Abridged method of calculating the altitude and longitude of the Nonagesimal, by the preceding Table. Add together the sun's right ascension, the apparent time at the place of observation, (counted from noon to noon) and 6 hours, the sum, rejecting 24 or 48 hours if greater than those quantities, is to be called the time T; this is to be sought for in the column of hours of Table XXVII. supposing the column marked A. M. to be increased 12 hours, as in the astronomical computation.* The corresponding log. co-tangent, added to the log. A of the Table, gives the log. tangent of the arch G; this added to the log. B of the Table, rejecting 10 in the index, will be the log-tangent of the arch F; these arches be ing less than 90 when T is found in the column A. M. otherwise greater. [This rule is general, except in places situated within the polar circles, which is a case that very rarely occurs. Within the north polar circle, the supplement of F to 360°, is to be used instead of F; within the south polar circle, the supplement of G to 180°, is to be taken instead of G, the other terms remaining unaltered.] Then the longitude of the Nonagesimal is equal to the sum of the arches F, G, and 90°, neglecting as usual 360° when the sum exceeds that quantity. To the tabular log. C, add the log. co-sine of the arch G, and the log. secant of the arch F, the sum, rejecting 20 in the index, will be the log-tangent of half the altitude of the Nonagesimal.‡ EXAMPLE I. Required the altitudes and longitudes of the Nonagesimal at Salem, June 16, 1806, at the times of the beginning and end of the eclipse, calculated in Problem VI. ? End of the Eclipse. 5 37 18.5 R. Ascension 6 A T 12 27 531 Co-tan. 8.78470 Beginning of the Eclipse. " Required the altitudes and longitudes of the Nonagesimal at the times and places mentioned in the Example of Problem VII.? Immersion. Emersion. 9.53607 Sec. N. 40 38 46 Tan. 81 17 32 9 88233 G 49° 50′ 18" Tan. 10.07370 Co s. B 9.80955 10.02718 Tan. 10.04504 9.88175 Altitude Nonages In these calculations it is usual to take the sun's right ascension, and the apparent times, to tenths of a second, and to take proportional parts for the seconds and tenths in finding the logarithms. Thus in Example I. in finding the log. co. tang. of 9h. 43′ 8′′.1; the nearest logarithms are 9.48849, 9.48304, corresponding to the 9h. 43′ 4′′, and 9ha 43′ 12". These logarithms differ 45, the times 8", and the difference between 9h. 43′ 4′′, and 9h. 43′ 8′′.1, is 4".1. Hence 8": 45 :: 4".1 23 the correction to be subtracted from the first log. 9.48849, (because it is decreasing) to obtain the sought log. co-tang. 9.48826. Thus if the time T is 5 hours, it must be called 5h. P. M. If T is 14 hours, it must be called 2h. A. M. In making use of a common table of logarithms, you must turn the time T into degrees, and make use of the log. co-tangent of its half. The arches F, G, are acute when the time T is found in the column A. M. otherwise obtuse. This is easily remembered from the circumstance that a is the first letter of acute and A. M. Some writers have not taken notice of the cases of the values of F, G, within the polar circles. Strictly speaking, the quantity thas obtained is the distance between the north pole of the ecliptic and the zenith of the place, which, in southern latitudes, and between the tropics, is frequently the supplement of the altitude of the Nonagesimal. The above form is made use of to simplify the rules for applying the parallaxes. It is immaterial whether the altitude of the Nonagesimal, or its supplement, is made use of in Table XLIV. PROBLEM V. Given the altitude and longitude of the Nonagesimal; the longitude, latitude and herizatal parallax of the moon, and the latitude of the place of observation, to find the moon': parallax in latitude and longitude. RULE BY COMMON LOGARITHMS. From the horizontal parallax of the moon, subtract its correction from Table XXXVIII. corresponding to the latitude of the place, the remainder, in occultations of a fixed star, will be the reduced parallax; but in solar eclipses this quantity is to be diminished by the sun's horizontal parallax, 8". 8* to obtain the reduced parallax. To the logarithm of the reduced parallax in seconds, add the log. sine of the alttude of the Nonagesimal, and the log. secant of the moon's true latitudef, the smm, rejecting 20 in the index, will be a constant log. From the moon's true longitude,† increased by 360° if necessary, subtract the longitude of the Nonagesimal, the remainder will be the moon's distance from the Nonagesimal, which if less than 180 is to be called the arch D, otherwise its supplement to 360° is to be called the arch D. To the constant logarithm, add the log. sine of D, the sum, rejecting 10 in the index, will be the logarithm of the approximate parallax in longitude in seconds, which add to the arch D, then take the log-sine of the sum, and add it to the constant logarithm, reject ing 10 in the index, and the logarithm of the corrected parallax will be obtained. This will in general be sufficiently exact, but when great accuracy is required, the operation may be again repeated, by adding the arch D to the corrected parallax ;§ then to the leg. sine of the sum add the constant logarithm, rejecting 10 in the index, and the logarithm of the parallax in longitude P will be obtained. This is to be added to the true longitude of the moon when her distance from the Nonagesimal is less than 180°, otherwise sub tracted to obtain her apparent longitude. If the true latitude of the moon is south, prefix the sign to it, if north the sign —. Then to the logarithm of the reduced parallax in seconds, add the log. co-sine of the altitude of the Nonagesimal, and the log. co-sine of the moon's apparent latitude, the sum, rejecting 20 in the index, will be the logarithm of the first part of the parallax in latitude in seconds, to which prefix the sign when the altitude of the Nonagesimal is less than 90°, otherwise the sign, this added to the true latitude of the moon, due regard being had to the signs, will give her approximate latitude. To the logarithm of the reduced parallax in seconds add the log. sine of the altitude of the Nonagesimal, the log. sine of the moon's approximate latitude, and the log. cosine of the sum of the arches D and P, the sum, rejecting 30 in the index, will be the logarithm of the second part of the parallax in latitude in seconds, to which prefix the sign when the arches DP, and the approximate polar distance¶ are both greater or both less than 90°, otherwise the sign+, this term connected with the approximate ⚫ latitude will give the apparent latitude of the moon,** which will be south if north if —. The moon's latitude subtracted from her apparent latitude, noticing the signs, wil give the parallax in latitude. BY PROPORTIONAL LOGARITHMS. The above rule will answer in calculating by proportional logarithms, with the following alterations. When the log. sine occurs, read log. co-secant; for log. sine read log. secant; for log. secant read log. co-sine; and for log. co-secant read *This is the mean value of the sun's parallax, and may be used instead of the true parallax, which varies from 8.7 to 8.9. The true solar parallax at any time may be found by subtracting the logaritha of the sun's distance, given in the Nautical Almanac, from the log. of 8.8, increasing the index by a when necessary, the remainder will be the logarithm of the soughi parallax in seconds. 4 Corrected for the errors of the tables, when known. This sum D+ cor. par. is nearly equal to D + P the apparent distance of the moon from the Nosagesimal, to be made use of in Table XLIV. in finding the augmentation of the moon's S. D. In solar eclipses the apparent latitude is so small that its log. sine may be put equal to 10.00000 is occultations you must calculate the first part of the parallax in altitude by approximation, making use of the true latitude instead of the apparent in the above rule, and deducing the approximate value of the first part of the parallax: this applied to the true latitude will give the approximate apparent latitude, with which the operation is to be repeated, and the first part of the parallax will be obtained to a sufficient degree of exactness. The apparent polar distance is found by adding + 90° to the approximate latitude, due regard being had to the sigus. To be perfectly accurate, the apparent instead of the approximate latitude ought to be made use of in this part of the calculation, and the logarithms of this term ought to be increased by the log. secant less radius of P; but these corrections are too small to affect the result. In calculating the second part of the parallax in latitude, it will be sufficient to take the logarithms to three or four places of the decimals. *This rule gives the apparent latitude in all cases, but it may not be amiss to observe, that in several late publications the cases where the moon is between the zenith and the elevated pole are by mistake neglected. log. sine. The parallaxes may be calculated to the nearest second by proportional logarithms. When greater accuracy is required, common logarithms must be made use of. To illustrate this rule, the following examples, corresponding to the times of the beginning and end of the total eclipse of the sun, of June 16, 1806, as observed at Salem, are given. The elements necessary for this purpose have already been calculated in Problems I. and IV. For greater accuracy the longitudes and latitudes of the moon are corrected for the errors-58′′.5 in longitude, and 11".4 in latitude, which were found by comparing several observations of the eclipse made at different places. EXAMPLE I. Given the altitude of the Nonagesimal 67° 58′ 50′′, its longitude 63° 22′ 31′′, the longitude of the moon 83° 49′ 3′′.5 her latitude 24' 27".4 N. her horizontal parallax 60′ 24′′.1, the latitude of the place of observation 42° 33′ 30′′: required the parallaxes in longitude and latitude? The correction in Table XXXVIII. corresponding to the latitude 42° 33′ 30′′, and parallax 60′ 24′′.1 is 5".6, this and the sun's horizontal parallax 8".8 subtracted from the D's horizontal parallax 60' 24".1 leaves the reduced parallax 60' 9".7=3609′′.7. The longitude of the Nonagesimal 63° 22′ 31′′ subtracted from the moon's longitude 83° 49′ 3′′, leaves the moon's distance from the Nonagesimal, 20° 26' 32" equal to the arch D, because less than 180°. Given the altitude of the Nonagesimal 70° 57' 46", its longitude 95° 26′ 36′′, the longitude of the moon 85° 29′ 32.6, her latitude 15' 10".4 N. her horizontal parallax 60 27".0, the latitude of the place of observation 42° 33′ 30′′. Required the parallaxes in longitude and latitude? The correction in Fable XXXVIII. corresponding to the latitude 42° 33′ 30′′ and parallax 60′ 27′′, is 5".6, this and the sun's horizontal parallax 8".8, subtracted from the moon's horizontal parallax 60′ 27′′.0 leaves the reduced parallax 60′ 12′′.6. The longitude of the Nonagesimal 95° 26′ 36′′, subtracted from the moon's longitude increased by 360°, viz. 445° 29′ 33′′, leaves the moon's distance from the Nonagesimal 350° 2′57′′, the supplement of which to 360° is 9° 57′ 3′′, equal the arch D. EXAMPLE III. Required the parallaxes in longitude and latitude at the time of the occultation of Spica, Dec. 12, 1808, at the times and place mentioned in the Example of Problem VIL? Reduced par. Constant Immersion. 0.4782 D app. lat. 04710 Sec. 0.914 Sec. 10.0003 P. L. 5935 Constant 4850 Par. long. P. true long. Dapp. long. Having thus explained the method of calculating the parallaxes of the moon, it now remains to give the rules for finding the longitude by eclipses and occultations. The main object in these calculations is to determine from the observed beginning or end of the eclipse or occultation, the precise time of the ecliptic conjunction of the sun, or star and moon, free from the effects of parallax, counted on the meridian of the place of observation, since the difference of the times of conjunction, obtained in this manner at two places, will be their difference of longitude. If the lunar and solar tables were perfeetly correct, the longitude might be determined by taking the difference between the time of conjunction, given in the Nautical Almanac, and that deduced from the observations of the eclipse or occultation; but it is much more accurate to compare the times deduced from observations actually made at the places for which the difference of longitude is sought. There are two different methods of finding the ecliptic conjunction, according as the latitude of the moon is supposed to be accurately known or not. If the latitude was given correctly by the lunar tables, or was accurately known by other observations, the ecliptic conjunction, and the longitude of the place, might be deter mined by each of the phases of the eclipse or occultation, by the method given in Problems VIII. and IX. But the moon's latitude not being generally given to a sufficient degree of accuracy, it is usual to combine together the observations of the beginning and end of the eclipse or occultation, or the beginning and end of total darkness in a total eclipse, or the two internal contacts of an annular eclipse, to ascertain the error of the moon's latitude, by the method given in Problems VI. and VII. In making the calculations in these Problems, it will be necessary to know nearly the longitude of the place, in order to find the supposed time at Greenwich, so as to take out the elements from the Nautical Almanac: and if the longitude, deduced from the observation, should differ considerably, the operation must be repeated with the longitude obtained by this operation. The moon's true latitude 1° 55' 11" must first be used, its log. secant being 10.000, which give the 1st part par. 9' 3', which, added to the true latitude of the moon, gives the app. lat. nearly 2 4 10 the log. secant of which is 10.0003, as above. The calculation for the emersion is made in a similar manner: |