Hence 1 circumference 400 grades =40 000 min. 4 000 000 sec.; which is commonly written Notwithstanding its greater convenience, this system never came into general use, owing to the difficulty of changing all the mathematical tables to correspond with it. 6. Decimals of Degrees or Minutes. Sometimes, instead of seconds, decimals of a minute are used. Both minutes and seconds may be dispensed with and decimals of a degree be used in their place. 7. General Measure of an Angle. The best way of thinking of angular measure is to conceive the side OA of the angle to turn round on until it reaches the position OB. In thus turning, a point A upon it will describe the circular arc which measures the angle AOB. The length of this arc will then be proportional to the amount by which OB turns in passing from OA to OB. The side may pass from OA to OB not only by describing the arc ab, but by moving through a whole revolution plus the arc ab, or through any number of revolutions plus the arc ab. When we consider the angle in the most general way, all these motions will equally measure the angle. Hence we may suppose, indifferently, B in which may be any integer whatever, positive or negative. We may consider this same form to include the negative measure a Mb. For, since ab + aMb = C, we have arc a Mb. By substituting this value in (1) it becomes Angle AOB = (i + 1)C — arc aMb. Since we have i ... 2, — 1, 0, 1, 2, 3, etc., ad infinitum, i+1 may go through the same system of integral values as i. In general, if an angle is no less than a circumference, we may call it, indifferently, Angle of (360-n)° or angle of — n°. The general measure of the angle expressed in the form (1) has its most convenient application in Astronomy. The heavenly bodies perform unceasing revolutions, and thus describe continually increasing angles; but each revolution brings them back to what we may consider the same position relative to the centre of motion. 8. In order to give entire algebraic precision to the measure ɔf an angle, we must suppose a distinction between the side from which we measure and the side to which we measure. In all the preceding examples we have supposed the measure to be from OA to OB. Had we measured from OB to OA, the arc ab would have been described in the negative direction, or aMb would have been described in the positive direction. Hence we should have had = ― Angle BOA arc ab or + arc aMb, which is the negative of the corresponding measure from A to OB. Hence : By interchanging the sides we change the algebraic sign of the angle. To give uniformity to this mode of measurement, the side OA, from which we measure, is supposed fixed, while the other side varies in direction according to the magnitude of the angle. When angles are represented in a general way, the side OA may be conceived as extending out horizontally towards the right. Then the other side, OB, will have a definite direction for every angle we choose to assign. For example: For 90° the side OB will point upward. Counting the angles in the negative direction, 90° the side OB will point downward. 66 to the left. For 66 etc. 66 270° 66 upward. etc.. 9. Division into Quadrants. The circle which measures angles is, for convenience, supposed to be divided into quadrants, as in the figure. An angle between 0° and 90° is in the first quadrant. b B 1. From the point O emanate a set of 5 lines making equal angles with each other, and another set of 6 lines making equal angles with each other, the line OA being common to the two sets. Compute the values in degrees of the ten angles AOb, bOB, BOc, etc., dto foA. 2. What is the value of that angle whose negative measure is numerically double its positive measure? D A 3. If a side starting from the zero point move through 1905°, in what quadrant will it be found, and what will be the smallest positive measure of the angle? 4. Two arms start together from the same position OA to turn round 0, the one going in the positive direction, so as to revolve in 60 seconds, the other in the negative direction, so as to revolve in 36 seconds. At what angle and in what time will they meet? 5. If two revolving arms start out together from the position 0° in the same direction, the one going 5° a minute and the other 8° a minute, through what arc will each have moved when they again come together? At what angle will they meet? If they continue turning, after how many revolutions of each will they be together at their starting point? What are 6. Four lines, a, b, c, d, emanate from the same point o, making angle boc = 2aob, cod= 2boc, doa 2cod. the values of the four angles which they form? 7. If an angle of 140° is multiplied successively by 4, 5, 6, 2, -3, 7, in what quadrants will the respective multiples fall, and what will be the smallest positive measures of the several angles formed? 8. Show that the following pairs of angles are supplementary : 10. Bisection. If the angle AOB = n°, and if Ob is its bisector, then AОb = n°. If the side OB revolves about 0, and the side Ob also revolves in the same direction half as fast, then Ob will continually bisect the angle AOB. When OB completes a revolution, returning to the position OB, the bisector Ob will have moved through 180°, and B с will therefore lie in the opposite direction, Oc. Another revolu tion of OB will bring the bisector to the position Ob again, yet another to Oc, and so on. Hence: The general measure of an angle has two bisectors 180° apart. 11. Trisection. If the side Ob is to continually measure one third the angle AOB as OB revolves, then we must have AОb=1AOB. If OB, starting from the position in the figure, goes through one revolution, Ob will go through 120° to the position Oc.. A second revolution of OB will bring Ob 120° farther, to Od, and a third to its first position, Ob, B after which it will repeat its movements. Hence: -8 One third the general measure of an angle has three special angular values differing by 120°. 12. Division into n Parts. If, as OB revolves, Ob continually 1 measures of it, then every revolution of OB will turn Ob 1 n through of a revolution. Hence: -- n The nth part of the general measure of an angle has n special angular values. 13. Analytic Deduction. It will be remarked that, in the preceding sections, what we take the nth part of is not the angle AOB, but the general measure of this angle. This will be clear from the following analytic deduction of the same result. Let the smallest measure of the angle AOB be a. Then the other measures of this angle (87) will be a + C, a + 20, a + 30... a + iC. Dividing these quantities by n, the quotients will be |