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Elementary Mathematics from an Advanced Standpoint: Geometry Paperback – June 18, 2004

by Felix Klein (Author)

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"Nothing comparable to it." — Mathematics Teacher. This comprehensive three-part treatment begins with a consideration of the simplest geometric manifolds: line-segment, area, and volume as relative magnitudes; the Grassmann determinant principle for the plane and the Grassmann principle for space; classification of the elementary configurations of space according to their behavior under transformation of rectangular coordinates; and derivative manifolds. The second section, on geometric transformations, examines affine and projective transformations; higher point transformations; transformations with change of space element; and the theory of the imaginary. The text concludes with a systematic discussion of geometry and its foundations. 1939 edition. 141 figures.

Elementary Mathematics from an Advanced Standpoint: Geometry

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Product Details

Paperback: 224 pages
Publisher: Dover Publications (June 18, 2004)
Language: English
ISBN-10: 0486474410
ISBN-13: 978-0486434810
ASIN: 0486434818
Product Dimensions: 5.5 x 0.5 x 8.5 inches
Shipping Weight: 10.4 ounces (View shipping rates and policies)
Average Customer Review: 4.8 out of 5 stars See all reviews (5 customer reviews)
Amazon Best Sellers Rank: #970,116 in Books (See Top 100 in Books)
- #912 in Books > Science & Math > Mathematics > Geometry & Topology
- #198449 in Books > Reference

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46 of 48 people found the following review helpful

The great master's way of looking at geometry

By Viktor Blasjo on April 12, 2005

Format: Paperback

About the set: Klein's Elementarmathematik lectures was intended as a survey of mathematics for those who already knew most of the technical detail, especially future teachers, but who perhaps lacked a good understanding of mathematics as a whole. The lack of a broad perspective is probably at least as big a problem today as it was then, so Klein's text is still valuable. Klein also frequently discusses historical and pedagogical aspects, and the tone is quite informal throughout.

About this volume: The first part is an elementary introduction to vector techniques in geometry, which is probably not very interesting for the modern reader. The second part of the book is more interesting. Klein has explained in his preface that he wants to present a unified view of geometry, and here in the second part he shows how many diverse concepts can be brought together under the notion of transformation. This approach paves the way for a discussion of Klein's own Erlanger Programm in part three. This third part is very nice. First there is a "systematic discussion of geometry", where the Erlanger Programm is one of the themes, then there is a discussion of the foundations of geometry. This part of the book makes great pleasure reading, as Klein speaks freely and informally, and sometimes voices his personal opinions.

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17 of 17 people found the following review helpful

A rare old gem of accessible yet important math

By Georg Essl on August 4, 2010

Format: Paperback

Klein in his later years decided that bringing new developments in mathematics into the high-school math curriculum was very important and to this end he gave a series of lectures to high-school teachers that explicated those aspects of recent developments of mathematics that would be accessible and useful for high-school teaching. This led to the three volumes on Elementary Mathematics from an Advanced Standpoint, originally published in German and luckily available in this fine English translation, dirt cheap from Dover.

This volume, on Geometry, is in my view the best of the series. This is not just an explication of linear geometry, it is an explanation of the powerful joint treatment of geometry and group theory of which Klein himself was a driving force (through his "Erlangen Program").

However this alone does not do this book justice. This is the only book I am aware off that gives a thorough yet accessible account of what we would today call Exterior Algebra in a very concrete and easy determinant representation (a very natural representation for this algebra). Incidentally we really should be calling Exterior Algebra also Exterior Geometry to highlight the deep relation of the two. Ultimately exterior algebra is the algebra of oriented lines, areas, volumes and higher-dimensional extensive quantities and rotating versions thereof (where Grassmann invented the word extensive go create a unifying term for everything that has some extension, be it a line, an area etc). Klein uses determinants to explain why the orientation matters, and how, by keeping the orientation alive one can naturally recover an algebra of determinants that allows one to construct a wealth of theorems of linear geometry, in fact invariance and group theory.Read more ›

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5 of 6 people found the following review helpful

Unification of Geometries

By Sitansu S. Mittra on June 15, 2009

Format: Paperback

Felix Klein's (1849-1925) famous Erlanger Programme was an address he prepared and delivered to a limited audience when he was appointed in 1872 to the philosophical faculty and the Senate of the University of Erlangen in Germany. He published it subsequently as a paper in July 1893 under the title "Recent Researches in Geometry". Later in 1908 Klein published this book as a much more enhanced and comprehensive version of his 1893 paper. The book consists of three parts, The Simplest Geometric Manifolds, Geometric Transformations, and Systematic Discussion of Geometry and Its Foundations. Klein defines a geometry as the study of the invariants of a group (in the algebraic sense) of transformations. As such, he defines and discusses in a rigorous manner projective, affine, and "plane" geometries corresponding to collineations, affine transformations, and motions (translations, rotations, similarities). By adding the concept of distance to the plane geometry he defines the metric geometries comprising the Euclidean and the two non-Euclidean (hyperbolic and elliptic) geometries. In Part 3 he combines the invariant-based definition of geometries with their axiomatic treatment known as foundations of geometries. The book is a classic and still very relevant to get a comprehensive view of a wide variety of geometries.

I used this book while giving a graduate level course on Foundations of Geometry at the University of Calcutta, India.

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Sadly the print quality is not good. Surprisingly the other volume (Arithmetic

By Arondes on May 10, 2015

Format: Paperback

It is no doubt a classic. Sadly the print quality is not good. Surprisingly the other volume (Arithmetic, Algebra, Analysis) is much better.

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0 of 1 people found the following review helpful

A Classic Geometry textbook for geometry students and geometry teachers.

By Daniel Martins on March 24, 2013

Format: Paperback Verified Purchase

This is a Classic geometry textbook for geometry students and teachers. This book helps mathematicians to underestand The Erlangen Program and Felix Klein's viewing of Mathematics.

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