Q.E.D. circles. quod erat demonstrandum (which was to be proved). quod erat faciendum (which was to be done). GEOMETRY INTRODUCTION 1. A physical body, such as a block of wood or iron, occupies a definite portion of space. The boundary which separates a body from space is called the surface of the body. If the material composing such a body could be removed without altering the surface, there would remain a portion of space called a geometrical solid or a solid. A B E H K FIG. 1. DEFINITIONS 2. A solid is a limited portion of space. It has three dimensions, length, breadth, and thickness. 3. Surfaces are the boundaries of solids, as ABED or ADHC (Fig. 1). They have two dimensions, length and breadth. 4. Lines are the boundaries of surfaces, as AB, AD (Fig. 1), etc., and have but one dimension, length. 5. Points are the boundaries or extremities of lines, and are without dimension, having position only. Surfaces may be conceived as existing independent of the solids whose boundaries they form. In like manner, lines and points may exist independently in space. FIG. 2. 6. Solids, surfaces, lines, and points are called geometrical magnitudes. 7. Geometry is the science that treats of the properties of geometrical magnitudes. A C. 8. A straight line is a line that has the same direction throughout its length, as AB. The word "line" is frequently Eused to denote a straight line. -B F FIG. 3. 9. A curved line changes its direction at every point, as CD. 10. A broken line is composed of several successive straight lines, as EF. 11. A plane surface or a plane is such a surface that a straight line joining any two points in the surface lies entirely in the surface. 12. A geometrical figure is any combination of solids, surfaces, lines, or points, as M or N. 13. A plane figure is a geometrical figure, all of whose points lie in the same plane, as N. 14. Plane Geometry treats of plane figures only. M N FIG. 4. 15. Solid Geometry treats of figures which are not plane. 16. When one figure can be placed upon another so that each point of the one lies upon the corresponding point of the other, the figures are said to coincide. 17. Equal magnitudes are those that can be made to coincide. 18. Proof by superposition is the method of proving the equality of two figures by means of coincidence. QUESTIONS 1. What is the path of a moving point? 2. What geometrical magnitude is, in general, generated by a moving line? by a moving surface? 3. What kind of a surface is represented by the walls of a room? LINES 19. From the definition of a straight line it appears that (a) two straight lines of unlimited length, coinciding in part, coincide throughout, (b) two straight lines can intersect only once, and (c) two points determine a straight line. The expression, straight line, is used to denote both an unlimited straight line and a part of such a line. 20. A line of definite length is also called a segment and is represented by a line whose ends are marked, as A and B (Fig. 5). C 21. The length of this line is called the distance from A to B. A line whose ends are not marked represents a line of indefinite length, as CD. HB D FIG. 5. 22. The direction of the line AB means the direction from A toward B; of BA, from B toward A. D B FIG. 6. 23. To produce the line AB means to prolong it through B; to produce BA means to prolong it through A. 24. To bisect a geometrical magnitude means to divide it into two equal parts. Thus, AC is bisected if AD equals A DC. D FIG. 7. 25. Two points, A and B, are equidistant from a third point, C, if CA CB. = ANGLES 26. An angle is the inclination of two intersecting lines to each other. B C FIG. 8. 27. The vertex is the point of intersection, and the lines are the sides of the angle. The lines BA and BC, meeting in B, form the angle ABC. B is the vertex, and BA and BC are the sides. |