# Lindemann, Carl Louis Ferdinand

# Lindemann, Carl Louis Ferdinand

(*b.* Hannover, Germany, 12 April 1852; *d.* Munich, Germany, 6 March 1939)

*mathematics.*

Lindemann studied in Göttingen, Erlangen, and Munich from 1870 to 1873. He was particularly influenced by Clebsch, who pioneered in invariant theory. Lindemann received the doctorate from Erlangen in 1873, under the direction of F. Klein. His dissertation dealt with infinitely small movements of rigid bodies under general projective mensuration. He then undertook a year-long academic journey to Oxford, Cambridge, London, and Paris, where he met Chasles, J. Berirand, C. Jordan, and Hermile. In 1877 Lindemann qualified as lecturer at Würzburg. In the same year he became assistant professor at the University of Freiburg. He became full professor in 1879. In 1883 he accepted an appointment at Königsberg and finally, ten years later, one at Munich, where he was for many years also active on the administrative board of the university. He was elected an associate member of the Bavarian Academy of Sciences in 1894, and a full member in 1895.

Lindemann was one of the founders of the modern German educational system. He emphasized the development of the seminar and in his lectures communicated the latest research results. He also supervised more than sixty doctoral students, including David Hilbert. In the years before World War I, he represented the Bavarian Academy in the meetings of the International Association of Academies of Sciences and Learned Societies.

Lindemann wrote papers on numerous branches of mathematics and on theoretical mechanics and spectrum theory. In addition, he edited and revised Clebsch’s geometry lectures following the hater’s untimely death (as *Vorlesungenüber Geometric* [Leipzig, 1876–1877]).

Lindemann’s most outstanding original research is his 1882 work on the transcendence of π This work definitively settled the ancient problem of the quadrature of the circle; it also redefined and reanimated fundamental questions in the mathematics of its own time.

In the nineteenth century mathematicians had realized that not every real number was necessarily the root of an algebraic equation and that therefore non-algebraic, so-called transcendental, numbers must exist. Liouville stated certain transcendental numbers, and in 1873 Hermite succeeded in demonstrating the transcendence of the base *e* of the natural logarithms. It was at this point that Lindemann turned his attention to the subject.

The demonstration of the transcendence of π is on the proof of the theorem that, except for trivial cases, every expression of the form , where A_{i} and a_{i}are algebraic numbers, must always be different from zero. Since the imaginary unity , as the root of the equation.x^{2} + 1 = 0, is algebraic, and since e^{πi} + *e*^{0} = 0, then πi and therefore also π cannot be algebraic. So much the less then is π root, representable by a radical, of an algebraic equation; hence the quadrature of the circle is impossible, inasmuch as πcannot be constructed with ruler and compass.

Lindemann also composed works in the history of mathematics, including a “Geschichte der Polyder und der Zahlzeiehen” (in *Sitzungsberichte der mathe-matisch-physikalischen Klasse der Bayrischen Akademie der Wissenschaften* [1896]). He and his wife collaborated in translating and revising some of the works of Poincaré. Their edition of his *La science et Phvpothése* (as *Wissenschaft und Hypo these* [Leipzig, 1904]) contributed greatly to the dissemination of Poincaré’s ideas in Germany.

## BIBLIOGRAPHY

I. Original Works. In addition to the works cited in the text, see especially “Über die Ludolphsehe Zahl,” in *Sitzungsberichte der Preussischen Akademie der Wissen schaften zu Berlin*, math-phys. Klasse, **22** (1882); and “Über die Zahl π,” in *Mathematische Annalen,***25** (1882).

II.Secondary Literature. See “Druckschriften-Ver-zeichnis von F. Lindemann,” in *Almanack der Königlich Bayrischen Akademie der Wlssenschaften zum 150. Stiftungsfest* (Munich, 1909), 303–306; “F. von Lindemanns 70. Geburtstag,” in *lahresberkhte der Mathenmtikerverei’ nigimg*, **31** (1922), 24–30; and C. Carathéodory, “Nekrolog auf Ferdinand von Lindemann,” in *Sitzungsberichte der mathematisch-naturwissenschaftlichen Abteiiung der Bayrischen Akademie der Wissenschaften zuMiinchen,* no. **I** (1940), 61–63.

H. Wussing

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