-Mi -- AN INTRODUCTION r to 2S( PROJECTIVE GEOMETRY BY *> -,c L; N. G. FILON, M.A., D.Sc. FELLOW AND LECTURER OF UNIVERSITY COLLEGE, LONDON EXAMINER IN MATHEMATICS TO THE UNIVERSITY OF LONDON LONDON EDWARD ARNOLD 41 & 43 MADDOX STREET, BOND STREET, W. [All rights reserved] <3A V7J F5V PREFACE object in writing the following pages lias been to supply the MYgrowing need of mathematical students in this country for a compact text-book giving the theory of Conic Sections on modern lines. During recent years increasing space has been allowed, in University syllabuses and courses of instruction, to the more powerful and general projective methods, as opposed to the more special methods of wdiat is still known as Geometrical Conies. The line of cleavage between the two has, however, been sharply maintained, with the result that the already much overworked mathe- matical student has to learn his theory of Conic Sections three times over: (1) analytically; (2) according to Euclidean methods; (3) according to Projective methods. The difficulty has been to reconcile the Euclidean and Projective of in fact to in the focal into definitions the curve ; bring properties Projective Geometry at a sufficiently early stage. The practice has usually been, in order to pass from the projective to the focal definitions, to introduce the theory of involution. But the latter requires for its fullest and clearest treatment the employment of imaginary elements. It seems undesirable that the more fundamental focal properties of the conies, e.g. the sum or difference of the focal distances and the angles made by these with the tangent and normal, should appear to depend upon properties of imaginary points and lines, even though this might introduce greater rapidity of treatment. The University of London has recognized this, for, while admitting Projective Geometry into its syllabuses for the Final Examination for a Pass Degree, it has excluded involution. Many teachers have felt that this exclusion amounted to a rigorous enforcement of the line of cleavage mentioned above. In the present book the difficulty has been met, it is hoped, successfully. Chapters I VII cover practically the whole field of Geometry of 3 IV PREFACE Conies which is required from the average mathematical student who is not reading for Honours. In these chapters no use has been made of involution. In Chapter VI a proof is given that any conic, projectively defined, can be cut from a real right circular cone. The foci are then obtained from the focal spheres and the rest of the focal properties follow. In Chapters 1 VII the only knowledge presupposed is that of Books I VI and Book also of Euclid, XI ; enough Analytical Geometry to understand the use of signs and coordinates and the meaning of the equations of the straight line and circle, so that the data of the drawing examples should be intelligible to the reader. Of plane Trigonometry the meaning of the sine, cosine and tangent and the formula for the area of a triangle (area = hob sin C) are all that is required. Instead of basing the treatment of the subject upon harmonic groups, I have introduced cross-ratio from the very beginning. Although fully appreciating the superior elegance of the former method, insomuch as it enables Projective Geometry to be developed without any appeal to metrical properties, I think it is hardly the one best suited to beginners. For this reason I have used metrical methods whenever their use was obviously indicated, although I hope it will be found that the spirit of the projective methods has been adhered to. The second part of the book is intended for students reading for Honours or desirous of making themselves familiar with the more advanced parts of the subject. It has been impossible, in such a short space, to do more than bring the reader to the threshold of the rich treasure house of Modern Geometry and to give him a glimpse of some of its more characteristic methods. No attempt has been made to give a complete account of the theorems obtained in this domain : those given have been chosen to illustrate the methods. Never- theless it is believed that an Honours student will find there most, if not all, of the fundamental results which he ought to know. The range of knowledge presupposed on the part of the reader is, of course, much wider in the later than in the earlier chapters. Thus the whole theory of imaginaries and of homography has been allowed to rest on an analytical basis. This should present no difficulty, for, by the time the reader reaches these chapters, he will almost certainly Lc've acquired sufficient knowledge of Analytical Plane and Solid Geometry to make his progress easy. On the other hand a purely geometrical development of imaginaries would have been too long and laborious for inclusion. But it will be found that those results which depend on analytical considerations are in every case broad generaliza- PREFACE V tions, such as those relating to the properties of conjugate imaginaries, to the operations which lead to nomographic relations and to the number of points in which curves and surfaces of given degree and order inter- sect. I have carefully abstained from using analysis to prove particular theorems. With regard to homography the method of one-one correspondence has been made fundamental. It is true that discrimination has to be used in applying the principle, but this may be said of almost any principle : and a student soon gets to know when a one-one corre- spondence geometrically given is really algebraic. The notion of nomographic involutions, which appears to be a powerful instrument, has been introduced in Chapter XI. Finally Chapters XIII and XIV deal with geometry of space. Many properties of cones of the second order, of sphero-conics, and of quadrics come most easily from purely geometrical considerations : and it seems a pity that the methods of Pure Geometry are not more frequently employed at this stage. In the preparation of the book the classical treatises of Cremona and Ileye and a more recent but very concise and instructive ex- position of the subject by M. Ernest Duporcq have been chiefly consulted. I have ventured to make certain changes in the recognized nomen- clature. Thus what is called by Mr Russell in his Treatise of Pure Geometry the axis and pole of homography I call the cross-axis and cross-centre, as the name seems to bring more vividly before the mind the fundamental property of the thing defined. For a similar reason I have used the term incident to denote two forms such that the elements of one lie in the corresponding elements of the other. The term perspective, which is employed by German writers in this connexion, appears misleading, since it would not then apply to what are universally known as perspective ranges and pencils. The examples are taken mostly from exercises set to classes at University College, London, and from College and University Examina- tion papers. For permission to use these my thanks are due to the Principal of the University of London and to the Provost of University College, London. The examples contain also many theorems which it has not been found possible to include in the text. A special feature of those on the first seven chapters is that they are divided into two are theoretical those marked are sets. Those marked (A) ; (B) drawing exercises. My own experience as a teacher leads me to believe that such actual drawing is of immense value in assisting beginners to understand the subject, as well as intrinsically useful in practice. VI PREFACE Considerable stress has therefore been laid upon drawing-board con- structions. I wish to express my deep sense of obligation to my friend and colleague, Mr J. H. Dibb, B.iSc, of University College, London, for the help he has given me both before and during the passage of the book through the Press. To Mr H. J. Harris, B.A., I also owe most hearty thanks for some valuable criticisms. L. N. G. FILON. Univeksity of London, University College. September, 1908. CONTENTS CHAP. PAGE I. Projection 1 II. Cross-Ratio; Projective Ranges and Pencils . 24 45 III. Projective Properties of the Conic .... IV. Pole and Polar 57 V. Non-Focal Properties of the Conic .... 76 VI. Focal Properties of the Conic 98 VII. Self-corresponding Elements 118 VIII. Imaginaries and Homography 126 IX. Transformation of Plane Figures .... 145 X. Involution 151 XI. The Homographic Plane Forms of the Second Order 173 XII. Systems of Conies 192 XIII. The Cone and Sphere 217 XIV. Quadrics 228 Index 245 PROJECTIVE GEOMETRY CHAPTER I. PROJECTION. 1. Corresponding figures. Geometrical properties may- be obtained by correspondence or transformation. Certain re- lations are assumed, which introduce a correspondence between the elements of one figure (namely its points, lines and planes) and the elements of another figure. The two figures are then said to correspond or to be transformable into one another. To any property of a set of elements of one figure corresponds a property of the corresponding set of elements of the corresponding, or transformed, figure. When the correspondence is such that to an element of either figure corresponds one element and one only of the other, the correspondence is said to be one-one. The study of the relations between such figures, when the correspondence is of a special type to be explained in Art. 3, constitutes what is called Projective Geometry. 2. Notation. Points will be denoted by Roman capitals A,B,C,...