EXAMPLE I. Required the sun's true amplitude at rising, in the latitude of 39° 0' N. on the 22d of December, 1820 ? BY LOGARITHMS. Latitude.......39° 0' log. sec. Sun's declin....23 28 log. sine True ampli.....30 49 log. sine BY INSPECTION. 0.10950 Under the declination 23° 29' and oppo9.60012 site the latitude 39 stands the true amplitude 30- 49. 9.70962 Hence the true bearing or amplitude of the sun at rising is E. 30° 49′ S. and at setting it is W. 30° 49' S. EXAMPLE II. Required the moon's true amplitude at setting, in the latitude of 35° 3′ N. when her declination is 130 N.? BY LOGARITHMS. Latitude ....35° 8' log. sec. Moon's declin. 13 0 log. sine Truc ampli....15 58 log. sine BY INSPECTION. 0.08734 Under the declination 13, and opposite 9.35209 the latitude 35 stands 15 56', which is nearly the true amplitude; the exact value 9.439-43 may be found by finding the amplitude for 36 latitude, and proportioning the differlence for the miles in the latitude. Hence the true amplitude at setting is W. 15° 53′ North, and at rising E. 15° 58′ N. EXAMPLE III. Required the sun's true amplitude in the latitude of 42° 30′ N. when his declination was 200 N.? BY LOGARITHMS. Latitude..... .42° 30′ log. sec. Sun's declin....20 N. log. sine True ampli.....27 38 log. sine BY INSPECTION. 0.13237 Under the declination 20 and opposite 9.53405 the latitudes 42 and 43, stand 27° 24′ and 27° 53'; the mean of these gives the 9.66642 true amplitude for the latitude of 42° 30′=27 38'. Hence the amplitude at setting is W. 27° 38′ N. and at rising E. 27° 38′ North. To find the true azimuth at any time. At the time of observing the magnetic azimuth, you must also observe the altitude of the object; this altitude must be corrected as usual for the dip, parallax, refraction,* &c. in order to obtain the true altitude; you must also find the declination of the object,† and the latitude of the place of observation, and then the true azimuth may be calculated by the following RULE. Add together the polar distance,‡ the latitude, and the true altitude, take the difference between the half sum and the polar distance, and note the remainder. Then add together the log. secant of the latitude, the log. secant of the altitude (rejecting 10 in each index) the log. co-sine of the half sum, and the log. co-sine of the remainder; half the sum of these four logarithms will be the log, co-sine of half the true azimuth, which being dou bled will give the true azimuth, reckoned from the north in north latitude, but from the south in south latitude. • In observations of the altitude of the sun's lower limb (by a fore-observation) it is usual to add 12/ for the effect of dip, parallax, and semi-diameter. The refraction is to be subtracted from the sum, and the remainder will be the true altitude nearly. The declination is to be found according to the directions in the note, in 11. last page. The polar distance of the sun, moon, or star, is the distance from the elevated pole, and is found by subtracting the declination of the object from 30°, when the latitude and declination are of the same name, but by adding to 9 when of different names. R EXAMPLE I. In latitude 51° 32′ N. the sun's true altitude was found to be 39° 28', his declination being then 16° 38' N.-required the true azimuth? The logarithm 9.72347 of this example is also the co-sine of 121° 56', which doubled gives another azimuth 243° 52', the former being 116° 8'. One of these corresponds to an observation in the forenoon, the other to arr afternoon observation. EXAMPLE II. In latitude 42° 16' S., the sun's true altitude was found to be 18° 40′, Heclination being then 7° 38′ N.-required the true azimuth? Polar distance......... 97° 38' his True Azimuth ........119 46 from the south. QUESTIONS TO EXERCISE THE LEARNER. Question I. Given the sun's altitude corrected for dip, refraction, &c. 20° 46', his de clination 17° 10' S. and the latitude of the place 40° 38' N. Required the true azimuth? Answer. 137° 50' from the north. Question II. What is the sun's azimuth in the latitude of 26° 30' N. in the forenoon, when his correct central altitude is 24° 28′ and his declination 22° 40' N.? Answer. 75° 44′ from the north. Question III. At the Island of St. Helena the sun's true central altitude was found to be 30° in the forenoon, his declination being then 22° 58' S. Required theazimuth at that time? Answer. 72° 21' from the south. Question IV. What point of the compass did the star Aldebaran bear on, in the latitude of 34° 23′ S. on January 1, 1804, when the correct altitude of that star was 22° 26'? Answer. 130° 16' from the south. Having the true magnetic amplitude or azimuth, to find the variation. Having found the true and magnetic amplitude or azimuth, the variation may be easily deduced therefrom by the following rule, in which the amplitude is reckoned from the east or west point of the horizon, and is called north when to the northward of those points, but south when to the south ward. The azimuth is reckoned from the north in north latitudes, but from the south in south latitudes, and is named east when falling on the east side of the meridian, otherwise west. If the observed and true amplitudes be both north or both south, their difference will be the variation; but if one be north and the other south, their sum will be the variation. If the true and observed azimuths be both east or both west, their difference will be the variation, other wise their sum; and the variation will be easterly when the point representing the true bearing is to the right hand of the point representing the magnetic bearing, but westerly when to the left hand; the observer being supposed to look di rectly towards the point representing the magnetic bearing. EXAMPLE I. Suppose the sun's magnetic amplitude at rising is E. 26° 12' N. and the true amplitude E. 14° 20′ N. Required the variation? From the greater E. 26° 12' N. Take the lesser E. 14 20 N. The variation in this example is easterly, because the true amplitude falls fo the right of the magnetic, EXAMPLE II. EXAMPLE III. The moon's true amplitude at rising The sun's true azimuth being N was found to be E. 15° 20' N. and her 800 E. and his magnetic azimuth N. magnetic amplitude E. 10° 0' S. Re- 600 E. it is required to find the varia quired the variation? tion? True amplitude ........ E. 15° 20' N. .... EXAMPLE IV. The star Aldebaran was observed at rising to bear by compass E. N. E. True azimuth. Diff. is the variation.. EXAMPLE V. .N. 80 E. .N. 60 E. 20 E The true amplitude of the planet Jupiter was E. 10° N. when his magwhen the true amplitude was N. E. by netic amplitude was E, 20° S.-Re E.-Required the variation? True amp. N.. E by E. or E. 33° 45′ N. 22 30 N. quired the variation? Sum is variation... .E. 100 N. E. 20 S. ......30 W. To calculate the variation by observing the sun's azimuth when at equal altitudes in the forenoon and afternoon. The variation of the compass may also be determined by observing the magnetic azimuths of the sun in the morning and evening when at the same altitude, the observer being supposed to be at the same place at both observations; for it is evident that if the declination of the sun did not vary during the time elapsed between the observations, the middle point of the compass between the two bearings would be the bearing of the true north or south point of the horizon, at the place of observation, and the difference between that bearing and the north or south point of the compass would be the variation. In this kind of observations it will be convenient always to estimate the magnetic azimuths from the south point of the compass, calling them east or west, as before directed, and this method is supposed to be made use of in the following rule. Then, if one azimuth be east and the other west, hall their difference will be the variation, otherwise their half sum, and the variation will be of the same name as their greater azimuth, excepting, however, where the half sum is taken and exceeds 90°, in which case its supplement will be the variation of a different name from the azimuth. The variation being always supposed less than 90°. If the declination of the sun yaries during the elapsed time between the observations, (as is generally the case) an allowance may be made for that variation by applying a correction to the afternoon azimuth, calculated by the following rule.* The rule given in Doctor Mackay's "Complete Navigator" is inaccurate. RULE. Find from Table IV. the daily variation of the sun's declination on the day of observation. Then to the constant logarithm 9.1249 add the log. co-sine of the latitude of the place, the log. sine corresponding to the elapsed time between the observations found in the column P. M. the Prop. Log. of the daily variation of the sun's declination, and the Prop. Log. of the elapsed time*, estimating hours and minutes as minutes and seconds, the sum, rejecting 30 in the index, will be the Prop. Log. of the correction to be applied to the western azimuth, by subtracting when the sun is approaching towards the northern hemisphere, otherwise by adding. The azimuth thus corrected is to be used in estimating the variation instead of the observed azimuth. It is not necessary in this calculation to find the latitude or declination to any great degree of accuracy, which is the greatest advantage of the method; another of the advantages consists in being able to take a great number of observations, and applying the correction at one operation to the variation deduced from the mean of all the observations, so that, when great accuracy is required, as in taking observations ashore, this method may be used with success; and it is evident that it is alike applicable to the moon or any heavenly body, but the observations must be taken in the same place, as it would increase the calculation considerably, to make an allowance for the change of place, as well for the change of declination; and it would be better in this case to calculate each observation separately by the rules before given. EXAMPLE. Suppose that on the 10th of April, 1820, in the latitude of 42° 29' N. long. 50 W. the sun's morning azimuth was observed to be S. 54° 24′ E. and in the evening, when the sun was at the same altitude, was S. 39° 46′ W. the elapsed time between the observations being 6h. 20m.-Required the variation? Difference 14.49 The half of which 7° 24' is the variation, which is easterly, because the greater azimuth S. 54° 24′ E. is easterly. The variation, thus found, is to be allowed on all courses steered by the compass to obtain the true courses. To make this allowance, you must look towards the point of the compass the ship is sailing upon, and allow the variation from it towards the right hand, if the variation be east, but to the left hand, if the variation be west. Thus, if a ship steer S. E. with one point westerly variation, the true course will be S. E. by E. If the variation is one point easterly, the course will be S. E. by S. In the following Table are collected a few observations of the variation, made at different times, and in different places. The elapsed time may be determined by any common watch, but if none was used in the observations, it may be determined as follows. If one of the observed azimuths was east and the other west, take half their sum, otherwise half their difference, and to the log, sine of this half sum (or half differ ence) add the log. secant of the sun's declination, and the log. co-sine of the sun's correct altitude at the time of taking the azimuth, the sum (rejecting 20 in the index) will be the log. sine to be used in the above calculation, and this fogarithm will correspond to the elapsed time, marked in the column P. M. of Table XXVII. In this rule it is supposed that the bearing of the sun, by the afternoon observation, is to the westward of the meridian by compass; but if there be a great variation, that bearing might be to the eastward of the meridian by the compass, and in that case the correction of the western azimuth must be applied in a contrary manner to the above directions. |