PROPORTIONAL LOGARITHMS.] N A VIG ATION N AUTI C AL ASTRONOMY. 



1105 



case, the error in longitude corresponding to an error of 

 10 in the lunar distance, is 250' = 4' 10'; and the error 

 in longitude caused by an error of 1' in the lunar distance, 

 is 25'. As every inaccuracy in the distance becomes 

 thus increased twenty-five fold in the longitude deduced 

 from it, even in the most favourable case, the reader will 

 at once perceive the importance of securing precision in 

 this element. Several distances ought to be carefully 

 observed, and the mean of them all employed ; and 

 scrupulous attention should be paid to the corrections 

 of the altitudes ; though, as already remarked, the 

 altitudes themselves need not be taken with the utmost 

 nicety. 



To obtain the time at Greenwich, corresponding to the 

 true distance determined at sea, requires an operation in 

 simple proportion. To facilitate this operation, Dr. 

 Maskelyue to whom, indeed, navigators are indebted for 

 originating the Nautical Almanac contrived the table of 

 Pruportional Logarithms, to be found in every collection 

 of nautical tables. As Proportional Logarithms have 

 not as yet been alluded to in the present section, it will 

 be necessary to say a word or two about them here. 



PROPORTIONAL LOGARITHMS. These logarithms are 

 derived from the logarithms in common use, thus : 

 From the logarithm of 10800, the number of seconds in 

 3 hours, subtract the logarithm of a, any portion of 

 time, in seconds, less than 3 hours : the remainder is 

 called the proportional logarithm of o ; in other words, 



10800 

 Prop. log. a = common log. 



Tf a = 10800, then Prop. log. a = com. log. 1 = 0; 

 so that proportional logarithms are, in fact, complements 

 of the ordinary logarithms to the number log. 10800 ; 

 just as the arithmetical complements of log. sines and 

 log. cosines, are complements to 10. 



As already remarked, the lunar distances, given in the 

 Nautical Almanac, are calculated for every three hours 

 of interval Suppose a distance is determined at sea, at 

 some Greenwich time within one of these intervals ; we 

 seek iu the Almanac for the nearest distance, preceding, 

 in onler of time, the given distance, and take the differ- 

 ence between it and the given distance : call this d, and 

 the difference between the two three-hourly distances, 

 intermediate to which the given distance occurs, call D : 

 then, by proportion, we should have for the time x h , cor- 

 responding to the given distance, 



D : d : : 3" : x" ; 



and therefore, in common logarithms, we should have 

 log. x h = log. 3" + log. d-log. D. 



But since 3" = 10800", the proportional logarithm of 

 which is 1, we should have, in proportional logarithms, 



Prop. log. x h = Prop. log. d Prop. log. D. 



In the Nautical Almanac, Prop. log. D is inserted be- 

 tween the distances there given, at the beginning and 

 end of every three hours ; so that by subtracting this 

 given proportional logarithm from the proportional loga- 

 rithm of d taken out of the table, we get a proportional 



i logarithm, answering to which, iu the table, is the por- 

 tion of time to be added to the hour of the earliest dis- 



i tance : the result is the Greenwich mean time corre- 

 sponding to the given distance. 



For example Suppose it "were required to find the 



j Greenwich mean time, at which the true distance between 



I the Moon and Pollux was 32 30' 25", on January 14, 

 1846. 



By inspecting the distances in the Nautical Almanac 

 for that year, we find against January 14, and opposite 

 the name of the proposed star, Pollux, the following row 

 of lunar distances : 



Midnight. 

 31 49' 25* 



Pr. log. 

 3388 



xv 

 33 12 7 18' 



Pr. log. 

 3343 



xviii h - 

 34' 35' 41* 



Pr. log. 

 3319 



xxi" 

 35 59' 31* 



Pr. log. 

 321)8 



From which it appears that the time at Greenwich cor- ' 

 responding to the given lunar distance was between mid- 

 night and XV hours ; the nearest distance preceding, in 

 order of time, the given distance, is therefore the distance 

 at midnight : we therefore proceed as follows : 

 Distance at midnight 31 49' 25" Prop. log. of diff. 33C8 - 

 Given distance 32 30 25 



Difference 41 . . Prop. log. 6423 



Portion of time ) 1V . , 

 after midnight ) lk WaL 2s ' 



Prop. log. 3057 



Hence the Greenwich mean time, when the distance 

 was as stated above, was Kill. 20m 2s. 



If the distance increased with perfect uniformity 

 during the interval within which the given distance is 

 found, the Greenwich time corresponding to that given 

 di.stauce, determined as above, would be strictly correct, 

 but, as such is not the case, a correction should be ap- 

 I to the time so found, for the variation of the dif- 

 ferences of the distances. A table for obtaining such 

 corrections of the approximate interval of time as found 

 above, is given in the Nautical Almanac. In the example 

 above, the correction comes out 8s. additive, so that the 

 correct Greenwich mean time is 13h. 29m. 10s.; the neg- 

 lect of this correction would, however, occasion an error 

 of only y in the longitude ; but in extreme, and there- 

 fore, of course, unusual cases, the error in longitude, 

 front a neglect of this correction, might amount to so 

 ninrh as 12' in the longitude. 



Besides the use of proportional logarithms in connec- 

 tion with the lunar problem, they also serve to point out 

 the star which is most favourably circumstanced for ac- 

 curate observation ; that star being to be preferred which 

 has the least proportional logarithm opposite to it : for, 

 as already shown (page 1104), the greater the velocity of 

 the moon from or towards a star, the greater is the re- 

 lianci; to b<; placed on an observation of the distance ; in 

 other words, the less is the effect of a small error in the 

 distance upon the longitude. It is a property of pro- 

 portional logarithms, to decrease as the natural numbers 



Vi PI.. I. 



answering to them increase ; a smaller proportional 

 logarithm, therefore, indicates a greater velocity of the 

 moon, or greater variation of distance iu the interval, 

 upon which the value of the observation depends. 



We shall add another example or two of finding the 

 Greenwich mean time, corresponding to a given lunar 

 distance on a given day. 



2. On August 2nd, 1836, the distance of the moon 

 from the planet Mars was found, from an observation at 

 sea, to be 56 30* 8": the Nautical Almanac gave, 

 Aug. 2, at3h. distance 57 43' 59" Pr. log of ditf. 2948- 

 Given distance . . 66 30 8 



Difference 



1 13 51 Pr. log. . 



3869 



Portion of time aft. 3h., 2h. 25m. 36s. Pr. log. . 921 

 Hence the mean time at Greenwich was 5h. 25m. 36s. 



3. On April 7, 1831, the true distance between the 

 sun and moon was found to be 65 54' 48": the Nautical 

 Almanac gave, 



April 7 at noon, Dint. 67 0' 8* P. L. of diff. 3051- 

 Given distance . . . 65 54 48 

 Difference. . , I 5 lo P.L. , 4412 



Portion of time aft. n., 2h. llm. 35s. P. L. . . 1301 



Hence the Greenwich mean time was 2h. llm. 35s. 



4. On November 22, 1853, the true distance of Saturn 

 from the moon was found to be 77 52' 45": the Nautical 

 Almanac gave, 

 Nov. 22, at 3h. Distance, 77 14' 40", and P. L- = 6745. 



Required the Greenwich mean time. 



Ans. 4h. 13m. 65s. 



The proportional logarithms annexed to the Lunar 

 Distances in the Nautical Almanac, and most of tho 

 tables of them inserted in books on Navigation, are 

 limited to four places of figures ; in certain parts of the 

 table, this number of places is too few to show any dif- 

 ference for two, or even three consecutive arguments; 

 thus the proportional logarithms of 2h. 41m., 2h. 41m. Is., 

 2h. 41m. 2s., are all the same namely, 484 : this is a 



