Solid Analytical Geometry And Determinants

SOLID ANALYTICAL GEOMETRY AND DETERMINANTS PREFACE A three-hour course in Solid Analytical Geometry is offered for students in the junior year in many of the colleges and universities in this country. Though books, on Plane Analytical Geometry frequently devote some chapters to the geometry of a space of three dimensions, the material covered in these chapters is, with few exceptions, not intended to do more than v prt vide a general introduction to the subject, so as to enable students tb understand the references to it which have to be made in courses on the cll culus. But it rarely goes far enough to acquaint them with the, more interesting and valuable methods of this field. For many years, while teaching this subject at the University of Wisconsin and at Swarthmore College, it has seemed to the author that, in the study of Solid Analytical Geometry, the young student of mathematics can find an excellent opportunity for an introduction to methods and principles which have an important part in various fields of advanced mathematics. Among these are the methods based on the theory of determinants and on the con cept of the rank of a matrix. In more advanced mathematical subjects these theories are developed and used with a great meas ure of generality they find relatively simple application in the subject to which this book is devoted. Unfortunately, there are not readily accessible for use in undergraduate classes treatments of these theories which are on the one hand adequate for the uses to be made of them here and on the other hand not too advanced to be available for an introductory purpose. For these reasons, the first chapter of this book presents an exposition of some of theproperties of determinants and matrices, followed in Chapter II by a treatment of systems of linear Aqua tions. The latter subject is not carried so far as to include the most general case, but, it is hoped, far enough to serve in the later chapters. The repeated applications of the results of the first two chapters which are made in the subsequent work as evidenced by the numerous references to Chapters I and II should indicate their importance. With the basis thus provided it becomes pos iii IV PREFACE sible to deal with the geometrical questions of the later chapters in a way which lends itself readily to extension to problems of a more general character. Thus the discussion of the theory of quadric surfaces in Chapters VII and VIII may be made to serve as an introduction to the theory of quadratic forms in n variables. Having studied these chapters, the reader should be able to proceed to the well-known excellent books, by Bocher and by Dickson, nientioned in the introductory paragraph of Chapter I. To these books the author owes a large debt. The spirit which pervades them has been a guide for him and it would be a source of grati fic, tion if the present book were to lead its readers to more extended sftidy of the subjects treated by these authors. Chapters III to X deal with the loci of equations of the first and second degree in three variables from the point of view of real, metric geometry. Elements at infinity and complex elements are considered as non-existent. This point of view has been taken because, in the authors judgment, a satisfactory treatment of the questions which arise through the inclusion of such elements can only be made after the explicit adoption of adequatebases on which projective geometry and complex geometry can be erected. Since this would involve quite a different orientation than the scope of the present book permits, it was deemed better to proceed on the implicit assumptions of real metric geometry on which the students earlier work in geometry may be supposed to have been founded...