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Elementary Mathematics from an Advanced Standpoint: Geometry

by Felix Klein (Author)
4.6 4.6 out of 5 stars 39 ratings
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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.
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  1. ISBN-10
    0486474410
  2. ISBN-13
    978-0486434810
  3. Publisher
    Dover Publications
  4. Publication date
    June 18, 2004
  5. Part of series
    Dover Books on Mathematics
  6. Language
    English
  7. Dimensions
    5.38 x 0.49 x 8.44 inches
  8. Print length
    224 pages
  9. See all details
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Product details

  • ASIN ‏ : ‎ 0486434818
  • Publisher ‏ : ‎ Dover Publications (June 18, 2004)
  • Language ‏ : ‎ English
  • Paperback ‏ : ‎ 224 pages
  • ISBN-10 ‏ : ‎ 0486474410
  • ISBN-13 ‏ : ‎ 978-0486434810
  • Item Weight ‏ : ‎ 9.9 ounces
  • Dimensions ‏ : ‎ 5.38 x 0.49 x 8.44 inches
  • Customer Reviews:
    4.6 4.6 out of 5 stars 39 ratings

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Felix Klein

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4.6 out of 5 stars
4.6 out of 5
We don’t use a simple average to calculate the overall star rating and percentage breakdown by star. Our system gives more weight to certain factors—including how recent the review is and if the reviewer bought it on Amazon. Learn more
39 global ratings

Top reviews from the United States

paul silberman
5.0 out of 5 stars purchased as gift
Reviewed in the United States on July 19, 2019
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classic math text.
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priyabrata mitra
5.0 out of 5 stars This book by F. Klein is well known. ...
Reviewed in the United States on September 23, 2014
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This book by F.Klein is well known .Klein is famous for his clarity and novelty of approach.
One person found this helpful
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Arondes
4.0 out of 5 stars Sadly the print quality is not good. Surprisingly the other volume (Arithmetic
Reviewed in the United States on May 11, 2015
It is no doubt a classic. Sadly the print quality is not good. Surprisingly the other volume (Arithmetic, Algebra, Analysis) is much better.
2 people found this helpful
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Milton Moon
5.0 out of 5 stars It's a classic. What more need be said?
Reviewed in the United States on July 25, 2014
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It's a classic. What more need be said?
One person found this helpful
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Rustam Issabekov
3.0 out of 5 stars good book badly written
Reviewed in the United States on November 19, 2013
Reading this book the first thing you will notice is that English is not native to the author. Almost every sentence is so badly written that they look like cryptic messages. Don't get me wrong, the material in the book is great but it so hard to read.
3 people found this helpful
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Amazon Customer
5.0 out of 5 stars Five Stars
Reviewed in the United States on January 30, 2016
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Just thank you.
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Jeremy Gross
5.0 out of 5 stars Great resource for high school math teachers with a strong mathematical background
Reviewed in the United States on June 17, 2016
This book (and its second volume on geometry) saved my sanity when I taught high school mathematics. Felix Klein is one of the greatest mathematicians of the 19th century, the first person to put geometry on a group-theoretical footing, which led to the geometry of modern physics. This book comes out of a course that Klein taught to high school math teachers in Germany in 1906-7. At that time, his audience would have had Ph.D.s in mathematics, and so he taught them high school mathematics from the perspective of a mathematician at the Doctoral level. The result is absolutely fascinating, and gave me so many resources for teaching math courses that would be interesting to my students and also interesting to myself.

I'll give you an example. When I was 15, I asked my teacher how an expression like 2^(sqrt 2) could be well-defined. Exponentiation is well-defined for rational numbers, but how do we know that such a definition extends to all real numbers? My teacher was stumped, and I was very disappointed. It wasn't until I took a real analysis course in college and proved that the exponential function exp(x) is continuous at every real value of x (and indeed at every complex number of finite modulus as well) that I understood why 2^(sqrt 2) is well-defined.

How does Klein approach the problem? He asks the student to draw the hyperbola xy = 1 on the blackboard. Then he asks the student to draw the line x = 1 on the same axes. Then he suggests using a yardstick as a slider to slide forward along the x-axis, or backwards towards x = 0, all the while noticing where the yardstick meets the graph. Then he asks the student to notice that the yardstick sweeps out area under the curve xy = 1, positive area to the right of x = 1, and negative area to the left of x = 1, Klein invites the student to consider that sweep of area to be a continuous function, zero at x = 1, negative between x = 0 and x = 1, and positive when x > 1. Klein notices that the value of that function at ab is equal to its value at a plus its value at b, and that its value at a squared is double its value at a.

Klein then says that this function is invertible since it is monotonic, and that its inverse has an interesting property, that its value at ab is equal to its value at a multiplied by its value at b. He then asks the student to name the original function the natural logarithm of x, and its inverse exp(x).

This demonstrates that both functions are continuous. From here, you can show that 2^x is equal to exp ((ln 2) x), and that 2^(sqrt 2) = exp ((ln 2) (sqrt 2)), and you have shown that 2^x is continuous at sqrt 2. This can obviously be made more rigorous, but it gives an easy introduction to natural logarithms and the exponential function, and I've used his demonstration in my classroom whenever I've introduced the subject since I read the book.

His understanding of mathematics is so beautiful and fluid and expressive. It is a great privilege to see how he views mathematics, and share it with my students.
28 people found this helpful
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Georg Essl
5.0 out of 5 stars A rare old gem of accessible yet important math
Reviewed in the United States on August 4, 2010
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. All this is treated in an immediately visualizable geometric setting.

Unfortunately it is probably fair to say that Klein's program of getting these ideas into high-schools and even undergraduate curricula largely failed - a rather stunning outcome considering the status Klein held. The geometric meaning is rarely mentioned in today's textbooks of linear algebra, and if it is mentioned, the natural progression into exterior algebra is omitted. The importance of orientation is largely lost, and proofs of simple geometric properties often follow complex algebraic steps because the deep intuitions that Grassmann and Klein tried to convey have not been assimilated.

Even more this particular treatment is unique to Klein. While Grassmann's work has been explained in other representations in other works, this direct treatment using determinants throughout is hardly to be found elsewhere (except in Klein's own encyclopedic work such as "Die Entwicklung der Mathematik im 19ten Jahrhundert"). In fact some linear algebra text books even regress, and intentionally downplay the central role the determinant can play in explaining the rich connection of linear (and later differential) geometry and linear (and multi-linear) algebra.

Too often do I see people ask: Why do I need the Jacobi determinant, what is exterior algebra popping up in all these fields, what does the determinant mean, how can I understand differential forms etc. Reading and propagating what is presented in this little volume would go a long way in alleviating much of this confusion that should long have found its way into contemporary linear algebra and analytic geometry textbooks. But there is still hope. At the super cheap price there is no excuse for any math educator to buy, and read this wonderful, and unique book, and hopefully restore a much more intuitive way of teaching linear algebra and linear geometry, and a much deeper understanding how differential forms really work (why we could generalize many core theorems of (multi-variate) calculus into just one, the generalized Stoke's Theorem).

After reading this, reading Grassmann's original books become more accessible (start with the second!), and reading more abstract treatments of exterior algebra (which often omit concrete linear geometry examples) become much clearer. Finally one will be ready with a deep geometric intuition that makes differential forms appear suddenly very concrete.

In short, this is one amazing little book about linear algebra and geometry. It's old but still unique and really good. Go read it!
37 people found this helpful
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Michail L.
5.0 out of 5 stars Felix Klein - the great Geometer of the 19th century
Reviewed in Spain on August 20, 2022
Verified Purchase
I really like this book.

Haven't finished it yet, but I really love prof. Klein's writing.

It was meant for math teachers so the level is high level (meaning: mainly exposition of ideas and concepts without the usual Definition-Theorem-Proof structure). But the insights go really deep. It even has a small chapter on Topology (at the time called "Analysis Situs").

Prof. Klein had studied in France, so the book has many algebraic concepts inside (groups, etc), which pretty much leads the way towards the revolutions that came later in the 1950s when the fields of Algebra and Topology were merged into Algebraic Topology.

The only annoying mishap I'd noticed was a translation error. The German word for "Set" ("Menge") was translated into "Aggregate" instead of "Set", so this could be a bit confusing. But if you replace the word "aggregate" with "set", the book reads just fine.
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VIDAL Diego
3.0 out of 5 stars Libro corretto, ma di scarsa utilità
Reviewed in Italy on June 10, 2021
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Vedasi recensione per l'analogo libro versione Algebra ed Analisi.
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Amazon Customer
5.0 out of 5 stars Geometry
Reviewed in the United Kingdom on April 15, 2014
Verified Purchase
This is a classic work by Felix Klein, the author of the Erlangen Program, comprising the third part of a series of lectures professor Klein delivered to prospective graduate mathematics teachers in 1908. It is NOT in fact concerned with the elementary mathematics taught in secondary schools but is designed to supply teachers with a (then) up to date advanced background to their subject which is still perhaps, in parts at least, well beyond the knowledge base of many of today's teachers.
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Moran Moueza
5.0 out of 5 stars Théorie des nombres
Reviewed in France on February 26, 2013
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C'est une façon très douce d'aborder la théorie des nombres qui est très formatrice d'un esprit de mathématicien. C'est une autre manière de voir les nombres, on n'étudie pas 3 kilos de pommes mais on recherche les propriétés du nombre 3; c'est du lourd.
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rentiere
5.0 out of 5 stars Klein 数学教授法: 算術、代数、解析
Reviewed in Japan on September 23, 2004
Verified Purchase
Elementarmathematik von Hoeheren Standpunkte aus Arithmetik, Algebra, Analysis
第三版(1924)の英訳である。初版は 1908 年だから一世紀前になる。
算術では整数、分数、無理数、複素数(Quaternion)を、代数では方程式を、解析では
指数函数、対数函数、三角函数(Fourier 級数と積分)、付録では e と π の
transcendence と Cantor の集合論を議論する。
本書の目的は教員が数学をいかなる心構えと準備をして、いかに教えるか、という教授
法を示すことにある。したがって上記の議論はその方法論に沿ったものとなっている。
抽象化よりも具象化、歴史的背景、他分野との連関に重点を置いていることにわたくし
は同感している。本書が書かれた時期は現代ほど抽象化も進んでいなかったはずだから
その観点は教育の本質といってよいし、学生の理解を深めるためには歴史背景と他分野
との連関は欠かせないものだと思う。扱う題材が初等的なものに限られていることもあ
るけれども、本書が時代遅れのものと感じさせないのは、本質にせまっている証拠であ
る。
原書は三巻本で、本書は第一巻の英訳。第二巻(幾何学)の英訳も Dover に入っている。
Dover 社に問合せたところ、第三巻(微分積分学の幾何学への応用)の英訳の計画はない
そうで残念。
ところで Dover のカバーデザインは、本文中の図を基調にしたどぎつい色づかいの
ものばかりで感心できなかったけれども、本書はパステルカラーを使ったしゃれたも
のとなっている。
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