(7) cot a=cos B cot c (8) sin c=sin a sin C (9) tan b=sin c tan B (10) cos B=cos b sin C. The relations of the quantities concerned in the equations may be seen more clearly by aid of the accompanying figure. B 8. The ten equations of the six cases of right-angled spherical trigonometry may be all embraced in one or two simple rules, called from their inventor, Napier's Rules of the Circular Parts. The parts of a triangle concerned in the equations are the three sides and two oblique angles of a rightangled spherical triangle. By the circular parts, however, are meant the two perpendicular sides, and the complements of the hypotenuse and the two oblique angles, instead of these three quantities themselves; the use of those complements having been found requisite in order to give rules of the simplest form. Supposing the hypotenuse and the two oblique angles of a triangle to be replaced by their complements, and the right angle to be left out of view, then of the five circular parts in the triangle, if either be taken for the middle part, the two that are next to this and separated by it are called the adjacent parts, and the other two the opposite parts. Napier's rules of the circular parts thus distinguished, are as follows: The SINE of the MIDDLE part, equals the product of the TANGENTS of the ADJACENT parts, and equals the product of the COSINES of the OPPOSITE parts. This proposition may be more easily remembered, when it is observed that the two prominent words in each of the three parts of which it consists, have their first vowels alike; this vowel being i in each of the words sine and middle, being a in the words tangent and adjacent, and o in the words cosine and opposite. 9. As each of the five circular parts may be taken for the middle one, and the proposition gives two equations for each case, it will furnish in all ten equations; and these will be found to agree with the ten previously stated. Thus if the parts of a rightangled triangle be represented as in the figure here given, where A is the right angle; and each of the five circular b parts, namely, b, c, comp. B, comp. a, comp. C, be in turn made the middle part, there will result the following equations, and sin b=tan c tan comp. C, (1) sin b=tan c cot C, that is, sin b=cos comp. B cos comp. a, that is, (2) sin b=sin B sin a (for the cosine of the complement of an arc or angle is the sine of that arc or angle). Also (3) sin c=tan b cot B (4) sin c=sin a sin C (5) cos B=tan c cot a (6) cos B=cos b sin C (10) cos C=cos c sin B, which differ only a little in form from those given in the preceding article, and may be made identical with them by applying, in a few cases, the principle that the tangent of an arc or angle equals the reciprocal of the cotangent, when radius is unity (Day's Trig., Art. 93). For example, the equation (1) of the seventh article may cot B tan C which is the seventh of the pre be stated thus, cos a= cos a=cot B cot C; ceding equations. 10. In applying Napier's theorem to any proposed case, it is best at first to make no distinction between the two parts given and the part required, but consider only which of the three circular parts concerned, must be made the middle part, in order that the other two may be either both adjacent or both opposite, and not one adjacent and the other opposite; and then by means of the theorem form an equation involving these three parts; afterwards reducing the equation, if necessary, so as to have the given parts on one side, and the part required on the other. Thus, if the sides a and c be given and the angle C be sought, the three circular parts concerned are c, comp. a, comp. C, of which it will be seen that c must be taken for the middle part, and the other two for opposite parts. By Napier's rule, the equation connecting these parts is the following, sin c sin a sin C. And if the quantities given were a and C, this equation would require no reduction: but as Cand not c is the quantity required, the equation must be reduced to the form mic calculation the value of C may readily be found. 11. One must first of all, however, gain some facility in stating, when any circular part of a right-angled triangle is taken for the middle one, which of the other four are adjacent parts, and which opposite. The statements that may be demanded are these; (1) If comp. a be the mid dle part, the adjacent parts are comp. B, comp. C, and the opposite parts are b, c. (2) If b is the middle part, a the adjacent parts are c, comp. C, and the opposite parts are comp. a, comp. B. (3) If c is the middle part, the adjacent parts are b, comp. B, and the opposite parts are comp. a, comp. C. (4) If comp. B is the middle part, the adjacent parts are comp. a, c, and the opposite parts are b, comp. C. (5) If comp. C is the middle part, the adjacent parts are comp. a, b, and the opposite parts are c, comp. B. 12. Now let it be required, with any three proposed circular parts (without regard to any distinction between them as known or unknown quantities), to state which must be taken for the middle part, in order that the other two may be either both adjacent or both opposite; and let it not be yet required to proceed to the use of Napier's rule in forming any equation. The several cases that will thus be presented, may be stated as follows: (1) If the three quantities concerned are b, c, C, then b is the middle part, and the parts c, comp. C, are adjacent. (2) If the three proposed quantities are a, b, B, then b is the middle part, and the parts comp. a, comp. B, are opposite. (3) If the three quantities are b, c, B, then c is the middle part, and the parts b, comp. B, are adjacent. (4) If the three quantities are a, c, C, then is c the middle part, and comp. a, comp. C, are opposite parts. (5) If the three quantities are a, c, B, then is comp. B the middle part, and the parts comp. a, c, are adjacent. (6) If b, B, C, are the three quantities, comp. B is the middle part, and the parts b, comp. C, are opposite. (7) If a, B, C, are the three quantities, comp. a is the middle part, and the parts comp. B, comp. C, are adjacent. (8) If a, b, c, are the three quantities, comp. a is the middle part, and the parts b, c, are opposite. (9) If a, b, C, are the three quantities, comp. C is the middle part, and the parts comp. a, b, are adjacent. (10) If c, B, C, are the three quantities, comp. C is the middle part, and the parts c, comp. B, are opposite. 13. The use of Napier's rules will now be shown by some examples. (1) Given the sides a and b; to find c. The circular parts here concerned are comp. a, b, c: of these comp. a is the middle part, and the other two, b, c, are opposite parts. Then by Napier's rule, sin comp. a=cos b cos c, that is, cos a=cos b cos c; and since c is the re |