• Mathematician Christian Felix Klein, 1849
• Mathematician Marius Sophus Lie, 1842

Klein was born in Düsseldorf, to Prussian parents; his father was a Prussian government official's secretary stationed in the Rhine Province. He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865 – 1866, intending to become a physicist. At that time, Julius Plücker held Bonn's chair of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest was geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868.

Plücker died in 1868, leaving his book on the foundations of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes, and thus became acquainted with Alfred Clebsch, who had moved to Göttingen in 1868. Klein visited Clebsch the following year, along with visits to Berlin and Paris. In July 1870, at the outbreak of the Franco - Prussian War, he was in Paris and had to leave the country. For a short time, he served as a medical orderly in the Prussian army before being appointed lecturer at Göttingen in early 1871.

Erlangen appointed Klein professor in 1872, when he was only 23. In this, he was strongly supported by Clebsch, who regarded him as likely to become the leading mathematician of his day. Klein did not build a school at Erlangen where there were few students, and so he was pleased to be offered a chair at Munich's Technische Hochschule in 1875. There he and Alexander von Brill taught advanced courses to many excellent students, e.g., Adolf Hurwitz, Walther von Dyck, Karl Rohn, Carl Runge, Max Planck, Luigi Bianchi, and Gregorio Ricci - Curbastro.

In 1875 Klein married Anne Hegel, the granddaughter of the philosopher Georg Wilhelm Friedrich Hegel.

After five years at the Technische Hochschule, Klein was appointed to a chair of geometry at Leipzig. There his colleagues included Walther von Dyck, Rohn, Eduard Study and Friedrich Engel. Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. In 1882, his health collapsed; in 1883 – 1884, he was plagued by depression. Nonetheless his research continued; his seminal work on hyperelliptic sigma functions dates from around this period, being published in 1886 and 1888.

Klein accepted a chair at the University of Göttingen in 1886. From then until his 1913 retirement, he sought to re-establish Göttingen as the world's leading mathematics research center. Yet he never managed to transfer from Leipzig to Göttingen his own role as the leader of a school of geometry. At Göttingen, he taught a variety of courses, mainly on the interface between mathematics and physics, such as mechanics and potential theory.

The research center Klein established at Göttingen served as a model for the best such centers throughout the world. He introduced weekly discussion meetings, and created a mathematical reading room and library. In 1895, Klein hired David Hilbert away from Königsberg; this appointment proved fateful, because Hilbert continued Göttingen's glory until his own retirement in 1932.

Under Klein's editorship, Mathematische Annalen became one of the very best mathematics journals in the world. Founded by Clebsch, only under Klein's management did it first rival then surpass Crelle's Journal based out of the University of Berlin. Klein set up a small team of editors who met regularly, making democratic decisions. The journal specialized in complex analysis, algebraic geometry, and invariant theory (at least until Hilbert killed the subject). It also provided an important outlet for real analysis and the new group theory.

Thanks in part to Klein's efforts, Göttingen began admitting women in 1893. He supervised the first Ph.D. thesis in mathematics written at Göttingen by a woman; she was Grace Chisholm Young, an English student of Arthur Cayley's, whom Klein admired.

Around 1900, Klein began to take an interest in mathematical instruction in schools. In 1905, he played a decisive role in formulating a plan recommending that the rudiments of differential and integral calculus and the function concept be taught in secondary schools. This recommendation was gradually implemented in many countries around the world. In 1908, Klein was elected chairman of the International Commission on Mathematical Instruction at the Rome International Mathematical Congress. Under his guidance, the German branch of the Commission published many volumes on the teaching of mathematics at all levels in Germany.

The London Mathematical Society awarded Klein its De Morgan Medal in 1893. He was elected a member of the Royal Society in 1885, and was awarded its Copley medal in 1912. He retired the following year due to ill health, but continued to teach mathematics at his home for some years more.

Klein bore the title of Geheimrat.

He died in Göttingen in 1925.

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 in 1870. In collaboration with Sophus Lie, he discovered the fundamental properties of the asymptotic lines on the Kummer surface. They went on to investigate W-curves, curves invariant under a group of projective transformations. It was Lie who introduced Klein to the concept of group, which was to play a major role in his later work. Klein also learned about groups from Camille Jordan.

Klein devised the 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 Euclidean space of dimensions 4 and higher.

In the 1890s, Klein turned to mathematical physics, a subject from which he had never strayed far, writing on the gyroscope with Arnold Sommerfeld. In 1894 he launched the idea of an encyclopedia of mathematics including its applications, which became the Encyklopädie der mathematischen Wissenschaften. This enterprise, which ran until 1935, provided an important standard reference of enduring value.

In 1871, while at Göttingen, Klein made major discoveries in geometry. He published two papers On the So-called Non - Euclidean Geometry showing that Euclidean and non - Euclidean geometries could be considered special cases of a projective surface with a specific conic section adjoined. This had the remarkable corollary that non - Euclidean geometry was consistent if and only if Euclidean geometry was, putting Euclidean and non - Euclidean geometries on the same footing, and ending all controversy surrounding non - Euclidean geometry. Cayley never accepted Klein's argument, believing it to be circular.

Klein's synthesis of geometry as the study of the properties of a space that is invariant under a given group of transformations, known as the Erlangen Program (1872), profoundly influenced the evolution of mathematics. This program was set out in Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion. The Program proposed a unified approach to geometry that became (and remains) the accepted view. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. Thus the Program's definition of geometry encompassed both Euclidean and non - Euclidean geometry.

Today the significance of Klein's contributions to geometry is more than evident, but not because those contributions are now seen as strange or wrong. On the contrary, those contributions have become so much a part of our present mathematical thinking that it is hard for us to appreciate their novelty, and the way in which they were not immediately accepted by all his contemporaries.

Klein saw his work on function theory as his major contribution to mathematics, specifically his work on:

• The link between certain ideas of Riemann's and invariant theory,
• Number theory and abstract algebra;
• Group theory;
• Geometry with more than 3 dimensions and differential equations, especially equations he invented, namely elliptic modular functions and automorphic functions.

Klein showed that the modular group moves the fundamental region of the complex plane so as to tessellate that plane. In 1879, he looked at the action of PSL(2,7), thought of as an image of the modular group, and obtained an explicit representation of a Riemann surface today called the Klein quartic. He showed that that surface was a curve in projective space, that its equation was x³y + y³z + z³x = 0, and that its group of symmetries was PSL(2,7) of order 168. His Ueber Riemann's Theorie der algebraischen Funktionen und ihre Integrale (1882) treats function theory in a geometric way, connecting potential theory and conformal mappings. This work drew on notions from fluid dynamics.

Klein considered equations of degree > 4, and was especially interested in using transcendental methods to solve the general equation of the fifth degree. Building on the methods of Hermite and Kronecker, he produced similar results to those of Brioschi and went on to completely solve the problem by means of the icosahedral group. This work led him to write a series of papers on elliptic modular functions.

In his 1884 book on the icosahedron, Klein set out a theory of automorphic functions, connecting algebra and geometry. However Poincaré published an outline of his theory of automorphic functions in 1881, which led to a friendly rivalry between the two men. Both sought to state and prove a grand uniformization theorem that would serve as a capstone to the emerging theory. Klein succeeded in formulating such a theorem and in sketching a strategy for proving it. But while doing this work his health collapsed, as mentioned above.

Klein summarized his work on automorphic and elliptic modular functions in a four volume treatise, written with Robert Fricke over a period of about 20 years.

His 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 traveled 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 traveled 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 decided then to visit Luigi Cremona in Milan but he was arrested at Fontainebleau under suspicion of being a German spy, an event which made him famous 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 (present day Oslo), in 1871, with a thesis entitled 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.

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).

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 professor at Leipzig, replacing Klein, who had moved to Göttingen. In November 1889 he suffered a mental breakdown and had to be hospitalized until June 1890. Lie resigned from his post in May 1898 and returned to Norway in September of that year.

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.

Sophus Lie died at the age of 56, due to pernicious anemia, a disease caused by impaired absorption of vitamin B12.

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. 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”.