The calculation of tables of logarithms may lie effected in many ways. Henry Briggs (1624), who suggested the common system with base 10. calculated to fourteen decimal places the loga rithms of numbers from 1 to 20.000 and from 90,000 to 100,000. From the logarithms of perfect powers of 10, he approximated the intermediate logarithms by continually computing geometric means between two numbers, one greater and the other less than the number required. Thus. to find the log 5 take the geometric mean between 1 and 10, or 3.162, the corresponding arithmetic mean (log 1 being O. and log 10 being 1) being 0.5; the geometric mean between 3.162 . . . and 10. or 5.623. corresponds to the arithmetic mean between 0.5 and 1, or 0.75, the geometric mean between 3.162 and 5.623, or 4.216, has its loga rithm = (0.75 + 0.5) or 0.625. This opera tion is continued until the result is obtained to the necessary degree of accuracy. More recent methods, however, are based upon the logarithmic series. If I he put for a in the formula, r 1 (u + 1) = u 2 5(2u + the Naperian logarithm of 2 is at once obtained to any degree of accuracy required; if 2 he put for u, the Naperian loga rithm of 3 can be calculated, etc.. and time com mon logarithms may he obtained by applying the modulus.
Vlacq (1628) supplied the logarithms for the numbers omitted by Briggs. Gellibrand and Vlacq published tables for the logarithms of the trigonometric functions for every minute of the quadrant.
For a full account of the construction of the early tables, consult the introduction to II utton's Mathematical Tables and Mathematiral Tracts (London. 1812). Gauss introduced addition and
subtraction logarithms and computed table,. N% hieh have been largely drawn upon by subsequent writers. Vega's Thesaurus Logarit hmorn Com plet us (Leipzig. 1794) had a wide circulation. Contributions to the rapid calculation of loga rithms have been made by Koralek (1851), and especially by R. Hoppe in hi-, Tufebi :Air dr( issig stelligra logarithmischen Rcehnung ( Leipzig. 1876). Other thoroughly reliable tables are Bremiker (Berlin, 1857; Ilth ed. 1890) ; Sehriin (Leipzig, 1800, 1886-90; English ed. by De Mor gan, London. 1865) ; Callet (Paris. 1795, and subsequent editions).
LOGAU, 16'g,on. FRIEDRICH, Baron (1604-55). A German epigrammatist, born at Broekut, in Silesia. and educated at and Frankfort. He entered the legal service of the Duchy of Lieg nitz as chancery councilor; but is far better known as a poet of the 'first Silesian school.' as the 'Detractor' of the Fruchtbringende Ucsell sehaft, which he joined in 1648, and, under the pseudonym Salomon von Golaw, as author of a collection of epigrams. Zweghundert tent seller leeimspriiche (1638). A second volume is en titled licutseher Sinugediehte drei-tausend ( 1654 ) . An edition of selections from Logan's epigrams. with linguistic commentary, was published by Lessing and Ramler (Leipzig. 1759) : a complete edition by Either (Stuttgart. 1872) succeeded his smaller edition with biography (Leipzig,