Elementary Mathematics from a Higher Standpoint

1. Front Matter

   Pages i-xvi

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2. Introduction

   • Felix Klein

   Pages 1-2

3. The Simplest Geometric Formations

   1. Front Matter

      Pages 3-3

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   2. I. Line-segment, Area, Volume as Relative Quantities

      • Felix Klein

      Pages 5-26

   3. II. The Grassmannian Determinant Principle for the Plane

      • Felix Klein

      Pages 27-35

   4. III. The Grassmannian Principle for Space

      • Felix Klein

      Pages 37-48

   5. IV. Classification of the Elementary Configuration of Space Accoring to Their Behaviour Under Transformation of Rectangular Coordinates

      • Felix Klein

      Pages 49-64

   6. V. Higher Configurations

      • Felix Klein

      Pages 65-79

4. Geometric Transformations

   1. Front Matter

      Pages 81-82

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   2. I. Affine Transformations

      • Felix Klein

      Pages 83-100

   3. II. Projective Transformations

      • Felix Klein

      Pages 101-113

   4. III. Higher Point Transformations

      • Felix Klein

      Pages 115-126

   5. IV. Tansformations with Change of Space Element

      • Felix Klein

      Pages 127-136

   6. V. Theory of the Imaginary

      • Felix Klein

      Pages 137-150

5. Systematic Discussion of Geometry and its Foundations

   1. Front Matter

      Pages 151-151

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   2. I. The Systematic Discussion

      • Felix Klein

      Pages 153-182

   3. II. Foundations of Geometry

      • Felix Klein

      Pages 183-235

6. Observations About the Teaching of Geometry

   1. Front Matter

      Pages 237-241

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   2. I. The Teaching in England

      • Felix Klein

      Pages 243-247

   3. II. The Teaching in France

      • Felix Klein

      Pages 249-257

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 liked to stress: “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.

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

• c/o Springer-Verlag, Heidelberg, Germany

  Felix Klein

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. 

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