PLANE GEOMETRY. INTRODUCTION. Section 1. — Elementary Definitions. THERE are certain terms which, although quite familiar to the student, are used in Geometry so commonly and with such exactness as to require careful definition or explanation at once. Like all simple terms, such as number, space, time, etc., they are difficult to define, but the explanations given will lead the student to a reasonable understanding of them. The terms are solid, surface, line, angle (with various kinds of each), and point. Space is divisible. Any limited portion of space is called a solid. That which separates one part of space from an adjoining part is called a surface. Every surface is divisible. That which separates one part of a surface from an adjoining part is called a line. Every line is divisible. That which B separates one part of a line from an adjoining part is called a point. A point is not divisible. Thus in the figure the surface of the block separates the space occupied by the block from all the rest of space. This surface is divisible in many ways; for example, it is divided into two parts by the line passing from A through B and C and back to A. This line is divisible in many ways; for example, it is separated into three parts by the points A, B, C. In the case of a line that returns into itself, a closed line, like the one just mentioned, two points are necessary to completely separate one part from the other. In geometry no attention is given to the substance of which the solid is composed. It may be water, or iron, or air, or wood, or it may Indeed, geometry considers only the space occupied by This space is called a geometric solid, or simply a solid, be a vacuum. the substance. while the substance is called a physical solid. It is impossible to draw mechanically a geometric line. A chalk mark, a thread, a thin wire, an ink mark, are all very thin physical solids used to represent lines; for this purpose they are very useful. So, too, a dot may be used to represent a point, and a sheet of paper may be used to represent a surface, although each is really a physical solid. A further explanation may be given of the point, line, surface, and solid. The point is the simplest geometric idea; it has position, but not magnitude. A moving point describes a line. This may be represented by a pencil point moving on a piece of paper. A moving line describes, in general, a surface. This may be represented by a crayon lying flat against the blackboard, and moving sidewise. How may a line move so as not to describe a surface ? A moving surface describes, in general, a solid. Thus the surface of a glass of water, as it moves upward, may be said to describe a solid. How may a surface move so as not to describe a solid ? Through two points any number of lines may be imagined A line is called a straight line if it possesses the following quality Through two points one straight line, and only one, can pass. In the figure, s represents a straight line, for only one such line can pass through P1 and P2. The expression straight line is used to mean both an unlimited straight line and a portion of such a line. The context generally determines which meaning is to be assigned. In case of doubt, line-segment, or merely segment, is used to mean a limited straight line. The word line, used alone, is to be understood to refer to a straight line, although for emphasis the word straight is sometimes used to modify it. From the definition it appears that two straight lines through the same two points coincide, or that Two straight lines can intersect but once. This may also be expressed by saying that Two points determine a straight line; and also that Two straight lines, in general, determine a point. As has been seen, a point is usually named by some capital letter. A segment is usually named by naming its end points, or by a single small letter. In the annexed figure, AB, AC, BC, A and o are marked off. B If three points, A, B, C, are taken in order on a line, then AC is called the sum of AB and BC, and AB is called the difference between AC and BC. If AB equals BC, then AC is said to be bisected at B, and B is called the mid-point of AC. If a segment is drawn out to greater length it is said to be produced. To produce AB means to extend it through B, toward C, in the above figure. To produce BA means to extend it through A, away from B and C. A line not straight, but made up of straight lines, is called a broken line. Through three points, not in a straight line, any number of surfaces may be imagined to pass. For example, through the points A, B, C the surfaces P and S may be imagined to P pass. B |