CHAPTER VII. ON THE USE OF SUBSIDIARY ANGLES. Subsidiary Angles are angles which, although not immediately connected with a given problem, are introduced by the computist in order to simplify his calculations. Their use, and the method in which they are employed, will be understood from what follows. When the two sides of a triangle, and the included angle, are given, according to the method pursued in the last chapter, we must determine the two remaining angles before we can compute the third side. It frequently happens, however, in practice, that the side only is required, and it therefore becomes desirable to have some direct method of computing the side independently of the two angles. Suppose that a, b, C are given, and c is required. By chap. III. prop. 4, c2=a+b2-2ab cos. C. the side c is determined theoretically at once by this expression, but the formula is not adapted to logarithmic computation, and would, if employed practically, lead to a very tedious and complicated calculation. We can, however, put this expression under a form adapted to logarithmic calculation, by having recourse to an algebraical artifice, and introducing a subsidiary angle. c2=a2+b2—2 ab cos. C Adding and substracting 2 ab on the right hand side. c2=a2+b2—2 ab+2 ab—2 ab cos. C c2 = (ab)2 (1+tan.' ) =(a-b)' sec.' C= (a-b) sec. q log. clog. (ab)+log. sec. -log. R The angle is known from the equation. C log.tan.q=log. 2+ + (log.a+log.b) + log. sin.-log.(a—b) being thus determined, log. sec. can be found from the tables, and the value of c becomes known. The angle o, which is introduced into the above calculation, in order to render the expression convenient for logarithmic computation, is called a subsidiary angle. The above transformation may be effected in a manner somewhat different, as before. c2=a2+b2-2 ab cos. C =a+b2+2 ab-2 ab-2 ab cos. C =(a+b)2—2 ab (1+cos. C). =(a+b)3—2 ab×2 cos.2 C c'=(a+b)' (1-sin.' ) =(a+b)2 cos.2 q c=(a+b) cos. ❤ log.c=(s+b)+log. cos. -log. R As before the angle & must be determined from the equation. in other words, that 2ab is always less than (a+b), this is easily done. But since (a-b) is necessarily a positive quantity, it must always be greater than 0 (except in the particular case a=b, C where it is=0), and therefore?✓ab cos. is always less than (a+b) unity, and consequently an angle may always be found whose sine is equal to it. In solving the same case of oblique-angled triangles, we de termined the difference of the angles A, B from the equation. A-B C -cot. C Whence log.tan. =log. (a—b)+log. cot.—log. (a + b) In the solution of certain astronomical problems, the logarithms of the sides a, b are given, but not the sides themselves, and these logarithms being given, we can very easily A-B calculate -without knowing the sides. and b, without calculating a and b. In the same way we have Whence A-B The angle thus becomes known from the logs. of a 2 C =tan. (45°+9) tan. 2 A great variety of geometrical problems may be solved with much elegance by the introduction of geometrical formulæ. We shall give a few examples. PROBLEM I. To express the area of a plane triangle in terms of the |