PLANE GEOMETRY CHAPTER I FUNDAMENTAL IDEAS 1. Geometry. - Of all the arts and sciences which man has discovered in the endeavor to meet his practical needs, geometry is one of the most ancient. History shows that a knowledge of geometry was used by the Egyptians in land surveying many centuries before the time of Christ. It is said by Herodotus, Greek historian who traveled in ancient Egypt, that because from time to time the overflow of a the Nile washed away the landmarks and some of the land, it became necessary to have the land measured frequently in order to determine the just distribution of taxes. The Egyptians must also have used considerable knowledge of geometry in the construction of the wonderful pyramids, obelisks, and temples. Such rules for practical measurement as the Egyptians knew were the result of centuries of observation and experience. Many Greek scholars traveled in this land of the pyramids and learned the Egyptian art of measurement. The Greeks subsequently discovered many new truths, and arranged all of them into a system or science, which they called geometry, meaning land or earth measurement. As civilization has advanced, geometry has come to have a much wider practical use than merely in land measurement or surveying. It is employed in architecture, designing, building, engineering, astronomy, navigation, the construction and use of measuring instruments and machinery, etc.; in short, it is needed in a vast number of the modern activities of man. 2. Solids, surfaces, lines, and points. - Geometry deals with solids, surfaces, lines, and points, and with combinations of these. It is devoted to the consideration of the properties of these elements and their combinations, and to the application of the truths relating to them in practical work. Any physical object, such as a book, a ball, or a block of stone, occupies space. Geometry is not concerned with the substance of the object, but with the properties of the space or room occupied by it. 1 Thus, the cylindrical object here represented is a physical solid, which can be touched or weighed. If one imagined this solid removed, the empty space formerly occupied by it would be of the size and shape represented by the diagram to the right. It is with the latter that geometry deals. The space occupied by an object is called a geometric solid. :: Every portion of space, or geometric solid, is separated from the adjacent or neighboring space by a surface. A portion of a surface is separated from the remainder of the surface by a line. A. line is separated into two parts by a point. For example, the above geometric solid is separated from the sur rounding space by a surface consisting of two flat parts and one round part. The round part of the surface is separated from each flat part by a line. The chief characteristic of a point is that it has position. If a point were imagined to move, it would describe a line as its path. Similarly, if a line were to move in any way, except along itself, it would describe a surface. And if a surface were to move in any way, except along itself, it would describe a solid. 3. Geometric figures. - Any geometric solid, surface, line, point, or combination of them, is called a geometric figure. Thus, the combinations of lines and points here represented, with which the student became familiar in the study of arithmetic, are geometric figures. Each is composed of one or more lines and points, all of which lie in one flat surface. 4. Plane surfaces. - A flat surface, such as the floor, blackboard, or top of a desk, is called a plane surface, or merely a plane. Plane geometry, to which the following pages are devoted, deals only with those geometric figures which lie in a plane. • A B 5. Representations of points and lines. - Since a point has no substance, it is invisible. It may be represented to the eye by a small dot or cross. It may be named by putting a capital letter near the dot or cross. Thus, we may speak of "point A" or "point B." Since a line has no substance, it is invisible, but may be represented to the eye by some kind of mark. It is not the lines nor points, then, that we see, but the physical representations of them. 6. Kinds of lines. - Three kinds of lines and combinations of them may be drawn in a plane, namely, straight lines, broken lines, and curved lines or curves. A straight line is represented by placing a ruler, or straightedge, upon a plane surface and marking along the edge of the ruler. STRAIGHT LINE BROKEN LINE CURVED LINE A broken line is a line which is not straight, but which is made up of parts all of which are straight. A curved line, or curve, is a line no part of which is straight. The principal curved lines used in elementary geometry are circles and arcs of circles. A circle is a closed curve all points of which are equally distant from a point within, called the center. The distances from all points of a circle to the center are called radii, which, therefore, are equal. Two radii in a straight line form a diameter. An arc of a circle is a portion of a circle. Radius Diameter A CIRCLE Since the time of 7. Drawing and measuring instruments. the ancient Greeks the figures considered in elementary geometry have been only those which may be drawn by the use of two instruments, the straightedge (unmarked ruler), and com passes (also called dividers). The straightedge, as pointed out in § 6, is used for drawing straight lines. The compasses are used for drawing circles and arcs of circles and for measuring and comparing distances. A good substitute for compasses, in case the latter are not accessible, consists of a cord with a loop tied in one end, into which the pencil or crayon may be inserted. The use of the cord in drawing circles or arcs, or in marking off distances, is easily seen. The uses of various other drawing and measuring instruments, such as are employed in practical work of the present day, will be pointed out farther along in the book. THE STRAIGHT LINE 8. Line, ray, and line-segment. - While the mark which represents a straight line must be of limited length, a straight line must be thought of as extending without end in two opposite directions. A straight line is named by writing two capital letters near it, as " line AB"; or by writing one small letter near it, as “line a." Hereafter when the word |