# Felix Klein

# Felix Klein

**1849-1925**

**German Mathematician**

Felix Klein's work had a profound effect on mathematical thought. He unified Euclidean geometry with the non-Euclidean geometries of Nikolai Lobachevsky (1792-1856) and Georg Riemann (1826-1866) by showing that they all could be derived as special cases of a larger system called *projective geometry*.

Projective geometry is more fundamental than Euclidean geometry because it deals with properties such as when points lie on the same line and when a set of lines meet in one point. These properties are *invariant* under a larger group of transformations, or mathematical manipulations, than the *congruence*, or equality of lengths, angles, and areas with which Euclidean geometry mainly concerns itself. "Projective geometry has opened up for us with the greatest facility new territories in our science," Klein wrote, "and has rightly been called a royal road to its own particular field of knowledge."

Klein, whose full given name was Christian Felix, was born on April 25, 1849, in Düsseldorf, Germany, and attended the *gymnasium*, or secondary school there. Later he attended the University of Bonn. As a postgraduate student sojourning in Paris, he worked on group theory with the Norwegian Sophus Lie (1842-1899), and became interested in its possibilities for unification of disparate mathematical situations.

In 1872, he became a professor at the University of Erlangen. It was there, in a lecture written to inaugurate his appointment, that he set forth his views on geometry. His system thus became known as the *Erlanger Programm*. It was not until 1916 that new geometries were discovered that did not fit the *Erlanger Programm*, thus requiring a wider-ranging synthesis. However, for the classes of geometries that it covers, Klein's analysis is still useful. It was especially influential in the United States for 50 years after its publication.

Klein taught at the Technical Institute in Munich from 1875 to1880. There he married Anne Hegel, the granddaughter of the great German philosopher Georg Wilhelm Friedrich Hegel. Later Klein moved to the universities of Leipzig (1880-1886) and Göttingen (1886-1913). Beginning in 1872, he edited the *Mathematische Annalen* of Göttingen. He began work on a major mathematical encyclopedia, the *Encyklopädie der Mathematischen Wissenschaften *("Encyclopedia of the Mathematical Sciences") in 1895, and supervised it for the rest of his life.

The eminent mathematician was the author of many popular books on the theory and history of mathematics and mathematical education, including *Elementare Mathematik von höheren standpunkte aus* ("Elementary Mathematics from an Advanced Standpoint," 1908). His view of mathematics was that it was "the science of self-evident things." Once a proof or calculation had been finished, no further marshaling of facts or opinions was necessary.

He was a leader in educational reform, and spearheaded the movement encouraging *functional thinking*. This advanced the idea that as part of a general education, students should learn to think in terms of *variables* and the *functions* that describe how one variable depends upon others. The importance of this concept is
that it is used in every field of mathematics and science as well as in daily life. For example, a function can be used to describe how the funds in a bank account grow over time.

Among Klein's other important works were *Vorlesungen über das Ikosaeder* ("Lectures on the Icosahedron," 1884), in which he showed how the rotation of regular solids such as the 20-sided icosahedron could be applied to solving difficult problems in algebra.

Always of delicate health, Klein tended to overwork. His physical and mental condition collapsed in 1882, and he suffered from depression for the next few years. This period essentially marked the end of his own career in research. However, during the second half of his professional life he established one of the world's finest mathematical research centers at Göttingen. He also concentrated on teaching and his efforts in improving mathematical education. He retired in 1913, but tutored students in his home during the First World War. He died on June 22, 1925, in Göttingen.

**SHERRI CHASIN CALVO**

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**Felix Klein**