SUGGESTIONS TO TEACHERS It is suggested to teachers that the introductory articles be read and discussed in class in an informal way, with the aim of drawing out and clarifying those ideas of spacerelations which the students may already possess. Some of the introductory matter can be passed over lightly on first reading, and returned to when necessary. Teachers may exercise their discretion with regard to articles in small print throughout the book. For a shorter course, any of the following groups of articles may be omitted without breaking the continuity of the subject: Book I. 180-186, 195-213, 232-247. Book III. 141-198. Book IV. 10. Most of the exercises that are given in immediate connection with the propositions should be solved by the student; but only a few of those placed at the end of sections need be taken on a first reading. They are all carefully graded, and many suggestions are given. The author will be glad to hear from any person who may meet with any error or difficulty. As some teachers may wish to use the Socratic or heuristic methods of instruction in certain parts of the work, the arrangement and development of the topics are such as to lend themselves easily to these valuable pedagogical methods, without interfering with the more formal presentation that is appropriate to a course in demonstrative geometry. The actual details of any such method are, however, left to individual discretion, as the skillful teacher has usually no difficulty in reconciling the claims of pedagogy and sound reasoning. viii ELEMENTARY GEOMETRY INTRODUCTION THE FOUR FUNDAMENTAL SPACE CONCEPTS 1. Geometry is that branch of mathematical science which treats of the properties of space. The space in which we live is divisible into parts. Every portion of matter occupies a part of space. The portion of space occupied by a body, considered separately from the matter which it contains, may be regarded as existing unchanged when the body moves into another portion of space. 2. Any portion of space capable of being occupied by a physical solid is called a geometrical solid, or simply a solid. 3. The common boundary of two adjoining solids, or of a solid and the surrounding space, is not a solid; it is a second kind of space element, called a surface. 4. Any surface is likewise divisible into parts; and the common boundary of two adjoining parts of a surface is not a surface; it is a third kind of space element, called a line. 5. Again, any line is divisible into parts; and the common extremity of two adjoining parts of a line is a fourth kind of space element, called a point. A point is not divisible into parts; hence, the point is the simplest space element. 6. A fine tracing point, or a dot on a sheet of paper, gives an approximate representation of the ideal geometric point. 1 |