]. These letters contain some acid-tongued remarks about Lindemann. In contrast to Minkowski, Hilbert esteemed Lindemann throughout his whole life.

  93. “Über den sog. letzten Fermat'schen Satz” [Leipzig 1909], “Untersuchungen über den Fermatschen Satz” [München 1928, published by the author].

  94. Alfred Loewy (1873–1935); see M. Pinl “Kollegen in einer dunklen Zeit” [Jber. DMV 71/1969, 167–228].

  95. Georg Faber (1877–1966, [Jbuch. Bay. Ak. 1966, 207–210]).

  96. Otto Volk (*1892) was Lindemann's assistant from 1919 to 1923. He now lives in Würzburg, where he became professor of mathematics and astronomy in 1930. He collected the money for Lindemann's bust in the University of München from colleagues and institutions.

  97. Martin Wilhelm Kutta (1867–1944 Neue Deut. Biog. 13/1982, 348–350). Today every student of maths becomes familiar with the Runge-Kutta-Method for numerical integration of differential equations.

  98. “Lehren und Lernen in der Mathematik” [München 1904].

  99. Johann Waldvogel: “Die Gymnasialmathematik in der Beleuchtung des Herrn Prof. Dr. Lindemann” [Blätter für das Gymnasialschulwesen 41/1905, 50–59]; J. Lenauer: “Über neuere Vorschläge zur Reform des mathematischen Unterrichts”. [ibid. 646–660].

  100. “Zur Geschichte der Polyeder und Zahlzeichen” [Sber. math. phys. Cl. Bay. Akad. Wiss. 26/1896, 625–758]; “Über einige prähistorische Gewichte aus deutschen und italienischen Museen I” [ibid. 29/1899, 71–136]; “Über einige Bleigewichte aus Pompeji” [Sber. math. nat. Abt. Bay. Akad. Wiss. 1935, 451–455].

  101. A communist movement proclaimed in Bavaria on April 7, 1919, the so called “Räterepublik”, copying Soviet Russia. After a month of struggle it was over-powered.

  102. Clemens Baeumker (1853–1924), professor of philosophy, had the “Catholic” chair; he is known for his work on medieval philosophy.

  103. described by Werner Heisenberg38 in “Der Teil und das Ganze” [München 1969]. However, the description which Heisenberg gives of his conversation with Lindemann is not correct. According to an eyewitness the dog did not bark!

  104. In one of his letters Minkowski wrote about the fairylike Italian nights which Lindemann organized as rector in Königsberg [l.c.93].

  105. It is described in [Jber. DMV, 31/1922, 24-30].

  106. Emil Hilb (1882–1929, [Jber. DMV 42/1932, 183–199]) received his PhD under the supervision of Lindemann in 1903 and became professor of mathematics at the University of Würzburg in 1909.

  107. Alfred Pringsheim (1850–1941); his daughter Katja (1883–1980) loved Perron27 but married the novelist and poet Thomas Mann (1875–1955). This fact is the reason for the description of the lecture notes on algebra and the appearance of Professor Lindemann (by name!) as a painter in Thomas Mann's novel “Königliche Hochheit” [Berlin 1909]; see also Peter de Mendelssohn: “Der Zauberer. Das Leben des deutschen Schriftstellers Thomas Mann” [Frankfurt am Main 1975].

  108. Bernhard Bleeker (1881–1968) was professor at the Academy of Fine Arts in München (since 1922). There is still a number of public monuments in München made by Bleeker, for instance, the statue of Prince Regent Luitpold in the entrance hall of the university (1908), the memorial of the unknown soldier (1924) and the fountain of Crown Prince Ruprecht (1961); see: Lothar Hennig: “Der Bildhauer Bernhard Bleeker” [Materialen-Dokumente zu Leben und Werk 5, Germanisches Nationalmuseum Nürnberg 1978].

  109. Erich v. Drygalski (1865–1949) oriented geographical research towards modern natural sciences.

  110. Karl Döhlemann (1864–1926, [Jber., DMV 37/1928, 209–212]) was a lecturer at the University of München when Lindemann arrived. According to a proposal of Lindemann he gave courses on descriptive geometry at the university, a subject which before was only taught at the Technical High School. He became such an expert on this subject that he later held the chair for geometry at the Technical High School (The former Technical School had been promoted meanwhile to a “High” School with status equal to a university).

  111. Walther Dyck (since 1901: W. von Dyck) (1856–1934, [Jber. DMV 45/1935, 89–98]) received his PhD under the supervision of Klein in 1879.

  112. Friedrich Hartogs (1874–1943); see M. Pinl: “Kollegen in einer dunklen Zeit III” [Jber. DMV 73/1971-72, 153–208].

  113. “Ferdinand Lindemann zu seinem 75. Geburtstage” [Forschungen und Fortschritte 3/1927, 88], “Ferdinand von Lindemann zum 80. Geburtstage” [ibid. 8/1932, 145]

  114. Constantin Carathéodory96: “Ferdinand von Lindemann” [Sber. math. nat. Abt. Bay. Akad. Wiss. 1940, 61–63].

  115. A list of the early publications: “Druckschriften-Verzeichnis von F. Lindemann' appeared in the “Almanach der Königlich Bayerischen Akademie der Wissenschaften zum 150. Stiftungsfest” [München 1909, p. 303–306].

  116. For tourists: The grave is situated in Section 43 of the Waldfriedhof, the nicest cemetery of München, near to the “Würmtalstraße”, and ornamented with the figure π.

  117. “Über arithmetische Eigenschaften gewisser transzendenter Funktionen” [Math. Ann. 22/1883, 211–229; 32/1888, 583–588].

  118. For a more detailed presentation of the development of this problem see R. Tijdeman “On the Gel'fond-Baker method and its applications” in “Mathematical developments arising from Hilbert problems. I” [Providence, R. I. 1976]. There one can also find the nice story of the famous number theorist Carl Ludwig Siegel (1896–1981) concerning Hilbert's judgement of the difficulty of this problem.

  119. Alexander Osipovich Gelfond (1906–1968, [Dictionary of Scientific Biography], not to be confused with the famous functional analysist Izrail Moiseevich Gelfand, *1913) taught maths at the Moscow university from 1931 until his death: “Sur les nombres transcendants” [C.R. Acad. Sci. Paris 189/1929, 1224–1226].

  120. Rodion Osievich Kuzmin (1891–1949, [Izvestija Akad. Nauk SSSR, Ser. Mat. 13/1949, 385–388]): “Ob odnom novom klasse transzendentnych chisel” [Izvestija Akad. Nauk. SSSR, 7. Ser. Otd. Fyz.-Mat. Nauk 3/1930, 583–597].

  121. “Sur le septiéme probléme de Hilbert” [Izvestija Akad. Nauk SSSR, 7. Ser. Otd. Mat. Estest. Nauk, 7/1934, 623–630].

  122. Theodor Schneider (*1911), now Professor Emeritus of the University of Freiburg: “Transzendenzuntersuchungen periodischer Funktionen I: Transzendenz von Potenzen” [J. reine angew. Math. 72/1934, 65–69].

  123. Alan Baker (*1939), Professor of Pure Mathematics at the University of Cambridge since 1974.

  124. “Transcendence Theory: Advances and Applications” ed. by A. Baker and D. W. Masser [London-New York-San Francisco 1977].

  125. “Journées Arithmétiques 1980” ed. by J. V. Armitage [Cambridge 1982].

  126. Mrs. Verholzer showed me unpublished autobiographical notes by Lindemann, which are one main source for this article.

  127. All remaining language mistakes are the responsibility of the author.

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    Rudolf Fritsch

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Fritsch, R. The transcendence of π has been known for about a century — but who was the man who discovered it?. Results. Math. 7, 164–183 (1984). https://doi.org/10.1007/BF03322501

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  • DOI: https://doi.org/10.1007/BF03322501

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