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Book DetailsAMS Chelsea PublishingVolume: 339; 1894; 109 ppMSC: Primary 01; 11; 13; 14; 22; 33; 34; 35; 51;
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 nonEuclidean 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 nonalgebraic means. Thus, as discovered by Hermite and Kronecker, the quintic can be solved "by elliptic functions". This also leads to Klein's wellknown 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 onehundred year retrospect, the musings of one of the masters of an earlier era.ReadershipGraduate students, research mathematicians, and mathematical historians.

Table of Contents

Chapters

Lecture I. Clebsch

Lecture II. Sophus Lie

Lecture III. Sophus Lie

Lecture IV. On the real shape of algebraic curves and surfaces

Lecture V. Theory of functions and geometry

Lecture VI. On the mathematical character of spaceintuition and the relation of pure mathematics to the applied sciences

Lecture VII. The transcendency of the numbers $e$ and $\pi $

Lecture VIII. Ideal numbers

Lecture IX. The solution of higher algebraic equations

Lecture X. On some recent advances in hyperelliptic and Abelian functions

Lecture XI. The most recent researches in nonEuclidean geometry

Lecture XII. The study of mathematics at Göttingen

The development of mathematics at the German Universities


Reviews

It is a noble example that Professor Klein has set all ages of mathematicians that, beginning his activity at a time when the contributions of the immediate past were so rich and so unrelated, he was able to uncover the essential bonds that connect them and to discern the fields to whose development the new methods were best adapted … His instinct for that which is vital in mathematics is sure, and the light with which his treatment illumines the problems here considered may well serve as a guide for the youth who is approaching the study of the problems of a later day.
William F. Osgood, President of the AMS, 1905–1906


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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 nonEuclidean 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 nonalgebraic means. Thus, as discovered by Hermite and Kronecker, the quintic can be solved "by elliptic functions". This also leads to Klein's wellknown 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 onehundred year retrospect, the musings of one of the masters of an earlier era.
Graduate students, research mathematicians, and mathematical historians.

Chapters

Lecture I. Clebsch

Lecture II. Sophus Lie

Lecture III. Sophus Lie

Lecture IV. On the real shape of algebraic curves and surfaces

Lecture V. Theory of functions and geometry

Lecture VI. On the mathematical character of spaceintuition and the relation of pure mathematics to the applied sciences

Lecture VII. The transcendency of the numbers $e$ and $\pi $

Lecture VIII. Ideal numbers

Lecture IX. The solution of higher algebraic equations

Lecture X. On some recent advances in hyperelliptic and Abelian functions

Lecture XI. The most recent researches in nonEuclidean geometry

Lecture XII. The study of mathematics at Göttingen

The development of mathematics at the German Universities

It is a noble example that Professor Klein has set all ages of mathematicians that, beginning his activity at a time when the contributions of the immediate past were so rich and so unrelated, he was able to uncover the essential bonds that connect them and to discern the fields to whose development the new methods were best adapted … His instinct for that which is vital in mathematics is sure, and the light with which his treatment illumines the problems here considered may well serve as a guide for the youth who is approaching the study of the problems of a later day.
William F. Osgood, President of the AMS, 1905–1906