Elements of Algebra

CHAP. XXIII. OF ALGEBRA. 73 have $log. 3650 = 3+x$; $log. 36500 = 4+x$; $log. 365000 = 5+x$, and so on. But we can also go back, and continually divide by 10; which will give us $log. 36.5 = 1+x$; $log. 3.65 = 0+x$; $log. 0.365 = -1+x$; $log. 0.0365 = -2+x$; $log. 0.00365 = -3+x$, and so on. 251. All those numbers then which arise from the figures $365$, whether preceded, or followed, by ciphers, have always the same decimal fraction for the second part of the logarithm: and the whole difference lies in the integer before the point, which, as we have seen, may become negative; namely, when the number proposed is less than $1$. Now, as ordinary calculators find a difficulty in managing negative numbers, it is usual, in those cases, to increase the integers of the logarithm by $10$, that is, to write $10$ instead of $0$ before the point; so that instead of $-1$ we have $9$; instead of $-2$ we have $8$; instead of $-3$ we have $7$, &c.; but then we must remember, that the characteristic has been taken ten units too great, and by no means suppose that the number consists of $10$, $9$, or $8$ figures. It is likewise easy to conceive, that, if in the case we speak of, this characteristic be less than $10$, we must write the figures of the number after a point, to shew that they are decimals: for example, if the characteristic be $9$, we must begin at the first place after a point; if it be $8$, we must also place a cipher in the first row, and not begin to write the figures till the second: thus $9.5622929$ would be the logarithm of $0.365$, and $8.5622929$ the log. of $0.0365$. But this manner of writing logarithms is principally employed in Tables of sines. 252. In the common Tables, the decimals of logarithms are usually carried to seven places of figures, the last of which consequently represents the $\frac{1}{10000000}$ part, and we are sure that they are never erroneous by the whole of this part, and that therefore the error cannot be of any importance. There are, however, calculations in which we require still greater exactness; and then we employ the large Tables of Vlacq, where the logarithms are calculated to ten decimal places*. * The most valuable set of Tables we are acquainted with are those published by Dr. Hutton, late Professor of Mathematics at the Royal Military Academy, Woolwich, under the title of, "Mathematical Tables; containing common, hyperbolic, and logistic logarithms. Also sines, tangents, &c. to which is prefixed a large and original history of discoveries and treatises relating to these subjects."