UM 



NAVIGATION NAUTICAL ASTRONOMY. 



[LfVAB DISTANCES. 



.'.oo. D 



- jcoa. <f+ com. 



001. o oo. a' 



(1) 



From the same two expressions for cos. Z, we also 

 have 



coo. A cos. A' + sin. A sin. A' cos. D 

 1-cos.Z-- com. A cos. A 



com. (A e/> A*) com. D 



cos. A c< 

 com. a cos. a' + sin. a sin. a' cos. d 



1 OOB.Z 



cos, (o 



cos. a cos. a' 

 P*) cos, d 



cos. a ooe. a 



. oos. (A c/> A') cos. P_ cos, (a c/i aQ cos, d 

 ' ' " cos. A <*.- "" " - "' 



.'. cos. D j cos. d cos. (o v> a,') 



0.-S. :l DOS- 



cos. A cos. JL' 

 OOB. a cos. a' 

 + cos. (A A") . . . . (2) 



The formula marked (1) and (2) sre> tolerably commo- 

 dious by help of the common logarithmic tables : ns an 

 exemplification of their advantages, we shall sol vis the 

 example at page 1101 by each. In this example the fol- 

 lowing quantities are given, namely 



d - 63 35' 14*, a - 24' 2^ 44" a' - 45 9 12* 

 A = 25 17' 45*, A' = 45' 8' 15*. 



Determination of D by formula (1). 

 d 63 35' 14* nat cos. -4443349 -f Logarithms. 



a 24 29 44 

 a 45 9 12 



comp. cos. -04<>:>n17 

 co in p. cos. -1510803 



a + o' 69 38 66 nat. cos. -3477722 + 



natural number -7926071 

 A 25 17 45 

 A' 45 8 15 



natural number -7877042 

 A-r-A 1 70 20 nat cos. -3349034 



. 1-8000579 

 cos. 9 {:,( ;__: ;i i 

 cos. 9 '8484402 



. V89G3G31 



D63 4' 35" nat. cos. -4528008 



With the exception of the middle logarithm in the 

 column on the right, the sum of the numbers in that 

 column may bo found at onco by a table of the Loga- 

 rithmic Difference, as already noticed at page 1100. Of 

 all the auxiliary table* employed in Nautical Astronomy, 

 or at least in that part of it with which we are here occu- 

 pied, the table alluded to is perhaps the most useful : 

 the element it f urnishes enters into nearly all the methods 

 of clearing the lunar distance. 



As far as the author knows, the method just given is 

 new ; but rules for clearing the distance are so numerous, 

 that it is more than probable it has appeared elsewhere. 

 In working by this method, it is best to take the formula 

 iUelf as a guide, and to attend to the algebraic sign of 



the factor | oos. d + cos. (a + a 1 ) ] : if this should be 



negative, the first natural number above, is still to be 

 treated as positive ; the second natural number, which is 

 the product of the two factors in the formula, will then, 

 however, be negative, and f rora this negative number the 

 nat. com. (A + A'} is to be subtracted as above. 



1-6909659 

 cos. 9-8484402 

 . 1-C882711 



natural number -4908693 

 \ . 



A' 46 8 18 . 



natural number -4R78329 . 

 AooA' 19 50 30 iiat. cos. -M06341 + 



D 63 4 35 nat. cos. -4528012 



The first of the preceding n.-itural numbers is, by the 

 formula, negative, though treated as positive. In con- 

 sequence of this, the second natural number which is 

 the product of the two factors in the formula U also 

 negative, so that cos. D is the difference between the last 

 nat. number and cos. (A </> A'). 



This last method of clearing the distance is tho same 

 as that proposed by Mr. Keith, in his Trfiititt on 7' 

 nometry, and, like the one previously given, is capable 

 of abridgment by means of the table of "Logarithmic 

 Differences." 



The reader will observe that the natural cosine of D, 

 arrived at above, differs by four units in thn seventh 

 place of decimals, from the natural cosine of D in the 

 former method, although the resulting arcs differ only by 

 a fraction of a second. Occasional insignificant dis- 

 crepancies of this kind must be expected in working the 

 same example by different methods with tables. In 

 looking at tho " differences" in tables, it will bo seen that, 

 in some parts, the difference between two consecutive 

 numbers is largo, and in other parts small. In propor- 

 tioning for an intermediate number, in tho former case 

 a comparatively large difference will have but small in- 

 fluence, while in the latter case a comparatively Miiall 

 difference may have a very sensible effect, lint a dis- 

 crepancy in the seventh place of decimals is of no moment 

 even in working to seconds. 



All tho preceding methods of clearing tho lunar dis- 

 tance are independent of subsidiary tables ; a large col- 

 lection of compendious rules for working the problem !>y 

 special tables will be found in Dr. Mackay's valuable 

 Treatise on the Lonyitude : and a very short form of 

 operation is also given by Mr. Woodhou.se in the Ap- 

 pendix to White's JSphemerii, 1855. 



Before leaving the present problem, it may be useful 

 to observe that the altitudes of the objects are not re- 

 quired to that precision with which the distance should 

 be taken : this is a desirable circumstance, because, from 

 the frequent obscurity of the sea horizon, it is more 

 difficult to get tho altitudes accurately than the distance. 

 If reference be made to any of the formulae given in the 

 preceding pages, it will be seen that d remaining the 

 same, a small alteration in the values of a, a', and 

 the same alteration in those of A, A', cannot produce 

 any sensible effect upon the value of D : the factor 



n the logarithm of which is, in nautical 

 cos. a cos a 



tables, called the Logarithmic Diflerence, is always very 

 nearly equal to unit, as is obvious ; and this is the prin- 

 cipal reason why a small error in tho altitudes does not 

 sensibly affect the distance. It is of importance, how- 

 ever, that the proper corrections be carefully applied to 

 the observed altitudes to obtain tho true altitudes, even 

 though the former should not have been taken with pre- 

 cision the relatirc values of tho observed and true alti- 

 tudes must still be preserved. With a view to the 

 determination of tho longitude from tho lunar dish. 

 accuracy in these corrections is of much importance ; and 

 the neglect even of those depending upon the state of the, 

 atmosphere, as indicated by tho barometer and thermo- 

 meter, will sometimes occasion an error of more than 

 thirty minutes of longitude. 



From what is hero said, the learner will perceive that 



