§ 2. Measurement of lines. A line may be regarded as traced by the movement of a point. Thus we may A FIG. 1. regard the straight line AD D (Fig. 1) as having been traced by the movement of a point from A, the beginning of the line, to D, the end of the line. Straight lines may be added by placing the beginning of the second upon the end of the first (so as to form a continuous straight line), the beginning of the third upon the end of the second, and so on. The straight line contained between the beginning of the first and the end of the last is a line equal to the sum of the lines to be added. Thus (Fig. 1) AD = AB+ BC + CD. § 3. a. There are many problems involving straight lines, in which it is neither necessary nor profitable to take into consideration any thing except the lengths of .these lines. There are, on the other hand, many cases where it will be useful to take into account direction as well as length. b. If we define any arbitrary direction to be positive, then the opposite direction will be negative. For, if AB is any line, -AB can only be that line which added to AB, by the method of § 2, produces 0; and that is the same line as AB taken in the opposite direction. The length of a straight line has no algebraic sign: it is always the same, whatever be the position or direction of the line. c. The signor prefixed to a number or letter denoting a line has reference merely to direction. Thus, if a is positive, the linea is a line whose direction is positive, and whose length is a units; and the line is a line whose direction is negative, and whose length is, a as before, a units. But, if a is negative, a is a line whose direction is negative, and a is a line of the same length, whose direction is positive. The terms "positive" and "negative" are used merely to enable us to distinguish between two opposite directions; and we can in any figure assume any directions (not opposite) to be positive, but, having once assumed certain directions to be positive, the opposite directions must be negative. (v., however, § 20, d.) d. It is possible to represent a straight line completely by an expression involving the points at its extremities. Thus AC (Fig. 2) is an expression for the straight A line which joins the points FIG. 2. A and c, whose length is the distance between the points, and whose direction is the direction from A, the point indicated by the first letter, to c, the point indicated by the second letter. AC denotes a line having the same length as the above, but an opposite direction, but this latter line is the line CA. .. CA — AC, or AC CA=0. The truth of this equation is evident if, as in § 2, we regard a line as traced by the movement of a point. For if a point moves from A to c, and then from c back to A, it is evident that its final distance from the startingpoint will be nothing. Similarly in each of the figures 3, 4, and 5, we have, This principle may be extended to any number of points in the same straight line. § 4. Measurement of angles.-A right angle is divided into 90 equal parts called degrees; a degree is subdivided into 60 minutes; and a minute, into 60 seconds. The degree, minute, and second are distinguished by the characters / // respectively. A degree of are is of the circumference of the circle to which the arc belongs. It evidently has different lengths for different circles. The degree of arc is subdivided in the same manner as the degree of angle. the circumference, or the arc intercepted by the sides of a right angle whose vertex is at the centre, is an arc of 90°, and is called a quadrant. The same term is applied to the sector bounded by two radii at right angles to each other. The four divisions of a plane separated by two lines perpendicular to each other are also occasionally called quadrants. Central angles contain the same number of degrees as their intercepted arcs. § 5. The complement of an angle or arc is the remainder obtained by subtracting the angle or are from 90°. The supplement of an angle or arc is the remainder obtained by subtracting the angle or are from 180°. Thus the complement of 25° is 65°; the supplement of 25° is 155°. Two angles or arcs are therefore complements of each other when their sum is 90°; and they are supplements of each other when their sum is 180°. According to these definitions, the complement of an angle or arc that exceeds 90° is negative. Thus the complement of 100° is 90° 100° = · 10°. In like manner the supplement of 200° is 180° - 200° —— 20°. § 6. Circular measure of an angle. As has been indicated above, an angle may be measured in degrees and subdivisions of a degree. There is, however, another method of measurement in use, called the Circular Method, by which an angle is measured by the ratio of its arc in the circle described from its vertex as a centre to the radius of that circle. To show that this method is legitimate, it must be proved that any two angles are proportional to the ratios by which we propose to measure them. Now, in the same circle, or equal circles, angles at the centre are proportional to their intercepted arcs. Therefore (Fig. 6), $ 7. COROLLARY. Any quantity can be put equal to its measure. We can therefore put 0- Now when the radius (r) is unity, we have ✪ r s; therefore the circular measure of an angle is equal to the length of the intercepted arc in a circle whose radius is unity. $1 § 8. Angular unit of circular measure. As in § 7 we can put have 0 = - 1. = Making 8r we shall Therefore the angular unit of circular measure is that angle whose intercepted arc is equal to the radius. § 9. To change from circular measure to degree measure, and the converse. We will first find how many degrees there are in the angular unit of circular measure; i.e., we will reduce to degrees when s = r. The ratio of circumference to diameter is constant for all circles, and is approximately equal to 3.1415927, which is represented by the Greek letter (v. Plane Geometry). Therefore letting C, D, and R, represent circumference, diameter, and radius respectively, we = ̃ ; .'. С = D; but D = 2 R ; .'. C = arc 360° = 2 π R. In the same circle, or equal circles, central angles are proportional to their intercepted arcs : C have D angle angle 360° arc C R are S S (when S⇒R) 2π Ρ 2 π R 360° .. angular unit= 57°.2957795. Conversely, the units of degree-measure have the following values in circular measure; and these values are also the lengths of the corresponding arcs of the unit cir |