Page:Encyclopædia Britannica, Ninth Edition, v. 14.djvu/801

logarithms omitted in Schulze’s work, and which Wolfram had been prevented from computing by a serious illness, were published subsequently, and the table as given by Vega is complete. The largest hyperbolic table as regards range was published by Zacharias Dase at Vienna in 1850 under the title Tafel der natürlichen Logarithmen der Zahlen. It gives hyperbolic logarithms of numbers from 1000.0 to 10500.0 at intervals of 1 to 7 places, with differences and proportional parts, arranged like an ordinary seven-figure table of Briggian logarithms. The table appeared in the thirty-fourth part (new series, vol. xiv.) of the Annals of the Vienna Observatory (1851); but separate copies were issued.

Hyperbolic antilogarithms are simple exponentials, i.e., the hyperbolic antilogarithm of x is e^x. A seven-figure table of e^x and its Briggian logarithm from x = .01 to x = 10 at intervals of .01 is given in Hülsse’s edition of Vega’s Sammlung, and in other collections of tables; but by far the most complete table that has been published occurs in Gudermann’s Theorie der potential- oder cyklisch-hyperbolischen Functionen, Berlin, 1833. This work consists of reprinted papers from Crelle’s Journal, and one of the tables contains the Briggian logarithms of the hyperbolic sine, cosine, and tangent of x from x = 2 to x = 5 at intervals of .001 to 9 places, and from x = 5 to x = 12 at intervals of 0.01 to 10 places. Since the hyperbolic sine and cosine of x are respectively 1/2(e^x − e^−x) and 1/2(e^x + e^−x), the values of e^x and e^−x may be deduced from the results given in the table by simple addition and subtraction.

Logistic numbers is the old name for what would now be called ratios or fractions. Thus a table of log a/x, where x is the argument and a a constant, is called a table of logistic or proportional logarithms; and since log a/x = log a - log x it is clear that the tabular results differ from those given in an ordinary table of logarithms only by the subtraction of a constant and a change of sign. The first table of this kind appeared in Kepler’s Chilias logarithmorum (1624) already referred to. The object of a table of log a/x is to facilitate the working out of proportions in which the third term is a constant quantity a. In most collections of tables of logarithms, and especially those intended for use in connexion with navigation, there occurs a small table of logistic logarithms in which a = 3600″ (−1 or 1{{sup|h), the table giving log 3600 − log x, and x being expressed in minutes and seconds. It is also common to find tables in which a= 10800″(= 3° or 3^h), and x is expressed in degrees (or hours), minutes, and seconds. Such tables are generally given to 4 or 5 places. The usual practice in books seems to be to call logarithms logistic when is 3600″, and proportional when a has any other value.

Gaussian logarithms are intended to facilitate the finding of the logarithms of the sum and difference of two numbers whose logarithms are known, the numbers themselves being unknown; and on this account they are frequently called addition and subtraction logarithms. The object of the table is in fact to give log (a±b) by only one entry when log a and log b are given. The utility of such logarithms was first pointed out by Leonelli in a book entitled Supplement logarithmique, printed at Bordeaux in the year XI. (1802-3); this work being very scarce, a reprint of it was published by M. J. Houel in 1876. Leonelli calculated a table to 14 places, but only a specimen of it which appeared in the Supplement was printed. The first table that was actually published is due to Gauss, and was printed in Zach’s Monatliche Correspondenz, vol. xxvi. p. 498 (1812). Corresponding to the argument $\mathrm {A}$ it gives, to 5 places, $\mathrm {B}$ and $\mathrm {C}$, where

$\mathrm {A} =\log x$,

$\mathrm {B} =\log \left(1+{\frac {1}{x}}\right)$,

$\mathrm {C} =\log \left(1+x\right)$.

so that $\mathrm {C} =\mathrm {A} +\mathrm {B}$.

We have identically

$\log \left(a+b\right)=\log a+\log \left(1+{\frac {b}{a}}\right)=\log a+\mathrm {B}$ (for argument loga/b);

and, in using the table, the rule is to take log a to be the larger of the two logarithms, and to enter the table with log a − log b as argument; we then have log (a + b) = log a + $\mathrm {B}$, or, if we please, = log b + $\mathrm {C}$. The formula for the difference is log (a − b) = log b + $\mathrm {B}$ (argument sought in column $\mathrm {C}$) if log a − log b is greater than .30103 and = log b − $\mathrm {A}$ (argument sought in column $\mathrm {B}$ ) if log a − log b is less than .30103.

The principal tables of Gaussian logarithms are (1) Mathiessen, Tafel zur bequemern Berechnung (Altona, 1818), giving B and C for argument A to 7 places,—this table is not a convenient one; (2) Peter Gray, Tables and Formulæ (London, 1849), and Addendum (1870), giving full tables of C and log (1 − x) for argument A to 6 places; (3) Zech, Tafeln dur Additions und Subtractions—logarithmen (Leipsic, 1849), giving 7-place values of B for argument A, and 7-place values of C for argument B. These tables appeared originally in Hülsse’s edition of Vega’s Sammlung (1849); (4) Wittstein, Logarithmes de Gauss (Hanover, 1866), giving values of B for argument A to 7 places. This is a large table, and the arrangement is similar to that of an ordinary seven-figure table of logarithms.

In 1829 Widenbach published at Copenhagen a small table of modified Gaussian logarithms giving log x + 1/x - 1(= D) corresponding to A as argument; A and D are thus reciprocal, the relation between them being in fact 10^{A +D} = 10^A + 10^D + 1, so that either A or D may be regarded as the argument.

Gaussian logarithms are chiefly useful in the calculations connected with the solution of triangles in such a formulæ as cot 1/2C = a + b/a - btan (A - B), and in the calculation of life contingencies.

Calculation of Logarithms.—The name logarithm is derived from the words λόϒων άριΘμός, the number of the ratios, and the way of regarding a logarithm which justifies the name may be explained as follows. Suppose that the ratio of 10, or any other particular number, to 1 is compounded of a very great number of equal ratios, as for example 1,000,000, then it can be shown that the ratio of 2 to 1 is very nearly equal to a ratio compounded of 301,030 of these small ratios, or ratiunculæ, that the ratio of 3 to 1 is very nearly equal to a ratio compounded of 477,121 of them, and so on. The small ratio, or ratiuncula, is in fact that of the millionth root of 10 to unity, and if we denote it by the ratio of a to 1, then the ratio of 2 to 1 will be nearly the same as that of a^301,030 to 1 and so on; or, in other words, if a denotes the millionth root of 10, then 2 will be nearly equal to a^301,030, 3 will be nearly equal to a^477,121, and so on.

Napier’s original work, the Descriptio canonis of 1614, contained, not logarithms of numbers, but logarithms of sines, and the relations between the sines and the logarithms were explained by the motions of points in lines, in a manner not unlike that afterwards employed by Newton in the method of fluxions. An account of the processes by which Napier constructed his table is given in the Constructio canonis of 1619. These methods apply, however, specially to Napier’s own kind of logarithms, and are different from those actually used by Briggs in the construction of the tables in the Arithmetica logarithmica, although some of the latter are the same in principle as the processes described in an appendix to the Constructio. It may be observed that in the Constructio logarithms are called artificials, and this seems to have been the name first employed by Napier, but which he subsequently replaced by logarithms. It is to be presumed that he would have made the change of name also in the Constructio, had he lived to publish it himself.

The processes used by Briggs are explained by him in the preface to the Arithmetica logarithmica (1624). His method of finding the logarithms of the small primes, which consists in taking a great number of continued geometric means between unity and the given primes, may be described as follows. He first formed the table of numbers and their logarithms:

Number.	Logarithm.
10.162277 …‍	1
3.162277 …	0.5
1.778279 …	0.25
1.333521 …	0.125
1.154781 …	0.0625

each quantity in the left hand column being the square root of the one above it, and each quantity in the right hand column being the half of the one above it. To construct this table Briggs, using about thirty places of decimals, extracted the square root of 10 fifty-four times, and thus found that the logarithm of 1.00000 00000 00000 12781 91493 20032 35 was 0.00000 00000 00000 05551 11512 31257 82702, and that for numbers of this form (i.e., for numbers beginning with 1 followed by fifteen ciphers, and

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