Hence, the characteristic of the logarithm of a decimal fraction having n - 1 cyphers after the decimal point, is n. Generally therefore, the characteristic of the logarithm of any number is the number of its digits minus one,—where if the number be a decimal fraction, the cyphers which follow the decimal point are alone counted, and are reckoned negatively. Conversely therefore, if logarithms be given having characteristics 1, 2, 3...ī, 2, 3... there are in the integral parts of the numbers to which these logarithms belong, 2, 3, 4...... 0, -1, -2...digits respectively. Thus the logarithms of 254 and 25400 have for characteristics 2 and 4, and the characteristics of the logarithms of 2:54, 25.4, .000254 are 0, 1, 4. The mantissa given in the tables for 110 3652 is ·5625308. (p. 165). 1103652 = 3.5625308, 11036.52 = 1.5625308, 110365200 = 5.5625308, 11•3652 = 1.5625308, 110.003652 = 3.5625308. 10. The peculiar advantages of Briggs' system of logarithms. From the last Article it appears, that if the base of the system be 10, it is requisite to register the mantissæ only in the tables, because the characteristics can be determined by counting the digits in the integral part of the number whose logarithm is required. This omission of the characteristics renders the common tables less bulky than those calculated to any other base. Also, from Art. 8, it appears, that in this system the N mantissa of N is also the mantissa of 10" X N, and of where n is any integer :—this circumstance renders the common tables more comprehensive than if any other base were taken ;—for if any other base were used, the mantissa of N N would not be the mantissa of 10”. N, or of 10"? 10" In the same way it might be shewn that if our arithmetic were duodecimal, instead of being decimal, tables calculated to the base 12 would possess the same advantages which have been here proved to belong to the tables in common use. 11. The tables of logarithms in common use register, either to five, or to seven, places of decimals, the mantissæ for numbers from 1 to 100000. There are printed at the end of this Appendix two pages of logarithms in which the mantissä are calculated to seven places of decimals. The line at the top of the second page begins with the number 3650, or 36500, and its mantissa •5622929 is placed opposite to it. And because the mantissæ of all numbers from 36500 to 36559, (comprised in the first six lines of the page), have the same initial three figures, viz. •562, these three figures are registered, once for all, opposite to the number 3650, and the four last figures of each succeeding mantissa are placed under the number to which they belong. Thus the first line of the page In the same manner, the next line gives the mantissä of numbers from 36510 to 36519 inclusive. 12. Since a change in the value of the three initial figures may not take place at the beginning of one of the horizontal lines, whenever a mantissa is to be taken from the tables, we must be sure to get the right initial figures. Thus, (see p. 165), the mantissa of 36643 is ·5639910, and the last four figures of the mantissa of 36644 are put down as 0029. Now if the mantissæ of these two numbers had the same initial three figures, the mantissa of 36644 would be less than that of 36643, which we know cannot possibly be the case. A change in the value of the three initial figures does, in point of fact, take place here,—and the mantissa of 36644, (as do those of the numbers immediately following 36644), begins with 564, and not with •563. Similar changes of the initial three figures of the mantissa occur at the numbers 36729, 36813, 36898, 36983, and are indicated by printing in a smaller type the fourth figure of the mantissæ of those numbers. The construction and use of the small tables in the last column of the page will be explained hereafter. (1). To multiply 23 by 6. Art. 4. Mantissa of 16 is .2041200; 11.16 = 1.2041200 2.5658478 And, p. 165, the significant digits corresponding to the mantissa •5658478 are 3680; 11,368•0 or 110368 = 2:5658478; and 368 is the product sought. And, p. 164, we find this mantissa to be that corresponding to the significant digits 72; ::: Z•8573325 = 110.072; and .072 is the quotient sought. (4). To find the values of (15-4), and (650)} Art. 6. 11015.4 = 1·1875207 p. 164. 3 3.5625621 = 1103652-3, nearly, p. 165. :: 3652-3 is the approximate cube of 15.4. Again, 110650 = 2.8129134, p. 164. - 1,0650 •5625826 = 1103•6524, nearly, p. 165. .. 3•6524 is the approximate fifth root of 650. (5). To find the values of (085)", and of (000065). 2.9294189 = 110.085 4 5•7176756 and this is the mantissa of 522006, nearly. Therefore ·0000522006 is the number sought, nearly. Again, 110.000065 = 5.8129134 = 6 + 1.8129134 ; 3) 5•8129134 2:6043044 = 110040207, nearly. 14. To expand 1,(1 + x) in a series ascending by powers of x. Now a' = 1, or 1,1 = 0; and 1,(1 + x) becomes 1.1 when x vanishes,—the series therefore which expresses the value of 1,(1 + x) cannot involve any negative power of x, (or it would become infinite when x vanished,) nor can it have a constant term, (or it would not vanish along with x). Let then A x + Bw? + C 203 + = 1,(1 + x); (1). :. also, A(x + h) +B(x + h)* + C(x + h)' + ... = 1,(1 + x + h), by writing x + h instead of w. By subtraction, A.{(x + h) – x} + B.{(x + h)' – x?}+C.{(x + h): – 28} + ... or Ah + B.(2xh + h) + C.(3x* h + 3x h2 + h) + ... = 1,(1 + x + h) - 1,(1 + x) h 1 + =1/(1+ ) Dividing both sides of this equation by h, we have, |