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When the mathematician Felix Klein first went to university, he was surprised at just how little what he had learned up to that point was relevant to his new studies.Professors had their own interests, and these they conveyed without regard for the math students of the future that these prospective secondary schoolteachers would one day instruct.Elementary Mathematics from an Advanced Standpoint was written to help remedy that problem.Though highly regarded as one of the finest mathematical minds of his day, Professor Klein took a great deal of interest in guiding teachers and "reducing the gap between the school and the university."Readers will come away impressed at the clarity of Klein's writing, and the ease with which he conveys complex mathematical ideas.Divided into three parts-arithmetic, algebra, and analysis-and covering such topics as complex numbers, real equations, and logarithmic and exponential functions, Klein's classic is essential reading for math instructors and students planning to become math instructors.German mathematician FELIX KLEIN (1849-1925), a great teacher and scientific thinker, significantly advanced the field of mathematical physics and made a number of profound discoveries in the field of geometry.His published works include Elementary Mathematics from an Advanced Standpoint:Geometry and Famous Problems of Elementary Geometry. ... Read more

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Still the best book for Math education today !
I read Felix Klein's [1849-1925] 3-volume Math book [publised in German, 1924] in Chinese translation bought recently in Beijing bookshop. The chinese translators are themselves mathematicians, so they corrected some translation mistakes made by the 1939 English translators from whom they translate into Chinese.

Klein was a great mathematician, being the Father of Modern Geometry based on Group Theory [Erlanger Program]. He was also a great Math educationist, who built the Gottingen University into the World Center of Mathematics before WW II, attracted the Germany's and the World's best mathematicians to teach and learn there. eg. David Hilbert, Noether, Laskar, etc.

This 3-volume Math books were compiled by his assistants from the lectures he gave over 20+ years in breaching the gap of Secondary School Elementary Math and University Modern Math, hence the title "Elementary Math from an advanced standpoint". Although written 80 years ago, his concerns and teachning are still valid today. Except for France where they teach modern math concepts in Lycee (secondary school from 3eme to Terminale), other countries' secondary schools don't introduce these modern concepts (Group, Ring, Field...), some with little Set theory at best. For Singapore and the English commonwealth countries, secondary and high school math are still using the same syllabus 100 years ago: 100% computational-bias (calculus, trigo, classical algebra).

Klein spotted this gap 80 years ago, as indicated in the volume 1, which he labelled as Syllabus A, Syllabus B and Syllabus C.

In Syllabus A (which is still used worldwide): all math subjects (algebra, geometry, trigo...) are taught as separate independent modules. The secondary school students don't see them as inter-connected.

In Syllabus B, Klein proposed to base whole math teaching on Functions, which is intuitive as functions can be visualised in Graphs, and link to all branches of math.
eg. Function exponential e: e^x = 1 + x + 1/2! x^2 + 1/3! x^3 +....
also e^ix = Cos x +i.Sin x
where the function exponential has linkage to trigo (cosine and sine), and algebra's complex number i. Same can be applied to calculus: integration and differentiation link to functions like Log, exponential, trigo, etc.

Although he didn't elaborate on Syllabus C (out of his life / book context), but I would presume he had predicted the Bourbaki school of Math based on Set Theory as math structures. This approach of teaching was very prominent in French Mathematics Education. It has proved with little success to students because of being too abstract, distancing the Math from real-life (pure math vs applied math). Klein, although a pure math par excellence, didn't encourage this approach which was advocated after his death by the French Bourbaki mathematicians.

I admire Klein's foresight in Syllabus B intuitive pedagogy which will introduce secondary school students gradually to the University modern math concepts.

After 80 years, his dream is still not fully appreciated by math educationists. It is hard to change the mindset in the math education bureaucrats worldwide.

Elementary Math Analysis
This book was just what I was looking for. Simple to understand, something to bring me up to speed on some long forgotten math.

Still a great resource
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 discusses basic arithmetic, with great emphasis on how it should be taught. Being such an enthusiastic lecturer, Klein cannot resist also including little sections on things like Fermat's last theorem and quaternions. The second part suddenly becomes much more sophisticated, as Klein sketches a rather non-standard approach to understanding algebraic equations, in particular incorporating a bit of Klein's trademarke icosahedron material. To keep up with this presentation one should probably consult his icosahedron book as well. For the third part, on analysis, we are back on familiar ground again, which makes it easier to appreciate this masterful overview, covering many of the great ideas of analysis and putting them in context with interesting discussions that are often partly historical. ... Read more

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A rare old gem of accessible yet important math
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!

Unification of Geometries
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.

The great master's way of looking at geometry
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. ... Read more

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In this classic of mathematical literature, first published in 1884, Felix Klein elegantly demonstrates how the rotation of icosahedron can be used to solve complex quintic equations. Divided into two parts-"Theory of the Icosahedron" and "The Theory of Equations of the Fifth Degree"-The Icosahedron covers:. the regular solids and the theory of groups. introduction of (x + iy). statement and discussion of the fundamental problem, according to the theory of functions. the algebraical character of the fundamental problem. general theorems and survey of the subject. the historical development of the theory of equations of the fifth degree. introduction of geometrical material. the canonical equations of the fifth degree. the problem of the A's and the Jacobian equations of the sixth degree. the general equation of the fifth degreeComplete with detailed equations and instructive material, The Icosahedron will be valued by experts in higher mathematics and students of algebra alike.German mathematician FELIX KLEIN (1849-1925) specialized in function theory, group theory, and non-Euclidean geometry. His published works include Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis; Elementary Mathematics from an Advanced Standpoint: Geometry; and Famous Problems of Elementary Geometry. ... Read more

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In the late summer of 1893, following the Congress of Mathematicians held in Chicago, Felix Klein gave two weeks of lectures on the current state of mathematics. Rather than offering a universal perspective, Klein presented his personal view of the most important topics of the time. It is remarkable how most of the topics continue to be important today. Originally published in 1893 and reissued by the AMS in 1911, we are pleased to bring this work into print once more with this new edition. Klein begins by highlighting the works of Clebsch and of Lie. In particular, he discusses Clebsch's work on Abelian functions and compares his approach to the theory with Riemann's more geometrical point of view. Klein devotes two lectures to Sophus Lie, focussing on his contributions to geometry, including sphere geometry and contact geometry. Klein's ability to connect different mathematical disciplines clearly comes through in his lectures on mathematical developments. For instance, he discusses recent progress in non-Euclidean geometry by emphasizing the connections to projective geometry and the role of transformation groups. In his descriptions of analytic function theory and of recent work in hyperelliptic and Abelian functions, Klein is guided by Riemann's geometric point of view. He discusses Galois theory and solutions of algebraic equations of degree five or higher by reducing them to normal forms that might be solved by non-algebraic means. Thus, as discovered by Hermite and Kronecker, the quintic can be solved ""by elliptic functions"". This also leads to Klein's well-known work connecting the quintic to the group of the icosahedron. Klein expounds on the roles of intuition and logical thinking in mathematics. He reflects on the influence of physics and the physical world on mathematics and, conversely, on the influence of mathematics on physics and the other natural sciences. The discussion is strikingly similar to today's discussions about ``physical mathematics''. There are a few other topics covered in the lectures which are somewhat removed from Klein's own work. For example, he discusses Hilbert's proof of the transcendence of certain types of numbers (including $\pi$ and $e$), which Klein finds much simpler than the methods used by Lindemann to show the transcendence of $\pi$. Also, Klein uses the example of quadratic forms (and forms of higher degree) to explain the need for a theory of ideals as developed by Kummer. Klein's look at mathematics at the end of the 19th Century remains compelling today, both as history and as mathematics. It is delightful and fascinating to observe from a one-hundred year retrospect, the musings of one of the masters of an earlier era. ... Read more

Customer Reviews (2)

This great book must be reprinted!
Unfortunately Birkhauser let it go out of print and there's currently only one used copy for sale for an exorbitant price. I could write a lengthy review to encourage you to buy a copy if it was available at a fair price.

Briefly, I. M. Yaglom has written a very accessible history of 19th century mathematics, with emphasis on interesting biographies of the leading protagonists and on the subjects most closely related to the work of Klein and Lie, whose own work is not discussed in detail until late in the book, despite the book's title.

A great history of geometry and groups in the 19th century
Starting with Galois and his contribution to the evolving subject of group theory Yaglom gives a beautiful account of the lives and works of the major players in the development of the subject in the nineteenth century:Jordan, who was a teacher of Lie and Klein in Paris and their adventures during the Franco-Prussian War. Monge and Poncelet developing projective geometry as well asBolyai, Gauss and Lobachevsky and their discovery of hyperbolic geometry. Riemann's contributions and the development of modern linear Algebra by Grassmann, Cayley and Hamilton are described in detail. The last two chapters are devoted to Lie's development of Lie Algebras and his construction of the geometry from a continuous group and Klein's Erlanger Programm unifying the different approaches to geometry by emphasizing automorphism groups. These last pages are definitely the climax of the book.

All of this only occupies the first half of the book, the second half consists of extensive notes to the former, intended to help the reader deepen her understanding of the subject in a second reading.

I think this book is very readable and should be accessible to any interested reader who has a slight understanding of the terms used so far in this review. In particular I enjoyed the details of the lives of the mathematicians involved and how the whole story seemed to culminate in the works of Lie and Klein. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Customer Reviews (1)

Riemann surfaces through fluid dynamics
Where a complex function is analytic it satisfies the Cauchy-Riemann equations, which implies that its real and imaginary part satisfy the Laplace equation, which means that the function corresponds to a steady fluid flow. Restricting ourselves to algebraic functions and their integrals, we can understand the corresponding flows by studying the infinities of these functions, and we can imagine electromagnetic set-ups that would produce such flows. "Now it seems possible, ab initio, to reverse the whole order of this discussion; to study the [flows] in the first place and thence to work out the theory of certain analytical functions" (p. 22). So we study this type of flows on general spheres with handles, and then show that this corresponds precisely to the algebraic functions and their integrals on their Riemann surfaces. In this way, without any of the usual analytic machinery, we reveal the qualitative insights of Riemann's theory. In particular, "among the numerous accidental properties of the functions, we distinguish certain essential ones" (p. 56), namely the infinities, the Riemann surface, and the topological types of the cross-cuts. This concludes the study of Riemann's theory proper, but for the last part of the book "we may bring forward the geometrical side of the subject and consider Riemann's theory as a means of making the theory of conformal representation of one closed surface upon another accessible to analytical treatment" (p. 56). ... Read more

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This scarce antiquarian book is included in our special Legacy Reprint Series. In the interest of creating a more extensive selection of rare historical book reprints, we have chosen to reproduce this title even though it may possibly have occasional imperfections such as missing and blurred pages, missing text, poor pictures, markings, dark backgrounds and other reproduction issues beyond our control. Because this work is culturally important, we have made it available as a part of our commitment to protecting, preserving and promoting the world's literature. ... Read more

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An interesting "unpretending little book"
The back cover claims that this book is "widely regarded as a classic of modern mathematics", but I think one should rather listen to Klein himself: in his preface, Klein calls it an "unpretending little book", apparently based on a few (unpretending?) little lectures, typed up into a book by some (unpretending?) little student.

Circumventing Galois Theory, Klein proves the impossibility of doubling a cube and trisecting an angle in the first ten pages. We then look at the solvable cases of circle division, especially the constructability of the regular 17-gon. Gauss proved algebraically that p-division of the circle is possible for Fermat primes p=2^2^k+1. We work through Gauss's proof in the special case p=17. Giving up on a fully geometric proof, "there remains, then, only the geometric translation of the individual algebraic steps". In the second part of the book, to prove that it is impossible to square the circle we prove the transcendence of pi, and in the process we must also show the transcendence of e (and then go through Euler's formula e^i*pi=-1). These proofs use only elementary mathematics but unfortunately they are still rather technical. Perhaps the history of these proofs--they are the results of successive reworkings of much more advanced analytic proofs--vouches for their artificiality. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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Publisher: London, A. MelrosePublication date: 1916Subjects: Hall, Richard Nelville, 1894-1915World War, 1914-1918 -- Personal narrativesNotes: This is an OCR reprint. There may be numerous typos or missing text. There are no illustrations or indexes.When you buy the General Books edition of this book you get free trial access to Million-Books.com where you can select from more than a million books for free. You can also preview the book there. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
This is a reproduction of a book published before 1923.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process.We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide.We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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The Theory of the Top. Volume II. Development of the Theory in the Case of the Heavy Symmetric Top is the second in a series of four self-contained English translations of the classic and definitive treatment of rigid body motion. Graduate students and researchers interested in theoretical and applied mechanics will find this a thorough and insightful account.Other works in this series include Volume I. Introduction to the Kinematics and Kinetics of the Top, Volume III. Perturbations. Astronomical and Geophysical Applications, and Volume IV. Technical Applications of the Theory of the Top. ... Read more

Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
This is a reproduction of a book published before 1923.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process.We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide.We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

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A Supplement To The Usual Treatises - An Exact Reproduction Of The 1893 Printing, To Include All Figures, From The Translation By Frances Hardcastle ... Read more

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General Books publication date: 2009Original publication date: 1905Original Publisher: A.C. McClurgSubjects: United StatesFiction / GeneralFiction / LiteraryHistory / United States / GeneralHistory / United States / 19th CenturyTravel / Essays ... Read more

Customer Reviews (1)

Riemann surfaces and algebraic curves
As in his book "On Riemann's Theory of Algebraic Functions and Their Integrals", Klein takes Riemann surfaces as the starting point of complex function theory. Complex functions are then introduced in terms of potential flows on these surfaces. Approaching algebraic functions from this foundation "has the great advantage that the existence and number of the algebraic functions on the Riemann surfaces is known in advance and we need only seek their algebraic presentations. The above mentioned authors [Weierstrass et al.] on the other hand face the double task of finding the algebraic presentations and from there to arrive at the theorems on the existence and the number of the functions. Because of this their approach is ... very laborious and abstract." (p. 72). This development of the theory of algebraic functions, abelian integrals, etc. is the first part of the book. The second part treats the deep harmony with the theory of algebraic curves. So for instance the Riemann-Roch theorem counting the number of linearly independent rational functions with prescribed poles (pp. 63-65) provides an approach to the "intersection point theorems" describing the interrelations of the intersection points of two algebraic curves (pp. 141-147), which "cuts through the entire subject of algebraic curves" (p. 147) and "from which the most remarkable geometric theorems can be derived" (p. 119). The a priori Riemann surface point of view is the way to go also in algebraic curve theory: "It is essential that we characterise the algebraic functions independently of their analytic presentations, and in particular that we understand the importance of the number p [the genus] as a number that measures the 'connectivity' of the algebraic object. When Brill and Noether ... introduce p otherwise, the theorem on the invariance of p under bijective transformations appears as a curious side effect, and not, as in Riemann, as something thoroughly obvious." (p. 192). ... Read more