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Imaginaries In Geometry
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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 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 : 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:
Book Synopsis The Imaginary in Geometry by : Ellery William DAVIS
Download or read book The Imaginary in Geometry written by Ellery William DAVIS and published by . This book was released on 1910 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Imaginary in Geometry by : Ellery Williams Davis
Download or read book The Imaginary in Geometry written by Ellery Williams Davis and published by . This book was released on 1910 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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. 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 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 Interpretation of Imaginaries in Projective Geometry by : Juna Marie Lutz
Download or read book Interpretation of Imaginaries in Projective Geometry written by Juna Marie Lutz and published by . This book was released on 1923 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Theory of the Imaginary in Geometry by : HardPress
Download or read book The Theory of the Imaginary in Geometry written by HardPress and published by Hardpress Publishing. This book was released on 2013-01 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: Unlike some other reproductions of classic texts (1) We have not used OCR(Optical Character Recognition), as this leads to bad quality books with introduced typos. (2) In books where there are images such as portraits, maps, sketches etc We have endeavoured to keep the quality of these images, so they represent accurately the original artefact. Although occasionally there may be certain imperfections with these old texts, we feel they deserve to be made available for future generations to enjoy.
Book Synopsis Application of Geometry to Imaginaries and to Imaginary Roots of Equations ... by : Herrick Ernest Herbert Greenleaf
Download or read book Application of Geometry to Imaginaries and to Imaginary Roots of Equations ... written by Herrick Ernest Herbert Greenleaf and published by . This book was released on 1925 with total page 56 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis An Essay on the Foundations of Modern Geometry by : Bertrand Russell
Download or read book An Essay on the Foundations of Modern Geometry written by Bertrand Russell and published by Courier Corporation. This book was released on 2003-01-01 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author, a Nobel Laureate and one of the 20th century's most important logicians, asks and answers basic questions about the intersection of philosophy and higher mathematics. 1897 edition.
Book Synopsis An Essay on the Foundations of Geometry by : Bertrand Russell
Download or read book An Essay on the Foundations of Geometry written by Bertrand Russell and published by . This book was released on 1897 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis An Essay on the Foundations of Geometry by : A. W. Russell
Download or read book An Essay on the Foundations of Geometry written by A. W. Russell and published by CUP Archive. This book was released on 2017-07-29 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: Trieste Publishing has a massive catalogue of classic book titles. Our aim is to provide readers with the highest quality reproductions of fiction and non-fiction literature that has stood the test of time. The many thousands of books in our collection have been sourced from libraries and private collections around the world.The titles that Trieste Publishing has chosen to be part of the collection have been scanned to simulate the original. Our readers see the books the same way that their first readers did decades or a hundred or more years ago. Books from that period are often spoiled by imperfections that did not exist in the original. Imperfections could be in the form of blurred text, photographs, or missing pages. It is highly unlikely that this would occur with one of our books. Our extensive quality control ensures that the readers of Trieste Publishing's books will be delighted with their purchase. Our staff has thoroughly reviewed every page of all the books in the collection, repairing, or if necessary, rejecting titles that are not of the highest quality. This process ensures that the reader of one of Trieste Publishing's titles receives a volume that faithfully reproduces the original, and to the maximum degree possible, gives them the experience of owning the original work.We pride ourselves on not only creating a pathway to an extensive reservoir of books of the finest quality, but also providing value to every one of our readers. Generally, Trieste books are purchased singly - on demand, however they may also be purchased in bulk. Readers interested in bulk purchases are invited to contact us directly to enquire about our tailored bulk rates.
Book Synopsis Graphs and Imaginaries by : J. G. Hamilton
Download or read book Graphs and Imaginaries written by J. G. Hamilton and published by . This book was released on 1904 with total page 64 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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.
