XI. MULTIPLE ANGLES, SUB-MULTIPLE ANGLES XVIII. RELATIONS AMONG THE SIDES AND ANGLES OF A SPHERI- 1. The primary object of Trigonometry was, as its name implies, to measure (or solve) triangles; i.e. having given the measure of certain parts of a triangle, e.g. two sides and its included angle, to compute the remaining parts. In a broader and now universally accepted sense, Trigonometry embraces, in addition to the solution of triangles, all investigations of the relations existing among certain ratios intimately associated with an angle. These ratios are defined in Art. 26. This branch of the subject is sometimes called Angular Analysis. 2. The figures with which we shall be concerned in our study of Trigonometry are, with the exception of the line and the angle, the same as those of Geometry; i.e. they are subject to the same limitations and possess the same properties as those of Geometry. For example, the sum of the interior angles of a triangle equals two right angles in Trigonometry as well as in Geometry. 3. The line of Trigonometry differs from the line of Geometry, in that, in Trigonometry, it is sometimes of advantage to distinguish between lines drawn in opposite directions. [See Art. 47.] For the present, however, we shall not make this distinction. 4. By the angle XOP (Fig. 1, Fig. 2, Fig. 3), in Trigonometry, is not meant the present inclination of the lines OX and OP as in Geometry, but the amount of turning which OP has done about the point O, in coming from its initial position OX, to its final position OP. ILLUSTRATION. Suppose a race to be run around a circular course. The position of any one of the competitors would be known, if we know that he has described a certain angle about the centre of the course. Thus, if the distance to be run is three times around, the line joining each competitor to the centre would have to describe an angle of 12 right angles. When we say that a competitor has described an angle of 63 right angles, we give not only his present position, but the total distance he has gone. He would, in such a case, have gone a little more than one and a half times around the course. It is evident from this definition that a trigonometrical angle. may have any magnitude however great. It is well to notice that angle XOP is the amount of turning that has been done. In other words, it is the result of the turning, not the process. 5. The geometrical representation of a trigonometric angle depends only on the initial and final positions of the line OP. Hence the figure XOP (Fig. 1, Art. 4) may be the geometrical representative of an unlimited number of trigonometrical angles. (i.) The angle XOP may represent the angle less than two right angles, as in Geometry. In this case, OP has turned from the position OX into the position OP by turning about O in the direction contrary to that of the hands of a watch. (ii) The angle XOP may represent the angle described by OP in turning from the position OX into the position OP in the same |