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 fig ures, and at the top, over the logarithm, the fourth figure, of the number required. This number, by pointing off decimals, 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 2.89115 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, 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 to the number 471.2 The number required is 471.244. The natural number belonging to 1.73698 is 54.57357, 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 48-24 X2=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. 21 And 3===Y=V=Y, &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. 233, 237, 280, 286.) 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 tables. The product is obtained, by finding, in the tables, the natural number belonging to the sum. (Art. 33.) 38. When any or all of the indices of the logarithms are negative, they are to be added according to the rules for the addition of positive and negative quantities in algebra. But it must be kept in mind, that the decimal part of the logarithm is positive. (Art. 10.) Therefore, that which is carried from the decimal part to the index, must be considered positive also. In each of these examples, +1 is to be carried from the decimal part of the logarithm. This added to 1, the lower index, makes it 0; so that there is nothing to be added to the upper index. If any perplexity is occasioned, by the addition of positive and negative quantities, it may be avoided, by borrowing 10 to the index. (Art. 12.) Here 10 is added to the negative indices, and afterwards rejected from the index of the sum of the logarithms. Here +1 carried to -4 makes it 3, which added to the upper index +1, gives 2 for the index of the sum. Multiply .00845 3.92686 or 7.92686 Into 1068. 39. Any number of factors may be multiplied together, by adding their logarithms. If there are several positive, and several negative indices, these are to be reduced to one, as in algebra, by taking the difference between the sum of those which are negative, and the sum of those which are positive, increased by what is carried from the decimal part of the logarithms. (Alg. 78.) Ex. 2. The prod. of 36.4 × 7.82×68.91 ×0.3846 is 7544. 3. The prod. of 0.00629 ×2.647×0.082 × 278.8×0.00063 is 0.0002398. 40. Negative quantities are multiplied, by means of logarithms, in the same manner as those which are positive. (Art. 16.) But, after the operation is ended, the proper sign must be applied to the natural number expressing the product, according to the rules for the multiplication of positive and negative quantities in algebra. The negative index of a log |