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The Topics in Hyperbolic Geometry handout. Course Description This is a course on Euclidean and non-Euclidean geometries with emphasis on i the contrast between the traditional and modern approaches to geometry, and ii the history and role of the parallel postulate.

This course will be useful to students who want to teach and use Euclidean geometry, to students who want to learn more about the history of geometry, and to students who want an introduction to non-Euclidean geometry. Euclid wrote the first preserved Geometry book which has traditionally been held up as a role model for logical reasoning inside and outside mathematics for thousands of years.

However, Euclid has several subtle logical omissions, and in the late s it was necessary to revise the foundations of Euclidean geometry. The need for such a revision was partly due to advances in mathematical logic and changes in the conception of an axiom system. The famous mathematician David Hilbert, building on work of several other mathematicians, was able to develop axioms that allow one to develop geometry without any overt or covert appeals to intuition.

His idea was that, although intuitions are important in discovering, motivating, communicating and appreciating the theorems, rigorous proofs should not appeal to them. With the more modern approach to the axiomatic method that is not logically dependent on intuition, mathematicians are free to develop more types of geometries than the traditional Euclidean geometry. We will discuss different types and models of geometry that are used today.

These include finite geometries with applications in discrete mathematics and number theory, spaces of more than three dimensions, geometries whose coordinates are not real numbers, and geometries where a line can pass through a circle without actually intersecting the circle.

Many of these geometries are useful, and not just curious examples. A second major theme of the course will be the history and role of the parallel postulate.

Modern geometry began in the s with the realization that there are interesting consistent geometries for which the parallel postulate is false. For example, hyperbolic and elliptic geometry do not satisfy the parallel postulate.

Since this postulate is less intuitively obvious than the other axioms of geometry, many mathematicians, especially medieval Arab mathematicians and later several European mathematicians of the s, tried to make the parallel postulate a theorem and not an axiom. This goes along with the traditional idea that axioms should be restricted to a few simple, self-evident propositions, and the rest of the subject should be built upon these using proof.

However, no mathematician was able to show that the parallel postulate followed as a theorem from the other axioms. Several prominent mathematicians thought that they had a proof of the parallel postulate, but subtle flaws were later discovered in their proofs. Finally mathematicians such as Lobachevsky and Bolyai started to believe that it is possible for there to be geometries where the parallel postulate fails, and they proved theorems about such non-Euclidean geometries.

After the discovery of Euclidean models of non-Euclidean geometries in the late s, no one was able to doubt the existence and consistency of non-Euclidean geometry. Also, these models show that the parallel postulate is independent of the other axioms of geometry: you cannot prove the parallel postulate from the other axioms.

Required Texts 2 There are two required texts.

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## MODERN GEOMETRY DURRELL PDF DOWNLOAD

The Topics in Hyperbolic Geometry handout. Course Description This is a course on Euclidean and non-Euclidean geometries with emphasis on i the contrast between the traditional and modern approaches to geometry, and ii the history and role of the parallel postulate. This course will be useful to students who want to teach and use Euclidean geometry, to students who want to learn more about the history of geometry, and to students who want an introduction to non-Euclidean geometry. Euclid wrote the first preserved Geometry book which has traditionally been held up as a role model for logical reasoning inside and outside mathematics for thousands of years. However, Euclid has several subtle logical omissions, and in the late s it was necessary to revise the foundations of Euclidean geometry. The need for such a revision was partly due to advances in mathematical logic and changes in the conception of an axiom system.

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## C. V. Durell

Most widely held works by Clement V Durell Readable relativity by Clement V Durell Book 35 editions published between and in 5 languages and held by WorldCat member libraries worldwide Precise, brief, and practical, this text is the work of a highly respected teacher with years of classroom experience, who sketches the mathematical background essential to a clear understanding of the fundamentals of relativity theory. Each subject-including the velocity of light, the measurement of time and distance, and the properties of mass and momentum-is illustrated with diagrams, formulas, and examples. All chapters conclude with a series of exercises, with solutions at the end of the book. Ideal for self-study, this text offers a clear, logical presentation of topics such as the properties of the triangle and the quadrilateral; sub-multiple angles and inverse functions; hyperbolic, logarithmic, and exponential functions; much more Advanced algebra by Clement V Durell Book 30 editions published between and in English and Undetermined and held by WorldCat member libraries worldwide 16 editions published between and in English and held by WorldCat member libraries worldwide Algebraic geometry by Clement V Durell Book 12 editions published between and in English and held by WorldCat member libraries worldwide Projective geometry by Clement V Durell Book 17 editions published between and in English and held by WorldCat member libraries worldwide Elementary calculus by Clement V Durell Book 28 editions published between and in English and Undetermined and held by WorldCat member libraries worldwide.

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## Modern geometry the straight line and circle,

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