Elementary Mathematics from an Advanced Standpoint: Geometry by Felix Klein (PDF)

Description

“Nothing comparable to it.”–Mathematics TeacherThis 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 ed. 141 figures.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐classic math text.

⭐Excellent! This book dispenses any comments. This a Felix Klein text! The text is clear, simple and full of relevants results in arithmetic, algebra and analysis. This is a book to study not to simply read. It helps us to underestand Klein’s integral viewing of mathematics.

⭐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.

⭐This book by F.Klein is well known .Klein is famous for his clarity and novelty of approach.

⭐Just thank you.

⭐It’s a classic. What more need be said?

⭐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.

⭐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!

⭐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.

⭐A classic, still very readable and not outdated.

⭐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.

⭐und mehr ist eigentlich kaum zu sagen. In dem ursprünglich dreibändigen Werk, von dem nun zwei Bände in englischer Übersetzung vorliegen (sieheElementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis (Dover Books on Mathematics)

⭐), hat Felix Klein seine Idee von einer Mathematikdidaktik dargelegt. Vielleicht ist Klein sogar als der Erfinder der Mathematikdidaktik – im Sinne von Herunterbrechen höherer Mathematik auf elementares Schulniveau – anzusehen. Wenn man heutige Machwerke zur “modernen” Didaktik liest und mit den Kleinschen Büchern vergleicht, kommt man nicht umhin, sich ob des Verlustes sehr, sehr traurig zu fühlen.Vedasi recensione per l’analogo libro versione Algebra ed Analisi.

⭐

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