The Theory of the Imaginary in Geometry: Together with the Trigonometry of the Imaginary (Classic Reprint) - Couverture souple

Hatton, J. L. S.

Couverture souple

ISBN 10 : 1330051076 ISBN 13 : 9781330051078

Editeur : Forgotten Books, 2019

Synopsis

The digital edition of all books may be viewed on our website before purchase. Excerpt from The Theory of the Imaginary in Geometry: Together With the Trigonometry of the Imaginary (Classic Reprint)



About the Publisher

Forgotten Books publishes hundreds of thousands of rare and classic books.

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. This text has been digitally restored from a historical edition. Some errors may persist, however we consider it worth publishing due to the work's historical value.The digital edition of all books may be viewed on our website before purchase.

A propos du livre

Présentation de l'éditeur

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]

Détails de l'édition

• Éditeur: Forgotten Books
• Date d'édition: 2019
• Langue: anglais
• ISBN 10: 1330051076
• ISBN 13: 9781330051078
• Reliure: Broché
• Nombre de pages: 230
• Coordonnées du fabricant: Copytech (UK) Ltd, Trading as Printondemand-worldwide.com; gpsr@podww.com; ; 9 Culley Court; Orton Southgate; Peterborough; Cambridgeshire; PE2 6WA; Royaume-Uni
• Personne responsable: Easy Access System Europe Oü, 16879218; +358405003575; gpsr.requests@easproject.com; ; Mustamäe tee 50; Tallinn; 10621; Estonie