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Elementary Mathematics from a Higher Standpoint

Volume II: Geometry

  • Book
  • © 2016


  • New translation of the classic volume on geometry
  • Invaluable treatise for any mathematics educator
  • Exhibits the connections between geometry and more formalistic mathematics
  • Includes supplementary material:

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Table of contents (17 chapters)

  1. Geometric Transformations

  2. Systematic Discussion of Geometry and its Foundations

  3. Observations About the Teaching of Geometry


About this book

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.

Authors and Affiliations

  • c/o Springer-Verlag, Heidelberg, Germany

    Felix Klein

About the author

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. 

Bibliographic Information

  • Book Title: Elementary Mathematics from a Higher Standpoint

  • Book Subtitle: Volume II: Geometry

  • Authors: Felix Klein

  • Translated by: Gert Schubring

  • DOI:

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Education, Education (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2016

  • Softcover ISBN: 978-3-662-49443-1Published: 06 July 2016

  • eBook ISBN: 978-3-662-49445-5Published: 29 June 2016

  • Edition Number: 1

  • Number of Pages: XVI, 315

  • Number of Illustrations: 157 b/w illustrations

  • Additional Information: Original German edition published by Verlag Julius Springer, Berlin, Germany, 1925

  • Topics: Mathematics Education, Mathematics, general

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