1 1 



NAVIC 



[LOGARITHMS. 



than Tolanx* of proacribed directions ; and even in the 

 -. winch form irt of the evory-day 

 routine on shipboard, time and figure* are often both 

 thrown *ry by the unscientific mariner, under the im- 

 pnMtiuu that be is increasing the accuracy of his work, 

 hen he w, in reality, only encumbering it with errors. 

 A Tory meritorious writer, Lieut Raper, justly remarks 

 that -'very indistinct and erroneous notions ju 

 among practical persons on the subject of accuracy of 

 computation, and much time is, in consequence, often 

 lout in coini'iitiii,- to a degree of precision wholly incon- 

 sistent with that of the elements themselves. The mere 

 habit of working invariably to a useloss precision, while 

 it can never advance the computer's knowledge of the 

 subject, has the unfavourable tendency of deceiving 

 those who are not aware of the true nature of such ques- 

 tions, into the persuasion that a result is always as cor- 

 rect as the computer chooses to make it ; and thus leads 

 them to place the same confidence in all observations, 

 provided onlv they are worked to the same degree of ac- 

 curacy."* These are very judicious observations ; aud 

 .1. Raper is, as far as we know, the only person who 

 has drawn attention, in print, to this customary waste of 

 calculation. 



The tables which accompany books on navigation niro, 

 in general, computed to an extent of decimals usually 

 six or seven places much beyond what the ordr 

 calculations of navigation require ; aud the unscientific 

 seaman uses them all, when half the number would, in 

 most cases, be amply sufficient : his fault is analogous to 

 that of the ill-taught school-boy, who, having to multiply 

 together two numbers of five decimals the final decimal 

 of each number being confessedly inaccurate is at the 

 trouble of computing ten decimals iu his product, and 

 fancies, if there be no error in his operation, that they 

 are all correct, whereas five of them at least are wrong, 

 aud, therefore, worse than useless. 



We shall now proceed to those preliminary matters 

 which furnish the suitable introduction to the different 

 sailings plane sailing, parallel sailing, middle latitude 

 sailing, and M creator's sailing. 



ON LOGARITHMS. Logarithms are a peculiar kind of 

 numbers, invented by Lord Napier, for the purpose of 

 shortening the calculations by common figures, when- 

 ever these calculations require the operations of multi- 

 plication or division, the raising of powers, or the ex- 

 tracting of roots. 



Logarithms are of great importance in the computa- 

 tions of Navigation and Nautical Astronomy ; and it 

 was mainly for the purpose of reducing the labour of 

 such computations, that Napier was led to construct a 

 tabU of logarithms. 



The peculiar character, and the practical use of these 

 important numbers may be briefly explained as follows, 

 the principles of Algebra, as given in our mathematical 

 ;ng previously understood. 



It is tli.-r. in .shown (see Algebra, page 465) that, a 

 being any bate, and x and y any exponents, 



o* X of a* + ', and o* -j- a* o' - ; 

 and, moreover, that whatever exponents m and n may 

 be, (o-)'- a-". 



lot us suppose that any number N could be 

 written in the form o* , a being some chosen number 

 fixud upon as a bate, and z the suitable exponent to 

 justify the equation o' - X ; thus, the base a might be 

 the number 2, or 3, or 4, or any other positive number 

 different from unity ; and it is plain that, whether we can 

 find it or not, tome value exists for z which would satisfy 

 the equation just written, whatever be the positive 

 number N. There is no doubt, for instance, that a 

 numerical value for z exists such that 2' 12 the 

 ralue of z must evidently lie between 3 and 4. Again, 

 the value of z, which satisfies the condition 1" = 0, must 

 evidently lie between 2 and 3. Now imagine two num- 

 xnrs thus represented, one of them N by a', and the 

 other ' by o; then, from tin- principle* cttUd above, 

 w should have, by multiplication aud division, 



TV frtrtl* ./ AM*tfM i4 Jlfutical Ailrnomv. Bj II 

 I-., SMnUrr to UM Ko/ri t^naom** Society. 



o. X a', tUatu, 0' + ' NN'; aud a' a', that is, a*- 



- N-r- ' 



where we see that the mu <>f the tiro numbers 



N,N' is replaced by the simple ad-l 

 ponents z and y, in the proposed notation fur t 

 iiiiiii1>r.4, and that t ! . \ i replaced l>v 



the simplo ml'lrnftion of the exponent y fro"i the ex'- 

 ponent z. The proper exponent z, to be placed OVIT 

 the base o, so that o* may replace the number N S 



therefore s z + y ; and the proper exponent, for the 

 M 



number^.., is t z y. It appears from this, that if a 



table were constructed in which the numbers 1, 2, 3. 



up to the highest number likely to occur in Miaul 



were inserted, and against each number n w 



the proper exponent s that is, such a value of .. that 



the equation a* = n should, in each case, be .* 



the operations of multiplication and division mi^ht. by 



aid of the table, be converted into the simpler . 



of addition and subtraction. 



Such a table was actually constructed by Napier ; it 

 was afterwards improved by Briggs, who found that 10 

 was the most convenient number to choose for tin- 

 a ; and the tables constructed to this base, and in which 

 are inserted against each number n the value of s proper 

 to satisfy the equation 10*= n, are the modern tablet of 

 logarithms: the exponent z, adapted to any proposed 

 number n, being called the lt*tni\tlnn of , so that a 

 table of logarithms is nothing more than a ta 

 ponents the base-number, to which each exponent a 

 conceived to be attached, being 10. 



1 ' >r example, suppose we want the logarithm of 67 : 

 in other words, that we require the value of the exponent 

 z, fitted to satisfy the condition 10* 67, we turn to the 

 tables, and find the value to be z = 1 -.-iv. .7:.. 



In like manner, if we want the logarithm of 58, or to 

 solve the equation 10* = 6 1 *, we find, from the table, 

 that 1=1-763428. We therefore S ;iy tUt !. 07 

 1 -826075, and log. 58 = l-703-tL'8 ; which is only another 

 way of expressing that 



67 = 10' >, and 58 = 10' ;u ' M . 



The product of these numbers is 67x68=10 l- " ttw H-' rtuv. 

 that is lO 1 -* 9103 . The quotient of the first by the second 

 is 67-5-58-10'-"""-'-'"" 1 ; that is 10"". In other w, 

 log. (67 X 58) = 3 -589503, aud log. (67 -3-f>S) = -Oli'-'tJir. 

 Turning now to the bible, we find, against the logarithm 

 3-589503, the number 3886, and against the logarithm 

 D62647, the number 1-155 ... : we conclude, therefore, 

 without actually performing the operation of multipli- 

 cation or division, that 



07 X 58=3880, and 07 -f- 58-1-155 .... 

 And in a similar way may the product or quotient of any 

 two numbers, within the limits of the table, be found : for 

 the product, we add the lo^s. of the factors ; for the 

 at, we subtract log. of divisor from log of dividend. 

 The result, in each case, is a log., against which, in the 

 table, is the product or quotient Bought ; and in this way 

 addition and subtraction is made to replace the more 

 lengthy operations of multiplication and division. 



It most be noticed, however, that the integral part of 

 the logarithm or exponent is not inserted in the table 

 only the decimal part : the insertion of the intend part, 

 or index, as it is called, is unnecessary, because this r 

 is at once known from inspection of the number to which 

 the logarithm belongs : thus, since 

 10<> = 1, 10>=.10, 10 s = 100, 10 s = 1000, 10* = 10000, <tc., 

 we know that the log. of a number between 1 and 10 

 must lie between and 1 ; we know, therefore, that tho 

 index of a positive number, anywhere between these 

 limits, is 0. We see, also, that the log. of a number be- 

 tween 10 and 100 must lie between 1 and 2 : the index 

 of such a number must therefore be 1 . In like manner, 

 since the log. of a number between 100 and Ki must 

 lie between 2 and 3, we know that tho index of such a 

 number must be 2 ; and so on. 



Ili-n'-i- the index of the log. of a number consistinu of 

 but one integer however many decimals may follow that 

 integer is ; the index of a log. of a number consisting 



