ISBN 0817633162 Vrij van inscripties e.d.
Translated by Sergei Sassinsky. Edited by Hardy Grant and Abe Shenitzer

Christian Felix Klein (German: [klaɪn]; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory. His 1872 Erlangen program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time.
Klein's dissertation, on line geometry and its applications to mechanics, classified second degree line complexes using Weierstrass's theory of elementary divisors.
Klein's first important mathematical discoveries were made during 1870. In collaboration with Sophus Lie, he discovered the fundamental properties of the asymptotic lines on the Kummer surface. They later investigated W-curves, curves invariant under a group of projective transformations. It was Lie who introduced Klein to the concept of group, which was to have a major role in his later work. Klein also learned about groups from Camille Jordan.
Klein devised the "Klein bottle" named after him, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space, but it may be immersed as a cylinder looped back through itself to join with its other end from the "inside". It may be embedded in the Euclidean space of dimensions 4 and higher. The concept of a Klein Bottle was devised as a 3-Dimensional Möbius strip, with one method of construction being the attachment of the edges of two Möbius strips.
During the 1890s, Klein began studying mathematical physics more intensively, writing on the gyroscope with Arnold Sommerfeld. During 1894, he initiated the idea of an encyclopedia of mathematics including its applications, which became the Encyklopädie der mathematischen Wissenschaften. This enterprise, which endured until 1935, provided an important standard reference of enduring value.

Marius Sophus Lie (/liː/ LEE; Norwegian: [liː]; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
He had his primary education in the south-eastern coast of Moss, before attending high school at Oslo (known then as Christiania). After graduating from high school, his ambition towards a military career was dashed when the army rejected him due to poor eyesight. Then he enrolled at the University of Christiania.
Sophus Lie's first mathematical work, Repräsentation der Imaginären der Plangeometrie, was published in 1869 by the Academy of Sciences in Christiania and also by Crelle's Journal. That same year he received a scholarship and travelled to Berlin, where he stayed from September to February 1870. There, he met Felix Klein and they became close friends. When he left Berlin, Lie travelled to Paris, where he was joined by Klein two months later. There, they met Camille Jordan and Gaston Darboux. But on 19 July 1870 the Franco-Prussian War began and Klein (who was Prussian) had to leave France very quickly. Lie left for Fontainebleau where he was arrested, suspected of being a German spy, garnering him fame in Norway. He was released from prison after a month, thanks to the intervention of Darboux.
Lie obtained his PhD at the University of Christiania (in present-day Oslo) in 1871 with a thesis entitled Over en Classe geometriske Transformationer (On a Class of Geometric Transformations). It would be described by Darboux as "one of the most handsome discoveries of modern Geometry". The next year, the Norwegian Parliament established an extraordinary professorship for him. That same year, Lie visited Klein, who was then at Erlangen and working on the Erlangen program.
In 1872, Lie spent eight years together with Peter Ludwig Mejdell Sylow, editing and publishing the mathematical works of their countryman, Niels Henrik Abel.
At the end of 1872, Sophus Lie proposed to Anna Birch, then eighteen years old, and they were married in 1874. The couple had three children: Marie (b. 1877), Dagny (b. 1880) and Herman (b. 1884).
Since 1876, he co-edited the journal Archiv for Mathematik og Naturvidenskab, together with the physician Jacob Worm-Müller, and the biologist Georg Ossian Sars.
In 1884, Friedrich Engel arrived at Christiania to help him, with the support of Klein and Adolph Mayer (who were both professors at Leipzig by then). Engel would help Lie to write his most important treatise, Theorie der Transformationsgruppen, published in Leipzig in three volumes from 1888 to 1893. Decades later, Engel would also be one of the two editors of Lie's collected works.
In 1886, Lie became a professor at Leipzig, replacing Klein, who had moved to Göttingen. In November 1889, Lie suffered a mental breakdown and had to be hospitalized until June 1890. Subsequently he returned to his post, but over the years his anaemia progressed to the point where he returned to his homeland. In 1898 he tendered his resignation in May, and left for home in September the same year. He died the following year in 1899 at the age of 56, due to pernicious anemia, a disease caused by impaired absorption of vitamin B12.
He was made Honorary Member of the London Mathematical Society in 1878, Member of the French Academy of Sciences in 1892, Foreign Member of the Royal Society of London in 1895 and foreign associate of the National Academy of Sciences of the United States of America in 1895.
Legacy
Lie's principal tool, and one of his greatest achievements, was the discovery that continuous transformation groups (now called, after him, Lie groups) could be better understood by "linearizing" them, and studying the corresponding generating vector fields (the so-called infinitesimal generators). The generators are subject to a linearized version of the group law, now called the commutator bracket, and have the structure of what is today called a Lie algebra.
Hermann Weyl used Lie's work on group theory in his papers from 1922 and 1923, and Lie groups today play a role in quantum mechanics.[5] However, the subject of Lie groups as it is studied today is vastly different from what the research by Sophus Lie was about and "among the 19th century masters, Lie's work is in detail certainly the least known today".
Sophus Lie was an eager proponent in the establishment of the Abel Prize. Inspired by the Nansen fund named after Fridtjof Nansen, and the lack of a prize for mathematics in the Nobel Prize. He gathered support for the establishment of an award for outstanding work in pure mathematics.
Lie advised many doctoral students who went on to become successful mathematicians. Élie Cartan became widely regarded as one of the greatest mathematicians of the 20th century. Kazimierz Żorawski's work was proved to be of importance to a variety of fields. Hans Frederick Blichfeldt made great contributions to various fields of mathematics.
(Wikipedia)