out, in a very short time. When great accuracy is not required, it will be easy to make an allowance sufficiently near, without formally stating a proportion. In the larger tables, the proportional parts which are to be added to the logarithms, are already prepared, and placed in the margin. 29. To find the logarithm of a DECIMAL FRACTION. The logarithm of a decimal is the same as that of a whole number, excepting the index. (Art. 14.) To find then the logarithm of a decimal, take out that of a whole number consisting of the same figures; observing to make the negative index equal to the distance of the first significant figure of the fraction from the place of units. (Art. 11.) The log. of 0.07643, of 0.00259, of 0.0006278, is 2.88326, or 8.88326, (Art. 12.) 3.41330, or 7.41330, 4.79782, or 6.79782. 30. To find the logarithm of a MIXED decimal number. Find the logarithm, in the same manner as if all the figures were integers; and then prefix the index which belongs to the integral part, according to Art. 8. The logarithm of 26.34 is 1.42062. The index here is 1, because 1 is the index of the logarithm of every number greater than 10, and less than 100. (Art. 7.) The log. of 2.36 is 0.37291, The log. of 364.2 is 2.56134, of 27.8 1.44404, of 69.42 1.84148. 31. To find the logarithm of a VULGAR FRACTION. From the nature of a vulgar fraction, the numerator may be considered as a dividend, and the denominator as a divisor; in other words, the value of the fraction is equal to the quotient of the numerator divided by the denominator (Alg. 110.) But in logarithms, division is performed by subtraction; that is, the difference of the logarithms of two numbers, is the logarithm of the quotient of those numbers. merator. (Art. 1.) To find then the logarithm of a vulgar fraction, subtract the logarithm of the denominator from that of the nuThe difference will be the logarithm of the fraction. Or the logarithm may be found, by first reducing the vulgar fraction to a decimal. If the numerator is less than the denominator, the index of the logarithm must be negative, because the value of the fraction is less than a unit. (Art. 9.) Required the logarithm of 4. The log. of the numerator is 1.53148 of the fraction 1.59196, or 9.59196. The logarithm of is 2.66362, or 8.66362. 7854 of 6329 3.04376, or 7.04376. 32. If the logarithm of a mixed number is required, reduce it to an improper fraction, and then proceed as before. The logarithm of 33-34 is 0.57724. 33. To find the NATURAL NUMBER belonging to any logarithm. In computing by logarithms, it is necessary, in the first place, to take from the tables the logarithms of the numbers which enter into the calculation; and, on the other hand, at the close of the operation, to find the number belonging to the logarithm obtained in the result. This is evidently done by reversing the methods in the preceding articles. Where great accuracy is not required, look in the tables for the logarithm which is nearest to the given one; and directly opposite on the left hand, will be found the three first figures, and at the top, over the logarithm, the fourth figure of the number required. This number, by pointing off dec imals, or by adding ciphers, if necessary, must be made to correspond with the index of the given logarithm, according to Arts. 8 and 11. The natural number belonging to 3.86493 is 7327, to 1.62572 is 42.24, to 2.90141 7.96.9, to 2.89115 0.07783. In the last example, the index requires that the first significant figure should be in the second place from units, and therefore a cipher must be prefixed. In other instances, it is necessary to annex ciphers on the right, so as to make the number of figures exceed the index by 1. The natural number belonging to 6.71567 is 5196000, to 3.65677 is 0.004537, 46840, to 4.59802 0.0003963. 34. When great accuracy is required, and the given logarithm is not exactly, or very nearly, found in the tables, it will be necessary to reverse the rule in Art. 28. Take from the tables two logarithms, one the next greater, the other the next less than the given logarithm. Find the difference of the two logarithms, and the difference of their natural numbers; also the difference between the least of the two logarithms, and the given logarithm. Then say, As the difference of the two logarithms, To the difference of their numbers; Required the number belonging to the logarithm 2.67325. Next great.log.2.67330. Its numb. 471.3. Given log. 2.67325. Next less 2.67321. Its numb. 471.2. Next less 2.67321. Differences 9 0.1 4 Then, 9 : 0.1 :: 4 : 0.044, which is to be added The natural number belonging to 4.37627 is 23783.45, to 1.73698 is 54.57357, to 3.69479 4952.08, to 1.09214 0.123635. 35. Correction of the Tables.-The tables of logarithms have been so carefully and so repeatedly calculated, by the ablest computers, that there is no room left to question their general correctness. They are not, however, exempt from the common imperfections of the press. But an error of this kind is easily corrected, by comparing the logarithm with any two others to whose sum or difference it ought to be equal. (Art. 1.) Thus 48=24X2=16X3=12X4=8X6. Therefore, the logarithm of 48 is equal to the sum of the logarithms of 24 and 2, of 16 and 3, &c. And, 3===14=18=44, &c. Therefore, the logarithm of 3 is equal to the difference of the logarithms of 6 and 2, of 12 and 4, &c. SECTION III. METHODS OF CALCULATING BY LOGARITHMS. ART. 36. The arithmetical operations for which logarithms were originally contrived, and on which their great utility depends, are chiefly multiplication, division, involution, evolution, and finding the term required in single and compound proportion. The principle on which all these calculations are conducted, is this: If the logarithms of two numbers be added, the sum will be the logarithm of the PRODUCT of the numbers; and, If the logarithm of one number be subtracted from that of another, the DIFFERENCE will be the logarithm of the QUOTIENT of one of the numbers divided by the other. In proof of this, we have only to call to mind, that logarithms are the EXPONENTS of a series of powers and roots. (Arts. 2, 5.) And it has been shown, that powers and roots are multiplied by adding their exponents; and divided, by subtracting their exponents. (Alg. 189, 193, 232, 239.) MULTIPLICATION BY LOGARITHMS. 37. ADD THE LOGARITHMS OF THE FACTORS: THE SUM WILL BE THE LOGARITHM OF THE PRODUCT. In making the addition, 1 is to be carried for every 10, from the decimal part of the logarithm, to the index. (Art. 7.) The logarithms of the two factors are taken from the |