Download or read book Foundations of Plane Geometry written by Harvey I. Blau and published by . This book was released on 2003 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ideal for users who may have little previous experience with abstraction and proof, this book provides a rigorous and unified--yet straightforward and accessible --exposition of the foundations of Euclidean, hyperbolic, and spherical geometry. Unique in approach, it combines an extended theme--the study of a generalized absolute plane from axioms through classification into the three fundamental classical planes--with a leisurely development that allows ample time for mathematical growth. It is purposefully structured to facilitate the development of analytic and reasoning skills and to promote an awareness of the depth, power, and subtlety of the axiomatic method in general, and of Euclidean and non-Euclidean plane geometry in particular. Focus on one main topic--The axiomatic development of the absolute plane--which is pursued through a classification into Euclidean, hyperbolic, and spherical planes. Presents specific models such as the sphere, the Klein-Betrami hyperbolic model, and the "gap" plane. Gradually presents axioms for absolute plane geometry.

Download or read book Axiomatic Geometry written by John M. Lee and published by American Mathematical Soc.. This book was released on 2013-04-10 with total page 490 pages. Available in PDF, EPUB and Kindle. Book excerpt: The story of geometry is the story of mathematics itself: Euclidean geometry was the first branch of mathematics to be systematically studied and placed on a firm logical foundation, and it is the prototype for the axiomatic method that lies at the foundation of modern mathematics. It has been taught to students for more than two millennia as a mode of logical thought. This book tells the story of how the axiomatic method has progressed from Euclid's time to ours, as a way of understanding what mathematics is, how we read and evaluate mathematical arguments, and why mathematics has achieved the level of certainty it has. It is designed primarily for advanced undergraduates who plan to teach secondary school geometry, but it should also provide something of interest to anyone who wishes to understand geometry and the axiomatic method better. It introduces a modern, rigorous, axiomatic treatment of Euclidean and (to a lesser extent) non-Euclidean geometries, offering students ample opportunities to practice reading and writing proofs while at the same time developing most of the concrete geometric relationships that secondary teachers will need to know in the classroom. -- P. [4] of cover.

Download or read book The Foundations of Geometry written by David Hilbert and published by . This book was released on 1902 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Axioms as the Foundation of Plane Geometry written by Rosetta Chess and published by . This book was released on 1918 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The Foundations of Geometry written by David Hilbert and published by Read Books Ltd. This book was released on 2015-05-06 with total page 139 pages. Available in PDF, EPUB and Kindle. Book excerpt: This early work by David Hilbert was originally published in the early 20th century and we are now republishing it with a brand new introductory biography. David Hilbert was born on the 23rd January 1862, in a Province of Prussia. Hilbert is recognised as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.

Download or read book The Foundations of Geometry written by David Hilbert and published by Prabhat Prakashan. This book was released on 1950-01-01 with total page 94 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Axiomatic Geometry written by Michael C. Gemignani and published by . This book was released on 1971 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The Foundations of Geometry written by David Hilbert and published by . This book was released on 2013-04-19 with total page 92 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Foundations of GeometryGeometry, like arithmetic, requires for its logical development only a small number of simple, fundamental principles. These fundamental principles are called the axioms of geometry. The choice of the axioms and the investigation of their relations to one another is a problem which, since the time of Euclid, has been discussed in numerous excellent memoirs to be found in the mathematical literature. This problem is tantamount to the logical analysis of our intuition of space.The following investigation is a new attempt to choose for geometry a simple and complete set of independent axioms and to deduce from these the most important geometrical theorems in such a manner as to bring out as clearly as possible the significance of the different groups of axioms and the scope of the conclusions to be derived from the individual axioms.Contents:CHAPTER ITHE FIVE GROUPS OF AXIOMS1. The elements of geometry and the five groups of axioms 2. Group I: Axioms of connection 3. Group II: Axioms of Order 4. Consequences of the axioms of connection and order 5. Group III: Axiom of Parallels (Euclid's axiom) 6. Group IV: Axioms of congruence 7. Consequences of the axioms of congruence 8. Group V: Axiom of Continuity (Archimedes's axiom) CHAPTER II. THE COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS. 9. Compatibility of the axioms 10. Independence of the axioms of parallels11. Independence of the axioms of congruence 12. Independence of the axiom of continuityCHAPTER III. THE THEORY OF PROPORTION. 13. Complex number-systems 14. Demonstration of Pascal's theorem 15. An algebra of segments, based upon Pascal's theorem 16. Proportion and the theorems of similitude 17. Equations of straight lines and of planes CHAPTER IV. THE THEORY OF PLANE AREAS. 18. Equal area and equal content of polygons 19. Parallelograms and triangles having equal bases and equal altitudes 20. The measure of area of triangles and polygons 21. Equality of content and the measure of area CHAPTER V. DESARGUES'S THEOREM. 22. Desargues's theorem and its demonstration for plane geometry by aid of the axioms of congruence. 23. The impossibility of demonstrating Desargues's theorem for the plane without the help of the axioms of congruence. 24. Introduction of an algebra of segments based upon Desargues's theorem and independent of the axioms of congruence. 25. The commutative and the associative law of addition for our new algebra of segments. 26. The associative law of multiplication and the two distributive laws for the new algebra of segments . 27. Equation of the straight line, based upon the new algebra of segments 28. The totality of segments, regarded as a complex number system 29. Construction of a geometry of space by aid of a desarguesian number system. 30. Significance of Desargues's theorem CHAPTER VI. PASCAL'S THEOREM. 31. Two theorems concerning the possibility of proving Pascal's theorem 32. The commutative law of multiplication for an archimedean number system. 33. The commutative law of multiplication for a non-archimedean number system . 34. Proof of the two propositions concerning Pascal's theorem Non-pascalian geometry. 35. The demonstration, by means of the theorems of Pascal and Desargues, of any theorem relating to points of intersection. CHAPTER VII. GEOMETRICAL CONSTRUCTIONS BASED UPON THE AXIOMS I-V. 36. Geometrical constructions by means of a straight-edge and a transferor of segments.37. Analytical representation of the co-ordinates of points which can be so constructed.38. The representation of algebraic numbers and of integral rational functions as sums of squares. 39. Criterion for the possibility of a geometrical construction by means of a straight-edge and a transferor of segments.

Download or read book The Foundations of Geometry and the Non Euclidean Plane written by G.E. Martin and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 525 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a text for junior, senior, or first-year graduate courses traditionally titled Foundations of Geometry and/or Non Euclidean Geometry. The first 29 chapters are for a semester or year course on the foundations of geometry. The remaining chap ters may then be used for either a regular course or independent study courses. Another possibility, which is also especially suited for in-service teachers of high school geometry, is to survey the the fundamentals of absolute geometry (Chapters 1 -20) very quickly and begin earnest study with the theory of parallels and isometries (Chapters 21 -30). The text is self-contained, except that the elementary calculus is assumed for some parts of the material on advanced hyperbolic geometry (Chapters 31 -34). There are over 650 exercises, 30 of which are 10-part true-or-false questions. A rigorous ruler-and-protractor axiomatic development of the Euclidean and hyperbolic planes, including the classification of the isometries of these planes, is balanced by the discussion about this development. Models, such as Taxicab Geometry, are used exten sively to illustrate theory. Historical aspects and alternatives to the selected axioms are prominent. The classical axiom systems of Euclid and Hilbert are discussed, as are axiom systems for three and four-dimensional absolute geometry and Pieri's system based on rigid motions. The text is divided into three parts. The Introduction (Chapters 1 -4) is to be read as quickly as possible and then used for ref erence if necessary.

Download or read book Foundations of Geometry written by David Hilbert and published by Open Court Publishing. This book was released on 1971 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: Along with the writings of Hilbert's friend and correspondent Frege, Hilbert's Grundlagen der Geometrie is the major prop that set the stage for Russell and Whitehead's Principia Mathematica. Hilbert presents a new axiomatization of geometry, the reduction of geometry to algebra, and introduces the distinction between mathematics and metamathematics, with a new theory of proof. This edition is translated from the tenth German edition, including all the improvements which Hilbert derived from his own reflections and the contributions of other writers. --Back cover.

Download or read book Foundations of Euclidean and Non Euclidean Geometry written by Ellery B. Golos and published by . This book was released on 1968 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Foundation of Euclidean and Non Euclidean Geometries according to F Klein written by L. Redei and published by Elsevier. This book was released on 2014-07-15 with total page 412 pages. Available in PDF, EPUB and Kindle. Book excerpt: Foundation of Euclidean and Non-Euclidean Geometries according to F. Klein aims to remedy the deficiency in geometry so that the ideas of F. Klein obtain the place they merit in the literature of mathematics. This book discusses the axioms of betweenness, lattice of linear subspaces, generalization of the notion of space, and coplanar Desargues configurations. The central collineations of the plane, fundamental theorem of projective geometry, and lines perpendicular to a proper plane are also elaborated. This text likewise covers the axioms of motion, basic projective configurations, properties of triangles, and theorem of duality in projective space. Other topics include the point-coordinates in an affine space and consistency of the three geometries. This publication is beneficial to mathematicians and students learning geometry.

Download or read book The Foundations of Mathematics written by Paul Carus and published by . This book was released on 1908 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Roads to Geometry written by Edward C. Wallace and published by Waveland Press. This book was released on 2015-10-23 with total page 510 pages. Available in PDF, EPUB and Kindle. Book excerpt: Now available from Waveland Press, the Third Edition of Roads to Geometry is appropriate for several kinds of students. Pre-service teachers of geometry are provided with a thorough yet accessible treatment of plane geometry in a historical context. Mathematics majors will find its axiomatic development sufficiently rigorous to provide a foundation for further study in the areas of Euclidean and non-Euclidean geometry. By using the SMSG postulate set as a basis for the development of plane geometry, the authors avoid the pitfalls of many “foundations of geometry” texts that encumber the reader with such a detailed development of preliminary results that many other substantive and elegant results are inaccessible in a one-semester course. At the end of each section is an ample collection of exercises of varying difficulty that provides problems that both extend and clarify results of that section, as well as problems that apply those results. At the end of chapters 3–7, a summary list of the new definitions and theorems of each chapter is included.

Download or read book Euclid s Elements written by Euclid and published by . This book was released on 2002 with total page 544 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The book includes introductions, terminology and biographical notes, bibliography, and an index and glossary" --from book jacket.

Download or read book An Axiomatic Approach to Geometry written by Francis Borceux and published by Springer Science & Business Media. This book was released on 2013-10-31 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focusing methodologically on those historical aspects that are relevant to supporting intuition in axiomatic approaches to geometry, the book develops systematic and modern approaches to the three core aspects of axiomatic geometry: Euclidean, non-Euclidean and projective. Historically, axiomatic geometry marks the origin of formalized mathematical activity. It is in this discipline that most historically famous problems can be found, the solutions of which have led to various presently very active domains of research, especially in algebra. The recognition of the coherence of two-by-two contradictory axiomatic systems for geometry (like one single parallel, no parallel at all, several parallels) has led to the emergence of mathematical theories based on an arbitrary system of axioms, an essential feature of contemporary mathematics. This is a fascinating book for all those who teach or study axiomatic geometry, and who are interested in the history of geometry or who want to see a complete proof of one of the famous problems encountered, but not solved, during their studies: circle squaring, duplication of the cube, trisection of the angle, construction of regular polygons, construction of models of non-Euclidean geometries, etc. It also provides hundreds of figures that support intuition. Through 35 centuries of the history of geometry, discover the birth and follow the evolution of those innovative ideas that allowed humankind to develop so many aspects of contemporary mathematics. Understand the various levels of rigor which successively established themselves through the centuries. Be amazed, as mathematicians of the 19th century were, when observing that both an axiom and its contradiction can be chosen as a valid basis for developing a mathematical theory. Pass through the door of this incredible world of axiomatic mathematical theories!

Download or read book Euclidean Geometry and its Subgeometries written by Edward John Specht and published by Birkhäuser. This book was released on 2015-12-31 with total page 537 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this monograph, the authors present a modern development of Euclidean geometry from independent axioms, using up-to-date language and providing detailed proofs. The axioms for incidence, betweenness, and plane separation are close to those of Hilbert. This is the only axiomatic treatment of Euclidean geometry that uses axioms not involving metric notions and that explores congruence and isometries by means of reflection mappings. The authors present thirteen axioms in sequence, proving as many theorems as possible at each stage and, in the process, building up subgeometries, most notably the Pasch and neutral geometries. Standard topics such as the congruence theorems for triangles, embedding the real numbers in a line, and coordinatization of the plane are included, as well as theorems of Pythagoras, Desargues, Pappas, Menelaus, and Ceva. The final chapter covers consistency and independence of axioms, as well as independence of definition properties. There are over 300 exercises; solutions to many of these, including all that are needed for this development, are available online at the homepage for the book at www.springer.com. Supplementary material is available online covering construction of complex numbers, arc length, the circular functions, angle measure, and the polygonal form of the Jordan Curve theorem. Euclidean Geometry and Its Subgeometries is intended for advanced students and mature mathematicians, but the proofs are thoroughly worked out to make it accessible to undergraduate students as well. It can be regarded as a completion, updating, and expansion of Hilbert's work, filling a gap in the existing literature.