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Books > Science & Mathematics > Mathematics > Geometry > General

Homological Algebra: In Strongly Non-abelian Settings (Hardcover): Marco Grandis Homological Algebra: In Strongly Non-abelian Settings (Hardcover)
Marco Grandis
R2,677 Discovery Miles 26 770 Ships in 12 - 17 work days

We propose here a study of 'semiexact' and 'homological' categories as a basis for a generalised homological algebra. Our aim is to extend the homological notions to deeply non-abelian situations, where satellites and spectral sequences can still be studied.This is a sequel of a book on 'Homological Algebra, The interplay of homology with distributive lattices and orthodox semigroups', published by the same Editor, but can be read independently of the latter.The previous book develops homological algebra in p-exact categories, i.e. exact categories in the sense of Puppe and Mitchell - a moderate generalisation of abelian categories that is nevertheless crucial for a theory of 'coherence' and 'universal models' of (even abelian) homological algebra. The main motivation of the present, much wider extension is that the exact sequences or spectral sequences produced by unstable homotopy theory cannot be dealt with in the previous framework.According to the present definitions, a semiexact category is a category equipped with an ideal of 'null' morphisms and provided with kernels and cokernels with respect to this ideal. A homological category satisfies some further conditions that allow the construction of subquotients and induced morphisms, in particular the homology of a chain complex or the spectral sequence of an exact couple.Extending abelian categories, and also the p-exact ones, these notions include the usual domains of homology and homotopy theories, e.g. the category of 'pairs' of topological spaces or groups; they also include their codomains, since the sequences of homotopy 'objects' for a pair of pointed spaces or a fibration can be viewed as exact sequences in a homological category, whose objects are actions of groups on pointed sets.

Morphological Models of Random Structures (Hardcover, 1st ed. 2021): Dominique Jeulin Morphological Models of Random Structures (Hardcover, 1st ed. 2021)
Dominique Jeulin
R4,875 Discovery Miles 48 750 Ships in 7 - 11 work days

This book covers methods of Mathematical Morphology to model and simulate random sets and functions (scalar and multivariate). The introduced models concern many physical situations in heterogeneous media, where a probabilistic approach is required, like fracture statistics of materials, scaling up of permeability in porous media, electron microscopy images (including multispectral images), rough surfaces, multi-component composites, biological tissues, textures for image coding and synthesis. The common feature of these random structures is their domain of definition in n dimensions, requiring more general models than standard Stochastic Processes.The main topics of the book cover an introduction to the theory of random sets, random space tessellations, Boolean random sets and functions, space-time random sets and functions (Dead Leaves, Sequential Alternate models, Reaction-Diffusion), prediction of effective properties of random media, and probabilistic fracture theories.

Automated Deduction in Geometry - 8th International Workshop, ADG 2010, Munich, Germany, July 22-24, 2010, Revised Papers... Automated Deduction in Geometry - 8th International Workshop, ADG 2010, Munich, Germany, July 22-24, 2010, Revised Papers (Paperback, 2011 ed.)
Pascal Schreck, Julien Narboux, Jurgen Richter-Gebert
R1,276 R389 Discovery Miles 3 890 Save R887 (70%) Ships in 12 - 17 work days

This book constitutes the thoroughly refereed post-workshop proceedings of the 8th International Workshop on Automated Deduction in Geometry, ADG 2010, held in Munich, Germany in July 2010.
The 13 revised full papers presented were carefully selected during two rounds of reviewing and improvement from the lectures given at the workshop. Topics addressed by the papers are incidence geometry using some kind of combinatoric argument; computer algebra; software implementation; as well as logic and proof assistants.

Optimal Control for Mathematical Models of Cancer Therapies - An Application of Geometric Methods (Hardcover, 1st ed. 2015):... Optimal Control for Mathematical Models of Cancer Therapies - An Application of Geometric Methods (Hardcover, 1st ed. 2015)
Heinz Schattler, Urszula Ledzewicz
R3,659 R3,273 Discovery Miles 32 730 Save R386 (11%) Ships in 12 - 17 work days

This book presents applications of geometric optimal control to real life biomedical problems with an emphasis on cancer treatments. A number of mathematical models for both classical and novel cancer treatments are presented as optimal control problems with the goal of constructing optimal protocols. The power of geometric methods is illustrated with fully worked out complete global solutions to these mathematically challenging problems. Elaborate constructions of optimal controls and corresponding system responses provide great examples of applications of the tools of geometric optimal control and the outcomes aid the design of simpler, practically realizable suboptimal protocols. The book blends mathematical rigor with practically important topics in an easily readable tutorial style. Graduate students and researchers in science and engineering, particularly biomathematics and more mathematical aspects of biomedical engineering, would find this book particularly useful.

Vector Analysis (Hardcover, 2001 ed.): L. Kay Vector Analysis (Hardcover, 2001 ed.)
L. Kay; Klaus Janich
R1,824 Discovery Miles 18 240 Ships in 7 - 11 work days

¿The present book is a marvelous introduction in the modern theory of manifolds and differential forms. The undergraduate student can closely examine tangent spaces, basic concepts of differential forms, integration on manifolds, Stokes theorem, de Rham- cohomology theorem, differential forms on Riema-nnian manifolds, elements of the theory of differential equations on manifolds (Laplace-Beltrami operators). Every chapter contains useful exercises for the students.¿ ¿ ZENTRALBLATT MATH

Chaos - An Introduction to Dynamical Systems (Paperback, 1st Corrected ed. 1996. Corr. 2nd printing 2000): Kathleen T.... Chaos - An Introduction to Dynamical Systems (Paperback, 1st Corrected ed. 1996. Corr. 2nd printing 2000)
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke
R1,759 Discovery Miles 17 590 Ships in 7 - 11 work days

Developed and class-tested by a distinguished team of authors at two universities, this text is intended for courses in nonlinear dynamics in either mathematics or physics. The only prerequisites are calculus, differential equations, and linear algebra. Along with discussions of the major topics, including discrete dynamical systems, chaos, fractals, nonlinear differential equations and bifurcations, the text also includes Lab Visits -- short reports that illustrate relevant concepts from the physical, chemical and biological sciences. There are Computer Experiments throughout the text that present opportunities to explore dynamics through computer simulations, designed for use with any software package. And each chapter ends with a Challenge, guiding students through an advanced topic in the form of an extended exercise.

Diophantine Geometry - An Introduction (Hardcover, 2000 ed.): Marc Hindry, Joseph H. Silverman Diophantine Geometry - An Introduction (Hardcover, 2000 ed.)
Marc Hindry, Joseph H. Silverman
R2,686 Discovery Miles 26 860 Ships in 7 - 11 work days

This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.

Geometry, Perspective Drawing, And Mechanisms (Hardcover): Talmage James Reid, Donald Row Geometry, Perspective Drawing, And Mechanisms (Hardcover)
Talmage James Reid, Donald Row
R1,748 Discovery Miles 17 480 Ships in 12 - 17 work days

The aim of this book is to examine the geometry of our world and, by blending theory with a variety of every-day examples, to stimulate the imagination of the readers and develop their geometric intuition. It tries to recapture the excitement that surrounded geometry during the Renaissance as the development of perspective drawing gathered pace, or more recently as engineers sought to show that all the world was a machine. The same excitement is here still, as enquiring minds today puzzle over a random-dot stereogram or the interpretation of an image painstakingly transmitted from Jupiter.The book will give a solid foundation for a variety of undergraduate courses, to provide a basis for a geometric component of graduate teacher training, and to provide background for those who work in computer graphics and scene analysis. It begins with a self-contained development of the geometry of extended Euclidean space. This framework is then used to systematically clarify and develop the art of perspective drawing and its converse discipline of scene analysis and to analyze the behavior of bar-and-joint mechanisms and hinged-panel mechanisms. Spherical polyhedra are introduced and scene analysis is applied to drawings of these and associated objects. The book concludes by showing how a natural relaxation of the axioms developed in the early chapters leads to the concept of a matroid and briefly examines some of the attractive properties of these natural structures.

Advanced Polyhedra 3 - The Compound of Five Cubes (Paperback): Gerald Jenkins, Magdalen Bear Advanced Polyhedra 3 - The Compound of Five Cubes (Paperback)
Gerald Jenkins, Magdalen Bear
R132 R110 Discovery Miles 1 100 Save R22 (17%) Ships in 10 - 20 work days

This beautiful model to cut out and glue together is the result of arranging five interpenetrating cubes in a highly symmetrical way Each cube is printed in its own colour and so it is easy to see that exactly two cubes meet at every vertex and that there are twelve surfaces defining each face of each cube. There is a strong internal frame and multiple divisions in the golden ratio to discover. Simple and logical instructions followed carefully yield extraordinary results.

Geometric Mechanics - Part Ii: Rotating, Translating And Rolling (2nd Edition) (Hardcover, 2nd Revised edition): Darryl D Holm Geometric Mechanics - Part Ii: Rotating, Translating And Rolling (2nd Edition) (Hardcover, 2nd Revised edition)
Darryl D Holm
R2,732 Discovery Miles 27 320 Ships in 12 - 17 work days

See also GEOMETRIC MECHANICS - Part I: Dynamics and Symmetry (2nd Edition) This textbook introduces modern geometric mechanics to advanced undergraduates and beginning graduate students in mathematics, physics and engineering. In particular, it explains the dynamics of rotating, spinning and rolling rigid bodies from a geometric viewpoint by formulating their solutions as coadjoint motions generated by Lie groups. The only prerequisites are linear algebra, multivariable calculus and some familiarity with Euler-Lagrange variational principles and canonical Poisson brackets in classical mechanics at the beginning undergraduate level.The book uses familiar concrete examples to explain variational calculus on tangent spaces of Lie groups. Through these examples, the student develops skills in performing computational manipulations, starting from vectors and matrices, working through the theory of quaternions to understand rotations, then transferring these skills to the computation of more abstract adjoint and coadjoint motions, Lie-Poisson Hamiltonian formulations, momentum maps and finally dynamics with nonholonomic constraints.The organisation of the first edition has been preserved in the second edition. However, the substance of the text has been rewritten throughout to improve the flow and to enrich the development of the material. Many worked examples of adjoint and coadjoint actions of Lie groups on smooth manifolds have also been added and the enhanced coursework examples have been expanded. The second edition is ideal for classroom use, student projects and self-study.

General Topology - Chapters 5-10 (Paperback, 1st ed. 1989, 2nd printing 1998): N. Bourbaki General Topology - Chapters 5-10 (Paperback, 1st ed. 1989, 2nd printing 1998)
N. Bourbaki
R1,603 Discovery Miles 16 030 Ships in 7 - 11 work days

This is the softcover reprint of the 1974 English translation of the later chapters of Bourbaki 's Topologie Generale. Initial chapters study subgroups and quotients of R, real vector spaces and projective spaces, and additive groups Rn. Analogous properties are then studied for complex numbers. Later chapters illustrate the use of real numbers in general topology and discuss various topologies of function spaces and approximation of functions.

General Topology - Chapters 1-4 (Paperback, 1st ed. 1989. 2nd printing 1998): N. Bourbaki General Topology - Chapters 1-4 (Paperback, 1st ed. 1989. 2nd printing 1998)
N. Bourbaki
R1,599 Discovery Miles 15 990 Ships in 7 - 11 work days

This is the softcover reprint of the English translation of 1971 (available from Springer since 1989) of the first 4 chapters of Bourbaki's Topologie générale. It gives all the basics of the subject, starting from definitions. Important classes of topological spaces are studied, uniform structures are introduced and applied to topological groups. Real numbers are constructed and their properties established. Part II, comprising the later chapters, Ch. 5-10, is also available in English in softcover.

Must Know High School Geometry (Paperback, Ed): Allen Ma, Amber Kuang Must Know High School Geometry (Paperback, Ed)
Allen Ma, Amber Kuang
R371 Discovery Miles 3 710 Ships in 12 - 17 work days

Publisher's Note: Products purchased from Third Party sellers are not guaranteed by the publisher for quality, authenticity, or access to any online entitlements included with the product. A UNIQUE NEW APPROACH THAT'S LIKE A LIGHTNING BOLT TO THE BRAIN You know that moment when you feel as though a lightning bolt has hit you because you finally get something? That's how this book will make you react. (We hope!) Each chapter makes sure that what you really need to know is clear right off the bat and sees to it that you build on this knowledge. Where other books ask you to memorize stuff, we're going to show you the must know ideas that will guide you toward success in geometry. You will start each chapter learning what the must know ideas behind a geometry subject are, and these concepts will help you solve the geometry problems that you find in your classwork and on exams. Dive into this book and find: * 250+ practice questions that mirror what you will find in your classwork and on exams* A bonus app with 100+ flashcards that will reinforce what you've learned* Extensive examples that drive home essential concepts* An easy-access setup that allows you to jump in and out of subjects* Geometry topics aligned to national and state education standards* Special help for more challenging geometry subjects, including proofs, transformations, and constructions We're confident that the must know ideas in this book will have you up and solving geometry problems in no time-or at least in a reasonable amount of time! The authors, between them, teach high school math courses including geometry, trigonometry, pre-calculus, calculus, and discrete math. Whew!

Understanding Nonlinear Dynamics (Paperback, 1st ed. 1995. Corr. 2nd printing 1997): Daniel Kaplan, Leon Glass Understanding Nonlinear Dynamics (Paperback, 1st ed. 1995. Corr. 2nd printing 1997)
Daniel Kaplan, Leon Glass
R1,512 Discovery Miles 15 120 Ships in 7 - 11 work days

UNDERSTANDING NONLINEAR DYNAMICS is based on an undergraduate course taught for many years to students in the biological sciences. The text provides a clear and accessible development of many concepts from contemporary dynamics, including stability and multistability, cellular automata and excitable media, fractals, cycles, and chaos. A chapter on time-series analysis builds on this foundation to provide an introduction to techniques for extracting information about dynamics from data. The text will be useful for courses offered in the life sciences or other applied science programs, or as a supplement to emphasize the application of subjects presented in mathematics or physics courses. Extensive examples are derived from the experimental literature, and numerous exercise sets can be used in teaching basic mathematical concepts and their applications. Concrete applications of the mathematics are illustrated in such areas as biochemistry, neurophysiology, cardiology, and ecology. The text also provides an entry point for researchers not familiar with mathematics but interested in applications of nonlinear dynamics to the life sciences.

Problems and Theorems in Analysis II - Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry... Problems and Theorems in Analysis II - Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry (Paperback, Reprint Of The 1st Ed)
C. E. Billigheimer; George Polya, Gabor Szegoe
R1,614 Discovery Miles 16 140 Ships in 7 - 11 work days

From the reviews: "... In the past, more of the leading mathematicians proposed and solved problems than today, and there were problem departments in many journals. Pólya and Szego must have combed all of the large problem literature from about 1850 to 1925 for their material, and their collection of the best in analysis is a heritage of lasting value. The work is unashamedly dated. With few exceptions, all of its material comes from before 1925. We can judge its vintage by a brief look at the author indices (combined). Let's start on the C's: Cantor, Carathéodory, Carleman, Carlson, Catalan, Cauchy, Cayley, Cesàro,... Or the L's: Lacour, Lagrange, Laguerre, Laisant, Lambert, Landau, Laplace, Lasker, Laurent, Lebesgue, Legendre,... Omission is also information: Carlitz, Erdös, Moser, etc."Bull.Americ.Math.Soc.

Algorithmic and Computer Methods for Three-Manifolds (Hardcover, 1997 ed.): A.T. Fomenko, S.V. Matveev Algorithmic and Computer Methods for Three-Manifolds (Hardcover, 1997 ed.)
A.T. Fomenko, S.V. Matveev
R3,404 Discovery Miles 34 040 Ships in 7 - 11 work days

One service mathematics has rendered the human race. It has put common sense back where it belongs. It has put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled discarded nonsense. Eric TBell Every picture tells a story. Advenisement for for Sloan's backache and kidney oils, 1907 The book you have in your hands as you are reading this, is a text on3-dimensional topology. It can serve as a pretty comprehensive text book on the subject. On the other hand, it frequently gets to the frontiers of current research in the topic. If pressed, I would initially classify it as a monograph, but, thanks to the over three hundred illustrations of the geometrical ideas involved, as a rather accessible one, and hence suitable for advanced classes. The style is somewhat informal; more or less like orally presented lectures, and the illustrations more than make up for all the visual aids and handwaving one has at one's command during an actual presentation.

Geometry: 1,001 Practice Problems For Dummies (+ Free Online Practice) (Paperback): Allen Ma, Amber Kuang Geometry: 1,001 Practice Problems For Dummies (+ Free Online Practice) (Paperback)
Allen Ma, Amber Kuang
R467 R362 Discovery Miles 3 620 Save R105 (22%) Ships in 12 - 17 work days

Practice makes perfect! Get perfect with a thousand and one practice problems! 1,001 Geometry Practice Problems For Dummies gives you 1,001 opportunities to practice solving problems that deal with core geometry topics, such as points, lines, angles, and planes, as well as area and volume of shapes. You'll also find practice problems on more advanced topics, such as proofs, theorems, and postulates. The companion website gives you free online access to 500 practice problems and solutions. You can track your progress and ID where you should focus your study time. The online component works in conjunction with the book to help you polish your skills and build confidence. As the perfect companion to Geometry For Dummies or a stand-alone practice tool for students, this book & website will help you put your geometry skills into practice, encouraging deeper understanding and retention. The companion website includes: * Hundreds of practice problems * Customizable practice sets for self-directed study * Problems ranked as easy, medium, and hard * Free one-year access to the online questions bank With 1,001 Geometry Practice Problems For Dummies, you'll get the practice you need to master geometry and gain confidence in the classroom.

Sheaf Theory (Hardcover, 2nd ed. 1997): Glen E. Bredon Sheaf Theory (Hardcover, 2nd ed. 1997)
Glen E. Bredon
R2,242 Discovery Miles 22 420 Ships in 7 - 11 work days

Primarily concerned with the study of cohomology theories of general topological spaces with "general coefficient systems", the parts of sheaf theory covered here are those areas important to algebraic topology. Among the many innovations in this book, the concept of the "tautness" of a subspace is introduced and exploited; the fact that sheaf theoretic cohomology satisfies the homotopy property is proved for general topological spaces; and relative cohomology is introduced into sheaf theory. A list of exercises at the end of each chapter helps students to learn the material, and solutions to many of the exercises are given in an appendix. This new edition of a classic has been substantially rewritten and now includes some 80 additional examples and further explanatory material, as well as new sections on Cech cohomology, the Oliver transfer, intersection theory, generalised manifolds, locally homogeneous spaces, homological fibrations and p- adic transformation groups. Readers should have a thorough background in elementary homological algebra and in algebraic topology.

Geometries and Groups (Paperback, 1st ed. 1994. Corr. 2nd printing 0): Viacheslav V. Nikulin Geometries and Groups (Paperback, 1st ed. 1994. Corr. 2nd printing 0)
Viacheslav V. Nikulin; Translated by M. Reid; Igor R. Shafarevich
R1,943 Discovery Miles 19 430 Ships in 7 - 11 work days

This book is devoted to the theory of geometries which are locally Euclidean, in the sense that in small regions they are identical to the geometry of the Euclidean plane or Euclidean 3-space. Starting from the simplest examples, we proceed to develop a general theory of such geometries, based on their relation with discrete groups of motions of the Euclidean plane or 3-space; we also consider the relation between discrete groups of motions and crystallography. The description of locally Euclidean geometries of one type shows that these geometries are themselves naturally represented as the points of a new geometry. The systematic study of this new geometry leads us to 2-dimensional Lobachevsky geometry (also called non-Euclidean or hyperbolic geometry) which, following the logic of our study, is constructed starting from the properties of its group of motions. Thus in this book we would like to introduce the reader to a theory of geometries which are different from the usual Euclidean geometry of the plane and 3-space, in terms of examples which are accessible to a concrete and intuitive study. The basic method of study is the use of groups of motions, both discrete groups and the groups of motions of geometries. The book does not presuppose on the part of the reader any preliminary knowledge outside the limits of a school geometry course.

Explorations In Geometry (Hardcover): Bruce Shawyer Explorations In Geometry (Hardcover)
Bruce Shawyer
R1,575 Discovery Miles 15 750 Ships in 12 - 17 work days

This book covers the basic topics in geometry (including trigonometry) that are accessible and valuable to senior high school and university students. It also includes material that are very useful for problem solving in mathematical competitions, from relatively easy to advanced levels, including the International Mathematical Olympiad.

McGraw-Hill Education Geometry Review and Workbook (Paperback, Ed): Carolyn Wheater McGraw-Hill Education Geometry Review and Workbook (Paperback, Ed)
Carolyn Wheater
R332 Discovery Miles 3 320 Ships in 12 - 17 work days

This engaging review guide and workbook is the ideal tool for sharpening your Geometry skills! This review guide and workbook will help you strengthen your Geometry knowledge, and it will enable you to develop new math skills to excel in your high school classwork and on standardized tests. Clear and concise explanations will walk you step by step through each essential math concept. 500 practical review questions, in turn, provide extensive opportunities for you to practice your new skills. If you are looking for material based on national or state standards, this book is your ideal study tool! Features: *Aligned to national standards, including the Common Core State Standards, as well as the standards of non-Common Core states and Canada*Designed to help you excel in the classroom and on standardized tests*Concise, clear explanations offer step-by-step instruction so you can easily grasp key concepts*You will learn how to apply Geometry to practical situations*500 review questions provide extensive opportunities for you to practice what you've learned

Fractional Order Systems: Modeling And Control Applications (Hardcover): Riccardo Caponetto, Giovanni Dongola, Luigi Fortuna,... Fractional Order Systems: Modeling And Control Applications (Hardcover)
Riccardo Caponetto, Giovanni Dongola, Luigi Fortuna, Ivo Petras
R2,204 Discovery Miles 22 040 Ships in 12 - 17 work days

This book aims to propose implementations and applications of Fractional Order Systems (FOS). It is well known that FOS can be applied in control applications and systems modeling, and their effectiveness has been proven in many theoretical works and simulation routines. A further and mandatory step for FOS real world utilization is their hardware implementation and applications on real systems modeling. With this viewpoint, introductive chapters on FOS are included, on the definition of stability region of Fractional Order PID Controller and Chaotic FOS, followed by the practical implementation based on Microcontroller, Field Programmable Gate Array, Field Programmable Analog Array and Switched Capacitor. Another section is dedicated to FO modeling of Ionic Polymeric Metal Composite (IPMC). This new material may have applications in robotics, aerospace and biomedicine.

Explorations In Geometry (Paperback): Bruce Shawyer Explorations In Geometry (Paperback)
Bruce Shawyer
R858 Discovery Miles 8 580 Ships in 12 - 17 work days

This book covers the basic topics in geometry (including trigonometry) that are accessible and valuable to senior high school and university students. It also includes materials that are very useful for problem solving in mathematical competitions, from relatively easy to advanced levels, including the International Mathematical Olympiad.

Curious Curves (Hardcover): Richard B. Darst, Judith A. Palagallo, Thomas E. Price Curious Curves (Hardcover)
Richard B. Darst, Judith A. Palagallo, Thomas E. Price
R1,647 Discovery Miles 16 470 Ships in 12 - 17 work days

Curious Curves is self-contained and unified in presentation. This book is suitable for a topics course, capstone course, or senior seminar; it is also intended for independent study by students and others interested in mathematics.Curves can often provide a better representation of natural phenomena than do the figures of classical geometry. Thus the content - presented with an emphasis on the geometric intuition characteristic of the study of curves - is highly relevant not only for people working in mathematics, but also those in other sciences. The explanations are detailed and illustrative to capture the interest of the reader, as well as complete to provide the necessary background information needed to go further into the subject.

Elementary Mathematics from a Higher Standpoint, Volume 2 - Geometry (Paperback, 2016 ed.): Felix Klein Elementary Mathematics from a Higher Standpoint, Volume 2 - Geometry (Paperback, 2016 ed.)
Felix Klein; Translated by Gert Schubring
R1,992 Discovery Miles 19 920 Ships in 7 - 11 work days

These three volumes constitute the first complete English translation of Felix Klein's seminal series "Elementarmathematik vom hoheren Standpunkte aus". "Complete" has a twofold meaning here: First, there now exists a translation of volume III into English, while until today the only translation had been into Chinese. Second, the English versions of volume I and II had omitted several, even extended parts of the original, while we now present a complete revised translation into modern English. The volumes, first published between 1902 and 1908, are lecture notes of courses that Klein offered to future mathematics teachers, realizing a new form of teacher training that remained valid and effective until today: Klein leads the students to gain a more comprehensive and methodological point of view on school mathematics. The volumes enable us to understand Klein's far-reaching conception of elementarisation, of the "elementary from a higher standpoint", in its implementation for school mathematics. This volume II presents a paradigmatic realisation of Klein's approach of elementarisation for teacher education. It is shown how the various geometries, elaborated particularly since the beginning of the 19th century, are revealed as becoming unified in a new restructured geometry. As Klein liked to stress: "Projective geometry is all geometry". Non-Euclidean geometry proves to constitute a part of this unifying process. The teaching of geometry is discussed in a separate chapter, which provides moreover important information on the history of geometry teaching and an international comparison.

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