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Book The Theory of the Imaginary in Geometry

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:

Book The Theory of the Imaginary in Geometry

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 The Theory of the Imaginary in Geometry

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 The Theory of the Imaginary in Geometry

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 The Theory of the Imaginary in Geometry

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 The Theory of the Imaginery in Geometry Together with the Trigonometry of the Imaginary

Download or read book The Theory of the Imaginery 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 215 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book The Theory of the Imaginary in Geometry Thogether with the Trigonometry of the Imaginary

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 The Theory of the Imaginary in Geometrytogether with the Trogonometry of the Imaginary

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 The Theory of the Imaginary

Download or read book The Theory of the Imaginary written by John Leigh Smeathman Hatton and published by . This book was released on 1920 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book The Journal of Philosophy

Download or read book The Journal of Philosophy written by and published by . This book was released on 1921 with total page 1136 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covers topics in philosophy, psychology, and scientific methods. Vols. 31- include "A Bibliography of philosophy," 1933-

Book Nature

Download or read book Nature written by Sir Norman Lockyer and published by . This book was released on 1920 with total page 984 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Technical Book Review Index

Download or read book Technical Book Review Index written by and published by . This book was released on 1920 with total page 1000 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Philosophical Magazine

Download or read book Philosophical Magazine written by and published by . This book was released on 1920 with total page 1028 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Subject Index of Modern Books Acquired

Download or read book Subject Index of Modern Books Acquired written by British Library and published by . This book was released on 1922 with total page 1228 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book The Journal of Education

Download or read book The Journal of Education written by and published by . This book was released on 1920 with total page 860 pages. Available in PDF, EPUB and Kindle. Book excerpt: