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.

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:

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.

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:

Download or read book The Theory of the Imaginary in Geometry written by J. L. S. Hatton and published by Forgotten Books. This book was released on 2015-06-12 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: Excerpt from The Theory of the Imaginary in Geometry: Together With the Trigonometry of the Imaginary The position of any real point in space may be determined by means 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. Such a Geometry contains as a particular case the Geometry of real points. 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. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

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]

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.

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.

Download or read book Theory of Parallels written by Nikolaj Ivanovič Lobačevskij and published by Independently Published. This book was released on 2019-05-22 with total page 52 pages. Available in PDF, EPUB and Kindle. Book excerpt: LOBACHEVSKY was the first man ever to publish a non-Euclidean geometry. Of the immortal essay now first appearing in English Gauss said, "The author has treated the matter with a master-hand and in the true geometer's spirit. I think I ought to call your attention to this book, whose perusal cannot fail to give you the most vivid pleasure." Clifford says, "It is quite simple, merely Euclid without the vicious assumption, but the way things come out of one another is quite lovely." * * * "What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobachevsky to Euclid." Says Sylvester, "In Quaternions the example has been given of Algebra released from the yoke of the commutative principle of multiplication - an emancipation somewhat akin to Lobachevsky's of Geometry from Euclid's noted empirical axiom." Cayley says, "It is well known that Euclid's twelfth axiom, even in Playfair's form of it, has been considered as needing demonstration; and that Lobachevsky constructed a perfectly consistent theory, where- in this axiom was assumed not to hold good, or say a system of non- Euclidean plane geometry. There is a like system of non-Euclidean solid geometry." GEORGE BRUCE HALSTED. 2407 San Marcos Street, Austin, Texas. * * * *From the TRANSLATOR'S INTRODUCTION. "Prove all things, hold fast that which is good," does not mean demonstrate everything. From nothing assumed, nothing can be proved. "Geometry without axioms," was a book which went through several editions, and still has historical value. But now a volume with such a title would, without opening it, be set down as simply the work of a paradoxer. The set of axioms far the most influential in the intellectual history of the world was put together in Egypt; but really it owed nothing to the Egyptian race, drew nothing from the boasted lore of Egypt's priests. The Papyrus of the Rhind, belonging to the British Museum, but given to the world by the erudition of a German Egyptologist, Eisenlohr, and a German historian of mathematics, Cantor, gives us more knowledge of the state of mathematics in ancient Egypt than all else previously accessible to the modern world. Its whole testimony con- firms with overwhelming force the position that Geometry as a science, strict and self-conscious deductive reasoning, was created by the subtle intellect of the same race whose bloom in art still overawes us in the Venus of Milo, the Apollo Belvidere, the Laocoon. In a geometry occur the most noted set of axioms, the geometry of Euclid, a pure Greek, professor at the University of Alexandria. Not only at its very birth did this typical product of the Greek genius assume sway as ruler in the pure sciences, not only does its first efflorescence carry us through the splendid days of Theon and Hypatia, but unlike the latter, fanatics cannot murder it; that dismal flood, the dark ages, cannot drown it. Like the phoenix of its native Egypt, it rises with the new birth of culture. An Anglo-Saxon, Adelard of Bath, finds it clothed in Arabic vestments in the land of the Alhambra. Then clothed in Latin, it and the new-born printing press confer honor on each other. Finally back again in its original Greek, it is published first in queenly Basel, then in stately Oxford. The latest edition in Greek is from Leipsic's learned presses.

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:

Download or read book Geometry Plane Solid and Spherical in Six Books written by Pierce Morton and published by Nabu Press. This book was released on 2013-09 with total page 572 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.

Download or read book The Axioms of Projective Geometry Primary Source Edition written by Alfred North Whitehead and published by Nabu Press. This book was released on 2014-03 with total page 74 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.

Download or read book Geometry of Einsteins Unified Field Theory Primary Source Edition written by Vaclav Hlavaty and published by Nabu Press. This book was released on 2014-03-13 with total page 384 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.

Download or read book Geometry and the Imagination written by D. Hilbert and published by American Mathematical Soc.. This book was released on 2021-03-17 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer—even after more than half a century has passed! The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. “Hilbert and Cohn-Vossen” is full of interesting facts, many of which you wish you had known before. It's also likely that you have heard those facts before, but surely wondered where they could be found. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in R 3 R3. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: π/4=1−1/3+1/5−1/7+−… π/4=1−1/3+1/5−1/7+−…. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem. One of the most remarkable chapters is “Projective Configurations”. In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader. A particularly intriguing section in the chapter on differential geometry is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? Many mathematicians are familiar with the plaster models of surfaces found in many mathematics departments. The book includes pictures of some of the models that are found in the Göttingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained! The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the “pantheon” of great mathematics books.

Download or read book Hume s Theory of Imagination written by Jan Wilbanks and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 187 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present work is, as its title indicates, a study of Hume's theory of imagination. Naturally, it is a study of a particular sort. It has a certain scope and limitations, takes a certain line of approach, exhibits certain emphases, has certain ends-in-view, etc. As an initial step in specifying the nature of this study, I shall indicate its central problem, i. e. , that problem to the solution of which the solutions of the various other problems with which it is concerned are merely means. The central problem of this study is that of determining how Hume's theory of im agination is related to, or involved in, the generic features and main lines of argument of his philosophy of the human understanding. The expression "philosophy of the human understanding" is obvious to allude to a restriction on the scope of this investigation. ly intended Actually, it is a title suggested to me by two of Hume's philosophical writings; and to anyone who is even modestly acquainted with these writings, its reference should be no mystery. Hume published the first two so-called "Books" of his A Treatise of Human Nature in 1739. The first of these two Books was entitled "Of the Human Understanding. " Nine years later, he published a work under the title, An Enquiry Con cerning Human Understanding.

Download or read book The Origin and Development of the Quantum Theory written by Max Planck and published by . This book was released on 1922 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The Journal of Education written by and published by . This book was released on 1920 with total page 852 pages. Available in PDF, EPUB and Kindle. Book excerpt: