19. To find the value of e, the base of the Napierian system of logarithms. If x=1, and the base be e, the series for am in the last Article becomes 1 1 + + 1.2 1.2.3 1.2.3...n Now, +1 2 1 + + 2:7182818 This gives the correct value of e, the base of the Napierian system, so far as the figures are put down. By taking more terms and a greater number of figures in each, we could determine the value of e to any required degree of accuracy. 20. On the construction of the common tables of logarithms. From one of the series (v), (vi), (vii), the Napierian logarithms of low primes may be found. The logarithm of a high number, which is not a prime, may be determined from the logarithms of its factors, by resolving the number into powers of its prime factors; Thus, le 288 = 16 23. 3= )2+13= 5 1,2 + 2 * 123. And the expressions (viii), (ix), (x), will greatly facilitate the finding of the logarithms of high numbers which are primes. The Napierian logarithms having been determined, the tables to base 10 are deduced from them by multiplying each by which is equal to ·434294819... Art. 3. 1 1,10 Some of the artifices used in computing the tables may be found in Sharpe's “Method of making Logarithms” prefixed to Sherwin's Tables. 21. In the common tables are registered the mantissæ of the logarithms of numbers of five places of figures, these mantissæ being computed to seven places of decimals; and at the side of each page is placed a “ Table of proportional parts,” by which, as it will be shewn, may be found the mantissa of the logarithm of a number containing six or seven places of figures: and conversely, if a logarithm be given whose mantissa is not contained exactly in the tables, the number corresponding to it may, by means of these additional tables, be determined to six or seven places of figures. 22. On the construction and use of the tables of proportional parts. Let m, and m, be the mantissæ of two consecutive integral numbers, n and n + 1, which contain five digits each, and let m be the mantissa of the number n + which contains 10 six digits, the last of which (a) is after the decimal point. a a Now, since n and n + have the same number of integral 10 places, their logarithms have the same characteristic; which, by expanding 110 (1 + ( nom) by (ii.) and neglecting the succeeding terms as being small compared with the first term, becomes Whence m may be found, if m,, my, a, be given,—or a may be determined, when my, my, m, are known. 23. By the Tables, page 165, Mantissa of 36633 = m, = •5638725 36632 = m = •5638606 .. ma – mi = .0000119 And in the expression m – m,= (my – m) o, writing for a the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 successively, we have Now, the “ difference” put down in the Tables for numbers near 36600, is 119, and the Table of proportional parts is as in the margin of the larger Table. We see then, that the significant digits only of the whole difference, and of the differences corresponding to the several digits, are inserted in the table of proportional parts. Hence, the following method of constructing tables of proportional parts is evident : Of the significant part of the whole difference point off the last digit as a decimal—this corresponds to the division of the whole difference by 10, when the whole difference is treated as an integer. an integer. Multiply this number by 1, 2, 3,...9 successively, and the whole numbers thus obtained (the last digit in the integral part being increased by unity where the decimal part is not less than •5) are the significant parts of the differences for the digits respectively. Thus, let the significant part of the whole difference be 156. 15.6 x 1 = 15.6 = 16 nearly. | 15.6 x 6 = 93.6 = 94 nearly. By constructing a table of this kind we see, that if the digit be given, the difference is immediately known, or vice versâ. To avoid the necessity of performing the operation of subtraction in any particular case, in order to find the whole difference, there is a line in the tables marked at the top with “Diff.”, in which the difference is placed opposite to that logarithm at which such difference begins. To know what the difference therefore is in any particular case, it is merely requisite to take the number in this line next above the logarithm in question. Ex. '1... To find the number whose logarithm is 3.5677766. 001 By the tables, p. 165, the mantissa next below the given mantissa is that of 11,36963, and the whole difference put down is 117. By the table of proportional parts to “ Diff.” 117, the difference 94 corresponds to the digit 8,—therefore the significant part of the number sought is 369638; also since the given logarithm has 3 for its characteristic, the number required is 3696-38. 24. To find the mantissa of the logarithm of a number which has seven places of digits. a nt + Let m2 be the mantissæ of n and n +1, two successive integers of five places each: let M be that of 6 which has the same number of integral places 10 100 as n and n + 1, and also one digit (a) in the place of the tenths, and another (6) in that of the hundredths. a b Now, since n and n + + have the same number of 10 -1003 integral places, their logarithms have the same characteristie ; & |