Which part being added to the less logarithm, before taken out, gives the whole logarithms sought very nearly. EXAMPLE. 532627. To find the logarithm of the number 34-0926. The diffs. are Then as 100 This added to 100 is 532754. and 127: 26: 33, the proportional part. Gives, with the index, 1.532660, for the log. of 34.0926. 4. If the number consist both of integers and fractions, or is entirely fractional; find the decimal part of the logarithm the same as if all its figures were integral; then this, having prefixed to it the proper index, will give the logarithm required. 5. And if the given number be a proper vulgar fraction: subtract the logarithm of the denominator from the logarithm of the numerator, and the remainder will be the logarithm sought; which, being that of a decimal fraction, must always have a negative index. 6. But if it be a mixed number; reduce it to an improper fraction, and find the difference of the logarithms of the numerator and denominator, in the same manner as before. EXAMPLES. 1. To find the log. of 3. | 2. To find the log. of 174. Log. of 37 1.568202 First, 17405. Log. of 23 Then, 2.607455 1.361728 Dif. log. of 1714 1.245727 Where the index 1 is negative. II. TO FIND THE NATURAL NUMBER TO ANY GIVEN LOGARITHM. THIS is to be found in the tables by the reverse method to the former, namely, by searching for the proposed logarithm among those in the table, and taking out the corresponding number by inspection, in which the proper number of integers are to be pointed off, viz. 1 more than the index. For, in finding the number answering to any given logarithm, the index always shows how far the first figure must must be removed from the place of units, viz. to the left hand, or integers, when the index is affirmative; but to the right hand, or decimals, when it is negative. EXAMPLES. So, the number to the log. 1.532882 is 34·11. And the number of the log. 1.532882 is ·3411. But if the logarithm cannot be exactly found in the table; take out the next greater and the next less, subtracting the one of these logarithms from the other, as also their natural numbers the one from the other, and the less logarithm from the logarithm proposed. Then say, As the difference of the first or tabular logarithms, Is to the difference of their natural numbers, So is the differ. of the given log. and the least tabular log. To their corresponding numeral difference. Which being annexed to the least natural number above taken, gives the natural number sought, corresponding to the proposed logarithm. EXAMPLE, So, to find the natural number answering to the given logarithm 1.532708. Here the next greater and next less tabular logarithms, with their corresponding numbers, are as below: Next greater 532744 its num. 341000; given log. 532708 Next less 532627 its num. 340900; next less 532627 81 Then, as 127: 100: 81: 64 nearly, the numeral differ. Therefore 34.0964 is the number sought, marking off two integers, because the index of the given logarithm is 1. Had the index been negative, thus ponding number would have been cimal. 1.532708, its corres340965, wholly de MULTIPLI MULTIPLICATION BY LOGARITHMS. RULE. TARE out the logarithms of the factors from the table, then add them together, and their sum will be the logarithm of the product required. Then, by means of the table, take out the natural number, answering to the sum, for the product sought. Observing to add what is to be carried from the decimal part of the logarithm to the affirmative index or indices, or else subtract it from the negative. Also, adding the indices together when they are of the same kind, both affirmative or both negative; but subtracting the less from the greater, when the one is affirmative and the other negative, and prefixing the sign of the greater to the remainder. 3. To mult. 3.902 and 597.16 | 4. To mult. 3.586, and 2.1046, and 0314728 all together. Numbers. Logs. 0.591287 2.776091 0314728-2-497935 and 0.8372, and 0.0294 all Prod. 73.3333 1.865313 0.8372-1.922829 0.0294-2.468347 Here the 2 cancels the 2, Prod. 0.1057618-1.268956 and the 1 to carry from the decimals is set down. Here the 2 to carry cancels the-2, and there remains the -1 to set down. DIVISION DIVISION BY LOGARITHMS. RULE. FROM the logarithm of the dividend subtract the logarithm of the divisor, and the number answering to the remainder will be the quotient required. Observing to change the sign of the index of the divisor, from affirmative to negative, or from negative to affirmative; then take the sum of the indices if they be of the same name, or their difference when of different signs, with the sign of the greater, for the index to the logarithm of the quotient. And also, when 1 is borrowed, in the left-hand place of the decimal part of the logarithm, add it to the index of the divisor when that index is affirmative, but subtract it when negative; then let the sign of the index arising from hence be changed, and worked with as before. EXAMPLES. 1. To divide 24163 by 4567. | 2. To divide 37.149 by 523.76 Numbers. Logs. Dividend Numbers. Logs. 4-383151 Dividend 37.149 - 1.569947 523-762-718132 24163 Note. As to the Rule-of-Three, or Rule of Proportion, it is performed by adding the logarithms of the 2d and 3d terms, and subtracting that of the first term from their sum. INVOLUTION INVOLUTION BY LOGARITHMS. RULE. TAKE out the logarithm of the given number from the table. Multiply the log. thus found, by the index of the power proposed. Find the number answering to the product, and it will be the power required. Note. In multiplying a logarithm with a negative index, by an affirmative number, the product will be negative. But what is to be carried from the decimal part of the logarithm, will always be affirmative. And therefore their difference will be the index of the product, and is always to be made of the same kind with the greater. Here 4 times the negative index being-8 and 3 to carry, the difference - 5 is the index Power 5 14932* of the product. 5850 0.711750 *This answer 5'14932 though found strictly according to the gene. ral rule, is not correct in the last two figures 32; nor can the answers to such questions relating to very high powers be generally found true to 6 places of figures by the table of logarithms in this work if any power above the hundred thousandth were required, not one figure of the answer found by the table of logarithms here given could be depended on. : The logarithm of 1.0045 is 00194994108 true to eleven places, which multiplied by 365 gives 7117285 true to 7 places, and the corresponding number true to 7 places is 5.149067. EVOLUTION Voz. I. 23 |