anomaly, and the mean longitude of her node at the beginning of the year 1800; because it is convenient in finding her mean longitude, &c. on any given day of any month, to add her longitude, &c. for that month and day to her longitude, &c. at the beginning of that year. And it is plain, that to find the longitude, &c. on the 1st day of January, the same for one day must be added to that of December 31st; and to find the longitude, &c. on the 15th of January, add the same for 15 days to that of December 31st; or to that of the beginning of January. And also to find it for the 20th of February, and the same for February, and for 20 days to that of December 31st, and so on. Having the longitude and anomaly at the beginning of 1800, the same at the beginning of any number of years after, may be found by adding the motion in longitude and anomaly for that number of years. But having the longitude of the node, at the beginning of 1800, to find the same at the beginning of any ne.nber of years after, subtract the motion of the node for that number of years, because the motion of the node is retrograde. Thus the Table is made, and continued at pleasure. The moon's motions are affected with many inequalities, and many equations are necessary to reduce her mean place to the true. The ecliptic orbits of the earth and moon, and the attraction of the sun and earth, it has already been observed, are the cause of these inequalities; and any further investigation or explanation of them, belongs to physical astronomy, and is not here necessary. The equations are contained in Tables 31, 32, 33, 34, 35, 36, and 37, pages 19, 21, 16, 18, 21, and 17; and the method of finding the arguments for these equations, and of taking out and applying them, is described in Problem 29. Tables 38 and 39, page 24 and 22, are for the moon's latitude, which depends principally upon her distanco from her node. Table 40, page 24, is the moon's hourly motion in latitude, depending on her distance from her node. Table 41, page 18, reduction of the latitude of places on the earth. This reduction is necessary on account of the spheroidal figure of the earth. Table 42, page 17, an equation to be applied to the ecliptic right ascension, in order to find the right ascension of the moon or a star, whose latitude does not exceed 6 degrees. Table 43, page 24, an equation for finding the declination of the moon, or a planet, or star, whose latitude does not exceed 6 degrees. Table 44, page 24; Mean new moons from 1724 to 1799, inclusive. Table 45, p. 24, the number of days difference between the Old Style and the New Style, at different periods of time. The number of days against any year, belongs to every year between that year and the next year in the table; thus, i1, the number against the year 1700, belongs to every year from 1700 to 1800; and 12 be longs to every year between 1800 and 2000. The days are to be subtracted from the Old Style to find the New, till the year 200 after Christ, then they are to be added. All the years in the Table are leap years by the Old Style, but none of them by the new. Table 46, page 25. The Dominical Letter. Table 47, page 19, shews the Golden Number for 3800 years after Christ; and is the same in the Old as in the New Style. Table 48, page 24, shows (by the Dominical Letter and Golden Number) the number of direction, or number of days to be added to the 21st of March, to find on what day of either March or April EASTER SUNDAY happens, New Style. Table 49, page 23, shows, by the Dominical Letters, on what days of the week the days of the months fall, and on what days of the months the days of the weeks fall. Table 50, page 21. The moon's mean angular distance from the sun. Table 51, page 16. The moon's mean angular motion from the sun. Table 52, page 25. The Solar system. Table 53, page 20. An equation or reduction, (found by the moon's true distance from her ascending Node,) to be applied to the time of true syzygy, to find the time of orbit syzygy. Table 54, page 25, the Constellations, with the degrees of right ascension and declination of the middle of each; the number of stars in each distinctly visible to the naked eye, and also the number that are of the sixth magnitude. The ancient constellations are in Roman letters; the modern, made out of unformed stars and parts of other constellations, are in italics. N. and S. show whether the declination is north or south. Table 55, p. 22. Logarithms of Numbers. The index of the logarithm of any number, being always known, is not put down in the Table. To find the logarithm of any number less than 1000; if the number is less than 100, find it in the left or right hand column, and next to it is its logarithm; if more than 100, look for the two left hand figures of the number in the column on the left or right, and for the figure at the top; against the two left hand figures, and under the other, is the logarithm. And having any given logarithm, to find its corresponding number, find the given logarithm or the nearest to it, in the table, and the number against it in the right or left column, together with the figure at the top, form the corresponding number: whether whole. mixed, or decimals, is determined by the index. Thus the logarithm for the number 397, is found to be 59879, whose index is 2, as there are three figures in the number; and the number corresponding to the logarithm 267486, is found to be 473; if the index had been 1, the number would have been 47,3 To find the logarithm of any given number that has in it more than three figures, find the logarithin of the three left hand figures of the number as just taught, subtract this logarithio from the next greater logarithm in the table, multiply the difference by the remaining part of the number, cut off as many of the right hand figures of the product as are in the multiplier; add the left hand part of the product to the logarithm of the three left hand figures of the given number, and the sum will be the logarithm required, to which prefix the proper index. Thus the logarithm of the number 7624 is found to be 3,8217. Having the logarithm of a number that has in it more than three figures, to find that number, or the nearest to it ;-divide, (according to division of decimals,) the difference between the given logarithm and the next less logarithm in the table, by the dif ference between the two nearest logarithms in the table to the given logarithm; add the quotient to the number corresponding to the logarithm in the table next less than the given logarithm, and the required number will he had; whether it is whole, mixed, or decimal, will be determined by the index of the given logarithm. Thus the number corresponding to the logarithm 3.88219, is found to be 7524. Table 56, p. 26. Logarithmic, or artificial sines, co-sines tangents, co-tangents, secants, and co-secants. To find the logarithmic sine, or co sine, &c. for any number of degrees and minutes in the table; if the degrees are less than 45, find them in the left hand column of one of the divisions of the table, and the minutes under them; against the minutes, and under the proper title, is the logarithm required; but if the degrees are more than 45, find them in the right hand column of one of the divisions of the table, and the minutes above them; against the minutes, and over the proper title, is the logarithm required. Thus, also, it is seen what degrees and minutes belong or correspond to a given logarithmic sine, or co-sine, &c. If the exact given logarithmic sine, or co-sine, &c. cannot be found in the table, take the nearest to it. When the logarithm for more than 90 degrees is required, subtract the given number of degrees from 180 degrees, and make use of the remainder. The Table is for only every 15m., but if greater accuracy is required, the logarithmic sine, or cosine, &c. for any number of minutes, may be found as follows: Take the difference between the two logarithms in the Table that are for the minutes nearest the given number of minutes; multiply this difference by the excess of the given number of minutes above om. 15m. 30m. or 45m.; divide the product by 15m. add the quotient to the logarithm of the number of minutes next less than the given number of minutes, and the sum will be the logarithm required. And to find the nearest minutes corresponding to a given logarithmie sine, or co-sine, &c. multiply 15m. by the difference between the given logarithm and the next less loga rithm to it in the proper coluna of the table; divide the product by the difference between the two nearest Jogarithms (in the proper column of the Table) to the given logarithm; add the quotient to the minutes cos Table 57, p. 23. Logistical or proportional logarithms to every 5th second for 60 minutes, or one dégrec or hour. To find the logistical logarithm for any number of minutes and seconds, find the minutes at the top of one of the divisions of the Table, and the seconds in the left hand column, against the seconds and under the minutes in the logarithm required; and thus it may be seen what minutes and seconds belong to a given logarithm; if the exact given logarithm cannot be found, take the nearest to it in the Table. A proportion by logistical logarithms, is solved as by other logarithms; that is, by subtracting the logistical logarithm of the 1st term from the sum of the logistical logarithins of the 2d and 3d terms; the remainder is the logistical logarithm required. The first term is frequently 1 degree, or 3600 seconds, whose logistical logarithm is 0; then, the sum of the 2d and 3d terms is the 4th terin. If one or more of the terms exceed 60 minutes, divide all the terms by some number that will make the largest term 60 degrees; then take out their logistical logarithms and work the proportion, and multiply the result by the number by which the terms were divided, and the answer or required term will be had. The Table is for only every 5th second, which is near enough in common cases; but if great accuracy is required, the logistical logarithm for any intermediate number of seconds may be found as directed in Table 56; as may also the seconds corresponding to any given logistical logarithm. responding to the next less logarithm (in the proper column of the Table) to the given logarithm, and the required minutes will be had. Table 58, p. 23. The visible ECLIPSES in the United States during the remainder of the present century. The figures on the right of the months, are the days, hours and ininutes of the middle apparent time of each eclipse at BOSTON: and the figures on the right of e. and m. show the digits eclipsed: e. stands for evening the earth, at different distances from the Equator. or afternoon, m. for morning, t. for total, and p. for partial. Table 59, p. 17. Length of a Degree of Longitude on TABLE 1.-Time in Mean Lunations; from 1-2 a Lunation, to 74212 Lunations; with the Sun's and Moon's Mean Motion in Anomaly, and the Distance the Sun and Moon's Ascending Node move from each other in these times. Time. TABLE 2-Mean Times of New Moon in March, with the Sun's and Ascend me anomalies, and the Sun's mean Distance from the Moon's he time, at the Greenwich Observatory. at DMS. O's anomaly.'s anomaly. fr. D's New in March. O's anomaly. D's anomaly. fr. D's So S 32739 S. 14 18 22 1 0 14 33 10 6 12 54 31 0 15 20 012237∞∞ "S. Yr. D. H. M. S. S. 7 1800 25 0 19 13 8 23 13 34 10 7 57 35 11 3 53 1810 5 6 36 52 8 3 11 18 3 26 10 4 26 1 38 34 8 12 14 48 6 24 43 45 11 19 1820 14 1836 17 2 59 38 8 15 6 22 9 6 28 11 10 1837 6 11 48 11 8 4 22 13 7 38 1838 25 9 20 47 8 22 44 23 6 33 11 236 5 52 22 7 22 50 34 6 26 32 6 8 5 21 52 265 18 36 25 8 21 56 53 7 22 21 7 9 6 2 5 3 13 8 13 10 8 10 6 32 9 13 59 42 19 11 7 22 33 1539 14 18 9 20 8 12 0 14 57 55 8 1 16 0617293 324 20 4 30 10 354 8 48 33 11 393 21 32 36 0 708 17 37 6 11 1092 15 9 42 11 15 16 10 9 48 9 0 8 2 47 19 36 18 0 16 5 34 0 11 10 21 37 2 1870 1 23 18 58 8 29 21 1880 10 15 20 40 8 8 24 52 9 708 6 1890 20 13 22 22 8 17 28 39 8 2 8 53 9 0 12 17 1900/30 8 24 4 18 26 26 32 26 6 23 27 27 3 23 21 TABLE 3.-Mean Time of New Moon, &c., in March, Old Style. 1800 13 0 19 13 8 23 13 34 10 7 57 35 11 3 53 TABLE 12.-Equation of the Sun from the inoon's Ascending Node. Argu ment. Sun's Mean Anomaly. 4 98 2569 3 52 0 0 2393 23 56 12 9 54 19 26 19 53 12 3 11 23 29 18 11 13 25 19 33 45 11 61843 5000 10 2 54 9 10 21 74212 6000] 23 22 58 3610 06307300 0 44 0 1 39 56 3 11 1 3 25 0 4963096 37 15 30 59 00 0 0 1 10 1 46 46 2 4 35 1 14 1 51 55 2 48 1 56 32 49 4 1 4 21 1 43 19 0 14 2 3 91 36 46 0 43 29 20 31 36 1 26 47 1 59 40 2 1 01 29 27 0 32 55 15 92 2 14 1 57 56 1 21 25 0 22 12 44 0 11 55 30 30 1 1 101 46 46 2 4 351 3 28 0 0 111 sig. +10 sig. + 9 sig. + 8 sig. +1 7 sig. + 6 Big. + +1 93 2322 TABLE 5.--The annual, or first Equation, of the mean, to the true Syzygy. Argument. 0 44 28 300 6 21 21 30) 10 5000 7 5 53 20 33 10 23 13 219 22 31 401 25 10 4 46 7 2610 16 57 33 6300 2 7 10 0 2011 For Months. 0000 0 17 54 48 Mar. 11 29 15 16 In Hours, in 0 17 10 3 Minutes, and in] Apr 0 22 53 23 Seconds. 1 10 48 11 1 16 31 22 11 sign. +10 sign. + 9 sign. + 8 sign. + 7 sign. + 6 sign. + ments. The Moon's passage over the meridian, either of the Moon from the Sun. ing the Moon's mean Anomaly from the double Distance TION. Argument 21. The remainder found by subtract on the Meridian. 54 .12 0 0 55 10 15 4 16 15 16 21 21 22 26 27 27 31 32 36 38 41 43 45 48 49 53 61 + Hourly Motions and Semidiameters, to every sign and tenth degree of their 1 2 3 4 5 60 61 30 0 39 44 1 9 11 1 20 25 1 10 12 0 40 45 0 0 0 0 11+ 10+ +6 6+ 0 0 0 46 45 1 21 32 1 35 00 10 1 37 0 48 10 1 22 21 1 35 20 3 13 0 49 34 1 23 10 1 35 3.0 4 520 50 53 1 23 57 1 35 40 6 28 0 52 19 1 24 41 1 508 60 53 40 1 25 24 60 9 420 55 0 1 26 70 11 20 0 56 21 1 26 48 80 12 56 0 57 38 9 1 14 44 90 14 33 0 58 56 1 28 6 1 34 10 0 16 10 1 0 13 1 28 43 1 33 53 1 " 11 23 40 48 19 30 21 22 14 0 46 51 29 11 21 24 0 45 23 28 TABLE 6-Equation of the Moon's Mean Anomaly. Argu ment. Sun's Men Anomaly. ¡Deg. Deg. 8 0 43 54 27 0 42 24 26 0 40 53 25 6 1 34 43 1 17 45 0 39 21 24 02 23 491 59 44 1 35 43 1 11 45 10 2 23 171 59 91 35 81 11 14 20 2 19 34 1 55 561 32 29 1 8 9 45 59 21 37 0 302 12 21 1 49 49 1 27 291 5 21 43 24 Lon. +5 signs with N. Lat. or +11 signs with S. Lat. 47 49 23 54 20 47 25 23 40 20 22 56 10 16 48 0 37 49 23 15 47 0 36 15 22 0 34 40 21 Lon. +0 signs with S. Lat. or+6 signs with N. Lat. Lat. 12' Lat.13 Lat. 14° Lat. 15° Lat.15 Lat. 47 49 0 11 451 35 43 1 59 412 23 49 30 47 12 1 10 45 1 34 181 57 50 2 21 23 20 45 34 1 8 141 30 501 53 23 2 15 51 10 42 50 1 4 31 25 111 46 132 7 10 0 Lon.-5 signs with S. Lat. or -11 signs with N.Lat. Lon.-1 sign with N. Lat. or 7 signs with S. Lat. 6° Lat. 15 Lat. (4° Lat. 13° Lat. 2° Lat./1° Lat. 1 4 32 0 20 7.12 ง 1 32 57 1 30 25 1 3 19 0 18 28 11 1 33 17 1 29 54 1 2 10 16 48 10 21 0 33 29 1 12 58 1 33 36 1 29 20 1 0 45 0 15 8 9 22 0 35 2 1 14 1 1 33 52 1 28 45 1 34 6 1 28 9 0 59 26 0 13 23 8 0 58 70 11 487 24 0 38 1 1 16 0 1 34 18 1 27 30 0 56 45 0 10 7 1 34 30 1 26 50 0 55 23 0 1 34 40 1 26 27 0 54 10 25 0 39 29 1 16 59 1 34 48:1 25 50 52 37 0 11 si.+10 si.+ 9 si. +8 si. s. 6 si. +1 TABLE 7.-Second Equation of the Mean to the true Syzygy. Argument. Sants-Mean Anomaly. 0 si. +1 si. +12 si. +3 si. 4 si. + si. 5+ H.M. S. H.M. S. H.M. S. H.M. S. H.M. S. H.M. S. P 0 0 0 0 5 12 48 8 47 89-46 418 8 594 34 33 30 3 12 4 26 1 29 10 10 58 5 21 56 8 51 45 9 45 38 20 21 56 5 30 57 8 56 10 9 45 12 7 57 23 4 17 25 28 30 32 54 5 39 51 9 0 25 9 44 1117 51 334 8 47 27 07.26 4 0 43 52 5 48 37 9 4 31 9 42 59 7 45 46 4 50 51 50 5 57 17 9 8 25 9 41 36 7 39 46 3 51 23 25 61 5 48 6 5 51 9 12 9 9 40 37 33 36 3 42 32 24 71 16 46 6 14 19 9 15 43 9 38 19 7 27 22 3 33 38 23 81 27 44 16 22 419 19 59 36 24 7 21 23 24 42 22 91 38 40 6 30 57 9 22 14.9 34 18 7 14 30 3 15 44 21 10 1 49 33 6 39 4 9 25 12 9 32 1 7 7 50 3 Lon. +1 sign with S. Lat. or +7 signs with N. Lat. 1 Lat.29 Lat.13 Lat. 4° Lat.15° Lat. (6° Lat. Lon. 31 25 11 46 13 2 7 10 30 31 17 71 36 31 54 54 20 31 6 251 22 39 1 38 46 10 01 5 511 18 41 0 38 51 0 58 33 31 0 50 26 48 0 39 580 53 4 signs with S. Lat. or 2 signs with N. Lat. or 10 signs with N.Lat. 8 signs with S.Lat. 16 Lat. 5° Lat. 4° Lat (3° Lat. 12° Lat. 1° Lat. 1 11 2 0 23 6 47 09 27 54 9 29 33 7 12 2 11 10 6 54 46 9 30 32,9 26 54 6 54 13 2 21 54 7 2 21 9 32 589 24 62 30 28 16 14 2 32 34 17 9 52 9 35 12 9 21 15 2 43 9 7 17 9 9 37 14 9 17 51 6 32 56 2 21 19 15 16 2 53 38 7 24 19 9 39 8 9 14 28 6 25 40 2 12 8 14 17 3 4 3 7 31 18 9 40 51 9 10 51 6 18 18 2 2 53 13 7 38 9 9 42 219 7 9 6 10 49 1 53 36 12 7 44 51 9 43 42 9 3 13 6 21 3 45 11 7 57 45 9 45 52 8 54 50 5 47 54 1 25 31 22 3 55 21 8 3 56 9 46 38 8 50 24 5 40 41 16 7 23 4 5 26 8 9 57 9 47 13 8 45 48 5 32 91 6 41 21 4 15 26 8 15 469 47 36 8 41 25 4 25 20 8 21 24 9 47 49 8 36 26 4 35 68 26 53 9 47 54 8 31 27 4 44 428 32 11 9 47 46 8 25 44 4 59 42 0 28 41 28 4 54 11 8 37 19 9 47 33 8 20 18 4 51 150 19 29 5 3 33 8 42 18 9 47 14 8 14 33 4 43 2 0 9 31 0 0 30 5 12 48 8 47 89 46 44 8 8 59 4 34 33 0 7 si. 6 si. 8 si. the axis Arguments. Her distance from 4 5 39 30 46 5 45 5 44 5 43 5 425 42 5 41 405 40 5 39 27 44 5 43 5 425 41 5 405 39 5 38 5 37 5 37: TABLE 14.-Angle of the axis of the Moon's Orbit with of the in Eclipses. her Eclipticode, in the right or left column; and her Hour Motion from the Sun at the top of the table. 29 | 30 | 31 | 32 05 47 5 465 45 5 Lon.-3 signs 15, 11 10, 6 0000 00 0 0 0 with S. Lat. or-9 signs with N. Lat 41 5. TABLE 59.Length of a Degree of Longitude at different Latitudes. Eng.Eng. miles miles 50/69.50 46 49.28 2 69.46 48 46.50 4 69.33 50 41.67 6 69.12 52 42.79 868.82 5140.85 10 68.44 56 38,86 12 67.98 58 36.83 1167.44 60 34.75 16 66.81 62 32.63 18 66.10 64 30.47 20 65.31 66 28.27 22 61.88 68:26.04 24 63.49 70 23.77 26 62.47 72 21.48 28 61.36 74 19.16 30 60.19 76 16.81 32 58.49 78 14.45 31 57.62 80 12.07 36 56.2282 9.67 38 54.77 84 7.26 40 53.21 86 4.85 4251.65 88 2.43 44 49.99 90 0.00 length of a degree on the Equator, so is the Co-Sine of the given Latitude reduced by Ta ble 41, page 18, to the length of a degree in that Latitude. '5, 11+ 4,10+ 3, 9+1 2* 14 05 29 3 58 015 28 28 38 70 80 O 0 2 30 11 28 50 90 0 100 O 0 0 36 39 11 29 13 55 0 11 39 il 28 38 36 0 45 48 11 29 2 24 200 O 1 31 36 11 28 4-48 300 0 2 17 25 11 27.7 12 400 3 3 13 11 26.9 36 500 0 3 49 26 11 25 12 .0 600 0 4 34 49 11 24 14 24 700 0 5 20 38 11 23 16.48 800 0 6 6 26 11 22 19 12 900 0 6 52 14 11 21 21 36 1000 0 7 38 52 11.20 24 0 2000 0 15.17 45 11 10 48 3000 0 22 54 7 11 1 12 4000 1 0 32 10 10.21 36 5000 1 8 10 12 10 12 10000 2 16 20 24! 8 24 36 1 4 1 32 1 TABLE 10. The Sun's mean Longitude and Anomaly at the beginning of current years. Also the Sun's mean motion in Longitude and Anomaly, in Julian Years; in Months; in Days; and in Hours, Minutes, and Seconds: with the mean distance the Sun and Moon's Ascending Node move from each other in Hours, Minutes, and Seconds, this being used in calculating eclipses. At the beginning of current years. 1890 9 10 6 111 29 45 40 11 29 44 38 2 11 29 31 21 11 29 29.17 3 11 29 17 111 29 13 55 4 0 0 1 50 11 29 57 42 5 11 29 47 30 11 29 42 20 6 11 29 33 11 11 29 26 59 711 29 18 51 11 29 11 37 8 0 0 3 40 11 29 55 24 9 11 29 49 20 11 29 40 2 10 11 29 35 111 29 24 40 20 0 0 9 10 11 29 48 29 30 11 29 44 10 11 29 13 9 40 0 0 18 19 11 29 36 58 50 11 29 53 20 11 29 1 38 60 0 0 27 29 11 29 25 26 0 7 The Sun's mean motion in Longitude and Anomaly, and the mean distance the Sun and Moon's Ascending Node move from each other, in Hours, Minutes, and Seconds. Time. FNS Sun from ( Com. 0 0 0 0 0000 Leap. 11 29 0 52 Moon's Feb. S Com. 1 0 33 18 34 0 31 9 33 45 36 21 57 Leap. 9 29 38 12 10 29 12 22 11 29 0 52 1 0 33 13 0 29 34 5 1 28 9 1 2 28 42 15 3 28 16 20 4 28 49 32 5 28 23 57 6 28 56 50 7 29 30 3 8 29 4 8 9 29 37 20 TO 29 11 25 As. Node! March "April May 2 36 June 5 12 July 7 47 August 0 29 34 10 1 28 9 11 2 28 42 30 3 28 16 40 4 28 49 58 5 28 24 8 6 28 57 26 23 September 7 29 30 44 59 October 8 29 4 54 35 November 11 December 20 46' 23 22 58 34 41 33 44 8 46 44 49 20 51 56 2-8 0 0 0 2 5 42 2 11 15 30 0 14 52 0 29 36 2 13 7 2 19 9 2 3 5 27 1 53 29 24 90 12 0 58 15 1 11 44 2 3 30 2 23 44 2 26 47 1 42 31 21 1 2 28 16 30 2018 1 17 3 15 1 2 49 12 2 22 55 16 12 3 2 11 15 Q 0 0 0 43 1 45 57 1 51 39 In Days, the same in Lon. as Anoin S.° 1800 11 5 43 12 1810 7 5 55 12 1820 3 19 17 47 11 0 5 23 1830 11 19 29 47 5 13 26 24 1834 5 10 12 42: 5 21 23 31 1835 9 19 36 2 8 20 6 53 2 12 9 42 6 21 32 47 1838 11 0 55 52 1839 3 10 18 58 8 28 4 3 11 28 57 54 1840 8 2 52 22 0 9 51 17 11 9 35 0 1850 4 3 4 22 6 23 12 16 4 26 11 26 1960 0 16 26 59 1 19 37 11 10 12 44 41 1870 8 16 38 59 8 258 10 3 23 21 7 1880 5 0 1 42 2 29 23 39 15 54 22 1890 1 0 13 42 9 12 44 1900 9 0 25 42 S.° 1 "1 9 20 19 29 S.° 4 3 40 30 1 3 15 29 6 19 52 5 06 25 20 5 23 1 46 3 5 39 42 2 16 20 1 0 1 54 6 1 26 57 4 0 27 25 1 7 37 41 5 29 20 44 0 18 18 36 2 2222 3333 כי 0 -13 3-18 |