7. Surface. The block shown in § 4 has six flat faces, each of which is called a surface. If the faces are made smooth by polishing, so that when a straight edge is applied to any one of them the straight edge in every part will touch the surface, each face is called a plane surface, or a plane. These surfaces are simply the boundaries of the solid. They have no thickness, even as a colored light shining upon a piece of paper does not make the paper thicker. A board may be planed thinner and thinner, and then sandpapered still thinner, thus coming nearer and nearer to representing what we think of as a geometric plane, but it is always a solid bounded by surfaces. That which has length and breadth without thickness is called a surface. 8. Line. In the solid shown in § 4 we see that two adjacent surfaces intersect in a line. A line is therefore simply the boundary of a surface, and has neither breadth nor thickness. That which has length without breadth or thickness is called a line. A telegraph wire, for example, is not a line. It is a solid. Even a pencil mark has width and a very little thickness, so that it is also a solid. But if we think of a wire as drawn out so that it becomes finer and finer, it comes nearer and nearer to representing what we think of and speak of as a geometrie line. 9. Magnitudes. Solids, surfaces, and lines are called magnitudes. 10. Point. In the solid shown in § 4 we see that when two lines meet they meet in a point. A point is therefore simply the boundary of a line, and has no length, no breadth, and no thickness. That which has only position, without length, breadth, or thickness, is called a point. We may think of the extremity of a line as a point. We may also think of the intersection of two lines as a point, and of the intersection of two surfaces as a line. 11. Representing Points and Geometric Magnitudes. Although we only imagine such geometric magnitudes as lines or planes, we may represent them by pictures. Thus we represent a point by a fine dot, and name it by a letter, as P in this figure. We represent a line by a fine mark, and name A it by letters placed at the ends, as AB. D B We represent a surface by its boundary lines, and name it by letters placed at the corners or in some other convenient way, as ABCD. We represent a solid by the boundary faces or by the lines bounding the faces, as in § 4. 12. Generation of Geometric Magnitudes. We may think of (1) A line as generated by a moving point; For example, as shown in the figure let the surface ABCD move to the position WXYZ. Then (1) A generates the line AW; (2) AB generates the surface AWXB; (3) ABCD generates the solid AY. Y Of course a point will not generate a line by simply turning over, for this is not motion for a point; nor will a line generate a surface by simply sliding along itself; nor will a surface generate a solid by simply sliding upon itself. X A W 13. Geometric Figure. A point, a line, a surface, a solid, or any combination of these, is called a geometric figure. A geometric figure is generally called simply a figure. 14. Geometry. The science of geometric figures is called geometry. Plane geometry treats of figures that lie wholly in the same plane, that is, of plane figures. Solid geometry treats of figures that do not lie wholly in the same plane. 15. Straight Line. A line such that any part placed with its ends on any other part must lie wholly in the line is called a straight line. B For example, AB is a straight line, for if we take, say, a half inch of it, and place it in any way on any other part of AB, but so that its ends lie in AB, then the whole of the half inch of line will lie in AB. This is well shown by using tracing paper. The word line used alone is understood to mean a straight line. Part of a straight line is called a segment of the line. The term segment is applied also to certain other magnitudes. 16. Equality of Lines. Two straight-line segments that can be placed one upon the other so that their extremities coincide are said to be equal. In general, two geometric magnitudes are equal if they can be made to coincide throughout their whole extent. We shall see later that some figures that coincide are said to be congruent. 17. Broken Line. A line made up of two or more different straight lines is called a broken line. For example, CD is a broken line. 18. Rectilinear Figure. A plane figure formed by a broken line is called a rectilinear figure. For example, ABCD is a rectilinear figure. 19. Curve Line. A line no part of which is straight is called a curve line, or simply a curve. For example, EF is a curve line. D B A E 20. Curvilinear Figure. A plane figure formed by a curve line is called a curvilinear figure. For example, O is a curvilinear figure with which we are already familiar. Some curvilinear figures are surfaces bounded by curves and others are the curves themselves. 21. Angle. The opening between two straight lines drawn from the same point is called an angle. Strictly speaking, this is a plane angle. We shall find later that there are angles made by curve lines and angles made by planes. The two lines are called the sides of the angle, and the point of meeting is called the vertex. m B An angle may be read by naming the letters designating the sides, the vertex letter being between the others, as the angle AOB. An angle may also be designated by the vertex letter, as the angle O, or by a small letter within, as the angle m. A curve is often drawn to show the particular angle meant, as in angle m. 22. Size of Angle. The size of an angle depends upon the amount of turning necessary to bring one side into the position of the other. One angle is greater than another angle when the amount of turning is greater. Thus in these compasses the first angle 火 is smaller than the second, which is also smaller than the third. The length of the sides has nothing to do with the size of the angle. 23. Equality of Angles. Two angles that can be placed one upon the other so that their vertices coincide and the sides of one lie along the sides of the other are said to be equal. For example, the angles AOB and A'O'B′ (read "A prime, O prime, B prime") are equal. It is well to illustrate this by tracing one on thin paper and placing it upon the other. B A 'B' A 24. Bisector. A point, a line, or a plane that divides a geometric magnitude into two equal parts is called a bisector of the magnitude. For example, M, the mid-point of the line AB, A is a bisector of the line. M 25. Adjacent Angles. Two angles that have the same vertex and a common side between them are called adjacent angles. For example, the angles AOB and BOC are adjacent angles, and in § 26 the angles AOB and BOC are adjacent angles. 26. Right Angle. When one straight line of meets another straight line and makes the adjacent angles equal, each angle is called a right angle. For example, angles A OB and BOC in this figure. If CO is cut off, angle AOB is still a right angle. IB C B A 27. Perpendicular. A straight line making a right angle with another straight line is said to be perpendicular to it. Thus OB is perpendicular to CA, and CA to OB. OB is also called a perpendicular to CA, and O is called the foot of the perpendicular OB. 28. Triangle. A portion of a plane bounded by three straight lines is called a triangle. The lines AB, BC, and CA are called the sides of the triangle ABC, and the sides taken together form the perimeter. The points A, B, and C are A B the vertices of the triangle, and the angles A, B, and C are the angles of the triangle. The side AB upon which the triangle is supposed to rest is the base of the triangle. Similarly for other plane figures. 29. Circle. A closed curve lying in a plane, and such that all of its points are equally distant from a fixed point in the plane, is called a circle. The length of the circle is called the circumference. The point from which all points on the circle are equally distant is the center. Any portion of a circle is an arc. A straight line from the center to the circle is a radius. A straight line through the center, terminated at each end by the circle, is a diameter. ARC DIAMETER CIRCLE RADIUS Formerly in elementary geometry circle was taken to mean the space inclosed, and the bounding line was called the circumference. Modern usage has conformed to the definition used in higher mathematics. |