1st cor. 2′′.205 log. 0.002372 0".00003 log 5.430786 0.000018 or 2d cor. insensible. 0.343447 Hence 41° 2′ 5′′.71+2′′.21=41° 2' 7"92-z, and the latitude, or λ=180°—(≈+8)=50° 37′ 23′′.22 N. To compute the Latitude by the Pole Star. Let Z be the zenith of the place of observation, P the pole, S, s the place of the star in its diurnal revolution round P. From S let fall the perpendicular SQ upon ZP. Then let ZS=z the zenith distance of the pole star, PS=p the north polar distance of the star; the angle ZPS=t the hourangle; ZP the colatitude of the place; and QP-u the segment formed by the perpendicular. Now z, p, and t being given, we have, by spherical trigonometry, (Playfair, prop. XXI.) in the triangle QPS, cos QPS: R:: tang PQ: tan PS, or tan PQ=cos QPSx tan PS, or tan u=cost. tan p. Also (prop. XXVI.) from the triangle PSZ, cos ZQ: cos QP :: cos ZS : cos cos QPX cos ZS PS or cos ZQ== COS PS cos (4+u)=cos u. cos z. sec p cos PQX cos ZS x sec PS, or (1.) Again, since the angle QPS is the complement to SPT, (prop. XVIII.) tan PT=cos SPT x tan PS, or if PT=r, then tan rsin t. tan p (2.) and consequently the azimuth may be determined by the pole star along with the latitude, which is frequently necessary in finding the bearing of the sides of triangles with respect to the meridian in trigonometrical surveying. TABLE for finding the Latitude by the Pole-Star.-BY CAPTAIN KATER. CORRECTION OF THE ALTITUDE OF THE POLE-STAR FOR Ex.8. At York Gate, Regent's Park, London, on the 22d of February, 1826, at 7h 42m 49s, mean time, the altitude of the pole star was observed by Captain Kater to be 51° 58′ 18′′.1; required the latitude? First to find the mean solar time when the star was upon the meridian. *'s App. R. A. `O's R. A. at noon, 36 56.9 True alt. 51 57 32 .6 2 36 31.2 2 50 21.9 7 42 49.0 -4 u= 0 27 48 .2 *tan =1° 32′ 45′′.6 = r = the azimuth W. from the pole. It seems unnecessary to extend our remarks farther with regard to these observations, more especially if the examples in the explanation of the table XXVIII. be consulted. If the observations are taken at sea with a reflecting instrument, on the principles of Hadley's quadrant, a correction must be made for the dip in addition to these already given. This may be taken from Table XI.; or the true altitude may be still more readily * Found by precept, page 10 of Explanation of the Tables. Were the author permitted to add any thing to what Captain Kater approves, it would be to employ the constant log 5.314425, the log of an arc = R", Table LXIII., and the sum of these logs would be the log u in seconds, which would save the trouble of finding the value of the log tangent of small arcs, or even by taking the log of p in seconds, the errors thence arising, not much exceeding half a second, when the star is in its most unfavourable position, or when t is about 90o. In the application of u, attention must be paid to the sign of the arc t, according to its situation in the circle which the star describes round the pole, in its diurnal revolution. If t is in the first or fourth quadrant, it is additive; but if in the second or third, it is subtractive. found from Table XIII. or XIV., sufficiently correct for all the usual purposes at sea. ان Height 10 15 20 25 30 35 of Eye. 2'.2 3'.1 3'.8 4'.4 5'.0 5'.5 5'.9 Ex. 1.-May 1st, 1825, in longitude 64° 25′ W., the observed me. ridian altitude of the sun's l. l. was 48° 34' 30", the zenith being north of the sun, and the height of the eye 14 feet; what was the latitude? It is unnecessary to push the calculations nearer than tenths of a minute, as any observation taken at sea is, from the indistinctness of the horizon and the uncertainty of the horizontal refraction, unless a dip-sector be used, liable to an error of at least one minute. Examples for Exercise. 2. On the 1st of September, 1824, in longitude 54° W., the meridian altitude of the sun's lower limb was 79° 44′ 15" S., the height of the eye being 24 feet; what was the latitude? Ans. 18° 30′.9 N. P 3. On the 1st of January, 1826, the meridian altitude of the star Arcturus was 60° 41′ S., the height of the eye being 24 feet; what was the latitude? Ans.--49° 29'.8. 4. On the 14th September, 1827, in longitude 103° 18′ E., let the meridian altitude of the moon's lower limb be 51° 4' N., and the height of the eye 20 feet; required the latitude? Ans.-19° 48'.4 S. 4. On the 29th September, 1827, in longitude 20° 40′ W., if the observed meridian altitude of the moon's upper limb be 83° 6' N., and the height of the eye 16 feet; required the latitude? Ans.-21° 25'.7 S. * u p N m As the meridian altitude may, by the interposition of clouds, or other causes, be lost at sea when a knowledge of the latitude is necessary for the safety of the ship, recourse must be had to other methods, particularly to that of H double altitudes, and the time between them, as being the most practicable. This method requires solutions in three spherical triangles. In the triangle ZPS there are given PS the sun's polar distance at the time of the first observation, PS' that at the second, and the angle S'PS measured by the elapsed time; to find the side S'S and the angle PS'S.t Again in the triangle ZS'S there are given the zenith distance ZS at the time of the observation, ZS' that at the second, and the side S'S already found to determine the angle ZS'S. But PS'S being already computed, ZS'P may be obtained. Whence there are in the triangle ZS'P, the sides ZS', and PS', and the contained angle ZS'P; to find the side ZP the colatitude. This is the regular method by spherical trigonometry; but if the polar distance PS be supposed to remain the same, that at the middle time, between the observations, or, as Professor Lax seems to think preferable, the same as at the time of the greater altitude, and, by combining the solutions of the several triangles in one, the operation becomes more simple. In order to render this method still more easy to practical seamen, Douwes proposed an approximate method by introducing the latitude by account, which, when properly restricted according to the rules of Maskelyne or the tables of Lax, will generally give the desired result sufficiently correct for nautical purposes, and the computations may be very readily performed by the tables of Lynn. When the common tables are used, Mr Ivory's solution is the best, particularly in the form of that Mr Riddle has given it, which we shall adopt here. Find the sun's declination for the time of the greater altitude, and the true altitudes, reducing the less if necessary for the ship's run to what it would have been had it been taken at the same place with the greater. This is accomplished by observing the sun's bearing by compass, at the time of taking the less altitude, and, finding the On the authority of a very distinguished practical navigator, I am informed, that double altitudes are not of such importance as is generally supposed; for if double altitudes can be successfully taken, meridian altitudes of the sun, moon, a planet, or a fixed star, may be generally obtained. A circle is supposed to pass through PS' P' similar to PSP'. |