LOGAKITHMS. 407 LOG-BOOK. the radii 0A„, OAi ( r = 1 ) at right angles, the sine S„Si parallel to OAi moves from to A„ at intervals forming an arithmetic progression, its value decreases in geometric progression. The segment 0S„ Napier originally called numerus artificialis, and later logarithmus (ratio num- ber ) . The first computers of logarithms did )iot understand the connection between logarithms and exponents, but modern investigations show that 2.7184593 is the base of 13iirgi'.s logarithms, and that j.^rTiVriu? i* the base of Napier's. Speidell ( 1G19. 1024), in adapting Naperian loga- rithms to positive integers, employed as a base 2.718281828. known since the time of Euler as e. This system was called by Halley the Nape- rian system, and this name has been retained, so that to-day 'Naperian logaritlmis' mean loga- rithms to the base e. Such logarithms are called ndtnral logarithms, and the relation between Naperian logarithms of a sine S and its natural logarithm is expressed by the equation Nap. log S =: 10' nat. log -^ , 10' being taken as the sine of 90" and its logarithm zero. According to the /dx — = log X -- h may serve as a definition for the logarithm of a number, and from it is derived a relation which gives to the natural logarithm the name hyper- bolic logarithm. The equation of a rectangular hyperbola (q.v.) referred to its asymptotes is a'l/ = c, hence 2/ = - , and ydx = — , an element of area of base dx and altitude y between the curve and the a;-axis ; the area between the curve, the a;-axiS| and two ordinates at a-j, x^ is propor- tional to log — ■ It appears from the definition that logarithms formed from one base must bear a constant ratio to those formed from another base. This ratio is called the modulus of the first system. The modulus of the common system is 0.43429448 . . . hence if the hyperbolic logaritlim of a number is /. its common logarithm is 0.43429448J. The calculation of tables of logarithms may be 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 appro.ximatcd the intermediate logarithms by continually computing geometric moans between two numbers, one greater and the other less than the number i-equired. Thus, to find the log .*) take the geometric mean between 1 and 10, or 3.102, the corresponding arithmetic mean (log 1 being 0. and log 10 being 1) being 0.5; the geometric mean between 3.162 . . . and 10, or 5.023, corresponds to the arithmetic mean between 0.5 and 1, or 0.75, the geometric mean between 3.162 and 5.623, or 4.210, has its loga- rithm = lo (0.75-4-0.5) or 0.625. This opera- tion is continued until the result is obtained to fhe necessary degree of accuracy. Jltu'e recent methods, however, are based upon the logarithmic series. If 1 be put for u in the formula, log, (» + l)=log„tt+2 [2^i+3{2u + lf+ i u . . . . , the Naperian logarithm of 5(2m -f 1)* J 2 is at once obtained to any degree of accuracy required ; if 2 be put for u, the Naperian loga- rithm of 3 can be calculated, etc., and the com- mon logaritlmis may be obtained by applying the modulus. Vlacq (1028) supplied the logarithms for the number.s omitted by Briggs. Gellibrand and Vlacq published tables for the logarithms of the trigonometric functions for e'ery minute of thf quadrant. For a full accovmt of the construction of the early tables, consult the introduction to Hutton's Mathematical Tables and Mathematical Tracts (London, 1812). Gauss introduced addition and subtraction logarithms and computed tables which have been largely drawn upon by subsequent writers. Vega's Thesaurus Logarithmorum Com- pletue (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 his Tafcln zur drcissig- stcllic/oi logarithmischen Rcchnung (Leipzig, 1876). Other thoroughly reliable tables are Bremiker (Berlin, 1S57; 11th cd. 1890); Schnin (Leipzig, 1860, 1886-90; English ed. by De Mor- gan, London. 1805) ; Callet (Paris, 1795, and subsequent editions ) . LOGATJ, lo'gou, Friedbicii, Baron (1604-55). A German epigrammatist, bom at Brockut. in Silesia, and educated at Brieg 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 Gesell- schaft, which he joined in 1048, and, under the pseudon.^nn Salomon von Golaw, as author of a collection of epigrams, Zweyhundcrt tcutscher Ucimspri'tchc (1038). A second volume is en- titled Deutscher Sinngedichte drci-tausend { 1654) . An edition of selections from Logau's epigrams, with linguistic commentary, was published by Lessing and Ramler (Leipzig. 1759) ; a complete edition by Eitner (Stuttgart, 1872) succeeded his smaller edition with biography (Leipzig, 1870). LOG-BOOK. The log-hook is the official record- book of a ship and contains a brief statement of the weather encountered, the speed made, positions of the ship as daily determined by astro- nomical observations or by dead reckoning, and a short account of all occurrences of importance at sea and in port. The logbook of a man-of-war is a large book of folio size and two pages facing each other are allotted to the records for each day. The left page is about half covered by the 'columns.' These are filled in every hour at sea and in ]iort, and the record consists of the speed of the sliip during each hour, the reading of the patent log, the course by compass, the leeway (if any), the height of barometer and thermom- eter, readings of the hygrometer, temperature of the sea-wat<>r, amount of clouds and their charac- ter and movement, and the state of the sea. Below the ruled spaces are spaces for recording the amounts of coal and distilled water on hand and received: the latitude, longitude, current, devia- tion of the compass, etc., as determined by ob- servations of heavenly bodies and by dead reck- oning; and other information. The right page is for remarks upon miscellaneous subjects and amplifying the data given in the columns when that is necessarv. The remarks are written up
