Geometry - Plane and Solid. There are two main reasons for the study of geometry at high school level. One of them is found in the material itself, the properties of figures in two and three dimensions. This material is both interesting and helpful in other areas of mathematics. In addition, it is sufficiently related to the development of our civilization that acquaintance with it is a reasonable requirement of an educated person.
The second reason is that the subject provides the best setting, at the elementary level, in which to show how a large body of results may be deduced from a comparatively small number of assumptions. For young students geometry has the advantage that the results are varied, in some cases unexpected, and can be represented in diagrams. Also, in a geometric problem there is often scope for ingenuity, intuition, and imagination.
The major purposes of this book, listed below, are such as to give full weight to each of the above reasons.
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To provide a thorough knowledge of the main topics of plane and solid geometry.
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To clarify the parts played in mathematics by undefined terms, definitions, and statements which are accepted without proof.
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To emphasize the meaning of deductive reasoning and of a mathematical proof.
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To give training in presenting an argument in clear, orderly fashion.
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To introduce the elements of coordinate geometry and the geometric aspect of trigonometry.
Every opportunity is taken throughout the book to use and maintain algebraic skills. The work with coordinates in Chapter 16 gives wide opportunity for the use of algebraic expressions and the solutions of equations. It also serves as a review of many of the results established for triangles and quadrilaterals in the earlier chapters.
Arrangement of Material
Chapters 1 through 15 provide a thorough study of rectilinear figures in two and three dimensions. Chapter 16, which introduces coordinates, provides an opportune break in the development of a succession of new theorems.
When circles are added to the figures that are to be investigated many new properties are discovered and the complexity of problems is increased. It is therefore desirable that Chapters 1-15 be thoroughly mastered before proceeding further. This is one reason for including the sets of review exercises at the end of Chapter 15. If they are not needed immediately, they can be used at intervals during the later work.
The study of similarity before circles enables us to get more quickly to the Pythagorean Theorem, and opens up a wide variety of problems, especially problems of a numerical nature.
The coordinate geometry precedes the formal approach to loci because experience has shown that the concept of a locus is more easily grasped when presented in a numerical context.
Flexibility
The book contains a great number of problems ranging from simple, one-step exercises to some which will provide opportunity for extended investigation by even the best students.
Sets of exercises labeled [A] are such that they should be attempted by all students, though toward the end of such a set there may be some of a more demanding nature. Topics and exercises labeled [B] are more difficult and may be omitted without interfering with the development of the course.
There is more material in the book than an average class can absorb. For such a class Chapters 1-16, 18, 20 with [B] exercises omitted would be a satisfying course. In such a class also it would be reasonable to postulate some of the more difficult theorems, for example, Theorem 4 and Theorem 8.
A good college preparatory section can cover all the material, using [B] topics and exercises with discretion. For such a class, Chapter 23 and topics from Chapter 24 might serve as sections for independent study.
The very best students will be stimulated by some of the [B] exercises, and by the challenging problems placed at the end of many of the chapters. It is hoped, too, that the references for additional reading listed at at the end of most chapters will be helpful to both student and teacher.
In general, the book is intended to be one from which the student can learn by his own efforts. This mean that not only is the language suited to the age of the student, but that proofs and discussions are given in sufficient detail to answer most questions. Some "Why's" are included, but only in situations which are quite familiar to the student or where economy of space is a factor.
In order that students may discover many of the results for themselves, exercises which anticipate the proof of a theorem precede the new theorem whenever possible. In the later chapters, proofs of theorems of a straight-forward nature are often left to the student. The aim of the presentation, in other words, has been to make it possible to use a wide variety of teaching procedures.
Above are extracts from the book Geometry (with permission of the publishers).
