**AMS Chelsea Publishing**

Volume: 339;
1894;
109 pp;
Hardcover

MSC: Primary 01; 11; 13; 14; 22; 33; 34; 35; 51; 53;

Print ISBN: 978-0-8218-2733-8

Product Code: CHEL/339.H

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# Lectures on Mathematics

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*Felix Klein*

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.

#### Readership

Graduate students, research mathematicians, and mathematical historians.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Lectures on Mathematics

- Cover Cover11
- Title page iii4
- Introduction v6
- Preface vii8
- Contents xi12
- Lecture I. Clebsch 114
- Lecture II. Sophus Lie 922
- Lecture III. Sophus Lie 1831
- Lecture IV. On the real shape of algebraic curves and surfaces 2538
- Lecture V. Theory of functions and geometry 3346
- Lecture VI. On the mathematical character of space-intuition and the relation of pure mathematics to the applied sciences 4154
- Lecture VII. The transcendency of the numbers 𝑒 and 𝜋 5164
- Lecture VIII. Ideal numbers 5871
- Lecture IX. The solution of higher algebraic equations 6780
- Lecture X. On some recent advances in hyperelliptic and Abelian functions 7588
- Lecture XI. The most recent researches in non-Euclidean geometry 8598
- Lecture XII. The study of mathematics at Göttingen 94107
- The development of mathematics at the German Universities 99112
- Back Cover Back Cover1130