*The Vision of Felix Klein*

**Author**: David Mumford,Caroline Series,David Wright

**Publisher:** Cambridge University Press

**ISBN:** 9780521352536

**Category:** Mathematics

**Page:** 395

**View:** 6604

Highly illustrated realization of infinitely reflected images related to fractals, chaos and symmetry.

In the late summer of 1893, following the Congress of Mathematicians held in Chicago, Felix Klein gave two weeks of lectures on the current state of mathematics. Rather than offering a universal perspective, Klein presented his personal view of the most important topics of the time. It is remarkable how most of the topics continue to be important today. Originally published in 1893 and reissued by the AMS in 1911, we are pleased to bring this work into print once more with this new edition. Klein begins by highlighting the works of Clebsch and of Lie. In particular, he discusses Clebsch's work on Abelian functions and compares his approach to the theory with Riemann's more geometrical point of view. Klein devotes two lectures to Sophus Lie, focussing on his contributions to geometry, including sphere geometry and contact geometry. Klein's ability to connect different mathematical disciplines clearly comes through in his lectures on mathematical developments. For instance, he discusses recent progress in non-Euclidean geometry by emphasizing the connections to projective geometry and the role of transformation groups. In his descriptions of analytic function theory and of recent work in hyperelliptic and Abelian functions, Klein is guided by Riemann's geometric point of view. He discusses Galois theory and solutions of algebraic equations of degree five or higher by reducing them to normal forms that might be solved by non-algebraic means. Thus, as discovered by Hermite and Kronecker, the quintic can be solved "by elliptic functions". This also leads to Klein's well-known work connecting the quintic to the group of the icosahedron. Klein expounds on the roles of intuition and logical thinking in mathematics. He reflects on the influence of physics and the physical world on mathematics and, conversely, on the influence of mathematics on physics and the other natural sciences. The discussion is strikingly similar to today's discussions about ``physical mathematics''. There are a few other topics covered in the lectures which are somewhat removed from Klein's own work. For example, he discusses Hilbert's proof of the transcendence of certain types of numbers (including $\pi$ and $e$), which Klein finds much simpler than the methods used by Lindemann to show the transcendence of $\pi$. Also, Klein uses the example of quadratic forms (and forms of higher degree) to explain the need for a theory of ideals as developed by Kummer. Klein's look at mathematics at the end of the 19th Century remains compelling today, both as history and as mathematics. It is delightful and fascinating to observe from a one-hundred year retrospect, the musings of one of the masters of an earlier era.

Graphical and geometrically perceptive methods enliven a distinguished mathematician's treatment of arithmetic, algebra, and analysis. Topics include calculating with natural numbers, complex numbers, goniometric functions, and infinitesimal calculus. 1932 edition. Includes 125 figures.

These three volumes constitute the first complete English translation of Felix Klein’s seminal series “Elementarmathematik vom höheren Standpunkte aus”. “Complete” has a twofold meaning here: First, there now exists a translation of volume III into English, while until today the only translation had been into Chinese. Second, the English versions of volume I and II had omitted several, even extended parts of the original, while we now present a complete revised translation into modern English. The volumes, first published between 1902 and 1908, are lecture notes of courses that Klein offered to future mathematics teachers, realizing a new form of teacher training that remained valid and effective until today: Klein leads the students to gain a more comprehensive and methodological point of view on school mathematics. The volumes enable us to understand Klein’s far-reaching conception of elementarisation, of the “elementary from a higher standpoint”, in its implementation for school mathematics. This volume I is devoted to what Klein calls the three big “A’s”: arithmetic, algebra and analysis. They are presented and discussed always together with a dimension of geometric interpretation and visualisation - given his epistemological viewpoint of mathematics being based in space intuition. A particularly revealing example for elementarisation is his chapter on the transcendence of e and p, where he succeeds in giving concise yet well accessible proofs for the transcendence of these two numbers. It is in this volume that Klein makes his famous statement about the double discontinuity between mathematics teaching at schools and at universities – it was his major aim to overcome this discontinuity.

This collection of essays by a distinguished mathematician and teacher examines important issues of dynamics from the viewpoint of the theory of functions of the complex variable. Based on a series of lectures delivered by Felix Klein in conjunction with Princeton University’s 150th anniversary, these presentations center on the problem inherent in the motion of a top—that is, a rigid body rotating about an axis—when a single point in this axis other than the center of gravity is fixed in position. The contents of this volume render discussions of dynamics-related issues simpler, more attractive, and relevant not only to mathematicians but also to engineers, physicists, and astronomers. Unabridged republication of the classic 1897 edition.

The Theory of the Top was originally presented by Felix Klein as an 1895 lecture at Göttingen University that was broadened in scope and clarified as a result of collaboration with Arnold Sommerfeld. The Theory of the Top: Volume III. Perturbations: Astronomical and Geophysical Applications is the third installment in a series of four self-contained English translations that provide insights into kinetic theory and kinematics.

Founder of the National Society for Graphology, Felix Klein began his study of graphology in his birthplace, Vienna, Austria, at the age of thirteen. He was a practicing graphologist all of his life and lectured and gave seminars throughout the United States and in Canada, England, Germany, Israel and Mexico. Mr. Klein came to the United States in 1940 after spending six months each in the concentration camps at Dachau and Buchenwald. While in those camps he formulated his theory of directional pressure as a result of studying changes in the handwriting of his fellow inmates. Mr. Klein did extensive work in personnel selection for major companies and banks, vocational guidance, and individual analyses, as well as forensic document examination for such entities as the U.N., AT&T, and a major political figure in Ghana, Africa. It was probably as a teacher that Felix was most known and loved. He held classes and offered correspondence courses in all levels of graphology: elementary, intermediate, advanced, Master Research, and Psychology for Graphologists. Wherever Felix spoke, his warm, caring personality and his naturalness and keen sense of humor generated enthusiastic responses from young and old alike.

The French priest offers his predictions for the country, which are largely concerned with religious groups; he also examines various social institutions, such as schools, and expresses concern regarding employment issues. He is also extremely interested in immigration and the ethnic groups coming to the U.S.

The lecture series on the Theory of the Top was originally given as a dedication to Göttingen University by Felix Klein in 1895, but has since found broader appeal. The Theory of the Top: Volume I. Introduction to the Kinematics and Kinetics of the Top is the first of a series of four self-contained English translations that provide insights into kinetic theory and kinematics.

Focusing on functions defined on Riemann surfaces, this text demonstrates how Riemann's ideas about Abelian integrals can be arrived in terms of the flow of electric current on surfaces. 1893 edition.

The Evanston Colloquium: Lectures on Mathematics by Felix Klein