# Elementary Mathematics from a Higher Standpoint

## Volume II: Geometry

• Felix Klein
Book

1. Front Matter
Pages i-xvi
2. Felix Klein
Pages 1-2
3. ### The Simplest Geometric Formations

1. Front Matter
Pages 3-3
2. Felix Klein
Pages 5-26
3. Felix Klein
Pages 27-35
4. Felix Klein
Pages 37-48
5. Felix Klein
Pages 65-79
4. ### Geometric Transformations

1. Front Matter
Pages 81-82
2. Felix Klein
Pages 83-100
3. Felix Klein
Pages 101-113
4. Felix Klein
Pages 115-126
5. Felix Klein
Pages 127-136
6. Felix Klein
Pages 137-150
5. ### Systematic Discussion of Geometry and its Foundations

1. Front Matter
Pages 151-151
2. Felix Klein
Pages 153-182
3. Felix Klein
Pages 183-235
6. ### Observations About the Teaching of Geometry

1. Front Matter
Pages 237-241
2. Felix Klein
Pages 243-247
3. Felix Klein
Pages 249-257

### Introduction

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 II presents a paradigmatic realisation of Klein’s approach of elementarisation for teacher education. It is shown how the various geometries, elaborated particularly since the beginning of the 19th century, are revealed as becoming unified in a new restructured geometry. As Klein put it: “Projective geometry is all geometry”. Non-Euclidean geometry proves to constitute a part of this unifying process. The teaching of geometry is discussed in a separate chapter, which provides moreover important information on the history of geometry teaching and an international comparison.

Felix Klein (1849-1925) was a leading German mathematician whose research interests included group theory, complex analysis, and geometry. His work influenced many areas of mathematics and related subjects, ranging from mathematical physics to mathematical didactics. To this day, Felix Klein is considered one of the most important mathematicians of the 19th century.

### Keywords

Analysis of Geometric Transformations Foundations of Geometry Geometric Manifolds Geometry Mathematics Education

#### Authors and affiliations

• Felix Klein
• 1
1. 1.c/o Springer-VerlagHeidelbergGermany

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-662-49445-5
• Copyright Information Springer-Verlag Berlin Heidelberg 2016
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages Education
• Print ISBN 978-3-662-49443-1
• Online ISBN 978-3-662-49445-5