The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. This book was first published in under the title A Course of Geometry for Colleges and Universities. At the time it appeared the author was part of a group of eminent geometers who worked on the College Geometry Project. This was a program funded by the National Science Foundation with the goal of producing short films for classroom use on specific topics in geometry. Twelve films were produced.
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By Dan Pedoe. This book is based on a course given for the past few years at the University of Minnesota to junior and senior students, to first-year graduate students and to a Year Academic Institute of College teachers of geometry who had returned to the University for a year to learn more geometry.
The main purpose of the course was to increase geometrical, and therefore mathematical understanding, and to help students to enjoy geometry. This is also the purpose of my book. I believe, and in this I am supported by many reputable mathematicians, that a reasonable knowledge and appreciation of the scope and methods of elementary geometry is desirable for those who wish to study and to relish more abstract and advanced notions.
I include projective geometry of two and three dimensions, with some geometry of n dimensions in my definition of elementary geometry. The full scope of my definition can be initially realized by a look at the Table of Contents.
In this book I discuss the algebraic methods available for a study of elementary geometry. I have tried not to be tedious, and on the whole only those theorems of elementary geometry are studied which are of significance for further study. Whenever possible, pointers to this significance are indicated, in the text or in the Exercises, of which there are more than The algebraic approach, although it is the main thread of my discourse, is never over-stressed.
If a theorem appears to be more attractive with a geometric proof, this is also given. In any case, geometric insights are stressed all through. That is, some acquaintance with the ideas of a vector space is assumed, including some theorems on the solution of linear homogeneous equations, and also some matrix theory. But in other algebraic matters the book is self-contained, and there is a Preliminary Chapter 0 with a summary of results the student will need.
The chapters of this book need not necessarily be taken in strict sequence. There is ample material for a year's course, or selections can be made which can be covered in one, two or three quarters. The second half of the book, which deals with the projective space of 2, 3 and n dimensions can be used independently of the first part. Chapter I, which need not be taken first, shows what kinds of theorem in real Euclidean and affine geometry can be proved by using vectors.
The student who has only come across vectors in a few sections of a book on the calculus may be surprised at the scope of the theorems which are amenable to the algebra of vectors. Later in the chapter the exterior algebra of Grassmann is introduced, and developed a little, and the power of the methods then available is demonstrated. Chapters II and III contain those parts of the theory of circles which are of importance for later developments in geometry. A fairly detailed study of coaxal systems is made.
After all, these are the simplest linear systems of curves, after linear systems of lines, and in Chapter IV coaxal systems of circles are mapped onto lines in space of three dimensions, and we therefore need to know something about the systems which have such a simple map. Inversion is also studied, not only for its theoretical importance, but because it never fails to fascinate the student.
The idea that geometry is an attractive study is not overlooked in this book. Chapter IV discusses the representation of circles by points in E 3, real Euclidean space of three dimensions, and here the interplay between theorems in space of different dimensions two and three in this particular case first becomes clear. The ability to regard theorems from different points of view is probably the hall-mark of a geometer, and in this chapter a student may receive his first training in this desirable art.
At the end of the chapter an algebra of circles is introduced. This is a further application of the ideas of Grassmann, and although the use made of the algebra is not new, the explanation is new and, it is hoped, no longer almost metaphysical.
As in the preceding chapters, there are numbers of Exercises, not completely trivial, on which the student can test his understanding of the text. Nobody has, as yet, discovered a better way of learning mathematics. Chapter V deals with mappings of the Euclidean plane, especially isometries. These mappings have already appeared in earlier chapters, but are now considered from the point of view of the algebra of complex numbers. This algebra has many advantages, and some disadvantages, but it is important for the student to become familiar with the field of complex numbers.
Wherever proofs using complex numbers are not completely satisfactory, aesthetically speaking, an alternative proof using Euclidean geometry is indicated.
Some group theory is used in this chapter. Elementary mappings of the Euclidean plane have, of course, been an important part of elementary geometry for many years.
Recently they have become important in the foundations of geometry. There is not too much development here, since once the student is convinced that Euclid's parallel axiom can be dispensed with, the rest is not necessary for our purpose, and can be found in other texts. The Appendix lists the Propositions of Euclid, as a useful reference. Chapter VII introduces projective geometry, the treatment being algebraic. Examples are given in two and three dimensions, some important constructs appearing.
Chapter VIII considers the projective geometry of n dimensions, but only essential topics are discussed. The main theorems of projective geometry are derived, and applied to geometric proofs of theorems treated algebraically in earlier chapters.
At the end of the chapter the geometrical interpretation of the Jordan normal form for a projective transformation of Sn is discussed. The algebraic derivation of the normal form must be considered to require more algebra than the readers of this book are expected to have, and so stress is laid on the geometrical meaning of the form when it is presented.
Chapter IX uses the material of the preceding chapters to develop the projective theory of conics and quadrics. This is a powerful and attractive theory, and there is constant interplay between theorems in two and three dimensions. Stereographic projection is one of the methods developed to show this interplay. Once the main theorems are established, the proofs and the thinking are geometrical.
Twisted cubic curves on a quadric are discussed, the curve of intersection of two quadrics is investigated, the theorem on eight associated points is proved, and applications made to theorems encountered much earlier, proved by different methods. The chapter ends with a discussion of oriented circles, and proves some attractive theorems which hold only for oriented circles.
The Appendix lists the Propositions of Euclid and gives a brief account of the Grassmann-Pluecker coordinates of lines in projective space of three dimensions. These coordinates occur in the text, but are not used in the development. There is also a pulling together of various threads of the discourse, with an indication of developments which are beyond the scope of this book. Bibliography and References list some of the books which have influenced me during the time spent in composing the lectures on which this book is based, and details are also given of books and articles which the student may wish to consult for further information.
Bold-face numerals refer to the Bibliography, p. I wish to thank Prof. Erwin Engeler for drawing my attention to Henri Lebesgue's comments, printed as a Frontispiece to this volume , on those he regarded as mathematical vandals. Without trying to explain why the teaching of geometry has slumped so badly in the United States during the past thirty years, and far more than in any other country, it is heartening to notice that there are signs of a revival, and I hope that my book may help in this.
I also wish to thank Mr. Wolfgang Nauck, who helped me with some of the drawings, and Prof. Wunderlich, who used the resources of his Institut fuer Geometrie in the Technische Hochschule of Vienna to supply other drawings.
Many of the Sections into which this book is divided contain at most one Theorem. If this is the case, the reference to the Theorem is given by the number of the Section containing the Theorem. In this chapter we introduce some of the notions which the reader will need when he begins his study of the main part of this text. We go into rather more detail in some cases than in others, our aim being to proceed fairly informally at first, and to introduce more precise notions as we proceed, and where necessary.
Some ideas will therefore be discussed more than once, but if the reader is already familiar with them, he will turn the pages until he comes across something new. The idea of a group, for example, is needed at the beginning of our work, but we do not go into more detail until Chapter V, when we need to know more about groups. Of course, we could assume that the reader is already familiar with all the algebraic concepts which we wish to use, or give references to another book, but we prefer to make this book fairly self-contained, and have inserted sections on algebraic topics where teaching experience has shown that the student needs to be reminded of these topics.
This manner of presentation may be less elegant, but we hope it will make this book more useful than a completely stream-lined presentation might be. This is the familiar plane of ordinary geometry, and we shall investigate it algebraically, using the real number system. The properties of the real numbers which we use are a that they form a field, which means that the ordinary processes of arithmetic are possible with real numbers, and b that they are ordered.
The points of our Euclidean plane are given by ordered pairs of real numbers x 1, x 2 , these being coordinates with respect to a pair of orthogonal cartesian axes OX 1, OX 2 Fig. We shall use upper-case letters P , Q , A , B ,.
We can set up a coordinate system on any given line m , so that if the point A has coordinate xA , the point B has coordinate xB , then.
Note that this definition implies that B lies on the line AC Fig. A half-line is sometimes called a ray. If A , B and C are on the same line the triangle is said to be degenerate. A non-degenerate triangle is a proper triangle. If ABC is a proper triangle, the sum of the lengths of any two sides is always greater than the length of the third side, and so we have. This inequality is often referred to as the triangle inequality.
In fact, we can say that if A , B , C are any three points ,. The point O is called the vertex of the angle. This can be done, for example, by drawing a circle, center O , and putting the half-lines through O into correspondence with the points in which half-lines through O intersect the circle Fig.
We call such angles sensed or oriented angles. We can also call them signed angles. Similar triangles occur frequently in our work. A point O on a given line m is the vertex of two half-lines. We may assign positive coordinates to the points of one half-line, taking O as origin, and negative coordinates to points of the other half-line.
If A , B are any two points on m , we define the sensed , or directed segment AB as. This ratio is independent of the position of the origin of coordinates O on the line, and independent also of which of the two half-lines vertex O is chosen to have positive coordinates. If P lies between A and B the division is said to be internal ; otherwise the division is said to be external.
If A and B are distinct and P coincides with A. If A and B are distinct and P coincides with B. As P moves along the line in either direction , the position-ratio r tends towards the value —1.
This suggests that the line has a unique point at infinity. Of course points at infinity have no place in Euclidean geometry, but it is very useful to extend our ideas of points so as to embrace this concept.
Geometry: A Comprehensive Course
By Dan Pedoe. This book is based on a course given for the past few years at the University of Minnesota to junior and senior students, to first-year graduate students and to a Year Academic Institute of College teachers of geometry who had returned to the University for a year to learn more geometry. The main purpose of the course was to increase geometrical, and therefore mathematical understanding, and to help students to enjoy geometry. This is also the purpose of my book.
Daniel Pedoe Geometry, A Comprehensive Course 1988
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