"A lucid and masterly survey." — Mathematics GazetteProfessor Pedoe is widely known as a fine teacher and a fine geometer. His abilities in both areas are clearly evident in this self-contained, well-written, and lucid introduction to the scope and methods of elementary geometry. It covers the geometry usually included in undergraduate courses in mathematics, except for the theory of convex sets. Based on a course given by the author for several years at the University of Minnesota, the main purpose of the book is to increase geometrical, and therefore mathematical, understanding and to help students enjoy geometry.Among the topics discussed: the use of vectors and their products in work on Desargues' and Pappus' theorem and the nine-point circle; circles and coaxal systems; the representation of circles by points in three dimensions; mappings of the Euclidean plane, similitudes, isometries, mappings of the inversive plane, and Moebius transformations; projective geometry of the plane, space, and n dimensions; the projective generation of conics and quadrics; Moebius tetrahedra; the tetrahedral complex; the twisted cubic curve; the cubic surface; oriented circles; and introduction to algebraic geometry.In addition, three appendices deal with Euclidean definitions, postulates, and propositions; the Grassmann-Pluecker coordinates of lines in S3, and the group of circular transformations. Among the outstanding features of this book are its many worked examples and over 500 exercises to test geometrical understanding.
"The collection, drawn from arithmetic, algebra, pure and algebraic geometry and astronomy, is extraordinarily interesting and attractive." — Mathematical Gazette
This uncommonly interesting volume covers 100 of the most famous historical problems of elementary mathematics. Not only does the book bear witness to the extraordinary ingenuity of some of the greatest mathematical minds of history — Archimedes, Isaac Newton, Leonhard Euler, Augustin Cauchy, Pierre Fermat, Carl Friedrich Gauss, Gaspard Monge, Jakob Steiner, and many others — but it provides rare insight and inspiration to any reader, from high school math student to professional mathematician. This is indeed an unusual and uniquely valuable book.
The one hundred problems are presented in six categories: 26 arithmetical problems, 15 planimetric problems, 25 classic problems concerning conic sections and cycloids, 10 stereometric problems, 12 nautical and astronomical problems, and 12 maxima and minima problems. In addition to defining the problems and giving full solutions and proofs, the author recounts their origins and history and discusses personalities associated with them. Often he gives not the original solution, but one or two simpler or more interesting demonstrations. In only two or three instances does the solution assume anything more than a knowledge of theorems of elementary mathematics; hence, this is a book with an extremely wide appeal.
Some of the most celebrated and intriguing items are: Archimedes' "Problema Bovinum," Euler's problem of polygon division, Omar Khayyam's binomial expansion, the Euler number, Newton's exponential series, the sine and cosine series, Mercator's logarithmic series, the Fermat-Euler prime number theorem, the Feuerbach circle, the tangency problem of Apollonius, Archimedes' determination of pi, Pascal's hexagon theorem, Desargues' involution theorem, the five regular solids, the Mercator projection, the Kepler equation, determination of the position of a ship at sea, Lambert's comet problem, and Steiner's ellipse, circle, and sphere problems.
This translation, prepared especially for Dover by David Antin, brings Dörrie's "Triumph der Mathematik" to the English-language audience for the first time.
If you accidentally try to order the same song twice from iTunes, you'll be warned that you already own it. Not because it would be illegal or unethical for Apple to profit from your forgetfulness. There's a clear business reason: the leaders of iTunes realize there's no better way to make you trust them than to be totally honest when you least expect it. In the age of the Web, smartphones, and social networks, every action an organization takes can be exposed and critiqued in real time. Nothing is local or secret anymore. If you treat one customer unfairly, produce one shoddy product, or try to gouge one price, the whole world may find out in hours, if not minutes. The users of Twitter, Yelp, Epinions, and similar outlets show little mercy for bad behavior. The bar for trust­worthiness is higher than ever and continuing to rise. Don Peppers and Martha Rogers argue that the only sane response to these rising levels of transpar­ency is to protect the interests of customers proac­tively, before they have a chance to spread negative buzz-even if that requires spending extra money in the short run to preserve your reputation and cus­tomer relationships in the long run. The payoff of gen­erating extreme trust will be worth it. The authors show how this trend is playing out in many different sectors. Among their insights: Banks will soon have to stop relying on overdraft charges, because depositers will expect advance warnings of low balances. Retailers will be expected to remind shoppers when they have unused balances on their gift cards. Credit card companies will have to coach customers on avoiding excessive borrowing. Cell phone providers will win more business by helping customers find the cheapest calling plans for their usage patterns. Health insurers will make recommendations based on improving long-term health, not increasing their revenue. The companies that Peppers and Rogers call “trustable” remember what they learn from each inter­action, and they use these insights to create better and better customer experiences. They focus on win­ning the long-term battle for trust and loyalty, even if the dollar value of that trust is hard to quantify. For instance, in 2009 Best Buy launched Twelp­force, a service that responds to customer questions and problems via Twitter. It's manned part time by more than two thousand employees. In its first year of operation Twelpforce responded to nearly thirty thousand inquiries-which not only improved cus­tomer service but also helped educate and motivate the associates who participated. The short-term profit might be small but the impact on trust is enormous. With a wealth of fascinating research as well as practical applications, this book will show you how to earn-and keep-the extreme trust of everyone your company interacts with.
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