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AJN INTRODUCTION TO PEOJECTIYE GEOMETRY By J. W fVTT VRA S. T and D. R. WARD, S. J. OXFORD AT THE CLARENDON PRESS 1937 OXFORD UNIVERSITY PRESS dMVHQUBB, B. C. 4 Xpndon Ediqbirgh Glasgow New York Toronto Sfelbourne Capetown Bombay Calcutta Madras HUMPHREY MILFORD PUBLISHER 10 THE UNIVERSITY HBIKTBD IN GREAT BBITAIN PREFACE - aim of this book is, as its title implies, to give the reader an introductory course in Projective Geometry. It aims, there fore, at removing the Euclidean prejudices which the study of elementary Geometry begets, anel at substituting for them what may be called the Projective mentality at the same time it seeks to familiarize the reader with most of the important methods used in the subject. Since these include not only what are called pure or Synthetic methods, but also the Algebraic method, the latter is included in the book. And in order that the subject-matter may be kept as simple as possible until facility in the use of these methods is attained, the work is con fined to two-dimensional Projective Geometry. In the first six chapters of the book, after a short historical introduction, the synthetic method is developed as far as the investigation of the more complex properties of the conic. In the next two chapters coordinate systems are introduced pro fectively, and the Algebyajc method is developed. This intro duction of coordinates mafies possible the definition of metrical concepts, and these are discussed in the ninth and tenth chap ters, their true place in the scheme of Geometry being shown. After a short treatment of the theory of transformations, the work is brought to a close by a chapter which indicates the possible developments of the subject from the point reached. It cannot with truth be said that the book has been written in order to supply a long-felt want. There seems, unfortunately, to be very little demand for the teaching of Projective Geo metry in this country. In default of this excuse, therefore, the authors must fall back on another, namely the hope that their work may do something to stimulate a demand for more wide spread familiarity with the subject. It is surely time that scholarship candidates in Mathematics and first-year Univer sity students should be allowed to know that the classical Geo metry which they assimilate occupies but a subsidiary place in the scheme of Geometry. An acquaintance with Projective vi PREFACE Geometry shows them what things are fundamental and what are subsidiary in that scheme it prepares them too for Geometries even more general than Projective Geometry, and for some at least of the subtleties of modern mathematical Physi S C. W. OH. 14 September 1936 D. B. W. CONTENTS CHAPTER I. HISTORICAL AND CRITICAL 1.1. Historical ....... 1 1.2. Critical ....... 6 CHAPTER II. THE PROPOSITIONS OF INCIDENCE 2.1. Undefined Elements and Initial Propositions . .12 2.2. Existence Theorems . . . . .14 2.3. First Deductions . . . . . .20 2.4. Extension . . . . . . .21 2.6. Notation . . . . . . .24 2.6. Figures, Theorems, Constructions . . . .25 2.7. Projective Geometry of Many Dimensions . . .27 CHAPTER III. PERSPECTIVITY AND PROJECTIVITY 3.1. Perspective Figures . . . . . .30 3.2. Projectivity . . . . . .34 3.3. Projectivity of Ranges and Pencils . . . .48 3.4. Cobasal Ranges and Pencils . . . .57 CHAPTER IV. THE FOUR-POINT AND THE FOUR-LINE 4.1. Definitions and Elementary Properties . . .62 4.2. Harmonic Tetrads . . . . . .70 4.3. Involutory Hexads . . . . . .78 4.4. Involutions . . . . . . .83 4.5. Concurrence and Collinearity in Triangles . . .90 CHAPTER V. THE CONIC 5.1. Introductory . . . . . .93 5.2. Definition and Basic Properties of the Conic . .95 5.3. The Incidence of Lines and Point-Conies and Dual . . 102 5.4. Desarguess Conic Theorem and Pascals Theorem . . 104 5.6. Pole and Polar . . . . . .112 5.6. Ranges and Pencils on a Conic . . . .119 CHAPTER VI, FURTHER THEOREMS ON CONICS 6.1. Pencils and Ranges of Conies .... 126 6.2
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An Introduction To Projective Geometry
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