Logarithmic Tables

tables of the logarithms of numbers; used to simplify calculations. Tables of common (Briggs’) logarithms are the most widely used. Since the common logarithms of the numbers N and 10^kN (for integral k) differ only in their characteristic and have the same mantissa (log 1 0^kN = k + log N), only the mantissas of logarithms of integers are given in the tables of common logarithms. We find the characteristic by the following rules: (1) the characteristic of a number greater than 1 is one less than the number of digits in the integral part of the number (thus, log 20,000 = 4.30103) and (2) the characteristic of a decimal less than 1 is equal to the number of zeros preceding the first nonzero digit in the decimal, taken with a minus sign (thus, log 0.0002 = 4.30103; therefore common logarithms of fractions are written in the form of a sum of a positive mantissa and a negative characteristic).

The number of digits in the mantissa may vary in different tables of common logarithms. Four-place and five-place tables are the most common. Seven-place tables, and in rare cases tables that permit logarithms having more digits to be calculated without great difficulty, are sometimes used. Logarithmic tables often include tables of antilogarithms—numbers whose logarithms are the given numbers—and tables of Gaussian logarithms, which are used for determining the logarithms of a sum or difference of two numbers in terms of the known logarithms of these numbers (without an intervening calculation of the numbers themselves). In addition to the logarithm of numbers, logarithmic tables usually also contain the logarithms of trigonometric values.

The first logarithmic tables were independently compiled by J. Napier and the Swiss mathematician J. Bürgi. Napier’s tables, Mirifici logarithmorum canonis descriptio (1614) and Mirifici logarithmorum canonis constructio (1619), contained eight-place logarithms of the sines, cosines, and tangents of angles from 0° to 90° at intervals of one minute. Since the sine of 90° was at that time considered equal to 10⁷ and it was often also necessary to multiply by it, Napier calculated his logarithms so that the logarithm of 107 would be equal to zero. The logarithms of the remaining sines less than 107 were then taken as positive. Napier did not introduce the concept of the base of a system of logarithms. His logarithm of the number N in modern notation is approximately equal to 10⁷ ln (10⁷/N). The properties of Napier’s logarithms are somewhat more complex than those of common logarithms since the logarithm of unity in his system differs from zero.

Bürgi, in Arithmetische und geometrische Progres tabulen (1620), gave the first table of antilogarithms (“black numbers”) and the values of numbers corresponding to equally spaced logarithms (“red numbers”). Bürgi’s red numbers are the logarithms of the black numbers divided by 10⁸ to the base (1 + (1/104)) 104.

Bürgi’s tables, and particularly those of Napier, rapidly drew the attention of mathematicians to the theory and calculation of logarithms. On the advice of Napier, the British mathematician H. Briggs calculated eight-place common logarithms (1617) for the numbers 1 to 1,000 and, subsequently, 14-place logarithms (1624) for the numbers 1 to 20,000 and 90,000 to 100,000 (common logarithms are sometimes called Briggs’ logarithms in his honor). The Dutch mathematician A. Vlacq published (1628) ten-place tables for the numbers 1 to 100,000. Vlacq’s tables are the basis of most subsequent tables, although later authors introduced many variations into the structure of their logarithmic tables and made corrections in the computations. (In fact, there were 173 errors in Vlacq’s tables and five errors in the tables published by the Austrian mathematician G. Vega in 1783. The first tables completely free of error were published in 1857 by the German mathematician K. Bremiker.) In Russia, logarithmic tables were first published in 1703 by L. F. Magnitskii. Tables of Gaussian logarithms were published in 1802 by the Italian mathematician Z. Leonelli. K. F. Gauss introduced these logarithms into general use in 1812.