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Elementary Mathematics from an Advanced Standpoint: Geometry (Dover Books on Mathematics)

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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.

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!

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.