GEOMETRY PRELIMINARY DEFINITIONS 1. Every material object occupies a limited portion of space and is called a Physical Solid or Body. 2. The portion of space occupied by a physical solid is identical in form and in extent with that solid, and is called a Geometrical Solid. In this work only geometrical solids are considered, and for brevity they are called simply solids. 3. Any limited portion of space is called a Solid. A solid has three dimensions, length, breadth, and thickness. The drawing in the margin is represented as having three dimensions. FIG. 1. 4. The limit of a solid, or the boundary which separates it from all surrounding space, is called a Surface. A surface has only two dimensions, length and breadth. A page of a book is a surface, but a leaf of a book is a solid. 5. The limit or boundary of a surface is called a Line. A line has only one dimension, length. It has neither breadth nor thickness. The edges of a cube are lines. 6. The limits, or extremities, of a line are called Points. A point has position only. It has neither length, breadth, nor thickness. The dots and lines made by a pencil or crayon are not geometrical points and lines, but are convenient representations of them. 7. Lines, surfaces, and solids are called Geometrical Magnitudes, or simply Magnitudes. 8. A line may be conceived of as generated by a point in motion. Hence a line may be considered as independent of a surface, and it may be of unlimited extent. A surface may be conceived of as generated by a line in motion. Hence a surface may be considered as independent of a solid, and it may be of unlimited extent. A solid may be conceived of as generated by a surface in motion. Hence a solid may be considered as independent of a material object. LINES AND SURFACES 9. 1. Select two points upon your paper and draw several lines connecting them. a. Which is the shortest line you have drawn? If this line is not the shortest that can be drawn between the points, what kind of a line is the shortest? b. What other kinds of lines have you drawn besides a straight line? 2. When a carpenter places a straightedge upon a board and moves it about over the surface, what is he endeavoring to determine regarding the surface? 3. If the straightedge does not touch every point of the surface of the board to which it is applied, what has been discovered about the surface? 4. How does he know whether or not the surface is an even or a plane surface? 5. If any two points on the surface of a ball or sphere are joined by a straight line, where does the line pass? 6. How much of the surface of a perfect sphere is a plane surface? 10. A line which has the same direction throughout its whole extent is called a Straight Line. A straight line is also called a Right Line, or simply a Line. In this work the term "line" always means a straight line. 11. A line no part of which is straight is called a Curved Line. Consequently, a curved line changes its direction at every point. 12. A line made up of several straight lines which have different directions is called a Broken Line. 13. A line made up of straight and, curved lines is called a Mixed Line. Any portion of a line may be called a segment of that line. FIG. 2. FIG. 3. FIG. 4. 14. A surface such that a straight line joining any two of its points lies wholly in the surface is called a Plane Surface, or a Plane. 15. A surface, no part of which is plane, is called a Curved Surface. 16. Any combination of points, lines, surfaces, or solids is called a Geometrical Figure. A geometrical figure is ideal, but it can be represented to the eye by drawings or objects. 17. A figure formed by points and lines in the same plane is called a Plane Figure. 18. A figure formed by straight or right lines only is called a Rectilinear Figure. 19. The science which treats of points, lines, surfaces, and solids, and of the construction and measurement of geometrical figures, is called Geometry. 20. That portion of geometry which treats of plane figures is called Plane Geometry. |