. 



LOO-BOARD AXD LOO-BOOK. 



LOGARITHMS. 



not a fixed point : it is moreover safer to have a ship behind the 

 r+^^ing than before it. which induces many commanders to 

 ihorten the distance* between the knot* to forty-eight and even 

 forty fivr feet. Whatever distance be taken, it is found convenient to 

 subdivide into ten parta for decimals of a mile. Careful commanders 

 I sm ii in n the log line frequently, to ascertain if it varies from iu 

 original length. In case of an alteration they apply a correction to the 

 rate found by a ******** process in the rule of three a* the length 

 which the commander reckons upon is to the real interval, BO U the 

 apparent rate to the true rate. A similar correction is required if the 

 half minute glass i* found to be wrong. 



In the best regulated vessel* the log i* hove every hour ; and in 

 calculating the ship's going it is supposed that the rate has not varied 

 between the interval* of heaving; but if the wind bos sensibly 

 varied, or more or lea* sail has' been set during the time, then an 

 allowance is made aeoord.ng to the discretion of the person who keeps 

 the account. 



A lew year* ago a very curious log was invented by Mr. Hookey, 

 which though ingenious was too complex to come into general use ; it* 

 object wa* to afford as great a resistance as possible to the pull of the 

 line, and at the same time to be easily drawn back to the ship when 

 it* work was done. This log is shaped like a fish, and the line is in its 



::. ' 



A more practically useful suggestion of Mr. Hookey was to soak 

 th l.ne in a mixture of three parts linseed oil and one part fish 

 nil, which prevented its shrinking; a matter of no Muall import- 

 ance when it is considered that a new line without preparation will 

 lose 60 or 60 feet of its length by contraction when wetted. 



Of late years Mamey's patent log has been used with great benefit 

 to navigation, as the tune being noted when it is put overboard, it may i 

 he allowed to tow a-*tern for hours, at the expiration of which the j 

 number of revolutions of a vane attached to it, may be read off from 

 a self-registering dial, and compared with the time, from which is 

 deduced the ship's velocity. Iu form is too well known to need 

 further description. 



LOO-BOAKD and LOG-BOOK. These contain the account of the 

 ship's progress, as deduced from observations of the log. The log-board 

 is either a large piece of plank, blackened, ruled, and prepared for 

 writing on with chalk, or else a slate with divisions scratched upon Ha 

 surface. As soon as the seaman has hove the log, and the rate of 

 motion i* ascertained, the number of knots, with the odd tenths, or 

 odd fathom*, eight of which equal one knot, are written on the 

 board, each in iu proper ruled column ; also the course of the vessel, 

 the direction of the wind, and any remarks made at the moment 

 This is repeated every time the log is hove, and once in twenty-four 

 hour* the whole is copied into a book called the log-book, which is 

 ruled for the purpose in the same way as the log-board, and in which 

 also all the transactions relative to navigation are inserted, such as 

 bearings and distance of lands, rocks, and shoals, the direction and 

 velocity of currents, the state of the weather, and also whatever principal 

 operations are performed in the ship, such as reefing topsails, tacking. 

 wearing, 4c. It is also usual to set down every day the whole course 

 and distance run. calculated from the results of all the several trials 

 made by the log, with the distance and bearing of some port to which 

 the ship is approaching. The account thus obtained is technically 

 termed dead- reckoning, and is never quite correct, being subject to all 

 the errors caused by changing the direction and velocity iu the intervals 

 of observing, by the sort of guess usually made at the course and 

 rapidity of currents, and at the amount of the falling off of the vessel 

 from its apparent course, technically called lee-way. The dead reckoning 

 i* however necessarily used until an opportunity U afforded of taking 

 observations for latitude and longitude, or until some place whose 

 position is known come* in sight ; the true place of the ship is then 

 substituted in the log-book for that obtained by dead reckoning, and 

 frm that place subsequent reckoning* are made until another 

 observation. 



by 17 & 18 Viet. c. 104, the Board of Trade sanction certain forms 

 of log-books (called uffieial logs) as adapted to certain kinds of voyages, 

 and it U rendered imperative on every commander to see that all 

 entries therein be made a* soon a* possible after the occurrences happen 

 which are therein noted. 



The official log-book is supposed to contain, in addition to the nautical 

 memoranda above named, a faithful register of every crime and . 

 with their punishment, on board the ship ; every case of accident or 

 illness, with particulars of treatment ; the cause of every death ; a 

 register of birth* ; every marriage ; the sale of deceased seamen's 

 effects ; the character of each of the crew, or sufficient reason* for 

 withholding the opinion thtreupon; a full account of all accidents to 

 the ship and stores; detail* uf collision*, Ac. : indeed the ship'. I.>K 

 book funns not only a complete history of the voyage, but adds to our 

 knowlrdgit of marine statuUcs, and thus furthers materially the 

 intemu of commerce, while it* records promote the ends of justice, 

 -il log are received as evidence in anv -urt 

 of law. 



N-r la commerce alone the gainer by attention to the log-book. 

 Remark* on temperature, winds, weather, and indeed all notable dim nal 

 phenomena, have, by indefatigable comparimns and well digested de- 

 ns, enlightened th* meteorologist; while the hydrograp! 



enriched with no less zeal the treasures of nautical science from the 

 mere notes of the observant nailer. These would seem to be the only 

 legitimate subjects which claim a navigator's attention in keeping hi* 

 log-book, but the progress of education among sea-officers is shown 

 year by year in a manner as pleasing as it is important. The zoologist, 

 the natural historian, and indeed the philosopher himself, perceive and 

 acknowledge the vast benefits which a judicious naval observer may 

 yet confer on science. The field for research is a wide on. 

 heavens, the atmosphere, the ocean, each yields its quota of alworbinx 

 interest and improving pastime to the intelligent sailor, and impression* 

 ax indelible as they are ennobling lead him insensibly towards social 

 elevation. Little indeed should we know of currents, trade) winds, the 

 gulf stream, dust showers, &c., but for the seaman's " log-book," < 

 or private. 



LOGARITHMIC < THYK and LOGARITHMIC SPIRAL. The 



former has for it* rectangular equation y = a x , and its most remarkable 

 property U that ite subtnngent is the same at every point of the cum-. 



The latter has r---r for it* polar equation, ami ita tangent 

 makes the same angle with it* radius vector; win nee it u called tlic 

 equiangular spiral. 



LOtiAUITHMS. The etymology of thi- word is \oyi,v iptO^t, tl< 

 nml>tr i if Ihr rali-a : and the reason for the appcll .;tion .vill appear in 

 the course of this article. We assume tli ; has the common 



knowledge of logarithm*, nnd of the method of IIMHI; them. 



\\'e have abandoned the intention of "jivim; n vi.-u of the rise and 

 progress of logarithms, for the following reasons. The subject is non- 

 one of Kuch wide extent, when its theory and piact; 

 includid, that it would be like writing the history of a complete 

 science to put together all that would be needed in an article pro- 

 fessing to show the past and present state of logarithmic algebra, as 

 well as of logarithmic computation. If we were to confine ourselves to 

 the latter only, the view of the subject would be too confined. And 

 since the elements of the subject now usually given are clothed in the 

 most modern algebraical form, it would take considerable space to 

 explain at length the process of the early writers in terms intelligible to 

 those who are not conversant with their writings. We shall th : 

 devote the first part of this article to such explanations as will enable 

 the student, fresh from modem books of algebra, to read the various 

 histories which exist with facility : and we shall then point out how to 

 deduce the principal formulae connected with logarithm*. 



The early history of logarithms will W found at length in the 

 preface to Dr. Mutton's Table* : in the ' History of Logn: 

 tained in the first volume of Dr. Mutton's Tracts, in. Del.n 

 ' Histoire de 1'Astronomie Moderne.' vol. i. pp. 4!'l-.1ti>. !-. 



. ami, in the Bloc. Div.. N.\m:i:. Hitu..;*. (irxTUi, KUT.FII, 

 MF.BCATOR, &c. 



The idea of logarithms originally arose (in the mind of Kapler) from 

 the desire to make addition and subtraction supply the place of multi- 

 plication and division. A table, in which ore registered 1, o, -', a 3 . &c., 

 supplies this desideratum to a certain extent ; for since n* multiplied 

 by a" gives o*" 1 -', we find the product of the first by adding their 

 exponents, and looking in the table for the (.r + i/)th power. Thus 

 for the set 1, 2, 4, 8, 16, lie., a table of logarithms is easily con- 

 structed, a specimen of which is as follows : 



Num. log. Num. log. Num. lo(t. 



10 32 5 MiM in 



J 1 64 ti 2048 11 



4 2 128 7 4096 12 



8 3 256 8 -I'.'J 13 



10 4 512 9 i-::i*l 14 



Thus, to multiply 64 and 128, that is, to find the product of the sixth 

 and seventh powers of 2, we must t.-ike the (0+ 7Hh or 13th power, 

 which, from the table, is 8192. 



Such a table would be useless for general purposes, since it omit.-. 

 more numbers than it contains. But if we take a very little greater 

 tii -n unity, the powers will increase but slowly, and every whole number 

 within given limits may be made either a power of a, or very near to a 

 power of n. Suppose, for instance, that we wish for a table of 

 logarithms which shall contain among its numbers either every whole 

 number under a million or a fraction within A of every number nn.l.-i- 

 a million. Kxtinrt the xquare root of one million, the kqimre root of 

 that square root, and so on, until, say the rth root of one million has 

 been extracted, and let this rth root be 1 + 1. It is obvious that this 

 extraction may lie carried on until ( is small as we please. Con- 

 sequently (1 -r tY in a million, and (very !..., |...\\<i ,.f 1 + ris less 

 than a million, so that im standing for a million) no two conw 

 powers ditli-i i; .** the o-lli icnce of m and tit (1 + <), or by so 



much an ml. It then we prorci-il ni'li (I, , until ml be lew 



than //, we sh.ill have I of the drgr f Mnallin ., ie,|uii.d : that is, 



.'. l.olo number less than m lie* I 



1 -M, having exponents Ivse than r, .-iy such w! 



I within li 1 +1. 



Thin is in fact the first view whi-h was token of the method of 

 constructing tables of logarithms; and it must be leun n, lured that 

 Napier was not in possession of the. modern way of expressing the 

 powers of quantities. On the methods of facilitating such enormous 



