This value of log sin B corresponds to 40° 52′ 21′′. The supplement of this, 139° 7′ 39′′, is inadmissible, because in this than b. Calculating C by equation (2), and c Case I., we find 4. Given b = case a is greater by equation (3), Ans. B 40° 52′ 21′′. C = 85° 47′ 33′′. c = 47.24556. 53, a = 47, A = 36° 42′ 30′′; find B (acute), C, = 312, a = 517, and A = 124° 32'; find B, C, and a. 29° 48′ 41′′. C = 25° 39′ 19′′. c = 271.71519. 5. Given b = 217, a= 199, and A = 62° 24′ 20′′; find B (obtuse), C, and a. Given two sides and included angle. Let the given sides be a and b, and the included angle C'; then ↓ (A + B) = 90° – † C. (1) By Prop. II. Chap. V., a+b: a-b:: tan (A + B) : tan (A – B) ; from which, 1+B)} (2) log tan (A-B) = log (a−b) + log tan (A- log (a + b) By means of the equations (1) and (2) we can calculate the values of (A+B) and (A-B), the sum and difference of which give respectively the values of A and B. The third side c may be calculated by means of equation (3), Case I. EXAMPLES. 1. Given a = 218, b = 156, and C = 38° 21′ 20′′; find A, B, and c. a+b=374 From which we find, by applying Rule XII. (Appendix), that (A – B) = 25° 29′ 3′′. From this value of } (A – B), and that already found for } (A + B), we find the values of A and B. 2. Given a = 516, b = 219, and C = 98° 54'; find A, B, and c. Ans. A = = 59° 37′ 18′′. B = 21° 28′ 42′′. C = 590.91723. 3. Given a 53.24, b = 31.27, and C = 126° 36′ 6′′; find A, B, and c. Ans. A 34° 8' 53'. B = 19° 15' 1". 4. Given a = 831, b = 536, and C = 16° 28′ 40′′; 5. Given a = Ans. A B: C = 76.14207. = C = 25° 37′ 17′′. 351.58431. 8214, b = 3732, and C = 61° 53'; find A, B, and c. = Ans. A 91° 5′ 57′′- c = 7246. 6. Given a = 1.73, b = 1.23, and C = 22° 13′ 30′′; find A, B, and c. Ans. A 119 34′ 58′′. = B= 38° 11' 32". c = 0.75245. CASE V. Given the three sides. If the sides be a, b, c, the values of A, B, C may be calculated by means of any of the groups of formulæ (5), (6), (7), or (8). In order to avoid the ambiguity which occurs in finding the value of an angle by means of its sine, any of the groups (6), (7), or (8), is to be preferred to (5); in using these no ambiguity can arise, because since any angle of a triangle is less than 180°, its half must be less than 90°. From these formulæ we derive the following, which are adapted to logarithmic calculations : log cos A= 10 + (log 8+ log (8-a) log sin A = 10 + {log (s - b) + log (sc) - log b - log c}. (1) -0)} (2) 1-0)} (3) log tan A 10+ (log (s - b) + log (s- c) In a similar manner B, and C may be calculated. Ans. A 36° 49′ 36′′.92. N. B. The sum of these angles is 180° 0′ 0′′.32, which is only o".32 in excess. 2. Given a = 15.32, b = 21.56, and c = 16.22; find the value of A. 3. Given a = 2134, b = 1617, and c = Ans. A 45° 9′ 16′′. = 815; find the value of B. Ans. B 41° 32′ 31′′. = 4. Given a = 1500, b = 1342, and c = 1110; find the value of C. Ans. C 45° 33′ 35′′. 5. Given a = 1, b = 1.32, and c = 0,75; find the value of A. Ans. A = 48° 46′ 24′′. 6. Given a = 27, b = 32, and c = 9; find the value of C. Ans. C 14° 37′ 35′′. APPENDIX. 1. Definition of Logarithms.-2. Properties of Logarithms.-3. Use of Logarithmic Tables.-4. Multiplication by Logarithms.-5. Division by Logarithms.-6. Involution by Logarithms-7. Evolution by Logarithms.-8. Calculation of Expressions.-9. Tables of Natural Sines.-10. Tables of Logarithmic Sines. DEFINITION OF LOGARITHMS. 1. If the number 10 be raised to the powers, 0, 1, 2, 3,-4, &c., we obtain the series of numbers, 1, 10, 100, 1000, 10000, &c. Thus : It is evident that for numbers intermediate to these, the powers to which 10 must be raised, so as to be equal to them, must lie between the numbers of the series, o, 1, 2, 3, 4, &c. Thus, for all numbers lying between 100 and 1000, the power to which 10 must be raised being greater than 2, and less than 3, will be 2, increased by some decimal fraction. For numbers lying between 1000 and 10000, will be 3, increased by some decimal fraction, and so on. These numbers are called logarithms; and in consequence of their great importance in numerical computation, their values have been accurately calculated for all numbers from 1 to 100000, and registered in tables, called Logarithmic Tables. DEFINITION.-The ordinary logarithm of a number is that power to which 10 must be raised, so as to be equal to it. Thus : a |