1066 



NAVIGATION NAUTICAL ASTRONOMY. 



[LATITCDE. 



tude was 35* 4' 7*. and hu declination 10 64' 20' N. 

 Required the latitude from the funmiln) (A). 



Ana. Latitude, 60* 48' 23* N. 



4. At 18m. 46*. from apparent IKMUI, in latitude - 

 by account, the (un't true altitude u 74 16', and his 

 i 23 27' 8. Required the latitude to tlio 

 nearest minute. Ana, Latitude, 8 23' S. 



6. In longitude a 46" W., at lib. 2m. 32*. apparent 



time, the observed altitude of the jx>le star was 61* 22*, 



;.-x correction being 2". tor apparent noon at 



Greenwich, on the day of observation, the Nautical 



Almanac gave the following particulars : 



Sun's right ascension, Oh. 61m. His. 

 Star's right ascension. lh. 1m. 41s. 

 Star's declination, 88" 26 66*. 



Required the latitude to the nearest minute by the 

 formulas (A). An*. Latitude, 61 47* N. ' 



ARTIFICIAL HORIZON. The problem just discussed 

 will be found very useful at sea, whenever the mariner, 

 on account of cloudy weather coming on near noon, is 

 disappointed of a meridian altitude. As the object may 

 be obscured though the horizon may be clear and well 

 defined, so, on the other hand, the celestial body may 

 be visible and in a position well suited for observation, and 

 yet a haze may obscure the horizon. In such a cose an 

 artificial horizon is employed : this is a shallow trough of 

 quicksilver, protected from agitation from the air by a 

 glass covering or roof. 



The observer, placing himself at a convenient distance 

 from tills horizon, so that both the celestial object and 

 its reflected image may be distinctly seen, measures with 

 his sextant the angular distance between the two ; and as 

 the real object is as much above the horizontal plane as the 

 image is below it, he thus gets double the altitude, and has 

 no correction to make foniip: the angle read off from the 

 instrument, being corrected for the index errorand divided 

 by 2, give* the apparent altitude of the point observed. 



To get a ni'ri</i<in altitude of the sun is one of the 

 principal item* in a "day's work" at sea, for correcting 

 the latitude by the dead-reckoning ; if the weather 

 interfere with this operation, then an observation off 

 . i Mian is sought for, and the latitude inferred by 

 help <>f tlio apparent time at the ship, as explained in 

 the preceding article. The reader will boar in remem- 

 brance that a ship carries the mean time at Greenwich 

 with her : the ship's chronometer, when the known daily 

 gain or loss is applied to it, supplies this important 

 information. The apparent time in deduced therefrom, 

 by means of the equation of time given in the Xnutical 

 Almanac; and the apparent time at the ship is ascer- 

 tained by turning the longitude into time ; and thus the 

 hour-angle of the object, observed from the meridian, is 

 found. The longitude by account may be somewhat in 

 error ; but the trifling inaccuracy in the resulting time 

 at the ship, will have no important influence on the 

 latitude deduced from it. 



But valuable as the chronometer is, yet, like all human 

 contrivances, it is subject to accidents and exposed to 

 derangements from circumstances beyond our control. 

 It is of great importance, therefore, to be able to find 

 the position of a ship at sea, independently of its aid ; 

 this we have seen, as far as latitwlr is concerned, may be 

 done by mean* of meridian altitudes. It may also be 

 done l.y aid of tn;, altitudes of the same celestial objects 

 ii off the meridian. This is technically called the 

 ).!.:.'.. n, of double altitudtt: we proceed now to investi- 

 gate it.-, principles. 



LATITUDE FBOM Two ALTITUDES or TUB SUN, AND 

 TUE TIME BETWEEN THE OBSERVATIONS. Scarcely any 

 I i .l.leui in nautical astronomy has received more atten- 

 tion from scientific men tl.ni the prolilem of i/i/ub/e /(i- 

 < ; and, as the calculations involved in it are mueh 

 T than those for finding the latitude from a single 

 Altitude, various tables, to facilitate the operations, have 

 been constructed. The determination of the latitude, 

 help of such tables, is what is called the indirect 

 method of solution ; and, like all such methods, it is not 

 so strictly correct as the direct method by trigonometry. 

 Ifelambr* carefully examined all the rules he was ac- 



quainted with for the solution of the present problem, 

 and he came to the conclusion that the rigorous process 

 by spherical trigonometry was to 1 1, as well 



for brevity as for accuracy of result The investigation 

 of the hod is as follows : 



Let P (Fig. 28) be the elevated polo, Z the zenith of 

 the ship, and therefore Z P its co-latitude. Let S, S' be 

 the two places of the sun when the altitudes are taken ; 

 then, drawing the great circle arcs, as in the diagram, or 

 in that of Fig. 27, we shall have the following quantities 

 given, namely 



The co-declinations P S, P S' \ 

 The co-altitudes 7. *, '/. X To find Z P. 

 The hour-angle S P S' ) 

 There are three spherical triangles to consider, viz. 



1. The triangle PSS', in which are given the two 

 sides PS, PS', and their included angle, to find the 

 third side SS', and one of the remaining angles, as, for in- 

 stance, the angle P S S'. 



2. The triangle Z S S', in which are given the three 

 sides, to find the angle S'S X ; Fig. 28. 



from which, and the pre- 

 viously-found angle PSS', 

 the angle Z S P becomes 

 known, so that we have, 



3. The triangle Z S 1', in 

 which are given two sides 

 and their included angle, to 

 find the third side Z P. 



Before proceeding to the 

 solution of these triangles, 

 the observed altitudes must, 

 of course, be reduced to the 

 true altitudes, as in the 

 former examples ; and since 



the ship most probably sails on during the interval be- 

 tween the two observations, an additional correction be- 

 comes necessary, in order to reduce the first altitude to 

 what it would have been, if taken at the place of the se- 

 cond observation. This correction for the ship's run will 

 become known, provided we know the number of minutes 

 the ship has sailed directly towards or directly from the 

 sun, in the time between the two observations ; and they 

 may be ascertained as follows : 



Take the angle included between the ship's course and 

 the sun's bearing at the first observation ; and consider- 

 ing this angle as a course, and the distance sailed as the 

 corresponding distance, find by the traverse table, or by 

 calculation, as in plane sailing, the difference of latitude ; 

 this difference of latitude, expressed in minutes, will be 

 the number of minutes by which the ship has approached 

 to or receded from the sun, so that we shall know how 

 many minutes must be added to or subtracted from the 

 first altitude, to reduce it to what it would have been if 

 taken by another observer, at the place of the second 

 observation. 



If the angle between the ship s track and the bearing 

 of the sun be leas than 90, the ship will obviously be 

 approaching towards the sun, in which case the correc- 

 tion of the altitude, determined as above, must be add. I : 

 but if the angle exceed 90, it must be subtracted : if it 

 be exactly 90, no correction of the altitude will be neces- 

 sary for the ship's change of place. 



But a correction of the elapsed time may be requisite 

 for the change of longitude ; this change, converted into 

 time, must be added to the time elui n the two 



observations, if the ship have sailed eastward, and sub- 

 tracted if she have sailed westward. 



These are the corrections necessary to fit the three 

 triangles above for trigonometrical calculation. And to 

 simplify the work of finding S S', without any material 

 sacrifice of accuracy, we may regard the declinations nf 

 the sun at the times of observation, as both equal to the 

 declination it has at the middle time between them ; the 

 shorter this time in, the less will the supposition affect 

 the precision of the result. 



Regarding the above mentioned corrections to have 

 been made, we shall now give an example of the trigo- 

 nometrical operation. 





