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The Theory Of The Imaginary In Geometry
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Book Synopsis The Theory of the Imaginary in Geometry by : John Leigh Smeathman Hatton
Download or read book The Theory of the Imaginary in Geometry written by John Leigh Smeathman Hatton and published by . This book was released on 1920 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author :J L S (John Leigh Smeathman) Hatton Publisher :Franklin Classics ISBN 13 :9780343095291 Total Pages :226 pages Book Rating :4.0/5 (952 download)
Book Synopsis The Theory of the Imaginary in Geometry by : J L S (John Leigh Smeathman) Hatton
Download or read book The Theory of the Imaginary in Geometry written by J L S (John Leigh Smeathman) Hatton and published by Franklin Classics. This book was released on 2018-10-14 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
Book Synopsis The Theory of the Imaginary in Geometry: Together with the Trigonometry of the Imaginary by : John Leigh Smeathman Hatton
Download or read book The Theory of the Imaginary in Geometry: Together with the Trigonometry of the Imaginary written by John Leigh Smeathman Hatton and published by . This book was released on 1920 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Theory of the Imaginary in Geometry by : J. Hatton
Download or read book The Theory of the Imaginary in Geometry written by J. Hatton and published by CreateSpace. This book was released on 2015-07-15 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: THE word theory in the title is to be understood in a very non-technical sense. Indeed, apart from the idea of the invariant elements of an elliptic involution on a straight line, no theory is found at all. The purpose of the book is rather to furnish a certain graphical representation of imaginaries under a number of conventions more or less well known. Three concepts run through the work: first, an incompletely defined idea of the nature of an imaginary; second, the analogy with the geometry of reals; third, the use of coordinate methods, assuming the algebra of imaginaries. Given a real point O and a real constant k, an imaginary point P is defined by the equation OP2 = -k - 2. The two imaginary points P and P' are the double points of an involution having O for center, and ik for parameter. The algebra of imaginaries is now assumed, and a geometry of imaginary distances on a straight line is built upon it. The reader is repeatedly reminded that in themselves there is no difference between real and imaginary points; that differences exist solely in their relations to other points. In the extension to two dimensions both x and ix are plotted on a horizontal line, while x and xy are plotted on a vertical line. Imaginary lines are dotted, and points having one or both coordinates imaginary are enclosed by parentheses, but otherwise the same figures are used for proofs, either by the methods of elementary geometry, or by coordinate methods.In the algebra of segments it is shown that an imaginary distance O'D' can be expressed in the form iOD, wherein OD is a real segment, or at most by OD times some number. Now follows a long development of the extension of cross ratios, etc., to imaginaries. In fact every word of this is found implicitly in any treatment of the invariance of cross ratios under linear fractional transformation.In Chapter II the conic with a real branch is introduced, beginning with involutions of conjugate points on lines having imaginary points on the conic. If the coefficients in the equation of a circle are real, the usual graph of x2 + y2 = a2 for real x and real y is followed by replacing y by iy, then proceeding as before. The former locus is called the (1, 1) branch, and the latter the (1, i) branch of the circle. Similarly, it has a (i, 1) branch, and another, (i, i) , but the latter has no graph. This idea is applied in all detail to ellipses, hyperbolas, and parabolas; in the case of the central conies it is also followed by replacing rectangular coordinates by a pair of conjugate diameters. The ordinary theorems of poles and polars, and the theorems of Pascal, Brianchon, Desargues, Carnot are shown to apply. Indeed, after having established the applicability of cross ratios in the earlier chapters, all these proofs can be applied in the same manner as to reals, without changing a word....-An excerpt from Bulletin of the American Mathematical Society, Vol. 27 [1921]
Book Synopsis The Theory of the Imaginary in Geometry by : J L S Hatton
Download or read book The Theory of the Imaginary in Geometry written by J L S Hatton and published by . This book was released on 2019-11-20 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: THE position of any real point in space may be determined by eans of three real coordinates, and any three real quantities may be regarded as determining the position of such a point. In Geometry as in other branches of Pure Mathematics the question naturally arises, whether the quantities concerned need necessarily be real. What, it may be asked, is the nature of the Geometry in which the coordinates of any point may be complex quantities of the form x + ix', y + iy', z + iz'? Such a Geometry contains as a particular case the Geometry of real points. From it the Geometry of real points may be deduced (a) by regarding x', y', z' as zero, (b) by regarding x, y, z as zero, or (c) by considering only those points, the coordinates of which are real multiples of the same complex quantity a+ib. The relationship of the more generalised conception of Geometry and of space to the particular case of real Geometry is of importance, as points, whose determining elements are complex quantities, arise both in coordinate and in projective Geometry. In this book an attempt has been made to work out and determine this relationship. Either of two methods might have been adopted. It would have been possible to lay down certain axioms and premises and to have developed a general theory therefrom. This has been done by other authors. The alternative method, which has been employed here, is to add to the axioms of real Geometry certain additional assumptions. From these, by means of the methods and principles of real Geometry, an extension of the existing ideas and conception of Geometry can be obtained. In this way the reader is able to approach the simpler and more concrete theorems in the first instance, and step by step the well-known theorems are extended and generalised. A conception of the imaginary is thus gradually built up and the relationship between the imaginary and the real is exemplified and developed. The theory as here set forth may be regarded from the analytical point of view as an exposition of the oft quoted but seldom explained " Principle of Continuity." The fundamental definition of Imaginary points is that given by Dr Karl v. Staudt in his Beiträge zur Geometrie der Lage; Nuremberg, 1856 and 1860. The idea of (α, β) figures, independently evolved by the author, is due to J. V. Poncelet, who published it in his Traité des Propriétés Projectives des Figures in 1822. The matter contained in four or five pages of Chapter II is taken from the lectures delivered by the late Professor Esson, F.R.S., Savilian Professor of Geometry in the University of Oxford, and may be partly .traced to the writings of v. Staudt. For the remainder of the book the author must take the responsibility. Inaccuracies and inconsistencies may have crept in, but long experience has taught him that these will be found to be due to his own deficiencies and not to fundamental defects in the theory. Those who approach the subject with an open mind will, it is believed, find in these pages a consistent and natural theory of the imaginary. Many problems however still require to be worked out and the subject offers a wide field for further investigations.
Book Synopsis The Theory of the Imaginary in Geometry by : J. L. S. Hatton
Download or read book The Theory of the Imaginary in Geometry written by J. L. S. Hatton and published by CreateSpace. This book was released on 2015-01-16 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the Preface. THE position of any real point in space may be determined by eans of three real coordinates, and any three real quantities may be regarded as determining the position of such a point. In Geometry as in other branches of Pure Mathematics the question naturally arises, whether the quantities concerned need necessarily be real. What, it may be asked, is the nature of the Geometry in which the coordinates of any point may be complex quantities of the form x + ix', y + iy' , z + iz'? Such a Geometry contains as a particular case the Geometry of real points. From it the Geometry of real points may be deduced (a) by regarding x', y', z' as zero, (b) by regarding x, y, z as zero, or (c) by considering only those points, the coordinates of which are real multiples of the same complex quantity a+ib. The relationship of the more generalised conception of Geometry and of space to the particular case of real Geometry is of importance, as points, whose determining elements are complex quantities, arise both in coordinate and in projective Geometry. In this book an attempt has been made to work out and determine this relationship. Either of two methods might have been adopted. It would have been possible to lay down certain axioms and premises and to have developed a general theory therefrom. This has been done by other authors. The alternative method, which has been employed here, is to add to the axioms of real Geometry certain additional assumptions. From these, by means of the methods and principles of real Geometry, an extension of the existing ideas and conception of Geometry can be obtained. In this way the reader is able to approach the simpler and more concrete theorems in the first instance, and step by step the well-known theorems are extended and generalised. A conception of the imaginary is thus gradually built up and the relationship between the imaginary and the real is exemplified and developed. The theory as here set forth may be regarded from the analytical point of view as an exposition of the oft quoted but seldom explained "Principle of Continuity." The fundamental definition of Imaginary points is that given by Dr Karl v. Staudt in his Beiträge zur Geometrie der Lage; Nuremberg, 1856 and 1860. The idea of (a, beta) figures, independently evolved by the author, is due to J. V. Poncelet, who published it in his Traité des Propriétés Projectives des Figures in 1822. The matter contained in four or five pages of Chapter II is taken from the lectures delivered by the late Professor Esson, F.R.S., Savilian Professor of Geometry in the University of Oxford, and may be partly traced to the writings of v. Staudt. For the remainder of the book the author must take the responsibility. Inaccuracies and inconsistencies may have crept in, but long experience has taught him that these will be found to be due to his own deficiencies and not to fundamental defects in the theory. Those who approach the subject with an open mind will, it is believed, find in these pages a consistent and natural theory of the imaginary. Many problems however still require to be worked out and the subject offers a wide field for further investigations.
Book Synopsis The Theory of the Imaginary in Geometry Thogether with the Trigonometry of the Imaginary by : J. L. S. Hatton
Download or read book The Theory of the Imaginary in Geometry Thogether with the Trigonometry of the Imaginary written by J. L. S. Hatton and published by . This book was released on 1920 with total page 215 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Theory of the Imaginary in Geometry by : J L S (John Leigh Smeathman) Hatton
Download or read book The Theory of the Imaginary in Geometry written by J L S (John Leigh Smeathman) Hatton and published by Nabu Press. This book was released on 2014-02-03 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book.
Book Synopsis Imaginaries in Geometry by : Pavel Alexandrovich Florensky
Download or read book Imaginaries in Geometry written by Pavel Alexandrovich Florensky and published by Philosophy. This book was released on 2021 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first complete English translation of Pavel Florensky's original and ambitious attempt to arrive at a geometric representation of imaginary numbers, in a context that had already captured the attention of other mathematicians, including Gauss, Argan, Cauchy and Bellavitis. Florensky did not limit his attempt solely to complex projective geometry, but extended it to encompass Ptolemaic-Dantean cosmology and Einstein's Principle of Relativity, as well as a new epistemological theory. The resulting treatise combines various disciplines and explores the relationship between an immanent realm of knowledge and a transcendent one.
Book Synopsis The Theory of the Imaginary in Geometrytogether with the Trogonometry of the Imaginary by : Jls Hatton
Download or read book The Theory of the Imaginary in Geometrytogether with the Trogonometry of the Imaginary written by Jls Hatton and published by Palala Press. This book was released on 2016-04-27 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
Book Synopsis New Foundations for Classical Mechanics by : D. Hestenes
Download or read book New Foundations for Classical Mechanics written by D. Hestenes and published by Springer Science & Business Media. This book was released on 2005-12-17 with total page 716 pages. Available in PDF, EPUB and Kindle. Book excerpt: (revised) This is a textbook on classical mechanics at the intermediate level, but its main purpose is to serve as an introduction to a new mathematical language for physics called geometric algebra. Mechanics is most commonly formulated today in terms of the vector algebra developed by the American physicist J. Willard Gibbs, but for some applications of mechanics the algebra of complex numbers is more efficient than vector algebra, while in other applications matrix algebra works better. Geometric algebra integrates all these algebraic systems into a coherent mathematical language which not only retains the advantages of each special algebra but possesses powerful new capabilities. This book covers the fairly standard material for a course on the mechanics of particles and rigid bodies. However, it will be seen that geometric algebra brings new insights into the treatment of nearly every topic and produces simplifications that move the subject quickly to advanced levels. That has made it possible in this book to carry the treatment of two major topics in mechanics well beyond the level of other textbooks. A few words are in order about the unique treatment of these two topics, namely, rotational dynamics and celestial mechanics.
Book Synopsis A History of Non-Euclidean Geometry by : Boris A. Rosenfeld
Download or read book A History of Non-Euclidean Geometry written by Boris A. Rosenfeld and published by Springer Science & Business Media. This book was released on 2012-09-08 with total page 481 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Russian edition of this book appeared in 1976 on the hundred-and-fiftieth anniversary of the historic day of February 23, 1826, when LobaeevskiI delivered his famous lecture on his discovery of non-Euclidean geometry. The importance of the discovery of non-Euclidean geometry goes far beyond the limits of geometry itself. It is safe to say that it was a turning point in the history of all mathematics. The scientific revolution of the seventeenth century marked the transition from "mathematics of constant magnitudes" to "mathematics of variable magnitudes. " During the seventies of the last century there occurred another scientific revolution. By that time mathematicians had become familiar with the ideas of non-Euclidean geometry and the algebraic ideas of group and field (all of which appeared at about the same time), and the (later) ideas of set theory. This gave rise to many geometries in addition to the Euclidean geometry previously regarded as the only conceivable possibility, to the arithmetics and algebras of many groups and fields in addition to the arith metic and algebra of real and complex numbers, and, finally, to new mathe matical systems, i. e. , sets furnished with various structures having no classical analogues. Thus in the 1870's there began a new mathematical era usually called, until the middle of the twentieth century, the era of modern mathe matics.
Book Synopsis Geometrical Researches on the Theory of Parallels by : Nikolaĭ Ivanovich Lobachevskiĭ
Download or read book Geometrical Researches on the Theory of Parallels written by Nikolaĭ Ivanovich Lobachevskiĭ and published by . This book was released on 1914 with total page 57 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book An Imaginary Tale written by Paul Nahin and published by Princeton University Press. This book was released on 2010-02-22 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them. In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i. In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots--now called "imaginary numbers"--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times. Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "numbers" in all of mathematics. Some images inside the book are unavailable due to digital copyright restrictions.
Book Synopsis Geometry and the Imagination by : David Hilbert
Download or read book Geometry and the Imagination written by David Hilbert and published by Chelsea Publishing Company, Incorporated. This book was released on 1952 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Visual Complex Analysis by : Tristan Needham
Download or read book Visual Complex Analysis written by Tristan Needham and published by Oxford University Press. This book was released on 1997 with total page 620 pages. Available in PDF, EPUB and Kindle. Book excerpt: This radical first course on complex analysis brings a beautiful and powerful subject to life by consistently using geometry (not calculation) as the means of explanation. Aimed at undergraduate students in mathematics, physics, and engineering, the book's intuitive explanations, lack of advanced prerequisites, and consciously user-friendly prose style will help students to master the subject more readily than was previously possible. The key to this is the book's use of new geometric arguments in place of the standard calculational ones. These geometric arguments are communicated with the aid of hundreds of diagrams of a standard seldom encountered in mathematical works. A new approach to a classical topic, this work will be of interest to students in mathematics, physics, and engineering, as well as to professionals in these fields.
Book Synopsis Complex Geometry by : Daniel Huybrechts
Download or read book Complex Geometry written by Daniel Huybrechts and published by Springer Science & Business Media. This book was released on 2005 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: Easily accessible Includes recent developments Assumes very little knowledge of differentiable manifolds and functional analysis Particular emphasis on topics related to mirror symmetry (SUSY, Kaehler-Einstein metrics, Tian-Todorov lemma)