SECTION II. THEIR NUMBERS DIRECTIOXS FOR TAKING LOGARITHMS AND FROM THE TABLES.* Art. 24. The purpose which logarithms are intended to answer, is to enable us to perform arithmetical operations with greater expedition, than by the common methods. Before any one can avail himself of this advantage, he must become so familiar with the tables, that he can readily find the logarithm of any number; and, on the other hand, the number to which any logarithm belongs. In the common tables, the indices to the logarithms of the first 100 numbers, are inserted. But, for all other numbers, the decimal part only of the logarithm is given ; while the index is left to be supplied, according to ihe principles in arts. 8 and 11. 25. To find the logarithm of any number between 1 and 100; Look for the proposed number, on the left; and against it, in the next colunin, will be the logarithm, with its index. Thus The log. of 18 is 1.25527. The log. of 73 is 1.86332. 26. To find the logarithm of any number between 100 and 1000 ; or of any number consisting of not more than three significant figures, with ciphers annexed. In the smaller tables, the three first figures of each number, are generally placed in the left hand column; and the fourth figure is placed at the head of the other columns. Any number, therefore, between 100 and 1000, may be found on the left hand; and directly opposite, in the next column, is the decimal part of its logarithm. To this the index must be prefixed, according to the rule in art. 3. * The best English Tables are Hutton's in 8vo. and Taylor's in 4to. In these, the logarithms are carried to seven places of decimals, and proportional parts are placed in the margin. The smaller tables are numerous; and, when accurately printed, are sufficient for common calculations. The log. of 458 is 2.66087, The log. of 935 is 2.97081, of 796 2.90091, of 386 2.58659. If there are ciphers annexed to the significant figures, the logarithm may be found in a similar manner. For, by art. 14, the decimal part of the logarithm of any number is the same, as that of the number multiplied into 10, 100, &c. All the difference will be in the index; and this may be supplied by the same general rule. The log. of 4530 is 3.66087, The log, of 326000 is 5.51322, of 79600 4.90091, of 8010000 6.90363. 27. To jind the logarithm of any number consisting of four figures, either with, or without, ciphers annexed. Look for the three first figures, on the lest hand, and for the fourth figure, at the head of one of the columns. The logarithm will be found, opposite the three first figures, and in the column which, at the head, is marked with the fourth figure.* The log. of 6234 is 3.79477, The log. of 783400 is 5.69398, of 5231 3.71858, of 6281000 6.79803. 28. To find the logarithm of a number containing MORE than four significant figures. By turning to the tables, it will be seen, that if the differences between several numbers be small, in comparison with the numbers themselves; the differences of the logarithms will be nearly proportioned to the differences of the numbers. Thus The log. of 1000 is 3.00000, Here the differences in the of 1001 3.00043, numbers are, 1, 2, 3, 4, &c. of 1002 3.00087, and the corresponding difof 1003 3.00130, ferences in the logarithms, of 1004 3.00173, &c. are 43, 87, 130, 173, &c. Now 43 is nearly half of 87, one third of 130, one fourth of 173, &c. Upon this principle, we may find the logarithm of a number which is between two other numbers whose logarithms * In Taylor's, Hutton's and other tables, four figures are placed in the left band column, and the fifth at the top of the page. are given by the tables. Thus the logarithm of 21716 is not to be found, in those tables which give the numbers to four places of figures only. But by the table, the log. of 21720 is 4.33686 and the log. of 21710 is 4.33666 The difference of the two numbers is 10; and that of the logarithms 20. Also, the difference between 21710, and the proposed num. ber 21716 is 6. If, then, a difference of 10 in the numbers make a difference of 20 in the logarithms : A difference of 6 in the numbers, will make a difference of 12 in the logarithms. That is, 10 : 20::6 : 12. If, therefore, 12 be added to 4.33666, the log. of 21710; The sum will be 4.33679, the log. of 21716. We have, then, this RULE. To find the logarithm of a number consisting of more than four figures ; Take out the logarithm of two numbers, one greater, and the other less, than the number proposed : Find the difference of the two numbers, and the difference of their logarithms : Take also the difference between the least of the two numbers, and the proposed number. Then say, As the difference of the two numbers, numbers, and the proposed number, the least of the two logarithms. It will generally be expedient to make the four first figures, in the least of the two numbers, the same as in the proposed number, substituting ciphers, for the remaining figures; and to make the greater number the same as the less, with the addition of a unit to the last significant figure. Thus, For 36843, take 36840, and 36850, The first term of the proportion will then be 10, or 100, or 1000, &c. Ex. 1. Required the logarithm of 362572. of 362500 5.55931 The differences are 100, and 12. Then 100: 12::72 : 3.64, or 9 nearly. 3. The log. of 143542 is 5.15698 4. The log. of 1129535 is 6.05290. By a little practice, such a facility, in abridging these calculations, may be acquired, that the logarithms may be taken 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, is 2.88326, or 8.88326, (Art. 12.) of 0.00259, 3.41330, or 7.41330, of 0.0006278, 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. Froin 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. 135.) 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. (Art. 1.) To find then the logarithm of a vulgar fraction, subtract the logarithm of the denominator from that of the numerator. The 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 34. of the denominator 1.93952 of the fraction 1.59196, or 9.59196. The logarithm of 2019 is 2.66362, or 8.66362. of u 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 31= 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 |