motion of the moon in 12 hours will be obtained,* which, being divided by 12, will give the horary motion. EXAMPLE I. Required the horary motions of the moon in longitude, Dec. 12, 1808, at 15h. 48m. 29s., and 17h. 1m. 29s., apparent time, at Greenwich. This corresponds to Example I., preceding, in which T is 3h. 48m. 29s., or 5h. 1m. 29s. The two first differences in longitude are 7° 6' 16", and 7° 11' 18"; their mean, 7° 8' 47", is the approximate motion in 12 hours, and the arc B is 4' 54".5. The rest of the calculation is as follows: In a similar manner, if the horary motion in latitude was required at 12d. 17h. 33m., the two first differences in latitude are 34 21", and -36′ 45′′; their mean, - 35' 33", is the approximate motion in 12 hours. The correction found by the above rule with the time T,5h. 33m., and the arc B-2′ 6.5, is -59", whence the true motion in 12 hours is -36′ 32", which, divided by 12, gives the horary motion-3′ 2.7. The negative sign indicates that the north polar distance is decreasing, the positive sign + that it is increasing. In the present example, the north polar distance was decreasing, and as the latitude was south, it was also decreasing, as is evident. EXAMPLE II! Required the horary motions of the moon in longitude, June 16, 1806, at 2h. 49m. 50s.1, and 5h. 34m. 6s.6, apparent time, by astronomical computation, at Greenwich. This corresponds to Example II. preceding, in which T is 2h. 49m. 50s.1, or 5h. 34m 63.6; the two first differences are 7° 17' 21", and 7° 20′ 53", the mean of which, 7° 19' 7/1 is the approximate motion in 12 hours. The arch B is +37". 1. When it is required to find the motion of the moon in longitude or latitude, for any given interval of time, the motion in 12 hours must be found for the middle of that interval. 2. In calculating an occultation of a star by the moon, the relative horary motion in longi tude is the same as the horary motion of the moon, because the star is at rest; but in calcu lating a solar eclipse, the sun's horary motion must be found from the Nautical Almanac in the manner mentioned below, and subtracted from the moon's horary motion in longitude. the remainder will be the horary motion of the moon from the sun in longitude. Thus, on the 16th of June, 1806, the sun's horary motion was 2 23.1, which, being subtracted from the horary motions found in Example II., 36′ 39′′ 2, and 36′ 42′′.8, leaves the corresponding horary motions of the moon from the sun in longitude 34' 16.1, and 34' 19.7. As the sun has no sensible motion in latitude, the relative horary motion of the moon from the sun in latitude, is the same as the true horary motion of the moon in latitude. *The motion in 12 hours thus obtained, which, for distinction, will be called the arc M, is not perfectly accurate, since the third and higher orders of differences are neglected; but the horary motion deduced there from is abundantly sufficient for the purpose of projecting an eclipse or occultation. When greater accuracy is required, the third differences may be taken into account in the following manner :-Having found the second differences as above directed, subtract the first of them from the second, noting the signs as in algebra, and call the remainder the arc b. Enter Table XLV. with this are at the top, and the time T at the side, and take out the corresponding correction, which is to be increased by one sixth part of the arc b, without noting the signs. To the quantity thus found is to be prefixed a sign different from that of the arc b, and then it is to be applied to the arc M, with its sign, to obtain the true motion in 12 hours. Thus, in the above example, the second differences of longitude are+52"+447". Subtracting the former from the latter, leaves the third difference or arc b=-15". Corresponding to this and the time T 3h. 48m 29s. in Table XLV. is 1.6, which, increased by one sixth of b=2.5, gives the sought correction 4".1 or 4, to which must be prefixed the sign + (because the sign of b is negative), making it +4. This, connected with the M=+7° 10′ 20, gives the true motion in 12 hours, 7° 10' 24", whence the horary motion is 35 52". In a similar manner, if the third differences were noticed in the above example for finding the horary motion in latitude, the two second differences - 24" and 1'49", the arc b+35", the correction of the motion in 12 hours-36 32", is 10"; making it — 36′ 42′′, or 3 3.5 per hour. arc 3. The horary motions of the sun in longitude were formerly given in page iii. of the Nautical Almanac ; but they are discontinued in its new form, so that we must now deduce the horary motion from the daily difference of longitude, by dividing it by 24. EXAMPLE III. Thus, if it were required to find the sun's horary motion in longitude, in the interval between July 1 and July 2, 1836, mean time, astronomical account, at Greenwich; we should have the longitude at noon, July 1, 99° 35' 03.0; July 2, 100° 32′ 13.7. Their difference is 57' 10".7; dividing it by 24, we get the sun's horary motion in longitude 2′ 22′′.9. The same method may be used in finding the horary motions of the planets, neglecting the second differences; but if we wish to notice the second differences, we may proceed as in the three preceding examples, making use of the arcs A, B, T, found as in Remark 2, Problem I. EXAMPLE IV. Required the horary motion of Venus in right ascension, 1836, August 23d. 16h. 40m., mean time, astronomical account, at Greenwich. Here we have, as in Example V. of the preceding problem, T=8h. 20m.; and the mean of the two first differences, 1m. 00s.54, and 1m. 08s.79, is the approximate motion, 1m. 048.66; also the arch B+88.13.......... Prop. Log. 3.194 158 28.93, which represents the horary motion of Ve nus in right ascension, corresponding to August 23d. 16h. 40m. The horary motion of the moon in right ascension or declination is found, by inspection, in the Nautical Almanac, taking the differences of the two successive numbers in the Nautical Almanac, the one before, the other after, the time for which the horary motion is wanted. EXAMPLE V. Required the horary motion of the moon in right ascension and declination, between the hours of 10 and 11, on the 4th of August, 1836, mean time, astronomical account, at Greenwich. 1836, August 4d. 10h. Moon's right ascension 3h, 07m. 20s.15 4 11 The differences are the horary motions. In R. A. Declination 17° 52′ 44.2 N. 3 09 19 .57 In declination 18 03 30 4 10 462 These horary motions correspond very nearly to the middle of the time between 10h. and 11h., that is to say, 10h. 30m. PROBLEM III. To find the time of the ecliptic conjunction or opposition of the moon with the sun, a planet, or a fixed star. The time of the ecliptic conjunction of the sun and moon is the same as the time of new moon given for the meridian of Greenwich in page xii. of the month of the Nautical Almanac. Thus, in January, 1836, the ecliptic conjunction is on the 17th day, at 20h. 27m.8, mean time, at Greenwich. The time of the ecliptic opposition of the sun and moon is the same as at the time of full moon given in the same page of the Nautical Almanac. Thus the full moon or ecliptic opposition in May, 1836, was 30d. 3h. 59m.7, at Greenwich. The time of the ecliptic conjunction is easily computed from the geocentric longitudes of the objects; and we have here inserted the rule, adapted to the calculation of the conjunc tion of the sun and moon, which, with a slight modification, will answer for any planet, or a fixed star. RULE. Take from the Nautical Almanac the two longitudes of the sun and moon at the noon and midnight preceding the time of the conjunction, and the two immediately following. Subtract the longitudes of the sun from those of the moon, noting the signs as in algebra; the remainders will represent the distances of the sun from the moon on the ecliptic. Subtract each of these from the following to obtain the first differences, and call the middle term the arch A; subtract each of these differences from the following for the second differences, and take their half sum or mean for the arc B, noting the signs as in algebra. To the constant logarithm 4.63548, add the arithmetical complement of the log, of the arch A in seconds, and the log. of the second of the above-found distances in seconds; the *The sun's longitude at midnight is the mean of the longitudes on the preceding and following noons, sum, rejecting 10 in the index, will be the logarithm of the approximate value of T in seconds. With this time T at the side of Table XLV., and the arc B at the top, find the equation of second differences, the logarithm of which, added to the two first logarithms used in finding T, will, in rejecting 10 in the index, give the logarithm of the correction of the approximate time T in seconds, to be applied to it with the same sign as the arc B, and the mean time of the conjunction at Greenwich, counted from the second noon or midnight, taken from the Nautical Almanac, will be obtained. From which the time of conjunction under any other meridian may be easily obtained, by adding to it the longitude in time when east, or subtracting when west. Remark 1. When the time of the ecliptic conjunction of the moon and a planet is required, the longitudes of the planet must be found by Problem I. for the noon and midnight immediately preceding, and those immediately following the time of the conjunction, and these are to be used in the above note instead of the sun's longitudes. If the ecliptic eonjunction of the moon with a fixed star is required, its longitude must be found in Table XXXVII., and corrected for the equation of the equinoxes and aberration by Tables XL. XLI., as shown in the explanation of those tables. This longitude is to be used instead of the sun's, in the above rule. The longitude and latitude of the star may also be computed more accurately, from the right ascension and declination, given in the Nautical Almanac, by the method in Problem XIX. of this Appendix, whenever the star used is one of the 100 stars, whose places are given for every 10 days in the Nautical Almanac. Remark 2. By the same rule, the time, when the moon is at any distance from the sun, may be found, by increasing the sun's longitudes given in the Nautical Almanac, by the quantity the moon is supposed to be distant from the sun, counted according to the order of the signs; then supposing a fictious sun to move so as to have these increased longitudes at the corresponding times, and finding by the above rule the time of conjunction of the moon with this fictious sun, which will be the sought time when the moon is at the proposed distance from the sun. Thus, to find the time of the first, second, or third quarter of the moon, the sun's longitudes must be increased 3, 6, or 9 signs respectively (rejecting, as usual, 12 signs when the sun exceeds that quantity). Thus, if the first quarter of the moon which happened in the afternoon, July 21, 1836, was required: The sun's longitudes increased by 3 signs give the longitudes of the fictious sun, July 20d. 12h.; 21d. Oh.; 21d. 12H., and 22d. Oh. respectively, 208° 11' 10".0; 208° 39′ 48.8; 209° 08′ 27′′.7, and 209° 37′ 06′′.7. The longitudes of the moon corresponding are 200° 22′ 15′′.8; 207° 03′ 18.4; 213° 49′ 32.4, and 220° 41' 13.8. Hence the time of the conjunction of the moon with the fictious sun found by the above rule, was July 21d. 3h. 5m. at Greenwich, which is the time of the first quarter required. In a similar manner, by increasing the longitudes of a planet or a star, the time may be found when the moon is at any proposed distance from it. EXAMPLE. Required the mean time of the ecliptic conjunction of the sun and moon in January, 1836. Conjunction 8h. 27m. 48s. past midnight, on January 17d. 20h. 27m. 48s., mean time at Greenwich; being the same as in the Nautical Almanac. The time of conjunction under any other meridian, as for example, 30° W., is found by subtracting the longitude 2h. from 20h. 27m. 48s., which leaves 18h. 27m. 48s. If the longitude had been 30° E., the time of conjunction would have been 22h. 27m. 48s. The usual method of calculating the parallaxes in eclipses of the sun or occultations, is that by using the longitude and latitude of the nonagesimal or ninetieth degree of the ecliptic above the horizon; or, in other words, the longitude and complement of the latitude of the zenith, relative to the ecliptic. Several methods have been proposed for calculating the altitude and longitude of this point, which are required at each of the phases. The following, which is an improvement I have made on that given in La Lande's Astronomy, seems well adapted to the purpose, since several of the logarithms are the same at each of the phases, which much abridges the calculation, and on this account it admits of considerable simplifications, by a table like that on page 403. The method of making these calculations will first be given at full length, and then in the abridged form, by means of the proposed table. The process of calculating the parallaxes with the right ascensions and declinations, instead of the longitudes and latitudes of the bodies, adapted particularly to the new form of the Nantical Almanac, will be given towards the end of this Appendix. PROBLEM IV. Given the apparent time at the place of observation, counted from noon to noon, according to the manner of astronomers, the sun's right ascension, and the latitude of the place, reduced on account of the spheroidal figure of the earth, by subtracting the reduction of latitude, Table XXXVIII.; to find the altitude and longitude of the nonagesimal degree of the ecliptic. RULE NOT ABRIDGED. Add 6 hours to the sum of the sun's right ascension and the apparent time of observation, and call the sum the time T, rejecting 24 hours when it exceeds that quantity. Seek for this time in the column of hours of Table XXVII., supposing that marked A. M. to be, increased by 12 hours, as in the astronomical computation. The corresponding log. cotangent being found, is to be marked in the first and second columns, as in the following examples. If the reduced latitude is north, subtract it from 90°; if south, add it to 90°; the sum or difference will be the polar distance. Take half of this, and half the obliquity of the ecliptic, and find their difference and sum. Place the log. cosine of the difference in the first column, its log. sine in the second column; the log. secant of the sum in the first column, its log. cosecant in the second column, and its log. tangent in the third. The sum of the logarithms in the first column, rejecting 20 in the index, will be the Jog. tangent of the arc G; the sum of these in the second column, rejecting 20 in the index, will be the log. tangent of the arc F; these arches being less than 90° when the time T is found in the column A. M., otherwise greater. This rule is general except in places situated within the polar circles. Within the north polar circle, the supplement of F to 360° instead of F, must be taken; within the south polar circle, the supplement of G to 180° must be taken instead of G; the other terms remaining unaltered. In all cases, the longitude of the nonagesimal is equal to the sum of the arcs F, G, thus found, and 90°; rejecting 360 when the sum exceeds that quantity. Place in the third column the log. cosine of G, and the log. secant of F; the sum of the three logarithms of this column, rejecting 20 in the index, will be the log. tangent of half the altitude of the nonagesimal. EXAMPLE. Required the altitude and longitude of the nonagesimal at Salem, in the reduced latitude 42° 22′ 4′′ N., June 15, 1806, at 22h. 6m. 188.1, apparent time, or 22h. 6m. 21s.5, mean time, by astronomical computation, when, by the Nautical Almanac, the sun's right ascension was 5h. 36m. 50s., and the obliquity of the ecliptic 23° 27′ 48′′. The sum of the apparent time, sun's right ascension, and 6 hours, rejecting 24 hours, is 9h. 43m. 88.1 T. The polar distance is 47° 37′ 56′′; its half is 23° 48′ 58′′, and the half obliquity 11° 43′ 54"; hence their difference is 12° 5' 4", their sum 35° 32′ 52". The rest of the calculation is as follows: The two upper logarithms of the first and second columns, and the upper logarithm of the third column, vary but little in several centuries; and as these numbers occur twice in calculating a partial eclipse or occultation, and four times in a total or annular eclipse or transit, it will tend considerably to abridge the calculations, to have a table like the following, con taining their values for various places, for the obliquity 23° 27′ 40′′, with the variations for an increase of 100" in the latitude or obliquity. The logarithms A, B, C, of the table, were calculated in the following manner : In north latitudes subtract the reduced latitude from 90°, in south latitudes add the reduced latitude to 90°, the sum or difference will be the polar distance: take half of this and half of the obliquity of the ecliptic, 11° 43′ 50′′, and find the sum and difference. Then, Log. A is equal to the log. cosine of the difference added to the log. secant of the sum, rejecting 20 in the index. Log. C is equal to the log. tangent of the sum. Log. B is equal to the log. tangent of the difference, increasing the index by 10, less the log. Ĉ. Thus, for Salem, in the reduced latitude 42° 22' 4", the half polar distance is 23° 48′ 58′′, the half obliquity 11° 43′ 50′′, the difference 12° 5' 8'', the sum 35° 32′ 48′′. Difference.. கஉ 12° 5/ Cosine 9.99027 Tangent +10=19.33065 In this way the logarithms may be found for places not included in the table. The changes for an increase of 100" in the latitude or obliquity, are found by repeating the operation with these increased values, and ascertaining the corresponding changes in the values of A, B, C. These logarithms are given to six places of figures, though, in general, five will be quite sufficient, since the latitude and longitude of the nonagesimal are rarely required to a greater degree of accuracy than 10′′. TABLE, calculated for the obliquity 23° 27' 40". + Albany, Berlin,... Natchez, 51 19 29 Oxford Obs....... 51 34 28 Philadelphia, Richmond Obs. 48 38 51 Rutland, Salem,... 43 24 32 Place Prob. VII.. 42 22 4 42 27 13 0.079670 53 97 9.475733 293 739 9.853328 223 223 9.771197 240 240 9.773925 | 240 240 9.855355 222 222 9.763705 242 | 242| 9.741011 249 249| 9.780232 238 238 10.003045 210 210 9.855411 222 222 9.874005 219 219 9.902005 216 216 9.779944 238 238 9.845648 221 224 9.854016 222 222 500 10.027183 211 211 These logarithms are calculated for the obliquity 23° 27′ 40′′. The columns marked Lat. represent the variations of A, B, C, for an increase of 100" in the reduced lat. The column Obl. represents the variations of A, B, C, for an increase of 100" in the obliquity of the ecliptic. The signs must be changed if the latitude or obliquity is less than 23° 27′ 40′′, which is used in calculating the table. EXAMPLE. Required the values of A, B, C, for Salem, when the obliquity is 23° 27′ 48′′. 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. cotangent, added to the log. A of the table, gives the log. tangent of the arc G: this added to the log. B of the table, rejecting 10 in the index, will be the log. tangent of the arc F; these arcs being 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 arcs F, G, and 90°, neglecting as usual 360° when the sum exceeds that quantity. To the tabular log. C, add the log. cosine of the arc G, and the log. secant of the are F: the sum, rejecting 20 in the index, will be the log. tangent of half the altitude of the nonagesimal.‡ *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. cotangent of its half. To prevent mistake, it may be proper to remark, that, in finding T, we must add the apparent time, and not the mean time; for if the mean time be used, we ought to use also the mean right ascension; whereas the apparent right ascension is given in the Nautical Almanac; and this must be added to the apparent time in finding T. The arcs 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 thus 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 |