hence, by (450), cotan. B'A'C' cotan. A Fifthly. The preceding equation between h, the angle A, and the opposite side a, leads to the following corresponding one between h, the angle B, and the opposite side b; sin. b sin. h sin. B. Sixthly. From triangles COA', B'A'C' and B'OC', by (5), (429), and (437), sin. C'OA'sin. b= A'C OC A'C BC B'C OC = tang. B'OC'tang. a= The product of (451) and (452) is A'C B'C A'C cotan. A tang. a= X B'C OC oci sin. bcotan. A tang. a. Seventhly. The preceeding equation between the angle A, the opposite side a, and the adjacent side b, leads to the following corresponding one between (445) (446) (447) (448) (449) (450) (451) (452) (453) (454) (455) (456) (457) (458) (459) (460) (461) (463) (464) (465) (466) the angle B, the opposite side b, and the adjacent side a; sin. a=cotan. B tang. b. (467) Eighthly. From (10), tang. a = tang. b The product of (460) and (461) is (462) sin. a sin. b cos. a cos. b = cot. A cot. B sin. a sin. b; which, divided by sin. a sin. b, becomes cos. a cos. bcotan. A cotan. B. = hence cos. b which, substituted in (454) and (455), give sin. a= cotan. B sin. b cotan. A sin. a But, by (436), sin. a Cos. a sin. b sin. b= cos. a Multiplying (458) by cos. b and (459) by cos. a, we have sin. a cos. b=cotan. B sin. b, cos. h=cos. a cos. b; cos. hcotan. A cotan. B. sin. B= Ninthly. We have, by (436) and (449), cos. h COS. a = cos. b' sin. b sin. h' the product of which is by (10) and (11) sin. b cos. h cos. a sin. B = = cos. b sin. h sin. b cos. h cos. h sin. h (468) But, by (442), cos. A = tang. b cotan. h; (469) Hence, from the equality of the second members of (468) and (469), cos. A cos. a sin. B. Tenthly. The preceding equation between the side a, the opposite angle A, and the adjacent angle B, leads to the following similar one between the side b, the opposite angle B, and the adjacent angle A; cos. B cos. b sin. A. (470) (471) 6. Corollary. The ten equations, (436), (442), (443), (448), (449), (454), (455), (465), (470), and (472) (471), have, by a most happy artifice, been reduced. to two very simple theorems, called, from their celebrated inventor, Napier's Rules. In these rules, the complements of the hypothenuse and the angles are used instead of the hypothenuse and the angles themselves, and the right angle is neglected. Of the five parts, then, the legs, the complement of the hypothenuse and the complements of the angles; either part may be called the middle part. The two parts, including the middle part on each side, are (478) called the adjacent parts; and the other two parts are called the opposite parts. The two theorems are as follows. (474) I. The sine of the middle part is equal to the product of the tangents of the two adjacent parts. (475) II. The sine of the middle part is equal to the product of the cosines of the two opposite parts. (476) (477) (478) (479) (480) (481) (482) Demonstration. To demonstrate the preceding rules, it is only necessary to compare all the equations which can be deduced from them, with those previously obtained (472). Let there be the spherical right triangle ABC (fig. 2.) right-angled at C. First. If co. h were made the middle part, then, by (473), co. A and co. B would be adjacent parts, and a and b opposite parts; and, by (474) and (475), we should have A sin. (co. h) or Fig 2 = or cos. h = cotan. A cotan. B, which are the same as (465) and (436). tang. (co. 4) tang. (co. B), =cos. a cos. b; B sin. (co. A) tang. (co. h) tang. b, cos. A cotan. h tang. b, a Secondly. If co. A were made the middle part; then, by (473), co. h and b would be adjacent parts, and co. B and a opposite parts; and, by (474) and (475), we should have = cos. A sin. B cos. a; which are the same as (442) and (470). In like manner, if co. B were made the middle part, we should have cos. B = cotan. h tang. a, cos. B = sin. A cos. b; which are the same as (443) and (471). Thirdly. If a were made the middle part; then, by (473), co. B and b would be the adjacent parts, and co. A and co. h the opposite parts; and, by (474) and (475), we should have sin. sin. a or = tang. (co. B) tang. b, sin. a sin. asin. A sin. h; which are the same as (455) and (448). In like manner, if b were made the middle part, we should have sin. b = cotan. A tang. a, sin. b = sin. B sin. h; which are the same as (454) and (449). cotan. B tang. b, Having, thus, made each part successively the middle part, the ten equations, which we have obtained, must be all the equations included in (474) and (475); and we perceive that they are identical with the ten equations of (472). 7. Corollary. When h is less than 90°, the first member of (436), cos. h = cos. a cos b, (483) (484) (485) (486) (487) (488) (489) (490) (491) (492) |