8. Required the product of .3817 ....025913 and .999 ? 12. Multiply .32 ... 4.08, and .12 together. Prod. .15667. 13. Multiply 1237... 12.37, and 1237 together. Prod. 1892.8. 14. Multiply .03 . . . .004 ... 157.8, and .0006 together. Prod. 000011362. ... 31. DIVISION BY LOGARITHMS. RULE I. From the decimal part of the logarithm of the dividend, subtract the decimal part of the logarithm of the divisor. II. Change the index of the divisor, if it be affirmative, to negative, and if it be negative, to affirmative. III. If after this change the indices have like signs, add them together, and prefix their sum, with its proper sign, to the decimal. IV. If the indices after this change have unlike signs, take their difference, and prefix the remainder, with the sign of the greater, to the decimal. V. Observe, that when 1 is carried from the decimal, it must be added to the index of the divisor, if affirmative, but subtracted, if negative; and this must be done in both cases before the index of the divisor is changed. it out in the table, and find its natural number to be 434.3, which is the quotient required. 2. What is the quotient of 28.5 divided by 301.23? sign,(because 3 is negative,)-2 therefore is the index to be prefixed to the decimal remainder. 3. Divide .05432 by .2345. 32. Division may be performed by using instead of the logarithm of the divisor, its arithmetical complement', whereby subtraction is changed to addition. The arithmetical complement of a logarithm is what that logarithm wants of 10. 33. To find the arithmetical complement of any logarithm. RULE. If the index of the given logarithm be affirmative, subtract it from 9; but if negative, add it to 9: then proceeding The use of the arithmetical complement was first introduced by Mr. Edmund Gunter, Professor of Astronomy at Gresham College, probably abouť the year 1620. See Briggs's Arithmetica Logarithmica, &c. cap. 15. from left to right, subtract each of the decimal figures from 9, except the last or right hand figure, which must be subtracted from 10; the result is the arithmetical complement required. To find the arithmetical complement of the logarithm 2.7817544. Here, beginning at the index 2, I subtract that and each of the figures (proceeding from left to right) from 9, except the right hand figure 4, which I subtract from 10, and the result is 7.2182456, the arithmetical complement required. To find the arithmetical complement of the logarithm -1.8893017. Beginning at the 1, I add it to 9, (because it is negative,) and subtract the other figures from 9, proceeding from left to right as before, except the last figure 7, which I subtract from 10. arithmetical complement therefore will be 10.1106983. The In like manner, the arithmetical complement of the logarithm 1.5250448, will be 8.4749552. The arithmetical complement of 0.8430458, will be 9.1569542. That of the logarithm -1.9854714, will be 10.0145286. That of 3.8653409, will be 12.1346591, &c. &c. 34. To perform division by addition. RULE I. Under the logarithm of the dividend, write the arithmetical complement of the logarithm of the divisor, and add both together. II. If the index of the sum be 10, or greater than 10, subtract 10 from it, and prefix the remainder (which will be affirmative) as an index to the decimal part of the sum. III. If the index of the sum be less than 10, subtract it from 10, and to the remainder (which is negative) prefix the negative sign and place it as an index to the decimal part of the sum, as before. To divide by any number, or to multiply by its reciprocal, produces the same result; and therefore, to add any logarithm, or subtract its reciprocal, must evidently do the same: now the arithmetical complement of a logarithm is the reciprocal of that logarithm increased or diminished by 10, according as the index is negative or affirmative; which accounts for the subtraction or addition of 10, as in Art. 34. The logarithm of the divisor 29.8 is 1.4742163; I find the arithmetical complement of this, and having placed it under the logarithm of the dividend 435, I add both together; and the index of the sum being 11, I subtract 10 from it, and place the 1 remainder for an index. Arith. comp. log. 31.024 8.5083022 14. Divide 1.4759 by 23.917. Log. of 1.4759 Arith. co. log. 23.917 = 0.1690569 .8.6212933 -2.7903502 35. The logarithm of a vulgar fraction is found by this rule. From the logarithm of the numerator, subtract the logarithm of the denominator, (or add its arithmetical complement,) and the result will be the logarithm required. If a mixed number be given, reduce it to an improper fraction, and then find its logarithm as above. 24. What is the logarithm of 124? Ans. 1.0969100. 36. PROPORTION BY LOGARITHMS. RULE Í. Prepare the terms as in the Rule of Three of Decimals, if they require it. II. Place the terms (so reduced) in order, one under another, with the logarithm of each opposite to its respective term. III. Add the logarithms of the two multiplying terms together, and from the sum subtract the logarithm of the dividing term; the remainder will be the logarithm of the answer. IV. Or find the arithmetical complement of the dividing term, and add it and the logarithms of the two remaining terms together, observing to take the difference of 10 and the index, as in division; the result will be the logarithm of the answer as before. EXAMPLES. 1. If the week's allowance for 5 seamen be 38lb. of biscuit, how many pounds will a ship's company of 224 men consume in the same time? VOL. I. |