For (Art. 30.) sin a cos b=1 sin (a + b)+1 sin (a−b) sin a sin b={cos (a - b) – į cos (a+b). And these equations, if a=30°, and therefore, sin a=1, become cos b=sin (30° + b) +sin (30°— b) sin b=cos (30° —b) - cos (30° +b) :. sin (30° +b) =cos b-sin (30°— b) cos (30° + b) =cos (30° – b) – sin b. So that if for b be substituted, successively, all the arches between 0° and 30°, there will be found the sines and cosines of all arches between 30° and 60°: and amongst the sines and cosines of all the arches between 0° and 45° will be found the sines and cosines of all arches whatever. Lastly, if the tangents and co-tangents be computed, of all arches between 0° and 30°, the tangents and cotangents of all arches whatever, that are measured by an even number of seconds may be found, without any other arithmetical operation, than a subtraction, and a division by 2. For (Art. 30.) cot. 2a= Ž (cot. a – tan. a). If, therefore, a=30°– b, and 2a = 60° — 2b, and cot (60° – 2b) ) — ) = tan (30°+2b)=cot (30°– b) – tan (30° — b) a а 2 Whence, the tangents and co-tangents of all arches between 30° and 60° may be found from the tangents and co-tangents of arches that are less than 30°; and thus the tangents and co-tangents of all arches whatever will be known, when those of all arches from 0° to 45° have been found. But it is manifest, that a and b in the two forms, I (cot a tan a) cot (30°—b) – tan (30° – ) tan (30° +26) cot 2a = 2 و denote whole numbers of seconds; and therefore, the arch (30° + 2b) of which the tangent is thus found, must always consist of an even number of seconds. The considerations, however, which have been suggested, in this last article, indicate a very great abridgement of labour in the actual computation of the tri- . gonometrical functions of circular arches. ON THE INVESTIGATION OF SUCH GENERAL PROPOSITIONS AS ARE APPLICABLE TO THE PURPOSES OF SPHERICAL TRIGONOMETRY. DEFINITION. a (227.) SPHERICAL Trigonometry is that part of Spherics, in which are investigated the relations existing between the sides, the angles, and the surface of a spherical triangle, with a view to the resolution of the following general problem :“Of the six first quantities, namely, those made up of the three sides, and the three angles, of a proposed spherical triangle, any three being given, to determine the rest." That this problem involves no impossibility, is very evident, from what has been already proved, in treating of Spherical Geometry, N |