2. Find the product of 3846, and 0.056214. Ans. 183406.96. 3 Find the product of 0.3467, and 0.0098563. log 0.03467 = 2.5399538 log 0.0098563 = 3.9937139 4.5336677 Ans. 216.199. Ans. 0.0003417178. 4. Find the product of 1576.3, and 0.85673. 5. Find the product of 18.54 and 735.6425. 6. Find the product of 21.357 and 6324.567. Ans. 1350.463. Ans. 13638.811. Ans. 135073.76. 7. Find the product of 18.21, 35.672, 2.847, and 11.256. Ans. 20816.56. 8. Find the product of 0.0873, 25.773206, and 0.005693. Ans. 0.0128092. DIVISION BY LOGARITHMS. 5. In division we proceed by the following-- RULE VIII. 1°. Find the logarithms of the numbers the quotient of which is required. 2. Subtract the logarithm of the divisor from that of the dividend; the difference will be the logarithm of the quotient. 3°. Find from the tables the corresponding number. This will be the required quotient. The reason of this rule appears from Prop. 11. sect. 2. We thus obtain for the figures of the quotient, the characteristic is 1, 2. Find the quotient of 0.32567 by 0.0129. 3. Find the quotient of 567.342 by 110.673. 4. Find the quotient of 0.01237 by 108.46. The additional figs. 26908216; and as Ans. 26.908216. Ans. 25.24573. Ans. 5.12629. Ans. 0.00011405. 5. Find the quotient of 385.27 by 10056789. 6. Find the quotient of 287.642 by 83456. Ans. 0.000038309. Ans. 0.00344663. INVOLUTION BY LOGARITHMS. 6. In involution we proceed by the following- RULE IX. 1°. Find the logarithm of the number which is to be raised to the given power. 2o. Multiply this logarithm by the index of the given power; the product will be the logarithm of the required power. 3°. Find from the tables the corresponding number. This will be the required power. The reason of this rule appears from Prop. III. sect 2. EXAMPLES, 1. Find the fourth power of 8.7. The figures of the power are 5728978947; and as the characteristic is 3, 2. Find the fifth power of 11. 3. Find the square of 82.56. 4. Find the cube of 196.3. 5. Find the twenty-first power of 1.5. log 0.03564 = 2.5519377 Ans. 5728.978. Ans. 48562. EVOLUTION BY LOGARITHMS. 7. In evolution we proceed by the following— RULE X. 1o. Find the logarithm of the number whose root is to be found. 2°. Divide this logarithm by the index of the given root; the quotient will be the logarithm of the required root. 3°. Find from the tables the corresponding number. This will be the required root. The reason of this rule appears from Prop. IV. sect. 2. 2. Calculate the value of 71 × (5.6)3 × (89)‡ (3.57)7 × (39) 9. The nature of Tables of Natural Sines and Cosines, &c., has been fully explained, Chap. II. sect. 4. In using these tables we have either to find the sine, cosine, tangent, &c., of an angle whose value is given in degrees, minutes, and seconds; or, being given the value of the angle in degrees, minutes, and seconds, to find the corresponding value of the sine, cosine, &c. If the value of the angle be given in degrees and minutes, the sine, cosine, &c., is found directly from the tables, in which are |