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AN INTRODUCTION TO PROJECTIVE GEOMETRY BY L. N. G. FILON, C. B. E., M. A., D. Sc., F. R. S. FELLOW OF AND PROFESSOR OF APPLIED MATHEMATICS AT UNIVERSITY COLLEGE, LONDON LATE VICE-CHANCELLOR OF THE UNIVERSITY OF LONDON. LONDON EDWARD ARNOLD CO. All rights reserved F F FIRST PUBLISHED IN 1908 SECOND EDITION . 1916 THIRD EDITION . 1921 FOURTH EDITION . 1935 MADE AND PKINTED IN GREAT BEITAIN BY WILLIAM CLOWES AND SONS, LIMITED, LONDON AND BEOCLES. PREFACE TO THE FOURTH EDITION DURING the quarter of a century which has elapsed since the first edition of this book was published, Projective Geometry has found new practical applications. In particular, the uses of photography in Air Surveying, as well as in Astronomy, involve the principles and methods of projection and it appears probable that, in both cases, the advantages of graphical constructions will be increasingly appreciated. Meanwhile the older applications to Cartography, Geometrical Optics and Engineering Drawing have lost nothing of their importance. No apology is therefore needed for the insistence on drawing board constructions, which was a feature of the earlier editions. Indeed this has been emphasized, in the present edition, by the addition, at the end of all the later chapters dealing with the geometry of the plane, of a set of drawing examples marked B, which had previously been restricted to the first seven chapters. From the purely didactic standpoint, actual drawing is even more valuable to clear up difficulties in the more advanced work than it is in the elementary parts of the subject. It still remains true, however, that the chief interest of project ve methods is for the pure mathematician, for whom they provide an instrument of remarkable range and power. The general scheme of the original edition has remained, save in one important respect, substantially unaltered. In particular I have not modified the lines on which the subject is introduced in Chapter I, though I have tried. to remove certain obscurities and have kept the graphical constructions concentrated towards the end of the chapter, so that they may be omitted by those who attach no importance to such constructions. I am aware that this will not satisfy certain critics, but I could not have met their objections without abandoning a conception of the genesis of the subject which I still believe to be the right one. The chief alteration involving the geometry of the plane has been a rearrangement of order, which brings in Involution before, VI PREFACE TO THE FOURTH EDITION instead of after, the discussion of foci and focal properties of the conic. This change was always desirable, for the introduction of foci by means of the focal spheres was never really the natural approach and had the defect of masking the true significance of foci from the projective point of view. The ban on the early introduction of Involution, which used to be imposed by certain University syllabuses, has now been generally abandoned, and the treatment of the whole subject gains thereby in clearness and coherence. The above modification of plan has necessitated a good many consequential alterations. Chapter VI now deals with ranges and pencils of the second order and self-corresponding elements, and this naturally leads to a discussion of Involution in Chapter VII, followed by the focal properties of the conic in Chapter VIII. Up to this point the whole treatment, although capable of interpretation in a wider sense, is based upon real elements and constructions actually possible on the drawing-board, as in my view this is essential to give confidence to the beginner. Chapter IX then introduces imaginary elements and the circular points at infinity. For this, appeal is made, as in the original edition, to algebraic considerations
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An Introduction To Projective Geometry
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