1081 



NAVIGATION NAUTICAL ASTRONOMY. 



[LATITUDE. 



3. Jon* let, 1823, the obterred altitude of the star 

 Cbpella, whou en the meridian below the pole, was 10 

 r5r ; it* declination WM 46 48' 27' N. ; and the height 

 of the eye IT feet Required the latitude. 



Ana. Latitude, 54 4' N. 



The foregoing example* exhibit the different mean* of 

 determining the latitude at tea, by taking the altitu.!<>f 

 a celestial object when uj-on the meridian of tho ship. 

 Whenever practicable, the sun is always the object ob- 

 served, and its mid-day altitude is that which is to be 

 preferred, became at mid-day it attains its greatest ole. 

 ration, and the refraction is less liable to variation from 

 the mnan state of the atmosphere. The altitude below 

 the elevated pole can be taken only in high latitudes, 

 where the sun is above the horizon during the whole 

 twenty-four hours for a part of the summer, and, there- 

 fore, the horizon clearly visible : the obscurity of the 

 horizon often precludes the possibility of accurately 

 measuring the altitude of a star ; and on account of its 

 rapid change in declination, the moon is less suited for 

 the purpose of deducing tho latitude, when there is 

 much uncertainty as to the longitude of tho ship, or the 

 time at Greenwich when the observation is made. In 

 preparing for a meridian altitude, the. observer holds j 

 himself in readiness, before the object attains its greatest 

 elevation, and continues to observe it till it ceases to rise 

 and appears for a moment stationary ; at this instant its 

 altitude is noted, and it is regarded as upon the meri- 

 dian. Strictly speaking, however at least as respects 

 the sun and moon the centre of the object is not neces- 

 sarily exactly on the meridian when its altitude is the 

 greatest or least ; for the change in declination may more 

 than counterbalance its change in altitude during the 

 few minutes which precede its meridian transit, especially 

 in the case of the moon ; but these differences are too 

 trifling to lead to any error of practical importance. It 

 nmy, however, happen that the object becomes hidden 

 by a cloud at tho time of transit, so that the meridian 

 observation cannot be made ; it is of importance, there- 

 fore, to take note of the altitudes before the transit, as 

 an altitude near the meridian may be made available for 

 the determination of the latitude, as we are now about 

 to show : we shall first, however, explain how to find the 

 latitude generally when the object observed is off the 

 meridian, provided the hour angle, which its declination- 

 circle makes with the meridian of the ship, is known. 



ToriND THE LATITUDE FROM THE DECLINATION, ALTI- 

 TUDE, AND HOUR ANGLE. Let Z be the zenith, and 

 ]' M Z the celestial meri- 

 dian of the ship, P the 

 elevated pole, and S tho 

 place of the heavenly body 

 off the meridian ; then P S 

 is the co-declination, Z S 

 the co-altitude, and PZ 

 the co-latitude. In the 

 spherical ' triangle P Z S, 

 there is supposed to be 

 known the three parts, 

 PS. Z S, and the hour 

 angle P, which, for the 

 sun, is the time from noon. 

 Hence, by Spherical Trigo- 

 nometry, PZ, the co-latitude, may be found (Mathematical 

 Srirnrn, page 601). But the following method, by right- 

 angled triangles only, will be more easily recollected. 



I'rr.v > M. p. rj-i-i;'li.-!i!:ir to th>' in.-n.li;iu : we shall 

 then have two riglit-anglud triangles P .M S and X M S ; 

 and applying Napier's rules to those (page 660), we Itave 



KI..III the triangle P M S, taking P f or middle part, 

 and I 1 S, 1' M, for adjacent parts, 



cos. P - cot. PS tan. P M /. tan. PM = 

 cos. P tan. P S . . (1) 



Also, from the same triangle, taking tho hypotenuse P 8 

 for middle part, and P M, S M, for opposite parta, 



con. PB - cos. PM cos. 8M . . (2) 

 And from the triangle Z M S, tmlring in like manner 



cos. ZM- 



Fig. 27. 



the hypotenuse Z S for middle part, and Z M, S M for 

 opposite parts, 



oos. ZS- cos. Z Moos. SM . . (3) 



Therefore, dividing (2) by (3), in order to got rid of S M 

 we have 



008. PS 008 I'M 



cos. ZS ~ COS.ZM 

 cos. P M cos. Z S sec. P S . . (4) 



As Z M is thus expressed in terms of the given quan- 

 tities, and as P M is also in like manner known from ( I ), 

 the difference (or the sum, if M fall between P and Z) 

 P Z, that is, the co-latitude becomes known. 



The formula; (1) and (4) to be combined are 



tan. P M cos. hour angle X cotan. dec. 

 oos. Z M cos. P M sin. alt. x cosec. dec. 



To know the hour-angle P, it is necessary to know the 

 time at the ship. The chronometer shows the moan 

 time at Greenwich ; and hence, by help of the longitude 

 by account, we may find the mean time of observation 

 nearly ; this, reduced to apparent time, bv applying tho 

 correction for the equation of time taken from tho 

 Nautical Almanac, wul make known the time from ap- 

 parent noon at the ship. The longitude by account is, 

 however, most likely affected with error, and it is there- 

 fore desirable that the altitude off the meridian be taken 

 at such a time as that a small error in tho hour angle 

 may have the least influence on the determination of tan. 

 P M. Now, whenever we have to employ the cosine of 

 a small angle, and have reason to suspect that tho angle 

 itself has not been accurately determined, the error in 

 the cosine will bo smaller as the anglo itself is smaller ; 

 for the cosines of arcs near the beginning of the quad- 

 rant differ very little from one another within tho limit? 

 of several seconds, and the difference becomes less as tho 

 arc approaches to zero (see / 'ii, page lulii). 



It follows, therefore, that when the time at tho ship 

 is only approximately known, tho altitude should bo 

 taken when the object is as near to the meridian as it is 

 likely to be, before being obscured by clouds. 



The formula; just established is, we see, generally ap- 

 plicable, however distant from tho meridian the observe. 1 

 object may bo, provided we know the apparent time at 

 the ship ; and we see, also, that they may be employed 

 when the time nearly is known, provided the object be 

 pretty close to the meridian- But for this latter case 

 there is a special method somewhat more convenient, 

 which may be investigated as follows : 



Referring to the preceding diagram, we have, by tho 

 fundamental theorem of Spherical Trigonometry, which 

 expresses the relations among the three sides and oue of 

 the angles of a spherical triangle, 



cos. ZS = cos. PZcos. PS sin. PZsin. PS cos. P. 



/. cos. P = 



cos. Z S -cos. PZcos. PS 

 sin. PZsin. PS 



Let z be the zenith distance that S would have when 

 upon tho meridian, and z' the difference between this 

 meridian zenith distance and that Z S actually observed ; 

 thiit is, let Z S = z + z'; then the equation above is 



cos, (g + z^-cos. PZcos. PS 



sin. P/ 



Subtracting each side from 1, we have 

 1-cos. P- 



sin. PZsin. PS + cos. PZoos. PS-cos. (z+ z 1 ) 

 sin. PZsin. PS 



COB. (PZoPS)-cos. (z + Q. 

 sin. PZsin. PS 



Now P Z </> P S, that is, tho difference between the 

 co-latitude and the co-declination is equal to Z, the 

 meridian zenith distance, because tho co-lntitudo 1' 7. is 

 always equal either to P S -f z, or PS-z, or z-PS, 



